Adaptive Detection of Multichannel Signals Exploiting Persymmetry

314 Pages·2022·9.283 MB·English

by  Jun Liu, Danilo Orlando, Chengpeng Hao, Weijian Liu

Adaptive Detection
of Multichannel
Signals Exploiting
Persymmetry
This book offers a systematic presentation of persymmetric adaptive detection, including
detector derivations and the definitions of key concepts, followed by detailed discussions relating
to theoretical underpinnings, design methodology, design considerations, and techniques
enabling its practical implementation.
The received data for modern radar systems are usually multichannel, namely, vector-
valued, or even matrix-valued. Multichannel signal detection in Gaussian backgrounds is
a fundamental problem for radar applications. With an overarching focus on persymmetric
adaptive detectors, this book presents the mathematical models and design principles neces-
sary for analyzing the behavior of each kind of persymmetric adaptive detector. Building
upon that, it also introduces new design approaches and techniques that will guide engineer-
ing students as well as radar engineers toward efficient detector solutions, especially in chal-
lenging sample-starved environments where training data are limited.
This book will be of interest to students, scholars, and engineers in the field of signal
processing. It will be especially useful for those who have a solid background in statistical
signal processing, multivariate statistical analysis, matrix theory, and mathematical analysis.
Jun Liu is an Associate Professor with the Department of Electronic Engineering and Infor-
mation Science, University of Science and Technology of China. Dr. Liu is a member of the
Sensor Array and Multichannel (SAM) Technical Committee, IEEE Signal Processing Society.
Danilo Orlando is an Associate Professor at Università degli Studi “Niccolò Cusano”. His
research interests focus on signal processing for radar and sonar systems. He has co-au-
thored more than 150 publications in international journals, conferences, and books.
Chengpeng Hao is a Professor at the Institute of Acoustics, Chinese Academy of Sci-
ences. His research interests are in the fields of statistical signal processing, array signal
processing, radar, and sonar engineering. He has authored and co-authored more than 100
scientific publications in international journals and conferences.
Weijian Liu is an Associate Professor with the Wuhan Electronic Information Institute,
China. His research interests include multichannel signal detection and statistical and
array signal processing.

Adaptive Detection
of Multichannel
Signals Exploiting
Persymmetry
Jun Liu
Danilo Orlando
Chengpeng Hao
Weijian Liu
This book is published with financial support from the National Natural Science Foundation of China
under Grants 61871469, 61971412, 62071482, the Youth Innovation Promotion Association CAS
(2019447), and the Anhui Provincial Natural Science Foundation under Grant 2208085J17.
First edition published 2023
by CRC Press
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and by CRC Press
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CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Jun Liu, Danilo Orlando, Chengpeng Hao, Weijian Liu
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lisher cannot assume responsibility for the validity of all materials or the consequences of their use.
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Trademark notice: Product or corporate names may be trademarks or registered trademarks and are
used only for identification and explanation without intent to infringe.
ISBN: 978-1-032-37424-6 (hbk)
ISBN: 978-1-032-37427-7 (pbk)
ISBN: 978-1-003-34023-2 (ebk)
DOI: 10.1201/9781003340232
Typeset in CMR10 font
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
In loving memory of my father, Houcheng Liu. To his
greatness and uniqueness.
– Jun Liu
To my lovely family.
– Danilo Orlando
To my beloved parents.
– Chengpeng Hao
To my wife and daughter.
– Weijian Liu

Contents
List of Abbreviations xiii
List of Symbols xv
1 Basic Concept 1
1.1 Multichannel Radar . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Adaptive Detection of Multichannel Signal . . . . . . . . . . 1
1.3 Persymmetric Structure of Covariance Matrix . . . . . . . . 5
1.4 Organization and Outline of the Book . . . . . . . . . . . . . 8
1.A Detector Design Criteria . . . . . . . . . . . . . . . . . . . . 9
1.A.1 Nuisance Parameter . . . . . . . . . . . . . . . . . . . 10
1.A.1.1 Rao Test . . . . . . . . . . . . . . . . . . . . 10
1.A.1.2 Wald Test . . . . . . . . . . . . . . . . . . . 11
1.A.1.3 GLRT . . . . . . . . . . . . . . . . . . . . . . 11
1.A.2 No Nuisance Parameter . . . . . . . . . . . . . . . . . 11
1.A.2.1 Rao Test . . . . . . . . . . . . . . . . . . . . 11
1.A.2.2 Wald Test . . . . . . . . . . . . . . . . . . . 12
1.A.2.3 GLRT . . . . . . . . . . . . . . . . . . . . . . 12
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Output SINR Analysis 19
2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 20
2.1.1 Unstructured SMI Beamformer . . . . . . . . . . . . . 21
2.1.1.1 Matched Case . . . . . . . . . . . . . . . . . 21
2.1.1.2 Mismatched Case . . . . . . . . . . . . . . . 22
2.1.2 Persymmetric SMI Beamformer . . . . . . . . . . . . . 22
2.2 Average SINR in Matched Case . . . . . . . . . . . . . . . . 23
2.3 Average SINR in Mismatched Cases . . . . . . . . . . . . . . 25
2.3.1 Homogeneous Case . . . . . . . . . . . . . . . . . . . . 25
2.3.2 Non-Homogeneous Case . . . . . . . . . . . . . . . . . 27
2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 28
2.A Derivation of E ρmatch . . . . . . . . . . . . . . . . . . . . . 31
p per q
2.B Proof of Theorem 2.3.1 . . . . . . . . . . . . . . . . . . . . . 35
2.C Derivations of (2.B.22) . . . . . . . . . . . . . . . . . . . . . 42
vii
viii Contents
2.D Derivations of (2.B.28) . . . . . . . . . . . . . . . . . . . . . 44
2.E Derivations of (2.B.36) . . . . . . . . . . . . . . . . . . . . . 44
2.F Derivation of E w˘w˘ in the Mismatched Case . . . . . . . . 45
:
p q
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Invariance Issues under Persymmetry 51
3.1 Preliminary Theory . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Homogeneous Environment . . . . . . . . . . . . . . . . . . . 52
3.2.1 Stochastic Representation . . . . . . . . . . . . . . . . 56
3.2.2 Invariant Detectors . . . . . . . . . . . . . . . . . . . . 58
3.2.3 Statistical Characterization . . . . . . . . . . . . . . . 60
3.2.3.1 LRT-Based Decision Schemes . . . . . . . . . 61
3.3 Partially Homogeneous Environment . . . . . . . . . . . . . 62
3.3.1 Invariant Detectors for Partially Homogeneous
Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.A Proof of Theorem 3.2.1 . . . . . . . . . . . . . . . . . . . . . 65
3.B Derivation of (3.38) . . . . . . . . . . . . . . . . . . . . . . . 69
3.C Proof of Theorem 3.2.2 . . . . . . . . . . . . . . . . . . . . . 70
3.D Proof of Theorem 3.2.3 . . . . . . . . . . . . . . . . . . . . . 71
3.E Proof of Theorem 3.2.5 . . . . . . . . . . . . . . . . . . . . . 76
3.F Proof of Theorem 3.3.1 . . . . . . . . . . . . . . . . . . . . . 77
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 Persymmetric Adaptive Subspace Detector 81
4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Persymmetric One-Step GLRT . . . . . . . . . . . . . . . . . 82
4.3 Threshold Setting . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.1 Transformation from Complex Domain to Real Domain 86
4.3.2 Statistical Characterizations. . . . . . . . . . . . . . . 87
4.3.2.1 Equivalent Form of XTPX . . . . . . . . . . 87
4.3.2.2 Equivalent Form of XTM 1X . . . . . . . . 88
(cid:1)
4.3.2.3 Statistical Distribution of Λ. . . . . . . . . . 89
4.3.3 Probability of False Alarm. . . . . . . . . . . . . . . . 89
4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 90
4.A Derivations of (4.13) . . . . . . . . . . . . . . . . . . . . . . . 91
4.B Derivations of (4.38) . . . . . . . . . . . . . . . . . . . . . . 93
4.C Derivations of (4.62) . . . . . . . . . . . . . . . . . . . . . . 96
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Persymmetric Detectors with Enhanced Rejection
Capabilities 101
5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Detector Design . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1 Persymmetric Rao Test . . . . . . . . . . . . . . . . . 103
5.2.2 Persymmetric GLRT . . . . . . . . . . . . . . . . . . . 104
Contents ix
5.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 106
5.A Derivations of (5.32) . . . . . . . . . . . . . . . . . . . . . . . 108
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 Distributed Target Detection in Homogeneous
Environments 113
6.1 Persymmetric One-Step GLRT . . . . . . . . . . . . . . . . . 114
6.1.1 Detector Design . . . . . . . . . . . . . . . . . . . . . 114
6.1.2 Analytical Performance . . . . . . . . . . . . . . . . . 116
6.1.2.1 Transformation from Complex Domain to
Real Domain . . . . . . . . . . . . . . . . . . 116
6.1.2.2 Statistical Properties . . . . . . . . . . . . . 117
6.1.2.3 Detection Probability . . . . . . . . . . . . . 119
6.1.2.4 Probability of False Alarm . . . . . . . . . . 122
6.2 Persymmetric Two-Step GLRT . . . . . . . . . . . . . . . . . 123
6.2.1 Detector Design . . . . . . . . . . . . . . . . . . . . . 123
6.2.2 Analytical Performance . . . . . . . . . . . . . . . . . 124
6.2.2.1 Statistical Properties . . . . . . . . . . . . . 125
6.2.2.2 Probability of False Alarm . . . . . . . . . . 126
6.2.2.3 Detection Probability . . . . . . . . . . . . . 127
6.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 130
6.A Derivations of (6.31) . . . . . . . . . . . . . . . . . . . . . . . 136
6.B Derivations of (6.39) . . . . . . . . . . . . . . . . . . . . . . . 139
6.C Derivations of (6.69) . . . . . . . . . . . . . . . . . . . . . . . 140
6.D Proof of Theorem 6.2.1 . . . . . . . . . . . . . . . . . . . . . 142
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7 Robust Detection in Homogeneous Environments 145
7.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 145
7.2 Detection Design . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2.1 GLRT Criterion . . . . . . . . . . . . . . . . . . . . . 148
7.2.1.1 One-Step GLRT . . . . . . . . . . . . . . . . 148
7.2.1.2 Two-Step GLRT . . . . . . . . . . . . . . . . 148
7.2.2 Wald Criterion . . . . . . . . . . . . . . . . . . . . . . 149
7.2.2.1 One-Step Wald Test . . . . . . . . . . . . . . 149
7.2.2.2 Two-Step Wald Test . . . . . . . . . . . . . . 151
7.2.3 Rao Criterion . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 153
7.A Derivations of (7.36) . . . . . . . . . . . . . . . . . . . . . . . 157
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8 Adaptive Detection with Unknown Steering Vector 161
8.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 161
8.2 Per-SNT Detector . . . . . . . . . . . . . . . . . . . . . . . . 162
8.2.1 Detector Design . . . . . . . . . . . . . . . . . . . . . 162