Ultra Wide Band Antennas

Ultra Wide Band Antennas

Ultra Wide Band Antennas

Edited by Xavier Begaud Series Editor Pierre-Noël Favennec

First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from Les antennes Ultra Large Bande published 2010 in France by Hermes Science/Lavoisier © LAVOISIER 2010 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 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

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© ISTE Ltd 2011 The rights of Xavier Begaud to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Antennes ultra large bande. English Ultra wide band antennas / edited by Xavier Begaud. p. cm. Rev. papers of the autumn school, GDR Ondes, organized in Valence, Oct. 2006. Includes bibliographical references and index. ISBN 978-1-84821-232-9 (hardback) 1. Ultra-wideband antennas--Congresses. I. Begaud, Xavier. II. Title: Ultra-wideband antennas. TK7871.67.U45A5813 2010 621.382'4--dc22 2010038273 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-232-9 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne. Cover photo: created by Atelier Isatis, Dijon, France

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1. Applications of Ultra Wide Band Systems. . . . . . . . . . . . . . Serge HÉTHUIN and Isabelle BUCAILLE

1

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. UWB regulation: a complex context . . . . . . . . . . . . . . . . . 1.2.1. UWB regulation in the USA . . . . . . . . . . . . . . . . . . . 1.2.2. UWB regulation in Europe . . . . . . . . . . . . . . . . . . . . 1.2.3. UWB regulation in Japan . . . . . . . . . . . . . . . . . . . . . 1.2.4. Emission mask in the United States, Europe and Japan . . . 1.3. Formal Ultra Wide Band types . . . . . . . . . . . . . . . . . . . . 1.3.1. Ultra Wide Band Impulse Radio (UWB-IR) . . . . . . . . . . 1.3.2. OFDM-ultra wide band (UWB-OFDM) . . . . . . . . . . . . 1.4. Non-formal ultra wide band types . . . . . . . . . . . . . . . . . . 1.4.1. Ultra wide band frequency hopping (UWB-FH). . . . . . . . 1.4.2. Chirp Ultra Wide Band (UWB-FM) . . . . . . . . . . . . . . . 1.5. Comparison between the different Ultra Wide Band techniques . 1.6. Typical UWB-OFDM applications . . . . . . . . . . . . . . . . . . 1.6.1. Peripheral connection to a PC . . . . . . . . . . . . . . . . . . 1.6.2. High speed applications in large structures with optical fiber backbone . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3. High speed UWB in a harsh indoor environment . . . . . . . 1.6.4. High speed UWB combined with other technologies . . . . . 1.7. Specialized UWB-OFDM applications . . . . . . . . . . . . . . . 1.7.1. Last mile radio applications . . . . . . . . . . . . . . . . . . . . 1.7.2. Information and video streaming applications . . . . . . . . . 1.8. Typical applications of the Impulse Radio UWB, UWB-FH and UWB-FM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.8.1. Professional geo-localization . . . . . . . . . . . . . . . . . . . . . . . 1.8.2. Geolocalization for private individuals . . . . . . . . . . . . . . . . . 1.9. Impact on the antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 31 32

Chapter 2. Radiation Characteristics of Antennas . . . . . . . . . . . . . . . . Xavier BEGAUD

33

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. What is an antenna and how can we define it? . . . . . . . . 2.1.2. Where does antenna radiation come from? . . . . . . . . . . 2.2. How can we characterize an antenna? . . . . . . . . . . . . . . . 2.2.1. Plane wave and polarization . . . . . . . . . . . . . . . . . . 2.3. Radiation fields and radiation power . . . . . . . . . . . . . . . . 2.3.1. Radiation fields . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Radiation power . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. The radiation pattern, the phase center . . . . . . . . . . . . 2.3.4. Directive gain, directivity . . . . . . . . . . . . . . . . . . . . 2.3.5. Radiation impedance and radiation resistance . . . . . . . . 2.4. Gain, efficiency and effective aperture . . . . . . . . . . . . . . . 2.4.1. Gain and efficiency . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Receive antenna effective aperture. . . . . . . . . . . . . . . 2.5. Budget link, transfer function . . . . . . . . . . . . . . . . . . . . 2.6. Equivalent circuits of the antennas . . . . . . . . . . . . . . . . . 2.7. Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Example of characterization: the triangular probe antenna in F 2.8.1. Description of the structure . . . . . . . . . . . . . . . . . . . 2.8.2. Impedance matching . . . . . . . . . . . . . . . . . . . . . . . 2.8.3. Radiation patterns. . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4. Optimization of the antenna . . . . . . . . . . . . . . . . . . .

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33 36 37 37 38 40 40 41 41 43 46 47 47 48 49 51 52 52 53 53 54 58

Chapter 3. Representation, Characterization and Modeling of Ultra Wide Band Antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christophe ROBLIN

61

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Specificities of UWB antennas: stakes and representation 3.2.1. Context and requirements of an effective and complete representation . . . . . . . . . . . . . . . . . . . 3.2.2. Transfer function in transmission . . . . . . . . . . . . 3.2.3. Transfer function in reception, reciprocity . . . . . . . 3.2.4. Transfer function and “conventional” quantities . . . 3.2.5. Elements on the measurement of transfer functions in the frequency domain . . . . . . . . . . . . . . . . . . . . . . 3.3. Temporal behavior, distortion . . . . . . . . . . . . . . . . .

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61 62

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63 64 71 75

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Table of Contents

3.4. Distortion and ideality . . . . . . . . . . . . . . . . . . . . . . . 3.5. Performance characterization: synthetic indicators . . . . . . 3.5.1. Energy gain and mean realized gain (MRG) . . . . . . . . 3.5.2. Synthetic indicators of distortion . . . . . . . . . . . . . . . 3.6. Parsimonious representation by development of singularities and spherical modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1. The singularity expansion method . . . . . . . . . . . . . . 3.6.2. Spherical mode expansion method (SMEM) . . . . . . . . 3.6.3. Parametric model with very high order reduction . . . . . 3.6.4. Examples of processing of measured ATF . . . . . . . . .

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80 82 83 86

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Chapter 4. Experimental Characterization of UWB Antennas . . . . . . . . Christophe DELAVEAUD

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4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2. Measurements of the characteristics of radiation . 4.2.1. Basic concepts . . . . . . . . . . . . . . . . . . . 4.2.2. Frequency methods . . . . . . . . . . . . . . . . 4.2.3. Time domain method. . . . . . . . . . . . . . . 4.3. Measurements of the electric characteristics . . . 4.3.1. Preamble . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Frequency domain measurements . . . . . . . 4.3.3. Time domain measurements . . . . . . . . . .

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113 114 114 117 127 156 156 157 159

Chapter 5. Overview of UWB Antennas . . . . . . . . . . . . . . . . . . . . . . Nicolas FORTINO, Jean-Yves DAUVIGNAC, Georges KOSSIAVAS and Xavier BEGAUD

163

5.1. Classification of UWB antennas . . . . . . . . . . . . 5.2. Frequency independent antennas . . . . . . . . . . . . 5.2.1. Equiangular antennas . . . . . . . . . . . . . . . . 5.2.2. Log-periodic antennas . . . . . . . . . . . . . . . . 5.2.3. Techniques of frequency-independent antennas performance improvement . . . . . . . . . . . . . . . . . 5.3. Elementary antennas . . . . . . . . . . . . . . . . . . . 5.3.1. The biconical antenna . . . . . . . . . . . . . . . . 5.3.2. The discone antenna . . . . . . . . . . . . . . . . . 5.3.3. The bowtie antenna . . . . . . . . . . . . . . . . . . 5.3.4. Planar monopoles antennas . . . . . . . . . . . . . 5.3.5. Performance improvement techniques of elementary UWB antennas . . . . . . . . . . . . . . . 5.3.6. Directive elementary antennas . . . . . . . . . . . 5.3.7. Antennas with progressive transition . . . . . . . 5.3.8. Horn antennas . . . . . . . . . . . . . . . . . . . . .

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163 164 164 170

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176 177 177 179 180 181

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190 195 196 201

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5.4. Miniaturization of UWB antennas . . . . . . . . . . . . . . . . 5.4.1. General principles of antenna miniaturization . . . . . . . 5.4.2. Miniaturization problems of UWB antennas . . . . . . . . 5.4.3. Miniaturization techniques applicable to UWB antennas 5.5. UWB antennas for surface penetrating radars. . . . . . . . . . 5.5.1. Presentation of SPR UWB technologies . . . . . . . . . . 5.5.2. Design of antennas for SPR radars . . . . . . . . . . . . . .

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202 202 203 204 206 206 207

Chapter 6. Antenna-Channel Joint Effects in UWB . . . . . . . . . . . . . . . Alain SIBILLE

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6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Recalls on the UWB radio channel . . . . . . . . . . . . . . . . . . . . . 6.3. Impact of the channel on the performance of UWB systems . . . . . . 6.4. Effective antenna performance in an ideal channel . . . . . . . . . . . 6.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2. Radiation patterns for various architectures . . . . . . . . . . . . . 6.5. Effective performance of non-directional antennas in dispersive channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1. Gain calculation for non-ideal antennas . . . . . . . . . . . . . . . . 6.5.2. Results on measured channels . . . . . . . . . . . . . . . . . . . . . 6.6. Effective performance of directional antennas in dispersive channels 6.7. Factorization of antenna patterns . . . . . . . . . . . . . . . . . . . . . . 6.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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213 214 218 220 220 221

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225 225 231 233 235 237

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239

Appendix A. Reciprocity of the Antennas in Reception and Transmission Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241

A.1. Reciprocity applied to waveguides . . . . . . . . . . . . . . . . . . . . . . A.2. Reciprocity applied to the passive antennas in transmission and reception . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243 245

Appendix B. Method of the Stationary Phase . . . . . . . . . . . . . . . . . . .

253

Acronyms and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275

Preface

Ultra wide band (UWB) has received a great amount of interest since the decision by the US Federal Communications Commission (FCC) in February 2002 authorizing the emission of very low power spectral density in a bandwidth going from 3.1 to 10.6 GHz. This technique of radio transmission consists of using signals whose spectrum is spread out over a wideband of frequencies, typically from about 500 MHz to several GHz. It was formerly used for military and radar applications, then transposed a few years ago to telecommunications, thus causing a growing interest within the scientific community and industry. This spectral availability makes it possible to consider the wideband communications and also leads to a fine space resolution for the radars. However, the current restrictions of the regulatory agencies on the emission power level limit the range of the UWB communications to a few meters for high data rates and up to a few hundred meters for low data rates. UWB technology thus seems naturally well positioned for short range communications (WLAN, WPAN), offering an alternative at the same time of low cost and low consumption to the existing standards in these networks. The acronym UWB gathers two standardized but distinct technologies today. The first is founded on the emission of impulses of very short duration; this is the mono-band or impulse radio approach. The second approach is based on the use of multiple simultaneous carriers where the bandwidth is subdivided into several subbands (multi-band approach).The modulation used in each sub-band is the OFDM (Orthogonal Frequency Division Multiplexing). The advantages and disadvantages of the mono- and multi-band approaches are delicate questions and have been the subject of debate by many regulatory agencies. A particularly important question is the minimization of the interference to the emission and reception of the UWB system.

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The multiple band approach is particularly interesting because the carrier frequencies can be suitably selected to avoid interferences with narrow band-based systems. This offers more flexibility but requires an additional layer of control in the physical layer. UWB signals in the impulse technique require very good RF components (very short switching time) and a greater accuracy of synchronization. UWB systems can then be developed at a relatively low cost. Contrary to the multi-band approach which is based on techniques which are tested and available already, the architecture of a telecommunication system in impulse mode has involved many developments and in particular has required the installation of new definitions. The antenna does not escape these changes and we will show that this interface between the propagation channel and the architecture of the transmitters/receivers must add other time-domain radiation characteristics to optimize the transmission and the reception of impulses. These characteristics naturally come to complement and not replace the conventional ones, making it possible to qualify the antennas. This is the method which we retained in this work; starting from the usual parameters necessary for characterization of the antennas in the spectral domain, we added to these the suitable definitions in the time domain. We will not consider the radiation characteristics which will not be used during the antenna design in time domain. We will thus look at the frequency and time-domain characteristics, by specifying each time the joint and specific definitions. This book, dedicated specifically to UWB antennas, provides the electromagnetic foundations to students and presents state of arts for engineers and researchers. The reader will notice some absences: the IRA (Impulse Radiating Antenna) and specialized UWB smart antennas which could not be detailed within the scope of this book. This book is one of the fruits of the autumn school, GDR1 Ondes, on UWB organized in October 2006 in Valence (France). Its role was to present the fundamental aspects, measurement, processing and architectures of UWB systems. The large majority of the authors of this book were already “on board” and took an active part in the GDR Ondes Working group “Ultra Large-Bande, Communications Hauts-Débits, Contrôle et Commande” (Ultra Wide Band, High Data Rates Communications, Remote and Control). Finally, the book is a summary of French work recognized at an international level on a subject which, still today, produces several hundred scientific articles

1. GDR ONDES 2451, created on 1st January 2002 by the CNRS, has the role of being an indispensible center for all specialists in electromagnetism, optics and photonics and acoustics.

Preface

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every year. The chapters were written by academic and institutional researchers and industrial specialists in the field. This book is composed of six chapters. Chapter 1 presents the definitions and the regulatory aspects of the UWB. A classification then a comparison of UWB approaches is proposed. The chapter is closed by a presentation of UWB target applications on fields as varied as broadband communications in multiple environments and geolocalization. Chapter 2 defines the radiation characteristics of the antennas usually used in the frequency domain. It is a restricted, rather than an exhaustive, presentation of the characteristics which will be then used throughout the book. Special attention has been brought to the validity of the definitions in the time and frequency domains. An example of a directive UWB antenna is then proposed to illustrate the characteristics defined in the chapter. Chapter 3 enriches the conventional characterization of the antennas. Through a functional approach, we define concepts, objects of reference and indicators appropriate for the analysis of time domain behavior of UWB antennas. In particular, we focus on the phenomenon of signal distortion and on the concept of an ideal antenna. Because of the significant amount of data (experimental or simulated) to be handled and analyzed, various indicators of performance are then proposed making it possible to synthesize information to better expose the behaviors and imperfections, in order to more easily compare the antennas. Then a parametric modeling approach based on drastic order reduction closes the chapter. Chapter 4 provides the necessary complement to the two preceding chapters and presents the experimental characterization methods allowing the validation of any design. The first part of this chapter takes the logic of the book, describing antenna radiation measurements in the spectral domain then the methods developed for the time domain characterization of UWB antennas. The methods presented are detailed and specificities of the instrumentation are also described. Measurements of a compact UWB antenna make it possible to illustrate the preceding definitions. The chapter is concluded by the measurement methods of the electric characteristics of the inputs of the antennas. Chapter 5 is devoted to a panorama of existing antennas with matching impedance characteristics on very wide bandwidths and to some techniques making it possible to improve their performances. The frequency-independent antennas which present the property to be dimensioned identically at all the frequencies are initially detailed. Then, the elementary antennas with a widened shape are also described, in particular for UWB communications. Directive antennas, then antennas

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with progressive transition and horns finish this non-exhaustive presentation. The second part of this chapter is devoted to the reduction of UWB antenna dimensions for mobile terminals and provides the main strategies. After detailing at lenght the solutions adopted for communication applications, this chapter presents some UWB antenna technologies for ground penetrating radars. Chapter 6 presents the joint antenna-channel effects in UWB. The objective is to show that the effective behavior of the antennas within a radio link cannot be analyzed separately. After some reminders on the propagation channel, the influence of the channel on the performance of the UWB systems is presented. The study of the antenna effective performances with ideal channel, then dispersive for directive antennas or not, is then detailed. The chapter ends with a factorization of the radiation pattern making it possible to show that, according to the architecture, it can be useful to evaluate the quality of a radio link in UWB. This final chapter concludes the volume and directs the reader towards the book by our colleagues Pascal Pagani, Friedman Tchoffo Talom, Patrice Pajusco and Bernard Uguen on the UWB propagation channel [PAG 08]. With the transmission channel closely associating the propagation channel with antennas, the book by Pagani et al. can also be referred to as the necessary complement to our book. Xavier BEGAUD October 2010

Chapter 1

Applications of Ultra Wide Band Systems

1.1. Introduction The first definition of the Ultra Wide Band Systems (UWB) for commercial applications was provided in February 2002 by the FCC (Federal Communications Commission) [FCC 02]. This definition, based on the occupied bandwidth, defines as Ultra Wide Band any system having a bandwidth higher than or equal to 500 MHz or having a ratio between its carrier frequency and the occupied bandwidth higher than 25%. This definition made it possible to count various forms of UWB waveforms with all the criteria. Among the most conventional UWB waveforms, we can note the Impulse Radio waveform and the MBOFDM (Multi Band Orthogonal FrequencyDivision Multiplexing) waveform. However, the definition stated by the FCC allowed other forms of less conventional waveforms to be created. Among these we can detail the Frequency Hopping waveforms on the one hand and the chirp waveform on the other hand. These various forms of UWB made it possible to define two main categories of applications: – High and very high data rate UWB applications enabling wireless communications with a data rate of 480 Mb/s or even 1 Gb/s. For these applications, the MB-OFDM solution was defined in various standards.

Chapter written by Serge HÉTHUIN and Isabelle BUCAILLE.

Ultra W ide Band Antennas Edited by Xavier Begaud © 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

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Ultra Wide Band Antennas

− Low data rate UWB applications having, in addition to communication services, localization services inside buildings, thus having a service comparable to GPS (global positioning system) but inside. 1.2. UWB regulation: a complex context 1.2.1. UWB regulation in the USA In February 2002, the FCC allocated a frequency band for UWB systems for communications applications, ground penetration radars, through-wall imaging, medical applications, security applications as well as radar applications for vehicles. For communications applications, the FCC differentiated the use of UWB systems inside and outside buildings. The frequency band authorized in the United States for communication and localization applications is between 3.1 GHz and 10.6 GHz with a maximum mean Equivalent Isotropic Radiated Power (E.I.R.P, see Chapter 2) of –41.3 dBm/MHz. Figure 1.1 presents the emission mask authorized by the FCC in 2002 for communications like those inside buildings and those for handheld devices used outdoors.

Figure 1.1. Emission mask authorized by the FCC in 2002

Applications of UWB Systems

3

For communication outside buildings, only mobiles are permitted and the authorized levels are 10 dB lower than those tolerated for communications inside buildings. The term “Part 15 limit” in Figure 1.1 relates to the limit tolerated by the FCC for non-intentional emissions, i.e. the radiation produced by electric household appliances for example. 1.2.2. UWB regulation in Europe In March 2006, the CEPT (Conference Européenne des Postes et Telecommunications – European Post and Telecommunications Conference) gave, through ECC TG3 (Electronic Communications Committee Task Group 3), the first European authorization for UWB systems [ECC 06a]. This first decision authorizes these devices in the 6-8.5 GHZ band without any mitigation techniques (used to reduce interferences) with a maximum mean E.I.R.P. of –41.3 dBm/MHz. In December 2006, the principle of progressive approach (phased approach) in the frequency band 4.2-4.8 GHz was accepted by the ECC. This allowed the introduction into Europe of a first generation of UWB equipment in this frequency band, with a maximum mean E.I.R.P. of –41.3 dBm/MHz without any mitigation techniques. The introduction of this first generation of UWB equipment is authorized until December 31st, 2010. Following this decision, in July 2007, the ECC amended its first decision for the generic UWB systems without license. Table 1.1 presents this decision which does not reveal the techniques of mitigation authorized in the 3.4-4.8 GHz band, as they are defined in another ECC decision. It should be noted that, in this decision, UWB systems are not authorized on aircraft, nor on fixed outside infrastructures, but are authorized in vehicles if they implement a Transmit Power Control such as the one defined in Table 1.1. The European commission published, in February 2007, a decision relating to the use of UWB equipment without license in Europe, taking again the points mentioned above. The European countries were required to apply this decision from September 2007 [ECC 07]. In the 3.1-4.8 GHz band and 8.5-9 GHz band, UWB equipment must apply mitigation techniques in order to protect the already existing services in these bands. If they implement mitigation techniques, UWB devices are authorized to transmit in these bands with a maximum mean E.I.R.P. of –41.3 dBm/MHz. In the 4.2-4.8 GHZ band, these methods are not mandatory until the end of December 2010.

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Ultra Wide Band Antennas

In December 2006, CEPT adopted the first decision relating to the mitigation techniques in the 3.4-4.8 GHZ band. This decision was amended in October 2008. This amendment defines at the same time both the LDC (Low Duty Cycle) mitigation technique in the 3.1-4.8 GHz band and the DAA (Detect And Avoid) mitigation technique in the 3.1-4.8 GHz and 8.5-9 GHz bands. These techniques were defined in order to protect WiMAX and radiolocation services while allowing a maximum mean E.I.R.P. of –41.3 dBm/MHz in these bands. If these mitigation techniques are not implemented, the levels of power authorized in these bands are defined in Table 1.1. Frequency range

Maximum mean E.I.R.P. spectral density) (dBm/MHz)

Maximum peak E.I.R.P. (measured in 50 MHz)

Below 1.6 GHz

–90 dBm/MHz

–50 dBm

1.6 to 2.7 GHz

–85 dBm/MHz

–45 dBm

2.7 to 3.4 GHz

–70 dBm/MHz

–36 dBm

3.4 to 3.8 GHz

–80 dBm/MHz

–40 dBm

3.8 to 4.2 GHz

–70 dBm/MHz

–30 dBm

4.2 to 4.8 GHz (Notes 1 and 2)

–70 dBm/MHz

–30 dBm

4.8 to 6 GHz

–70 dBm/MHz

–30 dBm

–41.3 dBm/MHz

0 dBm

8.5 to 10.6 GHz

–65 dBm/MHz

–25 dBm

Above 10.6 GHz

–85 dBm/MHz

–45 dBm

6 to 8.5 GHz (Note 2)

Note 1: UWB equipment placed on the market before December 31st, 2010 is authorized in the 4.2-4.8 GHz frequency band with a maximum mean E.I.R.P. spectral density of –41.3 dBm/MHz, and a maximum peak E.I.R.P. of 0 dBm measured in 50 MHz. Note 2: In case of devices installed in road and rail vehicles, operation is subject to the implementation of Transmit Power Control (TPC) with a range of 12 dB with respect to the maximum permitted radiated power. If no TPC is implemented, the maximum authorized mean E.I.R.P. spectral density is limited to –53.3 dBm/MHz. Table 1.1. Decisions of the ECC in July 2007

Applications of UWB Systems

5

Restriction LDC, which consists of limiting in time UWB emissions, is especially applicable to low data rate UWB applications. With this technique, the sum of all the transmissions (by equipment) must be less than 5% of the time over one second and less than 0.5% of the time over one hour. Moreover, the duration of each transmission should not exceed 5 ms. Note that UWB equipment operation on board vehicles is not subject to Transmit Power Control as defined in Table 1.1 if they implement LDC. Restriction DAA consists of detecting the presence of other possible radio signals (like WiMAX or radiolocation services) and reducing the transmitted power of UWB equipment to a level that will not cause interferences on the reception of other radio signals or quite simply changing the channel used by the UWB device. Thus, before initiating a communication, UWB equipment implemented with a DAA mitigation technique must be able to identify the electromagnetic radio environment in a minimum of time in order to detect the devices that are not to be disturbed. The equipment must also be able to detect the changes during the time of the electromagnetic radio environment in order to modify the UWB parameters if necessary. The DAA mechanism applies especially to short range high data rate UWB devices for which various channel models were defined in the ECMA-368 standard (European Computer Manufacturers Association) [ECM 07]. For the reduction of UWB equipment transmitted power in a given channel, the DAA mechanism was defined in a flexible way, thus making it possible to define several levels of power according to the area in which it is located. An area is defined by a range allowing separation between UWB equipment and another communications devices which can be subject to interference in the same band. The three areas defined for the DAA mitigation technique and their associated ranges correspond to the maximum power spectral density as defined in Table 1.2. Table 1.2 gives the various values to be applied for the DAA without changing channel. All the parameters and justifications of the LDC and DAA mitigation techniques are detailed in reports ECC 94 [ECC 06b] and ECC 120 [ECC 08].

6

Ultra Wide Band Antennas

Zone 1 for signal detection level S>A

Operation frequency (GHz)

3.1-3.4 GHz

Minimum initial channel availability check time (seconds)

14

Maximum mean E.I.R.P. spectral density (dBm/MHz)

–70

Default avoidance bandwidth (MHz)

300

200

500

–38

–38

–61

Maximum mean E.I.R.P. spectral density (dBm/MHz)

–41.3

–65

–41.3

Default avoidance bandwidth (MHz)

-

200

-

-

–61

-

-

–41.3

-

Signal threshold detection A (dBm) Zone 2 for signal detection level A>S>B

Signal threshold detection B (dBm) Zone 3 for signal detection level signal B>S

Maximum mean E.I.R.P. spectral density (dBm/MHz)

3.4-3.8 GHz

3.8-4.8 GHz

5.1

–80

8.5-9 GHz

14

–70

–65

Table 1.2. UWB equipment transmitted power applying the DAA

1.2.3. UWB regulation in Japan In Japan, the regulation organizations authorized UWB emission with a maximum mean E.I.R.P. of –41.3 dBm/MHz without mitigation techniques in the 7.25-10 GHz band. The common band to the USA, Europe and Japan is thus 7.258.5 GHz without any mitigation techniques and 7.25-9 GHz with DAA as used in Europe. This last band of 1.75 GHZ allows the use of three sub-bands as defined in the ECMA standard [ECM 07].

Applications of UWB Systems

7

In the lower band (3.4-4.8 GHz), Japan adopted mitigation techniques based on the European model. A “phased approach” allowing the marketing of a first generation of equipment in the 4.2-4.8 GHz band without mitigation techniques was also put into practice until the end of 2008. Figure 1.2 represents the mask in Japan with mitigation techniques between 3.4 and 4.8 GHz.

Figure 1.2. Japan emission mask

1.2.4. Emission mask in the United States, Europe and Japan Figure 1.3 summarizes the various emission masks in the USA, Europe and Japan. The mask used in Japan is more generally used in Asia. Certain UWB standards such as the ECMA 368 require a bandwidth of 3*528 MHz, which is more than 1.5 GHZ. The study of the various masks shows the difficulty of obtaining an identical band throughout the world. Even in the band above 6 GHz where mitigation techniques are not compulsary in Europe, it is necessary for the equipment to conform to standard ECMA 368, to apply mitigation techniques between 8.5 and 9 GHZ in order to be able to sell the same equipment in Asia where the high spectrum begins from 7.25 GHZ. This lack of agreement preventing the use of a common mask throughout the world for UWB systems is at the origin of the delays in starting mass production of this equipment.

8

Ultra Wide Band Antennas

Figure 1.3. Emission mask in the USA, Europe and Japan

1.3. Formal Ultra Wide Band types 1.3.1. Ultra Wide Band Impulse Radio (UWB-IR) The Impulse Radio UWB waveform is characterized by the periodic emission of a pulse of very short duration. The transmission interval of the pulses is defined using the PRP (Pulse Repetition Period) or PRF (Pulse Repetition Frequency) parameter. Classically, parameter PRP is about 200 nanoseconds (Figure 1.4).

Amplitude

PRP

Time

Figure 1.4. Principle of a pulse modulation (here Pulse Position Modulation)

Applications of UWB Systems

9

The duration of a pulse is typically 2 nanoseconds and is inversely proportional to the occupied bandwidth. Thus, the band at (–3 dB) bandwidth is defined by 1.16/τ (τ being the pulse width) and the band at –10 dB is defined by 1.8/τ. Figure 1.5 represents typical transmitted pulses as well as the bandwidth used according to the width of the impulse.

Figure 1.5. Impulses and band-width

It should be noted that UWB-IR equipment using the low band in Europe cannot apply the DAA mitigation technique as the entire spectrum is not cut out. The LDC (Low Duty Cycle) or TPC (Transmit Power Control) types of mitigation can be applied.

Amplitude

An elementary transmitted pulse corresponds to the first derivative of a Gaussian signal. Various modulations can be applied to the impulse radio waveform. The simplest of the modulations (Figure 1.6) is OOK (On Off Keying) modulation. This modulation, though very simple, does not allow us to obtain good performances. It is not implemented in the recent versions of equipment or prototypes.

Time Figure 1.6. OOK modulation with an impulse radio waveform

10

Ultra Wide Band Antennas

Amplitude

The modulation which is typically used for Impulse Radio waveform is the PPM (Pulse Position Modulation) modulation. As illustrated in the Figure 1.4, the bursts of pulses are transmitted at regular intervals. In order to be able to differentiate the data sent, one of the solutions consist of transmitting the bits with a shift (delta PPM) that is positive or negative compared to the nominal position to which the pulses must be transmitted. Figure 1.7 represents a PPM modulation with four states.

Time

(ps)

Figure 1.7. PPM modulation with an impulse radio waveform

The UWB Impulse Radio waveform has been standardized in IEEE 802.15.4a in order to define low data rate robust communications, with low power consumption and enabling very precise distance measurements inside buildings. In the standard, various data rates are possible: a nominal capacity of 851 kb/s is mandatory and optional data rates of 110 kb/s, 6.81 Mb/s and 27.24 Mb/s are also defined. It should be noted that the Impulse Radio UWB waveform allows very different radio data rates as the repetition time of the pulses is easily adjustable. In the standard, the modulation used for the impulse radio waveform is a combination of BPSK (Binary Phase Shift Keying) and BPM (Burst Position Modulation) modulations. The BPM modulation is comparable to PPM modulation but applied to a burst of pulses and not only to one elementary pulse. This modulation is used in order to support non-coherent receivers as well as coherent receivers. The combination of the two modulations corresponds to the modulation in BPSK of a burst of pulses themselves modulated in BPM (Figure 1.8).

11

Amplitude

Applications of UWB Systems

Time

Figure 1.8. BPSK modulation with an impulse radio waveform

The main interest of the Impulse Radio UWB waveform lies in the fact that it enables a localization with a precision of less than one meter (due to the very short length of the pulses). This precision cannot be reached with Wi-Fi devices. The few pieces of equipment that exist to date are thus for the moment mainly limited to professional applications. Companies like Time Domain or Ubisense developed systems based on tags allowing us to locate people carrying these tags, for example, in hospitals. In these systems, the tags are transmitters but with very low communication capability. The localization is carried out on the level of the infrastructure receiver and remains confined to this level. To date there are products based the on Impulse Radio waveform making it possible for individuals to locate themselves compared to other people present in their entourage. Indeed, this requires for each equipment, an UWB transmitter and receiver and involves a more important data rate at the level of each node decreasing de facto the link budget between two nodes. Prototypes resulting from the European project PULSERS II were developed in order to obtain localization information on each node of the network. The prototypes use an Impulse Radio waveform centered at 4.2 GHz with a bandwidth of 1 GHz. Two modulations are available: a DBPSK (Differential Binary Phase Shift Keying) modulation or a PPM. The radio data rate of each equipment is 387 kb/s. An ASIC (Application-Specific Integrated Circuit) including the baseband and analog parts was developed in the scope of the project. The size of the prototypes is 100*60*40 mm. They were optimized within order to offer a great autonomy since the whole platform consumes 500 milliwatts and the ASIC at the receiver only consumes 8 milliwatts. Figure 1.9 below represents a node developed within the framework of this project. It should be noted that the antenna is integrated into the casing, and that a card with a temperature sensor and PIR (passive infrared) sensor was integrated into

12

Ultra Wide Band Antennas

the front face to allow demonstrations within the framework of sensor network applications.

Figure 1.9. UWB-IR Node in PULSERS II project

Moreover, for the networked setting of these nodes, a Medium Access Control (MAC) was developed and integrated into the FPGA. The developed MAC layer is based on a TDMA (Time Division Multiple Access) protocol and enables the deployment of a centralized mesh network (the coordinator of the network can be any node of the network). The procedures necessary for the distance measurements were also implemented on UWB-IR PULSERS II nodes. It is thus possible to obtain a distance measurement between nodes by calculating the messages round trip time between these nodes. In the two way ranging procedure, two messages are exchanged for the round trip time calculation whereas in the three way ranging procedure, three messages are used in order compensate clock drifts. The choice between two way or three way ranging is made at the initialization of the nodes. The resolution on the distance measurement is of 1.1 nanosecond providing a granularity of 30 cm. 1.3.2. OFDM-ultra wide band (UWB-OFDM) The UWB-OFDM waveform was created due to the necessity to separate the 3.110.6 GHZ spectrum authorized in the United States into different sub-bands in order to fulfill the regulation defined in Europe and Asia. The basic idea consists of dividing the spectrum into sub-bands of 528 MHz. Thirteen 528 MHz sub-bands were defined between 3.1 and 10.6 GHz as shown in Figure 1.10.

Applications of UWB Systems

GROUP B

GROUP A

GROUP C

13

GROUP D

Band #1

Band #2

Band #3

Band #4

Band #5

Band #6

Band #7

Band #8

Band #9

Band #10

Band #11

Band #12

Band #13

3432 MHz

3960 MHz

4488 MHz

5016 MHz

5808 MHz

6336 MHz

6864 MHz

7392 MHz

7920 MHz

8448 MHz

8976 MHz

9504 MHz

10032 MHz

f

Figure 1.10. Sub-bands used for the UWB-OFDM waveform

Four distinct groups and two modes have been defined. The first mandatory mode uses the first three sub-bands of group A. The second (optional) mode uses the groups A and C. According to the various regulations that have been put in place throughout the world (see section 1.2), the bands can be used as described in Figure 1.11. The Multi-Band OFDMD UWB waveform used for high data rate applications was standardized in December 2005 by ECMA (ECMA 368). A second version of the standard was published in December 2007 in order to take into account the latest regulation rules (such as the DAA mitigation technique). Standard ECMA 368 allows us to support various radio data rates: 53.3Mb/s, 80 Mb/s, 106.7 Mb/s, 160 Mb/s, 200 Mb/s, 320 Mb/s, 400 Mb/s and 480 Mb/s. The data rates of 480 Mb/s, 200Mb/s and 80 Mb/s respectively enable link distances of 2, 4 and 10 meters.

Figure 1.11. Use of UWB-OFDM bands throughout the world (WiMedia Source)

14

Ultra Wide Band Antennas

The standard specifies a multiband OFDM (MBOFDM) waveform using 100 subcarriers for the data and 10 guard subcarriers. To these 110 subcarriers are added 12 pilot subcarriers enabling a coherent detection at the reception. The spread spectrum in time and frequency as well as convolutional coding (1/3, 1/2, 5/8 or 3/4) allows us to obtain various radio data rates. The standard also defines a Medium Access Control (MAC) layer allowing the communication between several UWB-OFDM nodes simultaneously. The architecture network is completely decentralized since no node plays the role of network coordinator. A TDMA protocol has been specified with this decentralized architecture. The coordination of the nodes in the network for the definition of the temporal TDMA slots is achieved by all the nodes, which all send beacons (contrary to a network having a coordinator who sends the beacons). The beacons contain information for network synchronization and temporal slot assignments. Authentication, encoding as well as a distance measurement procedure are also detailed in the standard. Equipment based on this standard is gradually appearing. This includes equipment by the companies Wisair and Staccato-Artimi. Intel is no longer developing a UWB-OFDM solution but Samsung announced in February 2009 that it would release a UWB product for the second quarter of 2009. All these companies were initially focused on a Wireless USB (WUSB) solution allowing us to connect the peripherals of a PC with a high speed wireless connection. However, other profiles can be added above the MAC layer of ECMA 368 standard. Indeed, Bluetooth and IP (WLP layer) profiles can be implemented using this standard. Certain companies have become interested in this since the WUSB has experienced difficulties getting off the ground with these modules. However, to date, no products corresponding to these other profiles have been delivered. 1.4. Non-formal ultra wide band types 1.4.1. Ultra wide band frequency hopping (UWB-FH) The principle of this technique consists of using fixed frequency hops (FH for frequency hopping) with a width of 20 MHz on a very wide frequency band (1.25 GHz between 3.2 and 4.8 GHz). The hops overlap at 50% thus having an overlap of 10 MHz. To adhere to regulation rules, the total band must be explored in

Applications of UWB Systems

15

less than 1 millisecond, which leads to a choice, in practice, of a speed of 62,000 jumps per second and a hop duration of less than 100 µs. On each elementary band of 20 MHz, a PN code of 20 Mchips/s is used, which makes, due to its classicism, the method extremely simple. However, due in particular to the speed and width of the spread spectrum technique used, the process is very robust. It is possible to use 25 orthogonal channels simultaneously, which makes the cohabitation of several devices in a close environment possible. Another advantage, inherent to the frequency hopping spread spectrum technique, is the possibility to reject the interfered frequencies or to avoid certain parts of the spectrum allocated to other services having higher priority (Figure 1.12). Spectral avoidance or reduced power levels possible for chosen bands

Relative Power dB

0 -10

Spectrum measured over ~1ms

Radio Relay band

-20 Narrow spectral regions can be avoided

-30 -40 -50 3000

3200

3400

3600

3800

4000

4200

4400

4600

4800

5000

Frequency, MHz

Figure 1.12. Frequency excision made possible by the UWB-FH

Figure 1.13. Localization of firemen using UWB-FH

5200

16

Ultra Wide Band Antennas

The main application for this type of UWB technique is the localization of first response teams (firemen, first-aid workers, etc.) during serious events, in particular inside buildings. Figure 1.13 shows a team of British firemen fitted with the currently available prototype (THALES TRT UK) allowing us to reach a precision better than 3 meters in 3D positioning. Figure 1.14 details the UWB-FH terminal for which the antenna is miniaturized [CHU 05], which is essential when being carried by first-aid workers in an operational scenario.

Figure 1.14. UWB-FH device and its miniaturized antenna

The first results obtained with UWB-FH technology are encouraging but it must not be forgotten that this technology is not supported by a standard. On the other hand, for some applications which do not require any particular standard like through-wall imaging and medical imaging, this technology can be applied. Indeed, for this application, the stress is laid on the use of 3D scanning UWB antennas, such as the one represented in Figure 1.15. The use of antennas with high gain allows the transmit power to be limited while obtaining a sufficient link budget.

Figure 1.15. UWB BOP antenna [CHU 05] with scanning

Applications of UWB Systems

17

Whilst the total bandwidth (1.25 GHz) is important, the limited instantaneous bandwidth (20MHz) makes UWB-FH a robust conventional technique. However the available radio data rate (15 kbits/s) limits Anchor Based Localization applications (detailed later on) and does not allow us to consider applications combining communication and localization. 1.4.2. Chirp Ultra Wide Band (UWB-FM) UWB-FM (FM for frequency modulation), still called UWB-CSS (CSS for Chirp Spread Spectrum), is extremely effective for communications applications as well as for localization applications.

Amplitude

The waveform developed for this technique has been used for decades in the radio altimeter or radar devices and for a much longer time by bats. In general, it is based on a linear frequency slope (Figure 1.16) with an excursion lower than or equal to the total available bandwidth.

Figure 1.16. Chirp waveform as used in UWB-FM

Within the framework of standardization 802.15.4a, the exploitation of the UWB-FM is authorized in the 2.45 GHz ISM band (industrial, scientific and medical) with 80 MHz of bandwidth and a maximum transmit power of 100 mW. The data to be transmitted are modulated on the slope (the chirp) according to a differentially bi-orthogonal (DBO) 8 M-ary modulation (8 states). The maximum capacity is 2 Mb/s. The chirps can be sent sequentially, but can also be superimposed, due to orthogonality properties of the successive slopes in order to increase aggregate rates. Thus, in Figure 1.17 we can compare a configuration with sequential slopes with a configuration with interleaved slopes (but at reduced power) for an increased radio data rate.

18

Ultra Wide Band Antennas

Figure 1.17. Interleaved slopes in UWB-FM

The distance measurement between two UWB-FM nodes is achieved by the common methods of two-way ranging or symmetrical double sided two-way ranging (Figure 1.18). The advantage of this technique is easily adaptable to the available frequency band from a minimum of 20 MHz to several hundreds of MHz. The main application of the UWB-FM is the distance measurement and by consequence the situation awareness of responders organizations. More precisely, this allows the deduction in real-time of the relative positioning of the team members due to the available date rate for data transmission. Node A

Node B Data

Propagation Processing

Ack

Propagation

Propagation Processing Propagation

Figure 1.18. Symmetrical double sided two-way ranging

Applications of UWB Systems

19

Indeed, contrary to a conventional technique based on the triangulation to anchors of known coordinates – ABL (for Anchor-Based Localization) technique presented in Figure 1.19 – the relative positioning of the members of a team, widely deployed inside a building, requires a technique known as AFL (Anchor-Free Localization) without anchors for which the distance measurement exchanges are numerous and must have the weakest possible latency (Figure 1.20).

Mooring

Point point 2 d’ancrage 2 (x2, y2) y2) (x2, d2

d1 Point Mooring point 1 d’ancrage 1 (x1, y1) (x1, y1)

d1 =

(x1 − xM )2 + ( y1 − yM )2

d2 =

(x2 − xM )2 + ( y2 − yM )2

d3 =

(x3 − xM )2 + ( y3 − yM )2

TRM (xM, yM)

d3

Point point 3 Mooring d’ancrage 3 (x3, y3) (x3, y3)

Figure 1.19. ABL (Anchor-Based Localization) technique

Figure 1.20. AFL (Anchor-Free Localization) technique

Possible distance measurements are then limited to the nodes having Line Of Sight connections. In the configuration presented in Figure 1.20, connectivity

20

Ultra Wide Band Antennas

between the nodes is obviously very reduced, however the algorithm related to the relative positions of the responders team can, within a certain limit related to the density and the number of inter-connected nodes, deduce the relative positions of all the group members. To be rigorous and reliable, this algorithm requires exchanges towards all the nodes with regard to distance measurements taken by each node with its neighbors and this must be done in a very short time. In nodes are not able to communicate this information in real-time (and thus with a sufficient speed), the restitution of the positions becomes useless and not in line with practical applications. This technique is operational right now, with the company NANOTRON proposing development kits (Figure 1.21).

Figure 1.21. UWB-FM development kit

1.5. Comparison between the different Ultra Wide Band techniques Excluding High Data Rate Ultra Wide Band waveforms (UWB-OFDM), other techniques like UWB FH, UWB-IR and UWB-FM can be differentiated in the following way: – UWB-FH: - limited noise bandwidth limited in reception due to the narrow instantaneous bandwidth (20MHz);

Applications of UWB Systems

21

- very good performance of distance measurements between nodes; - limited in communications capability; - not standardized; – UWB-IR: - significant noise bandwidth in reception due to the great instantaneous bandwidth (500 MHz); - very good performance of distance measurements between the nodes; - very good communications capability; - standardized (802.15.4a); – UWB-FM: - ultra wide band with the possibility of operation with a more reduced bandwidth and compatible with more traditional regulations; - noise bandwidth in reception according to the bandwidth used; - very good distance measurement performance between the nodes; - very good communications capability; - standardized (802.15.4a). 1.6. Typical UWB-OFDM applications The applications concerned by this waveform are high and very high data rate applications. 1.6.1. Peripheral connection to a PC The first applications for the UWB-OFDM relates to the connection of various peripherals such as printer, web-cam, Digital camera, HD video camera, digital photo frame. The companies working on the UWB-OFDM first manufactured equipment with a USB interface (no other interface is currently available). Thus, Wisair, Staccato Communications, Artimi and Belkin marketed Wireless USB equipment based on the ECMA 368 standard. At the CebiT 2009 show, Fujitsu-Siemens and Olidata presented Wireless USB adapters based on Wisair technology.

22

Ultra Wide Band Antennas

The suggested solutions enable fast data transfers via a USB interface, but only in point-to-point mode for the moment. Indeed, the MAC protocol specified in standard ECMA 368 being complex this is difficult to implement in the various solutions. In point-to-point mode, these wireless UWB adapters allow us to reach a radio data rate of 480 Mb/s with a link distance of 2 meters. Figure 1.22 represents a typical use case of WUSB equipment.

Figure 1.22. Wireless USB applications (source WUSB forum)

1.6.2. High speed applications in large structures with optical fiber backbone One of the high speed UWB applications relates to average or large jumbo jets. Indeed, in the future, modern aircrafts will have to support a greater flexibility and to adopt, consequently, a faster installation of the seats and various equipments into the aircraft, while enabling easier maintenance. Moreover, in order to facilitate communications between the crew members and to offer new services to the passengers, wireless gateways will be deployed on board aircraft. Equipment suppliers distinguish three categories of services: – management system of the cabin; – entertainment offered to the passengers; – maintenance or crew equipment.

Applications of UWB Systems

23

The management system includes all the services of first need as well as the additional functions available in the cabin. This includes the audio notices made during the flight, momentary calls, information for the passengers, cabin lighting, reading lights, specific lighting such as for the bar or the toilets, emergency lights, the signals of various sensors present in the cabin, the wireless transmission of information devices and radio devices dedicated to the crew. Extending the cabin management system (CMS) by a wireless infrastructure makes it possible to increase the specific adaptability of the cabin according to the needs, to optimize the services according to particular requests of the companies and the customers while authorizing a fast reconfiguration in answer to particular requests by the users (for example, the number of seats in business class versus the number of seats in economy class). The following diagram (Figure 1.23) represents the on-board network architecture enabling us to connect by radio the fixed or mobile equipment. In order to increase the reliability and the availability of the system, dual mode radios are envisaged. With such redundancy, the system can thus tolerate the loss of an access point and avoid interference problems.

Figure 1.23. Functionalities of a traditional IFE system (source: EADS Innovation Works, Munich, Germany)

The IFE (In-Flight Entertainment) system on board aircraft is still, for the moment, based on wired connections with the main disadvantages of a lack of flexibility, the weight as well as the high cost of installation. These disadvantages are mainly solved with a wireless IFE system. For its characteristics of fast attenuation and available bandwidth, the 60 GHz frequency band is a serious candidate for this variety of application.

24

Ultra Wide Band Antennas

An IFE system is typically made up of the following elements (Figure 1.24): – an IFE control center with content and file servers as well as Ethernet “switches”; – a functional IFE distribution network going from the central IFE server to the various seat rows; – the seat equipment for passengers; – video display devices; – a IFE control panel located in the IFE control center for the checking operations undertaken by the cabin crew. Remote Control Centre (RCC) : D V D

DVD or VTR

Control CD Drive

Printer

Panel

Keyboard Credit Card Reader

Notebook

PDA

Overhead Video Displays

Passenger Seat

Media/File Server

Handset Seat Display

Avionics Interface Unit Other

Ethernet Switch

Aircraft Systems Remote Maintenance

Ethernet

Distribution

Ethernet

Network 800VU

SEB

RS485/USB

IFEC Rack

ISPSS Power Outlet

SeatPeripherals

Figure 1.24. Traditional IFE system components (source: EADS Innovation Works, Munich, Germany)

Three basic approaches can be considered for radio coverage of the cabin using a UWB sub-network. The first topology consists of using radiating cables for the UWB backbone. Figure 1.25 shows a configuration with two radiating cables each having two access points. Each of these radiating cables are deployed on both sides of the cabin.

Applications of UWB Systems

25

Figure 1.25. Structure based on radiating cables (source: EADS Innovation Works, Munich, Germany)

The feasibility of this approach is based on the possibility of having radiating cables working with UWB signals. The second topology suggests covering the whole cabin as indicated in Figure 1.26. Each Access Point (AP), placed on the central axis of the cabin, covers a part of the cabin.

Figure 1.26. The second architecture with the AP installed on the central axis of the cabin (source: EADS Innovation Works, Munich, Germany)

This approach appears more adapted to radio coverage for single corridor aircraft. This conclusion is based on the fact that UWB nodes can provide a maximum capacity up to a distance of 3 meters. Considering that the width of the A380 cabin is approximately 6.35 meters and that the distance between two APs is 6 meters, the maximum distance to be covered by APs is 4.36 meters. In this case, by making the assumption of omni-directional antennas for APs, it is not possible to provide a sufficient connection quality in any point of the cabin. In the third suggested topology (Figure 1.27), APs are installed on the walls on both sides of the cabin. In this case, the AP antennas can be directive antennas and thus allow us to obtain a sufficient link budget with a maximum capacity at any point. In an IFE system deployed on an aircraft with a double bridge, we consider that the maximum number of radio nodes necessary to cover the bridge of a 70 meters long cabin is 500 nodes.

26

Ultra Wide Band Antennas

Figure 1.27. The third architecture with the AP installed on the walls of the cabin (source: EADS Innovation Works, Munich, Germany)

Concerning necessary data rates, we can draw up the list of the following requirements: – the IFE system must provide the diffusion of the audio and video services as well as the video on demand for each passenger; – the IFE system must be able to: - treat and distribute all the usual video formats (MPEG 1, 2, 4) and audio formats, - provide and transmit HDTV signals (video streaming), - provide Internet/email/Web-based 2.0 applications, - provide the capacity for interactive games between passengers; – the resolution of graphic outputs must be 720 pixels; – the down link data rate by passenger must be higher than 20 Mbit/s; – the up link data rate by passenger must be higher than 5 Mbit/s; –The latency must be equal to or better than 300 milliseconds. 1.6.3. High speed UWB in a harsh indoor environment In certain environment configurations, propagation is particularly difficult even while exploiting the usual parameters which are the transmit power, the desired data rate, the receiver sensitivity or the antenna gain. This is unquestionably the case in metal structures as we can find them in large cargo boats or in buildings with very thick reinforced concrete walls. In such situations, wireless product fitters (such as Wi-Fi) seek to cover the maximum number of cabins or offices with a minimal number of APs on optimal sites. This approach, in the case of high data rate UWB systems, is hopeless. Indeed, the radiated power is so low that it is illusory to think that it will be possible to cross

Applications of UWB Systems

27

the walls. Moreover, in completely metal closed environments, even by increasing the transmitting power, the hope to be able to cross the walls is vain, regardless of the technology. An alternative consists to deploy an optical fiber backbone. The principle is based on optical radio-electromagnetic converters. RF waves are carrying directly on optical fibers and the deployment is then ensured by the nodes in each of the cabins or in each office. This type of work is carried out in various European programs and in particular in UROOF (Ultra wide band Radio Over Optical Fiber) project. The radio cells are very limited but the radio data rate is very important (about 1 Gigabits/second). Moreover, the limitations in range allow us to ensure a re-use factor which is very important. In addition the security is strongly improved. 1.6.4. High speed UWB combined with other technologies UWB technology is not intended to be used separately. In particular, UWBOFDM can be combined with other communication technologies in order to provide a communication solution with the most favorable data rate throughout the situations encountered by a user in his work or leisure.

Figure 1.28. UWB-OFDM combination with WiMAXHSPA

28

Ultra Wide Band Antennas

Thus, the coupling between a high data rate short range UWB-OFDM system and a technology like WiMAX802.16 and/or HSPA having moderate data rates but much longer range, allows us to consider different stages as described in the scenario in Figure 1.28. In the first stage (position A), the user benefits from his proximity with his xDSL connection point to obtain the highest data rate in the car. During its move towards the shopping mall, the device switches into WiMAX or HSPA network. Lastly, at the time of arrival at the shopping mall, the device profits again from very high data rate due to UWB-OFDM access points installed on the site. 1.7. Specialized UWB-OFDM applications 1.7.1. Last mile radio applications This application consists in extending in cities the existing fiber optic backhaul to users inside buildings using very high data rate wireless technologies such as UWB-OFDM. Indeed, thanks to the millimeter waves, the radio transmission data rates can reach 1Gb/s and more. The maximum distances from wireless UWB-OFDM nodes to optical fiber backbone are about 300 m, and to arrive there – so as to distribute to all the buildings concerned – it is necessary to set up relays, installed in particular on the standard lamps (Figure 1.29).

Figure 1.29. High data rate wireless connection to optical fiber

Applications of UWB Systems

29

Each node must be able to transmit or receive in the various directions of the buildings or to neighbor nodes. Based on the TDMA access protocol, the directional antenna – for link budget reasons – must be able to be positioned in a very short time (< 250 nanoseconds) in order not to penalize the operational data rate and also not to call into question the standard used. This leads to the concept of FESA™ (Fast Electronically Steerable Antenna) which, with 16 beams each of a 16 dBi gain, can be contained at 60 or 40 GHz in a 10 cm height casing with a diameter of a few centimeters (Figure 1.30).

Figure 1.30. FESA™ antenna

1.7.2. Information and video streaming applications Many examples of high data rate information download or streaming video can be listed. Among the typical applications, one consists of downloading multimedia content in railways stations just getting on the train or waiting for the departure of the train.

Figure 1.31. High speed diffusion of information and video streams

30

Ultra Wide Band Antennas

The diffusion of these non-privative contents must be carried out in a very short time in order not to force the person who downloads the information to wait for a long time. This is carried out, of course, with very high data rate wireless technologies (about 500 Mb/s) especially for the transmission of photo albums, web sites, road maps or complete cinematographic films. The transmission range is considered at ten meters (Figure 1.31) thus enabling, with a radio data rate of 500 Mb/s, to download a 500 Mo video clip in less than 20 seconds. The same device can be used in bus garages for uploading into the buses different information, TV news or commercial spots. The maximum time assigned for this up-loading operation is deduced directly from the time necessary to fill a vehicle with fuel-oil (Figure 1.32).

Figure 1.32. Uploading of information and commercial spots inside buses

1.8. Typical applications of the Impulse Radio UWB, UWB-FH and UWB-FM 1.8.1. Professional geo-localization The professional applications of geo-localization mainly concern, in terms of market potential, firemen teams and to a lesser extent groups of policemen or infantrymen. These applications, largely focused on urban deployments, require radio transmissions with a sufficient link budget margin for indoor operation generally in harsh environments. The current regulation allows in the 3.2-4.8 GHz band, a mean E.I.R.P transmit power of –41.3 dBm/MHz, preventing link budget greater than 50 meters in free space conditions or limiting the transmissions inside a room.

Applications of UWB Systems

31

To partly solve this barrier, three elements can be considered, as individual or complementary actions: – The first solution consists of improving the receiver’s sensitivity by using modulations able to process the phase of the signal and by using coherent demodulators. Although significant, the gain obtained cannot considerably increase the range on its own. – The second solution is based on an increased RF power at the transmitter output, an increase which can be limited to certain types of deployment such as emergency operations which are concerned with the life of citizens and for which the mercantile aspect is secondary. It is in this spirit that action is undertaken by THALES at the ETSI (European Telecommunications Standards Institute) and CEPT to obtain a 20 dB increase in transmitted power in the 3.2-4.8 GHz band only for temporary emergency operations (Localization Applications for Emergency Services). – The third solution can be based on sophisticated antenna systems at the points of deployment which allow it. Thus, in an anchor-based configuration at the edge of the buildings, it is possible to consider modules for which the volume and the weight is higher than the portable equipment for individuals, but for which it is possible to implement intelligent UWB antennas like SUSA (specialized UWB smart antennas). Obviously, due to compatibility with the current regulations (expressed as radiated power), the antenna gain is used only in reception, which obliges us to modify the localization algorithms. With this type of antenna, we can hope for an improvement of about 15 dBi in the link budget margin. 1.8.2. Geolocalization for private individuals One of the headlight applications of UWB corresponds to the localization of individuals inside large commercial structures in order to be able to locate them and guide them towards particular items such as stores, automatic teller machines, payment terminals, etc. The technique generally used is based on an infrastructure with APs charged to perform the triangulation of the nodes. This system suffers from two main drawbacks: the first is the need to cable all the buildings concerned to deploy the infrastructure; and the second, which is even more problematic, is the weak range related to low authorized RF power within the framework of these generic UWB applications.

32

Ultra Wide Band Antennas

1.9. Impact on the antennas In conclusion to this chapter relating to the various UWB techniques and their potential applications, we can carry out the following report. In the lower band, one of the main requirements is the availability of effective and miniaturized antennas for individuals. This constitutes most of the demand. Nevertheless, for the applications for which anchors are tolerable, and in order to compensate for the low authorized transmit power, intelligent FESA™ or SUSA are strongly recommended. In the case of an exploitation in 2D, the gain of the elevation beams can be important and supplemented by a gain in azimuth. In 3D, the gain is more moderate because the elevation coverage to be supported is larger and the corresponding gain is weaker. For these antennas, the number of beams is inevitably not very large because the coverage in azimuth can be limited to a half-plane. On the other hand, the bandwidth required for UWB systems obliges us to consider delay lines and not phase-converters. One of the future studies for these applications is the integration of the commutation matrices of the various delays to apply beam forming matrices. For millimeter wave lengths (helping the wavelength), the antenna sizes with the same gain are reduced allowing us to consider FESA™ structures with a great number of beams while remaining within a reasonable volume.

Chapter 2

Radiation Characteristics of Antennas

2.1. Introduction Antenna designers are frequently confronted with two types of approach. The first where we describe their characteristics with the Maxwell’s equations and the assistance of operators like Hertz’s potential [STR 61], led to long calculations and was often not very instructive. In the second, a hand-waving approach using some formulas is more intuitive but often limited to well-known examples that cannot be easily generalized [LIZ 04]. In this chapter, we focus on the unified and rigorous approach of antenna design which was proposed by Per-Simon Kildal [KIL 99]. We will endeavor to reveal the physical phenomena characterizing the behavior of the antennas thanks to a compact formulation. The objective is to define all of the radiation characteristics which thereafter will be used and developed within the framework of ultra wide band (UWB) applications. As we will see in the following, a UWB antenna can be analyzed in the frequency and time domains and the discussion thread of this work is to present and connect these two domains. For an exhaustive and more academic presentation of the radiation characteristics of antennas, the authors propose to consult the following reference [BAL 05] which presents a conventional and well established formulation of the antenna theory. After defining the usual characteristics of the antennas, we will show an example of conventional characterization of a UWB antenna. This last part will give the reader the necessary criteria that the designer must check to validate a study. The antenna presented [LEP 08] will then be used in the following chapters on specific features in the time domain. Chapter written by Xavier BEGAUD.

Ultra W ide Band Antennas Edited by Xavier Begaud © 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

34

Ultrra Wide Band Antennas A

Radioo wave trannsmission in the environm ment and more m preciselyy in the propagattion channel led to the developmentt of multiplee uses and services: broadcassting, televisioon, radar, teleecommunicatiions, radio-naavigation, etc. In all of these applications, thee antenna inddicates this essential compoonent that trannsmits or receives electromagneetic waves. Thhe word anten nna is of Latiin origin and was used to indicaate the yard of the rigginng of Roman ships1. The rigging r of a schooner presented in Figure 2.1 2 reminds us a little off the geometrry of the logg-periodic antennass of which we will speak aggain a little later in this bookk.

Figure 2.1. Kruzhenstern K (R Russian schooner) photographhed at the time of the international maritime festival f Brest 088

Bothh visible and discrete d antennnas are increaasingly presennt in our enviironment. They maake it possiblle to establishh communication between,, at least, twoo devices 1. Encycllopédie Universsalis, Jean-Charrles Bolomey, 2008. 2

Radiation Characteristics of Antennas

35

while removing the connecting cables. These wireless applications provide users with flexibility, mobility and low installation costs. Antenna applications are thus increasingly numerous: − for terrestrial telecommunication systems: direct broadcasting, radio link, mobile phone and base station, WLAN (wireless local area network), radiofrequency identification (RFID), navigation, etc; − for satellite telecommunication systems: connection between ground stations for long distances, satellite TV, terrestrial mobile terminals/mobile phones, etc; − for civilian and military radars: weather radar, aviation radar, search and surveillance radar, synthetic aperture radar (SAR) (pollution, traffic), research on the ionosphere, etc; − in radiometry: radio astronomy, meteorology and monitoring of the Earth’s resources, etc The notations that will be used throughout this book are defined in the following. The International System of Units (SI) defines the units of measurement in m (meters), of time in s (seconds), of voltage in V (Volts) and of current in A (Amps). The sizes noted in bold face are vectors and when they are in small letters or surmounted by a hat, they are a unit vector. The units of the electric field E and the density of magnetic current equivalent M will thus be V/m, and for the magnetic field H and the density of electric current J, A/m. For the impedances, we will use Ohms and for the admittances Siemens. The unit of power will be Watts and finally, the partial waves used for calculation of S parameters will be in square root of Watt. In many cases, we will use terms without dimensions like the reflection and transmission coefficients. These coefficients will often be given in dB: A / A ref

dB

= 20 log A / A ref

where A is amplitude (voltage, current, intensity of the electric field, etc) and Aref the reference value. P / Pref dB = 10 log P / Pref

where P is a power or power density and Pref the reference power or power density.

36

Ultra Wide Band Antennas

For these two expressions, the result is identical because:

P / Pref = A / A ref ²

We often find a letter associated with the result in dB: dBi, dBm, dBW, etc. This letter simply indicates the selected reference. Here the reference is isotropic, a power of 1 mW or 1W. To qualify an antenna, we will calculate its efficiency. We will give the efficiency in dB because it is thus easier to obtain the “gain” brought by the antenna [KIL 99]. Relative efficiency

1

0.99

0.95

0.9

0.8

0.64

0.5

Efficiency in dB

0

–0.04

–0.22

–0.5

–1

–2

–3

Table 2.1. Relative efficiencies and corresponding values in dB

It is also useful to point out the following correspondences. Amplitude ratio

1

0.32

0.1

0.032

0.01

Power ratio

1

0.1

0.01

0.001

0.0001

Ratio in dB

0

–10

–20

–30

–40

Table 2.2. Amplitude and power ratio and corresponding values in dB

To define the radiation characteristics of an antenna, we will try to answer two simple questions in the next sections. 2.1.1. What is an antenna and how can we define it? An antenna is a device that ensures the coupling between a high frequency generator of current, voltage and an electromagnetic radiation. This coupling is reciprocal, the antenna creates a radiation when power is injected (transmit antenna); and a contrario to collect energy, when it is plunged into an electromagnetic field (reception antenna). We also speak about an aerial, a radiating element, a transducer between free space and a guided structure or about a “space filter”, etc.

Radiation Characteristics of Antennas

37

This transduction between the guided structure where energy is controlled and free space where energy is radiated, is an important characteristic of the antennas which will be developed for the UWB applications. The concept of wave impedance characterizes the medium inertia which presents the propagation of a wave. The free space, frequently compared to the vacuum for the study of the radiation of the antennas has a wave impedance equal to η0 = 120π Ω. The antenna can thus be represented like a impedance transformer ensuring the transition from a wave being propagated in a guided structure − generally of characteristic impedance equal to 50 Ohms − towards free space. The concept of impedance matching is a fundamental characteristic that we will reconsider. 2.1.2. Where does antenna radiation come from? Within the framework of a conventional description of antenna electromagnetic field, we show that it is the accelerating charges which produce the electromagnetic field. They are thus the time variations of the sources, here of the current distributions which produce the radiation. Conversely, an antenna plunged into an electromagnetic field is likely to give rise to current and charge distributions. 2.2. How can we characterize an antenna? The objective of this section is to define and describe in a general way the various definitions usually used to characterize the antennas. All of these are functions of the electromagnetic fields or the currents, which depend on the position and the frequency when they are observed in the frequency domain. The fields are written:

E( x , y , z , f ) and H( x , y , z , f )

[2.1]

In the majority of studies, E and H are complex time-harmonic vector fields and a term in e jωt is then present in all the expressions: E( x , y , z ) e

j ωt

and H( x , y , z ) e

jωt

[2.2]

However, our objective here is to present the radiation of antennas fed with harmonic signals but also by impulses of various forms.

38

Ultra Wide Band Antennas

The electric and magnetic fields are thus complex vector fields which depend on the position and frequency without another distinction.

Important remark – In the following, we will present a certain number of results obtained with time-harmonic dependences which we will specify each time. These results will be presented because it is possible to define well-known radiation characteristics or to illustrate the physical behavior of the antennas. In the same manner, the instantaneous power (or Poynting vector) crossing a closed surface S with a normal n is written: P (t ) = ∫∫ S

(E × H). n dS

[2.3]

When the fields are periodic (ω = 2π /T), it is possible to calculate the power averaged over a period T:

Pav (t ) =

1T 1 P(t )dt = ∫∫ Re {E × H*}. N dS ∫ T 2 0 S

[2.4]

1 Re {E × H*} is the time average power density vector Wav. Term 1/2 2 comes from the time average of the product of the time-harmonic dependences of E and H (in cos). If we consider effective amplitudes as root-mean-square (RMS) for the fields E and H, this factor disappears.

where

2.2.1. Plane wave and polarization Far from the sources, far from the antenna, the radiation field can be locally observed as the propagating plane wave. The incident field on a reception antenna can thus be considered as a plane wave. Let us consider a plane wave propagating in positive z-direction, an EM field in free space is thus written:

E = Et e − jkz = [Ex x+ Ey y] e − jkz

[2.5]

The wavenumber k= 2π/λ where λ is the wavelength. In the same way λ = c/f, f being the frequency and c the celerity (2.9979 x 108 m/s).

Radiation Characteristics of Antennas

39

With time-harmonic dependences, the power density vector becomes: Wav (x,y,z) =

2⎤ 2 2 1 1 ⎡ 1 Et z = Re {E × H*} = ⎢ E x + E y ⎥z 2ηo 2ηo ⎢⎣ 2 ⎥⎦

[2.6]

which means that there is propagation of power in the positive z-direction. η o is the wave impedance in free space ( η o =120 π). We have written the transverse electric field to the direction of propagation, with two components Ex and Ey and implicitly defined two polarizations.

According to the application, it is thus necessary to favor an orientation of the electric (or magnetic) field; and therefore we are led to define in a more general way the field components radiated by an antenna:

E = ⎛⎜ E co + E xp ⎞⎟ e − jkz xp ⎝ co ⎠

[2.7]

We call the desired field component the copolar component, and the undesirable field component the cross component. The unit vectors “co” and “xp” must satisfy the following: − the main component of the electric field is then:

E co = Et . co *

[2.8]

− the cross component of the electric field is then:

E xp = Et . xp *

[2.9]

When the orientation of the copolar vector does not change during the propagation, we speak about linear polarization. In many applications we speak about horizontal or vertical polarization. We define horizontal polarization as linear and parallel to the ground. Vertical polarization being, of course, perpendicular to horizontal polarization. In space telecommunications applications in particular, a circular polarization is used. This undefined polarization in the time domain will not be discussed in this book. If this state of polarization is the desired polarization then πˆ = Co.

40

Ultra Wide Band Antennas

2.3. Radiation fields and radiation power 2.3.1. Radiation fields

For the remainder of this book, we will only be interested in the far field. When the observation distance is higher than 2D²/λ, the electromagnetic field is locally considered to have the structure of a plane wave and it decreases like 1/r. D is the greatest dimension of the antenna and λ is the wavelength. This distance is called the Fraunhofer distance (condition or criterion). In most studies, we examine the electric or magnetic field in their complete form (without time-harmonic dependences, i.e. e jωt). The expressions then depend on the distance R and the direction of propagation. In the far-field region, the variables can be separate: 1 r

E( r , f ) = e − jkr Ψ( rˆ , f )

[2.10]

The far-field function thus defined Ψ (θ,ϕ) depends only on the angular variations of the electric field. The distance between the source and the observation point disappears. This means that there is no longer any need to calculate the term in 1/r (divergence coefficient) or the term e-jkr (factor of phase). This simplifies the calculations and eliminates the development of the terms that complicate the formulation. For reasons which will be detailed in Chapter 3, the electric field can also be written in the following form where A is called the vector amplitude of the field. 1 r

E( r , f ) = e − jkr

η0 4π

A ( rˆ , f

)

[2.11]

The polarization of the far-field function, is naturally defined after being projected on the copolar and cross unit vectors:

Ψ Co (θ,ϕ) = Ψ (θ,ϕ) . co*

[2.12]

Ψ xp (θ,ϕ) = Ψ (θ,ϕ) . xp*

[2.13]

The vectors co, xp and rˆ (unit vectors) respect the right-hand rule.

Radiation Characteristics of Antennas

41

2.3.2. Radiation power

The antenna transmits an electromagnetic field and it is important to know the radiated power. A direct way (conventional) of evaluating the radiated power consists of calculating the flow of the power density through a sphere surrounding the antenna:

Prad = ∫ ∫ PΩ ( rˆ )dΩ

[2.14]

where dΩ is a solid angle element and PΩ is the power density per solid angle element. Take note that this total radiated power Prad does not depend on the observation distance. With time-harmonic dependences, the calculation of the magnetic field can result simply from the calculation of the electric field using Maxwell’s equations: H (r, θ, ϕ) =

1

η0

rˆ × E (r, θ, ϕ)

[2.15]

The power density vector then becomes: p=

1 1 1 2 Re {E × H*} = rˆ = Ψ(θ , ϕ ) rˆ 2 η0 r ² 2

[2.16]

The power density per solid angle unit (or radiation intensity) then becomes: pΩ = r ²( p. rˆ ) = −

1 2η0

Ψ(θ ,ϕ )

2

[2.17]

This last expression, only valid with time-harmonic dependences, shows that the spatial distribution of the power is directly proportional to the square of the magnitude of the far-field function. 2.3.3. The radiation pattern, the phase center

The radiation pattern is a graphical representation of the power or the field radiated by the antenna. We define the copolar radiation pattern like the graphical representation of |Ψ Co (θ,ϕ) | and the phase copolar pattern like the graphical representation of Φ Co (θ,ϕ) with Ψ Co (θ,ϕ) = |Ψ Co (θ,ϕ) | e j Φ Co (θ,ϕ) co.

42

Ultra Wide Band Antennas

The cross component radiation pattern is the graphical representation of |Ψ xp (θ,ϕ) |. In most applications, we define a direction which will be used as a reference for the radiation pattern. The diagram is then presented in dB:

(Ψco (θ ,ϕ ))dB = 20 log ΨΨco ((θ0 ,,ϕϕ )) co

Ψ

[2.18]

(θ , ϕ )

⎛⎜ Ψ (θ ,ϕ ) ⎞⎟ = 20 log xp Ψ ( 0 ,ϕ ) ⎝ xp ⎠ dB co

The simultaneous plot of the copolar and cross components makes it possible to check the polarization purity of an antenna. In many applications, the level of the cross component must be 20 dB lower than the copolar component in order to be negligible. Many types of radiation pattern exist. The simplest is the first that we defined, which only represents the angular dependence of radiation from the antenna. According to the accessible information, this graph can be enhanced by the values of directivity or the gain if the antenna is completely known. We will define these two concepts in the next sections. We will see in the following chapters that the radiation pattern concept is also very useful in the temporal field to quantify and qualify the way in which the antenna radiates according to time. The two previous expressions represent the angular dependence of electric or magnetic field magnitude and are sufficient for a good number of applications. However, for the systems using parabolas or lenses, the phase information of the electromagnetic field is important. When we are sufficiently far from a transmit antenna, a sufficiently small reception antenna is illuminated by a wave front whose amplitude can be easily described as uniform. If the transmit antenna is a point source, the phase variation between all the points of the wave front arriving on the reception antenna is weak. Therefore, now the transmit antenna is no longer specific, the various points of the wave front do not have the same phase and can be out of phase. We define the position of the phase

Radiation Characteristics of Antennas

43

center of an antenna as the “origin point” of the electromagnetic radiation which would produce, on a portion of sphere, a wave front with a quasi-constant phase.

2.3.4. Directive gain, directivity The plot of the radiation pattern in the previous section highlighted the need to compare the radiation of the antenna with a reference (equation [2.17]). If we want to compare the antennas, it is also necessary to define a reference antenna. The reference antenna which was universally selected is the isotropic antenna. It is, in fact, not a very interesting antenna because there is no application which requires a electromagnetic radiation in all directions (4 π steradians). For an isotropic antenna:

Ψ co (θ , ϕ) = Ψ iso = constant The power density radiated by the isotropic radiator is: Piso = Prad /(4π r²)

[2.19]

This isotropic antenna is defined by a zero level of cross polarization. Without further precision, the directive gain might be calculated using the following equation: D (θ , ϕ) = pΩ / (Prad/(4 π)) = 4 π pΩ / Prad

[2.20]

With time-harmonic dependences and according to equations [2.14] and [2.17] we can write that the power radiated by the isotropic antenna is: Piso = Ψ iso

²

1

²

2π

∫∫ sin θ dθ dϕ = 2η Ψ iso 4π = η Ψ iso 0 0 4π

²

[2.21]

Directive gain is expressed in dBi (decibel relative to an isotropic radiator), we can also separate it into two terms:

⎛ Ψ (θ ,ϕ ) ² ⎞ ⎛ 2π Ψ (θ , ϕ ) ² ⎞ ⎟ ⎜ ⎟ co co 10 = log ⎟ ⎜ ⎟ η0 Prad ⎜ ⎜ ⎟ Ψiso ² ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ 2π Ψ (θ ,ϕ ) ² ⎞ ⎛ Ψ (θ , ϕ ) ² ⎞ ⎟ ⎜ ⎟ ⎜ xp xp ⎛⎜ D (θ ,ϕ ) ⎞⎟ ⎟ ⎟ = 10 log⎜ ⎜ 10 log = ⎝ xp ⎠ dBi η0 Prad ⎟⎟ ⎜⎜ ⎜⎜ Ψ iso ² ⎟⎟ ⎠ ⎝ ⎠ ⎝

(Dco (θ ,ϕ ))dBi = 10 log⎜⎜

[2.22]

44

Ultra Wide Band Antennas

Let us now assume that there is no cross component, the radiation power is thus written:

Prad =

1 2η0

[

]

∫∫ Ψco (θ ,ϕ ) ² sin θ dθ dϕ

4π

[2.23]

and with equation [2.22], we obtain:

Prad =

[

]

1 ∫∫ Dco (θ ,ϕ ). Ψiso ² sin θ dθ dϕ 2η0 4π

Prad = Prad

1 4π

[

]

∫∫ Dco (θ ,ϕ ) sin θ dθ dϕ

4π

[2.24]

[2.25]

which leads to:

[

]

4π = ∫∫ Dco (θ , ϕ ) sin θ dθ dϕ 4π

[2.26]

In addition: 4π =

∫∫ sin θ dθ dϕ .

4π

We can claim that the directive gain cannot be higher than 1 in all directions! A high directive gain in a given direction is obtained to the detriment of the other directions. The proverb, “let well enough alone” is good advice to follow when optimizing an antenna. The directive gain value in the direction of the maximum radiation of the copolar component is called directivity: ⎛ 2π Ψ ² ⎞ o ⎟ ⎟ P ⎜ 0 rad ⎟ ⎝ ⎠

(Do )dBi = 10 log⎜⎜ η

[2.27]

Radiation Characteristics of Antennas

45

It is necessary to be careful when speaking about directivity, directive gain or gain because the names are very similar but the characteristics are different. We can also plot the E or H-plane radiation pattern (for the antennas with two planes of symmetry). E-plane is the plane which contains the electric field, H-plane is orthogonal to E-plane. Let us assume for example that the polarization of the antenna is oriented in the y-direction: − then, co = y; − the E-plane copolar radiation pattern is: |Ψco (θ , 90°)|; − the H-plane copolar radiation pattern is: |Ψco (θ , 0°)|. Figure 2.2 gives an example of a radiation pattern. From this graph we can extract the following information: − the antenna directivity D0; − half-power-beamwidth (HPBW). This is the angle (in a given plane) inside which is contained more than half the radiated power; − the presence or absence of side lobes and back lobe. The higher the levels of these lobes are, the lower the directivity is.

Dco (θ,ϕ=constant) (dB) D max (dB)

D max – 3dB

-π

0 HPBW Figure 2.2. Example of a radiation pattern

π

θ

46

Ultra Wide Band Antennas

2.3.5. Radiation impedance and radiation resistance

We said previously (section 2.1.1) that the antenna could be considered as a impedance transformer between a guided structure and free space. But before this transformation, it is necessary to match the antenna to the generator or the device to which it is connected. Let us consider a generator connected to a transmission line which is itself connected to the input of the antenna (Figure 2.3).

Generator

Transmission line

Za

Antenna

Zc Zg

Γ

Zc

Figure 2.3. Matching the antenna to the generator

The transmission line has a characteristic impedance equal to Zc the antenna has an input impedance equal to Za and the generator has an output impedance equal to Zg. The matching of the antenna is evaluated by calculating or measuring the reflection coefficient at the antenna input; by definition this coefficient is:

Γ = (Za- Zc)/(Za+ Zc)

[2.28]

In many applications when the magnitude of the reflection coefficient is lower than 0.1, we consider that the antenna is well matched. The power coming from the transmission line is then transmitted into the antenna. In antenna design, we seek to minimize the stationary waves. We choose: Zg=Zc, i.e. Rg=Zc and Xg=0 and Za=Zc, i.e. Ra=Zc and Xa=0 and in most of the cases Zg=50 Ohms.

Radiation Characteristics of Antennas

47

The complex conjugate matching used in electronics, for example, is not used here because it is too difficult to implement on a frequency bandwith. The antenna input impedance is complex: Za= Ra+jXa. The real part Ra is the sum of two contributions, the radiation resistance and loss resistance. We also use the voltage standing wave ratio (VSWR) to represent the matching of an antenna to an access which has a characteristic impedance Zc. By definition this ratio is equal to: VSWR = (1 + |Γ| )/(1 - |Γ| )

[2.29]

When this ratio is close to 1, the input impedance of the antenna is close to Zc. If on the contrary, the VSWR is larger, even much larger than 1, the antenna is unmatching. 2.4. Gain, efficiency and effective aperture 2.4.1. Gain and efficiency

We previously defined the antenna as a device allowing us to transform guided energy into radiated energy (section 2.2.1). This function is quantified using the gain of the antenna. We defined directivity (section 2.4.2) as the relationship between the radiation intensity in the direction of the maximum and the total radiated power. Gain (or power gain or absolute gain) G has the same definition, but it is simply necessary to replace the total radiated power by the total power delivered to antenna port. We can thus express it according to directivity: G = erad D0

[2.30a]

where erad is the radiation efficiency. The radiation efficiency is the relationship between the radiation power and the power delivered to the antenna. We can separate it into two terms: epol and eohmic:

− epol is the polarization efficiency due to the crossed polarization in the axis. When there is no cross polarization epol = 1; − eohmic is the efficiency due to the Ohmic losses in the conducting and dielectric

parts.

48

Ultra Wide Band Antennas

The power budget at the transmitter is the following (see Figure 2.4), with Pg the power given by the generator (transmission line is matched) and Pin the power injected into the antenna. Prad is the radiation power.

Pg

Generator

Pin = (1 - |Γ|² ) Pg

Transmission line

Antenna

Prad = erad Pin

|Γ|² Pg Figure 2.4. Power budget at the transmitter

The matching losses between the generator and the antenna can also be modeled using an efficiency, er = 1-|Γ |² is the transmission efficiency due to the reflection at the input of the antenna (section 2.5). We can thus define the realized gain Gr. which takes into account all the losses by: Gr = er G = er erad D0

[2.30b]

The power losses are equal to (1 - erad) Pin. 2.4.2. Receive antenna, effective aperture

Receive antennas are often characterized by their effective aperture rather than by their gain. The effective aperture Ae is defined by the relationship between the total power Pr received by the antenna and the power density Wt illuminating the antenna: Ae= Pr Wt

[2.31]

In addition, it can be demonstrated that the maximum directivity of such an aperture is expressed as [ELL 81]: Dmax= (4π /λ ² ) A

[2.32]

Radiation Characteristics of Antennas

49

A is the real surface (aperture of a horn or a parabola for example), which we can also express using: Ae= (λ ²/4π) Gr. with Ae= eap A. These definitions of effective apertures are also valid for small antennas and linear antennas. The aperture efficiency is then given by: eap = D 0 / Dmax

[2.33]

The total gain or realized gain is then: Gr = eant Dmax with eant = er eohmic epol eap with eant the total efficiency of the antenna. In fact, efficiency demonstrates the ability of the antenna to radiate. This is a very important characteristic which requires more attention and a validation by measurement (see Chapter 4). 2.5. Budget link, transfer function

The objective of this section is to calculate the transmission losses between two antennas in a telecommunications system. Let us consider a transmitting antenna with a gain equal to Grt, and a receiving antenna with an effective aperture equal to Aer. The two antennas point towards each other (Figure 2.5).

r

Generator

Grt Pt

Receiver

Aer Pr

Figure 2.5. Budget link

50

Ultra Wide Band Antennas

It is assumed that the generator delivers a power Pt to the antenna (without any reflection). Then, if the antenna is isotropic, copolar and lossless, the radiated power density in front of the receiver is: Wiso= Pt /(4π R ²)

[2.34]

r is the distance between the two antennas. The receiving antenna has a gain relative to the isotrope Grt (it was assumed that there was no loss): Wo= (Pt/(4π R ²)) Grt

[2.35]

The power available at the port of receiving antenna is then easily calculable by using the definition of the effective aperture (equation [2.32]): Pr = Wo Aer = Wo (λ ²/4π) Grr = Pt (λ ²/(4π R ²)) Grt Grr

[2.36]

where Grr is the total gain of the receiving antenna. In fact we have used the principle of reciprocity. This principle states that a passive antenna composed of homogeneous and isotropic materials has the same gain at transmission as at reception [ELL 81]. We will reconsider this principle in the following chapter and Appendix 1. Consequently, we obtain the following result called the Friis equation (budget link or telecommunications equation) [FRI 46]: (Pr/Pt) = Grt Grr (λ/(4π R))²

[2.37]

The term (λ/4π R)² is called free space attenuation. The gains of the antennas at transmission and reception take into account the possible losses of matching, polarization mismatching and losses of the antennas. This information is contained in the term efficiency. It is easy to understand that relation [2.37] is true for two directive antennas pointed one towards the other. If the two antennas are now unspecified and laid out in a random way, it is easy to understand that the expressions of the gains that we used above are inaccurate. Thus it is necessary to use the gains relative to particular directions at both the transmission and the reception. We will see hereafter, that this is necessary to define multidimensional transfer functions for the antennas at transmission and reception (section 3.1). These antenna

Radiation Characteristics of Antennas

51

transfer functions will take into account the direction, polarization, efficiency and thus the gain of the antenna. The PtGrt product is called the equivalent isotropic radiated power (E.I.R.P.). E.I.R.P. is defined in the direction of the antenna where the transmit power is maximum: this is the power which it would be necessary to apply to an isotropic antenna to obtain the same field in this direction. Within the framework of the UWB regulation (see Chapter 1), this E.I.R.P. is a very important element because it is limited and dependent on the regions. The deployment of UWB solutions can be done only by respecting the rules given by the regulations agencies (Figures 1.1, 1.2, 1.3). 2.6. Equivalent circuits of the antennas

In a system, the antennas are connected to a transmitter or a receiver, using a transmission line (coaxial cable, etc.) or of guide. To analyze the system, it is necessary to define the equivalent circuit at transmission and reception.

Generator

Transmission line

Za

Antenna

Zc Zg Vg

Zg Zc

Zc

Za

Figure 2.6a. Equivalent circuit at transmission

In Figure 2.6a, Vg is the voltage and Zg is the internal impedance of the generator. The equivalent circuit is a Thevenin circuit model. Zc is the characteristic impedance of the transmission line. This circuit can also be represented by a Norton circuit model.

52

Ultra Wide Band Antennas

Antenna

Transmission line

Za

Zc

Receiver

Zr Zc

Za Zc

Var

Zc

Zr

Figure 2.6b. Equivalent circuit at reception

In Figure 2.6b, Zr is the input impedance of the receiver. Var is the voltage induced at the port of the reception antenna. 2.7. Bandwidth

The bandwidth of an antenna is the range of frequencies over which it is usually well matched. But it can also be specified using the other radiation characteristics (gain, lobes secondary, etc.). Without any precision, the bandwidth is defined for a magnitude of the reflection coefficient (or return loss) lower than –10 dB. Relative bandwidth and absolute bandwidth are the following: f0 - Δ F /2 < F < f0 + Δ F /2 ΔF is the absolute bandwidth and f0 is the frequency center. The relative bandwidth is defined by fmax/fmin. 2.8. Example of characterization: the triangular probe antenna in F

The object of this last section is to present some conventional results that it is necessary to have when one wishes to completely characterize an antenna. This conventional characterization makes it possible for any user to appreciate the performances of an antenna. The following chapters will illustrate the other characteristics of this antenna in the time domain.

Radiation Characteristics of Antennas

53

2.8.1. Description of the structure

As we will see in the following, much work has been carried out on the design of omnidirectional antennas and little has been done on directive antennas. This is due to the mobile applications and the regulatory context which limit the E.I.R.P. (see Chapter 1). During reception, for the access point, in particular, or for radar applications, a directive antenna is very interesting because it makes it possible to improve the budget link or the range of the system. The antenna presented below is complex and we will only present some of the radiation characteristics. You will find more details in the references [LEP 05] and [LEP 08]. The description of the geometry is given in Figure 2.7. The parameters which define the antenna are: LP = 36 mm, D = 0.6mm, H3 = 0.4mm, α = 84°, H = 14.9 mm, H1 = 10.05 mm, H2 = 5.6 mm, L1 = 9.9 mm, L2 = 10.8 mm, W = 3.6 mm, width of the horizontal strips = 1.2 mm, size of the ground plane = 67 x 67 mm, thickness of the probe = 0.4 mm.

Figure 2.7. Antenna description

2.8.2. Impedance matching

The first step which conditions the continuation of a study is to verify that the antenna is well matched (usually to 50 Ohms). The simulation result presented below is obtained using the Microwave Studio™ software from CST. In Figure 2.8, we plot the magnitude of the reflection coefficient. It is necessary to calculate the input impedance in the conditions of the antenna’s use (when that is possible) and in

54

Ultra Wide Band Antennas

most of the cases under the measurement conditions. The results presented below take into account the connector presented in Figure 2.7.

Figure 2.8. Antenna matching

When we note a good agreement between simulation and measurement, we conclude that the antenna matching is validated by measurement. All of the measurement techniques will be presented in Chapter 4. However, this is only a first step which ensures that the power given by the generator will be well injected into the antenna. That does not absolutely prejudge that we have the radiation of the antenna. The result presented in Figure 2.8 is partial because we only present the magnitude of the reflection coefficient. To know the behavior of the antenna it is necessary to plot this coefficient on Smith’s chart, which makes it possible to simply connect this complex coefficient to the antenna input impedance. The use of Smith’s chart is a good method to detect resonances, the inductive or capacitive behaviors of the antenna input impedance. 2.8.3. Radiation patterns

When the radiation pattern is plotted, it is necessary to specify the representation plans and to define precisely what is plotted: main component (copolar), cross component, directivity, gain, etc. For this antenna, we define the E-plane as plane XZ and the H-plane as plane YZ (see Figure 2.7).

Radiation Characteristics of Antennas

Copolar component

Crosspolar component 3.0 GHz

4.2 GHz

5.4 GHz

6.15 GHz

Figure 2.9. Radiation pattern measurements in E-plane

55

56

Ultra Wide Band Antennas

The radiation patterns in Figures 2.9 and 2.10 are normalized. They represent the ability of the antenna to focus (or not) in one direction rather than another.

Main component

Cross component 3.0 GHz

4.2 GHz

5.4 GHz

6.15 GHz

Figure 2.10. Radiation pattern measurements in the H-plane

Radiation Characteristics of Antennas

57

What can we deduce from these measurements? − In antenna bandwidth, i.e. between 3 and 6 GHz, the antenna is directive. In the E-plane, the direction of the main lobe is stable: maximum misalignment is about 15o. The cross component level is low: lower than −20 dB. Thus the antenna has a linear polarization. This low level is also due to the geometry of the antenna which is symmetrical around the OX axis. − In the H-plane, the main direction of the lobe is stable and due to the symmetry of the antenna around the OX axis. The cross component is weak in the direction of the lobe; its maximum reaches −7 dB for θ = ± 60°. Figure 2.11 presents the evolution of the gain in the direction θ = 0°, φ = 0° in the bandwidth. It plots the total gain, or realized gain (section 2.4.2.), which takes into account all antenna losses.

Figure 2.11. Measured and simulated realized gain of a F-probe triangular antenna

The maximum gain is from 7.6 dB to 4.5 GHz, which is high for such a compact antenna (λ0 X λ0 x 0.2 λ0, ground plane included, λ0 is the wavelength at the center frequency of the bandwidth). This is relatively stable up to 5.5 GHz, then it decreases. The average gain is 6.1 dB. The agreement between simulation and measurement is good, except at the high end of the bandwidth.

58

Ultra Wide Band Antennas

2.8.4. Optimization of the antenna

The prototype of the F-probe triangular antenna presented in Figure 2.12 remains delicate and difficult to realize.

Figure 2.12. F-probe triangular antenna

This antenna has been modified using a new technology that simplifies its realization and also minimizes its cost. We have used a particularly innovative solution developed at IETR (Rennes, France), which metallizes low permittivity foams with an arbitrary geometry [CHA 02].

Figure 2.13. F-probe triangular antenna realized with metallized foam

Radiation Characteristics of Antennas

59

In the following chapters, we will refer to this antenna more simply, by calling it the FFPTP (Foam F-Probe Triangular Patch) antenna. Thanks to Anne Claire Lepage for her scientific contribution on the design of the FFPTP antenna, to all the colleagues from IETR for their help in the realization of the prototypes and last but not least to Per-Simon Kildal who brought me his unified approach to antennas.

Chapter 3

Representation, Characterization and Modeling of Ultra Wide Band Antennas

3.1. Introduction Ultra wide band (UWB) antennas have particular characteristics and constraints, in particular in impulse radio − which operates on signals with extremely broad instantaneous band − which do not exist or wave no object in the narrow band: their design is in general more delicate because it is subjected to more constraints and because their behavior is more difficult to understand and to qualify. Their experimental characterization is much more involved and appreciably more demanding. Finally, the additional dimension − time or frequency − confers onto the antenna a new functional role in the communication system. This is why we have adopted a functional approach to the UWB antenna, presented in section 3.2. The physics of the antenna − in particular its temporal behavior in impulse radio − must be correctly analyzed and interpreted using concepts, reference objects and suitable indicators: sections 3.3 and 3.4 are thus devoted to the signal distortion phenomenon and the concept of an ideal antenna. Because of the additional dimension, the significant amount of data (experimental or simulated) to be handled and analyzed very naturally results in the definition of various performance indicators synthesizing information, allowing us to better understand the behaviors, to qualify the imperfections quantitatively (but simply) and to more easily compare the antennas to one another: section 3.5 is devoted to this. Lastly, faithful to this step of information synthesis, an approach of parametric modeling with high order

Chapter written by Christophe ROBLIN.

Ultra W ide Band Antennas Edited by Xavier Begaud © 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

62

Ultra Wide Band Antennas

reduction − based on the singularity expansion method (SEM) and the spherical mode expansion method (SMEM) − is presented in section 3.6. 3.2. Specificities of UWB antennas: stakes and representation In the radio communication chain, the antenna is of course an essential functional element. In the narrow or moderate band, once specified according to system (and technological) constraints, the antenna is usually designed more or less independent of the other elements of the chain (RF front-end, propagation channel, physical layer), because the interactions with the latter can be evaluated by rather simple criteria: impedance matching, gain, etc. As we will see, it is more intricate in UWB because the interactions cannot be uncoupled as easily, in particular in impulse radio. The temporal behavior of the antenna is indeed more complex; it is consequently more difficult to understand and qualify, and therefore to model and design − incidentally it is more difficult to measure and characterize (additional dimension, frequency or time, “absolute” measurements required, etc). The design of the UWB antennas is thus not only subjected to the “ordinary” constraints of antenna design (cost, size, integration, efficiency, etc.), but is subjected to “particular” constraints: a matching bandwidth typically greater than an octave, a control of the dispersion (the behavior of the insertion phase) and more generally of the distortion (linear1) − that is to say, of the deformation of the waveform – of the total efficiency, etc. All of these constraints will be tackled in an optimization step under a trade-off in particular, for reasons related to the foundations of electromagnetism, between the band width, the size and the total efficiency [MCL 96]. All these limitations can be the source of degradations of the radio system performance, which it is desirable to evaluate a priori, i.e. from the design stage. This is why it seems necessary to define a certain number of specific concepts and theoretical analysis tools, for the concept of an “ideal” antenna − providing references allowing comparisons by relative assessment − at the development of various quantitative indicators of performance, in particular making it possible to describe and then to compare the antennas with one another. A specificity related to the distortion must be mentioned now: the antenna effect on the waveform is different in transmission and reception; one will thus be brought to define different transfer functions in transmission and reception. That of course does not contradict, in any way, the reciprocity principle, which is precisely the very means to establish a relationship between these two transfer functions. We will see in particular that it is theoretically impossible to conceive an antenna without distortion both in transmission and reception.

1. Which is the frequency effect divided into amplitude distortion (gain variation) and phase distortion or dispersion (phase non-linearity).

Representation and Modeling of UWB Antennas

63

3.2.1. Context and requirements of an effective and complete representation The “ordinary” and “particular” constraints on the design of UWB antennas, discussed previously, is leading first to the development of a suitable formalism, making it possible to qualify all their radiation performances, which are specific as well as conventional. Here we will restrict the analysis to antennas used in the far field zone − i.e. essentially dedicated to communications, radars, measurement, localization or possibly to broadcasting − excluding thus in particular field probes or ground penetrating radars (GPRs). For clarity, we will subsequently consider the context of the radio communications which are at the origin of the renewed interest in UWB, observed over these last few years. The suggested approach is easily applicable to the other applications. Moreover we will limit further analysis to passive2, therefore linear and reciprocal antennas. In this context, the UWB antenna appears as a multidimensional linear time invariant (LTI) system, including, in transmission, one (or several) input port and an infinity of output ports − corresponding to the measurement of the field over 4π steradians − and the reverse in reception. In practice, the radiated field is angularly sampled in elevation and azimuth, i.e. it is measured (or simulated) for only a finite set of observation directions . The “input-output” functional relation can thus be written by means of a multidimensional transfer function or transfer matrix.

Figure 3.1. Block diagram of an UWB communication chain. The unshaded blocks are linear

2. Excluding any non-reciprocal material.

64

Ultra Wide Band Antennas

Any antenna transfer function will have, moreover, to account for the vector nature of the field (polarization), the effective energy transfer from the source to free space, the frequency and time behavior of the antenna (waveform and distortion) − and thus of the phase of the field − and of course of the directional characteristics. We must thus be able to establish simple relationships between the transfer function on the one hand; and polarization, efficiencies, gains and directivity on the other hand. Because they cover the main part of the applications in communication (and/or localization) considered over recent years, we are mainly interested in the microwave range3, in which the antennas are generally fed by waveguides − transmission lines or hollow waveguides − whose more successful (and most elegant) processing is based on the wave formalism and the scattering matrix4. This is why the following theoretical presentation is founded on the theory of partial waves and not on the voltage/current formalism. Unlike the narrow band frequency domain − for which it can be shown that the transmission and reception gains are equal − the behavior of an antenna with respect to the waveform of a non-harmonic signal is different in transmission and reception. The situation cannot thus be perfectly symmetrical, except by resorting to certain purely formal artifices. It is then necessary to define two transfer functions, one in transmission and the other in reception, the reciprocity offering a way to establish a relationship between the two. We will eventually limit the presentation to one-port antennas, the generalization to multi-port antennas not raising any major difficulties. All these considerations result in defining two directional and complex vector transfer functions called antenna transfer functions (ATFs), in transmission and reception. 3.2.2. Transfer function in transmission Following the usual practice in control, filtering or signal theory, we make the choice here to define a dimensionless transfer function in the transmitting mode: it just requires a suitable normalization of the field. In addition we will see that, if this normalization is judiciously chosen, we can establish an extremely simple relationship between this transfer function and the realized gain, which is a more “conventional” quantity. The ATF is an amplitude 3. Typically from UHF to centimeter waves. 4. Or more simply the S matrix.

Representation and Modeling of UWB Antennas

65

ratio (voltage, current, partial wave or field), i.e. a first order quantity, whereas the realized gain, of the same type as the power gain, is a ratio of powers, i.e. a quadratic quantity. What simpler relationship than a squared modulus can we imagine between the two? This will be one of our objectives. The main question at this stage in the definition of the two ATF relates to the choice of the input and output signals. In UWB, the evaluation of the distortion and the real energy transfer rests as much on the coupling of the source with the antenna as on the way in which the antenna transforms the coupled signal. Thus, for certain antennas, it is the reflection coefficient which contributes mainly to the distortion (or to the energy transfer), whereas for others it is the radiation itself. The case of the elementary doublet (infinitesimal or Hertzian dipole) is thus exemplary: it is wellknown that its radiation pattern (transmission) or its effective length (reception) are independent of the frequency, whereas the transmitted signal (radiated field or received signal) undergoes a severe distortion; this is only due to the strong frequency dependence of the input impedance and thus of the reflection coefficient. We wish moreover to develop a formalism directly related to the experimental characterization in the frequency domain, mainly founded on the measurement of the scattering parameters (S matrix), i.e. of partial wave ratios. k ZC ZC

(a)

a1

π

P1

Pa = (1−|S11|2) P1

b1

S11

Prad = erad Pa

-k a2 = 0 (b)

ZC

ZC

ζ

P2 b2

Figure 3.2. Schematic of the antenna system: (a) in transmission; source model including a transmission line of characteristic impedance Zc. The input signal of the system is the incident partial wave a1. The output signal A is proportional to the far field E (k,π) in the state (k,π); (b) in reception; the input signal is proportional to the electric field of the incident plane wave in the state (−k,ζ), of identical polarization if ζ = π*

66

Ultra Wide Band Antennas

All these considerations result in adopting the generator model in Figure 3.2 for the transmitting antenna, in which the source signal − i.e. the input signal from the system point of view − is the incident partial wave a1 measured in a suitably selected reference plane (the reflected wave b1 is also measured in this plane) − generally the antenna connector5. For the questions of normalization mentioned above, we write the far field as [ROB 03]: E ∞ ( f , r) =

e − jkr r

η0 ⋅ A ( f , rˆ ) 4π

[3.1]

where η 0 = 120π Ω is the free space impedance, k is the free space wave vector, k = |k| = 2π/λ = ω/c is the wavenumber, rˆ = r / r is the unit radius vector (to read “ rˆ = (θ , ϕ ) ” in the argument of the functions), A ( f , rˆ ) = A (k ) = A (k ) πˆ (k ) is classically called the vector amplitude of the field in state (k,πˆ ) 6 [LO 88] and the notation |X| represents the Hermitian norm, i.e. X = X ⋅ X∗ , where the dot represents the dot product and the symbol * represents the complex conjugation. Note that the quantities a1 and A have the same unit (W½), so that |a1|2 and |A |2 have the unit of a power. The vector amplitude of the A field carries all the relevant information of the far field radiation and depends linearly on the incident signal a1. The antenna vector transfer function in the transmitting mode H T is thus naturally defined by [ROB 03]:

H T ( f , rˆ ) =

A (rˆ , f ) a1 ( f )

=

4π

η0

⋅

re jkr E ∞ ( f , r ) a1 ( f )

[3.2]

In the following, the ATF in transmission H T will be simply written H most often. The time domain response of the transmitting antenna is then characterized by its directional vector impulse response − which will be called more simply antenna impulse response (AIR) − inverse Fourier transform of the ATF: 5. The definition of the reference plane is closely related to the measurement procedure (more precisely to the calibration for a measurement in the frequency domain) or to the definition of the source signal in electromagnetic simulation, and fixes the origin of time. 6. Giving the direction of propagation and polarization. Note that A, therefore πˆ , can be complex. Also let us specify that, in the usual definition of A, the numerical normalization is slightly different [LO 88]. We also have A ( πˆ ) = vector field pattern (see Chapter 2).

η0 Ψ( πˆ ) where Ψ (πˆ ) is a conventional 4π

Representation and Modeling of UWB Antennas

h (t, rˆ ) = F −1{H ( f , rˆ )}(t )

67

[3.3]

We will use the following determination for direct and inverse Fourier transforms: X ( f ) = F {x (t )}( f ) ≡ ∫

∞

−∞

x (t ) = F −1{ X ( f )}(t ) ≡ ∫

x (t )e − j 2πft dt

∞

−∞

[3.4a]

X ( f )e j 2πft df

[3.4b] −1

The time response of the antenna to any input waveform a1 = F (a1) is of course given by the inverse Fourier transform of equation [3.2], i.e. by a −1 convolution, given by, denoting a = F (A):

a (t, rˆ ) = [h (⋅, rˆ ) ∗a 1 ](t )

[3.5]

or, denoting e = F − 1(E): e ( t , rˆ ) =

1 η0 [h (⋅, rˆ ) ∗ δ r / c ∗ a 1 ] (t ) r 4π

[3.6]

It is natural at this stage to form the autocorrelation function of the preceding AIR, and to thus define a gain gr in the time domain: g r (t , rˆ ) = ℜhh (t , rˆ )

[3.7]

the correlation being applied to vectors, is given by7: ∞

ℜ vv (t ) = ∫− ∞ v (τ ) ⋅ v ∗ (τ − t ) dτ

[3.8]

It is thus straightforward to show that the Fourier transform of gr is nothing else than the realized gain Gr. We obtain: 7. This definition of the autocorrelation is applicable to deterministic signals of finite energy; it corresponds to the L2 norm in energy: ||x||² = ℜxx(0) and its FT is the energy spectral density 1 T ∗ (ESD). The expression ℜ xx (t ) = lim ∫−T x(τ ) x (t − τ ) dτ , which is more common, T → ∞ 2T corresponds to random signals − of infinite total energy and finite power (thus related to a power norm) − via the Wiener-Kintchine theorem.

68

Ultra Wide Band Antennas

F [g r (t, rˆ )] = F [ℜhh (t, rˆ )] = H ( f , rˆ ) 2

[3.9]

However: 2

H ( f , rˆ) =

2 ∞ 4π r E ( f , r )

η0

a1 ( f )

2

2

=

1 2η0 1 2

r 2 E ∞ ( f , r) 2

a1 ( f ) / 4π

2

=

PΩ ( f , rˆ ) P1 ( f ) / 4π

[3.10]

where P1 is the power incident on the antenna (Figure 3.2a) and PΩ the radiation intensity8. By definition, this quantity is nothing else than the realized gain, thus:

Gr ( f , rˆ ) = F [g r (t , rˆ )] = H ( f , rˆ )

2

[3.11]

We recall that the antenna gains − realized gain, power gain and directivity − are all proportional to the radiation intensity. The definitions differ by the choice of the reference power appearing at the denominator: incident power P1 for the realized gain Gr, accepted (or coupled) power Pa = (1 − |S11|²) P1 (see section 2.4.1) for the power gain G and radiated power Prad = erad Pa for the directivity D (Figure 3.2a); thus: 2 2 Gr ( f , rˆ ) = ⎛⎜1 − S11 ⎞⎟ G( f , rˆ ) = erad ⎛⎜1 − S11 ⎞⎟ D( f , rˆ ) = eant D( f , rˆ ) [3.12] ⎠ ⎠ ⎝ ⎝

erad being the efficiency9 and eant = erad·(1 −|S11|²) the total efficiency10 which is a more appropriate quantity than the efficiency in UWB. We can measure here the “superiority” of the transfer function concept in comparison with the gain concept: the ATF preserves the phase information because it is complex − whereas the gain is real − and, as a vector, it conveys all the information regarding the polarization. The reader will have however noticed that the ATF defined in this way is a total quantity from the polarization viewpoint, since it is related to the total radiation intensity − including all the field components − and therefore to the total realized gain. Following the example of the usual definition of partial gains, we can of course without any difficulty deduce from this partial transfer functions computed by 8. It is reminded that the radiation intensity is the power radiated per unit solid angle, see section 2.3.2. 9. Sometimes called radiation efficiency. 10. Sometimes called antenna efficiency.

Representation and Modeling of UWB Antennas

69

projection onto the orthonormal vectors defining the principal (or co-) and crosspolarizations components in any orthoradial plane − it is even more natural since the ATF is a vector. We will consequently define the partial transfer function for a “polarization state (k,πˆ ) ”11 as the projection: H π ( f , rˆ ) = (H ( f , rˆ ) ⋅ πˆ *) πˆ = H π ( f , rˆ ) πˆ

[3.13]

For example, for an antenna with linear polarization placed in an Oxyz frame of reference, we can write in spherical coordinates:

H ( f , rˆ ) = H θ ( f , rˆ ) + H ϕ ( f , rˆ ) = H θ ( f , θ , ϕ ) θˆ + H ϕ ( f , θ , ϕ )ϕˆ

[3.14]

The impulse response h being real, the transfer function H presents the Hermitian symmetry, i.e.: H (− f , rˆ ) = H ∗ ( f , rˆ )

[3.15]

In practice, the ATF is measured or simulated for the positive frequencies only. To calculate the AIR, we can directly apply equation [3.15] in order to form the function with Hermitian symmetry before computing its inverse Fourier transform. However, we can also, more simply, use the “analytical signal12” (complex) h+:

h + (t, rˆ ) = F −1{H + ( f , rˆ )}(t )

[3.16]

where H+( f ) = H( f ) + sgn( f )H( f ) = 2H( f )ϒ( f ) is the “unilateral” ATF (i.e. zero for f < 0) and ϒ is the function of Heaviside. We then have: h = ℜe(h + )

[3.17]

It should finally be underlined that for any antenna constituting a causal system, the AIR is of course also causal, which is written: h (t , rˆ ) = h (t , rˆ ) ϒ(t )

[3.18]

11. Corresponding to the polarization πˆ (unit vector) with frequency f in the direction of propagation k ↔ ( f , rˆ ) . 12. We also have: h + = h + jH H (h ) where H is the Hilbert transform.

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Ultra Wide Band Antennas

A well-known fundamental consequence, the ATF verifies the Bayard-Bode relationships13, of which we recall here only the most common form relating to real and imaginary parts, posing H = U + jV: U (ω ) =

V (ω ) =

2

π

2ω

π

ΩV (Ω)

∞

∫0

Ω2 − ω 2 ∞

∫0

dΩ

U (Ω ) Ω2 − ω 2

dΩ

[3.19a]

[3.19b]

These relationships mean, in particular, that the amplitude and the phase of the ATF cannot vary independently. In particular, we cannot significantly make the amplitude vary without creating a phase non-linearity. Let us note finally that there would be no fundamental obstacle to define the same way a near field transfer function, which would then depend on a fourth variable, the radial coordinate, and would no longer be transverse (Hr ≠ 0).

Figure 3.3. Example of ATF: realized gain of a biconical antenna (96 × 80 × 80 mm3) measured in an anechoic chamber at approximately 3 meters with a vector network analyzer (R & S ZVA40®) calibrated in the input planes of the antenna connectors [GHA 05]

13. Called the Kramers-Kronig relationships in physics.

Representation and Modeling of UWB Antennas

71

h

Figure 3.4. Example of AIR (biconical in Figure 3.3). The simulated IR was synchronized with the measurement in order to take into account the delay due to the internal feeding cable of the antenna

Figure 3.5. Example of time response: biconical antenna excited by a Rayleigh pulse GM1(t) (Gaussian derivative − see footnote 17). Simulation was synchronized with measurement as in Figure 3.4

3.2.3. Transfer function in reception, reciprocity As we already pointed out, the UWB antenna in reception behaves differently with respect to the signal. That of course does not violate the reciprocity theorem,

72

Ultra Wide Band Antennas

which is actually the very means to establish a relationship between the recently defined ATF in reception and the preceding ATF in transmission. This dissymmetry is due to the fact that in transmission the radiated wave is spherical in the far field region, whereas an incident plane wave illuminating the antenna from any direction is considered to characterize it in reception. The radiated power, finite, is spread over the sphere with a density decreasing as 1/r2, whereas the plane wave must be characterized by a power density, since it conveys, by construction, an infinite power. The sources are thus of a necessarily different nature, localized in transmission and by unit of area in reception. It is thus seen that the transfer functions − ratio of a response and a source − must be a priori different. In reception, we thus consider the model of an antenna illuminated by a plane wave in Figure 3.2. We will choose as input signal a vector quantity proportional to the incident electric field, and for the output signal the partial wave b2 − which is necessarily proportional to it. Precisely:

η0 A pw ( f ) 4π

E pw (k , r ) = e jk ⋅r

[3.20]

where A pw ( f ) = A pw ( f ) ⋅ ζˆ ∗ is the vector amplitude of the incident field in the 2 state14 ( −k , ζˆ ) . In fact, Π = − 12 A pw kˆ is none other than the Poynting vector, so that ½| Apw|2 represents the power density of the incident wave. As Apw carries all the relevant information of the incident wave (amplitude, polarization and signal waveform), we set by definition, in the frequency domain:

[3.21]

b2 ( f , rˆ ) = H R ( f , rˆ ) ⋅ A pw ( f ) or, in the time domain:

b 2 (t, rˆ ) = [h R (⋅, rˆ ) ∗a pw (⋅)](t )

[3.22]

&

with apw = F

−1

(A pw), h R = F

−1

(H R) and b2 = F

−1

(b2), the “contracted” notation ∗ &

representing the composition (in the operational sense) of the convolution with the dot product, i.e.: 14. Which means that the field belongs to the plane normal to the propagation direction − kˆ ,

the electric field Epw being collinear to the unit vector ζˆ fixing the polarization.

Representation and Modeling of UWB Antennas ∞

[u ∗ v](t ) = ∫− ∞ u(τ ) ⋅ v(t − τ ) dτ &

73

[3.23]

The application of the reciprocity principle, following for example Deschamps’ approach [LO 88], [DES 66], allows us to obtain the relationship between the ATFs in reception and transmission (see Appendix A for the proof):

λ T H R ( f , rˆ ) = − j H ( f , rˆ ) 4π h T (t , rˆ ) =

2 c

[3.24]

∂ t h R (t , rˆ )

[3.25]

k1

a1 1

P1

r

2

-k1 π2

P2

b2

π1

Figure 3.6. Sketch of the two-antenna radio link

We must now consider the radio link between two antennas (Figure 3.6), the first being in transmission and the second in reception. For a given geometrical configuration, the system thus made up is a linear two-port network characterized by its scattering matrix S, in particular its transmission coefficient S21. If, moreover, the receiver is matched, we have: b2 = S21 a1

[3.26]

signals being measured in the reference planes used for the definition of the ATF. By definition of the ATF, injecting [3.1], [3.2], [3.20] and [3.21] into [3.26], we obtain: S 21 =

e − jkr r

H 1T ⋅ H 2R =

e − jkr ⎛ -jλ ⎞ T ⎜ ⎟ H 1 ⋅ H 2T r ⎝ 4π ⎠

[3.27]

This equation is perfectly in line with the proposed functional approach: it is the product of three terms, each corresponding to a functional block − the transmitting antenna, the propagation channel and the receiving antenna.

74

Ultra Wide Band Antennas

The propagation channel is ideal here − in line of sight with a single path and being non dispersive − but we can easily imagine the generalization of equation [3.27] to any channel, with multipath components in particular (see Chapter 6). The transposition in the time domain by the inverse Fourier transform gives: s21 =

c T −1 T (h 1 ∗ ∂ t h 2 * δ r / c ) & 2r

[3.28]

the notation ∂ t−1 representing the integral operator: ∂ t−1 ≡ ∫− ∞ . t

If we form the square modulus of equation [3.27], we recover the well-known Friis formula: 2

2

b2 P2 2 ⎛ λ ⎞ ⎟⎟ Gr (k1 ) Gr ( −k1 ) e pol = = S21 = ⎜⎜ 1 2 2 P1 a1 ⎝ 4π r ⎠

[3.29]

in which we recognize the free space attenuation, the realized gains and the 2

polarization efficiency15 e pol = πˆ 1 ⋅ πˆ 2 . Note that [3.27] is actually the fundamental equation of the two-antenna system, because it is more general than quadratic equation [3.29] as it “contains it”, and in addition preserves the phase information. We also note that the S21 quantity is a scattering parameter, and is, for this reason, directly measurable with a vector network analyzer (with a possible translation of the reference planes, i.e. a simple phase shift). This is why we will call equation [3.27] the complex Friis formula or the generalized Friis formula. The reader may also refer to Shlivinski et al., who proposed in [SHL 97] comparable definitions (but different for some) for a transmitting and a receiving antenna, and the two-antenna link.

15. It must be stressed that, so that the antennas are polarization matched, we must take π 2 = π1∗ (and not π1): we thus have epol = 1. The complex conjugation is due to the fact that the polarization of the second antenna is considered in its own referential, thus for an opposite direction of propagation (−k1).

Representation and Modeling of UWB Antennas

75

3.2.4. Transfer function and “conventional” quantities A receiving antenna is usually represented by two conventional quantities: the effective height and the effective area (aperture). We will establish the relationship between these quantities and the ATF in reception. Let us consider an incident plane wave given by:

E pw ( f , r ) = e jkri ⋅r E 0 ( f ) ˆ

[3.30]

with E 0 ⋅ rˆi = 0 . By linearity (of any passive antenna in reception), the open circuit voltage VOC is proportional to the incident electric field, in other words: [3.31]

VOC = h e ( f , rˆi ) ⋅ E 0 ( f )

which defines the effective height he. The voltage across the load (assumed matched to the transmission line) ZL = Zref is written: VL =

Z ref Z in + Z ref

VOC =

Z ref b2

[3.32]

Zin being the antenna input impedance. We then deduce: b2 =

1 − S11 VOC 2

[3.33]

Z ref

Substituting [3.31] then [3.20] into [3.33], we obtain: b2 =

1 − S11

η0

1

2

Z ref

4π

h e ⋅ A pw

[3.34]

and finally: 1 − S11 HR = 2 4π

η0 Z ref

he

[3.35]

The effective area is defined as the ratio between the power accepted by the antenna (i.e. coupled with the load) and the incident power density (see section 2.4.2). This is related to the realized gain by:

76

Ultra Wide Band Antennas

Σe =

λ2 Gr e pol 4π

[3.36]

The relation with the ATF in reception is thus straightforward:

Σ e = 4π H

R 2

e pol = 4π H

R

⋅ πˆ i

2

[3.37]

3.2.5. Elements on the measurement of transfer functions in the frequency domain Compared with conventional frequency measurements of narrow band radiation, the measurement of the ATF in UWB is more intricate not only because it is necessary to carry out a broad band scanning for each direction of observation, but especially because it is necessary to carry out absolute measurements. The measurements performed with a vector network analyzer must therefore be calibrated in order to compensate for the frequency response of the measurement system (instrument, cables, etc.), as well as various mismatches. Several common methods of absolute characterization exist; these are generally called “gain measurement”: we can quote, in particular, the methods known as “three-antenna” and “two-antenna” methods (see section 4.2.2). The first consists of carrying out three transmission measurements for each couple of antennas (of unknown gain) among three: we then obtain a linear system of three equations with three unknowns. The second consists of measuring the transmission between a reference antenna, assumed known, and an antenna to be characterized as the antenna under test (AUT) as it is called. To be of good quality, the measurement is carried out in an anechoic chamber by positioning the antennas far enough away from each other to be in their respective far field zone16. Measuring the transmission parameter of S21 in well defined reference planes − generally connectors of antennas − we obtain, according to [3.27]: H AST ( f , rˆ ) = j 4π

re jkr

λH ref ( f )

S 21 ( f , rˆ )

[3.38]

16. Classically: r > 2D²/λ and r > 10 D (where D is the diameter of the minimum sphere).

Representation and Modeling of UWB Antennas

77

A practical difficulty appears here: a priori equation [3.38] is independent of the coordinate system and in particular of its origin; actually, measurement is not strictly performed in the far field so that the distance r considered in calculation has an influence on the result, in particular on the phase. It would be desirable to take “its” phase center as the “origin” of the AST, but it does not always exist; actually for the majority of antennas it does not exist. Experience shows that the best compromise consists of choosing the center of the minimum sphere [ROB 07a], [BAU 71] after moving the antennas as far away from each other as possible. It is nevertheless necessary to keep in mind that distance errors considered a priori as small (a few millimeters) in the range of centimetric waves can lead to significant phase errors being able to induce awkward time shifts in impulse radio (up to several tens of picoseconds).

3.3. Temporal behavior, distortion We recall that the term distortion refers to any deformation of a signal waveform. This term thus usually refers to the signals, but, through the misuse (or extension) of language, we also speak about distortion for a system or a device when we want to specify its nature. Having in addition restricted our analysis to passive − hence linear − antennas we will be interested only in the sources of linear distortion. Dispersion on the other hand, indicates the frequency variation of the propagation velocity of waves and thus characterizes a medium or a system. It constitutes in many cases one of the major causes of distortion − it is then called phase distortion − but it is not the only one. More precisely, the sources of linear distortion of a signal emitted by an antenna are: − the antenna mismatch; more precisely, the dispersive character of the input impedance ( ∂Z A / ∂f ≠ 0 ) − and thus of the reflection coefficient S11 − implies that the proportion (and the phase) of the wave coupled to the antenna depends on the frequency, which affects its waveform; − the frequency dependence of the gain G( f ) which is called the amplitude distortion; − the frequency dependence of the group delay τ g( f ) (or phase non-linearity) which is called the phase distortion.

78

Ultra Wide Band Antennas

Let us consider as an example the time response of two very different antennas to a so-called Laplace incident pulse, second derivative of a Gaussian17:

( ) t −t

1 − 2 2τ0 a 1 (t ) = 3τ 2

π

2

e

−

( )

t − t0 2 2τ

[3.39]

2

The first, a small biconical antenna (37 x 31 x 31 mm3) [GHA 05], induces a very low distortion of the signal in its main lobe. As shown in Figure 3.7, its realized gain (as well as its group delay − not represented) is indeed almost constant over the “spectrum support ” (bandwidth over which the spectrum takes significant values, in a loose sense). We can observe in Figure 3.8a that, apart from a slight time ringing, the waveform is essentially preserved. In particular, no derivative operation of the signal is observed; contrary to what is sometimes asserted here or there, or what definitions of transfer functions including an explicit term jω can lead us to believe, or, more basically, the expression of the far field as a function of the vector potential E∞ = −∂tA┴ (A┴ = A − ( A ⋅ rˆ )rˆ being the transverse potential).

Figure 3.7. Example of an incident signal (equation [3.39]): (a) waveform (τ = 20 ps); (b) normalized spectrum (and maximum realized gain (θ, ϕ) = (π /2, 0) in main polarization)

17. The n-th derivatives of a Gaussian GM n (t ) = d n [e − t ² / 2σ ² ] form a family of functions dt n −( 2π f )²σ ² / 2 – sometimes called the family of of spectrum GM n ( f ) = 2π σ ( j 2πf ) e n

Rayleigh pulses or Gaussian monopulses – with very interesting time and frequency domain properties and are therefore very often used in UWB. The term Rayleigh pulse is sometimes reserved for the first derivative.

Representation and Modeling of UWB Antennas

79

In this last expression, the delayed potential is expressed by a space integral on the sources so that if these sources are travelling waves the operations of temporal derivation and space integration can be compensated.

Figure 3.8. Time responses to a 2nd order Rayleigh pulse (waveform of equation [3.39]): (a) biconical antenna (π /2,0); (b) LPDA (π /2,0)

The second, a log-periodic dipole array (LPDA) is on the other hand extremely dispersive: the main cause of distortion is indeed its phase non-linearity. When the frequency varies, the active region (a few dipoles which actually contribute to the radiation at a given frequency) moves along the antenna so that the spectral components of the signal follow distinct paths corresponding to different propagation times. Contrary to the preceding case, the signal radiated by the LPDA is extremely distorted as expected. Let us confirm this interpretation by the following experiment: let us consider the frequency response of the LPDA and linearize its phase; we thereby produce an artificial antenna of ATF: H a ( f , rˆ ) = H mes ( f , rˆ ) e − j 2πfτ

[3.40]

We then calculate the AIR of this artificial LPDA by inverse Fourier Transform (FT): we then observe (Figure 3.9) that the residual distortion is very low in the absence of dispersion. However, it can be observed on the realized gain, that the ATF presents quite significant amplitude variations as function of the frequency (Figure 3.9b).

80

Ultra Wide Band Antennas

Figure 3.9. “Artificial” LPDA: (a) AIR (π /2,0), (b) Gr (π/2,0)

3.4. Distortion and ideality There are several ways of considering the “ideality” of an antenna from the point of view of distortion. We must consider the following three “canonical” configurations: − antennas without distortion in transmission; − antennas without distortion in reception; − antennas without distortion in the transmit-receive radio link (for an ideal channel).

In the first case, an antenna (in transmission) must be ideally a pure phase shifter, i.e.: ⎧⎪H T ( f ) = H 0 ⋅ e − j 2π fτ ⎨ T ⎪⎩h (t ) = H 0 ⋅ δ (t − τ )

[3.41]

The realized gain Gr = |H0|² is then independent of the frequency, and the received power varies as 1/f ²: Pr ∝ Σ e ∝ 1/f ². The ideal antenna in transmission is consequently − and according to [3.24] − a pure integrator in reception: H R ( f ) ∝ 1 / jω . Actually, any antenna necessarily presents a low frequency cut-off (the static field does not propagate...) so that [3.41] is not realizable in practice, or even in theory. We can, on the other hand, approach this ideality over a finite band BW = [f1,f2]: any incident band limited signal of spectrum support included in the antenna bandwidth will thus be radiated without any distortion (Figure 3.10).

Representation and Modeling of UWB Antennas

81

Figure 3.10. “Ideal” antenna in transmission

In practice, this type of behavior is approximately obtained with quasi-punctual or frequency-independent antennas (biconical antenna for example). In the second case, the antenna must also be a pure phase-shifter, but in reception: h R (t ) = H 0R ⋅ δ (t − τ )

[3.42]

The transmitting antenna then behaves as a pure differentiator (H T ∝ jω), the realized gain varying as f ² (Gr ∝ f ²). In practice, this type of behavior is roughly obtained with TEM horn antennas. In the last case, we make the assumption of a link between two identical antennas in free space (ideal channel). So that the transmission is carried out without any distortion, it is necessary that: ∂ ( P2 / P1 ) ∂f

[3.43]

=0

However: 2 2 4 − jλ G2 ∝ HT ⋅H R = H T ∝ r2 P1 4π f

P2

[3.44]

It is thus necessary that: Gr ∝ f ⇒

HT∝

f

⇒

H R ∝ 1/

f

Moreover it is also necessary for the ATF to have a linear phase.

[3.45]

82

Ultra Wide Band Antennas

This type of behavior was roughly obtained for example with thick monopole type planar antennas with dielectric “lens” (Figure 3.11) [DER 06]. We can observe that the transmission gain is almost constant between 4 and 9 GHz (Figure 3.11c). When a signal whose energy is primarily concentrated on this band is transmitted − here a Rayleigh pulse GM5 with σ = 55 ps − we must receive an almost undistorted signal if the phase of transmission remains sufficiently linear over the band: this is what is observed in practice (Figure 3.11d) when the normalized received signal is compared to the normalized incident signal, delayed for that purpose, the residual distortion mainly appearing in the form of a time ringing of a few hundred picoseconds.

Units [mm]

Figure 3.11. Two-antenna radio link: (a) DFMM-HL antenna, (b) boresight radio link, (c) Link budget (|S21|²), (d) time response b2(t) to an incident Rayleigh pulse (see footnote 17) a1(t) = GM5 (t) with σ = 55 ps (from [DER 06])

3.5. Performance characterization: synthetic indicators We saw that the fundamental quantities of the UWB antennas are the ATF or the AIR, all these functions furthermore resulting from the ATF in transmission H, which will therefore be considered as the fundamental quantity: all the radiation properties of the antenna in the far field zone result from this, if the matching information is associated with it (input impedance Z11( f ) or reflection coefficient S11( f )).

Representation and Modeling of UWB Antennas

83

However, it is easily conceivable that the representation of the antenna performance with a complex function of three variables is not very simple. We can of course represent these functions using various projections, in both domains (frequency and time), but it is rather cumbersome and only gives a “local” vision of the characteristics. It is for example possible to represent the ATF in transmission in the form of (Figure 3.12): the realized gain as function of the frequency (for various directions of observation) to qualify the energy transfer; 2D radiation patterns, normalized or not, in polar or Cartesian representation, for various frequencies; the phase as a function of the frequency (for various directions of observation), or even better, the group delay τg = −∂φ/∂ω in order to qualify the phase distortion; etc.

Gr

τ g ( f , rˆ )

f1

θ1

θ2 θ1

f2

θ2

f

f (a)

(b)

(c)

Figure 3.12. “Usual” representations: (a) realized gain , (b) radiation pattern (cut), (c) group delay

It is consequently desirable to grasp the antenna in a more global way by defining synthetic performance indicators, which will − in particular − make it possible to facilitate the comparison of the antennas between them. We will successively define indicators characterizing: − the power transfer and directionality: energy gain and mean realized gain; − the phase distortion: differential group delay and standard deviation; − the distortion, considered as a whole: antenna delay spread, envelope width and fidelities; − as well as some characteristic parameters which are derived from them.

3.5.1. Energy gain and mean realized gain (MRG) The AIR (or the ATF) is an intrinsic quantity, characteristic of the directional and temporal properties of the antenna, the antenna response to a given excitation being obtained by convolution. If we wish to give an overall characterization of the energy transfer from the source towards free space for a particular waveform, it is

84

Ultra Wide Band Antennas

natural to define an energy gain from the energy norms previously defined [ROB 03]:

Grt (rˆ )

∞

2

2

ℜaa (0, rˆ ) ∫− ∞ Gr ( f , rˆ ) ⋅ a1( f ) df = = = ∞ 2 2 ℜa1a1 (0) a1(t ) ∫−∞ a1( f ) df a (t , rˆ )

[3.46]

We see that this gain is none other than the mean realized gain weighted by the power spectral density of the signal of excitation. If we consider, for the latter, a rectangular frequency window corresponding to the antenna input bandwidth BWi, we obtain the MRG (mean realized gain), a synthetic, but intrinsic, indicator contrary to Grt : MRG(rˆ ) =

1 BWi

f2

∫ f1 Gr (rˆ, f ) df

[3.47]

From the MRG, we can, without any ambiguity, define the average direction of the main lobe (Θ M,Φ M) and the beamwidth, for example using the average 3 dB beamwidths Θ3dB and Φ3dB (see section 2.3.4) in the principal planes. Let us specify that these definitions do not have anything artificial (or purely mathematical), but have a clear physical interpretation. In particular, if we use the energy gain − a quantity which is more general and actually an MRG weighted by an incident spectrum − these quantities respectively represent the direction of the maximum of the radiated energy, and its corresponding beamwidths (Figure 3.15).

Figure 3.13. Example of MRG patterns in elevation and azimuth (FFPTP antenna [LEP 04]; see Figure 2.13)

Representation and Modeling of UWB Antennas

85

Monocone Ground plane

Figure 3.14. Elevation patterns (Mean Realized Gain) of a monocone for various frequencies: 3, 4.5, 6, 7.5 and 9 GHz (from [BOR 03])

The case of the monocone is a very good example of the interest of the preceding definitions. Indeed, we can observe that its radiation pattern in elevation varies very significantly with frequency, both for the direction of the main lobe and for its beamwidth (Figure 3.14). Assessing average quantities from these patterns without resorting to the preceding definitions would be purely arbitrary.

-30°

-15°

0°

15°

30°

-45°

45°

ΘM

-60°

60°

Θ-3db

-75°

75° 90°

-90° -15

-105°

105°

Θ-3db ~ 40 deg ΘM ~ 50 deg 120 -10

-120°

°

-5

-135°

135°

0 -150°

-165°

5 165°

150°

±180°

Figure 3.15. Example of determining the mean direction and width of the main lobe from MRG (previous monocone)

86

Ultra Wide Band Antennas

3.5.2. Synthetic indicators of distortion As already mentioned, the group delay is an important indicator of distortion characterizing dispersion more specifically. However, regardless of the considered linear system, it is not so much the value of the group delay which matters but its dispersion, i.e. its variations over a band of interest: a constant group delay corresponding to a simple propagation delay through the system. In the case of the UWB antennas, the directional variations of the group delay are also important, as we explained in section 3.3. Let us recall the usual definition of the group delay for a monodimensional system with a scalar transfer function H( f ) = |H( f )|e jφ ( f ):

τg(f ) = −

∂φ

(f) = −

∂ω

1 ∂φ 2π ∂f

[3.48]

(f)

The ATF being a vector field, it is necessary to consider one antenna group delay per polarization state (k,π):

τ πg ( f , rˆ ) = −

1 ∂φπ

2π ∂f

[3.49]

( f , rˆ )

with H π ( f , rˆ ) = H π ( f , rˆ ) e jφπ ( f , r ) (see [3.13] and [3.14]). ˆ

However, to lighten the notation, the reference to polarization will be omitted in the following. We now define two synthetic indicators characterizing these variations [ROB 03], the differential group delay Δτ g and the standard deviation of the group delay στ g : Δτ g (rˆ ) = max τ g ( f , rˆ ) − min τ g ( f , rˆ ) f ∈BW i

1

σ τ g (rˆ ) =

BWi

f ∈BWi

f 2 ∫ f12 [τ g ( f , rˆ ) − τ g (rˆ )] df

[3.50]

[3.51]

with – φ being the phase of H – τ g the average group delay:

τ g (rˆ ) =

1 BWi

f ∫ f12 τ g ( f , rˆ ) df = −

Δφ 2πBWi

[3.52]

Representation and Modeling of UWB Antennas

87

Following the example of the radiation patterns at a given frequency, of the MRG or the fidelity, we will be able to easily represent the angular variations of these two indicators with conical or meridian cuts − using either polar or Cartesian representations − or using “3D” representations. The differential group delay is more specifically adapted to the characterization of highly dispersive antennas, whose group delay varies significantly over the band, but essentially in a monotonic way. This is for example the case for log-periodic antennas (Figures 3.16 and 3.17).

Figure 3.16. Group delay of a LPDA (main lobe (θ,ϕ) = (0,0))

For slightly dispersive antennas, or those presenting a “fluctuating” group delay over the band, the standard deviation should be preferred. This is for example the case of biconical antennas (Figure 3.18). The range of the angular sectors − more generally of the solid angles − over which the preceding quantities present small variations (Figures 3.17 and 3.18) provide a first piece of quantitative information on the feasibility of any potential compensation of the dispersion, and more generally on the distortion as we will see it in what follows.

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Ultra Wide Band Antennas

Δτg(θ )

σ τ g (θ ) Elevation θ (deg) Figure 3.17. Angular pattern of the differential group delay of an LPDA

Figure 3.18. Standard deviation of the group delay of a biconical antenna

As already highlighted, dispersion is not the only cause of distortion. This is why we also consider an indicator in the time domain, which is more global in the sense that it takes into account all the sources of distortion. Following the example of the characterization of the propagation channel, we define [ROB 03] the antenna delay

Representation and Modeling of UWB Antennas

89

spread from the AIR like the second-order moment of the delay (weighted by the instantaneous power18), thus: ∞ 2 ∫0 (τ − τ ) p(τ , rˆ ) dτ

τ ds (rˆ ) =

[3.53]

∞ ∫0 p(τ , rˆ ) dτ

with p (τ , rˆ ) = h 2 (τ , rˆ ) and τ being the mean excess delay, defined as the first moment given by: ∞

τ (rˆ ) =

∫0 τ p (τ , rˆ ) dτ ∞ ∫0

[3.54]

p (τ , rˆ )dτ

z -30°

θ

ϕ

30°

-45°

y x

DELAY SPREAD (ns) 0° -15° 15°

45°

-60°

60°

-75°

75° 90°

-90° 0.1

-105° -120°

Planar LPDA Bicone

-135°

105°

0.2 120°

0.3 135°

0.4 -150°

-165°

0.5 165°

150°

±180°

Elevation

Figure 3.19. Delay spread: meridian cuts (ϕ = 0); Comparison between a bicone and a printed log-periodic

Figure 3.19 gives examples of the time spreading for antennas with low and high distortion. For the biconical antenna, we observe a time spreading lower than 150 ps (respectively 200 ps) over a beamwidth of approximately 60° (respectively 125°) in the neighborhood of the equatorial plane, whereas the 3 and 6 dB beamwidths of the corresponding MRG (calculated over [2.7, 25] GHz) are of 80° and 125° 18. For each polarization state, like the definition of the group delay.

90

Ultra Wide Band Antennas

respectively. For the planar log-periodic antenna, known as dispersive, the spreading is of course appreciably more important: from 400 to 450 ps over approximately 70° around the equatorial plane; the 3 and 6 dB beamwidths of the MRG being respectively 80° and 100°. The mean delay spreads and the related standard deviations in the 6 dB main lobe of the MRG are given in Table 3.1. < τ ds > − 6dB (ps)

σ τ ds

− 6dB

BICONE

142

20

PLPDA

415

45

(ps)

Table 3.1. Mean and standard deviation of delay spread in the 6 dB main lobe

We can finally characterize time spreading by the envelope width of the AIR. It is recalled that the envelope of a signal is the modulus of the analytical signal, that is to say:

henv = | h+ |

[3.55]

More precisely, we will use the normalized envelope:

hˆenv (t , rˆ ) = h env (t , rˆ ) / max[h env (t , rˆ )]

[3.56]

t

We will define the envelope width τ env as the full width at half maximum (FWHM) of henv: ⎧ ⎫ ⎧ ⎫ [3.57] τ env (rˆ ) = max ⎨arg hˆenv (t , rˆ ) = 1 / 2 ⎬ − min ⎨arg hˆenv (t , rˆ ) = 1 / 2 ⎬ ⎩ t ⎭ ⎩ t ⎭

[

]

[

]

We observe in the case of the small biconical antenna an envelope FWHM of approximately 50 ps in the main lobe (Figure 3.20), a value to be compared with the standard deviation of the group delay (Figure 3.18). It should be stressed that, contrary to the delay spread, the envelope width does not generally give any information about the potential time ringing which is observed most of the time, including for antennas with low distortion − only some antennas specifically designed for this purpose, such as “IRAs” (impulse radiating antennas) or some TEM horns, present almost no ringing.

Representation and Modeling of UWB Antennas

91

BICONE – Normalized IR Envelope (Main beam) h h

0.8

z

τenv(π/2,0)

0.6

env

θ

0.4

y

h & henv 0.2

x

ϕ

0

-0.2

-0.4 0.9

1

1.1

1.2 1.3 1.4 t (ns) (delayed by 1 ns)

1.5

1.6

1.7

Figure 3.20. Envelope width (biconical antenna)

Finally some authors use a “time of integration” of the energy to evaluate the “duration of the impulse”, for example between 10 and 90% (Figure 3.21). 1

BICONE - Normalized Cumulative Energy (Main beam)

0.9 0.8 0.7 0.6

E/ETot

0.5 0.4 0.3 0.2

τ10-90(π/2,0)

0.1 1.05

1.1

1.15

1.2

1.25

1.3

1.35

t (ns)

Figure 3.21. Cumulated energy and time of integration from 10 to 90% (biconical antenna)

92

Ultra Wide Band Antennas

As we saw, there are many ways to characterize the distortion: the concept of fidelity − based on intercorrelations − is notably employed by several authors, but no real consensus emerges for a definition. We propose the definition of two slightly different concepts concerning the transmitting antenna: absolute fidelity and relative fidelity. They both characterize the distortion of the radiated waveform (field observed in a given direction), but for the former, with respect to the incident waveform − i.e. typically that of the source − and for the latter, with respect to the waveform radiated in a privileged direction, typically that of the main lobe. The techniques of linear distortion compensation, well-known in signal or filtering theory, are indeed mature: it is sufficient in practice to use an “inverse” filter, either before the system (“pre-distortion”) or afterwards. But the specific difficulty is due here to the fact that the antenna distortion depends a priori on the direction of radiation, which makes any attempt of pre-distortion bound to fail, except if this dependence is weak (an approximate compensation is then possible). We see here the interest in characterizing the angular dependence of the distortion irrespective of the source, because this distortion is not exactly “compensable”. On the contrary, the (linear) distortion with respect to the source is compensable regardless of its magnitude, but for only one direction of radiation. Absolute fidelity is defined as the correlation of the vector amplitude of the field and the incident wave, these two quantities being normalized in energy; it measures the resemblance of the radiated signal and the incident signal for any direction of observation, and any state of polarization:

F(rˆ ) = max ℜaˆ(rˆ ),aˆ1 (τ )

[3.58]

τ

the normalization being given by: uˆ (t ) =

u(t ) ℜuu (0)

=

u(t )

[3.59]

u

For any complex vector u(t , rˆ ) belonging to an orthoradial plane, πˆ k being unit and orthogonal vectors belonging to this plane, the function “max” must be considered here vectorially, and be read: max u(t ) = t

⎛ ⎞ ∑ ⎜ max u(t ) ⋅ πˆ ∗k ⎟ πˆ k k

⎝ t

⎠

[3.60]

Representation and Modeling of UWB Antennas

93

In practice, the simplest thing is to calculate F in the spectral domain. We have indeed:

ℜaˆ(rˆ ),aˆ1 (τ ) = ∫

∞

−∞

aˆ(t , rˆ ) aˆ1(t − τ )dt =

[h (⋅, rˆ ) ∗ ℜa1a1 ](τ )

[3.61]

ℜaa (0, rˆ )ℜa1a1 (0)

then: H ( f , rˆ ) a1 ( f )

F [ℜaˆ(rˆ ),aˆ1 (τ )]( f ) =

∞

∫− ∞ Gr ( f , rˆ ) a1( f ) H ( f , rˆ ) aˆ1 ( f )

=

∞

2

2

∞ 2 a ( f ) df −∞ 1

df ∫

2

∫− ∞ Gr ( f , rˆ ) aˆ1( f )

2

[3.62]

df

The inverse Fourier transform of the preceding quantity finally gives the function to be maximized. Relative fidelity is defined as the correlation between the vector amplitude of the field observed in any direction rˆ and that observed in a privileged direction rˆ0 . It measures the variation of the distortion according to the direction of observation, for a given state of polarization πˆ :

FRπ (rˆ ) =

max ℜa π ( rˆ ),a π ( rˆ ) (τ ) τ

[3.63]

0

ℜa π ( rˆ

0 ),a

π

( rˆ0

(0 ) )

In practice, calculation is also carried out in the spectral domain, with:

F [ℜa π (rˆ ),a π (rˆ0 ) (τ )]( f ) = F ⎧⎨ ∫−∞∞ [(h π (⋅, rˆ ) ∗a1 )(t ) ⋅ (h π (⋅, rˆ0 ) ∗a1 )(t − τ )] dt ⎫⎬ ⎩

=∫

∞

−∞

⎭

[(h π (⋅, rˆ ) ∗ a1 )(t )] ⋅ [H π ( f , rˆ0 ) a1( f )]∗ e − j 2π f t dt

= [ H π ( f , rˆ0 ) a1 ( f )]∗ ⋅F [(h π (⋅, rˆ ) ∗ a 1 )( t )] ( f )

= [H π ( f , rˆ0 )]∗ ⋅ H π ( f , rˆ ) a1 ( f )

2

[3.64]

94

Ultra Wide Band Antennas

Note that these indicators are not intrinsic since they depend on the incident waveform. Figures 3.22 and 3.23 give absolute and relative fidelities of two antennas with respectively, a very low and high distortion. The fidelity of the biconical antenna remains very close to 1 on nearly 4π steradians whereas that of the log-periodic antenna does not typically exceed 0.7. In addition, note that relative fidelity takes more into account, in addition to the distortion effect, the directivity effect. BICONE FIDELITY (a1=GM5) -15°

0°

BICONE RELATIVE FIDELITY (a1=GM5) 0° 15° -15°

15°

-30°

30°

-45°

60°

-90°

-

120°

-165° ± 180°1 165°

0.4 Azimuth Elevation

-120°

θ

135°

0.8 -150°

105°

-105°

z

0.6 °

90°

-90°

105°

0.4 -120°

75°

-75°

90° 0.2

60°

-60°

75°

Elevation Azimuth

-105°

45°

-45°

-60° -75°

30°

-30° 45°

-135°

150°

-150°

y

(a)

x

120° 135°

0.8 150° -165° ±180° 165°

ϕ

(b)

Figure 3.22. (a) Absolute and (b) relative fidelity of a bicone for the incident waveform of Figure 3.11 (GM5) PLPDA RELATIVE FIDELITY (a1=GM5) 0° 15° -15°

PLPDA FIDELITY (a1=GM5) 0° -15° 15°

-60°

75° Azimuth Elevation

90°

0.2

-105°

105° 120°

0.6

-135°

135°

0.8 -150°

-165° ±180°1 165°

(a)

75°

-75°

90°

-90° 0.2

-105°

105°

0.4

0.4

-120°

60°

-60°

60°

-75°

45°

-45°

45°

-90°

30°

-30°

30°

-30° -45°

150°

-120°

120°

0.6

-135°

135°

0.8 -150°

-165° ±180°1 165°

150°

(b)

Figure 3.23. (a) Absolute and (b) relative fidelity of a printed LPDA for the incident waveform in Figure 3.11 (GM5)

Representation and Modeling of UWB Antennas

95

3.6. Parsimonious representation by singularity expansion and spherical modes 3.6.1. The singularity expansion method The SEM (singularity expansion method) was developed in the early 1970s by E.C. Baum [BAU 71], [FEL 76] primarily for modeling the transient response of scatterers and antennas19, one of the main applications being target recognition with impulse radars. The method is founded on the expansion of the transient response on a set of damped sinusoids whose counterpart in the Laplace domain is a rational function in particular characterized by its poles. The latter represent the natural frequencies of the considered object − i.e. frequencies for which the object can have a response in the unforced regime (i.e. with no excitation) − and can be real or appear as complex-conjugate pairs (if the time sequence is real). It is important to state from now on that these poles are characteristic of the considered object, irrespective of the direction of observation. It is also important to note that, although theoretically other types of singularities such as branch points or essential singularities may occur, it has been shown that for certain categories of objects − i.e. finite-size objects in free space made up of passive materials (conductors and/or other media) − the poles are the only singularities in the finite complex plane [BAU 71], [FEL 76]. These considerations apply to any electromagnetic field of the problem, i.e. not only to the quantities bound to the conductors − charge and current densities − but also to the fields [BAU 71], [BAU 73] (diffracted, i.e. re-radiated in the case of scatterers and radiated in the case of antennas). Although mainly applied to problems of diffraction in the literature, the method was also applied to antenna radiation, in particular, by Baum on a theoretical level [BAU 73] and by Licul with radiation measurement (in his PhD and in [LIC 05]). The radiated field can thus be expanded over an − a priori infinite − series of poles and residues, i.e.: E( s, r ) = w( s )∑ p

R (pE ) (r ) s − sp

+ W ( s, r )

[3.65]

where w is a function representing the waveform directly related to the source (a1). In this expression, we restricted ourselves to the most common cases for which the poles are simple. In addition, the latter are independent of the direction of observation. Lastly, W is an entire function of s (containing none of the poles of the 19. In order to study at the time high power (nuclear) electromagnetic pulses (EMP).

96

Ultra Wide Band Antennas

expansion in the finite s plane) which is generally not required for finite size conducting objects [BAU 71], which is the case of antennas in practice. This equation is applicable to the near field as well as to the far field, but we are only interested in the latter so that, by linearity, the ATF can be written as: H ( s, rˆ ) =

∑

R p ( s, rˆ )

[3.66]

p s − sp

The form of the residues is not unique (only the sum must be). However, it was shown that for perfectly conducting finite-sized bodies, a very simple entire function, representing a pure delay ( e − st 0 ) could be factored out, giving the simplified form: R p (rˆ ) ˆ H ( s, rˆ ) = e − st 0 (r ) ∑ p s − sp

[3.67]

an expression in which the residues Rp depend only on the direction of observation rˆ = (θ , ϕ ) and on the polarization, and are thus independent of the complex frequency. The expression of the AIR is deduced from [3.67] immediately: h (t , rˆ ) = ∑ R p (rˆ ) ⋅ e

s p [t − t 0 (rˆ )]

[3.68]

p

In practice, the preceding sums are truncated so as to be able to process them numerically. Moreover, for real antennas, which are by nature band-limited − at least because of losses − the pole number is finite. In fact, including for “ideal” antennas (comprising perfect conductors in free space) the responses can be represented with precision with a limited number of dominant poles. We will thus write: ˆ H ( s, rˆ ) ≅ e − st 0 (r )

P

p =1

h (t , rˆ ) =

P

R p (rˆ )

∑s−s

∑ R p (rˆ ) ⋅ e

[3.69]

p

s p [t − t 0 (rˆ )]

[3.70]

p =1

sums in which the poles are sorted out by ascending order of the resonance frequencies, i.e. ℑm( s2 p ) < ℑm( s2 p +1 ) .

Representation and Modeling of UWB Antennas

97

From the algorithmic point of view, although generally based on works as old as those of Cauchy about the functions of a complex variable or those of Gauss who invented the method of least squares, many techniques have been proposed over the last few decades ranging from polynomial methods based on Prony’s one [PRO 95], [BLA 75] to identification methods such as ARMA models [ROB 07b] [ROB 04]. These methods are satisfactory for high SNR20, but often prone to unacceptable inaccuracy in the presence of noise, including for moderate SNR. The total least squares method has also been applied directly in the frequency domain. Unfortunately, it is also very sensitive to noise, particularly because of illconditioning, and leads to erroneous results in practice. More sophisticated methods, based on the singular value decomposition (SVD) − whose interest is to perform a projection of the data on the “signal subspace” − are more promising [KUM 82]. In [KUM 90a] and [KUM 90b] the identification of the transfer function is directly carried out in the frequency domain. In practice, most often the frequency response is extracted from transmission measurements (see section 3.2.6) or from electromagnetic simulations. It is thus an interesting option since it avoids the calculation of an inverse Fourier transform − with the resulting problems of secondary lobes − but has not been thoroughly investigated yet with antenna measurement data. An interesting alternative method is the “matrix-pencil” (MP) algorithm [SAR 95], [HUA 89] which is applied in the time domain and therefore on the AIR (computed from the measured or simulated ATF by inverse FT after a suitable windowing). It was indeed shown that the MP method is significantly more robust to noise than the conventional polynomial methods of identification (Prony and others, ARMA/Steiglitz-MacBride [STI 77], etc.) [HUA 90a]. In the initial version of the algorithm, the poles, being theoretically independent of the direction of observation, are extracted from the response in a preferred direction (main lobe typically). Actually, in practice, it is not what is observed because of noise and measurement errors, inducing a dispersion of the poles − in particular the damping factor (real part of the complex pole) – which, de facto, depend on the direction of observation. A generalization that we will call the generalized MP-TLS or GMPTLS21 method, published by the same team, is based on the use of a set of transient responses radiated along various viewing directions to perform a single estimate for all the poles [SAR 00]. This is a question of dealing with a singular value problem using an “augmented” matrix, considering a judiciously selected subset of directions of observation − i.e. the more “energy-giving” − which reinforces the robustness to noise. Moreover, more basically, some poles may not be excited for some angles of observation (their residues being zero or very low), and are therefore unobservable 20. Signal to Noise Ratio. 21. Generalized Total Least Square Matrix-Pencil.

98

Ultra Wide Band Antennas

in the corresponding responses. However, the presentation of this method is beyond the scope of this book; so, the reader is referred to the references [SAR 95], [HUA 89], [HUA 90a], [SAR 00], [HUA 90b], [HUA 90c], [HUA 91], [YIL 06]. Note eventually that another virtue of this algorithm is its straightforward implementation, for example with Matlab®. One of the practical difficulties resides in the choice of the truncation order, i.e. parameter P. We will see that the observation of the decreasing of the singular values according to the model order is a piece of usable information, as well as the evaluation of the model error. It will be actually a question of finding a compromise between order reduction and accuracy. 3.6.2. Spherical mode expansion method (SMEM) In the frequency domain, the radiated field can be expanded over a complete orthogonal basis of vector spherical wave functions [HAR 61], [SCH 43], [HAN 88]. It consists of two series of transverse (to radial vector r) magnetic (TM) and electric (TE) modes. The expansion, valid for any distance r, is simplified in the case of the far field. By linearity, the ATF can be consequently expanded on the same basis [ROB 06], that is to say, with u = {1, 2} = {TE, TM}:

H ( f , rˆ ) =

∞

n

2

(u ) ˆ (u ) ψ nm ∑ ∑ ∑ H nm

[3.71]

n =1m = − n u =1

ˆ } series forming a complete orthonormal basis is given by: where the { ψ 1 ⎞ ⎛ ˆ ˆ TM ψ ⋅ ∂ϕ Ynm ⋅ ϕˆ ⎟ nm = ν nm ⎜ ∂θ Ynm ⋅ θ + sin θ ⎠ ⎝ ⎛ 1 ⎞ ˆ TE ⋅ ∂ϕ Ynm ⋅ θˆ − ∂θ Ynm ⋅ ϕˆ ⎟ ψ nm = ν nm ⎜ ⎝ sin θ ⎠

[3.72]

where Ynm are the spherical harmonics: Ynm (θ , ϕ ) = Pnm (cos θ ) ⋅ e jmϕ

[3.73]

Representation and Modeling of UWB Antennas

99

Pnm are the associated Legendre functions and νnm are normalization coefficients:

ν nm

⎡ 2n + 1 (n − m)!⎤ =⎢ ⋅ ⎥ ⎣ 4πn(n + 1) (n + m)!⎦

1

2

[3.74]

with: ∂θ Ynm (θ , ϕ ) = dθ Pnm (cos θ ) ⋅ e jmϕ ∂ϕ Ynm (θ , ϕ ) = jmPnm (cos θ ) ⋅ e jmϕ

[3.75]

(u ) The modal coefficients H nm are of course computed by projection of the ATF on the basis, the Hermitian inner product being given by:

∗

〈ψ a , ψ b 〉 =

∫∫ ψ

=

∫0 ∫0

∗

a

⋅ ψ b dΩ

π 2π

∗

ψ a ⋅ ψ b sin θ dθ dϕ

[3.76]

with the orthonormality conditions: ∗

∗

TM TE TE 〈ψ TM nm , ψ pq 〉 = 〈 ψ nm , ψ pq 〉 = δ npδ mq

∗

TE 〈 ψ TM nm , ψ pq 〉 = 0

[3.77]

[3.78]

and finally: ∗

TM ˆ TM ( f ) = 〈H , ψ H nm nm 〉

= ν nm ∫

π 0

2π

∫0

dPnm (cos θ ) ⎡ ⎤ − jmϕ m H θ − jmPn (cos θ ) H ϕ ⎥ e dθ dϕ ⎢⎣sin θ dθ ⎦ [3.79]

100

Ultra Wide Band Antennas ∗

TE ˆ TE ( f ) = 〈H , ψ H nm nm 〉

= ν nm ∫

π 2π ⎡

0 ∫0

m ⎢− jmPn (cosθ )H θ − sin θ

⎣

dPnm (cosθ ) ⎤ H ϕ ⎥ e − jmϕ dθ dϕ dθ ⎦

[3.80] (u ) The AIR being real, the modal coefficients H nm ( f ) (functions of the frequency) satisfy a relationship similar to the Hermitian symmetry of the Fourier transform of real functions:

[

(u ) H nm (− f ) = (−1) m H n(u,−) m ( f )

]

*

[3.81]

In the particular case of a “pure” polarization along θˆ , [3.79] and [3.80] are reduced to: TM ,TE H nm (f) =

π

∫0

~

,TE ρ TM (θ ) H θ ( f , θ , m) dθ nm

[3.82]

with: dPm (cosθ ) TM ρ nm (θ ) = ν nm sin θ n dθ

[3.83] [3.84]

TE ρ nm (θ ) = − jmν nm Pnm (cosθ )

~ H θ ( f , θ , m) =

2π

∫0

H θ ( f ,θ ,ϕ) e

− jmϕ

dϕ

[3.85]

which is actually the Fourier coefficient of H (ϕ ), a function which must obviously be single-valued and consequently, 2π-periodic in ϕ. This last point is not only formal, not only because it shows that the calculation can be efficiently implemented with an FFT, but especially because it facilitates the interpretation of the mode order m which is actually the Fourier conjugate variable of the azimuth ϕ, i.e. the “frequency” of the ATF variations with ϕ. In other words, we can anticipate that the number of m-modes (energy significant) should be low for

Representation and Modeling of UWB Antennas

101

omnidirectional or quasi-omnidirectional antennas. It is well-known that small size and/or low gain antennas present a small (total) number of modes. The interpretation of the transform is comparable for mode degree n, although it is the Fourier-Legendre transform, which is in fact a Fourier transform in spherical coordinates, which is involved − the two angles not playing symmetrical roles on the sphere. The degree n, conjugate variable of θ, therefore measures the frequency of the ATF variations in elevation. In practice we will operate a truncation as in the case of the SEM, keeping only the dominant modes, i.e. the most powerful. We can now write: [3.86]

H = H N + ΔH N + N ≈ H N

H N ( f ,θ , ϕ ) =

N

n

(u ) u) ˆ (nm ( f )ψ (θ , ϕ ) ∑ ∑ ∑ H nm

n =1m = − n u =1, 2

ΔH N ( f , θ , ϕ ) =

∞

n

∑ ∑

[3.87] (u ) u) ˆ (nm ( f )ψ (θ , ϕ ) ∑ H nm

n = N +1 m = -n u =1, 2

where N represents the noise and measurement errors. However, in practice we will merge this term and the remainder ΔHN in a single “term of error” of the model. For a truncation of degree N, the total number of modal coefficients is a priori NT = 2 N (N + 2). In the specific case of a linear polarization along θˆ , H nTE 0 =0, which reduces this number to NT = N (2 N + 3). The theory of the spherical modes states in addition that the number of dominant modes is related to the size of the antenna. More precisely, Rmin being the radius of the minimal sphere, k = 2π/λ being the wavenumber and ⎣⋅⎦ indicating the integer part: N ≈ ⎣kRmin ⎦

[3.88]

A more precise estimate considering the relative error (in dB) of the truncated power is given by Jensen et al. in [JEN 04]:

⎣

N ≈ kRmin + 0.045 3 kRmin ( − ΔPtr )

⎦

[3.89]

102

Ultra Wide Band Antennas

More generally, an estimate related to the mean square error, i.e. not only to the remainder but also to all the sources of error (see [3.86]), will be considered as:

⎣

N ≈ kRmin + 3 kRmin f ( MSE )

⎦

[3.90]

When the primary objective is not the accuracy, but a very high order reduction, we will be interested more specifically in the full number of coefficients, with: [3.91]

NT ≈ f (kRmin , MSE , Sym)

where Sym represents possible antenna symmetries [ROB 08].

3.6.3. Parametric model with very high order reduction The two preceding transformations being linear, it is possible to compose them (in the operational sense). This is particularly interesting when it is desired to obtain an order reduction − i.e. a “compression” of the representation data − which is extremely high [ROB 06]. A first possible procedure is to start from the ATF, to apply the SMEM, to analyze the energy distribution over the modes in order to determine a minimal order for a given accuracy, and eventually to apply the SEM to each mode with the GMP-TLS algorithm. The alternative procedure presented here is to start from the AIR, to apply the SEM, to analyze the reconstruction error in order to fix the required number of poles, and eventually to apply the SMEM to each residue. By identification of the two possible expansions of the ATF: H ( s, rˆ ) =

∞

R p (rˆ )

∑s−s

p =1

=

p

∞

n

(u ) u) ˆ ˆ (nm ( s )ψ (r ) ∑ ∑ ∑ H nm

[3.92]

n =1 m = − n u =1, 2

It can be shown that the residues can be expanded as: R p (rˆ ) =

∞

n

(u ) ˆ (u ) ˆ ψ nm (r ) ∑ ∑ ∑ Rnmp

[3.93]

n =1 m = − n u =1, 2

with for each TM or TE mode: ∗

(u ) u) ˆ (nm Rnmp = 〈R p , ψ 〉=

π 2π

∫0 ∫0

∗

u) ˆ (nm R p (θ , ϕ ) ⋅ ψ (θ , ϕ ) sin θ dθ dϕ

[3.94]

Representation and Modeling of UWB Antennas

103

Finally, the antenna response in the spectral and time domains can be expanded as: H ( s, rˆ ) ≈ H N,P ( s, rˆ ) =

h (t , rˆ ) ≈ h N,P (t , rˆ ) =

P

⎡

N

⎤

n

( u ) ˆ (u ) ˆ ψ nm (r )⎥ ⋅ ( s − s p ) −1 ∑ ⎢ ∑ ∑ ∑ Rnmp

p =1 ⎢⎣ n =1 m = − n u =1, 2

⎡

P

N

n

⎥⎦

⎤

(u ) ˆ (u ) ˆ ψ nm (r ) ⎥ ⋅ e ∑ ⎢ ∑ ∑ ∑ Rnmp

p =1 ⎢⎣ n =1 m = − n u =1, 2

s pt

⎥⎦

[3.95]

[3.96]

(u ) with, for both, the same set of coefficients { Rnmp } called modal residues − for the

pole p and the TM or TE (n,m) mode − the poles being sp = σp ± jΩp. The AIR being real, the modal residues verify a general relationship similar to [3.81] for the spherical modal coefficients [ROB 08]:

[

(u ) m (u ) Rnmp * = ( −1) Rn, − m, p

]

*

[3.97]

where p* is the index of the conjugate pole of index p. A priori, a model including P dominant poles and a truncation of the spherical modes expansion to order N require a total number of complex parameters NT = P/2 + N(N+2)P to be compared to the size Nf Nθ Nϕ of the initial dataset (also complex) − in general the ATF sampled in frequency and over the sphere {H(fn, θq, ϕm)}|n=1,…Nf, q=1,…Nθ, m=1,…Nϕ. As an example, for measurement over a bandwidth of 20 GHz, with a frequency step of 100 MHz and an angular step of 5°, the full number of measurement samples is Nf Nθ Nϕ = 535,464. In comparison, a model with P = 10 and N = 3 (corresponding to an acceptable accuracy for some cases) requires only a total of NT = 155 coefficients which corresponds to a compression rate of 99.97% (or a compression ratio of ~ 3,333).

3.6.4. Examples of processing of measured ATF 3.6.4.1. Comparison of various algorithms for the SEM The poles of the small biconical antenna in Figure 3.7 extracted with the Steiglitz-McBride (ARMA model), MP-TLS (for various elevations) and GMP-TLS algorithms, are compared in Figure 3.24 [ROB 06]. The variance of the results (with respect to the elevation angle) is clearly improved with the MP algorithm which highlights its best robustness to noise (wide

104

Ultra Wide Band Antennas

sense, i.e. including the various sources of measurement errors). The GMP algorithm appears as a kind of weighted average privileging the main lobe contribution, as can be verified in Table 3.2 [ROB 06]. GMP-TLS -4.084 ± i 82.20 -5.382 ± i 68.47 -5.344 ±i 51.91 -5.055 ± i 35.56 -3.599 ± i 20.98

MP-TLS (θ = 90°) ARMED (θ = 90°) -4.124 ± i 84.62 -5.833 ± i 85.01 -5.874 ± i 71.38 -9.445 ± i 70.26 -5.420 ± i 53.96 -8.495 ± i 51.72 -5.225 ± i 35.59 -6.567 ± i 31.98 -4.047 ± i 21.70 -4.176 ± i 18.16

Table 3.2. Comparison of the algorithms in Figure 3.24 (10 poles, in Grad/s) 100

80

60

40

20

ℑm ( s p )

0

-20

-40

-60

-80

-100 -12

-10

-8

-6

-4

-2

0

ℜe ( s p ) Figure 3.24. Comparison of SEM algorithms (10 poles, In Grad/s) for the biconical antenna in Figure 3.7: (c) Steiglitz-Mcbride, (8) MP-TLS and (š) GMP-TLS

3.6.4.2. Model from the SEM − GMP-TLS algorithm As an example, first of all let us consider an omnidirectional case (the small preceding biconical antenna) processed starting from “2D” measurement data, i.e. considering only a meridian-plane cut (function of the elevation and the frequency).

Representation and Modeling of UWB Antennas

105

The normalized singular values (with respect to the largest) are shown by decreasing order Figure 3.25. A slope break is observed around P = 9 which suggests that this zone of truncation offers an interesting precision/compression compromise. This is why we present now some results for a model with P = 10. 1 0.9 0.8 0.7 0.6

sv/svmax

0.5 0.4 0.3 0.2 0.1 0

5

10

n°

15

20

25

Figure 3.25. Normalized singular values of the GMP-TLS algorithm (small biconical antenna from Figure 3.7)

Figure 3.26 shows the comparison of the AIR measured in the azimuth plane (θ = π /2) and for an elevation of 30 degrees (θ = π /6) with the reconstructed AIR from a ten poles model (P = 10). 40

25

h (90) hr (90)

θ = 90° 30

θ = 30°

h (30) hr (30)

20

15 20

10

h (GHz)

10 5

0 0

-10

-20

-5

-10 0

0.2

0.4

0.6

0.8

t (ns)

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t (ns)

Figure 3.26. AIR measured (−) and reconstructed (−) with a model with 10 poles, for two directions of observation (small biconical antenna from Figure 3.7)

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The realized gains, measured and reconstructed (for the same model) are compared in a meridian-plane cut at a frequency of 3.125 GHz in Figure 3.27. 0 30

330

f = 3.125 GHz 60 300

5

0

-5

-10

-1 5

90

270

120 240

150

210 180

Elevation (deg) Figure 3.27. Realized gain (dBi), measured (—) and reconstructed (—) with a model with 10 poles: meridian cut for f = 3.125 GHz

Let us consider as a second example a balanced printed dipole (with an integrated balun (balanced-unbalanced transformer)) presenting a quasiomnidirectional behavior [PUL 05]. In Figure 3.28, the realized gain measured in the direction normal to the antenna plane (ϕ = 0) and for two elevation angles (θ = π /2 and π /6) is compared with the reconstructed gain for two models (P = 10 and 14). Contrary to the preceding case, here the measured data have not been subjected to any preprocessing22. In particular, no time gating was used which allows us to highlight the filtering effect of the modeling on the residual interferences observed on the frequency response. In particular, it can be seen that with the fourteen poles model, the responses are both smoothed and reconstructed very accurately.

22. Frequency windowing intended for attenuating the secondary lobes in the AIR and time gating intended for strongly reducing interferences (residual multipath components (MPCs) in the anechoic chamber) responsible for ripples in the frequency response.

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It is also interesting to observe the precision of the models on more global quantities such as the MRG. Figure 3.29 presents a meridian cut of the MRG averaged over 2–10 GHz and computed from measurement and from the ten and fourteen poles models.

Figure 3.28. Realized gain (dBi) versus frequency of a balanced dipole, measured and reconstructed with two models (with 10 and 14 poles) for (θ,ϕ) = (π /2,0) and (π /6,0)

5

0

-5

-10

-15

Figure 3.29. MRG (balanced dipole), measured and reconstructed for 10 and 14 poles

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3.6.4.3. Ultra compressed parametric model: Combined SEM and SMEM Let us take, to illustrate this modeling, the FFPTP antenna [LEP 04] from Figure 2.13, which is moderately directive (maximum realized gain of about 8 dBi). We will consider two SEM models with 10 and 14 poles, obtained with the GMP-TLS algorithm. These values correspond to the zone of clean break of the slope in the decreasing of the singular values (Figure 3.30).

Figure 3.30. Normalized singular values of the GMP-TLS algorithm (FFPTP antenna of Figure 2.13)

The MRG patterns computed over the −10 dB input bandwidth (BWi = 3.3 – 5.8 GHz) from preprocessed measurement (time gating, etc., [ROB 06]) and from the 10 and 14 poles models are presented in the two principal planes (Figure 3.31). The marginal improvement obtained on this global quantity with 14 poles instead of 10 is explained by the fact that the MRG thus calculated is only representative of the considered band (here BWi) over which it is correctly represented by the first poles23 which values are close for the two models. However, we can observe from the realized gain that the frequency response is appreciably improved while passing from 10 to 14 poles, in particular “out-of-band” (Figure 3.32): the choice of the model order depends of course strongly on the width of the considered frequency band.

23. I.e. the poles with the lowest resonance frequencies fp = |ℑm(sp)|/2π.

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Figure 3.31. MRG patterns computed over the input bandwidth (3.3 – 5.8 GHz) from measurement and from SEM models with P =10 and 14 (FFPTP antenna from Figure 2.13)

Figure 3.32. Realized gain (main lobe) measured, preprocessed and reconstructed from the SEM models (P =10 and 14) (FFPTP antenna from Figure 2.13)

Ultra Wide Band Antennas

|Rp|

FFPTP antenna : Residues (P=14) & reconstructed after SMEM (N=4) 25 p=1 θ = π/2 p=3 p=5 20 p= 7 p=9 p = 11 p = 13 15 p = 1, N = 4 p = 3, N = 4 |Rp| p = 5, N = 4 p = 7, N = 4 10 p = 9, N = 4 p = 11, N = 4 p = 13, N = 4 5

0

|Rp|

p

|R |

110

0

50

100

150 200 Azimuth (deg)

250

300

350

FFPTP antenna : Residues (P=14) & reconstructed after SMEM (N=5) 25 p=1 θ = π/2 p=3 p=5 20 p=7 p=9 p = 11 p = 13 15 p = 1, N = 5 p = 3, N = 5 p = 5, N = 5 p = 7, N = 5 10 p = 9, N = 5 p = 11, N = 5 p = 13, N = 5 5

0

0

50

100

150 200 Azimuth (deg)

250

300

350

Figure 3.33. SEM Residues (P = 14) and reconstructed after SMEM (N = 4 & 5) (FFPTP antenna of Figure 2.13)

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We can now apply the SMEM to residues Rp(θ,ϕ) of the preceding model (with 14 poles), then reconstruct them from the calculated modal residues. Figure 3.33 compares the residues and the reconstructed residues for models of order N = 4 and 5, with a significant improvement of the modeling for the latter. Nevertheless the “collected” energy in the first modes (dominant) is only marginally increased by 94.7% to 97.4% when passing from order 4 to order 5. We can observe in Figure 3.34 that the frequency response in the main lobe is accurately modeled in the band of interest, for N = 4 as well as for N = 5, and very correctly out-of-band. We can on the other hand observe on the MRG patterns (Figure 3.35) − in particular in comparison to Figure 3.31 − that if order 5 improves the pattern in azimuth, it does not improve the representation in elevation, in particular near the zenith(elevations from 0 to 20 degrees): we would probably need a significantly higher order to obtain a precise model in this angular range. Nevertheless, the main lobe is very accurately represented on a very broad solid angle for P = 14 and N = 4 (in the band of interest) and we could even further reduce the order (P = 10 and N = 3 [ROB 06]) while preserving a very acceptable precision in the “range of interest” (angular and frequency). The achieved compression rate in this last case is 99.8% (155 complex model parameters against 97,200 complex measurement data, corresponding to a compression ratio of ~ 627).

Figure 3.34. Realized gain (main lobe) measured and modeled (SEM, P = 14) and (SEM + SMEM, P = 14, N = 4 & 5) (FFPTP antenna from Figure 2.13)

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Figure 3.35. MRG calculated over the matching bandwidth (3.3 – 5.8 GHz) from measurement and models (SEM, P = 14 and SMEM, N = 4 and 5) (FFPTP antenna)

To finish, we will evoke the possibility of imposing certain antenna symmetries a priori − plane or axial − on the models, which can have been corrupted either by the measurement24 or by manufacturing defects. This procedure allows us not only, in some cases, to correct errors, but especially to achieve extremely high model order reduction, in particular for the omnidirectional antennas. It is thereby shown in [ROB 08] that all the far field radiation information of the “omnidirectionalized” biconical antenna in Figure 3.3 can be correctly represented by a model including only 20 complex coefficients25 (5 poles and 15 modal residues R n( 20)p ), and this over an extremely wide band (up to about 16 GHz) and 4π steradians26. This corresponds to a compression ratio of approximately 25,000 (i.e. a compression rate of 99.996%) if all the measurement data are considered (400 frequencies and 36 × 35 angles). The same model (P = 10, N = 3) applied to the initial dataset − i.e. without consideration of symmetry − would have led to a model with 125 parameters, that is to say six times more, for comparable precision.

24. Noise, calibration errors due to the rotation of the positioner, positioning imprecision of the AUT (not respecting the symmetries), etc. 25. P = 10 and N = 3. 26. Only the TM modes for m = 0 are non-zero in the omnidirectional case.

Chapter 4

Experimental Characterization of UWB Antennas

4.1. Introduction Experimental characterization is an essential stage in the design of antennas. Precise measurements of the characteristics of aerials are necessary for many applications using antennas with well established properties. In many cases, the properties of the antennas can be calculated theoretically. However, for the complex structures of antennas, these calculations are not always possible. Estimates of properties can be advanced by the means of increasingly powerful electromagnetic simulations, but are often carried out with certain simplifications of the structures and by making assumptions on the properties of materials. This is particularly the case for compact antennas integrated in their operational environment. Thus, measurement always appears essential to the characterization of the real performances of the antennas. Indeed, the processes and materials used for their manufacture generally introduce factors of dispersion which are not taken into account in a more or less ideal modeling. However, if measurements are often taken as values of reference performance, they are also prone to errors which it is then advisable to quantify. The experimental characterization of antennas generally aims at measuring the parameters describing the conventional performances of the antennas: the radiation pattern, polarization, directivity, gain, efficiency and impedance. These Chapter written by Christophe DELAVEAUD.

Ultra W ide Band Antennas Edited by Xavier Begaud © 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

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measurements are usually taken at discrete frequencies using specific instruments and installations. These techniques are largely described in many works [HOL 70], [BAL 05], [KRA 02] and reference documents [IEE 79]. They are briefly included in the first part of the chapter by studying their adaptability with the characterization of UWB antennas. The UWB antenna is characterized mainly by the measurement of properties on a very broad frequency band from an antenna. The other great specificity of the characterization of UWB antennas lies in the need to know the phase of the signals of interest with the aim of estimating specific properties to broad band operation. These needs then restrict the choice of the usable instrumentation during frequency type measurements, and encourage the use of measurement techniques in the time domain. The measurement techniques of antennas in the time domain are thus the object of a more detailed presentation in the second part of this chapter because they are less classically presented in the current works. 4.2. Measurements of the characteristics of radiation The measurement of antenna radiation properties generally consists of experimentally determining the parameters describing the way in which electromagnetic energy is radiated (or received) in all space. The parameters most often measured over a narrow frequency band are the radiation patterns, the diagrams or levels of gain, the polarization of the field and the efficiency of radiation. These parameters are not generally sufficient to characterize the properties of Ultra Wide Band (UWB) antennas, in particular within the framework of their use in communications systems. The specific criteria of characterization such as energy gain, the variation of the group delay or the factors of fidelity (see Chapter 3) are estimated in experiments starting from the measurement of a characteristic parameter of the antennas: the complex vectorial transfer function (amplitude and phase), which is proportional to the square root of the absolute gain of the antennas (equation [3.11], Chapter 3). Consequently, the measurement of the characteristics of radiation of UWB antennas can appear very similar to that of narrow band antennas. It requires however the measurement of radio frequency signals in amplitude and phase on a wide frequency band. 4.2.1. Basic concepts 4.2.1.1. Configurations of measurements The procedure of measurement of the characteristics of radiation generally consists of placing a measurement antenna in various positions (orientations) with respect to an antenna under test (or conversely) with the aim of obtaining a sufficient

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number of samples of the radiation in space. The various positions are classically obtained by putting in rotation the antenna under test (Figure 4.1a) or by moving the measurement antenna to a constant distance (Figure 4.1b). To ensure a correct measurement of the antenna under test, only one direction of electromagnetic propagation waves must preferentially exist between the two antennas. This condition is obtained using specific environments without reflection of electromagnetic waves such as the anechoic rooms or free space.

θ

ϕ

θ

ϕ

Figure 4.1. Configurations for the measurement of antenna radiation properties

The antenna under test can be characterized in experiments in reception or emission mode, in accordance with the reciprocity theorem (see Chapter 2 and Appendix A). The validity of the theorem of reciprocity is subjected to certain assumptions generally checked with the passive antennas but that it is advisable not to admit systematically [KRA 02]. In practice, we often prefer to place the antenna whose position changes, in reception mode. Indeed, the emitting antenna is generally connected to the radio frequency generators and amplifiers, which can be cumbersome, sensitive and delicate to put into motion. Thus, it is the measurement antenna, often selected with high directivity, which is fixed with the emission in case (a) whereas it is mobile in reception in case (b). 4.2.1.2. Types of antenna measuring systems Antenna radiation properties are often necessary in nominal conditions of use, i.e. in the far field. This type of measurement is thus often favored because it has many comparative advantages with other techniques. Indeed, the field diagrams remain valid regardless of the distance from measurement, only the intensity of the field varies conversely at the distance, the sensitivity is reduced compared to the variation of the position of the phase center, the coupling and the reflections between antennas are not very perturbing. The main drawback of measurement in the far field lies in the necessary distance between the antenna under test and the measurement antenna. The first ambiguity also lies in the determination of the

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distance of the far field of an antenna. The generally used limit is the Rayleigh distance delimiting the Fresnel and Fraunhofer areas of radiation (see section 2.3.1):

d=

2 D2

[4.1]

λ

− d: distance of the far field lower limit; − D: largest dimension of the physical aperture of the antenna; − λ: wavelength. This formula, resulting from the analysis of the radiation properties of an aperture supporting a uniform distribution of the fields, defines the distance for which the curvature of the spherical wave front at the ends of the antenna under test produces a phase error of π /8 radians compared to the center of the antenna. We also propose minimizing the errors to respect a distance calculated from adding together the far field distance of the antenna under test and that of the measurement antenna.

d '=

(

2

2 Dast + Dam

λ

2

) [4.2]

− Dast: greater dimension of the physical aperture of the antenna under test; − Dam: greater dimension of the physical aperture of the measurement antenna. The determination of these distances is always subjected to discussion. They can introduce errors during the measurement of an antenna with very broad frequency band. For a fixed dimension D, this distance varies proportionally with the frequency and can lead to significant spacing with the high frequencies. At the low frequencies of the spectrum where certain antennas can be regarded as electrically small, the Raleigh distance becomes lower than the distance from reactive field d" [4.3]. Other criteria must then be implemented [KRA 02]:

d "=

λ 2π

[4.3]

There are a great number of configurations of antenna measurement facilities. They can be categorized as indoor or outdoor test ranges. The frequency and the size of the antennas to be characterized often determine the type of measurement set-up configuration to be used. With the UHF frequencies, the characterization of compact

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antennas is done classically using the far field measurement technique realized in an anechoic room. An anechoic room simulates an environment without the reflection obtained by covering the walls of a closed chamber with high-quality absorbing materials. When the frequencies decrease, the conditions of far field lead to the realization of external antenna measurement facilities. Two types of external ranges particularly adapted to the characterization of large antennas geometrically exist: in the “elevated” ranges, the antennas are placed at a height, on towers, buildings or mountains to limit the effects due to the environment. With the VHF frequencies, the reflections on the ground are difficult to avoid (source antennas cannot be highly directive since they are cumbersome) and the measurement ranges with reflection controlled on the ground are used. For the measurement of large antennas functioning in particular at the millimeter-length frequencies, the recourse to the compact ranges or the measurement techniques in the near field becomes essential. A compact range of measurement simulates, at a reduced distance, a length of infinite propagation by creating a plane wave front using reflectors, lenses, horns, arrays of antennas or holograms. The measurement technique in the near field consists of measuring the field in the vicinity of the antenna under test on a known surface and using calibrated probes. The measured data are then calculated in the far field on the one hand using analytical formulations of the Fourier transform method adapted to the measurement surface, and on the other hand by carrying out corrections to take into account the properties of the measurement probe. The various types of measurement ranges allow the characterization of antennas on broad frequency bands. However, some require additional precautions for use, in particular when reflections or beam formation devices are used (test ranges with reflection on the ground, compact ranges). 4.2.2. Frequency methods This approach is the most traditional. It has been used classically within the framework of measurements of narrow band antennas for a long time to consider impedance (insertion losses) and characteristics of radiation (radiation patterns, gain, efficiency, polarization). The characterization of antennas on a broad frequency band is generally obtained by scanning frequencies on the band of interest. The sampling of the analysis frequencies can be critical for the analysis of the properties of UWB antennas. Indeed, according to the type of antenna, abrupt variations of certain characteristics can be missed when measurements are practiced at discrete frequencies. An antenna measurement facility classically consists of a signal emission device, a measurement antenna, a controlled environment of propagation, a system of reception, a system of positioning the antennas, instruments of recording and

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information processing. A modern example of instrumentation associated with a UWB antenna range of measurement in the far field realized in anechoic room is described in Figure 4.2. The inner faces of the anechoic room of rectangular shape (it can also have a tapered shape side of antenna source) are covered with pyramidal or wedge-shaped absorbing materials with high efficiencies. The measurement and test antennas are aligned according to the longest median of the rectangle. The measurement antenna is close to a wall. The antenna under test is positioned in the middle of the quiet zone. The quiet zone is the area where the reflections are minimized. The level of reflectivity in the quiet zone is estimated starting from the characteristics of the measurement antenna, the properties of absorption and the arrangement of the absorbers. The realization of precise measurements assumes the use of high quality radio frequency generators and receivers. The transmitter must present properties of frequency stability and spectral purity. The frequency must be controllable; the emitted power must be stable, adjustable in level and possibly modulable. For the measurement of antennas on a broad frequency band, the use of a frequency synthesizer is essential. The receiver must be sensitive, it must function as a narrow band to remove the possible interfering signals and to reduce the signal to noise ratio, to be linear and present great dynamic range. The vectorial network analyzers construct all of these emission reception functions and are classically used for the measurement of the radiation properties of microwaves antennas. They present in particular intrinsic dynamics able to go up to 120 dB, which confers on the antenna measurement set-up (including the other instruments and the losses in free space) dynamics often higher than 40-50 dB. Thus, for reasonable dimensions of an anechoic room, the properties of radiation of UWB antennas result from a conventional measurement of the complex S21 transmission parameter (see section 3.2.3). If spectrum analyzers or RF power meters (bolometers) can be used in reception for simple radiation measurements, they are not adapted to the characterization of UWB antennas. Indeed, they do not make it possible to obtain the phase of the measured signal, without a specific configuration (see section on phase measurement). According to the length of the radio frequency cables used and the frequencies of study, an amplifier can possibly be placed before the antenna source to increase the measurement dynamics of the antenna. This is often associated with a coupler which makes it possible to take the signal of reference for the analyzer. The time necessary for a measurement depends on the speed of tuning of the transmitter, the synchronization time of the receiver and the bandwidth of measurement of the receiver. The modern vectorial network analyzers carry out these measurements in a very fast way, for example a measurement in a few tens of

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microseconds. They authorize frequency scans with several thousand measurement points, guaranteeing a suitable discretization of the broad frequency band of analysis. shield

absorbers Antenna under test

Polarization

ϕ

θ

Measurement antenna

Coupler

Pr Vectorail network analyzer PC

Pref

Power amplifier GPIB bus Signal / generator receiver

Motion controler

Measurement control data acquisition Pre/post treatment

Figure 4.2. Antenna measurement set up in the frequency domain

The antenna under test is classically fixed on a support with one or two axes of rotation. Cut planes at constant elevation are possible. The measurement antenna is positioned on a support with only one axis of rotation. The measurement antenna is generally selected to be directive to limit low angle illuminations of the side walls of the room. It presents properties of polarization adapted to measurement (linear, dual linear or circular) to allow discrimination in polarization of the measured field. The band of operation of the measurement antenna is also selected in agreement with the frequency band of analysis, which we desire to be broad in UWB antennas. The antenna bases are controlled remotely by a control device. The automation of measurements is essential during the characterization of UWB antennas, mainly because of the quantity of data to be recorded. A

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microcomputer controls the vectorial network analyzer (scanning of frequencies) and synchronizes the motion controller automatically to carry out measurements of transmission according to various cut planes and field polarizations. The microcomputer also records the measured data, allows their processing and analyzes results. 4.2.2.1. Measurement techniques of physical quantities 4.2.2.1.1. Radiation pattern The radiation pattern is a normalized representation, in the cut plane of interest, of the power measured according to the observation angle (see section 2.3.3). This measurement is obtained by rotation of the antenna under test in the case of the base described in Figure 4.2. The field radiated by an antenna can be split up according to two orthogonal components. Consequently, the total power is distributed according to two diagrams, the co-polarized diagram and the diagram of cross polarization. The polarization of the measurement antenna must be adapted to main and cross polarization of the antenna under test during the measurement of the respective diagrams. Generally, the diagrams are measured in the main planes of the linearly polarized antennas (E plane and H plane, see section 2.3.4). When many cut planes are measured with θ and/or ϕ constant, the diagram in three dimensions can be built. The radiation pattern makes it possible to characterize the space distribution of the radiation of an antenna. The radiation pattern is useful to deduce the form, width and direction of the main beam, the level and the secondary direction of the lobes and the rear radiation. The measurement of radiation patterns of antennas on a broad frequency range can lead to a large volume of data according to the desired frequency step. The diagrams can be obtained by rotation of the antenna under test at selected fixed frequencies (a technique often favored as a narrow band) or then by carrying out a frequency scan in a given direction. According to the necessary time with the measurement of a point of frequency, rotation speed of the antenna mast, averaging options and delay times to stabilize the antennas, this type of characterization can generate consequent measurement times. 4.2.2.1.2. Gains According to the power taken as reference (and the associated possible losses), various definitions of the gain are possible. In experiments, measured gain is in general the realized gain, which takes into account all of the antenna losses, including loss of impedance mismatching. The gain is measured for a given

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polarization adapted to the polarization of the antenna under test. The gain in the total field is calculated by summation of the orthogonal components of the gain. In the case of far field measurement carried out in an anechoic room, two methods of gain measurement are classically used. The absolute method is based on the use of the Friis formula [4.4] presented in sections 2.5 and 3.2.3. The antennas are assumed to be aligned face-to-face, in the far field and adapted in polarization. The method with two antennas assumes the use of similar antennas having identical performances. The gain of the antennas is given starting from R, λ and the ratio of the powers. The ratio of the powers corresponds to the S212 transmission coefficient obtained with the vectorial network analyzer after adapted calibration (see section 3.2.3).

⎛ λ ⎞ Pr ⎟⎟ = Grt Grr ⎜⎜ Pt ⎝ 4π r ⎠

2

[4.4]

− Pr: received power; − Pt: transmitted power; − Grr: realized gain of the receiving antenna; − Grt: realized gain of the transmitting antenna; − λ: wavelength; − r: distance between the two antennas. If two identical antennas are not available, a third antenna can be introduced into the procedure. Three series of measurement are then carried out, respectively with a different pair of antennas. A system of three equations with three unknown factors is then obtained. The comparison method uses a standard antenna whose gain is known either theoretically or by measurement according to the absolute method. The principle of measurement lies in the comparison between the power measured with the antenna under test and the power obtained with the standard antenna [4.5]. Gaut =

Paut Gsta Psta

− Paut: power received with the antenna under test; − Psta: power received with the standard antenna;

[4.5]

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− Gaut: realized gain of the antenna under test; − Gsta: realized gain of the standard antenna. The dipole or horn antennas are often used as standard antennas because they have a predictable gain and a pure polarization. The horn antennas are better adapted to the measurement of UWB antennas thanks to their broad band properties. The gain is generally measured in the direction of the maximum of radiation of the antenna under test. The diagrams of gain result from the radiation patterns by translation of amplitude defined in the direction of measurement of the gain. A representation classically adopted to show the variations of the gain patterns (or radiation) with the frequency consists of building a 3D curve while carrying the frequencies on one of the axes (Figure 4.17). 4.2.2.1.3. Polarization As presented in Chapter 2, any electromagnetic field can be expressed as the sum of two orthogonal components that may be linear, circular or elliptical. The full description of the radiation of an antenna requires the measurement of the field polarization state according to the direction in all space. However, we often reduce this need to knowledge of the main and cross components of the field in the E and H planes. Various measurement techniques of the polarization of an antenna exist [KRA 02]. A common method consists of putting the antenna source into rotation with linear polarization. The amplitude measured according to the angle of tilt of the antenna source traced on a polar diagram describes the diagram of polarization. We can extract from this the axial ratio and the tilt angle of the ellipse. By carrying out a fast rotation of the antenna source and simultaneously a slow rotation of the antenna under test, a diagram of modulated radiation can be obtained. The maxima and minima correspond respectively to alignments of the antenna source with the major and minor axis of the ellipse of polarization. The axial ratio can be given starting from the high and low envelopes of the diagram. The sense of rotation is given by comparing the signals collected by two antennas of circular polarization with an opposite sense of rotation. This type of measurement applied to the UWB antennas can advantageously be carried out by privileging the frequency scanning in a given position of the antenna source and the antenna under test. 4.2.2.1.4. Phase Knowledge of the phase pattern or the center of phase of an antenna is important for a great number of applications. For the UWB antennas, the variation of the position of the phase center (see section 2.3.3) with the frequency can constitute

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123

important information to estimate its properties of dispersion. The search for the phase center of an antenna at a given frequency consists of moving the antenna under test with respect to the origin of the coordinate system until a constant phase is observed over the beam of interest. This measurement can prove to be delicate on the antennas of large size compared with the wavelength because they are likely to present several phase centers. The measurement of the phase pattern can take place in various manners according to the instrumentation of measurement used (direct method, with reference or differential antenna [KRA 02]). This measurement is directly accessible with the phase from the S21 parameter of transmission delivered by a vectorial network analyzer. Without special precautions, this phase is relative and can be exploited to determine the group delay. Obtaining an absolute phase requires a calibration in transmission. This calibration can be carried out in various ways: cabled or radiated with an antenna calibrated in phase. 4.2.2.1.5. Efficiency The total efficiency of an antenna is defined by the ratio of the total radiated power to the power delivered to the antenna. This definition can also be expressed in the form of the ratio of the gain to the directivity in a given direction (Chapter 2). According to the definition of the chosen gain, various types of efficiency are identifiable. The most conventional types are the total efficiency and the radiation efficiency. The total efficiency is deduced from the realized gain and takes into account all the losses of the antenna, including loss of impedance mismatching. The radiation efficiency uses the intrinsic gain of the antenna and takes into account only the intrinsic losses of the antenna, other than the loss of impedance mismatching. 4.2.2.1.6. Integration of the radiation Various methods of measurement of the antenna efficiency exist [KRA 02]. The common method used in the base of measurement described in Figure 4.2 consists of determining, in experiments, the directivity and the gain of the antenna under test. This method is generally called the radiation integration method because directivity is obtained by integrating the diagrams in power according to formula [4.6]. However, in the case of directional antennas, directivity can also be obtained using the aperture characteristics of the main beam: D=

4π 2π π

∫∫ p (θ ,φ ) ⋅ sin θ dθ dϕ n

0 0

[4.6]

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Ultra Wide Band Antennas

pn (θ , ϕ ) =

p(θ , ϕ ) pmax (θ0 , ϕ0 )

[4.7]

with pn(θ,ϕ) the normalized power per unit area, p(θ,ϕ) the power per unit area and pmax (θ0,ϕ0) the maximum power per unit area (see also section 2.3.2, equation [2.16]). Alternatively, the efficiency can also result from the integration of realized gain pattern [4.8]: 2π π

∫∫

2π π

Grr (θ , φ ) ⋅ sin θ dθ dϕ =

0 0

∫∫e

ant

D (θ , φ ) ⋅ sin θ dθ dϕ

0 0

[4.8]

= eant

If the principle is simple since it is copied on the definition, its implementation generally appears not to be very precise and long, even for narrow band antennas. The realization of this type of measurement with UWB antennas remains possible but very costly in time with far field test ranges.

Wheeler sphere [WHE 59] Measurement techniques of efficiency freeing themselves from measurements of the characteristics of radiation were developed mainly for the miniature antennas. At resonant frequency of resonance, the input impedance of an antenna can be modeled by an equivalent circuit made up of a radiation resistance in series with a loss resistance. The radiation efficiency then has as an expression [POZ 88]: erad =

R rad R rad + R losses

[4.9]

Rrad being the radiation resistance and Rlosses the loss resistance of antenna. Wheeler established that the efficiency can be deduced starting from two measurements of the input impedance of the antenna. The first is the measurement of the real part of the impedance of the antenna in free space, corresponding to the sum of two resistances Rrad and Rlosses. The second measurement consists of considering the real part of the impedance of the antenna when it is placed in a conducting cavity (“Wheeler cap”) at a constant distance from λ /2π of the walls. This sphere is placed at the transition between the energy stored from the antenna in the near field and the active energy radiated in the far field. The cavity eliminates the field radiated by canceling Rrad radiation resistance of the antenna, the real part

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125

of the input impedance then exclusively represents Rlosses. This method was then extended to the other shapes of cavity and antennas with large ground planes. The method, being based on measurements of circuit parameters, is precise and fast to realize. Alternatives of this technique were developed to solve certain specific problems such as the uncertainty of the assumptions related to the equivalent circuit of the antenna [MKI 97], [GEI 03]. In particular, the technique consisting of placing the antenna in a waveguide with sliding walls [JOH 98] proves more general. The main disadvantage of this method remains its narrow band of analysis. Indeed, this method is theoretically valid only at the frequency where the cavity has a size of λ/2π. Consequently, the measurement of efficiency of the UWB antenna obliges us to split up measurement into sub-bands using a panoply of cavities of adapted size.

Schantz sphere Taking the Wheeler method as a starting point, Schantz [SCH 01a], [SCH 02] developed another technique to quickly measure the efficiency of UWB antenna. This method was then supplemented by Hyunh in order to correct some inaccuracies in the work by Schantz [HYU 04]. Contrary to the Wheeler method, whose cavity is assumed to remove the radiation of the antenna, the Schantz method uses the radiation, its reflection on the walls of the sphere and its reabsorption by the antenna under test. Thus, dimensions of the sphere must henceforth be higher than λ/2π to make it possible for the antenna under test to radiate and receive its own radiation by reflection (the losses due to the conductivity of metallic walls of the cavity are assumed to be negligible). An incident signal applied to an antenna port undergoes a reflection which results in expressing the fraction of reflected power (P) in the form: q=

Preflected Pincident

q = S11EL

2

[4.10]

[4.11]

S11EL being the reflection coefficient of the antenna in free space. The other fraction (1−q) is delivered to the antenna. This portion is then radiated with a radiation efficiency er, including the losses of the antenna. The signal reflected by the widened Wheeler cavity is received with an efficiency er and undergoes a reflection by loss of impedance mismatching (1−q) at the port of the antenna.

126

Ultra Wide Band Antennas

However, because of the loss of impedance mismatching of the port, part of the received signal is retransmitted with a fraction q er. The assessment of the fractions of power emitted and received at the port of the antenna under test, as illustrated in Figure 4.3, makes it possible to express the reflection coefficient inside the Wheeler sphere S11SW in the form: S11SW S 11 SW

2 2

( )

= q + (1 − q)2 er 2 + (1 − q)2 er 2 = q + (1 − q ) 2 e r 2

2

( )

q + (1 − q)2 er 2

3 2

( )

q + (1 − q)2 er 2

4 3

q

1 1 − er 2 q

[4.12]

Figure 4.3. Schematic diagram of the measurement of efficiency according to the Schantz method

The radiation efficiency er can thus be deduced by solving equation [4.12] and by replacing q [4.10]. er =

S11SW

2

− S11 EL

2

1 − 2 S11 EL + S11SW

2

2

S11 EL

2

[4.13]

The total efficiency can be calculated starting from the radiation efficiency and antenna impedance mismatching losses:

(

eant = 1 − S11EL

2

) erad

[4.14]

At certain frequencies, the spherical resonant modes of the cavity are excited. These modes appear by local and sharp variations of the efficiency which it is advisable not to consider. This tends however to limit the precision of the method. This technique is fast and simple to implement. It requires two measurements of the reflection coefficient of the antenna (in free space and in the sphere) to estimate the

Experimental Characterization of UWB Antennas

127

efficiency of an antenna on a wideband of frequency. An illustration of the result is provided in Figure 4.22. 4.2.2.2. Synthesis The frequency methods thus make it possible to evaluate the performances of the UWB antennas by measuring their conventional properties on a broad frequency band but also by calculating characteristics more specific to the impulse field such as for example the effective height [LIC 04], the antenna factors, the factors of fidelity [ALL 93] and the group delay [MOH 03]. The measurement device is often based on the use of a vectorial network analyzer whose last evolutions reduce the disadvantages dependant on the scans on a wide frequency band, namely a great number of measurement points and an important acquisition time. The advantages of the frequency techniques are the dynamics of measurement brought by the vectorial network analyzer. The possible disadvantages are the slowness of measurements, volumes of data, the limitation of the spectral data (frequency sampling), impossibility of identifying certain sources of measurements error (reflections of the equipment in the room, faults of materials, dispersive behavior of the measurement antennas [HER 04]). In addition, the measurement of antenna radiation in the frequency domain requires the use of anechoic (to simulate infinite space and to eliminate the multipath propagation due to the reflections on the objects/walls/ground surrounding the antennas used for the link) and shielded rooms (to be immunized with respect to the potential interferers in the broad working band). It will also be noted that broad band measurement in an anechoic room can be used to improve the precision of frequency measurements by time windowing [KAL 04]. 4.2.3. Time domain method 4.2.3.1. Motivations, requirements The terminology of analysis in the time domain indicates the methods and techniques which treat physical quantities in their form of time dependence. Another possible and largely used designation is the transient analysis, which was originally reserved for the study of short transition phenomena between two established states. The time domain measurement techniques in electromagnetism are not recent, but their employment remained conditioned for a long time by the specific developments of instruments such as the generators of short impulses, the sensors and associated recording equipment [MIL 86]. The motivations for developing such measurement techniques for antenna characterization has received renewed interest

128

Ultra Wide Band Antennas

with the numerous recent studies dedicated to UWB communication systems, most particularly those using impulse radio technique. The time domain study indeed has unquestionable advantages for the analysis of physical phenomena and in particular transients. Thus, it is the domain in which the physical phenomena find their origin and where the required solution must be generally expressed (real solution). It is thus essential to the study of nonlinear or variable phenomena in time, in opposition to the frequency methods which assume stable established modes. The time domain approach also authorizes a better analysis and comprehension of the physical phenomena. The effects of a short impulse on a radiating structure become observable wherever the propagation delay of the signal is comparable to the rise time or the duration of the impulse. In such a situation, only certain portions of the antenna contribute to the mechanism of radiation at a given moment [SMI 01]. When the response of a system is limited in time, the analysis in the time domain appears most suitable for a compact description of the properties. For example, let us consider an ideal filter circuit with very narrow bandwidth. Its simplified transfer function can be represented at first approximation by the right-angled function: H( f ) =

with rect Δf

1 rect Δ f ( f − f o ) Δf

⎧ Δf Δf ⎤ ⎡ ; f0 − ⎪1 si f ∈ ⎢ f 0 − 2 2 ⎥⎦ ⎣ ( f − f o ) = ⎪⎨ ⎪0 si f ∉ ⎡ f − Δf ; f − Δf ⎤ 0 ⎢ 0 2 ⎪⎩ 2 ⎥⎦ ⎣

[4.15]

[4.16]

The impulse response obtained by the inverse Fourier transform in the time domain is: h( t ) =

sin( 2 π Δ f t ) − j 2 π f 0 t e 2π Δ f t

[4.17]

The duration of its impulse response is spread out over time, whereas in the frequency domain, its transfer function is described on a narrow frequency band. The time domain does not seem to be best adapted to describe the behavior of the system and to measure its properties. Such a device, on the other hand, is simply described in the frequency domain. Now let us take the case of a very wide band circuit accepting a frequency range larger than the range constituting the impulse of

Experimental Characterization of UWB Antennas

129

study. Its transfer function can be simplified in a constant module function presenting a phase varying linearly with the frequency:

H ( f ) = H 0 e − j 2π α f

[4.18]

Its impulse response obtained by the inverse Fourier transform in the time domain is then:

h( t ) = H

0

δ (t − α )

[4.19]

with δ the Dirac function. The impulse response of the device is simply the time delay function and the output signal of the system is the delayed impulse (with the H0 coefficient near). The time domain clearly seems the best adapted to a compact characterization of the system. 4.2.3.2. History of developments [MIL 86] The first developments of radar systems, being interested in reflectometry in the time domain or the analysis of electromagnetic impulses associated with nuclear explosions in the 1960s, mark the beginning of the development of tools for experimental analysis of electromagnetic phenomena in the time domain. At that time, the necessary tools for the experimentation in the time domain were still very limited with bandwidths not exceeding the octave. A significant stage was reached when the first sampling oscilloscope appeared [HEW 64]. This device made it possible to display short time signals by the “equivalent” method. In parallel, many electronic devices generating impulse or step function signals were developed. Each decade which precedes was in particular the occasion of a reduction of times characteristic of the transient signals in the instrumentation to reach 1 nanosecond in the 1960s. The introduction of avalanche transistors, of StepRecovery, tunnel or Schottky diodes made it possible to reach durations in the area of a picosecond. The 1970s testified to commercial applications of these devices offering a broad range of performance in rise time (several tens of picoseconds to the nanosecond) and maximum amplitude of the signals (respectively, voltages of a few volts to a hundred volts) [BEN 78]. More recently, the progress made in micro-electronics has allowed significant improvements of technologies for the generation, detection, digitization and recording of transient signals. Digital oscilloscopes, real-time or equivalent time, functioning on a broad analog frequency band have been available for a few years.

130

Ultra Wide Band Antennas

4.2.3.3. Principles of time domain analysis The techniques of experimental analysis in the time domain is based on the theory of modeling the response y(t) of a system to its excitation x(t) by the means of its impulse response h(t) [DAU 88].

y (t ) =

∫

+∞

−∞

x(t − τ ) dτ

[4.20]

y( t ) = x( t ) ∗ h( t )

x(t)

[4.21]

h(t) H(f)

X(f)

y(t) Y(f)

Figure 4.4. Representation of the system

In the frequency domain, the corresponding relation obtained by a Fourier transform is: Y( f ) = X ( f ) H( f )

[4.22]

If the system is excited by a Dirac function δ(t) impulse, the response of the system hδ(t) is then directly worth: yδ ( t ) = h( t )

[4.23]

enabling the determination of the impulse response. For an excitation of the step function u(t) type, the expression of the response of system hu(t) is then worth: +∞

yu ( t ) =

∫ h( t )

−∞

[4.24]

Experimental Characterization of UWB Antennas

131

The impulse response of the system results in: h( t ) =

∂ yu ( t ) ∂t

[4.25]

The basic experimental device giving access to these quantities consists of a generator of the signals of excitation and a device for recording the signal transmitted through the system to be characterized. The problems of generating the excitation signal are not inevitably simple, as far as the Dirac or step function are concerned, since we are ideally assuming non-physical null rise times. To circumvent the problem of non-perfect excitation, we generally carry out the deconvolution of the answer y(t) of the system by the real signal exciter x(t). The operation of deconvolution is a critical stage of the time domain techniques which will be developed thereafter. The recording of the signal transmitted through the system to be characterized is carried out classically using an oscilloscope with the performances adapted to the signals to be recorded. Two measurement techniques of short time signals exist: the real-time technique and the equivalent time technique. Real-time measurement acquires the signal data starting from only one occurrence whereas the equivalent time technique requires a certain number of repetitions of the signal to gather the data. These two types of technique can be used in adequacy with the characteristics of the signal of excitation. The real-time technique will be used for the isolated transient signals (for example of the modulated signals) whereas the equivalent time technique applies to the repetitive sequences of impulses. 4.2.3.4. Reminder on the studied time domain characteristics Specific descriptors were previously defined (see Chapter 3) to characterize the transient behavior of the broad band antennas. The choice of the descriptors depends partly on the approach adopted for their determination. In the case of an analysis in the time domain, two approaches are possible:

− An intrinsic (or absolute) approach: aims at knowing the behavior and the performances of an antenna without restriction on the operating frequency band, in other words, on an infinite spectrum. This excites the antenna, in the case of time domain characterization, by a waveform corresponding to the Dirac function. In practice, this characterization is limited by the instrumentation and access to the performances of the antenna with this type of excitation are only possible over a frequency band. The descriptors obtained are only related to the antenna and result from the complex transfer function (see Chapter 3). − An applicative approach: is a more specific approach to the impulse devices, which considers the access to the behavior of an antenna by assuming an applicative waveform (itself related to a system). With this waveform a frequency band in use is associated for which the antenna was designed. The measurement means used to characterize the antenna can also be used to limit the band to be considered. This

132

Ultra Wide Band Antennas

type of characterization loses its general information because it depends on the waveform used. The criteria of performance evaluation of an antenna in impulse mode can also strongly depend on the detection mode used within the system (detection of energy, detection of a voltage peak, cross-correlation of the received signal with a particular waveform, etc.). Among the various characteristic parameters identified in the literature, two quantities seem particularly appropriate to the characterization of the performances of the UWB antennas for telecommunications. The first makes it possible to evaluate the aptitude of the antenna efficiently to radiate a short impulse from the point of view of energy. The second characterizes the resemblance of this with respect to the initial form of the impulse which crosses it. These two quantities are defined relative to a given impulse. They are not intrinsic characteristics of the antenna and aim at characterizing the behavior of the antenna by integrating some systems properties. The first characteristic is related to the density of energy (energy per unit of solid angle) of the signal radiated in a direction (θ ,ϕ ) . For a given UWB signal, the density of energy of the radiated impulse is calculated then normalized compared to the density of energy of the impulse radiated by a perfect isotropic radiator (without losses and perfectly matched) [LAM 94]. Same manner as for the gain of an antenna defined in fixed frequency, we express this value in dBi where index i refers to the isotropic radiator. We will note this quantities Gimp ( θ ,ϕ ) , like gain in impulse mode. The impedance mismatching losses are voluntarily included in the calculation of this gain.

4π Gimp (θ , ϕ ) =

η0

∫

+∞

−∞

∫

r 2 E (t , r ,θ , ϕ ) r 2 dt

+ ∞Vg (t )

−∞

4Z g

[4.26]

2

dt

r where E( t , r ,θ ,ϕ ) is the radiated field, V g ( t ) is the fem generator, η 0 is the impedance characteristic of the vacuum and Zg is the internal impedance of the generator.

Gimp ( θ ,ϕ ) can be calculated thanks to the realized gain Gr (θ , ϕ , f ) expressed in its harmonic form (taking into account mismatching losses) and with the Fourier transform of the subjected impulse:

Experimental Characterization of UWB Antennas

Gimp

∫ (θ , ϕ ) =

+∞

0

133

2

Gr ( f , θ , ϕ ). Vg ( f ) df

∫

+∞

0

[4.27]

2

Vg ( f ) df

It will be noted that if Vg(f) is the Dirac function, we find the gain expressed in its harmonic form. The second quantity considered is characteristic of the deformation brought to the impulse by the antenna. In the emission mode, we calculate the maximum of the cross-correlation function of impulse subjected to the antenna and the impulse radiated in a direction (θ,ϕ). We normalize this value compared to the square root of the product of the energy of the subjected impulse and the density of energy of the radiated impulse. We previously defined this quantities as fidelity factor (see section 3.5.1) and will note it Fe(θ,ϕ) (e referring to emission). An ideal antenna not deforming the impulse in the emission mode will have an Fe equal to 1. A real antenna will have an Fe(θ,ϕ) ranging between 0 and 1. In the same manner, we can calculate the fidelity factor or function of resemblance in the receiving mode Fr (θ,ϕ). In this case, we calculate the maximum of the cross-correlation function of the incident field and of the impulse collected in the matched load of the antenna then we normalize according to a procedure that is identical to the previous one. ⎞ ⎛ + ∞ V g (t − τ ) .E ray (t ,θ , ϕ ) dt ⎟ maxτ ⎜ ⎟ ⎜ −∞ 2 ⎠ ⎝ Fe( θ ,ϕ ) = 2 + ∞ V g (t ) +∞ 2 dt . E ray (t ,θ , ϕ ) dt −∞ −∞ 4

∫

∫

− where

Vg

[4.28]

∫

is the emf of the generator of the transmitting antenna and E ray the

field radiated by this; − where E inc is the incident field on the reception antenna and Vr the voltage imposed by this on its load. ⎛ maxτ ⎜ ⎝ Fr( θ ,ϕ ) =

∫

+∞

−∞

∫

⎞ Vr (t − τ ).Einc (t ,θ ,ϕ ) dt ⎟ ⎠

+∞

−∞

2

∫

Vr (t ) dt.

+∞

−∞

2

Einc (t ,θ ,ϕ ) dt

[4.29]

134

Ultra Wide Band Antennas

4.2.3.5. Measurement techniques of the physical quantities One of the objectives of this type of measurement is the determination of the impulse response (in the time domain) or of the transfer function (in the frequency domain) of the antenna under test (AUT). Consequently, a time domain system of measurement is a device of transmission able to measure a signal emitted by a generator and the effects of the transmission of this signal through the device under test. The basic principle for the characterization of the antenna radiation properties in the time domain field is similar to the measure in the frequency domain. It consists of measuring powers associated with radiated (or received) electromagnetic waves by the antenna under test under conventional conditions of far field, for example. Experimental implementation is different since we are using instruments functioning in the time domain. An emission block generates the impulse of analysis and a reception block has to detect, acquire and record the transmitted signals. Figure 4.5 describes the elementary principle of an antenna measurement system in transient mode.

Recording

Figure 4.5. Basic principle of a time domain system of antenna measurement

If in the past, the specific instruments were studied and developed to carry out the emission and the acquisition of the transient signals in a more or less complex way, the evolution of the radio frequency instrumentation today allows the realization of these functions using two main families of devices available commercially: − pulses generators; − the real-time or equivalent time oscilloscopes. Synchronization is not inevitably necessary during the use of a real-time oscilloscope.

Experimental Characterization of UWB Antennas

135

The measurement of the antenna characteristics is never carried out directly. The characteristics result from the calculation of the complex transfer function obtained by the measurement of transmitted powers. Direct measurement from the oscilloscope is an electric voltage proportional to the power of the impulse which was propagated through the various components of the radio frequency link. This waveform is generally not characteristic of the antenna under test because it contains the influences of the various components of the transmission chain (propagation through the cables, connectors, free space, amplifiers, filtering, information of the spectrum of the impulse of origin generated by the pulse generator). A calibration phase is thus necessary to extract intrinsic information with the antenna under test. Various calibration methods are possible. Two methods are more particularly described in the following. The method with two identical antennas uses two models of antennas under test which are presumed to have identical properties. The estimate of the impulse response hast(T, θ, φ) (or the transfer function Hast (F, θ, φ) in the frequency domain) of the antenna under test requires four different voltage measurements as illustrated in Figure 4.6.

Amplifier

Amplifier

Amplifier Amplifier

Figure 4.6. Description of the four measurement configurations

136

Ultra Wide Band Antennas

A measurement of transmission including the entire chain with two presumedly identical and suitably directed antennas (a). A measurement using the emitting and receiving devices carried out by connecting them via a suitably dimensioned coaxial cable (b). A direct measurement of the impulse emitted by the generator (c) and finally, the measurement of the connection device (d) used in configuration (b). Among these measurements, only measurements of configuration (a) are specific to the antenna under test. The various voltages measured by the oscilloscope for each configuration can be written in the form of the system: s1 (t , θ , ϕ ) = V g (t ) ∗ he (t ) ∗ hast (t , θ , ϕ ) ∗ hel (t ) ∗ hast (t , θ , ϕ ) ∗ hr (t ) s 2 (t ) = V g (t ) ∗ he (t ) ∗ hc (t ) ∗ hr (t ) s 3 (t ) = V g (t )

[4.30]

s 4 (t , θ , ϕ ) = V g (t ) ∗ hc (t )

After transposition in the frequency domain, we can express the transfer function of the antenna under test: H ast ( f , θ , ϕ ) =

S1 ( f , θ , ϕ ) S 4 ( f ) 1 − j.λ0 − jk0d S 2 ( f ) S 3 ( f ) .e 4π d

[4.31]

− λ 0: wavelength in the vacuum (m); − k0: associated wavenumber; − d: distance between the two antennas under test. This technique has the advantage of not using a specific measurement antenna. However, it requires a delicate stage of calibration (movement and connections of cables, radio frequency component connections). It uses antennas whose properties are not inevitably ideally adapted to measurement (low directivity, discrimination of polarization, efficiency). The comparison method uses an antenna of reference for which the impulse response href (T, θ, φ) (or the transfer function complexes Href (F, θ, φ)) is known. The preceding method can be used to carry out the measurement of this transfer function. The estimate of the transfer function of the antenna under test requires two voltage measurements associated with the two configurations described in Figure 4.7.

Experimental Characterization of UWB Antennas

137

After measurement of the entire system incorporating the measurement antenna and the antenna under test (a), the second measurement is carried out by replacing the antenna under test by the antenna of reference (b). The third measurement can prove to be useful to check the form of the generated impulse (c). The antenna of reference can be used as measurement antenna for all of the experiments. Among these measurements, only measurements of configuration (a) are specific to the antenna under test.

Figure 4.7. Description of the three measurement configurations

The various voltages measured by the oscilloscope for each configuration can be written in the form of the system: s 1 ( t , θ , ϕ ) = V g ( t ) ∗ he ( t ) ∗ h mes ( t , θ mes , ϕ mes ) ∗ hel ( t ) ∗ h ast ( t , θ , ϕ ) ∗ h r ( t ) s 2 ( t ) = V g ( t ) ∗ he ( t ) ∗ h mes ( t , θ mes , ϕ mes ) ∗ hel ( t ) ∗ h ref ( t , θ ref , ϕ ref ) ∗ h r ( t ) s 3 (t ) = V g (t )

[4.32]

While expressing, in the frequency domain, the ratio of the voltages S1(f, θ, φ) on S2(f), we obtain the following expression of the transfer function of the antenna under test:

138

Ultra Wide Band Antennas

H ast ( f , θ , ϕ ) =

S1 ( f , θ , ϕ ) H ref ( f , θ ref , ϕ ref ) S2 ( f )

[4.33]

Compared with the preceding technique, this method has certain advantages. The different operations carried out during calibration are limited to a change of antenna, which limits the sources of errors. The use of a measurement antenna having optimized radiation properties for measurement (directivity, gain, low level of cross polarization) contributes to improving the measuring accuracy. However, this method requires a precise calibration of the reference antenna. 4.2.3.6. Associated instrumentation 4.2.3.6.1. Impulse generators Impulse generators are equipment which deliver in a repetitive way the short transient signals of more or less high energy. They are generally classified in two categories: impulse generators in current or in voltage. At the microwave frequencies, the impulses in voltage are classically carried out with analog technology with circuits made up of capacitors, resistances and fast commutation devices. The capacitors are initially in charge in parallel through a resistance with load using a high voltage or a direct current source. Then, once the capacitors are connected in series, a fast commutation device releases all of the energy stored in the load. The different types of impulse generators vary mainly by their switching system which is subjected to constraints of strong voltages and high levels of energy dissipation. We generally distinguish two types of generator according to the nature of the dielectric medium used to become conductive within the switch: generators with semiconductors and generators with gas. The most conventional generators delivering low voltage signals are produced containing switches with semiconductors. The principle implemented for commutation is then based on the abrupt passage of a state locked in an on-state. The progress made over the last 50 years on semiconductors, with the introduction of diodes DSRD (Drift Step Recovery Diodes), Tunnel and Schottky diodes associated with the evolution of architectures of generators (parallel configurations), makes it possible to today obtain impulses about one hundred picoseconds, with amplitudes of a few hundred volts and repetition rates in the MHz range. The selection criteria of a generator of transient signals for the characterization of antenna can be limited to: − rise time, duration and the amplitude of the generated signal; − the shape and purity of the time profile;

Experimental Characterization of UWB Antennas

139

− the repetition rate. It depends primarily on the technology employed for the switching function. The medium in which the change of state is carried out requires a time dependant on its physical nature to find its initial state; − the stability of the phase in time (jitter). It represents the fast, short and random fluctuations of the phase of a repetitive signal with instabilities in the time domain. This variation depends on the type of switch used in the generator. It is possible to identify three types of jitter: the jitter of repetition (period of repetition), the jitter of delay (with respect to the signal of release) and the jitter of duration (of the generated signal). According to the type of used acquisition, the jitter can strongly disturb obtained measurements. Thus, in the case of a sequential acquisition, the signal to be acquired is assumed to be reproduced identical to itself in time. Various types of transient generators of signals are commercially available to create short signals with low average energy adapted to the characterization of UWB antennas for applications in telecommunications (the specific issue of generation of high power short pulse signals for radar applications, for example, is not approached here). These generators classically make it possible to obtain in a repetitive way transient signals having the Gaussian, monocycle or step function shapes. Table 4.1 gathers the main features of the generators available commercially. Manufacturer

Model

Shape

Amplitude (v)

Rise time or width of impulse (ps)

Jitter

Repetition rate (MHz)

Duration (NS)

Kentech

APG1

Gaussian

100

150

10

0.01

0.15

Geozondas

GZ1118G N-01EV

Gaussian

40

100

2.5

1

0.1

Picosecond Pulse Labs

4050B

Level

10

45

1.5

1

10

Table 4.1. Example of impulse generator features

Other types of signal generators can also be used. Recent arbitrary waveform signal generators, in particular, make it possible to generate UWB signals of more than 500 MHz of bandwidth up to 9.6 GHz [TEK]. 4.2.3.6.2. Filters An impulse shaping circuit is often necessary to adapt the signals from the generators to the desired spectral profile, to study the system under test. Various types of impulse shaping circuits can be used to obtain the adapted spectral profile. In particular, a passband type filtering makes it possible to limit the spectral band of interest and a high-pass type filter used under its cut-off frequency carries out the derivative of the entry signal. An illustration of this principle is proposed in Figures 4.8 to 4.10.

140

Ultra Wide Band Antennas

Figure 4.8. Time evolution (a) and associated spectrum (b) with the step function delivered by the generator Picosecond 4050B

Figure 4.9. Gaussian signal obtained from step function signal (Figure 4.8) through the high-pass filter Picosecond 5208: time evolution (a) and associated spectrum (b)

Experimental Characterization of UWB Antennas

141

a)

b) Figure 4.10. Gaussian monocycle type signal obtained from step function signal (Figure 4.8) through two successive high-pass filters Picosecond 5208: time evolution (a) and associated spectrum (b)

4.2.3.6.3. Fast digital oscilloscopes The measurement of electrical signals in the time domains is generally split up into various stages successively carrying out the detection of the signal, its transmission, its prepacking, its capture and its recording. The digital oscilloscopes carry out this type of measurement by digitizing the signal on the level of the stage of capture. The important technical main features for the measurement of time domain UWB signals are as follows. The analog bandwidth: the oscilloscope sampling head acts as a filter whose bandwidth is limited. The acquisition of a short rise time signal is then possible if its spectrum remains lower than the analog band of the oscilloscope. The analog bandwidth is likely to modify the measurement of the rise time and the amplitude of the transient signal. The measured rise time (10%-90%) is typically worth [AND 98]:

tm( measured ) = tm( oscilloscope)2 + tm( signal )2

[4.34]

142

Ultra Wide Band Antennas

with tm (oscilloscope ) =

0.35 Δf

[4.35]

The rise time of the oscilloscope becomes negligible (1% error) when its bandwidth is 3 to 5 times higher than that of the measured signal. The sampling rate corresponds to the number of samples acquired per second (characteristic of the analog-to-digital converter). It also fixes the measurable maximum frequency within the limit of the analog band (Shannon’s theorem). The sampling rate makes it possible to define the number of points laid out on the fast transitions from the signal. The higher the number of samples, the more accurate the representation of the details of the fast signal. The vertical sensitivity indicates how much amplification a weak signal is given. The vertical sensitivity is usually given in millivolts per division. The vertical resolution determines the precision with which the captured signal is coded. A coding on n bits authorizes 2n intervals. The minimal quantification is given by the number of divisions occupied by the signal, multiplied by the caliber and divided by the 2n bits. The record length corresponds to the number of points which the oscilloscope can memorize. With the sampling rate Fe, the record length Pm fixes the time interval limits Δt on which the acquisition will be realized.

Δt =

Pm Fe

[4.36]

The measurement dynamics of the digitizer is an important criterion for the detection and recording of signals. We will seek to favor the important dynamics to better distinguish the signal from the noise. The acquisition of the transient signal can be realized in two ways: by direct (real-time oscilloscope) or sequential (equivalent time oscilloscope) sampling. A real-time measurement carries out acquisition starting from a single occurrence of the signal. The repetition of the signal to be measured can then advantageously be used to multiply direct acquisitions and to improve the signal-to-noise ratio by averaging. A measurement in equivalent time is obtained starting from signal samples taken with each successive repetition of the impulse and moments different with respect to the start of the impulse. When the entire signal is acquired, information is reconstituted according to a base of time called equivalent time. Sequential acquisition authorizes much broader analog bands than a direct acquisition. Indeed, the strategy of digitization of one sample per cycle allows the

Experimental Characterization of UWB Antennas

143

use of slower and broad band samplers. However, sequential acquisition is only realizable on repetitive signals and with weak jitter. Table 4.2 outlines the main features of examples of commercially available real-time oscilloscopes. Table 4.3 proposes the same type of outline for the equivalent time oscilloscopes.

Manufacturer

LeCroy

Tektronix

Agilent

Model

WM 820Zi

DSA7004B

DSO91304A

Analog band (GHz)

20

20

12

Rise time 10%-90% (ps)

20

19

32

Sampling rate (GS/s)

80 (2*) 40 (4*)

50 (4*)

40 (4*)

Record length (Mpts: std/max)

10/512

20/250

10/1,000

Vertical resolution (bits: std/moy.)

8/>11

8/11

8/>12

Voltage of input max (V rms)

5.5

5

+/-5

Sensitivity (mV/div)

10-500

20-500

1-1,000 0.2

Trigger Jitter (ps)

2

1

Number of channels*

4

4

4

Impedance (Ω)

50 or 1 MΩ

50 or 1 MΩ

50 or 1 MΩ

* according to the models, the sampling rate varies according to the number of channels used

Table 4.2. Real-time oscilloscopes

Manufacturer

LeCroy

Tektronix

Agilent

Model

WE 100H

DSA8200

86106B

Analog band (GHz)

20 to 100

20 (80E04)

18 and 40

Rise time 10%-90% (ps)

18 to 4

17.5

19.5 to 9

Vertical resolution (bits: std/moy.)

14/>17

14

14/>15

Voltage of input max (V rms)

+/-2,5

+/-3

+/ 2

Sensitivity (mV, total)

1-1,000

10-1,000

1-1,000

Jitter noise floor (ps)

0.23

0.8

1

Number of channels

4

4-8

2 RF (2 opti.)

Impedance (Ω)

50

50

50

Table 4.3. Equivalent time oscilloscopes

4.2.3.7. Experimental setups According to the application concerned, the type of signal to be measured and the use of the collected data, experimental implementation of tools for time domain analysis can be declined in various alternatives integrating the more or less specific

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and sophisticated devices. At the end of the 1970s, the National Institute of Standards and Technology (Boulder, Colorado, the United States) set up the first experimental studies to measure transient electromagnetic fields using laboratory instruments and two antennas laid out on a common metallic plane [LAW 78]. Antenna measurement facilities with reduced infrastructure (without an anechoic room) were then developed, in particular for radar applications using the high power signals. Indeed, the measure to the time domain authorizes a windowing and a calibration which make it possible to eliminate the influence from the environment of measurement [JON 97]. Generally using high power impulse generators, potential jammers, i.e. the signals with the powers comparable with that concerned for measurement, can be eliminated by average in particular on free space test ranges. Thanks to the developments of the time domain instrumentation, there was during the two decades which followed a significant multiplication of the developments of antenna measurement facilities in the time domain within the community of radar operators and in particular for the ground penetrating radars (GPR) [ATC 03], [DAN 04], [AND 05]. In the field of UWB telecommunications, the levels of emitted power being limited by the various standards, the conventional instruments can be advantageously used. The low levels of power concerned suggest the use of a shielded room to isolate the signals of interest from the high power narrow band signals, and most likely to blind the receiver. The shielded room can also be anechoic to simplify or eliminate the calibration procedures related to the taking into account from the context of measurement. A system of time domain measurement [BOR 07] specifically dedicated to the measurement of antenna for applications of UWB communication is described in Figure 4.11. This setup consists of a shielded anechoic room where the measurement and under test antennas are localized. A generator of repetitive impulses presenting a short rise time of 45 ps is used to generate the analysis signal applied to the antenna under test used in the emission mode. A set of filters (pass-band, derivative) is used to shape the impulse according to the considered analysis frequency band. Two frequency range configurations are classically used: the first covers the band 0.3-2 GHz and the second 2-12 GHz specifically for the UWB communications. The measurement antenna and the antenna under test are put into rotation for respectively changing polarization and carrying out a measure in a cut plane using positioners controlled by an acquisition software. Specifically designed wide band antennas or commercially available calibrated antennas are used to carry out measurements and in particular calibration (ridged TEM horn: EMCO 3115 from ETS Lindgren ®). On the reception side, the use of a horn-type antenna (with constant aperture) as a measurement antenna makes it possible to compensate for the attenuation in 1/f partially intervening with the reception. Also, except for the attenuation and the delay which are related to the propagation in free space, the

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145

signal received by the receiving antenna almost corresponds to the waveform radiated by the antenna under test submitted to a time domain excitation.

Figure 4.11. Measurement system of the antenna in the time domain

A low noise 0.05−12 GHz amplifier is then used to adapt the levels of the signals received by the measurement antenna to the dynamics of the receiver. The amplified signal is then detected, digitized, recorded and displayed by an equivalent time oscilloscope (Tektronix® CSA8200) synchronized with the impulse generator. The oscilloscope has a numerical analog converter of 8 bits and a capacity of recording 4,000 points maximum by acquisition. For certain requirements, in particular the study of the antenna behavior with respect to UWB modulated impulse signals, this oscilloscope can be replaced by a real-time oscilloscope with which synchronization is not obligatory.

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Ultra Wide Band Antennas

Measurement is automatically carried out using software which orders the orientation of the antennas, the acquisition and the recording of the signals via a GPIB connection. A dedicated procedure to measurement with the oscilloscope was specifically developed to optimize the tunings with respect to the measured signals (instant of analysis, times windows, sampling, measurement calibers, averaging). 4.2.3.8. Processing the time domain results (time-frequency transposition) The operation of convolution which binds the time domain signals to the impulse response of a system is often delicate to realize directly in the time domain. The operation of deconvolution, mathematically defined as inverse of the convolution operation is necessary for obtaining the impulse response of the system and obeys the same limitations. The calculation of a product of convolution (deconvolution) is carried out by establishing a set of linear equations obtained starting from the discrete representation of the signals within the integral of convolution. This system of equations is then solved to obtain the time domain evolution of the impulse response h (t ) . This resolution, obtained by matrix inversion, is prone to errors which multiply to deteriorate the result accuracy significantly. We generally prefer to operate a transposition in the frequency domain where the operation of convolution (deconvolution) is replaced by a multiplication (division). However, the transposition from the time domain to the frequency domain must be carried out with precaution during the processing of the UWB signals, in particular concerning the operation of inverse Fourier transform. The determination of a real time domain signal x(t) from its dual in the complex frequency domain X(f), presenting a Hermitian symmetry, is generally made using an inverse Fourier transform: +∞

x( t ) =

∫ X( f ) e

j2π f t

df

−∞

[4.37]

X * ( f ) = X ( − f ) ∀ f ∈ℜ

In experiments, we have a discrete representation of an analog signal in the time domain as well as the frequency domain. If we have a transfer function H ( f ) , defined on a limited frequency range going from f min to f max for N f frequency points, obtaining its dual h (t ) is made starting from a discrete inverse Fourier transform: h(t ) = Δf

N f −1

∑ H( fk ) e

k =0

j 2π fk t

[4.38]

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147

−f f with Δf = max min frequency sampling step and f k = f min + k Δf a given Nf value of frequency. H ( f ) then do not have Hermitian symmetry because it is defined only between f min and f max . Also, the signal obtained after the discrete inverse Fourier ∧

transform is complex and corresponds to h(t ) + j h(t ) . To obtain the real value h (t ) , it is thus necessary to use a transfer function H ( f ) which has by construction a Hermitian symmetry. In addition, to improve the

visual resolution of the rebuilt signal h (t ) , the spectral support of H ( f ) must be wide. We carry out for this an operation consisting of adding frequency points of null level (zero padding). The equality Δt Δf Nf = 1 indicates that the increase in

N f necessarily involves a reduction in Δt for Δf constant. The rebuilding of a h (t ) real value from H ( f ) is thus carried out by applying two successive operations to H ( f ) before the discrete inverse Fourier transform: the zero padding and the forcing of Hermitian symmetry. 4.2.3.9. Zero padding operation The zero padding operation consists of generating a new transfer function from H ( f ) . This new transfer function H 1 ( f ) comprises more samples than H ( f ) . H 1 ( f ) is built by using the following relation with f e =

⎪⎧ H ( f ) H1 ( f ) = ⎨ ⎪⎩ 0

if f ∈ [ f min ; f max ]

1 . 2 Δt

if f ∈[ Δ f ; f min [ ∪ ] f max ; f e ]

[4.39]

f − f max f H 1 ( f ) consists of N r + N f + N t where N r = e and N t = min . Δf Δf

We artificially reduce the time sampling step of the signal h (t ) when the zero padding operation is applied, which smooths the signal. However, h (t ) remains complex.

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Figure 4.12. Discretization of H ( f ) and zero padding operation

4.2.3.10. Hermitian symmetry operation The Hermitian symmetry operation consists of creating, from H 1 ( f ) , a transfer function H 2 ( f ) definite from − f e to f e as Figure 4.13 illustrates. if f > 0 ⎧ H 2 ( f ) = H1 ( f ) ⎪⎪ * if f ∈ ]Δ f ; f e ] ⎨ H 2 ( − f ) = H1 ( f ) ⎪H ( f ) = O if f = 0 ⎪⎩ 2

[4.40]

The time domain signal h (t ) , obtained starting from the inverse discrete Fourier transform transfer function H 2 ( f ) , is from now a real quantity.

Figure 4.13. Illustration of the operation of Hermitian symmetry

4.2.3.11. Examples of results with directional and omnidirectional antennas 4.2.3.11.1. Equivalence of time-frequency measurements The validation of the time domain measurement facility is based primarily on the comparison of the complex transfer functions (magnitude and phase) of antennas measured in the time domain and the frequency domain at the same time. The

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149

antenna measurement with moderate directivity is favored compared to that of compact UWB antennas. Indeed, while radiating mainly in a half space, these antennas are not very sensitive to the experimentation conditions, and in particular to the antenna pedestal and measurement cable positioned behind the antenna. Also, an antenna with moderate directivity presents a wide aperture, less sensitive to the problems of precise orientation of the measured lobe. An example of the comparison is proposed hereafter with commercial wide band antennas (ridged horn TEM: EMCO 3115 of ETS Lindgren®) with rectilinear polarization.

2

|H| (dBi)

Figure 4.14. Antenna couples EMCO 3115 (ETS Lindgren ®)

time method frequency method

f1.

Rn+1 f = 1 Rn f2

[5.9]

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171

The closer τ is to 1, the more the antenna will approach the ideal behavior of a frequency-independent antenna, however there will be more periodic elements (here “teeth” of the structure) to cover the same bandwidth:

− the ratio χ defines the width of the teeth:

χ=

rn Rn

[5.10]

The angles α and β define the length of the teeth and the minimum and maximum radius delimit the dimensions of the structure. The lowest operating frequency is fixed by the longest tooth length (approximately λ/4 at this frequency). That leads to an overall dimension for the antenna of about the size of a wavelength at the lowest frequency. The highest frequency is given by the smaller tooth dimension. This is similar for equiangular antennas, for which only part of the antenna, whose “teeth” measure a quarter wavelength, contributes to the radiation, and this area approaches the feeding point of the antenna as the frequency increases. The radiation pattern is also bidirectional, symmetrical compared to the planar antenna with a maximum in its normal direction and zero in this plane. The halfpower beamwidth, which strongly depends on the ratio τ, is around 60/70° which thus represents a rather low directivity. Polarization is linear with identical beam width in the two E- and H- planes. The maximum gain is typically 4 dB.

Figure 5.4. Parameters of a circular log-periodic antenna

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Ultra Wide Band Antennas

This antenna is generally realized with two symmetrical arms as in Figure 5.5, which implies the use of a balun. 5.2.2.2. The trapezoidal log-periodic antenna The trapezoidal log-periodic antenna is a circular log-periodic antenna for which all the edges (ends of the antenna, in the shape of “teeth”) present a flat profile rather than a curved profile (Figure 5.5).

Figure 5.5. Trapezoidal log-periodic antenna

This concerns a geometry that is easier to realize, in particular for the linear version, whose performances are identical to the circular version that it is in terms of bandwidth (several octaves), of dimensioning, or always bidirectional radiation in linear polarization and whose directivity is a function of the periodicity ratio of the structure. Finally, its feeding is also symmetrical. 5.2.2.3. The log-periodic dipole antenna The log-periodic dipole antenna corresponds to a trapezoidal log-periodic antenna whose angle β would tend towards 0 (Figure 5.4), which reduces the triangles thus supplying the parallel dipoles in simple lines. Another difference relates to the diameter of the various dipoles which remains constant instead of increasing periodically; only their length follows this evolution. This approximation is acceptable so that its performances remain about constant until relative bandwidths of 130%. As for the other log-periodic structures, the antenna is completely characterized by its flare angle α, which fixes the size of the dipoles according to their distance from the feeding point. In the same way, its periodicity ratio τ indicates the relationship between two successive dipole lengths (Figure 5.6). Like the other log-periodic structures, the antenna is fed at its apex, and waves propagate along the power cables until reaching quarter wavelength dipoles. Before these areas, the voltage remains constant along the antenna. It thus concern an area

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173

of transmission. The active area is the place of strong currents which contribute to antenna radiation; while beyond, the currents and the voltages are weak. This behavior thus has two consequences: the distance from the active area to the feeding point increases as the frequency decreases, and the antenna bandwidth is limited by the dimensions of its extreme dipoles.

Figure 5.6. Dipole log-periodic antenna

The performances of the log-periodic dipole antenna are related on its flare angle and the periodicity ratio which fix its geometry. For example, when τ decreases, the input impedance will increase (fewer elements will be connected in parallel to the line per unit length). This kind of antenna can be matched over a bandwidth of about five octaves. However, its performances vary slightly with the frequency, in particular in terms of radiation where directivity increases at the top of bandwidth. Indeed, the antenna radiation in linear polarization presents a directive behavior with a maximum in the direction of its apex. Its directivity varies typically from 5 to 11 dB [BAL 05] and increases with τ or when α decreases. 5.2.2.4. The sinuous antenna The Archimedes spiral and the log-spiral antennas have been used for several decades on extremely wide bandwidths, but Duhamel in his patent [DUH 87] shows that the preceding attempts to use four or more log-periodic elements to provide two orthogonal polarizations with radioelectric properties and similar dimensions were unfruitful. The geometry of the sinuous antenna can be presented as a hybrid structure between the spiral antenna and the log-periodic antenna. The geometry of the arms reminds us of the log-periodic antenna and provides double polarization. When the

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Ultra Wide Band Antennas

sinuous antenna is self-complementary, its input impedance is independent of the frequency, as we saw previously. Usually, this plane source is laid out above a cavity to remove the back radiation This cavity contains the absorbent and feeding system. The radiating part of the antenna is printed on a substrate of low permittivity.

Figure 5.7. Sinuous antenna with double polarization

The sinuous antenna consists of p cells; each cell is defined by two parameters

α p and τ p (Figure 5.8).

Johnson [JOH 93] presented a formula to calculate the lowest operating frequency of such an antenna: 2r (α p + δ ) ≈

λ

[5.11]

2

where the angles are expressed in radians. This low frequency is thus limited by the external radius of antenna R1, with λ L = 4 R1 (α1 + δ ) . The high frequency is also limited by dimensions of the feeding area. For obtaining good performances, it is necessary that the smallest segment of the feeding area is lower than λ H / 4 to provide a good transition between the feeding area and the active radiation area. The highest operating frequency can then be calculated by: 2 R p (α p + δ ) ≈

λH 2

[5.12]

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175

Figure 5.8. Construction of the geometry of the sinuous antenna

The bandwidth of the antenna can be tuned by adjusting the outside and inside diameters of the antenna. Dimensions of the feeding system limit the high frequency of use. The radiation pattern is bidirectional (without the cavity), symmetrical compared to the antenna plane with the maximum following its normal and zero in this plane. The half-power beamwidth varies between 60 and 100° in the E and H-planes. Polarization is linear. The maximum gain is typically 5 dB. As this antenna is self-complementary, its input impedance is independent of the frequency but remains high and close to 60 π Ohms. To carry out a feeding system with very wideband allowing the transformation of impedance of 50 Ohms towards 60 π Ohms is a difficult operation (see the end of section 5.1.1.1). The geometry of this sinuous antenna can be modified [BEG 00]. It is then no longer self-complementary but has an input impedance that varies slightly and allows the matching to a input impedance that is much lower and close to 100 Ohms (Figure 5.9). The other radiation characteristics are similar to the sinuous antenna.

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Ultra Wide Band Antennas

Metallic Plane

slot

Figure 5.9. Modified dual polarized sinuous antenna

5.2.3. Techniques of frequency-independent antennas performance improvement The frequency independent-antennas, previously studied in detail, are forsaken today to the profit of monopole antennas. Also, the principles of performance enhancement of these antennas are very few and relatively well-known. Thus, the bandwidth of these antennas can be increased toward low frequencies. The improvements generally made are frayed arm ends for a better adaptation at the end-of-line. This enhancement decreases the reflection of the highest wavelengths at the end of the arms. It makes it possible to obtain less input impedance variation. For the log periodic antennas, we will be able to look after a good matching by jointly exploiting their flare angle α and their periodicity ratio τ. A compromise will thus be found between overall dimension and matching: for great values of α or small values of τ, the structures will be more compact with fewer and more spaced elements. On the contrary, a weaker angle α or an increased ratio τ implies antennas made up of many more elements, and whose performances will be linear. In terms of radiation, it will also be advisable to avoid sharp truncations at the end of the structure, and to prefer widened forms which will avoid the reversal currents and will confer on the frequency-independent antenna a circular polarization even at low frequencies. In order to obtain acceptable time domain performances, it is advisable to choose a planar balanced antenna in order to avoid

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177

strong variations of the antenna phase center position with frequency, and thus to obtain a less dispersive behavior. It is usual to see a log-spiral antenna or any other of these planar antennas, used with absorbing cavity in order to obtain a directive pattern [REE 05]. This constraint, which, at low frequencies, led to heavy cavities because it was charged with absorbents, is no longer up to date. Indeed, it was shown that we could replace this cavity with a particular microstrip circuit composed of a ground plane, a dielectric substrate and an array of square patches connected to one another by lumped resistances. The performances obtained using this new absorbent are comparable with those obtained with a cavity on a wideband close to the decade but for a thickness ten times weaker [SCH 06], [SCH 07]. 5.3. Elementary antennas The antennas known as elementary are in fact an evolution of simple dipoles or monopoles whose behavior is well-known and is described in many works. Indeed, an exploited characteristic of these antennas is that their bandwidth increases with the diameter, and thus surface, of their radiating cylinder. This idea is developed and gives rise to antennas of flared conical, triangular, round, elliptic even rectangular shapes which have UWB properties. This class of UWB antennas is certainly represented and used in communications because they preserve the interesting characteristics of omnidirectional radiation pattern as well as the principles of dimensioning the monopole or dipole antenna, which are in fact relatively compact structures. 5.3.1. The biconical antenna The end of the 1930s brought a revival of interest in wide bandwidth antennas, in particular within the framework of research for television. Thus, the biconical antenna was then reintroduced by Carter in 1939 [SCH 03]. His concept was based on the fact that structures of thick linear antennas led to wider bandwidths. This could be even wider if the conductors flare to form the biconical structure. This is a balanced structure in which each opposite cone can be regarded as a transmission line flattening ad infinitum. The voltage difference to their apex produces spherical waves between the cones, mainly in TEM mode. The voltage generated between the cones and the currents on their surface make it possible to deduce the input impedance from an infinite structure. It is shown in [BAL 05] that the voltage between two symmetrical points on each cone at a distance r from the origin is written:

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Ultra Wide Band Antennas

⎡ ⎛ α ⎞⎤ V (r ) = η H 0 e − jkr ln ⎢cot ⎜ ⎟⎥ ⎣ ⎝ 4 ⎠⎦

[5.13]

In the same way, the currents on the surface of these cones are expressed by:

I (r ) = 2π H 0 e − jkr

[5.14]

Hence the impedance characteristic of the cones (equal to the input impedance of the antenna because independent of r): Zc =

⎡ ⎛ α ⎞⎤ V (r ) = Z in = 120 ln ⎢cot⎜ ⎟⎥ I (r ) ⎣ ⎝ 4 ⎠⎦

[5.15]

This impedance is purely real because there are only traveling waves on this antenna.

Figure 5.10. Infinite biconical antenna

However, these characteristics are obtained in the case of infinite structures, which are of course not realizable. When the cones are truncated, part of the energy is reflected at the end of the antenna. The cones can then be seen as lines of impedance equal to those described in the formula, but loaded across their ends. Introduced discontinuity thus causes higher modes, which induces a reactive component in the input impedance and increases the standing-wave ratio. This truncation of the cones also reduces the bandwidth of the antenna at low frequencies, since their oblique height is approximately equal to a quarter wavelength at the lowest frequency. The bandwidths obtained with biconical antennas typically range from 120 to 150%. Their radiation pattern is typically dipolar, omnidirectional in the plane perpendicular to the axis of the cones and zero according to this axis, and of linear polarization. The maximum gains reached are about 0 to 4 dB. The time domain characteristics of a biconical antenna are presented in detail in section 3.5.1.

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179

5.3.2. The discone antenna

The discone antenna is a biconical antenna for which one of the cones is replaced by an infinite ground plane or dimensions that can be regarded as such compared to the wavelength at the lowest frequency. This ground plane is generally of circular shape with the result that the total structure of the antenna is made up of a disc and a cone (from where its name comes). Disc

Cone

Figure 5.11. Discone antenna

Although it preserves part of the characteristics of the biconical antenna, the replacement of the second cone by an ground plane confers some differences to it. First of all, owing to the fact that the antenna is a monopole, its input impedance is divided by two compared to its equivalent dipole, i.e. the biconical antenna. Dimensions of the ground plane must also be optimized because this influences the bandwidth but also the antenna radiation [LIU 99]. Compared to biconical antenna, the excitation of the discone requires new adjustments. The geometry of the antenna is no longer symmetrical, which makes it possible to feed it directly by a coaxial cable whose mass will be connected to the apex of the cone and the central core to the circular ground plane. In order to allow this excitation, the apex of the cone is truncated. The radius of the plate thus formed strongly influences the highest frequency of the antenna bandwidth which is inversely proportional to it [KIM 04]. Another important factor exploiting the bandwidth is the existing difference between the ground plane and the disc which must thus be optimized. Once all these factors are adequately fixed, the antenna presents good performances in terms of bandwidth, since it can be adapted on bandwidths higher than 150%. It is then limited only by the oblique height of the cones (approximately λ/4 at the lowest frequency), and the truncation of the apex which reduces the bandwidth at the highest frequencies.

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Ultra Wide Band Antennas

The radiation is omnidirectional in the disc plane (azimuth), but supports the half space containing the cone of front elevation. Thus, it has a more significant directivity than the biconical antenna. 5.3.3. The bowtie antenna

Thanks to its characteristics in terms of bandwidth and radiation, the biconical antenna represents a good initial design to research new UWB antennas. The primary objective of these antennas is to preserve its ideal characteristics: light weight, simple geometry, low cost and especially compact. The bowtie antenna is a planar version of the finite biconical antenna, which can be printed on a dielectric substrate. It is thus a symmetrical structure. Just as for the latter, the currents are concentrated mainly along the edges of the structure, which also makes it possible to use linear realization techniques while keeping equivalent performances when the application concerned can be subjected to the wind. The bowtie antenna performances are not as good as the biconical antenna in terms of bandwidth, which is limited by the truncation of the antenna. However, its input impedance varies more with the frequency than that of the finite biconical antenna of the same dimensions [BAL 05]. As a result, matching quality is not as good and there is a smaller bandwidth, but which can reach values higher than 100% all the same.

Figure 5.12. Influence value of the flare angle (°) from the bowtie antenna on the real part of the input impedance

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181

In order to evaluate the influence of the flare angle α for a bowtie antenna made up of two triangles of height H = 3 cm (Figure 5.12) a study was realized. The results of this study were compared with those obtained by Brown and Woodward [BRO 52] and show the importance of the flare angle on the low frequency of matching and the stability of impedance of the antenna. If we plot the length L in terms of wavelength at the maximum of the real part of the impedance according to the flare angle (Figure 5.13) we see that this length is almost close to a halfwavelength regardless of α. In terms of radiation, the antenna presents a dipole-type radiation pattern, i.e. omnidirectional in the plane perpendicular to that of the antenna. The gain varies between 0 and 3 dB. The bowtie antenna can also be realized with slots in a metal plate or on a dielectric substrate [DIN 95], which makes it possible to obtain a structure which is no longer symmetrical. In this way, its feeding can be realized by a coaxial cable without a balun. However, the input impedance of such an antenna, remains close to 80 Ω.

Figure 5.13. Height H and side L of the antenna expressed in terms of wavelength at the maximum resistance according to the flare angle α

5.3.4. Planar monopoles antennas

Many studies were undertaken on various forms of planar monopoles on infinite ground plane. These antennas are formed by replacing one half of a dipole-like antenna with a ground plane at right angles. The triangular, circular and square structures are potential antennas with a wideband behavior and their optimization leads to various shapes. The radiating elements are mounted perpendiculary to the

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Ultra Wide Band Antennas

ground plane, a work [CHE 03] having shown that their slope, in order to limit the height, drastically reduced their bandwidth. 5.3.4.1. Circular monopole The circular monopole antenna consists of a disc with a ground plane at right angles. The dimensions of this ground plane must be at least equal to a wavelength at the lowest operating frequency of the antenna in order to avoid the reflections at the edge. However, satisfactory performances can also be obtained with a ground plane with a size lower than half-wavelength at the lowest frequency [POW 04]. The distance between the disc and this ground plane is a critical parameter to be optimized since it influences the antenna bandwidth.

Ground plane Connector

Figure 5.14. Circular monopole

The shape of the antenna element can be elliptic, but a great ellipticity ratio reduces the bandwidth, especially if the largest dimension is parallel to the ground plane [AGR 98]. If it is perpendicular to the ground plane, the bandwidth decreases to a lesser extent but has an advantage since the lowest matching frequency (equation [5.16]) decreases. This frequency can be calculated in the same manner as for a cylindrical monopole [AGR 98]: f = (30 * 0.24)/ (l + r )

[5.16]

with f in GHz, l the height of the monopole and r its equivalent radius in cm. The equivalent radius is calculated by equalizing the surface of the ellipse to that of a cylinder: 2π r l = π r1 r2

with r1 the minimal radius and r2 the maximum radius of the ellipse.

[5.17]

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183

These formulas imply a theoretical overall dimension for the antenna element height that is slightly lower than λ/4 at the lowest frequency. The circular monopole presents excellent performances in terms of bandwidth since this is about 160%. Moreover, its input impedance is around 50 Ω which facilitates its integration with feeding systems and thus avoids the use of an antenna matching transformer. Radiation is quasi-omnidirectional in azimuth on the entire bandwidth, but privileges the half-plane containing the disc in elevation as the frequency increases. The gains observed can be higher than 4 dB in certain directions. 5.3.4.2. Triangular monopole The triangular monopole (Figure 5.15) is the planar version of the conical antenna and consists of a triangle with a ground plane at right angles. The evolution of its behavior according to its dimensions follows overall that of the triangular (bowtie antenna) or conical (biconical and discone antennas) structures. However, a comparative study of this type of antenna was undertaken by Brown and Woodward [BRO 52], who indicate some differences between the planar structure and its 3D counterpart, in terms of matching and radiation.

Ground Earth plane plane

Connector Figure 5.15. Triangular monopole

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Ultra Wide Band Antennas

Thus, the increase of angle α of the triangular monopole decreases its input impedance, but this remains much higher (about 50% for α = 90°) than for the corresponding conical antenna. Moreover, the impedance variations of the planar antenna are more important. These variations will limit the bandwidth which could be as important as that of a biconical antenna, but values of about 120% are still reachable. Another influence of this angle is highlighted in [BRO 52]. Indeed, the more significant this is, the more the lowest matching frequency of the antenna decreases, and this phenomenon is amplified for the strong values of α. It is thus of double interest to increase this angle: to lower the input impedance (and to decrease the variations by them), as well as to decrease the overall length A of the antenna element which can be lower than λ/6 for values of α higher than 90°. However, its oblique height will remain equivalent to a quarter wavelength. In terms of radiation, the diagram is quasi-omnidirectional in the azimuth plane, even if the absence of symmetry of revolution brings some variations in this plane compared to the conical monopole. This phenomenon is more significant as the flare angle is important. In elevation, like all the structures with ground plane at rightangles, the directions in the half space containing the triangle are privileged. This behavior is especially sensitive to higher frequency, when the antenna is electrically larger. Some variations around the shape of the triangular monopole were studied and tested. The reversed triangle, in particular, presents interesting performances in terms of bandwidth. This antenna element shape involves taking into account the distance between the triangle and the ground plane because this parameter greatly influences the behavior of the antenna. An optimization of this structure was carried out in [KER 01] using genetic algorithms. The antenna proposed has a bandwidth higher than 80%. 5.3.4.3. Square and derived monopole The trapezoidal monopole (Figure 5.16) is proposed as a variation of a square monopole for which we vary the extreme widths L1 and L2 of the antenna element. Indeed, the square monopole introduces a quite limited wideband behavior [AGR 98] since it is about an octave, thus a bandwidth of 66%. A relative bandwidth higher than 80% is obtained for the trapezoidal monopole for values of L1 lower than L2, in particular when L1 = ¾ L2 [CHE 00a]. The height H of the trapezoid according to the frequency at the lowest matching frequency is approximately equal to λ/5 for the structures with the widest bandwidth.

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185

Figure 5.16. Trapezoidal monopole

Ground plane Figure 5.17. Butterfly monopole

The butterfly monopole (Figure 5.17) is also a variation around the square monopole in order to increase the bandwidth. The radiating element reduced at the middle height which decreases the L2 width compared to the constant length L1, drawing the shape of the butterfly. An optimization [CHE 00b] leads to a relative bandwidth higher than 75% for a ratio of L2/L1 = 0.8. All these results were obtained by feeding the antenna with a coaxial cable with 50 Ω characteristic impedance. It is necessary to note the importance of the distance between the monopole and the

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ground plane. Whatever the value of L2 chosen, the more significant this variation is, the wider the bandwidth becomes. Lastly, just as for the other monopoles, dimensions of the ground plane must be sufficient in order not to degrade the performances of the antenna in terms of bandwidth or radiation. 5.3.4.4. Printed monopoles with reduced ground plane The interest of printed antennas with the same performances as a monopole with the ground plane (matching, radiation) is obvious: the antenna is thus compact, light, low cost and easy to realize. These monopoles fed with coplanar lines present moreover a great capability of integration and make it possible to place the antenna on the same support as its feeding system. This is main reason why this type of structure has had such great interest these last years. The printed antenna benefits from a miniaturization compared to its counterpart with the ground plane since its substrate has a dielectric permittivity higher than the air. Lastly, compared to a printed UWB dipole type, the printed monopoles do not require a balun and their input impedance is divided by two (they are easily matched to 50 Ohms). However, the transformation of a monopole to its printed counterpart cannot be summarized with a simple homothety. The design of these antennas requires taking into account new parameters as dimensions of the feeder which can deteriorate its performances. Indeed, the coplanar microstrip lines or waveguides must theoretically have infinite ground planes. However, for these compact coplanar antennas, the overall dimension of the line becomes problematic. Dimensions around the wavelength not being acceptable, the feeders are thus truncated in width but also in length. However, a minimal length is necessary in order to remove the effects of the discontinuity of the field which change from a TEM mode in a coaxial connector to another mode according to the selected access. Consequently, it is no longer possible to regard these transmission lines as such, but rather as an intrinsic part of the antenna. A simple study on the matching of a printed monopole, made up of a fixed size antenna element and of a variable length coplanar line, makes it possible to measure the impact of the line on the performances of the antenna. Figure 5.18 gives the results obtained for a circular monopole with 15 mm diameter of which the length l of the coplanar feed waveguide varies from 10 to 18 mm. It is clear that the bandwidth as well as the antenna resonant frequencies depend directly on l: the low limit of matching decreases when l increases, and is inversely proportional to the overall length of the antenna.

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Figure 5.18. Influence length l of the coplanar waveguide on the reflection coefficient and the real part of the input impedance of the antenna

In order to take this influence into account, various approaches are planned. The first consists of isolating the radiating part of the feeder using a specific shape in T for the side ground planes of the coplanar waveguide [FOR 04] (Figure 5.19). This decoupling between the antenna element and the line limits the reverse currents in the direction of the connector, making it possible to limit the phenomenon of resonant length for the line and thus avoiding nuisances during the integration of the antenna. However, a minimal line length of approximately a quarter wavelength at the lowest frequency is necessary so that the mode can be established correctly in the line. Moreover, the overall dimension side of the antenna again becomes important (approximately two-thirds the wavelength at the lowest frequency).

Figure 5.19. Triangular monopole with T-shape ground planes

Another design approach consists of realizing the same ground plane shape as the antenna element, with almost the same dimensions (Figure 5.20) [ROB 05] [SIB 04]. Consequently, the line no longer has a ground plane which can be regarded as infinite. The latter thus play only the role of an image for the antenna element: the doublet thus formed is a dipole antenna with a whole part. For this type of structure, the feeder forms an integral part of the antenna, and its length influences the bandwidth of the antenna which behaves like a dipole. The operation of this type of structure partly explains the influence of the coplanar waveguide lenght on the matching frequency of the antenna observed for the circular monopole.

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Figure 5.20. Trapezoidal antenna with symmetrical ground planes

The main shapes of printed UWB monopole antennas fed into a coplanar are inspired by their 3D equivalent (Figure 5.21). The most represented are the triangular, round and square elements like their respective derivatives (elliptic, trapezoidal, rectangular, etc.). They are generally fed by a coplanar waveguide or microstrip line. Their performances are quasi-identical to their counterparts with a ground plane in terms of bandwidth, while their radiation patterns will of course be less symmetrical (in elevation compared to an element with infinite ground plane or in the azimuth plane compared with a monopole with a symmetry in revolution).

(a)

(b)

(c)

Figure 5.21. Main UWB monopoles with ground planes and their equivalents fed with coplanar waveguide

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It is the same for their aptitude to radiate short impulses: these antennas will present a less stable impulse response according to directions but will remain not very dispersive like the monopoles from which they result. In certain cases, the miniaturization of the ground planes can also involve currents reflections, and thus distortions of the radiated pulse. The round or elliptic printed monopoles (Figure 5.21a) present the widest bandwidth. Like their ground plane equivalent, the most important dimensioning parameters of these antennas are their diameter, their elliptic ratio and the distance between the line and monopole. The diameter d can be fixed according to the following formula:

d= 4

λ εr +1

[5.18]

2

with εr the dielectric permittivity of the substrate, and λ the wavelength at the lowest operating frequency. It should however be recalled that the length of line influences the matching of the antenna. This result is thus valid for dimensions of the feeding line close to d. By preserving the constant diameter horizontally, the ellipticity ratio is changed by decreasing or increasing the vertical radius r2. The best results in terms of bandwidth are obtained for values of r2 /r1 greater than 1, with in particular a notable reduction at the lowest matching frequency. However, the bandwidth decreases in high frequencies for too great ellipticity rates. A rate equal to 1,3 seems to constitute an optimal value. It is the same for the distance between ellipse and feeding line; a distance equal to λ/20 constitutes an optimal length to obtain the best results in terms of bandwidth. The triangular monopoles (Figure 5.21b) are the most well-known structures since they derive directly from the bowtie antenna from which they follow the rules of dimensioning. Their lowest matching frequency is fixed by the height h of the triangle (approximately λ/4 at this frequency), while a flare angle of 120° confers to them an input impedance equal to 50 Ohms. Like the monopoles with ground plane, the printed square structures (Figure 5.21c) fed with coplanar waveguide do not have very wide bandwidths (typically 66%). However, they have the great advantage of being the most compact structures

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for an identical lowest matching frequency. They remain interesting structures, in particular as basis element for develop miniature UWB antennas. 5.3.5. Performance improvement techniques of elementary UWB antennas

Since the 2000s, printed elementary antennas aroused a growing interest because of their low costs, their integration facility and their omnidirectional radiation. This interest was followed by many more or less effective innovations which mainly related to the improvement of the antennas matching on increasingly wider bandwidths. However, radiation qualities, evolution of the diagrams with the frequency, as well as the time domain behavior of the antennas were also studied. The geometry of a printed UWB monopole fed into a coplanar can be schematically separated into three parts (Figure 5.22). First of all feeder of the antenna composed of the line which is used to guide the electromagnetic wave of the connector to the radiating element. Then comes the transition zone, composed of the lower part of the antenna element, whose main function is the impedance matching of the element with the line. Lastly, the third part known as the radiation part is composed of the upper part of the antenna. A variation of each part has, of course, repercussions on all of the antenna performances; on the other hand these names summarize the main influence of each area. The transition part thus will be mainly studied in order to improve the bandwidth, while the radiation part will be optimized in order to obtain more stable diagrams and a better time domain behavior.

Radiation part

Transition part

Feeder

Figure 5.22. Main parts of a printed monopole

In spite of the number of antennas matched on several octaves, the interest in searching for techniques able to increase their bandwidth did not cease. Thus, a good

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part of the efforts focused on the transition part (Figure 5.22). For example, the “diamond” antenna is a bowtie-like antenna whose triangles have been turned over and are fed by the base [SCH 01b]. The bandwidth of a conventional diamond antenna is of approximately an octave, i.e. 66%. It is shown in [LU 04] that the smooth shapes in the transition part (Figure 5.23b) of the antenna increases its bandwidth by approximately 20% but only at highest frequencies. Better results can be observed by reusing this principle until the transition part becomes circular or elliptical (Figure 5.23c). The antenna thus obtained is a PICA (Planar Inverted Conical Antenna) whose bandwidth can exceed 160% [SUH 04]. It should be noted that the impedance matching is improved only at highest frequencies. The dimensioning rules being dependent on the desired lowest matching frequency, they are the same as for a triangular element.

λ/4

(a)

(b)

(c)

Figure 5.23. Evolution of the transition part from a diamond antenna to a PICA antenna

For a triangular monopole or a bowtie antenna, the degrees of freedom are less important. The process then consists of introducing notches or “steps” into the transition part in order to more finely regulate the impedance matching of the antenna [LEE 02], [CHO 06]. However, these techniques remain relatively delicate to control and the improvements in terms of bandwidth are, in the case of triangular elements (Figure 5.24a), relatively limited. For the rectangular elements on the other hand (Figure 5.24b), the results obtained are very interesting; the cutting of notches then makes it possible to optimize the coupling between the aerial element and its ground plane in order to match the antenna on a bandwidth up to four times higher [SU 04]. Other techniques were used in order to improve the impedance matching of rectangular elements on wide bandwidths. Thus, cutting in bevels of the transition part (Figure 5.24b) appreciably decreases the variations of impedances of the antenna [AMM 03]; but, because of a greater number of parameters of tuning, the cutting of notches remains more effective.

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(a)

(b)

Figure 5.24. Optimization parameters of the coupling between the antenna and the ground plane

The feeding technique, in particular of the rectangular elements, was also studied. Indeed, the use of dissymmetrical feeding makes it possible to excite higher modes in the square monopole. Thus, Ammann [AMM 04] shows that by feeding the monopole, not in the center, but at a quarter of its width, it is possible to increase its relative bandwidth from 70% (for the initial monopole) to 130%. This improvement is of course made with the detriment symmetry radiation pattern, in particular at the highest frequencies [WU 05]. On the basis of this principle, Wong [WON 05] proposed a new feeding technique with a three-pronged fork able to bring a uniform distribution of current on the monopole, to control the coupling with the ground plane, and consequently to considerably increase its bandwidth. With this intention, the three access points must be positioned regularly on a distance from approximately λ/12 at the lowest frequency, the distance between antenna and ground plane must be optimized. Fed in this way, a square element reaches a bandwidth of almost 160%. λ/5

λ/5

λ/5

Figure 5.25. Feeding techniques allowing us to increase the bandwidth

Other more conventional techniques of increasing bandwidth can also be used for elementary UWB antennas. Thus, it is possible to add new resonant lengths inside the structure, whose resonance frequencies will combine with the bandwidth of the antenna. These new resonances can be created with short-circuit [AMM 03], or with slots for which the length must be half-wavelength at the desired frequency [VAL 05]. The use of fractal-type slots can also be considered, giving the antenna a multiband behavior [PUE 98]. The insertion of an open, fractal-type slot in a triangular monopole was introduced by Fortino [FOR 04]. The elements not being connected, the higher triangles become parasitic elements; the optimization of the coupling with the fed triangle increase its bandwidth at the same time with the

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lowest and highest frequencies [FOR 08]. Other types of parasitic elements can also be used: a parasitic ring surrounding a simple quarter-wavelength monopole makes it possible [KAN 07] to cover a bandwidth higher than 80%.

Figure 5.26. Addition of new resonance frequencies

The elementary antenna radiation improvement techniques primarily consist of trying to preserve an omnidirectional diagram over the entire bandwidth. However, if the antenna can be considered as electrically small at the beginning of the bandwidth and thus radiate in an omnidirectional way, its directivity increases as the frequency increases. It is then advisable to improve the geometry of the antenna radiation part. The goal is then to avoid the reflections of current at the end of structures by choosing smooth shape or circles arcs instead of sharp truncations. Applied to triangular monopoles [BRO 52], these modifications do not have an influence on the bandwidth of the antenna, but this then radiates like an element of greater dimension and is thus less directive. In order to increase the half-power beamwidth of the radiation pattern of a printed monopole, i.e. to decrease its directivity, it is also possible to modify the geometry of its side ground planes by folding them up [FOR 08]. In this way, the radiation becomes at the same time more dipolar and less dependent on the frequency than with ground planes perpendicular to the line of transmission. Figure 5.28 compares the measured radiation patterns of a triangular monopole with T-shape (a) or folded (b) ground planes. Various forms can be possible according to whether we prefer to privilege the compactness of the structure, its radiation, or its bandwidth. In all the cases, the ground planes must be preserved constant to avoid antenna mismatch.

β

Figure 5.27. Folding up effect of the side ground planes on the radiation patterns

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(b) 5

0

0

-5

-5

G ain (dB )

G ain (dB)

5

-10 -15 -20

3 GHz 5 GHz 7 GHz 8 GHz

-25 -30

-10 -15 -20

3 GHz 5 GHz 7 GHz 8 GHz

-25 -30

-35

-35 0

φ (°)

45

90

-90

-45

0

φ (°)

45

90

Figure 5.28. Measured radiation pattern in the plane of the antenna without the folded ground planes (a) and with the folded ground planes (b)

It is usual to see, for radar applications [THA 01], antennas with a cavity in order to avoid back radiation. This technique is however used at the expense of radiation efficiency, which is then divided by two, the totality of the back radiation being absorbed in the cavity. The improvement of the time domain behavior of the antennas is closely linked to the radiation optimization. To a lesser extent, a very wide bandwidth can also improve the impulse response of an antenna. In this sense, the techniques previously described, in order to increase the bandwidth or the stability of the radiation patterns are also applicable. Thus it is shown in [FOR 08] that the fold of the side ground planes for a triangular monopole, associated with a work of adaptation optimization, made it possible to appreciably decrease the parasitic oscillations at the time of the radiation of short pulses and this, in the two main radiation planes. However, these two conditions are necessary but not sufficient. Another wellknown example is the log-periodic antenna which is at the same time very wideband, and for which the radiation patterns are independent of the frequency. It however remains a very dispersive antenna, with the impulse response which has very important parasitic oscillations as we saw in Chapter 3. A third paramount condition is thus necessary: the antenna must present a quasiinvariant phase center with the frequency set not to distort and spread out the radiated pulse. With this intention, the antenna should not change operating mode on the bandwidth, and the active region taking part in the radiation should not strongly depend on the frequency. Thus, the addition of resonant lengths can cause dispersion according to their location on the antenna.

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In the case of structures presenting good time domain performances intrinsically, the impulse response can still be improved by avoiding the reflections of current which can pose a problem, in particular in lowest frequencies. Thus, the bowtie antenna can constitute, from its compactness and its low costs of realization, a good base for the radiation of very broad band impulses; by applying a resistive load at the end or along the structure, the stability of impedance as well as the bandwidth are improved. That especially avoids the reflections of current at the end of structure and still improves the impulse response of antenna [SHL 94]. However, these improvements are made at the price of important losses in the resistive load, decreasing the radiation efficiency by the same amount. Other more innovative solutions were successfully proposed [YAR 04] by carrying out a capacitive loading of the antenna by discrete slots or elements. It is also possible to apply this principle to the ground planes in order to reduce their dimensions while avoiding reflections of current in the periphery. A monopole is presented in [WU 08] whose ground plane is associated with a resistive ring. 5.3.6. Directive elementary antennas

The omnidirectional antennas are matched to terminals for which no particular direction is privileged. On the other hand, directional antennas are beneficial when they are placed at the interface of two devices or are used in a sectorized topology of the local area network. For a terminal or an access point, they also authorize the implementation of angular diversity multi-sensors which makes it possible to gain in the budget link, to reduce the interferences, etc. A directive antenna can also be placed on the device which it is possible to request to the user to point in the desired direction. Even if they do not improve the budget link at transmission, because E.I.R.P is limited by regulatory agencies (see Chapter 1), the directive antennas reduce the parasitic radiations and thus power consumption. They also make it possible to limit the coupling with the electronic circuits which disturb the radiation characteristics of the antenna and the good performance of these circuits. Moreover, at reception, the use of directive antennas makes it possible to gain a few invaluable dB in the budget link and to thus increase the range of the system. The triangular F-probe antenna (FFPTP) is one of the rare compact antennas available to date [LEP 05]. We have already shown its usual characteristics given in

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the frequency field in Chapter 2. It should also be highlighted that this antenna has excellent characteristics in the time domain (see also sections 3.5.1, 3.6.4 and 6.4.2). If we are interested in the radiated pulse with this antenna, we can note that the antenna does not deform the shape of the pulse [LEP 08].

Figure 5.29. Transmit pulse ( in the main lobe (simulation (

) into the antenna and radiated ) and measurement ( )

5.3.7. Antennas with progressive transition

Antennas with progressive transition are based on the idea that an antenna can be seen as a transducer between a waveguide or any different system of feeding and free space. Thus, these antennas are inspired by the lines of transmission whose characteristic impedance is constant on a very wideband. They widen then to gradually bring the structure to the impedance of the vacuum and to radiate the wave hitherto guided [SCH 03]. They are generally made of a line matched to the feeding system of the antenna, followed by a transition part comparable to an impedance transformer, which increases the characteristic impedance of the structure up to that of free space. This flare shape leads to a radiation aperture matched to the free space impedance whose geometry is designed in order to minimize the reflections at the end of the structure. 5.3.7.1. Antennas with progressive transition Tapered Slot Antennas (TSA) are planar structures which present a transition starting from a printed line or waveguide. They were introduced in 1974 by Lewis and Gibson with the Vivaldi antenna. They are generally made up of a line with slot widening until a final discontinuity according to a given profile. The profile of these

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openings can be various forms according to the radiation specifications or complexity given by the designer. Thus, the Vivaldi antenna or ETSA presents an exponential or elliptical transition profile, and is characterized by a weak cross polarization. However, other types of transitions have been studied like the linearly tapered slot antenna (LTSA), with broken line (BLTSA), or exponential thus constant (CWSA). LTS

CWS

Vivaldi

BLTS

Figure 5.30. Various types of TSA antennas

Originally, TSA antennas are fed by a line with a gradually widening slot, as described in Figure 5.30. As long as the wave is propagated in the narrow slot, in a substrate with higher permittivity than air, the fields remain confined in the slot and the antenna behaves like a very efficient line. By widening the slot, then by opening it at its end, the antenna becomes a genuine antenna radiating − via this aperture − an electric field parallel with the substrate plane.

Figure 5.31. Representation of an antipodal Vivaldi antenna

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Other feeding types are also used for TSA antennas. Indeed, certain achievements propose a transition starting from a microstrip line [NOR 03] which offers the advantage of being a unbalanced feeding which can be connected to a coaxial cable. The antenna is then printed on the two faces of the same dieclectric substrate (Figure 5.31). The ground plane side of the microstrip quickly narrows in order to form a section with a pair of strips, which is used as a transition and comes to symmetrize the structure. The radiating part is thus realized with the flare shaped slot formed between the two metallic parts on both sides of the substrate. This type of antenna is called the Vivaldi. TSA are very high performance structures. The interest in these structures lies in the absence of resonant dimension and in the printed technology which facilitates the realization of it. They are matched over very wide bandwidths of about 125% to more than 170% for the version with metal outgrowths [GUI 98]. Their radiation pattern is directive in the plane of the substrate and has a low level of cross polarization. Their directivity increases with the frequency and the gains reached by these antennas lie between 7 and 10 dB according to the types of TSA. These qualities associated with a good time domain behavior make them ideal candidates for radar subsurface applications, in CEM or metrology, because they can be associated in array and allow a weak distance inter-antenna. They can also be used as a primary source for wideband reflectors. 5.3.7.2. Teardrop antenna The teardrop antenna was introduced by Lindenblad in 1941 [SCH 03]. It concerns a transition starting from a coaxial waveguide widening gradually. Indeed, Lindenblad improved the idea of the sheathed monopole by adding to it a gradual transformation of impedance along its structure in order to increase its bandwidth further. Thus, starting from a coaxial waveguide, the antenna widens in a first part by keeping a constant D/d ratio (Figure 5.32), then is closed again at the apex of the central conductor. Lindenblad undertook an experimental study which showed the importance of the shape of this conductor on the antenna performances, in terms of bandwidth. An optimized version with an input impedance of 110 Ω is described in Figure 5.33.

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d D

Figure 5.32. Example of progressive transition

Figure 5.33. Cross-section of the Lindelblad teardrop antenna

5.3.7.3. Omnidirectional coaxial horn The coaxial omnidirectional horn antenna (Figure 5.34) was introduced by Brillouin in 1948 [SCH 03] and falls under the continuity of the research into antennas built starting from coaxial transitions. Indeed, in the same way as the teardrop antenna, this antenna is developed starting from a coaxial waveguide and its structure evolves in such a way that the D/d ratio remains constant (Figure 5.32). By adopting a gradual evasive form, and when D is sufficiently large at the end of the antenna, this radiates with few reflections at the end of the structure, and this on a very wide band.

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Figure 5.34. Representation of the omnidirectional horn resulting from [BRI 48]

Its radiation pattern is omnidirectional in azimuth, because of the revolution symmetry of the antenna, but Brillouin also developed at the same time a directive version of this antenna. 5.3.7.4. Techniques for improving the performances of antennas with progressive transition Less used in UWB communications, the antennas with progressive transition have not experienced the same great interest as the elementary antennas in recent years. However, they remain of great importance in applications for which the radiation quality and the gain take precedence over the overall dimension (radar, reference antenna, etc.). The improvement techniques of these antennas are thus mainly related to these last aspects, and to a lesser extent, their bandwidth size. Indeed, the antennas with progressive transitions are intrinsically wideband since the form of the transition is sufficiently linear. The form of this transition mainly makes it possible to optimize their bandwidth but its length and the design of its end also influences their radiation. One of the characteristics and the advantages of ETSA antennas comes from the fact that their radiating part is well separated from the feeding system placed upstream, and can be optimized for the radiation by affecting the matching quality a little. In order to obtain an effective radiation, the width of the slot at the end of the antenna must at least measure a half-wavelength at the lowest frequency. However, to preserve a small opening and a less overall dimension, while adapting the antenna to a lower frequency, an increase in the metal outgrowths of the radiant part of the antenna is proposed by Guillanton [GUI 00]. This modification is very interesting because it makes it possible to widen the bandwidth of the antenna towards the low frequencies while preserving its

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performances at the top of the bandwidth. It should be noted however that this solution alters the guiding behavior of the antenna and confers on it a dipole-like radiation pattern at low frequencies. Concerning the radiation patterns, it is sometimes useful to have an antenna with a great polarization purity. It is then possible to feed a TSA antenna starting from a stripline [GUI 00], the microstrip and its prolongation then being closed between two thicknesses of substrate each comprising a metal ground plane with identical geometry (Figure 5.42). This technique improves the polarization purity by symmetrizing the electric field in the slot, which theoretically makes it strictly parallel to the plane of the dielectric substrate. With the same aim, a cutting of the metallic radiating part outgrowths in “palm tree” also makes it possible to decrease the level of cross polarization [GAR 07]. 5.3.8. Horn antennas

Another category of antenna also known for its wide bandwidth properties is the horn antenna; this chapter could not overlook this family. Indeed, these structures supplied with rectangular or circular waveguides have intrinsically wideband properties, with impedance matching on relative bandwidths going from 50% to 180%. This characteristic is partly due to the fact that the widening of the waveguide towards a circular or rectangular aperture allows an efficient radiation of the signal in free space, in the same way as for the antennas with progressive transitions. In addition, the horn antennas have very good radiation qualities and in general present a great purity of polarization as well as a phase center almost invariant with the frequency. A behavior completely appropriate to the transmission of short impulses results from this, because their far from dispersive behavior enables them to accurately radiate the impulse transmitted without widening it by parasitic oscillations. Various types of horns were studied [BAL 05], but their optimization always related mainly to the quality of their radiation rather than over their bandwidth. Thus, the horn antennas are divided into two big families: horns with smooth walls (sectoral horn in the E-plane, in H-plane, pyramidal horn, conical horn) and ridged horns (“corrugated horn” [TEN 02]). In the case of the horns with smooth walls, the radiated field is the combination in amplitude and phase of the field radiated by the aperture and the fields diffracted by the ends of the antenna if dimensions of the aperture are not sufficient. The corrugated horns, on the other hand, make it possible to preserve a non-zero tangential electric field on the walls to produce a uniform illumination. This leads, when the aperture is circular, to diagrams with a symmetry revolution.

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In spite of these qualities, the horn antennas are not used in the applications in time domain communications because of their integration difficulty and their great directivity. Indeed, these antennas are 3D structures, with strong overall dimension since their dimensions are generally higher than a wavelength at the lowest frequency, which involves a consequent weight which limits their integration in a communication system. Their manufacturing costs are consequently relatively high, especially compared with a printed antenna. Lastly, their very directive radiation pattern cannot be compliant for applications where all space must be covered. Moreover, these structures were already largely studied in literature [BAL 05]. This is why this family of antennas is introduced but not detailed. 5.4. Miniaturization of UWB antennas

Miniaturizing an antenna is often a compromise between lowest size which we wish to obtain and the radioelectric performances which can be accepted. The reduction in the volume of an antenna is accompanied by an increase of the quality factor and thus a reduction in the bandwidth. For this, it is necessary to add a degradation of certain radiation characteristics. The antenna efficiency then becomes an essential criterion for its performances. Any antenna contained in a sphere of radius r< λ/(2 π) is considered electrically small. 5.4.1. General principles of antenna miniaturization

Many techniques are used to reduce the size of antennas: − substrates with high dielectric permittivity: we use substrates with high permittivity to reduce the wavelength in the antenna elements and consequently the size of antennas [ZHA 95]. The theoretical factor of miniaturization is close to((ε r+1)/2)1/2. However, the losses in material are crucial because the quality factor of the antenna will increase and its bandwidth will decrease; − new materials: magnetic (composites) and electric (ferroelectric, piezoelectric) with high permittivities [HAN 00]; − metamaterials: they are materials for the naturally non-available manufactured physical properties: - we can judiciously place a metamaterial under the aerial element with a permittivity and (or) a negative relative permeability [TRE 05], - or, we can build a material with high impedance surface, which then behaves like an artificial magnetic conductor. The waves radiated by the antenna are reflected by this surface without being in opposite-phase, as they are naturally with a metal conductor. It is then possible to place the antenna very close to the artificial

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magnetic conductor and to thus build a directive antenna with very low thickness [THI 09]; − insertion of short-circuits between elements and the ground plane: this is the principle of the PIFA antenna [HIR 02]: the distribution of the electric field of the first resonant mode of the cavity formed by a patch and the ground plane present a zero in the middle of the resonant length. By positioning a perfect electric wall at this place, the distribution of the electric field is not perturbed. An antenna then results, with a physical length λ g/4, known as a quarter wavelength antenna; − folding in volume (geometry filling space) [ZAI 99]; − folding surfaces (creation of meanders): we preserve the electric length while optimizing the overall total surface dimension [KOS 89]. It should be noted that the creation of meanders generates slots on the resonator and thus on the face-to-face metal parts creating the capacitive effects; − insertion of slots: folding on the surface creates open slots. The introduction of the open slots (of various forms) can also lengthen the path of the currents and consequently obtain elements of low dimension [SKR 01]; − fractal shape [BEN 05]; − use of self-inductive or capacitive elements for localized purposes: a capacity positioned in the end of an aerial element makes it possible to artificially increase the electric length of the antenna, and thus to decrease the frequency of resonance. This capacity can be realized, for example, with a vertical translation of the element towards the ground plane [CIA 04]; − use of lumped components: lumped components can also be placed on the antenna. For example, to replace a short-circuit with a low resistance [WON 02]. All these techniques can, obviously, be combined until even more important size reductions are obtained. It also should be noted that the integration of the antenna in its environment, such as for example the use of part of an ground plane support, can take part in the reduction in the required overall dimension total. 5.4.2. Miniaturization problems of UWB antennas

It is of course more difficult to apply a miniaturization technique to a UWB antenna since its performances are required on the entire bandwidth, whereas these principles are generally in a close relationship with the wavelength. Generally, we try to decrease the low frequency of a very wideband antenna without degrading the performances at the highest frequencies.

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Lastly, it is important to note that the McLean limit remains valid in UWB, but the Chu-Harrington limit does not [SCH 03]. 5.4.3. Miniaturization techniques applicable to UWB antennas

In the following are a certain number of examples of UWB antenna miniaturization techniques. This list is not exhaustive because many of the research tasks on the subject are still underway: − use of materials with strong dielectric permittivity [KRA 05], new materials and metamaterials: the same principle and the same problems (low efficiency) as for narrow band antennas. This time, it is necessary to match the behavior of these materials to the wideband antennas; − use of reactive loading without discrete elements (Figure 5.35) [KRA 06];

Figure 5.35. Inductive strip of a spiral antenna [KRA 06]

− use of discrete loads: this method is less interesting because the increase in the losses remains problematic;

Figure 5.36. Insertion of discrete loads

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− suppression of a dimension: the principle is to pass from a dipole (λ /2) to a monopole (λ /4) with a ground plane and to integrate the ground plane in a planar feeding system (microstrip, coplanar, tri-plate or ground coplanar);

Figure 5.37. Evolution of an antenna 3D towards an element 2D

− conservation of constant perimeter of monopoles: to optimize the overall dimension by maintaining the perimeter of the constant monopole. However, it should be noted that the performances of the antenna in radiation do not remain constant and depend on the shape of the element;

Figure 5.38. Modification of the monopoles geometry

− optimization of the transition part: work within the ground plane (realization of the notches) at the level of the access to the monopole makes it possible to reduce the overall longitudinal dimension of the antenna [TOU 07];

Figure 5.39. Optimization of the profile of the ground plane on the level of monopole feeding

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− folding the surface: this relates primarily to the ground plane [FOR 08];

Figure 5.40. Optimization of the surface overall dimension by folding

− folding in volume: this can either relate to the ground plane of the antenna or the element. In Figure 5.41 we find an antenna with widened geometry in order to gradually matched to the impedance of free space then folded, [DEM 06a], and an antenna whose radiating element end is folded in order to gain in compactness [RUV 06];

Figure 5.41. On the left: folding the ground plane. On the right: folding the element

− shorted-circuit elements: insertion technique of short-circuits between aerial elements and the ground plane is usually applied to antennas weak bandwidth and much less to UWB antennas (Figure 4.40 on the right) [RUV 06]; − electromagnetic antenna: the electric combination dipole − magnetic dipole led to elements of lower dimensions [KWO 08]. 5.5. UWB antennas for surface penetrating radars 5.5.1. GPR and SPR technologies Since approximately 2002, with the growing interest for UWB technologies principally for high-data rate-short-range communications (OFDM, impulse radio) or for localization applications, the research and development of UWB antennas

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have seen a very intensive activity. Many works have been carried out in particular concerning the miniaturization of antennas imposed by the compactness of these new communicating objects. However, the use of UWB antennas goes back several tens of years beyond this, first of all, in metrology and characterizations in EMC (electromagnetic compatibility), and then with radars for the detection of targets located behind an interface . This type of radar, well-known under the name GPR (Ground Penetrating Radar) or SPR (Surface Penetrating Radar), is a system which makes it possible to locate targets hidden under the surface of the ground or inside an opaque medium. There are many and varied applications. The most important deal with the detection of anti-personnel and anti-tank mines, non-destructive testing in civil engineering (monitoring of bridges, buildings, roadways, detection of pipes, etc.), the detection of cavities in archeology, the monitoring of ground and rocks in geophysics or more recently in the field of civil security to rescue people trapped under a collapsed building after an earthquake. Two main technologies exist for this type of radar. The first one is based on impulse electronics for which the main block-systems are a) a pulse generator, b) UWB antennas, c) sampling and analog to digital conversion of the measured signal. The second technology is based on the sweeping frequency of a UWB source. The receiver gives access to the magnitude and the phase of the measured signals over a frequency band which can be controlled by the user. Each one of these technologies has advantages and drawbacks in terms of acquisition time, dynamics of measurement, signal-to-noise ratio, and image resolution. For more information on these systems, see [DAN 04]. 5.5.2. Design of antennas for SPR radars Frequencies used for SPR radar applications vary roughly from one hundred MHz up to ten GHz. The choice of this very large frequency range is dictated by the variety of applications presented previously. Indeed, if we wish to detect deeply hidden targets (several tens of meters), we need to use low frequencies (100-500 MHz) which will ensure good penetration of the electromagnetic waves if the medium is not too lossy. On another side, if we seek to detect targets hidden just a few centimeters under an interface or for example to measure the thickness (2 to 3 cm) of the first bituminous layer of a roadway, we can use higher frequencies (500 MHz to 10 GHz). However, we must choose the frequency range (and more particularly the lowest frequency) according to the configuration of the radar scene. It is very beneficial to work with the largest frequency range because it will guarantee the best resolution for the images obtained after the processing of the data. The antennas should not be too dispersive to avoid the time-spread of the radiated

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pulse. To obtain this behavior, we use travelling wave antennas which have the advantage of great radiation efficiency without important spread-time of the radiated pulse. We can also have recourse to resonant antennas (for example: dipole) which are intrinsically narrow band antennas, but for which we widen the bandwidth by loading the antenna with resistors or absorbing materials. We thus obtain not too dispersive UWB antennas, but with a poor radiation efficiency (most of the power injected into the antenna is dissipated in resistors or absorbing materials). It is also desirable to maintain the radiation pattern and the gain as stable as possible over the entire bandwidth. The frequency-independent antennas are a solution regarding these previous requirements, but present the main drawback of being dispersive. It is thus necessary to make a compromise between all these requirements. For applications using low emission power ((a few ten of mW to several W), it is preferable to use a less dispersive antenna with the highest radiation efficiency. For more detailed information, refer to the following works and articles [EID 98], [CAR 99], [CAR 98], [YAR 00], [MAR 00], [SCH 00], [EID 00a], [EID 00b], [DAN 04]. The antennas presented in the following articles deal with the use of a progressive transition based on the principle of the Vivaldi antenna [GIB 79], [GAZ 88], [LAN 96]. 5.5.2.1. Planar antennas in microstrip technology Figure 5.42 presents an exploded view of the generic antenna. The progressive transition has an exponential profile. The antenna includes 3 levels of metallization to ensure a high quality of the linear polarization. An air substrate (a foam) is used to minimize the losses inside the antenna in order to guarantee an optimal radiation efficiency. The layers of metallization are printed on epoxy resin plates with a thickness of 100 microns for the inner layer (line and progressive transition) and a thickness of 0.7 mm for the external layers (ground plane and progressive transition). The layers of epoxy provide only the rigidity of the structure and do not have an influence on the electromagnetic behavior of the antenna. The bandwidth of this antenna for a VSWR less than 2 is of 20:1 between the highest and the lowest operating frequencies. The main lobe of the radiation pattern is along the Y axis. Linear polarization is along the X axis. From this generic antenna working from 1.5 GHz up to more than 20 GHz, we build by homothety another antennas working respectively from 100 MHz and 500 MHz. At these frequencies, the use of foam as a substrate makes it possible to considerably reduce the weight of the antenna. The figure below presents ETSA (Exponential Tapered Slot Antennas) antennas manufactured for SPR UWB applications between 150 MHz and more than 20 GHz. We distinguish two types of profiles. The profile of ETSA, ETSA A4 and ETSA A0 antennas was optimized to minimize their size.

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ε

ε

Figure 5.42. ETSA-type antenna

The antenna presents a dipole-like behavior between fb and 2.fb (fb: low cut-off frequency), and a behavior of traveling wave antenna beyond approximately 2.fb. This results in a variable gain between 3 and 11.5 dB on the totality of the bandwidth and a larger dispersion of the radiated pulse due to the dipole behavior (in the case of a pulse which covers the entire bandwidth). The other profile (ETSA A5 and ETSA A3 antennas) is optimized in order to maintain a traveling wave behavior from the lowest operating frequency. This type of profile gives a more directive antenna with an optimal impulse response (see Figures 5.44 and 5.45 where the impulse response of an ETSA A3 antenna is compared with the impulse response of a UWB horn).

Figure 5.43. ETSA antenna developed with the LEAT

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Figure 5.44. Impulse response of ETSA A3 antenna

Figure 5.45. Impulse response of a ridged horn

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5.5.2.2. Antennas for impulse radar For impulse radar systems using signals of great amplitudes (>10 Kv), tapered antennas are also interesting solutions. Taking into account the levels of voltage concerned, matching circuits are necessary to connect the antennas to the pulsegenerators. Many achievements have been carried out by the department OSA of XLIM Laboratory. Below, various antennas are presented used in SPR radar systems.

Figure 5.46. Valentine antenna − 300-3,000 MHz (XLIM/OSA-Europulse)

Figure 5.47. Array of 4 Valentine antennas − 300-3,000 MHz (XLIM/OSA-Europulse)

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Figure 5.48. Cisor antenna 100-2,000 MHz (XLIM/OSA)

Figure 5.49. Dragonfly antenna − 250-3,000 MHz (XLIM/OSA - Europulse)

Chapter 6

Antenna-Channel Joint Effects in UWB

6.1. Introduction The ultra wide band (UWB) signals undergo a series of deformations between the output of the emission stage and the input of the reception stage of a radio link. These deformations are due to the antennas at the two ends of the link, but also to the radio channel because of the multiple events (diffraction, reflections, absorptions, etc.) which the waves undergo between the two antennas. They appear in the temporal domain as well as in the spectral domain and take all the more importance as the spectral width is large, or in a dual way, as the signals simultaneously contain fast and slow variations. The waveform of the emitted signal thus plays a big role in the deformations which it can undergo, as do the antennas and the propagation environment. The shape of the received signal is affected differently according to the various cases, and this very much influences the quality of detection. It is obvious that the architecture of the transmitter, through the emitted waveform, and that of the receiver, through the nature of the detector, impact the performance of the radio link. This results in a larger complexity than for a narrow band-based system, for which the antenna-channel block behaves like a simple dephasing attenuator. This complexity in the frequency or temporal domains makes the serialization of the antenna-channel-antenna blocks more difficult and less intuitive, and does not allow us to immediately evaluate the quality of the link by simple empirical engineering rules. It is nevertheless possible to define the effective performance of an antenna in statistical terms, but this depends at the same time on Chapter written by Alain SIBILLE.

Ultra W ide Band Antennas Edited by Xavier Begaud © 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

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the emission/reception architectures and on the angular and temporal characteristics of the channel. It is the goal of this chapter to show and analyze these issues.

Direct path

r

Indirect path Port 2

Port 1

Transmission channel

ri1

ri 2 Scatterer

Figure 6.1. Evolution of the signals during the transmission

6.2. Recalls on the UWB radio channel The data communication channel includes all the physical elements implied in the transmission of signals between the moment when they leave the transmitter and the moment when they reach the receiver. In a radio link, this concerns the radiating and receiving antennas but also the free space and all the objects interacting with the radio waves. The UWB radio channel is the subject of the book [PAG 08], which was recently published. Here we recall the essential characteristics of the UWB channel, the emphasis being put on the aspects discussed in the following sections. We limit ourselves to the case of the static channel, i.e. when neither the transmitter, receiver nor the scatterers move or evolve over time. Figure 6.1 schematically shows the deformation of the signals in the transmitting antenna, then the propagation in the medium including both free space zones and scatterers, and finally, the superposition and the deformation of the signals within the reception antenna. In the case of hypothetical ideal antennas (isotropic, non-dephasing, unit gain at any frequency), the transfer function in the frequency domain between the amplitude of the complex wave emitted at the input of the transmitting antenna and that at the receiving antenna output, is written1: H

21 ( f ) = − j

λ 4π

⎧⎪ e − jkr ⎫⎪ e − jkri 2 e − jkri1 +∑ H i ( f , Ω1i , Ω 2i ) ⎬ ⎨ 4π ri 2 4π ri1 ⎪⎩ 4π r ⎪⎭ i

[6.1]

1. Equation [6.1] only takes into account simple scattering, but it can easily be generalized to multiple scattering. It also assumes that the scatterers are in the far field of the antennas.

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215

In this expression, we separated the free space path (length R), and the scattered paths of lengths ri1 and ri2. The term Hi ( f , Ω1i , Ω2i ) which is homogeneous with a length, describes the bistatic scattering of the incoming plane wave towards the spherical wave radiated by scatterer i. Generally speaking, this term affects the amplitude and the phase differently, according to the frequency. Ω1i and Ω 2 i represent in compact form the angles (azimuth, elevation) of departure and arrival for the scatterer of numbering i.

Figure 6.2. Main propagation mechanisms (according to [PAG 08])

Notably, expression [6.1] carries the seed of its own limitation since the polarization is not expressed, meaning it is usable for a single polarization. Its generalization with an unspecified polarization and non-ideal antennas leads to2:

2. We will draw attention to the conventions of the angular representation. For example, if one antenna transmits in the direction (ϕ1 , θ1 ) , the other receives in direct view from a source of direction (ϕ 2 , θ 2 ) = (ϕ1 + 180 °,−θ 2 ) .

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⎧ e− jkr T e− jkri 2 e − jkri1 ⎫ T Ω ⋅ Ω + (f , ) (f , ) H H ⎪ ⎪⎪ 2 1 2 1 λ ⎪ 4πri2 4πri1 ⎬ H 21 (f ) = − j ⎨ 4π r i 4π ⎪ ⎪ T t HT ⎪⎩ 2 (f , Ω 2i ) ⋅ H i (f , Ω1i , Ω 2i ) ⋅ H1 (f , Ω1i ) ⎪ ⎭

∑

[6.2]

Here H i ( f , Ω 1i , Ω 2i ) is now a matrix, which connects any of the two components of the incoming wave field to any of the two components of the scattered field. H1T and H T2 are the vector transfer functions in emission (see Chapter 3) of each antenna. We notice in equation [6.2] the complexity contained in the transfer function of the UWB radio channel, since the antennas as well as the scatterers contain a frequency dependence which is specific to them. If we are interested in the deformation of the signals, the Fourier transform h 21 (τ ) =

∞ ∫− ∞ H 21 ( f )exp (2 jπfτ )df gives access to the temporal representation of

this transfer function, which we also call the (real valued) channel impulse response: ⎧ r +r ⎛ δ ⎜τ − i1 i 2 ⎪ c δ ′ −1 (τ ) ⎪ 1 h T (τ , Ω ) * h T (τ − r , Ω ) + ∑ ⎝ 2 1 2 1 *⎨ h 21 (τ ) = 4πri1ri 2 c 4π r i c π ⎪ Tt T ⎪ *h 2 (τ , Ω 2i ) * h i (τ , Ω1i , Ω 2i ) * h 1 (τ , Ω1i ) ⎩

⎞⎫ ⎟⎪ ⎠⎪ ⎬ [6.3] ⎪ ⎪ ⎭

The fundamental integration operator δ ′ −1 (τ ) comes from the reciprocity relation between transmission and reception (see Chapter 3). In the frequency domain this operator results in the λ term factored in equation [6.2]. A strong consequence is the decrease of the transfer function with f, in other words a lower contribution of the high frequencies in the received signal. Each path contributes to a propagation delay and a free space attenuation between two successive events of emission, reception or scattering. The total impulse response is a sum of terms which involves the convolution between the impulse responses of the antennas in the adequate directions, and possibly with that of the scatterer. In UWB, this observation is essential, since it means that the received signal will be the result of a composite deformation by these various elements. We could write an expression (with complex values) equivalent to [6.3] in “baseband” around a carrier f0, in the form h 21 (τ ) exp(− 2 jπf 0τ ) [PAG 08]. However, although it possible to get rid of the fast temporal variations stemming from the narrow band carrier, it turns out to be of more debatable interest in the ultra wide band case when the relative band exceeds several tens of percent. It is commonplace in experimental studies of the radio channel to use “neutral” antennas with respect to the propagation phenomena. In the narrow band, instead of fully isotropic radiators which are not permitted by the laws of electromagnetism it

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217

is conventional to use omnidirectional antennas which do not privilege any direction in the azimuth plane, and have a dipolar-type radiation pattern in elevation. At a given frequency the antenna contributes only by one constant dephasing, without any impact on the analysis of the results. In UWB, we also often use omnidirectional antennas but both the frequency variation of the gain in the azimuth plane and the non-linear frequency dependence of the phase are difficult to avoid. Ideally, it would be necessary to calibrate the antennas in all radiation directions, then to post-process measurements to eliminate the antenna contribution. Unfortunately this would require determination of the arrival and departure directions of the channel paths, which is an enormous and impractical effort in UWB. The calibration can only be imperfect, based for example on the assumption that the antenna contribution is independent of the direction. Assuming perfect calibration, we extract from measurements the data relevant to the propagation between the position of the transmitter and that of the receiver. However, taking into account the two possible polarization states on each side, we need four independent measurements to obtain complete information on the propagation. In the case of an ideal antenna calibration, we then obtain the specific propagation transfer function: h

21 (τ )

=

1 4π r

r

δ (τ − ) + ∑ c

i

1 4πri1ri 2

h i (τ −

ri1 + ri 2 c

, Ω1i , Ω 2i )

[6.4]

being aware that the integrator was built-into the vector transfer function in reception (Chapter 3). h 21 (τ ) is a matrix which can for example be written in the ⎛ hθθ (τ ) hθϕ (τ ) ⎞ 21 21 ⎟ by using polarizations ϕ and θ. The current form h 21 (τ ) = ⎜ ϕθ ⎜ h (τ ) hϕϕ (τ ) ⎟ 21 ⎝ 21 ⎠ practice requires that the impulse response of the propagation channel be rewritten in the form of a sum of terms for which each one represents a radio wave path, of amplitude Ai and delay τi , between the transmitter and the receiver: h (τ ) =

∑ Aiδ (τ − τ i )

[6.5]

i

This form is not correct in the general case, because it neglects the temporal deformation undergone by a scattered wave. However, it is always possible to resort to [6.5] by extending the number of paths towards infinity and assimilating the discrete sum to an integral, since the identity h (τ ) = ∫0 h (τ i )δ (τ − τ i ) dτ i is always true. Expression [6.5] can thus be seen as a generic radio channel model, for which the number of effective paths to retain in the sum depends on the desired accuracy. Thus a discrete time channel impulse response (tap model) simply taking into ∞

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account the Shannon criterion can be considered sufficient for the simulation of a wireless communication channel. In this case, τi will take all its discrete values resulting from this sampling, and Ai will represent the amplitude of the impulse response at time instant τi.

Figure 6.3. Example of UWB channel response (on the left LOS, on the right NLOS)

Starting from a model of a generic channel like that of equation [6.5], it is the

{

}

retrieval of the parameters Ai , τ i from experimental data which will finalize the model. This is a typical parametric estimation problem, for which one of the classical difficulties is the determination of the model order, i.e. the number of independent parameters. The more parameters we accept, the better the accuracy is at the price of the extraction and use complexity. In the present case, increasing the number of parameters amounts to increasing the number of paths, which enables better temporal resolution and fidelity of the signal shape. We thus expect the number of paths to grow with the considered band width. This was indeed shown in [SAA 07], where it was discussed that the number of degrees of freedom depends on the type of channel and increases rather sub-linearly with the band width. Figure 6.3 is an example of impulse channel responses measured between 2 and 10 GHz (LOS: with line of sight between the transmitter and the receiver, NLOS: no line of sight). It is clear that the complexity of such a response is important, and that precise modeling will require a large number of parameters to be estimated (several hundred). 6.3. Impact of the channel on the performance of UWB systems The channel is traditionally considered by communication systems designers as a cause of performance degradation because of various harmful effects such as fading or inter-symbol interference. In UWB this remains true, even if the characteristics of UWB communication techniques somewhat modify common intuition. Because of its great spectral width, the UWB signal has a strong variability of its temporal shape. Adequately taking into account this variability in the architecture of transceivers is highly likely to improve their performance, at the price of an increased complexity.

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219

Many studies have been devoted to the design of such architectures. In general the performance-complexity trade-off aims at reducing the latter rather than reaching a high performance. On the one hand, the great bandwidth forces us to call upon the latest generations of semiconductor technologies, which are expensive. In addition, the high performances often go hand in hand with high power consumption. These two disadvantages poorly comply with the anticipated requirements for anticipated UWB applications (see Chapter 1). We thus have to expect and accept suboptimality of UWB transceiver architectures. We traditionally distinguish architectures operating in the frequency domain from those operating in the temporal domain. The first are typically based on an OFDM (orthogonal frequencydivision multiplexing) modulation scheme. Note that this is a modern variant to the historical frequency multiplexing scheme, based on the discrete inverse Fourier transform and implemented very effectively in the form of the fast Fourier transform. The orthogonality of the various emitted frequency components allows their separation by numerical Fourier transformation within the receiver, which allows a dense spectral use by avoiding the old complication of analog filter banks. In the case of the ultra wide band, OFDM modulation is at the basis of the ECMA368 standard on which many equipment suppliers base UWB products. In addition to the traditional OFDM modulation inside a given sub-band, this standard imposes repeated hops from one sub-band to another. This hybrid form of OFDM and frequency hopping can thus be coined multiband-OFDM (MBOFDM).

Out

Out

Figure 6.4. Coherent architecture (top), incoherent (middle), MBOFDM (bottom)

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Architectures operating in the temporal domain can take two main forms: coherent architectures which take the shape of the signal into account, and incoherent architectures which do not. In the reception stage the first use either a matched filter, or a correlator based on a reference waveform. The second aims at detecting the received energy, irrespective of the waveform. They are thus less performing, since they exploit only part of the information available. We can therefore expect a particular sensitivity of coherent architectures to the shape of the received signals, and thus to the radio channel. This is what will be discussed in the following sections. 6.4. Effective antenna performance in an ideal channel 6.4.1. Introduction As has been explained in the previous chapters, the fundamental characteristic of UWB systems is the strong deformation that the signal undergoes at the time of its passage through the various parts of the channel. From the point of view of the radio channel, this deformation is altogether the result of the radiating and receiving antenna distortion and of the propagation channel distortion. The signal which arrives at the receiver is thus very different from the signal at the radio frequency transmitter output, in a way which depends both on the antennas and the propagation between antennas. This raises a difficulty compared to a narrow band system, for which the waveform is a moderately deformed sinusoid. Taking into account the dispersive character of the channel will be the subject of sections 6.5 and 6.6. Here, we will analyze the joint effect of the two antennas in free space, i.e. for a non-dispersive ideal channel. By again starting from the expressions of section 6.2, we can write in this case: H

21 ( f ) = − j

h 21 (τ ) =

1 2πrc

λe − jkr T H 2 ( f , Ω 2 ) ⋅ H 1T ( f , Ω1 ) 4πr r

δ ′ −1 (τ )* h T2 (τ , Ω 2 ) * h1T (τ − , Ω1 ) c

[6.6]

[6.7]

This last equation shows in particular the role of the convolution of the antenna impulse responses on the received signal. From the point of view of the engineer, the convolution is an operation that is difficult to represent mentally, particularly if the impulse responses of each antenna have a certain complexity. However, for a signal processing architecture operating in the temporal domain, the shape of these signals plays an obvious part and the deformation brought about by the antennas will inevitably impact the detection performance. To try and appreciate these effects, the

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221

analysis below relates to the behavior of an antenna alone and aims at defining radiation patterns which take into account the effect of the antenna dispersion, according to the architecture. 6.4.2. Radiation patterns for various architectures We consider the architectures depicted in Figure 6.4, where rec(t ) is the signal at the antenna output after amplification. Coherent architectures make use of a waveform reference ref (t ) with which the received signal is correlated and synchronized before detection. Incoherent architectures multiply the signal with itself before integrating it over its total duration. An MBOFDM architecture carries out a filtering by sub-band sequentially, then computes the inverse FFT to separate the subcarriers before parallel detection. We limit ourselves here to the case of an ideal reception antenna. For the three architectures considered, the received signal depends on the radiation of the transmission antenna in the direction (ϕ , θ ) of the receiver. We can thus define as many radiation patterns as there are architectures, by dividing the amplitude of the output signal of the real antenna by that of an ideal transmitting antenna. The definitions of the power gains are the following:

{

⎛ Max R τ rec, ref (τ , φ , θ ) TDG (φ ,θ ) = ⎜ ⎜⎜ Rref ,ref ( 0 ) ⎝

∫ rec ( t , φ ,θ ) dt IG (φ , θ ) = 2 ∫ ref ( t , φ ,θ ) dt

} ⎞⎟ ⎟⎟ ⎠

2

[6.8]

2

[6.9]

∑ rec (ωk ,φ ,θ )2

OG (φ , θ ) = k

∑ ref (ωk ,φ ,θ )2

[6.10]

k

Here, Ru ,v (τ ) =

∞

∫−∞ u (t ) v ( t + τ )dt

represents the correlation between the

signals u ( t ) and v ( t ) , and u (ω ) is the Fourier transform of u ( t ) . The above

definitions take as a reference ref (t ) that is the signal received for an ideal antenna, which ensures immediately a gain of one for an ideal antenna. The sum on

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k in [6.10] is carried out over all subcarrier pulsations ω k in all the considered subbands. Another pattern of interest is that of the antenna distortion, measured by the normalized correlation between the received signal and the reference waveform. This quantity equals 1 if the waveforms are identical within a multiplicative constant:

DIST (ϕ , θ ) =

{

}

Maxτ Rrec, ref (τ , ϕ , θ )

Rrec, rec (0, ϕ , θ ).Rref , ref (0 )

[6.11]

Quantities defined in equations [6.8]-[6.11] were evaluated for some UWB antennas differing in their characteristics (Figure 6.5), some being laboratory prototypes and others commercial antennas (Skycross™, Fractus™ [FRA 06]). Naturally, the band used for the calculation of the patterns was selected inside the matching band-width of the antenna, i.e. about 3.7-5.3 GHz for the quasiomnidirectional antennas shown in Figure 6.6. Note that in spite of the rather imperfect omnidirectionality of the TDG antenna gain, the distortion pattern deviates very little (less than 0,2 dB) from the perfect circle. The antennas thus minimally distort the signals radiated in their nominal band, and the deviation of the gain from omnidirectionality is comparable to that of a narrow band antenna. This observation is not very astonishing, and was also conjectured by the authors of [STA 06], on the basis of a electrical circuit model for such antennas.

Figure 6.5. UWB antennas: 1 bicone, 2 MDFS, 3 Skycross, 4 Fractus, 5 LPDA, 6 FFPTP, 7 horn

In the absence of signal deformation, few differences are to be expected between the various types of gain suggested previously. On the other hand, for directional antennas the situation is different. For example, Figure 6.7 shows the radiation patterns of a commercial log-periodic antenna (LPDA, Figure 6.5), which is

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223

considered to be quite dispersive. If the transmitted signal has a band located inside the nominal band of the antenna (figure on the left), patterns TDG, OG and IG are very similar, the latter being slightly higher in terms of gain. Outside this nominal band (figure on the right), the distortion is very strong (DIST=-6.5 dB in the main lobe) and the coherent gain TDG is very bad. On the other hand, the incoherent gain remains completely acceptable. We thus see here the direct effect of the distortion on the gain. 4 dBi

-0

-6 dBi

Bicône Biconical

-1

MDFS

Skycross

Fractus

Figure 6.6. Radiation patterns (TDG, left) and distortion patterns (DIST, right)

1 -5

TDG

0

2 5

IG

0

-5

5

OG

Figure 6.7. Azimuth patterns (dBi): 1) 3.7-5.3 GHz BW; 2) 4.2-7 GHz BW

Figure 6.8 is another example for two excited directional antennas inside their nominal band, which are on the one hand an air patch antenna (FFPTP) involving a broad band feeding probe [LEP 08], and on the other hand a horn antenna. According to the antenna and the type of architecture the gains differ, but the variation remains moderate. In a general way, it appears that the lowest distortion is obtained in the main lobe of radiation, and it remains relatively constant in this lobe (see Figure 6.9).

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The case of the LPDA is interesting, because it is considered to be a very dispersive antenna by construction. However, in the azimuth plane we find a distortion a little higher than 1 dB within the main lobe, which remains moderate. On the other hand, in the elevation plane degradation is significantly more pronounced. In short, it is noted that especially in the secondary lobes the antenna distortion contributes in a harmful way to the coherent gain. The incoherent gain is always higher than the coherent gain. In certain cases, the OFDM gain is higher than the incoherent gain, but this is not a general rule. This is due to the fact that the spectrum of the signal emitted in OFDM is flatter than in an impulse modulation, consequently spectral differences remain between OG and IG, which explain the residual differences between these two gains according to the antennas.

0

10

TDG

IG

5

10

OG

Figure 6.8. Azimuth patterns (dBi): FFPTP (left); horn (right)

Figure 6.9. Distortion patterns (DIST, 2dB full scale)

15

Antenna-Channel Joint Effects in UWB

225

6.5. Effective performance of non-directional antennas in dispersive channels 6.5.1. Gain calculation for non-ideal antennas In this section we will show, starting from a simplified model and some approximations (see [SIB 06]), how energy at the demodulator output is jointly affected by the antenna characteristics and the multipath density of the channel, according to the nature of the (pulse based) modulation pattern. We will adopt a temporal processing discretized at the Nyquist frequency FC = 1 / TC where TC is the duration between two samples, with purely real UWB signals. If we put aside the directional dependences in expression [6.3], the combined impulse response of channel and antennas is the convolution h = hTx * hC * hRx , where hTx and hRx represent the transfer functions in emission of both Tx and Rx antennas and where the integration operator is incorporated in the propagation part hC . By convenience, we can combine the antenna responses in the form hTxRx = hTx * hRx , that is to say K −1

in discrete time hTxRx , n = ∑ hmδ n − m , with K expressing the finished duration of i =0

these responses, and δ 0 = 1 , δ n ≠ 0 = 0 (Kronecker). We will agree that an ideal antenna (not physical because radiating in d.c.) verifies K = 1 and h0 = 1 . The channel response is expressed in a similar form hC , n = ∑ Am δ n − m , with Am zero when m < 0 or m > P − 1 , P being the number m

of paths. We make the assumption of a wide sense stationary and uncorrelated scattering channel (WSSUS, see [PAG 08]) such as E Ai . A j = 0 if i ≠ j , where

(

)

E (.) stands for the mathematical expectation. The paths’ amplitudes obey a centered Gaussian distribution, whose variance depends exponentially on the time ( α.TC : standard deviation on the propagation delay):

(

)

p ( Am ) = exp − Am2 / 2σ m2 / σ m 2π

with σ m2 = σ 02 . exp (− m / α )

[6.12]

This model describes a dense channel, since the amplitude is not zero regardless of the delay. A low density channel where the majority of the amplitudes are zero can be represented by the same type of model, with the additional condition p ( Am Am ≠ 0 ) = exp ( − Am2 / 2σ m2 ) / σ m 2π with P (Am ≠ 0 ) = 1 / λ . We will impose that λ is rather large so that the temporal difference between two paths is almost always higher than the duration of the antenna responses (extreme “UWB” mode).

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6.5.1.1. Coherent reception without Rake combining Let θ ( t ) be the impulse at emission stage output and ref ( t ) the reference waveform used by the correlator in reception. At the output of the receiving antenna we have r ( t ) = h * θ ( t ) + n ( t ) , where n ( t ) is the additive noise. The correlator being sampled at the Nyquist frequency FC , the output can be written (n + 1)T C After synchronization, we obtain s = r (t ).ref ((n + 1)T − t )dt .

∫nT C

n

C

S = Max n (sn ), with n = 1,..., N where N is the length of the search window of the maximum. For a pair of ideal antennas, we have hTxRx, n = δ n and with a

transmitting signal θ n = δ n this leads to s n = hC , n = ∑ Ai δ n − i = An . i

6.5.1.1.1. Channel with a strong multipath density In dense channels the WSSUS assumption makes it possible to write for the distribution function of the output signal after synchronization: CDF (ξ ) = P ( S < ξ ) = P ( s1 < ξ ,..., s N < ξ ) = P ( s1 < ξ ) × ... × P ( s N < ξ )

[6.13]

that is to say:

(

) )

(

CDF (ξ ) = ∏ 1 − erfc ξ / σ n 2 / 2 n

If

N = − < FI S FII >

[A.5a]

< FII S FI > = − < FI S FII >

[A.5b]

It is three-linear with respect to its factors FI, FII and S. Notably if the domain of integration can be separated into two disjoined parts S = S1 + S2: < FI S FII > = < FI S1 FII > + < FI S2 FII >

[A.5c]

If S is the boundary ∂V of a finite volume V, as div (FI × FII ) = 0 , we have: < FI S FII > = 0

[A.5d]

If we can continuously deform surface S in surface S’ without crossing any sources of FI or FII, then: < FI S’ FII > = < FI S FII >

[A.5e]

If a part SZ of S coincides with a material surface which imposes on the fields boundary conditions of surface impedance Z of type Etan = Z n × H tan , then: < FI S FII > = 0 2. The same surface with the opposite orientation will be denoted –S.

[A.5f]

Appendix A

243

In particular, the far field produced by a bounded source is transverse with respect to r and verifies E∞ = η0 rˆ × H∞ , η0 being the wave impedance of the medium

μ / ε , therefore, on a sphere of large radius:

< FI ∂V∞ FII > = 0

[A.5g]

When sources are present in the considered region, the result is modified. Introducing the vector of the sources C (of dimension 6) built from the electric J and magnetic K current density vectors: J⎤ C = ⎡K ⎢⎣ ⎥⎦

[A.6]

The volume integral denoted: < CI V FΙΙ > = ∫ CI ⋅ FΙΙ dV

[A.7]

V

is a quantity having the unit of a power called reaction of the field FII and the part of sources CI in V. The Lorentz reciprocity relation (in integral form), integrated over a volume V of boundary ∂V (its normal being oriented outside) is then written [DES 66]:

< FI ∂V FII > = < CI V FII > − < CII V FI >

[A.8]

A.1. Reciprocity applied to waveguides

Let us initially consider the case of waveguides3, closed and lossless, for simplicity. If we apply the Lorentz relation [A.8] to a volume V delimited by two cross-sections S1, S2 and the waveguide border Σ, for two solutions FI and FII of sources external to V, we obtain: < FI ∂V FII > = 0

[A.9]

3. In the sense of any wave guiding structure invariant by translation, i.e. uniform, thus including the specific case of closed transmission lines (coaxial for example).

244

Ultra Wide Band Antennas

with ∂V = S2 − S1 + Σ, all the surfaces being oriented towards the interior of V. In addition, according to property [A.5f], < FI Σ FII > = 0. The remaining terms are thus: < FI S2 FII > = < FI S1 FII >

[A.10]

a relationship particularly valid for a couple of modes. Considering a mode of order n propagating in the direction of increasing z (“incident” mode) Fn, with wavenumber kn, we can show that it has an associated mode propagating in the direction of decreasing z (“reflected” mode) denoted F−n, of wavenumber −kn, obtained by reversal of Fn (with respect to any cross-section, in particular in z = 0). It is shown then in [DES 66] that Fn can be normalized in any symmetry plane, in particular in the section S0 (z = 0) so as to make the average power crossing this section unitary, that is to say: ½ < F−n S0Fn > = 1

[A.11]

According to [A.10], that is also true in any cross-section S: ½ < F−n S Fn > = 1

[A.12]

It is otherwise straightforward to show the orthogonality4 of the modes (not degenerate, i.e. of different wavenumbers km ≠ kn for m ≠ n): < F−m S Fn > = 0

[A.13]

By their very construction, any field in the guide (in any source-free region) is expandable over the modal basis, that is to say: F=

∑ a n Fn + bn F− n

[A.14]

By using the orthonormality relationships [A.12] and [A.13], the power crossing any section S5 is expressed immediately by: 1 2

< F∗ S F >=

P

∑ an

n =1

2

− bn

2

4. In the sense of the sesquilinear form that the cross-flux constitutes. 5. I.e. the net active power propagating towards the direction of increasing z.

[A.15]

Appendix A

245

Here the sum is restricted to the traveling modes (non evanescent). Like partial waves used in transmission lines and microwaves circuits (to define the scattering matrix for example), the coefficients an and bn are actually the complex amplitudes of the “forward” and “backward” modes. We will restrict ourselves to the more common case in practice for which the waveguide is single-mode, i.e. only propagating its fundamental mode. Sufficiently far away from sources so that the evanescent modes can be neglected, the field is thus expressed by: F = aF+ + bF−

[A.16]

F+ and F− being the normalized fields of the fundamental mode.

Let us consider two field solutions in the guide and let us form the cross-flux: < FI S FII > = < ( a I F+ + bI F− ) S ( a II F+ + bII F− ) > a = −2 ( a I bII − a II bI ) = −2 I bI

a II

[A.17]

bII

A.2. Reciprocity applied to the passive antennas in transmission and reception

We consider an antenna fed by a waveguide (wide sense6), on the one hand, in transmission (source supplying the guide) and on the other hand, in reception (i.e. illuminated by a plane wave incoming from any direction, created by sources “at infinity”, the guide being “matched”, i.e. loaded by a reflectionless termination). Let us denote Fe and Fr the fields, resulting from each of these two states of the sources. In addition let us consider a volume V of boundary ∂V = S∞ − S (S∞ ∪ S within the meaning of the previously defined orientation of the surfaces), the normal n to ∂V being outward-oriented with respect to V (Figure A.1), where S is a surface surrounding the considered antenna and S∞ a sphere of very large radius centered on the origin O (any point in the vicinity of the antenna).

6. Including transmission lines.

246

Ultra Wide Band Antennas

In “state e”, the antenna, in transmission, creates a far field taking the form of [3.1], that is to say:

Fe∞ =

e − jkr r

η 0 ⎡ Ae ( πˆ ) ⎤ ⎢1 ⎥ 4π ⎣⎢η 0 rˆ ∧ Ae ( πˆ )⎦⎥

[A.18]

In “state r”, the antenna, in reception, is illuminated by a plane wave traveling along any direction kˆ i and taking the form of [3.20], that is to say:

Fri = e jk i ⋅r

⎤ Ari η0 ⎡ ⎢1 i⎥ ˆ 4π ⎢η k i ∧ Ar ⎥ ⎣

[A.19]

⎦

0

kˆ i

n

S∞

O

Sant

S

n

V

Figure A.1. Surfaces of integration of cross-flux

This incident field is of course perturbed by the presence of the antenna which creates in response a radiated field Frr taking the same form as [A.18] at large distance:

Frr

∞

=

e − jkr r

r η 0 ⎡ Ar ( πˆ ) ⎤ ⎢1 ⎥ r 4π ⎢ η πˆ ∧ Ar ( πˆ )⎥

⎣

0

⎦

[A.20]

Appendix A

247

The total field in the receiving mode is thus:

Fr = Fri + Frr

[A.21]

The reciprocity relationship of an antenna, respectively in transmission and reception will be established by forming the cross-flux of the fields in the two states “e” and “r” (through S − see Figure A.1) and by calculating it in two different ways. According to the property [A.5c] (“trilinearity”):

< Fe∞ S Fr > = < Fe∞ S Fri > + < Fe∞ S Frr >

[A.22]

It can easily be shown that the second term is null; indeed, according to [A.5c] and [A.5a]:

< Fe∞ ∂V Frr > = < Fe∞ S ∞ Frr > − < Fe∞ S Frr >

[A.23]

The field Frr verifying the Sommerfeld radiation condition (form [A.20]), is orthoradial and thus, according to the property [A.5g], the first term is null. In addition, V does not contain any source, so that, according to the relation of Lorentz [A.8], < Fe∞ ∂V Frr > = 0 and thus:

< Fe∞ S Frr > = 0

[A.24]

Finally, only one integral remains:

< Fe∞ S Fr > = < Fe∞ S Fri >

[A.25]

According to property [A.5e], this cross-flux is independent of S as long as S surrounds the antenna, therefore in particular on the very surface Sant of the antenna:

< Fe∞ S Fr > = < Fe∞ S ant Fri >

[A.26]

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Ultra Wide Band Antennas

ae Sant nΣ

zˆ

O

z

bri Figure A.2. Antenna feeding with a waveguide.

In the case of a waveguide feeding (Figure A.2), according to property [A.5f]:

< Fe∞ S ant Fri > = < Fe∞ Σ Fri >

[A.27]

where Σ is any cross-section of the guide, sufficiently far away from the source (transmission) and from the radiating part. However, according to [A.17], we have:

< Fe∞ Σ Fri > = −2(aebri − ari be ) = −2aebri

[A.28]

because it is assumed that the termination is matched to the guide in reception (i.e.

ari = 0 ). We finally have: < Fe∞ S Fr > = < Fe∞ S Fri > = Fe∞ × Fri ⋅ dS = −2aebri

∫

[A.29]

S

through any surface S surrounding the antenna, therefore in particular any sphere of sufficiently large radius r on which the field in transmission can be approximated by the far field. The cross-flux is then written: < Fe∞ S Fr > = Fe∞ × Fri ⋅ d S = (Ee∞ × Hri − Eri × He∞ ) ⋅ d S

∫ S

∫ S

[A.30]

Appendix A

249

z rˆ

θˆ

θ

ϕˆ

Ae θι kˆ i θˆ i

O

ϕ

S

ϕˆ i

A ri y

ϕι

x Figure A.3. Spherical coordinate system used in the calculation of integral [A.30]

However, (Figure A.3): Ee∞ × Hir − Eir × He∞ =

e − j ( kr − k i ⋅r ) [ Ae × ( kˆ i × Ari ) − Ari × (rˆ × Ae )] 4π r

[A.31]

and d S = dS rˆ = r 2d Ω rˆ . We thus calculate7: [ Ae × (kˆ i × Ari ) − Ari × (rˆ × Ae )] ⋅ rˆ = (kˆ i ⋅ rˆ − 1) Ae ⋅ Ari − ( Ae ⋅ kˆ i ) ( Ari ⋅ rˆ )

[A.32] taking into account that A e ⋅ rˆ = 0 . We thus obtain : < Fe∞ S Fr > = −

e − jkr i Ar ⋅ ∫ e jk i ⋅r [(1 − kˆ i ⋅ rˆ ) Ae + ( Ae ⋅ kˆ i ) rˆ ] dS 4π r S

r e − jkr i =− Ar ⋅ ∫∫ e jk i ⋅r [(1 − kˆ i ⋅ rˆ ) Ae + ( Ae ⋅ kˆ i ) rˆ ] d Ω 4π 4π 7. Using the vector identity: a × ( b × c) = (a ⋅ c) b − (a ⋅ b) c .

[A.33]

250

Ultra Wide Band Antennas

We have otherwise: k i ⋅ r = kr kˆ i ⋅ rˆ = kr cos(θ − θ i ) cos(ϕ − ϕ i ) so that: < Fe∞ S Fr > = −

r e − jkr i Ar ⋅ I 4π

[A.34]

with I a vector integral of the form: I=

∫∫ e

jkr g (θ ,ϕ )

f (θ ,ϕ )d Ω

[A.35]

4π

in which: f (θ , ϕ ) = (1 − kˆ i ⋅ rˆ ) A e + ( A e ⋅ kˆ i ) rˆ and g (θ ,ϕ ) = cos(θ − θ i ) cos(ϕ − ϕ i ) .

Except in the neighborhood of kˆ i = (θ i ,ϕ i ) , the argument krg(θ,ϕ) of the exponential is a rapidly oscillating function; we can thus use the so-called stationary phase method (Appendix B). We indeed have ∇ g (kˆ i ) = 0 (the phase is stationary at 2 2 kˆ i ) and in addition ∂θθ g = ∂ϕϕ g = − g , so that:

2 2 ∂θθ g ( kˆ i ) = ∂ϕϕ g ( kˆ i ) = −1

[A.36]

2 In addition: ∂θϕ g ( kˆ i ) = 0 .

We thus deduce, according to appendix B:

I=

2π kr

e jkrg (θ i ,ϕ i ) Det [ Hess g (θi , ϕ i )]

e jσπ / 4 f (θi , ϕi )

with f (θ i ,ϕ i ) = 2 A e ( −kˆ i ) , Det[ Hess g (θ i ,ϕi )] = 1 and σ = − 2.

[A.37]

Appendix A

251

Finally: < Fe∞ S Fr > = −

r e − jkr i 1 A r ⋅ I = − A ri ⋅ A e e − jπ / 2 = −2ae bri 4π k

[A.38]

Finally, by identification, introducing [3.2] and [3.21]:

λ T H R ( f , rˆ ) = − j H ( f , rˆ ) 4π

[A.39]

Appendix B

Method of the Stationary Phase

The method relates to the asymptotic calculation of integrals of the form:

I (r ) =

∫de

jrg ( x )

f (x) dx

[B.1]

R

g being a function of class C∞ on RD.

The objective is to find an equivalent of I as r → ∞. The complex exponential is in this case a function with fast oscillations whose integral, weighted by f, tends consequently towards 0. We will limit ourselves to the case interesting for us in practice for which g presents only one stationary point x0 in the domain of integration. It is then obvious that the only significant contribution to the integral comes from a neighborhood of x0. Hence we have ∇g(x0) = 0 and thus g (x) ≈ g (x 0 ) + Hessg (x 0 ) ⋅ (x − x 0 , x − x 0 ) . It can therefore be shown that:

⎛ 2π ⎞ I (r ) = ⎜ ⎟ ⎝ r ⎠

d /2

e jrg ( x 0 ) Det[ Hessg(x 0 )]

where σ is the Hessian signature of g at x0.

Ultra W ide Band Antennas Edited by Xavier Begaud © 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

(

e jσπ / 4 + O r − d / 2 −1

)

[B.2]

Acronyms and Abbreviations

ABL AFL AIR AP ASIC ATF AUT BLTSA BPM BPSK BW CEPT CMS CWSA DAA DBO DIST DSRD ECC ECMA E.I.R.P EM EMC ETSA ETSI FCC FESA FFT FFPTP FPGA GMP-TLS

Anchor-based localization Anchor-free localization Antenna impulse response Access point Application-specific integrated circuit Antenna transfer function Antenna under test Broken linearly tapered slot antenna Burst position modulation Binary phase shift keying Band width Conférence européenne des postes et telecommunications (European Postal and Telecommunications Conference) Cabin management system Constant width slot antenna Detect and avoid Differentially bi-orthogonal Distorsion diagram Drift step recovery diode Electronic communications committee European computer manufacturers association Equivalent isotropically radiated power Electromagnetic Electromagnetic compatibility Exponential tapered slot antennas European telecommunications standards institute Federal communications commission Fast electronically steerable antenna Fast fourier transform Foam F-probe triangular patch Field-programmable gate array Generalized matrix-pencil total least squares

Ultra W ide Band Antennas Edited by Xavier Begaud © 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

256

Ultra Wide Band Antennas

GPIB GPR GPS HD HSPA IEEE IFE IG IR IRA ISM LDC LOS LPDA LTI LTSA MAC MBOFDM MPEG MP-TLS MRG NLOS OFDM OG OOK PICA PIR PPM PRF PRP PSD RF RFID RMS SAR SED SEM SMEM SNR SPR SRD SWR SUSA TDG TDMA TE TEM

General purpose interface bus Ground penetrating radar Global positioning system High definition High speed packet access Institute of electrical and electronics engineers In-flight entertainment Gain in power in the case of incoherent reception Impulse response Impulse radiating antennas Industrial, scientific and medical Low duty cycle Line of sight Log-periodic dipole antenna Linear time invariant Linearly tapered slot antenna Media access control Multi band orthogonal frequency division multiplexing Moving picture experts group Matrix-pencil total least squares Mean realized gain Non line of sight Orthogonal frequency division multiplexing Gain in power in the case of a MBOFDM receiver On off keying Planar inverted conical antenna Passive infrared Pulse position modulation Pulse repetition frequency Pulse repetition period Power spectral density Radio frequency Radio frequency identification Root mean square Synthetic aperture radar Spectral energy density Singularity expansion method Spherical mode expansion method Signal to noise ratio Surface penetrating radar Step recovery diode Standing wave ratio Specialized UWB smart antennas Gain in power in the case of coherent reception Time division multiple access Transverse electric Transverse electromagnetic

Acronyms and Abbreviations

TM TPC TSA UHF USB UWB UWB-CSS UWB-FM UWB-IR UWB-OFDM UWB-FH VHF WiFi WiMAX WBAN WLAN WPAN WSSUS WUSB xDSL 802.15.3 802.15.4

Transverse magnetic Transmit power control Tapered slot antenna Ultra high frequencies Universal serial bus Ultra wide band Ultra wide band chirp spread spectrum Ultra wide band frequency modulation Ultra wide band impulse radio Ultra wide band orthogonal frequency division multiplexing Ultra wide band frequency hopping Very high frequencies Wireless fidelity Worldwide interoperability for microwave access Wireless body area network Wireless local area network Wireless personal area network Wide-sense stationarity uncorrelated scattering Wireless universal serial bus X digital subscriber line IEEE working group on high flowrate WPAN wireless networks IEEE working group on low flowrate WPAN wireless networks

257

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List of Authors

Xavier BEGAUD Institut Télécom Télécom ParisTech, CNRS France

Serge HETHUIN THALES Communications France Colombes France

Isabelle BUCAILLE THALES Communications France Colombes France

Georges KOSSIAVAS LEAT University of Nice-Sophia Antipolis CNRS Valbonne France

Jean-Yves DAUVIGNAC LEAT University of Nice-Sophia Antipolis CNRS Valbonne France Christophe DELAVEAUD LETI CEA Grenoble France Nicolas FORTINO LEAT University of Nice-Sophia Antipolis CNRS Valbonne France

Ultra W ide Band Antennas Edited by Xavier Begaud © 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

Christophe ROBLIN Unité Electronique-Informatique ENSTA-ParisTech & Institut Télécom Télécom ParisTech, CNRS France Alain SIBILLE Institut Télécom Télécom ParisTech, CNRS France

Index

A absolute fidelity, 92 analog band-width, 141 Anchor-Based Localization, 19 Anchor-Free Localization, 19 anechoic chamber, 70 room, 117, 118, 121, 127, 144, 155, 160 antenna delay spread, 83, 89 transfer function, 51, 64 antennas with progressive transition, 163, 196, 200 without distortion, 80 aperture or effective surface, 48 applicative approach, 131 Archimedean spiral, 169 ARMA, 97, 103 autocorrelation, 227 automation of measurements, 119 average delay, 229

B Babinet, 167 Balanis, 164

Ultra W ide Band Antennas Edited by Xavier Begaud © 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

bandwidth, 1, 6, 9, 11, 17, 20, 21, 23, 32, 52, 56, 57, 80, 219 bases of measurement, 117, 124 Baum, 95 bicone, 89 biconical antenna, 70, 71, 78, 81, 8891, 94, 103-105, 112, 164, 177-180 bistatic diffusion, 215 BLTSA, 197 Brillouin, 199, 200 Burst Position Modulation, 10 butterfly antenna, 180-183, 189, 191, 195

C calibration, 121, 123, 135, 136, 138, 144, 159, 160, 217 celerity, 38 characteristic impedance, 37, 46, 47, 51 Chirp Ultra Wide Band, 17 Chu-Harrington limit, 204 circular monopole, 182 coaxial omnidirectional horn, 199 coherent architectures, 220, 235 comparison method, 121 compensation of dispersion, 87 complex Friis formula, 74

276

Ultra Wide Band Antennas

conical spiral, 166-168 copolar, 39-42, 44, 45, 50, 54 correlation, 221 cross component, 39, 42, 44, 54, 57 CWSA, 197

D data communication channel, 214 directive elementary antennas, 195 dispersion, 62, 79, 86, 88, 97, 221, 229, 231, 233 dispersive antenna, 87 channel, 225, 232, 233 distortion, 61, 62, 63, 64, 77-83, 8690, 92-94, 220, 222-224, 237 Drift Step Recovery Diodes, 138 dynamics of measurement, 127, 207 Dyson, 164

E efficiency, 50 measurement, 125 electromagnetic radiation, 36, 43 elementary antennas, 177 envelope, 83, 90, 91 E-plane, 56 equiangular antennas, 164, 170, 171 equivalent time technique, 131 ETSA, 197, 200, 209, 210 examples of processing resulting from measured ATF, 103 experimental characterization, 113

F factorization of the antenna patterns, 235 far field, 40, 63, 65, 66, 72, 77, 78, 96, 98, 112, 214, 242, 245, 248 field polarization state measurement, 122

filter, 36, 128, 139-141, 219, 220, 228 Fraunhofer, 40, 116 free space attenuation,50, 74 frequency independent antenna, 164, 171 Fresnel, 116

G, H gain in impulse mode, 132 generalized MP-TLS method, 97 group delay, 78, 83 Half-Power Beam Width, 45 H-plane, 54, 56, 57 horizontal polarization, 39

I, L ideality, 80 impulses generator, 134 impulse response, 66, 69, 216, 217, 220, 225, 227-229, 231, 232, 236 incoherent architectures, 220 indicators of performance, 62 input bandwidth, 84 impedance, 46, 47, 52, 53, 54 integrator, 80 intrinsic (or absolute) approach, 131 isotropic antenna, 43, 51 Lindenblad, 198 link budget, 82 LOS, 218, 231, 232, 234, 236, 237 LTSA, 197

M, N main component (copolar), 54 component, 39, 54 lobe, 45, 47, 56 maximum directivity, 45 McLean limit, 204 mean

Index

excess delay, 89 realized gain, 83, 84 metamaterials, 202, 204 method of the stationary phase, 253 miniaturization of UWB antennas, 202 model of a generic channel, 218 modeling the response y(t) of a system to its excitation, 130 NLOS, 218, 231, 232, 234-237 Nyquist, 225, 226

O, P, Q One Off Keying, 9 order reduction, 102 phase shifter, 80, 81 plane wave, 38, 40, 215, 245 polarization efficiency, 47 power gain or absolute gain, 47 Poynting vector, 38 principle of reciprocity, 50 printed monopoles in reduced ground plane, 186 Prony, 97 propagation mechanisms, 215 quiet zone, 120

R radiated field function , 40, 41 power, 4, 26, 31, 40, 41, 44, 45, 47, 48, 50, 51 radiation measurement characteristics, 114 radiation integration method, 123 radiation efficiency, 47 radiation pattern, 113, 117, 120, 217 Rake combiner, 226, 228, 229, 234, 236 realized gain, 48, 49, 57, 67, 68, 69, 78, 79, 83, 84, 106, 108

277

real-time technique, 131 reciprocity principle, 73 reflection coefficient, 46, 52, 53, 65, 82, 126, 158, 159, 187 relative fidelity, 92-94 rise time, 128, 129, 138, 141, 142, 144, 160

S S matrix, 65 Schantz sphere, 155 Shannon, 218 side lobes, 45, 224 sinuous antenna, 173-175 SMEM, 98 Specific propagation transfer function, 217 spherical modes, 95, 98, 101, 103 spiral logarithmic curve, 165 stability of the phase, 139 standard deviation of the group delay, 88 standing wave ratio, 47 Steiglitz-MacBride, 97 surface penetration radars, 207 SVD, 97

T Tapered Slot Antennas, 196, 209 teardrop antenna, 198, 199 time domain system measurement, 134 time-harmonic, 39, 41, 43 total radiated power, 41, 47 transduction, 37 transfer function, 49, 50, 63, 64, 66, 68-70, 86, 97, 214, 216, 217, 225, 232 transient electromagnetic fields measurement, 144 transmission efficiency, 48 trapezoidal monopole, 184

278

Ultra Wide Band Antennas

triangular monopole, 183, 187

regulation, 2, 3, 6, 51

U

V, W, Z

ultra wide band, 1, 12, 21, 33, 213, 216, 219, 232 impulse radio, 8 units, 35 UWB, 1-18, 20-22, 24-28, 30-32, 33, 37, 51, 213, 214, 216-220, 222, 225, 227, 230, 232, 233, 235, 237 antennas, 16, 31, 164, 177, 180, 190, 192, 203, 204, 206-208, 222

vector amplitude of the field, 66, 92, 93 vertical polarization, 39 resolution, 142 sensitivity, 142 Vivaldi, 196, 197, 198, 208 Wheeler sphere, 124, 126 WiMAX, 4, 5 zero padding, 147, 148