Information-Theoretic Radar Signal Processing E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 678

Veröffentlichungsjahr: 2024

Beschreibung

A comprehensive introduction to the emerging research in information-theoretic radar signal processing

Signal processing plays a pivotal role in radar systems to estimate, visualize, and leverage useful target information from noisy and distorted radar signals, harnessing their spatial characteristics, temporal features, and Doppler signatures. The burgeoning applications of information theory in radar signal processing provide a distinct perspective for tackling diverse challenges, including optimized waveform design, performance bound analysis, robust filtering, and target enumeration.

Information-Theoretic Radar Signal Processing provides a comprehensive introduction to radar signal processing from an information theory perspective. Covering both fundamental principles and advanced techniques, the book facilitates the integration of information theory into radar signal processing, broadening the scope and improving the performance. Tailored to the needs of researchers and students alike, it serves as a valuable resource for comprehending the information-theoretic aspects of radar signal processing.

Information-Theoretic Radar Signal Processing readers will also find:

• Presentation of alternative hypotheses in adaptive radar detection
• Detailed discussion of topics including resource management and power allocation
• Direction-of-arrival (DOA) estimation and integrated sensing and communications (ISAC)

Information-Theoretic Radar Signal Processing is ideal for graduate students, scientists, researchers, and engineers, who work on the broad scope of radar and sonar applications, including target detection, estimation, imaging, tracking, and classification using radio frequency, ultrasonic, and acoustic methods.

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IEEE Press445 Hoes LanePiscataway, NJ 08854

IEEE Press Editorial BoardSarah Spurgeon, Editor-in-Chief

Moeness Amin

Jón Atli Benediktsson

Adam Drobot

James Duncan

Ekram Hossain

Brian Johnson

Hai Li

James Lyke

Joydeep Mitra

Desineni Subbaram Naidu

Tony Q. S. Quek

Behzad Razavi

Thomas Robertazzi

Diomidis Spinellis

Information-Theoretic Radar Signal Processing

Edited by

Yujie Gu

Aptiv, USA

Yimin D. Zhang

Temple University, USA

Copyright © 2025 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.

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To my devoted family for their endless love, unwavering support, and constant encouragement

—Yujie Gu

To my loving family whose unwavering support and boundless love have illuminated my entire academic career

—Yimin D. Zhang

About the Editors

Yujie Gu received his PhD degree from Zhejiang University, Hangzhou, China, in 2008. Currently, he works at Aptiv Advanced Engineering Center as a Senior Radar Scientist focusing on research and development of advanced automotive radar systems. His research experience was in statistical and array signal processing, including robust adaptive beamforming, information-theoretic compressive sensing, spatial spectrum estimation, and performance bound analysis, with applications to radar, wireless communications, and radio astronomy imaging. Dr. Gu is a senior member of the Institute of Electrical and Electronics Engineers (IEEE). He is an elected member of the IEEE Sensor Array and Multichannel (SAM) Technical Committee and the IEEE Signal Processing Theory and Methods (SPTM) Technical Committee. Dr. Gu is an Associate Editor for IEEE Transactions on Signal Processing, a Subject Editor-in-Chief for IET Electronics Letters, and an Editor for Elsevier Signal Processing. He was a Special Sessions Co-Chair of the 2020 IEEE Sensor Array and Multichannel Signal Processing Workshop. Dr. Gu was the recipient of the 2019 IET Communications Premium Award. He is the corresponding author of a paper that received the 2021 IEEE Signal Processing Society Young Author Best Paper Award.

Yimin D. Zhang received his PhD degree from the University of Tsukuba, Tsukuba, Japan, in 1988. He is currently an Associate Professor with the Department of Electrical and Computer Engineering, Temple University, Philadelphia, PA, USA. His research interests include array signal processing, compressive sensing, machine learning, information theory, convex optimization, and time-frequency analysis with applications to radar, wireless communications, and satellite navigation. Dr. Zhang is a Senior Area Editor for IEEE Transactions on Signal Processing and an Editor for Signal Processing. He was an Associate Editor for IEEE Transactions on Signal Processing, IEEE Transactions on Aerospace and Electronic Systems, IEEE Signal Processing Letters, and Journal of the Franklin Institute. He was a Technical Co-Chair of the 2018 IEEE Sensor Array and Multichannel Signal Processing Workshop. He was the recipient of the 2016 IET Radar, Sonar & Navigation Premium Award, the 2017 IEEE Aerospace and Electronic Systems Society Harry Rowe Mimno Award, the 2019 IET Communications Premium Award, and the 2021 EURASIP Best Paper Award for Signal Processing. He coauthored two papers that, respectively, received the 2018 and 2021 IEEE Signal Processing Society Young Author Best Paper Awards. Dr. Zhang is an IEEE Signal Processing Society Distinguished Lecturer. He is a Fellow of the Institute of Electric and Electronics Engineers (IEEE) and of International Society for Optical Engineering (SPIE).

List of Contributors

Pia Addabbo

Università degli Studi Giustino Fortunato

Benevento

Italy

Kristine Bell

Metron, Inc.

Reston, VA

USA

Daniel W. Bliss

Center for Wireless Information Systems and Computational Architectures (WISCA)

Arizona State University

Tempe, AZ

USA

Nianxia Cao

Advanced Safety and User Experience

Aptiv

Agoura Hills, CA

USA

Badong Chen

Institute of Artificial Intelligence and Robotics

Xi’an Jiaotong University

Xi’an

China

Lujuan Dang

Institute of Artificial Intelligence and Robotics

Xi’an Jiaotong University

Xi’an

China

Fuwang Dong

Department of Electronic and Electrical Engineering

Southern University of Science and Technology

Shenzhen

China

Alejandro C. Frery

School of Mathematics and Statistics

Victoria University of Wellington

Wellington

New Zealand

Gaetano Giunta

Department of Industrial, Electronic and Mechanical Engineering

University of Roma Tre

Rome

Italy

Hana Godrich

Department of Electrical and Computer Engineering

Rutgers University

Piscataway, NJ

USA

Nathan A. Goodman

Advanced Radar Research Center

School of Electrical and Computer Engineering

University of Oklahoma

Norman, OK

USA

Yujie Gu

Advanced Safety & User Experience

Aptiv

Agoura Hills, CA

USA

Sora Haley

College of Engineering and Computer Science

Syracuse University

Syracuse, NY

USA

Lei Huang

State Key Laboratory of Radio Frequency Heterogeneous Integration

Shenzhen University

Shenzhen

China

Zacharie Idriss

Radar Division

U.S. Naval Research Laboratory

Washington, DC

USA

Chris Kreucher

KBR Wyle Services

LLC

Ann Arbor, MI

USA

Bin Liao

Guangdong Key Laboratory of Intelligent Information Processing

College of Electronics and Information Engineering

Shenzhen University

Shenzhen

China

Fan Liu

Department of Electronic and Electrical Engineering

Southern University of Science and Technology

Shenzhen

China

Engin Masazade

Department of Electrical and Electronics Engineering

Faculty of Engineering

Marmara University

Istanbul

Türkiye

Ram M. Narayanan

School of Electrical and Computer Engineering

The Pennsylvania State University

University Park, PA

USA

Abraão D. C. Nascimento

Departamento de Estatística

Universidade Federal de Pernambuco

Recife

Brazil

Arye Nehorai

Department of Electrical and Systems Engineering

Washington University in St. Louis

St. Louis, MO

USA

Danilo Orlando

Department of Information Engineering

University of Pisa

Pisa

Italy

Bryan Paul

General Dynamics Mission Systems

Fairfax, VA

USA

Siyuan Peng

School of Information Engineering

Guangdong University of Technology

Guangzhou

China

Athina P. Petropulu

Department of Electrical and Computer Engineering

Rutgers University

Piscataway, NJ

USA

H. Vincent Poor

Department of Electrical and Computer Engineering

Princeton University

Princeton, NJ

USA

Jose C. Principe

Department of Electrical and Computer Engineering

University of Florida

Gainesville, FL

USA

Raghu G. Raj

Radar Division

U.S. Naval Research Laboratory

Washington, DC

USA

Muralidhar Rangaswamy

Air Force Research Laboratory

Dayton, OH

USA

Zhiguo Shi

College of Information Science and Electronic Engineering

Zhejiang University

Hangzhou, Zhejiang

China

Hing Cheung So

Department of Electrical Engineering

City University of Hong Kong

Hong Kong

China

Petre Stoica

Department of Information Technology

Uppsala University

Uppsala

Sweden

Bo Tang

College of Electronic Engineering

National University of Defense Technology

Hefei

China

Jun Tang

Department of Electronic Engineering

Tsinghua University

Beijing

China

Pramod K. Varshney

College of Engineering and Computer Science

Syracuse University

Syracuse, NY

USA

Peng Xiao

School of Marine Engineering and Technology

Sun Yat-sen University

Zhuhai

China

Yifeng Xiong

School of Information and Communication Engineering

Beijing University of Posts and Telecommunications

Beijing

China

Qianhui You

Guangdong Key Laboratory of Intelligent Information Processing

College of Electronics and Information Engineering

Shenzhen University

Shenzhen

China

Yimin D. Zhang

Department of Electrical and Computer Engineering

Temple University

Philadelphia, PA

USA

Zongyu Zhang

College of Information Science and Electronic Engineering

Zhejiang University

Hangzhou, Zhejiang

China

Preface

The roots of information theory can be traced back 100 years to the early works of Fisher [1], Nyquist [2], Hartley [3], and others. In [1], Fisher laid down a cornerstone by defining statistical information as the reciprocal of the variance of a statistical sample. This fundamental concept was later independently derived by Cramér and Rao, who utilized it to establish the lower bound on parameter estimation variance [4, 5]. Therefore, information theory has been naturally closely related to signal processing from the beginning.

After World War II, both disciplines of information theory and signal processing witnessed remarkable advancements. In 1948, Shannon published his groundbreaking article, “A mathematical theory of communication” [6], which served as a lodestar for the burgeoning discipline of information theory. In the same year, the Institute of Radio Engineers (IRE), the predecessor of the Institute of Electrical and Electronics Engineers (IEEE), inaugurated its first society, the Signal Processing Society. This historical coincidence further symbolized the close relationship between information theory and signal processing.

Information theory originally emerged to explore the communication of messages. Its core principles include quantification of information, source coding, and channel coding. Shannon’s contributions fostered a vibrant community of scholars, leading to the establishment of the IEEE Information Theory Society in 1951. His legacy continues to influence the fields of communications, information theory, and signal processing.

The earlier works of Woodward and Davies in the early 1950s [7, 8] marked the beginning of information-theoretic signal processing for radar applications. However, unlike communication systems, which aim to extract information from received signals in a cooperative framework, radar systems operate in noncooperative environments, seeking information about unknown targets, such as their range, velocity, and angle. This inherent disparity in understanding of information has resulted in a notable lag in the development of information-theoretic processing for radar systems compared with the communication counterparts.

Information theory experienced a resurgence within the radar signal processing community following a period of stagnation spanning over three decades. A key turning point was the publication of Bell’s PhD dissertation in 1988 [9], wherein mutual information was used for the first time as an optimization criterion to design radar waveforms that maximize the target information extracted from the received measurements. Since then, information theory has witnessed a renaissance in radar signal processing, especially in the era of multiple-input multiple-output (MIMO) radar. In 2005, Guo et al. [10] provided further insights into the theoretical connection between mutual information in information theory and minimum mean square error (MMSE) in estimation theory. Nowadays, information-theoretic criteria, including Fisher information, Shannon entropy, mutual information, and Kullback–Leibler divergence (also known as relative entropy), have become the foundation of adaptive radar systems and found great success in solving various radar signal processing problems. This book provides a comprehensive introduction to information-theoretic criteria, methods, and applications in radar signal processing with a focus on the latest theoretical and practical advances in selected important topics.

This book is intended not only for radar signal processing researchers, but also for postgraduate and PhD students, researchers, and engineers working on a broad spectrum of signal processing and its applications, including but not limited to radar systems. The readers are assumed to have some background in linear algebra, probability and statistics, compressive sensing, detection and estimation, control and optimization, information theory, as well as radar systems.

The book contains fourteen self-contained chapters, each focusing on a specific topic. These chapters, contributed by leading experts from academia, industry, and government, can be broadly categorized into six areas: target detection (Chapters 1–3), parameter estimation (Chapters 4 and 5), radar imaging (Chapters 6 and 7), target tracking (Chapters 8–10), resource management (Chapters 11 and 12), and performance bound analysis (Chapters 13 and 14).

Target detection

is the primary mission of many radar systems. In

Chapter 1

, the authors (Bo Tang, Jun Tang, and Petre Stoica) employ the relative entropy metric to design MIMO radar waveforms under various practical constraints in order to enhance the detection performance of radar systems. In

Chapter 2

, the authors (Pia Addabbo, Danilo Orlando, and Gaetano Giunta) utilize the Kullback–Leibler divergence criterion to develop adaptive detection architectures for solving multiple hypothesis testing problems in radar applications. In

Chapter 3

, the authors (Lei Huang and Hing Cheung So) develop a linear shrinkage minimum description length (MDL) criterion and two shrinkage coefficient-based detectors for source enumeration in a computationally efficient way.

Parameter estimation

is one of the fundamental functions of radar systems. In

Chapter 4

, the authors (Yujie Gu, Nathan A. Goodman, and Yimin D. Zhang) adopt the maximum mutual information criterion to optimize the compressive sensing kernel over the Stiefel manifold to achieve enhanced time delay estimation performance. In

Chapter 5

, the authors (Bin Liao, Qianhui You, and Peng Xiao) exploit the normalized ℓ1 Shannon entropy function to develop an entropy-enhanced one-bit compressive sensing algorithm for direction-of-arrival (DOA) estimation.

Radar imaging

becomes increasingly important in modern high-resolution radar systems, where man-made radar targets often extend over multiple range bins when illuminated by wideband radar signals. In

Chapter 6

, the authors (Zacharie Idriss, Raghu G. Raj, and Ram M. Narayanan) optimize radar waveforms for multistatic radar imaging by maximizing the mutual information between the received signal and the scene of interest, where natural scenes are sparsely represented by wavelet coefficients. In

Chapter 7

, the authors (Alejandro C. Frery and Abraão D. C. Nascimento) present information-theoretic results in conjunction with statistical inference for the processing and comprehension of synthetic aperture radar (SAR) and polarimetric SAR (PolSAR) imagery.

Target tracking

is another important task in radar applications to locate moving targets over time. In

Chapter 8

, the authors (Nianxia Cao, Pramod K. Varshney, Engin Masazade, and Sora Haley) propose an information-theoretic sensor selection algorithm for target tracking in sensor networks that strikes a balance between the computational complexity and the tracking performance. In

Chapter 9

, the authors (Siyuan Peng, Lujuan Dang, Badong Chen, and Jose C. Principe) introduce the minimum error entropy (MEE) criterion for the derivation of adaptive filters that exhibit robustness against non-Gaussian noise. In

Chapter 10

, the authors (Bryan Paul and Daniel W. Bliss) utilize the estimation information rate as a metric to dynamically manage radar systems, resulting in a target scheduling algorithm and a resource allocation scheme for the generalized radar tracking Bayesian filter.

Resource management

helps to improve cost efficiency of radar systems, including reducing system size, enhancing spectrum utilization, and minimizing power consumption. In

Chapter 11

, the authors (Hana Godrich, Athina Petropulu, and H. Vincent Poor) propose two power allocation strategies for target localization in distributed multiple-radar architectures that, respectively, prioritize estimation accuracy and power consumption. In

Chapter 12

, the authors (Kristine Bell, Chris Kreucher, and Muralidhar Rangaswamy) develop a fully adaptive radar resource management scheme that emulates the perception–action cycle of cognition and uses the mutual information criterion to design future radar waveforms for the optimized estimation of the state of a surveillance region.

Performance bound analysis

provides an efficient way to understand the performance of radar systems and insightful guidelines for radar system design and algorithm development. In

Chapter 13

, the authors (Yifeng Xiong, Fuwang Dong, and Fan Liu) derive the information-theoretic performance boundaries that govern the tradeoff between the sensing accuracy and communication rate in integrated sensing and communication (ISAC) systems. In

Chapter 14

, the authors (Zongyu Zhang, Zhiguo Shi, and Arye Nehorai) devise a global tight Ziv–Zakai bound as a linear combination of the

a priori

bound and the Cramér–Rao bound for evaluating the performance of multisource DOA estimation.

The organization of this book allows readers to gain a comprehensive and deep understanding of key radar signal processing topics through the lens of information theory, and to learn how information-theoretic criteria and methods can be applied to design and implement efficient and effective radar systems. The independence of each chapter also provides the flexibility for instructors to select a subset of chapters for university courses or short seminars.

References

1

Fisher, R.A. (1925). Theory of statistical estimation.

Mathematical Proceedings of the Cambridge Philosophical Society

22(5): 700–725.

2

Nyquist, H. (1924). Certain factors affecting telegraph speed.

Bell System Technical Journal

3(2): 324–346.

3

Hartley, R.V.L. (1928). Transmission of information.

Bell System Technical Journal

7(3): 535–563.

4

Rao, C.R. (1945). Information and the accuracy attainable in the estimation of statistical parameters.

Bulletin of the Calcutta Mathematical Society

37: 81–91.

5

Cramér, H. (1946).

Mathematical Methods of Statistics

. Princeton, NJ: Princeton University Press.

6

Shannon, C.E. (1948). A mathematical theory of communication.

Bell System Technical Journal

27: 379–423 and 623–656.

7

Woodward, P.M. and Davies, I.L. (1951). A theory of radar information.

Philosophical Magazine

41: 1101–1117.

8

Woodward, P.M. (1953).

Probability and Information Theory with Application to Radar

,

2

e. London: Pergamon.

9

Bell, M.R. (1988). Information theory and radar: mutual information and the design and analysis of radar waveforms and systems. PhD dissertation. Pasadena, CA: California Institute of Technology.

10

Guo, D., Shamai, S.(S.), and Verdú, S. (2005). Mutual information and minimum mean-square error in Gaussian channels.

IEEE Transactions on Information Theory

51(4): 1261–1282.

1Information-Theoretic Waveform Design for MIMO Radar Target Detection

Bo Tang1, Jun Tang2, and Petre Stoica3

1College of Electronic Engineering, National University of Defense Technology, Hefei, China

2Department of Electronic Engineering, Tsinghua University, Beijing, China

3Department of Information Technology, Uppsala University, Uppsala, Sweden

1.1 Introduction

Information theory is a fundamental tool to study how to communicate reliably in noisy channels [1]. The analysis of the ultimate data compression rate and the highest data transmission rate in communications is a main focus of information theory [2]. After nearly one century of research, many elegant results about the data compression and transmission rate have been developed in information theory. These studies have achieved tremendous success in many areas, including but not limited to communication system design, computer science, statistics, and stock market investment.

An early attempt at using information theory as a guideline for designing radar systems dates back to Woodward’s work in the 1950s [3–5]. These cited references pointed out that a well-established and easy-to-understand concept in the radar community, viz., the signal-to-interference-plus-noise ratio (SINR), did not measure information except for the Gaussian case and certain special elliptical contoured distributions [6]. Thus, an “information function” was defined, which was related to the posterior distributions of the observations in the radar receiver. In addition, a posterior receive filter was proposed based on the information function. Besides the receiver design based on information-theoretic approaches, there have been even more attempts to design radar transmit waveforms based on information theory. In [7], the author considered the design of radar waveforms to maximize the mutual information between the target response and the received signals. By connecting the mutual information to the rate-distortion function, the study showed that the mutual information maximization resulted in the reduction of the parameter estimation uncertainty. Therefore, it is meaningful to maximize the mutual information through waveform design. The results in [7] showed that the optimal waveforms maximizing the mutual information had a “water-filling” interpretation. In [8, 9], the authors investigated the design of clutter-resistant radar waveforms. The results therein also showed that the optimal waveforms admitted a “water-filling”-like solution. Note that only the spectral distribution of the radar waveform was derived in [7, 9]. In real-world radar systems, we need to synthesize radar waveforms under practical constraints. In [10, 11], several iterative algorithms, which leveraged fast Fourier transform, were proposed to synthesize waveforms with arbitrary spectral shape under some practical constraints, including the constant-modulus constraint, the low peak-to-average-power ratio (PAPR) constraint, and the similarity constraint. Different from the studies in [7, 9], which focused on the design of continuous-time waveforms, the authors of [12] considered the design of discrete-time waveforms to maximize the mutual information. Moreover, they showed the connection between the waveform maximizing mutual information and that minimizing minimum mean square error (MMSE).

Intuitively, the mutual information between the target response and the received signals stands for the amount of useful information that can be extracted from the received signals. Therefore, the waveforms designed based on mutual information maximization are beneficial for target parameter estimation as well as classification. However, the waveforms maximizing the mutual information might not have a satisfactory detection performance. To improve the target detection performance via designing waveforms, information-theoretic approaches have also been investigated [13–15]. Indeed, Stein’s lemma [2] states that, for a fixed probability of false alarm, the probability of miss detection asymptotically decays with an exponential rate equal to the relative entropy (aka Kullback–Leibler divergence) between the distributions of the observations under the two hypotheses (viz., the target is present/absent). Therefore, maximizing the relative entropy results in minimizing the probability of miss detection. In [13], the author studied the problem of detecting point-like targets in colored noise. He proved that the performance of Neyman–Pearson detector is determined by the relative entropy. Moreover, the optimal waveform based on relative entropy maximization was derived. Interestingly, the optimal waveforms also had a “water-filling” interpretation. In [15], the authors considered the detection of weak extended targets based on an information-theoretic approach. Different from the results in [13], the authors showed that the optimal waveforms achieving the maximum relative entropy place all the transmit energy into the frequency bin with the highest signal-to-noise-ratio (SNR) (a similar observation is also presented in [16]).

Note that the above works focus on information-theoretic methods for designing single-input single-output (SISO) radar systems. Unlike SISO radar systems, multiple-input multiple-output (MIMO) radar employs a multi-antenna array for transmitting diverse waveforms and receiving returned signals, showing significantly improved target detection and parameter estimation performance [17–19]. This improvement is partly attributable to the waveform diversity provided by MIMO radar. Therefore, waveform design for MIMO radar is an important topic, and there are many studies devoted to designing MIMO radar waveforms based on information theory [20–24]. In [20], the authors considered waveform design based on maximizing the mutual information for MIMO radar in the white noise case. It is shown that the optimal waveforms that maximize the mutual information also achieve the lowest MMSE. In [21], the authors used an assumption different from that in [20]. By assuming that the target response is uncorrelated (i.e., the target covariance matrix is a scaled identity matrix) and the receiver noise is colored, it is proved that the optimal waveforms maximizing the mutual information are identical to those achieving the Chernoff bound.

In this chapter, we consider information-theoretic approaches for MIMO radar waveform design. Our purpose is to boost the detection performance of MIMO radar in the presence of external disturbance. The external disturbance can be due to hostile jamming (which is independent of the transmit waveforms) or clutter (which is dependent on the waveforms). Considering that the detection probability associated with the detector has a complicated expression, we resort to the relative entropy between the distributions of the observations under the two hypotheses as the waveform design metric. We present an algebraic solution to the waveform design problem for a distributed MIMO radar (namely, the radar antennas are widely separated) in the absence of clutter, as well as a computational approach to design MIMO radar waveforms for more general cases. The performance of the proposed solutions is demonstrated through numerical simulations.

The remaining parts of this chapter are structured as follows. Section 1.2 introduces the signal model and formulates the waveform design problem. Section 1.3 investigates the design of waveforms for a distributed MIMO radar in the absence of clutter. Section 1.4 presents an iterative algorithm to design waveforms for MIMO radar (which can be either distributed or colocated) in the range-spread clutter case. Section 1.5 provides numerical examples to assess the performance of the synthesized MIMO radar waveforms. Finally, Section 1.6 concludes this chapter.

Notations: See Table 1.1.

Table 1.1 List of notations.

Symbol

Meaning

,

Domain of the real and complex numbers

real (complex)-valued matrix

, , and

Matrix, vector, and scalar

The th column of

Identity matrix

Matrix of zeros

The diagonal matrix formed by

The block diagonal matrix formed by and

, ,

Conjugate, transpose, and conjugate transpose

Square root of a positive semi-definite matrix

Determinant of a square matrix

Trace of a square matrix

Magnitude

Euclidean norm

Infinity norm

Frobenius norm

Expectation of the random variable

Vectorization

Kronecker product

, ,

The argument, real and imaginary part of a complex number

Complex Gaussian distribution with mean and covariance matrix

is positive definite (semi-definite)

The smallest eigenvalue of

The largest eigenvalue of

1.2 Signal Model and Problem Formulation

1.2.1 Signal Model

Consider a MIMO radar with transmit antennas and receive antennas. Let denote the transmit waveform matrix, where is the (discrete-time) transmit waveform of the th transmitter, , and is the code length. By assuming that the Doppler shift is negligible, the complex baseband signal of the target return received by the radar system can be written as [25–28]:

(1.1)

where denotes the target response matrix, whose th element, denoted as , represents the response from the th transmit antenna to the th receive antenna. The expression and the statistical properties of are determined by the array configuration. For a colocated MIMO radar in the presence of targets, the target response matrix can be described as , where is the amplitude of the th target signal, denotes the associated direction-of-arrival (DOA), and and are the transmit and the receive array steering vectors at . For a distributed MIMO radar, it is usually assumed (see, e.g., [23–25,29]) that the columns of are independent and identically distributed, such that the covariance matrix of is given by , where is the covariance matrix of each column of .

In radar systems, the target return is usually contaminated by clutter as well as disturbances from inside and outside the receiver. The clutter refers to unwanted returns, e.g., from ground, sea, and clouds, and it might be in the same range cell as the target or occupy the neighboring cells, as illustrated in Figure 1.1. By taking the clutter and disturbance into consideration, the received signals of a MIMO radar can be modeled by:

(1.2)

where is the shift matrix, whose th element equals 1 if , and equals 0 otherwise. denotes the response matrix of the clutter in the th range cell (note that the number of clutter range cells adjacent to the target range cell is ), and is the (signal-independent) interference (including thermal noise inside the receiver and disturbances outside the receiver).

Figure 1.1 MIMO radar target detection in clutter.

To facilitate the algebraic derivations in the sequel, we define , which can be written as

(1.3)

where , , , , and .

Remark 1.1  If the Doppler shift of the signals is taken into consideration, the signal model will be more complicated. Take a colocated MIMO radar as a case study. Denote the Doppler frequency of the th target by . Then the target return is given by

(1.4)

where , and is the sampling interval. Similarly, the clutter can be modeled by

(1.5)

where , , , and are, respectively, the amplitude, the Doppler frequency, and the DOA associated with the th clutter patch in the th range cell, and is the number of clutter patches in the th range cell. Note that if , and , , the target and the clutter models degrade to those that are valid in the absence of Doppler shift (see, e.g., (1.2)).

1.2.2 Problem Formulation

To determine whether a target is present in the cell under test (CUT), we consider the following binary hypothesis testing problem:

(1.6)

Assume that , , and . Moreover, the target response vector (i.e., ) and the clutter response vectors (i.e., ) are independently distributed. Given these assumptions, the probability density functions (PDFs) of under the two hypotheses are, respectively, given by

and

where and .

Using the Neyman–Pearson criterion [30], one can verify that the optimal detector for (1.6) is given by

(1.7)

where is the detection threshold. Moreover, the test statistic under either hypothesis is a weighted sum of chi-squared random variables (which follows from Lemma 8 in [25]). However, the probability of detection has a very complicated expression. Consequently, an information-theoretic approach is employed herein to design the waveforms, namely, the waveforms are designed to maximize the relative entropy between and .

The relative entropy between and is given by

(1.8)

Thus, the waveform design problem based on relative entropy maximization can be stated as:

(1.9)

where represents the constraint set of the transmit waveforms. The following constraints are considered in this chapter.

Energy constraint

: Considering that the energy of the transmit waveforms in practical radar systems is limited, we impose the following constraint:

(1.10)

where is the transmit energy.

Low PAPR constraint

: To facilitate the radio frequency amplifier to operate in the saturated mode and reduce unnecessary distortions, low PAPR waveforms are preferable. To make the transmit energy across the transmit antennas uniformly distributed and control the waveform PAPR, we enforce the constraint below

(1.11)

where , and . Particularly, if , the PAPR constraint becomes the constant-modulus constraint:

(1.12)

where .

Similarity constraint

: To achieve desired properties like some existing waveforms, a similarity constraint can be enforced on the waveforms:

(1.13)

where stands for the matrix of reference waveforms with certain favorable properties, , is the user-specified parameter determining the similarity between the designed waveforms and the reference waveforms.

Constant-modulus and similarity (CMS) constraints

: To ensure that the designed waveforms have a constant modulus and exhibit certain desirable properties, we consider the following constraint:

(1.14)

where represents the th element of , , denotes a reference waveform matrix (the modulus of each element of is equal to ), and controls the similarity between and .

In the formulation of the waveform design problem (1.9), we have assumed that the prior knowledge on , , and are available. To justify the assumption, we highlight that and can be estimated from the received signals in previous scans of the MIMO radar (see, e.g., [31–33]). Moreover, the knowledge on the statistical property of the external disturbance can be gained by configuring the radar in a passive mode.

1.3 Optimal Waveforms for Distributed MIMO Radar in the Absence of Clutter

In this section, we consider a simple case, in which the receive antennas of the MIMO radar are widely separated (i.e., a distributed MIMO radar), and the clutter in the received signals is ignored. Due to the widely separated receive antennas, the spatial correlation at the receiver end is negligible. Moreover, similar to [23, 25, 29], we assume that and , i.e., the columns of and are independent and identically distributed, where stands for the temporal correlation matrix of the interference. Based on this assumption, we have that and . Thus, is given by

(1.15)

Therefore, the energy constrained waveform design problem based on relative entropy maximization can be formulated as

(1.16)

where we have ignored the constant terms.

In [25], it is proved that the optimal solution to (1.16) is given by

(1.17)

where is a unitary matrix whose columns are the eigenvectors corresponding to the eigenvalues of in decreasing order, is a unitary matrix whose columns are the eigenvectors corresponding to the eigenvalues of in increasing order, , are the eigenvalues of , , () is given by

, is the solution to the equation (which can be obtained, for instance, using Newton’s method), and are the eigenvalues of . According to (1.17), the maximum relative entropy that can be achieved by an MIMO radar is given by

(1.18)

We can see from (1.17) that the left and right singular vectors of the optimal waveforms should be aligned with the eigenvectors of and , respectively. Moreover, the eigenvector corresponding to the th smallest eigenvalue of should be paired with the eigenvector corresponding to the th largest eigenvalue of . This pairing is referred to as “mode alignment” in the area of compressive sensing [34]. We highlight that the relevance of “mode alignment” is also revealed in waveform design for MIMO radar in spectrally crowded environments [28, 35].

In [25], it is also proved that the maximum relative entropy that can be achieved by a MIMO radar transmitting orthogonal waveforms (i.e., the waveform matrix is a semi-unitary matrix that satisfies ) is given by

(1.19)

and the orthogonal waveform matrix achieving the maximum relative entropy is given by

(1.20)

We proceed to give a simple case to showcase the performance of the optimal waveforms maximizing the relative entropy. In this example, the distributed MIMO radar has transmit antennas and receive antennas. The code length is . The eigenvalues of and are and , respectively. The eigenvectors of and are randomly generated. First, we comment on the optimal power allocation of the transmit waveforms. Figure 1.2 shows the allocated power on each mode of the optimal waveform matrix for dB and dB, respectively. We can observe that for the mode that corresponds to a larger (i.e., the target response is stronger and the interference is weaker), more power is allocated. Moreover, in the case of dB, only two modes are active, implying a higher coherency between the transmit waveforms. If we increase the transmit energy to dB, all the modes are activated, meaning a higher waveform diversity.

Figure 1.2 Optimal power allocation of the transmit waveforms. (a) dB. (b) dB.

Figure 1.3 Performance comparison between the optimal waveforms and the optimal orthogonal waveforms. (a) Relative entropy versus transmit energy. (b) versus transmit energy.

Now, we compare the performance of the optimal waveforms in (1.17) with that of the optimal orthogonal waveforms in (1.20). Figure 1.3(a) shows the relative entropy of the optimal waveforms and the orthogonal waveforms versus the transmit energy. We can see that, for a given transmit energy, the optimal waveforms achieve a larger relative entropy than the orthogonal waveforms. Figure 1.3(b) compares the detection probability () of the waveforms corresponding to Figure 1.3(a). In the analysis, the optimal Neyman–Pearson detector (see (1.7)) is used. In addition, the probability of false alarm is , and independent Monte Carlo trials are conducted to determine the detection threshold and the detection probability, respectively. The results in Figure 1.3(b) are consistent with those in Figure 1.3(a), demonstrating that the performance of the optimal waveforms is visibly superior to that of the optimal orthogonal waveforms. This is attributed to the fact that the optimal waveforms have a larger number of degrees of freedom and thus achieve additional waveform diversity.

1.4 MM-Based Waveform Design in the Presence of Range-Spread Clutter

In this section, we design waveforms to boost the detection performance of MIMO radar in the presence of range-spread clutter. To this end, we consider the optimization problem (1.9). In general, this problem is nonconvex. In the following, we derive a minorization–maximization (MM) algorithm [36] to tackle this nonconvex problem. To this end, we split the objective function in (1.9) into three parts:

(1.21)

where, , and . Next, we derive a quadratic surrogate function (i.e., a minorizer) for each , .

1.4.1 Surrogate Function Construction

Define , , and , where if , and . Then and . Following the derivations in [27, 37], we can show that

(1.22)

where , and

It follows from Lemma 1 in [27] that is convex with respect to . By using a property of convex functions [38], we have

where is the gradient of at (which can be shown using results from [39]). Let us partition as:

where , , and . One can verify that , , and [37]. As a result,

Define . Then a surrogate function of is given by

(1.23)

Next, we rewrite as

(1.24)

Using the fact that is convex with respect to [38], we can obtain

Define . Then one can verify that a surrogate function of can be written as

(1.25)

Finally, we note that . Moreover, by using the matrix inversion lemma, we have , where ,

, , and . Thus,

(1.26)

Note that is convex with respect to [27]. Hence, we have

where is the gradient of at , and . Let us partition as

where , , and . It can be checked that , , and [37]. Therefore, we have

Define . Then is minorized by

(1.27)

Based on the above results, it can be inferred that a surrogate function of is given by

(1.28)

Moreover, we can verify that and . Thus, is minorized by .

To show that can be written as a quadratic function of , we let , where , and let , where , . Partition as

where . Then according to Proposition 1 in [37],

(1.29)

where , , , , with being the th column of (), , , , , , , , and .

Therefore, we can formulate the minorized problem based on (1.29) at the th iteration as the following quadratic programming problem:

(1.30)

1.4.2 Efficient Solutions to the Quadratic Programming Problem (1.30)

Energy Constraint

We can formulate the quadratic programming problem under the energy constraint as

(1.31)

where we have used the fact that . The optimization problem (1.31) can be solved by the Lagrange multiplier method (see, e.g., [12, 27, 37] for details), and the solution is given by

(1.32)

where is the optimal Lagrange multiplier corresponding to the energy constraint, which can be obtained via solving the equation .

PAPR Constraint

We formulate the quadratic programming problem under the PAPR constraint as

(1.33)

The above quadratic programming problem can be tackled by MM. To this purpose, note that , where denotes the solution obtained by the proposed algorithm at the th iteration. Here, the index