UltraWideband Radio Propagation Channels A Practical Approach
Pascal Pagani Friedman Tchoffo Talom Patrice Pajusco Bernard Uguen Series Editor PierreNoël Favennec
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UltraWideband Radio Propagation Channels
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UltraWideband Radio Propagation Channels A Practical Approach
Pascal Pagani Friedman Tchoffo Talom Patrice Pajusco Bernard Uguen Series Editor PierreNoël Favennec
First published in France in 2007 by Hermes Science/Lavoisier entitled: “Communications ultra large bande : Le canal de propagation radioélectrique” First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 2737 St George’s Road London SW19 4EU UK
John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA
www.iste.co.uk
www.wiley.com
© ISTE Ltd, 2008 © LAVOISIER, 2007 The rights of Pascal Pagani, Friedman Tchoffo Talom, Patrice Pajusco and Bernard Uguen to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress CataloginginPublication Data Communications ultra large bande. English Ultrawideband radio propagation channels / Pascal Pagani ... [et al.]. p. cm.  (A practical approach) Includes bibliographical references and index. ISBN 9781848210844 1. Broadband communication systems. I. Pagani, Pascal. II. Title. TK5103.4.C6213 2008 621.382dc22 2008030345 British Library CataloguinginPublication Data A CIP record for this book is available from the British Library ISBN: 9781848210844 Cover image created by Atelier Isatis, based on an original photograph by Denis Stenderchuck. Printed and bound in Great Britain by CPI Antony Rowe Ltd, Chippenham, Wiltshire.
Contents
Foreword
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11
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Chapter 1. UWB Technology and its Applications . . . . . . . . . . . . .
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1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1.2. Deﬁnition and historical evolution . . . . . . . . . 1.2.1. Deﬁnition . . . . . . . . . . . . . . . . . . . . . 1.2.2. Historical evolution . . . . . . . . . . . . . . . 1.3. Speciﬁcities of UWB . . . . . . . . . . . . . . . . . 1.4. Considered applications . . . . . . . . . . . . . . . . 1.5. Regulation evolution . . . . . . . . . . . . . . . . . 1.5.1. Regulation in the USA . . . . . . . . . . . . . 1.5.2. Regulation in Europe . . . . . . . . . . . . . . 1.5.3. Regulation in Asia . . . . . . . . . . . . . . . . 1.6. UWB communication system and standardization 1.6.1. Impulse radio . . . . . . . . . . . . . . . . . . . 1.6.1.1. Pulse position modulation . . . . . . . . 1.6.1.2. Pulse amplitude modulation . . . . . . . 1.6.2. Direct sequence UWB . . . . . . . . . . . . . 1.6.3. Multiband OFDM . . . . . . . . . . . . . . . . 1.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . .
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21 22 22 23 24 26 30 31 32 33 34 35 35 38 39 40 41
Chapter 2. Radio Wave Propagation . . . . . . . . . . . . . . . . . . . . .
43
2.1. Introduction . . . . . . . . . . . . . . . 2.2. Deﬁnition of the propagation channel 2.2.1. Free space propagation . . . . . . 2.2.2. Multipath propagation . . . . . . 2.2.3. Propagation channel variations .
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UltraWideband Radio Propagation Channels
2.2.3.1. Spatial selectivity . . . . . . . . . . . . . . . . . . . . . . 2.2.3.2. Frequency selectivity . . . . . . . . . . . . . . . . . . . . 2.2.3.3. Doppler eﬀect . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Propagation channel representation . . . . . . . . . . . . . . . . . 2.3.1. Mathematical formulation . . . . . . . . . . . . . . . . . . . . 2.3.2. Characterization of deterministic channels . . . . . . . . . . 2.3.2.1. The time varying impulse response . . . . . . . . . . . . 2.3.2.2. The frequency domain function . . . . . . . . . . . . . . 2.3.2.3. The time varying transfer function . . . . . . . . . . . . 2.3.2.4. The delayDoppler spread function . . . . . . . . . . . . 2.3.3. Characterization of linear random channels . . . . . . . . . . 2.3.4. Channel classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . 2.3.4.1. Wide sense stationary channels . . . . . . . . . . . . . . 2.3.4.2. Uncorrelated scattering channels . . . . . . . . . . . . . 2.3.4.3. Wide sense stationary uncorrelated scattering channels 2.4. Channel characteristic parameters . . . . . . . . . . . . . . . . . . 2.4.1. Frequency selectivity . . . . . . . . . . . . . . . . . . . . . . . 2.4.1.1. RMS delay spread . . . . . . . . . . . . . . . . . . . . . . 2.4.1.2. Coherence bandwidth . . . . . . . . . . . . . . . . . . . . 2.4.1.3. Delay window and delay interval . . . . . . . . . . . . . 2.4.1.4. Exponential decay constants . . . . . . . . . . . . . . . . 2.4.1.5. Cluster and ray arrival rates . . . . . . . . . . . . . . . . 2.4.2. Propagation loss . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Fast fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4. Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48 48 50 51 51 52 53 53 54 54 54 55 55 56 57 58 58 59 59 60 61 61 62 63 64 64
Chapter 3. UWB Propagation Channel Sounding . . . . . . . . . . . . .
67
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Speciﬁcity of UWB channel sounding . . . . . . . . . . . . . . . 3.3. Measurement techniques for UWB channel sounding . . . . . . 3.3.1. Frequency domain techniques . . . . . . . . . . . . . . . . . 3.3.1.1. Vector network analyzer . . . . . . . . . . . . . . . . . 3.3.1.2. Chirp sounder . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Time domain techniques . . . . . . . . . . . . . . . . . . . . 3.3.2.1. Pulsed techniques . . . . . . . . . . . . . . . . . . . . . 3.3.2.2. Correlation measurements . . . . . . . . . . . . . . . . 3.3.2.3. Inversion techniques . . . . . . . . . . . . . . . . . . . . 3.3.3. Multipleband time domain sounder for dynamic channels 3.3.3.1. Principle of multipleband time domain sounding . . 3.3.3.2. Description of the SIMO channel sounder . . . . . . . 3.3.3.3. Extension towards UWB . . . . . . . . . . . . . . . . . 3.3.3.4. Experimental validation . . . . . . . . . . . . . . . . .
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67 67 70 71 71 72 73 73 75 78 78 80 81 81 84
Contents
3.4. UWB measurement campaigns . . . . . . . . . . . . . . . . . . . . 3.4.1. Overview of UWB measurement campaigns . . . . . . . . . . 3.4.2. Illustration of channel sounding experiments . . . . . . . . . 3.4.2.1. Static measurement campaign over the 3.1–10.6 GHz band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2.2. Static measurement campaign over the 2–6 GHz band 3.4.2.3. Dynamic measurement campaign over the 4–5 GHz band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 4. Deterministic Modeling of the UWB Channel
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4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Overview of deterministic modeling . . . . . . . . . . . . . . . . . 4.2.1. FDTD based approach . . . . . . . . . . . . . . . . . . . . . . 4.2.2. MoM based approach . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Ray based approach . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Speciﬁcity of deterministic modeling in UWB . . . . . . . . . . . 4.4. Overview of UWB deterministic modeling . . . . . . . . . . . . . 4.4.1. Qiu model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Yao model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3. Attiya model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4. Uguen and Tchoﬀo Talom model . . . . . . . . . . . . . . . . 4.5. Illustration of a deterministic model formalism . . . . . . . . . . . 4.5.1. Received signal synthesis . . . . . . . . . . . . . . . . . . . . . 4.5.2. Ray impulse response without delay . . . . . . . . . . . . . . 4.5.3. Ray channel matrix without delay . . . . . . . . . . . . . . . 4.5.4. Described model results . . . . . . . . . . . . . . . . . . . . . 4.5.4.1. Emitted waveform and considered scenario . . . . . . . 4.5.4.2. Channel matrix of each emitted waveform in the LOS case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4.3. Received signal with ideal antennas . . . . . . . . . . . 4.6. Consideration of real antenna characteristics in deterministic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Building material eﬀects on channel properties . . . . . . . . . . . 4.8. Simulation and measurement comparisons . . . . . . . . . . . . . 4.8.1. Evaluation of real antenna consideration . . . . . . . . . . . 4.8.2. Evaluation of impulse response reconstruction . . . . . . . . 4.9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99 99 100 100 101 101 102 102 102 103 104 104 105 105 108 110 110
Chapter 5. Statistical Modeling of the UWB Channel
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5.1. Experimental characterization of channel parameters . . . . . . . 5.1.1. Propagation loss . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.1. Frequency propagation loss . . . . . . . . . . . . . . . .
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UltraWideband Radio Propagation Channels
5.1.1.2. Distance propagation loss . . . . . . . . . . . . . . . . . 5.1.2. Impulse response characterization . . . . . . . . . . . . . . . 5.1.2.1. Delay spread . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2.2. Power delay proﬁle decay . . . . . . . . . . . . . . . . . 5.1.2.3. Ray and cluster arrival rate . . . . . . . . . . . . . . . . 5.1.3. Study of smallscale channel variations . . . . . . . . . . . . 5.1.4. Eﬀect of moving people . . . . . . . . . . . . . . . . . . . . . . 5.1.4.1. Observation of temporal variations . . . . . . . . . . . . 5.1.4.2. Slow fading . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4.3. Fast fading . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4.4. Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . 5.2. Statistical channel modeling . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Examples of statistical models . . . . . . . . . . . . . . . . . 5.2.1.1. IEEE 802.15.3a model . . . . . . . . . . . . . . . . . . . 5.2.1.2. IEEE 802.15.4a model . . . . . . . . . . . . . . . . . . . 5.2.1.3. Other models . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Empirical modeling principles . . . . . . . . . . . . . . . . . . 5.2.2.1. Propagation loss model . . . . . . . . . . . . . . . . . . . 5.2.2.2. Modeling the channel impulse response over an inﬁnite bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.3. Modeling the channel impulse response over a limited bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.4. Simulation results . . . . . . . . . . . . . . . . . . . . . . 5.3. Advanced modeling in a dynamic conﬁguration . . . . . . . . . . 5.3.1. Space variation modeling . . . . . . . . . . . . . . . . . . . . . 5.3.2. Modeling the eﬀect of people . . . . . . . . . . . . . . . . . . 5.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
136 137 137 141 145 148 151 151 152 153 156 157 158 158 159 160 162 162 163 166 166 169 169 172 175
Appendices A. Baseband Representation of the Radio Channel . . . . B. Statistical Distributions . . . . . . . . . . . . . . . . . . B.1. Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . B.1.1. Rayleigh distribution . . . . . . . . . . . . . . B.1.2. Rice distribution . . . . . . . . . . . . . . . . B.1.3. Nakagami distribution . . . . . . . . . . . . . B.1.4. Weibull distribution . . . . . . . . . . . . . . B.1.5. Normal distribution . . . . . . . . . . . . . . . B.1.6. Lognormal distribution . . . . . . . . . . . . B.1.7. Laplace distribution . . . . . . . . . . . . . . B.2. KolmogorovSmirnov goodnessofﬁt test . . . . . C. Geometric Optics and Uniform Theory of Diﬀraction . C.1. Geometric optics . . . . . . . . . . . . . . . . . . . C.1.1. Introduction . . . . . . . . . . . . . . . . . . .
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Contents
C.1.2. Field locality principle . . . . . . . . C.1.3. Field expression in geometric optics C.1.4. Change of local basis . . . . . . . . . C.1.5. Incident ﬁeld . . . . . . . . . . . . . . C.1.6. Reﬂected ﬁeld . . . . . . . . . . . . . C.1.7. Refracted and transmitted ﬁeld . . . C.2. Uniform theory of diﬀraction . . . . . . . C.2.1. Introduction . . . . . . . . . . . . . . C.2.2. Diﬀracted ﬁeld . . . . . . . . . . . . . C.2.3. UTD 2D coeﬃcient . . . . . . . . . . C.2.4. UTD 3D coeﬃcient . . . . . . . . . . D. Ray Construction Techniques . . . . . . . . . . D.1. Ray launching . . . . . . . . . . . . . . . . D.2. Ray tracing . . . . . . . . . . . . . . . . . D.3. Other techniques . . . . . . . . . . . . . . E. Description of the TimeFrequency Transform
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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
237
Bibliography
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Foreword
Although the origins of distant signal transmission are ancient, the theoretical foundations of modern telecommunication techniques are owed to Claude Shannon’s 1948 publications. Since then, the ﬁeld of telecommunications has not stopped evolving. In particular, spread spectrum techniques enabled unprecedented improvements in the quality and security of digital communications under harsh transmission conditions. Today, the main players in the telecommunication world are facing an increasing demand for multimedia wireless applications, linked to a real need for very high throughput radio communication systems. Among the most recent innovations in this ﬁeld, the scientiﬁc community is particularly interested in ultrawideband (UWB) technology. This technique consists of transmitting radio signals spreading over very large frequency bandwidths, typically in the order of 500 MHz to several GHz. Initially developed in the ﬁeld of radar localization, UWB technology is now seen as a promising candidate for future wireless transmission systems. As such, it is considered with an increasing interest in both scientiﬁc and industrial communities. UWB technology undeniably oﬀers numerous advantages and unprecedented possibilities in the design of radio systems. Its very large spectral bandwidth optimally exploits the beneﬁts of spread spectrum techniques, by increasing the transmission capacity and by improving the jamming immunity. Simultaneously, UWB presents a high resolution in the time domain, which may be used to eﬃciently process the multiple propagation paths or handle localization issues. In particular, the capacity of UWB pulses to travel through diﬀerent materials allows us to consider throughwall imaging applications. It is also noteworthy that UWB systems present a low power spectral density, which not only increases the discretion and security of wireless communications, but also reduces the potential jamming experienced by other spectrum users. Finally, the complexity of
12
UltraWideband Radio Propagation Channels
UWB transceivers may be considerably reduced with respect to traditional architectures. These characteristics are unique to UWB technology and enable the design of communication systems oﬀering very high data rates, up to several hundred Mbps. In 2002, the Federal Communication Commission (FCC) – an American regulation body – authorized the transmission of UWB signals in the 3.1–10.6 GHz band, encouraging the research eﬀorts in this ﬁeld in all continents. In Europe, a transmission mask for UWB signals has been deﬁned and the coexistence between UWB systems and other applications is under study. In the current context of high demand for wireless multimedia applications, UWB seems to be an innovative and attractive solution for future radio communication systems. As an illustration, important industrial consortiums such as UWB Forum and WiMedia Alliance are currently involved in the design of UWB based equipment. These systems are particulary well suited to ad hoc communication networks, but a number of other applications may be envisioned by exploiting the unique characteristics of UWB. The possibility of accurate localization enable the development of sensor networks known as radio frequency identiﬁcation (RFID), for industrial environment applications or for mass market geographical information services. Radar identiﬁcation techniques based on UWB signals may be exploited to design anticollision systems for vehicles, but may also be used in the ﬁelds of civil engineering, medicine and imaging. Numerous applications still need to be explored, associating UWB with advanced techniques such as multipleinput multipleoutput (MIMO) antennas techniques or time reversal techniques. In order to develop such systems, many challenges are still to be encountered. The short duration of UWB signals requires a high accuracy in the system synchronization procedure. Also, electronic components need to be adapted in order to process such a wide frequency band, while maintaining a tractable complexity. In particular, the antenna characteristics largely vary with increasing frequency and require an accurate characterization. Among these scientiﬁc challenges, a complete knowledge of the radio channel properties is fundamental. Indeed, the performance of a wireless transmission system is directly linked to the propagation conditions between the emitter and the receiver. These devices need to be designed in order to beneﬁt from the channel characteristics and to mitigate the channel impairments. For instance, modeling the propagation loss allows us to estimate the radio system coverage, while link level simulations may be used to assess the communication robustness. Owing to the width of its frequency band, the
Foreword
13
UWB propagation channel is intrinsically diﬀerent from traditional wideband channels. For instance, the interactions between the radio waves and their environment need to be described more accurately and the variations of the material properties with frequency need to be taken into account. It is thus necessary to closely study the propagation channel in order to evaluate the potential and the constraints attached to UWB communication systems. This textbook results from an intense collaboration between France Telecom’s Research and Development Division and the IETR–UMR CNRS 6164 (Institut d’Electronique et de T´el´ecommunications de Rennes/Institute of Electronics and Telecommunications in Rennes). These research teams conducted joint studies on UWB techniques and on the impact of the transmission channel on UWB communication systems. Through its didactic presentation and the detailed illustration of the discussed topics, this document is an excellent introduction for engineers and communication systems designers as well as for researchers and lecturers willing to expand their knowledge to the ﬁeld of the UWB transmission channel. The originality of this textbook lies in its experimental approach, which allows the reader to follow step by step the theoretical and practical aspects of radio channel characterization and modeling. This approach may easily be adapted to diﬀerent contexts: for diﬀerent applications, in other environments or in diﬀerent frequency bands, such as the available bands around 17 GHz and 60 GHz. It should also be noted that the accurate knowledge of the UWB channel in the 3.1–10.6 GHz band gives access to all the useful information for developing systems included in this frequency band, such as WiFi systems operating around 5 GHz. This book is divided into ﬁve distinct parts. Chapter 1 presents UWB technology. Its historical evolution, the envisioned applications and its main characteristics are detailed. UWB spectrum regulation issues and the proposed communications techniques are also discussed. Chapter 2 describes the propagation of electromagnetic waves in general. All large scale and small scale radioelectric phenomena at play are highlighted. The reader is then introduced to the area of mathematical representation and its characteristic parameters. This didactic presentation covers both indoor and outdoor environments and is applicable in both contexts of mobile radio and wireless networks. Chapter 3 presents an overview of the channel sounding techniques adapted to UWB technology. A distinction is made between frequency domain and time domain techniques, with a discussion on the application domains and the
14
UltraWideband Radio Propagation Channels
limitation of each solution. The main UWB channel measurement campaigns available in the literature, including those conducted by the authors, are listed and illustrated by a few examples. For the study of the propagation phenomena and the design of communication systems, the UWB propagation channel is simulated using deterministic or statistical models. These two modeling approaches are detailed and illustrated in the last two chapters. Chapter 4 focuses on deterministic UWB modeling. A literature review of the proposed deterministic models is given, comparing the advantages and drawbacks of each proposal. The fundamental issues and the theoretical formalism of a UWB deterministic model are then detailed. To illustrate this presentation, some examples are given, where the authors describe a comprehensive deterministic simulator for the UWB channel. Chapter 5 is dedicated to the statistical modeling of the UWB transmission channel. The characterization of the most representative radio channel parameters is ﬁrst presented. Diﬀerent results available in the literature are compared and commented upon, hence providing a rich experimental database on the UWB propagation channel characteristics. After a description of diﬀerent statistical models, the principles of statistical modeling and the diﬀerent related issues are presented following an experimental method. This approach is illustrated using a model designed by the authors, allowing for the simulation of the UWB channel in both static and dynamic environments. Finally, the interested reader will ﬁnd useful additional information regarding channel analysis and modeling in the appendices. The discussed technical material includes the ﬁelds of signal processing, statistics, geometrical optics and diﬀraction theory. It should also be noted that the numerous bibliographical references constitute an abundant source of information, which may easily be exploited to learn more about the last advances in this ﬁeld. This book examines all the characteristics of the UWB transmission channel and provides useful tools for designing eﬃcient UWB systems. Nonspecialists will be introduced to UWB technology in general and more particularly to the propagation channel, which is the key element in a communication system. Specialists will ﬁnd valuable information and a practical approach in order to design simulators, setup measurements, study the channel characteristics and deﬁne models for UWB or in other contexts. This reference document is also an excellent basis for further research on advanced techniques, such as time reversal or UWB transmission in a MIMO
Foreword
15
conﬁguration. I am convinced that this textbook will prove essential reading in future research in the ﬁeld of UWB communications. Professor Gha¨ıs El Zein Deputy Director at IETR
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Acronyms
ADC AGC BER BPSK CDF CDMA CEPT DAA DAC DCS DOA DOD DSO DSUWB ECC EIRP ESD ETSI FCC FDML FDTD GO GPS GSM GTD IDA IEEE IF IMST
Analogtodigital converter Automatic gain control Bit error rate Binary phase shift keying Cumulative density function Code division multiple access Conference of European Postal and Telecommunications Detect and avoid Digitaltoanalog converter Digital communication system Direction of arrival Direction of departure Digital sampling oscilloscope Direct sequence ultrawideband Electronic Communication Committee Eﬀective isotropic radiated power Energy spectral density European Telecommunications Standards Institute Federal Communication Commission Frequency domain maximum likelihood Finite diﬀerence time domain Geometrical optics Global positioning system Global system for mobiles Geometrical theory of diﬀraction Infocom Development Authority of Singapore Institute of Electrical and Electronics Engineers Intermediate frequency Institut f¨ ur Mobil und Satellitenfunktechnik
18
UltraWideband Radio Propagation Channels
IR ISB ISM ITU LDC LNA LO LOS MBOFDM MIC MIMO MoM NLOS OFDM OOK PAM PDA PDF PDP PLL PN PPM PSD QAM QPSK RF RFID RIR RSB SAGE SHF SIMO SISO SWR UHF UMTS UNII US UTD UWB VCO VNA WBAN
Impulse response Incident shadow boundary Industrial, scientiﬁc and medical International telecommunication union Low duty cycle Low noise ampliﬁer Local oscillator Line of sight Multiband orthogonal frequency division mutliplexing Ministry of Internal Aﬀairs and Communications Multipleinput multipleoutput Method of moments Nonline of sight Orthogonal frequency division mutliplexing Onoﬀ keying Pulse amplitude modulation Personal digital assistant Probability density function Power delay proﬁle Phase locked loop Pseudonoise Pulse position modulation Power spectral density Quadrature amplitude modulation Quadrature phase shift keying Radio frequency Radio frequency identiﬁcation Ray impulse response Reﬂection shadow boundary Space alternating generalized expectation Super high frequencies Singleinput multipleoutput Singleinput singleoutput Standing wave ratio Ultra high frequencies Universal mobile telecommunications system Unlicensed national information infrastructure Uncorrelated scattering Uniform theory of diﬀraction Ultrawideband Voltage control oscillator Vector network analyzer Wireless body area network
Acronyms
WiFi WLAN WPAN WSS WSSUS 802.11 802.15.3 802.15.4
Wireless ﬁdelity Wireless local area network Wireless personal area network Widesense stationary Widesense stationary uncorrelated scattering IEEE task group on WLAN IEEE task group on high rate WPAN IEEE task group on low rate WPAN
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Chapter 1
UWB Technology and its Applications
1.1. Introduction
By the end of the 20th century, studies in the telecommunication ﬁeld had made signiﬁcant progress. The advent of new radiocommunication technologies allowed telephony to change from a telegraphic transmission support to a radio transmission support. Over the last few years, the processing speed and storage size of the computers have increased considerably. This explains the general public’s passion for communicating objects, which require the high speed transfer of a great amount of information.
One of the current scientiﬁc challenges, where signiﬁcant research eﬀorts are engaged, is related to the use of very high data rate radio transmissions techniques on relatively short ranges. In this context, ultrawideband (UWB) technology, initially used in radar, appears to be an ideal candidate for future wireless communication systems.
This chapter presents the UWB technology and its applications for wireless communication systems. After a deﬁnition of UWB and of its historical evolution, its characteristics are ﬁrst outlined, then the considered applications and the regulation spectrum in the USA, Asia and Europe are detailed. The chapter ends by presenting the modulation techniques proposed for UWB and by a state of the art of standardization today.
22
UltraWideband Radio Propagation Channels
1.2. Deﬁnition and historical evolution 1.2.1. Deﬁnition The generic term UWB is used to represent a radio technique which was studied under various names. In the earliest writings on this ﬁeld, we can ﬁnd the terms impulse radio, carrierfree radio, baseband radio, time domain radio, nonsinusoid radio, orthogonal function radio and large relative bandwidth radio [BAR 00]. The relative bandwidth is deﬁned by: Bf,3 dB = 2 ·
fH − fL fH + fL
[1.1]
where fH and fL respectively represent the upper and lower cutoﬀ frequencies of the band deﬁned at −3 dB. Initially, UWB signals were deﬁned by a relative bandwidth of 25% or more [TAY 95]. In 2002, the American regulation authority, the Federal Communication Commission (FCC), extended this deﬁnition to a broader category of signals, by including signals with a relative bandwidth Bf,10 dB higher than 20% or with a frequency band higher than 500 MHz [FCC 02]. Typically, the bandwidth of UWB signals is about 500 MHz to several GHz. Thus, the denomination UWB not only includes impulse techniques, but also all the modulations presenting an instantaneous band higher than or equal to 500 MHz. GHz
Power spectral density (dBm.MHz 1)
MHz Conventional narrowband modulation
kHz
Spread spectrum Ultrawideband
41.3 dB m.MHz 1
Bandwidth (Hz) Figure 1.1. Comparison of various radio system spectrums
Figure 1.1 illustrates the comparison between conventional radio systems, which generally modulate a narrowband signal on a carrier frequency, wideband
UWB Technology and its Applications
23
systems, with spreading spectrum for example, and ultrawideband systems, which show a weak power spectral density. As a comparison, the bandwidth of UMTS signals is 5 MHz.
1.2.2. Historical evolution The study of electromagnetism in the time domain began 40 years ago. The ﬁrst research was concentrated on radar applications because of the broadband nature of the signals, which implies a strong resolution1 in the time domain. It was in 1960 that impulse radars were developed by the American and Soviet armies. Indeed, impulse systems have good space resolution properties. The resolution in distance of a system is conversely proportional to its bandwidth; the brevity of an impulse signal determines its spectrum width. In the 1970s, Bennett and Ross presented a complete study of the ﬁrst research carried out on UWB [BEN 78]. Two decades later, Taylor described the bases of the UWB technology applied to radar [TAY 95]. Since the middle of the 1960s, research on this ﬁeld regularly progressed, as mentioned in the historical bibliography published by Barrett [BAR 00]. However, the use of UWB signals for radio communication was not really considered before the end of the 20th century. In 1990, the Department of Defense of the US government published the results of its evaluation of UWB technology. These results were mainly concentrated on radar systems, since no application of UWB technology to communication systems was considered at that point [FOW 90]. More recently, research focused on UWB signals for radio communication [SCH 93, SCH 97a], by using the main characteristics of this technique: a time resolution around one nanosecond due to the huge frequency bandwidth, a short duty cycle allowing for modulations such as time hopping and the management of multiusers, as well as a possible transmission without carrier, which can lead to a simpliﬁcation of the radio system architecture [FOE 01a]. In 1998, the FCC launched the ﬁrst study on UWB. In February 2002, a ﬁrst regulatory report was published, allowing particularly for the transmission of signals in the 3.1–10.6 GHz band for wireless communications, with strong constraints on the power spectral density [FCC 02]. From this date, intensive academic and industrial research was undertaken with the aim of proposing a powerful communication systems using UWB technology.
1. The resolution of a system is its capacity to separate very close energy paths.
24
UltraWideband Radio Propagation Channels
1.3. Speciﬁcities of UWB With the need for increasing the data rate of wireless systems, the UWB technology seems to be an ideal candidate for future radio communication systems in various types of networks, which can be residential, oﬃce, ad hoc, etc. As shown in section 1.2.1, the main characteristic of the UWB signal is the width of the occupied frequency band, typically about 500 MHz to several GHz. Information theory teaches us that with the use of an appropriate code, it is possible to transmit data with a binary error rate (BER) lower than a ﬁxed arbitrarily low threshold, that is, if the data rate is lower than the theoretical value of the transmission channel capacity. Thus, channel capacity C gives an indication of the theoretical maximum data rate reachable with a given channel. It can be obtained using Shannon’s theorem [SHA 49]: S [1.2] C = Bw · log2 1 + N where C is the channel capacity (bit/s), Bw is the signal bandwidth (Hz), S is the signal power level (W) and N is the noise power level (W). We can note that for a given bandwidth the channel capacity increases in a logarithmic way with the signal to noise ratio. In the case of a constant signal to noise ratio, the capacity increases linearly with the signal bandwidth.2 In the context of an increasing demand for communication systems with high data rates, radio technologies using a broad spectral band are more able to propose convenient data rates. So, with a frequency band reaching several GHz, the UWB is well adapted to an increase of data rate than the systems showing a high constraint on the bandwidth [FOE 01a]. The main properties of UWB systems are described below: • High temporal resolution capability Because of their great bandwidth, UWB signals have a high temporal resolution, typically about one nanosecond. A ﬁrst implication of this property is related to the localization: knowing the delay of a signal with
2. We can note that for a given signal power level S, the capacity increases nonlinearly with the bandwidth and reaches the asymptotic value of Clim = NS0 · log2 (e), where N0 is the noise power spectral density.
UWB Technology and its Applications
25
a precision of about 0.1 to 1 ns, it is possible to obtain information on the position of the transmitter with an accuracy of 3 to 30 cm. • Robustness against fading related to multipath propagation In usual propagation channels, narrowband systems suﬀer from fading related to the multipath which combine in a destructive way. In the case of impulse signals, the transmitted waveforms can have a great bandwidth, so the multipath presenting delays lower than one nanosecond can be resolved and added in a constructive way. This recombination causes some complications on the system implementation, as it leads to the design of a receiver with a great number of diversity branches. • Low power spectral density This characteristic is not intrinsic to UWB signals as they were deﬁned (see equation [1.1]), but it is imposed by the radio spectrum regulation authorities. Indeed, as UWB signals present a wide spread spectrum, the occupied frequency band necessarily covers the frequencies already allocated to existing radio systems. To allow a peaceful coexistence of UWB with existing narrowband radio technologies, the FCC has limited the power spectral density of UWB signals to −41.3 dBm.MHz−1 , which corresponds to the power spectral density limit of authorized nonintentional radio transmissions.3 This low power spectral density improves the safety of UWB radio communications, since it becomes more diﬃcult to detect the transmitted signals. Another consequence of this characteristic concerns the propagation distance, which is thus limited to about ten meters. So, the applications considered for UWB systems are short range and high data rate, and are particularly adapted to the development of ad hoc networks. • Less sensitivity to jamming UWB systems oﬀer a great processing gain.4 Thus, the interference UWB systems may have on other systems is reduced, thanks to the low level of the power spectral density authorized by the FCC. On the contrary, the interference caused by narrowband systems on UWB systems is a priori minimized by the bandwidth covered by the impulse signals.
3. Thus, we can consider that UWB radio signals are transmitted “under the noise level”, although the imposed limits remain above the thermal noise. 4. The processing gain of a system is a parameter which gives an indication of the resistance of this system against the jamming caused by the other systems.
26
UltraWideband Radio Propagation Channels
• Protected and secured communications UWB signals are signals that are by nature diﬃcult to detect. Indeed, they are spread over a broad band and transmitted with a low power spectral density close to the noise ﬂoor level of traditional radiocommunication receivers. These characteristics enable secured transmissions with a weak probability of detection and a weak probability of interception. • Relative simplicity of the systems design In terms of implementation, the conventional radio systems are generally of heterodyne design: the signal which codes the transmitted data is generated in baseband and is then transposed to higher frequencies before being emitted. The UWB technology allows the use of impulse generated in baseband and directly transmitted on the radio channel without modulation. This possibility of transmission without carrier may simplify the architecture of the radio systems. Indeed, it is possible to design UWB transmitterreceivers without any synthesizer using a phase locked loop (PLL), any voltage control oscillator (VCO) or any mixer. Thus, this can lead to the design of systems with low manufacturing and marketing costs. • Good obstacle penetration properties UWB signals oﬀer good capabilities of penetration in the walls and the obstacles, in particular for the low frequencies part of the spectrum. This makes it possible to have a good precision in terms of localization and tracking [DEN 03b]. The American company Time Domain, pioneer in the topic of UWB for communication, has developed a broad activity around radar systems of vision through walls. 1.4. Considered applications For a few years, the world of telecommunications has faced an increasing demand for wireless numerical applications, in the industrial environment as well as from the general public. Moreover, we can add to this tendency an important need for total connectivity, the information having to be available for anybody, anywhere and at any time [POR 03b]. This increasing need for wireless connectivity leads to the development of many standards for wireless and short range communication systems. We can mention Bluetooth, the WiFi standards family (IEEE 802.11 a, b and g), Zigbee (IEEE 802.15.4) and the recent standard 802.15.3. We may note that the majority of these technologies for wireless local area networks (WLAN) and personal area networks (WPAN) uses free frequencies in ISM and UNII bands, with a maximum bandwidths of about 10 MHz. A comparative study of the characteristics of some wireless technologies is presented in Table 1.1. Taking into consideration other wireless systems like
UWB Technology and its Applications
27
Bluetooth or WiFi, the UWB technology presents a very low level of emission. However, we can reach transmission data rates that are deﬁnitely higher than the two other technologies. Figure 1.2 shows the position of UWB in comparison with the leading WLAN and WPAN standards in terms of rate and maximum achievable ranges. We can note that contrarily to WiFi standards, UWB technology mainly addresses short range WPAN networks. However, its potential data rate exceeds the performances of all current WLAN and WPAN standards.
Maximum bit rate (Mbps)
1000 UWB 100 802.15.3
WiFi 802.11g
WiFi 802.11a
WiFi 802.11b
10
1
Bluetooth
802.15.4a (Low rate UWB)
Zigbee
0.1 10
20
30
40
50
100
Maximum indoor range (m)
Figure 1.2. WLAN and WPAN main standards: rate and maximum ranges
In order to provide a high data rate anywhere, the future networks will have to be designed considering an optimization of the space capacity, namely the global available data rate per unit of area. In narrow band, we generally regard the spectral capacity of the systems in (bps/Hz) as one of the main parameters for a transmission. The various elements of a communication system, like the modulation, the coding, the implementation, etc. make it possible to improve it. By increasing only the transmission power or the signal band, the capacity also increases. However, the spectral resource for these systems is limited and their power cannot be increased indeﬁnitely. It is limited by medical or commercial considerations, for example the pollution of the spectrum or the life duration of the batteries. For UWB systems, the transmission level or the transmitted power spectral density must be kept suﬃciently low because these systems operate in already
3.1–10.6 GHz 3.1–10.6 GHz ISM 2.4 GHz 5 GHz ISM 2.4 GHz ISM 2.4 GHz
≥ 100 Mbps
≥ 500 kbps
≤ 700 kbps
≤ 54 Mbps
≤ 11 Mbps
≤ 54 Mbps
UWB
Bluetooth
WiFi
Modulation
0.1–1 W
0.1–2 W
0.2–1 W
type 1: 20 dBm type 2: 0 dBm
BPSK, 16QAM, QPSK, 64QAM, OFDM
CCK, BPSQ, QPSK, DSS
BPSK, 16QAM, QPSK, 64QAM
GMSK
−41.3 dBm/MHz PPM, OFDM, CDMA, . . .
−41.3 dBm/MHz PPM, OFDM, CDMA, . . .
EIRP
Table 1.1. Comparison of some wireless communication technologies
Frequency band
Data rate
Technology
IEEE 802.11g
IEEE 802.11b
IEEE 802.11a
IEEE 802.15.1
IEEE 802.15.4a
IEEE 802.15.3a
Speciﬁcation
28 UltraWideband Radio Propagation Channels
UWB Technology and its Applications
29
occupied bands. These low levels are compensated by the use of a broad band. So, compared to the existing wireless systems (see Table 1.2), UWB technology has a low spectral capacity. Thus, it is more correct to speak about the spatial capacity [GHA 04b]. This parameter corresponds to the maximum data rate of a system divided by the surface covered by this system. Rate
Distance
Spatial
Spectral
capacity
capacity
2
(Mbps)
(m)
(kbps/m ) (bps/Hz)
UWB
100
10
318.3
IEEE 802.11a
54
50
6.9
2.7
Bluetooth
1
10
3.2
0.012
IEEE 802.11b
11
100
0.350
0.1317
0.013
Table 1.2. Comparison of the spatial and spectral capacity of some wireless systems
Wireless and very high data rate radio technologies like UWB will make it possible to considerably increase the spatial capacity, by the development of dynamic ad hoc networks [POR 03b]. Finally, we can note that a standardization work is currently in progress in the IEEE 802.15.4a work group to use UWB spectrum within the framework of low data rate and short range radio links. The expected data rate is typically the same as that of the Zigbee standard, with a range of about a hundred meters. Thus, the potential applications of UWB radio technology are mainly related to two techniques: high data rate for short range systems (typically 200 Mbps up to 10 m), and low data rate for long range systems (typically 200 kbps at 100 m). These two ways of using the UWB radio spectrum allow us to consider a given number of typical applications for UWB systems [YAN 04, POR 03b]. Firstly, UWB technology will make it possible to increase the data rate of traditional personal wireless networks. So, it will be useful for WiFi networks which make wireless access to the Internet network possible, or for connections between various peripherals (printer, readers, etc.) in limited size environments of, for example, one or more rooms. Because of a potential of very high data rate in short range, applications requiring a higher data rate are also possible with a range from 1 to 4 meters, for example a high quality multimedia transfer between a DVD player and a screen. In the same manner, the UWB promoters also proposed a wireless alternative for the Ethernet standard.
30
UltraWideband Radio Propagation Channels
In addition, UWB applications are expected to be used for house automation, where a great number of devices able to communicate at a distance of several dozen meters are deployed in an oﬃce or residential environment. In this usage, the exploited characteristics of UWB systems are the low costs of the equipment and the possibility of obtaining localization information. Such potential house automation applications include detection of intrusion, or the electronic reception (owner detection and launching of services like the unlocking of doors). In outdoor scenarios, UWB technology will be used for pointtopoint communication applications. An example usage is the transfer of data between several personal devices. In addition, some studies are in progress concerning services of multimedia contents diﬀusion from electronic kiosks. An example of application typical of such electronic kiosks is the download of stockmarket information on a portable digital assistant (PDA) every time the user is waiting at his regular subway station. Finally, UWB applications are also envisioned in the industrial context. By exploiting the possibilities of long range localization combined with information transfer, sensor networks could be deployed in the productions lines or warehouses, in order to followup and automatically manage the operations. This kind of application is well suited for UWB low data rate and long range communication. The main challenge to be raised for this type of applications is the control of radio communication under diﬃcult propagation and interference conditions. 1.5. Regulation evolution The bandwidth of UWB signals requires a strict regulation of their transmission spectrum. Indeed, many systems, whether licensed or not, are presented in UHF and SHF bands, which are very favorable for radio systems deployment. To allow the use of UWB signals over several GHz, regulatory authorities imposed a strict limitation on the transmission power. Figure 1.3 shows some radio systems existing in UHF and SHF bands. We can note that there are reserved bands for several systems like the standards of cellular telephony GSM (900 MHz), DCS (1.8 GHz) and UMTS (2 GHz). The global positioning system (GPS) also occupies a reserved band around 1.5 GHz. Other frequency bands are already used for unlicensed communication systems. For example, the ISM band is used by systems such as Bluetooth, WiFi and DECT, and is also authorized for domestic devices such as microwave ovens. The UNII band is the frequency band where the WiFi 802.11a and HiperLAN standards operate. In order to limit the UWB signals eﬀects on the other radio systems, the regulatory agencies agree on the use of the 3.1–10.6 GHz band for UWB
UWB Technology and its Applications
31
Power spectral density (dBm.MHz1)
UNII
GPS DCS UMTS ISM
GSM
signals [AIE 03b].5 This part of the radio spectrum allows us to use a bandwidth as wide as 7.5 GHz, while avoiding telephony systems and GPS. The very low authorized power spectral density, located under the level of unintentional emission ﬁxed by the FCC (−41.3 dBm.MHz−1 ), is compensated by the bandwidth, allowing it to emit a total power of 0.6 mW.
Bluetooth, 802.11b, DECT, microwave ovens
802.11a, HiperLAN Unintentional radiation limit (41.3 dBm.MHz1) UWB
0.9 1.5 2.0 2.4 3.1 1.8
5
10.6 Frequency (GHz)
Figure 1.3. Radio systems in UHF and SHF bands
1.5.1. Regulation in the USA In the USA, the regulatory agency FCC launched its ﬁrst works on UWB radio technology as early as 1998 [MOR 03, POR 03a]. In May 2000, a ﬁrst proposal for a regulation was published (Notice of Proposed Rule Making), which led to the current regulatory text Report & Order of February 2002 [FCC 02]. The FCC rules of the UWB spectrum regulation enable it to emit signals mainly in the frequency band 3.1–10.6 GHz, by respecting a power spectral density lower than the one applied to nonintentional radio transmissions. Three diﬀerent classes of equipment are considered: • Visualizing systems: ground penetrating radars, throughwall visualizing systems, medical systems and monitoring systems. 5. We will see that in certain regions of the world (Europe, Asia), only a part of the 3.1–10.6 GHz band is considered.
32
UltraWideband Radio Propagation Channels
• Onboard radar systems: for example, the radars for cars in the 24–29 GHz band. • Communication and measurement systems.
UWB EIRP emission level (dBm.MHz1)
Each class of equipment has its own emission mask. Figure 1.4 presents the emission mask of the communication systems, for indoor use. The spectrum was deﬁned to ensure a protection of the sensitive systems, more particularly the GPS (1.2–1.5 GHz), and the bands dedicated to civil aviation. 35
FCC (indoor)
7.25 10.25
40
CEPT
45
MIC (proposal)
1.99
50
Singapore (UWB Friendly Zone)
10.6
3.1
55 60 65
GPS band
70 75
2.7 3.8 6
0.96
8.5
1.61
1
10
Frequency (GHz)
Figure 1.4. UWB systems emission masks
1.5.2. Regulation in Europe In Europe, the European Telecommunications Standards Institute (ETSI) has operated since 2001 on the attribution of the frequencies in Europe for UWB systems. The studies are carried out in close cooperation with group SE24 of the Conf´erence Europ´eenne des Postes et T´el´ecommunications (CEPT), which more particularly analyzes the possible impact of the UWB systems on the existing ones [POR 03b]. Normally, these European authorities aim to reach a given accord for all the European Union States, but each national regulatory agency keeps the right to manage its radio spectrum independently. Compared to the American regulation, a more restrictive position was adopted by the CEPT. In a decision presented in March 2006 [ECC 06], the regulatory organization Electronic Communication Committee (ECC) proposed a spectral mask limiting the emission of UWB signals to the
UWB Technology and its Applications
33
6–8.5 GHz band with a power spectral density of −41.3 dBm.MHz−1 , are shown in Figure 1.4. The emission is also authorized in the band 3.8–6 GHz with a power spectral density of −70 dBm.MHz−1 . However, we should note that such a power limitation does not make it possible to carry out reliable communication systems for a distance of about one meter. In order to encourage a fast emergence of UWB systems in Europe, the ECC currently considers the possibility of using mitigation techniques to ensure the compatibility of UWB systems with the other radio services in the 3.1–4.8 GHz band. Among these mitigation mechanisms, we can ﬁnd the detection and avoidance (DAA) systems which allow us to avoid bands already used by other systems and the use of low duty cycle (LDC). Hence, all UWB systems using satisfactory mitigation mechanisms could be authorized to emit with a power spectral density of −41.3 dBm.MHz−1 in the 3.1–4.8 GHz band. Temporarily – until June 2010 – the ECC authorizes the emission in the 4.2–4.8 GHz band with a power spectral density of −41.3 dBm.MHz−1 . 1.5.3. Regulation in Asia In Asia, the regulation of UWB is particularly well advanced in Japan and Singapore. In Japan, from September 2002, the Information and Technology Communication SubCouncil working group presented its ﬁrst investigations on UWB technology at the ministry for telecommunications, in order to prepare the regulation of UWB. Moreover, the Communications Research Laboratory (CRL) is developing a project with many industrial partners in order to design marketable UWB systems. However, the emission mask proposed by their ministry of internal aﬀairs and communications, MIC, remains more restricted than the American mask. As illustrated in Figure 1.4, a preliminary proposal presented in August 2005 suggests a limitation of UWB emissions to the 7.25–10.25 GHz frequency bands with a power spectral density of −41.3 dBm.MHz−1 . At the beginning of 2003, the Singaporean regulatory agency named Infocom Development Authority (IDA) created a UWB research area, called UWB Friendly Zone, which makes it possible to deploy tests and demonstrators in Singapore with experiments using an emission power of about 10 dB above the FCC limit and a band spreading from 2 GHz to 10 GHz [POR 03b]. With that, the IDA tries to give an signiﬁcant advance in Singapore to new telecommunication technologies, in order to remain scientiﬁcally and economically competitive. To conclude, it should be speciﬁed that the greatest constraint of regulation comes from management of the interference. The problems of UWB regulation
34
UltraWideband Radio Propagation Channels
do not come from the eﬀect of only one UWB system, but from the aggregation of hundreds of devices using this technology, creating a sum of signals which can possibly interfere with other systems, like the navigation or safety systems. So, the UWB scientiﬁc community currently works to test and deﬁne systems which remain inoﬀensive, even when using several colocalized equipments. 1.6. UWB communication system and standardization The emission mask of UWB radio signals established by the FCC allows the use of various signals. Figure 1.5 presents various solutions which can be considered. For each approach, the used frequency band as well as the emission mask of the FCC are presented in the lefthand graph. In each case, the righthand graph presents the time domain signal corresponding to the band represented in solid line. As we can observe, the duration of the obtained impulse is inversely proportional to the bandwidth used. The monoband approach consists of using all the frequency band available. It is characterized by very short impulses, therefore resistant to the eﬀects of superimposed multipaths, and the signals can be created from an arbitrary impulse modeled by an adequate ﬁlter. However, this approach allows little ﬂexibility in the use of the radio spectrum, and requires us to use very powerful radio frequency (RF) components. Another solution consists of dividing the allocated UWB spectrum into two parts: it is the dual bands approach. It makes it possible to use cheaper integrated circuit technologies, especially for the low band (typically 3–6 GHz), the high band being used according to the development of new RF components. The ﬂexibility of the spectrum radio remains moderate, but this solution allows it to avoid an arbitrarily sensitive band, as the UNII band around 5 GHz for example. Finally, the multiband approach consists of using frequency bands of minimal width about 500 MHz. This solution has a very great ﬂexibility for radio spectrum management. For example, if the emission mask is more restricted in a given country, we can avoid the subbands which are not authorized. The management of the multiuser communications is also simpliﬁed, as many frequency combinations or temporal multiplexings are possible. Many modulation techniques were developed from these various UWB signals. Historically, the ﬁrst form of modulation suggested for UWB was the impulse radio [SCH 93]. Regarding standardization, the American institute IEEE began the work of deﬁning a high data rate communication system
UWB Technology and its Applications
35
using the UWB spectrum in 2002. The choice of the type of modulation for this system was the subject of a procedure of very strict selection, within the 802.15.3a work group. The debate for a single solution was mainly articulated around two proposals: Direct Sequence UWB (DSUWB), proposed by the consortium UWB Forum [FIS 04] and MultiBand Orthogonal Frequency Division Multiplexing (MBOFDM), proposed by the consortium MultiBand OFDM Alliance/WiMedia Alliance [BAT 04b]. In January 2006, as no consensus could be found for a single solution, the IEEE 802.15.3a group was disbanded. Today, only one industrial standard exists concerning UWB systems: the standard ECMA368 [ECM 05], developed by the consortium WiMedia Alliance and based on MBOFDM. 1.6.1. Impulse radio The impulse radio concept, developed from radar studies, corresponds to the emission of impulses of very short duration (around 100 ps to 1 ns). Typically, this type of impulses occupies a very broad spectrum (about 1 to several GHz). It is thus a monoband approach. The impulse signals generally adopted for UWB communications include the Gaussian impulse, its ﬁrst derivative (Gaussian monocycle), and its second derivative, as represented in Figure 1.6. The drawback of the Gaussian impulse lies in its nonzero mean value, which corresponds in the frequency domain to an important continuous component. Thus, the Gaussian impulse cannot be propagated without deformation, and we generally prefer the Gaussian monocycle [BAT 03]. 1.6.1.1. Pulse position modulation Relation [1.3] gives the typical expression of a transmitted impulse radio signal, using a pulse position modulation (PPM) [SCH 97a]: [1.3] w t − j · Tf − cj · Tc − Δ · d j s(t) = Ns
j
where w(t) represents the waveform of a transmitted monocycle, normally starting from t = 0. Thus, the transmitted signal corresponds to a sequence of impulses transmitted at diﬀerent times, the j th impulse being transmitted at j · Tf + cj · Tc + Δ · d j . Ns
The term j · Tf allows a uniform spacing of the impulses. Indeed, the signal deﬁned by: [1.4] s(t) = w t − j · Tf j
36
UltraWideband Radio Propagation Channels
(a ) 1 Normalized signal
Power spectral density (dBm.MHz 1 )
40 50 60 70 80
0
5
10
0.5 0 0.5 1
15
0
0.1
0.2
0.3
0.4
(b) 1 Normalized signal
Power spectral density (dBm.MHz 1 )
40 50 60 70 80
0
5
10
0.5 0 0.5 1
15
0
0.5
1
1.5
2 Time (ns)
4
(c) 1 Normalized signal
Power spectral density (dBm.MHz 1 )
40 50 60 70 80
0
5 10 Frequency (GHz)
15
0.5 0 0.5 1
0
Figure 1.5. UWB spectrum and signals: monoband approach (a), dualband approach (b) and multiband approach (c)
2
UWB Technology and its Applications
37
1
Normalized signal
Gaussian pulse 0.5
Gaussian monocycle Second derivative
0
0.5
1
0
0.2
0.4 0.6 Time (ns)
0.8
1
Figure 1.6. UWB impulse waveforms
corresponds to a sequence of impulses uniformly distributed, with a pulse spacing equal to Tf seconds. The impulse sequence of relation [1.4] is represented in Figure 1.7(a). Tf is generally called “frame duration” and is about 100 to 1000 times the impulse duration. This makes it possible to obtain signals with a low duty cycle, and therefore with a low power spectral density. However, we should note that the periodicity of this signal generates parasitic spikes in the radio spectrum. In addition, impulses cyclically transmitted by the users of the network are very sensitive to collision when the signals access the channel. These two problems are solved by the use of a temporal pseudorandom hopping code. The frame of duration Tf is subdivided in a given number of time intervals (chip) of duration Tc . Each user is provided with a pseudorandom code {cj } of length Ns which indicates in which chip of each frame the impulse must be transmitted. The use of a pseudorandom code makes it possible to decrease the eﬀect of appearance of spikes due to the periodicity of the frame, and the spectrum appears much more smoothed [PEZ 03]. If the pseudorandom sequence is suﬃciently long, the UWB signal can be compared, on the occupied band, to a Gaussian white noise. In addition, the pseudorandom code allows the management of multiusers on the radio channel. The transmitted signal with the considered code is expressed by: [1.5] w t − j · Tf − cj · Tc s(t) = j
and its shape is given by Figure 1.7(b).
38
UltraWideband Radio Propagation Channels
Tf (a) t Tc (b) t
(c) t
Figure 1.7. Sequences of radio impulses: (a) sequence of uniformly distributed impulses, (b) spreading code at position 0, and (c) spreading code at position 1
In relation [1.3], the modulation used to transmit data is PPM. Indeed, the term Δ represents a time interval of about T2c , and the dk terms are the 0 and 1 symbols to be transmitted. The index d j , where · indicates the integer Ns part, shows that the same symbol is used over the whole length of the code. There is thus a redundancy of information, which makes it possible to increase the processing gain. Under these conditions, when zero is transmitted, there is no temporal shift in the emission of the data, while a duration shift of Δ is applied over the entire code duration when one is transmitted. These two states are shown in Figure 1.7(b) and (c). We can note that PPM modulations using a greater number of states are possible. Concerning impulse radio, the signals reception is done by correlation. The basic idea is to multiply the received signal by a model signal, which makes it possible to demodulate the transmitted data. So, the model signal corresponds to a given pseudorandom emitted code. Thus, the use of time codes allows the management of multiaccess [SCH 93]. If several users emit simultaneously by using orthogonal pseudorandom codes, only the signal corresponding to the selected code will be demodulated, the other users will appear as noise. This scheme corresponds to the traditional code multiple division access (CDMA). 1.6.1.2. Pulse amplitude modulation The pulse amplitude modulation (PAM) is an alternative to the pulse position modulation. This technique consists of varying the amplitude of the transmitted impulses to code the diﬀerent states. Theoretically, an unlimited number of diﬀerent values can be used for the signal amplitude. In practice, the PAM modulation is often reduced to two
UWB Technology and its Applications
39
states, 1 and 1. Under these conditions, 2PAM modulation may be regarded as a form of binary phase shift keying (BPSK). This BPSK modulation has a good robustness to the eﬀects of the channel and simpliﬁes the synchronization. Indeed, the position of the impulse remains ﬁxed, and it is only its phase which varies. Another alternative of the PAM modulation consists of transmitting two states: 1 and 0. This corresponds to an onoﬀ keying (OOK) modulation. In each deﬁned transmission timeslot, an impulse is emitted to code 1, and nothing is emitted to code 0. Finally, we may also consider hybrid modulations. We can for example create a modulation of 512 states by using a combination of 256PPM with a modulation 2PAM. The impulse radio modulation technique was implemented by the American company Time Domain, and was marketed in an electronic chip, named PulsOn 210. However, because of the diﬃculty of implementation and the low spectral ﬂexibility of this type of system, the standardization authorities have chosen other types of modulations, presented in the following subsections. 1.6.2. Direct sequence UWB The direct sequence ultrawideband (DSUWB) modulation is a solution proposed by the industrial group UWB Forum [FIS 04]. This modulation uses the frequency band allocated to UWB under two dual bands covering respectively 3.1–4.85 GHz and 6.2–9.7 GHz. This conﬁguration make it possible to avoid the UNII band at 5 GHz used by WiFi systems. On these dual bands, the transmitted impulses have a time duration of about 0.3–0.5 ns and present several cycles (see Figure 1.5(b)). In a ﬁrst step, only the low band is used, in order to simplify the architecture of the radio transmission systems. As with the PPM modulation, the DSUWB modulation uses a frame divided in chips. However, an impulse can be transmitted in each chip of the frame. So, the signal is transmitted continuously and there is no low duty cycle as in the case of the impulse radio. The transmitted symbols are represented by ternary spreading codes of one frame length, composed of a series of 1, 0 or −1. According to the speciﬁcations proposed by the UWB forum, all the DSUWB systems will be able to provide these codes using a BPSK baseband modulation. Optionally, a more eﬃcient modulation, named MBOK, is deﬁned in order to ensure a more robust transmission. The length of the spreading code varies from 1–24 chips according to the considered data rate. The transmission data rate proposed for the DSUWB systems
40
UltraWideband Radio Propagation Channels
range from 28–1,320 Mbps. The management of multiple users, clustered in subnetworks called piconets, is performed by the use of orthogonal codes. In this prospect, the DSUWB modulation is similar to the CDMA system used in UMTS. Finally, we can note that the isolation of users belonging to diﬀerent piconets is improved by the use of slightly diﬀerent chip frequencies into each piconet. More details on this technology can be found in [WEL 03]. Compared to impulse radio, the DSUWB systems seem easier to implement, because the considered frequency bands are narrower, which imposes less constraints on RF components. The used modulation still being based on impulses, this radio access technique is robust against the multipath eﬀects of the channel. In terms of regulation, the separation into two dual bands makes it possible to protect the sensitive frequency bands, but the transmitted spectrum is not very ﬂexible. In terms of standardization, we may notice that following the disbanding of the IEEE 802.15.3a working group, no oﬃcial norm is currently based on the DSUWB modulation. However, some industrial companies are developing this technical solution, starting from the speciﬁcations proposed by the UWB Forum group. As an example, the company Freescale Semiconductor, founder of the UWB Forum group, implemented the DSUWB modulation in the component Freescale XS110. 1.6.3. Multiband OFDM Multiband orthogonal frequency division multiplexing (MBOFDM) is a solution based upon the UWB spectrum proposed in standardization and used by the industrial groups MBOA and WiMedia Alliance [BAT 04b]. It is a multiband approach, where the FCC spectrum is subdivided into 14 subbands of 528 MHz. Figure 1.8 shows these subbands which are divided into diﬀerent groups. First of all, the industrial companies focussed their development eﬀorts on group 1 (3.1–4.9 GHz). The other groups should be quickly exploited in order to produce systems in conformity with the European and Asian regulation. In each subband, an OFDM modulation is applied, the signal being distributed over 100 carriers with narrow bandwidth. For each carrier, the baseband modulation is either BPSK or QPSK. This conﬁguration facilitates a very ﬂexible management of the radio spectrum. Indeed, to avoid the jamming of any frequency band, we can forbid a series of carriers or the totality of a subband. This management can possibly be performed according to the legislation of the country of use or in a dynamic way according to the potential jammers. The management of the multiusers of the same group of subbands is performed using timefrequency code techniques. In a group of subbands, the communication of a given user regularly hops from one band to another, according to a cycle of approximately 1 μs duration. The hopping pattern
UWB Technology and its Applications
41
from one band to another corresponds to the considered timefrequency code, which is unique to each user. The data rates oﬀered by this technology extend from 53.3–480 Mbps. References [BAT 04a, BAT 04b] give all the necessary details for the implementation of this system. Group 1 Gr. 2 Gr. 3 Gr. 4 Gr. 5 ← →← →← →← →←→
Power spectral density (dBm.MHz1)
40
50
60
70
80
0
2
4
6 8 Frequency (GHz)
10
12
Figure 1.8. Subbands for the MBOFDM solution
The advantages of the MBOFDM radio access technique are mainly related to its low complexity of implementation, as the OFDM modulation shows a high degree of maturity. Moreover, it is currently the only UWB communication technique developed as an international standard, ECMA – 368, available since December 2005. The restriction on the used frequency band to the ﬁrst group of subbands allows it to use existing systems and RF components. However, in order to achieve an international deployment of the MBOFDM solution, it will be necessary to develop systems operating under the European and Asian frequency regulations. In technical terms, it should be noted that signals are no longer impulsional, and the technology no longer beneﬁts from the advantages related to a very wide frequency band, such as the robustness to the radio channel eﬀects or the possibilities of localization. The integrated circuit UBLink proposed by the company Wisair is an example of commercially available MBOFDM design.
1.7. Conclusion Initially used for radar localization applications, UWB technology has been studied for the last 15 years in the ﬁeld of wireless communications. The main characteristics of this technology, like the width of the spectrum
42
UltraWideband Radio Propagation Channels
and its strong temporal resolution, made it possible for the scientiﬁc and industrial communities to propose a certain number of interesting applications: high data rate WLAN networks, home automation applications, etc. As developed in this chapter, various types of UWB modulations were proposed in standardization, in particular DSUWB and MBOFDM. UWB regulation has been eﬀective in the USA since 2002. In Europe, a mask of emission for UWB signals was established in 2006, but the coexistence of UWB systems with other applications is still under study. As for any radio access technique, an accurate knowledge of the propagation channel as well as the antennas is predominant for the development of UWB communication systems. Thus, the following section presents the radio waves propagation phenomena inside buildings, and describes the principal parameters for UWB radio channel characterization. In this book, we address antennas by including them in the channel. A thorough study of the antennas in the speciﬁc case of UWB could be the subject of a speciﬁc book. Indeed, the realization of an antenna presenting good characteristics in terms of adaptation and radiation on such a broad band is still a challenging task [SCH 03c, SCH 03b, OSS 03].
Chapter 2
Radio Wave Propagation
2.1. Introduction The existence of electromagnetic waves was theoretically predicted by J.C. Maxwell as early as 1855 [MAX 55]. The German physicist Hertz tried to demonstrate that electromagnetic waves travel at a ﬁnite speed, and in 1886 he performed the very ﬁrst radio propagation experiments. The oscillating circuit designed by Hertz consisted of two discharging metallic spheres, which produced an observable spark on an open wire loop [SCH 86]. Interestingly, the signal used by Hertz consisted of a short duration pulse, which could hence be regarded as an ultrawideband signal [AIE 03a]. In order to facilitate the industrial use of radio transmission, a large amount of research has been conducted to characterize the electromagnetic wave propagation mechanisms. This research ﬁrst focused on signals conﬁned in narrow frequency bands and then extended to wideband signals. This chapter details the deﬁnition of the propagation channel and its mathematical representation. The characteristic parameters of the radiomobile channel are then presented. 2.2. Deﬁnition of the propagation channel By deﬁnition, a radio transmission system transforms an emitted electrical signal e(t) into a received electrical signal s(t) by means of electromagnetic waves. The propagation channel corresponds to the system converting the signal e(t) into the signal s(t) and thus accounts for the interactions between the electromagnetic waves and their environment. At this point, a distinction should be made between the propagation channel and the transmission
44
UltraWideband Radio Propagation Channels
channel. The propagation channel represents the transformation of the electromagnetic waves throughout their propagation, while the transmission channel also includes antenna radiation patterns (cf. Figure 2.1). In the literature, some authors assimilate these two concepts, but the distinction comes fully into play when considering multipleinput multipleoutput (MIMO) channels [COS 04]. The full study of antennas in the speciﬁc UWB context would require a dedicated book. Indeed, the development of a UWB antenna presenting adequate characteristics in terms of adaptation and radiation over such a wide frequency band is still a challenging issue [SCH 03c, SCH 03b, OSS 03]. Transmission channel Emitted signal
Received signal
e(t)
s(t) Propagation channel
Figure 2.1. Propagation channel and transmission channel
2.2.1. Free space propagation Let us ﬁrst consider an ideal case where the transmission system is placed in free space, i.e. in an environment where no obstruction is present. Denoting by GE the emission antenna gain and PE the emitted signal power, the power density observed at a distance d is given by [PAR 00]: W =
PE GE 4πd2
[2.1]
The power PR collected at the output of a receiving antenna with gain GR relates to the power density W as follows: PR = W AR = W
λ2 GR 4π
[2.2]
where AR represents the eﬀective area of the receiving antenna, and λ represents the wavelength at the working frequency.
Radio Wave Propagation
45
Equations [2.1] and [2.2] yield the Friis formula, giving the signal attenuation in free space: 2 c PR = GT GR [2.3] PT 4πf d where we used the relation c = f λ existing between the wavelength λ, the frequency f and the propagation speed c. It should be noted that this relation holds for a distance d large enough for the receiving antenna to be considered in the far ﬁeld region with respect to the transmitting antenna [AFF 00]. A receiver is situated in the far ﬁeld if the distance d is larger than the Fraunhofer distance dF , which is related to the largest dimension of the transmitting antenna D and to the wavelength λ as follows:1 2D2 [2.4] dF = λ Free space propagation is a theoretical or reference situation. In realistic propagation conditions, the transmitted wave is aﬀected by the system environment through diﬀerent mechanisms, which are presented in the following section. 2.2.2. Multipath propagation In a realistic environment, signal transmission follows not only the direct path, but also a number of distinct propagation paths. These paths undergo various eﬀects depending on the type of interaction between the wave and the surrounding objects. At the output port of the receiving antenna, the observed signal corresponds to a combination of diﬀerent waves, each of them presenting a diﬀerent attenuation and a diﬀerent phase rotation. Moreover, each wave reaches the receiver with a distinct delay, linked to the length of the propagation path. Multipath propagation mechanisms may lead to a signiﬁcant distortion of the received signal. On the other hand, a direct path, known as the line of sight (LOS) path, is not always available, which is particularly frequent in indoor conﬁgurations. In this case, so called nonline of sight (NLOS) paths are necessary to enable eﬃcient radio communication. Figure 2.2 illustrates the concept of multipath propagation, as well as the main propagation phenomena.
1. It should be noted, though, that for systems operating at diﬀerent frequencies, the antenna dimension D is generally adapted to the wavelength. For a wire antenna, the antenna length is for instance deﬁned as D = λ2 , and hence the Fraunhofer distance follows dF = λ2 .
46
UltraWideband Radio Propagation Channels
Diffuse reflection Ȝ Diffuse scattering
Transmission
Receiver
Diffraction Specular reflection Transmitter
Waveguide effect
Figure 2.2. Main propagation mechanisms
• Reﬂection: reﬂection takes place on obstacles of large dimensions with respect to wavelength. When two diﬀerent materials are separated by a plane surface (i.e. a surface where possible rough spots are small with respect to the wavelength), the reﬂection is said to be specular. In this case, the direction and the amplitude of the reﬂected ray are governed by the SnellDescartes and Fresnel laws. When the surface separating the two materials presents nonnegligible random rough spots, the reﬂection is called diﬀuse reﬂection. Most of the energy is then directed along the reﬂected ray, but part of the energy is diﬀused in neighbor directions.
Radio Wave Propagation
47
• Transmission: when the medium where a reﬂection takes place is not perfectly radioopaque, part of the incident wave travels through the material following a socalled transmission mechanism. Most of the building materials used in indoor environments signiﬁcantly attenuate the transmitted wave. For a given material, the attenuation and the direction of the transmitted signal are related to the wavelength, since the refractive index of the material varies with the frequency. Finally, for layered materials such as plasterboard, multiple reﬂection may occur within the material. • Diﬀraction: diﬀraction takes place on the edges of large sized obstacles with respect to the wavelength. This mechanism accounts for the electromagnetic ﬁeld continuum on either side of the optical line of sight. The calculation of a diﬀracted ﬁeld uses Huygens’ principle, which considers every point of a wavefront as a secondary spherical source. Hence, diﬀracted waves are distributed along a geometrical cone, with an angle corresponding to the incident angle. • Diﬀusion: when an electromagnetic wave travels towards a group of obstacles of small dimensions with respect to the wavelength, the observed phenomenon corresponds to the superimposition of a large number of random diﬀractions. In this case, the behavior of the incident wave is handled in a statistical way and the resulting phenomenon is called diﬀusion. In general, the electromagnetic wave is directed in all directions with a variable attenuation. This phenomenon is generally observed in outdoor environments, for instance in the presence of tree foliage. In indoor environments, diﬀusion may occur on a group of small objects. • Waveguide eﬀect: the waveguide eﬀect occurs in indoor environments, between two corridor walls for instance. Successive reﬂections on two parallel obstacles lead to a global wave motion along the guiding direction. This phenomenon also occurs in urban environments, for instance between two buildings lining a narrow street. 2.2.3. Propagation channel variations Owing to the diﬀerent interactions between the radio waves and their propagation medium, signiﬁcant variations of the channel characteristics are observable at diﬀerent scales. When isotropic antennas are used in free space, the resulting power attenuation is proportional to the square of the distance d between the antennas (cf. section 2.2.1). However, actual observations present large scale power variations linked to the propagation environment. In practice, the obstacles within the environment as well as the combination of multiple propagation paths lead to an additional attenuation of the
48
UltraWideband Radio Propagation Channels
transmitted waves. In general, this attenuation is a function of the distance d between the transmitter and the receiver. It is characterized by the path loss exponent Nd , the received signal power decreasing proportionally to d−Nd . The parameter Nd is equal to 2 in free space, and varies between 2 and 5 in NLOS conﬁgurations. In LOS conﬁgurations, waveguide eﬀects may lead to a value of Nd lower than 2. Finally, we may note diﬀerences in the order from 1 to several dB between the received power and the mean trend following a d−Nd law. This is due to occasional obstructions known as shadowing or slow fading. Small scale ﬂuctuations are directly linked to multipath propagation mechanisms. Multiple versions of a single signal presenting diﬀerent attenuations and diﬀerent phase delays may eventually combine at the receiver, which produces signiﬁcant fast ﬂuctuations in the order of 10 to 20 dB. 2.2.3.1. Spatial selectivity Let us consider a signal composed of a single carrier frequency propagating along two paths, the direct path and a reﬂected path. If the reﬂection occurs in the vicinity of the LOS path, we may consider that these two paths will cause a similar attenuation. However, depending on the wavelength of the transmitted signal and on the length diﬀerence between the two paths, the two versions of the signal may arrive in phase or antiphase. This concept is illustrated in Figure 2.3. In the ﬁrst case, the signals are adding constructively and some power gain is observed. In the second case, the signals are adding destructively and the total received power is strongly attenuated. When the mobile device is displaced, the phase rotation linked to each path leads to a series of signal peaks and signal fades, which is called a fast fading signal. When this phenomenon occurs with a large amount of paths, the received signal may be considered as a random process. 2.2.3.2. Frequency selectivity Let us now consider a more realistic signal, spreading over a given frequency bandwidth. As explained in the previous section, the spatial selectivity phenomenon depends on the phase diﬀerence between multiple paths. Thus, spatial selectivity also depends on signal frequency. When the frequency band is narrow, all frequencies forming the transmitted signal undergo similar phase variations. Hence, the possible power fades are constant over the whole considered frequency band. This phenomenon is known as ﬂat fading. For signals occupying wider frequency bands, the various frequencies may be aﬀected in a diﬀerent way, and thus the received signal is somehow
Radio Wave Propagation
V
49
t1 direct path t
V
t2
reflected path (case 1) t
V
t2 + ǻt
reflected path (case 2) t
case 1 V
global signal t case 2
Figure 2.3. Constructive and destructive addition of two propagation paths
distorted with respect to the transmitted signal. In this case, we may observe frequency selective fading, which consists of a variation in the power received with increasing frequency. The bandwidth over which the frequency components are aﬀected in a similar way is called the coherence bandwidth or correlation bandwidth. In the delay domain, frequency selectivity corresponds to a delay between the various versions of the signal propagating along diﬀerent paths. This delay is in the order of a few nanoseconds in indoor environments, and in the order of a few microseconds in outdoor environments. Depending on the signal bandwidth, these echoes may superimpose on each other, which results in signiﬁcant fades. For signals presenting a very wide frequency band, and particularly for UWB signals, the time domain resolution is very high, which limits the possible interference between diﬀerent delayed versions of the transmitted signal. In this case, signal fading is less pronounced. It is then possible to apply advanced reception techniques, such as channel equalization or RAKE reception,2 in order to collect as much energy as possible from the multiple propagation paths [CRA 98, FOE 01b, GAU 03].
2. A RAKE receiver combines the signals from diﬀerent multipaths, by using several reception branches in parallel [HAY 01].
50
UltraWideband Radio Propagation Channels
Finally, it should be noted that in the case of wideband signals, frequency selectivity leads to a time domain spreading of the transmitted signal. Characterizing this spreading is essential to calibrate communication systems and mitigate intersymbol interference.3 2.2.3.3. Doppler eﬀect The spatial selectivity phenomenon demonstrates that the properties of the radio propagation channel may vary signiﬁcantly when the receiving antenna is positioned at diﬀerent locations. It is thus questionable how the propagation channel is aﬀected when the transmitting antenna, the receiving antenna or even the environment are moving. The Doppler eﬀect corresponds to an observed shift in the frequency of an electromagnetic signal due to the variation of its propagation path. As a simple example, let us consider a mobile receiver moving at a speed v and receiving a radio signal as a plane wave arriving with an angle α with respect to the mobile direction. In this case, the observed Doppler shift is [PAR 00]: v [2.5] ν = f cos(α) c where f is the signal frequency and c is the wave propagation speed. The statistical case where the ray distribution is represented by the probability density function (PDF) of their angle of arrival pα (α) was developed in [CLA 68]. For a simple case where the handset moves at a constant speed, the PDF of the Doppler shift pν (ν) is given by:
ν ν 1 [2.6] pν (ν) = ν 2 · pα − arccos νmax +pα arccos νmax 1− νmax where νmax is the maximum Doppler shift given by: v νmax = f c
[2.7]
This theoretical PDF of the Doppler shift may be linked to the Doppler spectrum of a measured signal. In particular, for a uniform distribution of the arrival angles, the PDF of the Doppler shift is proportional to the Jakes Doppler spectrum [JAK 93]: pν (ν) =
1 ν 2 π 1 − νmax
[2.8]
3. Intersymbol interference consists of the corruption of a transmitted symbol by the previous symbols due to the time domain spreading generated by the propagation channel. It is one of the main sources for the degradation of the transmission quality.
Radio Wave Propagation
51
In what follows, a distinction will be made between the concepts of temporal and spatial variations. Spatial channel variations are due to the displacement of at least one of the antennas in an otherwise static environment. Temporal variations are linked to a change in the close environment of a ﬁxed radio link [HAS 94a]. In the radiomobile context, both types of variation are generally observed, but one or the other may be considered as predominant, depending on the situation. In the indoor environment, temporal variations are mainly due to moving people. In both cases of spatial and temporal variations, propagation paths between the transmitter and the receiver may appear, disappear or undergo successive transformations. In the case of a stationary channel, only the existing paths are considered, and the channel variations consist of an evolution of the path length, which may be positive or negative. Hence, both spatial and temporal channel variations lead to an observed Doppler eﬀect.
2.3. Propagation channel representation 2.3.1. Mathematical formulation Owing to the multipath propagation phenomena, the received signal s(t) is composed of a number of superimposed replicas of the transmitted signal e(t), each of them presenting a diﬀerent delay and a diﬀerent attenuation. The propagation channel thus behaves like a linear ﬁlter. Hence, the propagation channel may be represented by its impulse response (IR) h(τ ), corresponding to the output of this ﬁlter when the excitation is a Dirac function. The received signal may thus be written as: s(t) =
∞ −∞
e(t − τ )h(τ )dτ
[2.9]
In most radio systems, the transmitted signal spreads over a frequency band which is not centered around zero. A given signal x(t) may thus be represented by its complex envelope γx (t) deﬁned as:
x(t) = γx (t)ej2πf0 t
[2.10]
where {·} represents the real part of a complex number and f0 represents a given frequency in the considered band. The complex envelope γx (t) is also named the equivalent baseband term of x(t).
52
UltraWideband Radio Propagation Channels
It can be shown that there are two ways of deﬁning the baseband ﬁlter heq (τ ) equivalent to the passband ﬁlter h(τ ) (cf. Appendix A): heq1 (τ ) =
1 γh (τ ) 2
[2.11]
−j2πf0 τ
heq2 (τ ) = h(τ )e
The three representations in Figure 2.4 are thus equivalent to observing the channel eﬀect on the transmitted signal. In the remainder of this book, we will use, unless otherwise stated, the channel impulse response in its complex envelope form γh (τ ), which will be denoted h(τ ) for the sake of simplicity. e(t)
s(t) 

h(τ )
γe (t)
γs (t) 
heq1 (τ ) =

1 2 γh (τ )
γe (t)
γs (t) 
−j2πf0 τ
heq2 (τ ) = h(τ )e

Figure 2.4. Equivalent representations of a static radiomobile channel
2.3.2. Characterization of deterministic channels The representation of the radio channel in the form of an IR h(τ ) is valid for static channels only. In practice, the environment or the antenna position may be modiﬁed, and hence the radio channel may vary with time. The IR h(t, τ ) is thus dependent on both time and delay. The inputs and outputs of a linear ﬁlter may be described in the time domain or in the frequency domain. This leads to a set of four transfer functions that may be used to describe the radio channel [BEL 63]. Figure 2.5 illustrates the relations between these functions.
Radio Wave Propagation
53
timedelay F delayDoppler
−1
h(t, τ )
F
@ I @ @ @ F −1
[email protected] @ @ @ @ @ frequencytime R @ @
S(τ, ν)
T (f, t)
@ I @ @ @ @ @ F −1
[email protected] @ @ @ @ R @
F −1 F H(f, ν)
frequencyDoppler Figure 2.5. Characteristic functions for a deterministic channel. Arrows represent a Fourier transform (F ) or an inverse Fourier transform (F −1 )
2.3.2.1. The time varying impulse response The function h(t, τ ) is called the timevariant impulse response. It relates the received signal s(t) to the transmitted signal e(t) according to the following ﬁltering operation: s(t) =
∞
−∞
e(t − τ )h(t, τ )dτ
[2.12]
The magnitude of this impulse response may be observed to distinguish between diﬀerent signal echoes according to their propagation delay. Equation [2.12] thus provides a physical representation of the channel as a continuum of scatterers that are ﬁxed – since their delay is constant – and scintillating – which corresponds to the channel’s temporal evolution. 2.3.2.2. The frequency domain function The function H(f, ν) is also called the output Dopplerspread function and reﬂects the Doppler shift phenomenon caused by the channel. It is the dual function of the function h(t, τ ) in the frequencyDoppler shift space. It thus
54
UltraWideband Radio Propagation Channels
relates the received signal spectrum S(f ) to the transmitted signal spectrum E(f ) as follows: ∞ E(f − ν)H(f − ν, ν)dν [2.13] S(f ) = −∞
Using this representation, the output signal spectrum S(f ) is considered as the sum of superimposed input signal spectrum replicas E(f ), each replica being Doppler shifted and ﬁltered. 2.3.2.3. The time varying transfer function Another approach to characterizing the radio channel consists of describing the relation between the time domain output signal s(t) to the input signal spectrum E(f ), using the time varying transfer function T (f, t): ∞ E(f )T (f, t)ej2πf t df [2.14] s(t) = −∞
The function T (f, t) is related to the functions h(t, τ ) and H(f, ν) through a simple Fourier transform. If the bandwidth of the considered channel is narrow enough, the timevariant transfer function may be directly measured using a network analyzer. 2.3.2.4. The delayDoppler spread function A ﬁnal approach consists of representing the radio channel in the delayDoppler shift space. The corresponding function allows us to simultaneously observe the channel dispersion in both the time domain and the frequency domain. For this reason it is referred to as the delayDoppler spread function. The function S(τ, ν) relates the output signal s(t) to the input signal e(t) through the following relation: ∞ ∞ e(t − τ )S(τ, ν)ej2πνt dνdτ [2.15] s(t) = −∞
−∞
Equation [2.15] represents the output signal s(t) as a sum of superimposed replicas the input signal e(t), each replica being delayed and Dopplershifted. The function S(τ, ν) is related to the functions h(t, τ ) and H(f, ν) through a simple Fourier transform. 2.3.3. Characterization of linear random channels In practical situations, the propagation channel ﬂuctuations are due to a number of superimposed phenomena, which cannot be measured individually.
Radio Wave Propagation
55
The radio channel variations may thus be regarded as a random process and it is no longer valid to describe them deterministically. The propagation channel is thus characterized in a statistical way. In practice, the statistical description of the channel is limited to the second order, and we consider the autocorrelation functions of the channel system functions only. These functions are deﬁned as follows: Rh (t, u; τ, η) = E h(t, τ )h∗ (u, η) RH (f, m; ν, μ) = E H(f, ν)H ∗ (m, μ) [2.16] RT (f, m; t, u) = E T (f, t)T ∗ (m, u) RS (τ, η; ν, μ) = E S(τ, ν)S ∗ (η, μ) where E[·] represents the mathematical expectation and (·)∗ represents the complex conjugation operation. These four autocorrelation functions are linked through double Fourier transforms in a similar scheme to the one linking the channel system functions (cf. Figure 2.5). The second order moments of the input and output signals are then given by: ∞ ∞ e(t − τ )e∗ (u − η)Rh (t, u; τ, η)dτ dη Rs (t, u) = −∞
Res (t, u) =
∞
−∞
−∞
[2.17]
Re (t, u − η)h(u, η)dη
The ﬁrst and second order moments are suﬃcient to completely describe the output signal s(t) in the case of a Gaussian signal. 2.3.4. Channel classiﬁcation The representation of the random radio channel may be simpliﬁed by considering diﬀerent assumptions about the channel characteristics. 2.3.4.1. Wide sense stationary channels A channel is considered as wide sense stationary (WSS) if its temporal (or spatial) variation meets the statistical conditions of second order stationarity. In other words, the expectation of the channel impulse response needs to be invariant with time, and its autocorrelation Rh (t, u; τ, η) will depend on the
56
UltraWideband Radio Propagation Channels
variables t and u through the diﬀerence ξ = t − u only. In practice, these conditions mean that the channel ﬂuctuation statistics are constant over a short time interval ξ, which is a reasonable assumption for traditional channels (e.g. indoor or urban environment). Under these conditions, the autocorrelation functions of the timevariant impulse response and the timevariant transfer function may be written as: Rh (t, t + ξ; τ, η) = Rh (ξ; τ, η) RT (f, m; t, t + ξ) = RT (f, m; ξ)
[2.18]
Considering the double Fourier transform relationship between Rh (t, u; τ, η) and RS (τ, η; ν, μ), and using ξ = t − u, this yields: RS (τ, η; ν, μ) =
∞
−∞
∞
−∞
= δ(ν − μ)
Rh (t, u; τ, η)ej2π(uμ−tν) dtdu ∞ −∞
Rh (ξ; τ, η)e−j2π(ξν) dξ
[2.19]
= δ(ν − μ)PS (τ, η; ν) where the term PS (τ, η; ν) may be identiﬁed as a power spectral density obtained from Rh (ξ; τ, η) by applying the WienerKinchine theorem. This last relation indicates that the spectral content of the signal is uncorrelated for diﬀerent Doppler shifts. Physically, this means that echoes generating diﬀerent Doppler shifts are uncorrelated. Similarly, we may write the autocorrelation of the frequency domain function as: RH (f, m; ν, μ) = δ(ν − μ)PH (f, m; ν)
[2.20]
2.3.4.2. Uncorrelated scattering channels Under the uncorrelated scattering (US) assumption, we consider that the contributions from elemental scatterers corresponding to diﬀerent delays are uncorrelated. This condition is equivalent to an assumption of second order stationarity in the frequency domain. As in the WSS case, we may simplify the autocorrelation functions of the frequency domain function and the timevariant transfer function as follows (and noting by Ω the frequency diﬀerence m − f ): RH (f, f + Ω; ν, μ) = RH (Ω; ν, μ) RT (f, f + Ω; t, u) = RT (Ω; t, u)
[2.21]
Radio Wave Propagation
57
The timevariant impulse response and the delayDoppler spread function may be rewritten in the form of power spectral densities: Rh (t, u; τ, η) = δ(η − τ )Ph (t, u; τ ) RS (τ, η; ν, μ) = δ(η − τ )PS (τ ; ν, μ)
[2.22]
2.3.4.3. Wide sense stationary uncorrelated scattering channels A wide sense stationary uncorrelated scattering (WSSUS) channel follows both the WSS and US assumptions. It corresponds to the simplest class of channels, which presents an uncorrelated spread in both the delay domain and the Doppler shift domain. In this case, the autocorrelation functions of the four channel system functions may be simpliﬁed as follows: Rh (t, t + ξ; τ, η) = δ(η − τ )Ph (ξ; τ ) RH (f, f + Ω; ν, μ) = δ(ν − μ)PH (Ω; ν)
[2.23]
RT (f, f + Ω; t, t + ξ) = RT (Ω; ξ) RS (τ, η; ν, μ) = δ(η − τ )δ(ν − μ)PS (τ ; ν) Figure 2.6 illustrates the simple Fourier transform relationships linking the four autocorrelation functions for a WSSUS channel. Two of the obtained functions are of particular interest. The function PS (τ ; ν) is called a scattering function. It physically represents the Doppler spectrum of radio wave paths as a function of their propagation delay. For ergodic signals, the function Ph (ξ; τ ) may be written as: Ph (ξ; τ ) = E h(t + ξ, τ )h∗ (t, τ ) 1 T →∞ T
= lim
T 2
− T2
h(t + ξ, τ )h∗ (t, τ )dt
[2.24]
Hence, the term Ph (0, τ ) corresponds to the temporal average of the impulse response power. This function is called the power delay proﬁle (PDP). In practice, it is generally assumed that observed channels fall under the WSSUS assumption. For measured channels, it is sometimes necessary to
58
UltraWideband Radio Propagation Channels
timedelay F −1 delayDoppler
Ph (ξ, τ )
F
@ I @ @ @ F −1
[email protected] @ @ @ @ @ frequencytime R @ @
PS (τ, ν)
RT (Ω, ξ)
@ I @ @ @ @ @ F −1
[email protected] @ @ @ @ R @
F −1 F PH (Ω, ν)
Dopplerfrequency Figure 2.6. Autocorrelation functions of a WSSUS random channel
conﬁrm this postulation. A thorough discussion about the validation of the WSSUS assumption for experimental measurements has been published in [BUL 02]. In particular, the RUN method is based on assessments of the process variance over successive subintervals [BEN 66]. 2.4. Channel characteristic parameters In order to evaluate the characteristics of a propagation channel, a set of measured impulse responses need to be analyzed. This section presents a number of parameters describing diﬀerent aspects of the radio channel. In general, their computation requires the WSSUS assumption. As it is not practically feasible to collect a set of statistical channel realizations, we generally assume that the channel system functions are ergodic. Hence, it is for instance possible to approximate the statistical expectation using a temporal (or spatial) average over a set of successive measurements. 2.4.1. Frequency selectivity Frequency selectivity is characterized by the presence of multiple paths, particularly observable on the channel’s impulse response. The power repartition as a function of the delay is given by the PDP, deﬁned by equation
Radio Wave Propagation
59
[2.24]. In order to mitigate the local eﬀects of fast channel fading, the PDP is calculated from a set of M impulse responses, successively measured over a path length in the order of 20–40 wavelengths [LEE 85]. This calculation is valid under the stationarity assumption. The following equation presents the PDP calculation: Ph (0, τ ) =
M 2 1 h tm , τ M m=1
[2.25]
2.4.1.1. RMS delay spread The root mean squared (RMS) delay spread τRMS , sometimes simply referred to as delay spread, represents the standard deviation of the PDP. It is calculated as follows: ∞ 2 τ − τm Ph (0, τ )dτ −∞ ∞ [2.26] τRMS = P (0, τ )dτ −∞ h where τm is the mean delay given by: ∞ τm = −∞ ∞
τ Ph (0, τ )dτ
−∞
Ph (0, τ )dτ
[2.27]
2.4.1.2. Coherence bandwidth The RMS delay spread is a signiﬁcant parameter for the analysis of the intersymbol interference. It is also closely linked to the correlation between the diﬀerent frequencies of the signal spectrum. In order to quantify this frequency dependence, the n% coherence bandwidth is deﬁned from the autocorrelation of the channel transfer function RT (Ω, ξ): RT (Ω, 0) = n = min Ω : RT (0, 0) 100
Bc,n%
[2.28]
The function RT (Ω, 0) is referred to as the frequency correlation function and is obtained from the PDP Ph (0, τ ) by a simple Fourier transform. The coherence bandwidth is thus the frequency lag above which the frequency autocorrelation function crosses a given threshold. Thresholds generally given in the literature are 90% and 50% [BAR 95].
UltraWideband Radio Propagation Channels
Relative magnitude (dB)
60
q% of the total energy
Noise level Ĳ0
Ĳ1
Ĳ2
Ĳ3
Delay (ns)
Figure 2.7. Power delay proﬁle deﬁning the delay window
2.4.1.3. Delay window and delay interval Two other parameters are also used to give a more precise idea of the PDP spread [COS 89]. The q% delay window is the duration of the central part of the PDP containing q% of the total energy. Referring to the delays represented in Figure 2.7, the delay window is given by: [2.29] Wq% = τ2 − τ1 q% The delays τ1 and τ2 are deﬁned by: τ2 τ3 q Ph (0, τ )dτ = Ph (0, τ )dτ 100 τ0 τ1
[2.30]
and the energy which is outside the window is split into two equal parts. The delays τ0 and τ3 are the delays at which the PDP ﬁrst rises and ﬁnally falls through the signal noise level. The p dB delay interval refers to a threshold positioned at p dB below the maximum of the PDP. It simply corresponds to the interval from the ﬁrst delay at which the PDP exceeds this threshold to the last delay at which it falls below this threshold.
Radio Wave Propagation
61
2.4.1.4. Exponential decay constants A number of analyses of the UWB radio propagation channel agree on a representation of the impulse response in the form of a discrete sum of individual contributions. Each contribution, known as ray, corresponds to a propagation path and is characterized by its distinct delay and amplitude. This representation is widely used for wideband channels [HAS 93]. In order to account for the possible presence of ray clusters, this discrete impulse response is represented using the formalism proposed by Saleh and Valenzuela [SAL 87]:
L(t) Kl (t)
h(t, τ ) =
βk,l (t)ejθk,l (t) δ τ − Tl (t) − τk,l (t)
[2.31]
l=1 k=1
where L(t) is the number of clusters, Kl (t) is the number of rays in the lth cluster, and Tl (t) is the arrival time of the lth cluster. Parameters βk,l (t), θk,l (t) and τk,l (t) respectively represent the magnitude, the phase and the arrival time associated with the kth ray within the lth cluster. Theoretically, all these parameters vary with time. Deﬁning the PDP from a set of measurements according to equation [2.25], and assuming that the delay of each ray is approximately constant during the measurements, the Saleh and Valenzuela formalism yields the following formula: Ph (0, τ ) =
Kl L
2 βk,l δ τ − Tl − τk,l
[2.32]
l=1 k=1
As shown in Figure 2.8, the magnitude of the PDP rays generally follows a decay which is close to an exponential function. This exponential decay may be observed at both the cluster level and the ray level within a single cluster. The inter and intracluster exponential decay constants, respectively denoted Γ and γ, are thus deﬁned so that the magnitude of the rays obeys the following rule: 2 2 = β1.1 e− βk,l
Tl −T1 Γ
e−
τk,l γ
[2.33]
These constants may be retrieved from measurements by applying a linear least squares ﬁtting procedure on the PDP represented in dB. 2.4.1.5. Cluster and ray arrival rates Without more accurate knowledge about the environment, it is generally assumed that ray arrivals correspond to independent events, and that the number of these events is dependent on the observation duration only. By
62
UltraWideband Radio Propagation Channels
Power (mW)
Ĳk1
Cluster eĲ/ī Ĳk2 Ĳk3
eĲ/Ȗ eĲ/Ȗ
T1
T2
eĲ/Ȗ
T3
Delay (ns)
Figure 2.8. Power delay proﬁle following the Saleh and Valenzuela formalism
deﬁnition, the ray – or cluster – arrivals may reasonably be regarded as a Poisson process. In the Saleh and Valenzuela formalism, it is thus proposed that the arrival probability for a new cluster or a new ray follows an exponential law, with respective parameters Λ and λ. Parameters Λ and λ are respectively called cluster and ray arrival rates, and the numbers Λ1 and λ1 represent the mean duration between two successive clusters or rays. These parameters are assessed by studying the intercluster and interray duration distributions.
2.4.2. Propagation loss When expressed in dB, equation [2.3] corresponding to the Friis formula for free space propagation is written: P L(f, d) = 20 log
4πf d c
− GT (f ) − GR (f )
[2.34]
where P L(f, d) represents the ratio between the transmitted power and the received power. The slow variations of the propagation channel are mainly due to propagation loss and shadowing eﬀect. For a practical channel, in order to
Radio Wave Propagation
63
characterize the double frequency and distance dependence of the path loss, the parameter P L(f, d) is approximated by the following formula: f d + 10Nd log + S(f, d) [2.35] P L(f, d) = P L f0 , d0 + 10Nf log f0 d0 where f0 and d0 correspond to an arbitrary frequency and an arbitrary distance.4 Nf and Nd are called frequency and distance dependent path loss exponents. Parameter Nd accounts for the interactions between the radio wave and the environment, such as the transmission phenomenon or the waveguide eﬀect, and can signiﬁcantly diﬀer from the theoretical value Nd = 2. Parameter Nf accounts for the frequency dependence of the propagation phenomena, and also includes the variations of the eﬀective area for an ideal isotropic antenna. By considering the propagation channel without antennas in a LOS situation, this parameter should be close to its theoretical value Nf = 2. Some authors consider antennas as a part of the propagation channel; in this case, additional variations of the parameter Nf may be observed, which is linked to the measurement antenna gain. Finally, the term S(f, d) corresponds to the slow variations of the propagation channel. As the parameters Nf and Nd are calculated by linear regression, the parameter S(f, d) presents a zero average. When expressed in dB, this variable is generally considered as Gaussian, and it is characterized by its standard deviation σS . In order to calculate the parameter Nf from measurements, we introduce the power transfer function PT (f ), equivalent to the PDP Ph (0, τ ) in the frequency domain. This takes the average received power as a function of frequency for a number of locally measured transfer functions: PT (f ) =
M 2 1 T f, tm M m=1
[2.36]
2.4.3. Fast fading The fast fading of the propagation channel corresponds to the magnitude ﬂuctuations in the received signal. It may be assessed for a narrowband signal or for a given delay in the case of a wideband signal. Mathematically, it is characterized by the statistical law of the random variable h(t, τ ) for a given τ . In practice, we study a set of samples h(tm , τ ) corresponding to M measurements taken at close locations (or in a timevariant environment),
4. We may use the central frequency in the considered band for f0 , and the distance d0 is generally 1 m.
64
UltraWideband Radio Propagation Channels
while verifying that the channel stationarity condition holds. The distributions generally used to assess fast fading include Rayleigh, Rice, Nakagami, Weibull, and lognormal laws (cf. Appendix B). In order to select a speciﬁc theoretical distribution, a goodnessofﬁt test is employed, such as the KolmogorovSmirnov test (cf. Appendix B.2). Some useful tools also exist to characterize the temporal behavior of a signal. The level crossing rate consists of determining the frequency at which the signal magnitude decreases below a given threshold. The average fade duration is an estimate of the time during which the signal magnitude stays below a given threshold [PAR 00]. 2.4.4. Spectral analysis Another way to characterize temporal variations in the channel consists of studying the Doppler spectrum for the main impulse response paths. These spectra may be observed on the scattering function PS (τ, ν). In order to characterize the channel variations independently of the delay, we use the mean Doppler spectrum deﬁned as: ∞ PS (τ, ν)dτ [2.37] PH (0, ν) = −∞
As in the PDP case, we may deﬁne the Doppler spread as: ∞ 2 ν − νm PH (0, ν)dν −∞ ∞ νRM S = P (0, ν)dν −∞ H where νm is the mean Doppler shift deﬁned by: ∞ νPH (0, ν)dν νm = −∞ ∞ P (0, ν)dν −∞ H
[2.38]
[2.39]
As explained earlier (section 2.2.3.3), the Doppler spectrum is closely linked to the arrival direction of the multiple propagation paths. Parameters such as the angular spectrum may be used to characterize the signal’s departure and arrival directions. This is of particular interest for MIMO applications [COS 04]. 2.5. Conclusion In order to design and optimize wireless communication systems, an accurate knowledge of the radio propagation channel is required. The performance of a
Radio Wave Propagation
65
wireless transmission system is indeed dictated by the propagation conditions between the transmitter and the receiver. These devices need to be designed in order to beneﬁt from the channel characteristics and to mitigate its negative eﬀects. In this chapter, the propagation channel characteristics have been presented in terms of power attenuation and multipath propagation. The spatial and frequency selectivity concepts as well as the Doppler eﬀect were introduced. Multipath propagation may be regarded as a ﬁlter causing a distortion and a temporal variation to the transmitted signal. A mathematical framework was presented in order to describe both deterministic and random channels. For this, the particular case of WSSUS channels has been thoroughly discussed. A number of parameters are available to characterize the propagation phenomena. We generally model the propagation loss with respect to frequency and distance, the channel impulse response’s general shape and its variations in terms of fast fading. The physical concepts and mathematical tools presented in this chapter will serve as a basis to describe the simulation and modeling techniques detailed in Chapters 4 and 5.
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Chapter 3
UWB Propagation Channel Sounding
3.1. Introduction Experimental characterization of the radio channel requires the analysis of a large number of propagation measurements. In order to set up a measurement campaign, a large number of sounding techniques are available [GUI 99]. Selecting one method or the other depends on the measurement environment, on the observed frequency band, and on the acquisition speed constraints. Sounding UWB channels raises some particular issues, which are developed in this chapter. The diﬀerent sounding methods for wideband channels are then exposed, by distinguishing between frequency domain and time domain techniques. In order to observe the UWB channel ﬂuctuations in more detail, it is necessary to employ a real time measurement technique. For this purpose, we present an advanced sounding technique, exploiting the performances of a wideband, singleinput multipleoutput (SIMO) sounder. The end of the chapter illustrates the diﬀerent channel sounding techniques by presenting a few measurement campaigns. First, the most signiﬁcant UWB channel measurement campaigns are listed. For each experiment, the equipment and the measurement conditions are described. We then present the setting up of a sounding campaign through a few examples. 3.2. Speciﬁcity of UWB channel sounding The purpose of channel sounding is to measure the impulse response h(t, τ ) linking the received signal s(t) to the transmitted signal e(t) through
68
UltraWideband Radio Propagation Channels
Delay (ns)
Relative magnitude (dB)
Sounder impulse response
Relative magnitude (dB)
Relative magnitude (dB)
a ﬁltering operation (see equation [2.9]). Such measurements characterize both the frequency selectivity and the time variability of the channel. In practice, there is no inverse convolution operation available, that would allow for the extraction of the impulse response from arbitrary signals e(t) and s(t). However, a number of techniques using speciﬁc exciting signals e(t) may be used, where the impulse response is obtained by processing the signals e(t) and s(t). The transmitted signal needs to fulﬁll some properties depending on the method used for calculating the impulse response. Channel impulse response over an infinite bandwidth
Delay (ns)
Convolution
Unresolved path
Measured impulse response
Delay (ns)
Figure 3.1. Time resolution of a wideband sounder
The main characteristics of a wideband sounder are the following: • Analyzed bandwidth: the analyzed bandwidth corresponds to the frequency band over which the impulse response is estimated. It generally corresponds to the bandwidth of the transmitted probe signal. Propagation channel measurements over bandwidths in the order of 100 MHz is well supported by the currently available sounders [GUI 99]. In the case of UWB signals, the bandwidth of several GHz may represent a challenge for channel measurements. • Time resolution: the time resolution characterizes the sounder’s capability to distinguish between two paths with very close delays. The impulse response estimated by the sounder corresponds to a convolution
UWB Propagation Channel Sounding
69
between the channel impulse response over an inﬁnite bandwidth and the sounder’s impulse response over the analyzed band (see Figure 3.1). The latter may be obtained by directly cableconnecting the transmitter and the receiver. The time resolution is generally deﬁned as half the width of the sounder impulse response peak. As an approximation, it may also be deﬁned as the inverse of the analyzed bandwidth. In order to avoid the shadowing of some paths by the side lobes of the sounder impulse response, a weighting window may be applied [HAR 78]. Such a window is selected as a compromise between the level of the side lobes and the width of the main lobe. • Maximum Doppler shift: when the propagation channel varies with time, we can measure its frequency dispersion by studying its Doppler spectrum (see section 2.4.4). For this purpose, the sounder needs to be able to quickly measure successive impulse responses. The measurement repetition duration ΔT (meas) is deﬁned as the duration separating two successive channel measurements. It encompasses the measurement acquisition duration t(meas) and some time for data processing and storage. It is then possible to measure a maximum absolute (meas) 1 . However, it should be noted that during the Doppler shift νmax = 2ΔT (meas) measurement of a single impulse response, the channel should be considered as static. Thus, a sounder capable of measuring the time varying channel needs to present a very low acquisition duration t(meas) . • Dynamic and length of the channel impulse response: the impulse response dynamic corresponds to the power ratio between the maximum impulse response and the noise level. A high dynamic allows the sounder to detect strongly attenuated paths. The length of the channel’s impulse response corresponds to the maximum delay that can be measured. In the case of the UWB propagation channel, the time resolution is generally high, due to the wide analyzed bandwidth. Consequently, developing UWB sounders with a low measurement duration is a challenging task. Some constraints are common in UWB channels, because of the wide measured frequency band. Over an analyzed bandwidth of several GHz, the behavior of the sounding equipment may vary strongly, and this needs to be accounted for in the measurement process. In particular, the antennas characteristics need to be stable across the measured frequency band. Narrowband antennas (dipoles, horns, etc.) and dispersive wideband antennas (spiral and logperiodical antennas, etc.) may thus not be used for this purpose [SCH 03a]. In practice, biconical, monoconical or planar UWB antennas are
70
UltraWideband Radio Propagation Channels
used. Some examples of these antennas are given in Figure 3.2. In any case, it is necessary to properly characterize the antennas used for the measurement [SIB 04]. The properties of some other pieces of equipment, such as cables or ampliﬁers, may also vary in frequency, and it is important to characterize each item accurately before the measurement. Monoconical antenna
Bowtie planar antenna
Ideal biconical antenna
Biconical antenna
Figure 3.2. UWB measurement antennas
Finally, a UWB receiver is sensitive to any radio emission at frequencies near or within the analyzed frequency band. Before the measurement starts, we need to make sure that the analyzed radio spectrum is free of any jamming from external systems. Outofband emissions, such as the GSM and WiFi signals, should also be ﬁltered. This can be done using a highpass ﬁlter. This ﬁlter should be taken into account during the calibration phase in order not to aﬀect the measured signal.
3.3. Measurement techniques for UWB channel sounding This section presents the main UWB radio channel measurement methods: frequency domain methods and time domain methods. For each method, the basic principles are presented, and the advantages and drawbacks of this method are detailed. We ﬁnally present an innovative hybrid method for the real time measurement of UWB channels.
UWB Propagation Channel Sounding
71
3.3.1. Frequency domain techniques The frequency domain method is the most commonly used UWB channel sounding technique, on account of its ease of implementation. It consists of sampling the channel transfer function T (f, t). This is done by transmitting a narrowband signal at a ﬁxed frequency, and by measuring the attenuation and the relative phase of the received signal [GUI 99]. In practice, the analyzed band is divided into N samples separated by a frequency step Δf (meas) . The impulse response is obtained using an inverse Fourier transform along the frequency axis. The obtained time resolution1 is: (meas)
Rt
=
1 N Δf (meas)
[3.1]
and the length of the channel impulse response is: (meas) τmax =
N −1 N Δf (meas)
[3.2]
3.3.1.1. Vector network analyzer The tool best suited to characterizing the channel using a frequency sweeping method is the vector network analyzer (VNA). This device is generally used to characterize high frequency quadripoles through the measurement of Sparameters. For the purpose of channel sounding, port 1 is connected to the transmitting antenna and port 2 is connected to the receiving antenna. The channel transfer function is derived from parameter S21 (f ). In general, a sine signal is used to sweep the analyzed band. The received signal is downconverted towards an intermediate frequency (IF), where it is passband ﬁltered around a ﬁxed frequency and analyzed. Hence, this device is capable of sweeping a very wide frequency band. Using narrowband ﬁlters at the receiver leads to a very good measurement dynamic, but also increases the overall measurement duration. In addition, the measurement of the transfer function phase requires a very good synchronization between the local oscillators (LO) at the transmitter and at the receiver. Figure 3.3 gives a schematic view of this device.
1. This time resolution is calculated without applying any frequency domain windowing.
72
UltraWideband Radio Propagation Channels
Propagation channel
Reference signal LO
LO
S21 measurement at IF Analysis and storage
Figure 3.3. Propagation measurement using a frequency domain sounder
This technique oﬀers advantages in terms of bandwidth and dynamic. As a result, it has been frequently used for the purpose of UWB channel sounding [GUN 00, OPS 01, GHA 02a, CHE 02, KEI 02, HOV 02, KUN 02a, GHA 03b, BUE 03, DAB 03, ALV 03, CHA 04, SCH 04, BAL 04a, KAR 04b, JAM 04, CHO 04b, CAS 04a, HAN 05]. However, the measurement duration is proportional to the number of measured frequency tones. For an analyzed bandwidth of several GHz, the measurement duration is in the order of 10 seconds. Hence, this technique cannot be used for time varying channels. When conducting VNAbased measurements, we should thus ensure that the environment remains static for the measurement duration. In addition, the distance from transmitter to receiver is limited to approximately 20 m, due to power attenuation in the feeding cables. 3.3.1.2. Chirp sounder A chirp sounder is a frequency domain method providing a solution to the issue of acquisition duration. Indeed, the sounding signal is no longer a pure sine wave, but a frequency chirp. Such a signal is generated using digital frequency synthesis components [COS 04].
UWB Propagation Channel Sounding
73
This frequency chirp may be characterized by the sequence duration Tc and the covered frequency range Bc , as illustrated in Figure 3.4. For a Bc Tc product larger than 100, over 98% of the energy is conﬁned in the analyzed bandwidth Bc . The channel impulse response may be calculated either by matched ﬁltering or using a heterodyne receiver and a lowpass ﬁlter. In the ﬁrst case, the impulse response of the matched ﬁlter corresponds to a time reversed version of the emitted chirp signal [ART 95]. The channel impulse response is directly obtained at the output of the matched ﬁlter. In the case of a heterodyne reception, the output of the lowpass ﬁlter provides the channel transfer function, represented on a compact frequency band. Its bandwidth depends on the parameters Bc and Tc [COS 04]. The signal acquisition may thus be performed at a reduced sampling frequency. However, generating a chirp signal with a bandwidth larger than a few hundred MHz is still a challenging task. For this reason, chirp sounders have not yet been used for UWB channel sounding.
Magnitude
Bc
Bc Magnitude
Tc
Frequency
Tc
Time
Time
> 98% energy Frequency
Figure 3.4. Chirp signal
3.3.2. Time domain techniques The common characteristic of all time domain techniques is the use of a wideband excitation signal at the emitter. This way, the receiver processes the whole frequency band simultaneously, which drastically reduces the measurement duration. This section presents the most common time domain methods, and in particular those that have been used for UWB channel sounding. 3.3.2.1. Pulsed techniques The pulsed technique is mathematically the most straightforward time domain measurement method. Indeed, if the transmitted signal e(t) corresponds to a Dirac pulse, the received signal s(t) is directly proportional to the channel impulse response h(t, τ ). However, such a Dirac pulse would present an inﬁnite ﬂat spectrum and is not physically implementable. In
74
UltraWideband Radio Propagation Channels
practice, we use pulse generators enabling the emission of short signals with a duration Δt in the order of 100 picoseconds. Denoting by ΠΔt (t) the emitted pulse, the received signal is given by: s(t) = h(t, τ ) ⊗ ΠΔt (t)
[3.3]
and represents a close approximation of the channel impulse response. At the receiver, a very fast acquisition of the signal is necessary. A digital sampling oscilloscope (DSO) is generally used, with sampling rates up to 20 Gsamples.s1 . Other types of receivers are also implemented: for instance, a correlation receiver has been developed by the company Time Domain [YAN 02]. This technique is particularly interesting for the measurement of UWB channels. Indeed, pulse generators capable of emitting directly in the FCC frequency band (3.1–10.6 GHz) are now available. Generally, UWB sounders based on this principle do not require an upconverter stage, which simpliﬁes the experimental setup. It should be noted, though, that the signals synthesized by the current pulse generators are limited to a maximum bandwidth of 1–2 GHz, with an upper frequency limited to 5 GHz. Hence, only parts of the FCC band may be measured using a direct pulse generator technique. In order to sound the upper part of the FCC spectrum, it is possible to add an upconverter stage, composed of an LO and a mixer at both the transmitter and the receiver. This solution is presented as a dashed line in Figure 3.5. The main advantage of the pulsed sounding technique is its low acquisition duration, the channel impulse response being recorded in real time. This technique should thus be selected for the measurement of space or time channel variations. However, this method also presents a number of drawbacks. Firstly, generating short duration pulses requires the ampliﬁers to deliver a high power directly followed by idle periods. The resulting low average power leads to a poor signal to noise ratio, and this method is not suited to large distances in NLOS conﬁgurations. Secondly, the power dynamic at the ampliﬁer stage does not allow for an accurate control of the pulse shape. Finally, this technique requires a perfect synchronization between the emitter and the receiver, which may be solved by connecting the terminals using a cable. The attractive simplicity of this solution led a number of scientists to use this method for UWB channel sounding, but with a sounded frequency band generally below the FCC band [WIN 97a, WIN 97b, GUN 00, OPS 01, CHE 02, YAN 02, TER 03, LI 03]. Two measurement programs have been established using this technique within the 3.1–10.6 GHz band [PEN 02, BUE 03].
UWB Propagation Channel Sounding
75
Propagation channel
Reference signal LO
LO
0° 90°
I Pulse generator
Synchronization
Q DSO
Analysis and storage
Figure 3.5. Propagation measurement using a pulsed technique
3.3.2.2. Correlation measurements A possible way to increase the signal to noise ratio at the receiver is to use the autocorrelation properties of pseudonoise (PN) sequences. Let Ree (t) and Res (t) respectively denote the input signal autocorrelation and the correlation function of the input signal e(t) with the output signal s(t). According to the convolution and correlation function properties, these functions are also linked by a convolution equation: Res (t) = h(t, τ ) ⊗ Ree (t)
[3.4]
If the input signal is close to a white noise, its autocorrelation function is close to a Dirac pulse. In this case, the identiﬁcation problem is similar to the one occurring in the pulsed technique. In practice, we use maximum length PN sequences, also called msequences. This corresponds to a good approximation of a colored noise with zero mean and limited bandwidth. An msequence may
76
UltraWideband Radio Propagation Channels
be generated using a shift register with m bits and an adder, which results in a sequence of length 2m − 1. Its autocorrelation function presents a theoretical dynamic of 20 log(2m − 1) dB. In the frequency domain, the −3 dB analyzed bandwidth is equal to the clock frequency fc used at the PN sequence generator. At the transmitter, a correlation sounder is thus composed of a PN sequence generator and an upconverter stage towards the working frequency. Typically, wideband correlation sounders transmit signals in the order of 100 MHz. For UWB channel sounding, the frequency band may be widened by increasing the clock frequency, but the sounder implementation becomes more complex. The University of Ilmenau, Germany, presented a MIMO UWB sounder working in the 0.8–5 GHz frequency band [SAC 02]. For this purpose, an integrated circuit shift register has been speciﬁcally developed. At the receiver, once the received signal has been downconverted in baseband, several acquisition techniques may be used. The ﬁrst consists of using an analog ﬁlter matching the transmitted PN sequence. This principle enables the measurement of the impulse response in real time. However, the analyzed bandwidth is limited by the passband of the ﬁlter, and the measurement dynamic is low. Another technique consists of sampling the received signal and computing the convolution function using digital ﬁlters. The maximum analyzed bandwidth is then limited by the speed of the analogtodigital converters. This solution is presented in Figure 3.6. It has been used by the University of Rome “Tor Vergata”, Italy, to sound the 3.6–6 GHz band [DUR 04]. The receiver was simply composed of a DSO. Time domain correlation sounders present the advantages of a high impulse response dynamic, and a possible functioning in real time. Their implementation is however complex, and the quality of the measured impulse response largely depends on the performance of the sounder components. In particular, an accurate synchronization between the transmitter and the receiver is necessary. This may be achieved using a stable reference (e.g. a rubidium oscillator) or a direct cable connection between the two terminals. Sliding correlation technique The sliding correlation sounder is a similar technique that proceeds the convolution operation in the analog domain while increasing the measurement dynamic. The convolution is performed using a replica of the original PN sequence with a clock frequency fc2 slightly diﬀerent from the initial clock frequency fc1 . This leads to a time domain sliding of the PN sequences with respect to each other. Hence, using an integrator, one point of the correlation
UWB Propagation Channel Sounding
77
Propagation channel
Reference signal 0° OL
OL
90°
I Arbitrary sequence generator (e.g. shift register)
Synchronization
Analysis and storage
Q
Fast acquisition
Correlation or inversion
Figure 3.6. Propagation measurement using a PN sequence. The illustrated acquisition technique consists of digitizing the received signal prior to convolution processing
function between the transmitted signal and the received signal is calculated at each instant. c1 in The obtained impulse response is expanded by a ratio k = fc2f−f c1 the time domain and compressed by a ratio k in the frequency domain. This compression enables the use of narrower ﬁlters at the receiver, which improves the dynamic. In addition, a lower sampling rate may be used for the analogtodigital conversion.
This technique has been used by the University of Kyoto Sangyo (Japan) to develop a UWB sounder operating in the 5.5–8.5 GHz frequency band [TAK 01]. The main drawback of this technique is the duration of the impulse response calculation, which precludes any Doppler analysis of fast time varying channels.
78
UltraWideband Radio Propagation Channels
3.3.2.3. Inversion techniques Inversion techniques are based on the transmission of an arbitrary sequence and are thus similar to correlation sounding techniques (see Figure 3.6). The main diﬀerence lies in the impulse response estimation, which is performed using an estimation ﬁlter. Filtering is generally applied in the digital domain after the acquisition of the analog signal. The used ﬁlter minimizes the squared error between the impulse response and the estimated response. In particular, such a ﬁlter accounts for the imperfections of the measurement device. The inversion technique is notably used in the sounder developed at France Telecom Research and Development [CON 03]. It uses a Wiener ﬁlter, with a frequency spectrum given by: GWiener (f ) =
E ∗ (f )I ∗ (f ) E(f )I(f )2 + α
[3.5]
where E(f ) represents the transmitted signal spectrum, I(f ) represents the sounder response spectrum, and α is a nonzero constant linked to the signal to noise ratio [BAR 95]. As a main advantage, the inversion technique improves the impulse response calculation by accounting for the imperfections of the sounder components. The used PN sequence is optimized in order to spread over a wide band with a relatively constant envelope. The limitations of this technique are directly linked to the performances of the digitaltoanalog and analogtodigital converters. In particular, the analyzed bandwidth is in the order of a few hundred MHz. For the sake of comparison, Table 3.1 presents the main advantages and drawbacks of the presented frequency domain and time domain techniques. 3.3.3. Multipleband time domain sounder for dynamic channels Classical frequency or time domain sounding techniques are not intrinsically well adapted to the measurement of time varying UWB channels. On the one hand, frequency domain techniques lead to a long measurement duration. On the other hand, time domain techniques are fast enough to enable the measurement of time varying channels, but their implementation within the FCC band is challenging. This section presents a hybrid solution consisting of using a wideband time domain sounder and successively measuring several adjacent partial bands. This principle was initially exploited by the University of Kassel (Germany). They developed a sounder capable of measuring a channel of 600 MHz divided into 10 partial bands of 60 MHz, with a measurement
Frequency
Time
signiﬁcant
Correlation
signiﬁcant
low
Pulsed
Inversion
signiﬁcant
Chirp
high
very high
VNA
Sliding correlation
Dynamic
Technique easy
complex
complex
easy
For practically implemented solutions.
500 MHz–1 GHz (limited complex by DAC and acquisition)
1–3 GHz (limited by clock frequency)
1–5 GHz (limited by clock frequency and acquisition)
1–2 GHz (limited by pulse generator)
no experiment
marginal
marginal
frequent out of FCC band
no experiment
frequent (static channel)
Implemen Use for UWB tation sounding
100–300 MHz (limited by complex signal generation)
10 GHz and above (limited by the measurement duration)
Maximum bandwidth a
Table 3.1. Comparison of the diﬀerent sounding techniques
a
possible
possible for low Doppler spreads
possible
possible
possible for low Doppler spreads
impossible
Doppler analysis
UWB Propagation Channel Sounding 79
80
UltraWideband Radio Propagation Channels
repetition duration of 300 μs [KAT 00]. In the following, we present the diﬀerent issues linked to the development of such a sounder, and we illustrate how it can be implemented on the basis of a wideband SIMO sounder [PAJ 03]. 3.3.3.1. Principle of multipleband time domain sounding The technique of multipleband time domain sounding exploits the performance of a time domain sounder, such as the sounder developed by France Telecom Research & Development. In its standard version, this sounder allows for the measurement of 250 MHz frequency bands. At the upconverter stage, an LO signal is responsible for the upconversion of the transmitted sequence towards the analyzed band. By appropriately driving this LO signal, it is possible to sound several adjacent bands. The diﬀerent challenges are the following: • Analyzed band: the number of measured partial bands determines the width of the global analyzed band. At both the transmitter the receiver, the global analyzed band needs to be ﬁltered and cleared from any interfering signal, such as the LO signal and its harmonics. • Measurement repetition duration: in an indoor conﬁguration, the maximum speed of mobile terminals is about 2 m.s−1 . Considering an extreme case, i.e. a mobile terminal and a maximum frequency of 10.6 GHz, the maximum Doppler shift is λv = 70 Hz. This maximum Doppler shift increases to 2v λ = 140 Hz if we consider mobile scatterers, taking into account the paths with one reﬂection as signiﬁcant paths. Thus, to characterize time varying channels, the equipment has to sample the channel every 3.5 ms. This time includes the necessary switching from one band to another, which needs to be synchronized with the transmitted sequence. • Calibration: when measuring adjacent bands, a simple process needs to be set up for an accurate calibration of the sounder in terms of phase, magnitude and absolute delay, for each of the adjacent bands. • Measurement dynamic: when working as a multipleband sounder, the dynamic performance of the original wideband sounder needs to be preserved, in order to obtain a reliable result. These diﬀerent issues are similar to those occurring when characterizing the wideband channel with multiple antennas at the receiver, in a SIMO conﬁguration. A SIMO sounder may thus be easily adapted to perform multipleband time domain sounding. For a better understanding, the next section reviews the general structure of the SIMO sounder available at France Telecom. A further description of this sounder is available in [CON 03].
UWB Propagation Channel Sounding
81
3.3.3.2. Description of the SIMO channel sounder The sounder developed at France Telecom Research & Development is a time domain sounder allowing for radio channel measurement over diﬀerent frequency bands between 2 GHz and 60 GHz. The sounding method is the Wiener inversion technique, applied on a PN sequence exhibiting a ﬂat spectrum over the analyzed bandwidth. For frequency bands below 17 GHz, a multiple sensor mode enables the simultaneous measurement of the radio channel over 10 receiving antennas. In terms of performance, the sensitivity of this sounder is about −85 dBm. The impulse response dynamic depends on the length of the PN sequence, the signal to noise ratio at the sounder input, and the oscillator phase noise. For a received signal of about −60 dBm over a band of 250 MHz around 5 GHz and a sequence of 8192 samples, the measurement dynamic is larger than 40 dB. Figure 3.7 presents a simpliﬁed block diagram of the receiver. To repeat the measurement as quickly as possible, a fast switch selects the following antenna every other sequence. The ﬁrst PN sequence is used for the measurement, while the antennas switch during the next PN sequence. The signal is then ampliﬁed using a wideband (3–18 GHz) low noise ampliﬁer (LNA). It is then passband ﬁltered around the RF carrier frequency. A ﬁrst downconverter stage driven by an external frequency synthesizer displaces the signal around an IF at 1.5 GHz. Automatic gain control (AGC) is then applied using variable attenuators to compensate for the power variations in the received signal. AGC is regularly performed, before the measurement of the signal received by the diﬀerent sensors. Thus, the AGC duration limits the measurement repetition duration. In a traditional conﬁguration with 10 antennas, the minimum repetition duration is about 1.2 ms. A second downconverter stage displaces the signal around a second IF at 250 MHz. The received signal is then sampled using an analogtodigital converter (ADC) at a sampling rate of 1 Gsample.s−1 . We may note that the transmitter and the receiver use the same IF at 1.5 GHz. Thus, the LO frequency is the same for the external synthesizers at the transmitter and the receiver. All LO are synchronized using a reference rubidium at 10 MHz. 3.3.3.3. Extension towards UWB In order to perform measurements over adjacent frequency bands, it is necessary to modify the LO frequency at the transmitter and at the receiver. Traditional oscillators present a locking time in the order of 10–50 ms. This transition duration is mainly due to the positioning duration of the internal phase locked loop, and cannot be reduced. Directly driving a single external synthesizer thus leads to a total measurement duration in the order of 40–200 ms for an analyzed band of 1 GHz. Taking into account the constraints
UltraWideband Radio Propagation Channels
Fast switch 10 1
82
RF
LNA
A External synthesizer
AGC
1.5
250 D MHz
GHz
LO
Figure 3.7. Receiver block diagram
linked to real time measurements (see section 3.3.3.1), this solution cannot be used. The extension of the SIMO sounder towards a real time UWB sounder exploits the duality between multipleinput measurements and multipleband measurements. The main idea is to reuse the fast antenna switching module, in order to switch between the carrier frequencies of each partial band to be measured. A practical implementation of this concept is presented in Figure 3.8. As can be seen, the input connector is now directly connected with the LNA, and a single antenna is available. The sounder is thus in a singleinput singleoutput (SISO) measurement conﬁguration. The fast switching module is used to sequentially feed the mixer of the ﬁrst downconverter stage with one of the 10 LO signals (f1 to f10 ) in turn. These signals are generated by external synthesizers and tuned so that the receiver actually sweeps the selected partial bands. As each synthesizer is locked at a ﬁxed frequency, no delay is experienced for the locking of the LO frequency. In this conﬁguration, the sounder is capable of sweeping up to 10 partial bands of 250 MHz each. Hence, UWB measurements are theoretically possible up to a bandwidth of 2.5 GHz. On account of its development concept, this sounder fulﬁlls most of the required criteria for real time UWB channel measurement. Indeed, the multipleinput architecture is directly used for the measurement over multiple bands. For the calibration of each partial band in terms of phase, magnitude
UWB Propagation Channel Sounding
External synthesizers
f10 Towards the emitter upconverter
Fast switch 10 1
f1
LNA
A
83
AGC
1.5
250 D MHz
GHz
LO
Figure 3.8. Block diagram of the modiﬁed receiver with UWB extension
and delay, the procedure used in the SIMO conﬁguration is still valid. In addition, regarding the fast switching of partial bands, the overall switching time through all partial bands may be as low as 1.2 ms, which allows for the measurement of channel variations with a maximum Doppler shift of 416 Hz. In the original version of the SIMO sounder, the ﬁlter preceding the ﬁrst downconverter stage was only a few hundred MHz wide to reject undesirable signals and reduce the noise level. The same type of ﬁltering was performed at the emitter. In the UWB conﬁguration, however, the sounded frequency band needs to be ﬁltered by one single passband ﬁlter, in order to avoid unnecessary ﬁlter switching. In its current version, the UWB sounder is equipped with ﬁlters of 1 GHz bandwidth. Unlike the SIMO channel sounding case, UWB measurements using the sweeping method necessitates a periodical modiﬁcation of the emitted signal central frequency. For this reason, the frequency switch performed at the receiver and at the transmitter need to be perfectly synchronized. To solve this problem, the same LO signal may be used to feed both the receiver downconverter and the transmitter upconverter stages. The main drawback of this solution is that the transmitter and the receiver need to be cable connected. However, in most indoor conﬁgurations, this connection is realizable using a cable of acceptable length. Advantageously, this solution allows us to use only one set of external synthesizer, which considerably reduces the global cost of the equipment.
84
UltraWideband Radio Propagation Channels
3.3.3.4. Experimental validation The modiﬁcations presented above have been applied to the sounder developed by France Telecom for its extension to real time UWB measurements. To sweep the 5–6 GHz frequency band, 5 external synthesizers have been connected to the fast switching module. In this conﬁguration, the impulse response dynamic was measured at 40 dB. Using 5 partial bands, the minimum measurement repetition duration is about 1 ms and the actual measurement duration is 20.5 μs. With its 256 MB memory, the acquisition card is able to sound the time varying UWB channel for a duration of 80 s in standard conditions (bandwidth of 1 GHz and observable Doppler spread of 150 Hz). Figure 3.9 presents the results obtained during a real time radio channel measurement in a dynamic environment. During this experiment, the transmitting antenna was ﬁxed and the receiving antenna was held by a moving person. This conﬁguration corresponds to the practical situation where a user with a mobile terminal walks in the proximity of a ﬁxed access point. 0 (a) 2
5
(b)
Distancedelay (m)
6
15 (c)
8
20
10
25
(d)
12 30 14
Relative attenuation (dB)
10
4
35 16
(e) 40
18 0
2
4
6 Time (s)
8
10
12
45
Figure 3.9. Time varying impulse response in a dynamic environment
The impulse response represented in Figure 3.9 corresponds to a 12.5 s record of the signal obtained at the receiving antenna moving during the experiment. Acquisitions were performed every 10 ms. For ease of
UWB Propagation Channel Sounding
85
interpretation, the path delay on the yaxis has been converted to path length in meters. In the ﬁrst part of the trajectory (between t = 0 s and t = 7 s), the person is moving towards the emitter antenna, reducing its relative distance from 6 m to 2 m. Hence, we can observe a main path with increasing power (a). In the second part of the trajectory (between t = 7 s and t = 11 s), the person is moving away from the transmitter antenna, partially obstructing the line of sight. This explains the shadowing experienced by the shortest path (b). In this part of the curve, three other main paths are observable, one with increasing length (c) and two with decreasing lengths (d, e). These two last paths might correspond to echoes transmitted via a reﬂection on a wall opposite the transmitter location. This feasibility study shows that it is possible to develop the UWB channel sounder by applying minor modiﬁcations to a wideband SIMO sounder. By exploiting the duality between multiple inputs and multiple bands, most of the SIMOcapable sounders could be extended towards UWB channel sounding. The prototype presented here allows for the measurement of UWB channels over a band of 1 GHz with a dynamic of 40 dB. 3.4. UWB measurement campaigns 3.4.1. Overview of UWB measurement campaigns A number of UWB radio channel measurement campaigns were cited in this chapter. These diverse campaigns used diﬀerent sounding techniques, but also diﬀer in the measured frequency band, the type of environment, the antennas used and the measurement setup. In order to compare the analysis results from these experiments, it is necessary to recall the conditions of each campaign. Table 3.2 summarizes the inventoried UWB measurement campaigns and provides some useful information for each of them. Among the ﬁrst UWB studies, the UltRaLab laboratory from the University of South California undertook two measurement programs in 1997, but most UWB experiments started from the year 2000 onwards. Most measurement programs used the VNA frequency domain method. Other experimental setups are based on a time domain method, using pulses or PN sequences. Diﬀerent indoor environments, such as the oﬃce or residential environments, and some outdoor environments (forest, urban) were sounded. We may also note some original experiments, on a metal ship or in an industrial environment. Finally, regarding the analyzed band, only seven measurement campaigns cover the entire FCC band [KUN 02a, BUE 03, ALV 03, CAS 04c, KAR 04b, HAN 05, PAG 06b]. Within the FCC band, three experiments only permitted real time UWB measurements [PEN 02, CAS 03, PAG 06a].
Pulse generator and DSO
Pulse generator and DSO; VNA; Pulse generator and correlation receiver
Pulse generator and DSO; VNA
1997 [WIN 97a, WIN 97b]
UltRaLab
UltRaLab SS 2000 [GUN 00] Curtis
Stanford University
Pulse generator and DSO; VNA
Intel Labs
2002 [CHE 02]
VNA
AT&T Labs  2002 [GHA 02a, MIT GHA 03c]
2001 [OPS 01]
Sounding method
Measurement Date Reference campaign
Spatial conﬁguration
Oﬃce; laboratory
Metal ship
LOS/ — NLOS
LOS/ — NLOS
LOS/ — NLOS
Forest; oﬃce LOS/ 49 locations NLOS on a 90 cm × 90 cm grid
4.375– Residential 5.625 GHz, with 3.1 MHz step
1–1.8 GHz, with 2 MHz step
700–1.8 GHz, with 1 MHz step
Width >1 GHz
Analyzed band Environment Link
Biconical 2–8 GHz, with Residential; LOS/ Two antennas (MEC) 3.75 MHz step oﬃce NLOS perpendicular and discone positioners antennas (length 25 cm) (Pulsicom with 20 Technol.) locations
Monoconical omnidirectional antennas
Wideband antennas
Patch diamond dipoles
Patch diamond dipoles
Antennas
86 UltraWideband Radio Propagation Channels
VNA
VNA
VNA
VNA
2002 [YAN 02]
2002 [KEI 02, KEI 03]
Time Domain Corporation (correlator)
UCAN CEA LETI
Ultrawaves  2002 [HOV 02] Oulu University
Whyless.com 2002 [KUN 02a]
AT&T WINLAB
2003 [GHA 03b]
Pulse generator and DSO
2002 [PEN 02]
Time Domain Corporation (DSO)
Pulse generator and correlation receiver [WIT 99]
Sounding method
Measurement Date Reference campaign
Analyzed band Environment Link
Spatial conﬁguration
NLOS —
1–11 GHz, Oﬃce with 6.25 MHz step
LOS/ 4500 locations NLOS on a 149 cm × 29 cm grid
LOS/ — NLOS
Omnidirectional 2–8 GHz, with Residential; LOS/ 25 locations monoconical 3.75 MHz step shopping NLOS on a 20 cm × antennas center 20 cm grid
Biconical antennas (ETH Z¨ urich)
Omnidirectional 2–8 GHz, with Oﬃce monoconical 3.75 MHz step antennas (CMA118/A)
Omnidirectional 2–3 GHz, with Residential; LOS/ 100 locations monoconical 2.5 MHz step oﬃce NLOS on a 90 cm × antennas 90 cm grid (CMA118/A)
Omnidirectional Width Oﬃce antennas >1.5 GHz around 2 GHz
Omnidirectional 3–5 GHz, with Residential; LOS/ — antennas 3.75 MHz step oﬃce NLOS
Antennas
UWB Propagation Channel Sounding 87
Pulse generator and DSO
2003 [TER 03]
2003 [PAG 03]
2003 [LI 03]
CNAM Thales
France Telecom R&D
Hong Kong University
VNA
2003 [DAB 03]
2003 [ALV 03]
New Jersey Instit. of Tech.
UCAN Cantabria University
VNA
Pulse generator and DSO; VNA
NETEX 2003 [BUE 03, Virginia Tech BAY 04]
Pulse generator and DSO
VNA
Sounding method
Measurement Date Reference campaign Corridor
Oﬃce; laboratory
Oﬃce
Omnidirectional 1–13 GHz, Oﬃce; monoconical with 2.5 MHz laboratory antennas step (EM6865)
Omnidirectional 2–6 GHz monoconical antennas and logperiodical antennas
Omnidirectional 100–12 GHz biconical antennas and TEM horns
Oﬃce; laboratory; meeting room
4–6 GHz, with Oﬃce 2 MHz step
0–3 GHz
—
Spatial conﬁguration
—
LOS/ 9 locations on NLOS a 6 cm × 6 cm grid
LOS
LOS/ 49 locations NLOS on a 90 cm × 90 cm grid
LOS/ xpositioner NLOS (length 120 cm) with 61 locations
LOS/ 60 locations NLOS on a rotating arm (27 cm radius)
LOS
Analyzed band Environment Link
Omnidirectional 1.2–1.8 GHz dipole
Patch omnidirectional antennas
Wideband antennas
Antennas
88 UltraWideband Radio Propagation Channels
PN sequence and DSO; VNA; PN sequence and correlation receiver
VNA
Ultrawaves  2003 [CAS 03, Rome Tor CAS 04c] Vergata University
2004 [CHA 04]
2004 [SCH 04]
2004 [BAL 04a]
2004 [KAR 04b]
ChiaoTung University
ETH Z¨ urich
Instit. for Infocomm Research
Lund University
VNA
VNA
VNA
Sounding method
Measurement Date Reference campaign Oﬃce; laboratory
2–8 GHz, with Corridor; 1.875 MHz lobby step
3–5 GHz, with Oﬃce 2.5 MHz step
3.6–6 GHz; 2–12 GHz, with 5 MHz step
Omnidirectional 3.1–10.6 GHz, Industrial monoconical with 6 MHz antennas step
Spatial conﬁguration
LOS/ Two NLOS xpositioners (transmitter/ receiver) with 7 locations along 30 cm
LOS/ 9 or 49 NLOS locations on a 10 cm × 10 cm or 30 cm × 30 cm grid
LOS/ 45 locations NLOS on a 28 cm × 56 cm grid
LOS/ 64 locations NLOS on a 26 cm × 26 cm grid
LOS/ 625 locations NLOS on a 48 cm × 48 cm grid; 73 positions on a nonuniform grid
Analyzed band Environment Link
Omnidirectional 3–6 GHz, with Urban; conical antennas 1.875 MHz oﬃce step
Planar omnidirectional antennas (Skycross)
Not communicated
Omnidirectional monoconical antennas (EM6865); prototype antennas (ENSTA)
Antennas
UWB Propagation Channel Sounding 89
VNA
2004 [JAM 04]
2004 [CHO 04b]
Oulu University
Samsung
VNA
2005 [PAG 06b]
2005 [PAG 06a]
France Telecom R&D
France Telecom R&D
Residential
LOS/ 25 locations NLOS on a 60 cm × 60 cm grid
Omnidirectional 4–5 GHz, with Oﬃce monoconical 2 MHz step corridor antennas
LOS
—
LOS/ 90 locations NLOS on a rotating arm (20 cm radius)
3.1–10.6 GHz, Residential; LOS/ Up to 10 × 10 with 10 MHz oﬃce NLOS × 7 locations step at transmitter and receiver
3–10 GHz, with 4.375 MHz step
Omnidirectional 3.1–11.1 GHz, Oﬃce monoconical with 2 MHz antennas step
Monopoles and biconical antennas
Planar dipoles
Spatial conﬁguration
LOS/ xpositioner quasi with 7.9 mm NLOS step
Analyzed band Environment Link
Omnidirectional 0.5–10 GHz, Oﬃce monoconical with 6.25 MHz antennas step (CMA118/A)
Antennas
Table 3.2. UWB propagation channel measurement campaigns
Multipleband time domain sounder
VNA
Tokyo Instit. 2004 [TSU 04, of Tech. HAN 05]
VNA
Sounding method
Measurement Date Reference campaign
90 UltraWideband Radio Propagation Channels
UWB Propagation Channel Sounding
91
3.4.2. Illustration of channel sounding experiments The previous study of the diﬀerent UWB channel sounding techniques shows that the frequency domain technique using a VNA is the only available method that enables a full characterization over the whole FCC band. In the following, two measurement campaigns using this method are presented. The ﬁrst campaign covers the 3.1–10.6 GHz band [PAG 06b], and the second one covers the 2–6 GHz band. A third campaign is then described, presenting real time measurements in the 4–5 GHz band. 3.4.2.1. Static measurement campaign over the 3.1–10.6 GHz band The measurement system used for the static measurement campaign over the 3.1–10.6 GHz band is presented in Figure 3.10.
Propagation channel
Rotating arm
Webcam
Rotating arm driver
USB GPIB
RS 232
Port 1
Port 2
VNA HP8510C Driving and storage
Figure 3.10. Equipment setup for the static UWB sounding over the 3.1–10.6 GHz band
Measurements were taken using a VNA (see Figure 3.11(a)) in the frequency band 3.1–11.1 GHz with a frequency step Δf (meas) of 2 MHz. The full acquisition process consists of measuring 4005 frequency tones in a duration of about 15 seconds. Using a Hanning window, side lobes may be reduced to −32 dB. This compromise corresponds to an impulse response (meas) of 0.25 ns. resolution Rt
92
UltraWideband Radio Propagation Channels
(a)
(b)
Figure 3.11. Measurement equipment: (a) HP8510C sounder and (b) rotating arm
In order to estimate the local PDP and assess the signal spatial ﬂuctuations, a rotating arm with radius 20 cm was used to measure the radio channel at 90 diﬀerent locations (see Figure 3.11(b)). This conﬁguration corresponds to a circular path of 45λ and to a distance between two successive sensors below λ2 at the maximum frequency of 10.6 GHz. Measurements were performed using two CMA118/A antennas from the company Antenna Research Associates. These antennas are monoconical antennas with a metallic ground plane. Their radiation pattern is omnidirectional in the azimuth plane and a standing wave ratio (SWR) lower than 2 in the 1–18 GHz frequency band. It should be noted that the radiation pattern of these antennas strongly varies, especially in the elevation plane, as the frequency increases several octaves. Their complex gain has thus been characterized in 3D in the 1–10 GHz band with a 1 GHz step. Figure 3.12 illustrates the antenna characterization process.
UWB Propagation Channel Sounding
93
Figure 3.12. Measurement antennas. CMA 118/A antennas (from the ARA company) during their characterization
This sounding equipment was complemented with up to three wideband power ampliﬁers. The sounder calibration consists of a measurement sample taken while the receiver and the transmitter are directly cable connected. In order to achieve an accurate time reference, all cables involved in the measurement process need to be taken into account at the calibration stage. As active components, the ampliﬁers are not included in this measurement. Instead, they are characterized separately and their transfer function is subtracted from the measurements at the digital postprocessing stage.
Measurements were taken in an indoor oﬃce environment. External walls are made of brick and concrete, and internal walls consist of thinner plaster and plastic boards. Two typical radio access conﬁgurations have been studied. In both cases, the transmitter was situated at a height of 1.40 m and was considered as the mobile terminal. The receiver was considered as the access point and was ﬁrst placed in a meeting room at a height of 2.19 m, and then in a corridor at a height of 2.45 m. Figure 3.13 presents these two conﬁgurations. In both cases, LOS and NLOS conﬁgurations were assessed. The transmitterreceiver distance varied between 1 m and 20 m. Over the whole measurement campaign, the rotating arm was placed at more than 120 diﬀerent locations, which corresponds to a set of over 10,000 UWB impulse responses available for statistical characterization. The analysis of these measurements is presented in Chapter 5.
Meeting room configuration
Figure 3.13. Measurement locations during the UWB sounding campaign over the 3.1–10.6 GHz band. Black squares represent ﬁxed locations (receiver) and white circles correspond to the rotating arm locations (transmitter)
Corridor configuration
4m
94 UltraWideband Radio Propagation Channels
UWB Propagation Channel Sounding
95
3.4.2.2. Static measurement campaign over the 2–6 GHz band The schematic description of the equipment used for the static measurement campaign over the 2–6 GHz band is presented in Figure 3.14. Measurements were performed using the VNA HP8753 D over the frequency band 2–6 GHz with a frequency step Δf mes of 2.5 MHz. The acquisition process for a full measurement consists of the measurement of 1601 frequency tones in a duration of 4 seconds. The CMA 118/A antennas have also been used for this measurement. The measurement did not require the use of power ampliﬁers at the transmitter or at the receiver. Indeed, the link budget without the ampliﬁers was high enough to support transmitterreceiver distances up to 12 m. CMA – 118/A MATLAB/SCILAB Post processing Analysis
Tx (Port 1)
Cable 20 m
Vector Network Analyzer
Tx Ant
Propagation channel
HP 8753 D PC Labview
GPIB
Cable 20 m
Rx Ant
Rx (Port 2)
CMA – 118/A
Figure 3.14. Equipment setup for the static UWB sounding over the 2–6 GHz band
The environments sounded during these measurements are residential houses: the ﬁrst house is a modern building and the second one is an older building. The external walls of the ﬁrst house consist of concrete blocks, bricks and insulation material made of glass wool. The external walls of the older house do not incorporate any insulating material. Internal walls of both houses are made of brick and plasterboards. Both dwellings are twoﬂoor houses. However, measurements have been performed at the ground level only. The transmitter and the receiver are placed at the same height of 1.40 m. Figure 3.15(a) represents the ﬁrst house with LOS measurement locations. Figures 3.15(b) and 3.15(c) represent the second house with both LOS and NLOS measurement locations. In these two environments, the number of measurements is respectively 89 and 230. 3.4.2.3. Dynamic measurement campaign over the 4–5 GHz band Finally, in order to study the eﬀect that mobile people have on the UWB channel, we present a real time measurement campaign performed over a bandwidth of 1 GHz [PAG 06a].
96
UltraWideband Radio Propagation Channels
(a)
(b)
(c)
Figure 3.15. Measurement locations during the UWB sounding campaign over the 2–6 GHz band. Squares represent ﬁxed locations (Tx: transceiver) and circles represent mobile locations (receiver): (a) modern house; (b) old house in LOS; and (c) NLOS
UWB Propagation Channel Sounding
97
A few channel sounders only are currently capable of measuring the UWB channel in real time. This experimental study has been performed using the UWB sounder presented in section 3.3.3. The channel impulse response was measured every 10 ms, which enables the measurement of Doppler shifts up to 50 MHz. The analyzed band extends from 4 GHz to 5 GHz, with a resolution of 2 MHz, which corresponds to a maximum delay of 500 ns. At the transmitter and the receiver, CMA118/A antennas were used, as presented in section 3.4.2.1. Rx antenna Rx
B
Tx B
F People motion
F
Tx antenna
Figure 3.16. Environment of the real time experiment. F: forward movement, B: backward movement, Tx: transmitter, Rx: receiver
The experiment took place in a typical indoor oﬃce environment. All measurements were taken near a bend of the building’s main hallway, as depicted in Figure 3.16. The receiving antennas was ﬁxed on a wall at a height of 2.10 m. The transmitting antenna was placed in the middle of the corridor at a distance of 11 m from the receiving antenna and at a height of 1.35 m. In order to assess the time variations of the UWB channel, measurements were taken over the ﬁxed radio link as groups of people walked towards the end of the hallway and back. During this displacement, people occasionally obstructed the LOS path and other main paths of the channel impulse response. The number of people within each group varied from 1 to 12. This experiment corresponds to a practical situation where the user of a WLAN is standing in a crowd of people, in a subway corridor for instance. The measurement campaign consisted of a collection of 27 records of the time varying UWB radio channel, containing about 3,000 impulse responses each. Chapter 5 presents a statistical analysis of this experimental data.
98
UltraWideband Radio Propagation Channels
3.5. Conclusion In this chapter, the diﬀerent channel sounding techniques adapted to UWB measurements have been presented. Frequency domain methods use either a vector network analyzer or a chirp sounder. Time domain methods include pulsed techniques, correlation measurements and inversion techniques. Due to the wide analyzed bandwidth, the vector network analyzer method seems to be the most appropriate technique for static UWB channel sounding. However, its main limitation lies in its long measurement duration. Several seconds are indeed required to measure the 3.1–10.6 GHz band. For the acquisition of real time UWB channel measurements, an innovative sounding technique based on the extension of a wideband SIMO sounder has been presented. This adaptation corresponds to minor modiﬁcations of the original equipment and allows for real time measurements over a 1 GHz band. A literature survey made it possible to inventory about 20 UWB measurement campaigns since 1997. Most of these campaigns used a VNAbased frequency domain method. The reported campaigns cover the indoor residential and oﬃce environment, as well as some outdoor environments. Finally, the experimental setup of a typical sounding campaign has been illustrated through three examples: two measurement campaigns using a VNA over the 3–10.6 GHz and 2–6 GHz frequency bands respectively, and a series of real time measurements performed over the 4–5 GHz band using a multipleband time domain sounder.
Chapter 4
Deterministic Modeling of the UWB Channel
4.1. Introduction Deterministic models are often used as site speciﬁc models allowing us to realistically predict signal propagation in a given environment using simulation software. Currently, little simulation software of this type is proposed for UWB in comparison to narrowband systems, where they are widely used to determine radio system deployment. The use of classical deterministic modeling software for UWB needs a particular adjustment in order to cover the entire frequency band of impulse signals. Concerning the total calculation ﬁeld, it is mandatory to consider not only the power loss amplitude but also the phase information in order to easily reconstruct in the time domain the received signal corresponding to the modeled UWB link. This chapter describes the UWB deterministic modeling. After the presentation of classical techniques used for deterministic modeling, the speciﬁcities of UWB applications are described. Then, the diﬀerent deterministic models proposed in the literature for UWB signal propagation are presented. Finally, theoretical formalisms are detailed and compared to measurement results in order to illustrate the deterministic modeling in the UWB context. 4.2. Overview of deterministic modeling Propagation models considered to be deterministic are obtained from simulations made in simpliﬁed propagation environments. They are based on
100
UltraWideband Radio Propagation Channels
electromagnetic wave propagation theory. Their use requires a good knowledge on the propagation environment and allows us to obtain precise as well as accurate predictions of signal propagation in the channel corresponding to the considered environment. For a theoretical point of view, the wave propagation characteristics can be calculated by solving Maxwell equations. However, this requires us to make complex mathematical operations and to use powerful numerical calculations. The most frequently used approaches are based on frequency diﬀerence in time domain (FDTD) techniques, methods of moments (MoM) or ray techniques. 4.2.1. FDTD based approach The FDTD based approach allows us to obtain a complete cartography of the ﬁeld on all points of a given map. It is typically used for electromagnetic ﬁeld coverage in a given environment [LAU 95, SCH 97b, KON 99]. The FDTD used for modeling consists of solving Maxwell equations using regular spatial discretization in time. All derivations are simpliﬁed by algebraic systems of equation. The studied volume is discretized in various elementary cells, the size of which is related to wavelength of typically λ/10 or λ/15. The solving method ﬁxes some restrictions in term of the considered cells sizes and time grid in order to ensure computation stability. The studied environment is limited by absorbing walls limiting the reﬂection phenomena on the boundaries [IBA 00]. These approaches demand huge memory spaces to obtain the solutions in all points of the considered environment and to update all iterative ﬁeld calculations for the diﬀerent instants. This approach is often used for small environments with respect to the wavelength or complementary to other approaches [YIN 00]. 4.2.2. MoM based approach Similarly to the FDTD approach, the MoM based approach is a numerical method which requires a large numerical memory for ﬁeld calculation and radio coverage in a given environment [DEB 96, YAN 99]. The MoM is based on the use of an integral form of Maxwell equations. The integral system is transformed by an impedance matrix discretization which represents the interactions between elementary cells of the environment. The accuracy of the solutions obtained with this approach depends on the size of the considered cells.
Deterministic Modeling of the UWB Channel
101
This approach is used when the structure size is roughly a few wavelengths. It can also be considered complementary to other approaches like the ray based approaches in order to obtain hybrid models. Nevertheless, the MoM based approach is mainly used to validate results obtained with other approaches like the ray based approach [YAN 98]. 4.2.3. Ray based approach Based on geometric optic (GO) combined with uniform theory of diﬀraction (UTD) (see Appendix C), the ray based approach is well suited for the study of radio wave propagation. The GO considers that the energy is radiated along inﬁnite tubes called rays. These rays deﬁne the propagation directions and can reﬂect or refract on all the surfaces they encounter. The use of UTD complements the GO as it introduces the diﬀracted rays and ensures the continuity of the ﬁeld on regions where the GO predicts a nonexistence of a ﬁeld. Before the determination of the propagated ﬁeld, we need to ﬁnd the rays from which the GO and the UTD will be applied. This step of ray ﬁnding can use various techniques. There are two main techniques of ray determination: ray launching is considered to be a forward technique and ray tracing is considered to be a backward technique [SAR 03] (see Appendix D). When the rays are obtained, the propagated ﬁeld can thus be calculated from the transmission to reception side. The ray based approach needs less numerical resources than the FDTD and MoM approaches for wave propagation prediction. Consequently, it is mostly used for deterministic propagation modeling [IKE 91, MCK 91, TAM 95, TAN 95, SAN 96, CHE 96, RIZ 97, VIL 99, ZHA 00]. Such approaches are not valid for low frequencies (typically lower than 100 MHz), when the size of objects interacting with rays become small or have the same order as the wavelength. An improvement of the propagation prediction with the ray based approach can be made by combining it with exact approaches such as those using FDTD or MoM. 4.3. Speciﬁcity of deterministic modeling in UWB When studying narrowband transmission techniques using deterministic software, much is made of the power loss of the transmitted signal. So, modeling allows the establishment of radio coverage maps on any considered environment at a given central frequency corresponding to the transmitted narrowband
102
UltraWideband Radio Propagation Channels
signal. Indeed, we consider that the bandwidth is narrow enough to only focus the propagation study on the central frequency. In UWB, the frequency components of the transmitted signal cover a vast bandwidth which can reach 7.5 GHz. So, the performed simulation will necessarily address all the phenomena appearing on the entire bandwidth. Thus, we cannot simply focus on the propagation around the central frequency. This implies that we have to consider in the synthesized link both antennas and material properties of the considered environment on the whole covered frequency band. Another particularity of the UWB deterministic modeling is the need to consider the phase information related to the propagation as well as combining the eﬀects of antennas and material properties. This phase information is important as it allows the identiﬁcation of potential deformations (such as attenuation, dispersion, etc.) appearing on the received signal during the transmission. Deterministic modeling can thus be used in speciﬁc environmental conﬁguration in order to better understand the eﬀects of physical phenomena observed on results of channel propagation measurement campaigns. So, the use of site speciﬁc tool for deterministic modeling allows us on the one hand to study the waves propagation in various types of environments and in speciﬁc conﬁgurations, and on the other hand to physically understand the eﬀects observed on measurements. 4.4. Overview of UWB deterministic modeling In this section, the four principal deterministic models which appear in the literature are described, with a main focus on their respective speciﬁcities. 4.4.1. Qiu model Some authors consider the study proposed by Qiu [QIU 02] on the diﬀraction undergone by an UWB signal as a UWB deterministic model. In the study made by Qiu, he mainly focuses on the distortions and dispersions introduced in a link and on system performances by single and multiple diﬀractions. 4.4.2. Yao model The particularity of the model proposed by Yao is that it takes into account the GO and UTD coeﬃcients in the time domain in order to account
Deterministic Modeling of the UWB Channel
103
for the eﬀects of the propagation channel interactions [YAO 03a, YAO 03b]. These time domain coeﬃcients allow us to directly express the impulse response of each interaction [VER 90, BAR 91, ROU 95, YAO 97]. This approach can appear appropriate for the UWB channel propagation modeling as the reconstructed signal is a sum of each ray contribution. The contribution of each ray at the receiver side is a successive convolution of the impulse responses of each interaction of the considered ray with the transmitted signal. Nevertheless, a deterministic modeling tool using this approach will need signiﬁcant numerical resources. In fact, a numerical operation consisting of a convolution needs great memory resources. So, although this model gives acceptable results, its use can become very restrictive in term of calculation time. 4.4.3. Attiya model The deterministic model proposed by Attiya [ATT 04] considers the GO and the UTD interactions coeﬃcients expressed in the frequency domain (see Appendix C). The received signal is obtained in the time domain thanks to the use of an inverse Fourier transform. This allows us to directly use all the classical formulations corresponding to the frequency behavior of the propagation phenomena. Another diﬀerence between Yao and Attiya’s deterministic model lies in the consideration of antennas, something Yao neglected entirely. Attiya’s proposed model takes into account the antennas by inserting analytical formulations of antenna radiation in the expression of the reconstructed signal. In his model, he focuses on the case of horn antennas for which he proposes a time domain characterization technique [ATT 03]. The same impulse response obtained from measurement in an anechoic chamber between a transmitting antenna and a receiving antenna is directly applied in the time domain on each ray. For each ray, the relation corresponding to the eﬀect of ray interactions is obtained after inverse Fourier transformation. So, the determination of the received signals after antennas are made by adding all the contributions of each ray. Each of these contributions is obtained by convolution of the impulse response between antennas with the relation corresponding to eﬀect of a channel without antenna. So, it appears that this method shows two limits. The ﬁrst limit is the consideration of the impulse response of antennas in order to insert their behavior in the model. The second limit is the application of the same impulse response for all rays, although they do not go and reach transmitting and receiving antennas respectively from the same direction.
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UltraWideband Radio Propagation Channels
4.4.4. Uguen and Tchoﬀo Talom model The other deterministic model proposed in the literature is that of Uguen and Tchoﬀo Talom [TCH 04, UGU 05]. The ﬁrst contribution of this model mainly presents the synthesis of a received signal which adopts a formalism considering the channel ray by ray [UGU 02]. In this ﬁrst contribution, some important elements of the transmission channel were not considered, such as the indoor multilayered materials and antennas. Eventually, a more complete description was made [TCH 04, UGU 05]. In comparison to the ﬁrst proposition, the introduced improvement concerns a better description of the consideration of each channel element in the model. This model is quite similar to the one proposed by Attiya [ATT 04]. However, unlike the Attiya model, the antennas are better taken into account in the model proposed by Uguen and Tchoﬀo Talom. Moreover, the construction of each ray contribution is made separately, which allows us to naturally access to the direction of departure (DoD) and direction of arrival (DoA) information. So, each ray is aﬀected by the antenna functions corresponding to the actual DoD and DoA of the ray, deﬁned in the frequency domain. 4.5. Illustration of a deterministic model formalism In this section, we illustrate the UWB deterministic modeling by presenting the theoretical formalism of the model proposed by Uguen and Tchoﬀo Talom [TCH 04, UGU 05]. This model allows us to synthesize a received signal for an indoor UWB link. We consider the application of an impulse signal p(τ ) on the transmitting antenna feeding port. As explained previously, it is not helpful to sum all the ray contributions before the transformation from the frequency domain to the time domain (see Appendix E). It is better to apply the transformation separately on each ray and then to sum each ray contribution directly in the time domain. This approach allows us to reduce the number of frequency points to consider and avoids unneeded frequency domain truncation of the complex exponential corresponding to the phase shift introduced by the propagation. The signal after the receiving antenna is obtained by summing all the vectorial and complex component of the ﬁeld coming from the rays arriving on the receiving antenna. After the frequency sweep on the band of interest, the signal is expressed in the time domain using an inverse discrete Fourier transformation. So, this signal can be considered as the convolution of the signal p(τ ) with the SISO impulse response of the propagation channel.
Deterministic Modeling of the UWB Channel
105
The presentation of the theoretical formalism of the deterministic model is made by the detailed synthesis of the received signal. This detail mainly focuses on the ray by ray treatment of the propagation and antenna information as well as the proper consideration of antennas in the modeling.
4.5.1. Received signal synthesis The received signal r(τ ) is constructed as the sum of the appropriately shifted contribution rk (τ ) of the Nray rays obtained from the ray tracing step (see Figure 4.1) by:
Nray
r(τ ) =
rk τ − τk
[4.1]
k=1
The parameter τk is the delay corresponding to the free space propagation of the k th ray. The nondelayed signal rk (τ ) (see relation [4.2]) corresponds to the convolution of the transmitted signal p(τ ) applied at the transmitting antenna port with the nondelayed ray impulse response (RIR) hk (τ ). rk (τ ) = hk (τ ) ∗ p(τ )
[4.2]
In relation [4.1], Nray is one of the important parameters for the reconstruction of the received signal. It determines the realism of the obtained synthesized signal r(τ ). Deterministic models using rays are strongly dependent on the number of signiﬁcant rays contributing to the total ﬁeld. So, it is important to use rapid and appropriate techniques allowing us to obtain the main contributors in LOS or NLOS situations. Most of the time, this last situation requires us to consider a great number of rays in order to improve the realism of the synthesized signal.
4.5.2. Ray impulse response without delay The nondelayed ray impulse response (RIR) is noted hk (τ ) and deﬁned in the time domain by: ck (τ ) f t τ, sˆtk [4.3] hk (τ ) = f r − τ, sˆrk
r˜k (τ − τk )
r˜2 (τ − τ2 )
r˜1 (τ − τ1 )
=
r(τ)
τ2 τk
τ
τ
r˜k (τ)
r˜2 (τ)
r˜1 (τ)
Figure 4.1. Building of received signal r(τ ) by the sum of shifted rk (τ )
τ1
τ
τ
T FFT
106 UltraWideband Radio Propagation Channels
Deterministic Modeling of the UWB Channel
107
f r (τ, sˆrk ) is a line vector corresponding to the impulse response of the received antenna in the arrival direction sˆrk of the k th ray. f t (τ, sˆtk ) is a column vector corresponding to the impulse response of the transmitted antenna in the departure direction sˆtk of the k th ray. ck (τ ) is a 2×2 matrix corresponding to the consideration of attenuations and distortions introduced by the reﬂection, transmission or diﬀraction interactions appearing on the k th ray when propagating through the channel. This term does not consider the delay introduced by propagation τk . This delay is directly used when adding all the rk (τ ) at the appropriated time position (see Figure 4.1). sˆtk and sˆrk are the directions of departure (t) and arrival (r) of the k th ray. They are respectively the couples of polar coordinates (θkt , φtk ) and (θkr , φrk ) in the entire spherical base. Nevertheless, as reﬂection, transmission and diﬀraction phenomena have a simple analytical expression in the frequency domain, rk (τ ) is obtained from k (f ): an inverse Fourier transform applied on its frequency expression R k (f ) Ft f, sˆt P (f ) k (f ) = Fr ∗ f, sˆrk C R k
[4.4]
P (f ) corresponds to the Fourier transform of the transmitted signal. Ft,r are the complex vectors corresponding respectively to the transmitting and receiving antenna frequency behavior. They allow us to consider the directivity Dt,r , the return loss Γt,r , the radiation eﬃciency η t,r and the antenna polarization state Ut,r [LO 88]: Ft f, sˆtk = Gt1 f, sˆtk Ut f, sˆtk
[4.5]
λ Fr f, sˆrk = −j Gr2 f, sˆrk Ur f, sˆrk 4π 2 Gt1 f, sˆtk = η t (f ) 1 − Γt (f ) Dt f, sˆtk
[4.7]
Gr2 f, sˆrk = η r (f ) Dr f, sˆrk
[4.8]
Ut,r f, sˆt,r = Uθt,r θˆt,r + Uφt,r φˆt,r k
[4.9]
[4.6]
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UltraWideband Radio Propagation Channels
The terms Ut,r are unitary vectors deﬁned in the basis corresponding to the directions of departure sˆtk and arrival sˆrk for a given ray. These terms follow the relation: U∗ U = 1
[4.10]
λ The term −j 4π in the expression of Fr corresponds to the integration operation made at the receiver side.
k is the 2×2 matrix corresponding to the frequency ﬁlter presented by the C propagation channel on the k th ray. This matrix is deﬁned from the GO/UTD ﬁeld and is expressed in a vectorial form: ⎤ ⎡ θ,θ θ,φ C C k (f ) = Ek = ⎣ ⎦ [4.11] C φ,θ Einc φ,φ k C C k the ﬁeld at the receiving antenna and Einc the ﬁeld coming from the with E k transmitting antenna in the direction sˆtk by: t t Einc ˆk P (f ) [4.12] k = F f, s k = Ek e+j2πf τk E
[4.13]
4.5.3. Ray channel matrix without delay The ray channel matrix without delay is expressed in a vectorial way by k . It depends on the expression of the GO and the UTD ﬁeld by the use C of the ﬁeld Ek (see relation [4.14]). The superscript tilde corresponds to the extraction of the delay related to the propagation in Ek expression. k Einc e−j2πf τk Ek = C k
[4.14]
Typically, there can appear on a k th ray approximately Nk interactions of reﬂection, transmission or diﬀraction nature. So, Ek is expressed by the relation: Ek =
1 s0k
1 Ak Gk e−j2πf τk Einc k s0k
[4.15]
corresponds to the spherical nature of the wave coming from the transmitting antenna.
Deterministic Modeling of the UWB Channel
109
τk is the delay related to the propagation of the k th ray. It is the sum of delays associated with the distances between the interactions appearing on the considered ray. τk =
Nk
τki
[4.16]
i=0
Ak corresponds to the product of all the divergence factors of each interaction appearing on the ray. The expression of each divergence Aik is related to the considered interaction’s nature: Ak =
Nk
Aik
[4.17]
i=1
Gk is the interaction matrix representing the consecutive reﬂections, transmissions or diﬀractions appearing on the k th ray. The term Gik of each interactions is related to the frequency, the polarization or ⊥ and the reﬂection, transmission and diﬀraction incidence angles. These terms correspond to 2 × 2 matrices and are deﬁned according to the incidence basis of reﬂection, transmission and diﬀraction interaction considered by: ⎤ ⎡ ,⊥ , G G ⎦ [4.18] Gik = ⎣ ⊥, ⊥,⊥ G G
In other terms, the matrix Gik corresponds to the R, T or D matrix according to the interaction nature (see Appendix C). So, it is necessary to insert, in the global expression of Gk [4.19], the matrix used to change the basis related to the possible existence of an angle between the incidence plane of an interaction and the reﬂection, transmission or diﬀraction plane of the previous interaction. N k out,Nk out,i−1 in,1 t Bk →Brk i Bk →Bin,i k G1k MBk →Bk Gk M [4.19] Gk = M i=2
t
in,1
MBk →Bk is a matrix used to change the description of a ﬁeld from a basis t Bk related to a given departure direction of a k th ray to the incoming basis of the ﬁrst ray interaction. Bin,1 k
110
UltraWideband Radio Propagation Channels out,Nk
MBk
→Brk
is a matrix used to change the ﬁeld from the outgoing basis the last interaction to the incoming basis Brk of the arrival direction ray.
k of Bout,N k th
of the k
out,i−1
in,i
→Bk MBk is a matrix allowing for a k th ray to express the ﬁeld, of the (i−1)th interaction in the incoming initially in an outgoing basis Bout,i−1 k in,i th basis Bk of the i interaction.
4.5.4. Described model results 4.5.4.1. Emitted waveform and considered scenario We consider the impulse waveform transmission p(τ ), which is obtained using the relation: p(τ ) =
Ep pˆ(τ )
[4.20]
with Ep the energy of the impulse signal p(τ ) and pˆ(τ ) a Gaussian impulse normalized in energy. pˆ(τ ) can be expressed using relation [4.21]: ! √ τ 2 2 2 √ sin 2πfc τ e−( β ) pˆ(τ ) = β π " √ 2 ln 10 α Bα,β = ≈ 0.216 α πβ 20 β
[4.21] [4.22]
where Bα,β is the band of pˆ(τ ) given at −α dB, β is a scaling factor allowing us to adjust the time domain support of pˆ(τ ) and fc is the central frequency of pˆ(τ ). In the frequency domain, the applied impulse signal is given by: P (f ) = Ep Pˆ (f ) ! √ 2 β π −[π β (f +fc )]2 √ e − e−[π β (f −fc )] Pˆ (f ) = 2
[4.23] [4.24]
So, P (f )2 is the energy spectral density (ESD) of the impulse signal. The impulse energy Ep is then obtained by integrating either the square of the
Deterministic Modeling of the UWB Channel
111
signal p(τ ) or P (f )2 , according to the following ParsevalPlancherel relation [PRO 83]: P (f )2 df p2 (τ ) dτ = [4.25] Ep = R
R
To specify Ep , relation [4.26] proposed by [UGU 04] can be used: Ep =
1MHz Tr Pmax 2 1MHz σ γmax
[4.26]
So, the impulse energy can be determined with respect to the emission 1 MHz , the pulse repetition period Tr , limits speciﬁed by regulation using Pmax 1 MHz the capability of the the considered modulation given by σ, and γmax transmitting antenna to emit an ampliﬁed signal in the channel over a 1 MHz band. According to the FCC speciﬁcations concerning the transmitted UWB 1 MHz signal, the authorized maximum PSD is Pmax = −41.3 dBm/MHz. In this case, Ep is directly expressed by: Ep =
Tr 10−4.13 σ 2 γ 1 MHz
[4.27]
max
Figure 4.2 shows two impulses obtained for a central frequency fc = 4 GHz and an ideal1 transmitting antenna. We can note that the signal time spreading conversely increases with the band as well as the maximum level. The previously described model is applied for an indoor link of a LOS conﬁguration in the environment represented in Figure 4.3. The rays reported in the ﬁgure are obtained using a 3D ray determination technique which combines ray launching and ray tracing [TCH 05a]. The material properties of the environment shown in Figure 4.3 are reported in Table 4.1. These properties are obtained from material characterizations [TCH 05a]. The frequency dependence of the materials is introduced by the permittivity expression in the interactions coeﬃcients (see Appendix C). In this table, we can note that the ﬂoor and the ceiling are in reinforced concrete because the environment is the ground ﬂoor of a house with two levels.
1. An ideal antenna is an omnidirectional isotropic antenna with a unitary gain in all directions.
112
UltraWideband Radio Propagation Channels
(a)
(b)
Figure 4.2. Impulses applied on an ideal transmitting antenna with fc = 4 GHz: B10 dB,β = 0.5 GHz (a) and B10 dB,β = 2 GHz (b)
Deterministic Modeling of the UWB Channel
113
Figure 4.3. 3D rays obtained for an indoor link in LOS conﬁguration
Elements
Material properties
Materials r
r
μr
μr σ (S/m) Δδ (m) e (cm)
Wall
brick
3.8
0
1
0
0.05
0
10
Windows
glass
3.1
0
1
0
0
0
1.5
Doors
wood
2.84 −0.02
1
0
0
0
3
Ceiling
reinforced concrete 7.7
0
1
0
1
0
10
Floor
reinforced concrete 7.7
0
1
0
1
0
—
Table 4.1. Properties and structure of various elements of the considered environment
4.5.4.2. Channel matrix of each emitted waveform in the LOS case Considering the two waveforms (see Figure 4.2) for p(τ ) and the LOS conﬁguration (see Figure 4.3), Figures 4.4 and 4.5 represent the four components of the channel matrix c(τ ):
cθ,θ (τ ) c(τ ) = φ,θ c (τ )
cθ,φ (τ ) cφ,φ (τ )
[4.28]
These illustrations are an artiﬁce of representation which allows us to report on the same time axis all the contributions of the rays associated with each component of the matrix c. Each component of c(τ ) is obtained by adding the electric ﬁeld contribution of each of the Nray for the considered component.
114
UltraWideband Radio Propagation Channels
This representation allows us to underline the time domain behavior of the Nray ray ﬁelds arriving at the receiver without considering any transmitting or receiving antenna. We can note in these ﬁgures that the ﬁeld level is weak on the cross components (cθ,φ and cφ,θ ). So, in order to obtain the received signal, the antennas considered at the transmitting and receiving sides will project these c(τ ) matrix components according to their characteristics [TCH 05b]. The shape of the ﬁnal obtained signal will strongly depend on the components of c(τ ) modiﬁed by the chosen couple of antennas. Moreover, when the frequency band of the impulse p(τ ) decreases, each contribution of all rays can no longer be clearly isolated. For a band B10 dB,β = 500 MHz and for each component of c(τ ), the obtained signal corresponds to a compact grouping of the ray contributions arriving at a time lap with approximately the same width as the transmitted signal. This does not allow us to isolate the contribution of each ray but underlines the progressive rise of shadowing as the transmitted signal bandwidth decreases. 4.5.4.3. Received signal with ideal antennas In this part, from the components of channel matrix c (see Figure 4.5) and for the LOS case as well as the transmitted pulse p(τ ) of Figure 4.2(b), we focus on the reconstructed waveform obtained in the case where the antennas at transmission (Tx) and reception (Rx) sides are ideal omnidirectional, with unitary gain. Figures 4.6(a) and 4.6(b) show the reconstructed signal r(τ ) obtained respectively for a couple of antennas polarized along θ and φ. In the considered conﬁgurations, the transmitting and receiving antennas show a good accordance in term of polarization. So, the reconstructed signals (see Figures 4.6(a) and 4.6(b)) are respectively a projection of the channel matrix c(τ ) components cθ,θ (see Figure 4.5(a)) and cφ,φ (see Figure 4.5(d)). In the case of a real antenna, the received signal r(τ ) will probably be a less simple projection of diﬀerent matrix c components. So, it will no longer be easily identiﬁed to one of the matrix c(τ ) components. One of the interests of the deterministic model detailed in section 4.5 is the ease of identifying DOA and DOD information. This is because the signal r(τ ) is reconstructed thanks to the rays obtained after ray tracing which directly gives the information corresponding to the direction of the rays. The other interest is that no sum is beforehand applied on the contributions of each ray. So, it is easy to extract the contribution associated with each ray.
(d)
(c)
Figure 4.4. Channel matrix component c(τ ) for the scenario of Figure 4.3 with the impulse p(τ ) of Figure 4.2(a): cθ,θ (τ ) (a), cθ,φ (τ ) (b), cφ,θ (τ ) (c) and cφ,φ (τ ) (d)
(b)
(a)
Deterministic Modeling of the UWB Channel 115
(d)
(c)
Figure 4.5. Channel matrix component c(τ ) for the scenario of Figure 4.3 with the impulse p(τ ) of Figure 4.2(b): cθ,θ (τ ) (a), cθ,φ (τ ) (b), cφ,θ (τ ) (c) and cφ,φ (τ ) (d)
(b)
(a)
116 UltraWideband Radio Propagation Channels
Deterministic Modeling of the UWB Channel
117
(a)
(b)
Figure 4.6. Received signal r(τ ) for the LOS scenario of Figure 4.3, the pulse p(τ ) of Figure 4.2(b) and the ideal omnidirectional antennas at Tx and Rx sides: TxRx antennas polarized along θ (a) and TxRx polarized antennas along φ (b)
118
UltraWideband Radio Propagation Channels
Figure 4.7 illustrates the extraction of DOA information corresponding to the signal reported in Figure 4.6(a). We can note that the obtained rays are well described in 3D. The direct path is clearly identiﬁed. It arrives at the receiving antenna side in the direction corresponding to the angles θ ≈ 95◦ and φ ≈ 245◦ . Here, the θ direction is not 90◦ because the transmitter and receiver antennas are not placed at the same height.
4.6. Consideration of real antenna characteristics in deterministic modeling Previously when ideal antennas were considered, we could easily identify the matrix c(τ ) component corresponding to the signal received after the Rx antenna. Here, real couple of antenna radiation characteristics are considered. In Figure 3.12, the used antennas which are called the CMA (Conical Monopole Antenna) are represented. The characteristic of these antennas are obtained from measurements performed on 4 π steradian in the Stargate 32 nearﬁeld antenna measurement engine of Satimo (see Figure 4.8) [TCH 05b]. The considered modeling conﬁguration is the case illustrated in Figure 4.3. The received signal is illustrated in Figure 4.9(a). We can note that the omnidirectional characteristic of the CMA allows us to obtain a signal diﬀerent from those obtained previously with ideal antennas. So, the obtained signal is no longer easily identiﬁed to one component of the channel matrix c(τ ) (see Figure 4.5). Figure 4.9(b) illustrates the amplitude and phase contribution applied by the antennas on each ray. The rays corresponding to the main contribution on the received signal are represented by a dark line. The contribution of a ray is thus proportional to the gray intensity used for its drawing. The darkest rays are those for which the combined directivity of transmitting and receiving antennas are important. If a rotation is applied along θ in order to ﬁt both antenna directions of maximum radiation in elevation, the received signal will have a higher level than in the previous case (see Figure 4.10) and the other directions will contribute less to the obtained signal.2 We can note that the consideration of real antennas characteristics in the deterministic model is crucial and contributes to the validation and realism of the received signal synthesized with the modeling.
2. By comparing the two ﬁgures, it seems that the second RI is less dense. In fact, it is the ﬁrst and main ray which has more energy in this case.
Figure 4.7. Illustration of r(τ ) DOA (see Figure 4.3) and the TxRx ideal antennas polarized along θ
Deterministic Modeling of the UWB Channel 119
120
UltraWideband Radio Propagation Channels
Figure 4.8. Satimo Stargate 32 nearﬁeld antenna measurement engine
4.7. Building material eﬀects on channel properties The previously described studies were made for materials whose properties are reported in Table 4.1. So, the received signal reported in Figure 4.11(a) is obtained for a LOS link (see Figure 4.3), the waveform transmission in Figure 4.2(b) and the use of a couple of CMA antennas. This signal is the same as that reported in Figure 4.9(a). Considering the properties reported in Table 4.2, the eﬀect of building material properties on the received signal synthesized by deterministic modeling is underlined. The new adopted properties correspond to a reduction of the previous wall thickness values (see Table 4.1), to a change of window structure and especially to an increase of ﬂoor and ceiling conductivity. In fact, the reinforced concrete used for constructing ﬂoors and ceilings at various stages of house building are made with metallic rods which can contribute to the increase in conductivity. The signal received with these new material properties shows important diﬀerences in comparison to the previous signals (see Figure 4.11). There are rays arriving after the direct path which show a higher level. These rays are those which connected the transmitter and receiver and reﬂected on the ﬂoor or the ceiling. So, the conductivity increase σ of the reinforced concrete introduces a high level contribution on the received signal for the rays which interact with the ceiling or the ﬂoor.
Deterministic Modeling of the UWB Channel
(a)
(b)
Figure 4.9. Received signal and ray contributions corresponding to the CMA antennas: received signal after the receiving antenna (a) and 3D rays with their corresponding contribution (b)
121
122
UltraWideband Radio Propagation Channels
(a)
(b)
Figure 4.10. Received signal and ray contributions corresponding to the CMA antennas (rotation of 35◦ in θ for both Tx and Rx antennas): received signal after receiving antenna (a) and 3D rays with their corresponding contributions (b)
Deterministic Modeling of the UWB Channel
123
(a)
(b)
Figure 4.11. Received signal obtained with CMA antennas for two conﬁgurations of building material properties: case of materials in Table 4.1 and case of materials in Table 4.2
124
UltraWideband Radio Propagation Channels
Elements
Material properties
Materials r
r
μr
μr σ (S/m) Δδ (m) e (cm)
Walls
brick
3.8
0
1
0
0.05
0
7
Windows
glass
3.1
0
1
0
0
0
0.5
air
1
0
1
0
0
0
0.5
glass
3.1
0
wood
2.84 −0.02
Doors
1
0
0
0
0.5
1
0
0
0
3
Ceiling
reinforced concrete 7.7
0
1
0
10
0
10
Floor
reinforced concrete 7.7
0
1
0
10
0
—
Table 4.2. Properties and structure of various building environment considered for the study of material inﬂuence
In addition to the antennas, the change of material properties strongly aﬀects the received signal shape. In order to increase the realism of the waveforms obtained with a deterministic modeling tool, the characterization of building materials is needed to insert into the modeling the good material properties for the environment considered in link modeling [KRA 93, SAT 95, HUA 96, COU 98, MUQ 03a].
4.8. Simulation and measurement comparisons In this section, comparisons are made between the results of the realized measurements (see section 3.4.2.2) and the performed simulations using the deterministic model described in section 4.5. These comparisons allow us to evaluate the impact of the antenna characteristics in the described model and the received signal building for various LOS links. These comparisons consider a transmitted signal covering a band of 2–6 GHz. As the measurements were made in the frequency domain, no impulse signal is applied. Nevertheless, the transmitted signal corresponds to a sinc function in the time domain.
4.8.1. Evaluation of real antenna consideration To evaluate the consideration of antenna radiation characteristics in the deterministic model described in section 4.5, direct link measurements were made in an anechoic chamber. This conﬁguration allows us to be free from multipath and to obtain a direct link with only one ray [TCH 06]. For
Deterministic Modeling of the UWB Channel
125
this evaluation, two antenna couples are used: directive horn antennas and omnidirectional CMA antennas. In the anechoic chamber, the antennas are placed at the same height and directed, in the case of horn antennas, so that both antenna directions of maximum gain face each other. This conﬁguration is reproduced in simulation for the same departure and arrival directions as in the measurement. The simulation uses antenna characteristics data obtained from wideband measurement made in the Stargate 32 engine [TCH 05b]. Figure 4.12 illustrates the impulse response and transfers functions obtained with measurement and simulation in the case of the couple of horn antennas. We can observe a good matching between measurements and simulation results. Although there is only one ray, the obtained transfer functions are not constant. This can be explained by the frequency dependencies of antenna characteristics which aﬀect the channel transfer function. Figure 4.13 illustrates the impulse responses and transfers functions obtained with measurement and simulation in the case of the couple of CMA antennas. We can also notice in this case a good matching between measurements and simulation results. The comparison of these two conﬁgurations testify to the relevance of antenna characteristics in deterministic link modeling. Indeed, according to their characteristics, antennas will aﬀect the shape of the received impulse responses. Moreover, the comparison between measurement and simulation validates the adopted vectorial insertion of antenna characteristics in the described deterministic model. 4.8.2. Evaluation of impulse response reconstruction To evaluate the impulse response reconstruction, the focus here is on LOS links for which the measurement conﬁgurations are described in section 3.4.2.2. In each of the shown ﬁgures, the measured and simulated impulse responses as well as transfer functions are superimposed. The measurement and simulation results are respectively represented in gray and black. These illustrations concern four typical LOS situations extracted from 126 measurements points made in the environment described in section 3.4.2.2. Figures 4.14 and 4.15 represent the results obtained considering the four following positions corresponding to 1.14 m, 4.38 m, 6.78 m and 8.94 m distances.
126
UltraWideband Radio Propagation Channels
The measured and simulated transfer functions show a great similarity in terms of level and ﬂuctuations in form (see Figure 4.14). The same high frequency decrease is observed in measurement and simulation. Nevertheless, some diﬀerences are presented on the transfer function reported in Figure 4.14(a). These diﬀerences can be explained by the fact that for the presented results the transmitter was placed near a stone made ﬁreplace. In simulation, a normal wall made with concrete was considered. Moreover, some diﬀerences can also be observed in phase information. This can be explained by the rays considered for the modeling, which are insuﬃcient to provide access to the degree of measurement detail. Although the transfer functions seem diﬀerent, we can note a great accordance between measured and simulated impulse responses (see Figure 4.15). Moreover, the simulated impulse responses are less consistent in term of rays compared to the measurement impulse responses. This can be explained by the simpliﬁed description of the environment used in the modeling. This simple description of the environment leads us to discard furniture and the inappropriate material properties for the considered environment walls. We may keep in mind that in such situations of very good concordance between simulation and measurement results, the simulation represents the beneﬁt of giving additional information like departure and arrival angles. The channel is here known on various dimensions and with less cost than using a sounder with a single sensor.
4.9. Conclusion For a long time, the deployment of narrowband technologies has used the deterministic modeling of the radio propagation channel to establish coverage maps. With the use of UWB technology, some deterministic models have been proposed for signal prediction in new context. These models allow signal prediction and the rapid study of various environments with low costs in terms of channel transmission eﬀects on the UWB link. From all the deterministic models presented, those which use frequency domain formalisms clearly seem better suited for the study of UWB wave propagation. More especially, the detailed Uguen and Tchoﬀo Talom model allows us to easily insert the antenna radiation information in the model. Moreover, this model naturally enables the access to DoD and DoA information as the electromagnetic calculations are performed ray by ray. The contributions of all the rays are then summed to obtain the overall received signal.
Deterministic Modeling of the UWB Channel
127
When deterministic modeling is used in UWB, it is important to take into account antenna characteristics and to use appropriate material properties for the considered environment. The reconstructed signal is strongly dependent on the number and relevance of the rays used in modeling. This ﬁrstly depends on the techniques used for ray determination and secondly on the details with which the environment is described.
(V)
UltraWideband Radio Propagation Channels
delay (ns)
(a)
(level)
Magnitude
Freq (GHz) Phase
(radian)
128
Freq (GHz)
(b)
Figure 4.12. Measurement (gray) and simulation (black) received signals for the couple of horn antennas: (a) impulse response and (b) transfer function: magnitude and phase
(V)
Deterministic Modeling of the UWB Channel
delay (ns)
(a)
(level)
Magnitude
(radian)
Freq (GHz) Phase
Freq (GHz)
(b)
Figure 4.13. Measurement (gray) and simulation (black) received signals for the couple of CMA antennas: (a) impulse response and (b) transfer function: magnitude and phase
129
(leve l)
(radian)
(leve l)
(radian)
(d)
Freq (GHz)
Freq (GHz) Phase
Magnitude
(b)
Freq (GHz)
Freq (GHz) Phase
Magnitude
Figure 4.14. Measured (gray) and simulated (black) transfer functions for 4 distances between Tx and Rx: (a) d = 1.14 m, (b) d = 4.38 m, (c) d = 6.78 m and (d) d = 8.94 m
(c)
Freq (GHz)
Freq (GHz) Phase
Magnitude
(a)
Freq (GHz)
Freq (GHz) Phase
Magnitude
(leve l) (radian) (leve l) (radian)
130 UltraWideband Radio Propagation Channels
delay (ns)
delay (ns)
(d)
delay (ns)
(b)
Figure 4.15. Measured (gray) and simulated (black) normalized impulse responses for 4 distances between Tx and Rx: (a) d = 1.14 m, (b) d = 4.38 m, (c) d = 6.78 m and (d) d = 8.94 m
(c)
(V) (V)
(V)
(V)
delay (ns)
(a)
Deterministic Modeling of the UWB Channel 131
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Chapter 5
Statistical Modeling of the UWB Channel
Deterministic modeling makes a relatively accurate reproduction of the UWB channel properties possible for a given conﬁguration in a known environment. Assuming that the elementary propagation phenomena are well modeled, the obtained impulse responses can be very realistic. The main constraints linked to this type of model lie in their long calculation time and the necessity of describing the considered environment in detail. Statistical models represent an interesting alternative for the simulation of UWB communication systems. They consist of reproducing a possible behavior of the propagation channel in a given type of environment. They are based on a large number of measurements, from which each parameter of the model is deﬁned using a statistical law. These models enable the random generation of diﬀerent impulse responses. This chapter presents the statistical modeling of the UWB channel through a practical approach. The principles of UWB propagation channel characterization is ﬁrst illustrated from a series of measurements performed in an indoor oﬃce environment. For each step of the characterization process, experimental results are compared to the main anaylses published in the literature. Following a description of the diﬀerent statistical models of the UWB channel, a full channel model is detailed. Its practical conception is based on a set of experimental characteristics. In particular, advanced techniques are presented for the modeling of spatial and temporal variations of the UWB channel.
134
UltraWideband Radio Propagation Channels
5.1. Experimental characterization of channel parameters In this section, the characterization process of the UWB propagation channel is illustrated from a measurement campaign covering the whole FCC frequency band. The description of this measurement campaign is given in section 3.4.2.1. This study also compares the experimental results from diﬀerent analyses of the UWB channel.
5.1.1. Propagation loss 5.1.1.1. Frequency propagation loss One of the channel properties characteristic to UWB signals is the power decay observed with increasing frequency. In an ideal free space conﬁguration, the propagation loss is given by the Friis formula, as recalled in equation [2.34]. This formula implies that for a given distance d, the channel power transfer function (see equation [2.36]), expressed in dB, should follow a frequency variation in the form of −20 log(f ). It should be noted that this frequency dependence is linked to the eﬀective area of an isotropic antenna and is not strictly speaking a characteristic of the propagation channel. In the literature, the attenuation of the received power as the frequency increases has been observed in diﬀerent studies dedicated to antennas [KOV 03, HOF 03], or to the UWB channel [CHE 02, KUN 02a, ALV 03, BAL 04b, HAN 04, CHO 05, PAG 05]. As an example, Table 5.1 presents diﬀerent estimates of the frequency dependent path loss exponent Nf . We may observe a relative diversity in the obtained values, as some of the authors even reported an increase in the received power with increasing frequency [KOV 03]. This may be explained by the high dependence between the parameter Nf and the measurement antenna. Measurement campaign
Nf
Whyless.com Aalborg University [KUN 02a] [KOV 03]
1.6 to 2.8
−0.3 to 5.5
Instit. for Infocomm Research [BAL 04b]
Samsung [CHO 05]
1.0 to 3.7
2.0 to 3.1
Table 5.1. Estimation of the frequency dependent path loss exponent for diﬀerent analyses of the UWB channel. The published values have been adapted to the deﬁnition of the parameter Nf used in this book
In order to estimate the frequency dependent path loss, we studied the channel attenuation at diﬀerent frequencies regularly spaced between 4 GHz and 10 GHz. For each measurement location, the path loss P L(f, d) was
Statistical Modeling of the UWB Channel
135
extracted from the power transfer function at the selected frequencies. In order to remove the distance dependency, each power transfer function was initially normalized by arbitrarily setting the attenuation of the total received power over the whole measurement bandwidth to 0 dB. The resulting normalized path loss P Lnorm (f ) can be compared to a model in the form: f + S(f ) [5.1] P Lnorm (f ) = P Lnorm f0 + 10Nf log f0 where S(f ) represents a residual term with zero mean expressing the diﬀerence in dB between the measurement and the model. 10
PL
norm
(f) (dB)
8 6 4 2 0 2 4 3
4
5 6 7 Frequency (GHz)
8
9
10 11
Figure 5.1. Average normalized path loss vs. frequency
The normalized path loss averaged over all measurement locations is represented as a function of frequency in Figure 5.1. It should be noted that the antenna gains GT (f ) and GR (f ), measured in an anechoic chamber, were subtracted from each power transfer function at the selected frequencies, prior to path loss calculation. For each antenna, the antenna gain was selected by taking the direction of the direct transmitterreceiver path into account. This approach may seem simplistic a priori, as the total power is not only received via the direct path, but also via numerous multipaths arriving from other directions. However, with no further knowledge of the departure and arrival direction of these secondary paths, this method makes a sensible compensation of the antenna eﬀect possible. We may observe that the dispersion of the measured plots around the linear approximation is reduced to a standard deviation of σS = 0.6 dB. The frequency dependent path loss exponent is Nf = 2.28. With respect to the error level inherent to the measurement
136
UltraWideband Radio Propagation Channels
and calculation methods, this value may be considered close to the theory (Nf = 2). Hence, we recommend using the theoretical frequency loss of 20 log(f ) in the modeling of UWB propagation channels, as suggested in [BUE 03]. 5.1.1.2. Distance propagation loss In the previous section, we observed that the used antennas may have a nonnegligible impact on the radio parameters. In the following analyses, in order to minimize this eﬀect, the collected data were corrected by accounting for the gain of both antennas, measured in the direction of the transmitterreceiver path. The antenna radiation patterns, measured every GHz, were interpolated in frequency. According to the previous analysis on the frequency path loss, all measurements were ﬁtted to a general formula as follows: f d + 10Nd log + S(f, d) [5.2] P L(f, d) = P L f0 , d0 + 20 log f0 d0 where f0 represents the central frequency of 6.85 GHz and d0 is an arbitrary distance of 1 m. 40 LOS NLOS
Attenutation (dB)
50 60 70 80 90 100 110 0.6 0.8 1
2
3 4 5 6 8 10 Distance (m)
20
30 40
Figure 5.2. Path loss vs. distance. Each point represents the median attenuation in the FCC band
For each measurement location, the accounted path loss is the median attenuation in the FCC band, after shifting each measurement to the reference frequency f0 . Path loss exponents were obtained by linear ﬁt in each LOS and NLOS situation. Figure 5.2 presents the obtained results. The measurement plots mainly follow a linear decay in logscale, which corresponds to an
Statistical Modeling of the UWB Channel
137
exponential decay of the received power with respect to the distance. In the LOS situation, a path loss exponent Nd = 1.62 was recorded, with a standard deviation σS = 1.7 dB.1 In the NLOS case, the measurement plots are somewhat more dispersed, with a distance dependent path loss exponent Nd = 3.22, and a standard deviation σS = 5.7 dB.2 The values of the parameter P L(f0 , d0 ) were respectively evaluated at 53.7 dB and 59.4 dB. Table 5.2 shows that these values are in line with other analyses of the UWB channel published in the literature. For comparison purposes, the UWB path loss parameters proposed by the standardization organization ITU [ITU 04] are also reported. More details regarding the inﬂuence of frequency on the path loss coeﬃcients are given in [PAJ 07]. In general, these parameters undergo a negligible variation when they are observed on partial frequency bands within the 3.1–10.6 GHz band. 5.1.2. Impulse response characterization Figure 5.3 presents typical PDP measured in LOS and NLOS conﬁgurations, as well as one of the 90 impulse responses. The delay on the xaxis has been converted in path length in meters, to ease the interpretation of the main paths. In both LOS and NLOS situations, we may observe one or several clusters, corresponding to a main echo followed by an exponential decay of diﬀuse power. In the LOS case, walls or pieces of furniture in the vicinity of the radio link generate signiﬁcant reﬂected or diﬀracted echoes, which explains the presence of peaks in the PDP. An attenuation of 10 dB to 20 dB has thus been observed between the power of the main path of each cluster and the power of the secondary paths. The general shape of the PDP is globally smoother in the NLOS case. 5.1.2.1. Delay spread The RMS delay spread τRM S was calculated for each measured PDP, over the whole 3.1–10.6 GHz band. In order to minimize the eﬀect of noise, a threshold placed 20 dB below the maximum value of the PDP was used.
1. In the LOS case, the value of Nd may be lower than in the theoretical case of free space (Nd = 2). This is due to the waveguide eﬀect, frequently observed in indoor conﬁgurations (see section 2.2.2). 2. This signiﬁcant value of the standard deviation in the NLOS case may be explained by the diversity of NLOS conﬁgurations, as the obstruction between the transmitter and the receiver may result from a single plasterboard or several concrete walls.
138
UltraWideband Radio Propagation Channels
Measurement campaign
Nd LOS
UltRaLab [CAS 01] AT&T Labs  MIT [GHA 02a]
σS (dB) NLOS
LOS
2.4 1.7
Time Domain Corporation [YAN 02]
NLOS
3.5
1.6
2.1 1.72
4.09
Whyless.com [KUN 02a]
1.58
1.96
UCAN  CEA LETI [KEI 03]
1.6 to 1.7
3.7 to 7.2
Ultrawaves  Oulu University [HOV 03]
1.04 to 3.18 to 1.80 3.85
2.7
1.48
47
51
3.63
1.4
New Jersey Instit. of Tech. [DAB 03]
1.55 to 1.72
0.77 to 1.98
1.58 to 2.41 to 1.60 2.60
1.6 to 1.9
3.3 to 6.1
2.3 to 2.4
2.8 to 3.6
2.8 to 5.4
AT&T  WINLAB [GHA 03b] 2.01 to 2.95 to 2.07 3.12
2.3 to 3.2
3.8 to 4.1
1.3
NLOS
3.55
UCAN  Cantabria University [ALV 03]
NETEX  Virginia Tech [BUE 03]
LOS
5.9
Intel Labs [CHE 02]
NETEX  Virginia Tech [MUQ 03b]
P L(f0 , d0 ) (dB)
3.2 to 4.1
43.7 to 47.3 to 45.9 50.3
Hong Kong University [LI 03]
1.8
3.4
0.6
3.2
Ultrawaves  University of Rome “Tor Vergata” [CAS 04a]
1.92
3.66
1.42
2.18
48.68
47.24
Instit. for Infocomm Research [BAL 04b]
1.8
1.8 to 2.1
1.5
2.4 to 4.2
36.6
46.4 to 52
ETH Z¨ urich [SCH 04]
1.2 to 1.6
2.2
49 to 51
45
Samsung [CHO 05]
1.18 to 2.18 to 0.93 to 1.43 to 46.5 to 41.3 to 2.48 2.69 1.50 4.69 50.1 47.3
France Telecom  INSA [PAG 06b]
1.62
3.22
1.7
5.7
ITU Recommendation [ITU 04]
1.7
3.5 to 7
1.5
2.7 to 4
53.7
59.4
Table 5.2. Estimate of the distance dependent path loss for diﬀerent analyses of the UWB channel
Statistical Modeling of the UWB Channel
139
(a)
Relative power (dB)
0
IR PDP
10 20 30 40 50
0
20
40 60 Distancedelay (m)
80
(b)
Relative power (dB)
0
IR PDP
10 20 30 40 50
0
20
40 60 Distancedelay (m)
80
Figure 5.3. Typical PDP and impulse response. (a) LOS and (b) NLOS conﬁgurations
Over the whole set of measurements performed in LOS conﬁguration, the mean delay spread is τRMS = 4.1 ns, with a standard deviation στ = 2.7 ns. In the NLOS conﬁguration, the mean delay spread is τRMS = 9.9 ns, with a standard deviation στ = 5.0 ns. Table 5.3 presents the results published in other analyses of the UWB radio channel. Values obtained for τRMS are in accordance with some of the previous experiments, although this parameter may vary from one experiment to the other. This is due to the high sensitivity of this parameter to the measurement environment and to the used experimental setup. The large scope of values in the published results may also be explained by the diﬀerent thresholds used when calculating τRMS [CAS 04a].
7.71 to 17.34 0.53 to 4.55 3.38 to 5.49 19.9 28 to 31 15.6 21.08 to 53.62 12.48 to 14 4.1
New Jersey Instit. of Tech. [DAB 03]
NETEX  Virginia Tech [BUE 03]
AT&T  WINLAB [GHA 03a]
Hong Kong University [LI 03]
Lund University [KAR 04b]
Instit. of Infocomm Research [BAL 04b]
ETH Z¨ urich [SCH 04]
Samsung [CHO 05]
France Telecom  INSA [PAG 06b]
2.7
1.53 to 1.87
1.63 to 3.37
1.8
3.3
5.22
1.4 to 9.5
NLOS
5
5.22 to 8.03
1.87 to 7.04
2.8
2.45 to 3.47
στ (ns)
1.58 to 1.63
2.3
1.9 to 2.1
LOS
Table 5.3. Estimate of the delay spread for diﬀerent analyses of the UWB channel
9.9
26.51 to 38.61
31.11 to 74.08
18.7 to 23.6
34 to 40
14.3
7.31 to 8.15
2.30 to 18.50
8.2
4.7
AT&T Labs  MIT [GHA 02a]
8.78 to 14.59 5.72
5.27
Time Domain Corporation (DSO) [PEN 02]
9.9 to 22.4
NLOS
Time Domain Corporation (correlator) [YAN 02]
10.9 to 12.2
UCAN  CEA LETI [KEI 03]
τRMS (ns) LOS
campaign
Measurement
140 UltraWideband Radio Propagation Channels
Statistical Modeling of the UWB Channel
141
In order to observe the evolution of these parameters with frequency, the mean values of τRM S calculated for seven partial bands of 528 MHz each are represented in Figure 5.4 as a function of the central frequency of each band. For each band, the value of στ is represented by the length of the vertical line. The values of τRMS and στ obtained in each partial band are relatively close to the values calculated from the global UWB frequency band. As can be seen, the delay spread is not aﬀected by the frequency. This was also observed by the University of Oulu from measurements performed in the 1–11 GHz band [JAM 04].
RMS delay spread (ns)
15
10
5 LOS 0
3
4
5
6 7 8 9 Central frequency (GHz)
NLOS 10
11
Figure 5.4. Mean delay spread for diﬀerent partial bands. The length of the vertical line represents the corresponding standard deviation
5.1.2.2. Power delay proﬁle decay Exponential decay constants The typical PDP presented in Figure 5.3 shows that the received power is grouped in diﬀerent clusters, corresponding to the main propagation echoes. The decay of the received power with increasing delay is generally characterized with the inter and intracluster exponential decay constants, respectively denoted Γ and γ (see section 2.4.1.4). To evaluate these parameters, the time intervals corresponding to the clusters of each measured PDP are identiﬁed by visual inspection. This technique was also used in [KAR 04a]. In each of the identiﬁed interval, a linear ﬁt is performed at the delays included between the maximum and the minimum values of the PDP (expressed in dB). This enables the extraction of the intracluster exponential decay constant γ. The intercluster decay
142
UltraWideband Radio Propagation Channels
constant Γ is obtained using a linear ﬁt on the maximum of each cluster. In all cases, only the parts of the PDP presenting a power larger than 5 dB above the noise level were considered. Figure 5.5 illustrates this parameter extraction.
Relative power (dB)
0
Intercluster decay Intracluster decay
10 20 30 40 50 60
0
30
60 90 Delay (ns)
120
Figure 5.5. Extraction of the inter and intracluster exponential decay constants
Among all PDP measured in a LOS situation, between 3 and 8 clusters (5.6 on average) were identiﬁed. The mean exponential decay constants have been evaluated as Γ = 15.7 ns and γ = 7.5 ns. In NLOS conﬁguration, the PDP encompass between 1 and 4 clusters (2.4 on average). The mean exponential decay constants have been assessed as Γ = 16.5 ns and γ = 12.0 ns. Table 5.4 compares these experimental values with results published from similar experiments. In some analyses [CAS 02, ALV 03], the whole PDP was considered as a single cluster, which explains the lack of results regarding Γ. The value of the parameters Γ and γ is generally between 7 ns and 30 ns, even if some higher values have occasionally been reported [CRA 02, ALV 03, CHO 04a]. The intercluster decay is generally stronger than the intracluster decay. Results published in [KAR 04b] are a particular case where this tendency is inversed, and where the values of Γ and γ are quite low. This may arise from the measurement environment, as the experiment took place in a factory. For this experiment, a dependency of the parameter γ with the delay has been observed. In [CRA 02], the authors suggest that the constant Γ is linked to the building architecture, while γ is determined by the objects in the vicinity of the receiving antenna. The diversity of the sounded environments may thus explain the variety of the obtained results.
4.94
2.63 22.1 to 24.0 27.8 15.7
Lund University [KAR 04b]
Samsung [CHO 04a]
Instit. of Infocomm Research [BAL 04b]
France Telecom  INSA [PAG 06b]
7.5
14.1
14.3 to 30.8
4.58
10.83 to 13.97
Table 5.4. Estimate of the PDP exponential decay constants for diﬀerent analyses of the UWB channel. The published values have been adapted to the deﬁnition of the parameters Γ and γ used in this book
16.5
24.6 to 30.4
36.9 to 51.5
13
Ultrawaves  University of Rome “Tor Vergata” [CAS 04b]
1.6
7.6
Intel Labs [FOE 03b]
16
2
21
7.1
6 to 8
NETEX  Virginia Tech [BUE 03, MCK 03]
9 to 20 100
14.5 to 21
UCAN  Cantabria University [ALV 03]
UCAN  CEA LETI [KEI 03]
84.1
UltRaLab [CRA 02]
27.9
16.1
LOS 13.6
NLOS
5.58
7 to 58
8.5
8
125 to 167
5 to 15
NLOS
12.0
25.3 to 33.8
27.4 to 38.6
γ (ns)
UltRaLab [CAS 02]
LOS
Γ (ns)
Whyless.com [KUN 02a]
campaign
Measurement
Statistical Modeling of the UWB Channel 143
144
UltraWideband Radio Propagation Channels
Power decay constants The assumption of an exponential decay for the cluster and ray amplitudes was ﬁrst introduced by Saleh and Valenzuela from their observation of the indoor wideband radio channel [SAL 87]. However, the analysis of the results from the measurement campaign shows that exponential decay is not completely satisfactory to model the slope of the PDP. Indeed, following this assumption, the PDP expressed in logscale should present a linear decrease with increasing delay. The whole PDP as well as each constitutive cluster should hence ﬁt into a triangular shape. This general shape is not representative of the experimental observations (see Figure 5.5). By considering successive echoes of the main path, we may identify two main sources of attenuation. On the one hand the propagation of the wavefront over a longer path induces a stronger power loss. On the other hand, delayed echoes undergo more propagation phenomena, which can be of a diﬀerent nature, such as reﬂection or diﬀraction. This physical interpretation leads us to model the multipath attenuation following a similar approach to that used for distance path loss. We may recall that in this case, the observed attenuation at a transmitterreceiver distance d is proportional to d−Nd , where Nd represents the distance dependent path loss exponent. Regarding the diﬀerent rays of the impulse response, the length of a propagation path is proportional to its delay. Hence, we suggest an adaptation of the Saleh and Valenzuela model, where the cluster and ray amplitudes decrease according to a power function. In the classical formalism presented in section 2.4.1.4, the amplitude βk,l of the k th ray in the lth cluster (see equation [2.33]) is replaced by the following formula: 2 βkl
=
2 β11
Tl T1
−Ω
τk,l + Tl Tl
−ω [5.3]
where Tl represents the delay associated with the lth cluster and τk,l is the delay of the k th ray within the lth cluster. The parameters Ω and ω are respectively called intercluster and intracluster power decay constants. As for the case of exponential decay, the values of the parameters Ω and ω have been assessed by linear ﬁt on the PDP clusters. In each case, the standard deviation σε of the error in dB between the model and the measurement has been calculated, in order to validate the proposed approach. Figure 5.6 illustrates the extraction of the parameters Ω and ω. It can be compared to the results reported in Figure 5.5. Regarding intercluster decay, using a power function instead of an exponential function leads to a decrease in the average standard deviation σε from 4.8 dB to 2.9 dB in the LOS case, and from 2.4 dB to 1.7 dB in the
Statistical Modeling of the UWB Channel
Relative power (dB)
0
145
Intercluster decay Intracluster decay
10 20 30 40 50 60
0
30
60 90 Delay (ns)
120
Figure 5.6. Extraction of the inter and intracluster power decay constants
NLOS case. Regarding the intracluster decay, the average modeling error σε decreases from 1.9 dB to 1.8 dB in the LOS case, and from 1.7 dB to 1.6 dB in the NLOS case. This validates the proposed model, which is closer to our experimental measurements. Finally, in the LOS conﬁguration, we observed a signiﬁcant power attenuation G between the main path of each cluster and the following rays, as may be seen in the example of Figure 5.6. This phenomenon was already observed for UWB channels in [CAS 02] and [KUN 03]. Over the whole set of experimental measurements, we observed the average values Ω = 4.4 and ω = 11.1 in the LOS case, and Ω = 3.9 and ω = 10.2 in the NLOS case. In the LOS conﬁguration, the average attenuation G was measured at 12 dB. 5.1.2.3. Ray and cluster arrival rate Study of the clusters In order to estimate the arrival time statistics for a new cluster, we consider the arrival time of the lth cluster, noted Tl , over the whole set of measured PDP presenting more than one cluster. This corresponds to the delay of the maximum value of the PDP in each time interval representing a cluster. The statistical distribution of the intercluster durations ΔT is then studied. ΔT is calculated as follows: ΔT = Tl+1 − Tl ,
l ∈ [1; L − 1]
[5.4]
where L represents the number of clusters in the PDP. Over the whole set of PDP presenting more than one cluster, the average intercluster duration was ΔT = 27.4 ns in the LOS case and ΔT = 40.1 ns in
146
UltraWideband Radio Propagation Channels
the NLOS case. The graphs in Figure 5.7 are percentilepercentile diagrams, representing the experimental distribution percentiles of ΔT on the xaxis, and the theoretical percentiles of an exponential distribution with parameter Λ on the yaxis. The best ﬁt leads to a value Λ = 27.41 ns = 36.5MHz in the LOS case, and Λ = 40.11 ns = 24.9 MHz in the NLOS case. In both cases, the alignment of the plots on the diagram diagonal shows that the exponential distribution is a reasonable approximation to modeling intercluster duration. (a) LOS
100
Theoretical ΔT (ns)
Theoretical ΔT (ns)
120
80 60 40
NLOS
150 120 90 60 30
20 0
(b) 180
0
20
40 60 80 100 120 Experimental ΔT (ns)
0
0
30
60 90 120 150 Experimental ΔT (ns)
180
Figure 5.7. Percentilepercentile diagrams for the intercluster duration. Experimental percentiles vs. theoretical percentiles corresponding to an exponential distribution with parameter Λ = 36.5 MHz in the LOS case (a) and Λ = 24.9 MHz in the NLOS case (b)
Study of the rays Individual echoes due to the diﬀerent multiple paths are not directly observable on the measured PDP. Because of the limited frequency band of the measurement, each echo is received in the form of an impulse with nonzero duration. The impulse response is a continuous waveform composed of the sum of all individual contributions, each one presenting a diﬀerent attenuation and a diﬀerent phase rotation. Diﬀerent methods are available to extract the delay and amplitude information of the main paths constituting an impulse response. The ¨ 74], was adopted CLEAN method, initially used in radio astronomy [HOG by diﬀerent researchers for the characterization of the UWB radio channel [YAN 02, PEN 02]. This method was modiﬁed by the University of South California for the study of the arrival direction [CRA 02]. The Tokyo Institute of Technology used an algorithm based on the highresolution method called space alternating generalized expectation (SAGE), also allowing the analysis
Statistical Modeling of the UWB Channel
147
of the arrival direction [HAN 03]. It should be noted that these two studies were based on measurements performed with a large number of colocated sensors. In order to illustrate the process of ray identiﬁcation we present here the frequency domain maximum likelihood (FDML) algorithm [DEN 03a, LEE 02]. The search of ray locations is performed in an iterative way, from the representation of the impulse response in its real form h (τ ). As an assumption, the channel is thus described in the form: h (τ ) =
K
βk δ τ − τk
[5.5]
k=1
where K represents the number of rays, and βk and τk are the real amplitude and delay linked to the k th ray. The parameter βk may take negative values, in order to account for the phase inversion linked to some interactions, such as the reﬂections. At each iteration of the process, a new ray is detected by ﬁnding the correlation peak between the measured impulse response and a template signal, corresponding to the sounding waveform. Contrarily to the CLEAN algorithm, the FDML method proposes to optimize the whole set of detected rays. This requires a minimization of the squared error between the measured frequency spectrum and the synthetic spectrum, built from the identiﬁed rays. This optimization presents two main advantages. First, it enables the detection of superimposed rays, which does not always lead to a correlation peak. Second, the processing is performed in the frequency domain, which avoids the resolution limitations linked to the sampling rate in the time domain. More details on adapting the FDML algorithm to accelerate the calculation time are available in [PAG 05]. The results of the FDML algorithm allow us to study the arrival time of the k th ray, over the whole set of selected impulse responses. As with the clusters case, we study the distribution of the interray durations, deﬁned as follows: Δτ = τk+1 − τk ,
k ∈ [1; K − 1]
[5.6]
where K represents the number of rays in the impulse response. Over the whole set of measurements performed in both LOS and NLOS conﬁgurations, the average interray duration was respectively evaluated at Δτ = 0.168 ns and Δτ = 0.161 ns. Figure 5.8 presents a percentilepercentile diagram used to compare the experimental distribution of Δτ to an exponential 1 distribution with parameter λ = 0.168 ns = 5.95 GHz in the LOS case and 1 λ = 0.161 ns = 6.19 GHz in the NLOS case. In this case, some diﬀerences may
148
UltraWideband Radio Propagation Channels
be noted between the theoretical and experimental data, but the exponential approximation still provides an acceptable ﬁt to the measurements. (a)
(b)
0.8
0.8 NLOS Theoretical ΔT (ns)
Theoretical ΔT (ns)
LOS 0.6
0.4
0.2
0
0
0.2 0.4 0.6 Experimental ΔT (ns)
0.8
0.6
0.4
0.2
0
0
0.2 0.4 0.6 Experimental ΔT (ns)
0.8
Figure 5.8. Percentilepercentile diagrams for the interray duration. Experimental percentiles vs. theoretical percentiles corresponding to an exponential distribution with (a) parameter λ = 5.95 GHz in the LOS case and (b) λ = 6.19 GHz in the NLOS case
The experimental values obtained for the cluster arrival rate Λ and the ray arrival rate λ are compared with the results available in the literature, shown in Table 5.5. Regarding the cluster arrival rate, the observed values are generally in the order of 10 to several hundreds of MHz. The average duration between two clusters is thus in the order of 10 ns to 100 ns. It should be recalled that a cluster within the PDP corresponds to a main path, arising from transmissions or reﬂections on walls, on the ceiling or on the building ground. The parameter Λ is thus dependent on the structure of the building where the measurement took place. The ray arrival rate λ presents variable values depending on the experiment. Indeed, values obtained depend highly on the ray identiﬁcation technique that was used in the analysis. For this reason, researchers from the German institute IMST advise to arbitrarily setting the interray duration at the temporal resolution value obtained [KUN 02a]. In this case, the ray arrival rate would be equal to the width Bw of the analyzed band. 5.1.3. Study of smallscale channel variations The last characteristic of the UWB radio channel studied from the experimental campaign (see section 5.1) is related to the fast fading of the impulse response. During the campaign, a rotating arm was used, which enabled the measurement of the channel impulse response at 90 locations
200 16.7 26 70.9 85 to 115 18.6 36.5
NETEX  Virginia Tech [BUE 03, MCK 03]
Intel Labs [FOE 03b]
Ultrawaves  University of Rome “Tor Vergata” [CAS 04b]
Lund University [KAR 04b]
Samsung [CHO 04a]
Instit. of Infocomm Research [BAL 04b]
France Telecom  INSA [PAG 06b]
24.9
2.4 to 13.4
47 to 64
89.1
59
90.9
100
10 to 800
5946
280
Bw
NLOS
6194
270 to 360
1390 to 1790
2857
714
1500 to 5500
434.78
1160 to 1960
2000
1429
45 to 180
LOS
λ (MHz)
Table 5.5. Estimate of the average cluster and ray arrival rates for diﬀerent analyses of the UWB channel. The published values have been adapted to the deﬁnition of the parameters Λ and λ used in this book
10 to 25
UCANCEA LETI [KEI 03]
21.98
UltRaLab [CRA 02]
NLOS 100 to 1000
LOS
Λ (MHz)
Whyless.com [KUN 02a]
campaign
Measurement
Statistical Modeling of the UWB Channel 149
150
UltraWideband Radio Propagation Channels
situated around a circle of 20 cm in radius. For each measurement, the space ﬂuctuation statistics were studied by comparing the 90 impulse responses collected locally. The study consisted of analyzing the statistical distribution of the amplitude of the received signal at each delay. In the literature, previous studies showed that this distribution was well represented using a Nakagami function [CAS 02]. The Nakagami m parameter was thus estimated for each measured PDP, at delays separated by 0.5 ns on the time scale (see Appendix B.1.3). An example of values for the parameter m is given for a speciﬁc measurement in Figure 5.9. As can be noted, the best ﬁtted Nakagami distribution presents a parameter m close to 1 for the vast majority of the delays within the PDP. Thus, the amplitude distribution of the impulse response for a given delay may be correctly described using a Rayleigh distribution. This was observed for most of the measured PDP. In addition, we may observe that the cases where the parameter m takes higher values correspond to the main paths of the PDP. In this case, the value of the parameter m may increase to m = 4 or m = 5, and up to m = 8 in the extreme cases. The characterization of amplitude distribution in the UWB impulse response is a controversial issue in the literature. The Rayleigh distribution has frequently been observed in the study of UWB radio channel fast fading statistics [CRA 02, KUN 02a, SCH 04, KAR 04b]. Researchers from ETH Z¨ urich [SCH 04] and from the Lund University [KAR 04b] also noted a modiﬁcation of this distribution for the main path. Other studies show that the fast fading is well represented by a Nakagami distribution [CAS 02, CAS 04b, BAL 04b]. In this case, the parameter m may vary with the delay. However, it has been demonstrated that a theoretical impulse response presenting a Rayleigh fast fading may be observed as following a Nakagami distribution, depending on the duration over which the received power is integrated in the time domain [KUN 03]. Finally, it should be noted that some analyses recommend that we use a lognormal distribution [KEI 03, FOE 03b, LI 03, BUE 03, MCK 03] or a Rice distribution [HOV 02]. In practice, we may distinguish between two sources of channel ﬂuctuations [HAS 94a]. Spatial variations arise when at least one of the antennas is moved in an otherwise static environment. Temporal variations are due to environmental modiﬁcations and may be observed in ﬁxed radio links. A study of the channel variations when the transmitting antenna is moved on a 1 m2 grid was presented in [PAG 04]. Results show that in the case of the UWB channel, the delay linked to the main paths of the impulse response signiﬁcantly varies while the antenna is moved. This shift of the main paths in the time domain is due to the high resolution of UWB signals and needs to be taken into account when modeling the channel spatial variations. More details on these observations are
Relative power (dB)
Statistical Modeling of the UWB Channel
0
151
PDP
10 20 30 40 50
0
20
40
60 80 100 Delay (ns)
120
140
0
20
40
60 80 100 Delay (ns)
120
140
Nakagami m
4 3 2 1 0
Figure 5.9. Nakagami m parameter analysis. PDP measured in an NLOS situation and parameter value m for each delay
available in [PAG 05]. In indoor environments, the temporal variations of the propagation channel are mainly due to moving people. This phenomenon is studied more particularly in the following section.
5.1.4. Eﬀect of moving people The study of the eﬀect of moving people on the UWB channel is based on the analysis of a measurement campaign performed in an indoor oﬃce environment. For this campaign, a UWB sounder enabling real time measurements over the 4–5 GHz frequency band was used. During the measurement process, a group of 1 to 12 people was walking in a corridor in the vicinity of a ﬁxed radio link. A complete description of this campaign is given in section 3.4.2.3. 5.1.4.1. Observation of temporal variations A typical impulse response collected during the experiment is given in Figure 5.10. In this graph, the delay has been converted to path length in meters to ease the interpretation of multipaths. After the direct path (a), whose length corresponds to the transmitterreceiver distance (11 m), we may distinguish two echoes (b) and (c) corresponding to reﬂections on the corridor walls.
152
UltraWideband Radio Propagation Channels
(a)
Power attenuation (dB)
70
(b) 80
(c)
90 100 110 0
10
20 30 40 Distancedelay (m)
50
60
Figure 5.10. Typical impulse response
A ﬁrst observation of the eﬀect that moving people have on the UWB channel is given in Figure 5.11. The time varying impulse response is represented in the case of 12 people walking back and forth through the radio link. Successive measurements are represented from left to right, while the vertical axis represents the excess delay converted in path length (m). The inﬂuence that moving people have on the CIR appears clearly on this graph. The main paths (a) and (b) are regularly obstructed by moving people, during both forward (t = 15 s to t = 27 s) and backward (t = 63 s to t = 74 s) displacements along the corridor. At other values of the excess path, we can observe strong signal ﬂuctuations, with respect to the stationary part of the diagram. 5.1.4.2. Slow fading As a ﬁrst step in the analysis, we observed the slow fading generated by human beings in the vicinity of the radio link. For this purpose, we studied the slow temporal evolution of the mean power received in the LOS path in the presence of people. Fast fading ﬂuctuations were eliminated by averaging the received power using a sliding window. Figure 5.12 presents the aggregate eﬀect of several people passing through the main signal path. In general, the progression of one person through the LOS path yields a maximum attenuation of about 8 dB, the shadowing eﬀect lasting for about 4 s. The obstruction duration increases with the number of people, up to about 15 s for a group of 12 people. In this case, the maximum attenuation of the mean power is about 15 dB. We clearly see that the shadowing pattern obtained for groups of people is composed of superimposed
Statistical Modeling of the UWB Channel
0
0 ← Forward mov. →
5
← Backward mov. →
5 10
15 15 20
(b) 20
25 30
25
35
30
40
(c)
35
45 0
Relative attenuation (dB)
(a)
10 Distancedelay (m)
153
20
40 Time (s)
60
80
40
Figure 5.11. Time varying impulse response in the case of 12 moving people
individual contributions. However, the particular eﬀect that each person has on the received signal is not always observable, as, for example, in the case involving 4 people. 5.1.4.3. Fast fading In addition to the largescale fading generated by moving people, smallscale fading is observed while people are walking in the vicinity of the radio link. The signal received at a delay corresponding to the main paths of the impulse response may be thought of as a vector summation of several multipath components, as would be the case in a typical situation exhibiting Rician fading. As depicted in Figure 5.13, we may distinguish between two components of the received signal. The dominant component accounts for the signal normally received in a static environment. The random component represents the summation of all waves scattered by moving people. In order to accurately analyze the fast fading observed during experimentation, the random component was isolated by ﬁltering. Figure 5.14 represents both magnitude and phase ﬂuctuations of the random component extracted from the signal received via the main path, when obstructed by
Relative power (dB)
15
20
25
30
15
20
25
30
0
4
8
12
16 20 24 Time (s)
Received power Slow fading
4 people
28
32
36
30
25
20
15
10
5
0
5
0
4
8
12
16 20 24 Time (s)
12 people
Figure 5.12. Typical largescale fading patterns for the LOS component. Eﬀect of 1, 4 and 12 people
10
10
8 12 16 Time (s)
5
5
4
0
0
0
5
1 people
5
28
32
36
154 UltraWideband Radio Propagation Channels
Statistical Modeling of the UWB Channel
155
Q Random component I
Dominant component
time
Figure 5.13. Vector decomposition of the received signal
12 moving people. This situation corresponds to the measurement depicted in Figure 5.12 (graph on the right). We may notice that the magnitude of the smallscale fading presents a high degree of regularity, compared to the nonstationary largescale fading pattern. Using a KolmogorovSmirnov testing procedure (see Appendix B.2), the amplitude distribution of the random component is well suited to a Rayleigh distribution. This observation could be made for most of the available measurements, independently of the number of moving people. The signal received via the main paths of the impulse response, composed of a Rayleigh random component and a dominant component of greater amplitude, the resulting signal hence follows a Rician distribution (see Appendix B.1.2). The Rician K parameter is deﬁned as the power ratio between the dominant component and the random ﬂuctuations. As the dominant component presents a slowly timevariant amplitude, the parameter K varies over time accordingly, depending on the obstruction of the main path. We may notice in this example that the mean signal power of the random component is about 12 dB below the unobstructed signal level. Depending on the attenuation level of the dominant signal, the parameter K varies from about 12 dB down to less than −20 dB. In this case, the total received signal follows a Rayleigh distribution. These results were observed on other measurement records, with maximum values of the parameter K varying between 8 dB and 13 dB. In addition, the phase of the random component depicted in Figure 5.14 presents a rather unstable behavior while people are interfering with the main path of the impulse response, with distinct periods of linear progress. However,
156
UltraWideband Radio Propagation Channels
Relative power (dB)
0 10 20 30 40 50
↑ 14
16
18
20 Time (s)
Phase (rad)
↓
↑
22
24
22
24
↓
2 0 2 14
16
18
20 Time (s)
Figure 5.14. Fast fading of the random component. LOS path, 12 people
we may note rapid and signiﬁcant phase shifts at instants corresponding to fading nulls (indicated by arrows on the ﬁgure), due to the rapidly decreasing power. 5.1.4.4. Spectral analysis This section presents a spectral analysis the analysis of the temporal signal variations received via the main paths of the impulse response. Figure 5.15 represents the average scattering function PS (τ, ν) of the random component, for the delay τ corresponding to the LOS path. Measurements involving 1 to 12 moving people are taken into account, and the collected records were distinguished according to the direction of the movement. The general shape of the scattering function is triangular, with an average Doppler shift centered around 0 Hz. The spectrum width was calculated in terms of Doppler spread νRM S , deﬁned in equation [2.38]. The calculated Doppler spread varied between 0.6 and 3.3 Hz, with no marked inﬂuence from the number of moving people. Similar 0 Hzcentered, triangular shapes of the Doppler spectrum are reported from continuous wave measurements of the channel temporal variations performed at frequencies around 1 GHz [HAS 94b, BUL 87].
Statistical Modeling of the UWB Channel
157
Relative power (dB)
0 Forward Backward
5 10 15 20 25 15
10
5 0 5 Doppler shift (Hz)
10
15
Figure 5.15. Average scattering function of the random component. LOS path, forward and backward motions
Some asymmetry can be observed in Figure 5.15 between measurements performed during forward and backward movements. This can be explained by the location of the antennas, which emphasize either the lengthening or the shortening of propagation paths, depending on the direction of the human motion. More details on the interpretation of these results can be found in [PAG 06a].
5.2. Statistical channel modeling In section 5.1, the characterization of the radio channel parameters was presented from a series of experimental measurements. The frequency dependent and distance dependent attenuation linked to the radio signal propagation was evaluated, as well as a number of parameters regarding the impulse response. The RMS delay spread, the PDP magnitude decay and the rays arrival rate are a few examples. Statistical models use these experimental parameters to realistically reproduce the channel eﬀects. This section presents a few statistical models for the UWB propagation channel available in the literature. The analyses presented in section 5.1 are then exploited to illustrate in more detail the design of a statistical model based on experimental data.
158
UltraWideband Radio Propagation Channels
5.2.1. Examples of statistical models In order to provide a unique channel model for the evaluation of diﬀerent UWB systems proposals during standardization meetings, the IEEE 802.15 work group made several calls for contributions. Two statistical models were deﬁned, the ﬁrst one for short range, high rate, indoor applications (IEEE 802.15.3a model), and the second one for applications with longer range in both indoor and outdoor environments (IEEE 802.15.4a model). These models are brieﬂy presented in the following sections. 5.2.1.1. IEEE 802.15.3a model The IEEE 802.15.3a model [FOE 03a, MOL 03] was developed from around 10 contributions, all referring to distinct experimental measurements, performed in indoor residential or oﬃce environments [GHA 02a, PEN 02, FOE 03b, HOV 03, KUN 02b, GHA 02b, CAS 02, CRA 02, SIW 02]. In order to reﬂect the phenomenon of ray clustering that was observed in several measurement campaigns, the model is based on the Saleh and Valenzuela formalism (see equation [2.31]). Parameters are provided to characterize the clusters and ray arrival rates (Λ and λ), as well as the inter and intracluster exponential decay constants (Γ and γ). Four sets of parameters are provided to model the four following channel types: • the channel model CM 1 corresponds to a distance of 0–4 m in a LOS situation; • the channel model CM 2 corresponds to a distance of 0–4 m in an NLOS situation; • the channel model CM 3 corresponds to a distance of 4–10 m in an NLOS situation; • the channel model CM 4 corresponds to an NLOS situation with a large delay spread τRMS = 25 ns. Regarding channel attenuation, the IEEE 802.15.3a model proposes a theoretical approach using a path loss exponent Nd = 2 for the LOS situation, which is equivalent to a free space propagation. The NLOS case was not addressed. Finally, the ﬂuctuations of ray amplitude modeled using a lognormal law (see Appendix B.1.6), and a random inversion coeﬃcient is introduced to simulate the phase inversion observed in the impulse response due to reﬂections. This comprehensive model is a reference for the study of UWB systems. It can be applied in indoor environments and short range conditions. However,
Statistical Modeling of the UWB Channel
159
modeling of the path loss is not practically addressed. It can also be noted that the measurements used for the model calibration [PEN 02, CHE 02] are limited to at most 6 GHz bandwidth (2 GHz for the models CM 1 and CM 2). 5.2.1.2. IEEE 802.15.4a model In order to be representative of a larger number of potential applications, the IEEE 802.15.4a working group proposed a model with a wider scope in terms of both frequency and environments [MOL 04]. The targeted applications are low rate communications (from 1 kbps to a few Mbps), in the following environments: indoor (residential and oﬃce), outdoor, industrial (factory, etc.) and onbody (for WBAN applications). Two UWB frequency bands are considered: 2–10 GHz and 0.1–1 GHz. We present here the model corresponding to the ﬁrst frequency band. The general structure of this statistical model is similar to the IEEE 802.15.3a model (see equation [2.31]). Some diﬀerences are however noteworthy regarding the shape of the impulse response: • The phase θk,l of each ray is no longer limited to the values 0 or π, but is uniformly distributed between 0 and 2π. Thus, this model reproduces the complex envelope of the baseband impulse response. • Ray arrival follows a dual law composed of two Poisson processes. Accordingly, the model proposes two ray arrival rates λ1 and λ2 , as well as a mixing parameter. • The exponential decay of each cluster increases with the delay. The intracluster exponential decay constant is thus of the following form: γl = kγ Tl + γ0
[5.7]
where Tl represents the time of arrival of the lth cluster and kγ accounts for the increase of the coeﬃcient γl with increasing delay. The main diﬀerence between the IEEE 802.15.4a model and the IEEE 802.15.3a model is that the former accounts for a realistic path loss, in both the distance and frequency domains. The proposed model is independent of the transmitter and receiver antennas. After some changes in the proposed variables, this path loss model can be expressed in the form given in equation [2.35], and the model provides the parameter values equivalent to Nf , Nd , P L(f0 , d0 ) and σS . In addition, the small scale variations of the ray amplitude are modeled using a Nakagami distribution (see Appendix B.1.3). As for the CassioliWinMolisch model, the parameter value m is a delay function.
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UltraWideband Radio Propagation Channels
Nine channel types were identiﬁed within the IEEE 802.15.4a work group. Each channel type is deﬁned by a given set of parameters as follows: • the CM 1 and CM 2 models correspond to the indoor residential environment (respectively in LOS and NLOS conﬁgurations); • the CM 3 and CM 4 models correspond to the indoor oﬃce environment (respectively in LOS and NLOS conﬁgurations); • the CM 5 and CM 6 models correspond to the outdoor environment (respectively in LOS and NLOS conﬁgurations); • the CM 7 and CM 8 models correspond to the indoor industrial environment (respectively in LOS and NLOS conﬁgurations); • the CM 9 model corresponds to the outdoor environment in an NLOS conﬁguration, in the speciﬁc case of a farm area or an area covered with snow. This model is more detailed than the IEEE 802.15.3a model, but is also somewhat more complex. The provided parameter sets are based on experimental measurements for each environment: residential [CHO 04b], oﬃce [BAL 04a, SCH 04], industrial [KAR 04b] and outdoor [BAL 04a, KEI 04]. A survey of the results from the literature complements these measurement campaigns. It should be noted, though, that the measurements performed in the oﬃce and outdoor environments covered frequency bands limited to 3 or 6 GHz. 5.2.1.3. Other models The CassioliWinMolisch model The CassioliWinMolisch model [CAS 02] is the result of a joint research work performed by the University of Rome “Tor Vergata” (Italy), Vienna University (Austria) and the UltRaLab laboratory from the University of South California (USA). This model is one of the ﬁrst statistical models describing the UWB propagation channel. Despite some limitations, it is thus frequently cited in the studies on UWB. This model is based on a series of measurements performed by the UltRaLab laboratory in an indoor oﬃce environment, over a frequency band of about 1 GHz [WIN 97b]. 686 impulse responses were used: 14 antenna locations were selected, and at each location, 49 measurements were taken over an area of about 1 m2 . The CassioliWinMolisch model is based on a discrete scale on the delay axis, with a delay bins deﬁned with a step Δτ of 2 ns. The overall impulse response power received between the delays kΔτ and (k + 1)Δτ is integrated, and it is assumed that a ray exists at each delay kΔτ . This corresponds to a 1 . The power of each ray follows an exponential decay, ray arrival rate λ = Δτ
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but a single cluster is observed. Following the Saleh and Valenzuela formalism (see section 2.4.1), the PDP model can be expressed in the form: Ph (0, τ ) =
K k=1
βk2 δ
d τ − − (k − 1)Δτ c
[5.8]
where d indicates the distance between the transmitter and the receiver. The PDP exponential decay is characterized by the coeﬃcient γ (see equation [2.33]), but the model also introduces an additional coeﬃcient r in order to account for a signiﬁcant attenuation between the 1st ray and the 2nd ray. The propagation loss is characterized as a function of distance according to a dualslope law: ⎧ ⎪ ⎪P L d0 + 20.4 log d + S(d) d ≤ 11 m ⎪ ⎨ d0 [5.9] P L(d) = ⎪ d ⎪ ⎪ + S(d) d > 11 m ⎩P L d0 − 56 + 74 log d0 The fast fading due to the antenna displacement is characterized by a ray amplitude distribution following a Nakagami distribution (see Appendix B.1.3). The Nakagami m parameter decreases with the delay to approach the value of 1 for the last rays in the PDP, where the ray amplitude follows a Rayleigh distribution. This model provides an accurate and reproducible description of the observed measurements. Its main limitations lie in the low number of measurements on which the statistical study is based, and in the reduced frequency band. Frequency domain approach Most of the research eﬀorts regarding statistical modeling for the UWB propagation channel concentrate on a time domain approach, where the model provides a description of the channel impulse response. However, other approaches are possible. The model proposed by the researchers from the AT&T Research Laboratory and MIT [GHA 04a] is an interesting example of a frequency domain approach. This model is designed from an extensive measurement campaign performed in 23 residential houses. The main concept of this model is to reproduce the channel transfer function T (f, t) in a statistical way. As for the time domain approach, each parameter
162
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can be described using a statistical law. The value of the frequency domain approach lies in its opening up the possibility of describing the transfer function components in a regressive way. The model can be expressed in the form of a ﬁlter with inﬁnite impulse response, which can be mathematically represented by: [5.10] T fi , t + a1 T fi−1 , t + a2 T fi−2 , t = ni where ni represents the input white Gaussian noise. The model can be represented using ﬁve variables: the parameters a1 and a2 , the input conditions T (f1 , t) and T (f2 , t), and the standard deviation of the Gaussian noise σn . Each of these parameters is then described in a statistical way as a function of the distance. The model is complemented with a power attenuation law. Other approaches were proposed regarding UWB channel modeling in the frequency domain, such as the Prony method used by the Virginia Polytechnic Institute (USA) [LIC 03]. The value of such a model lies in its low complexity: only a few parameters are required for its description. However, there is no direct knowledge of the traditional channel characterization parameters, such as the shape of the PDP or the delay spread. For this reason, the comparison between the two model types is not straightforward. 5.2.2. Empirical modeling principles This section aims at illustrating UWB channel statistical modeling based on experimental data. A statistical model based on the Saleh and Valenzuela formalism is designed from the characteristic parameters observed in section 5.1 [PAG 06d]. The following sections present a propagation loss model and an impulse response model. Finally, simulation results are compared to the experimental measurements. 5.2.2.1. Propagation loss model The ﬁrst step in the design of the UWB channel model is the deﬁnition of attenuation due to signal propagation. In section 5.1.1.1, it was shown that when the antenna eﬀect is correctly compensated, the channel attenuation in the frequency domain is close to the theoretical loss of 20 dB per decade. The path loss model can thus be expressed in dB in the following form: f d + 10Nd log + S(d) [5.11] P L(f, d) = P L f0 , d0 + 20 log f0 d0 where d0 = 1 m represents a reference distance, f0 = 6.85 GHz corresponds to the center frequency of the FCC analyzed band and S is a Gaussian random variable with zero mean. The parameters of this model are deﬁned by experimental characterization (see section 5.1.1) and are given in Table 5.6.
Statistical Modeling of the UWB Channel
LOS
NLOS
Nd
1.62
3.22
σS (dB)
1.7
5.7
P L(f0 , d0 ) (dB)
53.7
59.4
163
Table 5.6. Path loss model parameters. Values of P L(f0 , d0 ) are given for f0 = 6.85 GHz and d0 = 1 m
5.2.2.2. Modeling the channel impulse response over an inﬁnite bandwidth The principle of the UWB impulse response model consists of generating all constituent rays while maintaining the characteristics observed from the experimental measurements. Among these characteristics, we mainly aim at reproducing the clustering of multipath echoes, the ray and cluster arrival rates, and the decreasing magnitude of the received power with increasing delay. For a baseband representation, a ray is described by its delay τ , its amplitude β and its phase θ. By generating these parameters, it is possible to describe the impulse response over an inﬁnite bandwidth. We then process these parameters in the frequency domain in order to include the eﬀect of the limited observation bandwidth. In order to account for the clustering of multiple paths, the impulse response is modeled using the Saleh and Valenzuela formalism (see section 2.4.1). At a given instant, the UWB channel impulse response is thus described by the following formula: h(τ ) =
Kl L
βk,l ejθk,l δ τ − Tl − τk,l
[5.12]
l=1 k=1
where L represents the number of clusters, Kl is the number of rays within the lth cluster and Tl corresponds to the arrival time of the lth cluster. The parameters βk,l , θk,l and τk,l represent the amplitude, phase and arrival time associated with the k th ray within the lth cluster. The experimental observations given in section 5.1.2.3 show that the interclusters duration follows an exponential distribution. Hence, the arrival of a new cluster can be modeled by a Poisson process, and the number of clusters in the impulse response can be generated by drawing a random variable L according to the following law [MOL 04]: pL (L) =
¯ L exp(−L) ¯ (L) L!
[5.13]
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UltraWideband Radio Propagation Channels
¯ represents the average number of clusters. During measurement where L ¯ = 5.6 in the LOS analysis, we observed an average number of clusters of L ¯ case and L = 2.4 in the NLOS case. Having selected a transmitterreceiver distance d for the simulation, the time of arrival of the ﬁrst cluster is given by T1 = dc , where c is the speed of light. The time of arrival Tl of the L − 1 remaining clusters is then calculated by generating intercluster durations following an exponential law [SAL 87]: [5.14] p Tl  Tl−1 = Λ exp − Λ Tl − Tl−1 where Λ is the cluster arrival rate. The intercluster power decay is accurately modeled by a power function (see section 5.1.2.2). This approach diﬀers from the one followed by Saleh and Valenzuela [SAL 87], but presents a closer ﬁt to the experimental measurements. The amplitude of the ﬁrst ray within each cluster is thus given by: −Ω Tl 2 2 [5.15] β1,l = β1.1 T1 where Ω represents the intercluster power decay constant. In accordance to the results of the static UWB channel study, it is recommended to use the values Λ = 36.5 MHz and Ω = 4.4 in the LOS case. In the NLOS case, the recommended values are Λ = 24.9 MHz and Ω = 3.9. The rays are iteratively generated for each cluster. The arrival time of each ray is calculated using interray durations following an exponential law [SAL 87]: [5.16] p τk,l  τk,l−1 = λ exp − λ τk,l − τk,l−1 A power function is used to calculate the ray amplitude (see section 5.1.2.2): −ω G τk,l + Tl 2 2 = 10− 10 β1,l [5.17] βk,l Tl where ω represents the intracluster power decay constant and G accounts for the observed attenuation between the ﬁrst path of each cluster and the following multipaths. For each cluster, the ray generation stops when the ray amplitude reaches a given threshold D, ﬁxed at −50 dB. According to the characterization study presented in section 5.1 and to the observation of the measured PDP, we recommend using the parameters λ = 5.95 GHz, ω = 11.1 and G = 12 dB in the LOS case. The recommended values in the NLOS case are λ = 6.19 GHz and ω = 10.2.
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Finally, the phase θk,l of each ray is generated using a uniform law over the interval [0, 2π[. This approach was also adopted in the IEEE 802.15.4a model [MOL 04]. Figure 5.16 presents the rays obtained in a LOS situation for a transmitterreceiver distance of 6 m, where we may observe 5 clusters. It may be noted that this representation corresponds to an inﬁnite observation bandwidth, each ray being represented by a Dirac function.
Relative power (dB)
0 10 20 30 40 50
0
20
40
60 Delay (ns)
80
100
Figure 5.16. Impulse response simulated over an inﬁnite bandwidth. LOS situation, with the parameters d = 6 m, Λ = 36.5 MHz, λ = 5.95 GHz, Ω = 4.4, ω = 11.1 and G = 12 dB
Table 5.7 summarizes all experimental parameters used in our statistical model of the UWB channel. LOS
NLOS
fmin (GHz)
3.1≤ fmin < fmax
fmax (GHz)
fmin < fmax ≤ 10.6
D (dB)
50
Λ (MHz) ¯ L
36.5
24.9
5.6
2.4
λ (GHz)
5.95
6.19
Ω
4.4
3.9
ω
11.1
10.2
G (dB)
12
0
Table 5.7. Parameters of the UWB impulse response model
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UltraWideband Radio Propagation Channels
5.2.2.3. Modeling the channel impulse response over a limited bandwidth In practice, impulse response is observed on a limited bandwidth, situated within the 3.1–10.6 GHz band. We note by fmin and fmax the minimum and maximum frequencies in the observation band, and by fc = 12 (fmin + fmax ) the center frequency. The channel transfer function limited to the observation bandwidth hence appears (for positive frequencies): ⎧ Kl L ⎪ 1 ⎨fc βk,l ej(θk,l −2πf (Tl +τk,l )) if fmin ≤ f ≤ fmax 'L 'Kl 2 Tlim (f ) = f β k=1 k,l l=1 k=1 l=1 ⎪ ⎩ 0 otherwise [5.18] where the coeﬃcient
P L(fc ,d) ' L ' Kl 2 l=1 k=1 βk,l
normalizes the power received at the
frequency fc according to the proposed path loss model. In addition, the term ffc accounts for the power decrease in −20 log(f ) described previously. This transfer function normalization procedure was also proposed in [FOE 03a, ALV 03]. The impulse response observed over a limited bandwidth hlim (τ ) is simply obtained from the transfer function Tlim (f ) using an inverse Fourier transform. In order to limit the side lobes level, it is possible to use a Hanning window for instance at this stage [HAR 78]. Figure 5.17(a) represents an impulse response example observed over the 3.1–10.6 GHz frequency band. We may note that the low delay between two consecutive rays (see Figure 5.16) naturally induces constructive or destructive interferences. Figure 5.17(b) presents the same impulse response observed over 3.1–4.1 GHz frequency band: we may note that the multipath resolution is less accurate when the observation bandwidth decreases. 5.2.2.4. Simulation results A series of simulations was conducted using the static UWB radio channel model presented in section 5.2.2. A set of 119 PDP was generated using the same transmitterreceiver distances as in our measurement campaign (see section 3.4.2.1), for the LOS and NLOS situations. Each PDP was calculated from a set of 90 impulse responses, simulating the motion of the antenna around a circle of radius 20 cm. The generation of the fast fading characteristics for the 90 locally simulated impulse responses was performed using an advanced algorithm, taking the ray arrival angle into account. This algorithm is fully described in section 5.3.1. Figure 5.18 presents two typical PDP obtained by simulation. One of the 90 constituting impulse responses is also represented. Note that on the xaxis,
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167
(a) 60 Relative power (dB)
70 80 90 100 110 120 130
0
20
40
60 Delay (ns)
80
100
80
100
(b) 60 Relative power (dB)
70 80 90 100 110 120 130
0
20
40
60 Delay (ns)
Figure 5.17. Impulse response simulated over a limited bandwidth. Identical set of rays, observed over the 3.1–10.6 GHz band (a) and the 3.1–4.1 GHz band (b)
the delay was converted in path length. As a general observation, simulations are similar to the measured PDP (see Figure 5.3). Two parameters representative of the channel temporal dispersion were calculated for the entire set of generated PDP: the delay spread τRM S and the 75% delay window W75% . The average values obtained for these parameters in measurement and simulation are compared for the LOS and NLOS situations in Table 5.8. We may note that the proposed model correctly reproduces the dispersion parameters experimentally measured over the UWB radio channel. The delay spread of the simulated PDP is particularly close to the measurement, in both LOS and NLOS conﬁgurations. The values obtained for the delay window are somewhat diﬀerent from the experimental data, but stay in the same range as the measured characteristics. The proposed model therefore allows us to reproduce not only the impulse response structure, but also channel dispersion.
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UltraWideband Radio Propagation Channels
(a)
Relative power (dB)
0
IR PDP
10 20 30 40 50
0
20
40 Distancedelay (m)
60
80
(b)
Relative power (dB)
0
IR PDP
10 20 30 40 50
0
20
40 Distancedelay (m)
60
80
Figure 5.18. PDP and impulse responses obtained by simulation. (a) LOS and (b) NLOS conﬁgurations
LOS
NLOS
Parameter
Measurement
Simulation
Measurement
Simulation
τRMS (ns)
4.1
4.0
9.9
9.7
W75% (ns)
7.6
9.7
23.7
21.2
Table 5.8. Comparison of the dispersion parameters: measurement vs. simulation
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169
5.3. Advanced modeling in a dynamic conﬁguration Section 5.2.2 describes how to simulate an impulse response from statistical parameters obtained through experimental measurements, in the case of a static channel. In instances of practical use for future UWB systems, the antenna displacement or the motion of people may generate signiﬁcant variations in the channel impulse response. This section describes two adaptations to the statical model for an advanced modeling of the spatial and temporal ﬂuctuations of the UWB radio channel [PAG 06c]. 5.3.1. Space variation modeling Spatial variations are due to the displacement of at least one of the transmitting or receiving antennas. In this case, the lengths of most propagation paths undergo signiﬁcant variation. Only a few studies on the statistical modeling of the UWB channel spatial variations are available. The IMST model is one example where a simpliﬁed ray tracing algorithm is used to calculate a ﬁxed delay variation for each cluster [KUN 03]. We here propose to model these spatial ﬂuctuations by taking the angle formed between the antenna motion and each propagation path into account. In the remainder of this section, we will assume a displacement of the receiving antenna and consider the arrival angle of each path. For the sake of clarity, the presented model allows an antenna displacement in the (O, x, y) plane only, by taking the ray azimuth ϕ into account. A 3D model including ray elevation is forwardly derivable. The ﬁrst step of this spatial ﬂuctuation model consists of generating an impulse response over an inﬁnite bandwidth h(x0 , y0 , τ ), corresponding to a location (x0 , y0 ), as described in section 5.2.2. This impulse response observed over an inﬁnite bandwidth may be expressed as in equation [5.12]. In order to calculate the impulse response observed over an inﬁnite bandwidth at a location (x, y) close to the location (x0 , y0 ), we consider that each ray corresponds to a plane wave. This assumption holds when the receiving antenna is placed in the far ﬁeld with respect to the surrounding walls and furniture where the transmission, reﬂection or diﬀraction phenomena occur. Figure 5.19 illustrates the obtained conﬁguration for a given ray. The path elongation Δl of the incident ray is given by: [5.19] Δl = − x − x0 cos(ϕ) − y − y0 sin(ϕ) where ϕ represents the incident ray azimuth. A limited number of statistical studies are available in the literature regarding the departure and arrival angles for UWB channels [CRA 02,
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UltraWideband Radio Propagation Channels
y r ĳ Į (x0, y0)
x
Figure 5.19. Path elongation related to an antenna displacement
HAN 05, VEN 05]. To determine the azimuth of each ray in the impulse response, we can use the model derived by Spencer et al. from wideband indoor measurements [SPE 97]. In this model, the arrival azimuth of the k th ray within the lth cluster is decomposed in Φl + ϕk,l , where Φl is the mean arrival azimuth in the lth cluster. Without further knowledge about the environment, we may consider that Φl is uniformly distributed in the [0, 2π[ interval. ϕk,l represents the distribution of the ray azimuth within a cluster, and follows a Laplace distribution with zero mean and standard deviation σϕ :
pϕk,l ϕk,l
1 − =√ e 2σϕ
√
2ϕk,l σϕ

[5.20]
In [HAN 05], the standard deviation of the arrival azimuth σϕ is lower than or equal to 6.7◦ .3 As a ﬁrst approximation, we recommend using this value for the parameter σϕ . This value could be reﬁned as more experimental data becomes available. For instance, [VEN 05] proposes a 2cluster model, where the azimuth distribution follows a Laplace law in the ﬁrst cluster (σϕ = 5◦ ) and a uniform law in the second cluster. From the impulse response generated at the location (x0 , y0 ) and knowing the arrival azimuth Φl + ϕk,l of each ray, we can calculate the transfer function
3. For the indoor LOS and NLOS conﬁgurations on a single ﬂoor.
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171
over a limited bandwidth at a location (x, y) close to location (x0 , y0 ): Tlim (x, y, f ) ⎧ Kl L ⎪ 1 ⎨fc βk,l ej(θk,l −2πf (Tl +τk,l +Δτk,l )) 'L 'Kl 2 = f β l=1 k=1 k,l l=1 k=1 ⎪ ⎩ 0
if fmin ≤ f ≤ fmax otherwise [5.21]
with: Δτk,l = −
1 x − x0 cos Φl + ϕk,l + y − y0 sin Φl + ϕk,l c
[5.22]
The impulse response hlim (x, y, τ ) observed over a limited bandwidth is simply obtained using an inverse Fourier transform. Figure 5.20 presents a simulation of the space varying impulse response observed over the 3.1–10.6 GHz band. The delay, amplitude and phase of each ray correspond to the parameters already used in the previous examples. The spatial variations of the impulse response were calculated for a displacement of the receiving antenna along the Ox axis over a distance of 2 m with 1 cm step. We may clearly distinguish the evolution of the main echoes (ae), depending on their arrival direction. For instance, the length of the ﬁrst path shortens from approximately 6 m to 4 m. 0
0 5
(a)
5
10 (b)
15 20
15 (c) 25 20 (d)
30
Relative power (dB)
Distancedelay (m)
10
35 25 40
(e) 0
50
100 Location x (cm)
150
200
45
Figure 5.20. Simulation of a space variant impulse response. Displacement of the receiving antenna along the Ox axis over a distance of 2 m
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UltraWideband Radio Propagation Channels
5.3.2. Modeling the eﬀect of people This section describes a method to model the eﬀect of people, based on the experimental analysis given in section 5.1.4. The main idea is to reproduce the time variance observed on each ray of a given impulse response. This model is based on the results of a real time UWB measurement campaign performed over the 4–5 GHz band (see section 3.4.2.3). As a result, the presented model is valid for simulating UWB channels with bandwidths up to 1 GHz only. However, the modeling methodology may be used in other conﬁgurations as more experimental characteristics of the timevariant UWB channel will become available. The extension of the model to temporal variations is based on an initial impulse response h(t0 , τ ) corresponding to a static environment at a given instant t0 . The method described in section 5.2.2 can be used to obtain a description of the impulse response over an inﬁnite bandwidth. The initial impulse response can thus be expressed according to equation [5.12], where αk,l (t0 ) = βk,l eθk,l denotes the initial complex magnitude of the k th ray within the lth cluster. As observed in our experimental characterization, we will diﬀerentiate between (a) the main path of each cluster, where each person generates a shadowing pattern in addition to fast amplitude ﬂuctuations, and (b) the clusters of dense multipath where Rayleigh fading occurs [PAG 06a]. When observing a radio channel over a relatively low bandwidth, in our case lower than 1 GHz, the observed main path encompasses not only the ﬁrst ray in each cluster, but also several rays situated within the observation resolution R. This resolution R is equal to 2 ns for a bandwidth of 1 GHz when using a Hanning window [HAR 78]. For all rays within the main path of each cluster, the timevariant amplitude αk,l (t) is generated according to the algorithm below. A more detailed description of this algorithm may be found in [PAG 05]. Assuming that a group of Np people is interfering with the UWB radio link, the algorithm steps are as follows: i) Select Np instants {tn }n=1...Np corresponding to the instants where each of the Np people are crossing the main path of the cluster. These passing instants may be randomly chosen, or calculated according to the environmental geometry. ii) Generate Np slow shadowing patterns {sn (t)}n=1...Np corresponding to the individual eﬀects of Np moving people, using the following Gaussianshaped attenuation function: ( ) 2 2 [5.23] sn (t) = −As exp − 2 t − tn Ts
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173
where As represents the maximal shadowing attenuation in dB, Ts represents the shadowing duration in seconds, and tn represents the shadowing instant. According to our experimental characterization, the parameter values may be selected in the range of 5 dB to 10 dB for As , and in the range of 3 s to 5 s for Ts . iii) Calculate the amplitude variations of the dominant component dl (t) in the linear scale as follows: dl (t) =
Np
10
sn (t−tn ) 20
[5.24]
n=1
iv) Generate additional fast fading ﬂuctuations rk,l (t) following a Rayleigh amplitude distribution and having a Laplacian scattering function. As observed from the experimental data, the Doppler spread νRMS should be in the range of 1–3 Hz. The mean power Pr of the random component rk,l (t) should be 8 dB to 13 dB below the power level of the dominant component αk,l (t0 ). v) Finally, calculate the timevariant amplitude αk,l (t) for rays within the main path of the lth cluster as: αk,l (t) = αk,l t0 dl (t) + rk,l (t) [5.25] Regarding step iv), several methods for the generation of a random signal presenting a Rayleigh distribution and an arbitrary Doppler spectrum are thoroughly discussed in [PAE 02]. The algorithm presented above allows us to generate the eﬀect that moving people have on the main path of each cluster. For the remaining rays, corresponding to regions of dense multipath, we propose to model the amplitude αk,l (t) variations by using a Rayleigh distribution. More experimental analysis should be performed to study the exact shape of the Doppler spectrum in the regions of dense multipath. As a ﬁrst step, we propose using a Laplacian distribution. Thus, the amplitude αk,l (t) may be generated as in step iv) of the algorithm. Having calculated the amplitude αk,l (t) for all impulse response rays, the time variant transfer function is given as: Tlim (t, f ) ⎧ Kl L ⎪ 1 ⎨ fc αk,l (t)e−j2πf (Tl +τk,l ) 'L 'Kl 2 = f β l=1 k=1 k,l l=1 k=1 ⎪ ⎩ 0
if fmin ≤ f ≤ fmax otherwise [5.26]
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UltraWideband Radio Propagation Channels
The impulse response hlim (t, τ ) observed over a limited bandwidth is obtained using an inverse Fourier transform. The proposed model was used to simulate the motion of 4 people in the vicinity of a radio link in a LOS situation. For selected realistic passing instants with respect to the main path of each cluster, the delays Tl and the passing instants tn were calculated using a ray tracing tool, from which only the most signiﬁcant rays were extracted. Figure 5.21 illustrates the simulation conﬁguration. The ﬁve selected paths correspond to the main path and to the reﬂections on the four walls. The dashed arrow indicates the motion of the group of 4 people and circles present locations where the group crosses one of the main echoes.
Rx
Tx
Figure 5.21. Simulation conﬁguration for the eﬀect of people
The time varying impulse response was simulated according to this information, and the results are presented in Figure 5.22. The impulse response is observed over the 4–5 GHz frequency band, as in an experiment. The following parameter values were used for the temporal variations model: As = 5 dB, Ts = 4 s, Pr = −13 dB and νRMS = 1 Hz. As indicated by the circle locations on the graph, the group of people interferes with each main path at a diﬀerent instant. For each shadowing instant, we may observe a power attenuation of the corresponding path, for a duration in the order of 3 s to 6 s. In addition, the presence of mobile people generates fast fading, observable at all impulse response delays. It can be noted that on occasions, this fast fading signiﬁcantly interferes with the main paths, as for instance on the second cluster. This proposed model could be reﬁned by analyzing additional experimental results. In particular, further research could be conducted by sounding the UWB channel over a larger bandwidth.
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175
Figure 5.22. Simulation of a time variant impulse response. Circles indicate the group passing instants for the ﬁve main paths of the impulse response
5.4. Conclusion Statistical modeling of the UWB channel consists of reproducing the main channel characteristics observed during the measurements using a mathematical description. The experimental analyses described in this chapter illustrated the characterization of the UWB propagation channel in an indoor oﬃce environment. Path loss was analyzed in both frequency domain and distance domain. The measured PDP were characterized through the delay spread, as well as the arrival rate and amplitude decay of the rays and clusters. The fast fading analysis shows that the signal amplitude received at a given delay follows a Rayleigh distribution, except for the main paths, where the situation is more deterministic. Several models of the UWB propagation channel were presented in the literature. Among the statistical models, the proposals presented in the standardization groups IEEE 802.15.3a and IEEE 802.15.4a, based on a Saleh and Valenzuela approach, are frequently used for the analysis of UWB communication systems. The principles of statistical modeling were presented using a practical approach based on experimental data. The described path loss model can be used directly for dimensioning studies and to evaluate the jamming generated by an UWB terminal. A detailed model has also been given for the static impulse response. Simulations show that this model accurately reproduces the channel dispersion. It can be used in system simulations, for the design and development of UWB based radio transceivers.
176
UltraWideband Radio Propagation Channels
Two extensions of the impulse response were ﬁnally described to account for the spatial and temporal variations of the UWB channel. The spatial variations model emulates the eﬀects of antenna displacement, by including the arrival direction of the delayed wavefronts. The temporal ﬂuctuations model integrates the eﬀects linked to the motion of people, and can be used to simulate a UWB radio link in realistic conditions.
Appendix A
Baseband Representation of the Radio Channel
A real signal x(t) with spectral components covering a bandwidth centered on the frequency f0 = 0 can be represented by its complex envelope γx (t) deﬁned by:
[A.1] x(t) = γx (t)ej2πf0 t There are several ways to deﬁne a complex envelope γ(t) able to represent the signal x(t) according to relation [A.1]. We will use here the complex envelope deﬁned by the analytical form of the signal x(t) as follows [BAR 95]: γx (t) = x(t)e−j2πf0 t
[A.2]
where x(t) is the analytical representation of signal x(t). A real signal possesses a complex spectral structure with a Hermitian symmetry. All the information concerning this signal is known in the positive part of the frequency spectrum. The analytical signal is the signal representation using only its positive spectral frequency components. In order to keep the same signal power, the power spectral density (PSD) of the spectrum’s positive side is multiplied by two. The obtained analytical signal is thus deﬁned in the frequency domain by:
[A.3] X(f ) = F x(t) = 2U (f )X(f ) = 1 + sign(f ) X(f ) where F{·} is the Fourier transform operation and U (f ) represents the unit step function (Heaviside step function).
178
UltraWideband Radio Propagation Channels
By using the inverse Fourier transform, it leads to the following relation: x(t) = x(t) + j
1 ⊗ x(t) = x(t) + jH x(t) πt
[A.4]
where ⊗ represents the convolution operator and H{·} represents the Hilbert transform. The signal s(t) is obtained after a convolution of e(t) and h(t): s(t) = e(t) ⊗ h(t)
[A.5]
By keeping the same notations as previously, we have [BAR 95]: S(f ) = 2U (f )S(f ) = 2U (f )E(f )H(f ) = and then:
1 E(f )H(f ) 2
γs (t) = s(t)e−j2πf0 t = F −1 S(f + f0 ) 1 E(f + f0 )H(f + f0 ) = F −1 2 =
[A.6]
[A.7]
1 γe (t) ⊗ γh (t) 2
There is a second way to represent the equivalent baseband ﬁlter [GUI 96]. Indeed, relation [A.6] allows us to write: S(f ) = E(f )H(f ) So, γs (t) can also be expressed by:
γs (t) = F −1 E f + f0 H f + f0 = γe (t) ⊗ h(t)e−j2πf0 t
[A.8]
[A.9]
So, the baseband ﬁltered heq (t) which is equivalent to the real band ﬁlter h(t) can be expressed in two ways: heq1 (t) =
1 γh (t) 2
heq2 (t) = h(t)e−j2πf0 t
[A.10]
Baseband Representation of the Radio Channel
179
In the frequency domain, these two baseband ﬁlters Heq (f ) equivalent to the real band ﬁlter H(f ) are expressed by: Heq1 (f ) = U f + f0 H f + f0 [A.11] Heq2 (f ) = H f + f0 From a practical viewpoint, the ﬁlter Heq1 (f ) corresponds to a translation of the positive side frequency components of the ﬁlter H(f ) so that they are centered around 0 Hz. The ﬁlter Heq2 (f ) corresponds to a simple translation of all the ﬁlter component H(f ). These ﬁlters are similar when they are applied to signals expressed in baseband.
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Appendix B
Statistical Distributions
B.1. Deﬁnition This section deﬁnes the main distribution laws presented in the book. Unless otherwise mentioned, most of these laws are generally used to characterize the magnitude of the channel impulse response for a given delay. For each distribution of the random variable X, we give the probability density function (PDF) pX (x), the cumulative density function (CDF) F (x) = P (X ≤ x), the ﬁrst and second order moments E[X] and E[X 2 ], and the variance V ar[X] when it can be simply expressed.
B.1.1. Rayleigh distribution The Rayleigh distribution [PAR 00] is deﬁned from the parameter σ which is related to the standard deviation of the distribution by a constant value. ⎧x x2 ⎨ e− 2σ2 2 pX (x) = σ ⎩ 0
F (x) =
if x ≥ 0
⎧ x2 ⎨1 − e− 2σ2
if x ≥ 0
⎩ 0
otherwise "
E[X] =
[B.1]
otherwise
π σ 2
[B.2]
[B.3]
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UltraWideband Radio Propagation Channels
E X 2 = 2σ 2 4−π σ2 Var[X] = 2
[B.4] [B.5]
B.1.2. Rice distribution The Rice distribution [PAR 00, LAU 94] is deﬁned from two parameters, s and σ. ⎧ 2 2 ⎪ ⎨ x e− x 2σ+s2 I0 xs if x ≥ 0 σ2 pX (x) = σ 2 ⎪ ⎩0 otherwise
[B.6]
where I0 represents the modiﬁed Bessel function of type 1. ⎧ ⎪ ⎨1 − Q s , x if x ≥ 0 σ σ F (x) = ⎪ ⎩0 otherwise
[B.7]
where Q is the Marcum function given by [MAR 60]:
∞
Q(x, r) =
ye−
x2 +y 2 2
r
" E[X] =
π σL 12 2
I0 (xy)dy
s2 − 2 2σ
[B.8]
[B.9]
where L 12 is the Laguerre function, the solution of the diﬀerential equation: x
1 dy d2 y + y=0 + (1 − x) dx2 dx 2 E X 2 = s2 + 2σ 2
[B.10] [B.11]
A typical parameter of this distribution is the parameter k expressed by: k=
s2 2σ 2
[B.12]
Statistical Distributions
183
Numerous estimators exist for the k parameter. The k parameter estimator considered in the book is the one based on the second and fourth order moments, and uses the following m parameter: 2 x2 m= 2 x4 − x2
[B.13]
where (·) corresponds to the empirical mean value. So, k is expressed by [ABD 01]: 1 1− m [B.14] k 1 1− 1− m The Rice distribution tends to a Rayleigh distribution when s tends to 0. B.1.3. Nakagami distribution The Nakagami distribution [LAU 94] is deﬁned from two parameters, m and Ω.
⎧ m mx2 ⎪ ⎨ 2m x2m−1 e− Ω m pX (x) = Γ(m)Ω ⎪ ⎩0
if x ≥ 0
[B.15]
otherwise
where Γ represents the Gamma function deﬁned for x > 0 by: ∞ e−t tx−1 dt Γ(x) =
[B.16]
0
⎧ 2 ⎪ ⎨γ mx , m if x ≥ 0 Ω F (x) = ⎪ ⎩0 otherwise
[B.17]
where γ represents the incomplete Gamma function deﬁned for x > 0 by: a 1 e−t tx−1 dt [B.18] γ(a, x) = Γ(x) 0 " Γ m + 12 Ω [B.19] E[X] = Γ(m) m E X2 = Ω [B.20]
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UltraWideband Radio Propagation Channels
Numerous estimators exist for the parameter m. The one used in the book is the following [ABD 00]: 2 x2 2 x4 − x2
m
[B.21]
The Nakagami distribution tends to a Rayleigh distribution when m tends to 1.
B.1.4. Weibull distribution The Weibull distribution [LAU 94] is deﬁned from the two parameters, a and b. * b abxb−1 e−ax if x ≥ 0 [B.22] pX (x) = 0 otherwise * b 1 − e−ax if x ≥ 0 F (x) = [B.23] 0 otherwise 1 − 1b [B.24] E[X] = a Γ 1 + b 2 2 E X 2 = a− b Γ 1 + [B.25] b
The Weibull distribution tends to a Rayleigh distribution when b tends to 2.
B.1.5. Normal distribution The normal distribution is deﬁned from two parameters, the mean μ and the standard deviation σ. (x−μ)2 1 e− 2σ2 2πσ 1 x−μ F (x) = 1 + erf √ 2 2σ
pX (x) = √
[B.26] [B.27]
Statistical Distributions
where erf represents the error function deﬁned by: x 2 2 erf(x) = √ e−t dt π 0
185
[B.28]
E[X] = μ 2 E X = μ2 + σ 2
[B.29]
Var[X] = σ 2
[B.31]
[B.30]
B.1.6. Lognormal distribution The lognormal distribution [WIE 03] is deﬁned from two parameters, μ and σ. This distribution corresponds to the normal distribution of a signal complex envelope expressed in dB. ⎧ (10 log(x)−μ)2 ⎨ √ 10 − 2σ 2 e if x ≥ 0 2πxσ ln(10) pX (x) = [B.32] ⎩0 otherwise ⎧ −μ ⎨ 1 1 + erf 10 log(x) √ F (x) = 2 2σ ⎩ 0 μ
1
σ
2
E[X] = 10 · 10 10 + 2 ( 10 ) μ σ 2 E X 2 = 10 · 102( 10 +( 10 ) )
if x ≥ 0
[B.33]
otherwise [B.34] [B.35]
We can estimate the parameters μ and σ by calculating the ﬁrst and second moments of the variable expressed in dB: E XdB = μ 2 E XdB = μ2 + σ 2 Var XdB = σ 2
[B.36] [B.37] [B.38]
B.1.7. Laplace distribution The Laplace distribution [SPE 97] is often used to model the arrival angle associated with the cluster rays. In the book, this distribution is also used to describe the Doppler spectrum observed on the fast variations of the signals
186
UltraWideband Radio Propagation Channels
generated by moving people. This distribution is deﬁned from two parameters, μ and σ. √ x−μ 2 1 pX (x) = √ e− σ 2σ x−μ√2 1 F (x) = 1 + sgn(x − μ) e− σ 2
[B.39] [B.40]
E[X] = μ E X 2 = μ2 + σ 2
[B.41]
Var[X] = σ 2
[B.43]
[B.42]
B.2. KolmogorovSmirnov goodnessofﬁt test Considering an empirical CDF Fn (x) based on n samples and the theoretical CDF F0 (x) of the random variable from which the random draw is held, we can study the following variable:
[B.44] Dn = max Fn (x) − F0 (x) Figure B.1 illustrates the calculation of the variable Dn , for which the distribution has been studied by Kolmogorov [KOL 33]. For a decision threshold α and the critical value dα , we note: P (Dn > dα ) = α
[B.45]
P (Fn (x) − dα ≤ F0 (x) ≤ Fn (x) + dα , ∀x) = 1 − α
[B.46]
or:
The critical value dα has been tabulated for various values of α and n. We can demonstrate for example that for n > 80: 1
d0.05 1.3581 · n− 2 1
d0.01 1.6276 · n− 2
[B.47]
For a sample size n = 100, for example, we can conclude that the probability is: – 95% that F0 (x) is totally situated between F100 (x) − 0.13581 and F100 (x) + 0.13581;
Statistical Distributions
187
Fn(x) P(X x)
Dn
F0(x)
x
Figure B.1. KolmogorovSmirnov test: theoretical and empirical CDF
– 99% that F0 (x) is totally situated between F100 (x) − 0.16276 and F100 (x) + 0.16276. The KolmogorovSmirnov test consists of calculating the maximum deviation between an empirical CDF and a theoretical CDF, for a given decision threshold α: – if this deviation is smaller than the critical value dα , we conclude that the empirical CDF follows the same law as the theoretical CDF; – if this deviation is higher than the critical value dα , we conclude that the empirical CDF does not follow the same law as the theoretical CDF. Formally, a statistical decision process like the KolmogorovSmirnov test can lead to two types of errors. If the test erroneously concludes that the sample set does not follow the theoretical distribution law, we are faced with an error of type I. If the test erroneously concludes that the sample set follows the theoretical distribution law, we are faced with an error of type II. In the case of the KolmogorovSmirnov test, the type I probability of error is known (α), but the type II probability of error cannot be directly calculated. In other words, the probability of error is not known when we conclude that some samples follow a given law. So, this test is to be used sparingly.
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Appendix C
Geometric Optics and Uniform Theory of Diﬀraction
C.1. Geometric optics C.1.1. Introduction Geometric optics (GO) have been developed for the analysis of light wave propagation which corresponds to high frequencies. In propagation problems, the GO is valid only for frequencies higher than 100 MHz. If we consider a monochromatic electromagnetic pulsation wave ω, the s, ω) and magnetic ﬁeld H( s, ω) expressions at an observation electric ﬁeld E( point P, the spatial position of which is deﬁned by a vector s, correspond to the Maxwell equation expressed in an inhomogenous medium free from charges, with a permittivity (s) and a permeability μ(s) [PET 93] (see equation [C.1]). The Maxwell equations imply a time domain ﬁeld dependence with e−jωt . s, ω) + jωμ(s)H( s, ω) = 0 ∇ × E( s, ω) − jω(s)E( s, ω) = 0 ∇ × H( s, ω) = 0 ∇ · (s)E( s, ω) = 0 ∇ · (s)E(
[C.1]
For a homogenous dielectric medium, the permittivity and the permeability are not dependent on the position ((s) = and μ(s) = μ).
190
UltraWideband Radio Propagation Channels
The study of propagation problems in homogenous media is conducted by using Helmholtz vectorial equations which are derived from Maxwell equations: (s, ω) + k 2 U (s, ω) = 0 ∇2 U
[C.2]
(s, ω) the electric or magnetic ﬁeld and k = ω √μ = 2π = 2π √r μr the with U λ λ0 wavenumber of the propagation medium. λ and λ0 are the wavelengths in the propagation medium and in the free space respectively. r = r − j(r + 60σλ) and μr = μr − jμr correspond to the relative permittivities and the permeabilities of the medium. The conductivity of the medium is represented by σ. C.1.2. Field locality principle In a homogenous medium, energy propagates on paths which are straight and orthogonal to the wavefront. These wavefronts are deﬁned by the wave surfaces which are plane, spherical, cylindrical or any other shape. A group of rays is a beam or tube which starts and ends with two caustic segments (see AB and CD in Figure C.1). The energy carried by a ray is persistent in the time and space domains as well as in magnitude and phase.
Caustics φˆ
A
sˆ
ρ2
P(s)
P(0)
D B
θˆ
C
propagation direction
ρ1
s
Figure C.1. Ray tubes, caustics and local base
As there is always a scale for which a wave can be considered as locally plane, the properties of transverse electromagnetic plane waves can thus be generalized to all the electromagnetic waves [GAR 87]. This is the electromagnetic wave
Geometric Optics and Uniform Theory of Diﬀraction
191
locality principle. The electric and magnetic waves are strictly transverse to the propagation direction s. The electric ﬁeld vector is in the plane orthogonal to H, s) is orthonormal, direct and the propagation direction. The trihedron (E, follows relation [C.3]. " = E
μ (H × s)
[C.3]
which The study can thus be restricted to the case of the electric ﬁeld E ˆ ˆ is expressed in a local basis B = (ˆ s, θ, φ). This local basis depends on the propagation direction. For reasons of readability, a simpliﬁed notation is with its adopted in the following text which allows us to mix up the vector E vectorial representation E.
C.1.3. Field expression in geometric optics According to the work of Luneberg and Kline, the electric ﬁeld is expressed in GO by [GLO 99]: E(s) = A(s, ρ1 , ρ2 ) E(0) e−jkΨ(s) ⎡ ⎡ ⎤ ⎤ 0 0 E(s) = ⎣ Eθ (s) ⎦ = A s, ρ1 , ρ2 ⎣ Eθ (0) ⎦ e−jkΨ(s) Eφ (s) B Eφ (0) B
[C.4] [C.5]
A(s, ρ1 , ρ2 ) corresponds to the ratio between the ﬁeld magnitudes E(s) and E(0). It is also called the divergence factor: " A(s, ρ1 , ρ2 ) =
ρ1 ρ2 (ρ1 + s)(ρ2 + s)
[C.6]
– s is the covered distance between P (0) and P (s). – Ψ(s) is a phase function at the observation point P (s) and is equal to s. – E(0) is the electric ﬁeld at the point P (0). ρ1 and ρ2 are the two main curvature radii of the wavefront measured on the central ray at the reference point P (0). This point corresponds to the source position or the interaction point between the wave and a surface.
192
UltraWideband Radio Propagation Channels
As the electric ﬁeld is transverse to the propagation direction sˆ, the component along sˆ is zero. In the following, the ﬁeld will be expressed in its basis B without its component along sˆ (see equation [C.7]):
Eθ (0) Eθ (s) = A s, ρ1 , ρ2 e−jks [C.7] E(s) = Eφ (s) B Eφ (0) B The curvature radii ρ1 and ρ2 vary along the ray trajectory. They have to be recalculated whenever the ray is obstructed because the interactions modify the ray trajectory. An interaction can be a reﬂection, a transmission (or double refraction) or a diﬀraction. C.1.4. Change of local basis Considering an electric ﬁeld expressed in a given basis B1 , if the wave corresponding to this ﬁeld undergoes an interaction, the ﬁeld has to be expressed in a new local basis. This new local basis corresponds to the input basis B2 of the interaction. To obtain the new expression of the ﬁeld, we need to use the basis change matrix MB1 →B2 (see equation [C.8]).
Eα2 (0) B1 →B2 Eα1 (0) =M [C.8] E(0) = Eβ2 (0) B Eβ1 (0) B 2
1
s1 , α ˆ 1 , βˆ1 ) and B2 = (ˆ s2 , α ˆ 2 , βˆ2 ) are the direct orthonormal bases in B1 = (ˆ which the incident ﬁeld is expressed at the interaction input. MB1 →B2 is the transition matrix allowing us to express the incident ﬁeld in the new basis. This matrix corresponds to a projection of the vectors α ˆ 1 and βˆ1 in the basis B2 :
α ˆ2 · α ˆ1 α ˆ 2 · βˆ1 B1 →B2 [C.9] M = ˆ ˆ 1 βˆ2 · βˆ1 β2 · α C.1.5. Incident ﬁeld The incident ﬁeld is the ﬁeld radiated by a source S in the direction of an observation point P placed at a distance si . The expression of this ﬁeld is i r deduced from equation (C.7) by applying the transition matrix MB →B :
i i
i i Bi →Br Ei (0) i E s M = A s, ρ , ρ e−jks [C.10] Er si = 1 2 i i E⊥ (0) Bi si Br E⊥
Geometric Optics and Uniform Theory of Diﬀraction
193
ρi1 and ρi2 are the main curvature radii of the incident wavefront. si , eˆi , eˆi⊥ ) is a direct orthonormal local basis at the emission. It Bi = (ˆ corresponds to a basis described by the direction sˆi . If we consider a spherical frame, eˆi and eˆi⊥ can be assimilated to vectors eˆθ and eˆφ which form with sˆi a direct orthonormal basis. si , eˆr , eˆr⊥ ) is a direct orthonormal local basis at the receiver side. Br = (−ˆ It corresponds to the basis described by the direction −sˆi . If we consider a eφ which spherical frame, eˆr and eˆr⊥ can be assimilated to the vectors eˆθ and −ˆ ˆ i form with −s a direct orthonormal basis. i
r
MB →B is the transition matrix which allows us to express the incident ﬁeld at the point P in the good basis. This matrix corresponds to a projection of the vectors eˆi and eˆi⊥ in the basis Br .
eˆr · eˆi eˆr · eˆi⊥ 1 0 Bi →Br M = [C.11] = r i 0 −1 eˆ⊥ · eˆ eˆr⊥ · eˆi⊥ Most of the time, we consider that the incident wavefront is spherical. So, the ﬁeld is expressed by: Er (si ) =
i 1 Bi →Br i M E (0)e−jks si
[C.12]
C.1.6. Reﬂected ﬁeld The reﬂected ﬁeld is the ﬁeld received at a point P after the reﬂection of a ray at a point Qr placed at a distance si from the source point S and at a distance sr to the point P (see Figure C.2). C.1.6.1. Expression of the reﬂected ﬁeld in GO The expression of the reﬂected ﬁeld Er (sr ) is derived from the fundamental expression of the ﬁeld in GO (see equation [C.7]). The reﬂection law which comes from the Fermat principle deﬁnes the reﬂection point Qr . The incident sˆi and reﬂected propagation sˆr directions are ﬁxed by the interaction surface by the normal vector n ˆ (see equation [C.13]) [MCN 90]. 0 [C.13] n ˆ × sˆi − sˆr = ˆ The incidence and reﬂection planes are the same and the incidence angle θi and reﬂection angle θr are equal (see Figure C.2). The incidence plane is deﬁned
194
UltraWideband Radio Propagation Channels
P
S
sˆr
eˆ r eˆ i eˆ i⊥ local basis Bi
eˆ r⊥
local basis Br
nˆ
sˆi θi
θr Qr
tangent plane of a surface
Figure C.2. Incident and reﬂection local bases
by the normal n ˆ and the incident ray given by the vector sˆi . The reﬂection plane is deﬁned by the normal n ˆ and the reﬂected ray is given by the vector sˆr . The use of the locality principle in the case of the reﬂected ﬁeld corresponds to the introduction of a reﬂection dyad R which is expressed according to the reﬂection coeﬃcients R and R⊥ by:
R 0 [C.14] R= 0 R⊥ Thus, the reﬂected ﬁeld Er (0) at the point Qr and deﬁned in the basis B only depends on the dyad R and on the incident ﬁeld Ei (si ) deﬁned in the basis B at this same point. It is thus necessary to express the ﬁeld in the correct si , eˆi , eˆi⊥ ). incident local basis Bi = (ˆ i
i
Er (0) = R MB→B
Ei (si )
[C.15]
ˆ corresponds to the orthonormal basis in In relation [C.15], B = (ˆ si , α, ˆ β) i i which the incident ﬁeld E (s ) is deﬁned. α ˆ and βˆ are an arbitrary couple i of orthonormal vectors, such that B is direct. The transition matrix MB→B allows us to describe the ﬁeld Ei (si ) in the basis B i . ˆ eˆi · βˆ eˆi · α B→B i [C.16] = i M eˆ · α ˆ eˆi · βˆ ⊥
⊥
Geometric Optics and Uniform Theory of Diﬀraction
195
The reﬂected ﬁeld Er (sr ) at the observation point P is expressed in the sr , eˆr , eˆr⊥ ) by the equation [C.17]. The main curvature reﬂected local base B r = (ˆ r r radii ρ1 and ρ2 of the reﬂected wave depend on those of the incident wave. In the case of a planar surface, the following equalities are obtained: ρr1 = ρi1 and ρr2 = ρi2 . i r Er sr = A sr , ρr1 , ρr2 RMB→B Ei si e−jks
[C.17]
C.1.6.2. Incident and reﬂected ray bases The local bases Bi and Br are deﬁned according to the incident ray ˆ . In fact, eˆi and eˆr are parallel to propagation direction sˆi and the normal n the incidence and reﬂection planes, while eˆi⊥ = eˆr⊥ are orthogonal to the same planes (see Figure C.2). C.1.6.3. Reﬂection coeﬃcients The dyad R depends on the reﬂection coeﬃcient components which are parallel R and perpendicular R⊥ to the incidence plane (see equation [C.14]). These coeﬃcients are for the phase and the magnitude changes introduced by the reﬂection phenomenon on each component of the ﬁeld. For multilayer interfaces with one or more layers, these coeﬃcients have to be adapted in order to take the stratiﬁcation into account. Equation [C.18] corresponds to the reﬂection coeﬃcient in the case of a surface with a thickness e and a permittivity r . The permeability is considered to be set at 1.
R,⊥ =
1 − e−2jδ e2jδ Γ,⊥ 1 − Γ2,⊥ e−2jδ e2jδ
[C.18]
Γ,⊥ are the Fresnel coeﬃcients in the case of an inﬁnite plane surface with a permittivity r : Γ =
r cos θi −
r − sin2 θi
r cos θi +
r − sin2 θi
cos θi −
r − sin2 θi
cos θi +
r − sin2 θi
Γ⊥ =
[C.19]
[C.20]
In the case of a perfect conductor surface (σ → ∞), the reﬂection coeﬃcients are expressed with a phase diﬀerence multiple of π by: Γ = +1
Γ⊥ = −1
[C.21]
196
UltraWideband Radio Propagation Channels
δ = kr l and δ k0 d2 are the phase terms associated with the delays √ introduced by each reﬂected ﬁeld (see Figure C.3). kr = k0 r corresponds to the wavenumber of the medium with permittivity r and l = e/ cos θt is the oneway distance in the medium. The incidence θi and refraction θt angles are deﬁned by: 1 sin θt = √ sin θi r 0
[C.22]
0
r
Transmitted rays Reﬂected rays θi θt θi e Incident ray
0
r
0
d
d l
d = 2l sin θi sin θt
e
Figure C.3. Reﬂection of a plane wave on a dielectric interface with thickness e
The expressions of R,⊥ do not consider the possible roughness of the encountered surfaces (see equation [C.18]). The roughness is introduced by
Geometric Optics and Uniform Theory of Diﬀraction
197
multiplying the coeﬃcients Γ,⊥ by the roughness factor ρ (see equation [C.23]) [BEC 87] [COC 05]. h is the height standard deviation in the rough interface. ρ = e−2(kr h cos θi )
2
[C.23]
C.1.7. Refracted and transmitted ﬁeld The refracted (or transmitted) ﬁeld is the received ﬁeld at the point P after refraction (or multiple refractions) of a wave at the point Qt placed on an interface at a distance si of the source point S and at a distance st of the point P. S
eˆ i eˆ ⊥i local basis B
sˆ i
eˆ t
i
θi
θt
nˆ
Qt
eˆ ⊥t
sˆ t
local basis B
P
t
Tangent plane at the refraction surface
Figure C.4. Transmission and incidence local bases
C.1.7.1. Expression of refracted and transmitted ﬁeld in OG The expression of the transmitted ﬁeld Et (st ) at the observation point P, placed at a distance st of the transmission point, also follows from the GO ﬁeld main expression (see equation [C.7]). The refraction law, also called the SnellDescartes law, comes from the Fermat principle. It establishes the relation between the incidence angle θi and the transmission angle θt (see Figure C.4) at the interface and in the direction air → medium (see equation [C.22]). The Fermat principle implies that the incidence and transmission planes are the same and deﬁned by the incident sˆi ray and transmitted sˆt ray directions, and the reﬂection surface normal
198
UltraWideband Radio Propagation Channels
n ˆ . Identically to the reﬂection example, the transmission can be considered as a local phenomenon, so the transmitted ﬁeld is expressed by introducing a transmission dyad T. This dyad is obtained according to the transmission coeﬃcients (see equation [C.24]).
T 0 [C.24] T= 0 T⊥ The ﬁeld Et (0) after the transmission point Qt depends on the coeﬃcient T and on the incident ﬁeld Ei (si ) at this same point, taking care to deﬁne the si , eˆi , eˆi⊥ ) (see equation [C.25]). incident ﬁeld in the incident local basis Bi = (ˆ i Et (0) = TMB→B Ei si
[C.25]
ˆ corresponds to the orthonormal basis in which the incident B = (ˆ si , α ˆ , β) i i ﬁeld E (s ) is deﬁned. As in the reﬂected ﬁeld case, the transition matrix i MB→B allows the expression of the incoming ﬁeld Ei (si ) in the incident local basis B i . The transmitted ﬁeld Et (st ) at the observation point P is expressed in the transmission local basis Bt = (sˆt , eˆt , eˆt⊥ ) by: i t Et st = A st , ρt1 , ρt2 TMB→B Ei si e−jks
[C.26]
In the simple refraction case, st corresponds to the distance between P and Qt . As the point P is inside the interface, the propagation constant considers the propagation speed related to the dielectric properties at the interface. For a multiple refraction (or transmission), st corresponds to the distance between P and Qt at which the propagation distance in the interface is removed. This last distance is obtained for propagation in free space. Indeed, in the case of multiple refraction, the phase delay associated with propagation in the interface is directly introduced in T. The divergence of a transmitted ray is given by the following relation: ! t t t ρt1 ρt2 A s , ρ1 , ρ2 = [C.27] t (ρ1 + st )(ρt2 + st ) ρt1 is the ﬁrst main curvature radius of the transmitted wave. Its expression is diﬀerent from the case of simple refraction (see equation [C.28]) and the case
Geometric Optics and Uniform Theory of Diﬀraction
199
of double refraction (see equation [C.29]). ρt1 = ρi1 α−1
[C.28]
ρt1
[C.29]
=
ρi1
+ αl
ρt2 is the second main curvature radius of the transmitted wave. Its expression is also diﬀerent from the case of the simple refraction [C.30] and the case of the second refraction [C.31] [PLO 03]. ρt2 = ρi2 α−1 ρt2 l=
e cos θt
=
ρi2
[C.30] 2
+ αγ l
1 α= √ r
γ=
[C.31] cos θi cos θt
[C.32]
C.1.7.2. Incident and transmitted ray bases ˆ The knowledge of the incident ray propagation direction sˆi and the normal n allows us to deﬁne the local bases Bi and Bt . In fact, eˆi and eˆt are respectively parallel to the incidence and transmission planes and eˆi⊥ = eˆt⊥ are perpendicular ˆ (see to the same planes. eˆi and eˆi⊥ are deﬁned according to the normal n Figure C.2). eˆt⊥ is obtained from equation [C.33]. Generally, after a double refraction Bi and Bt are the same because sˆi = sˆt . eˆt = eˆt⊥ × sˆt
[C.33]
C.1.7.3. Transmission coeﬃcient In the expression of the transmission dyad, the two components T and T⊥ are respectively the transmission coeﬃcients parallel and orthogonal to the incidence plane. They correspond to the phase and magnitude changes introduced by the transmission phenomenon on each component of the ﬁeld. In the case of a simple refraction and a plane surface, their expressions are given by equations [C.34] and [C.35]. Γ and Γ⊥ correspond to the Fresnel reﬂection coeﬃcients deﬁned at equations [C.19] and [C.20]. √ 1 + Γ 2 r cos θi = [C.34] T = √ r r cos θi + r − sin2 θi T⊥ = 1 + Γ⊥ =
2 cos θi cos θi +
r − sin2 θi
[C.35]
200
UltraWideband Radio Propagation Channels
In the case of a double refraction (transmission), T and T⊥ are expressed by equation [C.36] (see Figure C.3). In [SAG 03], the expression of the transmission coeﬃcient for an interface with more than two materials is proposed. T,⊥ =
1 − Γ2,⊥ e−2jδ e2jδ 1 − Γ2,⊥ e−2jδ e2jδ
[C.36]
C.2. Uniform theory of diﬀraction C.2.1. Introduction Electromagnetic waves are continuous, in magnitude and phase, in the time and space domain. However, the GO does not ensure the total continuity ﬁeld. It predicts areas where the ﬁeld is zero (shadow zone). In 1953, Keller proposed a generalization of the GO in order to consider diﬀracted rays. This led to the birth of general diﬀraction theory (GTD) [KEL 62]. The GTD allows us to solve the problem of ﬁeld existence in the shadow zone of the GO. However, it presents a singularity at the boundary between a lighted zone and a shadow zone. So in 1962, the theory has been completed by Kouyoumjian and Pathak in order to ensure the total ﬁeld continuity in all space points [KOU 74]; henceforth we talk about UTD. Thereafter, Burnside and Burgener have proposed a formalism of the uniform theory of diﬀraction (UTD) in the case of the 3D diﬀraction on a corner of small thickness [BUR 83]. Other authors have also studied the simple diﬀraction, the double diﬀraction as well as the slope diﬀraction in the frequency domain [LUE 89, DES 84, ROU 96, ROU 99] and in the time domain [VER 90, ROU 95]. Then, only the expression of the diﬀracted ﬁeld in the frequency domain is addressed, considering the case of the diﬀraction by dielectric or metallic dihedron.
C.2.2. Diﬀracted ﬁeld The expression of the diﬀracted ﬁeld given by equation [C.37] corresponds to the diﬀraction by a dihedron. For this kind of diﬀraction, the discontinuity line is mixed up with the one of the diﬀracted beam caustics and so the curvature radius ρd2 , also called minor ray, becomes zero. i d Ed sd = A sd , ρd1 , ρd2 → 0 DMB→B Ei si e−jks
[C.37]
Geometric Optics and Uniform Theory of Diﬀraction
201
A(s) is the divergence factor of the ray diﬀracted by the dihedron and is expressed by: ! d d d ρd [C.38] A s , ρ1 , ρ2 → 0 = d 1 d d ρ1 + s s D is the dyad introducing the diﬀraction coeﬃcients. This dyad can be written diﬀerently according to the diﬀracted ray’s nature, which can be twodimensional (2D) or threedimensional (3D). i
The matrix MB→B makes it possible to describe the incident ﬁeld, initially expressed in the basis B, in the incidence basis of the diﬀraction sd , eˆi , eˆi⊥ ). The diﬀracted ﬁeld Ed (sd ) is deﬁned in the diﬀraction basis Bi = (ˆ d B = (ˆ sd , eˆd , eˆd⊥ ). The diﬀraction law, coming from the generalized Fermat principle, connects the Keller angle β0 to the incident ray direction sˆi and the diﬀracted ray direction sd as well as the tangent tˆ of the dihedron wedge (see Figure C.5) at the diﬀraction point Qd by: sˆi · tˆ = sˆd · tˆ = cos β0
[C.39]
The incidence and diﬀraction planes are thus deﬁned by the tangent of the wedge tˆ and by the incident and diﬀraction ray directions. Generally, these two planes are not merged, as illustrated in Figure C.5. The vectors of the bases Bi and Bd are obtained using the following relations: tˆ si × eˆi⊥ = sˆi × eˆi eˆi = −ˆ sin β0 [C.40] tˆ d d d d d eˆ = sˆ × eˆ⊥ = sˆ × eˆ sin β0 C.2.3. UTD 2D coeﬃcient In the case of a 3D ray, the dyad D is diagonal and directly expressed from the two components of the diﬀraction coeﬃcients D and D⊥ :
D 0 [C.41] D= 0 D⊥ The terms D and D⊥ are the diﬀraction coeﬃcients initially introduced by Keller and then modiﬁed by Kouyoumjian, Burnside and Luebbers.
202
UltraWideband Radio Propagation Channels
n tio ac r ﬀ Di
tˆ Incidence plane
eˆ d⊥
β0
eˆ i
local basis Bd Qd
sˆi β0
eˆ i⊥
P
sˆd
eˆ d
local basis Bi
S
ne pla
φd
φi (2 − n)π
Figure C.5. Incidence and diﬀraction local bases
C.2.3.1. UTD coeﬃcient – dihedron conductor Contrary to the GTD, the UTD introduced by Kouyoumjian and Pathak [KOU 74] ensures the ﬁeld continuity in all the space, especially at the optic boundaries ISB (incident shadow boundary) and RSB (reﬂection shadow boundary). These authors have taken an interest in the case of a perfect conductor dihedron and have deﬁned two new diﬀraction coeﬃcients which introduce a correcting factor compared to the GTD coeﬃcients:
D1 D2 D3 D4
D//,⊥ = D1 + D2 ± (D3 + D4 ) + , e−j π + (φd − φi ) F kLa+ (φd − φi ) =− √ cot 2n 2n 2πk sin βo + π , e−j 4 π − (φd − φi ) F kLa− (φd − φi ) =− √ cot 2n 2n 2πk sin βo + π , e−j 4 π + (φd + φi ) √ F kLa+ (φd + φi ) =− cot 2n 2n 2πk sin βo + π , e−j 4 π − (φd + φi ) F kLa− (φd + φi ) =− √ cot 2n 2n 2πk sin βo
[C.42]
π 4
[C.43]
Geometric Optics and Uniform Theory of Diﬀraction
203
β0 is a semiangle at the diﬀraction cone head. The normal incidence corresponds to βo = π2 (see Figure C.5). φi and φd are respectively the incidence and diﬀraction angles. They are speciﬁed from the wedge face 0 (see Figure C.5). n is a parameter deﬁned so that the dihedron inner angle is given by (2−n)π. ± corresponds to the reﬂection coeﬃcient in the case of a conductor [C.21]. F (x) corresponds to the transition function which uses the Fresnel integral [LEG 95]: √
jx
F (x) = 2j xe
∞
√
2
e−jt dt
[C.44]
x
L is a distance parameter which corresponds to the incidence type on the wedge. It brings into play the incident wave main curvature radii (ρd1 and ρd2 ) as well as the wedge of curvature radius (ρde ) (see equation [C.45]). In the case of a spherical wave, L is given by equation [C.46]. L = sin2 β0 sd
ρde + sd ρd1 ρd2 d d d d ρe ρ1 + s ρ2 + sd
L = sin2 β0
sd ρd1 + sd
ρd1
[C.45] [C.46]
a± is a function deﬁned by: a± = 2 cos2
2nπN ± − (φd ± φi ) 2
[C.47]
with N ± an integer corresponding (outside the optic boundaries) to relation [C.48]: 2πnN ± − (φd ± φi ) = ±π
[C.48]
When the cotangent (cot) of one of the Di (i = 1, 2, 3, 4) terms becomes singular for the optic boundary angles φi and φd , the corresponding term is √ L replaced by − 2 sign() [PLO 00].
204
UltraWideband Radio Propagation Channels
C.2.3.2. UTD coeﬃcients – dielectric dihedron Luebbers proposes a more general formulation for the diﬀraction coeﬃcients. This formulation allows us to consider the dielectric nature of the wedge as well as its roughness [LUE 89]. These coeﬃcients are heuristic solutions as they are not obtained by solving Maxwell equations: n 0 [C.49] D3 + G0//,⊥ D2 + ρ0 R//,⊥ D4 D//,⊥ = Gn//,⊥ D1 + ρn R//,⊥ Di (i = 1, 2, 3, 4) are terms reported on equations [C.43]. 0,n corresponds to the Fresnel reﬂection coeﬃcients [C.19] and [C.20]. R//,⊥ The incidence angles θi0 = π2 − φi and θin = π2 − (nπ − φi ) depend on the faces 0 and n of the considered wedge.
ρ0,n corresponds to the roughness coeﬃcient of the considered face [C.23]. G0,n //,⊥ are the correcting coeﬃcients which allow us to consider the grazing incidences on the faces 0 and n of a dihedron. ⎧ 1 ⎪ 0 ⎪ for φi = 0 and 1 + R//,⊥ >0 ⎪ 0 ⎪ ⎨ 1 + R//,⊥ G0//,⊥ = 1 [C.50] ⎪ for φi = nπ ⎪ ⎪ 2 ⎪ ⎩ 1 otherwise ⎧ 1 ⎪ n ⎪ for φi = nπ and 1 + R//,⊥ >0 ⎪ n ⎪ ⎨ 1 + R//,⊥ Gn//,⊥ = 1 [C.51] for φi = 0 ⎪ ⎪ ⎪2 ⎪ ⎩1 otherwise
C.2.4. UTD 3D coeﬃcient Contrarily to the case of a 2D ray, the dyad D of a 3D ray is no longer diagonal and is expressed only by two diﬀraction coeﬃcients, as in the case of 2D ray (see equation [C.49]). This 3D dyad has been introduced by Burnside and Burgener [BUR 83] and reused by Rouvi`ere [ROU 99] in the case of a dielectric halfplane with very small thickness. To obtain the 3D dyad of [BUR 83], ﬁeld continuity has to be ensured at each optic limit. So, the boundary conditions at the reﬂection in incidence
Geometric Optics and Uniform Theory of Diﬀraction
205
boundaries have to consider relations [C.52] and [C.53]. In these relations, Ui and Ur correspond to the incident and reﬂected ﬁelds respectively. * d −1/2Ur area I e−jks = RU (Qd )D(φ + φ ) √ area II 1/2Ur sd * d e−jks −1/2(1 − T)Ui area II (1 − T)Ui Qd D φd − φi √ = area III 1/2(1 − T)Ui sd i
d
i
[C.52]
[C.53]
with the areas denoted I, II and III illustrated by Figure C.6. RSB
Zone I incident + reﬂected + diﬀracted
Zone II incident + transmitted
ISB
Zone III diﬀracted + transmitted diﬀracted wedge
Figure C.6. Field area around a diﬀracting wedge
Using these boundary conditions, the following expression is obtained for the dyad D: D = (I − T )D(φd − φi ) + R D(φd + φi )
[C.54]
D(φd − φi ) = D1 + D2
[C.55]
D(φd − φi ) = D1 + D2
[C.56]
Each of the dyads (D, R and T ) are expressed in their respective local basis. It is thus necessary to calculate the base change matrices in order to express all the ﬁelds in the local basis of the diﬀraction. α is the angle between the incidentreﬂection plane as well as the incidencetransmission plane and the incident plane. The diﬀraction plane is given by π2 − α. In 2D,
206
UltraWideband Radio Propagation Channels
the incidencereﬂection and incidencetransmission planes are perpendicular to the incidencediﬀraction plane (α = 0). In 3D, these planes have no particular direction (see Figure C.7).
ce den inci
ion ﬂect  re e n pla
nˆ
−α eˆ i S
eˆ d⊥
θr
θi
sˆi
Qd β0
eˆ i⊥
ne pla
ne pla ce en cid In π 2
sˆd
eˆ d
n tio ac ﬀr Di
P
tˆ
φd
φi (2 − n)π
Figure C.7. Reﬂection and diﬀraction incidence planes
The relations between the reﬂection and diﬀraction local bases are illustrated at Figure C.8 allow us to establish the matrix of basis change necessary to express the reﬂection R (see equation [C.57]) and transmission T dyads in the diﬀraction local basis (see equation [C.57]). In these equations, the exponent terms correspond to the plane nature (i for incident and d for diﬀraction) as well as the interaction specifying the plane (r for reﬂection and d for diﬀraction). ⎡ MiR→D (α)
eˆi,d ˆi,r ·e
= M(α) = ⎣ i,d i,r eˆ⊥ · eˆ ⎡
MdR→D (α)
ˆd,r eˆd,d ·e
= M(−α) = ⎣ d,d d,r eˆ⊥ · eˆ
⎤
⎦ = cos α sin α eˆi,d · eˆi,r eˆi,d ˆi,r ⊥ ·e ⊥
⊥
eˆd,d ˆd,r ⊥ ·e eˆd,d ˆd,r ⊥ ·e ⊥
⎤ ⎦=
− sin α cos α
cos α − sin α
sin α cos α
[C.57] [C.58]
Geometric Optics and Uniform Theory of Diﬀraction
eˆ i,d ⊥
eˆ i,r ⊥
eˆ d,d ⊥
α
207
eˆ d,r ⊥
α eˆ i,d α
eˆ i,r
eˆ d,d
α
eˆ d,r before diﬀraction
after diﬀraction
Figure C.8. Incidence reﬂection and diﬀraction bases in 3D case
In the case of transmission, the same basis change matrices are used. They are denoted M(−α) because we turn conversely. The dyadic matrices are thus expressed by R and T by using equations [C.59] and [C.60]. −1 R = MdR→D (α) R MiR→D (α) = M(−α) R M(−α)
R cos2 α − R⊥ sin2 α (R + R⊥ ) cos α sin α = −(R + R⊥ ) cos α sin α R⊥ cos2 α − R sin2 α −1 T = MdT →D (α)T MiT →D (α) = M(α)TM(−α)
T cos2 α + T⊥ sin2 α (T − T⊥ ) cos α sin α = (T − T⊥ ) cos α sin α T⊥ cos2 α + T sin2 α
1 − T cos2 α − T⊥ sin2 α −(T − T⊥ ) cos α sin α (I − T ) = −(T − T⊥ ) cos α sin α 1 − T⊥ cos2 α − T sin2 α
[C.59]
[C.60]
[C.61]
Expressions [C.59] and [C.61] inserted in equation [C.54] underline the nondiagonal structure of the dyad D which is expressed by [BUR 83] [GLO 99]:
D=
Da Dc
Db Dd
[C.62]
Da = 1−T cos2 α−T⊥ sin2 α D φi − φd −[R cos2 α − R⊥ sin2 α D φi + φd [C.63]
208
UltraWideband Radio Propagation Channels
Db = cos α sin α [(T⊥ − T ) D(φi − φd ) + (R + R⊥ ) D(φi + φd )]
[C.64]
Dc = cos α sin α [(T⊥ − T ) D(φi − φd ) − (R + R⊥ ) D(φi + φd )]
[C.65]
Dd = [1−T sin2 α−T⊥ cos2 α]D(φi − φd )−[R⊥ cos2 α−R sin2 α]D(φi + φd ) [C.66] In equations [C.63] [C.64] [C.65] and [C.66], the terms R,⊥ and T,⊥ are the reﬂection and transmission coeﬃcients, the expressions of which are given by [C.18] and [C.36].
Appendix D
Ray Construction Techniques
D.1. Ray launching Ray launching is a forward approach which consists of sending rays in all the directions from a transmission position of coordinates (Tx) with an incremental angular step (Δα) which can be set to various values. Then, the receiver position coordinates (Rx) are speciﬁed (see Figure D.1) [CHE 96, KIM 97, STA 98, LIN 89]. The speed of ray determination with the ray launching approach is directly related to the chosen Δα. When the Rx position is ﬁxed, the rays connecting the transmitter Tx and the receiver Rx are obtained considering a receiver sphere of diameter Δd which can be set to various values. So, the launched rays connecting Tx to Rx will be considered if they intersect the receiver sphere (see Figure D.1). The number of rays obtained increases with the sphere diameter Δd (for a given Δα), but the precision of the calculated ﬁeld and the received signal decreases. D.2. Ray tracing Ray tracing is a backward approach which consists of applying the image principle from Tx and Rx position in order to obtain reﬂected rays (see Figure D.2) [HUM 03, MCK 91, CHA 03]. The speciﬁcity of this approach is the increase of the number of the potential images combinations with the complexity of the considered environment. This can lead to important ray calculation time.
210
UltraWideband Radio Propagation Channels
δα
Tx
(a)
δα δd
Rx
Tx
(b)
Figure D.1. Ray launching principle: (a) rays launched from the transmitter and (b) rays obtained considering intersection of receiver sphere
Im Im Tx /s1 /s2
Im Tx /s1
s1 Tx
Rx
s2
Figure D.2. Ray tracing principle
Ray Construction Techniques
211
D.3. Other techniques In the literature, other techniques are also proposed and used for ray determination. Some of these techniques are hybrid and combine ray launching and ray tracing [TAN 95, TCH 03, AVE 04], while others require a visibility tree to be built [AGE 97, AGE 00], or rely upon calculation time speed improvement techniques borrowed from the image processing domain. These last techniques use octree, raster or Voxel matrices for ray determination [FOL 84, LEG 05].
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Appendix E
Description of the TimeFrequency Transform
The determination of a time domain signal x(τ ) from its frequency domain expression X(f ) is generally made using an inverse Fourier transform (see equation [E.1]). As the frequency domain expression X(f ) of the signal x(τ ) shows a Hermitian symmetry (see equation [E.2]), the obtained signal x(τ ) is real. +∞ X(f ) ej 2 π f τ df [E.1] x(τ ) = −∞
∗
X (f ) = X(−f ) ∀ f ∈ R
[E.2]
From frequency domain measurements, the obtained channel transfer function H(f ) is deﬁned on a ﬁnite frequency band from fmin to fmax , for Nf frequency points. As H(f ) is discrete, the determination of h(τ ) is made min using an inverse Fourier transform (see equation [E.3]). δf = fmaxN−f is the f frequency sampling step and fk corresponds to a frequency value given by fk = fmin + k δf . Nf −1
h(τ ) = δf
H(fk ) ej 2 π fk τ
[E.3]
k=0
The transfer function H(f ) does not show a Hermitian symmetry because it is only deﬁned between fmin and fmax . The signal obtained after discrete inverse ˆ ) (see Figure E.1). Fourier transform is complex and corresponds to h(τ ) + j h(τ
40
50
10
20 30 delay (ns)
imaginary part
Figure E.1. Inverse Fourier transform of H(f )
0
−0.08
−0.08
20 30 delay (ns)
−0.06
−0.06
10
−0.04
−0.04
0
0
0.02
0.02
−0.02
0.04
0.04
0
0.06
0.06
−0.02
0.08
0.08
real part
40
50
214 UltraWideband Radio Propagation Channels
Description of the TimeFrequency Transform
215
To obtain a real signal h(τ ), it is mandatory to use a transfer function H2 (f ) presenting by construction a Hermitian symmetry. Moreover, to improve the precision of the reconstructed h(τ ), we can expand the spectral support of H2 (f ) in comparison to H(f ). This consists of making an operation of zero padding. Indeed, the product δτ δf Nf is equal to 1, so the increase of Nf necessary leads to a reduction of δτ and consequently improves the time domain precision of the signal h(τ ) obtained after the discrete inverse Fourier transform. The reconstruction of a real h(τ ) from H(f ) is made by applying two successive operations on H(f ) before the discrete inverse Fourier transformation: zero padding and Hermitian symmetric forcing. Operation of zero padding The operation of zero padding consists of obtaining a new transfer function H1 (f ) from H(f ) (see Figure E.3). The new transfer function H1 (f ) contains far more samples than H(f ). H1 (f ) is constructed according to the following relation with fe = 2 1δτ . * H(f ) f ∈ fmin , fmax [E.4] H1 (f ) = 0 f ∈ δf, fmin − δf ∪ fmax + δf, fe So, the new transfer function H1 (f ) is constructed as shown in equation min max − 1 and Nr = fe −f . [E.5]) with N1 = Nr + Nf + Nl samples where Nl = fδf δf H1 (f ) = zeros(1, Nl )
H(f )
zeros(1, Nr )
[E.5]
Figure E.2 shows the time domain signal obtained after discrete inverse Fourier transformation is applied on H1 (f ). We can notice that the sampling of the obtained time domain signal is improved. Nevertheless, it is still complex. Operation of Hermitian symmetric forcing The operation of forcing the Hermitian symmetry consists of creating a transfer function H2 (f ) from H1 (f ). This new transfer function H2 (f ) is deﬁned from −fe to fe (see equation [E.6]) as illustrated in Figure E.3. Figure E.4 represents a real impulse response h(τ ) obtained after a discrete inverse Fourier transform applied on H2 (f ). * H2 (f ) = H1 (f ) for f > 0 [E.6] ∗ H2 (−f ) = H2 (f ) ∀f ∈ δf, fe
40
50
10
20 30 delay (ns)
imaginary part
Figure E.2. Zero padding and inverse Fourier transform on H1 (f )
0
−0.08
−0.08
20 30 delay (ns)
−0.06
−0.06
10
−0.04
−0.04
0
0
0.02
0.02
−0.02
0.04
0.04
0
0.06
0.06
−0.02
0.08
0.08
real part
40
50
216 UltraWideband Radio Propagation Channels
−fe
−f max
0
f max
f min
f max
H (f ) + zero padding
f min
Figure E.3. Illustration of the Hermitian symmetric forcing
−f min
H *(f) + f lip zero padding
Operation of symmetric f orcing
±f
H (f ) + zero padding
fe f
fe f
Description of the TimeFrequency Transform 217
−0.04 −0.06 −0.08
−0.04
−0.06
−0.08 10
20 30 delay (ns)
imaginary part
Figure E.4. Reconstruction of h(τ ) after discrete inverse Fourier transform applied on H2 (f )
0
−0.02
−0.02
50
0
0
40
0.02
0.02
20 30 delay (ns)
0.04
0.04
10
0.06
0.06
0
0.08
0.08
real part
40
50
218 UltraWideband Radio Propagation Channels
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Index
A Antenna diversity, 102 eﬀective area, 44, 134 isotropic, 111 polarization, 107 radiation eﬃciency, 107 return loss, 107 Arrival rate cluster, 61, 145 ray, 146 rays, 61 Average fade duration, 64 B BER, 24 Bluetooth, 26, 27, 29, 30 BPSK, 39, 40 C CDMA, 38, 40 Channel characteristic parameters, 58, 134, 137 deﬁnition, 43 deterministic, 52 impulse response, 181 linear random, 54 matrix, 108 propagation, 25 representation, 51, 177 sounding, 67
transfer function, 213 US, 56 variations, 47, 148, 150 WSS, 55 WSSUS, 57 Channel capacity, 24 Channel equalization, 49 Channel modeling statistical, 133 Chirp sounder, 72 Coherence bandwidth, 49, 59 Correlation bandwidth, 49 Correlation sounder, 75 sliding correlation, 76 D DAA, 33 Delay interval, 60 Delay window, 60 DelayDoppler spread function, 54 Deterministic model, 99 Diﬀraction, 47 Diﬀusion, 47 DoA, 104 DoD, 104 Doppler eﬀect, 50 shift, 50, 69, 156 spectrum, 64, 173 spread, 64, 156, 173 DSUWB, 35, 39, 40, 42
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Duty cycle, 23
KolmogorovSmirnov test, 64
E ESD (Energy spectral density), 110 Exponential decay constants, 61, 141
L Laplace distribution, 170, 173, 185 Level crossing rate, 64 Lognormal distribution, 64, 150, 185 LOS, 105
F Fading fast, 48, 63, 148, 153, 172 ﬂat, 48 slow, 48, 152, 172 Far ﬁeld, 45 FCC, 22, 25, 31, 33, 34, 40, 111 FDTD, 100 Fourier transform, 103, 178, 213 Fraunhofer distance, 45 Free space propagation, 44 Frequency correlation function, 59 Frequency domain function, 53 Fresnel law, 46 Friis formula, 45, 134 G General theory diﬀraction, 200 Geometric optic, 189, 200 ﬁeld expression, 191 ﬁeld locality, 190 GO, 101 GPS, 31, 32 GSM, 30 H Hermitian symmetry, 177, 213, 215 Hilbert transform, 178 Huygens’ principle, 47 I Impulse radio, 35, 38–40 Impulse response, 51, 103, 137, 163, 166 time varying, 53 Intersymbol interference, 50, 59 Inversion sounding technique, 78 ISB, 202 K KolmogorovSmirnov, 186
M MBOK, 39 msequence, 75 Maxwell equation, 189, 204 MBOFDM, 35, 40–42 Mean delay, 59 Measurement campaigns, 85 examples, 91 overview, 85 Model statistical, 157 CassioliWinMolisch, 160 dynamic, 169 frequency domain, 161 IEEE 802.15.3a, 158 IEEE 802.15.4a, 159 principles, 162 MoM, 100 N Nakagami distribution, 64, 150, 159, 161, 183 NLOS, 105 Normal distribution, 184, 185 O OOK, 39 P PAM, 38, 39 Path loss, 62, 134, 162 exponent distance dependent, 48, 63, 136 frequency dependent, 63, 134 Power decay constants, 144 Power delay proﬁle, 57, 58, 137 Power spectral density, 25, 33, 37, 177 PPM, 35, 38, 39 Processing gain, 25
Index Propagation free space, 44, 134 model, 99 multipath, 45, 48 phenomena, 45 PSD (Power spectral density), 111 Pulse sounder, 73 Q QPSK, 40 R Radio channel, 26 RAKE reception, 49 Ray launching, 209, 211 Ray tracing, 209, 211 Rayleigh distribution, 64, 150, 155, 161, 173, 181, 183, 184 Rays launching, 101 Rays tracing, 101 Reﬂection, 46 Rice distribution, 64, 150, 155, 182, 183 RIR, 105 RMS delay spread, 59, 137 RSB, 202 RUN method, 58 S Sparameters, 71 Saleh and Valenzuela model, 61 Scattering function, 57, 64, 156, 173 Selectivity frequency, 48, 58 spatial, 48 Shadowing, 48 SHF, 30 SISO, 104 SnellDescartes law, 46 Sounding analyzed bandwidth, 68 CIR dynamic, 69 CIR length, 69 maximum Doppler shift, 69
239
measurement techniques, 70 frequency domain, 71 multipleband time domain, 78 time domain, 73 time resolution, 68 Standing wave ratio, 92 T Time hopping, 23 Transfer function, 125 time varying, 54 Transmission, 47 U UHF, 30 UMTS, 23, 30, 40 Uncorrelated scattering, 56 Uniform theory of diﬀraction, 200 UNII, 30, 34, 39 UTD, 101 UWB applications, 27, 29, 30 characteristics, 24 deﬁnition, 21, 22 history, 23 regulation, 30 V Variations spatial, 51, 150, 169 temporal, 51, 150, 172 Vector network analyzer, 71 W Waveguide eﬀect, 47, 137 Weibull distribution, 64, 184 Wide sense stationary, 55 WiFi, 26, 27, 29, 30, 39 WLAN, 27 WPAN, 27 Z Zero padding, 215 Zigbee, 26, 29