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ULTRAWIDEBAND RADAR TECHNOLOGY
© 2001 by CRC Press LLC
ULTRAWIDEBAND RADAR TECHNOLOGY Edited by
James D. Taylor, P.E.
CRC Press Boca Raton London New York Washington, D.C.
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Library of Congress CataloginginPublication Data Ultrawideband radar technology / edited by James D. Taylor. p. cm. Includes bibliographical references and index. ISBN 0849342678 (alk.) 1. Radar. 2. Ultrawideband devices. I. Taylor, James D., 1941TK6580 .U44 2000 621.3848dc21
00030423
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Preface Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information upon it. Samuel Johnson (1709–1784) My first book, Introduction to UltraWideband Radar Systems, gave engineers and managers a practical technical theory book about a new concept for remote sensing. UltraWideband Radar Technology presents theory and ideas for future systems development and shows the potential capabilities. Ultrawideband (UWB) radar systems use signals with a bandwidth greater than 25 percent of the center frequency. In this case, bandwidth means the difference between the highest and lowest frequencies of interest and contains about 95 percent of the signal power. Example waveforms include impulse (video pulse), coded impulse trains, stepped frequency, pulse compression, random noise, and other signal formats that have high effective bandwidths. UWB radar has advanced since I entered the field in 1987. Several years ago, I received an advertisement for a UWB radar intrusion alarm and bought some as birthday presents for my special ladies. A petroleum distributors’ trade magazine described how future gas stations will use UWB communication links and transponders to identify customers and vehicles. While robots fill the tank, the individual account will be properly charged and the transaction completed without the driver leaving the car. Some companies are proposing to build locating systems for factories that will attach a UWB transponder to each container. Personal locating systems are other potential applications. Other companies are designing UWB wireless links to connect homes, offices, and schools. When I was young, Uncle Scrooge McDuck, a Walt Disney character, had a special watch that computed his income tax by radar. Given the recent commercial development of micropower UWB technology, I expect to see a UWB wristwatch radar advertised soon. It sounds like a great idea for soldiers, policemen, hunters, or anybody who might want to detect hidden persons or objects behind cover. After adding some more technology, followon models could calculate your taxes by radar. Gyro Gearloose, Scrooge McDuck’s inventor friend, was right on target. When I organized a session at the 1999 International UltraWideband Conference in Washington, D.C., my speakers reported on UWB radar progress in groundpenetrating radar, airborne SAR systems, automotive radar, and medical imaging. In contrast to the SPIE and IEEE meetings that I usually attend, commercial development issues occupied almost half of the program. The main topic of discussion was about changing the Federal Communications Commission regulations covering lowpower UWB sensing and communication systems. It appears that legal issues will be the principal obstacle to future UWB systems development. Given this background, I think that UWB radar technology will develop along parallel tracks. Commercial lowpower shortrange systems, using integrated circuit technology for sensing and communications, will be one path. Highpower systems for remote fine resolution imaging and sensing will be the other. Microwatt power impulse radar systems can provide practical solutions to many shortrange sensing and communication problems. There is now a strong interest in using UWB signals for shortrange wireless interconnection and networking activities. Commercial applications will drive the development of lowpower devices once the many regulatory issues are settled. Because UWB signals can provide allweather sensing and communications over short ranges, it may appear in © 2001 by CRC Press LLC
smart vehicles and highway systems. I recommend visiting the Ultrawideband Working Group web site at www.uwb.org for the latest developments and news. Highpower systems for defense and environmental remote sensing will be the other systems development direction. The American Department of Defense Advanced Research Projects Agency (ARPA) sponsored UWB radar programs during the 1990s. Program objectives included highresolution sensing and mapping, foliage penetration for imaging hidden objects, and buried mine detection. The annual SPIE sponsored AeroSense conference has been one of the principal forums for reporting UWB radar technology activities and progress. In my opinion, longrange development goals for highpower systems will include remote environmental and biomass sensing, small target detection using longduration pulse compression signals and bistatic techniques, and using multiple short range UWB systems to provide highresolution surveillance for industrial and urban areas. Better uses of polarimetry and signal processing can enhance UWB radar capabilities. Propagation and media characterization studies will help develop better signal processing and imaging techniques. Discussions with my professional associates indicate that understanding UWB radar requires a new philosophical approach. Because many UWB circuits work with shortduration signals, the steadystate condition is never reached. This requires analyzing the system in the time domain and looking at transient conditions, as opposed to the steadystate frequency response that characterizes many electronic systems. Fine radar resolution means that targets are much larger than the signal resolution, so they can no longer be considered as point source reflectors. Many of the rules and descriptions used for continuous sinusoidal wave signals cannot be directly applied to UWB radar problems. Concepts such as radar cross section will have new meanings as range resolution becomes smaller than the target. Chapter 1, “Main Features of UltraWideband (UWB) Radars and Differences from Common Narrowband Radars,” was written by Dr. Igor Immoreev of the Moscow State Aviation Institute in Russia. He explains how UWB signals will produce effects not encountered in conventional lowresolution radar. This leads to the concept of signal spectral efficiency. Life is further complicated because fine range resolution turns the target into a series of point returns from scattering centers and creates major signal processing and target detection problems. A direct result of the timevariable UWB antenna and target characteristics is a timedependent radar range equation. Chapter 2, “Improved Signal Detection in UWB Radars,” by Dr. Igor Immoreev, expands the concepts of Chapter 1. He presents an approach to detecting overresolved targets by correlating multiple returns over an estimated spatial window about the physical size of the target. This solves the problem of reduced UWB radar returns from numerous scattering centers; however, it will present a new way of considering radar reflection characteristics and target radar cross section specifications. Chapter 3, “HighResolution UltraWideband Radars,” by Dr. Nasser J. Mohammed, of the University of Kuwait, presents a concept for identifying UWB radar targets. This method involves correlating the series of target returns against a library of known target signals. There is a close relationship with ideas of Chapter 2. Chapter 4, “Ultrawideband Radar Receivers,” by James D. Taylor, examines some major theoretical issues in receiver design. This chapter starts with concepts of digitizing and recording impulse signals in a single pass, which is a major problem area building impulse radars for material recognition. Pulse compression is another UWB radar technique that has potential applications where fine range resolution is needed at long ranges. Practical guidance for estimating the bandwidth of UWB signals is given by an explanation of the spectrum of pulse modulated sinewaves. Performance prediction for UWB systems remains a problem area, and the solution may have to be specific to radar systems and waveforms. While the question cannot be answered with a single neat equation, I have provided an approach to performance estimation as a starting point. Chapter 4 is complementary to Chapters 1, 2, and 3. © 2001 by CRC Press LLC
Chapter 5, “Compression of Wideband Returns from Overspread Targets,” by Dr. Benjamin C. Flores and Roberto Vasquez, Jr., of the University of Texas at El Paso, provides a look at how to use pulse compressed signals in radio astronomy. While the ambiguity function was mentioned in Chapter 4, this chapter shows what happens to longduration pulsecompressed signals when there are time or frequency shifts caused by target motion. Chapter 6, “The Micropower Impulse Radar,” by James D. Taylor and Thomas E. McEwan, examines lowpower system technology for shortrange applications. Recent advances in integrated circuit technologies will provide a wide variety of shortrange sensors and communication systems. Using micropower radar techniques can put radar sensors in places never thought of before. Chapter 7, “Ultrawideband Technology for Intelligent Transportation Systems,” by Dr. Robert D. James and Jeffrey B. Mendola, of the Virginia Tech Transportation Center, and James D. Taylor, shows how future smart highway systems can use UWB signals. Shortrange sensing and communications are two requirements for watching traffic conditions and then communicating instructions to vehicles. Additionally, we can expect to see some form of radar installed in automobiles and trucks for station maintenance and collision avoidance with other vehicles in traffic. Automotive radar and communications may be a primary UWB development area in the near future; however, it will require a large effort to build smart highways and vehicles. The question of infrastructure design, systems standards, highway control schemes, communication protocols and links, and other issues must be settled before any widespread smart highway system can be built. This chapter raises the potential for vehicle tracing and location, which may raise some serious constitutional privacy issues in a democratic country. Chapter 8, “Design, Performance, and Applications of a Coherent UWB Random Noise Radar,” by Dr. Ram Narayanan, Yi Xu, Paul D. Hoffmeyer, and John O. Curtis, shows how bandwidth alone determines range resolution. Dr. Narayanan and his University of Nebraska associates built and demonstrated a continuous random noise signal radar. By preserving the random noise signal in a delay line, this experimental 1 GHz radar achieved spatial resolution of 15 cm. Such a concept would be potentially valuable for building a stealthy, low probability of intercept radar or for operating at low power levels to avoid interference with other systems. Chapter 9, “New Power Semiconductor Devices for Generation of Nano and Subnanosecond Pulses,” by Dr. Alexei KardoSyssoev, of the Ioffe PhysicalTechnical Institute in St. Petersburg, Russia, describes the fundamentals of highpower impulse generation. Producing highpower impulse signals involves suddenly moving large amounts of current, which implies special switches that close or open in picoseconds. This chapter explains the theory of drift step recovery diodes and other highspeed switching devices. Dr. KardoSyssoev is the head of the Pulse Systems Group of the Ioffe PhysicalTechnical Institute. His engineers have provided advanced semiconductor switching devices to SRI International and other American organizations. Chapter 10, “Fourier SeriesBased Waveform Generation and Signal Processing in UWB Radar,” by Dr. Gurnam S. Gill, of the Naval Post Graduate School in Monterey, California, presents another approach to generating ultrawideband waveforms. While highspeed switching techniques are a straightforward approach to impulse generation, repeatability remains an issue. There is always a suspicion that each impulse may be slightly different from the others, which will affect signal processing. Generating UWB signals by adding many different waveforms together offers a more flexible approach to building highpower UWB radar systems, especially ones that need a highly accurate and coherent waveform. Chapter 11, “HighResolution StepFrequency Radar,” by Dr. Gurnam S. Gill, shows how to build a UWB radar using longduration narrowband radar signals. Processing many narrowband returns can give the same result as an instantaneous UWB signal. This is an approach to avoiding the regulatory issues that limit highpower UWB system development. Interference with narrowband systems may force the designer to notch out certain restricted frequency bands before the system can be used legally. Dr. Gill develops the theory of using stepfrequency waveforms, which transmit many longduration, narrowband signals and then process them to achieve the effect of a UWB signal. © 2001 by CRC Press LLC
Chapter 12, “CARABAS Airborne SAR,” by Dr. Lars Ulander, Dr. Hans Hellsten, and James D. Taylor, describes a stepfrequency UWB radar developed and tested by the Swedish Defence Ministry. The Coherent All Radio Band System (CARABAS) demonstrates how to build a highresolution SAR using stepfrequency radar. CARABAS demonstrated both highresolution imaging and foliage penetration expected from VHF signals. Chapter 13, “UltraWideband Radar Capability Demonstrations,” by James D. Taylor, describes the state of the art in UWB radar for precision imaging, finding targets hidden by foliage, and detecting buried mines. ARPAsponsored demonstrations showed the potential of highpower UWB radar as a practical sensing system for military applications. ERIM International, the Lawrence Livermore National Laboratory (LLNL), SRI International, MIT Lincoln Laboratory, and the Army Research Laboratory programs show the capabilities and problems of UWB radar. Chapter 14, “Bistatic Radar Polarimetry,” by Dr. AnneLaure Germond, of the Conservertoire National des Arts et Metiers in Paris, France, and her colleagues Dr. Eric Pottier, and Dr. Joseph Saillard, presents a new approach to understanding and analyzing bistatic radar signals. Bistatic radar will be an important future technology for detecting small radar cross section targets. Using sidescattered energy for target detection has several potential advantages, including the ability to locate transmitters in protected refuges and move the receiver freely over areas in which it would be dangerous to radiate. Polarimetric radars using orthogonally polarized signals to increase target detection will be a major future radar trend. Analyzing the measured polarization shifts of reflected radar signals may provide a future method for passive target identification. The future of remote sensing may be polarimetric UWB radar. My special thanks for my collaborators who gave their time and effort to make this book possible. We hope this will stimulate new ideas to advance UWB radar technology.
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Acknowledgments The editor and authors of this book wish to acknowledge our families, employers, friends, supporters, opponents, and critics. Special thanks to our colleagues who inspired, assisted, and gave their frank considered opinions and suggestions. There are too many to name without unfairly omitting someone, so we must thank you for your contributions this way. We also extend our thanks to the government, industry, and university representatives who made this book possible by supporting and encouraging ultrawideband radar technology related programs. My heartfelt thanks to all the writers for working with me and taking my lengthy critiques to heart during the revisions of their chapters. We wanted to make this book unique, useful, and readable. We thank our families and friends who supported us and provided us the time we needed to prepare this book. James D. Taylor January 22, 2000 Gainesville, Florida, U.S.A.
© 2001 by CRC Press LLC
In Memoriam Rachel Z. Taylor, 1937–1994
© 2001 by CRC Press LLC
Contributors John Curtis Environmental Laboratory U.S. Army Waterways Experiment Station Vicksburg, Mississippi, U.S.A. Benjamin C. Flores, Ph.D. Department of Electrical and Computer Engineering University of Texas at El Paso El Paso, Texas, U.S.A. AnneLaure Germond, Ph.D. Chaîre de Physique des Composants Conservatoire National des Arts et Metiers Paris, France Gurnam S. Gill, Ph.D. U.S. Naval Postgraduate School Monterey, California, U.S.A. Hans Hellsten, Ph.D. Swedish Defence Establishment (FOA) Department of Surveillance Radar Linkoping, Sweden Paul Hoffmeyer Department of Electrical Engineering University of Nebraska Lincoln, Nebraska, U.S.A. Igor I. Immoreev Doctor of Technical Sciences, and Professor Moscow State Aviation Institute Moscow, Russia
Thomas E. McEwan, MSEE McEwan Technologies LLC Pleasanton, California, USA Jeffrey B. Mendola, MS Virginia Tech Transportation Center Blacksburg, Virginia, U.S.A. Nasser J. Mohamed, Ph.D. Electrical Engineering Department University of Kuwait Safat, Kuwait Ram Narayanan, Ph.D. Department of Electrical Engineering University of Nebraska Lincoln, Nebraska, U.S.A. Eric Pottier, Ph.D. UPRESA CNRS 6075 Structures Rayonnantes Laboratoire Antennes et Télécommunications Université de Rennes 1 Rennes, France Joseph Saillard, Ph.D. Ecole Polytechnique de l’Université d Nantes Nantes, France James D. Taylor, MSEE, P.E. J.D. Taylor Associates Gainesville, Florida, U.S.A. Lars Ulander, Ph.D. Swedish Defence Establishment (FOA) Department of Surveillance Radar Linkoping, Sweden
Robert B. James, Ph.D. Virginia Tech Transportation Center Blacksburg, Virginia, U.S.A.
Roberto Vasquez, Jr. Raytheon Electronic Systems Bedford, Massachusetts
Alexi F. KardoSysoev Doctor of PhysicoMathematical Sciences Ioffe PhysicalTechnical Institute St. Petersburg, Russia
Yi Xu, Ph.D. Department of Electrical Engineering University of Nebraska Lincoln, Nebraska, U.S.A.
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About the Editor James D. Taylor was born in Tifton, Georgia, in 1941, and grew up in North Carolina and Maryland. After earning his BSEE degree from the Virginia Military Institute in 1963, he 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 Holloman Air Force Base, New Mexico, until 1975. He earned his MSEE in guidance and control theory from the Air Force Institute of Technology at WrightPatterson AFB, Ohio, in 1977. From 1977 to 1981, he was a staff engineer at the Air Force Wright Aeronautical Laboratories Avionics Laboratory. From 1981 to 1991, he served as a staff engineer in the Deputy for Development Planning at the Electronic Systems Division at Hanscom Air Force Base, Massachusetts. Upon retiring from the Air Force in 1991, he worked as a consultant to TACAN Aerospace Corp. in San Diego, California, and edited Introduction to UltraWideband Radar Systems for CRC Press. He has actively participated in PIERS symposiums radar workshops since 1998 and presented short courses in ultrawideband radar in America, Italy, and Russia. His professional achievements include Professional Engineer registration from Massachusetts in 1984. He is a senior member of the Institute of Electrical and Electronics Engineers and the American Institute of Aeronautics and Astronautics. He retired from the U.S. Air Force as a Lieutenant Colonel, and is now a gentleman engineer, consultant, technical writer, editor, and novelist.
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Contents Chapter 1 Main Features of UWB Radars and Differences from Common Narrowband Radars Igor I. Immoreev Chapter 2 Improved Signal Detection in UWB Radars Igor I. Immoreev Chapter 3 HighResolution UltraWideband Radars Nasser J. Mohamed Chapter 4 UltraWideband Radar Receivers James D. Taylor Chapter 5 Compression of Wideband Returns from Overspread Targets Benjamin C. Flores and Roberto Vasquez, Jr. Chapter 6 The Micropower Impulse Radar James D. Taylor and Thomas E. McEwan Chapter 7 UltraWideband Technology for Intelligent Transportation Systems Robert B. James and Jeffrey B. Mendola Chapter 8 Design, Performance, and Applications of a Coherent UWB Random Noise Radar Ram M. Narayanan, Yi Xu, Paul D. Hoffmeyer, John O. Curtis Chapter 9 New Power Semiconductor Devices for Generation of Nano and Subnanosecond Pulses Alexei F. KardoSysoev Chapter 10 Fourier SeriesBased Waveform Generation and Signal Processing in UWB Radar Gurnam S. Gill Chapter 11 HighResolution StepFrequency Radar Gurnam S. Gill © 2001 by CRC Press LLC
Chapter 12 The CARABAS II VHF Synthetic Aperture Radar Lars Ulander, Hans Hellsten, James D. Taylor Chapter 13 UltraWideband Radar Capability Demonstrations James D. Taylor Chapter 14 Bistatic Radar Polarimetry Theory AnneLaure Germond, Eric Pottier, Joseph Saillard
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1
Main Features of UWB Radars and Differences from Common Narrowband Radars Igor I. Immoreev
CONTENTS 1.1 Introduction 1.2 Information Possibilities of UWB Radars 1.3 How UWB Radar Differs from Conventional Radar 1.4 Moving Target Selection in the UWB Radar and Passive Jamming Protection 1.5 Short Video Pulse Features in UWB Radar References
1.1 INTRODUCTION The majority of traditional radio systems use a narrow band of signal frequencies modulating a sinusoidal carrier signal. The reason is simple: a sine wave is the oscillation of an LC circuit, which is the most elementary and most widespread oscillatory system. The resonant properties of this system allow an easy frequency selection of necessary signals. Therefore, frequency selection is the basic way of information channel division in radio engineering, and the majority of radio systems have a band of frequencies that is much lower than their carrier signal. The theory and practice of modern radio engineering are based on this feature. Narrowband signals limit the information capability of radio systems, because the amount of the information transmitted in a unit of time is proportional to this band. Increasing the system’s information capacity requires expanding its band of frequencies. The only alternative is to increase the information transmitting time. This information problem is especially important for radiolocation systems, where the surveillance time of the target is limited. Past radars have used a band of frequencies that does not exceed 10 percent of the carrier frequency. Therefore, they have practically exhausted the information opportunities in terms of range resolution and target characteristics. A new radar development is the transition to signals with a wide and ultrawide bandwidths (UWBs). For designing UWB radars, as with any other equipment, we must understand the required theory that will allow us to correctly design and specify their characteristics. The theory is also necessary for defining the requirements of radars and for developing the equipment needed to create, radiate, receive, and process UWB signals. In spite of recent developments and experimental work, there is no satisfactory and systematized theory of UWB radars available. The reason is that the process of radartracking and surveillance with UWB signals differs considerably from similar processes when using traditional narrowband signals. The study of these differences helps us to
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understand when the traditional theory of radartracking detection can and cannot be used for designing UWB radars. When traditional theory cannot be used, then we must develop new methods. In this chapter, we will examine the new information opportunities that result from applying UWB signals in radars, and the basic differences between UWB radars and narrowband radar systems.
1.2 INFORMATION POSSIBILITIES OF UWB RADARS The informational content of the UWB radars increases because of the smaller pulse volume of the signal. For example, when the length of a sounding pulse changes from 1 µs to 1 ns, the depth of the pulse volume decreases from 300 m to 30 cm. We can say that the radar instrument probing the surveillance space becomes finer and more sensitive. The UWB radar’s reduced signal length can 1. Improve detected target range measurement accuracy. This results in the improvement of the radar resolution for all coordinates, since the resolution of targets by one coordinate does not require their resolution by other coordinates. 2. Identify target classes and types, because the received signal carries the information not only about the target as a whole but also about its separate elements. 3. Reduce the radar effects of passive interference from rain, mist, aerosols, metallized strips, etc. This is because the scattering cross section of interference source within a small pulse volume is reduced relative to the target scattering cross section. 4. Improve stability when observing targets at low elevation angles at the expense of eliminating the interference gaps in the antenna pattern. This is because the main signal, and any ground return signal, arrive at the antenna at different times, which thus enables their selection. 5. Increase the probability of target detection and improved stability observing a target at the expense of elimination of the lobe structure of the secondaryradiation pattern of irradiated targets, since oscillations reflected from the individual parts of the target do not interfere and cancel, which provides a more uniform radar cross section. 6. Provide a narrow antenna pattern by changing the radiated signal characteristics. 7. Improve the radar’s immunity to external narrowband electromagnetic radiation effects and noise. 8. Decrease the radar “dead zone.” 9. Increase the radar’s secretiveness by using a signal that will be hard to detect.
1.3 HOW UWB RADAR DIFFERS FROM CONVENTIONAL RADAR 1.3.1
SIGNAL WAVEFORM CHANGES PROCESSES
DURING
DETECTION
AND
RANGING
Narrowband signals (i.e., sinusoidal and quasisinusoidal signals) have the unique property of keeping their sinusoidal shape during forms of signal conversions such as addition, subtraction, differentiation and integration. The waveforms of sinusoidal and quasisinusoidal signals keep a shape identical to that of the original function and may differ only in their amplitude and time shift, or phase. Hereinafter, shape is understood as the law of change of a signal in time. On the contrary, the ultrawideband signal has a nonsinusoidal waveform that can change shape while processing the above specified and other transformations. Let us assume that a UWB signal S1 shown in Figure 1.1 is generated and transmitted to the antenna in a form of a current pulse. The first change of the UWB signal shape S2 occurs during
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pulse radiation, since the intensity of radiated electromagnetic field varies proportionally with the derivative of the antenna current. The second change of the shape occurs when pulse duration in the space cτ (where c is velocity of light, τ is the pulse duration in the time domain) is less than linear size of the radiator l. When current changes move along the radiator, then electromagnetic pulses are emitted from radiator discontinuities. As a result, a single pulse transforms into a sequence of k pulses divided by time intervals τ1, τ2, . . . ,τk–1, shown as S3 in Figure 1.1. The apparent radiator length changes according to variations of the angle θ between the normal to the antenna array and the direction of the wave front. Therefore, interpulse intervals vary with this angle as follows: τ 1 cos θ,
τ 2 cos θ,…, τ k – 1 cos θ
The third change of the shape occurs when the signal is radiated by a multielement antenna array composed of N radiators with a distance d between them. The pulse radiated by one antenna element at the angle θ is delayed by the time (d/c) sin θ compared to the pulse radiated by the adjacent antenna element. The combined pulse will have various shapes and duration at different angles θ in the far field as equation S4 in Figure 1.1. Farfield pulse shapes at different angles θ are shown in Figure 1.2. Note that the combination of multiple square pulses radiated by the fourelement antenna array and shifted in time over different angles have waveforms very different from the radiated rectangular video pulse. The fourth waveform change is S5 in Figure 1.1, and it occurs when the target scatters the signal. In this case, the target consists of M local scattering elements, or bright points, located along the line L. If the UWB signal length is cτ 0.5Tav prevailing is the first factor, and at τt < 0.5Tav is the second. With the further decrease in τT, the MTS stops influencing the interference immunity because of complete decorrelation of the interference, and only the first factor remains—reduction of the pulse volume. The level of interference is once again decreased, and Q increases. Thus, the position of the second extreme (minimum) corresponds to the complete decorrelation of the interference and absence of the velocity selection. For the most effective rejecter (IPC2), these regularities are displayed more distinctly due to the stronger sensitivity to the correlation properties of the interference.
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Q/Q0 dB
IPC1
20
Trσ d
10
2π
0.03 0.06 0.12 0.25 0.5
1
= 0.02
0.04
τ
0.06 0.08 0.1
Tav
Q/Q 0
IPC2
dB
Trσ d
30
2π = 0.02
20 0.04
10
0.06
0.08 0.1 0.03
0.06 0.12 0.25 0.5
1
2
4
τ
Tav
FIGURE 1.9 Moving target selection depends on the pulse length and the effects of signaltointerference ratios at the system output; shown here for single (IPC1) and double (IPC2) interleaved periodic compensation for τ = Tav and σdTr/2π = 0.1.
Let us note that, with the decrease in the relative width of the passive interferences spectrum, σdTr /2π, the position of the second extreme is shifted to the left toward the lowerduration τ. It means that, with the decrease in the width of the Doppler spectrum of the interference and increase in the pulse repetition frequency, the complete decorrelation of the interference will occur with the shorter signal duration. Within the limits for the nonfluctuating passive interference the decrease in the pulse duration does not influence the MTS effectiveness. Thus, using MTS with respect to the UWB signal is advisable with rather narrowband interferences (local objects) in the radar with the relatively high repetition frequency (i.e., with the small range).
1.5 SHORT VIDEO PULSE FEATURES IN UWB RADAR One of the peculiar features of the UWB radar operating with short video pulses with duration τ is additional losses of energy. The point is that any antenna does not radiate in the range of frequencies lower than some fmin. On the other hand, the frequency spectrum of any video pulse
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has the maximum at the zero frequency. The basic energy of the pulse is concentrated in the band of frequencies ∆f and restricted by some fmax usually lying in the region of the first zero of its spectrum. As a result, the frequency characteristic of the antenna and spectrum of the signal appear to be unmatched. Part of the energy that did not fall into the antenna frequency band will be lost. This is seen well in Figure 1.10, which gives the frequency characteristic of the Hertzian dipole P (length l) and spectrum of the rectangular pulse S. With respect to the signal, the antenna is essentially a highfrequency filter. The notion of the spectral efficiency η∆f was introduced to account for these losses and is a part of the total efficiency of the transmitting device. This efficiency determines the relative share of the energy of the sounding pulse falling into the operating frequency band of the antenna. W ∆f η ∆f = Ws
where WS = full pulse energy Wf = the energy of that part of the pulse spectrum falling into the antenna frequency band For the singlepolarity pulses, the spectral losses can be rather significant. It is possible to reduce these losses by selecting an optimal duration, τopt, for each pulse form in the given band of frequencies that will have the maximum value η∆fopt spectral efficiency. The curve “a” in Figure 1.11 shows the dependence of η∆fopt on ∆f for three simple singlepolarity pulses: rectangular, bellshaped, and triangular. For all considered pulses at ∆f < 3, the maximum efficiency η∆f max < 50%, which essentially worsens the efficiency of a radar. The spectral efficiency η∆f can be improved by changing the radiating pulse spectrum. With this aim in mind, spectrum S2(f) of the correcting pulse u2(t) is subtracted from spectrum S1(f) of the basic singlepolarity pulse u1(t). The form and intensity of this spectrum were selected so that,
P
S 1.0
200
0.8
160
0.6
120
0.4
80
0.2
40
0 0.2
0.4
0.6
0.8
1.0
_l = _lf λ c
FIGURE 1.10 Antennas are a major cause of energy loss in transmitting UWB signals. This shows the frequency response of a dipole antenna P and the spectrum of a video pulse S. The antenna will distort the signal by passing only the higherfrequency components.
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Spectral efficiency
η ∆f opt
}
0.8
(b)
0.6
}
(a)
0.4 0.2 0
1
3
5
∆f
7
Spectral bandwidth(∆f ) FIGURE 1.11 Singlepolarity video pulses have lowfrequency components that do not radiate well through antennas, as shown for cases (a). Correcting the pulse shape to eliminate the low frequencies will increase the spectral efficiency and provide better performance, as shown in (b).
in the summation spectrum SΣ(f) = S1(f) – S2(t), the lowfrequency components for f < fmin were considerably smaller than in the basic spectrum S1(f), and, for f > fmin , the changes were insignificant. The corrected bipolar sounding pulse will be uΣ ( t ) = u1 ( t ) – u2 ( t )
Now the spectral efficiency will depend on the parameters of basic and correcting pulse. The possible maximum values of the spectral efficiency η∆fopt have been determined for the simplest corrected pulses, consisting of the difference between two singlepolarity pulses, each of which has a simple waveform of rectangular, bellshaped, or triangular. The dependencies of η∆fopt on ∆f, computed for different forms of the basic and correcting pulses, are shown in curve (b) of Figure 1.11. The introduction of a corrected pulse appreciably improved the radar efficiency. Figure 1.12 gives the curves indicating the dependence of the ratio of the maximum spectral efficiency of the pulses with the correction and without correction η∆fopt(with corr)/η∆fopt(without corr) on the signal spectrum ∆f. These curves make it possible to estimate the effectiveness of correction of the pulse form so as to increase the spectral efficiency. With the growth of ∆f, the correction of the pulse form becomes less effective, decreasing from 2 at ∆f = 3 down to 1.2 at ∆f = 10. In the general case, the correcting pulse u2(t) can be shifted relative to the basic pulse by the time t0. Then it can be written that uΣ ( t ) = u1 ( t ) – u2 ( t – t0 )
and we shall get for the summary spectrum 2
2
S Σ ( f ) = [ S 1 ( f ) – 2S 1 ( f )S 2 ( f ) cos 2πft 0 + S 2 ( f ) ]
1 2
Figure 1.13 shows the dependence of the maximum spectral efficiency η∆fopt on the relative delay γ = t0/τ and the correcting pulses of different forms on the basic rectangular pulse. Curves (a)
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η∆f opt(with corr) η∆f opt(without corr)
3
2
1
0
1
3
5
7
∆f
9
FIGURE 1.12 The dependence of the spectral efficiency of pulses with and without corrections to the signal spectrum.
Spectral efficiency
η∆fopt
0,8
∆f = 3
0,6
}b }a
0,4 0,2 0 0
2
4
6
8
10 12
Correcting pulse relative delay
14
16
γ = t0 I τ
γ = t0 I τ
FIGURE 1.13 Shifting the correcting pulse by some time increment can improve the system’s performance. This plot shows the effects of the relative time delay on spectral efficiency.
reflect the dependence on η∆fopt on the delay γ at the constant parameters of correction. Curves (b) in Figure 1.13 reflect the same dependence for the case where optimal correction parameters were sought for each value of γ. From a practical point of view, the greatest interest will be the corrected pulses, whose basic and correcting pulses do not overlap by time, i.e., when t0 > (τ1/2 + τ2/2). For the case γ > 1.5, the
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pulses follow each other, and η∆fopt becomes less than at γ = 0 but remains higher than in the uncorrected pulse case. Thus, when selecting video pulse UWB radar waveforms, it is necessary to take into account the spectral efficiency because, for the singepolarity pulses, it can be considerably less than 1. This is especially true of the pulses having the ratio of the high spectrum frequency to the lower one equal to ∆f < 3. In this case, the efficiency does not exceed 50 percent. By increasing ∆f, the spectral efficiency increases so that, at ∆f ≈ 10, it can reach 85 to 90%. Therefore, it is advisable to use the correction of sounding pulses at ∆f < 3, which provides higher values of spectral efficiency. The correction of the pulse waveform makes it possible to increase the spectral efficiency at ∆f ≤ 3 by two times, and about 1.2 times at ∆f ≈ 10.
REFERENCES 1. Harmuth, H., Nonsinusoidal Waves for Radar and Radio Communications. Academic Press, New York, 1981. Translation into Russian. Radio i Svyaz, Moscow, 1985. 2. Harmuth, H., “Radar Equation for Nonsinusoidal Waves.” IEEE Transactions on Electromagnetic Compatibility, No. 2, v. 31, 1989, pp. 138–147. 3. L. Astanin and A. Kostylev, Fundamentals of UltraWideband Radar Measurements, Radio i Svyaz, Moscow, 1989. (Published as Ultrawideband radar measurements: analysis and processing, IEE, UK, London, 1997.) 4. Stryukov B., Lukyannikov A., Marinetz A., Feodorov N., “Short impulse radar systems.” Zarubezhnaya radioelectronika No. 8, 1989, pp. 42–59. 5. Immoreev, I., “Use of UltraWideband Location in Air Defence.” Questions of Special Radio Electronics. Radiolocation Engineering Series. Issue 22, 1991, pp. 76–83. 6. Immoreev, I. and Zivlin, V., “Moving Target Indication in Radars with the UltraWideband Sounding Signal.” Questions of Radio Electronics, Radiolocation Engineering Series, Issue 3, 1992. 7. Shubert, K. and Ruck, G., “Canonical Representation of Radar Range Equation in the Time Domain.”SPIE Proceedings: UWB Radar Conference, Vol. 1631, 1992. 8. Immoreev, I. and Vovshin, B., “Radar observation using the Ultra Wide Band Signals (UWBS),” International Conference on Radar, Paris, 3–6 May, 1994. 9. Immoreev, I. and Vovshin, B., “Features of Ultrawideband Radar Projecting.” IEEE International Radar Conference, Washington, May, 1995. 10. Immoreev, I., Grinev, A., Vovshin B., and Voronin, E., “Processing of the Signals in UWB Videopulse Underground Radars,” International Conference, Progress in Electromagnetions Research Symposium. Washington, DC, 22–28 July, 1995. 11. James D. Taylor (ed.), Introduction to UltraWideband Radar Systems. CRC Press, Boca Raton, FL, 1995. 12. Osipov M., “UWB radar,” Radiotechnica, No. 3, 1995, pp. 3–6. 13. Bunkin B. and Kashin V. “The distinctive features, problems and perspectives of subnanosecond video pulses of radar systems.” Radiotechnica, No. 4–5, 1995, pp. 128–133. 14. Immoreev, I., “Ultrawideband (UWB) Radar Observation: Signal Generation, Radiation and Processing.” European Conference on Synthetic Aperture Radar, Konigswinter, Germany, 26–28 March, 1996. 15. Immoreev, I., “Ultrawideband Location: Main Features and Differences from Common Radiolocation,” Electromagnetic Waves and Electronic Systems. Vol. 2, No. 1, 1997, pp. 81–88. 16. Immoreev I. and Teliatnikov L. “Efficiency of sounding pulse energy application in ultrawideband radar.” Radiotechnica, No. 9, 1997, pp. 37–48. 17. Immoreev, I. and Fedotov, D. “Optimum processing of radar signals with unknown parameters,” Radiotechnica, No. 10, 1998, pp. 84–88. 18. Immoreev, I., “Ultrawideband radars: New opportunities, unusual problems, system features,” Bulletin of the Moscow State Technical University, No. 4, 1998, pp. 25–56.
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2
Feature Detection in UWB Radar Signals Igor I. Immoreev
CONTENTS 2.1 2.2
Introduction Brief Overview of Conventional Methods for Optimal Detection of Radar Signals 2.3 Quasioptimal Detectors for UWB Signals 2.4 Optimal detectors for UWB Signals References
2.1 INTRODUCTION Any radar signal scattered by a target is a source of target information. However, the returned scattered signal will combine with radar receiver frontend internal noise and interference signals. Each signal processing system must provide the optimal way to extract desired target information from the mixed signal, noise, and interference input. Optimum is a term relative to the radar system’s mission or function. What is optimal for one use will not be so for another. Information quality depends on the process that determines the algorithm for analyzing the mixture of signal, noise, and interference and sets the rules for decisions after the analysis is complete. This decision may be based on the detection of the echo signal, the value of the measured signal parameters such as the Doppler shift, power spectral content, or other criteria. The algorithm process efficiency is defined by the statistical criterion, which helps to determine if this algorithm is the best possible one for the application. An algorithm is called the optimum if the information is extracted in the best way for a particular purpose, and the resulting distortions of information resulting from processing operations are minimal. Radar functional requirements will determine the sophistication of the signal processing algorithms. Simple binary detection provides minimal information and shows only that some target is present. Distinguishing and resolving several targets requires additional information requirements and therefore a larger signal bandwidth. If the signal parameters are time variable, the quantity of the information received must be large enough for the recovery of echo signals. Sophisticated applications such as target imaging and recognition will require even more information. For passing information, the channel frequency bandwidth and the signal bandwidth are the determining factors. An advanced “smart” radar that can resolve multiple targets in a small space, or image and identify targets, will need a wider bandwidth signal than other systems. Target information begins with target detection. Therefore, primary attention will be given to this problem in this chapter.
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Barton, Skolnik, Shirman, Sosulin, Gutkin, and others have described the problems of detection for targets concealed by noise in Refs. 1 through 5. The problem is that past work was done mainly for narrowband signals, i.e., sinusoidal and quasisinusoidal signals. Mathematical treatment of sinusoidal signals is simplified, because they do not change their waveforms during the processing operations of summation, subtraction, differentiation, and integration, which occur during radiation, reflection from the target, and reception at the radar receiver. In this case, we mean that the signal amplitude, frequency, and initial phase of narrowband echo signals can change. Target reflection can modulate any of the signal parameters, but the shape of narrowband signals is unchanged during target location and remains a sinusoidal harmonic oscillation. For narrowband signal and target detection, the known signal waveform is a priori information. This feature allows using matched filters and correlators to process narrowband radar signals. In the case of highinformationcontent ultrawideband (UWB) radar signals, not only the signal parameters but also the signal shape will change during processing operations mentioned above. As shown in Refs. 6 and 7, the UWB signal changes shape during target locating many times. As a result, the shape of a signal at the processor input differs essentially from the shape of a radiated signal. The changed signal waveform contains target information, as shown by Van Blaricum and Sheby in Ref. 6. As a result, conventional optimal processing methods, such as matched filtering and signal correlation, are impossible to implement, because there is no a priori signal waveform information. Building successful UWB radars will require new processing algorithms. The objective of UWB radar optimal processing algorithms should be to give the maximum signaltonoise ratio at the processor output for signals with unknown shape. To solve this problem, first we consider the conventional narrowband signal processing algorithms in Section 2.2 then examine quasioptimal and optimal detection methods for UWB signals in Sections 2.3 and 2.4.
2.2 BRIEF OVERVIEW OF CONVENTIONAL METHODS FOR OPTIMAL DETECTION OF RADAR SIGNALS The majority of radar target detection prediction problems can be solved using the methods of statistical decision theory. Those methods analyze the receiver output voltage during a certain period of time and reach a decision about whether a target return signal is present or absent in the voltage. Because the signal must be described statistically, the quality of detection is expressed as the probability of detection and false alarm for given target conditions. Two conditions should be met to make a reliable target detection decision. First, we must have some preliminary (a priori) information about the constituents of receiver output voltage. A well known noise probability density W0(u) and signal + noise probability density W1(u) can be used as a priori information. Later, we will show that the shape of a desired signal can be also used as a priori information. Second, the output voltage processing and target presence detection must be performed according the particular algorithm. This process must increase the volume of the received (a posteriori) information on the constituents of output voltage to the maximum. Furthermore, we consider this algorithm. We can have two groups of events for binary detection. The first group comprises two events, which reflect the actual situation in the radar surveillance area. They are event A1, when the target is present, and event A0, when the target is absent. Each of these two events has a probability of occurrence described by integrated distribution functions P(A1) and P(A0). These events form a full group and are incompatible, because only one of them may happen at a time; therefore, P(A1) + P(A0) = 1. The second group is another two events, which reflect the actual situation at the signal processor output after the received voltage has been processed and the decision has been made. These two events are A′ 1 , meaning that the target is present, and event A′ 0 , meaning the target is absent. The © 2001 CRC Press LLC
probabilities of occurrence of these two events are P( A′ 1 ) and P( A′ 0 ). These events are incompatible, so they too form a full group P ( A′1 ) + P( A′ 0 ) = 1. One event of the first group and one event of the second group will be noted in every surveillance area cell when detecting a target. As a result, only one of four possible variations of simultaneous occurrence of two independent events appears in every volume cell. Two of these variants apply to the true case so that the events A1 and A′ 1 correspond to reliable target detection, and the events A0 and A′ 0 correspond to the case when targets are not detected because none is there. Another two variants are wrong decisions cases where the events A1 and A′ 0 correspond to the miss of a target and the events A0 and A′1 correspond to a false alarm where no target is present, but one is indicated. These wrong case variants result from the statistical (noise) characteristics of the receiver output voltage. As is known, the probability of a simultaneous occurrence of two compatible and dependent events P(An + A′k ) is determined by the multiplication of probabilities. It is equal to the product of the probability of one even P(An) and the conditional probability of the occurrence of the second event calculated under the assumption that the first event has already occurred P( A′k /An): (2.1)
P ( A n + A′ k ) = P ( A n ) ⋅ P ( A′ k /A n )
As shown in Figure 2.1, the conditional probability of false alarm, given the condition that the signal is absent, is the probability that noise voltage u(t) will exceed the threshold value u0. ∞
P ( A′ 1 A 0 ) = P [ u ( t ) ≥ u 0 ] =
∫ W0 ( u )du
(2.2)
u0
Then the probability of false alarm is ∞
P ( A 0 + A′ 1 ) = P ( A 0 ) ⋅ P ( A′ 1 A 0 ) = P ( A 0 ) ∫ W 0 ( u ) du
(2.3)
u0
+U
0
W0(u)
t
U0 = U
thresh
U
False alarm probability
FIGURE 2.1 False alarm probability for random noise, or the chance that random noise will exceed some threshold value.
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Figure 2.2 shows the conditional probability that the signal will be missed when it is present, or the probability that the signal + noise voltage u(t) will not exceed the threshold value u0: P ( A′ 0 A 1 ) = P [ u ( t ) ≤ u 0 ] =
u0
∫0 W1 ( u )du
(2.4)
The probability that the desired signal will be missed is determined by the following expression: u0
P ( A 1 + A ) = P ( A 1 ) ⋅ P ( A A 1 ) = P ( A 1 ) ∫ W 1 ( u ) du ′ 0
′ 0
(2.5)
0
The events (A0 + A′1 ) and (A1 + A′ 0 ) are incompatible. In accordance with the rule of composition of probabilities, the probability that one of two wrong decisions will be made is P [ ( A 0 + A′ 0 ) or ( A 1 + A′ 0 ) ] = P ( A 0 + A′ 1 ) + P ( A 1 + A′ 0 ) u0
∞
= P ( A 0 ) ∫ W 0 ( u )du + P ( A 1 ) ∫ W 1 ( u )du
(2.6)
0
u0
If we change the limits of integration, this expression will take the following form: u0
∞
P [ ( A0 + A′ 1 ) or ( A 1 + A′ 0 ) ] = 1 – P ( A 0 ) ∫ W 0 ( u )du + P ( A 1 ) ∫ W 1 ( u ) du 0
(2.7)
u0
signal
{
+U
probability of signal missing
Wo (u) 0 Uent
U = 0 U thresh
W1 (u)
U
FIGURE 2.2 Conditional probability that the signal will not be detected when present, or the probability that the signal plus noise voltage u(t) will not exceed the threshold u0.
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The probability of making true decision will be P [ ( A 0 + A′ 0 ) or ( A 1 + A′ 1 ) ] = 1 – [ P ( A 0 + A′ 1 ) or ( A 1 + A′ 0 ) ] u0
∞
= P ( A 0 ) ∫ W 0 ( u )du + P ( A 1 ) ∫ W 1 ( u )du 0
(2.8)
u0
To find the optimum threshold level u0, it is necessary to determine threshold value for which the probability of making true decision is maximum. For this purpose, we calculate the following derivative: dP [ ( A0 + A′ 0 ) + ( A 1 + A′ 1 ) ] du 0
(2.9)
and then set it to zero. As a result, we get P(A0)W0(u0) = P(A1)W1(u0) or P ( A0 ) W1 ( u0 )  = W0 ( u0 ) P ( A1 )
(2.10)
Figure 2.3 shows the noise probability density W0(u) and the signal + noise probability density W1(u). It is evident from the picture that, the larger the signal amplitude, the higher the threshold level must be. For P(A0) = P(A1) = 0.5, the optimum threshold level is defined by the point of crossing of two probability density W0(u) and W1(u). The necessary condition for making the decision on target presence is W1 ( u ) P ( A0 )  ≥ W0 ( u ) P ( A1 )
(2.11)
We can make a decision on target absence by reversing the inequality. This inequality is true for the value of the noise voltage and the signal plus noise voltage in one moment of time, and it comprises a onedimensional probability density W0(u) and W1(u). The inequality can be extended to the case where the decision is made from N voltage values, which
D W (u)
W (u)
0
1
F 0 FIGURE 2.3
U
thresh
u U
ent
The noise distribution function W0(u) and the signal plus noise distribution function W1(u).
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we can get from the ensemble of realization at one time moment or from one realization at different time moments: W 1 ( u 1 , u 2 , u 3 ,…, u N ) P ( A 0 ) ≥ W 0 ( u 1 , u 2 , u 3 ,…, u N ) P ( A 1 )
(2.12)
In this case, the probability density W0 and W1 become multidimensional. This most simple statistical criterion is called the ideal observer criterion. In practical cases, the disadvantage of this criterion is that we do not know the a priori probability P(A1) that a desired target is present, and the probability P(A0) that a desired target is not in the radar surveillance area. There is one further problem in that the ideal observer criterion does not consider the consequences of wrong decisions. To overcome the ideal observer criterion, we introduce weight coefficients B and C in the equation, which describes the estimation of probability of wrong decisions. These coefficients characterize the losses caused by false alarm and target miss: P [ ( A 0 + A′ 1 )or ( A 1 + A′ 0 ) ] = B ⋅ P ( A 0 + A′ 1 ) + C ⋅ P ( A 1 + A′ 0 )
(2.13)
In this case, the following inequality must be satisfied to make a decision on the target presence: W 1 ( u 1 , u 2 , u 3 ,…, u N ) B ⋅ P ( A 0 ) ≥ W 0 ( u 1 , u 2 , u 3 ,…, u N ) C ⋅ P ( A 1 )
(2.14)
This statistical criterion is called the minimum risk criterion. Its practical implementation is rather difficult, not only because the priori probabilities P(A1) and P(A0) are unknown, but also because the a priori estimations of weight coefficients B and C are unknown as well. This criterion, along with the ideal observer criterion, is referred to as the Bayes criterion. One more well known criterion is the maximum likelihood criterion. The probability density for N random voltage values at the receiver output W(u1, u2, u3,…, uN), which we mentioned above, is named the likelihood function. The maximum likelihood method helps to determine the maximum value of this function. To perform this operation, we must take the derivative of the likelihood function with respect to the desired signal and set it to zero. The solution of this equation helps to find the maximum likelihood estimation of the signal. For example, if random values of voltage at the receiver output: u1, u2, u3,…, uN are distributed normally, the estimation is equal to their average value. This method gives less dispersed estimations than other methods. Such estimations are called efficient, so the criterion of the optimum operations, which use the maximum likelihood method, is the estimation efficiency. If the maximum likelihood criterion is used, then the decision on target presence is made when the likelihood function W1 exceeds the likelihood function W0: W 1 ( u 1 , u 2 , u 3 ,…, u N ) ≥1 W 0 ( u 1 , u 2 , u 3 ,…, u N )
(2.15)
As was mentioned above, we need some a priori probabilities for making decisions on target presence, but in many practical cases these will be unknown. Another widely used criterion, which does not depend on these probabilities, is called the Neumann–Pearson criterion. This provides the maximum probability of detection D = P(A1 + A′ 1 ), at the constant false alarm rate F = P(A0 + © 2001 CRC Press LLC
A′ 1 ). According to this criterion, the threshold value u0 located in the right part of the likelihood expression is chosen for a given conditional probability of false alarm: ∞
P [ u ( t ) ≥ u0 ] =
∫ W0 ( u )du
(2.16)
uo
So, in many cases, the solution of the problem of target detection is reduced to the calculation of the following ratio: W 1 ( u 1 , u 2 , u 3 ,…, u N ) Λ = W 0 ( u 1 , u 2 , u 3 ,…, u N )
(2.17)
This ratio is called the likelihood ratio. We make a decision on target presence when this ratio exceeds some constant level u0, given according to the selected criterion. The calculation of the likelihood ratio helps to design the optimum receiver. Conventional methods for optimal radar signal detection use the noise probability density at the receiver output as a priori information. This noise is usually approximated by so called white noise, which has an equally distributed spectral power density N0 (W/Hz) within the receiver bandwidth ∆f and the normal time probability density of the voltage u: 2
1 u W 0 ( u ) =  exp – 2 2σ 2πσ
(2.18)
This probability density has zero average value, and its dispersion is σ2 = N0∆f. The samples of noise voltage are statistically independent if they are spaced at the ∆t = 1/2∆f. Then, the likelihood function for N noise samples is the product of N factors so that N
N
i=0
i=1
N 1 1 2 W 0 ( u 1 , u 2 , u 3 ,…, u N ) = ∏ W 0 ( u i ) =  exp – 2 ∑ u i 2πσ 2σ
(2.19)
The probability density for signal plus noise depends on a signal structure. We usually use a hypothetical signal to understand the general laws of optimal processing in the conventional radar theory. For this case, all the signal parameters are fully known except the time of arrival. Therefore, the signal plus noise probability density differs from the noise probability density only by a nonzero average value that is equal to the signal amplitude. 2 1 ( u – s ) W 1 ( u ) =  exp – 2 2πσ 2σ
(2.20)
The likelihood function for the signal plus noise is N
1 N W 1 ( u 1 , u 2 , u 3 ,…, u N ) = ∏ W 1 ( u i ) =  exp 2πσ i=0
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N
1 2 – 2 ∑ ( u i – S i ) 2σ i=1
(2.21)
The likelihood ratio for a fully known signal will be N
N
i=1
i=1
2 W 1 ( u 1 , u 2 , u 3 ,…, u N ) 1 2 Λ = = exp – 2 ∑ s 1 • exp 2 ∑ u i s i W 0 ( u 1 , u 2 , u 3 ,…, u N ) 2σ σ
(2.22)
Considering that σ2 = N0 × ∆f and ∆t = 1/2∆f, we can write 1/σ2 = 2 × ∆t/N0. Then, we have N
N
1 2 2 Λ = exp –  ∑ s 1 ∆t • exp  ∑ u 1 s i ∆t N0 N0 i=1
(2.23)
i=1
In conventional theory, the next step we use is to the transfer to the limits at ∆t→0. But it should be noted that here ∆f→∞ and therefore σ2→∞; that is, the noise power grows infinitely high. Nevertheless, such a model is used in practice. Then, we go from the summation to the integration in the time interval from 0 to T, where N random values of the receiver output voltage u1, u2, u3,…, uN are located: T
T
0
0
1 2 2 Λ = exp –  ∫ s ( t )dt • exp  ∫ u ( t ) ⋅ s ( t ) dt N N0 0
(2.24)
To remove the exponential member in this expression and simplify the design of the optimal receiver, a logarithm of Λ is calculated instead of the value of Λ, thus, T
T
2 1 2 lnΛ =  ∫ u ( t ) • s ( t ) dt –  ∫ s ( t ) dt N0 N0
(2.25)
0
0
In this equation, the second member is the ratio of signal energy to spectral power density, which does not depend on receiver output voltage u(t). For the known signal and given noise power density, this member has constant value, which can be considered when we determine the threshold level u0 or can be included in it. To get optimum detection algorithm, we must calculate the following integral: T
2u ( t ) • s ( t ) dt N0 ∫
(2.26)
0
and compare the received value to a threshold. This expression is a correlation integral, and it determines the association between the signal s(t) and the receiver output voltage u(t). We might say that this integral indicates how closely the received voltage resembles the desired signal. It is clear that the knowledge of the desired signal parameters is very important a priori information for this detection method. The circuit design that performs conventional correlation processing with the use of this integral is shown in Figure 2.4. It is clear from the circuit that two signals are used for processing; they © 2001 CRC Press LLC
U(T)
uecho(t)+un(t)
∫ uref(t)
FIGURE 2.4
uthresh = u0
Conventional correlation processing block diagram.
are a reference signal (a delayed radiated signal is usually used) and an echo signal that is received together with the noise and is fed into the correlator. Here, we introduce some notation so that uref(t) is reference signal voltage, uecho(t) is echo signal voltage, un(t) is noise voltage, and u(T) is the voltage at the correlator output at the end of accumulation period T. Considering this notation, the voltage at the correlator output is T
2 u ( T ) =  ∫ u ( t )u ref ( t ) dt N0
(2.27)
0
A target is considered as a detected one if u(T) > u0. The receiver output voltage u(t) is the correlator input. In various cases, this voltage will be as follows: u(t) = un(t) if the echo signal is absent, or u(t) = uecho(t) + un(t) if the echo signal is present. Now consider each of these cases. If an echo signal is absent, then the correlation integral has the following form: T
2 u ( T ) =  ∫ u n ( t )u ref ( t ) dt No
(2.28)
0
In reality, this integral indicates the variations of noise probability density while the noise is passing through the correlator circuits. This fact is of great importance for the target detection procedure, because the noise probability density at the correlator output determines the threshold level u0 if the echo signal is absent. For example, if for some reason the noise dispersion at the correlator output changes value, then to retain a constant value of false alarm rate, the threshold level must be changed according to the Neumann–Pearson criterion (see Figure 2.1). Let us consider the factors that influence the noise dispersion in this case. The noise voltage at the correlator input is normally distributed, so the normal law is preserved while the noise is passing through the correlator circuits. The average value of this noise distribution is zero, but the dispersion of the noise voltage at the correlator output differs from those at the correlator input. There are some reasons for this difference. First, the spectral noise density N0 can vary during the operation process, which will cause noise dispersion variations. Second, the multiplication of the noise voltage with the reference signal will cause an increase in the noise dispersion proportionally to the signal energy E. To retain the constant rate of false alarm with the increase in noise dispersion, we must raise the threshold level u0. But to retain the given detection probability, it is necessary to increase the signaltonoise ratio, that is, to increase the energy scattered from a target by increasing the transmitter power or antenna gain. © 2001 CRC Press LLC
In practical cases, some measures for stabilization of the threshold level and false alarm rate are taken. For this purpose, the noise voltage and the reference signal voltage are normalized, i.e., divided by 1/ N 0 and 1/ E correspondingly. In this case, the threshold level u0 will be constant, but to retain the given detection probability, the signaltonoise ratio must be increased. In a general case, the noise dispersion at the correlator output depends on the integration interval T as well. But this dependence is lacking if the receiver bandwidth is matched to the duration of the integrated signal. If the echo signal is present in the correlator input voltage, then the correlation integral takes the form
2 u ( T ) = N0
T
T
∫ uref ( t )uecho ( t ) dt + ∫ uref ( t )un ( t )dt
(2.29)
0
0
The first integral of this expression determines the nonzero average value of the probability density W1(u). The second integral determines the dispersion of this probability density; it will be equal to the dispersion of probability density W0(u) for the case when the echo signal is absent. Figure 2.5 shows the mutual arrangement of probability densities W0(u) and W1(u) at the conventional correlator output related to values of the reference and echo signals. The influence of a reference signal on dispersions of these probability densities and threshold levels is evident from the plot. By using the data of Figure 2.5, the detection characteristics can be plotted for this case. It is evident from the plot that the false alarm probability is ∞
F =
1
2
exp – 2 du ∫  2σ 2πσ u
(2.30)
u0
W(u)
W0(u), uecho=0, uref=1.
0.45 0.4 0.35 0.3 0.25 0.2
W0(u), uecho=0, uref=2. W0(u), uecho=0, uref=8. W1(u), uecho=2, uref=2.
0.15 0.1 0.05 0
W1(u), uecho=8, uref=8.
0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 8.8 9.6 u FIGURE 2.5 Dependence of the functions W0(u) and W1(u) at a conventional correlator output on the reference and return (scattered) signals. The value u is the value of the sum of the correlated reference and target signals plus the correlated reference and noise signals.
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and the detection probability is ∞
D =
(u – s)
1
2
 du exp – 2 ∫  2σ 2πσ
(2.31)
ν0
Using the probability integral, x
2 1 u du Φ ( x ) =  ∫ exp –  2 2π
(2.32)
–∞
we can write the expressions for F and D as follows: 2s D = 1 – Φ u 0 –  N 0
F = 1 – Φ ( u 0 ),
(2.33)
On the basis of these formulas, the detection characteristics (that is, the dependence of detection probability D on signaltonoise ratio q = s/N0 for the constant value of the false alarm probability F) are plotted in Figure 2.6. These characteristics were plotted for the fully known signal, and so they are the best among the detection curves that might be plotted for other types of signal. In all further discussion, we shall consider these characteristics as the standard. In addition to the scheme shown in Figure 2.4, there are some other ways to perform the procedure described by the correlation integral. One widely used solution is a matched filter that has an impulse response matched with the detected signal shape. The impulse response enters into the expression under the integral sign instead of the reference signal. The signaltonoise ratio at the matched filter output is similar to this ratio at the correlator output. But we will not discuss matched filters in this chapter, because they are designed for processing signals with a priori known shape and cannot be used for processing signals with unknown shapes, such as UWB signals.
D 0.8 10
10
0.6
8
10 6
10
0.4
4
F = 10 0.2
12
10 1
2
4
6
8
q
FIGURE 2.6 Detection probability D of a conventional correlator for a fully known signal for different probability of false alarm rates F vs. the signaltonoise ratio q.
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2.3 QUASIOPTIMAL DETECTORS FOR UWB SIGNALS As was shown previously, a real UWB signal scattered from a target has an intricate shape, as shown in Figure 2.7, and its parameters, such as a duration and a number, location and amplitude of signal maximums are unknown. The lack of a priori information on signal parameters makes it impossible to describe such a signal analytically and to introduce some a priori information about the signal into a signal processor. There are some other difficulties that can be added to those mentioned above. • Target multiple returns. The decrease in radiated pulse duration (nearly three orders of magnitude compared to conventional narrowband radar) increases the number of range resolution cells many times. A target will provide a series of returns from the combined scattering centers at each range resolution increment. The requirements for signal processor capacity and memory volume are increased correspondingly. • Target motion effects. Because of target movement, the scattered signals received in adjacent pulse repetition periods can arrive from different resolution cells. If a target has a radial velocity VR = 800 km/h and pulse repetition period Tr = 1 ms, target movement during this period is VR Tr = 22.2 cm. At the same time, the length of a resolution cell is only 15 cm when the pulse duration is 1 ns. This results in some difficulties such as accumulation and interperiod compensation when we use algorithms that process signals from various repetition periods. Target multiple returns and motion effects are the conditions under which UWB target return signals must be detected. In principle, it is possible to realize the procedure for optimal detection of an unknown target that has a large number of point scatters. The returns from point scatterers can be resolved into distinct “bright points.” Van der Spek first proposed such a processing algorithm in Ref. 8. Let us suppose that a target has the length L and occupies N resolution cells x1, x2, …, xN in space. The signals scattered by bright points are present in K cells, and the other cells are “empty.” Processing all combinations from N elements on K bright points can provide optimal detection of the unknown signal. This algorithm realizes the detection of a fully known signal, as one of these combinations must coincide with a signal scattered from a target. The schematic diagram for such an optimal detector is described in Ref. 8 and shown in Figure 2.8, which shows that a practical realization of this scheme requires many processing channels. For example, if the number of resolution cells is N = 40 within the observation interval, the signal bandwidth is 1 GHz, and the number of expected bright points is K = 8, then the number of processing channels required
t FIGURE 2.7 Power levels from an UWB signal scattered from a target when the range resolution is smaller than the target size. The target return becomes a series of lowenergy returns from scattering centers. This concept differs from that of the usual target radar cross section models in which the resolution is considerably larger than the target.
© 2001 CRC Press LLC
Const
( )
x1 x2
K
2
exp
exp
xk Threshold
xN
{
exp
Const FIGURE 2.8
N
Signal energy
CK channels
k
An optimal detector for a multiplepoint scatterer target.8
is 2.9 × 1010. The structure of such detector is very complex and cannot be realized using presentday electronic components. Van der Spek proposed two simpler algorithms that can realize quasioptimal processing of unknown signals.8 The first algorithm uses the changes of the energy at the detector output when a signal scattered by a target is received. It is shown in Ref. 8 that if N = K (where a scattered signal is present in all resolution cells within the observation interval), the optimal detector shown in Figure 2.8 is modified into a quadratic detector with a linear integrator as shown in Figure 2.9. In this case, the integration is performed over all N resolution cells, so there is no need to have a priori information on presence and location of K bright points. This detection scheme is called the energy detector. If we use this detector when K ≠ N, additional losses result from the summation of noise in “empty” resolution cells within an observation interval. By increasing the number of bright points K within this interval, the detection curve of energy detector approaches the detection curve for optimal detector for fully known signal. To reduce the losses in energy detector when K 1. This chapter will discuss signal correlation and signaltonoise ratio improvement, linear FM chirp signals and matched filtering, phasecoded waveforms used to provide greater performance improvement through time sidelobe suppression, and signal generation and detection methods.
4.8 SIGNAL CORRELATION AND SIGNALTONOISE RATIO IMPROVEMENT One strategy for signaltonoise ratio improvement is to multiply the received signal with a reference signal and then integrate the product over the known length of the signal, as shown in Figure 4.6. When the received and reference signals match, the integrated energy will give a high power output over an interval that is shorter than the signal duration. The ratio of signal length to correlator output signal length is called the compression ratio. The ratio of average received power to correlator © 2001 CRC Press LLC
x(t)
x(
h(t) H( )
)
y(t) Y( )
+
u1(t)
i(t) l( )
X u2(t)
Signal
2
S N
0
n1(t) N1( )
0
1 0
50
100
150
Time
x(t) X( )
+
n2(t) N2( )
Correlation value
Peak 2
Sidelobes 0
1 50
50
0
100
Time Shift Signal
Perturbing function
Additive noises
Multiplier
Integrating filter
Output
FIGURE 4.6 Generalized correlator block diagram. The received signal is multiplied by a reference signal and the product integrated over the signal time interval. The peak output power will be higher than the input power level depending on the ratio of the highest and lowest frequencies in the signal. © 2001 CRC Press LLC
output power depends on the methods of pulse compression or coding used. The pulse compression design problem is to find a waveform whose autocorrelation function provides a single peak output at one point, and a low output, or sidelobes, at all other times.
4.8.1
PULSE COMPRESSION THEORY
In 1957, Paul E. Green presented the analytical basis for using signal correlations to improve radar signal detection in “The Output Signal to Noise Ratio of Correlation Detectors,” and described a correlation detector that multiplies two waveforms and performs a smoothing, or integrating, function. The detector consists of a multiplier and integrating filter as shown in Figure 4.6.6 The two input signals u1(t) and u2(t) will have a nonzero correlation. For practical purposes, they are the signal waveform x(t) perturbed by additive noise n1(t) and n2(t), and possibly distorted other ways. Green’s analysis assumed that • All signal and noise components are independent, stationary, and ergodic random functions of time with Gaussian first and secondorder amplitude functions. • Signals are Fourier transformable and have power density spectra X(ω), N1(ω), and N2(ω), respectively, all confined to 2"W, a closed interval in ω. • All time functions have singlesided frequency spectra. • All network system functions use doublesided frequency spectra. • Only one correlator input includes a filter h(t). Multiple filters and effects can be lumped into the signal through h(t). • The instantaneous product of u1(t) and u2(t) is the ideal four quadrant multiplier output. • The integrating filter is a realizable two terminal device with a complex system function I(ω), which is Fourier transformable into the filter impulse response I(t). In the case where I(t) is a rectangular pulse of duration T, the filter is an ideal integrator with integration time T. The filter is tuned to ∆ so that the maximum frequency response is at ∆. Further ∆ is at least equal to 2πW, which is the bandwidth of significant values of the signal and noises. Other appropriate forms will have an effective integration time T equal to the reciprocal of the effective noise bandwidth of the filter, as shown in Figure 4.7.
4.8.2
PULSE COMPRESSION SIGNALTONOISE RATIO ANALYSIS
Green’s analysis can be summarized as follows. Given the correlator shown in Figure 4.6, determine the output signaltonoise ratio so that of the dc output of the integrating filterSNR = Square Fluctuation power at the same point This gives the general result for the bandpass detector case SNR where the effect of the integrating filter is W f = I max ( ω )
–2
∞
∫0
2
I ( ω ) ) dω
(4.7)
which is the effective noise bandwidth of the filter in radians per second. The resulting SNR expression is © 2001 CRC Press LLC
Output
2
T
0
2
0
50
100
150
time
a. Low pass filter output 2
Output
T 0
2π 2
0
50
100
150
time
b. Bandpass filter output Impulse response of ideal integrators used in the correlator analysis.
FIGURE 4.7
2
2
∞ ∞ ∫ X ( ω )Re [ H ( ω ) ] dω + ∫ X ( ω )Im [ H ( ω ) ] dw 0 0 S 1  =  ∞ N 0 W 2 2 2 f ∫ { X ( ω ) [ H ( ω ) ] + X ( ω )N1 ( ω ) + X ( ω ) H ( ω ) ( N2 ( ω ) + ∆ ) + N1 ( ω )N2 ( ω ) } 0
(4.8)
The reciprocal of Wf /2π is the effective integration time, since it produces the same (S/N)0 as an ideal integrating filter having a rectangular impulse response of duration 2π/Wf . For the lowpass filter case, the effective integration time is π/Wf . Applying Equation (4.8) to a restricted case and simplifying so that H(ω) = unity and so that the time function x(t), n1(t), and n2(t) have the same spectral shapes produces W S 1 1 1 = x ε +  +  +  N 0 FW ρ1 ρ2 ρ1 ρ2
–1
(4.9)
ε = a quantity equal to 2 for a lowpass filter or 1 for a bandpass detector Wx = the effective noise bandwidth of the signal and two noise values n1(t) and n2(t) ρ1 and ρ1 = signaltonoise power ratios X(ω)/N1(ω) and X(ω)/N2ω, respectively F(d) = a spectrum form factor where D is the density function and
where
© 2001 CRC Press LLC
∞
∫0 D ( ω )dω 2
F ( D ) = ∞ D max ( ω ) ∫ D ( ω ) dω
(4.10)
0
Table 14.1 shows F(D) form factor values. Simplifying further for the restricted case of two white noises N01 and N02 watts/radian/sec and H(ω) is unity, then W N 02 N 01 N 02 W′ N 01 S  = x εF +   +  + 2 N 0 Wf X max X max X max W x
–1
(4.11)
where Xmax = the maximum value of signal power density spectrum X(ω) W′ is the bandwidth of the two white (rectangular spectrum) noises and we assume that W′ is large enough to include all of X(ω). Note that Equation (4.11) is similar to Equation (4.10) except in defining the SNR in terms of power densities, where the signal power density is at the signal spectrum peak. The form factor enters a different way, and both these equations show that SNR depends on the ratio of the signal bandwidth to the filter bandwidth, or ratio of the filter integration time to the period of the signal. TABLE 14.1 Form Factor Values for Common Filters5 Type of Spectrum
Density Function D(ω) 1for ω in bandwith Ω 0 otherwise
Rectangular
– 2 ω – ω c /Ω for ω – ω c < Ω /2 1 + 0 otherwise
Triangular
Form Factor F(d) 1
2/3
Gaussian
exp [ – ( ω – ω c ) ]
12
Exponential
exp [ – ω – ω c ]
1/2
Firstorder Butterworth (single tuned RLC Circuit)
[ 1 – ω – ωc ]
nthorder Butterworth
[ 1 – ω – ωc
2
2
–1
2n
]
–1
1/2 1 – 1/2n
Correlator Output The correlator output for a signal x(t) correlated with a reference signal x(t – τ) will resemble the correlation function of the input waveform, expressed as 1 T lim   x ( t )x ( t – τ ) dt φ ( τ ) = T → ∞ 2T ∫–T
(4.12)
where T = integration time For practical computation in a sampled digital electronic correlator, the result of evaluating the average of a number of sample pairs of x(t) will be © 2001 CRC Press LLC
N
1 φ ( τ ) =  ∑ a n b n ( τ ) N
(4.13)
1
where N = the number samples and an and bn are samples of x(t) separated by the interval τ6 Figure 4.8 shows the autocorrelation waveforms of a single sine wave cycle, a square pulse, a four cycle square wave, and a linear chirped signal. The output reaches a maximum value at time shift τ = 0. The output has time sidelobes when the signal has many cycles, as shown in Figure 4.8c and d. In the case of Figure 4.8c, setting the detection threshold too low would result in multiple detections of the same target. Later sections will cover sidelobe size and minimization. There is no output value improvement unless the bandwidth is large, as shown in Figure 4.8d, where the chirp signal has a highesttolowest frequency ratio of 3, which gives an output signal three times larger than the input signal. The time sidelobes are small because the signal produces a high correlation only at one instant. This is the result expected from Equations (4.12) and (4.13). Figure 4.9 shows that the correlation process suppresses noise. When the input signal is mixed with noise, the correlated value of the noise and signal will be much lower, while the correlated output value of the signal will be higher. However, the correlated value of the signal will be higher than the noise level. If the noise level is large enough, then the correlated noise output can overcome the correlated signal output. This is what we would expect from Equation (4.11), which shows the output SNR as a function of the bandwidth and individual signal to noise ratios. The correlator output will also be smaller if the received signal is distorted.
4.8.3
PULSE COMPRESSION SUMMARY
Pulse compression increases the effective SNR by using a wide bandwidth coded signal. The SNR increase is limited by the ratio of the highest to the lowest frequency in the signal. The detector output will be the autocorrelation function of the signal waveform, which is a short, highpower peak and time sidelobes. The signal waveform affects the time sidelobes as will be shown later.
4.9 CORRELATOR OUTPUT TIME SIDELOBES AND PULSE COMPRESSION High range resolution radar applications may include target detection, object classification, imaging, terrain mapping, precision ranging, and distributed clutter suppression. There are two approaches to range resolution. The first is to use a simple shortduration, highpower pulse, or a lowpower, longduration wideband coded pulse. The detected coded pulse output will be a peak signal that occurs when the received and reference signals coincide in time, and accompanying time sidelobes that result from signal mismatch when they are not coincident, or τ > 0. When a compressed pulse waveform signal of duration T passes through a matched filter, the output will be 2T in duration and have a peak value proportional to the compression ratio as shown in Figure 4.9a. The responses outside of τ < τ are called range or time sidelobes. The range sidelobes from any given range bin may appear as signals in an adjacent range bin and must be controlled to avoid false alarms and multiple target indications. Sidelobe measures include peak sidelobe level (PSL), which is associated with the probability of a false alarm in a particular range bin due the presence of a target in a neighboring range bin. maximum sidelobe power Peak sidelobe level (PSL) = 10 log peak response © 2001 CRC Press LLC
(4.14)
20
Correlated value
Signal
2
0
−2
50
0
100
0
20
150
100
50
0
Time
50
100
50
100
50
100
Time shift
a. Signal sine wave cycle and correlator output 40 Correlator output
Signal
2
0
−2
50
0
100
20
0
20
150
100
50
0
Time
Time shift
b. Square pulse and correlator output 50 Correlated value
Signal
2
0
−2
0
100
50
0
50
150
100
50
Time
0 Time shift
c. Four cycle square wave and correlator output 20
Correlation value
Signal
2
0
−2 0
50
100
150
Time
10
0
10 50
0
50
100
Time shift
d. Chirped signal BW > 1/τ FIGURE 4.8 Correlation output examples. The correlator output will peak when the received and reference signal coincide. When they signals do not coincide, then the output is a lower value, called a time sidelobe. The objective in waveform design is to suppress the time sidelobe value as much as possible. © 2001 CRC Press LLC
2 Correlation value
Signal
2
0
2
0
50
100
0
2
150
50
0
50
100
Time shift
Time
a. Linear chirp signal and correlator output 2 Correlated noise output
Noise value
2
0
2
0
50
100
150
0
2
Time
50
0
b. Noise signal and correlator output
50
100
Time shift
2 Correlator output
Signal + Noise
2
0
2
0
50
100
150
Time
0
2
50
0
c. Signal plus noise and correlator output FIGURE 4.9
50
100
Time shift
Effects of noise on correlator output. Correlation reduces the output noise level.
The integrated sidelobe level (ISL) is a measure of the energy distributed in the sidelobes. It is important in dense target scenarios and when distributed clutter is present and quantifies the sidelobe level. Total side lobe power Integrated sidelobe level (ISL) = 10 log peak response
(4.15)
Loss in processing gain (LPG) is the loss in SNR when a the receiver has a mismatched, as opposed to matched, filter.7 CR Loss in processing gain (LPG) = 10 log peak response
(4.16)
LPG quantifies the loss in SNR performance due to using a mismatched filter in the receiver. © 2001 CRC Press LLC
Sidelobes are a natural product of correlation, but they can interfere with proper target detection or produce false targets, as shown in Figure 4.8c. Sidelobe suppression is a major consideration when selecting pulse compression waveforms.
4.9.1
LINEAR FM CHIRP SIGNALS
AND
TIME SIDELOBES
The linear frequency modulated, or linear FM chirp, is the simplest form of pulse compression signal. The linear FM chirp signal is widely used in radar applications and is a good place to start a discussion of practical pulse compression. Linear FM chirp detection commonly uses a technique called matched filtering, which produces an output described by the correlation function of the signal.8,9,10,11 Charles E. Cook presented an intuitive explanation of linear FM pulse compression in Ref. 8. Cook said that evolutionary trend of military radar is to extend detection range for a given size target. The obvious solution is to transmit more energy; however, this means a longer pulse length and less resolution. However, to increase detection range without degrading resolution means increasing the transmitter tube performance in terms of maximum and average output power. But more transmitted energy results in increased weight, prime power consumption, and system cost. The other approach is to enhance the SNR in the receiver by using pulse compression techniques to increase the detected signal power level. The pulse compression device became known as the matched filter and produces an output that is the autocorrelation of the transmitted signal. Looking at it from another way, the matched filter correlates the signal waveform against a reference stored as the filter transfer function. Matched filters are sometimes called North filters or conjugate filters. Early work by R.H. Dicke and S. Darlington proposed essentially identical approaches. Dicke reasoned that a linearly swept carrier frequency (as shown in Figure 4.10a, b, c, d, e, and f), when used with a matched filter with time delay vs. frequency characteristics shown in Figure 4.10g, would delay each frequency component and provide an output as shown in Figure 4.10h. The matched filter performs provides an output that is the correlation function of the input signal, so the correlator output does not follow the input signal form but gives a peak output value indicating the reception of a particular waveform. The matched filter output is proportional to the input signal cross correlated with a replica of the transmitted signal delay by time t1. For this case, cross correlation of signals y(t) and λ(t) is defined as R(t) =
∞
∫–∞ y ( λ )s ( λ – t ) dλ
(4.17)
The output of a filter with an impulse response h(t) when the input is yin(t) = s(t) + n(t) is y0 ( t ) =
∞
∫–∞ yin ( λ )h ( t – tλ ) dλ
(4.18)
For the matched filter case, then h(λ) = s(t1λ), so the previous equation becomes y0 ( t ) =
∞
∫–∞ yin ( λ )s ( t1 – t + λ ) dλ
= R ( t – t1 )
(4.19)
so that the matched filter output is the cross correlation between the received signal corrupted by noise, and a replica of the transmitted signal. The transmitted signal replica is built into the matched filter as its frequency response function. If the input signal is the same as the reference signal s(t) © 2001 CRC Press LLC
Amplitude
Aunc
T t t1
t2
(a) Amplitude vs time of linear FM chirp pulse. Frequency
fc f/2 f2
fc
fc + f/2
(e) Amplitude spectrum
f1 t1
t
t2
o
Amplitude
(b) Frequency vs time of linear FM chirp pulse
fc f/2
fc + f/2
(f) Phase spectrum Time delay
(c) Linear FM chirp pulse waveform
Amplitude
fc
t
A
t1
t2 t1
t
t2
f2 f 1 Frequency (g) Matched filter time delay characteristics
(d) Compressed linear FM chirp pulse output
o2(t)
4 dB T f
13.2 dB
Range sidelobes Small target response
time 1/ f 1/2f
+1/ f
1/ f
2T (h) Linear FM chirp matched filter output and sidelobe structure
FIGURE 4.10 Linear FM pulse compression: (a) transmitter waveform, (b) frequency of transmitted waveform, (c) representation of the time waveform, (d) output of the pulse compression filter, (e) amplitude spectrum, (f) phase spectrum, (g) chirp pulse output, and (g) matched filter response and sidelobe structure.
used to design the filter, then the output will be the autocorrelation function. Figure 4.8 shows some common autocorrelation functions.1 On a practical level, the compressed signal amplitude and phase spectra will look like Figure 4.11a. One approach to matched filter design is the bridgedT allpass network, shown in Figure 4.11b.8 © 2001 CRC Press LLC
AMPLITUDE
PHASE
13:1
COMPRESSION RATIO 13:1
wo
w/2 wo wo
wo
w/2
52:1
wo
w/2
wo
w/2
wo
wo
wo
w/2
52:1
wo
w/2
wo
w/2
130:1
wo
w/2
wo
wo
w/2
wo
w/2
130:1
wo
w/2
wo
w/2
wo
a. Post compression amplitude and phase spectra for various compression ratios
L1 C2 L2
L2
L3 R
R C1
b. General form of bridgedT allpass network used to implement matched filters for the above amplitude and phase spectra
FIGURE 4.11 Practical pulse compression spectra and a circuit for implementing them. (Reprinted with permission of IEEE from Ref 8.)
4.9.2
FM CHIRP SIGNAL TIME SIDELOBES
The linear FM chirp is a widely used pulse compression signal, because it is easy to generate, is insensitive to Doppler shifts, and has many ways to generate the signal. © 2001 CRC Press LLC
Linear FM chirp signals have disadvantages. First, excessive rangeDoppler cross coupling results when the Doppler shifted signal is correlated and shows a different time of arrival. Overcoming the range Doppler error requires knowing, or determining, the range, or Doppler, by other means. Figure 4.12 shows the effects of rangeDoppler coupling. Linear pulse compression also gives high time sidelobes, as shown in Figure 4.10g. The highest time sidelobe occurs at –13.2 dB below the peak output. This requires setting a high threshold level. Smaller targets may get lost in the time sidelobes. Nonlinear chirp waveforms are a way to suppress time sidelobes. For example, the waveform shown in Figure 4.13a has a frequency vs. time function of f ( t ) = W t + T where
K1 = K2 = K3 = K4 = K5 = K6 = K7 =
7
2πnt
 ∑ kn sin t
(4.20)
n=1
–0.1145 +0.0396 –0.0202 +0.118 –0.0082 0.0055 – 0.0040
This nonlinear FM chirp signal can produce a –40 dB time Taylor time sidelobe pattern. Nonlinear FM chirp signals have several advantages. They do not need time or frequency weighting for sidelobe suppression, because the FM modulation of the waveform provides the desired amplitude spectrum. They provide matched filter reception and low sidelobes that are
f1 + fd fd f1 FREQUENCY
Compresssible Segment
f2 ∆td
t1
t1 + ∆td
t2
Time
FIGURE 4.12 RangeDoppler coupling of linear FM. The signal frequency shift results in a miscorrelation and error in the estimated time of arrival. Also, less of the signal is correlated, which reduces the output signal strength. The miscorrelation will also produce higher time sidelobes. © 2001 CRC Press LLC
Frequency
3200
3100
3000 0
50
100
150
Time
a. A nonsymmetrical nonlinear chirp signal from Eqn 14 . This produces a.40 dB side lobe pattern
fo +B/2
Frequency
Frequency
fo +B/2
fo
foB/2
T
Time
b. Nonsymmetrical nonlinear chirp frequency vs time
fo
foB/2
T
Time
c. Symmetrical nonlinear chirp frequency vs time.
FIGURE 4.13 Nonlinear FM chirp signals: (a) Taylor series nonlinear chirp with a –40 dB time sidelobe pattern (b), nonsymmetrical nonlinear chirp, and (c) symmetrical nonlinear chirp.
compatible in design, and they eliminate signaltonoise losses associated with weighting. Figure 4.13b and c shows some typical nonlinear waveforms. However, nonlinear FM chirp signals also have disadvantages. They are more complex, and there has been limited development of nonlinearFM generator devices. They require a separate FM modulation design for each amplitude spectrum to achieve the required sidelobe level. Note that nonlinear FM signals can come in two nonsymmetrical and symmetrical forms, as shown in Figure 4.13. The nonsymmetrical form retains some of the Doppler cross coupling of the linear FM waveform. The symmetrical form frequency increases (or decreases) during the first half of the pulse (t ≤ T/2) and then decreases (or increases) during the second half, (t ≥ T/2).11,7
4.9.3
FREQUENCY STEPPED PULSE COMPRESSION
Frequency stepping is another pulse compression method that applies discrete frequency steps to the transmitted signal, as shown in Figure 4.14, with three different frequency stepping signals. All have the same pulse compression features at zero Doppler, but they have different characteristics when some Doppler shift is present. For the frequencystepped pulse of Figure 4.14a, τT = transmit subpulse length; τTc = compressed pulse length; N = number of subpulses; fk = frequency of the kth subpulse for k = 1, . . . ,N; ∆fs = fk – f k – 1 = subpulse frequency step for k = 1, . . . , N; and ∆f = N ∆fs = total frequency excursion. © 2001 CRC Press LLC
N
f1
f2
f3
fN1
Frequency
ττ fN
Time
Time
fN
f3
f2
f1
Frequency
(a) Discrete linear FM
fN1
Time
Time
Frequency
(b) Scrambled frequency stepping ττ f1
fN1
f2
fN
Time
T
Time
(c) Interpulse frequency stepping
tT = transmit subpulse length tc = compressed pulse length
N = number of subpulses f k = frequency of the kth subpulse for k = 1,...,N fs = f k – f k–1 = subpulse frequency step for k = 1,...,N Df = N Df s = total frequency excursion FIGURE 4.14 Frequency stepped waveforms: (a) Discrete linear FM, (b) scrambled frequency stepping, and (c) interpulse frequency stepping.
Assume that the transmit subpulse length ∆fs = 1/τT because ∆fs > 1/τT and ∆fs < 1/τT give undesirable compressed waveform characteristics.13 Then, the waveform approximates a linear FM waveform of duration T and with a total bandwidth B = N ∆fs = N/τc. The compressed pulse length becomes τ 1 τ c =  = T B N
(4.21)
Nτ 2 T CR =  = T = N τT τ T /N
(4.22)
and the pulse compression ratio will be
Frequency stepped waveform variations include a scrambled frequency coded pulse as shown in Figure 4.14b. This uses the same frequency components as the linearly stepped pulses of Figure 4.14a, except that they occur randomly in time. However the bandwidth is the same, which results in the same compression ratio CR = N2. The same analysis applies to the interpulse frequency stepping shown in Figure 4.14c. © 2001 CRC Press LLC
All the frequency stepped pulse waveforms respond the same way to zero Doppler conditions from zero radial velocity targets; however, when there is a Doppler shift, then their characteristics are different. The linearly stepped frequency pulse has characteristics similar to the linear FM chirp signal. The Doppler characteristics of the other two depend on the order and spacing employed. Edward C. Farnett and George H. Stevens discuss these waveforms and rangeDoppler characteristics in Skolnik’s Radar Handbook, 1990.13 Also see Nathanson.12
4.10
PHASECODED WAVEFORMS
Phase modulating a radar carrier signal with a square wave coded baseband signal is called phasecoded modulation and is another approach to getting low sidelobe values from a pulse compression signal. A phasecoded waveform has long similarly coded intervals at the beginning and progressively shorter intervals as the signal duration approaches its end. Figure 4.15 shows binary phase coding, transmitted waveform, correlator, and detector output waveforms. Only a few code sequences out of the many possible will produce all low sidelobes. Barker and complementary codes can produce minimal sidelobes.
4.10.1
BARKER CODES
Barker codes are binary phase codes that have autocorrelation, or sidelobe values less than or equal to 1/N in size, where N is the code length, and the maximum output is normalized to 1. Table 4.2A lists the known barker codes, including the longest known 13element code. The Barker code’s advantages are minimum possible sidelobe energy and uniformly distributed sidelobe energy. Therefore, Barker codes are sometimes called perfect codes. There are a number of nearly perfect longer codes with almost uniform and minimal energy sidelobes. McMullen gives a list of nearperfect codes in Ref. 14. TABLE 4.2A Known Barker Codes1 Code Length
Code Elements
PSL (dB)
ISL (dB)
+ – , ++
– 6.0
–3.0
++ – , + – +
–9.5
–6.5
4
++ – +, +++ –
–12.0
–6.0
5
+++ – +
–14.0
–8.0
1
+
2 3
7
+++ – – – + –
–16.9
–9.8
11
+++ – – – + – – + –
–20.8
–10.8
13
+++++ – – ++ – +
–22.3
–11.5
TABLE 4.2B A Combined Barker Code Example +
+
+
–
+
++ – +
++ – +
++ – +
– –+–
++ – +
Combined Barker Codes The Barker code’s 13bit maximum size limits the SNR improvement in radar applications. However, the known Barker codes can be combined to generate sequences longer than 13 bits with low © 2001 CRC Press LLC
T=13τ τ
+
1
+
+
+
+


+
+

+

+
Time
(a) Thirteen bit binary coded sequence coding and binary phase code waveform Tapped Delay Line
+
+
+
+
+


+
+

+

+
INPUT
Filter Matched to Pulse of Width τ
(b) Autocorrelation computation using a tapped delay line with weighting
Amplitude
13
1 13τ
τ 0 +τ T
+13 τ
Time
T
(c) Correlator output format for a 13 bit coded sequence
FIGURE 4.15 Binary phasecoded pulse compression signal and autocorrelation function.
sidelobes. For example, a 20:1 pulse compression ratio system can be made using either the 5 groups of fourelement, or four groups of fiveelement Barker codes. The five group of fourelement Barker code uses a fivebit Barker code, where each bit is a fourbit Barker code. Table 4.2B shows an example 20bit Barker code. The combined code correlator is a combination of filters matched to the individual codes. Individual codes (and filters) are called subcodes (or subsystems or components) of the full code (system). A directly implemented correlator would consist of a tapped delay line whose impulse response is the time inverse of the code as a combination of subcodematched filters. © 2001 CRC Press LLC
Figure 4.16 shows an example of a combined filter implementing a 5 × 4 combined Barker code. The first filter stage, on the upper right, is the matched filter for the inner (fourbit) code. The second stage is the fivebit code matched filter, with the active taps spaced four taps apart. This filter is the equivalent to the 20tapped delay line matched filter, which has an identical impulse response. However, the number of active arithmetic elements (+, –) in the combined filter is 9, the sum of the subcode lengths. The equivalent 20bit code filter would have 20 elements, the product of the subcode lengths. These general results apply to codes that are the combination of any number of subcodes. The decreased number of arithmetic elements for combined Barker matched filter has some advantages. First, there are many possible codes of various lengths appropriate for different modes of a radar system, such as surveillance, tracking, and identification. Second, the combined Barker code requires less arithmetic processing elements than a single tapped delay line matched filter. Third, the procedure for finding the tap weights of an nlength ISL optimized filter requires solving n linear equations in n unknowns. As n grows longer, the solution is more difficult. Sidelobe reduction techniques may be used for a long code by combining filters optimized for each subcode.15 Fourth, the designer needs to know only the sidelobe characteristics of each component of the code, so determining the combined sidelobe characteristics is easy. For example, the PSL of a system is approximately the PSL of its weakest component. The ISL is approximately the root sum of the squares of the subsystem’s ISLs. The LPG is approximately the sum of individual LPGs. However, the ISL and LPG show some sensitivity to order.16
4.10.2
PSEUDORANDOM CODES
Pseudorandom sequences (PN codes) are another approach to signal coding for pulse compression with lower but not minimal sidelobes. PN codes are easy to generate, have good sidelobe properties, and are easily changed algorithmically. Maximallength PN codes (maximallength binary shift register sequences, maximallength sequences, or simply msequences) are the most useful. These have many potential radar and spread spectrum communications system applications.2 A binary shift register with feedback connections that can generate a PN code is shown in Figure 4.17. The shift register is initialized in a nonzero state and the system clocked to circulate the bits, which are picked off at appropriate outputs. This process generates a 2n – 1 sequence where n is the number of shift registers. To get a maximal length, or nonrepeating, sequence, the feedback paths must correspond to the nonzero coefficients of an irreducible, primitive polynomial
+
+
+ 0 0 0 + 0 0 0 + 0 0 0

−
+
0 0 0 +
Output to Detector FIGURE 4.16 A correlator block diagram for a 5 × 4 combined Barker code. © 2001 CRC Press LLC
Code Input
modulo 2 of degree n. For example, a polynomial of degree five is (1) + (0)x + (0)x2 + (1)x3 + (0)x4 + (1)x5, as shown in Figure 4.17. Note that the constant 1 term corresponds to the adder feedback to the first bit of the register. The 1 value coefficients of the x3 and x5 terms correspond to feedback from the adder to the third and fifth in the register. Generally, the shift registers may be initialized in any nonzero state to generate a msequence. Figure 4.17 shows an initial state of 0,1,0,0,0. The properties of maximum length pseudorandom codes are given in Table 4.3. A designer can construct msequence PN code generators from degree 1 to degree 34 (output length 234 – 1) using the list of irreducible polynomials in Peterson and Weldon’s, Error Correcting Codes.17 There are some practical PN code considerations. First, notice that for a large N = 2n – 1, the 1⁄2 peak sidelobe is approximately ( 1 ⁄ N ) in voltage when the signal is normalized to 1. The values depend on the particular sequence. For example, with N = 127, (n = 7) the PSL varies between –18 and –19 dB, as opposed to the –21 dB predicted by the rule of thumb. The rule of thumb approximation improves as N increases. Second, for a continuous, periodic flow of PN codes through a matched filter, the output is a periodic peak response of N (in voltage) and a flat range sidelobe response of –1. Third, PN codes are appropriate for pulsed radar applications in which only a few closely spaced targets are expected in the field of view. These PM waveforms are unsuitable for hightargetdensity and extended clutter situations because of their relatively high ISL level. Fifth, PN codes must use either a fully tapped delay line or bank of shift registers for each code bit to compress the signal at all ranges. Therefore, PN msequences have been more popular in communications and CW radar applications than in pulsed radar applications.
4.10.3
POLYPHASE CODES
The Barker and combined Barker codes are biphase codes that carry information as the 180° phase shift of the carrier signal (ϕ). A more complicated wave for is possible using M multiple phase shift conditions such that 2π' φ k =  k M
(4.23)
for k = 0, . . . , M – 1, to code a long constant amplitude pulse. Proper design will produce a desired matched filter output with suitable range sidelobes and a sufficient peak value for detection above noise.There are several approaches including Frank, Welty, and Golay codes. (0)X 1
(0)X 2
(0)X 3
(0)X 4
(1)X 5
0
1
0
0
0
(1)X 0
(2)
FIGURE 4.17 Maximal length binary shift register with initial state. Used for generating a degree five pseudorandom sequence code for pulse compression. (Adapted from Ref. 7, Figure 159.) © 2001 CRC Press LLC
TABLE 4.3 Maximum Length Pseudorandom Codes and Their Properties12 Degree (number of stages) and length
Polynomial octal
Lowest peak sidelobe amplitude
Initial** conditions, decimal
Lowest RMS sidelobe amplitude
Initial conditions, decimal
1(1)
003*
0
1
0.0
2(3)
007*
–1
1,2
0.707
1,2
3(7)
013*
–1
6
0.707
6
4(15)
023*
–3
1,2,6,9,10,12
1.39
2,8
5(31)
045*
–4*
5,6,26,29 (9 conditions) 2,16,20,26
1.89 1.74 1.96
6,25 31 6
6(63)
10.3*
–6
1,3,7,10 26,32,45,54 (9 conditions) (9 conditions)
2.62
35 7
1
2.81 2.38
7(127)
203* 211* 235 247 253 277 313 357
–9 –9 –9 –9 –10 –10 –9 –9
1,54 9 49 104 54 14,20,73 99 15,50,78,90
4.03 3.90 4.09 4.23 4.17 4.15 4.04 4.18
109 38 12 24,104 36 50 113 122
8(225)
435 453 455 515 537 543 607 717
–13 –14 –14 –14 –13 –14 –14 –14
67 (20 conditions) 124,190,236 54 90 (10 conditions) (6 conditions) 124,249
5.97 5.98 6.1 6.08 5.91 6.02 6.02 5.92
135 234 246 218 90 197 15 156
*Only single modtwo adder required. ** Mirror images not shown. Source: Nathanson, Radar Design, p. 465.
Frank Codes Frank polyphase codes can provide a discrete approximation of a linear FM chirp waveform.13,19 For each integer M, there generally is a Frank code of length N = M2 that uses phase shifts 2π/M, 2(2π/M), . . . , (M – 1)(2π/M), 2π. The Frank code of length N will have peak signal to sidelobe approaching π N for large values of N (as opposed to N for pseudorandom codes). The Frank code is an alternative to pseudorandom codes when the radar application involves expected extended clutter or a highdensity target environment. Because the Frank code discretely approximates linear FM signals, the autocorrelation functions degrade due to Doppler shifting. However, the degradation is not as fast as in binary phase shift codes. So the Frank code has © 2001 CRC Press LLC
potential applications where binary code Doppler sensitivity is a problem. Frank codes show the rangeDoppler coupling inherent in linear FM waveforms. Because Frank codes are discrete, the smooth peak response degradation of linear FM waveforms may appear as a loss of detection at certain intermediate velocities, that is, blind speeds. Designers need to consider this property when designing Frank codes. Codes similar to Frank codes, but with better Doppler tolerance are described by Ketschmer and Lewis in Ref. 19. Welty and Golay Codes Welty and Golay codes are another approach to total sidelobe cancellation. They have a unique property of providing equal value but negative sidelobes. Adding the two correlator outputs, as shown in Figure 4.18, can reduce the sidelobes to zero and double the output peak. Welty codes are a general set of polyphase codes having this property. Golay codes are a subset of these sidelobecanceling codes that are binary phase codes. These advantages are purchased at the expense of increased signal processing complexity and the need for two sets of matched filters and a summing junction.
2
Code B
Correlator output
40
20
0
0 2 0
50
20
100
Time
50
0
50
Time shift
+
+
+



+

+ + +   + 
+
+
+

+
+

+
++++++
Summing Junction
100
1, 0 1 0 3 0 1 8 1,...
50
0
50 50
0
50
Time Shift
0,0,0,0,0,0,16 1,0,1,0,3,0,1,8,1,... Correlator output
40
Code B
2
20
2 0
50 Time
0
20 50
0
50
Time shift
FIGURE 4.18 An example of a Golay (sidelobecanceling) code pair of length 8. © 2001 CRC Press LLC
0
100
4.11
PULSE COMPRESSION WAVEFORM GENERATION AND PROCESSING
The advantages of pulse compression have associated costs in system complexity and losses. This section describes methods for generating phasecoded waveforms and waveform processing methods.
4.11.1
GENERATING PHASECODED WAVEFORMS
Radar transmitters may generate phasecoded waveforms using either active or passive methods, as shown in Figure 4.19. Passive Analog Filter It is theoretically possible to design an analog filter that will give any desired waveform hr(t) for an impulse input. Practically, this is limited to certain waveforms, the most important being the linear FM waveform, which is generated by sending an impulse into a dispersive delay line and then band limiting and gating the output. Figure 4.19a shows a passive analog filter. Memory Readout Figure 4.19b shows how a waveform can be digitized, or computed, at equal time intervals. The discrete values are clocked out of a shift register and converted to an analog signal that is then up converted to the transmitter frequency and transmitted. Active Generation Figure 4.19c shows a programmed control voltage driving a voltage controlled oscillator (VCO) to generate linear or nonlinear FM pulsed waveforms up to several hundred megahertz. Active Phase Coder Figure 4.19d shows how a codecontrolled signal directs a sine wave pulse signal between a 0 and 180° phase shifter to generate a biphase code. Multiple phase codes and other complicated waveforms would use additional phase shifters. Burst Generator Figure 4.19e shows a coherent comb generator producing a series of discrete frequencies. A code controller selectively transmits single frequencies or combinations.15
4.11.2
PASSIVE IF WAVEFORM PROCESSING
An analog filter can theoretically have any transfer function Hr(f), but there are practical limits on what can be done. Lumped constant LRC filters are primarily used for bandpass and spectrum weighting applications and combined with dispersive acoustic delay lines, as shown in Figure 4.20. Dispersive Delay Lines Dispersive delay lines can take several forms, as shown in Figure 4.21. Generally, these convert the electrical signal into sound waves. The signal is filtered by acoustic diffraction and transit times differences, and then by converting acoustic waves back into electrical signals. The problem is that large losses occur when converting electrical signals to sound and back again. Dispersive delay lines may require gains from 30 to 60 dB ahead of a dispersive acoustic filter. Table 4.4 shows typical delay line characteristics. © 2001 CRC Press LLC
Gate
h,(t) (a) Passive analog filter generation
D/A Storage registers
Control voltage
frequency
(b) Memory readout generation
voltage
Gate
(c) Active generation 180 0
Code Control
(d) Active phase coder f1, O1
f1, O1
f2 , O2
fn,On
f2 , O2 fn,On
Comb generator
Code Control
(e) Burst generator FIGURE 4.19 Different pulse compression waveform generators. (Adapted from Ref. 16, Figures 26 and 27.) © 2001 CRC Press LLC
Hr(f)
Envelope Detector
a. Lumped constant pulse detection
b. Dispersive delay (linear FM)
c. Surface wave tapped delay line FIGURE 4.20 Passive analog IF waveform processing methods. (Adapted from Ref. 16, Figure 28.)
TABLE 4.4 Table of Typical Dispersive Delay Characteristics20 f0 (MHz)
B x f0
τΒ
τ (µs)
Insertion Loss (dB)