Adaptive Antennas for Wireless Communications

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Author: George V. Tsoulos

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ADAPTIVE ANTENNAS FOR WIRELESS COMMUNICATIONS

IEEE Press 445 Hoes Lane, P.O. Box 1331 Piscataway, NJ 08855-1331 IEEE Press Editorial Board Robert J. Herrick, Editor in Chief M. Akay J. B. Anderson P. M. Anderson J. E. Brewer

M. Eden M. E. El-Hawary R. F. Hoyt S. V. Kartalopoulos D. Kirk

M. S. Newman M. Pagdett W. D. Reeve G. Zobrist

Kenneth Moore, Director of IEEE Press Catherine Faduska, Senior Acquisitions Editor Linda Matarazzo, Associate Acquisitions Editor Mark Morrell, Associate Production Editor IEEE Antennas & Propagation Society, Sponsor AP-S Liaison to IEEE Press, Robert Mailloux Cover design: William T. Donnelly, WT Design

Technical Reviewers Hans Steyskal, Rome Laboratory/ERA, Hanscom AFB, MA Robert Mailloux, Rome Laboratory/ERI, Hanscomb APB, MA

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ADAPTIVE ANTENNAS FOR WIRELESS COMMUNICATIONS

Edited by

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Library of Congress Cataloging-in-Publication Data

Adaptive antennas for wireless communications / edited by George V. Tsoulos. p. em. Includes bibliographical references and index. ISBN 0-7803-6016-8 1. Adaptive antennas. 2. Wireless communication systems--Equipment and supplies. I. Tsoulos, George V., 1968TK7871.67.A33 A33 2000 621.382' 4--dc21

00-040990

Contents

xi

Preface

Chapter 1

Introduction and Channel Models

1

Adaptive Antenna Systems 3 B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode (IEEE Proceedings, December 1967). Overview of Spatial Channel Models for Antenna Array Communication Systems 20 R. B. Ertel, P. Cardieri, K. W. Sowerby, T. S. Rappaport, and J. H. Reed (IEEE Personal Communications Magazine, February 1998). Antenna Systems for Base Station Diversity in Urban Small and Micro Cells 33 F. C. F. Eggers, J. Tcttgard, and A. M. Oprea (Journal on Selected Areas of Communication, September 1993). A Statistical Model for Angle of Arrival in Indoor Multipath Propagation 44 Q. Spencer, M. Rice, B. Jeffs, and M. Jensen (IEEE Vehicular Technology Conference, May 1997).

Chapter 2

Adaptive Algorithms

49

Highlights of Statistical Signal and Array Processing 51 A. Hero (IEEE Signal Processing Magazine, September 1998). Application of Antenna Arrays to Mobile Communications, Part II: Beamforming and Direction-of-Arrival 95 Considerations L. C. Godara (Proceedings of the IEEE, August 1997). High-Resolution Frequency-Wavenumber Spectrum Analysis 146 J. Capon (Proceedings of the IEEE, August 1969). An Algorithm for Linearly Constrained Adaptive Array Processing 157 O. L. Frost III(Proceedings of the IEEE, August 1972). The Application of Spectral Estimation Methods to Bearing Estimation Problems 167 D. H. Johnson (Proceedings of the IEEE, September 1982). On Spatial Smoothing for Direction-of-Arrival Estimation of Coherent Signals 178 T-J. Shan, M. Wax, and T. Kailath (IEEE Transactions on Acoustics, Speech, and Signal Processing, August 1985). Detection of Signals by Information Theoretic Criteria 184 M. Wax and T. Kailath (IEEE Transaction on Acoustics, Speech, and Signal Processing, April 1985). Multiple Emitter Location and Signal Parameter Estimation 190 R. O. Schmidt (IEEE Transactions on Antennas and Propagation, March 1986). Using Spectral Estimation Techniques in Adaptive Processing Antenna Systems 195 W. F. Gabriel (IEEE Transactions on Antennas and Propagation, March 1986). Implementation of Adaptive Array Algorithms 205 R. Schreiber (IEEE Transactions on Acoustics, Speech, and Signal Processing, October 1986). v

Contents

Steady State Analysis of the Generalized Sidelobe Canceller by Adaptive Noise Cancelling Techniques 213 N. K. Jablon (IEEE Transactions on Antennas and Propagation, March 1986). Adaptive-Adaptive Array Processing 221 E. Brookner and J. M. Howell (Proceedings of the IEEE, April 1986). ESPRIT-Estimation of Signal Parameters via Rotational Invariance Techniques 224 R. Roy and T. Kailath (IEEE Transactions on Acoustics, Speech, and Signal Processing, July 1989). Spectral Self-Coherence Restoral: A New Approach to Blind Adaptive Signal Extraction Using Antenna Arrays 236 B. G. Agee, S. V. Schell, and W. A. Gardner (Proceedings of the IEEE, April 1990). Sensor Array Processing Based on Subspace Fitting 250 M. Viberg and B. Ottersten (IEEE Transactions on Signal Processing, May 1991). Direction-of-Arrival Estimation via Exploitation of Cyclostationarity-A Combination of Temporal and Spatial Processing 261 G. Xu and T. Kailath (IEEE Transactions on Signal Processing, July 1992). Space-Alternating Generalized Expectation Maximization Algorithm 272 J. A. Fessler and A. O. Hero (IEEE Transactions on Signal Processing, October 1994). Unitary ESPRIT: How to Obtain Increased Estimation Accuracy with a Reduced Computational Burden 286 M. Haardt and J. A. Nossek (IEEE Transactions on Signal Processing, May 1995). Joint Angle and Delay Estimation (JADE) for Multipath Signals Arriving at an Antenna Array 297 M. C. Vanderveen, C. B. Papadias, and A. Paulraj (IEEE Communications Letters, January 1997).

Chapter 3

Performance Issues

301

Smart Antennas for Mobile Communication Systems: Benefits and Challenges 303 G. V. Tsoulos (Electronics & Communication Engineering Journal, April 1999). 314 An Adaptive Array in a Spread-Spectrum Communication System R. T. Compton, Jr. (Proceedings of the IEEE, March 1978). 324 On the Performance of a Polarization Sensitive Adaptive Array R. T. Compton, Jr. (IEEE Transactions on Antennas and Propagation, September 1981). Effect of Mutual Coupling on the Performance of Adaptive Arrays 332 I. J. Gupta and A. A. Ksienski (IEEE Transactions on Antennas and Propagation, September 1983). Optimum Combining in Digital Mobile Radio with Co-channel Interference 339 J. H. Winters (IEEE Transactions on Vehicular Technology, August 1984). On Optimum Combining at the Mobile 351 R. G. Vaughan (IEEE Transactions on Vehicular Technology, November 1988). The Performance of an LMS Adaptive Array with Frequency Hopped Signals 359 L. Acar and R. T. Compton, Jr. (IEEE Transactions on Aerospace and Electronic Systems, May 1985). An LMS Adaptive Array for Multipath Fading Reduction 371 Y. Ogawa, M. Ohmiya, and K. Itoh (IEEE Transactions on Aerospace and Electronic Systems, January 1987). Optimum Combining for Indoor Radio Systems with Multiple Users 378 J. H. Winters (IEEE Transactions on Communications, November 1987). The Performance Enhancement of Multibeam Adaptive Base-Station Antennas for Cellular Land Mobile Radio Systems 387 S. C. Swales, M. A. Beach, D. J. Edwards, and J. P. McGeehan (IEEE Transactions on Vehicular Technology, February 1990). Combination of an Adaptive Array Antenna and a Canceller of Interference for Direct-Sequence SpreadSpectrum Multiple-Access System 399 R. Kohno, H. Imai, M. Hatori, and S. Pasupathy (IEEE Journal on Selected Areas in Communications, May 1990). Direction Finding in the Presence of Mutual Coupling 406 B. Friendlander and A. J. Weiss (IEEE Transactions on Antennas and Propagation, March 1991). vi

Contents

Improving the Performance of a Slotted ALOHA Packet Radio Network with an Adaptive Array 418 J. Ward and R. T. Compton, Jr. (IEEE Transactions on Communications, February 1992). Signal Acquisition and Tracking with Adaptive Arrays in the Digital Mobile Radio System IS-54 with Flat Fading 427 J. H. Winters (IEEE Transactions on Vehicular Technology, November 1993). Effect of Fading Correlation on Adaptive Arrays in Digital Mobile Radio 435 J. Salz and J. H. Winters (IEEE Transactions on Vehicular Technology, November 1994). 444 Capacity Improvement with Base-Station Antenna Arrays in Cellular CDMA A. F. Naguib, A. Paulraj, and T. Kailath (IEEE Transactions on Vehicular Technology, August 1994). 452 Analytical Results for Capacity Improvements in COMA J. C. Liberti and T. S. Rappaport (IEEE Transactions on Vehicular Technology, August 1994). Adaptive Transmitting Antenna Arrays with Feedback 463 D. Gerlach and A. Paulraj (IEEE Signal Processing Letters, October 1994). Adaptive Antennas for Third Generation DS-CDMA Cellular Systems 466 G. V. Tsoulos, M. A. Beach, and S. C. Swales (Proceedings of 45th Vehicular Technology Conference, July 1995). The Spectrum Efficiency of Base Station Antenna Array System for Spatially Selective Transmission 471 P. Zetterberg and B. Ottersten (IEEE Transactions on Vehicular Technology, August 1995). Capacity Enhancement and BER in a Combined SDMA/TDMA System 481 J. Fuhl and A. F. Molisch (Proceedings of the 46th Vehicular Technology Conference, April 1996). Performance of Wireless CDMA with M-ary Orthogonal Modulation and Cell Site Antenna Arrays 486 A. F. Naguib and A. Paulraj (IEEE Journal of Selected Areas in Communications, December 1996). Smart Antenna Arrays for CDMA Systems 500 J. S. Thompson, P. N. Grant, and B. Mulgrew (IEEE Personal Communications Magazine, October 1996). Efficient Direction and Polarization Estimation with a COLD Array 510 J. Li, P. Stoica and D. Zheng (IEEE Transactions on Antennas and Propagation, April 1996). Upper Bounds on the Bit-Error Rate of Optimum Combining in Wireless Systems 519 J. H. Winters and J. Salz (IEEE Transactions on Communications, December 1998). The Range Increase of Adaptive Versus Phased Arrays in Mobile Radio Systems 525 J. H. Winters and M. J. Gans (IEEE Transactions on Vehicular Technology, March 1999). 535 A Comparison of Two Systems for Downlink Communication with Base Station Antenna Arrays P. Zetterberg (IEEE Transactions on Vehicular Technology, September 1999).

Chapter 4

Implementation Issues

551

Fundamentals of Digital Array Processing 553 D. E. Dudgeon (Proceedings of the IEEE, June 1977). A Novel Algorithm and Architecture for Adaptive Digital Beamforming 560 C. P. Ward, P. J. Hargrave, and J. G. McWhirter (IEEE Transactions on Antennas and Propagation, March 1986). Nonlinearities in Digital Manifold Phased Arrays 569 B. D. Mathews (IEEE Transactions on Antennas and Propagation, November 1986). Adaptive Beamforming with the Generalized Sidelobe Canceller in the Presence of Array Imperfections 579 N. K. Jablon (IEEE Transactions on Antennas and Propagation, August 1986). 596 An Efficient Algorithm and Systolic Architecture for Multiple Channel Adaptive Filtering S. M. Yuen, K. Abend, and R. S. Berkowitz (IEEE Transactions on Antennas and Propagation, May 1988). Mutual Coupling Compensation in Small Array Antennas 603 H. Steyskal and J. S.Herd (IEEE Transactions on Antennas and Propagation, December 1995). 607 A Unified Approach to the Design of Robust Narrow-Band Antenna Array Processors M-H. Er and A. Cantoni (IEEE Transactions on Antennas and Propagation, January 1990).

vii

Contents

Design Trades for Rotman Lenses 614 R. C. Hansen (IEEE Transactions on Antennas and Propagation, April 1991). Optimum Networks for Simultaneous Multiple Beam Antennas 623 E. C. DuFort (IEEE Transactions on Antennas and Propagation, January 1992). Direction Finding in Phased Arrays with a Neural Network Beamformer 630 H. L. Southall, J. A. Simmers, and T. H. O'Donnell (IEEE Transactions on Antennas and Propagation, December 1995). Application of Orthogonal Codes to the Calibration of Active Phased Array Antennas for Communication Satellites 636 S. D. Silverstein (IEEE Transactions on Signal Processing, January 1997). The Analogy Between the Butler Matrix and the Neural-Network Direction-Finding Array 649 R. J. Mailloux and H. L. Southall (IEEE Antennas and Propagation Magazine, December 1997). Forward-Backward Averaging in the Presence of Array Manifold Errors 655 M. Zatman and D. Marshall (IEEE Transactions on Antennas and Propagation, November 1998).

Chapter 5

Experiments

661

Multiple Source DF Signal Processing: An Experimental System 663 R. O. Schmidt and R. E. Franks (IEEE Transactions on Antennas and Propagation, March 1986). An Implementation of a CMA Adaptive Array for High Speed GMSK Transmission in Mobile Communications 673 T. Ohgane, T. Shimura, N. Matsuzawa, and H. Sasaoka (IEEE Transactions on Vehicular Technology, August 1993). A Four-Element Adaptive Antenna Array for IS-136 PCS Base Stations 680 R. L. Cupo, G. D. Golden, C. C. Martin, K. L. Sherman, N. R. Sollenberger, J. H. Winters, and P. W. Wolniasky (IEEE 46th Vehicular Technology Conference, May 1996). Ericsson/Mannesmann GSM Field-Trials with Adaptive Antennas 685 S. Anderson, U. Forssen, J. Karlsson, T. Witzschel, P. Fischer, and A. Krug (IEEE 46th Vehicular Technology Conference, May 1996). 690 Preliminary Measurement Results from an Adaptive Antenna Array Testbed for GSM/UMTS P. E. Mogensen, K. I. Pedersen, P. Leth-Espensen, B. Fleury, F. Frederiksen, K. Olesen, and S. L. Larsen (IEEE Vehicular Technology Conference, May 1997). Performance Evaluation of a Cellular Base Station Multibeam Antenna 695 Y. Li, M. Feuerstein, and D. O. Reudink (IEEE Transactions on Vehicular Technology, February 1997). Space Division Multiple Access (SDMA) Field Trials. Part 1: Tracking and BER Performance 704 G. V. Tsoulos, J. McGeehan, and M. Beach (lEE Proceedings of Radar, Sonar, and Navigation, February 1998). 710 Space Division Multiple Access (SDMA) Field Trials. Part 2: Calibration and Linearity Issues G. V. Tsoulos, J. McGeehan, and M. Beach (lEE Proceedings of Radar, Sonar, and Navigation, February 1998).

Chapter 6

Applications and Planning Issues

717

High Data Rate Indoor Wireless Communications Using Antenna Arrays 719 M. J. Gans, R. A. Valenzuela, J. H. Winters, and M. J. Carloni (6th International Symposium on Personal, Indoor and Mobile Radio Communications, September 1995). On Optimizing Base Station Antenna Array Topology for Coverage Extension in Cellular Radio Networks 726 J-W. Liang and A. J. Paulraj (IEEE 45th Vehicular Technology Conference, July 1995). Usage of Adaptive Arrays to Solve Resource Planning Problems 731 M. Frullone, P. Grazioso, C. Passerini, and G. Riva (Proceedings of the 46th Vehicular Technology Conference, April 1996). viii

Contents

Subscriber Location in CDMA Cellular Networks 735 J. Caffery, Jr. and G. L. Stuber (IEEE Transactions on Vehicular Technology, May 1998). On the Capacity Formula for Multiple Input-Multiple Output Wireless Channels: A Geometric Interpretation 745 P. F. Driessen and G. J. Foschini (IEEE Transactions on Communications, February 1999). Optimum Space-Time Processors with Dispersive Interference: Unified Analysis and Required Filter Span 749 S. L. Ariyavisitakul, J. H. Winters, and I. Lee (IEEE Transactions on Communications, July 1999).

Author Index

761

Subject Index

763

ix

Preface

O

technology. The key areas are separated in six chapters as follows:

VE R the last few years, the demand for service provision via the wireless communication bearer has risen beyond all expectations. This fact introduces one of the most demanding technological challenges: the need to increase the spectrum efficiency of wireless networks. Whereas great effort until today has been focused toward the development of modulation methods, coding techniques, communication protocols, and so forth, the antenna-related technology has received significantly less attention up to now. Nevertheless, in order to achieve the ambitious requirements introduced for future wireless systems, new "intelligent" or "self-configured" and highly efficient systems will most certainly be required. In the pursuit for schemes that will solve these problems, attention has recently turned to spatial filtering methods using advanced antenna techniques: adaptive or "smart" antennas. Filtering in the space domain can separate spectrally and temporally overlapping signals from multiple mobile units, and hence the spatial dimension can be exploited as a hybrid multiple access technique complementing the basic underlying multiple access technique. Adaptive antennas have been studied for many years by the sonar and radar research communities as interferenceresistant aids (the first known case of an adaptive antenna dates back to 1959: L. C. Van Atta, Electromagnetic Reflection, U.S. Patent 29080002, October 6, 1959), and their main application until recently has been military. Advances in processor cost and speed have only recently made it possible to overcome the major obstacle of hardware cost and complexity and start considering the possibility of applying this technique to commercial communications. This book targets a very wide audience. It can be used as a reference source (e.g., in conjunction with other texts on signal processing, antennas, mobile communications) for students at the undergraduate and/or postgraduate level, academics, researchers, professionals, and managers who either are specifically interested or want to understand general aspects of this technology. In order to achieve these goals, a large number of published works on a variety of issues related to adaptive antennas have been gathered. The papers included in this volume, along with the cited references, constitute a very detailed source of information dealing with almost all the important issues related to this

Chapter 1: One introductory paper that provides important background information on adaptive antennas, followed by three papers with the channel models necessary for simulations and material dealing with the spatial characteristics of the radio channel for different operational environments. Chapter 2: Nineteen papers with the most representative, widely used and researched adaptive methods and algorithms such as MUSIC, ESPRIT, and SAGE. Chapter 3: Twenty-seven papers dealing with the issue that has attracted most of the attention in terms of research up to now, the performance of adaptive antennas with different adaptive methods and algorithms under a variety of conditions in mobile communication environments. Chapter 4: Thirteen papers dealing with implementation issues for adaptive antennas: beamforming techniques, calibration, mutual coupling, nonlinearity problems, and so forth. Chapter 5: Eight papers presenting experimental results for issues related to this technology, mainly from adaptive antenna test beds. Chapter 6: Six papers that deal with more general issues related to adaptive antennas such as specific applications for user location, indoor wireless high data rate networks, planning issues for adaptive antennas, and novel techniques that seem promising to open new directions for this technology in the future. The work included in the different chapters is sorted chronologically except for papers that present overviews or comparisons for the issues of focus in a chapter. The latter are always at the beginning of the chapter. I sincerely hope that you find this source of reference useful. If it manages to stimulate you and as a result opens new horizons for you in this very exciting and promising area, then it will have succeeded in its purpose.

George V. Tsoulos xi

Chapter One Introduction and Channel Models

A

DAPTIVE antenna arrays have long been an attractive solution to a plethora of problems related to signal detection and estimation. An array of antenna elements can overcome the directivity and beamwidth limitations of a single antenna element, and when it is combined with methods from statistical detection and estimation and control theory, a self-adjusting or adaptive system emerges. This key capability was recognized in 1967 by Widrow and his colleagues in their publication in the IEEE Proceedings, with which this book opens. This paper offers a valuable introduction to the adaptive antenna concepts. A smart antenna system relies heavily on the spatial characteristics of the operational environment to improve the output signal. In order to study the performance of

adaptive algorithms in radio operational environments (Chapters 2 and 3), it is essential to employ suitable channel models that provide both spatial and temporal information. For that reason, three papers are included in this chapter. There is still a lot of work to be done in terms of characterizing the radio channel and producing propagation models capable of providing all the information needed to efficiently study wideband systems that also exploit the spatial dimension. This need was recently underlined by the international standardization organisations, and several research activities are already under way (e.g., subgroup on spatial propagation models of the COST-European Union Forum for Cooperative Scientific ResearchAction 259).

Adaptive Antenna Systems B. WIDROW,

MEMBER, IEEE,

P. E. MANTEY, MEMBER, IEEE, L. J. GRIFFITHS, B. B. GOODE, STUDENT MEMBER, IEEE

STUDENT MEMBER, IEEE, AND

Ahstract-A system consisting of an antenna array and an adaptive processor can perform filtering in both the space and the frequency domains, thus reducing the sensitivity of the signal-receiving system to interfering directional noise sources. Variable weights of a signal processor can be automatically adjusted by a simple adaptive technique based on the least-mean-squares (LJ\lS) algorithm. During the adaptive process an injected pilot signal simulates a received signal from a desired "Iook' direction. This allows the array to be "trained" so that its directivity pattern has a main lobe in the previously specified look direction. At the same time, the array processing system can reject any incident noises, whose directions of propagation are different from the desired look direction, by forming appropriate nulls in the antenna directivity pattern. The array adapts itself to fonn a main lobe, with its direction and bandwidth determined by the pilot signal, and to reject signals or noises occurring outside the main lobe as well as possible in the minimum meansquare error sense. Several examples illustrate the convergence of the L~IS adaptation procedure toward the corresponding Wiener optimum solutions. Rates of adaptation and misadjustments of the solutions are predicted theoretically and checked experimentally. Substantial reductions in noise reception are demonstrated in computer-simulated experiments. The techniques described are applicable to signal-reech'ing arrays for use over a wide range of frequencies.

T

INTRODUCTIO~

H E SENSITIVITY of a signal-receiving array to interfering noise sources can be reduced by suitable processing of the outputs of the individual array elements. The combination of array and processing acts as a filter in both space and frequency. This paper describes a method of applying the techniques of adaptive filtering! I} to the design of a receiving antenna system which can extract directional signals from the medium with minimum distortion due to noise. This system will be called an adaptive array. The adaptation process is based on minimization of mean-square error by the LMS algorithm.[2 1- [-+] The system operates with knowledge of the direction of arrival and spectrum of the signal, but with no knowledge of the noise field. The adaptive array promises to be useful whenever there is interference that possesses some degree of spatial correlation ~ such conditions manifest themselves over the entire spectrum, from seismic to radar frequencies. .

Manuscript received May 29. 1967; revised September 5, 1967. B. Widrow and L. J. Griffiths are with the Department of Electrical Engineering, Stanford University, Stanford, Calif. P. E. Mantey was formerly with the Department of Electrical Engineering, Stanford University. He is presently with the Control and Dynamical Systems Group, IBM Research Laboratories, San Jose. Calif. B. B. Goode is with the Department of Electrical Engineering, Stanford University, Stanford, Calif., and the Navy Electronics Laboratory, San Diego, Calif.

The term "adaptive antenna" has previously been used by Van Atta[5] and others!"! to describe a self-phasing antenna system which reradiates a signal in the direction from which it was received. This type of system is called adaptive because it performs without any prior knowledge of the direction in which it is to transmit. For clarity, such a systern might be called an adaptive transmittinq array: whereas the system described in this paper might be called an adaptive receiving array. The term "adaptive filter" has been used by Jakowatz, Shuey, and White[7] to describe a systern which extracts an unknown signal from noise, where the signal waveform recurs frequently at random intervals. Davisson!"! has described a method for estimating an unknown signal waveform in the presence of white noise of unknown variance. Glaser!"! has described an adaptive system suitable for the detection of a pulse signal of fixed but unknown waveform, Previous work on array signal processing directly related to the present paper was done by Bryn. Merrnoz, and Shor. The problem of detecting Gaussian signals in additive Gaussian noise fields was studied by Bryn, (lOl who showed that. assuming K antenna elements in the array, the Bayes optimum detector could be implemented by either K 2 linear filters followed by "conventional" beam-forming for each possible signal direction, or by K linear filters for each possible signal direction. In either case, the measurement and inversion of a 2K by 2K correlation matrix was required at a large number of frequencies in the band of the signal. Merrnoz! 11] proposed a similar scheme for narrowband known signals, using the signal-to-noise ratio as a performance criterion. Shor[l:!] also used a signal-to-noise-ratio criterion to detect narrowband pulse signals. He proposed that the sensors be switched off when the signal was known to be absent, and a pilot signal injected as if it were a noisefree signal impinging on the array from a specified direction. The need for specific matrix inversion was circumvented by calculating the gradient of the ratio between the output power due to pilot signal and the output power due to noise, and using the method of steepest descent. At the same time, the number of correlation measurements required was reduced, by Shor's procedure, to 4K at each step in the adjustment of the processor. Both Mermoz and Shor have suggested the possibility of real-time adaptation. This paper presents a potentially simpler scheme for obtaining the desired array processing improvement in real time. The performance criterion used is minimum meansquare error. The statistics of the signal are assumed

Reprinted from IEEE Proceedings, \'01 55, No. 12, pp. 2143-2159, December 1967.

3

to be known, but no prior knowledge or direct measurements of the noise field are required in this scheme. The adaptive array processor considered in the study may be automatically adjusted (adapted) according to a simple iterative algorithm, and the procedure does not directly involve the computation of any correlation coefficients or the inversion of matrices. The input signals are used only once, as they occur, in the adaptation process. There is no need to store past input data; but there is a need to store the processor adjustment values, i.e., the processor weighting coefficients Cweigh ts" ). Methods of adaptation are presented here, which may be implemented with either analog or digital adaptive circuits, or by digital-computer realization.

a cycle at frequency j~ (i.e., a 90° phase shift), denoted by 1/(4/0}. The output signal is the sum of all the weighted signals, and since all weights are set to unit values, the directivity pattern at frequency fo is by symmetry the same as that of Fig. l(a). For purposes of illustration, an interfering directional sinusoidal "noise" of frequency wfo incident on the array is shown in Fig. 2(a), indicated by the dotted arrow. The angle of incidence (45.5°) of this noise is such that it would be received on one of the side lobes of the directivity pattern with a sensitivity only 17 dB less than that of the main lobe at 0=0°. If the weights are now set as indicated in Fig. 2(b)., the directivity pattern at frequency j~ becomes as shown in that figure. In this case, the main lobe is almost unchanged from that shown in Figs. l(a) and 2(a), while the particular side lobe that previously intercepted the sinusoidal noise in DIRECTIONAL AND SPATIAL FILTERING Fig. 2(a) has been shifted so that a null is now placed in the An example of a linear-array receiving antenna is shown in direction of that noise. The sensitivity in the noise direction Fig. l(a) and (b). The antenna of Fig. l(a) consists of seven is 77 dB below the main lobe sensitivity, improving the noise isotropic elements spaced ;"0/2 apart along a straight line, rejection by 60 dB. where i.. o is the wavelength of the center frequency j~ of A simple example follows which illustrates the existence the array. The received signals are summed to produce an and calculation of a set of weights which will cause a signal array output signal. The directivity pattern.. i.e., the relative from a desired direction to be accepted while a "noise from sensitivity of response to signals from various directions, a different direction is rejected. Such an example is illusis plotted in this figure in a plane over an angular range of -n/2 < f) < rc/2 for frequency J~. This pattern is symmetric trated in Fig. 3. Let the signal arriving from the deabout the vertical line 8 = o. The main lobe is cen tered at sired direction 0 = 0 be called the "pilot" signal p( t) = P () = O. The largest-amplitude side lobe.. at V= 24 . has a sin wot., where Wo ~ 2rrj~, and let the other signal. the noise, be chosen as n(l) = N sin ((Jot, incident to the receiving array maximum sensitivity which is 12.5 dB below the maximum at an angle 0 = rt/6 radians. Both the pilot signal and the main-lobe sensitivity. This pattern would be different if it noise signal are assumed for this example to be at exactly were plotted at frequencies other than f~. the same frequency .f~. At a point in space midway between The same array configuration is shown in Fig. ltb): howthe antenna array elements, the signal and the noise are ever.. in this case the output of each element is delayed in assumed to be in phase. In the example shown, there are time before being summed. The resulting directivity pattern. two identical omnidirectional array elements. spaced i~o/2 now has its main lobe at an angle of'" radians.. where apart. The signals received by each element are fed to two variable weights., one weight being preceded by a quarter. - 1 (;~oc5j~) 1 ( 1) ~ = SID - - = sIn wave time delay of 1/(4j~). The four weighted signals are d d then summed to form the array output. The problem of obtaining a set of weights to accept p(t) in which and reject net) can now be studied. Note that with any set wfo = frequency of received signal of nonzero weights, the output is of the form A sin (wa t + 4», ;"0 = wavelength at frequency j~ and a number of solutions exist which will make the output b = time-delay difference between neighboring-element be p(t). However, the output of the array must be indepenoutputs dent of the amplitude and phase of the noise signal if the d = spacing between antenna elements array is to be regarded as rejecting the noise. Satisfaction of c = signal propagation velocity = ;"oj~. this constraint leads to a unique set of weights determined as The sensitivity is maximum at angle t/J because signals re- follows. The array output due to the pilot signal is ceived from a plane wave source incident at this angle" and delayed as in Fig. l(b), are in phase with one another and produce the maximum output signal. For the example illustrated in the figure, d=A.o/2, £5=(0.12941/ j~), and thereFor this output to be equal to the desired output of p(t)=P fore t/!=sin- 1 (2£5/0 ) = 15°. There are many possible configurations for phased arrays. sin wot (which is the pilot signal itself), it is necessary that Fig. 2(a) shows one such configuration where each of the antenna-element outputs is weighted by two weights in (3) parallel, one being preceded by a time delay of a quarter of 4

. - (C6)'

4

LOOK

DIRECTION

o·

DIRECTIVITY PATTERN

"S

NOSE

S·~RECTION /(NOISE AT FRED. t.)

ANTENNA ELEMENTS

/\ d

\

I a)

o· ,

I bl W, ""'

0 .0 99

w:t- - 1.255

w] = - 0 2 6 6

.....4 =- I 51 8 """,""' 0 . 18 2 w. ~- I 6 10 w.," O.OCX>

Fig . I.

Dir ecti vit y pattern fo r a linea r a rray . (a ) Sim ple a rr ay . (0 ) Dela ys added .

" PILOT" SIGNAL p(t)=PSltlw..t

F ig. 2.

l

....\. - - I 255

Direct ivity pa tte rn o f linea r a rray . (a ) With equal weighting. (b ) With weighting fo r no ise elimina tio n.

/ ' NOISE" / n (t ) =NSl n( £)..1

I

/--K

I i

.....- - 1 2 3 3 .... ... - 0 18 2 ""10=-- 1.6 10 '* 11 0 26 6 w1z· - 1 5 19 'IIIII [] _ -0999

6 ;.'

/

/

/

i

'-------o t~f~T Fig. 3.

Array confi guration for noise elimination example.

5

With respect to the midpoint between the antenna elements, the relative time delays of the noise at the two antenna elements are ± [1/(4fo)]sin n/6 = ± 1/(810)= ± Ao/(8c), which corresponds to phase shifts of ± n/4 at frequency.fo . The array output due to the incident noise at () = n/6 is then

N

[WI sin

(root -~) + W2sin (root - 34n) + W3 sin (root + ~) +

W4

sin

(wot - ~)]

(4)

For this response to equal zero, it is necessary that Fig . 4.

(5)

Adaptive array configuration for receiving narrowband signals .

Thus the set of weights that satisfies the signal and noise response requirements can be found by solving (3) and (5) simultaneously. The solution is

(6) With these weights, the array will have the desired properties in that it will accept a signal from the desired direction. while rejecting a noise, even a noise which is at the same frequency 10 as the signal, because the noise comes from a different direction than does the signal. The foregoing method of calculating the weights is more illustrative than practical. This method is usable when there are only a small number of directional noise sources, when the noises are monochromatic, and when the directions of the noises are known a priori. A practical processor should not require detailed information about the number and the nature of the noises. The adaptive processor described in the following meets this requirement. It recursively solves a sequence of simultaneous equations. which are generally overspecified, and it finds solutions which minimize the mean-square error between the pilot signal and the total array output.

;;:5 , Fig . 5.

0

Adaptive ar ra y contigurauou for receiving broadband signals.

Thus the two weights and the I 14};) time delay provide completely adjustable linear processing for narrowband signals received by each individual antenna element. The full array of Fig. 4 represents a completely general way of combining the antenna-element signals in an adjustable linear structure when the received signals and noises are narrowband. It should be realized that the same generality (for narrowband signals) can be achieved even when the time delays do not result in a phase shift of exactly nl2 at the center frequency }~. Keeping the phase shifts CONFIGURAnONS OF ADAPTIVE ARRA YS clo se to 7[/2 is desirable for keeping required weight values Before discussing methods of adaptive filtering and signal small. but is not necessary in principle. processing to be used in the adaptive array, various spatial When one is interested in receiving signals over a wide and electrical configurations of antenna arrays will be band of frequencies. each of the phase shifters in Fig. 4 can considered. An adaptive array configuration for processing be replaced by a tapped-delay-line network as shown in narrowband signals is shown in Fig. 4. Each individual Fig. 5. This tapped delay line permits adjustment of gain antenna element is shown connected to a variable weight and phase as desired at a number of frequencies over the and to a quarter-period time delay whose output is in band of interest. If the tap spacing is sufficiently close, this turn connected to another variable weight. The weighted network approximates the ideal filter which would allow signals are summed, as shown in the figure. The signal, complete control of the gain and phase at each frequency assumed to be either monochromatic or narrowband, is in the passband . received by the antenna element and is thus weighted by a ADAPTIVE SIGNAL PROCESSORS complex gain factor AeP. Any phase angle (x, (/).

mean-square-error function is to take the gradient of a single time sample of the squared error

(18)

When the choice of the weights is optimized, the gradient is zero. Then

....~2----:3 - ---:.

10 )

REL ATIV E FREOUENCY. f I I.

10 )

'0 DESIRED LOOK

08

a::

~ 06 w

'" 6 z

8--13'

DIRECTION-

SPECTRUM OF NOiSE INCIDENT AT - 70 ·

SPECTRUM OF NOISE INCIDENT AT 50'

w

~ 0.4

'"a: ..J

Fig. 19. Comparison o r optimum broadband dire cti vity pattern with experimental pattern after former has been adapted during 625 cycles o r .I~ . (Plo tted at frequency fo ') (a ) Optimum pattern. (b) Adapted with k , = -0.00025.

W

0.2

(b)

Fig. 17.

I 2 3 RELATI VE FREOUENCY, f / f.

4

Freq ue ncy spect ra to r br oadband experiments. (a I Pilot signa l a t 0= - 13 ' . (b) Incid ent noises at 0 = 50 ' and 0 = - 70 '

15

quency 10, and a total delay-line length of one wavelength 19(b), the broadband directivity pattern which resulted at this frequency. from adaptation (after 625 cycles of 10' with k s = -0.0025) The computer-simulated noise field consisted of two is plotted for comparison with the optimum broadband wideband directional noise sources/ incident on the array pattern. Note that the patterns are almost indistinguishable at angles ()= 50 u and 0 = - 70 u • Each source of noise had from each other. power 0.5. The noise at 8= 50° had the same frequency The learning curves of Fig. 18(a) and (b) are composed spectrum as the pilot signal (though with reduced power): of decaying exponentials of various time constants. When while the noise at 8 = - 70° was narrower and centered at k, is set to - 0.00025, in Fig. 18(b), the misadjustment is a slightly higher frequency. The noise sources were un- about 1.3 percent, which is a quite small, but practical value. correlated with the pilot signal. Fig. 17(b) shows these fre- With this rate of adaptation, it can be seen from Fig. 18(b) quency spectra. Additive "white" Gaussian noises (mutually that adapting transients are essentially finished after about uncorrelated) of power 0.0625 were also present in each of 500 cycles of"j~. If j~ is 1 MHz, for example, adaptation could be completed (if the adaptation circuitry is fast the antenna-element signals. enough) in about 500 ps. If j~ is 1 kHz, adaptation could To demonstrate the effects of adaptation rate, the experiments were performed twice, using two different values be completed in about one-half second. Faster adaptation (-0.0025 and -0.00025) for k s ' the scalar constant in (23). is possible, but there will be more misadjustment. These Fig. 18(a) and (b) shows the learning curves obtained under figures are typical for an adaptive antenna with broadband these conditions. The abscissa of each curve is expressed noise inputs with 25 adaptive weights. For the same level in cycles ofj~, the array center frequency ~ and, as before, of misadjustment, convergence times increase approxithe array was adapted at a rate of twenty times per cycle mately linearly with the number of weights.!!' The ability of this adaptive antenna array to obtain of j~. Note that the faster learning curve is a much more "frequency tuning" is shown in Fig. 20. This figure gives noisy one. Since the statistics of the pilot signal and directional the sensitivities of the adapted array (after 1250 cycles of noises in this example are known (having been generated in j~ at k.. = - 0.(0025) as a function of freq uency for the the computer simulation), it is possible to check measured desired look direction. Fig. 20(a), and for the two noise values of misadjustment against theoretical values. Thus directions. Fig. 20(b) and (c). The spectra of the pilot signal the «D(x, x) matrix is known, and its eigenvalues have been and noises are also shown in the figures. In Fig. 20(a). the adaptive process tends to make the computed.' Using (30) and (31) and the known eigenvalues, the mis- sensitivity of this simple array configuration as close as adjustment for the two values of k, is calculated to give the possible to unity over the band of frequencies where the pilot signal has finite power density. Improved performance following values: might be attained by adding antenna elements and by adding more taps to each delay line: or.. more simply. by bandTheoretical Experimental k.. limiting the output to the passband of the pilot signal. Fig. Value of.W Value of Jt 20(b) and (c) shows the sensitivities of the array in the direc0.1288 0.134 -0.0025 tions of the noises. Illustrated in this figure is the very strik0.Ol70 0.0129 -0.00025 ing reduction of the array sensitivity in the directions of the noises, within their specific passbands. The same idea is The theoretical values of misadjustment check quite well illustrated by the nulls in the broadband directivity patterns with corresponding measured values. which occur in the noise directions" as shown in Fig. 19. From the known statistics the optimum (in the least- After the adaptive transients subsided in this experiment, squares sense) weight vector Wl MS can be computed, us-ing the signal-to-noise ratio was improved by the array over (19). The antenna directivity pattern for this optimum weight that of a single isotropic sensor by a factor of 56. vector WLMS is shown in Fig. 19(a). This is a broadband directivity pattern, in which the relative sensitivity of the I MPLEMENTATION array versus angle of incidence is plotted for a broadband The discrete adaptive processor shown in Figs. 7(a) and received signal having the same frequency spectrum as the pilot signal. This form of directivity pattern has few side 8 could be realized by either a special-purpose digital apparatus or a suitably programmed general-purpose malobes, and nulls which are generally not very deep. In Fig. chine. The antenna signals would need analog-to-digital conversion, and then they would be applied to shift regis2 Broadband directional noises were computer-simulated by first ters or computer memory to realize the effects of the tapped generating a series of uncorrelated ("white") pseudorandom numbers. applying them to an appropriate sampled-data (discrete. digital) filter to delay lines as illustrated in Fig. 5. If the narrowband scheme achieve the proper spectral characteristics. and then applying the reshown in Fig. 4 is to be realized, the time delays can be sulting correlated noise waveform to each of the simulated antenna eleimplemented either digitally or by analog means (phase ments with the appropriate delays to simulate the effect of a propagating wavefront. shifters) before the analog-to-digital conversion process. 3 They are: 10.65,9.83, 5.65~ 5.43, 3.59, 3.44,2.68, 2.13. ) .45~ 1.35. 1.20. The analog adaptive processor shown in Figs. 7(b) and 8 0.99,0.66, 0.60,0.46, 0.29, 0.24, 0.20, 0.16, 0.12. 0.01,0.087,0.083.0.075, could be realized by using conventional analog-computer 0.069.

e

16

stru cture would be a capacitive voltage divider rather than a resistive one . Other possible realizations of analog weights include the use of a Hall -effect multiplier combiner with magnetic storage[241 and also the electrochemical memistor of Wid row and HoffYSI Further effort s will be req uired to impro ve existing weigh ting ' elements and to de velop new ones which are sim ple, chea p, a nd adaptable according to the requ irements of the various adaptation algorithms. The realization of the processor ultimately found to be useful in cert ain application s may be composed of a combination of analog and d igital techniques.

10

0.8 -,.- ARRAY SENSITIVITY 8~-13'

06

0 .4

0 .2

o o~'----'--_>-J._---'------' I 2 RELATIVE FREOUENC Y,

( a)

3

4

t / I.

R ELAXATION ALGORITHMS AND TH EIR IMPLEM ENTATI ON 10

ARRAY

/ SENSITIVITY .' AT~

08

>j-,

~

;;;

06

z

~

~

a:

04

"

02

k .~

- 0 .00025

O ;--~-7"':-----"~---~--~

Ib ) 0

08

I 2 3 RELATIVE FREOUENCY. 1 / t.

4

.sPECTRUM . OF NOISE AT - 70· J.RRAY

.' SENSITIVIT Y AT - 70 '

06

04

0.2

( c)

°O~-~--:---=-'->-::---:------' 4 I 2 3 RELA TIVE FREOUENCY. f / f.

F ig. 20. Array sens itivity vers us frequen cy, for broadband experime nt of F ig. 19. (a) Desired look direct ion. Ii= -13 . (b) Sensiti vity In o ne no ise di rectio n. Ii= 50' . (e ) Sen siti vity in the o ther noise d irectio n. IJ= - 70 .

apparatus, such as multipliers, integrators, summers, etc. More economical realizations that would , in add ition. be more suitable fo r high-frequency operation might use fieldeffect transistors as the variable-gain multipliers, whose control (gate ) signals could come from capacitors used as integrators to form and store the weight values . On the other hand, instead of using a va ria ble resistance struc ture to form the vector dot products, the sa me functi on could be achieved using variable-voltage capacitors, with ordinary capacitors again storing the weight values . The resulting

Algorithms other than the LMS procedure described in the foregoing exist that may permit considerable decrease in complexity with specific adaptive circuit implementations. One method of adaptation which may be easy to imp lement electronicall y is based on a relaxation algorithm described by Southwell. [26 ] This algo rithm uses the sa me erro r signal as used in the LMS technique. An estimated mean-squ are error formed by squaring and averaging this error signal over a finite time interval is used in determining the proper weight adjustment. Th e relaxat ion algorithm adjusts one weight at a tim e in a cyclic sequence, Each weight in its turn is adjusted to minimize the measured mean-squ are err or. This method is in contras t to the simultaneou s adj ustment procedure of the LMS steepest-descent a lgo rithm. The relaxation procedure can be shown to produce a misadjustment th at increases with the square of the number of weight s, as opposed to the LMS algorithm whose misadju stm ent increases o nly linearl y with the number of weights. For a given level of misadjustment, the ada pta tio n sett ling time of the relax ation process increases with the sq ua re of the number of weights. For implementation of the Southwell relaxation algo rithm. the configura tio ns of the array and adaptive processor remain the sa me, as does the use of the pilot signal. Th e relaxat ion a lgorithm will work with eith er the two-mode o r th e one-mode adaptation pr ocess. Savings in circu itry ma y result , in that cha nges in th e adj ustments of the weight values depend only upon error measurements and not upon confi gurations of error measurements and simultaneous input-signal mea surements. Circuitry for implementing the LMS systems as shown in Fig. 7(a)' and (b) ma y be more co mplicated. The relaxati on method ma y be applicable in cases where the adjustments are not obvious " weight" settings. For example, in a microwave system, the adj ustments might be a system of motor-dri ven apertures or tuning stubs in a waveguid e or a network of waveguides feeding an antenna. Or the adj ustments may be in the antenna geometry itself. In such cas es, th e mean-square error can still be measured, but it is likely that it would not be a simple quadratic fun ction of the adjustment parameters, In any event, some very interesting pos sibilities in automatic optimization are presen ted by relaxation adaptation methods.

17

has been computer simulated and shown to operate as expected. However. much work of a theoretical and experimental nature needs to be done on capture and rejection phenomena in such systems before they can be reported in detail.

OTHER ApPLICATIONS AND FURTHER WORK ON

ADAPTIVE ANTENNAS

Work is continuing on the proper choice of pilot signals to achieve the best trade-off between response in the desired look direction and rejection of noises. The subject of "nullsteering, where the adaptive algorithm causes the nulls of the directivity pattern to track moving noise sources" is also being studied. The LMS criterion used as the performance measure in this paper minimizes the mean-square error between the array output and the pilot signal waveform. It is a useful performance measure for signal extraction purposes. For sicnal detection, however, maximization of array output signal-to-noise ratio is desirable. Algorithms which achieve the maximum SNR solution are also being studied. Goode[27] has described a method for synthesizing the optimal Bayes detector for continuous waveforms using Wiener (LMS) filters. A third criterion under investigation has been discussed by Kelley and Levin[28] and . more recently" applied by Capon et ale [29] to the processing of large aperture seismic array (LASA) data. This filter. the maximum-likelihood array processor.. is constrained to provide a distortionless signal estimate and simultaneously minimize output noise power. Griffiths' and em are the angle of arrival measured relative to the line of sight 2UU m [16] . In [3]. it is s t a t e d th at t h e from the base station and the mobile, respectively. active scattering region around the mobile is a bo u t 100-200 wav el en gth s fo r 900

- --.{

(D

)(D

(D

)(D\ ·+"t

25

c: - 1 ~-2

G WSSUS. Figure 13 shows the geometry assumed for the GWSSUS model corresponding to £I = 3 clusters. The mean AOA for the kth cluster is denoted SOk. It is assumed that the location and delay associated with each cluster remains constant over several data bursts. b. The form of the received signal vector is

]' -3

'; -4 -5

.~

~ -6

~ -7

ii

'"o

.0

5

et

X,,(l) =

Time of arrival (us)

Angle of arrival (degrees)

where Vk.b is the superposition of the steering vectors during the bth data burst within the kth cluster. which mav be expressed as . v, "t." = I<XC ieJQ I.'a(SOk -Su).

• Figure 12.Joint TOA and ADA probability densityfunction.

elliptical model (log-scale).

i= 1

where N k denotes the number of scatterers in the kth cluster. <Xk.i is the amplitude. lPkj is the phase. Sk.i is the angle of arrival of the ith ret1ected scatterer of the kth cluster, and a( S) is the array response vector in the direction of S [9). It is assumed that the steering vectors are independent for different k. If v, is sufficiently large (approximately 10 or more [J 9]) for each duster of scatterers, the central limit theorem may be applied to the elements of Vk./> . Under this condition, the elements of vk.b are Gaussian distributed. Additionally. it is assumed that Vk./, is wide sense stationary. The time delays tk are assumed to be constant over several bursts. b. whereas the phases

-20

~

-25

c::

10-2

10 '!

...c,

~ II)

til >.

100

0

\

BS2

2

o

8

1.5

"

00

0 a

0

0

0

0 0

0

00 0

0

"

o0"__ 0

0

00

"

0

•

0.8

t

\

\ \ \ \

\

'

\

400

\

\

c= 0.5: c=8·7: c= .9:

\ \

0.4

c

"" ,,"" " " ,,"

500

-'

!

\

.e)( /
- -E [Q v.nilv.m l> -PIQV.H-2-·

(A-9)

+ 2abE[rVIlE[rHllE[cos(8v1-

E[P vl ]2 + R (V t , V 2) .

a~ '

The antenna branch voltages in (A-7) are a sum of two Rayleigh fading signals which produce a new Rayleigh fading signal as the quadrature components are Gaussian distributed). Then, the power relationship of (A-4) applies for PI and P2 as well, and the power correlation coefficient {Jpowl2 for the two tilted antennas is

E[P,l = a2E[P vd + b2E [PHI l

r

ponents of field components given in (A-I) as

Comparing the two polarization correlations by substituting (A-5) into (A-2), we have

+ brHl cos(8H1W + [arvi sin(8v,) + brHI sin(8 H1 W a2PVI + b2PHI + 2abrVIrHi cos(8 v, b2PV2+ a2PH2 - 2abrV2rH2 cos( 8V2 -

(a 4 + b4 )

E[PV1Pd

R( VI, V 2 ) can be expressed directly by the quadrature com-

Fig. 13. Square quadrature component correlation coefficients (pi, + pJQ) and power correlation coefficient versus square module of complex correlation coefficient (IP12I')·

VI = aE V1 + bE HI :

(1 + ;2)

(A-15)

For Rayleigh fading signals, it can be found that PIQ = - PQI and PII = PQQ [13]. Furthermore, the complex correlation is related as PIZ = 1/2(PII + PQQ) + j1l2(PIQ - PQI = PII + jPIQ' From (7], it follows that PII + PIQ = Ipd z = Ppowv . This is shown in Fig. 13 for the data presented in this paper. The quadrature component correlations (offset by 0.4 for clarity) have an extremely high correlation (cc = 0.995) with the squared complex correlation coefficients. The power

r:=

.9

1

~

0.... .... 0 u

)(

:rt?
C

>C

>C~

)(

... 0

N

I

cc=0.909

0

-1

-1

-0.5

ACKNOWLEDGMENT

I

>C

·C

~

~

XX

Xx

)(

tU -0.5 s::

:r:

x

)()(.

)(

.J::

0.. ,

into (A-II), the power correlation coefficient in (A-9) reduces to the expression (9) shown in Section VI.

)(

0.5

0

The authors acknowledge the participation of C. Jensen in the experimental part of this work during his employment at Aalborg University, and H. Fredskild from Telecom Denmark Telelaboratoriet for providing a second matching base-station antenna. The comments of the reviewers, who helped enhance the quality of this paper, are greatly appreciated.

REFERENCES

1

[1] B. Glance and L. J. Greenstein, "Frequency-selective fading effects in digital mobile radio with diversity combining" IEEE Trans. Commun., vol. 31, no. 2, pp. 1085-1094, Sept. 1983. [2] J. B. Andersen, P. C. F. Eggers, and B. L. Andersen, "Propagation aspects of datacommunications over the radio channel-A tutorial," in Proc. EUROCON '88, Stockholm, Sweden, pp. 301-307. [3] B. L. Andersen, P. C. F. Eggers, and J. B. Andersen, "Time and phase variations in the mobile channel," in Proc. Nordic Radio Symp. '89, Saltsjobaden, Sweden, pp. 297-304. [4) P. E. Mogensen, P. C. F. Eggers, C. Jensen, and J. B. Andersen, "Urban area radio propagation measurements at 955 and 1845 MHz for small and micro cells," in Proc. IEEE GLOBECOM'91, pp. 1297-1302. [5] P. E. Mogensen, "Preliminary results from short-term measurements in urban area," COST 231 TD(90)-88, Paris, France, Oct. 8-11, 1990. [6] A. A. M. Saleh and R. A. Valenzuela, "A statistical model for indoor multipath propagation," IEEE J. Select. Areas Commun., vol. 5, no. 2, pp. 128-137, Feb. 1987. [7] J. N. Pierce and S. Stein, "Multipath diversity with nonindependent fading," IRE Proc., pp. 89-104, Jan. 1960. [8] M. T. Feeney and J. P. Parsons, "Cross-correlation between 900 MHz signals received on vertically separated antennas in small-cell mobile radio systems," lEE Proc., vol. 138, no. 2, pp. 81-86, Apr. 1991. [9] S. Kozono, T. Tsuruhara, and M. Sakamoto, "Base station polarization diversity reception for mobile radio," IEEE Trans. Vehic. Technol., vol. 33, no. 4, pp. 301-306, Nov. 1984. [10] P. C. F. Eggers and J. B. Andersen, "Base station diversity for NMT900," in Proc. Nordic Radio Symp., Saltsjobaden, Sweden, pp. 77-85. [11] A. M. D. Turkmani and J. P. Parsons, "Characterisation of mobile radio signals: base station crosscorrelation," lEE Proc., vol. 138, no. 6, pp. 557-565, Dec. 1991. [12) R. G. Vaughan, "Polarization diversity in mobile communications," IEEE Trans. Vehic. Technol., vol. 39, no. 3, pp. 177-186, Aug. 1990. [13] D. E. Kerr, Propagation of Short Radio Waves. Boston, MA: Boston Tech. Pub. Inc., 1964.

Vertical inter-in-phase correlation (a)

e .9

1

I

I

I

)(

~

~

.... .... 0 u

....

~

)( >sc)(

~ )(

0.5 -

)(

)(

)( )(

~

~ ....

"0

~

='

':r :n

'"

0--

)(

:n

)(

0.... u

)(

co -0.5 ~

=

)(

cc=O.8809

0

.:N a

~ ~

)C'

-1

-1

-0.5

0

0.5

1

Vertical cross-quadrature correlation ( b) Fig. 14. Horizontal versus vertical (a) inter-in-phase (p,,) and (b) crossquadrature (PtQ) correlation coefficients for BS2.

correlation coefficients have a slightly poorer correlation (cc = 0.96), but are still very high. With the previous discussion, we can reduce (A-I2) (with the use of (A-14) and (A-I5) and rearranging) to

(A-16) It follows from Fig. 14, that it is reasonable to assume that the quadrature component correlations are approximately equal for the two polarizations, though the shown data only represent BS2. At BSl, the antenna cables were changed between the vertical and horizontal polarization experiment. Thus, the phase offset on the complex correlations will be changed from the vertical to horizontal experiment. Consequently, the PH and PIQ relations between the two polarizations at BSI are altered and not usable. With this discussion and inserting (A-16)

43

A Statistical Model for Angle of Arrival in Indoor Multipath Propagation Quentin Spencer, Michael Rice, Brian Jeffs, and Michael Jensen Department of Electrical & Computer Engineering Brigham Young University Provo, Utah 84602 [1], whose work was based on the work of Turin. Their work consisted of collecting temporal data on indoor propagation, from which they proposed a time domain model for indoor propagation. Most indoor propagation research has dealt with the time of arrival and paid little attention to the angle of arrival. In order to predict the performance of adaptive array systems, the angle of arrival is very important information. Some recent papers have begun to address the angle of arrival. Lo and Litva [3] found that multipath arrivals tend to occur at varying angles indoors, but were not able to arrive at any conclusions based on their limited data. Guerin [4] collected angular and temporal data separately, but did not correlate the two. Wang, et al [5], used a rectangular array to estimate both the elevation and azimuth angles of arrival for major multipaths, but did not measure the corresponding time of arrival. Litva, et al, [6] collected simultaneous time and angle of arrival data, similar to the format of the data used in this paper. They came to the preliminary conclusion that it is possible to make accurate measurements of this type and learn more about what is happening in the indoor multipath channel. However, their experiment was not extensive enough to make any conclusions about the channel. This paper presents an extension to the Saleh-Valenzuela model which accounts for the angle of arrival. This is based on data that includes information about both the time and angle of arrival, presented in [7]. The Saleh-Valenzuela J model is explained, and the new data is discussed. Model parameters based on the new data are derived and compared to the parameters found by Saleh and Valenzuela at a lower frequency.

Abstract- Multiple antenna systems are a useful way of overcoming the effects of multipath interference, and can allow more efficient use of spectrum. In order to test the effectiveness of various algorithms such as diversity combining, phased array processing, and adaptive array processing in an indoor environment, a channel model is needed which models both the time and angle of arrival in indoor environments. Some data has been collected indoors and some temporal models have been proposed, but no existing model accounts for both time and angle of arrival. This paper discusses existing models for the time of arrival, experimental data that were collected indoors, and a proposed extension of the Saleh-Valenzuela model [1], which accounts for the angle of arrival. Model parameters measured in two different buildings are compared with the parameters presented in the paper by Saleh and Valenzuela, and some statistical validation of the model is presented. I.

INTRODUCTION

There have been many different approaches for overcoming the problem of multipath interference, both in outdoor and indoor applications. Some of them include channel equalization, directional antennas, and multiple antenna systems, each being more particularly suited to different applications. The use of multiple antenna systems can be particularly useful for indoor applications such as local area networks, because they allow the possibility of communicating with multiple users simultaneously over a single frequency band, increasing throughput and making efficient use of frequency spectrum. The signals from different antennas can be combined in various ways, including diversity combining, phased array processing, and adaptive array algorithms. Adaptive array sytems are becoming increasingly feasible for high bandwidth applications with continuing improvements in digital signal processors. In addition, the availability of new, higher frequency bands has made wireless networks an increasinly attractive and feasible option. The effects of multipath interference have been studied extensively in various outdoor scenarios. However, the study of the indoor multipath channel is relatively new. In order to be able to predict the performance of indoor communications systems, models are needed that accurately model the behavior of radio transmissions in indoor environments. Several other researchers have already collected various types of data on indoor mulipath propagation. The foundation for much of today's work was by Turin, et al [2], which was a study of outdoor multipath propagation in an urban environment. The first model for indoor multipath propagation was proposed by Saleh and Valenzuela

II.

THE SALEH- VALENZUELA MODEL

The model proposed by Saleh and Valenzuela is based on a clustering phenomenon observed in their experimental data. In all of their observations, the arrivals came in one or two large groups within a 200 ns observation window. It was observed that the second clusters were attenuated in amplitude, and that rays, or arrivals within a single cluster, also decayed with time. Their model proposes that both of these decaying patterns are exponential with time, and are controlled by two time constants: I', the cluster arrival decay time constant, and " the ray arrival decay time constant. Fig. 1 illustrates this, showing the mean envelope of a three cluster channel.

Reprinted from IEEE Vehicular Technology Conference, pp. 1415-1419, May 1997.

44

Crabtree Building, constructed mostly of steel and gypsum board. Each data set can be viewed as an image plot, with angle as one axis, and time as the second axis. A typical data set is pictured in Fig. 2. The images were processed to remove blurring effects so that the precise time, angle and amplitude of each major multipath arrival is known. The data collection and processing is discussed in greater detail in [7]. Visual observation of the data showed that clustering like that observed by Saleh and Valenzuela was present in the data. The nature of the clustering tended to follow the model of Saleh and Valenzuela quite well. In general, the strength of clusters tended to decay with increasing delay times, and arrivals within each cluster showed a similar pattern of decay. One difference from the Saleh-Valenzuela data is the higher average number of clusters per data set.

The impulse response of the channel is given by:

L L f3 00

h(t)

00

k Lb(t

- Tl

-

rkL)~

(1)

l==O k==O

where the sum over l represents the clusters, and the sum over k represents the arrivals within each cluster. The amplitude of each arrival is given by {3kl, which is a Rayleigh distributed random variable, whose mean square value is described by the double-exponential decay illustrated in Fig. 1. Mathematically it is given by:

(2)

{32 (T l , Tkl)

,82 (0, 0) e -

T 1/

r e-

Ik I / , ,

(3)

where {32 (0, 0) is the average power of the first arrival of the first cluster. This average power is determined by the separation distance of transmitter and receiver. The time of arrival is described by two Poisson processes which model the arrival times of clusters and the arrival times of rays within clusters. The time of arrival of each cluster is an exponentially distributed random variable conditioned on the time of arrival of the previous cluster. The case is the same for each ray, or arrival within a cluster. Following the terminology used by Saleh and Valenzuela, rays shall refer to arrivals within clusters, so that the cluster arrival rate implies the parameter for the intercluster arrival times and the ray arrival rate refers to the parameter for the intracluster arrival times. The distributions of these arrival times are shown in equations 4 and 5:

p(T1\T1- 1 ) p( TkL IT(k-l)l)

Ae-;\(T/-T/- 1 )

(4)

Ae-A(Tld-T(k-l)/) ,

(5)

IV.

NluLTIPATH PROPAGATION

In this section we propose a statistical model for the indoor multipath channel that includes a modified version of the Saleh- Valenzuela model, and incorporates an angle-ofarrival model. In addition, methods of estimating parameters from the data are discussed. A.

Time of Arrival

The time and amplitude of arrival portion of the combined model is represented by h(t) in equation (1), where, as before, {3~l is the mean square value of the kth arrival of the lth cluster. This mean square value is described by the exponential decay given in equation (3) and illustrated in Fig. 1. As before, the ray arrival time within a cluster is given by the Poisson distribution of equation (5), and the first arrival of each cluster is given by T l , described by the Poisson distribution of (4). The inter-ray arrival times, Tkl, are dependent on the time of the first arrival in the cluster Tl · In the Saleh-Valenzuela model, the first cluster time T 1 was dependent on To which was assumed to be zero. With the estimated parameter in [1] of 1/ A ~ 300 ns, the first arrival time will typically be in the range of 200 to 300 ns, which is a reasonable figure. However, a problem with this was found when the A parameter in the new data was discovered to be very low, but the delay time to the first arrival was often still on the order of 200 ns. Under the Saleh-Valenzuela model, this would make any long delays which would occur at larger separation distances between transmitter and receiver highly improbable. To remedy this problem, it is proposed that To be the line of sight propagation time:

where A is the cluster arrival rate, and A is the ray arrival rate. In their data, Saleh and Valenzuela did not have any information on angle of arrival, and assumed that the angles of arrival were uniformly distributed over the interval

[0,271').

Other indoor multipath models have been proposed, such as the model proposed by Ganesh and Pahlavan [8], but they will not be discussed here. The data used in this paper fit the Saleh-Valenzuela model well, and as a result the model was chosen as the basis for the extended model presented here. III.

A PROPOSED TIME/ ANGLE MODEL FOR INDOOR

EXPERIMENTAL DATA

In order to analyze and model the indoor multipath channel, a data gathering apparatus was designed which was able to take simultaneous measurements of the time and angle of arrival. The frequency band was from 6.75 to 7.25 GHz. Using the system, a total of 65 data sets were collected in two buildings on the Brigham Young University Campus. In the Clyde building, a reinforced concrete and cinder block building, 55 data sets were collected. For comparison, ten additional data sets were collected in the

To

r , c

(6)

where c is the speed of light, and r is the separation distance. This allows for the time of the first arrival to be more directly dependent on the separation distance.

45

B.

Angle of Arrival

D.

It will be assumed that time and angle are statistically independent. If there were a correlation, it would be expected that a longer time delay would correspond to a larger angular variance from the mean of a cluster. This was not observed in the data, so at this point an assumption of independence is reasonable, but further study of the correlation structure may be warranted. The consequence of this independence is that the complete impulse response with respect to both time and angle, which we will call h( t, B), becomes a separable function:

h(t, B)

~

h(t)h(B).

The extended model for h(t, B) is useful for analysis or simulation of array processing algorithms that might be used in an indoor environment. In order, for example, to conduct a Monte Carlo simulation of an array antenna processor, it is necessary to generate a random channel using the statistical model. This section outlines the procedure for doing so. The first step is to choose the transmitter/receiver separation distance r, which can be chosen either randomly or arbitrarily. Knowing T, the next step is to determine {32 (0, 0), the mean power of the first arrival, which is given by

(7)

As a result, h(B) can be be addressed separately from h(t). We propose an independent angular impulse response of the system, similar to the time impulse response of the channel given in 1:

LL 00

h(B)

(10) where G(lm) is the channel gain at r == 1 meter, and Q is a channel loss parameter. , and B are respectively the ray decay parameter and ray arrival rate in the model for h(t). Equation (10) is derived and the characteristics of Q in the indoor environment are discussed in greater detail in [1]. After 13 2 (0, 0) is determined, the next step is to determine the cluster and ray arrival times. The corresponding distributions are given in equations (4) and (5), where To == r / c. After the times are determined, the mean amplitudes /3 kL are determined by equation 3. The actual amplitudes for each arrival, f3kl, are determined by sampling a Rayleigh distribution whose mean is ,Skl. The angles are determined by first randomly choosing the cluster angles, which are uniforrnly distributed from 0 to 21r. Relative ray angles are then determined by sampling a Laplacian distribution as given in equation (9).

00

fJkl c5 (O- 8l - Wkl),

(8)

l==O k=:.O

where, as before, f3kl is the ray amplitude for the kth arrival in the lth cluster, given in equations (2) and (3). 8 l is the mean angle of each cluster, which is distributed uniformly on the interval [0, 21r). We propose that the ray angle within a cluster, Wkl, be modeled as a zero mean Laplacian distribution with standard deviation a:

p(B)

(9)

The correlation of these distributions to the data is shown in the next section.

c.

Using the Model

Parameter Estimation

v.

This section outlines methods of deriving the distributions and estimating the parameter a given in the previous section. The distribution parameters of the cluster means, 8 l , is found by identifying each of the clusters in a given data set. The mean angle of arrival for each cluster is calculated. In order to remove the specific room geometry and orientation, the first arrival (in time) for each data set is taken as the reference. The relative cluster means are calculated by subtracting the mean of the reference cluster from all other cluster means. To estimate the distribution of cluster means over the ensemble of all data sets, a histogram can be generated of all relative cluster means, disregarding the first clusters (since their relative mean is always 0). The procedure to estimate a is similar. The cluster mean is subtracted from the absolute angle of each ray in the cluster to give a relative arrival angle with respect to the cluster mean. The relative arrivals are collected over the ensemble of all data sets, and a histogram can be generated. Using a least mean square algorithm, the histogram is fit to the closest Laplacian distribution, which gives the value for G'.

IvloDEL PARAtvlETERS FROM THE DATA

The intercluster time decay constant, I', was estimated by normalizing the cluster amplitudes (the amplitude of the first arrival) so that the first one had an amplitude of 1 and a time delay of O..A.lI of the cluster amplitudes were superimposed as shown in Fig. 3. The estimate for r was found by curve fitting the line (representing an exponential curve) to minimize the mean squared error. The values for rand , were estimated for both buildings in a similar manner. In this particular example, the fit is less than ideal, but it was better in the other cases, especially when there were more data points. In their data, Saleh and Valenzuela did not have exact amplitudes available, and as a result were not able to use curve fitting or generate plots as in Fig. 3. Their parameters were as a result very rough estimates, but they did observe the same general decay trend as in this data, which supports the exponential decay model. The Poisson parameters, A and A, representing the intercluster and intracluster arrival rates were estimated by subtracting each arrival time from its predicessor to produce a set of conditional arrival times p(TklIT(k-l)l). The

46

parameter

r !

l/A 1/),

a

Clyde Building 33.6 ns 28.6 ns 16.8 ns 5.1 ns 25.5°

Crabtree Building 78.0 ns 82.2 ns 17.3 ns 6.6 ns 21.5°

rive than the first arrival in the cluster, and are usually attenuated relative to this first arrival. The amplitudes of clusters and rays within clusters both follow the same pattern of exponential decay over time observed by Saleh and Valenzuela. The differences in model parameters are likely due to the difference in frequency (Saleh and Valenzuela used 1.5 GHz). The other discrepancy is in the markedly faster cluster arrival rate, which is most likely explained by the larger overall number of clusters resulting from a more sensitive data gathering apparatus. The model parameters for the Clyde and Crabtree Buildings were in general very similar. The most notable exception is the extremely slow amplitude decay of rays within a cluster in the Crabtree building. In general, the model seemed to be able to accurately describe the differing multipath characteristics in both buildings, regardless of their very different construction. This implies that the model could possibly provide a general representation for many different types of buildings, and model parameters could therefore be found for other types of buildings. The angle-of-arrival model presented here, though yet unconfirmed, is a strong alternative to only previous option for simulation: random assignment of angles or guessing at the anglular properties of the channel. The rnost important area for continued research is applying the model for its intended purpose-comparison of array processing algorithms. This can be done either by mathematical analysis or Monte Carlo simulation. A mathematical analysis is likely intractible due to the large number of variables in the model, but the model can be a very useful tool for the generation of random multipath channels for simulation.

SalehValenzuela 60 ns 20 ns 300 ns 5 ns

Table 1. A comparison of model parameters for the two buildings and from the Saleh-Valenzuela paper [1]

probability distribution of these with the best fitting pdf (for the Clyde Building) is shown in Fig. 4. Fig. 5 shows a CDF of the relative cluster angles for the Clyde Building, illustrating the relatively uniform distribution of clusters in angle. The same was true in the Crabtree Building. The distribution of the ray arrivals with respect to the cluster mean is shown in Fig. 6. The sharp peak at the mean is characteristic of a Laplacian distribution. The superimposed curve is a Laplacian distribution that was fit by integrating a Laplacian PDF over each bin, and matching the curves using a least mean square goodness of fit measure. The Laplacian distribution turns out to be a very close fit in both buildings. Table 1 shows a comparison of the model parameters estimated for the Clyde Building, the Crabtree Building, and those estimated by Saleh and Valenzuela from their data. The most obvious discrepancy is in the estimates for the value of A. This is due to the fact that there were significantly more clusters observed in both the Clyde and Crabtree buildings compared to an average of 1-2 clusters observed by Saleh and Valenzuela. This may be partly due to the higher RF frequency, but the more likely cause is the ability of our testbed to see clusters that were close together in time, but separated in angle. Another interesting phenomenon is that r is very low in the Clyde Building, and ! is larger than r in the Crabtree Building, meaning that the Clyde Building tends to attenuate more than the Crabtree Building. The values of a were close in both buildings, and there is no precedent for comparison with other data.

VI.

REFERENCES

[1J Adel A. M. Saleh and Reinaldo A. Valenzuela. A statistical

[2] [3] (4]

CONCLUSION

Many aspects of the model have plausible physical explanations. Because an absolute angular reference was maintained during the collection of the data, it was possible to compare the processed data with the geometry of each configuration. The strongest cluster was almost always associated with the direct line of sight, even when there were walls blocking the line of sight path. Apparent causes of weaker clusters were back wall reflections and doorway openings. It is likely that each cluster corresponds to a major path to the receiver, and the arrivals within each cluster are likely the result of smaller, closely associated objects that are part of a very similar group of paths to the receiver. These paths will take slightly longer to ar-

(5] [6]

[7]

[8]

47

model for indoor multipath propagation. IEEE Journal on Selected Areas of Communications, SAC-5:128-13, February 1987. George L. 'Turin et al. A statistical model of urban multipath propagation. IEEE Transactions on Vehicular Technology, VT21(1):1-9, February 1972. T. Lo and J. Litva. Angles of arrival of indoor multipath. Electronics Letters, 28(18):1687-1689, August 27 1992. Stephane Guerin. Indoor wideband and narrowband propagation measurements around 60.5 ghz. in an empty and furnished rOOID. In IEEE Vehicular Technology Conference, pages 160164, 1996. Jian-Guo Wang, Ananda S. Mohan, and Tim A Aubrey. Anglesof-arrival of multipath signals in indoor environments. In IEEE Vehicular Technology Conference, pages 155-159. IEEE, 1996. John Litva, Amir Ghaforian, and Vytas Kezys. High-resolution measurements of aoa and time-delay for characterizing indoor propagation environments. In IEEE A ntennas and Propagation Society International Symposium 1996 Digest, volume 2, pages 1490-1493. IEEE, 1996. Quentin Spencer, Michael Rice, Brian Jeffs, and Michael Jensen. Indoor wideband time/angle of arrival multipath propagation results. In IEEE Vehicular Technology Conference. IEEE, 1997. R. Ganesh and K. Pahlavali. Statistical modeling and computer simulation of indoor radio channel. lEE Proceedings-I, 138(3):153-161, June 1991.

10' ~--r--~--r--.,--~--r--.,..--~--r--.,..---,

10

time

F ig. 1. An illust ra t ion of ex p on entia l decay of me an cluster pow er and ray power within clusters

15

20

25

30

delay(ns)

35

40

45

so

F ig. 4. C O F of Relative Ar r ival T imes W ith in C luste rs in the Clyd e 5 .1n s) Bu ild ing (1/.\

=

00014 0.9 0.8 0.7 0.6

E0.5

x

/

0 .4

0.3 0.2

50

1SO

200

angle (degrees)

2SO

300

350

F ig. 5. COF of relative mean cluster a n gles in t he C lyde Bu ild ing wit h resp ect to the fir st clu ster in each set

F ig . 2. A typical raw data set

10'

100

/'

,r

0.12

.-----...-----.,----~---~---~---, x

0.1

10' O.OB 0

~

~ 0.06

..

'0

0.0'

0.02

x

x x

0 - 200

x 10-' L----'------'----'~--'---~~---_'c:_--:!.

a

20

40

60

relative delay (ns)

80

100

120

-1 50

- 100

100

150

F ig . 6. Histogram of relat ive ray arrivals with respect to the cluster mean for the Clyde Building. Superimposed is the best fit Laplacian d istribution (IT = 25 .5°) .

F ig . 3. Plot of normalized cluster amplitude vs, re lative delay for the Clyde Building, with the curve for r = 33 .6 ns superimposed.

48

Chapter 2 Adaptive Algorithms

O

F paramount importance for an adaptive antenna is the manipulation of the signals induced on its elements. Two overview papers on array processing set the scene at the beginning of this chapter and provide the reader with valuable background information on many signal processing issues related to adaptive antennas. They provide a comprehensive and detailed treatment of different beamforming methods, adaptive algorithms to adjust the weights of the antenna array elements according to some optimization criterion, and direction of arrival methods. Some of the adaptive methods that are included here are conventional beamforming, conjugate gradient, least

squares, recursive least squares, linear prediction, maximum likelihood, maximum entropy, minimum norm, constrained optimization, spectral estimation and eigenstructure methods, weighted subspace fitting, and well-known algorithms such as MUSIC (multiple signal classification), ESPRIT (estimation of signal parameters via rotational invariance techniques), SCORE (spectral self-coherence restoral), and MVDR (minimum variance distortionless response). Also, among the topics presented in this chapter are issues related to some of these algorithms, such as spatial smoothing and the estimation of the number of signals impinging on the antenna array.

49

Celebrating a Half Century of Signal Processing

Highlights of Statistical Signal and Array Processing

U

of SSAP . To provide readers with pointers for further study of the field, this article includes a very impressive bibliography-close to 500 references are cited. This is jusr one of the indications that the field of statistical signals has been an extremely active one in the signal-processing community. This article also introduces the recent reorganization of three technical committees of the Signal Processing Society. During the reorganization, the SSAP, Digital Signal Processing, and U nderwater Acoustics Signal Processing technical committees were restructured to form three new committees : Signal Processing Theory and Methods, Signal Processing for Communications, and Sensor Arrays and Multichannel Signal Processing. After the reorganization, research topics that used to belong to the SSAP TC are now distributed to the three new TCs. Therefore, although the name "SSAP'~ does not exist anymore, the research activities related to it have been given a new life and will continue to thrive in the Signal Processing Society . . Now, let me invite you to enjoy this article, which will give you a quick but comprehensive tour ofthe highlights of statistical signal and array processing.

nlike most other technical committees ofthe Signal Processing Society, which deal with signals of deterministic nature and process signals one at a time, the Statistical Signal and Array Processing (SSAP) Technical Committee deals with signals that are random and processes an array ofsignals simultaneously. This issue features the SSAP-TC's contribution to the Anniversary series, which covers this spedal field ofrandom signals and array processing. The field of SSAP represents both solid theory and practical applications. Starting with research in spectrum estimation and statistical modeling, study in this field is always full of elegant mathematical tools such as statistical analysis and matrix theory. The area of statistical signal processing expands into estimation and detection algorithms, time-frequency domain analysis, system identification, and channel modeling and equalization. The area of array signal processing also extends into multichannel filtering, source localization and separation, and so on. Work in SSAP areas has already made an impact in a large variety of applications, ranging from communication and radar/sonar processing to many medical imaging technologies, and even econometrics. This article represents an endeavor by the members of the SSAP -TC to review all these significant developments in the field

Tsuhan Chen) Guest Editor Carnegie Mellon University

Reprinted from IEEE Signal Processing Magazine, Vol. 15, No.5, pp. 21-64, September 1998.

51

any engineering applications require extraction ofa signal or paranleter of interest from degraded measurements. To accornplish this it is often useful to deploy fine-grained statistical models; diverse sensors that acquire extra spatial, temporal, or polarization information; or multidimensional signal representations, e.g., time-frequency or time scale. When applied in combination these approaches can be used to develop highly sensitive signal estimation, detection, or tracking algorithms that can exploit small but persistent differences between signals, interferences, and noise. Conversely, these approaches can be used to develop algorith111s to identity a channel or svstem producing a signal in additive noise and interference, even when the channel input is Un1Gl0\Vn but has known statistical properties. Broadly stated, the statistical signal and array processing (SSAP) area is concerned with reliable estimation, detection, and classification of signals that arc subject to random fluctuations. Opening a recent issue of the IEEE Transactions on Siqnal Processinq to a SSAP paper the reader will probably see one or more of the following: (1) description of a mathematical and statistical model for measured data, including models tor sensor, signal, «.111d noise; (2) careful statistical analysis of the fundamental limitations of the data including deriving benchmarks on performance. e.g., the Cramer-Rao , Ziv-Zakai, Barankin, Rate Distortion, Chernov, or other lower bounds on average estimator/detector error; (3) developruent of mathcrnaticallv optimal or suboptimal estimation/detection algorithms; (4) asymptotic analysis of error performance establishing that the proposed algorithm C0111CS close to reaching a benchmark derived in (2); and (5) simulations or experiments that C0111 pare algoritlun performance to the lower bound and to other competing algorithnls. Depending on the specific application, a SSAP algorithm nlay also have to be adaptive to changing signal and noise environments. This requires incorporating flexible statistical models, implementing low-complexirv real-time estimation and filtering aIgorirhrns, and on-line performance monitoring. Until recently the statistical signal and array processing area was cover~d by the SSAP Technical COlllnlitte~, which grew out of the Spectrum Estimation and Modeling Technical Committee (discontinued in 1991). At ICASSP-98 in Seattle, an administrative restructuring took place that eliminated the SSM, Digital Signal Processing (DSP), and Underwater Acoustics Signal Processing (UASP) Technical Committees, replacing them by three new C0I11111ittees: Signal Processing Theory and Methods (SPTM), Signal Processing for Cornmunications (SPCOM), and Sensor Arrays and Multichannel signal processing (SAM). The SSM areas described in this article have migrated to these new Technical C0I11rnittees and remain very active within the Signal Processing Society. In particular, the following workshops sponsored or co-sponsored by SSAP will continue to pro-

SSAP TC ~Jiernb~:r'i (as of 1\;1ay 199B) Alfred O. Hero III (SPTiVl), University of .Micliiaa», ~41111 Arbor (clutinnnn) Gcorgios R. Giannakis (SPCONl), University oj' Vi1Xfinia (picc-cIJai17J'/I.17J) Moeness Amin (SPTM), Villanova University Kevin Bucklev, Villanova Uuivcrsitv . Jcan- Francois Cardoso (SPTiVl), Eeoll' Natunutlc SlJPt~l'iCtJ1"C des Telecommunications Zhi Ding (SPCONl) ~ Auburn University Petar M. Djuric (SPTNl), State University OJ'NL1P York at Stonv Bl'()o/~ Hamid Krim (SPTM),NortlJ CarolinaStttte University Jeffrey Krolik (SAiVI), Duke University ., Fu Li, Portland State University Hugit Messer-Yaron (SP'fM), Tel-Aviv University Eric Moulines (SPTM)., Ecolc Natuntalc S1Jplrit:1~rc des Tclcconnnunications Arve Nchorai (SAM), Univ. oiIllinois at Chicano L(;uis Scharf (SPTi\tl), University 0f'(:olorndo, Boulder Ananthr arn Swami (5 PCONl), Artnv RCJcnn:1J Lf1b01'nt01~Y

A. Lee Swindlehurst (SPCONl), Briaham YOtHl1T Univcrsitv . ''David J. Tho111S0n (SPTM), Lucent Tcclmolo..n. ics [itcndra Tugnait (SPCOlVl), Auburn University Raghuvccr M. R.~10, Rochester Just. ojTcclmolo..C. f.Y Lang Tong (SPCO.t\tl)., Cornell University Mats Vibcrg (current Vice-Chair SPTM), Clmlnters University ()j'Tlx!J1l(}lo.!.~11 Mati \V~LX, Rnjhel, Israel (SANI) Guanghan Xu (SPCONl), Thc University oj'Tc.,\;(fs at Austin Michael Zoltowski (SPCOM), Purdue University (nell' 1,(~ affilintiollS

ill parc1lthesis';

based on rcstructurnut arc indicated

vide forums for researchers in the area: the Workshop on Higher Order Statistics (to be held in Caesaria, Israel, in 1999 (http://sig.ensr.ti-/-hos99)), the Workshop on Statistical Signal and Array Processing (to be held in the Poconos, Pennsylvania, in 2000), and the Workshop on Signal Processing Advances in Communications (to be held in Annapolis, Maryland, in 2000). Similar to other Technical Committees, SSAP ran workshops, recommended paper awards, and reviewed papers for ICASSP. To facilitate the paper review process and provide focus for award nominations, the scope of SSAP was divided into several subareas, called "'SP EDICS" categories. These categories were spectral analvsis; higher-order statistical analysis; cyclostationary signal analysis; statistical multichannel filtering; statistical modeling; paranleter estimation; detection; performance analysis; system identification; computational aJgorithms; and applications. These categories are covered in this article and continue to be represented in the aggregated EDICS of the SPTM, SPCOM, and SAM Technical Committees,

52

As the reader will see from this article, SSAP impacts a very wide range of applications. Among the applications mentioned in the sequel are: radar signal processing; sonar signal processing; geophysics and climate; radar and optical remote sensing; electrocardiography (ECG); electroencephalography (EEG); nlagnetoencephalography (MEl~); nuclear magnetic resonance (NMR) inlaging; radio-isotope inlaging (PET and SPECT); chemical sensing of the environment; physical oceanography; fractal internet traffic modeling; astronomy: biology; economctrics; speech; and TI1LlSic analysis/synthesis. Over the past several years the application of signal processing to communications has become a prevalent theme in SSAP. The pre-existence of many relevant core SSAP areas made communications a very ripe applications area. In particular, research in cyclostationaritv, higher-order statistics, and systenl identification was a springbuard to the development of novel methods for channel equalization in digital communications. Likewise, work in detection and estimation led naturally to iterative multiuser detection, source separation, and high-performance modulation classification algorithnls. As another example, deployment of phased antenna arrays and the associated signal processing has spearheaded 11111Ch recent activity in spatial diversity reception tC)1wireless communications. The sections by Giannakis, Tong, and others highlight some of these cornmunications applications of SSAP. Our article begins with a group of two sections on recent developments in detection/estimation algorithms written by Alfred Hero and Pctar Djuric, respectively. The section by Hero focuses on two areas of significant activity: constant-false-alarm-rate (CFAR) detection and iterative maximum-likelihood (ML) paranleter estimation using the expectation-maximization (EM) algorithnl. The section by Djuric describes the emerging area of Bayesian signal processing including estimation, detection, tracking and Monte Carlo Markov chain (MCMC) sampling, which is a technique that was largely impractical before the current generation of high-speed conlputers. The article continues with J section on time-delay estimarion written by Hagit Messer and Jason Goldberg and a section on multiwindow spectral estimation by David Thomson. From a historical perspective, time-dclav esrimarion and spectral estimation are two of the oldest areas of statistical signal processing, dating back at least to the late 19th century (see [42 J), yet they remain two of the most active areas today. Continuing along these lines are sections on the increasingly important problems ofdetection and estimation in the time-frequcncv domain, written by Moeness Arn i n , and the time-scale or rnultiresol ution domain, written by Hamid Krirn and Jcan-Christophe Pesquet. Next comes a section written by Georgios Giannakis on recent SSAP activity in channel estimation and equalization tor digit,-11 communications. This is followed by t\VO sections dealing with the critical problems of model-

The fundamental theory behind detection, classification, and estimation has its home in mathematical statistics and decision theory. ing, systenl identification, and the often overlooked area of data validation. Ananthram Swami starts off with a broad overview of non-Gaussian measurement models and higher-order statistical methods, followed by a section by [itendra Tugnait on advances in multichannel systenl identification and testing random processes for non-Gaussian or nonlinear behavior. These are followed by a section written by Arye N chorai on exciting opportunities in SSAP due to recent advances in sensor technology. Finally, the article turns to array signal processing with four sections written by Lee Swindlehurst, Jeff Krolik, Iean-Francois Cardoso, and Lang Tong, respectively. Swindlehurst provides a bird's-cvc view of sensor-array processing and its applications to source local ization, source separation, and channel estimation. Cardoso tallows up with a section focusing on developments in blind-source-separation algorithlTIs. Tong discusses the increasing importance of blind-source separation and diversity in multiuser communications systenlS design. The final section, written by Krolik, discusses the use of C0111putational propagation models for processing sonar and radar arra v data. It is essential to point out that) in a limited overview article such as this, one cannot possibly do justice to the large number of areas that comprise SSAP. Neither can we hope to cover but a fraction of the contributions of individuals who have had a role in the development of SSAP through the years. We offer our sincere apologies to iU1Y individuals who feel omitted from this overview. WWW links relevant to the area oj-'S~/!I): A The (old) SSAP horne page: http.z/www .eng. J.U btl r11. edu/- d ing/SSAP/ SSAP .h rml .. A database of "selected papers" that appeared in the

IEEE Transactions on Signall)1!"occssin4...1J 1988-1995:

http r//www .eng. au bu rn. cdu/- d ing/SSAP /1ntp. html The SPTM'l SPCOM and SAM Technical C0111111ittee horne pages can be accessed through the IEEE Signal Processing Society home page: http://www .ieee.org/sociery/sp/inclex.ht1111 A. A clearinghouse for information on lnany aspects of signal processing is the Signal Processing Information Base at: http://sPi b. rice.edu/sp i b.htrnl A. S0I11e other web pages of interest to those working in SSAP: -The IEEE Societies on Computers, Antennas and Propagation, Communications, Aerospace and Electronic Svstems, Information Theory., and the IEEE NeuA

53

ral Network Council, all have SSAP related activities and links can be found on the IEEE page:

hypothesis testing [227], invariant hypothesis testing [287J, and the generalized likelihood ratio (GLR) test [205J. For lack of space we fOClIS only on CFAR detection using the min-max, GLR, and invariant testing approaches. We regretfully must omit work in adaptive detection for assumed known noise backgrounds, nonpararnetric techniques, distributed detection, Huber robust detection, sequential detection, signal classification, and detection of number of signals. Min-max CFAR hypothesis testing seeks to maximize detection probability subject to a constraint on maximum false-alarm rate. The min-max approach was recently adopted in [20] and [21] in the context of simultaneous detection and classification of multiple signals. This produced optimal detectors that took the form of a weighted likelihood ratio (LR) test. It was also shown in [20] that this min-max CFAR test implicitly implements J variant of Rissanen's maximum data length (MDL) signal selection criterion, establishing that MDL is ruin-max optimal. It is sometimes possible to arrive at min-max optimal detectors through the method of similar tests [366]. Finally, the min-max CFAR optimal detector can be viewed from the point of view of Bayesian detection implernented with a least favorable prior on the unknown noise density. Thus, in principle, the Bayesian methods developed in [119], [30], and more recently in [61], call be manipulated to provide CFAR tests. In n1any cases direct min-max optimization is diffi..:ult, and simpler suboptimal CFAR alternatives are ofinterest. The conceptually simplest approach is the GLR "estimate and plug" procedure, which requires computing ML estimates for the unknown noise paranleters. In [204J the GLR principle produced an adaptive detector for detecting spatio-temporal signals or targets in Gaussian noise with unknown spatial covariance. A different GLR adaptive target detector was derived in [57] for the case of optical images, The Gl.R for a general multichannel measurement was derived in [205], which specializes to the cases derived in [204] and [57] by applying suitable coordinate transformations. A related and important result was presented in [333] where exact confidence regions for the Gl.Rvmaximizing signal vector were derived for unknown spatial covariance. Additional applications of the GLR strategy to multispectral infrared images were presented in [330J and [464J. In [48] the GLR test was applied to arbitrary subspace projections of the data under similar assumptions as [205]. Other notable CF AR applications of GLH.. have appeared in the following areas: signal detection in noise of slowly fluctuating power [100]; transient signal detection in Gaussian noise of unknown power [316]; signal detection in unknown Gaussian-Gaussian mixture noise [39J; colored autoregressive noise [202, 367]; spatia-temporal signal detection in Gaussian noise with unknown spatial covariance [120, 266, 341]; signal detection in unknown i m pu ls i ve noise [59]; rnultiwindow/Gl.R sinusoid detection [187, 296]; tests

http://\vvv\v.ieec.org/tab/clll"_sub_soc_sub_bps.htll1l

-The American Statistical Association: http://vvv./\,,.amstat.org/

-The Institute of Mathematical Statistics: http://vvvvVY'.inlstat.org/

-The International Association for Statistical Computing: http://\Vw\v.stat. un ipg. it/iasc,htm 1 -The Royal Statistical Society: http://ll1aths. ntu.ac.uk/ rS5/ index2. html -The Acoustical Society of America: http://a~a.aip.org/

-The International Union of Radio Science: http://V~l'vV\v.il1to, "Blind Signal scp;1r..n ion: stati~tic.ll principles," jJroc. oftlJl'

It"]-:!::'. Spccial i.\Jut:011 blind idclItijicntilJlJ (Juri t'StilJ'/ITtilm, 1998. To .1ppe4.1r.

IEEE, vol. 77, pp. 941-l)~1, 19X9.

51. J.-f. Cardoso, M. L1\'idle, and E. l\tlo11lin~s, "Un algorithme d'identification par lluximlllll de vr..lisembl.1nn: pour des dOllne~s incol11plctcs," CompteJ Rcudu.Idc l'Amdcwic des SciclI£:n, StTicJ I, vol. 320, !lO. 3, pp. 363-368, 1r Urth-urder

322. M. B. Priestley, lVvlI-Ll1lCflr m/ft NOJ/-Statummy TilJ/c Saic.\" J·IJ1n.~l'JiJ, Audcmic Pres~, San Diego CA, 198X.

346. B. M. .s~Kiler, G. Gi.lIl11akis, and K. Li, "E~t1n10, eo) > 0, Ve. If s(t) denote the signal induced, on an element present at the center of the coordinate system, due to a broad-band source of power density 5(f) then the output of the fth sensor pre-steered in (cPo, eo), is given by

xe(t) == s(t

+ T e(cP, e) - Ti (cPo, ()0)).

(64 )

For a source in (cPo, eo), it becomes

xe(t) == s(t - To) f == 1,2,··· ,L

(65)

yielding identical waveforms after pre-steering delays. The TDL structure shown in the figure following the steering delay on each channel is a FIR filter. The coefficients of these filters are constrained to specify the frequency response in the look direction. It should be noted that these coefficients are real compared to the complex weights of the narrow-band processor. Let w., defined by (66)

106

Steering Delay

Tapped Delay Line Structure

:nr\ Fig. 6.

~2

Broad-band beam-former structure using TDL filter.

denote LJ coefficients of the filter structure with '.W..nl. denoting the L coefficients after the (M - 1)th tap. The mean out power of the beam former for a given 'JQ is given by

It is related to the spectrum of the signal by the Fourier transform, that is (70)

(67)

Thus, from the knowledge of the spectra of sources and their DOA's, the correlation matrix may be calculated. In practice, this can also be estimated by measuring signals at the output of various taps. In situations where one is interested in finding coefficients such that the beam former cancels the directional interferences and has the specified response in the look direction, the following beam-forming problem is normally considered:

where the LJ x LJ -dimensional real matrix R denotes the array correlation matrix, with its elements representing the correlation between various tap outputs. The correlation between the outputs of the (f - 1)th tap on the mth channel and the (k - 1)th tap on the nth channel is given by

(Rrn,n)e.k == p[(m - n)T + Tt(cPo, eo) - Tk(¢o, eo) +Tk(¢,())-Tt(¢,())] (68) with p(T) denoting the correlation function

p(T) == E[s(t)s(t + T)].

(69) 107

minimize

(71 )

subject to

(72)

IDL

~

IDL lDL Fig. 7.

Structure of partitioned realization of the broad-band beam former.

where F is a J -dimensional vector that specifies the frequency response in the look direction and C is an LJ x J constraint matrix. For a point constraint in the look direction

1

1

point constraint in the known direction of the signal would cancel the desired signal as if it were an interference. The other directional constraints to improve the performance of the beam former in the presence of the look-directional constraints include multiple linear constraints [117], [118] and inequality constraints [119]-[ 121]. A set of nondirectional constraints to improve the performance of the beam former under look-direction errors is discussed in [122]. These are referred to as correlation constraints, which use the known characteristics of the desired signal to estimate an LJ -dimensional correlation vector 'f.d between the desired signal and the array signal vector. The beam-forming problem using these constraints becomes

o

C==.

(73) 0

1.

with 1 denoting the L-dimensional vector of Is. Let fll. denote the solution of the above problem. It is given by l25]

iu == R-1C(C T R-1C)-1 F.

(74)

The point-constraint minimization problem specifies J constraints on the weights such that the sum of L weights on all the channels before the jth delay is equal to F j . For all pass frequency responses in the look direction, all but one F j , j == 1, ... ,J are selected to be equal to zero. For j close to (J + 1)/2, F, is taken to be unity. Thus, the constraints specify that the sum of the weights across the array is zero except for one near the middle of the filter, which is equal to unity. The implication of these constraints is that the array pattern has a unity response in the look direction. This pattern can be broadened by specifying additional constraints, such as derivative constraints [114]-[ 116], along with the constraints discussed above. The derivative constraints set the derivatives of the power pattern with respect to () and ¢ equal to zero in the look direction. The higher the order of derivatives, that is, the first order, second order, etc., the broader the beam in the look direction normally becomes. A broader beam is useful when the actual signal direction and the known direction of the signal is not precisely the same. In such situations the processor with the

minimize

Y2.T Rw

(75)

subject to

r.~ w == Po

(76)

where Po is a scalar constant that specifies the correlation between the desired signal and the array output. Application of broad-band beam-forming structures using TDL filters to mobile communications has been considered in [56] and [123]-[125] to overcome multipath fading and large delay spread in a TDMA as well as a CDMA system.

H. Partitioned Realization The broad-band beam-former structure shown in Fig. 6 is sometimes referred to as an element-space processor or direct form of realization, compared to a beam-space processor or partitioned form of realization, as shown in Fig. 7. The structure shown in Fig. 7 is discussed below for a point constraint, that is, the response is constrained to be unity in the look direction. A discussion of partitioned realization for derivative constraints may be found in [126]. The steering delays are used to align the waveform arriving from the look direction, as discussed. The array 108

the maximum attainable SNR and depends upon the FBW of the signal. This range includes a quarter-wavelength spacing at the center frequency fo. The quarter-wavelength spacing produces a 90° phase shift at fa and is equal to 1/410. By measuring the tap spacing as a multiple of this delay, it is indicated that the intertap spacing with multiple around l/FBW yields close to the highest attainable SNR. With the multiple between l/FBW to 4/FBW, one needs a larger number of taps for an equivalent performance. A study of the jamming rejection capability [104] and the tracking performance of the array in a nonstationary environment [105] also indicates that when tap spacing is measured in terms of the center frequency of the signal, the best performance is achieved when the spacing is 1/4fo. For this tap spacing, R has less eigenvalue spread, which is the reason for this performance. The eigenvalue spread of a matrix indicates the range of values its eigenvalues take. A larger ratio of the largest eigenvalue to the smallest eigenvalue indicates a larger spread. The TDL filter tends to increase the degrees of freedom of the array, which may be traded against the number of elements such that an array with L elements is able to suppress more than L-1 directional interferences, provided their center frequencies are not the same and fall within the FBW of the signal [107]. Though the TDL structure with constrained optimization is the commonly used structure for broad-band array signal processing, alternative methods have been proposed. These include:

signals after the steering delays are passed through two sections. The top section consists of a broad-band conventional beam with required frequency response obtained by selecting the coefficients of the FIR filter. Signals from all of the channels are equally weighted and summed. For this realization to be equivalent to the direct form of realization, all the weights need to be equal to 1/ L, and the filter coefficients F j , j == 1, 2, ... ,J need to be specified as before. Furthermore, the output of the upper section is given by

L Jik+ly(t - Tk)

J-1

Yc(t) ==

(77)

k=O

with

_ ~T(t)l ( ) --L-. yt

(78)

The matrix prefilter shown in the lower section is designed to block the signal arriving from the look direction. Since these signal waveforms after the steering delays are alike, it can be achieved by selecting the matrix W s such that the sum of each of its rows is equal to zero. For the partitioned processor to have the same degree of freedom as that of the direct form, the L - 1 rows of the matrix ~TS need to be linearly independent. The output Xl (t) after the matrix prefilter is an L - l-dimensional vector given by

x'(t) == Ws~(t)

(79)

and can be thought of as the outputs of L - 1 beams, which are then shaped by the coefficients of the FIR filter of each TDL section. Let an L - I-dimensional vector fJ:.k denote these coefficients before the kth delay. The output of the lower filter is then given by

1) adaptive nonlinear schemes, which maxirmze SNR subject to additional constraints [127];

2) a variation of a Davis beam former [88], which adapts one filter at a time to speed up convergence [128L

J-1

(80)

3) a composite system, which also utilizes a derivative of beam pattern in the feedback loop to control the weights [129] to reject wideband interference:

These coefficients are selected by minimizing the mean output of the processor, that is

4) optimum filters, which specify rejection response [87L

Ya(t)

=:

'" T I LtQkK-(i-kT).

k=O

5) a master and slave processor with broad-beam capabilities without derivative constraints [130]; 6) a hybrid method that uses an orthogonal transformation on data available from the TDL structure before applying weights [131] to improve its performance in multipath environment;

The performance of the broad-band arrays as a function of the number of various parameters, such as the number of taps, tap spacing, array geometry, array aperture, and signal bandwidth, has been considered in the literature [101 ]-[ 108] to understand their influence on the behavior of the arrays. An analysis [101] of broad-band arrays using eigenvalues of R indicates that the product of the array aperture and the FBW of the signal is an important parameter of the broad-band array in determining its performance. The FBW is defined as the ratio of the bandwidth to the center frequency of the signal. It is shown that the number of taps required on each element depends upon this parameter as well as on the shape of the array, with more taps needed for a complex shape. A study [102], [103] of the SNR as a function of intertap spacing indicates that there is a range of intertap spacing that yields close to

7) weighted Chebyshev method [134]; 8) two-sided correlation transformation method [135].

I. Frequency-Domain Beam Forming A general structure of the element-space frequencydomain processor is shown in Fig. 8, where broad-band signals from each element are transformed into frequency domain using the FFT and each frequency bin is processed by a narrow-band processor structure. The weighted signals from all elements are summed to produce an output at each bin. The weights are selected by independently minimizing

109

x.(t)

ITP

@

[}!

@

F

T

x I. (t)

T

x L(t)

N

un T

Broadband Time Domain Signals Fig. 8.

N

N

F

y(t)

F yN(t)

T

@ Narrowband Processing on each Frequency Bin

Conversion to Time Domain

Element-space frequency-domain processor structure.

---~8-1

Fig. 9.

y.(t)

t(8) i

----"'~8 Output~

Delay-and-sum beam former.

the mean output power at each frequency bin subject to steering-direction constraints. Thus, the weights required for each frequency bin are selected independently, and this selection may be performed in parallel, leading to a faster weight update. When adaptive algorithms such as the LMS algorithm (discussed in Section III-B) is used for weight update, a different step size may be used for each bin, leading to faster convergence. Various aspects of frequency-domain beam forming are reported in the literature [136]-[ 150]. The performance of the time- and frequency-domain processors are the same only when the signals in different frequency bins are independent. This independence assumption is mostly made in the study of frequency-domain beam forming. When this assumption does not hold, the frequency-domain beam former may be suboptimal. Some of the tradeoffs and comparisons of the two processors may be found in [136] and [149]. A study of the frequency-domain algorithm [140] for coherent signals indicates that the frequency-domain method is insensitive to the sampling rate and may be able to reduce the effects of element malfunctioning on the beam pattern. A study in [141] shows that due to its modular parallel

structure, beam forming in the frequency domain is well suited for VLSI implementation and is less sensitive to the coefficient quantization. The computational advantage of the frequency-domain method for bearing estimation is discussed in [144], [146], and [150], and the advantage for correlated data is considered in [145] and [148]. A general treatment of time- and frequency-domain realization with a view to comparing the structure of various algorithms of weight estimation in a unified manner is provided in [139]. J. Digital Beam Forming

Consider the analog beam-former structure shown in Fig. 9, where the signals from each element are weighted, delayed, and summed to form the beam output

y(t) ==

L WiXi(t - Ti(O)). L

(82)

i=l

The delays are adjusted such that the signals induced from a given direction, where the beam needs to be pointed, are aligned after the delays. This aspect of beam steering was discussed in detail earlier. The weights are adjusted to shape the beam.

110

Samples from

Element

,

o.

(88)

This scheme of estimating weights using the inverse update is referred to as the RLS algorithm, which is further described in Section III-C. It should be noted that as the number of samples grows, the matrix update approaches its true value, and thus the estimated weights approach the optimal weights, that is, as n --+ 00, R(n) --+ Rand 1Q(n) --+ W or 1Q1\.1SE' as the case may be. More discussion on the SMI algorithm may be found in [40] and [181]. Procedures for estimating array weights with efficient computation using SMI are considered in [182], and an analysis to show how it

112

performs as a function of the number of snapshots is provided in [89]. Application of SMI to estimate the weights of an array to operate in mobile communications systems has been considered in many studies [56], [59], [60], [183]-[186]. The study in [183] considers beam forming for GSM signals using a variable reference signal as available during the symbol interval of the TDMA system. An application discussed in [184] is for vehicular mobile communications, whereas that presented in [186] is for inducing delay spread in indoor radio channels. A presentation in [59] is for mobile satellite communications systems.

B. LMS Algorithm

iteration. The array signal vector, however, is £( n + 1), the reference signal sample is 1'(n + 1), and the array output (92)

In its standard form, the LMS algorithm uses an estimate of the gradient by replacing R and ~ with their noisy estimates available at the (n + 1)th iteration, leading to

+ 1)~H (n + l)~(n) - 2£(n + 1)1'(n + 1) == 2~(n + 1)C:*('JQ(n))

fl(1Q(n)) == 2~(n

where C:(1Q(n)) is the error between the array output and the reference signal, that is

The application of the LMS algorithm to estimate the optimal weights of an array is widespread, and its study has been of considerable interest for some time now. The algorithm is referred to as the constrained LMS algorithm when the weights are subjected to constraints at each iteration. It is referred to as an unconstrained LMS algorithm when the weights are not constrained at each iteration. The latter is mostly applicable when weights are updated using a reference signal and no knowledge of the direction of the signal is utilized, as is the case for the constrained case. The algorithm updates the weights at each iteration by estimating the gradient of the quadratic surface and then moving the weights in the negative direction of the gradient by a small amount. The constant that determines this amount is normally referred to as the step size. When this step size is small enough, the process leads these estimated weights to the optimal weights. The convergence and the transient behavior of these weights, along with their covariance, characterize the LMS algorithm, and the way that the step size and the process of gradient estimation affect these parameters is of great practical importance. These and other issues are now discussed in detail. J) Unconstrained LMS Algorithm: A real-time unconstrained LMS algorithm for determining optimal weight 1Qf\'1SE of the system using the reference signal is [27], [187]-[ 199]

w.(n

+ 1) == w.(n) -

JLfl(w.(n))

Thus, the estimated gradient is a product of the error between the array output and the reference signal as well as the array signals after the nth iteration. For JL < 1/ A1ll a x , with An 1ax denoting the maximum eigenvalue of R, the algorithm is stable and the mean value of the estimated weights converges to the optimal weights. As the sum of all eigenvalues of R equals its trace, the sum of its diagonal elements, one may select the gradient step size JL in terms of measurable quantities using JL < 11Tr (R), with Tr (R) denoting the trace of R. It should be noted that each diagonal element of R is equal to the average power measured on the corresponding element of the array. Thus, for an array of identical elements, the trace of R equals the power measured on anyone element times the number of elements in the array. The con vergence speed of the algorithm refers to the speed by which the mean of the estimated weights (ensemble average of many trials) approaches the optimal weights. It normally is characterized by L trajectories along L eigenvectors of R with the time constant of the fth trajectory given by Tt

(90)

at the nth iteration with respect to w.(n), given by

V' ~MSE(w.)I~=~(n) == 2R1Q(n) - 2if.

1

==-2J-l At

(95)

with At denoting the lth eigenvalue of R. Thus, these time constants are functions of the eigenvalues of R, the smallest one dependent upon A1ll ax , which normally corresponds to the strongest source, and the largest one controlled by the smallest eigenvalue, which corresponds to the weakest source or the background noise. Therefore, the larger the eigenvalue spread, the longer it takes for the algorithm to converge. In terms of interference rejection capability, this means canceling the strongest source first and the weakest source last. The convergence speed of an algorithm is an important property, and its importance for mobile communications is highlighted in [200] by discussing how the LMS algorithm does not perform as well as some other algorithms due to its slow convergence speed in situations of fast-changing signal characteristics. The availability of time for an algorithm to converge in mobile communications systems depends not only on the system design, which dictates the duration of

(89)

where 1J2( n + 1) denotes the new weights computed at the (n + l)th iteration; JL is a positive scalar (gradient step size) that controls the convergence characteristic of the algorithm, that is, how fast and how close the estimated weights approach the optimal weights; and g(w.(n)) IS an unbiased estimate of the gradient of the MSE

MSE(1Q(n)) :=E[I1'(n + 1)1 2 ] + 'JQH (n )R1ll.( n) - 2'JQH (n)if

(93)

(91)

It should be noted that at the (n + 1)th iteration, the array is operating with weights w.( n) computed at the previous 113

and

the user signal present (such as the user slot duration in a TDMA system) but also on the speed of mobiles, which changes the rate at which a signal fads. For example, a mobile on foot would cause the signal to fade at a rate of about 5 Hz, whereas the rate would be on the. order of about 50 Hz for a vehicle mobile, implying that an algorithm needs to converge faster in a system being used by vehicle mobiles compared to one used by a hand-held portable device [47]. Some of these issues for an IS-54 system are discussed in [56], where the convergence of the LMS and SMI algorithms in mobile communications situations is compared. Even when the mean of the estimated weights converges to the optimal weights, they have finite covariance, that is, their covariance matrix is not identical to a matrix with all its elements equal to zero. The covariance matrix of the weights is defined as

(100) then the misadjustment M is given by (101 ) For a sufficiently small u; this results in M ~ 2J-t Tr( R). It follows from this expression that increasing J-t increases the misadjustment noise. On the other hand, an increase in J.L causes the algorithm to converge faster, as discussed earlier. Thus, the selection of the gradient step size requires satisfying conflicting demands of 1) reaching vicinity of the solution point more quickly but wandering around over a larger region and causing a bigger misadjustment and 2) arriving near the solution point slowly with the smaller movement in the weights at the end. The latter causes an additional problem, particularly in a nonstationary environment, say, when the interference and optimal solution move slowly, causing adapting estimated weights to lag behind the optimal weights. This phenomenon is referred to as the weight vector lag. Many schemes, including variable step size, have been suggested to overcome this problem [201 ]-[208]. Some of these schemes are now discussed. The adaptive algorithm estimates the weights by minimizing the MSE. Thus, in schemes where a variable step size is used, it reflects the value of the MSE at that iteration (going up and down as the MSE goes up and down) such that it stays between the maximum permissible value for convergence and the minimum value based upon the allowed misadjustment. It may be truly variable or it may be allowed to switch between a few preselected values for the ease of implementation, as well as to shift by one bit left or right where digital implementation is used. The step size may also be adjusted to reflect the change in the direction of the gradient of error surface at each iteration [207]. The optimal value of the step size at each step is suggested in [203] such that it minimizes the MSE at each iteration. This is a function of the value of the true gradient at each iteration and R. In practice, these may be replaced by their instantaneous values, leading to a suboptimal value. Instead of having a single step size for an entire weight vector, one may select a variable step size for each weight separately, leading to an increased convergence of the algorithm [204]. The convergence speed of an algorithm may also be increased by adjusting the weights such that interferences are canceled one at a time [209], [210] and by using a scheme known as block processing [211]. For broad-band signals, an implementation in the frequency domain may help increase the speed of convergence. The application of frequency-domain beam forming to estimate the weights using the LMS algorithm for the case when a reference signal is available [138], [139], [142], [143] shows how the frequency-domain approach yields improved convergence and reduced computational

where 'ill == E[1Q( n )] denotes the mean of the estimated weights at the nth iteration. This causes the average of the MSE not to converge to the MMSE and leads to the excess MSE. From the expressions of the MSE and MMSE, it follows that for a given 1Q( n), the MSE is given by

MSE(1Q(n)) == MMSE + V H (n)RV(n)

(97)

V(n) == 1Q(n) - fQ

(98)

where

is the difference between the estimated weights and the optimal weights at the nth iteration. Note that E[V(n)] ~ 0 as n - 7 00. As all elements of k·w·w{n) do not approach zero as n - 7 00, it follows that the average value of the excess MSE does not approach zero as n - 7 00, that is, lilnn~00 E[l/H (n)RV(n)] :f O. The transient and steady-state behavior of the weight covariance matrix and the average excess MSE are important parameters of the LMS algorithm and are discussed in detail in [188] and [198]. A study of the convergence of the LMS algorithm applicable to the PIC processor and a discussion on the gradient step size selection can be found in [75]. The difference between the weights estimated by the adaptive algorithm and the optimal weights is further characterized by the ratio of the average excess steady-state MSE and the MMSE. It is referred as the misadjustment. It is a dimensionless parameter that measures the performance of the algorithm. The misadjustment is a kind of noise and is caused by the use of the noisy estimate of the gradient. This noise is referred to as the misadjustment noise. For the present case when the gradient is estimated by multiplying the array signals with the error between the array output and the reference signal and the gradient step size is selected such that (99)

114

W2

Constant Power Surface Cotour

-Jl g(~(n))

Fig. 12.

Constrained LMS algorithm: pictorial view of the projection process.

signal sensitivity compared to the normal LMS algorithm. A discussion of its application to mobile communications can be found in [225]. 3) Constrained LMS Algorithm: A real-time constrained algorithm [7], [25], [226]-[233] for determining the optimal weight vector {Q is

complexities over the time-domain approach. Improved convergence normally arises from the use of different gradient step sizes in different bins. For the constrained LMS case, this is likely to cause deterioration in the steady-state performance of the algorithm. This deterioration, however, does not affect the performance of the unconstrained algorithm [212]. An algorithm known as a sign algorithm [208], [213], where the error between the array output and the reference signal is replaced by its sign, is computationally less complex than the LMS algorithm, as discussed. The algorithm is usually analyzed assuming that successive samples are uncorrelated. This assumption helps in simplifying the mathematics by allowing expectations of data products to be replaced by the products of their expectations. A discussion of situations of correlated samples and a nonstationary environment may be found in [214]-[216]. Applications of an unconstrained LMS algorithm to mobile communications systems using an array include basemobile communications systems [46], indoor-radio systems [47], and satellite-to-satellite communications systems [97]. 2) Normalized LMS Algorithm: This algorithm is a variation of the constant-step-size LMS algorithm and uses a data-dependent step size at each iteration. At the 'nth iteration, the step size is given by

J.L(n)

Mo

= ;rH(n);f(n)

u:(n + 1) == P {1Q(n) -

M9(ill (n ))}

-

+ ~o

~o ~o

( 103)

where H

P~I_~o~o

L

( 104)

is a projection operator, 9 ('!ld.(n )) is an unbiased estimate of the gradient of the power surface 'JQH ('n )R1Q(n) with respect to 1Q( n) after the nth iteration, /-L is the gradient step size, and 20 is the steering vector in the look direction. The algorithm is "constrained" because the weight vector satisfies the constraint at every iteration, that is, 1QH (n )~o == 1, Vn. The process of imposing constraints may be understood from Fig. 12, which shows how weights are undated and how a projection system uses a vector diagram for a two-weight system [25]. The figure shows constant power contours, the constraint surface (a line 1J2H ~o == 1 for a twodimensional system), a surface parallel to the constraint surface passing through the origin ('JQH ~o == 0), weight vectors 1Jl.( n), ~(n + 1), and {Q, and the gradient at the nth iteration. The point A on the diagram indicates the position of the weight after completion of the nth iteration. It is the cross section of the constraint equation 1Jl.H ~o == 1 and the power surface 'JJ!..H (n)R1Q(n) (not shown in the figure). The weights are perturbed by adding a small amount - M!l(1Q( n)) and then are projected on 1QH ~o == 0 using

( 102)

where /-La is a constant. The algorithm and its convergence using various types of data have been studied widely [217]-[ 2241. It avoids the need for estimating the eigenvalues of the correlation matrix or its trace for selection of the maximum permissible step size. The algorithm normally has better convergence performance and less 115

projection operator P. This point is indicated by B on the diagram. Note that P~o == O. Thus, the projection operator projects the weights orthogonal to ~o' The constraint now is restored by adding §.o/§.{f §.o and the updated weights 1Q( 11,+ 1) move to point C. The process continues by moving the estimated weights toward point D, the optimal solution. The effect of the gradient step size J.L on the convergence speed and the misadjustment noise may also be understood using this figure. A larger step size means that the weight vector moves faster toward point D, the solution point, but wanders around it over a larger region, not reaching close to it and causing more misadjustment. The gradient of w. H (n)Rw.(n) with respect to 1Q(n) is given by

speed of the algorithm depends upon the eigenvalue spread of PRNP. The discussion so far has concentrated on the convergence of the mean value of the weights to the optimal weights. The variance of these weights is an important parameter, and the transient and steady-state behavior of the weight covariance matrix k~ ~ ( n) are indicators of the performance of the algorithm, as discussed' previously for the unconstrained LMS algorithm. An expression for k~~ (n) indicates [228] that it is a function of the variance of the gradient estimate. For the standard algorithm, an expression for the variance of the gradient is given by

The steady-state value of the weight covariance matrix governs the misadjustment. For the standard algorithm, it is given by

and its computation using this expression requires knowledge of R, which normally is not available in practice. For a standard LMS algorithm, an estimate of the gradient at each iteration is made by replacing R by its noisy sample £(n+l)£H(n+l) available at time instant (n+1), leading to g(1Q(n)) == 2~( n + 1) y* (w(n)). Thus, the gradient estimate is the product of the array signals and the array output available after the nth iteration. The mean value of the weights estimated by the algorithm using this gradient converges to the optimal weights, provided that the gradient step size is small enough to satisfy

O<J.L
1

(47)

since the first term in (46) is always nonnegative. However, (47) implies that Qj=O, j= 1. .. " K~ which proves that F' is positive definite. ApPLICATIONS TO SEISMIC DATA

We now wish to describe the application of the conventional and high-resolution frequency-wavenumber spectrum estimates to seismic data obtained from LASA. The

150

-- - --N

/'

/

/

/

"....

/@

@

§

I I

@

~

\

\

\

F1

@

@)

81 AO 83 82

@

8

@ @ \

.............

'-@ -,

...

.. ~ ~.~

1r

,

.

~

..: , ~,":

- 0.0 4

. l~

~

..., ~.

:'

~

' j ,. ........ ;~ ~ 4-( '/ ~ j~ O'~~ L ,",.". ~

-

E

~

""" ~

"

P'

" -0.02

~

/,

.~

.~

i~

'; ~

J!1 ~ _.

I-

:::::i

Figs. 2(c) and 3(c) illustrate examples of this estimate. Using (17), the value of this spectral estimate for k = k o and Mo] /a~ 1 is easily shown to be

»

PLP(k

(L

~

1 or 0) when the source bearings are closely spaced (separated by less than one-half a beamwidth) [18]. However, the opposite result holds when the source bearings are not closely spaced (Fig. 4). Fig. 4 also illustrates the nonlinear nature of this spectral estimate. The resolution capability of an algorithm can be affected by the signal-to-noise ratio ~ some algorithms are affected more (rn o = 4 in Fig. 4, for example) than others. Furthermore, the increased resolution indicated in Fig. 4(a) is obtained at the expense of increased bias. A criterion for the "proper" choice of the predictive element rna has not been found to date. This is but one linear-prediction algorithm. While all of them start with the model given in I~ 21), the error criterion to be minimized differs. For example, in time series, the sum of the so-called forward and backward squared errors is sometimes used (7], [30], (36]. For 3. linear array, this choice 01 criterion corresponds to the sum of the squared errors for rn o = 0 and 111- 1. The approach presented here and these other algorithms can also be modified to allow the order of the prediction model to be different from M - 1. The tradeoffs involved in reducing the model order have not been studied extensively. These various algorithms do have different

171

resolution and bias properties [30), but these properties are not fully understood at this time.

IV.

01

COMPARISON OF THE MAXIMUM-LIKELIHOOD AND LINEAR-PREDICTIVE METHODS

These two spectral estimation methods provide spectra having better resolution properties than conventional beamformmg. Comparison between these two estimates are often drawn. The maximum-likelihood method is an adaptive beamforming algorithm while linear prediction does not yield weights for beamforming. The linear-predictive method has better resolution properties. However, this increased resolution is accompanied by a ripple in the power estimate PLP(k) when the direction-of-look is not equal to the actual signal bearing (Fig. 3). These spectra have been related to each other analytically by Burg [8] in the case of equally spaced linear array by 1

:fl

M-l

PML(k)

=

001

o 001

where

'cos(umo,E;~-1)12

I 10

I

I

I

i

20

39

t

I , I

I

I i Ph

49 50

99

= I I [M(Mo;

v;r,.

+o~)]

and when the direction-of-look is significantly different from

(27)

k o so that E is orthogonal to S, this cosine squared is approximately equal to I/Ma~. The precise value of the cosine will

oscillate about this quantity depending on the projection of E onto S. Considering Fig. 5, the amount of this projection diminishes as the direction-of-look departs more from the signal direction. The characteristic of the linear-predictive estimate that results in both better resolution and increased ripple when compared to the maximum-likelihood estimate is this cosine-squared term.

11011 =a'!a. Note that when a and fJ are complex, this expression is complex-valued. Despite this property, the magnitude of this cosine is bounded between o 'and 1 because of the Sch warz inequality. Consider the ratio of the high-resolution spectral estimates

v.

(28)

Comparing (27) and (28), we have 2 = lcos (u rn o, E", ~ -1 )1.

J

0

while vectors orthogonal to S are reduced in length by Therefore, when E =S.

2

--&.:.:.::..-

1 T\

VMo; +o~

II all ~ denotes the norm of a as generated by !

PML(k) PLP(k)

i

remaining eigenvalues equal l/o~. If the cosine were computed with respect to the identity matrix, 'cos (II rn o, E) 12 = 11M for all direction-of-look vectors E. When computing the cosine with respect to gt-l, the result depends on the relationship between E and S. The space induced by ~-1 reduces the lengths of vectors parallel to S by a factor of

P(k) as the former "averages-in" lower order linearpredictive models. Another result can be obtained by noting the expression for the generalized cosine of two vectors a and /J.

cos (a,

ii'

-50 -40 -30 -20 -10

BEARING-deg

1

o'/J

11'1 I I tIt I -90

(29)

The ratio of these spectral estimates is thus equal to the cosine of the angle between u m o and E with respect to the vector space generated by ~ -1. One consequence of this expression is that PML(k) ~PLP(k).

The linear-predictive spectrum will be much greater than the maximum-likelihood estimate when this cosine is small. The natural orthogonal basis for the vector space induced by 1 g{-l is comprised of the eigenvectors of this matrix. When the correlation matrix is given by (11), one eigenvector equals S while the remaining eigenvectors are the (M - 1) orthogonal vectors spanning the subspace orthogonal to S. The eigenvalue of ~-l corresponding to S is equal to 1/(Ma; +a~) while the lit is easily shown that these eigenvectors are mutually orthogonal because i-I is a Hermitian matrix. Therefore, the eigenvectors constitu te an orthonormal basis.

EIGENVECTOR METHODS

A class of spectral estimation procedures based on an eigenvector-eigenvalue decomposition of the spatial correlation matrix has been developed recently [3], [26], (431. These procedures are intimately related to the maximumlikelihood and linear-prediction methods just described. The motivation for this approach is to emphasize those choices for E which correspond to signal directions. As the expressions for the maximum-likelihood (16) and linear-prediction (24) estimates have E appearing only in the denominator, the rationale is to reduce the lengths of those E's corresponding to signals and increase those not corresponding to plane-wave signals. The problem is that one does not know, in general, which direction to emphasize; it is these directions that we are trying to determine from the spatial spectra. On the other hand, these directions determine the structure of the spatial correlation matrix, in particular the eigenstructure of matrix. By examining this structure, one can obtain algorithms which enhance the spatial spectra in an objective way so that peaks corresponding to propagating signals are made more prominent. The eigenvalues Ai and eigenvectors Yi of ~ are defined by the relationship i= 1,···,M

where Al ~ A2 co "0

~ o ...u

W

0 :J ~

o,

-10

(f)

~

(b)

j

-10l

~

0~

(b)

-20

II

-20~

~

0

ill

0-

(f)

-3~ ~ -'-II??'' ' ' '""--1...-,.. .--,"T"j""j--r-"j "'T"j~j""j~1--Y-I-'j~T '

-30

-90

0

-50 - 40 -30 -20 -10

a

I

I

10

I j I 20

30

I

I

I

I

40 50

I

t'

I ' Jln

90

BEARING-deg

Fig. 7. Result of applying eigenvector methods. The spatial correlation matrix used in this computation is the same as that used in Fig. 6. The eigenvector expansion (36) was truncated at 8 terms. (a) Maximum-likelihood spectral estimate. (b) Linear-predictive estimate. Note that for this figure, (25) was used in the computation of the linear-predictive estimate rather than that described by (24).

-10

~

(c)

-291~ ~

-3e~f'---"""'".,. . .1. . ,......., "",-Ir-TI~I-rj---r,---'-1rr-'I I "T"'"i

-90

-50 -40 -30 -20 -10

0

- ,

10

, , ii' I 20

30

i

I

40 50

I

orthogonal to the smallest eigenvector. One is then faced with determining the set of "best" signal vectors [6], [ 11].

I j 11ft 90

BEARING-deg

VIII.

Fig. 6. Effect of finite averaging on various spectral estimates. The array configuration and signal characteristics are as described in the caption to Fig. 3. The matrix i is given by (47); time-bandwid th product of the computation is 50. (a) Bartlett estimate. (b) M-aximum-likelihood estimate. (c) Linear-predictive estimate. The same correlation matrix was used in each spectral estimate.

where 1

N=K

L K

i=1

N i and

is an estimate of the noise spatial correlation matrix ~. The matrix ~ is Hermitian but is usually not Toeplitz, As the /\ ...., time-bandwidth product increases, ~ -4- ~ and N ~ O. If the noise is spatially white, then ~ = g and (11) results. However, in most applications of interest, K is not large enough to justify such a simple formula. The cross terms between signal and noise and the presence of ~ instead of ~ imply that spectral errors can occur (Fig. 6). It has been shown that the maximum-likelihood and linearpredictive estimates are sensitive to the cross terms [ 18]. Furthermore, the increased resolution capability of linear prediction is mitigated to some extent by its sensitivity to K. Roughly speaking, the time-bandwidth product for linear prediction must be M times that for maximum likelihood to result in the same statistical variability of the spectral estimate. When !l = g, the eigenvector methods are more sensitive to the cross terms than to the statistical variation present in~. A finite time-bandwidth product limits the resolution of the eigenvector methods [26]. Fig. 7 illustrates the spectra obtained when these eigenvector methods are used. The Pisarenko method is more sensitive to a finite value of K. As the matrix 3t is no longer Toeplitz, no signal vector may be

CONCLUSIONS

A cohesive methodology of deriving high-resolution spectral estimates for array processing problems has been presented. Each has been shown to be the solution of a constrained optimization problem. This approach is quite general and can be used to derive procedures applicable to time series (i.e., onedimensional data) and to multidimensional data. While arrays are usually multidimensional, the spectral estimation problem equivalent to the bearing estimation problem has very specific properties. Procedures designed to compute the spectrum of a signal sampled on a regular grid (such as rectangular and hexagonal ones) do not usually apply to array processing pro blems. The impact of array geometry on spectral estimation procedures is largely unknown. Many designers of arrays use unequally spaced sensors for a variety of reasons. Hence the generality of the theory presented here. Some work is emerging on the geometry question (14], [32). The underlying model used for the signal in these derivations can also be questioned. The wavefront of sound propagating from a point source is curved. Significant curvature of the wavefron t across the array aperture can significantly affect the quality of estimates which assume a plane wave. If the curvature were known, it could easily be taken into account; unfortunately, it rarely is known. A more serious problem is coherence between signals impinging on the array from different directions. In this case, a nodal pattern of peaks and valleys of signal power is established across the array. This effect results in a location-dependen t amplitude and phase variation beyond that assumed in the usual plane-wave model. Current research is directed towards methods which can cope with coherent signals [21], [22]. The linear-predictive estimate is more sensitive to its signal model than most of the other procedures described here. In

175

addition to the usual plane-wave assumption, the signal recorded at each sensor is also assumed to be modeled by a linear difference equation. While the plane-wave signal may obey this relationship, the noise usually does not. In practical problems, the signal-to-noise ratio at each sensor is small, usually being around 0 dB. Consequently, this method is sensitive to noise and to finite time-bandwidth products. This problem has been recognized in the time-series literature; in that context, so-called ARMA models emerge [15], [27]. These are pole-zero models where the poles describe the signal, and the zeros are due to the presence of noise. Procedures are being developed to measure parameters of such models [28], [42], [45], but the applicability of them to array processing problems is limited because of problems similar to those encountered in the multidimensional maximum-entropy spectral estimate. While further work is needed to find spectral estimation procedures for time-series problems and to quantify their behavior, the bearing estimation problem offers a different set of issues, which are apparently more challenging.

Let

be the M X M matrix

eM = [1~~1 ~] and ~M to be the matrix in the equivalent quadratic form in the second term of (AS)

1'i

XM~MXM =

1

1=0

aiXM_i!2

The elements of this matrix are j(~ f P k i=k+l

i=k+l

.( -

1

q "'2 > "'M IN > 0 1 0 2 S PAN THE SI G N AL SUBSPAC E 1, 8(",), _ ( 8

Fig. I.

2)

ARE T HE IN C I DENT S IGNAL MODE V E C T O R S

Gcomct n c portrayal for three-antenna case .

D=M- N

range system, 0 will be replaced by 8. : _' _~

'-"

SNR- Z4dB . , SNR .. lOdB

.,, '--- - - - »- - ·Wl

.~

3r1';

. ..

-.

SNA . 24 dB

x

SNR " lOdS

f~~

A

x

TRIANGLE ARRAY

min

~ I

,

,

'00

I~

10

,

J' "

SECOND PREDOMINANT PEA KI$ff'RONG

(A N AMB IGUITY )

1'00

" l-::=-.,-----'-::=-c"..---,,~1'00 - 100

~

XI

. 1\ 0

100

· '>(1

II

AOA

TO 57 dB

SO

100

I~

MUSIC

d B G L AMBIGUITY .10 '0

,

Fig. 3.

NO BIAS ERROR

OR

CONFUSION

_

"

Exampl e o f azimuth-only DF perfor mance .

MAX LIKELIHOOD

de

AOA

·1

8

which implies a D dimensional search (and plot !). PMEC(}) is based on selecting one of the M array elements as a " reference" and attempting to find weights to be applied to the remaining M - I received signals to permit their sum with a MMSE fit to the reference . Since there are M possible references , there are M generally different P M E( (}) obtained from the M possible column selections from S- I. In the comparison plots, a particular reference was consistently selected . An example of the completely general MUSIC algorithm applied to a problem of steering a multiple feed parabolic dish ante nna is shown in Fig . 5 . sin x /x pencil beamshapes skewed slightly off boresight are assumed for the element patterns. Since the six antennas are essentially colocated, the OF capacity arise s out of the antenna beam pattern diversity . The computer was used to simulate the " noisy" S matrix that would arise in pract ice for the conditions desired and then to subject it to the MUSIC algo rithm. Fig. 5 show s how three direct ional signals are distinguished and their polarizations estimated even though two of the arr iving signals are highly similar (90 percent correlated) . The application of MUSIC to the estimation of the frequen cies of mult iple sinusoids (arbitra ry amplitudes and phases ) for a ve ry limited duration data sample is shown in Fig . 6 . The figur e suggests that. even though there was no actual noise included . the rounding of the data samples to six decimal digits has already destro yed a significant port ion of the information present in the data needed to resolve the several frequencie s.

ADA BI AS ERR OR

I

8

~

dB: ~MAXENTRO~~l~~E~~I~~:S

100 (TIME BANDWIDTH I

>

'$0

MAX LIKELIHOOD

o 100 AOA

EQU ILATERAL

sr

dB "

>00

0

AUA

I P ML(8) = -;\-.-(A-*-S--'-A-)

NO ABILITY TO RESOLVE TWO SIGNALS

MAX ENTROPY

S UMMAR Y AND C ONCL USION de

LARGE BIAS ERROR : PEAK IS RELATIVELY SH ARP

AOA

MUSIC

de

NO BIAS ERROR ; PEAK IS VERY SHARP

AOA

Fig. 4 .

Exa mple of azimuth-onl y DF perfor mance (scale expanded abo ut weak er sig nal at 30° ) .

where c is a column of S - '. The beamformer expre ssion calculates for plotting the power one would measure at the output of a beamformer (summing the array element signals after inserting delays appropriate to steer or look in a specific direction) as a function of the direction. P ML ( (} ) calculates the log likelihood function under the assumptions that X is a mean zero , multivariate Gaussian and that there is only a single incident wavefront present. For

As this paper was being prepared. the works of Gething [I J and Davies [2] were discovered , offering a part of the solution discussed here but in term s of simultaneous equations and special linear relation ships without recourse to eigenstructure. However, the geometric sign ificance of a vector space setting and the interpretation of the S matri x eigen structure was missed. More recent work by Redd i [3J is also along the lines of the work presented here though limited to uniform , collinear arrays of omnidirectional elements and also without clear utilization of the entire noise subspace. Ziegenbein [4] applied the same basic co ncept to time series spectral analysis referring to it as a Karhunen-Loeve transform though treat ing aspec ts of it as " ad hoc. " El-Behery and MacPhie [5] and Capon [6] treat the uniform co llinea r arra y of omnid irect ional eleme nts using the maxim um likel ihood method . Pisarenko [7] also treats time series and addresses only the case of a full complement of sinuso ids; i.e. , a one-dimensional noise subs pace. The approach presented here for multiple signal classification is very general and of wide application. The method is interpretable in terms of the geometry of complex M spaces where in the eigenstructure of the measured S matrix plays the central role . MUSIC provides asymptotically unbiased esti-

193

mates of a general set of signal parameters approaching the Cramer-Rao accuracy bound. MUSIC models the data as the sum of point source emissions and noise rather than the convolution of an all pole transfer function driven by a white noise (i.e., autoregressive modeling, maximum entropy) Or maximizing a probability under the assumption that the X vector ' is zero mean, Gaussian (maximum likelihood for Gaussian data). In geometric terms MUSIC minimizes the distance from the 0(8) continuum to the signal subspace whereas maximum likelihood minimizes a weighted combination all component distances. No assumptions have been made about array geometry. The array elements may be arranged in a regular or irregular pattern and may differ or be identical in directional characteristics (amplitude/phase) provided their polarization characteristics are all identical. The extension to include general polarizationaIIy diverse antenna arrays will be more completely described in a separate paper.

METRIC CONTOURS OF 3 d B B E A M W IDTH S O F T H E SIX A N T EN NA FE E D EL EM E NT S L IN E A R L Y POLAR IZED AS P ER A R ROWS

EMITTER PAT H N U M BE R 2

IN T E R F E R ER

AZIMUTH , B E-AMy/IDTH S

0.50 EMITTE R PATH N UMB E R 1

SIGNAL PO LARIZATIONS:

o

RELAT IVE POWE RS ;

t dB

EMITTE R PATH N UMBE R 2

CJ o dB

IN T E R F E R E R

/ 3 dB

REFERENCES

[I) (2)

COR R E LAT I ON C OEF F IC IE NT S

',00 .91 .00] [ .91

1 .00

.00

.00

1 .00

.00

(3) (4)

Fig. 5.

MUSIC applied to a multiple feed. parabolic dish antenna system . [5) D A TA : 1 6 C O M PLE X TI M E S A M P L E S OF 6 CI S O I OS F RE Q (Hz)

R ELAMP (dB)

78 .1 134.1 138.6 142.9 152.9 16 6. 3

-62.33 - 1 1. 10

0.0 -11 .1 0 -40. 0 -2 6. 02

(6) [7J

P. 1. D. Gething, " Analysis of multicomponent wavefields ." Proc. Inst. Elec. Eng.• vol. 118. no . 10. Oct. 1971. D. E. N. Davies . "Independent angular steering of each zero of the directional pattern for a linear array ." IEEE Trans. Antennas Propagat.• vol. AP-15, Mar. 1967. S. S. Reddi, " Multiple source location-A digital approach . " IEEE Trans. Aerosp. Electron. Syst. , vol. AES-15 . no. I. Ian . 1979. 1. Ziegenbein, "Spectral analysis using the Karhunen-Loeve transform, " presented at 1979 IEEE Int. Conf. Acoust. Speech Signal Processing , Washington . DC . Apr . 2-4, 1979. pp. 182-185 . I. N. El-Behery and R. H. MacPhie . " Maximum likelihood estimation of source parameters from time -sampled outputs of a linear array." J. Acoust. Soc. Am., vol. 62 . no. I. July 1977. 1. Capon . " High resolution frequency-wavenumber spectrum analysis ," Proc. IEEE, vol. 57. no. 8. Aug. 1969. V. F. Pisarenko , " The retrieval of harmon ics from a covariance function." Geophys. J. Royal Astron. Soc., no. 33. pp. 374-366 . 1973.

Using Spectral Estimation Techniques in Adaptive Processing Antenna Systems WILLIAM

F.

GABRIEL, FELLOW, IEEE

Abstract-Improved spectral estimation techniques hold promise for becoming a valuable asset in adaptive processing array antenna systems. Their value lies in the considerable amount of additional useful information which they can provide about the interference environment, utilizing a relatively small number of degrees of freedom (DOF). The "superresolution" capabilities, estimation of coherence, and relative power level determination serve to complement and refine the data from faster conventional estimation techniques. Two conceptual application area examples for using such techniques are discussed; partially adaptive lowsidelobe arrays, and fully adaptive tracking arrays. For the partially adaptive area the information is utilized for efficient assignment of a limited number of nOF in a beamspace constrained adaptive system in order to obtain a stable main beam, retention of low sidelobes, considerably faster response, and reduction in overall cost. These benefits are demonstrated via simulation examples computed for a 16..e lement linear array. For the fully adaptive tracking array area the information is utilized in an all-digital processing system concept to permit stable nulling of coherent interference sources in the main beam region, efficient assignment/control of the available nOF. and greater Ilexibility in timedomain adaptive filtering strategy.

I

I. INTRODUCTION

M PRO VE D SPECTRAL estimation techniques are an emerging technology which derives largely from modern spectral estimation theory of the past decade and adaptive array processing techniques [2], [3]. [4]. Coupled with the phenomenal advances in digital processing [5]. these techniques are becoming a valuable asset for adaptive array antenna systems. Their value lies in the considerable amount of additional useful information which they can provide about the environment. utilizing only a relatively small number of degrees of freedom (DO F). For example. current spectral estimation algorithms can provide asymptotically unbiased estimates of the number of interference sources. source directions, source strengths, and any cross correlations (coherence) between sources [6], [7]. Such information can then be used to track and "catalogue" interference sources, hence assign adaptive DOF. These newer techniques are not viewed as a" superresolution' replacement for more conventional estimation methods such as main beam search, analog beamformers, or spatial discrete Fourier transforms (DFT). Rather. the new technology is considered complementary to the other methods and best used in tandem. For example, "superresolution" techManuscript received May 3. 1985~ revised September 16. 1985. This work Was supported by NAVAIRSYSCOM and the Office of Naval Research. This paper is a condensation of NRL Rep. 8920 [1). The author is with the Radar Division, Naval Research Laboratory, Washington. DC 20375. IEEE Log Number 8407018.

niques cannot compete with the speed of a DFT. Some comparisons of the various methods may be found described in the literature [4], [7], [8]. The purpose of this paper is to present two conceptual application areas for using spectral estimation techniques; partially adaptive low-sidelobe antennas, and fully adaptive tracking arrays. A partially adaptive array is one in which only a part of the OOF (array elements or beams) are individually controlled adaptively [9], [10], [11]. Obviously, the fully adaptive configuration is preferred since it offers the most control over the response of the antenna system. But when the number of elements or beams becomes moderately large (hundreds), the fully adaptive processor implementation can become prohibitive in cost, size, and weight. The paper is divided into three principal parts. Section II discusses partially adaptive. low-sidelobe antennas with the focus upon a constrained beamspace system; Section III considers source estimation and beam assignment from .. superresolution " techniques: and Section IV discusses an alldigital, fully adaptive tracking array concept.

II .

PARTIALLY ADAPTIVE LOW-SIDELOBE ANTENNAS

The antenna system addressed in this section is assumed to be a moderately large aperture array of low-sidelobe design wherein the investment is already considerable and one simply could not afford to make it fully adaptive. The assumption of low sidelobes (30 dB or better) is intended to give us good initial protection against modest interference sources and to reduce the problems from strong sources, i.e., in regard to the number of adaptive DOF required and the adaptive dynamic range of the processor. Thus. retention of the low sidelobes is considered a major goal in our adaptive system. In the discussion to follow, it is shown that using improved spectral estimation techniques in such a system can result in the following benefits over a fully adaptive array system. 1) Considerably faster adaptive response, reduction in computation burden, and reduction in overall cost because relatively few adaptive DOF are implemented. 2) Minimal degradation of both the main beam and sidelobe levels because simple adaptive weight constraints are made possible. 3) Compatible with a larger number of adaptive algorithms, including even analog versions. 4) Greater flexibility in achieving a "tailored" response due to greater information available. On the negative side, a partially adaptive system can never be guaranteed a cancellation performance equal to that of a

Reprinted from IEEE Transactions on Antennas and Propagation, Vol. AP-34, No.3, pp. 291-300, March 1986.

195

fully adaptive array and, in addition, will deteriorate abruptly in performance when the interference situation exceeds its adaptive DOF. These risks are an inherent part of the package and must be carefully weighed for any specific system application.

A . A Low-Sidelobe E igenvector Constraint

I

I

"

o

-28

(

C I B (

t

S

-J.

9. SPATIAL AI'lGLE IN DECREES

(a)

(1)

where W is the adaptive weights vector , R is the sample covariance matrix, S* is the quiescent main beam weights vector , the asterisk denotes the conjugate of a complex vector or matrix, and JL is a constant. Furthermore, compute the sample covariance matrix via the simple " block" average taken over N snapshots,

R=- ~ [E(n)E(n)*t], N n= 1

(

R N

We begin this section by reviewing that unconstrained adaptive arrays can experience very "noisy" sidelobe fluctuations and main beam perturbations when the data observation/ integration time is not long enough, even though the quiescent mainbeam weights are chosen for low sidelobes . Consider a linear array of K elements, with each element adaptively weighted , and let us compute the complex adaptive element weights W k from the well-known sample matrix inverse (SMI) algorithm [11], [12]. Expressed in convenient matrix notation ,

W=JLR-'S*

p

o -,

u

p

o

U

-18

E R

(2)

where E(n) is the element signal data vector received at the nth time sampling. (See Appendix I for description of snapshot signal model.) The data observation/integration time (b) in (2) is the parameter N. If R is estimated over a lengthy observation time. like thousands of snapshots , . then the Fig. I . Fully adaptive 16-e1emem linear array. SMI algorithm wrth R esurnated from :!56 snapshots per update . three 30 dB noncohereru sources sidelobe fluctuations from W updates will be relatively small . located al 14·. 18· and :!2·. (al Quiescent main beam pattern . 30 dB Taylor However, practical system usage often demands short obserweighting . (b) Typ ical adapted patterns . rune update tn a ls plotted . vation times on the order of hundreds of snapshots or even less. Fig. I illustrates typical adapted pattern behavior for t denotes the transpose of a vector or matrix . The {37 and ei independent estimates of R using N = 256 snapshots per are the eigenvalues and eigenvectors, respect ively. of the update for the case of three 30 dB noncoherent sources sample covariance matrix . and (3~ is equal to receiver channel located at 14 0 , 18 0 , and 22·. The antenna aperture chosen for noise power level. Equation (3) shows that W consists of two this example is a lb-element linear array with half-wavelength parts: the first part is the quiescent main beam weight S*: the element spacing and a 30 dB Taylor illumination incorporated second part. which is subtracted from S*. is a summation of in S*. Note that the adaptive algorithm maintains the main weighted , orthogonal eigenvectors . This is a clear expression beam region and successfully nulls out the interference of the fundamental principle of pattern subtraction which sources , but that it also raises the sidelobe levels elsewhere . . applies in adaptive array analysis. The reader is referred to The adaptive patterns are in continual fluctuation in the [13] for a more extensive discussion . sidelobe regions and may exceed the quiescent sidelobe level We introduce the term principal eigenvectors (PE) to mean by a considerable margin. Also, the main beam suffers a those eigenvectors which correspond to unique eigenvalues significant modulation which would degrade tracking perform- generated by the spatial source distribution ; and the teon ance . These effects worsen as the value of N decreases. "noise eigenvectors" to mean those eigenvectors which To understand the reason for this undulating pattern correspond to the small noise eigenvalues generated by the behavior, it is helpful to analyze the optimum weights in terms receiver channel noise contained in the finite R estimates. The of eigenvalue/eigenvector decomposition. A derivation of PE are generally rather robust and tend to remain relatively such a decomposition for (I) can be found in [I], and we stable from one data trial to the next, whereas the noise reproduce here (B18). eigenvectors tend to fluctuate considerably because of the K {3-{3o inherent random behavior of noise. This difference in behavior (3) is illustrated in Fig. 2 for the three source case described W=JL' S*- ~ - ' - 2 - a.e, 1= I {3, above, wherein there are three PE and 13 noise eigenvectors where (Xi = ejlS* and JL' = JLI {3~. associated with each R estimate. Fig . 2(a) shows the stability

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B. Low-Sidelobe Constraints for a General Beamformer Consider next a more intere sting configuration shown by the schematic diagram of Fig . 4, where we repre sent an adaptive array system operatin g in beamspace so as to have available some preadaption spatial filter ing. Applebaum and Chapman [10], (14) first described beamspace systems of this type. utilizing a Butler matrix (15) beamformer whe rein the vector of beamformer outputs E may be expressed.

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of the three PE for nine trial s. and Fig. 2(bl shows the random behavior of typica l noise eigenvectors for the exact same tria ls. Th us. we would expect that the sidelobe undulations in Fig . I(b) are associated primarily with the noise eigenvectors . This thesis is verified in Fig . 3. which illustrates the adapted patterns resulting from (3) whe n only the PE are subtracted . The above adaptive array pattern behavior leads to the following obse rvations for source distr ibutions which do not encroach upon the main beam and involve a small number of the available degrees of freedom . I) It is possible to reta in low sidelobes in the adapted patterns , even with short obse rvation times. by constraining our algorithm (3) to utilize only the PE . The weight solution is unique and therefore stable. 2) Utiliz ing only the PE is tantamount to operating our adaptive system in beamspace (as opposed to element space ) with a set of weighted orthogonal canceller beams. 3) The fully adaptive array automatically forms and "assigns" its PE canceller beams to cove r the interference source distr ibution, with one beam per each OaF needed. Therefore, we have set forth a low-sidelobe eigenvector constraint algorithm for this type of restricted interference situation.

where 8 is a K x K matrix contammg the beamformer element weights. Other descr iptions of beamspac e systems are also available in the literature [II] , (16) , (17), [18), of which Adams et al. [17] is particularly germane to our discussion . Chapman (10) pointed out that when utilized in a partially adapti ve confi guration . such beamspace systems are susceptible to aperture element errors and cannot arbitrarily compensate the random error compone nt of their sidelobe structure. Thi s makes it necessary to control element errors in accord ance with the quiescent main beam sidelobe level desired. and fits into our initial assumption of low-sidelobe design mentioned earlier. A separate weighted main beam summing is indicated which may be obtained either by coupling into the beamformer outputs as shown. or by coupling off from the ele ments and providing su itable phase shifters for steering plus a co rpo rate feed network . Our purpose here is to examine the sidelobe performance of such a partially adaptive beamspace sys tem in which element erro rs are kept low and beamforrner beam s are subjected to simple constraints. Spat ial estimation data on the interference source distribu tion shall determine which beamformer beams are to be adapti vely controlled. Such beams are defined here in as " assigned" beams, and the idea is to assign only enough beam s to accommodate the OaF required by the source distr ibution . Whenever the two are equal, the adaptive weight solution is unique and we avoid adding any extra " noisy " weight perturbations . The reader will recognize that we are attempting to replace the PE beams of Section II-A with assigned beams from our ge neral beam former. Thus, we are defining a partially adaptive array which will utilize only a relatively small number of its available OaF. In addition to

197

where the particular value of k must be selected for the jth assigned beam. Our J dimension adaptive weight solution thus becomes

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(8)

Equation (8) gives us the J assigned beam weights required in (5). The proposed constraint I W,. I ~ 'Y can be applied directly to the solution from (8) , but recognize that this is a "hard" constraint and the results will not be optimal when the limit is exceeded. A softer. more flexible constraint for our purposes is one suggested by Brennan I based upon Ow sley [19], where one selects weights which simultaneously minimize both the output and the sum of the weight amplitudes squared , i.e . . Fig. 4 . Beamspace partially adapt ive array with a separatel y weighted main beam and canceller beam s assigned by a source estimation processor.

minimize {I V - wry 1 2 + aWrW *} .

this assigned beam constraint. we seek to limit the adaptive weights of assigned beams to a maximum level 'Y chosen to exceed the mainbeam sidelobe level by only a few decibels . This prevents an excessive rise in adaptive sidelobe level. including the condition where the number of assigned beams exceeds the OaF required . 'Y actually represents the product of assigned beam gain and adaptive weight magnitude. such that we have the option of working with beam former beams which are considerably decoupled/attenuated . An equation formulation may be expressed in terms of the same pattern subtraction principle as utilized in (3) for K beams. Wo=

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where I W k I ~ 'Y for J assigned beams and W k = 0 for all other beams . b k is the kth Butler matrix beam element-weight vector. When Wk == 0, that beam port is essentially disconnected from the output summation and it is much to our advantage to reduce the OaF of the adaptive weight processor accordingly , i.e. , this processor reduction relates directly to the computational burden . response time. sidelobe degradation, and overall cost mentioned earlier. For example. utilizing the SMI technique described in ( 1) and (2) . we would now have the advantage that our sample covariance matrix of signal inputs R involves only the J assigned beams and its dimensions reduce from K x K down to J x J, thereby greatly easing the computation burden involved in obtaining its inverse (II]. The equivalent "steering vector " A per Applebaum [9] is also reduced to dimension J and consists of the cross correlation between the main beam signal V and the J ass igned beam outputs Y

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(6)

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W==[R+aIJ -1A

(9)

where a = (,,(2/1) trace [R] . Note that (9 ) adds a small percentage o f the average assigned beam power to the diagonal terms of R. i.e .. it is a " pseudono ise " add ition technique . Recall that 'Y was selected to be close to the main beam sidelobe level. Although 0: is a small percentage of the trace [R). it is generally much larger than the receiver noise level P~. and this domination over recei ver noise by a constant will tend to se verely dampen weight fluctuat ions due to no ise . Of course. (9 ) deviates from the optimum Weiner weights and will result in a slightly larger output residue . but the cost is neglig ible compared to the remarkabl y stable results achieved from this rather simple con straint . It essent ially permits the number of assigned beam s to exceed the DOF required . and yet reta in low sidelobe level s . Equations (5 )- (9 ) were util ized in computing the adaptive pattern examples which follow . The reader should recognize that the J dimension adaptive weight so lution may be arrived at via any of the current adaptive processing algorithms such as Howells-Applebaum [9] . Gram-Schmidt [II] . sample matri x inverse update [:!Ol. etc. Applying the se constraints to our three-source case of Fig. 1. we would assign beamformer beams 10. II. and 12 to co ver the sources . as illustrated in Fig . 5 . These assigned beams are then given a maximum ga in level about 5 dB above the - 30 dB main beam sidelobes, or I W k I ~ 0 .055 . All other W k are set to zero . Typical resultant adapted patterns are almost identical to Fig . 3 . The pattern stability is near-perfect for a unique solution like this, and note that the three sources have been nulled with very little perturbation of the rnainbearn sidelobes except in the immediate vicinity of the sources . Since we are inverting a matrix of only 3 x 3 dimension in (8) for this case. it follows that the number of snapshots processed per trial could be reduced by an order of magnitude [12] and still have excellent results. The adaptive weights will tend to become "noisy" if we include even one extra OaF beyond the

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unique solution , However, if we use the " soft" constraint of (9) in solving for the weights , stable performance is again restored despite the extra OOF. Although not shown here. another example of interest was the case of using a two-beam cluster ( I I and 12 in Fig. 5) to cancel a single 40 dB broad-band source located at 22· . It was found that the source could be adequatel y cancelled at bandwidths up to IS percent. Many other combinations of source distributions and assigned beams were tested to further verify the technique. and the partiall y adaptive performance was satisfactory provided

that the assigned beams were sufficient to cover the DOF demanded by the source distribution .

C. Interference Sources in (he Main Beam Region Extension of the foregoing partiall y adaptive array technique for main beam interference is straightforward. provided we relax the constraint upon the value of y in (5). Obviously, the low-sidelobe strategem becomes seco ndary to the greater menace of an interference source coming in thru our high-gain main beam . Low sidelobes could still be retained. if necessary , by implementing a beam former which is capable of producing a family of low-sidelobe assigned beams [17]. III.

SOURCE ES TlM ATl ON AND B E A~l A SSIGNME NT

Modern spectral estimation technique s are considered complementary to the conventional methods for tracking and cataloging interference sources. They do not interfere with any function s of the main beam. and they are capable of providing superior source resolution from fewer elements. The latter advantage is gained in part because we assumed low sidelobes for the main beam. i.e. . the only sources that require estimat ion are those few which are of sufficiently high SNR to get thru the mainbeam sidelobes . Resolution performance is always directly related to signal-to-noise ratio (SNR). of course [3], [7], [8] . The principle of achieving source estimation from a small fraction of the aperture OOF has been demon strated via many techniques, both conventional and optimal [2], [4], [21]. It is not within the scope of this paper to attempt a comprehensive

comparison of such techniques, but the point is important to our concept so that an example of a half-aperture linear array estimator is given in this section. The type of application envisioned is illustrated in Fig. 6, where we represent a K x K element aperture system in which the adaptive beam OOF are to be assigned on the basis of estimates derived from two orthogonal linear arrays of KI2 elements each. An extension of the two-dimensional beamspace adaptive array system of Fig. 4 to the three-dimensional system suggested by Fig. 6 permits several beamformer options , including I) two orthogonal two-dimensional beam formers of which one is coupled into a row and the other coupled into a column of elements; and 2) a complete three-dimensional beamformer [22] coupled into the aperture elements , perhaps on a thinned basis . The separate main beam must be summed from all K 2 elements in order to attain the desired low sidelobes. Although they involve relatively few elements from the aperture. the linear array estimators represent a significant increase in system expense because they are all-digital processing subsystems. Processing of the digital signals to estim ate the sources may be carried out in accordance with a number of spectral estimation algorithms available in the literature [I J-[8J . Several algorithms that were utilized in the simulations conducted for this paper are discussed in [I] . Once the source estimation information is available, then we can assign beam former beams via a computer logic program. A Fortran IV computer code named "BEAMASSIGN" was developed which accepts source information updates. compare s the new data against a source directory kept in memory. computes track updates for sources alread y in memory . determines priority ranking , and assign s beams to cover the sources of highest prior ity. An important point to note is that beam assignment does not require great accuracy . i.e .. a halfbeamwidth is usually close enough. Also, clusters of two or three adjacent beams may be assigned for doubtful cases. A demonstration of beam assignment was conducted with a moving source simulation involving the 16-element linear array of Fig . I . Four sources of unequal strength were set up in the far field . traveling in crisscrossing patterns . Two of the sources are of 30 dB strength with start angles of 3.0· and 39 .0 · , and two are of 43 dB strength with start angles of 5.0 · and 70 .0·. The estimation of the scanned main beam for this example is shown in Fig . 7(a). Each time-unit plot cut is computed from R averaged over 160 snapshots , (10)

where S* is the main beam steering vector used to generate the display plot. As expected, this simple Fourier output is dominated by the two stronger sources . In contrast, Fig . 7(b) shows the source estimation derived from eigenanalysis processing using only half of the aperture (eight elements). Note that the "superresolution" characteristics of this type of optimal estimation produces excellent source tracking , even in the vicinity of crossover of three of the sources. The results of using the source information data contained in Fig. 7(b) to continuously update beam assignments is illustrated in the adapted pattern cuts shown in Fig. 8(a). Note that the main beam remains steady and the sidelobes seldom exceed

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their quiescent 30 dB peak level, despite the drastic shifting of the nulls as the moving sources crisscross in the sidelobe region. In contrast, Fig. 8(b) illustrates the adapted pattern cuts obtained when we utilize the SMI algorithm weights with the array fully adaptive . Although the source cancellation is excellent, the main beam suffers significant modulation and the peak sidelobe levels rise considerably .

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AN ADAPTIVE ARRAY TRACKING ApPLICAnON

A second area where spectral estimation techniques can provide valuable assistance is that of adaptive array tracking systems. Here we are dealing with the problem of attempting to track targets under the condition of having interference sources present in the main beam region. Some early proposed solutions in this area evolved from the growing adaptive array

200

technology of the 1970's. For example , a paper by White [23] discusses the radar problem of tracking targets in the lowangle regime where conventional tracking radars encounter much difficulty because of the presence of a strong surfacereflected ray . The first extension of fully adaptive arrays to angle estimation in external noise fields is the contribution of Davis et al. [24], who developed an algorithm based on the outputs of adaptivcly distorted sum and difference beams. The adaptive beams filter (null) the external noise sources, and distortion correction is then .applied in the resultant monopulse output angle estimate. Their work is particularly appropriate as a starting point for this section, where we discuss the advantages of using spectral estimation techniques in an alldigital, fully adaptive, array tracking system; [17] is also pertinent .

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A . Coherent Spatial Interference Sources The existence of significant coherence between spatial sources as , for example, in multipath situations involving a specular reflection, continues to represent a serious problem area even for a fully adaptive tracking array. Reasons include

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To demonstrate these reasons. adaptive characteristics were computed for a Io-elemeru linear array for an interference case in which there are two 13 dB coherent sources in the main beam region. There is also a third source. noncoherent, in the nearby sidelobe region to act as a stable null comparison point. In Fig . 9. we illustrate the severe changes in our main beam caused by variation of the phase shift between the two coherent Sources located at - 7.6· and - 4.0·. The quiescent mainbeam has the same Taylor weighting as in Fig . I(a) . Note that for phasing of O· and 180·, the adaptive weights are not achieving cancellation by steering nulls onto the coherent Sources but, rather. by the weight phasing itself. The array Output was driven down to receiver noise level for all three phases. The plots for 90· phase are very similar to what one would obtain if all three sources were noncoherent, i.e . . cancellation is achieved by adaptive null steering in this instance . Such severe sensitivity to coherent source phasing in the mainbeam region produces different distortions in tracking estimates from adaptive L (sum) and .:l (difference) patterns, as shown in Fig . 10. The equation development for this type of plot is in [I], but the main point here is to show the considerable changes in track angle estimates just due to phase variation. Once again, if all three sources were noncoherent, the distortion plot would be stable and very similar to the one shown for 90· phase.

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8. All-Digital Tracking System Concept The separate estimation of interference source data (total number, power levels, location angles, coherence) and its

utilization to improve the output SNR of desired signal detections is a mode of system operation that has been addressed in the literature a number of times for various applications [7], [8], [17], [18]. In this section, we briefly review such a system wherein the estimated data is used to drive a fully adaptive tracking processor [1]. The concept is illustrated in Fig. 11. Starting on the left side, the system continuously computes/updates a sample covariance matrix R. Of particular significance is the fact that R may be dimensioned either equal to or less than the total number of array elements, i.e ., the model order of the estimate is selectable per subaperture averaging option choice. Off-line processing on Ris then conducted at periodic intervals to estimate the locations and relative power levels of interference sources via the most appropriate spectral estimation algorithms. The central processor unit (CPU) then applies these data to the computation of optimized adaptive spatial filter weights for the right side of

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Fig. 12. Comparison of main beam scan versus spatial smoothing processing for coherent source case of Fig. 9 . PEGS eigenanalysis . 256 snapshots per trial.

Fig. II . Separation of source estimation from adaptive filter weight computation can be done accurately only in an alldigital processing system, but it permits the following benefits :

this source estimation data, we can construct an equivalent covariance matrix dimensioned for the full aperture per the procedure given in Appendix I. and compute its inverse for obtaining the adaptive filtering . If we define the constructed covariance matrix as M , then its inverse may be viewed as a matrix set of adaptive' 'bearnforrner' filter weights to give us the filtered output nth snapshot vector Ef(n),

I) estimation of coherent interference source locations for

(II)

Fig . II.

All-digital adaptive array tracking system concept.

deliberate adaptive null filter placement; 2) remembering slowly changing or time-gated sources; 3) anticipating sources from a priori data inputs; 4) flexibility in time-domain control of the filtering to counter interference time strategies; 5) tracking/cataloging/ranking sources; 6) efficient assignment of available OaF; 7)compatible with fast-response adaptive algorithms, i.e ., parallel algorithm processing. The right side of Fig. II indicates a fast-memory storage capability which is intended to permit selected time delays of the snapshots for feeding into the filter weights . The idea is to synchronize selected snapshots with their filter weight updates if possible . Finally, the filtered signal output residue is fed into a beamformer which is weighted to produce the desired search and mono pulse track beams for target detection and tracking . The algorithms of Davis et al. [24] may be applied for estimating the target signal angle of arrival, based upon the outputs of adaptively distorted sum and difference beams. Reference [1] discusses the equivalence of such beams to the Fig . 11 concept. As an example, let us apply this concept to the coherent sources case utilized for Figs . 9 and 10 wherein we would utilize a 16-element linear array feeding into our all-digital processor. An appropriate estimation algorithm is that of forward-backward subaperture spatial smoothing [7] , [28], [29], [30] combined with eigenanalysis . The rudiments of this algorithm are described in [1], and the results are plotted in Fig . 12 in comparison with a scanned main beam output. From

Conventional beam weighting S* can then be applied to the filtered output residue to obtain the final output for the nth snapshot, Yo(n)

=Ej(n)S* = E'(n)M -IS*

(12)

or

where W 0 is the familiar optimum Wiener filter weight. Note that the constructed covariance matrix M permits options such as adding synthetic sources or changing power levels. Furthermore, since it is always Toeplitz, solutions may be simplified somewhat. For the current example. the computed adaptive characteristics would be very similar to those plotted in Figs . 9 and 10 for the 90° phase angle.

202

V.

CONCLUSION

Two conceptual application areas have been presented for using spectral estimation techniques; partially adaptive lowsidelobe arrays, and fully adaptive tracking arrays. In both cases, improved spectral estimation techniques are used separately to acquire information about the interference environment which is beyond that ordinarily available in a conventional adaptive array . Examples discussed included "superresolution" effects, relative power level determination, estimation of coherent sources, and the tracking/cataloging l ranking of sources. For the partially adaptive area, the information was utilized for efficient assignment of a limited number of DaF in a beamspace constrained adaptive system in

order to obtain the following benefits (as compared to a fully adaptive array): retention of low sidelobes plus a stable mainbeam; considerably faster adaptive response; reduction in overall cost; and greater flexibility. On the negative side of the coin, we incur the risk of possible inferior cancellation performance if the interference source situation is not adequately covered by the assigned DOF. For the fully adaptive tracking array area the information is utilized in an all-digital processing system to obtain the benefits of stable nulling of coherent interference sources in the main beam region, efficient assignment of the available DOF, and a far greater flexibility in the time-domain control of adaptive filtering strategy. ApPENDIX

assumed to be a random process with respect to both the time index n and the element index k. ) Equation (15) permits us to construct a convenient column vector of observed data in the form, E(n) = Vp(n) + ,,(n)

where V is a K x I matrix containing a column vector each of the I source directions; i.e.,

Consider a simple linear array of K elements. The received signal samples are correlated in both space and time, giving rise to a two-dimensional data problem, but we convert this to spatial domain only by assuming that narrow-band filtering precedes our spatial domain processing. Bandwidth can be handled when necessary via a spectral line approach [13] or tapped delay lines at each element [20). but we did not consider such extra complication essential to the basic purposes of this analysis. Thus. the postulated signal environment on any given observation consists of I narrow-band plane waves arriving from distinct directions AI' The RF phase at the kth antenna clement as a result of the ith source would be the product WIX~" where .¥k is the location of the element phase center with respect to the midpoint IJf the array in wavelengths. and WI is defined as ( 13)

This notation is deliberately chosen to have the spatial domain dual of sampling in the time domain. so that the reader may readily relate to the more farnil iar spectral analysis variables. Sin (J, is the dual of a sinusoid frequency /,, and the X k locations are the dual of time sampling instants t k • Note that if our elements are equally spaced hy a distance d, then X k may be written. ( 14)

where 'A is the common RF wavelength. The ratio d/»: becomes the dual of the sampling time T with the cut-off frequency equal to the reciprocal. The complex amplitude of the ith source at the array midpoint phase center is P" such that we can now express the nth time-sampled signal at the kth element as I

(}WiXk)

for

Note that (16) separates out the basic variables of source direction in the direction matrix V, source baseband signal in the column vector p (n), and element receiver channel noise in the column vector." (n). The vector E(n) is defined as the nth snapshot, i.e., a simultaneous signal sampling across all Karray elements at the nth time instant. These snapshots would nominally occur at the Nyquist sampling rate corresponding to our receiver bandwidth, so that a radar-oriented person may view them as range bin time samplings. However. for source estimation purposes. they need not necessarily be chosen from contiguous range bins and. in fact. for most applications it would be highly desirable to selectively time gate the snapshots used for source estimation. For this simple analysis . let us postulate that the snapshots are gated at more or tess arbitrary instants of time. Over typical processing intervals. the directions of arrival will not change significantly. so that V is a slowly changing matrix. In contrast. the signals Pi(n) will generally vary rapidly with time. often unpredictably. such that we must work with their statistical descriptions. It is assumed that the signals are uncorrelared with receiver noise. Proceeding then from (16). we can obtain the covariance matrix R via application of the expected value operator 8, or ensemble average.

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(17)

SNAPSHOT SIGNAL MODEL

Ek(n) = T/k(n) +

(16)

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( 18)

R== VPV*I+ N

( 19)

where N == 8['1(n).,,(n)*r], P = 8[p(n)p(n)*/], the asterisk is the complex conjugate, and t is the transpose. N is a simple diagonal matrix consisting of the receiver channel noise power levels. The diagonal elements of P represent the ensemble average power levels of the various signal sources, and off-diagonal elements can be nonzero if any correlation exists between the sources. Note that correlated far-field signals can easily arise if significant specular reflection or diffraction multipath is present.

(15)

i==1

Where gk «()i) is the element pattern response in the direction ()i, and 11k (n) is the nth sample from the kth element independent Gaussian receiver noise. (The receiver noise component is 203

REFERENCES

W. F. Gabriel. "Using spectral estimation techniques in adaptive processing antenna systems." Naval Res. Lab. Rep. 8920, Oct. 1985. [2] D. G. Childers, Ed.. Modern Spectrum Analysis. New York: IEEE Press. 1978. (31 W. F. Gabriel. "Spectral analysis and adaptive array superresolution techniques," Proc. IEEE, vol. 68. pp. 654-666, June 1980. [4] Special Issue on Spectral Estimation, Proc. IEEE, vol. 70, Sept. 1982. [5] S. Y. Kung, H. J. Whitehouse. and T. Kailath. Eds., VLSI and Modern Signal Processing. Englewood Cliffs, NJ: Prentice-Hall,

(1]

[6]

1985.

R. Schmidt. "Multiple ernitter location and signal parameter estima-

[7]

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

[181

r19] [20]

tion," in Proc. RADC Spectrum Estimation Workshop, RADC-TR79-63, Rome Air Development Center, Rome, NY, Oct. 1979, p. 243. J. E. Evans, J. R. Johnson, and D. F. Sun, "Application of advanced signal processing techniques to angle of arrival estimation in ATC navigation and surveillance systems," MIT Lincoln Lab. Tech. Rep. 582, (FAA-RD-82-42), June 1982. A. J. Barabell et al., "Performance comparison of superresolution array processing algorithms," MIT Lincoln Lab. Rep. TST-72, May 1984. S. P. Applebaum, "Adaptive arrays," IEEE Trans. Antennas Propagat., vol. AP-24, pp. 585-598, Sept. 1976. D. J. Chapman, "Partial adaptivity for the large array," IEEE Trans. Antennas Propagat., AP-24, pp. 685-696, Sept. 1976. R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. I. S. Reed, J. D. Mallett, and L. E. Brennan, "Rapid convergence rate in adaptive arrays," IEEE Trans. Aerosp. Electron. Syst., vol. AES10, pp. 853-863, Nov. 1974. W. F. Gabriel, "Adaptive arrays-An introduction," Proc. IEEE, vol. 64. pp. 239-272. Feb. 1976. S. P. Applebaum and D. J. Chapman. "Adaptive arrays with mainbeam constraints," IEEE Trans. Antennas Propagai., vol. AP24, pp. 650-662, Sept. 1976. J. Butler, "Multiple beam antennas," Sanders Assoc. Internal Memo RF 3849, Jan. 1960. J. T. Mayhan. "Adaptive nulling with multiple-Beam antennas," IEEE Trans. Antennas Propagat., vol. AP-26, pp. 267-273, Mar. 1978. R. N. Adams. L. L. Horowitz. and K. D. Senne. "Adaptive mainbeam nulling for narrow-beam antenna arrays, ., IEEE Trans. Aerosp. Electron. Syst., vol. AES-16. pp. 509-516, Jul. 1980. E. C. DuFort. "An adaptive low-angle tracking system:' IEEE Trans. Antennas Propagat., vol. AP-29, pp. 766-772, Sept. 1981. N. L. Owsley. "Constrained adaption." in Array Processing Applications to Radar. New York: Academic, 1980. E. Brennan. J. D. Mallett. and I. S. Reed .:: Adaptive arrays in airborne

[21] [22] [23] [24] [25] [26] [27] [28] [29]

[30]

204

MTI radar," IEEE Trans. Antennas Propagat., vol. AP-24, pp. 607-615, Sept. 1976. B. M. Leiner, "An analysis and comparison of energy direction finding systems," IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, pp. 861-873, Nov. 1979. J. P. Shelton, "Focusing characteristics of symmetrically configured bootlace lenses," IEEE Trans. Antennas Propagat., vol. AP-26, pp. 513-518, July 1978. W. D. White, "Low-angle radar tracking in the presence of multipath," IEEE Trans. Aerosp. Electron. Syst., vol. AES-10, pp. 835853, Nov. 1974. R. C. Davis, L. E. Brennan, and L. S. Reed, "Angle estimation with adaptive arrays in external noise fields," IEEE Trans. Aerosp. Electron. Syst., vol. AES-12, pp. 179-186, Mar. 1976. W. D. White, "Angular spectra in radar applications," IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, pp. 895-899, Nov. 1979. A. Cantoni and L. Godara, "Resolving the directions of sources in a correlated field incident on an array," J. Acoust. Soc. A,n., vol. 64, pp. 1247-1255, 1980. B. Widrow et al., "Signal cancellation phenomena in adaptive antennas: Causes and cures," IEEE Trans. Antennas Propagat., vol. AP-30, pp. 469-478, May 1982. T. J. Shan and T. Kailath, "Adaptive beamforming for coherent signals and interference," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 527-536, Jun. 1985. A. H. Nuttall, "Spectral analysis of a univariate process with bad data points, via maximum entropy and linear predictive techniques," Naval Underwater Syst. Center, New London, Cl', NUSC-TR-5303, Mar. 1976. L. Marple, "A new autoregressive spectrum analysis algorithm," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-28, pp. 441-454, Aug. 1980.

Implementation of Adaptive Array Algorithms ROBERT SCHREIBER

A. Notation

Abstract-Some new, efficient, and numerically stable algorithms for the recursive solution of matrix problems arising in optimal beamforming and direction finding are described and analyzed. The matrix problems considered are systems of linear equations and spectral decomposition. While recursive solution procedures based on the matrix inversion lemma may be unstable, ours are stable. Furthermore, these algorithms are extremely fast.

I

I. INTRODUCTION

N this paper we consider the computational procedures to be used in implementing some standard and some more recently proposed adaptive methods for direction finding and beamforming by sensor arrays. We discuss the computation of a minimum variance distortionless response (MVDR) beamformer and of several high-resolution methods (recently advocated by Bienvenue and Mermoz [1], Owsley [9], and Schm idt [10]) that are based on the spectral decomposition of the signal covariance matrix. We are especially concerned with recursive implementation of these procedures. Whenever the signal is sampled, an estimate for the covariance matrix is updated and the computed solution (a weight vector) changes in response to this new information. We shall propose and analyze some new, efficient. numerically stable algorithms. The computational procedures we advocate take advantage of this on-line character. We find methods for updating the solutions that are much less expensive than procedures that do not make use of the previously computed solution. For the MVDR method, some previous work has been done [8]. An update method based on the Sherman-Merrison-Woodbury formula (which is also known as the matrix inversion lemma) has been advocated. We show that this procedure can, in one common circumstance, be numerically unstable. We propose three new, stable methods here. For the high-resolution methods, we illustrate the use of some efficient procedures for updating eigenvalue and singular value decompositions. We show how to take advantage of the existence of multiple eigenvalues of the signal covariance matrix to further reduce the work. We also show that complex arithmetic can largely be avoided. Manuscript received June 26, 1984; revised December 18. 1985. This work was supported by the U.S. Office of Naval Research under Contract N00014-82-K-0703 and by the U.S. Army Research Office under Contract DAAG 29-83-K-OI24. The author is with the Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180. He is also a Consultant to Saxpy Computer Corporation, Sunnyvale, CA 94086. IEEE Log Number 8609623.

Let ((~n and ((;;m X n denote the spaces of complex n vectors and m X n matrices. We use upper case italic letters for matrices, lower case italic letters for vectors. The In X n matrix A has, by convention, the columns ra), Q2, . .. ,a,,], and the elements [ai.i]; the vector x has elements (~b ... , ~n)T; for A E (em x ", A T denotes the transpose and A H the conjugate transpose of A. If A E I,c;n x n is diagonal (a,.) == 0 for i j), we denote A by diag(al, I . . . an.,,)' We denote the r x r identity matrix by l.. For A E ":" X ": the Frobenius norm of A is given by

'*

In giving operation counts for algorithms. we use the terrn operation to mean one complex multiplication and one complex addition. One operation costs about as much as four real multiplications and four real additions. Note that computing .r + a)' with real a costs one-half an operation.

II.

AN ON-LINE ALGORITH~1 FOR ,-\DAPTIVE

BEA~1FOR"llNG

Let .r E "be a narrow-band signal received by an array of n elements. Let its covariance matrix be denoted R.

( 1) where E{ } denotes expected value. R is Hermitian and. if any noise is present, positive definite. Thus. R has a Cholesky factorization

(2) where L is lower triangular and has positive, real diagonal elements. The factorization can be computed in n 1/6 operations [12]. With the help of the Cholesky factorization, we can compute R- 1d with fl2 operations by solving two triangular systems: Lu == d. and

Thus, w == LFf-lu == (LLH)-Id == R-1d. Consider the adaptive control of an n-element array. The minimum variance distortionless response beamformer determines the output of the array by g(d) = ~vHX

Reprinted from IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-34, No.5, pp. 1038-1045, October 1986.

205

(3)

where g is an estimate of the signal arriving from some given bearing, d is a steering vector for the given array and bearing, x is the signal vector, and w = R-1dp(d),

(4)

where p(d) is an estimate of the average power arriving from the given bearing, p(d)

==

(dflR-1d)-I.

(5)

In practice, we have several bearing angles and corresponding steering vectors d., i = 1, 2, .... , m. Let D be the n x m matrix of these vectors D

== [d" ..... , d",].

The principal computational problem is, then, to find the n x m solution matrix

(6) In the on-line beamfonning problem, D remains fixed, but R is often changed to incorporate a new sample x of the signal:

(7) where J.l. E (O~ I). We refer to such a change as a rank-one update to R (since the rank of xx H is one) although . if J.L =f.:. 1, the change to R is, in general, of full rank. One must find the corresponding updated solution *W

= *R-1D.

(8)

The obvious method [81 is based on the Shennan-Morrison-Woodbury formula for *R - I:

*R- '

=

J.L -IR-

1

+ 13:::. H

c) for every column w of Wand corresponding col umn d of D, i) compute 0 : = B: Hd; ii) compute w : = J.L -lw + The cost of step 2a) is n 2 operations; of step 2b) i~ (~)n 2 operations; of step 2ei) is nm operations; of stej 2cii) is about (~)nm operations. The alternative algorithm, in which the updated Cholesky factorization of f is used to solve for R-1D, costs n 2m operations for solv ing triangular systems and (~)n ~ for updating the factorization. Unfortunately, the method is unstable. If 0 < p.< 1. then p.- I > 1. Any error in W is amplified by the factor 1 J.L- every time the update (12) is performed. These errors eventually render the computed solutions W uselessly inaccurate. Thus, correct solutions must occasionally be calculated directly from D and the Cholesky factorization of R according to (8). Fortunately, in some applications one may take J.l. ~ 1 (so that the estimate of R computed using (7) is a better approximation to the true covariance matrix.) Thus, #J. -I is only slightly larger than 1, and the "unstable" update (12) can be used for quite some time. In fact, the choice J.l. ~ 1 is appropriate for relatively stationary signal environments. And in this case, it may also be allowable to avoid updating the weights with every new sample. But, in a rapidly changing environment, one would take J.L substantially smaller than 1, to allow R to change fast enough. In that case" the update (12) would be useless" and a stable method would be essential. Are there equally efficient, stable methods? By (11) and

oz.

(6)

(9)

= XH\v.

where (10)

Thus, (12) is equivalent to the formula *~V

and

(11) Thus, if d is a typical column of D. and \v is the corresponding column of W, then by (6L (8). and (9) . __ IIl.J ( 12) *\V -- Jl -I ~v + fJ/~ ........ To make use of the method (12) one requires that R-1x be computed. This can be done with the aid of the Cholesky factorization of R. Moreover, Gill, Golub, Murray, and Saunders have suggested a method for updating the Cholesky factorization after the rank-one change (7) that uses about (~)n 2 + O(n) operations 15]. Fast computation of this algorithm by parallel processor arrays was considered by Schreiber and Tang [Ill· This suggests the following algorithm. 1) (Initialize.) Let R = I, L = l, and W = D. Thus, R = LL H , and W = R-1D. Compute (3 from (10). 2) Every time an update (7) is made to R, a) solve for z = R-1x by solving the two triangular systems Ly = x and L HZ = y~ b) update the Cholesky factor L of R; and 206

=

JL -I~,

+ I3zx H w .

( 13)

Notice that w now appears twice. Perhaps the second use of w has stabilized the method? It has. Theorem: The residual does not change when formula (13) is used: even when w only approximately satisfies Rw = d, the identity d - Rw = d - *R*~v holds. Proof: It suffices to show that *R*w = Rw. By Jirect computation

*R*,v = (JlR

+

(1 - Jl)XXH)(Jl-I~V

= Rw + xx

Hw[{3(J!

+

+

(3zxH~v)

(1 - p,)xHz)

+

(1 - Jl)j-t-l]

= Rw sinee (3 (j-t +

(1 - Jl)x HZ) = (Jl - 1) JA. - I . II This shows that the error is not amplified by (13). A similar analysis can be done for the method (12). It shows that the residual can increase by a scalar multiple of x, whose length is proportional to (xHw - zHd). An even more stable procedure can be devised. From (12) it follows that

*w -

W E

span {w, z}.

(14)

z tI '

Jl -

/I'

IV

1,=

/I '

I tl'

,=

/I '

+n

,1 I

+ -, z

Fig . 1. Computat ion of

a ,.

All update procedures seek in some way to find the linear combination of wand Z that, when added to w, gives *w . Among the many possible methods are a group of conjugate direction procedures that are especially desirable in that they make almost no assumptions other than (14) and use the available data to choose the coefficients of wand z in the linear combination. In these methods. one chooses an orthogonal basis for the span of wand ::. then takes a step from w in the direction of one of the basis vectors. go ing to the precise point in that direction closest to * w. Then another step, in the direction of the other basis vector. produces *w. The various possible algorithms differ in the choice of basis and the innerproduct used . The following procedure is computationally convenient. Define z == *R - Ix. Note that ; is a sca lar multiple of R- 'x . Choose 'Y so that z and z , == IV - 'YZ are *Rorthogonal. (T wo vectors u and u are *R-orthogonal if u H " Ru = 0 .) Starting fro m IV, take a step a l;: 1 so that w, == W + al Zl is as close as possible to *IV with respect to the *R-norm. (The square of the *R-norm of a vector u is given by u H *Ru.) Then take a step a 2 Z so that W2 == WI + a2 Z is as close as possible to *w . Now *w = W2 is the updated solution. It is well known [6] that if Rw = d exactly then, except for rounding errors, W2 exactly equals *w. On the othe r hand. suppose that w = R- 1d + e . where e is the current error in w. We can write e = el + e2, where el E span {w, z } and e2 is *R-o rtho go nal to both w and z. Then after application of the two conjugate direction steps as de scribed above , we will obtain *w = *R- Id + ez. In other words , the component of the error that lies in the span of wand z will have been annihilated . Computation of the step lengths in a conjugate direction method usually involves computing dot products. In this case , however, computation o f al can be greatly simplified by some geometric insight-see F ig . I. By (12 ) , *w - /l -IW is a sca la r multiple of z . Moreover, w, = (1 + al)w + a l'YZ. Thus, *w - WI is a scalar multiple of z if *w + al)w is. Thus , we should take al = /l- I -

I.

(I

Of course, the computed solution w does not exactly satisfy (12) . So we should really compute al by requiring

that *w - w, be *R-orthogonal to ZI ' But we would need to· compute a matrix-vector product to be able to do this without error. That would raise the cost of the method from O(n) to O(n 2) . In fact, our procedure is equivalent .to replacing the product Rw by the vector d in the innerproduct Z ~ Rw. Let r = Rw - d. The error we make is therefore Z ~ r. Now ZI is orthogonal to x. So the error will be rather small if r is close to the span of x. In view of the fact that r is a residual and R is gi ven by (I), it is likely that this is so. The method is as follows. Algorithm (Conjugate Direction): Given the Cholesky factor L of R, a new signal sample .r , the current computed solution W = R-1D , and /l, I) update the Cholesky factor; now *R = *L *L H; 2) so lve (*L*L H)Z = x; 3) compute Z HX (which is real); and 4) for every column w of Wand corresponding column d of D . a) compute xHw ; b) compute ;: Hd; c) compute v : = -x Hwl;:Hx : d) compute 2, : = w + 'Y;: ; e) cornpute ce . t > /l - I - I ; f) compute w l : = w + a ,;: ,: g) compute a2 : = (zHd - xHw)/;:HX: h) compute W2 : = WI + a2 ::: i) stop : W2 is the computed solution to " R" IV = d .

•

Re call that computing x + ay with complex .r , v. ami real a costs one-half an operation . Thus . this conjugate direction procedure costs (~)n 2 + (~ )11 11I operations . For m » n it is slightly less than twice as costly as the meth ods (\2 ) and (13 ) . In view of the experimental results given below. it does appear to be more accurate than ( 13) in some cases . But its superiority is not uniform : it de pends on a 2 and u : A. Experimental Tests

We have verified our claims by an experiment. Vectors x = (1 ,2 , 3,4, 5 , 6)T + s were generated. where shad random, independent, normally distributed components of mean zero and variance a 2 . We took d = ( I, I. I. I , I. I )T and initially R = I and w = d. We then used the three methods discussed above for 100 updates. Let W(I ) denote the solution vector given by the method (12), w(2 ) the solution given by the stable for mula (13), and w (3) the solution given by the conjugate direction formulas . At each step we updated the Cholesky factorization of R by computing Q

[(I -

Il) 1/2X T]

/l 1/2L T

=

[*L 0

r ]

where Q is the product of n plane rotations . We give four error statistics below . The first is a measure of the accuracy of the updated Cholesky factor L,

207

ER ==

IIR -

LLTIIFIIIRIIF'

RELATIVE

p. (}2

=

ER E1 E2

£3 (12

=

ER E, £2 E)

(}2

= 0.8

100 = 0.197( -6) = 0.408( +3) = O.448( - 5) = O.840( -6) 10- 2 = O.224( -6) = O.284( +4) = O.467( - 5) = O.850( -5)

= IO-h

ER

TABLE I ERRORS AFTER 100

= 0.206( -6)

£1 = 0.990( +5) £2 = O.159( -2) E, = O.402( -2)

p.

Lv

UPDATES

= 0.9

p.

= 0.501(-6) = O.172( -1) E2 = 0.184( -5) ER

£1

E) = O.252( -6)

g(d)

= 0.219(-5) = 0.144( -4) £2 = 0.761(-5) £) = 0.412(-6) ER

= 0.818(-6)

EJ

= O.240( -

£2

EJ

= O.720( -5) = O.702{-6)

5)

= O.677( -6) = 0.709(+0) £2 = 0.344(-3) E.\ = 0.872( -4) £R

£1

(16

= a.2IO( -5) = 0.113(-4) E'). = O.256( -4) E., = O.244( -4) ER

A theoretical analysis of this procedure is given in Section II-B below. Also, for j = 1, 2, 3, we give a measure of the error in the updated solution l-v( j),

EJ == Ihv(j) - (UT)-'dll/\I(LLT)-'dll.

wHx

= (v Hy) p(d)

( 17)

·where y is the solution to the triangular linear system

Ly

E, = O.129( -4)

£1

=

= dHR.-1xp(d)

ER

£1

= O.307( -6) = O.365( -I)

£2 = O.955( -5)

d.

And from (2), (4), and (3) we have that

= 0.99

E,

ER

=

= x.

(18)

If we are willing to solve the system (18) at a cost of

n 2 operations for every new signal x (and if there are many bearings d, this is reasonable), then we may use (17) to compute g(d). Thus, we no longer need the weight vector

w, but rather the vector u and the power estimate p. We now give a stable method for updating v and p after the change (7). This new algorithm is especially convenient in that it can be incorporated into the process of updating the Cholesky factor L of R. The systolic array devised by Schreiber and Tang [11] can be used to perform the necessary additional computations. By (7), we seek the Cholesky factor *L of

We took J.L = 0.8,0.9, and 0.99 and 0 2 = 102 , 10- 2 , and 10-6. The results were essentially unchanged for 0 '2 greater than 102 . In Table I we show the errors in the format

h were y == (1 - Jl) i,"-x. Let Q be an orthogonal matrix such that TQ == [0

E~ E~

for each pair (IJ.. 0 2). The notation O.123( -4) means 0.123 x 10- 4 • All computations were done in single pre-

x

*L]

*R = TQQflT

=

Il

+ 1 (19)

H

L LH

* *

so *L is the Cholesky factor of *R. This method is very stable. If there is some error in L, for example, if

B. Another Stable Method

We now discuss a third stable updating method that differs in two ways from those already considered. It avoids explicitly fanning w; and it can be viewed as an extension of the process for updating the Cholesky factor L-a process that we shall describe more fully in this section. From (2) and (5) we have that (p(d))-l

+ 1

where *L is lower triangular with positive real diagonal. It is easy to see that Q can be obtained as the product of n plane rotations. Now. clearly ~

cision on a VAX.

Note that the conjugate direction method (method 3) is distinctly more accurate in those cases where high accuracy is useful: low signal-to-noise ratio, which tends to make R well conditioned. and Jl ::::: 1. so that R is accurately estimated. Three such cases occur in the upperright-hand corner of Table I. In these cases, the conjugate direction method is ten times more accurate than the stable update method that uses (13).

11

= dHR-ld (15)

where v is the solution to the triangular linear system 208

LL H == R + E where E is an error matrix, then by (7) and (19), *L*L H = TQQHTH

= TT H = v:" + JlLL H = *R + J.LE. Since Jl < 1, the error is reduced. In this sense, this method of updating the Cholesky factor is self-correcting. It was used in the experiments of the previous section, which show that it is very accurate. It can, therefore, be strongly recommended. Now let d be a given steering vector, let p = p(d) be the corresponding power estimate, and let u = v(d) =

L -1 d be the corresponding solution to (16). Apply the rotations used in finding *L to obtain [0,

J.L -1/2VH]Q ==

[8,

*v H].

(20)

Now it follows, by (19) and (20), that d =

lY

=

[0

1r

l:-I12J

Jll/2 L

= TQQH

0

l.Jl

_1/2

and *p

Since

jl-I

-I

H

= *v *v

+

JL

-I

E.

> 1, this approach is unstable, and

*p should

H

be computed as *v *v. This procedure requires (~)n 2 operations for the Cholesky update, (~)nm operations for updating *v using (20), and (! )nm operations for recomputing *p.

1

•

III.

METHODS FOR UPDATING THE SPECTRAL DECOMPOSITION

*L1l:J

A number of modern, high-resolution methods make use of the spectral decomposition of R [1], [9], [10]: R == MAM

== *L*v, so that *v is the updated solution to (16). To show that (20) is both correct and stable. we assume that Lv is not exactly equal to d . but that

H

(21)

where A == diag ( AI ~ ... , A,,) is the matrix of eigenvalues of R, ordered so that

AI

Lv == d + ,.

~ A~ ~

...

~

Aft"

Here M == [/nl ~ . . . , Ill,,] where In; is a normalized eigenvector corresponding to Ai' Note that /vt is unitary (M H where r is a residual vector. But by (19) and (20), == MWe shall be concerned, therefore, with updating d + r == Lv the decomposition (21) after a rank-one change (7) to R. In practice, one uses an estimate of R that is the product XHX, where X is a matrix whose rows are (weighted) sampies of x. Therefore. the eigenvalues and eigenvectors of the estimate are the squared singular values and right singular vectors of X. Periodically. a new observation of .r is made and is appended to "Y as a new row. Thus. the problem of updating the spectral decomposition (21) is == *L*v. mathematically equivalent to that of updating the singular so that *v satisfies (16) as well as did u. This is therefore values and right singular vectors of X when a row is apa stable update method. pended. We now consider an efficient update formula for the Bunch and Nielsen recommend that to update the sinpower estimate p. From (20) it follows that gular value decomposition of X when a row 1':) added, one should update the corresponding spectral decomposition II - 1P - I == [0 - 1/2V H] I 0 I of R after a rank-one change (2). This can obscure the r: J.1. -1/2 I l f.1 v small singular values of X: a singular value of size A,\O + I

=

A.\" ~ 2

[~I

= . . . = An ==

:AJ

where SI = J.LA I + (1 - J,t)zlz7 and ZI = (rt, r.~ + I)T. Now, given the spectral decomposition

S.

=

TI*A 1 Tf,

we have that

T

S= [ I o

Si=,nix,

a 2•

l~i:5s.

We must first compute these values. Now we let

(23)

then

H'

s=

H

H

and let *M

is done, then

is the spectral decomposition of S. This has been observed in previous work [7]. Let us be specific about the choice of ms + .' . . . , m., Since z = MHx we have

Let S == JLA

r, = O. If this

In.\,+

I

== (x

-± 1=

I

r,m i

)/lI x -

1

±rim;ll. = I

Thus, m, ... I is the normalized orthogonal projection of x on N. Now add additional vectors Inj' j = s + 2~ .. " n, until an orthonormal basis for N is obtained. This makes t, = 0 for j == s + 2~ ... , n . In fact. it is not necessary to construct In" + 2' • . • , m.; (If it were, they could be taken as the columns of a certain Householder matrix [3].) Note that the number of eigenvalues (of the estimated covariance matrix) greater than o 2 can increase by 1 every time we update. On the other hand. the number of eigenvalues of R greater than a:' is determined by the number of linearly independent signals hitting the array. The updating (7) moves one of the eigenvalues a 1. to the right, but the remainder move to the left. reduced by the factor u, In the equilibrium state of this process there are s eigenvalues greater than a 2. and a cluster of n - s near a 1.. This is not completely satisfactory. Karasalo, Goetherstrorn, and Westerlin suggest an attractive method that can be used if we have an a priori upperbound s on the number of signals [7]. They replace the new matrix *R by its closest approximation by a matrix of the form A + a:'l, where rank(A) == s. They show that

(24)

(The space N == span {ms + I, . · . ~ ,n,,} is called the noise subspace. While R determines N~ any orthonormal basis for N can serve as the last Il - S columns of M.) To take advantage of the repeated eigenvalue, let AI = diag (AI' . · • , A~ + I) and A:! = diag (As +:!' . · . , An)' It is possible to choose the last n - s columns of M so as to make t\, + 2

210

and that the new noise level is *a 2 == [(12 -

S -

1)(12

+ *As + tl/(n - s).

To compute the eigenvectors of *R, note that

(25)

where M} = [m}, . · . , m, + I]. Because T is real, this multiplication costs n(s + 1)2/2 operations. How much more costly is it to recompute (21) every step? Direct computation of a Hermitian spectral decomposition requires approximately 5n 3 + O(n 2) operations. The work required by the method developed here is 1) form z1 at cost n(s + 1) operations; 2) compute the spectral decomposition of 51 at cost O(s 2) operations; and 3) compute *M = M 1 Tat cost (~)(s + 1)2n operations. The net cost is therefore (~)n(s + 1)2 + Oins + s 2) operations. The relative expense of the new scheme drops very rapidly as n/ s increases from 1, and is never more than 23 percent of the cost of a full spectral decornposition. Additional computational savings are possible. Let us consider the methods proposed by Owsley [9], which are typical. These methods all seek to estimate the power of the signal hitting the array at a given angle by g(d) == ~vHd

TABLE II OPERATION COUNTS AND STABILITY OF THE BEAMFORMING ALGORITHMS

Section II Nonrecursive solution of (4) Matrix inverse lemma:

J=1

mj/J'( A;)m JHd

c

==

MHd.

*c

==

*MHd.

311m

+

m nm

n.a.

(27)

and 5+1

*g

==2: ./=1

(3(*Aj) \ *~'r'j

2 1

To compute *g, we need only update But by (25) and (28) *c

n.a.

o

.\ + I

*MHd

THMHd

.

C1.

(29)

then use (29).

n.a.

+

Key

Denote the elements of these vectors by c == ('Y 1 ' 'Y n)T and *c = (* 'Y 1. • . . , *'Y n) T. Let C I == ('Y 1, v, + 1)T and *c\ == (* 'Y1' . . . . *v, + I)T. Then

(3/2)n

(3/2)n 2

p:

(28)

+

+

o

Recomputing p

and

nm

(5/2)nrn + m] (l/2)n{5n + 9m]

Section II-B Updating the Cholesky factor L using orthogonal transformations: (19) Updating the solution v using the same orthogonal transformations (20): Using the orthogonal transformations to update

where f3( A) is real valued. Various choices for (3 can be made that succeed in suppressing the effect of noise and enhancing the resolution of the method. In computing the output power g(d), we can do better than to use the definition (26) directly. We can instead use the relations (25), (26), and (27) to reduce the cost from n'2 operations to (5 + 1)212 operations for each vector d. To be specific, let

+

(10)-(11)-(13)

+

== .2:

2m

(5/2)nfn

Conjugate direction:

(26)

n

Stability

(10)-(11)-(12)

Stabilized matrix inversion:

where d is a steering vector for the given array and angle and w

Cost

Method

= o" v:

m]

+

stability is not an issue for nonrecursive methods: the method is unstable: errors are amplified by every step; the method is stable: errors are neither amplified nor damped; the method is stable: errors are damped by every step:

so that (30)

Thus. from (29) and (30). we can compute g with only (s + 1)212 operations for each vector do IV.

CONCLUSIONS

We have given three numerically stable and cornputationally efficient procedures for adaptive bearnforming that improve, either in speed or accuracy. on known procedures. These procedures make methods based on the inverse of the signal covariance matrix much more practical for real-time use. This is especially true for large sensor arrays, since the dominant cost of these procedures grows only linearly with the number of array elements (in this respect they are like the LMS method). Straightforward use of the matrix inverse or a triangular factorization incurs quadratic cost. For methods based on a spectral decomposition of the signal covariance matrix, we have obtained a similar economy. The resulting rather dramatic reduction in cost makes these methods, too, more practical for real-time use. To summarize the algorithms recommended, we give their operation counts and stability properties. In Table II, we give the results for the bearnforrning algorithms discussed in Section II. In Table III, we give the results for updating the spectral decomposition discussed in Section III. We have not made an issue of stability of the spectral decomposition methods. Because no factor of 1-L -I occurs in the methods , there is no reason to suppose that insta-

211

(4] P. A. Businger, "Updating a singular value decomposition," BIT, vol. 10, pp. 376-385, 1970. [5] P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders, "Methods for modifying matrix factorizations," Numerische Mathematik. vol. 28, pp. 505-535, 1974. [6] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: Johns Hopkins University Press, 1983. [7] I. Karasalo, L. Goetherstrom, and V. Westerlin, "Nagra signalbehandlingsfonner basered pa kovariansestirnat av signal och bros." (English title: ••Some signal processing methods based on an estimate of the covariance of signal and noise") Swedish Defense Res. Inst. (FOA), Huvudavdelning 3, S581 11 Linkoeping, Sweden, Tech. Rep. FOA Rep. C30343-El, ISSN 0347-3708, Oct. 1983. [8] R. A. Monzingo and T. W. Miller. Introduction to Adaptive Arrays. New York: Wiley, 1980. [9] N. L. Owsley, "High resolution spectrum analysis by dominant mode enhancement, " in VLSI and Modern Signal Processing, S. Y. Kung, H. J. Whitehouse, and T. Kailath, Eds. Englewood Cliffs. NJ: Prentice- Hall, 1984. [10] R. O. Schmidt. "A signal subspace approach to multiple emitter location and spectral estimation," Ph.D. dissertation. Stanford Univ., Stanford, CA. 1981. [11] R. Schreiber and W. P. Tang, . 'On systolic arrays for updating the Cholesky factorization," Swedish Roy. lost. Technol.. Dep. Numeric. Anal. Comput. Sci., Stockholm. Sweden, Tech. Rep. TRITANA-8313, 1983. also submitted to BIT. [12] G. W. Stewart, Introduction to Matrix Computations. New York: Academic. 1973. [13] I. Karasalo ... Estimating the covariance matrix by signal subspace averaging." IEEE Trans. Acoust., Speech, Signal Processing. vol. ASSP-34. pp. 8-12, Feb. 1986.

TABLE III OPERATION COUNTS OF ALGORITHMS FOR THE SPECTRAL DECOMPOSITION

Cost

Method Section III Full recomputation of the decomposition (21) Use of the Bunch-NeilsenSorensen method (21 )-(22) Exploiting repeated eigenvalues (24) when there are s < n

signals

Recomputing g(d) using the definition (26), when there are m different direction vectors d Use of the recursive method (29)(30) to compute g(d)

mns (1/2)ms

bility of the type encountered in the first beamfonning method of Section II will occur. Moreover, Bunch, Neilsen, and Sorensen [3] and Karasalo, Goetherstrom, and Westerlin [7], [13] give substantial experimental evidence for the stability of some fast update methods for the spectral decomposition. ACKNOWLEDGMENT

I thank P. Kuekes, N. Owsley, and G. Golub for many interesting discussions on this subject. I would also like t ) thank the referees, who made a number of very useful suggestions. REFERENCES

[I] G. Bienvenue and H. F. Mermoz ... New principle of array processing in underwater passive listening." 10 VLSJ and Modern Signal Processing. S. Y. Kung, H. J. Whitehouse. and T. Kailath. Eds. Englewood Cliffs. NJ: Prentice-Hall. 1984. [2] J. R. Bunch and C. P. Nielsen. "Updating the singular value decomposition:' Numerische Mathematik. vol. 31. pp. 111-130. 1978. . ~31 J. R. Bunch. C. P. Nielsen. and D. C. Sorensen. "Rank-one modification of the symmetric eigenproblern.' Numerische Mathematik. vol. 3 1. pp. 3 1-48. 1978.

212

Steady State Analysis of the Generalized Sidelobe Canceller by Adaptive Noise Cancelling Techniques NEIL K. JABLON,

(due to weight jitter) [6] will be different. However, as long as the GSC signal blocking matrix has dimension one less than the number of antenna elements and its columns are linearly independent, then the Frost and GSC implementations lead to the same steady state output signal-to-interference-plus-noise ratio (SINRo) in a stationary environment, based on a comparison of Wiener solutions (i.e., infinitely slow adaptation) [5]. In the first part of this paper, using adaptive noise cancelling techniques [6], exact expressions are derived for the GSC Wiener solution, SINR o, and performance improvement due to adaptation (PIA), defined as the ratio of SINRo after adaptation compared to SINRo before adaptation. The results are derived assuming

Abstract-Narrow-band adaptive noise cancelling techniques are used to study the generalized sidelobe canceller (GSC), a general form of linearly constrained adaptive beamforming structure. In an environment which consists of a look-direction signal, one jammer, and additive receiver noise, exact expressions are derived for the Wiener solution, the steady state output signal-to-interference-plus-noise ratio (SINRo), and performance improvement due to adaptation (PIA), defined as the ratio of SINRo after adaptation to SINRo before adaptation. These expressions are in terms of the signal directions and power levels, an arbitrary array geometry, and a general signal blocking matrix. Next, easily evaluated scalar equations for PIA are given for two classes of constraints. The first is constant gain in the look direction, and the second is constant gain plus a main beam zero first derivative in the look direction. Under the further assumption of an equally spaced line array, even simpler equations for PIA result, and are used to show that for jammers arriving outside the beamwidth between first nulls (BWFN) region of the unadapted beampattern, the introduction of the additional main beam zero first derivative constraint leads to negligible degradation in PIA.

T

I.

MEMBER, IEEE

INTRODUCTION

HE GENERALIZED sidelobe canceller (GSC) is an important adaptive antenna structure, for both theoretical and practical reasons. As Griffiths and Jim explained in [1], the GSC can be viewed as an alternate implementation and extension of Frost's [2], [3] algorithm, using a basic model due to Applebaum and Chapman (4]. Like the Frost beamformer, the GSC adapts to minimize mean square error (MSE) while implementing a look direction constant gain constraint, but in addition is easily generalizable to deal with main beam zero derivative constraints of any order [5]. "New methods of adaptive beamforrning are suggested by the GSC structure," [1] for example, combined temporal/spatial constraints. Since the GSC uses an unconstrained rather than a constrained algorithm to adapt the weights, it may be possible to adapt much faster. The GSC will also be less sensitive to coefficient quantization effects, since the dynamic range of the signals in the adaptive portion of the beamformer is compressed. In general, the Frost beamformer and GSC have different autocovariance matrices, so that algorithm performance measures formulated in terms of autocovariance matrix eigenvalues, such as transient response time and misadjustment Manuscript received March 5, 1985; revised July 8, 1985. This work was supported by the Naval Air Systems Command under Contracts NOOO 19-83C-0287 and NOOO19-85-C-0018, and by the Fannie and John Hertz Foundation Graduate Fellowship Program. The author is with the Information Systems Laboratory, Electrical Engineering Department, Stanford University, Stanford, CA 94305. IEEE Log Number 8406814.

• all antenna elements are omnidirectional with identical amplitude and phase: • the beamforrner is narrow band, so that instead of using tapped delay lines to form the adaptive weights. there is only a single column of complex weights: • the wanted signal is in the far field. and its angle of arrival is assumed to be known. Each antenna element contains a phase shifter so that the array can be steered to this look-direction. The wanted signal will hereafter be referred to as just the signal: • additive receiver noise of equal power is present at each antenna element and can be modeled as a Gaussian process, uncorrelated from element to element and from time sample to time sample; • the signal can be modeled as narrow band, and is assumed to have a planar wavefront. The narrow-band assumption means that the reciprocal of the signal bandwidth is large compared to the transit time of the wavefront across the array; • a single far-field jammer is present, which can also be modeled as narrow band and planar: • the signal, jammer, and receiver noise are zero-mean, wide-sense stationary, and statistically independent of one another: • the propagation medium is linear, homogeneous, and isotropic. The derived expressions thus will not take into account misadjustment [6], non-Wiener signal cancellation [6], multipath [6], [7], element amplitude/phase errors [I], [3j, [6][10], mutual coupling [11], and sky noise, often modeled as

Reprinted from IEEE Transactions on Antennas and Propagation, Vol. AP-34, No.3, pp. 330-337, March 1986.

213

spherically isotropic noise [12]. These effects all alter the Wiener solution, and thus degrade performance. The derived expressions are also exact, given the above assumptions. No further assumptions were made concerning relative power levels of the signal, jammer, and receiver noise. For a given array geometry at a fixed frequency, it will be demonstrated that one only has to evaluate a single equation involving matrix quantities, which is just a function of look direction, jammer angle, and the signal blocking matrix used. Since all signal blocking matrices meeting certain criteria lead to the same Wiener solution [5], [13], one can simply pick the most convenient one when evaluating these expressions. Although only a single jammer is considered, the results presented here could be extended to the case of multiple jammers, either by analytically inverting the more complicated expression for a multiple jammer autocovariance matrix, or by simply solving for the autocovariance matrix inverse numericully. The inverse of the autocovariance matrix is then multiplied by the crosscovariance vector to obtain the Wiener solution.. after which it is straightforward to calculate SINR o. The second part of this paper involves evaluating PIA for two classes of constraints. The first is that of constant gain in the look direction (zero-order constraint). One possible realization for the signal blocking matrix is then adjacent element differencing . an attractive choice for practical implementation. Due to the results in [13]. PIA calculated by this method will be the same as for a Frost beamformer. assuming stationary signal statistics [5). This expression for PIA is exact, and involves no matrix operations (i.e.; matrix rnultiplication and inversion). As will become clear later in the paper. knowing PIA for any signal blocking matrix provides all the information needed to compute SINR,), as long as the signal power level is known. The second class is the double constraint of constant gain and a main beam zero first derivative in the look direction tfirst-order constraint). which is useful for reducing the bearnformer sensitivity to perturbations such as amplitude/ phase errors [I}. A simple realization for the signal blocking matrix is then two columns of differencing in series. From [13) it also follows that PIA calculated for this particular signal blocking matrix applies to all signal blocking matrices which implement the first-order constraint. An equation for PIA with this second type of signal blocking matrix is presented. As with the equation for PIA using the zero-order constraint. no matrix operations are involved in this equation, either. Taking the example of a broadside equally spaced line array, both equations for PIA are plotted as a function of jammer angle for various numbers of antenna elements and input interference-to-noise ratios (INRI). It is easily seen from the graphs that the use of the additional main beam zero first derivative constraint in the look direction only degrades performance within the beam width between first nulls (BWFN) region of the unadapted beampattern , and outside that region the loss in PIA due to the use of the additional constraint is insignificant. Although Vural [9] and marc recently Er and Cantoni [14] also investigated the degradation in SINRo, or equivalently PIA, due to the introduction of an additional main beam zero

first derivative constraint in the look direction, their results were based on computer simulation and did not include explicit scalar equations for SINRo loss due to the additional constraint that a system designer could quickly evaluate. Er and Cantoni's work was focused on the wide-band case. Takao and Komiyama [15] investigated the use of an additional constraint to the Frost beamformer which consisted of a beam pattern zero first derivative constraint in the jammer's direction. As with Er and Cantoni, Takao and Komiyama focused on the wide-band case, and their results with respect to SINRo also consisted of computer simulations. Hudson [12] presented approximate results for beamformers implementing a firstorder constraint, based on a polynomial expansion to characterize the antenna main lobe response. Therefore, this is the first work which exactly quantifies the loss in SINRo due to the introduction of a main beam zero first derivative constraint in the look direction. The outline of this paper is as follows. Section II derives the Wiener solution of the GSC, Section III the steady state SINR o, and Section IV PIA. Section V presents equations for PIA in the two cases of zero-order and first-order constraints. In Section VI an equally spaced line array is assumed, and the two equations for PIA are plotted and compared. Several approximations are presented for PIA in certain situations. Section VII contains the conclusions. II.

WIENER SOLUTION

In this section. the Wiener weight vector will be derived for the GSC. It is well known (6) that this weight vector minimizes MSE under the assumption of stationary signal statistics. The GSC is shown schematically in Fig. 1. consisting of K omnidirectional elements having identical amplitude and phase. The array is electronically presteered to a known look direction. which in the narrow-band case can be done with a phase shifter at the output of each antenna element. The presteering delays - T s are given by I•

-

T i.s

_(dt-do )

==

-c-"-

.

SIn

On

;=1, ... , K

(1)

where d, represents the location of element i, do represents the location of the array phase center, c represents wave speed in the propagation medium.. and Os the look direction, measured as indicated in Fig. 1. A jammer coming from an angle OJ would in the absence of presteering, undergo a time delay at each element of

i= 1, ... , K.

(2)

Thus, in the presence of presteering, the signal can be treated as coming from the array broadside, and the jammer undergoes a total time delay at element ; of T;

~ T i .) - Ti,s'

i= 1, ... , K.

(3)

Equations (I) and (2) assumed a straight line array. This assumption was made only to illustrate the functional dependence of the presteering delays, and does not limit any of the results in this paper to that type of array.

214

CONVEN T IONA L DEL AY-A ND-SUM DEAMFOnMEn

ADAPTIVE NOISE CANCELLE n

r - ----- -- - - --- - - -,

r- ---- - -- ---- --,

I I

I I

1

I Des ired resp on se d k

I

I

L

_

I Outpu t Ok

~~'_.

+

: Prim a ry I I I

I I

I

I

I

I

I

Sidclobo

_ ___ _ _ _ _ ...J

1

ca nce lling

I

I

I I

I

I I

I

sig nal

I I

rrmr nOCEsson

I

I

I I

Uk

I

" I .k

I

I

I I

!l efer en ce I

B

"K.k Referen ce

I I I

I I I

I

I

I

I

I I I

I

I I

:

L

Fig . \ .

Co m ple x a d ap t ive

a l~o rl t 11 m

:

...J

Block d iagra m o f narrow-band GSC . The addit ive receiver noise s following the stee ring dela ys are not shown .

Following passage through the presteering delays , the signal received at each antenna element is corrupted by additive zeromean white Gaussian noise (WGN). as in Fig. 2. Other aspects of narrow-band adaptive antennas are discussed in [161. The main pan of the beamformer consists of two branches . The top one is termed the desired response branch, and its purpose is to form the signal db which is the primary input to the adaptive noise canceller. The manner in which the desired response branch is configured here constrains the look direction gain to be unity . In general, the desired response branch of the GSC will be the same as a conventional delayand-sum bearnformer, which has no adaptive weights . The bottom branch of the beam former is termed the sidelobe cancelling branch, and its purpose is to form the sidelobe cancelling signal Yb by providing K reference inputs to the adaptive noise canceller. Note the use of complex conj ugate weights Wi.k (i = I, . . . , K) in computing Yk. The se weight s can be updated by several different methods , for example the complex. least mean square (LMS) algorithm of Widrow et al. [6] . The sidelobe cancelling branch is preceded by the signal blocking matrix. B, a preprocessor designed to block the signal. The preprocessor has K inputs and K outputs . Although not absolutely necessary , in this paper it will be assumed that K < K. If only a zero-order constraint is present, then K = K - 1, and there will be K - I degrees of freedom available in the sidelobe cancelling branch to form nulls in jammer directions . If a first-order constraint is used instead, then K = K - 2, and one less degree of freedom will be available . The restrictions on B are [I]

B1 K = 0..: rank (B) =

(4)

K.

(5)

x~

T'~

I " '~k

Fig. 2 .

l~ TO'"'''~

Model of ith receiver channel (i

=

•

I • . . . • K) .

TK is the all l ' s vector of length

K , and 0..: the zero vector of length K. For the rest of this paper the all l ' s vector will simply be indicated by T. In the subsequent derivation . matrices and vectors will be denoted by bold uppercase and lowercase letters, respectively . Complex conjugates will be represented by overbars, transposes by a superscript T, Hermitian (complex conjugate) transposes by a superscript H, and stead y state quantities based on using the Wiener weight vector by a superscript aster isk. E [ . I represents time expectation. At time sample k, define the state vector u, as [u 1.k> " ', U":,k) T and the weight vector W k as [w i,k, "', w,u) T . The autocovariance matrix R"u and cross-covariance vector fud are

R UII

~ E[uku{il

(6) (7)

There is no k subscript on Ruu and rlld because the signal, jammer, and receiver noise were all assumed wide-sense stationary . The Wiener solution for the GSC. denoted by the vector w*,

215

where Qj is the normalized array (spatial) factor [18] in the jammer's direction

is then [6] (8)

The complex snapshot vector at the kth time sample Xk g [XI,k, ••• , XK,kl T comprises signal and jammer after presteering, plus receiver noise. B then transforms Xk into Uk: (9) Xk is the sum of three components: a component s, due to the signal, a component jk due to the jammer, and a component n, due to the receiver noise:

(15)

Equation (15) without the j subscripts defines the normalized array factor in any general direction (J, with A formed by substituting (J for (Jj in (2). a is the normalized voltage beam pattern of a conventional beamformer. Based on (6) and (13) and the independence of jammer and receiver noise R li u = o 2nBB T + (Jj2 (RAJ -1)(BA j -1 )H.

(10)

Utilizing the plane-wave assumption for the signal, It IS possible to represent s, purely in terms of the signal Sk at the kth time sample and the vector T. The same plane-wave assumption can be used to represent ik in terms of the jammer jk at the kth time sample, a diagonal matrix AJ which accounts for the phase shift of components of ik due to presteering, and the vector T. Notationally

Substituting (13), (14) into (7) and using (4) to eliminate

BE[DkD:lT results in rud = aJajBAjT, which is multiplied in front by R 1:U) to obtain 1

Iw*=wo(BBT)-IBAjl.

The complex constant

where AJ d diag {e - jWT I, " ' , e - JINT K} with w being the center frequency of si, ik' and the receiver noise in rad/s, n, ~ [n I.k, " . , n K,k ] T, where ni,k is the noise added to antenna element i at time sample k. s., i k' and ni.k can all be represented in complex envelope notation. A sampled signal x, which is represented in complex envelope notation is written ""(k = a,eJ( r,« + tit k ). ak is the random amplitude-modulated portion of xi, el w r» the noninformation bearing carrier frequency portion. and e!Vt k the random phase-modulated portion. T, is the sampling period, in seconds. (J2, the power as measured at any element" is defined as E[ 10k 1 2 ] . V;k must be - U(O, 21l") for Xk to be stationary [17], where - U(a, b) represents a random variable uniformly distributed over (a, b) on the real line. If a, and 'J;k are slowly varying, then x, is considered narrow band. Thus to represent si, )k, and ni,k in complex envelope notation, as,k, aj,k, and an,i.k are defined as their random and (J~ are defined as the signal, amplitudes. Secondly, jammer, and receiver noise power as measured at any element. Thirdly, "'s,k' "'j,k' and "'n,i,k are defined as the random phase of s., ik' and ni.k, respectively. Fourthly, Qs,k, Qj,k, and Qn,i,k, are all statistically independent, as are l/;s,k, "'j,k' and "'n,i,k' and the latter six quantities are further assumed to be varying slowly enough to satisfy the narrow-band assumptions. Using (9)-(12), and (4) to eliminate s, in (13) IN

(J;, (J;,

Wo

where INRi is

(17)

is given by

Wo

(11 ) (12)

(16)

INRia

J g ---I +INR,

(18)

aJ / a~, and the real quantity will be called the

signal blocking matrix factor (j

= (AjT)HBT(BBT)-IB(AjT).

(19)

The transformation (9) is an underdetermined system of equations, since rank (B) < K_ Thus, if one were to try to estimate Xk directly from Ukt there would be an infinite number of x, that would solve the estimation problem. The solution having minimum power is the one with minimum nonn. This minimum norm x, (denoted by ik) is given by either the left or right inverse solution to (9) [20] . Here the right inverse solution is appropriate, so that Xk = B T (DB T) -I Uk. Using (17) and the symmetry of(BBT)-I, (i.e., the steady state Yk) is Yk ~ (W*)H Uk

y:

= wo(A j T)H Xk .

(20)

Since Aj 1 represents the received jammer, including presteering, (20) demonstrates that the GSC sidelobe cancelling branch forms the minimum norm estimate of the jammer for use by the adaptive noise canceller. Y k then represents the output of a cancellation beam steered in the jammer's direction. This can be seen by applying (13) and (17) to the first line of (20):

(13)

Yk = WO[ikO + (A j T)HB T(BB T) -1 0 k].

(21)

The first term in (21), namely (woo»)k» is the amount of I

Inversion of Ruu is accomplished with the matrix inversion lemma [191

(Q+J-J'H)-I=Q-I- Q-ljjHQ-l l+jHQ-lj

(14)

216

where Q is a nonsingular matrix, j is a vector, and Q + jjH is also assumed to be a nonsingular matrix.

n

Recall that (27) involved no approximations with respect to relative power levels of signal, jammer, and receiver noise.

jammer in From (14), the amount of jammer in d k is (cx) )jk. Therefore, the closer that the quantity WOO is to ai' the better the jammer cancellation is. Since () is a function of both the array geometry through A) and the signal blocking matrix used through B, one must be careful about making generalizations as to when the jammer cancellation is best. However, one thing can be said for sure:

IV.

Therefore, regardless of the array geometry or signal blocking matrix used, as long as a degree of freedom is available, the beamformer approaches an infinite null in the direction of the jammer as the receiver noise approaches zero. Due to B, there are no terms involving the signal in the Wiener solution. Furthermore, due to the uncorrelatedness of the WGN in both time and space, there are no distinct terms in the Wiener solution that attempt to cancel receiver noise. However, the presence of the receiver noise power in the denominator of the term INRi does allow the receiver noise power to impose an inherent limitation on the ability of the GSC to cancel jammers. STEADY STATE

Calling the ratio SINRri / SINR Oc performance improvement due to adaptation (PIA) from (18) and (27), (28) PIA = 1 +

SINRo

~2 E[iZkI 2 ]

" -E[dkYk] - -E[dkYd - +E[IYkI 2 ]}. ="21 {E[ Idkl-]

(23)

When each term in (23) is evaluated, it turns out that Po can be expressed as the sum of three terms, one each due to the signal (Pos), jammer (P Oj), and receiver noise (POn). When w* is used in the first line of (20) to calculate y:, and then y i is used in (23) to calculate the steady state values of POs, P Ob and Pan (denoted by Po:' P and Po:)' the following expressions result: 1

V.

EXPRESSIONS FOR

-

K

(25)

p*

SINR * £ Os o - POj+P * O*n

SNR i

where SNR j is the input signal-to-noise ratio

(27)

a; /

a~.

1 + INRt(Klaj 1 2 + 0)

(29)

0

\VHEN ZERO-ORDER AND FIRST-ORDER

The zero-order constraint is easily implemented by the (K 1) x K matrix

0(0)

SINRti (i.e., the steady state SINRo) is then

l2 o

CONSTRAINTS ARE USED

(24)

(26)

INR7Klaj

This expression is exact, and is seen to be controlled by just four quantities. The only complications involved in computing PIA from (29) are in performing the matrix operations given by (19), the equation for O. PIA is the real quantity of interest to a system designer who has to make a decision whether to use an adaptive antenna or not, for it explicitly indicates how many dB in SINRri can be gained by using a particular adaptive antenna structure, as opposed to a conventional beamformer. It should also be noted that PIA is different from both cancellation ~ defined as the improvement in output interference-to-noise ratio due to adaptation [10], [21], and array gain, defined as the ratio of SINR~ to input signal-tointerference-pius-noise ratio [14].

of,

P*Os == -2 a s2

DUE TO ADAPTATION

(28)

In this section the Wiener weights calculated in Section II will be used to derive the steady state SINRo of the GSC. The output power Po of the GSC is Po d

SINRo

If all the adaptive weights are zero, as when the GSC is initialized, the only active part of the beamformer will be the desired response branch. In this case the output signal Zk will just be di, since Yk will be zero. SINRo for this situation will be symbolized by SINR oc, SINRo for a conventional beamformer. SINRoc can be derived simply by setting Wo to zero in (27):

(22)

III.

IMPROVEMENT IN

_[1-1. 0 ] o 1-1

(30)

where the subscript indicates the number of antenna elements, and the superscript (r - 1) the use of a main beam zero (r 1)st derivative constraint in the look direction. Fig. 3 shows a simple circuit for B~-l). Thus the zero superscript in (30) represents r = 1, which is just adjacent element differencing. The first-order constraint can then be implemented by the (K - 2) x K matrix B~) == B(~_l B~). B~) represents two columns of differencing in series, and as such entails one less degree of freedom than B~). Griffiths and Jim [1] first suggested B~) as a remedy for reducing system sensitivity to pointing errors caused by element amplitude/phase or other perturbations. In order to evaluate PIA, one must first be able to evaluate

217

where the constant interelement phase shift Uj is 27r(d/A)(sin OJ - sin Os), with d denoting the interelement spacing and A the wavelength. For the equally spaced line array, aj has the special form [16]

2

~

···~I

K Inputs

K - r

sin

Outputs

aj=

KUo) (T

K sin

(~)

(35)

VI. INTERPRETATION OF THE BEHAVIOR OF PERFORMANCE

Fig. 3.

Cascaded columns of differencing.

IMPROVEMENT DUE TO ADAPTATION

o. The most difficult part in the analytical evaluation of 0 is the calculation of (BB T) - 1. fJ corresponding to the zero-order constraint is written as Do, and is [22] 2

00 = K (I -

I

Ci)

1

2

(31)

).

For the first-order constraint, 0 is written as 01, and is [22]

ol=K

[1-2 (~:ll)

lajl2

(K+I) ( Ii3 1-., - a - a-i3 ]

- 3 K_1

j

j Pj -

j

j )

(32)

with the normalized (for the case of r, = 0; i = I, "', K) quantity (33) Ar this point, it is worthwhile to pause and reflect on the fact that PIA can now be computed for any arbitrary array geometry without having to perform a single matrix operation. as long as either a zero- or first-order constraint is chosen. The most complicated work involved is just the evaluation of a} according to (15) and its "cousin" /3J according to (33). Sometimes (3J can be written in terms of aj, which makes the computation even easier. An example is an equally spaced line array, and in [22] it is further shown that for this type of array

a;-2aj cos

KU') cos (u.) 1 +1 (T

(34)

2 Substitution of (18) and (31) into (27) yields an equation for SINRo which agrees with the one for a converged narrow-band Frost beamformer obtained .by Takao et 01. in [3. cq. (48)]. This result is consistent with Griffith and Jim's [I] assertion that the GSC implemented as in this paper and the Frost [2] beamformer should provide the same steady state performance in a stationary signal environment.

In this section, PIA for an equally spaced line array using the zero-order (PIA o) and first-order (PIA I) constraints will be plotted. PIA o and PIAl will be computed using (35) for aj, (31) for 00, (34) for 01, and then (29) for PIA o and PIA l : Plots of PIA and () as a function of jammer angle are presented in Figs. 4 and 5. Fig. 4 plots PIA for a three-element equally spaced line array with a broadside look-direction, and INRi = 0, 20, and 40 dB. Fig. 5(a) plots 0 for a ten-element equally spaced line array with a broadside look direction. Fig. 5(b) plots PIA for the latter case when INRI = 20 dB. It is assumed throughout that d/). = 0.5. Studying these plots and keeping (29) in mind. several points are apparent. 1) Signal power has no effect on PIA. This is due to at least two simplifications in the analysis. The first is the lack of element imperfections, which would cause the signal to "leak" into the sidelobe cancelling branch [6]. Jablon [22], [23] did a detailed study of this phenomenon, and found that although for ~ 'high" SNRi the GSC is hypersensitive to small element imperfections, the problem could be fixed by artificially injecting receiver noise, a la Zahm [24]. Hudson [12] also treated this subject. The second simplification is the use of a steady state analysis based on the Wiener solution, which does not take into account non-Wiener signal cancellation [6]. 2) PIA goes to 0 dB for several angles. At these angles, either the array factor approaches 0 or the jammer is actually coming from the look direction. The explanation is that when the jammer falls in a null of the array factor, the unadapted array naturally performs extremely well. The adapted array can match this performance without altering the weights from their initial all-zero values. On the other hand, when the jammer comes from the look-direction, the GSC is helpless, as forming a spatial null in the look direction is prevented by the constant gain constraint, so the adapted array is forced to perform as poorly as the unadapted array. 3) PL4 0 ~ PIA t. This is no surprise, as one has to give up something to get the robustness to perturbations that come from using the extra constraint. Fortunately, however, the degradation in PIA due to the extra constraint only appears to be evident in the inner half of the unadapted pattern BWFN region. Stutzman and Thiele [16] show that for K ~ 1 and d/). = 0.5, BWFN of a conventional beamformer near broadside is about equal to 4/K rad. Using this approximation for BWFN, the degradation in PIA is only experienced over an angle of roughly 21K rad. For a ten-element array BWFN is

218

approximately 20 so the degradation in PIA due to the additional constraint will only be significant when the jammer falls within ± 50 of the array broadside, which will be tolerable for many applications. 0

50,----------------------, K -3 Look-direction

----

~ ~o

_0.5

/ P IA / '/ ' PIA~

S., _ rjl

30

l

INR ;

PlAul PIA I INR

j

,

= . . 0 dO

= 20

4) As INRi increases, the degradation in PIA near the look-direction due to the extra constraint becomes more serious. The willingness of any adaptive beamformer to null a

an

10

o · 10 · 100

·50

0

50

100

Jammer angle or arrival, OJ (deg) -

Fig . 4. Narrow-band GSC performance improvement due to adaptation for three-element broadside line array having d /): = 0.5 and INR, = O. 20. and 40 dB.

12

i

jammer falling in any part of the unadapted pattern other than a null or a look direction is related to INRi' From Fig . 4, for a jammer falling in the BWFN region, when INRi is small, the first-order constrained GSC does not even bother to null the jammer, which accounts for the flatness of PIA, near broadside. However, when INRi reaches a certain critical level , the first-order constrained GSC decides to null the jammer. Due to the extra constraint, it has to work harder than the zero-order constrained GSC, and the degradation in PIA thus becomes especially pronounced.

5) The key to understanding the difference in PIA resulting from the use of different signal blocking matrices is to understand the behavior of o. In Fig. 5(a), 0, is flat near the look direction, so PIA, must also be flat there . Since 0 is the only quantity in (29) that changes when B changes, if two B have similar 0, they will also have similar PIA's. This is clearly demonstrated by the fact that the two PIA's in Fig . 5(b) are similar for the same jammer angles where the two 0 in Fig. 5(a) are similar. It is also interesting that 00 ~ 0,.

Look-direction

10

6) It is useful to approximate PIA, although one must be cautious. First consider PIA o. Using (31) and the fact that

almost surely INR, K

~

I

PIAo == I + INR,Ka; (I -

- 0.5

·50

0

50

100

Jammer angle or arrival, OJ (deg) _ (a)

3 0K,-10 ---------------------, ~

PIAol max

- 0.5

., _

8

(36)

This expression can be differentiated with respect to aj to find the angle(s) 8m. , where PIA o is maximum. It is straightforward to show that at 8ma. , aj = l /,fi. and from a visual inspection of a J versus 8j plots , 8m• • == ± (BWFN/4) for broadside arrays as considered here. Substituting aj = I/-li into (36) with INR,K ~ I:

K -10

~

a;).

0°

I

- 4 INR;K .

It is more complicated to analyze PIA, in the same way, as a comparison of (31) and (34) makes quite clear . However, easy results can be obtained outside the BWFN region, where it seems reasonable to approximate PIAl by PIA o, so that (36) applies with PIAl replacing PIA o. Unfortunately, inside the BWFN region , PIA, does not behave as "nicely" as PIAodid, which makes the analysis beyond the scope of this paper, except to say that PIAd ma, :S PIAolmax '

INRi - 20 dB

20

10

VII . CONCLUSION o l . - - L_ _---''"'----L---'_.L.-..L.---L.......l._L-...u-_ _~_-.J

· 100

-50

0

50

100

Jammer angle or arri val, OJ (deg) (b)

Fig. 5. Narrow-band GSC implemented with ten-element broadside line array having d/'A = 0.5 . (a) Signal blocking matrix factor (0). (b) Performance improvement due to adaptation (PIA) when INRi = 20 dB.

219

The narrow-band generalized sidelobe canceller was studied by applying adaptive noise cancelling techniques . The signal environment was assumed to consist of a look-direction signal , one jammer, and additive Gaussian receiver noise at the antenna elements. Exact expressions were derived for the Wiener weight vector, steady state output signal-to-interferencc-plus-noise ratio, and performance improvement due to

adaptation, defined as the ratio of SINRo after adaptation compared to SINRo before adaptation. These expressions were shown to all be critically dependent on a single quantity 0, called the signal blocking matrix factor, which is related to the amount of the jammer appearing in the beamfonner steady state output signal. For a general array geometry and GSC signal blocking matrix, the evaluation of () involves matrix inversion and matrix multiplication, but nothing more complicated than that. It was also shown that if one were willing to make certain assumptions about the nature of the linear constraints involved, namely the use of a constant gain constraint in the look direction, or a constant gain plus a main beam zero first derivative constraint in the look direction, that the matrix operations involved in the evaluation of the signal blocking matrix factor could be eliminated completely. In order to demonstrate the usefulness of these completeI y scalar equations for signal matrix blocking factor, an equally spaced line array was assumed, and PIA was plotted in several situations. It was seen from the graphs that for jammers arriving outside the beamwidth between first nulls region of the unadapted beampattem, the use of the additional main beam zero first derivative constraint resulted in negligible degradation in PIA. ACKNOWLEDGMENT

The author is grateful to his research supervisor, Dr. Bernard Widrow, and also to Dr. A. Paul raj , Dr. Richard Gooch, and Dr. William C. Newman for several enjoyable discussions on adaptive beamforming. In addition, Dr. Widrow, Dr. Paulraj, and the anonymous reviewers made several suggestions which significantly improved this paper. Thanks go to Mieko Parker for her careful typing.

[9] A. M. Vural, "Effects of perturbations on the performance of optimum/adaptive arrays," IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, no. 1, pp. 76-87, Jan. 1979. [10] J. T. Mayhan, "Some techniques for evaluating the bandwidth characteristics of adaptive nuJling systems," IEEE Trans. Antennas Propagat., vol. AP-27, no. 3, pp. 363-373, May 1979. [11] I. J. Gupta and A. A. Ksienski, "Effect of mutual coupling on the performance of adaptive arrays, " IEEE Trans. Antennas Propagat.• vol. AP-31, no. 5, pp. 785-791, Sep. 1983. [12] J. E. Hudson, Adaptive Array Principles. New York: Peter Peregrinus and The Institute of Electrical Engineers, 1981, ch. 2 and 6. pp. 56 and 160-191. [13] C. W. Jim, "A comparison of two LMS constrained optimal array structures," Proc. IEEE, vol. 65, no. 12, pp. 1730-1731, Dec. 1977. [14] M. H. Er and A. Cantoni, "Derivative constraints for broad-band element space antenna array processors," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, no. 6, pp. 1378-1393, Dec. 1983. [I 5] K. Takao and K. Komiyama, "An adaptive antenna for rejection of wideband interference," IEEE Trans. Aerosp. Electron. Syst., vol. AES-16, no. 4, pp. 452-459, Jul. 1980. [16] L. L. Horowitz and K. D. Senne, "Performance advantage of complex LMS for controlling narrow-band adaptive arrays," IEEE Trans. Circuits Syst., vol. CAS-28, no. 6, pp. 562-576, Jun. 1981. [17] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill. 1965, ch. 9, p. 303. [18] W. L. Stutzman and G. A. Thiele. Antenna Theory and Design. New York: Wiley, 1981, ch. 3, pp. 124-129. [19] T. Kailatb. Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980, appendix, p. 655, A. 20. [20] G. Strang, Linear Algebra and Its Applications, 2nd ed. New York: Academic. 1980. ch. 2, p. 79. (21] R. C. Johnson and H. Jasik, Antenna Engineering Handbook, 2nd ed. New York: McGraw-Hill, 1984, ch. 22, p. 22-6. [22] N. K. Jablon, "Adaptive beamfonning with imperfect arrays," Ph.D. dissertation. Elec. Eng. Dept.. Stanford Univ.~ Stanford, CA, Aug. 1985. [23] - -... Adaptive beamforming with the generalized sidelobe canceller in the presence of array imperfections." IEEE Trans. Antennas Propagat.. to be published. [24] C. L. Zahm, "Application of adaptive arrays to suppress strong jammers in the presence of weak signals," IEEE Trans. Aerosp. Electron. Syst., vol, AES-9. no. 2. pp. 260-271, Mar. 1973.

REFERENCES

[I]

[2] [3] [4] [5]

[6] [7] [8]

L. J. Griffiths and C. W. Jim. "An alternative approach to linearly constrained adaptive beamforming," IEEE Trans. Antennas Propagat., vol. AP-30. no. I. pp. 27-34. Jan. 1982. O. L. Frost, III. ,. An algorithm for linearly constrained adaptive array processing," Proc. IEEE, vol. 60, no. 8, pp. 926-935, Aug. 1972. K. Takao, M. Fujita. and T. Nishi. "An adaptive antenna array under directional constraint:' IEEE Trans. Antennas Propagat., vol. AP24, no. 5, pp. 662-669, Sept. 1976. S. P. Applebaum and D. J. Chapman, "Adaptive arrays with main beam constraints." IEEE Trans. Antennas Propagat., pp. 650-662, Sept. 1976. L. J. Griffiths, "An adaptive beamformer which implements constraints using an auxiliary array preprocessor." in Aspects of Signal Processing, pt. 2, G. Tacconi. Ed. Dordrecht, Holland: D. Reidel Publishing Co., 1977. pp. 517-522. B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs. NJ: Prentice-Hall. 1985. R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980, ch. 11, pp. 451-475. J. M. McCool, "A constrained adaptive beamformer tolerant of array gain and phase errors," in Aspects ,of Signal Processing, pt. 2, G. Tacconi, Ed. Dordrecht, Holland: D. Reidel Publishing Co., 1977, pp. 477-483.

220

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the equivalent array could be very small. For example, if there Jre only 10 jammers then J + 7 becomes 17. The total of multiplies needed to do the adaptive array processing reduces from something of the order of 2N J = 2 (70(X)()J) = 2 X 70'2 to about 105 a reduction in the computation complexity by seven orders of magnitude. In addition, the settling time for the adaptive-adaptive array is much faster. For the above example, the settling time for the full array is about 20CXJO samples, whereas for the adaptive-adaptive array it is only 22 time samples, for an improvement of three orders of magnitude. SUI\t'MAR'r

A technique is described for adaptive array processing which eliminates the complex computation problem (see Table 1) of a large fully adaptive array while at the same time provides the same optimum performance as obtained for the fullv adaptive array in [1]. The technique also has the advantage of not Significantly degrading the antenna sidelobe levels at angles where the Jammers are not present; see Fig. 1. This feature IS Important in the presence of Intermittent short pulse interference coming through the radar sidelobes and for ground radars which have clutter In the sidelobes

(a) -80.0 i'------''----r~-___._-~.......----&.--....... -90.0 -45.0 0.0 45.0 90.0 ANGLE FROM BROADSIDE (DEGREES)

Adaptive-Adaptive Array Processing

0.0

ELI BROOKNER AND JAMES M. HOWELL

iii'

(b) ;

A technique is described which provides the jammer cancellation

-40.0

C(

e

advantages of a fully adaptive array without its many disadvantages such as an excessively large number of computations, poor sidelobes in the directions other than the jammer locations, and poor transient response. This is done at the expense of the hardware complexity. The technique involves transforming a large array of N elements into an equivalent small array of J + 7 elements, where j is the number of jammers present. The technique involves estimating the number and locations of the jammers by a discrete Fourier transform of the array element outputs or by the use of standard maximum entropy methods (MEM) or by other super-resolution techniques. Once the number and locations of the jammers have been determined, beams are formed in the direction of the jammers using the whole array. The outputs of these jammer beams together with the output of the main signal beam from the transformed array now consist of J + 7 ports instead of N ports. The standard sample matrix inversion (SMI) or the Applebaum algorithm can be applied to the J + 1 ports of the equivalent adaptive-adaptive array. Whereas N may be very large, like 70CXXJ for large arrays, J + 1 for

-80.0,-+---r------ro-..l-_-_ -90.0 -45.0 0.0 45.0 90.0 ANGLE FROM BROADSIDE (DEGREES)

0.0

(c)

iii'

i

;(

-40.0

C)

-80.0f------,.---A._--'T""---"_......--'--~

-90.0

-45.0

0.0

45.0

90.0

ANGLE FROM BROADSIDE (DEGREES)

Fig. 1. Sixteen-element array having 40-dB antenna sidelobes (Chebyshev weighting). Jammer at 20 0 (peak of second sidelobe). (a) Unadapted antenna pattern. (b) Antenna pattern for fully adaptive array (SMI algorithm). M = 2N = 32 (M equals the number of time samples used to estimate the adaptive antenna weights). For the fully adaptive array, not only is there a degradation of the antenna sidelobes, there is also a degradation in the antenna main lobe peak gain. The peak gain degradation was found to be as much as 5 dB in the simulation carried out. (c) Antenna pattern for adaptive-adaptive array processing. M = 2(J+1) = 4.

Manuscript received January 3, 1985; revised April 15, 1985. The authors are with the Raytheon Company, Wayland, MA 01778.

Reprinted from Proceedings of the IEEE, Vol. 74, No.4, pp. 602-604, April 1986.

221

Comparison of Computations Required (Assumption : J = number of jammers ~ 10.)

Table 1

Jammer Cancellation Technique

Number of Complex Mult iples to Calculate We ights

One-D imensiona l ( N Ele me nts

Fully Adapt ive AdaptiveAdaptive (Improvement)

2"; - 2 x 10& 2(J + 1)3 + 7 Nr - 10 '

Square Two -D imens ional (N Elements

Fully Adapt ive Adapt iveAd aptive (Improvement)

Type Array An tenna

- 100)

_ 10' )

Complex Mult ipl ies to Form Array Transient Time Output per Signa l (Units of Signal Time Sample" Time Samples) 2 N - 10 2N - 200

J + 1 - 11

200 1012

2"; - 2 x 2(J + 1)3 + N log 2 .;, 7 x 10 ' _ l OS - 2 x 107

IN••

2(J + 1 ) - 22

- 10 10

- 10

N = 10'

2N = 2 X 10'

J + 1 - 11

2(J + 1)

,;" 10 10 3

=

22

- 10 3

#Does not incl ude co mp utatio ns of column three. 'Second term assumes MEM algorithm used to locate jammer. This term drops out if jammers located usmg sea rch beam. For th is case number o f mult iplies .. 2(J + 1)3 .. 2 X 10 3 and im p rov e me nt be comes - 103.

"Second te rm assume s fast Fourier transform (H T) algor ithm used to locate jam me r. Term drops out if Jammer located using search beam (or beams). In this case number of mult iplie s " 2(J + 1 ») .= 2 x 10 ) and im p ro ve me n t becomes - 109 .

[4J. Thus for the adaptive-adaptive array 2(1 + 1) time samples are needed instead of the 2N required for the full array; see Table 1.

and the mainlobe. The adaptiv e-adaptive array also has the advantage of a much faster settl ing time ; see Table 1. The technique uses a two-step process. First the number of .ntertering jammers and thei r locations are estimated by such ~,~ c h n i q u e s as a spatial discrete Fourier transform of the array outputs (d igitally or by use of a Butler matrix or Rotman lens), by maximum entropy method spectral estimation techniques [2]. [3), or just by a search in angle with an auxil iarv beam. On ce the numb er of jammers and their locat ions have been determ ined, auxiliary beams are form ed po int ing at these jammers, wi th one beam being po inted at each jammer; see Fig. 2. These beams are form ed usi ng

JAMMER

It is useful to physically understand wh y the adaptive-adapti ve array does not degrade the antenna sidelobes . The adaptive-adap tive array subtracts one auxil iary beam pointed at the jammer and containing the jammer signal from the main Signal channel beam as ill ustrated in Fig. 3. The gain o f the auxil iary beam in the dir ection LEGEND --

UNADAPTED MAIN BEAM PATTERN

• ••••••

AUXILIARY PATTERN

- -- - -

ADAPTIVe-ADAPTIVE PATTERN

JAMMER

NO .1

0 .0

NO .2

TARGET

!{1

~

iii

a

~

z :c o

JAMMER

NO . ..

-80.0 _---'L...,,......J'-~...:..-.;...J:~-l._...I, - 90.0

- 4 5. 0

0 .0

45.0

90.0

ANGLE FROM BROADSIDE (DEGREES)

MAIN ____ ARRAy

T1 Tl ··· ··· T.

Fig. 3.

Main unadapted array pattern, the auxiliar y jammer beam

pornted at the jammer that is subtracted from the mam beam at

the Jammer locat ion . and the resultant adapnve-adapnve pattern

MAIN BEAM

AUXILIARY

BEAM POINTING AT JAMMER NUMBER 2

Fig. 2. Adaptive-adaptive array configuration. N·element array is reduced to J + I element array where) is equal to t he number

of jammers.

the whole array. They are formed using beam-form ing networks parallel to the main signal beam network. The number of beams formed is equal to the number of jammers. These beams could be formed using amplitude weight ing to achieve low sidelobe levels if desirable . The outputs of the auxiliary jammer beam ports together with the main signal beam port form the adaptive-adaptive transformed array. The number of degrees of freedom in this transformed array is reduced from N, the number of elements in the orig inal array, to one plus the number of jammers j. Thus for the .i daptive- edaptive array a (1 + 1) (1 + 1) matrix has to be inverted instead of an N X N matrix . Furthermore , the conversion time for the adaptive-adapt ive array is much faster than for the full array. For the SMI algorithm, the number of time samples needed to form the weights is equal to two times the number of degrees of freedom in order to obtai n cancellat ion within 3 dB of the optimum

222

of the Jammer IS made to equal the gain of the main channel beam sidelobe in the direction of the jammer . As a result, the subtraction produces a null at the angle of the jammer in the main channel sidelobe. It IS apparent from Fig. 3 that the auxiliary antenna pattern subtraction does not signif icantly degrade the main antenna beam sidelobe levels. For the fully adaptive array, N retrod irect ive beams are formed based on the eigenvalues and eigenvectors of the fully adaptive array covariance matrix [5). Because of the presence of thermal no ise in the array elements the estimates of the covariance matr ix of the fully adaptive array and, in turn, the retrodirective beams are poor for M = 2N . Instead of forming onl y one retrodirecti ve beam as desired when one jammer is present, N retrodirective beams are formed for the full y adaptive array. The N - 1 retrodi rective beams for wh ich there are no jammers are the ones wh ich degrade the antenna sidelobe levels at the angles where no jam mers exist. It is found that even i f 3000 time samples are used, the sidelobe levels are still severely degraded for the fully adaptive array system although considerably improved. The adaptive-adaptive array technique first determines what jammers are present which will degrade the system performance . Once the locations o f these jammers are determined the array adapts to the situation by only placing retrodirective beams at these angles. Consequently, the beams at other angles where there are no jammers are not formed and do not, as a result , degrade the antenna sidelobes at these angles. A number of variations are possible on the above adaptive-adap-

tive array system. First, the MOSAR method of [6] can be used to locate the jammer positions based on a single time sample. Second, it is not necessary to use the whole array to locate the jammers. Third, if the jammers can be located so as to come through the backlobes, then an auxiliary array (or arrays) is needed which covers the backlobes or whatever angles are not covered by the main array. Finally, it is possible to use only one parallel beam-forming network instead of J with this beam-forming network being time multiplexed so as to produce the J beams pointed at the J jammers and in this way reduce the hardware complexity of the adaptive·adaptive array processor. The physical explanation given above together with Fig. 3 helps in understanding the performance of the adaptive-adaptive algorithm for nonperfect conditions and leads to the following insights. Even if the jammer location is in error by plus and minus a half beamwidth , jammer cancellation results similar to those in Fig. l(c) will still be obtained. There will only be a degradation of the sidelobe to the right or left of the null by about 3 dB. Furthermore, if the cancelor weights calculated using the SMI (or some other adaptive algorithm) is inexact, the null depth will be degraded but it IS apparent from Fig. 3 that the sidelobe level will be unaffected except for a small amount for the sidelobes just to the right and left of the null. Increasing M for the SMI computation will increase the null depth. If a jammer is not detected than it will not be canceled out. This, however, will tend to occur only if the jammer is weak, a case not of as much concern because the jammer will then only cause a small degradation in signal-to-interference ratio. If a jammer is estimated to be present when in fact it is not, the system will incur very little degradation in signal-to-Interference ratio and in antenna sidelobe level because the SMI weights for the channel pointing in the direction where no jammer actually exists will be very low, the weight being established by the correlation between the noise in the main channel and the noise in the auxiliary channel pointing at no jammer with these noises being Independent so that the correlation on the average is zero. If there are a large number of jammers then there can be antenna sidelobe level degradation If the auxiliary jammer beams have sidelobe levels that are not low enough If J jammers are present then, .n order to avoid sidelobe level degradation in the main channel, the auxiliary channel antenna sidelobe levels should be greater than 10 (loglo !) decibels down, a condition that can generally be met. ACKNOWLEDGMENT

The idea of pointing high-gain auxiliary antenna beams in the direction of the jammers appears to have first been suggested by P. W. Howells, the inventor of the IF sidelobe cancel or [7]. He did not, however, form multiple auxiliary high-gain beams in an array to achieve jammer nulling performance essentially that of a fully adaptive array while avoiding the associated sidelobe degradation problem as done in this letter. W. F. Gabriel of NRL has independently done this. Fig. 1 was obtained using a simulation written by C. D. Brommer (Raytheon). REFERENCES

[1]

[2]

[3] [4)

[5]

[6] [7]

S. P. Applebaum, "Adaptive arrays," IEEE Trans. Antennas Propeget, vol. AP-24, no. 5, pp. 585-598, Sept. 1976. S. M. Key and S. l. Marple, Ir., "Spectrum analysis-A modern perspective," Proc. IEEE, vol. 69, no. 11, pp. 1380-1418, Nov. '981. J. P. Burg, "Maximum entropy spectral analysts." Ph.D. dissertation, Dept. Geophysics, Stanford U., Stanford, CA, May 1975. I. S. Reed, J. D. Mallett, and L. E. Brennan, "Rapid convergence rate in adaptive arrays," IEEE Trans. Aerosp. Electron. Syst., vol. AES-10, no. 6, pp. 853-863, Nov. 1974. W. F. Gabriel, "Adaptive arrays-An introduction," Proc. IEEE, vol. 64, no. 2, pp. 239-272, Feb. 1976. M. A. Johnson, "Phased-array beam steering by multiplex sampling," Proc. IfEE, vol. 56, pp. 1801-1811, Nov. 1%8. P. W. Howells, "Explorations in fixed and adaptive resolution at GE and SURe," IEEE Trans. Antennas Propagat., vol AP-24, no. 5, pp.

575-584, Sept. 1976.

223

ESPRIT-Estimation of Signal Parameters Via Rotational Invariance Techniques RICHARD ROY

AND

THOMAS KAlLATH.

Abstract-High-resolution signal parameter estimation is a problem of significance in many signal processing applications. Such applications include direction-of-arril'al (DOA) estimation, system identification. and time series analysis. A novel approach to the general problem of signal parameter estimation is described. Although discussed in the context or dircction-of-arrival estimation, ESPRIT can be applied to a wide variet~ of problems including accurate detection and estlmanon of sinusoid~ in noise. It exploits an underlying rotational invariance among signal subspaces induced by an array or sensors wllh a translational invariance structure. The technique, when applicable, maniIests ~ignificant performance and computational advantages over previous algortthrns such as MEl\rI. Capon's l\tLM, and MUSIC.

I

I.

INTRODUCTION

N many practical signal processing problems, the objective is to estimate from measurements a set of constant parameters upon which the received signals depend. For example, high-resolution direction-of-arrival (DOA) estimation is important in many sensor systems such as radar, sonar, electronic surveillance, and seismic exploration. High-resolution frequency estimation is important in numerous applications. recent examples of which include the design and control of robots and large flexible space structures. In such problems, the functional form of the underlying signals can often be assumed to be known (e.g., narrow-band plane waves, cisoids). The quantities to be estimated are parameters (e.g .• frequencies and DOA's of plane waves, cisoid frequencies) upon which the sensor outputs depend. and these parameters are assumed to be constant. \

There have been several approaches to such problems including the so-called maximum likelihood (ML) method of Capon (1969) and Burg s (1967) maximum entropy (ME) method. Although often successful and widely used, these methods have certain fundamental limitations (esq

Manuscript received January 12. 1988: revised October 5. 1988. This work was supported in part by the Joint Services Program at Stanford University (U.S. Army. U.S. Navy. U.S. Air Forcej under Contract DAAG2985-K-0048. and the SOl/1ST Program managed by the Office of Naval Research. under Contract NOOO14-85-K-0550. The authors are with the Information Systems Laboratory. Stanford Universiry, Stanford. CA 94305. IEEE Log Number 892R125. I Extensions to situations in which the parameters may be time varying can be made. however. they rely on an Inherent time-scale or eigenvalue separation between the parameter dynamics and the dynamics of the signal process. Fundamentally. the assumption IS made that over time intervals long enough to collect sufficient information from which to obtain accurate parameter estimates. the parameters have not changed significantly.

FELLOW. IEEE

pecially bias and sensinvny in parameter estimates). largely because they use an incorrect model (e.g .• AR rather than special ARMA) of the measurements. Pisarenko (1973) was one of the first to exploit the structure of the data model, doing so in the context of estimation of parameters of cisoids in additive noise using a covariance approach. Schmidt (1977) and independently Bienvenu (1979) were the fi rst to correctly exploit the measurement model in the case of sensor arrays of arbitrary form. Schmidt, in particular. accomplished this by first deriving a complete geometric solution in the absence of noise, then cleverly extending the geometric concepts to obtain a reasonable approximate solution in the presence of noise. The resulting algorithm was called MUSIC (MUltiple SIgnal Classification) and has been widely studied. In a detailed evaluation based on thousands of simulations, M.1.T. 's Lincoln Laboratory concluded that, among currently accepted high-resolution algorithms, MUSIC was the most promising and a leading candidate for further study and actual hardware implementation. However, although the performance advantages of MUSIC are substantial, they are achieved at a considerable cost in computation (searching over parameter space) and storage (of array calibration data). In this paper., a new algorithm (ESPRIT) that dramatically reduces these computation and storage costs is presented. In the context of DOA estimation. the reductions are achieved by requiring that the sensor array possess a displacement in variance. i.e .. sensors occur in matched pairs with identical displacement vectors. Fortunately, there are many practical problems in which these conditions are or can be satisfied. In addition to obtaining signal parameter estimates efficiently, optimal signal copy vectors for reconstructing the signals are elements of the ESPRIT solution as well. ESPRIT is also manifestly more robust (i.e., less sensitive) with respect to array imperfections than previous techniques including MUSIC [1]. To make the presentation as clear as possible, an attempt is made to adhere to a somewhat standard notational convention. Lowercase boldface italic characters will generally refer to vectors. Uppercase boldface italic characters will generally refer to matrices. For either real- or complex-valued matrices. (.)* will be used to denote the Hermitian conjugate (or complex-conjugate transpose) operation. Eigenvalues of square Hermitian matrices are assumed to be ordered in decreasing magnitude, as are the

Reprinted from IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 37, No.7, pp. 984-995, July 1989.

224

singular values of nonsquare matrices. Knowiedge of the fundamental theorems of matrix algebra dealing with eigendecompositions and singular value decompositions (SVD) is assumed (cf. [2]). II.

THE DATA MODEL

Although ESPRIT is generally applicable to a wide variety of problems, for illustrative purposes the discussions herein focus on DOA estimation. In many practical signal processing applications, data from an array of sensors are collected, and the objective is to locate point sources assumed to be radiating energy that is detectable by the sensors (cf. Fig. 1). Mathematically, such problems are quite simply, although abstractly, modeled using Green's functions for the particular differential operator that describes the physics of radiation propagation from the sources to the sensors. For the intended applications, however, a few reasonable assumptions can be invoked to make the problem analytically tractable. The transmission medium is assumed to be isotropic and nondispersive so that the radiation propagates in straight lines, and the sources are assumed to be in the far-field of the array. Consequently, the radiation impinging on the array is in the form of a sum of plane waves. For simplicity, it will initially be assumed that the problem is planar, thus reducing the location parameter space to a single-dimensional subset of d.

em x d,

III.

E

THE GEOMETRIC ApPROACH

In 1977, Schmidt [4] developed the MUSIC (MUltiple SIgnal Classification) algorithm by taking a geometric view of the signal parameter estimation problem. One of the major breakthroughs afforded by the MUSIC algo-

225

rirhm was the ability to handle arbitrary arrays of sensors. Until the nlid-1970's. direction finding techniques required knowledge of the array directional sensitivity pattern in analytical form, and the task of the antenna designer was to build an array of antennas with a prespecified sensitivity pattern. The work of Schmidt essentially relieved the designer from such constraints by exploiting the reduction in analytical complexity that could be achieved by calibrating the array. Thus, the highly nonlinear problem of calculating the array response to a signal from a given direction was reduced to that of measuring and storing the response. Although MUSIC did not mitigate the computational complexity of solution to the DOA estimation problem. it did extend the applicability of high-resolution DOA estimation to arbitrary arrays of sensors. A. Array Manifolds and Signal Subspaces

To introduce the concepts of the array manifold and the signal subspace. recall the noise-free data model x (t) = A (0) s( t). The vectors a( 0,) E trill, the columns of A (9), are elements of a set (not a subspace), termed the array manifold" (a), composed of all array response (steering) vectors obtained as 8 ranges over the entire parameter space. (1 is completely determined by the sensor directivity patterns and the array geometry, and can sometimes

be computed analytically. However, for complex arrays that defy analytical description, (1 can be obtained by calibration (i.e., physical measurements). For azimuth-only DOA estimation, the array manifold is a one-parameter manifold that can be viewed as a rope weaving through ce III. For azimuth and elevation DOA estimation, the manifold is a sheet in cr m. To avoid ambiguities, it is necessary to assume that the map from 9 = {8 1 , • • • , Oil} to , and the columns of T are the eigenvectors of 'P. This is the key relationship in the development of ESPRIT and its properties. The signal parameters are obtained as nonlinear functions of the eigenvalues of the operator 'I' that maps (rotates) one set of vectors ( Ex ) that span an m-dimensional signal subspace into another ( E F) D, Estimating the Subspace Rotation Operator

In practical situations where only a finite number of noisy measurements are available, Es is estimate? from the covariance matrices of the measurements RZ7 or, equivalently, from the data matrix Z. The result is that CR { Es } is only an estimate of Sz, and with probabil ity one, CR { E s } *" ... >

"2ii) ,

and partition E into

'1'£'( ::::: E y . Since the problem 1\ undcrdeternuncd by construction t cf typically d < there I" no unique solution. although the parameter estimates. the d ergenvalucs of 'I' on the unit circle that arc associated with the cJ-dimensional subspace being rotated, arc urnquc. Imposing a minimum norm constramt on 'I' leads to a unique LS solution in which III - d eigenvalues are equal to zero. See 11] for funhcr detai I".

. . . Ie"1 =

~ A~""

d

x

asubmatrices,

11/ ).

~ISee [ l] and 151 for details on various techniques for esurnaung the number of sources

231

6) Calculate the eigenvalues of 'I' ~k

=

Ak(-E rl.E 22

J

) .

:=

vk =

-

DOA's is given by

E I2 E;l,

1, ...

W

,d.

7) Estimate fh = f - I ( J>k); e. g., for DOA estimation, lJk = sin - I {c a rg (~k) / ( Wo A ) }. For arrays with multiple invariances, such as uniform linear arrays. the decomposition of Es into Ex and Ey is not unique. See [14) and {17] for more details concerning multiple invariance ESPRIT. In many instances, it is preferable to avoid forming covariance matrices, and instead to operate directly on the data as discussed in Section III. This approach leads to (generalized) singular value decompositions (GSVD's) of data matrices . and a GSVD variant of ESPRIT discussed in detail in [I]. From the key relation (21), several other quite striking results can be derived. For example, not only is knowledge of the array manifold not required, but the elements thereof associated with the estimated signal parameters (DOA's) can be estimated if desired. The same is true of the source correlation matrix. knowledge of which is not needed in ESPRIT. F. Array Calibration Using the TLS formulation of ESPRIT . the array manifold vectors associated with each signal (parameter) can be estimated (to within an arbitrary scale factor). From (21).. the right eigenvectors of 'I' are given by Eo+ = T - I . This result can be used to obtain estimates of the array manifold vectors as

EsE tt =

A1T- 1 = A.

(23)

No assumption concerning the source covariance is re-

quired. Although simple to compute, this estimate will not in general conform to the invariance structure of the array in the presence of noise. In low SNR scenarios, the deviation from the assumed structure A = [A TI (A cD) T] r may be significant. In such situations, improved estimates of the array manifold vectors can be obtained by employing the formulation discussed in r18]. G. Signal COP)' In many practical applications . not only the signal parameters, but the signals themselves, are of interest. Estimation of the signals as a function of time from an estimated DOA is termed signal copy. The basic objective is 10 obtain estimates S( t) of the signals s (r ) from the array output . zt r) = As(t) + n(t). Employing a linear estimator, a squared-error cost criterion in the metric of the noise (which is ML if the noise is Gaussian), and conditioning on knowledge of A, leads to the estimate i (t), the vector of coefficients resulting from the oblique projection of z( t) onto the space spanned by the columns of A (cf. the Appendix). The resulting weight matrix W (i.e., the linear estimator) whose ith column is a weight vector that can be used to obtain an estimate of the signal from the ith estimated DOA and reject those from the other

= 1:;IA[A*1:,~'A]-'.

(24)

In terms of quantities already available. (24) can be written as W = I;,~I e, lEt}:;' ES]-' E;*, (25)

using (23) to estimate A. This equivalence is easily established since from (21) it follows that the right eigenvectors of 'I' equal T - I. Combining this fact with E s == A T and substituting in (25) yields

w- = E;' [E.~l:,~1 Esl -I Etr,;l == [A * £; I A ] - I A * 1:.; ,.

( 26)

Note that the optimal copy vector is a vector that is ~; I orthogonal to all but one of the vectors in the columns of A since W*A" = 1. There is, of course, a total least-squares alternative to conditioning on knowledge of A. Since only estimates of A are available, in low SNR scenarios where accurate signal estimates are desired, the TLS approach yields improved estimates at the cost of increased computation. Although not derived herein, S( I) can be obtai ned by performing a (generalized) singular value decomposition of [A Iz(1) ]. The right singular vector corresponding (0 the smallest singular value yields i (t) as the first d elements after normalizing the last element to unity. 22 H. Source Correlation Estimation

There are several approaches that can be used to estimate the source correlations. The most straightforward is to simply note that the optimal signal copy matrix W obtained above removes the spatial correlation in the observed measurements fcf. (25)]. Thus. W*CzzW == DSD* where S is the source correlation (not covariance) matrix, eZL = Rzz - 0- 21:", and the diagonal factor D accounts for arbitrary normalization of the columns of W. Note that when Rzz must be estimated, a manifestly rank d estimate CZl = e, [A ~cl) - &2lcd E can be used [cf. (6)], where A~d) = diag {AJ, .. · , Ad} and Ai is a GE of (R zz , 1:n ) . Combining this with E s = A T gives DSD* = T[A1 J - &~/d] T*. (27) If a gain pattern for one of the elements is known, specifically if the gain Ia I (Ok)' is known for all (Jk associated with sources whose power is to be estimated, then source power estimation is possible since the array manifold vectors can now be obtained with proper scaling.

t

V. SIMULATION RESULTS Many simulations have been conducted exploring different aspects of ESPRIT and making comparisons to ~~This approach is clearly suboptimal if the sampled signals are temporally correlated in the sense that E { S, ( t ) 5, (I + T)} -:I: 0 for r o. If. for example. the signals are known to be sinusoidally modulated RF and uniform temporal sampling is employed. then estimating the underlying signals requires only the estimation of the modulation frequency. another problem well suited to ESPRIT. Note that in general the modulation Irequency must be a small fraction of the carrier to satisfy the narrow-band assumption.

232

*"

other techn ique s (cf. [I ]) . Herein . only one of the sce narios. but one that addresses se veral issues that a rise in a practical implementatio n of ESPRIT, is presented. Thu s, sensor gain and phase errors , as well as sensor spacing errors , are included. Furthermore , unequal source powers and a high degree of source co rrelatio n are assumed . More speci fically . the array chos e n was a ten -element array with doublet spaci ng A/ 4 and the five doublet s ran domly spaced on a line resulting in a n ape rture of approximately 4 A. Tw o sources were located at 24° and 28° (approxi mately 0.3 Raylei gh or 3 dB beamwidth separa tion) , and were of unequ al po we rs, 20 dB and 15 dB . rcspectivcly . Sen sor errors we re introd uced by zero -me an norm al random additive e rrors with sigmas of O. I dB in am plitude and 2° In phas e" (inde pende nt of angle) . Senso r location error s (alo ng the axis of the a rray ) with sig ma 0.005 ( A/2) were incl ude d as well. The sources were 90 percent temporall y co rrelated and 5000 tr ials were run . A histogram of the result s is giv e n in Fig . 4 . The number of so urces was assumed to be known in the impleme ntatio n of both MUSI C and ESPRIT . The indi cated fail ure rate for MU SIC of 37 pe rcent is the perccn tage of trials in which the conve ntional MUSIC spectrum did not ex hibit two pea ks in the inte rval [20 ° , 32° ). This, of course , is not an issue in ESPRIT , where two parameter est imates are obt ained eve ry time . The sa mple mean s and sigmas of the ESPRIT estimates were 23.93 ° 1: lO? " and 28.06 ° ± 1.37°. wh ile those of the 3 175 successful MUS IC tr ials were 24.35° ± 0.28° and 27.48 ° ± 0.38° . Note that wit h refe re nce to Fig. 4. the re is an ove rlap in the distribut ion s of the ES PRIT estimates . Thi s has an effect o n thc sta tistics calc ulated , since a simple angle-o rderi ng sche me was used whe re in the larg er of the two angle estimates in eac h trial was assoc iated with the 28° source .:" Th e effec t is pre sum ed to be sma ll in this case . Th e results indicate the presen ce of a bias even in the successfu l co nve ntional MUSIC estimates . the source of which IS descr ibed In detail in [I) . On the othe r hand , the ESPRIT estimates are unb iased , altho ugh of larg e r variance since less info rmatio n co nce rni ng the array geometry IS being utilized . Note also that in comparing the estimate variances. there is no atte mpt to account for the 1825 trials in which MUSIC failed 25 to pro vide two DOA estimates! Howe ve r, as the suba rray sepa ratio n inc reases , the ESPRIT paramet er esti mate variances approac h those of MU SIC. The sa me expe rime nt was run fo r a suba rray sep-

1000

900

or

'2

t-8 "!

!? 0:

~ :f

SNR • (20,I5]dB 100 pointslttial 700 5000 trials

n

-

0( . The least-squares SCORE processor block diagram is shown in Fig. 3. The reference signal r(t) is generated by linearly combining, delaying, conjugating (if conjugate selfcoherence is being exploited), and frequency-shifting the data received by the array. The reference signal is then used as a training signal to adapt the processor vector w using a least-squares algorithm. The only control parameters used in the processor are the control vector c, the delay 1, the

B. The Least-Squares SCORE Algorithm

The simplest SCORE algorithm, referred to here as the

least-squares SCORE algorithm), is developed using the interpretation of spectral self-coherence given in (10)-(12).

We define a reference signal r(t) by

(17)

where the vector c is referred to as the control vector and the optional conjugation (Ifl) is applied if and only if con3The least-squares SCORE algorithm was first presented in [8].

239

vector as well as the processor vector to some appropriate value. This generalization leads to the cross-SCORE algorithmS, discussed in the next section.

C. The Cross-SCORE Algorithm

Fig. l.

An algorithm for adapting c can be developed by motivating the least-squares SCORE algorithm from a propertyrestoral viewpoint. The same value of Wscgiven in (21) results from maximizing the strength of the cross-correlation coefficient between y(t) and r(t)

least-squares Score processor.

conjugation control, and the frequency-shift a; however, only ex and the conjugation control are critical to the operation of the processor. For most communication waveforms much latitude can be allowed in the choice of c and T, because in theory these parameters need only be chosen to yield a nonzero value of Bsc in (26)4. In addition, the frequency-shift parameter ex need not be related in any way to the bandwidth or sampling rate of the receiver system; however, care must be taken to avoid aliasing effects if the processor is implemented in digital form and et is large. From Fig. 3 it is clear that the least-squares SCORE processor can be generalized in several ways. For instance, the delay operation can be replaced by a more general filtering operation, by generating r(t) using i(t) = h(t) ~ x(t),

Fsc(w; c) ~

(.hH

a

J CPSf' l

fR: v'R;'

= IwHAKrI2/[wHtxxw R"]

(29)

IwHalluc I2

(30)

= [wHRxlIw][CHRuuc]'

where u(t) is defined to be the control signal, (31)

The cost function f sc is an indirect measurement of the spectral self-coherence in y(t) at frequency separation et; it is lowered if x(t) contains interference that is not spectrally self-coherent at this frequency separation. In this sense, the least-squares SCORE algorithm can be interpreted as a method for restoring this spectral self-coherence to the processor output signal. The crcss-correlatlon coefficient is also degraded if interference is present in r(t). Consequently, maximizing ~sc with respect to wand c should restore this spectral self-coherence to both y(t) and «t). For this reason, (30) is referred to here as the cross-SCORE objective function, and methods for optimizing (30) are referred to here as cross-SCORE algorithms. From the Cauchy-Schwarz Inequality, it is clear that w is optimized for fixed c by

(27)

where h(t) is the control filter impulse response and ® denotes convolution. The optimum weight vector then converges to

Bsc = [ a

IR yrI2/l Ryy R,,1

(28)

where set) is the filtered SOL As Section V shows, a key parameter affecting the convergence rate of the SCORE processor is the strength of the spectral self-coherence P:Si-' being restored by the processor; appropriate design of the control fi Iter can improve the performance of the SCORE processor by increasing this strength. The critical dependence of the SCORE processor on the choice of target ex can also be eased somewhat by the particular choice of averaging window used to calculate the finite-time correlation matrices and i.T• If a growing rectangular window is used to calculate ill" for instance, then the processor will eventually reject a received SOl if there is any error between the self-coherence frequency of the sal and the target self-coherence frequency of the processor. In many environments, however, the self-coherence frequency of the 501 cannot be known exactly, for instance, if the SOl is subject to Doppler shift (which shifts the conjugate self-coherence frequency of the SOl). The SCORE processor can be made more tolerant to this error if a different choice of averaging window, such as an exponentially decaying window, is used to compute Ru . The greatest improvement in SCORE processor performance can be obtained by adaptively adjusting the control

W Op l

oc i~1Rxr = R;x1ixuc,

(32)

wh ich is the least-squares SCOREsolution (if the delay operation in Fig. 3 is generalized to a filtering equation). Similarly, c is optimized for fixed w by (33)

an

Substituting (33) into (30)yields a generalized Rayleigh quotient in w (34)

which is maximized by setting w equal to the dominant mode (eigenvector corresponding to the maximum eigenvalue> of

Al.. w = [l.uA;u'RuJw.

(35)

Similarly, the control vector is globally optimized by setting c equal to the dominant mode of

ARuuc =

[Au. l ;lI' .xulc.

(36)

Equations (35) and (36) are referred to here as cross-SCORf eigenequations; both of these eigenequations have the same eigenvalues, with the maximum eigenvalue equal to the maximized objective function value. Equations (35) and

41n practice, it is important to choose values of c and r that yield

a large value of 8se, as this parameter does have a strong effect on the convergence time of the SCORE algorithm. This can impose a serious constraint on c (for example, if the array is subject to strong co-channel interference), but does not impose a strong constraint on T in most communication applications.

SThe cross-SCORE algorithm was first presented in [8].

240

where gwand Be are used to normalize the power of yCt) and r(t) at each step in the algorithm. Equations (41) and (42) converge very rapidly to the dominant mode of (36) in the ra~k 1 spectral self-coherence environment, because of the Wide spread between the maximum and lesser eigenvalues of (37) that prevails in this case.

(36) can also be used to obtain an equivalent joint crossSCORE eigenequation

where ~ = /. and every solution (~,C' Wk, Ck) to (35), (36) has a palf of solutions (.J>.:;., W/.:;., Wk, -Ck) to (37). It is easily shown that the dominant modes of (35) and (36) both converge to the maximum-SINR solution given in (15) if set) IS the only received signal with spectral self-coherence or conjugate self-coherence at Q. In this environment, the Hermitian matrix on the right-hand side of (35) reduces to a rank-1 matrix as T --t 00,

D. The Auto-SCORE Algorithms

Although the cross-SCORE algorithm can be interpreted as a property-restoral algorithm, it is essentially an exte~ sion of the least-squares SCORE algorithm, and as such IS motivated more naturally from the interference-decorrelation viewpoint discussed in Section II. A more natural framework for developing a true property-restoral algorithm based on spectral self-coherence is to consider the problem of maximizing the spectral or conjugate selfcoherence strength at the output of a single linear combiner

(38)

For an M-element antenna array, the eigenvectors of (35) therefore converge to M - 1 signal-rejection solutions where w is orthogonal to a and A is equal to zero, and one signal-selection solution where w is equal to W max and ~ is approximately equal to I p~sl·)12, ~max

-+

H

(aHR~1 aHa (1

+

Ri-. a) 1

I p~s\·d2

)'x 2)(1

l~s(.d2

+ 'Yi-2) == Ip~~(.)

Fsc(w)

IwHR~~·'(T)w(·)1

wHRuw(·)

(39) 12

~ IP~YT, the elements of ',

[-Yh · .. ,'Yel], l'i

IIEs A'/2 - ATII}

Aea,l

where the set 8 is defined by 8 = {A \ A

Aeacl

= arg max tr {PAEsA£.;}

(24)

Ae0. T

arg min

e }. (25)

Lemma 1: Let the TLS-ESPRIT estimate of 'P be defined by (22), (23). Then the minimizing o.

256

.IN (11 - 9) = 0

in probability.

(57)

Proof' Write V( W) to stress the dependence of the criterion function on the weighting matrix. In view of (42) and (412 . the result follows if Vry(W) = Vry(W) + op(l/.JN) and Vl1 t; ( W ) = Vl1~(W) + op(l). But the first equality follows immediately from (45) and the second equality is trivial. 0 Some more convenient covariance formulae are obtained for the special choices of weighting matrix W that correspond to the methods of the previous section. Corollary]: Assume that S is invertible. Then, i) The asymptotic covariance of the unweighted MDMUSIC method (W = I) is given by (49) with.

1/" = -2 Re {(D*Pi D) Q=

2(12

Re {(D*P; D)

0

(A*A)-T}

0

[(A*ASA*A)-I

+ a:!(A*ASA*ASA*A)-I]T}.

(58)

ii) The asymptotic covariance of the deterministic ML method (W = A) is C = CRB DET

where

+ (V,,)-IQ(V")-l

V" = -2 Re {(D*Pi D)

0

ST}

(59) (60)

Q=

2a 4 Re {(D*Pi D) 0 (A*A)-T}

(61)

and where CRB DET is the asymptotic deterministic Cramer-Rae lower bound as derived in [18J, [29J. Proof: From the structure of the covariance R, straightforward manipulations give the following relations:

(63)

A == E;ASA*E,..

(64)

Proof: The matrix inequality follows immediately from [11, lemma A.2]. For proving (70), notice first from (54), (55) that V" = -a- 2Q. Hence, only the expression AtEs WoprE7AT* needs to be evaluated. The weighting matrix is Wopt

(68)

This in conjunction with Lemma 2 proves ii). C The expression for the asymptotic deterministic ML covariance ii) is consistent with the expressions derived independently in r 11] and r 121· Note that the deterministic ML variance is greater than the deterministic CRB. This was also observed in [291. Let us emphasize that this bound assumes deterministic signal waveforms and is different from the CRB for Gaussian signals. Notice also that when S is diagonal (uncorrelated signals) ~ the asymptotic covariance of ML coincides with the one of MUSIC [17],

[18], [29].

D. Performance Optimization Given the result of Theorem 2, it is natural to ask if there exists an optimal weighting for the subspace fitting method. In other words, is there a weighting Wopt which minimizes the estimation error variance?

Theorem 3: Let the matrix vaLued function C (W) be defined by (49), (54), (55). Then for all Hermitian matrices W (69)

where the matrix inequality means that the difference C (W) - C (A lA,;') is positive semidefinite. 171e asymptotic covariance for the optimal subspace fitting method is

.,

- o ' I.

(71)

Using this and the fact that A'~EsAE,:AT* = S from (63), in (73) gives A 7E\ WnptE; A 7* = ATE.\.(.\.

+

a~A,\~' - a 2I ) E.:A 7*

= S - (a- 2SA*A

+ I)-Is

= S - (I - SA*(ASA*

+

(74)

a 21)- IA) S

== SA*R -lAS.

i ng that P, = E." E =: shows i). The result ii) is obtained by noting that

S + a 2C4 *.4 ) - I.

-I

(73)

Applying the above expressions to (54) and (55) and not-

* ==

4

-A(a- 2SA*A + /)-'SA*.

(67)

7

-

== A + a As

Applying the matrix inversion lemma, the expression above can be written as

Notice that

7

-(

(72)

(66)

A EsA.\E7ft

-:2

A As

It follows immediately from (64) that

To prove i), assume that S is full rank, (64) then gives after some manipulation (65)

=

(75)

The implication of Theorem 3 is of considerable theoretical interest. It follows that the optimal subspace fitting method, referred to as the weighted subspace fitting (WSF) method. never performs worse than ML and in general it outperforms ML. As we will see, the result is also of practical interest since the difference can be large when the sources are highly correlated. It should be noted here that Stoica and Nehorai [11] have reported a special case when the one-dimensional MUSIC method gives smaller asymptotic variance than ML. V.

NUMERICAL EXAMPLES AND SIMULATIONS

In this section we present some numerical examples to compare the performance of the discussed methods. Simulations are also included to investigate the applicability of the theoretical results obtained in the previous section. For the theoretical curves, Theorem 2 is used in the pragmatic form COy (0) =::: N -1 C. The theoretical results assert the quality of the local maximum of the criterion function closest to 0 0 , The estimates are therefore calculated by initializing a Gauss-Newton type descent method at the true DOA's. In practice, other methods of initialization would, of course, have to be considered and the question of global convergence arises. This issue is not addressed here. The WSF method is implemented using the weights Wo Pt == (As - a2[ )2A .; 1, where the noise variance is estimated as the average of the m - d' smallest eigenvalues of R. In both examples, a uniform linear array (ULA) of half-wavelength element spacing is assumed, the array manifold vector having length and first element 1.

-;r;;,

257

Two emitters of equal power are symmetrically located with respect to the array broadside . For each case , 500 independent trials are run and the standard deviation of the first DOA estimate is calculated. The bias, observed in the simulations , is less than 20% of the standard deviation for all cases and methods. To obtain correct results in the region where the standard deviation is of the same order as the DOA separation , a DOA/signal association must be made . This was done by comparing the estimated signal waveforms, ob tained from (16), with the true ones. The CRB under the Gaussian signal assumption [I], [2] is also displayed in the figures.

MD·MUSIC: •• . • Det. ML: WSF : CAB :

. '. 10·

DOAs: 2' ·2'

Number of sensors: 6 Correlation : 0 SNA : 13 dB 102

10'

N

A. Example 5.1: Uncorrelated Signals In the first example, we examine how many snapshots that are needed for the asymptotic results to be valid for a "typical case ." A six element ULA is used . the signal waveforms are uncorrelated and both of power level 13 dB above the noise. The DOA' s are 2 0 and - 2 0, respectively. In Fig. I, the standard deviation of the first DOA estimate is plotted versus the number of snapshots . The theoretical values are displayed with lines . while the sim ulated values are represented by +. 0 , and x. The figure illustrates that the theoretical results agree very well from about 5 snapshots for this case . The theoretical variance of the WSF estimates is identical to the CRB up to numerical precision. Also, the ML variance is indistingu ishable from the CRB in Fig. I. but they are not identical. Clearly , the unweighted MD-MUSIC method performs notably worse . This is somewhat unexpected since the one-dimensional MUSIC is known to have the same asymptotic performance as ML for uncorrelated sources [10]. 0 B. Example 5.2 : Correlated Signals This example is chosen to demonstrate the extreme importance of the weighting matrix Win WSF when the signals are highly correlated. A four element ULA is assumed and the DOA 's are 50 and -5 0. The waveforms have a 99% correlation with correlation phase 0 0 (at the first sensor), i.e., the signal covariance matrix is

S =

lO(o. 'SNR) X

[

I 0.99

0.99] . I

(76)

The number of snapshots is 200 . In Fig. 2, the standard deviation of the first DOA estimate is plotted versus the SNR of the signals. Again, the theoretical standard deviation of WSF is equal to the CRB up to numerical precision . Deterministic ML performs notably worse for SNR's below 8 dB . Notice the severe degradation of the MD-MUSIC method for this scenario . 0 VI. CONCLUSIONS This paper attempts to collect several algorithms and versions of algorithms under a unified framework. Alge258

Fig . I. Standard deviation o f DOA estimate versus the number of snap shots .

deg

2 0r-~-~-~-,..--.,-~---~--~--'

18

16

MD·MUSIC: Det. ML: WSF : CA B:

DOAs: S' ·S·

Number of sensors: 4 Correlation : 0.99 Number of snapshots: 200

14 12 10

8

,

j

. j a

2

4

6

8

10

12

?

14

,

16

1 ,

,

18

20 dB

Fig. 2. Standard deviation of DOA estimate versus SNR.

braic relations are presented and based on this . asymptot ic expressions for the estimation error are derived. The proposed framework allows a unified derivation of the asymptotic properties of several subspace fitting based methods. A weighted subspace fitting method is introduced and its asymptotic properties are derived for a general weighting and arbitrary signal correlation (including full coherence) . This result includes the asymptotic distribution of the estimation error for deterministic ML and MD-MUSIC as special cases . The optimal weighting matrix is derived, resulting in the WSF method which always outperforms the deterministic ML method . Numerical examples presented herein and in [38], indicate that the asymptotic variance of the WSF estimates coincides with the CRB for the Gaussian signal and noise model. There are other advantages in viewing sensor array processing as a subspace fitting problem. Extensions of the ESPRIT algorithm become clear in this framework, e .g ., multiple array invariances [391 . Furthermore, the meth ods within the framework can be implemented by means of the same optimization algorithm. The examples also demonstrate that the difference in performance between the methods can be significant in many cases, i.e., the choice of subspace weighting can be crucial. The simulation study indicates that the theoretical

results can also be found in [27]. For ease of notation we write P instead of PAce). Consider the first derivative

variances are valid for a large range of information-tonoise ratios. ApPENDIX

A

TOTAL LEAST SQUARES AND SUBSPACE FITTING

Pry

Consider a system of linear equations

and let A and iJ represent noisy observations of the linearly related matrices A o and Bi; Let A and jj be the solution of ..\.8

II i1- B- IIi r

A - .4

min H. 'I'

I

11

I £~ £1

1-

i

f'

B I B'V i

1)

111F

min

ill

t:

1I'. (I>. t ;: II E' . . !I

,r I

I'

r

(

(B.3)

(B.4)

Using (B.2) and that P;- ==
(20)

== A(j~)[R~(O)AH(lo), R~(TJAH(io), R~ (lvTJA H (J())]

(21)

has the same column span as A (fo). If each specific R~ (7) is rank deficient, the combination of all these [R~(O)AH(/0)' R.~(TJAH(j~»)~ . . . , R~(NTJAH (fo)] is less likely to be rank deficient. Hence, the problems mentioned above can be resolved. -',I evertheless , although the suggested alternative approaches use all the cyclic correlation functions with T == 0, Ts ' • • • , (N - 1) T:" they are still not asymptotically exact especially for wide-band sources. Also, due to a much larger size of the matrix X (a) in the first approach and additional computational cost for S(\~ (J') involving matrices R~Y (T) in T := O. T,.. . . . in the second approach. both methods are not so computationally efficient as SC5sr described in Section III.

c.

Resolution

The resolution properties of direction finding algorithms depend on many practical factors. c. g.. the sensor separation distance D. the effective frequency J~, of the s(T)

-l (l)

+

s(t +

Ilj(t)

Td)

exp (j27rfn Tcf) +

induced manifold, as well as the range of the wave spatial coherency and the coupling between sensors. In this section. we attempt only a brief discussion of the resolution properties of existing cyclic methods and of our new SCSS:- techniques. 1"'0 achieve high resolution and to minimize the coupling effect, one needs to have a large sensor separation distance D. On the other hand. if D is too large, ambiguity problems may arise. For conventional MUSIC or cyclic MUSIC, the maximum D that does not cause ambiguity is c /2fo, where c is the wave speed and fa denotes the carrier frequency. Nevertheless, for the SC-SSF approach, the maximum D is c/2a. In many cases, the cycle frequency a that one may exploit is close to twice the carrier frequency fa, For example, the cycle frequency of AM or FM is 2io [10] and the a of BPSK or FSK is 2.1;) + ih lII], where fj} is the baUd rate. Therefore, the same resolution as conventional MUSIC or cyclic MUSIC can be achieved if D is reduced by half. In this case, there is no fundamental difference

n2(t)

(22)

where Ttl == D sin () / c. Let R.~ (r), R~ (T) be the conjugate cyel ie correlations of Xi ( .) and s ( . ), respectively. It is easy to see that

R(~I (T) ==

(s (t

+ 7/2 +

(i -

I) r d) ex P [j 27r foU - 1) T d]

. stt - 7/2 + (i - l)Td) . exp [j27rj(li - 1) Td] . e-J :'. 1ra c )

== R~(7) exp [j211'"(2fo + a)(i - l)Td].

(23)

Hence, (12) becomes

265

!1.~ (r)

= A (210 + ex) B.~ (r),

(24)

In the following examples of computer simulation, we use BPSK signals, and the conjugate cyclic property mentioned above is exploited.

D. Algorithm Implementation Conventional signal subspace algorithms such as MUSIC and ESPRIT start from the covariance matrix R, ==

liN E~=l -!(n)-!H(n). For cyclic MUSIC and ESPRIT, we need to evaluate a cyclic covariance matrix, i.e., R~(T) = (!(t - TI2)~H(t + iI2)e-j21rat), which is estimated by R~(m) = liN E~=l x(nTs + mTs)!.H(nTs)e-j21ranTs, where T, is the sampling time and N is the number of samples for averaging. If the calculation of the cyclic correlation is done by an analog circuit, then just replace the sum by an integral in the above equation. If noise is stationary but spatially correlated, then cyclic MUSIC and ESPRIT do not require the knowledge of R; (0) and the computation of the generalized eigendecomposition {Rx (0), e, (0) }. Since the original data are sampled at a very high rate to avoid aliasing, and the evaluation of the spatial cyclic correlation is a very simple (though time consuming) operation, the computation of the spatial cyclic correlation is usually implemented by digital or analog parallel processing hardware. The processed data, i.e., the cyclic correlation, will be sent to the central processing unit for further more complicated processing, e.g., an eigendecomposition or a MUSIC search. From the formulas noted above, it is not difficult to see that the evaluation of cross correlation required by the cyclic MUSIC and ESPRIT algorithms involves multiplication of data from different sensors, e.g., Xi (n)xj* (n). Therefore, for each sensor, we have to build interconnections to the other (M - 1) sensors, which may make the hardware design much more difficult. Problems such as cross talk or coupling may also arise. In fact, the computation of the covariance matrix required by the conventional MUSIC and ESPRIT algorithms also has the same problems. On the other hand, the SC-SSF algorithms only require the temporal cyclic correlation, i.e., the cyclic autocorrelation among the data at each sensor. Therefore, the computation of cyclic correlation can be achieved locally at each sensor and no interconnections are required. Figures 1-3 show a simplified diagram of the computation schemes for the three types of correlation matrices required by conventional SSF, cyclic SSF, and SC-SSF algorithms, respectively. V.

Fig. 1. Simplified covariance matrix computational flow graph for Conventional signal subspace algorithms.

e J 2 Jr4 ' Fig. 2. Simplified covariance matrix computational Row graph for cyclic SSF algorithms.

e

COMPUTER SIMULATIONS AND DISCUSSIONS

In order to show the effectiveness of the proposed approach, computer simulations for testing signal selectivity, high resolution, and sensitivity to spatially correlated noise were conducted. In these simulations, the conventional MUSIC, cyclic MUSIC, and SC-MUSIC algorithms were used to estimate the source DOA's. In the following examples, a seven-element uniform linear array was used, with the smallest sensor separation c 12fe being used to avoid any ambiguity in DOA estimation, where c is the wave speed and Ie is the center frequency of the effective array manifold. In order to show the relative performance of the above mentioned approaches in both narrow-band and wide-band scenarios, seven cases ranging from temporal narrow-band (BW/carrier = 1 %) to temporal wide-band (BW/carrier = 40%) 266

j2 . .,

Fig. 3. Simplified covariance matrix computational flow graph for the SCSSF algorithms.

were studied. In this paper, the measure of temporal narrow band or wide band is defined as the ratio of the bandwidths of the complex baseband signals and the carrier. The statistics of the DOA estimates were calculated based on 500 independent trials. For cyclic MUSIC and SCMUSIC, we evaluated the conjugate cyclic correlation to extract the SOl (BPSK). As explained in Section IV-C, although the cycle frequency a = Ib' the effective trequency I. in the induced array manifold is 2/0 + lb. The signal-to-noise ratio (SNR) for each source was defined as the ratio of the power of this source to that of the back" ground noise. In all the following examples, we started with the baseband (complex) signals. The background

J10ise in cases A and B is spatially uncorrelated Gaussian noise, while it is a correlated Gaussian process in case C . For Cyclic MUSIC, we always pick the optimal lag r at which the cyclic correlation R~ (r) achieves its maximum , although we may not know the optimal lag in reality. In order to make the comparison even fairer, SC-MUSIC only used 7 cyclic correlation values so that X(a) is of the same size as that of the R~ (r) used in the cyclic MUSIC algorithm. In fact, SC-MUSIC can use as long a cyclic correlation sequence as possible, although the maxinn .n useful r value is limited by the width of R~(r) that exhibits significant variation. The simulated data in this paper is all generated in the time domain instead of in the frequency domain .

Il9 0.•

0.6

~

I ~

BPSK

\

0.5 0.4

AM

\

0.3 0.2 0.1

Fnqucn(., t1§) - 0. The improved convergence rate of ML-SAGE-2 is closely related to this difference. To illustrate, Fig. 2 displays the likelihood 4>( Bi ) versus iteration for the ML-EM algorithm and for ML-SAGE-2 applied to a simulation of PET data . The image was an 80 x 110 discretization of a central slice of the digital 3-D Hoffman brain phantom (2 mm pixel size) . The sinogram size was 70

277

radial bins (3 nun wide) by 100 angles. A 900000-count noisy projection was generated using (6-mm-wide) strip-integrals for {ank} [29], including the effects of nonuniform head attenuation and nonuniform detector efficiency. A uniform field of random coincidences was added, reflectin~ a scan with 9% of the total counts due to randoms (i.e., ~'n=1 Tn ~ 0.12::=1 Yn(A)), a typical fraction for a PET study. Further details can be found in [8] and [30], including comparisons over a large range of Tn'S. Also shown in Fig. 2 is the LINU unbounded line-search acceleration algorithm described by Kaufman [23]. The ML-SAGE-2 likelihood clearly increases faster and reaches its asymptote sooner than both the ML-EM and ML-LINU algorithms.' (ML-SAGE-2 was also considerably easier to implement than the bent-line LIND method.) The convergence in norm given by Theorem 3 of Appendix A is inapplicable to this Poisson example when the ML estimate has components that are zero, Le., when the ML estimate lies on the boundary of the nonnegative orthant [33]. See [30] for a global convergence proof for ML-SAGE-I and ML-SAGE-2 similar to the proofs in [3] and [17]. The reader may wonder whether one can also find a better complete-data space for the classical EM algorithm. Because the EM update is simultaneous, one must distribute the background events among all pixels; therefore, the terms Zk are reduced by a factor of roughly [8], [30]. Since Vp is in the hundreds, the change in convergence rate is insignificant, which is consistent with the small reduction in Fisher information [8], [30]. Other simultaneous updates (17] similarly do not improve much [30]. Apparently one benefits most from this less informative hidden-data space by using a SAGE method with the parameters grouped into many small index sets. An alternative to SAGE is the coordinate-wise sequential Newton-Raphson updates recently proposed by Bouman and Sauer [19]. That method is not guaranteed to be monotonic, but when it converges it might do so somewhat faster than SAGE since it is even greedier. One can obtain similar (but monotonic) greediness by using multiple subiterations of the E- and M-steps in the SAGE algorithm, as indicated by Step 5 of the generic SAGE algorithm. However, for the few cases we have tested, we have not observed any improvement in convergence rates using multiple subiterations. Although further investigation of the tradeoffs available is needed, including comparisons with possibly superlinear methods such as preconditioned conjugate gradient [23], [34], it appears that the statistical perspective inherent to the SAGE method is a useful addition to conventional numerical tools.

vp

analysis of the convergence rates. In this section, we analyze the problem of estimating superimposed linear signals in Gaussian noise [2], [9]

where A = fa1 ... ap ] , and f. is additive zero-mean Gaussian noise with covariance II, Le., f. N(O, II). For simplicity we consider a quadratic penalty P(O) = ~O'PO, so the penalizedlikelihood objective function is f'"V

- cll(B)

= ~(y -

AB)'JI-l(y - AB)

+ ~B'PB.

Such objective functions arise in many inverse problems [9]. We assume A has full column rank, P is symmetric nonnegative definite, and the intersection of the null spaces of P and A is empty, in which case the (unique) penalizedlikelihood estimate is

fJ

= (A'rr- 1A + p)-l A'rr-1y.

(21)

If A is large, or if positivity constraints on () are appropriate, then (21) is impractical and iterative methods may be useful. (One can also think of (20) as a linearization of the more interesting nonlinear problem [2].) We present the linear version here since we can derive exact expressions for the convergence rates. We first present admissible hidden-data spaces for this problem, derive EM and SAGE algorithms, and then prove that the SAGE algorithm converges faster. Since the mean of Y is linear in 0, the conventional complete-data [2], [9] for the EM algorithm for this problem is also linear in f}. Here. we restrict our attention to hidden-data spaces ~yS whose means are also linear in f}, and for which the conditional mean of Y given X S is linear in ~Y 5 and OS' Considering a general index set 5, the natural hidden-data space for () 5 is

. ~ ~ ./V(BO s + BO s' C) S

YIX

5

-

=X~jV(Gx+G(}s~W)

which is admissible provided the two normal distributions are independent and consistent with (20), i.e., As = GB, As = GB + G, and II = W + GCG'. The log-likelihood for ~~s is given by

IV. EXAMPLE 2 LINEAR GAUSSIAN MEASUREMENTS

The Poisson problem has important practical applications, but the nonlinearity of the algorithms complicates a formal 3 Fast convergence is clearly desirable for regularized objective functions, but we advise caution when using "stopping rules" in conjunction with coordinate-based algorithms for the unregularized case, since for such algorithms the high spatial frequencies converge faster than the low frequencies [26].

where Cl and C2 are independent of Os. By standard properties of joint normal distributions

278

X

S

= E{X s I y" == u: Oi} = B(}~

+ B(}§ + CG'II-1(y -

A()i).

The ¢s function of (4) is thus i

c/>s(8 s; 8

)

SAGE Algorithm for Superimposed Signals

= (B 8s + B8§)'C- 1 ( X 1

- 2

[0 s ] 8§

I

[pP'2P3

1P 2 ]

S

-

Initialize: for i == 0,1, ...

~(B8s + B8§)),

k == 1 + (i modulo p),

[B s ]

S == {k}, O~+l :== (a~rr-lak + Pkk)-lpkO i + ( ak'IT-l ak + P kk )-1 ak'IT"E " . . (oi+l (}ik ) o,k, E :== E + k -

01 -

8§ + C2

which, maximized over Os, yields the generic combined Eand M -step

85+ 1

= (Fxs + Pd-1[B'C-1(X S -

fJ ki +

B8§) - P 28§]

== fJ~ + (F};s + Pl)-lA~rr-l[y - AtJl] - (F.ys + Pl)-1[P 1 P2]Oi

(22)

A. EM Algorithm

The ordinary EM algorithm [2], [9] is based on the following choices for the complete-data space: S == {1..... P}, ~ == diag{ak}, C == ~Ip I~ IT, G == (l/p.~ I p), and B ==

+

(ll~rr-lak

+ Pkk/p)-l(l~rr-l£LJ,J)~

ABl)

(23)

/ H==LH+DH+L H

B. SAGE Algorithm Because of the additive form of (20)~ it is natural for the SAGE algonthm to update each parameter Ok individually. i.e., S' == {k} where k == 1 + Ci modulo p). In light of the discussion in Appendix C. we would like the Fisher information of the hidden-data space for () k to be small, so we associate all of the noise covariance with the signal vector (Lk f'

r-.J

}v

(24)

where D H is a diagonal matrix with the diagonal entries of H, and L H is a strictly lower triangular matrix. Similarly. let

F == A'Il-1A == L F + D F + L~. DH==DF+D p

where D p == diag{ P k k } and F is the Fisher information for y. with respect to f}. Let II x II denote the standard Euclidian norm of a vector r, and for a nonsingular matrix T define !I~rjlT == IIT.rll, which induces the matrix nonn

IIAIIT =

Sk

+ 1~ ... ~p.

To establish convergence of the EM and SAGE algorithms, we use Definition 3 and Theorem 3 of Appendix A. A few definitions are needed. Let H == A'rr- 1 A + P be the Hessian for this problem, and decompose it by

for k == 1..... p.

.Y

1, k

C. Convergence

G == W == 0, where diag l} denotes a diagonal or blockdiagonal matrix appropriately formed. Ip denotes the p vector of ones. and :~ is the Kronecker matrix product. Note that these choices distribute a fraction 1.p of the noise covariance II to .ach signal vector ak. Thus, F.\ == p diag{a~I1-1(Lk}. which being a diagonal matrix is easily inverted. However. since S == {1..... p}, the penalized EM algorithm (22) requires inversion of F.\ + P. which could be just as difficult as inverting i\'TI- 1 A + P for a general P.Therefore. we consider the case where P == diag{ Pk k } . for which the EM algorithm simultaneouslv updates all parameters via + p kk / p)·-1 akIIJ-l(· .l/ -

e.j' J"-- 1, ... , k -

where P kk is the kth diagonal entry of P, and P k is the kth row of P. Note that unlike the EM algorithm, the SAGE algorithm circumvents the need to invert P by performing a sequential update, so a nondiagonal smoothness penalty P is entirely feasible.

where F.\s == B/C-1B is the Fisher information of ..~s for

iJl+l == -1.,\ akIrr- 1ak P

l . '-

}

Os·

ok

t==y-AOO

rn;x II~~~T

=

IITAT-ill.

In addition. let p( A) denote matrix spectral radius. the maximum magnitude eigenvalue of A. SAGE Algorithm: From the SAGE algorithm given above. one can show (cf. proof of Theorem 3) that g(i+l)p _ {)

== M p

.....

M

1

.

(Oi p

-

(25)

(})

where

M, == I - ekH;;klek/H~ = H- 1 / 2 Hl/2ek(Hkk)-lek'Hl/2)Hi/2,

(I -

(a k () k:» IT)

Y ==.oX" s' + ~ ~aj()j. J=I=k

Thus, F.\_.,.k == a~II-lak, which is p times le~s informative than the EM case. which associates only a fraction 1/p of the noise covariance with each signal. (This provides a statistical interpretation of the modified EM algorithms in [35] and [36].) The above choice for the hidden-data space corresp,?nds to B == o,k, C == IT, B == W == 0, G == I. and G == [al ... ak-l ak+l ... a p ], which, substituted into (22), yields the following algorithm. 279

== T- 1 (I - tk(t~tk)-lt~)T, == T-1ptT

pt

and where T = H 1j 2 , the kth column of T is tk, is the orthogonal projection onto tk, and ek is the kth unit vector of length p. Since an orthogonal projection is nonexpansive, IIM k liT ~ 1, which confirms condition 2 of Definition 3. To confirm condition 3, rewrite the SAGE algorithm using (24) as fjC'i+l)p _

0 == [1 - (D H + L H )-1 HJ((}ip

-

0)

which is the Gauss-Siedel iteration (see p. 72 of [37]). Condition 3 follows from p. 109 of [37] since

III-

(DH

+ L H )-lHIIT

= liMp'"

MIII T

< 1.

where x = H 1/ 2 v. Rearranging and multiplying both sides by x'

EM Algorithm: One can use (21) and (23) to show that (Ji+ 1 - {)

=M

. ((Ji

-

IIxl1 2 =

0)

+ P)-lH.

X'DHX

Thus, the EM algorithm is closely related to the simultaneous overrelaxation (JOR) iteration (p. 72 of [37]). To establish that IIMIIT < 1 for T = H 1 / 2 using Theorem 4, we must show that S + S' > H, where in this case S == pDF + P. Since H == LF + D F + L' r + P and P ~ 0, it suffices to show that pD r > LF + D r + L' F, or equivalently that pI > L + I + where L = Dpl/2LpDpl/2. Since A'rr- 1 A

is positive definite by assumption, x' (L + I + [')x > 0 for any nonzero z; therefore, using x == ej ± ek, we see that I i j E (-1,1). Thus, for any nonzero x, x'(E + 1+ r')x < (Lk I Xk 1)2 ~ pllxl1 2 , where the second inequality is a special case of Holder's inequality. The result then follows. We have thus established that both the EM and SAGE algorithm converge globally. The convergence is globally monotonic in norm with respect to the norm T = H 1 / 2 , i.e., R+ is all of lRP •

= p(I -

PEM

= p(I -

(pDp + P)-lH) == p(1 - ((p - I)D F + DH)-lH) == p(I - (DH + LH)-lH)

(26)

(27)

PEM

~

(Ilxll~

-

x'D Hx)/2.

l)D F + D H )- 1/ 2z 11 2 IIz11 2

1 - 1I((p - l)DF

= 1_

IIxll 2

+ DH)1/2H-l/2xII2 2 IIxII

x'[(p - 1)H-l/2DpH-1/2

> 1- IIxll

+ DH]x

2

X'DHX

= 1 - (1 +

>u =

V)~(l _ v)

= (1 ~ v )

u

PSAGE

where the last inequality follows from v E [O~ 1). 0 The inequalities in this proof are rather loose. and often the difference in convergence rate between EM and SAGE is more dramatic than the proof might suggest. To illustrate. consider the case where P = O. Then returning to (25), for the EM algorithm we have

< PEM < 1.

= x'L~x =

+ DH )-lH) + DH )-1/2H

for any z (by definition of spectral radius). In particular, for z = ((p - I)D F + DH )1/2H- 1 / 2 x :

Proof' The right inequality follows from PE:\t1 ~ IIMIIT < 1. From (24), 1 = LH + D H + L~ where L H = H- 1/ 2L HH - 1/ 2 and D H = H-l/2D HH --1/ 2. Thus, for any vector x

x'LHx

((p - l)DF

IIH 1/ 2 ((p -

~ 1-

since P = D p for diagonal P. Theorem 2: For linear superimposed signals in Gaussian noise with a diagonal penalty matrix, the SAGE algorithm asymptotically converges faster than the EM algorithm, i.e. PSAGE

I-v

x ((p-1)Dp+D H )- 1/2)

-

To compare the root-convergence factors of EM and SAGE. we focus on the case where P is diagonal, since otherwise the EM algorithm is in general impractical. Therefore, from the results above

= 1 + v llxlI 2 .

= p(1 - ((p - l)D F

D. Convergence Rates

PSAGE

+ DH)X.

By the invariance of eigenvalues under similarity transforms

r,

PE~1

(1 - v)x'(L H

Combining with (28)

for the EM algorithm, where (cf. (37))

M == I - (pDp

thus

(28)

Since I - G -1 H is similar to the real symmetric matrix 1 - G- 1 / 2 H G - l / 2, the eigenvalues of 1- (D H + L H )-lH are real. For u = PSAGE E [0,1) there exists v :I 0 such that

[I - (D H + LH)-lH]v = vu 280

M =

~

tM =

P k=1

k

T-

1

(~tPt)T. P k=l

Since eigenvalues are invariant to similarity transforms, it follows that root-convergence factors for the two algorithms are given by the spectral radii PEM

PSAGE

= p(~ p

=P

t Pt), k=l

(Ii Pt) k=l

i.e., for the EM algorithm we have a convex combination of orthogonal projections and for the SAGE algorithm we have the product of those projections. Thus, this SAGE algorithm is closely related to the method of alternating projections [38], [39]. In particular, if P = 0 and the columns of A are orthogonal, then PSAGE == 0 whereas PEM 2: 1 - lip, i.e.,

1,....--....---.,...--....-----.,,----

demonstrated that SAGE algorithms yield faster convergence than EM algorithms in two signal processing applications. The particular SAGE algorithms that we presented in this paper sacrifice one important characteristic of the EM algorithm: they are less amenable to a parallel implementation since they are coordinate-wise methods. However, the general SAGE method is very flexible, and work is in progress on more parallelizable algorithms using index sets S consisting of several elements of () [30]. The benefits of parallelization must be weighed against the convergence rates for each application. It is probably no coincidence that the applications we put forth are ones in which the terminology "incomplete-data" and "complete-data" as introduced in [1] are somewhat unnatural. In most of the statistical applications discussed in [I], there is a clearly identifiable portion of the data that is "missing," and hence one natural complete-data space. In contrast, there is nothing really "incomplete" about tomographic measurements; the problem is simply that the log-likelihood is difficult to maximize. The EM algorithm is thus just a computational tool. (To further illustrate this point, note that in classical missing data problems the estimates of the missing data may be of some intrinsic interest, whereas the "complete-data" for tomography is never explicitly computed and would be of little use anyway.) SAGE algorithms may be most useful in such contexts. We have emphasized that the SAGE algorithm improves the asymptotic convergence rate. The actual convergence rate will certainly depend on how close the initial estimate is to a fixed-point. In tomography and image restoration, fast linear algorithms can provide good initializers for penalized likelihood estimation. A greedy algorithm like SAGE is likely to be most beneficial in applications where such initializers are available.

0.9

•

Conventional EM

0.8

1 ...

0•7

~0.6

c

4D

~0.5

a0.4 4D

> u

~

80•3

a:

0.2

SAGE Algorithm

0.1

0'--.......---......- ----------.......- -......- - - - ' o 0.2 0.4 0.& 0.8 1 cos(Complementary Angle 8etw"n Sub_paces) Fig. 3. Comparison of root-convergence factors for conventional EM algorithm and proposed SAGE algorithm versus complementary angle between subspaces of superimposed signals. The SAGE algorithm has a significantly improved convergence rate.

the SAGE algorithm converges in one iteration, while EM converges very slowly. When p = 2, using a Gram-Schmidt argument one can show that t 1 = [1 0]' and t2 == [Q~)' where a = la~II-la21/(llallllla211) is the cosine of the complementary angle between al and a2. Thus PSAGE

= P(

[

~ ~])

=0 2

PE~I = p(~ [-~;l ~

(12

-a~]) 1+a2

_~ + ::

-

2

2'

Fig. 3 illustrates that the root-convergence factor of SAGE is significantly smaller than that of EM, which substantially reduces the number of iterations required. Not only is PSAGE < PEM, but also PSAGE < P~M' so one SAGE iteration is better than two EM iterations, at least when p = 2. Thus, even though the EM algorithm appears to have the advantage that one can parallelize the M -step using p processors that simultaneously update all parameters, in this case the convergence rate of the parallel algorithm is so much slower that a sequential update may be preferable. This depends, of course, on how difficult the M -step is; in the nonlinear case discussed in [2], the Mstep is presumably fairly difficult, so parallelization may be advantageous. Equations (26) and (27) help one examine these types of tradeoffs. V.

DISCUSSION

We have described a generalization of the classical EM algorithm in which one alternates between several hidden-data spaces rather than using just one, and updates only a subset of the elements of the parameter vector each iteration. By updating the parameters sequentially, rather than simultaneously, we

ApPENDIX A

MONOTONE CONVERGENCE IN NORM

Because the SAGE "algorithm" is so general, a single convergence theorem statement/proof cannot possibly cover all cases of interest (see, for example, the variety of special cases considered for the classical EM algorithm in [40].) Here we adopt the Taylor expansion approach of [4] since it directly illuminates the convergence rate properties and prescribes a region of monotone convergence in norm, However, this general approach has the drawback that it assumes the fixed point lies in the interior of e. This restriction is often not a necessary condition, and at least for some applications one can often find specific convergence results without the restriction, e.g., [3] and [30]. Readers who are satisfied with the assurance of monotonicity of the objective 4?((Ji), as provided by Theorem 1, may wish to simply skim this Appendix. For simplicity, we discuss only the case where the index sets s' are chosen cyclically with period K, i.e., Si = Sk where k == 1 + (i modulo K). We also assume that U~=l Sk == {1, ... , p} so that each parameter is updated at least once per cycle. Before stating the convergence theorem, several definitions are needed. Consider an index set S, and let m denote its

281

cardinality. Bearing in mind our notational convention that cPs (() S; 7J) cPs (() s: (j s, 7Js), we define the m x m matrices

=

Theorem 3: Assume i) Si = Sk where k = I+(i modulo K) and U:=l Sk = {I, ... ,p}, ii) iJ is a fixed point of the SAGE algorithm (5) in the interior of 8, iii) for all 7J E R+ the Sk maximum over ()Sk of cjJSk ((}SIc; 8) is in the interior of 8 (0), iv) s» ; 6) is twice differentiable in both arguments V8 E e Sk and V(}Sk E 8 (8), and v) the region of monotone convergence R+ for a nann II . liT is nonempty. 1. If E R+ then

«: (()

and

eo

and the m x (p - m) matrix

110 + 1 1

-

OIlT ~ IIIP - OIlT

Vi

and where \7 denotes the (row) gradient operator and \7' its transpose. Let {} be a fixed point of the SAGE algorithm, and define

1I()(l+l)K -

-1 1 1

('V 200¢S)(tf)s

+ (1 -

t)Os; to

V ((}s; 0) 1 = ('VllO¢S) (tf)s

+ (1 -

t)Os; to + (1 - t)O)dt

S

+ (1 -

t)O)dt

(29)

(30)

WS(f)s; 0) = 1\'V 101¢S)(tf)s

+ (1- t)Os; to + (1- t)O)dt. (31)

Let R S denote the p x p permutation matrix that reorders the elements of {S, S} into {I, .... p}. Then define the p x p composite matrix

MS(Os; 8) == R S [U S ( 0s: 0)-1 [Y S (

eS: (j) W S ( ()S; 0)] ] (R S )' Ip-

O(p-m)xm 11,

x

11,

(32)

m

(JIlT

(34)

p(MSK(OSK;B) ... MSI(Bsl;B)) which is bounded above by

and

where In denotes the In addition, define

~ QIIOll< -

where a < 1 is defined by (33). Therefore, IIBll< - 0IIT converges monotonically to zero with at least linear rate. 2. The root-convergence factor [41] of the subsequence {e,K } ~o is given by the spectral radius

US((}S; 0)

=

BIIT

Q

(35)

< 1.

Note that by the equivalence of matrix noons (p. 29 of [37]), monotone convergence with respect to the norm II . liT implies convergence with respect to any other norm, although probably nonmonotonically. Since the index sets are chosen cyclically, a "full iteration" consists of K updates; therefore, (34) bounds the convergence rate of the subsequence {8 i K }~o. Proof: Consider the ith iteration and let S == Sk where k == 1 + Ci modulo K). Define

and let ¢s ( z )

= cPs (f)S ~ 7]).

Let

d(z) == d(()s; 8) == (v~s(ps)(z)

identity matrix.

then by assumption iv) we can apply the Taylor formula with remainder [42] to expand d(z) about z

With the above definitions, we can define the following region of monotone convergence in norm to B. Definition 3: R+ c e is a region of monotone convergence in norm if there exists a nonsingular p x p matrix T such that R+ is an open ball with respect to the norm II . liT and Sk 1. For k == 1, ... ,K, U (OSk; B) is invertible for all () E R+ sk and for all f)Sk E (see J.7)). _ 2. For k == 1, ... ,K,JjM (Osk;(I)IIT ~ 1 for all (I E R+ sk and for all ()s» E (0),

e (!L e

3. There exists Q: < 1 such that for any () , ... , e -- k -k and ()Sk E s (0), k == 1, ... , K

~I
k will be on the unit circle. IL however. the number of snapshots .V is too small or if there is only noise present. the eigenvalues of \{J TLS might fail to satisfy

(26)

+J

Moreover, the eigenvectors of both matrices are identical. Proof' a) Assume, for the moment that the left iI-real matrix Q2 d is the one we have defined in (3). Then. (25) yields

After partitioning V and W as before, we therefore conclude .rom (23) tVTLS

== -(W 12

+ jW12)(Wl~

== - (( - W

1:2W

=f(YT LS )

221 )

-

with

- jW 21 l -

j I d )( ( - W

Y T LS

1

12 W

2} ) + j I d ) -

= -W 12W;}.

(27)

Here. f(.1") denotes the linear fractional transformation which is analytic for .1' # -j. Let TTLS

= -W12W~}

t

= TfIT- i

(26),

(30)

(28)

be the eigenvalue decomposition (EVD) of the real matrix

Y rt.s It is a well-known result from function theory that the eigenvalues of tV TLS can be obtained through the "arne linear fractional trans formation. i.e.

(t)]

where A is an amplitude constant, Wo is the carrier frequency, and (/)(t) is a binary waveform whose value switches between o and n, The waveform c/>(t) is made up of the (modulo 21T) SUm of t\VO bit streams, (/)data(t) and (/>code(t). (/)data(t) is the useful information sent over the communication system; it has a bit rate of fd bit/so code(t) is Ie bit/so The code rate is higher than the data rate by the ratio

(N is an integer), which we call the spreading ratio. Fig. 4 illustrates a typical (/)data(t) and (/)code(t) , and shows the resulting total phase modulation f!>(t). The bit transitions in 4>data (r) coincide with bit transitions in 4>code(t). It is assumed that the details of q,code(t) (i.e., the feedback connections, shift register length, and clocking rate Ie) used at the transmitter are also known at the receiving site, so a similar code can be generated in the receiver. With such a desired signal, a reference signal for the adaptive array can be generated as shown in Fig. 5. In this loop, the array output signal is first mixed against a coded local-oscillator signal '1 (t). '1 (r) is obtained by modulating a CW signal with the same PN code used to modulate the desired signal at the transmitter. This code is known at the receiving site, so it can be generated and used to produce '1 (r). (We assume for the moment that the local code timing is known. The method used to establish code timing in the receiver is discussed in Section IV.) The output from this mixer contains both desired and interference components. Because the code modulation in rl (t) matches that on the desired signal, the desired signal component at the mixer output is compressed to data bandwidth, while the interference is not. The filter bandwidth is chosen wide enough to pass the desired signal, which now has only data modulation, but not wide enough to pass the interference, which has full-code bandwidth. As a result, the filter removes all but the center portion of the interference spectrum. Next, the signal is passed through a limiter, which controls the amplitude of the reference signal. Finally, the limiter output is mixed again with the coded LO signal '1 (t), and the result is used as the reference signal. If we trace the desired signal through this loop, we find that it has the code .modulation removed at the first mixer, it passes through the data-bandwidth filter and through the limiter, which fixes its amplitude, and finally it has the code modulation put back on at the second mixer. The result is that the desired signal passes through this loop unchanged, except for the amplitude adjustment at the limiter and the envelope time delay associated with the filter. 3 An interference signal without the proper PN-code modulation, however, has its waveform drastically altered by this loop. For example, a CW interfering signal which has a single line spectrum at the array output produces a reference signal component with the full bandwidth of the PN-code modulation (and with lower power than at the array output). The correlation between the interference signal at the array output and the reference signal has been essentially destroyed by the loop. 3 The filter bandwidth has to be wider than a, conventional spreadspectrum filter to keep the envelope delay under ~ bit.

316

There are several reasons why the reference signal amplitude must be controlled. First, the amplitude of the reference signal determines the amplitude of the signals present at the array output, which should fall within a certain range for proper operation of the multipliers, etc., in the feedback loops. Second, the reference-signal amplitude should be fixed so the array will yield a maximum ratio of signal power to interference and noise power at the output, as discussed above. Third, the reference-signal amplitude cannot be linearly dependent on the array output amplitude, because then there is a problem with the operation of the array weights. For example, if the reference loop were linear and its gain (to the desir~d signal) were greater than unity, the loop would return a reference signal larger than the array output signal, causing the array weights to increase without limit. Conversely, a loop gain less than unity returns a reference signal smaller than the array output, causing the array weights to drop to zero. Thus stable operation requires a fixed reference-signal amplitude. Finally, when the reference-signal amplitude is fixed, the desired signal voltage at the array output will also be fixed, regardless of the incident power of the desired signal. This behavior is important for the delaylock loop used to track code timing (described in the next section); it makes the threshold setting for code acquisition independent of incoming signal strength. We note finally that the processing loop in Fig. 5 not only generates the reference signal, but also delivers the desired signal with the PN code removed at the output of the databandwidth filter. (The desired, signal is available at this point once the interference has been nulled.) Thus, the referencesignal loop incorporates the spectrum despreading, and the signal out of the data-bandwidth filter is ready for data-bit detection. Moreover, it is important to note that the interference protection afforded by the waveform processing (the PN coding) is available in addition to that due to array nulling. The two types of interference suppression are in cascade. A reference loop of the type shown in Fig. 5 was implemented and used to obtain the experimental data described in Section V. The desired signal was generated by using a slowspeed PN code to simulate data and a higher speed PN code for spectrum spreading. The sum bit stream was biphase modulated onto a 70-MHz carrier. The data and code frequencies (fd and Ie) could be varied (for different data modulation frequencies, different filter bandwidths were used in the reference signal loop) over a wide range up to about 5Mbit/s, so experiments could be performed with different spreading ratios. One of the goals of the experimentation was to determine the array performance that could be obtained with very modest spreading ratios. Most of the data in Section V are for a spreading ratio of 5 :I. The PN codes were obtained from a Hewlett-Packard 1930-A generator. The shift register length used to generate the codes could be varied from 3 to 20 bits. (However, as will be discussed in the sequel, PN-code sequence length appears to have no effect on array performance other than code lockup time.)

IV. THE DELAy-loCK TRACKING Loor The PN code used in the reference loop (Fig. 5) must be synchronized with the incoming desired signal code for the reference loop to operate properly . For large timing errors between the two codes, the desired signal will not pass through the reference loop. The array then regards the desired signal as interference and nulls it out. Experiments (6] - [ 8] have

shown that the array tracks the desired signal properly for timing displacements between the two codes up to about onehalf bit. Beyond one-half bit, the array nulls the desired signal. To keep the codes properly aligned, the delay-lock loop shown in Fig. 6 was used. This loop is a modified version of a conventional delay-lock loop [9] - [ 11]. (The modifications are discussed below.) This loop splits the array output signal into two channels and mixes it in each channel with a 100 MHz LO-signal biphase modulated with a PN code. The PN codes in the two La signals are displaced one bit in time. A 30 MHz zonal filter then selects the difference frequency output, and the output of this filter is shifted to 5 MHz and filtered to data bandwidth. The 5 MHz-filter outputs are squared, filtered to narrow bandwidth at 10 MHz, and envelope detected. The square root of each signal is then taken and the sum and difference of the two signals is derived. The sum voltage is used for timing acquisition, as described below, and the difference-voltage controls the veo frequency that clocks the PN code generator. The PN-code generator (not shown in Fig. 6) generates the two PN codes used in the two 100 MHz LO signals. These two codes are displaced one bit in time. It also generates a third version of the PN code timed halfway in between the two LO codes. This "in-between" code is used to provide the coded LO signal in the reference loop in Fig. 5. When the delay lock loop tracks, the code on one LO runs one-half bit ahead of the incoming signal code, and the code on the other runs one-half bit behind. The code used in the reference loop is then synchronized with the received signal code. The reason for using a delay-lock loop with squaring and square root operations was to enable the array system to lock-on and track a desired signal with data modulation present on the signal, i.e., we did not wish to send a datafree preamble for code acquisition. In a conventional delay lock loop [9] - [ 11], the incoming signal has only PN-code modulation. When this signal is mixed with a coded LO signal whose code timing differs by an amount ~T from the incoming signal code, the mixer output contains a spectral line whose amplitude is proportional to the autocorrelation function of the PN code, for delay ~T, i.e., the amplitude of this spectral line drops linearly with 6.T, up to one bit. The output of this mixer is then narrow-band-filtered to extract this line frequency, and the result is envelope detected. In the system described here, there is also data modulation on the signal. Hence, at the output of the first mixer the desired spectral line is now convolved wth a (sin x{X)2 datamodulation spectrum Therefore, it is necessary to retain an adequate bandwidth after the first mixer to pass this data modulation. However, the signal is then squared, which eliminates the biphase data modulation and collapses the spectrum. By filtering at the second harmonic, one can use a narrow-filter bandwidth to establish an adequate SIN. In this way a good estimate of the code autocorrelation function (squared) results. Finally, the square root is taken to eliminate the effect of squaring on the autocorrelation function. The major effect of using a wide-bandwidth filter after the first mixer is to increase the thermal- and self-noise power [9] - [ 11] over what they would be in a conventional delaylock loop. However, the adaptive array performance is not particularly influenced by minor amounts of code jitter, as long as the total timing error remains well below bit [6][8] . Furthermore, the amount of code timing jitter one

317

t

KEEPS

DIVISOR

FROM DROPPING BELOW AN ACQUISITION THRESHOLD

VOLTAGE

ADJUSTABLE MINIMUM

SWEEP

VOLTAGE

VOLTAGE

BIPHASE

MODULATORS

X

/

OUTPUT TO

PH GENERATOR

PH CODE I PM CODE 2 CENTER FREQUENCY

TRIM VOLTAGE

Fig. 6. Delay-lock tracking loop.

obtains is controlled by the bandwidth of the filter used at the second harmonic. Hence one can trade off this jitter against other system parameters, such as allowable frequency offsets or code-slewing speed (lockup time). Code acquisition with this delay-lock loop behind the adaptive array is rather interesting. Code timing is acquired by running the local PN -code generator faster (or slower) than the incoming signal code. This code slewing is continued until the sum channel voltage indicates that the two codes are aligned. During this slewing process, since the PN code used in the reference loop (Fig. 5) is slaved to the delay-lock loop .codes, the reference-loop code is also slewed. Thus before the -desired signal code and the reference-loop code are aligned, there is no correlation between the incoming desired signal and the reference signal. Hence the array nulls the desired signal during this time. As the two codes begin to align, the reference signal begins to correlate with the desired signal, causing the array weights to change, and the array pulls the desired signal up out of the noise. Thus the desired signal appears at the array output just as the local code timing approaches its correct value. The desired signal is present at the delay-Iockloop input when it needs to be, but not before. The sum channel output rises and trips a threshhold comparator, which removes the slewing voltage from the VCO input and allows the loop to begin tracking. This technique of slewing the PN code to acquire timing is well known, except that because the delay-lock loop interacts with the adaptive array while it slews, there are several novel features, which we discuss below. First, the array provides full interference protection during the lockup phase. That is, the array nulls interference regardless of PN-code timing in the array. (Interference nulling is not related to the local PN-code timing.) Because the array removes interference during slewing, the delay-lock loop does not have to contend with interference at all. Hence, the integration time in the loop, the slewing speed, and the time required for lockup do not need to change when interference is being received. Second, the desired signal that appears at the array output when the code timing is correct has a fixed amplitude, not

dependent on incoming signal strength. This behavior occurs because the array forces the output-desired-signal amplitude to match the reference-signal amplitude, which in turn is controlled by the limiter in the reference loop (Fig. 5). A fixed desired signal amplitude at the array output is helpful because the delay-lock loop does not have to operate over a range of signal levels. For example, a fixed threshhold value, not dependent on received signal power, can be used in the sum channel for acquisition. Also, circuit linearity problems are vastly simplified, e.g., a fixed signal level makes the squaring operation feasible. A third point of mild interest concerns how the array pulls the desired signal out of the noise when the timing approaches the correct values. Since the desired signal starts out in a null, many people have wondered how there will be any desired signal present at the array output to allow a reference signal to be generated when the timing is correct. The answer involves two factors. The first factor concerns the design of the reference loop in Fig. 5. When the array output signal is small, the voltage in the reference loop is too small for the limiter to clip the signal. Below the clipping level, the loop is linear; it produces a reference signal whose amplitude is linearly dependent on the array output signal. In this low-signal region, the loop must have a gain greater than unity. Then the reference signal will have a greater amplitude than the array output signal. This situation makes the array weight setting that nulls the desired signal a point of unstable equilibrium. In other words, if one weight changes away from this value slightly, so a small desired signal appears at the array output, a reference signal of larger amplitude will appear. This behavior reinforces the movement of that weight away from the null. In this way, the array output signal will grow until it is large enough that the limiter clips. When the clipping level is reached, the reference signal does not increase, and the array output signal settles into its steady-state value. The second factor is that the array weights in the adaptive array are random processes. They are derived from the product of two noisy signals, Xi(t) and e(t). Thus there is always a certain amount of weight jitter, so the movement away from the

318

.

I

I~--'

(a)

(a)

(b)

(b)

Fig. 8. Array interference rejection. (a) Spectrum on one array el ment. (b) Array output spectrumafter adaptation(S/Iimprovement_ 35 dB). Code rate = 2.5 Mbit/s; Data rate = 0.5 Mbit/s; Interference coherent (20 0 spatial): SIN =0 dB, S/I = -20 dB; Vertical Scale-

(c)

10 d Bjcm .

Fig. 7 . (a) Sum channel output with desired signal only (array inoperative). (b) Sum channel output with desired signal and coherent CW interference (array inoperative). (c) Sum channel output with desired signal and coherent CW interference (array adapting during acquisition).

v.

desired signal null has no difficulty starting. In practice, we have found no tendency for the array to keep the desired signal nulled once the code timing is nearly correct. The weight jitter always starts the weights away from this point. Finally, we remark that this lockup procedure requires the array speed of response to be fast enough for the desired signal to appear at the array output before the local code timing has passed by. In other words, the array time constants must be commensurate with the slewing speed . In the system used for experimentation, the adaptive array processor had time constants between 1 and 10 ms. (The array time constants depend on incoming signal power [121 .) The code slewing rate (the difference between the incoming code frequency fe and the local code frequency) could be varied . For a code rate f e of 2.5 Mbit/s and a 1023-bit code period (i.e., a lO-bit shift-register length [5)), lockup time was typically around 1 s. Of course , with this slewing technique, the code acquisition time is linearly proportional to the code length. Fig. 7 shows a typical scope trace of the sum channel output from the delay-lock loop as the array is being slewed . Fig.7(a) shows the sum channel output versus time with no interference and with the array not operating (with the weights fixed) . Fig. 7(b) shows the sum channel output when a CW interference signal 20 dB above the desired signal is added (the array is still inoperative). In this case, each channel of the delay-lock loop is saturated , and the desired signal pulse does not come through. Finally, Fig. 7(c) shows the sum channel output when the array is adapting during slewing. The interference is no longer present on the delay-lock loop, and the desired signal pulse reappears . The shape of this pulse has been altered somewhat from the shape in Fig. 7(a) because the array weights are changing during slewing (and hence they modulate the desired signal) . These figures illustrate the interference protection of the array during the code acquisition phase .

ARRAY PERFORMANCE

In this section we discuss the experimental operating char. acteristics of the adaptive array with the coded reference loop and delay-lock timing loop . The adaptive array processor in these tests was a four-element processor operating at 70 MHz, as described in [11 . In the experiments discussed here, the desired and interference signals were each split in 4-way power dividers and fed to the processor inputs after appropriate delays to simulate various arrival angles. Independent thermal noise was also included in each simulated element signal. We begin with a series of spectrum analyzer photographs illustrating typical interference-rejection experiments. Fig. 8(a) shows the power spectral density on one element of the array when PN-coded desired signal, thermal noise , and CW interference are present. The desired signal-to-thermal-noise ratio is 0 dB, and the desired signal-to-interference ratio is - 20 dB (as measured on each element). The PN-code rate is 2 .5 Mbit/s , and the data rate is 0 .5 Mbits/s, so the spreading ratio is 5 : 1. The desired signal is in-phase on each element (corresponding to a signal arriving from broadside), and the interference has a progressive phase shift of 60° between elements (corresponding to an arrival angle of 19.s° off broadside for half-wavelength element spacing) . The interference frequency is coherent with the carrier of the desired signal (70 MHz). Fig. 8(b) shows the array output spectrum after adapting. As may be seen, the interference has been nulled , and the SIN has been improved due to the array gain. Measurements taken of the output-signal powers separately, with the weights frozen , showed that the signal-to-interference ratio had been improved 35 dB in this case. Fig. 9 shows the same experiment when the interference frequency was 71.2 MHz. In this case, the improvement in signal-to-interference ratio measured 36 dB. Fig. 10 shows another case in which the interference was swept CW, swept over a 1.2-MHz band with a l-kflz triangularsweep waveform . The desired signal was modulated with a 1 Mbit/s PN code and 200 kbit/s data (5 : I spreading ratio). Fig. 10(a) shows the power spectral density on one element of

319

(a)

(b)

Fig. 9. Array interf ere nce rejection. (a) Spectrum on one element. (b) Ar ray output after adaptation (Sfl improvement = 36 dB). Code rate = 2.5 Mbit/s; Dat a rate = 0.5 Mbit/s; Interference freq. = 71.2 MHz; Signal Freq . = 70.0 MHz; SIN = 0 dB. SII = - 20 dB; Vertical scale = 10 dB /cm .

Fig. 11. Interference rejection : 31-bit PN-code sequence. (a) Spectrum on one element. (b) Array-output spectrum.

(a)

Fig. 12. Interference rejection: 7-bit PN-code sequence. (a) Spectrum on one element. (b) Array -output spectrum .

to-signal ratio , but the interference power is the quantity being varied. These curves were taken with a data rate of 250 kbit/s, a code rate of 1.25 Mbit/s (5:1 spreading ratio), and the array, and Fig. lO(b) shows the array output-power with a desired signal-to-thermal-noise ratio of 0 dB on each spectral density. In this test, no element noise was present. element . The desired signal arrives from broadside, so it is in Next we comment on the PN-sequen ce length . PN-sequence phase in all four elements. Three curves are shown , correlength appears to have no effect on the interfer ence suppres- sponding to three different angles of arrival of the interference . sion of the array. Figs. 11 and 12 illustrate the effect on the For the top curve, the interference has alSO element-tospectrum of changing code-sequence length. Fig. 11 shows the element phase shift , corresponding to a spatial angle of 4 .8° one-element spectrum and the arr ay output spectrum with for half-wavelength element spacing. The second curve shows a sequence length of 31 bit. The desired signal is modulat ed the case when the interference element-to-element phase shift at a code rate of 2 Mbitls with a spreading ratio of 10. The is 30° (a spatial angle of 9.6° for half-wavelength spacing) , interference is swept CW, swept over a 1.8-MHz band at a and the third curve shows the case when the interference 1200 Hz rate. The interference power is 6 dB above the de- element-to-element phase shift is 75° (a spatial angle of 25° sired signal power. for half-wavelength spacing) . Fig. 12 shows the same experiment repe ated with a 7 bit The trend of all three curves is similar. For low interference code sequence length. There is no measureable difference in power, the output interference increases linearly with input interference -suppr ession as a result of the change in code- interference. In this region, the interference power is small sequence length, although the effect on the spectrum may be compared with the desired signal and the thermal noise ; it seen in Figs. 11 and 12. has little effect on the array feedback . The feedback loops do Next. we show measurements of the interference-suppres- nothing to null the interference, so the output interference sion of the arra y and discuss its depend ence on various factors. power rises linearly with the input interference power . When Fig. 13 shows a typical measured curve of output inter ference the interference power increases enough, however, the interpower as a function of input intereference power , with CW ference begins to dominate the error signal, and the array interference. The abscissa is plotted as input interfer ence- feedback corrects against it . As the interference power in-

Fig. 10. Array reject ion of swept CW interference. (a) Spectrum on one element. (b) Arra y-output spectrum.

320

°r----4.......=.:::::=---

25 .-------------------"-~ INPUT 9/N • 0 dB CODE RA TE' 1 .25 MHz

15

en

." w u

:z w

a: w \L

m -e

DATA RATE' 0.25 MHz

-'

-S

INTERFERE NCE

ANGLE - O· ELE C. I S· ELEC . _ . 30· ELEC .

SIGNAL

x-x - 7 S'

z

....:::>

x

"'" x"

o w

'" Vl - 6

E LEC .

x............

DES I RED SIGN AL - O· ELEC INT ERF. SIGNA L - 30 ' EL EC

w o

....-0

SI N

- 3 dB

w

0-----0

SI N SI N

- 0 dB - -3 dB

>

~ - 8

- IS

x- x

x

-.

~x """ x /

-' w a:

o

-25

/

x

/X

/"

X_x"""

x

-,

- \0

x

I NPUT

o

- 20

- 10

0

INPUT SIGNAL /I NTERF ERENCE

lis

--""'x

10

...........

x

20

(dBl

Fig. 17. Output SIN versus in put lIS.

7 .8 dB . For a O-dB element SNR , the output should be +6 dB, and 5.5 dB was measured. For a -3-dB element SIN, the output should be +3 dB, and the result was 2.8 dB. We see that as the noise is decreased (or the element SIN is increased), the array performance departs more from the ideal. The reason for this is the presence of multiplier offset voltages [171, which alter the weights from their ideal values . The less noise , the more effect multiplier offset voltages have. The curves in Fig. 15 are for the case where the interference element-to-element phase shift is 30° . (The desired signal is in phase on all four elements.) Thus, Fig. 15 represents a case where the interference is too close to the desired signal for full performance from the array. The fact that the signals are too close is seen in the strong dependence of the output desired-signal-to-thermal noise ratio on int erference power.

ful suggestions of D. Townsend and D. Himes of NRL and R. Bauman of NASC.

Fig. 16 shows a similar set of curves, except that the spreading ratio has been increased to 10: 1. A comparison of Figs. 15 and 16 shows that there is little difference in the performance as a result of changing the spreading ratio. Next, Fig. 17 shows the same measurements as in Fig. 16 except that the interference phase shift between elements has been increased to 75°. (In Fig. 16 it was 30°.) With a phase shift of 7S°, it may be seen from Fig. 17 that the output desired-signal-to-thermal-noise ratio is less sensitive to the input interference power. Thus, at a 30° phase shift, the interference is too close to the desired signal; at 75°, the signals are far enough apart that the SIN is not degraded by the interference.

REFERENCES ( 1] R. T. Compton, Jr., "An experimental four-element adaptive array," IEEE Trans. Antennas Propagat., vol. AP-24, p. 697 Sept. 1976. ' (2] B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode,

"Adaptive antenna systems," Proc. IEEE, vol 55, p. 2143, Dec. 1967.

[3] R. S. Kennedy, Fading Dispersive Communication Channels. New

CONCLUSIONS

This paper has described the integration of an LMS adaptive array into a PN-coded spread-spectrum communication system. A method of reference- signal generation and code-timing acquisition were described that allow the array to distinguish the desired signal from interference. The hybrid system yields full interference protection during the code-lockup phase as well as after timing has been acquired, i.e., code lockup time is not affected by the presence of interference. Typical experimental results have been presented illustrating the interference suppression characteristics of the hybrid system. Most of the results presented are for a spreading ratio of 5: 1, a very modest value. With a 5: 1 ratio, interference suppressions of 35 dB are typical. This performance represents much greater protection than can be achieved by spectrum spreading alone with such a spreading ratio. The results presented illustrate the advantages of combining adaptive array techniques with waveform design. However, it is clear that for such systems the adaptive-array parameters and the signaling waveforms must be selected together to be compatible, because of the close interaction between the antenna and communication systems. In general, it is difficult to add an adaptive array to an existing spread spectrum system where the communication-system waveforms have been chosen independently of the array. ACKNOWLEDGM ENT

The author is grateful to Professor R. J. Huff and Professor A. A. Ksienski for many helpful discussions during this program. In addition, the author appreciates the numerous use-

York: Wiley-Interscience, 1969. (4) S. P. Appleboum, "Adaptive arrays," IEEE Trans. Antenna Propagat.; vol. AP-24, pp. 585-598, Sept. 1976. (5] J. J. Stiffler, Theory of Synchronous Communications. Englewood Cliffs, NJ: Prentice-Hall, 1971, p. 178. (6] K. L. Reinhard, "Adaptive array techniques for TDMA network protection," Section II of R. J. Huff, "Coherent multipleXing and array techniques," ElectroScience Lab., Dep. Electrical Eng., Ohio State Univ., Rep. 2738-3, Feb. 1971; prepared under Contract F30602-69-C-0112 for Rome Air Development Center, Griffis, AFB, NY. (7] K. L. Reinhard, "An adaptive array for interference rejection in a coded communication system," ElectroScience Lab., Dep. Electrical Eng., Ohio State Univ. Rep. 2738-6, Apr. 1972; prepared under Contract F30602-69-0112 for Rome Air Development Center, Griffis AFB, NY. [8] R. 1. Huff and K. L. Reinhard, "Coherent multiplexing and array techniques," ElectroScience Lab., Dep. Electrical Eng .• Ohio State Univ., Rep. 2738-9, June 1972; prepared under Contract F30602- 69- 0 1 12 for Rome Air Development Center, Griffiss AFB, NY. (9) J. 1. Spilker, Jr. and D. T. Magill, "The delay-lock discriminatoran optimum tracking device, Proc. IRE, vol. 49, p. 1403, Sept. 1961. [ 10] J. J. Spilker, "Delay-lock tracking of binary signals, IRE Trans. Space Electron Telem., vol. SET-9, p. 1, Mar. 1963. (11 ] V-'. J. Gin, "A comparison of binary delay-lock tracking loop implementations," IEEE Trans. A erosp. Electron. Syst., p. 41 S, July 1966. ( 121 R. L. Riegler and R. T. Compton, Jr., "An adaptive array for interference rejection," Proc. IEEE, vol. 61, p. 748, June 1973. (13 ) C. A. Baird, Jr., G. P. Martin, G. G. Rassweiler , and C. L. Zahm, "Adaptive processing for antenna arrays," Final Rep., Radiation Systems Division, Harris Intertype Corp., Melbourne, FL, June 1972. (141 K. L. Reinhard, "Adaptive antenna arrays for coded cornmunication systems," ElectroScience Laboratory, Dep. Electrical EnJ., Ohio State Univ., Rep. 3364-2, Oct. 1973; prepared for Rome Air Development Center under Contract F30602-72-C-0162. ( 151 B. S. Abrams, S. J. Harris, and A. E. Zeger, "Interference cancel ... lation ," RADC-TR-74-225, Final Rep., General Atronics Corp., Sept. 1974. (16) A. E. Zeger, B. S. Abrams, and C. Luvera, "Interference canc~· lation system for sensors," hoc. NRL Adaptive Antenna Systeml Workshop, Mar. 1974. [ 17] R. T. Compton, Jr., "Multiplier offset voltages in adaptive arrays," IEEE Trans. Aerosp. Electron. Syst., vol. AES-l~.. p. 616, Sept. 1976.

323

tt

On the Performance of a Polarization Sensitive Adaptive Array R. T. COMPTON, JR.

Abstract-The ability of a least mean square (LMS) adaptive array to adapt to the electromagnetic polarization of incoming signals is considered. An array of two pairs of crossed dipoles is studied. A desired signal and an interference signal are assumed to arrive from arbitrary directions with arbitrary elliptical polarizations. The output signal-to-interference-plus-noise ratio (SINK) from the array is computed as a function of the signal angles of arrival and polarizations. It is shown that as long as certain special desired signal polarizations are avoided, the array is difficult to jam with a single interference signal. To produce a poor SINK, an interference signal must both arrive from the same direction and have the same polarization as the desired signal.

INTRODUCTION

A

DAPTIVE arrays [1] -[ 3] are currently of great interest because of their ability to null interference and track desired signals automatically. Numerous papers have discussed the performance of adaptive arrays [4]. In spite of the extensive literature, however, for radio applications of these arrays (as contrasted with sonar applications), one aspect of this subject appears to have received little attention. We refer to the fact that an adaptive array can adapt to the electromagnetic polarization of signals, as well as their arrival angles. If an adaptive array uses elements responding to more than one polarization, the array feedback loops will automatically combine the signals from these elements to optimize reception, or provide a null, for particular signal polarizations. Such an array can automatically track a desired signal with one polarization while nulling interference with a different polarization. Most analytical studies of adaptive arrays have assumed isotropic elements. This assumption, although useful for certain purposes, tacitly eliminates any consideration of the effects of signal polarization on array performance. In essence one assumes all signals arrive at the array with the same polarization. If an array receives and uses more than one polarization its performance can be far superior to one that does not.. For example, an array of isotropic elements always yields poor performance if interference arrives too close to the desired SIgnal. When an array adapts to polarization, however, this difficulty occurs only if both signals have the same polarization as well as angle of arrival. When two signals arrive from the same direction, it is perfectly possible to null one signal and not the other, if their polarizations are different. The purpose of this paper is to examine the performance of a polarization-sensitive adaptive array. As a model, we will consider an array of two pairs of crossed dipoles. We will compute the output signal-to-interference-plus-noise ratio (SINR) Manuscript received February 15, 1980; revised October 16, 1980. This work was supported in part by the Naval Air Systems Command under Contract N00019-79-C-0291, and in part by the Joint Services Electronics Program under Contract NOOa 14-78-C-0049, both with the Ohio State University Research Foundation. The author is with the ElectroScience Laboratory, Department of Electrical Engineering, The Ohio State University, Columbus, OH43212.

from this array when a desired signal and an interference signal arrive with arbitrary polarizations and angles of arrivaj! We will show that in most cases interference has little effect on the array output SINR unless it arrives from the same direction and has the same polarization as the desired signal. However there are two exceptions. If the desired signal polarization is linear, oriented either parallel or perpendicular to the vertical dipoles, the array is susceptible to interference from other angles as well. These desired signal polarizations are ones that should be avoided in a system design. Finally we will find that when both signals arrive from broadside, the array output SINR is simply related to the separation between the signal polarizations on the Poincare sphere. Section II of the paper formulates the necessary equations. Section III contains the calculated results and Section IV the conclusions.

II. FORMULATION OF THE PROBLEM Consider a four-element adaptive array consisting of two pairs of crossed dipoles, as shown in Fig. 1. The signal from each dipole is to be processed separately in the array. The upper and lower dipole pairs have their centers at Z = + L/2 and Z = -L/2, respectively. Let _~ 1 (t) and x3(t) be the cornplex signals received from the upper and lower vertical dipoles, and X2 (r) and X4(t) the signals received from the upper and lower horizontal dipoles, respectively. Each signal Xj(t) is multiplied by a complex weight Wj and summed to produce the array output. We assume the weights Wj are controlled by an LMS processor [2], [5), so the steady-state weight vector, W = (wI, w2, ... , W4)T, is given by (1)

where rI> is the covariance matrix (2)

and S is the reference correlation vector S = E{X*r(t)}.

(3)

In these equations X is the signal vector

(4)

ret)

is the complex reference signal 2 used in the adaptive array feedback [2], (5), T denotes transpose, the asterisk denotes complex conjugate, and E(·) denotes expectation. Assume two continuous wave (CW) signals are incident on the array, one desired and the other interference. Let () and ¢ 1 By arbitrary polarizations, we refer to signals that are completely polarized (i.e., elliptically polarized). \Ve do not consider partially polarized signals [ 12] . 2 r(t) is called the "desired response" in [2J.

Reprinted from IEEE Transactions on Antennas and Propagation, Vol. AP-29, No.5, pp. 718-725, September 1981.

324

Fig. 2.

Polarization eIlipse.

x Fig. J.

Crossed dipole array.

denote standard polar angles , as shown in Fig . I . We assu m e th e: desired signal arrives from angular d irection «(J d , tPd) and th e interference from «(J i , tPi) ' Furthermore each signal is assumed to have an arbitrary electromagnetic polarization. To characterize the polarization o f each signal we make the follow ing de finit ions . G iven a transverse ele ctromagnetic (T E M) wave propagating in to the array, w e co nsid er t he polarizat ion ell ipse produced b y the tra nsve rse elec t ric fie ld as we view the inco m ing wave fro m the coo rd inate o rigin . N o te th at u n it vectors , ~, iJ , - ;, in that o rde r , fo rm a rig h t-ha nd ed coo rdi na t e sy ste m for an incorn ing wa ve . Suppose the elec t ric field ha s transve rse co rn po nen ts

I

(9a)

3 These relationsh ips are derived in [6]. Our definitions and notation correspond exactly to those in [61 if we substitute/:;q, - X, EO Y, 1) -!p.

(6)

r

where r is the a xi a l rat io r=

m ino r ax is

(7)

ma jor axis

In add ition Q is defi ne d po sitive w he n t he elec tric vector ro tates cloc kwise a nd negative w he n it rota te s co u n te rclock wise (w he n the incom ing wa ve is viewed fro m the coo rdi n ate o rigin, as in Fig. 2) . Q is alway s in the range - 71/ 4 ~ Q ~ ni«. F ig. 2 depicts a situation in which Q is positive . For a given state of polarization , specified by Q and (3 , the elec t ric field co m po ne n ts are given by ( as id e from a common ~ h a s e fact o r)

Eq, = A cos "( Eo

(8 a)

= A sin "(e i T)

Where "( and 1/ are related to cos 2"( = cos 2Q cos 2{3

Q

(9 b)

(8 b)

(We will call Eo the horizontal co m po ne n t and Eo the vertical co m po nen t o f th e f'ield.) In general, as time progresses Eq, and E.j wi ll describe a polarizat ion ellipse as shown in F ig. 2 . Gi ve n th is ellipse we d ef ine (3 to be the o rient a ti o n a n gle o f th e m ajo r a .is o f th e ellipse w ith res pec t to Eq" as sh own in F ig. 2. T o e.irnina te amb iq uities we d efi ne (3 to be in t he range 0 ~ (3 < 71 . We also de fine the elli p tic ity a n gle Q to have a magn itude given by

= tan -

tan 1/ = tan 2Q esc 2{3.

Poincar e sphere.

The re lation ship am ong the fo ur angu lar va ria bles Q, (3, "(, a nd 1/ is m ost easily visualized by mak ing use of the Poin car e sphere co nce pt [ 6] . This t echnique represents the sta te of polariza tion by a point o n a sphere , su ch as point M in Fig. 3. For a given M, 2"(, 2(3, and 2Q form the sides o f a right sp he rical triangl e , as sho w n . 2"( is the side o f th e tr iangle between M and a poin t la be lled II in t he figu re ; H is the point represent ing h o rizo n ta l linear polar izat ion . Sid e 2(3 ex tends alo ng t he eq ua to r and side 2Q is vertical , i.e ., perpend icular to side 2(3. The an gle 1/ in (8) and (9) is the angle between sides 2"( and 2{3.3 The sp ecial case whe n Q = 0 in (6) a nd Fi g. :; co rr es po nds to linear polarizat ion ; in th is case th e po int /11 lies o n the eq ua tor. If in ad di tio n, (3 = O. only l:;1j) is nonzer o a nd the w ave is horizontall y polarized . This case de fines the point H in Fig . 3. If instead (3 = 71/ 2 , o nly c'o is non zero and the wave is vertically polarized . Point ,\If th en lies o n the equat or di ametrically behind H. The po les of th e sp he re co rr es po nd to circ u lar pol ari zat ion (Q = ±4 So) , with clo ck w ise circ ula r pol a riz ation (Q = +45°) a t th e uppe r po le . Thus a n arb itra ry pla ne w ave co mi ng in to the a rray m ay be cha racte rized by fo u r angu lar param et e rs an d a n am plitu d e . For exa m pl e t he de sir ed signal w ill be charac te rized by its ar rival a ngles «(Jd, tPd), its polarizat ion ell ipt icit y an gle Qd an d orientat ion angle (3d , and it s amplitude A d ( i.e ., A d is the value of A in (8 ) for the d esired signa l) . We will say the de sired signal is defined b y «(J d , epd, (Xd , (3d , Ad )' Similarly th e in terfe re nce is defined by «(J i, epi, (Xi , (3i, A i) ' We assu me eac h d ip ole in th e array is a short dipol e, i.e. , th e o u t p u t vo lt age f ro m eac h d ipole is pr opo rt ional to the electric field co mpo ne n t alo ng th e d ipole . Ther e fore the vert ica l and hor izontal d ipo le o ut pu ts w ill be p ro portio nal to the z- and x-com po ne n ts, res pec t ively, o f t he electric fie ld . An inc o m in g signal, wi th arb itra ry ele ct ric fiel d co m po nents EIj) and Eo, has

(5 )

Q

Fig. 3.

and (3 by [ 6]

325

x, y, z components:

vector

E = E(j)~ + E(J8

= (Eo cos (J + (Eo cos

( 17)

cos ~ - E(j) sin ~)x (J sin ~

+ Erp cos

The covariance matrix in (2) is then given by

~)y

-(Eo sin 9)2.

(18a)

(10)

where When E rP and Eo are expressed in terms of A, 'Y, and 11 as in (8), the electric field components become

E = A[ (sin 'Y cos 8 cos epejTl

+ (sin 'Y cos 9 sin

l/Je i Tl

+ cos

I

y cos l/J)Y

(12)

(-sin "1 sin (JeiTl)e-jp

rrL

A

I

I

( 18d)

with I the identity matrix. To make the LMS array to track the desired signal, the reference signal ;(t) must be a signal correlated with the desired signal and uncorrelated with the interference [2], [5]. Several techniques have been described for obtaining such a reference signal [ 10] , [ 11 ] . Here we assume

(13)

w is the frequency of the signal, 1/1 is the carrier phase of the signal at the coordinate origin at t = 0, and p is the phase shift of the signals at the dipoles due to spatial delay

=-

I

For the reference correlation vector, (3) then yields

(sin 'Y cos 8 cos epeiTl - cos 'Y sin
I 120

( DEGREES )

10

~

Of-

8;

I 90

*'

A .2

8,

I 90 ( DEGREES I

I 120

I 150

180

(0

=40 dB. (a) 13; =0° . (b) 13; =

=-i=interference-to-noise ratio (INR) , a

(26b)

The derivation o f (25) from (21) may be found in [13, ap pendix] . Calculation of the SINR from (25) is much eas ie r th an from (22)-(24) because (25) does not require calculation of the weight vector. In the next section , we show typical curves of the arr ay performance based on (25) . III. RESULTS Bec ause o f t he large number of parameters required to speci fy both the desired and interference signals, many types o f curves can be plotted . Unfortunately space does not permit an exhaustive set of curves here. However, we will show a number of typical curves, in clu di ng those illustrating the worst performance . Figs. 4 and 5 show curves of output SINR when the desir ed signal arrives from broadside (0 d = rf>d = 90°). The de -

327

10

0 0

d,Pd)e iPd2

(10)

ZL

Antenna array as aN + 1 terminal network.

where (ed, ¢Jd) defines the desired signal direction. Pa is the polarization of the desired signal, fj(8, fJ>, p) is the pattern response of the jth element to a signal incident from direction (e, ¢J) with polarization P and Pdf is the desired signal phase at the jth element, measured with respect to the coordinate origin.

Substituting (2) and (3) into (1) one gets

Z11 1+_

(9)

f 1 ((J d , fJ>d' Pd) e d 1

BILATERAL N+I PORT NETWORK

Ud

Fig. 2.

«v.:

coordinate origin, l/J ik is the carrier phase of the kth jammer at the coordinate origin and Ud, Ui k are, respectively, the desired signal vector and the kth jammer vector defined as follows;

OUTSIDE SOURCE

__----...10..-------'---------. } A~~~~~A LINEAR

Aikei(wot+1/J

A 7k is the average power in the kth jammer, Wo is the carrier frequency, l/J d is the carrier phase of the desired signal at the }

_

=

(8)

In (8) and (9), A~ is the average power in the desired signal,

Basic adaptive array.

Vi •

= Adei(wot+l/Jd)Ud

Z 12

ZlN

ZL

ZL

vi

II (Oik, ¢Jik' Pik)e i Pik J Uik

J I

VON

f2COik' ¢Jik' Pik)eiPik2

(11)

NC8ik, ¢Jik' Pik )e i PikN

where the notation is analogous to that for the desired signal vector. Using (6) and (7), the input signal to the adaptive processor will be

(4)

Or, more compactly

V=ZOl(Xd+i;xik)'

(5)

In (5), Zo is the normalized impedance matrix and Vo represents the open circuit voltages at the antenna terminals. Since Zo is nonsingular, one can find the element au tput voltages from the open circuit voltages. The element output voltages will be given

v = Zo 1 Vo.

=

k=l

If thermal noise is also added to each element of the array then the total input signal to the processor will be X= V+X n

(6)

::=

It should be noted that the matrix Zo is a normalized impedance matrix, normalized to the load impedance. It acts like a transformation matrix, transforming the open circuit element voltages to the terminal voltages. What is normally assumed in analyzing adaptive antenna systems is that the element spacing is large enough .so that the mutual coupling between the elements is small and consequently the matrix Zo becomes diagonal. If one further assumes that the self-impedances (Z;;, i = 1,2, ... ,N) are equal, the input signal vector will be just the open circuit voltage vector multiplied by a trivial scaling factor involving the self and load impedance terms. Thus the array performance will be the same as calculated using the open circuit voltages as the input signals to an adaptive processor.

(12)

z; 1 [Xd+ i; xikJ + X n

(13)

k=l

where X n is the noise vector defined as Xn

= (n 1( t), n2 (t),

'.., nN (t)) T .

(14)

In (14), T denotes transpose. In the case of an adaptive array, the signal xi(t) from the jth element is multiplied by a complex weight wi(t). The signals are then summed to produce the array au tpu 1. Using the LMS algorithm [3], the steady state weight vector W of the array is given by W=-IS

(15)

where is the covariance matrix

333

= E {X* X T }

(16)

and S is the reference correlation vector

S

gets

= E{X*R(t)}.

(17)

ZT

~-l = a~ (R; 1 - iR; 1 ~U'SR; 1 X(Zo J )*)- I

In (16) and (17), R(t) is the complex reference signal in the adaptive array [3], [4], the asterisk denotes complex conjugate, and E{ •} denotes expectation. From (13) and (16)

where (26)

~=EI [ZoJ (xd+~ Xik) +XinT · [ZOI

(Xd+

= (ZOI)*£

I[(

tl Xik) +XnJTl

x, + ~ X ik) + ZoX n

The array will acquire and track the desired signal if the reference signal is correlated with the desired signal and is uncorrelated with interference signals. Assuming that the reference signal R(t) is given by

r

R(t)

S = A,.Ad(ZO 1 )*U~.

[a Z*Z T + ~ A ~ u: UT + A U*U 0

0

~

k=l

Ik

Ik

d2

Ik

d

T] d

W = KZ&R;; 1 ~

(19)

-vhere 0 is the thermal noise power. From (19)

+

~ ~

T

~ -1 must be computed. The following matrix inversion lemma [5] is used to compute -1: (A-aU*UT)-l =A- 1-{jA- 1U*U T A -

1

(23)

where A is a nonsingular N X N matrix, U is a N X 1 column vector and Q, t3 are scalars related by

a-I +/3-1 =UTA-1U*.

(24)

Using the matrix inversion lemma to invert in (22) one

(33)

and Pn is the output thermal noise power 2

Pn=-IWI 2 . 2

then = (Zo 1 )*a (R n

(31)

k==l

where ~d is the ratio of the desired signal power to the thermal noise power and ~ik is the ratio of the kth jammer power to the thermal noise power. Let

2

(30)

(32)

(20)

s, = z3zl; -+ ~ ~ikU!kUhc

(29)

is a constant. The weights given in (29) will lead to the maximum output SINR in the presence of multiple jammers (see Appendix) . Knowing the steady state weight vector, one can compute the output SINR of the array which is given by

t v ,,*UT]eZ-1)T + C;d d d 0

m

(28)

where

(Z-l)T 0

2

"" = (Zo-1)*a 2 [Z*OZOT 't'

(27)

Using (25) and (28) the steady state weights (15) of the array are given by

Assuming that the thermal noise voltages from the array elements are Gaussian with zero mean and are uncorrela ted with each other, and the carrier phases of the narrow-band signals .ire uniformly distributed on (0, 2n') and are statistically independent of each other and of the thermal noise voltages, the covariance matrix ~ is given by 2

= A,.ej(wot+ wd)

and using (13), (17) yields

(18)

= (Z-l)* o

(25)

(34)

Using (3l)-{34), (31) yields

SINR = ~d UJR;; J U;.

(35)

Equation (35) is used to compute the steady state output SINR of an adaptive array consisting of N half-wavelength, center-fed dipoles. All the dipoles are assumed to have similar radiation characteristics and are spaced at a distance d apart (Fig. 3). Note that H-plane arraying of the dipoles is done. The desired signal and all jammers are assumed to be theta polarized (Fig. 3). For the results presented in this 'paper, [k(f) , 4>, p) = 1.28142 (10). Fig. 4 shows the output SINR of an adaptive array of six dipoles as a function of the desired signal direction. The dipoles

334

o

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IV.

--.",__- -

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one interferer when there is also a second interferer with a 3 dB signal-to-noise ratio. The simulation used 100000 samples per data point. The results are shown for a 10- 3 BER, but as seen in Fig. 6, these results are similar to the results for other BER's less than 10- 2 • Fig. 7 also shows the maximum improvement possible if both interfering signals are completely nulled in the receiver output (i.e., the difference between the maximal ratio combiner performance with and without interference). The improvement is within about 2 dB of the maximum with six or more antennas. Fig. 8 shows the improvement versus the number of antennas with one to six equal power (fj = 3 dB) interferers. Again, 100000 samples per data point were used. The improvement is shown to be between 1-6 dB as M varies from 2 to 8. Thus, optimum combining has some improvement over maximal ratio combining even with a few antennas, and the improvement greatly increases with the number of antennas. Although the results of Fig. 8 are for equal power interferers with a particular value of Ij, they demonstrate the following characteristics of optimum combining that apply to other interference cases as well. First, when the number of antennas is much greater than the number of interferers, the improvement is limited. That is, in this case there is little improvement (relative to maximal ratio combining) with additional antennas. This can also be seen from Figs. 4 and 7. Second, except for the above case, the increase in the improvement (in decibels) with each additional antenna is approximately constant (about 0.6 dB for f J = 3 dB). Finally, the most interesting characteristic is that there is a large improvement even when the number of interferers is greater than the number of antennas. This implies that in analyzing systems we must consider many interferers individually even if there are only a few antennas. For example, consider the case of five antennas with six interferers , each with f J equal to 3 dB. From Fig. 8, the improvement is 2.7 dB. However, if only five interferers are considered individually, and the power of the sixth one is combined with the thermal noise, f J is - 1.8 dB and the improvement is only 1.6 dB. Thus, we must consider individually as many interferers as possible to determine accurately the actual optimum combining improvement.

...=tl

10

(dB)

Fig. 7. The improvement of optimum combining over maximal ratio combining versus the signal-to-noise ratio of one interferer when there is also a second interferer with a 3 dB signal-to-noise ratio. The improvement is within about 2 dB of the maximum improvement with six or more antennas.

better than maximal ratio combining with nine antennas (which, from (27), requires -1.7 dB for a 10- 3 BER). Fig. 7 shows the optimum combiner improvement over maximal ratio combining versus the signal-to-noise ratio of

PERFORMANCE IN TYPICAL SYSTEMS

This section studies the performance of optimum combining in typical cellular mobile radio systems. Using the techniques of Section III, we study optimum combining when the signals are subject to Rayleigh fading.' Optimum combining is studied only at the base station receiver because multiple antennas and the associated signal processing for optimum combining are less costly to implement at the base station than on numerous mobiles. (Adap tive retransmission with time division [1], [9] can be used to

345

s In an actual mobile radio system, the signals are also subject to shadow fading [9] which greatly complicates analysis. We therefore only consider Rayleigh fading so that system comparisons can easily be made.

10

r--,----,----,----,----,....----.

TABLE I

COMPARISON OF OPTIMUM AND MAXIMAL RATIO COMBINING IN TYPICAL MOBILE RADIO SYSTEMS- THE NUMBER OF ANTENNAS REQUIRED AND THE SINR MARGIN FOR A 10 - 3 BER

BER • 10- 3

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0.0 1.7

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0.1 0.7 0.3 2.9 1.0 5.5 0.0 1.0

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improve reception at the mobile with multiple base station antennas only (see Section V-B).) As before, all results for optimum combining are compared to maximal ratio combining. Analysis of optimum combining with numerous interferers requires a substantial amount of computer time. It is therefore nearly impossible to determine the average performance of the adaptive array in the typical cellular system with random mobile locations. Therefore, in this section we consider a worst case scenario only, i.e., the mobile transmitting the desired signal is at the point in the cell .fart~est from the .b ase station, and the interfering mobiles In the surrounding cells are as close as possible to the base station of the desired mobile. Furthermore, in the ~alysis we consider only the six strongest interferers individually. The power of the other interferers is combined and considered as thermal noise. .The systems studied involve two different cell geometries WIth hexagonal cells. In one geometry the base stations are located at the cell center, and in the other geometry the base stations are at the three alternate comers of the cell and are equipped with sectoral horns. In the latter geometry, each of the base station's three antennas has a 120° be~width and serves the three adjoining cells. We also consider both frequency reuse in every cell and the use of three channe~ sets. Furthermore, because in the typical system the SIgnal strength falls with the inverse of the distance raised to between the third and fourth power we ' also consider these two extremes," The performance of optimum combining and maximal ratio combining in typical mobile radio systems is shown in Table I. For each of the systems described above Table I lists the number of antennas required to achieve a 10- 3 BER and the average output SINR margin. We also show the margin with an additional antenna. The results show that with three-comer base station g~~etry and frequency reuse in every cell, optimum combining more than halves the required number of antennas. Furthermore, the increase in margin with an additional .6Th e calculation of the power of the signals in these cellular systems WIll not be descnbed here. The method is similar to that described ill [10J.

Antennas

Mlrsin. a (dB)

Optimum

Number or Intcnnu

Combinin•

SINR Matpn a (dB)

1.6

0.4 0.9

1.1

0.1 2.4 2.7 5.3

a Margins are a~urate to within a few tenths of a decibel and were determmed from Simulation results using 100 000 samples .

antenna is much greater. With the same geometry and three channel sets, even though only a few antennas are required with maximal ratio combining, optimum combining increases the margin by 2-3 dB. With centrally located base stations and frequency reuse in every cell, optimum combining substantially reduces the number of antennas. As few as 11 antennas are required with optimum combining as compared to more than 50 with maximal ratio combining. Finally, with three channel sets, optimum combining requires one less antenna and has higher margins. Thus, the improvement with optimum combining is the largest in systems where a large number of antennas is required because of low received SINR. However, even with high SINR and few antennas, the improvement is 2 dB or more. Therefore, the results for typical cellular systems agree with those of Section III (i.e., Fig. 8). In an actual system we would expect the optimum combining improvement to be even greater than that shown in Table I because of the following three reasons. First, all the channels in all the cells may not always be occupied. Thus , the total interference power will be less, and the power of the strongest interferers (when transmitting) relative to the power of the sum of the other interferers r will be higher. As shown in Section III, as r increases, the optimum combining improvement increases, Second, with random mobile locations rather than the worst case, the total interference power will be lower. Thus, rJ for the • strongest mterferers (those closest to the desired mobile's base station) will be higher, and therefore, so will the improvement. Third, for the results in Table I only the six strongest interferers were considered individually, and thus the results are somewhat pessimistic. Finally, we note that in actual systems the fading can be non-Rayleigh with direct paths existing between an interfering mobile and a base station (i.e., the fading might not be independent at each antenna). Under these conditions, the performance of maximal ratio combining can be significantly degraded while optimum combining can still achieve the maximum output SINR.

346

v.

IMPLEMENTATION

In this section we discuss the implementation of optimum combining in mobile radio. We consider the use of an LMS [3] adaptive array at the base station receiver and adaptive retransmission with time division for base-tomobile transmission. For the LMS adaptive array, we discuss the dynamic range, reference signal generation, and modulation technique.

'Oll)

ARRAY

OUTPUT

A. The LMS Adaptive Array 1) Description: Of the various adaptive array techniques [2]-[4] that can be used in mobile radio, the LMS technique appears to be the most practical one for mobile radio because it is not too complex to implement and it does not require that the desired signal phase difference between antennas be known a priori at the receiver. Fig. 9 shows a block diagram of an M element LMS adaptive array. It is similar to the optimum combiner of Fig. 1 except for the addition of a reference signal r( t) and an error signal eel). As shown in Fig. 9, the array output is subtracted from a reference signal (described below) r( t) to form the error signal e(t). The element weights are generated from the error signal and the x/,et) and xQ/(t) signals by using the LMS algorithm which minimizes the power of the error signal. The reference signal is used by the array to distinguish between the desired and interfering signals at the receiver. It must be correlated with the desired signal and uncorrelated with any interference. Under these conditions the minimization of the power of the error signal suppresses interfering signals and enhances the desired signal in the array output. Generation of the reference signal in digital mobile radio systems is described in Section V-A3). We now consider the weight equation for the LMS adaptive array in a mobile radio system. In the typical system the bit rate is 32 kbitsjs, and the carrier frequency is about 840 MHz. With the signal bandwidth 1.5 times the

+ r(t) REFERENCE SIGNAL

Fig. 9. Block diagram of an M element LMS adaptive array.

desired and interfering signals. However, the weights must also change much more slowly than the data rate so that the data modulation is not altered. It has been shown [11] that for PSK signals the maximum rate of change in the weights without significant data distortion is about 0.2 times the data rate. For the typical mobile radio system, the maximum fading rate is about 70 Hz (for a carrier frequency of 840 MHz and a vehicle speed of 55 mijh), and the code rate is 32 kbitsjs. Thus, the permissible range in signal power at the array input is given by . R _ 0.2x32xl0 3 D ynannc ange 70 ~

bit rate, the relative bandwidth of the mobile radio channel is only 0.006 percent, and we can consider the signal as narrow band. For narrow-band signals, the weight equation for the LMS array is given by [6, eq. (9)], i.e., the LMS adaptive array maximizes the output SINR. However, these are the steady state weights, and in mobile radio the signal environment is continuously changing. Therefore, we must consider the transient performance of the array. That is, because the weights are constantly changing, the performance will be degraded somewhat from that of the optimum combiner. (Analysis of the transient performance is not considered in this paper.) Also, we must consider the dynamic range of the LMS adaptive array. 2) Dynamic Range: One limitation of the LMS adaptive array technique is the dynamic range over which it can operate. In an LMS adaptive array, the speed of response to the weights is proportional to the strength of the signals at the array input. For the array to operate properly, the weights must change fast enough to track the fading of the

20 dB.

(42)

The received signals in a mobile radio system vary by more than 20 dB, however, and therefore automatic transmitter power control (which could add significantly to the cost of the mobile radio) is required to control the power of the strongest signals at the receiver. With this power reasonably fixed, the dynamic range determines the power ratio of the strongest to the weakest received signal that the array can track. A 20 dB dynamic range is certainly not large, but it is more than adequate for mobile radio for the reasons described below. In the mobile radio systems studied in this paper (see Section IV), the average received SINR at each antenna is relatively small. This is because an adaptive array is not needed when the received SINR is large. For example, for maximal ratio combining with two antennas, an average received SINR at each antenna of 11 dB [1] is required for coherent detection of PSK with a 10- 3 BER. For optimum combining the required SINR is less with two antennas and, of course, even lower with more antennas. Thus, the received SINR is much less than 20 dB for all cases of interest. (It is typically between - 5 and 5 dB.) A small received SINR affects array operation as follows. First, if the power of an interfering signal is more than 20 dB below the desired signal's power at an antenna, the array need not track the interfering signal at that antenna because it has a negligible effect on the output SINR. Second, if the power of an interfering signal is more than 20 dB higher than the desired signal's power at an

347

antenna, the array need not track the desired signal at that antenna because the resulting weight for the antenna will be almost zero. Thus, because the received SINR is small in the systems where the LMS adaptive array is practical, a 20 dB dynamic range is adequate. Note that if the received SINR is large' (e.g., greater than 20 dB, as in a lightly loaded system), the LMS adaptive array will have the same performance as maximal ratio combining. 3) Reference Signal Generation and Modulation Technique: The LMS adaptive array must be able to distinguish between the desired signal and any interfering signals. This is accomplished through the use of a reference signal as discussed in Section V-AI). The reference signal must be correlated with the desired signal and uncorrelated with any interference. A reference signal generation technique that allows for signal discrimination is described in [12] and involves the use of pseudonoise codes with spread-spectrum techniques. To generate the spread-spectrum signal the pseudonoise code symbols, generated from a maximal length feedback shift register, are mixed with lower speed voice (data) bits, and the resulting bits are used to generate a PSK signal. The code modulation frequency is an integer multiple of the voice bit rate, and this multiple is defined as the spreading ratio k. The reference signal is generated from the biphase spread-spectrum signal using the loop shown in Fig. 10. The array output is first mixed with a locally generated signal modulated by the pseudonoise code. When the codes of the locally generated signal and the desired signal in the array output are synchronized, the desired signal's spectrum is collapsed to the data bandwidth. The mixer output is then passed through a filter with this bandwidth. The biphase desired signal is therefore unchanged by the filter. The filter output is then hard limited so that the reference signal will have constant amplitude. The hard-limiter output is mixed with the locally generated signal to produce a biphase reference signal. The reference signal is therefore an amplitude scaled replica of the desired signal. Any interference signal without the proper code has its waveform drastically altered by the reference loop. When the coded locally generated signal is mixed with the interference, the interference spectrum is spread by the code bandwidth. The bandpass filter further changes the interference component out of the mixer. As a result, the interference at the array output is uncorrelated with the reference signal. Thus, with spread spectrum, a reference signal is continuously generated that is correlated with the desired signal and uncorrelated with any interference. Furthermore, since pseudonoise codes are used, every mobile can be distinguished by a unique code. Unfortunately, spread spectrum increases the biphase signal bandwidth by a factor of k and therefore increases both the total cochannel interference power and the number of interferers in cellular mobile radio. For example, with frequency reuse in every cell, the cochannel interference power and the number of interferers from surrounding cells are increased by factors of k and 2k -1,

ARRAY OUTPUT ADAPTIVE ARRAY

REFERENCE SIGNAL

Fig. 10. Reference signal generation loop with the adaptive array. When the desired signal is a biphase spread-spectrum signal, the reference signal is correlated with it but not with any interference.

respectively. This increase in interference power is canceled by the processing gain of spread spectrum, but the increased number of interferers degrades the performance of the LMS adaptive array. Furthermore, 2(k -1) cochannel interferers are now present within the desired mobile's cell. Thus, even with a small spreading ratio (e.g., 5 or less) the performance of the LMS adaptive array with the biphase spread-spectrum signal can be worse than that of maximal ratio combining, making the LMS system impractical. The bandwidth increase with spread spectrum and its associated problems can be overcome in the following way. The biphase spread-spectrum signal is combined with an orthogonal biphase signal modulated by the voice bits only (see [13]). The data modulation rate of the orthogonal biphase signal is the same as the code modulation rate of the biphase spread-spectrum signal. The resulting fourphase signal therefore has a bandwidth determined by the data rate only, i.e., the bandwidth is not increased by the spreading ratio. Furthermore, a reference signal for the four-phase signal can be generated from its biphase spread-spectrum signal component using the loop described earlier. As shown in [14], the performance of the LMS adaptive array with the four-phase signal is close to that with the biphase signal. Therefore, with this system, we can generate a reference signal without any increase in interference power or the number of interferers and achieve an improvement with an LMS adaptive array close to that for optimum combining which is shown in Sections III and IV. We now describe the modulation technique in detail by describing three possible ways to modulate the four-phase signal. The simplest technique is for the voice bits to modulate only the orthogonal biphase signal. The biphase spread-spectrum signal then contains the code plus data bits for transferring information from the mobile to the base station. With this first technique, the signal bandwidth corresponds to the voice bit rate r (e.g., 32 kbitsz's). However, the energy-per-bit-to-noise (interference) density ratio Eb / No is half that of a biphase signal. Thus, the improvement with an LMS adaptive array is 3 dB less than that shown in Sections III and IV. A data channel is also available, however, with an r/ k data rate. Furthermore, since the Eb / No for the data bits is k times that for the voice bits (because of the spread spectrum), the BER for the data bits is very low. If a data channel is not required, then voice bits can

348

TABLEII

replace the data bits. With this second technique, the voice bits are split into two channels, one modulating the biphase spread-spectrum signal and the other modulating the orthogonal biphase signal. The bit rate for the latter channel is k times that for the biphase spread-spectrum signal. The signal bandwidth is reduced by k I( k + 1) as compared to the first technique. However, the Ebl No of the voice bits on the biphase spread-spectrum signal is k times that on the orthogonal biphase signal. Through appropriate coding techniques, this difference can be used to improve the overall BER. We can equalize the BER for both channels by decreasing the power of the biphase spread-spectrum signal by II k. With this third technique the Ebl No for the voice bits is just k /tk + 1) times that for a biphase signal. For example, with k equal to 5, the improvement with an LMS adaptive array is 0.8 dB less than that shown in Sections III and IV. Table II summarizes the above results for the three modulation techniques. A block diagram of the four-phase signal generation circuitry for the three modulation techniques is shown in Fig. 11. The code symbols of duration ~ are mixed with either voice or data bits of duration kA. The resulting symbols modulate a local oscillator to generate a biphase spread-spectrum signal. As shown in the lower portion of Fig. 11, voice bits, also of duration ~, modulate the local oscillator signal shifted by 90° to generate the orthogonal biphase signal. This signal is then combined with the biphase spread-spectrum signal to obtain the four-phase signal. By adjusting the biphase spread-spectrum signal level with fJ and modulating this signal with either voice or data bits, we can generate any of the three four-phase signals listed in Table II.

FOUR-PHASE SIGNAL PARAMETERS FOR THREE MODULATION

TECHNIQUES IN AN Technique No.

Relative Biphase Silnal Powen

LMS ADAPTIVE ARRAy SYSTEM

Spread-Spectru m Biphue Sian_l Bit a Information E,,/N o b Biu Rile

Ortholonal Bipbuc Silnal Information Bit Bill Rate

1

1:1

Data

r fk.

2

1:1

Voice

rj(1e +1)

k /2

Voice

3

ljle :1

Voice

rj(1e +1)

lej(1e +1)

Voice

a The code modulation rate is b Relative to biphase signals.

le/2

Voice

E"INo b

r

0.5

Ie

(kF)r

O.S

(k+1 )r

k/(Ie+l)

k times the bit rate.

CODE

------.,,+ ) - - - - - - - - e . ( Y l - - - - l (~)

DATA

OR

of.

VOICE BITS

(A.6)

+

FOUR - PHASE

I,.......---.. SIGNAL

VOICE BITS

(td

Fig. 11. Block diagram of the four-phase signal generation circuitry for the LMS adaptive array. A biphase spread-spectrum signal, modulated by code symbols plus data or voice bits, is combined with an orthogonal biphase signal, modulated by voice bits, to generate the four-phase signal.

B. Base -to - Mobile Transmission

As we have shown, the LMS technique can significantly improve signal reception at the base station. This improvement is, of course, also desired at the mobile. However, since there are many more mobiles than base stations, it is economically desirable to add the complexity of the LMS technique (particularly multiple antennas) only to the base stations. Adaptive retransmission with time division (1], [9] can be used to improve reception at the mobile with multiple base station antennas only. With adaptive retransmission, the base station transmits at the same frequency as it receives, using the complex conjugate of the receiving weights. With time division, a single channel is time shared by both directions of transmission. Thus, with the LMS technique, during mobile-to-base transmission the antenna element weights are adjusted to maximize the signal-to-noise ratio at the receiver output. During base-to-mobile transmission, the complex conjugate of the receiving weights are used so that the signals from the base station antennas combine to enhance reception of the signal at the desired mobile and to suppress this signal at other mobiles. Therefore, by keeping the time intervals for transmitting and receiving

much shorter than the fading rate (e.g., transmitting in 10 bit blocks), we can achieve the advantages of the LMS technique at both the mobile and the base station. With adaptive retransmission using the LMS technique, each base station transmits in a way that maximizes the power of the signal received by the desired mobile relative to the total power of the signal received by all other mobiles. Thus, at the mobiles, interfering base station signals are suppressed and the improvement in the performance with the LMS technique as compared to maximal ratio combining should be similar to that at the base stations. The actual improvement for a given mobile, however, depends on the interference environment of every base station. Because of the complexity of the analysis, we will not study this improvement in detail. It should be noted, though, that for base-to-mobile transmission, spread spectrum on the signal is not required because a reference signal is not generated at the mobile. Therefore, without the degradation with the modulation scheme in the mobileto-base transmission (see Section V-A-3), the BER at the mobile may be lower than that at the base station.

VI.

SUMMARY AND CONCLUSIONS

In this paper we have studied optimum combining for digital mobile radio systems. The combining technique is optimum in that it maximizes the output SINR at the receiver even with cochannel interference. We determined the BER performance of optimum combining in a Rayleigh fading environment and compared the performance to that of maximal ratio combining. Results showed that with cochannel interference there is some improvement over maximal ratio combining with only a few receiving antennas, but there is significant improvement with several

349

antennas. With optimum combining, the typical cellular system was seen to have greater margins and require fewer antennas than with maximal ratio combining. Finally, we described how optimum combining can be implemented in mobile radio with LMS adaptive arrays. Thus, we have shown that optimum combining is a practical means for increasing the channel capacity and performance of digital mobile radio systems. REFERENCES

[10] Y. S. Yeh and D. O. Reudink, "Efficient spectrum utilization for mobile radio systemsusingspace diversity," IEEE Trans. Commun., vol. COM-3D, R. 447, Mar. 1982. [11] T. W. Miller, ' The transient response of adaptive arrays in TDMA systems," Electrosci. Lab., DeP. Elec. Eng., Ohio State Univ., Columbus, OR, Rep. 4116-1, p. 287, June 1976. [12] R. T. Compton, Jr., "An adaptive array in a spread-spectrum communicationsystem," Proc. IEEE, vol. 66, p. 289, Mar. 1978. [13] J. H. Winters, "Increased data rates for communication systems with adaptive antennas," in Proc. IEEE Inter. Coni. Commun., June 1982. [14] - , "A four-phase modulation system for use with an adaptive array," Ph.D. dissertation, Ohio State Univ., Columbus, OH, July 1981.

[1] W. C. Jakes Jr. et al., Microwave Mobile Communications. New York: Wiley, 1974. [2] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. . [3] B. Widrow, P. E. Mantey, L. 1. Griffiths, and B. B. Goode, "Adaptive antenna systems," Proc. IEEE, vol. 55, p. 2143, Dec. 1967. [4] S. R. Applebaum, "Adaptive arrays," IEEE Trans. Antennas Propagat., vol. AP-24, p. 585, Sept 1976. [5] B. Widrow,1. McCool,and M. Ball,"The complex LMS algorithm," Proc. IEEE, vol. 63, p. 719, Apr. 1975. [6] C. A. Baird, Jr. and C. L. Zahm, "Performance criteria for narrowband array processing," 1971 Coni. Decision Contr., Miami Beach, FL, Dec. 15-17, 1971, p. 564. [7] V. M. Bogachev and I. G. Kiselev, Optimum combiningof signals in space-diversity reception," Telecommun. Radio Eng., vol. 34/35, p. 83, Oct. 1980. [8] P. Bello and B. D. Nelin, "Predetection diversity combining with selectively fading channels," IRE Trans. Commun. Syst., vol, CS-10, p. 32, Mar. 1962. [9] P. S. Henry and B. S. Glance, "A new approach to high-capacity digital mobile radio," Bell Syst. Tech. J., vol. 60, no. 8, p. 1891, Oct. 1981. U

350

On Optimum Combining at the Mobile RODNEY G. VAUGHAN,

Abstract-Optimum combining for diversity antennas at the mobile is discussed. Effectively, the aim is to add the wanted signal vectors in a maximum ratio sense, while interferers are weighted so that their resultant is in a permanent deep fade. Even if there are not enough degrees of freedom available to accomplish this fully, an optimum solution can still be found. Many interpretations from conventional array technology do not apply to the mobile communications case and the mechanism of optimum combining for array branch signals rather than discrete spatial signals is reviewed. Physical interpretation of the formulation is emphasized tbroughout. Problems with the adaptive algorithm and its implementation are also identified. Sample matrix inversion is shown to be a likely algorithm to apply in vehicular mobile communications receivers. A worst case example gives an idea of the required computation rates.

MEMBER, IEEE

z r-'" t' ""'---X-" ~ \

\

\

\

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I

\

I

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I. INTRODUCTION

S

IGNAL COMBINATION in mobile communications is generally discussed (cf. [8], [9]) in terms of pre- and postdetection classes of selection, switched, equal-gain, and maximum ratio. To date, adaptive techniques have not been extensively discussed in the area of mobile communications. In the antenna literature, the interpretation of the adaptive techniques revolves around antenna patterns. In particular. the adaptation of nulls toward interferers has been the explanatory mechanism, which accounts for terms such as "null steering." The wanted signals and interferers are considered as discrete. usually resolvable, points in real space, as indicated in Fig. 1. This is not the case in a typical mobile communication scenario, where wanted sources and interferers easily outnumbered the degrees of freedom of the diversity array and, furthermore, are distributed and generally unresolvable, as suggested by Fig. 2. From the point of view of adapting antenna patterns, the situation appears impossible. Lee [9, p. 451] brietly discusses adaptive techniques for mobile communications but bases the discussion on antenna patterns and interferer directions. Bogachev and Kiselev [3] seem to be the first to have discussed optimum combining with respect to (space) diversity antennas. They derive curves for the probability of the signal-to-noise ratio (SNR) in the presence of Rayleigh fading. Winters [17] presents a good discussion of optimum combining at the base station and also provides simulation results. For the returns offered by optimum combining in the mobile communications case, the reader is referred to Winters' article. The best improvements (over other combining methods) occur when the interference Manuscript received April 1, 1987; revised August 20, 1988. The author is with the Physics and Engineering Laboratory, Department of Scientific and Industrial Research, P.O. Box 31313. Lower Hutt, New Zealand. IEEE Log Number 8927107.

0)

WANTED

88

UNWANTED

Fig. 2. Multipath scenario for array at mobile. Wanted signal and interferers arrive from many directions. Array pattern. after convergence to optimum combining solution, does not offer meaningful interpretation mechanism as in simple scenario case.

levels are high relative to the wanted signal, and when the number of diversity antenna elements is large. As far as the author is aware, there is no discussion of optimum combining applied at the mobile, where, as at the base station, it is necessary to consider branch signals rather than discrete spatial sources. The application is in high-density cellular systems where the cochannel interference is high. A theoretical worst case occurs in a corner of a hexagonal cell where the average interference power is less than 5 dB below average signal power [19]. Local shadow effects such as caused by large buildings can make this figure worse still, and Winters [17] suggests that a signal-to-interference-plus-noise ratio (SINR) of + 5 dB to - 5 dB is typical. The number of diversity antenna elements should be large, typically five or more, to obtain significant improvement (i.e., several dB in SINR) over maximum ratio combining. Glance and Greenstein [5] discuss up to six base station elements and Yeh and Reudink [19] go as far as discussing a system with 20-element mobile antennas with 24element base station antennas, each using predetection maxi-

Reprinted from IEEE Transactions on Vehicular Technology, Vol. 33, No.4, pp. 181-188, November 1988.

351

mum ratio combining. A disadvantage of optimum combining at the mobile rather than at the base station (using adaptive retransmission) is that the cost and complexity of implementation at every mobile is much greater over-all than that of implementing only at each base station. An advantage of many-branch systems at the mobile, however, is that the antennas can be realized in a compact manner. At the base station, even a six-element array becomes very expansive. In the following, the mobile diversity antenna is treated as an array antenna. The terminology is also array antennaoriented. For example, the multipath environment of mobile communications is referred to as a multiple source, or distributed source scenario since the distribution of sources effectively approaches a continuum in urban environments. The scenario is referred to as stationary, which is here taken to mean that its statistics are unchanging with time and the position of the mobile. In some sections, the scenario is referred to as static, indicating that the source positions are constant relative to the mobile. A real-world mobile scenario is rarely stationary or static, but the latter simplification does not seem too drastic for the short periods over which each optimum array solution is sought. Section Il contains the basic complex formulation for the array signals. The differences between the more familiar conventional array case and the mobile communications case are discussed with specific attention given to the physical interpretation of the formulation. Section III follows an approach similar to that in Section II but covers the signal combining aspects. The optimum antenna array (specific antenna elements are not within the scope of the article) is addressed in terms of conventional array parameters. Section IV contains a brief look at adaptive algorithms for the mobile communications case, and Section V looks at aspects of implementation. A worst case example gives some idea of the computational requirements and the lirnitations of adaptive algorithms. II.

source is defined as a single point source; any incident signal

which is derived (e.g., diffracted, reflected, or delayed) from the point source, is considered as a separate (point) source. In the mobile communications case, N is limited to less than, say, six. M is effectively unlimited since the sources are considered distributed. Many of the M sources bear the same signal because of the multipath. If there are P different signals (one wanted signal and P - 1 interferers) , then each of the P signals can be considered randomly allocated to many of the M sources. With the time factor ei wt understood, the complex envelope of the signal conveyed by the nth branch from the mth source is (1)

where mm(t) is the signal modulation of the mth source and

(j)nm is the carrier phase of the mth source, in the nth branch.

The total signal in the nth branch is thus .W

xn(t) = }:; xnm(t).

(2)

m=l

is real and represents the signal amplitude (or rather its mean over the modulation) in the branch from a particular source. The static scenario is characterized by the Q and cP being independent of time. Some physical insight is offered for the mobile communication case if a and cP are thought of as functions of distance moved by the mobile. After the antenna has (hopefully) adapted to a set of Q and (j), the mobile has moved, and adaptation to a new set of Q and q, is required.

Q nm

Array Branch Signals Define the total (sum over all sources) branch signals in the usual way

BASIC FORMULATION

The distributed source scenario around the mobile in an urban environment must be considered unchanging for each adaptation cycle of the combining algorithm. With this in mind, this section considers constant (in location) sources only. In particular, signal correlation measurement intervals or a sequence of them-to establish the covariance matrix with sufficient accuracy during convergence-are taken over a static scenario. In practice, this will not be the case (unless the mobile and its surroundings are still): the adaptation algorithm will be chasing a changing solution, and its performance will be correspondingly degraded. If the adaptation is fast compared with the rate of change of the sought solution, the degradation will not be too much. Some practical points are discussed in Section V. Where possible, the notation here follows that of Hudson [7]. However, the multipath scenario of mobile communications calls for a different formulation, and some designations common to this article and Hudson's are not interchangeable. The array antenna has N branches with M incident signals from M sources in the field of view of the array. Here, a

352

(3)

and a vector of the RF (or IF) phase terms weighted by the amplitude of the source signal in the branch

T m are the vector contributions of the mth source to the array. When only the phases are included (Qnm = 1, all n) Tm = Sm is the source vector to the mth source, a term from conventional array technology. Now let all the sources be included in the total weighted source vector

(5)

Note that a column of Q corresponds to a given source signal in all branches and a row corresponds to all the source signals in a given branch. Finally, define a vector containing the source modulations

AT = [ml (t) m2(t) · .. m.w-(t)]

(6)

products occur in R. Denote a source modulation correlation

so that X(t)

= QA (t)

(7)

is the vector formulation of (1). In the mobile communications case, let there be P different (uncorrelated) signals, with each signal conveyed by subsets of the M sources. The subsets contain MJ, M 2 , •• " M; sources. Now

Pmq

= m :(t)mq(t)

(15)

in which the normalization is understood. Obviously, PmQ = P * and Pmm = 1. Now the inner term of the covariance matrix (14) is

of

Pl2

pi2

A*AT=

••• PIM

1

(8)

(16)

i= I

and (2) can be written as xn(t) =

L L anmmM;(l)ej(j>nm P

,Wi

(9)

i=l m= I

L anMjmAfi(t)eJcPnMi p

~

1=

(10)

I

where the summation is over i for all Mi. In (9), the inner summation is over each source bearing the ith signal. and the outer summation is over the difference source subsets. i.e., over the different signals. The subscript M; serves as a reminder that the (subscripted) signal quantity is derived from several sources rather than an individual source in the scenario. Equation (10) shows that each branch can be considered to support P signal-bearing vectors, each of which correspond to the vector sum of the signals received from the appropriate sources. This is an important maxim as it allows branch signals to be combined such that certain signals are maximized and/or others minimized. The latter effect is the extension of maximum ratio combining to optimum combining. Equation (10) is used in the following in a vector formulation of the covariance matrix for the mobile communications case.

In the case where all the sources bear different (uncorrelated) signals, A *A T becomes the identity matrix. In the mobile communications case, the off-diagonal elements will be randomly placed (other than complying with the Hermitian construction) ones or zeros. Each source adds one to the rank of A *A T which makes it of mathematical interest only except in simple scenarios, i.e., those with a manageable number of sources. In the mobile communications case, not even the rank of A * A T is known. Inclusion of the weighted source vectors is facilitated by denoting

=

7r nmqpcPqpnm'

(Q)nm is the nmth element of Q. cPqpnm is the phase difference between the pth source in the qth branch and the mth source in the nth branch. 1r nmqp has the dimensions for power and denote 1r nmnm = 1r nm . From (10), (14), and (18),

L p

(R)nq

=

1r nM;qlvf;cPqMinMi

and (20)

The output signal is usually written

= WT X(t)

i= I

(11 )

where W is the column vector of weights. The output power is the Hermitian form (12)

in which the covariance matrix is defined as R=X*(t)XT(t)

(19)

i= I

The Covariance Matrix

y(t)

(18)

(13) (14)

The overbar means time averaging in the presence of the static scenario. It is evident that R must contain all cross products of both the weighted space vectors and modulations. Some physical insight can be obtained by examining where these cross

353

The off-diagonal elements are seen to be a summation of vectors, each vector corresponding to one of the P signals from one of the subsets M, of the M incident sources. Little can be remarked about simplifying the physical interpretation of R, its form being too involved. However, some progress is possible by performing a similarity transformation of the desired-signal-only covariance matrix (see the later discussion of prewhitening, below). In practice, the covariance matrix will also contain terms due to noise. The noise is uncorrelated between branches (any unwanted signal that is correlated between branches is defined as interference) so that only the principle diagonal in the covariance matrix becomes augmented. This is convenient because it ensures that R is always positive definite and the existence of R - I is assured. Conversely, if the noise level is very low relative to the source signal levels, numerical problems may sometimes arise in seeking R- I from measured

The optimality criteria can be expressed as the Wiener solution

values of R. Techniques for ill-conditioned R include taking the generalized inverse instead of the conventional inverse. III.

MWopt=kTZ,o

THE OPTIMUM ANTENNA

In public service mobile communications. the average (over several fades) power of interferers would rarely be above that of the wanted signal. In each antenna branch or a single port antenna, however, the instantaneous level of the interferer could be a few tens of decibels above the wanted signal owing to the Rayleighlike fading. The optimum antenna (here the antenna is regarded as the elements, weights, and summing network) is defined to maximize the SINR. If covariance matrices R, and M are defined to embody the wanted signals only and the interference-plus-noise, respectively, then the adaptive algorithm seeks an optimum weight vector Wopt which is well-known to satisfy

where k is a scaling constant ( =golk ' ) which has no effect on the SNIR. If M is invertible (which can be arranged, if necessary, by adding noise), then the scaled weights of the optimum antenna are given by (28)

Equations (27) and (28) are the generalized version (applicable to the mobile communications case) of the results for conventional arrays in which TMO is replaced by roSo. The difference is that the conventional array employs purely phase weighting. Here, both phase and amplitude weighting are necessary. In practice, o must be known a priori, or else it is measured, as is the covariance matrix. In mobile communications, a priori knowledge of 110 is not possible and instead, knowledge of a characteristic of the wanted signal is used to estimate To measu~e M, the wanted signal must first be removed from each branch. Measurement of R, which includes the wanted signal, is clearly more practical but its unentangled inclusion in the optimum solution of (28) requires special conditions. Specifically, if

Ttt

(21)

where k' is a constant which is, in fact. the maximum SINR. Denote a weighted space vector for the wanted source To; then

TZt .

(22)

In the mobile communications case, the wanted signal and interferers are distributed across many source vectors and the small size of the array will normally prevent the isolation of only one of these, the To above. However, R, can always be expressed as a dyadic when the sources are considered branch sources rather than spatial sources. Each branch source is the summation of the contributions from the actual sources which bear the same signal. Equation (19) depicts the situation. The weighted source vector of (4) becomes reformulated as branch signals, (23)

in which M;

TnM ; = ~ t;

(27)

(24)

(29)

then (21), (26), and (29) can be combined to yield the optimum scaled weights solution

Wopt=R-ITZto

(30)

which is known to be equivalent to (28) (e.g., Hudson [7, p. 74]) but features the more readily available total covariance matrix. The validity of (29) is the key resource: the desired signal and the interferers must be uncorrelated (Pmp = 0, m =1-= p) or else orthogonal in the sense

TTMm T*Mp =0 ,

m=l

m e p,

(31)

In the conventional array case, the analogy of (31) is the spatial orthogonality relationship

and then the form (25)

ST·~*=O fTtI p ,

is a valid representation even in the case of a spatially distributed set of (e.g., wanted) signal sources. Substituting (25) into (21) gives

k' MWopt = TttoT~oWopl =goTt,o

(26)

where go is a constant representing the amplitude gain of the array to the wanted signal or to the wanted source in the case of an isolated source. In the latter case, the right side of (26) is go1roSri . So is called the steering vector, and 1ro is the total power (in all branches) of the desired source, quantities familiar from the conventional array case.

m e p,

(32)

which is sought in null steering by retrobeams (eigenbeams for subtraction from the quiescent array pattern). In the mobile communications case, eigenbeams for preprocessing can be formulated mathematically (only at the expense of juggling signal powers since the TM; are not orthonormal), but the patterns are not in real space.

Prewhitening Preprocessing can offer increased convergence rates for adaptive combiners in conventional arrays. The power of the prewhitening is in the equalization and decorrelation of all interference (-plus-noise) in each branch. The combining algorithm can then just maximize the output power. White [15]

354

antenna. For example, in a conventional array, the modulus part of complex weighting allows the suppression of array space near-field sources while maintaining a response to farfield sources in the same direction. In the mobile communications case, the N complex weights provide 2N nOF, but each interferer must be dealt with in both amplitude and phase. Cancellation of an interferer thus requires two real OOF. In the context of mobile communications, it would be appropriate to consider the complex weighted N-branch array to have N complex OOF, each available one of which can be used to suppress at least one interferer.

discusses adaptive beam-forming in simple scenarios but does not consider transient performance. Accelerated convergence techniques (see, e.g., Monzingo and Miller [12, p. 188-192]) can help considerably for antennas with a relatively small number of elements, as would normally be the case in mobile communications. Preprocessing, in particular Gram-Schmitt orthogonalization (e.g., Monzingo and Miller [12, pp. 364383]), seems to offer considerable gains, but again, the scenario is assumed to be unchanging during adaptation. Since the rate of convergence is important in mobile communications, an investigation of the applicability of preprocessing is worthwhile. Since M is always positive definite, it can be uniquely factored using the Hermitial matrices CC = M, and since M is of full rank, C- 1 exists, so C-IMC- 1 =/.

Array Patterns The array pattern for a conventional array is defined as the amplitude gain for a given set of weights

(33)

g({}, ¢) = WTs

The operation is seen to decorrelate the interference signals between all branches (e.g., Hudson [7, p. 62]). The transformed covariance matrix of the desired signal is

B=C- IR sC- l •

(34)

Maximizing the SINR is now just a question of finding the maximum eigenvalue of B. The transformation from R, to B is between array space and eigenspace. However, the eigensources cannot be related to array space sources as is possible with conventional arrays. In fact, the eigensources are difficult to relate to the branch sources (see (4)) as well. The problem again lies with the weighted source vectors TM; not being orthonormal. In the conventional array case with a simple static scenario featuring M < N spatial sources, the array with a beam-former C- 1 provides a (transformed) covariance matrix which is diagonal and contains the powers of each of the sources as its element values. In practice, much effort is required to implement the beamformer of changing scenarios. (For constant scenarios, the beam-former can be hard-wired; indeed, most applications are for this case.) Generating C-I is not trivial because C is the square root of M. M must first be measured, its eigenvalues found numerically, their square roots taken (positive radicals are understood in the unique factorization of M), and then assembled into a diagonal matrix. Finally, the inverse taken to yield C- 1. The continuous measurement and processing required to implement the prewhitening is formidable, and any acceleration of convergence as a result of preprocessing will be well mitigated by the time taken for the preprocessing. Moreover, it is shown in Section V that the gradient search techniques are not very suitable for single-receiver systems in the vehicular mobile communications case.

(35)

where S provides the directive information. If the scaled weight vector corresponds to maximizing the array amplitude gain towards a source 51, then (in the absence of interferers) W = S and S I is the steering vector. If a second source 52 is introduced, the amplitude gain toward 52 using weights W is given by

i,

g({}, cP)=S~5~.

(36)

The array pattern then is a mapping of an inner product of 51 and S2' S1 could be referred to as a testing source, in keeping with the position of a probe antenna in pattern measurements. The nulls of g «(J, q,), if they exist, occur when the source 52 does not affect the array output, i.e., when the sources are spatially orthogonal,

Degrees of Freedom (DOF) Clearly, an N-branch conventional array with N phase-only weights has N DOF; or if the weights are complex, there are 2N DOF. In general, not all the DOF are available for suppression of an interferer, and the number available depends on both the scenario and the configuration of the array

355

(37)

Array space patterns are difficult, if not impossible to interpret for the mobile communications case. A signal amplitude gain quantity can be defined analogously to (35) (cf. (26)) by g = T'f.,; W. TM; carries the "directive" information but the direction is not in real array space. A test weighted source vector could be defined as above (cf. S2 above), but again there seems little point because the inner product analogous to (35), T'f,T2 , does not readily offer a mapping which is physically meaningful. The traditional use of array patterns to illustrate antenna adaptation does not apply to mobile communications. IV.

ADAPTIVE ALGORITHMS

The adaptive algorithms well-known from conventional arrays are all applicable for complex rather than phase-only weights. Some caution must be exercised with some conceptual and explanatory interpretations as noted in the previous section. The suitability and implementation feasibility of the various algorithms in the mobile communications case is the remaining issue. Ricardi [14, p. 6.2] suggests three classical

methods, under which all adaptive algorithms can be classified. These are power inversion, least mean squares (LMS), and direct sample matrix inversion (SMI). SMI is conceptually the simplest technique. The covariance matrix and weighted steering vector are estimated (in this case) and the weights are calculated directly from (38) where the hat denotes estimate. This offers the quickest technique for finding the weights and is independent of the makeup of R, as long as it is well-conditioned with respect to the word size of the implementation. The covariance matrix estimate is formed with K samples 1 K R(k)=- }: X*(k')XT(k')

K

(39)

k'=l

where X(k) is a sample of X(t) at the kth sample time. The estimate for the weighted steering vector is similar (cf. Monzingo and Miller [12, p. 300]):

where d(k) are the samples of d(t), which is a reference signal well-eorrelated with the desired signal. In some ways, SMI is not truly adaptive because it is an open loop system. However, it is considered adaptive in the sense that it will produce a new solution for a new scenario. As in any open-loop system, the accuracy of the implementation must correspond to the desired accuracy of the solution. SMI is discussed further at the end of the subsection. The power inversion and LMS algorithms employ a closedloop system that works to converge towards the solution. Power inversion is simply a minimization of output power, often with a constrained gain to the wanted signal (or spatial source, in simple scenarios). The weighted steering vector needs to be known a priori so the power inversion algorithm is not applicable to the mobile communications case. When the weighted steering vector is estimated with the aid of a reference characteristic of the wanted signal, the algorithm effectively becomes the LMS technique. Despite their differences in implementation, theLMS and power inversion techniques are very similar. Both algorithms are steepest descent gradient search techniques. The convergence rate of the gradient search techniques is very complicated to predict accurately, except in simplified situations. The convergence rate is known to be sensitive to the spread of eigenvalues in the covariance matrix. As far as the author is aware, no simulations have been reported for complex vectors are weights. Simulations for conventional arrays in simple scenarios usually involve very low signal-to-interference ratios, a situation which would be unlikely in the mobile communications case. Winters' [17] base station simulations include Rayleigh fading but assume complete convergence (no combiner losses) although no transient analysis is included. From conventional array simulations (see, e.g., Monzingo and

Miller [12]) convergence to within 3 dB of the optimum solution rarely occurs in less than several hundreds or even thousands of iterations. The eigenvalue spread of the covariance matrix in these simulations is generally more than would be found in mobile communications, so results will be pessimistic. On the other hand, the interference-to-signal ratio is much higher than would be found in the mobile communications case, which would make the results optimistic. Results from conventional array simulation can only give a rough guide. Some conceptually derived time constants allow some conclusions regarding implementation. An upper limit on the rate of change of the weights is given by the data rate. Winters ([17], p. 152], after Miller [11]) notes that the maximum rate of change in the weights is about 0.2 times the data rate before significant data distortion occurs for phase-shift keyed (PSK) signaling. 1 For a data rate rd = 16 kbits/s, the maximum rate of weight update for PSK signals is then about every 0.06 ms, and the weights cannot change completely within 0.3 ms. For good adaptive performance, it is best to try to attain this rate. In terms of a Rayleigh fading envelope, this upper limit is quite severe. A worst case fading rate, given by a vehicle traveling at, say, 140 kmlh with a 9OO-MHz carrier frequency and omnidirectional antenna, is about 120 fades/so If the ,'branch scenario" (the scenario describing the "positions" of the P signals in each diversity branch) can be assumed uncorrelated at intervals of half of the fading period, i. e. , every 2.1 ms, then there can be a maximum of less than 2.1/ 0.06 :: 35 iterations between independent scenarios. This worst case situation does not look favorable for the gradient search techniques. However, once the weights are close to optimum, the required number of iterations are less, but still well over 30. A dual receiver system appears necessary for the gradient search techniques. One receiver would be for establishing the weights and another, using only periodically updated weights, dedicated to data throughput. The 8MI technique offers a possible single-receiver solution. The processing required can be couched in the traditional units of number of multiplications. The covariance matrix is formed with K independent samples, and clearly, the larger K is, the better the estimate and the more accurate the solution since a static scenario is assumed. For an ensemble average solution to be within 3 dB of the optimum solution for 50 percent of the estimates, Reed et al. [13] suggest K > 2N 3. The usual role of thumb is to take K > 2N. Boronson [2] notes that K > 3N and K > 4N ensures that the solution will be within 3 dB of the optimum for 98 and 99.68 percent of the estimates, respectively. Formation of R(k) and Mo(k) require KN(N + 1)/2 and KN complex multiplications, respectively. Inversion of R(k) and calculation of W(k) require (N3 /2 + N2) and N2 complex multiplications respectively [13]. Thus for a three-element array, there are at least 86 complex multiplications. A real 16-bit multiplication can be performed within about 150 ns with current off-theshelf hardware. For the three-element array, the multiplica-

356

t

I In fact, the weights must respond slower than 0.2 times the data rate. In practice, the weights can be updated at the data rate, but cannot change completely in less than five data bit intervals (Winters [18]).

4. 63

6 .17

18

24

7 .71 30

TIME 1" \

)

9 .16

10. 80

12.3 4

13 .89

1\ . 43

36

41

48

\4

60

DIS TANCE TRAVELLED AT 140 \:mj h - 00

i ~ i:i

20

1&

SO !It IA"PLE NU"UIl:

40

after the processing is complete and the weights are calculated [18] .

~ cm )

V.

60

-' 0

- 10

- 10

tI

iI 1

/

Mea s ur e

"-

We lgh ts set Ca lc ul e t e

and set we t qnt. s

We Ights us ed

I i n tm s in te rva l e r i v ed f rom mea sureeent s he re

~

1 UPDATE CfCL[

Fig. 3. Example of worst case fading (carrier frequency = 900 MHz. vehicle speed 140 krn/h) with maximum weight response rate (for l o-kbit/s data rate and single receiver system) of about every 0.3 ms. At 140 km/h , weights respond about every 1.2 cm . Limitation in manageable dynamic range of signal is illustrated in fade 10 left. where installed weights correspond to "out of date" measurements. Limitation in realizing this with sufficient accurac y. maximum update rate is in estimat ing

Two

tions alone would take about 52 JJ.S, easily within the upper limit of 0 .3 ms for the weight updating . A six-element array requires 504 complex multiplications, taking about 0.3 ms, indicating the need for a second multiplier to maintain the maximum weight update rate. It is probable that an exponential deweighting for the covariance matrix update would be more suitable for tracking the solution . Hudson ([7, p. 125] after Lunde [10], and Monzingo and Miller [12, p. 314]) give formulas for directly updating the inverse of the deweighted covariance matrix and the steering vector, respectively . A fundamental limitation lies with the required measurement times . The K independent samples must be taken within a short enough time that the scenario appears static. On the other hand, the period taken to retrieve the K samples necessarily extends long enough for the correlations to be correctly detected . This places some lower limits on the crosscorrelation bandwidths of the reference signals. If pilot tones are used, for example, a correlation time interval equal to the maximum weight response rate of 0.3 ms calls for pilots to have a theoretical minimum spacing of about 3 kHz . (In practice, there would be a much larger pilot spacing requ ired.) This is rather inconvenient; for eight cochannels, for example, an entire 25 kHz bank is used only for pilots! A point of academic interest is that pilots should be spaced by more than the upper Doppler limit of about 120 Hz ; otherwise , an interferer may often be singled out as the wanted signal and vice versa . If it can be assumed that the weighted steering vector can be estimated accurately in less than about 0.3 ms, then the maximum weight response rate can be realized using SMI. At the worst case fading rate (corresponding to a vehicle traveling at 140 km/h), the response occurs about every 1.2 ern. The measured signals are taken over the preceding interval , or maybe even two intervals before weight update, if slower processing is used . The situation is depicted on a Rayleigh envelope in Fig. 3 . A way around dealing with "out of date" measurements is to delay the received signal and operate on it

IMPLEMENTATION

Winters ' [17] base station implementation proposal uses a spread-spectrum system with an LMS algorithm. It is based on Compton 's [4J description . While the system is claimed to be practical, the transient analysis of the algorithm is not addressed. A possible system could use Winters' spreadspectrum proposal with the SMI algorithm. The spreading factor should be kept modest, not only to keep the processor bandwidth down, but also because the cochannel interference increases with channel bandwidth in a cellular system, and the returns from the optimum combining system diminish . However, the spreading factor must be sufficient to get a pseudonoise sequence completed and preferably repeated within the time that the branch scenario changes, i.e ., about every 0 .3 ms in the worst case . With implementation at the mobile rather than the base station, shorter pseudonoise sequences are possible since there are far fewer base situations than mobiles . For example , for a sequence length of 16 bits with an information rate of 16 kbit/s , a spreading factor of at least three is required to attain a rate of one sequence per 0.3 ms . The implementation of optimum combining is obviously much more complicated and expensive compared to conventional combining . Switched or selection combining are the easiest techniques, especially when there are just two branches. Postdetection equal-gain combining is also easy to implement, since there is no need for cophasing the signals . Postdetection maximum ratio combining is more complicated because an amplitude weighting must be included, and calculating the weights requires measurement of SNR in each branch. Predetection maximum ratio combining requires both amplitude weighting and cophasing of the signals . This demands much extra hardware for each branch. Maximum ratio combining is more often used as a theoretical performance benchmark because much progress is possible in calculating the diversity returns. A loss budget must be introduced for imperfect combining . For example, the system proposal of Yeh and Reudink [19] assumes maximum ratio combining and allows I-dB .. modem loss " for the theoretical output SINR . Optimum combining, in the implementation considered here, requires not only the measurement and weighting hardware for predetection maximum ratio combining , but also powerful digital computation hardware for calculating the weights . Signal digitization would be one of the expensive aspects of an implementation. A possibility which has not received much attention is bandpass sampling, in which the IF (or RF) signal is digitized directly to baseband. The cost of the extra hardware necessary for optimum combining is very high indeed when compared to a switched , selection , or equal-gain system . Nevertheless, integrated packages for this type of signal processing are becoming increasingly available and digital hardware costs are still decreasing . If high spectrum efficiency becomes sufficiently important in cellular mobile systems, then optimum combining of diversity antennas at the mobile, with its significant

357

improvement in the channel capacity compared to conventional combining types, forces itself into consideration for the system architecture. V.

CONCLUSION

Optimum combining for diversity antennas at the vehicular mobile has been discussed in terms of traditional array parameters. The emphasis has been on physical interpretation of the mathematical formulation. A worst case example of a fast-moving vehicle operating with cellular communication frequencies indicates that the LMS, or other truly adaptive algorithms are not particularly suitable owing to the potentially short times available for adaptation. Also, it has been shown that preprocessing is not likely to be useful for accelerating convergence. However, for slower mobile speeds such as pedestrians and/or lower carrier frequencies, gradient techniques become feasible. Using a sample matrix inversion algorithm, the signal-processing requirements are not so severe and can be implemented using current hardware. In a system with conventional frequency divided 25 kHz bands for each channel, the measurement time for estimating the weighted source vector imposes serious limits on the situations for which the optimum combining is useful. Conversely, the rate of change of the scenario (effectively the speed of the vehicle or its immediate surroundings) limits the returns because of the required measurement time for the weighted source vectors. To derive the benefits of optimum combining, it is necessary to use wide-band pilots such as the pseudonoise codes in a spread-spectrum communications system.

digital mobile radio with diversity combining," IEEE Trans. Commun., vol. COM-31, no. 9, pp. 1085-1094, 1983. [6] P. W. Howells, "Intermediate frequency sidelobe canceller," U.S. Patent 3 202 990, 1965. [7] J. E. Hudson, Adaptive Array Principles. London: Peregrinus, 1981. [8] »: C. Jake~, Ed Microwave Mobile Communications. New York: Wiley, 1974. [9] W. C. Y. Lee. Mobile Communications Engineering. New York: McGraw-Hill, 1982. [10] E. B. Lunde, "The forgotten algorithm in adaptive beamforming," in Aspects of Signal Processing, G. Tacconi, Ed. Reidel, 1977. [11] T. W. Miller, "The transient response of adaptive arrays in TDMA systems," ElectroSci. Lab., Dep. Elec. Eng., Ohio State Univ., Columbus, Rep. 4116-1, 1976. (12] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. [13] I. S. Reed, J. D. Mallet, and L. E. Brennan, HRapid convergence rate in adaptive arrays," IEEE Trans. Aerospace Electron. Syst., vol. AES-IO, no. 6, pp. 853-863, 1974. [14] L. F. Ricardi, "Adaptive and multiple-beam antenna systems," in

Proc. Summer School on Satellite Communication Antenna Technology. The Netherlands: North Holland/Elsevier, ch. 6. [15] W. D. White, "Cascade preprocessors for adaptive antennas," IEEE Trans. Antennas Propagat., vol. AP-24, no. 5, pp. 670-684, 1976. [16] B. Widrow, P. E. Mantey, L. J. Griffiths. and B. B. Goode, "Adaptive antenna systems," Proc. IEEE, vol. 55, 1967. [17] J. H. Winters. "Optimum combining in digital mobile radio with cochannel interference," IEEE Trans. Veh. Technol., vol. VT-33, no. [I 8) [19]

REFERENCES

S. Applebaum, "Adaptive arrays." IEEE Trans. Antennas Propagat.. vol. AP-24. pp. 585-598. 1976. [2] D. M. Boronson. "Sample size considerations for adaptive arrays:' IEEE Trans. Aerospace Electron. Syst., vol. AES-16. no. 4. pp.

[I]

446-451, 1980. V. M. Bogachev and I. G. Kiselev, "Optimum combining of signals in space-diversity reception." Telecommun. Radio Eng., vol. 34/35, no. 10, p. 83, 1980. (4] R. T. Compton, HAn adaptive array in a spread spectrum communication system," Proc. IEEE. vol. 66, pp. 289-298, 1978. [5] B. Glance and L. J. Greenstein, ·'Frequency selective fading effects in

[3)

358

3, pp. 144-155, 1984. - - , private communication', Mar. 1988. Y -S. Yeh and D.O. Reudink, "Efficient spectrum utilization for mobile radio systems using space diversity, " IEEE Trans. Commun., vol. COM-30, no. 3, pp. 447-455, 1982.

The Performance of an LMS Adaptive Array with Frequency Hopped Signals

LEVENT ACAR R.T. COMPTON. JR., Fellow, IEEE The Ohio State University Electroxcience Laboratory

The performance of an Ll\-tS adaptive array with a frequency hopped, spread spectrum desired signal and a CW interference signal is examined. It is shown that frequency hopping has several

effects on an adaptive array. It causes the array to modulate both the amplitude and the phase of the received signal. Also, it causes the array output SINR (signal-to-interference-plus-noise ratio) to vary with time and thus increases the bit error probability for the received signal. Typical curves of the desired signal modulation and the time-varying SINR at the array output are presented. It is shown how the array performance depends on hopping frequency, frequency jump size. interference frequency, signal arrival angles. and si2nal powers.

Manuscript received January 19. 1984: revised December 27, 1984. This work was supported in part by the Department of the Navy, Naval Air Systems Command. Washington, D.C. under Contract NOOO 19-82C-0190 with the Ohio State University Research Foundation.

Adaptive arrays based on the least mean square (LMS) algorithm [1] are very effective for protecting communication systems from interference. These antennas can automatically track desired signals while also nulling interference [2]. Methods have been developed for using adaptive arrays with ordinary amplitude modulated (AM) signals [3], binary frequency shift keyed (FSK) signals [4, 5,61, binary phase shift keyed (PSK) spread spectrum signals [4, 7], and quadriphase PSK spread spectrum signals [8]. These techniques have all been demonstrated experimentally. In this paper we study the performance of an adaptive array with another type of spread spectrum signal, a frequency hopped signal [9]. Frequency hopping is a widely used method of spectrum spreading. Its purpose is to make a communication system less vulnerable to interference. For some applications, it may be desirable to combine adaptive arrays with frequency hopped signals to obtain the interference protection of both. However, very little information is available on the performance of adaptive arrays with frequency hopped signals. As we shall show in this paper, frequency hopping has several adverse effects on an LMS array. First, it causes the array to modulate both the amplitude and the phase of the received signal. Second. it makes the output SINR (signal-to-interference-plus-noise ratio) vary with time and hence increases the bit error probability for the demodulated signal. If an LMS array is to be used with frequency hopped signals. these effects must be taken into account in the design of both the array and the signal modulation. In this paper we consider an ordinary LMS adaptive array with continuous feedback loops. We do not consider various modifications of the LMS array (such as weight storage and recall algorithms) that might be used to improve its performance with frequency hopped signals. Our purpose here is to determine when the basic LMS array has problems and to characterize the array behavior with frequency hopped signals. We use a simple model to study this problem. We consider an array with three elements. and we assume the frequency hopped signal has only a few frequencies. Such a model is adequate to illustrate the effects of frequency hopping on the array, and it allows us to explore the interaction between the hopping characteristics and the array transients. Section II of the paper defines the problem and formulates a method for calculating array behavior with frequency hopped signals. Section III describes numerical results based on the technique in Section II. Section IV contains the conclusions.

Authors' current addresses: L. Acar, The Ohio State University,

II. FORMULATION

Electro-Science Laboratory. Department of Electrical Engineering. Columbus. OH 43212: R.T. Compton, Jr., Department of Electrical Engineering, Ohio State University, 2015 Neil Avenue. Columbus, OH 43210.

Consider an LMS adaptive array [1 J with three elements, as shown in Fig. 1. Let the clements be

Reprinted from IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-21, No.3, pp 360-371, May 1985.

359

hopped with a periodic hopping pattern. We suppose the hopping sequence repeats after p hops. We model the desired signal as a CW signal with constant frequency Wh on each hop interval Th - l :5 t < Ti; where h is an integer denoting the hop interval (1 :::; h s p) and This the time at the end of interval h. The duration of hop interval h, TIr - Th - 1 , is called the dwell time. We assume the dwell time is the same for each h. We refer to the separation between two Wit that are adjacent in frequency as the frequency spacing, and we assume the Wh are equally spaced across the band. (Two Wh that occur sequentially in a given hopping pattern are not necessarily adjacent.) We define the hopping frequency iii to be the number of hops per second (the reciprocal of the dwell time), j~ = (T 1 - To) - 1, and the pattern frequency f p to be the number of complete hopping periods (or .. patterns' ') per second ~ i.e. ,fp = (TfJ TO)-l = fH/P, We define the center frequency We of the desired signal to be the arithmetic mean of the hopping frequencies Wh. (The antenna elements in Fig. 1 are assumed to be a half wavelength apart at frequency we') We denote the difference between a specific signal frequency Wh and the center frequency We by .lw".

ARRAY

OUTPUT

i(U

REFERENCE SIGNAL r(t)

Fig. 1

Three-element adaptive array.

isotropic, noninteracting, and a half wavelength apart at the center frequency of the signals. The analytic signal y)(t) from the jth element is mixed with a local oscillator (LO) and then passed through a narrowband filter (NBF). The purpose of the LO and NBF is to dehop the desired signal, as discussed below. The filter output xJ(t) is the input to the jth channel of an LMS processor [1]. This processor multiplies each signal x)(t) by a complex weight w) and then sums the result to form the array output get). The weights wJ in an LMS processor are obtained by correlation feedback loops that minimize the average power in the error signal e(t) [3]. e(t) is the difference between the reference signal ret) and the array output set). The reference signal determines which signals are to be retained in the array output and which are to be nulled. Received signals correlated with ret) will be retained and signals uncorrelated with ret) will be nulled. In practical communication systems, ret) is usually derived from the array output by nonlinear signal processing operations [4-8]. In this paper, we do not address the problem of refernece signal generation. We simply assume ret) to be a signal correlated with the desired signal. Let yet) be a vector containing the element signals.

(5) Finally. we define the relative bandwidth B, of the desired signal to be its total bandwidth divided bv its . center frequency.

max{wh} - min{wh}

To dehop the desired signal. we assume that the LO in Fig. 1 is hopped synchronously with the received desired signal. 1 The LO signal is

(7) where WL is the center frequency of the LO. We assume that WL < We and that the NSF has a center frequency We WL' The bandwidth of the NSF ~ which we denote bv B,-. is assumed smaller than the separation between . adjacent hopping frequencies. All the NBFs in Fig. 1 are assumed to be identical. With this model, the vector YJ(t) is

(1)

and let X(t) be a vector containing the signals at the LMS processor input (2) where T denotes the tranpose. We assume below that the array receives a desired signal and an interference signal, and that there is also thermal noise in the element outputs. Thus, yet) and X(t) may be written yet) = Yd(t) + Yi(t) + Yn(t)

(6)

Yd(t)

=

AJ

eXP{j f(Wc + ~Wh)t + 4JJ)} ] exp{j[ (we + ~Wh) it - Tel) + wdl} [ exp{j[(w c + UWh)(t - 2Td) + l1J J 1}

,

(3)

and (4)

'We do not address the issue of timing synchronization here. Also. if the desired Signal amves from a direction other than broadside. its hopping will have a different timing on each element. because of the propagation time delay between elements. Thus. strictly speaking. the LO hopping cannot be synchronized exactly with the desired signal hopping on every element. However. we assume the interelernent propagation time to be very small compared with the dwell time. In this case, differences in desired signal timing on different elements may be neglected.

where Yd(t), Y,(t), and Yn(t) are vectors containing the desired, interference, and thermal noise signals from the elements, and Xd(t), Xi(t), and X'l(t) are the corresponding vectors at the processor input. Now let us define the signals and determine these vectors. First, we assume the desired signal is frequency 360

where Ad is the desired signal amplitude, 4Jd is the desired signal carrier phase angle, and Td is the propagation time delay between two adjacent elements,

Finally, we assume the element signals Yj(t) contain white, Gaussian noise. After mixing and filtering, the iJ(f) then contain narrowband Gaussian noise signals. The noise vector Xn(t) is

(9)

(19) where the ii/t) are zero mean, Gaussian random processes, each with variance tr', We assume the flj(t) are statistically independent of each other and of l/Jd and l/Ji' Once the signal vector X(t) is defined, the array weights may be found as follows. The weights satisfy the system of equations [1],

ad is the desired signal arrival angle with respect to

broadside. (8 is defined in Fig. 1.) $d is assumed to be a random variable uniformly distributed on (0, 21T). After dehopping, the desired signal vector Xd(t) is }(d(t) == Adexp{j[(Wc-WL)t+WJJ} Ud(h),

dW(t) dt

where

- - + k¢(t) ( 11 )

== kS(t)

(20)

where Wet) is the weight vector

and d(h) is the desired signal interelement phase shift during interval h,

(21)

( 12)

k is the LMS loop gain, ¢(t) is the covariance matrix,

Next, consider the interference. Suppose the interference is a CW signal at frequency Wi incident on the array from angle Sf' The interference signal vector f,(t) is

where

Iw, -

(23)

Then, from (23), Set) is ( 17)

Set) == A,A d VI (h).

with 4>; the interelement phase shift,

(26)

Also, we note that (t) and Set) depend only on h, because they are constant during each hop interval. Hence we denote their values during interval h by (h) and S(h). Thus, for one period of the hopping pattern, Wet) satisfies the sequence of equations,

( 18)

Note that the frequency hopping has converted the CW interference signal at the antenna element into a pulsed signal at the processor input. 361

d:?) + k-

; Sr = I :

. .J'"

co a: co

0"

cr:' ncr:

~=r

- - - - - -- - - - -- - -- - - - .. - ._. :. .~ --;'-:-:- --.....-.. _ ~ _ .i. _ ....:... _

---

ec w

... '", -I

g. 7.

0

1

2

3

~

PATTERN FREOUENCY . HZ XIO"

5

-I

6

Envelope variation versus pattern frequency . e" = 15°. ll, 45°. ~., = 6 dB .~. = 40 dB. p = 2 . '», = w, .

Fig. '}.

=

the way the desired signal envelope changes as the hopping frequency varies. In particular, the time at which am m occurs is in one hopping interval for low values of [p and in the other hopping interval for high values of [p. Typically the smallest envelope in one interval increases with [p, while the smallest envelope in the other interval decreases with [p. At the value of [p where the two minima become equal, the location of am rn in time changes from one hop interval to the other. At this change, the slope of am m versus [p reverses, so the slope of m changes in Fig. 10. Fig. 8 shows typical curves of phase variation f3 versus [p . These curves were computed for the same 0-r-

~

:.

'

~---~-~--~----~--

~~ ':c-_~:i},--:;;~l:l:1:~-c_=C '"'2

-I

0

I

2

3

PATTERN FREOUENCT. HZ

~

X\O"

5

3

~

5

6

Bit error probability versus pattern frequency . 9" = 15°. ll, = 45°. ~" = 6 dB. ~. = -to dB . P = 2. w, = w, .

intermediate J;, and then drops at higher f;, . However.

The larger the frequency jumps encountered by the array. the larger the variations m and f3 and the greater the SINR reduction . In a frequency hopped system, the size of the frequency jumps depends not only on the frequency spacing (the total bandwidth divided by the number of frequencies) , but also on the hopping pattern. For the same spacing. different hopping patterns will produce different frequency jumps. Moreover, bit error probability is affected not only by the size of the frequency jumps. but also by how oftell the jumps occur. since it is an integrated quantity. In general. to minimize 15" one should choose a hopping pattern that minimizes the number of large jumps and also reduces the frequency with which large jumps occur. The effect of frequency jump size may be seen in Figs. 7, 8 and 9. These figures each show several curves for different bandwidths Br • Since there are only two frequencies i p = 2), the total bandwidth is the same as the frequency spacing and the frequency jump size. As may be seen, as bandwidth increases, the variation m and 13 and the bit error probability Pe all increase . This behavior is easily understood . As the desired signal frequency jumps become larger, the jump in interelement phase shift at each hop becomes larger. A larger jump means that the array weights are farther from their optimal values at the new frequency . Thus, a larger weight transient is required after the jump. More

, . .

~

2

C. The Effect of Frequency

LMS AnnA T

\

I

PATTERN FREOUENCY . HZ X\O"

behavior is similar to what happens when an adaptive array receives pulsed interference and a desired signal with no hopping [Ill . (Frequency hopping converts the CW inteference into pulsed interference . The two problems differ, however. because frequency hopping also causes jumps in the desired signal interelement phase shifts .)

--,

~~

0

: 0.001 ,

P, is always higher at Iarge j, than at low!" . This

3 ELEMENT

\

t

: 0 .2 : 0 .1

6'

Fig. 8. Phase variation versus pattern frequency. e,/ = 15°, e, = 45°, ~d = 6 dB, ~; = 40 dB, p = 2, W; = W I '

parameters as in Fig. 7. In general, phase variation is highest at low hopping frequency and drops to a constant as the hopping frequency increases. At large Jp, the array weights are too slow to track the hopping. The nonzero asymptotic phase variation is caused by the jumps in interelement phase shift when the frequency hops. Finally, Fig. 9 shows typical curves of bit error probability Pe versus [p, again for the same param:!ers as in Fig. 7. As may be seen, for higher bandwidths P, simply increases with [p. At lower bandwidths, P, peaks

366

or-------------------,

equivalent angle will be closer to 6, if the interference is on W I than if the interference is on wz, since W3 - W I is greater than W 3 - Wz. Hence, with interference on w" the desired signal falls farther into the interference null, the SINR is reduced more and a higher P, results than with interference on Wz. Note that the edge of the band that is worse depends on the signal arrival angles. In the example above, we have 0 < 6d < 6" and the worst performance is obtained with the interference on W I ' If instead we have 0 < 6j < 6d , then interference on W 3 , the other band edge, will give the worse performance.

,

E. The Effect of Arr ival Angles

envelope and phase modulation is produced and the SINR is lower after the jump. D . The Effect of Interfer en ce Frequen cy

Interference near the edge of the hopping bandwidth is more harmful to the array than interference near the center of the band. Figs. 10 and 11 illustrate this point. These figures 15°. 6j 45°, ~d = 6 dB, show P, versus f p for 6;, z

o x_

.... ''" CD

The envelope variation and the bit error probability increase as the interference arrival angle approaches the desired signal arrival angle. Interference arrival angle has almost no effect on phase modulation except when the interference signal is extremely close to the desired signal. Figs. 12 and 13 show the envelope variation m as a function of pattern frequency for 6d = 15°, ~ d = 6 dB,

a:

CD

o~

~'

Q..

~

0' ~

.

.... ~

... '", CD

-I

Fig. 10.

R

o

O

Il

3

~

PA TTERN FREQUENC Y, HZ XIO·

5

Bit error proba bility versus patte rn freq uency. ij./ = 15' . = ~5 ° . ~.J = 6 dB. ~ , = ~O dB. p = 3. w , = w ,

°

0,

6

3 ELEHENT LHS ARRAY

E

3 ELEHENT LHS ARRA Y

x...... l

.... '"' . -CD

_

~

.

-

, 6 r=l.4

ILl"

;

...J

._

.

~

:

._.-

,

... '", ,- --~.-

...

._

._

._

._

:

.~

i

o

._

ILl

.

ILl

3

~

-,

., i l

3

~

PATTERN FREQUENCY, HZ X10"

Fig. 12. E nvelope variation versus pattern frequ enc y. 6" = 15°, ~ ,/ 6 dB. ~ , = 40 dB , p = 2, B, = 0.1, W , = W "

CD

Oil

,

O

-I

0 .00 1 /

PATTERN FRE QUENCY , HZ XIO"

.,

>'" 20

,

.... r ·0 .2

-1

... _._i

D.o

~: _:> ; ~-'~-l~-,]~t:=Fig. II .

---..

ij ,

=

5

0-.=-."-=,...,,..::-::-.,..-.,.....,..,...0-:-:-------------,

Bit erro r prob ab ility vers us patte rn freq uency . 6" = 15°. 6, = ~5 ° . ~.J = 6 dB. ~ , = ~O dB. p = 3. w , = w , .

E

= 40 dB, and p = 3. Fig. 10 is for W , = W I and Fig. 11 is for W = W :! (where WI < Wz < w 3)' The performance in Fig. 10, when the interference is at the edge of the hopping band. is much worse than that in Fig. 11 , when the interference is at the center of the band. The reason for this difference may be understood in terms of the equivalence between desired signal frequency and arrival angle discussed earlier. Suppose 0 < OJ < 6, as in Fig. 10. The array will produce a null in the pattern at 6j on the interference frequency, either W I or w, . Since o < 6J < a., the equivalent desired signal arrival -angle. as seen on the interference frequency, will be closest to 6, when the desired signal is on frequency w, . The ~,

j

"

ILl ' 0.0

o...J

ILl",

>. Zo

ILl

-I

O

i l

3

~

PATTERN FREQUENCY. HZ XIO"

Fig. 13. Env elop e variat ion ver sus pattern frequen cy. 6" 6 dB , ~, = 40 dB , p = 2, B, = 0.1, W , = W "

367

5

= 15°, ~ " =

~d = 40 dB, B, = 0.1, P = 2, W, = WI. and for different interference angles. Fig. 12 shows 6; = 0°, 30°, 45°, 60°, and 90°, and Fig. 13 shows 6; = 5°, 10°, 13°, 17°, 20°, and 25°. It may be seen that m increases as 16, - 6d l decreases. For 6, very near 6d , the variation m is quite large. The phase variation 13 is small unless 6, is very near 6". Fig. 14 shows a typical case, for 6" = 15°, ~d = 6 dB, ~; = 40 dB, B, = 0.5, p = 2, and W; = WI' Note

probability is primarily to shift the value of the hopping frequency for a given 15e : Bit error probability is very sensitive to the input SNR. Figs. 16 and 17 show the envelope variation and the bit error probability versus jp for 6J = 15°, 6, = 45°, ~ = 6 dB, P = 2, w, = WI ' B, = 0.1, and for several 0-r-

E

0_-----------.,----,-------, 3 ELEHENT

--,

3 ELEHEN T lHS ARRAY ,

..

zci

o

I-

L.HS ARRA Y

~

!

I !

..

:

irci

>

- -

;

. :-:--:- . .

jj ...

.

~

:/

t::-= ::.::---A

~25°

-J

' - - 130 I

2

3

PATTERN FREQUENCY , HZ

~

X10 N

Fig: . 16.

..o

30~ I

i

!

L....

cr

CD 0'"

a: '

"-

a: ~=r a:

....

;

1

,

/

I

. .# . . . , . . . . . " .",

2

3

.

~

PATTERN FREQUE NCY, HZ XIO N

S

2.8,

=

0.1. w ,

=

ad =

15°.

W"

e,

=

20

'

6~~ ·

II>

o

I

2

3

PATTERN FREQUENCY, HZ

~

X10 N

Bit error probabil ity versus pattern freq uency

=

45 °. ~"

=

6 dB. f'

=

2.8,

=

a"

0 .1 . w, = W"

S

=

6

15°.

a,

values of INR. Fig. 17 illustrates how the hopping frequency at which P. peaks varies with the INR. Fig. 18 shows 13 versus j~ for the same parameters as in Figs. 16 and 17 except that B, = 0.5 . Fig. 19 shows the bit error probability versus f p for aJ 15°,6, = 45° "~i = 40 dB, P = 2, Wi = WI' B, = 0.1, and for several values of input SNR. As may be seen, Pe is extremely sensitive to the SNR, as it would be even in a simple DPSK communication system without an adaptive array.

!

60 0

".

=

~ I N R = 10dS'

r ig . 17.

•

- ---- ----------

.

6 dB . P

~

-I

; : 45 i ___'__s> 0

t, =

6

S

... ,

I

,

~

X10 N

Envelope variation versus pattern frequency .

-' '" ' CD

• 0° ............

I

3

,

~----;--~- ~-

'Fig. 15.

I

3 ELEHENT LHS ARRAY

3 ELEHENT LHS ARRAY

=

2

o

0,-.--------------------. ; 8i l

J

0

)(-

also that 13 is much larger for 6, just above 6d than for 6, just below 6d • The reason is as discussed above: in one case the desired signal hops into the null left by the interference whereas in the other it hops away from the null. Fig. 15 shows curves of the bit error probability for the same parameter values as in Fig. 12. It is seen that P, is largest when 6, is near 6d •

o

i

i !

L...

20

.

t

,

PATTERN FREOUENCY , HZ

45 °.

6

5

Fig. 14. Phase variation versus pattern frequency. e" = 15°, I;d = 6 dB , 1;; = 40 dB, P = 2, B, = 0.5, w, = WI'

-I

;

!

. 50

- - -~--:-:~-----------j o

j.

> ,'""- ~/ . z 0 ~.~ w . IO

_ . -~'--....., .... "",,-=-. ~- .' -=--:-";':"';"=' '' -=-=-,-

~ IOO

·1

I

, INR i=4 0 dB l·~ ~ ·

~~ ..........--

.iI

.. .

j

~

6

Bit error probability versus pattern frequency. ed = 15°,1;" = 6 dB, 1;, = 40 dB, p = 2, B, = 0.1, w, = W I '

F. The Effect of Signal Powers

IV. CONCLUSIONS

The input INR has almost no effect on the phase modulation and very little effect on the envelope modulation. The effect of the INR on the bit error

A frequency hopped desired signal has several effects on an LMS array. It causes the array to modulate both the envelope and the phase of the output desired signal.

368

----,

o~

Also , it causes the array output SINR to vary with time below its optimal value and increases the bit error probability for the received signal. The signal parameters affect the desired signal modulation and the bit error probability as follows:

3 ELEMENT LMS ARRAY

z o

C)

1- ...

~.; et:

(1) Envelope and phase modulation are large for low hopping frequencies and drop as the frequency increases. Bit error probability is low at low hopping frequencies and increases with frequency . Both the envelope variation and the bit error probability may have local peaks at intermediate hopping frequencies. (2) Envelope and phase modulation increase with the size of the frequency jumps in the hopping pattern. Bit error probability is increased as the frequency jump size increases. (3) An interference frequency at the edge of the hopping bandwidth is more harmful to the array performance than an interference frequency at the center of the band . (4) Envelope modulation and bit error probability incre ase as the interference arrival angle approaches the desir ed signal arrival angle. Phase modulation is not affected by interference arrival angle unless the interference is extremely close to the desired signal. (5) Input INR has almost no effect on phase modulation and very little effect on envelope modulation. Input INR affects bit error probability by shifting the value of the hopping frequency required for a given bit error probability. Input SNR has a very large effect on bit error probability, as it would in any DPSK system, even without an adaptive array.

r

UJO

::I: '"

ll.';

.

I----~/_ · _

Fig, IX ,

30

. 40

.

Phase va riation versus pattern frequency. B" .J5° . f."

=

fldB .!, = 2 .8, = 0 5 .w. =

o ;---

>s + + \is; M i

M; eJ(P-q) 0,9. the null is shifted or lost. Now we discuss the case where C nearly equals I. When C = I. i is significantly less than the reciprocal of the frequency bandwidth of the signal. In this case, the fad ing is not frequenc y-selective but freque ncy -tlat. Furthermore . we may regard d ;)(r) defined by (2 1) as the approx imate output desired signal. d ;l (t) :== .~I) ( t)

([ 9)

When So 2:: Mo. we consider that .W ) is the desired signal and ,nU) is the undesired one. On the other hand. when .Vl o > So, we consider that ,nUl is the desired signal and sU) is the undesired one. Here. we represent the desired-to-unde sired-signal-ratio by OCR. Note that the output OUR is given by So;Ml) when Sf) 2:: Mo and that it is given by M ol 50 when Mf) > So. Moreover. we add that all of the numerical results which are shown later are computed values.

10

Let D ;\

.i,

In ,)(t) .

D:, denote the power of ii ;\(t).

i.e .. (22)

: a~ ([) I ~ ') , ~ ,

:== (

The output O':\R (D ' I)/ .Y,)l represents the approximate output desired-signal-to-noise-ratio when C = I. Fig. 5 shows the output O':-JR versus C tor several values of '1' . From these curves. it is seen that the LMS adaptive array prevents the output signal power from decrea sing. Namely. the frequency-nat fading is reduced , When C = I, the weights are determined in such a way

III. MULTIPATH FADING REDUCTION

Now we discuss the steady-state performance of the LMS adaptive array. In this section we assume that Tr:== 0 holds. Namely . we assume that the reference signal coincides with s( t) . In order to simplify the notation. we introduce the real-valued symbols C (0 ~ C ~ 1) and 'I' which satisfy (20).

--" I I J~

Fig.

.i

(20) From ( 17), it is seen that C and '1' are the magnitude and phase delay of the correlation coefficient of sd t) and rn\ (t ) , respectively. Fig. 3 shows the output OUR versus C for several values of '1'. Since Sf) 2:: Mo holds for these parameters. i( f) is the desired signal and 1iI(t) is the undesired one . It is seen that the output OUR depends on the correlation Coefficient (C e - )'1') . The undesired signal is, however. Suppressed significantly by the LMS adaptive array

Array pattern. N = 2, 8, = 0°, 8.. = 30°, i' = O°, Tr =O. S,IN, = M,/N, = 20 dB . 5 0 , --

-

-

- -- - ,

1) antennas and multiple remotes with one antenna each. The base station ~as, for every remote's transmitted signal, an optimum combiner that uses the signals received by each of the M antennas. Thus, the designation of the desired and interfering signals depends only on which optimum combiner is being considered. All the signals are, of course, desired at the receiver. The capacity of multiple users per channel systems ~as calculated by first using Monte Carlo simulation to determine the probability that (for a given received signal-to-noise ratio and number of antennas) a given number of users can use the same frequency channel simultaneously. From this probability, we then calculated the probability that, with a given number of simultaneous users, another user can be added to the channel. Finally, these results were used to determine the capacity of systems with a 0.01 blocking probability (i.e., 99 percent availability was considered in our study). The analysis uses the following notation. Let K be the number of simultaneous users per channel (all with BER < 10- 3) . Also, let r d and r j be the average received signal-tonoise ratio per antenna for the desired and jth interfering signals, respectively. Thus, r d = Prdl Ma 2 , and for the multiple users per channel system, I', = I' d for j = 1, L a~d !-= K - 1. Our results are given as a function of rd' This IS because T d determines the required transmit power of the remotes or, alternatively, with fixed maximum transmit power, the maximum range. Note that a 6.8 dB SIN is required for a 10- 3 BER, and assuming a cubic law of signal strength falloff with distance, a 9 dB increase in required r d with fixed transmit power implies a 50 percent range reduction. The probability P K that K users can simultaneously use the sanle channel was determined by computer simulation. A large

Figs. 4-7 show the probability that K users can use the same channel simultaneously versus the average received desired signal-to-noise ratio per antenna with two-nine antennas. Ten thousand cases per data point were used. To conserve computer time, only up to six simultaneous users were considered. The figures show that one user per channel is always possible if T d is greater than 7-10 log 10 M dB, and that for K > 1, the probability of accommodating K simultaneous users increases with rd. M users per channel with high probability are possible if I' d is increased by up to 20 dB, with higher values of K possible only at a much lower probability. Note that as the number of antennas increases. smaller increases in I' d are required for multiple users at a high probability. For example, with nine antennas, an increase in I' d of only 10 dB is required for a six-fold increase in capacity. For fixed transmit power in a typical building, this represents about a 50 percent reduction in maximum range. We now consider the probability P K/ K - 1 of being able to add the Kth user (with BER < 10- 3 for all K users). That is, P K / K -1 is the probability that one more user can use the same channel given that K - 1 users are using the channel. This probability can be derived from the previous results by noting that the BER for each of the existing K - 1 users can only be increased (not decreased) by adding an additional interferer. Thus the cases where BER < 10- 3 with K users are a subset of the cases where BER < 10- 3 with K - 1 users, and the probability of adding the Kth user is PKIPK - 1• Fig. 8 shows the probability that a Kth user can be added to a channel versus the received desired signal-to-noise ratio per antenna for six receive antennas. This probability is similar to the probability for K simultaneous users (Fig. 6) because the probability of adding the Kth user successfully is usually much less than that for the K - 1 user. Similar results were obtained for two, four, and nine receive antennas. The blocking probability for a single channel with capacity K is defined here as the probability that a K th user cannot be added to the system, 5 i.e., for a one-channel system (N = 1),

B= 1-PK / K -

1e

(12)

Thus, the call blocking probability for a single channel can be calculated directly from the above results. Fig. 9 shows the capacity (maximum number of simultaneous users) versus I' d for a single-channel system with a 0.01 blocking probability. The figure shows that the increase in r d required for each additional user becomes smaller as the

381

5 This is actually the worst case blocking probability for the capacity K system since the blocking probability is substantially less when there are fewer than K - 1 users.

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382

Fig. 9. The capacity (maximum number of simultaneous users) versus r d for a single-ehannel system with a 0.01 blocking probability for several values ofM.

Dumber of antennas increases. For example, five users with six antennas require r d = 17 dB, while with nine antennas, only 5 dB is required. Also, the results show that close to M users are possible, but only with a substantial increase in r d as compared to the single-user system. However, multiple users with a small I' d penalty are possible if the capacity is much less than M. \Ve now study the capacity of multiple channel systems (N > 1) where N is the number of channels. Because of dynamic channel assignment, the capacity for a given blocking probability is greater than just N times the capacity of a singlechannel system. In fact, with dynamic channel assignment, there may be many users in one channel and only a few in another. However, to simplify the analysis, we will assume that all channels have K users before any have K + 1 users. This is a worst case model since the capacity is greater if the number of users in each channel is more unevenly distributed. OUf results are, therefore, somewhat pessimistic. Consider an N-channel system with N - (I - 1) channels with K users per channel and I - I channels with K + I users per channel (0 < I ~ N). Then the total number of users is NK + (I - 1), and the blocking probability for the next user is given by

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1, the weights are adjusted to suppress the interference in the output to a level far below the noise. In this case, increasing the received interference power decreases the interference-to-noise ratio at the optimum combiner output. The optimum combiner can greatly suppress (far below the noise level) interferers and not greatly suppress the desired signal if the received desired signal phases differ somewhat from the received interference signal phases at more than one antenna. With multiple antennas and multipath, it is very unlikely that the phases will be the same. Therefore, the probability of the optimum combiner being unable to null the interference is negligible. However, interference nulling does reduce the output desired signal-to-noise ratio. Thus, call blocking occurs when SIN is reduced to less than 7 dB (i.e., BER > 10- 3) with high received interference power. The optimum combiner can therefore tolerate interference at any power" with high probability if I' d is large enough. These points are illustrated in Fig. 11 for M = 4. This figure shows the maximum rjlr d versus I' d for a blocking probability of 0.01 with eight channels. Thus, the probability of call blocking in one channel is 0.56 [(0.56)8 = 0.01]. Results show that the system can tolerate M - 1 (= 3) interferers at any power if r d is 7 dB greater than that required without interference. With M or more interferers, the optimum combiner can only tolerate interference that has power approximately equal to that of the desired signal even with very high rd' Similar results were obtained for M = 2 and 4 with N = 1 and 8. From the above results, the r d required for the system to tolerate L interferers at any power can be determined. Fig. 12 shows the maximum number of interferers at any power versus r d for a blocking probability of 0.01 with one channel. The figure shows that close to M - I interferers can be tolerated with large increases in rd. 6 In a hardware implementation, the maximum interference power that can be tolerated is usually limited to 40-80 dB.

383

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Fig . 13 shows the maximum number of interferers at any power versus r d for a blocking probability of 0.01 with eight channels. M - 1 interferers can be tolerated with M = 2, 4, and 6 and increases in r d of only 3, 7, and 8 dB, respectively . Thus, the results in this section show that M - 1 interferers at any power can be tolerated with a several dB increase in r d if M s 6. Since these results are similar to those for a nonfading environment (where up to M - 1 interferers can be nulled), we might again expect that our results would be valid, even if the fading were not Rayleigh and/or there were more than six antennas .

D. Implementation For the system with optimum combining to be practical, the antenna array at the base station must not require a large area. The separation for (nearly) independent fading at each antenna is one-quarter wavelength (A/4, e.g. , 8 em at 900 MHz and 1.5 mat 50 MHz) . Thus, with space diversity [9, p. 310] , an array of M antennas requires a A/4(M - 1) by A/4(M - 1) area. Furthermore, direction [9, p . 311, uj, polarization [9, p. 311, 12J, or field diversity [9, p. 148J can also be used. With these diversity schemes, antennas can be added without increasing the physical size of the antennas array . For example, with polarization diversity in addition to space diversity, the number of antennas can be tripled (three orthogonally polarized antennas for each space diversity antenna) without any change in the area of the array. Thus,

with a mixture of diversity techniques, a large number of antennas can be placed in a relatively small area. Optimum combining can be implemented for in-building systems in the same way as in mobile radio [7] . The optimum combiner can be implemented with an LMS [15], [16] adaptive array . Signals can then be distinguished at the base station by different pseudonoise codes, with these codes added to the biphase PSK signal with an orthogonal biphase PSK signal (see [17]) . The pseudonoise codes that are used to distinguish signals are also useful for carrier recovery . The received signal can be mixed with the code to generate a narrow-band signal for carrier recovery . Because of the processing gain with the code, the narrow-band signal will have a high signal to interference plus noise ratio. even when /IS at the receiver output is high . Therefore, the receiver can track the signal phase with little phase jitter even when /IS at the receiver output is close to 1. A major difference between in-building systems and mobile radio is the fading rate . In mobile radio, the fading rate is about 70 Hz . Thus, the weights must adapt in a few mill iseconds. In buildings, however, the fading rate is much less . For example. a 1.5 mls velocity (i.e., walking with the remote) produces a 4 .5 Hz fading rate at 900 MHz and a 0.25 Hz fading rate at 50 MHz. Thus, the weights can be adapted much more slowly, making implementation of the LMS algorithm on a chip much easier. Furthermore, because the fading rate is less, the dynamic range of the LMS adaptive array is greater. That is, the receiver can operate with higher interference to desired signal power ratios . Using the analysis of [7], we can show that the maximum interference to desired signal power ratio is on the order of 30 dB for a 4.5 Hz fading rate as compared to 20 dB for mobile radio. If greater dynamic range is required, other (more complicated) techniques [8] may be used because rapid adaptation is not required. As noted in Section Ilf-Al ), for adaptive retransmission to be completely effective (i.e., same BER at the remote as at the base station) , two requirements are placed on the systems. First, all transmissions must be synchronized. That is, all remotes must transmit at the same time , and all base stations must transmit at the same time . With one base station and multiple remotes, synchronization is not a problem. However, with multiple base stations, there should be synchronization between systems within the same building. A second requirement is that all base stations use optimum combining with adaptive retransmission . If another system did not use this technique, it could interfere with the base-to-remote transmis-

384

than integration. Unfortunately, the series has convergence problems (on a digital computer) for most of the cases of interest in this paper. Thus, (A -1) was used to calculate the BER, but only for L -s 5. Fig. 2 shows the results. Note that for large SIN with L = 5, there appears to be some error in the curve. (For L = 5, the error could not be determined because of the extensive computer time required.) However, this error does not affect our results for the reasons discussed below. We also considered two other BER equations. First, for large L, the interference can be considered to be the same as Gaussian noise [18], and therefore, the BER is given by

sio ns 7 of other systems on a channel. However, the system without optimum combining could suffer interference on both transmission paths. Therefore, in high-density multiple-user environments, systems could not operate without optimum combining, and would be required to use optimum combining with adaptive retransmission. In this paper., we have studied only the steady-state performance of the optimum combiner. In an actual system, the base station receiver must track both the desired and interfering signals. Although the dynamics of in-building radio communications are slow, the movement of the remotes will affect the performance of the LMS adaptive array (or any other implementation of the optimum combiner). Thus, the transient performance of the system should also be studied. Finally, in this paper, we have studied the performance of the base station receiver only. A brief analysis (not presented in this paper) shows that the BER at the remote should be similar to that at the base station (for adaptive retransmission with time division). Computer simulation is needed, however, to verify that when the BER is less than 10- 3 at the base station, it is also less than 10- 3 at the remote. IV.

1 BER=-2 erfc (

)

(A-3)

Results using this approximation are given in Fig. 2. Second, an upper bound on the BER with interference for any number of interferers is given by [20]

BER~exp [_

SUMMARY AND CONCLUSIONS

In this paper, we have studied multiple-user in-building radio communication systems. We described a multiple-user system and showed that optimum combining can be used to increase the capacity and interference tolerance of the system. Computer simulation results showed that with optimum combining, a system with one antenna at each remote and M antennas at the base station can achieve either an M-fold increase in capacity or tolerate M - 1 interferers. Finally, we discussed implementation of the system and showed that the system was practical for the office environment. ApPENDIX

Extending the results of Section II, we can see that with L interferers, the BER is

where

+···+-JIL/Scosf)L)2

1

(S/N)-l+//S

(A-2)

and Iii S is the interference to desired signal power ratio of the ith interferer. Note that the total interference to signal power ratio liS is ~f=l L/S, There are two problems with (A-I) and (A-2), however. First, the BER depends not only on the total interference to signal power ratio, but on the individual interference powers as well. However, it was concluded (although not proved) in [18] and [19] that for fixed total interference power, the highest BER is achieved with equal power interferers, i.e., Ii/S = (I/L)IIS for i = 1, L. Therefore, we considered equal power interferers as a worst case and generated an approximate lower bound for maximum liS versus SIN for a 10- 3 BER. A second problem is that for numerical evaluation of (A -1), computer time grows exponentially with L, and therefore, calculations are only practical for small values of L. Another formula for the BER is given in [18], which uses a series rather

1

(S/N)-l+//S

]

(A-4)

Results using this upper bound are also shown in Fig. 2. Note that this bound is not very tight for small liS; from this bound, the SIN is 8.4 dB at a 10- 3 BER (without interference, liS = 0), while the actual SIN required [from (1)] is 1.6 dB less. Fig. 2 shows that the maximum II S varies significantly with the BER equation used. (Equations (A-I) and (A-2) with equal power interferers were used for the results presented in Figs. 4-13.) However, our results for the optimum combining system (with M antennas and L interferers) for L < M in Figs. 4-13 and our conclusions do not depend on the BER equation used. This is because, for L < M, the number of degrees of freedom in the adaptive array using optimum combining is greater than or equal to the number of interferers, and therefore, the array can usually greatly suppress the interferers without affecting the desired signal. Therefore, the liS at the array output is small, and, if the SIN is large enough, the BER is less than 10- 3 • Thus, the array usually operates in the small liS region where the required SI N is about the same for all the BER equations (except for the upper bound (A-4) where the required SIN is 1.6 dB higher). We verified that our results for L -s M in Figs. 4-13 were not significantly changed by the liS curve used, except that the SI N was 1.6 dB higher for the liS curve from (A-4). For L ~ M, the number of degrees of freedom in the array is less than the number of interferers, and therefore, the array cannot greatly suppress all the interferers in most cases. Thus, the variation in maximum 1/S at high SIN has a dramatic effect on the results. As noted above, the results in this paper are based on (A-I) with equal power interferers, and thus, our results should be conservative for L ~ M. However, our conclusions (an M-fold increase in capacity and suppression of M - 1 interferers) are based on the L < M case, and therefore, do not depend on which BER equation is used.

7 It would not interfere with remote-to-base transmissions of systems with optimum combining, however, as optimum combining suppresses any mterference.

385

REFERENCES

[1]

K. Tsujimura and M. Kuwabara, "Cordless telephone system and its propagation characteristics," IEEE Trans. Vehic. Techno/., vol. VT26, pp. 367-371, Nov. 1977. [2] K. Yamada, S. Naka, A. Nishiyama, and T. Miyo, "2 GHz-band cordless telephone system," in Proc. 29th IEEE Vehic. Techno/. Conf., Arlington Heights, IL, Mar. 1979, pp. 159-163. [3] S. E. Alexander, "Radio propagation within buildings at 900 MHz," Electron. Lett., pp. 913-914, Oct. 14, 1982. [4] P. S. Wells, "The attenuation of UHF radio signals by houses," IEEE Trans. Vehic. Technol., vol. VT-26, pp. 358-362, Nov. 1977. [5] A. A. M. Saleh and R. A. Valenzuela, "A statistical model for indoor multipath propagation," in Proc. Int. Conf. Commun., 1986, pp. [6]

27.2.1-27.2.5. D. M. J. Devasirvatham, "The delay spread measurements of

[7]

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

wideband radio signals within a building," Electron. Lett., vol. 20, pp. 950-951, Nov. 8, 1984. J. H. Winters, "Optimum combining in digital mobile radio with cochannel interference," IEEE J. Select. Areas Commun., vol. SAC-2, pp. 528-539, July 1984 (also IEEE Trans. Vehic. TechnoI. , vol. VT33, pp. 144-155, Aug. 1984). R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. W. C. Jakes, Jr., Microwave Mobile Communications. New York: Wiley, 1974. H. Taub and D. L. Schilling, Principles of Communication Systems. New York: McGraw-Hill, 1971. K. H. Awadalla, "Direction diversity in mobile communications," IEEE Trans. Vehic. Technol., vol. VT-30, pp. 121-123, Aug. 1981. R. T. Compton, Jr., "On the performance of a polarization sensitive adaptive array," IEEE Trans. Antennas Propagat., vol. AP-29, pp. 718-725, Sept. 1981. P. S. Henry and B. S. Glance, "A new approach to high-capacity digital mobile radio," Bell Syst. Tech. J., vol. 60, pp. 1891-1904, Oct. 1981. C. A. Baird, Jr. and C. L. Zahm, "Performance criteria for narrowband array processing." in Proc. Con! Decision Contr., Miami Beach, FL, Dec. 1971, pp. 564-565. B. Widrow, P. E. Nantey, L. J. Griffiths, and B. B. Goode. "Adaptive antenna systems," Proc. IEEE, vol. 55, pp. 2143-2159, Dec. 1967. B. Widrow, J. McCool, and M. BaH, "The complex LMS algorithm." Proc. IEEE, vol. 63. pp. 719-720, Apr. 1975. J. H. Winters, "Increased data rates for communication systems with

adaptive antennas, " in Proc. Int. Coni. Commun., Philadelphia, PA, June 1982, pp. 4F.3.1-4F.3.5. [18] A. S. Rosenbaum, "Binary PSK error probabilities with multiple cochannel interferences," IEEE Trans. Commun. Techno/., vol. COM-18, pp. 241-253, June 1970. [19] V. K. Prabhu, "Error rate consideration for coherent phase-shift-keyed systems with co-channel interference," Bell Syst. Tech. J., vol. 48, pp. 743-767, Mar. 1969. [20] G. J. Foschini and J. SaIz, "Digital communications over fading radio channels," Bell Syst. Tech. J., vol. 62, pp. 429-456, Feb. 1983.

386

The Performance Enhancement of Multibeam Adaptive Base-Station Antennas for Cellular Land Mobile Radio Systems SIMON C. SWALES, MARK A. BEACH, DAVID J. EDWARDS, JOSEPH P. McGEEHAN, MEMBER, IEEE

Abstract- The problem of meeting the proliferating demands for mobile telephony within the confinements of the limited radio spectrum allocated to these services is addressed. A multiple beam adaptive basestation antenna is proposed as a major system component in an attempt to solve this problem. This novel approach is demonstrated here by employing an antenna array capable of resolving the angular distribution of the mobile users as seen at the base-station site, and then using this information to direct beams toward either lone mobiles, or groupings of mobiles, for both transmit and receive modes of operation. The energy associated with each mobile is thus confined within the addressed volume, greatly reducing the amount of co-channel interference experienced from and by neighboring co-channel cells. In order to ascertain the benefits of such an antenna, a theoretical approach is adopted which models the conventional and proposed antenna systems in a typical mobile radio environment. For a given performance criterion, this indicates that a significant increase in the spectral efficiency, or capacity, of the network is obtainable with the proposed adaptive base-station antenna.

T

I.

INTRODUCTION

HE FREQUENCY SPECTRUM is. and always will be. a finite and scarce resource, thus there is a fundamental limit on the number of radio channels that can be made available to mobile telephony. Hence, it is essential that cellular land mobile radio (LMR) networks utilize the radio spectrum allocated to this facility efficiently, so that a service can be offered to as large a subscriber community as possible. Indeed, a major consideration of the second generation cellular discussions in both the US and Europe has focused on this point. However, present and proposed future generation cellular communication networks which employ either omnidirectional, or broad sector-beam, base-station antennas, will be beset with the problem of severe spectral congestion as the subscriber community continues to expand. A measure often used to assess the efficiency of spectrum utilization is the number of voice channels per megahertz of available bandwidth per square kilometer [11. This defines the amount of traffic that can be carried and is directly related to the ultimate capacity of the network. Hence, as traffic demands increase, the spectral efficiency of the network must also increase if the quality and availability of service is not to be degraded. At present this is overcome in areas with a Manuscript received April 18. 1989. This work was supported by UK

SERe.

The authors are with the Centre for Communications Research, Faculty of Engineering. University of Bristol, Bristol, 858 lTR. UK. IEEE Log Number 9034227.

AND

high traffic density by employing a technique known as cell splitting. However, the continuing growth in traffic demands has meant that cell sizes have had to be reduced to a practical minimum in many city centers in order to maintain the quality of service. As well as increasing the infrastructure costs, the number of subscribers able to access these systems simultaneously is still well below the long-term service forecasts due to the reduced trunking efficiency of the network. This places great emphasis on maximizing the spectral efficiency, or ultimate capacity, of future generation systems. and thereby fulfilling the earlier promises of performance. There have already been significant developments in terms of spectral efficient modulation schemes, e.g., the proposed US narrow-band digital linear system [21, [3) and the second generation Pan-European cellular network [4}. Also. in the area of antenna technology. the use of fixed coverage directional antennas has been considered [51. In particular. the use of fixed phased array antennas. with carefully controlled amplitude tapers and sidclobe levels for the enhanced UK T:\CS network (ETACS) [6]. are currently under evaluation. However. the application of adaptive antenna arrays in civil land mobile radio systems has hitherto received little attention. in spite of the significant advances made in this field for both military and satellite communications. In this paper a multiple beam adaptive base-station antenna is proposed to complement other solutions, such as spectrum efficient modulation, currently being developed to meet the proliferating demands for enhanced capacity in cellular networks. The feasibility of such a scheme is demonstrated. and a comparison made with existing conventional antennas in a realistic mobile radio environment. Geometrical and statistical propagation models are used and a unique insight is given into the benefits of utilizing adaptive base-station antennas in a cellular radio system. Finally, the concept of such a scheme is discussed and the integration of adaptive antenna array technology into a mobile communications environment considered. II.

ADAPTIVE ANTENNA ARRAYS

An adaptive antenna array may be defined as one that modifies its radiation pattern, frequency response, or other parameters, by means of internal feedback control while the antenna system is operating. The basic operation is usually described in terms of a receiving system steering a null, that is, a reduction in sensitivity in a certain angular position. toward a

Reprinted from IEEE Transactions on Vehicular Technology, Vol. 39, No.1, pp. 56-67, February 1990.

387

source of interference. The first practical implementation of electronically steering a null in the direction of an unwanted signal, a jammer, was the Howells-Applebaum sidelobe canceller for radar. This work started in the late 1950's, and a fully developed system for suppressing five jammers was reported in open literature in 1976 by Applebaum [7]. At about the same time Widrow [8] independently developed an approach for controlling an adaptive array using a recursive least squares minimization technique , now known as the LMS algorithm. Following the pioneering work of Howells, Applebaum, and Widrow, there has been a considerable amount of research activity in the field of adaptive antenna arrays , particularly for reducing the jamming vulnerability of military communication systems. However, to date, there has been little attention to the application of such techniques in the area of civil land mobile radio. Adaptive antenna arrays cannot simply be integrated into any arbitrary communication system, since a control process has to be implemented which exploits some property of either the wanted. or interfering, signals. In general. adaptive antennas adjust their directional beam patterns so as to maximize the signal-to-noise ratio at the output of the receiver. Applications have included the development of receiving systems for acquiring desired signals in the presence of strong jamming. a technique known as power inversion [91 . Systems have also been developed for the reception of frequency hopping signals [10), [II). TDMA satellite channels [12) and spread spectrum signals [131. Of particular interest for cellular schemes is the development of adaptive antenna arrays and signal processing techniques for the reception of multiple wanted signals [14] .

A. Fundamentals of Operation The adaptive array consists of a number of antenna elements. not necessarily identical. coupled together via some form of amplitude control and phase shifting network to form a single output. The amplitude and phase control can be regarded as a set of complex weights, as illustrated in Fig. I. If the effects of receiver noise and mutual coupling are ignored . the operation of an N element uniformly spaced linear array can be explained as follows. Consider a wavefront generated by a narrow-band source of wavelength A arriving at an N element array from a direction fh off the array boresight. Now taking the first element in the array as the phase reference and letting d equal the array spacing. the relative phase shift of the received signal at the nth element can be expressed as ,T. 'l'nk

=

27rd(n - I)

A

.

Sin

(J

k·

(1)

Assuming constant envelope modulation of the source at Ok. the signal at the output of each of the antenna elements can be expressed as

(2) and the total array output in direction fh as

YkCt) =

L wnej("'I +'~n.) N

n ~1

(3)

Far Field Signa l Source Array Outpu t

Antenna Arra y

Fig. I.

An adaptive antenna array..

where wn represents the value of the complex weight applied to the output of the nth element. Thus by suitable choice of weights, the array will accept a wanted signal from direction I and steer nulls toward interference sources located at fh, for k # I. Likewise. the weighting network can be optimized to steer beams (a radiation pattern maxima of finite width) in a specific direction. or directions. It can be shown [15] that an N element array has N - 1 degrees of freedom giving up to N - I independent pattern nulls. If the weights are controlled by a feedback loop which is designed to maximize the signalto-noise ratio at the array output. the system can be regarded as an adaptive spacial filter. The antenna elements can be arranged in various geometries . with uniform line. circular and planar arrays being very common. The circular array geometry is of particular interest here since beams can be steered through 360 0 • thus giving complete coverage from a central base-station. The elements are typically sited A/2 apart. where A is the wavelength of the received signal. Spacing of greater than A/2 improves the spatial resolution of the array. however. the formation of grating lobes (secondary maxima) can also result. These are generally regarded as undesirable.

o

B. Adaptive Antenna Arrays for Cellular Base-Stations Multiple beam adaptive antenna arrays have been considered by Davies et al. [16] for enhancing the number of simultaneous users accessing future generation cellular networks. It is suggested that each mobile is tracked in azimuth by a narrow beam for both mobile-to-base and base-to-mobile transmissions. as shown in Fig. 2. The directive nature of the beams ensures that in a given system the mean interference power experienced by anyone user, due to other active mobiles, would be much less than that experienced using conventional wide coverage base-station antennas. It has already been stressed that high capacity cellular networks are designed to be interference limited, so the adaptive antenna would considerably increase the potential user capacity . This increase in system capacity of the new base-station antenna architecture was evaluated [17] by considering the spatial filtering properties of an antenna array. The results show that this type of base-station antenna could increase the spectral efficiency of the network by a factor of 30 or more. These results were obtained for a hypothetical fast frequency hopping

388

cellular systems. However, some well-established trends are becoming apparent in the quest toward higher spectrally efficient modulation schemes [I] for the systems of the year 2000 and beyond . It is thus vital during the initial stages of research to develop antenna architectures which are, in essence, modulation scheme independent, so that a figure of merit can be obtained for the rnultibeam base-station antenna .

Base-station

III . REDUCTION OF CO-CHANNEL INTERFERENCE USING ADAPTIVE ANTENNAS

Fig. 2.

Tracking of mobiles with multiple beams .

code division multiple access cellular network (18) . assuming uniform user distribution and complete frequency reuse for the omnidirectional antenna case. i.e .. adjacent cells are cochannel cells. Complete frequency reuse is then assumed for each of the beams formed by the adaptive array . i.e.. adjacent beams are co-channel beams . Further. it was shown that a similar enhancement of efficiency can be obtained for either an idealized multibeam antenna . or a realizable 128 element circular array (19] . It was recognized in the analysis. but not fully assessed. that this approach would greatly increase the level of co-channel interference. It was. therefore. suggested that this problem could be overcome using dynamic channel allocation to eliminate the so called common zones. This again introduces additional hand-offs , reducing the trunking efficiency and available capacity of the network. as the mobile circumnavigates the cell. The only study previous to the work discussed above considering the use of an adaptive antenna array in land mobile radio was by Marcus and Das (201 in 1983. The analysis assumed that the base-station. or repeater. sites could be placed closer together if an antenna array formed 20 dB nulls toward co-channel sites. This effectively reduces the amount of co-channel interference at the output of the base-station as explained in Section II . It was suggested that in this system the beam steering information could be derived from the squelch tone injection which is presently used in the US FM land mobile radio. In contrast with the null steering technique considered by Marcus and Das, here the ability of the adaptive array to steer radiation pattern maxima toward the mobiles is considered . In the limit it can be envisaged that individual beams will be formed towards each mobile as illustrated in Fig . 2. It has already been mentioned that adaptive antenna technology cannot be simply integrated into an arbitrary communication system. and at present no one particular modulation scheme , or access technique, has been selected for the third generation of

In this paper the integration of an idealized adaptive array into an existing cellular network is considered . In order to ascertain the benefits of this class of antenna system compared with that of conventional omnidirectional base-station antenna systems, the following network topology has been assumed . 1) A cellular network consisting of hexagonal cells. with channel reuse every C cells (C is the cluster size) . 2) The base-station transmitters arc centrally located within each hexagonal cell . 3) There is a uniform distribution of users per cell. 4) There is a blocking probability of B in all cells. 5) The omnidirectional base-station antenna has an ideal beam pattern. giving a uniform circular coverage . 6) The adaptive base-station antenna can generate any number. m, of ideal beams. with a bearnwidth of 2;r !m. and a gain equal to the omni-antcnna . 7) Each adaptive beam will only carry the channels that are assigned to the mobiles within its coverage area . 8) Any mobile (or group of mobiles) can be tracked by the adaptive base-station antenna. 9) The necessary base-station hardware is available to enable bcarnforrning and tracking. 10) The same modulation scheme can be used with each antenna system. The blocking probability of B in assumption -i) is the fraction of attempted calls that cannot be allocated a channel. If there are "a" Erlangs of traffic intensity offered. the actual traffic carried is equal to at 1 - B) Erlangs. The Erlang is a measure of traffic intensity. and measures the quantity of traffic on a channel or group of channels per unit time . This gives an outgoing channel usage efficiency (or loading factor) (211 of (-i) 1) = a ( l - B) /N where N is the total number of channels allocated per cell. Assumptions 6). 7). and 8) imply the deployment of a somewhat hypothetical adaptive antenna system. This approach can be justified since a uniform user population has been assumed for both categories of antenna system. It is recognized that the dynamic. nonuniform. user distribution will have a significant effect on the results presented here. This will be considered in a subsequent more rigorous study. Also, in the analysis which follows only the base-to-mobile link has been studied. however, it can be shown that the analysis is also valid for the mobile-to-base link. Two different categories of co-channel interference models are used as the basis for the study presented here. The first is the geometrical model adopted by Lee l51. followed by a more rigorous statistical analysis (211-(23) .

389

o

= Base station

Wanted cell

Interfering cell Region of interference

Region of no interterenc/ dw

0-------------- -------------------------------Wanted

Fig. 3.

IV.

Base-station

D

...

_ Interfering Base-station

Worst-case position

Two co-channel cells.

GEOMETRICAL PROPAGATION MODEL

This approach considers the relative geometry of the transmitter and receiver locations, and takes into account the propagation path loss associated with the mobile radio channel. A. One Co-Channel Cell

Consider one co-channel cell which forms part of a cellular network as shown in Fig. 3. By definition both the cells have the same channel allocation . and a reuse distance of D separating the base-station transmitters. The co-channel reuse ratio is defined as

Q

==

D iR.

(5)

This ratio has also been termed the co-channel interference reduction factor [5j since the larger it is (i.e ... the further apart the cells) the less the co-channel interference for J given modulation scheme. The level of acceptable co-channel interference governs the value of this parameter and the overall spectral efficiency of the network. The area mean signal level experienced at the mobile is assumed to be inversely proportional to the distance from the base-station raised to a power -y. With the advent of smaller cells . the propagation path loss is close to the free-space value [24 J.. however. it is envisaged that the proposed base-station will initially operate in larger cells. Therefore . as a starting point for the comparison to follow, the commonly used approximation that the received signal power is inversely proportional to the fourth power of range will be used [251. Hence . the area mean signal level (in volts) received from the wanted base-station at a mobile a distance d w from the transmitter is

(6) Similarly, the area mean signal level from the interfering basestation transmitter at a distance d, is (7)

assuming in each case identical radiated transmitter powers and signal propagation constants, as denoted by the constant k.

Co-channel interference will occur when the ratio of the received wanted signal envelope, s.,; to the interfering signal envelope, s., is less than some protection ratio . Pr, i.e.:

(8)

Fig. 4.

Contour defining interference regions.

The protection ratio is defined by the modulation scheme employed [1]. Considering only the propagation path loss, the received signal envelopes are equal to the area mean signal levels, hence: mw

d~

d~ S Pro w

(9)

So.. for a given protection ratio . a locus given by

d, [d ; == V!jJ;

( 10)

can be drawn. This defines a region where no interference will occur.. and where it will always occur, as illustrated in Fig. 4. For the worst-case position, which is in a direct line between the transmitters as shown, the co-channel reuse ratio is

Q ==D!R == 1 +di/dw == 1 + JJJ;.

(11)

For a given protection ratio and modulation scheme, this defines the minimum spacing between co-channel cells in order to avoid interference, and the maximum spectral efficiency obtainable. In this discussion it is assumed that the same modulation scheme is employed for both antenna systems under evaluation. This implies that the protection ratio and reuse distances are identical in both cases. Therefore, there would appear to be no apparent benefit from employing adaptive antenna technology at the base-station site. However, the occurrence of co-channel interference is a statistical phenomena. Hence, when comparing omni- and adaptive antennas, it is necessary to introduce the concept of the probability of co-channe/ interference occurring, i.e., Pis.; :::; p,Sj). This is often called the outage probability, which is the probability of failing to obtain satisfactory reception at the mobile in the presence of interference. If the cells are considered to be identical, i.e., have equal blocking probabilities, then on average, there will be N1J active channels in each cell (71 is as defined in (4». So, in the case of the omnidirectional antenna, given that the wanted mobile is already allocated a channel, the probability of that channel being active in an interfering cell is the required outage

390

0 ··· ··········· 0

J/ ~O

t~

i "Z i -o

,: c0 : u ,

11>

\U1

.0

o o L a.

.0

Wan ted ce ll

g

QC9

Q)

0'

o .....

'\~

:J

o

Fig . 6.

lnt er ter inq cel ls

~

Hexago nal cellular layout showin g tiers of inte rferers .

a ( I - B ) active channels (or user s). Th is is only rea lly valid if a ( I - B ) > m and that the use rs are uniformly dist ributed

5

10

15

20

25

30

Nu m b er of beams Fig . 5.

35

40

45

within the cell. If this were not the case, and the numbe r of beams formed was less than m, the outage probability would be reduced even further since the wanted mobile will not be covered by a co-channel beam all the time . This situati on will not be pursued further since thi s analysis can be regarded as worst -case situation.

50

m

Outage as a function of the number of beams .

probability. He nce. when the wanted mobile is in the region of co-c hannel interference the outage probability is given by P (s w

< Pr Si) = -

numbe r of active channels . total number at channels

NT] N

=

B . Six Co -Channel Cells

1) .

( 12)

Now consider the case of the adaptive antenna as previously desc ribed. with m beams per base-station providing coverage of the whole cell, and wit h N T/ 1m channels per beam. give n a unifo rm distribution of user s. T he same regions of co-channel interfe rence can be defi ned . however , whe n the wanted mobile is within the region where co-cha nnel interference may occ ur, the outage probability is reduced . The wanted mob ile is always covered by at least one beam from the co-channel cell. hence. the outage probability is equal to the probability that one of the channels in the aligned beam is the corresponding act ive co-channel ' and is given by

Pts; < P rs ; ) -

number channels per beam total number of channels

= ----,----.,.-'---NT/ 1m N

=

m

The previous ap proach can now be simply extended to JS sess the effect of six co-channel interfer ers. i.e .. the first tier of co-channel cells in a con ventional cellular scheme JS shown in Fig . 6 . It is co nsidered that further tiers of interfere rs will not significantly af fect the results except when reu se distances become small. Equat ion ( 9 ) can now be rewritten for this mor e reali stic representation o f the cell ular network s". s/

I The " active co-cha nnel " is the chan nel that has also been allocated to the wanted mobile .

In /

=

d,:c II

Ld,-c

'5:

o,

( 14)

1= 1

where the total mean signal le ve l from the interfering cell s. m, , is the sum of the mean level from each active cell. Thu s in a fully loaded system. the number of active users is six ( i.e . , n = 6). If all the d , are assumed to be equal and the wanted mobile is at the edge of a cell bou nda ry. as for the case described in Lee [5). then the co-c hannel reuse factor ca n be expressed as

(13)

where the omnicase is given by m = I . These results are pre sented graphically in Fig . 5, and show the strong influence of the number of beams, m , on the outage probability. The influence of the loading factor, 1] , is as expected, i.e ., the less the loading , the fewer the numbe r of active channels , and hence, a red uced chance of co-c hannel interfe rence . This assumes that there are still m bea ms formed even thoug h the re are only

I n ".

Q = [6(5 w I5dl" "'I .

( IS)

Subjective tests showed that over a mobile radio channel

swl5/ 2:: 18 dB (i .e. , PR = 18 dB) gave good speec h trans-

mission for a 25-kHz FM channel operation . A value of Q can now be calculated to defi ne the minimum cluster size, C. Using sim ple geometry it would be possible to evaluate the actual swls/ in the worst-case locations. From this, a conto ur may be d raw n defi ning region s with and without interference . For bo th classes of antenna sys tems the outage probab ility is still ze ro within the contour (i .e .. when s.. .. ls, > p,) , but outside,

391

in the region of interference: P(Sw 5: Pr S / ) = ( : )

6

(16)

where s I is the total co-channel interference and the omnicase is given by m == 1. Since it is assumed that all m beams per cell are formed, there are six beams aligned onto the wanted mobile at any time. The outage probability within the region of interference is then found by considering the probability that the active co-channel is in each of these beams. C. Analysis of Results The use of adaptive multiple beam-forming base-stations would, based upon the analysis presented so far, appear to give an improvement in performance with regard to the reduction of the probability of co-channel interference. The improvement depends on the degree of adaptivity used, i.e .. the number of ideal beams formed. However . the above approach is over simplistic and gives a rather optimistic view of the situation. Firstly, the beams are assumed to be ideal. giving an equal gain over the whole beamwidth. In practice this would not be the case. Also. a hypothetical situation could be envisaged where. if In is large enough to satisfy a given outage critcrion.? it would appear that the ultimate reuse distance (D R == 2) is possible for any modulation scheme. Hence. adjacent cells arc co-channel cells . the radius of which is decided by the rcqui red coverage area of the base-station site. In spite of this though. the analysis has been useful in introducing some of the important factors that affect the performance of a mobile radio network which exploits frequency reuse as a means of increasing spectral efficiency.

v . STATISTICAL

PROPAGATION ~loDEL

In the previous analysis only the path loss associated with the mobile radio environment was considered when calculating the level of co-channel interference. This was useful in demonstrating the principle benetits to be offered by adaptive antennas . although it is an over simplified approach and totally unrealistic of many land mobile radio environments. It was shown that a single contour defining regions of operation where co-channel interference would occur can be drawn., however. it is known that the signal levels fluctuate rapidly generating small isolated pockets of interference in an operational system. In some adverse environments these areas may be quite close to the base-station antenna. There is seldom a line of sight path between the base-station and the mobile . and hence, radio communication is obtained by means of diffraction and reflection of the transmitted energy. This produces a complicated signal pattern causing the field strength to vary greatly throughout the cell, and the received signal at the moving mobile to fluctuate very rapidly. This is generally attributed to the superposition of two different classifications of signal fading phenomenon: fast fading (or just fading) due to the multipath nature of the received signal. and slow fading (shadowing), the slower variations of 2

the received signal due to variations in the local terrain. In areas experiencing this type of signal variation, the area mean signal level is essentially constant. In order to model these propagation effects, the are included in a statistical fashion, the fading and shadowing described above being represented by Rayleigh and log-normal type distributions, respectively.

A. One Co-Channel Cell Various studies [26]-[28] have been undertaken to analyze co-channel interference originating from a single co-channel interfering cell in an attempt to characterize the mobile radio environment. In particular the rigorous analysis presented by French [22] has been adopted here. The fast fading is the rapid fluctuation of the signal level 5 about the local mean s (s == (s)), and is usually described by a Rayleigh type probability density function (pdf) . i.e.:

7rS

[ _ ;rs~ ] .

Pts Is) == ---:-;- exp 25-

( 17)

4s~

Shadowing of the radio signal due to the terrain, i.e., by buildings and hills, causes the local mean level s to fluctuate about the area mean. It has been generally accepted that this variation is log -normally distributed about the area mean md, where md == (Sci) . the mean of s in decibels. (Note, a subscript I".d' indicates that a signal is in decibels.) The area mean level is approximately proportional to the inverse of the distance from the base-station raised to the power 1, as described in Section IV-A. Hence. the log-normal shadowing pdf is given by ( 18) The standard deviation. (J . describes the degree of shadowing. This parameter typically varies from 6 to 12 dB in urban areas ~ the larger value being associated with very built up inner city areas. The combined pdf can now be expressed as

If there arc ten beams (m = 10) a loe;{J outage criterion could be satisfied

in a fully loaded system (13).

392

P(S)

= i:P(S/S).P(Sd)dSdo

By substituting s == IcY d / 2o (from Sd == 20 the combined pdf becomes P(s)

==

J

7r 18(J2

J

x

.

-x

S -dj'O

lOS ..

exp

. exp

(19)

10gIO s)

into (17),

2] [7rS -d no 4 x lOS

- (Sd -

.,

2(J-

[

1-

m d)2]

dSd.

(20)

1) Outage Probability With Fading and Shadowing: The outage probability with fading and shadowing is derived in French [22] and the resulting integral is

Pis; 5: PrSi) where, Zd = mdw internal variable.

-

1

=..Ji mdi -

j'oo

-00

PR,

exp ( _u 2 ) 1 + 1O(zr2aul/ 1O du (J

=

O"w

=

(Ji,

and

(21) u

is an

In many situations it is possible to greatly reduce the fading, e.g. , antenna diversity at the mobile, and a similar result to that above can be derived [22) for shadowing only. Note that the result in (21) is for the case of the omnidirectional basestation antenna . 2) Outage Probability with an Adaptive Antenna: The outage probability for an adaptive antenna can be simply ex pressed as

rts; :::; p-s] , m)

=

a = 6 dB n' = 0 .7

-,

10

-0

o o ....

et»; :::; P,Si)

-0

Q..

probability of an active co-channel) (

Q)

Q'l

...,o

in the aligned beam

= Pis; :::; p ,s,) ' ( ; )

:J

o (22)

i.e ., the probability that the ratio of the wanted signal to the interfering signal is less than some protect ion ratio (21) and the probability that the aligned beam actually contains the ac tive channel (13) . Again, the outage probability is redu ced by a factor m. This is illustrated graphically in Fig . 7. The loading factor is fixed at 70lJD (,., = 0.7) and the fading and shadowing case is considered for a = 6 dB . This represents a typical urban environment. The se results have been obtained by solving (21) and (22) numerically with In = l. 2. ..J.. 8. 16, and 32. Note that In = I gives the ornnicase . The outage probability varie s as expected with :::

Xl

"'CIS(

DDr-->0-~-'T--->@ibm---~:r=="'lRi!]}--r'G~'=ID--,-""RJf

X'.'->

_0

.-

_

IITEl' - - , . ; . :' - CHm'a --;,...,- - - -- - fUl' 1'9l -

Fig . '

=

W,, (k ) - /ldk).r,, (k) *

( 5)

where /l is the step size . If every complex weight is o ptim ized. the adapti ve a rray will yie ld automatic beam tracking o f the de sired SS sg na l and adequate su ppressi o n o f interfering SS s igna ls because o f d irecting its null s at them . As a result. th e interfering si g na ls will be attenuated by th e nulls w h ile the de sired SS signal will not he a null . How ever . an a rray antenna sys te m can rej e ct co m p lete ly o nly narrow -h and signa ls . In the adapti ve arra y antenna using DS -SSMA . the null depth ma y not be suffi cient to ac h ie ve the de sired interference rejection unless the syste m is modified to u peate over wide bandwidth s . Hen ce . there w ill be so me residual inte rfe re nc e [41. [51 . More o ver. the adapti ve ar ray antenna cannot suppress th e interfering SS sig na l who se arrival angle is the same as that o f the desired 55 signa l.

III. AN ADAPTIV E ARR A Y INTERF EREN CE FOR

I NCL UDIS G A C 'SCELLE R OF A

-

- - - - --

--

A DS ,SSMA sy ste m w ith an adapt ive arra y antenna incl udi ng a ca nce lle r of interference (pa ra lle l ca nce lle r struc tu re) .

for integer i , l- xR,,(k) = Re { x,,(k )} and xl,,(k) = 1m ( .r, (k) }. Since matrices R,,(O) and R" (0) cannot be obtained from the observed signal. LMS algorithm can be used in order to update complex weights chip by c h ip . suc h a s

W,, (k + I )

-

D5-SSMA 5 YSTE\1

Thi s sect io n propose s a D5 -5SMA rec eiving system using an adaptive arra y which can demodulate a de sired 55 sig na l robustl y even if he a vy inte rfe ring 55 signal s have the sa me arrival angle as that o f the de sired S5 s ig na l in a time-varying channel.

A. The Structure of the System Fig . 2 sho ws the structure o f the D5-55MA syste m with an adaptive array including a canceller of interference : it is po ssible to extend it to other S5 modulations . A prim itive 55 receiving sy stem ha ving neither an array antenna nor a canceller cannot demodulate the desired S5 sig na l even by using the inherent processing ga in of the 55 sy stem when the desired S5 s ig na l power is much lower than the interfering S5 signal power. The adaptive array antenna system (mentioned in the previous section) is effec tive in suppressing cochannel interference with arrival angles different from that of a de sired user. but the residual

interference and cochannel interference having the same arrival angle as the desired 55 signal are a major problem in S5MA . In order to solve thi s problem. we propose an adaptive array syste m including a canceller of interference . which can eliminate the interfering 5S s ig na ls having the sa me arri val angle as that of the de sired 55 s ignal by using ad apti ve digital filters (A D F ·s ). The proposed sys te m can al so ca nc e l the residual co channel interference having arrival angles different from that of a de sired user which a n adapti ve arra y antenna cannot completel y suppress. When th e int erfering si g nals from M undesired users rema in at the o utput of the arra y antenna. the ad apt ive canceller has ( W O typ e s of struc tur e. suc h as the parallel structure sho w n in F ig . 2 and the se ria l o ne in Fi g. 3. H for each antenna e le m e nt in F ig . 2 is a qu adrature hyb rid splitt ing the recc iv ed s ig na l into quadrature components . In the ca nc e lle r. th e int erfering 55 s ig na l from the ith user (i = I . 2 . . .. M ) is demodulated and respread by 55DEMi and 55MODi . respecti vel y . ADFi ( i = I. 2.. . . M ) is used to identify (he entire channel characteristics of both the channel for the i th user and the array and to generate a repl ica of the di storted inte rfe re nce component in the a rra y o utput. Then every replica of interference is subtracted from the delayed output s ig na l of the array . The o utput signal from the c a nce lle r is then fed to ADFO. whi ch compen sates the di stortion of the desired S5 signal from the Oth user. and demodulated by 5SDEMO . The final o utput data are respread by SSMODO in order to produce the reference s ig na l for the arra y and the ADF' s. In the se ria l st ruc tu re o f the canceller. interfering S5 s ig na ls are cancelled in the order of decreasing receiving po wer be cause it is ea sy to achieve acquisition, demodulati on , and ca nc e llat io n of an interfering 55 s ig na l ha ving g rea te r power and its can cellation makes it possible or ea sy to cancel o the r interfering 55 signals . Therefore . the se ria l structu re may perform more robust cancellation than the parallel one . However, the latter, shown in Fig. 2, can achieve more stab le adaptability of the array than the former. because time-delays within the feedback loop updating the array weights will result in instability unless the loop gain or the array speed of response is reduced . When there are a few strong interfering 5S signals, cancelling only those strong signals is sufficient to achieve

401

~=r::::r

' EIC>
Fig . 3. Serial canceller structure .

stable acquisition and reliable demodulation of the desired SS signal. Then the amount of hardware will not be very large. Note that this system can achieve more stable acquisition , demodulation. and cancellation of the interfering SS signals than a conventional array system without a canceller even when the D /1 ratio (of the desired to undesired SS signals' power) at the array output is small. This is because interfering SS signals can be demodulated more correctly in the case of such a small 0 /1 ratio . On the other hand . when the D / 1ratio is so large that the system can achieve stable acquisition and reliable demodulation without such a canceller. the funct ion of cancellation of ADFi (i = I. 2.... M) in Figs . 2 and 3 may be stopped . Moreover. it is possible to control ADFO so as to reject other intentional jamming and narrow-band interference similar to an interference rejection filter [7].

= /,2.: Re =0

-4l

8

20 10 •..

(

.

l .::::.:::J::.:::::.:::::t::.:::.:'.':. .

. ._

. . ..

:

.

:

.

·············[·············t·············;···

60 50

.:.

.+

·····t··············i·············;········

.... . ...•...... . ...... . . . .;_..

. . . . . . . . .:.

_.....

...\.·..··.······r···· ········i······

..........::::.:. ::::::::r:: .

.. _. .

':.

...;.-

.

3S

ITERATIONS

= 20·

.

SNR = 30 , SOO SNAPSHOTS 100

Fig. 5.

aa = 0 .2

.

ITERATIONS

dB

1 = 500 snapshots

a",

IS

o

Z

where 'Y; is a uniformly dist ributed random numbe r between zero and one and is the variance of the se nsors phase . Figs. 2-6 describe an experime nt with the follow ing parameters :

SNR = 10 log 10 ( a}/ a,;) = 30

:;;: o

ZO

Coupling coefficient error versus iteration number.

-- ---

10.1

SNR = 30. SOO S NAPSHOTS

/ I'---.. ~

0

w

'" ~ ~

2

ffi < 8

-4

o

-I--.. ~

C

·8

o

V

----

r-,

t-....

L....-

l---

~

w

~

10'

"'" 10

10

IS

20

2S

30

3S

10

IS

2S

20

~ 3S

30

Fig. 7. RMSE of the magnitude of the coupling coefficient versus SNR . The solid line depicts theoretical values comp uted by the Cramer- Rae lower bound . The point estimates are the means and 90% confidence intervals from Mon te Ca rlo experi ments for three values of the SNR . Each Monte Carlo experiment consisted of 30 runs of 500 snapshots each .

DOA errors versus iteration number.

coup ling coefficie nt = 0.2

.

to-....

SNR ldB J

ITERA TIO NS

+j .0

DOA I = -30·

10'

DOA 2 = _ 5· DOA 3

. 100 which is a measure of the relative gain /phase errors of all the sensors . I', is the gain /phase matrix at iteration i while I', is the true ga in/ phase matrix . Fig . 5 shows the relative coupling coefficient error as a function of iterations. Befor e the first iteration the err or is 100% and it reduce s to 1.9 %. Fig. 6 shows the DO A errors for the three sources as a function of the itera tio n number . To demo nstra te the statistical efficiency of the propo sed procedure we performed the following Monte Carlo experiments . The six sensor circular array described above was used, with three far-field narrow-band emitters . The gains and phases were selected as before, with UOI = 0.02 and Ucp = 2· . The DOA 's were 'YI = 0·, 'Y 2 = 120· , 'Y3 = 240· . The coupling coefficient between any two adjace nt senso rs was cc = 0.2(x + iy) where x and yare two i.i .d . random variables with uniform distribution over the interval [-0.5, 0.5] . The coupling coefficient for any nonadjacent sensors was assumed to be zero . We performed 30 experiments for each signal-to-noise ratio , for SNR = 10, 20, 30 dB. In each experiment 500 snapshots of data were collected and processed by the algo rithm. The values of the sensor gains and phases were kept cons tant throughout these simulations . For each SNR we used the results of the 30 experiments to compute the estimated root mean square error (RMSE) and the bias. Figs . 7 -11 depict the RSE' s and compares them to the corresponding Cramer- Rao lower bound. (computed as shown in Appendix II) . The se figures clearly indicate that the proposed algorithm is statisticall y efficient even for fairly low SNR 's, at least for the test case conside red here. The bias was small co mpa red to the RMSE in each case .

_.

= 21m {diag{ -C-IAPAHR-IA",.PAHR-IC

+tr {A~,-P44 HR-IAPA7R-l}}.

-

{AI£PAHR-IAPA~R-I}}

= 2 Re {jere-lAPA HR-1AIJ. PA HR-1Ce j}

I}

J rr = 2 Re {tr {A;-PAHR-IArPAHR-I}

JWI}

= 2 Re {tr {A IL PA H R - lAtPj PA H R - I } +tr

Jp.f/Jj

(92)

+tr{-jA lAo PAHR-IAPAHC-He.er:CHR-l} j }

+ tr { A)J. PA H R -

+ tr

(91)

J ra = 2 Re {diag {G - I C- lA QA H A r QA H C

we obtain

J~I-t

(90)

+CHR-1AIL(p - Q)AHe-HG- 1}}

To simplify the analysis we concentrate here on a circulant matrix with only a single coupling coefficient given by

where

}

JI1Q = 2Re [diag {G-IC-lAQAHAILQAHC

H. Derivatives with Respect to Mutual Coupling Coefficient

p.e

j

= 2Re {eJG-lC-JAQAHAp.QAHCej

x(eHR-1C)T - (G-1e-1AQAHC)

C 12 =

11

(89)

fA Q-j PA H R - I }

+tr {AI£PAHR-IAPA~R-I}} 416

A. Paulraj and T. Kailath, "Direction of arrival estimation by eigenstructure methods with unknown sensor gain and phase," in Proc. IEEE ICASSP'85, Tampa, FL, 1985, pp. 640-643. [21 A. M. Bruckstein, T.-J. Shan, and T. Kailath, "The resolution of overlapping echoes, " IEEE Trans. Acoustics, Speech, Signal Processing, vol. ASSP-33, pp. 1357-1367, Dec. 1985. [3] A. J. Weiss, A. S. WHIsky" and B. C. Levy, "Eigenstructure approach for array processing with unknown intensity coefficients," IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, Oct. 1988. [4] H. Wang and M. Kaveh, "Coherent signal subspace processing for detection and estimation of angles of arrival of multiple

[5]

wideband sources," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 823-831, Aug. 1985.

Y. Rockah and P. M. Schultheiss. "Array shape calibration using sources in unknown iocations - Part I: Far field sources," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 286-299, Mar. 1987. [6] - , " Array shape calibration using sources in unknown locations-Part II: Near-field sources and estimator implementation." IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP35, pp. 724-735, June 1987. [7] J. T -H. Lo and S. L. Marple, Jr., "Eigensrructure methods for array sensor localization," in Proc. IEEE ICASSP 1987. Dallas, TX, 1987, pp. 2260-2263. [8] R. O. Schmidt, '''A signal subspace approach to multiple emitter location and spectral estimation," Ph.D. dissertation, Stanford University, Stanford. California, 198 t . [9] A. J. Weiss and B. Friedlander, "Array shape calibration using sources in unknown locations - Maximum likelihood approach." IEEE Trans. Acoust., Speech, Signal Processing. vol. 37, pp. 1958-1966, Dec. 1989. [10] R. O. Schmidt, "Multilinear array manifold interpolanon;' Tech. Memo ESL-TM166J, ESL Inc., Sunnyvale. CA. Sept. 1983. [11] B. D. Steinberg, Principles of Aperture and Array System Design Including Random and Adaptive Array. New York: Wiley, 1976. [12] B. Friedlander and A. J. Weiss. "Eigenstructure methods for direction finding with sensor gain and phase uncertainty." in

Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing.

[13]

Apr. 1988. pp. 2681-2684. (Also. J. Circuits, Systems and Signal Processing, vol. 9. no. 3. pp. 271-300. 1990.) B. Friedlander. A sensitivity analysis of the MUSIC algorithm." IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp. H

1740-1751, Oct. 1990.

417

Improving the Performance of a Slotted ALOHA Packet Radio Network with an Adaptive Array James Ward, Member, IEEE, and R. T. Compton, Jr., Fellow, IEEE

Abstract-The use of an adaptive antenna array is presented as a means to improve the performance of a slotted ALOHA packet radio network. An adaptive array creates a strong capture effect at a packet radio terminal by automatically. steering the rece.ive antenna pattern toward one packet and nulling other contending packets in a slot. A special code preamble and randomized arrival times within each slot allow the adaptive array to lock onto one packet in each slot. The throughp~t and delay perform~nce of a network with an adaptive array IS computed by applying the standard Markov chain analysis of slotted ALOHA [1], [2]. It is shown that throughput levels comparable to CSMA are attainable with an adaptive array without the need for stations to be able to hear each other. The performance depends primarily on the number of adaptive array nulls, the array resolution, and the length of the randomization interval within each slot.

A

I. INTRODUCTION

LOHA packet radio communi.cation syste~s are ?f int.erest because they provide a Simple way of multiplexing many users into a single radio channel. In these systems radio terminals transmit packets to each other whenever they have information to send, regardless of whether other terminals may be transmitting at the same time. Because terminals do not coordinate their transmissions, packets from different terminals frequently collide. A collision destroys all packets involved, and these packets must then be retransmitted after a random delay. Collisions limit the maximum throughput at one receiver in an ALOHA system to 18% if the system is unslotted and to 36% if it is slotted [3]. Because of these low throughputs, much effort has been devoted to finding improved packet radio protocols. One wellknown improvement is carrier sense multiple access (CSMA) [4], in which terminals listen to the channel before transmitting to determine if it is busy. If the channel is busy, transmission is delayed until the channel becomes idle. Kleinrock and Tobagi have shown that choosing the retransmission probability carefully in a CSMA system can yield high throughputs [4]. However, the usefulness of CSMA depends on whether all terminals in the network can hear one another. When this is Paper approved by the Editor for CATV of the I~EE Communications Society. Manuscript received February 10, 1990. This work was supported in part by the U.S. Army Research Office, Research Triangle Park, NC. and by the Office of Naval Research. Arlington, VA, under Contracts DAAL0389-K-0073 and NOO014-89-J-I007 with The Ohio State University Research Foundation, Columbus, OH. 1. Ward was with the ElectroScience Laboratory, The Ohio State University. He is now with M.LT. Lincoln Laboratory, Lexington, MA, 02173. R. T. Compton, Jr., is with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212. IEEE Log Number 9106306.

not the case, as in satellite or mobile communications, CSMA is less effective. In the standard slotted ALOHA analysis, it is assumed that if two or more packets arrive in the same slot, none of them is received correctly. In reality. the correct reception of a packet depends not only on whether interfering packets are present, but also on the received power of each packet. Roberts [5] first noted that if one of the packets is of much higher received power than the others, it may still be correctly received. This "power capture" effect improves the throughput and delay performance of a packet radio system. Power capture has been studied by Abramson [3] and Namislo [7] when it occurs naturally as a result of different propagation distances from transmitt~r to receiver and/or channel fading. Lee [6] considered assigning random signal levels to the stations to induce the capture effect. Also. since the received power from a given direction is proportional to the receiver antenna response in that direction. directional antennas can he used to create the capture effect at the receiver. Binder ct al. (K J have considered using directional antennas to resolve potential crosslink conflicts in a multiple satellite packet system. In their work the direction to which an antenna is steered i~ obtained a priori from a form of scheduling used to set up each communication link. Their scheduling procedure. in addition to providing direction information. also reduces the contention somewhat at the expense of increased packet delay. In this paper, we examine the use of an adaptis'C antenna array to create a capture effect and thus improve the performance of a slotted ALOHA system. An adaptive array is an antenna system that controls its own pattern in response to the signal e~vironment [9], [10]. An adaptive array can capture a packet by pointing the peak antenna response toward that packet while simultaneously forming pattern nulls on other interfering packets [11]. An adaptive array can do this automatically without requiring any a priori direction information. Thus. there is no need for prearranged scheduling in a system with an adaptive array and the delay performance should be improved. Furthermore" an adaptive array provides a much stronger capture effect than an ordinary directional antenna, because pattern nulls are placed in the directions of contending packets. We shall show that the use of an adaptive array can provide throughput and delay performance comparable to that of CSMA. Moreover, with an adaptive array there is no need for users to be able to hear each other. In Section II we describe the communication system we

shall consider. Section III gives a brief overview of adaptive

arrays. Section IV describes how an adaptive array can acquire

Reprinted from IEEE Transactions on Communications, Vol. 40, No.2, pp. 292-300, February 1992.

418

ARRAY

OUTPUT

s;.V

I

r.;

:s :s

+ 1.

(21)

To find (I) for :2 l lV + 1. recall that Ol is the arrival angle of the acquired packet and define D 1 [H t - Hb/2. 01 + 01>/2]. Then [).~ia(l)

( 15)

where PI!(l) is the probability that a packet is acquired given 1 packets arc incident. and P.. . la (I) is the probability that a packet is successful given it is acquired and l packets are present in the slot. The ~L (I) depend on the arrival times and the length of the uncertainty interval. while the !)"lfL (/) depend on the arrival angles, the resolution capability of the adaptive array, and the number of available nulls. With the preamble code structure described in Section IV, the first packet in a slot is acquired as long as all subsequent packets in that slot arrive at least one bit duration Tv later than the first packet. If the first packet is not acquired, no packets are acquired for that slot. Thus,

Fa(I) = l P r { t 2 > i. 1

1 == 1 1

(1 _ ~ )1; 1 >

+ Ti; t J > f 1 + Ti; . . . . . t I > t 1 + Tv} ( 16)

== Pr{H 2 ~ D 1 · f) 3 ~ D l · · · · .H[ ~ D l } == E o1[Pr{ t1 2 ~ D 1·03 ~ D1 ... ·.O[ ri D1IB l

= Eli,

[g

Pr{ H,

~ D1IHd]

}]

(22)

where Eel [] denotes an expectation over the random variable H1, and we have taken advantage of the independence of the arrival angles. However,

Pr{ H t

~ D1IBd =

(1 - ~~ ).

(23)

which is independ-ent of Ol. Thus, (22) becomes 2~ j SN

+ 1.

(24)

Hence, from (15), (19), and (24), the success probabilities are

where the factor of I accounts for the fact that any of the l packets transmitted can be the first packet in the slot. If only a single packet is transmitted in a slot, it is acquired, so (17)

423

1

[=0

1~

l=1

O'

P~(l) ==

{

(1 - -1 )l( 1-~ (} )l-l~ U

27T"

0;

2~l~N+1·

1> N

+1

(25)

Given that the system is in state i, the probability of a successful packet transmission is the conditional throughput S(j) , given by

0.9

M

S(j)

=L

Qt(lJj )Ps(l ).

(26)

1=1

The average number of new packets entering the system state j is

In

(27) The Markov chain described above is irreducible. Since we assumed a finite population, all states are recurrent non-null. The states are also aperiodic. Consequently, this Markov chain has a limiting distribution denoted by 11"

= [1l" (0), 1l"(I), ·· · , 1l" (M )]

j, the number

(a)

0.9

(28)

where

1l"(j) = Pr{X=

.

= j} = lim,,_oc Pr{X k + n = jlXk = i }.

~

f

(29) The steady-state probabilities are found by solving the linear system of equations [21] 11"

= 1l"P

§

(30) 10

along with the constraint that

L 1l"(j) = 1.

(31)

25

(32)

) =0

First we examine the conditional throughput S(j ) of systems with and without an adaptive array. We consider a network of 50 users. We start with an example where P71 0.002 and p; 0.2. For this case. lvl p" 0.1, which is a low traffic situation where slotted ALOHA may typicall y be used. Fig . 8(a) shows the conditional throughput S(j ) and the new packet input rate Sin(j) versus the state j . Curves for various numbers of adaptive array nulls are also shown. For these curves we have Bb = 10° and u = 62. There is a significant increase in conditional throughput as the adaptive array is added and the number of nulls is increased. Also, note that there is a fixed number of nulls above which little further improvement is gained. The stability problems of ALOHA systems have been well documented [1], [2], [22]. The finite population ALOHA model is said to be stable if there is a single intersection point of the S(j ) and Sin(j) curves and this intersection point is in a region of low delay. In Fig . 8 we have intentionally chosen Pr high enough so that the system without an adaptive array is unstable. The curves with an adaptive array are stable. Moreover, for an adaptive array with 4 nulls or more. the

=

and the average throughput is M

S(j )1l" (j ).

(33)

j=O

In the steady state, the average input rate equals the average throughput, so (34) We use Little's theorem [23] to express the average delay D experienced by a new packet as

B

B

S in

S

D= =- = = .

-10

U» \ r~

VI. RES ULTS

AI

B = Lj1l"(j ).

35

:ilJ

Fig.8. Cond itional throughput comparison . For the curves with an adapt ive array: 0. = 10", U = 62. (a) M = 50, P» = 0.002, p, = 0.2. (b) M = 50, P. = 0.006. Without the adaptive arr ay, p, = 0.1; p, = 0.2545 with the adaptive array.

Once the 1l"(j ) are found. they can be used to determine the average throughput, delay, and backlog of the system. Given 7r(j), the average number of blocked terminals B is

-

:0

(b)

j =O

=L

15

J, Ihe num ber o f blocked

;\I

S

orblocked users

(35)

We now use these results to examine the performance of a slotted ALOHA system with an adaptive array .

424

=

=

0.9

200

S.(j)

180 160

~

!=

= ;; ....

.e

.

,., " Q

-=

~

r
"

"'~ 0.8). The effect of correlation on reducing the effectiveness of antenna arrays against interference suppression is as follows. With M antenna elements, the array has M - 1 degrees of freedom. Thus, as shown by theoretical and computer simulation results [1], [3], [4], [9], [10], an M antenna element

Fig. 4. for d>

Correlation of the real portion of the fading versus antenna spacing

= 90°.

8 , - - , - - - - - - , - - - - - - - . - - - -r - - - - - . ,

6

'" o v

~' .E

s

2

0'----'-------':------'------'------' 90

80

60

40

20

a

6 (Degrees)

Fig. 5. Antenna spacing required for the envelope correlation to remain below 0.5 as a function of t, located at an angle (P, we can write

(=1

. ( ) - ~1J..:j Ck.J W ~ J{ (,

C

HI

where ne is the set of integers such that I; ri f~4t· Denoting Lnr gne-iwotn == be and taking Fourier transforms of both sides of (A-3), we obtain the standard L-ray, or frequencyselective multipath description of fading channels,

So(w) == S(wn

D

== -

L

L set -

Tf)b~ktj)

f==1

(A-8)

where

(A-5)

1/

{=l

For this model to be useful, a statistical characterization of the set of M x N frequency functions (:~ ..J (w) must be provided. In our application, we shall assume that the terms in the various sums defining hi's arc random quantities and so it is reasonable to assert that the h(l' s are complex random variables. Furthermore. we assume that there are large number of terms in each sum and that each sum includes different random terms and. consequently, from the central-limit theorem. the hI" S, f == 1 ... L, ITIay be regarded as i.i.d. complex, zero-mean. Gaussian random variables. If we let wot n == f)n in the sums defining br, we write the real and imaginary parts as

1

where (/)1/ is the angle of arrival of the n-th ray. As we have already argued, the h;,(\) 's are complex i.i.d. Gaussian random variables associated with array numbers n, and therefore the sought-after correlations are determined by each h~(\) and different O"s. Thus, we seek the correlation coefficients between the following random variables: (k)

(k)

. (k)

h['

==:r p

+ I,Yr' l: ==

IJ (j) t

- ,,(.d -.L (

" (J). + 1,.tJ p •J

1.. ...J.\;1

and -·1 - ., .. 1"/ ~

where '1,(0:) == R(:"\b(n) ./(' " e

(A-6)

Now, it is reasonable to regard Hn IllOd ulo27f as i.i.d, unifonnly distributed random variables with the consequence that Ie and Ye are now independent and so lb, I is Rayleigh distributed and / be is uniform. This is then the rationale for regarding Ck] (w fin (A-5) as a complex Gaussian process in the frequency domain. For our application, the correlation among the elements of Ckj' s is of paramount importance. In order to facilitate the evaluation of these parameters, we must return to the basic definition of the br's in (A-6). We begin by considering the following geometrical model. This entails placing the users and the antenna array in a reasonable geometrical relationship, Without loss of generality assume that the antenna array is linear with M elements with identical spacing, D, between elements. We label the elements in ascending order. Users are located at arbitrary angles and distances with respect to the antenna array as depicted in Fig. 2. With each user, we associate a scattering angle of size 2~. This implies that all subpaths from the user to the antenna array are restricted to emanate from within this angle.

(A-9)

and ') L{ ?ie(0) -_ I III b(O:) P ,n - 1.-.- . ...ivl .

We note that since the en's are i.i.d. uniform, the real and imaginary parts of b~o) are independent for any n. We now calculate for any n

E[x~(»f = E[y;L\)f

=

~L

ELq;]·

(A-IO)

Tlf

It is now straightforward to calculate the four correlation coefficients

441

E [:E~k) .T~j)] = E

[y;k) y;j)]

= ~ L E [g; cos nf

[(I, - j)21r ~ sin ¢nJ J (A-II)

where the Jm's are Bessel Functions of integer order and

and

D

E[X~k)y~j)] = -E[x~j)y?)]

(A-I8)

z = 27r"I'

= ~ tE[g~Sin [(k - j)27r~ Sin¢n]] nt

(A-12)

we can integrate (A-I3) and (A-I4) and obtain the following convenient formulas for the desired correlation coefficients:

where

D

wo-

C

D

= 27rfoC

D = 21r,.

.

= Jo(z(k - J))

1\

~

+ 2 L..J

J 2m (z(k - j)) cos (2m Po)

=

Pr (

h

E < No + 1 0

Pr (II + f, > ~ _ (J~). S MP

350, the anal ytical result s for Pb are only smaller than the simulation results by a factor of 0 .3 or less . Unlike the omnidirectional. sectorized, and flat-topped pattern s. the binomial phased arra y did not exhibit constant two -dimensional gain as a funct ion of scan angle. Therefore . the use of the three-dimensional directive ga in as an " average " gain in (2 .23) is an approximation. By comparing Figs . 6 and 7 it may be concluded that a smaller value of average directive ga in might result in a better match between the simulated and analytical results for the binomial phased array . Nevertheless, these figures demonstrate the accuracy of (2 .23) when compared with extensive simulations . As noted in [13], use of a path loss exponent of n = 4 can result in overly optimistic estimates of system capacity and performance . The different base station antenna configurations demonstrate vary ing sensitivity to the path loss exponent , n. As illustrated in Fig . 8, the flat-topped beam system is highly sensitive to changes in the path loss

459

...,.........,.......,,.....---......,..--..-...

exponent. This is reasonable to expect since , when the CIR is large , the bit error rate is more sensitive to rela tively small changes in interference power. IV. SIMULAnON OF ADAPTIVE ANTENNAS AT THE

...,.......----..-.---,.-...,

Ie

PORTABLE UNIT TO IMPROVE REVERSE CHANNEL

Ie- ..

PERFORMANCE

100

In this section, we examine how the reverse channel is affected by using adaptive antennas at a portable transmitter. A flat-topped beam shape, as illustrated in Fig. 2 , was used to model an adaptive antenna at the portable transmitter. Since space is extremely limited on the portable unit, the gain achievable by the portable unit antenna will be considerably less than that at the base station. For this study, it was assumed that the portable unit could achieve a beamwidth of 60° with a side lobe level that was 6 dB down from the main beam. This corresponds to an antenna with a directivity of 4.3 dB . The pattern is similar to that shown in Fig. 5(c) except that the beamwidth is wider in this case. It was assumed that each portable unit was capable of perfectly aligning the boresight of its adaptive antenna with the base station associated with that portable unit. In this manner. portable units could radiate maximum energy to the desired base station. while reducing battery power proportional to the directivity of the portable antenna . Portable units with adaptive antennas were simulated for each of the five base statio n patterns described in Section III. As in Section III. average values of Ph were found by averaging the bit error rates of each user in the central cell, subjected to interference from the central cell and all immediately adjacent cells . The resulting bit error rates for these systems are shown in Fig . 9. Note that. com paring Fig. 6 and Fig . 9. the bit error rates for the reverse channel are improved when directive antennas are used at the portable unit. For omnidirectional base stations. the BER is only decreased by a small amount (20% or less) for K > 200 when steerable directive antennas are used at the portable unit. However, for highly directive base station antenna patterns such as the adaptive-sectorized pattern, the BER was decreased by an order of magnitude for K > 300 . In Fig . 10, we have defined the BER factor as the ratio of the BER with adaptive antennas at all portable units to the BER without adaptive antennas at the portable units. A small BER factor indicates that adding adaptive antennas improved the BER significantly. For example, a BER factor of 0.5 indicates that using an adaptive antenna at the mobile unit resulted in a reduction in BER of 50% compared with the case of omnidirectional antennas at the mobile unit. As shown in Fig. 10, the adaptive sectorized base station pattern improved greatly by adding adaptive antennas at the portable unit. The resulting BER for this base station configuration when using adaptive antennas at the portable unit was decreased by an order of magnitude

200

300

Numbct of Uan pct Ceq (K)

400

100 200 :JOO 0600 N.."bc, of UKr, per Cell f K )

SOO

(b) 0=3

(I) 0=2

I" ,.......-::-c,.....--

-

.....

,...-

-r--:-n

~1c. :

"::C lc-.

Ic ·J

101) 200 JOO ..00 Nllrnbcr of Ux n ptr Cell fK )

SOO

(c) 0=4

Fig. 9. BER for five different base stat ion configurations using adapti ve antennas at the portable unit for (a) n = 2, (b) n = 3, and (c) n = 4. These results were developed through simulation by averaging the BER of every user in the central cell.

-

0 ......1

o. .-

..e..,

S«toril~

- . Ad.,lIn F1at.lopped • Ad-.pli,,~Srr;'larfud

~

D.

,

., . D.J

..-r : . . ~

~

~

.

...

-~_

... .. ..... ",

*

....::,

~

I ..

...

. ....

-

~. -

D ~ "'-'- -'U." .... C. MIKI

:

"

~

4~

m

Fig. 10. BER factor, defined as the ratio of the BER with adapt ive antennas at the portable unit to the BER without adaptive antennas at the portable , for five different base station configurations using when using adaptive antennas at the portable unit. This comparison is made for n = 4.

compared with the BER when omnidirectional antennas were used at the portable unit. In general. the more directive base station configurations benefitted more from adding adaptive antennas at the portable unit. Using a 60 " beamwidth fiat-topped pattern with a -6 dB side lobe level at the portable unit, the reverse channel BER for omnidirectional base stations was only improved slightly over the case of omnidirectional antennas at the portable . For directive antennas at the base station, the improvements were more dramatic , as illustrated in Fig . 10. The relatively small improvements obtained by using adaptive antennas at the portable unit can be explained by the fact that when omnidirectional antennas are used at the mobile unit. no more than 1-0.455, or 0 .545, of the total interference power is due to users in adjacent cells (see Table III where f = 1/(1 + 8(1». When using adaptive antennas at the mobile unit, all users in the central cell will appear no different to the central base station than if they had used omnidirectional antennas. Thus, adaptive

460

TABLE

III

RATIO OF IN-CELL INTERFERENCE TO TOTAL INTERFERENCE, FUNCTION OF PATH

Loss

f,

more efficient reuse, and for more frequent reuse of signature sequences throughout a large coverage area.

AS A

EXPONENT, FOR FIVE BASE STATION ANTENNA

PATTERNS WITH OMNIDIRECTIONAL ANTENNAS AT THE PORTABLE UNIT

Base station antenna pattern

n=2

n=4

n=3

Omni

0.4535

0.6012

0.6927

Sectorized

0.4532

0.6008

0.6924

Adaptive

0.4524

0.6002

0.6920

Flat-topped

0.4534

0.6011

0.6926

Adapti ve-sectorized

0.4531

0.6007

0.6922

0.4552

0.6028

0.6939

1

1 + 8{3 (Eq.2.13) (values of

r3 from Table

2.1)

TABLE IV RATIO OF IN-CELL INTERFERENCE TO TOTAL INTERFERENCE, FUNCTION OF PATH

Loss

V. CONCLUSIONS

f,

AS A

EXPONENT, FOR FIVE BASE STATION ANTENNA

PAfTERNS WITH ADAPTIVE ANTENNAS AT THE PORTABLE UNIT. THIS DATA IS FROM THE SIMULATION DESCRIBED IN SECTION IV

Base station antenna pattern

n=2

n=3

n=4

Omni

0.6752

0.8155

0.8826

Sectorized

0.6749

0.8153

0.8824

Adaptive

0.6753

0.8152

0.8822

Flat-topped

0.6751

0.8154

0.8826

Adaptive-sectorized

0.6747

0.8150

0.8823

antennas at the portable unit will only reduce out-of-cell interference levels, Therefore. the maximum improvement in CIR. on the reverse link. that can be achieved by using adaptive antennas rather than omnidirectional antennas at the portable unit is only 3.5 dB. Table III shows several values of the reuse factor, f, defined in (2.12) as the ratio of in-cell interference to total interference. for several base station patterns when omnidirectional antennas are used at the portable unit. Similarly, Table IV shows values of f when steerable. directional antennas. with directivities of 4.3 dB, are used at the portable units. Comparing Tables III and IV. it can be concluded that the use of adaptive antennas at the base station does nothing to improve the reuse factor, f: however the use of adaptive antennas at the portable unit does allow f to be improved. When omnidirectional antennas are used at the portable unit, f is entirely determined by the cell geometry, the power control scheme, and path loss exponent, n, which is a function of propagation and not easily controlled by system designers. Using adaptive antennas at the portable unit, it is possible to tailor fto a desired value which is greater than the reuse factor obtained using omnidirectional antennas at the portable unit. Ideally, driving f to unity would allow system design to much less sensitive to the intercell propagation environment, when perfect power control is assumed. This is an important result for CDMA cellular systems because it indicates that use of adaptive antennas at the portable unit could help to allow greater capacity through

It was shown in this study that adaptive antennas, with relatively modest bandwidth requirements, and no interference nulling capability, both at the base station and at the portable, can provide large improvements in BER, as compared to omnidirectional systems. Analytical expressions which relate the average BER of a CDMA user to the antenna directivity and propagation environment were derived and used to determine capacity improvements offered by a number of antenna patterns. It was demonstrated in Section III that the linear phased array provided an order of magnitude of improvement over the omnidirectional base station. The low-gain (5.1 dB) flat-top pattern provided almost two orders of magnitude of improvement over the omnidirectional system. In addition, it was shown that up to three orders of magnitude of improvement can be achieved by adding a simple three element linear array to a three-sector base station. In terms of capacity, the results of Section III indicate that using adaptive antennas at the base station can allow the number of users to increase by a factor of 2 to 4, while maintaining an average BER of 10- 3 on the reverse link. The bit error tate on the reverse channel is further improved by adding adaptive antennas at the portable unit. Using a 4.3 dB gain antenna at the portable, the bit error rate for the directive base station configurations (but not the omnidirectional base station) was at least half of the bit error achieved without directive antennas at the portable unit. For the highly directive adaptive sectorized base station, the improvement was over an order of magnitude for user densities less than 425 users/cell when each user employed an adaptive antenna. Since the directivity of portable unit adaptive antennas is limited by the size of a handheld device, improvements achieved on the reverse channel at the portable are not as dramatic as gains achieved by adaptive antennas at the base station. In addition, cost issues may limit the application of portable unit adaptive antennas. However, the reduction in reverse channel BER may be critical in extremely high traffic environments. In addition, the portable unit is required to track the only current base station, while adaptive antennas at the base station must track every user in die cell. It should be noted, however, most importantly, Tables III and IV showed the increase in reuse efficiency which portable adaptive antennas provide. By using modest gains at the portable unit, such antennas ameliorate the loss in capacity due to intercell propagation through interference control. In short, adaptive antennas at the base station can have a major effect on bit-error-rate performance, but cannot impact the reuse factor" f. Conversely, it has been shown in this paper that adaptive antennas at the portable unit can provide no more than a 3.5 dB improvement in reverse channel CIR; however, they allow the reuse factor,

461

!,

to be altered. It should be noted, however, that the use of directional antennas at the portable unit can only result in an increase in reuse factor of approximately 1/3. It was assumed throughout this study that the adaptive algorithms and hardware could be designed to meet the specified requirements on beamwidth, side lobe level, and tracking ability. It should be noted that, unlike the arrays discussed in this paper, a properly designed adaptive array can null out interference. Conversely, tracking a large number of users with an adaptive array is nontrivial, and it was assumed that each of the base station arrays described here were able to track all of the portable units without error. The multipath channel was not considered in detail in this study; however, it will be significant in developing algorithms for successful adaptive antenna steering. Rather than tracking users, the adaptive array in a multipath environment must track the angle of arrival of multipath components in order to distinguish the maximum signal. This problem is currently under investigation. Furthermore, efforts are currently underway to develop bit error rate expressions which are accurate for small numbers of simultaneous CDMA users with non-identical power levels.

[12] R. K. Morrow and J. S. Lehnert, "Bit-to-bit error dependence in slotted OS/SSMA packet systems with random signature sequences," IEEE Trans. Commun .. vol. 37, Oct. 1989. [13] L. B. Milstein, T. S. Rappaport. and R. Barghouti, "Perfonnance evaluation for cellular CDMA," IEEE lSAC, vol. 10, May 1992. (14] B. Widrow, P. E. Mantey. L. J. Griffiths, and B. B. Goode, "Adaptive antenna systems." Proc. IEEE, vol. 55, no. 12. Dec. 1967. [15] R. Kohno, H. Irnai, M. Hatori, and S. Pasupathy. "Combination of an adaptive array antenna and a canceller of interference for directsequence spread-spectrum multiple-access system." IEEE lSAC, vol. 8, May 1990. [16] S. Anderson, M. Millnert, Mats Viberg. and Bo Wahlberg, "An adaptive array for mobile communication systems," IEEE Trans. Veil. Technol., vol. 40, Feb. 1991.

REFERENCES [1] T. S. Rappaport and L. B. Milstein, "Effects of radio propagation path loss on OS-COMA cellular frequency reuse efficiency for the reverse channel.·· IEEE Trans. Veh. Techno/ .. vol. 41. no. 3. Aug. 1992. [2] G. R. Cooper and R. W. Nettleton, "A spread-spectrum technique for high-capacity mobile communications." IEEE Trans. Veh. Technol .. vol. VT-27, Nov. 1978. [3] A. Salmasi ... An overview of advanced wireless telecommunication systems employing code division multiple access." Con! Mobile, Portable & Personal Commun., Kings College, England, Sept. 1990. [4] W. C. Y. Lee. Mobile Cellular Telecommunications Systems. New York: McGraw Hill, 1989. [5] K. S. Gilhousen et al., Han the capacity of a cellular COMA system:' IEEE Trans. Veh. Technol .. vol. 40, May 1991. [6] M. B. Pursley, "Perforrnance evaluation for phase-coded spread spectrum multiple-access communications with random signature sequences, " IEEE Trans. Commun., vol. COM-25, Aug. 1977. [7] W. A. Gardner, S. V. Schell, and P. A. Murphy, "Multiplication of cellular radio capacity by blind adaptive spatial filtering, " IEEE Con! Sel. Topics Wireless Commun. Mobile. Vancouver, B.C., Canada, Jun 1992. [8] S. C. Swales, M. A. Beach, D. J. Edwards. and J. P. McGeehan, "The performance enhancement of multibeam adaptive base-station antennas for cellular land mobile radio systems," IEEE Trans. Veh. Technol., vol. 39, Feb. 1990. [9] R. T. Compton, Adaptive Antennas. Englewood Cliffs, NJ: Prentice Hall, 1988. [10] B. Agee, "Solving the near-far problem: Exploitation of spatial and spectral diversity in wireless personal communication networks, " in Proceedings Third Virigina Tech Symp. Wireless Personal Commun., June 1993. [11] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1981.

462

Adaptive Transmitting Antenna Arrays with Feedback Derek Gerlach and Arogyaswami Paulraj

Abstract- We address the problem of transmitting multiple cochannel signals from an antenna array to several receivers so that each receiver gets its intended signal with minimum crosstalk from the remaining signals. In addition to the usual "information" mode, we propose a "probing" mode during which probing signals received at the mobiles are fed back to the transmitter. These probing signals are used to identify an unknown propagation environment, enabling the transmitter to form the necessary transmission beampatterns.

A

I. INTRODUCTION

DAPTIVE receiving antennas have been widely applied to military and communication problems to eliminate unwanted interference or separate multiple signals. The aim of receive beamforming is to form a spatial filter that passes the desired signals and suppresses unwanted components. A receiving beamformer can observe its own output and modify its spatial filtering to improve the signal suppression/enhancement [1] . By contrast, the aim of transmit beamforming is to launch a signal into a propagation environment so that each receiver gets its desired signal without crosstalk from the signals intended for other receivers. This task is complicated by the presence of reflecting bodies of which the transmitter has no knowledge. The proposed adaptive transmit beamforming approach uses feedback of the signals received at the mobiles. This feedback makes possible the transmission of multiple signals to multiple receivers with low crosstalk, even in the presence of an unknown multipath environment. While receiving adaptive antenna arrays have been widely studied [2], [3], the transmit problem is equally important and has received little attention so far, except in [4]. In this letter, we formulate the adaptive transmit array problem and present simulations of its signal separation performance. We consider antenna arrays at the transmitter only, and the receiver has a single omnidirectional antenna. II. PROBLEM STATE~IENT AND ASSUMPTIONS

The goal of adaptive transmit antenna arrays is to send multiple cochannel signals from an antenna array through Manuscript received April 18, 1994; approved July IS, 1994. This work was supported by the Army Research Office under Grant DAAH04-93-G0029 and by ARGOSystems, Inc. under Subcontract 59613. The associate editor coordinating the review of this letter and approving it for publication was Prof. Moura. The authors are with the Information Systems Lab, Stanford University, Stanford, CA 94305 USA. IEEE Log Number 9405627.

Fig. 1. Multiple infonnation bearing signals transmitted from an array to multiple mobile receivers.

a propagation environment to several receivers so that each receiver only gets its intended signal with minimum crosstalk. Let (1)

be the d information bearing signals intended for d remote receivers. Let the antenna array consist of tri transmitting elements, and let the complex vector channel from the array to the kth receiver be given by ak:

(2) where aik is the complex channel response from the ith element to the kth receiver. The channel vector ak represents the total channel including the transmitter electronics, antenna array, and reflections.within the medium. In order to ensure that the vector channel is adequately described by a single vector, we need the following assumption:

bm p

«

BW- 1

(3)

where bm p is the maximum differential delay due to multipath in the propagation medium, and BW is the information signal bandwidth (same for all information signals). This narrowband condition is present in today' s advanced mobile phone system (AMPS), which has a 25-KHz bandwidth. Digital systems which meet (3) do not suffer from lSI.

Reprinted from IEEE Signal Processing Letters, Vol. 1, No. 10, pp 150-152, October 1994.

463

We can now define a channel matrix (4)

where A is a m x d complex matrix in which the (i, k) entry gives the complex channel gain from the ith transmitting element to the kth receiver. The channel matrix provides a complete description of all reflections and scattering in the environment. In order to transmit the d signals to the receivers, let W j be the beamforming weight vector for the information signal. Let (5)

receivers form the entries of the probing response matrix B. Next, the complex amplitude data is fed back to the transmitter. Knowing B, the base can estimate the matrix A. Let the I probing signals be

Pl(t), ... ,pl(t).

(11 )

Unlike the usual information signals, each probing signal is sent on an orthogonal channel (time, frequency, code) so that the receivers may measure the response of each probing signal. As before, each probing signal is transmitted according to its own probing vector

If we consider the array output due to only the jth signal t) with its corresponding weight vector W i: then the signal recei ved at the kth receiver will be

(12)

Sj (

(6)

where CJk represents the information signal amplitude received at the kth receiver, which was intended for the jth receiver, and * denotes conjugate transpose. If we let [C]jk = Cjk, we have

W*A=C

(8)

Diagonal elements of C are desired signal levels, and offdiagonal elements of C are crosstalk amplitudes. To ensure that each information signal is received only by its intended receiver with unit amplitude, we would like

C =1.

(9)

Since each element adds a degree of freedom, to achieve (9), we must have m 2 d. III. INCORPORATION OF FEEDBACK The channel matrix A summarizes the channel including both the antenna array and the propagation environment, and matrix A is not known to the transmitter. We therefore cannot directly find W such that "W* A = I. We propose to use feedback from the receivers to estimate A and hence W. Once A is estimated, a W that achieves (9) is

W=A+*

(13)

where bj k is the amplitude received at the kth receiver due to the jth probing signal. If we let [B]jk = bj k , we have

(7)

Equation (7) is a vector version of the familiar statement

in). reCeiver ) = (transmitter) . . x (h c anne I gaIn ( amphtude amplitude

The response at the kth receivers due to the jth probing signal is given by

(10)

where + denotes psuedoinverse. To estimate A, we introduce the concept of probing and information modes. In the probing mode, the transmitter transmits probing signals, whose responses at the receivers are measured and fed back to the transmitter from which A is estimated. During probing, transmission of the usual (information) signals is temporarily halted. Instead, the array is excited in tum by several probing signals, and each receiver measures the relative complex amplitude response of each probing signal. These complex responses measured at the

B

= V* A.

(14)

The probing signals are agreed on by the transmitter and receiver. One choice for Pj ( t) is Pj

(

t) = {

~xp( ua; t )

o

if t E

[(j -

otherwise

1)f, j f]

(15)

where We is the carrier frequency, and T is the probing signal duration. In any case. since the 1probing signals are orthogonal using a bank of 1 matched filters matched to the probing signals, each receiver can measure a column of B corrupted by measurement noise: (16)

where the entries of E are assumed to be zero mean i.i.d. Gaussian random variables. Next, each mobile digitizes the received probing signal amplitudes and feeds these back to the base on its own reverse (digital modulation and assumed error free) channel. These reverse channels are assumed to be available, and for the purposes of this discussion, they can be on different frequency channels or even a wireline channel. The spatial reuse of the reverse channel is an independent problem and has been well studied [3]. The cell site assembles 13 and then computes A. Knowing V, which are the inputs to the channel, and B, which are the noise-corrupted probing responses, the transmitter identifies A using a least squares estimate

464

(17) where + denotes the pseudoinverse operation. Once A is in hand, the transmitter can determine W using (10). Each additional probing vector provides another equation involving A. Since A has m rows" the condition to uniquely determine A is 1 2: m. Additional probing vectors will improve the accuracy of the least-squares estimate (17).

Channel reuse in an access method therefore consists of the following five steps: 1) Transmit information signals to the receivers using information weight vectors W. 2) Monitor the level of crosstalk at the receivers periodically, and halt the information transmission when the crosstalk exceeds a threshold of acceptab ility. 3) Enter the probing mode: a) Choose probing vectors V. b) Transmit the probing vectors V , and measure the response matrix B at the receivers. c) Feed back the probing response matrix B to the transmitter. d) Estimate A via A = V+*B. e) Form information signal weights according to W =

A+* .

. 'l:OP: 8prObesIWavelength

~

:s .1l

£. 10.

IV. SIMULATIONS

Simulations were carried out to evaluate the performance of an adaptive array sharing a single channel among two receivers simultaneously . Using a six-element circular array with a 15.0° beamwidth, a beampattern for each signal was created according to (17) and (l0). The propagation environment contained 20 local scanerers for each mobile placed randomly in a 250 wavelength vicinity of the mobile. Energy arrived at the receivers only via local scatterers, and no line of sight was present. The receivers moved around the transmitter in a circular path of radius 5000 wavelengths (carrier = 900 MHz) at 2.5 inilhr, maintaining a fixed angular separation. To track A, the transmitter periodically alternated between probing and information mode. Because the channel varied as the receivers

2

~ '5 o

10-' L - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - - - - ' ' - - -,, - - . l 0 .4 0 .5 0 .6 0.7 0 .8 0.9 1 1.1 1.2 1.3 1.4 Mobile Spacing in Beamwidths (=15 degrees)

Fig. 2.

4) Resume information transmission with the new choice of weight vectors. 5) Go to step 2. The frequency with which it will be necessary to enter the probing mode will be determined by the receivers' speeds and the propagation medium complexity . If two mobiles sharing a channel approach each other, then the channel matrix will become singular. Since this method is not designed to accommodate singular channels, the transmitter should hand off one receiver to a new channel. To accommodate the probing signals in a TDMA system, a portion of each slot should be devoted to probing. If the mobile motion, and hence, the required probing rate is slower, probing could occur every n th slot.

.

Bottom .: 16 _ p robesiw aYe l e ng~

Outage probability versus mobile spacing.

moved, the interference was least immediately after each probe and worst just before the next probe. Two probing rates of 8 and 16 probes/wavelength were used, and each real entry of A was specified with 4.2 bits, for a net feedback rate of 1379 and 2753 bps, respectively. Fig. 2 shows the probability that the channel's SINR was below 7.3 dB for various mobile spacings. The 7.3 dB threshold is a BER of 10- 3 for B~SK. As the mobile spacing increased, the channel quality increased because the two channel vectors were less parallel. V. CONCLUSION

We have proposed an adaptive transmit antenna array that uses feedback to achieve low signal crosstalk at the intended receivers . Simulations show that at low mobile speeds (2.5 mi/hr), adequate signal separation requires feedback data rates in the thousands of kilobits per second, making the approach most applicable for static of slow-moving receivers. Methods of reducing the feedback rates are needed.

465

REFERENCES [1] B. Widrow and S. Stem. Adaptive Signal Process ing . Englewood Cliffs. NJ: Prentice-Hall. 1985. [2] A. Naguib and A. Paulraj, "Performance of COMA cellular networks with base-station antenna arrays," in Proc. Int. Zur ich Seminar Digital Commun. (Zurich. Switzerland ), Mar. 1994. [3) B. Sublett. R. Gooch, and S. Goldberg, "Separation and bearing estimation of co-channel signals," in Proc. MILCOM '89, May 1989, pp. 629--Q34. (4) O. Gerlach and A. Paulraj, "Spectrum reuse using transmitting antenna arrays with feedback," in Proc. Int. Conf Acoust.. Speech, Signal Processing (Adelaide, Australia), Apr. 1994, pp. 97-100.

Adaptive Antennas for Third Generation DS-CDMA Cellular Systems George V. Tsoulos, Mark A. Beach, Simon C. Swales Centre for Communications Research University of Bristol Bristol, UK Fax: +44 117 9255265, Tel: +44 117 9287740 e-mail: [email protected]

Abstract: This paper considers the perfonnance of a DS-CDMA system employing adaptive antenna technology at the base station site for both an Umbrella and a Micro-cell in a hierarchical cell structure. The possible advantages and problems from such a deployment are discussed. By exploiting the capabilities of Ray Tracing to provide the complex channel impulse response, a new adaptive antenna simulation model is presented along with some initial results for the perfonnance of well known adaptive algorithms in a multiple interference scenario. These provide insight into how the adaptive antenna operates when used in conjunction with DSCDMA and illustrate the potential benefits. Finally, propagation measurements are provided in order to validate some of the claimed capabilities. 1.

INTROD UCTION

Figure 1: Hiera rchical cell structure concept.

The need for mobile radio systems with increased spectrum efficiency is paramount in the drive towards third generation systems [1]. Currently favoured solutions in today's systems include the deployment of smaller cells as well as fixed sector, or multi-beam antennas, at the base stat ion (BS) site. In terms of modulation schemes and access techniques, application of spread spectrum modulation with Code Division multiple access (COMA) and especi ally Direct Sequence (OS) COMA, look to be amongst the favoured approaches. Recognising that the ambitious requirements of UMTS & FPLMTS can not be fulfilled with the known cellular architectures (macro, micro, pico cells) led to the conception of the idea of a hierarchical cell structure [2 - 3]. The key issue for this type of cell architecture is to apply multiple cell layers to each service area , with the size of each layered cell tailored to match the required traffic demand and environmental constraints (Fig. 1). In essence , microcells will provide the basic radio coverage but they will be overlaid with Umbrella cells to maintain the ubiquitous and continuous coverage required. Especially for the OS-COMA system, this mixed cell technique gives answers to situations where a possible performance degradation may occur, e.g. fast moving users requ iring handover, or black spots in coverage .

Advanced antenna techniques, such as adaptive antennas, is an area which seems to gather momentum recently [4 - 7], as another possible way to increase the efficiency of a given system. Adaptive antennas, based on the spatial filtering at the base station, separate the spectrally and temporally overlapping signals from multiple mobile units. This can be exploited in many ways such as: - Support a mixed architecture. - Comb at the near-far effect. - Support higher data rates. - Combine all the available received energy, (multip ath). In the following section , a brief discussion will be presented on the application of adaptive antennas in an Umbrella cell. The conclusions are taken from an earlier publication [5], but include some additional propagation measurements to support previous claims. The remaining sections focus on the use of adaptive antennas in a microcellul ar environment operating with OSCOMA. This work includes the development of a detailed Ray Tracing based simulation model and the pre sentation of some initial results .

Reprinted from Proceedings of 45th Vehicular Technology Conference, Vol. 1, pp. 45-49, July 1995.

466

gree of spati al selectivity that can be applied by the antenna system, i.e. whether to form a single narrow beam or adopt an optimum combining approach. In a large cell application, the use of an A DA based approach for a beamformer, would po tentially be more desirable since the ADA of the signals has a relatively narrow angular spread [9]. In a microcellular environment, th e angul ar spread of the signal from a single user is much greater , (figures 6b , 6c), due to the lower height of th e BS antenna and the close proximity of the scattering objects. Also, the ADA of the signa ls will change rapidly, with the do minant direction not always towards the desired user , as in the large cells case . Therefore , in the microc ellular case , the optimum combining approach see ms to be more flexible, providing increased capacity, as it will be shown in the following sections.

2. A N ADAPTI VE BASE STATIO N A NTENNA FOR THE UMBRELLA CELL OF A MIXED CELL STRUCTURE The potential advantages offered by employing an adaptive ante nna at an U mb rella BS site with a OSCOMA system, can be summarised as follows :

• • • • •

Mitigation of the near-far eff ect. Capacity enhancement. More efficient handover. "In-f ill" cove rage for the dead-spot s. A bility to support high data rates.

These wer e discussed in greater detail in an earlier publi cati on [5], although in orde r to support the last claim, some propagation measurements have been carried out. The measurements were performed with a Fast Fouri er Transform (FFT) D ual Channel Sounder at 1.823 GH z [8]. The RMS del ay spread was calculated using a 10

900 "0

750

'" 600 '" .s e-, '"

a; 0

Ul

.:

f:

450

..J

II:

300

distance (meters) Figur e 2: Wid eb and measurement s

dB power window on each measured impulse response profi le. The results are shown in figure 2, while figure 3 shows the map of the area where the mea sur ements were per formed . For the umbrella cell base sta tion which was at the roof of a building with approximat e height 50m, two ante nnas wer e used : on e omnidirectional end-fed dipole (identical to the mobil e ante nna) and one directional shro uded yagi with 15dBd gain. From the above figur e can be see n that th e RMS delay spread is much less for the case of the directiona l antenna, with a reduction which can be up to 1/5. The reduced de lay spread results in less int er symb ol interferenc e and, therefor e, provides the pos sibility of suppo rting higher bit rate services .

Figur e 3: Map of the area un der investigation

4. SIMULATION MODEL T he simulation model can be separated into two basic block s: a) The block which generates the impulse response of the channel under investigation. This is done with th e help of a Ray Tracing simulatio n tool developed by the Un iversity of Bristol [10]. Th e input parameters include the number of reflections and diffractions, th e tr ansmitted power, antenna radiation patterns, etc. Th e result ant output file includes the time delay, the angle of arrival and the power of each received ray. h) The block which simulates th e adaptive antenna array, illustra ted in th e next figur e 4.

3. AN ADAPTI VE BASE STATION A NTENNA FO R SM ALL CELLS The angle of arrival (ADA) of the radio signal, along with its multipath compon ents. dir ectly affect s the de -

467

-

,

X,

(k)

where N is the total number of antenna elements. The desired, or reference signal, roCk) is simply the PN sequence from on e user, (i.e. no data modulation is considered at the moment) , and the error signal is defined as the difference between the array output and the desired signal e(k) = y(k) - roCk). This model for the adaptive antenna offers the capability of selecting one from several adaptive processing algorithms, such as the LMS , NLMS, RLS, SQRLS and the OMI, [11 - 13].

x,(kJ

y (k) Array Output

J---r---~

An/enna Array I >-I....,+-T-+-+-~ (N elements)

Adap tive Co ntrol Pro cessor

5.

Reference '. (k)

The aim for the simulations is to investigate the performance of the adaptive algorithms on an environment basis and to provide insight into the mechanism followed by the adaptive antenna, when operating in conjunction with OS-COMA. Parameters used in the simulations include: averaging over 15 runs, 8 antenna elements with half wavelength spacing, 1023 chips M-sequence with 1.25 MHz chipping rate, step for the LMS and NLMS algorithms 0.01 and a value of 1 for the forgetting factor for the RLS and the SQRLS algorithms. From figure 5 can be seen that, as it was expected, the recursive least squares algorithms, converge very fast, (within around 50 samples, while neither of the LMS - NLMS have reached the same level even after ten times that time) . The RLS and the SQRLS algorithms have very similar behaviour, with the SQRLS giving the best output and being more robust. The choice of an adaptive algorithm must be made on the basis that the algorithm must be able to rapidly acquire and track the signals in a variety of mobile scenarios. Therefore the obvious choice is either of the RLS - SQRLS algorithms. In the following simulations the RLS algorithm is used.

Figure 4: Adaptive Antenna Arr ay

x.(k) is the sample of the total received signal at the nth element at instant t = kT, where T is the sampling interval, as well as being the chip duration of the PN sequence. x.(k) consists of the desired and interfering OS-COMA signals and random noise, and it can be expressed as:

x.(k) =

2: 2: hm,e Jkd(. -I )sin (~,)rm(k M

R

m=l r=1

t,)

SIMULATION RESULTS

+ N(k) (1),

where h"" and r",(k) are the elements of the vectors of the impuls e response and the OS-COMA signal from the mth user respectively:

h, = [hi'll' hm2," ' , hm,," ' , hmR]T, r m = [rm(k), rm(k - t l ) , " ' , rm(k - t,),· .. , rm(k - tRW. rm(k) = dm(k)· PNm(k) . ei~m, with dm(k) the binary data and 'Pm the carrier phase of user m. N(k) represents the

random Gaussian thermal noise. M is the total number of users, R is the total number of rays, d is the interelement distance, {J, and t, are the angle of arrival and the delay of each ray r respectively and [ ]T denotes the transpose. Although the total received signal at the n th antenna element is calculated by considering the interelement phase shift for each incoming ray , (n - 1)kd sine {J,), depending on the environment under investigation, it can also be calculated directly from the ray tracing to ol. The output from the adaptive array in vector notation is: y(k) = wT(k)x(k), where w(k) and x(k) are the weight and element vectors respectively. Using (1), this gives :

0 r---;--~-,",-.,..--"'---c----,-----,---,----, - 10

1.. .• _ tll-~==="'-l==='4-=~======"'"'-1 _ ~_.

.. •

100

:-. . _.

.:..

200

:_..

samples

J . _.

300

~

__

;.. _

400

Mea n weight for the second antenna eleme nt

Figure 5: Mean weight convergence for different algorithms, with 16 users.

468

soo

SINR_IN

-+-

;

~ a:

z

0;

.... _

1------.----;.--

- -.-

than steering it towards the first desired ray, because there is much more interference around the first ray which would be accommodated by the main lobe and hence would decrease the output SINR.

"_ ' . . .. .. •, . . ..

,

.. __

- ._

,~ _

__ _

,

,~

.

If better output SINR than the one depicted in figure 6a , is needed, then an increase in the antenna elements would offer great improvement, as it is depicted in figure 7.

_..

.

·30 ' - - - - - - '- - - - ' - - - -........- - - - ' - - - - 4 12 8 20 24 Users

( a)

18 . - - - - - . - - - - - - - - - , , . - - - - - - - - , - - - - - . ,

16

a:

14

;!; 12 CIl

10

8

·30

0

Angle

61!!-- - - - ' - - - - - " " ' - - - - - - - - - - l 20 16 12 4 8 No of elements

•

Ifl)ut SINA ... ·19.61 LIB

30

Figur e 7: Output SINR for the RLS algorithm and 16 users as a function of the number of a nte nna elemen ts,

(h i

Simulations showed that the influence of the thermal noise (modelled as White Random Gaussian noise), to the adaptive antenna performance is negligible. For example , for a microcellular environment with 16 users and the RLS or the SQRLS algorithm, there is a reduction of less than OAdB in the output SINR. This maximum reduction corresponds to the rather worst case situation of an input SNR of 3dB . The above behaviour can be explained on the ground that the influence of th ermal noise in a system can be neglected when traffic in the system is close to its capacity limit, because then interference power becomes a dominant factor for determining communication quality and channel capacity. This obviously is even stronger for the case of OSCOMA .

-10

·60

30

30

60

90

I.e)

Figure 6: (a) Output SINR for the RLS algorithm as a function of the number of users. (b) & (c) Produced radiation patterns for 8 and 24 users respectively.

6.

The results depicted in figure 6 show that the array is capable of adapting to the given user scenario even with as many as 24 users . It has to be mentioned here that the SINR values shown in figure 6a are the mean values after convergence. By comparing the results depicted in figures 6b and 6c, the concept of the " smart" antenna is revealed: Although the array should direct its main lobe towards the ray with the maximum incoming power, its first sidelobe towards th e next ray with the next maximum power and so on, it doesn 't do so for the case of figure 6c. The reason for this behaviour is that the criterion used by the adaptive algorithm is the optimum SINR. This is going to be achieved by steering the main lobe towards the seco nd desired ray rather

DISCUSSION

The advantage from using an adaptive antenna with a OS-COMA system is two-fold: First , the output SINR is greatly improved, which corresponds to an improvement on the capacity of OSCOMA, which can be substantial. Second, the produced radiation pattern has a directionalit y which varies according to the environment under invest igation. For an umbrella cell scenario, due to the small number of signals and their very narrow angular spread, the produced radiation pattern can be very directional, which can be exploited in a number of ways as it was described in [5]. Even for the microcells, where the number of users is great and the angular distribution

469

of the incoming signals very wide, the produced radiation pattern is going to be better than an omnidirectional pattern (even slightly). Obviously, the pattern oriented analysis for the benefits achieved with an adaptive antenna, (discussed in [5]), can not be applied for the case of microcells.

REFERENCES

[1] IBC Common Functional Specification, "Mobile Communications: General Aspects and Evolution", Specification RACE D731, Issue D, Dec. 1993. [2] Hakan Eriksson et aI, "Multiple Access Options for Cellular Based Personal Communications", 43rd VTC, Secaucus, New Jersey, USA, May 18 - 20 1993, pp. 957-962. [3] S. Chia, uThe Universal Mobile Telecommunications System", IEEE Communications Magazine, pp 54-62, December 1992. [4] J.S.Winters, "Signal acquisition and tracking with adaptive arrays in digital mobile radio system IS54 with flat fading", IEEE Transactions on VI: Vol. VT-42, No.4, November 1993, pp. 377-384. [5] G.V.Tsoulos, M.A.Beach, S.C.Swales, "Application ofAdaptive Antenna Technology to Third Generation Mixed Cell Radio Architectures", 44th VTC, June 8-10 1994, Stockholm, Sweden, pp. 615-619. [6] G.V.Tsoulos, M.A.Beach, S.C.Swales, "Adaptive Antennas for Third Generation Cellular Systems", 9th ICAP, 4 - 7 April 1995, Eindhoven, the Netherlands. [7] Race Tsunami Project, "Requirements for Adaptive Antennas for UMTS", R2108/ART/WP2.1/DS/I/ 004/bl, 22 April 1994. [8] M.A.Beach, S.Chard, J.Cheung, T.Martin and T.Wiltshire, "Description ofthe advanced handover experiment", PLATON R2007, 1993. [9] S.C.Swales and M.A.Beach, Direction Finding in the Cellular Land Mobile Radio Environment", lEE Fifth International Conference on Radio Receivers & Associated Systems, RRAS90, University of Cambridge, England, 23rd - 27th July 1990, pp.192-196. [10] G.E.Athanasiadou, A.R.Nix, J.P.McGeehan, "A Ray Tracing Algorithm for Microcellular Wideband Modelling", 45th VTC, Chicago, USA, July 1995. [11] Adaptive Filter Theory, S.Haykin, 2nd edition, Prentice Hall 1991. [12] Introduction to Adaptive Arrays, R.Monzingo, T.Miller, John Wiley, 1980. [13] Advanced Digital Signal Processing, J.Proakis et al, Macmillan Publications, 1992.

In a system like the DS-CDMA, the optimisation process must be repeated cyclically for each desired user. This can be done either in parallel with the help of a bank of beamformers or with one time shared beamformer. Considering as an example, the case of a channel which is sampled every 1ms, the following can be mentioned: • For the case of an umbrella cell with 10 users, the time available to the Beamformer to optimise its response for each user in a serial mode, corresponds to 125 samples for a 1.25MHz PN sequence. This means that if fast algorithms are used, the use of one Beamformer in a serial mode, can be possible for this kind of cell structures. • For the case of a microcell with 24 users, the samples available for convergence when one Beamformer is used, are limited to 52. This obviously indicates the need for a bank of Beamformers and parallel beamforming. 7.

CONCLUSIONS

Work presented in this paper discussed the application of adaptive antennas in a third generation DS-· CDMA mixed cell architecture system, at both the umbrella and the microcell base stations. It was shown that an adaptive antenna can be used in order to enhance the performance of a DS-CDMA system. In the microcellular environment, simulation results were presented which employed a Ray Tracing tool to provide the radio channel characteristics. Work currently under way is investigating the performance of an adaptive antenna in different cellular environments with moving users. Also, different forms of adaptive antennas are considered as a function of the environment they are operating, in an attempt to provide a unified approach for all the different environments. ACKNOWLEDGEMENTS

George V. Tsoulos wishes to thank the Centre for Communications Research (University of Bristol) for his postgraduate bursary. The authors would like to thank Professor J.P .McGeehan for his continuous encouragement and the provision of laboratory facilities. Also, the authors would like to thank C.M.Simmonds for the propagation measurements and the postprocessing of the results. Finally, many thanks to G .E.Athansiadou for her help with the Ray Tracing and M.P.Fitton for his help with the field trials.

470

The Spectrum Efficiency of a Base Station Antenna Array System for Spatially Selective Transmission Per Zetterberg, Student Member, IEEE, and ·Bjorn Ottersten, Member, IEEE Abstract- In this paper we investigate the spectrum efficiency gain using transmitting antenna arrays at the base stations of a mobile cellular network. The proposed system estimates the angular positions of the mobiles from the received data, and allows multiple mobiles to be allocated to the same channel within a cell. This is possible by applying a transmit scheme which directs nulls against co-channel users within the cell. It is shown that multiple mobiles per cell is an efficient way of increasing capacity in comparison with reduced channel reuse distance and narrow beams (without directed nulls). The effect of the spatial spread angle of the locally scattered rays in the vicinity of the mobile is also investigated.

U

1.

INTRODUCTION

SING antenna arrays, at the base stations, to perform spatially selective reception and transmission is a newly proposed way of increasing the capacity of a cellular network [2], [13], [15]. In [13] and [15], the reduction of the channel reuse factor is investigated as a means of increasing capacity. The analysis in [13] assumes that ideal sectorized beams are formed in the direction of the mobile, thereby reducing the probability of co-channel interference. In [15] the antenna array outputs are linearly combined to produce the least mean square error at the output. Since the weights are updated at the fading rate, not only is co-channel interference suppressed but the fading is also mitigated. For base to mobile communication reuse of weights adapted during reception is proposed in [15]. However, this requires a system which uses time duplex division (TDD); that is, contiguous timeslots are allocated for mobile to base and base to mobile communications (at the same frequency). Since outdoor systems such as TACS, GSM, DCS-1800 and IS-54 use different frequency bands for receive and transmit [12], the base to mobile scheme described in [15] cannot be applied in any of these systems. Since it is desirable to increase capacity also in the base to mobile link, we here investigate a transmit scheme which does not rely on reuse of weights adapted during reception. The technique here is based on array response and directional information. In the proposed scheme, the angular positions of the mobiles are estimated during reception and then used to calculate the transmit weights for array transmission. The problem of angle estimation is not addressed in this paper, and we refer to [71, Manuscript received May 25, 1994; revised February 8, 1995. This work was supported in part by the Swedish National Board for Industrial and Technical Development (NUTEK). The authors are with the Royal Institute of Technology, S-100 44 Stockholm, Sweden. IEEE Log Number 9413245.

[9], [10] and [14] where algorithms for solving this task may be found. In order to calculate the transmit weights, the antenna transfer function, at the transmit frequency, is assumed known. In this paper we introduce a new approach for increasing capacity. While [13] and [15] explore reduced reuse distances we here also investigate the reuse of channels within the cells (with unchanged distance between co-channel cells). This permits the use of a simple dynamic channel allocation scheme which avoids major interferers from getting close (in angle) to the desired mobile. Transmit weights are chosen such that a main beam is pointed at the desired mobile with nulls in the direction of co-channel interferers within the cell, but not outside the cell. It is possible to direct nulls against cochannel users in other cells also. However, the implementation of such a scheme in the downlink of a TDMA system might be difficult due to synchronization problems. In this paper, a comparison between reduced cluster sizes and multiple mobiles per channel is also made. Our results indicate that the latter technique is more effective. However, it should be kept in mind that the transmit scheme directs nulls only against co-channel users within the cell. In the analysis, the spread angle of the locally scattered rays in the vicinity of the mobile is a crucial factor. We find that it is possible to increase the capacity between two and twelve times using up to 20 antenna elements. The capacity is largely dependent on the spread angle of the locally scattered rays in vicinity of the mobile and on the number of antennas at the base stations. The paper is organized as follows: Section II explains the cellular network, the base station antenna array transmission system and the propagation modeling. The weight selection algorithm used in base to mobile communication is derived in Section III-A. In Section III-B, the channel allocation scheme is presented and finally, Monte Carlo studies are used in Section IV-B to determine the spectrum efficiency gain. II.

PRELIMINARIES

A. The Cellular Network

The coverage area of the mobile radio system is assumed to be divided into a network of hexagons [8], where each hexagonal cell is covered by a base station site. The channel reuse factor (cluster size) will be denoted with C, the cell radius with R, and the channel reuse distance with D. The parameters C, Rand D are related through D == J3CR. Thus, a large set of channels, C, implies a large distance

Reprinted from IEEE Transactions on Vehicular Technology, Vol. 44, No.3, pp. 651-660, August 1995.

471

Fig. 2.

Illustration of the transmission system.

Fig. 1. The Cellular Network with channel reuse factor four, C = 4.

between co-channel cells, D. Increase in C means decreased interference but also a decrease in the number of channels available in the cells . In this paper, the hexagonal cells are divided into three 1200 sector subcells . Each of these subcells uses a fixed third of the channels available in the hexagonal cell. The subcells are covered by 1200 base station antenna arrays. The sectorization reduces the number of interfering cells in the first tier of interferers from six to two. The concept is illustrated in Fig. I for the case C = 4. Interferers in the second tier and further away will be neglected .

B. The Base Station Antenna Array Transmission System

• • •

x (t ) = W ' (9 )s(t) x ,(t)

•

•

•

!xm(t j

/.

( >120 \ .,:

0

The base station transmission system is based on four Fig. 3. The Uniform Linear Array (ULA) and the spatial multiplexer. algorithmic blocks as depicted in Fig. 2. One of the building blocks is the direction finder. This algorithm estimates the mobile angular positions 4' and their angular spreads r; (to where c denotes complex conjugate. The resulting m dimenbe defined) of the mobiles in the subcell from the received sional vector, x(t), is the input to the m antenna elements. The data D . As mentioned in the introduction, multiple mobiles implementation of the transformation can be done in analog or will be allocated to the same channel within the subcell . digital hardware and at different intermediate frequency bands. The channel allocator determines which mobiles should be The exact interpretation of the messages S k (t) depends on this allocated to the same channel. This algorithm uses only the implementation, but the analysis of this paper is independent angular positions of the mobiles 4' as input. The channel of this. allocation is represented by the n c x d matrix e, where n c is the number of channels within each subcell and d is the number of C. The Antenna Array Configuration simultaneous mobiles on the channel within the subcell . Each The antenna array of the base station is assumed to be linear contains the angular positions of the mobiles on a row in with uniformly spaced antenna elements. This form of antenna corresponding channel. The elements of an arbitrary row of configuration is known as a Uniform Linear Array (ULA) . The will be denoted by the 1 x d vector (). The corresponding individual elements of the array are ideal sectorized antennas angular spreads will be denoted by a , with a sector of 1200 • The active sectors of the antennas The weight selector calculates matrices of weights to be are positioned towards the broadside of the array (see Fig. used in transmission. One matrix is calculated for each chan3) and the spacing between the antenna elements , is set to nel. The angular positions , (), and angular spreads a of A/ v'3 where A is the wavelength of the carrier wave. The the mobiles allocated on the channel in the cell are used number of antennas in the configuration, m, is an important in this calculation. In order to simultaneously transmit d parameter of the system. In Fig. 3 the polar coordinates (T, a ) different messages, {s( t) , .. . , Sd(t)} , to d different mobiles, are introduced. The elements of the 1 x d dimensional vector the messages are spatially multiplexed. This operation can be of mobile positions, 0, are given in terms of the angle a. represented by the multiplication of the m x d dimensional matrix, W C ((), u), with the d dimensional vector s(t) = D. Propagation Modeling [Sl (t ), " ' , Sd(t )V i.e. In this section, we define the channel model between the x(t) = W C(O, u)s(t) (I) antenna elements of the array and a receiver at the position

e

e

472

(T, a). The transfer function consists of three factors: path loss, shadowing, and fast fading. The path loss and the shadowing are common to all the antenna elements. The path loss is modeled as (1/ T ) T where ry is the path loss exponent. The shadowing is modeled by a factor L which has log-normal distribution [5]. The standard deviation of 10 log L is denoted as adB (the mean is zero). The fading gain and the phase of the m antenna elements of the array are stacked into a vector denoted v(a~ a). The vector v(a, 0") is a random vector with a distribution depending on a and 0" (where a is to be defined). When the receiver and transmitter are located in different cells the fading of the antenna elements is assumed to be fully correlated, or equivalently the local scatterers in the vicinity of the mobile have negligible radius (implies that 0" == 0°) in comparison with the distance between the base station and the receiver [6]. Mathematically we model this as v( a, 0°) == {response of a single ray}

== Fa(a)

Fig. 4.

Illustration of local scattering.

where

CJ

. 27r . ( )) = F [ 1, exp (-J J3sm a , " ' ,

~ sin (a))]T

~ak E~

(2)

where F is the common fading of the antenna elements and a(a) models the phase differences of the antenna elements due to propagation path differences. The complex random variable F has a Rayleigh distributed amplitude and is uniformly distributed in phase [0, 27r]. The Rayleigh distribution is normalized such that E{IFI 2 } == 1. The function a(o:) in (2) is assumed to be known, although in practice a(a) may have to be obtained by calibration. When the receiver is in the same cell as the base station, the fading in the antenna elements is not assumed fully correlated. The model used in this situation is discussed in more detail in the next section.

E{v ( a, a)} == E {ej ip k } E{a( a

==

21r sin E N ( J3

(o ), (j )

O.

(7)

u.;(a, a) == E{v(a, 0") V * ( a, a)}

E{ (teXPj;k) a(a + 6a k))

(t

(3)

21r sin (n + 6ak) v'3

+ ~a k ) }

The second equality follows since ip k is uniformly distributed [0, 21r] and thus E{ e j rpk } == O. The covariance matrix is derived next By definition

=

The phase shifts, 'Pk, of the rays are assumed to be uniformly distributed [0, 21r], and the angular perturbations, ~ak are assumed to be distributed

(6)

N(O, a)

where .6.ak is given in degrees. This means that the spatial spread of the energy which is received by the mobile has approximately normal shape with standard deviation a. The normal distribution has previously been used in the propagation study [1]. As will be seen in the simulations, a is a critical parameter for the system. The parameter CJ which is related to a through (5) represents the angular spread in terms of the beamwidth of the array. Since the beamwidth of a linear array increases with a, CJ decreases with a. We will refer to a as the physical spread and CJ as the spread in terms of the beamwidth. Since the number of rays, N, is assumed large, it is natural to assume that the entries of the vector v (a, a) are jointly normally distributed. The following expression is obtained for the mean

E. Modeling of Local Scattering In this section, the model of the fast varying factor of the transfer function between the antenna elements of the array and a mobile receiver in the cell is presented. Consider the situation when the signal received by the mobile is built up by N locally scattered rays in the vicinity of the mobile, as depicted in Fig. 4. Assume that each of these rays has an individual stochastic phase 100). The number

476

= a

,

0.09

I

~

,

~O .04

a'5

l

/

x: C=4

,+

+.. C-7 -

/

//

//

I

0.03 __ _ B.ef.e rence

-

.J -

//

(J :::::::

'0

x- - -

K

10

11

" 10 ~ ~

c

..., _

E

E

-

..J

,'"

'2 ~

_

0°

~

ro 12

"

.4"

a

~ 14

I

0- - / /

,I I

,)to -

,'"

-

->.)M2 B · (I:>.)x. (58) x { x * B(6.)B(()k)a(O) == 1.

-

Wk((J

0"0)

y*Af2~

= argm;x y* B(fh)a(O) = 1

{ where y == B*(6.)x.

(59)

w~((Ji,

D B

ao/ cos (81 ) , ... , aO / cos (()D) )v( ()1 \ ao/ cos (()1 ))

== w~((O, 82

-

81 , "

' ,

. V(Ol, ao/cos (B l ))

Bd

-

e

1 ],

ao)B( -HI) (61)

where (62)

Since v(B1 , ao/cos(e 1 ) ) is multivariate normally distributed with mean zero and covariance R vv (01 , ao/cos (81 ) ) we obtain from (46), (50) and (51) that B( -01)V(01' O"o/cos ((;II))

N(o, B( -B1)Rvv(iJ 1 , O"o)B*( -8 1 ) ) == N(o, ~Jv(O, 0'0)). E

(63)

Applying (56) and (63) to (61) yields w~((J,

ao/cos(e 1 ) , · · · ~O"o/COS((}D))V(Ol, ao/cos(Ol)) d ist _

*

-

== Wn([O, (02 -

-

--

el)27'i~"" (()d - ( 1 )27r L

ao)v(O, 0'0) (64)

where d~t denotes that the distributions of the left and right hand side are equal. Using the property (30), (55) and (56) on (64) yields w~((},

where

ao/cos (()1),··· ,{Jo/cos (()V))V(f)l, {Jo/COS (01)) dist == W n* ((}v ,(J'V) V (0 ~ (To ) (65)

lJ is given by (38-39) and

(67)

ACKNOWLEDGMENT

1

The proof here is based on the definitions and results of Appendix A. Without loss of generality the proof is given for mobile number one, that is k == 1 in (12)-(15). Applying (55) and (57) (with L\ = -2n/V3sin(t11)) to w~((J, ao/cos(e 1 ) ,. · · ~ao/cos(eD))V(()l, O"o/COS(Ol)) yields w~ ((J,

(01),···,ao/cos (eb))a(ai)

where al is uniformly distributed [0, 2n], {} is given by (38-39) and (f is given by (40). Applying (65) and (67) to (12)-(15) with (33) yields the desired result. 0

(60)

ApPENDIX

(JO/COS

~t w~(O, u)B(ai)a(O)

From (52) and (59) it is obvious that

PROOF OF THEOREM

(66)

=

1L& +~,

Oi)21r )a(O)

where g~ 27r / V3 sin (Oi) and Oi == 27r / J3sin ( Qi)' Since 01 is independent of Qi and uniformly distributed [0, 27r] from (32), the argument [01 - Qi]21r, will also be uniformly distributed [0, 27r] and independent of ()~. Thus using (66)

Introducing y == B* (6.)x yields

~

-

if is given by (40).

Special thanks to Dr. T. Trump, Dr. U. Forsen, and Dr. M. Almgren for valuable comments and discussions.

REFERENCES [1] F. Adachi. M. T. Feeney, A. G. Williamson, and J. D. Parsons, "Crosscorrelation between the envelopes of 900 Mhz signals received at a mobile radio base station site," lEE Proc., vol. 133, pt. F, no. 6, Oct. 1986 . [2] S. Anderson. M. Millnert, M. Viberg, and B. Wahlberg, "An adaptive array for mobile communication systems," IEEE Trans. Veh. Technol., vol. 40, pp. 230-236, Feb. 1991. [3] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore: Johns Hopkins Press, 1983. [4] S. W. Halpern, "Reuse partitioning in cellular systems," Proc. Veh. Technol. Con! VTC-85, 1985, pp. 322-327. [5] W. C. Jakes, Ed., Microwave Mobile Communication. New York: Wiley, 1974, pp. 79-131. [61 W. C. Y. Lee, Mobile Communications Design Fundamentals. New York: Wiley, 1993, pp. 157-159. [7] 1. Li and R. T. Compton, Jr., "Maximum likelihood angle estimation for signals with known waveforms," IEEE Trans. Signal Processing, vol. 41, pp. 2850-2862, Sept. 1993. [8] V. H. MacDonald, "The cellular concept," Bell Syst. Tech. J., vol. 58. pp. 15-41. Jan. 1979. [9] S. J. Orfanidis, Optimum Signal Processing, An Introduction. Singapore: McGraw-Hill, 1990. [10] B. Ottersten, M. Viberg, and T. Kailath, "Analysis of subspace fitting and ML techniques for parameter estimation from sensor array data," IEEE Trans. Signal Processing, vol. 40, no. 3, pp. 590-600, March 1992. [11] J. Salz and J. H. Winters, "Effect of fading correlation on adaptive arrays in digital wireless communications," in Proc. Centre International de Conferences de Geneve (ICC-93), Geneva. Switzerland. [12] R. Steele, Ed., Mobile Radio Communications. London: Pentech Press, 1992, pp. 82. [13] S. C. Swales, M. A. Beach, D. J. Edwards and J. P. McGeehan, "The performance enhancement of multibeam adaptive base station antennas for cellular land mobile radio systems," IEEE Trans. Veh. Technol., vol. 39, pp. 56-67, Feb. 1990. [14] T. Trump, Maximum likelihood estimation of nominal DOA and angle spread using an array of sensors, Tech. Rep., (IR-S3-SB-9422), Access: see [17] below. [15] J. H. Winters, "Optimum combining in digital mobile radio with cochannel interference," IEEE Trans. On Veh. Tech. , vol. 33. pp.

144-155, 1984.

[16] _ _ , "On the capacity of radio communication systems with diversity in a Rayleigh fading environment," IEEE Selected Areas Commun., vol. SAC-5, no. 5, pp. 871-878, June 1987.

479

[17] P. Zetterberg, "The Spectrum Efficiency of a Basestation Antenna Array System for Spatially Selective Transmission," Report Version, (IR-S3-SB-9403), available by Mosaic: Document URL: http://www2.e.kth.se/s3/signaVINDEX.html or by anonymous ftp to: elixir.e.kth.se directory/pub/signal/reports, [18] P. Zetterberg and B. Otters ten, "Experiments Using an Antenna Array in a Mobile Communications Environment, " Proc. 7th SP Workshop on Statistical Signal & Array Processing, 1994 (IR-S3-SB-9412) Access: See [17] above.

480

Capacity Enhancement and BER in a Combined SDMA/TDMA System Josef Fuhl and Andreas F. Molisch Institut fur Nachrichtentechnik und Hochfrequenztechnik, Technische Universitat Wien, Vienna Gusshausstrasse 25/389, A-I049 Wien, Austria Phone: (+43) 1 58801 3546; Fax: (+43) 1 587 05 83; email: [email protected] Many papers have been published on SFU, see e.g. [1],

Abstract - This paper considers the performance of a TDMA system employing smart antennas at the base station. Two adaptation schemes are analyzed - the switched beam approach and an adaptive array based on an adaptation algorithm to maximize the Signal-toInterference and Noise ratio. For an SDMA system the switched beam approach performs worse than the adaptive array. Adaptive arrays based on gradient-vector estimation (e.g. LMS) are not suitable for mobile radio. The class of Least Squares (LS, RLS, SQRLS) algorithms shows satisfactory performance. For a linear array with 8 elements a minimum angular separation of 100 between two users is sufficient for the adaptive array to achieve as good performance as a system serving one user per traffic channel.

[2], [3], [4]. They show that for a single user the bit error

rate (BER) can be decreased by pointing the "main beam" of the antenna towards the current location of the user. This contribution is devoted to a true SDMA scenario. We consider the canonic situation that two users are served on the same traffic channel, consequently the capacity of such a system will nearly be doubled. OUI simulations are based on a channel model including directions of arrival and on the air interface of the 2nd generation standards GSM and DCS1800. We show how the BER is changed by adding the second user, as a function of Signal-to-Noise Ratio (SNR) and the number of antenna elements. The paper is organized as follows: Section 2 discusses the channel model used for the simulations. Section 3 addresses the simulation setup for the whole system. In Section 4 we take a look at the performance of different adaptation schemes. Section 5 gives the simulation results for various parameter combinations. Section 6 concludes this work.

1. INTRODUCTION The growing number of users of cellular communication systems necessitates measures to increase the performance of such systems, i.e. their coverage and capacity. Currently, there is a considerable interest in adaptive base station (BS) antenna arrays for 2nd and 3rd generation mobile communication systems. A possible 2-step implementation procedure for smart antennas may be as follows:

II. CHANNEL MODEL

• Spatial Filtering at the Uplink and at the Downlink

(SFU-SFD): Smart antennas are used both at the uplink ( mobile station (MS) transmits, BS receives) and at the downlink (BS transmits, MS receives). Only one user is served in one traffic channel. The aim is increased coverage and decreased interference for cochannel cells.

• Space Division Multiple Access (SDMA): With the use

of adaptive directional antennas and additional hardand software at the BS, users in different angular positions can be served in the same frequency band and in the same timeslot, i.e. on the same traffic channel. The data intended for each user are separately processed in base-band in such a way as to give the user-specific antenna pattern. The signals are added (linearly superposed) and modulated onto the RFcarrier, which is radiated from the antenna. This approach leads directly to increased spectral efficiency of the system. However, it can be added to an existing 2nd generation mobile communication system only if there are also changes and redefinitions in the switching and signaling system. The concept of a cell in its traditional sense has has to be redefined.

\Ve utilize a channel model including directions of arrival (DOA) and fading [5], [6], [7], and [8] (Fig.I). It is suitable" for both rural and urban areas. Fading correlation at the receive array is automatically included by the model. Like all the references above, our considerations are restricted to a two-dimensional channel model (i.e. the horizontal plane), but as mentioned in [8] this does not impose severe restrictions for mobile radio applications. Many scatterers in the vicinity of the mobile combine to one fading signal, spread out in angle over several degrees dependent on the distance of the mobile from the BS. By this model we extend the concept of DOA to a nominal DOA associated with an angular spread in contrast to the widely used discrete DOAs. In order to model the propagation physically we partition the propagation area into two different regions [8], [7]: (1) Regions without scatterers; (2) Circular regions where the scatterers are located. The motivation for choosing circular regions where the scatterers are located lies in measurements conducted by [5] and [6]. The radius R of these regions is about 100A - 200"\, where A denotes the wavelength [6]. This models can be easily generalized as shown in [8]. The overall impulse response for this channel at the location of the m-th antenna element r-« = [x m , Ym, Zm]T

Reprinted from Proceedings of the 46th Vehicular Technology Conference, Vol. 3, pp. 1481-1485, 1996.

481

---

CDF( Q',)

" I

I

~

~

~~2R

Scatterer

""

1 ::~===--~ ", ,

0.7

\

I

.'

, "

0 .5

' ••..

": 1

I ; ··

., ...

0 .3

,

0 .2

BS

,, ,.

r

. . ..

0 .4

\ \ I

I I

. , .. ..

0 .6

\

.... . . . j ••.

r. .

0.1 - :

, ,r

Fig.! Chann el model

h(rm ,

T,

cell radius

t, cp)

L

= 2: h,(rm , T, t, cp), 1=1

- ' - : r2

(1)

where L is the number of scattering points, T is the delay (relative time), t is the absolute time, and cp is the azimuth angle. The quantity ht(rm , T, t, cp) denotes the impulse response of the l-th path at the location r m of the m-th antenna element. The impulse response is different for every location of an antenna element. A. Angular Spread

We define the angular spread a s as the angle under which the diameter of the scattering region is seen at the BS (Fig. I). It is given by

as = 2 arctan

(~),

(2)

where R is the radius of the scatterer circle and r m is the distance between BS and the mobile. We assume that the users are uniformly distributed within the cell area. The Cumulative Distribution FUnction (CDF) F(CLs,k) of the angular spread CLs,k of the incoming signals from user k, 1 ::s k -s K, where K denotes the number of users, seen at the BS can be calculated to [9)

(3) where r l (rz) are the distances from the BS to the nearest (farest) location of the user and 0'0 given by CLo =

= r l = 100,), a n d the oute r = 1000,),; - : r 2 = 5000,),; and

Fig.2 CDF of the angular spre a d for R

(1 ::s m -s M) is given by

2 arctan

(~)

(4)

is the angular spread associated with a user located at the cell fringe. Fig.2 gives the CDF of the angular spread with the outer cell radius rz as a parameter.

482

r2

as parameter. - - :

=20000,),.

r2

III. SIMULATION MODEL We use a mod el with a protocol closely related to t he 2nd generation standards GSM and DCS 1800. The only difference is that two or mor e users are served at t he same traffic channel (i.e. at th e same frequency and time slot ). Therefore each user 's mobile has to be assigned a unique training sequence, each of which must com ply with the GSM (DCS 1800) specifications. This 26-bit midam ble originally int ended for estim ation of t he impulse response of th e channel (equalizer t raining) is now used for user separation and identificati on also . We assume perfect time alignment of the received sequ ences. We consider a narrowband channel where transmission suffers from flat-fading only. As parameters for th e chan nel we assumed L 20 scattering points per user. All th ese points are randomly located within th e scattering circle. The radius of the scattering circle has been set to T\ = R = 100>' and th e outer cell radius to rz = 5000>'.

=

IV . ADAPTATION SCHEMES Two different adaptation schemes are investigated in this paper , th e switched beam approach and an adaptive array based on Least Squares (LS) adaptation. Sw itched beam uses a set of P (P 2: K) different beam positions (Fig .3) to separate t he users. The output signals from the different positions are demodulated and th e reference sequence (t raining sequen ce or user identifier) is compared to th e training sequences used . Different criteria like minimum Bit-Error-Rate (BER) , maximum received power, or a combination of both can be utilized to determine the best suited pattern . The signal containing th e training sequence which is closest to the desired sequence (in terms of the abovementioned criteria) is taken .for reception of the specific user .

2

~"~ ':_~w ,(n)

~...... j -.:.:.~-.......

wz(n) ~~: ---=-~---4C

~,..;-i

w3(n) .

-...-oGG--../

Fig.3 Switched beam

(a) 1 ~., .1. . . ._ _~~_ _.... L-I

Y

I

,< .. .:

I

2~_ ' -h--~-+- L-I

Y

I

I

I I

I 1

, :. :

3 ~_:-++r---01~-+-+--__""

L-I

I

1-

,

"

.......

"

""S..:.

Training

'

.

,.' \

.... ....

.. ,

.,.

... Co: :.

~

,

'0

,

~

,

::.:f! U HU! U Uf~~;HJ~;~: i l;U:

..

.:.

,,',

_I

5

...

"

\

Switch for Tracking

I

,

.., t ::::::.': :, ': ::: 10'

M~,., :-++h-_~H+-' y

....

'1

.10

Steps

10

'5

(b)

Fig.4 Adaptive array

-

-

.

~

20

25

Steps

Fig.5 Averaged mean power e2 of t he erro r for t he co nsidered adap-

Adaptive A rrays are based on maximization of the

Signal-to-Noise and Interference (SNIR) ratio upon a known sequence [10] (FigA). Algorit hms based on gradient-vector estimation (LMS) and/or Least Squares (LS), (LS, RLS, SQ RLS and its non- deterministic counterpart , the time- space domain Wiener filter) may be utilized. Fig.5 a shows t he averaged mea n power of the error for the LMS. T he numbe r of anten na elements is M = 8, their spac ing d = )../2. The step size J-l LM 5 of the LMSalgorithm was set to the ha lf of the maximum ste p size to obtain convergence. The Signal-to-Noise Ratio (SNR) was set to 20dB. Since th e LMS does not converge with in the used 26-bit traini ng sequence, we used it repeatedly. The slow convergence of the LMS-algorithm is remarkable . T his is due to t he eigenvalue sp read of the covariance matrix of the antenna outputs. For smart antenna applicatio ns t he corre lat ion mat rix is usua lly ill-conditioned (i.e. its eigenvalues are widely spread) , therefore t he convergence speed of t he LMS is low. Fig .5b shows the averaged mea n power of t he error for the RLS, SQRLS , and LS al-

tation sche mes. (a) LMS; (b) - - : LS, - : RLS, and 0 : SQ RLS . No te the di fferent scale on t he x - a xis .

gorithm. T he forgetting factors of bot h RLS and SQRLS were set to 1. They show that t he RLS and the SQRLS perform in the sa me way for infinite pr ecision arit hmetic . For finite precision arit hmetic the SQ RLS is preferable [11]. T hese graphs illustr ate that LMS is not suitable for a mobi le radio environment with short t rai ning sequences [12], [4], but LS (Wiene r filter) or (SQ)RLS algorith ms are well suited . V. SIMU LATION RESULTS A. Single User

Fig .6 shows the BER for t he considered adaptation st rategies for various SNRs wit h t he number of anten na elements M as parameter . The curve for M = 1 agrees well with t heory (BER 1/(2SNRlin), for SNRlin > 10, where SNRlin is the SNR on a linear scale) . Switched beam performs better tha n the adaptive array. T his may be att ributed to the non- st ationar y chan nel. T he nomi nal DOA of t he impi nging signa ls is the only quantity being st ationary . This gives adva ntages for DOA-based approaches for this specific scena rio.

483

=

1=: f- .__.

-

-

-- ..

-.

..

-

.-

~=1 -'- :;>... ~

~~

--

F '

f=.

-

~

~

.V

--

M-2 -

~-

""'-..;;

-............. ~

z~/m) (I)

•

••

cit) I !

-========(===>

)

dz . '">

(62)

However, due to the delay in the control loop, finite step size by which the mobile can increase or decrease its power. and errors on the downlink, power control cannot be ideal (see, e.g., (28]). Therefore, the symbol error probability obtained above needs to be averaged over the probability density function of 1t;, which is not known in general. However, an approximation to the uncoded BEP can be obtained as follows. Here, we can use the approach in [29] and [20] to get an approximation for the uncoded BEP. First, let C; denote the coefficient of variation of 1 s' defined as

c. ==

JVar{1'8 } ---e-{- } - .

(63)

Is

In [29] and [201, it is shown that for a low Cv , (less than O.~ [20)), a reasonable approximation of the symbol error probability is

P,'c' : : :; ~FAd7$) + ~PM(;s + J3(7,) 3

6

t-

~PMhs

-

v'3(7~)

(64)

Vv here is and a I are is the mean and variance of 1 ~. We can see that (J 'Y also represents the power control error. Then, the corresponding uncoded BEP is

" Pb

is given by (60)

;'\;/-1

l=O

fOG [fl(Z[2) < z I zP) = z)]M-l fz;I)(Z I 'Ys)dz

./0

(61)

where i is symbol energy to interference-plus-noise ratio per path per antenna. Also, the density function of z;n) for n == h given in (53) becomes an unconditional density. Therefore, the syrnbol error probability is P,f\".l

Y.

1 .L . K

2J - 1 " =: -J--PjH.

2 - 1

(65)

B. High Doppler Frequency

For high Doppler frequency and/or long loop delay, the fading statistics of the received signal after power control remain the same as those of the multipath fast fading with only perfect average power control. In this case, we have L

" 18 == 1- '~

'.9

lat,ll 2 .

(66)

l=1

The distribution of depends on the angle spread 6. through al,}- We consider the following three cases.

1) Small Angle Spread: For zero (or relatively small) angIe spread ~, the channel vector of the lth multipath component can be expressed as

(67)

493

where aZ,1 is a zero mean complex Gaussian random variable and for aULA Yr.i is a Vandermonde vector [30] given by Vi,l

==

where J..K

1rk=IT

1=1 L-:j=k

[1

e-rrrsinf:Jz,lD/A ... e-J7f~1Il6J,1 D(K-l)/-\]T,

rs

has a X2 distribution

with 2L degrees of freedom, i.e., £-1

' (ry) = f 'Ys, (ryK)L(L -

(II

ZL-l

"2) =

(J2L(1

Zl ) \

+ iK)L(L -

(69)

1)!

" e-z/((T~(l+l'K)). (70)

2) Large Angle Spread: For large angle spread, theLelements C

becomes uncorrelated and hence 2::£=1 lal,112 is a sum of K L i.i.d. random variables having a X2 distribution with two degrees of freedom. Therefore, '"Ys is distributed as a X2 random variable with 2K L degrees of freedom

of

al)

f-y,(-r)

=

/_

,KL-l

(71 )

C'Y)KL(KL-l),e-"'"

The corresponding unconditional probability density function of zln). for h

= ti is (see the

Appendix)

where

R == I

1

(0- 2 (1 + "y))KL-l X

((!(-l)L) l

(~)(K-l)L-l(_l

] + l'

1 + l'

)1.

(73)

3) Other Values of Angle Spread: For other values of ~, we can easily show (see the Appendix) that the syrnbol energy to interference-plus-noise ratio is L

/8 ==

K

L L ilil u

ZLI

2

(74)

l=J i=l

where 1111···'ULK are i.i.d. zero mean complex Gaussian random variables and i i i == l' )..li, where {Al,i} 1.=l,"', K are the eigenvalues of R Z,l , the covariance matrix of the first mobile's lth multipath component. Also. if the is

'8

=::

Let {i'h.h-:l. L,1.=1· K be equal to {il,}1.=l-- KD· \ve assume that the ilt' s are distinct (this is true angles of arrival are sufficiently different). Then, distributed as [101 (75)

KL {eL (2( '1=1

1

7ri

where

and

(]'

1+

_ )) * g(z)

"L .

t.::

g(z)

Hence, it can be shown that the corresponding unconditional pdf of z~n) for h == ii is (see the Appendix)

f

k=l, ... ,LK

~

Z/ (U2 ( 1+ i 1»)

fzH(Z)

e-,/Ci R-) 1)! -

_

(76)

and the corresponding unconditional pdf of z~n) for h == ti is (see the Appendix)

(68)

In this case, we can show that

,k ik - i1.

* denotes

= u 2 ( L _~)(; _

2)! e-

}

c o2 /

(77)

(78)

the convolution operation.

zi

Using the unconditional density of ) for h = n in (70), (72), and (77), the average symbol error probability is given by "

PM=::l-l

oc [

z u '2 l_e- /

n

1 z Lf!(u

£-1 l=O

2)

l ] M-l

fz~dz)dz

(79) and the corresponding average uncoded BEP is given by (65).

VI. NUMERICAL AND SIMULATION RES\.}Ll'S

First, we study the accuracy of the approximation that the MAl signal vector can be assumed to be a spatially white complex Gaussian random vector. The base station receiver in Fig. 2 was simulated. In our simulation, we assumed that the processing gain G == 256, L == 4,1'1 = 40,1\-1 == 64 and LJ == 0.375. We also assumed ideal power control. 'VJ.le assumed that the base station has three sectors, each with a five element ULA as shown in Fig. 5. The angles of arrival {8k ,2} were assumed to be uniform over [0,120°] (i.e., uniform over the sector). The angJe spreads {~kJ} were assumed uniform over [0,60°]. The results of 10000 post-correlation signal vectors were used to estimate the statistics of the Mi\I signal vector. Figs. 6 and 7 show the empirical PDP of both the I and Q component of the MAL at the first antenna. From both figures we can see the validity of the Gaussian approximation. Also, the covariance matrix of the MAl vector ft~~J.k.,l was estimated and is shown at the bottom of the next page. The

Frobenius norm [311 of the error IleilF == IIRS~:'k)l - IIIF was estimated as 0.0058 which also shows that the MAl can be assumed to be spatially white. Next, we study the system uncoded BEP. The closed loop power control was simulated according to the model in [28] and 1271. In this simulation, we assumed that the mobile can increase/decrease its transmitted power by 0.5 dB at a time and that the power control command was sent every 6Tu.We assumed that the loop delay IS also 6Tw and that the return channel error rate is 0.05. Figs. 8 and 9 show the power controlled signal level distribution versus the distribution of the simulated multipath Rayleigh fading at the RAKE output for f d == 5 and 100 Hz. From these two figures we can see that closed loop power control eliminates most of the channel variations at low Doppler frequencies, while at high Doppler

494

0.08 - ' _ .. ,.__._ -

-

..- ..- --- . ,.-

-

IE:] Simuinuo ns - "om. ',,'m,

0 07

-

-

, --

-

--,---,

I

~

0.06

----------"---1

'::':--

0.05

8

is e:

j

0 04 0.03 0.02

d

• F Ig. .)

008

000 0.05

•

•

0.01 0.00

Si rnulario n scenario

i'

I 007

•

•

-:._=-.....,. ".-

: ~ Siruulatiunv

Nurnm.l

,-

!

_." ---'-.._ -- '''''- - '- '- --,.-

f

0.9 0.8

~ E

\

r-

-e

V1

i

..

.s c

a· [

140

180

180

.. .

.-

---6-

-

-- ..

~

-o- K = l

I --0- KK =3 - 5

H-+-

Ideal Power Contro l

~

.. __.

...

1\

I---

- .-. _..

--

I I

I

/ /

1 0~

Power ControlJd = 5 Hz

150

7 .-

0..

1==

-

V /

- - 0 ~~

~

--_. _.

~

10-2

"3

1, C,

_..

a;

200

N . Number of Users

N, Number of Users

Fig to.

11.

W

-1- ----\

/

-#--r:~ K = 3, C,=0.077 ;? J--&-- K=5, C,=O.077

_.

~ --:-

~

K

..

f

..-J.'j

/

~l

-

--_.

PDF of : 1( " ; for

t==r=

-

-;::ft.

_.

._ - +-_.f--.

-' .

'>I

._-

Fine-. 11.

' 1( 1)

_rI:. ~

----_..

.-

IdealPower Control Fading, t. ~ D· 0 Fading, t. ~ 3 l-adin g, t. ; (fJ.

}-----A-~"'-'--...:+

Nor ma lized Signal Level (dB)

=

- -

}--- ---+-+- \ - -

"=>

. .•

·2

·4

0.05

I--

u,

. ..

,1

If

0.1

-

. ....

)

1--1

0 .0

-:

Fig . 12.

Hz and closed loop power control.

diversity for this multi path component. As the angle spread increases, the signal fading at different antennas becomes more and more uncorrelated which leads to less variations in the RAKE output. Figs. 12-14 show the uncoded BEP for 6. == 0° ,6. == 3°, and 6. - 600 for a high maximum Doppler frequency fd (high enough such that the statistics of the received signal after power control remain the same as those of the multi path fast fading). For 6. == 0° we can see that for Pb == 10- 3 the maximum number of allowable users reduces to (compared to the perfect power control case) 10, 29, and 55 for K == 1,3, and 5, respectively, which corresponds to a 65% reduction in system capacity. This capaci ty reduction is due to the multipath fading which was not eliminated by the closed loop power control. With a single antenna, the statistics of "Ys does not depend on the angle spread . Therefore , the maximum number of allowable mobiles for J{ -'= I is the same at ten mobiles per cell for any value of angle spread . However, for angle spread 6. = 3° this number goes up to 44 and 90 mobiles

Ph

for high /" and ~

~~

10' \

.--.

tJ

..n ~

..

-

1/ _.__ .

/

10"

I - ;---f-.

20

_.

I

'"

-_.

I

40

~

--. 60

80

..

-0- K =l ~--0- K =3 ---6- K -- 5

. ;.......-.

7'

_.

.--

"7

,/ ... -

..

I-----

r.--

.»:

..-

~

10"

. _---

power control.

J

H J. 0."

= 0" . and

100

120

Angle Spread t>-- 3" . -

140

160

180

200

N, Number of Users

Fig. 13.

Ph

for high / ,/ and 2l.

= 3° , and powe r contr ol.

for K == 3 and J( "..- 5, respectively. This is due to the additional space diversity gain provided by the array . For

496

_.

.-

-~

--

-

g

10-2

'"

.-

T-'---

==t --:-.- - =--= -_. r--P-- --

-

-- - f

,

.-

_." - -

/

J

-/I---j- -

-0-

-0-

--- -

-

-&-

--j- .- - -_..,- j ---~ t-::::::-:t:j

I

20

60

40

80

- r

.... --- ~-

• . _;---t.

_ .

~

- _.

C - -. _

'-

f--

(

_.

1 0~

./

-_. .

100

- --.-

-. - ..

,../'

~

---'

120

140

-

K=1 K=3 K=5

Antle SpreadIl. -

-

~ -

~. L

1aO

180

200

sen ted an approximatio n fo r the uncoded BEP as a function of the mea n symbol energy to interference-plus-noise ratio and the power co ntrol error. Fo r the case of high maximum Dopp ler shift, we derived exact expressions for the uncoded BEP for different cases of angle spread. In all cases, an improvement in sys tem performance pro portion al to the number of sensors is observed. Addi tiona l improvemen t is obtained due to space diversity gai n at hig h angle spreads . APPENDIX

To derive the uncon dition al pro bability density funct ions in (70), (72) , and (77), first we reca ll the chara cteristic function rep rese ntation of the condi tional density in (53) [10]

N. Number of Users

Fog 14.

P" for high

TABLE [ P ERCENT RED UCTION I" C APACITY AT

38 11 1

187

TABLE II

K 1 3 5

P" :::

50% 50% 50%

PFRCENT REDUCT ION IN CAPACITY AT

29

85 142

. , k l

wh ere 8 i denotes th e beari ng of th e ith multipath . xm(k . nt; + (k - \ )t e ) is the c ha n ne l t ap o u tp u t a t the mth a n te n na. Ak de notes th e signa l a mp litude. a nd xT de no tes th e vec to r tr an spose o pe ra tio n. The firs t e nt ry of q is spe cified to be a po sitive re al num be r. so th e term exp( 't',) represents the ca rr ier phase of the kth multipath a t th e first ante nna at tim e t. T he s ta tis tics of a give n c ha n ne l ta p vec to r x(k . I) a re of inte res t. T he p hase of eac h e ntry of x(k. I) is un iformly distribu te d ove r [0, Zn], so th e mean vec to r o f x(k . r) is the zero Extent of mu ltipath / scattering

Power

i

I i i

I

i ii

" ~i;~I~'ofscatterers ..

.:'..>. . ':....;.. a) • .. ·i...'.

Bearing . b) '.

,-:, '.

• Figure 5. a) The phy sical geometry f or the Salzlwinters m odel; b) the uniform distribution of multipatk ene rgy with angle.

503

vector. The second order moments of x(k, t) are specified by its M x M mean covariance matrix \Ilk, which is defined as: \}It

= E[x(k,t)x

== Ai

r

H

(k,t)]

9 0 + ca n be written as E = E[ (cos ; )e e + (sin;ej 'l)e o ].

sin dJ cos, ] sin ¢ sin , C)I,

27l'.-1.,1 Ao

L sd

An advantage of the COLD array is that its antenna elements are not sensitive to the azimuth angle () of the signal becau se both the loop s and dipoles have the same si n (b field pattern . as may be seen from (9) . Hence. the incoming signal described by (4) is independent of () . We assume that the antennas and the inci de nt signals are co planar. i.e.. ¢ = 90° . Thu s. (9) become s

(I)

£ (J E repre sent the comple x voltages induced at the loop and dipole outputs by a sig nal with a unit elect ric field parallel to the loop s and dipoles. respecti vely. Let s(t ) = E so(t )Ve cos -y. The ZI(t ) in (8) can be rewritten as Z(t )

= US(t )(ll

( 13)

1 ] = [ \'; F" ta n; e j'1

(14 )

where U

We ass ume that K signals. spec ified by incident angles Ih , l: = 1, 2. · · · . K. are incident on the arra y. In addition. we ass ume a thennal noise voltage vecto r 01 (t ) is present at each output vector z/(t ). The o /(t ) are assumed to be zero-mean cir cul arly-symmet ric complex-Gaussia n random processes that are statistically independent of eac h other and to have covariance matrix a 2 I, where I denotes the identity matrix . Under these assumptions. the total output vector recei ved by the COLD pair centered at y = 51 is given by

(4)

Let us define the spatial phase factor

(5) 1 For a narrowband BPSK (binary phase -shift keyed) signal. for example. so (t ) e}[-"ol + ..·(t }] . where ~'o is the carrier frequency and .;.(t) is [he modulating phase.

=

511

K

ZI(t ) =

L UkSk(t )qlk + o /(t ) ,

l = L 2. · · · . L

(15)

k=l

where U k and qlk are given by (14 ) and (5). respectively, with subscript k added to each angular quantity. Furthe r. Sk(t) = Eksok(t )Vo cos ; /c. where Eks odt.) denotes the kth narrowband signal. The incident signals may or may not be correlated (including completely correlated. i.e.. coherent) with each other.

Let z(t),s(t), and n(t) be column vectors containing the received signals, incident signals, and noise, respectively, i.e.,

z(t) ==

[:~:~g],

s(t) ==

ZL(t)

[:~~g],

n(t) ==

[:~m].

(16)

nL(t)

8l\-(t)

The received signal vector has the form

z(t) == As(t) where

A

+ n(t)

(17)

is a 2L x K matrix

A ==

[V8!)

r,k = -arg ( _r~9}

and

V u == I.

(40)

Then

(A t H ® I)V Thus,

12 (fr r)

==

Ah V h

+ Au V».

" 2

(41 )

+Tr [

A--lE~ s; Vh W]

v;H A hH"Esj\ ~

v

2

~ I"H AE s AvVvW ]

(42)

IV.

Since V h and V v are diagonal matrices, (42) can be written in the following matrix form

h(fJ. r) =

[v~

e T ] Q(fi)

[Veh

1

(43)

where (44), as shown at the bottom of the page, with (~) denoting the Hadmard-Schur matrix product (i.e., the elementwise multiplication) (45)

and

e == [1

(46)

Note that the polarization parameters are contained only in v h. By setting 8f2/8vh == 0, we obtain

Vh == -Ql 1(B)Q2(())e.

(47)

Using (47) in (43) gives

f3(()) == e T [Q3(()) - Q~ (8)Q 1l(B)Q2(8)]e

and using

e in

8

== 1,2.···.K.

(51)

(49)

STATISTICAL PERFORMANCE ANALYSIS

We present below the asymptotic (for large ;.V) statistical performance of MODE for both direction and polarization estimation with the COLD array. Before we present the analysis results, however, we first describe the method we use to describe the accuracy of the polarization estimates. For reasons discussed in [I), we define the polarization-estimation error to be the spherical distance between the two points M and JI on the Poincare sphere that represent the actual state of polarization (",(",) and the estimated state of polarization ('"Y. 'f)), respectively. Let ( be the angular distance between AI and .:.v[. Then [11

cos ( == cos 2, cos 21'

+ sin L--Y sin 21' cos( r} -

,))

(52)

where ( is always in the range 0 S ( ~ tt, Applying the first-order approximation to the left side of (52) yields ( k2

(48)

which is a concentrated function depending only on f). The MODE-estimates {{), r} can be obtained by

{) == arg min [11 (f)) + 13(8)]

k

For signals that are not highly correlated or coherent with each other, the initial estimates of 0 and r in Step I may be obtained by using MUSIC [14], which requires a onedimensional search over the parameter space. For highlycorrelated or coherent signals, the initial estimate of () may be determined by setting W == I and minimizing 11(8) + 13(B). as shown in (49). The initial estimate of r can be calculated by using the initial estimate of B in (47). The initial estimates obtained by using MUSIC for noncoherent signals or MODE with W == I are known to be consistent [6], [151·

in (35) can be rewritten as

I2(B~ r) == Tr[V~ Af:EsA

(50)

== 4(""t» -

+

. 2'2 ,k ) 2 SIll \ I k )(" 1Jk -

iT} k )? -.

(53)

The asymptotic variances of the polarization estimates are obtained with (53) and the accuracy results on :y and fry given below.

Let

(54)

(47) to obtain r.

" ?

(A~Es~: A-IE~ A v) 8 (A~EsA A-lE~ AI') 8 513

W W

T T

]

~ [Ql(()) Qlf(O)

(44)

It follows from [5], [8] that the asymptotic (for large N) statistical distribution of f is Gaussian, with mean T and covariance

MUSIC WIth COLD : NSF with COLD :

matrix equal to the corresponding stochastic Cramer-Rao bound (C R B ). The ij th element of CRB - 1 is given by [C R B-l j,.j

= ~ Re[tr{AfPiA iSAHR-l AS}]

CRB for COLD :

(55)

where A i = 8Aj8Ti with Ti denoting the i th element of T.

o o

V. N UMERICAL RESULTS

+ Lsd sin ~!ej " sin (} cos L sd 5111 . (/)" Sill Ie in

_ [LSd cos y cos (} -

¢] .

o

8

10

MUSIC Wll h COLD:

,e)"

(\Ve remark that if the antennas and the incident signals are

not coplanar, we will need two-dimensional CCD or COLD arrays for angle and polarization estimation, which is the case not considered herein. For this case, however, the COLD array will not always perform better than the CCD array.) First, we present two examples that illustrate how the angle separation between the two incident signals affects both the direction and polarization estimates. We begin with the case of two signals with identical circular polarizations (al = a 2 45°). Fig. 2 shows the root-mean-squared errors (RMSE' s) of the estimates of the first signal as a function of angle ,eparation t::.B when two correlated signals with correlation coefficient 0.99 arrive at the array from angles Bl t::.B /2 and B2 = t::.B /2. We note that MODE performs better than MUSIC and NSF. Further, MODE achieves the best possible unbiased performance, i.e. , the corresponding CRB , as the angle separation increases. Because the signals arrive from angles near the broadside of the arrays, the CRB' s for the COLD and CCD arrays are similar. This case corresponds to

=

=-

18

20

22

o

NSF w,th COLD : MODE wrth COLD : CRB for COLD:

o

.

o

o

,

(57)

.

12 14 16 Angle Separation (deg)

o

10 ' rf ---r--~-~----~---r--~--~-,

=

L sd cos I cos B] . ' [ - Lsd Sill

o

. o

.o

o

18

20

(56)

In the following examples, the antennas and the incident 90 0 , for both signals are assumed to be coplanar, i.e., ¢ 0 the COLD array and CCD array. For ¢ = 90 , (56) becomes

=

o

(a)

10

J.LCCD

o

.

We present below several examples show ing the performance of using the MODE algorithm with the COLD array and comparing the asymptotic statistical-performance analysis results with the Monte-Carlo simulation results . We compare MODE with MUSIC and NSF for both angle and polarization estimation. The simulation results were obtained by using 50 Monte-Carlo simulations. In the examples, we assume that there are K = 2 incident signals and both signals are assumed to have the same amplitude Ek, such that We Eki = W Ekl = 1, k = 1. 2. Hence, the signal-to-noise ratio (SNR) used in the simulations is -10 log 10 a 2 dB . The array is assumed to have L = 8 COLD pairs that are uniformly spaced with the spacing between two adjacent COLD pair s equal to a half wavelength. We also compare the est imation performance of using the COLD array with that of using a CCD array with the same array geometry. The CCD array consists of crossed yand z-axes dipoles . The counterpart of (9) for the CCD array can be written as f.LC C D -

°

MODE Wllh COLD : !l

,

12 14 16 Angle Separanon (

=

=

VlIU

vt:r~U:-i I

_,

lu r

Lue

= , >2 =

= 10 dB : ra)

(58)

-

[V~(:~~I) ][B 01 0

, .

,':,. 'c' -

-0 -

-

')

o

_

-0-- -

-

_ 0::)

---

II

!

-

..

0 ..

tb ) . E . s ) 01. estimates . ..ig. lI . xoor-mean-squareu errors (KMS versus .\ " t'or the sec ond of the two signals in the pre sence of contaminated Gaussian noise 0 50 and FI ~ 70 0 " I 1 \ '2 ~5 ° i l h 0° . when Ii, correlation coefficient = 0.99 . and SNR 10 dB : (a) Direction estimates and (h) polari zation estimates.

=

=

=

=

=

=

= 2L -

Thus ~ has full column rank 2(L - K ) + K dim[N(AH )]

= 2L -

=

K. Since

K

(61)

it follows that the columns of ~ span the entire null space

N (A H). Hence

(59 )

.

This result shows that the columns of .6. belong to Moreover

(B lI B ) ® I

o

10 L_ _"'--_--'-_ _'":-_=_-::-::-:---,:-:::-_~::____:_;:;:_-_;; 450 0 100 150 400 200 250 300 350 50

. 50

400

BlI ® I ] VH (A t :2lI) .

:::.HA = [(BHA ZI)U ] _O

= [

~_---,

11:.

vf!

H

,

~

o

We first show that the columns of the matrix .6. span the Uk = 0, we obtain null space of A H , Since B H A = 0 and

.6. ti =

~_~_ _~_-.,.-

TP for MODE wit h COLD'

Proof of (27J: Let B be an L x ( L - K ) matrix whose columns span the null space of A H . Also, let a (2 L - K ) x 2L matrix .6. H be defined as H _

_

TP lor MODE wIth ceo:

ApPENDIX

~

450

N

\l!RL,L,. "'}

=

o

second of the two signals when H, 50 0 and H ~ 70°. n ~5 ° . 11 1~ 0° , correlation coefficient 0.99 . and SNR Direction estimates and (b) polarization estimates.

=

400

MODE wrth CO LD:

i

150

100

350

MUSIC w,th COLD : NSF with COLD:

1

o

MODE with COLD:

o

300

(a)

MUSIC wrtn COLD: NSF wIth COLD:

cr - _ _ o

250

N

pi = P.:> =

.6.(D.H .6. )-1 D. H

= (B 0 I)[(BHB ) -

I

® I](B H ® I )

+ (AtH @ I)V {VH[(A H Af1 V

N(A H).

H

@

I]V} - l

(A t @ I ).

(62)

Since PBt~I = P:L3;I' we conclude that (27) must hold true.

(A tH ® I )V ]

REFERENCES

0 ] VH [(AHAf l @I]V . (60) 517

[Ij J. Li and R. T. Compton, Jr. , " Angle and polari zation estimation using ESPRIT with a polarization sensitive array," IEEE Trans. Anten/las Propagat ., vol. 39, pp. 1376-1383, Sept. 1991.

[2] Y. Hua, HA pencil-MUSIC algorithm for finding two-dimentional angles and polarizations using crossed dipoles," IEEE Trans. Antennas Propagat., vol. 41, pp. 37~376, Mar. 1993. [3] J. Li, "Direction and polarization estimation using arrays with small loops and short dipoles," IEEE Trans. Antennas Propagat., vol. 41, pp. 379-487, Mar. 1993. [4] J. Li and P. Stoica, "Efficient parameter estimation of partially polarized electromagnetic waves," IEEE Trans. Signal Processing, vol. 42, pp. 31 14-3 J25. Nov. 1994. (~; P. Stoica and K. C. Sharman, "Maximum likelihood methods for direction-of-arrival estimation." IEEE Trans. Acoust.. Speech, Signal Processing. vol. 38. pp. 1132-1143, July 1990. (6] _ _ , "Novel eigenanalysis method for direction estimation," in lEE Proc.. Pt. F, vol. 137, Feb. 1990, pp. 19-26. [7] A. Swindlehurst and M. Viberg, "Subspace fitting with diversely polarized antenna arrays," IEEE Trans. Antennas Propagat., vol. 41, pp. 1687-1694, Dec. 1993. [8] B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and large sample ML techniques for parameter estimation and detection in array processing," in Radar Array Processing, ch. 4, S. Haykin, 1. Litva. and T. J. Shepherd. Eds. New York: Springer-Verlag, 1993.

[9! C. A. Balanis, Antenna Theorv-i-Analvsis and Design.

[101

[11] [12]

l13~ [14] [15] [161

518

New York: Harper & Row, 1982. G. A. Deschamps, "Geometrical representation of the polarization of a plane electromagnetic wave." in Proc. IRE. May 1951. vol. 39, pp. 540-544. R. C. Johnson and H. Jasik. Antenna Engineering Handbook. New York: McGraw-Hill, 1984. M. Viberg and B. Ottersten. "Sensor array processing based on subspace fitting," IEEE Trans. Acoust., Speech, Signal Processing, vol. 39. pp. 1110-1121. Mav 1991. M. Wax and T. "Kailath. "Detection of signals bv information theoretic criteria." IEEE Trans. Acoust.. Speech. Sig..n. al Processing, vol. ASSP-33. pp. 387-392. Apr. 1985. E. Ferrara. Jr. and T. Parks, "Direction tinding with an array of antennas having diverse polarizations," IEEE Trans. Antennas Propagat., vol. AP-31. pp. 231-236. Mar. 1983. P. Stoica and A. Nehorai, ·'MUSIC. maximum likelihood. and Cramer-Rae bound:' IEEE Trans. Acoust., Speech. Signal Processing, vol. 37. pp. 720-741. May 1989. P. 1. Huber. Robust Statistics. New York: Wiley. 1981.

Upper Bounds on the Bit-Error Rate of Optimum Combining in Wireless Systems Jack H. Winters, Fellow, IEEE, and Jack Salz, Member, IEEE Abstract- This paper presents upper bounds on the bit-error rate (BER) of optimum combining in wireless systems with multiple cochannel interferers in a Rayleigh fading environment. We present closed-form expressions for the upper bound on the bit-error rate with optimum combining, for any number of antennas and interferers, with coherent detection of BPSK and QAM signals, and differential detection of DPSK. We also present bounds on the performance gain of optimum combining over maximal ratio combining. These bounds are asymptotically tight with decreasing BER, and results show that the asymptotic gain is within 2 dB of the gain as determined by computer simulation for a variety of cases at a lO-J BER. The closed-form expressions for the bound permit rapid calculation of the improvement with optimum combining for any number of interferers and antennas, as compared with the CPU hours previously required by Monte Carlo simulation. Thus these bounds allow calculation of the performance of optimum combining under a variety of conditions where it was not possible previously, including analysis of the outage probability with shadow fading and the combined effect of adaptive arrays and dynamic channel assignment in mobile radio systems. Index Terms- Bit-error rate, optimum combining, Rayleigh fading, smart antennas.

A

1.

INTRODUCTION

NTENNA arrays with optimum combining combat multipath fading of the desired signal and suppress interfering signals, thereby increasing both the performance and capacity of wireless systems. With optimum combining, the received signals are weighted and combined to maximize the signal-tointerference-plus-noise ratio (SINR) at the receiver. Optimum combining yields superior performance over maximal ratio combining, whereby the signals are combined to maximize signal-to-noise ratio, in interference-limited systems. However, while with maximal ratio combining the bit-error rate can be expressed in closed form [1], with optimum combining a closed-form expression is available only with one interferer [2], [3]. With multiple interferers, Monte Carlo simulation has been used [3]-[5], but this requires on the order of CPU hours even with just a few interferers. Thus the improvement of optimum combining has only been studied for a few simple Paper approved by N. C. Beaulieu, the Editor for Wireless Communication Theory of the IEEE Communications Society. Manuscript received September 21, 1993; revised November 28, 1996. This paper was presented in part at the 1994 IEEE Vehicular Technology Conference, Stockholm. Sweden, June 8-10, 1994. J. H. Winters is with AT&T Labs-Research, Red Bank, NJ 07701 USA. J. Salz, retired, was with AT&T Labs-Research, Crawford Hill Laboratory, Holmdel, NJ 07733 USA. Publisher Item Identifier S 0090-6778(98)09388-X.

y

User~

Fig. 1. Block diagram of an Jl-element adaptive array.

cases, and detailed comparisons (e.g., in terms of outage probability) have not been done. In [6], we showed that, with ]VI antenna elements, the received signals can be combined to eliminate L (L < M) interferers in the output signal while obtaining an M - L diversity improvement, i.e., the performance of maximal ratio combining with ]\II- L antennas and no interference. However, this "zero-forcing" solution gives far lower output SINR than optimum combining in most cases of interest and cannot be used when L 2: Ail. In this paper we present a closed-form expression for the upper bound on the bit-error rate (BER) with optimum combining in wireless systems. We assume flat fading across the channel and independent Rayleigh fading of the desired and interfering signals at each antenna. 1 Equations are presented for the upper bound on the BER for coherent detection of quadrature amplitude modulated (QAM) and binary phase-shift-keyed (BPSK) signals, and for differential detection of differential phase-shift-keyed (DPSK) signals. From these equations, a lower bound on the improvement of optimum combining over maximal ratio combining is derived. In Section II we derive the upper bound on the BER. In Section III we compare the upper bound to Monte Carlo simulation results. A summary and conclusions are presented in Section IV. II. UPPER BOUND DERIVATION

Fig. 1 shows a block diagram of an M -element adaptive array. The complex baseband signal received by the ith antenna element in the kth symbol interval Xi (k) is multiplied by a controllable complex weight ui, and the weighted signals are summed to form the array output signal So (k). I As shown in [7], the gain of optimum combining is not significantly degraded with fading correlation up to about 0.5. Thus our bounds, based on independent fading, are reasonably accurate and useful even in environments with fading correlation up to this level.

Reprinted from IEEE Transactions on Communications, Vol. 46, No. 12, pp. 1619-1624, December 1998.

519

With optimum combining, the weights are chosen to maximize the output SINR, which also minimizes the mean-square error (MSE), which is given by [8] MSE == (1 + U~R~~1£d)-l

(1)

where Rnn is the received interference-plus-noise correlation matrix given by

n.; =

0"

2

1+

L L

(2)

Uj1£}

j=l

CJ2 is the noise power, I is the identity matrix, Ud and

are the desired and j th interfering signal propagation vectors, respectively, and the superscript denotes complex conjugate transpose. Here we have assumed the same average received power for the desired signal at each antenna (that is, microdiversity rather than macrodiversity) and that the noise and interfering signals are uncorrelated, and without loss of generality, have normalized the received signal power, averaged over the fading, to 1. Note that the MSE varies at the fading rate. For coherent detection of BPSK or QAM, the HER is bounded by [9]

1.£j

t

r. ::; e(1/er;) E [e( -l/MSE)] = e((1/IT~)-l) E [e-U;,R;;-,:Ud]

(3)

the bound. Also, note that with only noise at the receiver, An = (1~, where O'~ is the variance of the noise normalized to the received desired signal power, and from (4) and (5) ., ((1~)~I 1 (6) Pe < -2- = = 2p AI where p is the received SINR, while the actual BER is 1/2(1 + p)}VI [1]. Thus even without interference, the bound differs from the actual BER, and this difference increases as the received SINR decreases. Let us consider the case of interference only. In this case, IRnnl, which is giyen by (2), may also be expressed as

IRnn I = IQ t QI

L

±DI Dm1D~Dm2 ... DtI D 1n M

(7)

where Q = (D 1 , · · · , D~I), D·, U = ((U1).m···(1£L)m)T, (Uj )In is the mth element of 1£j, the sum is extended over all M! permutations of the Il.;' s, D rn , is the ith element of the permutation of the D 111 's, the "+" sign is assigned for even permutations (i.e., an even number of swapping of DnJ.'s in the permutation), and the "-" sign for odd permutations. Now

"2 L

t E[D·m,D,u]

= L...-J

(8)

aj

j=l

where O'J is the average power of the .ith interferer normalized to the desired signal power, and

= L O'f· L

E[D!nDnD;"D.rn]

where now the expected value is taken over the fading parameters of the desired and interfering signals, and O"~ is the variance of the BPSK or QAM symbol levels (e.g., O'~ == 1 and 2 for BPSK and quaternary phase-shift keying (QPSK), respectively). For differential detection of DPSK, assuming Gaussian noise and interference.? the BER is given by [1] 1 [ e-udt B-1 Ud] . P = -E nn e 2

=

(9)

j=l

Similarly, from (7), it can be shown that

(4)

Thus the BER expression for both cases differs only by a constant, and we will now consider the term E[e-u~R;~Ud]. As shown in the Appendix, this term can be upper-bounded by (5)

where IRnnl denotes the determinant of Rnn, and An is the nth eigenvalue of Rnn. Since (5) is the key inequality in our bound (and is the only inequality we use in determining the bound for differential detection of DPSK), let us examine its accuracy. The bound is tight if An ~ 1, and since the An's are proportional to the interference signal powers, the bound is tight for large received SINR, i.e., low BER's. Although for all cases (1 + (l/An»-l < 1 and thus BER < 0.5, for An > 1 the BER as given by the bound may exceed 0.5. Thus with small received SINR, occasionally BER's greater than 0.5 may be averaged into the average BER, reducing the tightness of

where the sum is over all sets of positive integers ik and lk that exist such that M ~ ... > i 2 > iI, with Ek iklk ~ M. For example, when M = 5, there are 6 sets of {ik' lk} such that Ek iklk ~ M (see Table I). All sets are of the form {iI, II}, e.g., {i 1 = 3, II = I} for 3 ·1 < 5, except for the set {i 1 = 2, 11 = 1, i 2 = 3, 12 = I} for 2 . 1 + 3 . 1 = 5. Q~}VI) is an integer coefficient corresponding to the qth set with M antennas. Note that a~/)';l) is obtained by summing the coefficients (±1' s) for similar terms in E[ IQ t QI]. a.~Nf) can be determined as shown below. Since E~=l CJ; 1/ p, and a~lvI) 1 when iklk 0, (10) can also be expressed as

2 Since the stronger the interference, the more that optimum combining suppresses it, with the Gaussian assumption we overestimate the probability of strong interference. Note that this is consistent with the derivation of an upper bound on the BER.

520

E[ IQtQI]

Ek

=

=

= p-Al

[1 + L a.~AI) (t(p.aJ)i q

=

1

)

h

J=1

-(t,(pa;)i2)'2.. -]

(11)

TABLE I

VALUES OF

FOR

II

i2

AI

= 2 TO

5

V ALUES OF

TABLE II

0'

q

M

i}

I}

1

-1

6

2 2

2 3

1 1

-3 +2

1 2 3 1 2

2

1 2 1 1

-6 +3 +8 -6

1 2 1 1 1 1

-10 +15 +20 -20 -30 +24

i

2

2

3

4

I

2

3 4 2

2 3 2

4 5

/2

1

3

2 3 3 4

5

7

2 2

1 2 3

6 7 2 2 2 3

1 2 1 1 1 1 1 1 1 1

2

2

3

3

4

where now M 2: ... > ';'2 > '£1 > 1. To determine the a~ll'..t), s, first note that if a~) 1, "', L, then L~=l

aJk

== Lo?", and

E[IQtQIJ = (L M +

t,

(J''J.

' .J'

(11) becomes

IhLM-k+l)iT2M

(12)

where the 13k's and the a~Af),s can be seen to be closely related. From [6], P; == 0 for L < M, and thus the {3k's are the coefficients of the N/th-order polynomial in L, L(L - l)(L - 2)··· (L - M + 1). This result is not only useful when all interferers have equal power, but also serves as a consistency check on our calculated values of Q.~AI). (1\1) Q.q

.

were generated USIng a computer Th e va Iues 0 f program to examine every permutation in (7) for given M. The number of each type of iI, ll, i 2 , l2, ... term was calculated to determine Q.~lYf). Tables I and II list these values for M == 2-7. Note that only i 1 and II terms exist for M < 4 and i? and

l2 terms also exist for 5 ~ M Values for ;;~l\j) for higher M can also be easily calculated. However, since the amount of computer time to generate the values of a.~j\1) increases exponentially with M, our program could only generate these 0

p-M

[1 + l:= Q.~JvI) (t(PiT JY1) q

i2

12

1 1

3 4

3

1 1

4 3

1 1

4. 5

1

AND

7

a(M) q

-15 +45 -15 +40 +40 -90 +144 -120 -120 +90

-21 +105 -105 +70 +280 -210 +504 -840 +720 -420 +630 -504 -420 +210

and from (4), the upper bound on the BER with differential detection of DPSK is given by

r. < ~ p- M

[1 + l:= Q.~J\I) (t(PiTJ)i 1) q

II

J=1

o

(t,(PiTjYi2 ) 12 o' oj

0

(14)

For the case of noise with L interferers, consider the noise as an infinite number of weak interferers with total power equal to the noise. That is, let

values in a reasonable amount of computer time for up to M == 10 (where a hundred CPU hours on a SPARCstation20 would be required). From (3), the upper bound on the BER with coherent detection of BPSK or QAM is now given by

e(l/O"~)-l)

AI = 6

1

1 1 1

2

FOR

1

6 2 2

5

Fe :S

~l\t{)

a(M)

M

5

n~lV/)

2 aj

a;

== K - L'

j

== L + 1, ... , K,

(15)

II

J=l

.(t,(piTJ)i y2 .. oj 2

(13)

for i k > 1. Therefore, with noise, the BER bound is the same as in (13) and (14), but with p including the noise. In this case, if we define the received desired signal-to-noise ratio a;;2 and the jth interferer signal-to-noise ratio as as d

521

r

f

j

= aJj a~, then (14)

becomes [similarly for (13 )]

10 r - - - - - - - - - - -- - - - - - ,

...........-

8

_.---_ ---....

__-----

_----8::: M=5

f l=10dB

Coherent Detection of BPSK L=1

2;M

Since is the bound with maximal ratio combining , the tenn in the brackets is the improvement of optimum combining over maximal ratio comb ining based on the BER bound. Defining the gain of optimum combining as the reduction in the required p for a given BER, from (17), this gain in decibels is given by Gain (dB) 10

=- M

10' \ .5

Fig. 2. Gain versus BER for coherent detec tion of BPSK-compari son of analytical result s to the asym ptotic gain.

log10

12 , - - - - - - - - - - - - - - --, 10

- - Theoretical Results •••• Simula tion Results Asymptotic Ga in M=2

This gain is therefore independent of the desired signal power (because the bound is asymptotically tight as p ---+ 00 ). However, this is the gain of the BER bound with optimum combining over the BER bound with maximal ratio combining. Since the required p for a given BER with maximal ratio combining is less than the bound , the true gain may differ from (18) and to obtain a bound on the gain, the gain in (18) must be reduced accordingly . For example, with differential detection of DPSK, to obtain a bound the gain given in (18) is reduced by the factor (pj( l + p))IIJ . Note that as p ---+ 00 , this factor reduces to one and the gain approaches ( 18) . Thus we will refer to (18) as the asymptotic gain.

III.

COMPARISON TO E XACT THEORY AND SIMULATION

In this section, we compare the bound to theoretical results for L = 1 and simulation results for L ~ 2. Fig. 2 compares theoretical results (from [1]-[3]) for the gain to the asymptotic gain (18) versus BER with coherent detection of BPSK. Results are generated for M = 2 and 5, and I' 1 = 3 and 10 dB. In all cases the gain monotonically decreases to the asymptotic gain as the BER decreases . The gain approaches the asymptotic gain more slowly with decreasing BER for larger M and also, at low BER's, the accuracy of the asymptotic gain decreases with higher f l . Thus the accuracy of the asymptotic gain decreases as the p required for a given BER with optimum combining decreases, as predicted by the approximation in Section II. Fig. 3 compares theoretical and Monte Carlo simulation [5] results for the gain to the asymptotic gain with M = 2 and L = 1, 2, and 6. Results are plotted versus f j , where all L interferers have equal power , for coherent detection of BPSK

L=2

5

10

f j (dB)

15

20

Fig. 3. Gain with .\1 = 2 for I. 2, and 6 equal-powe r interferers versus signal-to-noise ratio of each interfe rer-s-co mparisc n of analytica l and Monte Carlo simulation res ults with coherent dete ction of BPSK [5] to the asymptotic gain.

at a 10- 3 BER,3 In all cases, the asymptot ic gain has the same shape as the gain and is within 1.7 dB for L = 1, 1.0' dB for L = 2, and 0.4 dB for L = 6. Since optimum combining gives the largest gain when the interference power is concentrated in one interferer and the least gain when the interference power is equally divided among many interferers, L = 1 and L = 6 represent the best and worst cases for the gain in an interference-limited cellular system. Thus from the results in Fig. 3, we would expect the asymptotic gain to be within 0.4-1 .7 dB of the actual gain for all cases in cellular systems with M = 2. 3Th is BER was used bec ause the result s in [5] were obtained for this BER. As shown in [5], the gain does not change sign ificantly for BER 's between 10- 2 and 10- 3 , the range of interest in most mobile radio sys tems.

522

of cases at a 10- 3 BER. These cases include interference scenarios that cover the range of worst to best cases for the gain of optimum combining in cellular systems with M == 2. The bound is most accurate with differential detection of DPSK and high SINR, corresponding to low BER and a few antennas. Because of the 2-dB accuracy, the bound is most useful where the optimum combining improvement is the largest, which is the case of most interest. The closedform expression for the bound permits rapid calculation of the improvement with optimum combining for any number of interferers and antennas, as compared with the CPU hours previously required by Monte Carlo simulation. These bounds allow calculation of the performance of optimum combining under a variety of conditions where it was not possible previously, including analysis of the outage probability with shadow fading and the combined effect of adaptive arrays and dynamic channel assignment in mobile radio systems,

6 -----------------., • • •• Simulation Results Asymptotic Gain

. .. .. ... . . ..... . ..... . . . ..•.. BER=10· 3

r j U=1,L)=3dB

'---------"""'

4

2

0'----.-..-----------"--------6 3 2

5

4

7

M

Fig. 4. Gain versus AI with two and six equal power interferers-comparison of Monte Carlo simulation results with coherent detection of BPSK [3] to the asymptotic gain.

ApPENDIX

Diagonalizing Now, consider the lower bound on the gain obtained from the BER bound (17), as compared to the asymptotic gain. Without interference, differential detection of DPSK with maximal ratio combining and All == 2 requires fJ ~ 13.3 dB (theoretically [10]) for a 10- 3 BER, while the BER bound (17) gives p ~ 13.5 dB. Thus the lower bound on the gain (from (17)) at a 10- 3 BER is 0.2 dB less than the asymptotic gain for any interference scenario-in particular, the lower bound on the gain is 0.2 dB less than the results shown in Fig. 3. Similarly, coherent detection of BPSK with maximal ratio combining and 1\;1 == 2 requires p ~ 11.1 dB for a 10- 3 BER, while the BER bound (13) gives 15.0 dB. Thus the bound is most accurate with differential detection of DPSK and low BER's. Fig. 4 compares Monte Carlo simulation results [3] for the gain to the asymptotic gain for L == 2 and 6. Results are plotted versus !vI with r j == 3 dB for all interferers and coherent detection of BPSK at a 10- 3 BER. Again the asymptotic gain has the same shape as the simulation results. The cases include both many more interferers than antennas and many more antennas than interferers, but in all cases the asymptotic gain is within 1.8 dB of simulation results.

Rnn by a unitary transformation W, we obtain (19)

where diag (.) denotes an M x !VI matrix with nonzero elements only on the diagonal, or

R n- n1 and

1 tRnn Ud

'Ud

-

,,/,t I.fI

di:lc1g (\/\ 1-1

t I}/,t - U d I.fI

di.rag (\Al-1

\ -1),1/ AI If),

A

\ -1),,/, " . /\;'1 If/Ud,

(20) (21)

Let (22) Then

tR- 1 u d nnU,d and

-

AI

j') An

~ ICn ... ~

n=l

(23)

E[e-U~R~~Ud] = E [exp(_~ Ic~2)] =E

IV, CONCLUSIONS In this paper we have presented upper bounds on the biterror rate (BER) of optimum combining in wireless systems with multiple cochannel interferers in a Rayleigh fading environment. We presented closed-form expressions for the upper bound on the bit-error rate with optimum combining, for any number of antennas and interferers, with coherent detection of BPSK and QAM signals, and differential detection of DPSK. We also presented bounds on the performance gain of optimum combining over maximal ratio combining and showed that these bounds are asymptotically tight with decreasing BER. Results showed that the asymptotic gain is within 2 dB of the gain as determined by computer simulation for a variety

."

[IT exp (J~~2) ].

(24)

Since with independent, Rayleigh fading at each antenna, the elements of U,d are independent and identically distributed (i.i.d.) complex Gaussian random variables, the elements of C are also i.i.d. complex Gaussian random variables with .the same mean and variance. Furthermore, the An's are independent of the c.,' s. Thus we can average over the desired and interfering signal vectors separately, i.e.,

523

E

[IT exp (_1~~2) ] E [IT E [exp(J~n~2) ]]. =

A

Cn

(25)

Since the en's are complex Gaussian random variables with zero mean and unit variance

[1]

(26)

[2]

E en [exp

IcnI2)] = 1 +1 } (- ~

REFERENCES

n

and

[3] [4]

Since the

An'S

1 1+

E[e-u~R;:~Ud] ~ E lR..nl

1

An

and, therefore,

where

[5]

are nonnegative

A

< An

(28)

[g An] =

denotes the determinant of

[6]

[7]

E A [ lR..nl]

R..n.

(29)

[8] [9] [l 0]

524

w. C. Jakes Jr. et al., Microwave Mobile Communications. New York: Wiley, 1974. V. M. Bogachev and I. G. Kiselev, "Optimum combining of signals in space-diversity reception," Telecommun. Radio Eng., vol. 34/35, no. 10, pp. 83, Oct. 1980. J. H. Winters, "Optimum combining in digital mobile radio with cochannel interference," IEEE J. Select. Areas Commun., vol. SAC-2, no. 4, July 1984. _ _ , "Optimum combining for indoor radio systems with multiple users," IEEE Trans. Commun., vol. COM-35, no. 11, Nov. 1987. _ _ , "Signal acquisition and tracking with adaptive arrays in the digital mobile radio system IS-54 with flat fading," IEEE Trans. Veh. Technol., Nov. 1993. J. H. Winters, 1. Salz, and R. D. Gitlin, "The impact of antenna diversity on the capacity of wireless communication systems," IEEE Trans. Commun., Apr. 1994. J. Salz and J. H. Winters, "Effect of fading correlation on adaptive arrays in digital wireless communications," IEEE Trans. Veh. Technol., vol. 43, pp. 1049-1057, Nov. 1994. R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley, 1980. G. 1. Foschini and J. Salz, "Digital communications over fading radio channels," Bell Syst. Tech. J., vol. 62, pp. 429-456, Feb .. 1983: J. H. Winters, "Switched diversity with feedback for f)PSK mobile radio systems," IEEE Trans. Veh. Technol., vol. VT-32, pp. 134-150, Feb. 1983.

The Range Increase of Adaptive Versus Phased Arrays in Mobile Radio Systems Jack H. Winters, Fellow, IEEE, and Michael J. Gans Abstract-In this paper, we compare the increase in range with multiple-antenna base stations using adaptive array combining to that of phased array combining. With adaptive arrays, the received signals at the antennas are combined to maximize signalto-interference-plus-noise ratio (SINR) rather than only form a directed beam. Although more complex to implement, adaptive arrays have the advantage of higher diversity gain and antenna gain that is not limited by the scattering angle of the multipath at the mobile. Here, we use computer simulation to illustrate these advantages for range increase in both narrow-band and spreadspectrum mobile radio systems. For example, our results show that for a 3° scattering angle (typical in urban areas), a 100element array base station can increase the range 2.8 and 5.5-fold with a phased array and an adaptive array, respectively. Also, for this scattering angle, the range increase of a phased array with 100 elements can be achieved by an adaptive array with only ten elements. Index Terms-Adaptive arrays, mobile communications, multipath channels, phased arrays.

M

1.

INTRODUCTION

ULTIPLE antennas at the base station can provide increased received signal gain and, thus, range in mobile radio systems. Two approaches for combining the received signals are the phased array, which creates an antenna beam directed at the mobile, and the adaptive array, which maximizes signal-to-interference-plus-noise ratio (SINR). Here, we compare the range increase of phased arrays to that of the more complex adaptive array technique for both narrow-band and spread-spectrum systems. Previous papers have studied the increase in gain with phased arrays [1]-[6]. With phased arrays, the signals received by each antenna are weighted and combined to create a beam in the direction of the mobile. The same performance can also be achieved by sectorized antennas, whereby a different antenna is used to form each beam. As the number of antennas increases, the received signal gain (range) increases proportionally to the number of antennas, but only until the beamwidth of the array is equal to that of the angle of multipath scattering around the mobile. Beyond that point, the increased gain of more antennas is reduced by the loss of power from scatterers outside the beamwidth. The range can even be reduced with narrower beamwidths because the resulting reduction in delay spread can cause a loss of diversity

Manuscript received September 19, 1994; revised July 19, 1998. J. H. Winters is with AT&T Labs-Research, Red Bank, NJ 07701 USA. M. J. Gans is with Lucent Bell Labs, Holmdel, NJ 07733 USA. Publisher Item Identifier S 0018-9545(99)01067-1.

gain in systems using equalization, e.g., in spread-spectrum systems using a RAKE receiver. This limitation in range increase can be overcome by the use of adaptive arrays [5]-[9]. With adaptive arrays, the signals received by each antenna are weighted and combined to maximize the output SINR. Although the most widely studied advantage of adaptive arrays is interference suppression [7J-[ 10], maximizing SINR also forms an antenna pattern matched to the wavefront (which is not a plane wave for nonzero scattering angle) and therefore provides a range increase that is not limited by the scattering angle. In addition, adaptive arrays can provide higher diversity gain than phased arrays, since all the receive antennas can be used for diversity combining. Thus, for a given number of antennas. adaptive arrays can provide greater range, or require fewer antennas to achieve a given range. In this paper, we describe the limitations of phased arrays for range increase and describe how these limitations can be overcome using adaptive arrays.' We use computer simulation to illustrate our results for the range increase in both narrowband and spread-spectrum mobile radio systems. For example, our results show that for a 30 scattering angle, a l Otl-elcmcnt array base station can increase the range 2.8 and 5.5- fold with a phased array and an adaptive array, respectively. Also. for this scattering angle, the range increase of a phased array with 100 elements can be achieved by an adaptive array with only ten elements. In Section II, we discuss the theoretical performance of phased and adaptive arrays. We present a mobile radio system model and illustrate the performance results by computer simulation in Section III. II.

DESCRIPTION OF PHASED AND ADAPTIVE ARRAYS

A. Phased Array

Fig. 1 shows a block diagram of a phased array with omnidirectional elements linearly spaced at >"/2, where X is the signal wavelength. The signals received by the antennas are weighted and combined to form a beam at angle ¢, i.e., the signal at the i th antenna is phase shifted by IT (i - 1) sin (P ~ 't == 1,··· .Ad.

For the mobile radio base station, the antenna beam should be narrow in elevation and the antenna characteristics should be independent of azimuth. A narrow elevation angle can be I Note that we consider range increase as a convenient way to express the effect of gain increase, and it also corresponds to a decrease in required number of base stations to cover a given area.

Reprinted from IEEE Transactions on Vehicular Technology, Vol. 48, No.2, pp. 353-362, March 1999.

525

signals should also be weighted by the voltage gain in the given direction to maximize signal-to-noise ratio (SNR) in the array output. These weighted signals are summed to generate the array output, with the output SNR for a beam with direction ¢ given by

L8 J.'1

r ec 1 .

si((p)

t=l

AI

Ll s i (¢ )1

2

(1)

2

•

•2

•

•

3

i=l

M

where

Fig. 1. Linear phased array with omnidirectional elements linearly spaced at ,\/2.

• • • •

• • • •

• • • •

• • • •

1~21 (a)

• • •

• • • •

• • • • 'JJ2

• • •

1

(b)

Fig. 2. (a) Array with linear elements on four panels in a square and (b) with elements on a cylinder.

created by using a vertical array of antenna elements for each horizontal element. The azimuth dependence can be reduced by placing the linear elements on four panels in a square, as shown in Fig. 2(a) [11]. However, a cylindrical array, as shown in Fig. 2(b), is usually used to create azimuth independence. Each antenna element is typically spaced at A/2, since smaller spacing reduces gain by creating a wider beamwidth with increased mutual coupling, while wider spacing can also reduce gain by decreasing the beamwidth and creating grating lobes, i.e., gain in directions other than the desired angle-ofarrival. The effect of antenna spacing on mutual coupling is studied in Appendix A. To create a beam in a given direction, the signals from the antenna elements are cophased, based on a plane wave arrival. Since to reduce mutual coupling between elements, each element should have higher gain in the direction pointing away from the center of the cylinder (see Appendix A), the

i,

8'i (

"rec,

is the complex received signal voltage at antenna

¢) is the expected (based on antenna location) antenna

voltage gain and phase (relative to the other antennas) for a signal arriving from angle rjJ, and the superscript * denotes complex conjugate. The weights can be implemented at radio frequency (RF) by different cable lengths for the fixed phase offsets and fixed attenuators for the amplitude weighting. The weighted signals for each beam are then combined, with a separate combiner and signal for each. beam. For each mobile radio user, the receiver then selects the beam output with the largest power to use for signal demodulation. However, this technique can require a large amount of hardware, including amplifiers, with large !VI, but the complexity can be reduced somewhat by combining only a portion of the antenna outputs-the signals from the antennas with the largest gain in a given direction-for each beam. Alternatively, the signal from each antenna can be brought to baseband and analog-digital (AID) converted, with the combining done in software. Although this method is similar to adaptive array processing, with the phased array the combining software needs to determine only one parameter, the angle-of-arrival ¢ (which changes slowly with time), for each mobile radio user. The same performance as the phased array can be achieved by using sectorized antennas, i.e., separate antennas for each beam, as is currently done at many mobile radio base stations. However, to create uniform coverage using sectorized antennas or phased arrays with predetermined (fixed) beams, overlapping beams should be used. (This is also useful for obtaining diversity-see below.) This doubles the number of antennas (with sectorized antennas) or the combining hardware (with phased arrays with fixed beams) without increasing the gain. Arrays increase the range by providing additional received signal gain due to two factors-antenna gain and diversity gain. With an M -element phased array and a point source, the antenna gain is lVI, neglecting mutual coupling (see Appendix A). The range increase is the gain raised to the inverse of the propagation loss exponent '"'(, typically a fourth power loss. Thus, with a point source, the range increase due to the antenna gain of an .1\11 -element array is MIlT. However, signal scattering around the mobile means that the signal received at the base station cannot always be considered as coming from a point source. As shown in Fig. 3, with scattering the signal arrives from a range of angles, called the scattering angle. Typically, the mobile signal is scattered mainly by objects within 1000 ft of the mobile,

526

•

Mobile

•

•

•

• •

• •

..

•

•

•

" •

•

•

•

Base Station

Base Station Fig. 3. Mobile radio environment with scattering around the mobile. where all signals from a mobile arrive within a scattering angle II

Fig. 4.

but this distance can vary widely, e.g.. with reflections off mountains 112]. Furthermore, this scattering angle increases with decreasing base-station height. Measured results for rural areas with 130-ft antenna heights show scattering angles of only a few tenths of a degree, while suburban and urban areas have much larger scattering angles r 131. Measured results in urban areas of Tokyo, Japan. for ranges up to 7 km [141, show a 3° scattering angle at a 50-m antenna height increasing to 360° at a l-m height (as on the mobile). In addition, digital mobile radio systems in North America (IS-136) and Europe (GSM) are designed to handle delay spreads up to 41 and 16 ItS, respectively, which, with an 8-mi cell radius. correspond to scattering angles of 52° and 21 0, respectively. Also, these scattering angles are for 900-MHz mobile radio systems, while at 2 GHz the range is reduced by about 509c (from the Hata model [15], for an antenna height of 50 m at the base station and 1 m at the mobile, medium-small city, and 8-mi cell radius), corresponding to a two-fold scattering angle increase. We expect that microcells will have even larger scattering angles because of the lower antenna height. Here. we do not consider what the likely distribution of scattering angles will be for any given system, but show results obtained for a wide range of scattering angles. Since receive signal power is lost when the beamwidth, which is approximately 360° /1\,1 (for a cylindrical array), is less than the scattering angle, the signal gain will be less than J.\;1 in the phased array with large enough 1\1. For example, for a uniform distribution of power within a scattering angle of a degrees, the maximum signal gain is given by an array with "Ai! == 360/ Q elements. Additional elements increase the antenna gain, but the power lost outside the beam reduces the signal gain by the same amount (under the uniform power distribution assumption). Thus, with phased arrays the signal gain, and the corresponding range increase, is limited. The other factor for receive signal gain is the diversity gain. Multipath fading results in a higher average output SNR required to achieve a given average receiver performance (e.g.,

Cylindrical array using of angle diversity.

BER in digital systems) than without fading. The fading in the output signal can be reduced by using multiple receive antennas and combining the received signals. We define diversity gain as the improvement in link margin beyond the factor of AI for array gain. For example, for a 10- 2 BER averaged over Rayleigh fading with coherent detection of PSK, a 9.5dB higher average output SNR is required than without fading. Two antennas provide up to a 5.4-dB diversity gain, while 3, 4, and 6 antennas provide up to 6.8,7.6. and 8.3 dB, respectively, with maximal ratio combining. Thus, six antennas can provide within 1.2 dB of the maximum diversity gain (i.e., the 9.5-dB gain achieved when the fading is eliminated). However, to achieve the full diversity gain, the fading at the antennas must be nearly independent. This requires that the spacing between antennas is at least the distance such that the beamwidth of an antenna with this aperture is approximately the scattering angle. For example, a spacing of lO-20A is used for the typical scattering angle of a few degrees [12], [14], [16]. For a cylindrical phased array, such an antenna spacing between elements is impractical and would create numerous grating lobes without providing the antenna gain commensurate with the diameter of the array (or providing diversity gain). However, when the beamwidth of the array is comparable to the scattering angle (i.e., the total array aperture size corresponds to a beamwidth given by the scattering angle), different beams can cover part of the same scattering angle and thereby angle diversity can be used [4], [13], as shown in Fig. 4. For the square array, another set of flat arrays could be spaced lO-20A apart on each side to provide diversity, as shown in Fig. 5. Note that this is not practical with cylindrical arrays, as the arrays would partially block each other. Similarly, to provide diversity with sectorized antennas, a separate set of antennas can be spaced lO-20A apart (as is used today) with overlapping sectors to provide more uniform coverage over all azimuth angles. In all cases, though, diversity gain requires additional hardware. To minimize the added cost, usually only dual diversity with selection combining is considered. Note that for the example case of a 10- 2 BER,

527

1\11 -element adaptive array is given by

I?J~II.....·--NA.-~

• • • •

• • • •

(2)

• • • •

• • • •

•

• •

•

• •

Fig. 5.

1-=1

• •

• • • •

Square array using space diversity.

• • • •

selection diversity with two antennas provides only about 3.9 dB of the maximum-possible 9.S-dB diversity gain (which is also I.? dB less than maximal ratio combining with two antennas). Frequency-selective fading due to delay spread can also be used to provide diversity by using equalization [9] in narrow-band systems, or a RAKE receiver in spread-spectrum systems [17]. In this case, the diversity gain of additional antennas is reduced. For example, a three-finger RAKE is used in the IS-95 CDMA system (three fingers on the downlink, but four fingers on the uplink). With received signal energy uniformly distributed over three code symbol periods (2.4 J-Ls), maximal ratio combining of the three fingers provides three-fold diversity, ora 6.8-dB diversity gain at a 10- 2 BER, and dual antenna diversity provides up to 1.5 dB (the overall combining is equivalent to six-branch maximal ratio combining) of the remaining 2.7-dB maximum diversity gain. Note, however, that, compared to a narrow-band receiver, one finger of this CDMA receiver is 4.8 dB lower in signal power, i.e., the RAKE receiver does not give any increase in average SNR (antenna gain). Finally, note that beamwidths smaller than the scattering angle can reduce the delay spread, and therefore the diversity gain, in systems with phased arrays.

B. Adaptive Array With an adaptive array, the received signals are combined to maximize the output SINR. Thus, the array can null interference in narrow-band systems/ (as discussed below), but here we consider only the increase in range due to higher antenna gain. Without interference, the output SNR of an 2 For spread-spectrum systems, nulling of all strong interferers is generally not possible since the number of interferers is typically much greater than the number of antennas.

Although (2) is simpler than the SNR equation for the phased array (1), the adaptive array is more complex to implement because the weights are not fixed, but depend on the received signals. Thus, variable gains and phase shifters are needed for each signal on every antenna. These can be implemented in hardware at RF or IF, or in software at baseband. For the software implementation, the signals from each antenna can also be digitized using block processing. Another complication is the need to acquire and track the weights. As compared to the phased array where the beam or the weights only need to track the angle of the mobile, the adaptive array weights must track the rapid fading of the signal. Algorithms to generate the weights include the constant modulus algorithm (CMA) [18], least-meansquared (LMS) algorithm [19], and the direct matrix inversion (DMI) algorithm [19]. It should be noted, though, that when interference is not a concern, i.e., when range increase is the issue as in this paper, simpler techniques may be possible for determining the weights. With the adapti ve array, though, the array pattern is matched to the multi path wavefront. That is, there is no antenna gain limitation due to multipath scattering angle, as with phased arrays, and an Ail-fold diversity gain can also be obtained. Achieving this diversity gain requires adequate antenna spacing however. With a base-station array oriented broadside to a small angle, a degrees, of scatterers around the mobile and with power arriving uniformly at the base from within n~, the magnitude of the correlation coefficient between two array elements spaced x wavelengths apart is approximately [see also [14], which approximates the envelope correlation Pc(x) by the square of the complex phasor correlation I p(:t) 12 ]

( )I ~ Ipx

sin( 1r 2ax /180) . (1r 2ax /180)

(3)

Thus, an antenna spacing of (360°/1ra) (>../2) is required for independent fading at each antenna, but spacings of about half of this still give low-enough fading correlation «0.7) that nearly the full diversity gain can be achieved. However, even with a spacing of (360° /(7fa))(>"/4), the required array size can be too large. For example, a 3° scattering angle requires a 10-ft antenna spacing at 900 MHz, and, thus, in particular, a 100-element cylindrical array would require a 330-ft diameter. However, since only a few-fold diversity is needed to obtain most of the maximum diversity gain, an array with a diameter of a few times the required antenna spacing (20-30 ft in the above example) should obtain almost all the maximum-possible diversity gain. Finally, we note that, although not studied in this paper, the adaptive array can also suppress interference. With the narrow beams of large arrays, the number of interferers is greatly reduced in both narrow-band and spread-spectrum systems. Since an M -element array can eliminate N interferers with an M - N diversity gain, large arrays can eliminate any significant

528

interference with little loss of di versity or antenna gain. Thus, these arrays can not only greatly increase the range when there is little interference, but they can also be used for future expansion by permitting the capacity to be greatly increased without increasing the number of base stations.

III.

RES ULTS

A. Model To verify and illustrate the above concl usions, we used Monte Carlo simulation with the following model (see Fig. 3). We considered transmission from a mobile to a base station. The multipath model consisted of 20 scatterers uniformly distributed in a circular area of radius T around the mobile. These scatterers had equal transmitted power, with a fourth law power loss from each scatterer to the base station. The phase of each multipath reflection at each antenna was determined from the path length. Recei ved power variation due to shadow fading was not considered. The base-station array was a cylindrical array of J\;1 equally spaced cardioid antennas [20], with each antenna pointing out from the center of the array, and one element at 0°. The mobile was at 90°. Note that for AJ == 2, the mobile at 90° results in equal gain from the two antennas, while with a mobile at 0° only one antenna has nonzero gain. Thus, for ~\1 == 2, the results depend strongly on the angle of the mobile (i.e., dual diversity at 90° versus no diversity at 0°). However. for l\lI 2 -1, the effect of angle is negligible, and therefore this angle was fixed at 90°. We considered spacings between elements of A/'2 or greater, and therefore neglected the effect of mutual coupling (see Appendix A). With the phased array, the weights were set to generate a beam that was pointed directly at the mobile. From (A-8) and (A-IO), these weights are given by

s; (!.l00) = /2 cos { ~ [Sill(21f(i . e- j ( 2 7r r / ).,) sin (2;"1 ( l -

1) II'v!) - l]}

1) j.'I [) .

i.

== 1. . . . . AI (4)

and the SNR is then given by (1). With the adaptive array, the weights are 8;ec i == 1.···.M and the SNR is given by (2). We consider coherent detection of phase-shift-keyed (PSK) signals, for which the BER is given by 1

'

BER == ~ erfc( jS/1V).

(5)

We used Monte Carlo simulation to determine the BER averaged over 10000 cases. Note that the BER depends on the ratio of transmit power to receive noise power. This ratio was adjusted to obtain a 10- 2 average BER for the baseline case of an omnidirectional transmit antenna with the mobile at a given range and scattering radius. With this ratio and the scattering angle fixed, we generated results for the 1\1[element phased and adaptive arrays, increasing the range until the BER exceeded 10- 2 , thus giving the range increase. All the following results for range increase and diversity gain are referenced to 10- 2 average BER. Note that the increase in range is not strongly dependent on the modulation and detection technique considered, but will vary significantly with the power loss exponent and the BER. Specifically, the range increase will be greater than we show

in the next section if the power loss exponent is less than four or the required BER is less than 10- 2 . We considered both the low data rate case (no delay spread) and the delay spread case. For the delay spread case, the signaldelay for each scattered signal depends on the distance from the mobile to the scatterer plus the distance from the scatterer to each base-station antenna. For the spread-spectrum system with delay spread, we studied the use of a three-finger RAKE receiver for both the phased and adaptive arrays. To simulate the RAKE receiver, the computer program first convolved the delayed impulse of each scatterer with the spread-spectrum correlation function given by

.f"( f'.) == { 1 O.

ltd - t 0.8

o.~\

.

for ltd -

tl ~

elsewhere

().~

/-LS

(6)

where tel is the time delay corresponding to the distance from the center of the base station to the mobile. The responses from the 20 scatterers were then summed to obtain the signal at each antenna. These signals were weighted and combined by the phased array weights or the adaptive array weights (s;{'c, . i == 1.···.1\1). Note that the adaptive array weights vary as a function of delay. We then determined the three largest peaks in the output response that were separated by integer multiples of the code rate and combined these three signals to maximize the output SNR. That is, these three peaks were cophased and weighted by their signal amplitudes before combining. For the phased array, we considered three different models. In the first model, we considered a single beam pointed at the mobile, i.e., the phased array weights as given in (4). Thus, our model corresponds to phased array combining with a RAKE receiver after the combiner. followed by maximal ratio combining of the RAKE output. To model the 15-95 CDMA system with a phased array, we also considered a RAKE receiver on each antenna, followed by phased array combining of the RAKE outputs, with the beam direction optimized for each delay [rather than set to 90° as in (4)]. Thus, a separate beam was fanned for each of the RAKE fingers. Finally, we modified the second model to consider the beam direction optimized over M different, equally spaced angles, which models sectorized antennas. For the adaptive array, our model corresponds to a RAKE receiver on each antenna branch, with adaptive array combining of the antenna signals followed by adaptive array combining of the three highest output peaks, with the receiver timing optimized to maximize the output SNR. For the no delay spread case, in our simulations we used a 40000-ft range as the baseline case, with the scattering radius given by the required scattering angle. However, our results can be generalized to any range, as they depend only on the scattering angle and not the absolute values of the range and scattering radius. Therefore, in the next section, we present our results only in terms of the normalized range. Similarly, although we generated results for a one foot wavelength, our results can be generalized to any wavelength. Therefore, our results on antenna spacings are only in tenus of A. Also, for the delay spread case, our simulations used a 1.25-Mbps data rate (as in the 1S-95 CDMA system). The scattering radius was

529

set to 1200 ft (which is typical in mobile radio in suburban and urban areas) which results in a delay spread of three symbols. This radius was chosen because, as shown in the next section, this is the minimum delay spread for which the maximum diversity gain is achieved with the three-finger RAKE receiver. Thus, the scattering radius was chosen to maximize the RAKE diversity gain as well as the effect of a narrow beam width on the performance. Again, our results do not depend on the absolute values of the range and scattering radius and are therefore presented in terms of normalized range and scattering angle. Finally , note that by keeping the scattering radius constant as we increase the range (which would be typical in mobile radio), the scattering angle decreases. For example, a 10° scattering angle with the baseline case is only about 3° with a three-fold range increase. With fixed scattering radius, the predicted range increase discussed in the previous section must therefore be modified. It was noted before that, for a given scattering angle 0', the maximum gain is 360 /0' , and therefore the maximum range R, normalized to the omnidirectionalantenna range R o, is given by

!i = (360) 1/4 Ro

n

(7)

But since the scattering radius is kept constant, the scattering angle at range R is less than the baseline scattering angle no at Ro. specifically

(8) Therefore, from (7) and (8) , the maximum range increase is given by

!i = Ro

(360) 1/ 3 0'0

(9)

=

(360/0'0)1 / 3]. This increase is [with the corresponding M greater than the maximum range increase of (360/n)1 /4 for the fixed scattering angle case, e.g., the range increase is 4.9 for 0'0 = 3° versus 3.3 for 0' = 3° .

B. Results for Range Increase Fig . 6 shows the normalized maximum range versus the number of antenna elements for phased and adaptive arrays with >./2 antenna spacing, neglecting the delay spread. Results are shown for different fixed scattering radii, with the scattering angle for the baseline case of one antenna element given. We also show the theoretical range due to the antenna gain (M 1 / 4 ) without diversity, and due to antenna gain and M -fold diversity . Also, the predicted maximum range with phased arrays is shown. With the phased array, the range is shown to be limited to the predicted range limitation. However, the range improvement is degraded due to the scattering angle for M less than the theoretical value corresponding to the range limitation, and it requires many times more antennas to actually reach this limitation. For example, with a 20° scattering angle, the predicted range limitation is 2.6, corresponding to 46 antennas, but with 46 antennas the range is only 2.3 . Note that at a range

6

- _ . Adaptive Array . . . . . Phased Array Theory

Phased Array Maximum Range Increase for

U

5

"0 =3

0

Ql

c: '" ee

a:

11

.~

4

(ij

E (;

z

3

..

"

0 20 .... .

2 . 4 5~ . . . .

....... 0.5

1.5

2

.... 60;···· 2.5

3

109 10 (M)

Fig . 6. Normalized maximum range versu s the number of antenna elements for phased and adaptive arra ys' with ),, / 2 antenna spac ing. neglectin g the dela y spread.

of 2.6, the scattering angle is reduced to about 80 for the 20° baseline curve. For the adaptive array , the range exceeds the no-diversity theoretical range for all scattering angles. due to antenna diversity . The diversity gain incre ases with the scattering angle and M, as expected. However, the diversity gain does not increase for scattering angles greater than about 20 0 • Thus, because the adaptive array has greater range with increased scattering angle, the difference between the adaptive and phased array increases dramatically with scattering angle. Next consider the effect of antenna spacing. With the phased array, our results show that the range does not increa se with wider spacing, and, in fact, the range decreases if the spacing is wide enough. With the adaptive array, the range increases with antenna spacing, up to that corresponding to the maximum diversity gain. Fig . 7 shows the increase in range with spacing for M = 2, 10, and 100 and baseline scattering angles of 3°, 10°, and 20°. Theoretical results for the range with maximum diversity gain are also shown. With baseline scattering angle s of 10° or more, the maximum range can be achieved with a spacing of about 10>'. Note that a baseline scattering angle of 10° corresponds to scattering angles of 6.20 , 3.4° , and 1.8° at the maximum range with J.\1 = 2, 10, and 100 , respectively. Consider the extreme example of a very large array. For-a baseline scattering angle of 30, with 100 elements a spacing of 10>' achieves a 5.IS-range increase versus the maximum 5.46 , even though the scattering angle at this range is only 0.58° (the array diameter would be 350 ft at 900 MHz and 160 ft at 2 GHz). Thus, with large arrays the antenna spacing can be much less than that required with two antennas to achieve nearly the full diversity gain . As a further example, a 100-element array increases the range about 2.8 times with a phased array and a scattering angle at the maximum range of 3° (about an 8.4°

530

7r--

I

- - - <Xo=20'

~ : ~ ::: ~~. 5

-

----

D M~~.~'~ :':-'::;'- _ ::-:

Theory

.... --:::"

,...

........ .. -

~ 4 c:

.- -

,...-~

,

,

-

.

l Oa-fOld ~

Drversuy

"~ , - ,- , - , ,

..... . .

6

.'

5

'"c:

OJ

a: '"

-0

.§

-0

'"

"iii

E

~

Ada ptive Array Pha sed Array Theo ry

. ;"

"

a: '"

- - - - , - -- - - - - , - - - - - - - - - ,

_~

3

10·l o ld

4

"iii

-=

E

Diversity

(;

z

3

2 2 -lo ld Diversity

3 Spacing (A)

10

20

2

n

2

Fig. 7. Increase in range of adapti ve arra ys with antenna spacing for J [ =2 . 10. and 100 and baseli ne scattering angles of 30 • 10° and 20 0 . neg lecting the de lay spread.

3

log10 (M)

Fig. 9. Normalized maximum range versus the number of antenna elements for phased and adap tive arrays with .\/2 antenna spacing and a three-finger RA KE recei ver.

1

2

3

Maximum Delay (Symbo ls)

Fig. 8. Diversity gain versus the maximum delay spread for a three- finger RAKE with a single antenna at the base station.

baseline scattering angle ) versus 5.5 times for an adaptive array with 10,,\ antenna spacing. Also, for this scattering angle , the range increase of a phased array with 100 elements can be achieved by an adapti ve array with only ten elements. For the delay spread case with the RAKE receiver , let us first consider the effect of the scattering radius on the diversity gain of the RAKE recei ver. Fig . 8 shows the diversity gain versus the maxim um delay spread for a three-finger RAKE with a single antenna at the base station . For our model, the maximum delay spread is given by twice the scattering

radius in symbol periods. That is, the minimum delay is given by the delay from the mobile to the base station. while the maximum dela y is given by a scatterer at the far edge of the scattering radiu s along the line between the mobile to the base station. The maximum delay is therefore the propagation time corresponding to twice the scattering radius. The diversity gain is seen in Fig. 8 to be within 0.1 dB of the maximum possible diversit y gain (three-fold diversity) for scattering radii corresponding to delay spreads of three symbols or greater. Therefore, in our simulation s, we set the scattering radius to three symbols. Note that with our model , the maximum delay spread does not decrease with the beam width of the array because the maximum delay variation is along the line between the mobile and the base station. Fig. 9 shows the normali zed maximum range versus the number of antenna elements for phased (with the IS-95 COMA system model ) and adapti ve arrays with ,,\ /2 antenna spacing and a three -finger RAKE receiver. As in Fig. 6, results are shown for differ ent fixed scattering radii, with the scattering angle for the baseline case of one antenna element given. However, in Fig. 9 the baseline case includes a three-finger RAKE with its 6.8-dB diversity gain. Thus, the actual range in the baseline case is 1.48 (= 106 . 8 /-G ru.,

fJ

1./

-

1

1.1.

T

I=p+l.-:j:.k

(J

,

~

:.0ell .D

0 •1

o

0:

S'o

0.08

ell

+" ~

0 0 .06

cl

c8

c3 X- - :: :: c2

*-

c4 __--.:~ __c8

0.04 c5

x-

0.02 c6 X" "

c7

0

0

234

Angula r spread

(J'

5

in degr ees

6

7

Fig. 6. Outage probability as a function of (J with geometric based handover. c l : RCS-WON , 1/1 analytical. c2: RCS-WON. 1/1. e = 1 simulation. c3: RCS-WON. 1/1, e 0 .3 simulation. c4: RCS-WIN. Ill . e 0 .3. simulation. c5: SCFR, 1/3, d 3. simulation. c6: RCS-WI N. Ill , e 1. simulation c7: RCS-WIN, Ill , e 1, analytical. c8: SCFR, 1/3, Ii 3. analytical. WIN with nulling, WON without nulling.

= =

=

=

=

=

e == 1), and the RCS without null s (e independent). Results using these expressions are also plotted in Fig . 6. Example 2: All the simulations and computations performed in Example I, are repeated but assumi ng signalstrength based handover. Thi s means that the mobiles are connected to the base station with the lowest path-loss (except for some hysteresis), see #1 of Appendix A. Figs . 7 and 8 are the counterparts of Figs. 5 and 6, respecti vely.

=

=

VI. CONCLUSIONS AND DISCUSSION The following sections list conclusions drawn from the ob servations made above, and discuss critical assumptions.

A. Power Control The results of Examples one and two show that the dynamic power control range in the mobiles must be larger than 50 dB in the e == 1 case, while ::::;30 dB is sufficient in the e == 0 .3

543

,=Jb= I 1~1 ~=~~::::B~, I -20

10

20

r~IL__-,-__p~ow=e, =-rt~"Eoon.

20

-30

-20

-20

-30

-10

0

-10

0

-10

10

0

10

30

I 30

(dB)

Power transmitted from the mobile (dB)

30

20

Fig. 7. Distribution of the power control settings: upper: SCFR. middle: RCS with (' = 1, lower: RCS with (' = D.3. 0 .06

r-----r----,---.---,-------r-r---.----.---, c2

c3 0.05

c4 c5 c6 c7

I I I

..,>, 0.04 ..0

2o

0'::

c2 /

0 .03

x c8

Q)

be

..,:tl

:: 0 0.02

0.01

.-'

c6

clx-

c3 c5 c4

x- - - _ ..

_

x-

,

- x

c7

c8

/

x- -

o

6

7

Fig. 8. Outage probability as a function of (7 with ~i gnal-.s tren gth based h~do.ver. cl : SCFR, 1/3, d = . 3, si m ulati~n. c2: RCS-W?~ I~- =_1 , S =:. 1, analytical. c3: RCS-WON, 1\ = 1, S = 1. e = 0.3 , simulation. c4: RCS-WIN, I~ = 1. S = 1. e = 0.3, simulation, c5. R,?S-W?N, I~ - 1.:-S -:- 1, e -:- 1, simulation. c6: SSFR, Il3,d = 3, analytical. c7: ReS-WIN, Ill, P = 1, analytical. c8: RCS-WIN, 1/1, e = 1, simulation, WIN - with nulling, WON = without nulling.

and SCFR case. As a reference, the GSM standard supports a power control range of 30 dB [9]. B. Dependence of Downlink Performance on Uplink Power Control The results of Examples I and 2 show that the uplink power control is critical for the downlink performance of systems with downlink intercell nulling . In particular, the results show

that the power control parameter e = 1 yields much better results than e = 0.3. Thus, the conjecture of Section IV-J~H appears correct, it stated that the base will be able to identify and null the mobiles with poor downlink quality if e = 1 is applied. Is this result general? If the identification threshold Pm in is made sufficiently small (i.e., the base can identify very weak mobiles), then e = 0.3 will perform equally well. This

544

means that the conclusion may not be true for any system. However, the result indicates the importance of an issue which is typically overlooked. It should also be noted that the beamfonning used in the paper takes the desired signal strength at the identified interfering mobiles into account in the criterion function (in order to achieve this, information has to be transmitted between the bases in the e = 0.3 case but not in the e == 1 case, Section IV -B3). If this is not the case, the effect may be worse since deep nulls will point toward users who already have a good signal to interference ratio. This problem does not arise in systems with only two users and two base stations, and analysis and experiments under such conditions can therefore be misleading. C. Downlink Outage Performance

The simulation results herein indicate that the RCS systems performs better than the SCFR system if signal-strength handover is applied, or if the uplink power control completely compensates for the path-loss, i.e., e == 1 (see Section IV -B 1) and nulling is applied. This would not have been so in the geometry handover case if more users and channels had been simulated [22]. The reason being that this would have separated the same frequency users in azimuth and thus made the SCFR system more robust against angular spreading. On the other hand, the simulation and analysis assume basically uniformally distributed users, which is favorable for the SCFR system. The simulation also assumes that all multipaths are confined to an area relatively close to the mobile. If this is not the case, a larger degradation is expected in the SCFR than in the RCS cases, since the SCFR system tries to separate mobiles in azimuth to avoid the influence of angular spreading. i\PPENDIX A SIMULATION PROCEDURE

The enumeration below describes the simulation procedure used in the paper. 1) The positions of the 11 x 3 x 48 users in cells 1-9. 12-13, are generated as follows: The position of user ~ is randomized with equal probability in the area 0

COS(30 ) ) 2/", ( COS((}i. i)

(~)

R:::; 2,

(61 'I

where ri,i and Oi,l are the distance and angle to the desired base station, respectively [the factor (cos(300)/cos( O',l) )2/1' models the cell radius as a function of angle 0 when the antenna patterns are given by p ( 0) == cos( 0)]. The lognormal fading to each neighboring base station is randomized and the corresponding path gain is calculated. The position of the user and the lognormal fading are regenerated (randomized) if a "mistake" is detected. In the geometryhandover case a mistake has occurred if the average path gain defined by (39), using L == 1, is larger for some other base than base i. In the signal strength handover case a failure has occurred if the strongest path gain, i.e., max, Gi,i is more than 3 dB stronger than the desired-

545

base path gain Gi, t. If there are n base stations which are stronger than O.5G·i , l (including the zth base), then a "mistake" is generated with probability (n - 1)In. Thus a random number is drawn to determine if a "failure" has occurred or not. This procedure is used to simulate the case with a fast handover and a hysteresis of 3 dB. 2) The channel allocation algorithms are invoked (Sections IV-A 1 and IV-B2). In the SCFR approach four power groups with 12 mobiles each are used. Group number one uses the first and second time-slot, group number two the third and fourth and so on. In the RCS approach, with e == 0.3 eight power groups (one for each time slot in the TDMA frame) are used. With e == 1 random channel allocation is employed. All simulations assume that the TDMA slots of the base stations are synchronized, although this is critical only for the reduced cluster size approach with directed nulls. However. the TDMA frames are desynchronized in the sense that each base has a random offset of one through eight bursts. 3) Weighting vectors (Section III) are calculated for all users in sectors I a-c, Zb-c, 3b, 4a, Sa. 6a. and 7c in the 1/1 reuse case and 1a, 4a, Sa, and 6a in the 1/3 reuse case. In the same-cell reuse approach only one weighting vector per user is necessary. This applies also to the reduced cluster size approach if nulling is not applied. With nulling however, multiple weighting vectors per user must be calculated. This is because frequency hopping is applied and the identified interfered users thus change between time slots. In order to calculate the weighting vectors it is therefore necessary to determ i ne which users are identified by the base. This requires, in turn, that the power control settings are calculated. Thus the power control at the mobiles are first calculated. Then it is determined which interfered users are stronser than Pmin' (and thus identified, see Section IV-B3). For subcell 1a only users inside subcells 1b, Ic, 2a-c, 3a-c, and 7a-c are candidates. Once the identified users have been determined the weighting vectors for all possible cases are calculated. 4) For each of the 48 users in subcell 1a it is investi sated whether they are experiencing acceptable speech q~lity or not. Based on the reasoning in Appendix B, we assume that this is obtained if the instantaneous signal to interference ratio exceeds 3 dB in at least 800/0 of the time slots. The fraction is calculated as follows: The mean desired power averaged over fadinzo G·I, I. for the considered user is calculated using (39). A random frequency hopping pattern is simulated by randomizing the cochannel user in cells 1-7 with nei bzhborin bo cells 10000 times. For each of the 10000 hops the cochannel users are drawn with equal probability among the mobiles allocated in the time slot. The mean interference (averaged over fading) at user 'i, is calculated for each hop using the formula

t, ==

L k

"7kGk,iw~Rvv(f)k,i,

ak,·dwk

(62)

where W k is the weighting vector of the kth user and G i, i, f)k, i, and o», i are the propagation parameters between the zth user and the kth users desired base (can be the same base in the SCFR case). The sectors selected in the sum of (62) are la-c, 2b-c, 3b, 4a, Sa, 6a in the 1/1 ReS case, and la, 4a, 5a, 6a in the 1/3 SCFR reuse case. Note that with SCFR, there is d cochannel users per sector (Section IV -A). To simulate discontinuous transmission the factor 'fJk is randomized independently for each hop (Pr{ 7]k = I} == 1 - Pr{ T}k == O} = 7]DTX == 0.5). The same cell cochannel users which are assumed to be active all the time constitute an exception. Note that the users which use the same frequency within the cell are the same in each time-slot (Section IV-A-I). When the mean desired and interfering signal has been calculated the probability for the instantaneous signal to interference ratio to exceed 3 dB is calculated using (63). This probability is averaged over the hops to produce the sought fraction. Finally, the number of users with acceptable speech quality are counted and the outage probability is estimated as the fraction of users in subcell 1a with unacceptable quality.

ApPENDIX C ANALYTICAL RESULTS

In this section, we derive analytical approximations of the outage probability for the SCFR system, the ReS system with intercell nulling (only the e == 1 case) and the ReS systems without nulling. The analysis uses the same assumptions as the simulations with a few exceptions. Among those are the antenna spacing, the number of users in the system and the spatial distribution of the users. The downlink antenna spacing is slightly increased to ~ == A/ J3, and the number of users is assumed large (infinite). The spatial probability density of the user positions (seen from the desired base) is assumed to be given by as shown in (65) at the bottom of the page, where the choice of TO is defined by the handover algorithm assumed. The reason for the choice of the distribution (65) is that it enables the derivation of an analytical solution for the outage probability while at the same time being very close to uniform at J == 3.5-4.0. In Appendix C-A-C below approximative expressions for the outage probability (probability of unacceptable speech quality) conditioned on the user position are obtained for the three cases. In order to obtain the unconditioned outage probability ~ the subcells are divided into "elements." ~1('il, 'i2 ) , defined by ~~('il ~ -l2) ==

ApPENDIX B INSTANTANEOUS OUTAGE PROBABILITY

Previous results have shown that 9 dB average signal to interference ratio is sufficient to provide reasonable speech quality in GSM (neglecting noise), [15]. We assume that the relevant property for the receiver is the probability that the instantaneous signal to interference is less than 3 dB. Assuming flat Rayleigh fading and one interferer this fraction can be computed using the formula (see [14]) 1

Pr{ SIRinstantaneous ::; SIRd = 1 + SIR/SIR

t

(63)

10 0 . 3 and SIR 10 0 . 9 . This yields Pr{SIRinstantaneous ::; SIR t } == 0.2. The formula (63) applies in the case of a single interferer only. However, with multiple interferers, the interference is usually dominated by the strongest interferer and we use (63) as an approximation in these cases. In Appendix C, analytical approximations of the outage probability are derived. In these derivations the following approximation of (63) is used with

SIR t

SIR o Pr{SIRinstantaneous ::; SIRt} = (1 + SIRo/SIRt)SIR

(64)

which is a "linearization" of (63) around SIR == SIRo. The natural choice of SIRo under the assumptions here is SIRo = 10°·9.

() .. )f( T 2.. , 2' 2,'1. -

const. x r.1.,2. COS(1-4f,) (6·2,1.' :)

{ 0,

cos(30 0 ) ) 2/ 1, O.05'i l ~ ( ~, (r 1.• { cos] f}i. 'L)

+ 0.05'£1

and 5'£2 - 60

t/ R)

:::; D.DS

< fI ::; S'i2 - 55}. (66)

This partitioning is illustrated in Fig. 9 using i 1 == 0, ... , 17, iz == o.... ~ 23, i.e., TO == 0.9. The outage probability is calculated for a central point in each element. Finally, the unconditioned outage probability is obtained as the sum of the central point outage probabilities, weighted by the fraction of users in the element. These fractions can be calculated analytically, [22]. The intention is that the elements should be small enough that the outage probability is approximately constant within an element. It is easily shown that the desired signal strength (disregarding lognormal fading) along the borders of the "annular elements" [where annular element i 1 is defined as Ui.., neil, 'i2)J is constant, when the element patterns are given -by p((}) == cos( fJ). If the user distributions of all subcells in the system are added, only small spots are left "empty" if TO = 0.9 is used. Thus TO == 0.9 will be used when "geometry based handover" is assumed. Previous results, [9], have shown that the gain of signal strength handover (described in item #1 of Appendix A) over geometry based handover is about 4 dB. We model this effect by choosing TO == 0.7, and thereby moving the mobiles (a distance corresponding to 4 dB), closer to the base. In Appendixes C-A, C-B, and C-C below, ReS with nulling, ReS without nulling, and SCFR are treated, respectively. The

if ~ (cos(300)/ cos((}i,i))2 f'(ri,i/R) ~ fa, elsewhere

546

I(}'i,il ~ 60°

(65)

0.8 0.6 0.4 0.2

o - 0.2 -0.4 -0.6 -0.8 -1 L.-_--'-_ _--L..._ _- ' - - - _ - - - '_ _::>.k=:--_ - ' -_ _L.-_--'-_ _-'--_---l 0.2 0.4 -1 -0.8 -0.6 -0.4 -0.2 0.6 0.8

o

Fig. 9.

The division of the subcell.

Res version with nulling is onl y considered in the case (' (e is defined in Section IV-8 I). A. Reduc ed Cluster Si;e (R CS) with Nulling and (:

=I

where D k . , and B (:r) are defined by

Dk

=1

We assume that all base s erroneously estimates their desired mobile to have zero angular spread. i.e., Il k . k = 0° for all /,; .5 Thi s yields

B (.I)

G i. , C ,' , R ,.,.(O . oi, cos(lh .,))

+ ( r - l )Pmi/lI

(71)

= Jiag (1.

ex p ] -):t) . . . . . exp ( - j (m - 1):1))

(72)

respectively . Using the equations above we obtain that the interfering power (averaged over fading) at the ith (from the kth base) is given by

= [1. exp ( -j J3Jr Sin( B)) . ex p ( - j J3( rn - I )Jr sin( I1) ) r

=

and

where arB ) is given by

arB)

,

G k . iwkR (B k ,. Il k . i )W k

=

Gk.,a *(O) B *(iid D;;l, R (O.

(68)

Il k . ,

cos(Bk , ;))Dk",ljB(O:k)a(O )

(a *(O) B *(a k )Dk',l,B (O:k)a( O))2

in (47) and (48) . With R k . k given by (67) it can be shown that the transmit vector at the kth base is given by

(73)

where

[(2Jr/J3) sin (Bk. d - (21l)J3) Sin Uh,j )] modulo 2Jr.

Assuming that only the kth and -ith user are identified by the kth base, the matrix M is given by M

(74)

=B (( 2JrI J3) sin( Bk, i) )Dk. jB* (( 2JrI J3) sin ( /h .;)) (70)

5 This is a pessimisti c assumpt ion if the angular spread of the desired user is large because in that case there exist possibilities to avoid transmitting toward the interfered mobiles by pointing the main beam toward the multipaths . However, the angular spreading considered herein is so small that this effect is negligible .

The impact of frequency hopping and discontinuous transmission is modeled by averaging (74) over the distribution of i'ik (which is shown to be uniform [0, 21r] in [22]) and assuming that the mobiles are active with probability TJDTX . Using the results of Appendix B, and assuming that all base stations have identified the -ith mobile but no other mobile, the probability

547

M, D k , i » and D(x, y) all be equal to the identity matrix i.e., M == Dk,i == D(x, y) == I.

that the instantaneous SIR is not exceeding SIR o is obtained as 1 - Pr{ Outage}

== Pr

{

SIRo SIR o

1+--

'TJDTX

c:

"

j

C. Same Cell Frequency Reuse (SCFR)

.

The derivation of the analytical approximation of the outage probability (conditioned on the user position) for the SCFR system is very similar to the derivation of [21, Theorem 11. The details of the derivation can be found in [22].

t

SIR t

(75)

~t

FOUR

::::: Pr { 10 log(G k , i) - 10 log(G i , , )

< 10 10 (

g 9

(-1

TJDTX

ApPENDIX

TO

R DL

D

TRANSLATIONS METHODS

1) If the up- and downlink manifolds are the same, i.e.,

(1 + SIRo/SIR t ) SIR t o

(81)

'Vi}

fJiCOS(Oi))),

R UL

(76)

it follows from (16) and (24) that the up- and downlink multipath covariance matrices are the same (except for the power scaling) i.e., RUL = pCLRDL, and the translation problem is thus eliminated. This requires two different antenna arrays for up- and downlink. The two arrays should have the same structure but scaled to their respective wavelength. This idea was first proposed in [16], and is referred to as "the matched array approach," in that paper. 2) If the same array is used in up- and downlink, i.e.,

where g(z , y) is defined as shown in (77)-(79), given at the bottom of the page, this approximation is possible as the interference usually is dominated by one base station. This is more true in systems with antenna arrays than otherwise. Assuming that the log normal fading (between a mobile and several base stations) is correlated with correlation coefficient, c, the outage probability at the 'lth user condition on its position is obtained as given in (80) at the bottom of the page. When (80) is used in Examples one and two, only neighboring cells are taken into account in the product. Furthermore, only the sector directed most closely toward the mobile of each cell is considered, since overly pessimistic results would otherwise be obtained. This is because (80) does not assume fully correlated lognormal fading between a mobile and the three sectors of a base station site. Also with ~ == A/2 there are fewer side lobes outside the ±60° region than with ~ = AI J3 [which (80) assumes]. The parameters SIR o and SIR t are set to SIR o == 9 dB and SIRt == 3 dB, respectively.

(82)

and (83)

and the relative duplex separation (fUL - fDL)/(fUL + is small then there may exist a compensation matrix Aconlpensate such that

f D L)

B. Reduced Cluster Size (ReS) Without Nulling see [22]. If (84) is valid R DL may be approximated as

By again assuming that all bases estimate the spreading of their desired mobile to be zero, i.e., o». k = 0° for all k,6 the results of previous section can be used by letting the matrices

pULRDL-"A -..

6 Simulations we have made have shown that the loss of neglecting the angular spreading in the ReS case without nulling is typically less than 1 dB, assuming linear arrays with eight-ten elements.

g(z, y) == max{f(x, y) z

~

compensate

RDLA*compensate'

3) If the spatial distribution of power is well approximated by a finite number of rays (which is less than the number

z}

(77)

f(x, y) =x (21f a*(O)B*(a)i>-l(x, Y)~v(O, y)f>-l(x, y)B(a)a(O) da }ii=O (a*(0)B*(a)D-1(x, y)B(a)a(O))2

(78)

D(x, y) == xRvv(O, y) + (r - l)Pmin I Pr{Outagelr"i,Oi,d=lx

~ roo

v 21r } x=O

II Q (m k'

exp ( i -

(85)

2-

(79) x2

20 dB (1 - c)

10 log

)

(g( 7JD~X (SIR ol + SIR;-l )t, O"k, i a dB v!1="C

k#i

548

cos( 8k ,i))) - mi, i-X)

dx

(80)

of antenna elements), i.e.,

lV

ftUL ~ ~ pULlhnI2aUL(fln)(aUL(enJ)*;

[10] T. Ohgane, "Spectral efficiency evaluation of adaptive base station for land mobile cellular systems," in Proc. IEEE Veh. Technol. Conf., 1994, pp. 1470-1474. [11] S. J. Orfanidis, Optimum Signal Processing, An Introduction. Singapore: McGraw-Hili, 1990. [12] B. Ottersten, M. Viberg, and T. Kailath, "Analysis of subspace fitting and ML techniques for parameter estimation from sensor array data," IEEE Trans. Signal Processing, vol. 40, no. 3, pp. 590-600. Mar. 1992. [13] B. Parlett, The Symmetric Eigenvalue Problem. Englewood Cliffs, NJ: Prentice-Hall, 1980. [ 14J R. Prasad and A. Kegel. "Improved assessment of interference limits in cellular radio performance." IEEE Trans. Veh. Technol., vol. 40, pp. 412-419, May 1991. r15] K. Raith and J. Uddenfeldt, "Capacity of digital cellular TDlVIA systems," IEEE Trans. Veil. Technol., vol. 40, pp. 323-332, May 1991. [16] G. Raleigh, S. N. Diggavi, V. K. Jones, and A. Paulraj, "A blind adaptive transmit antenna algorithm for wireless communication," in Proc. IEEE

N<m

n=l

(86) then the powers 1 hIT/. 12 and directions gri of these rays can be estimated from Rf~, using a conventional direction finding technique, e.g., [2], [11], [121, [17], and [18]. These estimates may then be used to calculate PtGLRDL using

JV

pULR D L ~ pUL ~ IlllnJ2aDL(Bn)(aDL(Hn))*.

(87)

Int. Conf Communications, 1995.

n=l

4) If a uniform linear array is used in the uplink, i.e., a UL (H) is given by (1), and the model described in Section II-B 1 applies, then the method of [19] may be employed to estimate the signal power, as well as H and a. With these estimates at hand, the transmit matrix pUL R DL may be explicitly calculated. REFERENCES [11 S. Andersson, U. Forsse n, and J. Karlsson, "Ericsson/Mannesrnann GSM field-trials with adaptive antennas," in Proc. IEEE Veh. Tee/mol. Conf., Phoenix, AZ. May 1997, pp. 1587-1591. f2J Y. Bresler and A. Mocovski, "Exact maximum likelihood parameter estimation of superimposed exponential signals in noise," IEEE Trans. Acoust., Speech, Signal Processing, vol. 34, pp. 1081-1089, Oct. 1986. [31 C. Carneheirn. S. O. Jonsson, M. Ljungberg. Nt. Madfors, and J. Naslund, "FH-GSM frequency hopping GSM," in Proc. IEEE Veil. Techno!' Conf., Stockholm. Sweden. June 1994, pp. 1155-1159. [4] C. Farsakh and 1. A. Nossek, "Channel allocation and downlink beamforming in an SDMA mobile radio system." in IEEE Int. Symp. Personal. Indoor and Mobile Radio Communications, Sept. 1995, pp. 687-691. [51 D. Gerlach and A. Paulraj, "Adaptive transmitting antenna arrays with feedback," IEEE Signal Processing Lett., vol. I. pp. 150-152, Oct. 1994. [6] _ _ , "Adaptive transmitting antenna arrays with feedback." IEEE Trans. Veh. Techno!., 1995, submitted. [7] _ _ , "Base station transmitting antenna arrays for multipath environments," Signal Processing, vol. 54, no. 1, pp. 59-73, 1996. [81 G. H. Golub and C. F. Van Loan. Matrix Computations. Baltimore. MD: The Johns Hopkins University Press. 1983. [91 M. Mouly and M. B. Pautet, "The GSM system for mobile communications," Michel Mouly and Marie-Bernadette Pautet, 49 rue Louise Bruneau, F-91120 Palaiseau France, 1992. ISBN 2-9507190-0-7.

[171 R. Roy, A. Paulraj, and T. Kailath, "ESPRIT-A subspace rotation approach to estimation of parameters of cisoids in noise." IEEE Trans. on Acoust., Speech. Signal Processing, no. 34, p. 1340. 1986. [181 R. O. Schmidt. "Multiple emitter location and signal parameter estimation," in RADC Spectral Estimation Workshop, Griffiths AFB. NY, 1979, pp. 243-258; reprinted in IEEE Trans. Antennas Propagat., vol. AP- 34, pp. 281-290. Mar. 1986. [19) T. Trump and B. Ottersten. "Maximum likelihood estimation of nominal direction of arrival and angular spread using an array of sensors." Signal Processing, vol. 50, nos. 1/2, pp. 57-69, Apr. 1996. [20J J. H. Winters. "Optimum combining in digital mobile radio with cochannel interference," IEEE Trans. Veh. Technol.. vol. 33. pp. 144-155, Aug. 1984. [211 P. Zetterberg and B. Ottersten, "The spectrum efficiency of a basestation antenna array system for spatially selective transmission." IEEE Trans. veh. Technol.. vol. 44. pp. 651-660, Aug. 1995. [22] P. Zetterberg, "Mobile cellular communications with base station antenna arrays: Spectrum efficiency, algonthms and propagation models." Ph.D. thesis, Royal Institute of Technology, Stockholm. Sweden. June 1997.

549

Chapter 4

Implementation Issues

P

OSSIBLY the most challenging problem related to adaptive antennas is their practical implementation, from both a technical and a cost point of view. In realworld adaptive antenna systems there are a number of sources of random errors, ranging from antenna element misplacement and mutual coupling to amplitude and phase mismatches and quantization errors. This chapter includes work that deals with these issues from both a theoretical and a practical implementation angle. It starts with a tutorial paper from Dudgeon that describes mathematically and intuitively the fundamentals of digital array processing. Then an adaptive algorithm and its efficient pipelined architecture in the form of a triangular systolic array, particularly applicable to VLSI design, are described. A multiple input, multiple output orthogonalization algorithm, its systolic implementation, and its comparison with the wellknown Gram-Schmidt orthogonalization procedure are discussed in the paper by Yuen et a1. DuFort considers the design of optimum beamforming networks, and Er and Cantoni et al. present a unified approach to designing robust array processors. The article by Hansen discusses

551

design trade-offs and a procedure for selecting design parameters for Rotman lenses. Neural beamforming has been suggested as a means to increase the performance of an adaptive antenna (it has been shown that neural networks can control arrays in an accurate manner even with element and network errors) and reduce manufacturing and maintenance costs. The paper Mailloux and Southall presents a comparison between a neural network and a Buttler matrix performing the same direction finding task, and the paper by Southall et a1. discusses a direction finding system implemented with a neural beamforming network and presents some test results. Several papers in this chapter deal with mismatch problems with adaptive antennas. Among the issues discussed are nonlinearity effects in digital manifold phased arrays (Mathews), array imperfections and methods to cope with the reduction of the nulling capabilities (Jablon), mutual coupling compensation (Steyskal and Herd), forward-backward averaging methods for array manifold errors (Zatman), and the use of orthogonal codes for remote antenna calibration (Silverstein).

Fundamentals of Digital Array Processing DAN E. DUDGEON,

Abstract-With the advent of high-speed digital electronics, it has become feasible to use digital compu ters and special purpose digital processors to perform the computational tasks associated with signal reception using an antenna or directional array. The purpose of this paper is mainly tutorial, to describe mathematically and intuitively the fundam~ntal relationships necessary to understand digital array processing. It 18 hoped that those readers with a background in antenna theory or array pr~essing will.see some of the advantages digital processing can offer, ~hile those WIth a ~ackground in digital signal processing will recognize the array processing area as a potential application for multidimensional signal processing theory.

M

I. INTRODUCTION

UCH of the theoretical work being done today in the area of multidimensional ~ignal proc.essin g is motivated by the need to process signals earned by propagating wave phenomena. For radar to be successful, it was necessary to develop directional transmitting and receiving antennas so that azimuth as well as range and range rate information could be ascertained from the radar return. Similarly, this problem is also encountered in active sonar and ultrasonics applications. In applications where the source signal is not precisely controlled (such as exploratory seismology) or where the received signal is externally generated (such as passive sonar, bioelectrical measurements, or earthquake seismology), it is desired to elicit characteristics of the received signal (its signature) as well as its direction and speed of propagation. In recent times, it has become more and more feasible to perform the signal processing operations associated with array processing using digital compu ters or special purpose digital processing hard ware. Correspondingly, digital signal processing theory has grown to encompass these various applications. The following references are representative of recent articles of digital processing in the fields of radar [ 1] , seismology [2] , sonar [3] , ultrasound [4] , and bioelectrical measurement [5] . The point of this paper is to examine the fundamental array processing techniques, in particular the concept of bearnforming to determine the speed and direction of propagation of an incoming wave, from the point of view of a multidimensional Manuscript received July 26, 1976; revised November 12 1976. The author is with the Computer Systems Division, Bolt' Beranek and Newman, Inc., 50 Moulton Street, Cambridge, MA 02138.

MEMBER, IEEE

signal processing problem. We shall see the close relationship between conventional sampled-data systems and the sensor array as a receiver sampling the waveform in space. Accordingly, Section II reviews some essential points about sampleddata systems and digital signal processing techniques. In Section III, a linear array of sensors is used as a basis for discussing the weighted delay-and-sum beamformer with attention given to how to choose the appropriate weights. In Section IV, the relationship between the computation of beam spectra and the computation of a two-dimensional (2-D) discrete Fourier transform is examined. Section V looks at extending the results of Section III to higher dimensions. As an example of results from digital signal processing which can be applied to digital beamforming, the problem of designing the sensor weights for a multidimensional beamformer is discussed in Section VI. In the case of a Cartesian array of sensors, an ingenious mapping due to McClellan [6) can be used to design and implement beams with nearly spherically symmetric main lobes in a computationally efficient manner.

II. IMpORTANT CONCEPTS IN DIGITAL PROCESSING In this section, several important concepts from digital signal processing theory will be reviewed. These concepts will be presented in terms of a one-dimensional (1-0) signal for ease of understanding, but they are easily generalized to multidimensional signals. The reader is directed to [71 as a text on digital signal processing and to [8] as a review of 2-0 filtering concepts. The fundamental assumption of digital processing is that input signals are bandlimited to frequencies below one-half the sampling rate. If a continuous-time signal is sampled at a rate too slow (undersampling) for the frequency content of the signal, the Nyquist sampling theorem tells us that frequencies above one-half the sampling rate in the continuoustime signal will act like frequencies below one-half the sampling rate. This phenomenon is known as aliasing, and it is explained in detail in [7] as well as in a variety of texts and papers on sampled-data systems. Although we have been speaking of a l-D time signal, the same statements apply to signals which are a function of distance or other continuous independent variables.

Reprinted from Proceedings of the IEEE, Vol. 65, No.6, pp. 898-904, June 1977.

553

We can represent a I-D digital signal by s(n) where n is an integer. By doing so we are effectively normalizing the sampling rate to be unity. The Fourier transform of such a digital signal is defined by S(w)

=L s(n)e-j w n . n

se:

) =

~s(n) exp (_i

2

:

k

).

(2)

If the signal s(n) is zero for n outside the range from 0 to N - 1, then the sum over n in (2) extends only from 0 to

N - 1. In this case (2) defines a discrete Fourier transform (DFT) which is invertible; that is, the samples s(n) may be recovered from the values S[(21fk)/N] by s(n)

Nt S (21rk) exp (i 21rnk) N N N

=.!.

l

k=O

for n = 0, N - 1.

The DFT of a signal may be computed by an efficient algorithm (an FFT), the details of which are contained in texts [7], [9], [10], and papers [11], [12]. The advantage of the FFT algorithm is that the computation of S[(21Tk)/N] is proportional to N log2 N rather than N 2 as in a direct evaluation of the OFT (see (2)] . One type of digital filter which is important to the understanding of digital array processing is the finite impulseresponse (FIR) filter. Again we shall briefly review the 1-0 case which is covered in detail in [7], [9], [13] , and [14]. The name FIR refers to the fact that the impulse response of the filter is nonzero only over a finite domain of the independent variable. For example, if a filter has an impulse response h(n) such that h (n)

=0

for n

o)

e+ tn1/!=e+ / (N -

l )", / 2

(50)

(51)

sin (Nl/;/2) _

(52)

sin (l/;/2)

because, from (49), (50), and (20), (21), all the terms} depend upon

(53)

Xj=(j-l)d.

For uniform illumination, {I (f)

QJ

=

1 and

.'V

= co(f)

~ [cos OJ - sin OJ] ;=1

'Zm-l

co

n

2; 2; Gm,n,k

+ 1/2 ~

(45)

m= 1 n=Q k=O Xj

sin

et>o)

(46)

.

(1

[ [

+

+ in- 2k + J )e- /(N -

[.

if

I

I

\

I

II

16 67 KH z

/ '\ em /: __ \_.../-...~ _ ~.~'~ _ \

I

__

/ .......

\.

10 KILOHERTZ

\

1

1/

I

The higher harmonic distortion peaks for near saturation levels will lie about - 145 dB relative to the mainbeam clutter peak . For extreme downlook dynamic ranges and long coherent integration , the distortion is well below the noise limited threshold level. Fig. 7 shows the distortion level at the processor for DBF has been significantly reduced. The effects of increasing the scan angle near broadside are shown in Fig . 8. The nonlinear case at 10 mrad scan is essentially identical to the conventional array analysis . as expected . When scanned to larger angles , the clutter width and the positioning error increase , due to the geometry change. as expected . The feature of interest is the relati ve level of the distortion peaks . At each harmonic peak, the directive gain is decreased for the increased scan . It may also be observed that the even harmonic terms ar e broader than the odd harmonics . The even harmonic distortion has two beams, each of which will peak at slightly different frequencies with the net effect of broadening the lobe. The higher odd harmonics further illustrate this peaking of directive gain . Due to the selection of m + I or m - 1, however, the relative max imum in directive gain translates the lobe maximum slightly from a pure harmonic of the positioning error. This effect tends to narrow the odd harmonics lobe . The m - I selection for the fifth harmonic leads to a directive gain equal to the third harmonic m + 1 terms, and the spectra differ only due to the convolution differences . In the derivation of these results, a number of ideal assumptions were made . Of particular interest in specifying the receivers are effects of channel imbalances . Phase and amplitude errors which vary randomly from element receiver to element receiver will produce the same sort of effects on array factors as root mean square (rms) errors due to manifold tolerances or phase shifter quantization in conventional phased arrays . Premixer imbalances will be multiplied by the harmonic number, and the sidelobe level for the distortion products will rise . If the rms sidelobe level for the conven-

0 .0

u. \\ .I

I I

(70) (71) (72) (73)

\

.--- 41- - _1-

5 RAO

::

,

I \ i \

60

8 ex:

8,

. I

OJ

a

190:;:;

.\

I .

i

which is the primary input to the adaptive noise canceller. In the absence of array imperfections, the desired response branch is constrained to have a unit look-direction gain. In general , this branch is a conventional delay-and-sum beamformer, with K nonadaptive weights being fixed in such a way that the array beamwidth and average sidelobe level are both satisfactory [6]. This paper assumes uniform 11K weighting, but other weightings could be considered with minor modifications to the analysis. The lower branch of the beamformer is the sidelobe cancelling branch. Its purpose is to form the sidelobe cancelling signal Yle by providing K reference inputs to the adaptive noise canceller. Yic contains estimates of the jamming components in the desired response , so that after subtracting Yic from di, the beamformer output Zle is a "cleaner" representation of the signal. Note the use of complex conjugate weights Wi.IeO = I, "', 1.. 2 aOs

= 30 dB =

50 dB

= 00

~-· - -=~~p--~:~e

':"

:: '.

" "" "

" "

I

-60 .00 . 00

-50 .00

-100 .00

50 . 00

100 .00

Jammer angle of arrival, 8j (deg) Fig. 8.

Wiener output signal-to-interference-plus-noise ratio versus jammer angle for Capon bearnformer implemented with imperf ect array.

, y{ \

20.00

Look-direction

. 00

......... ~ "0

-....J

d

.-ca-

- 20 .0 0

/

~

0

u

ctl ......

-40 . 00

" "

..

Jammer at 8j

=

170

ctl

0. CI.l

>.

-60.00

ctl

K

~

< ~

- 80 .0 0

-d

10

Ideal array

1

Imperfect array

x =

2

=

00

6s

CI r

>..

= 0.01

-1 00 . 00 - 100. 00

- 50. 00

. 00

50 .00

Far-field angle, 6 (deg) Fig . 9.

Array spatial factors of ideal and imperfect arrays.

589

100 . 00

10 .00

Look-direction signal

•

..

Jammer

20 .00

-

.00

CQ ~

-

......"

-20 .00

\(

>.

0> °BU

-10 .00

~

is

-d A

-60.00

1

= -2

as = 0° aj -- 17°

-80 .00

- 100 . 00

SNR i

= 30 dB

INR I.

= 50 dB

-sc .oo

-100.00

. 00

Far-field angle,

10. 00

Look-direction signal

CO

"Cl

-

..

..

-~~V~\(

10 .00

......"

(deg)

100 .00

~

Single jammer far-field direct ivity pattern of Capon beamformer implemented with ideal array .

Fig. 10.

-

a

so.co

[I

.00

Jammer

/>.»>:

V

~

'--

-20 .00

>.

0>

.~

-10.00

a

-60.00

.... U

K

=

d A

1 = -2

10 SNR I.

= 30 dB

INRi

= 50 dB

as = 0°

-80.00

0" r

aj = 17°

-100 .00 - I ~ O . OO

A

= 0.01

-sc.oo

.00

Far-field angle, Fig. 11.

a

so.co

100 . 00

(deg)

Single jammer far-field directivity pattern of Capon beamfonner implemented with imperfect array .

590

where R gg is defined as the covariance matrix of element imperfections:

It was previously shown that the degradation of the GSC performance in the presence of array imperfections was due to leakage lie of the signal into the sidelobe cancetling branch, as given by (36). In order to eliminate this leakage, consider the constrained optimization problem minimize Et,a[1 Zk 1 w

2

(70)

Thus it is hoped that (69) represents a good approximation to the penalty function. Unfortunately, in an on-line implementation of stochastic steepest descent, the computation of P(Wk) as given by (69) could still prove to be quite a mess, due to the presence of the time expectation operator. Consequently, it is proposed that the instantaneous value of the weight vector at the time sample k be used to generate an approximation to the expectation on the last line of (69), in the manner

]

subject to Ik = 0

(64)

where Et,a[·] represents a double expectation, taken over both time and an ensemble of i.i.d. antenna elements. Equation (64) is ill-posed, and therefore cannot be solved directly. The reason is that knowledge of the signal leakage in advance is the same as knowledge of the unknown array imperfections, so the only way to be sure of satisfying the constraint would be to set all adaptive weights to zero. The result would be a conventional beamformer. However, it is well known in optimization theory that the solution of a constrained optimization problem can be approximated by solving a corresponding unconstrained optimization problem. From Luenberger [39], one learns that for the method of steepest descent, the procedure is to convert the constrained optimization problem minimize

!(Wk)

P(Wk) = a;wfBRggB TWk

where P(Wk) represents the instantaneous estimate of P(Wk). The approximation used to obtain (71) becomes succesthe biased Wiener weight sively better as W k converges to vector when the penalty function is used. If the element imperfections are assumed zero-mean i.i.d., then from (70)

w:'

R gg = a;I.

(73)

(65)

The complex LMS algorithm, which can be used to update the weights in the GSC, was shown by Widrow et al. [2J to be

to the unconstrained optimization problem minimize (j(w k) + CP(Wk» w

Wk..-'

(66)

= Wk + 2j.LE-k Uk = (I -

2J.LUkU~)Wk + 2J.Ldl\Uk

(74)

where J.L is called the adaptation constant, and fA; is the error signal at time sample k, which for the GSC is chosen to be the beamformer output z; = d, - Yk' If (74) is used for the GSC, the mean weight vector will converge to W* [2], and all the analysis done so far in this paper concerning hypersensitivity to array imperfections will apply. The complex LMS algorithm can be derived by considering the squared error function i; or performance surface estimate at time sample k of the form

where f(wk) , q(Wk), and P(Wk) all represent suitable functions for the problem. In system optimization theory, P(Wk) is known as a penalty junction, and c as a penalty constant, the latter generally chosen to be "large." For a stochastic problem, one usually chooses P(Wk) as E[lq(Wk)\2], with the expectation taken as appropriate. Then by making the penalty constant large in (66), the minimization of CP(Wk) dominates the minimization of (!(Wk) + CP(Wk». Therefore, the solution to (66) must have small q(Wk). Indeed

{k=~*+(Wk-W*)HUkU~(Wk-W*)

(67)

which leads back to the corresponding constrained minimization problem (65). Based on the above argument, the logical choice for the constrained optimization problem is the one in (64), namely (68)

By using (36) and the conjecture that for large penalty constant, the weight vector will be fairly independent of the array imperfections, the penalty function p( W k) can be written as [35] P(Wk) = Ec,a[\lk 1 2]

== a;Et[wfBRggBTWk]

(72)

Substitution of (72) into (71) results in

=0

subject to q(wk) = 0

(71)

(69)

(75)

with ~k denoting the MSE (i.e., performance surface) at time sample k, and ~* representing the minimum (attainable) mean-square error (MMSE). By taking Jl times the negative gradient of t, with respect to Wk, and adding it to Wk, Widrow et al. [2] developed the complex LMS algorithm. Due to the form of (75), the MSE is a K -dimensional concave upward parabolic bowl, which means that taking the negative gradient of tk on the average leads to descent in the bowl's steepest direction. Use of the estimated penalty function P(Wk) yields the modified performance surface estimate Pk, which is Pk={k+CP(Wk)=~*

(76)

with Pk denoting the MSE at time sample k when the modified performance surface is used.

591

Taking p. times the negative gradient of Pk with respect to and adding it to Wk in the manner of Chestek [40]:

weight vector and MSE by choosing p. in accordance with

Wk,

1 O .ij 0 ....

-40.00

'-"

/

~ .--e

0

-60.00

K

= 10

d A

=

SNR i = 30 dB

1 2

INRi = 50 dB

-80.00

u,

= 35 dB

= 0.01

-100.00 -100.00

.00

-50.00

Far-field angle, Fig. 12.

50.00

a

100.00

(deg) ...

Single jammer far-field directivity pattern of Capon beamfonner implemented with imperfect array, when approximately 3 dB more than the optimum amount of artificial receiver noise is injected.

direction signal as if it were a jammer. Modeling the array

B. Wide-Band Adaptive Array Processing The extension to the wide-band case is trivial, assuming that wide-band adaptive noise cancelling techniques are now used, which give the optimal Wiener weightings as a function of frequency. Although these optimal Wiener weightings are, "ideal, based on the assumption of an infinitely long, twosided (noncausal) adaptive transversal filter," Widrow et al. [2] showed that their performance could be closely approximated by using all-zero filters. Gooch and Shynk [41] recently demonstrated the potential for even better synthesis of the Wiener weightings by using pole-zero filters. When applying the results of this paper to the wideband case, one only needs to keep track of the change in wavelength Aas a function of frequency (since it affects both presteering and any possible random element misplacement), and 2 I CXj 1 , and 0-; must all be interpreted as functions of i», These conditions mean that the array imperfections must be viewed as frequency-dependent, and at some frequencies certain assumptions may no longer hold.

0';, 0';,

a;, 0';,

VI.

CONCLUSION

This paper tackled the problem of hypersensitivity of linearly constrained adaptive beam/arming to array imperfections for' 'high" input signal-to-noise ratio, by considering a particularly simple and general structure known as the generalized sidelobe canceller. The aforementioned hypersensitivity manifests itself as nulling of the friendly look-

imperfections as random element amplitude and phase errors constant during the period of adaptation, the hypersensitivity phenomenon was discussed in detail using Wiener filter theory to analyze steady state behavior, and computer simulations to check the results. Artificial receiver noise injection algorithms were derived for the generalized sidelobe canceller, and simulations were carried out to demonstrate their ability to provide the beamformer with robustness to array imperfections. For the special case of the Capon maximum-likelihood beamformer, simple approximations were presented for the Wiener output signal-to-interference-plus-noise ratio, the random element gain (amplitude and phase) error variance which leads to a 3 dB degradation in this Wiener output signal-tointerference-plus-noise ratio from its value when an ideal array is assumed, and the optimal amount of artificially injected receiver noise. Suggestions for how the theory could be extended to the two important cases of multiple jammers and wide-band adaptive array processing were discussed. Ideas for further investigation can be found in [35]. ACKNOWLEDGMENT

The author is grateful Widrow, for suggesting fect arrays" as a Ph.D. during the course of this 594

to his principal advisor, Dr. Bernard "adaptive beamfonning with imperthesis topic, and for his supervision research. The experience of working

with Dr. Arogyaswami Paulraj, who served as associate advisor, was equally rewarding.

[22] [23]

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[1] [2] [3] [4] [5]

[6] [7] [8] [9] [10] [11] [12] [13] [14]

[15] [16] [17] [18] [19] [20] [21]

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[29] [30]

[31]

[32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

595

I. J. Gupta and A. A. Ksienski, "Effect of mutual coupling on the performance of adaptive arrays," IEEE Trans. Antennas Propagat., vol. AP-31, no. 5, pp. 785-791, Sept. 1983. L. C. Godara, "The effect of phase-shifter errors on the performance of an antenna-array beamformer," IEEE J. Ocean. Eng., vol. DE-IO no. 3, pp. 278-284, July 1985. ' C. L. Zahm, "Application of adaptive arrays to suppress strong jammers in the presence of weak signals," IEEE Trans. Aerosp. Electron. Syst., vol. AES-9, no. 2, pp. 260-271, Mar. 1973. W. D. White, "Artificial noise in adaptive arrays," IEEE Trans. Aerosp. Electron. Syst., vol. AES-14, no. 2, pp. 380-384, ~1ar. 1978. J. G. Charitat, I r. , "The effects of error in the adaptive antenna reference," Proc. IEEE, vol, 70, no. 9, pp. 1128-1129, Sept. 1982. - - , " An algorithm for adaptive antennas and superresolution systems with faulty steering vectors," IEEE Trans. Antennas Propagat., vol. AP-34, no. 3, pp. Mar. 1986. B. Widrow and J. M. McCool, "A comparison of adaptive algorith~ based on the methods of steepest descent and random search," unpublished manuscript. J. M. McCool, "A constrained adaptive beamformer tolerant of array gain and phase errors," in Aspects of Signal Processing, pt. 2, G. Tacconi, Ed. Dordrecht, Holland: Reidel, 1977, pp. 517-522. M. H. Er and A. Cantoni, "Derivative constraints for broad-band element space antenna array processors," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, no. 6, pp. 1378-1393, Dec. 1983. M. H. Er and A. Cantoni, ., A new approach to the design of broadband element space antenna array processors," IEEE J. Ocean. Eng., vol. OE-IO, no. 3, pp. 231-240, Iuly 1985. K. M. Ahmed and R. I. Evans, HAn adaptive array processor with robustness and broad-band capabilities," IEEE Trans. Antennas Propagat., vol. AP-32, no. 9. pp. 944-950, Sept. 1984. R. T. Compton, Jr., "An adaptive array in a spread-spectrum communication system," Proc. IEEE, vol. 66, no. 3, pp. 289-298, Mar. 1978. N. K. Jablon, "Steady state analysis of the generalized sidelobe canceller by adaptive noise cancelling techniques," IEEE Trans. Antennas Propagat., vol. AP-34, no. 3. pp. 330-338. Mar. 1986. - - , "Adaptive beamforming with imperfect arrays," Ph.D. dissertation, Elec. Eng. Dept., Stanford Univ., Stanford, CA, Aug. 1985. A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1965, ch. 9, p. 303. T. Kailath, Linear Systems. Englewood Cliffs. NJ: Prentice-Hall, 1980, Appendix, pp. 655 and 658. H. E. Schrank, "Low sidelobe phased array antennas," IEEE Antennas Propagate Soc. Newsletter, vol. 25, no. 2, pp. 5-9, Apr. 1983.

D. G. Luenberger, Introduction to Linear and Nonlinear Program-

ming,2nded. Reading, MA: Addison-Wesley, 1984, ch. 7, pp. 221222. R. A. Chestek, "The addition of soft constraints to the LMS algorithm," Ph.D. dissertation, Elec. Eng. Dept., Stanford Univ., Stanford. CA, May 1979, chs. 5, 8, and 9, pp. 13, 19, and 30. R. P. Gooch and J. J. Shynk, "Wide-band adaptive array processing using pole-zero digital filters," IEEE Trans. Antennas Propagat., vol. AP-34, no. 3, pp. 355-368, Mar. 1986.

An Efficient Algorithm and Systolic Architecture for Multiple Channel Adaptive Filtering STANLEY M. YUEN, KENNETH ABEND,

SENIOR MEMBER, IEEE, AND

A bstracl-A multiple input-multiple output orthogonalization algorithm and its efficient systolic implementation are presented. The processing architecture is developed using a basic two input-two output decorrelation processing element (PE) as the primitive building block. Its features are discussed and compared to the recently published approach based on the well-known modified Gram-Schmidt (MGS) orthogonalizalion procedure.

' A

I. INTRODUCTION

DAPTIVE FILTERING has been a subject of intense research since the early work of the stochastic gradient or the least mean square (LMS) algorithm of Widrow [1], [2]. The virtue of the LMS algorithm lies in its computational simplicity. However. it suffers slow initial convergence and thus poor adaptivity in a rapidly time-varying environment. This drawback has served as motivation for deriving other methods of adaptive filtering which can provide faster cc nvergence and are not so sensitive to the signal statistics. One method of obtaining faster convergence is to adopt an exact least squares (LS) approach rather than the statistical approach. The family of recursive least squares (RLS) algorithms represents one group of techniques which are theoretically less sensitive to the statistical propenies of the data [31-[6]. There are basically two important differences between the gradient-based type of algorithms and the RLS (a.so known as Kalman) family. First of all. Kalman-type algorithms minimize an exact error criterion constructed from the actual input data . in contrast to the statistical error criterion for the gradient-based methods. Secondly, the error criterion is satisfied at every point in time for the Kalman family of algorithms. whereas the error criterion is achieved at convergence or steady state for the gradient -based techniques, as in the case of the LMS algorithm. Although the Kalman-type algorithms possess attractive convergence property, they have two major drawbacks. One is their large computational complexity and the second is their sensitivity to round-off noise. The latter may cause an algorithm to become unstable after a large number of iterations. To be more specific, the Kalman-type algorithms require O(N2) operations per time update for computing an Nth order filter, whereas only O(N) operations are needed for the LMS algorithm. To remedy the problem of round-off noise, a Manuscript received February 12. 1987: revised September 1~ 1987. S. M. Yuen is with the Electronic Systems Department, RCA Government Electronic Systems Division. Moorestown, NJ 08057. K. Abend and R. S. Berkowitz are with the Department of Electrical Engineering. University of Pennsylvania. Philadelphia, PA L9104. IEEE Log Number 8819825.

RAYMOND S. BERKOWITZ,

FELLOW, IEEE

family of algorithms based on orthogonal transformations can be used. They include the Givens, Householder, and modified G~am-Schmidt (MGS) transformations. These algorithms deal WIth data matrices with condition numbers equal to the square root of the condition number of the input signal covariance matrix. The condition number of a matrix of interest is defined as the ratio of its largest and smallest nonzero singular values and has an interpretation of being an error magnification factor. Consequently, these orthogonalization-based algorithms are less sensitive to round-off noise. In many advanced signal processing applications, the use of regularly structured processing is considered as the most feasible approach to obtain real time performance. The development of versatile processing nodes by sophisticated very large-scale integration (VLSI) design can lead to a new generation of adaptive processors which can achieve real time throughput rate as well as flexibility. As a result, researchers have investigated various implementation aspects of orthogo~alization-based algorithms in the context of parallel processmg. For example, time-recursive versions of the Givens transformation and the modified Gram-Schmidt algorithm have been developed and discussed in the context of systolic array implementation [7]-[10]. The MGS approach, in particular, h~s received a tremendous amount of attention in many radar SIgnal processing applications. Besides having a modular and regular processing architecture, the MGS algorithm possesses both time and order recursive properties [10]. Furthermore, it has been shown to yield good performance simultaneously in arithmetic efficiency, stability, and convergence times [11], [12]. The MGS procedure has been considered in the literature as an orthogonalization preprocessor for the LMS algorithm [13] , as a linear predictor for temporal input [14], as a sidelobe cancellor [15], and for clutter rejection in a nonstationary radar environment [16], [17]. More recently, the MGS orthogonalization algorithm and its corresponding triangular processing architecture have been generalized for efficient multiple channel adaptive filtering [18], [19]. The purpose of this paper is to introduce an alternative orthogonalization algorithm which results in a more efficient architecture for filtering applications in which there are as many output channels as there are input channels. One example is adaptive pulse Doppler processing in radar. For completeness, the basic theory and the architecture proposed by Gerlach [19] based on the MGS procedure are reviewed in Section II. We then systematically develop the new alternative algorithm and the corresponding efficient processing structure in Section III, starting with a.simple two

Reprinted from IEEE Transactions on Antennas and Propagation, Vol. 36, No.5, pp. 629-635, May 1988.

596

The vector X I - YW is called the residual vector and is orthogonal to the columns. of Y. Hence we can compute W from the normal equations

input-two output decorrelation processing element as the primitive building block. Finally, the differences between the two approaches are discussed and future work relevant to the new algorithm and processing architecture is addressed in Section IV. II.

The LS Problem

Y'YW=Y'X I

REVIEW OF BASIC THEORIES

9

(5)

Equation (5) is often called the LS estimator and is akin to the Wiener-Hopf equation derived using the criterion of least mean squares error [20]. Equation (4) can also be written explicitly as follows:

The LS problem is known by different names in different scientific disciplines. In the IEEE literature, the solution of the

K K K

L X!(k)Xl(k)- Wz L xi(k)X2(k)-··· -

WN ~ xi(k)XN(k) = 0

2: Xj(k)XI(k)- W L Xj(k)X2(k)-··· -

W N ~ Xj(k)XN(k)=O

k=O

k=O 1\

K

2

k=O

k:O A

k=O

k=O

K K K

2: X~(k)XI (k) -

W2

k=O

2: X:t(k)X2(k) -

[XI (k), x~ (k), ... ,

Since X. and X, - YW are not orthogonal, their inner product can be expressed as K

2: xi(k)x, (k) -

/':=(l

K

W2

2: xi(k)X2(k) /.:=0

A'

- . . . - ~V v ~ Xi ( k ) X\,( k ) =; }l

k=O,l'··, K.

where

(1)

that

J.1.

(4)

is a nonzero quantity. Next we combine (6) and (7) so is replaced with the matrix equation

We desire to determine a weight vector \V which minimizes the sum of squared errors defined as €(K)

==

L 1\

1

Rxxw ==

e2 ( k )

}l

o

(8)

o

1.:=0

==

(7)

k:..:O

xy(k)] ,

=[x\(k), yJ(k)]',

(6)

k~O

LS problem is associated with a number of equations and vector space concepts. The purpose of this section is to review brietly the essential equations and fundamental concepts, and to demonstrate that they are indeed interchangeable in the interpretation of the LS solution. Assume that we have an N channel system with the measurement vector at a given time instant k represented as X (k) =

... - W N ~ X~(k)XN(k) = o.

k=O

where w == [1, -W']' and R xx is the N x lv sample (2) covariance matrix of the input channels calculated based on K + 1 observations.

2: (xI(k)-y'(k)W)2 K

k=O

Wy'J.

The minimization of (2) is equivalent to the solution of the LS problem of minimizing the Euclidean length

According to the theory of signal-to-noise (SIN) optirnization in the field of adaptive arrays. the optimum weight vector W Op l is the value of w that satisfies

IIXI-YWII

Rxxw=J.lS*.

where W == [W2 , W 3 t

.",

(3)

where

XI =

Xl

(0)

X2(O)

X3(0)

XN(O)

XI

(1)

x2(1)

x3(1)

xN(l)

x2(K)

x3(K)

xN(K)

xI(K)

and y=

It is well known that the solution W satisfies the condition Y' (X, - YW) =0.

(4)

(9)

5 == [5., 52, ... , 5 N ) ' is often called the steering vector and J.L in this case can be an arbitrary constant. Equation (9) is known as the Applebaum maximum signal-to-noise criterion (21]. In a linear array antenna with equally spaced elements, the components of 5 are determined by the direction of the desired signal. Although Jl in (9) can be arbitrary ~ Jl in (8) is not, and is chosen so that the first element of w in (8) is a one. The key point in the derivation of (9) is the application of the CauchySchwartz inequality. The similarity between (8) and (9) is obvious. Although the approaches to deriving (8) and (9) are

597

and 12 ,1 is calculated so that (10) is satisfied. It is easy to see that

==== .

YIY1 12 1 , \Y2\2 DP

(12)

In an actual application, a finite number of samples would be taken for each input channel, thus (12) is estimated as s

I

v:1 Fig. 1.

Decorrelation processor [14].

~ Yl (k)Yi(k)

-x-----

_k=O

2,1 -

(13)

~ \Y2(k)1 2

k=O

LEVEL 1

LEVEL 2

··•

LEVEL N·2

LEVELN·l

Fig. 2.

~

DP

~

+OUTPUT

Modified Gram-Schmidt N-channel decorrelator [15].

somewhat different, it is clear that they provide the same solution of the LS problem. In fact. (8) can be considered as a special case of (9) with S = [1, 0, ... , 0]'. An alternative interpretation of (8) is that it is obtained by transforming the input signals of (9) such that the effective steering vector in the transformed signal space has the same simple form S = [I, 0, ·'·,0]'.

Multiple Channel Adaptive Filtering Using MGS The direct implementation of (8) corresponds to the inversion of the covariance matrix. However. it is well known that problems occur in the solution of the weights if R xx is illconditioned. A better approach is the use of the MGS orthogonalization technique which has been reported to have good numerical properties [22], [23]. Its processing architecture is shown in Fig. 2, using the simple decorrelation processor (DP) of Fig. 1 as the building block. To understand the operation of a single DP, we consider two channels of complex values input data: YI and Y2. The objective here is to form an output channel which is decorrelated with Y2. This is equivalent to setting

y;

Y; Yi=O

(10)

Where the overbar and asterisk denote the time average and the cOlnplex conjugate, respectively. We can also express (11)

In Fig. 2, X N is decorrelated with X., X 2 , " . , X N - I in the first level of the processing structure. In the second level, the output channel which results from decorrelating X N with X N - 1 is decorrelated with the other outputs of the first level of DP's, and the process continues as indicated in the figure. At the end, a final output is generated and it is totally decorrelated with the input X 2 , X 3 , ••• , X N • It is clear that the MGS decorrelation (orthogonalization) procedure is not unique with respect to the order in which X 2 , X 3 , • • " X N are decorrelated from XI. In multiple input-multiple output adaptive filtering, there can be as many output channels as input channels. Specifically, given there are N input channels, it might be desirable to generate N output channels so that each one of the N output channels is totally decorrelated with the rest of the N - 1 input channels. For example, this concept can be applied to radar Doppler processing in which a bank of filters is used to cover the entire Doppler band, and each Doppler sub band is processed such that it is totally decorrelated with the rest of the other subbands. Mathematically, this corresponds to solving the matrix equation

Rxxv=[

(14)

where V is the optimal weighting matrix and I is simply an N x N identity matrix. The SIN associated with each of the output channels is maximized by the corresponding column vector of if, and the nth output channel has a desired signal vector:

(0, 0, ... , 0, 1, 0, .'., 0)

I

i nth position. Based on the fact that there is no logic behind the ordering of the input channels in the MGS procedure, Gerlach was able to develop an efficient processing architecture for multiple channel adaptive filtering [15]. The design is illustrated in Fig. 3 for the case of eight input and eight output channels. The key point in the design is that arithmetic efficiency is achieved by taking advantage of computational redundancies and substructure sharing that can occur for different output channels. III.

DERIVATION OF THE NEW MULTIPLE CHANNEL ORTHOGONALIZATION ARCHITECTURE

Using the basic decorrelation processor of Fig. 1, it is possible to configure other orthogonal networks. To derive a

598

Fig. 3.

Complete realization of an eight-channel decorrelation network [15].

tive defined, we then proceed to construct an orthogonalization processing network . One structure which naturally takes advantage of the symmetry property of the PE is the tree-like network of Fig. 5. An eight input channel structure is illustrated in this example . The extension to an arbitrary number of input channels is obvious. Furthermore, the use of eight input channels also provides a one-to-one comparison between the newly derived tree-like architecture and the architecture based on MGS discussed in the previous section. The numbering system N 1(N 1 , N 3, • • • ) used at the output of each PE gives a clear picture of the orthogonalization procedure carried out by the tree-like processing architecture. The notation implies that the N1th output channel is totally decorrelated with the input: N 2 , N J , ••• etc . In the first row or y: y~ I J level of decomposition, XI is decorrelated with X 2 (and vice versa), X 2 is decorrelated with X J (and vice versa), and so on. E (Yj· V i) _"':'---Yj Next , the output channel which results from decorrelating Xl with X 2 is decorrelated with the proper output channel which E Y j \2) results from decorrelating X J with X 2• The decorrelation E (Y i • Yj) process continues as seen in Fig . 5 until two final output ----Vi channels are generated. It is important to emphasize that the E!lYi\2) two inputs to any given PE must be compatible, i.e. . the set of Fig. 4. Two input-two output building block. input channel indices enclosed by the two pairs of parenthe ses must be identical. Thus the first channel XI and the last more regular architecture for multiple channel adaptive channel X s are totally decorrelated with the input set (Xl , X J, filtering, we begin by considering possible modifications at the . . . , X s) and (X" X 2 , " ' , X 7 ) , respectively, in Fig. 5. primitive or building block level. An intuitive approach of The processing architecture of Fig. 5 generates two of the N achieving structural compactness is to employ the two input- desired output channels . In the case of multiple input-multiple two output processing element (PE) of Fig . 4 as the primitive output adaptive filtering , the remaining N - 2 channels can be building block. The only difference between the DP and the efficiently generated using the technique of " sliding-window" PE is the second orthogonal output associated with the PE substructure sharing as shown in Fig. 6. Four windows in our building block. This second output , however, requires a example of eight input channels correspond to the following smaller number of arithmetic operations than the only output four ordered input sequences : of the DP. This is especially true in the case of batch processing in which E{Yj*(t)Y;(t)} and E{Yi(t)Yj(t)} are obtained by time averaging. Once one of the two expectation operations is estimated by summing N time samples , the other is easily obtained by taking the conjugate . As N becomes large, the use of the PE as the basic building block would result in improved arithmetic efficiency . With the PE prirniy.

y. J

I

u

599

x,

PE

1 12~121

31~141

213'M131

V3

1l2 .~2.3 1

213

.t~(5)

31• . ~•. 5'

.• ,

' 15~5.61

",.,%?,.M,~.." "..~s." 112.3.• ~ 612.3'.51

~

5t6~6'

617~817 1

5 16'~16 .7I

".)d".'"

2 13 .4~.4 .5.6,31~1• .5.6.71

M,

~

112.3.•. 5.6~• .5.61 213 .~ 813.'.5 .6.71

~

~

112.3.• . 5.6~2.3 .•. S.6.71

/::~

8(1 ,2.3 .4 .5.6.11

112.3,4,5.6 .7.8'

CHANNEL I

Fig. 5.

CHANNEL 1

Tree-like orthogonaJization network.

x,

x,

\

\ \

\ \

. \

\ \

\

CHANNEL

8

CHANNEL 1

Fig. 6.

CH AN N EL

2

CHANNEL 3

CHANNEL

CHANNEL

•

5

CHANNEL

CHANNEL

6

7

" Sliding window" substructure sharing.

The first window generates the eighth and the first decorrelated output channels , the second window generates the second and the third decorrelated output channels, and so forth. In other words, the two output channels generated correspond to the first and the last elements of the ordered input sequence associated with a given window. The concept of substructure sharing using the "sliding window" approach is a useful tool in reducing the total number of PE's as well as the total number of arithmetic operations. Besides the "sliding window" technique, we can achieve further sharing of substructures by exploiting one other structural symmetry in Fig. 6. This symmetry is illustrated by the two dashed triangles

enclosing two identical substructures. As a result, one of the two substructures can be completely eliminated, and the final processing architecture requires JV2 - 3N12 (JV2 - (3N - 1)12) PE's if N is even (odd), where N is the number of input channels. The final design is given in Fig. 7. In the case of batch processing, we see that as the data are processed through a given row , the input data may be discarded and the output data become the new input data set for the next row of PE' s. Hence the two-dimensional structure of Fig. 6 can be collapsed to just a single row of PE's using simple feedback as shown in Fig. 8. Fig. 7 illustrates that as outputs leave at one side of the parallelogram structure they

600

CHANNEL CHANNEL CH A N N EL CH AN N E L 8 CH AN N EL 2 CHAN N E L 4 CH AN N EL 6 CH A N N E L 1 3 S 7

Fig. 7.

CHANN EL

8

Efficient orthogonalization architecture for multiple channel adaptive filtering .

CH ANNEL

1

CHANNEL

2

Fig . 8.

CHANNEL

CHANNEL

3

4

CHANNEL

5

CHA N NEL

6

CH AN N EL 7

Hardware compaction for batch processing.

enter at the other side. Thus we can imagine that Fig. 7 and Fig. 8 represent a cylindrical systolic architecture and a simple ring structure, respectively , in three dimensions . IV. SUMMARY AND FUTURE RESEARCH The main features of the newly developed multiple channel orthogonalization architecture are summarized as follows . 1) In contrast to the architecture based on MGS orthogonalization, it requires no broadcasting of data and any given processing node in the structure only communicates with its neighboring nodes in a pipelining fashion . Hence the design is "purely" systolic . 2) In terms of the total number of arithmetic operations, it is at least as efficient as the MGS approach. A detailed comparison will be given in a future paper. 3) The new architecture is developed in a very systematic and bottom-up fashion, starting with a simple two input-two output decorrelation processing element as the building block. 4) It is an extremely regular and compact processing structure. This is particularly true for batch processing, since in this case the original two-dimensional systolic array can be collapsed into a linear array with just N processing elements , where N is the number of input channels. 5) No unscrambling of the output channels is needed . The

MGS approach , on the other hand , requires a commutatio n algorithm so that the final output channels are properly aligned with the input channels . 6) The technique based on the MGS approach is most efficient for 2 m input channels , where m is a positive integer. The architecture presented in this paper , however, places no restriction on the number of input channels. In the field of adaptive filtering and estimation using VLS: parallel processing, many problems still exist. We are currently investigating the following research topics relevant to the new multiple channel adaptive filtering technique: 1) This paper focuses on the development of an efficient and compact processing architecture for multiple channel adaptive filtering based on the concept of orthogonalization. For simplicity of illustration in the development, batch processing is emphasized. The time-recursive version is yet to be investigated in detail. One advantage of the time-recursive version of the algorithm is that it only has a latency of N computing cycles for the first output to be generated , whereas N ·N, cycles are required in the case of batch processing. /If and N, correspond to the number of processing channels and the number of time samples used for decorrelation, respectively. 2) In real time applications, divisions take more time than

601

multiplications, since most of the special hardware components are optimized to perform multiplication and addition. It is desirable to modify the new algorithm and architecture so that the number of divisions is minimized. 3) It has been reported that the MGS algorithm possesses good numerical properties. It is important to examine the numerical properties of the new algorithm and compare them with those of the MGS algorithm. 4) The application of geometrical vector space concepts for deriving the rapidly converging recursive least squares adaptive filters is well known. Although the new algorithm derived in this paper is done from an architectural perspective, it is worthwhile to rederive the new algorithm using the geometrical approach.

[23]

REFERENCES

B. Widrow and M. E. Hoff. "Adaptive switching circuits," in 1960 WESCON Conv. Rec., pt. 4, pp. 96-140. [2] B. Widrow et al., "Stationary and nonstationary leamingcharacteristics of the LMS adaptive filter." Proc. IEEE, vol. 64, pp. 1156-1162, [!~

Aug. 1976.

[3] Lee et al., "Recursive least squares ladder estimation algorithms." IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. [4]

[51

[6] [7] [8] [9]

[lC: [11] [12] [13] [14 1 [I5] [16]

[17]

[I8: [19] [20] [21] [22]

627-641. June 1981. J. M. Cioffi and T. Kailath .: 'Fast, first-order, least squares algorithms for adaptive filtering." IEEE Proc. ICASSP '83, Boston. MA. Apr. 1983, pp. 679-682. F. Long and J. G. Proakis, "A generalized multichannel least squares lauic algorithm with sequential processing stages." IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, no. 2, pp. 381389. Apr. 1984. 1. D. Pack and E. H. Satorius, "Least squares adaptive lattice algorithms." NOSC Tech. Rep. TR423. Apr. 1979. J. Mcwhirter. ."Recursive least-squares minimization using a systolic array ... SPIE paper 431-15. 1983. H. T. Kung and W. M. Gentleman. "Matrix triangularization by systolic arrays," Proc. SPIE, vol. 298. 1981. S. Y. Kung. "VLSI array processors." IEEE Acoust., Speech, Signal Processing Mag., vol. 2. pp. 4-22. July 1985. F. Ling et al., .. A recursive modified Gram-Schmidt algorithm for least-squares estimation." IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34. no ...L pp. 829-835. Aug. 1986. B. Friedlander. "Lattice filters for adaptive processing," Proc. IEEE, vol. 70. no. 8. pp. 829-867. Aug. 1982. I. S. Reed. J. D. Mallet. and L. E. Brennan.: 'Rapid convergence rate in adaptive arrays:' IEEE Trans., Acoust., Speech Signal Processing, vol. AES-IO. pp. 853-863. Nov. 1974. R. A. Monzingo and T. W. Miller. Introduction to Adaptive Arrays. New York: Wiley. 1980. Ch. 9. N. Ahmed and D. H. Youn. "On a realization and related algorithm for adaptive prediction;' IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-28, pp. 493-497, Oct. 1980. F. F. Kretschmer and B. L. Lewis, "A digital open-loop adaptive processor:' IEEE Trans. Acoust., Speech, Signal Processing, vol. AES-14. pp. 165-170. Jan. 1978. F. F. Kretschmer, B. L. Lewis, and F. L. C. Lin .: 'Adaptive MTI and doppler filter bank clutter processing." in Proc. IEEE 1984 Nat. Radar Conf., Atlanta. GA, Mar. 1984. A. Farina and F. A. Studer, "Application of Gram-Schmidt algorithm to optimum radar signal processing," Proc. lnst. Elec. Eng., vol. 131, pt. F, no. 2, Apr. 1984. K. Gerlach. "'Multiple channel adaptive filtering using a fast orghogonalization network: an application to efficient pulsed doppler radar processing, " NRL Rep. 8840. 1984. - - , "Fast orthogonalization networks." IEEE Trans. Antennas Propagat., vol. AP-34, no. 3. pp. 458-462, Mar. 1986. S. Haykin, "Nonlinear methods of spectral analysis," Topics in Appl. Phys., vol. 34, 1983. S. P. Applebaum, "Adaptive arrays," IEEE Trans. Antennas Propagat., vol. AP-24, no. 5, Sept. 1976. C. L. Lawson and R. J. Hanson. Solving Least Squares Problems. Englewood Cliffs. NJ: Prentice-Hall, 1974.

602

A. Bjorck, "Solving linear least squares problems by Gram-Schmidt

orthogonalization." BIT, vol. 7, pp. 1-21, Jan. 1967.

Mutual Coupling Compensation in Small Array Antennas

aperture distribution is obtained in the presence of these parasitics, the mutual coupling can be compensated for. This compensation principle has been reported for a slot array [1] and dipole arrays [2]-[4]. The former is the only one that considers the case of scanning and presents some experimental data; all four rely on computed coupling coefficients. The present study differs in that it rephrases the approach for the receiver mode, appropriate for a digital beamforming antenna where the technique is most practical, and it describes an alternative method to determine the mutual coupling coefficients, that does not require analytically simple or reciprocal array elements. It also presents experimental data for a scanned waveguide array.

HANS STEYSKAL, MEMBER, IEEE, AND JEFFREY S. HERD, MEMBER, IEEE

Abstract-A technique to compensate for mutual coupling in a small amy is developed and experimentally verified. Mathematically, the compensation consists of a matrix multiplication performed on the received signal vector. This, in eifect, restores tbe signals as received b) tbe isolated elements in tbe absence of mutual coupling. The technique is most practical for digital beamforming antennas where tbe matrix operation can be readily implemented.

THEORY INTRODUCTION

We consider an array of single-mode elements, meaning that the element aperture currents (electric or magnetic) may change in amplitude but not in shape, as a function of radiation direction. In the receive mode, the signal at the output of the individual antenna element has several constituents: a dominant one due to the direct incident plane wave, and several lesser ones due to scattering of the incident wave at neighboring elements. As depicted in Fig. 1, we can write the received signal at element m as

The radiation pattern of an array of identical antenna elements is usually taken to be the product of an element factor and an array factor, based on the presumption that all elements have equal radiation patterns. Unfortunately, this may not be true for a practical array, where, due to mutual coupling, each element ., sees" a different environment. The nature of the error thus incurred can be displayed by expressing the individual array element pattern f n ( u) as the sum of one average array element pattern fa(u) and a pattern deviation o!n(u), which leads to the total array pattern

F(u) =

Vm(u) = cmmEm!i(u) +

LQnfn(u)eJnkdu

=f

n Q

(

u) L QneJnkdu + L Qnofn( u) eJnkdu. n

n

(1)

Here an = I a; I exp (jet>n) denotes the complex element weight, k the wavenumber, d the uniform element spacing and u the sine of the angle 8 from broadside, respectively. The first term on the right side of (1) represents the idealized pattern, and the second represents the error. One effect of this error pattern is to introduce a noise floor that precludes synthesis of high-quality patterns with very low sidelobes or deterministic pattern nulls. Other effects appear in signal processing arrays, such as adaptive or superresolution systems, which can be extremely sensitive to small errors due to the nonlinear processing involved. Since real-life signal processing arrays usually are comparatively small arrays, where element pattern differences are relatively large, this is a significant problem. It is clear from (1) that the element coefficients {an} always can be chosen such as to compensate for the pattern error at one particular angle. It is less obvious that the error normally can be corrected for all angles simultaneously. Furthermore, since this correction is scan independent, it also applies in the case of electronic scanning. It is the purpose of this communication to discuss such a technique and to present some experimental results. The key to the technique is an alternative formulation to (1), which recognizes that 1) any composite array pattern can be considered as a weighted sum of the isolated element patterns and 2) the effect of mutual coupling is simply to parasitically excite all elements, even though only one element is driven. Thus, by driving the array with modified element excitations, such that the desired array Manuscript received November 29, 1989; revised June 12, 1990. The authors are with the Electromagnetics Directorate, Rome Air Development Center, Hanscom AFB, MA 01731. IEEE Log Number 9038579.

L

n , m,*n

cmnEnfi(u).

(2)

The incident field Em at element m impresses an aperture current amplitude Em!i(U), where fi(u} is the. isolated element pattern, i.e., the pattern of the current mode assumed in the element aperture. This aperture current will produce an element output voltage cmmEmfi(u), where cmm denotes the coupling from the aperture to the output transmission line. The effect of the neighbor-

Vm Um at element m consists of a directly transmitted and several scattered components.

Fig. 1. The received signal

ing elements is described similarly, with Cm n denoting the COupling of aperture mode n to element output m. From a mathematical point of view, (2) simply expresses the linear relationship between the aperture excitations and the element output voltages. The physical meaning of the Cmn will be discussed below. We introduce the notation

(3) since this represents the desired, coupling-unperturbed signal received by the single element at the aperture. Thus for our unifonnly spaced array of identical elements Eoejnkdufi(u)

= u~(u)

(3a)

where Eo is the amplitude of the plane wave incident from direction

u.

Reprinted from IEEE Transactions on Antennas and Propagation, Vol. 38, No. 12, pp. 1971-1975, December 1995. U.S. Government work not protected by U.S. Copyright.

603

Substituting (3) in (2) leads to

iog antenna system. It then allows all subsequent beamforming operations to be performed with ideal element signals. such as are usually assumed in pattern synthesis.

(4)

DETEIUIlINA1l0N OF THE MU11JAL COUPUNG COEFFICIENTS

On the left side, the vector v represents the coupling perturbed signals { lin} at the element output ports, which via the coupling matrix C is related to the vector y d, representing the unperturbed desired signals {u:l. Thus compensation for the mutual coupling can be accomplished by simply multiplying the received signal v by the inverse coupling matrix C - I , yd

=

(5)

C-1y.

This concept is depicted in Fig. 2, where a network corresponding to C- I is attached to the array antenna. Note that the coupling. compensation is scan independent, i.e., the same matrix C - I applies universally for all directions of the incoming wave, as a consequence of our single-mode assumption. Multimode elements, as considered in [2]-[4], would require a scan dependent coupling compensation. When the received and compensated signals vd are weighted and summed in the conventional beamforming network, shown in Fig. 2, we obtain the array pattern F{u), defined as the ratio of the output voltage and the incident wave amplitude Eo,

L anv~ = f'{u) L anelnkd". o

1

F{u) = E

n

(6)

n

The array pattern (6) now has the desired form of a product of an element factor and an array factor. A comparison with (1) shows that, with the transformation performed, we have succeeded in dissolving the error pattern, the second term on the right side of (1). The matrix C- 1 may be difficult or impractical to realize by an analog network, but it can be readily realized in a digital beamform-

mutual coupling and feed line errors

n

11

coupling compensation desired signals

cf. (2).

and, recognizing that the cmn are the Fourier coefficients of these patterns, determine these coefficients numerically according to

(8) In order to do this, ji(U) must not have a null in the integration

interval. However, since the isolated element pattern normally is very wide, this is no serious limitation. Another restriction on (8) is that the element spacing be larger than A/2. Otherwise the integration interval extends beyond visible space, i.e., beyond the interval - 1 < u < 1 where gm(u) and li(u) are known. For the case of element spacings d < A/2 we can still perform a Spectral analysis of

gm(u) fi(U) -

LC n mn

11n d

scanning

beam shaping

A-

output beam Fig. 2. Illustration of coupling compensations and beam forming in an array antenna. Interelement coupling at the array face. represented by (c",n)' leads to received signals I} n at array element outputs" that are linear combinations of the desired, coupling-unpenurbed signals Multiplication by (c m n ) - I restores these signals. which are then weighted and summed to fonn the desired beam.

I):.

(9)

ejnkdu

to determine the coefficients Cm n' but the convenient orthogonality of the harmonic functions is lost and accuracy becomes a major issue. An advantage of this method is that it does not require reciprocal antenna elements. Thus, it is applicable to receive-only arrays. such as used for digital beamforming, where the element includes an entire microwave receiver. Furthermore, any channel imbalances, i.e., differences in insertion amplitude and phase between the element aperture and the element output terminal, manifest themselves in the self-terms em m and are also compensated for. In this sense the technique is similar to a conventional array calibration. In the second method, the matrix C is obtained from the related scattering matrix S ::: (smn) of the array. This relation is developed

desired signals vn d received on isolated elements

measured signals

There appear to be two di fferent methods to determine the coupling coefficients-one by Fourier decomposition of the measured array element patterns and another by coupling measurements between the array ports. The former requires driving the antenna only in one mode" either transmit or receive, and thus applies to nonreciprocal antenna systems. The latter requires driving each element in both modes and therefore is less practical, as discussed below. In the Fourier decomposition method we measure the complex voltage patterns g m( u) of the elements in their array environment.

8 -

-A'

- -

T T- - - T- - - - - - - 8' o~ b~

b~

Fig. 3. lliustration of the scattering matrices 5 and 5' of the array. Line sections between aperture plane AA' and terminal plane BB' are matched and reciprocal, with transmission coefficients tn.

604

here for the simplest case of a waveguide array fed by matched generators . For the general case the relation is complicated and not very useful. We consider a uniformly spaced array of waveguide elements, shown in Fig. 3, and determine the array element pattern of element m. This element is excited with a wave of amplitude am ' all other elements are passive. Assuming a reference plane A A ' for the antenna element terminals that coincides with the element apertures, the aperture voltages thus are ( 10) where the Kronecker delta wise. The radiated far field

0 mn

=

I for n

= m.

and

=0

other-

complicated than pattern measurements with Fourier decompos itions, in reality often is the less practical method. E XPERIMENTS

The coupling compensation technique outlined in the preceding section was applied to an eight-element linear array of X-band rectangular waveguides in a ground plane. Each element was in turn a column of 8 rectangular waveguides in a common H-plane . combined via a fixed 1:8 power divider. The array axis thus was parallel to the E-plane and in this plane the element spacing d = 1.25 cm = 0.5 17 A. The isolated element pattern corresponds to a normalized uniform apertur e distribution kl sin - u 2 kl

P (u )

( 17)

-u 2

where rn is the distance of element n to the observation point and the usual far-field approximations have been made. Comparing this expression with (7) and requiring that the transmit and receive patterns are identical, shows that

apart from a constant factor of no interest. Thus

C

=I+S

(12)

where I denotes the identity matrix. In real arrays , the scattering matrix cannot normally be measured directly at the element apertures , as assumed above, but only from a reference plane a certain distance behind the apertures. A more realistic case therefore is as shown in Fig . 3, where sections of transmission line are included between the apertures and the reference plane BB' , from which the modified scattering matrix S' is measured. These feed lines have different insertion phase and loss. However , for simplicity , we still assume them to be matched and reciprocal. so that they can be characterized by single transmission coefficients t m : Defining a diagonal matrix T , ( 13)

where I is the interior waveguide height. in our case I = 1.02 cm = 0.417 A. The complex voltage patterns g m( u) of the array elements were measured under matched load conditions , and recorded at 1/ 2-degree intervals with a digital rece iver. The coupling coefficients cm n were then numerically evaluated accord ing to (8) and the inverse matrix C- I was computed. In a second , similar measurement, the received voltages lI m ( u) were again recorded . Then , in an off-line simulation of a digital beamforming system, they were multiplied with C - I for coupling compensation, and amplitude and phase weighted for pattern shaping and scanning, as shown in Fig. 2. Examples of element patterns for a central and an edge element are shown in Fig. 4. We note that there is indeed a considerable difference in shape, which is attributable to mutual coupling effects , and also in overall power level , which mainly is due to a difference in feed line losses . In Figs. 5(a) and 5(b) we show synthesized 30-dB Chebyshev patterns as obtained without and with the mutual coupling compensation. Apparently the compensation technique gives about 10 dB improvement in sidelobe level , with the result that the actual pattern is quite close to the theoretical one. The remaining difference indic ates that the array excitation tolerance erro rs equ al ab out - 35 dB in ampllitude and 1° in ph ase. Figs. 6(a) and 6(b) show th e same 30-dB Chebyshev patt erns scanned to - 30°. Without compe nsation the side lobe level is still

it is easy to show that

(14)

S' = TST

o

and, from (4) and (12), that the received signals at plane BB' are /

( 15)

~

aI ~ II:

The modified coupling matrix C' at plane BB' thus is

C' = T + S'T -

1

-5

W

(16)

which shows that in this case we need to measure not only the scattering matrix S' but also the transmission coefficients {t m} . This mayor may not be possible, depending on the design of the actual array . Clearly for the general case , where the feed lines between the element apertures and output terminals are not matched, the required measurements become still more extensive . Thus , measurement of the network parameters, which intuitively would seem less

~Q. ·10

I

I I /

I

/

l/ ""\

""-

/ ./

./

,,- '"

-

-'

domain can be achieved by integrating the mean-square-deviation between the desired unity response (A = 1) and the response of the beam

608

• - o·

former over a spatial region of interest [cPl, cPu] as follows:

Source Direct:1on

e = ;4>1::' 11 - G(fo, 8,4>)12 dd: 2

== WH QW - (PHW + WHP) + 1

(17)

where (18)

p=

;4>I::S(fo,8,4»d4>

+---+-_ _ • • 90·

ac;.

(19)

where cPu == ¢o + 6.4>/2, cPl == 4>0 - 6.4>/2, and 6.<jJ defines the spatial region in the look direction over which the desired unity response is to be preserved. Fig. 2.

B. Element Spacing Deviation

The second example is the design of a beamformer with robustness against element spacing errors. An optimum beamformer can be very sensitive to errors in the assumed array element spacing. Recall that the steering vector S is a function of element spacing through the phase relation of the signal. Thus any deviation of the element spacing from its assumed design value could create an erroneous steering vector which is different from the one assumed in the constraint on the weight vector. Signal suppression may occur when the mean output power is minimized. Robustness in the design can be achieved by integrating the mean-square-deviation over some tolerance bound ~r about the nominal value rl as follows:

where

Azimuth plane of a double- ring circular array having ten sensor elements with five elements equally spaced on each ring.

5 Ii cos (cPO

-

(Xi) -

5 Ii COS (4)0 -

4 V cos (0

d i ==

)

where

P

1

== (Llr)L

Cij

Iff-

+M/2

f

L

-~f/2

...

f

r , + ~r/2 r,-~r/2

S([o, 00 , ¢o) ds, dr, ... dr.:

(22)

For example, for the double-ring ten elements circular array as shown in Fig. 2, the (i, j)th element of Q matrix when the mean-square-deviation is integrated over some tolerance bound Sr about the nominal value To is given by [Q l.. = exp [j 21T fOTOhij ]sinc (1T fO~Thij), i, j == 1, 2, ... , L

(23)

-

Cii),

1 S:. i

" has amplitude taper for the outside beam as shown in Table II. Also shown is the taper for apex pointed horns. Use of apex pointing produces appreciable improvement. Gain is slightly improved.

Amplitude errors are calculated using beam port and element port hom patterns of sine ;r u, where horn widths are all set to a nominal >../2. Each port hom has its axis normal to the port curve. Fig. 14 shows amplitude error, normalized to 0 dB for the axial ray, for a lens with ex = 30, {:j = .94. Curves for ray angles of 0° , 15°, and 45° are given. Similarly, Fig. 15 is for a lens with ex = 40, {:j = 0.9, for ray angles of 0° , 20°, and 50° . These examples are two of the ex - {:j pairs of Table 1, and thus the amplitude errors are similar. As expected, for wide ray

618

V . CALCULATION OF LENS GAIN

Element and beam port spillover, phase and amplitude errors, port impedance mismatches, and transmission line loss all contribute to reducing lens gain. Note that, as in the case of a horn feeding a reflector antenna, there is no feed hom spreading loss, due to the equal path property through the foci. The small inequality of other paths is subsummed in the path phase and amplitude errors. Gain will be calculated here based on port

2. 0

1.

1. >

'"

n. Incident waves zn and r~ are scattered by couplers each of which is characterized by an angular parameter an (a in Fig. 4). Referring to the typical coupler in Fig. and zn by 6, the outputs Zn+1 and 'n are related to inputs the following equations derived with the aid of Fig. 4:

csc

2a

n

= [

I .1 + ~ I r ]/1 r, 1~ esc? ««. ZN+

2

l 1

2

2

(46)

'n

r:

or

I ZN+I 12 =

(40)

max[ I r, 1

2

2 CSC

amx

(47)

n

The following choice minimizes the wasted power

(39) These equations are applied at each bank of couplers in one of two ways. First, assume the C network is to be designed and all the values of c/l n and an for coupler banks 1,2 q - 1 which generate vectors £),. ··, Eq _ 1 have been determined. The qth bank is designed by recursively calculating r~ from

N

L 1',1 2 •

IZN+112~ IrnI2csc2a.mx-

-

~ 1',1

1 2

Z N + 1 I 2: ] .

(48)

The coupling now can be determined from (46), and the input power is determined from (45) with n = 1. Since the Eq are

628

2 Although 4N+ 1 is not needed, it can be calculated and should be zero.. Thus curious result also follows from energy conservation.

nonnalized to unity, the efficiency is efficiency

= I Eq I 2 / I Z I I 2 = 11/ Z I I 2 .

[6] ( 49)

Thus, starting with q = 1 and r n = (E1)n, the entire rff network can be built recursively until q = M. The U network may be constructed similarly if desired. Note from (48) when Ct m x = 7r /2, I z.!V+ I 1 2 = 0 and the networks 1/ and U t will be lossless. Generally, the load losses will require adjustment of the attenuators in Fig. 2. Combine the efficiencies of the U t and {/ networks for the qth interconnecting channel to form an overall efficiency 11' q I 2. Then the attenuators in Fig. 2 are I Tq I 2 as follows:

I T q I 2 I Tq I 2 = 'A q Since

I Tq I 2 1 > ~ (T - 1T - 1 * ) kk - (T*T) kk - /\k \ .

nt==l

x

Itl

L

with (42) and (44), we obtain

CX)

+

AT

jt-1(k, m)1 2

== 1

+ [dJl(k)d~(k') + E{BiL(k)B;(k')}]

d (k)d* (k') [ tL tL

(44)

== I(T- 1T)kkI 2

(R~JE)kkl == E{ BiL(k)B: (k')}

1T- 1 *

== (T*T)kk < NEln ax

where the terms B,(k) ~f L:~~=l t-1(k, m)ii,(m)/s(k). The estimation mean, covariance, and asymptotic form of the

L

1,2, ···.1'1. (43)

Combining the Schwarz inequality

== [dJL(k) + Btl (k)][l - Bo(k) + B6(k) - ...]

x

==

Hence,

f\/

BJL(k) == diL(k)s(k)

(42)

As the variances are monotonic functions of the (T-1T- 1*)kk's, they will be minimized by the class of encoding matrices T that minimize the (T-1T-1*)kk'S. The condition that the power emitted from an encoded single element is less than En1ax. is represented by the inequality

IV

L

L It-1(k, 'm)12 ~ O.

nt=l

== S + T-1iio,

s(k) ==s(k) +

==

(41)

The methods for generating the asymptotic form given above are discussed in Appendixes A and B. The factor of N in the correction terms arises from Theorem 6.2 proven below, which proves that the asymptotic minimum variance form of (T-1T- 1 * )kk is 8kk, N- 1. These error terms are extremely small for the practicable system, e.g., for a IO-dB SNR, with N == 16 and a == 0.9, the correction terms are r-v7 x 10- 6 . We now prove the theorem that renormalized unitary encoding matrices are optimal for the UTE process.

1

(T*T)kk

1 .

Ak

(47)

This lower bound is satisfied when the Schwarz inequality (45) becomes an equality. These conditions are jointly satisfied iff t*(m, k) == Akt-1(k, m)~ t-1*(k, m) == Aklt(m~ k). The orthogonality and normalization of the column vectors of T follows from

(T*T)k1k ==

L t*(m, k')t(m, k)

l

643

rn==l

== Ak(T-1T)kfk == Akbk.tk·

(48)

This proves part a). The proof of part b) follows from combining the minimum variance normalization condition 2 Lrn It(m, k)1 == Ak with the power constraint on the maximum allowed values of It(m, k)\ from (43) Nlt(m, k)l~lax ~ Ak- These two relations dictate that the minimum variance encoding matrix elements must satisfy N

L

2

It(m, k)t ~ Nlt(m, k)l~lax·

The kth components of these vectors are given by

Zp.o(k) == [1 - dp.(k)]s(k)

+L

Zp.v(k) == [1 - dp.(k)]dv(k)s(k) + t- 1(km )[n3(m ) - n4(m)].

L

(49)

Tn=l

This equation can be satisfied only as an equality and, then, iff {It(i, k)1 2 } == AkiN. 0 Corollary 6.2: If the straight-through-path signals have equal power, then the minimum variance UTE encoder is a renormalized unitary matrix with all matrix elements having the same magnitude. Proof: If all the elemental powers are equal, the Ak' s are all equal to a constant A. From (48), T*T == AT-iT =} T- 1 == A-IT*, which is the definition of a renonnalized unitary matrix. At equal elemental power, from Theorem 6.2b), {It(i, k)1 2 } == AIN; hence, all matrix elements of the minimum variance form of T have equal magnitude. D Substituting the minimum variance form for T into (41) gives the asymptotic minimum variance form of the UTE covariance associate with a maximum received single element power E Illax

t-1(km)[nl(m) - ii 2 (m )],

rn

(53)

nl.

Let us define

Av(k) ~f [1 - dv(k)]s(k); Bvi(k) ~f Av~k)

L

(54)

C1(k, m)ni(m);

(55)

m

Cvij(k) ~f Bvt(k) - B vj (k)

for i

=I j.

(56)

Take the ratio of the components in (53) to get the estimate of dp.(n) and expand in an asymptotic Taylor series

dJ1.(k) = zVJ1.(k)

zvo(k) == [dp.(k) + C v34(k)] x [1 - C v12(k)

+ C;12(k)

- ...].

(57)

Proceeding in a manner similar to Section VI-B,

E{dJL(k)} = dp.(k) ,

(R~~~v)kkl ==E{Cv34(k)C~34(k')}

+ [dp.(k)d~(k') + E{Cv34(k)C:34(k')}]

L 00

x

E{[Cv12(k)C~12(k,)]nl.}.

(58)

nl.=l

Therefore, the minimum variance form of the UTE covariance matrix satisfies the physical symmetry conditions enumerated in Section II.

From Appendixes A and B, the asymptotic form of the CCE covariance is

C. CCE Algorithm

Consider a general CCE encoding case where the control matrices T are bipolar, invertible, but not necessarily of the Hadamard form. In the presence of ATGN receiver noise, the demodulated signals vectors in the CCE algorithm are represented by

Y~o =D:S +nl;

y:o ==

D~S + n2

D:

+ n3; Y~v == D:dvS + D.4. Y~v ==

dvS

(51)

The signal estimates used to estimate the delay circuit parameters are obtained from decoding with the inverse of the bipolar control matrix

ZJLO == T

"

-1

"

-1

F

R

(YJLO - Y JLO)

== (I - dJL)S + T- 1(nl - n2),

Zp.v ==T

F

Theorem 6.3: The minimum variance bipolar CCE matrix is a Hadamard matrix. Proof- The CCE variances, which are similar to the UTE variances, are represented as a nonnegative series in powers of the positive semi-definite diagonal matrix elements (T-1T-1*)kk. CCE variances are minimized by the class of encoding matrices that minimize the (T-1T- 1 * )kk 'so The required bipolar character of the CCE encoding matrix It(m, k)\ = 1 mandates that N

R

(Yp.v -¥p.v)

== (I - dp.)dvS + T- 1(n 3 - n4).

(52)

644

(T*T)kk ==

L

nl.=l

t*(m, k)t(m, k) == N.

(60)

From Schwarz's inequality (45)

ESNR - DELAY CIRCUIT ESTIMATE - THEORY & MC SIMULATION simula tion statistics based on N X 10K random trials

( T - 1 T - 1 *) . . > kk

1 - (T *T )kk 1 - N'

28

26

(6 1)

24

Again, the minimum variance form of T renders the Schwarz inequality an equality. Thi s equality is satisfied iff t * (m , k) = NC 1( k, m ). As T is a real -bip olar matrix, the minimum variance condition can be expressed as the matrix relati on

T-

1

=N -

1

=N - T T

(R/l /l,vv)kk' =t5,.:k'

+

2(T 2

NElll ax ll

-

d",(k)1 2

+

N

Ell axl~(T~ d", (k )12]

[ ld/l(kW

X{At, m =l

_ 0(

m! [

2 (T2

N Elllaxil

-

2

0.04

. x - theory - CCE + - theory - UTE

o-

Me Simulations

I

for the

7r

for the

"2 7r

2] m} (63)

64

ApPENDIX A

.

phase shift

NUMBER OF ELEMENTS

element power to the rec eiv er noise power E lll ax / (T2 of 10 dB . Both the theoretical and simulation results given in Fig . 2 co rres pond to the ESNR for the complex gain estim ate s of the 7r /2 phase shifter delay circuit. The MC sim ulation and theoretical results for the single element ESNR ' s of the 7r /2 phase shifte r are 6.121 and 6.120 dB , respectivel y for 320k random trials. The close correspondence between the the ory and the MC simulations serves to validate the theoret ical ana lysis . Th ese results also illu str ate the dramati c incre ase in the ESNR ' s that ean be obtained using the orthogon al codes , for d irect encod ing in the UTE pro cess, and for co ntro l co des in the CCE process.

phase shift

7r

32

Fig. 2. Theory versus MC simulation of the effective estimation SNR (ESNR) for the estimate of the complex gain of the 1l' /2 phase shift circuit.

d", (k)1

Qc-E~,./(2 ) }.

d",(k)1 = { :

e

16

Here again, the optimal form of the cov ariance matri x satisfies the physical symmetry co nditions enumerated in Secti on III. We note that the expansion parameter for the CCE vari ance has a fac tor of 11- d", (kW in the den ominator. As

11-

.

..

(62)

.

Thi s relati on is the definition of the class of bipolar Hadamard matrices. 0 For maximum elemental power levels , the asymptotic , optimal CCE covariance assumes the form {

.

SNR 10dB

.

T*

1

CC E

SE ESNR . theo ry 6 120 dB • simu lat ion 6 121 dB 320K trials

(64)

.

for the 16 phase shift

lower vari ance parameter estimates are obtained when the CCE encoding is performed with the phase shifter delay circuit that produces the largest magnitude phase shift.

D. Comparisons: Theory and Simulation We now compare the asymptotic ESNR ' s, which are represe nted by the ratios of the square of the mean s to the variance s of the parameter estimates, using the the oretical result s for the SE , UTE, and CCE varia nces given by (36) , (50), and (63), respecti vely, with Monte Ca rlo (MC) simulations of the corres pond ing ESNR ' s. For these examples, the erro rs induced by the trun cat ion of the asy mptotic expansions are so small that we truncated the expan sion s in (36), (50) , and (63) at their ninth -order expansion term s. Fig . 2 illustrates these comparative results. Hadamard matrices were used for the unitary encoding matrix in the UTE simulations. Hadamard matrices, as required, are also used in the CCE simulations. For these simulations, we have used a value of the single element SNR , which is defin ed as the ratio of received single

NOISE MODELS FOR COHERENT DETECTION

A typical approach to performance analyses of estimation algorithms is to develop analytical expressions for the variances of the parameter es timates under the assumption that the information bearing signals are co ntam inated by AWGN . Th e wid e use of AWGN has effec tively raised the model to the level of a de fa cto standa rd. However, one mu st pro ceed with caution, as the model is basically ad hoc , and generic applica tion can , at times, produce problematic results. The primary reasons for the popularity of the AWGN mod el are a) AWGN fairl y acc urately models thermal noise for ana lytica l expre ssion s invol ving low-ord er mom ent s o f the noise energy, and b ) the model is mathematicall y simple for both anal ytical analysis and Monte Carl o simulations . Unfo rtunatel y, the literal form of the AWGN model is inappropri ate for performance analysis of demodulated signal coherent det ecti on algorithms, which are the main topi c of thi s pap er. In the coherent detection proces s, the noi se-contaminated signals are passed through a series of filters in the process of demodulating the signals from RF down to a low en ough intermediate frequency for the signals to be sampled digitally . Demodulation both band limits and energy limits the signals. If one neglects the physical constraints on the noise imp osed by

645

the demodulation process and heuristically uses the AWGN model for the performance analysis of a coherent detection system, the mathematical results as shown below will exhibit analytical singularities. These difficulties can be resolved by picking a noise model that is more appropriate for the physical process. The physical energy constraint imposed by the demodulation filters can be mathematically captured by cutting off the noise energy at a large fraction of the signal energy. The signals are sampled at a Nyquist rate commensurate with the final demodulation filter bandwidth, and hence, the individual bandlimited noise samples will be statistically independent. Accordingly, the demodulated noise, for which the samples are represented by {ii( k) }, is modeled by independent amplitude and phase random variables with probability density functions (pdf's) characterized, respectively, by a truncated Rayleigh amplitude distribution and a uniform phase distribution

for 1'2 < oe: a otherwise

< 1; ()

. The moments of the ATGN noise energy WIth x clef == e / a-? are given by

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These moments are mathematically more complex than the comparable AWGN moments E{lnI 2 A1 }A\VGN == M!a 2 1\1.. By changing variables in (A.2), 1oo( o , x) can be alternatively expressed in terms of exponential integral functions [6]

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L

Gaussian noise (ATGN). As discussed in Section VI-A, the variance of the singleelement complex gain estimate is proportional to

[Ei(x) - Ei([l - alx)].

(A.4)

nt=O

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-ax

For moderate to high SNR, x asymptotic form

L

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(A.2)

The closed contour in the above contour integral is counter clockwise on the unit circle. As the domain of t is 0 S t ~ a < 1 in the integral, the contour integral is equal to the residue at the pole at z == - Vi; the resulting integral is well behaved. On the other hand, for the AWON case, the domain of t is 0 ~ t < 00; the evaluation of the contour integral in this case produces a symmetric singularity in the neighborhood of t == 1 that causes a divergence in the subsequent t integral.

1, and foo(ax) assumes the

== fix(x) -

1.

(A.6)

The asymptotic form of the expansion is significantly simplified as it is an expansion in terms of the moments for the simplified AWGN model. One must truncate the expansion at an integral order Me for the asymptotic representation to be valid as the high order AWGN moments diverge. The maximum cutoff Me is chosen commensurate with the minimum value of £-nlE{\nI 2n1} A\ VG N , i.e., the lowest order that satisfies the inequality £-(1\1+1) E{lnI 2 ( 1\1 + 1 ) } A\VGN ~ M 2M E- E{lnI } AWGN. For Me == fix(x) - 1, using Stirling's formula for large M, In M! M In M - M, we see that the last term in the truncated expansion, e-1\1 E{lnI2A1 } A\ VG N ' is "-' exp (-x). For Q == 0.9, the fractional errors for different SNR's arising from the use of the asymptotic form for relatively low values of M are illustrated in Fig. 3. It is apparent from these results that the asymptotic form of the expansion is valid down into the moderate SNR regime, with SNR's down to "-'10 dB. In addition, for large effective SNR's, the errors are so small and fall off so precipitously that one can truncate the expansions at much lower orders than the maximum Me cutoff point described above. With these results in mind, we revisit the expression E{lsI 2Is+nl- 2 } and perform a Taylor series expansion in the ratio of the complex receiver noise amplitude random variable I"V

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-5.5,0.0,5.5.11.0,16.6,22.6,28.5,35.0,41 .9,49.9,59.4,72 .7,88.2] The power evaluation for each of the 22 directions is carried out based on the steering vector w(e )

= [1, exp(- jn sinCe )),... ,exp(- j(N -1)1t sin(e)) f

0

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and the estimated impulse response vector as follows

o

(6)

100

150 Time [sec]

200

250

Fig. 8: Estimated DoA when DTX is off using a moving average window of 21 bursts. The DoA calculated from the GPS information is the dotted line.

In this expression B is the number of bursts over which averaging is performed and ht,b denotes the estimated impulse response vector for delay f in burst b. The logarithm of the power over 3 taps is calculated for ke [1...9]. The direction and delay tap corresponding to the maximum power is then expressed as (Bo ,ko) =argee9.ke[I•...9] maxP(B.k) .

50

80 60

--- - Estimated DoA CalculatedDoA

40

(7 )

Calculating (Bo,ko) necessitates an exhaustive search over 9

c. CD

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delay taps and 22 directions. The value B, which determines the averaging period, can be varied. Simulation results presented in [2) have been obtained using B=2I. This value was selected because in discontinuous transmission (DTX) mode it coincides with the number of received bursts during a period of almost one SACCH frame period. The burst structure when DTX is off or on is illustrated below.

';;;' 0 C> c:

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Fig. 9: Estimated DoA using DTX on a block of impulse responses. It can be seen from Fig. 8 that the DoA algorithm occasionally estimates a direction quite different from that of the mobile. Inspection of the PAS at these points reveals that this function exhibits a deep fade in the direction of the mobile in combination with a component off that direction. This result indicates that averaging over 21 bursts is insufficient in the NonFH case. Applying a moving average window of 104 bursts (one

(b)

Fig. 7: Burst structure when (a) DTX is on and the SACCH is active and (b) where DTX is off.

The DoA algorithm proposed in [2] is based on a moving average over the bursts. An average over a block of 104 bursts

693

SACCH) instead, the estimated DoA is always close to the direction of the mobile station. Using DTX on the same route and averaging over a fixed block of impulse responses (21 out of 104) yields the result depicted in Fig. 9. The DoA estimation algorithm shows almost identical performances when DTX is on (see Fig. 9) and off. Results for test route 1 and 3 are reported in Fig. 10 and Fig. 11 respectively. In both cases, a moving average over 21 bursts is performed and DTX is off. The large angular deviation in Fig. 11 between GPS information and the DoA estimate in the range 350-500 s is due to poor GPS information and not a DoA estimation error.

period (0.48 s) is sufficient for the algorithm to track the temporal changes of the DoA.

VI. ACKNOWLEDGEMENTS The work presented is partly financed by the EU, ACTS programme as part of the TSUNAMI II project, Technology in Smart Antennas for Universal Advanced Mobile Infrastructure (Part 2). Nokia Tele Communications has also kindly cosponsored the work. Celwave and Analog Devices have sponsored the antenna array and DSP boards respectively. 80 - - - Estimated DoA : 60 ...... 'Calculated DoA , ..

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Fig. 11: Estimated DoA for route 3. The GPS position was lost on some parts ofthis route. and therefore differences occur.

400

Fig. 10: Estimated DoA on route 1. It can be seen that the algorithm looses track when the mobile is behind the antenna (see maps).

VII. REFERENCES [1]

It is observed from Fig. 10 that there is a good agreement between the estimated and calculated DoA's except when the true DoAs is close to -9Odeg. In this case the algorithm looses track since the mobile moves "behind" the antenna. The direct path is lost. The sudden large deviation between the estimated and calculated DoA arise because the magnitude of the angular response of the array (see Eq. 1.) takes large values for directions near +90 and -90 deg.

v.

...

[2J

[3] [4]

CONCLUSION AND FUTURE WORK

[5]

Preliminary results are presented which have been derived based on first measurements conducted in an urban area employing an adaptive 8 element antenna array testbed. A method for estimating small angular spreads is proposed. It is found that the angular spread is confined within 0-5deg. A correlation of -0.56 between the angular spread and slow fading is also observed. Based on the measurement data the performance of a DoA algorithm designed for GSM related systems is verified in a real interference-free environment. The results show that the algorithm is capable of tracking the DoA towards the mobile station. An averaging time of approximately one SACCH frame

[6] [7] [8J

[9]

[10]

694

Zetterberg. Per. "A comparison of two systems for down link communication with antenna arrays at the base station. 1995. submitted to IEEE Trans. on Vehicular Technology. Mogensen. Preben, P. Zetterberg, H. Dam. P. Le\h-Espensen, F. Frederiksen. " Algorithms and antenna array recommendations (Part I)", TSUNAMI 2 Technical report AC020/AUClA1.2IDRlP/005/al . 27 Oct. 1996. Mogensen. Preben, F. Frederiksen and H. Dam. "A DSP and dataacquisition architecture for an adaptive antenna array testbed". Proc. of DSP ' 96 Scandinavia. pp. 99-106. Copenhagen. June 18-19. 19%. Mogensen. Preben E.• Frank Frederiksen. Henrik Dam. Kim Olesen. and Sten Leth Larsen." TSUANMI 11 Stand-alone Testbed", Proc. of ACTS Mobile Summit, Granada, Spain, Nov. 1996, pp. 517-527. Eggers. Patrick. "Angular Dispersive Mobile Radio Environment Sensed by Highly Directive Basestation Antennas," Proc. Personal. Indoor and Mobile Radio Communications (PIMRC·95). September 1995. pp. 522526 . Krim, Hamid, M. Viberg, "T wo Decades of Array Signal Processing Research."IEEE Signal Processing Magazine, July 1996. pp. 67-94 . Bello. P.• "Characterization of Randomly Time-Variant Linear Channels," IEEE Trans. on Comm. Syst. , vol. CS-ll. no. 12, Dec. 1963. pp. 360-393. Zetterberg, Per. and Poul Leth-Espensen, "A Downlink Beam Steering Technique for GSMlDCS 1800IPCS 1900". IEEE Proc. of Personal. Indoor and Mobile Radio Communications . Taipei. Taiwan. Oct. 1996. pp. 535539 . Zetterberg, Per. Poul Leth Espensen, and Preben E. Mogensen, "Propagation Model. Direction of Arrival and Uplink Combining for Use in Mobile Communications", Proc. of ACTS Mobile Summit. Granada. Spain. Nov. 19%. pp. 500-509. Clarkson, 1.. "An Advanced Antenna System - Unplugged". Proc. of ACTS Mobile Summit. Granada, Spain, Nov. 1996. pp 302-307 .

Performance Evaluation of a Cellular Base Station Multibeam Antenna Yingjie Li, Member, IEEE, Martin J. Feuerstein, Member, IEEE, and Douglas O. Reudink

Abstract- Experimental test results are used to determine the performance that can be achieved from a multibeam antenna array, with fixed-beam azimuths, relative to a traditional dualdiversity three-sector antenna configuration. The performance tradeoffs between the hysterisis level, switching time, and gain improvement for a multibeam antenna are also examined. The multibeam antenna uses selection combining to switch the signals from the two strongest directional beams to the base station's main and diversity receivers. To assess the impact of beamwidth on overall system performance, the following two multibeam antennas were tested: a 12-beam 30° beamwidth array and a 24-beam 15° beamwidth array. Both multibeam antennas were field-tested in typical cellular base station sites located in heavy urban and light urban environments. Altogether, the system performance is evaluated by investigating three fundamental aspects of multibeam antenna behavior. First, the relative powers of the signals measured in each directional beam of the multibeam antenna are characterized. Then, beam separation statistics for the strongest two signals are examined. Gain improvements achievable with a multibeam antenna compared to the traditional sector configuration are determined in the second phase of the analysis. Results indicate that in excess of 5 dB of gain enhancement can be achieved with a 24-beam base station antenna in a cellular mobile radio environment. Finally, the effects of hysterisis level and switching time are characterized based on gain reductions relative to a reference case with no hysterisis and a 0.5-s switching decision time. Useful approximations are developed for the gain effects associated with varying hysterisis levels and switching times. The resulting design curves and empirical rules allow engineers to quantify multibeam antenna performance while making appropriate tradeoffs for parameter selection.

A

1.

INTRODUCTION

s THE 900-MHz cellular subscriber population continues to grow at a rapid pace, service providers are forced to find new methods of enhancing the coverage and capacity of their networks. One approach is to simply build additional, smaller cell sites to increase capacity or to fill coverage holes (i.e., cell splitting). Unfortunately, building new cellular base stations is a time-consuming, expensive process. In many high-

Manuscript received July 12, 1995~ revised October 27, 1995. This work was supported by U S WEST NewVector Group Inc. Y. Li was with U S WEST NewVector Group Inc., Bellevue. WA 98008 USA. She is now with Lucent Technologies Bell Laboratories, Whippany, NJ 07981 USA. M. Feuerstein was with U S WEST NewVector Group Inc., Bellevue. WA 98008 USA. He is now with the Radio Performance Group at Lucent Technologies Bell Laboratories, Whippany, NJ 07981 USA. D. Reudink was with U S WEST NewVector Group Inc., Bellevue, WA 98008 USA. He is now with MetaWave Communications Corporation, Redmond, WA 98052 USA. Publisher Item Identifier S 0018-9545(97)01323-6.

traffic areas, cell sizes are already about as small as can be accommodated given real-world handoff delays. Sectorization is another method of increasing capacity, but with associated tradeoffs in trunking efficiency. Once again, in many cellular markets, sectorization has been utilized ~o its practical limit. Most cellular service providers are choosing to adopt digital modulation techniques to cope with increasing traffic demands and to accommodate the drive to reduce per-subscriber infrastructure costs. However, an existing customer base of analog-only subscriber radios must still be serviced, especially during the transition period to a digital air interface. As analog radio channels are converted to digital service, the blocking rates on the remaining analog channels inevitably will increase. The situation may well worsen in the future because, at least in the United States, analog advanced mobile phone service (AMPS) cellular service is the only universal standard available for customer roaming from one market to another. Thus, the transition to digital cellular air interfaces may, for a time, actually exacerbate the capacity problems of an already strained analog system. In the 1.9-GHz frequency band, new personal communications service (peS) entrants are faced with the daunting task of building a complete wireless network from scratch. pes operators must quickly achieve coverage and capacity parity with today's cellular carriers in a frequency band where propagation path loss is significantly higher. To create such a network and to simultaneously minimize the number of required cell sites, system operators must take advantage of every possible improvement in link margin. For both 900-MHz and 1.9-GHz wireless systems, one economical approach to the problem of increasing capacity and coverage is to use multibeam adaptive base station antennas. By using adaptive control to keep a narrow beam pointed in the direction of each subscriber served by the cell, the effective gain and carrier-to-interference ratio can be dramatically improved compared to a typical sector configuration [1]-[5]. In addition, for established systems, such a technique can make use of existing cell sites, thus eliminating the costly step of deploying new sites. In this paper, the gain improvement of a low-complexity multibeam antenna system is investigated. The multibeam antenna uses 12 or 24 narrow beams, each with fixed pointing directions. Selection combining [6]-[8] is used to switch the strongest two directional beams for a given subscriber to the base station's main and diversity receivers. In the proposed switched-beam method, the complexity associated

Reprinted from IEEE Transactions on Vehicular Technology, Vol. 46, No.1, pp. 1-9, February 1997.

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with adaptively scanning the beam-pointing azimuth (e.g., by varying complex weights in a beam-forming network [1]-[5]) is avoided by switching between fixed-beam azimuths [6]. The beam-switching approach allows the multibeam antenna and switch matrix to be easily integrated with existing cell site receivers as an applique. Multibeam antenna performance is measured using experimental field tests from existing cellular sites located in heavy urban and light urban environments. The paper examines only reverse link improvements on the base station receive path; however, similar concepts can be used to provide improvements to the forward link base station transmit path as well [1], [3], [4]. Subsequent sections of the paper are organized as follows. Section II describes the experimental setup and the two multibeam antennas that were tested. In Section III, the field test environments and test methodologies are presented. The experimental results for both multibeam antennas are discussed in Section IV. Finally, Section V summarizes the key results and draws relevant conclusions . Acknowledgments are noted in Section VI.

II. EXPERIMENTAL SETUP The experimental test setup included a data-acquisition receiver system located at the cellular base station site in conjunction with an 8S0-MHz continuous wave transmitter placed in a roving test vehicle. Functional block diagrams of the test transmitter and receiver systems are shown in Fig. I. The base station data-acquisition system was developed to measure received signal strength simultaneously from each beam of the multibeam antenna array and also from each antenna of a dual-diversity three-sector configuration. The data-acquisition system was capable of digitally sampling up to 30 receive input ports at a sampling frequency of 1 kHz per port. The receive chain for each sampling port consisted of the following: a bandpass filter (for image frequency rejection) and a directional coupler (for injection of a calibration signal) feeding a measurement receiver with a 3D-kHz intermediate frequency bandwidth. The receive signal-strength indicator (RSSl) outputs from each of the 30 measurement receivers were fed to a 3D-channel 12-b analogto-digital (AID) conversion card. The AID card was located

696

in an MSDOS computer, which also contained a large hard disk for data storage. All field data from drive tests were stored in binary files as raw A/D samples; later post processing introduced the effects of calibration offsets. averaging, and other signal processing. The drive test vehicle contained a signal generator, power amplifier, and a mobile whip antenna. The test vehicle also included a global positioning system (GPS) receiver connected to a logging computer that recorded time versus position information for later analysis. During the course of the experimental work, two multibeam antenna arrays were field tested. The first multibeam antenna had 12 beams, each with approximately 30° beamwidth. The antenna physically consisted of three panels with four beams per panel. The three panels were oriented to provide nonoverlapping 120° azimuth coverage (i.e., the three panels in concert provided complete 360° coverage in azimuth). The actual beamwidths and antenna gains for the four beams on each panel were 35° (14.1 dBd), 30° (15.3 dBd), 30° (15.3 dBd), and 35° (14.1 dBd), with the variations due to inevitable beam broadening off the boresight of the panel. Front-to-back ratios for the beams ranged from 20-35 dB. The second multibeam antenna was a 24-beam array with approximately 15° beamwidth per beam. Again, the antenna physically consisted of three panels, except in this case, each panel had eight individual beams. The actual beamwidths for the eight beams on each panel were 23° (15.2 dBd), 18° (17.7 dBd), 15° (18.7 dBd), 14° (18.4 dBd), 14° (18.4 dBd), 15° (18.7 dBd), 18° (17.7 dBd), and 23° (15.2 dBd). Front-to-hack ratios for the beams ranged between 20-39 dB. Typical worstcase sidelobe levels for both the 12- and 24-beam arrays were down approximately 13-20 dB from the main peak of each beam. The reference antenna system for each of the tests was a traditional dual-diversity three-sector configuration. Each of the three sector faces consisted of two 92° beamwidth (11dBd gain) antennas separated by approximately 4.5 m for spatial diversity. During drive testing, received signals were simultaneously recorded from each of the six sector antennas (three sectors x two antennas/sector). III. DESCRIPTION OF TESTS To investigate the influence of the propagation environment on multibeam antenna performance, two locations were tested: a heavy urban site and a light urban site. The heavy urban cell site was located on the roof of a seven-story office building on the fringe of the downtown core of Seattle, Washington. The cell site was situated near a dense urban environment with a number of tall office buildings in the vicinity. Streets in the downtown core were lined with tall buildings, creating the typical urban canyon prevalent in many modern cities. A major commuting highway passed close to the cell site. The light urban cell site was located on the roof of an II-story office building in the central business district of Bellevue, Washington (a suburb of Seattle). The business district consisted of a cluster of approximately a dozen office buildings, ranging from 10 to 25 stories, interspersed with

commercial business and shopping areas. Locations further from the center of the business district were mixed with typical residential neighborhoods. The cell site was located close to the intersection of two main commuting highways. Four separate measurement campaigns were conducted: 12and 2~4-beam antennas in heavy and light urban environments. During each measurement campaign, the test vehicle was driven along selected drive routes that extended to the edge of the measurement system coverage range. Drive routes were selected to provide a thorough mix of speeds and conditions, as might be experienced by a representative cellular subscriber. Emphasis was placed on highway routes commonly used for commuting. Attempts were made to uniformly cover all azimuth directions around the cell sites. In this manner, an extensive database of measurements was collected for each of the antenna configurations.

IV. EXPERIMENTAL RESULTS To derive the results presented in this paper, the raw signal-strength samples were post processed as follows. First, redundant samples obtained when the measurement vehicle was stationary for extended periods of time (e.g., street lights, stop signs, etc.) were removed. Second, signal-strength readings near the noise floor of the measurement system were removed to eliminate the influence of limited receiver dynamic range. Third, the raw I-kHz samples were averaged over 500ms intervals to remove the effects of fast fading. The resulting local mean received power measurements were then used to provide estimates of the relative performance differences between the dual-diversity sector antennas compared to the multibeam arrays. A. Relative Power Differences and Beam Separations The received power, normalized to the strongest beam, versus rank, based on received power for the 24-beam antenna, is shown in Fig. 2. For comparison purposes, results for both Bellevue (light urban) and Seattle (heavy urban) are shown. Each curve in the figure represents an average of the differences for all the measurements in that environment. As expected, there is a significant difference in received power from the strongest to the weakest beam. For the light urban environment, the total range was 20 dB, while for the heavy urban, the range was 17 dB. As a point of reference, the worst-case sidelobe and backlobe levels for the antennas were approximately 13-20 dB down from the main beam. The reduction in signal range for the heavy urban compared to the light urban was due to the additional reflection and shadowing present in the more dense urban setting. In Fig. 3~ the cumulative distribution function (CDF) of the received power difference between the first and second strongest beams is shown for the 24-beam antenna. For the heavy urban case, 90% of the measurements had a received power difference between first and second strongest beams of 7.5 dB or less. For the light urban case, the difference was 10 dB or less for 90% probability. As expected from Fig. 2, the difference was smaller for the heavy urban, compared to

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propagation environment. From the similar slopes of all the curves, one can conclude that the sensitivity of gain reduction to changes in the hysterisis level is independent of the environment type and the switching time ; hence , as a rule of thumb, each decibel increase in hysteris is level results in approximately a l/lO-dB gain reduction . which hold s true for the 24-beam antenna. Another meaningful observation is that the difference in gain reduction between the two environmental types is only significant for switching times greater than about I s. In other words, if the switching time is held to less than approximately 1 s, there is little additional degradation when moving from the more benign light urban environment to the harsher heavy urban setting. As a final comparison, in Fig. 13 the 12- and 24-beam antennas are compared by plotting gain reduction versus switching time. Based on the slopes of the two curves , the performance difference s between the two antennas can be noted . As might be expected, the gain reduction for the 24beam antenna is a stronger function of switching time when compared to the 12-beam antenna . The larger reduction in performance for the 24-beam antenna is expected given the 15° beamwidth relative to the 30° beamw idth of the 12-beam antenna. One would expect to have to switch faster for the narrower beamwidth array as the mobile move s. V . C ONCL USION S

A low-complexity multibeam antenna, based on switching between beams with fixed boresight orientations, has been evaluated in this paper. Switching between fixed beams dramatically reduces the complexity of the associated signal processing hardware and also allows the antenna system to be readily integrated with existing cell site receivers. The reverse link performance of both 12- and 24-beam arrays has been field tested in light and heavy urban propagation environments.

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Based on an extensive experimental measurement database, three fundamental aspects of the performance of cellular base station multibeam antennas have been investigated : I) relative power differences and beam separations for the individual directional beams; 2) gain improvements achievable relative to a traditional three-sector configuration; and 3) the influence of switching time and hysterisis level on the effective gain improvement. Based on experimental observations in heavy urban environments, the relative power difference between the strongest and weakest beams is approximately 3 dB lower than for a light urban environment. Such an observation is consistent with the extra multipath reflection and shadowing that are prevalent in a dense urban setting. This interpretation is also borne out by examining the received power differences between the first and second strongest beams, where the difference is significantly less for the heavy urban than for the light urban case. A calculation of the probability of beam separation shows that, with a high probability, the first and second strongest beams will be located in physically adjacent antenna beams. In fact, in over 80% of the measurements for the light urban environment, the first and second strongest beams were either adjacent or separated by only one beam. Based on the calculated correlation coefficients, the strongest two signals are more highly correlated in the heavy urban environment , where building shadowing effects and multipath are prevalent. The gain improvement obtainable from a multibeam antenna relative to a dual-diversity sector configuration has been examined. Experimental results in heavy and light urban environments show that linear average gain improvements of approximately 2.9 and 5.2 dB can be achieved from 12and 24-beam antennas, respectively. CDF's and probability density functions of gain relative to a dual-diversity sector configuration have been presented. Based on goodness-of-fit measures, the probability density functions for gain improve-

ment in decibels are well modeled as Gaussian. The design curves presented in this paper can be used to estimate the coverage improvement that can be achieved with 12- and 24-beam antennas in representative urban environments. Experimental results have been used to characterize the tradeoff between hysterisis level, switching time. and gain for a multibeam base station antenna. As expected, increasing hysterisis level results in less ping-ponging between antennas ; however, the effective gain of the multibeam antenna is correspondingly reduced. As a rule of thumb for the 24-beam antenna, each decibel increase in hysterisis level results in a I/lO-dB reduction in effective gain for the multibeam antenna, compared to an ideal no-hysterisis case, due to the suboptimum switching operation associated with hysterisis [8). Increasing the switching decision time is one method of reducing the burden on the computer that controls antenna assignment; however, increasing the switching time also reduces the effective gain of the multibeam antenna. As an approximation for the 24-beam antenna, each I-s increase in switching time results in a 113-dB reduction in effective gain, compared to a switching time of 0.5 s, for the multibeam antenna in the heavy urban environment. Once again, the effective gain reduction is due to the fact that the receiver remains connected for longer periods of time to antenna beams that do not contain the strongest signal. As expected. due to the more complex propagation environment, the gain of the multibeam antenna is a stronger function of switching time in the heavy urban environment than in the light urban case. Even though the probability of switching to a new beam is higher in the dense urban environment, if the switching is accomplished rapidly enough. then the effective gain will not be markedly reduced. Results have shown that the difference in gain performance between the light and heavy urban environments is only significant for switching times greater than approximately I s. Because of the smaller beamwidth , the gain of the 24-beam antenna is more sensitive to the switching time than the 12-beam antenna. As a final point, several relevant topics, other than gain enhancement effects, should be mentioned when assessing the overall performance of multibeam antennas. The ability of multibeam antennas to combat fast fading through angular diversity influences the effective coverage area of the cell [2], [7], [8]. The correlation of fast fading on the two strongest beams from a multibeam antenna should be contrasted against that of a typical spatial diversity antenna pair, thereby determining the relative effect of the angular diversity benefit [9]. For capacity-constrained and interferencelimited circumstances, the issues of interference reduction [i.e., carrier-to-interference (CII) improvement) and, hence, frequency reuse efficiency are significant concerns [I], [2], [4). These topics represent important aspects of multibeam antenna performance that are not within the scope of this paper. ACKNOWLEDGMENT

The authors would like to thank D. Jones, P. Perini, M. Harrison, D. Ellingson, and J. O'Connor for performing the experimental tests described in this paper.

702

REFERENCES

[1] S. C. Swales, M. A. Beach, DJ. Edwards,and 1. P. McGeehan, "The performance enhancement of multibeam adaptive base-station antennas for cellular land mobile radio systems," IEEE Trans. Veh. Techno!., vol. 39,pp.56-67,Feb.1990. [2] S. P. Stapleton and G. S. Quon, "A cellular base station phased array antenna system," in IEEE Veh. Technol. Conf, 1993, pp. 93-96. [3] M. Goldburg and R. H. Roy, "The impacts of SDMA (spatial division multiple access) on pes system design," in Int. Con]. Unio. Personal Comm., 1994, pp. 242-246. [4] P. Zetterberg and B. Ottersten, "The spectrum efficiency of a basestation antenna array system for spatially selective transmission," in IEEE Veh. Technol. Conf, 1994, pp. 1517-1521. [5] G. Y. Delisle and A. T. Denidni, "Experimental investigations of phased array characteristics for pes applications," in Int. Conf Universal Personal Comm., 1993, pp. 49-53. [6] T. Aubrey and P. White," "A comparison of switched pattern diversity antennas," in IEEE Veh. Technol. Conj., 1993, pp. 89-92. [7] D. G. Brennan, "Linear diversity combining techniques," Proc. IRE, vol. 47, pp. 1075-1102, June 1959. [8] W. C. Jakes, Ed., Microwave Mobile Communications. New York: Wiley, 1974. [9] 1. R. Pierce and S. Stein, "Multiple diversity with nonindependent fading," Proc. IRE, vol. 48, pp. 89-104, 1960.

703

Space division multiple access (SOMA) field trials. Part 1: Tracking and BER performance G.Tsoulos J.McGeehan M.Beach

Indexing terms: Digitalbeamforming, Adaptive antennas, SDMA

suppress interference [3--6], and beam steering to focus energy towards desired users [7-10]. By exploiting the spatial domain via an adaptive antenna, the operational benefits to the network operator can be summarised as follows [5]: • Capacity enhancement • Coverage extension ('smart' link budget balancing) • Ability to support value added services (e.g. high data rates, user location, etc.) • Increased immunity to 'near-far' problems • More efficient handover • Ability to support hierarchical cell structures

Abstract: An adaptive antenna testbed for mobile communication applications is briefly described and results from field trials presented. The goal is to provide an experimental demonstration of both transmit and receive digital beamforming supporting SOMA user access. Trials are presented for a typical urban environment with different combinations of user positions and the ability of the employed adaptive algorithm to establish the link and track the channels is investigated alongside the link BER performance. The tracking performance of the adaptive algorithm used for SOMA is also tested for an indoor environment against maximum ratio combining and a fixed grid of beams.

1

2

SDMA testbed

The experimental testbed that was used in the field trials was that developed under the RACE TSUNAMI project [11]. The DECf radio standard was selected as the operational wireless bearer since it could be readily integrated with the adaptive antenna platform and, furthermore, DECT can be operated in an isolated radio cell mode, thus allowing networking aspects (e.g. handover) to be addressed at a later phase. Some key characteristics of the OECT system [12] are: • Frequency band: 1880-1900MHz • Number of carriers: 10 • Carrier spacing: 1.728MHz • Peak transmit power: 250mW • Carrier multiplexing: TOMA - 24 slots per frame • Frame length: lOms • TOD with two slots (up-down links) on the same carrier • Gross bit rate: 1.152 Mbit/s • Net channel rates: 32kbiUs traffic (B-field) and 6.4kbit/s control/signalling (A-field), per slot The testbed hardware consisted of an eight-channel system employing a patch antenna array which could be deployed in various configurations and eight independent linear up and down conversion chains which transform the signals to quadrature baseband (see Fig. 1). The baseband system provides two independent bidirectional wideband beamfonner outputs to the DECT radio system. The digital beamforming devices were two DBFII08 chips [13], each providing 32 million complex operations per second processing rate and 8 bit complex data, 11 bit complex weighting coefficients. The two DECT radios were modified such that they only operated on a single fixed frequency and timeslot

Introduction

Over the last few years the demand for service provision via the wireless communication bearer has risen beyond all expectations. The extraordinary fact that some half a billion subscribers to mobile networks are predicted by the year 2000 worldwide [1], introduces the most demanding technological challenge: the need to increase the spectrum efficiency of wireless networks. Filtering in the space domain can separate spectrally and temporally overlapping signals from multiple mobile units, and hence the spatial dimension can be exploited as a hybrid multiple access technique complementing FDMA, TOMA and COMA. One such approach is usually referred to as space division multiple access (SDMA) and enables multiple users within the same radio cell to be accommodated on the same frequency and time slot. Realisation of this technique is accomplished using an adaptive antenna array which is effectively an antenna system capable of modifying its temporal, spectral and spatial response by means of amplitude and phase weighting and internal feedback control. Numerous approaches using adaptive antennas have been considered to exploit the spatial domain, for example null steering to isolate cochannel users [2], optimum combining to reduce multipath fading and ©IEE,I998 lEE Proceedings online no. 19981782 Paper first received 29th April and in revised form 14th November 1997 The authors are with the Centre for Communications Research, University of Bristol, Merchant Venturers Building, Woodland Road, BristolBS8 1UB, UK

Reprinted with permission fcom lEE Proceedings of Radar, Sonar, and Navigation, G. Tsoulos, J. McGeehan, and M. Beach, "Space Division Multiple Access (SDMA) Field Trials Part 1: Tracking and BER Performance," Vol. 145, No.1, pp. 73-78, February 1998. © 1998 by Institute of Electrical Engineers.

704

,'antenna array

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arrival associated with the beamformer. The angle of arrival associated with the steering vector, which produces the maximum inner product with the calculated weight vector, is taken to be the notional angle of arrival of the maximum ratio combiner when an angular comparison is required. After demodulation the signal is compared with a known reference sequence to calculate the bit per symbol errors that have occurred. The reference sequence is a 320 bit sequence in the Bfield of the data field of the DECT burst.

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=

w{fw Al. 0 (1) where ( )8 represents complex conjugate transpose. Note that the Tx weights are always the complex conjugate of the Rx weights, i.e. adaptive retransmission is used. During the call setup procedure for establishing the DECT links to the two handsets for SDMA operation the initial weight vectors for the two beamformers are loaded for both transmit and receive modes and form two initial beams over which the two users communicate with the base station. Hence, during acquisition the azimuthal position of each user is aligned with the orientation of each initial beam . The MUSIC algorithm is waiting for the appearance of the first strong signal, which it will assume is user A (synchronisation must be first for user A). The tracking algorithm assigns this signal to track 1, and then performs the same steps for user B. The second signal is assigned to track 2 by the tracking algorithm. After the beamformer has acquired both users, two orthogonal beams are computed, and are set up in the two DBFs. Since the adaptive algorithms used to perform optimisation operations will be required to cope with nonstationary environments, one issue which needs attention is the required processing update rate to perform the adaptive beamforming operation. This depends both on the algorithm used and the rate of change of the environment. An indoor picocellular enviroriment is characterised by slowly moving users and high scattering. The fading envelopes at the different antenna elements will be uncorrelated if they are spaced at least half a wavelength apart at the carrier frequency and the low mobile speed results in a relatively long coherence time for the channel. Thus , it is feasible for an adaptive antenna to track the fading envelopes of the signals at each of the antenna elements. Such a scheme suppresses unwanted signals and also minimises the effects of fading , and requires an algorithm update rate which is fast compared to the channel coherence time. On the other hand, a macrocellular environment is characterised by high mobile speeds and very low scattering at the basestation. Since in such environments it is difficult to obtain spatial diversity at the basestation unless large arrays are used (more than lOA. interelement spacing (17)), if a small array is used (e.g. eight elements with 0.5A. spacing), the fading will be more or less constant across the array. Furthermore, if the adaptive processing tracks the mobile signal while simultaneously forming nulls in the direction of cochannel interference sources, then in this case the required update rate is not governed by the coherence time of the channel but rather by the angular velocities of the signal sources. 3

3. 1 Outdoor urban environment

The results are from the trials performed in Bristol during early 1996. The area is a typical outdoor urban environment around the engineering building of the University of Bristol (Fig. 2). The base station antenna is a "}J2 linear deployment at a height of approximately 30 m above the ground.

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706

3.1.1 SOMA for two stati onary users: User A is at - 300 (or point 2), while user B is at - -3 50 (or point 1) and both users are stationary. The beamformer was able to spatially separate and support the users throughout the whole experiment with a HER which was much less than 10-3. Fig. 3a is a plot of the cumulative distribut ion function (CDF) of the power from each antenna element and the output of the beamformer, for the period of the experiment. More than IOdB improvement in power has been achieved with the adaptive antenn a against the power achieved over a single element. From Fig. 3a the antenn a element with the best CDF curve is selected and in Fig. 3b the power received from that element is plotted along with the power at the output of the beamforme r. 60

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Reprinted with permission from lEE Proceedings on Radar, Sonar and Navigation, G. Tsoulos, J. McGeehan, and M. Beach, "Space Division Multiple Access (SOMA) Field Trials Part 2: Calibration and Linearity Issues" Vol. 145, No.1, pp. 79-84, February 1998. © 1998 by Institute of Electrical Engineers.

710

demanding. Here the quality of the calibration will be evident from the noise floor of the MUSIC spectrum, which is an indicator of the null depth that is achieved in the field.

weight controller. An alternative approach is calibrating only the steering vector of the weight controller. This approach is based on the observation that the only difference between a calibrated and an uncalibrated output, is that for the former the steering vector p must be multiplied by the inverse of the calibration matrix (C)

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To obtain an approximation of the calibration accuracy required, a simple model of two antenna elements where the second element introduces amplitude and phase mismatches, can be considered. Fig. I shows the variation of the residual cancellation error due to the amplitude and phase mismatch. Ideally, the output should be zero (i.e. a perfect null steered towards an interferer) which occurs when there is no amplitude or phase mismatch. A maximum phase mismatch of I ° and amplitude mismatch of 0.1 dB must be maintained if a -30dB null depth level [3] is required to secure sufficient interference cancellation for mobile communication applications.

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As outlined in Fig. 2, calibration of the receivers in the RACE TSUNAMI testbed was implemented offline (not continuously), using an external signal generator to produce a carrier tone which was split eight ways and directly injected into the multiple receiver ports. Using this approach the tone need not be phase locked to the receiver local oscillators, since it is only the relative phase and amplitude of the signals in each channel that needs to be determined, Also, the amplitude and phase relationship of the 1:8 power splitter was calibrated in the laboratory before the system level results could be obtained and it is assumed that these parameters do not vary significantly with time. Calibration was performed before the trials and at regular intervals during the trials to account for thermal drift in the receiver and transmitter subsystems, as well as connector wear and cable rerouting necessary for the numerous field deployments. In summary, this process was achieved by measuring the transfer functions of all eight paths and applying a (complex) correction coeffi-

711

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Fig.8 Rx calibration offset as function of temperature a Amplitude mismatch b Phase mismatch after calibratio n Mean values are plo tted. Maximum and minimum values for each element are indicated by bars separated by dashed lines

Fig.6 Mismatch for differ ent gain settings and for each antenna element a Amplitude b Phase Mean values are plotted . Maximum and minimum values for each element are indicated by bars separated by dashed lines

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day 3

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-100

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"C :>

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100

0.04

(a)-(9)

t

:

:

-100

-50

~ .

l

I

:

:

:

:

o

50

100

I(LSBs)

Fig.9 Example of vector diagram before and after calibration and with oldcalibration factors for the sixth antenna element

is limited within a window of 0.057dB and 0.58°, respectively, (Figs. 7 and 8). This level of performance can be translated to an approximate cancellation level of -35dB from Fig. 1. Further, Fig. 9 shows the I, Q diagram produced from the unmodulated signals of the sixth antenna element. The offset ellipse is the uncalibrated output, the inner noisy circle is the calibrated output but with calibration factors calculated in a previous day, and the outer centred to zero circle is the calibrated output. With the old calibration factors the ellipse is corrected and centred to zero, but only after the new calibration the system appears to have reached

-0.4 Ll--72--3~;"";""-4~-~5'---~6--~-~-7 8 b element

Fig.7 Rx calibration as function of temperature a Amplitude mismatch after calibration b Phase mismatch after calibration

After calibration, the mean amplitude and phase variation is limited to 0.02dB and 0.08° across the array, while the worst case dynamic variation for the elements

713

an acceptable level of purity. The only problem that was noticed during the experiments was that although all the circles should have the same radius, there are small deviations as it can be seen from Fig. 9. This is because the array elements can still introduce small imbalances which the calibration loop employed in this testbed cannot correct. 4

Linearity issues for adaptive antennas

The spurious-free dynamic range of the transceivers also plays an important role in the performance of the adaptive antenna and it is usually specified in terms of the maximum level of intermodulation products (IMP) that can be tolerated. The creation of the IMPs is a complex nonlinear process. For two signals at frequencies!1 and!2 present at the input to a device or system, the output frequency spectrum can be represented as the sum of discrete frequencies

E E A(i,j) . (i· /1 ± j · /2) 00

spectrum ~

00

(8)

i=1 j=1

where the amplitude of each intermodulation product is given by A(i, J). The majority of the IMPs are at frequencies outside the bandwidth of the desired user. Especially when a signal without constant envelope passes through nonlinear devices, the signal is spread, which effectively means that adjacent channel interference and eventually capacity reduction is caused. These effects will be even worse when multiple carrier base station architectures are employed alongside adaptive antenna signal processing to support handover in the frequency domain whenever users can not be maintained on SDMA mode. The use of digital baseband beam forming techniques poses high linearity demands on both the RF/IF upand down-conversion chain. This is because the weights of the beam former are carefully calculated and constructed at digital baseband and distortions in the up or down conversion chains can alter the antenna beam pattern [1]. Some of the fixed mismatches between channels can be calibrated out using calibration techniques, as already discussed, but nonlinearity effects in the transceiver chain cause intermodulation distortion which cannot be calibrated out in any practical way. In [4] an example for the impact of inband intermodulation distortion on sidelobe levels, null depth, main beam width and change in null direction, was given for an eight-element linear array with Dolph-Tchebyscheff weighting. It was shown that as the level of nonlinearities increases, the sidelobe level also increases and there is considerable shift of the nulls. These changes have serious system implications as the degree of interference suppression will be severelyimpaired. The intermodulation products will increase the level of spurious radiation and this will ultimately reduce the performance of the adaptive antenna. Generally, there are two solutions to the problem • use of backed-ofTclass-A amplifiers. This solution is not power efficient, and furthermore the required levels of IMD for such systems cannot be easily achieved. • use linearisation techniques [5, 6]. RF power amplifier linearisation is a classic problem to wireless system designers. The use of feedback or feedforward from a nonlinear system to provide a correction signal is a well established technique for

714

improving the linearity of amplifiers. The feedforward linearisation technique [5] was considered to be the most suitable approach for adaptive antenna applications mainly because of the high degree of linearity attainable, broad frequency bandwidth and unconditional stability. The feedforward amplifier operates by comparing the distorted main amplifier output-with an undistorted reference signal; the error signal generated is suitably combined with the main amplifier output such that the distortions are cancelled. Eight amplifiers based on this architecture were built for the TSUNAMI testbed system. Each module includes an IQ up converter and fully adaptive control (using a TMS320C50 DSP card) of the amplitude and phase control in the main amplifier signal path. When the signal passes through the main amplifier, the nonlinearity of the amplifier generates IMPs at about -27 dBc while at the output of the operating feedforward amplifier the IMPs are at about -65dBc. The use of linear power amplifier technology will also support the deployment of multiple carrier base station architectures alongside adaptive antennas, with capability of supporting IMD levels better than -80 dBc using multiple adaptive loops. This is important, as mentioned, since the SDMA operation cannot always be maintained owing to the spatial location of the users, thus handover to the frequency domain will be necessary. desired user

interferers

! !!! !

8

!

!

4

m 'C

8c

0

!

~ -4

'C

-8

-60

Fig. 10

-40

-20

0

20

40

60

angle, deg Difference of measured radiation patterns with and without lin-

ear power amplifiers

In Fig. 10 the difference of the measured radiation patterns with and without power amplifier linearisation is shown. The environment where these measurements were performed was an open field test area. The testbed transmits with a radiation pattern which is the response to a simulated scenario where the desired user is approximately at 32° and there are interfering users at -3, -32, -37, -43, -45, and -50°. Power measurements were performed every 5° along the circumference of a circle with radius -20m which had the adaptive antenna array at its centre. Without linear power amplifiers the sidelobe level is generally increased and in particular for the nulls there is a decrease which ranges from 5 to 12dB. 5

Discussion

In the pursuit for schemesthat will efficiently utilise the spectrum, attention has turned into spatial filtering methods using advanced antenna techniques, adaptive or smart antennas. An experimental demonstration of

both transmit and receive beamforming supporting SDMA user access was performed in Part 1 of this paper [7] where it was demonstrated that it is possible with the adaptive antenna system to establish the required links and track the channels with an acceptable BER performance (better than 10-3) , for a variety of operational scenarios. In Part 2, some implementation issues critical to the performance of an adaptive antenna system have been investigated. Results showed that the amplitude and phase distortion of the transceiver paths vary over time and for different gain settings and indicated the need for regular calibration. After calibration the amplitude and phase errors were reduced to 0.057dB and 0.58°, respectively. The last values can theoretically give an approximate cancellation level of -35dB. Finally, the sensitivity of the system to the linearity of the transmit power amplifiers was discussed. It was shown that the produced radiation patterns are affected from the intermodulation distortion introduced by the power amplifiers and that this problem can be combated with the use of linear power amplifiers. To further develop adaptive antenna technologies for third-generation systems, a follow-on project ACTS TSUNAMI II is under way. Here, within the family of the GSM derivatives, it was decided to use the DCS 1800 standard because arguably it is closer to third generation than GSM, but most importantly, it utilises very similar frequency bands to UMTS/ IMT2000 proposals. The TSUNAMI II field trial will exercise the adaptive antenna system to fully identify the performance of the adaptive antenna relative to the performance of the existing trisectored DCS 1800 base

stations. Of particular interest are parameters which impact on cell sizing, fading protection and, most importantly, spectral efficiency gains using adaptive antenna technology. These will be assessed both analytically and via field trial experimentation. 6

Acknowledgments

The authors thank the CEC for funding the RACE TSUNAMI project and the partners of RACE TSUNAMI for their contributions to this activity. Special thanks are due to R. Arnott of ERA Technology and R. Wilkinson and C. Simmonds of Bristol University. Finally, the authors thank the reviewers for their useful comments. 7

2

4

5 6

7

715

References LANGSTON, J., SHASHIKANT, S., HINNMAN, K., KEISNER, K., and GARCIA, D., Design definition for a digital beamfonning processor, Rome Air Development Centre, NY, 1988 SHAN, T., WAX, M., and KAILATH, T.: 'On spatial smoothing for direction-of-arrival estimation of coherent signals', IEEE Trans., 1985, ~-33, pp. 806-811 TSUNAMI project final report, R2108IERAlWP1.3IMRIPI0961 b2, 1996 XUE, H., BEACH, M., and MCGEEHAN, J.: 'Non linearity effects on adaptive antennas'. April Proceedings of the 9th international conference on Antennas and propagation, ICAP'95, Eindhoven, Netherlands, 1995, pp. 352-355 KENNINGTON, P., and BENNET, D.: 'Linear distortion correction using a feed-forward system', IEEE Trans. Veh. Techno/., 1996,45, pp. 74-81 TSUNAMI, I., Baseline technology, R2108/ERAlWP5.IIDS/l1 0221b1, 1994 TSOULOS, G., McGEEHAN, J., and BEACH, M.: 'Space division multiple access (SOMA) field trials. Part 1: Tracking and BER performance', lEE Proc.-Radar Sonar Navig., 1998, 145, (I), pp. 73-78

Chapter 6 Applications and Planning Issues

T

HIS book concludes with six papers that deal with more general issues related to adaptive antennas including specific applications for user location, indoor wireless high data rate networks, planning issues, and novel techniques that have attracted a great deal of attention recently, such as space-time processing and multiple input, multiple output systems, that seem promising to open new directions for this technology in the future. The first paper demonstrates how very high data rates (in excess of 1 Gbps) can be achieved in an indoor environment with the use of antenna arrays at the transmitter and receiver ends. The work from Stanford University studies the performance of linear and circular base station antenna arrays with different topologies, angular spread, and number of elements in order to extend the coverage of a net-

work. The next work considers the use of adaptive antennas in order to solve resource planning problems. In particular, improvement of the reuse distance is considered in conjunction with the fractional loading technique in the context of improving spectrum efficiency and reducing the effort for network planning. A recent mandate by the Federal Communications Commission that requires all cellular communications service providers to provide user location services for emergency calls has triggered much of the work in this area. Caffery and Stuber consider the application of time and angle of arrival methods to achieve this goal in CDMA systems. Finally, the last two papers of the book present work on multiple input, multiple output systems and processing methods to achieve high data rate communication.

717

High Data Rate Indoor Wireless Communications Using Antenna Arrays Michael J. Gans, Reinaldo A. Valenzuela, Jack H. Winters, and Manny J. Carloni

AT&T Bell Laboratories Holmdel, New Jersey 07733 USA Abstract

typically 60 dB (relative to 1 meter, averaged over the multipath fading), while the nns delay spread is typically on the order of 100 ns (2,3]. This rms delay spread limits the maximum data rate to about 1 Mbps. Current proposals consider equalization or multicarrier processing to increase this data rate to 20 Mbps, but the circuitry is near the complexity limit for an economical system and maintaining reasonable outage probability with a 60 dB propagation loss (relative to 1 meter) may be difficult to achieve. Here we consider the use of phased arrays (tested in our experiment by using directive antennas) to increase the power margin and decrease the delay spread of the signal at the receiver, thereby permitting data rates in excess of 1 Gbps. Note that if the multipath in a building generated signals at the receiver that were uniformly distributed in power and delay spread with respect to angle-of-arrival, antenna arrays would not be useful. : However, results using the propagation-prediction techniques of [1] for in-building propagation over an entire floor of the Crawford Hill Building show that this is not the case. In particular, our results show that arrays at the transmitter and receiver with 25 0 beamwidths can isolate one ray, with high probability, and thereby achieve nearly the full gain of the antennas and eliminate delay spread. To support this conclusion, we present experimental results for 622 Mbps at 19 GHz from several locations within the Crawford Hill building, using manually-scanned directive (15 0 beamwidth) hom antennas. We have investigated ways of economically fabricating antenna arrays with these beamwidths, which, based on our results, would make entire floor coverage at high data rates economically feasible. In Section 2 we discuss the data rate limitations due to power margin and delay spread. The effect of antenna arrays is studied in Section 3, using both propagationprediction results and experimental data.

In this paper, we consider the feasibility of indoor wireless communications at very high data rates (up to multi-Gbps). In particular we wish to use one base station to cover the entire floor of an office building, which may have in excess of 60 dB propagation loss relative to 1 meter. This feasibility depends on two factors: received signal power margin and delay spread. Based on results using the propagationprediction techniques of [1], supported by experimental results up to 622 Mbps, we conclude that neither multicarriers, equalization, nor antenna arrays with less than 1600 elements at one end of a communication link are economical methods for increasing the data rate substantially above 10 to 20 Mbps for multiple room indoor wireless coverage. However, based on the propagation-prediction techniques of [1] and verified by our experimental measurements using directive antennas (15 0 beamwidth) at both ends of a link between the center of the Crawford Hill building to an end laboratory, we have shown that high-speed ubiquitous communication is feasible. Using antenna arrays with 50 to 200 elements at both the transmitter and receiver, we expect to obtain entire floor coverage at data rates in excess of 1 Gbps.

1. Introduction In this paper, we consider the feasibility of indoor wireless communications at very high data rates (up to multi-Gbps). In particular we wish to use one base station to cover an entire floor of an office building. This feasibility depends on two factors: received signal power margin and delay spread. Previous measurements have shown that the maximum propagation loss for a single floor in several office buildings, including the Crawford Hill building, is

Reprinted from 6th International Symposium on Personal, Indoor and Mobile Radio Communications, pp. 1040-1046, 1995.

719

2. Data Rate Limitations Second Floor Averaged Propagation Loss 4 feet into Offices, Crawford Hill

Consider first the received signal power margin. The margin is given by

(Dm MMSUred by OL RSR & RAV at18 GHz.2I10194)

10

r----------------

(1)

o Eb

where N

o

is the energy per bit to noise density ratio at

,,

the receiver and -

is the ratio required to achieve a

,

e,

P rec -=-No N

-0

.,.,

(2)

where Prec is the received signal power given by [6, p. 490]

,

\

\

\ ----- ----,- --------------------------------

.E -30

...o

Fixed at -50.1' dB beyond 30 met....

\

m

given bit error rate (BER). Now,

,,

, --------------,----------------- -------------------------

-20

req

Rcvd Power dBm Propaglltlon Loss ----

---------'""~--------------- -34.34Lgt(r)up to 30 meters range

"

Eb

No

-10

,,

-

-~--------------------------------

""

""

"",

,,

-40

------------~------------------------

-50

---------------\\-;'\~,,."'i--

,

'~\\

"

-60

-----------------------------------------

(3) -70 - - - - - - - - - - - - -

1

10

100

Range in Meters

N is the noise power given by N =kTB·NF

--.J

(4)

Figure 1 and Pa is the power out of the transmit amplifier, Lcr is the loss of cable to the transmit antenna, G, is the transmit antenna gain, A is the wavelength, Lp is the propagation loss relative to 1 meter free space, Gr is the receive antenna gain, L CR is the loss of cable from the receive antenna, k is Boltzmann's constant (l.38xlO-2o mWIH:zJ°K), T is the system noise temperature, B is the bandwidth, and NF is the noise figure of the receiver. In (4), we assume that the signal bandwidth is equal to the data rate. Let us consider typical values for the following parameters: Pa=23 dBm, LCT=l dB, LcR=l dB, T 290 OK, and NF - 6 dB. To operate with data rates up to Gbps, the carrier frequency must be in the range of 19 GHz (or higher), or A - 3xl08/1.9xl010 m. Now, the prediction techniques of [1], along with experimental measurements [1], have shown that in several buildings, including Crawford Hill (which has sheetrock interior walls), with a suitably-placed base station, the maximum propagation loss, Lp , on one floor is equal to 60 dB. In particular, Figure 1 shows propagation loss

Propagation loss versus range for the Crawford Hill building.

measurements of the Crawford Hill building made by R. S. Roman, O. Landron, and R. A. Valenzuela at 18 GHz using omnidirectional antennas. These results show the loss, averaged over the multipath fading by moving the antenna over an area with a radius of several wavelengths, for transmission from the center of the main hallway to 4 feet inside each room (i.e., no line-of-sight). The maximum loss is seen to be less than 60 dB. Thus, with a 60 dB loss, from (2)

E

Nb

o

= 71 dB - IOloglO(B) + Gt + Gr

(5)

where Gt and Gr are in dB. If we assume that a BER of

10~8 is ~Uire~, IWith coherent detection of binary phase

shift keying

0-

720

-

N

req

12 dB [6, p. 380], and the margin

receive antenna gain is cancelled by the loss of power from the signal outside the beamwidth. Similarly, transmit antenna gain would not increase the margin. However, results using the propagation-prediction techniques of [1], for transmission to users up to 4 rooms away, show that even though the signal received by a user can arrive via hundreds of rays at different angle-ofarrivals, about 50% of the signal energy, i.e., total power from all rays, which is approximated by the multipathaveraging in the measurements mentioned before, is usually concentrated in one ray. Thus, directive antennas should provide an increase in multipath-averaged received signal power over isotropic antermas within 3 dB of their directive antenna gain. For an antenna with a beamwidth in azimuth and elevation (assumed equal) of e in degrees, the gain for small eis given by (see also [7])

is given by (from (1) and (5)) Margin

= 59 dB

- lOloglO(B) + G, + Gr.

(6)

Thus, with isotropic antennas (Gt=Gr=O dB), the maximum data rate is about 800 kbps. (Note that to increase the data rate limitation due to power margin, coding could be used to permit a higher raw BER. For example, with BER = 10-2 , and coding to reduce this to 10- 8 , an additional 4.3 dB margin can be obtained. Thus, from (6), the maximum data rate would be 2 Mbps.) This is the maximum data rate considering the loss averaged over the multipath fading, however. Multiple paths from various directions produce fades in signal strength which vary with distance at wavelength intervals. In practice, additional margin (with a correspondingly lower data rate) must be considered because of this fading. For example, with a single receive antenna and Rayleigh fading (note, however, that our results in Section 3 indicate that the fading is Rician in our building), 10 dB of additional margin is required for 90% availability and 20 dB is required for 99% availability, which lowers the data rate limit to 80 and 8 kbps, respectively, for full, single-floor, building coverage. Of course, at millimeter wavelengths a user would only have to move the receive antenna by a fraction of an inch to move out of a fade, and therefore might not need this margin. However, even though the user antenna may be stationary, the environment changes, which makes this method not practical, and may mean that even a 99% availability would be unacceptable due to frequent, but short, outages. Diversity can be used to greatly reduce this additional margin, though, with two receive antennas cutting the margin required for a given availability in half (in dB). Next consider delay spread. For many buildings, the rms delay spread is on the order of 30 to 250 ns [2,3]. Since without equalization, a BER of less than 10-8 requires an rms delay spread less than about 10% of the symbol period, this rrns delay spread results in a data rate limitation of about 1 Mbps. Note that this is similar to the limitation due to power margin (without the multipath fade margin), and thus both factors need to be reduced to operate at very high data rates.

G :::: 10loglO [

J 360}2 1 1 1-81t J

.

(7)

Note that this beamwidth and gain can,~_~El~!!l~Y an array of M antennas, with the gain, \G=10Iog 10M. For example, from (6) (which assumes L p ':: 60 dB), to~obtain enough receive power to support 155 Mbps (with a ray with power 3 dB less than the total received signal power) requires an antenna gain G=26 dB, or, from (7), a 400-element (e= 10 0 ) base station array with omnidirectional antennas at the users. Note that the required gain is given by the product of the gain of the receive and transmit antennas. Thus, we could also use a 100-element (8=20 0 ) base station antenna with a fourelement handset, or a 20-element (6=45°) array at both ends. Thus, for example, antenna arrays with 15° beamwidths (183 elements) at the transmitter and receiver should support up to 10 Gbps (if delay is not an issue). Note that these results do not consider additional fade margin due to multipath fading as in the omnidirectional antenna case. We do not consider the multipath fading to be a concern with directive antennas for two reasons. First, an isolated ray should have no fading, since multiple rays are required for fading and the single ray should remain relatively constant in amplitude over many wavelengths. However, although the prediction techniques of [1] may generate only a single ray, the environment may actually generate multiple rays that are closely spaced, rather than a single ray, which could result in fading, albeit with longer fading intervals than with omnidirectional antennas. This fading would typically be Rician, though, with a large K, which greatly reduces the required fade margin. Second, the prediction technique of [1] shows that, at least with the Crawford Hill building, there are typically seven isolated rays, with

3. Antenna Arrays First, consider the use of higher gain antennas to increase the margin. Note that if the multipath causes the received signal to be uniformly distributed in power with respect to angle-of-arrival, increased receive antenna gain does not increase the margin. In this case the increase in

721

low delay spread and sufficient power. In this case, we obtain seventh order diversity. which reduces the required fading margin even further. Therefore. we expect the required fade margin to be substantially less than with omnidirectional antennas and will therefore not consider it. Next. consider antenna arrays to reduce the delay spread problem (see also [4], which also considers directive antennas at both ends of the link to reduce delay spread. but for line-of-sight systems). Since the data rate limitation due to margin (without an additional fade margin) and delay spread are about the same and arrays are needed to increase the maximum data rate due to the margin limitation. we would hope that an M-element array would increase the data rate limitation due to delay spread by the same factor as that due to power margin. Similar to our above results. if the distribution of received power and delay spread was uniform with respect to the angle-of-arrival. directive antennas will not increase the data rate limitation due to delay spread. However. as stated above. the prediction techniques of [1] show that the power of the received signal is not uniformly distributed in angle-of-arrival. However, the range of signal delays can remain large even for small beamwidths as shown below. An example of this problem is shown in Figure 2, where with omnidirectional transmit and directive receive antennas the delay spread is large even when the receive beamwidth is small. As a result. higher directivity reduces the total power of the weaker rays with respect to the strongest ray within each beam. but not necessarily the delay spread of the signals in the beam . Therefore, the maximum data rate remains below 1 Mbps until the power of the weaker rays becomes small enough. At this point the delay spread limitation is essentially removed. Thus. as the antenna directivity is increased. the data rate is limited to 1 Mbps by the delay spread until the directivity exceeds some value at which point the data

rate dramatically increases to the power margin limitation. Furthermore. as discussed above. additional margin due to multipath fading is no longer needed. Note that under these conditions. neither multicarrier techniques nor equalization can significantly increase the data rate. Until the beamwidth is sufficiently narrow . the number of carriers or the length of the equalizer must increase linearly with the data rate. independent of the antenna directivity. These techniques become very complex and expensive for data rates greater than 20 Mbps. The critical parameter is therefore the directivity required for high data rates. Consider first an omnidirectional transmit and directive receive antenna. as in Figure 2. Using [1]. Figure 3 shows the beamwidth required at the receiver versus the number of equalizer taps (see below) for 90% coverage at a 1 Gbps data rate. These results were generated for the model of the Crawford Hill building (see Figure 4). using ray tracing to determine all rays received with up to 3 reflections. To show the most optimistic results for the

25

!

20

'0 15

~

10

5

Eye p8lllIIly • 20 dB

21.-.:Dons 100 150 200 250 Beamwidlh (0egrMs)

Figure 3

300

350

The beamwidth required versus the number of equalizer taps for 90 % coverage with omnidirectional transmit and directive receive antennas.

~~'IIIII[I[lliDJ • I [/I J I 1

14

118m

~

R-I46

Figure 2

An example case of omnidirectional transmit and directive receive antennas where the delay spread is large even when the receive beamwidth is small.

Figure 4

722

The bottom floor of the Crawford Hill building.

effect of equalization and arrays, we only considered delay spread and used a coverage requirement of a 20 dB eye penalty (nearly closed eye) as given by a 1/.9 ratio of the power of the strongest ray (plus other rays within ±.5 of the symbol period) to the sum of the powers of the other rays (with delays outside of ±.5 of the symbol period) within the beamwidth. Furthermore, we considered an N-tap decision feedback equalizer that eliminated the rays in either the strongest N precursor or N postcursor symbol intervals. Figure 3 shows that. even under these overly optimistic conditions, coverage within a distance of four rooms requires 5° beamwidths, and equalization does not significantly reduce the required beamwidth. Thus, a 1650-element array is needed. which appears to be impractical with today's technology. Therefore, consider using directive antennas at both the transmitter and receiver. Note that the example given in Figure 2 would benefit greatly from this strategy. On the other hand, the example given in Figure 5 shows that even with directive antennas at both ends a small be~width may not always reduce the delay spread. However, we would hope that these cases are rare. Therefore, to further improve performance we consider searching over all rays to find the beam direction with the lowest BER due to thermal noise and delay spread. Thus, we could choose a beam with a ray with lower power than the strongest ray, but with less delay spread. To illustrate these results. and determine the critical antenna size. consider the data rate limitation given by directive beams for one floor of Crawford Hill (see Figure 4) with a 10-8 BER requirement. Specifically, for each ray, using the prediction techniques of [1] with rays with up to 3 reflections, we determined if the receive

E

_b

No

..c._

Figure 5

An example case of directive transmit and receive antennas where the delay spread is large even when the beamwidth is small.

1000"'"

0.8

t

- -

_';:1"

0.6

~ 0.4 0.2

400

15

Figure 6

20

25

30

Beamwidlh (Degrees)

35

40

45

Availability versus beamwidth for several

data rates for transmission to the edge of

the coverage area using the propagationprediction techniques of [1] for the Crawford Hill building with rays with up to 3 reflections.

was

greater than 12 dB and, for all the rays within the beamwidth, the rms delay spread was less than 10% of the symbol period. For a given receiver location, an outage occurs if no ray can be found that meets these requirements . For each beamwidth, we chose 60 locations, at the edge of the coverage region, and determined the availability at a given data rate. Figure 6 shows our results for the availability versus beamwidth (with data points taken at 2.5° intervals) for several data rates. These results show that for data rates greater than 20 Mbps, the availability depends primarily on the beamwidth. Availability greater than 90% requires a beamwidth less than 30° (:::50 elements) for 45 Mbps, but 1 Gbps requires only a 25° beamwidth. Thus. if the beamwidth is narrow enough to isolate at least one ray for 45 Mbps operation, data rates up to 1 Gbps and higher are also feasible. For a 13° beamwidth (244 elements), the maximum data rate exceeds 1 Gbps with 100% availability. Thus, in all 60 locations 13° antennas find

an isolated ray with enough power to support Gbps data rates. In fact, our results show that about 7 isolated rays (in different 13° beams) with sufficient power are usually found for each location. To support our conclusions, we performed the following experiment, which is summarized in Table 1. Using the LuckyNet [5] setup, we transmitted up to 622 Mbps at 19 GHz within Crawford Hill. For Table I, radio link factors are given in [5-7]. The propagation loss is determined as 3 dB less than free space (assuming half the total power in one beam, as above) minus 3.4 power law excess loss (from Figure 1). The transmitter was located in the hallway near the library and reception area on the first floor and the receiver was positioned about 12 feet inside room RI46 at the end of the corridor (see Figure 4). Although this is the short end of the corridor

723

(about half the length from the reception are to the other end of the corridor), propagation measurements (see Figure 1) show that the propagation loss (L p ) is similar to that for the longer end, i.e., an average of about 50 dB. The transmit and receive antennas were 15 0 beam width hom antennas, which could be manually scanned. BER measurements were made at a combination of 6 locations by moving the antenna height or lateral position within a few feet at both ends of the link. At each location, both antennas were manually scanned to try to jointly find the best transmit and receive angles. Note that there are over 33,000 possible transmit/receive angle combinations with 15° beamwidths, and therefore it was not practical to exhaustively search for the best angle. However, since the transmitter was located down a long corridor, we assumed that pointing the transmit antenna toward Rl46 would be most likely to give the good performance, With this general direction for the transmit antenna, the receive antenna was manually scanned to find a reasonable BER, and the transmit angle was then adjusted slightly to try to improve this performance. We

found that good receive angles could not be determined a

priori, e.g., pointing at the door did not always result in a satisfactory BER. The strongest receive signal had a

propagation loss of 51 dB, compared to the predicted propagation loss with omnidirectional antennas of 50 dB. This is in agreement with our expected result of the strongest ray containing about half of the total receive power. The BER results for 6 locations are shown in Table 1 and range from 3xlO-8 to 10-3 • Note that even a 10-3 BER is acceptable since with coding the error rate could easily be reduced below 10- 8 • Table 1 also shows the variation in BER with bit rate at one location. Except for the highest data rate measured, 622 Mbps, we did not have a clock recovery circuit and used a coaxial line to feed the clock to the receiver. Data rates were adjusted slightly to synchronize the coaxial-line-fed clock to that of the signal received by radio. The BER is nearly constant for data rates greater than 210 Mbps, which implies an irreducible BER (albeit, low BER, ~10-7) that is independent of the data rate, i.e., the received signal consisted of one strong ray with much weaker rays with delay spreads in excess of 5 nsec. Thus, with sufficient receive power, data rates in excess of 1 Gbps should be possible. This experiment only presents anecdotal results at a few closely-spaced locations to support our conclusions. The propagation loss was 10 dB less than is required for full floor coverage (with a maximum 60 dB propagation loss), but we did not exhaustively search all transmit/receive angles. Even a computer-controlled exhaustive search with directive homs would take many hours for each location, so experimental measurement of availability awaits the construction of phased array antennas.

1. Transmitter Radio Frequency Power Amplifier Output Power Transmitter Cable Losses OmDidirectional Antenna Gain Directional Antenna Gain

19.00GHz +23dBm 1 dB 4.5 dB 22 dB

2. Signal: Data Rates, Mbls BPSK, Coherent:, Piiol Aided

622.08, 340,210,110.50& 10 2'·1 PRBS NRZ

3. Receiver: Noise Figure Bandwidths, MHz Reqd. Output SNR, for 10-IBEK, dB Antenna Gains & Cable Loss

6dB Approximately equal to Bit Rate 12 dB Same as for the Tnnsmitter

4. Propagation: r-J.4 Below 1 meter Free Space, Half of Power in Main Ray. Assumes obstructed paths. W,

4. Conclusions

A)2 ('; )34 (G G) where '0 = 1 meter. = W, (4x,o T'

Economically-fabricated antenna arrays with about 100 elements which determine and track the optimum combination of transmit and receive beams, along with networking issues, are complex problems that require further research. However, assuming such technology were available, we make the following conclusions. Based on the propagation-prediction techniques of [1], supported by experimental measurements, neither multicarriers, equalization, nor antenna arrays at one end of a communication link are economical methods for increasing the data rate substantially above 20 Mbps for multiple room indoor wireless coverage. However, based on propagation-prediction techniques and verified by our experimental measurements using directive antennas at both ends of a link between the center of the Crawford

Expected Omni Revd Power@l34ft. -8S.5dBm Expected Orelni Revd Power@l34ft -SOJldBm

s. Margin @ 622Mbls, (neglecting intersymbol Omin:Exp.-Reqd for 10- 3 BER, dB Direta:Exp.-Reqd for 10-1 BER, dB

interferences due to delay spread): -17.7 +17.3

6. Error Rate Measurements:

6 dearly spaced locations @ 622.08 Mbls gave sweet spots of 3*10-1, 6*10-1, 1*10-7,2*10-4,3*10-4, & 1*10-3 BEL At One location the BER varied with bit Rate as shown: 621.06 Mbls 6*10-5 BEK 340 Mbls 1*10-7 BER 210 MBls 1*10-1 BER 110 MBls 0 BER SO Mbs 0 BER 10 Mbs 0 BER

Table 1

Indoor Wireless Error Rate Measurement

724

Hill building to an end laboratory, we have shown that high-speed ubiquitous communication is possible. Using antenna arrays with 50 to 200 elements at both the transmitter and receiver, we can expect to obtain entire f oor coverage at data rates in excess of 1 Gbps.

[2]

A. A. M. Saleh and R. A. Valenzuela, "A Statistical Model for Indoor Radio Propagation," IEEE J. Selected Areas Commun., Vol. SAC-5, pp. 128-137, Feb. 1987.

(3]

D. M. 1. Devasirvatham, "Tune delay spread measurements of wideband radio signals within a building," Electronic Letters, Vol. 20, pp. 950-951, Nov. 8, 1984.

[4]

P. F. Driessen, "High-speed wireless LANs with directional antennas," Proc. of VTC'94, Stockholm, Sweden, June 7-10,1994.

[5]

M. J. Gans, T. S. Chu, P. W. Wolniansky, and M. J. Carloni, "A 2.5 Gigabit 23~Mile Radio Link for LuckyNet," Proc. of GLOBECOM'91, pp. 1065-1068, Dec. 2-5, 1991.

[6]

H. Taub and D. L. Schilling, Principles of Communication Systems, McGraw-Hill, New York, 1971.

[7]

J. D. Kraus, Antennas, 2nd edition, McGraw-Hill, 1988.

Acknowledgements We gratefully acknowledge R. S. Roman and O. Landron for providing the experimental results presented in Figure 1, and A. A. M. Saleh for proposing the highspeed data measurements.

References [1]

R. A. Valenzuela, "A ray tracing approach to predicting indoor wireless transmission," Proc. of VTC'93, pp. 214-218, Secaucus, NJ, May 18-20, 1993.

725

On Optimizing Base Station Antenna Array Topology for Coverage Extension in Cellular Radio Networks JEN- WEI LIANG, AROGYASWAMI

J.

PAULRAJ Information Systems Laboratory, Stanford University Stanford, CA 94305

Abstract

sults from two factors: array gain and diversity gain. In [5], the authors show that using maximal ratio combining maximizes the array gain, and the average array output SNR is given by

Use of higher frequencies (1.8 GHz) for the US upper tier PCS cellular service and the FCC regulations on the network build out have resulted in significant interest in improving coverage of cellular networks. Networks whose coverage is limited imply that thermal noise is the limiting factor. Also, since the for.ward link (base station to mobile) has higher power than the reverse link, cell coverage is usually limited by the reverse link. This coverage can be extended by improving the reverse link budget. Use of receive antenna arrays for boosting array gain on the reverse link is therefore of great interest [1, 2]. When receive antenna arrays are used at the base station, several conflicting choices affect system performance and cost. Some of these aspects are: the number of antenna elements (and channels) improves coverage but also increases system cost; the maximum span of the array increases diversity but must be limited for convenient deployment on a tower; large inter-element spacing can increase diversity but cause grating lobes at the same time. These conflicting requirements mean that a careful design of the array topology can minimize the cost. In this paper, we study performance of linear and circular base station antenna arrays with different topologies, angle spread, and the number of elements. We compare alternate topologies using maximal ratio combining for narrowband systems such as AMPS and IS-54.

I

(1) where M is the number of antennas and SN R; is the element SNR. Diversity gain comes from various sources such as polarization, multipath in spread spectrum systems, interleaving and coding, and spatial decorrelation. In this paper, we study how the base station antenna topology affects spatial diversity and hence changes the system performance. Maximum diversity gain can be achieved from uncorrelated fading at each antenna. Large inter-element spacing decorrelates signal fading but it also creates grating lobes which cause direction ambiguity and power transmission to the undesired directions. These conflicting requirements reveal that a careful design of the base station antenna array topology can effectively maximize the spatial diversity at minimum cost.

II

Signal Model

We consider transmission from a single mobile to a base station. The base station receiver performance is only limited by thermal noise and no co-channel interference is present. We assume Rayleigh fading and fourth power loss, and do not consider slow fading such as shadowing. CDMA systems can benefit from multipath diversity provided by resolved paths received at the RAKE receiver [6]. However, in this paper we assume that delay spread is much smaller than a symbol period and therefore the channel undergoes flat fading. As mentioned earlier, other diversity dimensions such as polarization, multipath, are neglected. Signals transmitted from the mobile are assumed to be scattered by 20 scatterers around the mobile,

Introduction

Recent studies [3, 4] have shown that antenna diversity can substantially improve the performance of most wireless communication systems. In this paper, we are interested in using base station antenna arrays to extend the coverage of cellular networks. The improvement in performance using antenna array re-

Reprinted from IEEE 45th Vehicular Technology Conference, Vol. 1, pp. 866-870, July 1995.

726

... ." .." " ."l~}

\

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o

.......................

0

b d ;.......0-...

_c~~>· ry , ..............

i-J2'i

1

(A ······/ ·.,··········· ···O\ \. -,

5A

0 .... ...........0.

(a)

(b)

(c)

(d)

0:

j

....

- :SCATfERS Figure 1: A constant angle scatterer model.

~

0 ·-Q-·0··0·-0-·0

(e)

-------.-Et~==========~ ir

~

0 ·-Q-'0··········-Q-·O··0

I

7.5A

I

(I)

Figure 3: Different topologies under consideration .

'(

LOCAL SCATIERERS

BA SE STATION DOMINANT SCATTERERS

Figure 2: A simple scattering scenario.

We now conside r the com bining schemes for the bas e st at ion ant enna array. The traditional beamforming assumes planar impinging wavefront , and signals are cophased and weighted to maximize SNR at th e array output . The array SNR for the planar beamformer is given by

(2) all of equal power . In Fig. 1, we illust rat e a constant angle scatterer model where the signal angle spread () is independent of the distance between scatterers and the base station . This model is applicable when there are dominant scatterers confining the scattering angle as shown in Fig . 2. The phase angle of scatterers have uniform distribution in·{O:"~;'l and scatters ar e randomly distributed on a const ant radius scattering arc centered at the base station . This is a simplified version of a scattering 'disc' model, but it is adequate to illust rate the topology tradeoff studied in this paper . Fig . 3 shows 6 different base station antenna topologies we have studied , which are (a)single circular array with >../2 and (b)large inter-element spacing , (c)single circular array plus one sensor , (d)dou ble circular arrays , (e)single linear array, and (f)double linear arrays. Different antenna patterns ar e used in circular and linear arrays: a cardioid pattern pointing away from the cent er for circular arr ays and an omnidirectional pattern for linear arrays .

where A1 is the number of antennas , Xi is the received signal at antenna i , Wi (¢) is the expected complex antenna gain for a signal arriving from angle ¢ , and O"~ is the noise power. However, the planar wavefront assumption is not valid in a cellular environment beca use of th e presence of angle spread in multipaths. On the other hand , maximal ratio combining exploits th e fact that th e impinging signals have arbitrary wavefront, and signals are com bined to maximize the output SINR. The array output SNR, ignoring the interference , is given by

(3) To study the coverage extension, , we set up the range, angle spread , and mobile power for scatterers and combine signals as the array output for each sna pshots. We obtain the BER of the i t h snapshot

727

SINGLE BLOOM CIRCULAR ARRAY

r:~I~

2.8

2.6 2.4

, 10

12

14

NUMBER OF ANTENNAS

16

18

20

22

Figure 4: Coverage extension for a single circular array. from the AWGN BER curve, and it is given by

BER;

= 2erfc(VSNR;). 1

(4)

We average over snapshot BERs to determine the BER for a Rayleigh fading channel, which is given by 1 N

BER= N LBER;,

(5)

i=l

where N is the number of snapshots. With the scattering angle and the target BER 10- 2 fixed, we increase the number of antennas and BER will improve. We then increase the range until the BER exceeds 10- 2 and thus obtain the coverage extension. All our results are normalized with respect to the range obtained from using a single omnidirectional antenna.

Simulation Results

Figure 4 shows the maximum coverage extension for a single circular array as a function of the number of antennas and angle spread. Using 12 antennas at 0 degree angle spread can provide 50% coverage extension. We should note that when we use cardioid antenna patterns in a circular array, approximately only about half of the elements participate in receiving signals. As we increase the angle spread, signals at the sensors become more uncorrelated and larger coverage improvement can be obtained from the increased diversity gain. However, diversity gain is saturated when signals at the sensors are totally uncorrelated.

10 20 Number of Antennas Scattering Angle=30

30

i:l~ z

0

10 20 Number of Antennas Scattenng Angle=40

30


30

1:1/ I [I~I 1:1/ I i:1/

1.8

III

0

z

2.2

1.2

Scattering Angle=20

Scattering Angle=l 0

z

0


30

z

0

z

0

10 20 Number of Antennas

30

z

0

10

20

Number of Antennas

Figure 5: Coverage extension comparison between single and double circular arrays. Therefore, as the angle spread becomes larger, the corresponding range increase is less pronounced. The comparison between single and double circular arrays appears in Fig. 5. Simply dividing sensors into 2 circular arrays with 5A apart, as shown in Fig. 3(a) and (d), we can further increased the coverage from 60% to 100% using 10 antennas and 10 degrees angle spread. However, when the angle spread increases substantially, the difference between single and double circular arrays becomes less obvious and both topologies perform in a similar way. Diversity is mostly provided by angle spread and hence the topology of the base station antennas has less influence. Figure 6 has one additional result for one large circular array as shown in Fig. 3(b). We find that a large circular array has slightly better coverage than double circular arrays when the angle spread is large. However, with 6 elements and 5-X diameter in a circular array, the inter-element spacing is 2.5A and grating lobes become a serious problem for transmission and direction-finding, i. e. , grating lobes cause ambiguity in the angular position of the desired mobile and it also causes interference by transmitting power in the undesired directions. In Fig. 7, we compare a single circular array with one circular plus one sensor as shown in Fig. 3(a) and (c). Placing one sensor 7.5-X apart, receive antennas acquire more diversity from this separated lone sensor and thus increase the coverage from 70% to 100%

728

30

r:~12:10 z

0


?20 r:~1 L I [I ;;;;20 O

1:1 30

z

0

z

0


30

z

0


30

z

0

z

0


1:17 1:17 11:17 [17 z

0

~ 31

10

20

Number of Antennas scat:tering Angle=50

t:~ z

0

10

20

Number of Antennas

I

30

~ 31

30

10

20

Number of Antennas

0

10

20

Number of Antennas

z

0

30

10

20

Number of Antennas

I

30

10

20

Number of Antennas

t7 t.7 ~ 31

Scattenng Angle=60

t:~ z

30

z

0

~3

Scattering Angle=50


30

z

1

0


Figure 7: Comparison between single circular and single circular+ 1.

for 10 antenna elements and 10 degrees angle spread. Thus, it is reasonable to use the 5-element circular array in Fig. 3(c) for transmission to avoid large grating lobes and use all 6 sensors for reception to obtain more diversity. We also find that when the number of antennas is large, the difference of the range increase between 2 curves becomes smaller. The comparison between a single linear array and double linear arrays is shown in Fig. 8. With the distance between the center of two linear arrays fixed (7.5A in Fig. 3(f)), both topologies achieve the same coverage extension at a large number of antennas.

when the system lacks diversity, for instance, there are few antennas and small angle spread. In a practical cellular system, only a limited number of antennas can be deployed and hence topology plays a more important role in the base station antenna design.

References [1] B. Khalaj, A. J. Paulraj, and T. Kailath, "Antenna arrays for CDMA systems with multipath," in MILCOM'93, pp. 624-628, 1993.

[2] J. H. Winters, "Presentation from First Workshop

on Smart Antennas in Wireless Mobile Communications," tech. rep., Stanford University, June 1994.

Concluding Remarks

In the previous section, we showed the coverage extension results for six different base station antenna topologies. Beyond the array gain described in Eq. (1), using multiple antennas at the base station provides different diversity gain with different topologies. We studied three diversity sources in this paper: the number of antennas, angle spread, and topology. The results show these three sources interact with one another. Topology becomes less important when large angle spread or a large number of antennas are present in the system. A careful design of the base station antenna topology can effectively provide more diversity and hence extend the coverage for a cellular network, especially

[3] J. H. Winters, J. Salz, and R. D. Gitlin, "The

impact of antenna diversity on the capacity of wireless communication systems," IEEE Trans. Commun., vol. COM-42, pp. 1740-1751, February 1994.

[4] A. Jalali and P. Mermelstein, "Effects of multipath and antenna diversity on the uplink capacity of a CDMA wireless system," in GLOBECOM'93, vol. 3, pp. 1660-1664, 1993.

[5] J. William C. Jakes, Microwave Mobile Communications. John Wiley & Sons, 1974.

729

I

30

Scattering Angle~o

Figure 6: Comparison between small circular, large circular, and double circular arrays.

IV

30

30

f~I;10 I z

0

&4

1:1 z

0

10 20 Number of Antemas Scattering Angle=30

i~l~ z 0

10

20

Number of Antennas Scattering Angle=40

~. f~l~

30

I

10

20

Number of Antennas Scattering Angle=50

f~l~

z 0

30

Scattering Angle=20

10

20

Number of Antennas

30

30

z

0

10

20

Number of Antennas

i~17 z 0


30

30

Figure 8: Comparison between single and double linear arrays.

[6] J. S. Lehnert and M. B. Pursely, "Multipath diversity reception of spread-spectrum multiple-

access communications," IEEE Trans. Commun., vol. COM-35(11), pp. 1189-1198, November 1987.

[7] P. Balaban and J. Salz, "Optimum diversity combining and equalization in digital transmission with applications to cellular mobile radio," IEEE Trans. Commun., vol. COM-40, pp. 885-894, May 1992.

[8] D. Parsons, The Mobile Radio Propagation Channel. John Wiley & Sons, 1992.

730

USAGE OF ADAPTIVE ARRAYS TO SOLVE RESOURCE PLANNING PROBLEMS M. Frullone*, P. Grazioso*,

c. Passerini", G. Riva*

* Fondazione Ugo Bordoni,I DEIS - Universita' di Bologna

Villa Griffone - Pontecchio Marconi, 1-40044, Bologna Tel.: +39 51 846854; Fax: +39 51 845758 e-mail: [email protected] Abstract - The usage of adaptive antenna arrays in cellular systems is currently being investigated by many researchers. In this paper we show by means of simulation that adaptive antenna arrays allow to reduce the reuse distance, and hence to increase spectrum efficiency. Furthermore, it is shown that, by adopting a fractional loading factor, it is possible to adopt a cluster size equal to 1, which avoids the need of frequency planning altogether, and allows a further improvement in spectrum efficiency.

II ADAPTIVE ARRAY In the following we assume to use a planar circular array of

N=8 elements situated at the base station. The elements are

supposed to be half-wavelength dipoles, and the radius of the array is half-wavelength. It has been supposed that no mutual coupling exists between array elements: it can be demonstrated that the presence of the mutual coupling does not affect the behaviour of the adaptive array under certain, widely satisfied, conditions.

I INTRODUCTION

s;

signals Sk,i(t) (O 10

j/

-

/

/

1/

/

/

I

! I

SN R

15

25

20

30

Fig 1. CDF of SNR for CS = 12 (non-adaptive BSs) The interference analysis is reported in figure (2), as a function of the cluster size . Omnidirectional BS antennas are assumed, and the minimum cluster size that satisfies the constraint on SIR is 12. This is a reasonable value compared with the CS = 3x3 of the actual GSM where the BSs are not situated in the centre of the served cell.

=Ck6)p~ .(I _ Po )36- k .

The radio links are affected by: 1) path -loss, depending on the distance, with a = 3.5 inverse power-law; 2) long-term fading, log-normally distributed with standard deviation of 6 dB; 3) short-term fading, Rayleigh distributed, uncorrelated between two subsequent frames. The active users (both in the target cell and in the cochannel cells) are regarded as the sources of the signals

0,4

0,35 0,3

ProbISIR<x}

_______._L. !

.•L. I

I

.

.~::-:!:-...-+

0,25 0.2 0, 15 0, 1

Sk -i (r), as defined in section II; therefore 15 N u 5 37 For each snapshot, Rayleigh fading is averaged over the 50 frames. This is different from GSM quality measure, which is averaged over a time period of 100 frames. The Cumulative Distribution Functions of the SIR and SNR are evaluated over 3000 snapshots . In this study it is assumed that a SIR less than 10 dB is tolerable in 5 % of the cases.

0,05

°·10

_._._+_ -5

_+"L-~+

o

10

«:

SIR 15

20

Pig. 2. CDP of SIR for different CSs (non-adaptive BSs) With the adopted numerical values (Co = 124, Pb =0.01), it is straightforward to evaluate the system capacity by a

733

simple inversion of Erlang B formula: we obtain that the carried traffic is about 4.5 Erl/cell. As final remark of this subsection, it is worth noting that the use of the fractional loading technique at non-adaptive BS doesn't produce any performance improvement.

Of course, a mixed solution involving adaptive arrays , intermediate CS and fractional loading could be put into practice. However, carried traffic results .are worse than the CS= I solution. VI. CONCLUSIONS

B. Results for the adaptive BS

In this paper we have discussed the usage of adaptive antenna arrays to solve frequency planning problems and to improve spectrum efficiency. The analysis was carried out by means of a Monte-Carlo simulation; we have shown that the adoption of adaptive antenna arrays allows a drastic reduction in the minimum cluster size which fulfils the requirements on SIR. In the numeric example here developed, the minimum cluster size could be reduced from 12 to 4; this, along with the higher trunking efficiency , led to an increase of almost 500% in the carried traffic. We have also shown that using a fractional loading factor it is possible to adopt a cluster size equal to 1, which avoids the need of frequency planning altogether, and allows a further improvement in spectrum efficiency . These guidelines can be applied in planning advanced cellular systems, even based on already existing air interfaces such as GSM.

In this section we will report the SIR results for the adaptive BS antennas. The curves in fig. 3 refer to the dimensioning criterion based on blocking probability, above introduced (see section lILA) and are directly comparable to those shown in fig. 2. We note that, by adopting adaptive BSs, the cluster size can be as low as 4. In this case, the carried traffic is about 21 Erllcell, with an improvement in spectrum efficiency of almost 500% with respect to the one obtained with non-adaptive BS antennas. 0,4

0,3S 0.3

0,2S 0,2 O,IS 0,1

REFERENCES

O,OS

°

-10

-S

°

10

IS

[1] T. Ohgane, T. Shimura, N. Matsuzawa and H. Sasaoka, "An Implementation of a CMA Adaptive Array for High Speed GMSK Transmission in Mobile Communications", IEEE Trans on Veh , Tech., vol. VT-42, No.3, August 1993, pp.282-288. [2] O. Norklit and J. Bach Andersen, "Mobile Radio Environments and Adaptive Arrays", PIMRC '94, The Hague, September 1994, pp.725-728 . [3] M. Frullone, C. Passerini, P. Grazioso and G. Riva, "Advanced frequency planning criteria for second generation cellular radio systems", ICf '96, Instambul, April 1996. [4] I.S . Reed, J.D. Mallett and L.E. Brennan, "Rapid Convergence Rate in Adaptive Arrays", IEEE Trans. on Aerospace and Electronic Systems, vol. AES- 10, No.6, November 1974, pp. 853-863

20

Fig. 3. CDF of SIR for different CSs (adaptive BSs) Finally, we have analysed the effect of dimensioning the system adopting CS = 1 and using a fractional loading factor (see fig. 4). We obtain that a fractional loading factor F = 0.3 is achievable. This means that every cell uses in average about one third of the available 124 channels, that is 37 channels; in other words, the carried traffic is now about 37 Erl/cell, with a further improvement of 75% with respect to the previous case. 0,4 0,3S

Prob(SIR <xj

0,3

0,2S 0,2 O,IS 0,1 O,OS

°·10

.._-- ---- .r':·'"!;.~ .~ .. ~?-r ·5

°

10

15

20

Fig. 4. CDF of SIR for CS = 1 and fractional loading

734

Subscriber Location in CDMA Cellular Networks James Caffery, Jr., Student Member, IEEE, and Gordon L. Stuber, Senior Member, IEEE

Abstract~ubscriberradio location techniques are investigated for code-division multiple-access (CDMA) cellular networks. Two methods are considered for radio location: measured times of arrival (ToA) and angles of arrival (AoA). The ToA measurements are obtained from the code tracking loop in the CDMA receiver, and the AoA measurements at a base station (BS) are assumed to be made with an antenna array. The performance of the two methods is evaluated for both ranging and two-dimensional (2-D) location, while varying the propagation conditions and the number of BS's used for the location estimate.

Index Terms-Cellular CDMA, position location.

O

I. INTRODUCTION

VER THE past decade, considerable attention has been given to vehicle location technology, and numerous applications have been proposed. Recently, in a few testbed areas, rental cars outfitted with location devices and map displays have aided visitors in unfamiliar territory [1]. Taxi and delivery drivers have utilized location technology in Tokyo to navigate the myriad of streets. Fleet operators use location technology to improve product delivery times and to improve the efficiency of the fleet management process. Emergency and police dispatchers have also utilized location technology to locate dispatch vehicles and emergencies for improved response times, In cellular telephone networks, location technology could be used for radio resource and mobility management [2], [3]. For example, a service provider who may have multiple agreements with personal communication services (Pf'S's), cellular, or satellite carriers, could offer its customers the ability to choose a carrier that best suits their needs at a given time and location [4]. Also, the Federal Communication Commission has recently released an order, to be implemented in two phases, requiring cellular service providers to provide a mechanism for generating subscriber location estimates for Enhanced-vl l (E-911) services [5]. A further application of location technology is in the rapidly emerging field of intelligent transportation systems (ITS's), which are designed to enhance highway safety, system operating efficiency, environmental quality, and energy utilization in transportation [6], [7]. Each of the above applications requires a method for determining and relaying the location of vehicular and pedestrian mobile stations (MS' s).

Manuscript received April 19. 1996; revised March 3, 1997. This work was supported by GTE Mobilnet. Portions of this paper were presented at the 5th IEEE International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), Toronto, Ont., Canada. September 1995. The authors are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. Publisher Item Identifier S 0018-9545(98)02477-3.

Automatic vehicle location (AVL) techniques have been studied thoroughly in the literature for the purpose of vehicle location. AVL systems entail the acquisition of information about the location of MS' s operating in an area, and all require the processing of that information to form location estimates. There are three basic AVL methods: dead reckoning, proximity systems, and radio location [8]. Dead reckoning computes the direction and distance of travel from a known starting position [8]. In proximity systems, the nearness of an MS to fixed detection devices is used to determine its position. The devices can be anything from magnetic sensors to conventional radio transmitters and receivers. Radio location systems attempt to locate an MS by measuring the radio signals traveling between the MS and a set of fixed stations (FS' s). The signal measurements are first used to determine the length or direction of the radio path, and then the MS position is derived from known geometric relationships [8]. Radio location can be implemented in one of two ways-either the MS transmits a signal which the FS's use to determine its location or the FS' s transmit signals that the MS' s use to calculate their own positions [e.g., the global positioning system (GPS)]. There are several fundamental approaches for implementing a radio location system including those based on signal-strength [9]-[12], angle of arrival (AoA) [13], and time of arrival (ToA) [3], [14], (15]. It is important to note that line-of-sight (LOS) propagation is necessary for accurate location estimates. Many of the existing location technologies use dead reckoning, radio location with GPS, or hybrids which require specialized subscriber equipment, the cost of which can severely limit their availability to the average consumer, With these technologies, the MS formulates the location estimate which may be relayed to a central site. Another approach for providing location services is to use the cellular telephone networks. A method has been proposed in [1] which incorporates the cellular network into the location process. However, this service requires a GPS receiver in the MS to determine the location, and the cellular network is only used to relay the location information to a central site. Only one previous work has examined a subscriber location technique that relies solely on the cellular network, that is based on signal attenuation measurements [12]. This paper examines the feasibility and performance of radio location techniques in code-division multiple-access (CDMA) cellular networks. CDMA is the chosen access scheme, since it appears to be the leading candidate for third generation cellular networks. The cellular network is used as the sole means to locate the MS' s, and the location estimates are determined through reception of signals that are transmitted by the MS at

Reprinted from IEEE Transactions on Vehicular Technology, Vol. 47, No.2, pp. 406-415, May 1998.

735

a set of base stations (BS's). This approach has the advantage of requiring no modifications to the subscriber equipment. Specifically, radio location methods based on AoA and ToA are studied. We concern ourselves with performance in terms of absolute accuracy with no concern given to the rate of location updates that can be achieved. The remainder of this paper is organized as follows. Section II outlines the methods employed for the AoA and ToA techniques that will be used for the performance evaluation. The propagation models for macrocellular and microcellular systems are discussed in Section. III, followed by simulation results in Section IV. A discussion of some practical issues for subscriber location is given in Section V, followed by some concluding remarks in Section VI.

BS

II. RADIO LOCATION SYSTEM

Fig. 1. MS-BS geometry assuming a ring of scatterers for macrocells.

A. Angle of Arrival AoA techniques estimate the location of an MS by using directive antennas or antenna arrays to measure the AoA at several BS's of a signal that is transmitted by the MS [13], [16]. Simple geometric relationships are then used to form the location estimate, based on the AoA measurements and the known positions of the BS' s. With the AoA method, a position fix requires a minimum of two BS's in a 2-D plane. In this paper, we consider the error due to multipath propagation, but do not consider angle estimation errors. Multipath propagation, in the form of scattering near and around the MS and BS, will affect the measured AoA. For macrocells, scattering objects are primarily within a small distance of the MS since the BS's are usually located well above surrounding objects [17], [18]. This results in reception of signals from all directions at an MS while the BS receives signals from a small azimuthal spread. For microcells, it has been suggested that the BS' s be placed below rooftop level (lamppost height) in order to confine the signal coverage to a small area [18]. As a result, the BS becomes surrounded by local scatterers and signals can arrive at the BS from a much broader range of angles. Consequently, the AoA approach, which may be used for macrocells, is impractical for microcells. Gans [14] and Jakes [17] have modeled the macrocellular propagation environment as a ring of scatterers about the MS, with the BS well outside the ring. Fig. 1 illustrates this geometry, where the primary scatterers are assumed to be on a ring of radius a about the MS. The distance between the BS and MS, d, is assumed to be much greater than a. We assume that the MS uses an omnidirectional antenna, so that

1 p(,) = 27r'

o :S ,

< 27r.

(1)

The distribution of the AoA at the BS, 0, is given by

d,

dO p(O) = 2p(,).

(2)

From the geometry of Fig. 1, we find that [14] 2

d-y ~ [ (~) - ({3 - B)2

] -(1/2)

dB.

(3)

r(t)

1\

Timedelayestimate, 't Fig. 2. The DLL used for time-based subscriber location.

Therefore, p(0) is

p(0) == {

K [

(~/ -

({3 - B)2] -(1/2) ,

0,

{3 - BM 5: B $ {3 + BM otherwise

(4)

where

OM == arctan( a/d) 1

K= 2 arcsin (dO ). aM

Note that for d ~ a, a small angle approximation can be invoked, with the result that OM ~ aid and K ~ l/tr. The model p(8) provides the AoA distribution for signals arriving at a BS. Our model goes one step further by assuming that a measured AoA at a BS also has the distribution p( 0). Since the measured angles are not equal to the true angles to the MS, the lines of position from the BS's will not intersect at the same point. This problem is resolved by deriving the location estimate from the centroid of the set of points defined by the intersecting lines of position. With three BS' s, for example, the lines of position intersect at

736

three points: (Xl: Yl), (X2' Y2), and (X3' Y3). The location estimate (x: y) is obtained by averaging the coordinates of the points of intersection. i.e., ;r == (Xl + X2 + x3)/3 and Y == (Yl + Y2 + Y3)/3.

B. Time of Arrival Many popular radio location techniques are based upon the measurement of the arrival times of a signal transmitted by an MS at several BS"s. These methods determine the distance between an MS and BS by measuring the time a signal takes to travel from the BS, to the MS, and back again. Geometrically, this provides a circle, centered at the BS, on which the MS must lie. Using at least three BS' s to resolve ambiguities in two dimensions. the intersection of circles provides the MS"s position.' This method is often called the ToA method and has the disadvantage that it requires the MS to act as a transponder in which processing delays and non-LOS propagation can introduce error. To overcome these limitations, time difference measurements rather than absolute time measurements can be used. Since the hyperbola is a curve corresponding to a constant time difference of arrival (TDoA) for two BS's. the time differences define hyperbolas, with foci at the BS' s, on which the MS must lie. The intersection of hyperbolas provides the location of the MS. This method is often called the TDoA method. Methods for obtaining the ToA or TDoA estimates include phase ranging [19], pulse ranging [3], [19], and spread-spectrum techniques [20], [21]. Since the cellular system being considered is CDMA, methods for determining the ToA's from the spread-spectrum signal are of interest. The two methods for determining time delays in spread-spectrum communications systems are coarse timing acquisition with a sliding correlator or matched filter and fine timing acquisition with a delay-locked loop (DLL) or taudither loop (TDL) [22]. Previous subscriber location studies have used coarse timing acquisition to obtain the ToA estimates [20], [21]. Since the DLL finely tracks the time delay, it is better suited for a location system. The DLL is an essential part of time estimation used for GPS and provides reasonable accuracy over the satellite-earth propagation channel. Here, the DLL-based location system will be investigated for its performance in cellular propagation environments. The DLL shown in Fig. 2 allows fine synchronization of the local spreading code with the incoming code. It operates by correlating the received signal with the early and late spreading codes c(t - f- + ~Tc) and c(t - f- - ~Tc), respectively, where f is an estimate of the delay between the local and incoming codes. The code phase error signal e(t) is obtained by squaring and differencing the correlator outputs. The squaring operations serve to remove the effects of data modulation and carrier phase shift. The loop is closed by applying e(t) to a low-pass filter, whose output is used to drive the voltagecontrolled clock (VCe) and correct the code phase error of the locally generated code. The parameter ~, 0 < ~ < 1, is called the early-late discriminator offset. The output of the vec provides the ToA estimate f. 1 In

C. Time-Based Location Algorithm

Two approaches are generally used to calculate the location of an MS from ToA or TDoA estimates. One approach uses a geometric interpretation to calculate the intersection of circles or hyperbolas, depending on whether ToA or TDoA is used. This approach becomes difficult if the hyperbolas or circles do not intersect at a point due to time measurement errors. A second approach calculates the position using a nonlinear least-squares (NL-LS) solution [3], [19], [23], which is a more statistically justifiable approach. The algorithm assumes that the MS, located at (xo, Yo), transmits its sequence at time TO. The N BS receivers located at coordinates (Xl, Yl), (X2, Y2), "', and (XN, YN) receive the sequence at times T1, T2, ... , TN· As a performance measure, we consider the function [19]

j"i(X) = C(Ti - T) - J(x t

x)2 + (Yi - y)2

-

(5)

where c is the speed of light, and x == (x, Y, T)T. This function is formed for each BS receiver, i == 1, ... , N, and all the Ii(x) could be made zero with the proper choice of x ; y, and T. However, the measured values of the arrival times t. are generally in error due to multipath and other impairments, and non-LOS propagation introduces errors into the range estimates that are derived from the arrival times. 1) Unconstrained NL-LS Approach: To obtain the location estimate from the raw ToA data, the following function is formed:

F(x) ==

'""' Qi ? f,;?(x) c:

(6)

i=l

where the O:i' S are weights reflecting the reliability of the signal received at BS i. The location estimate is determined by minimizing the function F(x). A simple approach for solving the nonlinear least squares problem in (6) is the steepest descent method, where successive location estimates are updated according to the recursion

where J.L is a constant (scalar or diagonal matrix), (Xk, Yk, Tk)T, V'x == d/dx, and

8FI 8x

\JxF(Xk) == \7 xF (X) lxk

=

737

==

Xk

8FI by Yk

(8)

bPI

N

2f- ( ) 2'""' ~Qi t Xk i=l

8T

Tk

x, - Xk J(Xi - Xk)2 + (Yi - Yk)2

N

2LO:; h(Xk) i=l

J(Xi -

Yi - Yk + (Yi - Yk)2

Xk)2

N

-2cLfi(Xk) i=l

general, locating an MS in n dimensions requires n + 1 measurements.

Xk

(9)

Since T is small (microseconds) compared to x and y (meters), the scalar step size J..L should be small enough to allow T to converge to a solution. Consequently, J.l. is chosen to be the diagonal matrix J.-L

==

/-Lx [

0

o

0 00]

J..Ly 0

BSI

(10)

J-L-r

where J-Lx, J.Ly ~ J..L-r. The recursion in (7) continues until lI\7x F (xk)1I is smaller than some prescribed tolerance f. One drawback of the steepest descent method is its slow convergence. Other algorithms have been investigated [19], [23], which form the solution to (6) by linearizing Ii(x) with a Taylor series expansion about Xk and keeping only the first order terms, i.e.,

BS2

(11)

where 6 == (6x , by, b-r)T = x and solving

Xk.

Substituting (11) into (6) (12)

for 6, the vector

Xk

is updated by (13)

This new estimate is substituted back into (11) and the process 8yl+cI6-r1< E, where f is a prescribed is reiterated untilI8 x l+1 tolerance. When the MS is either close to the BS' s or near the perimeter of the area defined by the polygon with the BS's as its vertices, then the linear approximation approach has convergence problems [3], [19]. For microcells, the MS is always within a short distance of the serving BS, so this method is not appropriate. The convergence problem arises from the approximation of fi(X) with the linear terms of the Taylor series expansion. Other objective functions F(x) can be formed replacing, for example, fl(x) with Ifi(X)I. However, these methods usually do not perform as well as minimizing the sum of squares [3]. 2) Constrained NL-LS Approach: It may be possible to improve the time-based location algorithm due to the fact that the range error is always positive [24]. This is because the ToA estimates are always greater than the true ToA values due to multipath propagation and other impairments. Also, the range estimates derived from the ToA estimates are greater than the true ranges due to non-LOS propagation. Therefore, the true location of the MS must lie inside the circles of radius Ti == C(Ti - r), i = 1, .", N, about the NBS's, since the MS cannot lie farther from a BS than its corresponding range estimate (Fig. 3). Mathematically, this implies r;

= c(ri - r) 2 J(Xi - x)2 + (Yi - y)2

(14)

where (x, y) is the position of the MS. Since the unconstrained NL-LS algorithm does not take this restriction into account, a constrained NL-LS approach can be used to force the estimate at each iteration to satisfy (14). However, the LS solution is complicated by the nonlinear functionals Ii(x) as well as the

Fig. 3. The location of the MS is constrained to the intersection area (shaded region) of circles of radius C(Ti - T) centered at each BS.

nonlinear inequality constraints of (14). Note that (14) implies that

J(Xi - x)2 + (Yi - y)2 - C(Ti - r) ~

o.

(15)

We recognize from (5) that the left side of the inequality in (15) is simply 9i (x) = - Ii (x). Hence, the restrictions Ii(x) 2 0 are formed, where the area within the constraint boundaries is known as the feasible region. There are many approaches to forming numerical solutions for NL-LS problems with nonlinear inequality constraints of the form gi(X) :::; 0 [25]. One simple, yet effective, method uses penalty functions to modify the objective function F (x) and form a solution using an unconstrained approach as in the previous section. The penalty functions provide a large penalty to the objective function when one or more of the constraints are violated. The objective function in (6) is modified to include the penalty functions 9i(X) as follows [25]:

F(x) ==

N

L

i=l

N

at fl(x) - P L[9i (X)]- 1

(16)

i=l

where P is positive for minimization. As any constraint is approached during the search, the penalty term forces F toward infinity, thus forming a natural optimum within the feasible region. This approach requires that the initial guess be placed within the feasible region. A method for doing this is described in [25]. The search procedure can be viewed as the optimization of a sequence of surfaces which tend toward the true value of the objective function. Initially, an unconstrained search method is used to provide an artificial optimum x 1 with a large value of P == PI- The next stage is initialized with the previous estimate x, and uses a smaller P = P2 to provide a better approximation to the true optimum. In this

738

COST207

where nk(dB) is the local mean envelope (or square envelope) level (in decibels) that is experienced at location k, ~ is a parameter that controls the spatial decorrelation of the shadowing, and {'T/k} is a zero-mean discrete-time Gaussian random process with autocorrelation ¢"P7 (n) == (j2 8(n ). The autocorrelation of nk(dB) is given by

TABLE I SIx-TAP REDUCED TYPICAL URBAN POWER DELAY PROFILE

COST 207 Model Fractional Power 0.189 0.379 0.239 1.6 0.095 2.3 0.061 5.0 0.037

Delay-To (J.lS) 0.0 0.2 0.5

(19)

way, the solution approaches the constraints more closely, if the optimum happens to lie close to one of the constraints. The penalty constraints become smaller at each stage, forming a monotonic-decreasing sequence PI > P2 > '.', and the sequence of artificial optima x. , X2, ... tends toward the true optimum. The search continues until several iterations fail to produce a change in the objective function. This formulation essentially replaces a constrained optimization by a sequence of unconstrained optimizations. III. PROPAGATION MODELS A three-stage model is used for the radio propagation environment, that includes multipath-fading, shadowing, and path loss. The particular models used in this paper for macrocellular and microcellular propagation environments are now described.

where as is called the shadow standard deviation. Typical values of the shadow standard deviation range from 5 to 12 dB in macrocells [17], [18], [28]. If we assume that the local mean is sampled every T s, then the autocorrelation can be expressed as

where ED determines the correlation between two points separated by a spatial distance D and v is the velocity of the MS. The simulations in the sequel assume a shadow decorrelation of 0.1 at a distance of 30 m. Several empirical path loss models have been presented in the literature, one of the most useful being Hata's model [28], which expresses the path loss in terms of the carrier frequency, BS height, MS antenna height, and the type of environment (urban, suburban, or rural). Rata's model for medium or small city urban areas is used in the sequel with a carrier frequency of f == 850 MHz, BS antenna height of 100 m, and an MS antenna height of 2.5 m.

A. Macrocells

For wideband spread-spectrum systems, the channel can be modeled by the M -tap tapped delay line

h(t) ==

M

L

ui, (t)8(t

-

Ti)

( 17)

i=O

where the {Ti} are the tap delays and the {W·t } are the tap gains, assumed here to be complex Gaussian random processes. For numerical convenience, the tap delays can be chosen to be an integer multiple of some small delay T, i.e., Ti == krr, i == 1,,'" M. The first tap delay TO is determined from the MS-BS geometry of Fig. 1 by calculating the distance traveled by a signal transmitted from the MS in a random direction according to p('"Y) and reflected from the ring of scatterers to the BS. The remaining delays are chosen according to the six-tap reduced typical urban delay profile defined in COST207 [26] (see Table I). The model deviates slightly from the COST207 model by assuming a classical Doppler spectrum for all taps, i.e., in the simulations the taps gains are all generated by using Jakes' method [17]. Shadow fades have been described from measurements as being lognormally distributed with a standard deviation that depends on the frequency and the environment [18]. Gudmundson [27] has suggested a simple Markovian model to describe variations in the local mean envelope (or squared envelope) level due to shadow variations. With this model nk+1(dB)

== ~rlk(dB)

+ (1 -

~)TJk

(18)

B. Microcells

The wideband channel and shadowing models discussed above can also be used to model microcellular propagation. However, the power delay profiles are different, and the standard deviation of shadowing in microcells typically ranges from 4 to 13 dB. Further differences in the propagation models for microcells and macrocells are discussed in the following. Microcellular path loss is often described by a two-slope characteristic, where the area mean == E[n] is given by [29]

n

n = 10 loglO [da(l: d/g)b ]dBm

(21)

where A is a constant, d is the radio path length, 9 is the break point (that ranges from 150 to 300 m), and a and b determine the slopes before and after the break point. In the simulations, we assume 9 == 150 m and a == b == 2. An important consideration for microcells is the corner effect, which occurs in microcellular scenarios when an MS rounds a street corner. To account for this effect, LOS propagation is assumed to the MS until it rounds the comer. The non-LOS propagation after rounding a street comer is modeled by assuming LOS propagation from an imaginary transmitter that is located at the street comer having a transmit power equal to the received power at the street corner from the serving BS. The area mean (in dBm) is given by (22), at the bottom of the next page, where de is the distance between the serving BS and the comer.

739

was used for the loop filter [i.e., F( s) = 1]. For the vee, the output time delay estimate and input waveform are related by

f(t)

.8S1

Fig. 4. Manhattan street microcell deployment.

Due to the site-specific nature of the microcellular propagation environment, techniques such as ray tracing have been developed. In this study , ray tracing concepts are used to calculate the propagation delays for the wideband channel model. A Manhattan street microcell BS deployment is assumed as shown in Fig. 4. When the MS is LOS with a BS, a four-path model is used, consisting of a direct path, a roadreflected path, and two wall-reflected paths. The taps of the wide band channel model are generated using Jakes ' method, appropriately modified for Rician fading. When the MS is non LOS with a BS, i.e., around the comer, a different approach is taken to determine the propagation delays . Since the literature prov ides no results that describe the power delay profile for an MS that is around a comer from a BS, a simplistic model is chosen. A four-path non-LOS propagation model is used that includes two paths that arrive from diffractions at the building comers in the street intersection and two remaining paths whose delays are generated by adding random delays to the first two paths . All paths are assumed to be Rayleigh faded. The model chosen here is inconsequential, because the extra time delay for non-LOS BS's introduces a large amount of error into the location algorithm. Hence, accurate modeling of multipath propagation on non-LOS streets is not necessary; only a means of introducing the excess propagation delay around the street comer is needed .

IV. SIMULATIONS The location techniques described in Section II were simulated in the macrocellular and microcellular environments described in Section III to determine their performance. The spreading code used was an m sequence of length 127 and chip rate Te- 1 = 1.2288 Mcps . In the DLL, an all-pass filter

= KveeTe

it

u(x) dx

(23)

where K vee is the gain of the vee, T; is the chip period, u(t) is the output of the loop filter, and the vee is assumed to begin .operating at time t = O. A simple accumulator models the operation of the vee in the computer simulations with the constant K veeTe = 0.003. Note that there is a limitation in the accuracy that can be achieved when simulating the DLL on a computer. As a result, we limit the resolution of the DLL to 1/120 of a chip to limit the simulation time. Consequently, the ranging resolution is limited to approximately 2 m, which causes all range estimates to be in error even in the absence of propagation impairments. However, with such a fine resolution, propagation impairments will be the predominant source of location error .

A. Range Estimation Ranging measures the I-D distance between an MS and BS. Only the time-based method is employed for ranging since AoA ranging does not make sense . For macrocells, our ranging results assume that the first path to arrive from the COST 207 model is a LOS path . Consequently, the ranging results for macrocells are very optimistic by disregarding the extra propagation caused by non-LOS propagation when a direct path does not exist. For microcells, the Manhattan street microcell deployment in Fig. 4 is assumed. 1) Effect of Standard Deviation of Shadowing, as : Fig. 5(a) shows the effect of the shadow standard deviation on the mean and standard deviation of the range estimation error with an early-late discriminator offset ~ = 1/2 and a chip-energy-tonoise ratio E e / No = 10 dB. The mean ranging error increases by approximately 10 m as a s increases from 4 to 12 dB. The standard deviation of the ranging error also increases due to the increased variability of the shadowing process. 2) Effect of Ec/N o: Fig.5(b) shows the effect of E e/No on the ranging error with ~ = 1/2 and as = 6 dB . The increase in ranging error for decreas ing E e/ No is expected in any system . The effect is not as pronounced in the microcellular environment due to the smaller delay spreads. 3) Effect of d : The effect of multipath on the tracking ability of the DLL can be explained by observing the distortion that multipath causes on the correlation function of the spreading code, which has a triangular shape for a rectangular chip-shaping pulse. Fig. 6 shows an example of a distorted loop S curve for the case of two multipath components, the second having half the power of the first and delayed by T e/2. Observe that the tracking error introduced by multipath propagation is reduced by using smaller ~. However, the

(22)

60.0 ~

4° l' -~ _~

:g 30 ~ ~~ l .~20

,

0--- - -- - ---0-- ----- --

~

~

~...: 40 .0 ~,i " o

0- - -

-.- - - - - - - - - ..

e ---------.-- -

g 50.0 ~

-- -------1

Ot-t-t-O

f

e

§

I

1

Macrocell

+--+ Microcell (LOS)

gf

.-

1

10

gf

I r 30 .0 ~

~

~ Macrocell

+--+ Microcell (LOS)

"

"

I

'0"

6

8

10

Standard deviation of shadowing, cr, (dB)

------

Il.l

~

Fig. 5. Effect of (a) shadow standard deviation (E r /N o (dashed lines) of the ranging error.

6

12

= 10

dB) and (b) Eo/ N u (a.

=

~

j

i1

- 1.0

L

Fig. 6.

- ' - -_

- 1.0

15.0

dB) on the mean (solid lines) and standard deviation

- - L -_

0.0

_

---'_~ 1.0

Error in delay estimate (chips)

V ARIOVS V ALVES OF LI. . V ALVES ARE IN METERS

t. 1/2 1/4 1/8

I

~

- 2.0 - 2.0

=6

FOR

g

o

1

-.J

M EAN R ANGING ERROR AND S TANDARD D EVIAT ION

~

..J

10.0

1

TABLE II

=1/2

6 = 118 6 1120

1.0

5.0

E/No (dB)

6 = 1/4

.g 'C '"

-- - - ---

-'--__

' -_ _......L_ _- - '

0.0

..

· ·0·------0

(b)

(a)

2.0

' . 0 ..

20.0 ~~_......Ac

o '------'---~---'------'--- ~ 4

'0-

2.0

Distortion of the S curve due to multipath for different values of LI. .

minimum size of Do is limited by hardware considerations (such as the clock rate) and the precorrelation bandwidth in the DLL. Band limiting tends to round the autocorrelation peak which limits the discrimination between the early and late correlation when using small Do [30]. Simulation results for various Do are presented in Table II with a, = 6 dB and Ec/No = 5 dB. The results show that the ranging error mean and standard deviation can be signifi cantly reduced by using a smaller Do . B. Two-Dimensional Location

Two-dimensional (2-D) location estimates the MS location by using several BS' s. Here, we focus on the accuracy of the location estimates as a function of the number of BS' s. This is an important consideration, since using more BS' s means more processing and an increased load on the network. Assuming a transmit power of I dBW (the maximum for Class III IS95 MS's) , a noise power of - 100 dBm was added to each BS receiver for the time-based method. The macrocell and microcell deployment scenarios are as follows.

Microcell Mean Std. Dev 15.6 15.5 10.8 12.1 4.5 6.1

Macrocell Mean Std. Dev 30.8 23.6 25.3 19.1 16.9 18.9

J) Mac rocells: We assume a distance of 6000 m between BS' s, i.e., the cell radius is 3000 m. Assuming known BS positions , the MS is randomly placed among the BS' s and the nearest BS' s are used for the location process. For macrocell s, ToA and AoA approaches are compared when using two-fi ve BS' s in the location process as a function of the scattering radius a about the MS. The simulations examined both the unconstrained and constrained location algorithms of Section II-C, which almost always converged with a < 1 for each BS. The mean and standard deviation of the location error for the ToA method using the unconstrained NL-LS algorithm are shown in Fig. 7(a). For a given scattering radius, the mean and standard deviation decreases when more BS' s are used. As expected, a larger scattering radius increases the location error due to nonLOS propagation. Recall that LOS propagation is necessary for accurate ranging and location estimates. The mean and standard deviation of the location error for the ToA method with the constrained NL-LS algorithm are shown in Fig. 7(b). Unlike the unconstrained NL-LS case, the performance is not improved significantly when more BS' s are used. Table III compares the performance of the unconstrained and constrained NL-LS methods, where both algorithms are initialized with the same location estimate. The mean location error is reduced up to 30% by using the constrained NL-LS algorithm. The constrained NL-LS with three BS' s perform s nearly as well as the unconstrained NL-LS with five BS's. This implies that the constrained NL-LS algorithm can result in less network loading.

741

800

S

'B

r~3 BS

_ _ 4BS

600

'" "CI

600 ,-... 500

S '-'

6-----6 5 BS

-r-----.-------,

_ _ 4BS

"'-----6 5 BS

"0

§

.

u u

....

20

'"

15

'0". u

(y , z) =( 1 5 , O)m

T

10

O'---_.---L_ _-..L_ _--'-_ _- ' -_ _-'--_----'_ _-..L_ _---L_

a

20

_

...l-_--l

40 60 80 100 1 20 140 160 180 R location index 0 : (y . z ) =( 5 . 30) m. 200: (y , z)=(6 , 40) m

200

Fig. 2. Capacity versus R location in street canyon T and R height .r = 1 rn, p = 20 dB. >. = 1/3 m.

0 .9 0 .8

-;;;

....''""

u

0.7

'"

0 .6

A

0 .5

.Q

'"

>.

....u u

'"

0.4

"

0.3

0.

.''":: c,

0. 2 0 .1 0

,

10

log(l+n rho)

Fig. 3. Capacity on Ricean channels. n

12

14 16 18 capacity (bits/cycl e)

20

r

22

nl og(l+rho )

= 3.

nn = n antennas. The base and subscriber ends of the link are designated as T and R, respectively, but reciprocity applies. For LOS propagation, and a narrow-band channel at fixed carrier frequency Ie = c] A; ray-tracing from T to R yields the channel transfer function matrix H = H LOS with complex scalar entries

where Ti , R; are coordinate vectors for the ith element of

T , R. H ik is normalized by the distance between the reference locations T I , R I , so that H I •I = 1 and the absolute attenuation need not be calculated. If the antennas are spaced less than A/2 apart at both T and R , H ik = ei 6•k ~ ei 6 for fixed () for all i , k, (H H*)ik ~ n , and C = !og2(1 + n p) = C log, so that Clog increases logarithmically with n . For this case, H = H LOS = H I is of rank 1, and the capacity gain is essentially due to the n -fold array gain in p. For arrays of n more widely spaced antennas at T and/or R, the complex scalars Hik all have magnitude near one but different phases (}ik. For (}ik so that H H* = nln l H =

746

......../ .

3 .?~ts

.., ...... daisy chain to spac e-time processor

eell slte et

center 01 cell

cell .ltea.t ed geo! cell •

cell . ite••t ce nter 01cell covlHlIgaarea 7ceUreuse

~ cell aitea at edga 01 cell cover age area

A = channel group

120 degree .ectora at ea ch cell alte

Fig. 4.

Ce ll sites at center and edge of coverage area.

H LO S = H " is of rank II , and C = n. log2 (l + p ) = G lin . so that G lin increases linearl y with n, An example is Bi k = ~[ (i - -o) - (k - /.;o)f where for II. = 2 and i o = /.;0 = 1 . ll . H lll a x = ( j ~ ) , and the corresponding array geometry is two linear arrays broadside to each other. In what follows. we show three mor e examples of geo metric arran gements for which HLos ~ One such arrangement is an n.-element T array spread along an arc at angles (/Jk - 1 for l: = 1. . . . . u, and a linear R arra y oriented broadside to the center of the arc . From ( I ). for arc radius D » A, and interelement spacing Z'" H i k = ex p(j 2 '-~= r [U - 1) - ":;- 1] sillfPI.'_ Jl which corresponds to the

n;

autocorrelation R:u U ~ /.;) = 2~Re .r~6 e "· (i- I.:)sin.l d/3 from [8. eq. (A- l3) 1 specialized for the n. discrete angle s of arri val (Pk - l ' G ~ G lin when the an gle subtended by the arc 2.6 = ¢"- 1-¢o is consistent with the beam width 2.6 = AI-Z,. at which Ru U - k) = O. The radiation pattern of the R array with all elements in phase E(¢ ) = T1sin( '~""j;)) , where ":;Ill "), _ j 2\

and a sim ilar R arra y on a circle of radiu s D,. S A at the center of the T arra y. From (1 ), H ;k = ex p (j 2"f " cos[U - k) 2,;]). For D,. A/2, the T elements are not in the nulls of the R array. but the elements of H H* for which i - k is odd are zero. and the off-diagonal even elements are approximately O.3n. Nonetheless, the capacity approaches G lin, con sistent with the observation [8] that small correlation « 0.3) has negligible effect on performance . Further calculations confirm that the capacity is robust in the presence of rotation or lateral movement of R or perturbations in the placement of the T elements. The third example is an urban street geometry with two parallel reflectors representing the building walls separated by the street width a (Fig. I). l ±nt represents an image due to m. specular reflections from the wall s. and 10 is the " ground reflection" image not visible in the figure. For thi s street geometry. the elements of H may be written

For z; = A/ 2. and the 11. array elements of T placed along the arc at ¢ k - l exactl y in the nulls of R, H i k = exp(j2r~ (-i - -io)(k - /';0 ) ) where i o = /.;0 = n! I , for which H H* = 11.1" so that H = H n and G = Glin. The second example arrangement is an n-element T array with elements spread evenly around a circle of radius D » A.

=

"( = 21fz,. sin ¢/ A.

747

ITl

Hi k

ex p( - j27f IT; -

- Rl i

+

f

IT; -

k= - m

Rk l/A)

Rkl

!l f - Rkl/A) II; - Rkl

rl.: ex p (- j 2:

(2)

r

where is the amplitude reflection coefficient.? m is the maximum number of reflections considered, and we have assumed isotropic array elements. For eight-element linear arrays with >../2 spacing oriented perpendicular to the street, Fig. 2 shows how the capacity increases as more images are added (and the angular spread of rays increases) and approaches Glin with seven images (Iml :S 3) plus T. Furthermore, the received signal envelope looks increasingly Rayleigh-like as more images are added.

B. Ricean Channels Next we consider the capacity for Ricean channels, where the deterministic component HLo S is fixed as either HI or H n . We follow the simulation methods of [1] using the normalized Ricean channel matrix H == (aHLos + bHRayleigh) with a 2 + b2 == 1 and Ricean K-factor K == a 2 / b2 . The results for n == 3 with p == 100 (20 dB) (Fig. 3) quantify how for closely spaced (~>..) array elements at both T and R, and no reflectors or scatterers such that H LOS ~ HI, the capacity decreases with increasing K toward C == Clog. However, for array geometries such as the above examples, where HLo S ~ H n , the capacity increases with increasing K toward C == Glin . For K == O. C corresponds to that obtained in [1].

cellular system can be combined to form one "edge-excited" (inward-facing) cell (Fig. 4) to enhance capacity for R not close to a base station. This is reminiscent of soft handoff in CDMA systems where multiple base stations serve one mobile, except that in this case, each base station carries a different substream of the transmitted data. The results of the third example suggest that in the absence of reflectors, we may use n antennas at each of n sites, thus replicating the effect of images of the n-element T array. The ray-tracing channel model for H == H LOS + H Rayleigh described here may be useful for the performance evaluation of MIMO wireless systems which use spatial diversity through space-time coding to exploit the available capacity with no bandwidth penalty (e.g. [2], [5]-[7]). REFERENCES [ 11 G. J. Foschini and M. J. Garis, "On limits of wireless comrnunicanon in a fading environment when using multiple antennas," Wireless Personal

Commun., vol. 6. no. 3. pp. 311-335. Mar. 1998. [21 G. J. Foschini, "Layered space-time architecture for wireless commu-

[3] [41

III.

DISCUSSION

Capacities approaching Glin == ti 10g2 ( 1+ p) can be achieved for MIMO channels in an LOS (non-Rayleigh) environment by spreading out the elements of T either explicitly (by placing one element of T at each of ti sites), or implicitly (by adding reflectors which create images of T). The results of the second example suggest that three sectors in a conventional general, the r are different for each image. since they depend on the angles of incidence and reflection, and the surface characteristics. Here we assume has the same constant value 0.6 for all reflections. except the ground reflection [0 which is set to -1. This approximation is sufficient to tllustrate the capacity gain. 2 In

r

748

[5] [6]

[7]

[8]

nication in a fading environment when using multi-element antennas." Bell Labs Tech. J .• vol. 2. no. 2. pp. 41-59. Autumn 1996.V. Weerackody, "Diversity for the direct-sequence spread spectrum system using multiple transmit antennas." in IEEE Int. Conf, Communications. Geneva. Switzerland. May 1993. pp. 1775-1779. N. Seshadri and J. H. Winters, "Two signalling schemes for improving the error performance of frequency-division-duplex (FDD) transmission systems using transmitter antenna diversity." lnt.T. Wireless lnformatton Networks. vol. 1. no. I. pp. 49-60. 1994. D. Agrawal. V. Tarokh. A. Naguib. and N. Seshadri, "Space-time coded OFDM for high rate wireless communication over wideband channels:' in Proc. IEEE Vehicular Technology Conf.. 1999, pp. 2232-2236. D. Agrawal, V. Tarokh. A. Naguib. N. Seshadri. and A. R. ':alderbank. "Space-time codes for high data rate wireless communication: Performance criteria and code construction:' IEEE Trans. Inform. Theory. vol. 44, pp. 744-765, Mar. 1998. D. Agrawal. V. Tarokh. A. Naguib. N. Seshadri. and A. R. Calderbank, "Space-time codes for high data rate wireless communication: Practical considerations," IEEE Trans. Commun .• to be published. J. Salz and J. H. Winters. "Effect of fading correlation on adaptive arrays in digital mobile radio." IEEE Trans. Veh. Technol.. vol. 43, pp. 1049-1057. Nov. 1994.

Optimum Space-Time Processors with Dispersive Interference: Unified Analysis and Required Filter Span Sirikiat Lek Ariyavisitakul, Senior Member, IEEE, Jack H. Winters, Fellow, IEEE, and Inkyu Lee, Member, IEEE Abstract- In this paper, we consider optimum space-time equalizers with unknown dispersive interference, consisting of a linear equalizer that both spatially and temporally whitens the interference and noise, followed by a decision-feedback equalizer or maximum-likelihood sequence estimator. We first present a unified analysis of the optimum space-time equalizer, and then show that, for typical fading channels with a given signal-to-noise ratio (SNR), near-optimum performance can be achieved with a finite-length equalizer. Expressions are given for the required filter span as a function of the dispersion length, number of cochannel interferers, number of antennas, and SNR, which are useful in the design of practical near-optimum space-time equalizers. Index Terms- Equalization, interference suppression, multipath channels, space-time processing.

I

I. INTRODUCTION

N WIRELESS communication systems, cochannel interference (CCI) and intersymbol interference (lSI) are major impairments that limit the capacity and data rate. These problems can be mitigated by spatial-temporal (S- T) processing, i.e., temporal equalization with multiple antennas [1]-[11]. In typical wireless systems where the cochannel interferers are unknown at the receiver, optimum S-T equalizers, either in a minimum mean square error (MMSE) or maximum signal-to-interference-plus-noise ratio (SINR) sense, consist of a whitening filter, i.e., an equalizer that whitens the CCl both spatially and temporally, followed by a decision-feedback equalizer (DFE) or maximum-likelihood sequence estimator (MLSE) [12]. However, under some channel conditions with dispersive CeI, the whitening filter requires an infinite span to achieve near-optimum performance, even with reasonable signal-tonoise ratios (SNR's). For typical fading channels, though, such channel conditions occur only occasionally, and the required filter span for near-optimum performance is finite in most cases. Since the filter span, specifically both the causal and anticausal portions, determines the required memory of the Paper approved by K. B. Letaief, the Editor for Wireless Systems of the IEEE Communications Society. Manuscript received May 20, 1998; revised October 20, 1998. This paper was presented in part at the IEEE Intemaional Conference on Communications, Vancouver, BC, Canada, June 1999. S. L. Ariyavisitakul is with Home Wireless Networks, Norcross, GA 30071 USA (e-mail: [email protected]). J. H. Winters is with AT&T Labs-Research, Red Bank, NJ 07701 USA. I. Lee is with Bell Labs, Lucent Technologies, Murray Hill, NJ 07974 USA. Publisher Item Identifier S 0090-6778(99)05236-8.

DFE and MLSE, these filter spans determine the required complexity of near-optimum S-T processors. In this paper, we first present a unified analysis of the optimum infinite-length S-T processor, considering three receiver types: 1) MMSE linear equalizer (LE); 2) MMSE-DFE; and 3) MLSE. The unified analysis includes both previously published results [12], [17]-[ 19] and additional new material. The objective here is to provide a consistent and comprehensive framework for expressing all these results in a form that is descriptive of the functions and properties of individual filter elements. We then present filter length analyzes for all three receivers by analyzing the z-transform expressions. We show that, with fading channels, the filter spans of these receivers can be truncated such that the average effect of the truncation is small compared to the effect of thermal noise. We then determine the required filter span to achieve near-optimum receiver performance. These expressions for the required filter span as a function of the dispersion length, number of cochannel interferers, and SNR are useful in the design of practical near-optimum space-time equalizers. Using computer simulation, we study the effect of thermal noise on the required filter span for specific fading channels. In Section II, the system model and notation is defined. The unified analysis is presented in Section III, and in Section IV the finite filter span analysis is presented. Section V shows numerical results. A summary and conclusions are given in Section VI. II. SYSTEM MODEL We consider a system where L + 1 cochannel signals are transmitted over independently fading multipath channels to an M -branch diversity receiver. The time-domain complex baseband expression of the received signal on the jth antenna is

Tj(t) ==

L

L L 00

xnthij(t - nT) + nj(t)

(1)

i=O n=-oo

where {Xni} is the transmitted data sequence from the ith source, with the desired source being indexed by i == 0; h ij (t) is the overall impulse response of the transmission link between the ith source and the jth antenna; T is the symbol period; and nj (t) is the additive white Gaussian noise at the jth antenna. The data {Xni} are independently identically,

Reprinted from IEEE Transactions on Communications, Vol. 47, No.7, pp. 1073-1083, July 1999.

749

Fig. 1. A space-time DFE receiver.

Fig. 2.

distributed complex variables with zero mean and unit symbol energy and are uncorrelated between sources. The frequency-domain expression of the above received signal is

Our analysis also includes the use of z-transfonns. The ztransform of a sampled sequence of a continuous-time function g(t) is G(z) ~ ~kgkz-k, where gk == T· g(kT) [we multiply g(kT) by T so that {gk} and g(t) have the same average energy per symbol interval]. The relation between z-transform and Fourier transform is given by the following using the Poisson sum formula [13]:

RJ(f) ==

L

L

X·i(f)Hij(f)

i=O

+ Nj(f)

(2)

where RJ(f), ..: ri(f) , Hil(f)~ and Nj(f) are the Fourier transforms of rj(t)~ {Xnl}~ h'ij(t), and nl(t)~ respectively. Since the data have unit symbol energy E[I~Y'i(f)lr~- == 1 for If I < 1/(2T) where E[·] denotes expectation. The noise at each antenna has two-sided power spectrum density No. The general space-time receiver using a DFE is shown in Fig. 1 (an LE receiver model can be obtained by setting the feedback filter response to zero). It consists of a linear feedforward filter, W J (f), j == O. 1 ~ ... , M -1, on each branch, a combiner, symbol-rate sampler, slicer, and synchronous linear feedback filter B(f). The feedforward filters {W) (f)} are shown as continuous-time filters, but they can be implemented in practice using fractionally-spaced tapped delay lines. The input to the feedback filter is the decided data {x nO} for the desired source. We assume correct decisions (x na == xnO) throughout this study. The input to the slicer (i.e., the space-time processor output) is denoted by sequence {Yn}, with its Fourier transform Y(f) given by Y(J)

=

1\1-1

L L 00

1=0 rn=-oo

G(ei 2rrf T ) =

2

== E[lYn - x no/ ] == T

j

1/ 2T

-(1/2T)

9ke-j2trkfT

=

f

G(f - ; )

'rn=-cx:,)

(5)

where G(f) is the Fourier transform of g(t). and j == It is easy to show that

Tj·l/2T

G'(f)df

-(1/2T)

= ~ 1 G(z) 21rJ

.r

R.

dz z

(6)

where G' (f) ~ 2::=-00 G(f - (miT)) is the Fourier transform of the sequence {9k}. The contour of the integration on the right side of (6) is the unit circle. For convenience, we omit the tilde sign from our z-transform notation throughout the rest of this paper, e.g., G(z) will be written simply as G (z). Furthermore, if G(f) is the Fourier transform of a symbolspaced sequence (instead of a continuous-time function), then

T

j

l / 2T

-(lj2T)

G(f)df=~ 21rJ

f G(z)~. d Z

(7)

III. UNIFIED INFINITE-LENGTH THEORY

(3)

E

f

k=-oc

w, (J - '; )RJ (f - ;) - B(J)Xo(J).

The summation with respect to m in the above equation is a result of spectrum folding due to symbol-rate sampling. Based on the MMSE criterion, the filters are optimized by minimizing the mean square error (MSE)

A space-time MLSE receiver.

A. Optimum Filter Expressions for DFE and LE Receivers The MMSE solution for the feedforward filters {W) (f) } with unconstrained length can be derived by using (2)-(4) and setting the derivatives {aE I aWj (f - (miT))} to zero. This yields

E[lY(f) - X O(f )/2]df·

(4)

where

Fig. 2 shows a space-time receiver using an MLSE. Here, the goal of optimization is to maximize the signal power (without suppressing lSI) to CCI plus noise power ratio, while whitening the CCI and noise components of the .input {Yn} to the MLSE. 750

w~

[Wo(f

-~)

WM-1(f

-~) ...

WO(f

+~)

WM-l (f

+~)]T

(9)

H,

~

[fI (f - ~)

Hi, M -1

iO

HiO(.f+~)

(f - ~) ... ~)r

H"M-1(f+

Rs ~ H~H6

~

L

(10)

H:H,;

+ iVoI.

R s is the correlation matrix of the desired signal, R 1 + 1V

is the correlation matrix of the interference plus noise, I is the identity matrix, and the superscripts * and T denote complex conjugate and transpose, respectively. We assume that the desired and CCI sources are strictly band limited to f == ±.J' / (2T ) (JI is a positive integer), and therefore J == (.I' -1) /2 when J' is odd, and .J == JI/2 when .I' is even (e.g., JI == 1 and .J == 0 when there is no excess bandwidth). Note in (9) and (10) that excess bandwidth provides additional diversity which can be exploited when there is sufficient transmit power outside the Nyquist band. e.g .. in a spread spectrum system [14] (or see also [171). Using the matrix inversion lemma [12. Appendix OJ, it can be shown that

+ R1+,v ] - IH*0

== 1

+

J+N

0

HTR-1 H*' 0

/+.V

()

(1 1 ) -I

Therefore, (8) becomes

W ==

R- 1 H* ~+ lV_ 1 0 * (1 + B (f) ). 1 + H o R 1 + 1yH o

['(1) =

WtR s W WtRI+~VW

(15)

superscript t denotes conjugate transpose W is the output signal power density, and WtR] +lV W is the output interference-plus-noise density at frequency f). Substituting (14) into (15) yields where

.wtn,

( 17)

Equation (17) gives the form of the MMSE solution, well known in array processing [12] (except for the consideration of spectrum folding and feedback filtering). This equation indicates that the optimum feedforward filter consists of a space-time filter R~NHo~ which performs spatial prewhitening (R ~: {; is the whitening filter of CCI and noise) and matching to the desired channel, followed by a temporal filter (1 + B(f)) / (1 + f(f)), which can be regarded as a post-whitening filter under some zero-forcing condition, as described below,

( 18)

+ r(f) jO (1 + B(z))(l + B*(z-l)) 1

.r

dz z

1 + r (z )

1 + I'(e) == SoG(z)G*(z-l)

( 19)

(20)

where the constant So is given by

So ==

e(ln(l+r(f)))

and

(-) ~ T

j

.l / 2T

. -(1/2T)

(21 )

[.J df

(22)

and G(,~) is canonical, meaning that it is causal (Yk == () for I: < 0). monic (gO == 1). and minimum phase (all of its poles are inside the unit circle, and all of its zeros are on or inside the unit circle). Using the Schwarz inequality, it can be shown that the MSE in (18) is minimized when

1 + 8(f) == G(f):

1 + B(z) == G(z).

(23)

Substituting (20)-(23) into (17) and (18), we obtain

*

-1

1

W OF E =RI+NHO SoG*(f)

(24)

and 1 _ e-(ln(l+r(!))) EDFE - - -

(25)

So

Using (16), (20), and (24), the CCI plus noise power density at frequency f is

- 1 H*0 H 0T R I+iV W RI+NW SZIG(f)1 2 t

( 16)

-1 * 1 + B(f) W = RI+NHO 1 + f(f) .

. df

1 + r(f) == SoIG(f)/2;

the

Thus, we can rewrite the optimum feedforward filter solution as

/1 + B(f)/2

where B(z) and I'(z ) are the z-transfonn equivalents of B(f) and r (f). Using spectral factorization theory [19J, 1 + r (f) and 1 + r (z) can be written as

( 14)

Furthermore, we can define the signal-to-interference-plusnoise power density ratio T'( f) at frequency f as

l/ 2T

. -(1/2T)

21rj

i=l

[R s

== T

== _1_

( 12)

R- 1 H*

j O

f

( 11)

L

R]+lY

The optimum feedback filter B ( f) can be determined through spectrum factorization. Substituting (17) into (3), and using (4) and (7), we obtain

r(f) 5 0 (1 + r(f)) .

_

(26)

Note that as f(f) --t DC for If I :S 1/(2T)~ the Cel plus noise power density becomes a constant 1/50 over f. Under this 'condition, we can regard 1/ SoG*(f) in (24) [or 1 + B(f)/l + f(f) in (17)] as a post-whitening filter. As will be relevant later, we can write 1 + B (z) also as

1 + B(z) ==

e[1 + f(z)]

(27)

where C[·] denotes canonical factor. The corresponding Fourier transform is given as

1 + B(f) == C[l + f(f)]·

(28)

Using the above expression, we can write (17) as

751

W

OF E

-1

* C[l

= RI+NHO

+ r(f)]

1 + f(f)

.

(29)

Fig. 3.

An equivalent model of the space-time DFE receiver in Fig. 1.

The optimum LE is obtained by setting B(f) to zero in (17) and (18)

where P is an (L + 1) x (L + 1) correlation matrix whose (a, b)th element Pub is given by

(30)

(34)

and

Furthermore, we can write

Do(f) == VTpTU

(31)

where U rows, and

B. An Alternative Solution for DFE and LE Receivers The optimum space-time filter solution in (17) is based on a general model which does not make any prior assumptions regarding the filter structure. Without loss of optimality, an analytical receiver model suggested by many in the literature (e.g., [15]-[18]) assumes the use of a bank of matched filters {Hij (f)}, each corresponding to the signal source i on diversity branch i, which, after diversity combining, is followed by a bank of T -spaced transversal filters {Vi (f) }, each corresponding to the signal source i (see Fig. 3). This analytical model leads to a different form of solutions which are important to our filter length analysis. The following derivation is similar to the LE receiver derivation in [18], but here we also provide the solution for the DFE. In Fig. 3, the Fourier transform of the input to the slicer can be written as

L Di(f)Xi(f) + N(f) - B(f)Xo(f)

L

vtr-tr-v.

(36)

The MSE for this receiver is given by E

==

EISI

+ ECCI + Enoise

(37)

where EISI

fCCI

== (IDo(f) - (1 + B(f))2)

(38)

=

(39)

(f; IDi(fW)

and cnoise

== (Novtr-v,

= (NoytptV).

(40)

Using (35) and (36), the MSE becomes

(32)

== (ytptpv

- 2Re[(1

i=O

D==PY

IDi (f )12 ==

+1

i=O

E

where Do(f) and Di(f) are the overall channel and feedforward filter responses for the desired signal and the ith interference, and N(f) is the noise at the combined output of the feedforward filters. Let D == [Do(f) ... DL(f)]T and Y == [Vo(f) VL(f)]T. We then have the following relationship:

... , O]T is a column vector with L

O~

L

L

Y(f) ==

== [I,

(35)

+ Nov'r-tv + 11 + B(f)1 2

+ B*(f))yTpTU]).

The MMSE solution for Y

(aE/a~(f)) = 0 for i = 0 to

V = (P

(41)

is obtained by solving

L. We then obtain

+ NoI)-lU(l + B(f)).

(42)

It can be shown that this receiver achieves the same MMSE as (25) (or (31) in the case of an LE receiver) and that

(33)

752

1

1

l+r(f) = No UT(P+NoI)-lU'

(43)

which satisfies

wtR1+NW == 1{3(f)121\lf(f)12H6RI~NH~

IC[f(f)]1 2 f (f )

{I\ }

{ynJ

+t=n~ ~~

-

If(f) /2

1

31

= constant where the overall filter response W is given by

Fig. 4. An equivalent model of the space-time MLSE receiver in Fig. 2.

Again, 1 + B(z) is the canonical factor of 1 Accordingly, we can rewrite (42) as

+

W == W'\lf(f) == R -1 H* C[f(f))

f(z).

I+IV

For an LE receiver

WMLS E

(45)

Fig. 4 shows an equivalent model of the MLSE receiver in Fig. 2. The front-end filters are now represented by spatial filters {W;(f)}~ which maximize SINR of their combined output, followed by a post-whitening filter 'I!(f). Let WI denote the vector of {W; (1)} similar to (9). The signal-tointerference-plus-noise power density ratio f(f) is then [ef. ( 15)]

w'tn-w

= W/tR I+iV. WI·

VVLSE=

(46)

*

opt.

WoptH O

(49)

(3(f)f(f)

(54)

.

1 + f(f) C[f(f)] C[l + f(f)]' f(f)·

(55)

(P+NoI)

-1

.

U(l+f(j))

C[f(f)]

f(j)·

(56)

(57)

(50)

Our filter length analysis is based on counting the number of zeros and poles in the z-transform expression of the optimum space-time filter. For all three receivers (LE, DFE, and MLSE), there are two forms of optimum filter solutions:

we can find a post-whitening filter

W(f) = C[f(f)]

f(f)·

IV. FILTER LENGTH ANALYSIS

Equation (48) has the same form as (17). As a result, we obtain the same expression for r(f) as (16). By factoring f(f) as

r(f) == SlIC[f(f)] 12

C[f(f)]

Thus, the desired signal at the output of the front-end filter has a canonical impulse response; this is the known desired property of the input to an MLSE [21] (in addition to the noise being white).

where

W'

+ f(f)]

H6 W == Do(f) == C[f(f)]·

(48)

,I+N

1 + f(f)

C[l

W DFE .

In (54), when f(f) ~ 1 such that 1 + f(f) ~ f(f), then WrvILSE ~ W DF E , i.e., the optimum front-end filter for an MLSE receiver is equivalent to the optimum feedforward filter of a DFE receiver. This is usually the case when there is no eel and the input SNR is sufficiently high [20]. However, it is generally not true in the presence of strong eel's. Also, using (53) and (56), it can be shown that

The maximum f(f) is, therefore, given by the maximum eigenvalue of RI~NRs. Let W~Pt be the eigenvector corresponding to this maximum eigenvalue, and substitute WI == W~pt into (46). We obtain

opt

(53)

Accordingly

(47)

W'R

=

V l\ILSE = V DFE

The optimum W' is obtained by solving (ar(f)j8Wj(f)) == 0, for j == 0 ~ 1, ... , M - 1; this gives the relationship

(3(f) ==

.

This relationship is extendable to the case where we use the analytical feedforward filter model in Fig. 3 to represent the front-end filter of the MLSE receiver. Thus, we can also write

C. Optimum Linear Filtering for MLSE

.

f(f)

0

Comparing (53) to (29), we find that

(44)

f(j)

(52)

(51)

753

1) one based on the general model (with a linear filter on each branch); 2) one based on the analytical model (with a bank of matched filters on each branch, followed by common filters).

The filter length determined by each solution is valid under different assumptions. The filter length based on the general model is valid when M < L + 1, i.e., when the number of interferers is equal to or greater than the order of diversity due to multiple antennas and excess bandwidth for convenience, we write the overall order of diversity as M, instead of M~. J The filter length based on the analytical model is valid when M ~ L + 1. The reason for these different conditions will become apparent later. Since the general analytical approach for determining the filter length is the same for both solution forms, we only provide details for one of them below. We choose to work on the analytical model case because it is slightly more complicated than the other case, and because the condition under which it is valid (the order of diversity exceeding the number of dominant interferers) is where the most interference suppression is achieved, i.e., an array with M antennas can null up to M - 1 interferers [23]. We begin by working on the MMSE solution for the DFE receiver. The z-transfonn equivalent of (42) is

+ N oI )- l U (l + B(z)) ~ Q- 1 U (1 + B(z))

V == (P

(58)

where V ~ [Vo(z) ... VL(z)]T and Q ~ P + Nol. The element Pab of matrix P is the z-transfonn of the sampled sequence of 1\1-1

~

Loo oc

haj(T)hbAr - t) dr.

(59)

Thus, each element qab of matrix Q has a two-sided response such that it includes both a causal factor and an anticausal factor of equal length. If we assume that all channels {h i j (t) } have a finite memory of K symbol periods (h i j ( t) == 0 for t < 0 and t > KT), then the causal and anticausal factors of qab will be polynomials of order K. Using the matrix identity [22]

Q-l

= Qadj IQI

(60)

where Qadj is called the adjugate matrix of matrix Q, and the (a, b)th element Qab of the transpose matrix of Qadi is called the cofactor corresponding to the (a, b)th element qab of matrix Q [Qab = (-l)a+bIQabl, where Q ab is obtained by deleting the ath row and bth column of Q], we can rewrite (58) as

v = ~ljU(l + B(z)) [Qoo where 1

+ B(z)

QOl

IQI

QOL]T (1 + B(z))

Q01

...

QOL]T

(IQI/O[lQID. O[Qoo]

Vo(z

(63)

)

Qoo/C[Qoo]

= IQI/O[lQIJ .

(64)

Thus, this filter is anticausal (cf. [17]). We now focus on each term in (63). Defining a permutation a as a one-to-one mapping a: (0, 1, ... , L) ---+ (aQ, 0"1, ... , a L), the determinant of Q is [22] (65)

where sgnl c ) == +1 or -1 depending on whether the number of exchanges in permutation (1 is even or odd, and the summation is taken over all (L + I)! permutations (1. Note that the product of two polynomials of order (J, and b results in a polynomial of order a + b, while the sum of them gives a polynomial of order max] (J" b]. Since each element £jab of matrix Q includes a causal factor and an anticausaI factor, each of order K, IQI will, in general, have a causal factor and anticausal factor, each of order K(L + 1). Similarly, Q ab will, in general, have a causal factor and anticausal factor, each of order K L. Accordingly, IQI/C[lQIJ will be anticausal (and maximum phase) with order K(L + 1)~ and C[Qoo] will be causal (and minimum phase) with order K L. Combining these results together, the causal part of each filter ~ (z) (for 'i > 0) will have K L zeros and K L poles, and its anticausal part will have K L zeros and K(L + 1) poles. Since each front-end matched filter hij ( - t) is anticausal with length K (we can always set the synchronization timing such that h i j (t) is a causal function), the overall feedforward filter on each branch will have a causal part with K L zeros and K L poles and an anticausal part with K (L + 1) zero and K(L + 1) poles. In general, a pole filter has an infinite impulse response. Nevertheless, we can always truncate a pole filter which is causal and minimum phase, or anticausal and maximum phase, such that the effect of truncation is small compared to the background noise. Thus, the lengths in units of T of the causal and anticausal parts (denoted as C and A, respectively) of the optimum feedforward filter can be given as

C == KL(l + a) A = K(L + 1)(1 + a).

(62)

(66)

Here, a determines the truncated length of a pole filter 1 (1 - €z)-l or (1 )-1. Note that C = 0 when L == 0; this agrees with the known result that, in the absence of CCI, the optimum feedforward filter of a DPE is anticausal. Note that the required length of the feedback filter is K + C, since the optimum feedback filter completely cancels the postcursors of the desired signal.

ez-

= 0 [NoUT1Q-lU]

C[\Qt]

[Qoo

Note that the common filter for the desired signal is

(61)

== e[l + f(z)]

= C[Qoo]·

V=

DFE (M ~ L + 1):

is given by [see (43)]

1 + B(z)

Thus, we obtain

754

Similarly, we can estimate the filter length for an LE receiver using (6]), with B (z) set to zero; this gives LE (Ai 2: L

C == K(L A == K(L

+ 1): + 1)(1 + (~) + 1)(1 + o ).

K

(67)

Note that the causal length of the LE receiver is greater (by Kc'i) than that of the feedforward filter of the DFE receiver. The above results are valid under the condition that the MMSE sol ution in (61) is compact, meaning that there is no cancellation of highest order terms in the summation in (65) for all determinants. By working on specific examples, we found (6 I) to be compact when 11/[ ~ L + 1. Otherwise, the MMSE solution based on the general filter model [given in ( 17)] is compact. Using the same analytical approach as above, we estimate the filter length for the case M < L + 1 as LE and DFE (J\;1

("1 )

A

~

K

+ K (L + 1)1>("1)

(77)

where (78)

and "1 is in decibels. The good agreement between the filter length s predicted by (77) and the simulatio n results are shown in Tables I-III. Although not shown here, we also found good agreement when testing the empirical formula e against simulation results with other sets of parameter values . Despite its empirical nature, (77) has meaningful analytical justifications. First, it gives the same form of expression for C as the analy tical result in (66), except for the dependence on the SNR [which is also expected of 0: in (66)]. Second, when "1 -; cc such that 1>("1) » 1, the expression for A in (77) becomes A -; K (L + 1)4>("1); thus, we also obtain the same form of expression for A as (66) . As discussed earlie r, (66) is also valid when "1 -; co . Finally, (77) gives the length A as

of a linear filter on each antenna branch, followed by a DFE or MLSE . In this analysis, we derived explicit expressions for the linear filter [e.g., (29), (44), and (54)], which are novel to the best of our knowledge. Using z-transform analysis, we also derived expressions for the linear filter length showing that the required span is proportional to the channel dispersion length and the number of interferers. We then used computer simulation to derive empirical expressions for the required filter span which show that the span is also proportional to the input SNR in decibels. The derived empirical expressions for the required span are in good agreement with simulation results with Rayleigh fading and a uniform-delay spread profile. These expressions are useful in the design of practical near-optimum space-time equalizers.

10-1

-

10-2

Q)

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