Introduction to ultra-wideband radar systems [1st ed.] 9781003068112, 0849344409, 9780849344404, 9781000102987, 100010298X, 9781000120158, 1000120155, 9781000141481, 1000141489, 1003068111

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INTRODUCTION to

ULTRA-WIDEBAND RADAR SYSTEMS Edited by

James D. Taylor Retired, U.S. Air Force

CRC Press Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data Taylor, James D. Introduction to ultra-wideband radar systems / editor, James D. Taylor p. cm. Includes bibliographical references and index. ISBN 0-8493-4440-9 1. Radar. I. Taylor, James D., 1941. II. Title: Ultra-wideband radar systems. TK6580.I58 1995 621.3848—dc20 94-30451 CIP This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 0-8493-4440-9/95/$0.00+$.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit our Web site at www.crcpress.com. © 1995 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-4440-9 Library of Congress Card Number 94-30451 Printed in the United States of America 6 7 8 9 0 Printed on acid-free paper

PREFACE “Dictionaries are like watches; the worst is better than none at all, and the best do not run quite true. ” -Samuel Johnson This book is an introduction to ultra-wideband radar technology system concepts for those who need to learn about, design, or evaluate them. Because ultra-wideband technology is a developing area of technology, the writers have emphasized theory, concepts, and hardware and have presented basic principles and concepts that can guide ultra-wideband radar system design. Radar encompasses many technologies and operational functions; here, we present radar as any system that detects objects by reflected electromagnetic energy for any purpose. The term ultra-wideband (UWB) describes radio and radar systems that transmit and receive wave­ forms with instantaneous bandwidths greater than 25 percent of center frequency. For comparison, conventional narrowband radio and radar systems generally have bandwidths less than 1% of center frequency. Ultra-wideband was first applied in 1988 to systems including impulse, nonsinusoidal, baseband, video pulse, super wideband, time domain, carrierless, and other related radar and radio concepts. The definition of ultra-wideband is not fixed and has been loosely used to include narrowband systems with wide coverage ranges, such as electronic support measures (ESM) receivers and microwave devices. If readers are in doubt about how a writer has used the term ultra-wideband, they should determine how the term is being used and what the subject is under discussion. Ultra-wideband is also spelled ultrawideband in the literature, because there was no universally accepted spelling when the articles were written. Ultra-wideband has become the generally used form. Ultra-wideband signals include both short duration (impulse) and longer duration (nonsinusoidal) waveforms. Ultra-wideband signals have demonstrated potential applications for fine range resolution, foliage penetration, ground penetration, low probability of intercept, remote sensing, and frequency spectrum sharing applications. Ultra-wideband radar technologies and concepts are extensions and variations of established radar technology. The key concept is that UWB signals can be more recogniz­ ably affected by the transmissive and reflective media than narrowband signals, and the resulting signal changes can be analyzed to extract information about either the media or the reflector. The signal information is the waveform in UWB systems, not a modulation on another carrier waveform. Ultrawideband radar technology may offer remote sensing capabilities not attainable by narrowband or passive systems. Consider UWB radar as another way to look at electromagnetic waves and radar technology by extending what you already know about conventional narrowband technology. This book can serve as a reference on ultra-wideband radar, or as a textbook for senior or graduate level engineering courses in UWB radar. The authors assumed an undergraduate level background in electronics, physics, and communications theory. In a new, developing technology, no book can be expected to contain the last word or to provide easy solutions in simplified tables. Our objective was to provide a guide to UWB radar for those who need to design, evaluate, analyze, or consider UWB signal technology for any application. The book is a guide to the general features of UWB technology and a source for more detailed information. We have included discussions and information that a systems or subsystems designer can use. The best approach for the reader is to combine the book with his or her own technical judgment. As Dr. Johnson put it over 200 years ago, a book is like a watch... and the worst watch is better than none at all if you use it with some common sense and have enough judgment to know when you should trust it.

ACKNOWLEDGMENTS This book would have been impossible without the help of friends, supporters, opponents, and critics who inspired, assisted, and gave their frank considered opinions and suggestions. There are too many to name individually without unfairly omitting someone. You know who you are and we publicly thank you for your contributions. We also extend our special thanks to the government, industry, and university officials who made this book possible by supporting and encouraging ultra-wideband radar technology related programs. Special thanks and recognition to the supervisors and managers of the Deputy for Advanced Technology and Development Planning at the Air Force Electronic System Division, Hanscom Air Force Base, Massachusetts, for their vision and encouragement of ultra-wideband radar when I was planning and editing this book. My heartfelt thanks to all the writers for working with me and taking my lengthy critiques and commentaries to heart during the revisions of their chapters. The authors gave their time and effort generously to make their work as readable and accurate as possible. We all thank our families and friends who supported us and provided us the time we needed to prepare this book.

James D. Taylor Gainesville, Florida

THE EDITOR James D. Taylor was bom in Tifton, Georgia, in 1941 and has lived in North Carolina, Maryland, and Florida. He earned his BSEE degree from the Virginia Military Institute in 1963 and entered active duty in the U. S. Army as an artillery officer. In 1968 he transferred to the U.S. Air Force as a research and development electronics engineer and worked for the Central Inertial Guidance Test Facility at Hollomon Air Force Base, New Mexico, until 1975. He earned his MSEE in guidance and control theory from the Air Force Institute of Technology at Wright Patterson Air Force Base, Ohio, in 1977. From 1977 to 1981, Mr. Taylor was a staff engineer at the Air Force Wright Aeronautical Laboratories Avionics Laboratory. From 1981 to 1991, he was a staff engineer for the Deputy for Development Planning at the Electronics System Division at Hanscom Air Force Base, Massachusetts, where he worked on advanced concepts, including unmanned vehicles, airborne radar platforms, and ultra-wideband radar. He organized the first DoD ultra-wideband radar symposiums in 1988 and 1989. He received his Professional Engineer registration from Massachusetts in 1984. He is a senior member of the Institute of Electrical and Electronics Engineers and American Institute of Aeronautics and Astronautics. He retired from the U. S. Air Force as a Lieutenant Colonel in June 1991 and is now a gentleman engineer, consultant, technical writer, and editor.

THE CONTRIBUTORS Terence W. Barrett, Ph.D. BSEI Vienna, Virginia R. N. Edwards, Ph.D. Associate Applied Physical Electronics Research Center University of Texas at Arlington Arlington, Texas

Iain A. McIntyre, Ph.D. Manager E/O Energy Compression Research San Diego, California Vasilis Z. Marmarelis, Ph.D. President Multispec Corporation Los Angeles, California

H arold F. Engler, Jr., MSEE Principal Research Engineer Electronic Systems Laboratory (CAD) Georgia Tech Research Institute Georgia Institute of Technology Atlanta, Georgia

William C. Nunnally, Ph.D. Director Applied Physical Electronics Research Center University of Texas at Arlington Arlington, Texas

Todd Erdley Paragon Technology, Inc. State College, Pennsylvania

David Platts, Ph.D. Project Engineer Los Alamos National Laboratory Los Alamos, New Mexico

P. R. Foster, Ph.D., CEng, FDEE Managing Director Microwave and Antenna Systems Great Malvern, Worcestershire United Kingdom

M uralidhar Rangaswamy, Ph.D. Rome Laboratory Hanscom Air Force Base Burlington, Massachusetts

D. V. Giri, Ph.D. Pro-Tech Lafayette, California J. Doss Halsey Senior Engineer Information Systems Laboratories, Inc. Vienna, Virginia Henning F. Harm uth, Ph.D. Professor Emeritus Department of Electrical Engineering Catholic University Washington, D.C. M alek G. M. Hussain, Ph.D. Professor Department of Electrical Engineering and Computer Engineering Kuwait University Kuwait Elizabeth C. Kisenwether, MSEE President Paragon Technology, Inc. State College, Pennsylvania

Robert Roussel-Dupre, Ph.D. Staff Scientist Space and Atmospheric Sciences Group Los Alamos National Laboratory Los Alamos, New Mexico Tapan K. Sarkar, Ph.D. Department of Electrical Engineering and Computer Science Syracuse University Syracuse, New York David Sheby, Ph.D. Chairman Multispec Corporation Cherry Hill, New Jersey Michael L. VanBlaricum, Ph.D. Vice President Toyon Research Corporation Goleta, California Oved S. F. Zucker, BEE President Energy Compression Research San Diego, California

TABLE OF CONTENTS Chapter 1 Ultra-Wideband Radar Overview.................................................................................................................1 James D. Taylor

Chapter 2 Technical Issues in Ultra-Wideband Radar Systems................................................................................ 11 Harold F. Engler, Jr.

Chapter 3 Analytical Techniques for Ultra-Wideband Signals.................................................................................51 Muriladhar Rangaswamy and Tapan K. Sarkar

Chapter 4 Transmitters.............................................................................................................................................. 109 David Platts, Oved S. F. Zucker, and Iain A. McIntyre

Chapter 5 Ultra-Wideband Antenna Technology.................................................................................................... 145 P. R. Foster, J. Doss Halsey, and Malek G. M. Hussain

Chapter 6 Direct Radiating Systems........................................................................................................................ 287 William C. Nunnally, R. N. Edwards, and D. V. Giri

Chapter 7 Propagation and Energy Transfer............................................................................................................ 325 Robert Roussel-Dupre and Terence W. Barrett

Chapter 8 Transmitter Signature and Target Signature of Radar Signals.............................................................. 435 Henning F. Harmuth Chapter 9

Radar Cross Section and Target Scattering............................................................................................457 Michael L. VanBlaricum

Chapter 10 Ultra-Wideband Radar Receivers............................................................................................................ 491 James D. Taylor and Elizabeth C. Kisenwether

Chapter 11 High-Order Signal Processing for Ultra-Wideband RadarSignals...................................................... 579 Vasilis Z. Marmarelis, David Sheby, Elizabeth C. Kisenwether, and Todd A. Erdley

Chapter 12 Performance Prediction and Modeling....................................................................................................609 Terence W. Barrett

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

657

Chapter 1

Ultra-Wideband Radar Overview James D. Taylor CONTENTS I. Introduction ........................................................................................... II. UWB Radar Terminology and C o ncepts.............................................. III. Potential Applications of UWB Radar ................................................ IV. UWB Systems Frequency Spectrum Sharing and Interference Issues V. Book Contents ...................................................................................... VI. Conclusion ........................................................................................... References

1 1 4

6 7 9 9

I. INTRODUCTION This book is about radar systems using wide relative (proportional) bandwidth signals called ultrawideband (UWB) waveforms. The potential advantages of using UWB waveforms for radar include better spatial resolution, detectable materials penetration, easier target information recovery from reflected signals, and lower probability of intercept signals than with narrowband signals. Designing UWB radar systems requires considering what happens when a signal is no longer a single, long duration sinusoidal wave. This book presents principles needed for understanding UWB radar concepts and their potential capabilities and provides a basis for further investigation, design, analysis, and fabrication. Radar systems use radiated and reflected electromagnetic waves to detect, locate, and identify targets. Radar systems is a broad term including everything from small police and ground-probing systems to the large radars for ballistic missile defense and airspace surveillance and tracking. Radar targets may include ground discontinuities, buried objects, stationary objects for navigation, and moving objects including vehicles from automobiles to reentry vehicle systems. Each reader’s experience and professional interest slants his personal concept of radar systems. The radar system designer’s problem is to balance the user’s needs and desires with available technology, achievable performance, and affordable cost. The designer’s objective is to satisfy the radar user’s needs effectively and cheaply. The radar system user is the final judge of acceptable radar performance and cost. The second section of this chapter is about UWB concepts and the differences between UWB radar and conventional narrowband radar. The third section is about potential applications for UWB radar. The fourth section is about approaches to handling an UWB system’s electromagnetic compatibility and interference issues. The fifth section is an introduction to and summary of each chapter of this book.

II. UWB RADAR TERMINOLOGY AND CONCEPTS Ultra-wideband terminology and definitions are not standardized as of this writing, and this may cause some confusion in literature searching. Terms such as narrowband and wideband can have several meanings depending on the subject, i.e., communications, radar, etc. This section will discuss what UWB radar is generally accepted to be and some of the alternative, but related terminology. Assump­ tions concerning meanings can be misleading. When in doubt, see how the writer uses the term and determine what is being described in basic functional concepts. The best advice is to be aware that ultra-wideband may also be called impulse, time domain, nonsinusoidal, baseband, video pulse, ultrahigh resolution, carrierless, super wideband, and other terms. 0-8493-4440-9/95/$0.00+$.50 © 1995 by CRC Press, Inc.

1

2

UWB DEFINED The Defense Advanced Research Project Agency’s (DARPA) 1990 Assessment of Ultra-Wideband Radar advised that “definitions need liberal interpretation and that mathematical definitions are difficult to achieve and seldom useful in a practical sense. ” The following definitions were given. Energy Bandwidth, BE: The energy bandwidth is the frequency range within which some specified fraction, say 90 or 99%, of the total signal energy lies. This must be defined for a single pulse, if all pulses are the same, or for a group of pulses that are processed together to yield a single decision. The upper limit of this range is denoted here by fH and the lower limit by fL. Time-Bandwidth Product, TB: The time-bandwidth product of a signal is defined as the product of the energy bandwidth and the effective duration of a single pulse or pulse group. It is the measure of the increase in peak signal-to-noise ratio that can be achieved in the radar receiver by appropriate signal processing.1 Bandwidth is defined as fractional and relative:

Fractional Bandwidth = -f^ H— ItL [fu + f L)

Relative Bandwidth =

— — (fn +fi)

(1.1)

(1.2)

The DARPA panel accepted the following definition: “Ultra-wideband radar is any radar whose fractional bandwidth is greater than 0.25, regardless of the center frequency or the signal timebandwidth product.”1 The term ultra-wideband refers to electromagnetic signal waveforms that have instantaneous fractional bandwidths greater than 0.25 with respect to a center frequency. There is no accepted standard usage for UWB terms; writers also use percent bandwidth and proportional bandwidth instead of fractional bandwidth. There are two other radar classes identified by signal fractional bandwidth: narrowband, where the fractional bandwidth is less than 1 %, and wideband, with a fractional bandwidth from 1 to 25% .1 These terms were specifically proposed for describing radar systems in 1989. Some confusion results because narrowband and wideband have very different meanings when describing communications channel bandwidths. Most narrowband systems carry information, also called the baseband signal, as a modulation of a much higher carrier frequency signal. The important distinction is that the UWB waveform combines the carrier and baseband signal. Baseband or impulse radar (or radio) are other names for UWB radar and radio signals.2 The UWB signals generally occur as either short duration impulse signals and as nonsinusoidal (e.g., square, triangular, chirped) waveforms. The rule of thumb is that sinusoidal wave signal bandwidth (Bw) for pulse signals are inversely proportional to pulse duration (r), or Bw ~ 7 /r.3 When the duration of a short sine wave pulse signal approaches several periods, then the relative bandwidth starts becoming a larger fractional value. There are also long duration nonsinusoidal waveforms having significant power at multiples of its fundamental frequency. Figure 1.1 shows some typical UWB waveforms and power spectral density plots based.

UWB TERMINOLOGY AND USAGE The term ultra-wideband (also ultrawideband) is a new term associated with radio and radar technolo­ gies called impulse, nonsinusoidal, baseband, video pulse, super wideband, time domain, carrierless,

3

a N a r r o w b a n d p u ls e s i g n a l a n d

b C h ir p e d r a d a r s ig n a l a n d

c

PSD

PSD.

E x p o n e n t i a ll y d a m p e d s in e w a v e im p u ls e s i g n a l a n d

PSD

Figure 1.1 Waveforms for comparison: (A) narrowband; (B), (C), (D) UWB waveforms.

and other related concepts. Before about 1989, UWB technology literature generally used one of the associated terms. Because ultra-wideband is a new term, it is best to look for the writer’s definition or to determine the meaning in context and the accompanying details, and then apply the mathematical descriptions loosely. Some physical reason for assigning the breakpoints between narrowband, wideband, and UWB would be much more satisfying. Interpretation of systems as UWB should be kept loose. For example, an argument that a system with a 24% fractional bandwidth signal is wideband and not UWB defies common sense and engineering judgment. Other examples of UWB usage include describing narrowband receivers and devices with a broad tuning range4 or broad proportional band­ width RF amplifiers with a 1 to 18 GHz bandwidth.5

4 UWB AND SPREAD SPECTRUM SIGNALS The term UWB may be confused with spread spectrum, thus we need to discuss the difference. Spread spectrum systems have a transmitted signal that is spread over a frequency band much wider than the minimum bandwidth required to transmit the information being sent. A spread spectrum system takes a baseband signal with a bandwidth of only a few kilohertz, such as a voice channel, and modulates it with a wideband encoding signal that distributes it over a larger bandwidth. The resulting signal might be a megahertz modulation signal on a several hundred megahertz carrier signal with a fractional bandwidth near 1 %. Because the spread spectrum signal has a much larger bandwidth than narrowband receivers in the same range and requires a knowledge of the wideband encoding method to demodulate it, it does not interfere with, or is not intercepted by, narrowband systems because so little power falls in the bandpass of any given narrowband receiver. Some general types of spread spectrum signals include 1. Direct sequence modulated systems which modulate the carrier with a digital code sequence whose bit rate is much higher than the information signal bandwidth. 2. Frequency hopping systems change the transmitter frequency within some predetermined order set by a code sequence. The signal never stays on any one frequency long enough to interfere with or be intercepted by a receiver without the frequency sequence. 3. Pulsed-FM or “chirp” modulation in which the carrier is swept over a wideband during a given pulse interval. The receiver follows the frequency change. 4. Time hopping systems, which have a time of transmission with a low duty cycle and short duration governed by a code sequence. Time-frequency hopping systems control both the time and frequency of transmission by a code sequence.6 While spread spectrum signals have a wide bandwidth with respect to other signals, they generally do not fit the UWB definition because their fractional bandwidth is well below 25 %. Radar signals such as binary-coded and chirped waveforms are sometimes misleadingly called spread spectrum.3

III. POTENTIAL APPLICATIONS OF UWB RADAR Fine spatial resolution, extraction of target feature characteristics, and low probability of interception and noninterfering signal waveform are some of the features that make UWB radar appealing. Thus, UWB radar offers possible solutions to defense requirements such as passive target identification, target imaging and discrimination, and signal concealment from electronic warfare equipment and antiradiation missiles. Frequency spectrum sharing with other radar and communications systems is another potential use. Future UWB radar applications will depend on the ability of a particular UWB system to perform a given detection or remote sensing function competitively with available alternative systems or to provide some operational advantage, such as a low probability of intercept signal.

TARGET SIGNAL INTERACTION AND FEATURE EXTRACTION The large signal bandwidth and information carried by the UWB radar return signal may provide sensing capabilities beyond simple target detection. The waveform content of reflected UWB impulse signals has been shown to change depending on the target shape and materials. Experimental work in singularity expansion method (SEM) radar using single “impulse” signals indicated that radar return waveforms were changed by target structure and electrical characteristics. Target information process­ ing for impulse signals is like determining the characteristics of a system from its impulse response. The reflected impulse signal characteristics seen in experiments appear unique enough to permit target identification.7'12 All radar signals will have some target-related change when reflected; the problem is to detect that change and uniquely relate it to the reflector. For example, compressor blade, or fan modulation, of radar return signals offers a potential identification method for narrowband systems. However,

5 determining more complicated information such as shape or specific materials from the signal target interaction may be done more easily by identifying distinct resonances with SEM or examining the higher order characteristics by bispectral processing. SEM and bispectral processing lend themselves to UWB signals. Some useful information is in any radar return signal; the technical problem is signal processing to turn data into useful, timely, and reliable information.

TARGET IMAGING AND DISCRIMINATION The UWB radar’s fine spatial resolution gives a potential capability for target imaging and discrimina­ tion of targets from background clutter. Promising work has been done with UWB synthetic aperture radar (SAR).13-14 The UWB SAR has the capability of imaging reflectors concealed in a forest.15 Some success has resulted from look-down tests to detect boats by using UWB signals.16 Target clutter separation is a major problem in look-down radar and limits the ability to detect small radar cross section (RCS) low altitude flying targets or surface targets. Moving surface targets can be detected by radar systems such as the Joint STARS, which uses SAR and Doppler filtering. Target extraction by moving target improvement (MTI) using Doppler shift can permit some small target detection in clutter; however, reducing target RCS can keep the target clutter ratio beyond the threshold limit needed for detection. There is a possibility that UWB signals reflected from background clutter might be different enough to permit discrimination of target signals based on reflected waveform higher order signal analysis, as discussed in Chapter 11. Geophysical surveying uses impulse ground-probing radar for buried and concealed object detection and subsurface mapping in mining, agriculture, highway and building construction, archeology, and ice field surveying. Multiple impulse radar returns can provide a picture of subsurface conditions and buried objects. The capability of impulse waveforms to penetrate solid structures and return signals from discontinuities in the index of refraction is what makes the radar useful.17'19 There is no free lunch in sensor systems. Fine spatial resolution in narrowband radar comes at a price, e.g., chirped waveforms that create antenna sidelobes, transmission losses, false targets, extra processing, while wide signal bandwidths mean high noise levels, etc.3 The UWB radar can provide fine spatial resolution by using short duration impulse or coded impulse train waveforms and correlation detection, which bring their own technical problems, as shown in Chapters 8 and 10. Discrimination of targets using higher order signal processing of impulse signals can distinguish between materials that would not be otherwise distinguishable by narrowband signals, again at the cost of complex signal processing, as discussed in Chapter 11.

LOW PROBABILITY OF INTERCEPTION UWB WAVEFORMS There are many military requirements that only radar can satisfy, but a radar must radiate to be useful and this is a disadvantage for military surveillance and detection systems. Generally, a receiver tuned to a radar’s frequency can detect a radar set from its emissions farther than the radar can detect the target from its return signal. If a radar set can be an asset (source of information) or a liability (“shoot here” sign) depending on the enemy’s electronic warfare capability, then any radar system with a difficult-to-detect radar signal can offer military advantages. Limiting radar and radio emissions until absolutely necessary is a practical operational solution; however, it defeats the purpose of owning the radar. Any radar system that has a difficult-to-detect signal is worth considering for military applications. The decision to use a UWB signal in preference to some other method will involve a tradeoff of technology availability, costs, perceived military advantage, and the need to replace existing systems and to revise operating procedures. The UWB radar could provide such advantages as a detection surveillance or tracking systems with a low probability of detection by the spectral characteristics of the signal. When a designer decides to use some UWB format, the question is can a UWB system provide adequate performance for the intended role. Assuming that a UWB radar system is introduced, then the operational issue will be how long can the particular radar system remain undetectable to enemy systems. Silent, undetectable, stealthy, unobservable, low probability of intercept, low probabili­ ty of detection, etc. are relative terms, because given enough time, resources, and incentive someone will build a UWB radar intercept receiver and the formerly silent radar will become a beacon again.

6 The name of the game in electronic warfare is to buy time and some temporary advantage. Today’s countermeasure buys an advantage until the enemy finds a way to get around that countermeasure, and then the process starts again. The military advantage of a new system depends on maintaining security about the operational details to keep the enemy ignorant as long a possible. This UWB radar could offer operational advantages by providing a hard-to-detect signal, as long as security is kept. Consider police radar for detecting speeders as a practical electronic warfare example. The speeder’s countermeasure was the police radar detector, which gave a driver time to slow down before coming into effective range of the radar. Police radar detectors proliferated at a price any driver could afford. Police response to the countermeasure was to add transmitter frequencies and use false speed radar transmitters. The new frequencies were outside the range of radar detectors for awhile. The false transmitters made drivers slow down on receiving a radar alarm and decreased the drivers’ confidence in their radar detectors. Another police innovation was laser radar speed measurement. The speeder’s countermeasure was to add the new frequency bands and laser detectors. While the example is familiar, the same principles apply to military electronic warfare.

IV. UWB SYSTEMS FREQUENCY SPECTRUM SHARING AND INTERFERENCE ISSUES FREQUENCY SPECTRUM SHARING There is limited available frequency spectrum and demands for communications, and radar-based sensors may continue to grow. Any electronic system that permits sharing the same frequency band in the same location without interference will eventually be used when it becomes profitable to do so. Ultra-wideband signals have a low probability of intercept (LPI) signal with respect to narrowband systems and may be able to share frequency spectra with narrowband and other UWB systems with proper design. These UWB systems may be able to share the same spectrum by means of waveform coding schemes to exclude unwanted UWB signals. The proliferation of personal communications systems and demands on the available frequency spectrum may create a demand for special UWB radar and radio communication links. Economic incentives for UWB radio and radar for civil applications such as private communications and short range sensors may be good.

ELECTROMAGNETIC COMPATIBILITY AND INTERFERENCE ISSUES Electromagnetic compatibility (EMC) and electromagnetic interference (EMI) must be considered early in UWB radar or communications system design to avoid potential interference problems. If the designer does not consider EMI and EMC issues immediately, then assuredly someone else will before any further serious design continues. The UWB signal definition can give the impression of a continuous wide spectrum signal, which is not correct. Many people have heard the definition and concluded that any UWB signal will jam everything in some portion of the frequency spectrum and be prohibited by regulatory agencies. Some reflection indicates that many UWB signals will be short duration, low duty cycle signals. The design objective for narrowband equipment is to build a set that is most sensitive to some narrowband of frequencies and attentuates all other frequency signals. A receiver acts like an integrator with a time constant of 1/Bw. If a UWB signal has a 1-ns duration and a receiver has a 1-MHz bandwidth, then the receiver’s integration time constant is 1 \xs, or 1000 times as long as the UWB signal duration, and the UWB signal power will be attenuated by 30 dB by being spread over 1000 times its normal duration. Now, if the UWB signal energy is high enough or has enough power to be detected after integration over a long time period, then the resulting power level in the receiver may be high enough for detection and interference. Interference will depend on the particular UWB signal, strength, and emitter location with respect to specific narrowband equipment characteristics. This book does not explicitly cover electromagnetic interference; however, the materials in Chapter 10, Appendix A, show how to estimate the impulse signal response of receivers. Given the impulse strength for detection (or interference) in watt-seconds, then a range can be determined at which

7 different receiver bandwidths can detect the signal. This initial estimate should give some indication of potential interference problems with narrowband electronic systems. The best approach appears to be to take each case and evaluate it using the methods from Chapter 10. For UWB radar designs using array antennas, the resulting UWB waveform off-axis can turn into a stretched or repeated waveform with different characteristics requiring an off-axis analysis evaluation for narrowband systems interfer­ ence. Chapter 5, Section 3, discusses the effects of array antennas on UWB signal waveforms and duration. There are enough signal format and power possibilities resulting from array antennas to make specific case analysis necessary. Interference and low probability of interception are relative, and any UWB or narrowband systems properties or claims will only be valid for specific cases and conditions. The best advice is to evaluate each case on its own merit to determine if a particular UWB system will operate without interference with particular equipment in a specified environment. Chapter 10, UWB Radar Receivers, provides the background for estimating impulse signal strength necessary for detection in narrowband receivers.

V.

BOOK CONTENTS

TECHNICAL ISSUES IN UWB RADAR SYSTEMS Chapter 2, by Harold Engler, discusses UWB radar systems in overall system terms and provides a guide to the remainder of the book.

UWB ANALYSIS Chapter 3, by Dr. Tapan Sarkar and Dr. M. Rangaswamy, discusses Fourier and Laplace transform analytical techniques and applications and limitations of Fourier and Laplace transforms. Both techniques are valuable and need to be understood before applying them to UWB signal analysis. Signal analysis is a continuously evolving subject and best followed through its own literature. Chapter 3 reviews the basics and provides background for further reading.

UWB TRANSMITTERS Chapter 4 is about several approaches to generating UWB signals. Any pulse radar systems must store energy over long periods and then release it over short periods. Shorter discharge (transmission) intervals present more problems then longer ones, because the frequency components are higher and more subject to dissipative effects. When the discharge interval starts to approach the time constant of the storage device, which is set by its physical dimensions, then energy storage systems become a sensitive part of the design. This chapter presents two approaches to energy storage and release. Dr. David Platts discusses Marx banks which can provide high voltage discharges over short intervals. The Marx bank charges capacitors in parallel and then uses spark gap switches to connect and discharge them in series. Dr. Oved Zucker and Dr. Iain McIntyre give an introduction to signal synthesis using photoconductive switches to generate UWB signals. Photoconductive switches are another approach to generating high power electromagnetic impulses. Semiconductors can change from insulators to conductors when subjected to intense illumination from a laser.

UWB ANTENNAS Chapter 5 is about coupling UWB signals into space. Ultra-wideband signals can cover the range of conditions from short duration impulses to long duration nonsinusoidal waveforms. The point to remember is that in UWB systems the modulation (information) is the waveform and must be detected as such. The UWB system antenna must be able to transmit or receive the range of frequencies in the signal without distorting any important part of the signal. Dr. P. R. Foster provides an overview and introduction to UWB antennas theory and discusses impulse radiation from common types of antennas. Dr. Malek Hussain describes new approaches to transmitting and receiving impulse signals called the large-current radiator and loop sensor antennas. These are radiating and receiving elements specifi­

8 cally designed for impulse signals. The theory of these elements and experimental measurements are presented. Dr. Hussain also describes the concepts and theory of impulse array antennas, which are different from the narrowband array theory for phased array radar. Mr. Doss Halsey provides an introduction to UWB coded pulse waveform array antennas and provides practical theory for array antenna design. The coded UWB pulse train waveform can provide fine spatial resolution with a long duration UWB signal. Determining antenna patterns for coded pulse train waveforms requires a change in the way we think about antenna patterns. The conventional antenna pattern concept is spatial distribution of radiated power at a given frequency. When we consider nonsinusoidal signals such as coded (or chirped waveforms) or time-coded impulse trains, the antenna pattern concept will be the distribution of power that may be correlated with a reference waveform. If the received signal is detected by correlation, then what is the path of a correctable signal from antenna array to receiving antenna. Transmitting a correctable signal (e.g., coded waveform or impulse) from an antenna array means that the interference of signals off-boresight will produce a signal with a different waveform than the reference signal.

DIRECT RADIATING ANTENNA SYSTEMS Chapter 6 is about how the dispersive effects of electronic components such as cables on UWB signals create a requirement for direct radiating systems or combined UWB signal generators and antennas. Dr. David Giri describes a nuclear electromagnetic pulse installation and the problems of building short pulse, high power systems. Dr. William C. Nunnally describes a transverse electromagnetic (TEM) horn emitter which uses the light-activated semiconductor switching described in Chapter 4.

PROPAGATION AND ENERGY TRANSFER Chapter 7 discusses how electromagnetic waves travel through the atmosphere. We included this chapter as background on how UWB and impulse propagation is a special case of steady-state narrow­ band propagation. These sections provide insight into the properties of short, UWB signals compared to long duration narrowband signals. Dr. Robert Roussel-Dupre discusses electromagnetic (EM) propagation and UWB waves. Dr. Terrence W. Barrett discusses UWB waves as an energy transfer phenomenon.

RADAR TRANSMITTER AND TARGET SIGNATURE Chapter 8 is about waveforms and the target signature of radar signals. Dr. Henning Harmuth discusses the results of a coded pulse train waveform reflecting from a target much larger than the spatial resolution of the signal.

RADAR TARGET CROSS SECTION Chapter 9 is about radar target reflection concepts. Dr. Mike VanBlaricum begins with power scattering and RCS concepts conventionally used in the radar equation. Radar scattering characteristics depend on the ratio of the target dimensions to the incident wavelength. A section covers relationships of CW, wideband, and transient scattering in terms of linear system theory. The final section discusses the singularity expansion formulation for describing electromagnetic scattering. Natural and forced response scattering components can be expressed in terms of the singularity expansion and make target identifica­ tion possible based on singularity expansion parameters. Singularity expansion formulation is the basis for SEM radar.

UWB RECEIVERS Chapter 10 discusses UWB receivers as an extension of the conventional receivers. Mr. James D. Taylor and Mrs. Elizabeth C. Kisenwether discuss threshold and correlation detection of UWB signals. One issue in UWB receiver design is waveform preservation vs. detection, which is driven by post­ processing use of the signal. Threshold detection preserves the waveform and is simple, but requires high signal-to-noise ratios (SNR). Any receiver can detect a UWB signal, or impulse, if it is strong enough. Narrowband receivers can detect strong UWB signals, and specifically designed UWB receivers can detect weaker

9 UWB signals. The section on threshold detection includes a discussion of receiver bandwidth and estimating received impulse strength for detection. Correlation detection indicates the presence of a signal which resembles the reference signal and can work with lower SNRs than threshold detection. Signal processing and target identification schemes such as Fourier analysis and SEM require a high SNR, which implies a shorter range than simple signal detection. Correlation detection is a detection method for weak signals which do not require preserva­ tion. Either case will require some minimum signal strength and this chapter provides guidance for estimating it. The chapter ends with some concepts for applying photonic technology to UWB receivers and advanced signal processing concepts.

HIGH ORDER SIGNAL PROCESSING FOR UWB RADAR SIGNALS Chapter 11 is about using kernel and bispectral analysis methods to characterize radar signal returns. Dr. Vasilis Z. Marmarelis, Dr. David Sheby, Mrs. Elizabeth C. Kisenwether, and Mr. Todd A. Erdley present high order signal processing concepts applied to impulse radar test results. This chapter describes the results of applying higher order signal processing to impulse signals reflected from steel plates, radar-absorbing material (RAM), and clutter materials. Applying kernel and bispectral analysis techniques can produce unique target signatures for each type of reflector. This chapter demonstrates an advanced concept for recovering information from reflected waveforms for target detection and identification.

UWB RADAR PERFORMANCE PREDICTION Chapter 12 is about UWB radar performance prediction using the same principles as the classical radar equation. Dr. Terrence Barrett discusses advanced concepts in signal reception and processing and how they can affect radar performance.

VI. CONCLUSION The UWB radar systems will evolve as technology can support UWB radar construction and as functional requirements demand some advantage that only UWB signals can provide more efficiently than other methods. In all cases there will be requirements, cost, and performance tradeoffs when deciding whether to use UWB radar in a particular role. The potential advantages of UWB radar systems are low probability of interception signals and the capability of sensing target shapes and materials through advanced signal processing. However, UWB radar is only one of many potential solutions to remote sensing and surveillance, and the user must make the final choice.

REFERENCES 1. OSD/DARPA, Ultra-Wideband Radar Review Panel, Assessmentof Ultra- Wideband (UWB) Technolo­ gy, DARPA, Arlington, \A, 1990. 2. Vickers, R., Ed., Ultrahigh Resolution Radar, SPIE Proceedings, Vol. 1875, SPIE, Bellingham, WA, 1993. 3. Skolnik, M.I., Introduction to Radar Systems, 2nd ed., McGraw-Hill, New York, 1980, 16, 432. 4. Sullivan, W.B., The Evolution of ultra-wideband receivers, J. Electr Defense, \bl. 14, No. 7, 1991. 5. MITEQ Corp., AFS Series Amplifiers, Hauppauge, NY 1992. 6. Dixon, R.C., Spread Spectrum Systems, 2nd ed., John Wiley & Sons, New York, 1984, chapter 2. 7. Moffatt, D.L., Target impulse response-historical development, Ultra-Wideband Radar: Proceedings o f the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991, 125—139. 8. Morgan, M.A. and Larison, P.D., Natural resonance extraction for ultra-wideband scattering signa­ tures, Ultra-Wideband Radar: Proceedings o f the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991, 125.

10 9. VanBlaricum, M.L. and Larry, T.L., Systems considerations of resonance based target identification, Ultra-Wideband Radar: Proceedings o f the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991, 393-403. 10. Baum, C.E., On the Singularity Expansion Method for the Solution of Electromagnetic Interaction Problems, AFWAL Interaction Note 88, December 11, 1971. 11. Turhan-Sayan, G. and Moffatt, D.L., K-pulse estimating and target identification for geometrically complicated low-Q scatters, Ultra-Wideband Radar: Proceedings of the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991, 435-462. 12. Jouny, 1.1, and Walton, E.K., Target identification using bispectral analysis of ultra-wideband radar data, Ultra-Wideband Radar: Proceedings o f the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991, 405-416. 13. Vickers, R.S., Gonzalez, YH., and Ficklin, R.W., Results from a VHF impulse synthetic aperture radar, UltrawidebandRadar, LaHaie, I. J., Ed., SPIE Proceedings, Vol. 1631, SPIE, Bellingham, WA, 1992, 219-226. 14. Swedish-developed radar to penetrate foliage, ground, Aviation Week Space Technol., 52-55, 1993. 15. Sheen, D.R., Wei, S.C., Lewis, T.B., and deGraff, S.R., Ultrawide-bandwidth polarimetric SAR imagery of foliage obscured object, Ultrahigh Resolution Radar, Vickers, R. S., Ed., SPIE Proceed­ ings, Vol. 1631, SPIE, Bellingham, WA, 1992, 106-113. 16. Pollock, M.A., Pusateri, VP, Tice, T.E., and Wehner, D.R., Ultrawideband radar facility and measured results at the Naval Ocean Systems Center, UltrawidebandRadar, LaHaie, I.J., Ed., SPIE Proceedings, Vol. 1631, SPIE, Bellingham, WA, 1992, 206-219. 17. Harmuth, H.F., Nonsinusoidal Waves for Radar and Radio Communications, Academic Press, New York, 1981, 30-46. 18. Lucius, J.E., Olhoeft, G.R., and Duke, S.K., Eds., Third Int. Conf. Ground Penetrating Radar: Abstracts of the Technical Meeting, U.S. Geological Survey, Open File Report 90-414, May 1990. 19. Kolcum, E.H., GPS, other new technologies help clear ordnance from Kuwaiti desert, Aviation Week Space Technol., 54-55, 1992.

Chapter 2

Technical Issues in Ultra-Wideband Radar Systems Harold F. Engler, Jr. CONTENTS I. Introduction ....................................................................................................................... II. Fundamental Radar P rin c ip le s.......................................................................................... III. Classification of Radar Waveforms.................................................................................... IV. Technical Issues in UWB Radar System Design ............................................................ V. S u m m a ry ............................................................................................................................. References ............................................................................................................................................ Appendix 2A: Signal Characteristics Governing Range and Velocity Measurement R esolution................................................................................................... Appendix 2B: Range Accuracy Requirements for Velocity Estimation from Differential Time D e la y ............................................................................................................ Appendix 2C: The Concept of Nonlinearity......................................................................................

11 14 28 31 41 43 44 48 49

I. INTRODUCTION CHAPTER OVERVIEW This chapter introduces the basic concepts of ultra-wideband (UWB) radar waveforms and systems. It begins with a discussion of the capabilities afforded by conventional systems which, in turn, motivate the study of UWB waveforms. This set of existing capabilities also serves as a baseline for comparison of capabilities offered by UWB systems. Following that discussion is a brief review of key research areas which have brought UWB technology into focus. At the time this book was published, the appropriate areas for application of UWB technology were still fluid and to some extent uncertain; to aid the reader in discerning which applications may be appropriate, this chapter continues with more detailed material on fundamental radar principles and a discussion of the current technical issues concerning the use of UWB signals for radar.

COMMENT ON SIGNAL ANALYSIS METHODS The common tools of waveform analysis and synthesis such as the Fourier and Laplace transforms carry some implicit assumptions about the nature of the system under consideration. One of these assumptions is that the system which generates and/or processes the signal of interest does not change within the duration of the signal. This property is generally called time-invariance. Another assumed property is that the system is linear (see Appendix 2C for a discussion of the concept of nonlinearity). Basically, linearity means that the system response does not change in response to changes in the magnitude or the complexity of the signal being processed. For narrowband systems, the requirement for linear time invariance (LTI) is usually met. It turns out that the LTI property is also met in most cases for UWB systems as well. However, due to the very high resolution properties of some UWB waveforms, there is an increased likelihood that the LTI assumption may not hold, particularly for time invariance. Hence, it is important to recheck these assumptions on a case-by-case basis when applying conventional analysis tools.

EXISTING TECHNOLOGY CAPABILITY There are a number of fundamental physical principles that bound the performance of any radar system. The approach to, and the feasibility of, attaining improvements in radar performance must always be 0-8493-4440-9/95/$0.00+$.50 © 1995 by CRC Press. Inc.

11

12 viewed in light of these boundary conditions. Some examples of radar performance dependencies are presented in Table 2.1. In many areas, current technology has already reached the lknits of achievable performance. In light of these limitations, it is important to seek out new approaches which sidestep these limitations, or at least move them out of the way a little more. The UWB signaling approaches are being examined to see if they offer any unique advantages in this regard. While the conclusion on this issue is not yet definite, Section IV of this chapter provides some discussion of the possibilities which make continued investigation worthwhile. Before proceeding, it is necessary to be more precise about the meaning of the word ultra-wideband (also ultrawideband). There are many similar sounding names in the radar and electronic counter­ measures literature (e.g., ultrabroadband, wideband). Foreign literature adds more (e.g., superwideband). The key feature of a UWB waveform is its large instantaneous percent bandwidth. The percentage bandwidth is defined as follows:

%BW = ~ 7- J

h

■*100

+ /l

where f H and f L are the upper and lower band edges of the signal, respectively. (The percentage bandwidth is also referred to as fractional bandwidth when not converted to percent.) Hence, it is also important to define the term band edge. In conventional narrowband systems, it is common to define the band edges as the frequencies at which the power spectral density is 3 dB below what it is at the center of the spectrum. This is convenient in the narrowband case because the spectrum is generally symmetrical about the center frequency and because the spectral region between the 3 dB points contains approximately 90% of the spectral energy. We will see later in this chapter that the spectrum of a UWB signal is not symmetric; in fact, in some cases the majority of the energy lies below what might be called the center frequency (see Section III). A Defense Advanced Research Projects Agency (DARPA) panel on UWB technology published one definition for UWB which has come into somewhat common use. The panel’s criterion for UWB was 25% .9 However, a more supportable limit is 20%, because this is the percentage bandwidth above which the angle and time/frequency resolution properties become coupled (see also the section on Measurement Resolution and Reference 11, p. 2096).

HISTORICAL BACKGROUND Conventional radar waveform design utilizes a small percent-bandwidth signal in order to take advantage of sinusoidal resonance effects. Resonance increases frequency selectivity and antenna efficiency. Strictly speaking, a circuit or an antenna is only resonant at a single frequency, but near­ resonance conditions can persist in many designs at frequencies up to 10% from the resonant frequency. However, this percentage is still quite confining on the allowable combinations of center frequency and bandwidth. To avoid this percent bandwidth limitation, some radar waveform designs incorporate multiple center frequencies (or a frequency sweep). However, the instantaneous percent bandwidth used in these designs is still confined to obtain the benefit of sinusoidal resonance. There are some radar applications that inherently require a larger percent bandwidth. The earliest known application was an Army need for detection of buried objects (a need that persists today). Research reports on this topic date back to the early 1960s.1 A relatively long wavelength is required for propagation into the Earth’s surface, and a relatively large bandwidth is needed to get acceptable resolution of the measured depth of the buried object. The nominal set of parameters for this application is a center frequency of 1 GHz with a bandwidth of 1 GHz for a percent bandwidth of 100. Other potential applications, which are discussed in more detail later in this chapter, are target imaging, foliage penetration and rejection of certain types of clutter, and the detection of low cross-section targets. At the time that UWB requirements were emerging, there were (and still are) a number of research and development activities which made it possible to test some of the theories of UWB systems. One

13 Table 2.1

Examples of Current Performance Limits

Performance Parameter

Limitation

Detection range

Depends on average effective radiated power (ERP), target response, propagation medium, and clutter

Average ERP

Depends on antenna gain, transmit power, and duty cycle; current systems already approaching 100% duty cycle limit; transmit power limited by hardware technology; antenna gain limited by acceptable aperture size

Target response

Generally decreasing in size; can approach the size of a small bird

Target identification

Increasing need to perform this function; methods include imaging and natural resonances; resonance method requires low frequencies; imaging requires large (1 GHz) bandwidth; neither of these require­ ments is easily met with existing technology.

Propagation

Propagation is medium sensitive; need for transmission through earth or water requires low frequencies, while maintaining good resolution properties requires wide bandwidth

Clutter suppression

Achieved using moving target indicator (MTI) or Doppler processing methods, both of which measure target and clutter velocity; perfor­ mance of both depends on the stability of signal phase; stability (short term coherence) of sinusoidal oscillators limits this; other clutter signals are received through the antenna sidelobes; tradeoffs among beamwidth, aperture size, and average sidelobe level limit the sidelobe clutter suppression

Velocity measurement

Resolution is dependent on the time available for target observation; since the observation time required to obtain the required accuracy generally exceeds one pulse width, ambiguities are introduced with conventional methods

Range measurement

Resolution is dependent on the bandwidth of the signal; for ordinary pulses, bandwidth is inversely proportional to pulse width; shorter pulses have less energy, leading to reduced detection range.

of these research activities is in the area of high power/short pulse radiation. Sources and radiating mechanisms for these pulses were developed for simulating nuclear electromagnetic pulse (EMP) effects to test the susceptibility of electronic components and systems. Another related research area was concerned with the study of what has been called time-domain electromagnetics.2'5The main purpose of this work was to develop more complete methods of character­ izing the reflection properties of radar targets. In the process, however, facilities were developed to test new theories, and these facilities included new methods for generation and radiation of UWB signals. A third related research area originally started with the investigation of alternative kernels for waveform analysis. The fast Fourier transform (FFT) algorithm was published in 1965 ;6 however, the

14 capability of rapidly multiplying digital words of the required size was not generally available at that time. For this reason, other basis functions, particularly binary-valued basis functions such as Walsh functions, were being investigated for use in alternatives to the Fourier transform.7These binary-valued basis functions were orthogonal, of course, and recognizing that resonance can be viewed in terms of preferred dimensions in an orthogonal function space, a more generalized theory of resonance was defined (see Reference 8, pp. 319 to 320). This work eventually grew into the largest published volume of work on baseband radar. As the results of these research efforts grew, so did the number of claims for potential improve­ ments in radar performance. Recently, a DARPA panel conducted their own study of the potential value of several radar techniques in the general category of UWB radar. The results of the study panel were published9 and a brief summary of their findings was also published by the IEEE.10 The panel’s report set aside many of the previous claims; however, there are some areas worthy of further exploration.9 In the end, the decision to use a UWB waveform will be dependent on whether that approach is more cost-effective than a modified conventional approach.

II. FUNDAMENTAL RADAR PRINCIPLES The current interest in UWB radar is based on the expectation that this type of radar will provide improvements in one or more of the following areas: Target detectability through radar cross-section RCS enhancements and improved clutter suppression Target identification through improved measurement resolution Reduced cost through employment of high power switches as transmitter sources To be able to recognize when a UWB waveform does offer unique performance advantages, it is important to understand how radar performance capabilities are related to the radar waveform character­ istics. This section presents a discussion of those relationships. Figure 2.1 shows a generic system block diagram for a radar, including the involvement of the target, antennas, and the propagation medium. This section presents the fundamental radar performance measures for each of these required blocks. Having these fundamentals in mind should make it easier to recognize when there is an advantage to using a UWB waveform.

MEASUREMENT RESOLUTION It can be shown (see Appendix 2A) that the resolution of a range and Doppler measurement is dependent only on the signal bandwidth and duration, respectively, regardless of the design of the actual waveform. Appendix 2A also shows that ambiguities in the measurement of range and Doppler are dependent upon waveform periodicities in the frequency and time domain, respectively, regardless of the waveform design. Hence, questions concerning measurement precision can be addressed strictly in terms of signal bandwidth, duration, and periodicity. The choice of carrier frequency and, hence, the percent bandwidth are irrelevant to the resolution and ambiguity in measuring range and Doppler shift; further, the achievement of a desired resolution does not necessarily require a UWB waveform. The coupling between the time- and frequency-domain resolutions is commonplace in conventional waveform design. (See Reference 28, p. 126, for a discussion of the radar uncertainty relation.) For example, a longer duration signal will generally have poorer range resolution but better Doppler resolution. Interestingly, when the percent bandwidth is large enough, a coupling arises between the time and frequency resolution capabilities and the angular resolution as well. In Reference 11, the expression for a four-dimensional ambiguity function is developed. The four dimensions are range, velocity (Doppler), azimuth, and elevation. When the signal percent bandwidth is sufficiently small (less than 20%), the four-dimensional function can be separated into the product of two two-dimensional functions: (1) range and Doppler, and (2) azimuth and elevation. Conversely, when the percent

15

Figure 2.1 Generic radar system block diagram showing components that may be influenced by application of UWB waveforms.

bandwidth is large, there is a coupling among all four resolution coefficients. We will pursue this matter further in the discussion of antennas in Section IV where it will be shown how the time-domain characteristics of the signal have an impact on the angle measurement capabilities of the radar.

TARGET CHARACTERISTICS Reflectivity Concepts The reflection of an electromagnetic wave occurs when (1) a wave meets a discontinuity in the characteristic impedance of the propagation medium (e.g., a target object in free space) causing currents to be generated in the object, and (2) the currents flowing in the object cause a signal to be re-radiated. The overall reflection from a complex target is, therefore, dependent on several factors, including •

Electrical size (relative to wavelength) of all reflectors Spatial relationships among all reflectors Angle of reflection of re-radiated signal

A target can be considered to consist of a collection of standard geometric shapes such as spheres, disks (or plates), or ellipsoids. The reflectivity of each of these reflector types contains three reasonably distinct regions. For example, Figure 2.2 shows the RCS of a sphere. In the figure, we see the Rayleigh region where the signal wavelength is much longer than the size of the reflector. In this region, the reflector is very inefficient, and the reflected energy is inversely proportional to the fourth power of the wavelength. In the resonance region, the wavelength of the signal is of the same order of magnitude as the size of the reflector. At certain frequencies in this region, the reflector can be caused to resonate, which enhances the reflected signal strength relative to that at other nearby frequencies. In the optical region, the wavelength becomes considerably shorter than the reflector and the reflectivity becomes much less sensitive to changes in wavelength. For some target shapes, the reflectivity becomes invariant with frequency in this region. The perceived reflectivity of a target is also heavily dependent on the angles of incidence and reflection of the radar signal. For example, the reflection from the fuselage of an aircraft can be quite large, but this high reflectivity is not available to the radar unless the receiver is positioned at the angle of reflection which results from the angle of incidence. For a monostatic system (where the transmit and receive antennas are collocated), this condition occurs only when the aircraft fuselage is nearly broadside to the radar beam. Under other conditions, this large return is reflected in a direction away from the receiver.

16

CIRCUMFERENCE/WAVELENGTH -2 n o /X Figure 2.2 Target RCS for sphere vs. circumference/wavelength. (From Skolnik, M. I., Radar Hand­ book, McGraw-Hill, New York, 1970. Copyright 1970 by McGraw Hill. With permission.)

In general, the reflectivity of a target for monostatic systems can be reduced by manipulating the angle of reflection so that even this broadside reflection is not large in the monostatic direction. Two methods for doing this are (1) design in as many curved surfaces as possible and (2) construct the target surface from randomly oriented facets. Table 2.2 presents the mathematical relationships for the RCS of many common reflector shapes. Note that in the last five entries in the table the RCS increases with wavelength X. The curvature of the target produces a reduction in the monostatic RCS as the frequency increases. Where it is not practical (e.g., for aerodynamic reasons) to employ one of these curved surfaces, it is also possible to reduce the reflectivity by faceting the surface. Faceting produces a surface roughness which makes the reflection more diffuse (i.e., scattered in many directions). An example RCS plot for a notional target that employs these RCS reduction techniques is shown in Figure 2.3. Note that, at high frequencies, the reflectivity is low for most aspects; however, there are also some narrow angular regions where the reflectivity is quite large. Note also that, as the frequency is reduced, the tall spikes become shorter but much wider; also, the low regions become higher. The relative size and location of the reflectors that make up a complex target also influence its composite reflection. The physical size of a reflector (relative to the signal wavelength) controls the amplitude of the response. The position of a reflector (relative to the others) controls the relative time delay of its reflection. In conventional systems where the wavelengths are such that operation is all in the optical region of the target and the range of wavelengths is small (i.e., small percent bandwidth), the impact of these two target features is seen in the variation in amplitude and phase of the reflected signal. The amplitude fluctuation is generally referred to as scintillation and the phase fluctuation is referred to as glint. When the wavelength is increased such that operation in the Rayleigh and resonance regions is present and when the percent bandwidth is increased, the target can impose some additional alterations on the illuminating signal. For example, when the signal bandwidth includes the target resonant frequencies, the reflecting surfaces will return energy at the resonant frequencies more strongly than

17 Table 2.2

Radar Cross Section of Selected Geometrical Shapes

GEOMETRY

TYPE

FREQ. DER

SIZE DER

FORMULA

REMARKS

MAXIMUM SQUARE TRIHEDRAL CORNER RETROREFLECTOR

\2jta4 X2

STRONGEST RETURN; HIGH RCS DUE TO TRIPLE REFLECTION

MAXIMUM

&

RIGHT DIHEDRAL CORNER REFLECTOR

L4

%na2b2

X2

SECOND STRONGEST; HIGH RCS DUE TO DOUBLE REFLECTION, TAPERS OFF GRADUALLY FROM THE MAXIMUM WITH CHANGING 0 AND SHARPLY WITH CHANGING

MAXIMUM

FLAT

L4

PLATE

4 jta 2b2

X2 (NORMAL INCIDENCE)

MAXIMUM

L3

CYLINDER

2jia b 2

X2

(NORMAL INCIDENCE) MAXIMUM

SPHERE

FO (NORMAL INCIDENCE)

THIRD STRONGEST; HIGH RCS DUE TO DIRECT REFLECTION, DROPS OFF SHARPLY AS INCIDENCE CHANGES FROM NORMAL.

PREVELENT CAUSE OF STRONG, BROAD RCS OVER VARYING ASPECT (0), DROPS OFF SHARPLY AS AZIMUTH (Cj)) CHANGES FROM NORMAL. CAN COMBINE WITH FLAT PLATE TO FORM DIHEDRAL CORNER REFLECTOR.

PREVALENT CAUSE OF STRONG, BROAD RCS PEAKS OTHER THAN THOSE DUE TO LARGE OPENINGS IN TARGET BODY ENERGY DEFOCUSED IN TWO DIRECTIONS.

Note: Adapted and revised from Knott, E.F. et al., Radar Cross Section, Artech House, Dedham,

MA, 1985, 178-179.

18 Table 2.2

GEOMETRY

Radar Cross Section of Selected Geometrical Shapes (continued)

TYPE

FREQ. SIZE DER DER

FORMULA

f( * s

&

STRAIGHT EDGE NORMAL INCIDENCE

CURVED EDGE NORMAL INCIDENCE

0 = FO

L2

Q

=

REMARKS LIMITING CASE OF 2-DIMENSIONAL CURVED PLATE MECHANISM AS RADIUS SHRINKS TO 0. PREVALENT CAUSE OF STRONG, NARROW RCS PEAKS FROM SUPER­ SONIC AIRCRAFT

O in t)^

ASPECT INTERIOR d i ­ ANGLE BETWEEN FACES MEETING AT EDGE

h ed r a l

F -1

ftm,) y

/( 9 .

L1

LIMITING CASE OF 3-DIMENSIONAL CURVED PLATE MECHANISM AS PRINCIPAL RADIUS SHRINKS TO 0. THE FUNCTION f IS THE SAME AS FOR SHAPE ABOVE.

a >

A

LIMITING CASE OF PREVIOUS MECHANISM AS a SHRINKS TO 0. FOR a = P, THE TIP IS THAT OF A CONE. FOR a = 0, THE TIP IS THE CORNER OF A THIN SHEET, OR FIN.

X2g(a, P, 0 , (j>) APEX

F-2

LO

a ,p = 0,(p =

DISCONTINUITY OF CURVATURE ALONG A STRAIGHT LINE, NORMAL INCIDENCE

ASPECT ANGLES

*

64 jt3 m \a f Y I

F-2

a >

LO jr = “ ,

-j- =

dx

DISCONTINUITY OF CURVATURE OF A CURVED EDGE

INTERIOR ANGLES OF TIP

F-3

- \dx K f TJ A

JUMP IN RECIPROCAL OF THE RADIUS OF CURVATURE SLOPE OF SURFACE w.r.t. INCIDENT RAY

IMPORTANT MECH­ ANISM FOR TRAVELING WAVE BACKSCATTER WHERE RCS OF DIS­ CONTINUITY IS AUG­ MENTED BY GAIN OF TRAVELING WAVE STRUCTURE. DEPEND­ ENCES ARE BASED ON DIMENSIONAL CON­ SIDERATIONS.

L-1

f(6,(p) b

=

FUNCTION OF ASPECT

= RADIUS OF EDGE > X

_3

DISCONTINUITY OF CURVATURE ALONG AN EDGE

F“ 4

L -2

g

(0 ,0 ) =

FUNCTION OF ASPECT

STRONGEST OF AN IN­ FINITE SEQUENCE OF DISCONTINUITIES. VERY WEAK MECHANISM WHICH TOGETHER WITH THE SHAPE ABOVE SHARES DOMINANCE OF NO SE-O N RCS OF CONE SPHERE.

IMPORTANT MECH­ ANISM FOR TRAVELING WAVE BACKSCATTER WHERE RCS OF DIS­ CONTINUITY IS AUG­ MENTED BY GAIN OF TRAVELING WAVE STRUCTURE. DEPEND­ ENCES ARE BASED ON DIMENSIONAL CON­ SIDERATIONS.

19 o 40

30

90

270

24

120 10 GHz 1 GHz 300 MHz

180 Figure 2.3 Example RCS from notional low-RCS target. At common radar wavelength, this target exhibits very strong reflectivity over very narrow angular sectors. Note, however, as the wavelength is increased that reflectivity is reduced, while the angular region is increased.

than at others. Also, since these resonant surfaces are likely to be at different physical locations on the target, each of the resonant frequency components will receive a different time delay before retransmis­ sion. This frequency-selective distribution in time delay means that the target becomes frequency dispersive. In general, dispersion in the signal frequency components produces distortion in the time domain characteristics of the signal. This distortion may be usable for identification of the target, but it may also make the signal less distinguishable from the accompanying clutter signal.

Target Identification Target identification can be accomplished by radar in several ways. The two methods that may require the use of a UWB waveform are addressed in this section. The first method illuminates the target with a sufficiently broadband signal to be able to estimate its resonant frequencies. These estimates can be used as a template for identification purposes. The other approach is to use sufficiently precise range resolution so that all of the major scatterers of the target can be resolved individually, thereby generating something of an image of the target. These two methods have very different demands on the waveform design. We will consider the target resonance method first, based on Reference 14. Because the natural body resonances of the target are related to its physical size, this method requires that the minimum frequency of the signal be such that its wavelength is 2 to 4 times the length of the resonant scatterer of the target. The entries in the center column of Table 2.3 show the implications of this requirement. Higher frequencies in the waveform will excite higher modes of these same resonances, but the fundamental resonant frequency generally provides the strongest response. To enhance the reliability of target identification, it is desirable to excite as many resonances as possible, and it is therefore necessary to use a signal of sufficient bandwidth to do so. From Table 2.3 it would seem unlikely that the upper frequency limit for this purpose would be much above 100 MHz, because the smallest resonators on a complex target will probably not be smaller than a small missile. Nonetheless, even if the signal must only cover 2 to 4 MHz, this is still a large (66%) percent bandwidth signal.

20

Table 2.3 Waveform Requirements for Target Identification3

Target Type

Minimum Frequency for Resonance (MHz)

Minimum Bandwidth for Imaging (MHz)

Bomber** Fighter0 Small missile0

2-4 5-10 50-100

20 50 500

a All waveform requirements based on composite values taken from Table 2.4. b Waveform requirements for the bomber are based upon the wingspan dimension. c Waveform requirements for the fighter and the missile are based upon the fuselage dimension.

It is important to point out an operational limitation of this technique. It may not be feasible to produce the signal-to-noise ratio (SNR) necessary to obtain sufficiently accurate measurements for this technique to work.15 The resonance extraction algorithms will always provide estimates of resonances based on the measured data, but these estimates may not be relevant to the target features. There are some ways in which a UWB signal can be avoided. If the target responds in a linear manner, it does not have to be illuminated with all frequencies in the band of suspected resonances simultaneously; a swept waveform could be used. Also, if the approximate location of the resonant frequencies is known, a multifrequency waveform could be used. However, if good range resolution is also needed (e.g., to isolate the target response from clutter signals), the required bandwidth, in combination with the low center frequency needed for resonance excitation, will result in a UWB waveform. Since good range resolution would almost certainly be required, this method of target identification would, in general, always require a UWB waveform. The second target identification method produces a one-dimensional “image” of the target (which is often referred to as range profiling). This technique requires the signal bandwidth to be at least equal to the value in the third column in Table 2.3. However, the spectral region in which the signal is located is not important to this target identification capability. Unless a low center frequency is needed for some other reason, then this target-imaging capability would not require a UWB waveform. Table 2.4 provides dimensions of typical targets.

Clutter Rejection Part of the original interest in UWB waveforms was the anticipation that they might provide some improved capability to separate small targets from clutter. This section describes three methods for clutter rejection/target enhancement to show where a UWB waveform may offer unique advantages: resolution cell size reduction, increased wavelength, and waveform polarity discrimination. The usual method for achieving clutter rejection is to isolate the clutter reflections from the target in one or more of the four measurement domains (range, Doppler, azimuth, and elevation). However, reducing the center frequency (as is done in some UWB waveforms) may also reduce clutter reflection from foliage. Also, a baseband type of UWB waveform may provide a unique capability of isolating a certain class of targets (high permeability) from clutter. The method for doing this is described later in this section.

Clutter Cell Size Reduction Generally, any steps taken to reduce the size of the resolution cell in any of the measurement domains will provide an improvement in clutter rejection. Note that if the radar makes ambiguous measurements of range or velocity, the ambiguous clutter returns may diminish the utility of a reduced resolution cell.

21

Table 2.4

Dimensions of Typical Targets1617

Target Type B-52 bomber F-15 fighter F-16 fighter 737-400 transport Tomahawk missile Standard ARM

Wingspan (ft)

Fuselage Length (ft)

185 42 32 93 9 3

160 63 49 100 21 15

Also, note that if the cell size is made so small that the target return is distributed across several resolution cells, a degradation in detection sensitivity will occur. In any case, range or Doppler resolution is only dependent on signal bandwidth or duration, respectively (see Appendix 2A). Angle resolution can be used to suppress clutter by reducing the angle cell size. Figure 2.4A shows the relationship between the illuminated clutter cell size and the radar antenna beamwidth. In Figure 2.4B the resolution provided by the waveform exceeds that provided by the antenna. For small percent bandwidth waveforms, angle resolution is completely determined by the antenna design and the center frequency. Ultra-wideband waveforms may offer unique advantages here because the resolution properties of range and velocity are coupled to the resolution in angle (see Section II and also Reference 11, p. 2096). With this coupling it may be possible, for example, to use excess range resolution to improve azimuth or elevation resolution. The rejection of clutter outside the resolution cell is not perfect — this is true for all of the measurement dimensions: range, Doppler, and angle. This imperfection is due to additional smaller cells, called sidelobes, which are displaced from the desired cell. Examples of sidelobes are shown in the ambiguity function in Figure 2.5. The center lobe in the function represents the resolving capability of the waveform. The additional lobes away from the center are sidelobes: regions of imperfect clutter suppression. In small percent bandwidth systems, the size and location of the range and Doppler sidelobes are controlled by the waveform parameters; the angle sidelobes are controlled by the antenna design. In UWB systems, the angle sidelobes are influenced by both the antenna design and the waveform design. Hence, UWB waveforms may offer some unique advantages for sidelobe clutter rejection.

Increased Wavelength Clutter rejection also may be improved by taking advantage of the fact that longer wavelengths are better able to penetrate foliage and natural vegetation.18 A fairly extensive source of measured clutter reflectivity for narrowband signals is provided in Reference 19. These data show that for some types of clutter the reflectivity is inversely proportional to wavelength. For other types, it is actually directly proportional to wavelength, and in others there is no monotonic behavior with wavelength. Hence, this data would not suggest an optimal frequency for clutter rejection. However, some other data obtained from measurements made with a UWB waveform (Reference 20, Appendix F) show that the reflection at these frequencies for heavily forested terrain can be dominated by the reflection from the trunks of trees, not the foliage. It could also turn out that the dominant clutter source may be from the ground beneath the foliage, rather than the foliage itself. If this turns out to be the case, then whatever attenuation is provided by the foliage will actually tend to suppress the return from the ground reflection. From these data, it seems that the performance of a system which must detect targets within or above foliage could probably be improved by moving the signal to a lower center frequency. Choosing a low center frequency (e.g., at UHF or below) for good foliage rejection while maintaining good range resolution requires a UWB signal.

22

a) MLC AREA - BEAM-LIMITED

b) MLC AREA - PULSE-LIMITED

Figure 2.4 Radar resolution effects on clutter cell size. In (A) the cell size in the elevation dimension is limited by the beamwidth of the antenna. In (B) the cell size is limited by the resolution of the waveform.

Waveform Polarity Discrimination Another possible method for clutter suppression which may be possible for baseband UWB waveforms (see Baseband Waveform in Section III) exploits the reflectivity property of the target itself. The reflection coefficient at a boundary between two media, p, is (Reference 21, p. 151)

P

g r _ T)2-T)l £, 112+ 11,

(2 . 1)

where (Reference 21, p. 128)

(2 .2 )

and = characteristic impedance in ohms a = the medium conductivity in mho/meter p = magnetic permeability, in Ilenry/meter s = electric permittivity or dielectric constant, in Farad/meter When the reflecting medium is a conductor, a is large. In a perfect conductor it is infinite, and then for that medium is zero. Suppose that in Equation 2.1, medium 1 is free space and medium 2 is a perfect conductor. Under these conditions, p becomes -1, which means that the orientation of the Efield vector is reversed on reflection. However, note from Equation 2.1, that it is possible to prevent the orientation of the E-field vector from reversing by making r/2 larger than Recognizing that tj

Figure 2.5 Example of ambiguity function showing time and frequency sidelobes. The center lobe in the function represents the resolving capabil­ ity of the waveform. The additional lobes away from the center are sidelobes — regions of imperfect clutter suppression. In small percentbandwidth systems, the size and location of the range and Doppler sidelobes are controlled by the waveform parameters, and the angle sidelobes are controlled by the antenna design. In UWB systems, the angle sidelobes are influenced by both the antenna design and the waveform design. Hence, UWB waveforms may offer some unique advantages for sidelobe clutter rejection.

0.8

co

24 we see by comparing to Equation 2.2 that for r)2>ril requires that

(2.3)

In most natural nonmagnetic materials, ju » /*0 and therefore for Equation 2.3 to hold would require the material to have a relative dielectric constant less than one (which is impossible). However, for high permeability materials, it may be possible for Equation 2.3 to hold and therefore produce a reflection without reversing the orientation of the E-field vector. The type of material with these properties is referred to as magnetic RAM (radar-absorbing material) and has the advantage of a high index of refraction \f\0T at lower frequencies. This property causes the material to exhibit an electrical thickness which is many times the actual thickness. For example, one nickel-zinc ferrite has an index of refraction above 50 at 100 MHz, making the material appear to be 50 times as thick electrically as it actually is (Reference 13, p. 255). This type of material, then, makes an effective radar signal absorber for longer wavelength signals. Conversely, it provides the capability for using much thinner coatings to absorb higher frequencies. However, the interesting feature of this type of material in the present discussion is that it may not produce a reversal of the Efield vector, even if ]}

(2A. 2)

where eft) is a composite measurement error process due to errors in measurement of the signal phase and errors in measurement of the time at which the phase was measured. The first term on the right side of Equation 2A.2 is the actual velocity v. The second is the total measurement error which can be made arbitrarily small choosing a sufficiently large value for T. The observation interval is limited in practical monostatic systems (systems that employ a common transmit and receive antenna) in order to operate the transmitter and receiver on a time-shared basis. The isolation achievable between the transmitter and receiver is such that, if the receiver had to operate at the same time as the transmitter, a significant reduction of receiver sensitivity would result. Hence, in a time-shared operating mode a contiguous observation interval would be limited to the round-trip propagation time to the closest target of interest. Typical minimum times are a few tens of microseconds. Thus, a velocity estimate (based on measurement of frequency shift) cannot be made in a single observation with acceptable accuracy, but, continuous observation of the changing phase is unnecessary since the expression for the velocity estimate based on a sequence of N discrete pulses can be written as

V= ^

m

„

y - T ) + t(tn) - e(r„ - Ts)]

(2A.3)

where Ts is the interval between successive phase measurements = T/N. Since (tn-TJ = (j>ftn_}) and eftn-Ts) = e(tn_j), the estimate collapses to

. X v = ^ W t„) ~ (/,) may be determined unambiguously. To do so requires that 0 (f) be sampled at least twice for each rotation through 2ir radians. For example, a Doppler frequency shift of 50 kHz would rotate through 2ir in 20 /*s. Therefore, the maximum value for Ts would be 10 ^s, or the sampling rate (which is usually referred to as the PRF) would be 100 kHz.

Range Measurement A relationship analogous to Equation 2A.2 can be written by making the following substitutions measurement time t 1 are not equal to zero, then the system is nonlinear. Substituting the sum of inputs from the examples above into this polynomial function will show that the terms for i > 2 will cause cross products of the terms in the input sum of functions.

Chapter 3

Analytical Techniques for Ultra-Wideband Signals Muriladhar Rangaswamy and Tapan K. Sarkar CONTENTS Preface P art 1: Fourier Analysis of Signals I. Introduction ....................................................................................................................... II. Information and B andw idth............................................................................................... III. Fourier Analysis of S ig n a ls............................................................................................... IV. Properties of the Fourier Transform.................................................................................. V. Dirac-Delta F u n ctio n .......................................................................................................... VI. Fourier Transform of Periodic Signals ............................................................................ VII. Numerical Computation of the Fourier Transform .......................................................... VIII. Spectral D e n sity .................................................................................................................. IX. Time Correlation of Power Signals .................................................................................. X. Power Spectral Density of Random Signals .................................................................... XI. C o n clu sio n .......................................................................................................................... References ............................................................................................................................................

52 55 56 62 65 68 70 71 73 74 74 74

P art 2: Laplace Transforms and Signal Analysis I. Introduction ....................................................................................................................... II. Laplace Transform s............................................................................................................. III. Inverse Laplace Transform s............................................................................................... IV. Properties of Laplace Transform ....................................................................................... V. One-Sided Laplace Transform .......................................................................................... VI. Applications of the Two-Sided Laplace Transform ......................................................... VII. Pulse Propagation in a Long Medium ............................................................................ VIII. C o n clu sio n.......................................................................................................................... References ............................................................................................................................................

75 75 80 81 83 84 88 90 90

P a rt 3: Limitations of Time and Frequency Approaches I. Introduction ....................................................................................................................... 91 II. Considerations in Performing Time Domain M easurem ents......................................... 92 III. Data Acquisition.................................................................................................................. 93 IV. Processing Considerations.................................................................................................. 94 V. Transformations from the Discrete to the Continuous Domain .......................................104 VI. Experimental V erification.................................................................................................... 105 VII. Characterization of Objects in the FrequencyDomain ......................................................105 VIII. C o n clu sio n.............................................................................................................................106 References .............................................................................................................................................. 106

0-8493-4440-9/95/$0.00+$. 50 © 1995 by CRC Press, Inc.

51

52 PREFACE For solving problems of science and engineering in a systematic fashion it is necessary to have a mathematical formulation, a mathematical solution, and finally a physical interpretation. First, the physical problem is described by a mathematical equation. The next step involves a successful solution to the mathematical equation. Finally, one needs to interpret the results of the mathematical equation and relate it to the physical problem. The area of ultra-wideband (UWB) technology is no exception.7 Conventionally, in electrical engineering one deals with steady-state solutions to physical problems. A steady-state solution is generally a solution to a given physical problem when the system is excited by a narrowband or specifically by a single frequency signal. However, when the system is excited by a wideband of signals, the problem becomes quite different. For example, typically an antenna pattern is defined at a single frequency. What is the antenna response to a pulse? Even though the mathematics for such an analysis has been available for over 100 years, it is difficult to find this information. Alternately, what is the impulse response of an antenna or, for that matter, the propagation of an impulsive field through a lossy medium. These are interesting problems that have to be addressed in order to develop a basic understanding of developing a consistent comprehensive treatise on UWB technology. All of these physical problems can be transformed into a mathematical equation. The next step is to obtain a solution to the mathematical equation. To this end, Part 1 describes the Fourier techniques that can be used to carry out broadband analysis of systems which are excited by signals with finite energy. Impulsive waveforms that carry infinite energy have to be analyzed utilizing the Laplace transform techniques as described in Part 2. The physical implications of the solution tech­ niques are presented in Part 3. Finally, it needs to be clearly stated that analysis of all systems excited by wideband waveforms can be adequately treated by the two-sided Laplace transforms (Fourier transforms are a special case of the Laplace transform) and has been available in the mathematical literature for over 200 years. One does not need new mathematics to analyze wideband systems; therefore, time domain results contain equivalent information as frequency domain results, only if the researcher is capable enough to interpret the data. Moreover, relative bandwidth (17%, 25%, or 90% of center frequency) is totally irrelevant to the analysis as the Fourier and Laplace transforms are quite general in nature and require no such bandwidth restrictions. Those who fa ll in love with practice without science is like a ship without a rudder moving with the waves not knowing where she is going. -Leonardo da Vinci

Part 1 FOURIER ANALYSIS OF SIGNALS I. INTRODUCTION A signal can be defined as a single-valued function of time that conveys information. Time is the independent variable, and for every value of time the function which is the dependent variable must have a unique value. The function can be real or complex and accordingly the signal is called a real or complex signal. We must note that the independent variable, i.e., time, is always real. We can also look upon the signal as a mapping which gives us a unique value for every value of the independent variable. Signals may be classified as 1. 2. 3. 4.

Periodic and nonperiodic signals Deterministic signals and random signals Energy signals and power signals Analog, discrete, and digital signals

53 PERIODIC AND NONPERIODIC SIGNALS A signal g(t) is said to be periodic if it satisfies the condition g(t) = g(t+nT0)

n = ± 1, n = +2 ...

(3.1)

where T0 denotes the fundamental time period and t denotes time. T0 is the time elapsed before the signal repeats. A signal that does not satisfy Equation 3.1 is called a nonperiodic signal. Figure 3.1 illustrates the definition of periodic and nonperiodic signals. A periodic signal therefore reproduces itself after every time period T0, whereas a nonperiodic signal does not. A periodic signal may have a large bandwidth or may contain a single frequency component.

Deterministic and Random Signals A deterministic signal is one in which there is no uncertainty about the value of the function (dependent variable) at any time instant. Hence, a mathematical function whose value is known for all values of time can be used to describe a deterministic signal. A random signal is one in which there is uncertainty about the value of the function (dependent variable) until it occurs. We can think of such a signal as one of an ensemble of signals, where each signal in the ensemble is different. An ensemble is defined as the collection of all possible signals in a particular class. Each signal has an associated probability of occurrence; therefore, it must be emphasized here that even when we talk about a random signal, the signal is always deterministic. The uncertainty lies in which deterministic waveform from the collection of signals ensemble occurs. In the ensemble, the probability of occurrence of a particular waveform is determined by the probability of occurrence of each of the waveforms.

Energy and Power Signals In an electrical system the two available measurements are current and voltage. Both of these quantities can be time varying and hence we can associate time functions with them. Accordingly we can speak of a voltage signal or a current signal. We can consider the following circuit shown in Figure 3.2 where u(t) = input voltage, i(t) = current in the circuit, and u0(t) = voltage across R. The output power (power dissipated in the resistance R) is given by

(3.2)

or (3.3)

It must be noted that Equations 3.2 and 3.3 give the instantaneous power. Clearly, from Equations 3.2 and 3.3 the instantaneous power is directly proportional to the square of the amplitude of the instantaneous voltage or current. Convention in signal processing is to work with a 1T2 resistor. This is only a matter of convenience so that the instantaneous power has the same mathematical form regardless of whether we work with a voltage or current signal. Let g(t) denote a voltage or current signal. Assuming that R = 112, we can write from Equations 3.2 and 3.3 the instantaneous power as P = |g (0 | 2

(3.4)

54 g ( t ) = g ( t + n TJ

A.

B

Figure 3.1 Periodic and nonperiodic signal examples. (A) Periodic signal g(t) repeating itself every interval (period) T0. (B) Nonperiodic signal with no apparent repetition over any interval.

+

Figure 3.2 An RC circuit. v(t) = input voltage; i(t) = current in the circuit; v0(t) = output voltage.

The total energy of the signal g(t) can then be defined as

E = | |g(t) |2 dt

(3-5)

The average power of the signal g(t) is given by

P =

lim T

1 oo

W

I

g(t)f dt

(3.6)

55 In order for g(t) to be called an energy signal, it must satisfy the condition 0 < E < oo. For a signal that has finite energy, E, it is called an energy signal. For a power signal, however, the power P has to be finite, i.e., is 0 < P < oo. We maintain that an energy signal has zero average power (the integral would be oo if the signal has an average value) and that a power signal has infinite energy. In most cases, periodic and random signals are power signals while signals that are both deterministic and nonperiodic are energy signals.

ANALOG, DISCRETE, AND DIGITAL SIGNALS A signal which is a continuous function of time and whose amplitude is continuous as well is called an analog signal. Consider the conversion of sound and light waves into an electrical signal such as the output of a carbon microphone or a photovoltaic cell. These signals are analog signals. A discrete signal is one in which the independent variable, i.e., time, takes on discrete values; however, the signal amplitude is continuous. Such signals arise when analog signals are sampled. A digital signal is one in which the independent variable, i.e., time, and the dependent variables take on a finite set of discrete values. The output of a digital computer is an example of such signals.

II. INFORMATION AND BANDWIDTH Information is a measure of uncertainty about the occurrence of an event. Let us say that we are interested in the occurrence of a particular event A. If an experiment is performed and the results concerning A become known, uncertainty is removed. Thus we say that the experiment provides information about the event A. The more the likelihood of the occurrence of an event, the lesser is the information content and vice versa. The largest frequency contained in a signal is called the bandwidth. Shannon’s theorem relates information and bandwidth. The maximum information that can be transmitted per second over a communication channel is called channel capacity. The channel capacity is dependent on two factors, i.e., bandwidth and signal-to-noise ratio (SNR). In a channel with certain specific noise characteristics (such as white Gaussian), the channel capacity is related to the bandwidth and signal to noise ratio by

C = W log(l+S/N)

(3-7)

where C W S/N

= channel capacity (bit/sec) = channel bandwidth = signal-to-noise ratio

It would appear that using a channel that has infinite bandwidth would result in its having infinite channel capacity from Equation 3.7; however, we will show that this is not true. We assume here that the signal-to-noise ratio S/N can be written as S/WN0 where W is the channel bandwidth and N0 is the power spectral density of the noise and S is the average signal power. Thus we can rewrite Equation 3.7 as

C = Wlog 1+ WN„

Taking the limit of the right-hand side of Equation 3.8 as W tends to infinity, we have

(3.8)

56

lim

W -*00 Wlog

1+ -

lim W-»oo

WNn

, *°S

1+. WN„

N„

since lim X-^OO

l+ i

Thus we have lim W -*o o

C = S/N

Hence we conclude that an infinite bandwidth channel does not have infinite channel capacity and that the channel capacity is fundamentally limited by the signal power to noise power spectral density (PSD) ratio.

III. FOURIER ANALYSIS OF SIGNALS There are several methods available for representation of signals. However, the Fourier method is the most popular because it allows us to represent a finite power signal in terms of its sinusoidal compo­ nents.

FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS Let us consider a signal g(t) obtained by taking the weighted sum of sinusoidal signals at angular frequencies 0 , co0, 2co0 ... kco0. co0 is called the fundamental angular frequency and is expressed in radians/second. co0 is given by co0 = 2w/T0 where T0 is the fundamental period and w = 3.141592... (a constant). Consequently, we may write g(t) = Zq + aj cos co0t + a2 cos 2w0t + ... + ak cos ka;0t

+ b, sin co t + b2 sin 2coot +

+ bk sin kcoot

Let us now consider the function after sometime t + T0,

Sp*TJ

2 tt

2l. c o s — (t

+ T0) + a2 cos 2

lit (t+ T o)

(3.10)

57

(3.11)

+ ... + bk sin k]2JL(t+T0) ’ 0n

Clearly g(t+T 0) = g(t). Hence, Equation 3.1 is satisfied for n = 1. Similarly we can show that Equation 3.1 is satisfied for n = -1, ± 2, ±3.... Hence we conclude that the signal g(t) given by Equation 3.11 is a periodic signal. We have thus established that a signal obtained by a weighted sum of sinusoids at frequencies 0 , cj0, 2oj0, ... koj0 is a periodic signal. It can be shown that the converse of this statement is also true. It is also clear from Equation 3.10 that appropriately specifying the coefficients (i = 0, 1, 2 ... k) and b4(i = 1 ,2 , ... k), we can obtain a variety of periodic signals. Equation 3.10 can be written in a more compact form as follows: K K g(t) = a, + £ am cos mo),/ + Y , m =1

bm sin mw,f

(3.12)

m=l

Later on, we will make use of the form given by Equation 3.12. In order to specify the coefficients ao, am(m = 1, ... k) and bm(m = 1, ... k) we use the following orthogonality relationships. Orthogonality makes possible the superposition of individual frequency components. O cos pcj(t cos qwot dt

= '

P ^ q

X()

0

p * q

(3.13)

sin pa)ot cos qwot dt

Orthogonality physically implies that when two functions (for example, cos pw0t and cos qco0t) are multiplied and provide a zero average over the time period T0, the result is zero.

Integrating both sides of Equation 3.12 over the interval (0, T0) we see that

(3.14)

Multiplying both sides of Equation 3.12 by cos mco0t and integrating over the interval (0, T0) we have

58

am = j - | 8(0 cos " H 1 dt

^3' 15^

Finally, multiplying both sides of Equation 3.12 by sin mw0t and integrating over the interval (0, T0) we have T0 bm =

y | S(t) sin ma)0t dt

(3-16)

The representation of this form of Equation 3.12 is called trigonometric Fourier series representa­ tion. It is worthwhile pointing out that the limits of integration in Equations 3.14 through 3.16 need not necessarily be between 0 and T0. Any interval such that the values of the signal over one entire period is available can be used, for example, between the interval (to, T0 + to).

COMPLEX EXPONENTIAL FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS We recognize that a sinusoidalsignal of angular frequency nw0 can be expressed interms of complex exponentials ejn 1

(3.20)

G -nn = -2 (an + jbn) n n

Therefore, to obtain the response of a system excited by an arbitrary shaped periodic waveform, the waveform needs to be decomposed first into the Fourier components; co0; 2co0; 3w0, etc. Then each component response of the system is obtained and finally the total response is obtained by linear superposition of each component response. The problem of determining the response of a system to an arbitrary shaped periodic waveform can be solved in either of two ways. One way is to solve the differential equation governing the response in the time domain. This will be illustrated in the next section on Laplace transform. The problem can also be solved in the frequency domain. This section illustrates how to use the Fourier techniques which are more suited to the solution of a problem in the frequency domain than in the time domain. The mathematical subtleties will be discussed later. Often there is a misconception that time domain solutions (1) are better, (2) are more accurate, or (3) contain more information than frequency domain solutions. However, with an adequate mathemati­ cal knowledge of Fourier and Laplace techniques it can be shown beyond any reasonable doubt that both time and frequency domain techniques provide equivalent information. Even for a nonlinear system this statement holds. That is why we have discussed only these two mathematical transform techniques in this presentation.

COMPLEX EXPONENTIAL REPRESENTATION OF NONPERIODIC SIGNALS AND FOURIER TRANSFORMS We next consider the representation of nonperiodic signals using complex exponentials. We first construct a periodic signal gp(t) such that the nonperiodic signal g(t) defines one cycle of gp(t). This is illustrated in Figure 3.3. T0 denotes the period of gp(t). We can allow the period T0 to become infinitely large so that there will be no overlap between successive pulses. Hence we can write

g(t) = T -.00 gp(t)

(3.21)

We have already seen that gp(t) can be represented as the weighted sum of complex exponentials. In order to obtain a similar representation for g(t) we need to consider the representation of gp(t) under the condition T0->oo. We can write

(3.22a)

where

60

The nonperiodic signal g(t)

defines one cycle of gp(t). which has a period T0

If T0 becomes infinitely large so that there is no overlap between successive pules, then

g(t) = lim g (t) T0>00

^

which can be modeled as weighted sum of complex exponentials

gp (t) = Gn e inatf Figure 3.3 A nonperiodic signal.

V2

3n = ir J 8p'(t) 0

e _jna,ot dt

(3.22b)

-T 0/2

Let us define Aai = — ■ con = r p ’ n

T

and G(w) = GnTo

Making this change of notation and using Equations 3.22a,b we have the following relationships for the interval T T 0 < t
oo, Equations 3.23 and 3.24 become oo

g(t) = J _ 2ir

f J

G(w)

doj

(3-25)

and

G(w) = (

g(t) e-J"1 dt

(3.26)

and G(co) is called the Fourier transform of g(t). Equations 3.25 and 3.26 give us the method for obtaining the Fourier transform and inverse Fourier transform of a transform pair. The following symbolic representation is used to describe this relationship. g(t)

G(co)

(3'2T)

The Fourier transform described by Equation 3.26 makes sense only if the function is absolutely integrable; therefore, we conclude that if a function is magnitude integrable, i.e., t)| dt

(3-28)

then the Fourier transform exists. We can also show that if a function has finite energy, i.e.,

( k(of dt ^ oo

(3.29)

62 the existence of the Fourier transform is guaranteed. Hence for signals such as step-, ramp-, and impulse-like functions, the Fourier transform is not clearly defined. In order to obtain the Fourier transform of these signals we need to use the theory of generalized functions. The Laplace transform is used for the problems in which Fourier techniques cannot be applied. Hence, any signal g(t) that satisfies either of the above conditions is said to be Fourier transformable; otherwise, the function may not be Fourier transformable.

IV. PROPERTIES OF THE FOURIER TRANSFORM Here we will briefly study some of the properties of the Fourier transform including duality, scaling, time shifting, frequency shifting, frequency differentiation, area under g(t), and convolution.

DUALITY Duality means that if g(t) ±5 G(w), then G(t) ** 2% g(-w)

(3'30)

This property is useful in obtaining Fourier transforms of signals for which the integral given by Equation 3.26 is cumbersome to evaluate. Consider the signal given by

g(t) -

1

|t| < r/2

0

elsewhere

(3.31)

Then using Equation 3.26 we have G(co) =

= r s in c J ifll, \ 2ir)

oj r/2

0 .32)

w here

sinc(x) =

sin 7rx 7rx

Now, if we need to obtain the Fourier transform of a signal given by g(t) = sine (at), the integral given by Equation 3.26 cannot be evaluated easily. However, using the duality property we see that 7r/a

G(u) =

0

|u | < a elsewhere

(3.33)

SCALING If g(t) ^ G(oj), then for a real constant a, we have g(at) ±5

JL

G(w/a)

(3.34)

la l i.e., if the function g(t) has a change of time scale, then its Fourier transform has both a change of scale and a scaling of the amplitudes.

63 TIME SHIFTING If g(t) * G(co), then g(t-a) f* ej“a G(oj)


t dt

=

lim a-*°

_ _1_1 = _2_ a-joi J jco

Finally, we can write sgn(t)

*

f_ l_ | a+jco

(3.61)

The signal is shown in Figure 3.6.

Unit Step Function The unit step function denoted by u(t) is defined as

u(» - 1 >

;

; »

0) as

(a > 0)

(3.100)

From Equation 3.95 we have the Laplace transform of g(t) as

dt

i

e (s+a)t dt

Letting s = a + jo>, we have G(a+jo>) =

Comparing with Equation 3.100 we can write

[

e ~(a+a)t e"jcjt dt

(3.101)

77

G(ff+jco) = _— I — _ (ff+a) + jcj

,

(ff+a) > 0

or alternatively G(s) = —1— (s+a)

,

Re{s} > -a

(3.102)

when a = 0, G(s) is nothing but the Laplace transform of u(t) and is given by

G(s) = J. s

,

Re{s} > 0.

Just as the Fourier transform does not converge for all signals, the Laplace transform may converge for some values of a and not for others. Here, the Laplace transform converges for Re{s} > -a only {real part of complex s, i.e., a}. If a > 0, we can obtain the Fourier transform of g(t) by letting a = 0 and thus we have

G(0+jw) = — -— jw + a

(3.103)

If a is negative or zero, the Laplace transform still exists but the Fourier transform does not. The shaded region in Figure 3.8 represents ROC.

Example 2.2 We now consider the case when g(t) = e"at u(-t). Then we have,

G(s) = - f

e"at e"st u(-t)

dt

“ 00

1 (s+a)

o = -j

e"(s+a)t dt

-oo

Re{s+a} < 0 or Re{s} < -a

Hence we have -e "at u(-t)

*

^

—1— * (s+a)

Re{s} < -a

(3.104)

Clearly, the Laplace transform obtained in Equations 3.104 and 3.102 have identical algebraic forms. However, the region of the complex s-plane for which the transform is valid is entirely different in the two examples. The ROC for this example is shown in Figure 3.9. The shaded region indicates the ROC.

Example 2.3 Next we consider the case when g(t) = e_t u(t) + e‘2t u(t). Clearly

78 lm(s)

JO

g(t) = ea'u(t)

G(s) =

- I

Re{s] > -a

Figure 3.8 Region of convergence for Re{s} > - a.

Figure 3.9 Region of convergence for Re{s} > - a.

G(s) = |

g(t) e "st dt = |

e “(s+1)t u(t) dt + J e"(s+2)t u(t) dt

or

G(s) =

_L

(s+1)

+

1

(s+2)

(3.105)

79 Equation 3.105 gives us the algebraic expression for G(s). We now need to specify its ROC. We have

e-

u(t)

e"2‘ u(t)

^

^

_ i_ (s+1)

Re{s} > -1

^ — «------" (s+2)

Re{s} > -2

where Re{x} denotes the real part of x, clearly G(s) is the sum of the two transforms indicated above. Thus we need to consider the region of the s-plane when both of these transforms converge. This region is Re{s} > -1. Thus we can write e _t u(t) + e 2t u(t)

^ - ^ s + ,j *------* (s2+3s+2)

Re{s} > -1

(3.106)

In all of the three examples seen thus far, the Laplace transform is a ratio of polynomials in the complex variable s, that is of the form G(s)

=

D(s)

(3.107)

where N(s) and D(s) are the numerator and denominator polynomial, respectively. If G(s) is in this form it is called a rational function. This is always the case when g(t) is a linear combination of real or complex exponentials, such functions frequently arise when we deal with linear systems. The roots of the numerator polynomial are called zeros of G(s), since G(s) = 0 at those values of s. The roots of the denominator polynomial are called poles of G(s) since G(s) becomes unbounded at those values of s. The poles and zeros of G(s) are plotted on the complex s-plane using X for poles and O for zeros. Figure 3.10 shows the pole-zero plot for Example 2.3 along with its ROC.

THE REGION OF CONVERGENCE FOR LAPLACE TRANSFORMS We have pointed out in the previous section that mere specification of the algebraic expression for the Laplace transform is not sufficient. In order to establish the uniqueness of the Laplace transform, we need to specify the ROC of the transform in the complex s-plane. We will now present some properties of the ROC that are of relevance. The proof of most of these properties is carried out in an intuitive manner rather than by a rigorous procedure.

Property 1 The ROC of G(s) is made up of strips parallel to the jco axis of the complex s-plane. This is evident from the fact that the ROC of G(s) consists of those values of s = a + jco for which the Fourier transform of g(t) e_ot converges. Hence the ROC is dependent only on the real part of s.

Property 2 If the Laplace transform G(s) is a rational function, then the ROC does not contain any poles. We maintain that G(s) becomes unbounded at the poles and hence the integral given by Equation 3.95 does not converge, and therefore Property 2 holds.

Property 3 If g(t) is of finite duration, as shown in Figure 3.11, and the Laplace transform converges for at least a single value of s, then the ROC is the entire s-plane.

J3 e 'fu(t) + e 2tu(t)


-1

1

Figure 3.10 Region of convergence and pole-zero plot for example 2.3

g(t)

Ay\A

Figure 3.11 A finite-duration signal.

Property 4 If g(t) is a right-sided signal and if the line Re{s} = cr0 is in the ROC, then all the values of s for which Re{s} > (70 will also be in the ROC. We consider the right-sided signal shown in Figure 3.12. We assume that the Laplace transform of g(t) converges for some value of 0, for example, 0Oin the complex s-plane.

Property 5 If a signal g(t) is left-sided and if Re{s} = 0Ois in the ROC, then all the values of s for which Re{s} < 0Owill also be in the ROC.

III.

INVERSE LAPLACE TRANSFORM

We discussed the Laplace transform and pointed out its relationship to the Fourier transform in Part 1. Here we will point out the similarity between the inverse Laplace transform and the inverse Fourier transform. We have seen in Part 1 that we can write

81

Figure 3.12 If a right-sided signal has a line Re{s} = s0 in the region of convergence (ROC), then all values of s for which Refs} > s will also be in the ROC. oo

g(t) =

_L 2 ir

[ G(a+jw) ej“‘ da)

J

(3.108)

Replacing a + joo by s and noting that a is constant so that ds = j dco, we have the inverse Laplace transform equation

g(t) =

JL

2 ttJ\o-

[ G(s) e st ds JJOO

(3.109)

Thus we conclude that given G(s), g(t) can be obtained by evaluating the integral of Equation 3.109 using a contour that is a straight line parallel to the co axis in the complex s-plane for any value of a such that the transform G(s) converges. The evaluation of the integral of Equation 3.109 requires the theory of contour integration in the complex s-plane and this is beyond the scope of this book. However, for the class of functions for which G(s) is a rational function, the inverse Laplace transform may be obtained by using a partial fraction expansion of G(s) of the form

G(s)

E i =1

A, (s+b,)

(3.110)

Then using the ROC of G(s) we can obtain the ROC for the individual terms of Equation 3.110. Implicit herein is the assumption that there are no repeated roots in the denominator polynomial of G(s). A good discussion of the case where there are repeated roots in the denominator polynomial of G(s) can be found in Reference 3.

IV. PROPERTIES OF LAPLACE TRANSFORMS Here we will present some of the useful properties of Laplace transforms. We will merely state the properties; the proofs are left as an exercise for the reader.

Property 1 — Linearity If g /t)^ ____^ ( s ) with ROC denoted by R u and g2(t) 1. (C) Pole location ? < 0.

Using the Laplace transform pair listed in row 4 of Table 3.1, we can write the impulse response in this case as

h(t) =

u(t)

(3.138)

2v/F T We now recognize that to obtain the step response we need to obtain the inverse Laplace transform of

V0(s) = H(s) V.(s) =

(3.139)

vo(t) = | h(r) dr = | h(r) dr

(3.140)

since Vj(t) = u (t). Using Equation 3.118 we have

since h(r) = 0 for r < 0.

88 Here for the three cases, we have (!)

vo(t) = u(t) / V - F

-

e"K ‘ cos [wn \ / l - f 2 t - fl] u(t)

where 6 = tan"

V0(t)

(2)

= u(t) - u(t)

and f < 1.

{t W„

(3.141)

e"""' + e ' “■*} and (3.142)

f = I-

(3) C,t

v„W

1+

(3.143)

c 2(t)

u(t)

with

2 /F T

t =

1.

The impulse response h(t) and the step response v0(t) have been plotted for several values of f in Figure 3.15A,B.

VII- PULSE PROPAGATION IN A LOSSY MEDIUM Consider the propagation of a UWB signal impulse propagating in a medium with an effective lossy permittivity of e and a conductivity a. The magnetic permeability of the medium is fi. The objective is to find the transient electric and magnetic fields if the excitation is an impulse. The mathematical equation satisfied by the electric field E, for example, is given by d 2E ^ dE d 2E n ue ----- + aa — - ----- = 0. dt2 P dt dz2 Now, it turns out the solution to this problem is similar to the solution of a pulse propagation in a lossy transmission line. For example, if R, L, G, C are the resistance, inductance, conductance, and capacitance per unit length of a lossy transmission line, then the voltage, v(z,t), on the transmission line is given by LC d 2v(z,t) + (RC+LG) dvVd) + RG y(z t) _ d 2v(z,t) = Q d t2 dt dz2 Here the transmission line is assumed to be oriented in the z-direction. An elegant solution to this problem has been given by Pipes in 1958 in his classic textbook1 utilizing the Laplace transform. The solution for the current on the transmission line satisfies a similar differential equation satisfied by the electric field. The original solution first developed more than 30 years ago is given as

89

Figure 3.15 Impulse and step response of a series RLC circuit with different values of f from +.1 to 1.5. (A) Impulse response for different values of ?. (B) Unit step response for different values of

t-_

v(z,t) = 'H*

i(z,t) =

_

t-£ V

+ — e * I.(ax) vx

t-_

+ c * f * I,(ax) - al (ax) 1

lx

J.

. u t -1 V

where u is the unit step function and 6 is the delta function, as have been defined earlier. I0 and the modified Bessel functions of the first kind and of zeroth and first order, respectively. Also,

are

Because of the existence of the step function in the above expressions, the solution is guaranteed to be causal. Now, the propagation of a pulse in a lossy medium is exactly similar to the pulse propagation in a lossy transmission line. Therefore, setting R = 0 in the above expression leads to the following equivalent v(z,t) i(z,t)

=> E(z,t) => H(z,t)

L

P

C

=» e

G

a

The electric and magnetic fields propagating in a lossy medium due to an arbitrary shaped waveform excitation can easily be obtained by convolving the above expressions with the excitation waveform.

VIII. CONCLUSION In conclusion, knowledge of the Laplace transform is extremely important in time domain analysis. Ignorance of this important topic may lead researchers to nonsensical conclusions such as Maxwell’s equations have no causal solutions or that UWB technology is something extraordinary which requires new mathematics to treat this subject. An inadequate knowledge of fundamental mathematics may lead one to conclude that time domain solutions contain more information and are better than frequency domain techniques or vice versa. The information in either domain is equivalent if the researcher knows how to interpret the results.

REFERENCES 1. Pipes, L. A., A pplied Mathematics fo r Engineers and Physicists, 4th ed., 1977, p 777. 2. Nahin, J. R, Behind the Laplace transform, IEEE Spectrum, March, 60, 1991. 3. Oppenheim, A. V., Willsky, A. S., and Young, I. T., Signals and Systems, Prentice Hall,Engelwood Cliffs, NJ, 1983. 4. Zemanian, A. H., DistributionTheory and Transform Analysis: An Introduction to GeneralizedFunctions, McGraw-Hill, New York, 1965.

91 Part 3 LIMITATIONS OF TIME AND FREQUENCY APPROACHES

I. INTRODUCTION The objective of this chapter is to obtain broadband characterization of objects. The broadband characterization can be done either in the time domain or in the frequency domain. Each method has both advantages and disadvantages. In the time domain, the objective is to measure the impulse response of a system. The impulse response represents a “ finger print” and can be used to characterize the system. In this approach a broadband pulse is transmitted (ideally an impulse), and the impulse response of the object is measured. A basic problem with time domain measurements is that a broadband characterization is desired from bandlimited measurements; therefore, an extrapolation problem exists which needs to be addressed. Further, the dynamic range of time domain measurements is very poor. Only four effective bits of data are practically available beyond a spectrum of 100 MHz. In addition, broadband, stable, jitter-free, high power sources are generally not available in the upper microwave region. On the other hand, time domain measurements involve real signals and the required measurement equipment is rather inexpen­ sive (approximately $30,000). A good anechoic chamber is not required, since undesired reflections and multipath effects can be “gated” out. To carry out time domain measurements it is necessary to deconvolve the measurement system response from the measured signal. A large ground plane is generally placed in front of the transmit/ receive antenna to obtain the response of the system. The large ground plane actually images the transmit/receive system and provides the desired measurement response. Next, the ground plane is removed, the object is put in its place, and the transient response is measured. A deconvolution is performed to extract the object response from the measured waveform, which includes the system response plus the object response. For simplicity, most time domain measurements are monostatic instead of bistatic, because the impulse response of an antenna is a function of the azimuth angle. The waveshape that is transmitted from an antenna varies in different directions. Unless the impulse response is known as a function of each angle, it is difficult to calibrate bistatic measurements. Broadband characterization can also be performed in the frequency domain. The measurements are generally done by sweeping a narrow band of frequencies across the entire spectrum of interest. In this way the complex (both magnitude and phase) transfer function of the system can be generated. Since the bandwidth of the measurement system can be made arbitrarily small (less than 0.01 %), the noise floor of the measurement system can be brought down drastically. Approximately 100 dB of dynamic range, equivalent to greater than 16 bits of data, can easily be obtained; however, the price paid for this is a very expensive and complex measurement setup. To perform the complex amplitude and phase measurements, one typically needs a vector network analyzer which itself costs approximately $250,000. In addition, an anechoic chamber is needed to eliminate the undesired responses and multiple reflections. The chamber itself can cost several hundred thousands to millions of dollars depending on the size. On the other hand, the measurement environment can be controlled quite accurately in an anechoic chamber contributing to a large dynamic range of the system. If one is only interested in making RCS measurements, then there is no need to deconvolve the system transfer function since the vector network analyzer measures the transfer function scattering parameters and the system response is assumed to be almost constant over the bend of interest. Therefore, in this case it is enough to calibrate the network analyzer at the connector ends. However, if one wants to use the same setup to perform broadband measurements, then certain additional criteria need to be introduced. To obtain a broadband response of the object, it is necessary to record both magnitude and phase. It is essential to have a reference of phase over the entire band of frequencies to obtain a time domain result. Also, since the measurement system response (namely of the connectors and the antenna) changes over a large bandwidth, it is necessary to deconvolve out the system response as mentioned earlier for the time domain. So, all the problems associated with time domain processing of data are also included in this methodology. There is also another problem. Time domain measure­

92 ments measure a causal time domain response, and it is quite easy to obtain a causal time domain response. From the causal response one can obtain information about the rise time and fall time of pulses and the like. However, in order to obtain a time domain response from a bandlimited frequency domain measurement, it is not at all clear that the inverse transformed complex bandlimited frequency domain data is going to provide a causal response (i.e., h(t) = 0 for t < 0). This is because a bandlimited system is not guaranteed to provide a causal response. In summary, broadband character­ ization can be made either in the time domain in the form of an impulse response or in the frequency domain in the form of a complex transfer function. In principle, information in both domains should be equivalent; however, the measurement setup and the processing requirements are quite different for each system. Particularly, to have the system in either domain calibrated to the desired accuracy, it is necessary not only to understand the different sources of error in hardware, but also what software is being utilized and how the data is processed. In both techniques, signal processing plays a major role in establishing the calibration standards.

II. CONSIDERATIONS IN PERFORMING TIME DOMAIN MEASUREMENTS The basic objective in this section is to provide analysis guidelines and measurement procedures for obtaining transient data from real devices. In order to achieve the above objective, it is necessary to provide guidance on: 1. how to acquire data 2. how to analyze the data in a nonparametric fashion, and 3. the errors associated with nonparametric processing. Almost all electromagnetic measurements deal with decaying exponentials that are not bandlimited. In this case, the conventional Nyquist sampling criteria does not hold. It is therefore necessary to establish some sampling criteria so that the signals can be interpolated and extrapolated without aliasing. Another characteristic of a realistic electromagnetic signal is that it may not be sampled uniformly. Of course, in practice one may want to avoid a nonuniform sampling strategy because it is quite complicated to implement and process. The problem that is addressed here may require nonuniformly sampled data records because of finite register lengths or other hardware limitations, hence the problem of how to analyze nonuniformly spaced data is also outlined in this chapter. A third problem associated with data acquisition of electromagnetic signals is determination of record length. Typically, one is dealing with decaying signals which in principle are not time-limited. If a short data record is used, then a portion of the signal is truncated. On the other hand, recording the signal for a long time after the signal has decayed down to noise level is simply measuring noise. The signal-to-noise ratio is then degraded and too much memory of the measurement system is unnecessarily consumed; therefore, an optimum strategy needs to be developed as to how long a data record needs to be measured. Once the data is acquired it will be analyzed to extract certain important features of the object. Since identification is made utilizing various feature extraction procedures, it is necessary that the procedures be reliable and it is also important to know a priori the errors associated with such processing. The data can be analyzed in either nonparametric or in parametric form. There are advantages and disadvantages associated with both procedures. The nonparametric form (such as obtaining the impulse response or applying the Fast Fourier Transform to compute the spectra) of analysis is much more robust to noise, but for nonparametric FFT techniques resolution is not very high. For example, if we have two frequencies, fj and f2, in a data record, then from the Rayleigh limit we know that to resolve both f, and f2 by taking the Fourier Transform of the data it is necessary to have a record length which is greater then l/(f2 - f{) (assuming f2 > fx). Parametric models, on the other hand, have higher resolution; their resolution is not dictated by the Rayleigh limit and they can achieve super resolution. However, their robustness in the presence of noise is poor.

93 In this subsection, we discuss the nonparametric form of processing and perform an error analysis to observe what types of accuracy can be obtained. The specific problems treated are 1. 2. 3. 4.

Tolerance of spectral estimation Error in the location of a spectral peak due to finitedata length Error in the location of a spectral peak due to additiverandom noise Error analysis of fixed point FFT computation

III. DATA ACQUISITION Since the accuracy of the analysis is totally dictated by the quality of the data, it is essential that proper care be exercised in collecting the data. This chapter discusses the three common problems that are associated with the acquisition of transient data: 1. Establishing a sampling criteria for decaying exponentials which are not bandlimited 2. Interpolating and extrapolating nonuniformly spaced data (which is not bandlimited) 3. Determining the length of transient data to be recorded This section summarizes the three problems and provides references for obtaining additional material.

ESTABLISHING A SAMPLING CRITERIA FOR NONBANDLIMITED SIGNALS If a signal is limited in time, it cannot simultaneously be bandlimited. The classical Shannon theorem, which is strictly valid for bandlimited signals, does not apply. Bolgiano found that if a signal of the form e(- | ,then all the necessary information that exists in the signal can be obtained. This result is strictlyvalid for signals that are decaying exponentials. If the signal contains a sum of complex exponentials, then the signal must be sampled at least six times the highest frequency component in the exponent to record all the relevant information without bandlimiting.1

INTERPOLATING AND EXTRAPOLATING NONUNIFORMLY SAMPLED DATA THAT IS NOT BANDLIMITED The signals being considered in this work may need to be nonuniformly sampled due to both memory limitations of the experimental setup and for theoretical reasons. At the sampling rates that will be required to adequately represent the waveforms, the number of samples may be too long for the available register lengths. Hence, it is necessary to have a high sampling rate on the leading edge of the pulse and a low sampling rate on the trailing edge. This can be achieved by either using two different transient digitizers sampling at different rates and combining the results or by using one transient digitizer and programming it to use varying sampling rates. In many cases, it is required to interpolate or extrapolate the data without aliasing. The Gregory Newton quadrature formula can do this on nonuniformly spaced, nonbandlimited data. The details are available in References 1-3.

DATA LENGTH TO BE ACQUIRED In most measurements, it is necessary to obtain sampled signals of decaying exponentials. Theoretically, the decaying exponential function becomes zero only at infinity, but it obviously is not possible to measure the data for infinite time. The question then arises as to what length of record is needed so that the “ windowing effect’’ on the data is negligible. This also addresses the question of what length of record is needed to resolve sums of decaying exponentials. For a bandlimited signal, the Nyquist rate determines the minimum number of samples over a given interval. In general, this implies that the minimum number of basic functions to represent a bandlimited

94 signal should equal the Nyquist rate times the length of the sampling interval for meaningful reconstruc­ tion. For example, we know that in order to resolve two signals of frequency fj and f2 by the FFT, we need a time record of length no less than l/(f2 - f,) with f2 > fx. However, the decaying exponential functions are neither time limited nor bandlimited. Under these circumstances, how long a record is needed in order to resolve decaying exponentials? The answer to this question may lead to a suitable design of experiments which will yield data for a meaningful analysis. In this analysis we limit ourselves to the case of two decaying exponentials, although the analysis is also applicable to a sum of decaying exponentials as will be shown later. In this case we transform the resolution between two complex exponentials to a correlation problem between two decaying exponentials. As the data length increases, the correlation between the two exponentials decreases and reaches an asymptotic value as the data length approaches infinity. The objective then is to find a data length for which the correlation between the two exponentials is within ±0.1% of its asymptotic value. If we consider two decaying exponentials of complex frequencies vx + jc^ and v2 + jco2 such that | (j2 | > | | , then it can be shown that for a data length T greater than 3/ | ctj the correlation function between two complex exponentials is within ±0.1% of its asymptotic value at infinity. That is:

The mathematical details are quite complex. The result is summarized here, but the details can be obtained from Reference 4. The above result can now be generalized to analyze a sum of decaying exponentials. A rule is provided as to how this can be achieved in real time as data is being recorded. Since ax has the smallest decaying coefficient, the signal corresponding to ax exists for a long time. As data is being recorded, the decay rate can be estimated and this will correspond to the more slowly decaying signal and will approximate ax. Once ox is estimated, the length of the data record is then given by 3/ | ox | .

IV. PROCESSING CONSIDERATIONS This section deals with the various types of errors that may be encountered in processing the time domain data.

TOLERANCE OF SPECTRAL ESTIMATION When time domain data is recorded, there is both a systematic and a random error. The systematic errors may arise due to a baseline shift, d.c. offset, etc. The random errors could be due to quantization effects or simply thermal noise. These errors are reflected in a variation of the amplitude and phase of the data. The question that arises here is how the errors reflect in the spectrum, which is the Fourier transform of the data. Specifically, one may be interested in transforming the time domain data to frequency domain to observe the spectral peaks. How does the error in the data translate to errors in the spectral peaks? This problem is addressed here. The error in the data is considered to be random. To this end, a statistical expression for the mean square error of a spectrum estimation has been derived in terms of the variances and covariances of the amplitude and phase errors of a complex data sequence. The various sources of error that can exist in the data are quantization errors, aperture error in the sampling device, jitter in the sampling device, noise and nonlinearities of the digitizer, and noise introduced by the active devices such as amplifiers and various filters used to preprocess the signals. A statistical expression for the mean value of the frequency deviation is provided. The frequency error is linearly proportional to the phase error; however, as the number of time samples are increased, the frequency and phase errors decrease.

95 In our data, the samples are real and the phase error is zero. In that case, the error in the frequency (spectrum) is linearly proportional to the amplitude error; however, for complex signals with a large phase error, the amplitude error has a diminishing influence. The details are available in Reference 10.

ERROR IN THE LOCATION OF A SPECTRAL PEAK DUE TO FINITE DATA LENGTH A common process in signal analysis is to estimate frequency through the Fast Fourier Transform (FFT) by the location of the peak in the computed spectrum of a windowed sample of the time function. In general, though, this estimate will be in error due to “ leakage” effects caused by interaction of the windowing function and the actual carrier frequency. Here we quantitatively evaluate the effect of window shape and length on the accuracy of the method of estimation. Nyquist windows are considered in this analysis.5 If we consider the time signal of infinite duration and take the Fourier Transform of the signal, the spectrum will consist of two delta functions. The signal and its Fourier Transform are shown in Figures 3.16 and 3.17, respectively. Thus, there would be infinite resolution and the two delta functions would be located at ± oj0. However, if the function f(t) = cos cj0t is truncated so that we have the function g(t) instead, as shown in Figure 3.18, then the spectrum of the function g(t) would be quite different from that of f(t). This is equivalent to viewing the ideal signal f(t) through a window w(t) (a rectangular window in this case) to obtain g(t). Thus, g(t) is obtained by multiplying f(t) by a window function w(t). Since multiplication in time domain is equivalent to a convolution in the frequency domain, the spectrum of g(t) will be the convolution of the spectrum of the window function w(t) (Figure 3.19) with that of the spectrum of f(t) (Figure 3.17). The resultant spectrum g(t) is shown in Figure 3.20. Observe that this is a sum of two sin(x)/x functions located at ± cj0. The spectrum of g(t) will be different from the spectrum of f(t) due to the leakage effects of the sin(x)/x functions. The problem is how does the spectrum of the windowed sample relate to the ideal spectrum. Since the error in the spectrum is due to the leakage effects produced by two functions located at ±co0, it is clear that for certain frequencies and/or data lengths the error would be zero. However, here we present an upper bound of the error. We also look at the effect of various windows on this error. The class of windows that we consider are those which have a Nyquist symmetry. A window of this kind is formed by the convolution of a rectangular pulse of duration (T-a) with a symmetric positive pulse of duration a. This is shown in Figure 3.21. A detailed analysis of error is available in Reference 5. Here we present a summary of the results. Figure 3.22 shows the upper bound of the error in the spectral shift due to time truncation of the data. The x-axis represents the number of cycles of the signal in the given observation interval T, v represents the shift in the spectral peak, and the y-axis represents +log10(vT). Observe that the smoothing of the shoulders of the windows can significantly reduce the error in the spectral peak. The parameter a defined in Figure 3.22 represents a fraction of the observation interval T; that is, a = aT. As an example, if T = 1.71 msec, fc = 25 kHz, and we utilize the rectangular window (oo = 0 ) , then the error in the spectral peak = 2.07 Hz. The point here is that this shift in the spectral peak may first appear trivial when assessed using the actual frequency, but may become unacceptable if heterodyning measures are taken to reduce memory. To maintain the size of the error in carrier frequency estimation to the levels shown in Figure 3 .2 2 , the windowed carrier must be sampled at rates on the order of four times the carrier frequency. Heterodyning would only increase the leakage shift v. The study in Reference 5 quantitatively evaluated the technique of carrier frequency measurement by means of the location of spectral peaks in a “ windowed” transmission sampled.

ERROR IN THE LOCATION OF A SPECTRAL PEAK DUE TO ADDITIVE RANDOM NOISE In the previous section the shift in spectral peak due to interaction of the window and carrier frequency was discussed. Addition of noise will also shift the spectral peak. Since noise is considered to be a random phenomenon, the error introduced in the spectral peak due to addition of noise will also be a random phenomenon. In this section we consider the variance of the error in the location of the spectral peak as a function of the signal to noise ratio in the data.

Figure 3.16 An infinite duration time function: f(t) = cos (w0t). Frequency domain

F(iQ)) u «

' t1

05

t1

►

Figure 3.17 Spectrum of the infinite time duration function shown in Figure 3.16, represented by ELQ, which has discrete single frequency spectrums at ± uj0.

Figure 3.18 Windowed time function of f(t) defines as g(t).

sinz

function

Figure 3.19 Spectrum of w(t). functions

3 CO

Figure 3.20 Spectrum of g(t).

Since the analysis is mathematically quite complex, we will omit it, but an interested reader may refer to References 5 and 6. The final result is presented here. We obtain

Figure 3.21 Formation of windows with "Nyquist symmetry". (A) Rectangular pulse of duration (T a). (B) Symmetric positive pulse of duration a. (C) Rectangular and symmetric pulses convolve to form a window, w(t), with "Nyquist symmetry".

where m) or 26! (7ra)m) depending on the nature of the decaying exponentials, i.e., whether h(t) is of the e'* cos0t or e * sin0t. Numerical results have been presented to illustrate the results for the bounds. Since most signals from electromagnetic systems can be characterized by a sum of complex exponentials, the bounds given here provide a limit on the maximum accuracy that can be obtained in the characterization of the impulse response from a given input and output. It is also interesting to note that the degree of accuracy is related to the sampling frequency. If an input x(t) applied to a linear system having an impulse response h(t) yields the output y(t), then we know for a causal signal y(t) = | X (t - r)h(f)df

and in the transformed domain, Y(jaj) = HGo,)X(jw)

If h(t) is real, then oo

h(t) = — | P(cj)cos wt dcj, for t > 0

where P(a>) is the real part of H(jw).

102

To obtain the impulse response h(t) numerically, the upper limit of the integral cannot be infinite but is limited to a number com (bandlimited function). The estimate of the impulse response is

h (t)

= —

f P ( w ) c o s w t dco

* I If we define the error as e(t) = h(t) - h(t) Then e(t) = —

f P (c o ) 7T J

< 7T

f |P(w)| J

coscot dco

| cos cot | dco < 7T

f IP(gj)

J

dco

It is also obvious that e(t) -» 0 as com oo. We now assume that the impulse response can be approximated by a sum of complex exponentials, i.e., h(t) = 2b e _

** ii "..:i♦.:. ; .

•< *• ’ . ' • :* • . if •• .. . *• i i * } > * f i f i 4 * * » :» * 4 * * i ■*. . . : • * ; .- * • .

If * '4 k ». 4 .4

* •• . . ? ‘ i * * * M 4

if:M.* U ha

f >

t *■■. .j'-' :,4: ' • : * ♦ M > 1 *:ifc ♦

I |» i 5

: ♦¥ ‘ 4-ijf* *Vl: i.l ♦*:* - ♦ ♦:
w; d, s are given3 by:

E =

di(t) _ s 2 d 2i(t)p r x (rxs) FTPn + -S dt rc dt 1 r Ls

471re

(5.101)

di(t) F _ s d 2i(t) dt 0 c dt2 2cos0

H =

Airrc

COS0

di(t) + s d 2i(t) F ] 1 r l dt c d t2 J rj

J

dt

_ s 2 d 2i(t) F ’ s x r sr rc dt2 '

FQ

r

(5.102)

In Equations 5.101 and 5.102, Z0 is the wave impedance of free space, s a vector of length s pointing in the z direction; c, speed of light; F0 and F,, parameters that are functions of the dimensions w, d, s of the metal sheet radiator in Figure 5.72, and the coordinates ry 0 , 0. These two parameters are defined in terms of integrals as follows: an F 0(r,0,0) = J L f "/2 ( WdS J -vv/2 -dn

( wds1

w n - w /2

r

2

an

r

r

r

f

1

_ d% dr) d£

(5.103)

J -1/2 r w

i w /2

J -d /2 J -

s /2

c

d ll

c /2 d%rvdt] - r d$ + f S

(5.104)

245

Figure 5.72 Geometry of the metal sheet radiator of Figure 5.70D. For the geometry of the sheet radiator in Figure 5.72, one obtains from Equation 5.102 the following components of the magnetic field strength H : the field component Hr = He = 0. The magnetic flux is defined as follows:

H

*

*

=

5 47T/r

Z orsin0 j

r^ ) + c i dt r

F oa

(5.105)

sind

dt (5.106)

h i 47rc In the limit r farfield,

i(t) + £

oo, the parameter F Q

J

i(t) dt F _ J 2 di(t) ° rc dt

sin20

1 and Equation 5.106 yields the magnetic flux

* m g = Urn *

= —Z—_ s in 20 47rc

in the

(5.107)

The dimension of is Vs, and it is measured in Weber (Wb). The components Ee and £ r of the electric field strength E given in Equation 5.101 can be written in terms of ^ m as follows:

Ee = - £ sin0 e r

E. =

r 2sin0

dd

(5.108) dr

E= 0

(5.109)

246 The expression of ^ m in Equation 5.106 can be rewritten in a normalized form that is more convenient for numerical evaluation by a digital computer. Let the current i(t) in Equation 5.106 be retarded by r/c and have a general time variation i = Ii(t - r/c). Introducing the normalized variables r and u, where

r = _ , t; = I

(5.110)

s

s

the driving current can be defined as follows.

i =

> 0

Ii(r - v)

T - V

0

T ~ U
s used in the derivation of E, H, and >km. The density of the flux lines in the plots of Figure 5.74 is proportional to the magnitude of the produced field strength. Hence, the regions of dense flux lines indicate the existence of a strong field. The plot of Figure 5.74 for the observation time r = ct/s = 1 0 shows the propagation of flux lines while the current pulse in Figure 5.73 is still flowing through the surface of the radiator in Figure 5.72. The plots of Figure 5.74, for r = ct/s = 20, 30, and 40, show the propagation of flux lines to the farfield at times greater than the duration of the current pulse in Figure 5.73. The propagation of the Gaussian pulses in Figure 5.73 from the large-current radiator of Figure 5.72 is well presented in Figure 5.74; the (dense) closed contours propagate further away from the origin at v = r/s = 0 as time elapses. For a periodic current consisting of only positive pulses with period T Q, the closed contours of flux lines in Figure 5.74 will be radiated at the end of each period. Flux lines of a current consisting of a positive as well as a negative pulse will be presented shortly. Figure 5.75 shows the current pulse of Figure 5.74 followed by the same pulse amplitude reversed. The derivative d i(j - v)/d(r - v) and the integration { i(X) 2 ^ P4>

(5.156)

where the small angle approximation, sin = , is applied to Equation 5.156. The peak-power pattern P() is defined as the square of the peak-amplitude pattern A(4>), P () = [A()]2 = 1,

for psin)]2 dt (5.158)

where y x(f) and y 2(t) are given in Equations 5.142 and 5.145, respectively. For on-axis reception, = 0, the energy U(0) of the Gaussian voltage signal v(t,0) given in Equation 5.146 is

273

U(0) = | *°° [v{t,0)\2dt

2mE ~AT

i :

exp

-Sir

AT

dt

(5.159)

_ s/2(mE)2 AT The energy pattern W(4>) is defined as the ratio U($)/U(0):

W(0 ) =

|

[erfc(7 ,(r)) - erfc(y2(/)) ]2dt

(5.160)

v/32"Ar(psin0)2 The effective duration of v(t,) in Equation 5.14 is 2a, where

t + psin0 AT 2

(5.161)

Thus, it will be sufficient to evaluate the integral of Equation 5.160 over the time interval - a < t < a. Each of the Gaussian pulses in Figure 5.89 that represent the voltage signal v{t,) in Equation 5.146, for different values of psin^, can be approximated by a triangular pulse by using a statistical linear-regression algorithm (LRA). The LRA approximates the two rising and falling sections of each Gaussian pulse by two least-squares linear ramps. The rising and the falling linear ramps have equal slopes of the magnitude:

N Y , t/ „ - Y , tn Y , rn S =

n= 0

(5.162) N

E '. where rn is a sample value of v(t,) taken at the time instant tn Since the peak amplitude of v(t,) is a rapidly decreasing function of psin,0) = S(a))H(o),4

S(u>) = E 2 exp

H(o},) = 2m sine

coAT

-(coAT)2 2 tt

(5.174)

(5.175)

psin0

where S(co) is the energy spectral density of the received Gaussian pulse and #(oj,0) the transfer function (or frequency response) of the linear array of sensors. The transfer function tf(o ,0 ) can be expressed in terms of wavelength X and array length L = 2md, where 2m is the number of array elements and d interelement spacing:

H(k,(j)) = 2m sinc(-!^l sin0) (5.176) ttL

2m sinc(— sin0) , X

m > 1

Note that H(k,) is of the same form as the array factor Ffl3) given in Equation 5.129 and shown in Figure 5.83. According to Equation 5.173, a linear array of sensors (or radiators) can be modeled as a linear, time-invariant, system with frequency response #(w,) and impulse response

=JL f 2ir J

"f/(co,0)exp(-ycdOdoi (5.177)

aw

n

n)

The impulse response h(t,) is a rectangular pulse of amplitude A ^ ) and duration 7,(0):

AM

=— f L 7rA7psm0

(5.178)

283 7)(4>) = TrA7>sin)/2

0

\t\ > T ' W 2

(5.180)

For on-axis reception 0 = 0, the impulse response of the linear array is a Dirac-delta function d(t) of weight 2m: = H tfi) = 2m8(t),

m > 1

(5.181)

The product i4,()r,(0), which results in the weight of /*(t,), is equivalent to the array gain G for =

0:

G = A ^ T ,{) = 2m,

m > 1

(5.182)

The transfer function H(k,(j)) and the impulse response h(t,(f)) can be considered as design tools for UWB array antenna.

VI.

SYNTHESIS OF ANTENNA ARRAY BEAM PATTERNS

According to the principle of pattern multiplication,9given the directivity pattern of an isotropic antenna element, one may synthesize a beam pattern for a linear array of such antenna elements by using the concept of array factor. The above principle can be used to synthesize beam patterns for a linear array of large-current radiators and a linear array of closed-loop receivers of the types illustrated in Figure 5.77. Let alcr( 2 5 %) are appropriately chosen such that the resonance regime of the class of targets is covered by the incident field illuminating the target. As an example, if one were dealing with identification of aircraft, the resonance regime is in the HF band, and one is required to design an HF antenna system with some frequency agility. One can think of a resistively loaded cylindrical post or, alternatively, a cylindrical post with a tuned circuit at its base. In order to obtain directivity, one can then consider an array of cylindrical posts, each with its own tuned circuit at its base. A certain amount of directivity is clearly feasible by controlling the amplitude and phase of each feed. Specific antenna systems may be designed with a well-defined set of requirements. Once again, the problems of connecting the generators to the antenna “terminals” are similar to what has been discussed earlier in this chapter.

ACKNOWLEDGMENTS The author would like to thank the many investigators who have contributed to the theory and develop­ ment of pulse-radiating antennas, including the pulse generators. Several of them are identified by references and the results of their work are used in composing this chapter. This author is especially grateful to Dr. C. E. Baum for his generosity in permitting the use of his extensive work in this field in a summarized form for the purposes of this chapter.

REFERENCES* 1. Baum, C. E., EMP simulators for various types of nuclear emp environments: an interim categori­ zation, IEEE Trans. Electmmagn. Compat., EMC-20, 35-53, 1978; also IEEE Trans. Antennas Propag., AP-26, 35-53, 1978. 2. Giri, D. V, Liu, T. K., Tesche, F. M., and King, R. W. P., Parallel Plate Transmission Line Type of EMP Simulators: A Systematic Review and Recommendations, Sensor and Simulation Note 261, April 1980, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 3. Baum, C. E. and Giri, D. V., The Distributed Switch for Launching Spherical Waves, Sensor and Simulation Note 289, August 1985, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 4. Giles, J. C., A survey of simulators of EMP outside the source region, some characteristics and limita­ tions, presented at NEM 84, Baltimore, MD, July 1984. 5. Baum, C. E., Emerging technology for transient and broad-band analysis and synthesis of antennas and scatterers, Interaction Note 300; also published in Proc. IEEE, Vol. 64, No. 11, November 1976. 6. Longmine, C. L., Hamilton, R. M., and Hahn, J. M., A Nominal Set of High-Altitude EMP Environ­ ments, Theoretical Note 354, January 1987, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 7. Baum, C. E., Some Limiting Low-Frequency Characteristics of a Pulse-Radiating Antenna, Sensor and Simulation Note 65, 28 October 1968, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 8. Baum, C. E., Review of hybrid and equivalent-electric-dipole EMP simulators, Sensor and Simulation Note 277, October 1982 and Proc. 5th Int. Zurich Symp., EMC 1983, March 1983, 147-152. 9. Brown, T. L., Giri, D. V, and Schilling, II., Electromagnetic field computation for a conical plate transmission line type of simulator, DIESES Memo 1, November 1983. 10. Simcox, G., Pulser for Vertically Polarized Dipole Facility (VPD-II), Physics International Company, Rep. No. PIFR-900, San Leandro, CA, May 1978. 11. Baum, C. E., Resistively Loaded Radiating Dipole Based on a Transmission-Line Model for the Antenna, Sensor and Simulation Note 81, April 1969, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 12. Schelkunoff, S. A. and Friis, H. T., Antennas: Theory and Practice, Wiley, New York, 1952, 425-431. 13. Wu, T. T. and King, R. W. R, The cylindrical antenna with non-reflecting resistive loading, IEEE Trans. Antennas Propag., AP-13, 369-373, 1965.

323 14. Shen, L. C. and King, R. W. R, The cylindrical antenna with non-reflecting resistive loading, IEEE Trans. Antennas Propag., AP-13, 998, 1965. 15. Giri, D. V and Baum, C. E., Airborne platform for measurement of transient or broadband CW electromagnetic fields, Electromagnetics, 9, No. 1, 69-84, 1989. 16. Singaraju, B. K. and Baum, C. E., A Simple Technique for Obtaining the Near Fields of Electric Dipole Antennas from their Far Fields, Sensor and Simulation Note 213, March 1976, AFWL, Kirtland AFB, NM. 17. Giri, D. V and Sands, S. H., Design of B Sensor for the Nose Boom of F-106B Aircraft, EM • Platform Memo 1, September 1983, AFWL, Kirtland AFB, NM. 18. Giri, D. V and Sands, S. H., Processing, Evaluation and Analysis of the Magnetic Field Data Acquired by the F-106B Nose Boom Sensor, EM Platform Memo 2, August 1985, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 19. Smith, I. D. and Aslin, H., Pulsed power for EMP simulators, IEEE Trans. Antennas Propag., AP-26, 1153-1159, 1978. 20. Barnes, P. R., Pulse Radiation by an Infinitely Long, Perfectly, Cylindrical Antenna in Free Space Excited by a Finite Cylindrical Distributed Source Specified by the Tangential Electric Field Associated with a Biconical Antenna, Sensor and Simulation Note 110, July 1970, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 21. Baum, C. E., General Principles for the Design of ATLAS I and II, Part IV: Additional Consider­ ations for the Design of Pulser Arrays, Sensor and Simulation Note 146, March 1972, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 22. Baum, C. E., General Principles for the Design of ATLAS I and II, Part V: Some Approximate Figures of Merit for Comparing the Waveforms Launched by Imperfect Pulser Arrays onto TEM Transmission Lines, Sensor and Simulation Note 148, May 1972, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 23. Pyne, Z. L. and Tesche, F. M., Pulse Radiation by an Infinite Cylindrical Antenna with a Source Gap with a Uniform Field, Sensor and Simulation Note 159, October 1972, Phillips Laboratory (AFSC)/ NT, Kirtland AFB, NM. 24. Tesche, F. M. and Pyne, Z. L., Approximation to a Biconical Source Feed on Linear EMP Simulators by Using N Discrete \foltage Gaps, Sensor and Simulation Note 175, May 1973, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 25. Baum, C. E., Early Time Performance at Large Distances of Periodic Planar Arrays of Planar Bicones with Sources Triggered in a Plane-Wave Sequence, Sensor and Simulation Note 184, August 1973, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 26. Liu, T. K., Admittances and Fields of a Planar Array with Sources Excited in a Plane Wave Sequence, Sensor and Simulation Note 186, October 1973, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 27. Baum, C. E. and Giri, D. V, Early Time Performance at Large Distances of Periodic Arrays of Flat Plate Conical Wave Launchers, Sensor and Simulation Note 299, 1987, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 28. Baum, C. E., Coupled Transmission-Line Model of Periodic Array of Wave Launchers, Sensor and Simulation Note 313, 1988, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 29. Giri, D. V, Impedance Matrix Characterization of an Incremental Length of a Periodic Array of Wave Launches, Sensor and Simulation Note 316, April 1989, AFWL, Kirtland AFB, NM. 30. Baum, C. E., Canonical Examples for High-Frequency Propagation on Unit Cell of Wave Launcher Arrays, Sensor and Simulation Note 317, April 1989, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM 31. Giri, D. V, A Family of Canonical Examples for High-Frequency Propagation on Unit Cell of Wave Launcher Arrays, Sensor and Simulation Note 318, June 1989, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 32. Baum, C. E., Radiation of Impulse-Like Transient Fields, Sensor and Simulation Note 321, November 1989, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 33. Baum, C. E., Giri, D. V, and Gonzales, R. D., Electromagnetic Field Distribution the TEM Mode in a Symmetrical Two-Parallel-Plate TransmissionLine, Sensor and Simulation Note 219, April 1986, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM.

324 34. Yang, F. C. and Lee, K. S. H., Impedance of a Two-Conical-Plate Transmission Line, Sensor and Simulation Note 221, November 1976, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM. 35. Smythe, W. R., Static and Dynamic Electricity 3rd ed., Hemisphere Publishing, 1989. 36. Brown, T. L., Giri, D. V, and Schilling, H., Electromagnetic Field Computation for a Conical Plate Transmission Line Type of Simulator, DIESES Memo 1, November 1983. 37. Baum, C. E. and Stone, A. R, Transient Lens Synthesis: Differential Geometry in Electromagnetic Theory Hemisphere Publishing, 1990. 38. Rahmat-Samii, Y, Reflector Antennas, Antenna Handbook, Lo, Y T. and Lee, S. W., Eds., Van Nostrand Reinhold, 1988. 39. Baum, C. E., Rothwell, E. J., Chen, K. M., and Nyquist, D. P., The SEM and its application to target identification, Proc. IEEE, (special issue on electromagnetics). 40. Tesche, F. M., private communication. 41. Baum, C. E., The singularity expansion method, in Transient Electromagnetic Fields, Felsen, L. B., Ed., Springer-Verlag, 1976, chap. 3. * Note that the Sensor and Simulation Notes, Theoretical Notes, EM Platform Memos, and DIESES Memos referenced above are available from the Defense Documentation Center, Cameron Station, Alexandria, YA 22314, and from the Editor, Dr. Carl E. Baum, Phillips Laboratory (AFSC)/NT, Kirtland AFB, NM 87117.

Chapter 7

PROPAGATION AND ENERGY TRANSFER CONTENTS Part 1: RF Propagation in the Atmosphere I. Introduction...................................................... 325 II. UWB Propagation........................................................................................................................ 335 III. Low Power Linear Propagation through a Background Plasma................................................305 IV. High Power Propagation in Nonlinear M edia.............................................................................337 V. Summary and Conclusions........................................................................................................... 360 References.................................................................................................................................................361

Part 2: Energy Transfer Through Media and Sensing of the Media I. Introduction to Energy Transfer Concepts.................................................................................. 365 II. Advanced Theory of Dielectrics and Transmission Through Media.........................................367 HI. Pulse Envelope Effects................................................................................................................. 385 IV. Soliton Waves, Group Theory, and Electromagnetic Missile Concepts....................................404 References................................................................................................................................................ 413 Appendix 7A: Further Developments in Self-Induced Transparency...................................................425 Appendix 7B: The Nonlinear Wave Equations and Solitons................................................................ 427 Appendix 7C: Relation of U (l) and SU(2) Symmetry Groups.............................................................428

P arti RF PROPAGATION IN THE ATM OSPHERE Robert Roussel-Dupre

I. INTRODUCTION Any application of ultra-wideband (UWB) technology involves in some form or another the interaction of an electromagnetic wave with the Earth’s atmosphere, with conducting layers such as the Earth’s surface, sea water, and metallic structures, and with various types of dielectric materials that are, for example, the targets of radar interrogation. The magnitude of phase and amplitude changes imparted to an electromagnetic wave as a result of such interactions depends strongly on the frequency of the wave, and the superposition of such effects over a broad frequency range and over large distances can result in significant alteration of the transmitted electromagnetic pulse. In this sense UWB propagation differs substantially from narrowband propagation. For example, in some cases it is possible for the atmosphere to transform an electromagnetic pulse into a chirp in which the high frequency components of the pulse arrive first at the receiver. This kind of distortion is not observed with narrowband radio waves and makes it difficult to extract pertinent information about the wave source or a radar target that is being interrogated, unless compensation can be made. Our ultimate goal in this chapter is to provide the reader with the basic tools necessary to develop a model capable of predicting the amplitude and temporal shape of a UWB signal after it has propagated through the atmosphere and interacted with various dielectric materials. While the area of broadband radio wave propagation is receiving considerable interest of late, narrowband propagation and the precise nature of the distortions introduced by the Earth’s atmosphere have been the subject of intense investigation ever since transatlantic communication was first achieved by Marconi in 1901. The notion that a conducting layer capable of reflecting radio frequency (RF) waves 0-8493-4440-9/95/$0.(X)+$.50 © 1995 by CRC Press, Inc.

325

326 existed in the atmosphere was proposed soon after an experimental verification for the existence of this layer, later termed the ionosphere, came in 1924 with the reception of ionospheric echoes from a remote, pulsed high frequency (HF), transmitter.1 Subsequent theoretical23 and experimental45 work lead to a better understanding of the detailed properties of this layer and ultimately to the development of a new field of research dedicated entirely to the study of the ionosphere (see Rishbeth and Garriott6 for a brief historical review of this subject). The role of the ionosphere in disrupting radio wave transmission and the information carried by these waves continues to be a topic of considerable interest (see Goodman and Aarons7 for a recent review). Activity in this area has been spurred on by technological advances made in communications, navigation, surveillance, remote sensing, radar technology, ionospheric sounding, ionospheric modifications, and the deployment of space-based detectors and transmitters. The implementation, in turn, of more sophisticated technologies has necessitated a more detailed investigation into the effects of the ionosphere on radio wave propagation and into the detailed characteristics of the ionospheric channel. The ionosphere, however, is a complex medium subject to diurnal, seasonal, and long term variations driven primarily by solar activity but also by local disturbances, the neutral wind, and unstable plasma configurations. These concerns have been addressed both experimentally and theoretically over the past several decades and a tremendous body of literature exists on the subject. Nevertheless, even the most detailed ionospheric models have not been very successful in predicting the characteristics of any given propagation channel for extended periods of time (exceeding tens of minutes to hours), and significant research remains to be done before adequate models are developed. In addition to the transionospheric channel, other conducting layers such as the Earth’s surface, sea water, and the metal structures that surround an RF transmitter can also affect the signature of radio waves detected at a remote receiver. Gaseous absorption and scattering by clouds, fog, and various forms of precipitation such as rain, snow, sleet, and hail can be important at microwave frequencies above 10 GHz. Transient ionization caused by meteorites or lightning can also reflect, scatter, and absorb radio waves. Finally, at very high powers, an electromagnetic wave can significantly alter the properties of the propagation channel, which in turn affects the amplitude and phase of the transmitted wave. Self-focusing and self-absorption are just two manifestations of this nonlinear regime.

CHAPTER OBJECTIVES In this chapter we examine in some detail the key issues associated with the propagation of an electro­ magnetic wave through the Earth’s atmosphere. Our objectives are to provide a theoretical framework on which more detailed investigations can be based and at the same time to present results sufficiently detailed to allow a direct application to problems of interest. The special aspects of UWB signals associated with their inherent high power levels and large spectral range are also addressed. While a comprehensive treatment of this subject is beyond the scope of this manuscript, references are used extensively to both compliment our discussions and to address special topics that are more carefully reviewed elsewhere. In order to set the stage for the more technical discussions that follow, the remainder of this section presents a brief description of important properties of the propagation medium (the Earth’s atmosphere) together with a heuristic description of the interaction of electromagnetic waves with matter. The latter discussion is divided into two parts, one treating the interaction with bound systems of charged particles and the other treating the interaction with free charges. In Section n, we define the UWB signals considered in this chapter and identify the important effects of transatmospheric propagation. In Section III, we define the low power linear regime and develop the mathematical models necessary to treat the deterministic effects of a stratified atmosphere and to characterize the statistical effects of a randomly structured atmosphere. In Section IV, we address the high power nonlinear regime, including the hydrodynamic equations necessary to characterize the plasma motion, the convenient form of Maxwell’s equations that permits a self-consistent and rapid numerical solution for the evolution of a high power electromagnetic pulse propagating through the atmosphere, and frequency scaling laws that are useful for defining effective air-breakdown and attenuation parameters applicable to a broad class of UWB pulse shapes. We do not discuss the effects of gaseous absorption and scattering by water droplets and hydrometeors (ice and fog), but instead refer the reader to available references that treat this subject in some detail. Throughout this manuscript we present concrete examples of the propagation effects under discussion. These results can be used, for example, to test propagation models.

327

Neutral Gas

Temperature (K)

Figure 7.1 Atmospheric temperature as a function of height. (From Kelley, M., Earth’s Ionosphere, International Geophysics Series, Academic Press, Orlando, 1989, 5. With permission.)

THE EARTH’S ATMOSPHERE AND PROPAGATION To a first approximation the Earth’s atmosphere can be considered to be horizontally stratified. For the neutral atmosphere, the distinction between various layers can be seen in a plot of atmospheric tempera­ ture vs. height (Figure 7.1). Radiation transport in the infrared coupled with turbulent convection cools the atmosphere near the Earth’s surface, resulting in a decrease in temperature with altitude through the troposphere at a rate of approximately 7K/km. At the tropopause (approximately 12 km altitude), the temperature gradient reverses and the temperature continues to rise through the stratosphere where solar ultraviolet (UV) radiation is absorbed by ozone. At 50 km (the stratopause) the temperature starts to decrease rapidly due to radiative cooling until a minimum of 130 to 190 K is reached at 80 km (the mesopause). The subsequent rise in temperature through the thermosphere is due to absorption of solar UV and extreme ultraviolet (EUV) radiation. The strong thermal conductivity of the thermosphere above 300 to 400 km leads to a nearly isothermal atmosphere at a temperature of approximately 1000 K. The charged and neutral composition of the Earth’s atmosphere above 100 km is shown in Figure 7.2. Below 100 km, turbulent mixing in the atmosphere maintains a uniform relative composition of the major constituents; namely, molecular nitrogen (N2) and oxygen (0 2), argon (Ar), and carbon dioxide (C 02). Photodissociation of molecular oxygen above 100 km combined with atomic and molecular diffusion lead to the observed dominance of atomic oxygen at altitudes above 200 km. Helium eventually takes over as the dominant species above approximately 600 km. The ionosphere is the charged component of the atmosphere between 70 and 1000 km. The electron density profile forms the basis for dividing the ionosphere into various layers. Figure 7.3 shows typical mid-latitude day- and nighttime profiles with the various regions identified. The term ionosphere arises in connection with the photoionization processes that create this layer. The solar EUV radiation incident on the Earth’s atmosphere is absorbed down to approximately 110 km and is primarily responsible for the formation of the ionospheric F-region (150 to 1000 km) and part of the E-region (90 to 150 km). The exponential increase in neutral density with decreasing altitude attenuates the solar EUV radiation and

Figure 7.2 Daytime ionospheric and atmospheric composition based on mass spectrometer measurements. Ion and neutral distributions below 250 km are from two daytime rocket measurements above White Sands, New Mexico (32°N, 106°W). The helium distribution is from a nighttime measurement. Distributions above 250 km are from the Elektron II satellite results of Istomin (1966) and Explorer XVII results of Reber and Nicolet (1965). (Johnson, C.Y., U.S. Naval Research Laboratory, Washington; reprinted from Ion and neutron composition of the ionosphere, by C.Y. Johnson, in Annals of the IQSY, Vol. 5, 1969, by permission of the MIT Press, Cambridge, MA. Copyright 1969 by MIT. With permission.)

328

329 10 8

10 9

1010

10 11

ELECTRON DENSITY (M ‘3 )

Figure 7.3 The daytime and nighttime ionospheric profiles shown in this figure were generated with the International Reference Ionosphere 1986 code. These profiles are typical of the mid-latitudes during spring and high solar activity. The D-region does not show up very well in these profiles but lies between 70 and 90 km. The E- and F-regions are shown between 90 to 150 and 150 to 1000 km, respectively. Note that the D- and E-regions virtually disappear at night. prevents further penetration below this altitude. The D- and lower E-regions between 70 and 110 km form by direct ionization of N2 and 0 2 by solar X-ray radiation and cosmic rays and by ionization of NO by solar Lyman-a radiation. The variation in ion composition with height also reflects the various dominant ionization processes (see Figure 7.2). The electron loss mechanisms that balance photoionization to produce the steady-state ionosphere include radiative recombination of 0 +above 110 km and dissociative recombination of N2+, 0 2+, and NO+ below this altitude. The gradual change in solar illumination with time and geographic location is largely responsible for the gross morphology and evolution of the ionospheric profile. At night the source of ionizing radiation is turned off and the ionosphere relaxes by recombination, resulting in a reduction in the F-region peak density and the disappearance of the D- and E-regions (Figure 7.3). The much longer radiative recom­ bination time scale relative to dissociative recombination causes the layer above 110 km to persist until solar illumination is restored. Seasonal variations dictated by the proximity of the sun and by changes in the angle of incidence of solar radiation are also observed in the ionospheric profile. Additional variations are driven by changes in solar activity (the solar cycle as well as transient eruptions on the solar surface) and by local disturbances (such as traveling ionospheric disturbances, or TIDs), the neutral wind, and unstable plasma configurations (e.g., Rayleigh-Taylor instabilities manifested as spread-F). Fluctuations in the solar wind-magnetosphere interaction cause magnetic substorms to develop, which ultimately result in high energy particle precipitation into the auroral oval. Impact ionization and excitation of the neutral gas in turn lead to enhanced ionization in the E- and F-regions and to the observed air-glow of the aurora. The lower atmosphere (below 70 km) is composed of molecular species with vibrational and rotational states capable of absorbing RF radiation and of various particle types and aerosols capable of scattering RF waves. When resonant interactions occur or when one is only interested in the energy scattered out of an RF beam, the radiative transfer equation discussed below forms the basis for treating RF propaga­ tion. When interference effects are of interest and weak, nonresonant interactions dominate then geo­ metrical optics prevail. Both methods of analysis have been used to describe electromagnetic wave

330 propagation in the lower atmosphere (see Handbook o f Geophysics and the Space Environment*). Most of the scattering and absorption occurs in the troposphere where the atmospheric density is highest and where high concentrations of water vapor, ice crystals, and hydrometeors exist. Scatteiing in the presence of precipitation is also isolated to tropospheric regions. The atmospheric region between 10 and 70 km has enough seed ionization so that an avalanche process driven by high power electromagnetic pulses is possible. RF propagation effects in this altitude range are therefore dominated by self-absorption and limited to high power pulses. Transionospheric propagation is best described in terms of geometrical optics and macroscopic scattering. Each of these processes is treated in more detail below.

IMPORTANT ASPECTS OF PROPAGATION In order to appreciate the limitations and approximations inherent to existing treatments of radio wave propagation in the Earth’s atmosphere, it is necessary first to understand how electromagnetic waves interact with matter. This subsection presents a simple, physical description of wave-particle interactions for both bound and unbound systems of particles. Consider the situation depicted in Figure 7.4A of an electromagnetic wave incident on a particle of charge q, whose motion may or may not be constrained by the presence of other charges as, for example, in the case of internal forces exerted within an atomic or molecular system. The incident wave exerts a force on the particle given by the Lorentz expression FL = q (E i + v x B i/c)

(7.1)

where q is the particle charge, E. and B; are the electric and magnetic field components of the incident electromagnetic wave, v is the particle velocity, and c is the speed of light. The energy transferred in this way from wave to particle is re-radiated by the accelerating charge, and the resulting radiation then interferes with the incident wave to define the final radiation field. A sketch of this process is shown in Figure 7.4B. In general, both the phase and amplitude of the scattered and incident radiation are important. This fundamental picture of single particle scattering forms the basis for all discussions and computations of wave-particle interactions. In the linear regime, the superposition principle, together with results from detailed calculations of single particle scattering, can be invoked to compute the radiation field resulting from scattering by all charges in the propagation medium. In most cases, however, it is impractical to follow the dynamical evolution of every particle as well as that of the radiation field, and, in general, some approximation is made either in the treatment of the electromagnetic waves or in the characterization of the medium. The analysis and description of propagation through any medium depends substantially on whether or not the scattering is phase coherent. The term phase coherent here means that the phase front of the incident wave as depicted in Figure 7.4 is virtually undistorted by the scattering medium over spatial scales much larger than a wavelength. This condition is generally met when the charged particles that compose the scattering medium can respond coherently to the driving electric field of the incident wave over large scale lengths, much larger than a wavelength. In general, there will always be an incoherent part to the net radiation field because of the discrete nature of the scatterers that make up the medium (e.g., Figure 7.4B) and because of their random distribution and motion. However, in the case where a large number of scattering centers are distributed uniformly over large scales, the superposition of the scattered radiation results in a canceling of the incoherent part and the production of a uniform phase front resulting from the coherent acceleration of the medium as a whole. For the most part resonant scattering by bound systems is incoherent, while propagation through a medium where the induced dipole and magnetic moments align with the incident electric field (as in the case of free charges) can result in coherent scattering. Although the general physical picture of single particle scattering described above applies to both cases, it is possible to make certain approxi­ mations that lead to different physical and mathematical prescriptions in each case. To illustrate this point we derive the scattered radiation field for both bound and unbound systems using a simple but heuristic model.

Scattering by Bound Systems Assuming a classical oscillator model with an external driving force, the nonrelativistic equation of motion for a bound particle of mass m can be written

331

constant phase

incident wave

Figure 7.4 Single particle scattering. (A) A plane wave with wavevector kinc and wavelength X is shown incident on a scattering center. (B) The incident wave with scattered radiation.

d x dt2

= q li-m y

dx dT

(7.2)

where qEi is the nonrelativistic limit of the Lorentz force (Ivl/c « 1 ), x is the particle displacement, co{) is the natural frequency of oscillation of the harmonic oscillator, and y is the classical damping constant given by ..

2q 2(°o2

7

3m c3

(7.3)

The damping constant represents the rate at which the particle momentum decays as a result of the electromagnetic radiation emitted by the particle in its unforced or natural state of oscillation. Assuming that the field of an incident wave of frequency co is given by Ei - Eoi e,a)t, the solution to Equation 7.2 can be written x = Re

(q/m)E0ieiM (co2 -c o 02) + i7io

(7.4)

where Re [arg] indicates that the real part of the argument should be taken and we have assumed that the wavelength of the incident wave is much larger than the dimensions of the bound system (hence the spatial variation of the wave amplitude eikr has been dropped). The result, Equation 7.4, indicates that the wave-particle interaction is resonant at co = co0, and that a frequency-dependent phase factor is introduced as a result of radiation damping. These results are also apparent in the expression for the scattered radiation field (see Jackson,9 p. 658),

332

Figure 7.5 Scattering geometry. (A) The geometry associated with the instantaneous and retarded positions of a charge in uniform motion. (B) The scattering geometry for an accelerating charge n equal to the location of the observer and a equal to the acceleration vector of the charge.

n x (n x a ) Es= 4 -

-

c

R

ret

(7.5)

where n is a unit vector in the direction from the charge to the observation point where the scattered radiation is measured (the appropriate geometry is illustrated in Figure 7.5), the quantity in brackets is evaluated at the retarded time (see Jackson9 for details), R is the distance at the retarded time from the charge to the observation point, and a [= d2x/dt2, can be obtained from Equation 7.4] is the particle acceleration. In addition, Equation 7.5 shows that the angular distribution of the scattered radiation depends on the direction of particle acceleration. In this derivation it was assumed that the restoring force of the harmonic oscillator was spherically symmetric so that the resulting acceleration and induced dipole moment (P = qx) align with the incident electric field. In reality, atomic and molecular systems possess discrete sets of angular momentum states in which the electron is constrained to move in particular orbits about the nucleus. The angular distribution of the scattered radiation in these cases is more complicated and depends on the orientation of the dipole moment of these states relative to the incident wave (see Griem10 for a detailed quantum mechanical treatment of scattering). Two additional effects not included in this simple model are the Doppler shift in frequency of the incident wave caused by the motion of the bound system as a whole and absorption caused by collisions among the scattering centers. The Doppler effect can be incorporated into our results by replacing the frequency 0) in Equation 7.4 by (co - V lc/kc), where V is the scattering center velocity and k is the wave vector of the incident wave. Collisional damping is included by replacing y in Equation 7.4 by y + yc where yc is the collision rate. The Doppler effect leads to frequency redistribution of the incident radiation while collisions can cause both frequency redistribution (pressure broadening) and absorption.

333 To arrive at a macroscopic description of wave-particle interactions, it is necessary to combine the microscopic, single particle scattering results with a model for the distribution of scattering centers. For our present purposes, we will assume that the scattering medium is composed of atomic and molecular systems that are uniformly distributed over scales large compared to a wavelength, and that the various electronic configurations occupied by these systems have dipole moments randomly oriented relative to the incident wave. In addition, we will assume that the various atomic and molecular quantum states are populated according to a Boltzmann distribution defined by the temperature of the medium. From the microscopic results obtained above, it is apparent that the resonant structure of bound systems can lead to significant absorption and scattering at incident wave frequencies corresponding to the transition energies (represented by co0) of the various quantum states. Frequency redistribution is important when natural (due to radiative damping), Doppler, and/ or pressure broadening become comparable to bandwidths of interest. Furthermore, because of the random orientation of the atomic and molecular configurations, the resonant nature of the interac­ tions, and the compact spatial distribution (scales much smaller than radio wavelengths) of scattering centers, the scattered radiation appears isotropic over macroscopic scales and all phase information associated with the incident wave is lost. As a result, it is only necessary to describe the resulting radiation field in terms of its intensity Iv, i.e., the energy per unit area, per unit time, per unit frequency, directed into a particular solid angle. In this formulation the interaction (collisions) between waves (photons) and matter is described in terms of scattering and absorption cross-sections and the radiative transfer equation given by (chv)"'[8lv/3t + (n •V)lv] = (tiv - XvI v)/(hv)

(7.6)

where h is Planck’s constant, v is the frequency of the wave, n is the direction of propagation of the intensity beam, V is the spatial gradient operator, r|v is the emissivity of the medium, and %v is the extinction coefficient which can be used to compute the temporal evolution and spatial and angular distribution of the intensity of the radiation field (cf. Mihalas11 for a thorough discussion of radiative transfer). This treatment represents a tremendous simplification over the more general analysis in which both the phase and amplitude of the waves are computed. In Figure 7.6, a monochromatic beam of radiation is shown incident on a layer composed of molecular and atomic species that absorb and scatter the radiation. The emerging beam is reduced in intensity as a result of angular and frequency, redistri­ bution, and absorption.

Scattering by Free Charges The microscopic results for scattering by free charges are obtained by setting co0 in Equation 7.4 to zero. In this case frequency redistribution is caused only by Doppler and pressure broadening. More impor­ tantly, the acceleration of the charge results in a dipole moment that is always oriented in the direction along the electric field of the incident wave (as opposed to bound systems which undergo transitions between states with dipole moments that are randomly oriented). In a medium composed of a uniform distribution of charges, the acceleration vectors line up along a phase front of the incident wave and the scattered radiation from the individual charges sums to produce a phase front that lags the incident wave in time by some fixed amount (equal to 180° in a collisionless medium). Given that the amplitude of the incident wave varies spatially as eik 1 and that there exists a relative phase between the incident and scattered wave fronts, the net radiation field (incident plus scattered) can be described as a wave whose phase propagates at a speed other than the vacuum speed of light. This fact will become evident in the derivation given in Section HI. Assuming that the scattered radiation is coherent at the incident frequency, the change in phase speed translates into a change in the wavenumber of the incident wave (i.e., c'ph = w/k').

The description of wave propagation in terms of the effects of the medium on the incident phase front and wave number forms the basis for the development of geometrical optics (see Born and W olf12 for a rigorous treatment of this topic). This description differs substantially from the radiative transfer approach usually adopted for resonant interactions with bound systems. Among the effects treated in geometrical optics are dispersion, refraction, reflection, and macroscopic scattering. These are illustrated in Figure 7.7. In a dispersive medium the phase and group velocity of an electromag­ netic wave depends on frequency and this causes a localized (in time and space) electromagnetic

334

Figure 7.6

Scattering from the perspective of radiative transfer.

pulse, for example, to spread out as shown in Figure 7.7A. Reflection simply refers to the backscattered part of the radiation field. When the number density of the scatterers is sufficiently large, it is possible for the forward component of scattered radiation to cancel the incident wave resulting in nearly 100% reflection, as occurs in the case of highly conducting metal surfaces (see Figure 7.7B). Refraction is a process where the direction of propagation of a wave front is altered as a result of the change in phase speed in the propagation medium across the wave front. In Figure 7.7C, a line of constant phase is shown at some instant in time (t,) spanning a region consisting of vacuum and a conducting medium. Because of the different phase speeds in these regions, the phase front becomes tilted or distorted as time progresses (t2, t3). Macroscopic scattering, illustrated in Figure 7.7D, occurs when the propagation path of different parts of an incident wave front are redirected in random directions by the presence of local inhomogeneities in the propagation medium. The net result is an interference phenomenon in which random variations in the amplitude and phase of the incident wave called scintillations occur. Each of these effects and their impact on the propagation of UWB signals will be discussed below. In summary, the particular treatment adopted for studying the propagation of electromagnetic waves through matter depends to a large extent on the approximations that can be made regarding the details of wave-particle interactions and the macroscopic description of the propagation medium. In the case of resonant scattering by bound charges, propagation is best modeled by a radiative transfer approach that keeps track of the intensity of the radiation field. For a medium composed of free charges, the simple picture of geometrical optics prevails. The frequency dependence of the scattering processes in both cases is an important factor in analyzing wave propagation, particularly in the case of UWB pulses. This factor will be treated in more detail in Sections III and IV.

j

INCIDENT WAVE

D is p e rs io n

R e fle c tio n

constant phase

R e fra c tio n

density contours

ilin,

M a c ro s c o p ic s c a tte rin g

Figure 7.7 Geometrical optics. The effects of pulse dispersion, reflection, refraction, and macro­ scopic scattering are depicted in (A), (B), (C), and (D), respectively.

II. UWB PROPAGATION To illustrate the detailed effects of propagation through the Earth’s atmosphere, the model UWB pulse shown in Figure 7.8 will be used for the calculations described below. This pulse is assumed to be in the farfield of an unspecified antenna and have a peak amplitude of 1 MV/m projected to 1 m from the antenna. The corresponding peak power density and total fluence are 2.7 GW/m2 and 20 J/m2, respectively. The peak amplitude at a distance r meters from the antenna, if reduced by beam divergence alone, is found by dividing the amplitude at 1 m by r. Similarly, the power density and fluence are reduced by a factor of r2. The spectral content of this pulse is shown in Figure 7.8B and is seen to extend over an 80-dB dynamic range to approximately 500 MHz. Many of the properties of this pulse are shared by a certain class of UWB pulses, however, the power level was chosen arbitrarily high and the spectral range was chosen to be somewhat narrow in order to accentuate the essential aspects of propagation through the atmosphere.

336

Time (^is)

Frequency (MHz) Figure 7.8 Ultra-wideband pulse. (A) The amplitude of the UWB pulse adopted in this chapter for illustrative purposes is plotted as a function of time. (B) The corresponding frequency spectrum of the pulse.

As noted previously the physical processes that describe wave propagation and the mathematical prescriptions used to model them are strongly dependent on frequency. This point, when coupled with the fact that most of the energy in UWB pulses is concentrated in the high frequency band above 30 to 50 MHz and below 10 GHz, leads to the conclusion that propagation effects at the dominant UWB frequencies are primarily associated with transionospheric propagation and nonlinear (high power) propagation through the lower atmosphere (20 to 70 km). In the linear regime, propagation is carried out for the individual frequency components of the pulse, which are then recombined to yield the resulting postpropagation pulse shape. Thus, the theoretical formulation is equivalent to that employed for narrowband propagation, except that the analysis is carried out in the Fourier domain. In the nonlinear regime, a detailed time-dependent solution of the coupled fluid and electromagnetic equations (Maxwell’s equations) is necessary.

337

III. LOW POWER LINEAR PROPAGATION THROUGH A BACKGROUND PLASMA In this section the equations necessary to treat the propagation of a low power electromagnetic pulse (EMP) through an existing plasma imbedded in a background magnetic field are examined. The resulting analysis is applicable primarily to propagation through the Earth’s ionosphere at altitudes above 70 km where the plasma density is sufficiently high and scale lengths are sufficiently long to significantly affect the EMP. Between 15 and 70 km altitude the electron density is small and only those pulses with sufficient power to substantially ionize the air are affected. This corresponds to the high power, nonlinear regime that is treated in Section IV. Between 70 and 100 km, the electron-neutral collision rate is sufficiently high so that waves with frequencies below several MHz can experience attenuation due to ohmic dissipation. At high power levels the plasma can be heated by electromagnetic pulses with sufficient pulse lengths (much greater than collision time scales), raising the collision rate (and therefore the frequency of affected waves) and enhancing absorption of energy out of the pulse. In the latter case “thermal runaway” can occur as attenuation efficiency increases with electron temperature, and in some cases 25 to 30 dB absorption is possible. At sufficiently high-power levels it is also possible to initiate various plasma instabilities that cause the ionosphere to structure and that ultimately absorb energy out of the EMP. In fact a number of high power RF facilities dedicated specifically to the study of ionospheric modifications exist around the world. The power density necessary to electrically breakdown air is approximately 24 kW/m2/torr2 (14 GW/m2 at sea level) for continuous waves (CW) and increases for shorter duration pulses, as will be discussed in Section IV. The electron thermal energy at these power levels is approximately 2 to 3 eV or approximately 100 times the ambient temperature. Absorption thresholds over atmospheric scale lengths (~7 km) are approximately a factor of two lower than breakdown thresholds. Ionospheric modification facilities gener­ ally operate in the frequency range of 1 to 15 MHz with power levels ranging from 10 MW CW to 1 GW pulsed effective radiated power (ERP). Below these power levels and at frequencies above approximately 30 MHz, an EMP will experience little attenuation and the analysis presented in this section will apply. Constitutive relations for the plasma current in terms of the applied electric field are derived in the Constitutive Relations section. An appropriate solution to Maxwell’s equations for wave propagation through a collisionless plasma is obtained in the Solution to Maxwell’s Equations section as well as a transfer function for the ionosphere, assuming the ionosphere can be treated as a high pass filter.

CONSTITUTIVE RELATIONS For low power (as defined above) UWB pulses, it is sufficient to treat the plasma as a fluid. In addition, the short duration and high frequency content of UWB pulses make it possible to ignore the ion motion and to neglect, to first order, the convective transport of electrons, of their momentum, and of their energy compared to the impulse generated by the EMP. With these assumptions, the plasma equations of motion reduce to

3v at"

eE rn

3T * =“

(7.8)

2v„(v,T)T 3

(7‘9)

where n() is the electron density, v is the electron velocity, T is the electron temperature, E is the electric field component of the applied EMP, B is the sum of a background magnetic field ( B0) and the applied magnetic field b, vm is the sum of the electron-ion momentum exchange rate (vei m) and the electronneutral momentum exchange rate (venm), and vc is the sum of the electron-ion energy exchange rate (vei e) and the electron-neutral energy exchange rate (vene). Ohmic dissipation is contained in the ve term which depends on the mean velocity explicitly and the electric field implicitly. The ion and neutral components of the gas are assumed to be stationary. In the limit that the energetically dominant pulse frequencies are

338 10"

10°

6x10

Electron Density ne(crrr3)

Figure 7.9 Collision frequency. The total electron collision frequency and electron density are plotted as a function of height between 100 and 1000 km.

large compared to the collision rates, the inertial terms dominate and the plasma can be treated as collisionless. A plot of the total electron collision rate as a function of altitude through the ionosphere is shown in Figure 7.9. The maximum value is less than 10 kHz and occurs at an altitude less than 100 km. For UWB pulses whose frequency content is primarily in the HF, VHF, and above, propagation through the ionosphere is essentially collisionless. In this limit both the electron density and temperature are constants in time; Equation 7.8 becomes linear and can be solved with Fourier techniques. With the plasma current given as J = -n 0 ev, and assuming that |b| |B0| or |v|/c « 1, a constitutive relation for the temporal Fourier transform of the current in terms of the transform of the electric field can be found from the transform of Equation 7.8: J=aE

(7.10)

where the plasma was assumed to be collisionless, g is a tensor representing the plasma conductivity given by Op °H 0 a = -c Hop 0 0

0 O//

with icoX ° p ~~ 4rc(l-Y e2) -toXYe ° H “ 47t(l- Ye2) and C/i = icoX/47t; q p, g h, and a„ are Pedersen, Hall, and parallel conductivities; X = cope2/o)2; Y e = (Oce/co; 0>pe(= ^47tn0e2/ me ) is the electron plasma frequency; (Oce (= eBo/mec) is the electron gyrofrequency; and co is the Fourier transform variable. A Cartesian coordinate system (x,y,z) with its z-axis in the direction of the

339 geomagnetic field was assumed. The 90° phase shift between the Hall conductivity and the Pedersen and parallel conductivities arises because the current in the former case is perpendicular to the applied electric field in contrast to the parallel and Pedersen currents. The Hall current results from an E x B drift of electrons relative to ions which in the present analysis occurs because we have allowed for inertial effects; otherwise, the plasma as a whole would E x B drift and no currents would develop. In the absence of a background magnetic field, gh goes to zero as expected while a,/ and o pbecome equal and are nonzero. In this case the conductivity tensor becomes diagonal and the plasma is isotropic. The constitutive relations in Equation 7.10 serve as input to Maxwell’s equations and allow for the derivation of a single set of equations for the propagation of an electromagnetic wave through a cold, collisionless plasma imbedded in a background magnetic field.

SOLUTION TO MAXWELL’S EQUATIONS The self-consistent evolution of the electromagnetic fields is obtained by transforming Maxwell’s equations into a single vector wave equation for the electric fields: - c2V2E + c2V(V•E) = - 4 j3

ot

ot

(7.11)

Taking the Fourier transform of this expression and substituting the constitutive relations for the current density yields, co2

|

E

-

c 2V

2E

+

C2 V ( V

E)

=

0

(7 .1 2 )

where E denotes the Fourier transform in time of the electric field vector and where | is the plasma dielectric tensor defined as £j 1£2 0 £ = -i£, £, 0 0

0

(7.13)

£V

where ex = 1 + 47tioP/co, £2 = 4rcaH/cD, and £3 = 1 + 47iia///co. Evidently the dielectric tensor is a function of both the electron density and the strength and direction of the geomagnetic field. In general, these parameters are characterized by small amplitude fluctuations superimposed on a mean value that varies smoothly as a function of altitude (e.g., in a slab atmospheric model). In this limit the dielectric tensor itself can be written as the sum of a smoothly varying part and a fluctuating part, i.e., g = | 0 + 5 | . Propagation through the deterministic part of the atmosphere with g = | 0isusually treated in the limit of geometrical optics, as discussed in the introduction. The stochastic part involves forward scattering of the electromagnetic wave and is treated as a perturbation on the deterministic solution. Solutions for both regimes are discussed below.

Deterministic Propagation Following the procedure outlined in Bom and Wolf12 and Krall and Trivelpiece,13 we substitute an expression for the electric field of the form E(r,co) = e(r,co)

(7.14)

into Equation 7.12, with the result ——y-£ •e + {k2e - k ( k •e)} -{i(V •k)e + 2 i k •Ve - i kV •e - V(k •e)}

-{V2e - V(V •e)} = 0.

(7.15)

340 Assuming that the wave number in the plasma Ikl and that the amplitude coefficient ( |e |) vary over scale lengths, of order the plasma dimensions and that the latter is large compared to the vacuum wavelength (k = 27tc/co) of the waves, the first line of Equation 7.15 dominates over the others. The zeroth order limit of geometrical optics corresponds to setting the first line of this equation without the k(k • e) term to zero (cf. Bom and Wolf,12 p. 112). In deriving the plasma dispersion relation, however, this term is retained and the zeroth order equation can be written 2

(7.16) Equation 7.16 forms the basis for nearly all studies of radio wave propagation through a deterministic ionosphere. The general solution to this equation yields the well-known Appleton-Hartree dispersion relation, which identifies two normal modes (ordinary and extraordinary) that differ in their polarization relative to the geomagnetic field and in their dispersion relations. The Appleton-Hartree dispersion relation is written (7.17)

where \i (= ck/co) is the plasma index of refraction; X = co^/co2; Ye = coce/co; YT = Ye sin\|/; YL= Ye cost)/; and \|/ is the angle between the wave vector k and the background magnetic field B0. The plus (minus) sign in Equation 7.17 corresponds to the ordinary (extraordinary) mode, as defined in ionospheric physics (cf. Rishbeth and Garriott6). In the limit Yj! /4 (1 - X)2 Y* (quasi-longitudinal approximation), this equation further simplifies to (7.18) Assuming that YL is small, the group velocity can be written

(7.19) As is evident from Equations 7.17 through 7.19, the net effect of dispersion is to introduce a time delay in the arrival of a particular frequency component of the incident pulse at a specified detector such that the high frequencies arrive first. Thus, as mentioned in the Scattering by Free Charges section and shown in Figure 7.7A, the interaction of an incident wave with a uniform distribution of free charges is such that the interference with the re-radiated wave results in a frequency-dependent phase (group) speed that, in the case of a plasma, is greater (less) than the speed of light. The accuracy of the quasi-longitudinal approximation and the magnitude of the temporal dispersion for typical ionospheric conditions are discussed below. The general form for the transionospheric signal in the Fourier domain for a given mode (ordinary or extraordinary) is given by Equation 7.14 where e(r,co) is taken to be the launched signal. In the time domain, the transionospheric signal can be written (7.20) where Hj is the ionospheric transfer function which can be derived from the results obtained above (cf. Roussel-Dupre and Kelley,14 Roussel-Dupre and Argo15). In the limit that the quasi-longitudinal approximation applies and when X 1, the transfer function for the deterministic ionosphere can be written

341

Hr(co) = J-Jei2W(a±tDceCosv) + e-i27ta/(-o)±o)cecosv) j

(7.21)

where a (in MHz) = 8.430 x 104 TEC, TEC is the total electron content along the propagation path in units of 1013 cm-2, all frequencies are expressed in MHz, the + (-) corresponds to the ordinary (extraor­ dinary) mode, and where we have omitted the effects of refractive bending and assumed a densityweighted gyrofrequency (coce) and angle (\j/) between the propagation vector and the magnetic field. Generally the deterministic ionosphere is well represented by the transfer function Equation 7.21 for frequencies above 20 to 30 MHz. In this case, the ionosphere is characterized in terms of its TEC and an average value for coce and cos\|/. However, at frequencies approaching the plasma frequency (typically 5 to 15 MHz) and in cases of oblique incidence, it is necessary to include higher order terms in an expansion about small X and to incorporate refractive bending. In addition, it is necessary to include Earth curvature effects in order to model the refractive bending accurately. A more detailed form for the transfer function that incorporates refractive effects in a spherically symmetric ionosphere, arbitrary polarization mixtures, and a dipole model for the geomagnetic field is discussed in Roussel-Dupre and Argo.15 The ionospheric TEC depends on geographical location, time of day, season, and solar and geomag­ netic activity. Diurnal variations in TEC during the spring season and high solar activity for three geographical locations corresponding to equatorial, middle, and polar latitudes are shown in Figure 7.10A through C, respectively. These results were obtained by integrating the electron density profiles generated using the International Reference Ionosphere (IRI) code.16 This code generates electron density profiles by interpolating a series of tables developed from synoptic measurements obtained with ionosondes, topside sounders, and incoherent scatter radars. The database in some cases constitutes more than 20 years of measurements taken on an hourly basis. Nevertheless, a recent study of the accuracy with which various ionospheric codes predict the TEC value for any given set of parameters indicated that factors of two deviation from measurement were not uncommon. Thus, the plotted magnitudes of TEC should be taken as indicative of a range of possible values. The impulse response function for a deterministic ionosphere is obtained from Equations 7.20 and 7.21, assuming that the transmitted signal is a unit impulse whose Fourier transform is given by e(r,co) = ei(0t() where ^ is the time at which the pulse is launched. The net result is given by

E d

= (s (t

-

t„) + ^ ^ } ™ S K c o s V( t - t 0)]

(7.22)

where Jj is a first order Bessel function; the argument x = 2 a 1/2 (t - 10)1/2; and the cos (sin) is used for the ordinary (extraordinary) mode. A plot of EDvs. time for a TEC = 1, fce = 0.8 MHz, \|/ = 20°, and to = 4 its is shown in Figure 7.1 IB. Note that the initial impulse (Figure 7.11 A) becomes stretched out in time over many microseconds by the ionosphere. A spectrogram of EDshown in Figure 7.12 illustrates the fact that the ionosphere produces a frequency-dependent group delay such that the high frequency components of the impulse arrive first. When the time signature of a UWB pulse is convolved against the impulse response of the ionosphere, we obtain the transionospheric pulse. Performing the convolution on the UWB pulse discussed in Section II for the ionospheric conditions described above yields the pulse shown in Figure 7.13. The frequency content of the pulse (shown in Figure 7.8B) is observed to be dispersed in time with the high frequencies arriving first. One obvious effect of dispersion is to lower the amplitude of the low frequency components relative to that of the high frequencies. Assuming that the ionosphere can be approximated geometrically as a slab and integrating the inverse of the group velocity over the propagation path from a transmitter to a receiver yield the following frequency dependent time delay t - 1 Zr , 1 Az mp2 d c cos0r c cos0r 2co2 v

± 2 ^ cosy ©

(7.23)

where zr is the vertical distance from the transmitter to the receiver (see Figure 7.14), 0r is the elevation angle to the receiver, Az is the thickness of the ionospheric slab, cop is the plasma frequency with ne equal to the slab plasma density, and ne Az corresponds to the total electron content in this case. The first term

342 LAT = 0.0

LONG = -61.5

SP O T NO. = 160.0

EL = 90.0

Hour (local lime) LAT = 40.0

LONG = - 6 1 .5

LAT = 75.0

LONG = -61.5

SP O T NO. = 160.0

EL = 90.0

Hour (local time) SP O T NO. = 160.0

EL = 90.0

Hour (local time) Figure 7.10 Total electron content, (a) The total electron content for an equatorial ionosphere (0°N, - 6 1 .5°E), with a sunspot number of 160 and an elevation angle of 90°, is plotted as a function of local time, (b) The total electron content for a mid-latitude ionosphere (40°N, -61.5°E), with a sunspot number of 160 and an elevation angle of 90°, is plotted as a function of local time, (c) The total electron content for a high-latitude ionosphere (75°N, -61.5°E), for a sunspot number of 160 and for an elevation angle of 90°, is plotted as a function of local time.

in Equation 7.23 is the delay in the arrival of the signal corresponding to vacuum, direct line-of-sight propagation while the second and third terms are frequency-dependent corrections introduced by the ionospheric plasma assuming cop2/co2 « 1. The second term is proportional to the TEC and generally

343 T"■» '—' » I • ' ' ' I • • 1 '—I—r""' ' ■ I

10.10

8.06

CN J O

(a) D e lta fu n c tio n .

X

f

6.02

CO

o o» E

3.98

Q. E

< 1 .9 4

-

0.10

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time (ps)

1.40

0.84 CM

o x

f*

0.28

CO

o CD

E

-0.28

Ql E


a> LU

Px (torr-s) Figure 7.21 Air breakdown thresholds. Theoretical calculations of pulsed microwave breakdown thresholds compared to experimental results plotted as an effective electric field divided by air pressure Eefl/P vs. air pressure times pulse length Px. These data were obtained from the following reports: AIL (artificial ionization layer) data. (From Armstrong et al., AGARD Conf. Proc. No. 485, 18A-1, 1990; Byrne, D.P., LLNL report UCRL-53764,1986; and August, G., DNA tech. rep. DNA-TR-88-80-VI, 1988. With permission.)

ne (x) = n0 exp (va x)

(7.49)

PT = loge(nb/n 0)/(v a/p)

(7.50)

which can be recast in the form

where nb is the electron density achieved in the breakdown process and x the time to attain this density. Characterization of pulsed breakdown proceeds from Equation 7.50 using the fact that va/p is a function of EeffA' or equivalently Eeff/p only. From Equation 7.50, one can solve for the effective electric field necessary to create a plasma of density % at a given air pressure in terms of the duration of the pulse x and the initial density no- For sufficiently high values of nb this effective field will represent the threshold necessary to generate a strong plasma capable of absorbing significant amounts of energy out of the tail of the pulse and/or reflecting the incident pulse. In this way we obtain a definition for the threshold electric field necessary for air breakdown or attenuation in terms of the duration of the pulse. A plot of Eeff/p vs px derived from fluid theory combined with experimental data is provided in Figure 7.21, where we have chosen a value of 108 for 1^ / ^ based on experimental work on microwave air breakdown. Errors introduced by this approximation are small because of the weak natural log dependence on this ratio. We see that for a given pressure the threshold field increases with decreasing pulse length as expected. At low px the curve turns over because there is a peak in the ionization rate and any further increase in the electric field causes an increase in the time to breakdown. The accuracy and validity of the Ecff/p vs px curve are discussed in detail in Roussel-Dupre.81

358

Px (torr-s)

Figure 7.22 UWB pulse trajectory. The model UWB pulse is mapped onto the Eeff/p - px space (see also Figure 7.21) at a pressure of 10 torr, corresponding to 30 km altitude.

The applicability of this analysis to high power UWB propagation through the atmosphere can be seen as follows. Given the amplitude as a function of time at a given altitude it is possible to map a UWB pulse onto the Eeff/p - px space (see Figure 7.21). In this way one can determine whether or not the pulse will cause breakdown of the air at that altitude. If the UWB pulse crosses the curve at some time x into the pulse then breakdown will occur, and it is a good approximation to assume that total absorption of the pulse beyond the time x will occur as a result of tail erosion over the large atmospheric scale height («7 km). By carrying out this operation over the entire range of propaga­ tion, one can determine the shape and amplitude of the transmitted pulse. The example shown in Figure 7.22 makes use of the UWB pulse defined in Section II at an atmospheric pressure of 10 torr (corresponding to an altitude of 30 km). The effective electric field plotted corresponds to the instantaneous electric field of the pulse with co = 0. We see that the pulse crosses the breakdown curve at time x = 2 ns (px « 20 torr-ns). By plotting the instantaneous field with co = 0, corresponding to DC breakdown, we are neglecting the effects associated with the finite risetime of the pulse. The net result is that we underestimate the time to breakdown and therefore the transmitted fluence (equal to the power density of the pulse times the length of the pulse). There are, however, certain techniques that can be used to circumvent these difficulties and still make use of the breakdown curve. A more accurate and automated technique for performing this type of analysis is discussed in Tunnell and Roussel-Dupre.89 Results for the UWB pulse are shown in Figure 7.23A and B. We find that the final pulse extends approximately 2 to 3 ns into the initial pulse in good agreement with the trajectory analysis (Figure 7.22) and the detailed hydrodynamic calculations. The propagation efficiency, defined to be the final pulse energy divided by the initial pulse energy, shows that 94% of the pulse energy is attenuated, again in good agreement with the detailed hydrodynamic calcula­ tions. As noted previously high power, nonlinear propagation occurs primarily below an altitude of 100 km simply because the minimum ionization time scale («400 ns at 100 km) becomes large compared to typical UWB pulse lengths. Beyond 70 km altitude, dispersion becomes important and it is necessary to include the effects discussed in the linear propagation section. Finally, we note that while propagation of a UWB pulse upwards from the ground utilizes the approximation co = 0 for

359

0.0

4.0

8.0

12.0

16.0

20.0

24.0

Time (s)

28.0

32.0

36.0

40.0 _..v9 x10

Altitude (km)

Figure 7.23 Frequency scaling analysis of tail erosion. The propagation of the model UWB pulse from 0 to 70 km is simulated using frequency scaling techniques and a simplified form of the nonlinear wave equation. The initial and transmitted pulse are plotted in (A) as a function of retarded time. Note the strong effect of tail erosion. The propagation efficiency, defined to be the final pulse energy divided by the initial pulse energy, is plotted in (B) and shows that 94% of the pulse energy is attenuated in good agreement with detailed hydrodynamic calculations.

the effective field, propagation of a UWB pulse downwards from space necessitates passage through the ionosphere which transforms the pulse into a chirp characterized by an instantaneous frequency. In this case it is necessary to incorporate this frequency in calculating the effective field for subsequent propagation through the lower atmosphere. All of these effects are contained in the more

360 accurate treatment, which involves a solution of the hydrodynamic equations simultaneously with Maxwell’s equations as described above. The calculations, however, are computationally more laborious.

V. SUMMARY AND CONCLUSIONS The physical processes inherent to the propagation of an electromagnetic wave through the Earth’s atmosphere depend strongly on the frequency content of the wave and on its power level. In this sense UWB signals are unique. Not only does their short duration translate into significant spectral content over a broad frequency range from 0 to 10 GHz and above, but the concentration of pulse energy into such small time scales (of order nanoseconds) can also lead to intrinsically high power densities (of order GW/ m2). In the low power regime (as defined in Section III) propagation is characterized by wave-particle interactions that are either resonant or nonresonant. In the case of resonant interactions, absorption, angular scattering, and frequency redistribution are among the dominant processes and the equations of radiative transfer are used to model propagation. For nonresonant interactions, the prevailing mathemati­ cal prescription is given by geometrical optics while the intrinsic physical processes are dispersion, refractive bending, reflection, mode splitting (polarization effect), and macroscopic scattering. In the linear regime, propagation is carried out for the individual frequency components of the UWB pulse, which are then recombined to yield the resulting postpropagation pulse shape. At highpower levels, an electromagnetic wave can significantly alter the properties of the propagation channel in turn affecting the amplitude and phase of the transmitted wave. Self-absorption (tail erosion), self-focusing, thermal runaway, and the development of plasma instabilities are just a few manifestations of high power propagation. In the nonlinear regime, modeling of UWB pulse propagation requires a solution of the kinetic or hydrodynamic equations of motion for the plasma in conjunction with Maxwell’s equations. Further characterization of wave propagation through the atmosphere leads to a division of the atmos­ phere into three distinct regions. In the troposphere (between 0 and 12 km), high concentrations of water vapor, ice crystals, and hydrometeors can cause significant scattering and absorption (0.1 to 30 dB/km) in the frequency range above 10 GHz. Scattering in the presence of precipitation may also be important in this frequency range. Resonant absorption due to fine structure in the ground state of 0 2 becomes important (>10 dB/km) around 60 GHz. While this subject was not treated here, a thorough review is provided in the Handbook o f Geophysics and the Space Environment. 8 In this altitude range the distribution of electrons is determined primarily by cosmic ray ionization which is statistical in nature. As a result, air breakdown and the corresponding process of tail erosion occurs over limited dimensions (centimeters) around cosmic ray showers (producing seed electrons), and ohmic dissipation becomes “spotty” across the radar beam width. This process has only recently been studied both in the laboratory and theoretically,90’91 In the middle atmosphere (between 10 and 70 km) the electron density increases sufficiently to permit initiation of air breakdown over large scales (hundreds of meters to tens of kilometers depending on the radar beam width). Ultra-wideband pulses with enough power (>24 kW/m2/torr2) and duration (tens of torrnanoseconds) can accelerate seed electrons to sufficient energies to drive an avalanche that leads to the formation of a plasma and to enhanced ohmic dissipation of the pulse. This nonlinear process of self­ absorption (termed tail erosion) is well known and has been measured in the laboratory. Detailed modeling proceeds from a solution of the kinetic or hydrodynamic equations that describe the plasma evolution together with Maxwell’s equations. Sample calculations presented for a UWB pulse with a power density of 2.7 GW/m2 and a duration of 8 ns indicate that over 90% of the pulse energy is absorbed in the altitude range from 15 to 50 km. Thus, atmospheric transmission at high power levels can severely limit the utility of UWB radars. The fact, however, that tail erosion affects only the tail of the pulse and that the threshold electric fields necessary to initiate this process increase with decreasing pulse length suggest that shorter pulses would be more readily transmitted (a concept termed sneak through). It is clear, therefore, that in designing efficient UWB radars one must consider the details of nonlinear propagation. In the upper atmosphere (above 70 km altitude), the electron density becomes sufficiently high (>10* cm-3) over long enough scales (hundreds of kilometers) so that even low power UWB pulses are severely distorted after propagation through this region. To illustrate the linear effects associated with transionospheric propagation, we propagated the UWB pulse defined in Section 13 assuming circular polarization through a deterministic ionosphere with a total electron content of 1 x 1017 n r 2 and a gyrofrequency of 0.8 MHz. The dominant effect was a dispersion of the initial pulse (~8 ns in duration) over tens of microseconds (high

361

frequencies arriving first) together with an additional small frequency-dependent delay introduced by the geomagnetic field. These results represent the veiy least distortion to be expected from deterministic propagation through the ionosphere. Additional effects result from refractive bending, reflection of the low frequency components (frequencies below the plasma frequency), scattering off of large scale electron density structures, and mode splitting. Effects of stochastic propagation associated with large, random variations in electron density (or index of refraction) were also studied for the same UWB pulse. Results showed severe distortion of the temporal signature of the initial pulse depending on the coherence bandwidth of the ionosphere and the frequency regime under study. This effect is particularly important for UWB signals in that it limits the bandwidth over which useful, coherent information can be extracted. Amplitude variations called scintillations also result from scattering. The impact of scintillations on received signals is to introduce additional uncertainties in signal measurement and time-tagging because of reductions in the SNR from fading and distortions in the temporal structure of the signal. The magnitude of these effects (both temporal distortions and scintillations) is random and can only be characterized statistically. As a result one cannot compensate for them (as in the case of deterministic propagation) but must include them in assessing a system’s performance. In the high power regime, the ionosphere will introduce additional pulse distortion due to nonlinear processes such as thermal runaway, self-focusing, ohmic dissipation, and the initiation of plasma instabilities. Depending on the frequency content and power level of UWB pulses and on atmospheric conditions, the effects of transatmospheric propagation can be severe both in terms of pulse distortion and attenua­ tion. At high frequencies (e.g., UHF and microwave bands up to 10 GHz) and low power levels, the atmosphere becomes effectively transparent. In certain regimes, however, the performance of UWB radars will be greatly inhibited. In any case, the propagation effects elucidated in this chapter will most certainly need to be considered in the design and implementation of UWB radars.

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364 88. Murphy, T., Total and Differential Electron Collision Cross Sections for 0 2 and N2, Los Alamos National Laboratory Rep., LA-11288-MS, 1988. 89. Tunnell, T. and Roussel-Dupre, R.A., Quick, Reliable Estimates of High-Power Microwave Fluences: The SNEAKY-II Program, Los Alamos National Laboratory Rep., LA-11680-MS, 1990. 90. Alvarez, R.A., Bolton, P.R., Sieger, G.E., Fittinghoff, D.N., Sparse Breakdown and Statistical “Sneakthrough” Effects in Low-Altitude Microwave Propagation, Lawrence Livermore National Laboratory Rep., UCRL-101807, preprint, 1990. 91. Yee, C.L. and Johnston, R.R, Theoretical Support of Sparse Air Breakdown Experiment, Science Applications International Corporation Rep., SAIC-U-136-PA, Preprint, 1988.

365

Part 2 ENERGY TRANSFER THROUGH MEDIA AND SENSING OF THE MEDIA Terence W. Barrett

I. IN T R O D U C T IO N T O E N E R G Y T R A N S F E R C O N C E P T S

How does short pulsed electromagnetic energy travel through a dielectric medium, reflect off a target, and return to a receiver? This section describes energy transfer and media sensing concepts that are important in pulsed UWB systems. Energy transfer is the movement of radiated energy through a medium with minimum dissipation. Media sensing is the selective interaction and characterization of the media. We will look at several efficient energy transfer methods through normally absorptive or dispersive media such as self-induced transparency, the Crisp zero degree pulse, and classical solitonic conduction. As we should not assume that the medium is a passive carrier, we will consider what happens when incident energy interacts with dynamic properties of the medium. Why is it necessary to study the properties of the medium? There are a number of reasons all related to the advantages of not using a general pulse or waveshape for all media and targets, but rather crafting the waveshape to specific media and targets. The advantages lie in obtaining: • Increased radar range • Transmittance through normally absorptive or dispersive media to either communicate or produce backscatter from a target • The capability of selectively sensing or imaging media, strata, or targets Section II discusses the classical Debye model of dielectric theory and contrasts it with the DissadoHill model of dielectric properties. The Dissado-Hill model is more complete and includes inertial effects, as well as being able to address cooperative effects in materials and media. To examine radio frequency (RF) dielectric properties in detail, a comprehensive quasi-quantum mechanical, or semiclassical, model is required which addresses RF wavelength media properties and such cooperative effects. High power pulsed RF sources are now available which permit the pulsed probing of media, the exciting of ultrafast pulse-duration-dependent nonlinear properties that might permit RF self-induced transparency, or, if not, classical Crisp zero degree pulses. Experimental work in this area is just beginning. At energies in the radar frequency range one should not expect intensity-dependent nonlinearities to be induced in a medium. However, there are other nonlinearities which are dependent on minimum energy thresholds and the duration of a pulse or signal. These are envelope effects. Section III examines transient pulse effects in transmission and reflection of electromagnetic energy due to the pulse envelope including: 1.

Self-induced transparency, which permits penetration of media normally absorbing and involves quantum mechanical effects. This effect is a function of both the media conditions (the relaxation time and inhomogeneous broadening of the medium) and pulse conditions (envelope and duration). Self­ induced transparency has rarely been seen experimentally or recognized as such, except at optical wavelengths. At optical wavelengths this is due to a rigorous requirement that the frequencies of a wave packet remain in phase. Microwave electron paramagnetic resonance self-induced transparency is the exception. We will show why we should look for the effect at radio frequencies. 2. The Crisp zero degree pulse is another media penetration method based on classical mechanics. Self­ induced transparency reduces to the Crisp zero degree pulse at low field strengths. Like quantum mechanical self-induced transparency, the zero degree pulse requires a matching of the pulsed radiation to the relaxation characteristics of the medium, i.e., it is a pulse envelope effect. 3. Classical solitonic conduction is a third method for media penetration also addressed in Section IV. A soliton is a mathematical entity in which, during propagation through the medium, either (1) the pulse dispersion is matched to the nonlinearity induced in the medium or (2) the pulse envelope is matched to the dispersion in the medium. At RF frequencies, it is the latter mechanism that is important. Both self-induced transparency and the Crisp zero degree pulse are versions of solitons. Matching an RF pulse to the medium to create a soliton would enable more efficient media penetration.

366 At energies in the radar frequency range, one should not expect intensity-dependent nonlinearities to be induced in a medium permitting the generation of a soliton by matching a short pulse to that nonlinearity. However, solitons can be induced by matching a short pulse and its envelope to the dispersion characteristics of the medium. The pivotal concept is the matching of pulse envelope, frequency, amplitude, phase, and polarization to the media characteristics. The classical and quantum mechanical area theorems address this matching and are discussed in Section HI. In some instances, these concepts have only been demonstrated at optical frequencies, which is appropriate for laser radar (LADAR) sensing. However, the concepts are also transferable to the IR and RF frequency range, if it is at the same time realized that (1) the media mechanisms polarized by radiation are quite different in the same medium at such widely different frequencies — electronic dipole polarization (optical), ionic dipole polarization (IR, RF), and orientational charge dipole (RF) — and (2) that those mechanisms’ coupling to the environment is actually increased at IR and RF frequencies due to cooperative and nonlocal effects. Such coupling to cooperative mechanisms prevents the dephasing of pulse frequencies — a condition for self-induced transparency to occur. There is the opportunity to investigate such effects for the first time with recently available IR and RF sources. Section III covers precursor phenomena or pulsed signal envelope effects which also permit media penetration. Such effects have been demonstrated at RF frequencies. These effects are a function of signal envelope rise and fall times and occur when the media dielectric characteristics react to rapidly changing excitation. The important dielectric characteristics of the media include excitation time constants, relaxation time constants, and specific polarization requirements. Reflection, absorption, and transmis­ sion of short duration pulsed radiation depend on the transient media dielectric characteristics and the temporal characteristics of the radiation determined by the pulse envelope. If the medium does not severely dephase the signal, and if the signal’s duration matches to the characteristics of the medium, we can pass radiation through media that would normally absorb or disperse the radiation. Section IV covers soliton waves, group theory, and electromagnetic missile concepts. It should be realized that the conditions for obtaining a match of the pulse to the medium to obtain increased penetration are severe. At present, we do not understand complicated media conditions sufficiently to predict the reaction of such media to electromagnetic energy at RF frequencies in all instances. However, we intuitively know that pulse length, amplitude, envelope, frequency content, phasing, and polarization all play a role in matching the energy wave to solid, liquid, and gaseous media. Optimum propagation of energy results when there is a match of pulse and medium properties. Thus if the medium is considered as a filter that conditions the pulsed signal, then crafting or matching the pulsed signal to the medium will maximize propagation through it. Therefore, the media properties have to be known precisely in order to appropriately craft or match the incident radiated signal properties. These crafted energy concepts are shown in Table 7.1. The group transformation rule description of fields is an important theme to emphasize in energy propagation. This arises because entities, like a soliton, or theories, like Maxwell’s theory, are the result of an underlying set of group transformations. If an entity, like a soliton, is to be described by a theory, such as Maxwell’s, then there has to be compatibility between the group representation of the entity and that of the theory. Now, Maxwell’s theory (at least the one that is used in current electrical engineering) has a very simple group representation known, in the jargon of group theory, as the U(l) symmetry group. On the other hand, in the pursuit of dissipationless propagation with matched signals or solitons, we are dealing with entities, or signals, which have a more complicated group representation known, again, in the jargon of group theory, as the SU(2) symmetry group. So now we have a problem: the well-known and well-used theory which we would like to use to describe the media penetrating pulses has a different group representation than the pulses we wish to describe; we have apples and pears. Therefore we examine the group theory characteristics of pulses in Section IV. It turns out that whereas any attempt to amend Maxwell’s theory causes consternation among electrical engineers, the extension of Maxwell’s theory into higher symmetry forms has been commonplace among physicists for years! Sometimes one sees schisms or cultural differences between science and engineering. In this section we present the groundwork for an advanced approach to Maxwell’s equations, i.e., Maxwell’s theory is extended.182*183 It is necessary for electrical engineering to move in this direction in order to be able to handle ultrafast pulse effects. In so doing, engineers will be in agreement with their physicist colleagues, who long ago placed Maxwell’s theory within a broader context, that of Yang-Mills theory.256 It is also little realized that Maxwell himself, would be astonished at what is today called “Maxwell’s theory”.184

367 Table 7.1

Efficient Energy Transfer Conditions

Type of Energy Transfer

Media Conditions

(1) Inhomogeneous broadening > homogeneous broadening, permitting preservation of signal phase • Semi-classical (RF, IR): classical area theorem applies. • Quantum mechanical (optical): quantum mechanical area theorem applies

Self-induced Transparency (optical, IR, RF)

Effects

Increased medium penetration, selective interaction with targeted media

(2) Threshold intensity: • Semi-classical (RF, IR): threshold intensity determined by dipole reorientation and charge separation • Quantum mechanical (optical): threshold intensity determined by optical dipole moment • Semi-classical (RF, IR): medium acts cooperatively and nonlocally • Quantum mechanical (optical): medium acts locally (3) Pulse temporal length must be shorter than relaxation time of medium Crisp Zero Degree Pulse (RF)

(1) Classical area theorem applies (2) Pulse temporal length must be shorter than relaxation time of medium

Classical Soliton (optical, IR, RF)

(1) Dispersion in the medium and/or intensity-dependent nonlinearity in the medium

Increased medium penetration, selective interaction with targeted media Increased medium penetration, selective interaction with targeted media

(2) Pulse temporal length such that pulse dispersion is matched to pulse nonlinearity; soliton is a function of both medium and pulse characteristics Finally, in this section we discuss the new field of focus wave modes or electromagnetic missiles. The objective of workers in this field is to launch pulses or wave packets which have localized energy in order to achieve greater range by beamforming or pulse crafting. This is an important but developing field and we merely report the major themes at the present time. As an overview, Table 7.1 gives some major conditions for efficient energy transfer through media and for media sensing at optical, IR, and RF frequencies.

II. ADVANCED THEORY OF DIELECTRICS AND TRANSMISSION THROUGH MEDIA To understand the transmission through, and reflection from, various media and surfaces it is necessary to understand the characteristics of media so that radiation can be matched to the media to produce

368

dynamic interactive effects enabling radar and LADAR surveillance and sensing. However, we cannot proceed directly to an established dielectric theory in order to obtain that understanding. Presently, the theory of dielectrics is undergoing a reappraisal and reevaluation. It is especially important to realize this; whereas with steady-state sinusoidal waves the conventional theory of dielectrics is appropriate, with pulsed wave packet media excitation, only the newer approaches to dielectric theory address the ultrafast dielectric responses of media. The new approach is more comprehensive and recaptures the conventional approach as a limiting case. We will commence with a review of the conventional dielectric theories. C LA S S IC A L M O DELS O F ENER G Y TR A N SM ISSIO N , R EFLECTIO N

Figure 7.24 shows the reaction of a steady-state continuous wave and a transient wave at plane interfaces of different materials. At a plane interface between dielectrics there are the two well-known classes of reflection and refraction caused by kinematic and dynamic properties of the medium (cf. Reference 1, p. 278). The kinematic properties are independent of the nature of the boundary conditions, i.e., material composition. Snell’s law describing the angle of refraction and reflection depends on kinematic proper­ ties. The dynamic properties are a function of the amplitude of incident, reflected and refracted radiation, temporal duration of the radiation, frequencies, and phase under the envelope of radiation, as well as polarization. The dynamic properties are also associated with changes in those aspects of the radiation. They also depend on the material composition of the boundaries. The steady-state simple model of the dielectric constant e = 1 + 4n%,

(coulomb2/Newton, meter2)

(7.51)

is restated as

e(co) = 1 + {47tNe2)/m } |z ifi((a.2 -iGJYj)''},

(7.52)

where N is the number of molecules per unit volume with Z electrons per molecule; e is the charge of the electron (= 1.602.10~19 C); is the number of electrons per molecule with binding frequency (ty and damping constant y, m is electron mass, and given that a collection of oscillators contributes to the macroscopic dielectric constant, Zjf* = Z is the sum rule for oscillator strengths. (The binding frequencies, C0j, and damping constants, y, can be read from the absorption spectra of atoms and molecules. The American Chemical Society publishes comprehensive absorption spectra.) This equation gives the dispersion and absorption curves for steady-state medium excitation (see Figure 7.25 below). Normal dispersion is an increase in Re e(co) with co; anomalous dispersion is a decrease. Only where there is anomalous dispersion is the Im e(co) appreciable and indicates resonant absorption. However, this description does not apply at low frequencies when some fraction of the electrons per molecule are free and at frequencies far above the highest resonant frequency limit. At the high frequency limit the dielectric constant is e(co) = l - { ( 0p2/ro2J,

(7.53)

cop ={4JtNZe2}/m.

(7.54)

where cop is the plasmon frequency

The term “plasmon frequency” does not refer to the creation of plasmas, but the frequency at which steady-state radiation canpropagate with the minimum of distortionor absorption.(For a discussion of the different plasmon forms, particularly at optical wavelengths and above, seeReference 2). At a frequency removed from the region of anomalous dispersion, i.e., removed from the region of absorption, a pulse travels undistorted in shape with a group velocity given by vg = dco/dk,

(7.55)

369 (a)

STEADY STATE

(b) “TRANSIENT (MATCHED)

Figure 7.24 (a) Kinematic properties of radiation traversing changes in a medium, i.e., media with differing dielectric constants. When the dielectric constant changes, part of the radiation is transmitted with a new velocity in the medium. The angle of incidence and the angle of reflectance change according to Snell’s law. Part of the radiation is reflected, (b) Dynamic properties. When the medium dielectric constant is a function of the pulse amplitude, pulse envelope shape, frequency content, temporal duration, and polarization, then steady-state narrowband assumptions need no longer hold for a pulsed crafted or matched to the medium. In the case of specially crafted pulses, the medium can become nonlinear in more than one way and can permit penetration denied uncrafted waves. Thickness of the arrows indicates the amplitude of the radiation.

where k is the wavenumber. As co(k) = ck/n(k), where c is the velocity of light, the phase velocity of a wave packet is vp = co(k)/k = c/n(k),

(7.56)

which will be greater or smaller than c depending on whether n(k), the refractive index, is smaller or larger than unity (see Figure 7.26). The reason a wave can appear to travel faster than the speed of light is really due to a shift in the peak of the wave to earlier components of the wave. For example, suppose a bell-shaped Gaussian wave is composed of three parts: an early part A, a middle part B, and a latter part C. For a true bell-shaped Gaussian, the peak of the wave will be in B, the middle part. Suppose that after passing through a medium parts B and C are diminished, and part A, the first part, is now the peak of the wave. Someone observing this might measure the peak, B, of the wave entering the medium and its time of departure and the peak, A, of the wave exiting the medium and its time of arrival, and conclude that the wave traveled faster than the speed of light due to the change in the position of the peak in the total length of the wave. This person measured the phase velocity. However, if another person measured part B on the wave entering and its time of departure, and part B again and its time of arrival — no matter that it was no longer the peak of the wave packet — on the wave exiting, he would find that the speed was always less than the speed of light. This person measured the group velocity. We turn now to consider the question of what happens when an ultrafast pulse interacts with targets. First, we will consider how an ultrafast and UWB pulse is like a narrowband wave. This is an important consideration, for it raises the question of when does a short duration pulse, when lengthened in duration, become identical in its effects to a narrowband pulse shortened in duration? Further, what are the comparative effects of (1) short duration, but less than widebandwidth, wave packets and (2) long duration, but less than narrow bandwidth, wave packets? The relation between the two types of signals — one of large time bandwidth and narrow frequency bandwidth, the other of large frequency bandwidth and narrow time bandwidth — is seen in the field of squeezed states of light.3 Although this field of study addresses quantum mechanical effects only

370 A.

ANOMALOUS DISPERSION

Figure 7.25 (A) Real and imaginary parts of the dielectric constant e((q) in the neighborhood of two resonances. The region of anomalous dispersion is also in the frequency interval where absorption occurs. (From Jackson, J.D., Classical Electrodynamics, 2nd ed., John Wiley & Sons, New York, 1975, 286.) (B) Typical RF dielectric response of materials which is characterized by losses significant only in restricted ranges of frequency. Regions of negligible loss make negligible contributions to the dispersion so that the real part of the dielectric permittivity is almost independent of frequency outside regions of significant loss, remaining constant at frequencies below a loss peak and decreasing at frequencies above the peak. Such profiles are seen in organic materials25 and biological materials, both animal353 and plant.7677 (C) The dielectric response of single crystal sodium p alumina — a ceramic material and also an ionic conductor plotted in logarithmic units (Adapted from Jonscher, A.K., Dielectric Relaxation in Solids, Chelsea Dielectrics Press, London, 1983.) The loss peak is very broad and symmetric over a total frequency range of eleven powers of ten. However, it is believed that this loss peak is due to finite moisture content in the material (drying decreases the peak amplitude).

described by non-Poissonian statistics, these states exhibit a quantum nature only in the few-photon regime,4 and the coherent quantum mechanical state is a close quantum counterpart to a classical field. The effect of squeezing a state can be given a classical representation,5*6 and although we are not going

371

Log[Frequency CO]

Figure 7.25 (continued)

Figure 7.26 Typical index of refraction n(co) as a function of frequency co at a region of anomalous dispersion; phase velocity vp and group velocity vg as functions of co. (Schematic from Jackson, J.D., Classical Electrodynamics, 2nd ed., John Wiley & Sons, New York, 1975, 302. ©1975 John Wiley & Sons. With permission.)

to address squeezing a state per se, we wish to use here the phase space representation as an aid in addressing the question of how UWB waves are like narrowband waves. Figure 7.27 shows a phase space representation of a wave packet. With constant volume, as in the case of the ambiguity function, the shape of the packet can be altered at will to provide either greater bandwidth (with decreasing temporal duration) or greater duration (with decreasing bandwidth). At the limit of either of these operations, we have the ultrafast/UWB pulse and the narrowband continuous wave. Shown in this figure is a wave packet intermediate in form. The choice of the exact form and polarization of the

372

Figure 7.27 Frequency/time, phase/space representation of signal envelope. The frequency domain components of a signal are represented along the x^axis; the time domain components of a signal are represented along the x2-axis; and the amplitude is represented in the z or Q(x1>x2) direction. Notice that the envelope is not symmetrical but “squeezed” so that the frequency bandwidth is wider than the time bandwidth. This signal is thus a short duration pulse or time domain signal. If the envelope were squeezed in the orthogonal direction it would be of narrow frequency bandwidth and wide time bandwidth. Such a signal would be a frequency domain signal. If the amplitude is E/t or power, then a constant volume of energy is maintained during the reshaping or squeezing, and the representation could be an ambiguity function diagram. (Permission of H. P. Yuen, Phys. Rev., A13, 2231, 1976.)

wave packet to be used in any particular situation will depend on intention and the medium with which the packet will interact. We can also ask the question: how are short duration, UWB pulse effects different from those of narrowband long duration waves? Time domain radiation-matter interactions require an understanding of dielectric theory which surpasses the usual steady-state frequency domain understanding. One might assume that the theory of dielectrics for both the transient and steady-state response is well studied and understood. However, this is far from being the case for any real medium of interest. A major difference between the steady-state or frequency domain probing of media and the transient-state or time-domain probing is the nature of any medium’s transient response. That response is very much a function of the solid-state picture chosen to describe the medium. Propagation velocities also change in the vicinity of a medium’s absorption band, as we will see in Section III. Even the entity of importance considered to be propagating: i.e., whether it is the group, phase, or signal properties of the wave packet, is debatable. * However, the phase representation of a continuous wave and a wave packet can be seen to be quite different. This can be seen with the introduction of a few definitions. With CDas the radian frequency of a wave and a the phasor, or neper, frequency defining the envelope shape, a complex signal can be defined as

373 (7.57) and an analytic signal (also known as the pre-envelope function) can be defined as S( t) = S(t) + iS(t) = exp[S0] = A(t)exp[ico0 + 25 This many-body viewpoint has resulted in the development of a rigorous quasi-quantum mechanical theory of dielectrics by Dissado and Hill.20-22 Another recent observation departing from the Debye picture is that of abrupt or discontinuous dipolar or charge carrier transitions between their preferred “stationary” orientations or positions. This abruptness is in contrast to the smooth transitions of the classical Debye dipoles, which are presumed “floating” in a continuous viscous medium. It is implicit in this many-body picture that there exist two very different time scales corresponding to microscopic and macroscopic processes. The interactive nature of a many-body system implies that any sudden changes of charge or a dipole produces a “chain” response stretching both in time and in space beyond the time and position of the initiating transition. Microscopic transitions are the very rapid oscillations of individual dipoles or charges and these would be the only processes at play in a noninteractive (Debye) medium. Due to the particle-particle interactions, these microscopic oscilla­ tions slowly transmit to other regions of the medium which itself adjusts more slowly to the rapid microscopic movements. These interactions may be either directly between the active particles, dipoles, or charges, e.g., through coulombic interactions, or, alternatively, they may take place through the intermediary of the “inert” matrix or lattice in which the active species are embedded. In the case of the latter transfer, the interplay between dielectric polarization and lattice-strain relaxation facilitates transfer through the lattice, i.e., the system shows cooperativity. The dynamics of these interactions are related to recent work on phase transitions.319'322 This, then, is the post-Debye picture based on the work of Dissado and Hill,22 which commences with the simplest description of a system responding to an external influence, called a probe. The system observable, or response, can be described by a power series in the probe field with the coefficients defined as the susceptibilities of varying order. In the limit of a response which is linear in the probe field, the formal theory of Kubo26'28 shows that the susceptibility is determined by the spontaneous fluctuations of the observable property in the absence of the probe. However, when the unprobed system is in equilib­ rium, the fluctuations die away, giving a relaxation which is determined by the dynamics of the system. Energy losses are a direct consequence of Kubo’s fluctuation-dissipation theorem. In condensed phase systems, relaxation is defined as the reorganization of the relative orientations and positions of constituent molecules, atoms, or ions and can only be possible in systems possessing some form of structural disorder. The microscopic mechanism of relaxation is determined by the local potential surface traversed by the relaxing entity and the modification of its vibrational and translational velocities, or both, as it moves between alternative equilibrium positions.29This more comprehensive picture of dielectric behavior has not yet fully influenced engineering design of sensors and radars for surveillance and communications. Central to the Dissado-Hill picture is the concept that relaxation must be based on a realistic picture (which the Debye picture is not) of the physical nature of the structure of an imperfectly ordered state and its consequences for the dynamics of its constituent molecules or atoms. The picture, which is of the Kubo type, is of gases, liquids, and materials composed of many clusters, which are spatially limited regions over which a regular structural order of individual units extends. The ordering in clusters is local; however, in any sample of material many clusters will exist. If some coupling interaction between them occurs, then an array will be formed possessing at least a partial long-range regularity. Now this partial long-range order in the limit of a perfectly regular array would be a superlattice. On the other hand, the absence of coupling between clusters leads to a cluster gas where clusters may collide without assimi­ lating and dissociating. Thus the extreme limits of the theory describe a gas at one limit and a superlattice or perfect crystal at the other limit (cf. Figure 7.31); any intermediate structure can be described without losing the essential features of microscopic structure and macroscopic averages. In this model, the probe field is spatially uniform over the sample and couples to the orientation or position fluctuations, or both, that are also spatially in phase. The relaxation of the fluctuations will be the same as if it were produced by some spontaneous fluctuation of sample energy.26 27The displacement fluctuations perturb the equilibrium geometry and alter the partial long-range order of the array. Then, any spatial uniformity obtained is lost on relaxing with the irreversible return of fluctuation excess energy

379

(b )

average structure

°I

8?S £« V§•-| o •e « t? 3

ideal liquid _ _ _ _ _ _ kinetic energy = one quantui kinetic energy of unbound component of average structure unbound ground

Figure 7.31 Schematic representation of the formation of a structural fluctuation of the average structure. The vertical axis is energy and the horizontal axis is distance in the material on either side of a vibrating oscillator or vibrational mode, (a) Solid, with fluctuations in the form of an ideal crystal. The average cluster-binding energy hE is required to raise the actual ground state to the ideal crystal zero-point level, h£, for each fluctuation, and a thermal population will exist in equilibrium, (b) Liquid, with fluctuations in the form of unbound constituents. The unbound component of the kinetic energy must increase by the average cluster-binding energy for each fluctuation formed, and again a thermal population will exist at finite temperatures. (Adapted from Dissado, L.A. and Hill, R.M., A cluster approach to the structure of imperfect materials and their relaxation spectroscopy, Proc. R. Soc. London, A390, 131-180, 1983. With permission.) to the thermal heat bath. Thus, relaxation is in terms of a many-body cooperative regression of a spatially uniform site displacement comprising partial local and long-range regularity. In general, the time scale of the displacement motions for modes in each of these groups is determined by the strength and binding appropriate to the group and will range from short time (about 10~13 s) of molecular vibrations to the relaxation time scale of macroscopic variables (>10-9 s) and therefore of long enough duration for RFmedium dynamic interactive effects to occur. It should be noted that the bulk relaxation time scale is orders of magnitude longer than the fastest motions because relaxation involves a large number of such vibrations either spatially concentrated to give a large positional change in the region of specific sites, or a large number of small positional adjustments at different sites. If the time scales of different groups contributing to the macroscopic relaxation do not overlap, then the displacements produced by the modes in the short time scale, high frequency groups will affect the macroscopic variables in a random manner, and the effect will be that of a random thermal noise on the macroscopic variable. In the Dissado-Hill model, fluctuations of the macroscopic variables are denoted by the spatial Fourier transform of the site fluctuations. The relaxation of the macroscopic fluctuation has a time development of the general form30 4 aaq ]/dt =

- Y q)a aq( 0 + * « , ( 0 + Y , [d»«, / ® « , ]E «q(0 .

(7 -7 6 )

where Eaq(t) is the field conjugate to the variable a 1, X "M = X'(°) - X 'M s ®m, a>/Yq < 1,

(7.78)

where the indices 0 < n, m >1 are shape parameters, already introduced above, of a two-parameter model. The equations in 7.78 describe nonexponential relaxation which is observed in dielectric studies, me­ chanical relaxation, acoustic attenuation, and optical turbidity. On the other hand, the Debye equation (7.77) describes behavior which is extremely rare. A refinement of the general equation (7.76) to account for interacting fluctuations of different wave vector and type32 produces nonlinear mixing resulting in a wave packet extending over a restricted spatial region. The eigenmodes of the localized packet describe the cooperative motions of the cluster defined by the localization of the packet. With increase in wavelength, q, cooperativity causes the unperturbed modes to approximate the macroscopic modes.33 The whole of the frequency range from short to long wavelengths is thus available for coupling to the probe field. This cooperative behavior results in an extra absorption at frequencies greater than the relaxation rate, the source of which is the cluster eigenmodes. The nonlinear mixing modifies the site relaxation rate, yq, making it an effective rate, the time dependence of which is due to the continually changing displacements in the site environment during relaxation. Thus, nonexponential relaxation is due to the progressive involvement of an increasing number of sites as the sites become coupled up to a limit determined by the cooperativity or localization of the cluster. The relaxation rate of each of the coupled groups cannot be assumed to belong to a separate independent relaxor, nor can the nonexponential behavior be simulated by a suitable (and arbitrary) distribution of relaxation rates — for this would lose the essential physical features of the theory; namely, that it is the coupled motions in the form of cooperative (localized) cluster modes that physically determine the nonexponential behavior, with each cluster unit participating in all modes. In order to place the Dissado-Hill picture in perspective, we may describe it in terms of two other but different examples of power law behavior. The first is a formal Green’s function approach. The equation of motion for the Green’s function Goq(t) contains a memory kernel resulting from the nonlinear mixing terms:34 {d/dt - icoq + Y , ) G q ( t ) = Y q S (t) - £ kVkqVkqJd sG k(t - s)Gq(s) o

(7.79)

The dynamic susceptibility can be obtained by Fourier transformation of G(t), giving x(«.q) = [xo_1(« .q )+ 2:(® .q)rl = Gq(u>).

(7.80)

381 where X(co,q) is the solution to the unmixed (linear) Equation 7.76 and is also a self-energy generated by nonlinear terms 1(G),q) = [-l/7 c]£ “dco'i;kVkqVki)G k(G)-G)')

(7.81)

It is the term, X(0),q), which shifts the eigenmodes of the system. The second example of power law behavior is the relaxation of a system close to a critical point predicted to have the form r n, 0 < n < 1.33 It is well known that in the theory of critical pheomena the order parameter fluctuations are localized within a correlation length, which increases toward macro­ scopic size as a transition temperature is approached and as the order parameter fluctuations decrease. The equation of motion is nonlinear and generated from a thermodynamic force obtained from the free energy gradient. The mixing term is related to the intrinsic site probability density for the order parameter, permitting the order parameter to adopt a continuum of values at a site, even though only a restricted number are possible in the site coordinate system.35 As the fluctuation rate decreases and approaches a transition temperature, the mixing term ensures no gap opens between its time scale and that of the other modes. At the same time an infrared divergence in the self-energy occurs. There is thus a parallel with the Dissado-Hill picture, the major difference being that the correlation length is divergent only in the critical systems. The unifying theme of these examples is mode-mixing via a nonlinear rate equation and an infrared divergence in the self-energy, the origin of which is the localization of a fluctuation within an imperfectly structurally ordered region as a cluster excitation.36The result is that the long wavelength fluctuation both evolves into a packet of localized modes and decays as r n, where n is determined by the degree of coupling of motions within the cluster. The power laws described in the equations in 7.78 above, which distinguish the Dissado-Hill approach from that of simple Debye theory, are thus due to the modecoupling of short- and long-range order. This new theoretical approach implies that, whereas in perfectly crystalline or ordered materials, i.e., limiting cases, elementary excitations are formed of the normal mode kind, in structurally disordered materials; on the other hand, there is an effect of macroscopic structure on microscopic relaxation behavior. Thus the presence of disorder requires the recognition that most materials are organized into different levels corresponding to different size scales ranging from macroscopic to microscopic. The effect of structural organization upon the normal mode structure of a crystalline sample is to factorize it into manifolds or large-size functioning entities made up of joined molecules.22 Weakly coupled and interpenetrating manifolds result, each of which spans the frequency range from site vibrations to relaxation without a gap. The Dissado-Hill model is able to describe disordered materials by including the effects of local cluster structure and the weak coupling of its eigenmodes to a coarser level of structure through intercluster exchanges. The equilibrium structure adopted by any cluster of atoms or molecules will be determined by the minimization of the net potential energy resulting from the short range attractive and repulsive potentials under the restraints of formation. The degree of structural ordering is then given by the cluster-binding energy expressed as a fraction of the energy required to form the same constituents into a perfect crystal. This fraction is equal to the ratio of the cluster average eigenmode frequency to the crystal zero-point frequency, which is also the cluster maximum eigenmode. The Dissado-Hill model thus provides a quasi-quantum mechanical picture of RF classical material behavior (Figure 7.31). The ideal structure of a material or liquid is a background whose dynamic properties are assumed formally understood and all displacements are referred to this ideal state, which acts as a template. The language of second quantization37 is used, and displacements are created from the ideal state by the application of operators a«p+ and eliminated via the annihilation operators, aop". The relaxation behavior is determined by the evolution of a specified fluctuation calculated to be in existence at t = 0. The ideal state defines a ground state and a fluctuation or spatially uniform displacements with respect to this ground state defines the excess energy (hE) or the cluster-binding energy below this state. Turning now to the relaxation behavior, what has become clear recently is that relaxation is determined by the annihilation of independent clusters through an irreversible exchange of energy with the heat bath. Thus, relaxation must be referred to the properties of the cluster as a whole rather than those of its constituents.2933 In fact, there is some experimental evidence for group behavior in yp, the site relaxation rate.29 Furthermore, the shifts in cluster ground-state energy and entropy determine yc. A consequence of this characterization of a bulk sample in terms of submacroscopically sized clusters is that the coarser

382 level of structural organization defines a superstructure in which the constituents are individual clusters, and in coarser materials with complex morphologies, this coarser level may even itself be embedded within other levels, i.e., there is a nesting of structural levels. With this picture in mind it is possible to predict that coupled motions can change form from a cluster mode to an intercluster mode, as the spatial extent of the coupling (wavelength) increases beyond the cluster size. Furthermore, an increase or decrease of cluster size, by permitting the number of constituents involved in intracluster motions to differ from the average, will affect the intercluster modes. Thus, the long wavelength (cluster size) cluster modes are weakly coupled to the intercluster manifold. These processes are called intercluster (IC) exchanges. In liquids, IC exchanges are produced by molecular transport as demonstrated in computer simula­ tions.38 These simulations demonstrated that a chosen atom could transfer between clusters as its fluctuating environment generated and removed the barrier to its transport. IC exchanges are also possible in solids. The index, m, introduced above as a shape parameter, may be defined as the degree of structural change between a cluster and the average array structure. The intracluster relaxation stochastically represents a non-Markovian process (a process with memory), and the evolution of the steady-state causes fluctuations in the relaxation process itself. The dynamic susceptibility of the array is X(co) = {r(l - n)/m}N(lA cF(co/Yc),

(7.82)

where T () is the gamma function, N^is the number density of constituents, Acis the amplitude per cluster constituent, and F(G)/yc) defines the shape of the susceptibility as a function of the reduced frequency CO/yc20 f (w/Yc) =

[{r 0 + m - n)}/{r(2 - n)r(m)}] [l + i((o/Yc)]' " x 2F ,( l- n , 1- m; 2 - n; [l+i((o/Yc)]"‘],

(7.83)

in which 2F , ( , ; ; ) is the Gaussian hypergeometric function. Using Equation 7.83, curves can be fit to empirical data (real and imaginary susceptibility data) with estimates of n and m. The excellence of fit gives confidence that the n and m chosen are correct. In conclusion, whereas the Debye theory of dielectric response is inappropriate to describe the dielectric response of most materials, the Dissado-Hill many-body approach has provided conceptual clarity and agreement with experiment. The latter theory also indicates that whereas a calculation from first principles of relaxation time constants would be very difficult and merely a gross estimate, by defining the shape parameters n and m with respect to spatial correlation and IC exchanges, a semiempirical approach to relaxation time estimation is, nonetheless, possible. The Dissado-Hill picture permits the treatment of a the full spectrum of materials from liquids to solids in terms of n (the degree of structural ordering in an average cluster) and 1 - m (the degree of ordering within the cluster array). The Debye model (which addresses independently relaxing units) can be recovered for Dissado-Hill parameters n = 0 and m = 1. In fact, n = 0, m = 1 also defines an ideal liquid with ideal transport providing a white noise source. Thus, n = 1, m = 0 defines an ideal crystal with no internal relaxation and zero loss. All other media lie between these limits. For example, fluids, for which n = 1/2, m ~ 1/2, are plastics, waxes, plastic crystals, or viscous liquids; solids, for which n -» 0, m —» 1, are ferroelectrics and interstitial impurities; and solids, for which n —>1, m —> 0, are topological impurities occurring in imperfectly crystallized materials, glasses, and vitreous polymer systems.22 What this means is that the medium at RF frequencies has far more complexity in its relaxation behavior than is conventionally believed. This also means that there are many ways to couple pulses to the medium at RF frequencies for purposes of efficient propagation through the medium, or for precise sensing of aspects of the medium. Of most importance is that it means that nonlocal and cooperative media effects can predominate at RF frequencies. Such effects are, however, difficult to calculate.

383

COOPERATIVE EFFECTS In Section III below we discuss the 1907 Sommerfeld proposal that when a wavefront with a fast risetime enters a dispersive medium, the wavefront proceeds undisturbed.39-40 This is explained on the basis of point electrons having an inertial mass and that they cannot react to the beginning of the wave. This was the first departure from a steady-state oscillation description of medium dispersive behavior, in favor of a time domain description, although the Lorentz dispersion law derived from the forced harmonic oscillator was still followed.4144 Brillouin45 expanded this approach, which is the linear coherent superposition of all frequency components of the dielectric susceptibility of the medium, x(w) P(co) = x(©)E(co),

(7.84)

where P(co) is the polarization, E(cd) is the electric field, and the SI units of P(co) are FJ/Asm2 (meter, m; Farad, F; joule, J; ampere, A; and second, s). The well-known relations to the dielectric permittivity, £, and the electric displacement, D, are D = E + 4tcP

(7.85A)

D(cd) = e(co)E(co)

(7.85B)

P(co) = x(to)E(co)

(7.85C)

£ = 1 + 471%.

(7.85D)

According to this conventional approach, P(t) in the transient regime is characterized by the singularities of %(co), whereas the steady state is characterized by the singularities of E(co) near the real co-axis.46’50 However, this approach, the conventional approach, assumes that each particle in the medium is in a steady state with its immediate environment. It is, in effect, merely a local description of the response of the medium: All existing models of dielectric relaxation and dispersion ... account for the cooperative effect of relaxation by a damping term in the equation of motion of a single particle, while it is clear that ... absorption and dielectric relaxation are cooperative phenomena, and therefore manybody problems.51 As we have seen in the discussion of the classical Debye theory of dielectrics, the conventional approach assumes a local steady state and that relaxation follows an exponential (Beer’s law) decay law, despite evidence to the contrary.11*22,52 On the other hand, at RF frequencies most media exhibit global (cooperative) effects. In fact, two dielectric relaxation functions53-54 can be described: one defined as the transient current generated by the application of a unit voltage; the other defined as the transient voltage generated by the application of a unit current. The relaxation function is generally believed to contain only two types of parameters: one energy dissipative, the other energy storing. However, a relaxation function can contain three types of parameters: one energy-dissipative parameter representing resistive elements, and two energy-storing parameters representing capacitive and inductive elements. If global (cooperative) effects, as well as local effects, can be damped, i.e., cooperative effects are not assumed to be steady-state effects, then the polarization is P(co) =

P(t) =

j

J* x(t,t')E (t')dt' 3c(©, T*, permitting the phase of the packet to be preserved and self-induced transparency to occur If 1 through 3 occur, Beer’s law will not hold. (There are, of course, other instances in which Beer’s law also will not hold.)

387 (a) Homogeneously broadened line. Af = 1>T

Frequency

Inhomogeneously broadened line. AF= 1/T*

T»T*

Frequency

Figure 7.32 Inhomogeneous and homogeneous broadening. In (a) the individual emission lines associated with the different oscillating dipoles are shown. These lines are homogeneously broad­ ened with a relaxation time T. In (b) the same collection of oscillating dipoles are shown in a dielectric medium. The medium’s emission line is shown inhomogeneously broadened with a relaxation time T*. In this example T > T*, i.e., inhomogeneous broadening is greater than homogeneous broaden­ ing. (Adapted from Allen, L. and Eberly, J., Optical Resonance and Two-Level Atoms, John Wiley & Sons, New York, 1975.)

ZERO DEGREE PULSES The zero degree pulse of Crisp56-59is another non-Beer’s law case (Figure 7.33). This type of packet is a small area pulse which enters a resonant medium, where the area referred to is the area under the wave envelope in an energy vs. time plot. Its leading edge excites a thin slice of the medium into a polarization state which radiates 180° out of phase with respect to the leading edge for a time of the order of T after the edge has passed. If the trailing edge of the packet drops off faster than the decay of the macroscopic polarization, then the trailing edge of the wave packet leaving this slice will go through zero and become negative. The next slice of material then sees a field envelope whose trailing edge drops off faster than before and this becomes negative. The resulting induced polarization in this next slice by the positive lobe of the the packet envelope radiates 180° out of phase with respect to that field and adds to the negative lobe. We can describe the zero degree pulse as a linear effect, but we cannot get the total result by a continuous wave as it is a signal envelope or pulse effect. If the packet duration, t, is defined as t > T, i.e., this is the conventional steady-state case, then the conventional complex dielectric constant is, as we have seen (7.97) for oscillator a, which is the usual dispersion relation and applies when packet times are so long that every dipole’s transient response has decayed to zero. However, in the T > t > T* range there is a different result due to the classical area theorem (7.98)

where K is the wave vector in the medium; A(t,z) is the area of the packet; k is the wave vector in a vacuum = co/c ;

388

E

E=

0, EA2 >0 distance x

E = 0, EA2 >0 distance x

Figure 7.33 Crisp zero degree pulse. Classical area theorem permits media penetration if E = 0 (i.e., XAE = 0), but E2 > 0 (i.e., power > 0). Area theorem permits increase in wavelength as seen with ground penetrating radars.

cop is the plasmon frequency for the dielectric; and g(0) is the inhomogeneous lineshape detuning function. The area theorem implies that the incident field first loses energy during propagation in exciting dipoles of the medium, but that loss is recoverable. After a time, T*, the dipoles will trade stored energy with each other — that energy being taken from the packet — but that energy still is not yet dissipated. If T > T* and T > t, then that stored energy need not be traded among the dipoles but can be “restored’’ to the packet. This effect, of course, only occurs with pulses of duration less than the T relaxation time and if inhomogeneous broadening is greater than homogeneous broadening. Under such conditions, the pulse and the medium can function together as a dynamic system. Other “stored-restored” energy effects include photon echos, optical mutation, free induction decay, optical adiabatic inversion, and self-induced transparency. Notice that the implication of the area theorem — that the electric field area must decay exponentially — does not imply that the energy must also decay.5760 This is because phase changes of n in the electric field, E, cause part of the envelope of E to change sign, resulting in the integral (area) of E to approach zero, but the packet energy, E2, can remain large. This analysis gives the E field as (7.99A) where (7.99B) and A is bandwidth detuning. In the case t < T for packets shorter than the relaxation times, T and T*, calculations show57 that in the case of Gaussian pulses, although the pulse area drops to zero in penetrating the medium and the frequency also lowers (Figures 7.34A,B), the corresponding absorption of energy, E2, occurs much more slowly (Figure 1 3 AC). These numerical simulation results are reminiscent of empirical results obtained using ground penetrating radar shown in Figure 7.35.

389

-J

I -2

L.

!

_i

0

2

4

i

L_

L

J

6

[t-z/cyr

t - T

Figure 7.34

Non-Beer’s law penetration by classical methods: numerical example. The field strengths required with classical RF pulses are many orders of magnitude less than those required with quantum mechanical optical pulses. This is because whereas the optical pulse induces an electronic polarization, the IR and RF pulses induce an ionic, dipole or orientational polarization. For bulk effects to occur at quantum mechanical optical energy levels high intensities are required. For bulk effects to occur at IR and RF classical or quasi-classical energies requires many orders of magnitude less energy but are difficult to calculate. (Permission of Crisp, M.D., Phys. Rev., A1, 1604-1611, 1970.)

SELF-INDUCED TRANSPARENCY Self-induced transparency is a quantum mechanical effect in which an otherwise absorbing medium becomes transparent or conducts without decrement. Pulses which penetrate by self-induced transparency when interacting with the medium are solitons. The conditions for this to occur are

Distance (ft)

Water Surface

Figure 7.35 RF imaging by serial probing using a Geophysical Survey Systems, Inc., impulse radar towed across the surface ice on a lake. These groundprobing radars transmit a pulse 1 to 5 ns in duration with a 500-MHz carrier. The returned signal is dispersed to a frequency much less than this initial carrier frequency, indicating behavior similar to that shown in Figures 7.33 and 7.34A and B, i.e., wavelengthening while penetrating. (Photograph courtesy of Geophysical Survey Systems, Inc.; North Salem, New Hampshire.)

LAKE BOTTOM

o

391 1. The incident radiation intensity or pulse must exceed an energy threshold and the pulse must be in resonance with that threshold. (Threshold energies are defined in terms of hco units; therefore, the energy of a wave is defined quantum mechanically in terms of Planck’s constant, Js, and frequency, cycles/s.) 2. Inhomogeneous broadening must be greater than homogeneous broadening in the medium so that phase is preserved in the pulse, i.e., T* < T. 3. The pulse must have a shorter duration than the relaxation time of the excited state, i.e., t < T. The only conclusive self-induced transparency demonstrations have been at quantum mechanical wavelengths. As radiation of wavelength 10-3 m can be produced by either microwave techniques (microwave oscillators) or by infrared techniques (incandescent sources), we might consider that quan­ tum mechanical wavelengths commence at about that wavelength. Theoretically, however, the effect is completely general and could occur quasi-classically. That is, a better description of the RF material dielectric response requires the Dissado-Hill quasi-quantum mechanical treatment described above. Investigations of self-induced transparency at RF wavelengths are a worthwhile pursuit because of recent developments in short duration RF sources. Furthermore, as explained above, many media behave cooperatively at RF vibrational wavelengths, which should produce greater inhomogeneous broadening than occurs with the precisely defined electronic energy levels at optical wavelengths. Specifically, in self-induced transparency a short pulse of coherent traveling-wave radiation propa­ gates with anomalously low energy loss while at resonance with a two-quantum-level system of absorb­ ers.61*62 The pulse width must be short with respect to dissipative relaxation times and the pulse amplitude must be at least as great as the energy separation of the two levels. Ideal transparency persists when coherent absorption of pulse energy during the first half of the the pulse is followed by coherent emission of the same amount of energy back into the beam direction during the second half of the pulse. The usual situation occurring when an EM field penetrates a dielectric medium is that the field amplitude decays exponentially with increasing penetration into the dielectric; this is Beer’s law as discussed above. For self-induced transparency to occur, the medium through which the pulse travels must not possess an internal spread of resonating dipole frequencies that will result in dephasing among the dipoles and consequent absorption due to the driving field being opposed by electric fields radiated by the dipoles. Inhomogeneous broadening must be greater than homogeneous broadening and too much spread of resonating dipoles (homogeneous broadening) will result in pulse dephasing. Given this condition, if the pulse excites resonant dipoles into a predominately inverted or “pumped” state before the pulse has subsided, some energy of induced emission radiation will be returned coherently into the remaining portion of the pulse. That is, the induced polarization will radiate an electric field which will then add to the driving field. An equilibrium condition is reached whereby the energy of induced emission transferred from the medium to the last half of the pulse becomes equal to the energy previously absorbed from the first half of the pulse by the medium. Thus for self-induced transparency to occur, attenuation by damping of the resonant dipoles and scattering losses must be small or absent and this can be achieved with a short duration pulse. If the pulse has a width shorter than the damping time, but longer than the inhomogeneous lifetime, damping attenuation is reduced. Therefore, the conditions for self-induced transparency are severe at optical frequencies, and for the effect to occur an optimum pulse length matched to the medium, not merely a short pulse, is required. The pulse polarization must also match the medium (cf. Reference 63). In the simplest case, the initial pulse evolves to a final symmetric shape which is a 2n hyperbolicsecant function. In the more complicated case, n - k pulse areas are required. Continual absorption of energy from the pulse leading edge and emission of energy into the trailing edge make the pulse velocity less than that of nonresonant radiation in the medium. The usual model for radiation interacting with a two-level atom starts by observing that such an atom is analogous to a spin half particle in a magnetic field. The model defines a fictitious electric spin vector, or pseudospin vector. The pseudospin vector has components related to the atom’s dipole moment and inversion. Bloch developed the spin formalism for electromagnetic resonance,64 which we can apply, with certain assumptions, to the pseudospin vector of the two-level atom case. The equations describing the components of the vector are known as the optical Bloch equations. We need some assumptions for the analogy between magnetic resonance and field-atom interaction to work. The first assumption is the rotating wave approximation or RWA.65 Using RWA, the pseudospin vector equations take the form of equations for the solid body precession acted on by a known torque.

392 The rotating wave approximation only considers the cumulative effect of the positive torque on the spin. We consider the negative torque as reversing itself and ignore it. A nearly stationary vector can then be defined in the rotating frame based on the components of the spin. The rotating wave approximation is not an exact description because it neglects the negative components of the torque mentioned above. The negative torque can be important and the Bloch-Siegert frequency shift of dipoles267 is due to this counterrotating torque. However, most workers accept the simple model as a framework for understanding transport of radiation through a dipole medium. The next assumption is the nonlinear area theorem of McCall and Hahn. This nonlinear area theorem replaces Beer’s law of linear absorption and accounts for (1) lossless propagation in an absorbing medium, (2) the breakup of large pulses into smaller ones, and (3) pulse compression by coherent absorption. It is the quantum mechanical equivalent of the classical area theorem examined earlier. We arrive at the nonlinear area theorem by way of the Bloch vector tipping angle in the rotating wave approximation. We start by describing the behavior of a pulse propagating without loss through a dielectric medium. Consider the Bloch vector tipping angle, a dimensionless quantity, in the RWA picture described above. The hyperbolic secant pulse of McCall and Hahn61*62 (see below) describes the tipping angle behavior of the electric field envelope. The tipping angle of the Bloch vector traces out a classical dimensionless area. When the Bloch vector tips completely up, over, and down, it traces out 360° or 2n radians, and arrives back at the energy level where it started. We call a pulse producing this behavior in a system a 2tc pulse. In the quantum mechanical case, the absorptive part of the on-resonance dipole amplitude is a nonlinear function of area traced by the Bloch tipping angle. McCall and Hahn first derived the nonlinear area theorem as an equation of motion for the pulse area.61 In the weak field limit, the Bloch tipping angle area is small and the quantum mechanical nonlinear area theorem is equivalent to a classical linear area theorem. That is, at this limit, the nonlinear quantum mechanical area theorem approximates the classical linear area theorem. However, in the high field case the quantum mechanical nonlinear area theorem and the linear classical area theorem differ. For example, the nonlinear area theorem makes no distinction between 2n and 2n7C pulses. Furthermore, the areas which are even multiples of n are more stable than those which are odd multiples. The 2tc pulse, which is both area and shape stable, is a solution to the lossless coupled Maxwell and Bloch equations. The envelope equation for the pulse in its simplest form is (7.100) where

M 2 = [4 /k 2] |k 2/(jli2F - A2)} K = 2d/(h/2n),

(7.101)

where d denotes the magnitude of the dipole matrix element, which will be large for optical wavelengths and less for IR and RF; A = co() - co is the detuning frequency; F(A) is a dipole response function; and h is Planck’s constant. The constant, M, exercises a restriction on the area of the pulse, regulates the eventual maximum pulse intensity and controls the pulse growth to its maximum rate. Equation 7.100 is of the same form as the equation for a pendulum for which the solutions are well known to be elliptic functions. Thus, a pendulum gives us a direct physical model for the Bloch vector behavior in the RWA picture. The rate at which transitions are coherently induced between two atomic levels is known as the Rabi frequency. At this frequency, there is a signal-and-medium resonance. A coherent light wave will exactly invert a ground state atom if the pulse is a “tc pulse” so that KE(t) = 71, w here K = 2d/(h/27C).

(7.102)

393 The postulated method of energy storage in the medium is a major difference between the classical, “quasi-quantum mechanical” Dissado-Hill type picture, described in Section III and quantum mechanical absorption picture. A classical oscillator can oscillate with arbitrary amplitude, but a two-level system can store energy only up to anamount, hco/27C. This is due to a conservationrelationshipbetween u, the dispersive component of the dipolemoment, v, the absorptive component of the dipole moment, and w the population inversion u2 + v 2 + w 2 = 1.

(7.103)

This relation requires the dipole moment to vanish when the dipole’s energy is maximum (w = +1) as well as minimum (w = -1). McCall and Hahn proposed what is now called the McCall-Hahn6162 area theorem in direct analogy with the classical case 3A(t,z)/3z = (-l/2)asinA (t,z),

(7.104)

where A is the area under the pulse envelope and a is the absorption coefficient. In the limit of weak fields, this relation is 3A(t,z)/3z = (-l/2)aA (t,z),

(7.105)

which is identical to the classical case. The area theorem shows that slowly varying wave packets with an area equal to an integer multiple of 2 k will be stable. A 2 k pulse which is both area-stable and shape-stable is a solution of the lossless coupled Maxwell and Bloch equations. The hyperbolic secant pulse of McCall and Hahn is E(t,z) = (2/KT)sech|(t~t0)/t},

(7.106)

where K = 2d/(h/27t)

d is the magnitude of a dipole matrix element, T is the pulse length,

E(t,z) is the electric field envelope corresponding to the tipping angle, 0 behavior of a Bloch vector (itself related to packet “area”). This is simply

320/3t2 - ( l / t 2)sin0 = 0,

(7.107)

which, as in the case of Equation 7.100, is the form of the equation for a pendulum for which the solutions are elliptic functions. The above is the theory in broad outline. It is, of course, more complicated for specific systems. Appendix 2A looks at the more complicated aspects of self-induced transparency. The following sections reviewexperimental results at optical, IR, RF,and acoustical wavelengths and fields. We must emphasize thatthe radiation-matter coupling mechanisms (dipoles, permanent and induced; electron energy levels; charge separation; vibronic energy levels; exciton/polaron energy levels, etc.) are different in the various cases considered at optical, IR, and RF energies.

Optical and IR Self-Induced Transparency: Experimental Results There are many examples of self-induced transparency in the optical region. McCall and Hahn,6162 Armstrong and Courtens,66 and Armstrong67performed the early work using ruby laser pulsed light acting on a passive ruby sample tuned to the driving pulse. Patel and Slushei68 showed the effect in a gaseous

394 medium of resonant SF6 with 10.6-mm radiation from a C 0 2 laser. Yariv69 and Slusher70 reviewed early and recent work and the theoretical underpinning. Patel and Slushed8 showed self-induced transparency and delay times of 200 ns in SF6. The same effect has been demonstrated by Gibbs and Slusher71'74’343344 in dilute rubidium vapor at 7944.66 A and by Asher & Scully75 in ruby. Ironically, despite the fact that self-induced transparency has been more clearly demonstrated in the optical, i.e., electronic, regime, the conditions for which the area theorem (both classical and quantum mechanical) and self-induced transparency apply might be expected to be more frequently encountered in the IR and RF regimes, as the incidence of inhomogeneous broadening can be higher in those regimes due to the prevalance of nonlocal and cooperative effects at those frequencies. The term “nonlocal” refers to the situation where activity at a point, e.g., x, to which all equations of motion refer, is influenced by activity at another point removed from x, e.g., y. Therefore, the activity at x cannot be explained completely by activity local to x but only by including nonlocal influences (from y). The lack of IR and RF equipment for generating transient signals and the absence until recently of an adequate theory of dielectrics may have precluded these conditions from being noted.

RF Self-Induced Transparency: Experimental Results When considering the RF spectral region there are two main points to keep in mind. First, analogous vibrational energy levels do exist, but they are not precisely defined as in the electronic regime (optical and IR; see below). Secondly, also unlike the electronic region, there is a greater degree of cooperativity, that works in favor of more inhomogeneous broadening (see below). In fact, there is much experimental evidence7677 showing that simple lumped circuit models involving one or two capacitors or resistors can describe the dielectric response of such complicated material as wood and compacted leaves. The exhibition of cooperativity at the vibrational level works toward greater inhomogeneous response times. In the laboratory, it has been shown that a microwave magnetic field pulse propagates with self-induced transparency in electron paramagnetic resonance with a free radical sample of electron spins distributed along the guide.103 Geophysical Survey Systems, Inc. (GSSI) manufactures ground- and fresh-water penetrating radars using 2- and 5-ns signals. Figure 7.35 shows an example of the images which may be built up over time. The interpretation of this data is debatable. Some point to the increase in wavelength of the returned signals as showing that long wavelength continuous wave signals would produce the same effects, if only there was no overlap of the emitted and returning signal. However, consider the following: 1. The phenomenon of non-Beer’s law behavior and the area theorem requires wavelength broadening (Figures 7.33 and 7.34). Therefore the increase in wavelength is certainly not unambiguous evidence that long wavelength continuous wave signals could penetrate as well. 2. On the other hand, the evidence is that the longer wavelengths present in the returned signals are certainly not present in the emitted signals; therefore, no linear Fourier decomposition of the signal before entering the medium explains the evidence — rather the evidence would indicate that the medium is either dispersive or active and performing frequency or wavelength conversion. 3. When discussing both the classical and quantum mechanical area theorems, it is of no consequence in self-induced transparency that the wavelength increases and area decreases; in fact, the area (which is to E by its integral) must decrease in order to preserve a finite E2 classical 2n pulse. What is of consequence is that the energy (which is E2-related) remains as constant as possible. Points 1 through 3 argue, therefore, that the evidence from ground-penetrating radars indicate the efficacy of transient signals over continuous waves, but what would be a definitive test? A definitive test is to see which system — time domain or frequency domain — returns the maximum E2energy and at a range for which the emitted and received continuous waves do not overlap. The frequency and wavelength are inconsequential in deciding on this particular application. Why has this test not been done? First, the range must be fairly long to obtain nonoverlapping continuous waves and there has been no long-range time domain system built. Second, the problem has never had an appropriate conceptualization or application. Other uses of media-penetrating pulses are for (1) ground- and fresh-water penetration;78'81-337 (2) vegetation penetration;82 84(3) ice penetration;85 90337 and (4) ground, clay, and peat penetration.819199The steady-state theory of electromagnetic ground penetration is treated by Wait.100102

395

a?

>

Figure 7.36 The functions c dk/doo = c/vg and n(co) = ck/co = c/vp vs. frequency for a one-resonance Lorentz model without damping, where k is the wavenumber = MX and amplitide is arbitrary. (From Jackson, J.D., Classical Electrodynamics, 2nd ed., John Wiley & Sons, New York, 1975, 318, ©1975 John Wiley & Sons. With permission.)

Acoustical Self-Induced Transparency: Experimental Results Beginning with early examinations of microwave ultrasonic pulses interacting with spin S = ±1 sys­ tems,1(Wthere have been a number of more recent theoretical and experimental treatments. (S = ±1 means that the two states of spin provide two energy levels with which the pulses may interact.) The effects of propagation of short acoustic pulses through a resonant-absorbing medium have been calculated.105 The pulse phasing was a random Gaussian process. The calculations showed that the presence of a random phase modulation decreases the pulse intensity theshold at which there appears a self-induced transparency by an amount proportional to the phase standard deviation. The pulse takes on a hyperbolic secant form while propagating. The acoustic self-induced transparency situation has been examined under the conditions of a “forbid­ den” transition when the pulse propagates through the medium by excitation of electron and nuclear spins simultaneously.106 The pulse velocity was found to increase as compared with the case of “allowed” transitions. In some semiconductors containing impurities, the probability of the simultaneous reorien­ tation of the spin of the electron and nucleus of the impurity is much larger than that of the processes occurring with the change of only the projection of the spin, for example, in silicon crystals containing 7Li impurities.

396

The influence of transverse relaxation on acoustic self-induced transparency in a spin system has also been assessed.107 A more exact description of obtained results demands an account of both transverse and longitudinal relaxation. Acoustic self-induced transparency in anisotropic media has been examined where a so-called extraordinary elastic wave can propagate;108*110 and the phase modulation of the self-induced transparency soliton in the phonon range of the spectrum has been derived by accounting for the nonlinear dependence of the phase on the dipole-active phonon density, e.g., on the phonon-phonon interaction.111

PRECURSORS Precursors are responses transmitted through media and targets due to the rapid, or ultrafast, rise and fall time of the envelope of probing pulses. Such media and target responses will be important in future optical, IR, and RF radar- and LADAR-pulsed ultrafast sensing and surveillance systems. They have been detected at RF wavelengths.112 How do they come about? The transferral of energy to any system, whether an antenna, a classical oscillator, or a quantum mechanical oscillator, cannot be instantaneous. If it were instantaneous, then the system to which the energy is transferred, would be inertialess, and such a system could not be said to have an independent existence. All systems have inertia. To be sure, the collision times of electrons in a metal are exceedingly fast, such that only radiation at X-ray wavelengths can probe such times. The collision times or time-offlight of the carriers of semiconductors — electrons and holes — are also quite fast, in the low picosecond realm. Acting according to the Franck-Condon principle, the electron cloud of semiconductors can respond even faster, almost without inertia, but not quite. On the other hand, ion conductors have long intercollision times and large inertia. There are two ways to obtain a reasonable conductivity in a material. One way is to have many carriers but poor (short duration) intercollison times. This is the case of a metal. At the other end of the scale, however, are those materials/liquids which have relatively very few and massive carriers, ions, but which have high mobility (long intercollision times). These materials/liquids have, relatively speaking, consid­ erable inertia, and their intercollision times are in the nanosecond range. Furthermore, when incident radiation induces a dipole in such media, the excitation of contributing excitons, polarons, etc. is nonlocal, again resulting in considerable inertia in “bringing up” the oscillators which interact with the steady-state radiation. One may imagine this bringing up to be analogous to the filling of a cavity. Prior to the establishment of standing waves within the cavity, i.e., the steady-state condition, the cavity responds in a qualitatively different way. Therefore one can state that the dielectric response of media to transient signals of sufficiently short duration (with respect to the relaxation time of the medium) is distinctly different from the dielectric response to steady-state signals. In the case of many materials (e.g., with ion conductivity, cooperativity, and nonlocal interactions) RF wave packets of picosecond duration are short enough. If one asks what qualitative differences in propagation and reflectance are occurring during this period before steady-state conditions are established, then there are some number, but their description is complicated. For example, the (Sommerfeld) precursor (wavefront) velocities through dispersive media are initially at the speed of light, c.1’45112*116 The group velocity (which can be greater than c, negative, or undefined in absorptive media) is a concept which applies to sinusoidal waves (i.e., the steady state) but can convey no information.113 In the case of anomalous dispersion, n < 1, the phase velocity can be greater than c, but, again, can convey no information. Also, the specialized concept of energy transfer velocity applies only to sinusoidal (steady-state) waves. The signal velocity, on the other hand, which applies to transient signals is identical to the group velocity in dispersive media, but differs from the group velocity in absorptive (lossy) media. Mathemati­ cally, the signal velocity is a nonperiodic, nonholomorphic function. That is, it is transient and not differentiable at all points (not analytic). The classical solutions for signal velocities are based on integrations in the complex plane and apply only to dispersive, but not lossy media 4244 113Solutions for lossy media for signal velocities have recently been derived based on the telegrapher’s equation and a continuum approximation.117*126 Besides con­ tinuum approximation solutions, which approximate the solution for a rectangular pulse with a linear combination of corresponding solutions for shifted step functions, there are solutions due to either the classical or the quantum mechanical area theorem.3755127 The bringing up of the media oscillators is reflected in the behavior of both the Sommerfeld and the Brillouin precursors. When radiation interacts transiently with those oscillators for short durations with respect to their relaxation time, their behavior, once brought up, is described by the area theorems, both

397

Figure 7.37 Dielectric dispersion characteristic. (Permission of Pleshko, P. and Palocz, I., Experi­ mental observation of Sommerfeld and Brillouin precursor in the microwave domain, Phys. Rev. Lett., 22, 1201-1204, 1969.)

classical and quantum mechanical. Therefore we will address these topics in the following order: theoretical evidence for precursor effects, empirical evidence for precursor effects, the classical area theorem, and, finally, propagation velocities.

THEORETICAL EVIDENCE FOR PRECURSOR EFFECTS The treatment of precursor effects began in 191440.128,129 an(j turns out t0 been largely correct. The motivation for seeking an understanding of such effects more than 50 years before their experimental observation112 stems from the valid theoretical notion at the turn of the century that, “The modem theory of dispersion and absorption uses the assumption of point electrons having a finite mass... The assump­ tion of an inertial mass results immediately from the fact that these particles can in no way react upon the beginning of a wave.”130131 The assumption still holds today ; in fact, we apply it to carriers other than electrons, such as holes, charge density waves, and ions, as well as induced dipoles. In order to provide an intuitive feel for the dynamic interactions occurring during transient pulse propagation, we show a plot of time of signal arrival — normalized with respect to distance and c (i.e., 0 = ct/x) — vs. frequency (Figure 7.36) and also a frequency wavelength plot (Figure 7.37). Addressing Figure 7.36 at a set distance, as time passes, the horizontal line (0) moves from unity, crossing the group velocity curve in the high frequency range (cos). This is when the Sommerfeld precursor appears. With passing time, the horizontal line crosses the n(0) and the low frequency end of the group velocity curve and this is when the Brillouin precursor appears. Stated differently, and addressing Figure 7.37 in which the two branches of the dispersion curve are explicitly shown, the steepest tangent to the dispersion curve is a measure of the velocity of the propagating wave. At early times this velocity is c for the high frequency (Sommerfeld precursor) components. At later times, the velocity is c/[ l+(cop/cao)2] 1/2 for the low frequency components, where cop= 47cNe27t/m (the plasmon frequency) is related to the charge, mass, or number of electrons in a unit volume. At subsequent times, the speed is given by co(), the characteristic frequency of the carrier. To describe these events, a signal is defined at position x in the medium s 0 + i»

E(t,x) = (l/27ii) Je(s,0 ) expj^st-(x/c)^s2 +(Op2j / jds,

(7.108)

s 0 -i° °

where E(s,0) is a unit step sine wave = co(/(s2 + cop2) or the Laplace transform of the applied signal at the origin, E(t,x) = U(t)sin t; U(t) = 1, t > 0; U(t) = 0, t < 0 Using the following definitions x = (00t, r| = co0 x/c = 27tx/A,, 0 = x/r| = ct/x, P = cop/co0, z = s/co0 = x + iy, 2~ iy/2, with co22 = co02Therefore n(co) = [{(co - coa)((o- cob)}/{(o) - coc)(co - cod)}]1/2. Im co describes the absorption and damping in the medium and Re co describes the dispersion. (From Jackson, J.D., Classical Electrodynamics, 2nd ed., John Wiley & Sons, New York, 1975, 313-327. ©1975 John Wiley & Sons. With permission.)

'f/A.

The signal arrival occurs at the instant when the real exponential behavior associated with the simple pole singularity appearing in the integrand becomes less (in magnitude) than the real exponential behavior associated with the interacting saddle point. The signal velocity does not peak to the vacuum speed of light near a resonance, but rather attains a minimum there.132*138 The signal velocity does not differ from the group velocity except in the region of anomalous dispersion. The velocity of energy transport is also equal to the group velocity when approximations such as neglect of absorption are made.

EMPIRICAL EVIDENCE FOR PRECURSOR EFFECTS It is possible to experimentally separate the two precursor effects in the laboratory. Such a separation could occur in an engineered RF sensor probing with analogous conditions of probe pulse and medium. Using a ferrimagnetically filled coaxial line with two propagation branches, for which the characteristic frequencies, (00 and (cop2+ co02)1/2, are dependent upon an externally applied magnetic field, Pleshko and Palocz112 confirmed the theoretical calculations made prior to that time regarding the Sommerfeld and Brillouin precursors and the signal arrival. In Figure 7.41 the experimentally obtained Sommerfeld and Brillouin precursors (top), the Brillouin precursor (middle), and the Brillouin precursor and signal (bottom) are shown at a carrier frequency of 0.625 GHz. The experimentally obtained results agree qualitatively with those predicted by Sommerfeld and Brillouin.40’45*128129 The experimental evidence indicates that 1. 2. 3.

Sommerfeld precursors travel at light speed. Brillouin precursors penetrate more deeply into a medium and decay at a rate proportional to l/z1/2 as compared with exponentially 1/e. Precursors are wave packet envelope effects and are elicited by the leading, and trailing, edges of the envelope; therefore, they are true transient, not steady-state, effects.

Precursors are linear effects, i.e., Fourier transformation applies; however, the descriptive function of Fourier transformation should always be emphasized. Furthermore, signal envelope-elicited linear re­ sponses cannot be recaptured by the summation of individually applied frequency components of the

\

-i P

iP

coSB, only the first precursor field evolves prior to the signal arrival, with the second precursor field occurring during the signal evolution. (Permission of Sherman, G. C. and Oughstun, K. E., Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves, Phys. Rev. Lett., 47, 1451-1454, 1981.)

signal regardless of the phasing of those components. One might decompose a short pulse exciting precursors into Fourier components. The signal and its Fourier components — with the exact phase relations required — would give the same complex Fourier spectrum. All of this is true, but one may not then proceed to take individual frequencies and expect to obtain the same linear dielectric response which was obtained with the pulse envelope without those exact phase relations. This is because the envelope, i.e., risetime or initial conditions, is a function of the summation of all of the Fourier components only at the correct phase relations. Even to compete a pulse against the same pulse’s swept frequency components would require those components to be started at the correct phasing, and arrangements would have to be made so that the risetime of each frequency, i.e., the initial conditions, were of no consequence. That is a very tall order empirically. Furthermore, in a fieldable system one must know those precise phase arrangements a priori, and that alters the causality conditions. Stated another way, even though it is possible to predict the behavior of pulsed signals by a complex Fourier combination of the amplitude and phased responses of the absorbing material at discrete CW frequency components, it is empirically difficult or impossible to add the individual responses at precisely the required phase relations, to commence sampling before the signal has arrived (causality), or to field a manageable system which could do so. Furthermore, even if one knows a priori the complex frequency response of the medium in order to empirically apply individual frequencies at the correct phasing, the individual CW signal generators must be turned on. In turning the individual frequency generators on, the risetime of the envelope of the waves contains frequency components which are other than that at steady-state CW. The Cauchy initial conditions thus preclude an empirically exact comparison of pulse and empirically obtained summed CW results (see Chapter 12).

200 Gauss

(c)

Figure 7.41 Pulsed sine wave response (5 ns/div) with carrier frequency of 0.625 GHz of ferrimagnetically filled coaxial line. The characteristic frequencies, cqq and (a2 + a>o2)1/2 are both dependent on an externally applied magnetic field permitting isolation of the signal components (see Figure 7.40). Sommerfeld and Brillouin precursors are shown in (A); the Brillouin precursor in (B); and the Brillouin precursor and signal in (C). (Permission of Pleshko, P. and Palocz, I., Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain, Phys. Rev. Lett., 22,1201-1204,1969.)

Even with knowledge of the precise phase relations and with these initial conditions inconsequential (which they are not), a sequential launch of such frequencies is only a substitute for the envelope, which is described by the simultaneous additions and subtractions of the Fourier components, if the medium is linear over a broad bandwidth. If the initial conditions, i.e., risetimes are considered of importance —

403

which they are — then there is more trouble. In selecting individual frequencies from the Fourier decomposition of the pulse, one would be involved in an infinite regression of (1) selecting Fourier components from the original expansion, which extends -«> < t < +°°, (2) applying initial conditions which destroy the same narrow band signal components, therefore necessitating reapplication of Fourier decomposition to the broadband components appearing during the risetimes of the the narrowband components, (3) selecting again Fourier components from this second expansion of the rising compo­ nents, (4) applying initial conditions to components of this second expansion, etc. for third and fourth iterations. The point is that complex Fourier decomposition of a pulse is a steady-state description of a transient state. When passing from this description, which is exact, to practical demonstration, transient effects are introduced at times greater than t = -oo in turning on the component generators, thus destroying the exact description.

PROPAGATION VELOCITIES The initial propagation velocity of the first (Sommerfeld) precursor is largely independent of the properties of the medium. The propagation velocities of the later part of the first (Sommerfeld) precursor, the second (Brillouin) precursor, the signal, the group, the phase, and the energy transport are dependent on (1) the crafting of the incident radiation, i.e., the risetime of the radiation, the initial conditions, the carrier or average frequency of the radiation, the fall time of the radiation, the final conditions, and the duration of the radiation and (2) the properties of the medium, i.e., the plasmon frequency, the mobility of the carriers, the absorption bandwidth, and the ratio of inhomogeneous to homogeneous broadening. Table 7.2 is a tabulation of the formulae for the velocities of radiated signal characteristics. However, these formulae are only valid at frequencies other than absorbing frequencies. When the wavenumber, k, becomes complex, i.e., when the medium absorbs, the formulae are invalid. Thus, there are no absolute velocities but rather a dynamic relativism between (1) the crafting of the incident radiation and (2) the properties of the medium. This table is, therefore, only half the story, the other half being the material properties. For example, if a dispersive medium is not too thick, a Gaussian pulse will emerge at an instant given by the speed of the classical group velocity, even if that instant is earlier than the instant at which the peak of the input pulse entered the medium.118This velocity can be less than or greater than c and even negative.122 (A negative pulse velocity occurs when the peak of the pulse emerges from the medium at an instant before the peak of the pulse enters the sample. Causality is not violated; the effect is due to the leading edge of the pulse being attenuated less than the trailing edge.122) For sufficiently thick media, the concept of group velocity no longer applies and only numerical techniques can be used for estimating the velocity.

SUMMARY OF THEORETICAL AND EXPERIMENTAL EVIDENCE FOR PRECURSORS The theoretical and experimental evidence indicates that 1.

Sommerfeld precursor effects, at optical, IR, and RF, frequencies are only dependent on the gross properties of the medium, because the electrons first react to a transient process and they are, at the beginning, quasi-free and described by the plasma refractive index. Brillouin precursor effects, at optical, IR, and RF frequencies depend on the characteristic frequencies and relaxation times of the medium. Under certain circumstances (e.g., steepness of the wave front,48 short total duration of the pulse), precursor effects can be larger than the main transmitted signals and decay more slowly. Experimental evidence of precursors agrees remarkably well with theoretical prediction and simulation at all frequencies. Depending on the oblique incidence of radiation on an interface, the precursors can be reflected/refracted, while the signal is refracted/reflected 48 2. The transmission speeds of signals, which are due to the forced oscillations of the medium, vary depending on the frequency of the incident radiation, 0^, the duration of the radiation, the onset and offset of the radiation, the resonance frequency of the medium, 0)(), the plasmon frequency, cop, the mobility of the carriers, the ratio of inhomogeneous to homogeneous broadening, and the dielectric dispersion and absorption. The energy transport velocity and group velocity slow down near resonance in spatially dispersive (exciton-polariton) media.140 3. The classical area theorem for a dispersive and absorptive medium permits E to decay while E2 is preserved. 4. All of the above indicate that the transmission characteristics of a medium are relative to the crafting of the incident radiation.

404 Table 7.2

Formulae for Wavelet Propagation Velocities

Dispersive Medium

Lossy Medium First (Sommerfeld) precursor velocity

c->[co02 + G)p2]"2

C -> [(D 02

Second (Brillouin) precursor velocity

c/[l + cop2/(0o2],/2 cA £

C /[l

Signal velocity

Vs

Group velocity (V g)

v s = Vg A cdabs,

=

Vg

= c{d(D/dk} with k complex or imaginary:

Vg

Vg > C, Vg

= ±oo,

Vg

=

c ( i- ( o y ® o ) 2)1/2 vg = c{dco/dk) = c/[n(co) + co(dn/dco)]

0

t> 0

t>0

Figure 7.42 Surface wave representation of soliton in terms of surface nonlinearity and dispersion. The horizontal axis is distance. In the top drawing, the various frequency components of the wave are not keeping together but are being dispersed. In the bottom drawing, the wave is breaking, i.e., showing an intensity-dependent nonlinearity. In the middle drawing, the dispersion and nonlinearity are precisely matched so that energy is conserved. A soliton can be obtained by matching either (1) a pulse’s duration (and thereby its dispersion) to an intensity-dependent nonlinearity in the medium, or (2) a pulse’s envelope (amplitude “nonlinearity”) to the dispersion in the medium. We are addressing (2) at RF energies. WHAT IS A SOLITON? A definition of a soliton is that it is a solitary wave which preserves its shape and speed in a collision with another solitary wave.141151 What is a solitary wave? A solitary wave is one which travels without change in shape or profile. They are both mathematical objects which can describe physical phenomena. How is a soliton obtained? It is obtained by matching either the radiation pulse area to the dispersion or to the intensity-dependent nonlinearity in the medium (Figure 7.42). Here we will be concerned only with matching to the dispersion.

WHAT IS NEW ABOUT SOLITONS? The following quotation answers this question: “The last century of physics, which was initiated by Maxwell’s completion of the theory of electromagnetism, can, with some justification, be called the era of linear physics. With few exceptions, the methods of theoretical physics have been dominated by linear equations (Max­ well, Schrodinger), linear mathematical objects (vector spaces, in particular Hilbert spaces), and linear methods (Fourier transforms, perturbation theory, linear response theory). “Naturally the importance of nonlinearity, beginning with the Navier-Stokes equations and continuing to gravitation theory and the interactions of particles in solids, nuclei, and quantized fields, was recognized. However, it was hardly possible to treat the effects of nonlinearity, except as a perturbation to the basis solutions of the linearized theory. “During the last decade, it has become more widely recognized in many areas of “field physics” that nonlinearity can result in qualitatively new phenomena which cannot be constructed via perturbation theory starting from linearized equations. By “field physics” we mean all those areas of theoretical physics for which the description of physical phenomena leads one to consider field equations, or partial differential equations of the form:

406 tt = F( at.141 In other words, the sine-Gordon equation is invariant under certain precisely defined matched compensatory transformations. In labora­ tory coordinates it is also the unique equation which is Lorentz invariant. Certain solitons are solutions to the sine-Gordon equation. Equations with soliton solutions are examples of infinite dimensional completely integrable Hamiltonian systems with an infinity of polynomial conserved densities. If uniqueness is not special enough, the work of Kruskal and associates, which produced the inverse scattering transform (defined below), has been deemed a major achievement of twentieth century mathematics,144 for reasons which will become apparent below.

HOW WERE SOLITONS DISCOVERED? The soliton was discovered quite recently, but the solitary wave, or the “great wave of translation”, was discovered in 1834. Only after that was studied for 130 years was the ground set for the discovery of the soliton. The solitary wave was first observed on the Edinburgh to Glasgow canal in 1834 by John Scott Russell.152 Russell (1808-1882) reported his discovery to the British Association in 1844 as follows: “I believe I shall best introduce this phenomenon by describing the circumstances of my own first acquaintance with it. I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped — not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation, a name which it now generally bears: which I have since found to be an important element in almost every case of fluid resistance, and ascertained to be of the type of that great moving elevation of the sea, which, with the regularity of the planet, ascends our rivers and rolls along our shores. “To study minutely this phenomenon with a view to determining accurately its nature and laws, I have adopted other more convenient modes of producing it than that which I just described, and have employed various methods of observation. A description of these will probably assist me in conveying just conception of the nature of the wave.”153 Russell undertook an extensive series of experimental investigations of water waves and divided them into two classes: the “great primary wave of translation” — later to be called a solitary wave — and all other waves that belong to the second or oscillatory order of waves. He established that the speed of solitary waves, which are long (shallow water) waves of permanent form in a channel of uniform depth is (7.114) where r| is the crest of the wave above the stationary water level, h is the depth of the channel, and g is gravity.153

407 No mathematical theory, however, was available to describe the solitary wave, and a controversy existed for some 50 years concerning its description or even existence. Nonlinear equations for long waves were derived by Boussinesq154*155 and Rayleigh268 with solitary wave solutions, but, it was not until 1895 that Korteweg and de Vries156 found an equation governing the two-dimensional motion of weakly nonlinear long waves in shallow water. This equation, now known as the Korteweg-de Vries equation, or KdV equation, can be written in the form of the transformations of a Lie group. In its original form the equation is dri/dt = (3 /2 )^ g /tj d/8x {(l/2 Ti2) + (2/3)ocr| + (l/3)a d 2r|/ax 2},

(7.115)

where x is the variable along the one-dimensional channel, t is time, rj(x,t) is the elevation of the water surface above the equilibrium level €, g is the gravitational constant, a is a constant related to the uniform motion of the liquid, and o a constant defined by (7.116) where T is the surface capillary tension and p the density. No new applications of the equation were found until 1960 when the equation was rediscovered as a model for the analysis of collision-free hydromagnetic waves.157 However, the greatest attention was paid to the equation when Zabusky and Kruskal158 demonstrated the equation’s relevance to the Fermi, Pasta, and Ulam problem.159 The Fermi-Pasta-Ulam problem involves the consideration of a one-dimensional lattice. The hypothesis was that, due to the nonlinear coupling, any smooth initial state would eventually relax to an equipartition of energy among the various degrees of freedom of the system. However, this is not what the results showed. Rather, “Instead of a gradual, continuous flow of energy from the first mode to the higher modes.. .the energy is exchanged, essentially among only a certain few...There seems to be little, if any, tendency toward equipartition of energy among all degrees of freedom at a given time. In other words, the systems certainly do not show mixing.”159 In their reconsideration of the Fermi-Pasta-Ulam problem, Zabusky and Kruskal158 considered a continuum model which reduced to a form directly related to the Korteweg-de Vries equation. They realized that the KdV equation had a special permanent wave solution u = 2k2sech2k(x - 4k2t - xo)

(7.117)

where k and xo are constant and the velocity of the wave 4k2 is proportional to the amplitude 2k2. Moreover, for the first time, it was realized that two of these waves interact elastically: after collision two waves, for example, regain their shape and velocity. The only effect of the interaction is a phase shift. Zabusky and Kruskal referred to these waves which regain their initial shape and velocity (with phase shift) after collision as solitons. The wave of the Russell type, which possesses a uniform shape but which does not interact elastically, is a solitary wave. It was later found that the existence of the soliton is dependent on the existence of an infinite number of conservation laws relating the density and the flux.160161 From 1965 onward, interest in the solitary wave and solitons picked up. It was noticed that the KdV equation Ut + 6U U X + Uxxx

= 0 = K(u)

(7.118)

and the modified Korteweg deVries equation (mKdV) Vt

are related by

-

6 v 2Vx + Vxxx

= 0 = M(v)

(7.119)

408 u=

(7.120)

as solutions,16() so that K(u) = -(2v + djd\) M(v).

(7.121)

Then it was noticed that the this u - v relation is a Ricatti equation for v in terms of u which may be linearized by the transformation: v = \|/x/\|/ yielding162’163 Vxx + u\|f = 0.

(7.122)

As the KdV equation is Galilean invariant, the following equation was considered164 \|/xx + (X + u)\j/ = 0,

(7.123)

which linearizes the KdV equation and is also the time-independent Schrodinger equation of quantum mechanics. By postulating a time evolution equation similar to this linear equation, it was possible to calculate eigenvalues specific to the problem considered.162,163 This procedure is the inverse scattering transform (1ST) method.323*326 Using 1ST, Zakharov and Shabat165 solved the nonlinear Schrodinger equation: iqt = q Xx + Kq2q*, K > 0

(7.124)

and the sine-Gordon equation was solved:166 uxt = sin u.

(7.125)

Finally, it was shown that the KdV, mKdV, nonlinear Schrodinger, and sine-Gordon equations are all related to one master eigenvalue problem and that given a suitable scattering problem, the nonlinear evolution equations can be derived which can be solved by 1ST.167,168 Furthermore, the scattering problem and associated time dependence which constitute an inverse scattering transform also constitute a Backlund transformation.

WHAT IS THE BACKLUND TRANSFORM? The Backlund transform can be viewed as as a technique for constructing solutions to a partial differential equation in the Pfaffian form:169 d0 = Pdx + Qdt

(7.126)

and the requirements: 2) /2] —0,T

(7.129)

and integration gives , = 4 tan [exp(ai; + T/a)],

(7.130)

which was a function known as the gudermannian to nineteenth century mathematicians.169 Under transformation Equation 7.128 can be written xx - u = sin 0. The inverse scattering method is the following procedure: associate to u(x,t) a wavefunction \|f(x,t;A.), a linear scattering problem L\j/ = X\j/, and a time evolution of \jr: A\|/ = \|/t. Then the differential operator L contains u(x,t) linearly as a scattering potential and A is a nonlinear functional of u. As u evolves in time the eigenvalues X of the scattering problem depend on time. Under the condition h = 0, Lt\|/ = [A,L]\|f. As L depends linearly on u and on time only through u, the operator relation Lt= [A,L] can be identified as the evolution equation ut = K[u] if A and L are suitably chosen. Thus the 1ST treats the soliton entity, u(x,t), as a scattering potential dependent on initial conditions. This transform sets the stage for describing the optimum pulsed radiation penetrating a medium, being backscattered from a target, and then re-penetrating the medium to return to the source by matching pulse envelope to media dispersion.

FROM U(1) TO SU(2): SYMMETRIES IN THE ELECTROMAGNETIC FIELD We turn now from the mathematical foundations of solitons to consider the physics of the electromagnetic field. When conditioned to conform to a certain symmetrical, e.g., solitonic, form (SU(2)), the algebra of which is that of the group of all unitary 2 x 2 matrices336 — the EM field — conventionally considered

410 to be of U (l) form, the algebra of which is that of the group of all 3 x 3 matrices, must be in SU(2) form. Because solitons which are known to exist are of SU(2) form and because Maxwell’s formulated theory is of U (l) form, there is a compatibility problem. (The term, “Maxwell’s formulated theory” is used intentionally, as careful study reveals great differences between Maxwell’s original formulation and its later interpretation, cf References 338 to 341). In this section, the problem is defined, but in Appendix 2C, the problem is analyzed more completely. Although electrical engineering has seen no reason to extend conventional Maxwell’s theory, physics has extended electromagnetism to higher order symmetry forms. (Indeed, much heat is caused in electrical engineering circles if such an attempt is made. There is a difference between electrical engineering and physics in this regard!) We can describe the reconciliation between the group form of conventional Maxwell’s theory and the group form of pulses conditioned to the medium (solitons) as follows. The electromagnetic potential is an integral part of the canonical momentum of the field when described by the Hamiltonian formalism; for example, by formulating electromagnetism as a classical Hamilton-Jacobi field theory, the momentum is replaced by the canonical momentum P n -* p ^ -e A n .

(7.134)

The form of the canonical momentum is due to Hamilton’s principle or the principle of least action obtaining both Maxwell’s equations and the equations of motion for charged particles from a single physical principle. The interaction of a charged particle with an electromagnetic potential is described by the Lagrangian density L = (l/2)[pM- eA(l]2 - ( l / ^ F ^ F r

(7.135)

The first term contains the canonical momentum or kinetic energy, and the second term is the energy density of the electromagnetic field with the Maxwell field tensor Fnv = 3^iAv —5vAn.

(7.136)

Finding the minimum of the action and setting the action equal to zero results in the Euler-Lagrange equations: a ^ a L / a ^ q , ,) ] - aL/aq(l = o.

(7.137)

If the coordinate, q^, is identified with the electromagnetic potential, A^,Maxwell’sequations are obtained. Thus, within the Hamiltonian formalism, the electromagnetic potential isan integral part of the canonical momentum and acts like a generalized coordinate (or better stated, operates on the generalized coordinate by generating phase changes) in the Euler-Lagrange equations. With the realization early in this century that the phase of a wavefunction could be a local variable, a gauge transformation was reinterpreted from H. Weyl’s initial meaning to a mean change in the phase of a wave function e_i\

(7.138)

The gauge transformation for AMbecame A ^ A ^ -3 ^ .

(7.139)

In quantummechanics, gauge invariance means that any phase change and change inpotential cancel each other exactly, resulting in constant equations of motion under changes in thelocal space-time variables. Thus, the vector potential becomes a physical operator, or generator, affecting the affine connection between two space time points. (An affine connection defines parallel transport or movement over a surface and provides a measure of curvature over a global space. An affine connection can be

411 contrasted with a metric connection which defines changes in length and angle at local points on a surface.) In summary, the set of all gauge transformations in electromagnetism as conventionally formulated in electrical engineering forms a one-dimensional unitary group, the U(l) group. The classical Maxwell’s theory of electromagnetism is considered to be of this kind. However, “pulses-matched-to-media” are solitons and of SU(2) symmetry. By reformulating Maxwell’s theory within the Hamiltonian formalism and defining the electromagnetic potential as part of the canonical momentum, we obtain an SU(2) formulation of Maxwell’s theory which is compatible with matched pulses. The vector potential then becomes a physical operator,178 184 not a mathematical convenience as presently stated in electrical engineering.

ELECTROMAGNETIC MISSILES This field of active research is motivated by the desire to obtain both focus waves and thereby propagation which decrease in energy more slowly than r 2. Essentially, the proponents construct theoretically what they would like to have as energy forms in the farfield, and then search for a source which will provide those desired farfield results. Therefore, in a sense, the goal of this research field is beamforming to obtain maximum antenna gain. This goal will impact the design of antennas and antenna arrays for optical, IR, and RF radar and LADAR. However, the field of focused wave transmission for maximum energy propagation, although important, is in the early stages of development. Therefore, we will report a blowby-blow account of its development, which is still far from complete. The major themes studied are as follows. In 1971, DesChamps showed that a Gaussian beam is a bundle of complex rays, or, that a Gaussian beam is equivalent paraxially to a spherical wave with a center at a complex location.185 This paper was significant for later developments. When the location of an oscillating point source is assigned a complex value, the isotropic spherical wave field is converted into a directive field which remains a solution of a wave equation.186Furthermore, in the paraxial region about the field maximum, the generated field behaves like a Gaussian beam. Gaussian field expansions have been demonstrated to yield the Fresnel fields of a number of antennas as well as the farfields of antennas characterized by very low sidelobes.187 Weber-Hermite beams have also been shown to play a role in the characterization of optical resonators. In 1983, Brittingham published his landmark paper describing “photonlike” classical focus wave modes (FWM).188 Focus wave modes are source-free wavefields which act like well-collimated beams. There are a variety of claims concerning how such beams can be produced; for example, by 1. A field radiated by an oscillating source with modes on a straight path in a complex coordinate space189 2. An oscillating source and antisource moving with the speed of light on a straight line trajectory in complex coordinate space190 3.

Superposition o f causal and anticausal time-dependent conventional Green’s function solutions191

4.

A transient plane wave synthesis over a real angular spectrum comprising forward and backward propagating plane waves with angle-dependent frequency of oscillations192-193 Spinor waves based on solutions to a homogeneous spinor wave equation194198

5.

Brittingham’s waves had infinite energy, which is not a drawback as plane waves also have this property. It does indicate that in their ideal form (as in the ideal form of standing wave sinusoids) the problem of causality will arise and that their physical realization will necessitate approximate solutions. This is also not a drawback because a strict interpretor of Fourier analysis would observe that sinusoids are never radiated (they have no temporal beginning nor end), but approximations to “continuous” waves are radiated. Brittingham attempted to obtain finite energy by truncating integrals, but Maxwell’s equations are not satisfied across discontinuities.199 Brittingham’s waves are considered linear wave packets, not solitons, because they are transverse to the axis of propagation and solutions to linear, not nonlinear wave equations. However, because of the amendments required to obtain finite energy in the original formulation (see below), the resemblance to solitons becomes increasingly manifest. Although the objective was to derive macroscopic EM “pho­ tons”, it has been known for some time269 and recently re-emphasized200 that solutions of Maxwell’s equations do not necessarily have a probability interpretation. There is, in fact, a “photon localization problem” or no covariant position operator for photons.269 The question of whether useful beams can be produced by physical sources remains under lively discussion. It is significant that the FWM exhibits photon-like interference behavior.201-202

412 The Brittingham pulse turns out to be a Gaussian beam which travels through space with only local deformation.189 Ziolkowski189 has obtained electromagnetic directed energy pulse train (EDEPT) solu­ tions to Maxwell’s equations thereby linking Brittingham’s work with the earlier work of DesChamps185 and Felsen,186’203’204 as mentioned above. These solutions, in contrast to plane wave decomposition, utilize basis functions which are more localized in space and therefore are better suited to describe the directed transfer of EM energy in space. The modified power spectrum (MPS) pulse is a Laplace transformderived EDEPT solution obtained by scaling and truncating a power spectrum.205As this pulse is localized well beyond the classical farfield distance, the diffraction limit is said to have been defeated.206207 The criticism was made that for FWM-type sources distributed over two parallel transverse plates that the FWM field between these plates is described by essentially backward radiation from the forward surface.208*209 Essentially, the criticism is that the FWM is generated by superposition of backwardpropagating, time-harmonic Gaussian beams and, when tightly collimated, are unphysical and noncausal. That is, unphysical in that the waves can be described by global mathematics but not generated as a wavelet or wave packet; and noncausal in that the description commences at minus infinity in time and progresses to plus infinity in time, with no commencement at time, t = 0, except mathematically. However, this criticism that the backward-propagating plane wave portion of the original Brittingham focus wave modes leads to noncausality,210 is obviated by a particular choice of spectral parameters.205 Intuitively, the result is very similar to that obtained with a large-current radiator.270 Seeking to develop a large-current antenna with efficient low driving voltage, Harmuth turned to the Hertzian magnetic dipole antenna.270 However, the radiation from this antenna is largely canceled by returning loop currents. The cancellation effect is then overcome by size differences between the size of the return current and radiating current elements, as well as by backing the antenna with lossy material. In this instance, again, the backward propagating wave is canceled, leaving the forward propagating wave. Hermite-Gaussian and Laguerre-Gaussian beams with complex arguments have been introduced.346 These functions arise naturally in correction terms of a perturbation expansion whose leading term is the fundamental paraxial Gaussian beam and as paraxial limits of multipole complex-source point solutions of the reduced-wave equation.211347 349 Ziolkowski has investigated the localized transmission of wave energy by an acoustic realization of the EDEPT or acoustic directed energy pulse train (ADEPT). In contrast to conventional arrays, the ADEPT and EDEPT driven arrays are driven by a spatial distribution of specified broad bandwidth time histories. The different sources of an array are driven by different pulses. The MPS pulse can be tailored to a particular application with a straightforward change in parameters and shows better behavior in propagation than the Hermite-Gaussian pulse.2(16Their physical realization will be due to a finite array with an aperture distribu­ tion which is modulated spatially and temporally with individual elements of large bandwidth. Such an array would not be of large size as upper elements can be folded into lower elements.206The experimental evidence of superior focusing and energy retention for ADEPTs was shown by Ziolkowski et al.207 Hillion196 obtained spinor focus wave modes using a procedure similar to Brittingham’s but based the analysis on a homogeneous wave equation. Bessel weight functions were used to gain finite energy solutions. Both Hillion’s and Ziolkowski’s FWM solutions are known as “splash modes”. The approach of Wu is quite different from that of Brittingham. Whereas the focus wave mode is a photon-like packet that is intended to conserve energy indefinitely, the Wu approach is the claim that, while admitting (or proving) that for any finite-energy solution of Maxwell’s equations the energy in an infinite slab of finite thickness approaches zero as time tends to infinity, that approach to infinity can, nevertheless, be as slow as one wishes.212,350 The missile approach commences with the time integral of the Poynting vector and solves for the Fresnel term and then looks for source distributions with a l/r2e dependence, where e can be made small. Wu’s goal is to construct a slowly decreasing wave by choosing an aperture and a pulse shape and risetime so that suitable current distributions and complete integrability is obtained. In that a match is sought between aperture form and pulse risetime and shape, the solution amounts to a soliton, at least at the source, if not after the pulse leaves the source. By increasing the risetime of an excitation pulse in an aperture, an energy decay of 1/r26 can be achieved, where e is related to the risetime of the pulse. The missile achieves slower than average decay over large distances by extending the Fresnel far-boundary zone away from the aperture.330 335 Wu212 discovered that some source models (e.g., uniform current distribution over a circular disk) predicted the generation of focused waves. Using a Mellin transform in the study of the time domain

413 radiation from a circular disk to obtain exact integrals over the source distribution, Lee213 confirmed the finding of Lee and Lee214 that current switched on at t = 0 must vary as (t/T)25 with 8 positive for 1/r behavior. The term electromagnetic missile for this effect was coined by Wu212 with the definition of a transient wave, the energy of which decreases much more slowly than r 2. The patterning of energy on the emitter, e.g., a disk, and the risetime of the current on the emitter are crucial in obtaining EM missiles. The electromagnetic field from the current in a circular disk showed that the transverse distribution of energy around the disk is stable. This transverse energy pattern has a cusp in the axis215216 for linear propagation. The pulse does not have to propagate linearly forward; Myers et al.217271 272 have shown that electromagnetic missiles could follow a path which is strongly curved. The paraxial region is important in obtaining FWM or missile propagation.218 222 (The paraxial approximation is a simplification whereby the optical axis of a system is made equal to the z-axis, i.e., radiation is assumed to make an arbitrarily small angle with respect to the axis (cf. Reference 223). Sezginer224 showed this close relation of focus wave modes to solutions of the paraxial wave equation using a decomposition into TE and TM components. In the strong focusing regime, the forward moving FWM is generated primarily by backward propagating contributions from the front plane. The largecurrent antenna270 also achieves the paraxial approximation: the backward modes are absorbed. Wu and Lehman350 showed that any finite solution will involve spreading of energy, thus an approximation is needed for any real solution which achieves focusing. However, what is not appreciated is that the paraxial approximation is incompatible with the exact Maxwell’s equations.225 This is because the paraxial approximation (the requirement of Ex, 0, 0, so that div E = 3Ex/3x * 0; Ex = vj/exp[ikz]; l0\|/2/9z2l I3\|//3xl) neglects the gradient of the diversion terms of the electric field (grad div E terms) and seeks a plane wave solution of Maxwell’s equations. By doing so, a paradox arises, because 3\|//3x = 0, but the paraxial approximation of Gaussian beams assumes a lowest mode Gaussian in x and y. This paradox is obviated by introducing nonlinear dielectrics and longitudinal components in the field which are out-of­ phase with the transverse components. The result of this application of the paraxial approximation is that Maxwell’s equations are expanded in terms of w/1, where w is the beam waist and 1 is the diffraction length. Zeroth-, first-, second-, and third-order fields can then be obtained with the zeroth-order field transverse, but the first-order field is longitudinal.225 A new directional decomposition of exact solutions to the scalar wave equation into bidirectional, forward, and backward traveling wave solutions has been provided by Besieris et al.226 The new representation forms a natural basis for synthesizing pulse solutions which can be tailored to give directed energy transfer in space. Shaarawi et al.201 have shown theoretically that it is possible to launch Brittingham-like pulses from a highly dispersive waveguide. These pulses are initiated in a Cauchy-initial value fashion, with the initial value of the pulse and its rate of change at t = 0 specified. Apart from electromagnetic missile or focused wave mode concepts, there are other approaches to more efficient energy transmission. For example, it has been known for some time115*227 that Bessel beams are eigenmodes of the wave operator. Dumin demonstrated that the Helmholtz equation has a class of diffraction-free mode solutions228 based on the lowest J0 Bessel mode. Dumin et al.229 further showed experimentally that these modes describe well-defined beams with narrow beam radii and that they possess a depth of field in the nearfield of the aperture. These beams have infinite energy but finite energy density. Other approaches include using a refined radon transform as a method of finding exact, causal, threedimensional solutions from essentially one-dimensional solutions in the wave zone,230'234 new approaches to aperture theory,327 329 and, more empirically, locating an ultrafast Hertzian dipole source at the focal point of a collimating lens to demonstrate single cycle of 0.6 ps or 0.5 THz pulses which preserved shape over 1 m in air.235

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APPENDIX 7A: FURTHER DEVELOPMENTS IN SELF-INDUCED TRANSPARENCY The field of self-induced transparency (SIT) at optical, IR, and RF frequencies naturally encompasses both the dynamics of radiation-medium interaction and also the dielectrical properties of the medium across an enormously broad range. An understanding of SIT across that range will result in efficient pulsed radar and LADAR. Although under development since the 1960s, the field of SIT, across that range, still contains areas of virgin territory. The following studies are paths through the field but still leave areas to be developed. The pioneering work of McCall and Hahn was generalized by abandoning the assumption that the phase of a pulse has no temporal dependence.236 If the host medium possesses significant nonresonant nonlinearities, then undistorted lossless single pulses are possible in such a medium. Also, the lossless single pulse must have frequency modulated, or chirped, form. A crucial relation can be derived between the envelope of the field and the modulation of its phase. The implication is that it is only necessary to control the pulse envelope to control other parameters such as phase. Some conclusions are 1. 2.

Undistorted or steady-state pulses can propagate in resonant absorbers even if the nonresonant host medium is nonlinearly dispersive. If the most important nonlinearity in the medium is of the Kerr-effect type, then all such pulses propagating through the medium must have a chirped form.

Slowly varying 2n pulses are not frequency modulated;237 however, the collision of two 2n pulses that have two different carrier frequencies leads to a 4rc pulse which can be frequency modulated. Other more recent findings are

426 1. The 2k pulse contains no frequency sweep for the Bloch equations with phase terms included. 2. The phase terms contain only spatial dependence for an asymmetric spectrum of inhomogeneous broadening. 3. The Bloch equations (cf. Section III) with phase terms included are equivalent to the equations constructed by Zakharov and Shabat165in their application of the 1ST method of Lax175to the equations governing self-focusing and self-modulation. The vector model of the two-level Bloch equation of Feynman et al.65 was generalized to an N-level system.238 The solutions are periodic and permit complete population inversion. That is, if all the population resides in level 1 at time t = 0, then at the time t = n/co (where 0) is the Rabi frequency) the population is entirely in level N. The combination of periodicity and complete inversion leads to an Nlevel area theorem. The Rabi frequency serves as a measure of interaction strength between the levels and also of population oscillation frequency. We obtain a generalization of the Feynman et al. two-level atom by postulating a second three-dimensional space within which a vector represents the population ampli­ tudes of the N-level atom. Within this second space, quantitized angular momentum vectors can be introduced. A physical interpretation of self-induced transparency can be made with an analogy with parametric amplification and three-wave mixing.5 Manley-Rowe relations can be derived. The resonance rotating frame for all transitions considered is at the Larmor frequency. The damping terms are of two types: spinlattice relaxation with time constant Ti and spin-spin relaxation with time constant Ts. Measurements were made on MgO crystals with (1) Ni2+ impurities (T2 = 0.76 to 1.20 x 1(H s) and (2) Fe2+ impurities (T2 = 5.8 to 11 x KHs). Self-induced transparency was demonstrated as well as pulse reshaping to a hyperbolic secant or even a Gaussian shape. Close to 100% transparency was observed at high input intensities. The initial studies conducted on SIT addressed one-photon processes. Later studies have considered two-photon absorption SIT239’240 Self-induced transparency with frequency modulation or chirping236241 has also been studied. Numerical investigations of pulse propagation in a medium with two-photon absorption show that under certain situations of propagation in an inhomogeneously broadened medium, coherent pulses can propagate accompanied by phase modulation due to nonresonant excitation.239 Theoretical studies have been made of the propagation of two short different-wavelength optical pulses in three-level absorbers.242 Solving the three-level Bloch-Maxwell equations produces new ana­ lytic solutions having the form of simultaneous different-wavelength optical solitons. Simulton is the name given these pairs of solitons. There have been attempts to find exact rotating wave approximation solutions for the N-level atom’s evolution under the influence of several fields with time-dependent envelopes.238243 Self-induced transparency in degenerate magnetic dipole transitions in an iodine absorber have been demonstrated.244 The relaxation time of the absorber was greater than 45 ns and the duration of the laser pulse was 13 ns. The case of a step-function input pulse with infinite duration in time has been examined. With the stepfunction input, there is a constant generation of pulses and the asymptotic solution cannot decompose into well-separated solitons, as the area theorem does not apply. An asymptotic area theorem fixes the time average of the electric field as equal to that of the input field. The mean separation of the pulses generated cannot change and the pulses become narrower with larger amplitude. This situation contrasts with the norm in that for a finite pulse the area theorem applies and the number of pulses generated by interaction with the medium is fixed. The degenerate Maxwell-Bloch model describing the interaction between two-level atoms and the electromagnetic field for degenerate transitions has been examined theoretically.245 This permits inves­ tigation of the polarization properties of SIT. The integrability of the Maxwell-Bloch model has been demonstrated237 and hence the possibility of the application of the 1ST method.246 Therefore, the pulse which induces SIT in an appropriate medium, due to that interaction with that medium, becomes a soliton. Furthermore, the presence of SIT in a medium indicates the presence of a soliton.. In any real atomic system, the dipole allowed transition must always be degenerate.245 The transition from a ground state whose angular momentum is zero, to an excited state whose angular momentum is one, is threefold degenerate with respect to the angular momentum projection. Theoretical treatment of S J J 5 .55 ,62.247 h as a i w a y S treated the nondegenerate case (the two-level approximation). This means that the radiation polarization was always treated as fixed, i.e., that the radiation does not depend upon space/time

427

variables. More recently, a generalization of the Maxwell-Bloch model has been made to include degenerate transitions.245 Important observations made by Gibbs et al.63-74-248 showed experimentally at optical wavelengths that choice of radiation polarization allows near-ideal SIT in all degenerate or nondegenerate systems, provided there is no destructive overlapping of transitions. Therefore, circularly polarized radiation will propagate as solitons under SIT conditions. The SIT of Frenkel excitons in a unimolecular crystal has been examined theoretically for an arbitrary direction of pulse propagation.108 The angle made by the wave vector with the optic axis determined the characteristic parameters of the resulting soliton pulse. Furthermore, the pulse velocity depends not only on the pulse width — as occurs in isotropic media — but also on the direction of the wave vector relative to the optic axis. Recently, constant wavelength transparency, if not SIT which is a pulsed envelope effect, has been shown to occur at optical wavelength in strontium vapor354 using a preconditioning wave. This occurs in a three (or more) energy level system and the absorbing third level can be excited from either the first or the second level. In the strontium vapor study, the third level could be excited (populated) by either green light at 570-nm wavelength, or UV light at 337 nm. Under normal conditions the vapor absorbs the UV light due to the excitation of electrons from the first energy level to the third energy level. However, if the third energy level is fully populated by electrons excited from the second energy level to the third energy level by the preconditioning green light, then the UV light is not absorbed, the third level being “unavailable”. This experiment has also been performed on lead vapor and hydrogen vapor. Whether the technique could be applied to molecular gases, liquids, and solids, as well as atomic gases, and at wavelengths other than optical, e.g., IR and RF, has yet to be investigated. Whether the preconditioning wavelengths and the nonabsorbed, normally absorbed, wavelength can exist in the same wave packet, wavelet, or pulse also has not yet been investigated.

APPENDIX 7B: THE NONLINEAR WAVE EQUATIONS AND SOLITONS There is an analogy between three-wave parametric amplification and the interaction of a single wave with a two-level resonant system.249 For example, three-dimensional ion-sound solitons can exist in a low pressure magnetized plasma.250 The three-dimensional solitons, which decrease in all directions, propa­ gate along the magnetic field and waves propagating in opposite directions weakly interact. There are other physical models. For example, the solution for one-dimensional, three-wave resonant interaction for bounded envelopes permits an interaction in which all solitons contained in the envelope of the wave with the middle group velocity will “split” and be added to the original number of solitons contained in both the fast and slow envelopes;251 there are exactly two soliton interactions on a nonlinear spring mass system.143252 These studies indicate the “internal” nonlinearity of the solitons. Bullough176 tabulated the areas of nonlinear physics in which solitons play a significant role: 1. 2. 3. 4. 5. 6. 7.

Theory of water waves151*305-317 Theory of plasmas and the interaction of radiation with plasmas301 Theory of Josephson junctions288290-310 Resonant and nonresonant nonlinear optics300-301-345-352 Nonlinear crystal physics: theory of dislocations; anharmonic crystals;recurrencephenomena in thermal transport; ferrodistortive phase transitions in crystals in thedisplaciveregime295-296-299-302-303-309 Ferromagnetics: Bloch wall motion296-309 Theory of fundamental particles292-306

One may also add: 8. 9. 10. 11. 12. 13. 14.

Charge transport in polymers286-287 Possibly: high temperature superconductivity The long Josephson junction: numerical computations on a sine-Gordonmodel of theJosephson junction fluxon oscillator showed good agreement with experiment253-273*284 The soliton laser285-297 Classical field theory291-293-294-304-307-308 Relativity298 Acoustic solitons311*315

428 15. Polarization solitons316 16. Emmission from nonlinear waveguide318 The three most important nonlinear wave equations with soliton solutions are: 1. The Korteweg-de Vries (KdV) equation in one space variable x and one time variable t and where subscripts indicate differentiation in the indicated variables Ut + 6uux + Uxxx = 0.

2.

(7.140)

This equation is relevant to water wave theory, plasma theory, and anharmonic lattice theory and describes a weakly nonlinear dispersive wave system. The KdV equation was the first equation for which the inverse scattering method was developed The nonlinear Schrodinger equation i ut + 2ulul2 + Uxx = 0.

(7.141)

This equation is relevant to nonlinear, optical self-focusing and self-phase modulation, plasma theory, and deep water wave theory. The cubically nonlinear Schrodinger equation is an important model of nonlinear phenomena in fluids and plasmas.254255 In the case of strong Langmuir turbulence, the dimensionless form of this equation in a spatially periodic system with periodicity length L is idtE(x,t) + 3x2E + IEI2E = W0E,

(7.142)

where t is the time, x is distance, E(x,t) is the slowly varying envelope of the electric field which is varying rapidly in time with frequencies near the plasma frequency, and the spatially averaged electric field density is Wo = (1/L)

f |E|2dx.

Jo

(7.143)

The most common types of numerical solutions of the nonlinear Schrodinger equation involve a truncation of the electric field to a finite number of spatial Fourier modes. Only three of the infinite number of conserved quantities survive the truncation, and these are termed “rugged invariants.”255 Furthermore, while the three-mode truncation remains integrable, the five-mode, seven-mode, etc. truncations are not integrable and stochasticity and the inability to predict the state of the system at a later time results. This stochasticity is signaled by a positive Lyapunov exponent; however, increasing the number of modes to over 100 reduces the exponent to one tenth of its maximum value. 3. The sine-Gordon equation U x x - u tt=

sinu.

(7.144)

This equation is relevant to any context in which bistability exists, e.g., pendula lines, SIT, spin one half systems, and Josephson junctions. More importantly, it is the equation of motion which describes SIT in media.

APPENDIX 7C: RELATION OF U(1) AND SU(2) SYMMETRY GROUPS An affine connection defines parallel transport or movement over a surface and provides a measure of curvature over a global space. An affine connection can be contrasted with a metric connection which defines changes in length and angle at local points on a surface. In 1954, Yang and Mills proposed an isotopic spin SU(2) groupwith anaffine connection similar to the vector potential of the U (l) group of the theory of electromagnetism,but whose influence produces a rotation in internal symmetry space256 R(0)\|f = ei0L\|f,

(7.145)

429

where 0 is the angle of rotation and L is an angular momentum operator. Thus the proposal called for the an isotopic spin connection and a potential acting in accordance with an SU(2) symmetry group. To ensure that the angle of rotation is proportional to the potential, as required for the Schrodinger equation to remain gauge invariant, the Yang-Mills potential is a linear combination of angular momentum operators An = XiAV(x)Li,

(7.146)

where the coefficients AV(x) depend upon the space-time position. Thus, the Yang-Mills potential generates rotation. The Yang-Mills potential, although possessing zero mass, carries electric charge. Belonging to the SU(2) group, its local symmetry transformations are non-Abelian. The Yang-Mills potential also acts as a connection on the internal symmetry space351 as may be seen by the following analysis. Separate a wave function into external and internal parts, where we mean, in this instance, by “external parts”, the two vectorial components of a polarized wave, and by the “internal parts” the phase modu­ lations between those two components 'F(x) = Z'F (x)u a

(7.147)

In this equation, the ua form a set of “basis vectors” in an internal space. The external part \|/a(x) is then a “component” of a polarization modulated wave \jr(x) in the basis ua. Under an internal symmetry transformation, they transform in the following way: yp = Upaya,

(7.148)

where Upa is some matrix representation of the symmetry group. The potential of this SU(2) field is also constrained by the SU(2) symmetry and can be defined as an operator ( A u)«p = I k O n 0 k) ( F k)ap ,

( 7 .1 4 9 )

where the Fk are the generators of an internal symmetry group and satisfy the commutation relations [Fi,Fj] = iCijkFk

(7.150)

in which the structure components Cijkdepend on a particular symmetry group, in the case of the isotopicspin rotation group SU(2); the generators Fk are the angular momentum operators; and the 0k(x) are parameters which are continuous functions of x. The change in a field acted upon by this operator is d\|f = Xp(d\|/)pup = Zp(D^\|/p)dx^up,

(7.151)

where is a generalized form of the gauge covariant derivative describing the changes in both the external and internal parts of \j/(x) DM \gp= Xa[8pa0n —iq(A^)pa]V|/a

(7.152)

which is the definition for an SU(2) spinor field. In contrast, for a U(l) field it is Dn\|fp = (3h- iqAn)\|/,

(7.153)

which is the canonical momentum. Thus, both the U(l) field, which possesses a one-dimensional internal space, and the SU(2) field, which possesses an isotopic spin internal space, also possess gauge co variant derivatives that can be interpreted as canonical momenta. Therefore the generalized version of the

430 external potential field, (An)ap, is a connection operator on the internal symmetry space. The potential is both an external field and an internal space operator.351 The non-Abelian version of the Maxwell field tensor is the operator F^v = 3^iAy —3vA^i —iq[A^,Av],

(7.154)

(contrast this with Equation 7.136) and the non-Abelian version of the homogeneous Maxwell equation is: D mFvX+ DXF^V+

= 0.

(7.155)

The differences between the Abelian and non-Abelian Maxwell theory equations arise because after polarization modulation the gauge field components no longer commute with each other at different points in space-time. In order to calculate the divergence of the field Fjw in Yang-Mills fields, its value must be known at two nearby positions, x and x + dx, so that the rate of change (e.g., in polarization modulation) can be calculated. However, a gauge field like F^v, or any of its derivatives, is not only a function of space-time position but also has a direction that can change between x and x + dx, depending upon the phase modulation rates at x and x + dx. Because the gauge fields do not commute, the internal space direction of F^v at x + dx will depend on the particular path taken between x and dx. This is not the case with Abelian or U (l) Maxwell theory. Furthermore, the electric and magnetic fields in a general non-Abelian gauge theory do not obey the principle of superposition. This can be seen in the Yang-Mills wave equation D,F p, = t t A v - a v(a pA J ^ { K ’Fp ] + 3J \ ’A v |

(7.156)

where the commutator terms indicate that non-Abelian waves do not obey the principle of superposition. The same observation has been made in the case of propagating pulsed signals.179181 This leads to the conclusion that the U (l) Maxwell theory is valid in the case of steady-state fields but needs extension (to SU(2) form) in the case of pulsed or transient fields179181 184as well as polarization modulated fields.178182183 The gauge theory representation of polarization modulation based on local internal symmetry can be related to the representation of the global representation of spaces, i.e., to topology. The motivation in doing so is to provide insight into EM emitter design. The requirement, in the case of the SU(2) electromagnetic field, would be for a configuration permitting two types of closed paths: one for internal and one for external space. The torus exactly describes such a topology which represents a multiply connected (as opposed to infinitely connected) space. The conventional theory of electromagnetism is the simplest example of a gauge theory in (3 + 1) dimensions and belongs to the gauge group U(l). This involves only linear Maxwell’s equations which do not yield solitary waves or solitons. If, however, the electromagnetic field is generalized to a nonAbelian gauge group, e.g., SU(2), then a triplet of gauge fields known as the Yang-Mills fields is created.256 If, then, these Yang-Mills fields are coupled to a triplet of scalar fields a nonsingular localized static soliton — or, at least, a solitary wave — solution is created257'259 with which magnetic monopoles can be associated.26(1 The conventional theory of electromagnetism, as we have emphasized, is an Abelian U (l) gauge theory. However, it is possible to embed the electromagnetic field within a SU(2) field. This amounts to a gauge-invariant definition of the electromagnetic field in terms of the parent fields258 F h¥ =

- (1/g)£abcaD^0(l - l/4 0 2) ^ (1 - ll4 Q 2r

(8.20)

for t > 0. The normalized voltage v(t)JZ0 1 is plotted in Figure 8.23 for various values of Q and 7j as a function of co0 t/2ir = f 0 t. We recognize that the transient time required to reach the steady state decreases with decreasing quality factor Q and increasing relative half-power bandwidth rj. Hence, a short transient time calls for a small value Q and a large value of rj, but with the resonance effect according to Figure 8.22 calling for a large value of Q and a small value of rj. These contradicting requirements are a basic feature of the conventional small-relative-bandwidth technology. One must generally make a compromise between time resolution and selectivity. No such contradictory requirement exists for the large-relative-bandwidth technology. The selec­ tivity depends on the number of pulses in a character or signal, while the time resolution depends on the duration and risetime of the individual pulses, as discussed in Section I. In order to develop a simple formula for the number of cycles in Figure 8.23 required to reach the steady state, consider the values of the exponential function e~“ot/2Q in Equation 8.20 after n cycles e ~u>0t / 2 Q

_

e ~ 2 ir f(/ l 2 Q

_ ^

- T n /Q

(8.21) n =U

455

0

5

10

15

20

Figure 8.23 Resonant circuit step input response. The normalized voltage v(t)/Z0 1 according to Equation 8.20 as function of fQ t = w0t/2n for the various values of the quality factor Q and the relative half-power bandwidth n •

456 For Q — n we get g-tnlQ _ £ - t

04321

(8-22)

Hence, after g = n = / 0 r cycles the exponential factor in the transient term in Equation 8.20 has decreased to about 4% of its original value. In this approximation we may say that the voltage has reached the steady-state value after n = Q cycles.

Chapter 9

Radar Cross Section and Target Scattering Michael L. VanBlaricum CONTENTS I. Introduction ......................................................................................................................... 457 II. Radar Cross Section ............................................................................................................457 III. The Scattering M a t r i x ......................................................................................................... 461 IV. Frequency Dependence of RCS ......................................................................................... 462 V. Relationships Among CW, Transient, and Wideband Scattering...................................... 467 VI. The Singularity Expansion Formulation ............................................................................472 VII. Resonance-Based Target Identification .............................................................................. 481 References ..............................................................................................................................................489

I. INTRODUCTION This chapter defines and develops the vernacular and tools for understanding scattering from a target when it is illuminated with an incident radar waveform. We begin with power scattering by introducing and defining radar cross section (RCS) as it is conventionally used with the radar range equation. We then present scattering from a transfer function point of view and define the more general scattering matrix for a target. The characteristics of radar scattering are dependent on the ratio of the target’s dimensions to the wavelength of the incident wave. We present a description of each of the three wavelength-dependent electromagnetic scattering regimes (Rayleigh, resonance, and optical) and discuss the tools used for predicting the scattering in each of these regimes. We then discuss the relationships among CW, wideband, and transient scattering in terms of linear systems theory. Finally, we look at the singularity expansion formulation for describing electromagnetic scattering. We look at the physical characteristics of the natural and forced response components of scattering in terms of this expansion and discuss the possibilities of target identification based on parameters derived from this description.

II. RADAR CROSS SECTION As the term radar cross section implies, it is a measure of equivalent surface area of a target that a radar sees. Formally, it is an equivalent area that, if it were to intercept the power incident from the radar and scatter it uniformly in all directions, would produce a return at the radar receiver equal to that from the target. In other words, RCS is a measure of the power scattered per unit solid angle by a target in a given direction and normalized with respect to the power density in the incident field. With this normalization, the decay due to the spherical spreading of the scattered signal is factored out. This later normalization removes the effect of the range, R, from the radar to the target. This definition of RCS of a target is based on the assumptions that the target is in free space and is in the farfield of the radar transmitter so that the incident field on the target is effectively a plane wave. Radar cross section is often alternatively referred to as effective echo area or just echo area. Radar cross section (conventionally represented as a) can be defined mathematically as 0-8493-4440-9/95/$0.00+$.50 © 1995 by CRC Press, Inc.

457

458

= 4>tt

power per unit solid angle scattered back to the receiver power per unit area incident on the target

2 I H I2 - = 4 tr lim R 2 I E: I2 l H; I 2

0 = 4tr lim R 2 1

(9.1)

where Es Ej Hs Hi R

= = = = =

scattered electric field at the receiver electric field incident on the target scattered magnetic field at the receiver magnetic field incident on the target distance between the radar and the target

Radar cross section has the units of area and is usually given in square meters. It is common to express RCS in decibels relative to one meter squared (dBsm): 10).

There is ambiguity in the above generalized definitions because of the definition of the term “target dimension” or L. For long thin targets, such as missiles, L is easily taken as the maximum length of the body. For fatter bodies the maximum target length is usually not as indicative of scattering performance as is the maximum circumference of the target. The extreme case is the sphere where the dimension that controls the scattering performance is the circumference. Obviously, the circumference of a sphere is related to the maximum linear dimension by 7r. Therefore, the inequalities in the above three definitions can be off as much as ir depending on one’s definition of L and should be only taken as guidelines.

463 Figure 9.2 displays these three general scattering regimes in terms of the characteristic dimension of a body, in meters, and the wavelength and frequency of the radar. The lines separating the dilferent radar regimes are dotted to indicate that the definition of the boundary is a little fuzzy. Because most radar targets are larger than a few meters long and because many surveillance radars and virtually all fire-control radars work at wavelengths less than 0.3 m, the optical or high frequency scattering has gotten the most attention over the years.

THE RAYLEIGH OR LOW FREQUENCY REGION In the Rayleigh region the size of the body is much smaller than the wavelength of the illuminating waveform. This region is named after Lord Rayleigh, who, in the late nineteenth century, studied scattering of light by small particles. Historically, dimensions of most radar targets have been larger than the radar’s wavelength. Therefore, modeling in the Rayleigh region has mainly been focused on scattering from rain, dust, and other small particles. However, with the increased interest in radars operating at HF (< 30 MHz) such as over-the-horizon (OTH) systems and with interest in baseband and other impulsive radar waveforms, the knowledge of the Rayleigh cross section of conventional targets is becoming more important. In the Rayleigh region, since the wavelength is much larger than the body size, there is essentially no variation in phase of the incident wave over the scattering body; all portions of the target are exposed to the same incident field level (magnitude and phase) at the same time. Except for the temporal variations, this is almost the same as a statics problem and is commonly referred to as a quasi­ statics problem. This static approximation was recognized by Rayleigh. The quasi-static incident field causes the charge on the body to polarize to the two ends of the body creating a dipole moment. The strength of this induced dipole moment is a function of the size of the body relative to the vector direction of the incident field. The scattered field is then proportional to this dipole moment and to the square of the frequency. The scattered power and the RCS are then proportional to the fourth power of the frequency and the square of the dipole moment.

THE RESONANCE REGION In the resonance region, the incident wavelength is on the order of the size of the body. As in the Rayleigh region, it is primarily the gross body features that affect the scattering characteristics and not the small details. Because of the target size relative to wavelength, the radar-induced currents on a target in this regime behave in an oscillatory or resonant manner. The scattered fields then also have this resonant characteristic. The analog between a target illuminated in the resonance region by an incident radar wave and that of a vibrating string or a vibrating drumhead radiating a musical signal are very close. In both cases the objects have been struck and driven into oscillation. In the case of a drum it is a mallet striking the head producing mechanical vibrations which in turn radiate a musical sound. For the radar target it is the radar’s incident electromagnetic signal that induces the resonant or “standing wave” currents that produce the radar returns. Resonant scattering phenomenology has gained more and more interest over the last decade as radar frequencies have gotten lower. Solving for the RCS of a resonant body is not as simple as in the Rayleigh regime, or for that matter as the optical regime, because simple approximations usually are not valid. The normal way of solving for the resonant RCS of a target is to solve an integral equation relating the induced current on or in the body to the incident field. Solutions of these integral equations have gained a lot of attention in the last 20 years with the advent of high speed computers. Resonance region RCS for many canonical shapes has been documented in the literature and in several reference texts. It will be shown in a later section of this chapter that the induced current and the scattered field can be represented in terms of a series of complex (damped sinusoids) exponentials in much the same way that the solution of a resonant circuit is written. This expansion gives more insight into the scattering physics of resonant scattering in both the time and frequency domains.

464

CHARACTERISTIC TARGET SIZE, meters

Figure 9.2 Relationship of the three general scattering regions.

THE OPTICAL OR HIGH FREQUENCY REGION In the optical or high frequency (not to be confused with the HF radar band) scattering region, the body is much larger than the wavelength of the incident wave. Unlike the Rayleigh and the resonance regions, the scattering here is not a result of the field interacting with the total body. In the optical region it is useful to think of the scattering as coming from a collection of independent scattering centers distributed over the target. Most of the scattering contribution comes from specular points, edges, and shadow boundaries. Hence, small target dimensions and details are important in this region. In fact, edges, metal seams, and discontinuities as small as rivets can be big contributors to the RCS in the optical region. Because the scattering in this regime is due to the return from a large number of individual scatterers, the RCS depends on the way all of these returns sum in phase. If the returns from these scattering centers all add in phase, then the overall return will be large. Similarly, if the returns tend to cancel each other then the overall return will be small, possibly even zero. The relative phases of the signals returned from each of the scattering centers depend on many parameters. Small changes in aspect angle, due to target vibration, for example, can cause the overall sum of the returns to vary rapidly or scintillate. Most realistic radar targets in the optical regime appear different from different aspect angles. Hence, even small changes in aspect angle can cause large overall changes in the RCS. Typically, the RCS plotted as a function of aspect angle at a fixed frequency looks very much like a “fuzzball” due to these fluctuations. Figure 9.3 shows the measured RCS of a QF-100 aircraft. The target measured was a third scale model that was measured at Ka band to represent a full scale target at X band. The scale factor has been accounted for so the plot represents the full scale RCS in dBsm. Notice how the RCS has the “fuzzball” characteristic. Although a target’s RCS pattern is a deterministic function of angle, frequency, and polarization, for many radar calculations it is convenient to treat the RCS as if it were a random variable characterized by a probability distribution function and its correlation from pulse to pulse or scan to scan. One could attempt to fit a probability-distribution function to the data as from the RCS plot of

Figure 9.3 RCS of QF-100 measured at Naval Air Warfare Center — Weapons Division Radar Signature Branch at Point Mugu, California.

Figure 9.3. However, it is usually not practical to obtain the experimental data necessary to calculate the statistical distribution of the RCS. A more reasonable method to account for the fluctuations in a target’s RCS is to postulate a model. Swerling7 has calculated the detection probabilities for four different models of RCS which differ in the assumed rate of fluctuation and the assumed statistical distribution of the RCS. These four Swerling models are quite useful in describing most practical situations. In the four cases, two different target models to determine the distribution functions are assumed. In the first model, he assumes that the target is made up of many independent scattering elements so that the RCS has an exponential density function. The second distribution model assumes that the target has one main scattering element that dominates plus many smaller independent elements yielding a Rayleigh density for a. The first model is typical for large complex targets such as aircraft. The second model is more useful for small targets of simpler shape.

466 In addition, Swerling assumed two different rates of fluctuation: (1) a relatively slow fluctuation, such that the values of RCS for successive scans of the incident radar beam past the target are statistically independent but the RCS stays constant from one pulse to another during a scan and (2) a relatively fast fluctuation, such that the RCS is independent from pulse to pulse within one beamwidth of a scan. By combining the two models of the scatterer-based distribution functions with the two fluctuation models one comes up with the four Swerling models. These four cases are summarized in Table 9.1. There are many different solutions to solving the scattering problem in the optical regime. For the most part these are approximate techniques. The type of technique used and the detail of the model and approximations required depend on the complexity of the body and the accuracy sought. Classical techniques which are used in this regime include, but are in no way limited to: •

Geometricaloptics(GO), where the scattering phenomenon is treated by classical ray theory; however, this approach does not apply to edges, tips, corners, tangent points, or shadow regions which, as stated already, are major contributors to the optical scattering region Geometricaltheory o f diffraction(GTD), where geometrical optics are extended to include edges, tips, corners, tangent points, and shadow regions by including an additional class of rays called diffracted rays Physical optics (PO), where the approximate currents predicted by geometrical optics are used in the exact integral representation of the scattered field

RCS OF THE SPHERE The sphere, one of the few bodies for which the RCS can be analytically determined (others include all ellipsoids of revolution), has a monostatic RCS that is independent of aspect angle. In addition, the scattering characteristics of the simple sphere are exemplary of the three scattering regions described above. Figure 9.4 presents the RCS of a sphere as a function of its circumference measured in wavelengths, 27ra/X = ka, where a is the radius of the sphere and k = 2x/X is the wavenumber. The RCS is plotted normalized to the geometric cross section of the sphere, 7ra2. The RCS in the Rayleigh region for the sphere varies with frequency to the fourth power or to X"4. In the optical or high frequency region, the sphere’s RCS converges to the physical cross section of ira2. Since the sphere has no edges, joints, or discontinuities there is no other contribution in the optical regime. The resonance region RCS of the sphere has several peaks which correspond to the natural electromagnetic resonances of the sphere. These resonances actually continue into the optical region but converge asymptotically to the physical cross section. The location of these resonances in frequency are seen to be directly related to the circumference of the sphere. The first resonant peak, and the highest RCS, corresponds to the point where the circumference equals exactly one wavelength. At this frequency, a wave traveling completely around the sphere’s surface will constructively add to a wave scattered from the specular point at the sphere’s front surface. In the resonance and optical regions of a sphere, the RCS varies around the physical cross section, o = 7ra2. One could think of this as the nominal cross section that is independent of frequency. The frequency variation comes in from the constructive and destructive interactions of the currents induced on the body. At the first resonance, the maximum cross section is approximately given as o = 0.286 X2 which occurs at the point, as mentioned above, where the circumference equals exactly one wavelength and we get constructive interference. It is interesting to compare the sphere’s mean RCS with frequency to that of two other standard scattering shapes, the flat plate and the right circular cylinder. For monostatic scattering, the broadside return of the flat plate is Q

0.67

LLI

EE

0.33

0 L-J 0

1------------- 1------------- 1 4.0

8.0

12.0

TIME, nano seconds

FREQUENCY GIGAHERTZ

(b) Figure 9.9 Time history and frequency spectrum of impulsive signal used. (From Van Blaricum, M.L., in Ultra-Wideband Radar: Proceedings of the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991. With permission.)

H ( r , s ) = £ , , ( s , p ) M,(r) (s - s,) "1' ♦ W( r , s , p ) i-1

(9-23)

where the above terms are defined as: Si

=

natural frequency, pole singularity, natural resonances. This is a complex frequency for which the system has a response when no forcing function is applied. The poles must appear in complex conjugate pairs or lie on the real axis. The poles also must not lie in the positive half of the s-plane.

475

TIME, NANO SECONDS

Figure 9.10 Transmitted time history for propagation into A dispersive medium at a 1-m depth. (From Van Blaricum, M.L., in Ultra-Wideband Radar: Proceedings of the First Los Alamos Sym­ posium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991. With permission.)

^ ----------0.5 m ----------►

Figure 9.11 Schematic of a 0.5-m slab over a PEC surface. (From Van Blaricum, M.L., in UltraWideband Radar: Proceedings of the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991. With permission.)

Mj(r)

= natural mode. This is the response of the system atSj which depends on theposition F on the structure and the object parameters only. r|i(s,p) = coupling coefficient. This is the strength of the natural oscillation s; in terms of the system and the incident wave parameters. It is independent of position, p = polarization of incident plane wave. r = the position vector. This is the position on the structure atwhich the transient response is being measured or observed, mj = the multiplicity of the i* pole.

476

FREQUENCY - GIGA HERTZ

Figure 9.12 Transfer function for a 0.5-m slab over a PEC surface. (From Van Blaricum, M.L., in Ultra-Wideband Radar: Proceedings of the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991. With permission.)

TIME, NANO SECONDS

Figure 9.13 Reflected time history for a 0.5-m slab over a PEC surface. (From Van Blaricum, M.L., in Ultra-Wideband Radar: Proceedings of the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991. With permission.)

477 The term W (r,s,p) is an entire function of s and dependent on the form of the coupling coefficient and the incident wave. In the most general case, this term is required by the Mittag-Leffler theorem21 in order to guarantee convergence of the infinite series. The coupling coefficients, ^(s), are the strength of the natural oscillations, sh in relation to the incident applied field. If these coupling coefficients are defined as constant values at each pole

v, = ,»i(s ) I , . , then we have what is known as class-one coupling coefficients. If the coefficients are defined as functions of frequency, then they are referred to as class-two coupling coefficients. The inverse Laplace transform of Equation 9.23 gives the time domain version or the impulse response as

h ( r , t ) = £ iji ( t , p ) M j ( f ) e S|t + W( r , t ) i =1

(9-24)

If class-two coupling coefficients are used, then the coefficients in Equation 9.24 are time varying. Class-one coefficients give constant coupling coefficients. The choice of the form of the coupling coefficients and the existence or necessity of the entire function is dictated by the physics of the problem and the intended use of the expansion. In practical problems, such as building a database for resonance-based target identification, it is necessary to obtain the SEM parameters from measured data. Realistic objects cannot be efficiently and, in most cases, accurately modeled numerically. As mentioned above, in order to extract the SEM parameters from data it is necessary to use resonance extraction algorithms such as the Eigen-Prony method.20 When any resonance extraction method is used, the exponential coefficients or residues are required to be constant. Therefore, from Equation 9.24 the expansion needs to be of the form

h(r, t) = £

Ai( r ) e S|t + W( t )

(9-25)

i

where the tj, and M are now combined as residues, Aj (r), which are constant in time. The question now arises as to the ability of the constant coefficient form of the pole expansion in Equation 9.25 to represent real transient electromagnetics data and whether or not the entire function, W(t), is required. While attempting to extract the poles, S;, from transient scattering data, it was observed that it was only possible to accurately do so after the driving function (incident waveform) had passed entirely over the body.20 That is, a portion of the measured induced current or scattered field had to be skipped before resonance extraction procedures could extract the proper poles. This observation implies that the early time cannot be represented completely by a constantcoefficient pole expansion. Hence, a time-limited, time-varying function must be added tothe constant coefficient exponential series to represent the early time data. The form of the pole expansion for measured scattered data now commonly used is

R ( r , t ) = u ( t - to) £

A j ( r ) e s,t + W ( f , t ) [ u ( t ) - u ( t - to) ]

(9.26)

where u(t) is a unit step function and t0 is, at a minimum, twice the time the driving function takes to pass over the body. That is, t0 = 2d/c, where d is the line-of-sight extent of the body. Morgan21 has

478 done an excellent job of mathematically proving and describing the form of Equation 9.26 and is suggested for a more formal derivation. Here we attempt to gain insight into the physics of both the early time and resonant components of transient scattering by presenting a few numerical experiments. Before presenting the numerical examples, we must first look at the temporal form of the electric and magnetic field integral equations (EFIE and MFIE) in order to get a better feel for the scattering physics. The equations for the induced current J (r,t) on a body due to an incident field are

MFIE: J ( r , t ) = 2n x H

(r, t )

1 ^ 1 9 R1 Rc“ J i ~

J ( r , r ) x aR | ds'

(9.27)

where = t- - , R = | f - f / | , a R=

r - r R

and EFIE: (ON WIRE - w) ALL TERMS INVOLVING w • E|nc ( r , t ) + f ■ RETARDED TIME 0 < r < t n

_

_

where HINC and EINC are the incident magnetic and electric fields respectively and in Equation 9.28 DlSP is the inverse operator resulting from the singular value (present time) part of the integral equation. In both equations, the right-hand side is broken into two parts. The first part represents the current induced due to the incident field and the second part represents the current induced by currents flowing on other parts of the body at times before the present time, t. Note that the incident field-induced term in Equation 9.27 is 2fi x H the “extended” physical optics (PO) solution for the current which has value even in the shadow region of the body. This PO term is the forced response while the second term is the natural response of the system. A heuristic time domain description of the scattering which results from these induced currents is as follows. Consider an impulsive field incident on a body. This field gives rise to local impulsive currents on the body in order to satisfy the boundary condition that the tangential E-field be zero. These currents are the PO currents of Equations 9.27 and 9.28. Hence, as the field passes over the body (including the shadow region) impulsive PO currents are excited. The impulsive currents then act as local sources to drive traveling waves of current on the body (the second terms on the right side of Equations 9.27 and 9.28). Traveling waves propagate in all directions from these local PO sources. Eventually, the traveling waves see the limits of the body and reflect back. At this point in time, the traveling waves begin to exhibit a standing wave nature, which are the well-known resonances of the body. This travel­ ing/standing wave response is the natural response. Now, from this heuristic argument and Equations 9.27 and 9.28, it is clear that the forced PO response is time-limited to the time the driver is on the body. Hence, it contributes to the entire function, W (r,t) of Equation 9.26. The natural response can be written as a sum of complex expo­ nentials. However, for the early times while the currents are establishing the extent of the body, this sum of exponentials must have time-varying coefficients (class-two coupling coefficients). That is, before the traveling waves become standing waves, the modal coefficients must be time varying. This time-varying term can also be thought of as contributing to the entire function W(r,t) of Equation 9.26.

479 Equations 9.27 and 9.28 are for the induced currents on a body. The currents induced on the body radiate producing a scattered field which will have the same general form as Equation 9.26. Writing this more explicitly, we have

E (t ) = [ u ( t ) - u ( t - to) ] [ Epo(t ) + ENRE( t ) ] + u ( t - O E

A ieS|‘

(9'29a)

or E ( s ) = E po( s ) + E NRE( s ) + Y

A| s - s,

(9.29b)

where Equation 9.29a is the time domain and Equation 9.29b is the frequency domain representation. Taking Equation 9.29 term by term, we have first EP0 which is the scattered field component due strictly to the PO contribution. This term is limited to the time the driving function interacts with the body. These fields result from the currents induced on the body to match the boundary conditions. In the low resonance region, these fields are usually a small part of the total scattered field. The forced PO currents act as local sources producing traveling waves that emanate in all directions from the local source. The traveling wave contribution to the scattered field is the ENRE term of Equation 9.29. (NRE stands for the natural resonance entire function.) After the forced response term turns off and the traveling waves turn into standing waves, the currents on the body are simply damped standing waves which can be represented as a sum of constant coefficient exponentials. This current produces the pure resonant portion of the scattered signal, which in the lower resonance region contains most of the energy. It is the driving of these resonances or standing waves that causes the resonance-region, cross-section enhancement commonly seen in air vehicles in the 10 to 100 MHz regime.

NUMERICAL EXAMPLES In order to show the validity of Expressions 9.29a,b and the strength of each term, a numerical example is presented. The thin-wire time domain (TWTD) scattering code22 was used to generate induced currents and scattered fields. A thin cylinder of length-to-diameter ratio L/d = 100 was modeled. The cylinder was driven with a narrow Gaussian pulse at 90° (broadside) incidence with the E-field polarized along the axis of the cylinder. A narrow Gaussian pulse was used so that the calculated response would be near the true impulse response of the system. The response of the system to a more realistic radar pulse can be calculated by convolving the Gaussian response with the desired driving waveshape. When using a time domain method of moments (MOM) code, it is possible to isolate the forced PO portion of the scattered field (the first term on the right-hand side of Equation 9.28). Figure 9.14A shows the total backscattered time domain electric field due to the incident pulse overlaid with the isolated PO response. For the total field, notice the initial spike followed by the damped oscillatory response. The PO portion is shown as the dotted line in Figure 9.14A and is very similar in characteristics to the early part of the total response. The scattered field is a spatial integral over the first temporal derivative of the body current: H ( r, t ) =

J_ 47rrc

f J L 7 ( r , r ) x aR ds'

Js

d

r

Because the driving function is a Gaussian pulse (an approximation to a delta function), the resulting PO scattered field is its first time derivative (a doublet). Figure 9.14B presents the frequency spectrum

480 0.070

0.049 0.028 DC LLI

2 DC

LLI CL CO

tj o >

0.007 -0.014 -0.035 -0.056 -0.077 -0.198 -0.119 -0.140

0

27

54

81

108

135

-L

-L

162

189

216

243

270

TIME STEPS (At = 6.9444E-11) seconds

(a)




§

LU DC

00 ~o Q Z) O


\

NRE E (s)

(a) Total vs Physical Optics and Traveling Waves

»______ )

+

Figure 9.15 Cylinder reflection components. Spectral contributions of each scattering component as a function of frequency for broadside backscatter from a 1-m long cylinder with L/d = 100. (From VanBlaricum, M.L., in Ultra-Wideband Radar: Proceedings of the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991. With permission.)

MAGNITUDE IN DECIBELS

PO E(s)

ANGULAR FREQUENCY IN UNITS OF cre/L)

)

=

ANGULAR FREQUENCY IN UNITS OF crc/L)

E(s)

482

483

LLI

Q D

CL




Received signal UWB single cycle with noise.

Figure 10.13 Superheterodyne and homodyne receiver block diagrams. (A) Narrowband receivers convert a higher received frequency to a lower, more easily handled intermediate frequency signal. The modulation is the important part of the signal, not the carrier. The super-heterodyne could be used in UWB detection if the signal waveform is not important. (B) Homodyne receivers preserve the received waveform. The detector would be a nonlinear threshold detecting device or a correlator to compare the signal with a known waveform. For some UWB signals, the waveform carries information about the reflector. The homodyne receiver preserves the signal waveform for detection and analysis. Both have potential UWB applications, but must be designed for specific purposes. Appendix 10A provides guidance on bandwidth and impulse signal sensitivity.

ind icates reception o f the reference waveform or som ething resem bling it w hich produces a detectable output. S ection IV w ill d iscuss correlation detection in detail.

UWB Radar Receiver Design Objectives A radar system design m ust m eet som e system perform ance level sp ecified by the user or buyer. For radar system s the overall system ’s perform ance sp ecification is generally probability o f detection and false alarm for a sp ecified target radar cross-section at som e range and target/background conditions. T he receiver bandwidth sets the m inim um detectable signal, and that drives the rem ainder o f the radar d esig n , including antenna gain and transmitter power. System m ission and function sets the perform ance requirem ents. For exam ple, a security intrusion detection radar or ground-probing radar w ill have different requirem ents than a m ilitary or aerospace surveillance or tracking radar. A practical radar system receiver design w ill be a continuous analysis, design , sim ulation, and testing process.

16

T

-o

Signal 0 Processor

C3

- 6 8 0 pF

R 9 -2 7 0 Q

' t

>

\f

f 2 K

V °-iv

>

f to

Voltage from antenna.

t

—20 ns

Incident pulse at antenna.

h -

X

Voltage at 2 from tunnel diode D1

J l

Collector transistor T 1 Base of T2 (avalanche transistor) Detector output 6

------------

Figure 10.14 UWB impulse threshold detection circuit. The circuit depends on the antenna, connecting cable, and components forming a filter sensitive to the frequency content of the received impulse. The incident pulse causes a tunnel diode to saturate which produces a longer pulse that can cause other transistors to conduct and produce a longer duration (low frequency) output. (Adapted from Figures 3 and 4 of Kenneth Robbins, U.S. Patent 3,662,316 (Short Base-Band Pulse Receiver), May 9, 1972.)

THRESHOLD DETECTION Narrowband Threshold Detection Radar books discuss narrowband threshold detection receivers and radar perform ance. D iscu ssion s usually cover narrowband detectors using linear or square-law com ponents to rectify a continuous signal voltage from the IF amplifier. W hen the vid eo (detected signal voltage or pow er) exceed s the noise voltage (power) in the system by som e margin, then detection occurs. Setting the detection threshold w ith respect to the noise determ ines probability o f detection and false alarm for a receiver, as show n in Figure 10 .5 . The problem is to determ ine analytically what received signal strength w ill be adequate to produce detection in a giv en bandwidth system , w hich m eans finding the SN R for a g iv en probability o f d etection and false alarm. W hile the plots in Figure 10.5 are for narrowband detection, the sam e princip les apply to estim ating probability o f false alarm and probability o f detection for U W B radar

512 receivers with threshold detection. The difference w ill be in the signal and noise probability distribution a ssu m p tion s.14 Figure 10.1 2 show ed the U W B threshold detection problem in term s o f the directly received signal. D etection occurs w hen the received voltage instantaneous value, or the distribution o f the signal voltage (pow er) and frequency distribution, exceed s som e positive or negative threshold.

UWB Threshold Detection Example O ne approach to U W B threshold detection is to convert the short duration signal into a longer duration signal. P ossible receiver designs include w ideband h om od yne receivers that can preserve the received waveform and superheterodyne receivers that distort the waveform but preserve it in another form at su ch as a longer, lower pow er signal. The short baseband receiver, described in K enneth W. R obbins’ patent15 and shown in Figure 1 0 .1 4 , is a good case study in practical U W B threshold detection and h om od yn e receivers. T his receiver u ses a tunnel d iod e, w hich is a nonlinear d evice driven tem porarily unstable by a received im pulse w ith enough signal strength. T he R obbins’ short baseband receiver works as follow s. The detector is the first elem en t after the antenna (1). N ote that there is no heterodyning and IF am plification as found in narrowband receivers. Variable resistor R1 sets the bias o f D l . If the incident 1-ns pulse at the antenna (1) is strong enough, then it drives the voltage at jun ction (2), and the voltage across the tunnel diode D l suddenly negative and h old s that condition for about 2 0 ns w hich turns transistor T1 off. (The tunnel diode characteristics are show n next to D l in Figure 1 0 .1 4 .) The base o f the avalanche transistor T 2 turns p ositive, w hich p rodu ces a long, negative going step and decay signal at the output to the signal processor. T he T E M horn antenna, coupling capacitor C l , and the characteristics o f the connecting line from the antenna to the detector make a filter sensitive to the tim e characteristics o f the incident pulse. The detection p rocess turns a short im pulse signal into a long duration signal for indicating target presence in a radar.15 T he R obbins’ im pulse threshold detection circuit is a nonlinear circuit and depends on an adequate change in the tunnel diode bias from the signal from the antenna. The circuit is set to trigger w hen the incom in g signal produces a - 0 .1 - v and 0 .5 -n s im pulse at jun ction (2). The change in voltage sends the tunnel diode into its unstable region and produces a long square wave into the base o f transistor T l . In the patent description, Robbins states: “The amplitude o f the received impulse at the receiving antenna may be, for exam ple, about 200 m illivolts in a typical operating circumstance, a value several orders o f magnitude greater than the signals present in an urban environment due to conventional radiation sources, such as interfering signals normally being at the m icrovolt level. Accordingly, although the novel receiver essentially accepts all signals over a very wide pass band, it is substantially immune to interference from conven­ tional radiation sources, including electrical noise signals, such as internal combustion engine ignition n oise.” 15 E ssen tially the sam e tunnel diode and avalanche transistor circuit design shows up in later baseband and U W B radio and radar patent concep ts. Short duration im pulse radar system s for intrusion detection and U W B (im pulse) radio system s concep ts use threshold d e te ctio n .16

Estimating the UWB Threshold Probability of Detection and False Alarm Analytical Approach and Objectives We w ill exam ine the general case o f a receiver where the perform ance prediction objective is to determ ine the probability o f detecting a given duration incident electrom agnetic pu lse, polarity, and field strength signal at the antenna. D eterm ining the probability o f detection w ill require an estim ate o f the m inim um size pulse n eeded to trip the detector, w hich is determ ined by the receiver circuit characteristics. D eterm ining the probability o f false alarm w ill require estim ating the probability that random n oise and incident environm ental signals w ill produce a pulse signal exceed in g som e level in the receiver. If the threshold level is set high, as describ ed in the previous Robbins short baseband

513 radio, it appears that the probability o f false alarm w ill be sm all or negligible. Both o f these require a kn ow ledge o f the signal, interfering signal, and noise environm ents.

UWB Performance Estimation T his section describes conventional probabilistic analyses only. There is existing work on characterizing U W B signals in the tim e dom ain only, resulting in conventional distribution characteristics. L et us assum e the U W B noise has a Gaussian pdf. T he next step is to determ ine the signal-plusn oise distribution for the U W B case. G iven the case o f a U W B single sine wave in G aussian noise, such as show n in Figure 1 0 .1 2 , what is the probability o f the signal exceeding som e detection threshold level? The Gaussian distribution is the preferred ch o ice in detection probability analysis because it p rovides a tractable m odel. If you stray from the Gaussian distribution you may get expressions with either no closed form solution or untractable m athem atics.3 We w ill show the results o f two cases from the literature. C a se 1: Dr. H enning F. Harmuth takes the case o f determ ining Pfa' and the thresholding calcu lation for nonsinusoidal pulse trains as part o f a perform ance prediction analysis for nonsinusoidal wave radar.17 Both the noise and signal-plus-noise distributions are assum ed to be Gaussian. The noise distribution WN(x) is defined as

WN(x)

\J2tt ac

exp

—x

( 10. 10)

~2oi

T he signal-plus-noise p d f Ws+n(x) is a G aussian p d f with m ean value shifted to a norm alized value o f 1 and m ean-square deviation still oG:

exp

-1(* - I )2

, where oG = —

( 10. 11)

2 a\

2

g

avg. noise energy received during signal duration T avg. signal energy received

H arm uth defines threshold as som e m ultiple aG, term ed to be

'

P

P robability o f detection,

=

2

r i 1 -

( 10. 12)

FaG. Probability o f false alarm is determ ined

-

F o r ] G

e rf

"

(1 0 .1 3 )

\/2 < 7 s

PD, is calculated as

(10.14)

1 + erf '/ 2 ° g

514 A

threshold FOg

Figure 10.15 Detection using Gaussian noise and signal plus noise distributions. (A) Probability distributions of Gaussian noise and signal plus noise. (B) Probability of detection and false alarm for values of F from 1.0, 1.33, ..., 2.66 plotted against SNR. The lower plots of probability of detection and false alarm show the effects of setting the threshold value F. About +12 to +15 dB SNR is needed for both good probability of detection and a probability of false alarm less that 0.001. (Adapted from Harmuth, H. F., IEEE Trans. Electromagn. Compat., Vol. 31, No. 2, 138-147, 1989. With permission.)

515

P,a

Pd>

Figure 1 0 .1 5 A shows the graphical diagrams o f and and the threshold for two values o f crG. Figure 1 0.15B shows Pd and Pfa plotted against SN R for values o f F betw een 1 and 2 .6 6 . For the range o f thresholds F aG show n, an SN R o f at least 12 dB is needed for a high (0 .9 9 9 ) probability o f detection and a low (0 .0 0 1 ) probability o f false alarm. Other plots o f probability o f d etection vs. SN R for giv en probability o f false alarm , such as shown in S k oln ik ’s Introduction to Radar Systems, Figure 2 .7 , illustrate this. For exam ple, to go from a probability o f detection o f 0 .9 9 to 0 .9 9 9 for a given probability o f false alarm requires an additional 1.1 dB o f received signal power, or an additional 1.1 dB transmitted signal pow er for detection at the sam e range. M ost U W B signal sources must trade o ff repetition rates and output power, thus the repetition rate m ust drop i f output pow er is increased. C a se 2: W halen (R eference 3, pp. 99 to 100) presents the case o f a single sam ple o f the sum o f a random ly phased sine wave and a Gaussian process:

r{t) = A sin(cof + 0) + n (0

(1 0 .1 5 )

w here 6 is uniform (0, 2w) and n(t) is zero m ean G aussian with a variance a2. A ssum e that the phase 6 and noise are statistically independent. The result is the expression

P(r)

=

1

( - r 2/2cr2)t

V27TC72 *=0

k\

1F1

1 1 A 2 kt + —; 1; - —

2

(1 0 .1 6 )

2 a2

w here 1F1 is the confluent hypergeom etric function

1FJ ( a b x ) = 1 + f l * + a(a+1>* 2 + «(«+l)(a+2) * 3 + b 1! b{b+1)2! i>(ft+l)(fc+2) 3!

(10.17)

Plotting the curve produces Figure 1 0 .1 6 A , w hich can take on a number o f shapes depending on the relation o f the sine wave am plitude to the variance o f the noise. In this case, we are looking at the probability that a signal w ill ex ceed som e positive or negative voltage value. R eview ing the parts o f Ps + g iv es insights into its shape: (1) the cusp shape from the sinusoidal com ponents is still evident and (2) this p d f w ill approach + 0 0 as x a. For this exam ple, Figure 1 0 .1 6 A shows the UW B noise p d f superim posed on the transient signal-plus-noise pdf, where we have assum ed a = A. Rem em ber that the p s + n(x) = p(z) p d f is present only for the tim e span T o f the U W B signal. Figure 10.16B show s the com parable pdfs for conventional narrowband detection. Several points stand out: (1) T hresholding to determ ine P f a and P D is different for the U W B and narrowband cases; the U W B c ase has two thresholds; ±V1, w hile the narrowband case has only Vt. For the U W B case (Figure 1 0 .1 6 B )

n(x)

PD =

f.V .W * +

Pfa = \ " P n ^

\

= 2J

~P,Jx)dx

+( J

Pl&k= 2 J ” P'QM*

(1 0 1 8 ) (1 0 1 9 )

For the narrowband case show n in Figure 10.16B

Pfa =

Pn^)dX

PD =

f ” PSJ X)dX

(10.20)

516 (2)

Shifting thresholds ±

VI or Vt still affects PD and Pfa in the sam e manner.

Change in Threshold

Effect

Increase IWI or increase Vt Decrease IWI or decrease Vt

Pfu and PDdecrease Pf(l and PDincrease

T he threshold must still be selected as a tradeoff in false alarm and detection su ccess. B efore m oving on, som e observations are helpful. T he U W B noise and signal-p lu s-noise pdfs in this d iscu ssion were the co llected values; no rectification or squaring were intentionally perform ed. This fact exp lain s why working w ith U W B pdfs requires two thresholds. The collected signal does, however, inclu de the transfer function effects.

RECEIVER IMPULSE SIGNAL SENSITIVITY Estimating Impulse Signal Output of Conventional Receivers* A s d iscu ssed earlier, a narrowband receiver can detect a U W B signal if the signal is strong enough. T h is sectio n addresses how to determ ine what im pulse pow er level w ill affect a narrowband receiver w ith g iv en noise figure and detectability (SN R for detection) characteristics. M ost radar engineers work daily w ith continuous signal characteristic term s such as P S D expressed in w atts/hertz. W orking with U W B im pulse signals w ill require som e mental readjustment. The designer m ust understand that voltage im pulses are quantified in term s o f volts-secon d or volts/hertz and pow er im pulses quantified in term s o f w atts/hertz2. 1819 W hile we refer to an im pulse in this case, the im pulse could be an extended w aveform or coded pulse train, as shown earlier. The m ain consideration is that the signal is com p osed o f transient, U W B short duration pulses. For our discussion, narrowband signals are those that have a long duration and bandw idth lim ited to about 1 % o f center frequency. The com plete derivation o f m inim um detectable im pulse is giv en in A p p en d ix 10A o f this chapter and the results presented below. U W B signals for this case are short duration im pulses, such as several sinusoidal cycles with exponential decay. C ontinuous U W B signals such as long duration triangular waveform s, square waves, etc. are not considered here. G iven an im pulse signal and a receiver with a noise bandw idth Bn and an im pulse bandw idth Bh the ob jective is to find the strength o f an im pulse signal that w ill produce a detectable signal. T he noise bandw idth is proportional to a receiver with a 3-dB bandw ith, with the exact proportion depending on the ro ll-o ff characteristics, as show n in Figure 10.3 5 . The im pulse bandwidth is proportional to the 6dB bandw idth. The m inim um detection level for a U W B signal is determ ined by an equation involving the detectability factor D (SN R for detection and false alarm ), the bandwidth Bn, B oltzm an n’s constant k, tem perature T (K elvin), and resistance Rs (nom inally 50 0 ); Z, and Zs are the source and input im pedances, and the noise figure is Favg The derivation in A ppendix 10A giv es the m inim um detectable U W B signal as Mhbv in volts-secon d or volts/hertz, w hich satisfied (S0/N0) = D. From Equation 1 0 .7 8 , Mhhv is where Zt(f) is the linear network im pedance and Zs(f) is the im pulse sou rce im pedance show n in Figure 10.39 in A ppendix 10A .

(1 0 .2 1 ,1 0 .9 6 )

Z,100 ns duration. For the transient pulse detected, the receiver sensitivity was degraded to - 5 7 dBm. The channelized receiver had 64 20-M Hz channels. The target signal was roughly six sinusoidal cycles long and was detected with all channels responding simultaneously, indicating that the receiver response was dominated by the individual channel’s response. N o valid measurement was possible unless the PRF was reduced to less than 1 Hz. The estimated channelized receiver sensitivity was - 6 8 dBm for conventional signal reception, but was degraded to -2 1 dBm for the six-cycle transient.

T h ese tests o f conventional receivers dem onstrated a U W B detection capability at degraded sen sitivity and without the capability o f characterizing the U W B signal in any manner. In these cases, the conventional receivers had w ide bandwidths and could react to the short duration signal. T hese c a ses can m ake a starting point for analysis o f U W B im pulse effects on conventional system s.

Conclusions on Threshold Detection T hreshold detection is the sim plest U W B radar receiver detection m ethod. However, it requires an SN R o f at least 10 dB to give 80% probability o f detection with any reasonable (10-3 or less) probability o f false a la rm .1,17 Perform ance and prediction m odeling dependis on selectin g distribution m odels with c lo sed form convolution solutions, for both signal and signal plus noise. G enerally, this m eans that it is b est not to stray far from the G aussian m odel. W halen3 is one good source o f ideas on m odelin g and detecting signals in noise. T he U W B signal level for threshold detection depends on the receiver bandw idth. In the case o f narrowband receivers, w e show ed that UW B signals are detectable, but at higher signal strengths. As a U W B receiver designer, a solid logical approach is to design for the signal bandw idth to get the best perform ance and to preserve the signal waveform. There w ill be a tradeoff o f sensitivity and bandw idth w h ich depends on the need to preserve the signal or to just detect it with a nonlinear d evice such as a tunnel diod e, as shown in Figure 10.1 4 .

IV. CORRELATION DETECTION T h is section is about autocorrelation and cross-correlation detection applied to U W B signals. T he previou s section discussed U W B threshold detection, w hich requires a signal that ex ceed s the receiver n oise and interference level. C orrelation detection is a way to detect the presence o f a signal (or som ething c lo sely resem bling the signal) in noise at som e m inim um SN R . The advantage o f correlation detection for radar is that it can provide spatial resolution shorter than the actual signal duration. T he fine spatial resolution capability with long duration signals m eans that lower signal pow er levels are p ossib le. Radar perform ance depends on transmitted and reflected energy, as an exam ination o f the radar equation shows. C orrelation provides a way to integrate the received energy o f a long, low pow er signal into a short, high pow er signal that exceed s the receiver noise level for detection.

521 T he next sections introduce correlation detection concep ts, give inform ation about cross-correlation and radar signal detection; describe practical tim e dom ain correlation, exam ine correlator perform ance, exp lore classical correlation d etection analysis, analyze the correlation with Barker and com plem entary (G olay) cod es, show the relationship betw een cross-correlation and convolution, discuss UW B correlation detection thresholding, summarize the m ain points o f correlation d etection, and list co n clu sio n s about correlation detection.

SIGNAL CORRELATION OVERVIEW C orrelation is a process that com pares an interval o f a signal with a reference waveform and produces an output proportional to the integral o f the product over that interval. Correlators, or correlation-type receivers, are electronic d evices that detect weak signals in noise by perform ing an operation approxi­ m ating the com putation o f a correlation fun ction.21 A related term is a correlation detection (modula­ tion) system w hich d escrib es detection based on the averaged product o f a received signal and a locally generated function possessin g som e known transmitted wave characteristics.18 The averaged product can be form ed by m ultiplying and integrating or by using a m atched filter w hose im pulse response, w h en reversed in tim e, is the locally generated function. Strictly speaking, the foregoing definition applies to cross-correlation detection. Correlation detection also applies to autocorrelation, where the lo cally generated function is m erely a delayed or duplicated form o f the received signal. A ssum e a signal expressed as voltage vs. tim e or P S D sam ples from a signal x(t) and a reference signal s(t) is available and both signals are made up o f n sam ples. Then the m athem atical correlation co efficien t o f x(t) and s(t) r is

Y , xs o

(1 0 .2 8 )

T he correlation coefficien t r value can have values from -1 < r < 1, where + 1 indicates exact correspondence betw een the signals, - 1 indicates a polarity (phase) reversal, and 0 indicates no relationship betw een the signals. Adding enough noise to a signal can reduce the correlation coefficient to near 0 . Figure 10.18 shows the overall correlation concep t with the signal show n in both tim e and frequency dom ains. In both cases the objective is to com pare the received signal characteristics with a m od el waveform or PSD. C orrelation may be either a real-tim e or nonreal-tim e process depending on the radar application. The important concep t here is the instantaneous frequency content changes w here the waveform consists o f received signals plus noise. C onsidered during its essentially random occu rren ces, the m easured signal would have a power spectral distribution as show n in Figure 10.18B that is different from the noise and interference PSD. The problem is to find those occurrences in a field o f m ultiple continuous narrowband interfering sources and noise with som e w ide distribution o f frequencies. Figure 10.1 9 shows b lock diagrams o f two different signal correlators. Figure 1 0 .1 9 A shows an analog correlator approach, and Figure 10.19B shows a finite im pulse response (FIR) filter correlator architecture that can be either digital or analog. The correlator output is the denom inator o f the correlation coefficient and produces an output signal proportional to the size, duration, and waveform o f the signals x(t) and s(t) in Figure 10.19A .

522

p /A

■L

t

L y

Analog Correlator

UWB Amp

-

YCorrelatof output‘

±

>t

ah

Threshold detector output

Correlation Detection Threshold ----- > Comparison

Average background noise PSD Narrowband signals

Reference Waveform

t

X —

Reference waveform

j? \< f

UWB signal instantaneous power vs frequency and time distributiion.

Y ~

^ 7 ->

t

Received waveform Rcvd signal PSC

X

j

UWB Amp

FFT & PSD

->

Correlator outpu

Detector output

Correlation Detection PSD - > Threshold Correlation Comparison

Figure 10.18 UWB correlation detection methods. (A) Time domain correlation detection and (B) PSD correlation detection. Waveform correlation and PSD correlation are two approaches to correlation detection. Waveform correlation compares the received signal plus noise with a reference signal (A). PSD correlation requires taking the PSD of sampled sections of received signal and comparing them with a reference PSD (B).

523

t+ T

x(t)

> (X > > S(t) ...........I

H -------------1-----------

2 -----------

r

—

1-----------1-----------

10 -----------1----------- 11 ----------- 1-----------

A

A V

1

x(t)

o •

M jJ \

k n lt M U

P

'

-

J \f

S(t)0

....

2 -----------'----------- '----------- 1----------20 40 60 80

t

2

1

y(t)

’ ^

___1___ 1___ 40

60

80

t

3

jl f\

1

\/V

-

11

1

40

60

t

x(t) = Input signal + noise s(t) = Locally generated referenceyft) = Correlator output

y ( t ) = Z X(' f'n r',wn y (t)

Digitized input signal x(t) = s(t) + n(t)

delayed samples.

Figure 10.19 Practical correlator concepts. (A) Analog correlator for detecting specific waveforms in noise. (B) Digital correlator for detecting specific signals in noise. The correlator detects the presence of a signal resembling the reference signal. If information (e.g., 0,1) is transmitted by different waveforms, the receiver will require a correlator for each waveform. The analog correlator (A) multiplies the received signal with a locally generated reference signal. Synchronization of the reference and received signal will affect the correlator output. The transversal filter (B), or finite impulse response filter, correlator uses the digitized signal and multiplies each sample by a reference signal weighting factor. Synchronization is not a problem with this type filter.

524

y(t) = j^ r x(t)s(t)dt

(10.29a)

and for the finite im pulse response correlator in Figure 1 0.19B the output is

y(0 = "

nr)wn

(1 0.29b )

w here wn is the set o f w eighting coefficients applied to each sam pled signal section , and y(t) is the correlator output. (R eference 3 , p. 168) The correlator output is a signal indicating the presence o f a w aveform plus noise w h ich m atches the reference waveform . The correlator d oes this by m ultiplying the n oise signal by the know n waveform sequence to produce an output larger than the signal or the n oise con cealin g it. N o te that the correlator output is proportional to received signal strength; therefore, there exists a m inim um signal strength for detection for a given reference signal strength or w eighting factors. C orrelation reception increases the receiver com p lexity and only detects the p resence o f signal w aveform s c lo sely resem bling the reference waveform . In other words, correlation cannot extract inform ation directly from the waveform. The inform ation content o f a correlation detected waveform is its presen ce or absence and tim ing with respect to other events, e .g ., previous waveform s, or a local clock-generated signal. C orrelation destroys the received signal unless it is sp ecifically stored in a digital m em ory or delay line. Correlation detection includes two processes: the correlation process, w hich integrates the received waveform with a reference waveform , and a nonlinear detection process, w hich com pares the correlator output with som e threshold level. We w ill show how correlation detection is a way to provide radar with finer tim e m easurem ent resolu tion w hile using signal durations longer than the tim e resolution. The advantage o f using correlation detection reception o f U W B coded pulse train waveform s is to reduce U W B transmitter pow er w h ile transmitting the sam e energy and givin g fine spatial resolution and low probability o f intercept signal advantages. One approach to U W B radar is to build and transmit a unique nonsinusoidal w aveform cod ed pulse train and to detect the return signal by correlation; how ever, correlation detection for w eak signals m eans increased receiver com plexity, com putational power, and strong dep en d en ce on the assum ptions built into the detection processor. It is im portant to note that correlation is an energy m anipulation process. T he receiver m ust receive en ou gh energy to exceed som e threshold. In threshold detection, the energy is processed in clo se to its original form and com pared to a detection level. In correlation detection energy is spread out in tim e (low er pow er) and then com pressed to produce a short, large (higher power) signal w hich exceed s the system noise level by som e predeterm ined amount. A ssum ing the sam e signal energy is received for threshold and correlation processing, correlation integrates the signal energy in a long, low pow er signal into a short high pow er signal w hich is greater than the receiver noise level. O ne o f the original u ses o f correlation detection was in system s designed to transm it and receive w eak signals. Inform ation was sent by increasing the energy in each bit by increasing the transm ission tim e at the sam e power, w hile proportionally slow ing the bit rate, i.e ., spread spectrum .

CROSS-CORRELATION AND RADAR SIGNALS DETECTION C ross-correlation is the correlation o f a received signal with som e reference signal waveform . The best p lace to start is through the work o f P. M . W oodward on inform ation theory.24 W oodw ard’s work from 4 0 years ago is relevant to U W B signal detection and processing and introduces som e correlation d etection concep ts and objectives. W oodward points out that the signal receiver should extract wanted signal inform ation that is received with unwanted noise. T he reception problem is to elim inate as m uch unw anted noise as possib le w ithout destroying the signal-of-interest information. In addition, W oodward d iscu sses the fact that m ost reception m ethods are aim ed at m axim izing output signal to noise. N o ise

525 is view ed as the lim iting factor in sensitivity; the less noise, the better.24 Som e caution is needed b ecau se there is no general theorem w hich shows that m axim um output signal to noise ensures m axim um inform ation gain, although the concept is intuitively appealing and looks like a practical place to start. T he ideal radar receiver, from inform ation theory fundam entals, is one that takes the ech o signal y(t) = x(t) + n(t) into the receiver, with the output being the conditional probability distribu tion py(x). S om e reference texts term

p y(x) as the inverse probability, and it is described as the probability o f

presen ce o f signal x(t) g iv en the received signal y(t). Woodward shows a detailed analysis o f such a receiver where the desired ech o signal is a doublet, term ed u(t - t ). The ech o signal is therefore y(t)

= u(t - t ) + n(t). Figure 1 0 .2 0 shows the (A) signal, (B) noise, (C) received signal, (D ) correlator output o f signal and noise, and (E) nonlinearly post-processed signals in W oodward’s analysis. The delay tim e t is the desired value, related to the roundtrip ech o travel tim e. T he first step in the detection process is to form the product o f y(t) and u(t - r), follow ed by integration over the time T w h ich the received signal is available. Woodward term s this product q(r) where N0 is the m ean noise pow er per unit bandwidth, so that

q(j) = JL \ T y(t)u(t -

t

) dt

(1 0 .3 0 )

q(r) effectively reduces the bandwidth o f the noise, as seen in Figure 10.20D . The noise 0.35ITu, where Tu is the tim e duration o f the signal u(t). N e x t W oodward form s exp [#(r)] = eq(T) and shows that if the distribution o f signal delay p{r) is u niform , th e n p y(j) = K exp[^(r0)], where K is a proportionality factor. The desired output o f the ideal receiver p y(t) is a nonlinear expression o f the cross-correlation o f the received signal y(t) and the received signal u(t). In m any radar receiver applications, it is the signal q(r), or som e sim ilar form o f q(r), w hich is presented to the radar operator or to the post-detection processor. The peak o f q(r) Form ing

is now bandlim ited to approxim ately

corresponds to the correct delay value, but the radar operator or post-detection processor m ust do this interpretation step. As show n in Figure 10.20E , p y(t) = Kexp[q(r)] giv es the correct value o f t 0 with little error. N ote that the peak in py(t) is at the signal start tim e or at the leading edge o f the signal u(t - r).

It is interesting to note that the SN R never entered into this analysis, but that the operation y(t) to form q(r) and py(r) d oes, in fact, m axim ize SN R . The concep t illustrated in Figure 10.2 0 can be exten ded for the im pulse ech o response o f a target. If the ech o is com posed o f m ultiple pulses, then q(r) and exp[#(r)] can be valuable tools in determ ining the exact intrapulse tim ing in the ech o response. T he waveform q{r) is now devoid o f the exact fine grain signal features; only relative delay tim es r 7, r 2, etc. in the ech o pulse packet are directly available. Figure 10.21 shows the sam e analysis for a case o f three c lo sely spaced signals in noise. The ech o pulse was already used in converting y(t) to q(r). W oodward goes on to exam ine the thresholding effects o f p y(r) and the behavior o f p y(r) for various signal-to-noise energy ratios. W oodw ard’s approach show n in Figures 10.2 0 and 10.21 brings out the follow ing im portant points about correlation detection. 1.

Correlation is a way to detect the presence o f a given waveform in noise. A correlator integrates the received waveform against a reference waveform and produces an output larger than the average value o f the input wave and shorter in duration. N otice that in Figure 10.20 the 1 unit maximum amplitude signal produces a correlator output o f almost 5 units, and in Figure 10.21 the 3 units maximum am­ plitude wave produces a unit correlator output value o f 27. The correlator output indicates that either the desired signal is present at som e level, or that som e signal which resembles the desired signal is present. Figure 10.21 also shows that correlation can find and identify each signal event in several overlapping signals. Note the effect o f N0 in Equation 10.30 on the output.

526

y(t) the cojnbined signal + noise

q(t) the correlajtor output of u(t-to) and y(t).

Py (t) = K exp [q(t)]t the nonlinear cross correlation

Figure 10.20 Detection of noisy signals. This shows the information theory approach to detection described by Woodward.24 Gaussian noise (A) is mixed with a doublet signal (B) of interest at time t = tQ. to form the received signal y(t) (C). The integral of the y(t) and u(t - to) is the signal q(t) (D). The last signal py(t) (E) is a nonlinear expression of the cross correlation of the received and source signals displayed or sent for post-detection processing.

527

10

^0

I

1

30 J 1

40

50

60

70

1

a. Thr^e signal^ separated by a 10 and 12 interval with noise.

,20

30

40

50

60

I 1 I b. The Icorrelatof output q(j) showing peaks at beginning of each received signal.

c. The nonlinear output py(x) showing peaks at each signal start time.

Figure 10.21 Demonstration of how signal correlation and nonlinear processing can detect overlap­ ping signals with the same waveform and different amplitudes. (A) Three signals separated by a 10 and 12 interval with noise. (B) The correlator output q(r) showing peaks at beginning of each received signal. (C) Nonlinear output py(r) showing peaks at each signal start time. The correlator output provides good spatial resolution. The nonlinear output provides a high signal level for display. The saturation value of the system will limit nonlinear signal output value.

528 2.

3.

The correlator output depends on the received and the reference signal level and duration. The correlator output includes an output from noise or other signals that are present. The output level from noise or false signals may approach the same level as the output from a low level desired signal, as shown in trace q{x) in Figure 10.20C where the noise peaks approach the size o f the desired signal peak. The correlator output must be evaluated to determine if the signal is present, which is the detection process shown in Figures 10.20D and 10.21C. The process py(t) = k exp( abs[Rx(t)] for all t , so the maximum autocorrelation value is at t = 0, i.e., when the •

signals coincide in time (synchronization). If px is the mean value o f time series x(t), then [ix = [/?*(). Figure 10.26C show s the absolute values o f the FFTs of| Vi(oj)| and |V 2(co) | w hich are the sam e as |V7(a>)| = |V2(w )|. The P S D s o f | V^co)! and | V 2(co)| are the sam e as show n in Figure 10.26D . A utocorrelation rem oves any inform ation revealing w hich doublet, Vj(t) or v2(t), was received; however, the correlation w ill indicate the tim e o f occurrence. The tim e occurrence o f ea ch pu lse is random , but the pulse shape is definite — a reasonable description o f the desired U W B p u lse train. In general let v(t) describe the individual pulse, ta being the tim e o f the ath pulse in the set and Na the total num ber o f pu lses. Therefore, the full pulse set w(t) is

MO = £ *

(1 0 .3 4 )

Na v(t - 0

W ainstein and Zubakov22 perform their analysis stating that for out at t oo, then the spectrum o f v(t) is V(u)\

v(t) — 0 for t < 0 , and with v(t) dying

(1 0 .3 5 )

(1 0 .3 6 )

T h e autocorrelation function o f the process

w(t) is (1 0 .3 7 )

w here

a is found by IN

~ 2

a = — 1 where N a = m ean square num ber o f pulses a Ata = tim e interval during w hich N a pu lses occu r

(1 0 .3 8 )

U sin g Equation 1 0 .3 6 into 10.3 7 yield s

Rw(r) = * [°°V (r - r)dt f “ Z ir

J

e » 'd t*

J

(10.39)

f

Figure 10.26 Pulse packet processing by autocorrelation. (A) Given two doublet pulses v^t) and v2(t) of opposite initial slope and occurring at random intervals, as shown. (B) The FFT real and imaginary components of v^t) and v2(t). (C) The absolute value of the FFT V^a/) and V2(c«/) are the same, as shown by the difference line in the center. (D) The PSD of v^t) and v2(t) will appear identical and are identical. Given random signals v^t) and v2(t) with the same waveform, but reversed polarity, the PSD of each will be identical. Correlation of the PSDs of signals v^t) and v2(t) against a reference PSD can detect the pulse packet occurrence.

f

538 Taking the inverse transform to obtain the P SD

SW(u) g iv es (10.40)

Sw(o) = a | V(oi) | 2

T herefore, the P SD o f the autocorrelated ech o pulse train is proportional to the absolute value o f the Fourier transform o f the original pulse elem en t

v(t). I f |^(co) |2 is unique enough from the autocorre­

lation o f the received noise and signals not o f interest, then the autocorrelation technique for detection is a pow erful tool. H owever, note that the phase inform ation in p rocess.

V(cj) is lost in the autocorrelation

S ^ u ) does not have a unique solution.

PSD Correlation T he logical application o f the previous discussion o f wave Fourier transform and autocorrelation P SD eq u ivalen ce is to use P SD correlation for signal detection. T he approach is to correlate a reference P SD and received signal PSD. The practical problem here is to get the received signal Fourier transform

S(u>) and the P SD S(u>)2. The d ecision to use real-tim e P SD correlation w ill depend on the digital sam pling rates, com putational sp eed , and the radar operational requirem ents. To dem onstrate P SD correlation principles, w e used the sam e waveform and n oise conditions as the exam ple show n in Figure 1 0 .2 2 A where the received signal

r(t) = s(t) + n(t). Figure 1 0 .2 7 A

show s the P SD plots for the sam e waveform s u sed in Figure 1 0 .2 2 A . N ote that the n oise is essen tially w hite for practical purposes. Figure 10.27B shows the results o f correlating the F FT and P SD o f the reference signal with the received signal plus noise and n oise for the sam e range o f S N R values. In this ca se, the correlator output was the absolute value product o f each reference, received and noise signal F F T term s. The resulting correlator output results show n in Figure 10.27B are sim ilar to the results o f tim e dom ain correlation show n in Figure 10.22B . The correlation coefficients o f signal P S D with reference signal P S D and the reference with the noise P S D vs. SN R shown in Figure 10 .2 7 C give sim ilar results to Figure 10.24C . The ch oice betw een tim e waveform correlation and P S D correlation w ill depend on the radar m ission , need for real-tim e processing, and com putational expense.

CORRELATION WITH BARKER AND COMPLEMENTARY CODES O ne ob jective in radar perform ance is to increase the range resolution. If w e use a sine wave pulse o f duration r, then the range resolution w ill be c r/2 . Short pulse lengths m ean high output pow er for a g iv e n radiated energy, w h ich m eans practical im plem entation problem s. One approach to increasing range resolution and accuracy is to use a long waveform w h ich is a nonsinusoidal wave, or a cod ed pu lse train, w hich w ill on ly correlate w hen there is an exact coin cid en ce o f reference and received waveform . We show ed this earlier in Figure 10.23C with the autocorrelation o f the chirped waveform . Now , let u s look at the approaches using Barker c o d es and com plem entary (or G olay) cod es. T he Barker co d e only correlates at one instant and is a discrete interval version o f the chirped waveform . T he com plem entary (G olay) co d e uses two coded seq u en ces w hich have sidelobes o f the sam e size, but op p osite sign. W hen com plem entary correlations add, the result is a large value output and zero tim e sid elob es. D iscu ssio n s o f Barker and com plem entary (G olay) co d es applied to conventional radar are in S k oln ik ’s

Introduction to Radar Systems}

The d iscussions apply the cod es as pulse com pression

sch em es to conventional radars, i.e ., m odulation to a narrowband signal where the coding w ould give som e increase in range resolution over unm odulated waveform s. C oded pulse train w aveform s proposed for U W B radars would use cod e as the transmitted signal waveform , not as a m odulation o f another waveform . The potential advantages o f transmitting a Barker or com plem entary coded U W B baseband signal directly include

539

c ® o £

®

-30

-25

-20

-15

-10

-5 0 SNR (dB)

10

15

20

25

Figure 10.27 PSD correlation and SNR. (A) The PSD of the signals used in Figure 10.22: reference signal r(t), received signal plus noise s(t), and the noise signal at 4.72 dB SNR. (B) (B) Correlator output from PSD r(t) with PSD s(t) and PSD r(t) with PSD n(t) vs. SNR (dB). (C) Correlation coeffi­ cient of PSD r(t) with PSD s(t) and PSD r(t) with PSD n(t) vs. SNR (dB). The results generally agree with the Figure 10.22. Figure 10.42 shows a block diagram for computing the results.

540

Code Length 2 3 4 5 7 11 13

Code elements +-, + + +++ + -+ , + + + + + +-+ + + +--++ + + -+ --+ + + + + --+ + -+ -+

Sidelobe level, (dB) 6.0 -9.5 -12 -14 -16.9 -20.8 -22.3

B

K-

m v /v v v A + + + +

-

1

+ +

Chip

Figure 10.28 Barker code pulse compression. (A) Barker code table showing phase shift coding (0,/7) for each chip; (B) 13-element Barker code and autocorrelation; (C) Barker code generator and correlator block diagram. Barker codes are a formal way to design a low autocorrelation sidelobe waveform for correlation detection. (After Skolnik, M.I., Introduction to Radar Systems, Second Ed., McGraw-Hill, New York, 1980.)

Reducing instantaneous power levels by spreading the signal energy over a longer interval. Radar performance depends on energy transmitted and received. U sing short duration signals for fine spatial resolution means generating short duration high power pulses which creates practical hardware problems. Obtaining better tim e and spatial resolution, because the autocorrelation produces a sort interval spike at one point and provides spatial resolution equal to the “length” o f the elem ents, or chips, as shown in Figure 10.28. R eceiving noise-corrupted return signals by correlation. The output signal is proportional to the number o f elem ents in the code because o f the integration which is part o f the correlation process. The correlation process integrates a signal o f n chips o f signal strength w and x seconds duration into an autocorrelation spike o f nw amplitude and x seconds duration, as shown in Figure 10.28B. The same minimum energy must be received for signal detection, but it can be taken over a longer

541 interval. Note that the computational and implementational challenges o f correlation vs. threshold detection must be considered. Creating a signal that w ill not interfere with other equipment, especially narrowband systems. The noninterfering signal argument is a valid performance objective. Building a low probability o f interception radar system is frequently a suggested defense system objective. The coded pulse train UW B signal would be difficult to detect with conventional radar warning and electronic intelligence receivers. A s in all previous acts o f the continuing m easures-and-countermeasuressaga, this requires a combination o f security and surprise for effectiveness. The advantage will last only as long as the other side cannot build a UW B intercept receiver.

Barker Codes H arm uth’s Nonsinusoidal Wives fo r Radar and Radio Propagation describes the use o f Barker cod es for nonsinusoidal U W B sig n a ls.14 In our book , Chapter 5 , Part 3, and Chapter 8 give introductions to signal coding and correlation applications. Figure 10.28 sum m arizes the construction, waveform , generation, and reception o f Barker codes. The table in Figure 1 0 .2 8 A shows coding sequences o f 2- to 13-elem ent cod e lengths. Longer Barker cod e sequ en ces are giv en in Chapter 10 o f the Radar Handbook, 2nd ed .45 Figure 10.28B shows a signal waveform for a 13-elem en t Barker code with each cycle o f length r, with a 0 or 7r phase shifted in accordance with the cod e. T he resulting coded sequence achieves the sam e effect as the chirped waveform shown earlier in Figure 10.23C . With Barker cod es providing a sharp autocorrelation peak, there is only one point where the correlator w ill have a large output. However, there w ill be sm all tim e sid elob es as shown. T he Barker coded sequence provides a way to get the advantages o f a UW B chirp ed waveform without the attendant w ide bandwidth o f a chirped signal. The Barker cod e generator and correlator use the sam e circuit, as shown in Figure 1 0 .2 8 C .1

Complementary Codes C om plem entary cod es are another approach to reducing tim e sidelobes by using two cod es with m utually canceling sidelobes. Work done in the 1950s by W elti30 and G olay31 introduced the concep t o f c o d e pairs. C om plem entary (or Golay) codes y ield an autocorrelation function with a single spike and zero sidelobes. L et A and B be a com plem entary cod e set, each o f length L-bits, then

(AO A) + (BOB) = 2L d(t)

(1 0 .4 1 )

T he autocorrelation functions (AOA) and (BOB) each have sidelobes with m agnitudes up to 10% o f the (AOA) or (BOB) autocorrelation peak. However, w hen the com plem entary autocorrelations are su m m ed, the sidelobe levels are reduced and the base o f the autocorrelation spike is narrower resulting in better tim e resolution. To dem onstrate this, a pair o f 16-elem ent code com plem entary cod es given by G olay31 were generated and correlated. Figure 1 0 .2 9 A shows the com plem entary cod es a(t) and b(t) w ith the sequence show n where “ 1 ” is given a value o f + 1 and “0 ” is given a value o f - 1 . Each elem en t is 4 units long, givin g an L o f 64. Figure 10.29B shows the autocorrelations AO A and BO B w h ich produce low, but distinct sidelobes. Figure 10.29C shows the sum o f AO A and BO B w hich g iv e s a zero level sidelobe and m axim um value o f 2Lb(t). The com plem entary cod e is a way to establish tim e resolution m uch shorter than the duration o f the signal. Practically speaking, a radar using com plem entary code m odulated baseband signals, i.e ., transmitting a waveform as the cod e, w ould transm it the two cod es sim ultaneously and then apply each reference code to the returned signal by splitting the signal and using two correlators. T here is reason to be cautious about what can be done with com plem entary codes. In Figure 10.29 we show ed a Barker cod e using phase coding to give the 0 and 1 conditions. We applied the sam e technique o f phase coding to the com plem entary code sequ en ce o f Figure 10.29; the results are in Figure 1 0 .3 0 . U sing the phase coding does give zero tim e sidelobes; however, the m axim um peak value o f the correlator output show n in Figure 1 0 .3 0 A shows the Lb(t) function shifted to give a sm aller m axim um output value. T he tim e resolution effect is the sam e; however, the m axim um output w ill be different.

542

A

Code A: 1000110110000010

Code B: 0100000101001110

C 200

CO

*

CD ioo

+

100

Figure 10.29 Complementary code pairs. (A) Complementary 16-bit Golay code pair a(t) and b(t) with 16 elements, and each chip equals 4 for length L = 64. (B) Autocorrelation functions of AO A and BOB. (C) Plot of A o A + B OB = 2L6(f) showing sidelobe cancellation and doubling of output size. Complementary codes have sidelobes of equal size and opposite polarity which cancel each other. The sum of the autocorrelation of two complementary low time side signals produces a single correlation peak value twice the individual autocorrelation peaks.

Complementary Codies, Cross Correlation, and SNR Improvement S ischk a, N ew ton , and N azarthy’s work also provides som e insight into the issue o f return signal averaging to im prove SN R . It can be shown that each tim e the num ber o f m easurem ents is dou bled, the S N R w ill im prove by

y/2t or approxim ately 1.5 dB. O ptical tim e delay reflectom etry (O TDR)

transm its a short im pulse through an optical fiber to determ ine length and locate discontinuities. P rocessin g the reflected optical pulse may be useful in determ ining the type o f break. In typical O TDR eq uip m ent (w here G olay coding is not used ), only single im pulses are transmitted. For this case, there is an equation term ed the “O TDR m aker’s form u la.”32 T he SN R o f an O TDR m easurem ent for a fiber d efect at a range z(km ) is

543

B

-100

-100 -5 0

50

-50

0

50

100

100

Figure 10.30 Effects of waveforms on complementary codes. (A) Complementary 16-bit Golay code pair a(t) and b(t) with 16 elements, and each chip equals 4 for length L = 64. (B) Autocorrelation functions of AO A and BOB. (C) Plot of A o A + BOB = 2 L6(t) showing sidelobe cancellation and doubling of output size. This shows the code from Figure 10.29 with a positive or negative sine wave associated with each element. The result here is to keep the absolute size of the correlation summation but displaced to a smaller maximum value.

544

SMR(dB) = PMl - 2az - NEP + 1.5 Noa

(1 0 .4 2 )

w here NEP is receiver n oise equivalent pow er (dBm ) and Noct is the number o f probe signal shots, N, w h ich are averaged and expressed in octaves: Noa = log2N. Pinit is the initial value o f the backscatter pow er in dBm such that Pinit * const x PinT, and Pin is the peak input power and T is the probe pu lse width. U sin g com plem entary correlation techniques, Equation 1 0 .4 2 is m odified to form the “correlation O T D R m aker’s form u la” :32

SNR(dB) = Pinil - 2az - NEP + 1.5(Noct + L J

(1 0 .4 3 )

T he additional term Loct is the octal representation o f the com plem entary code length, L : Lnct = log2L. E quation 10.43 shows that the detected SN R is im proved by an extra 1.5 dB for each octave o f the p robe sig n a l’s code length. Equation 1 0 .4 0 also shows that integration tim e (represented by Noct) can be reduced with the addition o f the new term LflCt with a desired SN R m aintained. U sing L = 6 4 length com plem entary code pairing as show n in Figure 1 0 .2 9 , the resulting autocorrelations and sum o f autocorrelations produced the effect o f a 8(t) function w ith an am plitude o f 2L = 128. In this case, long com plem entary code pairs produce the resolution o f a short im pulse without the associated problem s o f generating an im pulse. The benefits seen in O TDR for com plem entary cod e use also apply to U W B w aveform transm ission, recep tion , and processing.

CROSS-CORRELATION Theory of Cross-Correlation N u m erous texts23 describe cross-correlation and the c lo sely related operation o f convolution. Som e key factors to consider about the effects o f the correlation operation are 1.

Cross-correlation o f two sets o f random data describes the general dependence o f one data set with respect to the other, where the time lag (x) is varied. Mathematically, cross-correlation is done over a tim e period T, and the cross-correlation function R ^ t) is defined as

Rxy(r) = lim T-+oo JL f TJ

x(t) y(t + i)dt

(1 0 .4 4 )

2.

Time domain correlation results in a time domain waveform which is significantly longer than the original signals. For example, if signal x(t) has signal features in time duration 0 < t< T h and signal y(t) has signal features in tim e duration 0 < t< T 2, the correlation or convolution o f x(t) and y(t) w ill result in a waveform with nonzero features over duration (Tj + T2).

3.

Cross-correlation, in short-hand notation, is written as

• •

R^fr) = x(t)y(t- t) and:

Cross-correlation is a real, odd function with symmetry about the ordinate R^fr) = R ^ - t). If R^(x) = 0 for all values o f t, then x(t) and y(t) are statistically independent (provided they are zero mean). |/ ^ ( x ) 2 < Rx(0)Ry(0) < 0.5[Rx(0)R/0)] K (t) Taking the Fourier transform o f R ^ t) gives the cross spectral density ^ (c o ), which is generally com plex, where 5„(co) = = 5^*((d), and | S^(co) | 2 < Sx(a )S (a ).

545 4.

Cross-correlation,

x), can also be defined in terms o f a joint probability density function. A ssum e is known, then the cross-correlation function is

x(t) an dy(t) are stationary random processes, terms xk(t) and yk(t), and the joint pd f

Rxy(r) = f ” ( y ^ p i x ^ d x . d y ,

(10.45)

(N ote that, in practice, com plete knowledge o f the joint pdf for a real-world application is unlikely, or as a minimum, difficult to determine.)

Applications for Cross-Correlation Function Measurements Time Delay Measurement C ross-correlation can provide a m easure o f tim e delay for a signal to pass through a linear system . If the linear system input is x(t) and the system output y(t), then the cross-correlation R ^ t ) w ill have a p eak value at r = tim e delay through the system . For radar applications, the transmitted radar signal is x(t), and the received ech o signal is y(t). N ote that if the transfer function is frequency-dependent, the cross correlogram (plot o f R ^ t ) v s . t ) w ill have a broad, nondistinct peak that could exist in representing the UW B radar transfer function.

Determination of Multipath M ultipath is the signal transm ission phenom ena where the transmitted signal returns to the receiver through more than one desired path. In general, the different signal transm ission paths (such as e ch oes o ff buildings, m ountains, or level ground) w ill cause y(t) to be com posed o f several multipath signals. T he cross-correlation o f x(t) with received signal y(t) w ill result in cross-correlation R ^ t) having several peaks at various values o f delay tim e r. If the expected tim e for the various undesired transm is­ sion paths can be calculated , their processing techniques can be used to tim e gate out undesired m ultipath signals.

Detection of Signals in Noise T his application is the focus o f the UW B cross-correlation detection approach. If a noise-free copy o f the desired UW B signal o f interest is available, the cross-correlation o f the pulse signal x(t) with the noise-corrupted received signal y(t) — x(t) + n(t) w ill extract the close replica o f the correlation function o f the desired U W B signal. N ote that the U W B signal can be periodic or aperiodic, w hich is an advantage o f cross-correlation function over the autocorrelation function, w hich is the m ost pow erful against period ic signals.

UWB CORRELATION DETECTION THRESHOLDING D etectio n using cross-correlation or autocorrelation techniques has distinct advantages in sensing the p resence o f a signal in noise. H owever, the issue o f processing time is critical in transient U W B processing. For exam ple, suppose the received U W B signal has N pulse elem en ts in it (N < 10), spanning a total o f Techo = 25 ns. A lso assum e that N pu lses are random ly distributed in Tech0, each with a duration o f roughly 1 ns. T he individual pulses m ust be known with high resolution, say 10 sam ples per pu lse, or an effective sam ple rate o f 10 G H z. In addition, this fine grain processing is to be perform ed continually so that no ech o pulses are m issed. The processing burden for continuous processing at these rates and the number o f integer m ultiplications per nanosecond im plies parallel digital signal processing fast enough to handle the signal, or electro-optic processing. C onsidering this hardware com plexity, other detection and post-detection processing techniques may be better options. Suppose the U W B ech o c o llectio n scenario has the ech o signals with positive SN R over the duration o f the individual pulse elem ents. I f the detector reliably sen ses the transient energy event and can queue the post-detection processor to process a selected w indow o f data, the enorm ous processing burden for continuou s autocorrelation or cross-correlation can be significantly reduced. N ote that this technique

546 rem oves the requirem ent from the detection stage to m aintain all signal features. The detection stage’s on ly purpose is to first detect the large positive AE/At event associated with the transient U W B signal, then queue the fine grain ech o signal processor. C onventional radar signals w ould not trigger the transient energy detection stages because receiver signal energy risetim es are too slow. The noise and signals not o f interest that may cause false triggers include shot noise, lightning at clo se range, and electron ic (arcing) discharges. A s discussed earlier, m inim um detectable signal levels for broadband (transient) signals have different definitions than the m inim um detectable signal for narrowband (sinu soid) signals. Figure 10.31 shows a con cep t for a short-term integrating threshold detection for U W B transient signals. T he co llected signal would be squared, then routed to a tapped delay and sum , w here the tim e sp acings betw een taps would be on the order o f 1/10 o f the duration o f the shortest transient pulse o f interest. T his

AE/At signal w ill be com pared continuous to a tim e-sm oothed version o f the original

co lle cte d signal, w here the integration tim e would be related to 2 7 ^ , w here is the duration o f the lon gest transient pulse o f interest. The integrated signal into the ( - ) input o f the com parator acts as a real-tim e, noise-riding threshold for the AE/At signal. N ote that this approach distorts the U W B transient signal, but the goal here is detection o f a g iv en length event. The detection event can be used to trigger a co llectio n o f the U W B pulse with optim al hardware bandwidth over a lim ited tim e window.

CORRELATION DETECTION SUMMARY T h is section sum m arizes the m ajor points about correlators and correlation detection. C orrelation refers to the general process o f com paring and integrating received signals with a reference signal. C orrelation is a way to detect the presence o f a signal in the presence o f noise. A utocorrelation correlates against the sam e signal. C ross-correlation com pares received signals with a known reference waveform . If we on ly need to know that a signal is present and w hen it occurs, then correlation offers som e advantages in detecting signals in noise. B elow are major points to rem em ber about correlation.

Conversion of Signals to Usable Formats T he radar receiver function is to convert an electrom agnetic signal into another usable form for display and processing. The objective o f signal detection is to pick the signal out o f a continuous background o f interfering signals and random noise. Threshold detection depends on the signal pow er being greater than the noise level. A correlation receiver w ill use integration to com pare arriving signals with a sp ecific waveform . The correlator output goes to som e nonlinear threshold detector, and the threshold detector output is the trigger for a long duration display. Signal inform ation w ill be the absence or presen ce o f the signal w ith respect to som e tim e reference, not the waveform itself.

Correlation Determines Signal Presence and Time of Occurrence C orrelation only detects the presence o f signals that w ill integrate with the reference signal to produce an output exceed in g the detection threshold. The correlator integrates the product o f reference and receiv ed signals over the duration o f the signal, w hich produces a large, short duration output signal. T he received waveform is lost in the correlation process and the detector output is a new signal indicating that the correlator output exceed ed som e threshold. The m axim um correlator output only occu rs at coin cid en ce o f the reference and received signal. The detection threshold may be set at any poin t b elow the m axim um correlator output. The correlator detection threshold setting w ill determ ine probability o f detection and false alarm, just like the threshold detector.

Correlator Outputs T he correlator output depends directly on the values o f the reference and received signal and the integration tim e (duration) as show n in Equation 1 0 .3 0 . Correlator outputs w ill fall under the follow ing general categories: 1.

M aximum correlator output results from the coincidence o f the reference waveform with a signal having the same waveform plus additive noise.

547

UWB signal

Timing control for fine grain signal collection

Figure 10.31 Transient signal detection block diagram. Generator of noise-riding threshold and estimate of AE/Af provides a means to detect transient signal, perform even counting, intrapulse time measurement, and switch control to select time window for fine grain signal analysis.

2. Less than maximum correlator outputs result from correlation of the reference waveform with a signal having a waveform that resembles the reference or has enough strength to produce an output signal approaching detection threshold. This would include the changed reflected waveform case, or interference cases. The correlator output from a waveform with a lower correlation coefficient with respect to the reference signal will be less than output for the same strength signal having the reference waveform. 3. Minimum correlator output results from the correlation of the reference waveform with noise. There will always be some correlator output from the inherent system noise. As the SNR decreases, the correlation coefficient between the signal and reference will approach the same level as the reference and noise correlation coefficient, as shown in Figure 10.22.

Correlation in Time or Frequency Domains Correlation detection can use either the received signal waveform or PSD. Waveform correlation is the most direct approach and can use analog devices for real-time correlation and detection. Power spectral density correlation would most likely be done by digital means; however, analog correlators may be possible. PSD correlation has the advantage of giving only positive outputs and is insensitive to the slope (polarity) of the waveform, which means lost signal information.

Minimum SNR for Reliable Correlation While correlation detection can find signals in noise, reliable correlation requires a minimum SNR. When the SNR level is low, then the correlation coefficient of the signal and reference approaches the same level as the correlation coefficient of the reference signal and noise, as shown in Figures 10.22

548 and 10.27. There will be some minimum SNR for reliable detection with a specified probability of false alarm.

Correlation Reference Signals The correlation reference signal waveform will be a major problem in UWB systems where distortion can occur at each stage of the signal trip from transmitter to receiver due to frequency-sensitive effects. The design of UWB reference signals for cross-correlation could be a major UWB radar receiver performance driver.

Minimum Detectable Signal There will be some minimum detectable signal for correlation. Dependable detection may be set based on the SNR level where the correlation coefficient of the signal and reference signal approaches the correlation coefficient of the reference signal and noise. For cross-correlation, the minimum detection criteria could be the signal level and waveform that produce a minimum correlation coefficient. The factors which will determine minimum detectable signal power include 1. Receiver noise from kTBw, the inherent noise due to receiver bandwidth and minimum SNR, will set the receiver performance; both signal power and waveform will affect receiver performance 2. The number of bits used in AD conversion 3. The correlator reference signal size or correlator weighting function values 4. The deviation of the received signal waveform from the reference signal; deviation may be due to path losses or interference from other signals 5. The nonlinear detector threshold value setting; the setting will drive the probability of detection and false alarm for a given signal strength 6. Reference signal synchronization, or wave signal sampling, with the received signal; any deviation from synchronization will reduce the output, as shown in Figure 10.23B; setting the nonlinear detector threshold lower will increase the probability of false alarm

Correlation Reference Signal Synchronization Synchronization of the reference and received signal will affect correlation detection. Failure of a strong received signal to be in proper phase with a reference signal will produce a no detection indication when the target should have been detected. A moving target may produce cyclic detection and no detection indications. Synchronization problems may be accepted if the radar can provide some other useful performance feature. The FIR architecture correlator is a way to eliminate synchronization problems.

Desirable Waveforms for Correlation Waveforms are important for determining time of signal occurrence. Regular waveforms, such as sine waves, will produce a series of autocorrelation peaks as shown in Figure 10.23B. The larger autocor­ relation time sidelobes could produce an off-time detection of signal arrival. An irregular waveform, as shown in Figures 10.23C and 10.28 to 10.30, has a form that only produces a large correlator output at one point. Barker codes and complementary (Golay) codes provide a formal method for finding waveforms with low time sidelobes and large output peaks.

V.

UWB RADAR RECEIVERS AND SIGNAL PROCESSING

INTRODUCTION A UWB radar receiver design will require large absolute bandwidths to preserve signal waveforms and high speed circuits for signal detection. This section is about potential directions in receiver design and signal processing. The next three sections are about practical receiver limitations that accompany large absolute bandwidths, photonic receiver concepts, and advanced signal processing concepts.

549

PRACTICAL UWB RECEIVER DESIGN AND LIMITATIONS The radar receiver’s function is to convert a short duration, wideband, pseudorandom signal into a signal for display, range and bearing computation, target presence, reflector characteristics, or identification. All UWB receivers use the same principles as narrowband receivers; however, the increased bandwidth associated with shorter signal durations, nonsinusoidal signal waveforms, and signal waveform preservation presents practical design and fabrication problems.

Bandwidth, Dissipation, and Distortion Earlier, in the section Practical UWB Receiver Design and Limitations, we discussed the requirements for receiver bandwidth based on signal risetime (rr). The rule of thumb for bandwidth, given in Equation 10.7, was f upper 3dB = 0.35 !tr. There are UWB RF amplifiers advertised with 0.1 to 18 GHz bandwidths and noise factors of about + 4.3 dB which could accommodate risetimes on the order of 19 x 10'12 s (19 ps).33 The low-end specification of 0.1 GHz implies a minimum risetime of 700 ps. Practically speaking, we would probably want to use a Barker or complementary coded waveform or impulse sequence described in the previous section on correlation detection. The 10 to 700 ps risetime sets the waveform slope limits that can be handled.

Bandwidth, Noise, and Minimum Detectable Signal Levels The 17.9-GHz bandwidth may sound extreme to many readers used to dealing with 5 to 10 MHz bandwidths in communications or radar applications. Now let us see what this implies for system noise from N = kTBFn so that Noise (dB) = -204 (dB watt/second) + B (dB second-1) + Fn(dB) which comes out about -97.147 + Fn dBw (-67.147 + Fn dBm) for the UWB amplifier described previously. For comparison consider a 1-MHz amplifier where the noise will be -204 + 60 + Fn = -138 + Fn dBw (or -108 + Fn dBm) which is the signal power level usually encountered in narrow­ band systems. Assuming that we are going to use correlation, then we could get a minimum signal level as much as -9 8 + SNR + Fn dBw, or -113 + Fn dBw for reliable correlation based on our previous estimates of -15 dB for the limit for reliable correlation of noisy signals. Careful design of the waveform and knowledge of the noise PSD might provide some improvements in detection. Probability of detection and false alarm requirements will drive the SNR and minimum detectable signal computation. The radar application remains the driving receiver design consideration.

PHOTONIC UWB RECEIVERS AND PROCESSING Photonic (combined optical and electronic devices) UWB receiver concepts may be an approach overcoming component dissipative bandwidth limitations of electronic signal processing and AD conversion components. The photonic receiver uses fiberoptic, semiconductor laser amplifier, fiberoptic modulator and photodiodes characteristics for signal processing, and preserving with fiberoptics (photonics) that were not available earlier. Fiberoptic links are now in use for carrying received microwave signals from remote aircraft antennas to electronics systems inside an aircraft and for long period delay lines for microwave signals. Optoelectronic mm-wave sources now make it possible to generate microwave signals optically and transmit them on fiber to antennas for conversion to RF signals.35 Once an electronic signal is modulated on a fiberoptical signal, it can be preserved in long fiberoptic delay lines and becomes a “narrowband signal” in the wider bandwidth of optical signals.

Electro-Optical Modulation The dissipative character of circuits for transient and short wavelength signals will be a major practical problem in UWB receiver design. The parasitic capacitance in circuits will look like a short circuit to gigahertz signal components. In the past, the practical solution to microwave signal dissipation was heterodyning the signal from a higher frequency modulated signal to a lower IF with the same modulation. Because the narrowband modulation (amplitude for AM and frequency shift for FM) was

550 the useful characteristic of the signal, and the modulation bandwidth was still considerably smaller than the IF, this worked well and became a standard receiver feature. The photonic signal processing concept is to convert the UWB signal into an amplitude modulated fiberoptical signal, which will be a small relative bandwidth signal relative to the frequency of the laser light beam. Advances in fiberoptic components and amplifiers may make the electro-optical UWB receiver a practical concept. Figure 10.32 shows the basic concept of a UWB electro-optical receiver. The approach is to make electrical and optical signal processing concepts work together. The photonic receiver design concepts include 1. Electrical signals interact in summingjunctions and linear or nonlinear elements; however, light waves will pass through each other with no reaction. 2. Fiberoptical signals are intensity-modulated from a laser light signal, i.e., the electronic signal will ride on the laser light signal and must have a smaller negative value than the amplitude of the laser light signal. In other words, the electronic signal cannot cut off the laser signal. 3. Laser light signals can be amplified directly using semiconductor laser amplifiers or erbium-doped fiber amplifiers. 4. Each time a fiberoptical signal is split, one half the power goes in each new direction. 5. Fiberoptical signals modulated with UWB signals can travel long distances in constrained paths with little dissipation and with a bandwidth greater than the UWB signal instantaneous bandwidth. This property means that we can preserve a signal in a long delay line for comparison, correlation, or other processing with signals that arrive later. Tapping or splitting the delay line lets us add signals arriving at different times for functions such as correlation and noise cancellation. 6. An 8-in. length of optical fiber will provide about a 1-ns delay, which makes precision multiple nanosecond delays easy to accomplish. The actual delay depends on the light wavelength and fiberoptical material. 7. At some point the processed amplitude modulated optical signal must be reconverted to an electronic signal. The characteristics of the electronic-to-optical and optical-to-electronic conversion components will determine the overall system performance.

Photonic Signal Processing Concepts The following sections illustrate some ways to use fiberoptic delay lines in signal processing of short duration, random UWB signals.

Signal Repetition with Delay Lines Figure 10.33 shows an electro-optical concept for turning a single transient UWB signal into a regularly repeated signal for some predetermined number of repetitions. The repeated impulse train output can be used for interleaved AD conversion as shown in Figure 10.11. The impulse-to-pulse train convertor is a series of fiberoptic delay line modules. Each delay line module uses a semiconductor laser amplifier to split and boost the laser signal at the output of each delay element. The design can use as many delay lines as needed to repeat the signal. The monopulse signal shown in Figure 10.33 can be any desired signal format of any length as long as the delay interval exceeds the signal length. The delay line must be longer than the signal duration, but shorter than the expected interval between the arrival of the next signal. For interleaved AD conversion, the delay must consider the AD conversion rate. The advantage of the photonic design is preservation of the waveform. The disadvan­ tage is the difficulty in changing the characteristics of the impulse-to-pulse train convertor.

Photonic FIR and Transversal Filters for Correlation Figure 10.34 shows a photonic UWB signal correlator for intensity modulated fiberoptic signals. The received impulse (or coded pulse train signal) s(t) is converted into a fiberoptic signal in the fiberoptic modulator. The correlator uses optical splitters and photonic delay elements to continuously time sample and correlate the received signal. The use of different lengths of optical fiber permits either regular or variable interval sampling of the waveform at intervals down to 100 ps for a 0.8-in. delay line. Optical delay lines permit much shorter sampling intervals than acoustic delay lines. The coefficients w l,...,w n

551

E9t)* >

J

*

Received UWB signal.

>

t

Fiber optical signal with UWB signal. Figure 10.32 Photonic UWB receiver concept for preserving UWB signal waveforms as fiberoptic signals. The received signal comes through a wideband amplifier which drives a fiberoptic (FO) intensity modulator. The continuous optical signal from a semiconductor laser diode is modulated with the received UWB signal. Once the signal is in fiberoptical format there will be little dissipation and the signal can be stored in delay lines for further processing, AD conversion, etc. A photodiode converts the signal back into electronic form.

are set either based on the reference waveform for cross-correlation or from sampling the transmitted waveform and adjusting the coefficients for autocorrelation.2628 For an advanced correlation concept, consider the correlator of Figure 10.34 with continuous adjustment of the coefficients based on some performance criteria as a neural network correlation detector.

ADVANCED SIGNAL PROCESSING AND RECEPTION PROCESSING Significant progress in new signal processing techniques is occurring, along with advances in hardware technology. This section summarizes some of the analysis and advanced signal processing concepts available for UWB signal processing and detection. As with any new technology area, the list is partial and the latest progress is found in the technical journals.

Fourier Transform and Analysis The Fourier transform provides a set of coefficients that are the amplitudes of a set of long sine and cosine waves which model a signal. Narrowband and wideband signals are typically long in duration, containing thousands of cycles of the component sinusoids, and they lend themselves to discrete Fourier transform (DFT) or FFT use. However, the UWB transient signal has the opposite characteristics: extremely short time duration, nonstationary, and not easily defined by the sum of weighted infinitely long duration sinusoids. The point to remember is that the Fourier transform gives you a mathematical model of the portion of a sampled signal. If the signal varies randomly, then the Fourier coefficients will also vary. Fourier series and PSD can be used to describe a UWB waveform if it meets the radar

552

t Single impulse signal from antenna Signal type Electrical Fiber optical,

Repeated impulse train for signal detection and processing

Figure 10.33 An electro-optical UWB receiver signal repetition concept. The design objective is to convert the electromagnetic signal into a fiberoptic signal as soon as possible to prevent dissipative losses in electronic circuits. The fiberoptic signal can be delayed and turned into a train of repeating signals for processing and detection. Potential optical or photonic processing possibilities include optical analog techniques to perform correlation or autocorrelation, instantaneous frequency spectrum analysis, higher order signal analysis, etc.

functional objective. If Fourier series cannot meet the functional objective, then there are other approaches. The design issue is appropriate technology and analytical approach for the particular radar function, and that will depend on the radar user and designer’s judgment. Chapter 3, Part 1, has a discussion of Fourier transforms.

Singularity Expansion Method (SEM) Signal Analysis The SEM is a way to model short duration transient waveforms as a series of sine wave amplitudes, frequencies, and damping coefficients. Chapter 8 covers SEM radar signal processing as applied to characterizing reflected waves from targets. The advantage of SEM representation is that it can describe a transient UWB signal compactly and with fewer numbers than a Fourier transform. Singularities and damping coefficients are a natural way to describe transient signals. Research using model targets and impulse signals indicates that the SEM model of a reflected signal is reasonably aspect free and unique to a given target and pulse excitation waveform.36'39

Higher Order Spectra Signal Processing Higher Order Spectra Processing Concept Power spectrum estimation provides a good method for looking at signals and has been the basis for much practical signal processing. However, power spectrum estimation emphasizes the frequency content, but suppresses the phases relations between signal components. In other words, two different signals may have the same power spectrum and different waveforms. While the power spectrum is an adequate description of a Gaussian signal, there are situations that require looking beyond the power spectrum and look at deviations from Gaussianity and the presence of phase relations. Going beyond power spectrum to recover phase relations gets in to high order spectrum signal processing.

553

JL Optical Signal Splitter

Z>

v 1 At

z> wO

2 At

3 At

n At

D

Z>

w2

W1

Optical Signal Summer

—

f

tim e

Optical Signal Output

Figure 10.34 Photonic signal correlator. Using fiberoptic delay lines permits sampling signals at small intervals. The coefficients (attenuators) w0, ..., w„can be fixed attenuators or made of electroni­ cally controlled, variable transmissibility materials. The optical output goes to a nonlinear detector (optical or electronic) for detection.

The higher order spectra, or polyspectra, defined in terms of cumulants, or higher order statistics, do contain signal phase relations which can distinguish one waveform from another and find nonlineari­ ties which aid signal processing and identification. The motivations for using higher order signal processing include techniques to: (1) suppress additive colored Gaussian noise; (2) identify or recon­ struct nonminimum phase signals; (3) find information due to deviations from Gaussianity; and (4) detect, characterize, and identify nonlinear properties in signals and systems. Nikias and Mendel provide a good introduction and overview in IEEE Signal Processing Magazine.40

554

Bispectral Processing Bispectral processing is the first step in higher order signal processing. The advantage of bispectral processing is that it has some immunity to noise and can bring out waveform characteristics that Fourier analysis cannot. Bispectral processing determines nonlinear terms associated with the waveform model. The conceptual leap here is that most signal processing approaches linearize responses for simplicity and assume that nonlinear terms are either not important or are negligible for most design and many processing purposes. Determining nonlinear terms invokes a considerable computational cost and is not readily suited to many real-time processes.41 Chapter 11 presents the theory behind bispectral processing and results from anechoic chamber tests which show how bispectral processing can extract distinct signatures from impulse radar returns.

Wavelets The wavelet transform uses short duration, typically orthogonal, basis functions to describe a signal. The Fourier transform produces coefficients which completely define a signal, where the coefficients are the weighting values for the sinusoidal basis functions. The wavelet transform also produces coefficients which completely define the signal, but these coefficients are the weighting values for short duration basis functions. Intuitively, expressing a transient signal as a weighted sum of short basis functions instead of infinitely long sinusoids is logical and could improve signal processing accuracy or reduce the number of coefficients to express the time domain signal. Wavelet transforms are well suited to transient signal analysis because the basis function in the transform is not a sinusoid, but a short, finite duration “wavelet.” Wavelets are a new subject and the journal literature is the best source of information. Some sources include Ten Lectures on Wavelets by Ingrid Daubechies,42 An Introduction to Wivelets, 4 3 and Wivelets: A Tutorial in Theory and Applications44 both by Charles K. Chui.

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555 15. Robbins, K.W., Short Baseband Receiver, U.S. Patent 3,662,316, May 9, 1972. 16. Ross, G.F. Base-Band Pulse Object Sensor System, U.S. Patent 3,772,697, November 13, 1973. 17. Harmuth, H.F., Radar equation for nonsinusoidal waves, IEEE Trans. Electromagn. Compat., Vol. 31, No. 2, 138-147, 1989. 18. The IEEE Standard Dictionary o f Electrical and Electronic Terms, 4th Ed., IEEE, New York, 1988, 462. 19. IEEE Standard fo r Measurement o f Impulse Strength and Impulse Bandwidth, ANSI/IEEE Std. 376-1975 (reaffirmed 1981), IEEE, New York, 1981, 5. 20. O’Laughlin, J.P and Copeland, R.P, Subsection A2: Hind 2 (H2), Gas Switched High Power Impulse Transmitter Technology at Phillips Laboratory, PLYWSES, May 9, 1991. 21. McGraw-Hill Dictionary o f Science and Technology 4th ed., McGraw-Hill, New York, 1989, 439. 22. Wainstein, L.A. and Zubakov, VD., Extraction o f Signals From Noise, (translated from Russian by Richard A. Silverman), Prentice-Hall, Englewood Cliffs, NY, 1962, 61-69. 23. Bendat, J.S. and Piersol, A.G., Random Data: Analysis and Measurement Procedures, McGraw-Hill, New York, 1953, 56-98, 344-380. 24. Woodward, PM., Probability and Information Theory, with Applications to Radar, McGraw-Hill, New York, 1971, 62-79. 25. Sischka, F.S., Newton, A., and Nazarthy, M., Complementary correlation optical time domain reflectometry, Hewlett-Packard J., December, 14-21, 1988. 26. Moslehi, B. and Goodman, J.W., Novel amplified fiber-optic recirculating delay line processor, J. Lightwave Tech., 10, No. 8, 1142-1147, 1992. 27. Psaltis, D. and Athale, R.A., High accuracy computation with linear analog optical systems: a critical study, Appl. Opt., 25, 3071-3077, 1986. 28. Ghosh A. and Paparao, P., High speed matrix preprocessing on analog optical associative processors, Opt. Eng., 28, 354-363, 1989. 29. Lee, S.H., Optical analog solutions of partial differential and integral equations, Opt. Eng., 24,41-47, 1985. 30. Welti, G.R., Quaternary codes for pulsed radar, IRE Trans. Inf. Theory, June, 400-408, 1960. 31. Golay, M.J.E., Complementary series, IRE Trans. Inf Theory, April, 82-87, 1961. 32. Newton, S.A., A New Technique in Optical Time Reflectometry, Hewlett Packard Publication No. 5952-9641, 1987. 33. MITEQ, Amplifiers, AFS Series, MITEQ Corporation, Hauppauge, NX 1992. 34. Plant, D.V, Scott, D.C., and Fetterman, H.R., Optoelectronic mm-Wave Sources, Microwave J., April, 62-72, 1993. 35. Nelson, G., RF Optical Links, IEEE AES Mag., July, 12-15, 1992. 36. Kennaugh, E.M. and Moffatt, D.L., Transient and impulse response approximations, Proc. IEEE, Vol. 53, No. 8, 893-901, 1965. 37. Bennet, C.L. and Ross, G.F., Time-domain electromagnetics and its applications, Proc. IEEE, Vol. 66, No. 3, 299-318, 1978. 38. Moffatt, D.L. and Mains, R.K., Detection and discrimination of radar targets, IEEE Trans. Antennas Propag., Vol. AP-23, No. 3, 358-367, 1975. 39. Baum, C.E., Rothwell, E.J., Chen, K.M., and Nyquist, D.P., The singularity expansion method and its application to target identification, Proc. IEEE, Vol. 79, No. 19, 1481-1492, 1991. 40. Nikias, C.L. and Mendel, J.M., Signal processing with higher-order spectra, IEEE Signal Proc. Mag., Vol. 10, No. 3, 10-37, 1993. 41. Jouny, 1.1, and Walton, E.K., Target identification using bispectral analysis of ultra-wideband radar data, Ultra-Wideband Radar: Proceedings o f the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991, 405-416. 42. Daubechies, I., Ten Lectures on Wavelets, Society of Industrial and Applied Mathematics, Philadel­ phia, PA, 1992. 43. Chui, C.K., An Introduction to Wavelets, Academic Press, New York, 1992. 44. Chui, C.K., Wavelets: A Tutorial in Theory and Applications, Academic Press, New York, 1992. 45. Farnett, F.C. and Stevens, G.H., Pulse compression radar, in Radar Handbook, Skolnick, M.I., Ed., McGraw-Hill, New York, 1990, 10-26.

556

ADDITIONAL REFERENCES Nonsinusoidal Radar and Transient Signal Processing Basseville, M. and Benvenisk, A ., Sequential detection of abrupt changes in spectral characteristics of digital signals, IEEE Trans. Inf. Theory, Vol. IT-29, No. 5, 709-724, 1983. Harmuth, H.F., Nonsinusoidal Wives for Radar and Radio Communication, Academic Press, New York, 1981, chapters 4, 6. Harmuth, H.F., Maxwell’s equations, steady state equations of electron theory and causality, Proc. RADARCON VO, Adelaide, Australia, April 18-20, 1990. Heinish, M.J. and Wilson, G.R., Detection of non-gaussian signals in non-gaussian noise using bispec­ trum, IEEE Trans. Acoust. Speech Signal Process., Vol. 38, No. 7, 1126-1131, 1990. Hussain, M.G., Principles of high-resolution radar based on nonsinusoidal waves. Part III. Radar target reflectivity model, IEEE Trans. Electromag. Compat., Vol. 32, No. 2, 144-152, 1990. Mohamed, N.J., Resolution function of nonsinusoidal radar signals I: range-velocity resolution with rectangular pulses, IEEE Trans. Electromag. Compat., Vol. 32, No. 2, 153-160, 1990. Papoulis, A ., Probability, Random \hriables, and Stochastic Processes, McGraw-Hill, New York, 1965, chapters 9, 10. Riegger, S. and Wiesbeck, W., Wide-band polarimetry and complex radar cross section signatures, Proc. IEEE, Vol. 77, No. 5, 649-658, 1989. Wong, K.M. and Jin, Q., Estimation of the time-varying frequency of a signal: the Cramer-Rao bound and the application of Wigner distribution, IEEE Trans. Acoust., Speech Signal Process., Vol. 38, No. 3, 519-535, 1990. Young, J.D., Approximate image reconstruction from transient signature, Acoustic, Electromagnetic and Elastic Wive Scattering-Focus on the T-Matrix Approach, Varadan, V.K. and Varadan, V.V., Eds., Int. Symp., June 25-27, 1979, Pergamon Press, New York, 1980, 631-653.

Receiver Design Concepts Barrett, T.W., Energy transfer and propagation and the dielectrics of materials: transient vs. steady state effects, Ultra-Wideband Radar: Proceedings of the First Los Alamos Symposium, Noel, B., Ed., CRC Press, Boca Raton, FL, 1991, 1-19. Cohen, J.D., A proposed microwave and millimeter-wave spectrum analyzer, IEEE Trans. Microwave Theory Tech., Vol. 37, No. 6, 1054-1057, 1989. Dudley, B., Basic Phenomena of Electronics, Electronics Engineers Handbook, 3rd ed., Fink, D. and Christiansen, D., Ed., McGraw-Hill, New York , 1988, 1-15. Hewlett Packard, HP Test and Measurement Catalog 1993, Hewlett-Packard Company, Santa Clara, CA, 1992. Light Motif: an optical computer stores its program in space-time, Sci. Am., April, 116, 1993. Murthy, U.M.S. and Rao, V.M., Analysis reveals vulnerable areas in EW receivers, Microwaves RF, August, 107-113, 1990. Sampaio-Neto, R. and Scholtz, R.A., Precorrelation filter design for spread-spectrum code tracking in interference, IEEE J. Selected Areas Commun., Vol. SAC-3, No. 5, 662-675, 1985. Scherer, D ., Designing sensors to read peak power of pulsed waveforms, Microwaves RF, February, 113-117, 1990. Skolnik, M., Ed., Radar Handbook, Second ed., McGraw-Hill, New York, 1989. Urkowitz, H., Energy detection of unknown deterministic signals, Proc. IEEE, Vol. 55, No. 4, 523-531, 1967.

Electro-Optical Techniques and Hardware Alferness, R.C., Waveguide electro-optic modulators, IEEE Trans. Microwave Theory Tech., Vol. MTT-30, No. 8, 1121-1137, 1982. Fleisher-Reumann, M. and Sischka, F., A high speed optical time domain reflectometer with improved dynamic range, Hewlett-Packard J., Vol. 39, No. 6, 6-29, 1988.

557 Jungerman, R. et al., Highspeed optical modulator for lightwave instrument applications, HewlettPackard Lightwave Symposium and Exhibition 1990, 112-120. Jungerman, R.L. and McQuale, D.J., Development of an optical modulator for a high speed lightwave component analyzer, Hewlett-Packard J., Vol. 42, No. 1, 1991. Kano, E , Fukuda, M., Sata, K., and Oe, K., High-speed intensity modulation of 1.5 ^m DBR lasers with wavelength tuning, IEEE J ’ Quantum Optics, Vol. 26, No. 8, 1340-1345, 1990. Kamiya, T. et al., A new scheme of resolution improved electro-optic sampling, Ultrafast Phenomena

VI, Proceedings o f the 6th International Conference, Mt. Hiei, Kyoyto, Japan, July 12-15, 1988, Springer-Verlag, New York, 1988, 192-194. Kobayashi, T., Guo, G.-G., Morimoto, A., Sueta, T., and Cho, Y., Novel method of waveform evaluation of ultrashort optical pulses, Ultrafast Phenomena IV Proceedings o f the Fourth Inter­ national Confcrrence, Monterey CA, June 11-15, 1984, Springer-Verlag, New York, 1984, 93-95. Nees, J., Williamson, S., and Mourou, G., Greater than 100 GHz traveling wave modulator, Ultrafast

Phenomena VI, Proceedings o f the 6th International Conference, Mt. Hiei, Kyoyto, Japan, July 12-15, 1988, Springer-Verlag, New York, 1988, 205-208. Rivoir, J. and Pless, W., Data processing in the correlating optical time-domain reflectometer, HewlettPackard J., Vol. 39, No. 6, 29-34, 1989. Valdmanis, J.A., Mourous, G., and Gabel, C.W.,Electricaltransientsampling system with two picosecond resolution, Picosecond Phenomena II, Proceedings of the Third International Conference on Picosecond Phenomena, Eisenthal, K.B., Hochstrasser, R.M., Kaiser, W., and Lauberau, A., Eds., Springer-Verlag, Berlin, 1982, 101-102. Wong, R.W., Hernday, P.R., and Hawkins, D.R., High-speed lightwave component analysis to 2 GHz, Hewlett-Packard J., Vol. 42, No. 1, 6-13, 1991.

APPENDIX 10A: NARROWBAND RECEIVER SENSITIVITY TO UWB SIGNALS INTRODUCTION AND OBJECTIVES Sensitivity, or the minimum detectable signal for a given signal to noise ratio (SNR) expressed in either volts or watts, is the receiver designer’s bottom line. The receiver bandwidth and noise figure determine a receiver’s sensitivity and minimum detectable signal. This appendix is about narrowband receiver sensitivity to transient wideband (impulse) signals. It can be shown theoretically and demonstrated practically that narrowband systems will detect strong UWB impulse signals and that the detectable impulse strength (volts-seconds) increases as the band­ width decreases. The receiver sensitivity analysis uses the input and output relationships of a linear two-port network and introduces the concepts of device noise bandwidth, impulse bandwidth, and average noise factor (or noise figure) to estimate detectable impulse strength for a given bandwidth. The analysis starts with the noise signal response of a two-port network without internal noise sources. A noise bandwidth proportional to the network 3-dB bandwidth and an impulse bandwidth proportional to the 6-dB bandwidth are introduced to simplify the analysis. The network’s internal noise is described by the network’s average noise figure. Minimum detectable impulse signal for narrowband and broadband cases is determined based on the network response and average noise figure concepts. Pay particular attention to the units used for continuous signals (volts, watts) and impulse signals (volts-seconds, volt/hertz, and watts/hertz2).

RECEIVER NOISE BANDWIDTH Narrowband Noise Model Our analysis starts with the linear two-port network which has network characteristics and impulse response shown in Figure 10.35. The network has no internal noise sources for analytical simplicity. Later we introduce an average noise figure to account for internal noise sources. Start by considering

558

Al0(f) Input noise power density

Output noise power density

The linear 2-port network used for developing inpulse sensitivity. Internally generated noise is not considered here and is expressed as an average noise figure Favg when considering the sensitivity to impulse signals.

Network a5 (t)

/

V:(t)

he

/1y ' / x » i'c

"1 c

A:(f)

1 A 0(f)

Vk «(f)

H

0

r -fC

Network amplitude and phase response characteristics

Impulse input signal

fr

Output response signals

Figure 10.35 Two-port linear network used for characterizing noise, noise bandwidth, and impulse bandwidth. The effects of internal noise are included in an average noise factor applied in sensitivity analyses.

an input noise signal N ffl and output signal N0(f) noise power densities in watts/hertz. For a linear network with an amplitude response function H(f), the input and output noise power densities are related by:

N0(f) = | H(f) | 2 Nt(f )

(watts/hertz)

(10.46)

The average power (in watts) is determined by 7V0

= Jf “

oo

N0( f ) d f = [ “ \H ( f) \2N ,(f)d f J -o o

(watts)

(10.47)

If the input noise is white where Niff) = NiP and the amplitude response \Hff)\2 is an even function (symmetric about / = 0), then N0 = IN, j j " \H (f) 12d f

(watts)

(10.48)

559 Equation 10.47 or 10.48 should be used to retain the frequency-dependent behavior of the input and output noise densities when required. However, another simplifying assumption of an equivalent noise bandwidth is often made, as shown in Figure 10.36. To remove the integral in Equation 10.48, replace | H(f) | by a rectangular amplitude response Hc with constant midband BN and centered at frequency f c . Noise bandwidth BN is therefore

BN = - L f ” \H (f)\2 d f Hc J0

(hertz)

(10.49)

combining Equations 10.49 and 10.50 gives

N0 = 2Ni H 1c Bn

(watts)

(10.50)

The noise bandwidth BN in a linear two-port network is well approximated by the 3-dB or half-power bandwidth. Figure 10.37 shows some relationships between 3-dB bandwidth and noise bandwidth of tuned circuits from Lawson and Uhlenbeck.4

Impulse Bandwidth Let the signal presented to the two-port network input now be defined as an impulse with strength a, (volts/hertz or volts-second) v.(f) = afi(t)

(volts)

Type of circuit

Noise Bandwidth 3 dB Bandwidth

Single-tuned

1.57

Double- tuned

1.11

Triple-tuned

1.05

Quadruple-tuned

1.02

Quintuple-tuned

1.01

Figure 10.36 Graphical interpretation of noise and impulse bandwidth. To simplify impulse response analysis use equivalent bandwidths. The noise bandwidth Bn is proportional to the 3-dB bandwidth. The impulse bandwidth B, is proportional to the 6-dB bandwidth. (A) Noise bandwidth is shown for a double tuned circuits; (B) impulse bandwidth is shown for a single tuned circuit; and (C) the table shows the equivalent noise bandwidth proportions for different types of circuits. (Adapted from Lawson, J.L. and Uhlenbeck, G.E., Threshold Signals, McGraw Hill, New York, 1950.)

560

* H (f) \ 1 -

H(f) =

1 for (fc-b/2) < f < (f +b/2) 1 for(-fc -b/2) < f< £ -f +b/2)

-fc 0(1) = 27raf over bandwidth segments, width b

Figure 10.37 Example transfer function H(f) for bandpass, linear phase.

The spectral density of Vfi) is constant for all frequencies:

Aiff) = aif for all frequencies, -oo < / < oo

(10.51)

The output spectral density is related to At(f) by

A0ff) = Atff) m \ = 0'\H(f)\

(10.52)

The output signal v0(t) for impulse input a,6 (if) is

v„(0 = f “

m

(10.53)

e * '* d f

The network transfer function H(f) can be expressed as a magnitude and phase H(f) — |H(f) | exp(/^(jf)). It can be shown that \H(f)\ is an even function, and phase is an odd function, i.e. = W - f) \

*ff)

(10.54)

Then using these properties

v0(r) = 2 f \H(F)\ cos [lirft + *(f)] d f

(volts)

(10.55)

Our goal in this analysis is to understand the relationship of the transfer function’s phase and bandwidth to output signal v0(t). Let H(f) = 1 over a bandwidth b as shown in Figure 10.37. Assume the impulse

561 input vft) = atb(t) into the network, where is in volts/hertz (volts-second). For the linear phase, constant amplitude transfer function, Hff), the output signal can be found from:

vo(0 = 2jff,|tf(/")|cos[27T/(r - a ) W (10.56)

f c+b!2 = 2oi

f

cos[2tt/(/ - a)]df

fM After integration, the following function-product can be used: 2 cos(jfc) sin (b/2) = sin(jfc + b/2) - sin{jfc - b/2)

(10.57)

The final solution for v jt) has the form:

2a.sin[7r(/ - ot)b1 vo(t) = — !-------------------- c o s [2 7 rf(t - a)] ir(t - a) JcK = £ ,0(Ocos[2tt/c(/ - a)]

(10.58)

(volts)

The first term E0(t) describes the envelope of the impulse response. The second term represents the oscillatory signal within the envelope, with frequency f c, as shown in Figure 10.38. Examining Equations 10.56 and 10.58 shows some important results, even for the simplest transfer function, H(f). First transfer function phase a is seen as a time delay in output response. Second, as the bandwidth b increases, the output response envelope shape becomes more narrow. Third, the amplitude of the envelope is directly related to bandwidth. These features of impulse response are keys to understanding the difference between narrowband (steady-state) sensitivity and broadband (impulse) sensitivity. Figure 10.38 shows the effects of bandwidth on output duration and amplitude. Returning to Equation 10.56, let tm = time at maximum voltage for v0(t) = iV^I^.Then the maximum response

1^01 * = 2a,Jo"|//(0|cos[27r/(/m - a)\df (volts)

(10-59)

occurs when cosine term is 1, or tm = a . Maximum input response is now directly related to \H(f) | :

=2 °o exists. This quantity is zero if f(t) is deterministic transient, (2) above, and has a Fourier transform. It is finite if f(t) is either deterministic periodic, (1) above, or continuing messages in noise, (3) above. If it is finite, then the integrated Fourier transform exists:

( 12.2)

Now the quadratic variation of the integrated Fourier transform is

(12.3) and according to the theorem of quadratic variation

613 The Cauchy Problem & Fourier Transformation

a+b+c+d +e

=P

A+B+C+D+E

Figure 12.1 The signal shown at the top, P, can be Fourier decomposed into a sum of standing waves -oo < t < + 00. A representative sample of those waves is shown in A through D. The addition resulting in augmentation resulting in the pulse and cancellation before and after the pulse may be said to constitute the Fourier description. If, however, an attempt is made to empirically generate those Fourier standing waves as signals, the initial conditions, or wave onset and offset, introduce other frequency components (shaded area in the following figures). The Fourier description states that a + b + c + d + e = P, the pulse shown, but these waves have no onset or offset, i.e., no envelope. However, summation of the real-life waves generated is A + B + C + D + E * P. Therefore, the reallife, or empirically generated, description involves a never-ending regression to another Fourier description of another pulse, generation, etc... If a medium is a nonlinear transducer in the intensitydependent sense, this nonlinearity might be probed by a series of pulses: P + 2P + 3P + 4P + 5P + 6P + ..., but each of these has the same Fourier description, which is insufficient to characterize the nonlinearity. Fourier analysis presumes that instrumentation can be turned on and turned off at t = -00 and t = oo, respectively. The Cauchy problem is to address the real-life situation of replicating the wave packet under restricted conditions of time of signal onset/offset and signal amplitude, i.e., the signal rise- and falltime and rates of change in amplitude. Whereas Fourier analysis approaches the problem of packet description from the viewpoint of harmonic frequency analysis, the Cauchy analysis approaches the same problem from the viewpoint of instantaneous frequency analysis. The Fourier approach is descriptive but the Cauchy analysis is realistic.

where O u (co) is the power density spectrum (Appendix 12A). This theorem states that the quadratic variation of the integrated Fourier transform of a function is equal to the mean square value of the function. The theorem holds for signals of type (1), (2), and (3), above, i.e., both deterministic and random.

614

The important point for our purposes is as follows. If the derivative of G(co) exists for all values of co, then f(t) must be a transient function, signal (2) above, with a mean value of zero and cannot be a continuing random function, signal (3) above, whose mean square value is finite and which is the signal o f our interest. If the derivative of G(co) becomes infinite for some discrete values of co, then f(t) is a function with a discrete power density spectrum. However, the power density spectrum of signal (3) is continuous by definition. Therefore, G(co) does not have a derivative for any value of co for the signals (3) of interest, i.e., does not have a Fourier transform. The conclusion is, then, that if a continuing random function has a continuous power density spectrum, it has an integrated Fourier transform, but not a Fourier transform. In other words, it is impossible to express a continuing function with a continuous power density spectrum as

(12.5) Although the continuous random variable has no Fourier transform, it does have an autocorrelation:

|G(to + e) - G(to - e)|2 coscoxdco,

£-»0

(12.6)

i.e., the autocorrelation function of a continuing function or random variable, f(t), is the cosine transform of the power density spectrum of f(t). The implications of this analysis for hardware systems are that any TD receiver-processor should not have frequency decomposition on its front end (because a local time analysis is required and local information is lost with a direct Fourier global analysis) and the return signal should not be detected by threshold methods (this discards the information in the envelope fine structure) but should be represented in an analog fashion by its power density spectrum. The definition of autocorrelation, power density spectrum and energy spectrum for periodic, aperiodic and random signals is described further in Appendix 12B. We can also consider the relation of Fourier series representation and the Fourier transform. A periodic function f(t) is expressed in a Fourier series as

(12.7A) n=l

= a.0/2 + ( l / 2 ) ^ ( a n -ib „ )e x p [in (0 1t] + ( l / 2 ) ^ ( a n + i b n)exp[-in

A / A t 2 ,A t,
A t2

Whole Body Response

Individual Scatterers

Figure 12.15 A comparison of the return echo signals obtained in the situation described in Figure 12.14. The information from individual scatterers is projected on the whole body response, but due to sampling restrictions and lacking the sequency information contained in the interpulse arrivals t1f *2>*3. *4> *5 its extraction is not possible.

FREQUENCY DOMAIN PHASOR AND TIME DOMAIN COMPLEX ANALYTIC SIGNALS In the case of FD narrowband sinusoidal signals of long duration, At, any point of discontinuity, 8t, e.g., at the beginning and end of the signal, is small compared with the duration At. For such signals, local time methods relate the global nature of the signal, i.e., its envelope, to its spectrum. This procedure is followed in Fourier analysis, as well as in Taylor series and other expansions. In contrast, in the case of TD pulse or wave packet signals, any point of discontinuity, 8t, is large or of the same order compared with the signal duration At. Therefore TD signals must be treated differently using real-time methods to preserve the local nature, or fine structure, of the signal. Periodic or aperiodic (transient), i.e., FD approaches to TD signal processing, cannot be used due to the fact that for TD signals St « At. The implication of this is that TD signals and their derivatives do not possess discontinuities, indicating their analytical signal properties. Analytic signals are not deterministic or causal but are continuous and require local or real-time methods for characterization. Whereas the FD signal is characterized by its harmonic frequency, the TD signal is characterized by its instantaneous frequency. That being the case, what is the required referent for a TD receiver corresponding to the frequency (phasor) referent of FD receivers? To answer this, the definitions of Chapter 7 are repeated here again. With co as the radian frequency of a wave and a the phasor, or neper, frequency defining the envelope shape, a complex signal is s« —

+ ico()

(12.39)

and an analytic signal (also known as the pre-envelope function) is29 C(t) = SCO+ iS(t) = exp[s0] = A (t)exp[ico0t + t] = exp[i(00]e x p [c 0]

(12.40)

S(t) = exp[a0]cosco()t

(12.41)

S(t) = exp[a0]sinco0t

(12.42)

with real part

and imaginary part

636

S (t) is the Hilbert transform of S(t)

S(t) = (l/7 t)J^ S (t)/(t-t)d t

S(t) =

- ( l/7t)J

S ( T )(t-t)d x

Therefore, S (t) can be considered the output of a quadrature filter with input S(t). The quadrature filter has an impulse response: h(t) = l/(7Ct) and system function H(ico) = -i, 0) > 0; i, co < 0 The envelope of a wave packet (also known as the absolute value of the pre-envelope) is defined as (12.43) and the carrier, average, or midfrequency is defined as

(12.44)

For example, suppose S(t) = A(t)cosco0t, then using the Hilbert transform the analytic signal is £(t) = A(t)cosco0t + iA(t)sinco0t and the envelope of £(t) is env
) + in (x)exp[-icox]dx.

(12.95)

Random Functions (Probabilistic Conception of Signals and Noise) The autocorrelation of the random function fj(t) (i.e., is a member function of an ensemble) is

4>u ( t) = l i m ( l / 2 T ) j \ ( t ) f ,( t + x)dt.

(12.96)

The power density spectrum of the random function f^t) is

(J)jJ(x) exp[-icox] dx.

(12.97)

Why power density spectrum? Let x of the autocorrelation be zero. Then, if fj(t) is a voltage or current and if there is a 1 Q of resistance assumed, the autocorrelation for a random function (with x = 0, i.e, the autocorrelation is the summed mean square of fj(t)), is the mean power taken by the load. The autocorrelation-power spectrum pairs for random functions are

^ n (x)= [ d)u (co)exp[icox]dco

(12.98)

O n (co) = (l/2ic)J (|)1|(x)exp[-icox]dT.

(12.99)

J— oo

APPENDIX 12B: THE GAUSSIAN APPROXIMATION: HETERODYNE VS. HOMODYNE RECEPTION The backscattered response from any target is composed of individual scattering elements. Convention­ ally, these individual elements are not seen because the backscattered response, even without multiple detections, is a time average:

J 'to+T dt A(t),

(12.100)

‘0 where t0 is the time at which the measurement is initiated and T is the time taken for surface currents to be established on the target. The average is meaningful only if T is large compared to the target elementary scatterers. The ideal FD sounding of a target would be when A is averaged over infinite time: M(j+T A (t0,T )= lim l/T J dt A(t). T_*°° •'to

(12.101)

652

Then this time average is independent of ^ and the target is said to be stationary. With independence from 1$ the average can be expressed as

< A > = lim l/T [ Tdt A(t),

(12.102)

Jo

where the “< > “ notation indicates an average. The autocorrelation is defined as:

= lim 1/T fTdt A(t)A(t + x),

(12.103)

where A(0) is the amplitude at zero time, and the spectral density (or power spectrum), I(co), of a time correlation function is

(12.104)

where A* is the complex conjugate of A. The autocorrelation and the power density spectrum are related as Fourier transform pairs:

(12.105)

In RF mixing methods, no “filter” is inserted between the scattering target and the receiver. In the homodyne (or self-beat) method, only the scattered RF signal is processed by the receiver, while in the heterodyne method an LO (sometimes the original emitted signal) is mixed with the scattered RF signal. If the autocorrelation function for the electric field, E, is defined as (12.106) where B is a proportionality constant, then the two scattered field autocorrelation functions can be defined for heterodyne and homodyne detection: I /t) =

(12.107)

I2(t) = .

(12.108)

Now, the scattering volume, V, can be subdivided into subregions of volume which are small compared to the wavelength of the incident radiation. Then the scattered field, E, can be regarded as a superposition of fields from each of the elementary scatters, so that (12.109) n

where E(n) is the scattered subfield from the nth subregion or elementary scatterer. As targets move, E(n) fluctuates. If the subfields are sufficiently large, then motions of the elementary scatterers in one subregion are independent of those in another and E can be regarded as a sum of independent random variables (E(1), E(2),...)* In this case, the central limit theorem implies that E, which is itself a random variable, must be

653

distributed according to a Gaussian distribution. A Gaussian distribution is completely characterized by its first and second moments and it follows that all higher moments of this distribution are related to the first two moments. Therefore, the homodyne correlation function, I2(t), which is a fourth moment of the distribution, is related to the heterodyne correlation function, I^t),which is the second moment, by the equation l 2(t) = |l,(0)|2 + |l1(t)|2.

(12.110)

This result, i.e., the Gaussian approximation, applies only if the scattered volume can be divided into a large number of statistically independent regions, which is the case when conventional radar signals are used. However, if the scattered volume is small, or made up of a number of separate individual scattered volumes, then the central limit theorem and the Gaussian approximation does not apply. When the Gaussian approximation no longer applies, the homodyne correlation function I2(t) contains more information than the heterodyne correlation function I^t). In the case of the heterodyne method, the correlation function is - B .

(12.111)

If (1) the amplitude of the LO is much greater than the amplitude of the scattered field; (2) fluctuations of the LO field are negligible; and (3) the LO field and the scattered field are statistically independent — then this equation yields 16 terms, 10 of which are zero, 3 are DC, and 1 is negligible. The remaining 2 terms give - B[lL02 + 2 I LOR eI,(t)],

(12.112)

where IL0= is the intensity of the local oscillator and Relj(t) is the real part of I,(t). We now examine those conditions when the Gaussian approximation does not apply. Define a quantity having the following properties: bj(t) = 1, if j e V; = 0, if j g V,

(12.113)

where bj(t) indicates whether or not a returned wave packet or impulse j is in V at time t. In the case of a short duration impulse interacting with a target, there will be a series of impulses returning from the elementary scatterers on the target. The function bj(t) will describe this series and the total return from the target is N

N(t) = £ b j(t),

(12.114)

j=l

where N(t) is the number of pulses in the return at time t. If q is a wave vector of the incident surface wave packet, then the homodyne function I2(t) and the heterodyne function Ij(t) are proportional to F2(q,t) = and

(12.115)

F,(q,t) = ,

(12.116)

where N

v(q-t)= X 'exp[iqrjW]j=l

(12.117)

6 54

is the center position of wave packet j at time t and the prime denotes that the sum is only over pulses in the volume of scattered radiation, V. Another way of writing \|/(q,t) is ij(t)

N

'Hq>0 = ] ^ b J(t)exp[iq • rj(t)].

(12.118)

j=l

Now, the only elementary scatterers which contribute to the heterodyne function Fj(q,t) are those which are being irradiated by V, i.e., those for which bj(t) = 1. The bj(0)bj(t) product is initally 1 and changes to 0 when thewavefrontof V moves on to other scatterers. Therefore, the time scale for the variation of bj(0)bj(t)is given by the time it takes for the irradiating wave packetto interact with the surface of the target of length L. A wave packet moves over the target length L on the time scale t = L2/D,

(12.119)

where D is the target surface current diffusion time. x is therefore the characteristic time of bj(0)bj(t). If the heterodyne function is written N

F,(q.t) =

b j (0)b j (t) exp[iq •

r,(0)}]>,

(12.120)

j=l

then the quantity [iq* {r^t) - rj(0)}] deviates substantially from 1 only for times such that the interval rj(t) - rj(0) becomes comparable to the length q_1. For a surface current this is Tq = (q2D)->.

(12.121)

x/xq = (qL)2.

(12.122)

Comparing the two time scales, we have

For a 1-ns incident wave packet, q « 1/2 foot, and for a target of 10 feet, x/xq = 25. For a 1-ms incident wave packet, x/xqis 1012 times 25. Therefore, for the long duration FD,1-ms wave packet bj(0)bj(t) varies on a much slower scale than exp[iq-{rj(t) - rj(0)}], but for a 1 ns short duration TD wave packet, bj(0)bj(t) varies almost commensurately with exp[iq{rj(t) - rj(0)}]. In the former case, it is permissible to set bj(0)bj(t) equal to bj(0)bj(0), but in the latter case it is not. Another way to write the heterodyne correlation function is F,(q,t) = exp[-q2Dt] = exp[-t/'Cq],

(12.123)

indicating that the heterodyne correlation function is an exponentially decaying function of time with time constant xq. On the other hand, the homodyne correlation function is

N

F 2(q>‘) =
.

(12.124)

j,k,l.m=l

As N

^ j* k = l

< bj(°)bk(t)> = < N ( N - 1 ) >

(12.125)

655

and N

X < b j ( 0 ) b k(t)> = < N (N - 1 )>

(12.126)

j* k = l

the homodyne correlation function is F2(q,t) = + |F(q,t)|2.

(12.127)

Now the number of separate wave packets in the return from the target can be expressed as N(t) = + 5N (t),

(12.128)

where 5N(t) is the deviation of the number of particles from the average number ( = 0). Therefore, the homodyne correlation function becomes F2(q,t) = < N > 2 jl + |F (q ,t)f} + .

(12.129)

The first term is the Gaussian approximation; the second, is an extra term deviating from that approxi­ mation. This second term depends on the separations occurring between the number of individual wave packets in the scattered return from the target. These separations occur on a time scale x which characterizes the time required for the currents set up by a wave packet to traverse the target. The Gaussian term decays on a time scale xq, which characterizes the time required for a wave packet to traverse the distance q 1. As xq x, F2(q,t) decays in two stages. Thus the homodyne function can be defined as follows with respect to the duration, t, of the incident wave packet on the target: F2(q,t) = 22 + for t = 0, i.e., for a Dirac impulse; F2(q,t) = 2 + for xq t x, i.e., for a wave packet of spatial length less than the length of the target, but longer than that of individual scatterers on the target; F2(q,t) = 2 for t > x, i.e., for a wave packet of spatial length greater than the length of the target. The heterodyne correlation function, F^q.t), on the other hand, does not include the term in all three instances. Therefore, when the Gausisian approximation does not apply, i.e., in the first two instances, the heterodyne correlation function is not equivalent to the homodyne correlation function and contains less information than the homodyne function.

APPENDIX 12C: BOUNDARY DIFFRACTION A situation used for studying diffraction involves two edges or screens combined to form a slit or aperture. The diffractions can be categorized into primary diffractions and secondary diffractions formed from re­ reflections delayed in time between the two edges. Besides diffraction influences, there is, first, a precursor signal, which penetrates the aperture within a time short compared with diffraction components and is composed only of the undiffracted incident field. The conventional approach to the aperture problem is to use a Kirchhoff integral equation which is based on a linear operator approach and obscures the fine structure in return scattering. This approach only permits the sum of boundary waves, which is adequate for combinations of monochromatic fields, i.e., for FD signals. With this FD approach, the precursor is not described by the Kirchhoff integral and is assumed not to exist. The second and third solutions of the aperture problem correspond to (1) the classical solution of geometric diffraction theory which is a wave decreasing in amplitude as l/V x ; and (2) a wave that diminishes as 1/x. These two solutions are the far (Frauenhofer) and the near (Fresnel) zone components. In the far zone, partial field components interfere; in the near zone, partial field components can be resolved, for a TD signal with a finite duration At. In FD monochromatic wave diffraction theory, the boundary of the Fresnel and Frauenhofer zones is given by (2a)2/A. = 1

(12.130)

6 56

where a is one half the aperture size. However, on the axis of the tube of rays through the aperture (for y = 0), the nearfield-farfield boundary is (12.131) Thus the boundary of the nearfield, which is formed by the interaction of boundary waves with large path differences, and the farfield, which is formed by the interaction of the same boundary waves with small or asymptotic path differences, is a function of the pulse length A,eff = cAt (where At is pulse duration). Primary, secondary, and higher diffraction components are studied on a conducting strip with edges A and B. The primary diffraction components produce echos spaced in time as a function of the orientation of the target. The secondary diffraction component is due to the scattering of wave A by edge B, and vice versa, and produces echos further spaced in time. Tertiary diffraction is due to waves traversing again before being scattered, and, again, the spacing in time is lengthened. The physical description of these component diffractions is similar to those components having passed through successive frequency filters. In order to detect the dimension and the orientation of a target, it is necessary to distinguish the separate diffraction components. These components are separated in time in the case of TD sounding, but are overlapping in the case of FD sounding and not accessible. Thus TD sounding provides a capability not present with FD sounding.

INDEX A Abelian commutation relation, 432 Absolute bandwidth, 239 Absorption, 337, 356, 357, 359 ACHILLES I, 303 Active antenna, 164 Active dipole, 168 Acoustic directed energy pulse train (ADEPT), 412 ADEPT, see Acoustic directed energy pulse train AD, see Analog-to-digital conversion rates Aerospace systems, resonance region of, 303 Affine connection, 428 Air breakdown, 361 Air breakdown, frequency scaling for, 356, 357 Ambiguity, 45 function, 610, 622, 623, 372 space, 621 Analog correlator, 521, 530, 534 Analog-to-digital (AD) conversion rates, 494, 501, 548 Analytic signal, 625 Anomalous dispersion, 368 Antenna capacitance, 300, 311 conductors, 303 conventional use for UWB, 35 current, 299, 300, 307 as differentiator, 33 energy patterns, 265 field, 146 gain, 506 impedance, 304, 312 loading, 311 losses, 506 parameters, 300 path loss between isotropic, 506 pattern, 8, 216, 273 radiation, 317 resistive loading of, 307 resonances, 299-301 as transducer, 33 transient and broadband electromagnetic response of, 299 UWB beam formation, 35 Aperiodic signals, 649 Aperiodic variables, 609, 612 Aperture formation, 317 Array(s), 195 antenna, 7, 146 element, 200 factor, 224, 261, 263, 272, 285 gain, 283 length, 263

parameters, frequency dependence of, 197 patterns, 216, 227 response, 282 of switches, 317 Aspect-independent target discrimination, 484 ATHAMAS II, 303, 314 Atmosphere, 325 deterministic part of, 339 losses, 494 temperature, 327 Atmosphere, propagation through, 361 Attenuation control in switches, 31 due to moisture content, 26 Autocorrelation, 218, 520, 533, 540 function, 50, 278, 530, 535, 536 PSD, 538 specific properties, 535 time sidelobes, 548 triple, 584 Avalanche rate, 350 transistor, 512 Average noise factor, 557, 563 Azimuth antenna patterns, 256

B Background noise level, 508 Backlund transformation, 408 Backscatter RCS, 460 Balun, 181 Bandlimiting, 509 Bandwidth, 303, 322, 492 higher effective, 501 imaging, 20 instantaneous, 373, 505 operational, 501 receiver design for, 501 signals, proportional, 1 wide relative, 505 Barker codes, 218, 521, 541, 548 Baseband pulse, 470 receiver, 512 signals, 541 BASS, see Bulk avalanche semiconductor switches Beamforming, 205, 261 Beamwidth to half-power (HPBW), 158, 159 for mono- and polycyclic waveforms, 38 for UWB waveforms, 39 Beer’s law, 383, 386

658 Biconical antenna, 171, 307-309, 316 Biconical radiator, 302 Binary pseudorandom sequences, 587 Binary-valued basis functions, 14 Bispectral analysis, 9, 587 Bispectral processing, 5, 554, 579, 583 Bispectrum, 581, 584 Bistatic cross section, 460 Bistatic RCS, 459 Bistatic system, 151 Black-box system identification, 580 Bloch equations, optical, 391 Blockage, 207 Blumlein line pulse generator construction of, 292 design of, 288 Boundary diffraction, 655 Bound charges, scattering by, 334 Bounded wave simulators, 301 Bound systems, 326, 330, 333 Breakdown -electric-field strength, 316 process, 352 Brillouin precursor, 396, 470 Brittingham’s waves, 411 Broadband antenna, 303 dipoles, history of, 164 sensitivity, 561, 570 signal case, 518 Bulk avalanche semiconductor switches (BASS), 646 Butler matrix, 205

C Capacitance parameter, 311 Capacitive discharge circuits, 112 Capacitive generator, 300, 307, 312 Capacitor circuit, peaking, 304 Carrier frequency, 239 Cauchy initial conditions, 401 Cauchy-initial value, 413 Cauchy problem, 609, 610 Center frequency, 40, 322 Channel, ergodic, 373 Characteristic impedance, 307, 309 Charging, 306 Chirp modulation, 4 Chirp radar, 120 Circular arrays, 203 Circular biconic pulse-generator/wave launching systems, 317 Classical area theorem, 387 Classical soliton, 367 Class one coupling coefficients, 477 Class two coupling coefficients, 477

Closed-loop sensor, 147, 255, 257 Clutter suppression, 13, 20, 445 Clutter-type materials (CTM), 579, 581 Coherence bandwidth, 345, 347, 348, 361 Coherent oscillator (COHO), 494, 616 COHO, see Coherent oscillator Collecting area, 151, 187 Collision rates, 338, 349, 350 Communications, technological advances in, 326 Complementary code, 446, 449, 521, 538, 541, 548 Complex frequency, 320 Complex natural frequencies, 473 Complex natural resonances, 322 Conductivities, 338, 339 Conical dipole antenna, radiated field from, 314 Conical systems, 317 Conical TEM wave launcher, 317 Conical transmission, 306, 319 Constant aperture antenna, 176 area feed, 208, 210 collecting areas, 213 gain feed, 208, 209 Constitutive relation, 338, 339 Continuous wave (CW), 337 Convolution integrals, 469 Cooperative effects, in materials, 365 Correlatable energy, 146, 216, 226 Correlating receiver, 216 Correlation, 216, 498 coefficient, 521, 572 complementary, 544 detection, 9, 146, 147, 498, 506, 509, 534, 548, 572 analysis, classical, 534 of signal by, 8 system, 521 synchronization and, 530 function, 521 minimum detectable signal for, 548 receiver, 500, 521, 546 reference signal, 530, 548 time sidelobes, 530 Correlator(s), 521 architecture, 528 output, 498, 524, 528, 530, 531, 546 performance, 530 receiver, 278 reference signal, 548 weighting function, 548 Coupling ambiguities, 14 coefficient, 475 in narrowband arrays, 198 in UWB arrays, 198 Crisp zero degree pulse, 365, 367 Cross-correlation, 524, 542, 545

659 autocorrelation and, 533 detection, 520, 521 function, 544, 545 theory of, 544 Cross-polarization coefficients, 461 Crystal radio set, 509 CTM, see Clutter-type materials Current density, 125, 129-131, 141 CW, see Continuous wave Cylindrical reflector, 213 Cylindrical systems, 317

D Damping coefficients, 552 Debye dielectric theory, 374 Debye model, 365 Decay time, 299 Delay line, 550 fiberoptic, 549 signal repetition with, 550 Delay value, 535 Detectability factor, 516 Detection, 493 bandwidth, 504 circuit, 564 impulse strength for, 9 minimum strength for detection, 524 probability of, 497, 510, 512-514, 534, 546 range, 13, 492 receiver probability of, 494 sensitivities, 502 system, 564 threshold, 495, 497, 528, 533, 546 unambiguous, 530 Detector deployment of space-based, 326 performance, 499 Device impulse bandwidth, 572 DFT, see Discrete Fourier transform Dielectric gas enclosure, 306 Dielectric oil tank, 304 Dielectric properties, Dissado-Hill model of, 365 Dielectric relaxation, 132 Dielectric tensor, 339, 345 Diffraction effects, 507 Digital synthesis, 123 Digital waveform, 534 Digitizer sampling rate, 314 Dipole antenna, 164, 311 Dipole moments, 301, 318 Dipole polarization, 386 Dirac-delta function, 283 Direct radiating systems, 287-324 basic UWB transmitter design and experiments, 287-299 basic impulse radiation considerations, 293-296

description of transverse electromagnetic transmitter, 288-293 experimental results, 296 practical UWB generators and radiators, 287-288 design and analysis of example of NEMP radiating antenna, 299-324 antenna concepts for UWB radar, 317-322 bounded or guided wave simulators, 301 hybrid simulators, 301-302 radiating NEMP simulators, 302-317 radiating simulators, 301 Direct sequence modulated systems, 4 Discharge interval, 7 Discrete Fourier transform (DFT), 580 Dispersion, 187, 333, 335, 340, 341, 346, 359 Dispersion, anomalous, 399 Dissado-Hill model, 365 Dissado-Hill theory of dielectrics, 376 Dissipative elements, 493 Distributed circuits, 110 Distributed reactances, 146 Distributed source, 316, 317 Distributed switch, for launching spherical waves, 316 Doppler, 645 frequency, 497 measurement, 14 processing, 443, 451 shift, 5, 443 Duality, 621

E Early time behavior, of antennas, 319 Echo area, 457 Echo pulse train, 538 EDEPT, see Electromagnetic directed energy pulse train EE, see Electrical engineering Effective aperture, 459 Effective duration, 265, 268, 273 Effective echo area, 457 Effective radiated power (ERP), 337 Effective receive aperture, 459 EFIE, see Electric field integral equation Eigen-Prony method, 477 Electric dipole antenna, 307, 314 Electric dipole moment, 318 Electric field integral equation (EFIE), 478 Electrical boresight, 152 Electrical breakdown, 306 Electrical engineering (EE), 75 Electromagnetic (EM) compatibility (EMC), 6 fields, 314 interference (EMI), 6, 29 propagation, 8, 316 pulse (EMP), 13, 301, 483

660 simulators, 301, 316 testing, 314 response, 299 waves, launching transient, 316 Electromagnetic directed energy pulse train (EDEPT), 411 Electromagnetic field, symmetries in, 409 Electromagnetic missile concepts, 366 Electromagnetic missile(s), 367, 411, 413 Electronic polarization, 386 Electronic support measures (ESM), 147, 187 Electronic warfare, 6 Elementary scatterers, 631, 653 Element spacing, 227 EM, see Electromagnetic EMC, see Electromagnetic compatibility EMI, see Electromagnetic interference EMP, see Electromagnetic pulse Energy bandwidth, 2 correlatable, 146-147 coupling, 506 pattern, 265, 273, 279 spectral density, 278 storage systems, 7 transfer phenomenon, 8 Energy transfer velocity, 396 Energy transport, velocity of, 399 Envelope effects, 365 Envelope function, 507 E-pulse, see Extinction pulse Equivalent transmission line, 309 Ergodicity concept of, 609, 624, 629 definition, 623 transformation, 627-628 ERP, see Effective radiated power Error function, 248, 269 ESM, see Electronic support measures EUV, see Extreme ultraviolet radiation External noise, 26 Extinction-pulse (E-pulse), 484 Extreme ultraviolet (EUV) radiation, 327

F False alarm performance, 494 probability of, 146, 495, 498, 510, 512-415, 520, 534, 546 Farfield, 299 regions, 222 risetime, 317 waveform, 299 Fast Fourier transform (FFT), 13, 580 FD, see Frequency domain

Feed network, 200 Ferrite-absorbing material, 255 FFT, see Fast Fourier transform Fiberoptic delay lines, 530, 549 modulator, 549 signals, 550 Field strengths, 243 Finite impulse response (FIR) filter, 521 architecture correlator, 548 filter correlator, 530 FIR, see Finite impulse response filter First-order kernel, 585, 591 Flared transmission line radiator, design of, 288 Fluctuation-dissipation theorem, 378 Fluid equations, Flux lines, 247, 248 Focus wave mode(s) (FWM), 367, 411 Foliage penetration, 21 detection of target in, 320 Forward scattering, 460 Fourier analysis, 9, 580 Fourier domain, transionospheric signal in, 340 Fourier transform, 148, 468, 538, 544, 551, 614,615 algorithm, 71 description, 611 coefficients produced by, 554 of original pulse element, 538 properties of, Dirac-delta function, 65 Free charges, 326, 333, 340 Frequency analysis, instantaneous, 613 bandwidth, 274, 348 content, 338, 359, 361 -dependent path effects, 494 domain (FD), 609, 615, 639 array beamforming, 277 detection, 528 receiver processor design, 615 return signal, 320 hopping, 4, 26, 438 instantaneous, 373, 396, 625 range, 147 resonance, 20 response, 282 scaling laws, 326, 349, 356 spectrum sharing, 4, 6 Free-space mode, 302 Frozen wave generation, 125 Full width at half maximum (FWHM), 290 Fuzzball, 464 FWHM, see Full width at half maximum FWM, see Focus wave mode

661 G Gain limit on, 202, 211 of receive antenna, 151 of transmit antenna, 151 Gas-filled tubes, 118 Gas switches, 306 Gaussian approximation, 609, 610, 626, 651, 653 distribution, 513 noise, 497, 498 pulse, 248, 265, 479 Gaussons, 431 Generator capacitance, 300 Geometrical optics (GO), 333, 334, 346, 352, 359, 466 Geometrical theory of diffraction (GTD), 466 Geophysical surveying, 5 Global frequency decomposition, 645 Global time event, 645 GO, see Geometrical optics Golay codes, 521, 548 Grating lobes, 198, 229 Ground penetration, radar, 32 reflection, 302 Group therapy, 366 Group velocity, 396, 399 GTD, see Geometrical theory of diffraction Gudermannian function, 409

H Half-power bandwidth, 559 Half-power beamwidth, 264, 276 Hall conductivity, 339 Hardware distortions, 506 Harmonic frequency analysis, 645 Hertzian electric dipole, 240, 242, 243 Heterodyne receivers, vs. homodyne receivers, 609, 626 HF, see High frequency High frequency (HF), 326, 462 High order kernels, 585 signal processing (HOSP), 579, 587 spectral (HOS) measurements, 580 High power microwave, 151 High resolution radar, 435 Hilbert transform, 373 Homodyne receivers, 512 Homodyne UWB receiver, 508 Homogeneous broadening, 386 HOS, see High order spectral measurements HOSP, see High-order signal processing HPBW, see Beamwidth to half-power Hybrid simulators, 301

Hybrid simulators, 301 Hydrodynamic equations, 326, 349, 350, 353, 354

I IC, see Intercluster (IC) exchanges IF, see Intermediate frequency IFM, see Instantaneous frequency measurement Impedance, 307 characteristic, 307, 309 loading, 300, 301, 307 matching, 121 Impulse bandwidth, 516, 518, 557, 559, 568, 572 echo response, 525 function, 469 -to-impulse train converter, 550 -like transient field, 317 -like waveform, noise bandwidth of, 571 propagation, 8 radar, 317, 579 radiation, 146 response, 148, 282, 283, 472, 521, 561, 562 signal, 2, 4, 146, 501, 508, 512, 516, 518, 572 output of conventional receivers, 516 radiation, 147 strength, 9 -type nonsinusoidal waves, 261 waves, 147 Index of refraction, 340, 345 Information theory, 525 Inhomogeneous broadening, 386 Inhomogeneous broadening, of medium, 365 Input noise, 498 wave launcher, 301 Instantaneous frequency measurement (IFM), 520 Intercept signal, probability of, 4 Intercluster (IC) exchanges, 382 Interference analysis, 520, 572 Intermediate frequency (IF), 493, 495, 508 Intermediate time behavior, of antennas, 319 International Reference Ionosphere (IRI), 341 Intrapulse timing, 525 Inverse scattering transform (1ST), 406, 408, 409 Inverse transform, 538 Ionic polarization, 386 Ionization, 351, 353, 361 Ionosphere, 326, 327 chirped dispersion introduced by, 343 coherence bandwidth of, 347 deterministic, 34, 346 impulse response of, 341 modifications, technological advances in, 326 profiles, 329

662 sounding, technological advances in, 326 total electron content for, 341 transfer function, 340 IRI, see International Reference Ionosphere Isolated dipole, unidirectional pattern for, 166 1ST, see Inverse scattering transform

J Jungle cover, detection of target in, 320

K KdV, see Korteweg-de Vries equation Kernel analysis, 9, 579, 585, 588 functions, 580 Kill-pulse (K-pulse), 484 Kirchhoff-Huygens diffraction integral, 373 Kirchhoff integral equation, 655 Korteweg-de Vries (KdV) equation, 407, 428 Kronig-Kramers transform, 376 K-pulse, see Kill-pulse Kuwait University, 255

L Laplace transform, 309, 468 Large-current radiator, 240, 244, 255, 256, 283, 412 Large relative bandwidth technology, 437, 438, 439 Largest linear dimension, 436 Large target signature, 439 Laser light signals, 550 LASS, see Light-activated semiconductor switches Late time behavior, of antennas, 319 Lenses, 317, 318 Light-activated semiconductor switches (LASS), 646 Light, squeezed states of, 369 Linear array, 265, 268 Linear-regression algorithm (LRA), 273 Linear time invariance (LTI), 11 LIP, see Low probability of intercept LO, see Local oscillator Loaded dipole, 164 Localized wave transmission, 42 Local oscillator (LO), 626, 627, 645 Lock-on process, 141 Loop antenna, 192, 242, 243 Lorentz gauge, 235 Low frequency radiation, 318 Low observable target, 458 Low probability of intercept (LPI), 6, 7, 9 LRA, see Linear-regression algorithm LTI, see Linear time invariance Lumped element equivalent circuit model, 304 Lunneberg lens, 646

M Macroscopic scattering, 333 Magnetic dipole moments, 318 field integral equation (MFTE), 478 flux, 245, 247 lines of force, 247, 248 switching, 119 vector potential, 233 Mainlobe, 446 Manley-Rowe relations, 426 Marchand balun, 190 Marx banks, 115 Marx generator, 304, 306 Marx pulsers, 302, 303 Match bandwidth, 163 Matched filter, 221, 278, 495, 507, 521 Match to transmitter/receiver, 181 Maxwell’s equations, 146, 232, 238, 326, 336, 349, 354, 359, 361 Median frequency, 148 Method of moments (MOM) code, 479 MFIE, see Magnetic field integral equation Minimum detectable signal, 518, 557, 564-566 levels, 549 power, 517, 548 mKdV, see Modified Korteweg-de Vries equation Model prediction, 593 Mode splitting, 359 Modified Korteweg-de Vries (mKdV) equation, 407 Modified power spectrum (MPS), 412 Modulation, 110, 492 MOM, see Method of moments code Monocone, resistively loaded, 314 Monopole antenna, 194 Monostatic radar equation, 459 Moving target improvement, 5 Moving target indicator (MTI), 40 MPS, see Modified power spectrum MTI, see Moving target indicator Multipath signals, 545

N Narrowband, 2 equipment, 6 planar array, 195 pulse, 497, 507 radar systems, 502 receivers, 492, 494, 516 requirement, 34 sensitivity, 561, 570 signals, 494 sinusoid, 497 spatial resolution in, 5

663 superheterodyne receiver, 495 systems, 505 threshold detection, 510 Natural resonance entire (NRE) function, 479 Natural response, 478 Navigation, technological advances in, 326 Nearfield/farfield distance, 153 Negative after-pulse, 28 NEMP, see Nuclear electromagnetic pulse Noise autocorrelation, 535 bandwidth, 516, 518, 557, 559, 562, 571 energy, 500 factor, 562 figure, 557 probability density function, 497 suppression, 437 Nominal frequency bandwidth, 268 Non-Abelian commutation relation, 433 Nonergodic transformations, 373 Nonlinear area theorem, 392 Nonlinearity, 583, 585 characterization, 593 device, 520 effects, 32 scattering, 468, 585 signal processing, 40 threshold detection, 509 Nonlinear Schrodinger equation, 408, 428, 433 Nonsinusoidal pulse trains, 513 signals, 146, 257, 541 waves, 241, 538 Nonstationary signals, 609, 623 Nonstationary wave, 373 Nonuniform resistive loading, 311 Normalized retarded time, 309 Normalized slope pattern, 273 Normalized waveform, 312 Notch antenna, 183 NRE, see Natural resonance entire function Nuclear electromagnetic pulse (NEMP), 299 pulse, 300 simulators, 302

o Object detection, 5 responses, 320 Obscured environments, detection of target in, 320 Off-boresight performance, 160 One-way pattern, 216 Optical delay lines, 550 gain, 137, 139

region, 462, 464 time delay reflectometry (OTDR), 542 Orientational polarization, 386 Oscillogram, 256 OTDR, see Optical time delay reflectometry Output pulse, risetime of, 306 signal, 509, 521, 540 switch, 304, 306, 316 wave launcher, 301

P Parabolic equation, 346 reflector, 207, 317 surface, 318 wave equation, 347 Parallel conductivity, 339 Parametric amplification, 426 Parasitic capacitance, 549 Paraxial approximation, 413 Passive loading, 164 Passive target identification, 4 Path loss, frequency-dependent, 506 Pattern multiplication, 225, 283 Peak-amplitude pattern, 272, 273 Peaking capacitor circuit, 304 Peaking capacitor, 304, 306 Peak-power pattern, 272, 273 Pedersen conductivity, 339 Percentage bandwidth, 239 Performance predicting and modeling, 609-656 boundary diffraction, 655-656 comparison analysis of frequency domain and time domain signals, 634-639 frequency domain phasor and time domain complex analytic signals, 635-637 signal-to-noise ratios, 639 system noise, 638-639 Gaussian approximation, 651-655 periodic, aperiodic, and random signals, 649-651 aperiodic functions, 650-651 periodic functions, 649-650 practical example of time domain radar system and analysis, 646-647 radar performance prediction principles, 630-634 energy radiated and incident on target, 630 minimum received energy, 632-634 target-reflecting characteristics, 630-632 rules for time domain radar performance prediction, 645-646 theoretical background for time domain signal processing, 610-630 Cauchy problem, 610-612 concept of ergodicity, 629-630

664 concept of stationary and nonstationary signals, 623-625 duality of frequency domain and time domain receiver processors, 620-623 heterodyne vs. homodyne receivers and Gaussian approximation, 626-627 periodic, aperiodic, and random variables, 612-615 philosophy of frequency domain receiver processor design, 615-616 philosophy of time-domain receiver processor design, 616-620 time domain receiver-processors gain concept, 627-629 time domain radar performance prediction, 639-644 Performance specification, 510 Periodic variables, 609, 612 Personal communications systems, 6 Phase center, 161 Phase shifters, 201, 205 Phase velocity, 396 Phenomenon of resonance, 238 Photoconductive switch, 7, 137, 140, 142, 288 Photonic receiver, 548, 550 signal processing, 550 technology, 9 UWB receiver, 549 signal correlator, 550 Physical optics (PO), 466, 478 Plane properties, 320 waves, 317 Plasma current, constitutive relations for, 337 Plasmon frequency, 368 PO, see Physical optics Poisson impulse sequences, 587 Polarization, 161, 340, 345, 352, 359 Pole(s), 322 aspect independence of, 484 singularities, 473, 474 Police radar, 6 Polyspectra, 553 Post-Debye picture, 378 Post-detection processor, 525 Power -distance relationship, 149 handling, 190, 191 levels high, 318 instantaneous, 540 management, 213 spectral density (PSD), 55, 73, 468, 493, 521, 538, 547, 580 supplies, 109, 306 Poynting relation, 131

Precursor effects, empirical evidence for, 399 Precursor phenomena, 366 Precursors, 396 Precursor signal, 655 PRF, see Pulse repetition frequency Primary diffraction, 630, 655, 656 Primary diffraction components, 373 Probability density function, 495, 535 distribution, 525 Processing gain, 501 Prony’s method, 473 Propagation, 13 atmosphere and, 327, 335 constant, 309 deterministic, 339, 361 into Earth’s surface, 12 electromagnetic, 8 energy transfer and, 8 impulse, 8 linear, 326, 336, 359 models, 326 narrowband, 325 nonlinear, 326, 336, 351, 356 path, 341 RF, 330 scattering, 330 stochastic, 345, 361 transatmospheric, 326 transionospheric, 336, 351 UWB, 325, 334 Propagation, energy transfer and, 325-434 developments in self-induced transparency, 425-427 energy transfer through media and sensing of media, 365-425 advanced theory of dielectrics and transmissions through media, 367-385 energy transfer concepts, 365-376 pulse envelope effects, 385-404 soliton waves, group theory, and electromagnetic missile concepts, 404-413 nonlinear wave equations and solitons, 427-428 relation of U (l) and SU(2) symmetry groups, 428-434 RF propagation in atmosphere, 325-364 high power propagation in nonlinear media, 349-359 low power linear propagation through background plasma, 337-349 UWB propagation, 335-336 Propagation, nonlinear, 361 Propagation, soliton, 434 Propagation velocities, 403 PSD, see Power spectral density Pseudorandom code, 216 Pulse(d) alteration, 199

665 charging, 110 chirped, 351 compression, 218, 240, 538 Doppler radar, 492 duration, 502 element, 538 generator, 302 length, 494 packet processing, 536 power, 109 -radiating antenna, 299, 302, 307, 314 repetition frequency (PRF), 32, 501 risetime, 494, 501 shape off-axis, 203 signal detection, 502 waveform array antennas, coded, 8 Pulsed signal, crafting, 366 Pulse envelope, 365 Pulser, 302, 304

Q Q-factor, 436 Quadratic nonlinearities, 585

R Rabi frequency, 392, 426 Radar -absorbing material (RAM), 9, 579, 581 LA-1, 588 sample, 587 carrier-free, 437 characters, 436 cross section (RCS), 5, 435, 457, 510 backscatter, 460 bistatic, 459 monostatic, 460 detection, 491 equation, 458 ground-probing, 437 intercept receiver, 6 military applications, 5 performance prediction principles, 609 range equation, 457, 458, 609, 639 receiver, 491, 525, 546, 549 signal bandwidth, 120 detection, 524 processing, 579 target signature of, 8 surveillance, 534 systems, 1, 299 targets, I, 8, 580 technology, technological advances in, 326 transmitter, target signature and, 8

Radar cross section, target scattering and, 457-490 frequency dependence of RCS, 462-467 optical or high frequency region, 464-466 Rayleigh or low frequency region, 463 RCS of sphere, 466-467 resonance region, 463 radar cross section, 457-460 monostatic and bistatic RCS, 459-460 radar equation, 458-459 relationships among CW, transient, and wideband scattering, 467-472 resonance-based target identification, 481-489 history, 483-484 issues of resonance based identification, 484-487 systems related questions, 487-489 scattering matrix, 461-462 singularity expansion formulation, 472-481 mathematical foundation, 473-479 numerical examples, 479-481 Radar signals, transmitter signature and target signature of, 435-^56 character coding and large target signature, 443-451 continuous and transient response of resonant circuits, 452-456 features of carrier-free radar signals, 437-439 position coding and large target signature, 439-443 Radial velocity estimation, 48 Radiated field, 299, 301, 311 Radiated spectrum, 301 Radiated waveform, 300, 303, 313, 318 Radiation bandwidth, 163 pattern, 157 resistance, 38 scattered, 333, 334 Radiative transfer, 333, 359 equations, 329, 333 scattering from, 334 Radio frequency (RF), 325, 365 RAM, see Radar-absorbing material Random functions, 651 Random intervened sampling, 501 Random signals, 649 Random variables, 609, 612 Range measurement, 13, 14, 45, 48 profiling, 19 resolution, 299, 492, 499, 538, 645 Rayleigh distribution, 497 Rayleigh region, 462, 463 RCS, see Radar cross section Receiver aperture, 645 bandwidth, 499, 506, 520, 549

666 definition, 491 design, 572 function, 508 hardware, 500 impulse sensitivity, 518 noise, 540, 548 equivalent power, 544 level, 508 performance analysis, 494 -processor gain, 628 sensitivity, 557 Reciprocity principle, 630 Recombination times, 141 Rectangular-loop antenna, 243 Rectangular pulse, 503 Reference signal, 146, 498, 509, 524, 546, 548 waveform, 493, 525 Reflection, 359 Reflectors, 317 antennas, 206, 213, 318 characteristics, 491 power, 179 Refraction, 333, 335 Refractive bending, 346, 359 Region of convergence (ROC), 76, 77, 81 Relative bandwidth, 239 Relaxation time, 365 Remote sensing, technological advances in, 326 Resistive loading, 303 Resistively loaded cylindrical antenna, 303 Resolution, 436 angle, 264, 275, 284 concept of, 436 Resonance, 15, 436 effects, 12, 25 generalized theory of, 14 radar, 317, 320 regime, 299, 322 region, 462, 463 Resonant absorption, 368 Retarded time, 310 Return signal shape, 645 RF, see Radio frequency Ricatti equation, 408 Ricean distribution, 497 Ridged horn, 186, 506 Rim diffraction, 207 Risetime, 132, 133, 299, 306, 492, 507 limited, 316 measurement error, 501 slowed, 318 Rising pulse, fast, 317 ROC, see Region of convergence Rotating wave approximation, 391 Rotman lens, 201, 646

s Sand, spectra for sample of, 587 SAR, see Synthetic aperture radar Scattering, 320, 326, 330, 359 coefficients, 461 dynamics, 585 forward, 339 geometry, 332 macroscopic, 334, 335, 359 matrix, 457 regimes, 457 transient and broadband electromagnetic response of, 299 of wave energy, 346 Scintillation index, 348 Scintillations, 361 Secondary diffraction, 630, 655, 656 Secondary diffraction components, 373 Secondary impedance, 112 Second-order kernel, 585, 591, 593 Selective reception, 439, 443, 444 Selectivity, 436 Self-absorption, 326, 360 Self-focusing, 326, 360, 433 SEM, see Singularity expansion method Semiconductor, 7 laser amplifier, 549 switches, 8, 118 Separate arrays, 199 Sequential switching, 123, 129 Shape recognition, 446 Short baseband receiver, 512 Sidelobes, 446, 541 Sidewall radiation, 179 Signal autocorrelation, 535 bandwidth, 2, 299, 505 collection components, low dispersion, 501 detection, 494, 538, 540 distortion, 530 duration, receiver design for, 501 envelope, 497, 645 format, 7 intercept, 26 -noise cross-correlation, 535 -to-noise ratio (SNR), 8, 20, 55, 217, 492, 508, 528, 639 degradation of, 564 enhancement, 40 improvement, 542 input, 563 minimum detectable signal for, 557 parameter conflicts, 48 processing, 5, 549, 579 waveform, 146, 508

667 Signal velocity, 396 Simulton, 426 Sine-Gordon equation, 408, 428 Sine-Gordon soliton equation, 406 Single shot digitizing rate, 501 Singularity expansion method (SEM), 4, 320, 472, 492, 552, 580 Sinusoidal carrier, 436 field, damped, 317 pulse, 437 signals, 147, 564 waves, 261 Sliding correlation, 216 Slope pattern, 273 Small angle approximation, 263 Small-relative-bandwidth technology, 438 Sneak through, 356, 361 SNR, see Signal-to-noise ratio Solitary wave, 405, 406, 407 Soliton(s), 366, 389, 405, 407 Solitonic conduction, 365 Soliton propagation, 434 Soliton propagation, in optical fibers, 433 Sommerfeld precursor, 396, 470 Spark gaps, 7, 118, 119, 306 Spatial resolution, 520, 524, 540 Spectral content, 359 Spectral density, 544 Sphere, RCS of, 466 Spherical wave, 317, 320 Spillover, 207 Spinor focus wave modes, 412 Spiral antenna, 187, 506 s-plane transfer function response, 473 Splash modes, 412 Spread spectrum, 4, 26, 216 coherent signal processing, 39 signaling, 28 Square-law components, 510 Stable local oscillator (STALO), 494 STALO, see Stable local oscillator Standing wave, 478 Stationary process, 612 signals, 609, 623 target, 651 Stationary wave, 373 Steady-state amplitude, 437 Stealthy target, 458 Step input, 317 Step-function aperture illumination, 317 Storage capacitors, 116 Stray impedances, 110 Superheterodyne receivers, 508, 512 Surveillance, technological advances in, 326

Swerling models, 465 Switches, 314, 318 Switching devices, 31, 32 matrix, 205 Synchronization, 548 Synthesized antenna patterns, 286 Synthetic aperture radar (SAR), 5, 265 System design driver, 502 identification, 579, 583 noise, 638

T Tail erosion, 349, 354, 356, 358, 360, 361 Tail erosion, frequency scaling analysis of, 360 Tangential electric field, 317 Tapped delay lines, 530 Target(s) characteristics, 492, 583, 645 class, elements of, 320 detection, 5, 579, 645 identification, 5, 13, 299, 317, 320, 579 imaging, 4, 19 reflectivity, 15-17, 22 frequency dispersive, 19 glint, 16 high permeability, 24 resonance, 19 response, 13, 501 sensing range, 492 signature, 436, 444, 446, 451 size of, 435 steady-state response of, 468 waveform interactions, 494 TD, see Time domain TEM, see Transverse electromagnetic Temperature noise, 499 Terminating impedances, 318 Ternary pseudorandom sequences, 587 Tertiary diffraction, 630 Tertiary diffraction components, 373 Test object, characteristic dimension of, 301 Thin-antenna approximation, 310 Thin-wire frequency domain (TWFD) code, 484 time domain (TWTD) scattering code, 479 Third-order kernel, 585 Three-wave mixing, 426 Threshold detection, 9, 493, 498, 499, 509, 510, 520, 645 signal detection, 508 voltage, 497 Time bandwidth product, 2 Time delay, 190, 191, 545

668 Time domain (TD), 313, 500, 502, 513, 609, 615, 639 correlation, 544 detection, 528 electromagnetics, 13 fine grain, 501 pseudorandomly coded signals, 146 receiver processor design, 616 gain concept, 609, 627 return signal, 320 waveform, 300 Time hopping systems, 4 Time measurement resolution, finer, 524 Time resolution, 541 Time scale conversion, 628 TL, see Transmission line Total electron content, 341, 361 Total transmitted energy, 645 Transfer function, 148, 282, 341, 468, 585 Transformers, 112 Transient data, 320 Transient plane EM wave, 301 Transient pulse radiation, 303 Transient signals, 552 Transmission line (TL), 114, 121, 161, 288, 307, 309-311 Transmitters, 109-144 light-activated semiconductor switches for UWB radar, 120-144 digital synthesis, experimental results, 125-128 digital synthesis of UWB signals by sequential switching, 123-125 evolution of requirements, 121-123 switch choice, 140-142 switches and limits, 128-137 switch optical requirements, 137-140 power supply design, 109-120 considerations for power supply design, 110-111 pulsed power supplies, 111-118 switching techniques for pulsed power supplies, 118-119 Transmitting antenna resonances, 300 Transparency, acoustical self-induced, 395 Transparency, RF self-induced, 394 Transparency, self-induced, 365, 367, 389, 425 Transverse coaxial line, 181 Transverse electromagnetic (TEM) hom, 8, 176, 202, 319, 506 transmission line, 121 wave, 317 spherical, 301 inhomogeneous, 317 Traveling waves, 478 TRF, see Tuned radio frequency receiver Triple autocorrelation, 584 Trispectra, 585

Tuned radio frequency (TRF) receiver, 509 Tunnel diode, 512, 520 TWFD, see Thin-wire, frequency domain code Two-port network, 557 Two-way patterns, 216 TWTD, see Thin-wire time domain scattering code

u Ultra-wideband (UWB), 1, 52, 325 analysis, 7 antenna(s), 7-8, 146 design, 152, 162 isolated dipole as, 164 pattern, 146 power handling in, 162 transient performance of, 162 baseband signal, 538 correlation detection thresholding, 545 detection circuit bandwidth, 504 detection receiver bandwidth, 504 impulse generators, 287 intercept receiver, 541 noise, 500 power supplies, 109 propagation, 349, 353 pulse, 334-337, 341, 352, 358 propagation of, 354 spectral content of, 347 radar, 287, 579 antenna, 303 detection, 491 performance prediction, 9 receiver, 508, 509 signals, high order signal processing for, 9 systems, technical issues in, 7 waveforms, 11 receivers, 8-9, 498 RF amplifiers, 549 signal(s), 326, 334, 345 definition, 6 minimum detection level for, 516 reception sensitivity in conventional receivers, 520 structure, 505 transient signal response, 536 transmitters, 7 Ultra-wideband antenna technology, 145-286 antennas and UWB signals, 147-215 antenna elements, 163-195 aperture antennas, 195-214 special antenna requirements for UWB radar systems, 147-152 UWB antenna design parameters, 152-163 array antenna calculations in time domain using pseudorandomly coded signals, 216-232 correlating receivers, 219-221

669 fundamentals of beamforming in time domain, 221-226 impulse signal design, 217-219 numerical simulation, 226-231 discussion of time-domain field equations, 232-238 linear array beamforming with nonsinusoidal waves, 286 array beamforming with sinusoidal waves, 261-265 beamforming with nonsinusoidal waves, 265-277 frequency-domain array beamforming, 277-281 synthesis of antenna beam patterns, 283-285 transfer function and impulse response of linear array, 282-283 UWB impulse antennas, 238-260 field strength and magnetic flux of large-current radiator, 244-255 large-current radiator, 240-243 measured antenna patterns, 255-260 relative bandwidth, 239-240 Ultra-wideband radar overview, 1-10 potential applications of UWB radar, 4-6 low probability of interception UWB waveforms, 5-6 target imaging and discrimination, 5 target signal interaction and feature extraction, 4-5 terminology and concepts, 1-4 UWB defined, 2 UWB and spread spectrum signals, 4 UWB terminology and usage, 2-3 UWB systems frequency spectrum sharing and interference issues, 6-7 electromagnetic compatibility and interference issues, 6-7 frequency spectrum sharing, 6 Ultra-wideband radar receivers, 491-577 computation of correlator output vs. SNR, 572-577 algorithm for computing correlator output and correlation coefficient vs. SNR, 573-576 computing PSD correlator output and correlation coefficient vs. SNR, 576-577 correlation detection, 520-548 classical correlation detection analysis, 534-538 correlation with Barker and complementary codes, 538-544 correlation detection summary, 546-548 correlator performance, 530-534 cross-correlation, 544-545 cross-correlation and radar signals detection, 524-528 practical time domain correlation, 528-530 signal correlation overview, 521-524 UWB correlation detection thresholding, 545-546 narrowband receiver sensitivity to UWB signals, 557-572 average noise factor, 563-564

comparison of broadband and narrowband sensitivities, 570-571 minimum detectable impulse signal, 567-570 minimum detectable sinusoidal signal, 564-567 receiver noise bandwidth, 557-563 narrowband and UWB receiver performance analysis, 494 narrowband and UWB signal receiver concepts, 494-520 frequency-dependent effects, 505-507 narrowband receivers, 494-498 narrowband and UWB receiver comparison, 500-501 receiver design for bandwidth, pulse risetime, and signal duration, 501-505 UWB signal detection, 498-500 UWB and narrowband receiver design objectives, 492-493 UWB radar detection and sensing, 491-492 UWB radar receivers and signal processing, 548-554 advanced signal processing and reception processing, 551-554 photonic UWB receivers and processing, 549-551 practical UWB receiver design and limitations, 549 UWB threshold signal detection, 508-520 receiver impulse signal sensitivity, 516-520 threshold detection, 511-516 UWB receiver design, 508-511 Ultra-wideband radar signals, high order signal processing for, 579-607 background, 579-582 experiment, 587 methodology, 583-586 bispectral analysis, 583-585 kernel analysis, 585-586 results, 587-595 Ultra-wideband radar systems, technical issues in, 11-50 classification of radar waveforms, 28-31 baseband waveform, 28-30 monocycle waveform, 31 polycycle waveform, 31 concept of nonlinearity, 49-50 fundamental radar principles, 14-28 detection, 27-28 interference susceptibility, 26-27 measurement resolution, 14-15 propagation medium, 25-26 target characteristics, 15-25 range accuracy requirements for velocity estimation from differential time delay, 48-49 signal analysis methods, 11 signal characteristics governing range and velocity measurement resolution, 44-48 range measurement, 45-48 signal parameter conflicts, 48 velocity measurement, 44-45, 47

670 technical issues in UWB radar system design, 31-41 antennas, 33-39 receivers, 39-41 transmitters and sources, 31-33 technology capability, existing, 11-12 Ultra-wideband signals, analytical techniques for, 51-107 Fourier analysis of signals, 52-75 analog, discrete, and digital signals, 55 Dirac-delta function, 65-68 Fourier analysis of signals, 56-62 Fourier transform of periodic signals, 68-70 information and bandwidth, 55-56 numerical computation of Fourier transform, 70-71 periodic and nonperiodic signals, 53-55 power spectral density of random signals, 74 properties of Fourier transform, 62-65 spectral density, 71-73 times correlation of power signals, 73-74 Laplace transforms and signal analysis, 75-90 applications of two-sided Laplace transform, 84-88 inverse Laplace transform, 80-81 Laplace transforms, 75-80 one-sided Laplace transform, 83-84 properties of Laplace transforms, 81-83 pulse propagation in lossy medium, 88-90 limitations of time and frequency approaches, 9 1 -1 0 7

characterization of objects in frequency domain, 105-106 considerations in performing time domain measurements, 92-93 data acquisition, 93-94 experimental verification, 105 processing considerations, 94—104 transformations from discrete to continuous domain, 104-105 Unit step function, 307 Usable bandwidth, 163

UWB pulse, propagation of model, 359 UWB, pulse trajectory, 359 UWB, see Ultra-wideband UWB signals, 359

v Variable correlation detection threshold, 534 Velocity measurements, 13, 32, 44, 47 Video detector bandwidth, 502 Voltage multiplication, 112 standing wave ratio (VSWR), 161, 162, 506 Volterra functionals, 585 VSWR, see Voltage standing wave ratio

w Wave equations, 339, 346 impedance, 240 signal sampling, 548 transmission, localized, 35 Waveform, 31, 498 chirped, 530 information content of correlation detected, 524 received, 508 UWB chirped, 541 Waveguiding system, 302 Wavelet packet transform, 625 Waveshape, crafting, 365 Weighting coefficients, 524 Weighting factors, 524 Wideband, 240

z Zero crossing, 299 Zero degrees pulses, 387