Special Design Topics in Digital Wideband Receivers (Artech House Radar Series)

Special Design Topics in Digital Wideband Receivers

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Special Design Topics in Digital Wideband Receivers James Tsui

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Contents Preface  Chapter 1  Introduction 1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8 

Introduction Purpose of This Book Predicated Requirements on Receiver Performance Overall EW Receiver System Operation Encoder Designs Approaches and References Criterion of the Software Approaches Organization of the Book References

 Chapter 2  Amplification Required in Front of the ADC

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1 1 1 3 4 5 6 6 7 7

9

  2.1  Introduction   2.2  Basic Design Criterion   2.3  Inputs to the Computer Program   2.3.1  The Inputs Related to the RF Amplifier   2.3.2  The Inputs Related to the ADC   2.3.3  The Inputs Related to the FFT Operator   2.4  Constants Generation   2.5  Equations Derived   2.6  Modification from the Previous Program   2.7  An Example   2.8  Nominal Sensitivity and Single-Signal Dynamic Range   2.9 Generate Nominal Values for ADC with Different Number of Bits 2.10  Noise Floor and the Number of Bits 2.11  Another Example 2.12  Discussions of Results   References

18 19 21 23 23

 Chapter 3  Dynamic Range Study Through Eigenvalue and MUSIC Methods

25

3.1  Introduction 3.2  Basic Definitions of Dynamic Range

9 9 11 11 12 13 13 14 15 15 17

25 25 

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Contents

  3.3  Prerequisite for Dynamic Range Measurements   3.4  Single-Signal Receiver Dynamic Range (SDR)   3.5  Dynamic Range for Receiver with Multiple Signal Capability   3.5.1  Single-Signal Dynamic Range   3.5.2 Two-Signal Third-Order Intermodulation Spur Free Dynamic Range   3.5.3  The Two-Signal Instantaneous Dynamic Range (IDR)   3.6 A Brief Discussion on the Eigenvalue Decomposition and MUSIC Methods   3.7  Define the Processing Procedure   3.8  Eigenvalues Generated with Noise and Noise Plus Signals   3.9  IDR Determination Through Eigenvalues 3.10  MUSIC Method 3.11  IDR Determined by Frequency Identification 3.12  Amplification Required in Front of the ADC 3.13 Digitization Effect on Sensitivity as a Function of a Number of Bits 3.14 Digitization Effect in the Instantaneous Dynamic Range Calculation 3.15  Curve Fitting for the Instantaneous Dynamic Range 3.16  IDR Calculated with 128 Data Points and Digitization 3.17  Generating Very High IDR Using Long Data Length 3.18  Conclusion   References  Chapter 4  Dynamic Range Study Through Fast Fourier Transform (FFT)   4.1    4.2    4.3    4.4    4.5    4.6    4.7    4.8    4.9  4.10  4.11  4.12  4.13  4.14  4.15  4.16  4.17 

Introduction Using Simulation Approach to Find the IDR Local Peaks Simulation Procedure Threshold Determination Windows and Input Frequencies IDR Results IDR with a Rectangular Window IDR with a Rectangular Window and Close Spaced Frequencies IDR with Hamming Window IDR with Blackman Window IDR with a Chebyshev Window IDR with a Park-McClellan Window Data Length and IDR Receiver Design Considerations Conclusion Remarks   References

26 27 27 27 28 28 29 30 31 33 36 37 42 43 45 47 48 51 51 52

53 53 53 54 56 57 58 59 60 62 63 65 67 69 71 72 73 75 75

Contents

 Chapter 5  In-Phase and Quadrature Phase (IQ) Study

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77

  5.1  Introduction   5.2  Approach to Find the IQ Imbalance   5.3  FFT Output Imbalance Measurement Procedure   5.4  Results from Measuring FFT   5.5  Imbalance Results of FFT Outputs   5.6  FFT Outputs from Imbalanced Inputs   5.7  Windowed FFT Output Imbalance Study   5.8 Procedure for Finding Phase Tracking After the FFT Operation   5.9 Procedure for Finding an IQ Imbalance of the Hilbert Transform 5.10 Results of an IQ Imbalance of a Hilbert Transform with a Rectangular Window 5.11 Results of IQ Imbalance of the Hilbert Transform with a Blackman Window 5.12  IQ Imbalance of Polyphase Filters 5.13 IQ Imbalance from a Special Sampling Downconversion Scheme 5.14  Conclusion   Reference

104 106 107

 Chapter 6  Signal Detection from Fast Fourier Transform (FFT) Outputs

109

  6.1  Introduction   6.2  Rayleigh Distribution Obtained from Noise Output   6.3  Signal-to-Noise (S/N) Distribution   6.4  Probability of Detection   6.5  Probability of Detection with a Blackman Window   6.6  Threshold Through the Convolution Approach   6.7  Threshold Obtained by a Gaussian Approximation   6.8  Probability of Detection with Summations   6.9 Threshold and Probability of Detection of the Polyphase Filter Approach 6.10 Summary of Sensitivity Calculations and Discussion and Final Adjustment by Considering the Number of Channels 6.11  Approach for Phase Comparison 6.12 Results from the 64-FFT Operation and Phase Comparison Aided with Amplitude Comparison 6.13  Create Additional Artificial Output Frequency Bins 6.14  Polyphase Phase Comparison Study and Basic Idea 6.15 Frequency Measurement Through Phase Comparison of a Polyphase Filter 6.16  Added Artificial Frequency Bins for the Polyphase Filter

77 78 79 80 82 86 89 91 92 94 97 99

109 110 112 113 115 116 119 121 121 123 124 125 128 131 133 135

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Contents

6.17  Decrease Shifting Time for Polyphase Filter (Long Short Shift) 6.18 Comparison of the Three Approaches for Finding a Fine Frequency of a Polyphase Filter 6.19  Conclusions References

139 141 142

 Chapter 7  Time-Domain Detection with 1-Bit ADC

143

  7.1  Introduction   7.2  Reduce Time Resolution and the Number of Windows   7.3 Conventional Time-Domain Measurement with Amplitude Information   7.4  Using Phase to Detect the Presence of a Signal   7.5 The Amplitude of the Correlation Output Is a Function of Frequency   7.6 Correlation Amplitude Change with a Specific Frequency and an Initial Phase   7.7  Moving Average Method with Different Window Lengths   7.8  Differential Moving Window   7.9  TOA and PW Calculation 7.10  Threshold Setting 7.11  Detailed Output Shape 7.12  Matched Window Determination 7.13  Ratio Method to Determine a Matched Window 7.14  Selecting a Short Window from the Match Window 7.15  TOA and PW Results 7.16  Sensitivity Test Results 7.17  Conclusion   References  Chapter 8  Eigenvalue and Related Operations   8.1  Introduction   8.2  Input Parameters to the Eigenvalue Problem   8.3  Simplified Approach   8.4  Matrix Formulation and Noise Eigenvalue Distribution   8.5 One Complex Signal and Noise Eigenvalue Distribution and the Probability of Detection   8.6  Matrix Order Effect   8.7  Two Complex Input Signals   8.8  Data Length Effect   8.9 Data Length Increase Through Summations of Shorter Matrices 8.10  Analytic Eigenvalue Solutions of a Low-Order Matrix 8.11  Eigenvalues Versus Initial Phase Difference 8.12  Eigenvalues and Frequency Separation

137

143 144 144 146 147 150 151 154 155 159 162 164 165 169 169 170 170 171

173 173 173 174 174 176 178 179 181 185 186 187 189

Contents

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8.13 Eigenvalue Threshold Method to Determine the Number of Signals 8.14  AIC and MDL Approaches 8.15  False Alarm Test 8.16  Input with One Signal and Two Signals 8.17  Effect of IQ Imbalance on Number of Signal Detections 8.18  Time-Domain Detection Using the Eigenvalue Method 8.19 Simulation of the Time-Domain Detection Using the Eigenvalues Method 8.20  Conclusion   References

198 201 202

 Chapter 9  Signals Close in Frequency Study and the MUSIC Method

203

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  9.1  Introduction   9.2 Input Signal Frequency Separation and Signal-to-Noise Ratio (S/N)   9.3  Study of the Order of the MUSIC Method for One Signal   9.4  Study of the Order of the MUSIC Method for Two Signals   9.5 Using the FFT Approach to Read Signals with a Close Frequency Separation   9.6 Detection of the Existence of Two Signals Close in Frequency from FFT Outputs   9.7 Detection of the Existence of Two Signals Close in Frequency from Eigenvalues   9.8  Frequency Identification with Close Frequency Separation   9.9  Conventional MUSIC Method 9.10  Low-Order MUSIC Method 9.11  Results from the Low-Order MUSIC Method 9.12  Frequency Selection for MUSIC Method 9.13  Conclusion   Reference

215 218 220 222 223 224 225 226

 Chapter 10  Digital Instantaneous Frequency Measurement (IFM) Receiver

227

  10.1    10.2    10.3    10.4    10.5    10.6    10.7    10.8    10.9  10.10  10.11  10.12 

227 228 229 230 231 234 236 238 240 241 243 245

Introduction Basic Concept of an Analog IFM Receiver Basic Digital IFM Receiver Hardware and Concept 1-Bit ADC Effect Number of Phase Difference Counts and Manipulations Signal-to-Noise (S/N) Effect on Angle q Ambiguity Resolution Simulation Results Threshold and Confirmation Performance of Two Simultaneous Signals Frequency Folding Time Resolution Improvement and Threshold with Hysteresis

203 204 205 206 210 210



Contents

10.13  10.14  10.15  10.16   

Imbalance of IQ Channels Hilbert Transform Converting a Real Signal to Complex Special Sampling Downconversion Transform Conclusion References

 Chapter 11  Receiver Designed Through a Conventional FFT Approach   11.1  Introduction   11.2  Requirements   11.3  FFT Length Selection and Frequency Resolution   11.4 Threshold Determined by the Probability of False Alarm Rate and the Probability of Detection   11.5  Threshold Adjusting   11.6 Frequency Reading Improvement Through Amplitude Comparison   11.7  Frequency Resolution on Two Signals   11.8  Detection of a Second Signal in a Receiver   11.9  PA Measurement 11.10  TOA and PW Measurements 11.11  Combine All the Information on One Input Pulse 11.12  Some Possible Improvements on an FFT-Based Receiver 11.13  Receiver Measurements 11.14  Summary   References  Chapter 12  Receiver Designed Through a Multiple FFT Operation   12.1  Introduction   12.2  Cascaded Filter Banks Through FFT Operations   12.3  Cascaded Filter Banks Through Polyphase Filters   12.4  Half Band Filter   12.5  Selection of FFT Lengths   12.6  Threshold Determination and Probability of Detection   12.7 Additional Detection Scheme to Improve Pulse Width Capability   12.8  Short Pulse   12.9  Long Weak Signal 12.10  Signals Detected by Multiple Numbers of Windows 12.11  Selection of Certain Windows 12.12  Two Signals in One Frequency Bin 12.13  Parameter Measurements 12.14  Conclusion   References

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255 255 255 256 258 261 262 265 268 269 271 272 273 274 276 276

277 277 277 279 281 283 285 287 289 290 291 293 294 296 297 297

Contents

 Chapter 13  Receiver Through a Polyphase Filter

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  13.1  Introduction   13.2  Methodology   13.3  Polyphase Filter Design   13.4  Property of the Polyphase Filter   13.5  Time-Domain Detection and Sensitivity   13.6  Rabbit Ear Generation   13.7  Detection Sequence and Rabbit Ear Effect   13.8  PW and FFT Operation   13.9  Determine the Number of Signals 13.10  Odd and Even Complex Receiver Outputs 13.11  Determine the Input Frequency 13.12  Frequency Resolution as a Function of Pulse Width 13.13  Amplitude, TOA, and PW Measurements 13.14 Minimum PW Limitation and Reduction of False Detection 13.15  Conclusion   References

319 320 320

 Chapter 14 Detection of Biphase Shift Keying (BPSK) Signals

321

  14.1  Introduction   14.2  Basic Barker Code Properties   14.3  Generation of BPSK Signals and Their FFT Outputs   14.4 Using FFT Outputs to Determine the Existence of a BPSK Signal   14.5 Study of Two Eigenvalues on Complex BPSK and CW Signals   14.6 Study of Three Eigenvalues on Complex BPSK and CW Signals   14.7  Define the Chip Time Limits of the BPSK Signals   14.8 Using Conventional FFT Receiver Outputs to Detect a BPSK Signal   14.9  Using Two Frames to Detect BPSK Signals 14.10  Using an Eigenvalue After the FFT Operation 14.11 Detecting BPSK with a Phase Comparison After the Polyphase Filter 14.12 Detecting BPSK with an Eigenvalue Ratio After the Polyphase Filter 14.13  Find the Phase Transition Locations in the BPSK Signal 14.14  Conclusion   References

299 299 300 304 306 308 310 310 311 312 314 316 318

321 321 323 326 331 331 333 333 336 339 341 342 343 347 348

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Contents

 Chapter 15  Frequency Modulated (FM) Signals   15.1  Introduction   15.2  A Chirp Signal in Time- and Frequency-Domain Outputs   15.3 Using FFT Outputs to Determine the Existence of a Chirp Signal   15.4 Using Eigenvalue to Determine the Existence of a Chirp Signal   15.5 Compare Chirp Signal Detecting Results from the FFT and Eigenvalue Methods   15.6  Recognizing Chirp Signals from Receiver Outputs   15.7  Frequency Measured Through an Amplitude Comparison   15.8 Signal Conditions for Chirp Detection After the Conventional FFT Operation   15.9  Chirp Output in One Frequency Bin 15.10  One CW Signal at Boundary of Two Frequency Bins 15.11  Chirp Output in Two Adjacent Frequency Bins 15.12  Chirp Signal After the Polyphase Filter Receiver Approach 15.13 Detecting a Chirp Signal After an FFT or a Polyphase Filter Operation 15.14  Phase Comparison After the Conventional FFT Operation 15.15  Phase Comparison After a Polyphase Filter Operation 15.16  Differentiating Against Two CW Signals 15.17  Summary   Reference

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 Chapter 16  Angle of Arrival (AOA) and Frequency Measurements

369

  16.1    16.2    16.3    16.4    16.5    16.6    16.7    16.8    16.9  16.10  16.11  16.12  16.13  16.14  16.15  16.16  16.17  16.18 

369 369 370 371 373 374 376 378 380 382 382 385 385 389 390 392 393 396

Introduction Define the Problem Signal Generation Normal and Simplified Approaches Base Line Performance Angle Measurement Processing Gain from Two-Dimensional Coherent Processing Frequency Conversion and Filtered Output Calibration 16-Element Antenna with Uniform Spacing Angle Measurement Through 16 Antenna Elements Frequency and AOA Conversion Two Examples for AOA Measurement Amplitude Comparison Two Simultaneous Signals Eigenvalue and MUSIC Method Nonuniformed Antenna Spacing AOA Measurement Through Nonuniformed Antenna Spacing Potential Front-End Design

Contents

16.19  16.20  16.21   

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Frequency Independent AOA Measurements AOA Operation Followed by the Frequency Operation Conclusion References

397 399 400 401

Appendix About the Author

403 409

Index

411

Preface This book can be considered as a sequel to Digital Techniques for Wideband Receivers. In that book many different approaches that might be applicable wideband were discussed. The main problem in building an electronic warfare (EW) receiver is in the encoder design. This subject is not discussed in Digital Techniques for Wideband Receivers, Second Edition, because the technology was not mature at the time. The recent advancement in field programmable gate array (FPGA) not only makes not only the encoder discussion viable, but also opens several areas for receiver designs. This book has four main goals. The first goal is to present a complete receiver design including the encoder. Although all the details cannot be described, the major elements are discussed. Four types of receiver designs are included: digital instantaneous frequency measurement (IFM), conventional FFT operation, multiple FFT operations, and polyphase filter. The digital IFM receiver should be a better approach, more robotic and lower cost than its analog counterpart. The conventional FFT receiver produces a base line receiver performance. The multiple FFT operation and polyphase approaches are to fulfill a dream design that the sensitivity and frequency resolution are pulse width dependent. The angle of arrival (AOA) obtained from an array of antennas is studied to obtain additional sensitivity. The second goal of this book is to introduce ideas in other fields to EW receiver designs. For example, eigenvalues from a low-order correlation matrix are used to determine the number of signals, which is a very important factor in EW applications. Also, Rayleigh distribution of the FFT outputs is used to determine threshold. The threshold of the FFT output amplitude summation is found by the convolution method. The multiple signal classification (MUSIC) method is used to identify signals with close frequency separation. The third goal of this book is to generate detailed information on some of the ideas discussed in Digital Techniques for Wideband Receivers, Second Edition. The dynamic range problem is studied through the MUSIC method with the digitization effect. The in-phase and quadrature (IQ) outputs imbalance of the FFT and the polyphase operation of the Hilbert transform and of the special sampling frequency are studied. These results are important for receiver designs. If the IQ channels are not well balanced, the dynamic range of the receiver will be limited. The last goal of this book is to discuss the detection of exotic signals. The signals are limited to biphase shift keying (BPSK) and frequency modulated (FM or chirp) signals. The characteristics of these two types of signals are discussed based on detection approaches. The detection will be performed after the FFT or polyphase filter operation to simulate the receiver effect. The existence of a signal must be confirmed first. The characteristics of the signal, such as chip rate of the BPSK and the chirp rate of the chirp signal, will be measured. xv

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Preface

All the topics presented in this book are closely related to an EW receiver design and some of them are discussed in detail. Therefore, the name special design topics is used in the book’s title. Most of the discussions are based on simulation results rather than on pure theoretical predications. Several of the often-used subroutines are listed in this book. In order to minimize simulation errors, similar problems are presented in different chapters. For example, the sensitivity of a receiver through different designs is listed in various sections. If all the results are comparable, the simulations can be considered correct. Unexpected results will be further studied and explained. This book is written at an undergraduate senior or graduate-school engineering level. It is written for researchers and managers working in the electronic warfare and communication fields. In almost all chapters examples are given. In order to compare the results from the different chapters, there are examples using similar conditions such as sampling frequency and input frequency ranges. In writing this book, I am in debt to my colleagues, Mr. David Lin and Dr. Liyeh Liou. Through discussions, they help me better understand the problems and avoid mistakes. Dr. Maan Broadstock worked with me on many subjects, and she produces many detailed results through simulations to reveal the inside details; I very much appreciate her assistance. I would like to thank Mr. Chuck Ward, a receiver engineer with more than 30 years of experience, who has given me a lot of good advice. I would also like to thank Dr. Stephen Hary, Mr. Luis Concha, and Mr. Todd Kastle for supporting some of the studies and reviewing my manuscript. I also thank Mr. Keith Graves for his help in reviewing my manuscript. I am in debt to many managers and engineers at AFRL—Mr. Chris Lesniak, Mr. Douglas Abner, Mr. Bob Blumgold, Mr. Jon Buck, Mr. Peter Buxa, Mr. Thomas Dairymple, Dr. Dan Janning, Mr. Matt Longbrake, Mr. John McCann, Mr. Bob Neidhard, Mr. John Norton, Dr. George Simpson, Dr. Dale Stevens, Mr. Kyle Zeller, and Ms. Desney Burrick—for their encouragement and support. I also thank Mr. John Zaccaria, Mr. Peter DiTore, Mr. Stuart Lopata, Mr. Rich Barozzi, Mr. Dan Jacobson, Mr. Brain Mayhew, and Mr. Dan Henry from ITT for their support of some research topics, valuable discussions on technical subjects, and measurement procedures for EW receivers.

Chapter 1

Introduction

1.1  Introduction In Digital Techniques for Wideband Receivers, Second Edition, many receiver design ideas were presented but not incorporated in a complete receiver design. In this book some new ideas are added and discussed in detail with simulation results to illustrate their performance. One of the most important functions of electronic warfare (EW) receiver is to generate the pulse descriptor word (PDW) for every input signal through some encoding schemes referred to as the encoder. The encoder design has not been discussed in previous books because of its complexity. Most of the performance limitations of a receiver are due to the encoder design. This book will study this subject and present some ideas. The advances in digital signal processing move at a tremendous speed. Some of the processing previously considered impractical can now be used in wideband digital receiver designs today. For example, if one digitizes the input signal at 3 GHz and wants to perform a 256-point fast Fourier transform (FFT), the processing speed cannot support such an operation in a continuous mode. In order to solve this problem, two approaches have been used in the past. First, the polyphase approach, also referred to as multirate processing, has been used. This approach decimates the input data by 32 and uses 256 input points of data to perform a sequence of 32-point FFTs. This approach widens the filter bandwidth of the receiver and results in reduced sensitivity and capability of separating signals close in frequency. The second approach uses the monobit idea that the kernel function of the FFT is reduced to a few bits to avoid multiplications in the FFT operation. This operation can be performed at high speed. Although the sensitivity and the single signal dynamic range of the monobit receiver are relatively high, the two-signal instantaneous dynamic range produced is rather low, only a few decibels. This property limits the receiver capability to receive simultaneous signals. Today’s technology can support the regular FFT operation. The performance of such a receiver built through an FFT operation can have both high sensitivity and wide single-signal dynamic range and two-signal instantaneous dynamic range.

1.2  Purpose of This Book The intention of this book is to answer several questions. The first one is to find the limits on the performance of receivers. For example, the instantaneous dynamic range is always a term argued by receiver engineers. Engineers have often claimed 



Introduction

results that are beyond realizable values. Someone may claim a high instantaneous dynamic range, but the result is obtained through observation on an oscilloscope display. Others may claim values from a system of receivers that may include several receivers. These claims generate great difficulties in receiver designs because the users always want some performance beyond what the technology can provide. Some people provide the dynamic range information through ignorance. Others provide the information to compete with a competitor. The receivers claimed usually cannot provide the advertised performance. In some rarely special condition, the receiver may generate a certain performance value but the value cannot represent the receiver performance in general. A modern electronic warfare (EW) receiver must have pulse descriptor words (PDWs) as outputs. Most receivers do provide PDW as outputs because EW processor requires this information to sort and deinterleave pulse trains and to identify the threats. In a receiver system it is possible that the receiver only indicates where a signal is at in a certain radio frequency (RF) range. This receiver may not require PDWs as outputs. The performance of this type of receiver should not be considered typical. In this book the performance is based on a receiver with PDWs as output. This book spends two chapters finding the maximum achievable instantaneous dynamic range. However, the theoretical limit is not available. Simulation results are used as the limit. In these simulations, ideal components are assumed. For example, the analog-to-digital converter (ADC) used in the simulation does not have unequal quantization levels and system clock does not have jitter. Thus, the ADC does not produce undesired spurs. The spurs produced will be caused by saturation or a lower number of bits. Although the limit is not obtained theoretically, years of observations on measured receiver show that it is difficult to achieve the simulated limits. The second goal is to obtain some detailed information on some methods discussed in [1]. For example, the fast Fourier transform (FFT) and Hilbert transform can generate complex outputs from real input data. The detailed performances of the complex outputs are not discussed. In this book the balance between the in-phase and quadrature phase of the complex outputs will be presented. This information is very important in actual receiver design. If the outputs are not balanced, the amplitude will vary with time, which will cause a detection problem and limit the dynamic range. In the frequency domain spurs will be produced due to the imbalance. The third goal is to provide additional receiver design techniques. For example, 1-bit data can be used to build a digital instantaneous frequency measurement (IFM) receiver. Although a monobit receiver [1, Chapter 12] can produce similar results, the program implemented in the field programmable gate array (FPGA) is much simpler. It is anticipated that this method can be used to build very wideband digital receivers. The fourth goal is to study the receiver sensitivity through the probability of detection and the probability of false alarm problem. It is a common practice since the days of the analog receiver to assume a signal-to-noise ratio (S/N) of 14 to define the desirable performance. However, it is difficult to measure the probability of false alarm. Using digitized data, one can actually find the noise distribution through simulation. From the distribution function, the probability of false alarm can be better defined. The fifth goal is the main one of this book: to provide a complete receiver design. The receiver design will start from digitized data through time-to-frequency conversion and finally produce the PDW. Three types of designs will be presented.

1.3  Predicated Requirements on Receiver Performance



One approach is considered as the general approach through a windowed FFT operation, and most EW receivers are designed through this approach. The second one is to use multiple FFTs with different lengths. Another is through polyphase filters and time-domain detection. The last two approaches can make the receiver sensitivity and frequency resolution pulse width (PW) dependent, which is a long-desired goal in EW receiver design. The sixth goal is to study modern radar signals. Although new radars may use very complicated signals, this study is limited to two types of signals: the biphase shift keying (BPSK) and frequency modulations (FM or chirp), because they are probably the most commonly used signals in threat radars. The intention is to detect these signals in a receiver through some special techniques. The seventh goal is to provide some references for receiver designers to use. For example, the balance of the I-Q outputs from the FFT operations is a function of the data length. On this subject many figures are generated from which the designers can select the FFT length to generate the desired results. The eighth goal is to introduce two-dimensional signal processing. Because of the advance in radio frequency (RF) design, building an antenna array with many elements, each followed by an RF chain and ADCs, are becoming a reality. The outputs of all these outputs must be properly processed to generate angle of arrival (AOA) information and improve receiver sensitivity through additional antenna gains. The overall goal is to produce a software package that can be used to evaluate a receiver design before actually manufacturing it. If a receiver can be sufficiently modeled, its performance can be predicted. This approach can reduce the risk in receiver design and speed up the overall design process. Such a program is not available in this book because the program needs desired design goals. Many of the subjects discussed in this book can be used as building blocks to accomplish it.

1.3  Predicated Requirements on Receiver Performance It is important to provide some idea of the performance of an ideal receiver so that an engineer can know what to expect in designing a receiver. However, this is also a debatable subject because some designers can surpass some aspects of performance but miss others. Under this condition, they will claim that they can exceed the performance of an ideal receiver. Table 1.1 lists the overall performance of an ideal receiver. The goal is to achieve all the requirements. Table 1.1  Specifications of an Ideal EW Receiver Input instantaneous bandwidth Minimum pulse detected with full sensitivity Minimum number of signals Receiver outputs Achievable frequency resolution on one signal Achievable frequency resolution on two signals Achievable time resolution Minimum detectable signal at output S/N Maximum two-signal instantaneous dynamic range Maximum one-signal dynamic range *Will be further clarified in Chapter 3.

1,000 MHz 100 ns 2 simultaneous PDW 1 MHz 5 MHz 50 ns About 14 dB 45 dB* 50 dB



Introduction

The receiver performance in Table 1.1 is based on experience or can be considered as an arbitrary assumption. The first four lines are requirements that the receiver must fulfill. One can surpass some of the performance if not all the requirements are fulfilled. For example, an IFM receiver can have wider input bandwidth, better minimum pulse width, high single signal dynamic range, and better frequency resolution, but cannot process two simultaneous signals. The achievable quantities are from a conventional receiver design, which will be discussed in Chapter 11.

1.4  Overall EW Receiver System Operation An EW system can be illustrated in Figure 1.1. There are three basic portions: the EW receiver, the EW processor, and the technique generator. The EW receiver generates the PDW, which usually includes five parameters: signal carrier frequency, pulse amplitude (PA), pulse width (PW), time of arrival (TOA), and AOA. Every input radar pulse should generate a corresponding PDW. The information from the EW receiver will be processed by an EW processor, which will separate all received radar signals into pulse trains. Each pulse train is supposed to belong to a specific radar. This processing is called deinterleaving. Except for the AOA, the other four parameters can be controlled by the radar operator who can change them. The AOA information is the most important one to perform deinterleaving because the radar cannot change is position easily even it is an airborne one. The TOA information can be used to generate pulse repetition frequency (PRF) or its reciprocal, the pulse repetition interval (PRI). The pulse train information such as RF, PW, and PRF can be used to compare with known stored radar data to determine the radar type. If the received radar is a threat, countermeasures may be needed. The technique generator will produce the proper jamming signals. It is interesting to know that although AOA is the most valuable information for deinterleaving, it is not useful in identifying radars because it does not contain information on the radar itself. If the EW generates false information, the EW processor may spend a fair amount of effort to figure it out. Any receiving system can generate false alarms under certain conditions. The general rule for EW receiver design is that it should miss one signal than generate a false one. The missing signal may be picked up at a later time but the false signal will be processed by the EW processor.

Figure 1.1  A typical EW system.

1.5  Encoder Designs



1.5  Encoder Designs An EW receiver can be considered as a unique type of receiving system. Not only it does not know the input signal, but the input signals are also trying to evade interception by the receiver with low probability intercept (LPI) radars. The receiver is required to intercept the input signal and provide characteristic information on it. The PDW is the information describing the input signal. An EW receiver is shown in Figure 1.2. In this figure there are five important parts: the antenna, the RF chain, the ADC, the time-to-frequency conversion unit, and the encoder. The PDW is the final product. Antenna research is a well-established field. There are researches on RF chains, and many times they are referred to as RF receivers. This reference is appropriate because many analog receivers are designed from RF input to intermediate frequency (IF) output. The IF is converted into a video signal through a crystal video detector for further processing. There are continuous researches on ADC to achieve high sampling frequency with a large number of bits. The output data from the ADC are in the time domain. In order to generate frequency information, usually the first step is to change the time domain information into frequency domain information through a time-tofrequency conversion process. There are several approaches to the time-to-frequency conversion such as finite impulse response (FIR) filters and fast Fourier transform (FFT) operations. These conversion processes are well established. Since this processing is continuous, the outputs contain both frequency- and time-domain information. In Figure 1.2, four of the five blocks have been studied. However, it is difficult to find information on the encoder designs. The majority of the receiver design issues are in generating the PDW from the time and frequency outputs. This processing is often referred to the encoder design. In order to design a receiver with high sensitivity, high dynamic range, and high frequency resolution, the encoder design can be very complicated. One major problem or obstacle is that most engineers cannot appreciate the difficulty of encoder design unless they try themselves. From past experience, it is found that most (about 90%) of the receiver problems are in the encoder designs. Hopefully, from some of the detailed design procedures, one can start to recognize the problem. The encoder design has been a difficult task since the analog receiver time. Before the field programmable gate array (FPGA) was introduced, the encoder was designed with hardware logic and based on the video outputs. The logic is rather tedious and

Figure 1.2  Basic building blocks of an EW receiver.



Introduction

there is no basic rule to follow. Since the development of the FPGA, the encoder has become a manageable task. Thus, this subject will be discussed in this book.

1.6  Approaches and References In this book many of the subjects are based on recent studies. Very few subjects are obtained from other references; references on wideband digital receivers are sparse. Most of the results have been obtained from computer simulations and the results are considered as expected. Unfortunately, some of the results cannot be explained theoretically. Under this situation, the results will be presented as obtained. For example, the eigenvalue varies as a function of input frequency and the length of the lags in making the correlation matrix to find the number of signals. Although many references are available on the eigenvalue, even related to stock market analysis, it appears that the applications on EW receivers are difficult to obtain. Thus, the number of signals is determined by observing the eigenvalue variation empirically. Using computer simulation has one risk: If the program has an error, the results can be erroneous. In order to minimize the errors, the results will be compared with similar results obtained with a slightly different approach. A more important approach is to check against the expected values. For example, if the data length is doubled, the processing gain should increase by 3 dB. In most examples in this book a sampling frequency of 2.56 GHz is used. The input data are usually real, and the corresponding Nyquist bandwidth is 1.28 GHz. Since the data are real, there are two frequency outputs, which will interfere with each other at the ends of the band: near 0 and 1,280 MHz. In order to minimize this interference effect, both ends of the band will be eliminated. The minimum bandwidth eliminated at the ends is arbitrarily chosen as 10 MHz. If the study is closely related to a receiver design, the band is typically selected as from 141 to 1,140 MHz to cover a 1,000-MHz bandwidth. The goal is to design a receiver with 1,000-MHz bandwidth at a 2.56-GHz sampling frequency.

1.7  Criterion of the Software Approaches In developing new methods to design a receiver, one should consider whether the methods can be implemented in hardware. In this book this concern will not be stressed. From past experience, many of the techniques considered impractical in the past become common practices in receiver design, such as high-speed ADC with many bits and relatively long real-time FFT operations. The methods considered in this book will not be limited to those that can be supported by today’s technology. Most of the FFT-related operations, irrespective of their length, are considered feasible to implement. Many repetitive simple operations are also considered feasible. Some highly computational intensive operations such as high-order (about 20) multiple signal classification (MUSIC) method, are not considered in the receiver design algorithms. Although the MUSIC method is used to generate some reference data such as instantaneous dynamic range, the operation is not in real time. It is believed that the technology may not support such operations in the near future unless

1.8  Organization of the Book



a breakthrough is developed. However, a low-order MUSIC method up to the third order is considered in the design because analytic solutions are available. With this consideration, the algorithms developed should be hardware applicable in the near future.

1.8  Organization of the Book Many different independent subjects are discussed in this book. The majority of subjects are the techniques of receiver designs. Chapter 2 is the overview of the RF chain design, discussed in [1, Chapter 7] with minor modifications and insightful information such as noise and quantization level. This information is repeated here for convenience of receiver designers. Chapter 3 discusses dynamic range through the MUSIC method, which is the subject that causes most receiver design confusions. Chapter 4 continues the discussion on dynamic range through FFT operations. Chapters 3 and 4 are devoted to instantaneous dynamic range because it is a loosely defined term. Chapter 5 concentrates on the performance information of in-phase (I) and quadrature (Q) phase processing. Chapter 6 studies the output of an ADC, the noise distribution, and the probability of detection. Chapter 7 discusses the time-domain detection for a 1-bit ADC. The outputs of windows with different lengths will be used to determine the desired results. This method has the potential to build a receiver with a very wide bandwidth. These three chapters provide detailed information on some well-known subjects. Chapter 8 uses the eigenvalue method to determine the number of input signals. Chapter 9 presents the MUSIC method and its capability of identifying two input signals with close frequency separation. These two chapters discuss new approaches that might be included in receiver designs. The remaining chapters discuss different methods of receiver construction. Chapter 10 presents a new approach to build an instantaneous frequency measurement (IFM) using a 1-bit ADC. Chapter 11 presents a basic receiver design with an FFT operation. Chapter 12 presents a receiver with multiple numbers of FFT operations to improve the sensitivity and signal separation on long pulses. Chapter 13 uses polyphase filters to build a receiver with goal similar to that in Chapter 12: to improve performance on long signals. Chapters 14 and 15 discuss the detection and processing of biphase shifting keying (BPSK) and frequency-modulated (FM) signals, respectively. Chapter 16 studies the angle of angle (AOA) problem with a linear antenna array. Many different subjects are presented; sometimes there are similarities between them. In order for the readers not to go back and forth to find the subject, each issue will be presented briefly when presented. Therefore, minor duplications might occur in some sections.

References   [1]  Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Norwood, MA: Artech House, 2001.

Chapter 2

Amplification Required in Front of the ADC

2.1  Introduction This chapter is a summary of [1, Chapter 7]. This discussion is presented here because the results are used in several chapters later in this book. The detailed discussion and derivation will not be included. Readers who are interested in the details can look in the reference. Ward used the results and added a few modifications [private communication with C. Ward, retired engineer from ITT, 2004]. One of the modifications is that when the input impedance of the analog-to-digital converter (ADC) and the output impedance of the radio frequency (RF) chain are not matched, an impedance mismatch applies. The second modification is that when a window is introduced in the fast Fourier transform (FFT) operation, a window factor will be introduced. In previous discussions of this subject, a certain gain value is chosen from the noise figure and dynamic range (DR) curves. The noise figure is directly related to the receiver sensitivity. The dynamic range discussed in this chapter is the singlesignal dynamic range (SDR). The subject of dynamic range will be discussed in detail in the next chapter. In general, higher sensitivity means lower dynamic range, or vice versa. In some receiver designs a higher sensitivity might be selected at the cost of slightly lower SDR. Each person may select a different combination of sensitivity and SDR. A nominal condition, where the product of sensitivity and SDR are maximized, is introduced here. The term nominal condition rather than optimum condition is used, because although the nominal condition may be the best choice from a mathematical point of view, it may not provide the specific desired result such as a certain sensitivity or dynamic range. The main goal of this chapter is to outline the computer program and how to use it. The discussion is based on the computer program.

2.2  Basic Design Criterion Although the details’ derivation is not discussed, the basic ideal will be briefly mentioned here. The lower limit of the sensitivity of a receiver is set by the noise floor or by the spurs generated in a nonlinear effect in the RF chain. In designing a receiver, the spurs generated by the third-order intermodulation are taken into consideration. The third-order intermodulations are generated by two equally powered 

10

Amplification Required in Front of the ADC

Figure 2.1  Third-order intermodulation in the frequency domain.

strong signals separated by a proper frequency such that the intermodulations are also in the input band of the receiver. The ratio of the third-order intermodulation level to the two-signal level is referred to as the two-signal third-order intermodulation dynamic range. This dynamic range is not a very important parameter in receiver performance because the required signal conditions seldom occur in a signal environment but it is important in receiver designs. However, this dynamic range

Figure 2.2  Third-order intermodulation from the output versus input plot.

2.3  Inputs to the Computer Program

11

Figure 2.3  Block diagram of a simple digital receiver.

can be related to the SDR. When the gain in the RF chain is high, the sensitivity of the receiver is high and it easily generates spurs limiting the SDR. When the gain is low, the sensitivity is low and the SDR is high. This is a common trade-off in receiver designs. The key issue in designing a receiver is selecting the RF gain such that when the input reaches the maximum allowable level of the ADC, the third-order intermodulation products equal the noise level. The third-order intermodulation is shown in Figures 2.1 and 2.2. Figure 2.1 shows the result in the frequency domain. For one signal to reach the saturation level of the ADC, the voltage swing must equal to the ADC saturation voltage. For two signals combined to reach the maximum allowable ADC input, each signal will be 6 dB below the desired value. In Figure 2.1 the intermodulation is equal to the noise level. Figure 2.2 shows the output versus the input and the intermodulation level. The relationships in this figure are used in deriving the required amplification. Figure 2.3 shows a simple block diagram of a typical receiver. It contains only three components: the RF amplifier, the ADC, and the FFT operator.

2.3  Inputs to the Computer Program The computer program is used to process real input data. The explanation of the program will follow the actual steps in the program. 2.3.1  The Inputs Related to the RF Amplifier



The thermal noise floor when the input temperature of the RF amplifier is at room temperature is -174 dBm/Hz. This value can be obtained from the thermal noise power as (2.1) N1 = kTB = −174 dBm where k = 1.38 × 10-23 J/K is Boltzmann’s constant, T = 290K is the room temperature, and B = 1 Hz is the unit bandwidth. In the following equations when dB or dBm are labeled, it indicates that a logarithmic operation is used. For example, in (2.1), N1 = 4 × 10-21, W = 4 × 10-18 mW, and 10log(N1) = -174 dBm, or simply N1 = -174 dBm.

12

Amplification Required in Front of the ADC

The noise figure (F) of the amplifier is given in decibels, but the gain (G) and third-order intercept point (Q3) are values to be obtained from the program. The definition of third-order intercept point can be observed from Figure 2.2 labeled Q3 on the y-axis. The sampling frequency is fs. The input RF bandwidth Br is usually equal to or less than half the sampling frequency for real input data. The RF system is assumed to have an impedance R of 50 ohms. Let us assume a threshold for determining the sensitivity of the designed receiver. A nominal required signal-to-noise ratio (S/N) is 14 dB. Under this condition, a threshold can be assigned so that the input signal can be detected approximately 90% of the time with a probability of false alarm of 10-7, which is a commonly accepted detection criterion. 2.3.2  The Inputs Related to the ADC

The number of bits of the ADC is b and the maximum voltage is Vs, where the unit of Vs is usually in mv. The maximum voltage is defined as half the peak-to-peak value. From these values the quantization level can be calculated. The input impedance of the ADC is Ra. If Ra = R = 50 ohms, where R is the RF system impedance, the RF and the ADC impedance are properly matched. If Ra is not equal to 50 ohms, the mismatch is considered as an insertion loss in the RF chain. The insertion loss can be calculated through Figure 2.4. The power delivered to Ra is 2



 V  P=  Ra  50 + Ra 

(2.2)

The power transfer is maximum when Ra = R. Under this condition the power V2 . Using this as the reference, the power transfer in decibels for transfer is Pm = 200 Ra is

Mismatch = 10log

 200Ra  P = 10log   dB Pm  (50 + Ra )2 

(2.3)

This term is considered as the insertion loss in decibels in the RF chain. When Ra = 50 ohms, the mismatch equals zero.

Figure 2.4  Source and ADC impedance representing insertion loss.

2.4  Constants Generation

13

2.3.3  The Inputs Related to the FFT Operator

The data length N and the window function are used in the FFT operation. The window function is represented by a window factor, which is the output main lobe width compared with that for a rectangular window. For example, the main lobe of a rectangular window is 0.027344, and a Blackman window is 0.050781. These values can be obtained from the “wintool” command in MATLAB. The window factor can be calculated as 1.857 (0.05078/0.02734), where the rectangular window is used as the reference. Another value for the Blackman window factor is 1.73 obtained from an equivalent noise bandwidth; see Table 4.1. For one special case, the smaller window factor of 1.73 has slightly (about 0.3 dB) better sensitivity and DR. In the following discussions, the value of 1.73 is used for the Blackman window factor. For further information, see [2]. In general, the window factor is greater than or equal to 1. A nominal S/N of 14 dB is assumed for determining the sensitivity of the receiver. Finally an M value is used as an input. This input parameter is defined as the ratio of RF chain noise output No to the quantization noise Nb as

M=

No Nb

or

M = No − Nb

dB

M1 ≡ M + 1 Md ≡

M M1

Md ≡ M − M1



(2.4)

dB

Since Nb is a fixed value for a certain ADC, M is proportional to No, which is related to RF gain G. The value of M is selected as from (1/2)5 to (1/2)-9 in (1/2)0.1 steps, and M1 and Md are defined for convenience.

2.4  Constants Generation From the inputs, several constants will be generated. First, the maximum power to the ADC will be obtained.



Ps =

Vs2 V2 W, for Vs in V, Ps = s ×10−3 mW, for Vs in mV 2R 2R

(2.5)

where Vs is the maximum input voltage to the ADC and R is the system impedance. Since the input is assumed a sine wave, the effective voltage value will be Vs /Ö2. The quantization noise Nb is

Nb = Ps − 6.02b − 1.76

dB

(2.6)

14

Amplification Required in Front of the ADC

where b is the number of bits and 1.76 is a constant from 10log(3/2). The equivalent video bandwidth Bv is Bv = w ×



fs N

(2.7)

where fs is the sampling frequency, w is the window factor, and N is the data length. The output noise from the RF amplifier No is No = Nb + M



dB

(2.8)

where M is the input variable. The gain of the RF chain is

G = No − N1 − F − Br − Mismatch

dB

(2.9)

where N1 is the input thermal noise at room temperature for 1 Hz, F is the amplifier noise figure, and Br is the RF bandwidth. Since No is a function of M, the G will be a function of M as mentioned in the previous section. The corresponding input power PI to reach the saturation power Ps is PI = Ps − 6 − G



dB

(2.10)

This equation indicates that the input and output powers are related by the gain G. The factor 6 is introduced because two signals of equal amplitude are required to reach the desired voltage Vs as discussed in the previous section.

2.5  Equations Derived From the input and the generated constants, the following equations can be obtained. The derivation of the equations is included in [1]. The overall noise figure of the system is Fs = F + Md



dB

(2.11)

where F is the noise figure of the amplifier and Md is introduced in (2.4). The third-order intermodulation P3 can be written as

P3 = N1 + G + Bv + Fs

dBm

(2.12)

where N1 is from (2.1), G is the RF gain, Bv is the video bandwidth from (2.7), and Fs is the overall noise figure. The required third-order intercept point (Q3) is



Q3 =

3PI − N1 + 2G − Bv − Fs 2

dBm



(2.13)

where PI is the input power from (2.10). The values of Fs and Q3 will be used to determine the G of the RF chain. In obtaining Q3, the relation of the third-order intermodulation equal to the noise floor is used.

2.7  An Example

15

Finally, the SDR and the sensitivity of the system can be obtained from (2.11) to (2.13) as DR = PI + G − P3 −



S N

dB

(2.14)

The SDR is the ratio of input power reflected to the output (PI + G) to the third intermodulation level, taking the threshold into consideration. The required S/N is usually 14 dB. The sensitivity SEN is Sen = N1 + Bv + Fs + S / N



dBm

(2.15)

which is the noise floor (N1 + Bv + Fs) plus the threshold level. The gain of the RF chain can also be obtained from SDR and SEN, which will produce the same results as obtained from Fs and Q3.

2.6  Modification from the Previous Program Let us refer the program in [1] as the original program. The basic program in the previous sections stays the same. Ward added three modifications. The window function is incorporated into the video bandwidth, Take the RF and ADC impedance mismatch as an insertion loss in the RF chain. Ward also suggested adding the sensitivity calculation and a required S/N value for both the SDR and sensitivity calculations. In the original program there is an arbitrarily chosen value (Vn) related to the noise. This value is subtracted from the maximum voltage Vs to prevent the ADC from saturation. This adjustment is deleted from the present program because by using an ADC with low number of bits, the sensitivity and SDR have abnormal behaviors caused by Vn.

2.7  An Example An example is used to illustrate the outputs of the results. The input conditions are: · · · · · · · · ·

F = 3 dB, noise of RF amplifier; fs = 2,560 MHz; sampling frequency; Br = fs/2, RF bandwidth; b = 8 [ADC has 8 bits]; ADC Vs = 1,000 ms, maximum voltage; Ra = R = 50 ohms, ADC input impedance and RF system impedance are equal; n = 256, FFT length; Window factor = 1 [rectangular window]; S/N = 14 dB.

The noise floor is at 1,280 MHz is -83 dBm (-114 dBm/MHz +10 × log (1,280)). With a 3-dB noise figure, the noise floor is at -80 dBm.

16

Amplification Required in Front of the ADC

Figure 2.5  Overall noise figure and minimum required Q3 versus RF gain.

The results are shown in Figures 2.5 through 2.7. Figure 2.5 shows the overall noise figure and the minimum required third-order intercept point versus the RF gain. Figures 2.6 and 2.7 show the SDR and sensitivity, respectively, versus RF gain. One can see that the selection of gain affects the SDR and the sensitivity. Higher gain has better sensitivity but lower SDR, or vice versa.

Figure 2.6  SDR versus RF gain.

2.8  Nominal Sensitivity and Single Signal Dynamic Range

17

Figure 2.7  Sensitivity versus RF gain.

The designer can pick a set of either Q3 and Fs or SDR and sensitivity and determine the required gain. It should be noted that the RF gain must be equal to the selected value. However, Q3 is the minimum required value, which means a higher Q3 value will not have an adversary effect.

2.8  Nominal Sensitivity and Single Signal Dynamic Range In the previous section each designer may pick different combinations of sensitivity and SDR depending on the requirements. However, one can find the rate of change of Q3 and Fs, and find the maximum of the product of them. In other words, find dQ3 dFs and use the maximum as the desired value. In this calculation, it is im× dG dG portant to choose the M value in (2.4) correctly. If the M value is chosen incorrectly, the curve will not have a peak. The M value is chosen as 2n, where n starts from 0 and increases monotonously. If the M is selected this way, the gain value in decibels on the x-axis is uniform. dQ3 dFs . At the maximum value, the following Figure 2.8 shows the plot of × dG dG results are obtained. For m = 1, the gain G = 40 dB, the required Q3 = 35 dB, the overall noise figure Fs = 6 dB, the SDR = 48 dB, and the sensitivity = -84 dBm. Since the receiver may have some definite requirements, this selection method is named nominal rather than optimum. One can obtain similar results by taking the differential of SDR and sensitivity, for example, the location of the peak, measured possibly in the value M. The M value is either 0.933 or 1, probably caused by a round-off error. The M value of

18

Amplification Required in Front of the ADC

Figure 2.8 

dQ3 dFs × versus gain. dG dG

1 is used as the nominal condition as suggested by Ward. Since Q3 and Fs are primary quantities and SDR and sensitivity are derived quantities for the program, Q3 and Fs will be used for the nominal value selection.

2.9  Generate Nominal Values for ADC with Different Numbers of Bits In this section, let us find the RF gain, the minimum required third-order intercept point Q3, the overall noise figure Fs, the SDR, and the sensitivity again. All the inputs stay the same except the ADC. The number of bits and the maximum voltage of the ADC will be changed. The ADC number of bits changes from 1 to 16, and two maximum voltages, 270 and 1,000 mv, are used. For ADC bits from 1 to 8, the voltage is 270 mv, and from 8 to 16 the voltage is 1,000 mv. For the 8-bit ADC two voltage values are assigned. The window factor is assumed to be 1 (rectangular window). The results are shown in Table 2.1, with all the parameters listed as in Section 2.7. In Table 2.1 the nominal condition used is M = 1. The ADC number of bits does not affect the overall noise figure because under the nominal condition, the overall noise figure is about double the noise figure of the RF amplifier or increased by 3 dB. Since the noise figure is constant, the sensitivity does not change with the number of bits either. This is a reasonable conclusion because the receiver sensitivity should be independent of the ADC used. It is interest to note that the DR calculated is very close to 6 dB per bit. These are expected results and also agree with results obtained in the next chapter.

2.10  Noise Floor and the Number of Bits

19

Table 2.1  Calculated Results as a Function of the ADC Number of Bits Number of Bits 1 2 3 4 5 6 7 8 (270 mv) 8 (1,000 mv) 9 10 11 12 13 14 15 16

G dB 70.78 64.76 58.74 52.72 46.70 40.68 34.66 28.64 40.01 33.99 27.97 21.95 15.93 9.91 3.89 –2.13 –8.15

Q3 dBm 2.55 5.56 8.57 11.58 14.59 17.60 20.61 23.62 34.99 38.00 41.01 44.02 47.03 50.04 53.05 56.06 59.07

Fs dB 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01 6.01

SEN dBm –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99 –83.99

SDR dB 5.84 11.86 17.88 23.90 29.92 35.94 41.96 47.98 47.98 54.00 60.04 66.04 72.06 78.08 84.01 90.12 96.14

2.10  Noise Floor and the Number of Bits [1] It is interesting to find the noise level at the input of the ADC such that the number of bits triggered by noise can be determined. The two maximum voltages Vs will be used to study this problem. The noise floor is at -80 dBm at the input of the ADC. Although the overall noise of the receiver is 6.01 dB, the noise figure from the input to the ADC can be considered as 3 dB because at nominal condition, the quantization noise doubles the noise of the receiver. The equivalent noise at the ADC input can be found from Table 2.1 and listed in Table 2.2. In Table 2.2, the second column is the nominal gain. The third column is the noise at the ADC input, which is obtained by -80 plus the gain. With an ADC of 15 and 16 bits, the gain is a negative value. This means that one may add an attenuator to reduce the overall gain, but the noise floor cannot go below -80 dBm. The equivalent noise voltage is listed in column 4 and the quantization level Q is listed in column 5. The ratio of the equivalent noise voltage to the Q value is shown in column 6, and the value is close to 0.28. The noise distribution in the time domain is assumed to be a normal distribution, and the equivalent noise to Q value ratio can be considered as the standard deviation. From this value of 0.28 the input noise and the first two thresholds are shown in Figure 2.9. The noise will trigger the lowest two levels. Another example is from [1]. The bandwidth is 30 MHz; thus, the noise floor figure is 3.3 dB, the ADC has 8 bits with maximum voltage of 270 mv, the sampling frequency is 250 MHz, and the length of the FFT is 1,024. The selected gain is 57 dB and the nominal gain is 45 dB. The noise floor is at -96 dBm (-114 + 3.3 + 14.7). With a 57- and a 45-dB gain the noise floors are at -39 and -51 dBm and the equivalent noises are 0.0025 and 6.3 × 10-4v. The 8-bit ADC with a maximum voltage of 270 mv and the corresponding Q value is 2.11 × 10-3v. The noise equivalent voltages to Q ratios are 1.18 and 0.30, respectively. The quantization is shown in Figure 2.10. Figure 2.10(a) shows the 57-dB gain case and the noise will

20

Amplification Required in Front of the ADC Table 2.2  Equivalent Voltage at the ADC Input Number of Bits

G dB

Noise dBm

1 2 3 4 5 6 7 8 (270 mv) 8 (1,000 mv) 9 10 11 12 13 14 15 16

70.78 64.76 58.74 52.72 46.70 40.68 34.66 28.64 40.01 33.99 27.97 21.95 15.93 9.91 3.89 –2.13 –8.15

–9.22 –15.24 –21.26 –27.28 –33.30 –39.32 –45.34 –51.36 –39.99 –46.01 –52.03 –58.05 –64.07 –70.09 –76.11 –80 –80

Equivalent noise v 0.0774 0.0387 0.0193 0.0097 0.0048 0.0024 0.0012 6.046 × 10-4 0.0022 0.0011 5.60 × 10-4 2.8 × 10-4 1.40 × 10-4 7.00 × 10-5 3.50 × 10-5 2.24 × 10-5 2.24 × 10-5

Q value v 0.27 0.135 0.068 0.034 0.017 8.44 ´ 10-3 4.22 ´ 10-3 2.11 ´ 10-3 7.81 ´ 10-3 3.91 ´ 10-3 1.95 ´ 10-3 9.77 ´ 10-4 4.88 ´ 10-4 2.44 ´ 10-4 1.22 ´ 10-4 6.10 ´ 10-5 3.05 ´ 10-5

Equivalent noise v/Q 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.37 0.8

trigger about three or four levels. Figure 2.10(b) shows the nominal gain where the noise triggers about two levels. From Figure 2.10, one can see that noise can be at a level to trigger two to three levels. If the noise can reduce a spurious response as discussed in [1], one might increase the noise to trigger several levels of the ADC.

Figure 2.9  The quantization level at nominal gain and the noise distribution.

2.11  Another Example

21

Figure 2.10  (a) Gain of 57 dB (b) gain of 45 dB with a bandwidth of 30 MHz and noise distribution.

2.11  Another Example In this section another example, one is applicable to global positioning system (GPS) receiver, is used to illustrate the application. The parameters are: · · · · · · ·

F = 1 dB noise of RF amplifier; fs = 5 MHz sampling frequency; Br = fs/2 RF bandwidth; b = 16 ADC has 16 bits; ADC Vs = 1,000 ms maximum voltage; Ra = R = 50 ohms [ADC input impedance RF system impedance are equal]; n =5,000 FFT length.

The SDR and sensitivity are shown in Figures 2.11 and 2.12, respectively. Since the nominal GPS signal is about -130 dBm, it is desirable for the receiver to achieve this result. A gain value of about 35 dB can achieve -129 dBm, which is close to the desired value. From this example, one can see that no gain is required of a 16-bit ADC to build an EW receiver with the desired parameters given in Section 2.7. However, for a different type of receiver, such as that for GPS applications, gain is required for a 16-bit ADC. In this example, the nominal gain is about 21 dB. If this value is used, the sensitivity is close to -126 dBm, which is slightly worse than the desired value of -130 dBm. In a GPS receiver, the sensitivity is in general of primary interest. Thus, the nominal choice is not the desired one. In an actual commercial GPS receiver the ADC used only has 1 or 2 bits. The required gain is very high. The use of a 16-bit ADC in the GPS receiver can be

22

Amplification Required in Front of the ADC

Figure 2.11  SDR versus gain for GPS receiver.

Figure 2.12  Sensitivity range versus gain for GPS receiver.

2.12  Discussions of Results

23

considered for special purposes such as receiving the GPS signal under strong jamming conditions.

2.12  Discussions of Results From the above discussions there are some interesting results. The overall noise figure and the sensitivity of a receiver are basically independent of the ADC used, as discussed in Section 2.9. The sensitivity of a receiver depends only on the noise figure, the bandwidth, and the required S/N; it is independent of the ADC number of bits. The gain required is decreased when the ADC number of bits is increased. In the first example the required gain reduced from over 70 dB for a 1-bit ADC to -8 dB for a 16-bit ADC. When the gain values become negative, it means that the noise is already high enough in the receiver and the lower number bits are triggered by noise. It is interesting to note that when the number of bits is high, gain in the RF amplifier is no longer needed. In contrast, one can build a receiver without RF gain. From this same argument, if one can decrease the saturation voltage of an ADC, it may also not require an RF gain. This subject is further discussed in Section 16.18. The SDR increases by increasing the ADC number of bits. The SDR is very close to 6.02 dB per bit. It can reach 96 dB for a 16-bit ADC. This SDR is from the required S/N to the saturation of the ADC input. Two 8-bit ADCs are used in this study: one with a maximum voltage of 270 mv and the other one with a maximum voltage of 1,000 mv. It is interesting to note that the sensitivity and SDR are almost independent of the maximum voltage. The only difference is the RF gain required. For lower maximum voltage ADC, the RF gain required is less. It appears that if a 13-bit ADC is built with a maximum voltage of 270 mv, RF gain may not be needed to build a receiver with the specifications given in Section 2.7.

References   [1]  Tsui, J., Digital Techniques for Wideband Receiver, 2nd ed., Chapter 7 Norwood, MA: Artech House, 2001.   [2]  Meikle, H., Modern Radar Systems, Norwood, MA: Artech House, 2001.

Chapter 3

Dynamic Range Study Through Eigenvalue and MUSIC Methods

3.1  Introduction The purpose of this study is to produce a guide on dynamic range (DR). Theoreti­ cally, dynamic range should be a very clearly determined term for electronic warfare (EW) receiver engineers. The dynamic range relates to the capability of a receiver to measure strong and weak signals. An EW receiver is a unique type of receiver that measures noncooperative signals. EW receivers are required to measure time coincident signals without apriority information, a requirement that most com­ munication receivers do not have. Let us refer to time coincident pulses as simulta­ neous signals. Some receivers are designed to receive up to four simultaneous signals, a considerably large number. Because of this special property of EW receivers, they have in general three definitions of dynamic range. Engineers talking about dynamic range without a specific definition can cause confusion. Worse yet, some engineers deliberately mislead others to believe that they have achieved a high dynamic range. The purpose of this chapter is to try to clarify different definitions of dynamic ranges. Although Chapter 2 discusses dynamic range, it concerns only the one signal case. In this chapter, dynamic range is concentrated on the two-signal instantaneous dynamic range (IDR). Simulations are used to find the limitations on the IDR. The IDR is obtained through two methods: the eigenvalue decomposition and the MUl­ tiple SIgnal Classification (MUSIC) methods. The IDR is obtained as a function of input frequency separation. Finally, the digitization effect will be added to the calculation of IDR, which is a function of an analog-to-digital converter (ADC) bit number. One important factor is that the dynamic range is pulse width (PW) dependent. PW should be associated with the dynamic range value.

3.2  Basic Definitions of Dynamic Range There are, in general, three definitions of dynamic range; each one will be discussed in a later section. The basic definition will be presented here [1]. A dynamic range is bounded by two limits: the lower limit and the upper limit. The lower limit is the weakest signal that can be detected. In some definitions of dynamic range, the sen­ sitivity level of the receiver is often the lower limit. The sensitivity is defined as an 25

26

Dynamic Range Study Through Eigenvalue and MUSIC Methods

input signal that is strong enough to break a certain threshold. The upper limit is the strongest signal that the receiver can measure without error or spurious output. Even this basic definition may have different meanings. For example, if a re­ ceiver measures the amplitudes of the input signals, its capability of measuring the weakest to the strongest signals may be considered the dynamic range. Under this situation, the maximum dynamic range is usually about 6b dB, where b is the number of bits of the ADC. Sometimes the dynamic range is determined by the receiver’s frequency measurement capability. If a receiver can measure both the sig­ nal amplitude and frequency, the receiver may have two different dynamic ranges. For example, if an instantaneous frequency measurement (IFM) receiver measures both signal amplitude and frequency, and the signal amplitude can be measured from -60 to 0 dBm and the frequency can be measured from -70 to 10 dBm, the two dynamic ranges of the receiver are 60 and 80 dB. In modern receivers the dynamic range is often defined by the receiver’s fre­ quency measurement capability. The weakest input signal whose frequency can be encoded correctly (say, 90% of the time with 10-7 probability of false alarm, which implies that the signal-to-noise ratio is about 14 dB [1]) and is the lower dynamic range limit. The correct output frequency is often defined by the error frequency, which is the output frequency minus the input frequency. Usually the acceptable er­ ror frequency is measured by the frequency resolution such as one or two frequency resolutions. A strong signal can drive some components in the receiver into satura­ tion. When this happens, spurious signals may be generated. The level at which a strong signal causes the receiver to report a spurious signal is considered to be the upper limit. The dynamic range in this case is the difference between these two limits. For example, if the lower limit is -60 dBm and the upper limit is +10 dBm, the dynamic range is 70 dB. This high value usually occurs to receivers that can measure only one signal. In general, both the lower and upper limits are functions of input frequency. As a result, the dynamic range is a function of frequency. If one value is used to repre­ sent the dynamic range, the average value used should not be the best value.

3.3  Prerequisite for Dynamic Range Measurements For a meaningful discussion on dynamic range, the receiver must be able to generate a pulse descriptor word (PDW). The PDW is generated on a pulse-by-pulse basis, including information such as frequency, pulse amplitude, pulse width, time of ar­ rival (TOA), and so forth. In several cases, engineers have reported the dynamic range of a receiver that did not generate a PDW. In other words, the dynamic range has been observed on an oscilloscope or a spectrum analyzer. This type of measure­ ment cannot perform on a pulse-by-pulse base, which is the desired result. The hu­ man brain is much more powerful than an encoder and the eyes can integrate the displayed signals. As a result, the reported data can be very good. An encoder needs to determine the number of signals from only one set of out­ puts usually in the frequency domain. While the eyes can identify input signals from many pulses, the encoder must determine whether a signal is true and false from one short set of data. In reality an “electronic warfare receiver” without an encoder

3.5  Dynamic Range for Receiver with Multiple Signal Capability

27

to generate PDW cannot be considered a complete receiver and the dynamic range quoted does not have much meaning. In the following discussions, the dynamic range is based on the frequency read­ ing capability of the receiver because this information is often considered to be the most important. Strictly speaking, a correct reporting should include all the four parameters: frequency, pulse amplitude, pulse width, and TOA. If all four param­ eters are included, the dynamic range should be lower because the requirements are difficult to meet. The three definitions of dynamic range will be explained based on two types of receivers: single and multiple signals.

3.4  Single Signal Receiver Dynamic Range (SDR) If a receiver is designed to receive only one signal such as an IFM receiver, there is only one definition of dynamic range from the frequency measurement capability and that is the single signal dynamic range (SDR). This is the receiver’s capability to only measure one signal frequency correctly from the minimum input to the maxi­ mum input levels. For this type of receiver it is rather easy to achieve a relatively high dynamic range. Since the receiver can only encode one signal, it will not pro­ duce false alarms on a second signal. It is anticipated that even a 1-bit ADC can pro­ duce a very high dynamic range. It has been demonstrated from existing hardware that a receiver with a 1-bit ADC can achieve about 70 dB of dynamic range. Thus, the dynamic range of this type of receiver is not of interest for this discussion.

3.5  Dynamic Range for Receiver with Multiple Signal Capability If a receiver is designed to process simultaneous signals, there are three definitions of dynamic ranges: the single-signal dynamic range, the two-signal third-order in­ termodulation spur free dynamic range, and the two-signal instantaneous dynamic range [2–5]. 3.5.1  Single-Signal Dynamic Range

The definition of SDR is the same as above. In general, this dynamic range is smaller than that of a receiver with single signal measurement capability because the receiver must detect a second signal. When the signal is strong, it will generate spurious outputs. When the receiver detects a spur as a signal, it is the upper limit of the SDR. The SDR is usually limited by the ADC and the PDW encoder following the frequency analysis operation, often the fast Fourier transform (FFT) outputs. If the ADC is driven into satu­ ration, the output will contain spurious responses such as harmonics. If the spurs are higher than a certain threshold, the receiver will detect them as signals. As a result, the receiver will generate false alarms. Usually, the spur generation will be the upper limit of the dynamic range of the receiver. This is why the SDR of a receiver with multiple signal capability is usually lower than that of a receiver designed for only one signal. The DR calculated from the previous chapter is the SDR.

28

Dynamic Range Study Through Eigenvalue and MUSIC Methods

The SDR can be improved with different threshold methods. For a conventional analog receiver the threshold is a fixed voltage level. When an output crosses the threshold, the signal is detected. Under this condition, a strong input signal can generate spurs. When a spur crosses the threshold, a false alarm is produced. This determines the upper limit of the SDR. With digital receiver a fixed threshold can be used to detect the existence of a signal. Once a signal is detected, a variable threshold can be used to detect a second signal. This threshold can be set based on the amplitude of the first signal. The spur level can be predetermined from a strong input signal. The threshold can be set above the spur level. In other words, the threshold is set below the amplitude of the strong signal. Under this condition, the spur should not be detected even if the input signal is very strong; thus, the cor­ responding SDR can be very high. In order to detect a second signal, the signal must be strong enough to cross the variable threshold. Base on this approach, a high SDR is not a difficult requirement to achieve. However, there is also limit on SDR with this detection method, especially when the signal is digitized. If the strong signal reaches the saturation level of the ADC, the amplitude of the strong signal will stay constant. Increasing the signal amplitude beyond saturation will increase the spur level. Thus, the spur can be detected as a signal. 3.5.2  Two-Signal Third-Order Intermodulation Spur Free Dynamic Range

This is also known simply as the two-signal spur free dynamic range. The spur free dynamic range was defined in Chapter 2. To measure this dynamic range, one must inject two signals of the same ampli­ tude and separate them in frequency so that the two third-order spurs are also in the input band of the receiver. When the signal amplitude increases, the third-order spurs will be detected as false signals. The upper limit of the dynamic range is the strong signal and the lower limit is either the required threshold above noise or the spur level generated through the FFT operation. The concept of third-order spurs is very important in receiver design. The am­ plification required before the ADC in an EW receiver is determined by the thirdorder spur level as discussed in the previous chapter. The two-signal spur free dynamic range is a receiver design criterion; however, it may not be a very important parameter in receiver performance. For example, one needs two signals of the same amplitude and their summation must be close to the ADC saturation level. The frequency separation of the two signals must be close enough that the intermodulation spur must be in the receiver bandwidth. Under this rarely occurring condition, two-signal spur free dynamic range can be measured. Because this condition may occur very rarely under real operational conditions, this dynamic range does not have much usage in real-world applications and will not be included as a receiver performance requirement. 3.5.3  The Two-Signal Instantaneous Dynamic Range (IDR)

The definition of instantaneous dynamic range is the capability of the receiver to simultaneously process a strong and a weak signal. The maximum difference be­ tween the strong and weak signals is considered the instantaneous dynamic range.

3.6 A Brief Discussion on the Eigenvalue Decomposition and MUSIC Methods

29

The theoretical bound of the instantaneous dynamic range has been studied in [2–5] as the Cramer-Rao (CR) bound. In [5], the variances of the frequency, amplitude, and phase measurements are used to evaluate the CR bound. It is concluded that the CR bound of each signal depends on the frequency separation, the phase differ­ ence between the two signals, and their signal-to-noise ratio (S/N). The amplitude of one signal will not affect the CR bound of the other. In general, when two signals are close in frequency, the measured frequency variance has large value reflecting a low instantaneous dynamic range. When the frequency separation increases, the instantaneous dynamic range increases. Beyond a certain frequency separation the CR bound becomes a constant because it depends only on the S/N of the measured signal; therefore, the corresponding instantaneous dynamic range also reaches a constant value. From the above discussion, it appears that the simple use of dynamic range can be a rather confusing term because there are several different definitions. In this book in order to minimize the confusion, the SDR and IDR are used. The IDR is measured from the frequency measurement capability of the receiver. If the ampli­ tude measurement is used to determine the dynamic range, it will be specified. Only the instantaneous dynamic range (IDR) will be discussed in this chapter. One more important parameter affecting the IDR is the PW. The upper limit of the IDR is at the saturation level of the ADC. The lower limit depends on the input data length (or the PW). Longer data processing can improve the receiver sensitivity and detect a weaker signal. Under this situation the upper limit stays at a constant level and the lower limit depends on the PW. Therefore, the IDR is PW dependent. This argument can also be applied to SDR and IDR with an amplitude measure­ ment capability. Thus, in defining IDR one must provide the PW and the frequency separation. If the frequency separation is far apart, the IDR will be independent of the separation; however, it still depends on the PW.

3.6  A  Brief Discussion on the Eigenvalue Decomposition   and MUSIC Methods In order to find the limit of instantaneous dynamic range, the eigenvalue and the MUSIC methods are used. Here the methods are briefly discussed. The eigenvalue decomposition method can be considered as a general high-resolution spectrum es­ timation method. It takes the input data and forms correlation matrix. Eigenvalues will be found from the correlation matrix. The amplitudes of the eigenvalues can be used to determine the number of signals. The MUSIC method was invented by Ralph Schmidt in 1981. It uses the eigenvalue decomposition approach to separate the input into the signal and noise spaces. This method is known for its capability to separate signals with a close frequency separation. In EW receivers, the number of simultaneous signals (time overlapping signals) is unknown. As a rough estimation, the number of signals in a certain time frame and a certain frequency range could be varied from 0 up to 4. If the number of input signals is assumed too high, the MUSIC process will produce spurious outputs. If the number is too low, it may miss signals. Therefore, assuming a certain number

30

Dynamic Range Study Through Eigenvalue and MUSIC Methods

of signals is not a very practical approach. However, during the MUSIC method, the eigenvalues will be obtained. The eigenvalues will provide information on the number of signals. In the MUSIC method, the order of operation must be selected. When a low order such as 5 or 6 is selected, the computation required is reasonable. It appears that a low-order MUSIC method does not have the capability of separating signals close in frequency but higher orders such as 80 to 100 can separate signals with close frequency. When a higher order is used, the computation is rather complicated and at present time is not suitable for an EW receiver application because the EW receiver requires a real-time response. The higher-order MUSIC method is used only to determine the limits of the IDR. In this study, the input number of signals is limited to two. Their relative ampli­ tudes and frequency separation are the input variables. From this study it appears that the eigenvalues may determine the number of signals under certain conditions, but their frequencies cannot be identified through the MUSIC method. For example, when two signals are separated by 1 MHz and the strong signal has S/N = 100 dB with 60 dB in amplitude difference, the eigenvalues can indicate there are two input signals. However, when the MUSIC method is applied to determine the frequency, the output contains only one peak rather than two. Under other input signal condi­ tions the MUSIC method may show two frequencies, but one of the frequencies can be wrong. In both cases it should be considered that the MUSIC method cannot identify the correct frequencies. Thus, the instantaneous dynamic range may be determined by two methods. One method is to find the existence of two signals through the eigenvalue method. This is very important because the EW receiver requires the determination of the number of signals. The other approach is not only to find the existence of two sig­ nals but also to determine their frequencies through the MUSIC method. The IDR of these two methods can be considered as the limit case or a guide for the EW receiver design. In the simulation, the frequency identification is decided by the frequency separation rather than the individual frequencies. For example, if two signals are separated by 1 MHz and the frequency separation measured is also 1 MHz, the measurement is considered correct. Under the highest amplitude separation (say, 39 dB), this condition can be achieved with a given S/N of the strong signal, and the IDR is 39 dB. It is observed that when the frequency separation is measured correctly, each individual frequency is also correct.

3.7  Define the Processing Procedure [6] The simulation method is used to study the IDR problem and 256 points of input data are used because this is one of the common data lengths used in EW receiver designs. If another number of data points is desired, the same approach will be used to determine the instantaneous dynamic range. The input contains two continuous wave (cw) signals and noise. In the first case, the input data are not digitized. In the second case, the input data are digitized by different number of bits. The instanta­ neous dynamic range of a different number of bits will be decided.

3.8  Eigenvalues Generated with Noise and Noise Plus Signals

31

The noise in the input signal is a function of receiver input bandwidth. In this study, it is assumed that the sampling frequency is 2.56 GHz and the input data are real. The corresponding bandwidth is 1.28 GHz, which has a noise floor at about -83 dBm. Arbitrarily assume that the receiver noise figure is 3 dB and the overall noise floor is at -80 dBm. This value is used for this study. The order of the MUSIC method is arbitrarily chosen as 80 and 100 because the general rule [6] is that the optimum order is about one-third to one-half of the input data points of 256. Another reason to choose these high orders is due to experiments with an order of 10, 20, and 40. These lower orders do not provide a satisfactory solution. For orders of 80 and 100, there are 80 and 100 correspond­ ing eigenvalues, respectively. If the input contains only one complex signal, only one eigenvalue represents the signal and the rest represent noise. Since the input of this study is limited to two complex signals, only the three largest eigenvalues (two corresponding to the signal and one corresponding to noise) will be evaluated. The purpose is to find the signal effect on the noise eigenvalue. The basic idea is to de­ termine the number of input signal from the three largest eigenvalues.

3.8  Eigenvalues Generated with Noise and Noise Plus Signals The first step is to test the effect on the three largest eigenvalues. Since the input sig­ nal is real, the corresponding eigenvalues are in pairs. For example, if there is only one input signal, there will be two large eigenvalues. If the eigenvalues are sorted by their amplitude, the first one will be the largest, and the last one (80th) will be the smallest. The eigenvalues selected for this test are the first, the third, and the fifth, and they are referred to as the largest three eigenvalues. The second- and the fourthlargest eigenvalues are ignored. In later discussions only the first, third, and fifth eigenvalues are used and are referred to as first, second, and third for simplicity. The eigenvalues are found from the following matrix Rm:



 R(0)   R(1) Rm =   � R(p)

R(1)*

�

R(0)

�

R(p − 1)

�

R(p)*   R(p − 1)*    R(0) 

(3.1)

where R is the correlation results obtained with different lag p such as R(p) = ∑ z*i z j

where

i−j=p

ij



zi = A cos(2π fi ti + θ ) + n

(3.2)

where * represents complex conjugate and zi is input data at time ti, A is the signal amplitude, fi is the input frequency, q is the initial phase angle, and n is the noise. In this study, the data are real and the complex conjugate operation in not needed.

32

Dynamic Range Study Through Eigenvalue and MUSIC Methods Table 3.1  Eigenvalues Generated from Noise Alone Mean 2.08 1.73 1.55

Eigenvalue 1 Eigenvalue 2 Eigenvalue 3

Standard Deviation 0.43 0.26 0.21

The eigenvalues and eigenvectors are defined as  R(0)   R(1)   � R(p)



 λ 0v00 λ v 0 10 =  �   λ 0v p0

R(1)*

�

R(0)

�

R(p − 1)

�

λ1v01 λ1v11

� �

λ1v p1

�

R(p)*   v00  R(p − 1)*   v10  �  R(0)  v p0

v01 v11

� �

v p1

�

v0 p  v1p    v pp  (3.3)

λ pv0 p  λ p v1p    λ pv pp 



T

where the v00 v10 � v p0  is one of the eigenvectors and λ0 is the corre­ sponding eigenvalue and the superscript T represents transposing a matrix. For an 80 by 80 matrix, there are 80 eigenvalues and eigenvectors. A MATLAB program (eig) can be used to find the eigenvalues and eigenvectors. When there is only noise, the three highest eigenvalues and their standard devia­ tions are listed in Table 3.1. The noise used is pseudo-randomly generated with a Gaussian distribution with a zero mean and a variance of 1. These values are generated from 1,000 runs. In every run 256 points of noise are generated. It shows that the higher the mean, the higher the standard deviations. One strong signal of S/N = 100 dB is used as input. The purpose of this opera­ tion is to check the signal effect on all three largest eigenvalues. Their values are listed in Table 3.2. These results are also obtained from 1,000 runs with random Gaussian noise with zero mean and variance of 1 and a strong signal. The S/N of 100 dB can be considered as an extremely strong signal. Under normal conditions, the input signal may not be this high. In order to avoid signal overlapping at the end of the input bandwidth (0 to 1,280 MHz), 10 MHz are avoided from both ends of the input

Table 3.2  Eigenvalues Generated from One Strong Signal (S/N = 100 dB) and Noise Eigenvalue 1 Eigenvalue 2 Eigenvalue 3

Mean 1.42 × 1011 2.04 1.70

Standard Deviation 7.14 × 109 0.42 0.26

3.9  IDR Determination Through Eigenvalues

33

Table 3.3  Eigenvalues Generated from Two Strong Signals (S/N = 100 dB) and Noise Eigenvalue 1 Eigenvalue 2 Eigenvalue 3

Mean 3.7 × 1010 4.37 × 108 2.06

Standard Deviation 2.14 × 109 2.23 × 107 0.43

bandwidth. The frequency of the signal is randomly selected between 10 and 1,270 MHz. The input frequency is randomly selected between these two values. The first eigenvalue increases tremendously. It is interesting to note that the second and third eigenvalues are approximately equal to the first and second values in Table 3.1. These are the desired results. One strong signal only affects one eigenvalue. If one signal affects more than one eigenvalue, it is difficult to use this method to deter­ mine the number of signals. The final test uses two strong signals. Both signals are 100 dB above noise. The results are shown in Table 3.3. The first signal is randomly selected between 12 and 1,270 MHz, and the second signal is 2 MHz lower than the first one. The first two largest eigenvalues are quite different and they are different by about two orders in magnitude. The important factor is that the third eigenvalue stays about the same value as eigenvalue 1 as shown in Table 3.1. From these results one can conclude that by measuring the eigenvalues one can determine the number of signals. Thresh­ olds are usually required in the process to determine the number of signals. There are other methods to determine the number of signals without thresholds and they will be discussed in Chapter 8.

3.9  IDR Determination Through Eigenvalues In order to determine the IDR, through the eigenvalues, thresholds are needed. It is difficult to find the distribution of the noise eigenvalues; thus, a theoretical thresh­ old is not available. It is arbitrarily chosen the threshold to be the mean value plus 10 times the standard deviation. This approach provides a very high threshold and the false alarm generated should be extremely low. The thresholds for the three largest eigenvalues are [6.45, 4.23, 3.54]. Comparing with the mean values in Table 3.1, they are quite large. In order to find the IDR, the following steps are taken. Select the first signal (the strong one) randomly in frequency and phase with S/N = 100 dB. The S/N is very strong for the first signal in order to minimize the noise effect on the second signal. The frequency is randomly selected between 60 to 1,270 MHz instead of 10 to 1,270 MHz. When the frequency of the second signal is 50 MHz below the first one, the lower frequency limit is still 10 MHz. The second signal is at a fixed frequency below the first one. The initial amplitude of the second signal is the same as the first one. The amplitude of the second signal is decreased by 1 dB steps and at each step all the eigenvalues are calculated. If the second eigenvalue is above the threshold, the amplitude of the second signal is decreased by 1 dB again. The procedure continues. When the second eigenvalue is below the threshold, the previous amplitude value is used as the dynamic range. Since this method is rather

34

Dynamic Range Study Through Eigenvalue and MUSIC Methods

Figure 3.1  Two signal dynamic range obtained with an 80-order eigenvalue method using eigenvalues as threshold.

time-consuming, the calculation at each frequency is repeated only 10 times. The results of the 10 calculations are averaged into one data value and it is the reported IDR. The results are shown in Figure 3.1, which can be referred to as the twosignal IDR. The results are amazingly good. When two signals are separated by only 1 MHz and their difference is separated by over 60 dB, they can be detected. This result is far beyond FFT capability, which is the expected result from an EW receiver. In the above simulations, the 80-order Rm matrix is used. In order to test the effect on the order on the instantaneous dynamic range, orders 50 and 100 are tested. The thresholds must be generated for these two tests. The results are shown in Tables 3.4 and 3.5. From these tables, it is shown that the eigenvalues are order dependent, and the higher the order, the larger the eigenvalues. Table 3.4  Eigenvalues Generated from Noise Alone Order = 50 Eigenvalue 1 Eigenvalue 2 Eigenvalue 3

Mean 1.84 1.54 1.39

Standard Deviation 0.35 0.21 0.16

3.9  IDR Determination Through Eigenvalues

35

Table 3.5  Eigenvalues Generated from Noise Alone Order = 100 Eigenvalue 1 Eigenvalue 2 Eigenvalue 3

Mean 2.19 1.81 1.61

Standard Deviation 0.47 0.30 0.23

Similar plots are obtained from the MUSIC method with the order equaling 50 and 100. The results are shown in Figures 3.2 and 3.3. There is very little differ­ ence between the results in Figures 3.1 and 3.3. In Figure 3.1, when the frequency separation is about 20 MHz, the IDR reaches the constant value of about 105 dB. While in Figure 3.3, the IDR reaches the constant value of 105dB at about 15 to 20 MHz. However, comparing Figures 3.1 and 3.2, the difference is slightly bigger and the IDR reaches the constant value of 105 dB at about 30 to 35 MHz. The constant dynamic range of 105 dB is likely limited by the noise level rather than the first signal. Since the strong signal is arbitrarily assigned to 100 dB about noise, when the weak signal is 105 dB below the strong one, the weak signal is about -5 dB below the noise. This is about the limit that the signal can be detected with 256 points of data. Since the difference between orders 80 and 100 is rather small, in order to save calculation time, the MUSIC method with an order equal to 80 will be used for the following studies.

Figure 3.2  Two signal dynamic range obtained with a 50-order eigenvalue method using eigenvalues as a threshold.

36

Dynamic Range Study Through Eigenvalue and MUSIC Methods

Figure 3.3  Two signal dynamic range obtained with a 100-order eigenvalue method using eigenvalues as a threshold.

3.10  MUSIC Method The MUSIC method [7, 8] is a very unique way to find signals close together in frequency. In order to keep this discussion simple, complex data are used for the explanation. Under this condition only one signal affects the amplitude of one ei­ genvalue, not two as in the case of a real signal. The operation is based on the eigenvalues and eigenvectors. From the above discussion, one can see that the am­ plitudes of the eigenvalues are signal dependent. If there are only two signals, two eigenvalues have relatively large values and the rest correspond to noise. In gen­ eral, the eigenvalues are divided into two groups: the signal and the noise eigen­ values. If there are M signals, λ0 to λM-1 are the signal eigenvalues, and λM to λp are the noise eigenvalues, where the subscript p represents the correlation matrix size. The corresponding eigenvectors are



 v00 v 10 Vs =  �  v p0

v01 v11

� �

v p1

�

v0M −1  v1M −1    v pM −1 

 v0M v 1M Vn =   �  v pM

v0M +1 v1M +1

� �

v pM +1

�

v0 p  v1p    v pp 

(3.4)

3.11  IDR Determined by Frequency Identification

37

where Vs and Vn represent the signal and noise eigenvectors, respectively. The idea of the MUSIC method is to use the orthogonal property of the signal and noise subspaces. Assume that the input signal vector s is

s = [1 e − j 2π f

� e − j 2 π(N −1)f ]

(3.5)

This vector is orthogonal to the noise subspace. One can write a function Pmus with the frequency f as the variable



Pmus (f ) =

1 sVnVnH s H

(3.6)

where the superscript H is the Hermitian operation. If the value f is equal to the input signal frequency, the denominator of Pmus(f ) is very small, and Pmus(f ) has a very sharp maximum. From this maximum the input frequency can be identified. It is interesting to note that the frequency resolution in (3.5) can be arbitrarily chosen. In other words, the frequency chosen can be very fine and is not limited by the signal length, as in the FFT operation.

3.11  IDR Determined by Frequency Identification The IDRs obtained through the eigenvalues in Section 3.10 are quite high. At these values the frequencies of the two signals are measured. In order to separate two signals by 1 MHz, the frequency resolution of the MUSIC method must be higher than the frequency separation. In the FFT operation, the frequency resolution de­ pends on the total number of points or the signal length. In this study, the sampling frequency is 2.56 GHz, and 256 data points is 100 ns in time. The correspond­ ing frequency resolution is 10 MHz if the FFT operation is used. For the MUSIC method the frequency resolution does not depend on the total number of points. In order to separate two signals by 1 MHz, a minimum of 4 data points are generated per megahertz. Since the total frequency coverage is 2,560 MHz, the total output data in the frequency domain are 10,240 (4 × 2,560). From Figure 3.1, when the two signals are separated by 1 MHz and 62 dB, the eigenvalue method can identify two signals. However, under this input signal con­ dition, only one peak instead of two can be found through the MUSIC method, as shown in Figure 3.4. Figure 3.4(a) shows the overall frequency plot, which contains only one peak. Figure 3.4(b) is a close-up plot of the peak and it confirms that there is only one frequency. This means that although the eigenvalues can indicate two signals being pres­ ent, the MUSIC method can identify only one of them. In order to find the two sig­ nals, the frequency separation is kept at 1 MHz and the amplitude separation must be decreased. The new definition of IDR used in this section is two signals being identified by their frequencies. The amplitude separation is reduced to 39 dB by keeping the frequency sepa­ ration at 1 MHz. Under this condition, the frequency plot is shown in Figure 3.5. Figure 3.5(a) shows the overall frequency plot, which contains only one peak.

38

Dynamic Range Study Through Eigenvalue and MUSIC Methods

Figure 3.4  Frequency plots of two signals separated by 1 MHz and 62 dB: (a) over the entire frequency range, and (b) a frequency close to the signals.

3.11  IDR Determined by Frequency Identification

39

Figure 3.5  Frequency plots of two signals separated by 1 MHz and 39 dB: (a) over the entire frequency range, and (b) a frequency close to signals.

40

Dynamic Range Study Through Eigenvalue and MUSIC Methods

Figure 3.6  Frequency plots of two signals separated by 2 MHz and 51 dB: (a) over the entire frequency range, and (b) a frequency close to signals.

3.11  IDR Determined by Frequency Identification

41

Figure 3.5(b) is a close-up plot of the peak area of Figure 3.5(a) and it clearly shows two peaks separated by about 1 MHz. It should be noted that the peaks generated by the MUSIC method do not represent amplitude of the signals and they only rep­ resent the position of the signal in frequency domain. Similar results are generated for a frequency separation of 2 MHz and an amplitude difference of 51 dB, as shown in Figure 3.6. Figure 3.6(a) shows the overall frequency plot and Figure 3.6(b) is a close-up plot of the peak area of Figure 3.6(a). Using the frequency reading capability of the MUSIC method, a new set of dynamic range can be obtained and the results are shown in Figure 3.7. The pro­ cedure is similar to the generation of Figure 3.1. The first signal (the strong one) is 100 dB above noise and the frequency is arbitrarily selected between 60 and 1,270 MHz with a random phase. The second signal is at a fixed frequency below the first one. The amplitude of the second signal is decreased by 1-dB steps. At every step, the MUSIC method is used to determine the frequencies. The frequencies are deter­ mined by measuring the peaks in the frequency domain. The peak is defined as a value, which is higher than both its adjacent values by more than 0.1 dB. The value of 0.1 dB is arbitrarily chosen. If two peaks are found and their frequency difference equals the input value, the MUSIC method is considered to find both frequencies correctly. The condition for qualifying the correct frequency separation is that the

Figure 3.7  Two-signal IDR obtained with an 80-order MUSIC method using frequency identification.

42

Dynamic Range Study Through Eigenvalue and MUSIC Methods

measured frequency separation and the input frequency separation are within ±3 MHz, which is also arbitrarily chosen. At every frequency this approach is repeated 10 times; the averaged value is plotted in Figure 3.7. The results obtained from this method cannot be easily related to the probability of false alarm and the probability of detection. This approach is used in order to save the program running time. Figure 3.7 is similar to Figure 3.1. At saturation, which occurs at a frequency separation of 15 MHz and beyond, the IDR in Figure 3.7 is about 5 dB lower than that in Figure 3.1. At a closer frequency separation such as at 1 or 2 MHz, the dif­ ferences are greater. Comparing the results of Figures 3.1 and 3.7, the difference is about 24 dB (62 - 38) for 1 MHz and 18 dB (75 - 57) for a 2-MHz separation. If the two frequencies cannot be obtained through the MUSIC method, the fre­ quency may not be obtained through the FFT operation because the FFT operation has less capability of separating two signals close in frequency. Since the IDR defined in Sec­ tion 3.5 is based on the frequency measurement capability, the IDR obtained through frequency separation by the MUSIC method will be used in future discussions. The IDR obtained above does not include the digitization effect. The digitiza­ tion must be introduced in the study so that the IDR obtained can be used as a guideline for receiver designs.

3.12  Amplification Required in Front of the ADC The amplification required on the ADC [9] has been discussed in Chapter 2. In Table 3.6, two sets of RF gains are selected. One is arbitrarily chosen and the gain values are greater than the nominal values obtained from Table 2.1. These gains provide better receiver sensitivity but smaller SDR. The second set of gains is the nominal values. Table 3.6 shows the selected gain values. In Table 3.6, the number of bits starts from 4 rather than 1. In a practical receiver design a low number of bits may not require a linear amplifier. A linear amplifier operating in saturation becomes nonlinear. For example, in the monobit receiver design [9], a 2-bit ADC is used. The amplifier in front of the ADC is a lim­ iting amplifier, which is a nonlinear component. It is assumed that when the ADC used in a receiver design has at least 4 bits, linear amplifier in front will be consid­ ered. That is why Table 3.6 starts from 4 bits. Two different maximum voltages are assumed for the ADCs: 270 and 1,000 mV. The 8-bit ADC appears twice in the table with different maximum voltages. Table 3.6  RF Gain Values Selected for Different Number of Bits Peak-peak volts (Vpp) Maximum volts (VM) Number of bits Amplifier gain (dB) Peak-peak volts (Vpp) Maximum volts (VM) Number of bits Amplifier gain (dB)

Arbitrarily Selected Gain 0.54V 2V 270 mV 1V 4 6 8 8 60 45 35 45 Nominal Gain 0.54V 2V 270 mV 1V 4 6 8 8 52.7 40.7 28.6 40.0

10 34

12 25

14 10

16 0

10 28.0

12 15.9

14 3.9

16 –8.1

3.13 Digitization Effect on Sensitivity as a Function of a Number of Bits

43

3.13  D  igitization Effect on Sensitivity as a Function of   a Number of Bits An ADC with a low number of bits can cause the sensitivity to decrease slightly. The decrease in sensitivity will affect the dynamic range value [10]. A simulation is used to illustrate this point. Using the same sampling frequency of fs = 2.56 GHz and real input signal, the equivalent bandwidth is 1.28 GHz, which corresponds to a noise floor at -83 dBm. Assuming a receiver noise figure of 3 dB, the noise floor is at -80 dBm. The input frequency is put on a frequency bin such as at 640 MHz with a rectangular window, and a 256-point FFT is performed on the input data. Since the frequency is on one of the frequency bins, it is easy to separate the signal from the noise in the frequency domain. The output S/N can be calculated as the signal power on the specific bin divided by the average noise power in the rest bins. The term noise is used in this calculation but for an ADC with a low number of bits, the noise includes many spurious outputs. This operation produces 21 dB of gain for a real signal because the input bandwidth to processed bandwidth is from 1,280 to 10 MHz and 10×log(128) is about 21 dB. If the input signal is real at -87 dBm, then, after processing, the signal is at -66 dBm (-87 + 21). Since the noise is at -80 dBm, the output S/N = 14 dB (-66 (-80)), which is the required value to produce the desired detection condition. It is interesting to know that using a 1-bit ADC can degrade the processed out­ put signal-to-noise ratio (S/N) [9]. When the signal is digitized by an ADC with a low number of bits, the processed output S/N will suffer. Figure 3.8 shows these re­ sults. The results are obtained from the following procedure. The input frequency is fixed at 640 MHz with a fixed amplitude and the noise is random. Each data point is obtained from the average of 1,000 noise runs. The S/N is calculated from the frequency domain outputs. Two sets of S/N are calculated. One set is that the input signal and noise are not digitized and in the other set the inputs are digitized. The digitization process does not follow an ordinary operation in a receiver. The digitization process is to find the maximum value of the input signal and use this value and the number of bits to find the digitized levels. For example, for a 3-bit ADC, the following procedure is used.

s3d (n) = ceil[4s(t) / sm ] − 0.5

(3.7)

where s3d(n) is the digitized output level of s(t), ceil is a MATLAB operator to find the lower integer of the division, 4 is half of the quantization levels (3 bits have 8 levels), sm is the maximum value in the input signal s, and 0.5 is a factor to adjust the digitized output. Using this discussion, all the levels of the ADC are exercise. The purpose of this process is to study the number of bits problem. In conventional receiver design, when the noise is low, only a few levels are triggered. The input level is from -89 dBm to -79 dBm in 0.5-dB steps. The x-axis is the output S/N without digitization, and the y-axis is the output S/N with digitization. With a 1-bit ADC, when the S/N on the x-axis is 14 dB, the y-axis reads about 12 dB in Figure 3.8(a). The exact degradation is 1.96 dB, which is the result given in [10]. However, the 2-bit ADC has a degradation of 0.86 dB, which is different from the 0.55 dB listed in [10]. A 3-bit ADC only has a 0.22-dB degradation. The degra­ dation of an ADC with 4 or more bits is insignificant.

44

Dynamic Range Study Through Eigenvalue and MUSIC Methods

Figure 3.8  S/N of output versus digitized output: (a) near detection level, and (b) over a wide input range.

3.14 Digitization Effect in the Instantaneous Dynamic Range Calculation

45

Figure 3.8(b) shows a wider range of input power. The same procedure is used, only the input is from S/N = -94 to -34 dBm covering the 60-dB range. It is inter­ esting to note that when the signal is strong, the S/N suffers more degradation as shown in Figure 3.8(b). The degradation at high S/Ns will not affect the detection process, but it will affect the signal amplitude measurement capability. For example, a 1-bit ADC can only generate a processed output S/N of 21 dB, which is above the desired value of 14 dB. The receiver has an amplitude measurement capabil­ ity of about 7 (21 - 14) dB, which is close to the expected value of 6 dB/bit. If the frequency measurement capability is used to determine the dynamic range, a 1-bit ADC can achieve a very high single signal DR as discussed in Section 3.4. For a 1-bit ADC the degradation might be considered as the difference between the area of a sine wave and the digitized area; the degradation can be estimated through the ratio of the areas. For example, the area under a sine function with am­ plitude of 1 from 0 to π is 2. The area of a rectangle with amplitude of 1 and width π is π. The degradation D can be considered as



æπö D = 10log ç ÷ = 1.96 è 2ø

dB

(3.8)

If the receiver is designed so that the minimum signal only triggers 1 bit of the ADC, then its performance is equivalent to a 1-bit ADC. Under this condition, the receiver sensitivity should be the same as that of a 1-bit ADC. Only when a receiver is designed so that at the sensitivity level the noise input crosses more than 1 bit, the sensitivity of the receiver is better than a 1-bit ADC. At the nominal gain level discussed in Section 2.10, the noise triggers about 1 bit.

3.14  Digitization Effect in the Instantaneous Dynamic Range Calculation In building digital microwave receivers, the input signal is digitized. It is desirable to add digitization to the input signal to calculate IDR [11]. In other words, it is desirable to find the IDR as a function of ADC bit number. The IDR obtained by using frequency identification is the same that as discussed in Section 3.11. The only difference is that each data point is obtained through 20 runs rather than 10. The IDR is a function of number of bits and frequency separa­ tion. The results are shown in Figure 3.9. In this figure, the arbitrary gain values listed in Table 3.6 are used. When the number of bits is low, such as from 4 to 10 bits, the MUSIC method cannot detect two signals separated by 1 MHz even if they have the same amplitude. When the number of bits is 12 or higher, signals separated by 1 MHz can be detected. One should note that the input data have 256 points, or are 100 ns long. When longer data are used, two signals with 1-MHz separation can be properly processed. The IDRs obtained from the two 8-bit ADCs (with 0.27V and 1V maximum voltage) are very close. The IDR from the ADC with 1-V maximum voltage is slightly higher. Thus, the IDR is mainly determined by the number of bits, and the maximum voltage can be considered as a secondary effect. A similar conclusion is drawn in Chapter 2.

46

Dynamic Range Study Through Eigenvalue and MUSIC Methods

Figure 3.9  IDR with the digitization effect and using a frequency identification method for 256 data points.

The saturation values of the IDR are close to the expected value of 6 dB per bit. These values can be listed in Table 3.7. These values are obtained by taking the average IDR values from frequency of 25 to 50 MHz. In Table 3.7 the first row gives the number of bits. In the second row the numbers are obtained by the number of bits times 6 and they can be considered as the expected IDR values. The third row lists the IDR from the arbitrarily chosen amplification gain. The fourth row lists the IDR from the nominal gain. Since the arbitrarily chosen gain is higher than the nominal value, the IDR will be smaller and the sensitivity is better. The results shown in rows 3 and 4 confirm these expectations that the IDR is about 6 dB per bit. Just for argument, one can further sacrifice the sensitivity and increase the IDR to a higher value.

Table 3.7  Saturated Instantaneous Dynamic Range Versus Number of Bits Number of bits Number of bits × 6 dB

4 24

IDR in dB

20.5

IDR in dB

24

6 36

8(0.27) 48

8(1v) 48

Arbitrarily Chosen Gain 34.9 45.4 46.7 Nominal Gain 36.1 48.1 48.1

10 60

12 72

14 84

16 96

58.1

67.5

81.7

91.9

60.2

72.1

84.1

96.3

3.15  Curve Fitting for the Instantaneous Dynamic Range

47

Table 3.8  Parameters for Curve Fitting for 256-Point Results

Df0 B A

4 bits

6 bits

4.8108 4.8364 20.5364

2.1360 4.8161 34.7555

8 bits (0.27V) 1.6009 4.1224 45.1147

8 bits (1.0V) 1.4839 4.2348 46.4121

10 bits

12 bits

14 bits

16 bits

0.3984 4.4089 57.8047

0.3910 3.8268 67.0290

–0.2343 3.6528 81.2997

–0.6969 3.6524 91.6628

3.15  Curve Fitting for the Instantaneous Dynamic Range In this section the data shown in Figure 3.9 will be curve-fitted to obtain a smooth results that is easier to use. The equation used for the curve fitting, provided by L. Liou of AFRL, is

DR(∆f − ∆fo ) = A{1 − exp[ −(∆f − ∆fo ) / B]}

(3.9)

The three parameters to be determined are A, B, and Dfo. Depending on the number of bits, these parameters can be obtained from the curve fitting program “fmin­ search” from MATLAB. These parameters are listed in Table 3.8. The results from curve fitting are shown in Figure 3.10. Similarly, the IDR obtained with the nominal gain are also curve-fitted as shown in Figure 3.11. These curves can be considered as a guide for a digital receiver de­ sign with 256 points of input data (or PW = 100 ns).

Figure 3.10  IDR from curve-fitted results of Figure 3.8 arbitrary gain.

48

Dynamic Range Study Through Eigenvalue and MUSIC Methods

Figure 3.11  IDR from curve-fitted 256 data points with nominal gain.

3.16  IDR Calculated with 128 Data Points and Digitization In this section, the effect of data length will be investigated. The procedure is simi­ lar to the previous approaches. Since the data length is shorter, the order of the MUSIC method is also reduced. Instead of the order of 80, the order of 60 is used in the evaluation. When the data length changes, the required gain calculated from Chapter 2 does not change significantly. The two sets of gains listed in Table 3.6 are used in these simulations. The results are shown in Figure 3.12, and each point is obtained through 20 runs. Comparing to Figure 3.11, the IDR for some special cases can be significant. For example, with 1-MHz frequency separation and 16-bit ADC, the IDR is about 1 dB for the 128 data points and about 31 dB for the 256 data points. The differ­ ence is about 30 dB. For the 128-point case, IDR reaches saturation with a wider frequency separa­ tion. In Figure 3.9 the IDR reaches saturation at about a 25-MHz frequency separa­ tion. In Figure 3.12 the IDR saturation is beyond 35 MHz. The saturated IDRs in Figure 3.12 are also lower. A similar curve-fitting method is used for the 128 points of data. The same equation, (3.7), is used and the constants obtained are listed in Table 3.9. The results are shown in Figure 3.13. In this figure, for b = 14 and 16, the saturated IDRs are about 77 and 87 dB, and they do not reach the final measured values of about 79 and 89 dB. This is due to the error introduced in the curve-fitting processing. The IDR curve for the nominal gain is shown in Figure 3.14. These figures can be considered as a guide for digital receiver design with 128 points of input data.

3.16  IDR Calculated with 128 Data Points and Digitization

49

Figure 3.12  IDR with digitization effect and arbitrary gain using 128 points of data.

Comparing the results of Figures 3.11 and 3.14, one can see that the saturated IDR is different by about 3 dB. Since the data length is doubled, the sensitivity level should improve by 3 dB. As discussed in Section 3.5, the IDR is a function of fre­ quency separation as well as pulse width. The saturated IDRs from Figures 3.11 and 3.14 are about 3 to 5 dB higher than single signal dynamic range (SDR) that can be calculated from Chapter 2. The difference may be caused by the simulation approaches. In Chapter 2 an S/N of 14 dB is required on the sensitivity. In this study, no specific probability of false alarm or probability of detection are required. The IDR is obtained through the methods discussed in Sections 3.9 and 3.11. In the procedure, in order to save calculation time, a small number of runs such as 10 and 20 are used to determine the IDR. Thus, the two ways of defining the DR are different.

Table 3.9  Parameters for the Curve Fitting of 128-Point Results

Df0 B A

4 bits

6 bits

11.59 12.13 18.04

5.00 10.67 31.97

8 bits (0.27V) 3.17 9.27 42.14

8 bits (1.0V) 3.04 8.94 42.76

10 bits

12 bits

14 bits

16 bits

1.69 8.39 54.03

0.86 7.71 63.19

0.63 7.48 77.39

0.12 5.80 86.73

50

Dynamic Range Study Through Eigenvalue and MUSIC Methods

Figure 3.13  IDR from the curve-fitted results of Figure 3.11 with arbitrary gain.

Figure 3.14  IDR from the curve-fitted 128 data points with nominal gain.

3.18  Conclusion

51

Table 3.10  IDR in Decibels as a Function of Data Length Using the MUSIC Method Data length/bits 4 8 IDR with arbitrary gain in dB 128 16.8 41.3 256 20.8 47.1 512 22.7 50.3 1,024 24.6 51.4 2,048 26.6 52.5 IDR with nominal gain in dB 128 17.5 41.6 256 25.8 49.8 512 29.3 53.2 1,024 31.3 53.8 2,048 31.1 56.1

16 87.1 92.3 95.8 98.3 98.5 90.8 98.8 101.8 102.8 104

3.17  Generating Very High IDR Using Long Data Length In Sections 3.14, 3.15, and 3.16, it was shown that when the input data length in­ creases, the IDR also increases. The general argument can be discussed through the FFT operation as follows. Suppose a strong signal is at the saturation level of the ADC and the weak signal is below noise. When the data length increases, the strong signal will not generate a spurious response but the weak signal will be integrated above noise. As a result, the IDR will increase as data length increases. The MUSIC method can be used to illustrate this phenomenon. The IDR is cal­ culated through an 80-order MUSIC method. The data length is from 128 to 2,048 points. The strong signal is at S/N = 100 dB and the weak signal is 50 MHz below the strong one. Under this condition the IDR should be the saturated values. The results for the two amplification values are listed in Table 3.10. Each data point in this table is obtained from one run to save calculation time. To calculate the IDR of long data can be rather time consuming. Since the data is obtained from one run, it is difficult to find the relations between the data length and the IDR, which should increase by 3 dB when data length is doubled. For a long data length, the 80-order MUSIC method may not be high enough. Table 3.10 provides the needed information, that is, when the data length increases, the IDR also increases, which is the only message Table 3.10 intended to provide. At a long data length, the IDR can be higher than 6 dB/bit; however, if the probability of false alarm and the probability of detection are taken into consideration, the IDR should be lower than the listed values.

3.18  Conclusion In this chapter different definitions of dynamic ranges are presented. For evaluating EW receivers, the terms SDR and IDR can be used and the IDR is the important pa­ rameter to determine the performance of a receiver. The IDR is calculated through the high-order MUSIC method. The results are not obtained from a theoretical

52

Dynamic Range Study Through Eigenvalue and MUSIC Methods

study but from the simulation results. The IDR is signal length dependent. Higher data length provides higher IDR. Since digital EW receivers are usually built from 128 to 256 points of data per processing window limited by the minimum PW, the IDR guides are generated by using 128 and 256 points. The digitization effect caused by the number of bits is also included in the study. The results from Figures 3.10, 3.11, 3.13, and 3.14 can be used a guide for receiver designs. The saturation values of the IDR in these figures are close to 6 dB per bit. Although a longer data length can increase the IDR beyond 6 dB per bit, the signal must be very long. Often an EW receiver is designed for minimum pulse width, which is close to 100 ns. It is anticipated that when the probability of false and the probability of detec­ tion are taken into consideration, the actual IDR will be lower than the values in Figures 3.11 and 3.14. In the actual building of an EW receiver, spurs will be gen­ erated from the RF chain such as from mixers and saturated amplifiers. The ADC used in the receiver will not be an ideal one; as a result, spurs will be produced. Thus, for an actual receiver, the IDR might be less than the ideal case of 6 dB per bit. In order to give a meaningful IDR, the PW and the frequency separation must be stated. If only one IDR value is provided, the saturated value can be used. Under this condition, the frequency separation should be given.

References   [1]  Skolnik, M. L., Introduction to Radar Systems, New York: McGraw-Hill, 1962.   [2]  Rife, D. C., and G. A. Vincent, “Use of the Discrete Fourier Transform in the Measurement of Frequencies and Levels of Tones,” Bell System Technical Journal, Vol. 49, February 1970, pp. 197–228.   [3]  Rife, D. C., “Digital Tone Parameter Estimation in the Presence of Gaussian Noise,” Doc­ toral dissertation, Electrical Eng. Dept. Polytechnic Institute of Brooklyn, 1973.   [4]  Rife, D. C., and R. R. Boorstyn, “Multiple Tone Parameter Estimation from Discrete Time Observations,” Bell System Technical Journal, Vol. 55, May 1976, pp. 1389–1410.   [5]  Smith, B. E., “Enhancing the Instantaneous Dynamic Range of Electronic Warfare Receiv­ ers Using Statistical Signal Processing,” Thesis, Air Force Institute of Technology, Wright Patterson Air Force Base, OH, 2004.   [6]  Ulrych, T. J., and R. W. Clayton, “Time Series Modeling and Maximum Entropy,” Phys. Earth Planetary Interiors, Vol. 12, August 1976, pp. 188–200.   [7]  Schmidt, R., “A Signal Subspace Approach to Multiple Emitter Location and Spectral Es­ timation,” Ph.D. thesis, Stanford University, Stanford, CA, August 1981.   [8]  Schmidt, R., “Multiple Emitter Location and Signal Parameter Estimation,” Proc. of the RADC Spectrum Estimation Workshop, Rome Air Development Center, 1979, pp. 243–258; reprinted in IEEE Trans. Antennas and Propagation, Vol. AP-34, March 1986, pp. 276–290.   [9]  Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Chapter 12, Norwood, MA: Artech House, 2001. [10]  Spilker, J. J., Digital Communication by Satellite, Englewood Cliffs, NJ: Prentice Hall, 1995, pp. 550–555.

Chapter 4

Dynamic Range Study Through Fast Fourier Transform (FFT)

4.1  Introduction In Chapter 3, the instantaneous dynamic range was obtained through the highresolution MUSIC method. The instantaneous dynamic range is a function of frequency separation between the two signals. When the frequency separation reaches a certain value, the instantaneous dynamic range becomes a constant. In this chapter the dynamic range is found through the FFT operation. At present most digital receivers determine the input signal frequency through the FFT operation. Although the signals can be easily transformed into the frequency domain, determining the number of signals and correctly encoding their frequencies is the most challenging task. Most of the receiver design effort is to perform this task. The main purpose of this chapter is to let the readers have some feeling on the usage of the FFT operation. In order to keep this problem simple, the two input signals are separated far in frequency. When the two signals are close in frequency the problem becomes more complicated. When two signals are close in frequency, the two frequency beats with each other, the output amplitude changes, and in some receiver designs even the pulse width (PW) is difficult to measure. Since the purpose here is not to design a receiver, only the signals with faraway frequency will be considered. In a receiver design one must detect the number of signals. In this chapter the number of signals is two. If two signals are detected, the frequencies must be correct; otherwise, one wrong frequency will be considered to be a false alarm. Most of the discussion in this chapter will be based on simulation results.

4.2  Using Simulation Approach to Find the IDR The simulation uses a sampling frequency of 2.56 GHz. As discussed in Section 2.7, the receiver bandwidth is 1,280 MHz and the noise floor is around -80 dBm if the receiver has a noise figure of 3 dB. A 256-point FFT is performed. The frequency resolution is 10 MHz. Before the study of the FFT operation, let us define some terms to simplify the description of the problem. The center of a frequency bin is that the input frequency is at the center of a particular bin. The boundary of two bins is that the input frequency is at the center of two adjacent frequency bins. For example, if the 53

54

Dynamic Range Study Through Fast Fourier Transform (FFT)

256-point FFT is performed on the input data sampled at 2.56 GHz, the frequency bin width is 10 MHz. The centers of the frequency bin are at 0, 10, 20, . . . , 1,280 MHz. The boundaries between two bins are at 5, 15, 25, and 1,275 MHz. In this special study, the strong signal is arbitrarily chosen either at 320 MHz, which is centered on a frequency bin or at 325 MHz, which is at the boundary of two adjacent frequency bins. The weak signal is arbitrarily chosen either at 800 MHz, which is centered on a frequency bin, or at 805 MHz, which is at the boundary of two adjacent frequency bins. The frequency separation is about 480 MHz from 320 to 800, which is wide enough such that the IDR should not be frequency dependent. Under this condition, the IDR obtained should be close to the maximum value. In this simulation, the frequency measurement is the only parameter used to determine the IDR, while signal amplitude is not taken into consideration. Five window functions are used to evaluate the IDR performance. First, a rectangular window is used. When the input signals are both centered on the frequency bins, they should produce the highest possible IDR. With a rectangular window when the input signals are centered on the frequencies bins, the signals will not generate any sidelobes. However, when the signals are at the boundary between two adjacent frequency bins, many of the neighbor bins have high outputs. When a rectangular window is used in an actual receiver design, the encoder, which is used to generate the pulse descriptor word (PDW) as discussed in Section 5.1, must be able to handle these high sidelobes. In most designs, the sidelobes are suppressed by using a window rather than designing an encoder to avoid the sidelobes. Second, a Hamming window is used. This window has sidelobes about 43 dB below the main lobes and not enough for actual receiver design. It can be used to illustrate the window effect on the IDR. The third and fourth windows are Blackman and Chebyshev, respectively. These windows have low sidelobe levels, which are suitable for receiver design. The last window used in the simulation is ParkMcClellan. This window has low sidelobes, but the output frequency is decimated. This window is used for receiver design and will be further discussed in Chapter 13. In this simulation, the number of bits of the ADC is changed from 4 to 16, as in Chapter 3. Only the even numbers of bits are simulated; for example, bits 4, 6, 8, . . . , 16 are studied. The maximum input voltage for the 4- to 8-bit ADCs is assumed to be 270 mV and from 8 to 16 is assumed to be 1V. The 8-bit ADC is simulated twice, one with a 270-mV maximum voltage and one with 1-V maximum voltage. The maximum voltage is half the peak-to-peak voltage (Vpp).

4.3  Local Peaks Before the simulation process, a local peak must be defined in the frequency outputs. In order to qualify an output as a signal, it must be a local peak. This requirement is needed to avoid identifying every output crossing a certain threshold as a signal. If an output is a local peak and crosses a certain threshold, it can be considered as a legitimate signal. Figure 4.1 shows the amplitude outputs in the frequency domain designated as Xi-4, Xi-3, Xi-2, and so forth. A peak must be higher than its neighboring outputs so that Xi is higher than Xi-1 and Xi+1, Xi-3 is higher than Xi-4 and Xi-2. In this figure, there are four peaks at

4.3  Local Peaks

55

Figure 4.1  Amplitude in frequency domain.

Xi-5, Xi-3, Xi, and Xi+4. These peaks may not be local peaks. A local peak is defined as one needing to be higher than its neighboring outputs by a certain value. If the data on either side of a peak is monotonically decreasing, these points will be taken into consideration for determining the local peak. For example, if Xi is higher than Xi-1 and Xi+1, then Xi is a peak but not necessary a local peak. In this figure, Xi-2 < Xi-1 < Xi > Xi+1 > Xi+2 > Xi+3, then Xi-2 will be the minimum on the left side of Xi and Xi+3 will be the minimum on the right side of Xi. Under these conditions, Xi-2, Xi, and Xi+3 will be used to determine whether Xi is a local peak. Similarly, Xi-2 and Xi-4 can be used to determine whether Xi-3 is a local peak. The difference between the peak Xi and the minimum on both sides can be written as



DX l = X i - X i - 2

and

DX r = X i - X i + 3



(4.1)

where DXl is the amplitude difference on the left side and DXr is the amplitude difference on the right side. Similarly, for peak Xi-3, one can write DX l = X i -3 - X i - 4

DX r = X i -3 - X i - 2

and

(4.2)

Quantities DXl and DXr must be larger than a threshold (6 dB) in order to qualify as a local peak. Let us refer to this threshold as the local peak threshold because there

56

Dynamic Range Study Through Fast Fourier Transform (FFT)

is another threshold to qualify certain local peaks as signals. The value of 6 dB is determined empirically from the simulation results. In actual receiver design, this same method can be used to determine the local peak, but the local peak threshold must be determined for the specific receiver. If the local peak threshold is too high, some peak produced by a signal may be missed (or missing the signal). If the threshold is too low, some false local peak maybe generated. This is the same trade-off problem between the probability of detection and the probability of false alarm. In this special case, Xi is the only local peak, but Xi-5, Xi-3, and Xi+3 are not. Once all the local peaks are found, those that cross another threshold will be considered as the input signals. This is a different threshold, which is used for determining whether a local peak is signal or not.

4.4  Simulation Procedure Two signals, one strong and one weak, are used for this simulation. The strong signal is at 320 (or 325) MHz, and the weak one is at 800 (or 805) MHz with -80dBm noise. The RF gain is applied to both signals and noise. The input signal S to the ADC can be written as

S = A1 sin(2π f1t + θ1) + A2 sin(2π f2t + θ 2 ) + n

(4.3)

where A1, A2 are the amplitudes of the strong and weak signals, respectively, and n is the noise, f1 (= 320 × 106 or 325 × 106), f2 (= 800 × 106 or 805 × 106) are the strong and weak signal frequencies, and q1 and q2 are the phases and are randomly selected with uniform distribution between 0 and 2π. The noise is white Gaussian with zero mean. The combined amplitude of the strong and weak signals and the noise must be less than the maximum voltage of the ADC. This approach will provide the best achievable IDR. If the ADC is saturated, the output will have a strong output at 960 MHz, which is the third harmonic of 320 MHz. In this study, sometimes the third harmonic will limit the IDR. The radio frequency (RF) gain used in this simulation is the arbitrary gain shown in Table 3.6. The amplitude of the strong signal A1 after the RF amplification can be calculated as

A1 = VM − A2 − 1.5n

(4.4)

where VM is the maximum voltage of the ADC. This equation indicates that sum of the strong and the weak signals is almost equal to the maximum voltage of the ADC. It has been experimentally determined that with noise levels 1.5 times n (the standard deviation of the noise) the ADC can reach saturation occasionally; however, the average value of the third harmonic observed in the FFT output is not strong enough to disturb the measurement. The input S is digitized with the calculated quantization level q. The desired window function is digitized with the same number of bits as the digitized input

4.5  Threshold Determination

57

signal. If a rectangular window is used, it can be considered as no window weighting. A total of 256 points of data is generated and a fast Fourier transform (FFT) is performed on the input data. Since the input is real, only the first 128 amplitude outputs are used. The Monte Carlo simulation method will be used to determine the IDR. The criterion for correct detection is that both signals will be detected above 90% with approximately a 0.1% (10-3) probability of false alarm. In general, the desired probability of a false alarm is about 10-7. However, using a simulation method to achieve such a small number (10-7) takes extremely long time to calculate. With the same probability of detection (90%), when the probability of false alarm decreases from 10-3 to 10-7 approximately, an additional 3 dB of signal-to-noise (S/N) is required [1]. Thus, if the desired probability of false alarm is 10-7, the IDR calculated can be reduced by 3 dB. With the strong signal fixed, the weak signal is measured 1,000 times with random noise and random phase q1 and q2. Among these 1,000 runs, if the probability of detection is less than 90%, the weak signal is increased by 1 dB until 90% or higher is measured correctly. The IDR is calculated as the difference between the strong signal’s amplitude and the last weak signal’s amplitude in decibels. When the weak signal is increased, the strong signal may be decreased as shown in (4.4) in order to avoid ADC saturation. Since this operation is only carried out 1,000 times, it is difficult to measure the probability of false alarm with much confidence. Therefore, the number of false alarms is only recorded. In order to have a slightly better result, the above operation will be repeated 10 times to calculate 10 IDR values. The overall IDR is the average value of the 10 measurements. The number of false alarms is also the average of these 10 runs. In the detection process, the outputs must be a local peak obtained from the previous section and must cross a certain threshold. When a local peak is detected as a signal, the frequency must be compared with the input signal. The four input signal frequencies are 800, 805, 320, and 325 MHz. If the detected signal frequency is 10 MHz (the frequency resolution) or more of the input signal, it will be considered as a false detection. The threshold is used to determine the probability of false alarm.

4.5  Threshold Determination Usually the noise alone in a receiver determines a threshold, and this threshold, in turn, determines the sensitivity of the receiver. The threshold is generated based on the desired probability of a false alarm. In this simulation, before two signals are applied, a threshold must also be determined. This threshold is determined not by the noise alone but by the noise and one strong signal. The noise level is -80 dBm at the input of the receiver, and at the ADC input the noise power is increased by the amplifier gain. The input signal for finding the threshold is close to the saturation level. Its amplitude is obtained from (4.4). In this case, there is no weak signal or A2 = 0. The input signal is generated through (4.3) with A2 = 0. This input is digitized using the desired quantization level to generate 256 points. If a window is applied, the digitized signal is multiplied by the digitized window. By computing the FFT of the

58

Dynamic Range Study Through Fast Fourier Transform (FFT)

digitized input, the values of the first and second largest local peaks in the frequency domain are found. In this threshold determining process the concept of the local peak discussed in Section 4.4 is used. From simulation it is noted that sometimes there are high peaks (not local peaks) that occur near the main lobe. If these peaks are used to determine the threshold, sometimes the threshold is very high. Thus, the local peaks are used to determine the threshold. From the simulation results it is determined that using local peak to determine the threshold can produce better results than using just the peak. The largest local peak should be the frequency of the input signal. The secondlargest local peak is caused by the noise or the window sidelobes. The purpose of this operation is to find the sidelobes and spurs caused by either the window or the strong signal. These outputs might be detected as the weak signal. Therefore, the threshold is set above them. This sequence of steps, from generating the input signal to finding the secondlargest local peak, is repeated 1,000 times. The maximum value of the second-largest local peak from the 1,000 runs is taken as a candidate threshold, which should provide a false alarm rate of about 10-3. This process was repeated 100 times. The resulting 100 candidate threshold values are averaged to find the final threshold value. The final threshold in each number of bits is rounded up to the nearest whole number. For example, if the averaged threshold is 38.1, the actual threshold will be 39. This roundup process should make the probability of false alarm slightly lower than 10-3. This threshold is used against the local peaks discussed in Section 4.4 to determine the IDR. Since this threshold is generated with a strong signal and noise, it is usually higher than the threshold generated by noise alone. This threshold is not used to determine the receiver sensitivity. When the actual IDR is determined, because of the small number of runs, the probability of false alarm measured has a rather low confidence level. For each window two thresholds will be determined: one for the strong signal on a frequency bin, and the other for the signal at the boundary between two frequency bins. The threshold settings are input signal dependent and used only for IDR study. It should generate higher IDR because the input frequency on a frequency bin and at the boundary of two bins generates different thresholds. The thresholds are not suitable for actual receiver designs.

4.6  Windows and Input Frequencies The five windows mentioned in Section 4.2 are listed as follows: 1. 2. 3. 4. 5.

Rectangular window; Hamming window; Blackman window; Chebyshev window; Park-McClellan window.

All the windows are digitized and normalized (divided by the maximum digitized value so that the maximum value of the window is always 1). All windows are 1.

Information from this section has been taken from the MATLAB wintool function and Wikipedia.

4.7  IDR Results

59

Table 4.1  Parameters of Window Window Type

Highest Sidelobe dB

Rectangular Hamming Blackman Chebyshev McClellan***

–13.3 –42.5 –58.1 –100 –70

3-dB BW* Reading/ Normalized 0.0273/1 0.0391/1.43 0.0508/1.86 0.0547/2.0 0.0435/1.59

Equivalent Noise BW** 1 1.37 1.73 — —

* From the wintool function of MATLAB. ** From Wikipedia. *** From the actual plot.

applied to the input signal by taking the product of the digitized window and the digitized signal in the time domain. Some important parameters of the windows are listed in Table 4.1. In this simulation the sampling frequency is 2.56GHz with a 256-point FFT, and the frequency resolution is 10 MHz (2.56 GHz/256). Since the frequencies of the input signals can affect the dynamic range and sensitivity, three cases are studied in this report: •  •  • 

 ase 1: Both signals are centered on the frequency bins. The strong signal is C 320 MHz, and the weak signal is 800 MHz. Case 2: The strong signal is centered on the frequency bin at 320 MHz and the weak signal is at the boundary between two adjacent bins at 805 MHz. Case 3: Both signals are at the boundary of two adjacent bins. The strong signal is 325 MHz and the weak signal is at 805 MHz.

In all three cases the frequency separation is about 480 MHz, which is far enough that the frequency separation is not a factor in this study. In the following discussions, the case number is used to refer the input frequency conditions.

4.7  IDR Results The IDR results will be reported according to the five different windows because the window affects the IDR. In each window the three input frequency conditions will be reported. The limitations of the IDR will be explored. For example, if a high threshold limits the IDR, the reasons for the high threshold (usually caused by a certain spur) will be discussed. If the number of the false alarm rate is high, its cause will be investigated. For example, as mentioned in the end of Section 4.5, the probability of false alarm measured may not be accurate. Thus, at each run, the number of false alarm among these 1,000 runs is recorded. This process will repeat 100 times and the results of the 1,000 runs will be averaged. Usually, the number of false alarms is less than 1 as expected. If this number is consistently higher than 1, for example, the number of false alarm is 5, then its cause will be discussed. The threshold values also have a trend. Sometimes if the threshold value is exhibiting an idiosyncrasy, a detailed look in the frequency domain will be performed.

60

Dynamic Range Study Through Fast Fourier Transform (FFT)

An explanation will be given. However, sometimes it is difficult to give an explanation. These investigations should be useful for future receiver designs. The overall purpose is that the reasons of having low IDR will be studied. Hopefully corrections should be made to avoid these problems. In an actual receiver design, if a low IDR is encountered, the cause will be studied to see whether the problem can be remedied.

4.8  IDR with a Rectangular Window The amplitude of a rectangular window is set to 1, which does not change the amplitude of the input signal. The sidelobe is about 13.3 dB down and the frequency resolution is 10 MHz. Figure 4.2 shows the IDR with rectangular window. As expected, the IDR increases by about 6 dB per bit when the strong signal is on the frequency bin. The higher the number of bits, the larger the IDR is. The IDR of case 2 is about 2 to 3 dB lower than that of case 1. This is due to the frequency of the weak signal being at the boundary of two bins, which causes the amplitude of the weak signal in the frequency domain to drop about 3 dB. In order to detect the weak signal, the minimum amplitude of the weak signal needs to be increased about 2 to 3 dB, which reduced the IDR by 2 to 3 dB. In case 3, the thresholds for 14- and 16-bit ADCs are zero because a secondlargest peak cannot be found. Figure 4.3(a) shows the frequency domain output when the input is at 320 MHz (on the frequency bin). In this figure besides the main peak there are many peaks, which are caused by noise. The second-largest peak will

Figure 4.2  IDR versus the number of bits with a rectangle window.

4.8  IDR with a Rectangular Window

61

Figure 4.3  Frequency domain outputs with one strong signal: (a) input frequency on a bin, and (b) input frequency between two bins.

be used as the threshold. Figure 4.3(b) shows the frequency domain output with the input at 325 MHz (at the boundary of two frequency bins). In this figure there are no other peaks than the strongest one. Since the signal amplitude is very strong, its sidelobes are also high and overshadow the noise effect. Therefore, the second local peak can not be found. In these cases (14 and 16 bits), the threshold is arbitrarily set to be the same as the thresholds in cases 1 and 2. The IDR for case 3 is 17 dB for a 4-bit ADC and about 30 dB for an ADC with 6 or more bits. The low IDR is due to the sidelobe of the strong signal. When the strong signal is at the boundary of two bins, increasing the stronger signal will also increase the sidelobes. Since the amplitude of the weak signal needs to be 6 dB higher than the neighboring sidelobes in order to be qualified as a local peak and detected as a signal, these high sidelobes limit the IDR to lower values. The average number of false alarms that occur in 1,000 runs is less than 1 except for the 14- and 16-bit ADCs in case 3. Under this condition, the average number of false alarms occurring in 1,000 runs for the 14- and 16-bit ADCs are 3.6 and 3.9, which are much higher than 1. The reason for the higher false alarm rate is the sidelobe next to the weak signal being detected as a local peak. Figure 4.4 shows the frequency domain of the strong signals and the weak signal using a 16-bit

62

Dynamic Range Study Through Fast Fourier Transform (FFT)

Figure 4.4  Two signals using a 16-bit ADC in frequency domain: (a) having no false alarm, and (b) having a false alarm.

ADC. Figure 4.4(a) illustrates these two signals in the frequency domain with no false alarm, and Figure 4.4(b) illustrates these two signals with a false alarm being detected next to the weak signal. If the criteria for a local peak are changed to consider only five monotonically consecutive points on either side, then these side­ lobes next to the weak signal will not be detected as false alarms.

4.9  IDR with a Rectangular Window and Close Spaced Frequencies Since a frequency for a rectangular window is on a bin and there are no sidelobes, the IDR for close frequency separation will be discussed under this condition. Let us use a few examples to illustrate these situations. When the strong signal is at 320 MHz, the weak signal at 330 MHz (one frequency bin away) cannot be detected, because under this condition, there is no local peak at 330 MHz. This result shows that when two signals are very close the weak signal can not be detected. When the weak signal is at 340 MHz (two frequency bins away), only the 8- and 16-bit ADCs are simulated. The IDR values obtained are very close to the values for the weak

4.10  IDR with Hamming Window

63

signal at 800 MHz. This result illustrates that when the weak signal is only two bins away from the strong one, the strong signal will not disturb the weak signal measurement. Just for the purpose of argument, one can say that a receiver can have very high IDR even if the two signals are close in frequency. When the strong signal is at 325 MHz, the nearest weak signal can be detected at 350 MHz (2.5 bins away). The IDR values obtained are 5, 9, and 12 dB when the weak signal is at 350, 360, and 370 MHz, respectively. The strong signal generates strong sidelobes and affects the detection of the weak signal. From these few data one can see that the IDR increases when the frequency separation increases because the sidelobes of the strong signal decrease. This agrees with the results obtained in Chapter 3. Although the rectangular window can produce high IDR under certain input conditions even with close frequency separation, it also generates very poor IDR under slightly different input conditions. Because of the low IDR produced for case 3 the rectangular window is not suitable for receiver design.

4.10  IDR with Hamming Window Figure 4.5 shows the Hamming window in the time domain and the frequency domains. The relative sidelobe attenuation of the Hamming window is about -42.5 dB below the main lobe, and the 3-dB bandwidth is about 14.3 MHz. Figure 4.6 shows the IDR versus the number of ADC bits using a Hamming window. Overall, the IDRs for cases 1 and 2 are less than the IDR with a rectangular window. The results show that the IDRs in case 1 and case 2 increased about 5–6

Figure 4.5  Hamming window: (a) time domain and (b) frequency domain.

64

Dynamic Range Study Through Fast Fourier Transform (FFT)

Figure 4.6  IDR versus the number of ADC bits using a Hamming window.

dB per bit. The IDR in case 1 is 1–2 dB higher than that in case 2 due to the weak signal in case 2, which is at the boundary of two bins. Applying the Hamming window reduced the sidelobe of the strong signal when the frequency of the strong signal is at the boundary of two frequency bins. Figure 4.6 also shows the IDRs of the Hamming window for case 3. For 4- and 6bit ADCs they are 20 and 34 dB, respectively. The IDRs stay roughly the same, between 38 and 39 dB for the 8- to 16-bit ADCs. This low IDR is due to the sidelobe of the Hamming window. The relative sidelobe attenuation of the Hamming window is -42.7 dB. A threshold must be used to reduce the false detection. Taking the threshold into consideration, the IDR is less than the highest sidelobe of -42.7 dB. Figure 4.7 shows the outputs of the strong signal for cases 1 and 2. Figure 4.7(a) shows that the strong signal is on the frequency bin. Under this condition, the sidelobes are relatively low and the window effect does not show. Figure 4.7(b) shows that the frequency of the strong signal is at the boundary of two bins, the Hamming window has sidelobes next to the strong signal, and the highest peak is about 40 dB down. When the threshold is calculated, these sidelobes are detected as the secondlargest local peaks and generate a high value of the threshold. Therefore, the amplitude of the weak signal needs to be higher in order to exceed this threshold and be detected. Thus, the IDR for case 3 is lower. The IDRs for case 3 using 8 bits and above are limited by the sidelobes of the window. The average number of false alarms that occurred in 1,000 runs is less than 1.

4.11  IDR with Blackman Window

65

Figure 4.7  Strong signal output in the frequency domain with a Hamming window: (a) frequency on the bin, and (b) frequency at the boundary of two bins.

4.11  IDR with Blackman Window Figure 4.8 shows the Blackman window in the time and the frequency domains. The sidelobe is about –58.1 dB below the mainlobe and the 3-dB bandwidth of about 18.6 MHz. Although the maximum sidelobe is about –58 dB down when the frequency is away from the main lobe (signal), the sidelobes keep decreasing. Figure 4.9 shows the IDR versus the number of ADC bits using a Blackman window. The results show that the IDRs in cases 1, 2, and 3 are very close, up to 8 bits. In case 3 with 10- and 12-bit ADCs, the IDRs are less than in cases 1 and 2 due to the higher thresholds generated. These are the expected results. However, for the 16-bit ADC, the measured IDRs are 81, 80, and 87 dB for cases 1, 2, and 3, respectively. These high IDRs indicate that the sidelobes of the Blackman window should be lower than 87 dB at the frequency separation of about 480 MHz (from 320 to 800 MHz). The IDR of case 3 is higher than cases 1 and 2, which sounds unreasonable. In order to explain this phenomenon, the frequency domain output of the strong signal digitized with a 16-bit ADC is shown in Figure 4.10. The Blackman window causes two strong peaks next to the strong signal when the frequency of the strong signal is on the frequency bin, as shown in Figure 4.10(a). These two high sidelobes

66

Dynamic Range Study Through Fast Fourier Transform (FFT)

Figure 4.8  Blackman window: (a) time domain and (b) frequency domain.

Figure 4.9  IDR versus the number of bits using a Blackman window.

4.12  IDR with a Chebyshev Window

67

Figure 4.10  Frequency domain output of a strong signal (16 bits) with a Blackman window: (a) frequency on the bin, and (b) frequency at the boundary of two bins.

require a high threshold to avoid them. When the frequency of the strong signal is between two frequency bins, the sidelobes next to the strong signal smoothly decline and are not detected as peaks. The cause of these outputs is not clear and requires further study. From these two figures one can see that the thresholds for cases 1 and 2 are higher than the threshold in case 3. The thresholds values are 126, 126, and 52 for cases 1, 2, and 3, respectively, averaged from one hundred 1,000 runs. A higher threshold requires a higher weak signal to exceed it; thus, the measured IDR is low. The average number of false alarms that occurred in 1,000 runs is less than 1.

4.12  IDR with a Chebyshev Window Figure 4.11 shows the Chebyshev window in the time and the frequency domains. The relative sidelobe attenuation of the Chebyshev window is about –100 dB below the mainlobe and the 3-dB bandwidth of about 20 MHz. Figure 4.12 shows the IDR versus the number of ADC bits using a Chebyshev window. The results show that the IDRs in cases 1, 2, and 3 are very close in each number of bits up to 14 bits. The IDRs increased about 5–6 dB per bit up to 14 bits. The increasing rate for 16 bits is less than 5 dB per bit.

68

Dynamic Range Study Through Fast Fourier Transform (FFT)

Figure 4.11  Chebyshev window: (a) time domain and (b) frequency domain.

For case 3, the IDRs are the same as the IDRs for cases 1 and 2, except for the 16-bit ADC. The IDR for 16 bits in case 3 is about 83 dB, which is about 3 dB less than the IDRs for cases 1 and 2. The limitation for the lower IDR is due to missing the weak signal detection. The Chebyshev window has a relatively wide main lobe width, which increases the width of the signal in the frequency domain and also

Figure 4.12  IDR versus the number of bits using a Chebyshev window.

4.13  IDR with a Park-McClellan Window

69

Figure 4.13  A Chebyshev window causes the weak signal to spread into three frequency bins.

increases the noise effect. When a weak signal is at the boundary of two frequency bins, sometimes the output spreads across three frequency bins as shown in Figure 4.13. If the middle value of the three highest points is less than the values of its neighbors on each side and the difference between the middle point and each of its neighbors is less than 6 dB, a local peak can not be declared. Thus, the weak signal cannot be detected. In order to be able to detect the weak signal, the amplitude of the weak signal needs to be higher to exceed the threshold. The average number of false alarms that occurred in 1,000 runs is less than 1.

4.13  IDR with a Park-McClellan Window There are receivers using the concept of multirate processing (or polyphase filters) to build [2]. Chapter 13 discusses a receiver designed through polyphase filters. These receivers usually use the Park-McClellan window; thus, it is studied here. Figure 4.14 shows one of the Park-McClellan windows in the time and the frequency domains. In this filter, there are 256 data points and the relative sidelobe attenuation of the Park-McClellan window is about –70 dB below the main lobe. The 3-dB bandwidth is about 127 MHz (80 × 1.59), where 80 MHz is the bandwidth due to the 32-point FFT operation and 1.59 is from Table 4.1. After the signal is multiplied by the Park-McClellan window in the time domain, a 256-point FFT is performed on the input. The frequency output per channel is rather wide and the output is decimated by 8 (selecting 0, 8, 16, . . . , 248 outputs) to obtain 32 outputs for FFT. Since the input signal is real, only 16 outputs are used. In an actual receiver design, the multirate approach is used to perform a 32-point FFT operation instead of the 256-point operation mentioned here to save time.

70

Dynamic Range Study Through Fast Fourier Transform (FFT)

Figure 4.14  Park-McClellan window: (a) time domain and (b) frequency domain.

Figure 4.15  The Park-McClellan window frequency-domain output with one strong signal: (a) case 1 and (b) case 3.

4.14  Data Length and IDR

71

Figure 4.16  IDR versus the number of bits with a Park-McClellan window.

For 14- and 16-bit ADCs in case 1, the second local peak cannot be found among these 16 points; therefore, the threshold is calculated as zero. Figure 4.15 shows the frequency domain output for a strong signal. Figure 4.15(a) shows the frequency on the bin and the outputs are smoothly increasing and no second local peak can be found. Figure 4.15(b) shows that the frequency is at the boundary of two bins and the frequency output shows a peak on the third bin that does not qualify for a local peak. Since the local peak cannot be found in 14- and 16-bit ADCs using the Park-McClellan window, the threshold used to determine the local peak is changed from 6 dB to 1 dB for the simulation. The definition of the local peak with an amplitude of 1 dB is applied to both finding the threshold and finding the IDR. Figure 4.16 shows the IDR versus the number of bits using Park-McClellan window. The results show that the IDRs in cases 1, 2, and 3 are very close. However, the wide window width causes low S/N and reduces the IDR. Overall, the IDRs with the Park-McClellan window are lower compared to other windows. The IDR increased about 4–5 dB per bit. The average number of false alarms that occurred in 1,000 runs is less than 1.

4.14  Data Length and IDR As discussed in Section 3.17, the IDR depends on the data length. In this section a similar result can be obtained. For example, one can use the rectangular window

72

Dynamic Range Study Through Fast Fourier Transform (FFT)

with the strong input frequencies at 320 MHz and the weak one at 800 MHz. The weak signal is about 50 dB below the strong one for the 8-bit ADC. This value is used because in Figure 4.2 the IDR is about 50 dB. However, the value of 50 is not very critical and the purpose is to keep the second signal small enough not to saturate the ADC and strong enough to be detected as a signal after the FFT operation. In this simulation an 8-bit ADC is used with the noise at –80 dBm and the gain is 45 dB, the arbitrary gain in Section 3.12. The simulation procedure is modified slightly, and there is no threshold generated. After the FFT operation the number of total outputs is 128 and roughly 10% of the bins at both ends will be eliminated. The highest two peaks are located from the remaining bins and they represent signals. Since the input frequencies are at the center of frequency bins, with rectangular window the rest frequency bins are noise. The noise power can be found from averaging noise bins. The S/N of the weak signal can be found from the weak signal and the noise power. The detection criterion is that the weak signal is assumed to be 14 dB above noise. The IDR can be found as



IDR = P1 - P2 +

S - 14 Nc

dB



(4.5)

where P1, P2 are the two highest peaks in decibels in the frequency domain, S/Nc is the measured weak strength, and 14 is the required threshold. When the input signal amplitudes are given in such a way that the ADC is not saturated, the two output peaks P1 and P2 are close to the input values. For example, if the two input signals have a 50-dB difference in amplitude, the output peaks are about 50 dB apart. If the calculated weak signal S/Nc equals 14 dB, the IDR will be 50 dB. If the S/Nc value calculated is 24 dB, it means that the weak signal can be decreased 10 more decibels. Thus, the corresponding IDR will be 60 dB. In this operation the amplitude of the signal needs not be adjusted. When the data length changes, the S/Nc changes and the corresponding IDR can be found. The result runs 1,000 times with a different noise but the same distribution and all the outputs averaged to obtain the final IDR. The input power difference is 50 dB. For the 256-point FFT operation, the IDR is about 51 dB, which is close to the result of 50.99 dB shown in Figure 4.2. When the length of the input data increases to 1,024 points, the measured IDR is about 56.93 dB, which corresponds to an improvement of 6 dB (56.93 – 51). Therefore, when the data length increases four times, the IDR increases about 6 dB, which is the expected result.

4.15  Receiver Design Considerations One of the objectives of this chapter is illustrate the problems in designing a receiver. In order to achieve high sensitivity and IDR, the receiver designer must be able to avoid false detection and detect a weak signal in the presence of a strong signal. When the signal is weak, it is difficult to differentiate a signal from a noise spike. From the above simple simulations, three idiosyncrasies are discovered. The first idiosyncrasy is a sidelobe near the weak signal as shown in Figure 4.4(b). This

4.16  Conclusion

73

problem causes a higher threshold and reduces the probability of detecting a weak signal. The second idiosyncrasy is sidelobes near a strong signal as shown in Figure 4.10(a). This problem can cause a false alarm by the high peaks. The third idiosyncrasy is that the signal is spread into several frequency bins and no local peak is formed as shown in Figure 4.13. Thus, the signal is missed. When the input signal varies continuously through the frequency range, many of these problems and other ones unknown from this simple frequency selection can occur. Whenever an idiosyncrasy is observed, the problem must be studied to see there is any remedy. One must deal with many different output conditions to make a decision. Ward indicates that when the input signal is close to a frequency bin but not on the bin, the outputs may have many high sidelobes near the main output, which can limit the IDR [private communication with C. Ward, engineer from ITT, 2004]. In order to have a high sensitivity, a threshold must be set to detect the weak signal. This threshold is often set by using noise alone. Once an output crosses this threshold, a signal is detected. In this study this threshold has not been discussed. From this simple study it appears that a strong signal must be detected first. Once the strong signal is found, a threshold will be set to detect the weak one and this threshold setting probably should be strong signal–dependent. In order to achieve a high IDR, not only the amplitude of the strong signal but also the frequency and the phase should be considered. When 14- to 16-bit ADCs are used in the simulation, the outputs in the frequency domain have sidelobes, which are not present for 4- to 12-bit ADCs. It appears that with a high number of bits the outputs may reveal detailed information on the windows. If an ADC with a high number of bits is used in building a receiver, the high sidelobes must be properly investigated. These high numbers of bits (14 to 16) may not be available for a wideband operation in near future. Furthermore, the definition of local peak in this study is arbitrarily chosen 6 dB above the neighboring bins. This criterion could vary when different windows or numbers of bits are used. Therefore, the results from this study provide only a reference for the receiver design. In the real receiver design, more details need to be investigated for a specific window and the number of bits used. In this study, the ADCs are assumed ideal, that is, the quantization level is uniformly distributed. In reality even the input frequency is exactly placed on a frequency bin, and performing long FFT (214 points), the outputs have many high spurs [2]. These high spurs will further limit the IDR, but not the single signal DR. Finally from this chapter, if the FFT is used to build a receiver to identify the input signal frequency, the answer is not simply picking the highest outputs as the desired signals. One must study the outputs carefully and make the correct decision. In real receiver design through the FFT operation, most of the effort is spent on after the FFT operation. From this study knowing the number of input signals is also very important.

4.16  Conclusion The IDR obtained for all the cases are listed in Table 4.1. The IDR is obtained from knowing the strong input signal and the threshold is based on the spurious

74

Dynamic Range Study Through Fast Fourier Transform (FFT)

responses generated from the signal. For the rectangular and Hamming windows, when the input frequency is on a bin or at the boundary between two bins, the sidelobes are different. As a result, the IDR measured for cases 1 and 2 are close, but for case 3 the value is very low. It appears that for the Blackman and Chebyshev windows the sidelobes are about the same amplitude for frequency on the bin or at the boundary of two bins. The IDRs obtained for all three cases are about equal. In Table 4.2 the IDR values obtained from the 8-bit ADC with different maximum saturation voltages (270 mV and 1V) are very close to the rectangular window. Thus the 1-V, 8-bit ADC is not calculated for the other types of windows. The IDR value strongly depends on the window selected. Therefore, the window selection has a large impact on the IDR. The window also governs the frequency resolution. In this study, only very limited cases are investigated such as how special frequencies are used. The IDR depends on the input frequencies as shown in Chapter 3, but this study limits the separation to about 480 MHz, which is a very large value. In an actual receiver one should report the IDR as a function of input frequency. It is difficult to compare the results obtained in this chapter versus the results obtained in Chapter 3. They have different procedures. For example, the input frequency, the probability of false alarm, and the probability of detection are all different. Both studies use the same data points (256). By examining Table 4.2, except for case 1 of the rectangular window, the IDRs are lower than 6 dB/bit. The results from Chapter 3 can be used as a guide for receiver designs. The results obtained in this chapter cannot be used for this purpose. It reveals the possible problems in designing a receiver with very high IDR. Table 4.2  Measure IDR Using FFT Approach Maximum 270 mV 270 mV 270 mV VM gain 60 dB 45 dB 35 dB Number 4 6 8 of bits Rectangular Window Case 1 23 39 49.9 Case 2 20 36 47 Case 3 17 30 31 Hamming Window Case 1 21 36.8 47 Case 2 20 35.1 46 Case 3 20 34 38 Blackman Window Case 1 20 34 46 Case 2 19 34 46 Case 3 18.8 35 45 Chebyshev Window Case 1 19.6 35.5 45.3 Case 2 19 35 46 Case 3 18 34 45 Park-McClellan Window (local peak is 1 dB) Case 1 10 27 36 Case 2 10 27 36 Case 3 10 26 37

1,000 mV 45 dB 8

1,000 mV 34 dB 10

1,000 mV 25 dB 12

1,000 mV 10 dB 14

1,000 mV 0 dB 16

50 47.5 30.1

62 59 29

71 68 30

86 83 30.2

96 93 30.1

59 58 38

68 67 39

80 78 39

89 88 39

58 57 53

67 67 64

78 77 78

81 80 87

57 56.6 57

67.7 67 67

80 81 79.5

86 86 82.7

48 48 48

55 55 55

59 59 59

62 62 62.8

4.17  Remarks

75

4.17  Remarks This and the previous chapters are devoted to DR and concentrations on IDR. The following phenomena are observed. It is obvious that rectangular is not suitable for receiver designs through the FFT operation because of the high sidelobes. For the MUSIC method, the input data are not windowed. However, under very special conditions one can obtain a very high IDR, higher than that obtained from the MUSIC method. It is not appropriate to report these high values as achievable IDR. The IDR is a function of frequency separation as well as data length. If the frequency separation is not specified, one can assume that the frequencies are far apart and the reported value is the saturation value in Figures 3.10 and 3.11. The IDR must associate with data length (or PW). The results obtained in Chapter 3 are for 100-ns or 50-ns pulses. As discussed in Chapter 3 IDR is a function of frequency separation. Since the IDR is input frequency and frequency separation dependent, it is appropriate to report the IDR as a function of frequency separation. This information can provide the receiver capability of reporting two closely separated frequencies. From Table 4.1 it appears that the Blackman window provides decent results. In the frequency domain, the width is about 1.73 compared with the rectangular window. The Blackman window is used from most receiver studies in the following chapters. Another very important phenomenon has been reported on a receiver measurement [private communication with D. Jacobson and B. Mayhew, engineers from ITT, 2008]. The single signal dynamic range (SDR) is measured on a receiver and the result is opposite to the conclusion in Section 4.13. The SDR for a pulsed signal is higher than a CW signal. The explanation is as follows. In this special receiver the FFT length is fixed; thus, the lower limit of the SDR is determined. The upper limit is determined by the RF analog chain rather than the ADC. A CW signal can saturate the analog chain at a lower power level than pulsed signals. Therefore, the SDR is lower for a CW signal than for pulsed signals. All the IDR studies in this and the previous chapters are based on the ADC performance and the digital signal processing (DSP) following it. The RF chain, which includes amplifiers and mixers, is not included in the studies. If the RF chain is included, the problem becomes even more complicated because a mixer will generate many spurs and they usually affect the IDR. One can assure that when the RF chain is included in a receiver design, the IDR is usually lower than that obtained from the ADC effect alone.

References   [1]  Skolnik, M. L., Introduction to Radar Systems, New York: McGraw-Hill, 1962.   [2]  Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Chapter 12, Norwood, MA: Artech House, 2001.

Chapter 5

In-Phase and Quadrature Phase (IQ) Study

5.1  Introduction In some receiver applications complex input signals are preferred. For example, the amplitude of a complex signal can be easily obtained and the phase from sample to sample can be easily compared. If the input is a frequency modulated (FM) complex signal, the frequency modulation can be removed by a delay, a conjugation, and a multiplication process. However, the signal received by a receiver is usually real. The real input signal can be made complex through hardware manipulation; however, it is difficult to obtain a well-balanced result over a wide frequency range. Well-balanced means that the amplitudes of the IQ channels are equal and the phases are 90° apart. Because of these problems, it is a common practice that the IQ channels are generated after digitization. In other words, the IQ channels are generated digitally [1]. The FFT operation will automatically generate complex output data. A Hilbert transform is used to convert from real to complex data. The purpose of this chapter is to find the degree of balance between the IQ channels. If the IQ channels are not well balanced, spurious outputs will be generated, which limits the dynamic range of the receiver. The spur level versus imbalance can be found from Figure 5.1, which is a duplication of Figure 8.6 in [1]. The imbalance problem will be studied with this required criterion. Table 5.1 lists a few dynamic range values versus required amplitude and phase balances. These values are obtained from Figure 5.1. In order to achieve a 30-dB spur free dynamic range, the amplitude imbalance should be less than 0.28 dB and the angle imbalance should be less than 3.7°. The applications of this figure will be explained in Section 5.6. The amplitude of a signal can be used to either measure the signal strength or to detect a signal. The amplitude can be obtained through squaring and summing the real and imaginary parts (or the outputs of the IQ channels). If the IQ channels are not well balanced, the amplitude calculated will vary with time. Thus, the amplitude measurement or the detection problem will be adversely affected.

77

78

In-Phase and Quadrature Phase (IQ) Study

Figure 5.1  Image amplitude as a function of amplitude and phase imbalances.

5.2  Approach to Find the IQ Imbalance A certain process such as a Hilbert transform or an FFT operation can be applied to a real input signal to obtain two outputs channels. For the Hilbert transform, the original input is one channel (arbitrarily identified as the I channel) and the other channel will be created as the Q channel. For the FFT operation, from the real data two channels will be generated. The complex outputs obtained from the real signal are designed as I(xi) and Q(xi), where xi is the data in the time domain. In an ideal case, the amplitudes of the I and Q channels should be equal and the phase between them should be 90° off. The IQ outputs are evaluated through another FFT, here referred to as the measuring FFT. The length of the measuring FFT should be long because long FFT produces good IQ balance. From the measuring FFT outputs, the amplitudes and initial phases of the two channels will be found. Table 5.1  Required Dynamic Range Versus Required Amplitude and Phase Balances Required Dynamic Range (dB) 30 35 40 45 50

Required Phase Balance (°) 3.7 2 1.2 0.65 0.37

Required Amplitude Balance (dB) 0.28 0.17 0.085 0.05 0.027

5.3  FFT Output Imbalance Measurement Procedure

79

In this approach there is a dilemma that an FFT is used to evaluate the imbalance of another FFT’s outputs. If the measuring FFT is inaccurate, it will affect the accuracy of the results. From the simulation results it is shown that longer FFT has a better balance. Thus, in the following simulations, the length of the measuring FFT is 2,048 points. This length is selected to provide decent results and in the meantime to save calculation time. Hopefully, most of the imbalances are generated by the real-to-complex procedure and not by the measurement method. In the following simulations, the sampling frequency is assumed to be 2.56 GHz. Thus, the Nyquist bandwidth is 1.28 GHz. The input frequency changes from 100 to 1,180 MHz to avoid the frequency close to the edges. Since this study is to determine the angle between the real and imaginary parts, input noise is not introduced. The initial phase of the input signal is arbitrarily chosen to be zero.

5.3  FFT Output Imbalance Measurement Procedure It is well known that the FFT outputs are complex values. Although the outputs contain real and imaginary parts, the amplitude of them may not be equal and the angle between them may not be 90° output of phase. The following steps are used to find the amplitude and angle between the real and imaginary parts. One important assumption is made in the study: long FFT outputs can produce well-balanced IQ outputs. From the following studies it appears that this assumption is correct. Based on this assumption, the following steps are used for IQ imbalance. 1. Choose the input frequency at 100 MHz as the beginning test frequency to generate the desired signal. The length of the signal must be long enough to perform both the FFT length under test and the testing FFT operations. Perform FFT on a frame of real data with the desired length, such as 64 points. The 64 data points will be used to illustrate the following procedures. In this case the imbalance of 64-point FFT will be studied. Shift the window one data point and perform FFT again on the new data. This operation can be referred to as a sliding FFT and it can be shown mathematically as

Xm (k) =

m + N −1

∑

n=m

x(n)e

− j 2π kn N

(5.1)

where m = 0, 1, 2, . . . , 2,047 is the initial data point, N is the window or the FFT length. The length of m must be longer than the FFT under tests. In this special case, N equals 64 and m equals 2,048. From this operation the frequency outputs Xm(k) have the same time resolution as the input signal because every time only one data point is shifted. The k values are from 0 to 31 because the input data are real. 2. Find the peak value of the amplitude of X0(k) in the frequency domain and designed the output frequency bin as X0(k0). For a fixed input frequency, the peak amplitude from X(k) is obtained only once from X0(k) (the first FFT output frame) rather than many time frames from Xm(k). The reason for this

80

In-Phase and Quadrature Phase (IQ) Study

operation is that if a frequency is at the boundary of two frequency bins, it is possible for the peak to occur in two adjacent bins for a different m value due to the round-off error in the computer. For example, for the 64-point FFT, the frequency resolution is 40 MHz. If the peak of X(k) oscillates between two adjacent frequency bins, the bin number of the maximum amplitude will change with time. For example, if the input frequency is 220 MHz (boundary of bins 5 and 6), the maximum amplitude output may occur in either bin 5 or 6 as m increases. If these outputs are used for the imbalance study, the result may not be accurate. If X0(k) is used to find the peak, this problem can be avoided. For the same example, if X0(6) is the maximum output of the first frame, all the following maximum will be assumed in bin 6. Under this condition, the amplitudes of X(5) and X(6) are basically equal. 3. Two thousand and forty-eight Xm(k0) outputs are collected. Although these outputs are obtained from the frequency domain, these data Xm(k0) are in the time domain. Perform a 2,048-point FFT (the measuring FFT) on both the real and the imaginary parts of Xm(k0) and label the outputs as Yr(k) and Yi(k) to represent the results from the real and imaginary input data. This k value is from 0 to 2,047 because the input data are complex. The measuring FFT reduces the frequency resolution from 40 MHz to about 19.53 kHz (40 MHz/2,048). Find the peak from the real part FFT and label it Yr(k1). It should be noted that the k values obtained from step 1 and this step are different. The same k1 value is applied to the imaginary part as Yi(k1). The peak obtained from the imaginary part should be the same as in Yr(k1); however, the computer round-off error may select an adjacent bin. To avoid this potential problem, the k1 value from the real part is used for both Yr(k1) and Yi(k1) outputs. 4. Calculate the phase angles of the real Yr(k1) and imaginary Yi(k1) parts of the measuring FFT. If the outputs are 90°, they are considered to be the desired results. The calculated results minus 90° are considered the phase error. If the calculated phase is 90°, the phase error will be zero. Take the ratio between the amplitudes of Yr(k1) and Yi(k1) to represent the amplitude imbalance between the original N point FFT outputs. The amplitude imbalance is expressed in decibels, which is 20 times the logarithm of the ratio. 5. Repeat steps 1 through 4 for different input frequency. The frequency changes from 100 to 1,180 MHz in 1-MHz steps. In some studies, the frequency range changes from 141 to 1,140 to cover only 1,000-MHz bandwidth.

5.4  Results from Measuring FFT Before testing the FFT output results, it is interesting to know the results generated from the measuring FFT. In this test the FFT under testing is 2,048 points and the measuring FFT is chosen as 4,096 points. The testing procedure is similar to the above-mentioned approach. The results are shown in Figure 5.2. It is assumed that the errors are contributed by the 2,048 point FFT operations. In Figure 5.2 the worst results occur at both ends of the input bandwidth. The phase imbalance is less than 0.15° and this value causes a spur of less than 50 dB down. The worst amplitude imbalance is close to 0.05 dB and this value generates a

5.4  Results from Measuring FFT

Figure 5.2  Results of a 2,048-point FFT measured by a 4,096-point FFT.

Figure 5.3  The 4-point FFT output amplitude and phase imbalance.

81

82

In-Phase and Quadrature Phase (IQ) Study

spur level of about 45 dB, as shown in Figure 5.1. Thus, the spur level is controlled by the amplitude imbalance. These values can be considered as the limit of the testing results because they are the results of the testing FFT operation. If these output data are further processed, the spur level should be close to 45 dB. The results from shorter FFT operations, which are worse than the results in Figure 5.2, can be considered as caused by the short FFT length.

5.5  Imbalance Results of FFT Outputs For an FFT length N where N = 4, 8, 16, 32, and 1,024, the results are shown in Figures 5.3 to 5.7. In Figure 5.3, the angle error from frequency 100 to about 470 MHz is a constant of 45°. The explanation is as follows. For a 4-point FFT, there are 4 outputs (0, 1, 2, 3), but only two outputs 0 and 1 are used because the input data are real. The output from channel 0 is real because it is the summation of all the input data. It is anticipated that when the input frequency is about 320 MHz (1,280/4), the output will switch from channel 0 to channel 1. The means that the output from channel 1 is greater than from channel 0. However, this switching occurs at about 470 MHz rather than the expected 320 MHz. The switching from channels 0 to 1 depends on the initial phase and the frequency of the input signal, and in this simulation, the initial phase is assumed to be zero. Since the input is real, there are outputs in both channels 0 and 3. When the input signal frequency is

Figure 5.4  The 8-point FFT output amplitude and phase imbalance.

5.5  Imbalance Results of FFT Outputs

83

low, the signal in channels 0 and 3 will interact with each other, and the FFT output changes from channels 0 to 1 depend on the interference of these two signals. When the FFT output contains only real parts such as in channel 0, the imbalance calculation does not make much sense. Under this condition, the phase calculated can be either 0° or 45° depending on the programming; the explanation is as follows. The measuring FFT operating on the real part generates a complex quantity but the measuring FFT operating on the imaginary part is zero because the imaginary part is zero. If the angle is calculated from the real part divided by the imaginary part, the result is 45° because large (infinite) real and imaginary values are used for angle calculation. If the measured phase is 45°, the error phase is also 45° due to the error phase defined in Section 5.3. If the angle is calculated by the imaginary divided by the real, the result is 0° because in MATLAB the angle(0) is 0. The error phase is 90°. When the error phase is either 45° or 90°, it means that the FFT output has only a real component and the imaginary component is zero. Under this condition, the amplitude imbalance cannot be calculated because taking the logarithmic of the amplitude ratio will produce meaningless data. The output data only makes sense when the output of the FFT is in channel 1. Similar arguments can be applied to the results in Figures 5.4 and 5.5. From these figures the imbalance is quite obvious. When the input frequency is at 640 MHz, which is the center of the input band, the balance is very good. When a 32-point or longer FFT is performed, the output from 100 MHz on is complex.

Figure 5.5  The 16-point FFT output amplitude and phase imbalance.

84

In-Phase and Quadrature Phase (IQ) Study

Figure 5.6  The 32-point FFT output amplitude and phase imbalance.

Figure 5.7  The 128 point FFT output amplitude and phase imbalance.

5.5  Imbalance Results of FFT Outputs

85

From Figures 5.3 to 5.6, the general information is as follows. In Figures 5.3 and 5.4 some points at the ends of the bandwidth produce a 45° phase error (or generate a real output). These results cannot support the desired bandwidth. In Figure 5.5 the real output occurs close to 100 MHz. At this frequency if the initial phase of the signal changes from 0° to 360° in 1° step, at some initial phases the FFT outputs are real. If the receiver input bandwidth is reduced to 1,000 MHz (141 ~ 1,140), the outputs from the 16-point FFT are all complex. Since the initial phase of the input signal also influences the phase error, the phase effect must be evaluated. Placing the input frequency at 140 MHz and change input phase from 0° to 360° in 1° steps, the outputs are all complex. Therefore, for a 1-GHz input frequency range, the 16-point FFT can be considered as the minimum length. In Figure 5.6 the FFT length is 32 points and there is no 45° phase error for a frequency range of 1,080 MHz (101 ~ 1,180). Thus, the 32-point FFT is considered as the shortest FFT that can produce decent results. The other FFT lengths of 64, 128, 256, 512, and 1,024 produce similar but better results. Only the results of the 128- and 1,024-point FFTs are shown in Figures 5.7 and 5.8. In general, when the frequency is farther away from the center, which is 640 MHz with a 2,560-MHz sampling rate, the angle is farther away from 90°. A similar effect appears on the amplitude imbalance. When the input frequency is on a frequency bin, the angle is 90° and the amplitude is perfectly balanced. When the FFT length increases, the angle is very close to 90° and the amplitude balance is also better.

Figure 5.8  The 1,024 point FFT output amplitude and phase imbalance.

86

In-Phase and Quadrature Phase (IQ) Study

Figure 5.9  Worst phase and amplitude imbalance for an FFT length of 32 to 1,024.

Figure 5.9 shows the worst cases of angle and amplitude imbalances for all the data lengths. For FFT length equal to 4, 8, and 16, the worst amplitude imbalance cannot be found as explained previously. If the frequency range decreases or increases the lower frequency limit above 100 MHz, the worst case can be found. From this study it shows that longer FFT operation produces a better IQ balance, which is the expected result. The effect of the measuring FFT length is shown in Figures 5.10 and 5.11. In Figure 5.10, the FFT length is 32 points and the measuring FFT length is 256 points. It appears that results are slightly worse than the results in Figure 5.6. In Figure 5.11, the FFT length under test is 128 points and the measuring FFT length is 256 points. In comparison to Figure 5.7, the results are significantly worse. The worst angle imbalance in Figure 5.7 is less than 1.5°, but the worst angle imbalance in Figure 5.11 is greater 3°. Thus, when the measuring FFT length is comparable to the FFT length to be measured, the error is caused by both FFTs.

5.6  FFT Outputs from Imbalanced Inputs In this section the imbalance effect will be illustrated. The results will be compared with the results in Figure 5.1. The imbalance is generated from relatively short FFT operations. As shown in Figure 5.6, the amplitude and phase imbalances generated

5.6  FFT Outputs from Imbalanced Inputs

Figure 5.10  The 32-point FFT output imbalance measured by 256 FFT.

Figure 5.11  The 128 point FFT output imbalance measured by 256 FFT.

87

88

In-Phase and Quadrature Phase (IQ) Study

from the 32-point FFT operation are frequency dependent. When the input frequency is at 362 MHz the amplitude imbalance is about 0.84 dB and the phase imbalance is about 5.54 degrees and these are considered as relatively high. With these values the spur is slightly lower than 20 dB down, which can be obtained from Figure 5.1. When the input frequency is at 561 MHz, the amplitude and phase are well balanced. To measure the imbalance, a 2,048-point FFT operation is used. An input signal is generated at 362 MHz and the total data length is 32 + 2,048 points. The 32-point FFT operation is performed on the signal 2,048 times by sliding one data point each time. The highest output in the frequency domain from the first frame is used as the desired frequency output. There are 2,048 samples at this frequency value and these data are complex and in the time domain. The study procedure is to find the frequency-domain responses of the 2,048-point FFT operation on these data points. Since the outputs from the first 32-point FFT operation are complex, all the FFT outputs have independent information. The frequency-domain result is shown in Figure 5.12. Figure 5.12(a) shows the results of the input frequency at 362 MHz, and the output has a spur lower than 20 dB down. This value is close to the results shown in Figure 5.1. Figure 5.12(b) is obtained in a similar manner for input frequency at 561 MHz. At this frequency the 32-point FFT operation generates well-balanced IQ outputs. The spur generated from these data is about 60 dB down. From these illustrations, it is obvious

Figure 5.12  FFT outputs of imbalanced and balanced results: (a) input frequency at 362 MHz (poorly balanced), and (b) input frequency at 561 MHz (well balanced).

5.7  Windowed FFT Output Imbalance Study

89

that when the imbalance FFT outputs are further processed such as performing additional FFT operations, the imbalanced results generate poor IDR. From this figure one can see that the location of the spur is symmetric with respect to the middle of the input frequency outputs. One may suggest that the image spur can be calibrated out, that is, if the input frequency is measured, the output at the spur location can be neglected. This approach has one obvious deficiency: a true signal falling on the spur location will be missed. When there are multiple signals, the calibration may become complicated.

5.7  Windowed FFT Output Imbalance Study In receiver design window is usually added onto the input data points to reduce the sidelobes. In this section a Blackman window is chosen as an example. The imbalance effect will be studied with the Blackman window. A Blackman window can be written as



 2π n   4π n  win = 0.42 − 0.5cos   + 0.08cos    N   N 

0≤n≤N



(5.2)

The maximum sidelobe of this window is about 58.1 dB down.

Figure 5.13  Worst phase and amplitude imbalance for FFT length of 32 to 1,024 with a Blackman window.

90

In-Phase and Quadrature Phase (IQ) Study

A similar test procedure as stated in Section 5.3 is used on the windowed data. In these tests window is only applied to the short FFT operation and not applied to the measuring FFT. The results have a similar trend as data with a rectangular window. In general, the Blackman windowed FFT results have a better balance than the rectangular window case. Since the results are similar to the rectangular windowed cases, the results are not shown here. The worst imbalance is shown in Figure 5.13. Because of the real output produced from the 4- to 16-point FFT over the lower frequency range, the worst imbalance measured is difficult to present. Thus, the worst-case results start from the 32-point FFT. Except the results from the 32-point FFT, the rest of the results are very good. The phase and amplitude imbalances are less than 0.3° and 0.1 dB, respectively, if the FFT length is 64 points or longer. Figure 5.14 shows that the results of a 32-point FFT with frequency ranges narrowed to 1,000 MHz (141 to 1,140). The phase imbalance will drop for about 1.4° (Figure 5.13) to about 0.24° and the amplitude imbalance drops from close to 1 to 0.3 dB. Because the worst imbalances are at the ends of the input frequency range, reducing the bandwidth can eliminate the worst imbalance. Thus, with a Blackman window, a 32-point FFT can be used for a receiver design with a 1,000-MHz frequency bandwidth. In the following receiver design examples, this frequency range is often used.

Figure 5.14  The 32-point FFT output amplitude and phase imbalance with a reduced bandwidth (141–1,140 MHz).

5.8  Procedure for Finding Phase Tracking After the FFT Operation

91

5.8  Procedure for Finding Phase Tracking After the FFT Operation In order to obtain angle of arrival (AOA) information, multiple antennas and receivers must be used. One approach to obtaining AOA is to measure the phase between two antenna outputs. One can perform FFT on the output from one antenna to find the peak value in the frequency domain. This operation obtains the input frequency. Comparing the phase between the peak outputs from the two antenna outputs, one can find the AOA information. In order to perform the above operation, one must make sure that the phase difference at different frequencies must track. In this section the phase tracking is studied. The approach is similar to the imbalance study discussed in the previous section. Only two antennas are used and each one will produce digitized outputs in time domain. The steps are as follows. 1. Perform FFT on the real input data in the time domain from one antenna with the desired FFT length, such as 32 points, to obtain the input frequency. Shift one point of data and perform FFT again on the new data to accomplish the sliding FFT operation. This operation must be performed again on the outputs of the second antenna. The two data sets have the same input frequency with a fixed phase difference between them. The input signals from the two antennas and receivers can be written as



x1 = sin(2π ft) x2 = sin(2π ft + θ )

(5.3)

where f is the input frequency and θ is phase difference between the two antennas. In the following study, the phase angle θ is arbitrarily chosen as 20°. The results from the FFT operations on signals x1 and x2 are X1(k) and X2(k). The subscripts 1 and 2 are used here to indicate the signals obtained from different antennas, while the subscript m in (5.1) references different times. 2. Find the peak value of X1(k0) in the frequency domain as discussed in the previous section. Once the frequency k0 is determined, find X2(k0). Collect 2,048 points of data from X1(k0) and X2(k0) to form two new time series. Perform a 2,048-point FFT on the two new time series. Let us refer to these outputs as Y1(k) and Y2(k), where k is the frequency obtained from the measuring FFT. 3. Find the peak from the real part measuring FFT and label it k1. The frequency k1 should be closer to the true input frequency than k0 because the second FFT uses longer data. The angles obtained from Y1(k1) and Y2(k1) are represented by q1 and q2, respectively. The difference between q1 and q2 should equal the input phase q. 4. Repeat the operations in steps 1 to 3 for different frequencies. The frequency changes from 100 to 1,080 MHz in 1-MHz steps. Since the phase may have a wraparound problem, adjustment is needed in some outputs. The difference between the measured phase and the input phase is

92

In-Phase and Quadrature Phase (IQ) Study

Figure 5.15  Phase tracking after a 32-point FFT operation.

recorded as the phase error. Since for the rectangular windowed input 32 data points are the minimum for FFT length to obtain the desired IQ balance as discussed in the previous section, this study will evaluate this case. The results are shown in Figure 5.15. Figure 5.15(a) shows the phase error and Figure 5.15(b) shows the amplitude imbalance and the results are very good. When the input frequency is farther away from the center frequency, the tracking error increases. The worst phase imbalance is about 0.015°. These results are expected because both 32-point FFT operations will produce similar errors and these errors will cancel. Thus, the results should be sufficient to support phase AOA measurements. Since the longer and window FFT operations can produce better results, they are not evaluated. In the above operations the results obtained from an antenna are frequencies k0 and k1. Since the second antenna will use the same frequency component, one component discreet Fourier transform (DFT) can be used to find the needed information. This operation saves calculation time, which might be considered in real-time receiver designs.

5.9  Procedure for Finding an IQ Imbalance of the Hilbert Transform The purpose of this study is to find the performance of the Hilbert transform [3]. The evaluation criterions are also the phase and amplitude balances. The two input parameters for this study are the input frequency range and the filter length.

5.9  Procedure for Finding an IQ Imbalance of the Hilbert Transform

93

In this study 2,048 points of real input data are used. The Hilbert transform is performed on the data to obtain the imaginary part. The coefficients of the Hilbert transform with a length of 11 sections are:



é -2 c=ê ë5π

0

-2 3π

0

-2 π

é -1 c=ê ë5

0

-1 3

0

-1 0 1 0

0

2 π

0

2 3π 1 3

0 0

2 ù 5π úû

1ù 5 úû

or (5.4)

In this equation, the first line is the values obtained from the discrete Hilbert transform. The second line is the value used in the computer program, because only the ratios are of interest. For another Hilbert filter length, the same pattern follows. The filter length must be an odd number, and the minimum increment is four sections because the two zero coefficients must be included. The shortest filter is three sections, the next one is seven sections, and so forth. After the imaginary portion is obtained, the transient effect in the filter must be compensated. The compensation length cl is a half filter length. Since the filter length is odd, the compensation length is

cl =

fl − 1 2

Figure 5.16  Phase and amplitude imbalance for a 3-section Hilbert filter.

(5.5)

94

In-Phase and Quadrature Phase (IQ) Study

where fl is the filter length. To compare the output of the Hilbert transform, the real signal (the input) will start from the filter length in order to obtain a filter with full data or avoid the transient effect. The imaginary signal will start from the filter length plus the compensation length cl. The input signal does not contain noise and the initial phase is arbitrarily chosen as zero. The measurement of the real and imaginary is similar to the procedure described in Section 5.3. An FFT with 2,048 points will be used to find the maximum frequency components. From the peak values the amplitude and phase are compared.

5.10  R  esults of an IQ Imbalance of a Hilbert Transform   with a Rectangular Window In the previous study in which the outputs are produced from a windowed FFT operation, the amplitudes of the I and Q channels are approximately equal. However, in the Hilbert transform, the amplitude of the imaginary part can be quite different from the real part. If this amplitude difference is constant across the testing frequency range, the amplitude difference can be compensated. If the amplitude imbalance is a function of frequency, a simple compensation method is difficult to apply. Since the results of different filter lengths are similar, only a few cases are shown. The results of the Hilbert transform imbalance are shown in Figures 5.16 to 5.18

Figure 5.17  Phase and amplitude imbalance for a 27-section Hilbert filter.

5.10 Results of an IQ Imbalance of a Hilbert Transform with a Rectangular Window

95

Figure 5.18  Phase and amplitude imbalance for a 39-section Hilbert filter.

Figure 5.19  Worst phase and amplitude imbalances of a Hilbert filter for a frequency range of 100 to 1,180 MHz.

96

In-Phase and Quadrature Phase (IQ) Study

for filter lengths of 3, 27, and 39. The phase results are very good and usually less than 1°. The amplitude imbalance is rather poor. In Figure 5.16, the amplitude imbalance is from about -6 to 6 dB. In Figure 5.17, the amplitude imbalance is from about -3 to -5.5 dB. The amplitudes of the real and imaginary parts may be adjusted by some constant. Thus, the worst amplitude imbalance should be the difference between the maximum and minimum values. For example, the worst amplitude imbalance in Figure 5.16 should be close to 12 dB (6- (-6)). The worst amplitude imbalance in Figure 5.17 should be around 2.5 dB (-3-(-5.5)). As in the case of FFT outputs, the worst phase and amplitude imbalances occur at the low and high frequency ends. Figures 5.19, 5.20, and 5.21 show the worst phase imbalance versus the input frequency range. Three frequency ranges are shown in these figures and they are from 100–1,180, 200–1,080, and 300–960 MHz. Even in Figure 5.19, the widest frequency range, the largest phase imbalances (slightly over 0.3°) are very good for different Hilbert filter lengths. Since the amplitude imbalance varies with input frequency, it is difficult to calibrate them out. The phase imbalances are all very good and are not the limiting factor. The amplitude imbalance as a function of the frequency range improves slightly with a smaller frequency range. In Figure 5.21 even with a 39-section Hilbert filter and the frequency reduced to 300 to 960 MHz, the worst imbalance is still at 0.7 dB, which corresponds to a spur level of less than 25 dB. The high spur level is not suitable for

Figure 5.20  Worst phase and amplitude imbalances of a Hilbert filter for a frequency range of 200 to 1,080 MHz.

5.11 Results of IQ Imbalance of the Hilbert Transform with a Blackman Window

97

Figure 5.21  Worst phase and amplitude imbalances of a Hilbert filter for a frequency range of 300 to 980 MHz.

receiver applications. In order to achieve better amplitude imbalance, the length of the Hilbert transform should be longer than 39 sections.

5.11  R  esults of IQ Imbalance of the Hilbert Transform   with a Blackman Window In this section the Hilbert transform is studied again and the only difference is that a Blackman window is used to replace the rectangular window. Since the phase and amplitude imbalances versus input frequency are similar to the results in the previous section, they are not shown. Only the worst phase and amplitudes imbalances are shown in Figures 5.22 to 5.24. As expected, the phase imbalance is not a problem and will not be discussed. In Figure 5.22, even the longest Hilbert filter (39 sections) has an amplitude imbalance of 0.78 dB, which corresponds to a spur level of 20–25 dB (see Figure 5.1). When the frequency range narrows, the amplitude imbalance gets better. For example, for the 200- to 1,080-MHz frequency range, the 27 section filters have an amplitude imbalance of 0.12 dB. For the 300- to 980-MHz frequency range, the 18 section filters have an amplitude imbalance 0.067 dB. From these results, it is obvious that the Blackman window improves the IQ amplitude imbalance. As a general rule, if the frequency range of the Hilbert transform is narrowed, the worst amplitude imbalance improves. In order to design a

98

In-Phase and Quadrature Phase (IQ) Study

Figure 5.22  Worst phase and amplitude imbalances of a Hilbert filter with a Blackman window for a frequency range of 100 to 1,180 MHz.

Figure 5.23  Worst phase and amplitude imbalances of a Hilbert filter with a Blackman window for a frequency range of 200 to 1,080 MHz.

5.12  IQ Imbalance of Polyphase Filters

99

Figure 5.24  Worst phase and amplitude imbalances of a Hilbert filter with a Blackman window for a frequency range of 300 to 980 MHz.

1,000-MHz bandwidth receiver, which is from 141 to 1,140 MHz, the 39-section Hilbert filter with Blackman window can produce an amplitude imbalance of 0.12 dB. This data point is obtained separately and is not available from the figures. This frequency range is used later in receiver design examples.

5.12  IQ Imbalance of Polyphase Filters Polyphase filter is also referred to as the multiple rate filter. The main idea can be found in [2]. Only the results will be very briefly mentioned here. An example will be used to illustrate the idea. A polyphase filter uses a short FFT such as 32 points to process longer data such as 256 points. A window will be used on the input data (256 points) and the input will be divided into eight 32 sections. The Park-McClellan window is used in this study and can be found in [1, 2]. All the 8 sections are added together to obtain 32 resulting points. The FFT operation will be performed on the 32 input data points. The data will shift 32 points each time. For real input data this operation will produce 16 output channels. The testing procedure is similar to the FFT output testing. The measuring FFT has 2,048 points. In the evaluation procedure, the polyphase filter shifts only one data point each step instead of the FFT length. Thus, the total input data length

100

In-Phase and Quadrature Phase (IQ) Study

Figure 5.25  (a) Time-domain and (b) frequency-domain response of an eight-point Park-McClellan filter.

used in this process is the filter length plus the 2,048 points. In this test the filter length is 8, 16, and 32 points. The frequency range is from 100 to 1,180 MHz. The results are shown in Figures 5.25 to 5.31. Figure 5.25 shows the time- and frequency-domain responses of a 256-point Park-McClellan window for the 8 data point FFT operation and the corresponding frequency-domain output. It is interesting to note in Figure 5.25(a) that there are about eight data points in the center of the window. These eight data points make the major contribution to the frequency outputs. It is reasonable, in actual filter design, the input data only shift eight points each frame so that there is no data lost in the processing. In the frequency domain, the sidelobes are very low and the highest sidelobe is about 160 dB down, as shown in Figure 5.25(b). Figure 5.26(a) shows the phase imbalance and Figure 5.26(b) shows the amplitude imbalance. The results are very good. It should be noted that in this figure the input frequency range is not from 100 to 1,180, but from 200 to 1,180 MHz. The lower frequency limit is reduced to 200 MHz to make sure that the FFT outputs are complex at the lowest frequency. The limit of the higher frequency range stays at 1,180 MHz. Figure 5.27 shows the time- and frequency-domain responses of a ParkMcClellan window for a 16-point FFT operation. There are about 16 data points in the center of the window. In actual application, the input data shift 16 points each frame. The highest sidelobe is about 105 dB down. Figure 5.28(a) shows

5.12  IQ Imbalance of Polyphase Filters

101

Figure 5.26  (a) Phase and (b) amplitude imbalances for an eight-point polyphase filter.

Figure 5.27  (a) Time-domain and (b) frequency-domain response of a 16-point Park-McClellan filter.

102

In-Phase and Quadrature Phase (IQ) Study

Figure 5.28  (a) Phase and (b) amplitude imbalances for a 16-point polyphase filter.

Figure 5.29  (a) Time-domain and (b) frequency-domain response of a 32-point Park-McClellan filter.

5.12  IQ Imbalance of Polyphase Filters

103

the phase imbalance and Figure 5.28(b) shows the amplitude imbalance. The results are also very good. The input frequency range for this simulation is from 100 to 1,180 MHz. Figure 5.29 shows the time- and frequency-domain responses of a Park-McClellan window for the 32 point FFT operation. There are about 32 data points in the center of the window. In actual application, the input data shift 32 points each frame. The highest sidelobe is about 59 dB down. The sidelobe for the 32-point polyphase filter can be fine-tuned to further suppress to sidelobe. In [2], the highest sidelobe is over 70 dB down for the same input condition. Since in this study the imbalance is the main subject, fine tuning of the filter is not performed. Figure 5.30(a) shows the phase imbalance and Figure 5.30(b) shows the amplitude imbalance. The results are very good. It is interesting to note that for the polyphase filter, the short sectioned filter has lower sidelobes. As expected, the worst imbalances for both the phase and amplitude are at the lower and higher frequency ranges. Figure 5.31 shows worst phase and amplitude imbalances. All the three filter lengths have good results. In a receiver design the number of input data may not be 256 points. Shorter or longer data can be used to modify the performance. This study only concentrates on the IQ balance. In an actual receiver design the transient effect is a major issue. A longer transient will limit the short pulse capability of the receiver.

Figure 5.30  (a) Phase and (b) amplitude imbalances for a 16-point polyphase filter.

104

In-Phase and Quadrature Phase (IQ) Study

Figure 5.31  Worst phase and amplitude imbalances for different polyphase lengths.

5.13  IQ Imbalance from a Special Sampling Downconversion Scheme This approach is to assume that there are IQ mixers with a local oscillator frequency at one-fourth the sampling frequency. In this example, the sampling frequency is at 2,560 MHz; thus, the local oscillator frequency is at 640 MHz. The detailed operation can be found in [2]. Only the procedure will be briefly presented here. 1. Decimate the input data alternatively into two channels. The results will be

I = x0 , 0, − x2 , 0, x4 , 0, x6 Q = 0, x1, 0, − x3 , 0, x5 , 0

(5.6)

where x is the individual data point and subscripts are the time reference. In this operation 0 will be added between two data points and the data change signs alternatively. 2. Ideally there are two output frequencies from a mixer. They are the sum and difference frequencies of the input and the local oscillator frequencies. The outputs generated from (5.6) must have a lowpass filter to eliminate the sum frequency component. The desired filter outputs are the difference frequency.

5.13  IQ Imbalance from a Special Sampling Downconversion Scheme

105

The testing procedure is similar to the previously discussed method. The input data is 2,148 points long. The filter has 16 taps and is generated through a Kaiser window. Since both the decimated outputs must pass a filter, there are transient effects at both ends of the signal. The measurement avoids the transients. At the IQ outputs 50 points are neglected at both the leading and trailing edges. The 50 points are arbitrarily chosen so that long enough data are removed from the outputs to avoid transients. The final results under test are 2,048 (2,148 - 2 × 50) points. The FFT operation is performed on both the IQ data to find the maximum output. The phase and amplitude are compared at the maximum output. The input frequency changes from 100 to 1,080 MHz. The results are shown in Figures 5.32 and 5.33. Figure 5.32(a) shows the phase imbalance and Figure 5.32(b) shows the amplitude imbalance. The results are well balanced except at the input frequency of 640 MHz. Figure 5.33 shows the same result without the frequency around 640 MHz. The frequency range neglected is from 623 to 643 MHz. This frequency region is decided by observing Figure 5.32 in detail. With this selection, the worst phase imbalance is less than 1.5° and the amplitude imbalance is less than 0.3 dB. The poor imbalance that occurs at 640 MHz can be explained as follows. Since the local oscillator frequency is at 640 MHz, when the input signal is close to this frequency the mixer output is close to the direct current (dc). When the outputs

Figure 5.32  IQ imbalance of the special frequency selection method: (a) phase imbalance and (b) amplitude imbalance.

106

In-Phase and Quadrature Phase (IQ) Study

Figure 5.33  Narrowed frequency range of IQ imbalance of the special frequency selection method: (a) phase imbalance and (b) amplitude imbalance.

are dc, the FFT output becomes real and the phase measured is similar to the lowfrequency phenomenon discussed in Section 5.5.

5.14  Conclusion Although an FFT-related operation can generate IQ channels, these channels may not be perfectly balanced. A longer FFT operation can improve both the amplitude and phase balances. Windowed FFT operation can improve the IQ balance. Thus, windowed FFT not only improves the dynamic range of a receiver by suppressing the sidelobes in the frequency domain, but also produces better balanced outputs. Using the FFT operation on different antennas produces a good phase tracking among the different outputs. The imbalances are similar between various outputs and the errors may be cancelled. It is interesting that the Hilbert transform causes mainly amplitude imbalance. The phases are well balanced even with a short filter length. Thus, if a system is designed only to use the phase measurement such as in an instantaneous frequency measurement (IFM) receiver, the Hilbert transform may produce satisfactory results with short filter length. Polyphase filter can produce well-balanced IQ outputs with a very short FFT operation. If a receiver is designed with a few channels, the polyphase filter should be the desired operation.

5.14  Conclusion

107

Table 5.2  Approach with an Achievable Dynamic Range in Decibels for 141 to 1,140 MHz Approach DR dB

32-Point FFT Blackman* Close to 50

39 Hilbert Blackman 35

8-Point Polyphase FFT** 50

* Figure 5.16 ** Longer FFT has a slightly worse balance.

Although IQ generated from the special sampling scheme can produce decent results, there is a hole in the input frequency range. The hole is at the center of the input bandwidth; this is an undesired result. It was illustrated in Sections 5.1 and 5.6 that the imbalance outputs limit may affect the receiver IDR if the outputs are further processed. If the output is only used for detection as discussed in chapter, the imbalance does not affect the IDR. It only affects the output amplitude such as the amplitude changes with time rather than staying at a constant value. Table 5.2 provides some brief guidelines for the three methods to generate IQ outputs.

Reference [1]

Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Norwood, MA: Artech House, 2001.

Chapter 6

Signal Detection from Fast Fourier Transform (FFT) Outputs

6.1  Introduction The purpose of this chapter is to find the probability of detection and probability of false alarm. This is a well-known problem for receiver engineers. In this study the outputs from the fast Fourier transform (FFT) operations will be used for detection. One common approach to building a receiver is to perform detection from the FFT outputs. Since Chapter 5 shows that when the FFT length is shorter than 32 points, the IQ outputs are not well balanced, the shortest FFT length in this chapter will start from 32 points. In order to perform this study, it is necessary to find the noise and signal distributions. Once the distributions are found, the probability of false alarm and probability of detection can be determined. Another related study is to sum the amplitudes of the FFT in the time domain. This is a noncoherent processing to improve receiver sensitivity if the input signal is long. By summing the FFT outputs will not improve frequency resolution but can only improve sensitivity for detecting long weak signals. This is a common practice in radar detection. After summing the amplitude of the FFT many times, the probability density changes from Rayleigh to Gaussian. It is important to find the probability density function of the transition period from Rayleigh to Gaussian. The study will include all three cases: the Rayleigh, the normal distributions, and the distribution during the transition stage. The window effect on the signal and noise will also be studied to see whether the window affects the receiver sensitivity. Most of the study in this chapter is based on simulation results. The conclusions drawn are also based on simulation results rather than theoretical prediction. Theoretical prediction is only used to check the simulation results. The main purpose of this study is not to determine the sensitivity of a receiver but to find the relative sensitivities from several different cases. For example, doubling the FFT length should improve the sensitivity about 3 dB. Adding two amplitudes together, which is referred to as noncoherent integration, should improve the sensitivity about 2.7 dB [1–3]. Many of these subjects will be discussed in this chapter. In additional to the detection problem, the phase comparison method will be studied. This method is critical for receiver design. It can provide very good frequency resolution such as with the kilohertz resolution. In order to detect the characteristics of a spread spectrum, biphase shifting keying (BPSK) and chirp signals, the phase comparison method can provide the chip rate and the chirp rate. If a 109

110

Signal Detection from Fast Fourier Transform (FFT) Outputs Table 6.1  Number of Frequency Bins Selected Versus FFT Length and Frequency Range FFT length Megahertz per channel 101~1,080 (1,080) 141~1,040 (1,000)

16 polyphase 160

32 80

64 40

128 20

256 10

— 7 (1–7)

15 (1–15) 13 (2–14)

29 (2–30 ) 27 (3–29)

57 (4–60) 53 (6–58)

109 (10–118) 101 (14–114)

receiver is designed to receive a spread spectrum signal, it appears that the phase comparison method is a necessity. In the following simulations, the sampling frequency is assumed at 2.56 GHz. The data lengths for the FFT operation are 64, 128, and 256 points. For the polyphase study the FFT length is limited to 16 and 32 points. The input frequency has two ranges: 101 to 1,080 with a bandwidth of 1,080 MHz and 140 to 1,040 MHz with a bandwidth of 1,000 MHz. The first frequency range is intended for testing the widest frequency coverage, and the second frequency range is intended for building a practical receiver. For both cases, the total output frequency bin numbers are listed in Table 6.1 for convenience. The FFT output channels are designed as 0 to N/2. This notation is different from the MATLAB notation because the first channel is 1 instead of 0. The overall number of channels is odd. In Table 6.1, the number of channels selected is slightly higher than the desired bandwidth. For example, the 64-point FFT has a 40-MHz frequency resolution as indicated in column 4. If 27 and 25 channels are selected, the total frequency range will be 1,080 (27 × 40) and 1,000 MHz (25 × 40), which equals the desired frequency bandwidth. Since it is desirable to select a bandwidth slightly higher than the desirable frequency range, the next higher numbers 29 and 27 are selected. The number of channels affects the probability of false alarm. Although this effect should be obvious, it is seldom discussed in the electronic warfare (EW) receiver designs. In order to keep the following discussion through the traditional expectation such as to generate a probability of false alarm of 10-7 with a 90% probability of detection, an S/N of 14 dB is required. The reason is to check whether the simulation follows the theoretical prediction. The impact on the number of channels will be discussed in Section 6.10.

6.2  Rayleigh Distribution Obtained from Noise Output In this section, the input contains only noise. The noise has a Gaussian distribution with 0 mean and standard deviation of 1. The purpose is to approximate the amplitude of the FFT outputs with a Rayleigh distribution and find a threshold. The procedure is to perform the FFT operation on a real noise input signal because real data will be used for receiver designs. Take all the frequency bins except the ones close to the edges to limit the output frequency range. For example, the 32-point FFT produces 16 independent outputs. Only the frequency bins from 3 to 14 will be taken into consideration. The output frequency range is less than 1,000 MHz. Similarly for the 256-point FFT, the frequency bins from 17 to 110 will be taken into consideration. In the following simulations, the sampling frequency is 2.56 GHz and the FFT length varies from 32 to 1,024 points with a rectangular window. One thousand

6.2  Rayleigh Distribution Obtained from Noise Output

111

runs with random noise are carried out. Since each run generates many frequency outputs, the total number of data used are more than 1,000 points. For example, the 32-point FFT will generate 12,000 useful frequency bins and the 256-point FFT will generate 94,000 frequency bins. The amplitude of the selected bin is recorded and it can be obtained from the real and imaginary portions of the FFT output as

A = Xr2 (k) + Xi2 (k)

(6.1)

where X(k) is the output with the kth frequency and the subscripts r and i represent the real and imaginary of the FFT output. The distribution of A should be Rayleigh, which is given as −r2



r 2 p(r) = 2 e 2σ σ

(6.2)

In (6.2) there is only one constant s 2 and r is a variable. This value s is related to either the mean or the standard deviation as σ=

m π /2

σ=

2v 4−π





(6.3)

where m is the mean and v is the variance of measured noise distribution. The mean and variance are obtained from the above simulation. The s values obtained by these two relations are very close. In this study the average from the two values is used. Once the noise distribution is approximated by the Rayleigh distribution, the threshold can be set as function of probability of false alarm

thr = −2σ 2 ln(Pfas )

(6.4)

where Pfas is the probability of false alarm, which is arbitrarily set to 10-7. Figure 6.1 shows the simulated results of the 256-point FFT operation. The symbol “o” is obtained from the histogram of the amplitude distribution. The Rayleigh distribution is shown as a solid line. Since all the noise amplitudes from different FFT lengths are well fitted to the Rayleigh distribution, only one of the noise distributions (the 256 points) is shown. The thresholds obtained from different FFT length are listed in Table 6.2. In Table 6.2, the threshold from the 256 point FFT is calculated 100 times and the standard deviation is about 0.35, which is about 0.54% (0.34/64.25) variation. The thresholds listed in Table 6.2 will be used later in the probability of detection study. For the Rayleigh distribution, this threshold is about 12 dB above noise. The threshold value t_dB in decibels can be obtained through

112

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figures 6.1  Noise distribution fitted by Rayleigh distribution with 256-point data with a threshold.







2



thr thr_ dB = 10log  N −1   1  ni2   ∑ N 0 



(6.5)

where thr is the calculated threshold, N is the total noise components, and ni is the individual noise component. If the noise is known as a Rayleigh distribution, the threshold can be calculated from (6.5) by setting t_dB = 12.1 dB.

6.3  Signal-to-Noise (S/N) Distribution When there is an input signal in the data, the output distribution should be Rician as r − p(r) = 2 e σ



r 2 + A2 2σ 2 I

 rA    σ2 

(6.6)

o

Table 6.2  Threshold for Probability of False Alarm of 10-7 with a Rectangular Window FFT length Threshold

32 22.92

64 32.07

128 45.26

256 64.24

512 90.66

1024 128.33

6.4  Probability of Detection

113

Figure 6.2  Rician distribution.

where A is the amplitude of the input signal, s 2 is the variance of noise, r is a variable, and Io is the modified Bessel function of the zero order. The Rician distribution is plotted in Figure 6.2. In this figure there are two S/N values: -100 dB (noise only) and 14 dB. When the input has only noise, the distribution should behave like a Rayleigh distribution. The results agree with the expectation. While the input signal increases, the distribution is somewhat like a Gaussian distribution. In this plot a threshold is also is provided and it has a false alarm rate of 10-7. This threshold is obtained through the Rayleigh distribution. With a 14-dB S/N, the probability should be close to 90%. This figure can provide some idea on the probability of a false alarm and the probability of detection. When the threshold moves higher, both the probability of a false alarm and the probability of detection decrease. On the other hand, when the threshold moves lower, both the probability of false alarm and the probability of detection increase. Thus, with a given S/N, one can obtain different combinations of the probability of a false alarm and the probability of detection. In general, it is difficult to use simulation data to match a Rician distribution. However, using simulation to find probability of detect is a straightforward approach. In the following sections, the probability of detection is obtained through simulation.

6.4  Probability of Detection With a sampling frequency of 2.56 GHz and a 256-point FFT, the frequency resolution is 10 MHz. All the frequencies such as 10, 20, 30 MHz, and so forth are on

114

Signal Detection from Fast Fourier Transform (FFT) Outputs

a frequency bin and frequencies at 5, 15, 25 MHz, and so forth are the boundary between two bins. In this section, the probability of detection will be studied only for two input conditions. In the first case the input frequency is fixed on the center of a frequency bin. If the input frequency is at 240 MHz, it is the center of a frequency bin for all FFT lengths from 32, 64, . . . , 1,024 data points. In the second case the input frequency is randomly selected between 141 and 1,140 MHz to cover an input bandwidth of 1,000 MHz. For both cases the initial phase of the input signal is arbitrarily chosen to be 0 and the thresholds listed in Table 6.1 are used for detection. FFT is performed on the input data for 1,000 times. The maximum output bin is selected. If the amplitude is higher than the threshold, a signal is detected and frequency accuracy is not checked. The results for the fixed frequency with a rectangular window are shown in Figure 6.3. In this figure, all the curves are separated by about 3 dB. This is the desired result that when the data length is doubled the sensitivity improves by 3 dB. For simplicity, only the 128- and 256-point FFT results will be discussed. The probability of detection is about 90% at about -4.3 and -7.4 dB for the 128- and 256-point results. Since the input is real, the Nyquist input bandwidth is 1,280 MHz. After the 128- and 256-point FFT operations the output bandwidth is 20 and 10 MHz, respectively. The bandwidth is improved 64 and 128 times, which corresponds to a gain of 18.1 and 21.1 dB. The output S/N is about 13.7 dB (21.1 7.4) and 13.8 dB (18.1 - 4.3), which is close to the required value of 14 dB. This simple calculation confirms the required S/N of 14 dB to achieve the 90% probability of detection and the 10-7 probability of false alarm. Figure 6.4 shows the results of the probability of detection for the random input frequency with a rectangular window. The separation between two adjacent curves

Figure 6.3  Probability of detection versus input S/N for a probability of false alarm 10-7 with a frequency of 240 MHz.

6.5  Probability of Detection with a Blackman Window

115

Figure 6.4  Probability of detection versus input S/N for a probability of false alarm 10-7 with a random input frequency between 141 and 1,140 MHz.

is also close to 3 dB. The required input S/N is slightly higher. For 128 and 256 points the required input S/N equals -2.8 and -5.7 dB, respectively, which is about 1.5 and 1.7 dB higher than the input frequency on a frequency bin. The sensitivity decrease is also the expected result, because when the input frequency is off a frequency bin, the amplitude decreases.

6.5  Probability of Detection with a Blackman Window In building a receiver, a window is usually used to reduce the sidelobes in the frequency domain. In this section a Blackman window will be applied to the input data. The probability of detection will be evaluated based on this window. Introducing the window also affects the noise distribution and threshold. A new set of thresholds is generated. Figure 6.5 shows the noise output and the corresponding threshold for the 256 data points. As expected, the results match the Rayleigh distribution closely. The outputs from other data length have similar results and are not shown. The threshold values are listed in Table 6.3. These values are considerably lower than those in Table 6.1. These values are used for calculating the probability of detection. The results are shown in Figure 6.6. For the 128- and 256-point FFTs, the S/N = -1.5 and -4.6 dB for a 90% probability of detection. Comparing with the results in Figure 6.4, the sensitivity drops slightly over 1 dB. From this study it is shown that the receiver sensitivity can be predicted quite accurately through the probability of false alarm and the probability of detection. With digital receivers the noise power and its distribution can be calculated

116

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figure 6.5  Noise distribution fitted by a Rayleigh distribution with a 256-point windowed data with a threshold.

through simulations rather than observed through a scope, as in the case of analog receivers.

6.6  Threshold Through the Convolution Approach One way to improve the sensitivity of a receiver is through noncoherent processing, which is summing the output from many frames. A frame of data is defined as the data in one FFT length. If the pulse width is several frames long, by summing the output can improve the sensitivity of a receiver and this is a common practice in radar detection. There are many studies on this subject [1, 2, 4] and graphs to determine the threshold. However, the results are difficult to use. The threshold can be obtained from numerical integration. This section discusses the approach. The amplitude outputs from the FFT outputs with different types of window can be considered Rayleigh distribution. Many of these amplitudes summed together can be approximated by a Gaussian distribution. If the distribution can be determined, the threshold can be found. However, if a small number of amplitudes are added, the Gaussian distribution may not a good approximation. Marcum [4] suggested that 1,000 summations can be approximated by a Gaussian. This result will be verified through the discussion in Section 6.8. Table 6.3  Threshold for a Probability of False Alarm of 10-7 with a Blackman Window FFT length Threshold

32 12.42

64 17.63

128 25.11

256 35.18

512 50.06

1024 70.74

6.6  Threshold Through the Convolution Approach

117

Figure 6.6  Probability of detection versus input S/N for the probability of false alarm 10-7 with a Blackman window and a random input frequency between 141 and 1,140 MHz.

The basic approach will be as follows. Assume that the outputs from the FFT outputs are Rayleigh distribution, the summation of two amplitude is equivalent to the convolution of two Rayleigh function. If the summation is n times, the Rayleigh function will be convolved n times. This convolution can be performed through numerical integration. Once the desired convolution result is obtained, the next step is to find the threshold. Since the convolution result is numerical form, the threshold is obtained through try and error. The overall approach can be summarized as follows: 1. Use FFT outputs to find the Rayleigh distribution. The FFT outputs will be used to find the Rayleigh distribution and also used for the summation process. Find the Rayleigh distribution through the approach discussed in Section 6.2. A threshold, referred to as the initial threshold, is calculated from the Rayleigh distribution. Simulation results indicate that the initial threshold is always smaller than the threshold obtained from convolution, the desired threshold. 2. Perform numerical convolution on the obtained Rayleigh function. The length of the Rayleigh function [or the r value in (6.2)] is selected from zero to 1.2 times the initial threshold. If the r value in (6.2) is not large enough, the convolution result will be inaccurate. Simulation results show that when the length of the Rayleigh function is from zero to the initial threshold the desired threshold calculated stays about constant. This means that choosing the Rayleigh from zero to the initial threshold is long enough to cover most the information. From Figures 6.1 and 6.5 this condition can be further illustrated. The length of the Rayleigh function is selected as 1.2 times the initial threshold to further guarantee the calculated results. The length of the numerical convolution function is twice the length of the Rayleigh function minus one. The number of convolutions performed equals the number of summations.

118

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figure 6.7  Two summations and the convolution output.

3. Use trial and error to find the threshold. Perform a numerical integration from the initial threshold to the end of the convolution function. If the integration result is greater than the desired probability of false alarm, which is 10-7, the initial point will increase 0.005 from the initial threshold and perform the integration again. Until the integration result is smaller than 10-7.

Figure 6.8  Four summations and the convolution output.

6.7  Threshold Obtained by a Gaussian Approximation

119

The integration step is selected as 0.005 because it is intended to keep two decimal points on the threshold value. The results of summation twice and four times are shown in Figures 6.7 and 6.8. In these simulations, the input data are generated from a random number generator rather than the FFT outputs for simplicity. There are 10,000 complex input points. The amplitude of these are summed together twice and four times. With these inputs the threshold generated from the Rayleigh distribution is 5.7 and the thresholds for two and four summations are 8.5 and 13.2. The thresholds are also shown in the figures. It is anticipated that the ratio of the summation threshold to the Rayleigh distribution can be used for an actual receiver design.

6.7  Threshold Obtained by a Gaussian Approximation In this section the Gaussian distribution is used as an approximation to find the threshold. The Gaussian function can be written as

p(r) =

1 e 2π σ

−(r − µ)2 2σ 2

(6.7)

where m is the mean and s is the standard deviation of the distribution. Once these two values are found, the Gaussian distribution can be obtained. In order to find the probability of the false alarm, integration is required. The integration of a Gaussian function is the error function, which can be written as erf (r) =



2 π

r

∫0 e

− t 2dt



(6.8)

From the “erfcinv,” the inverse complementary error function in the MATLAB program, the probability of a false alarm Pfa of a Gaussian distribution can be obtained from the following equation.

Pfa =

∞

th

∫ th pn (x)dx = 1 − ∫ −∞ pn (x)dx

(6.9)

where pn(x) is the noise Gaussian distribution as shown in (6.7). If the value of Pfa is given, a threshold th can be found as

th = 2 erfcinv(2Pfa )σ + m

(6.10)

If the Pfa is 10-7, the threshold th can be approximated by

th ≈ m + 5.2σ

(6.11)

This threshold generates a Pfa » 0.9964 × 10-7, which is slightly less than the desired value. The result from two summations is shown in shown in Figure 6.9. In this figure, one can see that the data and the Gaussian have a slight mismatch.

120

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figure 6.9  Two summations and the Gaussian distribution.

Finally, the thresholds are calculated from both the numerical integration method and an approximated Gaussian distribution. The results are shown in Table 6.4. In order to obtain the threshold through these two methods, the same noise input data are used. In order to obtain the convolution function, the noise data must be used to obtain the Rayleigh distribution first. To obtain the Gaussian distribution, only the summations of the amplitude data are required. The number of summed data used in these simulations is 10,000 points. For example, for the summation 8 case, 10,000 summed data are used. In order to obtain these data points, 80,000 points are required initially, and these 80,000 points are used to generate the Rayleigh function. In Table 6.4, the first column is the number of summations. The second and third columns are the thresholds obtained through the convolution and the Gaussian methods. The fourth column is the percentage error, which is the difference between the two calculated thresholds divided by the threshold obtained through the convolution method. Table 6.4  Thresholds Calculated from Both the Gaussian Distribution and the Convolution Methods Number of Summations 2 4 8 16 32 64 128

Thr by Numerical 8.54 13.19 21.12 35.18 61.00 109.04 200.75

Thr by Gaussian 7.37 11.88 19.55 33.61 59.41 107.39 199.12

Difference in Percent 13.67 9.92 7.43 4.44 2.62 1.51 0.81

6.9 Threshold and Probability of Detection of the Polyphase Filter Approach

121

In Section 6.2, the standard deviation of the calculated threshold is about 0.35%. In order to reduce the difference in Table 6.4 to 0.35%, the number of summations required is about 250 to 300. This is also a large number, although not as much as claimed in [4]. Perhaps the criterion of determining the required error is different.

6.8  Probability of Detection with Summations In this section the probability of detection from summation results of a 128-point FFT with a Blackman window will be studied. First, the threshold must be calculated from the FFT outputs with noise as input. In the previous sections, in order to save computation time, the noise was generated from a random generator without going through the FFT operations. To find the threshold for an FFT operation, the noise is Blackman windowed and a 128-point FFT is performed on the data. Only the data in the output frequency bins from 9 to 56 (narrower than those listed in Table 6.1) are used for determining the threshold. Each threshold is obtained from 1,000 runs and each run generates 48 data points (9 to 56). The results are shown in Table 6.5. For the summation number 1 the noise distribution is Rayleigh and the result in Table 6.2 is used. These threshold values are used for measuring the probability of detection. The outputs from the windowed FFT are summed up to six times for the same input frequency. For each run there are six outputs: summations 1 to 6. The maxima of the outputs are compared to the thresholds. If the maximum is greater than the threshold, it is referred to as signal detected. A total of 1,000 runs with randomly selected frequencies from 141 to 1,140 MHz are used to determine the probability of detection. The input S/N changes from -15 to 5 dB and the results are shown in Figure 6.10. When there is no summation, the S/N required to provide 90% detection is about -1.5 dB, which is close to the results obtained from Figure 6.6. For the summation of 2, the required S/N is about -4.3 dB, which improves by about 2.8 dB (4.3 1.8). This is also close to the expected improvement of about 2.7 dB. Further addition decreases the gain advantage, which is referred to as the integration loss [1, 2]. In all the probability of detection simulations, the S/N changes in 0.5-dB steps. If the S/N step is changed to 0.1 dB, 1,000 runs cannot generate smooth results. Thus, it is estimated that the accuracy of this simulation is close to 0.3 dB, which is slightly over the step of 0.5 dB.

6.9  T  hreshold and Probability of Detection of the Polyphase   Filter Approach In this section, the threshold and probability of detection will be studied. The approach is identical from the above method. Use Rayleigh and convolution to find the threshold and then the probability of detection. The polyphase filter approach to Table 6.5  Threshold for Summing the 128-Point FFT Outputs Number of sums Threshold

1 25.11

2 37.20

3 48.08

4 58.33

5 66.99

6 75.80

122

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figure 6.10  Sensitivity plot from the summations of the FFT output.

design a receiver will be discussed in Chapter 13. The polyphase filter uses 16-point FFTs and the input signal contains 128 points. There are 8 output channels, each with a bandwidth of 160 MHz. Only the 2 to 8 outputs are used in the receiver because the outputs from channel 1 contain real rather than complex output. The detailed design will be discussed in Chapter 13. The sensitivity will be considered

Figure 6.11  Sensitivity plot from summations of polyphase filter output.

6.10  Summary of Sensitivity Calculations and Discussion and Final Adjustment

123

for three conditions: no summation, 16-point summation, and 128-point summation. With noise only as input the thresholds obtained from the polyphase outputs are 1.06, 6.56, and 37.32 for the 1, 16, and 128 summations. The probability of detection is shown in Figure 6.11. From this figure, at a 90% probability of detection, the required S/N is about 4.8, -4.4, and -10 dB.

6.10  S  ummary of Sensitivity Calculations and Discussion and Final Adjustment by Considering the Number of Channels The sensitivity of the above calculated results is listed in Table 6.6 as a quick reference. In the previous discussion, the probability of false alarm and the probability of detection are based on one channel. However, in Table 6.6 the probability of false alarm Pfar is referred to as the receiver and it is the channel number multiplied by the single-channel probability of false alarm of 10-7. The simulation results follow the theoretical predications with no surprise. It is also interesting to note that when the pulse width is 100 ns, three different methods produce very close results as shown by the * mark. The maximum difference of 0.3 dB is close to the predicted simulation error. In the actual building, a receiver window function must be used. A rectangular window has very high sidelobes and the dynamic range will be low. The results obtained from Chapter 2 can be different from the above results. For a Blackman window the window factor can be found as 1.73, as discussed in Section 4.6. Using a 256-point FFT, the nominal sensitivity calculated is -81.5 dBm. In this special example, the noise floor is at -80 dBm. Thus the S/N is about -1.5 dB, which is about 3 dB less sensitive than the values listed in Table 6.6. The explanation of this result is that the sensitivity calculated in this chapter does not take the dynamic range and the digitization effects into consideration. In order to have a decent dynamic range, the sensitivity may be sacrificed. All the thresholds in the above calculations are based on probability of false alarm (Pfa) of 10-7. This Pfa value is calculated from one FFT output. As discussed in Section 6.1, the actual Pfa of the receiver should be the number of channels multiplied by 10-7 [personal communication with Professor C. H. Cheng at Miami Table 6.6  Sensitivity Calculated from Different Approaches 1

Approaches

2 3 4 5 6 7 8 9 10 11

128 pt rectangular freq on bin 128-point rectangular random freq 128-point Blackman * 256-point rectangular freq on bin 256-point rectangular random freq 256-point Blackman * 128-point Blackman 2 frame sum* 16-point polyphase 16 frame sum* 16-point polyphase ** 16-point polyphase 128 frame sum**

* Approach can be applied to receiver designs. ** Used in receiver design in Chapter 13.

Data length in ns 50

100

6.25 800

Pfar

S/N in dB

53 × 10-7 53 × 10-7 53 × 10-7 101 × 10-7 101 × 10-7 101 × 10-7 53 × 10-7 7 × 10-7 7 × 10-7 7 × 10-7

-4.3 -2.8 -1.5 -7.4 -5.7 -4.6 -4.3 -4.5 5.2 -10

124

Signal Detection from Fast Fourier Transform (FFT) Outputs

University, Oxford, Ohio, 2009]. These values are listed in column 3 in Table 6.6. The number of channels is listed in Table 6.1. The impact of the number of channels on the sensitivity will be discussed here. If the overall probability of false alarm is 10-7, the largest Pfa listed in Table 6.6 is about 10-5. Based on [1, 2], at a 90% probability of detection, when the Pfa changes from 10-7 to 10-5, the input S/N value changes slightly above 1 dB. In other words, the sensitivity of the receiver decreases by about 1 dB. When the Pfa changes from 10–7 to 53 × 10-7, the sensitivity decreases less than 1 dB. The sensitivity listed in Table 6.6 can be modified slightly through results in [1, 2]. One can also use the new Pfa such as 101 × 10-7 to find a new threshold and then use simulation to find the corresponding probability of detection. In Chapters 11 and 12, sometimes, the Pfa calculation is based on a single channel because receiver designers are familiar with this approach. For example, the 14-dB required S/N to produce a Pfa of 10-7 and a Pd of 90% is widely used by designers.

6.11  Approach for Phase Comparison The phase comparison method must use complex input data. The outputs from the FFT operation are complex; thus, the phase comparison will be performed after the FFT operation. The phase comparison method usually only operates on one signal. If more than one signal exists, the phase will generate erroneous information. It is assumed that each FFT frequency bin contains only one signal. Two FFT operations will be studied. The first one is the windowed FFT operation and the second one is the polyphase approach. Simulation will be used for these studies. In the first study, the input data are Blackman windowed and a 64-point FFT is performed. The short time frame is intended to find the frequency change in a signal. For example, if the input is frequency modulated (FM or chirp), the short frame may find the frequency changes. In the second study, a polyphase filter will be used to perform the FFT. The FFT length is 16 points and the data used in the FFT operation are 128 points. The reas­ on the polyphase is used because of its short time shift, such as only 16 points. The short time shift hopefully can determine biphase shift keying (BPSK) with high chip rates. Polyphase is used because it can produce good balanced IQ channels. Two frames of FFT data are collected. Since the FFT outputs are complex, the angle between them can be easily found. The fine frequency ffine can be found through φ = tan −1[X1(k)] − tan −1[X2 (k)]

f fine = ts =

φ 2π × 64t s

Hz



(6.12)

1 fs

where j is the calculated phase from the two consecutive frames X1(k) and X2(k) at peak frequency k, fs is the sampling frequency, and ts (1/fs) is the corresponding

6.12  Results from the 64-FFT Operation and Phase Comparison

125

sampling time. The factor 64 in the equation is the number of samples separating the two frames. The coarse frequency can be obtained from the FFT outputs, and the coarse frequency fcoarse can be obtained as

fcoarse = (ind1 − 1) × 40 × 106

Hz

(6.13)

where ind1 is the position of the maximum FFT amplitude output. The factor 40 is the frequency resolution of the 64-point FFT operation. The final frequency fout and the error frequency ferr are obtained as

fout = fcoarse + f fine ferr = fin − fout



(6.14)

where fin is the input frequency.

6.12  R  esults from the 64-FFT Operation and Phase Comparison Aided with Amplitude Comparison The input frequency changes from 101 to 1,080 MHz in 1-MHz steps. The error frequency defined in (6.14) versus the input frequency is presented in this section. Figure 6.12 shows the coarse error frequency. In this figure the input S/N = 100 dB,

Figure 6.12  Coarse frequency errors with S/N = 100 dB.

126

Signal Detection from Fast Fourier Transform (FFT) Outputs

which implies no noise. The maximum frequency error is ±20 MHz, which is the expected results because the coarse frequency resolution is 40 MHz. The output of the fine frequency error with S/N = 100 dB is shown in Figure 6.13. In this figure at some input frequency the error frequency is close to 40 MHz. These errors occur at the boundaries of two frequency bins. When an input signal is at the boundary of two frequency bins as shown in Figure 6.14, the differential phase is close to ±p. If the wrong sign is selected, the frequency will be off by 40 MHz. This is an intrinsic property of this phase comparison method. This problem can be mitigated by augmenting with amplitude comparison method. The approach is to find the phase angle close to ±p. As shown in Figure 6.14, if the input frequency is at the boundary of frequency bins k and k + 1, and k is the higher output, the amplitude at bin k + 1 should be much higher than that of k - 1. By comparing k - 1 and k + 1, the phase can be determined. The differential frequency calculated by using the phase angle must be added to the frequency bin k. Figure 6.15(a) shows similar results as Figure 6.13. These two figures should be but not identical because of the computer round-off error even with S/N = 100 dB. Figure 6.15(b) shows the results incorporated with an amplitude comparison. In this simulation, the threshold is the angle, which is empirically chosen at 2.3 radians. If the angle is greater than 2.3, the amplitude information is used to make the correction. The results are very good. The 40-MHz errors are eliminated and the worst frequency error is close to 5 kHz. When the input S/N = 2 dB, the results are shown in Figure 6.16. In Figure 6.16(a) the amplitude comparison is not incorporated and the results have many

Figure 6.13  Fine frequency error with S/N = 100 dB.

6.12  Results from the 64-FFT Operation and Phase Comparison

127

Figure 6.14  An input frequency at the center of two frequency bins.

errors close to 40 MHz. Figure 6.16(b) has an amplitude comparison, and many of the 40-MHz errors are eliminated, but not all of them. When the angle threshold is decreased arbitrarily to 1.6 radians, the results are shown in Figure 6.17. In Figure 6.17(b), more 40-MHz errors are eliminated, but still not all of them. Some of the errors are partially corrected. The results show some large errors, which are less than 40 MHz but still large such as 10 to 30 MHz. Examining Figures 6.16 and 6.17, one can find that some of the large errors are created by the amplitude comparison method because without an amplitude comparison some errors in Figures 6.16(b) and 6.17(b) do not appear in Figures 6.16(a) and 6.17(a).

Figure 6.15  Fine frequency error with S/N = 100 dB: (a) phase comparison only and (b) with amplitude comparison.

128

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figure 6.16  Fine frequency error with S/N = 2 dB with 2.3 radians threshold: (a) phase comparison only and (b) with amplitude comparison.

The conclusion of this study indicates that when the signal is strong the amplitude comparison method augmenting the phase comparison method can produce very good results. However, when the signal is weak, the amplitude comparison method only has limited improvement.

6.13  Create Additional Artificial Output Frequency Bins In this section, in addition to the 32 frequency bins generated by the 64-point FFT operation, other frequency bins are generated, which will be referred to as the artificial frequency bins. These frequency bins are generated from the 32 frequency outputs. Referring to Figure 6.14, the basic relation is shown in the following equation

X(m) =

X(k) − X(k + 1) C

(6.15)

where m represents the frequency bin at the center of X(k) and X(k + 1) and C is a constant, which can be used to adjust the X(m) output. For example, if k = 300 MHz, then k + 1 = 340 MHz and m = 320 MHz at the boundary of k and k + 1. If the input signal is at 310 MHz, it is anticipated that X(k) and X(m) have the same amplitude because this frequency is at the boundary

6.13  Create Additional Artificial Output Frequency Bins

129

Figure 6.17  Fine frequency error with S/N = 2 dB with 1.6 radians threshold: (a) phase comparison only and (b) with amplitude comparison.

of these two frequency bins. The value of C is adjusted according to this condition and it is determined empirically. For a different window the C value will be different. For the Blackman window, C = 1.78. Since there are 32 outputs, there are 31 differences and the overall frequency outputs are 63. The phases of the created outputs are also used to determine the fine frequency. When the input frequency is at X(k) values, the phase difference is 0 because after 64 sample shifts there are integer number of cycles between X1(k) and X2(k). The subscripts 1 and 2 represent two consecutive FFT outputs. When the input frequency is at the center of X(m), the phase difference between X1(m) and X2(m) is p. This difference must be taken into account to find the fine frequency. The coarse frequency error from this method is shown in Figure 6.18 with S/N = 100 dB. As expected, the coarse frequency is within ±10 MHz because of the higher-frequency resolution. The fine frequency error is shown in Figure 6.19. The worst error is about 7 kHz and there is no intrinsic phase problem at the center of two frequency bins. When the S/N of input signal is at 2 dB, the results are shown in Figure 6.20. In Figure 6.20(a) there are some frequency errors close to 40 MHz. These errors are mainly caused by noise. A similar amplitude comparison is used to augment the phase measurement with artificial bins and the results are shown in Figure 6.20(b). The threshold for the phase correction threshold is also chosen empirically at 2.3 radians. The results are improved and the maximum frequency error is close to

130

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figure 6.18  Coarse frequency errors with added frequency bins and S/N = 100 dB.

Figure 6.19  Fine frequency error with S/N = 100 dB with added frequency bins.

6.14  Polyphase Phase Comparison Study and Basic Idea

131

Figure 6.20  Fine frequency error with added frequency bins and S/N = 2 dB with 2.3 radians threshold: (a) phase comparison only and (b) with amplitude comparison.

4 MHz. However, in some runs the 40-MHz errors are not totally eliminated (not shown). When the angle threshold is changed, similar results are obtained as the previous section. In general, the frequency calculated with the additional frequency bins produces better results than the phase comparison method without artificial bins.

6.14  Polyphase Phase Comparison Study and Basic Idea The purpose of this study is to find the fine frequency of a polyphase filter through the phase comparison method. One of the reasons to study this problem is that the polyphase filter has a very good IQ balance as listed in Section 5.12. Another important factor is that the polyphase filter has very few points between two consecutive frames. This short data length hopefully will retain the phase transition in a BPSK signal because the polyphase will be used to detect the phase transition and chirp signals. In finding the fine frequency there are some ambiguity frequencies where the frequency will have larger frequency errors as discussed in the phase comparison method. Three approaches, discussed in Sections 6.16 through 6.18, are used to eliminate these ambiguities and their results will be compared. In order to simplify the discussion, examples will be used to illustrate the operations. The input data are sampled at 2.56 GHz and 128 data points will be used in making the polyphase filter. The filter is performed through 16-point FFT operations.

132

Signal Detection from Fast Fourier Transform (FFT) Outputs

The time and frequency domains of the Park-McClellan window are shown in Figure 6.21. Figure 6.21(a) shows the time-domain response. Although there are a total of 128 data points, a horizontal line is drawn below 16 data points. These points are considered to be the main contributors to the filter output. The rest of the points are highly attenuated. Of course, this assessment is a bit arbitrary. Figure 6.21(b) shows the frequency-domain response. This response is obtained from dividing the windowed data into a 16-point section and summing them together to obtain 16 points. An FFT is performed on these 16 data points to obtain the frequency-domain response. The top of the filtered output is rather flat and this specific filter has sidelobes of about 59 dB down. These operations can be expressed in the following equations. y j = x16 j + i w 16 j + i



z=

where i = 0 ~ 15; j = 0 ~ 7

7

å yj



j =0

Y (k) =

15

å zn

- j 2π kn e N

N = 16

(6.16)

n = 0 ~ 15

k =0

where x is the input data points consisting of 128 points from x0 to x127 and w is the digitized window of the Park-McClellan filter and consisting of 128 points from

Figure 6.21  (a) Time and (b) frequency domains of a 16-point FFT polyphase filter with 128 input data points.

6.15 Frequency Measurement Through Phase Comparison of a Polyphase Filter

133

w0 to w127. The i value changes from 0 to 127 and the j = 0 to 7. For i = 0 and j = 1, the corresponding x and w values are 16, 32, . . . , 112. There are 8 yj values and each one consists of 16 points. The z value, consisting of 16 points, is obtained from the summation of the yi values and zn represents the nth point of z. More detailed discussions of the polyphase filter can be found in [3]. Under normal operating conditions, the input data shift 16 points per time frame. Since the time domain is mainly contributed from 16 points and the frame shifts about 16 points, all the input data are considered to be sufficiently used. The goal of using this short data point is that hopefully in these data points only one BPSK phase transient occurs. If there is more than one phase transient, it might be difficult to detect them. If the sampling frequency is 2.56 GHz, 16 data points are 6.25 ns long. It is assumed that the shortest time between two consecutive phase transitions in a BPSK is longer than this time of 6.25 ns, which means in 16 points there is only one phase transition. Since the input data are only 6.25 ns long, the corresponding filter width in the frequency domain is about 160 MHz (1/6.25 ns) and there are total 8 output channels to cover 1,280 MHz. Among these 8 channels only 7 channels will be used for the receiver design as listed in Table 6.1.

6.15  F  requency Measurement Through Phase Comparison of a Polyphase Filter The frequency resolution generated by this specific filter is 160 MHz and it is too coarse to measure the frequency of an input signal. The frequency error produced by this approach is shown in Figure 6.22. In this figure the S/N = 100 dB and the

Figure 6.22  Coarse frequency error measured from the frequency bins for polyphase filter.

134

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figure 6.23  Fine frequency error with a phase comparison for a polyphase filter.

Figure 6.24  Fine frequency error with a phase comparison and augmented by an amplitude comparison for a polyphase filter.

6.16  Added Artificial Frequency Bins for the Polyphase Filter

135

input frequency is from 100 to 1,080 MHz. The maximum error frequency is at 80 MHz, which is the expected result with a 160-MHz frequency resolution. Since this filter top is relatively flat, it may not be suitable for an amplitude comparison to obtain a fine frequency resolution. The fine frequency can be obtained from the two consecutive FFT outputs. The error frequencies are shown in Figure 6.23 with an input frequency from 100 to 1,080 MHz and S/N = 100 dB. As expected, there are ambiguity regions when the input frequency is at the boundary of two frequency bins. The frequency error at the ambiguity region is about 160 MHz or one frequency resolution bin. These ambiguity regions can be mitigated by adding amplitude comparison method similar to the conventional phase comparison method discussed in Section 6.12. The error frequencies are shown in Figure 6.24 for frequency from 140 to 1,040 MHz and S/N = 100 dB. The angle threshold is also at 2.3 radians. If the angle is above this value, an amplitude comparison will be used to determine the fine frequency. In the simulation, the bandwidth is narrowed to 1,000 MHz, because at low- and high-frequency ranges there are some large error frequencies. The bandwidth of 1,000 MHz should be wide enough for EW applications. With the amplitude comparison the maximum error is less than 60 kHz.

6.16  Added Artificial Frequency Bins for the Polyphase Filter In this section artificial frequency bins are created for the polyphase filter. For the 16-point FFT operation on real signals, there are 7 output frequency bins and another 6 additional frequency bins will be created. A similar method discussed in Section 6.13 is used to generate the additional frequency bins, but the results are not satisfactory. Sometimes the output of the frequency bin with the signal is lower than the neighboring ones. Even this problem is eliminated through some trial and error; if the artificial frequency bins are used in augmenting the fine frequency calculation, large frequency errors can be obtained. Another approach to create the artificial frequency bins is through zero padding. Using the 16 outputs obtained from the time domain and padded with 16 zeros, a 32-point FFT is performed on the z values. The operation can be expressed as z = [z

Z (k) =

0�0 ] 31

å z ne

- j 2π kn N

N = 32

n = 0 ~ 31



(6.17)

k =0

where 0 . . . 0 contain 16 zeros. With this padding the frequency output components will increase from 16 to 32. Since the input data are real, only 16 frequency bins will kept. When the input frequency is at 400 MHz, bin number 6 should be the maximum output; instead, a minimum is shown in Figure 6.25(a). When the input frequency is at 480 MHz, the results are shown in Figure 6.25(b). This is the expected result. When the additional frequency bins are generated from the existing frequency bins as discussed in Section 6.13, this phenomenon also appears. Different

136

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figure 6.25  Frequency bins generated with zero padding at end: (a) 400 MHz and (b) 480 MHz.

approaches can eliminate this problem; however, the fine frequency calculated still has frequency ambiguity regions. Therefore, those methods will not be discussed here. There is another approach to padding with zero. This approach is shown here. z = [z0 ~ z7

Z(k) =

31

∑ zn

0~0 − j 2π kn e N

z8 ~ z15 ] N = 32

n = 0 ~ 31



(6.18)

k=0

The 16 zeros are placed in between the z outputs. With the same input frequencies at 400 and 480 MHz, the results are shown in Figure 6.26, where Figure 6.26(a) shows the expected results. With this type of zero padding the outputs have 16 outputs. Phase relation from all these 16 outputs is used to find the fine frequency. The coarse frequency and the fine frequency errors are shown in Figure 6.27. The input S/N = 100 dB and the input frequency range is from 100 to 1,080 MHz. From Figure 6.27(a), the coarse frequency error plot, it is obvious that the widths of the 16 frequency bins are not uniform. The added artificial frequency bins have a narrow frequency range. Figure 6.27(b) shows with the fine frequency error that additional frequency bins eliminate

6.17  Decrease Shifting Time for Polyphase Filter (Long Short Shift)

137

Figure 6.26  Frequency bins generated with zero padding in between: (a) 400 MHz and (b) 480 MHz.

the ambiguous frequency range. The worst fine frequency error is close to 2 MHz and occurs periodically at a 160-MHz frequency range. This indicates that the fine frequency calculated through the phase of the artificial frequency bins has some error. This artificial frequency bin creation method will be used to compare two other methods.

6.17  Decrease Shifting Time for Polyphase Filter (Long Short Shift) Another approach to eliminate the ambiguity regions is to increase the output sampling rate. As discussed in Section 6.14, the time frame consisting of 16 data points and each time the frame is also shifted by 16 data points. The corresponding output sampling rate is 160 MHz (2,560/16). Since the data are complex through the FFT operation, the output processing bandwidth is 160 MHz. If the frame is shifted by 8 points, the output processing bandwidth will increase from 160 MHz to 320 MHz. Let us refer to the 16–data point shift as the long shift and to the 8-point one as the short shift. The phase angles between two frames separated by 16 and 8 samples are shown in Figure 6.28. In this figure the input frequency is from 100 to 1,080 MHz and the S/N = 100 dB. Figure 6.28(a) shows the phase variation for the long shift. The unambiguous frequency range is 160 MHz and Figure 6.28(b) shows the phase variation for the short shift. The corresponding unambiguous frequency range is 320 MHz.

138

Signal Detection from Fast Fourier Transform (FFT) Outputs

Figure 6.27  Frequency error with added artificial frequency bins: (a) coarse and (b) fine.

Figure 6.28  Phase angle versus input frequency: (a) long shift and (b) short shift.

6.18 Comparison of the Three Approaches for Finding a Fine Frequency of a Polyphase Filter

139

Figure 6.29  Frequency error plot with frequency calculated from long and short shifts.

When the phase angle is in the long shift is close to ±p, it is close to the ambiguity regions. From Figure 6.28, at these regions phase angles of the short shift do not have ambiguities. For the long shift the phase angle is divided into two regions with a threshold of ±p/2. If the phase angle is within ±p/2, the phase angle of the long shift is used to calculate the fine frequency. If the phase angle is beyond ±p/2, the phase angle of the short shift is used to calculate the frequency. The fine frequency can be found in a similar way as was stated in (6.12), only the delay time should changed from 64 ts to 16 ts. For the fine frequency calculated from the short shift, not only the delay time should be changed from 16 ts to 8 ts, but also the phase angle must also be modified sometimes. If the maximum frequency bin is an odd number, the fine frequency can be calculated from the phase angle divided by 2p8ts. If the maximum frequency bin number is even, the phase angle should be adjusted by p. Using the phase angles of both the long and short shifts, the fine frequency can be calculated. The frequency error is shown in Figure 6.29 with S/N = 100 dB and a frequency range from 100 to 1,080 MHz. The maximum error is less than 0.15 MHz, which is about double the worst value in Figure 6.24. These are the expected results because by using the short shift the delay time is short and the frequency resolution will degrade.

6.18  C  omparison of the Three Approaches for Finding a Fine Frequency of a Polyphase Filter In this section, the results from the three approaches will be compared. The data obtained in the previous figures have very strong signals. As indicated in Section

140

Signal Detection from Fast Fourier Transform (FFT) Outputs

6.13, when the S/N is decreased, the frequency error should be compared. In the following simulations identical data are used in the three methods. Identical data mean that the same data including the random noise are processed through all three methods. In order to accommodate the amplitude comparison approach, the input frequency range is from 140 to 1,040 MHz rather than 100 to 1,080 MHz. With input S/N = 10 dB, the results are shown in Figure 6.30. Figure 6.30(a) shows the amplitude comparison method; Figure 6.30(b) shows the artificial frequency bins method; and Figure 6.30(c) shows the long, short shift method. The amplitude comparison method has one large error caused by ambiguity. The error frequency in Figure 6.30(b) appears slightly lower than in Figure 6.30(c). This indicates that although the added frequency bin method has a relatively large frequency error at high S/N, the error does not increase much when the S/N decreases. When the input S/N is decreased to 5 dB, the results are shown in Figure 6.31. Figure 6.31(a–c) shows the amplitude comparison, the artificial frequency bins, and the long short shift methods, respectively. It is obvious that the amplitude comparison is more sensitive to noise. Although Figure 6.31(c) does not show error caused by ambiguity, the error is increased to about ±30 MHz. With a limited number of runs, it appears that S/N = 5 dB is about the lower limit. When the S/N is below this value, there are many large frequency errors. The amplitude comparison method is the simplest approach with fewer calculations. Both the added frequency bin and long short shift methods increase the

Figure 6.30  Frequency errors for the polyphase filter with S/N = 10 dB: (a) amplitude comparison, (b) added frequency bins, and (c) long short shift.

6.19  Conclusions

141

Figure 6.31  Frequency errors for the polyphase filter with S/N = 5 dB: (a) amplitude comparison, (b) added frequency bins, and (c) long short shift.

operation complexity, either by adding more FFT operations such as in the long short shift method or by performing longer FFTs as in the added frequency bin method. The improvement versus operational complexity should be further analyzed. In Chapter 13 the polyphase method is used to build a receiver. Additional FFT will be used to improve the frequency resolution after the polyphase operation. The short data shift must be used to avoid missing signal when two signals are separated by 160 MHz.

6.19  Conclusions This chapter discusses some methods to process the FFT outputs. The methods for threshold setting are discussed. These methods base on the theoretical noise distribution function. The parameters in the theoretical functions are obtained from measured noise distribution. Since the probability of false alarm is very low such as 10-7, in order to obtain a dependable threshold from a simulation, a large number of runs is required. Using the known probability density function should be a better approach compared to large number simulation runs. The probability of detection is performed based on the obtained theoretical threshold and the results from different methods appear close to the expected values.

142

Signal Detection from Fast Fourier Transform (FFT) Outputs

Another way to process the FFT outputs is to compare the phase between two consecutive frames of data. This approach can only apply to one signal in the channel. It provides a better frequency resolution. In Chapters 14 and 15, it will be shown that the phase comparison method is essential to process spread spectrum signals.

References   [1] Skolnik, M. I., Introduction to Radar Systems, New York: McGraw-Hill, 1962.   [2] Barton, D. K., Modern Radar System Analysis, Norwood, MA: Artech House, 1988.   [3] Tsui, J., Microwave Receivers with Electronic Warfare Applications, Raleigh, NC Scitech Publishing, Inc., 1986, reprint 2005.   [4] Marcum, J. I., “A statistical theory of target detection by pulsed radar, mathematical appendix,” IRE Trans. Information Theory, Vol. IT-6 April 1960, pp. 145–267.

Chapter 7

Time-Domain Detection with 1-Bit ADC

7.1  Introduction If a 1-bit ADC can be used to build a receiver, the sampling frequency can be extremely high, close to 10 GHz. If the receiver can be built with in-phase and quadrature phase (IQ) channels, the bandwidth of the receiver can reach 10 GHz. In general, it is difficult to build a multiple-bit IQ channel receiver because it is difficult to balance the IQ outputs. Therefore, most receivers with multiple numbers of bits are single input channels. When the input has only 1 bit, the balance is not an important problem, which will be discussed in Chapter 10. With a 1-bit ADC, two problems will be studied. The first problem is the time-domain detection and will be studied in this chapter. The second problem is an instantaneous frequency measurement (IFM) receiver, which will be discussed in Chapter 10. If an electronic warfare (EW) receiver can be designed to match the input pulse width (PW), the sensitivity and the frequency resolution can be optimized to achieve the highest sensitivity and frequency resolution. Since the time of arrival (TOA) and PW are unknown, one must measure these two parameters. If the signal is strong, these two values are relatively easy to obtain when the ADC has a multiple number of bits. Examining the digitized data in the time domain, one can find the approximate TOA and PW. If the signal is weak, one must search the input data in the time domain to find the TOA and PW. The basic idea is discussed in [1], but no detailed information is given. One obvious approach is to use many processing windows with different widths to slide through the data. When the window width and the input signal are approximately matched, the output will be close to a maximum (or a matched filter output). The window length will be close to the PW and the beginning of the input data in the window will be the desired TOA. In order to have many windows sliding through the input data, the window operation must be very simple; otherwise, it will be not suitable for EW applications. Many window lengths sliding through the input data require many operations. The operation should be limited to simple additions and subtractions. The offend of the chapter concentrates on some detailed simulations on how to generate the PW. Although many approaches are studied, only the one with the best results will be presented. The detailed approach is rather tedious. Hopefully, the presentation can provide some idea on the problems encountered in a receiver encoder design. Before the actual method is discussed, let us first define the boundary of the problem. 143

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Time-Domain Detection with 1-Bit ADC

7.2  Reduce Time Resolution and the Number of Windows Before the actual discussion of the time-domain detection, the number of filters will be discussed first. In an EW receiver, the received signal can have a very wide range of PW from less than 100 ns, say, to a few hundred μs. Usually, PW beyond a certain length can be assumed to be a continuous wave (CW). 40 μs is arbitrarily chosen as the upper limit for the following discussions. If the PW range is expressed in units of seconds, the results will be 100 × 10-9 s to 40 × 10-6 s. The ratio of the maximum to minimum PW is 400 to 1. If a linear time resolution (100 ns) is used to cover the entire time range, approximately 400 different time windows are needed; this is a very large number. In measuring PW the scale is usually nonuniform. For a short PW finer time resolution is needed and for a long PW coarse time resolution is satisfactory. For example, 100 ns is used to measure PW up to 1 μs, a 1-μs time resolution is used to measure PW up to 10 μs, and a 10-μs time resolution is used to measure PW up to 40 μs. It is a common practice in analog EW receiver design to reduce the bit number of the pulse descriptor words (PDWs). With this nonuniform time resolution, the total number of windows is 22 (10 for PW = 100 ns to 1 μs, 9 for PW = 2 μs to 10 μs, and 3 for PW = 20 to 40). Although 22 is still a large number, compared to 400, it is a manageable number. The TOA and PW may not have the same time resolution. The TOA is used to find the pulse repetition frequency (PRF) or its inverse, the pulse repetition interval (PRI). A fine time resolution will provide better PRF results. On the other hand, when the PW is long, a coarse time resolution usually is satisfactory. In the following discussion, the window lengths change nonuniformly (the longer window has coarse resolution), but the TOA resolution stays constant. Let us use a simulation example to explain this problem. Assume that the sampling frequency is 2,560 MHz and the corresponding sampling interval (or time resolution) is approximately 0.39 ns (1/2,560 MHz). In general, such a fine time resolution is not needed for EW applications. For some special processing, a very fine time resolution may be desirable. Usually this kind of information is obtained from many pulses or a pulse train rather than from one single pulse. Using the assumed sampling frequency of 2,560 MHz, in 1 μs there are 2,560 data points. In 40 μs there are 102,400 data points, also a large number. If a 0.39-ns time resolution is not needed, the input data can be grouped together. This operation decreases the time resolution but in the meantime also decreases the number of operations. In the following examples every 32 data points will be combined into one point. The time resolution will be decreased to 12.5 ns (32/2,560), which correspond to a sampling rate of 80 MHz. The corresponding number of operations will also be decreased by a factor of 32.

7.3  C  onventional Time-Domain Measurement   with Amplitude Information If the ADC has a multiple number of bits, one common approach to measure the TOA and PW is through amplitude information. The receiver usually has input IQ channels. The outputs and their amplitude can be calculated as

7.3 Conventional Time-Domain Measurement with Amplitude Information

145

x(n) = A cos(2π ft) y(n) = A sin(2π ft) A = x(n)2 + y(n)2



or (7.1)

y(n) A ≈ x(n) + 2

if x(n) > y(n)

x(n) 2

if x(n) < y(n)

A ≈ y(n) +



In this equation, the amplitude can be calculated using the squaring and square root process, but this process might be calculation-intensive. The approximation approach should be relatively easy to implement and the error in decibels is plotted in Figure 7.1. The worst error is at |x(n)| = |y(n)/2|. Under this condition, the error is about 0.97 dB. After the amplitude is found, the amplitude from many frames will be added together. This process, referred to as a noncoherent integration, is well discussed in radar operations as in [2, 3]. Comparing to coherent integration, there is a noncoherent integration loss. The noncoherent loss increases as the number of summations increases. The coherent gain is

G = 10log(N) dB

(7.2)

where N is the number of coherent integration. The noncoherent integration gain is the coherent gain minus the noncoherent loss. Figure 7.2 shows the noncoherent

Figure 7.1  Amplitude error in decibels caused by a simplified operation.

146

Time-Domain Detection with 1-Bit ADC

Figure 7.2  Noncoherent gain for Pd = 0.9 and Pfa = 10-7.

gain for the probability of detection Pd = 0.9 and the probability of false alarm Pfa = 10-7. For example, if N = 1,000, the coherent gain from (7.2) is 30, but the noncoherent gain is about 20 dB (30 - 10), where the noncoherent loss is10 dB. This result indicates that the noncoherent operation is not as efficient as the coherent integration. To find the amplitude of the FFT outputs, (7.1) is used. In actual hardware, it is difficult to obtain well-balanced IQ channels over a wide frequency range. If the two channels are not balanced, the amplitude varies with time for even a relatively small imbalance in amplitude and phase. If the amplitude varies with time, it will complicate the detection problem. In an actual digital receiver it is a common practice to use only one digitized channel, and the Hilbert transform or an FFT operation is used to convert the real input into complex ones.

7.4  Using Phase to Detect the Presence of a Signal If a receiver has IQ channels, the outputs can be written as a complex quantity of z(n), where n is the discrete time. The phase P of the signal and the summation of phases can be written as z(n) = Ae j 2π nfts P = z(n)z(n − m)* = Ae j 2π mfts

C = ∑ P = ∑ z(n)z(n − m)* n

n

(7.3)

7.5 The Amplitude of the Correlation Output Is a Function of Frequency

147

where * is complex conjugate, f is the input frequency, ts is the sampling time related to sampling frequency fs as (ts = 1/fs), and m is the time delay. The phase P obtained is independent of time n. When the input frequency is a constant, the phase is a constant complex quantity. The summation of the phases is expressed as C, which will be referred to as correlation output for convenience. Since P is complex, the summation of P can be considered to be coherent processing. Using the phase information to detect the existence of a signal might be better than using the amplitude information. However, there is a penalty in obtaining the phase. To obtain the phase, one must multiply one data point with the complex conjugate of a consecutive one as shown in (7.3). When the S/N is low, multiplication can further reduce the S/N. Thus, even the processing can be considered coherent, and the results should be inferior to an FFT operation, in which the reference signal for the multiplication is noise free. An EW receiver with hardware IQ outputs of multiple bits of ADC has difficulty producing balanced results over a wide frequency range. If the ADC has only 1 bit, the phase resolution is coarse and under this condition, the phase imbalance will not have a severe impact on the performance of the receiver. Besides the phase tolerance, another advantage is that 1-bit output is the simple to process. For 1-bit IQ channels there are only four pairs of outputs (1, 1; 1, -1; -1, 1; -1, -1); therefore, there are only four phase angles. The angle calculation can be rather simple. The disadvantage is that the angle resolution of 90° is poor. This coarse angle resolution will further decrease the processing gain. In the following sections, an ADC with only 1 bit will be used for all the discussions. Using the 1-bit ADC the correlation obtained from phase has the following properties, which will be discussed in Sections 7.5 and 7.6.

7.5  T  he Amplitude of the Correlation Output   Is a Function of Frequency It is desirable to have the correlation output as a constant independent of time and frequency. Although the phase obtained from correlation is independent of time, it is a function of frequency as shown in Figure 7.3. In this figure, there are four plots corresponding to four delays. These plots are generated from the following approach. For the one-delay case, the phase P is calculated from (7.3) by substituting m = 1. Afterward 32 phase points are summed together into one point. Its delay becomes 32 data points corresponding to a 12.5-ns time resolution as discussed in Section 7.2. Since the P value is complex, the summation is also complex. The amplitude of summed P value is obtained. This procedure is performed on every input frequency, which changes from 1 to 2,560 MHz. This frequency change is from 1 to 2,560 MHz without eliminating the end frequencies because for complex data, the input signal at the end frequencies (1 and 2,560 MHz) has no interference. In actual receiver design, the overall frequency range is usually less than 2,560 MHz because the two end frequencies are ambiguous. The amplitude of the summed P value (or C) is plotted in Figure 7.3. For the 2 delay case, the only difference is that the initial P value is obtained from (7.3) with m = 2. Each data point is also obtained from summing 32 phase values.

148

Time-Domain Detection with 1-Bit ADC

Figure 7.3  Amplitude variations versus frequency for strong signal: (a) one delay, (b) two delays, (c) three delays, and (d) four delays.

In all the four plots the maximum value is 64. For the 1 delay case, the maxima occur at frequency of multiple of 640 MHz. For the results with the two-delay case, the first maximum occurs at a multiple of 320 MHz (640/2). For three and four delays, the first maximum occurs at multiples of 640/3 and 640/4 MHz, respectively. This phenomenon can be explained as follows. When the input signal is at 640 MHz, for the one-delay case, the phase shift is 2π × 640 × 106/(2,560 × 106) = π /2. This phase shift is represented by a value of 2 on the imaginary axis as shown in Figure 7.4(a). The value of 2 comes from the 1-bit ADC. Since 32 samples are summed together, the value of 64 is obtained. Also for the one-delay case, when the input frequency is at 320 MHz, the corresponding phase shift is π /4, represented by Figure 7.3(b). When 32 samples are summed, the 2 2 result is 32 + j32, and the corresponding amplitude is 32 + 32 ≈ 45.25. This coarse digitization makes the amplitude changes with frequency. This amplitude variation can cause a detection problem. A frequency close to the peak value can be detected more easily than a signal at the valley frequency. This amplitude variation can be mitigated by using the summation of multiple delays. Figure 7.5(a) shows the sum of one and two delays. Figure 7.5(b) shows all four delays summed together, where the amplitude variation is reduced slightly. However, in actual operation, this phenomenon does not cause much problem. Figures 7.4 and 7.5 are generated with a very strong signal. When the input signal

7.5 The Amplitude of the Correlation Output Is a Function of Frequency

149

Figure 7.4  Vector representation of phase: (a) π/2 and (b) π/4.

is strong, the signal can be detected easily anyway and the amplitude variation is not a concern. When the signal is relatively weak such as at S/N = 0 dB, the results of Figures 7.4 and 7.5 are shown in Figures 7.6 and 7.7, respectively. In these figures, the amplitude variation is not significant. Of course, it is difficult to detect weak signals; however, under weak signal conditions, the amplitude variation as a function of the frequency decreases.

Figure 7.5  Summation of amplitudes with different delays for strong signal: (a) one and two delays summed, and (b) all four delay outputs summed.

150

Time-Domain Detection with 1-Bit ADC

Figure 7.6  Amplitude variation versus frequency for S/N = 0 dB: (a) one delay, (b) two delays, (c) three delays, and (d) four delays.

7.6  C  orrelation Amplitude Change with a Specific Frequency   and an Initial Phase There is another problem with this phase correlation method using a 1-bit digitizer. When the frequency is at 640, 1,280, 1,920, and 2,560 MHz and the initial phase of the input signal is at 0, π /2, π, 3π /2, the correlation amplitude will drop, which causes measurement inaccuracy. The cause of this phenomenon is due to the fact that at these frequencies, the phase shift is a multiple of π /2, which matches the phase resolution generated by two frames of the 1-bit ADCs. If the initial phase of the signal is any value other than the four specific values, the phase change will be π /2 and the results will be added to a normal value that is 64 for a very strong signal. If the initial phase of the signal is one of the four specific values, sometimes the phase difference will be π /2, and other times the phase difference can be 0 or π. The result of this variation causes the correlation amplitude to drop. Figure 7.8(a) shows the correlation result of signal without noise, and Figure 7.8(b) shows that the S/N = 0 dB. This phenomenon exists for strong as well as weak signals. It appears that noise will only mitigate the amplitude reducing problem but cannot eliminate it. When the signal is strong, the range of the initial phase affecting the correlation output is very narrow. For example, when there is no noise and the initial phase is at 10-4 radius rather than 0, the correlation output does not have the dip at frequencies of 640, 1,280, 1,920, and 2,560 MHz.

7.7  Moving Average Method with Different Window Lengths

151

Figure 7.7  Summation of amplitudes with different delays for S/N = 0 dB: (a)one and two summed and (b) all four outputs summed.

When the signal is weak, the initial phase angle has a wider range that can impact the correlation output. For example, when S/N = 0 dB even with an initial phase = 0.1 radius, the correlation output can still have a smaller amplitude. This amplitude dipping will affect the detection of a signal. When a signal appears at the specific frequency and the initial phase angle, the probability of detection may be low due the decreasing amplitude. In the following sections, in order to simplify the signal detection problem, the correlation output amplitude fluctuation discussed in Sections 7.5 and 7.6 will not be treated specifically. In the simulations, the frequency and initial phase of the input signal are randomly selected. These problems will not be recognized in the operations.

7.7  Moving Average Method with Different Window Lengths A moving average method uses a time window to average input data. The window length is fixed. It takes input data and sums them together. When the data points summed equal the window length, the operation changes to add one new data and delete the oldest data. This operation continues to the end of the data. Assume that the input PW is 4 ms. The window lengths are from 100 ns to 1 ms in 100-ns steps and from 1 to 11 ms in 1-ms steps. In this example, some of the windows are shorter than the PW and some are longer than it.

152

Time-Domain Detection with 1-Bit ADC

Figure 7.8  Correlation output versus frequency with initial phase equal to 0: (a) strong signal and (b) S/N = 0 dB.

Figure 7.9 shows the results of the filter outputs. In Figure 7.9(a) the window lengths are from 600 ns to 1 ms in 100-ns steps. In Figure 7.9(b), the window lengths are from 2 ms to 6 ms in 1-ms steps. In Figure 7.9(c), the window lengths are from 7 to 11 ms in 1-ms steps. The outputs can be clearly identified because the signal is relatively strong. In the center plot of Figure 7.9(b), the window length and PW are both 4 ms and they are matched. This case can be considered as the optimum condition. However, even looking at all the outputs from all the windows it is difficult to determine the matched condition. The encoding scheme is to find the matched condition and find the correct PW, especially when the signal is weak. From Figure 7.9 one can expect that measuring the PW and TOA can have some problems. The difficulties in decoding these moving average outputs will be discussed as follows. First, it is difficult to tell whether the window is longer or shorter than the PW because the outputs are all tropsoidal shape. Thus, the correct PW is difficult to determine. Second, if the output is relatively flat, it is difficult to find a peak value. Without a point on the output, it is difficult to find the TOA and PW. When the window length and PW are matched, the output is triangle-shaped theoretically and it should be recognizable. The encoder must detect the matched case and generate the TOA and PW. To recognize the triangle-shaped output is

7.7  Moving Average Method with Different Window Lengths

153

Figure 7.9  Moving average outputs from different window lengths with PW = 4 ms: (a) window length from 0.6 to 1 ms, (b) window length from 2 to 6 ms, and (c) window length from 7 to 11 ms.

154

Time-Domain Detection with 1-Bit ADC

Figure 7.9  (continued)

not obvious either. One can try to find the slope change from the tropsoidal or the triangle-shaped output. To find the slope, differentiating is required, which almost always decreases the S/N. It is especially difficult to find the TOA and PW for a relative weak signal such as S/N = 0 dB with the moving average method. A simple application of the moving average window is to detect the existence of a signal. As mentioned in Section 10.1, an IFM receiver usually depends on another receiver such as crystal video receiver to provide a trigger signal. One-bit ADC data can be used to build an IFM receiver as discussed in Chapter 10. The moving average can be used to detect a signal and make the receiver an independent subsystem.

7.8  Differential Moving Window Due to the difficulties mentioned in the previous section, a differential moving average is used. In this approach there are two windows of the same length but of different signs—one positive and one negative [private communication with D. Lin, engineer at AFRL/RYSD, 2008]. These two windows are consecutive to each other in the time domain as shown in Figure 7.10. The amplitude of each window is obtained and the results are summed together. Since the windows have different signs, the outputs have positive and negative values. The results are shown in Figure 7.11. In Figure 7.11(a), the window lengths

7.9  TOA and PW Calculation

155

Figure 7.10  Differential windows.

are from 100 ns to 500 ns in 100-ns steps. In Figure 7.11(b) the window lengths are from 600 ns to 1 ms also in 100-ns steps. In Figure 7.11(c) the window lengths are from 2 ms to 6 ms in 1-ms steps. In Figure 7.11(d) the window lengths are from 7 ms to 11 ms in 1-ms steps. The PW is still at 4 ms. The outputs have positive and negative peaks. When the window length is shorter than the PW, both the positive and negative outputs are triangle shaped. As a result, the peak values can be relatively easy to pick up. When the window length is longer than the PW, the outputs are tropsoidal shaped. Under this condition, the maximum and minimum are difficult to find. However, it is relatively easy to determine whether the window is longer or shorter than the PW by observing the output shapes. When the window length and the PW are matched, as shown in the third plot of Figure 7.11(c), the output should be the optimum case. If the signal is relatively strong, the narrow peaks should be able to produce the desired results as shown in Figure 7.11(a, b). If the signal is weak, the window length matching the PW case should produce a better result. If the signal is weak and short, it is difficult to detect it through any approach. Because it is easier to determine the TOA and PW through the differential moving windows, these windows with different lengths will be used in the following discussion.

7.9  TOA and PW Calculation From the differential moving average, the TOA and PW can be calculated as follows. If the positions of the positive and negative peaks are determined, the TOA is the window length before the positive peak and the PW is the distance between the positive and negative peaks. Figure 7.12 is the expanded version of the second plot 1.

M. Brookstock, an adjunct professor at Miami University, Oxford, Ohio, generated all the simulation data.

156

Time-Domain Detection with 1-Bit ADC

Figure 7.11  Differential moving average outputs from different window lengths with PW = 4 ms: (a) window length from 100 to 500 ns, (b) window length from 0.6 to 1 ms, (c) window length from 2 to 6 ms, and (d) window length from 7 to 11 ms.

7.9  TOA and PW Calculation

Figure 7.11  (continued)

157

158

Time-Domain Detection with 1-Bit ADC

Figure 7.12  From positive and negative peaks to obtain TOA and PW.

in Figure 7.11(a). In this figure, the relations between the TOA and PW and positive and negative peaks are illustrated. If the signal is strong, the results obtained from Figure 7.11(a) to the third plot of Figure 7.11(c) should provide similar results. It is difficult to obtain good results from the fourth plot of Figure 7.11(c) to Figure 7.11(d) because there is no obvious peak values in the plots. Table 7.1  Measured TOA and PW with S/N = 0 dB Window Length (ms) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10 11

TOA (ms) 187.8 187.6 187.8 187.8 187.7 187.6 187.6 187.7 187.7 187.6 187.8 187.8 187.6 181.3 181.3 187.3 185.8 147.6 151.8 154.6

PW (ms) 39.38 39.75 39.38 39.38 39.50 39.63 39.75 39.50 39.63 39.75 39.00 39.38 40.00 52.00 53.88 65.75 42.38 128.88 139.75 145.38

7.10  Threshold Setting

159

Table 7.2  Measured TOA and PW with S/N = -10 dB Window Length (ms) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10 11

TOA (ms) 491.7 564.2 206.5 188.9 188.5 189.0 189.4 189.3 189.9 190.5 189.9 187.9 189.5 190.5 187.3 189.1 189.9 188.4 187.4 142.6

PW (ms) –353.88 –444.50 3.13 25.50 26.00 27.88 25.13 27.63 26.50 26.88 36.63 39.75 37.38 47.88 60.75 52.88 60.38 89.00 83.00 150.3

When the window is longer than the input signal, the outputs have flat tops and it is difficult to determine a peak’s location. Even though both a positive and a negative peak can be obtained, the distance between them is usually longer than the true PW. Based on this simple observation, one can realize that the PW obtained through the shortest window should provide the correct answer. For the 100-ns pulse the only correct answer will be provided by the shortest window. The PW measured from all the other windows will be longer than the true value. Based on the above discussion one can calculate many different values for TOA and PW from different window lengths. One of these sets of data is listed in Table 7.1. In Table 7.1, the S/N = 0 dB and the input TOA= 187.55 ms and the PW = 4 ms, which are arbitrarily chosen. With this S/N value, the TOA and PW values are close to the input until the window length is 4 ms. The TOA error is probably caused by tropsoidal output shape and the measured PW is stretched by the window length. If the input S/N = -10 dB, the measured outputs are shown in Table 7.2. The only TOA and PW values closing to the inputs are when the window length and the PW are approximately equal. In this special case, the three window lengths are 2, 3, and 4 ms. Now the problem is how to determine the correct TOA and PW from all the output values.

7.10  Threshold Setting From Table 7.2, one can see that when the input signal is weak and window length is very short such as in a few hundred nanoseconds, the TOA and PW values are

160

Time-Domain Detection with 1-Bit ADC

far from the input values such as the window lengths are 0.1 to 0.3 ns. This phenomenon can be explained that when the input signal is weak, the outputs from the windows are also weak, especially for short windows. These weak outputs cause the erroneous outputs data. When the input is only noise, the peaks obtained from the different window lengths are shown in Figure 7.13. The output increases modestly with window length. When the input S/N = 10 dB, the outputs are shown in Figure 7.14. Comparing the peak values from these two figures, one can see that when there is a relatively strong signal, the output is much higher. From these two figures, a threshold can be set to limit the noise outputs. The threshold is obtained from 1,000 runs with only noise. From these runs, the mean and standard deviation values are obtained. The threshold is empirically chosen as the mean plus five times the standard deviation. For each output from a different window length there is a threshold. Since in Tables 7.1 and 7.2 there are 20 windows, there are 20 thresholds. When the thresholds are applied, the outputs for S/N = -10 dB and the 4 ms PW are listed in Table 7.3. In this table, only the peak values from window of 1 ms and longer are higher than the threshold. The thresholds eliminate many outputs from the short windows. When the window length is close to the PW, the measured PW is close to the input value. The value in Table 7.3 is slightly different from that of Table 7.2 because although the input conditions are the similar, the noises are randomly generated in the simulation program. When the input PW is small, some of the outputs from

Figure 7.13  Peak outputs with only noise: (a) positive peaks and (b) negative peaks.

7.10  Threshold Setting

161

Figure 7.14  Peak outputs with PW = 4 ms and S/N = 10 dB: (a) positive peaks and (b) negative peaks.

longer windows will not cross the threshold. The use of a threshold can limit the number of TOA and PW outputs. In Figure 7.14 the peak values increase with the window length. When the window length is equal to the PW, that is, 4 ms, the outputs slope changes. When the input PW is long, such as 12 ms, the peak values keep increasing, as shown in Figure 7.15. In this figure it shows that even though the window sizes increase in 0.1- and 1- ms steps, the output slope is approximately the same. This slope and its change can be used to determine the TOA and PW.

Table 7.3  Measured TOA and PW with S/N = -10 dB with Threshold at Outputs Window Length (ms) 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10 11

TOA (ms) 185.9 185.3 185.8 185.8 185.4 170.5 172.2 164.7 153.6 161.4 142.1

PW (ms) 39.88 41.25 39.38 40.25 43.75 60.75 58.75 67.63 79.63 94.38 110.25

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Time-Domain Detection with 1-Bit ADC

Figure 7.15  Peak outputs with PW = 12 ms and S/N = 10 dB: (a) positive peaks and (b) negative peaks.

7.11  Detailed Output Shape In Sections 7.8 through 7.10, the general idea of finding TOA and PW is discussed. In order to actually determine the TOA and PW, the output shape will be discussed in more detail. To keep the discussion simple, it is assumed that the input is a rectangularshaped pulse. The filtered outputs can be divided into four cases: 1. 2. 3. 4.

The window length (WL) is less than half the PW (WL < PW/2). The window length is between a half and a full PW (PW/2 < WL PW).

The outputs of the four cases from the differential filters are shown in Figure 7.16. Figure 7.16(a) shows the output for case A (WL < PW/2). Under this condition the output has a very sharp peak and it is relatively easy to determine the peak location. The shorter the window length is the sharper the output peak is. Thus, it is desirable to use this case to measure both the TOA and PW. Figure 7.16(b) shows the results of case B (PW/2 < WL < PW). The peak is not as sharp as that in Figure 7.16(a). The line connecting the positive and negative peaks actually has two different slopes. Figure 7.16(c) shows the results of case C (WL = PW). It is interesting to note that although the window length is longer than the previous case 2, the peak is

7.11  Detailed Output Shape

163

Figure 7.16  (a) The output for case A (WL < PW/2), (b) the results of case B (PW/2 < WL < PW), (c) the results of case C (WL = PW), and (d) the results of case 4 (WL > PW).

sharper. This is the result of matching the window length to the PW. In general this case seldom occurs in real signal conditions. Figure 7.16(d) shows the results of case 4 (WL > PW). Since there is no peak in this output, it should not be used to measure TOA and PW. From this simple illustration, one can see that if the signal is strong enough, it is obvious that the shortest window that can produce the best answer and should be used to determine the TOA and PW.

164

Time-Domain Detection with 1-Bit ADC

7.12  Matched Window Determination From the above observation, a set of rules is used to determine the TOA and PW. These rules must take into consideration of both short and long pulses. The TOA is slightly easier to measure than the PW. When the TOA is measured wrong the measured PW usually will be wrong. When the TOA is correct, the measured PW can still be wrong. The TOA depends on the maximum output position, but the PW depends on both the maximum and minimum positions. In other words, the PW measurement depends on TOA as well as a time of departure (TOD). From this argument the following approaches will be concentrated on determining the PW. Through the PW measurement the TOA can also be obtained. Before actually finding the PW, intuitively one needs to find the matched window because it produces the highest S/N of the window outputs. Under normal condition, an exactly matched window to the PW is difficult to generate. In this study, the closest window to the PW is considered the desired one. Four possible approaches can be used to determine the matched window: 1. From the results in Tables 7.1 to 7.3, one can find that the input PW values are close to the majority of the calculated values. It appears that the majority of the calculated values can be used to determine the PW. It appears that one can generate a histogram from the all the calculated PW. The maximum can be considered as the correct PW. There are two major difficulties in this approach. First, when the input PW is short, such as 100 ns, only the first window can generate the correct value; the rest of the outputs will be longer than 100 ns. Under this condition, the maximum of the histogram will not provide the desired result. Second, when the input PW is rather long, the outputs can spread over a very wide range, and it is difficult to find a definite peak in the histogram. 2. From Figure 7.15, it appears that the output slope is a constant when the PW is less than or equal to the window length. From Figure 7.14, the slope of the output decreases when the window length is longer than the PW. By detecting a slope change, one can decide that the PW and the window length approximately match. From this window output one can determine the PW. The slope can be obtained from the Dy/Dx of Figure 7.14, where Dy is the differential window output and Dx is the corresponding differential window length. In general, taking a ratio increases the noise effect. Under noisy conditions at a short window length, the slope measured may not be dependable. Another difficulty is that taking a ratio loses the first data point. When the PW = 100 ns, the slope test cannot detect the signal. Under this condition, an additional shorter window is needed. One short window of 50 ns is added. The outputs of this 50-ns window and the 100-ns window can be used to generate the slope to find a 100-ns PW. However, a 50-ns window only contains 8 data points and it is sometimes difficult to detect the output. 3. Comparing the results of Figures 7.13 and 7.15, it is easy to see that the window output is a function of signal strength. When the signal is strong, the output is also strong. Based on this observation, a set of window outputs without noise is obtained as a reference; let us refer to them as maximum window

7.13  Ratio Method to Determine a Matched Window

165

outputs. The measured window outputs are divided by the maximum window outputs, and the maximum outputs (or the value closest to one) should be the matched window. Simulation results show that although the maximum output is close to the matched window, it may not be the correct one. 4. This method is also a ratio method but involves the threshold. Let us refer to it as the ratio method. This method is chosen as the basic step to determine the TOA and PW and will be discussed in the next section.

7.13  Ratio Method to Determine a Matched Window As discussed in Section 7.12, only the PW measurement will be discussed here. The ratio R is defined as

R=

wo − thr wo

(7.4)

where wo is the window output and thr is the threshold that is obtained in Section 7.10. Simulation results show that at the maximum value of R the PW is very close to the window length. In general, the PW does not exactly match the window length. When the PW is longer than a certain window, the maximum value often indicates the next window length. As discussed in Section 7.8, when the window length is longer than the PW, it is difficult to determine the exact peak position because the window output becomes tropsoidal. Thus, the matched window in this study is defined as the window length closest to the PW but less than or equal to the PW. For example, if the PW is 4.5 ms and the maximum R occurs at the 5-ms window instead of the 4-ms window, the 4-ms window will be assigned as the matched one. Of course, when the PW exactly matches the window length, such as a 5-ms PW, the matched window will also be selected as the 4 ms window. Table 7.4 shows the R ratio as a function of the PW and window length at S/N = 3 dB. When PW = 4 ms, the correct window is selected. When the PW is 4.8 ms, a window of 5 ms is selected. However, using the window selection rule in the previous paragraph, window lengths of 3 and 4 will be selected as the matched ones for PW = 4 and 4.8 ms, respectively. Whether the matched window is selected correctly must be evaluated. Table 7.5 shows the accuracy of using the R values to detect the matched window. In Table 7.5, when the PW is between 4 and 5 ms, both 4 and 5 are considered as the correct window sizes. The error run is defined as the number of the test runs on which the maximum R is not at the best matched WL. Each simulation is obtained with 100 test runs. At each run the S/N and PW range are given. The TOA is generated randomly, and the PW is also generated randomly in the specified range. When the signal is very weak (-10 dB), only a long pulse (10 to 40 ms) can be detected. When the signal is weak (-5 to -3 dB) and PW is relatively short (0.2 to 1 ms), some error will occur, but these errors are less than 10%. Table 7.5 also shows that when the signal is strong, such as S/N = 5 dB, no errors are detected. Thus, the output ratio R can be considered as a good indicator for finding the most matched window.

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Table 7.4  R Values with Different PW S/N = 3 WL in ms 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4* 5** 6 7 8 9 10 20 30 40

PW = 4 ms R values 0.618 0.726 0.766 0.800 0.816 0.835 0.847 0.858 0.868 0.875 0.915 0.932 0.941 0.934 0.927 0.920 0.915 0.910 0.908 0.879 0.850 0.830

PW = 4.8 ms R values 0.552 0.708 0.763 0.792 0.816 0.836 0.847 0.858 0.867 0.874 0.913 0.928 0.939 0.944 0.938 0.935 0.929 0.924 0.921 0.893 0.875 0.860

* Maximum R value for PW= 4 mms ** Maximum R value for PW=4.8 mms

The problem with a near matched window is that the PW cannot be determined accurately. Figure 7.17 shows the difference between an exactly matched output and the output with PW longer than the window through simulation and under strong signal condition. Figure 7.17(a) shows the exactly matched case where PW = WL = 4 ms and in Figure 7.17(b) the PW = 4.8 ms and the window equals 4 ms. The peak at Figure 7.17(a) is slightly sharper than that of Figure 7.17(b). When the signal is weak, the same conditions are shown in Figure 7.18(a, b). These situations are clearly illustrated in Figure 7.16(b, c). From these two figures, it is easy to see that the peak location can be determined more accurately for the exactly matched window. The less sharp peak from the near matched window can cause inaccuracy in PW measurement. From the differential outputs of many window lengths, the general observation is that the short window or the matched window provides better results. If the window length exactly matches the PW, it is close to the matched filter case and theoretically will provide the best result. The simulation result agrees with this argument. However, the exactly matched condition may not be determined. One should assume that the window is only a near match rather than an exact match. Under this assumption the output from a shorter window usually provides a better result. When the window is shorter, the peak is sharper and its position can be measured Table 7.5  Matched Windows Detection S/R dB PW (ms) Error runs

–10 10 ~ 40 0

–5 0.8 ~ 1 8

–5 1 ~ 40 0

–3 0.2 ~ 0.8 5

–3 1 ~ 40 0

5 0.2 ~ 0.8 0

5 1 ~ 40 0

100 0.1 ~ 40 0

7.13  Ratio Method to Determine a Matched Window

167

Figure 7.17  Outputs from window length = 4 ms strong input: (a) PW=4 ms and (b) PW = 4.8 ms.

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Time-Domain Detection with 1-Bit ADC

Figure 7.18  Outputs from window length = 4 ms weak input: (a) PW = 4 ms and (b) PW = 4.8 ms.

7.15  TOA and PW Results

169

more accurately. This result can be shown by comparing the first plot of Figure 7.11(a) and the third plot of Figure 7.11(c). In general, the narrower peak provides better position measurement. However, when the signal is weak, the matched filter should provide the output with the highest S/N. The general rule is that the output from a shorter window will be used to determine the PW.

7.14  Selecting a Short Window from the Match Window The following sections are methods of using a shorter window for the PW measurement. One must first select this short window. There are several possible ways to select a short window that can generate TOA and PW better than that of a matched window. The shorter window determination is based also on the matched window. Several methods are tried and only the approach providing the best result will be presented here. The following approach is based on empirical trial and error. Instead of using the window with the highest R value (7.4), the R value is used to determine a short window. The approach is to use the maximum value to generate a threshold, which is used to compare with all the outputs from windows shorter than the window with the maximum R value. For example, if the maximum R is 0.9 and the window length is 3 ms, one can empirically choose R/2 (0.45) to be a threshold. This threshold is used to compare with all the outputs from windows shorter than 3 ms. The shortest window that passes this threshold will be selected as the desired one short window. This window will be used to determine the PW. If all the outputs from the shorter windows do not pass the threshold, the window with the maximum R value will be used to calculate the PW.

7.15  TOA and PW Results The results from the above-mentioned method are listed in Table 7.6. These simulation data are generated from 100 test runs. The TOA and PW are randomly selected. The PW are divided into two groups from 0.2 to 0.8 ms and from 5 to 40 ms. Each group has several different S/N values. The standard deviation indicates the variation of the measurement error. As expected, a strong signal has a small error. For weak signals or long signals, the errors usually are large. Two ways are used to show the estimation error: an error in absolute value in nanoseconds or a percentage of an error relative to the PW. For example, in Table 7.6 at S/N = -10 dB the main TOA error is about 390 ns and the standard deviation is about 523 ns. The PW error is about -840 ns and the standard deviation is about 1,224 ns. These values are obtained from a 100 run with random PW from 5 to 40 ms. At each run the input PW is known; thus, the TOA and PW error can be divided by the known PW to produce the percentage error. For the TOA the main percentage error is about 2% and the standard deviation is about 3%. For the PW, the main percentage error is about 4.3% and the standard deviation is about 6.7%. As discussed in Section 7.12 the TOA measured should be better than the PW because the PW measurement depends on the TOA and (TOD). Table 7.6 confirms the expectation that the measured TOA results are better than the PW results.

170

Time-Domain Detection with 1-Bit ADC Table 7.6  Measured TOA and PW Errors Using Half R Threshold Approach SNR dB

TOA error (ns)

PW = 0.2 ~ 0.8 ms Mean –3 12.88 0 5.25 3 3.50 5 2.00 10 2.88 PW = 5 ~ 40 ms –10 389.75 –5 50.88 –3 26.50 0 7.38 3 3.88 5 4.38 10 2.63

(TOA error)/ PW (%)

PW error (ns)

(PW error)/ PW (%)

Std 29.91 12.20 7.76 6.58 6.37

Mean 2.78 1.23 0.76 0.40 0.69

Std 6.93 2.91 1.89 1.73 1.59

Mean –29.68 –11.97 –7.66 –5.71 –7.48

Std 44.51 17.38 9.74 7.84 6.89

Mean 6.09 2.76 1.67 1.36 1.70

Std 0.29 4.27 2.46 1.97 1.81

522.63 80.25 34.78 13.54 7.47 6.97 5.97

1.97 0.29 0.15 0.04 0.03 0.03 0.02

3.01 0.60 0.22 0.07 0.05 0.05 0.06

–839.55 –101.51 –52.95 –16.95 –7.91 –8.04 –6.40

1,223.94 105.00 54.17 17.06 7.47 8.72 7.52

4.31 0.56 0.29 0.09 0.05 0.05 0.05

6.70 0.71 0.37 0.12 0.06 0.07 0.08

7.16  Sensitivity Test Results One of the main purposes of the time-domain detection is to detect a weak signal when the signal is long. The results are shown in Table 7.7. In this simulation, S/N ranges from -16 to -3 dB. As in the previous cases, each simulation contains 100 test runs. At S/N = -16 dB, a 40-ms PW can achieve 96% detection, and a 30-ms PW can achieve an 84% detection. At each S/N value, different PWs are tested. The shortest PW is defined as that in which the pulse can be detected at 90%. The mean and standard deviations of the TOA and PW are measured in percentages of PW. The results in Table 7.7 show that at a given S/N, the longer PW, the smaller TOA and PW error. For a weak signal, such as the S/N of -5 dB, even though the signals can be detected at 99% with PW = 0.5 ms, the average TOA and PW errors are large. When the PW increases, the error decreases.

7.17  Conclusion In this chapter 1-bit ADC is used as the input data from the time-domain detection. Since 1 bit has no amplitude information and very coarse phase information, the IQ balance is not a severe problem. One can build a very simple IQ channel output through hardware. Since the input is only 1 bit, the data contain only one signal. It can not detect simultaneous signals. When simultaneous signals with unequal amplitude are present, usually the strong one will capture the weak one and the correct results will be obtained. When simultaneous signals have comparable amplitudes, the results can be erroneous. From this simple approach to detect only the TOA and PW, the encoding process can be quite tedious. Although the approach is relatively simple, using the differential moving windows to determine the correct outputs is the most difficult

7.17  Conclusion

171 Table 7.7  Sensitivity Test Results S/N

PW (μs)

Percentage of Detection

–16 –16 –15 –15 –14 –14 –14 –13 –13 –13 –12 –12 –12 –11 –11 –11 –10 –10 –10 –10 –5 –5 –5 –3 –3 –3

40 30 40 30 40 30 20 40 30 20 30 20 10 20 10 5 20 10 5 3 3 1 0.5 1 0.5 0.2

96 84 100 100 100 100 99 100 100 100 100 100 100 100 100 99 100 100 100 99 100 100 99 100 100 99

TOA Error in % of PW Mean 2.77 3.6 1.33 –1.55 1.64 –1.47 –3.07 1.22 0.02 –4.26 0.58 3.74 3.02 1.97 2.69 –2.5 1.55 1.98 –0.03 –4.10 1.67 3.13 -0.93 3.93 2.45 –6.83

Standard 10.20 15.72 9.36 15.79 6.57 10.40 24.14 4.92 9.32 24.88 0.59 9.24 10.32 6.10 13.33 21.43 5.40 7.63 18.67 22.90 2.44 5.39 25.27 8.69 8.60 21.58

PW Error in % of PW Mean 7.07 11.55 3.18 1.68 4.34 –0.91 –0.61 2.40 2.35 –3.10 2.05 6.45 8.12 4.33 7.39 –3.13 3.50 5.20 –0.68 –3.72 2.87 6.93 -3.39 7.20 5.56 –8.12

Standard 20.86 32.79 10.98 25.94 8.64 16.83 31.14 6.03 11.10 33.69 7.88 10.07 13.76 7.01 17.67 32.21 6.98 11.66 25.17 31.85 3.47 8.16 45.79 10.30 12.77 30.17

problem. Among all the window outputs the chosen outputs can be limited by using a threshold. One must decide the correct window size to be used to obtain the final results. After the window is selected, the TOA and PW must be estimated by taking the window length into consideration. These discussions can provide an example of an encoder design. Although the idea of detection is simple for 1-bit ADC outputs, determining the correct window and measuring the PW correctly are not obvious. This phenomenon generally happens in encoder designs, which is why the majority of efforts in a receiver design is in the encoder.

References   [1]  Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Chapter 9, Norwood, MA: Artech House, 2001.   [2]  Barton, D. K., Modern Radar System Analysis, Norwood, MA: Artech House, 1988.   [3]  Tsui, J., Fundamentals of Global Positioning System Receivers, 2nd ed., Section 10.7, New York: John Wiley & Sons, 2005.

Chapter 8

Eigenvalue and Related Operations

8.1  Introduction Some basic operations of eigenvalue and the multiple signal classification (MUSIC) have been discussed in Chapter 3 to find the limit of instantaneous dynamic range. In this chapter, the purpose is to use the eigenvalue to detect the existence of signals. The eigenvalue method is very effect in detecting the existence of multiple signals. One major concern is using this method in actual receiver design. Thus, the order of the correlation matrix must be very low so that it can be implemented in hardware with a reasonable effort. With these low-order correlation matrices, some basic properties of eigenvalues will be studied. Most of the study will deal with matrices of two or three orders with one input signal. Some limited discussion on two input signals will be included. Hopefully, from these simple studies some fundamental information can be obtained and used in receiver designs. Eigenvalue will be used to determine the number of signals in the polyphase receiver design in Chapter 13 and determine the existence of exotic signals in Chapters 14 and 15. In this chapter, time-domain detection is performed on input signals. Time-domain detection using 1-bit input data and phase summation is discussed in Chapter 7 and the approach can only apply to one signal. Eigenvalue is used for the detection of multiple signals. If one can use a rather simple method to determine how many signals are in the input data, the receiver design problem can be simplified.

8.2  Input Parameters to the Eigenvalue Problem Input data can be used to produce correlation matrix. The eigenvalues obtained from this correlation matrix may provide important information for the input signals. This problem contains many input parameters and a general solution appears rather complicated. Let us divide the input parameters into two groups: the input signal and the formation of the correlation matrix. The input data include the number of signals and number of data points. If there is only one continuous wave (cw) signal, the input signal can contain three parameters, the signal-to-noise ratio (S/N), the input frequency, and the initial phase. If there are two cw signals and they are totally overlapped, the number of parameters will be doubled to six. The relative amplitude and frequency separation of the two signals will become critical information. When two signals are close in 

M. Broadstock, adjunct professor at Miami University, provided the results in parts of this chapter.

173

174

Eigenvalue and Related Operations

frequency, the relative initial phase should also be considered. Taking the data length into consideration, it is difficult to select a limited number of input data to cover all different input conditions. The order of the correlation matrix is another parameter. With the same order of matrix, there can be different lags. In forming the matrix, the autocorrelations (or lag with zero delay) are always used in the main diagonal elements of the matrix. If the matrix order is 2, since one lag has zero delay, only one lag is a selectable parameter. If the matrix order is 3, two lag values can be selected. The combination of lags can be numerous. Combining the parameters in formatting the matrix with the input signals, the overall possible combinations of input conditions might be very complicated to deal with. Therefore, a few very simple conditions will be selected for this study.

8.3  Simplified Approach In order to keep the problem in a manageable condition but to still illustrate the basic properties of the eigenvalues, very simple cases will be studied. The order of the matrix is chosen to be 2; thus, there is only one lag to choose. Under this condition, only two eigenvalues can be obtained. Although the input signal can contain two input signals, the first study will be limited to only one signal. The frequency and initial phase are randomly selected; thus, the only controlled parameter is the S/N. If the input data are real, one input signal affects two eigenvalues and both eigenvalues are related to noise and the signal; thus, the two-signal case can not be easily studied. If the input signal is complex, the eigenvalues will be affected by only one signal and the noise. The complex signals obtained through the receiver hardware usually have amplitude and phase imbalance between the real and imaginary outputs. Since this study is only interested in the basic property, the imbalance will be ignored. For receiver design with fast Fourier transform (FFT) operation, the outputs are complex and usually well balanced. Thus, the complex signal study can be applied directly. Complex signals are generated through simulations; therefore, their amplitude and phase are well balanced. The distribution of eigenvalues will be studied through simulation. In order to save processing time, 1,000 runs with random frequency and initial phase but fixed S/N will be performed for each case. It is desirable to approximate the results by a known distribution function. The effect of the following parameters will be studied. First, the length of the lags will be observed. In this study, only one lag will be chosen because one lag is always 0. Second, the effect of S/N on the eigenvalue distribution will be studied. Third, the effect of data length on eigenvalues will be studied. It is anticipated that when the data length increases, it is easier to separate the signal from the noise eigenvalues.

8.4  Matrix Formulation and Noise Eigenvalue Distribution The correlation matrix is discussed in Section 3.8. The only difference is that the input signal is complex and can be written as

8.4  Matrix Formulation and Noise Eigenvalue Distribution

175

z = Ae j(2π f1t + θ) + n



(8.1)

where A is the amplitude of the signal, f1 is the input frequency, q is the initial phase, and n is the noise. The noise is Gaussian with zero mean value and standard deviation of 1. With these input signals, a 2 × 2 correlation matrix R can be obtained as

 z(k − 1) R(1, k) =   z(0)*

*



 R11 = R21

*

z(k)

...

z(1)*

...

R12  R22 

 z(k − 1)   z(k) z(N)  z(N − k + 1)*   :   z(N) *

z(0)   z(1)   :  z(N − k + 1)

(8.2)



where 1 is the first lag, which is a fixed value, and * represents complex conjugate. The second element k can be any value less than N, where N is the total data length. It should be noted that 1 stands for zero delay and k stands for k – 1 delay. In the computer program the value [1, k] is used to represent the matrix. For simplicity, it is referred to as the matrix R(1, k) with lag [1, k]. However, the corresponding delay times are [0, k – 1]. This notation will be used in later discussions. The simulation is performed with 256 complex data points and a sampling frequency of 2,560 MHz. In order to detect the existence of the signal, one of several

Figure 8.1  Eigenvalue distribution for noise only with different lags: (a) lag = 1, 2; (b) lag = 1, 8; and (c) lag = 1, 32.

176

Eigenvalue and Related Operations Table 8.1  Mean, Standard Deviations, and Threshold for Eigenvalue Distribution Lag 1, 2 1, 8 1, 32

E1 Mean 2.10 2.05 1.86

E1 Standard 0.14 0.14 0.13

E2 Mean 1.88 1.84 1.65

E2 Standard 0.13 0.13 0.12

Threshold 2.84 2.78 2.54

E1 is the large eigenvalue and E2 is the smaller eigenvalue.

approaches is to set a threshold from a noise-only input. To obtain smooth outputs, a large number of simulations are required, such as 10,000. The distributions of the two eigenvalues are shown in Figure 8.1. In this figure, three difference lags—1, 2; 1, 8; and 1, 32—are arbitrarily chosen. The plots in Figure 8.1 are approximated against a Gaussian distribution. It appears that the results match relatively well. The Gaussian distribution is discussed in Section 6.7. The threshold is obtained from (6.11) and rewritten here as

th ≈ m + 5.2σ

(8.3)

The threshold is set against the larger eigenvalue because the eigenvalue must be greater than these values to be detected. In Figure 8.1 the “o” is simulation data points and the lines are generated from the Gaussian function. The vertical lines are the thresholds obtained from the error function. As the lag increases, the threshold decreases because with a large lag, the number of data points forming the correlation matrix decreases. Less data points generate smaller noise eigenvalues. From this simple illustration, a threshold must be determined for different lag. The mean and standard deviation of the eigenvalues with threshold obtained from the larger eigenvalues are listed in Table 8.1.

8.5  O  ne Complex Signal and Noise Eigenvalue Distribution   and the Probability of Detection In this section, let us add one complex signal to the input and observe the variation of the eigenvalue distributions. It was discussed in Section 3.8 that a very strong signal such as S/N = 100 dB will not affect the amplitude of the noise eigenvalue. However, the signal will bring the smaller eigenvalue to a value close to the larger noise eigenvalue without the input signal. In this section, the distribution of the eigenvalues will be studied through simulations. With the sampling frequency of 2,560 MHz, the Nyquist bandwidth is 2,560 MHz because the data are complex. In each run, the input frequency is randomly selected between 100 and 2,550 MHz with a random initial phase between 0 and 2p. The input frequency is the Nyquist bandwidth of 100 MHz on both ends to avoid signals close to the edges, although there should be no interference when a complex signal is near the band edge. The S/N is arbitrarily chosen to be 3 dB. The same three lags—1, 2; 1, 8; and 1, 32—are used. The results are from 10,000 runs and shown in Figure 8.2. It appears that the Gaussian approximation fits the data points reasonably well for both the noise and signal eigenvalues. The vertical lines in

8.5 One Complex Signal and Noise Eigenvalue Distribution and the Probability of Detection

177

Figure 8.2  Signal and noise eigenvalues distributions for S/N = 3 dB with different lags and the corresponding Gaussian distribution: (a) lag = 1, 2; (b) lag = 1, 8; and (c) lag = 1, 32.

Figure 8.3  Probability of detection versus an input S/N with a probability of false alarm of 10–7.

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Eigenvalue and Related Operations

Figures 8.1 and 8.2 are the thresholds obtained from Section 8.4. In these figures the small eigenvalues do not cross the threshold; this is the expected result. For S/N = 3 dB, the signal eigenvalues are far above the threshold. The probability of detection can be found through simulation. The frequency of the input signal is randomly selected between 100 and 2,550 MHz with a random initial phase and a fixed S/N. One thousand runs are performed, and the larger eigenvalue is compared with the threshold determined in Section 8.4. All three lags are tested and the results are shown in Figure 8.3. As expected, the probability of detection is almost independent on the lag selection. At a probability of detection of 90%, the required S/N is about –5.3 dB. Comparing with the results of Table 6.6, the probability of detection is about –5.7 dB for a real signal with a rectangular window. Since the data used to obtain Figure 8.3 is complex, the sensitivity should be 3 dB higher. It appears that the eigenvalues method provides inferior results. This problem will be further studied in the following section.

8.6  Matrix Order Effect So far in this study the correlation matrix is restricted to 2 × 2. A high-order correction matrix is not likely used in the near future for an EW receiver because of calculation burdens. However, the effect of a higher-order matrix will be studied. Four more correlation matrices will be investigated. They are the 5 × 5, 10 × 10, 20 × 20, and 40 × 40 matrices. The lowest matrix order to process two real signals and find their frequencies is 5 × 5 because one real input signal affects two eigenvalues.

Figure 8.4  Probability of detection versus S/N for different orders of matrix.

8.7  Two Complex Input Signals

179

Table 8.2  Eigenvalue Mean and Standard Deviations, Threshold, and Required S/N to Generate a 90% Probability of Detection for a Different Matrix Size Matrix Size 5×5 10 × 10 20 × 20 40 × 40

Eigenvalue Mean 2.32 2.59 2.98 3.62

Eigenvalue Standard 0.18 0.24 0.34 0.53

Threshold 3.28 3.83 4.74 6.35

Required S/N (dB) –7.3 –8.4 –9.2 –9.6

Two input signals will produce four signal eigenvalues. In order to find the signal frequencies through the MUSIC method, at least one noise eigenvalue and the corresponding eigenvector are needed. This is the reason to choose the 5 × 5 matrix. The higher-order matrices are selected to find the sensitivity of the eigenvalue method through the probability of false alarm and the probability of detection. For these high-order matrices there are many possible ways to choose the lags. Since there is no known optimum way to select the lags and for one signal the lag length does not make much difference on probability of detection, the lags selected are contiguous, such as 1 to 5, 1 to 10, and so forth. First, a threshold will be determined by using noise-only input data. The largest noise eigenvalue will be used to find the threshold. Equation (8.3) will be used to find the threshold from 1,000 runs. With this threshold the probability of false alarm is about 10–7. The probability of detection for the four matrices is shown in Figure 8.4. From this figure, the S/N required to achieve a 90% probability of detection can be found. The results are shown in Table 8.2. The required S/N values are lower for a higher-order matrix. When the matrix order increases from 5 × 5 to 10 × 10, the improvement is about 1 dB, but from 20 × 20 to 40 × 40 it only improves about 0.4 dB. There is a limit in the improvement and the required S/N shows this trend. For the 40 × 40 matrix, the required S/N is –9.6 dB. This result should compare to the result with 256 data points and a rectangular window in Table 6.6 where the required S/N for real data is –5.7 dB. The corresponding complex result should be –8.7 dB because the sensitivity of complex data is 3 dB higher. From these two results the eigenvalue method has a slightly higher (close to 1 dB) sensitivity.

8.7  Two Complex Input Signals In this section, it is intended to study the detection of two complex signals with the two lag matrices. The input signals have many different parameters, such as the relative amplitude, frequency, and phase differences between the two signals. It is difficult to cover most of the possibilities. Here the study will be limited to a very few cases. Hopefully, from these limited cases certain patterns can be revealed. Let us make the two signals the same amplitude with S/N = 0 dB. One input signal has a zero initial phase and the other one has a random initial phase. The frequency difference between them is kept as a constant. Three arbitrarily frequency separations are chosen: 30, 220, and 1,000 MHz.

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Figure 8.5  Two equal amplitude signals with S/N = 0 dB and a frequency separation of 30 MHz: (a) lag = 1, 2; (b) lag = 1, 8; and (c) lag = 1, 32.

The results are shown in Figures 8.5 to 8.7. The threshold values are the values listed in Table 8.1. The results are obtained from 1,000 runs with random input frequencies but the difference is a constant. Both signals are restricted in the frequency range of 100 to 2,550 MHz. For the two input signal case, both eigenvalues are signal eigenvalues and there is no noise eigenvalue. If the eigenvalue crosses the threshold, the signal is detected. In Figure 8.5, the frequency separation equals 30 MHz and the best result is from the lag 1, 32 combinations because both signals are detected. In Figure 8.6, the frequency separation equals 220 MHz and the best result is from the lag 1, 8 combinations. In Figure 8.7, the frequency separation equals 1,000 MHz and the best result is from the lag 1, 2 combinations. From these three figures, it is shown that there are relations between the frequency separation and the lag length. In an actual application it is desirable to use more than two eigenvalues and there are many different ways to select the lags. It is desirable to know the relationship between the frequency separations and lags. In the following study the selections of lags are obtained from empirical observations. The Nyquist input bandwidth of this study is 2,560 MHz, as presented in Section 8.5. Half of the bandwidth (1,280 MHz) can be used to estimate the second eigenvalue. Let us define a term as the frequency difference lag product, which

8.8  Data Length Effect

181

Figure 8.6  Two equal amplitude signals with S/N = 0 dB and a frequency separation of 220 MHz: (a) lag = 1, 2; (b) lag = 1, 8; and (c) lag = 1, 32.

equals the input frequency separation times the second lag length. The frequency difference lag product is divided by 2,560 MHz and we will refer to the remainder as the lag remainder for simplicity. If the lag remainder is close to 1,280 MHz, the second eigenvalue has a large value. For example, if the frequency separation is 40 or 120 MHz for the lag 1, 32 combinations, the frequency difference lag products are 1,280 (40 × 32) and 3,840 (120 × 32). The lag remainders are both 1,280. Under this condition, the second eigenvalue is the largest compared with the lags 1, 2 and 1, 8 combinations. If the frequency separation is 80 MHz, for the lags 1, 32 the lag remainder is zero and the second eigenvalue will be small. In order to cover a range of difference frequencies in actual applications, many correlation matrices with different lags must be used. The number of matrix should stay low to save calculation time. The lags selected for all the matrices should have the property to cover most of the frequency separations.

8.8  Data Length Effect In this section, the effect of data length to form the correlation matrix will be studied. In the previous study, 256 points of data are used. In this study the data will change from 32 to 1,024 points increased in binary-based numbers. To keep

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Figure 8.7  Two equal amplitude signals with S/N = 0 dB and a frequency separation of 1,000 MHz: (a) lag = 1, 2; (b) lag = 1, 8; and (c) lag = 1, 32.

this problem simple, only the 2 × 2 matrix with one complex input will be studied. Since for one signal the lag length does not have significant influences, only the 1, 2 combinations will be used in the study. First, let us study the eigenvalue distribution. One signal of S/N = 0 dB is used as the input and 1,000 runs are performed. The input data points are 64 and 256; Figure 8.8 shows the results. The means of both signal and noise eigenvalues appear at the same positions; however, the standard deviation of the 64-point case is larger than that of the 256-point case. This phenomenon indicates it is easier to detect a signal with 256 data points than with 64 data points, which is the expected result. The next step is to calculate the probability of detection for a different data length. Before calculating the probability of detection, a threshold must be determined. The threshold is based on (8.3), which should have a probability of false alarm of about 10–7. The thresholds obtained are listed in Table 8.3. The mean and standard deviations and threshold for the 256-point case listed in this table is slightly different from the lag 1, 2 combinations listed in Table 8.1 because different noise sets are used. Using these thresholds the probabilities of detection versus S/N are shown in Figure 8.9. For a 90% probability of detection and a 10–7 probability of false alarm, when the data length increases from 32 to 1,024 points, the S/N required decreases from 0 to close to –9 dB, as shown in Figure 8.10.

8.8  Data Length Effect

183

Figure 8.8  Eigenvalue distributions for data lengths of 64 and 256 points.

If, considering that every time the data length doubles, the required S/N will drop 3 dB for coherent processing, the overall improvement will be 15 dB rather than 9 dB. In this study, the average sensitivity improvement is close to but slightly less than 2 dB when the data length doubles. The reason for this degradation might be caused by the low-order correlation matrices. It is noted that when the order of the correlation matrix increases, the sensitivity also increases as shown in Section 8.6. The probability of detection in the present study is limited by the order of the matrices rather than by the S/N. Thus, the 3-dB increase in sensitivity by doubling the data length may not apply.

Table 8.3  Eigenvalue Mean and Standard Deviations and Threshold for Different Data Length Number of Data Points 32 64 128 256 512 1,024

Eigenvalue Mean 2.27 2.20 2.14 2.10 2.07 2.05

Eigenvalue Standard 0.42 0.30 0.20 0.15 0.10 0.07

Threshold 4.47 3.74 3.17 2.86 2.59 2.41

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Eigenvalue and Related Operations

Figure 8.9  Probability of detection versus S/N with a 10–7 false alarm for a different data length.

Figure 8.10  Required S/N to achieve a 90% detection with a probability of false alarm of 10–7 versus data length.

8.9  Data Length Increase Through Summations of Shorter Matrices

185

8.9  Data Length Increase Through Summations of Shorter Matrices In Section 8.8, when the data length doubles, the correlation matrix is calculated individually. For example, when 64 data points are used for calculation, the correlation matrix is generated through (8.2) from the selected data. In this section, it is intended to test a different approach to form the correlation matrix. This approach is to combine matrices with short data length to form a long one. For example, two sets of 32 correlation matrices can be summed together to form a matrix with 64 data points. The intention is to save calculation complexity. The matrix with long data does not have to start with (8.2). Once a short matrix is obtained, the result can be used to form the matrix with the long data. Mathematically, the matrix with the long data length formed from short ones does not equal the matrix formed from (8.2). However, if the eigenvalue distributions are similar, it can be used to detect signals. Figure 8.11 shows the eigenvalue distributions of matrices with 64 and 256 data points. The results are from 1,000 runs with S/N = 0 dB. Four matrices are formed each with 64 data points. The first matrix is used to find the eigenvalue distribution of the 64 data point case. The elements of the matrix with 256 data points are obtained from the average of the 4 matrices. It is interesting to note that this figure is very similar to the result in Figure 8.8 where the matrices are formed through (8.2). From this simple illustration, it is anticipated that this summation approach can be used in the time-domain detection,

Figure 8.11  Eigenvalue distributions for data lengths of 64 and 256 points in a summation method forming matrices.

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where many different windows lengths will be applied to match the signal in the time domain.

8.10  Analytic Eigenvalue Solutions of a Low-Order Matrix From the previous study it is obvious that the eigenvalues can be used to determine the number of signals. Although a high-order correlation matrix can produce good results as discussed in Section 8.6, it might not be implemented in the hardware in the near future. The following study will limit the correlation matrix order to 3. The solutions are available in many references. It is presented here as a quick reference. Assume that the covariance matrix is 2 × 2; the result can be written as a R= c



b d 

(8.4)

The eigenvalues can be found as [1] a − λ  c 

b  =0 d − λ 

λ 2 − (a + d) λ+ (ad − bc) = 0 λ=



(8.5)

(a + d) ± (a + d)2 − 4(ad − bc) 2

Assume that the covariance is 3 × 3; the result can be written as a R = d  g



b e h

c f  k

(8.6)

In this equation, k is used instead of i because i may be used for imaginary notation. The eigenvalue can be found as

R

a λ d g

b e λ h

λ 3 (a e k) λ2

c f

0

k λ (ae ak ek cg

afh bdk cge aek bfg cdh

(8.7) fh bd)λ 0



which can be written as

λ 3 + a2 λ 2 + a1 λ + a0 = 0

(8.8)

8.11  Eigenvalues Versus Initial Phase Difference

187

The result can be obtained from a conventional handbook [1] q=

1 1 a1 − a22 3 9

r=

1 1 3 (a1a2 − 3a0 ) − a 6 27 2

s1 = [r + (q3 + r 2 )1 / 2 ]1 / 3 s2 = [r − (q3 + r 2 )1 / 2 ]1 / 3

(8.9)

λ1 = (s1 + s2 ) − (a2 / 3) 1 j 3 λ2 = − (s1 + s2 ) − (a2 / 3) + (s1 − s2 ) 2 2

1 j 3 λ3 = − (s1 + s2 ) − (a2 / 3) − (s1 − s2 ) 2 2 These results show how the simple eigenvalues of a low-order MUSIC method can be found. The formation of the 3 × 3 matrix can be obtained in a similar manner as in (8.2): éz (l - 1)* ê R(1, k , l) = êz (k - 1)* ê ê z (0)* ë éz (l - 1) ê z (l) ê êë z (N )



é R11 = êêR21 êë R31

z (l)*

�

z (k)*

...

z (1)*

...

z (k - 1) z (k) z (N + k - l) R12 R22 R32

ù ú *ú z (N + k - l) ú z (N - l + 1)* úû

z (N )*

� � �

z (0) ù ú z (1) ú z (N - l + 1)úû

R13 ù R23 úú R33 úû

(8.10)



where 1 < k < l. Similar to the discussion in Section 8.4, the lag of the matrix R(1, k, l) is [1, k, l] with a delay of [0, k – 1, l –1].

8.11  Eigenvalues Versus Initial Phase Difference Some basic properties that may affect the signal detection problem will be presented here. In this section, the eigenvalues versus the initial phase difference of the two complex signals will be studied. The results are obtained from simulation results. When the two signals are close in frequency such as 10 MHz, the initial phase

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Eigenvalue and Related Operations

difference affects the eigenvalues. When the frequency separation is greater than 10 MHz, the eigenvalues are independent of the initial phase difference. In this simulation, the initial phase and frequency of the first signal are randomly selected. The second signal is kept at a fixed frequency with respect to the first one and the phase changes from 0 to 2p in p/10 steps with respect to the first signal. Both signals have an S/N of 40 dB, which are arbitrarily chosen. The first frequency is randomly selected and the second one has a fixed difference frequency with respect to the first one. Both frequencies are limited in the range of 100 to 2,550 MHz. In order to separate two frequencies close together, a relatively long lag is needed. The lags of this study are 1, 128, and the longer lag is arbitrarily chosen. Figure 8.12(a) shows the results when the frequency separation is 1 MHz. When the initial phase difference is p, the larger eigenvalue reaches a minimum and the smaller eigenvalue has a maximum. Figure 8.12(b) shows the results when the frequency separation is 5 MHz. The minimum and maximum of the two eigenvalues occur at about 3p/2. It should be noted that the y-axis in this plot is logarithmic, which indicates the eigenvalues have a large range of values. This figure is obtained from 1,000 runs with random input frequency and the eigenvalues are the average value for 1,000 runs. When the figure is obtained from only one run, very similar results are obtained (not shown). This indicates that the result is independent of the input frequency. From this simple illustration, the detection problem is affected by the initial phase difference.

Figure 8.12  Eigenvalues versus initial phase difference with a frequency fixed difference with a lag = 1, 128 and S/N = 40 dB: (a) frequency separation 1 MHz and (b) frequency separation 5 MHz.

8.12  Eigenvalues and Frequency Separation

189

8.12  Eigenvalues and Frequency Separation In Section 8.7, it was shown that the eigenvalues depend on the lag length and the frequency separation between two input signals. Since this phenomenon affects the detection problem, this problem will be further investigated here. With the fixed lag length, when the frequency separation changes, the eigenvalues vary almost periodically. Two lags are used in the simulations to illustrate this phenomenon: 1, 2 and 1, 128. The strong signal is at S/N = 40 dB and the weak signal is 5 dB below the strong one. The frequency difference changes from 1 MHz to 2,530 MHz. These input conditions are arbitrarily chosen. The results are shown in Figure 8.13. Each data is obtained from the average of 10 runs. With a short lag, the eigenvalues change smoothly over the range of the frequency separation range as shown in Figure 8.13(a). With a long lag, the eigenvalues change rapidly over the frequency range as shown in Figure 8.13(b). Figure 8.13(c) shows the fine structure of Figure 8.13(b). When the large eigenvalue decreases, the small eigenvalue increases. Figure 8.14 shows similar results with lag = 1, 4 and 1, 9. The purpose of these two plots is to find a relation between the lag and the frequency separation. Observing from these figures it appears that for the lag 1, 4, the eigenvalue amplitude variation has three cycles and for the 1, 9 lag, the variation is eight cycles. From these figures, one can find that the number of cycles of the eigenvalue variation and the lag length can be related by

Figure 8.13  (a–c) Eigenvalues versus frequency separation: (a) lag 1, 2 and (b) lag 1, 128.

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Figure 8.14  Eigenvalues versus frequency separation: (a) lag 1, 4 and (b) lag 1, 9.

C = k - 1 or

P=

fs f = s C k - 1

(8.11)

where C is the number of cycles, k is the second lag, and P is the period measured in frequency. For example, for the lag 1, 9, there are eight cycles and each cycle has a frequency range of 320 MHz (2,560/8).

8.13  E  igenvalue Threshold Method to Determine the Number   of Signals Three methods using eigenvalues will be used to determine the number of signals. The first one uses several lags and thresholds to perform the detection; we will refer to it as the eigenvalue threshold method or simply eigenvalue method. Two other methods, the Akaike information criterion (AIC) and the minimum description length (MDL) methods, will be discussed. These two methods do not need a threshold and mathematically are rather elegant. The first method, the eigenvalue threshold method, is easy to understand. A threshold is used to determine the number of signals. If no eigenvalue crosses the

8.13 Eigenvalue Threshold Method to Determine the Number of Signals

191

threshold, there is no signal. If one eigenvalue crosses the threshold, there is one signal. If two eigenvalues cross the threshold, there are two signals. However, not only the signal amplitudes affect the eigenvalues; the frequency separation also affects them. As discussed in Section 8.12, the frequency separation affects the eigenvalues differently for various lag lengths. In order to detect a weak signal, two lag matrices are needed. This phenomenon can be explained in Figure 8.15. In this figure, the strong signal is at S/N = 3 dB and the weak signal is at S/N = 1 dB. The frequency of the strong signal is randomly selected and the difference frequency changes in 1-MHz steps. One data point is obtained by one run. The lags used are 1, 2; 1, 18; 1, 95; and 1, 128. These lags are by no means the optimum selections. They are selected empirically. The horizontal straight lines in the plots are the threshold. All the larger eigenvalues cross the threshold, but the weak eigenvalue does not. In Figure 8.15(a), the center portion of the curve crosses the threshold; however, the end portions do not. This indicates that when lag 1, 2 is used, two signals are close in frequency or far apart, and the weak one cannot be detected. The other three [Figures 8.15(b–d)] indicate that at some difference frequencies the eigenvalues cross the threshold, but not all of them. The idea behind the selection of lags is that at any difference frequency, at least one of the lower eigenvalues generated from the four matrices will cross the threshold. If at a certain difference frequency all the small eigenvalues cannot cross the threshold, one signal will be missed at that difference frequency. Of course, if more matrices are used, the difference frequency coverage will be better; however, the calculation will be more tedious.

Figure 8.15  Eigenvalues versus frequency separation with a threshold: (a) lag 1, 2, (b) 1, 18, (c) 1, 95, and (d) 1, 128.

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Eigenvalue and Related Operations

8.14  AIC and MDL Approaches Two approaches for determining the number of signals are based upon the application of the information theoretic criteria for model selection. The first approach is introduced by Akaikeas as the Akaike Information Criterion (AIC) [2, 3]. The second approach is introduced by Schwartz and Rissanen [2, 3] as minimum description length (MDL). The advantage of these approaches is that no subjective threshold is needed. The number of signals is determined as the values for which the AIC or the MDL criteria are minimized. The form of AIC for detecting the number of signals is given by 

p





(p − k)N

 ∏ li i = k +1   AIC(k) = −2 log p  1  li   ∑  p − k i = k +1  

1 /( p − k)

+ 2k(2 p − k)

(8.12)



The MDL criterion is given by 





p

∏

li1 /(p −k)



(p − k)N



 MDL(k) = − log  i = k +1 p  1  li   ∑  p − k i = k +1 

+

1 k(2 p − k)log N 2

(8.13)

where l is the eigenvalue, p is the number of eigenvalues, and N (=256) is the number of data in the time series for eigenvalue calculation. The number of the signals is determined as the values of k, where k = 0, 1, 2, …, p – 1, for which either the AIC or the MDL is minimized. In order to detect up to two signals, a minimum of three eigenvalues are needed so that p = 3 and k = 0, 1, and 2. Thus, from (8.12) one can obtain AIC(0), AIC(1), and AIC(2). If AIC(2) is the minimum among these three values, there are two signals. If AIC(1) is the minimum among these three values, there is one signal. If AIC(0) is the minimum among these three values, there is no signal. The same argument holds for (8.13). It is noted that the frequency separation effect also occurs in these two methods because the AIC and MDL use eigenvalues, which are affected by the frequency separation and lags in low-order correlation matrices. In order to mitigate the frequency separation effect, more than one set of two lag matrices are needed as discussed in the previous section. Here three three-lag correlation matrices are used, and their lags are 1, 2, 3; 1, 18, 128; and 1, 53, 96. These lags are determined empirically. If the largest number of signals detected is used as the desired result, the results have a false alarm. The number of signals is determined by the majority results from these sets. For example, if the result from the first set of matrix indicates that there are two signals,

8.15  False Alarm Test

193

but the results from the second and the third sets of the matrices show that there is only one signal; then the final result should be one signal. This approach reduces the false alarm to zero with limited tests. In order to compare the eigenvalue threshold methods against these two methods, the same correlation matrices are used. Thus, the eigenvalue threshold method also uses three 3 × 3 matrices. The two larger eigenvalues are used to determine the number of signals. The detection criterion is that the largest number of signals measured is the desired answer. This approach does not generate a false alarm with limited tests. For simulations performed in the following sections, all the three methods use identical input data.

8.15  False Alarm Test In this section the input is only noise. If a certain method detects a signal, the results will be considered as a false alarm. The input S/N = –100 to –180 dB. With these S/Ns, one can consider that the input contains only noise. The purpose of using these S/N values is for plotting the results. Each data point is obtained from 100 runs.

Figure 8.16  Probability of detection with a noise-only input: (a) zero signals detected, (b) one signal detected, and (c) two signals detected.

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Eigenvalue and Related Operations

The AIC and MDL methods in this study only detect up to two signals. Although the eigenvalue method can detect up to three signals with the 3 × 3 matrices, only the two larger eigenvalues are considered to detect two signals. Figure 8.16 shows the results. The number of detections for 0, 1, and 2 signals is plotted according to the detection methods. The correct results are zero signals detected. In Figure 8.16(a), both the eigenvalue threshold and the MDL methods show that zero signals are detected 100%. The AIC method indicates that most of the time zero signals are detected but they are not 100%. Sometimes the AIC method only detects about 95% of no signals. Figure 8.16(b) shows the detection of one signal. The correct results should be 0%. Both the eigenvalue threshold and the MDL methods produce this result, but the AIC sometimes detects one signal. All three methods never detect two signals. This study results indicate that both the eigenvalue threshold and the MDL methods do not produce any false alarm with these limited tests. The AIC method produces false alarms by detecting one signal, but it does not detect two signals.

8.16  Input with One Signal and Two Signals In the first study only one signal is present. The signal S/N changes from –5 to 80 dB. Figure 8.17 shows the results of the percentages of detection for different numbers

Figure 8.17  Probability of detection with one input signal: (a) zero signals detected, (b) one signal detected, and (c) two signals detected.

8.16  Input with One Signal and Two Signals

195

of signals. Similar to Figure 8.16, Figure 8.17 contains three plots with zero, one, and two signal detections. The correct result is 100% for one signal detection. However, when the input signal is weak such as at S/N = –5 dB, the signal is difficult to detect. The eigenvalue method can reach about 90% detection for one signal when the S/N = –3 dB and above. The MDL method reaches about 90% detection for one signal when the S/N = –2 dB and above. The difference between the eigenvalue method and the MDL method is very small. The AIC method can reach about 90% of detection for one signal when the S/N = –5 dB and above. However, the AIC method overestimates the number of signals throughout the whole S/N range. The AIC method detects 1% to 5% of two signals, one of which is a false alarm. For the two-signal test, the S/N of the first signal changes from –5 to 80 dB and the amplitude of the second signal is chosen randomly from 0 to –S/N dB with respect to the strong one. With this arrangement, when the S/N of the first signal is 50 dB, the second signal varies from S/N = 50 to 0 dB so that the amplitude difference is from –50 to 0 dB. If the first signal has a negative S/N value such as –5 dB, the second signal varies from –5 to 0 dB so that the amplitude difference is from 0 to 5 dB. Thus, both signals have a minimum S/N = –5 dB. The input frequency is from 10 MHz to 2,550 MHz. The frequency separation between the two signals is randomly selected between 2 to 2,540 MHz.

Figure 8.18  Probability of detection with two input signals: (a) zero signals detected, (b) one signal detected, and (c) two signals detected.

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Eigenvalue and Related Operations

Each data point is obtained from 100 runs. Figure 8.18 shows the results of the percentages of detection for different numbers of signals. When the S/N = –5 dB, the eigenvalue method cannot detect both signals and it can detect one signal 79% of the time and zero signals 21% of the time. The MDL method cannot detect both signals either and it can detect one signal 34% the time and no signals 66% of the time. The AIC method detects two signals 23%, one signal 61%, and zero signals 16%. Both the eigenvalue and MDL methods cannot detect two signals when the S/N = –5 dB. When the S/N increases, the percentages of detection of two signals increase with all the approaches. When the S/N reaches 0 dB, both the AIC and the eigenvalue methods reach a correct detection of 90% or better. The MDL method reaches a 90% correct detection when the S/N is 3 dB and above. For detecting two signals, the AIC and the eigenvalue approaches have a slightly better perform­ ance than MDL, but the relatively high false alarm of the AIC method limits its application.

8.17  Effect of IQ Imbalance on Number of Signal Detection Since the input in the above study is complex, the imbalance of the in-phase (I) and quadrature phase (Q) can be very critical. The imbalance can be considered separately as the amplitude and phase imbalances. When there is only one signal, the phase and amplitude imbalances increase the amplitude of the eigenvalues. This causes all the approaches to overestimate the number of the signals, especially when the signal is strong. Simulations with 100 runs per data are performed with an input data containing only one signal with a random initial phase to find the number of signals. When the amplitude imbalance 0.99 (or less than 0.1 dB) and the input S/N = 36 dB, all three methods overestimate and detect two signals. When the amplitude imbalance is 0.8 (–1.9 dB), the overestimation occurs when the S/N = 10 dB. The phase imbalance also causes an overestimated detection. With one signal and a 10° phase imbalance all three methods detect two signals at S/N above 20 dB. The eigenvalues can be used to detect one or two complex signals and the results are rather sensitive. Imbalance in the IQ outputs can cause overestimation, which means that the number of signals estimated is higher than the actual input signal number. This phenomenon is expected because when IQ channels are not balanced image will appear output as shown in Section 5.1. Since FFT outputs can generate well-balanced IQ outputs, these methods can be used to determine the number of signals after the FFT operations. Since the AIC method generates a false detection, it is not suitable for EW receiver designs.

8.18  Time-Domain Detection Using the Eigenvalue Method In the previous discussions, the input signals are complex. In this section, the input signal is real and the eigenvalue method is used to perform the time-domain detection. The time-domain detection is discussed in Chapter 7 for only one complex signal. The method used here is for two real signals. Almost all the discussions in Chapter 7 are applicable here. As mentioned in Section 8.3, one real signal affects

8.18  Time-Domain Detection Using the Eigenvalue Method

197

two eigenvalues. Two real signals will affect four eigenvalues. On the other hand, a 3 × 3 matrix can have an analytic resolution, which might be easily implemented in the hardware. Thus, a 3 × 3 matrix will be used in this study. If there are two real signals, all three eigenvalues are related to signals and there is no noise eigenvalues. The first signal affects the largest two eigenvalues and the second signal affects the third signal. To detect two signals, only the first and the third eigenvalues will be examined and the second eigenvalue is ignored. As discussed in Section 8.8, a long data length provides a better sensitivity but increases the calculation burden. To solve this problem, the approach in Section 8.9 is used. Instead of using all the data points to form a correlation matrix, the alternative approach is to generate the correlation matrix by averaging two matrices, each formed with half of the data points. The eigenvalues generated by the two approaches are very close. For example, if the correlation matrices for two 32 data points are calculated, the average of these two matrices can be used to represent the correlation matrix with 64-point data. Similarly, the average of two 64-point matrices can be used as the 128-point correlation matrix. This approach can be used to form a correlation matrix with longer data. Different data lengths are similar as different window lengths in Chapter 7; thus, let us refer to them as different window lengths. Theoretically, when the window length closely matches the input signal length, the signal can be detected, which briefly resembles the matched filter approach. The limitation of this approach is to select the length of the lags. For example, for 32 points of data the longest lag is 32 (or a 31-point delay) and the matrix formed from 128 data points directly can select longer lags. Figure 8.19 explains how the matrices are calculated in this approach.

Figure 8.19  Block diagram of making a correlation matrix with long data through the summation of short matrices.

198

Eigenvalue and Related Operations

8.19  S  imulation of the Time-Domain Detection Using   Eigenvalues Method In this section the simulation procedure is discussed. With a sampling rate of 2.56 GHz, 25,600 data points are used, which corresponds to 10 ms of time. Since 32 real data points are used to form the shortest matrix, the time resolution is 12.5 ns. Matrices of longer data can be formed by these short matrices. The window lengths used are 32, 64, 128, . . . , 2,048. The input data contains one or two real sinusoidal waves with a given time of arrival (TOA), a pulse width (PW), and a pulse amplitude (PA). The eigenvalue method is used for the time-domain detection study to determine the TOA and PW on two signals. Although the eigenvalue method provides slightly better results over the MDL method, this test uses limited data and may not be considered as a general case. The previous study suggests that the eigenvalues are affected by the frequency differences. In order to detect most of the input signals, four different lags (1, 2, 3; 1, 3, 14; 1, 5, 15; 1, 7, 16) are selected empirically. Any one of the four eigenvalues is greater than the threshold, and one signal is detected. Comparing the threshold with the first and third eigenvalues, one can find the time frame sections with no signal, one signal, and two signals. When two signals are present, the eigenvalue amplitudes can be used to separate them. For example, the large eigenvalue is for one signal and the small one is for the second signal. The first window crossing the threshold can be considered as the TOA and the last one as the time of departure (TOD). The PW can be found from the difference of TOD and TOA. In order to obtain a better TOA, the actual TOA can be modified by a half window length and this adjustment is determined empirically. In the first simulation the two signals are partially overlapped. The frequencies of these two signals are random selected but with a minimum of a 50-MHz difference. The S/N of the first signal is 0 dB, and the S/N of the second signal is –3 dB. The TOA of the first signal is at 3 ms (240th time frame) with a PW of 2 ms; the TOA of the second signal is at 4 ms (320th time frame) with a PW of 2 ms also. Figure 8.20 shows the simulation results with window length of 512 points. Figure 8.20(a) shows the first eigenvalues, Figure 8.20(b) shows the third eigenvalues, and Figure 8.20(c) shows the one-signal and two-signal sections in time frames. In Figure 8.20(a) the eigenvalue has two values close to 4 and close to 2. In Figure 8.20(b) the eigenvalue approximately equals 2. Thus, one can determine that amplitude 4 belongs to one signal and amplitude 2 belongs to another. From these sections, one can determine the TOA, TOD, PW, and the relative position of these two signals. Table 8.4 shows the simulation results of TOA, TOD, and PW for both signals for all window lengths. The final result is obtained in two steps. The first step is to group all data with similar values together and the most frequent occurrence is used as a guide. The second step is to determine the final value. The result closest to the guide and generated by the shortest window length is the desired TOA, TOD because in Chapter 7, it is shown that short windows produce better results. In Table 8.4 only the 256-point window length generates all the data close to the guide. The 64-point window provides good TOA1, PW1, but TOA2 and PW2 are off. The

8.19 Simulation of the Time-Domain Detection Using Eigenvalues Method

199

Figure 8.20  Eigenvalues of two signal with window length of 512: (a) large eigenvalue, (b) small eigenvalue, and (c) signal overlap.

512-point window provides slightly better TOA and PW than that of the 256-point results, since the 256 point is shorter and its results are selected. The estimation errors of TOA, TOD, and PW are less than 10%. In the second simulation the two signals are partially overlapped but with a short PW. The first signal has a 3-ms TOA, a 0.2-ms PW, and an S/N of 5 dB. The

Table 8.4  Estimated TOA, TOD, and PW for a Two-Signal Detection with S/N = 0, –3 dB (PW1 = 2 ms, PW2 = 2 ms, TOA1 = 240, TOD1 = 400, TOA2 = 320, TOD2 = 480) Data TOA1 Length 32 241.5 64 241 128 240 256 239 512 235 1,024 228 2,048 213 The final results 256 239

TOD1

PW1

TOA2

TOD2

PW2

Description

398 400 400 387 387 385 387

1.9563 1.9875 2 1.85 1.9 1.9625 2.175

0 435 409 325 321 313 303

0 457 474 474 474 472 469

0 0.275 0.8125 1.8625 1.9125 1.9875 2.075

One signal only Two signals no overlap Two signals no overlap Two signals partially overlap Two signals partially overlap Two signals partially overlap Two signals partially overlap

387

1.85

325

474

1.862

Two signals partially overlap

200

Eigenvalue and Related Operations

Figure 8.21  Eigenvalues of two signals with a window length of 512 data points: (a) large eigenvalue, (b) small eigenvalue, and (c) signal overlap.

second signal has a 3.1-ms TOA, a 0.2-ms PW, and an S/N of 0 dB. These two signals overlap by 0.1 ms. A short window length cannot detect the weak signal (the second one), but a longer data length can detect both signals. Figure 8.21 shows the outputs from the 512-point window. Figure 8.21(a) shows the first eigenvalues, Figure 8.21(b) shows the third eigenvalues, and Figure 8.21(c) shows the numbers

Table 8.5  Estimated TOA, TOD, and PW for a Two-Signal Detection with S/N = 5, 0 dB (PW1 = 0.2 ms, PW2 = 0.2 ms, TOA1 = 240, TOD1 = 248, TOA2 = 256, TOD2 = 264) Data TOA1 Length 32 241.5 64 241 128 240 256 239 512 235 1,024 227 2,048 212 The final results 128 240

TOD1

PW1

TOA2

TOD2

PW2

Description

264 264 255 255 255 254 254

0.28125 0.2875 0.1875 0.2 0.25 0.3375 0.525

0 0 250 250 248 243 230

0 0 264 263 263 261 260

0 0 0.175 0.1625 0.1875 0.225 0.375

One signal only One signal only Two signals partially overlap Two signals partially overlap Two signals partially overlap Two signals partially overlap Two signals partially overlap

255

0.1875

250

264

0.175

Two signals partially overlap

8.20  Conclusion

201

Table 8.6  Estimated TOA, TOD, and PW for a Two-Signal Detection with S/N = 5, 2 dB (PW1 = 1 ms, PW2 = 0.5 ms, TOA1 = 240, TOD1 = 320, TOA2 = 256, TOD2 = 296) Data TOA1 Length 32 241.5 64 241 128 240 256 238 512 234 1,024 226 2,048 211 The final results 32 241.5

TOD1

PW1

TOA2

TOD2

PW2

Description

320 320 320 295 294 294 319

0.98125 0.9875 1 0.7125 0.75 0.85 1.35

257.5 257 256 255 251 244 228

296 296 295 320 319 319 294

0.48125 0.4875 0.4875 0.8125 0.85 0.9375 0.825

One signal is inside the other One signal is inside the other One signal is inside the other Two signals partially overlap Two signals partially overlap Two signals partially overlap One signal is inside the other

320

0.98125

257.5

296

0.48125

One signal is inside the other

of signal sections. Figure 8.21(c) also shows that there are two one-signal sections and one two-signal section. The eigenvalue amplitudes of these two one-signal sections are compared. If the two amplitudes are close, it is assumed that one signal is inside the other one. If the amplitudes are different, it is assumed that the two signals are partially overlapped. Table 8.5 shows the results of TOA, TOD, and PW for the two signals. The results show that the estimated errors for TOA, TOD, and PW are within 13%. In this table the results from the 128-point window are used as the final results. In the last example, there are two signals; the one short signal is inside a long signal. The first signal has a TOA of 3 ms, a PW of 1 ms, and an S/N of 5 dB. The second signal has a TOA of 3.2 ms, a PW of 0.5 ms, and an S/N of 2 dB. The results are listed in Table 8.6, where four sets of windows report the description as “One signal is inside the other,” and three sets of windows report the description as “Two signals partially overlap.” The reason for the different descriptions is because of the different amplitudes of averaged eigenvalues. The calculations of averaged amplitude for the short windows and the long windows use different lengths of eigenvalues according to the points in the specific sections. Since in this example, the majority indicates one signal inside another, it is the selected results. The results show that the estimated errors for TOA, TOD, and PW are within 5% for the correct description. When the signal is short, the amplitude of average eigenvalues might not be accurate. This causes the TOD and PW calculations to be incorrect. Instead of determining that one signal is inside the other, it might estimate that the two signals partially overlap. In these cases, the TOA is correct, but the TOD and PW will be incorrect. In all the three examples the S/N of the two signals are different. If the two S/Ns are the same or closer, the eigenvalue method may not be able to separate the two signals correctly. Under this condition, the PW calculated could be wrong.

8.20  Conclusion This chapter uses simulations to study the properties of eigenvalues. In order to use the eigenvalues in EW receiver design in the future, the order to the correlation matrix is limited to 3 × 3. It appears that the eigenvalue is very sensitive to detecting

202

Eigenvalue and Related Operations

weak signals comparable with an FFT approach. Long data length and a higherorder matrix can improve the detection sensitivity. In order to avoid complicated computations, long correlation matrices can be obtained through summations of shorts ones. Several methods can be used to determine the number of input signals. It appears that using an eigenvalue against a threshold has slightly better results than the MDL method. This method is used to perform the time-domain detection. This approach can detect more than one signal. In this chapter, the input is limited to two real-time signals. If a higher-order eigenvalue approach such as a correlation matrix of over a 3 × 3 order can be implemented in hardware, it should be studied. Not only can the approach detect the number of signals, but it can also be extended to detect the frequencies of the input signals. Chapter 9 will study this problem.

References   [1] Reference Data for Radio Engineers, 5th ed., New York: Howard W. Sans & Co. Inc., 1956.   [2] Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Chapter 14, Norwood, MA: Artech House, 2001, p. 479.   [3] Mati, M., and T. Kailath, “Detection of Signals by Information Theoretic Criteria,” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP.33, No. 2, April 1985.

Chapter 9

Signals Close in Frequency Study and the MUSIC Method

9.1  Introduction The fast Fourier transform (FFT) is the most common algorithm implemented in digital electronic warfare (EW) receivers. Because of its assumption of periodicity and finite data length, it has an inherent limitation of spectral resolution. When multiple signals are concerned, the signal with a smaller energy cannot be resolved without some sophisticate algorithm added onto the FFT. For the case of two signals within one FFT frequency resolution bin, it is very difficult to identify these two signals. High-resolution methods were developed in the hope of compensating the deficiency of the FFT. The multiple signal classification (MUSIC) is one of the popular high-resolution methods. The approach is briefly discussed in Section 3.10. It requires the calculation of the correlation matrix from the input data and subsequently performs eigenvalue analysis, like most of the other high-resolution methods. There are two important parameters in the MUSIC method. One is the number of the signal, and the other is the order of the correlation matrix. If the number of signals is wrongly estimated, the spectral result shows either missing signals or false signals. The order is usually suggested to be the number of 1/3 to 1/2 of total data points [1]. For a modern EW receiver using a data block process with 256 data points as one block, it requires a correlation matrix of at least 80 orders. To compute the eigenvalues and eigenvectors, it exerts a heavy computation load, and hardware implementation may not be practical at the present time. Since the simple rule of 1/3 is not practical for an EW receiver application, the MUSIC method performance with a low order deserves a close look. The high-order MUSIC method such as an 80 × 80 matrix cannot be implemented in a real-time operation with the present technology. It might be possible to implement a moderate high-order approach in the future. Thus, the order to the correlation matrix will be studied. The MUSIC method can provide a high resolution with a short data length. This is very attractive for angle measurement. In an EW receiver the angle of arrival (AOA) is the most valuable parameter because the hostile radar cannot change it easily. The antenna elements for EW applications cannot be very large due to the cost and the available space for antenna installation limitations. With the low number of antennas such as 16 elements the MUSIC method might provide decent AOA information.

203

204

Signals Close in Frequency Study and the MUSIC Method

Another potential application is to find the multiple frequencies such as 2 in one FFT frequency bin. For this operation, the input signal is complex; theoretically, a 3 × 3 correlation matrix can be used to find two frequencies. To find two real input signals, a 5 × 5 correlation matrix is needed. These simple subjects will be discussed in this chapter.

9.2  Input Signal Frequency Separation and Signal-to-Noise Ratio (S/N) In an EW receiver, the number of signals occurring simultaneously is rather small, such as two or three. Usually the FFT-based receiver can detect both signals when their frequency separation is in the tens of megahertz. However, sometimes the requirement is to detect signals with a close frequency separation such as a few megahertz. It is important to generate a similar guide for a signal-to-noise ratio (S/N) and a frequency difference as in Section 3.9. Since there are numerous combinations of two input signals, a very simple case will be used as the guide. The two signals will be time coincident and have equal amplitude. The guide will be generated with a very high MUSIC method such as 80 × 80. In order to evaluate the minimum separation of 1 MHz, the frequency resolution must be less than 1 MHz. In the FFT approach the frequency resolution is determined by the length of the FFT, while in a MUSIC method, the frequency can be chosen independently of the data length. However, a fine frequency resolution requires a longer processing time. In this case the frequency of 0.25 MHz is chosen such that in 1 MHz there are four frequency outputs. The approach is to fix the frequency separation and increase the S/N from an arbitrarily chosen starting point of -15 dB. At each input S/N value the MUSIC method is used to find the two frequencies. The process repeats 10 times with the same S/N but a different noise input. The measured frequencies are evaluated against the input signals. If the results are correct for nine times or more, the S/N value is the desired one at that frequency separation. If the correct results are less than nine times, the input S/N is increased by 1 dB, and the process repeats again until the correct results are obtained. There are two methods to compare the outputs with the input conditions. One method is similar to the requirement of Section 3.11: to compare the difference frequency of the input and measured results. If the two results are within ±3 MHz, the correct result is considered to be obtained. The other method is to compare the two measured signals and the input signals separately. If both signals are within ±3 MHz, the correct result is considered to be obtained. The two methods obtain very similar results. To be consistent with Chapter 3, the results from the difference frequency comparison method are presented in Figure 9.1. The input signals are sampled at 2.5–6 GHz, and 256 points are used for the simulator. The equivalent window length is 100 ns. Since this figure is generated from a small number of testing such as 9 out of 10, the curve is not smooth, but the trend is shown. This small number is used to save the calculation time. In this figure the minimum frequency separation is 1 MHz and under this condition, the required S/N is rather high over 50 dB.

9.3  Study of the Order of the MUSIC Method for One Signal

205

Figure 9.1  Frequency separations versus the required S/N to be identified by an 80-order MUSIC method.

9.3  Study of the Order of the MUSIC Method for One Signal The purpose of this section is to compare the MUSIC and the fast Fourier transform (FFT) methods. The FFT is the conventional way used in designing electronic warfare (EW) receivers. The properties of FFT are well known. Whether the MUSIC method can have a similar performance as the FFT operation remains to be seen. Since the MUSIC method is computational intensive in comparison with the FFT operation, finding the lowest possible order MUSIC operation is intended. The FFT operation discussed in the following sections uses the Blackman window in order to build a receiver. Simulations will be used for the study. As usual the sampling frequency is 2.56 GHz and 256 data points are used in the simulation. The data used in the study is real because a real input signal rather than a complex signal is anticipated to be used in receiver designs. It should be noted that from the limited data in the following discussion that an accurate comparison is difficult. Only a visual display will be used for a rough comparison. For one signal, the input frequency changes from 100 to 1,180 MHz in 1-MHz steps. The input bandwidth is wider than the desired 1,000 MHz. Before studying the order of the MUSIC method, the performance of a relatively low-order performance will be shown. The frequency error is plotted in Figure 9.2 with only one input signal. The S/N = 16 dB, which can be considered relatively high. The MUSIC method is 5 orders with lags of 1:5, and the frequency resolution is arbitrarily chosen as 2.5 MHz. Figure 9.2(a) shows the FFT results, which have 128 output frequency bins with a 10-MHz resolution. Figure 9.2(b) shows the MUSIC method. The frequency error generated by the MUSIC method is about half that generated by the FFT method, which is why the MUSIC method is rather attractive.

206

Signals Close in Frequency Study and the MUSIC Method

Figure 9.2  Error frequency versus input frequency for S/N= 16 dB, lags 1:5: (a) FFT outputs and (b) MUSIC outputs.

In Section 6.10, the required S/N equals -4.6 dB for a 256-point FFT with a Blackman window to achieve the desired detection condition. Thus, the S/N = -4 dB is selected to compare the FFT output and MUSIC method. For the lags 1:12, the results are shown in Figure 9.3. In order to compare the two plots, the frequency resolution for the MUSIC method is decreased from 2.5 MHz in the previous example to 10 MHz. There is no clear-cut way to compare these two results. Sometimes higher or lower lag numbers than 12 also produce similar results. Figure 9.4(a) shows the outputs from an 8-order MUSIC method, and Figure 9.4(b) shows the outputs of 26 orders. Although the frequency resolution is improved from the 8 ordered results to 12 orders, the frequency resolution does not change much from 12 to 26 orders, which is limited by the 10-MHz frequency resolution. Since the noise is random, each run will have slightly different results. Another way to compare the two methods is to reduce the S/N to -7 dB. Under this condition, sometimes the both the FFT and the MUSIC methods have large frequency errors. When the MUSIC method order is low, the occurrence of a large frequency error is high. From these two very coarse comparison methods as shown in Figure 9.5, it is decided that the 12-order MUSIC method is comparable to the FFT operation, although the FFT outputs appear to have better frequency readings.

9.4  Study of Order of the MUSIC Method for Two Signals We will compare the FFT operation and the MUSIC method for two signals. Since there are many possible arrangements for the two signals, only a very simple case

9.4  Study of Order of the MUSIC Method for Two Signals

207

Figure 9.3  Error frequency from the FFT and 12-order MUSIC methods: (a) FFT outputs and (b) 12-order MUSIC outputs.

Figure 9.4  Error frequency from 8- and 26-order MUSIC methods: (a) 8-order outputs and (b) 26-order outputs.

208

Signals Close in Frequency Study and the MUSIC Method

Figure 9.5  Frequency error for S/N = -7 dB: (a) FFT outputs and (b) MUSIC outputs.

will be evaluated. The two signals have the same amplitude of S/N = 0 dB with a frequency separation of 100 MHz. The input signal with the higher frequency changes from 241 to 1,140 MHz in 1-MHz steps to cover 900 MHz. The MUSIC order is kept at 12. The results are shown in Figure 9.6. Figure 9.6(a) shows the FFT results and Figure 9.6(b) shows the MUSIC result. As expected, since the two frequencies are separated far apart, both frequencies are measured correctly by the FFT approach. The outputs from the MUSIC method have larger frequency errors with order = 12. Sometimes the MUSIC method outputs have large frequency errors, but the FFT outputs are robotic even at S/N = 0 dB. When the frequency separation decreases to 50 MHz with S/N = 0 dB, the MUSIC method with 12 orders can no longer produce decent results and there are large frequency errors. The order needs to increase to 20 for the MUSIC method to process signals separated by 50 MHz. When the input frequency difference is reduced to 25 MHz, sometimes the FFT cannot separate them correctly and the outputs have large frequency errors. For a 256-point FFT and a 2.56-GHz sampling frequency, the frequency resolution is 10 MHz, and usually to separate two signals it requires more than two frequency bins. Under this frequency separation, the MUSIC method order needs to be increased to 44 to produce good frequency results. The results are shown in Figure 9.7. In this figure, the frequency separation is 25 MHz and both S/N = 0 dB. Figure 9.7(a, b) shows the error frequency for the FFT operation and there are many large errors. Figure 9.7(c, d) shows the error frequency of a 44-order MUSIC method and the results are reasonably well.

9.4  Study of Order of the MUSIC Method for Two Signals

209

Figure 9.6  Frequency error plots for two input signals of same amplitude. (a, b) FFT outputs, (c, d) music outputs.

Figure 9.7  Comparison of a 256-point FFT and a 44-order MUSIC method with both S/N = 0 dB: (a) FFT outputs for signal 1, (b) FFT outputs for signal 2, (c) MUSIC method outputs for signal 1, and (d) MUSIC method outputs for signal 2.

210

Signals Close in Frequency Study and the MUSIC Method

From the above trial-and-error study, it appears that the FFT is a very powerful method. It can obtain good frequency readings on single and two signals. In order to obtain comparable results for two signals, the MUSIC method must have high orders such as 44. If it is desirable to separate two signals close in frequency so that the FFT approach cannot distinguish them, a MUSIC method with an order higher than 44 is needed. In Chapter 3 a MUSIC method with 80 orders is used to separate signals close in frequency. It is anticipated that such a high order may not be implemented in hardware in the near future.

9.5  U  sing the FFT Approach to Read Signals with a Close   Frequency Separation It is well known that the FFT approach cannot separate two signals close in frequency. For a 10-MHz frequency resolution the frequency separation should be more than 20 MHz. If the frequency domain does not show multiple peaks, there is no obvious way to detect two signals. In order to separate two peaks, there should be a valley between the two peaks. The valley must be deep enough so that the two peaks can be detected faithfully. If there is apriority information on the input signals such as two input signals, the problem can be simplified tremendously. For example, if it is known that there are two input signals, one does not need to find separate peaks because one peak may contain two signals. Under this condition one only needs to find the highest two peak in the frequency-domain outputs. This approach is extremely simple and decent results can be obtained. In the following two examples, it is assumed that the input contains two signals. To simplify the discussion, two signals of the same amplitude will be used. Figure 9.8 shows that the two signals are separated by 5 MHz and the S/N = 20 dB. The higher input frequency changes from 146 to 1,140 MHz in 1-MHz steps and the second signal is from 141 to 1,135 MHz; thus, the entire frequency of 141 to 1,140 MHz is covered. The worst frequency error is about 7 MHz for the first signal and -7 MHz for the second signal. Figure 9.9 shows that the two signals are separated by 15 MHz and the S/N = 0 dB. The higher input frequency changes from 156 to 1,140 MHz. The worst frequency error is about ±5 MHz, which is limited by the FFT resolution of 10 MHz. From this simple illustration it is further demonstrated that the FFT is a very powerful tool. If more information can be provided to the input signal conditions such as two signals, the frequencies can be found through a relatively easy operation.

9.6  D  etection of the Existence of Two Signals Close in Frequency from FFT Outputs In the previous section, it is assumed that the number of signals is known. In this section, it is intended to find out whether there are multiple signals in the main lobe of the FFT. A simple approach is to measure the width of the main FFT output lobe. If the main lobe is wide, it can be considered to contain two signals. However, the

9.6 Detection of the Existence of Two Signals Close in Frequency from FFT Outputs

211

Figure 9.8  Frequency error for two signals separated by 5 MHz, S/N = 20 dB: (a) Signal 1 and (b) Signal 2.

Figure 9.9  Frequency error for two signals separated by 15 MHz, S/N = 0 dB: (a) Signal 1 and (b) Signal 2.

212

Signals Close in Frequency Study and the MUSIC Method

main lobe is a function of signal amplitude. If the input signal is strong, the main lobe will be wide. A threshold must be built based on the maximum amplitude of the FFT outputs. An initial threshold is used to measure how many frequency bins crossing this threshold. For convenience, this threshold is 36, obtained from the Rayleigh distribution and listed in Table 6.3. The threshold is obtained from the following procedure. Increase the S/N value from 0 to 45 dB in the 1-dB step because in the actual receiver simulation the maximum S/N is about 45 dB. Since the S/N of the input signal is unknown, the only way to obtain the amplitude information is from the peak of the output. Thus, the peak value will be used to determine threshold rather than the input S/N. At each step the input signal frequency is randomly selected from 141 to 1,140 MHz with s random initial phase from 0 to 2p for 1,000 runs. At each run the maximum FFT output will be recorded and there are 1,000 amplitudes. The average of these amplitudes will be expressed in decibels (20 times the log of the amplitude). At each run the number of threshold crossing is registered. The maximum of numbers of crossing is recorded and the results are shown in Figure 9.10. In this figure, the x-axis is the measured amplitude rather than the input S/N. Let us refer to the threshold obtained from Figure 9.10 as threshold 1 in order to differentiate with the original threshold. In Figure 9.10 the number of bin crossings is a constant over a certain range of amplitude. For example, for a measured amplitude from approximately 48 to 65 dB, the number of crossings equals 5. To set threshold 1 the number of crossings is also 5. When the amplitude of the output is above this value, detection is counted. Threshold 1 has the values listed in the first two rows of Table 9.1, which follows

Figure 9.10  Maximum number of frequency bins crossing the threshold versus input S/N.

9.6 Detection of the Existence of Two Signals Close in Frequency from FFT Outputs

213

Table 9.1  Threshold 1 Values 20 log(Amp_max) Threshold 1 20 log(Amp_max) Modified Threshold 1

=79 7

the pattern in Figure 9.10. The threshold is one unit above the number of crossings in the figure. Once threshold is set, the next step is to test false detection. The procedure is to set a certain input S/N value and change the input frequency randomly from 141 to 1,140 MHz with a random initial phase from 0 to 2p for 1,000 times. At each time the number of bins crossing threshold 1 is recorded. If threshold 1 is crossed, it means two signals are detected. Since the input contains only one signal, detecting two signals is considered a false alarm. The input S/N is changed from 0 to 45 dB. The results are shown in Figure 9.11(a). In some S/N regions there is one false output per 1,000 runs. The region of false detection usually occurs at the right end of the step functions shown in Figure 9.10, or the upper limit in Table 9.1. Threshold 1 can be modified as shown in the third and fourth rows of Table 9.1. Using these new threshold 1 values, the false detection is usually zero as shown in Figure 9.11(b). The next test is the probability of detection. The input of two signals has many different combinations such frequency separation and amplitude difference. To limit this study, the two signals are assumed to have the same amplitude at S/N = 15 dB. This S/N value is selected based on Figure 9.1. At this S/N value two signals separated by 5 MHz or more can be detected. Each data point is generated by 1,000 runs with a random input frequency but a fixed difference frequency. The results are

Figure 9.11  False detection for 1,000 runs: (a) original threshold 1 and (b) modified threshold 1.

214

Signals Close in Frequency Study and the MUSIC Method

shown in Figure 9.12, where when the frequency separation is 18 MHz or beyond, the probability of detection is over 90%. Figure 9.12 shows that when the difference frequency is less than 18 MHz the probability of detection decreases rapidly. When the frequency separation is less than 15 MHz, only less than 50% can be detected. When the input S/N changes from 5 to 45 dB, the frequency separation that can be detected varies from 17 to 22 MHz. Because the threshold is measured amplitude dependent, the frequency separation does not vary much with an input S/N. Another approach can be used to detect the FFT outputs for two signals close together. If two signals are separated, sometimes they generate two peaks and a valley between them. This approach is to detect the valley between two peaks. The first step is to find all the FFT amplitude outputs that cross the initial threshold of 36. Usually the total number of bins is less than 10. The second step is to find the amplitude difference between adjacent frequency bins and convert them into ±1. If the difference is positive, a 1 is assigned; otherwise, it is -1. The output pattern will become [1 1 1 -1 -1]. If there is a peak, the patterns of [1 -1 1] or [1 -1 -1 1] will exist. For the pattern [1 -1 1] there is one point between two peaks, and for the pattern [1 -1 -1 1] there are two points in the valley. The last step is to detect these two patters. One simple approach is to use the correlation method. If either of these patterns can be detected, it is considered that a second signal is detected. The results are shown in Figure 9.13. The input conditions are similar to those used for generating Figure 9.12. Both signals have S/N = 15 dB. At this S/N value two signals separated by 10 MHz or more are detected. Each data point is generated

Figure 9.12  Probability of detection versus the difference frequency at S/N = 15 dB using the threshold crossing method.

9.7 Detection of the Existence of Two Signals Close in Frequency from Eigenvalues

215

Figure 9.13  Probability of detection versus a difference frequency at S/N = 15 dB using the peak valley detection method.

by 1,000 runs with random input frequency but fixed difference frequency. For a 90% detection, the frequency separation is about 23 MHz, which is inferior to the previous method. These studies show that although using the FFT outputs to detect multiple signals is very simple, it is not effective when two signals are close together. This conclusion will be used as a guide for the study in the next section.

9.7  D  etection of the Existence of Two Signals Close in Frequency   from Eigenvalues In this section we will use the eigenvalue method to detect a signal closer than about 19 MHz. In Section 8.12 it was illustrated that the frequency separation and the lag length are related. In order to cover a wide frequency separation range, many lags are required. This problem can be limited to two frequencies separated by about less than 19 MHz. If the lower frequency separation limit is arbitrarily chosen as 5 MHz and the upper limit is 19, the entire frequency separation range is about from 5 to 19 MHz. In this frequency range the eigenvalue will be used to find two signals. We will use the lowest-order correlation matrix and only one set of lags to reduce the calculation complexity. Since the input signal is real, the lowest order is 3 × 3. The largest and the third eigenvalues are used to detect the signals. It was shown in Section 8.12 that the frequency separation and the lag length are related for the 2 × 2 matrix. The average frequency from 5 to 19 MHz is about

216

Signals Close in Frequency Study and the MUSIC Method

12 MHz. It is desirable to make the third eigenvalue a peak at this frequency. From (8.11), the corresponding longest lag is about 106.7 (2,560/24). The three lags are 1, 3, and 107 and the selection is based on the peaking of the third eigenvalue between 5 and 19 MHz. The second lag length is 3 and it is not very critical. The first (largest) and the third (smallest) eigenvalues are shown in Figure 9.14. Each point is obtained from the average of 1,000 runs and the input frequency is randomly selected between 141 and 1,140 MHz with input S/N of 30 dB. This high S/N value is used to illustrate the trend of the eigenvalue variation. The large eigenvalue shown in Figure 9.14(a) is not critical because it always crosses the threshold. Figure 9.14(b) shows the third eigenvalue and there is a peak between 5 and 19 MHz, which is the desired result. The third eigenvalue is higher at 19 MHz than at 5 MHz. Increasing the length of the third lag will decrease the eigenvalue at 19 MHz, but it does necessarily increase the eigenvalue at 5 MHz. This operation can fine-tune the third eigenvalue versus the frequency between 5 and 19 MHz. Once the lags are determined, a threshold will be decided through a noise input and the largest eigenvalue. The threshold is obtained through the Gaussian approximation as discussed in Section 8.5. The threshold of 1.05 is obtained from 10,000 runs with a probability of false alarm of 10-7. The next step is to find the probability of detection. Because the input signal conditions are numerous, a sort of random test will be performed. The two signals are assumed of the same S/N, or the two signals have equal amplitudes. Both the input frequencies are randomly selected from 141 to 1,140 MHz, but the difference frequency is randomly selected between 5 and 19 MHz. The S/N is changed from 1

Figure 9.14  Eigenvalue versus frequency separation for S/N = 30 dB with lags 1, 3, and 107: (a) largest eigenvalue and (b) smallest eigenvalue.

9.7 Detection of the Existence of Two Signals Close in Frequency from Eigenvalues

217

to 20 dB in a 1-dB step. At each S/N level 1,000 runs will be performed and tested against the threshold. These 1,000 runs have different input frequencies and different frequency separation. The number of signals crossing the threshold will be counted and the results are shown in Figure 9.15. It appears that when the input S/N is above 14 dB both signals can be detected over 90%. However, from Figure 9.15 the probability of detection cannot be measured against the frequency separations. Another test is to find the probability of detection versus frequency separation. The S/N = 15 dB is selected and the input frequency is randomly selected between 141 MHz and 1,140 MHz, but the frequency separation is kept constant. The results are shown in Figure 9.16. When the frequency separation is greater than 8 MHz, the probability of detection is over 90%. When the frequency separation increases, the probability of detection decreases slightly and this trend matches the third eigenvalue amplitude change shown in Figure 9.14. Let us summarize the previous two sections and this section for detecting two signals close in frequency. The detection can be achieved in three steps: 1. Test the FFT amplitude and the number of frequencies crossing a certain threshold. There are actually two approaches in this step. Either one can be used to find the existence of two input signals. The threshold method delivers a slightly better result than the peak valley detection method. The selection of the method can be determined by the complexity of the hardware. 2. Use one 3 × 3 correlation matrix to calculate three eigenvalues. Compare the smallest eigenvalue with a predetermined threshold to decide whether there is a second signal.

Figure 9.15  Number of signals detected from 1,000 input signals.

218

Signals Close in Frequency Study and the MUSIC Method

Figure 9.16  Probability of detection versus difference frequency at S/N = 15 dB.

3. Use the FFT outputs to find the first and the second signals. The highest output is one frequency and the second highest one is another one. The second highest frequency bin can be adjacent to the first one. This approach will be further discussed in the next section. If step 1 is positive, step 3 is used to find the two frequencies and step 2 will be skipped. If the step 1 result is negative, step 2 will be applied. If step 2 is positive, step 3 will be used to find the two frequencies. If both steps 1 and 2 are negative, there is only one signal in the FFT outputs.

9.8  Frequency Identification with Close Frequency Separation In step 3 in the previous section, the second peak is used as the second signal. In Section 9.5 the second frequency is obtained in a similar manner. This is an intuitive approach but in reality the results produced are not totally correct. Several ways are used to evaluate this approach. The first test is to use a Blackman window and perform FFT on 256 points of data. In this test the input signals do not contain noise. The frequency of the first signal is randomly selected and the second signal is at a fixed frequency separation (say, 5 MHz) below the first one. The amplitude of the second signal is 10 dB below the first one. These input conditions are obtained from empirical results. Both frequencies are in the range from 141 to 1,140 MHz. Perform the FFT on the input data to obtain 128 frequency outputs. The highest two peaks are used to represent

9.8  Frequency Identification with Close Frequency Separation

219

the two input frequencies. The problem is that the second-highest peak may not be at the correct side of the strong signal. Since the second signal is 10 dB down and at 5 MHz lower than the strong one it is anticipated that the second peak is at a lower frequency of the first one. This prediction is not always correct. The second peak may appear at a higher frequency than the first one. It appears that the phase relation between the two signals plays an important role. The histograms of the error frequencies are shown in Figure 9.17. In this simulation, the frequency is read from the index of the FFT output and the frequency resolution. Figure 9.17(a) shows the frequency error of the first signal. This is the expected result, which is about ±6 MHz because the frequency resolution is 10 MHz. This indicates that the largest peak represents the strong signal. Figure 9.17 (b) shows the frequency error of the second signal. A numerical example will be used to illustrate this figure. If the input frequency of the strong signal is at 600 MHz, the weak one will be at 595 MHz. Let us assume that the strong signal is encoded correctly at 600 MHz. If the second peak is on the correct side of the strong one, the frequency reading should be 10 MHz below the strong one at 590 MHz. The error frequency will be 5 MHz (595 - 590). If the second peak is one wrong side of the strong one, the second signal will be read as 610 MHz. The corresponding error frequency will be -15 MHz (595 - 610). The results in Figure 9.17(b) show that the errors are divided into two regions. The right region is the correct one and the left region in the wrong side. From this plot, it is interesting to note that the second peak on the correct side only slightly outnumbered the ones on the wrong side.

Figure 9.17  Error frequency of two signals separated by 15 MHz and 10 dB: (a) first signal and (b) second signal.

220

Signals Close in Frequency Study and the MUSIC Method

In the above approach, if the strong signal is close to the boundary between two bins, both signals will be detected from the main bin. The error frequency of the first signal will be close to 5 MHz. There are several possibilities affecting the second frequency error and reflect in Figure 9.17(b). There is another way to find the second peak. The approach is to ignore the two neighboring bins of the maximum then find the second peak. In other words, the first and the second peaks are 20 MHz apart. With this modification, when the second signal is about 30 MHz and 10 dB down the correct frequency can be read correctly most of the time.

9.9  Conventional MUSIC Method In the following sections, we will use low-order MUSIC method to separate two signals close in frequency to save the computation load. The MUSIC method is considered a high spectral resolution method, which can detect signals with very close frequency separation. This property is used in Chapter 3 to find the instantaneous dynamic range. Before the discussion of a low-order MUSIC method, the outputs of a high-order MUSIC method will be discussed. In Sections 9.3 and 9.4, the order of the MUSIC method is studied against the FFT output by comparing sensitivity. Here the order is tested against two signals of close frequency separation. A simple case is used to examine this property. The two

Figure 9.18  Frequency plot from an 80-order MUSIC method for S/N = 20 dB, input frequency = 600 MHz, and frequency separation 5 MHz: (a) entire frequency range and (b) near peak value.

9.9  Conventional MUSIC Method

221

signals are of the same amplitude at S/N = 20 dB with input frequencies arbitrarily selected at 600 MHz and random initial phases but the frequency separation is 5 MHz. This selection is based on Figure 9.1 so that at these conditions the two signals can be identified. For the 80-order MUSIC method the results are shown in Figure 9.18. In this figure the frequency resolution is 0.25 MHz. Figure 9.18(a) shows the entire frequency range and the amplitude is in logarithmic scale. The frequency spectrum is rather flat except near the peak value. Figure 9.18(b) shows the results near the peak of Figure 9.18(a). Two peaks can be clearly shown. Figure 9.19 shows the results of a 44-order MUSIC method, which is determined from Section 9.4 with identical input conditions as Figure 9.18. The input data are identical to that producing Figure 9.18 and the outputs have only one peak. Most of the time the results show two distinct peaks but in Figure 9.19 there is only one peak. The purpose of this selection of displays is to show that the 44 order is not high enough to produce two peaks faithfully. It should be noted that under certain input conditions, even the 80-order MUSIC method cannot produce two clearly defined peaks. This condition is assumed that the two signals are not separable. From these two MUSIC methods it is clearly shown that in order to separate two signals close in frequency, the order must be very high. The following example will show a low-order MUSIC method that can detect signals with a close frequency separation.

Figure 9.19  Frequency plot from a 44-order MUSIC method for S/N = 20 dB and frequency separation 5 MHz: (a) entire frequency range and (b) near peak value.

222

Signals Close in Frequency Study and the MUSIC Method

9.10  Low-Order MUSIC Method This subject will be discussed from two types of input signals: real and complex. The real signal approach can be used to process digitized signals after the analogto-digital converter (ADC) because it is difficult to obtain well-balanced in-phase and quadrature phase (IQ) outputs from hardware over a wide frequency range. The complex data can be obtained from the FFT outputs. The FFT outputs can be either from one ADC output or multiple ADC outputs. From one ADC the consecutive FFT outputs can be further analyzed to find a fine frequency resolution and signals close in frequency through the MUSIC method. An antenna array can produce many parallel ADC outputs. If the MUSIC method can be applied to the FFT outputs obtained from all antenna elements, the angle of arrival (AOA) can be measured. To separate two real signals, the lowest-order MUSIC method is 5 because one signal will affect two eigenvalues. Two signals will generate four signal eigenvalues. To find the frequency through the MUSIC method, at least one noise eigenvalue is needed. For the complex input, one signal only affects one eigenvalue; therefore, the minimum order MUSIC method is 3. In Section 9.9 it was illustrated that a low-order MUSIC method has difficulty separating two signals close in frequency if the lags are selected in a continuous manner such as [1:44] for the 44 order matrix. In this section the low-order MUSIC method has nonuniform distributed lags. The correlation matrix can be obtained through the following relation. Rij =



mmax

∑

xm + p xm + q

(9.1)

m=0

The relationship among i (row index), j (column index), and (p,q) is expressed in Table 9.2. The lag selection is empirical and by no means the optimum case. The scanning signal vector must match the selected lags and the results are t = [t0 , t1, t2 , t3 , t 4 ] = [0,1, 43, 64, 92]ts



(9.2)

To obtain the frequency output, (3.5) and (3.6) are rewritten here as

s

[e

j 2 π ft

]

[1 e 2π fts e 2 π f 43ts e 2π f 64ts e 2 π f 92ts ]

where t is a vector as shown in (9.3). Table 9.2  Correlation Matrix with Lags 1, 2, 44, 65, and 93 j 0 1 2 3 4 .

i (p,q)

0

1

2

3

4

(0,0) (1,0) (43,0) (64,0) (92,0)

(0,1) (1,1) (43,1) (64,1) (92,1)

(0,43) (1,43) (43,43) (64,43) (92,43)

(0,64) (1,64) (43,64) (64,64) (92,64)

(0,92) (1,92) (43,92) (64,92) (92,92)

L. Liou, an engineer at AFRL, worked with the author on this subject.

(9.3)

9.11  Results from the Low-Order MUSIC Method

223

The frequency Pmus can be obtained as

Pmus (f ) =

1 sVnVnH s H

(9.4)

where the superscript H is the Hermitian operation. For detecting two complex signals, the minimum order of the MUSIC method is 3, which is even simpler than the order 5 MUSIC method. Detecting the complex two signals will not be repeated again.

9.11  Results from the Low-Order MUSIC Method The results of MUSIC method will be presented here. The first example is to show the frequency outputs. In order to compare with the results in Figures 9.18 and 9.19, identical data are used. The results are shown in Figure 9.20. Figure 9.20(a) shows the entire frequency spectrum. There are many high peaks. If this method is used in receiver design, it might be difficult to differentiate the correct peaks. In this special case the highest two peaks are selected. Figure 9.20(b) shows the close in plot at the two peaks. The results provide the correct answer. It is interesting to note that when the MUSIC method with a matrix of 44 orders cannot separate two signals close in frequency, the low-order MUSIC method with nonuniformed lags can separate them.

Figure 9.20  Frequency plot from a 5-order MUSIC method for S/N = 20 dB frequency separation 5 MHz: (a) entire frequency range and (b) near peak value.

224

Signals Close in Frequency Study and the MUSIC Method

Figure 9.21  Error frequency versus input frequency for a 5-order MUSIC method with S/N = 20 dB and a frequency separation of 9 MHz.

In order to obtain a better idea on the low order MUSIC method, the input frequency is changed in a 1-MHz step from 141 to 1,140 MHz at S/N = 20 dB and a frequency separation of 5 MHz. Although there are many small frequency errors, there are also many large frequency errors. When the frequency separation is increased to 9 MHz, the results are shown in Figure 9.21, where the frequency resolution is 1 MHz because a 0.25-MHz resolution takes a long calculation time. These results can be considered rather well; the largest frequency error is 3 MHz. Most the errors are 0 and a few 1- and 2-MHz errors. Because the frequency resolution is 1 MHz and the input frequency also increases by 1 MHz, most of the time the frequency error is exactly zero. These simple illustrations indicate that a low-order MUSIC method may have some EW receiver applications. Further study is needed in this area.

9.12  Frequency Selection for MUSIC Method In the MUSIC method forming the correlation matrix will be calculation intensive. If the matrix is 3 × 3, the eigenvalues will be solved analytically. For matrices larger than 3 × 3, a numerical method might be used to solve the eigenvalues. Once the . M. Emmert, a professor in the Electrical Engineering Department at Wright State University, used the monobit receiver design.

9.13  Conclusion

225

eigenvalues are found, the eigenvector will be calculated. All these operations can create heavy calculation loads. In addition, the frequency resolution of the MUSIC method does not depend on the data length, which is why high frequency resolution can be obtained. From a simulation operation through MATLAB, when the frequency resolution is doubled, the calculation time is more than doubled; in a special case the calculation time is increased by a factor of about 2.3. One possible approach to search the frequency peak is through some kind of binary search. A coarse search is performed first to the high values and concentrates on the fine frequency search on the high amplitude region. This method has been implemented in a field programmable gate array (FPGA) in the EW receiver designs. This method might be applicable to a high-order MUSIC method such as in Figure 9.18. However, the binary search definitely is not applicable to the low-order MUSIC method such as in Figure 9.20. There many local peaks and the binary search will generally converge on a wrong value. It is possible that the MUSIC method can be used in conjunction with the FFT operation. If two frequencies are detected by the FFT approach, the same input data can be processed again through the low-order MUSIC method. Under this condition, the frequency range searched can be rather narrow and close to the main lobe of the FFT outputs. The many local peaks produced through the low-order MUSIC method can be ignored, maybe outside of the searching frequency range.

9.13  Conclusion From the simple studies it appears that using the MUSIC method to match the performance of the FFT operation requires a rather high matrix order. If the applications are limited to a low order matrix, it might be used to augment the FFT operation. If the wrong number of signals is assumed, the MUSIC might produce erroneous frequency reading or false output. Since the eigenvalue value can be used to determine the number of signals rather accurately, the MUSIC method might be used to find the frequency of the signals. All the study is limited to two input signals close in frequency. If the number of signals is increased to three, the order of the MUSIC method will increase and complicate the calculations. Since the lower MUSIC method generates many peaks, it might be difficult to process more than two signals. When the FFT operation only shows one peak, one can test the number of frequency bins crossing a certain threshold, or in other words to test the width of the main lobe. From the main lobe width one can determine multiple signals in the lobe. Even if the number of signals can be determined the frequency the weak signal might be difficult to determine. The second peak from the FFT operation may not represent the correct frequency of the weak signal when the two frequencies are very close. The second peak appears as a function of relative phase between the signals. Although this chapter suggests several methods to several methods to detect the existence of simultaneous signals, the studies are preliminary. The study concentrates on detecting two signals close in frequency but short in data length such as 100 ns. This appears to be a challenge problem.

226

Signals Close in Frequency Study and the MUSIC Method

Usually, it requires to separate two signals close in frequency when the input signals are relatively long. Intuitively, one can argue that long FFT operations can be used to find signals with close frequencies. However, in a receiver design, the frequency resolution is often based on the minimum pulse width and the FFT length is short. It might not be practical to apply a long FFT in a continuous mode. A long FFT may miss short pulses because the signal is spread in many parallel outputs. Long FFT also degrades the time resolution. If the existence of long signals can be detected through some simple algorithm, applying a long FFT operation can improve the performance. This is one reason to apply the time-domain detection in receiver designs.

Reference [1]

Ulrych, T. J., and R. W. Clayton, “Time Series Modeling and Maximum Entropy,” Phys. Earth Planetary Interiors, Vol. 12, August 1976, pp. 188–200.

Chap te r 10

Digital Instantaneous Frequency Measurement (IFM) Receiver

10.1  Introduction This chapter introduces a new approach to building a digital IFM receiver. A conventional analog IFM receiver can cover a very wide input bandwidth such as 16 GHz (from 2 to 18 GHz). The IFM receiver can report a fine frequency on a short pulse and the frequency resolution is usually much better than a fast Fourier transform (FFT)–based receiver. The digital receiver will use a 1-bit ADC, which can provide a very high sampling rate such as 10 GHz. The high sampling rate means a wide bandwidth of several gigahertz. However, in order to keep the discussion consistent with other examples in this book, the sampling frequency is still assumed to be 2.56 GHz. An IFM receiver has one major deficiency: the receiver can report only one frequency at a time. Even worse, sometimes the frequency reported could be wrong. Two possibilities cause this problem. One possibility is that when there is no input signal the IFM receiver reports frequency from noise measurement. The other possibility is simultaneous signals. When two signals of approximate the same amplitude arrive at the receiver, the frequency reported may be unrelated to neither input frequency. To solve the first problem, usually a crystal video receiver works with the IFM receiver. The output of the crystal receiver will control the frequency reading of the IFM receiver. When there is no output from the crystal receiver, the IFM will not report a frequency from the noise input. However, the simultaneous signals problem is a very difficult one to solve in an IFM receiver. Although an IFM receiver has this major deficiency, it is still popular in an EW system because of its wide input band, accurate frequency reading, and simple structure. A 1-bit ADC can provide enough information to make an IFM receiver. The approaches discussed in Chapter 7 can be used to detect the existence of a signal and can be used as a trigger circuit. Whenever a digital IFM receiver reports a frequency, the frequency can be confirmed as either correct or wrong. The input frequency can be confirmed by some threshold crossing method. This same procedure can also be used to detect the existence of a signal and avoid noise triggering. This approach can be considered as a different detection circuit from the approach in Chapter 7. In other words, if properly designed, an IFM receiver can be a standalone receiver without a crystal video receiver used for triggering. In addition, the structure of the receiver can be very simple and implemented relatively easily in a field programmable gate array (FPGA). 227

228

Digital Instantaneous Frequency Measurement (IFM) Receiver

Two types of inputs will be discussed in this chapter. One is that the input signal is complex and the other is real. When the input is complex, the effect of the imbalance between the in-phase (I) and quadrature (Q) will be discussed. When a real signal is used as input, the signal will be converted into a complex signal through digital signal processing. Two approaches will be used to convert the input to complex: the Hilbert transform and the special sampling method. Both methods are presented in Chapter 5.

10.2  Basic Concept of an Analog IFM Receiver In order to understand a digital IFM receiver, the basic concept of an analog IFM receiver will be discussed because the digital approach is based on the analog concept. The basic unit in an IFM receiver is the correlator. There are many different ways to build a correlator. Figure 10.1 shows one of the arrangements. In this figure there is one delay line with delay time t, three power dividers, two hybrids (one 90° and one 180°), four detectors, and two differential amplifiers [1]. The video detectors are nonlinear devices, which perform the multiplication function. The outputs are Asin(2pfit) and Acos(2p fit). Since the delay time t is known, the input frequency fi can be calculated as  A sin(2π fi τ )  θ = tan −1   = 2π fi τ  A cos(2π fi τ) 



fi =

θ 2πτ

(10.1)

where fi is the input frequency. In building an IFM receiver, multiple correlators are needed with different delay line lengths. There are two common approaches to select the delay lines for a 2-GHz bandwidth analog IFM receiver. One is t, 2t, 4t, 8t, 16t, 32t, 64t with seven correlators. The other is t, 4t, 16t, 64t with four correlators. The decoding circuitry following the seven correlators is simpler than that of the four correlators;

Figure 10.1  Analog correlator.

10.3  Basic Digital IFM Receiver Hardware and Concept

229

therefore, both approaches are used. For the 2-GHz IFM receiver the t value is usually chosen as 1/2,560 MHz or approximately 0.391 ns. The longest delay line is used to generate the desired frequency resolution and the short delay lines are used to resolve ambiguity.

10.3  Basic Digital IFM Receiver Hardware and Concept The basic concept of digital IFM receiver is similar to the analog one. However, the hardware design and signal processing are quite different. Figure 10.2 shows the hardware of the digital IFM receiver. In this figure, there is one 90° hybrid, two high-speed 1-bit ADCs, and one digital signal processor. Because an IFM receiver can process only one signal, an ADC with only one bit will be satisfactory. Mathematically, the outputs from the 90° hybrid can be written as xn = cos(2π fi tn ) yn = sin(2π fi tn )

(10.2)

zn = xn + jyn = exp(jπ fi tn )



where fi is the input frequency and n is the discrete time tn. In these equations the amplitude of the signal is ignored and the digitization effect is not included either. Noise is not added to the input signal because it will affect the discussion on the IFM receiver approach. The autocorrelation value can be obtained from z as zn = exp(j 2π fi tn )

zn − m = exp[ j 2π fi (tn − mt s )]

R(m) = ∑ zn z'n − m = ∑ exp(j 2π fi tn )exp[− j 2 π fi (tn − mt s )] = exp(j 2π fi mt s ) n



where

(10.3)

n

t s = t n +1 − t n



In this equation, ts is the sampling interval related to the sampling frequency fs through ts = 1/fs and m is the correlation lag. The correlation R(m) is independent

Figure 10.2  Hardware for a digital IFM receiver.

230

Digital Instantaneous Frequency Measurement (IFM) Receiver

of time tn, but only is a function of mts, which can be considered as the delay time t in (10.1). From R(m) the input frequency fi can be calculated. In an analog IFM receiver many delay lines are needed. In a digital IFM receiver, different delay times are also required. The short delay lines resolve the ambiguity and the longest delay line provides the frequency resolution. These delay times can be obtained from the same output data. To change the delay time, one only needs to change the m value. Thus, from a hardware point of view, a digital IFM receiver can be much simpler.

10.4  1-Bit ADC Effect Since the ADC has only 1 bit, one cannot use it to measure the angle very accurately. As a matter of fact, the angle resolution is only 90° as shown in Figure 10.3. In this figure, the two real and imaginary outputs from the two ADCs are ±1 and ±j, respectively. The angle difference has five values: 0, p/2, -p/2, p, and –p. From this coarse angle resolution, it is difficult to obtain an accurate angle. This problem can be solved from many angle values. Let us use an example to explain this problem. This same example will be used in the following sections. Assume that the sampling frequency fs = 2,560 MHz. The sampling corresponding sampling time ts = 1/fs » 0.391 ns. Since the input signal is complex, the maximum bandwidth of the receiver will be 2,560 MHz. Let us use 256 points to perform the IFM receiver operation, which covers 100 ns. The delay times chosen will be based on an analog IFM delay line selection. The shortest delay time is ts and the other three delay times are 4ts, 16ts, and 64ts. In order to obtain a

Figure 10.3  Angle resolutions from 1-bit ADC.

10.5  Number of Phase Difference Counts and Manipulations

231

better estimation of the angle, all the samples are used to find the phase difference between z0 and z1, z1 and z2, …, z255 and z256. There are a total of 255 phase differences and all these values will be used in the receiver design. This operation will generate R(1) in (10.3). A similar approach can be used to obtain R(4), R(16), and R(64). In general to generate R(m), the phase differences between z0 and zm, z1 and zm+1, and zn and zn+m are used. If the input frequency is very low (or very high) the phase difference between zi and zi+1 will be very small. Since the angle resolution is 90°, most of the time the phase difference between consecutive sample pairs is zero. If the true phase difference is 120° in 100 ns, most likely in all the 255 phase differences obtained, only one phase difference is 90° and the rest are all zeros. If one adds all the phase differences, the result will be 90°, and the error between the measured and the true phase values will be –30° (90 – 120). This is the price that the coarse angle resolution pays. In this approach, instead of measuring the phase difference, the number of phase difference will be counted. Since there are four values of phase differences, 0°, 90°, –90°, and ±180°, there are four corresponding counts: N1, N2, N3, and N4. In other words, the number of phase differences of 0°, 90°, –90°, and ±180° is labeled as N1, N2, N3, and N4.

10.5  Number of Phase Difference Counts and Manipulations The measured N1, N2, N3, and N4 values versus the input frequency for a very strong input signal is shown in Figure 10.4. Figure 10.4(a) shows the N1 results. For

Figure 10.4  N1, N2, N3, and N4 versus input frequency for ts delay.

232

Digital Instantaneous Frequency Measurement (IFM) Receiver

this plot, since the input is complex, the frequency changes from 0 to 2,560 MHz, although in an actual IFM receiver design it may not cover this entire range. In this figure when the input frequency is very low or high almost all the phase differences measured are zero except N1. When the very low frequency increases (or the very high frequency decreases), the N1 value decreases. Between frequencies of 640 and 1,920 MHz, the N1 value reduces to zero. Figure 10.4(b–d) represents N2, N3, and N4, respectively. Figure 10.5 shows the plot with a 4ts delay. As expected, the outputs repeat the results in Figure 10.4 four times. Similar results can be expected from the 16ts and 64ts delays, which are not shown. The input frequency can be found from these N values. The following discussion is one of several possible approaches to find the signal input frequency from the N values. Let us define N14 ≡ N1 − N 4

and

N23 ≡ N2 − N3



(10.4)

The plots of N14 and N23 are shown in Figure 10.6. They are somewhat like sinusoidal curve plots, which are similar to the video outputs from an analog IFM receiver. Therefore, the analog IFM receiver decoding scheme can be applied. Let us also define that



N  θ = tan −1  14   N23 

Figure 10.5  N1, N2, N3, and N4 versus input frequency for 4ts delay.

(10.5)

10.5  Number of Phase Difference Counts and Manipulations

Figure 10.6  N14 and N23 versus the input frequency for a ts delay.

Figure 10.7  q versus input frequency with ts delay time.

233

234

Digital Instantaneous Frequency Measurement (IFM) Receiver

Figure 10.8  q versus input frequency with 4ts delay time.

The results of q versus the input frequency can be plotted in Figures 10.7 and 10.8 with delay times of ts and 4ts, respectively. In Figure 10.7, if the angle q can be obtained, the input frequency can be uniquely determined and this is the basic idea of building a digital IFM receiver. The frequency resolution from the angle measurement in Figure 10.7 will be relatively coarse. High frequency resolution must be obtained from the longest delay time of 64ts. The three shorter delay times of ts, 4ts, and 16ts are used to resolve ambiguity. The real receiver design issue is to obtain the high-frequency resolution by combining all the delay outputs.

10.6  Signal-to-Noise (S/N) Effect on Angle q The plots shown in Figures 10.7 and 10.8 are obtained from a strong signal with (S/N = 100 dB). Figure 10.9 shows a closer look of Figure 10.7 at a frequency of 1,280 MHz. In this plot, the input frequency is changed by 1 MHz. One can see that the angle q changes from p to –p when the frequency changes by 1 MHz at 1,280 MHz. If the signal is weak (S/N = 2 dB), the angle q transition will not be as clean. Figure 10.10 shows the angle q transition for a weak signal. There is a range of frequencies around 1,280 MHz where the angle q can be either p or –p from one frequency to the next. This phenomenon will affect the frequency decoding process. If the signal is at a certain frequency with angle q close to the p to –p transition and falls in the wrong angle, the frequency encoded could be wrong.

10.6  Signal-to-Noise (S/N) Effect on Angle q

Figure 10.9  Angle q transition with a delay time ts of a strong signal.

Figure 10.10  Angle q transition with a delay time ts of a weak signal.

235

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Digital Instantaneous Frequency Measurement (IFM) Receiver

Figure 10.11  Angle q transition with a delay time of 64ts of a weak signal.

It is noted that the noise has less effect on long delay times. With the same signal strength (S/N = 2 dB), the angle q at the p to –p transition is shown in Figure 10.11. In this figure, the delay time is 64ts and the phase transition is clean. Thus, in receiver design, it is assumed that the angle q measured with a 64ts delay time is correct. If this measurement is wrong, the frequency will not be encoded correctly. This delay time provides the fine frequency resolution. The other three shorter delay times are used to resolve frequency ambiguity.

10.7  Ambiguity Resolution In this section the basic idea of resolving ambiguity is discussed. Figure 10.12 shows the angles q plot from the frequency 150 to 330 MHz for an input S/N = 0 dB. In order to simplify the discussion, let us designate q3 as the phase angle for a delay time of 16ts and q4 for the 64ts delay. It is assumed that the frequency reading from the 64ts is correct. For the 16ts delay time, when the input frequency changes from 160 to 200 MHz, the angle q3 changes approximately from 0 to p/2 as shown in Figure 10.12(a). This frequency range is divided into four regions—0, 1, 2, and 3—in Figure 10.12(b), which is the q4 result. From the frequency 160 to 320 MHz, q4 repeats four times and the frequency reading from the corresponding q4 must be determined by q3. In other words, one of the four regions of q3 must be assigned to determine the input frequency. These four regions can be decided as follows in Table 10.1.

10.7  Ambiguity Resolution

237

Figure 10.12  Angle q transition with S/N = 0 dB: (a) 16ts delay time and (b) 64ts delay time.

One should note that a frequency at a boundary could be assigned to the wrong region due to noise. For example, this can occur between 0 and 1, 1 and 2, and 2 and 3. This phenomenon can cause an error in the frequency reading and the frequency error will be 40 MHz, since each region covers 40 MHz. This kind of frequency error caused by assigning the q to a wrong region can be considered as catastrophic error because the error is large. From Figure 10.12(a), if the input frequency is at 160 MHz or 320 MHz, the q3 values should be close to 0. Because of noise, the value of q3 can change from positive to negative or vice versa at the boundaries. If this change appears, region 0 can be assigned as 3 or vice versa and the corresponding frequency reading will have an error of 160 MHz. In order to avoid this occurrence, the results of q4 will be used to determine the region of q3. From Figure 10.12(b), one can see that if q4 is in region 0 and the value of q3 is close to 0, q3 can only be in region 0 and cannot be in region 3. The corresponding input frequency is close to 120 MHz. When q4 is in region 0 and q3 is in region 3, the value of q3 should be close to –p/2 with a frequency close to 280 MHz. At the q4 value close to 0, the q3 value can be allowed in a delta angle range. When the value of q3 is in these delta ranges, the region will

Table 10.1  Region Assignment of q3 Region 0 1 2 3

Condition 0 £ q3 < p/2 p/2 £ q3 < p –p £ q3 < –p/2 –p/2 £ q3 < 0

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Digital Instantaneous Frequency Measurement (IFM) Receiver

be determined from q4. For example, if q4 is in region 0, even if q3 is close to +p, it cannot be in region 1. It should be designated as region 2. In order for q3 to be in region 1, q4 must be in region 3. These are the arguments to assign q3. In the following simulation the delta angle range is assumed to be ±p/8. Once region q3 is determined, q2 can be determined from q3 in a similar manner and q2 can be used to determine q1. When all the regions of q1, q2, and q3 are determined, the frequency can be calculated. The fine frequency can be calculated from q4.

10.8  Simulation Results In the following simulation, the input frequency changes from 1 MHz to 2,560 MHz and the input S/N = 0 dB. Figure 10.13 plots the error frequency versus input frequency. The error frequency is defined as the measured frequency minus the input frequency. In this plot the first and last two input frequencies are not included, because near this frequency range, the frequency may wrap around. For example, the receiver cannot differentiate 0 MHz to 2,560 MHz. When the input is at 1 MHz, it may generate an output frequency close to 2,560 MHz and the error will be very large. Without the four ending points, the error is within ±3 MHz, which are respectable results. In this plot, a bias of 1.5 MHz is artificially corrected to show an unbiased plot.

Figure 10.13  Error frequency versus input frequency with S/N = 0 dB.

10.8  Simulation Results

239

Figure 10.14  Error frequency versus input frequency with S/N = –3 dB.

Figure 10.15  Error frequency versus input frequency with S/N = –3 dB excluding catastrophic errors.

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Digital Instantaneous Frequency Measurement (IFM) Receiver

When the input is decreased to S/N = –3 dB, the plot is shown in Figure 10.14. In this figure, eight outputs have catastrophic errors. When these eight errors are taken out of the plot, the rest of results are as shown in Figure 10.15. Under this condition, the errors are less than ±7.5 MHz. These results are comparable with an analog IFM receiver.

10.9  Threshold and Confirmation An analog IFM receiver measures frequency constantly. When there is no signal, the receiver measures frequency on noise and generate meaningless data. Thus, an analog IFM receiver usually works with a crystal video receiver as shown in Figure 10.16. When a pulsed signal arrives and crosses a certain threshold, the crystal receiver will generate a leading edge signal. The leading edge will trigger the IFM receiver to read the frequency. The crystal receiver also generates amplitude information. Another major deficiency of the analog IFM receiver is that when simultaneous signals with frequencies f1 and f2 and comparable signal strengths arrive at the receiver, the receiver may generate erroneous frequency information. The erro­ neous frequency means that the output frequency is neither f1 nor f2. If the output frequency is either f1 or f2, the frequency reading is considered correct. The worst scenario is that the IFM receiver generates an erroneous frequency, but there is no way to know the frequency is wrong. In a digital IFM receiver this problem can be mitigated. One can determine whether the receiver output frequency is equal to f1 or f2. A confirmation method is introduced here. The confirmation method is based on the discrete Fourier transform (DFT). It might be time limited to perform a fast Fourier transform (FFT) on 256 points. However, this confirmation works only on one component of the DFT. The calculation can be written as X(fo ) =

255

∑ x(n)e

n=0

−j

2π fon N

(10.6)

Figure 10.16  An analog IFM receiver triggered by a crystal vide receiver.

10.10  Performance of Two Simultaneous Signals

241

where fo is close to the frequency measured from the IFM receiver. If the X(fo) value is high, it means a signal exists; otherwise, there is no signal. In order to determine whether there is a signal with frequency near the measured value, a threshold must be set. One way to determine the threshold is through an FFT operation as discussed in Chapter 6. Even though the input is only 1 bit, the FFT output appears to be Rayleigh. This threshold is not used to detect a signal but to confirm a signal. The frequency must be first generated by the digital IFM receiver. This frequency is used to find a one frequency component output from a DFT operation and the result is compared to the threshold. Therefore, the conventional thought on the probability of false alarm may not be applicable. This problem may require a detailed study if a receiver is actually designed and built. The main purpose of this discussion is to illustrate the basic concept on the signal confirmation. For simplicity, the following empirical method is used to find the threshold. The input signal is noise alone and digitized to 1 bit. Perform FFT on these data for 1,000 times and find the maximum value as the threshold. The threshold value obtained in this simulation is about 81, which should have a probability of false alarm of about 10-3. There are two approaches to calculate the X(fo) value. One is to use the actual measured frequency. For example, if the measured frequency is 1,275 MHz, fo should be 127.5 because a 256-point DFT has a 10-MHz resolution. It should be noted that fo is not necessary an integer as in the FFT operation. The other approach is to use an integer number for fo. If fo is selected as an integer, the rounded-off frequency output might not be the correct one and the amplitudes of two neighboring frequencies need to be calculated also. As a result, three frequency bins rather than one will be calculated. The highest value among the three frequency bins is used to compare with the threshold. If the calculated X(fo) is higher than the threshold, the measured frequency is assumed to be correct. At the same time this threshold can also be used to determine the arriving of a signal. The IFM will report a frequency every 100 ns, and DFT output can be used to determine whether there is a signal. With this operation, the triggering signal generated by the crystal video receiver is no longer needed. The possible triggering method from Section 7.7 will not be needed, if the DFT output is used for triggering. Although the phase summation discussed in Chapter 7 can be used to determine the existence of a signal, it cannot be used to confirm the correct frequency output; therefore, the DFT is used.

10.10  Performance of Two Simultaneous Signals In the simulation, the three-frequency component method is used to confirm the output frequency. Since the input signal only has 1 bit, the Kernel function in (10.6) only has 1 bit also and they are ±1 and ±j. There are two signals and their S/N = 10 dB. One signal has a fixed frequency at 1,250 MHz, which is close to the center of the band, and the other frequency changes from 10 to 2,550 MHz in a 1-MHz step. This input condition is arbitrarily chosen. If the highest one of the three frequency bins is higher than the threshold of 81, the frequency is recorded. The measured frequency is compared to both the input frequencies and

242

Digital Instantaneous Frequency Measurement (IFM) Receiver

Figure 10.17  Frequency errors from two input signals with signal confirmation: (a) frequency error of valid signals, (b) DFT outputs of valid signals, (c) frequency error of invalid signals, and (d) DFT outputs of invalid signals.

10.11  Frequency Folding

243

generates two frequency errors. The smaller frequency error is considered the correct one. The results are shown in Figure 10.17. Among the 2,540 input frequencies, 787 data points cross the threshold for this simulation and they are considered as the valid signals. Figure 10.17(a) shows the frequency error versus input frequency of the valid signals. The worst frequency error is about 36 MHz and two data points have this value. The rest of the frequency errors are less than 15 MHz. Figure 10.17(b) shows the amplitude of the DFT outputs. Figure 10.17(c) shows the frequency error versus input frequency of the invalid signal readings. Most of the frequency error is around 30 MHz. It is difficult to read the fine frequency values from this figure. Figure 10.17(d) shows the corresponding DFT outputs. From this figure, it appears that a simultaneous signal problem is mitigated. When two signals of comparable amplitude are present, the IFM receiver can report an erro­neous frequency, which cannot be recognized previously. The confirmation method based on the DFT operation can process simultaneous signals. Thus, the erroneous frequency generated will be removed through the confirmation. If an IFM receiver can report one of the input frequencies correctly, it will be considered to be functioning properly.

10.11  Frequency Folding For a 1-bit ADC sampling at 2,560 MHz, the input bandwidth of the ADC can be very high. For example, a 2-bit ADC operating at this sampling frequency can digitize an input signal at 10 GHz directly [personal communication with D. Schwab, Mayo Clinic, 2008], even though the Nyquist bandwidth is only 1.280 GHz. This is a very attractive feature for receiver design. To receive a signal at 10 GHz a downconverter is not needed and the signal can be digitized directly. Let us discuss this problem from two digitizing approaches: real and complex. If the digitizing data are real with a sampling frequency fs, the Nyquist bandwidth of the receiver will be fs/2 and the bandwidth folding is shown in Figure 10.18. In this figure, two signal bandwidths are labeled: desired and undesired. For the desired case, after sampling,

Figure 10.18  Band folding for real data sampling.

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Digital Instantaneous Frequency Measurement (IFM) Receiver

Figure 10.19  Signal downconversion arrangement.

the input will be transferred to baseband without band folding. In other words, the input and output bandwidths are equal. For the undesired bandwidth, after sampling the portions of input bandwidth will be folded on top of each other. In other words, the input bandwidth is wider than the output bandwidth. From this discussion one can see that the input frequency and the sampling frequency must be carefully selected, otherwise, band overlapping may occur after sampling. When band overlapping occurs, simulta­ neous signals may not be properly measured. Assume that the input frequency is between 2fs and 5fs/2. If the input band of the ADC does not cover up to 5fs/2, the input signal must be downconverted to, say, fs/2 and fs as shown in Figure 10.19. This approach requires a mixer and a local oscillator and the cost will be high. If the ADC has a high enough input bandwidth, the signal can be digitized directly without downconversion. For a complex signal, if the sampling frequency is fs, the input bandwidth is also fs. The band folding after sampling is shown in Figure 10.20. In this figure, as long as the input bandwidth is less than fs, there is no band overlapping. The input band can be placed at any frequency range. In other words, no special relation is required between the input and the sampling frequencies. If the input bandwidth is selected as shown in this figure, one must adjust the output frequency

Figure 10.20  Band folding for real data sampling.

10.12  Time Resolution Improvement and Threshold with Hysteresis

245

Figure 10.21  Output frequency versus input frequency (1,000 to 3,560 MHz) at S/N = 0 dB.

range. In this figure, the lower input band folds into the higher portion of the output band and the higher input band folds into the lower portion. Since the input to the digital IFM receiver is complex, it can contain any input band range as long as the bandwidth is less than 2,560 MHz. Figure 10.21 shows the input frequency range across a boundary. In this figure, the input is from 1,000 to 3,560 MHz. The output frequency versus input frequency is shown in Figure 10.21. In this figure, the output frequency is from 1,000 to 2,560 MHz and then from 0 to 1,000 MHz. If a constant frequency of 2,560 MHz is added to a 0- to 1,000-MHz frequency range, the output frequency becomes 2,560 to 3,560 MHz, which are the correct values. The application of this idea can be illustrated as follows. Assume that the input of an ADC only has a maximum sampling rate of 4 GHz, but the input frequency bandwidth is much higher. Using IQ channels as shown in Figure 10.2 can achieve a bandwidth of 4 GHz. For example, if the desired input frequency band is from 2 to 6 GHz, the direct digitization method can be used. The input frequency above 4 GHz can be read as beyond the sampling frequency fs and can fold into 0 to 2 GHz. With this design the analog input bandwidth must be wide enough to accommodate the highest input frequency.

10.12  Time Resolution Improvement and Threshold with Hysteresis The time resolution from the crystal video receiver can be rather fine such as in the tens of nanoseconds. The confirmation method discussed in Section 10.11 uses

246

Digital Instantaneous Frequency Measurement (IFM) Receiver

100 ns of data. Sometimes a finer time resolution is desirable for the time of arrival (TOA) and pulse width (PW) measurements. To achieve this goal with the IFM receiver, the following steps can be taken. The input signal frequency can be calculated through 256 data points as discussed in previous sections. This data length can provide the desired frequency resolution. To confirm the output frequency, only 64 points can be used rather than the 256 points discussed in Sections 10.9 and 10.10. This low number of data points will decrease the sensitivity of the receiver. However, from simulated results it appears that the sensitivity is comparable with the frequency reading capability of the IFM receiver. Figure 10.22 is used to explain the operation. In this figure, each block labeled A, B, C, … contains 64 points. If D through G are used to generate a frequency value, then one can start the confirmation from block B (two blocks before D) because it is possible that the signal starts there, but the IFM receiver does not provide the correct frequency in the previous 256 points of data. This operation is based on an arbitrary assumption that if a signal starts from block A, the 256 data points before D should generate the correct frequency. If B is confirmed, then C is tested. If C is confirmed, then D is tested, and so on. Under this case, the beginning of block B is considered the TOA. If B is not confirmed, then C, D, E, F, and G will be tested. If all confirmations fail, the signal is not found in these data. The threshold for the confirmation is obtained in a similar way as discussed in Section 10.10. The only difference is that only 64 points of data are used. The threshold obtained from this simulation is about 38; we will refer to this value as threshold 1. If a signal is confirmed in a certain block, the following blocks will be tested for the same frequency, but a slightly lower threshold (threshold 2) is used. The threshold of the following blocks should be about 2 dB lower than threshold 1 or about 30 (20log(30/38) » 2). When a weak signal is tested, a block may pass the threshold, but the following blocks may not all pass the same threshold. If a following block does not pass the threshold, the signal will be chopped into pieces. Lowering the threshold for the following blocks will lower the probability of this phenomenon occurring. This type of threshold, testing the TOA with a higher value and the time of departure (TOD) with a lower value, is often referred as the threshold with hysteresis. One can use three or four blocks to make a decision on whether a signal is detected. This procedure can decrease the probability of false alarm. Since for each block, the probability of false alarm is about 10–3, for three blocks, the overall probability of false alarm is about 10–9 (10–3 × 10–3 × 10–3). However, for the second and

Figure 10.22  Blocks of data for frequency calculation and time measurements.

10.12  Time Resolution Improvement and Threshold with Hysteresis

247

Figure 10.23  Average TOA error.

third blocks, the threshold is reduced; the overall probability of false alarm will be higher than 10–9. In calculating the X(fo) value from (10.6), two approaches are used to find the Kernel function exp(–j2pfon/N). One is to calculate the actual value. The other one is to use only four values ±1 and ± j to represent the Kernel function. This approach might be considered in hardware design. Simulations are used to find the results. The input data has 2,560 data points. The TOA and PW are randomly selected between 512 and 1,280 data points. The frequency is random between 80 and 2,480 MHz. The results are obtained from 1,000 runs.

Figure 10.24  Average PW error.

248

Digital Instantaneous Frequency Measurement (IFM) Receiver

The simulated results are shown in Figure 10.23 for TOA and in Figure 10.24 for PW. In each figure, there are two curves: one for the calculated Kernel and the other for the 1-bit Kernel. As expected, the 1-bit Kernel provides slightly inferior results at low S/N. The average TOA error at S/N = –3 dB is less than 30 ns, but the average PW error is larger and less than 100 ns. A larger PW error is due to the fact that the PW must be calculated from both the TOA and the time of departure (TOD). Thus, there are more sources of error. For 64 data points, the time resolution is 25 ns. Theoretically, the time error should be ±12.5 ns; thus, the average time should be about 6 to 7 ns. The TOA average errors are close to this value when the input signal is strong and the PW error is close to 10 ns. The results indicate that the TOA and PW resolutions can be improved through short confirmation data.

10.13  Imbalance of IQ Channels In Figure 10.2 and the previous discussions, it is assumed that the IQ channels are perfectly balanced. This means the amplitudes of IQ outputs are equal and the phases between them are exactly 90° output of phase. However, if an actual hybrid is used in the IFM receiver design, the outputs will not be perfectly balanced. The effect of the imbalance must be evaluated. The imbalance can be divided into two categories: amplitude and phase. This problem is discussed in Chapter 5. The test procedure is to introduce the amplitude and phase imbalance in the complex input signal before digitization. The digitized data are used as the input to the IFM simulations. The frequency of the receiver is measured. The variance of the frequency is measured against the imbalance. First, let us discuss the amplitude imbalance. Since the ADCs following the 90° hybrid only have one bit, an amplitude imbalance should not cause any difference. For example, if the output is greater or equal to zero, it will generate a +1 and if the output is less than zero, it will generate a –1. The amplitude of the output will make no difference. Simulated data are used to test the imbalance. As expected, the amplitude imbalance makes no difference in the output data. The phase imbalance is also tested through simulation. The results are shown in Figures 10.25 and 10.26. In these figures, the results are obtained from 100 runs with random noise and the S/N varies from –3 to 10 dB. Figure 10.25 shows the average frequency error and Figure 10.26 shows the number of bad data, which is defined as the measured frequency is more than 5 MHz away from the input frequency. In both figures, there is little difference when the phase imbalance is within 30°. Since most of the phase balance of a wideband 90° hybrid is better than 30°, the phase imbalance is not a problem in an actual receiver building. From this simple test, one can conclude that the IFM receiver using a 1-bit ADC is rather robust. 1.

Broadstock, an adjunct professor at Miami University, Oxford, Ohio, provided data for parts of this chapter.

10.14  Hilbert Transform Converting a Real Signal to Complex

249

Figure 10.25  Average frequency error versus phase imbalance. 

10.14  Hilbert Transform Converting a Real Signal to Complex In this and the following sections, instead of using complex data generated from a 90° hybrid and two ADCs, a real data is used for simulation. The real data also have only 1 bit. In order to use the IFM receiver approach, the data must be converted into complex. Two conversion methods will be used [2]. The first one uses the Hilbert transform and the second one uses downconversion through a special sampling frequency. After the conversion, the real data become multiple bits. In order to keep the input to the IFM receiver 1 bit, after the transforms the output data will be digitized again into 1 bit. The entire conversion approach including a real to complex transform and generating a 1-bit output can be achieved through a simple hardware operation. The Hilbert transform will be discussed in this section. The Hilbert transform is discussed in Sections 5.9 and 5.10 and also in [2]. In Section 5.10 it is shown that the IQ channels generated through the Hilbert transform have a very good phase balance but a poor amplitude balance. Since the input to the IFM receiver only requires 1 bit, the amplitude imbalance is not of concern. Intuitively, the Hilbert transform should produce satisfactory results. A few of the coefficients of the Hilbert transform are listed as: –2/5p, 0, –2/3p, 0, –2/p, 0, 2/p, 0, 2/3p, 0, 2/5p. In order to keep the operation simple, only seven 2.

Lopata, engineer at ITT, generated the original simulation program.

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Digital Instantaneous Frequency Measurement (IFM) Receiver

Figure 10.26  Number of bad data versus phase imbalance.

terms are used: –2/3p, 0, –2/p, 0, 2/p, 0, 2/3p. In addition, only the ratio is taken into consideration and the resulting coefficients become –1, 0, –3, 0, 3, 0, 1. They can be labeled as in Table 10.2. The output can be found as a finite impulse response (FIR) filter shown in Figure 10.27. The coefficient of 0 means that there is no summation or the signal delays two units as shown in the figure. The input signal is digitized into 1 bit before it passes through the Hilbert transform filter. After the Hilbert transform, the output signal is digitized again into 1 bit. It is interesting to note that the coefficients h(2) = –3 can be replaced by other values than –3 as long as its value is less than –1. Similarly, h(4) = 3 can be changed to another value greater than 1. This change can be made because the output is digitized again into 1 bit. When h(2) = –2 and h(4) = 2 are used, it might be easier to perform multiplication in hardware. The results generated this way have same number of complex data points as the real input data points. Since the input bandwidth cannot increase through this operation, the number of output points can be reduced to half through decimation. In other words, half of the output data can be eliminated. Simulated results2 with S/N = 0 dB are shown in Figures 10.28 and 10.29. Since the input is real, the overall bandwidth is only 1,280 MHz. The input frequency range is from 1 to 1,280 MHz. There are relative large errors at both ends of the input frequency range. This is the expected result because at low and high frequencies the signal interferes with the signal in its neighboring region, which is the Table 10.2  Hilbert Transform Coefficients h(0) –1

h(1) 0

h(2) –3

h(3) 0

h(4) 3

h(5) 0

h(6) 1

10.14  Hilbert Transform Converting a Real Signal to Complex

251

Figure 10.27  Hilbert FIR filter.

fundamental property of real signal. Excluding the big errors at both ends of the input band, the results are shown in Figure 10.29. Comparing with Figure 10.13, the frequency error is slightly larger at low or high frequencies. This is the expected result because for complex data, the coherent processing gain is 3 dB higher than real data. One can conclude that by using real data with low S/N, the input bandwidth is less than 1,280 MHz and the frequency error is slightly inferior to the complex data result.

Figure 10.28  Frequency error of real data through a Hilbert transform at S/N = 0 dB.

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Digital Instantaneous Frequency Measurement (IFM) Receiver

Figure 10.29  Frequency error by eliminating big errors at both ends of the input band of Figure 10.28.

10.15  Special Sampling Downconversion Transform As discussed in [2] and Section 5.13, there is another method of converting real to complex data referred to as the downconversion method. The basic method is briefly mentioned in Section 5.13. In this section, the lowpass filter can be designed through a very simple approach. In order to illustrate this concept, (5.6) is rewritten here. The input data x(n) will be divided into even and odd outputs: I(n) = x0 , 0, − x2 , 0, x4 , 0, − x6 �

Q(n) = 0, x1, 0, − x3 , 0, x5 , 0, − x7 �

(10.7)

In this equation the subscript expression is used to represent individual data points. A lowpass filter must be added to the outputs. One very simple lowpass filter is the moving average of two data points. The I(n) Q(n) outputs after the moving average can be obtained as I(n) = x0 , x0 , − x2 , − x2 , x4 , x4 , − x6 �

Q(n) = 0, x1, x1, − x3 , − x3 , x5 , x5 , − x7 �

(10.8)

These are the desired results. Since the input is digitized into 1 bit, the resulting data from this operation are also 1 bit. The output data no longer need to be digitized again. This overall operation is rather simple.

10.15  Special Sampling Downconversion Transform

253

Figure 10.30  Frequency error of real data through downconversion at S/N = 0 dB.

Simulated results are shown in Figures 10.30 and 10.31. The input frequency range is also 1 to 1,280 MHz. There are relatively large errors at both ends of the input frequency range. Excluding the big errors at both ends of the band, the results are shown in Figure 10.31. The results obtained from the Hilbert transform and the downconversion methods can be considered roughly comparable. It appears that

Figure 10.31  Frequency error by eliminating big errors at both ends of the input band of Figure 10.30.

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Digital Instantaneous Frequency Measurement (IFM) Receiver

the Hilbert transform has slightly better results with limited test data. However, the operation appears slightly complicated. If one plans to use real data to build an IFM receiver, these methods need to be further studied. In Section 5.13, the IQ outputs from this special sampling downconversion scheme have poor results around 640 MHz, the center of the input band. It is interesting to note that the frequency measured from the IFM receiver using data obtained through this method does not produce large errors at 640 MHz. This problem should be further investigated.

10.16  Conclusion In this chapter, it is illustrated that by using a 1-bit ADC one can built a very simple IFM receiver. A signal confirmation circuit can be added to the receiver to reduce the error generated from simultaneous signals, which is a major problem of the IFM receiver. The confirmation circuit can also be used as a triggering circuit and make the IFM receiver a self-contained unit avoiding triggering from another receiver. Although the discussion is based on a sampling frequency of 2.56 GHz, a 1-bit ADC can achieve very high sampling frequencies in the tens of gigahertz. For a 1-bit ADC the analog input bandwidth can be rather wide; thus, one can build a very wideband receiver without downconverting the input frequency. Using complex data generated from hardware is a better approach because the IQ channel imbalance is not a problem in this design. Using IQ data can cover the entire bandwidth of sampling frequency. For example, with a 2.56-GHz sampling rate the receiver can have a bandwidth of 2.56 GHz. The only problem is that the receiver cannot distinguish the frequencies at 0 and 2.56 GHz. Eliminating a very narrow frequency range at either ends of the input frequency band should get rid of this problem. With real data the bandwidth cannot reach 1.28 GHz because the frequencies at the ends of the band will interfere with each other. About 100 MHz in the end zones cannot be used in the receiver. Comparing the analog approach with several correlators, the digital approach should be much simpler. An IFM receiver should be compared with a monobit receiver [2] to determine the performance against complexity.

References   [1] Tsui, J., Microwave Receivers with Electronic Warfare Applications, New York: John Wiley, 1986.   [2] Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Chapter 8, Norwood, MA: Artech House, 2001.

C h a p t e r 11

Receiver Designed Through a Conventional FFT Approach

11.1  Introduction When wideband analog receivers were built, 90% of the effort was on the encoder design. First, the RF is converted into video signals and the video signals are digitized. The encoder takes the digitized data and converts into the pulse descriptor words (PDW). Most the receiver problems are also in the encoder designs. The encoder designs are very complicated and it is difficult to provide a good description because the logic designs are based on digital hardware. Since the advance in field programmable gate array (FPGA), the encoder design is becoming a manageable job. Due to the increase in the FPGA speed, the channelization can also be performed through it. The filters built through the digital approach are much better than the analog approach in performance. Thus, the receiver performance should improve and the entire receiver design procedure becomes a manageable task. There are many different approaches to building a wideband digital receiver. In this book only three approaches will be discussed. The first approach is through a conventional fast Fourier transform (FFT) and will be presented in this chapter. This approach is used in actually building a wideband digital receiver and the channelization can be achieved in the FPGA. Before the FPGA can perform the desired channelization, a polyphase filter approach is used and the application specific integrate circuit (ASIC) is used for channelization. Several receivers have been built through the conventional FFT approach. This approach is similar to the analog receiver design but uses blocks of data in a continuous mood rather than taking data point by point continuously. In general, the block processing method has a relatively poor time resolution, but it reduces the transient effects at both the leading and trailing edges of a pulsed signal. As a result, the encoder design can be simplified. The receiver design will start from the receiver requirements. The FFT operation is based on the requirements. The main discussion will be concentrated on the encoder design. The performance limitations will also be included.

11.2  Requirements As with most subjects discussed in this book, a numerical example will be used for this presentation. The sampling frequency is assumed to be 2,560 MHz; thus, the 255

256

Receiver Designed Through a Conventional FFT Approach

Nyquist bandwidth is 1,280 MHz. Taking off 140 MHz at the beginning and the ending of the Nyquist bandwidth, the receiver bandwidth is from 141 to 1,140 MHz, which covers 1,000 MHz. For simplicity, the digitizing effect will not be included in this discussion. In most receiver designs, the minimum pulse width (PW) is often the primary requirement. In this study let us use 100 ns as the minimum required PW. The performances of the receiver, such as the sensitivity and the frequency resolution, are determined by the minimum PW in this type of design. This is an undesired result. The desired resolution is that the minimum PW only determined the receiver sensitivity and frequency resolution of this particular PW. For a longer pulse the receiver should have a higher sensitivity and frequency resolution. In other words, the sensitivity and frequency resolution should be PW dependent. The sensitivity of the receiver is determined by the probability of false alarm (Pfa) and the probability of detection (Pd). A reasonable set of values is Pfa = 10-7 and Pd = 90%, which requires an output signal-to-noise ratio (S/N) equal to about 14 dB. Once an FFT window length is fixed, the time resolution is also determined. For the same applications it is desirable to have a fine time resolution. However, for this kind of receiver design, basically the minimum PW determines the time resolution; however, additional processing may improve the time resolution as discussed Section 11.12.

11.3  FFT Length Selection and Frequency Resolution Let us refer to the FFT length as the window length (or frame length). Before the selection of window length, it is assumed that there is no window overlap, which means that the beginning of a window is adjacent to the end of the previous one. In this approach the beginning or the ending of a pulse can only occur in one window because the window selected must be shorter than the PW. A window containing a partial pulse will have a spectrum spreading effect, which is referred to as rabbit ears by receiver designers. Rabbit ears usually cause encoder design problems. Since the rabbit ears happen only in one window, this problem is minimized and the encoder design can be simplified. Based on the minimum PW of 100 ns, the FFT length can be determined. For the 2,560-MHz sampling frequency, 100 ns contains 256 samples. If 256 samples are used to perform the FFT operation, the chance of exactly matching the minimum PW and the window is 1 out of 256. When the window and the minimum PW are not exactly matched, the receiver sen­sitivity will decrease. In order to use the FFT outputs to measure an input signal, the FFT window should be filled with a signal. If a window is partially filled with input signal, the output data could be erroneous, such as the pulse amplitude (PA) measured will be reduced and the frequency will spread into neighboring channels. In order to completely fill a window of 100 ns, the minimum PW will be 200 ns. This is double the minimum PW requirement. If the output of only one frame passes the threshold, a signal will be detected. This frame may not fill with signal data and the receiver may report incorrect PA information. The TOA resolution is 100 ns.

11.3  FFT Length Selection and Frequency Resolution

257

In order to guarantee one full window of data, a reasonable approach is to use two windows for detection. The first window usually is partially filled with a signal and the second window is often fully filled with signal data, if the PW is reasonably long. Even if the first window is partially filled with data, the output frequency from the FFT operation is usually correct. Comparing the frequency outputs of the two consecutive windows can confirm the detection of an input signal. If a detection can be obtained only in one window, it is usually considered a noise spike. In short, this approach uses the assumption that the second frame is full of signal data. Using the above argument, the window length of 128 points is selected, which corresponds to 50 ns. To achieve the maximum receiver sensitivity, the two consecutive windows should fill completely with signal data. If one window is partially filled with a weak signal, it may not be detected. Under this condition, the receiver will miss the signal. In order to detect a weak signal at a minimum PW, the minimum PW should be 150 ns to guarantee obtaining two full windows of signal data. In other words, to receive a signal with a full sensitivity (or at the sensitivity level of the receiver), the PW must be 150 ns or longer. This assumption, strictly speaking, cannot fulfill the requirement. Figure 11.1 shows the minimum PW versus the FFT window. In this special case, the PW is three times the window length. It is clearly shown that under any input condition, the signal can completely fill two consecutive windows with signal data. From this window selection the minimum PW is 150 ns, which is longer than the required minimum value. The time resolution is 50 ns for this window length. When the signal is strong, the receiver may even detect it even when the PW is less than 100 ns. For example, an 80-ns pulse may be evenly divided into two consecutive windows and both windows can generate the correct frequency and pass the detection threshold. By comparing the outputs of the two windows, the signal is detected. Thus, even if the minimum PW is 150 ns, the receiver may detect pulses shorter than 100 ns if the signal is strong. This is one of the reasons to choose the 128 data point window. The 128-point FFT operation will generate 128 frequency outputs. Since the input data are real, only 65 (from 0 to 128/2) of the 128 outputs carry independent information. Each channel is 20 MHz wide, which represents the frequency resolution. Note that channels 0 and 64 only cover half the frequency resolution (or 10

Figure 11.1  Probability of matching the window to the input signal.

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Receiver Designed Through a Conventional FFT Approach

MHz). To cover the desired bandwidth of 1,000 MHz, 50 channels will be enough. Usually at each end of the input bandwidth, a guard channel is used. The guard channel may be used to work with its neighboring channel to improve frequency reading. Including the two end guard channels, the total channels should be 53. In other words, from the 65 channels, 4 channels at each end will be eliminated. This idea was also discussed in detail in Section 6.1.

11.4  T  hreshold Determined by the Probability of False Alarm Rate and the Probability of Detection [1, 2] There are two ways to determine the receiver sensitivity, the absolute and the relative. In the absolute way, the receiver sensitivity is given in dBm. In the relative way the sensitivity is given in decibels with respect to the noise floor. If the noise floor is calculated, the relative sensitivity can be expressed in dBm. Thus, these two methods are interchangeable. Since, as mentioned in Section 11.2, the digitization effect will not be considered in this chapter, the sensitivity calculation in Chapter 2 will not be applied. As a result, in this study, the relative sensitivity is used. The sensitivity is determined by the probability of detection and probability of false alarm rate. Before the probability of false alarm rate will be determined, the detection scheme will be discussed. As mentioned in Section 11.2, once a window length is selected, the sensitivity of the receiver is determined. This section is determining a threshold from the given probability of false alarm Pfa of 10-7. The threshold will be used to determine the Pd. Based on the detection from two consecutive windows, three slightly different conditions will be used to evaluate the receiver sensitivity. The first one is considering only one channel, which is the conventional approach used by receiver engineers. Suppose that only one frequency bin is of interest such as the tenth and the probability of false alarm of this one window is Pfaw. In the second frame the same channel is observed and the Pfaw is the same. Under this condition, the Pfaw of each window will be

2 Pfaw = Pfa = 10−7

Pfaw = 10−7 = 3.162 × 10−4

(11.1)

In this definition, the false alarm will occur at the same channel. In the second case, also only one channel will be studied. The difference is that when one channel in the first window crosses a threshold, three adjacent bins in the second frame will be considered as a signal. Therefore, the probability of false alarm will be caused by three bins. The Pfaw per frame can be written as

2 3Pfaw = Pfa = 10−7

Pfaw = 10−7 / 3 = 1.826 × 10−4

(11.2)

This is a better definition than the previous definition because in the probability of detection study, if the two adjacent channels are not taken into consideration, a signal at the boundary between two frequency bins may not be detected. To simplify the notation, Pfaw will be used for all three cases.

11.4  Threshold Determined by the Pfa and the Pd

259

The last case will consider the probability of false alarm for all 53 channels; thus, the probability of false alarm is 53 times higher. In order to keep the overall probability of false alarm, the Pfaw per frame can be written as

2 3Pfaw = Pfa = 10−7 / 53

Pfaw = 10−7 /(3 × 53) = 2.508 × 10 −5

(11.3)

From these three probabilities of false alarms, three thresholds are determined: 17.67, 18.26, and 20.26. In obtaining these data, the input noise is modified by a Blackman window to reflect the receiver design. The first threshold is shown in Figure 11.2. Once the threshold is determined, the Pd can be found. The desired Pd is 90%. Since it requires two consecutive windows to cross the threshold, the Pd for each window should be higher. The Pd for one window is

Pd = 0.9 ≈ 0.949

(11.4)

The Pd plot is shown in Figure 11.3. In this figure, each data curve is obtained from 1,000 runs with a random frequency from 141 to 1,140 MHz and a random initial phase. A Blackman window is used for the precondition of the input signal. To achieve about a 95% detection, the required S/N is close to -3.6 dB. In this test the output frequency is not compared because only one frame is used in the evaluation. The probability of detection for one channel by comparing frequencies of two consecutive frames is shown in Figure 11.4, which is obtained from a simulation

Figure 11.2  Noise distributions after an FFT operation with a Blackman window and a threshold.

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Receiver Designed Through a Conventional FFT Approach

Figure 11.3  Probability of detection obtained for one window frame.

with the following conditions. Not only must the two windows cross the threshold, but the output frequency from the two windows must also be matched. This means that the two output frequencies are either equal to or have a difference of one frequency bin to be qualified as a detection. In this figure at a 90% detection the required S/N = -3.3 dB, which is different from the above method by about

Figure 11.4  Probability of detection obtained from two window frames.

11.5  Threshold Adjusting

261

Figure 11.5  Probability of detection obtained from two window frames for an overall probability of false alarm of 10-7.

0.3 dB. This method of detection should be considered as the desired one because it matches the actual detection in an actual receiver. These values are about 1 dB lower than the values listed in Table 6.6 where the values are obtained through coherent processing. Here the window length is half of 256 points, but the detection scheme uses two frames to make a decision. It is expected that the sensitivity calculated should be slightly lower. Finally, if the overall receiver probability of false alarm is at 10-7, the result is shown in Figure 11.5, and at a 90% detection, the required S/N = -2.7 dB, which is about 0.6 dB lower than the one-channel case. It is difficult to obtain a good reading from [1, 2] because the figure is not large enough for this application. It appears when the probability of false alarm is changed by 53 that the sensitivity decreases by close to 0.5 dB, which is close to the simulated result of 0.6 dB. The three sets of sensitivity obtained are listed in Table 11.1.

11.5  Threshold Adjusting In order to detect a signal, the maximum amplitude of the frequency bins of a certain frame must be found. The maximum amplitude must be above the threshold Table 11.1  Sensitivity Obtained from Different Methods Method S/N dB

One Data Frame* -3.6

Two Frames, One Channel -3.3

*Actual uses two frames with the same frequency on both frames.

Two Frames, 53 Channels -2.7

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determined from the noise distribution. The maximum frequency bin of the consecutive frame must also be above the threshold. In addition, the bin number of the two maxima from the two consecutive frames must be the same or within ± bin. Once a signal is detected, it is important to find the length of the signal to obtain the PW. Theoretically, this should be rather simple. When the amplitude of the same FFT bin crosses the threshold, it should be the same signal. However, in reality there are two problems. The first one is that the input signal can straddle two frequency bins. This problem has been discussed in Section 11.4. The second problem is caused by a weak signal. When the signal is weak and close to the threshold of the receiver, if the signal is long, one frame may cross the threshold and the next one may not. Thus, one long pulse can be reported as many short ones. This phenomenon was first discovered in a field test. For a searching radar rotating in the azimuth direction, the main beam will point toward the receiver and the radar signal will cross the threshold being detected. When the beam is turning away from the receiver, the signal will be at the threshold level again. In both cases the receiver may produce many short pulses if the radar pulse is long. In analog receivers a hysteresis loop can be added to the receiver threshold to eliminate this phenomenon. A hysteresis threshold actually has two levels. The input signal must cross the higher one to be detected and cross the lower one to be declared as the end of the pulse. The difference between the two thresholds is usually close to 2 to 3 dB. This circuit can degrade the receiver sensitivity by 2 or 3 dB if the receiver has only one detector such as in a crystal video receiver. In a digital receiver the same technique can be applied. However, in a digital receiver the threshold can be set at the output of each channel. Once a signal is detected in a certain channel, the threshold of this channel alone is lowered, which should not affect the Pfa and Pd of the overall receiver. In this study the threshold is dropped by 80%, which is close to 2 dB (20log0.8). This value is determined empirically. Another way to remedy this problem is to fill holes, for example, if the input crosses the threshold at frames 1, 2, and 3 but misses 4 and 5 and crosses again at 6, 7, and so forth. The output will be 1110011…, where 1 represents a threshold crossing and 0 represents a miss. Under this condition, one can consider that the two 0s will be replaced by two 1s and the pulse is continuous. This approach may have some shortcomings, such as when the last few frames of the pulse do not cross the threshold, there is nothing to remedy the zeros. Under this condition, the PW measured will be shorter than the actual PW. This problem will be further discussed in Chapter 13.

11.6  F  requency Reading Improvement Through Amplitude Comparison [3, 4] In this receiver design it takes two frames to detect a signal. It is assumed that the PW is long enough so that the second frame is filled with signal data. Thus, the data in the second frame is used to calculate the frequency of the input signal. The channel width of this FFT design is 20 MHz, which corresponds to a frequency resolution of ±10MHz. A finer frequency resolution is usually desirable. The coarse frequency is determined by the maximum of the FFT outputs. The frequency reso-

11.6  Frequency Reading Improvement Through Amplitude Comparison

263

lution can be improved by comparing phases between two consecutive frames as discussed in Chapter 6. In this section the fine frequency will be obtained by comparing the amplitudes of the maximum output to its neighboring one. The maximum frequency bin has two neighboring frequency bins. The fine frequency can be determined by the maximum value and the higher neighboring output. For different windows there are various approaches to find the fine frequency resolution. For rectangular and Hanning windows there are close form solutions [3]. However, for these two windows the sidelobes are not low enough to provide a decent dynamic range and the Blackman window will be used in the present study. For the Blackman window the fine frequency can be obtained as [4]

k=

3X1 − 2X0 X0 + X1

(11.5)

where X0 is the maximum amplitude value in the frequency-domain output and X1 is the higher one of its two neighboring frequency bins. The actual frequency can be calculated as

F = (k0 − 1 ± k)20

MHz

(11.6)

where k0 is the index of the maximum peak and 20 MHz is the resolution of each frequency cell without a fine frequency adjustment. Let us assign k-1 and k1 as the two neighboring indexes of k0. The amplitudes at these indexes are X(k1) and X(k-1). If X(k1) is greater than X(k-1), the plus sign is used; otherwise, the negative sign is used in (11.6). One of the advantages of using an amplitude comparison method against the phase comparison method in Chapter 6 to obtain fine frequency resolution is that only one frame of data is required. The phase comparison method needs two consecutive frames. Figure 11.6 shows the simulation results. In this figure the input signal does not contain noise. The plots show the error frequency versus the input frequency. The frequency changes from 141 to 1,140 MHz in a 1-MHz step. As expected, without the fine frequency resolution the frequency maximum frequency error is about ±10 MHz. With the fine frequency adjustment, the maximum frequency error decreased to about ±2.5 MHz, an improvement of four times. When noise is added, it is possible that the fine frequency may introduce more frequency errors. For example, if an input frequency is exactly at the center of a certain frequency bin, because of noise the two neighboring frequency bins may not be the same amplitude and the fine frequency adjustment will increase the error frequency. However, this argument can also be applied to different fine frequency adjustment methods. Another approach to obtaining the fine frequency is to use a numerical approximation [personal communication with D. Lin, AFRL/RYDR, 2009]. An empirical equation will be generated from this method. In generating the empirical equation, the input frequency is moved from the center of a frequency bin to the boundary between two bins. At each frequency, the ratio of the higher neighboring amplitude to the highest one is taken and referred to as r. The results are shown in Figure 11.7.

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Receiver Designed Through a Conventional FFT Approach

Figure 11.6  Frequency errors versus input frequency: (a) coarse frequency and (b) fine frequency.

Figure 11.7  Amplitude ration r versus input frequency from the center of a frequency bin to the center between two bins.

11.7  Frequency Resolution on Two Signals

265

In this figure, the “*” is the data point. When the input frequency is at the center of a frequency bin, the ratio r is about 0.64 and at the boundary between two bins the ratio r is about 1. The solid line is a curve fitting using a second-order function generated through the MATLAB function “polyfit.” The second-order function can be written as Df = -14.3516r 2 + 47.8773r - 23.5451 or

Df -14.3516r 2 + 47.8773r - 23.5451 = 20 f



(11.7)

where Df is the adjusting frequency and r is the amplitude ratio. For a given r a fine frequency adjustment can be made. The first portion of (11.7) can be applied to this specific 128-point FFT operation. The second part of (11.7) can be applied to a general case, for example, when r = 1, Df/f » 0.5, which is at the center between two frequency bins. When r = 0.64, Df/f » 0. The error frequency with an input frequency from 141 to 1,140 MHz is shown in Figure 11.8. In Figure 11.8(a) there is no noise and the maximum frequency error is less than 25 kHz, which is much better than the approximation method in Figure 11.6(b). Figure 11.8(b) shows the results at S/N = 2 dB. There is no catastrophic error in this plot and the result is comparable to the phase comparison discussed in Chapter 6. The amplitude comparison method has the following advantages: (1) it is simpler to implement, (2) it uses shorter data that is only one frame, and (3) it is more robust than the phase comparison method because no catastrophic frequency error is generated. When a signal is long, many frames of data can be used to calculate the frequency. The same frequency measurement is applied to each frame. The frequencies obtained from all the frames can be averaged to obtain a final frequency. This approach usually can further improve the frequency measurement.

11.7  Frequency Resolution on Two Signals In this section, the frequency reading on two signals will be discussed [personal communication with C. H. Cheng, professor at Miami University, Oxford, Ohio, 2009]. This type of frequency resolution is related to the window shape used in the receiver design. For a 128-point FFT with Blackman window, the FFT results are shown in Figure 11.9. In this plot the signal does not contain noise. In Figure 11.9(a) the input frequency is at 1,010 MHz, which is at the boundary between two frequency bins. Figure 11.9(b) shows the frequency at 1,005 MHz, which is at one-fourth bin away from the center of a frequency bin. From these two plots, one can see that in order to obtain a better dynamic range such as close to 50 dB, the neighboring frequency bins of the maximum output must be discarded. For example, in Figure 11.9(a) once a maximum is detected, two frequency bins on one side and three frequency bins on the other side must be discarded. Since at

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Receiver Designed Through a Conventional FFT Approach

Figure 11.8  Frequency errors versus input frequency with a numerical approximation: (a) no noise and (b) S/N = 2 dB.

11.7  Frequency Resolution on Two Signals

267

Figure 11.9  Frequency plot of one signal with a Blackman window: (a) frequency at 1,010 MHz and (b) frequency at 1,005 MHz.

1,010 MHz the highest two peaks are close in amplitude, it is difficult to predict which one is a peak; therefore, it is difficult to determine which side should discard two frequency bins and which side should discard three frequency bins. Under this situation, the amplitudes of the neighboring frequency bins must be tracked. All the high sidelobes close to the peak must be discarded. A simpler approach is to discard three frequency bins from both sides of the peak frequency bin. This operation degrades the frequency separation capability. If only two frequency bins are discarded from both sides of the peak frequency bin, one remaining frequency bin may have a relatively high output. In order to detect a second signal, its output must be higher than the remaining frequency bins. Thus, the dynamic range will suffer, but the frequency resolution will be improved. This simple discussion can provide a frequency resolution and dynamic range tradeoff. Simulation shows that when two frequencies are separated by 40 MHz, the receiver cannot read both of them. When two signals are separated by 60 MHz, the receiver can obtain the frequencies of both signals. The above discussion is based on traditional receiver design. If the results of Section 9.5, to detect two signals close in frequency, are used, the result may be different and signals close in frequency can be detected. First, the width of the main lobe will be measured based on the highest FFT output. If the width of the main lobe is wider than a predetermined threshold, there are two signals in it and the second highest output will be the second frequency. This approach may cause a relative large frequency error because the second peak may be on the wrong side of the strong signal as discussed in Section 9.8. When the frequency separation is beyond a certain value, the frequency reading should be correct. If there is only one signal

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Receiver Designed Through a Conventional FFT Approach

in the main lobe, all the outputs crossing the detection threshold will be eliminated to detect a second signal outside the main lobe.

11.8  Detection of a Second Signal in a Receiver The discussion in Section 11.7 to detect a second signal by crossing the threshold can generate decent results in simulation. However, it has a problem when it is used in real receiver designs. The main difference between simulated results and real data is that in a real FFT outputs generated from digitized data may have spurs. Figure 11.10 illustrates a signal with several spurs. Some of the spurs can be easily identified as harmonics; however, some of them are difficult to identify. The spurs can come from the digital circuit as well as the analog circuits. When the input signal amplitude increases, the spur amplitudes also increase. The spurs may cross the threshold and be detected as an input signal. These spurs may limit both the single signal and instantaneous dynamic ranges. A better approach to detect a second signal can be based on the amplitude of the strong signal. This approach is discussed in Section 4.5 and will be further discussed here. The approach is to generate a table to determine the threshold. The amplitude of the strong signal is kept at constant amplitude and the frequency is varied across the entire bandwidth or randomly. At each input frequency the highest spur level is recorded. From many input frequencies the highest spur is used to determine a threshold. This process continues for different amplitudes. Thus, the threshold is set with respect to the signal amplitude, or the threshold is set a certain decibel below the input signal as shown in Figure 11.10. One can consider this threshold to be a variable one. These results can be stored in a table. Once the highest amplitude is measured, from the table the threshold of the second signal can be obtained. Although in simulation it is easier to set a threshold for the second signal from the noise level, in reality, the fixed threshold may generate many false alarms. The threshold determined by experimental data may reduce the IDR but eliminate spurs.

Figure 11.10  Threshold setting for a real receiver.

11.9  PA Measurement

269

11.9  PA Measurement Usually, the frequency measurement in a receiver is the most complicated one. Once the frequency measurement method is properly designed, it is anticipated that the rest of the three parameters—PA, time of arrival (TOA), and PW—can be measured in a straightforward manner. The PA measurement can be considered in two steps. The first step is to relate the FFT output to the input power and the second step is to take the pulse shape into consideration. In the first study three approaches are tested and their relative results will be presented. The three approaches use one-, two-, and three-frequency outputs to calculate power. The first approach uses the highest frequency bin to calculate power, the second approach uses the highest two frequency bins to calculate power, and the third approach uses the highest three frequency bins to calculate power. The relative results will be compared. The test is to change the input frequency from the center of a frequency bin to the boundary of two frequency bins in a 1-MHz step. At each frequency the power will be represented by the square of the highest frequency bins as P1 = A12

P2 = A12 + A22



(11.8)

P3 = A12 + A22 + A32 where A1, A2, and A3 represent the highest three frequency bins and P1, P2, and P3 are the power calculated from the three approaches. The variations of the P values versus input frequency are shown in Figure 11.11. Figure 11.11(a–c) shows the results of P1, P2, and P3 versus input frequency, respectively. In this figure P2 has the smallest variation. Thus, the highest two frequency bins are used to power the PA. The actual power measurement should be calibrated and a table will be generated for conversion. Cheng suggested that the PA measurement can be achieved through similar approach as frequency measured through amplitude comparison as discussed in Section 11.6 [personal communication with C. H. Cheng, professor at Miami University, Oxford, Ohio, 2009]. The inverse ratio of the maximum to its higher neighbor is defined as r as in (11.7). Another quantity can be defined as an amplitude factor, which is also a ratio as:

ra =

Am Amb

(11.9)

where Am is the measured maximum FFT output with any input frequency and Amb is the maximum FFT output when the input frequency is on a frequency bin. When the frequency is on a bin, most of the energy is on one bin. Thus, the ra is less than unity. It should be noted that if a rectangular window is used, the energy will be on one bin when the input frequency is one bin. When a Blackman window is used, there are sidelobes at neighboring bins even if the input is on a frequency bin.

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Receiver Designed Through a Conventional FFT Approach

Figure 11.11  Relative power variation versus input frequency: (a) P1, (b) P2, and (c) P3.

The Amb can be related to input S/N. The procedure is to related r to ra. The ra versus r is shown in Figure 11.12. The * is the calculated points and the solid line is the approximation of the points. The relation between ra and r can be written as

Figure 11.12  ra versus r plot.

11.10  TOA and PW Measurements

271

Figure 11.13  Typical pulse shape.



ra = 0.89r 3 − 2.67 r 2 + 2.22r + 0.43

(11.10)

If r is measured, the value of ra can be obtained from this equation. From ra and Am, Amb can be obtained from (11.9). The S/N can be obtained from Amb from a predetermined table. It appears that this approach can produce decent results. Once the relation between the input power in a frame to the FFT outputs is established, the pulse shape will be taken into consideration. Figure 11.13 shows a typical pulse shape with leading and trailing edges. The frames labeled with A to G represent the FFT outputs crossing the threshold. It is usually desirable to measure the PA at the steady state of the pulse. Frame B is the second frame and is filled with signal data; however, it contains data in the leading edge. Using the maximum output from this frame will not generate an accurate PA result. A method to select the proper frames for the PA measurement should be studied. The simplest approach is to ignore the outputs from the end frames. The PW plays an important role in PA measurement for this simple approach. If the signal is long, the amplitude can be measured from many consecutive frames. In Figure 11.13 if the outputs from frames D to E are averaged, the PA measured should be more accurate. If the input signal is irregularly shaped, even defining the PA becomes a problem. Under this condition compromise must be made. Although the PA appears to be an easy parameter to measure, to obtain an accurate value some effort is needed.

11.10  TOA and PW Measurements A simple-minded approach to measure TOA and PW will be discussed first. The TOA can be measured from the first frame in which a signal is detected. In this approach the time resolution is 50 ns, which can be considered the maximum TOA error. The signal will keep being detected in the following frames if the PW is over many windows. Since threshold of this channel is lowered to accommodate a weak signal, the pulse should be measured continuously through the entire PW. When the

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Receiver Designed Through a Conventional FFT Approach

Figure 11.14  Correct approach to define PW.

signal is no longer detected in a certain frame, the previous frame should be used as the time of departure (TOD). The TOD value minus the TOA value should be the PW. In some applications, a finer TOA is desirable. Let us refer the frame producing the TOA data as the TOA frame. A finer TOA might be achieved by reanalyzing the TOA data frame. If the signal is strong, by examining the amplitude of each data point in the TOA frame, one might find a certain point starting to cross a certain threshold. Since the input data are real, the amplitude of each individual data point depends on the input frequency and the initial phase. Even taking all these problems into consideration, the TOA resolution should be better because it measures the TOA in the 50-ns time frame and each data point has a time resolution of less than 0.4 ns. The PW measured through the above method is PA dependent. When the signal is strong, it can cross the threshold at an earlier time in the leading edge and at a later time in the trailing edge. The PW measured is longer for a strong signal and shorter for a weak signal. This is an undesirable result. The correct way to define a PW is shown in Figure 11.14. The PW is measured 3 dB below the amplitude of the pulse. With this method the PW measured should be more or less independent of the PA but the time resolution will be 50 ns and the fine resolution from the previous paragraph cannot be applied. From Sections 11.9 and 11.10 one can learn that even to obtain a good PA and PW measurement, the answer is not straightforward. Efforts are needed to handle many different input conditions.

11.11  Combine All the Information on One Input Pulse An EW receiver should report PDW on a pulse-by-pulse basis, not on a frame-byframe basis. In order to generate the PDW on a certain pulse, one must wait until the end of the pulse. In other words, the TOD must be obtained before the PDW can be generated. In most EW receivers there is an upper limit for PW. If the input signal PW is longer than the limit, it is usually declared as continuous wave (cw) and the PDW will be reported.

11.12  Some Possible Improvements on an FFT-Based Receiver

273

The general operation of a receiver can be summarized in the following manner. 1. The receiver will take 128 input data and perform an FFT. The amplitude of the FFT outputs will be obtained. 2. Compare all the FFT outputs with the predetermined threshold. If the threshold is not crossed, the data is considered as containing no signal and will be discarded. 3. Count all the frequency bins crossing the threshold. If several adjacent frequency bins, such as five frequency bins, cross the threshold, they can be considered as one signal. The frequency bin with the maximum amplitude will be considered as the desired one. When the frequency bins crossing the threshold are separated into two or more groups, they can be considered as two or more signals and their maximum will be identified. The local peak discussed in Section 4.3 can be used to determine the number of signals. The FFT outputs of the following frame must cross the threshold at the same frequency or within ±1 frequency bin. If the FFT outputs of the following frame do not cross the threshold at the same frequency, it is considered to be no detection and the FFT output data will be discarded. 4. When a signal is successfully detected, the second frame output will be used to determine frequency and amplitude. A fine frequency can be obtained either from the amplitude or the phase comparison method. The first frame can be used to provide the TOA information. All the information must be stored temporarily until the TOD is reached. 5. The threshold of the frequency bin with the signal will be lowered. 6. FFT will be performed on all input data in a continuous manner. As long as the FFT output crosses the frequency bin with the new threshold, the frequency and amplitude will be stored. 7. When the FFT output does not cross the new threshold, it is considered as the end of the pulse. The last frame with FFT output crossing threshold will be the TOA. 8. One can average all the frequencies and amplitudes measured to report as the measured frequency and PA. The TOA is already obtained and the PW can be obtained from the TOD-TOA. From the above simple discussion, one can see that to build a receiver many tedious operations and data storage are required.

11.12  Some Possible Improvements on an FFT-Based Receiver Several ideas can be used to improve the performance of the FFT-based receiver. Four parameters can be improved on the receiver: the sensitivity, the frequency resolution, the time resolution, and the close-in frequency detection. They are discussed as follows. 1. S ensitivity improvement: This approach is through noncoherent integration. The consecutive FFT outputs of all the channels can be summed together. The

274

Receiver Designed Through a Conventional FFT Approach

operation is noncoherent integration. Summing a small number of frames together can produce an efficient improvement. For example, with Pfa at 10-7 and a Pd of 90%, summing 2 and 4 samples will improve the sensitivity of about 2.7 and 5 dB, respectively. Comparing with a coherent integration of 2 and 4 samples, the improvements are 3 and 6 dB, respectively. If amplitudes of several FFT outputs in time are summed, the receiver sensitivity can be improved. 2. Frequency resolution: The frequency resolution can be improved through phase as discussed in Chapter 6. If the phases between two consecutive outputs are compared, the frequency resolution can be improved. Although the amplitude comparison method might be a better approach, the phase comparison operation is needed to detect both biphase shifting keying (BPSK) and chirp, which are discussed in Chapters 14 and 15. 3. Time resolution: The time resolution depends on the window length, which is 50 ns. It usually happens that the first and last data frames are partially filled with signal data and the second frame is fully filled with signal data. One way to better estimate the TOA and TOD is to compare the amplitude of the FFT outputs from the first and the last frames to the second frames. If the ratio of the second frame output to the first frame is taken, extrapolation can be used to estimate a finer TOA time. If the ratio is close to 1, the TOA will be close to the beginning of the first frame. If the ratio is small, the TOA will be close to the ending of the first frame. A similar method can be used for the TOD measurement. This method can be applied to the simple approach discussed in Section 11.10. 4. Two-signal resolution: Two signals with a frequency separation beyond 60 MHz can both be detected. This performance can be considered as relatively poor. Methods discussed in Chapter 9 can be used to improve the performance. Another clear way to improve this performance is through squaring the raw input data then performing an FFT operation [personal communication with D. Lin, engineer at AFRL, 2008]. The squaring operation will produce a summing and difference frequencies. If two signals exist and are separated close in frequency, the difference frequency will be close to zero. By searching the output frequency close to the direct current (dc), the difference can be found. If one frequency is found through the regular FFT operation, the other frequency can be either below or above the measured frequency. If two signals are totally overlapped in time, the frequency of the second signal might be difficult to determine. However, if two signals are partially overlapped, the frequency of the second signal can be determined from the nonoverlapped portion of the second signal.

11.13  Receiver Measurements The purpose of this section is to provide some receiver testing input conditions. It is desirable to have a software program that can evaluate the performance of a digital receiver before it is actually built. The approaches in this chapter can be put into such a program. The performance of the designed receiver can be obtained. The

11.13  Receiver Measurements

275

most important information is that the limitations and the possible deficiencies can be revealed before hardware money is spent. In order to test the receiver performance, a set of standard input conations are desirable. Such standards are hard to come by. The receiver testing procedure in [1] still can provide useful information. Usually a receiver is tested through trial and error. The direction of testing is based on the receiver performance. If a special problem is discovered, the evaluation will concentrate on it and either find a solution or an explanation. Experienced receiver testing engineers [personal communication with D. Jacobson and B. Mayhew, engineers at ITT, 2009] suggested the outlines of the following tests. The tests are limited to one or two signals. They will be briefly presented as follows.

1. One signal test is used to obtain the following information: Sensitivity and single signal dynamic range; Minimum PW capability; Frequency accuracy; PA accuracy; TOA and PW accuracy.

Because of the possible number of input combinations (such as input frequency, PA, PW), it is difficult to obtain a representative result. Even with this problem the test results should provide some valuable information. 2. A two-signal test will include the following three input conditions. Two cw signals: Parameters included in this test are input frequency, frequency separation, amplitude value, and amplitude difference.

Figure 11.15  Pulse overlapping conditions: (a) exact overlap, (b) partial overlap, (c) one pulse inside another.

276

Receiver Designed Through a Conventional FFT Approach

One cw and one pulse: The input parameters are similar to the condition listed above, but TOA and PW will be the additional parameters. Two pulsed signals: The two pulse input conditions are similar to a single pulse test and only double the number of input parameters. Three pulse overlapping conditions, shown in Figure 11.15, will be included: total overlap, partial overlap, and one pulse in another one. From all the possible cases, it is obvious that the receiver can only be tested under very limited conditions. One possible approach is to use random input signals. Another problem that needs to be answered is the meaning or the significance of the random test. If two receivers are tested, there should be some criteria to tell which one is better. This is one of the difficulties in receiver research. Another possible approach to evaluate a receiver is mission-specific, that is, whether the receiver under test can fulfill the required mission. If this approach is adopted, several sets of standard signal environments should be available. The receiver should be designed an evaluated against specific signal environments. Under this test it might be better to have an electronic warfare (EW) processor to sort the PDW and identify the individual signals.

11.14  Summary The receiver designed in this chapter is rather straightforward. The receiver is designed against the minimum PW. The FFT length is based on the minimum PW. Once the FFT length is determined, the receiver sensitivity and frequency resolution are determined. The receiver performance depends on the minimum PW. A longer pulse will produce similar frequency accuracy and resolution as a short one. The long pulse and the short pulses should have the same sensitivity. Amplitude and phase comparison methods can improve the frequency measurement capability. If decent PA, PW, and TOA are required, special efforts might be needed to produce the required results. One major problem is receiver design is the testing procedure. Effort must be spent on this subject so that a general accepted receiver performance standard can be established. The dynamic range discussed in Chapters 3 and 4 is an especially confusing term.

References [1] [2] [3] [4]

Tsui, J., Microwave Receivers with Electronic Warfare Applications, New York: John Wiley, 1986. Slolnik, M. I., Introduction to Radar Systems, New York: McGraw-Hill, 1962, p. 34. Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Chapter 8, Norwood, MA: Artech House, 2001. Xue, H., and R. Yang, “Optimal Interpolating Windowed Discrete Fourier Transform Algorithms for Harmonic Analysis in Power Systems,” IEE Proceedings-Generation, Transmission and Distribution, Vol. 150, No. 5, September 15, 2003, pp. 583–587.

C h a p t e r 12

Receiver Designed Through a Multiple FFT Operation

12.1  Introduction In Chapter 11 fast Fourier transform (FFT) is used to build a receiver. The length of the FFT is a fixed value to process the desired minimum pulse width (PW). The sensitivity and the frequency resolution depend on the minimum PW requirement. In this chapter multiple FFT lengths will be used to build a receiver in the hope that the sensitivity and frequency resolution will be PW dependent. This is a common idea shared by several electronic warfare (EW) receiver designers. The processing of multiple FFTs with different lengths should be computation intensive. This method may be equivalent to several receivers operating in parallel. There are several ways to accomplish a multiple numbers of filters. One obvious approach is to use multiple FFT lengths against the input data, such as FFT operations with 64, 256, … points. Another approach is to use a cascade operation, that is, to build a filter bank from the outputs of a previous filter bank. To accomplish this design there might be several possible approaches such as using windowed FFT operations and polyphase filters. The cascaded filter designs will be briefly discussed in the following sections. There are many different combinations one can select to form a filter bank such as the number of outputs and their frequency resolutions. The appropriate approach is to design against a certain performance goal. In this book only the technology is of interest. The only requirement discussed in Chapter 11 is to measure minimum PW of 100 ns. The design in Chapter 11 cannot really fulfill this requirement with a full sensitivity. Since in this chapter multiple filter banks can be used, it is desirable to measure the minimum PW with fully designed sensitivity. For longer pulses the receiver sensitivity will improve. Based on this assumption, three filter banks will be used to illustrate a receiver design idea. The sensitivity and frequency resolution dependent on PW will be illustrated.

12.2  Cascaded Filter Banks Through FFT Operations This is to illustrate the performance of filter banks built through FFT operations. Only two filter banks are simulated. The frequency resolutions of the two filter banks are arbitrarily selected. It is desirable to have the second filter bank to produce 1,024 frequency bins or 512 frequency bins with independent information for a real 277

278

Receiver Designed Through a Multiple FFT Operation

input signal. With a 2.56-GHz sampling frequency, the frequency resolution is 2.5 (2,560/1,024) MHz. In order to keep only a 1,000-MHz input bandwidth, 401 frequency bins will be kept. The overall input bandwidth is 1,002.5 MHz (2.5 × 401) to just cover the desired frequency range. In a real receiver design, the frequency range should be wider than the selected value to accommodate the possible analog bandpass filter shape. Since this is only a study, the minimum frequency is selected. There are several ways to accomplish this goal, such as a 32-point FFT followed by a 32-point FFT or a 64 point FFT followed by a 16-point FFT. In the following example, the 32-point FFT followed by a 32-point FFT example will be used for illustration. The input frequency is arbitrarily chosen as 600 MHz and 1,024 real data are generated without noise for a sampling frequency at 2.56 MHz. The input data are divided into 32 blocks in the time domain and each block has 32 points. A Blackman window is applied on the input data and a 32-point FFT is performed to generate 32 frequency bins in the frequency domain. Since the input data are real, only 13 (2 ~ 14) frequency bins are kept (see Section 6.1). This operation is repeated 32 times in the time domain. The output data are two-dimensional with 13 frequency bins and 32 points in the time domain. The second FFT with a Blackman window is performed on the 32-point time-domain data 13 times to generate all the data in the frequency domain with 416 (13 × 32) frequency bins. This is considered as the second filter bank. This filter bank has more output frequency bins than discussed in the previous paragraphs. The operation can be illustrated in Figure 12.1. Since the first FFT outputs are complex, the second FFT operation has 32 outputs instead of 16. The first filter bank has 13 frequency outputs and the second filter bank has 416 outputs. The frequency-domain outputs of these filter banks are shown in

Figure 12.1  Illustration of two 32-point FFT operations.

12.3  Cascaded Filter Banks Through Polyphase Filters

279

Figure 12.2  Two filter banks from two approaches: (a) first filter bank, (b) second filter bank from cascade, and (c) 1,024-point FFT.

Figure 12.2. The first filter bank has 32 time-domain outputs and only the amplitude of the first time domain is shown. Figure 12.2(a) shows the 13 frequency outputs of the first filter bank and the results are the expected results. For 32 FFT the frequency resolution is 80 MHz and a 600-MHz input signal is at the boundary of two bins; thus, the output at Figure 12.2(a) has four outputs and they are adjacent. Figure 12.2(b) shows the outputs of the second filter bank. There are four visible outputs. These outputs are different from one 1,024 FFT output, which is shown in Figure 12.2(c). Each frequency component in Figure 12.2(a) produces 32 outputs in the frequency domain and all the outputs are arranged in order, resulting in Figure 12.2(b). In Figure 12.2(b) there are four outputs and each output is generated from the one output frequency bin of Figure 12.2(a). As a result, the frequency outputs are not close together but separated by 32 bins. From these plots, it is obvious that the cascade approach can produce more problems in designing the encoder. In Figure 12.2(c) there is only one peak value, which means that all the output energy is at one frequency bin. In detecting weak signals, the cascade approach should be less effective because the output energy is spreading into several frequency bins.

12.3  Cascaded Filter Banks Through Polyphase Filters In this section the cascade approach will use the polyphase filter for frequency separation. If a filter bank with sidelobes is close to 60 dB, which is close to the

280

Receiver Designed Through a Multiple FFT Operation

performance of a Blackman window, from limited simulations it appears that the polyphase filter is not suitable for a long FFT operation. For example, to obtain 59-dB sidelobes, a 16-point FFT operation requires 128 input data points; for a 32-point FFT operation, it requires 256 points of data. The long data required may restrict the minimum pulse requirement. In the following example, two 16-point FFT operations will produce 128 channels. Because the input is real after the first 16-point FFT, only 8 of the 16 outputs have independent information. After the second 16-point FFT there are 128 outputs and only 101 (14 ~ 114) channels are kept to accommodate a 1,000-MHz input bandwidth. These outputs do not fulfill the required 1,024 output frequency bins. For the first FFT operation 128 input data points are required to perform one FFT. The input data will slide 16 points to generate the second frame of outputs. To generate 16 frames of data, the total number of data points needed is 2,160 (16 × 127 + 128). For the second filter bank, only 128 data points are required. In this simulation the input frequency is at 700 MHz. The outputs of the two filter banks are shown in Figure 12.3. These results are similar to the results in Figure 12.2. The first filter produces two adjacent outputs as shown in Figure 12.3(a). The outputs from the second filter banks are shown in Figure 12.3(b). The two outputs are separated by 16 frequency bins. This approach is not suitable for receiver designs either. These simple-minded cascade filter bank approaches produce undesired frequency outputs.

Figure 12.3  Cascade filter banks through polyphase approach: (a) polyphase outputs from a 16-point FFT operation and (b) polyphase outputs from the second FFT operation.

12.4  Half Band Filter

281

12.4  Half Band Filter One of the possible of filter designs that can fulfill the cascade approach is through a half band filter [personal communication with C. Ward, engineer at ITT, and D. Lin, engineer at AFRL, 2009]. The approach is to divide the input into low and high bands. One simple illustration is shown in Figure 12.4. In this figure, the entire input bandwidth is from 0 to fs /2 for real signal sampled at frequency fs. The first step is to build two filters covering this bandwidth, thus, one filter is from 0 to fs /4 and the second one is from fs /4 to fs /2. One simple approach is to design a lowpass filter as shown in Figure 12.4(a) with passband from 0 to fs /8. Since it is a lowpass filter, the entire bandwidth will be fs /4 from –fs /8 to fs /8 and this provides the desired bandwidth. The second step is to shift this filter to the desired frequency range. To shift the input frequency to the right by fs /8 is to multiply the filter outputs by exp(j2pfs /8). The result is shown in Figure 12.4(b). It should be noted that this operation changes output from real to complex because exp(j2pfs /8) is complex. The third step is to shift the output to the upper frequency range by exp(j2p3fs /8) and the result is shown in Figure 12.4(c). This filter structure can be used repetitively to the outputs of each filter output. A 25-tap filter is used and the lower bandpass filter is shown in Figure 12.5, where the sidelobes are more than 70 dB down. Four sets of data are generated from the filter coefficients and the quantities cos(2pfs /8), sin(2pfs /8), cos(2p3fs /8), and sin(2p3fs /8) for the repetitive operations. The filter responses are shown in Figure 12.6,

Figure 12.4  Basic concept of half band filters: (a) lowpass filter, (b) lower bandpass filter, and (c) higher bandpass filter.

282

Receiver Designed Through a Multiple FFT Operation

Figure 12.5  Responses of a 25-tap FIR filter.

Figure 12.6  Two consecutive half filter banks: (a) seventh-level outputs and (b) eighth-level outputs.

12.5  Selection of FFT Lengths

283

where there is no noise added in the simulation and the input frequency is arbitrarily chosen at 615 MHz. The amplitude scale is arbitrarily limited from -100 to 0 dB and the spur responses below -100 dB are not shown. The dynamic range is about 70 dB, which agrees with the results in Figure 12.5. Figure 12.6(a, b) shows the outputs from the seventh-level (128 outputs) and the eighth-level (256 outputs) operations, respectively. One important factor in this figure is that for one input signal the outputs may have one peak or two adjacent peaks of the same amplitude and this is the property for this type of cascade filter banks. These outputs are considered accepted for receiver design but the approaches in the previous sections do not provide the desired results. The half band filters have some unique properties. For example, for the 128 outputs the frequency resolution is 10 MHz and the input frequency of 615 is at the boundary between two frequency bins. Under normal filter outputs the signal will split equally between two adjacent frequency bins. Figure 12.6(a) shows that there is only one peak output. For the 256 outputs, the frequency resolution is 5 MHz, and the input frequency is on a frequency bin. However, Figure 12.6(b) shows that there are two outputs of equal amplitude. Not only are the amplitudes equal, but the complex quantities of these two bins are also identical. In building the half band cascade filter banks, the input data are shifted by 1 point per calculation. If every output is taken into consideration, a good time resolution can be obtained. However, the transient effect, which causes rabbit ears, will occur in many output frames. This phenomenon affects the receiver encoder design and makes the task very complicated. To mitigate this problem one can avoid processing every output frame and process only selected frames. This problem is equivalent to the FFT operation. For example, one can perform a 256-point FFT operations by shifting the input data every input point to obtain a fine time resolution. This operation not only is time-consuming but also generates long rabbit ears, which are difficult to encode. The general approach is to shift 256 points and simplify both the FFT operations and encoder design as the approach discussed in Chapter 11. If the outputs from the half band filters are processed in a similar manner, it seems that the FFT might be a simpler approach. This is why the half band filter is not further discussed for receiver designs in this book.

12.5  Selection of FFT Lengths Since Sections 12.3 and 12.4 illustrated that the FFT-related cascade approach does not provide the desired results, an individual FFT operation with different lengths will be used for the receiver design. In the design we will use three different FFT lengths to process the input signal. The FFT length selection is based on the minimum pulse width (100 ns) requirement. A similar argument is presented in Section 11.3. The FFT length selection in Section 11.3 cannot satisfy the 100-ns requirement under certain conditions. Since multiple FFT lengths will be used, it is desirable to fulfill the minimum PW requirement; therefore, the minimum FFT length is 64 points, which cover a total time of 25 ns with a sampling frequency of 2.56 GHz. If a minimum of two frames full of data is required to process a signal, the minimum data length is 75 ns, which is shorter than the minimum PW

284

Receiver Designed Through a Multiple FFT Operation

requirement. Under the worst mismatch between frames and the input signal, it requires three frames to generate two full frames with data as discussed also in Section 11.3. The other two FFT lengths are arbitrarily selected as 256 and 1,024 points, which are 4 and 16 times the minimum FFT length. The corresponding frames time are 100 and 400 ns. Using these three frames and the worst scenario assumption, the pulse widths that can be processed are 300 and 1,200 ns. The corresponding frequency resolutions from the three FFT operations are 40, 10, and 2.5 MHz, respectively. The sensitivity will improve by about 3 dB whenever the FFT length is doubled. The total number of channels is 27, 101, and 401 (see Section 6.1), if the input frequency is from 141 to 1,140 MHz. One easy way to check the number of channels is that the number of channels times the frequency resolution should be greater than 1,000 MHz, the required total bandwidth. The minimum window (or frame) of 64 points will be shifted by 64 data points, which means that there is no overlap in time. The two longer windows of 256 and 1,024 points can move in a nonoverlapping or overlapping manner. The overlapping move provides two advantages: a better match to the signal and a better time resolution. As discussed in Chapter 7, when the window approximately matches the signal, it is close to a match filter and the signal can be easier detected. Overlapping window shifting has a better time resolution. The overlapping move can cause a transient effect or rabbits, which complicates the encoder design. This phenomenon can be illustrated in Figure 12.7. In this figure the data window overlapping three-fourths of the window length, such as moving the 256-point window 64 points each time. The leading edge of a pulse is shown and it appears in 4 data windows. As a result, 4 windows are partially filled with data as illustrated. For a partially filled window, the spectrum after the FFT operation will spread into neighboring frequency bins, which are referred as a

Figure 12.7  Illustration of overlapping time shifting.

12.6  Threshold Determination and Probability of Detection

285

rabbit ear. When the signal is strong, the energy can spread into many neighboring bins and these rabbit ears may cross the predetermined threshold. The encoder must avoid the rabbit ears and detect the input signal in the correct frequency bin. This phenomenon always occurs in analog channelized receivers. The conventional approach is to detect the signal after the rabbit ears. In this special case the rabbit can be 4 frames long. If the input signal is weak, the rabbit ear may be shorter than 4 frames because windows filled with less signal data will spread less energy to its neighboring frequency bins. This similar phenomenon occurs at the trailing edge of a pulse. If the 1,024-point window moves 64 points every time, 16 consecutive windows will contain a leading or a trailing edge of a pulse. In other words, the rabbit ear can be 16 frames long. If nonoverlapping shifting is used, the rabbit effect only appears in one frame. This means that the rabbit is very short. In Chapter 11 the receiver is designed with nonoverlapping shifting and the encoder is relatively simple. In Chapter 13 the receiver is designed with overlapping shifting and the encoder is rather complicated. In this chapter the nonoverlapping is used to simplify the designs.

12.6  Threshold Determination and Probability of Detection [1–3] For each FFT length, there must be a threshold. The probability of false alarm is based on two consecutive detections of two frames and the frequency reading from the two frames must be in the same bin or in the neighboring bins to be declared as detection. In other words, to generate a false detection, only three output bins in the second frame are accountable, regardless of the channel width. In Chapter 11 the probability of false alarm is based on one channel analysis because this is a familiar term to receiver engineers. For simplicity, let us refer each FFT operation as one receiver. In this chapter since there are three receivers. If the probability of false alarm is based on individual channel, each receiver will have slightly different probability of false alarm. It is reasonable to assign the three receivers with the same probability of false alarm and we will use Pfar = 10-7 to represent the receiver probability of false alarm. Thus, the probability of false alarm is different from all the three cases in Section 11.4. The probability of each frame can be written as



Pfa =

Pfar N

Pfa =

2 Pfar 3Pfaw = N N

Pfaw =

(12.1)

Pfar 10−7 = 3 3

where Pfa is the probability of false alarm per channel and Pfar is the probability of false alarm of the receiver (10–7), Pfaw is the probability of false alarm per frame. This equation is different from equations in Section 11.4 because of a different definition.

286

Receiver Designed Through a Multiple FFT Operation Table 12.1  Sensitivity Calculation Results 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

FFT length Equivalent data length (ns) Mini PW can be detected 100% Frequency resolution (MHz) Number of channels (channel number) N Probability of false alarm per receiver Probability of false alarm per channel (Pfa) Probability of false alarm per frame (Pfam) Thresholds Sensitivity (required S/N) (dB) Probability of false alarm per receiver Probability of false alarm per channel (Pfa) Probability of false alarm per frame (Pfaw) Thresholds Sensitivity (required S/N) (dB) Sensitivity difference dB (row 10–row15)

64 25 75 40 27(3 ~ 29) 10-7 3.70 × 10-8 1.83 × 10-4 12.83 -0.3 2.7 × 10-6 10-7 9.48 × 10-4 11.56 -1 0.8

256 100 300 10 101(14 ~ 114) 10-7 9.90 × 10-9 1.83 × 10-4 25.90 -6.3 1.01 × 10-5 10-7 1.83 × 10-3 22.14 -7.25 0.95

1,024 400 1,200 2.5 401(56 ~ 456) 10-7 2.49 × 10-9 1.83 × 10-4 51.72 -12.3 4.01 × 10-5 10-7 3.66 × 10-3 41.85 -13 0.7

The probability of false alarm calculated from (12.1) is listed in the Table 12.1. A Blackman window is used on all three window lengths. From these probabilities of false alarm the threshold values can be found. The threshold value is also listed. Based on the threshold the probability of detection can be found and the results are shown in Figure 12.8. At 90% of detection, the required S/N is listed in the tenth row of Table 12.1. Comparing with the sensitivity of -4.2 dB calculated in Section 11.4, the result fits well between the 64- and 256-point FFT lengths and the differ-

Figure 12.8  Probability of detection for three different FFT lengths.

12.7  Additional Detection Scheme to Improve Pulse Width Capability

287

ence is about 3 dB. In this table the sensitivity improves about 6 dB when the data length increases 4 times and this is the expected result. If the probability of false alarm of one channel is given, the results are listed in rows 11 to 15 of Table 12.1. The sensitivity listed in row 15 is slightly better than that listed in row 10. These are the expected results because the false alarm is also higher. The difference in sensitivity is listed in row 16 (difference between rows 10 and 14), which is less than 1 dB. These results agree with the discussion in Section 6.4, when the FFT length increases by four times and the sensitivity improves by 6 dB. The results for the same FFT length need more explanation. For example, for the 1,024-point FFT, row 6 shows the probability of false alarm at 10-7 but in row 11 the probability of false alarm is 401 times higher. From [1–3], it is expected that the sensitivity difference is about 2 dB due to this change of probability of false alarm. A single frame detection is simulated separately to check this sensitivity change and the result is close to 2 dB. In this simulation the difference is only about 0.7 dB. It appears that the small difference is caused by the two-frame detection method. In the two-frame detection the probability of false alarm per frame is listed in row 8 as 1.83 × 10-4 and in row 13 as 3.66 × 10-3 and their difference is 20 times rather than 401 times. When the probability of false alarm decreases 20 times from 10-7, for a single frame detection, the sensitivity difference is about 1 dB, which is close to the simulation results. The encoding scheme for the three FFT lengths are similar to that discussed in Chapter 11 and will not be repeated here. Additional encoding is needed and that is how to combine all three outputs together.

12.7  Additional Detection Scheme to Improve Pulse Width Capability In this section let us further discuss the sensitivity and the PW of the receiver. For the 100-ns and 400-ns frames listed in Table 12.1, row 3, the minimum PW is 300 ns and 1.2 ms and they are relatively long. In order to detect shorter pulses, the single frame detection will be used to the 256 and 1,024 data point frames in addition to the two frame detections. For the single frame detection with 256-point FFT frames, the time is 100 ns. For this type of detection, when the pulse width is 200 ns, one full frame of data can be obtained. It is interesting to determine the sensitivity with this type of detection when the input signal is split into two windows. For example, a PW of 100 ns can be divided into two frames in a different manner. Intuitively, when the input signal of 100 ns is equally divided into two consecutive windows, the probability of detection should be the worst. Under this condition, the output of either one of the two frames crossing the threshold is considered as a signal detected. The threshold is 35.18 listed in Table 6.3. The detection results are shown in Figure 12.9. The sensitivity (90% detection) is at S/N of above -0.2 dB. This value is very close to the two frame (64 points) detection result listed in row 10 of Table 12.1. When the signal is divided by ¾ (192) and ¼ (64) into two frames the sensitivity is at S/N = -4 dB. When the signal is all in one frame, the sensitivity is at S/N » -4.6 dB, which agrees the results listed in row 7 of Table 6.6. This value can be compared with the sensitivity of -7.25 listed in row 15 of Table 12.1.

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Receiver Designed Through a Multiple FFT Operation

Figure 12.9  Probability of detection for one frame of data split into two frames.

It is interesting to note that when the signal splitting changes from 50/50 to 75/25 in two consecutive frames, the sensitivity improves by about 4 dB because the 75/75 splitting can be considered as the average between the 50/50 and 100/0 cases. When the signal changes from 75/25 to 100/0, the sensitivity only improves by about 0.6 dB. From this simple illustration, one can consider that using detection with one frame (256 points), the average sensitivity can improve by about 4 dB. In this operation, no additional FFT operations are required. The only additional operation required is to compare with an additional threshold. Similar argument can be applied to the 1,024-point FFT operations. In order to detection a signal, by using two frames of 1,024-point FFT, the minimum pulse width for 100% detection is 1,200 ns. If detection with one frame is used, the minimum pulse width can be reduced to 800 ns. For a signal of 400 ns, similar argument can be applied as the previous example. The threshold is 70.74 (Table 6.3) obtained with a probability of false alarm rate of 10-7. The signal can be detection with by two frames of a 256-point FFT at S/N of -7.25 dB (row 15, Table 12.1). When the signal is split evenly between two consecutive frames, the sensitivity is at S/N of -6.4 dB. When the signal is split 75/25, the sensitivity is at S/N of about -10.2 dB, which is about 3 dB better than that detected by two 256-point frames. The study results can be listed in Table 12.2. In the detections for the 256 and 1,024 FFT outputs in Table 12.2 there are two groups of thresholds, referred to as the one-frame and two-frame thresholds. The one-frame threshold is higher than the two-frame threshold. This additional detection can improve the detection sensitivity over a certain PW range without additional FFT operations.

12.8  Short Pulse

289 Table 12.2   Detection Results from One Frame 1 2 3 4 5 6 7

Data length Minimum pulse fully detected (ns) Threshold required for probability of false rate of 10-7 Sensitivity (S/N dB) 100 ns equally divided (dB) Sensitivity (S/N dB) 100 ns 75/25 divided (dB) Sensitivity (S/N dB) 400 ns equally divided (dB) Sensitivity (S/N dB) 400 ns 75/25 divided (dB)

256 200 35.18

1,024 800 70.74

-0.2 -4.0 -6.4 -10.2

12.8  Short Pulse In the following sections some input signal conditions will be studied. The purpose is to find out the outputs of the three FFT lengths under different signal conditions. From these outputs an encoder can be designed. For the sensitivity study on a short pulse discussed in Section 12.7, the input signal is short and weak. Under this condition, the signal usually cannot be detected from the output of the 1,024-point FFT operation. To detect a short pulse, the outputs from the 64- and 256-point FFTs will be used. Two thresholds with one and two frames can be applied to the 256 FFT outputs. When a short pulse is strong, all three FFT outputs may pass the one-frame threshold. Thus, there are three detections. For example, when the input signal is at S/N = 15 dB and PW = 100 ns, the outputs from the 1,024 FFT outputs are shown in Figure 12.10. In this figure, the 100-ns pulse occurs only in one frame. Figure 12.10(a) shows all the 401 outputs. Figure 12.10(b) shows the fine structure of Figure 12.10(a). These figures show that the outputs have many peaks, which are caused by noise and spectrum spreading. When the input signal is strong, usually there is only one peak. If a short strong input signal is split into two consecutive frames, it might be detected as a signal with several output peaks. From this figure, one can see that if the 1,024 FFT outputs are used for detection, it may cause a false alarm because of the possible multiple peaks. If a short strong pulse is divided into two frames in two 1,024-point FFT frames, it might be detected in both frames. Thus, it is possible that a strong short pulse can be detected in all three FFT outputs, although the chance might be low. The following simple rule can be applied to find the input signal. It is possible that a weak short signal sometimes can only be detected by the one-frame 256-point FFT operations but not by the two-frame 64 FFT operation such as S/N from -1 to -4 dB (Tables 12.1 and 12.2) and the signal split ¼ and ¾ between two consecutive frames. Under this situation, the pulse width will be encoded as 100 ns and the TOA will be the beginning of the frame. The amplitude and frequency can be found from the amplitude and the location of the frequency outputs. A better frequency reading might be obtained from the amplitude comparison as discussed in Section 11.6. When a short signal is detected only by the one-frame 256-point FFT operation, the detection might be ignored to reduce

290

Receiver Designed Through a Multiple FFT Operation

Figure 12.10  Frequency domain outputs of a 100-ns pulse in a 400-ns window: (a) output or entire frequency domain and (b) detailed outputs near the peak frequency bin.

the probability of false alarm. This decision is based on the requirement of the receiver. If a short signal is detected by the 64-point FFT operation, these outputs should be used to generate the TOA and PW information because they provide better time resolution. The amplitude can be obtained from the frequency outputs and frequency resolution from an amplitude comparison method. When the PW of the input signal measured is less than a certain value (say, 400 ns), the outputs from the 1,024-point FFT operation should be ignored. Although the 1,024-point FFT outputs have a better frequency resolution, it cannot separate two signals close in frequency when the input signal is short. Only when the signal is longer than 400 ns, will the outputs from the 1,024-point FFT be used.

12.9  Long Weak Signal It is arbitrarily assumed that the weak long pulse width is around 400 ns because under this condition the signal may either detected by the 256- or 1,024-point FFT operations. If the S/N is from the range of -7.25 and -10.2 dB and the signal is properly divided into two consecutive 1,024-point frames, it might be detected only by one of the 1,024-point FFT operations. Under this condition, the PW will be encoded as 400 ns and the TOA will be the beginning of the frame. The amplitude and frequency can be measured from the frequency domain outputs. Since the time

12.10  Signals Detected by Multiple Numbers of Windows

291

resolution is rather poor, that is, 400 ns, both the TOA and PW measured will not be accurate. However, when a signal is weak, it is difficult to measure the time-domain data accurately through any method. If the input signal is detected by only the 256-point window or only by the 1,024-point window, the information will be obtained by that output. If the input signal is detected by both the 256 and 1,024-point FFT operations, the TOA and PW will be obtained from the short window (256 points) and the frequency will be obtained from the long window (1,024 points). When the signal is weak and detected only in one frame, only one signal should be reported even there are closely spaced peaks. If there are two simultaneous weak signals, this approach will miss one of the signals. This argument is derived from Figure 12.10. Since the input may have multiple peaks produced by only one input signal, reporting multiple numbers of signals may produce a false alarm. When two peaks are far apart in the frequency domain, two signals may be reported. When a signal is only detected once by a one-frame detection, the information obtained may not be dependable. Only frames filled with signal data can report good amplitude and frequency data. When only one frame detects a signal, the output is difficult to confirm. When there is a weak cw signal, the 1,024 FFT output should produce decent results. According to row 10 of Table 12.1, the sensitivity can achieve -12 dB, which is much better than a receiver designed with only one FFT output.

12.10  Signals Detected by Multiple Numbers of Windows In general, an input signal with a reasonable strength and PW will be detected by all three FFT outputs. The general rule is that a short window is used to measure TOA and PW. The PA amplitude should also be measured by the short frame. If the leading and trailing edges of the pulse are long, they can spread into a few frames and the PA should be measured at the steady PA value. The short window may detection of PA as a function of time. Once the PA value is obtained, a threshold can be set to measure the PW. As mentioned in Chapter 11, it is desirable to measure the PW independently of the PA. If a fixed threshold is used to determine PW, the PW will be PA dependent. The frequency resolution can be improved through the amplitude and phase comparison discussed in Section 11.6 and Chapter 6. For a single signal the frequency can be read accurately through these methods. Two main problems remain with the short frame: the sensitivity and the capability of separating two signals close in frequency. The sensitivity improvement is discussed in Sections 12.6 and 12.7. The long window will be used to find multiple frequencies. One main task is to determine the number of signals after the FFT operation. One simple approach in determining the number of input signals is to find the number of peaks in the FFT outputs. One example shows all the three FFT outputs in Figure 12.11. In this figure, both signals have S/N = 10 dB, they are time coincident with a PW of 800 ns, and the frequency separation is 100 MHz. Since the PW is long, there are many outputs from all the three windows. Figure 12.11(a) shows one of the outputs in which the two frequency peaks are well identified. However, not all the 64-point windows generate clearly identified peaks. In some

292

Receiver Designed Through a Multiple FFT Operation

Figure 12.11  Two signals with S/N = 10 dB, PW = 800 ns, and frequency separation = 100 MHz: (a) 64-point window, (b) 256-point window, and (c) 1,024-point window.

of the outputs the two peaks cannot be easily identified. In all the outputs from the 256-point and 1,024-point windows two frequency peaks are clearly shown, as in Figure 12.11(b, c). When the two input signals are close in frequency such as 10 MHz, sometimes the 256-point window can show two peaks. From the results of Chapter 11, when the frequency separation is about three times the frequency resolution, the two frequencies can be encoded. Based on these results, the 256-point window can resolve two signals with a frequency separation of 30 MHz. The 1,024-point window can resolve two signals with a frequency separation of 7.5 MHz. The results of two signals with S/N = 10 dB, PW = 800 ns, and a frequency separation of 10 MHz are shown in Figure 12.12. In this case, the 256-point window only has one peak, but the 1,024-point window clearly shows two peaks. From these simple illustrations, it appears that two signals can be easily identified through the three FFT outputs. If all three windows all show one output signal, one signal will be reported. If two signals are very close in frequency, the short FFT may not distinguish them, but the long one has a better chance. In these illustrations, the signals are the same amplitude. When the two signals have a different amplitude, the problem becomes more complicated. As discussed in Chapters 4 and 11, a strong signal can generate spurious output. The detection approach must avoid the detection of spurs. Thus, for the second signal detection a dynamic threshold depending on the amplitude of the first signal might be used as discussed in Section 11.8.

12.11  Selection of Certain Windows

293

Figure 12.12  Two signals with S/N = 10 dB, PW = 800 ns, and frequency separation = 10 MHz: (a) 64-point window, (b) 256-point window, and (c) 1,024-point window.

The PW is a very important parameter in determining input frequency. If the PW measured is short, the pulse can partially fill a long FFT window and the output can spread into several frequency bins. Simulation shows that only the strong short signal can cause severed spectrum spreading. A strong signal can be detected by short frames. From the short frame result one can determine the status of the long frames.

12.11  Selection of Certain Windows As discussed in the previous sections it is desirable to process the data in the second window after detection because it is anticipated that the window is full of signal data. This assumption can be applied to a short window but may not be suitable for a long window. For example, for the 25-ns window (64 points), a minimum anticipated pulse of 100 ns can fill at least three windows. Thus, once a signal is detected, it is reasonable to assume that the second window is full of signal information. For the long window operation the second frame may not fill with signal data. Both the first and the second windows can be partially filled with signal data even if the signal is detected by two consecutive windows. If the outputs of more than three windows pass the threshold, the second window can be considered as full of signals. The signal length measured by the shortest window will be used to determine which long frame contains more signal data. The frame containing the longer

294

Receiver Designed Through a Multiple FFT Operation

Figure 12.13  Determine the frames filled with data.

signal length will be used to calculate the frequency resolution. Figure 12.13 shows a simple example. In this figure, 9 short frames (64 points) cross the threshold and the PW is at least 175 ns (7 × 25) long, as shown in Figure 12.13(a). The two end frames might not be full of signal data. The frames in Figure 12.13(b) are 100 ns long and in this case the two frames might be both partially filled with signal data. Since the second frame has longer signal data measured from Figure 12.13(a), this frame is used to calculate the frequency information. If there are simultaneous signals, the problem will not be as simple. If the two frequencies are close enough that the shortest frame cannot distinguish them, the simple observation presented here may no longer be true.

12.12  Two Signals in One Frequency Bin In Chapter 11, it is assumed that one frequency bin only contains one frequency. If two frequencies are in one bin, only one frequency will be encoded. The coarse frequency should be correct, but one signal will be missed. The amplitude and phase comparison methods to find the fine frequency can produce erroneous results. For the 64-point FFT the frequency resolution is 40 MHz. Two signals can appear in one frequency bin. If this situation occurs, it will cause a detection problem. To illustrate this problem, two signals without noise are used for the simulation. The two signals have same amplitude—one frequency is at 655 MHz and the second one is at 625 MHz, and they are 30 MHz part. Both signals have random initial phases. The two frequencies are in one frequency bin centered at 640 MHz. The input signals are shown in Figure 12.14. The two frequencies beat against each other and the beat frequency is the difference frequency of 30 MHz. Since the two signals have the same amplitude, the minimum input is close to zero. This special case can be considered as the worst scenario because it causes the most problems in receiver designs. The input data are divided into 64-point frames. Each frame is multiplied by a Blackman window and FFT is performed. Figure 12.15 shows the maximum

12.12  Two Signals in One Frequency Bin

Figure 12.14  Two input signals with frequencies at 625 and 655 MHz with equal amplitude.

Figure 12.15  Maximum amplitude output from 64-point FFT operations.

295

296

Receiver Designed Through a Multiple FFT Operation

amplitude of the FFT outputs from consecutive frames. The amplitude is expressed in decibels and it varies with time. This phenomenon can cause the following problems in receiver design. 1. Detection problem: If the two signals are weak, the peak may pass the threshold and be detected as a signal. The valley may be below the threshold. The 2-dB decrease in threshold once a signal is detected may not work properly under this condition. From Figure 12.15 the amplitude can vary close to 12 dB. Under this condition, the signal will be detected as a bunch of short pulses. The hole filling method discussed in Section 11.5 may produce a better result. 2. PW measurement problem: Because the true outputs from the FFT operation are short pulses in the time domain, the signal will be detected as a short pulse. The PW measured is incorrect. 3. PA measurement problem: Due to the amplitude variation, it might be difficult to make a correct PA measurement. Even when the peak value is measured, it should be the amplitude of two signals combined. In this receiver design with three FFT operations, the longer FFT operation should solve this problem. Once the two signals are separated into two FFT outputs, the frequency beating phenomenon does not occur. The frequency beating in the short FFT must be detected and the outputs from the longer FFT should be used to measure the PW and PA.

12.13  Parameter Measurements Once the outputs from all the different frames are detected, the encoder can be designed to measure the four parameters: frequency, pulse amplitude (PA), PW, and TOA. All the criteria discussed in the previous chapter are applicable in the multiple FFT approach, such as the threshold decrease after a signal is detected to avoid a long weak signal being chopped into short pulses. The frequency can be measured from the frequency output bin number and a better accuracy can be obtained either from the amplitude comparison or from the phase comparison on two consecutive frames. The PA measurement can sum the outputs of two adjacent outputs to obtain the signal strength. A curve-fitting method discussed in Section 11.9 should theoretically produce a better amplitude estimation. A table will be used to determine either the relative measured amplitude in dB or the absolute amplitude in dBm. The PW and TOA measurements can be accomplished through the number of frames crossing the threshold. The short window will generate the most desirable results. To obtain a better time resolution, the amplitude of the first and last frames can be compared with the amplitude of a full signal-filled frame. These approaches are all discussed in Chapter 11 and will not be repeated here. For the 256- and 1,024-point windows one-frame detection is introduced to improve the sensitivity in a certain PW range as discussed in Section 12.7. When only the one-frame detection output crosses threshold, a signal is considered to be

12.14  Conclusion

297

detected. Usually these signals are weak. The time resolution and PA measurements may not be accurate, which is the expected result.

12.14  Conclusion Although a cascade filter can be used to improve the time and frequency resolutions, the filter calculation may not be simpler than the multiple FFT operations. The cascade filter is usually made through continuous data shifting. This approach will provide a better time resolution, but it also generates long rabbit ears. These rabbit ears, in general, cause encoder design problems. A simpler approach using nonoverlapping multiple FFT lengths is studied. This approach should provide the capability of making the sensitivity and frequency resolution PW dependent.

References   [1] Skolnik, M. I., Introduction to Radar Systems, New York: McGraw-Hill, 1962.   [2] Barton, D. K., Modern Radar System Analysis, Norwood, MA: Artech House, 1988.   [3] Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Norwood, MA: Artech House, 2001.

C h a p t e r 13

Receiver Through a Polyphase Filter

13.1  Introduction The goal of the receiver design in this chapter is similar to that Chapter 12: to have variable sensitivity and frequency resolutions. The requirements will be the same as in Chapter 12: input bandwidth and minimum pulse width (PW). When the signal is longer than the minimum PW, the receiver can intercept the signal with a higher sensitivity and the frequency resolution will be better also. There are many similarities between the receiver in this chapter and in Chapters 11 and 12. If the same ideas are used, their discussion will be brief and will not be repeated. In order to compare the performances of all receiver designs, the sampling frequency will stay the same as 2.56 GHz and the input bandwidth is from 141 to 1,140 MHz. Although the design idea is introduced in this chapter, the actual receiver design may have many trade-offs. For simplicity only limited design cases will be discussed. For example, in the time-domain detection the number of summation windows is limited to two. The signals contained in each polyphase filter is also limited to two. If more processing can be implemented, the performance of the receiver can be adjusted.

13.2  Methodology The basic idea to obtain the sensitivity as a function of PW is through time-domain detection. When the input signal PW is relatively long, a longer matched filter can be used to increase the sensitivity. Since there is no a priori information on the input signal PW, many filters of different lengths will be used to match the input signal. There are two ways to perform time-domain detection, using either the amplitude or the relative phase of the input signals. The relative phase detection was discussed in Chapter 7. Using amplitude to perform time-domain detection is discussed in [1]. Although eigenvalues can be used as a time-domain detection to determine the number of signals, it is simpler to use amplitude information. In order to use amplitude information for time-domain detection, the input signal must be complex. Because a large number of filters are used in the time domain, the receiver cannot have many channels. For example, in Chapter 11 the receiver has 52 channels and if each channel has 4 filters of different lengths performing the time-domain detection, there will be too many filters. Considering the requirements 299

300

Receiver Through a Polyphase Filter

of complex input data and the limited number of channels, a polyphase filter will be used to achieve these goals. One can build 8 output channels by using the polyphase approach; the outputs are complex. A 16-point conventional FFT operating on real data can produce 8 outputs, but the in-phase and quadrature phase (I and Q) of the conventional 16-point FFT are not well balanced (Chapter 5). The polyphase filter achieved through a 16-point FFT operation has well-balanced IQ channels. The balance of the IQ channels was discussed in Section 5.12. A polyphase filter is used in the receiver design because the approach can generate a small number of frequency outputs with well-balanced IQ channels. All these characteristics are suitable to perform the time-domain detection.

13.3  Polyphase Filter Design The detailed design of a polyphase filter is discussed in [1] and will only be discussed very briefly in this section. There are two important parameters in polyphase design: the FFT length and the total data points used. The selection of these parameters will be discussed. Once these parameters are determined, the mathematical approach to generate the outputs will be presented because it is easier to illustrate with a numerical example. First, the FFT length will be discussed. If the FFT length is 8 points, because the inputs are real data, there are only four output channels. The output of the first channel will be real because of the FFT property. The useful channels do not have the bandwidth to cover the desired 1,000 MHz. If the FFT length is 16 points, there will be eight independent channels. By eliminating both the low and high frequencies, seven channels can be used to build the receiver. The seven channels can cover the desired frequency range of 141 to 1,140 MHz. If the channelization is designed through a polyphase filter, a 16-point FFT is a minimum length to fulfill the required bandwidth. Let us consider the number of data points. If 64 input data points are used in the polyphase filter design, the sidelobes are only about 30 dB down. These high sidelobe levels limit the dynamic range of the receiver. If 128 input data points are used in the polyphase filter design, the sidelobes are about 59 dB down, which is a respective value. When the polyphase contains 256 input data points, the sidelobes are lower than 90 dB. The shapes of the Parks-McClellan filters and their corresponding frequency responses of these three filters are shown in Figures 13.1 to 13.5. Figure 13.1 shows the results of 64-point data. Figures 13.2 and 12.3 show the 128- and 256-point data, respectively. In these figures the data points near the edges of the time domain are heavily attenuated. The corresponding frequency-domain responses have relatively low sidelobes. If the outputs of all the seven channels are plotted, the results are shown in Figures 13.4 and 13.5 with 128 and 256 data points, respectively. With all the channels, the sidelobes with 128-point input data are about 54 dB down, which is higher than the single filter sidelobes at 59 dB. For the 256-point results, the sidelobes are about 94 dB down, which is higher than the single value of close to 100 dB. From

13.3  Polyphase Filter Design

301

Figure 13.1  Time- and frequency-domain responses for a 16-point FFT with 64 input data points.

Figure 13.2  Time- and frequency-domain responses for a 16-point FFT with 128 input data points.

302

Receiver Through a Polyphase Filter

Figure 13.3  Time- and frequency-domain responses for a 16-point FFT with 256 input data points.

Figure 13.4  Filter bank with a 16-point FFT with 128 input data points.

13.3  Polyphase Filter Design

303

Figure 13.5  Filter bank with a 16-point FFT with 256 input data points.

Figures 13.4 and 13.5 it appears that the 256 data point results should provide a better two-signal instantaneous dynamic range. After a simple frequency determination study, it reveals that although the 256 data points produce a better frequency-domain result, they adversely affects the time-domain result. If 256 data points are used for the polyphase filter operation, the transient effects are long on both ends of a pulse. Measuring the transient region of a pulse will produce erroneous results because both the frequency and amplitude have errors during the transient period. Avoiding the transient regions on both ends of a pulse and the middle portion of the pulse carries the correct information. With long transient regions, the middle portion will be short and it is difficult to obtain the information on a short section of a pulse. Considering both the time- and frequency-domain performances is a common practice in receiver design. In this design, the 128-point case is selected. Once the 16-point FFT and 128 data points are determined, the polyphase filter design can be achieved. A special hardware approach can be used to implement the polyphase filters. In the approach the filter is decimated and the name of polyphase is used to describe the filter. The detailed polyphase filter design can be found in [1]. In simulation the programming of the filter can be understood through Figure 13.6. In this figure, the input data from 0 to 127 are multiplied by a Parks-McClellan window function and represented in a long rectangular box. The data are divided into 8 sections, each containing 16 points. The result are summed together to obtain 16 points and they are represented by z. Mathematically, the operation can be written as

304

Receiver Through a Polyphase Filter

Figure 13.6  Operation of input data to obtain the polyphase filter outputs.

y = (x0 , x1, ... x127 )(w0 ,w1, ... w127 ) = x0w0 , x1w1,... x127 w127 y0 = x0w0 , x1w1,... x15w15 ;

y1 = x16w16 ,... x31w31;

...;

y7 = x112w112 ,...x127 w127

z0 = x0w0 + x16w16 + ... + x112w112 ;

...;



(13.1)

z15 = x15w15 + x31w31 + ... + x127 w127 Z=

15

∑ zn e

−

2π nk N

n=0

This equation can be explained as follows. First, the digitized input signal x is multiplied by the window function w term by term and the results are referred to as y. Divide y into 8 sections and each section contains 16 input data. Summing the proper terms of y, the results are represented by z0 to z15. Each z value contains 8 data points. From these results one can see that z0 is the summation of all the first elements in the y function. Finally, a 16-point FFT is performed on the z function to obtain 16 outputs in the frequency domain, which are the polyphase filter outputs in the frequency domain. Among these 16 outputs, only 8 carry independent information. By limiting the input bandwidth, only seven channels will be used.

13.4  Property of the Polyphase Filter The filter designed in the previous section has a bandwidth of 160 MHz. With seven channels (1 to 7), the total bandwidth is 1,120 MHz. The shape of the filter can be ad-

13.4  Property of the Polyphase Filter

305

justed to change the skirt on both sides of the filter. This special filter shape is obtained from the Parks-McClellan filter in the MATLAB program. In actual receiver design, it might be desirable to change the filter shape slightly after detailed TOA and frequency encodings are evaluated. In other words, the filter might be fine-tuned to improve receiver performance. With a bandwidth of 160 MHz, it is too coarse to generate the frequency information. The main purpose of the polyphase filter is to separate the input signals into a different frequency range for further signal processing. With this operation, the number of simultaneous signals in each output can be reduced. Another important factor in designing the polyphase filter is the data output rate or the output time-domain resolution. When the data point shift equals the FFT points, which is a common operation mode, it is referred to as the critical sampling rate. In this case, the FFT is 16 points with the input data shifting 16 points every frame, which becomes the critical sampling rate. In Figure 13.2(a) the time domains of the 16-point FFT with 128 data points are shown. The data at the ends of the window are highly attenuated. One can consider that only 16 points at the center of the window make the major contribution of the FFT outputs. In other words, the center portion of the window consists of 16 data points and each time also shifts 16 points. This is equivalent to having a nonoverlapping shifting of 16 points, in which a rather interesting argument can be applied to the discussion of spread spectrum detections in Chapters 14 and 15. Since the sampling frequency is 2.56 GHz, 16 samples cover 6.25 ns. This is also the time resolution. However, the outputs from the polyphase filter outputs must be further processed through another FFT operation. With a data rate of 6.25 ns, the bandwidth equals the polyphase filter pass bandwidth of 160 MHz because the data are complex. Under this condition, when two signals are separated close to 160 MHz, the two signals will fold into one location and the receiver will miss one signal. In general, this situation is unacceptable. In order to remedy this shortcoming, the filter output rate must be increased. In this special case the output data is doubled. The corresponding time resolution is 3.125 ns. In forming these outputs, the input data are shifted by 8 points as shown in Figure 13.7. This is referred to as an oversampling ratio of 2. This time resolution

Figure 13.7  Polyphase filter data shifting.

306

Receiver Through a Polyphase Filter

will be used to measure the TOA and PW, which is better than the time resolution of 25 and 50 ns used in Chapters 11 and 12. Every input frame contains 128 data points. If the frame is partially filled with data, that is, there is a leading or trailing edge in the frame, the output in the frequency might spread into neighboring channels and generate rabbit ears in a receiver design. The rabbit-ear generating process is discussed in Section 12.5 and Figure 12.7. Since the data points are heavily attenuated at both ends of the ParksMcClellan window, when the leading and trailing edges are close to the ends, the transient effect will be subdued. Taking care of the rabbit ear phenomenon is one of the major tasks in encoder design. In the basic FFT approaches discussed in Chapters 11 and 12, the data are shifted in such a manner that only one frame contains partially filled data. The rabbit ear only occurs in one frame. The rabbit ear problem will be further discussed in Section 13.6.

13.5  Time-Domain Detection and Sensitivity [1–4] The major effort in this design is to process the seven polyphase filter outputs. The signal will be detected and further processed through conventional FFT operations. There are two FFT outputs: from polyphase filters and from conventional FFT operations. In order to simplify this discussion and avoid confusion between these FFT outputs, let us refer each polyphase output as a complex receiver because the data to be processed are complex. Each complex receiver has an input bandwidth of 160 MHz. The sampling rate is 320 MHz (1/3.125 ns). With complex data the processing bandwidth is also 320 MHz. This wide processing band is used to separate two signals by 160 MHz at the edges of the input band. The complex receiver is different from the overall receiver, which has real input, and the sampling frequency is 2,560 MHz. The complex input data can be used to detect the existence of signals. The amplitude of a signal can be found by the amplitude of the signal obtained from the real and imaginary parts. The basic idea of time-domain detection is to use different summation lengths to process the input signal. If the summation length is close to the signal length, it can provide a better signal-to-noise ratio (S/N). In this case the short summation length is 1 or a single output. The probability of detection of one sample can be found through the same procedure discussed in Chapter 12. The probability of false alarm (Pfa) is set at 10-7 as usual. Using one output data point as the detection criterion, the threshold is at 1.06 and S/N = 5.2 dB is required to achieve the 90% probability of detection (Pd). Performing detection on only one complex input sample, the time resolution is at 3.125 ns. In order to figure out the Pfa and Pd for other summation lengths, an individual threshold must be obtained and Pd must be calculated. This topic is discussed in Section 6.6. A simpler approach is to use the noncoherent integration used in radar books [1–3] to improve the sensitivity on long signals. One problem that must be taken care of in using the noncoherent integration is data overlapping. Generally when the FFT length is 16 points long and the shifting is also 16 points, there is no overlap. When the shifting is 8 samples, there is data overlapping. Considering the noncoherent integrations, the nonoverlapping case should be used.

13.5  Time-Domain Detection and Sensitivity

307

For 100-, 200-, 400-, and 800-ns filters (also referred to as the summation window or length), the corresponding summation lengths are 16, 32, 64, and 128. If the PW is long enough to fill the summation window, the corresponding non­coherent gains are 9.6, 11.6, 13.4, and 15.2 dB, respectively, for Pfa = 10-7 and Pd = 90% [1, 2]. With these gains comparing the sensitivity of S/N = 5.2 dB, the sensitivity of the summations can be found from the noncoherent gain. For example, for the 100-ns pulse the required S/N = -4.4 (5.2 - 9.6) dB. Comparing with the result listed in row 8 of Table 6.6 (S/N = -4.3 dB). This calculated sensitivity appears a little high. The required S/N of 16 summations is obtained from another approach, the conventional way. The noise output is approximated by the Rayleigh distribution and the summation is obtained through the convolution method as discussed in Section 6.6. The threshold obtained is 6.28 and the sensitivity is about -5 dB, which is even lower than -4.4 dB. Maybe the polyphase window causes this phenomenon. For a conventional FFT approach, 100 ns contain 256 data points. In the polyphase approach the 128-point window is shifted 15 times and there are a total of 368 (15 × 16 + 128) data points. In the simulation, the input signal is 368 points long. These extra data might contribute to the sensitivity improvement. If this argument is true, it should apply to the other summation results, which make the sensitivities slightly better. This argument violates the minimum PW requirement. Since the minimum PW is only 256 points long, using 368 points will imply that the minimum PW is close to 150 ns. The required S/N values obtained from the noncoherent integration gain are listed in Table 13.1. In Table 13.1, the PW is assumed long enough to fill the summation window length. The sensitivity indicates that for a signal with S/N > 5.2 dB, the signal can be detected through one data point and with a very good time resolution. From Table 13.1 one can see that the sensitivity is a function of PW, which is one of the desired results. The summation window can move one complex input data at a time. In other words, the window will slide through the complex input data. The shortest summation window is 100 ns, and the time resolution will be 3.125 ns as shown in Figure 13.8. This window should match the minimum PW at some time with the fine time resolution. Theoretically, the minimum PW will be detected with a sensitivity very close to the expected value. In the time-domain detection several filters with a different length can be used to improve the detection on longer pulses. When a large number of summation lengths are used, the receiver design will be more complicated. The summation window lengths can be chosen as base 2 numbers, because different FFT lengths can be applied to the complex input data to generate finer frequency resolutions. Base 2 numbered Table 13.1  Sensitivity Versus PW Filter Length (ns) 3.125 100 200 400 800

Summation Lengths    1*   16*   32   64 128*

* Used in the actual receiver design.

Noncoherent Gain  0   9.6 11.6 13.4 15.2

Sensitivity (S/N) (dB)   5.2 -4.4 -6.4 -8.2 -10

308

Receiver Through a Polyphase Filter

Figure 13.8  Shortest summation window motions with respect to the minimum PW.

FFT operations can be built more efficiently. In this discussion, let us use only three summation windows for simplicity: the one sample detection and the 100-ns and 800-ns summation windows. For this arrangement, the receiver sensitivity will be of S/N = 5.2 for the 1-point detection. For a PW from 100 to 800 ns the required S/N = -4.4 dB. When the PW is longer than 800 ns, the S/N = -10 dB. As discussed in Chapter 12, once a signal is detected, the threshold of the channels should be decreased slightly to avoid chopping a weak input pulse into many pieces. Since the polyphase filter output rate is doubled, in order to obtain 100 ns, 32 samples are required to sum together. To obtain a 800-ns window, 256 complex input data are needed. If the input signal is strong with S/N > 5.2 dB, the one data detection will find it. The signal strength must be less than 5.2 dB to be detected by the 100-ns window and less than -4.4 dB to be detected by the 800-ns window. For a long weak signal it is difficult to detect the leading and trailing edges. Thus, a time resolution of 3.125 ns may not be required. Under this assumption, 32 summations can be saved as one point (referred to as the sum point). This process will reduce the program complexity and simplify receiver design. In other words, the summations of 32 points (100 ns) can be saved every 100 ns in a nonoverlapping manner. The 256-point window can be obtained by summing 32 sum points. These summation results can be used to find the weak signal. The 800-ns window can move 100-ns steps and the time resolution will decrease to 100 ns. The outputs of the three filters will have the following time resolution. The 1-point detection and the 32-point windows both have a time resolution of 3.125 ns (8 data points). The 256-point window is obtained from 32 summations rather than 256 points and has a time resolution of 100 ns.

13.6  Rabbit Ear Generation It was discussed in Chapter 12 that when the input signal does not fill a time window completely, after the FFT operation, the frequency will spread into neighbor-

13.6  Rabbit Ear Generation

309

ing frequency bins. This phenomenon is referred to as rabbit ears in receiver design [personal communication with M. Broadstock, adjunct professor, Miami University, Oxford, Ohio, 2009]. In Chapter 12, it was assumed that the input signal will fill the second window frame, and there are no rabbit ears. In the polyphase filter, it takes 128 samples to fill a window frame and the frame moves every 8 samples. Ideally it takes 16 moves to reach 128 samples. Therefore, the transient will be 16 complex input data long, or 50 ns. However, from the Parks-McClellan window (Figure 13.2), only the center portion will make a major contribution to the input signal. From the simulation result, the rabbit ear occurs at S/N about 10 dB and

Figure 13.9  Rabbit ears caused by a strong signal.

310

Receiver Through a Polyphase Filter

the length is short, only 1 or 2 data points long. For a strong signal of S/N = 45 dB, the rabbit ear can be 13 samples long, or about 40 ns, which is just short of the 16 input data points. Since a strong signal will have a significant rabbit ear, 40 ns can be considered as the longest rabbit ear. In detecting a signal from the polyphase filter output, only crossing threshold cannot be considered a signal. The time crossing the threshold must be longer than the longest rabbit. This criterion will eliminate the rabbit ears and detect the desired signal. From simulation results it is shown that when the input signal is weak, such as below S/N = 5.2 dB, the outputs do not have rabbit ears. Therefore, if a signal crosses the threshold of one data point detection, the rabbit ear problem needs to be considered. Figure 13.9 shows a strong signal in complex receiver 6 and a weak signal in complex receiver 5. The strong signal produces two rabbit ears in all six receivers and the weak signal does not generate rabbit ears in its neighboring receivers.

13.7  Detection Sequence and Rabbit Ear Effect In order to understand the problem caused by the rabbit ears, the detection method will be discussed. Figure 13.9 is used to help illustrate the explanation. The 1-point detection is performed first to find a relatively strong signal such as above 5.2 dB. The output not only needs to cross the predetermined threshold but also must be longer than 40 ns (the expected longest rabbit ear). In this case only complex receiver 6 detects a signal. The summation of 32 points will be performed on the other six complex receivers without detection to find weak signals in them. If there are strong rabbit ears in the receiver, the summation method will detect them as signals. In other words, although the first time detection (1 point) can avoid the rabbit, the second time detection (32 points) will generate a false alarm. In order to eliminate these false alarms, once a rabbit ear is detected, it must be eliminated. One simple way is to reduce the rabbit ear to zero. This approach may affect the summation 32-point detection because even the outputs caused by noise alone do not have zero outputs. Reducing the rabbit ear to zero will lower the output from the 32 summation. This operation in turn may miss the detection of a weak signal. Thus, a reasonable approach is to reduce the rabbit ear to a constant level, the average noise level. With this modification, the false detection by summing 32 samples will be greatly reduced. From the complex receiver 5 of Figure 13.9, the second rabbit ear in the receiver should be reduced to the weak signal level rather than the noise level. Although this is supposed to be the desired approach, the actual implementation could be complicated.

13.8  PW and FFT Operation Once a signal is detected, the length of the signal needs to be determined. This is determined by observing the length of the signal crossing the threshold. If the signal is

13.9  Determine the Number of Signals

311

longer than a 32 complex data length but less than 64 points, a 32-point FFT will be performed to obtain the fine frequency resolution. Thirty two points cover 100 ns and the corresponding frequency resolution will be 10 MHz (1/100 × 10-9). If the PW is longer than 200 ns but less than 400 ns, a 64-point FFT will be used to obtain a frequency resolution of 5 MHz. For a PW between 400 and 800 ns, a 128-point FFT will produce a 2.5-MHz frequency resolution. If the PW is longer than 800 ns, 256 points will produce a 1.25-MHz frequency resolution. From this simple operation, the frequency resolution is PW width dependent, which is the desired result. In detecting the length of the output signal, the beat frequency phenomenon discussed in Section 12.12 must be taken into consideration. Since the complex receiver has a wider input bandwidth of 160 MHz, the chance of having two simultaneous signals in one receiver is increased. When there is frequency beating in a receiver, the amplitude of the output will vary with time. Decreasing the threshold 2 to 3 dB after a signal is detected does not produce a satisfactory result, that is, the signal will be chopped into short pulses. One way of mitigating this problem is to adopt the hole filling method discussed in Section 11.5. When the frequency separation between two signals is small, the beat frequency has a long period. As a result, many holes can occur in the output. This problem must be carefully studied for the polyphase receiver design.

13.9  Determine the Number of Signals Once a signal is detected in the complex receiver, the signal frequency resolution is 160 MHz, which is usually too coarse as a final frequency information. Further processing is needed to obtain finer frequency information. The selected approach is to measure the length of the signal and apply another FFT operation, which is referred to as the second FFT. The input data for this FFT operation is multiplied by a Blackman window. The first FFT operation is included in the polyphase filter operation. It is possible that the detected signal may contain more than one signal. If FFT is used to determine the fine frequency, a threshold must be set to detect whether there are multiple signals. However, for simplicity, only two signals per complex receiver are the limit in this study. The threshold can be set as using noise as the input. A different approach can be used to find the number of signals in the detected signal. This is through the eigenvalues. A 3 × 3 matrix can be used to determine the number of signals. Since it is known that there is at least one signal in the complex, only the second and the third eigenvalues are of interest. By using eigenvalues, thresholds are also needed. The thresholds are also obtained from noise input. If the eigenvalues detect three signals, one knows that one signal is missed by the receiver in this special design because one complex receiver only detects two simultaneous signals. Setting the threshold for the second FFT operation might be an easier approach because no further processing is required. However, the eigenvalue method might detect a signal close in frequency. In the FFT operation the output must be separated by certain frequency bins in order to detect them. In other words, peaks must be detected in the FFT operation. If the eigenvalue method detects two signals but only one can be found in the FFT outputs, one can assume that the second signal is close to the first one in frequency.

312

Receiver Through a Polyphase Filter

Simulation results indicate that the input data used for the number of signal tests are very important. If the input data contain a rabbit ear, usually both methods show multiple signals. These are the expected results because a short pulse in the time domain is a spread spectrum in the frequency domain. Under this condition, the test will produce a false detection. To avoid this problem, one approach is to take the input data between two rabbit ears. In conclusion, there are at least two approaches to determine the number of signals from the outputs of a complex receiver: the FFT outputs and the eigenvalue method. Further study is needed to determine which method is more suitable for a receiver design.

13.10  Odd and Even Complex Receiver Outputs In Chapters 11 and 12, the frequency could be found rather easily. When the outputs of two adjacent channels cross the threshold, the channel with the higher output will be used to determine the frequency and the other channel will be ignored. Thus, only two signals with a frequency separation greater than two frequency bins can be detected. As usual in this receiver design one signal can also appear in either one or two complex receivers. However, one cannot compare amplitudes and suppress one channel because in this approach the bandwidth is 160 MHz. Neglecting the output from a neighboring complex receiver may miss a signal separated far apart in frequency from the first signal. All the signals detected in the complex receivers must be further processed through the second FFT operation. The length of this FFT operation is PW dependent. If signals appear in two adjacent complex receivers, after finding the frequencies, their frequencies must be compared. If the two frequencies are close, they are considered to be the same signal; otherwise, there are two signals. In the following sections, the frequency encoding after the second FFT operation will be discussed. Figure 13.10 shows the frequency range of the polyphase filter bank. In this figure the centers and edges of all the 7 complex receivers are

Figure 13.10  Polyphase filter bank.

13.10  Odd and Even Complex Receiver Outputs

313

listed. Since the 0 output is neglected, the 7 complex receivers are designed as 1 to 7. This figure can help the discussion of the frequency encoding scheme. First, let us assume that the polyphase filter is shifted by 16 samples. The reason of this practice is to indicate the frequency output difference between the 8 and 16 shift approaches. If the complex receiver is required to measure only one signal, the shift 16 approach is satisfactory. Under this condition, both the input and output bandwidths are 160 MHz. When two signals are separated by 160 MHz, the receiver can only see one signal because both signals are folded into the same frequency bin. Let us examine the frequency output at the center of a frequency bin. Figure 13.11(a, b) shows the input frequency at 160 and 320 MHz, respectively, after a 32-point FFT operation. In both figures the maximum occurs at the zero frequency. In order to better show the peak value, a linear scale is used for the amplitude. For all the input frequencies at the center of a frequency bin, the maximum always occurs at the zero frequency bin. Figure 13.12 shows the results for an 8-point shift. In Figure 13.12(a) the input is at 160 MHz, where the maximum is at the center of the frequency output rather than at the zero output. In Figure 13.12(b) the input frequency is at 320 MHz, where the maximum output at the frequency is at the zero frequency bin. At the center frequency of the seven complex receivers, the maximum outputs are alternatively at zero or center. At frequencies 160, 480, 800, and 1,120 MHz, the maximum outputs are at the center frequency bins. These are the odd number complex receivers. At frequencies 320, 640, and 960 MHz, the even complex receivers, the

Figure 13.11  Input frequency at 160 MHz and 320 MHz in complex receivers 1 and 2 with a 16-data point shift: (a) complex receiver 1 and (b) complex receiver 2.

314

Receiver Through a Polyphase Filter

Figure 13.12  Input frequency at 160 MHz and 320 MHz in complex receivers 1 and 2 with an 8-data point shift: (a) complex receiver 1 and (b) complex receiver 2.

maximum outputs are at the zero frequency bins. These differences must be included in the frequency encoding.

13.11  Determine the Input Frequency It is necessary to input signal at different frequency in different complex receivers to observe the output location for frequency encoding. In Figure 13.13, the two inputs are at frequencies 140 and 180 MHz. Comparing with Figure 13.12(a), the lower frequency is shifted to the left and the higher frequency is shifted to the right. When the input frequencies are at 300 and 340 MHz, the results are shown in Figure 13.14. Comparing with Figure 13.12(b), when the input is below 320 MHz, the input appears at the high-frequency region, which is lower than the zero frequency. When the input is higher than 320 MHz, the output is at the lower-frequency range and about zero. These are simple rules to encode frequency. The outputs from the odd and even complex receivers are different and the frequency encodings are also different. First, the odd number complex receiver will be discussed. In an odd number complex receiver, the frequency will be adjusted to zero because the center bin frequency is at the center rather than at zero. Since the second FFT length used in the polyphase filter approach depends on the PW, the FFT length is not predetermined as discussed in Chapter 12. Let Nf represent the FFT length and the maximum output index will be adjusted to zero as

ind = indm − N f / 2

(13.2)

13.11  Determine the Input Frequency

315

Figure 13.13  Frequency outputs in complex receiver 1: (a) input at 140 MHz and (b) input at 180 MHz.

Figure 13.14  Frequency outputs in complex receiver 2: (a) input at 300 MHz and (b) input at 340 MHz.

316

Receiver Through a Polyphase Filter

where indm is the measured index. This adjustment is only applied to the odd number complex receivers. After this simple adjustment, the fine frequency ffin within the complex receiver bandwidth of 320 MHz of both the odd and even complex receivers can be found as ffine = ffine

320 ´ ind Nf

for ind f + 1 2



(13.3)

The final frequency f can be obtained from the number of complex receivers as

f = 160k + f fine

(13.4)

where k is the number of a complex receiver. If signals are detected in a certain complex receiver, the frequency will be calculated. When two signals are in two adjacent receivers, the frequencies in both receivers will be calculated. However, it is possible that one signal is close to the boundary between two receivers. In other words, one signal can appear in two adjacent complex receivers. Under this condition the two adjacent receivers will detect two signals. Since it is only one signal, the two frequencies measured should be very close together. Thus, when two signals are measured from two adjacent complex receivers, the frequency must be compared. If the two signals have the same PW and appear in two adjacent complex receivers, the FFT lengths in both receivers are the same. If the two frequencies are very close within a certain range (say within ± 1 resolution cell), the receiver will assume them as one signal and one signal will be missed. Under this condition, the output with the stronger output will be used to determine the frequency. If the two signals have a different PW, the FFT lengths will be different. Under this condition, the two frequencies will still be compared. If they are close in frequency, the frequency obtained from the longer FFT will be used as the output because the longer FFT produces a better frequency resolution. The frequency range used for comparison determines whether there are one or two signals. If the frequency range is too small, one signal might be identified as two signals. If the range used for comparison is too large, two signals will be read as one signal. This is the fundamental trade-off between generating spurious signals and missing signals. It is believed that in an electronic warfare (EW) receiver, it is missing a signal rather than generating a spurious one.

13.12  Frequency Resolution as a Function of Pulse Width In this section some preliminary frequency measurement results will be presented. The PW is used as a variable and the results are shown in Figures 13.15 and 13.16. In Figure 13.15 the S/N = 10 dB. Under this condition the signal can be detected by a single output and the PW is 100 ns. Sometimes the result shows that the PW is

13.12  Frequency Resolution as a Function of Pulse Width

Figure 13.15  Frequency measured on a 100-ns pulse.

Figure 13.16  Frequency measured on a 600-ns pulse.

317

318

Receiver Through a Polyphase Filter

100 ns. Sometimes the PW detected is less than 100 ns. The reason is that in this special example the PW and the window length are identical. Since the window moves in 3.125-ns steps, depending on match between them, different PW values will be generated. If the PW is less than 100 ns, the second FFT length is 16 points, which provide a frequency resolution of 20 MHz because 16 output points equal 50 ns. Under this condition the frequency error is ±10 MHz. If the PW is detected as 100 ns, a 32-point FFT will be performed, which provides a frequency resolution of 10 MHz with a frequency error of ±5 MHz. Figure 13.15 reflects these results. In this figure the frequency changes at 1-MHz steps. In performing the various FFT operations of different lengths, Blackman windows are used to match the data length to suppress the sidelobes in the frequency domain. The amplitude comparison method discussed in Section 11.6 can be used to improve the frequency reading. Since the FFT length matches the signal length, the entire signal will be processed by one FFT operation and there are no consecutive outputs. The phase comparison method discussed in Sections 6.12 and 6.13 cannot be applied here because there is only one frame of output. The amplitude comparison works fine; however, the results are not shown in Figures 13.15 and 13.16, which are used to illustrate the FFT length effect of the FFT lengths. If an amplitude comparison is used, it will mask the FFT length effect because the frequency resolution will be independent of the FFT length. Figure 13.16 shows the results of a 600-ns pulse with S/N = 10 dB. The maximum PW in this special program is set at 400 ns. A 128-point FFT operation can be performed on the signal to create a 2.5-MHz resolution with an error frequency of ±1.25 MHz. The frequency changes at 1 MHz per step. In this figure, the results show large errors at some frequencies and these frequency ranges are some what periodic. Initial investigate shows that at some frequencies the pulse width measured are shorter than 400 ns and a 64-point FFT will be performed on the input signal. The resulting frequency resolution will be reduced. Further study may remedy this shortcoming.

13.13  Amplitude, TOA, and PW Measurements Although the polyphase filter outputs are complex and amplitudes can be obtained from the complex receiver outputs, the amplitude measurement should not perform at these outputs. Because the complex receiver bandwidth is rather wide at 160 MHz, the probability of containing simultaneous signals can be high. If there are simultaneous signals in a complex receiver, the amplitude measurement will not be accurate. The amplitude should be measured after the second FFT operation. For the polyphase filter receiver the FFT length depends on the PW. For each length of the FFT operation, the amplitude must be calibrated. In this receiver design, there are four different FFT lengths: 32, 64, 128, and 256, although there are only two summation filters: 32 (100 ns) and 256 (800 ns) points. For each FFT length the amplitude must be related to the input signal power level. The pulse amplitude (PA) can be obtained from the maximum FFT amplitude output. Two methods to improve the PA measurement are discussed in Section 11.9. There is usually only one second FFT operation to obtain the amplitude of a signal, instead of multiple

13.14  Minimum PW Limitation and Reduction of False Detection

319

FFT operations, and the rising and trailing edges will be included the measurement results. In order to measure the PA at the steady amplitude and to avoid the leading and trailing edges, further processing is needed. The time of arrival (TOA) can be obtained from the time-domain detection results. If the input signal is strong, it can be detected by one output detection. Under this condition, the time can be obtained rather accurately. When the input signal is weak, the two window lengths (100 and 800 ns) must be used to match the PW to improve the probability of detection, in other words, to improve the receiver sensitivity. However, under this condition the length of the summation window must be taken into consideration. From the time-domain detection study in Chapters 7 and 8, the TOA can be obtained by subtracting the window time from the time that the amplitude reaches a constant value. When the input signal is weak, it is rather difficult to determine the shape of the window output. As a result, the TOA measurement will not be accurate. The time of departure (TOD) is measured in a similar manner. When the signal is strong, the TOD can be measured accurately. When the signal is weak, it is difficult to obtain an accurate TOD. The difference between the TOD and TOA is the PW. Thus, the measured PW accuracy depends on the accuracy of the TOA and TOD measurements.

13.14  Minimum PW Limitation and Reduction of False Detection When an input signal is short and strong, the two rabbits generated in the neighboring receivers may merge together as shown in Figure 13.17. This is a well-known problem in analog receiver design. Those complex receivers detecting this signal will produce a false output. This is the limitation of a short pulse. For the purpose of eliminating false detection, let us assume that receivers 2 and 4 generate the false outputs and the true signal is in receiver 3. One approach to reducing this problem is to measure the TOA, PW, and PA of the strong signal. If the PA is large, this signal will generate rabbit ears in adjacent receivers and may be detected as a false signal. The TOAs and the TODs of the real and the false signals

Figure 13.17  Two rabbit ears of a short pulse can merge into one.

320

Receiver Through a Polyphase Filter

will be very close. By comparing the PA, TOA, and TOD of the outputs from the three adjacent receivers, the false signal can be eliminated. In other words, if two signals are detected in adjacent receivers with the same TOA and TOD, the weak signal will be considered as a false alarm. Although this approach may eliminate some false signals, under certain input conditions such as two signals—one strong short and one long weak, with the same TOA and TOD, the receiver may miss the weak signal. In this discussion, the frequency measured is not used as a comparison parameter because it is difficult to assess the frequency measured from a transient period. In a transient period the frequency varies. The frequency will change from the center frequency of the filter to the signal frequency.

13.15  Conclusion One important issue discussed in this chapter is the problem caused by the rabbit ear, which can cause false detection if not properly treated. In order to minimize the rabbit ear effect, the performance of the receiver may degrade. The rabbit ear can be considered as a physical problem. In order to generate a finer time resolution, other parameters measured may be degraded by the rabbits. From limited simulation results, it appears that the polyphase filter receiver design can achieve a better performance, that is, the receiver sensitivity and frequency resolution are PW dependent. However, the process is rather complicated in comparison with the design through conventional FFT operations. Many trade-offs must be performed. Due to the slow processing speed in the software receiver designed in the chapter in comparison with a hardware receiver, only limited input signal conditions can be tested. A hardware receiver means that the software used for designing the receiver is programmed in an FPGA. The hardware receiver should have a much faster processing speed and will be able to evaluate more input conditions. If a deficiency is found, a trade-off will be used to improve the receiver performance. A general rule for a trade-off performance is to reduce the signal detection rather than producing false detections. Some occasionally missed pulses may not miss a certain radar detection because the missed pulses usually can be detected at a different time. Generating a false signal will cause the EW processor following the EW receiver to try to identify the pulse and waste a fair amount of time. Since the pulse information is false, it is difficult to sort into a pulse train. The false data is either discarded or, worse yet, generates erroneous information.

References   [1] Tsui, J., Fundamentals of Global Positioning System Receivers, 2nd ed., New York: John Wiley & Sons, 2005.   [2] Barton, D. K., Modern Radar System Analysis, Norwood, MA: Artech House, 1988.   [3] Skolnik, M. I., Introduction to Radar Systems, New York: McGraw-Hill, 1962.   [4] Tsui, J., Microwave Receivers with Electronic Warfare Applications, New York: John Wiley & Sons, 1986.

Ch a p t e r 14

Detection of Biphase Shift Keying (BPSK) Signals 14.1  Introduction Modern radars use phase modulated signals to spread their spectrum to improve the processing gain. There are several approaches of using the phase modulation. The phase change can be π, π/2, π/4, and so forth. When the phase change is π, it is referred to as biphase shift keying (BPSK). Communication signals use various types of phase modulations. Although some radars use a π/2 phase shift, which is referred to as quadrature phase shift keying (QPSK), this study only concentrated on BPSK because it is more popular in radar applications. It is important to detect the existence of a BPSK signal for an electronic warfare (EW) receiver. The detection of a BPSK signal can help identify the radar type, which is important information. If more information can be obtained from the signal, the identification approach can be easier. In this study the requirements are to detect the existence of a BPSK signal and find the frequency and its chip time (or chip rate). The chip time is the shortest time between two consecutive phase shifts in a BPSK signal and the chip rate is the inverse of the chip time. This study is limited to one BPSK signal. For simplicity, simultaneous signals with one BPSK and one pulsed signal will not be discussed. This assumption is further explained as follows. If the input signal passes through the filter bank (FFT operation), it is assumed that the output frequency bin contains only one BPSK signal. For the polyphase receiver design, each frequency output bin has a bandwidth of 160 MHz, referred to as a complex receiver. The assumption means that in each complex receiver there is only one BPSK signal. Other signals can appear in other complex receivers. In this study the BPSK signal will pass through a receiver such as discussed in Chapters 11 through 13. In other words, the FFT operation will be performed on the signal to generate outputs in the frequency domain. Detection and analysis will be performed after the signal passes through the filters of the receiver to see whether the signal can still be recognized. Hopefully, from this study, some basic properties of the BPSK signal can be revealed, that will provide design ideas for an electronic warfare (EW) receiver to detect and identify this type of signal.

14.2  Basic Barker Code Properties [1] There might be many kinds of radars with different types of BPSK signals. There are maximum length sequences (MLS) that can be used to generate BPSK signals. The 321

322

Detection of Biphase Shift Keying (BPSK) Signals

MLS length can generate a long sequence and produce a higher process gain. The MLS signal is probably more popular for communication applications. The global positioning system (GPS) uses the BPSK signal with Gold codes and the sequence length is 1,023. The Barker code is a short sequence and popular for radar applications. The Barker codes have different sequence lengths. The sidelobes generated from the autocorrelation of the Barker code have a constant amplitude. The maximum length of a Barker code is limited to 13. There are several different Barker codes. In order to limit the scope of this study, only two types of signals will be used in the simulation: the 11 and 13 length Barker codes, and they are: B11 = [1 − 1 1 1 − 1 1 1 1 − 1 − 1 − 1]

B13 = [1 1111

− 1 − 111

− 11 − 11]

(14.1)

The 1 and –1 represent 0 and π phases in a sinusoidal wave. The Barker code of the 13 length is shown in Figure 14.1. Figure 14.1(a) shows the code itself and Figure 14.1(b) shows the autocorrelation of the code. In this figure all the sidelobes are of the same amplitude, which is an unique property of the Barker codes. Each of the Barker codes studied will have two different chip times: 20 and 500 ns. The 20-ns chip time is select to study the high chip rate effect, such as two phase transitions in one FFT frame. The 500-ns case does not represent the longest chip time of a realistic radar signal because some radar may have a longer chip time such as several microseconds. The 500-ns chip time is chosen in order to avoid using a long digitized data because it will take longer to run simulations. The sampling rate is still assumed to be 2,560 MHz and the ADC will have a large number of bits. In

Figure 14.1  Barker code of the 13 length: (a) time domain and (b) autocorrelation.

14.3  Generation of BPSK Signals and Their FFT Outputs

323

other words, the digitization effect will be neglected. The receivers in Chapters 12 and 13 have a sensitivity of S/N about slightly below of -4 dB. Thus, it is arbitrarily chosen that the weakest BPSK to be detected is about S/N = 0 dB. A well-known approach to obtain a BPSK signal is through squaring the input data. With this operation, the biphase transition will be eliminated and the input frequency becomes a cw signal with a frequency equal to twice the input frequency. Doubling the input frequency will increase the noise level and reduce the receiver sensitivity. Although this approach may be effective, its implementation in a receiver will not be considered.

14.3  Generation of BPSK Signals and Their FFT Outputs [2] First, a BPSK signal is generated and the basic properties will be studied. In this study, FFT will be performed on the signal. The purpose of this operation is to find the time and frequency domain of a BPSK signal. A BPSK signal can be written as x = cos(2π ft + φ )

where

φ = 0 or π

(14.2)

When φ = 0, x = cos(2πft); φ = π, x = –cos(2πft). One phase transition is shown in Figure 14.2. The BPSK signal with length 11 is shown in Figure 14.3. Figure 14.3(a) shows the only Barker code and Figure 14.3(b) shows the entire signal. The input frequency is arbitrarily chosen at 200 MHz and the phase transition can be seen in Figure 14.3(b).

Figure 14.2  One 180° phase transition.

324

Detection of Biphase Shift Keying (BPSK) Signals

Figure 14.3  Barker coded signal with length 11: (a) Barker code and (b) signal.

Two BPSK signals with different chip times are generated and their FFT outputs will be observed. Both are Barker codes of length 11. When the chip time is 20 ns, 561 data points are generated and the total signal length is about 2.19 ms. The input signal to noise ratio (S/N) = 100 dB is used to avoid the noise effect and the input frequency is arbitrarily chosen at 800 MHz (the center of a frequency bin). Since it is desirable to perform FFT with a base 2 number, 1,024 will be used for the FFT. The input is filled with 463 (1,024 - 561) zeros. Zero padding can increase the frequency resolution and show the fine structure of the input signal. Figure 14.4 shows the results. Figure 14.4(a) shows all the 512 outputs and Figure 14.4(b) shows the results near the peak value. In the second example, the chip time is 500 ns. For a Barker code of length 11, the total the total output data points are 14,080, which are about 5.5 ms long. The base 2 number used for the FFT operation is 32,768 rather than 16,384 because the high number of points can provide a better display. The window fills with 18,688 zeros. Figure 14.5 shows the results. While Figure 14.5(a) shows the entire spectrum, Figure 14.5(b) shows the results near the peak. From Figures 14.4 and 14.5 one can see that even though the chip time is 25 times difference (20 versus 500 ns), the spectrums are quite similar. There is a dip at the center of the main spectrum lobe and this is the nature of BPSK signal; the carrier is suppressed. This phenomenon can be simply explained as follows. If a sine wave of limited length is divided into two sections of equal length, the sign of the second is changed from +1 to -1 and this is equivalent to a π phase shift. Under this condition the spectrum generated from the first half and the second half will cancel each other and the carrier will be suppressed.

14.3  Generation of BPSK Signals and Their FFT Outputs

325

Figure 14.4  Frequency domain of a length 11 Barker code with a 20-ns chip time: (a) all spectrum and (b) spectrum around the maximum.

Figure 14.5  Frequency domain of a length 11 Barker code with a 500-ns chip time: (a) all spectrum and (b) spectrum around the maximum.

326

Detection of Biphase Shift Keying (BPSK) Signals

Since the purpose is to illustrate the spectrum shape, no particular window (or rectangular window) is applied to the input data. The high sidelobes are generated from the rectangular window. The highest one is about 13 dB down.

14.4  Using FFT Outputs to Determine the Existence of a BPSK Signal This detection is based on a long FFT operation and may take a special effort to implement. The first requirement on the detection of a BPSK signal is to detect the existence of it and generate an indicator. When two continuous wave (cw) signals are present, the indicator should not generate a report. In other words, the method must be able to differentiate one BPSK signal and two cw signals. If two cw signals cause the indicator to generate a report, it will be considered as a false alarm. Based on this requirement, the following study is performed. The two signals with 20- and 500-ns chip times used in Section 14.3 will be the input signal. For the 20-ns chip time 561 data points are generated and a 512-point FFT is performed. For the 500-ns chip time case, 14,080 data points are generated and a 8,192-point FFT is performed. Selecting the window length shorter than the data length can fill the window with signals. A Blackman window is used in this study to limit the width of the spectrum output. Usually, a threshold determined by the noise must be obtained first. With the Blackman window the noise distribution is still Rayleigh. The noise threshold is obtained from a 1,000 run. The desired probability of false alarm (Pfa) is the usual value of 10-7. For the 512-point FFT, 233 (256 - 23) output frequency bins are used in the study. With 1,000 runs there are 2.33 × 105 data points. For the 8,192-point FFT operation, 3,897 (4,096 - 199) frequency bins are selected. For 1,000 runs the total data points are 3.897 × 106 and the inverse of this value is close to the desired Pfa. The thresholds are shown in Figures 14.6 and 14.7 for the 512- and 8,192point FFTs and the actual threshold values are 46.34 and 185.64, respectively. It is interesting to note that the threshold in Figure 14.7 is very close to the highest data point. This is reasonable because the number of total noise outputs is close to 10-7, the inverse of the Pfa. Before the detection of the input signal, let us take a look at the output shape of the 512 and 8,192 points with a BPSK signal. Figure 14.8 shows the results with the input frequency arbitrarily chosen at 800 MHz and only the frequency bins near the peak value are shown. In obtaining the output, a Blackman window is used. It is interesting to note that in Figure 14.8(a) the center frequency bin becomes a peak rather than the expected valley. This phenomenon is caused by truncating the Barker code, not by the Blackman window. The following procedure is used to determine the existence of a BPSK signal. FFT will be performed on the 512 input data points and a Blackman window is used. The Blackman window will limit the sidelobes of a cw signal. If the maximum of the FFT outputs passes the threshold, a signal is detected. Next to the maximum output of nine frequency bins are selected and four on each side of the maximum. This selection of nine frequency bins is determined empirically. Among these nine elements, the ratio of the minimum to maximum will be calculated. If the signal is a BPSK, the ratio should be large because the frequency bins all have relatively high

14.4  Using FFT Outputs to Determine the Existence of a BPSK Signal

327

Figure 14.6  Noise distribution and threshold for 512-point FFT outputs with Blackman window.

values. An empirical value of 0.3 is selected as the threshold. If the calculated ratio is higher than the threshold, it will be identified as a BPSK signal or other types of spread spectrum signals. However, in this study only the BPSK signals are of interest. If the maximum of the FFT output does not cross a threshold of 46.34 obtained in Section 14.3, the ratio is set to zero and the signal will not be detected.

Figure 14.7  Noise distribution and threshold for 8,192-point FFT outputs with Blackman window.

328

Detection of Biphase Shift Keying (BPSK) Signals

Figure 14.8  FFT outputs of a Barker code with a Blackman window: (a) 512 points and (b) 8,192 points.

An important factor is to test whether this approach will classify multiple cw signals as BPSK. If this approach is good, the number of false detections will be low. Since the combination of two signals can be numerous, the two signals used in this test are of the same amplitudes. The S/N used in the simulation is the same the BPSK signal. One frequency is randomly selected and the second frequency is lower than the first one by 1 to 30 MHz in 1-MHz steps. Both frequencies are limited to 141 to

14.4  Using FFT Outputs to Determine the Existence of a BPSK Signal

329

Figure 14.9  Detection of BPSK signal and false detection of two cw signals at S/N = -5 dB.

1,140 MHz. At each frequency separation 1,000 random frequencies are selected. If the maximum frequency output does not cross the threshold of 46.34, the ratio will be set to zero and will not be counted as a false detection. The test results are shown in Figures 14.9 to 14.11. In Figure 14.9 the x-axis represents the frequency separation in megahertz for the two cw signals and the

Figure 14.10  Detection of BPSK signal and false detection of two cw signals at S/N = 0 dB.

330

Detection of Biphase Shift Keying (BPSK) Signals

Figure 14.11  Detection of BPSK signal and false detection of two cw signals at S/N = 5 dB.

y-axis represents the number of signal and the false signal detected. In Figure 14.9 the S/N = -5 dB, the average detection of the BPSK signal is 211/1,000, and the false detection of two cw signals is 0.13/1,000. In Figure 14.10 the S/N = 0 dB, the average detection of the BPSK signal is 735/1,000, and the false detection of two cw signals is 0.13/1,000. In Figure 14.11 the S/N = 5 dB, the average detection of the BPSK signal is 925/1,000, and the false detection of two cw signals is 0.067/1,000. From these three figures, one can see that at different frequency separations between the two cw signals the false detects stay about the same. When the S/N = 10 dB, the average detection of the BPSK signal is 994/1,000 and the false detection of two cw signals is 0/1,000. Increasing the S/N beyond 15 dB, usually 1,000 BPSK is detected and the false signal detection is still 0. The last two results are not shown graphically. From these results, the BPSK signal can be detected with above a 90% probability of detection. The false detection usually is rather small, such as 0.01%. These results can be shown in Table 14.1. For the chip time of 500 ns, the BPSK has 10,480 data points, but only 8,192 points are used in the FFT operation. A similar method is used to detect the BPSK signal and the two cw signals. The only change made is to the thresholds. The maximum FFT output must be higher than 185.64 to be considered as a detection. The ratio threshold is changed to 0.05 rather than the 0.3 values used in the previous detection. In order to reduce the computation time caused by the long FFT opera-

Table 14.1  Detection of BPSK Signal Using FFT Outputs S/N (dB) Pd False detection

–5 21.1 0.013

0 73.5 0.013

5 92.5 0.007

10 99.4 0

15 100 0

14.6  Study of Three Eigenvalues on Complex BPSK and CW Signals

331

tion, at each frequency difference 100 random frequencies are selected, rather than 1,000. With this threshold at S/N = 0 dB, 96.9/100 BPSK signals and 3.5 false signals are detected. Since the result is similar to the short chip time, no further study is made.

14.5  Study of Two Eigenvalues on Complex BPSK and CW Signals The purpose of this study is the same as that in Section 14.4 to detect the existence of BPSK and avoid detection of two cw signals. In this section the eigenvalue method will be studied. Only the approach will be presented because it is interesting. If the input signal is real, one input signal will affect two eigenvalues. In order to study two input signals, a minimum of three eigenvalues will be needed. If this approach is used at the outputs of a filter bank, the signal will be complex such as through an FFT operation. Since both the BPSK and the cw signals used in this study are complex, two eigenvalues are required to detect two signals. The concept of using a 2 × 2 matrix with two eigenvalues to differentiate a BPSK and two cw signals is intriguing. However, the results will not be presented because they appear slightly inferior to the FFT method. The approach is to find two sets of two eigenvalues [personal communication with L. Y. Liou, engineer at AFRL, 2009. The first set uses the lag (1, 2) and the second sets use the lag (1, 17). These two lags are determined empirically. Let us refer to the two sets of eigenvalues as (e1, e2) and (f1, f2,), while subscript 1 means the first (larger) eigenvalue set and subscript 2 means the second set. The symbols e and f represent the two different matrices. If the input signal is a BPSK, the difference between e2 and f2 that is (e2 - f2) is close to a constant K. This constant K depends on the input signal power, which is related to the sum of the first two eigenvalues e1 and f1 that is (e1 + f1). From the (e1 + f1) value the constant K can be determined. Near this K value, a range of K ± k, where k is some predetermined value, can be considered as the threshold. If the (e2 - f2) value is within K ± k value, a BPSK signal is detected. From this study, it is interesting to note that for the two sets of two eigenvalues with different lags there are certain relations between the large and small eigenvalues. The relation is signal type dependent. Using only two sets of two eigenvalues, the true detection is very high, but the false signals detected are also relatively high. To reduce the false detection, a third set of eigenvalues can be added. The lag used is (1, 6). With this additional two eigen­ values of (g1, g2), a similar detection approach on (e1, e2) and (f1, f2) is performed by replacing (f1, f2) by (g1, g2). With the third eigenvalue set the false detection is reduced; however, the result is still slightly worse than with an FFT approach.

14.6  Study of Three Eigenvalues on Complex BPSK and CW Signals Using three eigenvalues to detect a BPSK signal is rather easy to understand. Two cw signals only affect two eigenvalues. If the third eigenvalue is large for a BPSK signal, the BPSK can be detected. It appears that the only problem is to select the combination of lags to enhance the third eigenvalue. The threshold is obtained from

332

Detection of Biphase Shift Keying (BPSK) Signals

the correlation matrix with lags (1, 2, 3). The largest eigenvalue from the noise input is used as threshold. The distribution of the largest eigenvalue is obtained from 10,000 runs and is shown in Figure 14.12. The distribution is approximated by a Gaussian function and the threshold of 2.88 can be obtained from (6.11). The next step is to find a lag combination that can produce a high third eigenvalue (the smallest one). The lags are selected through a sequence search. For a 3 ´ 3 matrix the lags can be represented by (1, x, y). The input signal is a BPSK signal at an arbitrarily selected frequency but without noise. The x value changes from 2 to 200 and the y value changes from 3 to 200 for 561 points of input data. Both changes are in unit steps. The third eigenvalue is monitored in every step and there will be 39,402 outputs. The highest value occurs at x » 45 and y » 115, which is about 0.08 (45/561) and 0.2 (115/561) of the total data length. It appears that the maximum has a plateau and the x and y values are not very critical. The center frequency of the BPSK signal is also not very important. When the frequency changes from 141 to 1,140 MHz, the x y values do not have large variation. The final step is to detect the existence of BPSK signal. The lags of (1, 45, 115) are used. It appears that the false detection of two signals is not a problem, because two cw signals only affect two eigenvalues, which is the expected result. The BPSK has a 20-ns chip time with a random frequency from 141 to 1,140 MHz. The input data contain 561 data points. The results are shown in Table 14.2. In comparing these results with the results in Section 14.3, one should pay attention to the input signal. In Section 14.3 the input signals are real, while in this section the signals are complex. For the same S/N, the S/N of the complex data is 3 dB higher than the real data. For example, the S/N = 0 dB in Table 14.2 is equivalent to S/N = 3 dB in Table 14.1. It appears that using three eigenvalues for detection of

Figure 14.12  The largest eigenvalue distribution and the threshold.

14.8  Using Conventional FFT Receiver Outputs to Detect a BPSK Signal

333

Table 14.2  Detection of BPSK Signal Using Three Eigenvalues S/N (dB) Pd False detection

–1 4.1 0

0 62.1 0

1 99.7 0

2 100 0

BPSK signal is slightly better. However, this method of detection can only apply to a complex signal; it is difficult to apply to real signal. For a real signal five eigenvalues are needed for the detection. The discussion up to this point has been based on the assumption that the length of the BPSK signal can be measured. The time-domain detection method in Section 8.18 can serve this purpose. It is desirable to detect the BPSK signal through a receiver output. The two types of receivers are the conventional FFT operation and polyphase filter approaches. In other words, it is desirable to detect the existence of BPSK signal after the FFT operation.

14.7  Define the Chip Time Limits of the BPSK Signals In Section 14.4 long FFTs are performed to cover the entire Barker code. In actually detecting the signal, a single phase transition in an FFT frame is anticipated to be found. Based on this assumption, the limit of the BPSK chip time will be selected. These limits of signals are based on the conventional FFT operation receiver design discussed in Chapter 11 and they do not represent realistic parameters in existing radars. The lower and upper chip time limits will affect the FFT outputs. The FFT length used in Chapter 11 is 128 points, which corresponds to a window time of 50 ns. If the chip time is shorter than 50 ns, it is possible to have two phase transitions in one data frame. If the chip time is longer than 100 ns, at most every two data frames can contain one phase transition or two phase transitions cannot occur in two consecutive data frames. Based on this argument, the chip times are selected between 20 and 100 ns for this study. For a 20-ns chip time, two phase transitions can occur in one time frame. For the chip time from 50 to 100 ns two transitions can occur in two consecutive windows. If the chip time is 100 ns or longer, two phase transitions can not occur in two consecutive frames. The purpose of the study is to cover all possible phase transition combinations with respect to the frame time. Before characterizing a BPSK signal, it is important to detect the existence of a signal. In order to keep this subject simple, only one BPSK will appear in one receiver output channel and simultaneous signal conditions will not be considered.

14.8  Using Conventional FFT Receiver Outputs to Detect a BPSK Signal In Chapter 11, the conventional FFT receiver design is discussed. Each frame has 128 points and a Blackman window. With a 2.56-GHz sampling rate, 128 points take 50 ns and produce 64 outputs. Among the outputs 53 channels are selected as shown in Table 6.1. The channel with highest output crossing a detection threshold will be used for the following studies.

334

Detection of Biphase Shift Keying (BPSK) Signals

The Barker code of length 11 and the modulated RF signal are shown in Figure 14.3, where there are five phase transitions at data points 52, 103, 205, 256, and 409. If 128 data points are the window length, the first window (1-128) contains two phase transitions. The second window (129-256) contains also two phase transitions, but the second phase transition is at the last data point. The third window (257-384) contains no data transition, and the fourth window (385-512) contains one. The input frequency is chosen at the center of a frequency bin of 600 MHz and the S/N of 10 dB. Figure 14.13 shows the frequency outputs of the four frames. In Figure 14.13(a, b) the spectrum spreading can be easily identified. From these figures, the detection scheme discussed in Section 14.4 should be applicable. In Figure 14.13(c) there is no phase transition in the data and the result looks like a cw signal, which is the expected result. In Figure 14.13(d) there is one phase transition, but the result is only slightly different from a cw signal. The reason might be that the phase transition is too close to one end of the data and that is at the 24th point. When the input frequency is shifted to the boundary of two frequency bins, such as at 610 MHz, the results are shown in Figure 14.14. These results are similar to Figure 14.13. When the chip time is long, such as over 50 ns, it is only possible to have one phase transition in one data frame. If the phase transition is close to the center of the data frame, the frequency domain will show the spectrum spreading and the phase transition can be detected. If the phase transition is close to the leading or trailing edge of the data frame, the spectrum will be close to a cw signal and the transition is difficult to detect. Figures 14.15 and 14.16 show the frequency-domain outputs as a function of the transition position. In both figures from (a) to (d) the phase transition changes

Figure 14.13  FFT outputs of four windows with input at 600 MHz and S/N = 10 dB: (a) with two phase transitions, (b) with two phase transitions at one end, (c) without a phase transition, and (d) with one phase transition at the 24th point.

14.8  Using Conventional FFT Receiver Outputs to Detect a BPSK Signal

335

Figure 14.14  FFT outputs of four windows with input at 610 MHz and S/N = 10 dB: (a) with two phase transitions, (b) with two phase transitions at one end, (c) without a phase transition, and (d) with one phase transition at 24th point.

Figure 14.15  FFT Four windows with only one phase transition at 600 MHz and S/N = 10 dB: (a) phase transition at 28, (b) phase transition at 29, (c) phase transition at 30, and (d) phase transition at 31.

336

Detection of Biphase Shift Keying (BPSK) Signals

Figure 14.16  FFT Four windows with only one phase transition at 610 MHz and S/N = 10 dB: (a) phase transition at 28, (b) phase transition at 29, (c) phase transition at 30, and (d) phase transition at 31.

from the 28th point to 31st point, which is about one-fourth of the data length. The input frequency in Figure 14.15 is at 600 MHz, the center of a frequency bin. The input frequency in Figure 14.16 is at 610 MHz, the boundary between two frequency bins. Comparing with Figures 14.13(c) and 14.14(c), the results of cw signal, one can see that the spectra in Figures 14.15 and 14.16 are slightly wider. Also when the phase transition moves from position 28 to 31, the spectrum is slightly wider, although the results are not very clear. It appears that if one takes four frequency bins from either side of the maximum, from the maximum to minimum ratio the BPSK signal can be detected. The method has been discussed in Section 14.4. Thus, using the conventional FFT output is not a dependable approach to detecting a BPSK signal. When the phase transition is close to the center of the data frame, it is can be detected.

14.9  Using Two Frames to Detect BPSK Signals In the following two sections the discussion is based on one assumption: in one FFT output, there is only one cw signal. If there are two cw signals in one FFT output, the conclusion will be erroneous. The phase comparison method can be used to determine the phase between two consecutive data frames. If the signal is cw, the phase differences between the two consecutive data frames are close. If the phase difference is different, a phase transition can be found in the data. The phase transition is assumed to be caused by the BPSK signal.

14.9  Using Two Frames to Detect BPSK Signals

337

Figure 14.17 shows the results. The input is Barker code of length 11 and the initial phase is randomly selected. The S/N equals 0 dB with an input frequency arbitrarily chosen at 600 MHz and chip time is 100 ns, which equals two frame times. Under this condition, 2,816 data points are generated, which contain 22 data frames. The most frequent phase transition is that every two frames contain one phase transition. Figure 14.17(a) shows the Barker code. Figure 14.17(b) shows the phase difference between two consecutive frames. When there is no phase shift between two consecutive frames, the phase difference stays at a constant value, which depends on the carrier frequency and the initial phase. Figure 14.17(c) shows the difference of phase difference. For simplicity, let us refer to the difference of phase difference as the double phase difference. It might be easier to find the phase transition from Figure 14.17(c) because when there is no phase transition, the difference stays close to zero. Either the positive or the negative peaks are close to the phase transition points. When the chip time is shorter than 100 ns, it is possible to have two phase transitions in two adjacent frames. If the chip time is still shorter, it is possible for one data frame to contain two phase transitions as discussed in Section 14.8. Under both conditions, the position of the phase transition is difficult to locate. However, if the double phase difference is not close to zero, it is can be considered as a spread spectrum signal. Figure 14.18 shows the double phase difference. The input frequency is at 600 MHz with 0 initial phase. The S/N = 0 dB, which is high enough to show the difference between a cw and BPSK signal. The chip time is 40 ns, which is less than the window frame time of 50 ns. The Barker code of length of 11 under this condition

Figure 14.17  Phase comparison of a BPSK signal after the FFT operation with random input frequency and initial phase, S/N = 0 dB: (a) Barker code, (b) phase difference, and (c) difference of phase difference.

338

Detection of Biphase Shift Keying (BPSK) Signals

Figure 14.18  Double phase difference chip time = 40 ns S/N = 0 dB random input frequency and initial phase: (a) cw signal and (b) BPSK signal.

generates 1,122 data points, which are more than 8 frames of 128 points. There will be seven phase differences and only six double phase differences. Figure 14.18(a) shows the results of a cw signal and Figure 14.18(b) shows the results of a BPSK signal. The difference between these two is obvious. When this input is a cw signal, the double phase difference is close to zero [Figure 14.18(a)]. When the input signal is a BPSK, the double phase difference can be far away from zero [Figure 14.18(b)]. If a threshold is set around zero for the double phase difference, a BPSK should be detected against a cw signal. Figure 14.19 shows the maximum double phase difference for cw and BPSK signals. The input frequency changes from 141 to 1,140 MHz in a 1-MHz step. The BPSK uses the Barker code of length 11. Two chip times are used: 20 and 100 ns with S/N = 10 and 0 dB, respectively. At each input frequency the input data are divided into 128-point frames. The phase differences and the double phase difference are also calculated. When the double phase difference is about 3π/2, which is considered close to 2π and is set to zero. This operation takes care of the phase wrapping problem. The maximum double phase difference is selected for every input frequency. At each input frequency a cw signal with the same data length is generated. A similar phase test is applied. The results are plotted in Figure 14.19. The upper results are from the BPSK signal and lower result is from the cw signal. Figure 14.19 shows that the BPSK signal has a larger double phase difference than the cw signal. From these results a threshold should be easily chosen to determine the existence of the BPSK signal. Figure 14.19(a, b) shows the chip time of 20 and 100 ns, respectively.

14.10  Using an Eigenvalue After the FFT Operation

339

Figure 14.19  Maximum double phase difference of cw and BPSK signals: (a) chip time = 20 ns, S/N = 10 dB and (b) chip time = 100 ns, S/N = 0 dB.

These results show that both short and long chip times can be separated from a cw signal. Detecting a short chip time BPSK signal requires a higher S/N than a signal with a longer chip time. From this and the previous sections, the BPSK signal can be detected in two steps. The first one is detect the width of the spectrum on a frame-by-frame base. If a phase transition can be detected in a certain frame, its location can be determined. Of course, there is the possibility of having two transitions in one frame. If the phase transition occurs close to the boundary of a frame, the spectrum detection method may not detect them. The phase comparison method between two consecutive frames can be used. Simulation shows that this approach is rather effective. If the chip time is long enough that no phase transition occurs in two consecutive frames, the location of the phase transition can be estimated through the phase comparison method. If the chip time is short and phase transition can occur in two consecutive frames or even two phase transitions can occur in one frame of data, the double phase difference can be used to indicate that the signal is BPSK. When the chip time is short such as 20 ns, a higher S/N is required for the detection of a BPSK signal.

14.10  Using an Eigenvalue After the FFT Operation Since it is assumed that one FFT output channel contains only one cw signal, if two signals are found in one channel, the result can be considered a spread spectrum.

340

Detection of Biphase Shift Keying (BPSK) Signals

The eigenvalue method discussed in Chapter 8 will be used to determine the existence of a BPSK or a spread spectrum signal. This method will be discussed only briefly as an alternative one to the phase comparison method. The highest FFT output amplitude will be used to form the correlation matrix. Since the FFT output data are complex, a 2 by 2 matrix with two eigenvalues can be used to determine the number of input signals. It is assumed that only one signal is in one FFT output. The lag of the correlation matrix is determined by the output data length. One lag is 1 and the second one is about 1/3 of the data length. Using the chip time of 20 ns can produce 561 data points and the outputs can be divided into four 128 data frames, which are considered as 4 complex data points for the eigenvalue test. The second lag is selected as 2. The signal input conditions are S/N = 10 dB, chip time = 20 ns, Barker code length = 11, and frequency changes from 141 to 1,140 MHz in a 1-MHz step. Two eigenvalues are obtained. The larger eigenvalue represents the signal and the second eigenvalue represents either noise or spectrum close to the main spectrum. The ratio of the smaller eigenvalue to the largest one is calculated and referred to as the eigenvalue ratio. Figure 14.20(a) shows the eigenvalue ratio of a cw and a BPSK signal. The upper results are from a BPSK signal and the lower ones are the results of a cw signal. As expected, when the input signal is cw, the eigenvalue ratio is small. When the input is a BPSK signal, the eigenvalue ratio is large. From this figure, it is obvious that the two different types of signals can be separated at S/N = 10 dB. Since there are only four output points, the sensitivity is low and a relatively high S/N is required.

Figure 14.20  Comparison of eigenvalue ratio of a BPSK and a cw signal generated from a 2 ´ 2 correlation matrix: (a) chip time = 20 ns and S/N = 10 dB and (b) chip time = 100 ns and S/N = 0 dB.

14.11 Detecting BPSK with a Phase Comparison After the Polyphase Filter

341

Figure 14.20(b) shows similar results. The only difference is that the chip time is 100 ns and the S/N = 0 dB. These are the same signal conditions used as examples in Section 14.9. Both the phase comparison and the eigenvalue methods are effective in separating the BPSK signal from a cw signal.

14.11  D  etecting BPSK with a Phase Comparison   After the Polyphase Filter The outputs from the polyphase filters are processed differently from the approach presented in Chapter 13. In addition to the previous processing, the following process will be performed to find spread spectrum signals. The detection methods of a BPSK signal from the polyphase filter outputs is similar to the methods discussed in Sections 14.9 and 14.10. One method is to use the double phase difference and the other method is to use the eigenvalues. The output from the polyphase filter may contain two signals because the channels are wider. If there are two signals in one output, the phase difference measurement may not be dependable. However, the simultaneous signal problem will not be addressed as indicated at the beginning of this chapter. Although each time frame contains 128 data points, only about 16 points in the center of the window make a major contribution to the output as discussed in Chapter 13. The phase transition will be affected by the polyphase filter, because the phase transition has a wide spectrum and a filter will reduce the bandwidth of the signal. In many cases the results depend on the position of transition in the data frame and the input frequency. Simulation results indicate that the phase transition can still be found from the polyphase filter outputs. Figure 14.21 shows the double phase difference method used in Section 14.9. In this study the outputs from the polyphase filters are used as input. In order to have a constant phase relation between two data frames, the outputs from the polyphase filter are decimated by 2. In other words, the phase comparison is performed on data frames separated by 16 data points rather than the 8 points. The 8-point shift is only used to increase the bandwidth from 160 to 320 MHz as discussed in Chapter 13. The input frequency changes from 141 to 1,140 MHz in a 1-MHz step with S/N = 8 dB. Since the polyphase filter is 160 MHz, which is 8 times wider than the conventional FFT operation, the sensitivity will be lower. It appears that a minimum S/N of about 8 dB is required to separate the BPSK and a cw signal. For the conventional FFT operation, the S/N is about 0 dB as shown in Figure 14.19. In Figure 14.21 when the input frequency is at about 240, 400, 560, 720, 880, and 1,040 MHz, the separation is near a minimum, where it is difficult to separate the BPSK from a cw signal. These frequencies are at the edge of the polyphase filters. In Figure 14.21(a, b), both input S/N are 8 dB and the chip times are 20 and 100 ns, respectively. It shows that it is easier to separate the two types of signal when the chip time is 20 ns than 100 ns. For the 100-ns chip time one frame can contain only one phase transition and the phase transition might be smoothed by the polyphase filter. The causes of these small separations at the filter edge as shown in Figure 14.21 and the easier detection of shorter chip time need to be further studied.

342

Detection of Biphase Shift Keying (BPSK) Signals

Figure 14.21  Maximum double phase difference to separate cw and BPSK signals for polyphase filter outputs and S/N = 8 dB: (a) chip time = 20 ns and (b) chip time = 100 ns.

14.12  D  etecting BPSK with an Eigenvalue Ratio After   the Polyphase Filter Because the high probability of have two signals in one polyphase filter, a 2 ´ 2 correlation matrix will not be a good approach to detect a BPSK signal after the polyphase filter. If two signals are detected, they may be two cw signals. In order to remedy this problem, a 3 ´ 3 matrix will be used to find the existence of three signals. If the eigenvalue method detects three signals, it is assumed that a BPSK signal is detected. Of course, if three signals are in one complex receiver, this assumption will be wrong. The lags used to generate the 3 ´ 3 matrix are determined by the results in Section 14.5. One lag is approximately 0.08 and the other is about 0.2 of the total data length of the polyphase filter outputs. The ratio of the smallest to the largest eigenvalue is calculated. The results of the Barker code of length 11 signal with S/N = 8 dB and the frequency changes from 141 to 1,140 MHz are shown in Figure 14.22. In Figure 14.22(a) the chip time equals 20 ns and in Figure 14.22(b) the chip time equals 100 ns. In Figure 14.22(a) the separation between of a BPSK and cw signal varies slightly with the input frequency. The minimum separations are close to the same frequency range as in Figure 14.21. In Figure 14.22(b), the variation of eigenvalue ratio versus frequency disappears. Contrary to the results in the double phase difference method, it is easier to detect the signal with a chip time of 100 ns rather than 20 ns. The eigenvalue method

14.13  Find the Phase Transition Locations in the BPSK Signal

343

Figure 14.22  Comparison of eigenvalue ratio of a BPSK and a cw signal by a 3 ´ 3 correlation matrix for polyphase filter outputs at S/N = 8 dB: (a) chip time = 20 ns and (b) chip time = 100 ns.

uses all the phase transitions from the polyphase filter outputs rather comparing the phases between two consecutive frames. When the phase transitions are separated into different frames, the eigenvalue method can detect them better.

14.13  Find the Phase Transition Locations in the BPSK Signal In this section the locations of the phase transition from the FFT and the polyphase filter outputs will be determined. The approach is to use the phase comparison of two consecutive frame outputs. This is the phase difference method, not the double phase difference. When the chip time is short, it is difficult to find the phase transitions in the FFT outputs because multiple transitions may be in one data frame. In the first example, the input frequency is arbitrary selected at 800 MHz and the initial phase is randomly selected. The Barker code of length 11 is used and S/N = 10 dB and chip time is 20 ns. The BPSK data generated may not be a binary-based number. In order to process all the data, 128 zeros are added. For example, with this input condition 561 points of data are generated. Only 512 points will be processed by four 128 data frames. If 128 zeros are added at the end, all 561 points will be processed by 5 frames. Padding with zeros at the end may create an additional phase transition in the end. Thus, in Figures 14.23 sometimes the last few phases are discarded.

344

Detection of Biphase Shift Keying (BPSK) Signals

Figure 14.23  Phase difference for chip time = 20 ns, S/N = 10 dB, frequency = 800 MHz: (a) Barker code, (b) FFT outputs, and (c) polyphase filter outputs.

Figure 14.23 shows the phase difference of both the FFT and polyphase filter results. Figure 14.23(a) shows the Barker code. Figure 14.23(b) shows the phase difference of the FFT outputs, which does not contain much information. Since the chip time is short, there are total of 5 output data points and 4 phase differences. Figure 14.23(c) shows the phase difference for the polyphase filter outputs. At this special frequency the phase difference is close to zero. The two lines above and below 0 are the threshold for detecting phase transitions. The threshold is arbitrarily chosen as 1.2 above and below the average phase difference. In Figure 14.23 there are four points crossing the threshold and the first phase transition is missed. It should be noted that the time in Figure 14.23(a–c) is not properly aligned. The x-axis is adjusted to match the output data points. This type of adjustment is also used in the Figures 14.24 and 14.25. In an actual receiver, when the positions of the phase transitions are found, their time must be calculated from the filter outputs. Figure 14.24 shows similar outputs as those in Figure 14.23. The only difference is that the chip time equals 100 ns. The initial phase is still randomly selected. Under these conditions both the FFT and polyphase filter outputs can properly detect the phase transition. When the input signal is weak, such as at S/N = 0 dB with a chip time = 100 ns, both methods can still detect the phase transitions as shown in Figure 14.25.

14.13  Find the Phase Transition Locations in the BPSK Signal

345

Figure 14.24  Phase difference for chip time = 100 ns, S/N = 10 dB, frequency = 800 MHz: (a) Barker code, (b) FFT outputs, and (c) polyphase filter outputs.

Since the detection depends on the phase difference, the results are frequency dependent. For S/N = 10 dB, frequency = 400 MHz (boundary of two bins for the polyphase filter) with an initial phase and a chip time of 100 ns, the results are shown in Figure 14.26. In Figure 14.26(b) the phase transition can be identified. In Figure 14.26(c) the phase transition can be visually identified, but automatically detection requires more processing. The same situation can occur in the FFT output. If the input frequency is changed to 410 MHz (boundary of two bins for the FFT operation), the outputs are shown in Figure 14.27. In Figure 14.27(b) the phase transient is difficult to identify, but in Figure 14.27(c) the phase transient can be identified. The problem in Figures 14.26 and 14.27 is that when the phase difference is near π or –π, the result is difficult to read. This can be modified by change the sign of –π to π. Figure 14.28 shows similar results in Figure 14.26. In Figure 14.28(b) the phase transient is difficult to identify but after the –π or π change the results are shown in Figure 14.28(c), where the phase transient can be easily identified. From this simple comparison, it appears that the FFT outputs are less immune to noise because longer data are used in the operation, but the FFT operation cannot process a BPSK signal with a short chip time. A signal with a short chip time can be detected after the polyphase filter outputs but is more sensitive to noise. The

346

Detection of Biphase Shift Keying (BPSK) Signals

Figure 14.25  Phase difference for chip time = 100 ns, S/N = 0 dB, frequency = 800 MHz: (a) Barker code, (b) FFT outputs, and (c) polyphase filter outputs.

Figure 14.26  Phase difference for chip time = 100 ns, S/N = 10 dB, frequency = 400 MHz: (a) Barker code, (b) FFT outputs, and (c) polyphase filter outputs with problem.

14.14  Conclusion

347

Figure 14.27  Phase difference for chip time = 100 ns, S/N = 10 dB, frequency = 410 MHz: (a) Barker code, (b) FFT outputs with a problem, and (c) polyphase filter outputs.

double phase difference results are not shown in this section because the results are rather noisy due to the difference operation.

14.14  Conclusion In this chapter several methods such as the double phase difference or eigenvalue are shown to detect the existence of a BPSK signal. The BPSK signal can be detected at both the FFT and polyphase filter outputs. Therefore, one can provide additional processing at the filter outputs to detect the BPSK signal. The locations of the phase transitions can also be detect with the phase difference or double phase difference. In performing a phase difference it is assumed that there is only one signal in a channel. The FFT outputs with a relatively narrow bandwidth have a higher probability of achieving this assumption. When the chip time is short, it is difficult to detect the locations of the phase transitions from the FFT outputs because of the relatively long frame time used. Both the FFT output and the eigenvalue methods can be used to determine the existence of a spread spectrum signal. Because the eigenvalue method uses all the information in the signal, it has a better sensitivity than observing the FFT outputs or the phase comparison method. However, it is difficult to use the eigenvalue method to find the location of the phase transition location.

348

Detection of Biphase Shift Keying (BPSK) Signals

Figure 14.28  Phase difference for chip time = 100 ns, S/N = 10 dB, frequency = 400 MHz: (a) Barker code, (b) polyphase filter outputs with a problem code, and (c) polyphase filter outputs modified.

In building a receiver, all the methods discussed in this chapter—the phase difference, double phase difference, observing the FFT outputs, and the eigenvalue methods—should be further studied to compare their performance as well as implementation complexity. Finally, an encoder must be designed to incorporate the detected information into the pulse descriptor word (PDW).

References   [1]  Wikipedia “Barker code.”   [2]  Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Chapter 8, Norwood, MA: Artech House, 2001.

C h a p t e r 15

Frequency Modulated (FM) Signals

15.1  Introduction In this chapter the frequency modulated (FM) signal will be studied. This type of radar signal is often referred to as a chirp signal because its frequency changes with time. There are several types of chirp signals. When the frequency of the signal changes from high to low, it is referred to as a down chirp and from low to high is the up chirp. The radar signals may include both up and down chirp signals. In some radar the frequency changes linearly with time. In others the frequency changes nonlinearly with time. The chirp rate, which is the frequency change per unit time, also varies for different radars. Among all the different types of chirp signals only the linear frequency changed chirp will be studied because it is probably the most popular one. The study is limited to the up chirp signal. It is anticipated that the down chirp signal should produce similar results. Two chirp rates will be studied—one high and one low—and hopefully they represent the limits of the chirp signals. This study will be similar to the BPSK signal detection discussed in Chapter 14. The time- and frequency-domain responses of the chirp signal will be studied first. Then the signal will be detected by using the FFT method and the eigenvalue methods. Finally the signal will be passed through both the conventional FFT and polyphase filter receivers. Special detection schemes can be used to detect both the existence of the signal and the chirp rate. In a BPSK signal, if the chip time is beyond a certain value such as one phase transition per two data frames, the chip time variation does not affect the detection process. However, the chirp signal does not have this characteristic, and the high and low chirp rates do affect the detection results. If a chirp signal is detected through a receiver, only one signal in an FFT output is assumed. It is not our intention to study the chirp signal in the presence of other signals. Since both the chirp and the BPSK are spread spectrum signals, there are similarities between them. Once the existence of spread spectrum signal is confirmed, one important task is to identify them. In chirp signal detection, it is desirable to find the chirp rate.

15.2  A Chirp Signal in Time- and Frequency-Domain Outputs [1] A chirp signal can be expressed from

x = cos[2π (f + ∆ft)t + φ ]

(15.1) 349

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Frequency Modulated (FM) Signals

where f + Dft is an instantaneous frequency. If Df is a constant, the frequency changes linearly with time. An example of chirp signal is shown in Figure 15.1. In this figure, the initial and final frequencies are arbitrarily chosen as 600 and 930 MHz, respectively, and the pulse width (PW) is chosen as 110 ns. The corresponding chirp rate, which is the frequency change per unit time, is 3,000 MHz/ms ((930-600)/0.11 ms). It is convenient to express the chirp rate in terms of MHz per microsecond. This high chirp is chosen to illustrate the time-domain plot because it is difficult to show a slow chirp in the time domain. In this plot the input contains 281 data points, which is not a base 2 number. In order to perform FFT with a base 2 number, the input data length is increased to 512 points through zero padding. Figure 15.1(a) shows the time-domain plot of the input signal. Carefully examining the plot, one can see that the cycles are close together as time goes on. Figure 15.1(b) shows the corresponding frequency domain plot, and the spectrum shape of a chirp signal is clearly shown. It should be noted that the x-axis in this figure only represents the frequency bin number and does not match the input frequency range. Although the spectrum width is shown, the information of the up down chirp is lost in this frequency-domain plot. Figure 15.1(c) shows a Blackman windowed version. The window only applies to the signal portion of the data, not the zeros added at the end of the time domain. The window smoothes the frequency-domain outputs as expected. If the window covers the entire signal including the zero padded input, the spectrum will be skewed.

Figure 15.1  Illustration of a chirp signal with chirp rate = 3 GHz/ms: (a) time domain (b) frequency domain, and (c) Blackman windowed.

15.3  Using FFT Outputs to Determine the Existence of a Chirp Signal

351

From these simple illustrations, it is obvious that the spectrum is different from a continuous wave (cw) signal or a few cw signals. It is also different from a BPSK signal. When the chirp rate is not so high, the time and frequency plots will not be so prominent.

15.3  Using FFT Outputs to Determine the Existence of a Chirp Signal Although biphase shift keying (BPSK) and chirp signals are both spread spectrum, they are basically different. If the phase transition in a BPSK signal can be detected, the signal can be identified. In a chirp signal if the chirp rate is very low or the starting and ending frequencies are close, the chirp signal is difficult to distinguish it from a cw signal. In this section a similar approach, observing the FFT outputs as discussed in Section 14.4, is used to detect the existence of a chirp signal or distinguish the signal from one cw signal. In Section 14.4 the BPSK is distinguished against two cw signals. Here it is intended to find the minimum chirp rate that can be separated from one cw signal. The FFT length is limited to 8,192 points to save computation time. Using the sampling frequency of 2.56 GHz, 8,192 digitized points are equal to 3.2 ms. There are two parameters that are important to determine the detection of a chirp signal: the signal-to-noise ratio (S/N) and the chirp rate. Let us define a delta frequency. For a linear chirp signal, there is a starting frequency f1 and an ending frequency f2. The frequency difference between f2 and f1 or (f2 - f1) is referred as the delta frequency, which is different from Df in (15.1). The unit of delta frequency is megahertz and the unit of Df is megahertz per unit time. The delta frequency equals Df times PW. The detection method is the same as that discussed in Section 14.4. In order to keep the sidelobes low, a Blackman window is applied to the input signal. The maximum of the FFT outputs is identified. The nine frequency bins next to the maximum will be selected and that is 4 on each side of the maximum. The ratio of the minimum to the maximum of these nine outputs is displayed. The selection of nine frequency bins is determined empirically. An example will be used to illustrate the test procedure. The starting frequency f1 is randomly selected between the 141 MHz and 1,140 – delta frequency and the ending frequency f2 = f1 + delta frequency. With these frequency assignments, both the starting and ending frequencies are in the frequency range of 141 and 1,140 MHz, which is the input band of the receiver. In the following studies, the delta frequency is used as a variable. Figure 15.2 shows the calculated ratios. In Figure 15.2(a) the input S/N = 100 dB and the delta frequency is 0.8 MHz. This delta frequency is selected through trial and error and it is the smallest value for which the chirp and the cw signals can be separated, as shown in Figure 15.2. The upper plot is the chirp outputs and the lower plot is the results from a cw signal. The frequency of the cw signal is randomly selected between 141 and 1,140 MHz. Both the chirp and the cw signals are repeated 1,000 times. The corresponding chirp rate is about 0.25 MHz/ms (0.8/3.2). In Figure 15.2(b), the input S/N = 0 dB and the delta frequency is selected at 3.5 MHz. The corresponding chirp rate is about 1.09 MHz/ms (3.5/3.2). Similarly, the

352

Frequency Modulated (FM) Signals

Figure 15.2  Ratio of the minimum to maximum of the nine frequency outputs neighboring to the maximum for PW = 3.2 ms: (a) S/N = 100 dB, delta frequency = 0.8 MHz and (b) S/N = 0 dB, delta frequency = 3.5 MHz.

ratio from the cw signal is the lower plot, and these results are also obtained from 1,000 runs. From this figure it is obvious that when the S/N is high, a low chirp rate signal can be separated from a cw signal. When the input signal is weak, a higher chirp rate is needed in order to separate a chirp signal and a cw signal. If the PW is reduced from 3.2 ms to 100 ns, 256 data points will be generated. The starting and ending frequencies of the chirp signal is selected the same way as mentioned in previously. If the signal is strong (S/N = 100 dB), the delta frequency is 26 MHz, which is close to 32 times the delta frequency of 0.8 MHz required in Figure 15.2(a). This corresponds to a chirp rate of 260 MHz/ms. Figure 15.3(a) shows the results of the minimum to maximum ratios for both the chirp and cw signals. When the S/N = 0 dB, an increase in the chirp rate will not separate the chirp and cw signals. In order to separate the two signals, the minimum required S/N is about 10 dB. Under this condition, the delta frequency of 135 MHz is required and the corresponding chirp rate is 1,350 MHz/ms. The results are shown in Figure 15.3(b). From these illustrations, it is shown that to detect a chirp signal three parameters are important: the S/N, PW, and chirp rate. The causes of S/N and chirp rate are understandable in that it is easy to detect a stronger signal and a wider spectrum. The PW can be explained through the frequency resolution. For example, if PW = 3.2 ms, the frequency resolution of the FFT outputs is 312.5 kHz (1/3.2 × 10-6). The Blackman window will limit the spectrum to about 8 output bins, with a minimum amplitude at about 57 dB below, when the input is at the boundary between two frequency bins. The ninth point is about 58 dB down. The 0.8-MHz delta frequency

15.4  Using Eigenvalue to Determine the Existence of a Chirp Signal

353

Figure 15.3  Ratio of the minimum to maximum of the nine frequency outputs neighboring to the maximum for PW = 0.1 ms: (a) S/N = 100 dB, delta frequency = 26 MHz and (b) S/N = 10 dB, delta frequency = 135 MHz.

just increases the ninth point slightly so that the minimum to maximum ratio of the nine frequency outputs also increases slightly. When the PW = 100 ns, the frequency resolution increases to 10 MHz. A similar phenomenon occurs when the delta frequency is 26 MHz.

15.4  Using Eigenvalue to Determine the Existence of a Chirp Signal In this section the existence of chirp signal will be detected by using eigenvalues. The input conditions are the same as in Section 15.3. Since one cw signal affects two eigenvalues, a correlation matrix must be by 3 × 3 to generate three eigenvalues. The third eigenvalue is used to represent either noise or a spread spectrum signal. Let us refer to the ratio of the smallest to the largest eigenvalues as an eigenvalue ratio, which is used as the measuring parameter. Before the actual test, we will select a combination of lags so that the third eigenvalues can be large. If there are two cw signals, the lags depend on the frequency separation between the two signals. Since there is only one cw signal, the two lags are arbitrarily chosen as the integer at 0.2 and 0.4 of the total data length. In the following example, the starting frequency is randomly selected, but the delta frequency is fixed and both frequencies must be within the frequency range of 141 and 1,140 MHz. Figure 15.4 shows the results of 1,000 runs for PW = 3.2 ms. The upper plot is the ratio for a chirp signal. The lower plot is the ratio for a cw signal.

354

Frequency Modulated (FM) Signals

Figure 15.4  The eigenvalue ratio obtained from a 3 ´ 3 correlation matrix for a chirp and cw signals for PW = 3.2 ms: (a) S/N = 100 dB, delta frequency = 0.001 MHz and (b) S/N = 10 dB, delta frequency = 1.8 MHz.

When the S/N = 100 dB, the largest eigenvalue for the cw signal is very large, the smallest to largest eigenvalue ratio is very close to zero. The results are shown in Figure 15.4(a). The delta frequency is 1 kHz. In order to show the details in this figure, the ratio amplitude is limited to about 10-8. Most of the data points above this limit are not shown. Only 2 outputs of 1,000 ratios are shown and they are above the ratio of the cw signal. The chirp rate is about 0.3125 kHz/ms (1 kHz/3.2 ms). Figure 15.4(b) shows the results of weak signals of S/N = 0 dB. In this figure, the minimum delta frequency is 1.8 MHz and the corresponding chirp rate is about 5.63 kHz/ms. For PW = 0.1 ms and S/N = 100 dB, when the delta frequency is at 1 kHz, the chirp signal cannot be separated from the cw signal. The two signals can be separated when the delta frequency is at about 2 kHz. This indicates that even at this high S/N value the PW still can play an important role. For the S/N = 0 dB and PW = 0.1 ms, it is difficult to separate the chirp signal completely from the cw signals in all 1,000 runs. It is interesting to note that once the delta frequency reaches a certain value, increasing the delta frequency does not increase the separation of ratios. Figure 15.5 shows the results of 1,000 runs with S/N = 0 dB and PW = 0.1 ms. Figure 15.5(a) shows the delta frequency of 78 MHz and Figure 15.5(b) shows the delta frequency of 1,000 MHz. The value of 78 MHz is obtained through trial and error. The 1,000-MHz value is arbitrarily selected to represent a large delta frequency. There is basically no difference between these two figures.

15.5 Compare Chirp Signal Detecting Results from the FFT and Eigenvalue Methods

355

Figure 15.5  The eigenvalue ratio obtained from a 3 ´ 3 correlation matrix for chirp and cw signals for PW = 0.1 ms: (a) S/N = 0 dB, delta frequency = 78 MHz and (b) S/N = 0 dB, delta frequency = 1,000 MHz.

This phenomenon can be explained as follows. For a stronger signal, when the delta frequency increases, the smallest eigenvalues of the chirp signal will increase, but the smallest eigenvalue of the cw signal stays about constant. When the input signal is weak, the smallest eigenvalue is partially affected by the signal and partially by the noise. At a certain S/N level, the signal effect has only a limited effect on the smallest eigenvalue. Thus, changing the chirp rate does not have much effect on the smallest eigenvalue; this is especially true for a short signal. In order to separate a chirp from a cw signal at PW = 0.1 ms, the S/N must be increased to about 10 dB. Under this condition the minimum delta frequency required is about 45 MHz and the corresponding chirp rate is about 14.1 MHz/ms.

15.5  C  ompare Chirp Signal Detecting Results from the FFT  and Eigenvalue Methods In this section the results obtained from the FFT and eigenvalue methods are summarized. In Table 15.1 the values are obtained from observation rather than from statistical calculations. Thus, these values can only be treated as a ball part estimation. From Table 15.1 one can see that the eigenvalue method can detect a signal with a

356

Frequency Modulated (FM) Signals Table 15.1  Minimum Required Delta Frequency or Chirp Rate to Separate Chirp and CW Signals

FFT method Delta frequency Chirp rate Eigenvalue method Delta frequency Chirp rate

PW = 3.2 ms S/N = 100 dB

S/N = 0 dB

PW = 0.1 ms S/N = 100 dB

S/N = 10 dB

0.8 MHz 0.25 MHz/ms

3.5 MHz 1.09 MHz/ms

26 MHz 260 MHz/ms

135 MHz 1,350 MHz/ms

0.001 MHz 0.31 kHz/ms

1.8 MHz 56 kHz/ms

0.002 MHz 20 kHz/ms

45 MHz 450 MHz/ms

slower chirp rate. For a strong signal and a long PW (3.2 ms), the delta frequency can vary 800 times, such as 0.8 MHz versus 0.001 MHz for the FFT against the eigenvalue methods. For a short PW (0.1 ms) the difference is 26 MHz versus 0.002 MHz. For a weak signal the difference between frequency changes is not as drastic, such as 3.5 MHz versus 1.8 MHz and 135 MHz versus 45 MHz. The eigenvalue has a better sensitivity in detecting a chirp signal than the FFT operation. This is also true for the BPSK against two cw signal tests, discussed in Chapter 14. When the signal is weak, a chirp signal is difficult to detect. It appears that the chirp rate is not an important parameter to determine whether the signal can be detected. The PW is a more important parameter. In the following discussion, the outputs are generated from FFT outputs and they are complex. For complex signal a 2 ´ 2 matrix can be used to differentiate a chirp signal and a cw signal. It should be noted that when an EW receiver cannot recognize or encode a chirp signal, it does not necessarily mean that the signal will be missed. For example, if the chirp rate is too low to be detected as a chirp signal, the receiver will still produce a frequency reading and a pulse width. The frequency reading will be a constant value as a cw signal and the PW measured usually is correct. From this information the EW processor may still be able to determine and classify the input signal. Under this condition, one can consider that the receiver still provides enough information.

15.6  Recognizing Chirp Signals from Receiver Outputs In this section, we will detect a chirp signal by simple observation from the simulated receiver output. From the receiver designs discussed in Chapter 11, the frame time is 50 ns and the corresponding channel is 20 MHz. When the delta frequency of the chirp signal is 20 MHz or less, the signal can stay in one frequency bin or change from one frequency bin to a neighboring one. In one frequency bin the receiver will detect it as a cw signal. Even if the signal changes from one frequency bin to a neighboring one, it may not be identified as a chirp signal because a cw signal close to the boundary of two frequency bins may have the same behavior. When the delta frequency is over 20 MHz, it may appear in three adjacent frequency bins. For example, if the starting frequency is close to the boundary of two frequency bins such as 409 MHz, It only takes 2 MHz (411 MHz) to enter the neighboring frequency bin. If the total delta frequency is 22 MHz, the final frequency will be 431 MHz. The entire frequency ranges from 409 to 431 MHz,

15.6  Recognizing Chirp Signals from Receiver Outputs

357

which will show in three channels with center frequencies of 400, 420, and 430 MHz. Figure 15.6(a) shows a signal that is 3.2 ms long with S/N = 100 dB, starting at 409 and ending at 431 MHz. From the frequency bin number change, one can declare that the signal is a chirp. If the starting frequency is changed from 409 to 411 MHz, even a 39-MHz delta frequency may only occupy two frequency bins. The results are shown in Figure 15.6(b). If it is required to sweep three frequency bins to be declared as a chirp signal, this signal condition can not be identified as a chirp. For the above discussion there is a minimum PW required. It is assumed that the minimum PW matches three consecutive FFT frames. If the PW is shorter than the requirement, the outputs can only be in two consecutive frames and do not provide enough information. From this simple illustration, one can see that when the delta frequency is from slightly above 20 to below 40 MHz, the chirp signal may be or may not be identified from the FFT outputs. It depends on the starting frequency. Thus, a wide delta frequency (close to but less than 40 MHz) may not guarantee detection. The combination of a delta frequency and the starting frequency is the determining factors. When the chirp rate is rather high, the output frequency bins from two consecutive frames may not be adjacent to each other, for example, if the PW = 0.2 ms, which provides four frames. If the delta frequency is arbitrarily chosen as 400 MHz, with the S/N = 100 dB, the results are shown in Figure 15.7. The output bin number changes by 5 from frame by frame.

Figure 15.6  Output frequency bin number for a chirp signal with PW = 3.2 ms and S/N = 100 dB: (a) frequency from 409 to 431 MHz and (b) frequency from 411 to 441 MHz.

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Frequency Modulated (FM) Signals

Figure 15.7  Frequency starting at 400 MHz with a delta frequency of 400 MHz, PW = 0.2 ms, and S/N = 100 dB.

For the example shown in Figure 15.6, the signal will continuously cross the threshold frame after frame. One can conclude that there is one signal in every consecutive frame. The encoder must recognize that during the time period of the pulse, the frequency changes. In Figure 15.6(a), the frequency bins have the same value for many frames. The chirp may be detected from the frequencies of the first and the last frames. The pulse in Figure 15.7 may not be detected from the two-frame detection method discussed in Chapter 11. The frequency measured from two consecutive frames may have phase wrapping and report erroneous frequency information. It can only be detected through a single frame detection. The encoder may report four short pulses with different frequencies. It might be the job of the electronic warfare (EW) processor to recognize it as a chirp signal.

15.7  Frequency Measured Through an Amplitude Comparison It was shown in Section 11.6 that amplitude comparison can improve the frequency resolution. The same scheme is applied to the frequency reading on the chirp signal. The ratio of the highest two frequency outputs is used in (11.7) to find the fine frequency reading. The results are shown in Figure 15.8. Figure 15.8(a) shows a signal with PW = 3.2 ms, S/N = 100 dB, and the delta frequency equal to 0.001 MHz, as listed in Table 6.1. This figure clearly shows the frequency change versus time. When PW = 3.2 ms, S/N = 0 dB, and the delta frequency is 3.4 MHz, as listed in Table 15.1, a frequency variation trend is difficult to notice. When the delta frequency is increased to 10 MHz, Figure 15.6(b) shows the results. A clear frequency

15.8 Signal Conditions for Chirp Detection After the Conventional FFT Operation

359

Figure 15.8  Frequency versus time plot for PW = 3.2 ms: (a) S/N = 100, delta frequency = 0.001 MHz and (b) S/N = 0 dB, delta frequency = 10 MHz.

trend is still difficult to show. If the starting and ending frequencies are observed, it might indicate a delta frequency of 10 MHz. A better approach is to fit all the output points into a straight line and use all the outputs to make a decision. From this simpler illustration it is shown that when the signal is strong, it is relatively easy to find a chirp signal. When the signal is weak it is difficult to see the frequency trend. These results also show that the eigenvalue is a better approach to detect a chirp signal than observing the frequency outputs from the frequency versus the time plot. If the input can be classified as a spread spectrum signal, the delta frequency and chirp rate will be easier to find.

15.8  S  ignal Conditions for Chirp Detection After the   Conventional FFT Operation The only goal of this study is to determine whether a chirp can be separated from a cw signal at the output from the FFT operation with a Blackman window. First, the output signals must cross the receiver threshold. In other words, the FFT outputs must be strong enough to be detected as a signal. If a signal is detected, the next task is to determine whether it is a cw or a spread spectrum signal. If the signal is a spread spectrum signal, the next step is to determine whether it is a BPSK or chirp signal. After the decision is made, the characteristics of the signal will be determined. In this section a chirp signal passing through a receiver built through a conventional FFT operation will be discussed. The purpose is to detect the existence of a

360

Frequency Modulated (FM) Signals

chirp signal after the FFT operation. Since the eigenvalue method provides a better result, this method is used for detection. Since the outputs from FFT operation are complex, a 2 ´ 2 correlation matrix can be used to find two signals. The lag of the correlation matrix is arbitrarily selected as 0.3 of the total data length. As discussed in Section 15.7, when the delta frequency is very large, the input frequency can cross several frequency bins either in a continuous or a discontinuous manner. Under this condition, the chirp signal should be recognized by observing the FFT outputs. Thus, this study will limit the delta frequency to under 40 MHz. In order to limit the input signal conditions, in this study, the PW is chosen as 0.8 ms, which generates 2,048 data points. For a 128-point FFT operation there are 16 data frames. Thus, the data length is 16 or less. When all the chirp outputs are in one frequency bin, there are 16 outputs. When a portion of the chirp signal is in adjacent bins, the data length will be less than 16. One frame detection will be performed to simplify the operation, which has a sensitivity of about S/N = -1.5 dB (Table 6.6, row 4). The signal with S/N = 0 dB is selected for this study. The purpose is to find the lowest delta frequency that can be detected. This study will be divided into two cases. The first case is all the outputs from one frequency bin. The second case is the chirp outputs being divided equally into two adjacent bins.

15.9  Chirp Output in One Frequency Bin If the Blackman windowed FFT outputs of the chirp signal are all in the same bin, there are 16 complex points. The starting frequency is chosen as 611 MHz, which is close to the lower edge of a bin boundary. Since the chirp signal is limited to an up chirp, the FFT outputs will be in one bin. The lag value is 5, about 0.3 times the data length. The eigenvalue ratio obtained will be compared with a cw signal. The input frequency of the cw signal is randomly selected between 141 and 1,140 MHz. Sixteen FFT operations with a Blackman window will perform on the cw signal. The output from the first frame is used to determine the bin number. If the cw signal is at the boundary between two frequency bins, the outputs in time can change from one bin to its neighboring one. For example, if the first bin is at 16 and the rest are at 17, bin 16 will be used for the eigenvalue calculation. The testing results are similar to Figures 15.2 to 15.4 and are not shown. The conclusion is obtained from visual observation. For 1,000 runs it appears that when the delta frequency is about 2.2 MHz, the ratios of the two eigenvalues can be separated. This value is between 3.5 and 1.8 MHz, obtained in Table 15.1. It is anticipated the results should be smaller than 1.8 MHz because after the FFT operation, the S/N should improve. However, the results do not match this anticipation; maybe the 128 Blackman window reduces the spectrum width of the chirp in each frame.

15.10  One CW Signal at Boundary of Two Frequency Bins Before the discussion of a chirp moving from one frequency bin to the neighboring one, let us discuss one cw at the boundary of two frequency bins. When the signal

15.10  One CW Signal at Boundary of Two Frequency Bins

361

is at the boundary of two bins, the outputs from frame to frame can change from one frequency bin to the next in a random manner. For example, if the input is at 610 MHz, the outputs will be either bins 31 or 32. If the highest bins are used to form the 16 data points and eigenvalue method is used to determine the number of signals, the results are similar to a spread spectrum signal. This phenomenon can be explained as follows. The outputs between two adjacent bins have different signs and this sign change is equivalent to a phase shift between two consecutive frames. There are two approaches to fix this problem. The first one is to use only one frequency bin instead of two. For example, the outputs are in 31 and 32 bins. One can choose the outputs from either 31 or 32 to form the 16 data points. The results will behave like a cw signal. Since one does not have a priori information on the signal, forcing the use of all the inputs from one bin may lose the true information. A second approach is to change the sign of the output signal. For example, if the maximum outputs and the corresponding indices are x(i) and ind(i) where i = 0 ~15, if ind(i) = 31 for i = 0 ~ 5 and ind(i) = 32 for i = 6 ~ 15, then y(i) = x(i) for i = 0 ~ 5 and y(i) = -x(i) for i = 6 ~ 15, where y(i) are the new data obtained from x. The sign change will eliminate the phase transition between two consecutive points. Figure 15.9 shows the ratios of the two eigenvalues. The input signal is cw at 610 MHz and S/N = 0 dB. The upper plot is the results from the peak values of 16 outputs. Since the peak values are in both bins 31 and 32, the signal made from these peak values has a 180° phase shift and the results behave as a spread spectrum. The lower plot is obtained from identical data. The only difference is that the signs of some elements are changed by the rule stated above. Obviously, this simple operation can make a cw signal behave correctly.

Figure 15.9  Ratio of two eigenvalues from a cw signal at a boundary of two bins S/N = 0 dB: upper plot from peak values and lower plot from peak values with a sign change.

362

Frequency Modulated (FM) Signals

15.11  Chirp Output in Two Adjacent Frequency Bins In this study, the chirp signal will start in one bin and end in an adjacent bin. The delta frequency is given and the middle frequency of the chirp signal is selected as 610 MHz. The 610 MHz is at the boundary of two frequency bins. If the chirp starts at 610 – half delta frequency, it will end at 610 + half delta frequency. There are three approaches to compare the chirp to a cw signal. The first method uses only a portion of the data. For example, only 8 data are in bins 31 and 32. The data in bin 31 can be used to perform the identification. The result is that the delta frequency required to separate from a cw signal is 15 MHz. This value is much higher than the 2.2 MHz obtained in Section 15.10, because less data points are used. The second method is to change the sign of data in the neighboring frequency bin as discussed in Section 15.11. The result obtained through this approach is also about 2.2 MHz, which matches the results in Section 15.10. The third method is to use all the data in bin 31. Even though about half of the data are not the peak outputs, the same delta frequency of 2.2 MHz is obtained. With the delta frequency of 2.2 MHz, the start frequency is at 609.9 (611 – 2.2/2) MHz and the end frequency is at 611.1 MHz. These frequencies are very close to the boundary of two bins; the two outputs from adjacent frequency bins are very close in amplitude. Thus, using the data in either frequency bin will produce the desired results. The detection of a chirp signal after the FFT operation can be summarized as follows. For PW = 0.8 ms and S/N = 0 dB, a delta frequency of a 2.2-MHz chirp can be separated from a cw signal. To form the correlation matrix, the data from one frequency bin will be used. Although using the maximum output with a sign correction should produce better results, using the output from one frequency bin does not show much difference. A similar method can be used for the cw signal on the boundary of two frequency bins. The bin number with the most outputs can be used as the desired one. For example, if the 16 outputs have 5 in bin 31 and 11 in bin 32, then bin 32 will be used. If the results are 8 and 8 for both bins 31 and 32, then either one can be used. The discussions from Sections 15.10 and 15.11 can be concluded as follows. The study limits the frequency of the chirp to two bins. If the signal is spread more than two bins, the chirp signal can be identified from the FFT outputs. The data used to form the 2 ´ 2 matrix can be from the frequency bin with the most peak values. Another approach is to use the all the peak values. When they are not in the same frequency bin, the sign of some of the values must be changed.

15.12  Chirp Signal After the Polyphase Filter Receiver Approach A similar approach will be applied to a chirp passing through a polyphase filter. The chirp signal stays the same: PW = 0.8 ms, S/N = 0 dB. The total data generated are 2,048 points. For a window length of 128 points and shifting by 8 points, the total outputs are 240 points. The lag is arbitrarily selected as 72 about 0.4 of the

15.13 Detecting a Chirp Signal After an FFT or a Polyphase Filter Operation

363

data length. The center frequency of the input chirp signal is placed at the boundary between two frequency bins. Thus, the spectrum of the chirp signal is equally divided into two complex receivers. If this case can be solved without difficulty, it is anticipated that when the chirp signal is in one complex receiver, similar results should be obtained. Simulation results confirm this expectation. The initial frequency and the initial phase of the cw signal are randomly selected. For these input conditions, the minimum delta frequency is about 3.2 MHz to be separated from a cw signal. The value is larger than the 2.2 MHz obtained from the conventional FFT outputs. This result appears reasonable because the polyphase has a wider bandwidth and the S/N after the filter is lower. When a signal is at the boundary of two bins (complex receivers), the output from the polyphase filter can randomly be in one of the two adjacent frequency bins. It is interesting to note that the outputs are different from the conventional FFT output. From the conventional FFT output, the two adjacent outputs have opposite signs. From the polyphase filter output, the two adjacent outputs have the same signs. When the chirp frequency is centered at the boundary of two frequency bins, the chirp signal will move from one bin to the next. The maximum from all the outputs is used to form the correlation matrix. The same approach is applied to the cw signal: the maximum value will be used to form the correlation matrix. When the input signal passes through the polyphase filter with 8 data shifts as stated in Section 13.8, the frequency outputs are different from odd and even channels. It is anticipated that when the input data are shifted by 16 samples rather than 8, the outputs may have better controlled characteristics. However, the simulation results show that when the shift is increased to 16 data points, the results are similar.

15.13  D  etecting a Chirp Signal After an FFT or a Polyphase   Filter Operation In previous sections, it was illustrated that when a chirp signal passes through a conventional FFT operation or passes through a polyphase filter, the eigenvalue method can be used to detect the its existence. The signal must be detected through the filter outputs. It is also assumed that the chirp signal is limited in two adjacent bins. If the chirp is spread into more than two bins, the FFT output can detect it. Another assumption is that each frequency bin contains only the chirp signal. This assumption is reasonable for FFT outputs because of the narrow bin. For the polyphase filter, this assumption might be difficult to meet because of the relatively wide bandwidth. The detection method uses one frequency or two adjacent frequency bins to form a 2 ´ 2 matrix and the two eigenvalues are obtained. If the minimum to maximum eigenvalue ratio is above a certain level, a spread spectrum is detected. For the FFT operation, outputs in the majority number of bins will be used to form the correlation matrix. If the maximum is each bin is used, outputs with a different bin number must be adjusted by a sign change. For a polyphase filter, the maximum in each bin will be used to form the correlation matrix. A threshold must be set for the detection process. This threshold setting might be a tedious task because it depends on the signal strength. For example, a strong

364

Frequency Modulated (FM) Signals

cw signal will have a very large signal eigenvalue, the noise eigenvalue stays at a constant, and their ratio is low. Thus, for a strong signal the threshold is low, which is different from the conventional sense that a stronger signal requires a higher threshold. The threshold setting will not be discussed here. Once a spread spectrum is detected, the type of signal still cannot be determined, that is, this method cannot tell the difference between a BPSK and a chirp signal. In the following sections, the frequency of the chirp will be measured. The method will compare the phases between two consecutive frames as discussed in Chapter 14 to find a BPSK signal.

15.14  Phase Comparison After the Conventional FFT Operation In this section the phase comparison after the conventional FFT operation is used to find the delta frequency. Although the amplitude comparison can produce a very good frequency reading versus time, as shown in Section 15.8, this method is still needed to detect the BPSK signal. As discussed in Chapter 14, if a phase transition is in the middle of a frame, from the shape of the FFT output a BPSK signal can be detected. If the phase transition occurs near the boundary of two frames, a phase comparison can indicate the transition. The operation is straightforward in finding the difference phase between two consecutive frames. A detailed discussion can be found in Chapter 6. First, the maxima of the FFT outputs from all the frames are detected and referenced by their bin numbers. The phase angles at these bin numbers are obtained and the phase difference between two consecutive frames is calculated. When two frequency bins are the same, the phase difference is calculated by subtracting the two phases. If the frequency index values are different by 1, the phase difference calculated must be adjusted by π or by changing the sign of one of the consecutive outputs. The following equation may better explain the operation. æ x i , j1 ö θ = ang ç ÷ è x (i +1), j2 ø



æ x i , j1 ö θ = ang ç ÷ è - x (i +1), j2 ø

if j1 = j 2 æ x i , j1 ö or θ = ang ç ÷ - π if j1 ¹ j 2 è x (i +1), j2 ø

(15.2)

where θ is the phase difference between two consecutive frames, ang represents the angle (or phase) calculation, the subscript i represents the frame time index, and j represents the frequency bin number. When two consecutive outputs have the same frequency bin, the first portion of the equation is used; otherwise, the second portion is used. Figure 15.10 shows the phase difference measurement results. In this figure, the S/N = 10 dB, PW = 0.8 ms, and delta frequency = 8.5 MHz. In Figure 15.10(a), the middle frequency of the chirp is at 600 MHz (the center of a frequency bin); thus, the chirp signal is in one frequency bin. The phase change is clearly illustrated. In Figure 15.10(b), the middle frequency of the chirp is at 610 MHz (the boundary

15.15  Phase Comparison After a Polyphase Filter Operation

365

Figure 15.10  Phase plot of a chirp signal with S/N = 10 dB, frequency change = 8.5 MHz, and PW = 0.8 ms through an FFT operation: (a) frequency in one bin and (b) frequency split into two bins.

of two adjacent frequency bins); thus, the chirp is split into two frequency bins. In this figure there is a phase shift of 2π, which can be eliminated through the phase-unwrap. A higher S/N of 10 dB is used in this figure to avoid many 2π phase changes. From the phase difference and the time between two consecutive data frames, the frequency can be calculated. Since the phase difference changes with time as shown in Figure 15.10, the frequency of the signal is also a function of time.

15.15  Phase Comparison After a Polyphase Filter Operation For a polyphase filter the amplitude comparison method to find the fine frequency is difficult to apply because the frequency-domain response of the filter is rather flat. A phase comparison method can be used to improve the performance. When a chirp signal passes a polyphase filter, a phase comparison can determine the delta frequency. Since the outputs from two adjacent frequency bins have the same phase, the phase value of the maximum output can be compared. The phase difference calculation is as shown in the first portion of (15.2). The frequency bin numbers selected from two consecutive outputs do not have to match. The same signal is used in the following example, with S/N = 10 dB, PW = 0.8 ms, and delta frequency = 8.5 MHz. When the chirp is centered at 640 MHz, the center of one frequency bin and the results are shown in Figure 15.11. A total of 239 data

366

Frequency Modulated (FM) Signals

Figure 15.11  Phase difference of a chirp signal with S/N = 10 dB, delta frequency = 8.5 MHz, and PW = 0.8 ms output in one frequency bin through an FFT operation: (a) phase difference and (b) averaged phase difference.

Figure 15.12  Phase difference of a chirp signal with S/N = 10 dB, delta frequency = 8.5 MHz, PW = 0.8 ms output in two frequency bins through an FFT operation: (a) phase difference and (b) averaged phase difference.

15.17  Summary

367

points are obtained from 240 phase data. Since the frame time is short, the outputs are noisy. Figure 15.11(a) shows the phase difference between two consecutive time frames. Since the results are difficult to read, eight outputs are averaged into one point. The results are shown in Figure 15.11(b). In this figure, the change in the phase difference can be better shown. When the chirp signal is centered at 560 MHz, the signal is divided into two frequency bins. The results are shown in Figure 15.12. Basically, there is no difference between this figure and Figure 15.11 because the phase difference calculation used stays the same. The 239 phase difference data are shown in Figure 15.12(a) and the 8-point averaged outputs are shown in Figure 15.12(b). From these simple illustrations one can see that the phase comparison method used after the polyphase filter can determine the phase variations.

15.16  Differentiating Against Two CW Signals In all the discussions on detecting the spectrum signals after the FFT or polyphase filters, a 2 ´ 2 matrix is used to generate two eigenvalues. The ratio of the two eigenvalues is used to determine the difference between one cw and a spread spectrum. This limitation on one cw signal may be good for the FFT operation, but it might be too loose for the polyphase operation. This problem can be extended to differentiate against two cw signals. The approach is to form a 3 ´ 3 matrix and generate three eigenvalues. For the 3 ´ 3 matrix, the mathematic operation is still very simple and analytic solutions are available. Since the outputs from the FFT and polyphase operations are complex, three eigenvalues can represent three signals. It is illustrated in Chapter 8 that two strong cw signals will only affect the amplitude of two eigenvalues. The third (smallest) eigenvalue will represent noise. However, if the signal is spread spectrum, the third eigenvalue is relatively large. By taking the ratio of the third to the first eigenvalues, the result can be used to test against two cw signals. For a 3 ´ 3 matrix two lags are needed and are arbitrarily chosen as about 0.2 and 0.4 of the data length. A minimum of four output data points are needed to form the correlation matrix. Simulation results show that this operation provides satisfactory results. Under the same input conditions such as PW and the S/N, the delta frequency needed to separate from two cw signals might be different. With this approach, if there are three cw signals existing in one polyphase output, the result will be erroneous. However, one can make a reasonable assumption that only two cw signals are present in one frequency bin during the time of the spread spectrum signal.

15.17  Summary From the study in this and the previous chapters, a spread spectrum signal can be detected and identified by a receiver built through either the conventional FFT operation or the polyphase filter approach. The general method is to find the existence of the spread spectrum signal through the eigenvalue method. This method has a

368

Frequency Modulated (FM) Signals

limitation that one filter output can contain a maximum of one or two cw signals. If there are more than two cw signals, the method will declare a spread spectrum signal. For a chirp signal if the PW is long, it is relatively easy to detect. If the PW is short, it requires a very large delta frequency to be differentiated from a cw signal. Once the spread spectrum signal is identified, the amplitude and phase comparisons will be used to determine whether it is a BPSK or a chirp signal. The phase comparison method can also estimate a phase transition location, if the chip rate is relatively low, such as one phase transition in one frame time. For a chirp signal the amplitude and phase comparison methods can determine the initial and the ending frequencies of the signal or determining the chirp rate. From these studies it is necessary to implement the amplitude and phase comparison methods for a receiver built through either conventional FFT or polyphase filter operations. If the input is a cw signal in one channel, the amplitude and phase comparison methods can provide a finer frequency resolution. One problem that needs to be studied is the encoder design. If the receiver is designed to recognize BPSK and chirp signals, the information must be incorporated in the PDW. Although the existence of phase transient and frequency change can be recognized visually in some of the figures, the information must be obtained by the encoder. It is desirable for the encoder to report the following information: 1. It should flag the existence of a BPSK signal. 2. It should provide the carrier frequency, the PW, and the total number of phase transition, which can be used to determine the type of code by an EW processor such as a Barker code with a certain length. The chip time (time between shortest phase transitions) is also needed. 3. It should flag the existence of a chirp signal. 4. It should provide the PW and the initial and ending frequencies of the pulse, which can be used to calculate the delta frequency. The delta frequency and the PW can be used to find the chirp rate.

Reference   [1]  Wikipedia, “Frequency modulation.”

Ch a p t e r 16

Angle of Arrival (AOA) and Frequency Measurements

16.1  Introduction In this chapter a receiver with AOA and frequency measurement capabilities will be discussed. Since frequency measurement has been discussed in many chapters, the discussion will be concentrated on the AOA measurement. There are many approaches to measure the AOA such as amplitude, phase, and time of arrival (TOA) comparisons. Most of these approaches are discussed in many references such as [1, 2]. The frequency measurement is to convert information from time to frequency information. The AOA measurement is to convert spatial information to AOA information. The two approaches are mathematically identical; thus, all the frequency measurement methods such as FFT and multiple signal classification (MUSIC) can be applied to solve the AOA problem. This chapter concentrates on one approach and generates detailed information. This approach uses antenna array instead of using only two antennas to compare amplitude or phase. The antenna array is to be used to improve the processing gain and detecting weak signals. The arrangement is based on digital signal processing to find the AOA information. The bandwidth of the antenna is assumed to be very wide, from 2 to 8 GHz. Achieving the actual antenna design is an antenna problem and will not be discussed in this book.

16.2  Define the Problem The frequency of the input signal is from 2 to 8 GHz (or a 6-GHz bandwidth), but the receiver only covers a 1-GHz instantaneous bandwidth with a sampling frequency of 2.56 GHz. The antenna bandwidth is arbitrarily chosen and the receiver bandwidth is used in most discussions in this book. The concept discussed should be applicable to other frequency ranges. The input bandwidth will be covered in a time sharing approach. Each time the receiver will look for only one-sixth of the input bandwidth. The input bandwidth must be properly converted into the baseband, which covers 141 to 1,140 MHz, as discussed in many previous examples. The hardware of converting from the front end to the baseband will not be discussed here because many possible ways can achieve this goal. The highest input frequency is 8 GHz, which determines the maximum space between the antenna elements. At 8 GHz, the wavelength (λ) is 3.75 cm and the 369

370

Angle of Arrival (AOA) and Frequency Measurements

maximum distance between adjacent elements is λ/2 or 1.875 cm to avoid ambiguity. At 2 GHz, the λ is 15 cm and λ/2 is 7.5 cm. The antenna array is limited to one dimension. Although two-dimensional arrays can provide both the elevation and the azimuth angles, the processing might be elaborated. A very simple example will be used to illustrate the number of operations. If the antenna array contains 128 by 128 elements and 128 time-domain data points are used to find frequency, it takes 16,384 (128 × 128) 128-point FFTs to find the frequency data from all the antenna elements. Since the input data are real, there are 64 frequency bins with independent information. These data will be used to find the AOA information. It takes 8,192 (64 × 128) 128-point FFTs to find the angle in the x direction and the same number of operations to find the angle in the y direction. The overall number of operations is 32,768 128-point FFTs. Not only is the number of operation high, but the total outputs are also large: 1,048,576. To search over all the output is not a simple operation either. Therefore, in this chapter, only a one-dimensional antenna array will be studied. Of course, one can use a 128 time domain from one antenna element to obtain the input frequency. Once the frequency is found, the AOA can be found at this frequency. However, this approach cannot take the advantage of the antenna gain. The number of elements in the antenna array is limited to 16 for a possible future receiver building. The AOA coverage is arbitrarily chosen as 120°. In addition to obtaining the AOA information, the purpose of using an antenna array is to provide an additional processing gain to detect weak signals. One obvious way of obtaining the necessary information is through the FFT operation in the spatial domain. In order to improve the AOA resolution, the MUSIC method can be used to replace the FFT operation. The number of input signals is limited to two. The two signals may have different input frequencies and different AOA. In order to simplify the discussion, two cw signals are studied.

16.3  Signal Generation In this section a reference case will be discussed. In this case, the complexity of operation is not a concern. The probability of detection will be obtained to verify the improvement in sensitivity. The sampling frequency is 2.56 GHz, and 128 points are used for the FFT operation to produce the frequency-domain information. The antenna array contains 128 elements. In this configuration, 128 128-point FFTs are performed to obtain the frequency. In the frequency domain, only 53 (6 - 58) frequency bins are kept. To obtain the angle information, 53 128-point FFTs are performed and the total output data are 6,784 (53 × 128) points. From one input signal, the signal can be generated from the following equation with reference to Figure 16.1. x(q, n) = A cos(2π f (n − 1)∆t + φ + φq ) + nq

φq =

2π fd(q − 1)sinθ 2π d(q − 1)sin θ = C λ

(16.1)

16.4  Normal and Simplified Approaches

371

Figure 16.1  One dimensional antenna array.

where x(q, n) is the signal collected from the qth antenna element, n is time, f is the input frequency, φ is the initial phase of the signal, φq, often referred to as the electrical angle, is the phase difference between the 0 element to the qth element, d is the distance between the antenna elements, θ is the incident angle (or the AOA) shown in Figure 16.1, λ is the wavelength of the input signal, C is the speed of light, and nq is the noise of the qth element. In order to find the AOA information, the frequency of the input signal must be known. If there are M input signals, the input can be written as x(q, n) =

M



∑ Am cos  2π fm (n − 1)∆t + φm +

m =1

2π fm d(q − 1)sin θm   + nq  C

(16.2)

where the subscript m is for the mth signal. The notations are the same as in (16.1).

16.4  Normal and Simplified Approaches In all the studies in this book, the input frequency is usually lower than 1,280 MHz for real signals and 2,560 MHz for complex signals. Under this condition, the sampling frequency of 2,560 MHz will be high enough to digitize the input signals. In this study, the input signal is from 2 to 8 GHz. The input signal must be downconverted into a lower frequency through an analog circuit and digitized at 2,560 MHz. If there are 128 antennas, all the outputs will be downconverted. One of the downconversion channels is shown in Figure 16.2. In this figure, for simplicity amplifiers are not included and only the frequency conversion is shown through the mixer. The input frequency is f1 and the local oscillator (LO) is f0. If the input frequency range is from 2 to 3 GHz, the LO frequency can be 1,860 MHz, which will convert the input frequency to 140 to 1,140 MHz. The selection of the LO frequency is convenient to explain the downconversion process and may not be practical because an actual receiver design often uses one LO to feed multiple mixers to save hardware. The output of the mixer actually contains two frequencies, f1 - f0 and f1 + f0, and many spurs, and the LO frequency must be selected to minimize the spur outputs. In this simple study the spurs will be neglected and only the

372

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.2  Analog circuits for one receiver channel.

two frequency outputs are of concern. The high-frequency component must be filtered output through a bandpass filter. The above analog processing can be simulated. In order to digitize the input frequency at 8 GHz, the sampling frequency must be higher than 16 GHz, such as 25.6 GHz. For this operation, in order to cover 100 ns of a signal, 2,560 points are needed. After digitization, the data will be mixed by multiplying a cosine function with the proper frequency (the LO frequency). A bandpass filter will be used to filter out the high-frequency component. The output can be decimated by 10 to produce the 2.56-GHz sampling rate. Using 128 antenna elements, the above processing must be repeated 128 times. Thus, the operation can be rather tedious. A simplified approach can be used to illustrate most of the operations without the high sampling rate of 25.6 GHz, but the input frequency range is limited in certain ranges. If the input frequency is selected as from 2.56 to 3.84 GHz and from 5.12 to 6.4 GHz, with a 2.56-GHz sampling rate the input frequency can be folded into the baseband of 0 to 1,280 MHz. Figure 16.3 shows the frequency folding. In order to limit the input bandwidth to 1,000 MHz, the two input bands are chosen as 2,701 to 3,700 MHz and 5,121 to 6,120 MHz. The frequency obtained from the FFT operation will be added to either 2,560 or 5,120 MHz to get the actual input frequency. This is the approach to evaluating some of the basic properties of the two-dimensional FFT processing to obtain the AOA and frequency information.

Figure 16.3  Frequency folding to baseband.

16.5  Base Line Performance

373

16.5  Base Line Performance In order to suppress the sidelobes in the FFT operations, Blackman windows will be applied to both the time and the spatial domains. Let us assume that there is no noise in the system and the input frequency is arbitrarily chosen at 3,200 (2,560 + 640) MHz. Since the sampling frequency is 2,560 MHz, the input frequency is equivalent to 640 MHz, which is at a bin center in the baseband. The incident angle θ is arbitrarily chosen at 30°. A 128-point FFT with Blackman window is performed on the time-domain output from all the antenna outputs. After the FFT operation only 53 (6 ~ 58) frequency bins are kept. These frequency bins are used to perform a 128-point FFT to obtain AOA information, also with a Blackman window. The output is shown in Figure 16.4. This is a three-dimensional plot. The x-axis represents the measured frequency; however, the input frequency must be adjusted by knowing the input bandwidth. The y-axis is an angle, but it is not the incident angle. The incident angle must be calculated through the measured angle and the input frequency. The z-axis is the amplitude of the signal in decibels. In this figure there are sidelobes in both the frequency and the spatial domains. Figure 16.5 shows the frequency and angle plots passing through the maximum value in Figure 16.4. Figure 16.5(a) shows the frequency plot. The actual frequency can be obtained as

f = [(mf − 1) + 5]∆f

(16.3)

where mf is the index of the maximum output in the frequency domain and Df is the frequency resolution. In (16.3) the number 5 is added because the first five frequency bins are not included in the calculation and the Df = 20 MHz. This equation

Figure 16.4  Three-dimensional plots of amplitude, frequency, and angle for input at 3,200 MHz and an incident angle of 30°.

374

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.5  Frequency and angle plots passing through the maximum output for input at 3,200 MHz and an incident angle of 30°: (a) frequency plot and (b) electrical angle plot.

is the conventional way to calculate frequency. The incident angle θ can be obtained as A=

(ia − 1) Qd

sin θ = A λ =



AC f

(16.4)

 (i − 1)C  −1  (ia − 1)λ  θ = sin −1  a  = sin    Qdf   Qd  where A is a constant, ia is the index of the maximum output in the angular domain, and Q is the total number of antenna elements.

16.6  Angle Measurement The frequency measurement method in (16.3) can provide a 1-GHz frequency range with a uniform frequency resolution; in this case the frequency resolution is 20 MHz. The angle measurement will not provide similar results. An example will be used to illustrate the results. If the input frequency is at 5,760 (5,120 + 640) MHz and the incident angle at 30°, the frequency and AOA plots are shown in Figure 16.6.

16.6  Angle Measurement

375

Figure 16.6  Frequency and angle plots passing through the maximum output for input at 5,760 MHz and an incident angle of 30°: (a) frequency plot and (b) electrical angle plot.

In this plot the frequency is at 640 MHz, which is the correct value. The angle is at index 24, but the angle in Figure 16.5 is at 14. For the same incident angle of 30°, the two angles are different. To further illustrate the angle measurement, let use three extreme case as examples. The input frequencies are at 320, 2,880, and 8,000 MHz, although the input of 320 MHz is outside the desired frequency range of the receiver of 2 to 8 GHz. All the three frequencies will be folded into the baseband at 320 MHz. The angle is at 70°, which is also outside the desired range of -60 to +60°. This angle can better illustrate the angle variation with input frequency. Figure 16.7 shows the results of the angle plots. In Figure 16.7(a), the peak value occurs at 2, in Figure 16.7(b) the peak value occurs at 13, and in Figure 16.7(c) the peak value occurs at 33. These plots can illustrate the angle resolution. For the very low frequency of 320 MHz, the angle changes from index 1 to 2 when the incident angle changes from 0° to 70°. For 2,880 MHz, the angle changes from 1 to 13 and for 8,000 MHz, the angle change from 1° to 33°. From these simple illustrations one can see that when the frequency is high, the angle measurement provides a decent angle resolution. When the input frequency is low, the angle measurement has a rather poor angle resolution. It is desirable to have uniform angle resolution over the frequency range. As one of the major problems in antenna design, this must be taken into consideration and will be discussed in Section 16.16.

376

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.7  Angle plot for incident angle of 70° with different input frequencies: (a) 320 MHz, (b) 2,880 MHz, and (c) 8,000 MHz.

16.7  P  rocessing Gain from Two-Dimensional   Coherent Processing [2, 3] In this section, the sensitivity of the receiver will be evaluated. The sensitivity will be considered through two approaches. One is the conventional way of considering only one channel and the second way is to consider the entire receiver. For both cases, the one channel and the overall receiver, the probability of false alarm will be 10-7. Since there are 6,784 (53 × 128) outputs after the two-dimensional FFT operations, the one channel case will reflect to the overall receiver false alarm of 6.784 × 10-4. This discussion is similar to that of Section 11.4. First, the desired probability of false alarm per window pfaw will be calculated as 2 9 pfaw = 10-7



2 9 pfaw =

10-7 6784

pfaw = 1.054 ´ 10-4 pfaw = 1.280 ´ 10-6

for one channel (16.5) for overall receiver

In this equation, the 6,784 is the total number of outputs. The value of 9 is obtained from a two-dimensional plot: the frequency and the angle. If two frames of data are used, the maximum of the frames must be within ±1 bins for both the frequency and the angle domains to be classified as detection. In a two dimensional

16.7 Processing Gain from Two-Dimensional Coherent Processing

377

plot, one point has 8 neighboring bins including the same point itself there are 9 points. Of course, a bin on a corner of the two-dimensional plot has only 4 neighboring bins and a bin on an edge has only 6 neighboring bins. The special situations will be neglected in this discussion. From these obtained probabilities of false values, the thresholds can be found as shown in Figure 16.8. As expected, the noise outputs are Rayleigh distribution and the threshold is 117.24 for pfaw = 1.054 × 10-4 and the threshold is 142.99 for pfaw =1.28 × 10-6. In order to achieve the desired probability of detection of 90%, there are two methods. One method uses two frames of data; the maximum of both frames must cross the threshold and the two maxima must be within ±1 bin in both frequency and angle. The second method uses only one frame of data. If the probability of detection can achieve 0.949(√0.9)%, the overall detection can achieve 90%. Only the two-frame detection method is simulated in this study. In this simulation, the input frequency cannot be randomly selected between 2 and 8 GHz because in certain frequencies the input will fold into the undesired frequency range and be eliminated in the output frequency bin. In order to avoid this problem, the input frequency is limited to 1,000 MHz. The range is arbitrarily chosen between 2,701 (2,560 + 141) and 3,560 (2,560 + 1,000) MHz. The results are shown in Figure 16.9. If the two output frequencies are close by ±1 bin, it is considered a successful detection. The only difference between the generations of these two curves is the thresholds. Since the curves are not smooth, the sensitivity can only be read approximately. For the single-channel case, the input S/N required is about -21.2 dB and for all the channels it is -20 dB and the difference is about 1.2 dB. For the one channel and the 6,784 channels, the difference is about 7 × 103. From [2, 3] when the probability of false alarm changes from 10-7 to

Figure 16.8  Noise distributions and the threshold at 117.24.

378

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.9  Probability of detection versus input S/N.

10-11 at a probability of detection at 90%, the required S/N change is slightly below 2 dB. The difference between the predicated change and the simulated change is about 0.8 dB. In Section 12.6 a similar sensitivity discrimination is explained. Compared with the results in Section 11.4, the improvement is about 17.9 (21.2 3.3) and 17.3 (20 - 2.7) dB for the one channel and overall channels, respectively. The desired gain of a 128-point FFT with a Blackman window is about 15 dB for a real signal as discussed in Section 6.11. Since the input data to the FFT are complex because they are obtained from the time-domain FFT operations, the gain should be close to 18 dB (15 + 3). The improvements of 17.3 and 17.9 dB are close to the desired results. From this simple operation, it proves that the FFT in the spatial domain produces an additional 18 dB of gain. Thus, by proper antenna design and signal processing, the antenna array can provide angle information and additional gain to receive weak signals.

16.8  Frequency Conversion and Filtered Output Calibration In the above example, the simulations are performed in a simplified way, that is, the input signal is folded into the baseband through the sampling processing. The input frequency is limited in a certain range such as from 2,701 to 3,560 MHz. In this section, the entire input frequency of the receiver will be properly covered through the analog circuit simulation discussed in Section 16.4. The input signal must be sampling at a very high frequency. In the following study, the sampling frequency is 25.6 GHz. If the sampling frequency of 2.56 GHz

16.8  Frequency Conversion and Filtered Output Calibration

379

is used, any input frequency higher than 1.28 GHz will be folded into the baseband of 0 to 1.28 GHz through the sampling process. Once the input signal is sampled at 25.6 GHz, the input is converted into 141 to 1,140 MHz through the mixing function. The conversion process is briefly discussed in Section 16.4. The local oscillator (LO) frequency of the mixer is selected at n + 860 MHz, where n is an integer. For input frequency of 2 to 3 GHz, n =1 and the input frequency will be converted into 141 (2,001 - 1,860) to 1,140 (3,000 - 1,860) MHz. For the rest of the input frequency bands of 3 ~ 4, 4 ~ 5, . . . , 7 ~ 8, the n value is 2, 3, . . . , 6. In this downconversion process, only the frequency conversion is considered and the spur products are not taken into consideration. After the conversion, the output signal contains two frequencies fi + fo and fi - fo, where fi is the input frequency and the fo is the LO frequency. Since the desired output frequency is fi - fo, the high frequency term fi + fo must be filtered out. The filter will cause a transient effect on the leading and trailing edges of the input signal. Since it is only desirable to process the signal in a steady state, the transient will be eliminated through a long data length. As a result, the input data must be longer than two frames, the desired signal length. After the filter the leading and trailing portions of the output will be eliminated and the resulting output will be two frames long. One important factor in this study is to find the gain in this conversion process. Simulations will be used to make the evaluation. Two examples are compared to find the gain. The first example uses a 2.56-GHz sampling frequency and the input is at 640 MHz, which is on a frequency bin. In this example 128 data points are collected and the FFT is performed on the input data with a rectangular window. In the output, the frequency bin 33 is the signal and the rest are noise. The bins from bins 6 to 58, excluding 33, are used for noise calculation. The noise power is the average of the square of each noise bin. The S/N can be found from 10 times the log of signal power to the noise power. One thousand runs are performed and the averaged gain is 18.16 dB, which is close to the expected value of 18 dB. For 128 points the gain is 21 dB, but if the input is real, the overall gain is decreased by 3 dB to 18 dB. In the second case the signal is sampled at 25.6 GHz with an input frequency at 7,500 MHz downconverted into 640 MHz, In this example 1,536 (128 × 12) samples are collected and filtered by a five-section Chebyshev filter. The bandwidth of the filter is from 1 to 1,280 MHz. After the filter the outputs are decimated by 10 and there are 154 output points. The equivalent sampling frequency is also at 2.56 GHz. In order to avoid the transient effect by the filter, 128 data are collected from point 14 to 141 and FFT is performed on the data. The S/N is calculated identical to the above approach and the resulting gain is 24.98 dB. The gain difference between these approaches is about 6.83 dB (24.98 - 18.16) with a standard deviation of 0.91 dB. The large value of the standard deviation is caused mainly by the noise variation. The ideal difference should be 7 dB. For a sampling frequency of 25.6 GHz, the bandwidth is 12,800 MHz, which is 10 times wider than 1,280 MHz, the bandwidth of 2.56 GHz, and the noise is 10 times higher. After the filter, the noise is reduced 10 times, but the signal is not affected. Thus, for the same input S/N, after the filter the input for the 25.6 sampling frequency should be 10 dB higher. The conversion process uses a real signal (i.e., cosine) as an oscillator, which has two frequency components. The two components split the input into two portions; thus,

380

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.10  Output in frequency domain: (a) fs = 2.56 GHz and (b) fs = 25.6 GHz.

the signal is dropped 3 dB. Considering the gain of 10 dB and loss of 3 dB, the net gain should be 7 dB. The difference of 0.87 dB (7 - 6.13) is probably caused by noise not filtered properly. By changing the bandwidth of the bandpass filter from 1 ~ 1,280 to 30 ~ 1,250 MHz, the gain difference is very close to 7 dB. This bandwidth is obtained through trial and error and will be used in later studies. After this adjustment, the results are shown in Figure 16.10.

16.9  16-Element Antenna with Uniform Spacing Since to study the high frequency signal, high sampling frequency must be used, the antenna will be reduced to 16 elements. In this study the antenna elements will be uniformly spaced with distance of 1.875 cm (Section 16.2). The probability of false alarm and probability of detection will be studied first. For the one-channel case, the probability of false alarm is still at 1.054 × 10-4 as obtained from (16.5). For the overall receiver, the overall outputs are 848 (16 ´ 53) and the probability of false alarm per frame can be obtained as



2 9 pfaw =

10−7 848

pfaw = 3.198 × 10−6

for overall receiver

(16.6)

The input signals are downconverted to baseband, filtered, and decimated as discussed in Section 16.8. The thresholds determined based on these probabilities of false

16.9  16-Element Antenna with Uniform Spacing

381

alarm are 8.92 and 10.48, respectively. These thresholds are used to determine the probability of detection. In these tests the input frequency changes randomly from 2 to 8 GHz and the initial phase is random between 0 and 2π; the input AOA is randomly from 0° to 60°. In the time domain 128 points are used for an FFT operation with a Blackman window to obtain 53 frequency outputs. Since there are a total of 16 antenna elements, there are a total of 16 × 53 outputs. Sixteen point FFTs with a Blackman window are performed on these data 53 times to obtain a 16 × 53 angle and the frequency data. Two frames are used to find the probability of detection. The maximum of both frames must cross the thresholds and the maximum of both frames must be neighboring each other. The detection is similar to that discussed in Section 16.7. The results are shown in Figure 16.11. Since the S/N is 7 dB higher than the direct conversion case as shown in Section 16.8, the input S/N is artificially adjusted by 7 dB. For example, at -12 dB shown in Figure 16.11, the actual input for the simulation is at -19 dB. For the one-channel case, at a 90% probability of detection, the S/N is about -12 dB and for the overall receiver, the S/N is -10.9 dB. The difference between the two S/N values is about 1.1 dB; this result is reasonable because the probability of false alarm is about 33 times different. For the one-channel case, the sensitivity is at -21.2 dB from Figure 16.9. The difference is about 9.2 (21.2 - 12) dB. The sensitivity obtained from Figure 16.11 uses 16 antenna elements and from Figure 16.9 there are 128 antenna elements and the ratio is 8 times. The processing gain due to the number of elements is 9 dB, which is very close to the measured value of 9.2 dB. From these simulation results, it is further illustrated that the overall frequency downconversion, bandpass filtering, and decimation produce the desired results.

Figure 16.11  Probability of detection of downconverted processing.

382

Angle of Arrival (AOA) and Frequency Measurements

16.10  Angle Measurement Through 16 Antenna Elements In this section, the AOA information will be measured through the 16-element antenna array. The approach is as follows. Probability of detection is not used in this study. It is assumed that the signal is already detected; therefore, only one frame of data is used in the simulation that is 128 points in the time domain and 16 points in the spatial domain. The time-domain signal is processed first through the FFT operation to obtain 16 × 53 point data. The spatial-domain data are processed through three approaches. The first one is through a 16 FFT with a Blackman window to obtain 16 output points. The second method is padding the 16 windowed frequency domain data with 112 zeros and performing 128 point spatial FFT. The third method is padding the 16 unwindowed frequency-domain data with 112 zeros and performing 128-point spatial FFT. In the third approach the Blackman window is not applied to the frequency-domain data. The peak value from the two-dimensional outputs is used to determine the frequency and the angle information. In the following simulation, all the input parameters are arbitrarily chosen. The input S/N is set at 10 dB (equivalent to 17 dB), the frequency is at 7,500 MHz, and the AOA is at 10°. The results are shown in Figure 16.12. Figure 16.12(a) shows the frequency-domain plot and the peak value is at x = 28, which corresponds to 640 MHz, which can be calculated as 7,500 MHz through modeling the downconversion process. Figure 16.12(b) shows the angle plot obtained from a 16-point FFT with a Blackman window and the peak is at x = 2. Figure 16.12(c) shows the angle plot with a 16-point Blackman window padded with 112 zeros through a 128-point FFT operation. This plot is similar to Figure 16.12(b); the only difference is that there is a fine resolution generated from zero padding. Figure 16.12(d) shows the 128-point FFT outputs without Blackman window. As expected, there are many high sidelobes, but the main lobe is narrower. If there are multiple signals, the high sidelobes will decrease the instantaneous dynamic range. For Figure 16.12(c, d) the maximum occurs at x = 11. The narrow beam width of the rectangular window does not produce a better angle data. The x-axis in the above plots is in the units of FFT outputs. They do not represent the input frequency and AOA.

16.11  Frequency and AOA Conversion It is desirable to change the x-axis information in the above plots into frequency and AOA information. The frequency conversion is discussed in many previous chapters and will not be repeated here. Only the downconversion portion will be illustrated as

fout = 860 × 106 + (k − 1) × 106 + (if − 1 + 5) ×



 f  k = floor  in3   10 

fs 128 (16.7)

16.11  Frequency and AOA Conversion

383

Figure 16.12  Frequency and angle outputs of a 16-element antenna array: (a) output frequency, (b) angle output for Blackman windowed 16 points, (c) angle output for Blackman windowed 16 points padded with 112 zero points, and (d) angle output for rectangular windowed 16 points padded with 112 zero points.

384

Angle of Arrival (AOA) and Frequency Measurements

In this equation, 860 is the oscillator frequency given in Section 16.8, k is a constant determined from the input frequency in megahertz, where floor is a MATLAB notation to take the lower integer number of the division, if is the location of the maximum value in the frequency domain, and fs is the sampling frequency. If the input frequency fin equals 7,500 MHz, the fin/103 equals 7.5, and the floor gives the k value of 7. In an actual hardware receiver the k value is determined through the input frequency band selection. The receiver can only process 1 GHz of input. If the band of 7 to 8 GHz is selected, the k value is 7. If 4 to 5 GHz is selected, k = 4, and so on. The k value is determined through (16.7) only for simulations. The sampling fs equals 2.56 GHz (not the initial sampling of 25.6 GHz). The AOA conversion has three steps. The first step is to change the index ia into  (ia − 1)λ  an electrical angle φ through the third equation in (16.4) as sin −1  . The  Qd  AOA can be found through the arcsine of this quantity. In order to provide a meaningful AOA, the electrical angle, which is the quantity in the parenthesis of sin-1 must be less than 1. Thus, the second step is to sort the electrical angle and eliminate all the values greater than unity. In Figure 16.11(b) the input AOA is at 10° and the index of the maximum output is at 2. When the input AOA increases, the index will move to the right. When the input AOA is a negative value, the index will start from the left

Figure 16.13  AOA outputs of a 16-element antenna array: (a) angle output for Blackman windowed 16 points, (b) angle output for Blackman windowed 16 points padded with 112 zero points, and (c) angle output for rectangular windowed 16 points padded with 112 zero points.

16.13  Amplitude Comparison

385

and moves toward the center. Similar results are applicable to Figure 16.12(c, d). The third step is to adjust the output through the following relation as

ia = ian − N / 2

(16.8)

where ian is the negative index and N is the total number of data points. In Figure 16.12(b) N = 16 and in Figure 16.12(c, d) N = 128. After these three adjustments, the results are shown in Figure 16.13. The plots are identical to Figures 16.12(b–d) except some of the outputs are eliminated to maintain a meaningful AOA value. The x-axis is the AOA in degrees and the zero degree is shifted to the center of the plots.

16.12  Two Examples for AOA Measurement Two examples will be shown to reveal some of the important properties of the information produced by the 16-element antenna array. Since the frequency information is well studied in the previous chapters, these examples will only show the AOA outputs. The input conditions are still arbitrarily chosen at S/N = 10 dB (equivalent to 17 dB), and the input AOA is still at 10°. Since the rectangular window creates high sidelobes as shown in Figures 16.12 and 16.13, they are not suitable for receiver design. Only the results with the Blackman window will be illustrated. The two input frequencies are at 2,000 and 8,000 MHz, the lowest and the highest of the receivers. The results are shown in Figure 16.14. From this figure, it is obvious that the AOA measurement is input frequency dependent. In Figure 16.14(a) the AOA only has three outputs: -30°, 0°, and 30°. Using an amplitude comparison method might improve the AOA resolution. With 128-point outputs, the AOA resolution is improved as shown in Figure 16.14(b). When the input frequency is at 8,000 MHz, the results are shown in Figure 16.14(c, d). The AOA resolution is improved especially for the 128 output case. It appears that the AOA outputs lack the capability to separate two signals, when the two signals have a close input frequency. If the two signals can be separated in the frequency domain, their AOA can be obtained separately. The AOA error, which is defined by the difference of the output AOA minus the input AOA, can be obtained through simulations. The input AOA changes from -60° to 60° in a 1° step. The results are shown in Figure 16.15. In Figure 16.15(a) the error is rather large due to the poor AOA resolution shown in Figure 16.14(a). The error can reach -60°. However, for the 128-point case, the error is less than ±3°. For the 8,000-MHz input, the AOA error is reduced. For the 16 outputs, the worst error is about -10° at input of 60° and most the errors are less than 5°. For the 128 output case, the error is usually less than 1°, which is a very respectable result.

16.13  Amplitude Comparison In the previous section, the AOA are calculated from the resolution of the angle measurements. The 16-point results provide coarse AOA and the 128-point results

386

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.14  AOA outputs with input AOA at 10° using a Blackman window: (a) 16 outputs at input frequency = 2,000 MHz, (b) 128 outputs at input frequency = 2,000 MHz, (c) 16 outputs at input frequency = 8,000 MHz, and (d) 128 outputs at input frequency = 8,000 MHz.

16.13  Amplitude Comparison

387

Figure 16.15  AOA errors versus input AOA using a Blackman window: (a) 16 outputs at input frequency = 2,000 MHz, (b) 128 outputs at input frequency = 2,000 MHz, (c) 16 outputs at input frequency = 8,000 MHz, and (d) 128 outputs at input frequency = 8,000 MHz.

388

Angle of Arrival (AOA) and Frequency Measurements

provide a finer AOA resolution with more computation load. Since Blackman is used in performing the spatial FFT operation, the amplitude comparison method can be used to find the AOA. The identical relation used in (11.7) is applied and rewritten here as



∆a −14.3516r 2 + 47.8773r − 23.5451 = a 20

(16.9)

where a and Da are the electrical angle and the differential electrical angle and r is the ratio of the higher neighboring to the maximum. Once the fine electrical angle is obtained, the actual AOA is calculated as previously discussed. Let us use the 16-point FFT and 2,000 MHz as input to find the AOA. The electrical angle plot is shown in Figure 16.16(a). Only three points in this figure can generate real AOA information and they are shown in Figure 16.14(a). Although the rest of the points cannot generate a meaningful AOA, they can be used in the amplitude comparison method. This is a very interesting phenomenon. Points 2 and 3 can be used to find the fine electrical angle between the two points. The AOA errors are plotted in Figure 16.16(b). The results are slightly better than that of Figure 16.15(b), which is obtained by a 128-point FFT. From this simple test, one can conclude that the amplitude comparison method is very powerful in obtaining fine frequency and AOA information. The 16-point FFT can provide adequate information with amplitude comparison and zero padding may not be needed.

Figure 16.16  Results from a 16-element antenna array with the amplitude comparison method: (a) electrical angle and (b) AOA error measured.

16.14  Two Simultaneous Signals

389

16.14  Two Simultaneous Signals In this section the two-input signal case will be illustrated. It is not intended to actually obtain the frequency or the AOA information but show the outputs. Two input signals with different frequency and AOA should produce two peaks in the frequency and angle domain. Once the peaks are located, the frequency and the AOA can be obtained from the previous results. In the three-dimensional plots the frequency is from 120 to 1,160 MHz, in 20-MHz steps, which is the baseband frequency and the AOA is from -60° to 60°. In order to distinguish the two signals in the plot, the input frequencies are in the 7- to 8-GHz range because they have narrow electrical angle peaks. The 16 spatial data are Blackman windowed and padded with 112 zeros because this type of plot has low sidelobes and shows the fine structure of the output. The amplitudes of the signals are in a linear scale rather than a logarithmic scale. The two signals are of the same amplitude, and one input is at 7,200 MHz and 30° and the second signal is at 7,600 MHz and -30°. These input conditions are selected so that the plot can show the two signals clearly separated. Figure 16.17 shows the results. In the next two examples, in the first one the two input signals are at same frequency but at different AOAs and in the second example the two signals are at the same AOA but at different frequencies. Figure 16.18(a) shows two signals with the same input frequency of 7,200 MHz and the AOAs are at -30° and 30°. Although this figure shows two peaks, when the input frequency low, it is difficult to distinguish two peaks because the AOA resolution is poor. Figure 16.18(b) shows two signals at 30° with input frequencies at 7,200 and 7,600 MHz. Both plots clearly show two peaks. These figures illustrate that two signals can be separated. From

Figure 16.17  Two input signals with different AOAs and frequencies.

390

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.18  Two input signals with only one different parameter: (a) same input frequency and (b) same AOAs.

these simple illustrations, one can see that the two-dimensional processing, which is time to frequency and spatial to AOA, is a very powerful approach.

16.15  Eigenvalue and MUSIC Method In Section 16.14, two signals can be either separated by their frequency or the incident angle. When the input signals are close in frequency at a lower frequency range such as both signals at 2 GHz, the frequency domain may not separate them. For example, Figure 16.19 shows that for the two signals both at 2 GHz, even though the AOA angles are separated by 65°, only one peak is shown in the AOA domain. The separation of the two signals depends on the difference of the initial phase. In this special case, both signals have 0 initial phases. This plot is produced through zero padding of a 16-point Blackman window to generate 128 angle outputs and the S/N = 10 dB (equivalent to 17 dB). Thus, the FFT operation does not have the capacity to separate two signals close in angle, which is the expected result. Eigenvalues are used to determine whether two signals exist. Since the input data for the spatial operation are complex because they are the outputs from the FFT operation in the time domain, to detect two signals, a 3 × 3 matrix with lags [1 2 3] can be used. This approach produces three eigenvalues. The eigenvalue generated through the noise input is used as a threshold. Since it is difficult to determine the

16.15  Eigenvalue and MUSIC Method

391

Figure 16.19  Incident angle plot of two signals, both at 2 GHz, and an angle difference of 65° with both initial phases = 0.

distribution of eigenvalue, a large number of outputs (53,000) from the noise input are used to find the threshold. Each run generates three eigenvalues and the results are shown in Figure 16.20. The threshold is arbitrarily chosen as 10. Keeping the input conditions of the two signals the same, such as S/N = 10 dB (equivalent to 17 dB) and input frequency of 2,000 MHz, both initial phases are zero. The only difference is that the incident angle difference is at 12° rather than 65°. The three eigenvalues generated from one special run are about 1.5, 11.9, and

Figure 16.20  Eigenvalue distribution from FFT outputs with lags [1 2 3] with noise as the input.

392

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.21  Electrical angle output for two signals with an incident angle difference of 20°.

2.45 × 104, which indicates that there are two input signals. As discussed in Section 3.11, when the eigenvalues can identify two input signals, their input frequency may not be identified through the music method. In this case it is their incident angles that cannot be identified through the music method. In order to separate two input signals their incident angle difference must be around 20°. The result is shown in Figure 16.21. In this figure the only difference is that the incident angle difference is at 20° and the rest of the conditions stay the same. Figure 16.21 only provides the electrical angle; the conversions to incident angles are still required.

16.16  Nonuniformed Antenna Spacing [5, 6] From the above discussion it is obvious that when the input frequency is low, the AOA resolution measured is coarse because the antenna length is short compared to the wavelength. An antenna array with nonuniform spacing will be discussed in this section [personal communication with L. Y. Liou, physicist at AFRL, 2009]. The total elements are still limited to 16 elements. The shortest distance between two elements must be less than λ/2 at the highest input frequency and another set of spacing is based on λ/2 at the lowest frequency. In the example discussed in the previous sections, the highest frequency is at 8 GHz and the corresponding spacing is 1.875 cm as presented as d1 in Section 16.2. At 2 GHz the corresponding λ/2 is 7.5 cm referenced as d2. In this special case, the two spacings selected are 1.875 and 7.5 cm. The antenna pattern is shown in Figure 16.22.

16.17  AOA Measurement Through Nonuniformed Antenna Spacing

393

Figure 16.22  One type of nonuniform antenna arrangement.

For processing the outputs, the antennas are divided into two groups: A and B. Each group contain 8 antenna elements. An 8-point FFT is performed on each group. Since the spacing is 7.5 cm at a high frequency, there are ambiguities, which are referred to as grating lobes for antenna engineers. The correct angle can be obtained from the relations between groups A and B because the spacing is only 1.875 cm and the angle calculated has no ambiguity. First, the sensitivity of the receiver will be discussed. The detection approach will be using the amplitudes of two 8-point FFT outputs and summing them. This operation is noncoherent integration and a threshold will be set as discussed in Section 5.6. The gain of two summations [5, 6] is about 2.7 dB which is slightly lower than the coherent integration gain of 3 dB. Since the sensitivity problem has been studied, it will not be repeated here. One can consider that the sensitivity is about 0.3 dB less than the values obtained in Section 16.9.

16.17  AOA Measurement Through Nonuniformed Antenna Spacing There are two approaches to calculate the fine AOA values from the nonuniformed antenna spacing [personal communication with L. Y. Liou, physicist at AFRL, 2009]. Both methods use the same basic idea. The basic approach is to find the FFT of two 8-point FFTs from the A and B group antenna elements. Each group has 424 outputs (53 frequency bins and 8 angle bins). If the signal can be detected by the amplitude of either group, the frequency and the electrical angle can be determined. If the signal is weak, the amplitudes of the two groups can be summed together to perform noncoherent integration to find the input signal. It is important that the peak location from both groups must be same. If they are not the same, the higher value will be used as the frequency and the electrical angle. For example, if from group A the frequency is at index 10 and the angle at 4, and from group B the frequency is at 11 and the angle at 3, and the amplitude from group A is higher than group B, then 10 and 4 will be used as the peak values for both groups A and B. It should be noted that the frequency and angle obtained from both groups A and B usually have the same value. If a signal is close to the boundary between two bins, the peak can fall in one bin in one group and the in a neighboring bin in another group. The phases from groups A and B at the peak value are compared to obtain a differential phase φ. Since the spacing between antenna elements A and B is less than λ/2 at 8 GHz, there is no ambiguity in the differential phase. Both methods use the same data to determine the AOA. The first approach uses the amplitude comparison method to find the fine electrical angle. When the input frequency is high, there is ambiguity in the electrical

394

Angle of Arrival (AOA) and Frequency Measurements

angle. The differential phase φ and the fine index can be used to determine the ambiguity range or the number of the grating lobe. The fine index is obtained from the peak value of the electrical angle with the amplitude comparison. Figure 16.23 shows the variation of a differential phase and a fine index versus AOA at 8 GHz without noise. One can see that from these two plots the ambiguous ranges labeled as -2 to 2 can be resolved. If the phase is calculated, the AOA can be uniquely determined. When the input frequency is lower, there are less ambiguous ranges. Mathematically, the ambiguous range ar and the AOA can be calculated as dl i 2ϕ a − 1 − 2 ar = − π 8  

dl  

 λ  8ar + ia − 1 − 2   180 θ = sin −1

  

8d2

(16.10)

  π 

where ia is the fine index value, dl is the data length, and in this special case dl = 8 because there are 8 antenna elements per group. Once the ambiguity range is found, the AOA can be calculated from the second portion of the equation.

Figure 16.23  Differential phase and fine index versus input AOA: (a) differential phase and (b) fine AOA value.

16.17  AOA Measurement Through Nonuniformed Antenna Spacing

395

Figure 16.24  AOA error plot for nonuniform antenna pattern: (a) 2-GHz phase comparison, (b) 2-GHz amplitude comparison, (c) 8-GHz phase comparison, and (d) 8-GHz amplitude comparison.

396

Angle of Arrival (AOA) and Frequency Measurements

The second approach to calculate the AOA is relatively straightforward. Since the differential phase measured is unambiguous, the value can be used to calculate the AOA. The input S/N = 10 dB (equivalent to 17 dB) and the input frequencies are at 2 and 8 GHz. The results are shown in Figure 16.24. From this figure, it appears that the amplitude comparison method produce better results. Comparing the results of Figure 16.24(b) with Figure 16.23(b), the results from the nonuniform antenna array are slightly better. This simple illustration demonstrates that the nonuniform antenna element produces slightly better results. When two input signals are close in frequency but separated in AOA, the nonuniform spacing antenna may have a problem separating them because the AOA ambiguity is resolved through phase comparison, which cannot process simultaneous signals.

16.18  Potential Front End Design In Section 16.5 it was discussed that an input frequency can be folded into the baseband through sampling. The conventional approach to building a multiple input band receiver is to convert the input frequency bands into a certain intermediate frequency (IF) band through local oscillators and mixers as shown in Figure 16.2. This approach is complicated and expensive. A different approach might be applicable: to choose a certain sampling frequency and change the desired input band into a baseband through sampling. An example will be used to illustrate the idea. If the desired input band is from 2 to 8 GHz in a 1-GHz band as discussed in this chapter, it is desirable to have a bandwidth wider than 1 GHz to minimize the aliasing signal fold in. Let us use 1,280 MHz as the desired bandwidth, which equals to adding 140 MHz at both ends of the 1,000 MHz input bandwidth. The minimum sampling frequency required is 2,560 MHz, or twice the desired bandwidth. The lower and higher edges of the input bands are referred to as f1 and f2 for the convenience of this discussion. A simple program with trial and error can be used to find the sampling frequency. The operation is based on aliasing. An input frequency fi can be aliased to the baseband frequency fo through the following relation.

fo = fi −

nfs 2

(16.11)

where fs is the sampling frequency to be determined and n is an integer. The approach is to find two output frequencies fo1 and fo2 corresponding to the edge frequencies f1 and f2. If the difference between fo1 and fo2 is equal or greater than 1,280 MHz, the fs is the desired sampling frequency. The sampling frequency fs starts from 2,560 MHz and increases by 1 MHz each step until the desired condition is met. For example, for the input bandwidth from 2 to 3, the actual bandwidth should be from 1,860 to 3,140 MHz. The sampling frequency of 3,140 is the lowest sampling frequency. With this sampling frequency, 1,860 MHz is aliased to 1,280 MHz and 3,140 MHz to 0, which is the desired result. For the entire front end, the input bands are listed in Table 16.1.

16.19  Frequency Independent AOA Measurements

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Table 16.1  Sampling Frequency for Receiver Front End Input Band (GHz) 2–3 3–4 4–5 5–6 6–7 7–8

Desired Range (MHz) 1,860–3,140 2,860–4,140 3,860–5,140 4,860–6,140 5,860–7,140 6,860–8,140

fs (MHz) 3,140 2,760 2,571 3,070 2,856 2,713

Output Aliased (MHz) 1,280–0 10–1,380 1,289–2,569 255–1,535 148–1,428 77.5–1,357.5

Although this approach can eliminate the mixer and local oscillator in the receiver front-end design, it creates several problems: 1. One obvious problem is that the sampling frequency changes. In this example the largest sampling frequency is 3,140 MHz and lowest one is 2,571 MHz and their ratio is 1.22. Different sampling will produce different performances such as the minimum PW and the frequency resolution. However, the difference of 22% based on the ratio of 1.22 might be an acceptable variation. 2. The analog input bandwidth of the analog-to-digital converter (ADC) must have the bandwidth to accommodate the high input signal. If the input bandwidth is not wide enough, this approach cannot be used. 3. Although the local oscillator for the mixer used in the analog frequency conversion can be eliminated, the clock frequency of the ADC must be changeable to digitize signals from different input bands. If the saturation voltage of an ADC can be lowered so that the receiver noise is comparable to the quantization as discussed in Section 2.10, the amplifiers before the ADC can be eliminated also. When the ADC has a low saturation voltage and a wide input bandwidth, it is possible to design a digital receiver with only analog filters between the antenna and the ADC. The analog filters are used to select the input frequency band of interest.

16.19  Frequency Independent AOA Measurements In some applications it is desirable to look into a certain angle to find all the frequencies from that direction. Such devices are available in analog receivers. For example, the microwave lens [2] (Rotman lens) can be used to fulfill this requirement. For a linear antenna array different length delay lines can be added after each antenna element to accomplish this goal. The basic idea is shown in Figure 16.25. In this figure there are two frequencies; each frequency has two outputs. These second outputs are delayed for a fixed time. If the time delay can be compensated, the two signals will add up in phase. Thus, this approach is independent of frequency. Mathematically, one of the signals can be written as x1 = sin(ω t)

x2 = sin[ω(t −τ )]

(16.12)

398

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.25  Time delay is frequency independent: (a) low frequency, (b) delayed low frequency, (c) high frequency, and (d) delayed high frequency.

where x1 and x2 are two outputs from two antennas. The τ is the time delay equal to dsinθ in Figure 16.1. If the second signal is advanced by τ, the two signals can be added in phase. By changing the delay time τ, the beaming can point to a certain direction. This idea cannot be applied after the mixer. After a mixer with angular frequency ωo, the two outputs can be written as x1 = sin[(ω − ωo )t ]

x2 = sin[ω (t − τ) − ω ot ] ≠ sin[(ω − ω o )(t − τ)]

(16.13)

If x2 = sin[(ω - ωo)(t - τ)], introducing a constant delay can make the two signals in phase. Since in x2 is not in this form, adding a simple delay time will no longer point to the desired angle independent of frequency. If another phase term cos(ωoτ) is introduced in (16.13), the output becomes x2 = sin[(ω - ωo)(t - τ)], which is the desired result. The phase term cos(ωoτ) is independent of input frequency. Thus, the overall operation after the mixer is to introduce the desired delay line and adjusted by a phase angle of cos(ωoτ). This operation can be performed digitally after the ADC.

16.20  AOA Operation Followed by the Frequency Operation

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16.20  AOA Operation Followed by the Frequency Operation In the previous study the time-domain data are processed first to obtain frequency information, and then the spatial domain is processed to obtain the AOA information. Let us refer to this operation as the frequency-AOA processing and use the 16 antenna elements for this discussion. From this type of operation the results are 64 frequency bins and 16 AOA bins. One reason that this is the preferred approach is that a longer FFT operation can be performed in the time domain to improve detection. If the signal can be detected from the frequency output, the signal frequency is known. Only one FFT is required from one of the antenna outputs. A discrete Fourier transform (DFT) can be performed in the time domain on the remaining 15 antenna outputs on the known frequency. This operation can save the calculation load. The other approach is to perform the spatial FFT first and then the timedomain FFT; this is referred to as the AOA-frequency processing. This type of operation generates 8 AOA bins because there are 16 antennas with real data and 128 frequency bins. However, these outputs can be considered as identical to the first approach. Figure 16.26 can be used to illustrate the results. In this figure there are 8 AOA bins and 128 frequency bins. The 128 frequency bins can be divided into two 64 bins. When the AOA is positive, the results appear in the first group. When the AOA is negative, the results appear in the second group. The two groups can be properly put together and the results are identical to the frequency-AOA operation. Figure 16.27 shows the results of both the AOA-frequency and frequency-AOA operations. In this simulation the S/N = 100 dB to minimize the noise effect. The input frequency is at 7,880 MHz, which can be digitized directly without downconversion and the input AOA is at 10°. In Figure 16.27(a) the results are obtained through AOA-frequency operations. The results are obtained by manipulating the

Figure 16.26  Illustrating the results of AOA and then the frequency operation.

400

Angle of Arrival (AOA) and Frequency Measurements

Figure 16.27  Comparing AOA - frequency versus frequency – AOA operations: (a) AOA then frequency operation (manipulated results) and (b) frequency then AOA operation.

8 by 128 outputs into 16 by 64 outputs. Figure 16.27(b) shows the results obtained by performing frequency and then an AOA operation. These two plots are identical. Since the input data are identical, different ordered operations should produce identical results. The only difference is in the output formats. Since the sampling frequency is 2.56 GHz, the input bandwidth is 1.28 GHz. For 128 input data in the time domain, the frequency resolution is 20 MHz, and for 16 antenna elements there should be 16 AOA beams. Both operations produce these predicated results.

16.21  Conclusion This chapter studies the two-dimensional receiver problems that are frequency and AOA. The basic approach uses the FFT operations in both time to frequency and spatial to AOA conversion. If the signal is properly processed, the sensitivity of the receiver can be improved through the antenna gain. The amplitude comparison method can be used to improve the AOA measurement accuracy even with a short antenna array. Since the antenna array increases the processing problem from onedimensional to two-dimensional, the computation increases several times, which may cause a hardware problem. Since the problem becomes two-dimensional, its capability also increases. For example, when two signals are close in frequency, a simple frequency encoding may not separate them. If the two signals have different AOAs, the angle resolution may separate them. Once they are separated, their frequencies can be obtained. An

16.21  Conclusion

401

antenna array with a nonuniform spacing may generate a better AOA resolution at a lower input frequency. Finally, a new approach for the receiver front end design is discussed. If the performance of ADC can be improved, it is possible, at least on paper, to design a receiver front end without radio frequency (RF) amplifiers and mixers.

References   [1]  Tsui, J., Digital Techniques for Wideband Receivers, 2nd ed., Norwood, MA: Artech House, 2001.   [2]  Tsui, J., Microwave Receivers with Electronic Warfare Applications, New York: John Wiley & Sons, 1986.   [3]  Skolnik, M. I., Introduction to Radar Systems, New York: McGraw-Hill, 1962, p. 34.   [4]  Skolnik, M. I., Introduction to Radar Systems, New York: McGraw-Hill, 1962.   [5]  Barton, D. K., Modern Radar System Analysis, Norwood, MA: Artech House, 1988.   [6]  Tsui, J., Fundamentals of Global Positioning System Receivers, 2nd ed., New York: John Wiley & Sons, 2005.

Appendix

List of Programs The programs can be divided into two groups. Eq2_1 is used to calculate the receiver sensitivity and dynamic range. The rest programs are subroutines used frequently in the book to perform simulations.

% Eq2_1.m provides the design between an amplifier and ADC. % JT 24 June 1992 % JT Modified Feb 9 2006 clear all close all % ******** INPUT **************************************************** % ** AMP ** n1_db = -174; % noise at input of amplifer per unit bandwidth f_db = 3; % noise figure of the amplifier % ** ADC ** fs = 2.56e9; % sampling frequency in Hz br = fs/2; % RF bandwidth Eq(2.3) br_db = 10*log10(br); b = 8; % # of ADC bits vs = 1000; % saturation voltage in mv q = vs/(2^(b-1)); % voltage per quantization level Eq(2.4) R = 50; % input impedance assumed 50 ohms Ra = 50; % RF system impedance mismatch_db = 10*log10(200*Ra/(Ra+50)^2); % mismatch insertion loss n = 256*1; % FFT length window_factor = 1.73; %Blackman 1.73 noise BW req_thr = 14; % 0 is the noise floor m = .03125*2.^[0:.1:14]; %input variable % ******** GENERATE CONSTANT **************************************** m_db = 10*log10(m); m1 = m+1; m1_db = 10*log10(m1); md_db = m1_db - m_db; %Maximum input power ps = (vs*vs)*1e-3/(2*R); ps_db = 10*log10(ps);

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Appendix nb_db = ps_db - 1.76 - 6.02*b; %Quantization noise bv = window_factor*fs/n; %video bandwidth bv_db = 10*log10(bv); no_db = nb_db + m_db; %RF output noise g_db = no_db - n1_db - f_db - br_db -mismatch_db; %RF gain pi_db = ps_db - 6 - g_db; %Input power for one of two signals % ******** CALCULATION **********************************************      %Overall noise figure fs_db = f_db + md_db; p3_db = n1_db + g_db + bv_db + fs_db;   %Third or intermodulation q3_db   =(3*pi_db   -   n1_db   +   2*g_db   -   bv_db   -   fs_db)/2;   %Rqrd    3rd    order    intercept    pt dr_db = pi_db + g_db - p3_db - req_thr; %Dynamic range sen_db = n1_db + bv_db + fs_db + req_thr; %Sensitivity en = length(m); % ********* PLOT **************************************************** figure; plot(g_db, q3_db,’-*’); hold; plot(g_db, fs_db,’-*’); grid; figure; plot(g_db, dr_db,’-*’); grid; figure; plot(g_db, sen_db,’-*’); grid; figure; out_nom = diff(q3_db).*diff(fs_db); out_nom1 = diff(dr_db).*diff(sen_db); plot(g_db(1:en-1), out_nom,’-*’); % plot(m(1:en-1), out_nom,’-*’); grid; [amp ind] = max(out_nom1); % [m(ind) g_db(ind) q3_db(ind) fs_db(ind) sen_db(ind) dr_db(ind)] [m’ g_db’ q3_db’ fs_db’ sen_db’ dr_db’]

% barker_gen_fct.m generate bpsk signal % JT April 10, 2007 % generate only one complete barker code function [x, n, bark] =  barker_gen_fct(fs, f1, bark_in, chip_time, snrdb) % fs (sampling frq), f1 (input frq) in Hz % bark_in either 11 or 13 % chip_time in ns % n: total number of output points chip_time=chip_time*1e-9;

Appendix ts = 1/fs; npt_per_chip = round(chip_time/ts); n_per_chip = ones(1,npt_per_chip); b11 = [1 -1 1 1 -1 1 1 1 -1 -1 -1]; b13 = [1 1 1 1 1 -1 -1 1 1 -1 1 -1 1]; bark = []; if bark_in = = 11; for ii = 1:11; bark_chip = b11(ii)*n_per_chip; bark = [bark bark_chip]; end; n = length(bark); elseif bark_in = = 13; for ii = 1:13; bark_chip = b13(ii)*n_per_chip; bark = [bark bark_chip]; end; n = length(bark); else; disp(‘not specified’); end; nn = [0:n-1]; amp = sqrt(2)*10^(snrdb/20); noise = randn(1,n); x = cos(2*pi*f1*ts*nn); x = bark.*x; x = amp*x+noise; % subplot(211), plot(bark,’-*’);

% chirp_gen_fct % JT modified D. Lin program April 10, 2007 % pw in us % f_st f_end in Hz function [x, n]=chirp_gen_fct(fs, pw, snrdb, f_st, f_end) ts = 1/fs; pw = pw*1e-6; n = floor(pw/ts); nn = [0:n-1]; phi = 2*pi*rand; amp = sqrt(2)*10^(snrdb/20); chirp_rate = (f_end-f_st)/pw; xnoise = randn(1,n); x = cos(2*pi*(f_st*ts*nn+0.5*chirp_rate*nn.*nn*ts*ts)+  phi); x = x*amp + xnoise;

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Appendix % eigenvalue_fct.m % JT Dec 13 2007 find all eigenvalues % modified March 10 2008, to normalized the outputs function lambda = eigenvalue_fct(z,m_ind) % z: Input data % m_ind: ord = length (m_ind); K = length(z); klength = K - max(m_ind)+  1; C2 = []; for k = 1:ord, C1 = z(m_ind(k):m_ind(k) + klength - 1); C1 = C1(:); C2 = [C2 C1]; end; Ca = fliplr(C2); C3 = C2’; Cb = flipud(C3); rmat = Cb*Ca/K; [v d] = eig(rmat); lambda = sort(real(diag(d))); % lambda = lambda*length(z)/(length(z) - m_ind(ord) + 1);

% thr_normal_fct.m find threshold from input % JT May 15 2008 % x: input % ck: pfa calculated function[thr ck x_m x_s] = thr_normal_fct(x,pfa) [x1 y1] = hist(x,20); x1_max = max(x1); x_m = mean(x); x_s = std(x); tmp1 = mean((x-x_m).^2); figure; plot(y1,x1/x1_max,’o’); hold; thr = sqrt(2)*erfcinv(2*pfa)*x_s + x_m; plot([thr thr],[0 1]); r1 = [-thr - 2:.01: thr + 2]; % for plot pr = (1/sqrt(2*pi*x_s^2))*exp(-(r1-x_m).^2/(2*x_s^2)); %Normal pr_m = max(pr); plot(r1,pr/pr_m); % r = [thr:.001: x_m + 10*x_s]; %for integration % pr = (1/sqrt(2*pi*x_s^2))*exp(-(r-x_m).^2/(2*x_s^2)); %Normal % ck = trapz(r,pr); % thr_db = 10*log10((thr-x_m)^2/tmp1); % thr_db = 20*log10(thr/x_s)

Appendix % thr_rayleigh_fct.m find threshold from input % JT May 14 2008 % x: input % r: radius [0:a:b] % v: average of two variable % pfa: probability of false alarm function [thr thr_db] = thr_rayleigh_fct(x,pfa); [x1 y1] = hist(x,20); x1_max = max(x1); x_mean = mean(x); x_std = std(x); tmp1 = mean(x.^2); v1 = x_mean*sqrt(2/pi); v2 = sqrt(2*(x_std^2)/(4-pi)); v = (v1 + v2)/2; % Avg two approximates figure; plot(y1,x1/x1_max,’o’); hold; thr = sqrt(-2*v^2*log(pfa)); r1 = [0:.1:round(thr*1.5)]; %for plot pr = (r1/v^2).*exp(-r1.*r1/(2*v^2)); %Rayleigh pr_max = max(pr); plot(r1,pr/pr_max); plot([thr thr],[0 1]); r = [thr*1:.001: thr*2]; %for integration CCCCCCCCCCCC pr = (r/v^2).*exp(-r.*r/(2*v^2)); %Rayleigh ck = trapz(r,pr); thr_db = 10*log10(thr^2/tmp1);

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About the Author James Tsui is a consultant for several companies. He is a retired electronics engineer from the Sensors Directory, Air Force Research Laboratory, Wright Patterson Air Force Base, Dayton, Ohio. His work has been primarily devoted to microwave receivers. He has authored six books on microwave receivers including the Global Positioning System (GPS) receiver. He holds many patents and has been widely published in technical journals and conferences. Dr. Tsui received a B.S.E.E. from the National Taiwan University, an M.S.E.E. from Marquette University, and a Ph.D. from the University of Illinois. He is a fellow of the AFRL and the IEEE.

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Index A Akaike information criterion (AIC), 190, 192–93, 194, 195, 196 Ambiguity region, 135 Ambiguity resolution, 236–38 Amplitude comparison angle of arrival (AOA), 385–88, 393–95, 400 fast Fourier transform (FFT), 125–28, 262–65 improving frequency resolution, 358–59 Amplitude information, time domain, 144–46 Analog instantaneous frequency measurement (IFM), 228–29, 240 Analog-to-digital converter (ADC) constants generation, 13–14 design criterion, 9–11 equations derivation, 14–15 global positioning system (GPS), 21–23 inputs to computer program, 11–13 modifications, 15 noise floor, 19–21 nominal sensitivity, 17–18 nominal values, 18–19 one-bit, 143, 227, 230–31, 243–45, 254 output data, 5, 15–17 quantization levels, 2 results analysis, 23 time domain, 5 Angle of arrival (AOA), 369–401 amplitude comparison, 385–88 angle measurement, 374–76, 382, 385 antenna array, 370 baseline performance, 373–74 conversion, 382–85, 384–85 eigenvalue method, 390–92

frequency conversion, 378–80, 382–84 frequency independence, 397–99 frequency operation, 399–400 frequency processing, 399–400 front end design, 396–97 input frequency, 369–70, 372 input signal downconversion, 371–72 multiple signal classification (MUSIC), 222, 392 nonuniformed antenna spacing, 392–96 overview, 4, 369, 400–1 phase tracking, 91–92 processing gain, 376–78 signal generation, 370–71 16-element antenna, 380–82 two simultaneous signals, 389–90 value of, 203 Application specific integrated circuit (ASIC), 255 Autocorrelation, 174

B Bandpass filter, 372 Barker code, 321–23, 334, 337, 342 Biphase shift keying (BPSK), 321–48 Barker code, 321–23 chip time limits, 333 eigenvalue method, 331–33, 339–41 eigenvalue ratio, 342–43 fast Fourier transform (FFT) outputs, 323–31 fast Fourier transform (FFT) receiver, 333–36 overview, 109, 133, 274, 321, 347–48 phase comparison, 341–42 phase transition locations, 343–47 411

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Biphase shift keying (BPSK) (continued) polyphase filter, 341–43 signal generation, 323–26 two frames, 336–39 Bit number, 18–21 digitization effect, 43–45 Blackman window, 13 angle of arrival (AOA), 373–74, 382, 386–88 biphase shift keying (BPSK), 326–31 chirp signal detection, 350, 351 fast Fourier transform (FFT), conventional, 259 fast Fourier transform (FFT) imbalance, 89–90 fine frequency, 263 frequency resolution, 265, 267 frequency separation, close, 218–20 Hilbert transform, 97–99 multiple fast Fourier transform (FFT), 294–96 phase amplitude (PA), 269 polyphase filter, 311 signal detection probability, 115–16 summations, 121 Broadstock, M., 309 C Cascaded filter banks half band filter, 281–83 through fast Fourier transform (FFT), 277–79 through polyphase filters, 279–80 Channel number, receiver sensitivity, 123–24 Cheng, C. H., 123, 265, 269 Chip rate, 109 Chip time, 324, 333, 334, 337, 338, 339, 340, 344–47 Chirp rate, 109 high and low, 349 Chirp signal, 131, 349–58 after polyphase filter, 362–64 after fast Fourier transform (FFT), 359–60 eigenvalue method, 353–56 in fast Fourier transform (FFT) output, 351–53, 355–56

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in frequency-domain outputs, 349–51 output in one frequency bin, 359–60 output in two frequency bins, 360–62 phase comparison, 364–65, 364–67 in time-domain output, 349–51 receiver output, 356–58 types, 349 Complex biphase phase shift keying (BPSK), 331 Complex continuous wave (CW) signal, 331 Complex receiver, 321 Computer simulation risk, 6 Continuous wave (CW), 272 biphase phase shift keying (BPSK), 326, 336–39 chirp signal detection, 353–60 signal differentiation against, 367 time-domain detection, 144 Convolution approach, 116–19 Correlation amplitude change in, 150–51 as function of frequency, 147–50 Correlation matrix data length increase, 185–86 low-order, 173, 178–79, 186–87 multiple signal classification (MUSIC), 203 noise eigenvalue distribution, 174–76 order effect, 178–79 Cramer-Rao (CR) bound, 29 Curve fitting, 47–48, 49–50 D Data length eigenvalue, 181–86, 197 instantaneous dynamic range (IDR), 71–72 Deinterleaving, 4 Delta frequency, 351, 356 Differential moving window, 154–55, 156–57 Differential phase, angle of arrival (AOA), 396 Digital instantaneous frequency measurement (IFM), 227–54

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ambiguity resolution, 236–38 analog receiver, 228–29 downconversion transform, 252–54 frequency folding, 243–45 Hilbert transform, 249–52 in-phase and quadrature (IQ) channels, 248–49, 250 one-bit analog-to-digital converter (ADC), 230–31 overview, 227–28, 254 phase difference counts, 231–34 receiver hardware and concept, 229–30 signal-to-noise (S/N) effect, 234–36 simulation results, 238–40 simultaneous signals, 241–43 threshold and confirmation, 240–41 threshold with hysteresis, 246–48 time resolution, 245–46 Digital signal processing (DSP), 1, 75 Digitization effect bit number function, 43–45 instantaneous dynamic range (IDR), 45–47 instantaneous dynamic range (IDR), 128 data points, 48–51 Discrete Fourier transform (DFT), 92, 240–41, 242, 243 Down chirp signal, 349 Downconversion angle of arrival (AOA), 371–72, 382–83, 382–85 in-phase and quadrature (IQ) imbalance, 104–6 instantaneous frequency measurement (IFM), 252–54 Dynamic range (DR) analog-to-digital converter (ADC) amplification, 9, 42 curve fitting, 47–48 definitions, 25–26 digitization effect, 43–47 eigenvalue decomposition, 29 eigenvalue generation, 31–33 eigenvalue method, 33–36 frequency identification, 37–42 long data length, 51 introduction, 25

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measurements, 26–27 multiple signal capability, 27–29 multiple signal classification (MUSIC), 29–30, 36–37 one-signal case, 9–23 128 data points, 48–51 processing procedure, 30–31 two-signal instantaneous dynamic range (IDR), 25–52 See also Instantaneous dynamic range (IDR); Single-signal receiver dynamic range (SDR) E Eigenvalue, 173–202 after fast Fourier transform (FFT), 339–41 Akaike information criterion (AIC), 192–93 angle of arrival (AOA), 390–92 biphase shift keying (BSPK) , 331–33, 347 chirp signal detection, 353–56, 361 complex signals, 179–81, 331–33 data length effect, 181–84 data length increase, 185–86 false alarm test, 193–94 frequency selection, 224–25 frequency separation, 189–90 generated with noise/noise plus signals, 31–33 imbalance effect, 196 initial phase difference, 187–89 input parameters, 173–74 instantaneous dynamic range (IDR), 29, 30, 33–36 low-order matrix, 173, 186–87 matrix formulation, 174–76 matrix order effect, 178–79 minimum description length, 192–93 noise distribution, 174–78 one- and two-signal input, 194–96 overview, 173, 201–2 signal detection, 173–202 threshold method, 190–91 time-domain detection, 196–201 two-signal detection, 215–20

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Electronic warfare (EW) receiver approach references, 6 encoder designs, 5–6 functions, 1 ideal performance, 3–4 operation summary, 273 pulse descriptor word (PDW), 2 software approaches, 6–7 system operation, 4 Encoder design, 5 F False alarm biphase shift keying (BPSK), 326 eigenvalue operation, 193–94 fast Fourier transform (FFT), 57, 109–42, 213–14, 256 fast Fourier transform (FFT), conventional, 258–61 instantaneous frequency measurement (IFM), 246–47 multiple fast Fourier transform (FFT), 285–87 pitfalls, 4 polyphase filter, 319–20 probability, 2 rabbit ear elimination, 310 two-dimensional coherent processing, 376–78 Fast Fourier transform (FFT), 53–75 balanced output, 2, 174 biphase shift keying (BPSK) signal determination, 326–31 biphase shift keying (BPSK) signal frequency, 323–26 chirp signal, 351–53, 355–56 conventional receiver design, 255–76 data length, 71–72, 300 dynamic range (DR) study, 53 frequency identification, 37 instantaneous dynamic range (IDR), 59–71 in-phase and quadrature phase (IQ), 77, 78–79 local peaks, 54–56, 58, 73 output imbalance measurement, 79–90

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overview, 1, 6, 9, 13, 73–75 pulse width (PW), 310–11 receiver design, 72–73 rectangular window, 71–72, 75 second operation, 311–12, 318 signal confirmation, 240–41 signal detection, 109–42 signal number, 311–12 signal separation, 42 simulation approaches, 53–54, 56–57 sliding, 79 spectral resolution limitation, 203 threshold determination, 57–58 time-domain detection, 146 time-to-frequency conversion, 5 used with multiple signal classification (MUSIC), 225 versus multiple signal classification (MUSIC), 205–15, 225–26 window and input frequencies, 58–59 See also Fast Fourier transform (FFT), conventional; Multiple fast Fourier transform (FFT) Fast Fourier transform (FFT), conventional, 255–76 amplitude comparison, 262–65 biphase shift keying (BPSK), 333–39 false alarm rate, 258–61 frequency resolution, 257–58, 265–68 length selection, 256–57 overview, 255, 276 performance improvement, 273–74 pulse amplitude (PA) measurement, 269–71 pulse width (PW) measurement, 271–72 receiver measurements, 274–76 reporting pulse descriptor word (PDW), 272–73 requirements, 255–56 second-signal detection, 268 threshold, 258–62 time of arrival (TOA), 271–72 Field programmable gate array (FPGA), 5, 225, 227, 255, 320 Filling holes method, 262 Filtered output calibration, 378–80 Finite impulse response (FIR) filter, 5

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Frequency-angle of arrival (AOA), 399–400 Frequency bin creating artificial, 128–31 two signals in one, 294–96 Frequency bin boundary, 53 chirp signal detection, 360–62, 363 fast Fourier transform (FFT), 265–67, 269 Frequency bin center, 53, 269 Frequency domain analog-to-digital converter (ADC), 5 chirp signal in, 349–351 Frequency folding, 243–45, 378, 396 Frequency identification, instantaneous dynamic range (IDR), 37–42 Frequency independent angle of arrival (AOA), 397–98 Frequency measurement after angle of arrival (AOA), 399–400 and angle measurement, 374–76 in dynamic range definition, 26 frequency conversion, 378–80, 382–85 frequency folding, 372, 396 overview, 369 Frequency modulated (FM) signal, 349–68 amplitude comparison, 358–59 chirp signal, 349–64 continuous wave (CW) signal, 367 eigenvalue method, 353–56 fast Fourier transform (FFT), 351–53, 355–56 overview, 77, 349 phase comparison, 364–67 receiver outputs, 356–58 See also Chirp signal Frequency resolution amplitude comparison, 358–59 fast Fourier transform (FFT), 256–58 improvement in, 262–65 multiple fast Fourier transform (FFT), 292, 294–96 performance improvement, 274 as pulse width (PW) function, 316–18 Frequency selection, multiple signal classification (MUSIC), 224–25 Frequency separation eigenvalue operation, 189–90, 215–20 fast Fourier transform (FFT), 210–15

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multiple fast Fourier transform (FFT), 292 multiple signal classification (MUSIC), 204–5, 220–24 Front end receiver, 396–97

G Gaussian distribution eigenvalue operation, 175, 176–77 signal detection, 116, 119–21 Gaussian probability density function, 109 Global positioning system (GPS), 21–23, 322 H Half band filter, 281–83 Harmonics, 27–28, 268 Hilbert transform in-phase and quadrature phase (IQ), 77, 78, 92–99, 106–7 instantaneous frequency measurement (IFM), 249–52 time-domain detection, 146 Hysteresis threshold, 245–48, 262 I Imbalance effect instantaneous frequency measurement (IFM), 248–49, 254 signal detection number, 196 Impedance mismatch, 9 In-phase and quadrature phase (IQ), 77–107 balancing, 78–79, 222 downconversion, 104–6 fast Fourier transform (FFT), 79–80, 82–89 Hilbert transform, 92–99 imbalance, 79–80, 82–89, 92–104, 248–49 overview, 77–78, 106–7 phase tracking, 91–92 polyphase filter, 99–104 windowed output imbalance, 89–91 Input frequency dependency, 385

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Index

Instantaneous dynamic range (IDR) calculated with 128 data points, 48–51 curve fitting, 47–48 data length function, 71–72 digitization effect, 45–47 eigenvalue method, 25, 33–36 frequency identification, 37–42 frequency separation, 53 input data length increases, 51 multiple signal classification (MUSIC), 25, 35, 36–37, 45 results, claims on, 2 two-signal, 28–29 Instantaneous frequency measurement (IFM) analog, 228–29, 240 building with 1-bit data, 2 receiver deficiency, 227 Instantaneous frequency measurement (IFM) (continued ) two dynamic ranges, 26 See also Digital instantaneous frequency measurement (IFM) Intermediate frequency (IF), 5, 396 J Jacobson, D., 75, 275 Jitter, 2 K Kernel function, 241, 247–48 L Lin, D., 274 Liou, L. Y., 331, 392, 393 Local oscillator (LO), 371, 371–72 Local peaks, 54–56, 58, 73 Long short shift, 137–39 Lowpass filter, 252–54, 281 Low probability intercept (LPI), 5 M Matched window ratio method, 165–69 time-domain detection, 164–69

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416

Maximum length sequence (MLS), 321–22 Mayhew, B., 75, 275 Minimum description length (MDL), 190, 192–93, 194, 195, 196, 198 Missing signal, 4 Mission-specific test, 276 Monobit approach, 1, 2 Monte Carlo simulation, 57 Moving average method, 151–54 Multiple fast Fourier transform (FFT), 277–97 cascaded filter banks, 277–80 half band filter, 281–83 improving pulse width capability, 287–88 length selection, 283–85 long weak signal, 290–91 multiple windows, 291–93 overview, 277, 297 parameter measurements, 296–97 probability of detection, 285–87 short pulse, 289–90 threshold determination, 285–87 two signals in one bin, 294–96 window selection, 293–94 Multiple rate filter. See Polyphase filter Multiple signal classification (MUSIC), 203–26 angle of arrival (AOA), 392 bit number function, 45 conventional method, 220–21 data length function, 51 eigenvalue method, 179, 187, 215–20 frequency selection, 224–25 high-order method, 203–4 implementation, 6–7 input data length increases, 51 input signal frequency separation, 204–5 instantaneous dynamic range (IDR) , 29–30, 35, 36–42 low-order method, 203, 222–24 one signal, 205–6 optimum order, 31 overview, 203–4, 225–26 signal-to-noise ration (S/N), 204–5 two signals, 206–210 versus fast Fourier transform (FFT), 205–15, 225–26

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417

N Noise eigenvalue distribution, 174–78 Noise figure analog-to-digital converter (ADC), 14 Rayleigh distribution, 110–12 receiver sensitivity, 9 Noise floor, bit number, 19–21 Nominal condition bit number variable, 18–19 term usage, 9 Noncoherent processing, 109, 116, 306 Nonoverlapping shifting, 285, 306 Nonuniformed antenna spacing, 392–96, 401 Nyquist bandwidth, 6, 243, 256 O Odd and even complex receiver, 312–16 One-signal test, 275 Oversampling ratio, 305 P Padding with zero, 136–37, 324, 350, 382 Park-McClellan window, 58, 69–71 polyphase filter, 100–4, 300–3, 309 polyphase phase comparison, 132–33 Phase, in signal detection, 146–47 Phase comparison after fast Fourier transform (FFT), 364–65 after polyphase filter, 365–67 aided by amplitude comparison, 125–28 biphase shift keying (BPSK), 336–39, 341–42 fast Fourier transform (FFT), 124–28, 142 Phase difference eigenvalue operation, 187–89 instantaneous frequency measurement (IFM), 231–34 Phase error, recording, 92 Phase modulation, 321. See also Biphase shift keying (BPSK) Phase transition, biphase shift keying (BPSK), 336–39, 341–42, 343–47

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417

Polyphase filter, 1, 299–20 adding frequency bins, 135–37 biphase shift keying (BPSK), 341–43 cascaded filter banks, 279–80 chirp signal detection, 362–64 data output rate, 305 decrease shifting time, 137–41 design parameters, 300–4 detection sequence, 310 false detection, 319–20 fast Fourier transform (FFT), 1 frequency measurement, 133–35 frequency resolution, 316–18 in-phase and quadrature phase (IQ), 99–104, 106–7 input frequency, 314–16 odd and even outputs, 312–14 output frequency bin, 321 overview, 304–6, 320 phase comparison, 365–67 pulse amplitude (PA), 318–19 pulse width (PW) , 310–11, 318–19 rabbit ear generation, 308–10 receiver sensitivity, 299–300 signal number, 311–12 threshold, 121–23 time-domain detection, 306–8 time of arrival (TOA), 318–19 Polyphase phase comparison, 131–42 Probability density function fast Fourier transform (FFT), 109 signal detection, 109–10, 141 Processing gain, coherent, 376–78 Pulse amplitude (PA) measurement, 269–71, 296 fast Fourier transform (FFT), 269–71 multiple fast Fourier transform (FFT), 291–92, 296 as pulse descriptor word (PDW) parameter, 4, 256 polyphase filter, 318–20 Pulse descriptor word (PDW) biphase shift keying (BSPK), 348 dynamic range (DR) measurements, 26–27 five parameters, 4

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418

Index

Pulse descriptor word (PDW) (continued) generation of, 1, 255 as output, 2 Pulse repetition frequency (PRF), 4, 144 Pulse repetition interval (PRI), 4, 144 Pulse width (PW) chirp signal detection, 360 eigenvalue method, 198–201 fast Fourier transform (FFT), 310–11 frequency resolution, 316–18 instantaneous dynamic range (IDR), 29 long and weak, 290–91 measurement, 271–73, 318–19 minimum, 319–20 minimum requirement, 256, 257 multiple fast Fourier transform (FFT), 287–88, 296 output shape, 162–63 polyphase filter, 316–20 as pulse descriptor word (PDW) parameter, 4 receiver sensitivity, 299–300 short, 289–90 time-domain detection, 144, 155, 158–59, 169–71 Pulse width (PW) dependency, 25 R Rabbit ears, 256, 283, 284, 285, 320 polyphase filter, 306, 308–10 Radio frequency (RF) amplifier, 11–12 Radio frequency (RF) gain, 11 Ratio method, matched window, 165–69 Rayleigh distribution, 109 Blackman window, 115–16 noise output, 110–12 signal detection, 117–21, 241 Receiver fast Fourier transform (FFT), 72–73. See also Fast Fourier transform (FFT) sensitivity. See Sensitivity, receiver Rectangular window, 58, 60–63, 71–72, 75 Hilbert transform, 94–97 phase tracking, 92 Rician distribution, 112–13

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418

S Sampling frequency, 6, 396–97 Saturation voltage, 397 Schmidt, Ralph, 29 Second fast Fourier transform (FFT), 311–12, 318 Sensitivity, receiver channel number, 123–24 defined, 25–26 digitization effect, 43–45 equation, 15 fast Fourier transform (FFT), 256 performance improvement, 273–74 polyphase filter, 299–300 single-signal dynamic range, 17–18, 23 threshold determination, 57–58 time-domain detection, 306–8 two-dimensional coherent processing, 376–78 window effect, 109 Shifting time, polyphase filter, 137–39 Short pulse, 289–90 Short window, 169 Sidelobes fast Fourier transform (FFT), 73, 75 Park-McClellan window, 100–4 Signal carrier frequency, 4 Signal detection using phase, 146–47 See also Signal detection, fast Fourier transform (FFT) Signal detection, fast Fourier transform (FFT), 109–42 amplitude comparison, 125–28 artificial output frequency bins, 128–31 channel number adjustment, 123–24 detection probability, 113–16, 121, 122 overview, 109–10, 141–42 phase comparison, 124–25, 133–35 polyphase filter, 121–23, 131–33, 135–41 Rayleigh distribution, 110–12 signal-to-noise (S/N) distribution, 112–13 threshold, 116–21

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419

Signal-to-noise (S/N) ratio bit number function, 43–45 biphase shift keying (BPSK), 328–31, 340–41 chirp signal detection, 351–356 defining desirable performance, 2 eigenvalue operation, 184, 194–96, 216–18 fast Fourier transform (FFT), 112–13, 206, 208, 209, 212, 214–15 higher-order correlation matrix, 179 instantaneous frequency measurement (IFM), 234–36, 237 multiple signal classification (MUSIC), 204–5, 206, 208, 209 polyphase filter, 140–41, 306 Simultaneous signals angle of arrival (AOA), 389–90 electronic warfare (EW) receiver, 29 instantaneous frequency measurement (IFM), 241–43 Single-signal dynamic range (SDR) analog-to-digital converter (ADC), 9, 15 defined, 27–28 fast Fourier transform (FFT), 75 receiver sensitivity, 17–18, 23 threshold methods, 28 16-element antenna, 380–82, 385 Sliding fast Fourier transform (FFT), 79 Spread spectrum signal, 339–41, 351. See also Biphase shift keying (BPSK); Chirp signal Spurious response fast Fourier transform (FFT), 268 instantaneous dynamic range (IDR), 52 low number of bits, 43 multiple fast Fourier transform (FFT), 292 single-signal dynamic range, 27–28 Square input data, 323 Summations, signal detection, 121, 185, 306–8, 310 Summation window, 307–8, 319 T Third-order intermodulation, 9–11, 14 two-signal spur free dynamic range, 28

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419

Threshold eigenvalue operation, 182, 183, 190–91 fast Fourier transform (FFT), 57–58, 73, 212–13 fast Fourier transform (FFT), conventional, 258–62 Gaussian approximation, 119–21 instantaneous frequency measurement (IFM), 240–41, 246–48 multiple fast Fourier transform, 285–87 single-signal dynamic range (SDR), 28 time-domain detection, 159–62 Time-domain detection, 143–71 amplitude information, 144–46 analog-to-digital converter (ADC), 5 chirp signal, 349–51 correlation output amplitude, 147–51 differential moving window, 154–55 eigenvalue method, 196–201 matched window, 164–69 moving average, 151–54 output shape, 162–63 overview, 143, 170–71 phase comparison, 146–47 polyphase filter, 319 pulse amplitude (PA), 299–300 pulse width (PW), 155–59, 169–70 receiver sensitivity, 306–8 sensitivity test, 170 short window selection, 169 threshold setting, 159–62 time of arrival (TOA), 155–59, 169–70 time resolution, 144 Time of arrival (TOA) eigenvalue method, 198–201 fast Fourier transform (FFT), 271–73 instantaneous frequency measurement (IFM), 246 long weak pulse, 290–91 multiple fast Fourier transform (FFT), 296 output shape, 162–63 as pulse descriptor word (PDW) parameter, 4 polyphase filter, 318–20 pulse repetition frequency (PRF), 144

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Index

Time of arrival (TOA) (continued) short pulse, 289–90 time-domain detection, 155, 158–59, 169–71 Time of departure (TOD), 169 eigenvalue method, 198–201 fast Fourier transform (FFT), 272 instantaneous frequency measurement (IFM), 246 matched window, 164 polyphase filter, 319–20 Time resolution instantaneous frequency measurement (IFM), 245–46 performance improvement, 274 Time-to-frequency conversion, 5 Transient effect, 283, 284, 303 Two-dimensional coherent processing, 376–78 Two-signal instantaneous dynamic range (IDR), 28–29 Two-signal resolution, 274 Two-signal spur free dynamic range, 28 Two-signal test, 275–76 Two-signal third-order intermodulation spur free dynamic range, 28 U Uniform spacing, antenna, 380–82 Up chirp signal, 349

Art_Tsui_Index.indd

420

W Ward, C., 73, 281 Weak signal, 262 White Gaussian noise, 56 Windowed fast Fourier transform (FFT) imbalance, 89–90 Window factor, 13 receiver sensitivity, 109 Window functions Blackman window, 58, 65–67, 89–90 Chebyshev window, 58, 67–69 close spaced frequencies, 62–63 Hamming window, 58, 63–65, 74 instantaneous dynamic range (IDR), 54, 58–59, 59–71, 74 Park-McClellan window, 69–71 rectangular window, 60–63, 71–72, 74 time-domain detection, 143, 144 Window length, fast Fourier transform (FFT), 256–58 Window number, multiple fast Fourier transform (FFT), 291–93 Window selection, multiple fast Fourier transform (FFT), 293–94 Z Zero padding, 136–37, 324, 350, 382

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