Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms [2 ed.] 1119521335, 9781119521334

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Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms

Wiley Series in Microwave and Optical Engineering Kai Chang, Editor Texas A&M University A complete list of the titles in this series appears at the end of this volume.

Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms With Advanced SAR/ISAR Imaging Concepts, Algorithms, and MATLAB Codes

Second Edition

Caner Özdemir, Phd Mersin University Mersin, Turkey

This second edition first published 2021 © 2021 John Wiley & Sons, Inc. Edition History John Wiley & Sons, Inc. (1e, 2012) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Caner Özdemir to be identified as the author of this work has been asserted in accordance with law. Registered Office John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA Editorial Office 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This work’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Özdemir, Caner, author. Title: Inverse synthetic aperture radar imaging with MATLAB algorithms: with advanced sar/isar imaging concepts, algorithms, and matlab codes / Caner Ozdemir, PhD Mersin University, Mersin, Turkey. Description: 2nd edition. | Hoboken, NJ, USA : Wiley, 2021. | Series: Wiley series in microwave and optical engineering | Includes bibliographical references and index. Identifiers: LCCN 2020031216 (print) | LCCN 2020031217 (ebook) | ISBN 9781119521334 (cloth) | ISBN 9781119521365 (adobe pdf) | ISBN 9781119521389 (epub) Subjects: LCSH: MATLAB. | Synthetic aperture radar. | MATLAB. Classification: LCC TK6592.S95 O93 2020 (print) | LCC TK6592.S95 (ebook) | DDC 621.3848/5–dc23 LC record available at https://lccn.loc.gov/2020031216 LC ebook record available at https://lccn.loc.gov/2020031217 Cover design by Wiley Cover image: Courtesy of Caner Özdemir, (background) © Maxiphoto/Getty Images Set in 9.5/12.5pt STIXTwoText by SPi Global, Pondicherry, India

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To: My wife Betül, My three daughters, My brother, My father, and the memory of my beloved mother

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Contents

Preface to the Second Edition Acknowledgments xix Acronyms xx 1 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 1.2.8 1.2.9 1.2.10 1.2.11 1.2.12 1.3 1.3.1 1.3.2 1.3.3

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Basics of Fourier Analysis 1 Forward and Inverse Fourier Transform 1 Brief History of FT 1 Forward FT Operation 2 IFT 3 FT Rules and Pairs 3 Linearity 3 Time Shifting 3 Frequency Shifting 4 Scaling 4 Duality 4 Time Reversal 4 Conjugation 4 Multiplication 4 Convolution 5 Modulation 5 Derivation and Integration 5 Parseval’s Relationship 5 Time-Frequency Representation of a Signal 5 Signal in the Time Domain 6 Signal in the Frequency Domain 6 Signal in the Joint Time-Frequency (JTF) Plane

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1.4 1.5 1.6 1.7 1.7.1 1.7.2 1.7.3 1.8 1.9 1.10 1.11

Convolution and Multiplication Using FT Filtering/ Windowing 12 Data Sampling 14 DFT and FFT 16 DFT 16 FFT 17 Bandwidth and Resolutions 17 Aliasing 19 Importance of FT in Radar Imaging 19 Effect of Aliasing in Radar Imaging 23 Matlab Codes 26 References 33

2 2.1 2.2 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1 2.4.2 2.5 2.5.1 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.6.5 2.7 2.7.1 2.7.2 2.7.3 2.8

Radar Fundamentals 35 Electromagnetic Scattering 35 Scattering from PECs 38 Radar Cross Section 39 Definition of RCS 40 RCS of Simple-Shaped Objects 43 RCS of Complex-Shaped Objects 44 Radar Range Equation 44 Bistatic Case 46 Monostatic Case 49 Range of Radar Detection 50 Signal-to-Noise Ratio 51 Radar Waveforms 53 Continuous Wave 53 Frequency-Modulated Continuous Wave 56 Stepped-Frequency Continuous Wave 59 Short Pulse 61 Chirp (LFM) Pulse 62 Pulsed Radar 69 Pulse Repetition Frequency 69 Maximum Range and Range Ambiguity 69 Doppler Frequency 70 Matlab Codes 74 References 82

Contents

3 3.1 3.2 3.3 3.4 3.5 3.5.1 3.5.1.1 3.5.1.2 3.5.2 3.5.2.1 3.5.2.2 3.5.2.3 3.5.2.4 3.5.3 3.5.3.1 3.5.3.2 3.5.3.3 3.6 3.6.1 3.6.2 3.6.3 3.7 3.8 3.8.1 3.8.1.1 3.8.1.2 3.8.1.3 3.8.1.4 3.8.1.5 3.8.1.6 3.8.2 3.8.3 3.8.4 3.9 3.10 3.10.1 3.10.2 3.10.3

Synthetic Aperture Radar 85 SAR Modes 86 SAR System Design 87 Resolutions in SAR 88 SAR Image Formation 91 Range Compression 92 Matched Filter 92 Computing Matched Filter Output via Fourier Processing 95 Example for Matched Filtering 96 Ambiguity Function 99 Relation to Matched Filter 100 Ideal Ambiguity Function 101 Rectangular-Pulse Ambiguity Function 102 LFM-Pulse Ambiguity Function 102 Pulse Compression 105 Detailed Processing of Pulse Compression 105 Bandwidth, Resolution, and Compression Issues for LFM Signal Pulse Compression Example 110 Azimuth Compression 110 Processing in Azimuth 110 Azimuth Resolution 116 Relation to ISAR 117 SAR Imaging 118 SAR Focusing Algorithms 118 RDA 119 Range Compression in RDA 120 Azimuth Fourier Transform 126 Range Cell Migration Correction 128 Azimuth Compression 129 Simulated SAR Imaging Example 130 Drawbacks of RDA 133 Chirp Scaling Algorithm 133 The ω-kA 133 Back-Projection Algorithm 134 Example of a Real SAR Imagery 135 Problems in SAR Imaging 136 Range Migration and Range Walk 136 Motion Errors 137 Speckle Noise 140

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3.11 3.11.1 3.11.2 3.12

Advanced Topics in SAR 140 SAR Interferometry 140 SAR Polarimetry 142 Matlab Codes 143 References 158

4 4.1 4.2 4.3 4.4 4.5

Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts 162 SAR versus ISAR 162 The Relation of Scattered Field to the Image Function in ISAR 166 One-Dimensional (1D) Range Profile 167 1D Cross-Range Profile 172 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle) 176 Resolutions in ISAR 180 Range Resolution 181 Cross-Range Resolution: 181 Range and Cross-Range Extends 181 Imaging Multibounces in ISAR 182 Sample Design Procedure for ISAR 185 ISAR Design Example #1: “Aircraft Target” 189 ISAR Design Example #2: “Military Tank Target” 193 2D ISAR Image Formation (Wide Bandwidth, Large Angles) 197 Direct Integration 198 Polar Reformatting 201 3D ISAR Image Formation 205 Range and Cross-Range resolutions 209 A Design Example for 3D ISAR 210 Matlab Codes 217 References 243

4.5.1 4.5.1.1 4.5.1.2 4.5.2 4.5.3 4.5.4 4.5.4.1 4.5.4.2 4.6 4.6.1 4.6.2 4.7 4.7.1 4.7.2 4.8

5 5.1 5.1.1 5.1.2 5.2 5.3 5.3.1 5.3.2

Imaging Issues in Inverse Synthetic Aperture Radar Fourier-Related Issues 246 DFT Revisited 246 Positive and Negative Frequencies in DFT 250 Image Aliasing 252 Polar Reformatting Revisited 255 Nearest Neighbor Interpolation 255 Bilinear Interpolation 258

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5.4 5.5 5.6 5.6.1 5.6.1.1 5.6.1.2 5.6.1.3 5.6.1.4 5.6.1.5 5.6.1.6 5.6.1.7 5.6.2 5.7

Zero Padding 260 Point Spread Function 264 Windowing 269 Common Windowing Functions 269 Rectangular Window 269 Triangular Window 269 Hanning Window 272 Hamming Window 272 Kaiser Window 272 Blackman Window 276 Chebyshev Window 277 ISAR Image Smoothing via Windowing Matlab Codes 280 References 304

6 6.1 6.1.1 6.1.2 6.2 6.2.1 6.2.2 6.3 6.3.1 6.3.2 6.4 6.5 6.5.1 6.5.2 6.6 6.6.1 6.6.2 6.7 6.8 6.8.1 6.8.2 6.8.3 6.8.4

Range-Doppler Inverse Synthetic Aperture Radar Processing 306 Scenarios for ISAR 306 Imaging Aerial Targets via Ground-Based Radar 307 Imaging Ground/Sea Targets via Aerial Radar 309 ISAR Waveforms for Range-Doppler Processing 312 Chirp Pulse Train 312 Stepped Frequency Pulse Train 314 Doppler Shift’s Relation to Cross-Range 316 Doppler Frequency Shift Resolution 317 Resolving Doppler Shift and Cross-Range 318 Forming the Range-Doppler Image 319 ISAR Receiver 320 ISAR Receiver for Chirp Pulse Radar 320 ISAR Receiver for SFCW Radar 321 Quadrature Detection 323 I-Channel Processing 324 Q-Channel Processing 324 Range Alignment 326 Defining the Range-Doppler ISAR Imaging Parameters 327 Image Frame Dimension (Image Extends) 327 Range and Cross-Range Resolution 328 Frequency Bandwidth and the Center Frequency 328 Doppler Frequency Bandwidth 328

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6.8.5 6.8.6 6.8.7 6.9 6.10 6.11

Pulse Repetition Frequency 329 Coherent Integration (Dwell) Time 329 Pulse Width 330 Example of Chirp Pulse-Based Range-Doppler ISAR Imaging 331 Example of SFCW-Based Range-Doppler ISAR Imaging 336 Matlab Codes 339 References 347

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Scattering Center Representation of Inverse Synthetic Aperture Radar 349 Scattering/Radiation Center Model 350 Extraction of Scattering Centers 352 Image Domain Formulation 352 Extraction in the Image Domain: The “CLEAN” Algorithm 352 Reconstruction in the Image Domain 355 Fourier Domain Formulation 362 Extraction in the Fourier Domain 362 Reconstruction in the Fourier Domain 364 Matlab Codes 368 References 382

7.1 7.2 7.2.1 7.2.1.1 7.2.1.2 7.2.2 7.2.2.1 7.2.2.2 7.3

8 8.1 8.2 8.2.1 8.2.1.1 8.2.1.2 8.2.2 8.3 8.3.1 8.3.1.1 8.3.2 8.3.2.1 8.3.2.2 8.3.3 8.3.3.1 8.3.3.2 8.3.3.3

Motion Compensation for Inverse Synthetic Aperture Radar 385 Doppler Effect Due to Target Motion 386 Standard MOCOMP Procedures 388 Translational MOCOMP 389 Range Tracking 389 Doppler Tracking 390 Rotational MOCOMP 390 Popular ISAR MOCOMP Techniques 392 Cross-Correlation Method 392 Example for the Cross-Correlation Method 394 Minimum Entropy Method 398 Definition of Entropy in ISAR Images 398 Example for the Minimum Entropy Method 399 JTF-Based MOCOMP 402 Received Signal from a Moving Target 403 An Algorithm for JTF-Based Rotational MOCOMP 404 Example for JTF-Based Rotational MOCOMP 406

Contents

8.3.4 8.3.4.1 8.4

Algorithm for JTF-Based Translational and Rotational MOCOMP A Numerical Example 410 Matlab Codes 415 References 436

9 9.1 9.2 9.2.1 9.2.2 9.3 9.3.1 9.3.2 9.3.3 9.4 9.5 9.5.1 9.5.2 9.6 9.6.1 9.6.2 9.7

Bistatic ISAR Imaging 440 Why Bi-ISAR Imaging? 440 Geometry for Bi-Isar Imaging and the Algorithm 444 Bi-ISAR Imaging Algorithm for a Point Scatterer 444 Bistatic ISAR Imaging Algorithm for a Target 448 Resolutions in Bistatic ISAR 449 Range Resolution 449 Cross-Range Resolution 450 Range and Cross-Range Extends 451 Design Procedure for Bi-ISAR Imaging 452 Bi-Isar Imaging Examples 455 Bi-ISAR Design Example #1 455 Bi-ISAR Design Example #2 457 Mu-ISAR Imaging 465 Challenges in Mu-ISAR Imaging 467 Mu-ISAR Imaging Example 468 Matlab Codes 472 References 483

10 10.1 10.1.1 10.1.2 10.1.3 10.2 10.2.1 10.2.2 10.2.3 10.2.4 10.2.4.1 10.2.4.2 10.2.4.3 10.3

Polarimetric ISAR Imaging 484 Polarization of an Electromagnetic Wave 484 Polarization Type 485 Polarization Sensitivity 486 Polarization in Radar Systems 487 Polarization Scattering Matrix 488 Relation to RCS 490 Polarization Characteristics of the Scattered Wave 491 Polarimetric Decompositions of EM Wave Scattering 493 The Pauli Decomposition 494 Description of Pauli Decomposition 494 Interpretation of Pauli Decomposition 495 Polarimetric Image Representation Using Pauli Decomposition Why Polarimetric ISAR Imaging? 497

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10.4 10.4.1 10.4.2 10.5 10.5.1 10.5.1.1 10.5.1.2 10.5.1.3 10.5.1.4 10.5.1.5 10.5.2 10.5.2.1 10.5.2.2 10.5.2.3 10.5.2.4 10.5.2.5 10.6 10.7

ISAR Imaging with Full Polarization 497 ISAR Data in LP Basis 497 ISAR Data in CP Basis 498 Polarimetric ISAR Images 499 Pol-ISAR Image of a Benchmark Target 499 The “SLICY” Target 499 Fully Polarimetric EM Simulation of SLICY 499 LP Pol-ISAR Images of SLICY 500 CP Pol-ISAR Images of SLICY 502 Pauli Decomposition Image of SLICY 503 Pol-ISAR Image of a Complex Target 507 The “Military Tank” Target 507 Fully Polarimetric EM Simulation of “Tank” Target LP Pol-ISAR Images of “Tank” Target 508 CP Pol-ISAR Images of “Tank” Target 510 Pauli Decomposition Image of “Tank” Target 512 Feature Extraction from Polarimetric Images 515 Matlab Codes 515 References 529

11 11.1 11.1.1 11.1.1.1 11.1.1.2 11.1.2 11.2 11.3 11.3.1 11.3.2 11.3.2.1 11.3.2.2 11.3.2.3 11.4 11.5 11.5.1

Near-Field ISAR Imaging 533 Definitions of Far and Near-Field Regions 534 The Far-Field Region 534 The Far-Field Definition Based on Target’s Cross-Range Extend 534 The Far-Field Definition Based on Target’s Range Extend 535 The Near-Field Region 537 Near-Field Signal Model for the Back-Scattered Field 537 Near-Field ISAR Imaging Algorithms 540 “Focusing Operator” Algorithm 540 Back-Projection Algorithm 541 Fourier Slice Theorem 542 BPA Formulation (3D Case) 543 BPA Formulation (2D Case) 544 Data Sampling Criteria and the Resolutions 546 Near-Field ISAR Imaging Examples 547 Point Scatterers in the Near-Field: Comparison of Far- and Near-Field Imaging Algorithms 547 Near-Field ISAR Imaging of a Large Object 552

11.5.2

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Contents

11.5.3 11.6

Near-Field ISAR Imaging of a Small Object 555 Matlab Codes 560 References 569

12 12.1 12.1.1 12.1.2 12.1.3 12.1.3.1 12.1.3.2 12.1.3.3 12.2 12.2.1 12.2.2 12.2.3 12.2.4 12.3 12.3.1 12.3.2 12.4 12.4.1 12.4.2 12.4.3

Some Imaging Applications Based on SAR/ISAR 571 Imaging Subsurface Objects: GPR-SAR 572 The GPR Problem 572 B-Scan GPR in Comparison to Strip-Map SAR 577 Focused GPR Images Using SAR 577 GPR Focusing with ω-k Algorithm (ω-kA) 579 GPR Focusing with BPA 582 Other Popular GPR Focusing Techniques 589 Thru-the-Wall Imaging Radar Using SAR 590 Challenges in TWIR 591 Techniques to Improve Cross-Range Resolution in TWIR 591 The Use of SAR in TWIR 592 Example of SAR-Based TWIR 594 Imaging Antenna-Platform Scattering: ASAR 596 The ASAR Imaging Algorithm 597 Numerical Example for ASAR Imagery 603 Imaging Platform Coupling Between Antennas: ACSAR 605 The ACSAR Imaging Algorithm 606 Numerical Example for ACSAR 609 Applying ACSAR Concept to the GPR Problem 611 References 615 Appendix 619 Index 628

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Preface to the Second Edition In the first edition of the book, I tried to cover most of the aspects of inverse synthetic aperture radar (ISAR) imaging starting from Fourier analysis to some advanced ISAR concepts such as range-Doppler ISAR processing and ISAR motion compensation techniques. The main goal was to present a conceptual description of ISAR imagery and the explanation of basic ISAR research topics. Although the primary audience would be graduate students and other interested researchers in the fields of electrical engineering and physics, I hoped that colleagues working in radar research and development or in a related industry might also benefit from the book. It has been more than eight years since the publication of the first edition. Since then, I have been really grateful that I have received positive responses from the researchers and colleagues that are interested and/or involved in radar imaging, and especially ISAR imaging. Undoubtedly, ISAR has been gaining more attention among researchers, scholars, and engineers as emerging new developments in ISAR research have been reported by various colleagues day by day. In this second edition of the book, I have tried to include the recent progress made in ISAR imaging research and also give insights to more advanced concepts. Therefore, in this edition of the book, I have made the following alterations and additions:

• •

All the chapters in the first edition have been revised including all the texts, equations, and figures with some additions. Typos in the first edition have also been corrected. Chapter 3 that is devoted to the issues of synthetic aperture radar (SAR) has been extended to include the SAR focusing/processing algorithms such as rangeDoppler algorithm (RDA), back-projection algorithm (BPA) and frequencywave number algorithm (ω-kA). The Matlab codes for these algorithms are being provided with the associated numerical examples. Brief explanations of

Preface to the Second Edition

• •

other SAR focusing/processing algorithms including chirp scaling algorithm (CSA) and phase shift algorithm (PSA) have also been mentioned. In Chapters 4, 5, 7, and 12 where we handle various aspects of ISAR imaging technology, new scattered field raw data and the corresponding ISAR images that are more visually attractive are presented together with the associated Matlab codes that can be used to generate these images. A total of three new chapters have been written to cover the topics that were not considered in the first edition and also to include more detailed subjects of ISAR imaging to be able to reflect the recent research studies. These are listed below: – The “Bistatic ISAR (Bi-ISAR) Imaging” concept is covered in Chapter 9. While the ISAR imaging algorithms presented in previous chapters are based on monostatic usage of ISAR imaging, we introduce the formulation of ISAR imaging for the bistatic usages by presenting key aspects such as resolutions in range/cross-range directions and usage limitations. Also, extension of BiISAR to multistatic ISAR (Mu-ISAR) imaging is derived with the associated Matlab examples. A general assessment of Bi-ISAR and Mu-ISAR imaging to conventional monostatic ISAR imaging is being made throughout the chapter by comparing the outcomes of quantitative metrics and giving the concluding statements about their advantages and disadvantages based on these measurable evaluations. – In Chapter 10, we have added a new and exciting research topic of ISAR called “Polarimetric ISAR Imaging.” As the traditional ISAR imaging algorithms are based only on a single polarization of the backscattered electric field, we demonstrate in this chapter that very exciting features of the target can be extracted with the use of other possible polarizations for the reflected wave. Polarization decomposition techniques are being introduced and Pauli decomposition scheme is taken as the tool to be applied to the different polarization ISAR images in this book. The formulation and the usage of Pauli decomposition technique are presented together with its Matlab codes. Various realistic simulation examples based on linear polarization, circular polarization, and also Pauli decomposition are given together with obtained polarimetric ISAR images. It has been demonstrated through the examples that polarized ISAR images definitely increase the recognition and classification of targets by providing increased number of extracted target features. – Thanks to the recent development in the microwave circuit technology and antenna design, ISAR imaging algorithms have been started to be used in the near-field region. Therefore, I have added a new part entitled “Near-field ISAR imaging” as Chapter 11. The near-field ISAR imaging algorithms are being introduced. Two of them called “Focusing operator” and the backprojection based focusing algorithms are given by presenting their theoretical formulation and algorithm steps together with corresponding Matlab codes.

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Preface to the Second Edition

•

Also, numerical and measured examples based on real scenarios are being shared. In Chapter 12 in which some examples based on SAR/ISAR imaging technologies are provided, I have previously introduced algorithms called antenna SAR (ASAR) and antenna coupling SAR (ACSAR) as the unique radar imaging algorithms to image antenna mounted on a platform-to-radar receiver interaction over the target and to image platform coupling over the antennas mounted on a target, respectively. In this edition of the book, I have added some new applications such as ground-penetrating radar (GPR) and through-the-wall imaging radar (TWIR) that also make use of SAR/ISAR imaging algorithms. Measured examples of GPR and TWIR radar images are provided to demonstrate how SAR/ISAR imaging algorithms can be effectively used in some popular radar imaging applications.

I hope that, with the new edition of the book “Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms,” the reader would benefit more in terms of abovementioned new ISAR imaging topics and also from the Matlab codes provided at the end of chapters. All MATLAB files may be accessed on the following FTP site: ftp://ftp.wiley. com/public/sci_tech_med/inverse_synthetic.

Mersin, October 2020

Caner Özdemir

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Acknowledgments I would like to address special thanks to the people below for their help and support during the preparation of this book. First, I am thankful to my wife, Betül and my three children for their patience and continuous support while writing this book. I am very grateful to Dr. Hao Ling, Emeritus Professor in Engineering of the University of Texas at Austin for being a valuable source of knowledge, ideas, and also inspiration throughout my academic carreer. He has been a great advisor since I met him, and his guidance on scientific research is priceless to me. I would like to express my sincere thanks to my former graduate students; Dr. Şevket Demirci, Dr. Enes Yiğit, Dr. Betül Yılmaz, Dr. Deniz Üstün, Özkan Kırık, and Dr. Hakan Işıker who have helped carrying out some of the research presented in this book. I would also like to thank my graduate student Rasheed Khankan for his help in preparing references. Last but not least, I would like to convey my special thanks to Dr. Kai Chang for inviting me to write the first and then second edition of the book. Without his kind offer, this book project would not have been possible. Caner Özdemir

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Acronyms 1D 2D 3D ACSAR ADC ANN ASAR ATC ATR Bi-ISAR BPA CAD CDF CFAR CO Co-pol CP Cross-pol CSA CW DCR DFT DTV EFIE EM ESM FM FMCW

One-dimensional Two-dimensional Three-dimensional Antenna coupling synthetic aperture radar Analog-to-digital converter Artificial neural network Antenna synthetic aperture radar Automatic target classification Automatic target recognition Bistatic inverse synthetic aperture radar Back-projection algorithm Computer aided design Cumulative density function Constant false alarm rate Contrast optimization Co-polarization Circular polarization Cross-polarization Chirp scaling algorithm Continuous wave Dihedral corner reflectors Discrete Fourier transform Digital television Electric field integral equation Electromagnetic Exploding source model Frequency modulated Frequency modulated continuous wave

Acronyms

FT GO GPR GPS GWN H HH HSA HV I IDFT IFT IMU InSAR ISAR JTF KB KMA L LFM LFMCW LHCP LHEP LL LOS LP LR MB MDA MFIE MIMO MOCOMP Mu-ISAR PEC PGA P-ISAR PO PolSAR PPP PRF

Fourier transform Geometric optics Ground-penetrating radar Global positioning system Gaussian white noise Horizontal Horizontal–horizontal Hyperbolic summation algorithm Horizontal–vertical Inphase Inverse discrete Fourier transform Inverse Fourier transform Inertial measurement unit Interferometric SAR Inverse synthetic aperture radar Joint time-frequency Kbytes Kirchhoff migration algorithm Left Linear frequency modulated Linear frequency modulated continuous wave Left-hand circular polarized Left-hand elliptically polarized Left–left Line of sight Linear polarization Left–right Mbytes Map-drift autofocus Magnetic field integral equation Multiple-input multiple-output Motion compensation Multi-static inverse synthetic aperture radar Perfect electric conductor Phase gradient autofocus Passive inverse synthetic aperture radar Physical optics Polarimetric synthetic aperture radar Prominent point processing Pulse repetition frequency

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Acronyms

PRI PSD PSF PSLR PSMA PSR Q QD R RCM RCMC RCS RDA RF RGB RHCP RHEP RL RLOS RR Rx [S] SAC SAR SBR SFCW sinc SNR SPU STFT TCR TFDS TWIR TWR Tx V VH VNA VV ω-kA

Pulse repetition interval Power spectral density Point spread function Peak-to-sidelobe ratio Phase-shift migration algorithm Point-spread-response Quadrature Quadradure detection Right Range cell migration Range cell migration correction Radar cross section Range-doppler algorithm Radiofrequency Red green blue Right-hand circular polarized Right-hand elliptically polarized Right–Left Radar line of sight Right–right Receiver Polarization scattering matrix Shift and correlate Synthetic aperture radar Shooting and bouncing ray Stepped frequency continuous wave Sinus cardinalis Signal-to-noise ratio Signal processing unit Short-time Fourier transform Trihedral corner reflector Time-frequency distribution series Through-the-wall imaging radar Through-the-wall radar Transmitter Vertical Vertical–horizontal Vector network analyzer Vertical–vertical Frequency-wavenumber algorithm

1

1 Basics of Fourier Analysis 1.1

Forward and Inverse Fourier Transform

Fourier transform (FT) is a common and useful mathematical tool that is utilized in innumerous applications in science and technology. FT is quite practical especially for characterizing nonlinear functions in nonlinear systems, analyzing random signals, and solving linear problems. FT is also a very important tool in radar imaging applications as we shall investigate in the forthcoming chapters of this book. Before starting to deal with the FT and inverse Fourier transform (IFT), a brief history of this useful linear operator, and its founders are presented.

1.1.1 Brief History of FT Jean Baptiste Joseph Fourier, a great mathematician, was born in 1768, Auxerre, France. His special interest in heat conduction led him to describe a mathematical series of sine and cosine terms that could be used to analyze propagation and diffusion of heat in solid bodies. In 1807, he tried to share his innovative ideas with researchers by preparing an essay entitled as On the Propagation of Heat in Solid Bodies. The work was examined by Lagrange, Laplace, Monge, and Lacroix. Lagrange’s oppositions caused the rejection of Fourier’s paper. This unfortunate decision cost colleagues to wait for 15 more years to meet his remarkable contributions to mathematics, physics, and especially on signal analysis. Finally, his ideas were published thru the book The Analytic Theory of Heat in 1822 (Fourier 1955). Discrete Fourier transform (DFT) was developed as an effective tool in calculating this transformation. However, computing FT with this tool in the nineteenth century was taking a long time. In 1903, C. Runge has studied on the minimization

Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms: With Advanced SAR/ISAR Imaging Concepts, Algorithms, and MATLAB Codes, Second Edition. Caner Özdemir. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

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1 Basics of Fourier Analysis

of the computational time of the transformation operation (Runge 1903). In 1942, Danielson and Lanczos had utilized the symmetry properties of FT to reduce the number of operations in DFT (Danielson and Lanczos 1942). Before the advent of digital computing technologies, James W. Cooley and John W. Tukey developed a fast method to reduce the computation time of DFT operation. In 1965, they published their technique that later on has become famous as the fast Fourier transform (FFT) (Cooley and Tukey 1965).

1.1.2

Forward FT Operation

The FT can be simply defined as a certain linear operator that maps functions or signals defined in one domain to other functions or signals in another domain. The common use of FT in electrical engineering is to transform signals from time domain to frequency domain or vice-versa. More precisely, forward FT decomposes a signal into a continuous spectrum of its frequency components such that the time signal is transformed to a frequency domain signal. In radar applications, these two opposing domains are usually represented as “spatial-frequency (or wave-number)” and “range (distance).” Such use of FT will be often examined and applied throughout this book. The forward FT of a continuous signal g(t) where −∞ < t < ∞ is described as G f =

gt ∞

=

gt −∞

exp − j2πft dt

11

where represents the forward FT operation that is defined from time domain to frequency domain. To appreciate the meaning of FT, the multiplying function exp(−j2πft) and operators (multiplication and integration) on the right of side of Eq. 1.1 should be examined carefully: The term exp − j2π f i t is a complex phasor representation for a sinusoidal function with the single frequency of “fi.” This signal oscillates with the single frequency of “fi” and does not contain any other frequency component. Multiplying the signal in interest, g(t) with exp − j2π f i t provides the similarity between each signal, that is, how much of g(t) has the frequency content of “fi.” Integrating this multiplication over all time instants from −∞ to ∞ will sum the “fi” contents of g(t) over all time instants to give G(fi) that is the amplitude of the signal at the particular frequency of “fi.” Repeating this process for all the frequencies from −∞ to ∞ will provide the frequency spectrum of the signal represented as G(f). Therefore, the transformed signal represents the continuous spectrum of frequency components; i.e. representation of the signal in “frequency domain.”

1.2 FT Rules and Pairs

1.1.3 IFT This transformation is the inverse operation of the FT. IFT, therefore, synthesizes a frequency-domain signal from its spectrum of frequency components to its time domain form. The IFT of a continuous signal G(f) where −∞ < f < ∞ is described as −1

gt =

G f

∞

=

G f −∞

exp j2πft df

12

where the IFT operation from frequency domain to time domain is represented by −1 .

1.2

FT Rules and Pairs

There are many useful Fourier rules and pairs that can be very helpful when applying the FT or IFT to different real-world applications. We will briefly revisit them to remind the properties of the FT to the reader. Provided that FT and IFT are defined as in Eqs. 1.1 and 1.2, respectively, FT pair is denoted as gt

13

G f

and the corresponding alternative pair is given by −1

G f

14

gt

Based on these notations, the properties of FT are listed briefly below.

1.2.1 Linearity If G(f) and H(f) are the FTs of the time signals g(t) and h(t), respectively, the following equation is valid for the scalars a and b. a g t +b h t

a G f +b H f

15

Therefore, the FT is a linear operator.

1.2.2 Time Shifting If the signal is shifted in time with a value of to, then the corresponding frequency signal will have the form of g t − to

exp − j2πf t o

G f

16

3

4

1 Basics of Fourier Analysis

1.2.3

Frequency Shifting

If the time signal is multiplied by a phase term of exp j2π f o t , then the FT of this time signal is shifted in frequency by fo as given below exp j2π f o t

1.2.4

gt

G f − fo

17

Scaling

If the time signal is scaled by a constant a, then the spectrum is also scaled with the following rule 1 f , G a a

g at

1.2.5

a

R, a

0

18

Duality

If the spectrum signal G(f) is taken as a time signal G(t), then, the corresponding frequency domain signal will be the time reversal equivalent of the original time domain signal, g(t) as Gt

1.2.6

g −f

19

Time Reversal

If the time is reversed for the time-domain signal, then the frequency is also reversed in the frequency domain signal. G −f

g −t

1.2.7

1 10

Conjugation

If the conjugate of the time-domain signal is taken, then the frequency-domain signal conjugated and frequency-reversed. g∗ t

1.2.8

G∗ − f

1 11

Multiplication

If the time-domain signals, g(t) and h(t) are multiplied in time, then their spectrum signals G(f) and H(f) are convolved in frequency. gt

ht

G f ∗H f

1 12

1.3 Time-Frequency Representation of a Signal

1.2.9 Convolution If the time-domain signals, g(t) and h(t) are convolved in time, then their spectrum signals G(f) and H(f) are multiplied in the frequency domain. g t ∗h t

1.2.10

G f

1 13

H f

Modulation

If the time-domain signal is modulated with sinusoidal functions, then the frequency-domain signal is shifted by the amount of the frequency at that particular sinusoidal function. gt

cos 2π f o t

gt

sin 2π f o t

1.2.11

1 G f + fo + G f − fo 2 j G f + fo −G f − fo 2

1 14

Derivation and Integration

If the derivative or integration of a time-domain signal is taken, then the corresponding frequency-domain signal is given as below. d gt dt t −∞

1.2.12

2πf G f

g τ dτ

1 G f + πG 0 j2πf

1 15 δ f

Parseval’s Relationship

A useful property that was claimed by Parseval is that since the FT (or IFT) operation maps a signal in one domain to another domain, their energies should be exactly the same as given by the following relationship. ∞

gt −∞

1.3

2

∞

dt

G f

2

df

−∞

1 16

Time-Frequency Representation of a Signal

While the FT concept can be successfully utilized for the stationary signals, there are many real-world signals whose frequency contents vary over time. To be able to display these frequency variations over time; therefore, joint time–frequency (JTF) transforms/representations are being used.

5

1 Basics of Fourier Analysis

1.3.1

Signal in the Time Domain

The term “time domain” is used while describing functions or physical signals with respect to time either continuous or discrete. The time-domain signals are usually more comprehensible than the frequency-domain signals since most of the real-world signals are recorded and displayed versus time. Common equipment is to analyze time-domain signals is the oscilloscope. In Figure 1.1, a timedomain sound signal is shown. This signal is obtained by recording of an utterance of the word “prince” by a lady. By looking at the occurrence instants in the x-axis and the signal magnitude in the y-axis, one can analyze the stress of the letters inside the word “prince.”

1.3.2

Signal in the Frequency Domain

The term “frequency domain” is used while describing functions or physical signals with respect to frequency either continuous or discrete. Frequency-domain representation has been proven to be very useful in innumerous engineering applications while characterizing, interpreting, and identifying signals. Solving differential equations, analyzing circuits, signal analysis in communication systems are few among many others where frequency-domain representation is much more Time domain signal 0.2 0.1 0 Amplitude

6

–0.1 –0.2 –0.3 –0.4 –0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time, s

Figure 1.1

The time-domain signal of “prince” spoken by a lady.

0.8

1.3 Time-Frequency Representation of a Signal

Frequency domain signal

0.035 0.35

Amplitude

0.025 0.25 0.015 0.15 0.01 0.005 0

0

5

10

15

20

Frequency, kHz

Figure 1.2 The frequency-domain signal (or the spectrum) of “prince.”

advantageous than the time-domain representation. The frequency-domain signal is traditionally obtained by taking the FT of the time-domain signal. As briefly explained in Section 1.1, FT is generated by expressing the signal onto a set of basis functions, each of which is a sinusoid with the unique frequency. Displaying the measure of the similarities of the original time-domain signal to those particular unique-frequency bases generates the Fourier transformed signal or the frequency-domain signal. Spectrum analyzers and network analyzers are the common equipments to analyze frequency-domain signals. These signals are not as quite perceivable when compared to time-domain signals. In Figure 1.2, the frequency-domain version of the sound signal in Figure 1.1 is obtained by using the DFT operation. The signal intensity value at each frequency component can be read from the y-axis. The frequency content of a signal is also called the spectrum of that signal.

1.3.3 Signal in the Joint Time-Frequency (JTF) Plane Although FT is very effective for demonstrating the frequency content of a signal, it does not give the knowledge of frequency variation over time. However, most of the real-world signals have time-varying frequency content such as speech and music signals. In these cases, the single-frequency sinusoidal bases

7

8

1 Basics of Fourier Analysis

are not considered to be suitable for the detailed analysis of those signals. Therefore, JTF analysis methods were developed to represent these signals both in time and frequency to observe the variation of frequency content as the time progresses. There are many tools to map a time domain or frequency-domain signal onto the JFT plane. Some of the most well-known JFT tools are short-time Fourier transform (STFT) (Allen 1977), Wigner–Ville distribution (Nuttall 1988), Choi–Willams distribution (Du and Su 2003), Cohen’s class (Cohen 1989), and time-frequency distribution series (TFDS) (Qian and Chen 1996). Among these, the most appreciated and commonly used one is the STFT or the spectrogram. STFT can easily display the variations in the sinusoidal frequency and phase content of local moments of a signal over time with sufficient resolution in most cases. The spectrogram transforms the signal onto two-dimensional (2D) timefrequency plane via the following famous equation: STFT g t

≜ G t, f ∞

=

−∞

gτ

w τ − t exp − j2πf τ dτ

1 17

This transformation formula is nothing but the short-time (or short-term) version of the famous FT operation defined in Eq. 1.1. The main signal, g(t) is multiplied with a shorter duration window signal, w(t). By sliding this window signal over g(t) and taking the FT of the product, only the frequency content for the windowed version of the original signal is acquired. Therefore, after completing the sliding process over the whole duration of the time-domain signal g(t) and putting corresponding FTs side by side, the 2D STFT of g(t) is obtained. It is obvious that STFT will produce different output signals for different duration windows. The duration of the window affects the resolutions in both domains. While a very short-duration time window provides a good resolution in the time direction, the resolution in the frequency direction becomes poor. This is because of the fact that the time duration and the frequency bandwidth of a signal are inversely proportional to each other. Similarly, a long duration time signal will give a good resolution in frequency domain while the resolution in the time domain will be bad. Therefore, a reasonable selection has to be bargained about the duration of the window in time to be able to view both domains with fairly good enough resolutions. The shape of the window function has an effect on the resolutions as well. If a window with sharp ends is chosen, there will be strong side lobes in the other domain. Therefore, smooth waveform type windows are usually utilized to obtain well-resolved images with less side lobes with the price of increased main

1.3 Time-Frequency Representation of a Signal

Signal in time-frequency plane 16 14

Frequency, kHz

12 10 8 6 4 2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time, s

Figure 1.3 The time-frequency representation of the word “prince.”

beamwidth, i.e. less resolution. Commonly used window types are Hanning, Hamming, Kaiser, Blackman, and Gaussian. An example of the use of spectrogram is demonstrated in Figure 1.3. The spectrogram of the sound signal in Figure 1.1 is obtained by applying the STFT operation with a Hanning window. This JFT representation obviously demonstrates the frequency content of different syllables when the word “prince” is spoken. Figure 1.3 illustrates that while the frequency content of the part “prin…” takes place at low frequencies, that of the part “..ce” occurs at much higher frequencies. JTF transformation tools have been found to be very useful in interpreting the physical mechanisms such as scattering and resonance for radar applications (Trintinalia and Ling 1995; Filindras et al. 1996; Özdemir and Ling 1997; Chen and Ling 2002). In particular, when JTF transforms are used to form the 2D image of electromagnetic scattering from various structures, many useful physical features can be displayed. Distinct time events (such as scattering from point targets or specular points) show up as vertical line in the JTF plane as depicted in Figure 1.4a. Therefore, these scattering centers appear at only one time instant but for all frequencies. A resonance behavior such as scattering from an open cavity structure shows up as horizontal line on the JTF plane. Such mechanisms occur only at discrete frequencies but over all time instants (see Figure 1.4b). Dispersive mechanisms, on the other hand, are represented on the JTF plane as slanted curves. If the dispersion is due to the material, then the slope of the image is

9

1 Basics of Fourier Analysis

Frequency

10

b e

d

f c a Time

Figure 1.4 Images of scattering mechanisms in the joint time–frequency plane. (a) Scattering center, (b) resonance, (c and d) dispersion due to material, (e and f ) dispersion due to geometry of the structure.

positive as shown in Figure 1.4c,d. The dielectric coated structures are the good examples of this type of dispersion. The reason for having a slanted line is because of the modes excited inside such materials. As frequency increases, the wave velocity changes for different modes inside these materials. Consequently, these modes show up as slanted curves in the JTF plane. Finally, if the dispersion is due to the geometry of the structure, this type of mechanism appears as a slanted line with a negative slope. This style of behavior occurs for such structures such as waveguides where there exist different modes with different wave velocities as the frequency changes as seen in Figure 1.4e,f. An example of the use of JTF processing in radar application is shown in Figure 1.5 where spectrogram of the simulated backscattered data from a dielectric-coated wire antenna is shown (Özdemir and Ling 1997). The backscattered field is collected from the Teflon-coated wire (εr = 2.1) such that the tip of the electric field makes an angle of 60 with the wire axis as illustrated in Figure 1.5. After the incident field hits the wire, infinitely successive scattering mechanisms occur. The first four of them are illustrated on top of Figure 1.5. The first return comes from the near tip of the wire. This event occurs at a discrete time that exists at all frequencies. Therefore, this return demonstrates a scattering center-type mechanism. On the other hand, all other returns experience at least one trip along the dielectric-coated wire. Therefore, they confront a dispersive behavior. As the wave travels along the dielectric-coated wire, it is influenced by the dominant dispersive surface mode called Goubau (Richmond and Newman 1976). Therefore, the wave velocity decreases as the frequency increases such that the dispersive returns are tilted to later times on the JTF plane. The dominant dispersive scattering mechanisms numbered 2, 3, and 4 are illustrated in Figure 1.5 where the spectrogram of the backscattered field is presented. The other dispersive returns with decreasing energy levels can also be easily

1.4 Convolution and Multiplication Using FT

2 4

3 30°

1

1 2 3 4 0

Frequency, GHz

16 –10

14 12

–20

10 –30 8 0

5

10

15

–40 dB

Time, ns

Scattering center

Dispersive returns

Figure 1.5 JTF image of a backscattered measured data from a dielectric-coated wire antenna using spectrogram.

observed from the spectrogram plot. As the wave travels on the dielectric-coated wire more and more, it is slanted more on the JTF plane, as expected.

1.4

Convolution and Multiplication Using FT

Convolution and multiplication of signals are often used in radar signal processing. As listed in Eqs. 1.12 and 1.13, convolution is the inverse operation of multiplication as the FT is concerned, and vice versa. This useful feature of the FT is widely used in signal and image processing applications. It is obvious that the multiplication operation is significantly faster and easier to deal with when compared

11

12

1 Basics of Fourier Analysis

to the convolution operation, especially for long signals. Instead of directly convolving two signals in the time domain, therefore, it is much easier and faster to take the IFT of the multiplication of the spectrums of those signals as shown below: −1

g t ∗h t =

−1

=

gt

ht 1 18

G f

H f

In a dual manner, convolution between the frequency-domain signals can be calculated in a much faster and easier way by taking the FT of the product of their time-domain versions as formulated below: G f ∗H f = =

1.5

−1

G f

−1

H f 1 19

gt

ht

Filtering/Windowing

Filtering is the common procedure that is used to remove undesired parts of signals such as noise. It is also used to extract some useful features of the signals. The filtering function is usually in the form of a window in the frequency domain. Depending on the frequency inclusion of the window in the frequency axis, the filters are named low-pass (LP), high-pass (HP), or band-pass (BP). The frequency characteristics of an ideal LP filter are depicted as dashed line in Figure 1.6. Ideally, this filter should pass frequencies from DC to the cut-off frequency; fc and should stop higher frequencies beyond. In real practice, however, ideal LP filter characteristics cannot be realized. According to the Fourier theory, a signal cannot be both time limited and band limited. That is to say, to be able to Filter amplitude 0 dB –3 dB

Ideal Real

Pass-band

Stop-band

fR

Figure 1.6

fC

An ideal and real LP filter characteristics.

Frequency

1.5 Filtering/ Windowing

achieve an ideal band-limited characteristic as in Figure 1.6, then the corresponding time-domain signal should theoretically extent to infinity which is of course not possible for realistic applications. Since all practical human-made signals are time limited, i.e. it should start and stop at specific time instants, the frequency contents of these signals extent to infinity. Therefore, an ideal filter characteristic as the one in Figure 1.6 cannot be realizable; but, only the approximate versions of it can be implemented in real applications. The best implementation of practical low-pass filter characteristic was achieved by Butterworth (Daniels 1974) and Chebyshev (Williams and Taylors 1988). The solid line in Figure 1.6 demonstrates a real LP filter characteristic of Butterworth type. Windowing procedure is usually applied to smoothen a time-domain signal, therefore, filtering out higher frequency components. Some of the popular windows that are widely used in signal and image processing are Kaiser, Hanning, Hamming, Blackman, and Gaussian. A comparative plot of some of these windows is given in Figure 1.7. The effect of windowing operation is illustrated in Figure 1.8. A time-domain signal of a rectangular signal is shown in Figure 1.8a and its FT is provided in Figure 1.8b. This function is, in fact, a sinc (sinus cardinalis) function and has major side lobes. For the sinc function, the highest side lobe is approximately 13 dB lower than the apex of the main lobe. This much of contrast, of course, may not be sufficient in some imaging applications. As shown in Figure 1.8c, the original rectangular time-domain signal is Hanning windowed. Its 1

Amplitude

0.8

0.6

Kaiser

0.4

Hanning Hamming

0.2

0

0

20

40

60

Frequency

Figure 1.7 Some common window characteristics.

80

100

13

14

1 Basics of Fourier Analysis

(a)

(b)

1

0.3

0.8 0.2

0.6 0.4

0.1

0.2 0 –50

0

50

0 –40

Time, s

–20

0

20

40

Frequency, Hz

(c)

(d)

1

0.3

0.8 0.2

0.6 0.4

0.1

0.2 0 –50

0

50

Time, s

0 –40

–20

0

20

40

Frequency, Hz

Figure 1.8 Effect of windowing. (a) Rectangular time signal, (b) its Fourier spectrum: a sinc signal, (c) Hanning windowed time signal, (d) corresponding frequency-domain signal.

corresponding spectrum is depicted in Figure 1.8d where the side lobes are highly suppressed thanks to the windowing operation. For this example, the highest side lobe level is now 32 dB below the maximum value of the main lobe which provides better contrast when compared to the original, non-windowed signal. A main drawback of windowing is the resolution decline in the frequency signal. The FT of the windowed signal has worse resolution than the FT of the original time-domain signal. This feature can also be noticed from the example in Figure 1.8. By comparing the main lobes of the figures on the right, the resolution after windowing is almost twice as bad when compared to the original frequencydomain signal. A comprehensive examination of windowing procedure will be presented later on, in Chapter 5.

1.6

Data Sampling

Sampling can be regarded as the preprocess of transforming a continuous or analog signal to a discrete or digital signal. When the signal analysis has to be done using digital computers via numerical evaluations, continuous signals need to be

1.6 Data Sampling

Figure 1.9 Sampling. (a) continuous time signal, (b) discrete-time signal after the sampling.

(a) s(t)

t

(b) s[n]

Sampling instants 0Ts Ts 2Ts 3Ts 4Ts . . . . . 0 1 2 3 4 ... . .

t n

Sampling numbers

converted to the digital versions. This is achieved by applying the common procedure of sampling. Analog-to-digital (A/D) converters are common electronic devices to accomplish this process. The implementation of a typical sampling process is shown in Figure 1.9. A time signal s(t) is sampled at every Ts seconds such that the discrete signal, s[n], is generated via the following equation: s n = s nT s , n = 0, 1, 2, 3, …

1 20

Therefore, the sampling frequency fs is equal to 1/Ts where Ts is called the sampling interval. A sampled signal can also be regarded as the digitized version of the multiplication of the continuous signal, s(t) with the impulse comb waveform, c(t) as depicted in Figure 1.10. According to the Nyquist–Shannon sampling theorem, the perfect reconstruction of the signal is only possible provided that the sampling frequency fs is equal or larger than twice the maximum frequency content of the sampled signal

Figure 1.10 Impulse comb waveform composed of ideal impulses.

c(t)

...

... 0 Ts 2Ts 3Ts 4Ts . . . . .

15

16

1 Basics of Fourier Analysis

(Shannon 1949). Otherwise, signal aliasing is unavoidable and only distorted version of the original signal can be reconstructed.

1.7

DFT and FFT

1.7.1

DFT

As explained in Section 1.1, the FT is used to transform continuous signals from one domain to another. It is usually used to describe the continuous spectrum of an aperiodic time signal. To be able to utilize the FT while working with digital signals, the digital or DFT has to be used. Let s(t) be a continuous periodic time signal with a period of To = 1/fo. Then, its sampled (or discrete) version is s[n] ≜ s(nTs) with a period of NTs = To where N is the number of samples in one period. Then, the Fourier integral in Eq. 1.1 will turn to a summation as shown below. N −1

S kfo =

s nT s

exp − j2π k f o

s nT s

exp − j2π

s nT s

exp − j2π

nT s

n=0 N −1

=

k NT s

n=0 N −1

= n=0

nT s

1 21

k n N

Dropping the fo and Ts inside the parenthesis for the simplicity of nomenclature and therefore switching to discrete notation, DFT of the discrete signal s[n] can be written as N −1

Sk =

s n exp − j2π

n=0

k n N

1 22

In a dual manner, let S(f) represent a continuous periodic frequency signal with a period of Nfo = N/To and let S[k] ≜ S(kfo) be the sampled signal with the period of Nfo = fs. Then, the IDFT of the frequency signal S[k] is given by s

n fs

N −1

=

S kfo

exp j2π k f o

n fs

S kfo

exp j2π k f o

n Nf o

S kfo

exp j2π

k=0 N −1

s nT s = k=0 N −1

= k=0

n k N

1 23

1.7 DFT and FFT

Using the discrete notation by dropping the fo and Ts inside the parenthesis, the IDFT of a discrete frequency signal S[k] is given as N −1

sn =

Sk k=0

exp j2π

n k N

1 24

1.7.2 FFT FFT is the efficient and fast way of evaluating the DFT of a signal. Normally, computing the DFT is in the order of N2 arithmetic operations. On the other hand, fast algorithms like Cooley-Tukey’s FFT technique produce arithmetic operations in the order of N log(N) (Cooley and Tukey 1965; Brenner and Rader 1976; Duhamel 1990). An example of DFT is given is Figure 1.11 where a discrete time-domain ramp signal is plotted in Figure 1.11a and its frequency-domain signal obtained by an FFT algorithm is given in Figure 1.11b.

1.7.3 Bandwidth and Resolutions The duration, the bandwidth, and the resolution are important parameters while transforming signals from time domain to frequency domain or vice versa. Considering a discrete time-domain signal with a duration of To = 1/fo sampled N times with a sampling interval of Ts = To/N, the frequency resolution (or the sampling interval in frequency) after applying the DFT can be found as Δf =

1 To

1 25

The spectral extend (or the frequency bandwidth) of the discrete frequency signal is B = N Δf N = To 1 = Ts

1 26

For the example in Figure 1.11, the signal duration is 1 ms with N = 10 samples. Therefore, the sampling interval is 0.1 ms. After applying the expressions in Eqs. 1.25 and 1.26, the frequency resolution is 100 Hz and the frequency bandwidth is 1000 Hz. After taking the DFT of the discrete time-domain signal, the first entry of the discrete frequency signal corresponds to zero frequency and negative frequencies are located in the second half of the discrete frequency signal as seen in Figure 1.11b. After the DFT operation, therefore, the entries of the discrete

17

1 Basics of Fourier Analysis

(a) 1.2

Figure 1.11 An example of DFT operation: (a) discrete timedomain signal, (b) discrete frequency-domain signal without FFT shifting, (c) discrete frequency-domain signal with FFT shifting.

1

s[n]

0.8 0.6 0.4 0.2 0

0

2

4

6

8

10

800

1000

Time, ms

(b)

6 5

s[k]

4 3 2 1 0

0

200

400

600

Frequency, Hz

(c)

6 5 4

s[k]

18

3 2 1 0 –500

0 Frequency, Hz

500

1.9 Importance of FT in Radar Imaging

frequency signal should be swapped from the middle to be able to form the frequency axis correctly as shown in Figure 1.11c. This property of DFT will be thoroughly explored in Chapter 5 to demonstrate its use in ISAR imaging. Similar arguments can be made for the case of IDFT. Considering a discrete frequency-domain signal with a bandwidth of B sampled N times with a sampling interval of Δf, the time resolution (or the sampling interval in time) after applying IDFT can be found as Δt = T s 1 = B 1 = NΔf

1 27

The time duration of the discrete time signal is To =

1 Δf

1 28

For the frequency-domain signal in Figure 1.11b or c, the frequency bandwidth is 1000 Hz with N = 10 samples. Therefore, the sampling interval in frequency is 100 Hz. After applying IDFT to get the time-domain signal as in Figure 1.11a, the formulas in Eqs. 1.27 and 1.28 calculate the resolution in time as 0.1 ms and the duration of the signal as 1 ms.

1.8

Aliasing

Aliasing is a type of signal distortion due to undersampling the signal of interest. According to Nyquist–Shannon sampling theorem (Shannon 1949), the sampling frequency fs should be equal to or larger than twice the maximum frequency content; fmax of the signal to be able to perfectly reconstruct the original signal: f s ≥ 2 f max

1 29

Since processing the analog radar signal requires sampling of the received data, the concept of aliasing should be taken into account when dealing with radar signals.

1.9

Importance of FT in Radar Imaging

The imaging of a target using electromagnetic waves emitted from radars is mainly based on the phase information of the scattered waves from the target. This is

19

20

1 Basics of Fourier Analysis

because of the fact that the phase information is directly related to the range distance of the target. In the case of monostatic radar configuration as shown in Figure 1.12a, let the scattering center on the target be at R distance away from the radar. Then, the scattered field E s from this scattering center on the target has a complex scattering amplitude, A and a phase factor that contains the distance information of the target as follows: Es

A exp − j2kR

1 30

As is obvious from Eq. 1.30, there exists a Fourier relationship between the wave number, k, and the distance, R. Provided that the scattered field is collected over a bandwidth of frequencies (Figure 1.12b), it is possible to pinpoint the distance R by

(a) Rmin

R

Target Rmax

Radar

(b) Es

fmin

fmax

f

(c) Range profile

R Rmin

Rmax

Figure 1.12 (a) Monostatic radar configuration, (b) scattered field versus frequency, (c) range profile of the target.

1.9 Importance of FT in Radar Imaging

Fourier transforming the scattered field data as depicted in Figure 1.12c. The plot of scattered field versus range is called the range profile, which is an important phenomenon in radar imaging. Range profile is, in fact, nothing but the onedimensional range image of the target. An example is illustrated in Figure 1.13

Figure 1.13 Simulated range profile of an airplane.

1

× 10–3

0.8

0.6

0.4

0.2

0

0

2

4

6 Range, m

8

10

12

21

1 Basics of Fourier Analysis

where the range profile of an airplane is shown. The concept of range profiling will be thoroughly investigated in Chapter 4, Section 4.3. Another main usage of FT in radar imaging is the ISAR imaging. In fact, ISAR can be regarded as the 2D range and cross-range profile image of a target. While the range resolution is achieved by utilizing the frequency diversity of the backscattered signal, the cross-range resolution is gathered by collecting the backscattered signal over different look angles of the target. An example of ISAR imaging for the same airplane is demonstrated in Figure 1.14 where both the CAD view and the constructed ISAR image for the airplane are shown. The concept of ISAR imaging will be examined in great detail in Chapter 4. The FT operations are also extensively used in synthetic aperture radar (SAR) imaging as well. Since the SAR data are usually huge and processing this amount of data is an extensive and time-consuming task, the FTs are usually utilized to speed up signal processing procedures such as range and azimuth compression. An example of SAR imagery is given in Figure 1.15 where the image was acquired by spaceborne imaging radar-C/X-band synthetic aperture radar (SIR-C/X-SAR 1997) onboard the space shuttle Endeavour in 1994 (www.jpl. nasa.gov/radar/sircxsar/capecod2.html). This SAR image covers an area of Cape Cod, Massachusetts. The details of SAR imagery will be explored in Chapter 3.

ISAR image

8 6 4 Cross-range, m

22

2 0 –2 –4 –6 –8

Figure 1.14

0

2

4

6 Range, m

8

Simulated 2D ISAR image of an airplane. Source: Caner Ozdemir.

10

12

1.10 Effect of Aliasing in Radar Imaging

Figure 1.15 SAR image of the famous “hook” of Cape Cod, Massachusetts, USA. Source: www.jpl.nasa.gov/radar/sircxsar/capecod2.html.

1.10

Effect of Aliasing in Radar Imaging

In radar applications, the data are collected within a finite bandwidth of frequencies. According to the sampling theory, if the radar signal is g(t) and its spectrum is G(f), the frequency components beyond a specific frequency B is zero, that is G f =

0 0

; f RCM_index RCM_filter(round(PRF*dur/2)-RCM_index:round(PRF*dur/2) +RCM_index)=1; else RCM_filter(1:size(RCM_filter,1))=1; end %% ————————Measurement Parameters———————————— rbins = round((Tf-Ts)/dt) + 1; % # of time (Range) samples t = Ts+(0:rbins-1)*dt; % Time array for data acquisition s = zeros(PRF*dur,rbins); % Echoed signal array %% ————————Target Initialization————————————— load TANKS2.mat % Target Intialization Variables ntarget = length(Xind); xn = zeros(ntarget,1); yn = xn; Fn = xn; for m=1: ntarget xn(m)=round(X_focus-5*X0/6+Xind(m)); yn(m)=round(vp*dur/3+Yind(m)); end %% range vector reconstruction load Stanks2.mat

% load SAR raw data

y = eta*vp; % Azimuth vector r = linspace(X_focus-X0,X_focus+X0,rbins); % Range vector tau = r/c/2; % Fast time vector % Figure 3.22a............................................. figure; plot(xn,yn,'k.','MarkerSize',6); axis([920 1060 20 140]); grid minor; axis equal set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight','Bold');

3.12 Matlab Codes

xlabel('range, m'); ('\itscene')

155

ylabel('azimuth, m'); title

% Figure 3.22b............................................. figure; matplot2(tau*1e6,eta,s,dy_range); ax = gca; ax.YDir = 'reverse'; ax.YDir = 'normal'; colormap(gray) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight','Bold'); title('\its_r_e_c(\it\tau, \eta )'); xlabel('range (fast) time, \mus'); ylabel('azimuth (slow) time, s'); axis tight; %% —————RANGE DOPLER ALGORITHM (RDA)———————————————————————— td0 = 0:dt:Tp; pha20 = pi*Kr.*((td0.^2)-td0*Tp); s0 = exp(cj*pha20); fs0 = fftshift(fft(fftshift(s0.'))).'; %Reference signal in freq. domain deltaR = (lambda^2*(X_focus).*((fa).^2))/(8*vp^2); % amounts of RCMs rbins = size(s,2)-size(s0,2)+1; x =linspace(X_field_c-X0,X_field_c+X0,rbins); cells = round(deltaR/((x(2)-x(1)))); rcm_max = max(cells); % maximum RCM fs = zeros(PRF*dur,size(s,2)-size(s0,2)+1); fsm=fs; %% ——————Fast Range Compression with Convolution—————— smb = conv2(conj(s0),1,s.','valid').'; %% range compressed data FMCW % Figure 3.22c......................................... figure; matplot2(r,eta,smb,dy_range); ax = gca; ax.YDir = 'reverse'; ax.YDir = 'normal'; colormap(gray); %colorbar; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight','Bold');

156

3 Synthetic Aperture Radar

title('\its_r_c(r, \eta )'); xlabel('range, m'); ylabel('azimuth (slow) time, s'); rbins = size(smb,2); fsmb = fftshift(fft(fftshift(smb,1)),1); FMCW

% Azimuth FFT

%% —————Azimuth FFT FMCW——————————————————————— % Figure 3.22d............................................. figure; matplot2(x,Fa,fsmb,dy_range); colormap(gray); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight','Bold'); ax = gca; ax.YDir = 'reverse'; ax.YDir = 'normal'; title('\itS_r_c(r, \itf_a )'); xlabel('range, m '); ylabel('azimuth frequency, Hz'); axis tight; % FMCW Ka Ka = (2*(vp^2))./(lambda*(X_field_c-X0:dr/2:X_field_c+X0)); %azimuth FM rate smb0=exp(-cj*(pi*Ka.*eta1.^2)).*(eta1 >= -(0.886*lambda*X_focus)… /(La*vp)/2 & eta1 fsmb0=ftx(smb0); % Azimuth Matched Filter Spectrum % Range Cell Migration Correction (RCMC) fsmb2=fsmb; for k=1:dur*PRF fsmb2(k,1:rbins-rcm_max)=fsmb(k,1+cells(k):rbins-rcm_max +cells(k)); end % Figure 3.22e............................................. figure; matplot2(x,Fa,fsmb2,dy_range); colormap(gray); % colorbar; set(gca,'FontName', 'Arial',

3.12 Matlab Codes

157

'FontSize',12,'FontWeight','Bold'); ax = gca; ax.YDir = 'reverse'; ax.YDir = 'normal'; title('\its_r_c_m_c(r, \itf_a )'); xlabel('range, m '); ylabel('azimuth, Hz'); axis tight; fsac1 = fftshift(ifft(fftshift(fsmb2,1)),1); Azimuth IFFT

%

% Figure 3.22f............................................. figure; matplot2(r,eta,abs(fsac1),dy_range); colormap(gray); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight','Bold'); ax = gca; ax.YDir = 'reverse'; ax.YDir = 'normal'; title('\its_r_c_m_c(r, \it\eta )'); xlabel('range, m '); ylabel('azimuth (slow) time, s'); axis tight; %% Azimuth Compression fsac = fsmb2.*conj(fsmb0); % Azimuth Matched Filtering sac = fftshift(ifft(fftshift(fsac,1)),1); sac=sac/max(max(abs(sac))); % Normalize SAR Data y=(eta*vp); % corresponding azimuth vector %% Focused SAR (range compressed - azimuth compressed) % Figure 3.22g............................................. figure; matplot2(x,y,abs(sac),dy_range); colormap(gray); ax = gca; ax.YDir = 'reverse'; ax.YDir = 'normal'; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight','Bold'); title('\its_r_c_a_c(\tau, \it\eta )'); title('\its_r_c_a_c(r, x)'); xlabel('range, m '); ylabel('azimuth, m'); axis([940 1040 30 130]);

158

3 Synthetic Aperture Radar

References Bamler, R. (1992). A comparison of range-Doppler and wavenumber domain SAR focusing algorithms. IEEE Transactions on Geoscience and Remote Sensing 30 (4): 706–713. Bauck, J. and Jenkins, W. (1988). Tomographic processing of spotlight-mode synthetic aperture radar signals with compensation for wavefront curvature. ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing, New York, NY, USA (11–14 April 1988). https://doi.org/10.1109/ICASSP.1988.196812. Berizzi, F. and Corsini, G. (1996). Autofocusing of inverse synthetic aperture radar images using contrast optimization. IEEE Transactions on Aerospace and Electronic Systems 32 (3): 1185–1191. https://doi.org/10.1109/7.532282. Berizzi, F., Martorella, M., Haywood, B. et al. (2004). A survey on ISAR autofocusing techniques. International Conference on Image Processing, Singapore, Singapore (24–27 October 2004). https://doi.org/10.1109/ICIP.2004.1418676. Bezvesilniy, O.O., Gorovyi, I.M., Sosnytskiy, S.V. et al. (2010). Multi-look stripmap SAR processing with built-in geometric correction. 11th International Radar Symposium, Vilnius, Lithuania (16–18 June 2010). Bezvesilniy, O.O., Gorovyi, I.M., and Vavriv, D.M. (2016). Autofocusing SAR images via local estimates of flight trajectory. International Journal of Microwave and Wireless Technologies 8 (6): 881–889. https://doi.org/10.1017/S1759078716000180. Bityutskov, V.I. (2001). Bunyakovskii inequality. In: Encyclopedia of Mathematics. New York, NY: Springer. Cantalloube, H.M.J. and Nahum, C.E. (2011). Multiscale local map-drift-driven multilateration SAR autofocus using fast polar format image synthesis. IEEE Transactions on Geoscience and Remote Sensing 49 (10): 3730–3736. https://doi.org/ 10.1109/TGRS.2011.2161319. Caputi, W. (1971). Stretch: a time-transformation technique. IEEE Transactions on Aerospace and Electronic Systems AES-7 (2): 269–278. https://doi.org/10.1109/ TAES.1971.310366. Carrara, W.G., Goodman, R.S., and Majewski, R.M. (1995). Spotlight Synthetic Aperture Radar: Signal Processing Algorithms. Boston, London: Artech House. Cumming, I.G. and Wong, F.H. (2005). Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation. Norwood, MA: Artech House. Cutrona, L., Leith, E., Palermo, C., and Porcello, L. (1960). Optical data processing and filtering systems. IEEE Transactions on Information Theory 6 (3): 386–400. https:// doi.org/10.1109/TIT.1960.1057566. Dall, J. (1992). A fast autofocus algorithm for synthetic aperture radar processing. IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, CA, USA (23–26 March 1992). https://doi.org/10.1109/ICASSP.1992.226290.

References

Evans, D.L., Alpers, W., Cazenave, A. et al. (2005). Seasat — a 25-year legacy of success. Remote Sensing of Environment 94 (3): 384–404. Fortuny-Guasch, J. and Lopez-Sanchez, J. (2001). Extension of the 3-D range migration algorithm to cylindrical and spherical scanning geometries. IEEE Transactions on Antennas and Propagation 49 (10): 1434–1444. https://doi.org/10.1109/8.954932. Gabriel, A.K. and Goldstein, R.M. (1988). Crossed orbit interferometry: theory and experimental results from SIR-B. International Journal of Remote Sensing 9 (5): 857– 872. https://doi.org/10.1080/01431168808954901. Goldstein, R.M. and Zebker, H.A. (1987). Interferometric radar measurement of ocean surface currents. Nature 328 (6132): 707–709. https://doi.org/10.1038/328707a0. Graham, L. (1974). Synthetic interferometer radar for topographic mapping. Proceedings of the IEEE 62 (6): 763–768. https://doi.org/10.1109/PROC.1974.9516. Held, D., Brown, W., and Miller, T. (1988). Preliminary results from the NASA/JPL multifrequency, multiploidization synthetic aperture radar. Proceedings of the 1988 IEEE National Radar Conference, Ann Arbor, MI, USA (20–21 April 1988). https:// doi.org/10.1109/NRC.1988.10921. Lacomme, P., Hardange, J.P., Marchais, J.C., and Normant, E. (2001). Air and Spaceborne Radar Aystems: An Introduction. Norwich, NY: William Andrew Publishing. Lanari, R., Hensley, S., and Rosen, P. (1998). Modified SPECAN algorithm for ScanSAR data processing. IGARSS 98. Sensing and Managing the Environment, Seattle, WA, USA (6–10 July 1998). https://doi.org/10.1109/IGARSS.1998.699535. Li, H.J. and Lin, F.L. (1989). Generalized interpretation and prediction in microwave imaging involving frequency and angular diversity. Digest on Antennas and Propagation Society International Symposium, San Jose, CA, USA (26–30 June 1989). Li, H.J., Lin, F.L., Shen, Y. et al. (1990). A generalized interpretation and prediction in microwave imaging involving frequency and angular diversity. Journal of Electromagnetic Waves and Applications 4 (5): 415–430. https://doi.org/10.1163/ 156939390X00636. Lopez-Sanchez, J. and Fortuny-Guasch, J. (2000). 3-D radar imaging using range migration techniques. IEEE Transactions on Antennas and Propagation 48 (5): 728– 737. https://doi.org/10.1109/8.855491. Luminati, J.E. (2005). Wide-angle multistatic synthetic aperture radar: focused image formation and aliasing artifact mitigation. Ph.D. thesis. Air Force Institute of Technology, Wright-Patterson Air Force Base. Mahafza, B.R. (2000). Radar Systems Analysis and Design Using MATLAB. Boca Raton, FL: Chapman and Hall/CRC. Medeiros, F.N.S., Mascarenhas, N.D.A., and Costa, L.F. (2003). Evaluation of speckle noise MAP filtering algorithms applied to SAR images. International Journal of Remote Sensing 24 (24): 5197–5218. https://doi.org/10.1080/0143116031000115148.

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Munson, D., Obrien, J., and Jenkins, W. (1983). A tomographic formulation of spotlight-mode synthetic aperture radar. Proceedings of the IEEE 71 (8): 917–925. https://doi.org/10.1109/PROC.1983.12698. Oliver, B.M. (1951). Not with a bang, but a chirp. Bell Telephone Laboratories Technical Memorandum, MM-51-150-10, case 33089 (8 March 1951). Özdemir, C., Yigit, E., and Demirci, S. (2011). A comparison of focusing algorithms for ground based SAR system. Proceedings of the Progress in Electromagnetics Research Symposium (PIERS) (20–23 March 2011). Marrakesh, Morocco: The Electromagnetics Academy. 7(1):21–26. Özdemir, C., Demirci, Ş., Yiğit, E., and Yilmaz, B. (2014). A review on migration methods in B-scan ground penetrating radar imaging. Mathematical Problems in Engineering 2014: 1–16. https://doi.org/10.1155/2014/280738. Raney, R., Runge, H., Bamler, R. et al. (1994). Precision SAR processing using chirp scaling. IEEE Transactions on Geoscience and Remote Sensing 32 (4): 786–799. https://doi.org/10.1109/36.298008. Rigling, B. and Moses, R. (2004). Polar format algorithm for bistatic SAR. IEEE Transactions on Aerospace and Electronic Systems 40 (4): 1147–1159. https://doi.org/ 10.1109/TAES.2004.1386870. Saepuloh, A., Koike, K., Urai, M., and Sumantyo, J.T.S. (2015). Identifying surface materials on an active volcano by deriving dielectric permittivity from polarimetric SAR data. IEEE Geoscience and Remote Sensing Letters 12 (8): 1620–1624. https://doi. org/10.1109/LGRS.2015.2415871. Samczynski, P. and Kulpa, K. (2010). Coherent mapdrift technique. IEEE Transactions on Geoscience and Remote Sensing 48 (3): 1505–1517. https://doi.org/10.1109/ TGRS.2009.2032241. Sullivan, R.J., Nichols, A.D., Rawson, R.F. et al. (1988). Polarimetric X/L/C-band SAR. Proceedings of the 1988 IEEE National Radar Conference, Ann Arbor, MI, USA (20–21 April 1988). https://doi.org/10.1109/NRC.1988.10922. Ustuner, M., Sanli, F.B., Bilgin, G., and Abdikan, S. (2017). Land use and cover classification of Sentinel-IA SAR imagery: a case study of Istanbul. 25th Signal Processing and Communications Applications Conference (SIU), Antalya, Turkey (15–18 May 2017). Wahl, D., Eichel, P., Ghiglia, D., and Jakowatz, C. (1994). Phase gradient autofocus-a robust tool for high resolution SAR phase correction. IEEE Transactions on Aerospace and Electronic Systems 30 (3): 827–835. https://doi.org/10.1109/7.303752. Wang, R., Loffeld, O., Nies, H. et al. (2010). Focus FMCW SAR data using the wavenumber domain algorithm. IEEE Transactions on Geoscience and Remote Sensing 48 (4): 2109–2118. https://doi.org/10.1109/TGRS.2009.2034368. Wang, G., Zhou, D., and Bo, Y. (2013). Modified prominent point processing in ISAR imaging based on minimum entropy method. Sixth International Symposium on

References

Computational Intelligence and Design, Hangzhou, China (28–29 October 2013). https://doi.org/10.1109/ISCID.2013.71. Wiley, C.A. (1954). Pulsed doppler radar method and means. US Patent 3,196,436, filed 13 August 1954 and issued 20 July 1965. Yang, R., Li, H., Li, S. et al. (2018). High-Resolution Microwave Imaging. Singapore: Springer. Yu, W. (1997). Study for SAR signal processing. Doctoral dissertation. Nanjing University of Aeronautics and Astronautics. Zuo, S., Sun, G., Xing, M., and Chang, W. (2015). A modified fast factorized back projection algorithm for the spotlight SAR imaging. In: 2015 IEEE 5th Asia-Pacific Conference on Synthetic Aperture Radar (APSAR), 756–759. Singapore: IEEE. https:// doi.org/10.1109/APSAR.2015.7306315.

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162

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts Inverse synthetic aperture radar (ISAR) is a powerful signal processing technique for imaging moving targets in range-Doppler (or range and cross-range) domains. ISAR processing is normally used for identifying and classifying targets. As range (or slant range) is defined as the axis parallel to the direction of propagation from radar toward the target, cross-range is defined as the perpendicular axis to the range direction. An ISAR image has the ability of successfully displaying the dominant scattering regions (hot points), i.e., scattering centers on the target. The classical 2D ISAR image is constructed by collecting the scattered field for different look angles and Doppler histories. Although ISAR processing is similar to SAR processing, ISAR imaging procedure has some conceptual differences when compared to the SAR imagery.

4.1

SAR versus ISAR

SAR generally refers to the case when radar platform is moving while the target stays stationary (see Chapter 3, Figures 3.1 and 3.4). The required spatial (or angular) diversity is accomplished by the radar movement around the target or terrain. On the other hand, the term ISAR is used for scenarios when the radar is stationary and the targets are in motion such as airplanes, ships, and tanks as illustrated in Figure 4.1. As similar to the SAR operation, the required range resolution is achieved by using finite frequency bandwidth of the transmitted signal for the ISAR case as well. As depicted in Figure 4.1, stationary radar collects the scattering data from the target for different look angles by utilizing the target movement. While the target is moving, the look angle of target is assumed to be changing with respect to radar line of sight axis to attain a successful ISAR image. This angular diversity in the ISAR data set is used to resolve different cross-range points. The details of these concepts will be thoroughly explained in the forthcoming Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms: With Advanced SAR/ISAR Imaging Concepts, Algorithms, and MATLAB Codes, Second Edition. Caner Özdemir. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

4.1 SAR versus ISAR

θ1 θ2

θ3

Radar

Figure 4.1 Inverse synthetic aperture radar (ISAR) geometry.

v

θ3 θ2 Target

θ1

Figure 4.2 Spotlight SAR geometry with circular flight path is analogous to ISAR geometry.

sub-sections. In terms of the collected echo data-set, ISAR geometry, in fact, can also be thought as the same as spotlight SAR geometry with circular flight paths as illustrated in Figure 4.2. A more detailed comparison can be made through the Figure 4.3 where the ISAR problem and the analogous spotlight SAR problem are illustrated for comparison. As it can be seen from Figure 4.3a, the radar moving along a circular path collects the backscattered field data from the stationary target for an angular span of Ω for

163

164

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

(a)

Moving radar along a circular path

R

Ω

Stationary target

(b) Ω Stationary radar R

Figure 4.3

Rotating target

SAR-to-ISAR transition: (a) spotlight SAR with circular flight path, (b) ISAR.

the spotlight SAR geometry. In Figure 4.3b, however, the stationary radar collects the backscattered field data from a rotating target. The same set of the reflectivity data can be obtained if the target rotates the same angular width of Ω provided that the radar, in either case, is tracked to the target and have the same frequency bandwidth. In most SAR scenarios, of course, the radar’s moves along a straight path rather than a circular path as depicted in Figure 4.4. Therefore, there will be a path length difference dR when compared to the ideal case shown in Figure 4.3a. When the integration angle is small and the target is at a sufficient range distance, R, away from the radar, the path length difference of the received signal will be relatively small as well. Assuming that the path length difference dR is smaller than the wavelength, the phase of the received signal for the straight path operation will have the following extra delay term when compared to the circular path operation: φ y = − 2k dR = − 2k y sin ϕy

41

4.1 SAR versus ISAR y

Radar is moving along straight path Øy R

Ø

y’ x

y R dR

Stationary target

Øy

Figure 4.4 Spotlight SAR with straight flight path.

The factor 2 stands for the two-way propagation between the radar and the target. As seen in Figure 4.4, sin ϕy = y R

42

Furthermore, since R is already assumed to be much larger than y, ϕ is quite small and therefore y

ϕ R

43

So, substituting Equations 4.2 and 4.3 into 4.1, we get φϕ

− 2k ϕ y 2ϕ = − 2π λ

y

44

From the above equation, one can easily realize that the Fourier relation between the aspect variable ϕ and the target’s cross-range variable y is evident. Therefore the scattering point along the target’s cross-range axis can be resolved by the following cross-range resolution δy =

1 width 2ϕ λ

λ 2 width ϕ λ = 2Ω

=

45

165

166

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

where Ω is the total viewing angle (or the angular width) of the aspect variable, ϕ. This clearly states that if the target’s reflectivity is collected over a larger aspect angle (or over a longer synthetic aperture), a finer resolution in the cross-range dimension can be achieved. In real-world practices, of course, the target may have also translational motion components such that the range R is changing while the target is rotating. This situation causes shifting of scatterers in the range from profile to profile and may result in image blurring which will be studied in Chapters 6 and 8 later.

4.2 The Relation of Scattered Field to the Image Function in ISAR We will now demonstrate how the scattered field or the reflectivity function from a target can be related to the image function of this target. We first start with the formula that have been presented in Chapter 2, Section 2.2 at which the scattered electric field from any perfectly conducting object is shown to be equal to Es r

= −

jk o Eo exp − jk o r 4πr s

j k −k

exp

i

r

i

2n r

× k ×u

Slit

d r 2

46

Here, k i and k s are the incident and scattered wave number vectors, n r is the outward surface unitary normal vector, k i is the unit vector in incident wave direction, Eo and u are the magnitude and the polarization unit vector of the incident wave and Slit is the illuminated region of the object’s surface. Now, let’s assume that the receiving antenna has a particular polarization such that it collects the scattered field in the v direction. Then, we can rewrite Eq. 4.6 as given below v Es r

= −

exp j k s − k i where O r

∞

jk o Eo exp − jk o r 4πr

−∞

r

O r 47

d3 r

can be treated as the scalar object shape function (OSF) (Chu et al.

1989; Bhalla and Ling 1993) that is specified as O r

=v

2n r

i

× k ×u

δ S r

48

In the above equation, the argument of the impulse function is defined as the following way

4.3 One-Dimensional (1D) Range Profile

S r

r ϵ Slit

0,

=

49

r ϵ Sshadow

0,

Also, notice that the surface integral in Eq. 4.6 is replaced by a volume integral over the entire three dimensional (3D) space as given in Eq. 4.6. If we define the 3D Fourier transform (FT) of the OSF O(r ) as ∞

O k

=

−∞

O r

exp j k

r

d3 r

4 10

then the scattered electric field in the v direction can be rewritten as v E

s

r

=

−

jk o Eo exp − jk o r 4πr

s

O k −k

i

4 11

This result clearly shows that the scattered electric field from a target is directly proportional to 3D Fourier transform of its OSF. An ISAR image is, in fact, can be regarded as the display of this OSF onto the 2D plane or in the 3D box. It is also worthwhile to mention that OSF varies with respect to look angle and the frequency of operation. As we shall demonstrate in Sections 4.5 throughout 4.7, ISAR image is directly related to the 2D or 3D inverse Fourier transform (IFT) of the scattered electric field as similar to OSF.

4.3

One-Dimensional (1D) Range Profile

ISAR image can be regarded as the display of range/cross-range profiles of the target on the 2D range cross-range plane. Before understanding the meaning of the ISAR image, therefore, it is fundamentally important to appreciate the meaning of the range profile and the cross-range profile concepts. A range profile is the returned waveform shape from a target that has been illuminated by the radar with sufficient frequency bandwidth. If the illuminating wave is a time-domain pulse, then the reflected signal collected by the receiver will have 1D characteristic, typically field intensity (or radar cross section area) versus time (or range) as illustrated in Figure 4.5. If the illuminating signal is collected in frequency domain, then the IFT of the received signal characterizes the 1D range profile of the target. The physical meaning of the range profile is clarified thru the case in Figure 4.5 where a range profile of an airplane is illustrated: As the incident waveform passes along with the target, some of the energy will reflect back toward the radar from so-called scattering centers on the target. If these scattering centers are located at different range distances from the radar, they will reach at different time instants

167

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

Reflected waves from different scattering centers Incident wave front Radar

Intensity

168

Range profile

Time (range)

Figure 4.5

Range profile of a target.

to the radar receiver so that they can be distinguished in the corresponding 1D range profile. As shown in Figure 4.5, the sources of backscattering points may lie on cockpit, motor duck, wings, tails, etc. Of course, it will not be possible to resolve the scattering centers that are in the same range by utilizing the range profile concept as they show up in the same range bin (or time location). The range profile is also named in the radar literature as the radar signature since the returned waveform shape is unique for a specific target as different targets provide different range profiles. Instead of displaying the range profile versus time, it is more physically meaningful to present it versus distance, or range. Then, the range axis can easily be scaled via r=c t

4 12

where r is range, t is time, and c is the speed of the electromagnetic (EM) wave. In air, it is equal to the speed of light. Let us examine how we can get the range profile by processing the frequencydiverse returned (or backscattered) wave. Let us assume that there exist K point scatterers along the downrange (assumed along the x-axis) each located at different xi location. Then, the backscattered electric field at the far-field can be approximated as the following summation over each of the point targets K

Ai exp − j2k x i

Es f i=1

4 13

K

=

Ai exp i=1

2f − j2π c

xi ,

4.3 One-Dimensional (1D) Range Profile

where Ai is the backscattered field amplitude for the point scatterer at xi and k = 2πf/c is the corresponding wave number for the frequency f. The number “2” in the exponential accounts for the two-way propagation between the radar and the point scatterers. Assuming that the backscattered field is collected at the far-field along the −x direction and the phase center of the scene is taken as x = 0, the sign of the exponential should be the same as the sign of the points “xi”s. With this construct, the range profile can be constructed by taking the IFT of this frequency diverse field with respect to (2f/c ≜ α) as given below: −1 α

Es x =

∞

=

Es f K

−∞

Ai exp − j2πα x i

4 14

exp j2πα x dα

i=1

Here, α− 1 denotes the IFT operation with respect to α. In the above equation, both the summing and the integration operators are linear, and therefore they can be interchanged as ∞

K

Es x =

Ai i=1

−∞

exp

j2πα

x − x i dα

4 15

Then, the integral in Eq. 4.15 perfectly vanishes to impulse (or Dirac delta) function, δ( ) as given below: K

Ai δ x − x i

Es x =

4 16

i=1

Here, Es(x) represents the range profile as a function of range, x. Therefore, the point scatterers located at different xi locations are perfectly pinpointed in the range axis with their associated backscattered field amplitudes of “Ai”s. Of course, the result in Eq. 4.16 is valid for the infinite bandwidth. In real applications, however, the backscattered field data can only be collected within a finite bandwidth of frequencies, say ranging from fL to fH. Then, the limits of the integral in Eq. 4.15 should be changed to αL = 2fL/c and αH = 2fH/c to give αH

K

Es x =

Ai i=1

αL

exp

j2πα

x − x i dα

4 17

One can proceed with Es(x) by taking the definite integral in Eq. 4.17 as N

Es x =

Ai i=1

1 j2π

exp

j2π

2fH c

x − xi

− exp

j2π

2fL c

x − xi 4 18

169

170

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

Defining the center frequency as fC = (fL + fH)/2 and the frequency bandwidth as B = fH − fL, the result in Eq. 4.18 can be simplified to give K

Es x =

Ai exp

j2k c x − x i

i=1

exp

j2π

B c

x − xi

− exp

− j2π

B c

x − xi

j2π 4 19 Here, kc = 2πfc/c is the wave number corresponding to the center frequency. The above result can be simplified as Es x =

2B c

K

Ai exp i=1

j2k c x − x i

sinc

2B c

x − xi

4 20

In the above equation, sinc( ) is the sinc (or sinus cardinalis) function already defined in Chapter 3, Eq. 3.32. The exponential in Eq. 4.20 is just the phase term and has the unitary amplitude. The second term, i.e., the sinc is the amplitude term that specifies the shape function of the point scatterer at xi. Therefore, the scattering centers on the range are centered at the true locations of “xi”s with their corresponding field amplitudes, “Ai”s. According to the Fourier theory, sinc defocusing around the scattering centers in the range profile pattern is unavoidable due to finite bandwidth of the radar signal. This defocusing is also known as point-spread-function (PSF) or point-spread-response (PSR) in the radar literature as it will be investigated later in various parts of this book. A very common way of constructing the range profile of a target is accomplished by illuminating it by the SFCW signal. In SFCW set-up, the radar transmits a continuous wave signal modulated at N different stepped frequencies of f1, f2, …, fN and collects the scattered field intensities, Es[ f] for these N discrete frequency values. Then, the time-domain range profile can easily be constructed by applying the inverse discrete Fourier transform (IDFT) operation as Es t = IDFT E s f

4 21

Afterward, the time axis can readily be transformed to range axis by the simple relation of x = c t to get Es[x]. If the frequency bandwidth is B, then the resolution in range, Δx can be taken one over the argument in front of (x − xi) in the sinc function as Δx =

1 c = 2B c 2B

4 22

4.3 One-Dimensional (1D) Range Profile

Each sample, distributed by Δx in the range, is called a range bin or a range cell. The total viewed range, Xmax is then equal to X max = N Δx N c = 2B

4 23

This is the “window frame” in the range or the range extent that can be viewed by the range profile. Therefore, Xmax should be greater than the length of the target in the range to avoid any image ambiguity that may cause aliasing in the ISAR image. An example of the range profile concept is demonstrated in Figure 4.6 where the range profile of a commercial airplane model is obtained. The EM simulation of the backscattered electric field is carried out by a physical optics (PO) and shooting and bouncing ray (SBR) (Ling et al. 1989) code called PREDICS (Özdemir et al. 2014; Kırık and Özdemir 2019) that can estimate the scattering from complex targets at high frequencies with good fidelities. The backscattered electric field is

11 10

Range profile intensity

9 8 7 6 5 4 3 2 1 –40

–30

–20

–10

0 Range, m

Figure 4.6 Range profile of a model airplane.

10

20

30

171

172

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

collected at the far-field from the nose of the airplane between 3.97 and 4.03 GHz for a total of 32 discrete frequencies. Corresponding range profile is obtained as shown in Figure 4.6 by taking the 1D IDFT of the frequency diverse backscattered electric field. As observed from the figure, the major scattering occurs from the nose of the airplane, the engine ducts, the wings, and the tails. The concept of range profiling plays an important role in radar imaging. It can be used as a standard tool for extracting the scattering centers and also determining the length of a target. It may also provide the essential information for the classification of the objects for automatic target recognition (ATR) applications.

4.4

1D Cross-Range Profile

While the range profile of a target can be obtained by processing the frequencydiverse radar return from a target, a cross-range profile can be formed in a dual manner by collecting the radar returns from a target for different look angles as illustrated in Figure 4.7. The aspect width of the look angles is used to resolve the required cross-range points to form the 1D cross-range profile. While the range

y M

ΦM x

Φ1

Φ2

2 1

Figure 4.7 Collecting radar returns at different look angles to form the cross-range profile of a target.

4.4 1D Cross-Range Profile

profile is obtained by treating the backscattered field at single look angle, but different frequencies, the cross-range profile is analogously acquired by processing the backscattered field at one frequency, but different look angles. Let us assume that there exist K point scatterers located at different (xi, yi) points. Our goal is to get the cross-range profile so that we would like to resolve the yi locations of these scatterer points. The backscattered electric field at the far-field for different look angles can be approximated as K

Es ϕ =

Ai exp − j2 k

ri ,

4 24

i=1

where Ai is the backscattered field amplitude for the each point scatterer and r i is the vector from the origin to the point scatterer at (xi, yi). The k r i argument in Eq. 4.24 can be written as follows: k

x i x + yi y r i = kx x + ky y = k x x i + k y yi

4 25

= k cos ϕ x i + k sin ϕ yi Therefore, the backscattered field is equal to K

Es ϕ =

Ai exp − j2k cos ϕ x i

exp − j2k sin ϕ yi

4 26

i=1

As applied in most ISAR applications, cos ϕ and sin ϕ are approximated to 1 and ϕ, respectively for small values of ϕ. Therefore, this equation reduces to K

Es ϕ =

Bi exp − j2kϕ yi i=1

4 27

K

=

Bi exp i=1

2f ϕ yi − j2π c

Here, Bi is a constant and equal to “Ai exp(−j2kxi).” In the above equation, there exists a Fourier relationship between (2f/c)ϕ ≜ γ and yi. Therefore, taking the 1D IDFT of Eq. 4.27 with respect to β, it is possible to resolve “yi”s in the cross-range using the following way Es y =

−1 γ ∞

= =

Es ϕ K

Bi exp − j2πγ yi

−∞

i=1

∞

K

−∞

Bi exp − j2πγ i=1

exp j2πγ y dγ y − yi

dγ

4 28

173

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

In the above equation, linear operators of integration and summation can be interchanged to give ∞

K

Es y =

Bi i=1

−∞

y − yi dγ

exp j2πγ

4 29

K

Bi δ y − yi

= i=1

Here, Es(y) represents the cross-range profile function as a function of y. Therefore, point scatterers located at different cross-range locations yi are perfectly pinpointed in the cross-range axis with corresponding backscattered field amplitudes. The result in Eq. 4.29 would be valid if the backscattered field is collected at infinite number of look angles. Of course, this is not possible in real-life applications. For practical implementation of cross-range profiling, therefore, the limits in the above integration should be changed to finite values of “ϕ” as given below γm 2

K

Es y =

Bi i=1

− γm 2

y − yi dγ

exp j2πγ

4 30

Here, γ m = 2fΩ/c where Ω is the total angular width in collecting the backscattered field. The definite integral in (4.30) can be calculated as K

Es y =

Bi i=1

1 exp j2πγ m j2π

K

= γm 2f Ω c

y − yi

sin 2πγ m y − yi 2πγ m

Bi i=1

=

y − yi − exp − j2πγ m

K

Bi sinc i=1

2f Ω c

y − yi 4 31

In the above equation, the impulse function in Equation 4.29 distorts to sinc functions owing to finite width of look angles as expected. An example of 1D cross-range profiling is illustrated in Figure 4.8 where the cross-range profile of the same airplane model is obtained. The scattered electric field is collected at the far-field around the nose of the airplane at the single frequency of 4 GHz with the help of PREDICS code (Özdemir et al. 2014). The look angle is varied between −1.04 and 1.01 in the azimuth plane for a total of 64 discrete frequencies. Corresponding cross-range profile is acquired as depicted in Figure 4.8 by taking the 1D IDFT of the aspect diverse backscattered electric field. This cross-range profile maps the scattering points from the nose of the airplane, from the engine ducts, and from the wings.

4.4 1D Cross-Range Profile

Cross-range profile intensity

7 6 5 4 3 2 1

–30

–20

–10

0

10

20

30

Cross-range, m

Figure 4.8 Cross-range profile of a model airplane.

The cross-range resolution can be obtained by taking the inverse of the argument in front of (y − yi) of the sinc function in Eq. 4.31 as Δy =

1 c λ = = 2f Ω c 2f Ω 2Ω

4 32

Therefore, the wider look angle span provides better cross-range resolution. Furthermore, the higher frequency of operation yields enhanced cross-range resolution.

175

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle) For bistatic ISAR configuration, the transmitter and the receiver are positioned at different locations in space. If the radar operates as both the transmitter and the receiver, this scenario is called monostatic ISAR which is the common practice in most real-world applications. In this section, we will present a simplified ISAR imaging theory for the monostatic case. Since the 2D ISAR image is nothing but the display of range profile in one axis and the cross-range profile in the other axis, the scattered field should be collected for various frequencies and aspects (i.e., look angles) to be able to generate the 2D ISAR image as illustrated in Figure 4.9a. In this figure, k vector is assumed to lie on the 2D kx–ky plane. Collected data set is generated in the spatial-frequency domains, namely kx and ky. If the backscattered electric field data are gathered within the finite bandwidth of frequencies, B, and within a finite span of angles, Ω. Then, the 2D data occupy a nonuniform grid in the kx–ky space (see Figure 4.9a). However, if both B and Ω are sufficiently small, the data grid in kx–ky space approaches to equally spaced linear grid as illustrated in Figure 4.9b. This situation makes it possible to make use of IDFT in forming the ISAR image as it will be explained in detail below. The algorithm for 2D ISAR imaging is provided for the monostatic case. Let us start with the algorithm such that a point scatterer P(xo, yo) is assumed to be situated on the target as illustrated in Figure 4.10. Taking the origin as the phase center of the geometry, the far-field backscattered field from the point scatterer at an azimuth angle ϕ can be approximated as Es k, ϕ

Ao exp − j2 k

r0

4 33

Here, Ao is the amplitude of the backscattered electric field intensity, k is the vector wave number in the propagation direction, r 0 is the vector from origin to point P. Equation 4.33 has a phase lag of amount 2 k

r 0 . This is because

of the fact that the EM wave hits the point P and reflects back in the same direction, it will travel an extra trip distance of 2 k r 0 when compared to the reference wave that hits the origin and reflects back. Notice that the k vector can be written on the 2D space in terms of the wave numbers in x and y directions as the following k =k k = k x cos ϕ + y sin ϕ ,

4 34

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

(a) BWk

ky

M M–1

kL

BWϕ

kx

2 2 N

kH 1

N–1

(b) ky kx

M M–1

2 N N–1 2 1

Original data Reformatted data

Figure 4.9 (a) Collection of ISAR raw data in Fourier space for the monostatic case (2D case), (b) ISAR data collection for small-bandwidth and small-angle case.

177

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

y P(xo, yo) →

ro

Radar

x

Ø kˆ

→ kˆ · ro

Figure 4.10

Geometry for monostatic ISAR imaging (2D case).

where k, x, and y are the unit vectors in k, x, and y directions, respectively. Then, the argument in the phase term of Equation 4.33 can be reorganized to give k

r 0 = k x cos ϕ + y sin ϕ = k cos ϕ x o + k sin ϕ yo

x x o + y yo 4 35

= k x x o + k y yo Therefore, we can rewrite Equation 4.33 as Es k, ϕ = Ao exp − j2k cos ϕ x o

exp − j2k sin ϕ yo

4 36

This equation offers two separate phase terms as a function of both the spatial frequency variable k and the look angle variable ϕ. If these phase terms are carefully examined, the Fourier relationships between (2k cos ϕ) and x, and (2k sin ϕ) andy can be easily noticed. Therefore, the ISAR image can be generated in range and cross-range domains by the convenience of the 2D IFT. When the practical ISAR imaging is concerned, the standard procedure is to use a small frequency bandwidth of B and a small aspect bandwidth of Ω while collecting the echoed data set. This is called small-bandwidth narrow-angle ISAR imaging. In this standard procedure of ISAR, the frequency bandwidth, B, is small compared to center frequency of operation, fc. In practice, the bandwidths that are less than one-tenth of the center frequency are considered to be sufficiently small. Then the wave number in the second phase term of Eq. 4.36 can be approximated as k

kc 2π f c , = c

4 37

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

where kc is the wave number corresponding to the center frequency, fc. In a similar manner, if the look angle width Ω is narrow such that the following approximations hold true: cos ϕ sin ϕ

1 ϕ

4 38

In practice, angular widths that are at most 5 –6 are generally considered to be small. Then, the scattered electric field from point P can be approximated to E s k, ϕ = Ao exp − j2k x o

exp − j2k c ϕ yo

4 39

To be able to use the advantages of FT, we reorganize Eq. 4.39 as 2f c exp − j2πα x o

E s k, ϕ = Ao exp = Ao

− j2π

xo

exp

− j2π

kc ϕ π

yo

4 40

exp − j2πγ yo

Then the ISAR image in x–y plane can be obtained by taking the 2D IFT of Equation 4.40 as −1 α,γ

= Ao

−1 α

Es x, y = Ao

∞ −∞

E s k, ϕ

exp − j2πα x o exp − j2πα x o

∞ −∞

exp − j2πγ yo = Ao δ x − x o , y − yo

−1 γ

exp − j2πγ yo

exp j2πα x dα

exp j2πγ y dγ

4 41

≜ ISAR x, y where α, and β are already defined in Sections 4.3, and 4.4, respectively. Here δ(x, y) represents the 2D impulse function on the x–y plane. As obvious from Equation 4.41, the point P manifests itself in the ISAR image as a 2D impulse function located at (xo, yo) with the correct EM reflectivity coefficient of Ao. The backscattered electric field from a target can be approximated as the sum of scattering from a finite number of single point scatterers, called scattering centers, on the target as shown below K

E s k, ϕ

Ai exp − j2 k

ri

4 42

i=1

Here, the backscattered electric field from a target is approximated as the sum of backscattered field from a total of K different scattering centers on the target. While Ai is representing the complex backscattered field amplitude for the ith scattering center, r i = x i x + yi y is the displacement vector from origin to

179

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

the location of the ith scattering center. Then, the ISAR image of the target can be found by taking the 2D inverse Fourier integral of the 2D backscattered field data as ∞

ISAR x, y =

−∞

E s k, ϕ

exp

j2παx exp

j2πγy dα dγ

4 43

Assuming that the backscattered signal can be represented by a total number of K scattering centers, the small-bandwidth small-angle ISAR image can then be approximated as ∞

ISAR x, y

K

−∞ i = 1

∞

K

=

Ai

−∞

i=1 K

Ai i=1 K

= i=1

Ai exp − j2 k

∞ −∞

− j2 k

ri

ri

exp j2παx

exp j2παx

exp j2πα x − x i

exp j2πγy dα dγ

exp j2πγy dα dγ

exp j2πγ y − yi

dα dγ

Ai δ x − x i , y − yi 4 44

Therefore, the resultant ISAR image is composed of nothing but the sum of K scattering centers with their EM reflectivity coefficients. Of course, the limits of the integral in Equation 4.44 have to be finite in practice due to the fact that the field data can be collected within a finite bandwidth and a finite aspect width. Therefore, the practical ISAR image response distorts from the impulse function to the sinc function for a finite frequency bandwidth of B and a finite look-angular width of Ω. Consequently, Equation 4.44 will convert to the following as analogous formulation has been given in Sections 4.3 and 4.4: K

ISAR x, y =

Ai sinc i=1

4.5.1

2B c

x − xi

sinc

2fc Ω c

y − yi

4 45

Resolutions in ISAR

Range and the cross-range resolutions in ISAR determine the quality of the resultant image. Therefore, these parameters should be taken into account while applying the ISAR imaging procedure. When the 2D backscattered field data are collected and numerically stored, the Fourier integral in Equation 4.43 is calculated numerically with the help of discrete Fourier transform (DFT).

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

4.5.1.1

Range Resolution

The first sinc term in Equation 4.45 can be used to retrieve the resolution metric for the range. The −4 dB width of this sinc function that is responsible for the range resolution that can be obtained by taking the inverse of the argument within the sinc as Δx =

c 2B

4 46

Therefore, higher frequency bandwidth offers better resolution in the range direction. To achieve a range resolution value of 15 cm, for example, the backscattered electric field data should be collected for a frequency bandwidth of 1 GHz. 4.5.1.2

Cross-Range Resolution:

In a similar manner, the cross-range resolution Δy can be calculated by the using the second sinc term in Equation 4.45. The −4 dB width of this sinc term is equal to c 2 f cΩ λc = 2Ω

Δy =

4 47

where λc corresponds to the wavelength for the center frequency, fc. Equation 4.47 suggests the higher the angle bandwidth the better the resolution in the crossrange direction. To acquire a resolution of 15 cm in the cross-range direction, for instance, the backscattered field data should be collected within an angular width of 5.73 for the center frequency operation of 10 GHz.

4.5.2 Range and Cross-Range Extends Once the range and cross-range resolutions are determined, selection of the number of sampling points determines the spatial extends in these domains, that is, how much the image window extends in range and cross-range directions in the ISAR image. If the frequency bandwidth is sampled by Nx times and the angular span is sampled by Ny times, corresponding image domain extends are given as X max = N x Δx Nx c 2B = N y Δy

= Y max

=

N y λc 2Ω

4 48

181

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

For the same example above, if the 2D frequency-aspect data are collected over 256 sampling points in each domain, the image size becomes 38.4 m by 38.4 m that is more than enough to image a fighter aircraft.

4.5.3

Imaging Multibounces in ISAR

ISAR imaging is based on single-bounce assumption of the scattered waves. On the other hand, it is no doubt that there may be some multiple bounce mechanisms as the EM wave hits the target and bounces around the target. Since the conventional ISAR imaging procedure is based on single-bounce situation, these multibounces will not be correctly mapped in the ISAR image to the actual scattering locations of the target. As it will be shown below, higher-order scattering mechanisms will simply be delayed in the down-range and dislocated in the cross-range. To illustrate how multibounces are mapped in the ISAR image, let us consider a total of N-bounce mechanism as illustrated in Figure 4.11. Assuming that the phase center of the geometry is the origin, the corresponding scattered electric field can be written in the following form Es k x , k y = A exp

−j k

N −1

ri+k

tripn + k

rN

,

4 49

n=1

where A is the complex scattered field intensity after N reflections, tripn = r n + 1 − r n . We can also call

N −1 n=1

tripn

as (triptot) which is the total

trip between the first and the last hit points. Note that displacement vectors r 1 y P2(x2, y2) Trip1

→ r1

From radar

P1(x1, y1)

Trip2 P3(x3 , y3)

x

Ø ˆk

→ rN

PN(xN, yN)

To radar

Figure 4.11

Geometry for imaging multibounce mechanisms in ISAR.

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

and r N equal to r 1 = x 1 x + y1 y

4 50

r N = x N x + yN y

Also, the wave number vector can be written in terms of axis variables and the incident angle variable ϕ as k = k cos ϕ x + k sin ϕ y

4 51

Substituting Equations 4.50 and 4.51 into 4.49, we can rewrite the scattered electric field with a more organized phase term as shown below E s k x , k y = A exp − jk x 1 + x N cos ϕ + y1 + yN sin ϕ + triptot 4 52 Under small bandwidth and narrow-angle ISAR case, the following approximations can be made k x 1 + x N cos ϕ k y1 + yN sin ϕ

k x1 + xN k c y1 + yN ϕ

4 53

Therefore, the final approximated scattered electric field becomes A exp − jk x 1 + x N + triptot

Es kx , ky

exp − jk c y1 + yN ϕ 4 54

Now, we apply the 2D IFT procedure to this scattered field to get the ISAR image. Then, E s x, y = IFTkx ,ky Es k, ϕ =

∞ −∞

Es kx , ky

exp jk x x

e jk y y dk x dk y

4 55

For the small-bandwidth and narrow-angle ISAR, kx 2k and ky 2kcϕ as previously shown. Therefore, the final ISAR image for an N-point multibounce mechanism will be obtained via E s x, y = A exp

jk y y −

∞ −∞

exp

y1 + yN 2

jk x x −

x 1 + x N + triptot 2

dk x dk y = A δ x −

x 1 + x N + triptot y + yN ,y− 1 2 2 4 56

This result shows that the image of a multiple-bounce scattering is delayed in the range and dislocated in the cross-range as illustrated in Figure 4.12. As easily

183

184

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

(a)

(b)

Figure 4.12 (a) Single-bounce mechanisms are correctly mapped in ISAR. (b) Multibounce mechanisms are delayed in range and dislocated in cross-range as their images may show up out of the target.

observed from Equation 4.56, the location of a multibounce scattering in the ISAR image occurs at the following down-range and cross-range points: x 1 + x N + triptot 2 y1 + yN y = 2

x =

4 57

Therefore, multiple bounce features in the ISAR image, if properly interpreted, do carry useful information for understanding the physical phenomenon behind the scattering of the EM wave from the target. When there is only single bounce, of course, xN = x1, yN = y1 and triptot = 0. Therefore, the result in Equation 4.56 readily reduces to the result of a single bounce as Es x, y = A δ x − x 1 , y − y1

4 58

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

Figure 4.13 All double bounces from a 90 corner reflector have the same travel distance as that of a single bounce from an imaginary point scatterer that is supposed to be present at the corner of the plates. 90° corner reflector

Imaginary point scatterer

An interesting case of multibounce occurs from a 90 dihedral corner reflector as demonstrated in Figure 4.13. All the multibounces from this geometry have the same travel distance (or time) that the fictitious point scatterer at the corner of both plates would do. A simulation of a perfectly conducting 90 corner reflector of two identical plates of 1 m × 1 m is carried out around 10 GHz using the PREDICS code (Özdemir et al. 2014, Kırık and Özdemir 2019). The simulated scattered field is carried out along the direction of symmetry line of the reflector for an angular bandwidth of 9.2 and the frequency bandwidth of 0.8 GHz. The resulting ISAR image is shown in Figure 4.14, where all the multibounces coincide at the corner in the image. An example of an ISAR image that contains multibounces is shown in Figure 4.15. The simulation of this airplane model is performed around 45 from the nose-on using PREDICS software. The backscattered electric field is collected between 5.8154 and 6.1731 GHz in frequencies and 41.47 and 48.42 in azimuth angles. The corresponding 2D ISAR image is plotted in Figure 4.15. While hot spots that correspond to single bounce cases occur within the outline of the airplane, some multibounce mechanisms are observed to be located at the outside of the outline of the target.

4.5.4 Sample Design Procedure for ISAR The basic flowchart for designing an ISAR image is given in Figure 4.16. The steps of the algorithm are given briefly in order: Step 1: The key point for a successful ISAR image is to start the procedure by selecting the ISAR image size, that is, range and cross-range window extends. If the range window extend is Xmax and the cross-range window

185

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

1.5

0 –5

1

–10 –15

0.5 x-range, m

–20 0

–25 –30

–0.5 –35 –40

–1

–45 –1.5 –1.5

–1

–0.5

0 Range, m

0.5

1

–50 dBsm

Figure 4.14 ISAR image of a 1 m × 1 m corner reflector at 10 GHz. The images of all multibounces show up at the corner.

0

6

–2 –4

4

–6 2 x-range, m

186

–8 –10

0 –12 –14

–2

–16 –4

–18 Images of multi-bounces

–6 –6

–4

–2

0

2

4

–20 –22 6 dBsm

Range, m

Figure 4.15 ISAR image of a plane model from 45 from the nose on. Some multibounces show up themselves outside of the plane’s outline.

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

Figure 4.16 Flowchart for the basic ISAR imaging algorithm.

Start

Image size: Xmax, Ymax resolutions: Δx, Δy

Nx = Ny =

Δf = Δϕ =

Xmax Δx Ymax Δy

c/2 Xmax λc/2 Ymax

B = Nx Δf Ω = Ny Δϕ

Collect Es(f,ϕ) for the determined frequencies and angles

Take 2D IFT of Es(f,ϕ) ISAR(x, y) Stop

extend is Ymax, then the size of the ISAR image, Xmax by Ymax, should be selected to cover the actual size of the target to be imaged. It is important to note that the size of the target changes in the ISAR image according to the look angle of the radar.

187

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

Step 2: The other key selection is the range and cross-range resolutions, Δx and Δy, respectively. These numbers are so critical that they define how many pixels will lie on the target. Therefore, these resolutions are directly linked to the quality of the ISAR image. After the resolutions in the ISAR image are decided, the sampling points in range, Nx, and the sampling points in cross-range, Ny, can be calculated using the formulas below: Nx = Ny

X max Δx

4 59

Y max = Δy

If the target’s range size is 15 m and the cross-range size is 12 m (which are nominal figures for a fighter aircraft) and the resolutions in both domains are selected as 15 cm, then the target’s range will be displayed with 100 range pixels (or bins) while the target’s cross-range will be displayed by 80 cross-range pixels (or bins). Step 3: Once the ISAR size is determined, the resolutions in frequency, Δf, and aspect, Δϕ, can be determined by utilizing the Fourier relationships between frequency-and-range and angle-and-cross-range in Equation 4.48 as demonstrated below: Δf =

B c 2 = Nx X max

4 60

Ω λc 2 Δϕ = = Ny Y max

Then, the frequency bandwidth, B, and the angular width; Ω will be equal to B = N x Δf =

Nx c 2 X max

4 61 N y λc Ω = N y Δϕ = 2 Y max Step 4: If the frequencies will be centered around fc and the radar look angles will be centered around ϕc, then the backscattered electric field should be collected for the following multiple frequencies and angles: f =

fc −

ϕ =

ϕc −

N x Δf 2 N y Δϕ 2

fc − ϕc −

Nx 2

− 1 Δf

fc

fc +

Nx 2

Ny 2

− 1 Δϕ

ϕc

ϕc +

Ny 2

− 1 Δf + 1 Δϕ

1 × Nx

1 × Ny

4 62

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

Then, collect the backscattered electric field for those frequencies and angles as Es(f, ϕ). Step 5: At this final step, we can take the 2D IFT to get the final ISAR image. If the backscattered field data are collected within a small frequency bandwidth and the angles, then IDFT can be readily applied. Next, we will demonstrate some numeric examples for the construction of ISAR images by applying the steps listed in the above algorithm. 4.5.4.1

ISAR Design Example #1: “Aircraft Target”

The CAD view of an aircraft model whose ISAR image is going to be constructed is shown in Figure 4.17a. The CAD file is composed of many triangle patches. The dimensions of the plane are 14.2, 8.46, and 5.22 m in x, y, and z directions, respectively. The simulation of this aircraft model was carried by the high-frequency EM scattering simulator tool of PREDICS (Özdemir et al. 2014; Kırık and Özdemir 2019). We would like to get a 2D ISAR image of the airplane on the x–y plane. Therefore, we start applying ISAR design steps: Step 1: Since the target size is 14.2, 8.46 m, we should choose an image window extend that should cover the whole airplane on the 2D x–y plane. So, we select the size of the image extends as 18.75 and 11.94 m in x and y directions, respectively. The backscattered data are to be collected looking −15 in elevation from the nose-on direction (105 in elevation and 0 in azimuth) around the center frequency of 12 GHz. The ISAR scenario for the scene is illustrated in Figure 4.17b. Therefore, the range (x) axis and the cross-range (y) axis directions are as indicated in the Figure 4.17b. Step 2: We select the range and cross-range resolutions as Δx = 10.42 cm and Δy = 9.95 cm, respectively. Therefore, the sampling points in range (Nx) and the sampling points in cross-range (Ny) will be equal to Nx =

18 75m 0 1042 m

= 180 Ny =

11 94 m 0 0995 m

4 63

= 120 Step 3: Now, we can determine the frequency resolution, Δf, and aspect resolution, Δϕ, as

189

190

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

(a)

(b) z 15°

x e

y Cr os ran sge

ng Ra

e

l lin

’s raft Airc

axia

15°

Radar look direction

Figure 4.17 situation.

(a) CAD view of a fighter plane, (b) ISAR simulation scenario for the monostatic

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

3 108 2 18 75 = 8 MHz 0 025 2 Δϕ = 11 94 = 0 001 rad 0 06 Δf =

4 64

The frequency bandwidth and the angular width should then be equal to B = 180 8 MHz = 1 44 GHz Ω = 120 0 01rad

4 65

= 0 1257 rad 7 2 Notice that the frequency bandwidth should be around or smaller than the one-tenth of the center frequency that satisfies the small-bandwidth approximation. We select the center frequency to a X-band frequency of 12 GHz which is about 8.33 times larger than the bandwidth that can be still considered to be practically valid for the small bandwidth approximation. The look angle of radar in ϕ direction varies from −0.0625 rad (3.58 ) to 0.0615 rad (3.52 ). Within this angular width, all azimuth angles satisfy sin(ϕ) ϕ; therefore, narrow-angle approximation also holds true. Step 4: Once the above quantities are defined, the backscattered electric field should be collected for frequencies from 11.25 to 12.74 GHz for a total of 180 discrete frequencies and from −3.58 to 3.52 for 120 distinct azimuth angles. Therefore, the simulation of the airplane model is obtained by calculating the backscattered electric field Es(f, ϕ) for these frequencies and angles. At the end of simulation, 2D multifrequency multiaspect backscattered field data of size 180 by 120 are obtained for both VV (vertical transmit – vertical receive) and HH (horizontal transmit – horizontal receive) polarizations. Step 5: In the last step, the 2D IFT of the collected data is taken for both VV and HH polarizations to form the ISAR images depicted in Figure 4.18a and b, respectively. As stated previously, an ISAR image displays the 2D profiles in range and cross-range plane. The ISAR images are plotted in logarithmic scale with the dynamic ranges of 45 dB. For referencing purposes, the silhouette of the aircraft model is also included as a shaded object on the plots. While generating the ISAR images, a Hanning type of windowing function was used to suppress the the sinc-type distortion due to finite size of collected data (finite frequency bandwidth and the finite angular width). The detailed examination of windowing operation for ISAR

191

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

(a)

ISAR image (VV polarization) –10

–5

x–range, m

–15 –20 –25

0

–30 –35 5

–40 –8

–6

–4

(b)

–2

0 2 Range, m

4

6

8

dBsm

ISAR image (HH polarization) –10

–5

–15 x–range, m

192

–20 –25

0

–30 –35 5

–40 –8

–6

–4

–2

0 2 Range, m

4

6

8

dBsm

Figure 4.18 2D ISAR images of the fighter model for (a) VV-polarization, and (b) HHpolarization.

images will be covered in Chapter 5. In both constructed ISAR images, we observe the main backscattering centers located around the canopy, jet engines, wings, and the landing gears of the aircraft while strong scattering centers around these portions of the aircraft can be easily observed.

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

Although the images in Figure 4.18a and b provide similar ISAR images, there are some different radar signatures that can be observed if the images are carefully examined. For instance, we see a straight-line signature at the trailing edge of the wing that extends thru the cross-range (around x = 4 m) that is not available in the HH-polarization image. This is very good example of classification signature that leading and trailing edges of planar surfaces react differently to VV and HH polarized waves. It has been demonstrated in Saynak et al. (2010) that horizontal leading edge of a surface support VV-polarized wave while horizontal trailing edge of a surface support HH-polarized wave. This and similar kind of judgments on this particular ISAR image can be used for classification studies.

4.5.4.2

ISAR Design Example #2: “Military Tank Target”

In this example, we will investigate the design parameters for generating the ISAR image of a military tank model whose CAD file is shown in Figure 4.19a. This tank model has x, y, and z extends of 9.75, 3.36, and 3.24 m, respectively. To calculate the backscattered electric field from this tank model, high-frequency EM scattering simulator tool of PREDICS was used (Özdemir et al. 2014; Kırık and Özdemir 2019). 2D ISAR image of this model can be got by following the ISAR design steps: Step 1: We would like to get the range cross-range image of the tank by looking at 30 above the nose of the airplane (75 in elevation and 0 in azimuth) at the center frequency of 12 GHz, so, the center wavelength is 2.5 cm. From this look angle, the tank model has the range extend of 9.75 m and the cross-range extend of 3.36 m. Therefore, we select the ISAR window size as 11.5385 m by 11.9366 m to be able to cover the whole airplane. The ISAR scenario for the airplane is shown in Figure 4.19b. Step 2: We select the range and cross-range resolutions as Δx = 9.01 cm and Δy = 9.33 cm, respectively. Therefore, the sampling points in range (Nx) and the sampling points in cross-range (Ny) equal to 11 5385 m 0 0901 m = 128

Nx =

11 9366 m Ny = 0 0933 m = 128

4 66

193

194

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

(a)

(b)

z

θ = 75° 0°

ϕ= 3

x y Radar look direction

Figure 4.19 (a) CAD view of a military tank model, and (b) ISAR simulation scenario for the monostatic situation.

4.5 Two-Dimensional (2D) ISAR Image Formation (Small Bandwidth, Small Angle)

Step 3: Now, we can determine the resolutions in frequency and azimuth angle domains as Δf =

3 108 2 11 5385

= 13 MHz 0 025 2 Δϕ = 11 9366 = 0 001 rad 0 06

4 67

Therefore, the frequency bandwidth and the angular width becomes equal to B = 128 13 MHz = 1 664 GHz Ω = 128 0 001 rad

4 68

= 0 134 rad 7 68 Since the frequency bandwidth is a little bit higher than the one-tenth of the center frequency that does not produce a problem in practice. Also, sin (Ω/2) = sin(0.067) = 0.067, therefore, this ISAR set-up satisfies the narrowangle ISAR approximation. Step 4: Once the above quantities are defined, the backscattered electric field should be gathered for frequencies between 11.2 and 12.788 GHz for a total of 128 discrete frequencies and from 26.18 to 33.76 for 128 distinct azimuth angles. Therefore, we simulated the military tank model to get the backscattered electric field Es(f, ϕ) for those frequencies and angles at two polarization cases of namely VV and HH. At the end of simulation, the 128 by 128 2D multifrequency multiaspect backscattered field data for each polarization channel were obtained. Step 5: In this last step, the 2D IFT of the collected data was taken to acquire the ISAR image. The resultant 2D ISAR images for VV and HH polarization cases are given in Figure 4.20a and b, respectively. The dynamic ranges of the ISAR images are set to 50 dB. In both images, the key scattering centers on the tank palettes, gun barrel, cupola, and various parts of the body can easily be observed with good resolution metrics. As we shall see in Chapter 10, the human-made objects of canonical shapes generally provide similar ISAR signatures for the co-polarized (either VV or HH) waves. Therefore, VV- and HH-ISAR images of the tank model produce similar scattering centers as easily observed from the constructed images. Still, there are some discrepancies between the two if the images are carefully inspected. For instance, the backscattering signatures from the tank

195

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

(a)

ISAR image (VV polarization) –10 5 –15

4 3

–20

x-range, m

2 –25

1 0

–30

–1 –35

–2 –3

–40

–4 –45

–5 –5

(b)

0 Range, m

5 dBsm

ISAR image (HH polarization) –10 5 –15

4 3

–20

2 x-range, m

196

–25

1 0

–30

–1 –35

–2 –3

–40

–4 –45

–5 –5

0 Range, m

5 dBsm

Figure 4.20 2D ISAR images of the military tank model for (a) VV-polarization, and (b) HHpolarization.

4.6 2D ISAR Image Formation (Wide Bandwidth, Large Angles)

palettes are more enhanced and more localized such that each of all nine palettes can be easily extracted as the precious feature from the HH-ISAR image in Figure 4.20b. Of course, such an information is quite valuable especially in target classification studies.

4.6 2D ISAR Image Formation (Wide Bandwidth, Large Angles) ISAR systems generally use narrow angular integration widths that may typically extend to only a few degrees while collecting the reflectivity data from the target. This is mainly due to the fact that using narrow look angle apertures provide major simplifications in signal processing and image formation procedures. The planewave illumination assumption can be made under most narrow-angle situations and therefore direct FT procedure can be efficiently employed in forming the final ISAR images as presented in Section 4.5. Besides these benefits, noting that the cross-range resolution is inversely proportional to the look angle span, highresolution ISAR images of targets which have large cross-range extents cannot be possible with the narrow-angle data. Therefore, collecting the backscattered data over a wide angular width will improve the cross-range resolution (Wehner 1997). Furthermore, wide-angle data collections can also provide high range resolutions even with relatively narrow waveform bandwidth (Luminati 2005). On the other hand, wide-angle systems significantly face with the problem of unfocused images. Plane-wave illumination assumption is no longer valid in the wide-angle set-up and hence the imaging algorithm must take the effect of wavefront curvature into account. One possible solution is the sub-aperture approach, which assumes planar wavefronts in smaller sub-apertures of the wide-angle data (Özdemir et al. 2009). This procedure exhibits resolution degradations because it does not use the whole angle aperture for the same integration time. If the frequency bandwidth and the angle width are not small, then the small bandwidth and narrow-angle ISAR imaging procedure cannot be employed. Then, the double integration in forming the ISAR image should be carried out numerically. There are usually two common different ways to acquire the wide bandwidth large angles ISAR imagery: 1) Direct integration 2) DFT based integration after polar reformatting In the former method, the 2D ISAR integral is numerically carried out by applying a numerical integration scheme such as Simpson’s integration and Gaussian quadrature integration procedure. Although this method provides better-resolved

197

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

images in range and cross-range domains for wide frequency bandwidth and large angles, the main drawback is the considerable computation time in evaluating the 2D ISAR integral. This disadvantage, of course, can be somewhat mitigated by applying parallel computing procedures while calculating the whole integration. In the latter method, the collected data are transformed into a uniformly spaced grid such as the ISAR integral is computed with the help of DFT. This transformation is known as polar formatting in radar literature. Next, we will explore the details of both imaging algorithms with numerical demonstrations.

4.6.1

Direct Integration

This method is based on the fact that the ISAR image is proportional to the following integral ϕ2 k 2 ϕ1 k 1

ISAR x, y =

ϕ2 k 2 ϕ1 k 1

E s k, ϕ Es k, ϕ

exp j2 k x x + k y y dk dϕ exp j2 k cos ϕ x + k sin ϕ y dk dϕ 4 69

Here, the backscattered electric field is assumed to be collected for the spatial frequencies from k1 to k2 and for the angles from ϕ1 to ϕ2. Let us investigate how this integral is capable of showing the locations of dominant scattering points of a target, that is, the ISAR image: For a single point scatterer at (xo, yo), the backscattered electric field can be approximated as Es k, ϕ

A exp − j2 k cos ϕ x o + k sin ϕ yo

4 70

Substituting Equation 4.70 into 4.69, we get ϕ2 k 2

ISAR x, y

A ϕ1 k 1

exp j2k cos ϕ

x − x o + sin ϕ

y − yo

dk dϕ 4 71

While this integration is carried out for different values of x and y, the result of Equation 4.71 is maximized only for x = xo and y = yo since the phase of Es and the phase of the integration argument are fully matched to sum up the energy contained in every pixel on the frequency-aspect domain as the integral is evaluated as shown below: ISAR x o , yo

A

ϕ2 k 2 ϕ1 k 1 dk

dϕ

= A k2 − k1 = A BW k Ω,

ϕ2 − ϕ1

4 72

4.6 2D ISAR Image Formation (Wide Bandwidth, Large Angles)

where BWk ≜ k2 − k1 is the spatial frequency bandwidth and Ω ≜ ϕ2 − ϕ1 is the angular width while collecting the backscattered data. For the other values of x and y different than x = xo and y = yo, the result of Equation 4.71 is quite small since the phase of Es and the phase of the integration argument are not matched; therefore, the integration value comes out to be very small when compared to the value at Equation 4.72. This means that the point scatterer at (xo, yo) is pinpointed with an appropriate integration routine with the wide-bandwidth large-angles ISAR imaging integral by normalizing Equation 4.69 as ISAR x, y =

1 κ Ω

ϕ2 k 2 ϕ1 k 1

Es k, ϕ

exp j2 k cos ϕ x + k sin ϕ y dk dϕ 4 73

The resolution of this ISAR image is improved by selecting wider bandwidth and larger aspect width. This, of course, in return, necessitates more computation resources as a result of wider integration ranges. The resolution of the ISAR image can also be visually improved by selecting a smaller numerical integration discretization. This is again at the expense of increasing the computational load in terms of the computing memory and the calculation time. Now, we will demonstrate an application of the wide-bandwidth large-angles ISAR imaging concept via a numerical example. For this purpose, a finite number of perfect point scatterers are assumed to be located as shown in Figure 4.21. A total

8 6 4

y, m

2 0 –2 –4 –6 –8 –8

Figure 4.21

–6

–4

–2

0 x, m

2

4

6

8

The locations of point scatterers around a fictitious airplane.

199

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

of 110 point scatterers are placed to emulate the outline of a fictitious airplane. These point scatterers are assumed to scatter the EM energy in all directions and in all angles with unitary amplitude. Before going into the wide-bandwidth large-angles ISAR imaging routine, it may be useful to get the small-bandwidth narrow-angle ISAR image of this geometry for comparison reasons. For this purpose, we select an ISAR image window of 18 m in the range direction and 16 m in the cross-range direction. Then, applying the basic ISAR design procedure, we end up with getting a frequency bandwidth of 525 MHz around the center frequency of 8 GHz and the angular width of 4.23 around the nose of the airplane-like geometry. After collecting the backscattered electric field for 64 different frequencies and 64 different aspects, the conventional smallbandwidth small-angle ISAR imaging algorithm is applied to get the final ISAR image as shown in Figure 4.22. Now, we will increase both the frequency bandwidth and the aspect width to employ the wide-bandwidth large-angles ISAR imaging procedure. For this purpose, the backscattered electric field from this geometry is collected around the nose of this airplane-like geometry from −30 to 30 in the azimuth angles. While doing this, the frequency is also altered from 6 to 10 GHz providing a 50% bandwidth around the center frequency of 8 GHz. This data collection set-up, of course, does not meet the regular ISAR imagery specifications of small bandwidth and narrow-angles. After collecting the backscattered field for those angles and frequencies, the regular ISAR algorithm based on small bandwidth and narrow

–8

0

–6 –5 –4 Cross–range, m

200

–2

–10

0 –15

2 4

–20 6 8 –8

Figure 4.22

–6

–4

–2

2 0 Range, m

4

6

–25 8 dBsm

Small-bandwidth small-angle ISAR image of the hypothetical airplane model.

4.6 2D ISAR Image Formation (Wide Bandwidth, Large Angles)

0

–5

–2

–4

–4

Cross–range, m

–3

–6

–2

–8

–1

–10

0

–12

1

–14

2

–16

3

–18

4

–20

5 –6

–4

–2

2 0 Range, m

4

6

–22 dBsm

Figure 4.23 Aliased ISAR image after applying a 2D IFT to a wide-bandwidth large-angle backscattered field.

angles is applied using the conventional DFT based ISAR imaging. The resulting image is depicted in Figure 4.23. Of course, the image is highly distorted since the collected data do not lie on a rectangular grid on the spatial frequency plane and therefore, 2D IFT operation vastly spreads the locations and PSRs of the scattering centers on the target as expected. This result clearly shows that wide-bandwidth and/or wide-aspect backscattered data should be treated differently. The same data are processed through the wide-bandwidth large-angles ISAR imaging integral in Eq. 4.73 with the use of Simpson’s integration rule. The resultant ISAR image is plotted in Figure 4.24 where the scattering centers are almost perfectly imaged thanks to direct integration method of ISAR. Here, the resolution cell is determined by the integration discretization value which is quite small which in turn provides almost perfect resolution values both in range and cross-range domains. It is worthwhile to mention that there always exists numerical noise due to approximations in the numerical integration scheme. The numerical noise in the ISAR image in Figure 4.24, for instance, can be visible after −25 dB below the maximum pixel in the image.

4.6.2 Polar Reformatting Another way of treating the wide-bandwidth wide-aspect backscattered data is by utilizing the polar reformatting algorithm. The main idea is to reformat the data in the spatial frequency domain to make use of DFT for fast formation of the ISAR

201

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

0 –5

–2

–4

–4

–3 Cross–range, m

202

–6

–2

–8

–1

–10

0

–12

1

–14

2

–16

3

–18

4

–20

5 –6

–4

–2

0 2 Range, m

4

6

–22 dBsm

Figure 4.24 Wide-bandwidth large-angle ISAR image of the airplane-like geometry (after direct integration).

ky

kx

Original data Reformatted data

Figure 4.25

Rectangular reformatting of polar ISAR data.

image. In this sub-section, wide-bandwidth large-angles ISAR imaging based on polar reformatting routine will be explored. Since the data, Es(k, ϕ), are collected in the frequency-aspect domain, it is in equally spaced rectangular form in this domain. However, the data are, in fact, in polar format in the spatial frequency domain on kx–ky plane as seen in Figure 4.25. It is obvious from Equation 4.73 that the Fourier relationship exists between kx and x and between ky and y. Therefore, fast computation of ISAR

4.6 2D ISAR Image Formation (Wide Bandwidth, Large Angles)

integral can be done by the help of DFT only if the data are transformed on a uniform grid on kx–ky plane. The use of DFT/IDFT requires the data should be in a discrete, uniformly spaced rectangular grid form. Therefore, the data need to be transformed from the polar format to the Cartesian format as illustrated in Figure 4.25. This process is known as polar reformatting. To minimize the error associated with this reformatting procedure, several interpolation schemes such as four-nearest neighbor approximation (Özdemir et al. 1998; Li et al. 1999) can be employed. After putting the data is in its proper format, the ISAR image can be generated in a similar manner as explained in Section 4.4 by applying 2D DFT operation. Polar reformatting algorithm is demonstrated with the same wide-bandwidth large-angles data-set collected for the hypothetical target in Figure 4.21. The backscattered field data in the frequency-aspect domain is depicted in Figure 4.26. Then, the data are reformatted on 2D spatial frequency plane of kx–ky by using the four-nearest neighbor algorithm. The details of this algorithm will be presented later in Chapter 5, Section 5.3. The resultant reformatted data are shown in Figure 4.27. Once, the field data are transformed uniformly onto the kx–ky plane, DFT processing can then be applied to transform the data onto x–y plane, that is, the image plane. The result is nothing but the 2D ISAR image as depicted in Figure 4.28. As in the case of direct integration, polar reformatting followed by IDFT processing provides very fine resolutions depending both in range and cross-range dimensions. The numerical noise due to reformatting the data from –30

0 –5

–20

Angle, degree

–10 –10

–15 –20

0

–25 10 –30 20

30

–35

6

6.5

7

7.5

8

8.5

9

9.5

–40 10 dBsm

Frequency, GHz

Figure 4.26

The backscattered field data in frequency-aspect domain.

203

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

0

–100

–5

–80 –60

–10

ky, rad/m

–40 –15

–20 0

–20

20

–25

40 –30 60 –35

80 100 110

120 130 140 150

160 170

180 190 200

dBsm

–40

kx, rad/m

Figure 4.27

The backscattered field data on spatial frequency plane of kx–ky.

0

–6

–4

Cross–range, m

204

–5

–2 –10 0 –15 2 –20

4

6 –8

–6

–4

–2

0 2 Range, m

4

6

–25 8 dBsm

Figure 4.28 Wide-bandwidth large-angle ISAR image of the airplane-like geometry (after polar reformatting and DFT).

4.7 3D ISAR Image Formation

polar format to the rectangular format is unavoidable. For this particular example, the numerical noise for this ISAR image starts to show up after −30 dB below the maximum point in the image.

4.7

3D ISAR Image Formation

In the conventional operation of ISAR, the coherent system tracks the target and collects the frequency-diverse reflected energy for different look angles of target, usually by utilizing the rotational motion of the target. Displaying the target’s cross-sectional area over different range points and Doppler frequencies (or cross-range points) yields the 2D ISAR image of the target. In a similar manner, a 3D ISAR image of the target may be obtained if the target’s translational and rotational motion components can be measured or at least estimated accurately such that the target’s profiles for two different orthogonal look angles are obtained. Recent studies on 3D ISAR imaging concept have been mainly focused on two different approaches (Knaell and Cardillo 1995; Fortuny 1998; Mayhan et al. 2001; McFadden 2002; Lord et al. 2006): The first approach is based on interferometric ISAR set-up, where multiple antennas are used at different heights so that the second cross-range dimension is tried to be resolved (Xu et al. 1999; Xu and Narayanan 2001; Wang et al. 2001; Zhang et al. 2004). This approach has many limitations including its high sensitivity with respect to glint noise of the target (Wu 1993) and the lack of multiple height (z) information for a selected range cross-range (x, y) point. On the other hand, the second approach that utilizes single ISAR antenna can provide radar cross section (RCS) profiles at multiple heights and multiple range and cross-range points. Some of the 3D ISAR imaging algorithms are listed in (Knaell and Cardillo 1995; Fortuny 1998; Mayhan et al. 2001; McFadden 2002; Lord et al. 2006). Here, we will present 3D ISAR imaging for small frequency bandwidth and narrow-angle approximation while collecting the backscattered field data. A 3D ISAR image, in fact, presents a 3D profile in range (say x) and two crossrange domains (say y and z). Therefore, the scattered field should be collected for various frequencies and different azimuth and frequency angles as depicted in Figure 4.29. As illustrated in this figure, the collected data occupies a space in kx, ky, and kz domains. If the backscattered electric field data are collected within a small bandwidth of frequencies, B, width of viewing azimuth (ϕ) angles, Ω, and width of elevation (θ) angles, ψ, the data points in kx − ky − kz space almost lie onto an equally-spaced linear grid. As similar to 2D ISAR, this assumption makes it possible to use DFT in forming the 3D ISAR image.

205

206

4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

kz 1 N N–1

2

ψ

21

M–1

M T T–1

kx

Ω 2 1

BWk

Figure 4.29

ky

Collection of raw ISAR data in Fourier space (3D monostatic case).

z P(xo, yo,zo) y →

ro

θ

Radar

A

ϕ

A’

x

α kˆ

Figure 4.30

α

→

kˆ · ro

Geometry for monostatic ISAR imaging (3D case).

The algorithm is based on the model in which a point target is assumed to be present (see Figure 4.30). The point scatterer to be imaged is located at P(xo, yo, zo). The radar illuminates the target with a wavenumber vector of k = k k . The direction of illumination makes an angle of θ with the z-axis. It also makes an angle of α with the x–y plane. The line A–A indicates the projection line of k vector on the x–y plane. So, one can easily notice that θ + α = 90 . Taking origin

4.7 3D ISAR Image Formation

as the phase center of the geometry, the far-field backscattered electric field from a point scatterer P at an azimuth angle ϕ and at an elevation angle θ can be written as E s k, ϕ, θ = A0 exp − j2 k

r0

4 74

Here, A0 is the amplitude of the backscattered electric field intensity, k is the vector wave number in k-direction, r 0 is the vector from origin to point P. The multiplier “2” in the exponential accounts for the two-way propagation between the radar and the scatterer. The reason Equation 4.74 has a phase lag of amount 2k

r 0 is that the EM wave hits point P and reflects back in the same direction

will travel an extra the trip distance of 2k r 0

when compared to reference

wave that hits the origin and reflects back. k vector can be written in terms of the wave numbers in x, y, and z directions as k =k k = k x sin θ cos ϕ + y sin θ sin ϕ + z cos θ =k

4 75

x cos α cos ϕ + y cos α sin ϕ + z sin α ,

where k, x, y and z are the unit vectors in k, x, y, and z directions, respectively. Then, the argument in the phase term of Equation 4.74 can be reorganized to give k r o = k x cos α cos ϕ + y cos α sin ϕ + z sin α x x o + y yo + z zo = k cos α cos ϕ x o + k cos α sin ϕ yo + k sin α zo = k x x o + k y yo + k z zo 4 76 Therefore, we can rewrite Equation 4.74 as E s k, ϕ, α = Ao exp − j2k cos α cos ϕ x o exp − j2k cos α sin ϕ yo

4 77

exp − j2k sin α zo This equation offers three separate phase terms as a function of the spatial frequency k and the angles α and ϕ. At this point we assume that the backscattered signal is collected within a frequency bandwidth, B, that is small compared to the center frequency of operation, fc. We further assume that the angular bandwidths in both α (elevation) and ϕ (azimuth) directions are also small. If the radar is situated around the x-axis, following assumptions are valid:

207

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

k cos α cos ϕ

kc 1

cos α sin ϕ

ϕ

sin α

α,

4 78

where kc is the wave number corresponding to the center frequency of operation and is given by kc = 2πfc/c. Then, the scattered electric field from point P is approximated as Es k, ϕ, α = Ao exp − j2k x o

exp − j2k c ϕ yo

exp − j2k c α zo 4 79

To be able to use the advantages of FT, we reorganize Equation 4.79 as 2f kc ϕ exp − j2π exp xo yo c π exp − j2πγx o exp − j2πβyo exp − j2πζzo

E s k, ϕ, α = Ao exp = Ao

− j2π

− j2π

kc α zo π

4 80 Here, we let (2f/c) ≜ γ, (kcϕ/π) ≜ β and (kcα/π) ≜ ζ for simplicity in the formulations. Then the 3D ISAR image in 3D x–y–z space can be obtained by taking the 3D IFT of Equation 4.80 as ISAR x, y, z =

−1 γ,β,ζ

E s k, ϕ, α −1 γ

= Ao −1 ζ

−1 β

exp − j2πβyo

exp − j2πζzo ∞

= Ao

exp − j2πγx o

−∞ ∞ −∞ ∞ −∞

exp − j2πγx o

exp

j2πγx dγ

exp − j2πβyo

exp

j2πβy dβ

exp − j2πζzo

exp

j2πζz dζ

= Ao δ x − x o , y − yo , z − zo

4 81

Here δ(x, y, z) represents the 3D impulse function in x–y–z space. As obvious from Equation 4.81, the point P manifests itself in the ISAR image as the 3D impulse function located at (xo, yo, zo) with the correct reflectivity coefficient of Ao. The backscattered electric field from a target can be approximated as the sum of scattering from a finite number of single point scatterers, called scattering centers, on the target as shown below

4.7 3D ISAR Image Formation K

E s k, ϕ, α

Ai exp − j2 k

ri

4 82

i=1

Here, K is the total number of scattering centers on the target. While Ai represents the complex backscattered field amplitude for the ith scattering center, r i = x i x + yi y + zi z is displacement vector from origin to the location of the ith scattering center. Then, the ISAR image of the target can be found by taking the 3D inverse Fourier integral of the 3D backscattered field data as ∞

ISAR x, y, z =

−∞

Es k, ϕ, α

exp j2πγx exp j2πβy exp j2πζy dγ dβ dζ 4 83

By putting the scattering center representation for the backscattered field, the small bandwidth narrow-angle ISAR image can then be approximated as K

Ai δ x − x i , y − yi , z − zi

ISAR x, y, z

4 84

i=1

Therefore, the resultant ISAR image is the sum of K scattering centers with their EM reflectivity coefficients. Of course, the limits of the integral in Equation 4.83 cannot be infinite in practice due to the fact that the field data can only be collected within a finite bandwidth for frequencies, B, finite width for azimuth angles,Ω, and a finite width for elevation angles, Γ. Therefore, the practical ISAR image response distorts from the impulse function to the sinc function in a similar manner that we have already found in 2D ISAR case (see Equation 4.45 for comparison): K

ISAR x, y, z =

Ai sinc i=1

sinc

2fc Γ c

2B c

x − xi

z − zi

sinc

2fc Ω c

y − yi 4 85

4.7.1 Range and Cross-Range resolutions Range and the cross-range resolutions in 3D ISAR are determined in the same way as they are decided in 2D ISAR. Since the same phase terms for the distance variables x and y show up in the 3D ISAR Equation 4.85, the range resolution in x and the cross-range resolution in y is the same as in the case of the 2D case: Δx =

c 2B

4 86

Δy =

λc 2Ω

4 87

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

The other cross-range resolution in the z-direction can be found by making use of the Fourier relationship between the elevation angle variable α and the crossrange distance variable, z. If the backscattered data are collected within finite angular width of Γ in the elevation angles, the cross-range resolution of ISAR in the z-direction can be retrieved from the last sinc term of Equation 4.85 as 1 2 f cΓ c c = 2 f cΓ λc = 2Γ

Δz =

4.7.2

4 88

A Design Example for 3D ISAR

In this example, the 3D ISAR imaging for the airplane whose model is shown in Figure 4.31a will be demonstrated. As the perspective, front and side views of the airplane are depicted in the figure, the perfectly electric conducting model has the extends of 15.931, 21.60, and 4.895 m in x, y, and z, respectively. To calculate the backscattered electric field from this airplane model, high-frequency EM scattering simulator tool of PREDICS was utilized (Özdemir et al. 2014; Kırık and Özdemir 2019). During the simulation, HH polarized electric field was gathered. 3D ISAR image of this airplane model can be obtained by following design steps: Step 1: The 3D ISAR imaging geometry of this problem is illustrated in Figure 4.31b. In this example, we will select center of the radar look angle as from the nose of the airplane (θc = 90 or αc = 90 , ϕc = 0 ). The center frequency of operation is designated as 8 GHz which means that the center wavelength is equal to 3.75 cm. The 3D ISAR image window size is selected as (Xmax = 37.5 m, Ymax = 25.5785 m, Zmax = 7.9577 m) to be able to include the whole airplane. Step 2: We select the range and cross-range resolutions as Δx = 26.79 cm, Δy = 24.59 cm, and Δz = 24.87 cm, respectively. Therefore, the sampling points in range (Nx) and the sampling points in two cross-range dimensions (Ny, Nz) will be equal to 37 5 = 140 0 2679 25 5785 Ny = = 104 0 2459 7 9577 Nz = = 32 0 2487

Nx =

4 89

4.7 3D ISAR Image Formation

(a)

(b)

z y

Cross– range

Cross– range

x Range α Radar look direction

ϕ

Figure 4.31 (a) CAD view of a bomber airplane, (b) 3D ISAR simulation scenario for the monostatic situation.

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

Step 3: Now, we can determine the resolutions in frequency, Δf, elevation angles, Δα, and azimuth angles, Δϕ as 3 108 2 = 4 MHz 37 5 0 0375 2 Δϕ = = 0 000733 rad 0 042 25 5785 0 0375 2 = 0 0024 rad 0 135 Δα = 7 9577 Δf =

4 90

Step 4: Now, we collect the backscattered electric field in 3D frequency–elevation– azimuth domains as follows: Frequencies: ranges from 7.70 to 8.296 GHz for a total of 140 discrete frequencies. Azimuth angles: varies between −2.149 and 2.107 for a total of 104 distinct angles. Elevation angles: starts from −2.107 (θ = 92.107 ) to 2.149 (θ = 87.851 ) for a total of 32 equally spaced angles. Provided that the center frequency of operation is 8 GHz, the frequency bandwidth becomes 560 MHz, the angular spans in both azimuth and elevation are 4.368 that clearly assures the small bandwidth, small-angle approximation for this particular example. Therefore, backscattered electric field Es(f, ϕ, α) from the airplane model is simulated for those frequencies and angles. At the end of simulation, 140 × 104 × 32 multifrequency multiaspect backscattered field data are obtained. Step 5: In the final step, the 3D backscattered data Es(f, ϕ, α) for the airplane model is inverse Fourier transformed to get the final 3D ISAR image. Since the image is three dimensional, the 2D range (x) cross-range (y) slices of this 3D ISAR image for different values of the other cross-range dimension variable (z) are displayed in Figure 4.32. Although there are 32 slices on x–y plane overall, only 8 of them are displayed starting from z = − 3.48 m and rise up to z = 3.48 m as 2D ISAR images with dynamic ranges of 40 dB. For referencing purposes, the shaded silhouette of the airplane model is also included in each ISAR image. By looking at these figures, we can observe different scattering centers for different z values in each of 2D ISAR slices. In an alternative representation, the 2D projections of the 3D ISAR image are presented in Figure 4.33a–c where projections on principal x–y, x–z and y–z plane are shown, respectively.

4.7 3D ISAR Image Formation

(a)

ISAR(x,y) @z = –3.48 m 10

–5 –10 –15 –20

0

dBsm

Cross–range, m

5

–25 –5

–30 –35

–10 –40 –10

–5

0 Range, m

5

10

(b) ISAR(x,y) @z = –2.49 m 0

10

–5 –10 –15 0 –20

dBsm

Cross–range, m

5

–25

–5

–30 –10

–35 –10

–5

0 Range, m

5

10

Figure 4.32 2D ISAR(x, y) slices for different z values of the 3D ISAR image of the airplane model. at z is equal to: (a) −3.48m, (b) −2.49 m, (c) −1.49 m, (d) −0.5 m, (e) 0.5 m, (f) 1.49 m, (g) 2.49 m, and (h) 3.48m.

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

(c)

ISAR(x,y) @z = –1.49 m 10

10 5 0 –5

0

dBsm

Cross–range, m

5

–10 –5

–15 –20

–10 –25 –10

–5

0 Range, m

5

10

(d) ISAR(x,y) @z = –0.5 m 15 10 10 5

5

0 0

–5 –10

–5 –15 –20

–10 –10

Figure 4.32

(Continued)

–5

0 Range, m

5

10

dBsm

Cross–range, m

214

4.7 3D ISAR Image Formation

(e)

ISAR(x,y) @z = 0.5 m 15 10 10 5 0 0

–5

dBsm

Cross–range, m

5

–10 –5 –15 –20

–10 –10

–5

0 Range, m

5

10

(f) ISAR(x,y) @z = 1.49 m 10

10 5 0 –5

0

–10 –5

–15 –20

–10 –25 –10

Figure 4.32

(Continued)

–5

0 Range, m

5

10

dBsm

Cross–range, m

5

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

(g)

ISAR(x,y) @z = 2.49 m 0

10

–5 –10 –15 0

dBsm

Cross–range, m

5

–20 –25

–5

–30 –10

–35 –10

–5

0 Range, m

5

10

(h) ISAR(x,y) @z = 3.48 m 10

–5 –10

5 –15 –20

0

–25 –5

–30 –35

–10 –40 –10

Figure 4.32

(Continued)

–5

0 Range, m

5

10

dBsm

Cross–range, m

216

4.8 Matlab Codes

(a) ISAR image (HH polarization) 15

10

Cross–range (y), m

10 5

5 0

0

–15 –5

–10 –15

–10 –20 –15

–10

–5

0

5

10

15

dBsm

Range (x), m

Cross–range (z), m

(b)

ISAR image (HH polarization) 10 2

5

0

0

–2 –4

–5 –10

–5

Cross–range (z), m

(c)

0 5 Cross–range (y), m

10

dBsm

ISAR image (HH polarization)

–4

–15

–10

–5

Range (x), m 0 5

10

15

10 5

–2

0

0

–5

2

–10 –15 dBsm

Figure 4.33 2D projections of the 3D ISAR image of the airplane model on (a) x–y plane, (b) x–z plane, and (c) y–z plane.

4.8

Matlab Codes

Below are the Matlab source codes that were used to generate all of the Matlabproduced figures in this chapter. The codes are also provided inside the CD of this book.

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

Matlab code 4.1 Matlab file “Figure4-6.m” %——————————————————————————————————————————————————————— % This code can be used to generate Figure 4-6 %——————————————————————————————————————————————————————— clear close all c = .3; % speed of light fc = 4; % center frequency phic = 0*pi/180; % center of azimuth look angles thc = 80*pi/180; % center of elevation look angles %________________PRE PROCESSING________________ BWx = 80; % range extend M = 32; % range sampling dx = BWx/M; % range resolution X = -dx*M/2:dx:dx*(M/2-1); % range vector XX = -dx*M/2:dx/4:-dx*M/2+dx/4*(4*M-1); % range vector (4x upsampled) %Form frequency vector df = c/2/BWx; % frequency resolution F = fc+[-df*M/2:df:df*(M/2-1)]; % frequency vector k = 2*pi*F/c; % wavenumber vector % load backscattered field data for the target load Es_range %zero padding (4x); Enew=E; Enew(M*4)=0; % RANGE PROFILE GENERATION RP = M*fftshift(ifft(Enew)); h = plot(XX,abs(RP),'k','LineWidth',2); grid minor set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('range profile intensity'); xlabel('range, m'); axis tight

4.8 Matlab Codes

Matlab code 4.2 Matlab file “Figure4-8.m” %——————————————————————————————————————————————————————— % This code can be used to generate Figure 4-8 %——————————————————————————————————————————————————————— clear close all c = .3; % speed of light fc = 4; % center frequency phic = 0*pi/180; % center of azimuth look angles thc = 80*pi/180; % center of elevation look angles %________________PRE PROCESSING________________ BWy = 66; % x-range extend N = 128; % x-range sampling dy = BWy/N; % x- range resolution Y = -dy*N/2:dy:dy*(N/2-1); % x-range vector YY = -dy*N/2:dy/4:-dy*N/2+dy/4*(4*N-1); % range vector (4x upsampled) %Form angle vector kc = 2*pi*fc/c; % center wavenumber dphi = pi/(kc*BWy); % azimuth angle resolution PHI=phic+[-dphi*N/2:dphi:dphi*(N/2-1)]; % azimuth angle vector % load backscattered field data for the target load Es_xrange %zero padding (4x); Enew = E; Enew(N*4) = 0; % X-RANGE PROFILE GENERATION XRP = N*fftshift(ifft(Enew)); h = plot(YY,abs(XRP),'k','LineWidth',2); grid minor set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight','Bold'); ylabel('cross-range profile intensity'); xlabel('cross-range, m'); axis tight

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

Matlab code 4.3 Matlab file “Figure4.14.m” %——————————————————————————————————————————————————————— % This code can be used to generate Figure 4.14 %——————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % Escorner.mat clear all close all c = .3; % speed of light fc = 10; % center frequency phic = 180*pi/180; % center of azimuth look angles %________________PRE PROCESSING OF ISAR________________ BWx = 3; % range extend M = 16; % range sampling BWy = 3; % xrange extend N = 32; % xrange sampling dx = BWx/M; % range resolution dy = BWy/N; % xrange resolution % Form spatial vectors X = -dx*M/2:dx:dx*(M/2-1); Y = -dy*N/2:dy:dy*(N/2-1); %Find resolutions in freq and angle df = c/(2*BWx); % frequency resolution dk = 2*pi*df/c; % wavenumber resolution kc = 2*pi*fc/c; dphi = pi/(kc*BWy);% azimuth resolution %Form F and PHI vectors F=fc+[-df*M/2:df:df*(M/2-1)]; % frequency vector

4.8 Matlab Codes

PHI=phic+[-dphi*N/2:dphi:dphi*(N/2-1)];% azimuth vector K=2*pi*F/c; % wanenumber vector %________________GET THE DATA____________________________ load Escorner %________________POST PROCESSING OF ISAR________________ ISAR=fftshift(ifft2(Es)); h=figure; matplot(X,Y,abs(ISAR),50); line([-0.7071 0 ], [-0.7071 0 ],'LineWidth',2,'Color','w'); line([-0.7071 0 ], [0.7071 0 ],'LineWidth',2,'Color','w');hold off; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('range, m'); ylabel('x-range, m'); colormap (hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ];

Matlab code 4.4 Matlab file “Figure4-15.m” %——————————————————————————————————————————————————————— % This code can be used to generate Figure 4.15 %——————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % PLANORPHI45_Es.mat % planorphi45_2_xyout.mat clear close all

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c = .3; % speed of light fc = 6; % center frequency phic = 45*pi/180; % center of azimuth look angles %________________PRE PROCESSING OF ISAR________________ BWx = 13; % range extend M = 32; % range sampling BWy = 13; % xrange extend N = 64; % xrange sampling dx = BWx/M; % range resolution dy = BWy/N; % xrange resolution % Form spatial vectors X = -dx*M/2:dx:dx*(M/2-1); Y = -dy*N/2:dy:dy*(N/2-1); %Find resoltions in freq and angle df = c/(2*BWx); % frequency resolution dk = 2*pi*df/c; % wavenumber resolution kc = 2*pi*fc/c; dphi = pi/(kc*BWy);% azimuth resolution %Form F and PHI vectors F = fc+[-df*M/2:df:df*(M/2-1)]; % frequency vector PHI = phic+[-dphi*N/2:dphi:dphi*(N/2-1)];% azimuth vector K=2*pi*F/c; % wanenumber vector %________________GET THE DATA____________________________ load PLANORPHI45_Es.mat; % load E-scattered load planorphi45_2_xyout.mat; % load target outline %________________ POST PROCESSING OF ISAR________________ %windowing;

4.8 Matlab Codes

223

w=hanning(M)*hanning(N).'; Ess=Es.*w; %zero padding; Enew=Ess; Enew(M*4,N*4)=0; % ISAR image formatiom ISARnew=fftshift(ifft2(Enew)); h=figure; matplot(X,Y,abs(ISARnew),22); % form the image hold; colormap (hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; plot(xyout_yout,xyout_xout,'w.','MarkerSize',1); hold; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('range, m'); ylabel('x-range, m');

Matlab code 4.5 Matlab file “Figure4-18.m” %————————————————————————————————————————————————————————— % This code can be used to generate Figure 4.18 %————————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % M_fighter.mat clear;

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

close all % Load the data load M_fighter M = length(Freq); P = length(Theta); N = length(Phi); c = 0.3; % speed of light % organize axes df = Freq(2)-Freq(1); BWf = M*df; dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); dphi = pi*(aspect(2)-aspect(1))/180; % in radians BWphi = N*dphi; fc = Freq(round(M/2+1)); % center freq dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1); h=hanning(M)*hanning(N).'; % 2D window % ———————————— Form ISAR vectors ——————————————— E_VV = 2*E_VV.*h; % 2 for windowing E_VV(4*M,4*N)=0; ISAR_VV = 16*abs(fftshift(ifft2(E_VV.'))); ISAR_VVdB = 20*log10(ISAR_VV); E_HH = 2*E_HH.*h; % 2 for windowing E_HH(4*M,4*N)=0; ISAR_HH = 16*abs(fftshift(ifft2(E_HH.'))); ISAR_HHdB = 20*log10(ISAR_HH);

4.8 Matlab Codes

CMAP='hot'; rd = 35; % dynamic range of display Th1 = mean(Theta); Ph1 = mean(Phi); % ———————————— Draw ISAR images ——————————————— figure imagesc(X,Y,ISAR_VVdB); grid on; axis equal; axis tight; colormap(hot); caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; set(cc,'FontName', 'Arial', 'FontSize',10,'FontWeight','Bold'); figure imagesc(X,Y,ISAR_HHdB); grid on; axis equal; axis tight; colormap(hot); caxis([max(max(ISAR_HHdB))-rd,(max(max(ISAR_HHdB)))]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (HH Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; set(cc,'FontName', 'Arial', 'FontSize',10,'FontWeight','Bold');

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

Matlab code 4.6 Matlab file “Figure 4-20.m” %——————————————————————————————————————————————————————— % This code can be used to generate Figure 4.20 %——————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % L_tank.mat clear; close all % Load the data load L_tank M = length(Freq); P = length(Theta); N = length(Phi); c = 0.3; % speed of light % organize axes df = Freq(2)-Freq(1); BWf = M*df; dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); dphi = pi*(aspect(2)-aspect(1))/180; % in radians BWphi = N*dphi; fc = Freq(round(M/2+1)); % center freq dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1); h=hanning(M)*hanning(N).'; % 2D window % ———————————— Form ISAR vectors ——————————————— E_VV = 2*E_VV.*h; % 2 for windowing E_VV(4*M,4*N)=0; ISAR_VV = 16*abs(fftshift(ifft2(E_VV.')));

4.8 Matlab Codes

ISAR_VVdB = 20*log10(ISAR_VV); E_HH = 2*E_HH.*h; % 2 for windowing E_HH(4*M,4*N)=0; ISAR_HH = 16*abs(fftshift(ifft2(E_HH.'))); ISAR_HHdB = 20*log10(ISAR_HH); CMAP='hot'; rd=40; Th1 = mean(Theta); Ph1 = mean(Phi); % ———————————— Draw ISAR images ——————————————— figure imagesc(X,Y,ISAR_VVdB); grid on; axis equal; axis tight; axis xy; axis equal; colormap(hot); %grid minor caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; set(cc,'FontName', 'Arial', 'FontSize',10,'FontWeight','Bold'); hold on;

figure imagesc(X,Y,ISAR_HHdB); grid on; axis equal; axis tight; axis xy; axis equal; colormap(hot); %grid minor caxis([max(max(ISAR_HHdB))-rd,(max(max(ISAR_HHdB)))]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold');

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image HH Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; set(cc,'FontName', 'Arial', 'FontSize',10,'FontWeight','Bold'); hold on;

Matlab code 4.7 Matlab file “Figure4-21and4-22.m” %——————————————————————————————————————————————————————— % This code can be used to generate Figure 4.21 and 4.22 %——————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % fighterSC.mat clear close all c = .3; % speed of light fc = 8; % center frequency phic = 0*pi/180; % center of azimuth look angles %________________PRE PROCESSING OF ISAR________________ BWx = 18; % range extend M = 64; % range sampling BWy = 16; % xrange extend N = 64; % xrange sampling dx = BWx/M; % range resolution

4.8 Matlab Codes

dy = BWy/N; % xrange resolution % Form spatial vectors X = -dx*M/2:dx:dx*(M/2-1); Y = -dy*N/2:dy:dy*(N/2-1); %Find resoltions in freq and angle df = c/(2*BWx); % frequency resolution dk = 2*pi*df/c; % wavenumber resolution kc = 2*pi*fc/c; dphi = pi/(kc*BWy);% azimuth resolution %Form F and PHI vectors F = fc+[-df*M/2:df:df*(M/2-1)]; % frequency vector PHI = phic+[-dphi*N/2:dphi:dphi*(N/2-1)];% azimuth vector K = 2*pi*F/c; % wavenumber vector %________________ FORM RAW BACKSCATTERED DATA___________ %load scattering centers load fighterSC l = length(xx); %—Figure 4.21————————————————————————————————————————————— figure plot(xx,yy,'k.','MarkerSize',10) grid minor set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); colormap(gray); xlabel('\itx, m'); ylabel('\ity, m'); axis([-8 8 -8 8]) %form backscattered E-field from scattering centers Es = zeros(M,N);

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

for m=1:l Es = Es+1.0*exp(-1i*2*K'*(cos(PHI)*xx(m)+sin(PHI)*yy (m))); end %_____ POST PROCESSING OF ISAR (small-BW small angles) ________ ISAR = fftshift(ifft2(Es.')); %—Figure 4.22————————————————————————————————————————————— h = figure; matplot2(X,Y,ISAR,25); colormap(hot); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; xlabel('\itrange, m'); ylabel('\itcross-range, m'); axis([-8 8 -8 8])

Matlab code 4.8 Matlab file “Figure4-23and24.m” %——————————————————————————————————————————————————————— % This code can be used to generate Figure 4.23 and 4.24 %——————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % fighterSC.mat clear all close all

4.8 Matlab Codes

c = .3; % speed of light fc = 8; % center frequency fMin = 6; % lowest frequency fMax = 10; % highest frequency phic = 0*pi/180; % center of azimuth look angles phiMin = -30*pi/180; % lowest angle phiMax = 30*pi/180; % highest angle %—————————————————————————————————————————————— % WIDE-BW AND LARGE ANGLES ISAR %—————————————————————————————————————————————— % A- INTEGRATION %—————————————————————————————————————————————— nSampling = 300; % sampling number for integration % Define Arrays f = fMin:(fMax-fMin)/(nSampling-1):fMax; k = 2*pi*f/.3; kMax = max(k); kMin = min(k); kc = (max(k)+min(k))/2; phi = phiMin:(phiMax-phiMin)/(nSampling-1):phiMax; % resolutions dx = pi/(max(k)-min(k)); % range resolution dy = pi/kc/(max(phi*pi/180)-min(phi*pi/180)); % xrange resolution % Form spatial vectors X = -nSampling*dx/2:dx:nSampling*dx/2; Y = -nSampling*dy/2:dy:nSampling*dy/2; %________________ FORM RAW BACKSCATTERED DATA________________ %load scattering centers

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

load fighterSC l = length(xx); %form backscattered E-field from scattering centers clear Es; Es = zeros((nSampling),(nSampling)); for m=1:l; Es = Es+1.0*exp(-j*2*k.'*cos(phi)*xx(m)).*exp (-j*2*k.'*sin(phi)*yy(m)); end axisX = min(xx)-1:0.05:max(xx)+1; axisY = min(yy)-1:0.05:max(yy)+1; % take a look at what happens when DFT is used %—Figure 4.23——————————————————————————————————————————————————— ISAR1 = fftshift(ifft2(Es.')); matplot2(axisX,axisY,ISAR1,22); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); colormap(hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; xlabel('\itrange, m'); ylabel('\itcross-range, m'); % INTEGRATION STARTS HERE % Building Simpson Nodes; Sampling Rate is nSampling % Weights over k h = (kMax-kMin)/(nSampling-1); k1 = (kMin:h:kMax).'; wk1 = ones(1,nSampling); wk1(2:2:nSampling-1) = 4;

4.8 Matlab Codes

wk1(3:2:nSampling-2) = 2; wk1 = wk1*h/3; % Weights over phi h = (phiMax-phiMin)/(nSampling-1); phi1 = (phiMin:h:phiMax).'; wphi1 = ones(1,nSampling); wphi1(2:2:nSampling-1) = 4; wphi1(3:2:nSampling-2) = 2; wphi1 = wphi1*h/3; % Combine for two dimensional integration [phi1,k1] = meshgrid(phi1,k1); phi1 = phi1(:); k1 = k1(:); w = wk1.'*wphi1; w = w(:).'; newEs = Es(:).'; newW = w.*newEs; % Integrate b = 2j; ISAR2 = zeros(round((max(xx)-min(xx)+2)/0.05+1),round ((max(yy)-min(yy)+2)/0.05+1)); k1 = k1.*b; cosPhi = cos(phi1); sinPhi = sin(phi1); tic; x1 = 0; for X1 = axisX x1 = x1+1; y1 = 0; for Y1 = axisY y1 = y1+1;

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

ISAR2(x1,y1) = newW*(exp(k1.*(cosPhi.*X1+sinPhi. *Y1))); end end time1 = toc; %—Figure 4.24——————————————————————————————————————————————————— matplot2(axisX,axisY,ISAR2.',22); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); colormap(hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; xlabel('\itrange, m'); ylabel('\itcross-range, m');

Matlab code 4.9 Matlab file “Figure4-26thru4-28.m” %——————————————————————————————————————————————————————— % This code can be used to generate Figure 4.26 thru 4.28 %——————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % fighterSC.mat clear close all c = .3; % speed of light

4.8 Matlab Codes

fc = 8; % center frequency fMin = 6; % lowest frequency fMax = 10; % highest frequency phic = 0*pi/180; % center of azimuth look angles phiMin = -30*pi/180; % lowest angle phiMax = 30*pi/180; % highest angle %—————————————————————————————————————————————— % WIDE BW AND WIDE ANGLE ISAR %—————————————————————————————————————————————— % B- POLAR REFORMATTING %—————————————————————————————————————————————— nSampling = 1500; % sampling number for integration % Define Bandwidth f = fMin:(fMax-fMin)/(nSampling):fMax; k = 2*pi*f/.3; kMax = max(k); kMin = min(k); % Define Angle phi = phiMin:(phiMax-phiMin)/(nSampling):phiMax; kc = (max(k)+min(k))/2; kx=k.'*cos(phi); ky=k.'*sin(phi); kxMax kxMin kyMax kyMin

= = = =

max(max(kx)); min(min(kx)); max(max(ky)); min(min(ky));

MM=4; % up sampling ratio clear kx ky;

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

kxSteps = (kxMax-kxMin)/(MM*(nSampling+1)-1); kySteps = (kyMax-kyMin)/(MM*(nSampling+1)-1); kx = kxMin:kxSteps:kxMax; Nx=length(kx); ky = kyMin:kySteps:kyMax; Ny=length(ky); kx(MM*(nSampling+1)+1) = 0; ky(MM*(nSampling+1)+1) = 0; %________________ FORM RAW BACKSCATTERED DATA________________ %load scattering centers load fighterSC l = length(xx); %form backscattered E-field from scattering centers Es = zeros((nSampling+1),(nSampling+1)); for n=1:length(xx); Es = Es+exp(-j*2*k.'*cos(phi)*xx(n)).*exp (-j*2*k.'*sin(phi)*yy(n)); end %—Figure 4.26——————————————————————————————————————————— figure matplot2(f,phi*180/pi,Es,40); colormap(hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itfrequency, GHz'); ylabel('\itangle, degree'); newEs = zeros(MM*(nSampling+1)+1,MM*(nSampling+1)+1); t = 0; v = 0;

4.8 Matlab Codes

for tmpk = k t = t+1; v = 0; for tmpPhi = phi v = v+1; tmpkx = tmpk*cos(tmpPhi); tmpky = tmpk*sin(tmpPhi); indexX = floor((tmpkx-kxMin)/kxSteps)+1; indexY = floor((tmpky-kyMin)/kySteps)+1; r1 = sqrt(abs(kx(indexX)-tmpkx)^2+abs(ky(indexY)tmpky)^2); r2 = sqrt(abs(kx(indexX+1)-tmpkx)^2+abs(ky (indexY)-tmpky)^2); r3 = sqrt(abs(kx(indexX)-tmpkx)^2+abs(ky(indexY +1)-tmpky)^2); r4 = sqrt(abs(kx(indexX+1)-tmpkx)^2+abs(ky(indexY +1)-tmpky)^2); R = 1/r1+1/r2+1/r3+1/r4; A1 A2 A3 A4

= = = =

Es(t,v)/(r1*R); Es(t,v)/(r2*R); Es(t,v)/(r3*R); Es(t,v)/(r4*R); newEs(indexY,indexX) = newEs(indexY,indexX)

+A1; newEs(indexY,indexX+1) = newEs(indexY,indexX +1)+A2; newEs(indexY+1,indexX) = newEs(indexY +1,indexX)+A3; newEs(indexY+1,indexX+1) = newEs(indexY +1,indexX+1)+A4; end end

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% down sample newEs by MM times newEs=newEs(1:MM: size(newEs),1:MM: size(newEs)); %—Figure 4.27——————————————————————————————————————————— % reformatted data figure; Kx = kx(1:Nx-1); Ky = ky(1:Ny-1); matplot2(Kx,Ky,newEs,40); colormap(hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itk_x, rad/m'); ylabel('\itk_y, rad/m'); % Find Corresponding ISAR window in Range and X-Range kxMax = max(max(kx)); kxMin = min(min(kx)); kyMax = max(max(ky)); kyMin = min(min(ky)); BWKx = kxMax-kxMin; BWKy = kyMax-kyMin; dx = pi/BWKx; dy = pi/BWKy; X = dx*(-nSampling/2:nSampling/2); Y = dy*(-nSampling/2:nSampling/2); %—Figure 4.28——————————————————————————————————————————— % Plot the resultant ISAR image

4.8 Matlab Codes

figure; tt = nSampling/4:3*nSampling/4; ISAR3 = fftshift(ifft2(newEs)); matplot2(X,Y,ISAR3(:,tt),25); axis([-8 8 -6 6]) colormap(hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itcross-range, m');

Matlab code 4.10 Matlab file “Figure4-32and4-33.m” %——————————————————————————————————————————————————————— % This code is for producing 3D ISAR %——————————————————————————————————————————————————————— %——————————————————————————————————————————————————————— % This code can be used to generate Figure 4.32 and 4.33 %——————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % B_plane.mat clear; close all; clc c=0.3; load

B_plane.mat

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

M P N %

= length(Freq); = length(Theta); = length(Phi); organize axes df = Freq(2)-Freq(1); BWf = M*df; dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); fc = Freq(round(M/2+1)); % center freq dphi = pi*(Phi(2)-Phi(1))/180; % in radians BWphi = N*dphi; dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1); dTheta = pi*(Theta(2)-Theta(1))/180; % in radians BWTheta = P*dTheta; dz = c/2/BWTheta/fc; Z = -P/2*dz:dz:dz*(P/2-1);

%-2D window functions in 3 principal planes————— hFreqPhi = hanning(M)*hanning(N).'; hFreqTheta = hanning(M)*hanning(P).'; hThetaPhi = hanning(P)*hanning(N).'; hFreq = hanning(M); for mm = 1: M win3(mm,:,:) = hFreq(mm)*hThetaPhi; end % -ISAR ———————————————————————————————————————— for mm = 1: M for nn = 1: P for pp =1 : N index=P*N*(mm-1)+N*(nn-1)+pp; EEHH(mm,nn,pp) = Es_HH(index)+1i*Es_HH(index); end;end;end %-2D ISAR in 3 principal

4.8 Matlab Codes

planes———————————————————————————————— EEFreqTheta = sum(EEHH, 3); EEFreqPhi = squeeze(sum(EEHH, 2)); EEThetaPhi = squeeze(sum(EEHH, 1)); EEFreqThetaw = 4*EEFreqTheta.*hFreqTheta;% 4 for windowing EEFreqPhiw = 4*EEFreqPhi.*hFreqPhi; EEThetaPhiw = 4*EEThetaPhi.*hThetaPhi; EEFreqThetaw(4*M,4*P)=0; EEFreqPhiw(4*M,4*N)=0; EEThetaPhiw(4*P,4*N)=0; ISAR_xy = 20*log10(16*abs(fftshift(ifft2 (EEFreqPhiw.')))); ISAR_xz = 20*log10(16*abs(fftshift(ifft2 (EEFreqThetaw.')))); ISAR_yz = 20*log10(16*abs(fftshift(ifft2 (EEThetaPhiw.')))); %—Figure 4.33a————— imagesc(X,-Y,ISAR_xy); grid on; axis equal; axis tight; axis xy; axis equal; colormap(hot); %grid minor caxis([max(max(ISAR_xy))-40,(max(max(ISAR_xy)))]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange (x), m'); ylabel('\itcross-range (y), m'); colormap (hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; title('\itISAR Image (HH Polarization)'); %—Figure 4.33c—————

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4 Inverse Synthetic Aperture Radar Imaging and Its Basic Concepts

imagesc(Z,X,ISAR_xz.'); grid on; axis equal; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); axis tight; axis xy; axis equal; colormap(hot); %grid minor caxis([max(max(ISAR_xz))-30,(max(max(ISAR_xz)))]); xlabel('\itcross-range (z), m'); ylabel('\itrange (x), m'); colormap (hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; title('\itISAR Image (HH Polarization)'); %—Figure 4.33b————— imagesc(Y,Z,ISAR_yz.'); grid on; axis equal; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); axis tight; axis xy; axis equal; colormap(hot); %grid minor caxis([max(max(ISAR_yz))-20,(max(max(ISAR_yz)))]); xlabel('\itcross-range (y), m'); ylabel('\itcross-range (z), m'); colormap (hot); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; title('ISAR Image (HH Polarization)');

%-3D ISAR for different z-values ——————————————————————— ii=0; for nn = 3:4:P ii=ii+1; EE_Freq_Phi = squeeze(EEHH(:,nn,:)); EE_Freq_Phiw = 4*EE_Freq_Phi.*hFreqPhi;

References

EE_Freq_Phiw(4*M,4*N) = 0; ISAR_xy2D = 20*log10(16*abs(fftshift(ifft2 (EE_Freq_Phiw.')))); %—Figure 4.32 a - h ————— h=imagesc(X,Y,ISAR_xy2D); grid on; axis xy; colormap(hot); %grid minor caxis([max(max(ISAR_xy2D))-40,(max(max(ISAR_xy2D)))]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('Range [m]'); ylabel('Cross - range [m]'); zp = num2str(round(Z(nn)*100)/100); zpp = ['ISAR(x,y) @ z = ' zp 'm']; title(zpp); c = colorbar; set(c,'FontName', 'Arial', 'FontSize',10,'FontWeight','Bold'); ylabel(c,'dBsm'); axis([-12 12 -12 12]) pause end

References Bhalla, R. and Ling, H. (1993). ISAR image formation using bistatic data computed from the shooting and bouncing ray technique. Journal of Electromagnetic Waves and Applications 7 (9): 1271–1287. https://doi.org/10.1163/156939393X00255. Chu, T.H., Lin, D.B., and Kiang, Y.W. (1989). Microwave diversity imaging of perfectly conducting objects in the near field region. Digest on Antennas and Propagation Society International Symposium, San Jose, CA, USA. 1, pp. 82–85. https://doi.org/ 10.1109/APS.1989.134617. Fortuny, F. (1998). An efficient 3-D near field ISAR algorithm. IEEE Transactions on Aerospace and Electronic Systems 34 (4): 1261–1270. https://doi.org/10.1109/ 7.722713.

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Kırık, Ö. and Özdemir, C. (2019). An accurate and effective implementation of physical theory of diffraction to the shooting and bouncing ray method via PREDICS tool. Sigma Journal of Engineering and Natural Sciences 37 (4): 1153–1166. Knaell, K.K. and Cardillo, G.P. (1995). Radar tomography for the generation of threedimensional images. IEE Proceedings – Radar, Sonar and Navigation 142 (2): 54–60. https://doi.org/10.1049/ip-rsn:19951791. Li, J., Wang, Y., Bhalla, R. et al. (1999). Comparison of high-resolution ISAR imageries from measurement data and synthetic signatures. SPIE’s International Symposium on Optical Science, Engineering, and Instrumentation 3810: 170–179. https://doi.org/ 10.1117/12.364077. Ling, H., Chou, R.C., and Lee, S.W. (1989). Shooting and bouncing rays: calculation the RCS of an arbitrary shaped cavity. IEEE Transactions on Antennas and Propagation 37 (2): 194–205. https://doi.org/10.1109/8.18706. Lord, R.T., Nel, W.A.J. and Abdul Gaffar, M.Y. (2006). Investigation of 3-D RCS image formation of ships using ISAR. EUSAR 2006 – 6th European Conference on Synthetic Aperture Radar, Dresden, Germany (16–18 May 2006). Luminati, J.E. (2005). Wide-angle multistatic synthetic aperture radar: focused image formation and aliasing artifact mitigation. Ph.D. thesis. Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, USA. Mayhan, J.T., Burrows, M.L., Cuomo, K.M., and Cuomo, J.E. (2001). High resolution 3D “snapshot” ISAR imaging and feature extraction. IEEE Transactions on Aerospace and Electronic Systems 37 (2): 630–642. https://doi.org/10.1109/7.937474. McFadden, F.E. (2002). Three dimensional reconstruction from ISAR sequences. Proceedings of SPIE 4744, Radar Sensor Technology and Data Visualization 4744: 58– 67. https://doi.org/10.1117/12.488289. Özdemir, C., Bhalla, R., Trintinalia, L.C., and Ling, H. (1998). ASAR – antenna synthetic aperture radar imaging. IEEE Transactions on Antennas and Propagation 46 (12): 1845–1852. https://doi.org/10.1109/8.743822. Ozdemir, C., Kirik, O., and Yilmaz, B. (2009). Sub-aperture method for the widebandwidth wide-angle inverse synthetic aperture radar imaging. 2009 International Conference on Electrical and Electronics Engineering – ELECO 2009, Bursa, Turkey (5–8 November 2009), pp. 288–292. Özdemir, C., Yılmaz, B., and Kırık, Ö. (2014). pRediCS: a new GO-PO based ray launching simulator for the calculation of electromagnetic scattering and RCS from electrically large and complex structures. Turkish Journal of Electrical Engineering & Computer Sciences 22: 1255–1269. https://doi.org/10.3906/elk-1210-93. Saynak, U., Çolak, A., Bölükbas, D., Tayyar, İ.H. and Özdemir, C. (2010). Utilizing ISAR imagery to analyze the diffraction effects from leading and trailing edges of a target. 10th Mediterranean Microwave Symposium, Guzelyurt, Cyprus (25–27 August 2010), pp. 393–396. https://doi.org/10.1109/MMW.2010.56050e95.

References

Wang, G., Xia, X.G., and Chen, V.C. (2001). Three-dimensional ISAR imaging of maneuvering targets using three receivers. IEEE Transactions on Image Processing 10 (3): 436–447. https://doi.org/10.1109/83.908519. Wehner, D.R. (1997). High Resolution Radar. Norwood, MA: Artech House. Wu, W.-R (1993). Target racking with glint noise, IEEE Transactions on Aerospace and Electronic Systems, 29 (1): 174–185. https://doi.org/10.1109/7.249123. Xu, X., Luo, H., and Huang, P. (1999). 3-D interferometric ISAR images for scattering diagnosis of complex radar targets. Proceedings of the 1999 IEEE Radar Conference. Radar into the Next Millennium, Waltham, MA, USA, pp. 237–241. doi: https://doi. org/10.1109/NRC.1999.767328. Xu, X. and Narayanan, R.M. (2001). Three-dimensional interferometric ISAR imaging for target scattering diagnosis and modeling. IEEE Transactions on Image Processing 10 (7): 1094–1102. https://doi.org/10.1109/83.931103. Zhang, Q., Yeo, T.S., Du, G., and Zhang, S. (2004). Estimation of three-dimensional motion parameters in interferometric ISAR imaging. IEEE Transactions on Geoscience and Remote Sensing 42 (2): 292–300. https://doi.org/10.1109/ TGRS.2003.815669.

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5 Imaging Issues in Inverse Synthetic Aperture Radar While applying the inverse synthetic aperture radar (ISAR) imaging algorithms, there are some design parameters, such as reformatting the raw ISAR data and applying window functions, that should be carefully handled such that the resultant ISAR image quality can be improved. In this chapter, we will describe and discuss how to tweak these parameters to obtain a good quality ISAR image.

5.1

Fourier-Related Issues

The concept of Fourier transform (FT) is very important in synthetic aperture radar (SAR)/ISAR imaging as has been demonstrated in many different places in this book. The classic way of forming the ISAR image is accomplished by transforming the scattered field data from the frequency-angle domain to the image domain by applying the FT operation. In SAR/ISAR processing, the collected data sequence is usually digitized such that the digital signal processing algorithms are used to form the final image (Sullivan 2000; Knott et al. 2004). The process of digitizing the signal has already been studied in Chapter 1, Section 1.6. After the data are converted to the digital form, the Fourier processing can be readily applied with the help of discrete Fourier transform (DFT) operations. Next, we will briefly explain the very basic but also very critical issues regarding getting a well-implemented and good quality ISAR image.

5.1.1

DFT Revisited

The sampling and digitizing processes of an analog signal are demonstrated in Figure 5.1. Let x(t) be the continuous time-domain signal as shown in

Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms: With Advanced SAR/ISAR Imaging Concepts, Algorithms, and MATLAB Codes, Second Edition. Caner Özdemir. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

5.1 Fourier-Related Issues

(a) x(t)

to

t

to + T

(b) s(t)

t 0

T

(c) s[n]

0 Ts 0

Ts 1

3 Ts 4 Ts . . . . . . 3 4 ......

2 Ts 2

(N – 1)Ts

t n

N–1

(d) Previous period

Next period

s[n]

... ...

0

1

2

3

4 ......

N–1

N

N+1N + 2 . . .

T

Figure 5.1 Digitizing process: (a) original continuous time signal, (b) observed signal with a duration of T, (c) sampled signal with N discrete points, (d) corresponding discrete-time signal s[n] is periodic.

Figure 5.1a, and let s(t) be the observed (or recorded) time-domain signal with a duration of T as depicted in Figure 5.1b. Therefore, s(t) may represent a portion of the original signal x(t). After digitizing s(t) with a sampling interval of Ts = 1/fs, a discrete set of s[n] is obtained, as in Figure 5.1c. Here, fs is the sampling frequency and also represents the fundamental frequency of a periodic version of the

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5 Imaging Issues in Inverse Synthetic Aperture Radar

observed signal s(t). In fact, as demonstrated in Figure 5.1d, the resultant discretetime signal s[n] is periodic in time. As its DFT will be shown later, S[k] is periodic in frequency as well. It can be easily observed from Figure 5.1d that the period is equal to N times the sampling interval as T = N Ts

51

It is important to note that only the s[n] values for n = 0, 1, 2, …, N − 1 is included in the discrete-time base signal. The Nth datum, s[N], belongs to the next period and is the same as s[0]. This observation is important when selecting the correct duration (or bandwidth) of the signal in time (or frequency). This phenomenon is clarified with the following examples. Example 5.1 Suppose that a discrete time-domain signal is collected for a time interval of 1 ms with 128 samples. Here, the first sample is taken at t = 0 s, and the last sample is taken at t = 1 ms for a total of 128 points. Therefore, the sampling interval is equal to 1 ms 127 = 7 874 μs

Ts =

52

Therefore, the total time interval is equal to T = N Ts = 128 7 874 μs = 1 0078 ms,

53

which is greater than the observation time of 1 ms. This is because of the fact that DFT is periodic, and the next period starts one sampling interval later than the last sampling point. This kind of calculation becomes important when dealing with the scaling in the other domain, especially when finding the resolutions and the bandwidths. The frequency resolution for this example is, therefore Δf =

1 T

1 1 0078 ms = 992 1875 Hz =

54

Then, the frequency bandwidth becomes equal to B = N Δf = 128 992 1875 = 127 kHz

55

5.1 Fourier-Related Issues

To check the accuracy of the calculations, let us find the time resolution as Δt =

1 B

1 127 kHz = 7 874 μs,

=

56

which is exactly the same as a sampling interval of Ts. Example 5.2 Now, let us consider a case for the ISAR bandwidth and resolution calculation. We suppose that a stepped frequency radar system collects the frequency-domain backscattered field data from a scene. The radar records the discrete frequency signal from 8 to 10 GHz for a total of 201 uniformly sampled discrete points. Therefore, the frequency resolution is 2 GHz 200 = 10 MHz

Δf =

57

Therefore, the total frequency bandwidth becomes B = N Δf = 201 10 MHz = 2 01 GHz

58

This represents a range resolution of c 2 B 03 = 2 2 01 = 7 4627 cm

Δr =

59

The total range can be calculated by multiplying the range resolution by the number of samples as R = N Δr = 201 7 4627 cm

5 10

= 15 m To check the accuracy of the calculations, let us determine the frequency resolution via the following Fourier equality:

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5 Imaging Issues in Inverse Synthetic Aperture Radar

c 2 R 0 3 109 m s = 2 15 = 10 MHz,

Δf =

5 11

which is exactly the same as in Eq. 5.7.

5.1.2

Positive and Negative Frequencies in DFT

The definition and the formulation of DFT have been already given in Chapter 1. The forward and inverse DFT pair is defined as given below: N −1

Sk =

s n exp − j2π

n=0 N −1

Sn =

Sk

exp j2π

n=0

k n N

n k N

k = 0, 1, 2, …, N − 1,

5 12a

n = 0, 1, 2, …, N − 1,

5 12b

where s[n] and S[k] present the time-domain and the frequency-domain signals, respectively. While each n step in Eq. 5.12b represents a Ts increment in the time axis, each k step in Eq. 5.12a represents an fs = 1/T increment in the frequency axis. It is important to note that DFT representation includes both the positive and the negative frequencies or the harmonics. Table 5.1 explains the indexing of positive and negative frequencies. While the first N/2 index entries are for the positive Fourier harmonics, the next N/2 index entries stand for the negative Fourier harmonics. This phenomenon can be obtained from the definition of DFT as follows: Let us investigate the index terms in the second N/2 index entries by putting (N – k) instead of k in Eq. 5.12a as S N −k =

N −1

s n exp − j2π

n=0 N −1

=

s n exp − j2πn

N −k n N exp − j2π

n=0

=

N −1

s n exp − j2π

n=0

−k n N

5 13

−k n N

≜ S −k Therefore, S[N − k] is identical to S[−k]. This phenomenon is demonstrated in Figure 5.2. Since negative frequencies come after the positives ones, they have to be interchanged to correctly arrange the frequency axis.

5.1 Fourier-Related Issues

Table 5.1 Index allocation of positive and negative frequencies in DFT representation. Indexing

0≤k≤

Corresponding harmonic frequencies

N −1 2

Positive frequencies: k fs

N ≤ k ≤ N −1 2

Negative frequencies: (k – N)fs

S[k]

Positive frequencies

Negative frequencies

k 0 1 2 3 .....

N/2

.....

N–1 N

S[k]

–(N/2–1) –(N/2–1)fs

. . . . . . –2 –1 0 1 2 . . . . . . . . . . . . –2f –f 0 f 2f ...... s s s s

(N/2) k (N/2)fs f

Figure 5.2 Demonstration of positive and negative frequencies in DFT: first half and the second half of the DFT sequence should be swapped to organize the frequency axis.

If the data are collected in the frequency domain, the signal in time domain, after inverse DFT, includes both the positive and the negative time indices, analogously. This can be easily observed from the definition of inverse DFT by testing the

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5 Imaging Issues in Inverse Synthetic Aperture Radar

following. Let us investigate the index terms in the second N/2 index entries by putting (N − n) instead of n in Eq. 5.12b as S N −n =

N −1

Sk

exp j2π

Sk

exp j2πk

Sk

exp j2π

k=0

N −n k N

N −1

=

exp j2π

k=0 N −1

= k=0

−k n N

5 14

−n k N

≜ S −n Therefore, Eq. 5.14 clearly shows that indices in the second N/2 entries, in fact, correspond to negative time indices. In ISAR imaging, the raw data are usually collected in the frequency-angle axis and are transformed to range cross-range (or range-Doppler) domain after Fourier transforming. To correctly position the ISAR image, the data indices corresponding to positive and negative range cross-range quantities should be swapped. To demonstrate the use of this property in ISAR imagery, let us reconsider the plane example in Chapter 4, Figure 4.22. As listed in the Matlab script of that example, the ISAR image is swapped by the use of “fftshift” command to correctly display the ISAR image. This is because of the fact that DFT and inverse discrete Fourier transform (IDFT) operations are periodic, which means that the image repeats itself in every image window size of (Δx Nx) – by −(Δy Ny), as shown in Figure 5.3a. After DFT operation, the discrete data entry for the first indices in range and cross-range domains corresponds to “0 m” as shown in the Figure 5.3b. After swapping the negative and positive range and cross-range distances as demonstrated in Figure 5.3, the final image is correctly positioned in the image domain as shown in Figure 5.3c.

5.2

Image Aliasing

ISAR image aliasing occurs when the backscattered field data are not collected with the minimum required frequency and/or angle sampling rates. This is, of course, the nature of the DFT and shows up when applying the ISAR signal processing. This phenomenon will be explained with an example. Let us consider the point scatterers whose locations are already plotted in Chapter 4, Figure 4.21. To be able to get the ISAR image of these scatterers, the ISAR image size is selected as [18 m × 16 m]. This selection of ISAR image size

5.2 Image Aliasing

(a)

y

Δy∙Ny

Image window after DFT

Δy∙(Ny–1)/2

0

Δx∙Nx

Corrected image window after swapping

–Δy∙Ny/2 –Δx∙Nx/2

Δx∙(Nx–1)/2

(b)

(c)

–8

0

0

–8

–6

–6 –5

–2

–10

0 –15

2 4

–20

6

–5 Cross–range (m)

–4 Cross–range, m

x

–4 –2

–10

0 –15

2 4

–20

6 –5

0 Range, m

5

–25 –5

0 Range, m

5

–25

Figure 5.3 (a) The image window is periodic in range and cross-range axes, (b) ISAR image after DFT with first indices in both domains corresponds to “0 m,” (c) ISAR image with correct indexing after 2D swapping operations.

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5 Imaging Issues in Inverse Synthetic Aperture Radar

imposes the following sampling ratios (or resolutions) in frequency and aspect domains: c 2 X max 3 108 2 = 18 = 8 333 MHz

5 15a

λc 2 Y max 0 0375 = 16 = 0 001 171 875 rad 0 0671

5 15b

Δf =

Δϕ =

If the data were collected with the above sampling ratios, the correct ISAR image as seen in Chapter 4, Figure 4.22 would be obtained. However, if the data are undersampled, then we will have an aliased ISAR image. To demonstrate this property, the data are sampled in frequency and aspect with twice the ratio that are calculated in Equations 5.15a and 5.15b, selecting Δf = 16.666 MHz and Δϕ = 0.1342 . This selection of resolutions in frequency and angle will correspond to a smaller ISAR window. In Figure 5.4, the corresponding ISAR image 0

–8 –6

–5 –4 Cross–range, m

254

–2

–10

0 –15

2 4

–20

6 –5

0

5

Range, m

Figure 5.4

An example of aliased ISAR image.

–25

5.3 Polar Reformatting Revisited

of 9 m × 8 m in size is plotted. As is obvious from the figure, the image suffers from the aliasing phenomenon due to undersampling of the collected data in the Fourier domain. This is due to the fact that the new Δf and Δϕ values correspond to an ISAR image size of 8 m × 8 m. Therefore, any scattering center beyond this image frame leaks back to the image from the opposite side of the frame according to the Fourier theory.

5.3

Polar Reformatting Revisited

Polar reformatting is the common mapping technique that is widely used in SAR/ ISAR processing (Mersereau and Oppenheim 1974; Ausherman et al. 1984). The problem of polar reformatting was already studied in Chapter 4, Section 4.6.2. What is mainly done in polar reformatting is to reformat the collected backscattered field data that correspond to polar data in spatial frequency onto a rectangular grid so that DFT can be applied for fast formation of the ISAR image, as depicted in Figure 5.5. In this section, we will look into the polar reformatting process more closely by investigating some common interpolation schemes.

5.3.1 Nearest Neighbor Interpolation Since the collected frequency-aspect data do not lie on a rectangular grid on the kx– ky plane, most of these data points will not coincide with the grid points in the Fourier domain as seen in Figure 5.5. One of the most popular interpolation techniques used in polar reformatting is the nearest neighbor scheme (Mersereau and Oppenheim 1974; Özdemir et al. 1998). It is a very general technique and can be applied to any data that are not required to be in the form of a uniform grid. For SAR/ISAR applications, however, it is commonly used to interpolate the polar formatted data to a uniformly sampled rectangular grid. In the first-order interpolation scheme, the closest four neighboring points on the uniform grid are linearly updated depending on their distances from the original data point at kxi, kyi as demonstrated in Figure 5.5. When three-dimensional (3D) data are collected, the algorithms are extended such that each point in kx–ky–kz space is interpolated to the closest eight neighboring points on the uniform grid as demonstrated in Figure 5.6. For the two-dimensional (2D) case, if the original data point has an amplitude of Ai and happens to be between [n Δkx, m Δky] and [(n + 1) Δkx, (m + 1) Δky], then

255

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5 Imaging Issues in Inverse Synthetic Aperture Radar

kx

Original data Reformatted data

(m + 1)˙∆ky r2 r1

r3 r4

m˙∆ky

n˙∆kx

(n+1)˙∆kx

Figure 5.5 Interpolation can be employed in different ways to reformat the polar ISAR data to rectangular data (2D case).

5.3 Polar Reformatting Revisited

5

8 r8

r5

6

7 r6

r7

r1

r4

1 r2

4 r3

3

2

Figure 5.6 First-order nearest-neighbor interpolation (3D case): eight nearest data points are updated.

the nearest four grid points on the kx–ky plane are updated using the standard interpolation scheme as follows: s

E n Δk x , m Δk y s

E n Δk x , m + 1 Δk y E E

=

Ai

R r1

=

Ai

R r2

s

n + 1 Δk x , m + 1 Δk y

=

Ai

R r3

s

n + 1 Δk x , m Δk y

=

Ai

R r4

5 16

where R=

1 1 1 1 1 + + + r1 r2 r3 r4

5 17

and rk’s (k = 1, 2, 3, 4) are the distances from the four grid points to the original s

data position as shown in Figure 5.5. E is the uniformly sampled reformatted data on the kx–ky plane. It can be deduced from Eqs. 5.16 and 5.17 that the amplitude share of a grid point is inversely proportional to its distance to the original point. If

257

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5 Imaging Issues in Inverse Synthetic Aperture Radar

it is close to the original data point, it will have a large portion of the amplitude of the original data. If not, it will share only a small fraction of the amplitude of the original data. In the second-order interpolation scheme, 16 nearest neighbor data points on the uniform grid are updated in the same way (see Figure 5.7).

5.3.2

Bilinear Interpolation

Another popular data interpolation scheme used in polar reformatting is the bilinear interpolation technique (Abramowitz and Stegun 1970). In contrast to the nearest neighbor scheme, the original data grid should be uniformly sampled in this type of interpolation technique. Since the collected, raw backscattered data are in rectangular format on the frequency-aspect plane but in polar format in Fourier space, the interpolation should be done in frequency-aspect space.

(m + 2)·Δy r5

r6

r8

r7

(m +1)·Δy r16

r1

r2

r9

r3

r15

r10

r4 m ·Δy

r14

r13

r11 r12 (m –1)·Δy

(n + 2)·Δx

(n + 1)·Δx

n·Δx

(n –1)·Δx

Figure 5.7 Second-order nearest-neighbor interpolation (2D case): 16 nearest data points are updated.

5.3 Polar Reformatting Revisited

(m +1)·∆ϕ c a

b ~ E s(f, Ø) d m·∆ϕ Original data Reformatted data

(n +1)·∆f

n·∆f

Figure 5.8 Implementation of bilinear interpolation (2D case).

The implementation of bilinear interpolation is illustrated in Figure 5.8 where the original data (shown as black dots) are collected uniformly on the f–ϕ plane. To be able to apply the DFT, the data should be uniformly sampled on the kx–ky plane. If this uniformly sampled grid on the kx–ky plane is transformed to the f–ϕ plane, it corresponds to a nonuniform, polar grid, demonstrated in Figure 5.8 as empty dots. According to the bilinear interpolation, any point in between the uniformly sampled data points can be interpolated in the following way: s

b d a+b c+d a d s + E n + 1 Δf , m Δϕ a+b c+d b c + E s n Δf , m + 1 Δϕ a+b c+d a c + E s n + 1 Δf , m + 1 Δϕ a+b c+d

E f , ϕ = Es n Δf , m Δϕ

5 18

where a = (n + 1)Δf − f, b = f − nΔf, c = (m + 1)Δϕ − ϕ and d = ϕ − mΔϕ. For the interpolation of 3D uniform data onto a nonuniform 3D grid, trilinear interpolation can be applied (Kohler et al. 2000). It is also possible to use cubic approximation to interpolate the points in between the regular grid points. If this is the case, bicubic (Keys 1981) and tricubic (Lekien and Marsden 2005) interpolation schemes are applied for interpolating 2D and 3D data sets, respectively.

259

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5 Imaging Issues in Inverse Synthetic Aperture Radar

5.4

Zero Padding

Zero padding is also regarded as an interpolation technique that is used to enhance the visual image quality in SAR/ISAR imaging. To express how zero padding in one domain means interpolation in the other domain, we start with the FT pair shown in Figure 5.9. It is well known that a rectangular pulse of duration T has the FT of a sinc waveform that has a zero crossing at integer multiples of 1/T, as depicted in Figure 5.9. To be more precise, the FT pair can be represented as the following: FT rect t T

= T sinc fT

5 19

Now, let us consider that the time-domain pulse in Figure 5.9a is sampled to have a discrete signal such that it forms a comb waveform as depicted in Figure 5.9c. The DFT of this discrete sequence happens to have a single nonzero entry in the Fourier domain as shown in Figure 5.9d. As mentioned in previous sections, if the bandwidth of the time-domain signal is T, then the resolution in the frequency domain is 1/T, which is the same as the zero-crossing interval of the continuous wave sinc function seen in Figure 5.9b. This situation is illustrated in Figure 5.9d where the critically sampled sinc signal is plotted. Next, this discrete signal is zero-padded to have time duration of 3T as shown in Figure 5.9e. Its DFT is drawn in Figure 5.9f where the frequency sampling interval is now three times smaller than the original one. With this construct, sinc waveform can now be visible after DFT processing, provided that the new frequency-domain signal has a DFT length three times larger and therefore has three times more samples of the original frequency signal. This whole process of zero padding in one domain and therefore having more sampled version of the original signal in the other domain is often called DFT interpolation. The use of interpolation via zero padding is often used in SAR/ISAR imagery. Since the data in a SAR/ISAR image are 2D or 3D, the zero padding should be done in all directions. Some examples of zero-padded ISAR images are shown in Figure 5.10. On the left side of the figure, some original ISAR images for three different airplanes are shown. In the middle of the Figure 5.10, ISAR imaging geometries of the simulated models are given. The radar look direction is indicated with three little arrows in the middle figures. On the right side of the figure, the interpolated ISAR images with more sampling points after applying a four-times zeropadding procedure in the frequency-aspect domain are plotted. It is easily noticed from the right-hand side images that the distribution of the scattering centers is more pronounced, to produce more visually satisfying images. It is important to note that zero padding does not really improve the resolution of the image. Rather, it just interpolates the data in between to show a smoother data transition.

5.4 Zero Padding

(a) T 1

Amplitude

0.8

0.6

0.4

0.2

0 –40

–30

–20

–300

–200

–10

0 10 Time, ms

20

30

(b) 0.02

Amplitude

0.015

0.01

0.005

0

0 100 –100 Frequency, Hz

1/T

2/T

200

300

3/T

Figure 5.9 Illustration of interpolation with zero padding: (a) a rectangular pulse in time, (b) its Fourier transform, a sinc, (c) discrete time-domain pulse, (d) its DFT which is a critically sampled sinc, (e) zero-padded version of the discrete time-domain pulse, (f ) its Fourier transform, an interpolated sinc.

261

5 Imaging Issues in Inverse Synthetic Aperture Radar

(c) 1.2

T

1

S[n]

0.8 0.6 0.4 0.2 0

–10

–5

0 Time, ms

5

10

(d) 0.02

0.015

S[k]

262

0.01

0.005

0 –400

–200

0 Frequency, Hz

1/T Figure 5.9

(Continued)

2/T

200

3/T

400

5.4 Zero Padding

(e) 1.2

3T 1

S[n]

0.8 0.6 0.4 0.2 0 –30

–20

–10

0

10

20

Time, ms

(f) 0.02

S[k]

0.015

0.01

0.005

0

–400

–200

0 Frequency, Hz

1/(3T)

Figure 5.9 (Continued)

2/(3T)

200 1/T

400

263

5 Imaging Issues in Inverse Synthetic Aperture Radar

(a)

(b)

(c)

ISAR image (HH polarization)

ISAR image (HH polarization)

x–range, m

4

4

–10

2

–10

2

0

–20

–2

–30

–4

–40 –5

0 Range, m

dB

5

0

–20

–2

–30

–4

–40 –5

0 Range, m

–15

8

–15

6

–20

6

–20

4

–25

4

2

–25 x–range, m

8

x–range, m

dB

5

ISAR image (VV polarization)

ISAR image (VV polarization)

–30

0 –35

2 –30

0 –2

–35

–4

–40

–4

–40

–6

–45

–6

–2

–45 –8

–8

–50 –5

0 Range, m

–50 –5

5 dBsm

0 Range, m

5 dBsm

ISAR image (VV polarization)

ISAR image (VV polarization) –25

–25

4

4 –30 2

–35

–30 x–range, m

x–range, m

264

2 –35 0

0

–40

–2

–45

–2

–4

–50

–4

–6

–4

–2 0 2 Range, m

4

6

dBsm

–40 –45 –50 –6

–4

–2 0 2 Range, m

4

6

dBsm

Figure 5.10 Interpolation using zero padding: (a) original ISAR images, (b) target models with indicated radar look directions, and (c) interpolated ISAR images after applying fourtimes zero-padding procedure in the Fourier domain.

5.5

Point Spread Function

In its general usage, the point spread response (PSR) or point spread function (PSF) defines the response of an imaging system to a point source or point object. In SAR/ISAR nomenclature, it is the impulse response of a SAR/ISAR imaging system to a point scatterer. As will be demonstrated in this section, its effect is more meaningful with the application of zero padding.

5.5 Point Spread Function

As defined in the previous chapter, a 2D ISAR image is described as the following double integral: 1 π2

ISAR x, y =

∞

∞

−∞

−∞

Es kx , ky

exp j2 k x x + k y y dk x dk y 5 20

It is a common practice that any scattered field can be approximated as the response of the sum of finite point scatterers on the target. As will be explained in detail in Chapter 7, this is called the point-scatterer model. Under this assumption, Es(kx, ky) can be represented as K

Ai exp − j2 k x x i + k y yi ,

Es kx , ky

5 21

i=1

where K represents the total number of point scatterers, Ai is the scattered field amplitude, and (xi, yi) is the spatial location for the ith point scatterer. Ideally, perfect ISAR imaging of these scatterers is possible, assuming that the scattered field is collected for the infinite values of kx and ky (or frequency and aspect). Then, the ISAR image converts to 2D ideal impulse functions, as demonstrated below: K

ISAR x, y =

1 2 π i=1

∞

∞

−∞

−∞

Ai exp j2 k x

x − xi + ky

y − yi

dk x dk y

N

Ai δ x − x i , y − y i

= i=1

5 22

In practice, however, the frequencies and angles (or kx and ky) are finite. L H Let, k Lx , k H x , k y and k y be the lower and upper limits of the spatial frequencies defined as kH x = k xo + BWk x 2 k Lx = k xo − BWkx 2 kH y = k yo + BWk y 2

5 23

k Ly = k yo − BWky 2 Here, kxo and kyo are the center spatial frequencies, and BWkx and BWky are the bandwidths in kx and ky domains, respectively. Then the result of the ISAR integral can be calculated in the following way:

265

266

5 Imaging Issues in Inverse Synthetic Aperture Radar K

ISAR x, y =

1 π2 i=1 K

=

Ai i=1

kH kH x y k Lx

k Ly

Ai exp j2 k x

x − x i + ky

y − yi

dk x dk y

L exp j2k H x x − x i − exp j2k x x − x i j2π x − x i

L exp j2k H y y − yi − exp j2k y y − yi j2π x − x i K

=

Ai

exp j2k xo x − x i

i=1

exp j2k yo y − yi

Ai

exp j2k xo x − x i

i=1

exp j2k yo y − yi

5 24

exp jBWkx y − yi − exp − jBWky y − yi j2π y − yi

K

=

exp jBWkx x − x i − exp − jBWky x − x i j2π x − x i

BWkx BWkx sinc x − xi π π

BWky BWky sinc y − yi π π

The result of Eq. 5.24 can be reorganized to reflect the physical meaning of PSF as k

Ai δ x − x i , y − yi ∗ h x, y ,

ISAR x, y =

5 25

i=1

where h(x, y) is the so-called PSF and is given by h x, y =

exp j2kxo x

BWkx BWkx sinc x π π

exp j2k yo y

BWky BWky sinc y π π

5 26 According to Eq. 5.25, PSF can be regarded as the impulse response of the ISAR imaging arrangement to any point scatterer on the target. This physical meaning of PSF is illustrated in Figure 5.11. A finite number of point scatterers that have different scattering amplitudes exist in 2D x–y space as shown in Figure 5.11a. The resultant ISAR image (Figure 5.11c) is nothing but the convolution of the point scatterers with the 2D PSF function in Figure 5.11b. In the common display format of ISAR, the resultant image is displayed in the 2D range and cross-range planes, as shown in Figure 5.11d. Because of the finite bandwidths in kx and ky domains, the PSF has tails that correspond to sidelobes of sinc functions in both range and crossrange directions. The common way of suppressing the sidelobes of PSF is to use windowing, as we shall explore next.

5.5 Point Spread Function

(a) 5

Amplitude

4 3 2 1 0 5

5 0

0

y, m

–5

–5

x, m

(b) 1

Amplitude

0.8 0.6 0.4 0.2 0 5

5 0

0 y, m

–5

–5

x, m

Figure 5.11 The physical meaning of PSF: (a) point scatterers, (b) the PSF, (c) the ISAR image is constructed by the convolution of point scatter with the PSF, (d) the effect of PSF: sinc sidelobes are noticeable in the 2D image plane.

267

5 Imaging Issues in Inverse Synthetic Aperture Radar

(c)

ISAR

4 3 2 1

5

5 0

0 –5

Cross–range, m

–5

Range, m

(d) ISAR 6 4

Cross–range, m

268

2

0

–2 –4

–6 –6

–4

–2

0 Range, m

Figure 5.11

(Continued)

2

4

6

5.6 Windowing

5.6

Windowing

5.6.1 Common Windowing Functions In radar imaging applications, windowing is a common practice to tone down the magnitudes of the sidelobes of the PSF such that the resultant SAR/ISAR image looks much smoother and more localized in terms of the point scatterers. A windowing (or tapering) function has nonzero entries within a chosen interval and zero outside this interval as defined below: wn =

h n ; n = 0, 1, 2, , N − 1 0

elsewhere,

5 27

where the entries of h[n] are equal to or less than 1. Here, the N-point w[n] is to be applied to a signal length of To. In many practical applications, the window functions that have smooth “bell-shaped” characteristics are used to successfully suppress the undesired sidelobes. Hanning, Hamming, Kaiser, Blackman, Gaussian, Nuttall, and Chebyshev windows fall into this category. While these windows provide better-focused images due to decreased sidelobe levels (SLLs), they, unfortunately, present lower resolution compared to windows types such as rectangular or triangular due to increased width of main lobe. 5.6.1.1

Rectangular Window

If h[n] = 1 for the chosen interval, this window is commonly known as a rectangular window, as depicted in Figure 5.12a. Applying a rectangular window produces the same data with the case when truncating the same portion of the data of the chosen interval. The spectrum of rectangular window has the maximum SLL of about −13 dB, which is the highest when compared with any other windowing function (see Figure 5.12b). It has, however, the narrowest main lobe width compared with the others. While the −3 dB width of the main lobe is about 88% of the one FFT bin, single FFT bin corresponds to the −4 dB width. This single frequency bin is equal to one fundamental frequency resolution of 1/(ToN). 5.6.1.2

Triangular Window

The triangular window function (Figure 5.13a) has the following formula within the chosen integral: h n = 1−

2 N

n−

N −1 2

5 28

Since a triangular window of length N can be obtained by convolving two identical rectangular windows of length N/2, the spectrum of a triangular window of

269

5 Imaging Issues in Inverse Synthetic Aperture Radar

(a)

Rectangular window 1

Amplitude

0.8

0.6

0.4

0.2

0

0

5

10

15

20

25

30

35

Samples

(b)

Spectrum of rectangular window 0

Normalized amplitude[dB]

270

–20 –40 –60 –80 –100 –120

Figure 5.12

100

200

300 Samples

400

500

(a) Rectangular window, (b) spectrum of rectangular window.

5.6 Windowing

(a)

Triangular window 1

Amplitude

0.8

0.6

0.4

0.2

0

0

5

10

15

20

25

30

35

Samples

(b)

Spectrum of triangular window

Normalized amplitude [dB]

0 –20 –40 –60 –80 –100 –120 100

Figure 5.13

200

300 Samples

400

500

(a) Triangular window, (b) spectrum of triangular window.

271

272

5 Imaging Issues in Inverse Synthetic Aperture Radar

length N is equal to the square of the spectrum of a rectangular window of length N/2. Therefore, the main lobe width of a triangular window is twice as big as the main lobe width of a rectangular window of the same length (see Figure 5.13b). The spectrum of triangular window has the maximum SLL of about −26 dB that is twice to that of decibel value of the rectangular window as expected. 5.6.1.3 Hanning Window

The shape of the Hanning or Hann window looks like half of a cycle of a cosine waveform (Blackman and Tukey 1958). The equation for the definition of this windowing function is equal to hn =05

1 − cos

2πn N −1

5 29

As is obvious from Eq. 5.29, the Hanning window function is, in fact, the normalized version of an up-shifted cosine waveform as plotted in Figure 5.14a. That is why the Hanning window is also known as raised cosine window. The spectrum of this window has the maximum SLL of −32 dB. 5.6.1.4 Hamming Window

A Hamming window (Figure 5.15a) is a modified version of the Hanning window as its wave equation is as below: h n = 0 53836 − 0 46164 cos

2πn N −1

5 30

Richard W. Hamming (Enochson and Otnes 1968) proposed the above particular coefficients for the two terms of the original Hanning equation that results in the maximum SLL of −43 dB in the spectrum of the window. This proper selection of coefficients helps to decrease the sidelobes closer to the main lobe and increase the sidelobes that are far to the main lobe as plotted in Figure 5.15b. 5.6.1.5 Kaiser Window

The equation for the Kaiser window (Kaiser 1974) is given below: I0 α 1 − hn =

2n −1 N −1 I0 α

2

1 2

,

5 31

where Io(α) is the zeroth order modified Bessel function of first kind. The spectrum of the Kaiser window has the maximum SLL of −36 dB for α = 1.5π. The window waveform and its spectrum for this windowing function are drawn in Figure 5.16.

5.6 Windowing

(a)

Hanning window 1

Amplitude

0.8

0.6

0.4

0.2

0

0

5

(b)

10

15 20 Samples

25

30

35

Spectrum of hanning window

Normalized amplitude [dB]

0 –20 –40 –60 –80 –100 –120 100

200

300

400

500

Samples

Figure 5.14

(a) Hanning window, (b) spectrum of Hanning window.

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5 Imaging Issues in Inverse Synthetic Aperture Radar

(a)

Hamming window 1

Amplitude

0.8

0.6

0.4

0.2

0

0

5

(b)

10

15 20 Samples

25

30

35

Spectrum of hamming window 0

Normalized amplitude [dB]

274

–20 –40 –60 –80 –100 –120

100

200

300

400

500

Samples

Figure 5.15

(a) Hamming window, (b) spectrum of Hamming window.

5.6 Windowing

(a)

Kaiser window, β = 1.5 π 1

Amplitude

0.8

0.6

0.4

0.2

0

0

5

10

15 20 Samples

25

30

35

(b) Spectrum of kaiser window, β = 1.5 π 0

Normalized amplitude [dB]

–20 –40 –60 –80 –100 –120

100

200

300

400

500

Samples

Figure 5.16

(a) Kaiser window, (b) spectrum of Kaiser window.

275

5 Imaging Issues in Inverse Synthetic Aperture Radar

5.6.1.6 Blackman Window

The equation for the Blackman window (Oppenheim et al. 1999) is given below: 2πn 4πn + 0 08 cos N −1 N −1

h n = 0 42 − 0 5 cos

5 32

The spectrum of the Blackman window has the maximum SLL of −58 dB. The window function and its spectrum for the Blackman are plotted in Figure 5.17.

(a)

Blackman window 1

Amplitude

0.8

0.6

0.4

0.2

0

0

5

(b)

10

15 20 Samples

25

30

35

Spectrum of blackman window 0

Normalized amplitude [dB]

276

–20 –40 –60 –80 –100 –120

Figure 5.17

100

200

300 Samples

400

500

(a) Blackman window, (b) spectrum of Blackman window.

5.6 Windowing

5.6.1.7

Chebyshev Window

Chebyshev window function (Harris 2004) is characterized by the following equation:

hn =

r+2 N

N −1 2

CN − 1 t 0 cos k=1

kπ N

cos

2kπ n − N

N −1 2

,

5 33

where r is the ratio of the main lobe to sidelobes in decibel, Ck(α) is the kth order Chebyshev polynomial given below,

Ck α =

cos k cos − 1 α cosh k cosh − 1 α

α ≤1 α >1

5 34

and the parameter to is given as

t o = cosh

cosh − 1 r N −1

5 35

The Chebyshev window waveform and its spectrum are shown in Figure 5.18. For this example, r is selected as 80 dB. As is shown in Figure 5.18b, the spectrum makes many ripples providing that the sidelobes have a maximum value of −80 dB. Table 5.2 summarizes the characteristics of several windowing functions used for smoothing the data of interest. Since higher resolution means greater sidelobes, a trade-off between the main lobe width and the maximum SLL should be taken into account when applying any smoothing window to the data. Table 5.2 presents the −3 dB width of the main lobe and the maximum SLL for the windowing functions that are considered in this book.

5.6.2 ISAR Image Smoothing via Windowing Applying a window before displaying the final ISAR image is a common procedure for smoothing the PSF around the scattering centers on the image. This process enhances the image visual appearance by suppressing the sidelobes around the scattering centers at the price of losing some resolution.

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5 Imaging Issues in Inverse Synthetic Aperture Radar

(a) Chebyshev window 1

Amplitude

0.8

0.6

0.4

0.2

0

0

5

10

15 20 Samples

25

30

35

(b) Spectrum of chebyshev window 0

Normalized amplitude [dB]

278

–20 –40 –60 –80 –100 –120

100

200

300

400

500

Samples

Figure 5.18

(a) Chebyshev window, (b) spectrum of Chebyshev window.

5.6 Windowing

Table 5.2

279

Comparative characteristics of different windowing functions.

Window function

Expression

Rectangular

h[n] = 1

Triangular

2 h n = 1− N

Hamming

h n = 0 5 1 − cos

Hamming

h n = 0 538 36 − 0 461 64 cos

N −1 n− 2

I0 α 1 − hn =

Blackman

h n = 0 42 − 0 5 cos

Chebyshev

2πn N −1

1 2 2πn 2 N −1

Kaiser

hn =

2πn N −1

–3 dB main lobe width (FFT bins)

Maximum SLL (dB)

0.88

−13

1.24

−26

1.40

−32

1.33

−43

1.30

−36 (for a = 1.5π)

1.69

−58

1.68

−80 (r = 80)

I0 α

r+2 n

2πn 2πn + 0 08 cos N −1 N −1

N −1

CN − 1 to cos k=1

2kπ N − 1 2 cos N

kπ N

To demonstrate the use of windowing functions in ISAR imaging, the ISAR images that are given in Figure 5.10 are used. The original ISAR images are shown along the left column in Figure 5.19. Then, a 2D Hamming window is applied to these images in the Fourier domain. A four-times zero-padding scheme is also applied to interpolate the data points. The resultant smoothed 2D ISAR images are plotted along the right column of Figure 5.19. New images do not suffer from the tails (sidelobes) of the 2D PSF around the scattering centers. Since the tails of the scattering centers are suppressed, scattering mechanisms that fall beneath the sidelobes of the nearby scattering centers’ PSF are now visible in the new windowed ISAR images. Yet, the decrease in the image resolution is also apparent in the new ISAR images, as we discussed before.

5 Imaging Issues in Inverse Synthetic Aperture Radar

(a)

(b)

ISAR image (HH polarization)

4

2

2

0

0

–2

–2

x-range, m

4

–4

ISAR image (HH polarization)

–4 –8

–6

–4

–2 0 2 Range, m

4

6

8

–8

–4

–2 0 2 4 Range, m ISAR image (VV polarization)

8

8

6

6

4

4

2

2

0

6

8

0

–2

–2 –4

–4

–6

–6

–8

–8 –5

0 Range, m

–5

5

ISAR image (VV polarization)

–5

–6

x-range, m

x-range, m

ISAR image (VV polarization)

x-range, m

280

–4

–3

–3

–2

–2

–1

–1

0

0

1

1

2

2

3

3

4

5

ISAR image (VV polarization)

–5

–4

0 Range, m

4 –6

–4

–2

0 2 Range, m

4

6

–6

–4

–2

0 2 Range, m

4

6

Figure 5.19 Effect of using smoothing windows: (a) original ISAR images, (b) interpolated ISAR images after applying four-times zero-padding procedure and Hanning window in the Fourier domain.

5.7

Matlab Codes

Below are the Matlab source codes that were used to generate all of the Matlabproduced figures in this chapter. The codes are also provided in the CD that accompanies this book.

5.7 Matlab Codes

281

Matlab code 5.1 Matlab file “Figure5-9.m” %———————————————————————————————————————————————————————————— % This code can be used to generate Figure 5.9 %———————————————————————————————————————————————————————————— clear close all clc %______________Implementation OF FT Window/Sinc______________ M = 500; t = (-M:M)*1e-3/5; E(450:550) = 1;E(1001)=0; T = t(550)-t(450); index=300:700; %—Figure 5.9a————————————————————————————————————— h = area(t(index)*1e3,E(index)); set(h,'FaceColor',[.5 .5 .5]) axis([min(t(index))*1e3 max(t(index))*1e3 0 1.15]) grid minor set(gca,'FontName','Arial','FontSize',12,'FontWeight','Bold'); xlabel('\ittime, ms'); ylabel('\itamplitude'); %—Figure 5.9b————————————————————————————————————— index = 430:570; df = 1/(max(t)-min(t)); f = (-M:M)*df; Ef = T*fftshift(fft(E))/length(450:550); figure; h = area(f(index),abs(Ef(index))); grid minor set(h,'FaceColor',[.5 .5 .5]) axis([min(f(index)) max(f(index)) 0 .023]) set(gca,'FontName','Arial','FontSize',12,'FontWeight','Bold'); xlabel('\itfrequency, Hz'); ylabel('\itamplitude');

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5 Imaging Issues in Inverse Synthetic Aperture Radar

%________________Implementation OF DFT________________ clear ; % TIME DOMAIN SIGNAL t = (-10:9)*1e-3; N=length(t); En(1:N) = 1; %—Figure 5.9c————————————————————————————————————————— figure; stem(t*1e3,En,'k','LineWidth',3); grid minor axis([min(t)*1.2e3 max(t)*1.2e3 0 1.25]) set(gca,'FontName','Arial','FontSize',12,'FontWeight','Bold'); xlabel('\ittime, ms'); ylabel('\its[n]');%grid on; %—————————FREQ DOMAIN SIGNAL——————————————— dt = t(2)-t(1); BWt = max(t)-min(t)+dt; df = 1/BWt; f = (-10:9)*df; Efn = BWt*fftshift(fft(En))/length(En); %—Figure 5.9d————————————————————————————————— figure; stem(f,abs(Efn),'k','LineWidth',3); grid minor % axis([min(f ) max(f ) 0 1.15]) set(gca,'FontName','Arial','FontSize',12,'FontWeight','Bold'); xlabel('\itfrequency, Hz'); ylabel('S[k]');%grid on; colormap(gray);hold on %———this part for the sinc template clear En2; En2(91:110) = En; En2(200) = 0; Efn2 = BWt*fftshift(fft(En2))/length(En); f2 = min(f ):df/10:(min(f )+df/10*199);

5.7 Matlab Codes

283

plot(f2,abs(Efn2),'k-.','LineWidth',1); axis([min(f2) max(f2) 0 .023]); hold off %———————ZERO PADDING ————————————————————— %TIME DOMAIN clear En_zero; En_zero(20:39) = En; En_zero(60) = 0; dt = 1e-3; t2 = dt*(-30:29); %—Figure 5-9(e)———————————————————————————————— figure; stem(t2*1e3,En_zero,'k','LineWidth',3); axis([-dt*30e3 dt*29e3 0 1.25]) grid minor set(gca,'FontName','Arial','FontSize',12,'FontWeight','Bold'); xlabel('\ittime, ms'); ylabel('\its[n]'); %FREQUENCY DOMAIN Efn2_zero = BWt*fftshift(fft(En_zero))/length(En); f2 = min(f ):df/3:(min(f )+df/3*59); %—Figure 5-9(f )———————————————————————————————— figure; plot(f2,abs(Efn2_zero),'k-.','LineWidth',1);hold on stem(f2,abs(Efn2_zero),'k','LineWidth',3); grid minor set(gca,'FontName','Arial','FontSize',12,'FontWeight','Bold'); xlabel('\itfrequency, Hz'); ylabel('S[k]'); axis([min(f2) max(f2) 0 0.023]); hold off

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5 Imaging Issues in Inverse Synthetic Aperture Radar

Matlab code 5.2 Matlab file “Figure5-10ac.m” %———————————————————————————————————————————————————————————— % This code can be used to generate Figure 5.10a and 5.10c %———————————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % M_fighter2.mat clear close all load M_fighter2.mat

% load the raw backscattered E-field

c=0.3; M = length(Freq); P = length(Theta); N = length(Phi); % organize axes df = Freq(2)-Freq(1); BWf = M*df; dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); dphi = pi*(aspect(2)-aspect(1))/180; % in radians BWphi = N*dphi; fc = Freq(round(M/2+1)); % center freq dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1); % ————————————Figure 5.10a—————————————— E_HH = vec2mat(Es_HH,N); ISAR_HH = abs(fftshift(ifft2(E_HH.'))); ISAR_HHdB = 20*log10(ISAR_HH);

5.7 Matlab Codes

285

rd = 40; figure; imagesc(X,-Y,ISAR_HHdB); grid on; colormap(hot) caxis([max(max(ISAR_HHdB))-rd,(max(max(ISAR_HHdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (HH Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; % ————————————Figure 5.10c—————————————— figure; E_HH = vec2mat(Es_HH,N); E_HH(4*M,4*N)=0; ISAR_HH = 16*abs(fftshift(ifft2(E_HH.'))); ISAR_HHdB = 20*log10(ISAR_HH); imagesc(X,-Y,ISAR_HHdB); grid on; colormap(hot) caxis([max(max(ISAR_HHdB))-rd,(max(max(ISAR_HHdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (HH Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ];

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5 Imaging Issues in Inverse Synthetic Aperture Radar

Matlab code 5.3 Matlab file “Figure5-10df.m” %———————————————————————————————————————————————————————————— % This code can be used to generate Figure 5.10d and 5.10f %———————————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % BM_fighter30.mat clear close all load BM_fighter30.mat

% load the raw backscattered E-field

c=0.3; M = length(Freq); P = length(Theta); N = length(Phi); % organize axes df = Freq(2)-Freq(1); BWf = M*df; dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); dphi = pi*(aspect(2)-aspect(1))/180; % in radians BWphi = N*dphi; fc = Freq(round(M/2+1)); % center freq dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1); % ————————————Figure 5.10d—————————————— E_VV = vec2mat(Es_VV,N); ISAR_VV = abs(fftshift(ifft2(E_VV.'))); ISAR_VVdB = 20*log10(ISAR_VV);

5.7 Matlab Codes

287

rd = 30; figure; imagesc(X,-Y,ISAR_VVdB); grid on; colormap(hot) caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; % ————————————Figure 5.10f—————————————— figure; E_VV = vec2mat(Es_VV,N); E_VV(4*M,4*N)=0; ISAR_VV = 16*abs(fftshift(ifft2(E_VV.'))); ISAR_VVdB = 20*log10(ISAR_VV); imagesc(X,-Y,ISAR_VVdB); grid on; colormap(hot) caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ];

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5 Imaging Issues in Inverse Synthetic Aperture Radar

Matlab code 5.4 Matlab file “Figure5-10ef.m” %———————————————————————————————————————————————————————————— % This code can be used to generate Figure 5.10g and 5.10i %———————————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % Pl_Plane.mat clear close all load Pl_Plane.mat

% load the raw backscattered E-field

c=0.3; M = length(Freq); P = length(Theta); N = length(Phi); % organize axes df = Freq(2)-Freq(1); BWf = M*df; dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); dphi = pi*(aspect(2)-aspect(1))/180; % in radians BWphi = N*dphi; fc = Freq(round(M/2+1)); % center freq dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1); % ————————————Figure 5.10g—————————————— E_VV = vec2mat(Es_VV,N); ISAR_VV = abs(fftshift(ifft2(E_VV.'))); ISAR_VVdB = 20*log10(ISAR_VV);

5.7 Matlab Codes

289

rd = 40; figure; imagesc(X,-Y,ISAR_VVdB); grid on; colormap(hot) caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; % ————————————Figure 5.10i—————————————— figure; E_VV = vec2mat(Es_VV,N); E_VV(4*M,4*N)=0; ISAR_VV = 16*abs(fftshift(ifft2(E_VV.'))); ISAR_VVdB = 20*log10(ISAR_VV); imagesc(X,-Y,ISAR_VVdB); grid on; colormap(hot) caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ];

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5 Imaging Issues in Inverse Synthetic Aperture Radar

Matlab code 5.5 Matlab file “Figure5-11.m” %———————————————————————————————————————————————————————————— % This code can be used to generate Figure 5.11 %———————————————————————————————————————————————————————————— clear close all clc % Prepare mesh [X,Y] = meshgrid(-6:.1:6, -6:.1:6); M = length(X); N = length(Y) ; Object = zeros(M,N); % Set 3 scattering centers hh = figure; Object(101,95)=5; Object(30,96)=2; Object(100,15)=3; %—Figure 5.11a———————————————————————————————————————————————— surf(X,Y,Object); colormap(1-gray) grid minor axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itx, m'); ylabel('\ity, m'); zlabel('amplitude') view(-45,20)

dx = X(1,2)-X(1,1); dy = dx;

% range resolution % xrange resolution

%Find Bandwidth in spatial frequencies BWkx = 1/dx;

5.7 Matlab Codes

291

BWky = 1/dy; % PSF h = sinc(BWkx*X/pi).*sinc(BWky*Y/pi); %—Figure 5.11b———————————————————————————————————————————————— hh = figure; surf(X,Y,abs(h)); grid minor axis tight; colormap(1-gray); axis([-6 6 -6 6 0 1]) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itx, m'); ylabel('\ity, m'); zlabel('amplitude') view(-45,20) %Convolution hh = figure; ISAR = fft2(fft2(Object).*fft2(h))/M/N; %—Figure 5.11c———————————————————————————————————————————————— surf(X,Y,abs(ISAR)); grid minor axis tight; colormap(1-gray); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itcross - range, m'); zlabel('ISAR '); view(-45,20) %—Figure 5.11d————————————————————————————————————————————— hh = figure; matplot(X(1,1:M),Y(1:N,1),ISAR,30); colormap(hot); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itcross - range, m'); title('ISAR ');

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5 Imaging Issues in Inverse Synthetic Aperture Radar

Matlab code 5.6 Matlab file “Figure5-12thru5-18.m” %———————————————————————————————————————————————————————————— % This code can be used to generate Figure 5.12 – 5.18 %———————————————————————————————————————————————————————————— % Comparison of windowing functions %————————————————————————————————— clear close all N=33; %—Figure 5.12(a)—————————————————————————————————————————————— %—Rectangular window rect = rectwin(N); h = figure; h1 = area(rect); set(h1,'FaceColor',[.5 .5 .5]) grid minor colormap(gray) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itamplitude'); title(' Rectangular Window') axis([-2 N+2 0 1.1]) saveas(h,'Figure5_12a.tif','tif'); %—Figure 5.12(b)—————————————————————————————————————————————— rect(16*N)=0; Frect = fftshift(fft(rect)); Frect = Frect/max(abs(Frect)); h = figure; plot(mag2db(abs(Frect)),'k','LineWidth',2); grid minor axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itnormalized amplitude[dB]'); title ('Spectrum of Rectangular Window')

5.7 Matlab Codes

293

axis([1 16*N -120 3]) saveas(h,'Figure5_12b.tif','tif'); %—Figure 5.13(a)—————————————————————————————————————————————— %—Triangular window tri = triang(N); h = figure; h1 = area([0 tri.']); set(h1,'FaceColor',[.5 .5 .5]) grid minor colormap(gray) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itamplitude'); title (' Triangular Window') axis([-2 N+4 0 1.1]) saveas(h,'Figure5_13a.tif','tif'); %—Figure 5.13(b)—————————————————————————————————————————————— tri(16*N)=0; Ftri = fftshift(fft(tri)); Ftri = Ftri/max(Ftri); h = figure; plot(mag2db(abs(Ftri)),'k','LineWidth',2); grid minor hold off; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itnormalized amplitude [dB]'); title ('Spectrum of Triangular Window') axis([1 16*N -120 3]) saveas(h,'Figure5_13b.tif','tif'); %—Figure 5.14(a)—————————————————————————————————————————————— %—Hanning window han = hanning(N); h = figure; h1 = area(han);

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5 Imaging Issues in Inverse Synthetic Aperture Radar

set(h1,'FaceColor',[.5 .5 .5]) grid minor colormap(gray); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itamplitude'); title (' Hanning Window') axis([-2 N+2 0 1.1]) saveas(h,'Figure5_14a.tif','tif'); %—Figure 5.14(b)—————————————————————————————————————————————— han(16*N) = 0; Fhan = fftshift(fft(han)); Fhan = Fhan/max(Fhan); h = figure; plot(mag2db(abs(Fhan)),'k','LineWidth',2); grid minor hold off; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itnormalized amplitude[dB]'); title ('Spectrum of Hanning Window') axis([1 16*N -120 3]) saveas(h,'Figure5_14b.tif','tif'); %—Figure 5.15(a)—————————————————————————————————————————————— %—Hamming window ham = hamming(N); h = figure; h1 = area(ham); set(h1,'FaceColor',[.5 .5 .5]) grid minor colormap(gray); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itamplitude'); title ('Hamming Window') axis([-2 N+2 0 1.1]) saveas(h,'Figure5_15a.tif','tif');

5.7 Matlab Codes

295

%—Figure 5.15(b)—————————————————————————————————————————————— ham(16*N)=0; Fham = fftshift(fft(ham)); Fham = Fham/max(Fham); h = figure; plot(mag2db(abs(Fham)),'k','LineWidth',2); grid minor hold off; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itnormalized amplitude [dB]'); title ('Spectrum of Hamming Window') axis([1 16*N -120 3]) saveas(h,'Figure5_15b.tif','tif'); %—Figure 5.16(a)—————————————————————————————————————————————— %—Kaiser window ksr = kaiser(N,1.5*pi); h = figure; h1 = area(ksr); set(h1,'FaceColor',[.5 .5 .5]) grid minor colormap(gray); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itamplitude'); title ('Kaiser Window, \beta = 1.5\pi') axis([-2 N+2 0 1.1]) saveas(h,'Figure5_16a.tif','tif'); %—Figure 5.16(b)—————————————————————————————————————————————— ksr(16*N) = 0; Fksr = fftshift(fft(ksr)); Fksr = Fksr/max(Fksr); h = figure; plot(mag2db(abs(Fksr)),'k','LineWidth',2); grid minor

296

5 Imaging Issues in Inverse Synthetic Aperture Radar

hold off; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itnormalized amplitude[dB]'); title ('Spectrum of Kaiser Window, \beta = 1.5\pi') axis([1 16*N -120 3]) saveas(h,'Figure5_16b.tif','tif'); %—Figure 5.17(a)—————————————————————————————————————————————— %—Blackman window blk = blackman(N); h = figure; h1 = area(blk); set(h1,'FaceColor',[.5 .5 .5]) grid minor colormap(gray); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itamplitude'); title ('Blackman Window') axis([-2 N+2 0 1.1]) saveas(h,'Figure5_17a.tif','tif'); %—Figure 5.17(b)—————————————————————————————————————————————— blk(16*N) = 0; Fblk = fftshift(fft(blk)); Fblk = Fblk/max(Fblk); h = figure; plot(mag2db(abs(Fblk)),'k','LineWidth',2); grid minor hold off; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itnormalized amplitude[dB]'); title ('Spectrum of Blackman Window') axis([1 16*N -120 3]) saveas(h,'Figure5_17b.tif','tif');

5.7 Matlab Codes

297

%—Figure 5.18(a)—————————————————————————————————————————————— %—Chebyshev window cheby = chebwin(N,80); h = figure; h1 = area(blk); set(h1,'FaceColor',[.5 .5 .5]) grid minor colormap(gray); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itamplitude'); title ('Chebyshev Window') axis([-2 N+2 0 1.1]) saveas(h,'Figure5_18a.tif','tif'); %—Figure 5.18(b)—————————————————————————————————————————————— cheby(16*N)=0; Fcheby = fftshift(fft(cheby)); Fcheby = Fcheby/max(Fcheby); h = figure; plot(mag2db(abs(Fcheby)),'k','LineWidth',2); grid minor hold off; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itsamples '); ylabel('\itnormalized amplitude [dB]'); title ('Spectrum of Chebyshev Window') axis([1 16*N -120 3]) saveas(h,'Figure5_18b.tif','tif');

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Matlab code 5.7 Matlab file “Figure5-19 ab.m” %———————————————————————————————————————————————————————————— % This code can be used to generate Figure 5.19a and 5.19b %———————————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % M_fighter2.mat clear close all load M_fighter2.mat

% load the raw backscattered E-field

c=0.3; M = length(Freq); P = length(Theta); N = length(Phi); % organize axes df = Freq(2)-Freq(1); BWf = M*df; dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); dphi = pi*(aspect(2)-aspect(1))/180; % in radians BWphi = N*dphi; fc = Freq(round(M/2+1)); % center freq dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1); % ————————————Figure 5.19a—————————————— E_HH = vec2mat(Es_HH,N); ISAR_HH = abs(fftshift(ifft2(E_HH.'))); ISAR_HHdB = 20*log10(ISAR_HH); rd = 40; figure; imagesc(X,Y,ISAR_HHdB); grid on;

5.7 Matlab Codes

299

colormap(hot) caxis([max(max(ISAR_HHdB))-rd,(max(max(ISAR_HHdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ]; % —2D hanning window ———————————— h=hanning(M)*hanning(N).'; % ————————————Figure 5.19b—————————————— figure; E_HH = vec2mat(Es_HH,N); E_HH = 2*E_HH.*h; % 2 for windowing E_HH(4*M,4*N)=0; ISAR_HH = 16*abs(fftshift(ifft2(E_HH.'))); ISAR_HHdB = 20*log10(ISAR_HH); imagesc(X,Y,ISAR_HHdB); grid on; colormap(hot) caxis([max(max(ISAR_HHdB))-rd,(max(max(ISAR_HHdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ];

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5 Imaging Issues in Inverse Synthetic Aperture Radar

Matlab code 5.8 Matlab file “figure5-19cd.m” %———————————————————————————————————————————————————————————— % This code can be used to generate Figure 5.19c and 5.19d %———————————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % Pl_Plane.mat clear close all load Pl_Plane.mat c=0.3; M = length(Freq); P = length(Theta); N = length(Phi);

% load the raw backscattered E-field

% organize axes df = Freq(2)-Freq(1); BWf = M*df; dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); dphi = pi*(aspect(2)-aspect(1))/180; % in radians BWphi = N*dphi; fc = Freq(round(M/2+1)); % center freq dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1); % ————————————Figure 5.19c—————————————— E_VV = vec2mat(Es_VV,N); ISAR_VV = abs(fftshift(ifft2(E_VV.'))); ISAR_VVdB = 20*log10(ISAR_VV); rd = 35; figure; imagesc(X,Y,ISAR_VVdB); grid on; colormap(hot)

5.7 Matlab Codes

301

caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ];

% —2D hanning window ———————————— h=hanning(M)*hanning(N).'; % ————————————Figure 5.19d—————————————— figure; E_VV = vec2mat(Es_VV,N); E_VV = 2*E_VV.*h; % 2 for windowing E_VV(4*M,4*N)=0; ISAR_VV = 16*abs(fftshift(ifft2(E_VV.'))); ISAR_VVdB = 20*log10(ISAR_VV); imagesc(X,Y,ISAR_VVdB); grid on; colormap(hot) caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ];

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5 Imaging Issues in Inverse Synthetic Aperture Radar

Matlab code 5.9 Matlab file “Figure5-19ef.m” %———————————————————————————————————————————————————————————— % This code can be used to generate Figure 5.19e and 5.19f %———————————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % Pl_Plane.mat clear close all load Pl_Plane.mat c=0.3; M = length(Freq); P = length(Theta); N = length(Phi);

% load the raw backscattered E-field

% organize axes df = Freq(2)-Freq(1); BWf = M*df; dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); dphi = pi*(aspect(2)-aspect(1))/180; % in radians BWphi = N*dphi; fc = Freq(round(M/2+1)); % center freq dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1); % ————————————Figure 5.19e—————————————— E_VV = vec2mat(Es_VV,N); ISAR_VV = abs(fftshift(ifft2(E_VV.'))); ISAR_VVdB = 20*log10(ISAR_VV); rd = 35; figure; imagesc(X,Y,ISAR_VVdB); grid on; colormap(hot)

5.7 Matlab Codes

303

caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ];

% —2D hanning window ———————————— h=hanning(M)*hanning(N).'; % ————————————Figure 5.19f—————————————— figure; E_VV = vec2mat(Es_VV,N); E_VV = 2*E_VV.*h; % 2 for windowing E_VV(4*M,4*N)=0; ISAR_VV = 16*abs(fftshift(ifft2(E_VV.'))); ISAR_VVdB = 20*log10(ISAR_VV); imagesc(X,Y,ISAR_VVdB); grid on; colormap(hot) caxis([max(max(ISAR_VVdB))-rd,(max(max(ISAR_VVdB)))]); axis xy; axis equal; axis tight; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itx-range, m'); title('\itISAR Image (VV Polarization)'); cc = colorbar; tt = title(cc,'dBsm'); tt.Position = [ 8 -15 0 ];

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5 Imaging Issues in Inverse Synthetic Aperture Radar

References Abramowitz, M. and Stegun, I.A. (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9e. New York: Dover Publications. Ausherman, D.A., Kozma, A., Walker, J.L. et al. (1984). IEEE Transactions on Aerospace and Electronic Systems AES-20 (4): 363–400. https://doi.org/10.1109/ TAES.1984.4502060. Blackman, R.B. and Tukey, J.W. (1958). The measurement of power spectra from the point of view of communications engineering — part II. The Bell System Technical Journal 37 (2): 485–569. https://doi.org/10.1002/j.1538-7305.1958. tb01530.x. Enochson, L.D. and Otnes, R.K. (1968). Programming and Analysis for Digital Time Series Data. Washington, DC: Shock and Vibration Information Center, United States Department of Defense. Harris, F.J. (2004). Multirate Signal Processing for Communication Systems. Upper Saddle River, NJ: Prentice Hall. Kaiser, J.F. (1974). Nonrecursive digital filter design using the I0-sinh window function. 1974 Proceedings of IEEE International Symposium on Circuits & Systems, San Francisco, CA, pp. 22–23. Keys, R. (1981). Cubic convolution interpolation for digital image processing. IEEE Transactions on Acoustics, Speech, and Signal Processing 29 (6): 1153–1160. https:// doi.org/10.1109/TASSP.1981.1163711. Knott, E.F., Shaeffer, J.F., and Tuley, M.T. (2004). Radar Cross Section, 2e. Raleigh, NC: Scitechpub Publishing. Kohler, T., Turbell, H., and Grass, M. (2000). Efficient forward projection through discrete data sets using tri-linear interpolation. 2000 IEEE Nuclear Science Symposium. Conference Record (Cat. No.00CH37149). https://doi.org/10.1109/ NSSMIC.2000.950067. Lekien, F. and Marsden, J. (2005). Tricubic interpolation in three dimensions. International Journal for numerical methods in engineering 63 (3): 455–471. https:// doi.org/10.1002/nme.1296. Mersereau, R.M. and Oppenheim, A.V. (1974). Digital reconstruction of multidimensional signals from their projections. Proceedings of the IEEE https://doi. org/10.1109/PROC.1974.9625. Oppenheim, A.V., Buck, J.R., and Schafer, R.W. (1999). Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall.

References

Özdemir, C., Bhalla, R., Trintinalia, L.C., and Ling, H. (1998). ASAR – antenna synthetic aperture radar imaging. IEEE Transactions on Antennas and Propagation 46 (12): 1845–1852. https://doi.org/10.1109/8.743822. Sullivan, R.J. (2000). Microwave Radar Imaging and Advanced Concepts. Norwood, MA: Artech House.

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6 Range-Doppler Inverse Synthetic Aperture Radar Processing In Chapter 4, the base algorithm for inverse synthetic aperture radar (ISAR) imaging is provided. This algorithm is based on the assumption that the target is stationary and the data are collected over a finite number of stepped look angles. In real scenarios, however, the target is usually in motion and therefore, the aspect diverse data can only be collected if the target’s motion allows different look angles to the radar during the coherent processing time of the radar. The radar usually sends chirp (linear frequency modulated [LFM]) pulses or stepped frequency continuous wave (SFCW) pulses to catch different look angles of the target. After the radar receiver collects the echoed pulses from the target, the ISAR image can only be formed in the two-dimensional (2D) range-Doppler space since the radar line of sight (RLOS) angle values with respect to target axis are unknown to the radar. This phenomenon will be explained in Sections 6.2–6.4. In this chapter, we will examine the ISAR imaging techniques for real-world scenarios when the target is not stationary with respect to radar and the Doppler frequency shift-induced backscattered data are collected by the radar. In particular, commonly used ISAR waveforms, namely the chirp (LFM) and the SFCW pulse waveforms, are utilized. The 2D range-Doppler ISAR imaging algorithms that employ these waveforms are presented.

6.1

Scenarios for ISAR

As thoroughly mentioned in Chapter 4, the ISAR provides an electromagnetic (EM) image of the target that is moving with respect to radar. The backscattered signal at the radar receiver is processed such that this signal is transformed to time (or range) and Doppler frequency (or cross-range). The time (or range) processing is accomplished by utilizing the frequency bandwidth of the radar pulse such that the points in the range (or RLOS) direction can be resolved. The movement of the target with respect to radar provides Doppler frequency shifts as the target moves Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms: With Advanced SAR/ISAR Imaging Concepts, Algorithms, and MATLAB Codes, Second Edition. Caner Özdemir. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

6.1 Scenarios for ISAR

and therefore the radar can collect the scattering from the target. The Doppler frequency analysis makes it possible to resolve the points along the cross-range axis, which is defined as the axis perpendicular to the RLOS direction. In real-world applications, the target can be aerial, such as an airplane or helicopter, or ground or sea based, such as a ship or a tank. In most scenarios, aerial targets are usually imaged with the help of a ground-based radar, whereas ground/ sea-based platforms are usually imaged via an airborne radar.

6.1.1 Imaging Aerial Targets via Ground-Based Radar As illustrated in Figure 6.1, this case represents when the radar is stationary on the earth’s surface and the target is an aerial one that has a general motion with respect to radar. As mentioned in previous chapters, the range or the line-of-sight resolution is achieved by using an adequately wide frequency band. If the target is rotating, the angular diversity of the target can be readily constituted between the received pulses. When the target is not rotating and is moving straight, as illustrated in Figure 6.1, its motion can be devised into radial translation motion, the motion along the RLOS axis, and tangential motion, the motion along the axis that is perpendicular to the RLOS. If this is the case, angular diversity of the target is realized for a longer time as the target’s tangential motion produces Doppler frequency shifts slower than the case of a rotating target. If the target’s distance from the radar is R and moving with a speed of v as depicted in Figure 6.1, it has a tangential speed of. vt = v sin φ. Therefore, the corresponding angular rotational speed becomes equal to ω=

νt R

61 Vt V

ϕ Vr R

Radar

Figure 6.1 For the ISAR operation, the aspect diversity is constituted target’s rotational and/or tangential motion with respect to radar.

307

308

6 Range-Doppler Inverse Synthetic Aperture Radar Processing

If the coherent integration time, also called the coherent processing time, the image frame time, or the dwell time, is T, the total angular width seen by the radar is Ω = ωT

62

How this angular width produces the required frequency Doppler shift will be presented in Section 6.3 together with the associated signal processing to resolve the points along the cross-range dimension. The aerial targets usually move on a straight path and rarely make rotational movements. Therefore, the necessary angular diversity required for a possible ISAR image can be obtained by the target’s tangential motion with respect to radar as depicted in Figure 6.1. Table 6.1 lists some values of angular width for different range distances and different tangential speed values for an aerial target. Noticing the fact that most jet fighters have a typical speed of about 800–900 km/h, Case #1 in Table 6.1 corresponds to the case where an airplane is flying mostly in the tangential direction. If the radar is 13 km away, the radar observes a rotational angular speed of 0.73 /s according to Eq. 6.1. Considering a nominal coherent integration time of 3 s, the radar look-angle width of target becomes 2.2 , which is good enough to form a good ISAR image. As listed in Case #2, however, the tangential speed of the target can be relatively smaller in some cases such that the look-angle width happens to be much lower than 1 . For this scenario, a good quality ISAR image may not be possible since the scattering centers on the target along the cross-range direction may not be resolved with such a small look-angle width. If the target is much closer, as listed in Cases #3 and #4, the integration time should be chosen accordingly to have a logical value for the angular width to be seen by the radar. Angular

Table 6.1 Angular speed and the total look-angle width observed by radar for a target that has a motion component in the translational direction.

Case ID

Target’s distance (R) (km)

Target’s tangential speed (vt) (km/h)

Corresponding angular rotational speed (ω) ( /s)

Integration time (T) (s)

Total angular width seen by radar (Ω) ( )

1

13

600

0.73

3

2.2

2

13

40

0.05

3

0.15

3

4

600

2.39

1

2.39

4

4

40

0.80

4

3.2

5

4

600

2.39

4

9.56

6.1 Scenarios for ISAR

widths between 2 and 7 are practical to get a fast ISAR image. If the integration time is not taken into account, as in Case #5, where the total angular width seen by radar becomes as big as 10 , the resultant ISAR image may have unwanted motion effects such as blurring and defocusing due to the fact that the scattering centers on the target may occupy several range bins during the integration time of radar. Furthermore, the use of discrete Fourier transform (DFT) and therefore fast formation of the ISAR image will not be possible since the small-angle approximation will not be valid for this case. It is also important to note that the target may maneuver during the integration time of the radar such that it may yaw, roll, or pitch while progressing at the same time. If this is the case, the target’s rotational motion with respect to radar would be mostly governed by the target’s rotational motion in its own axis. For such cases, the target’s look-angle width will be much wider when compared with the case shown in Figure 6.1. Therefore, very small integration time values will be sufficient to construct a good quality ISAR image for such scenarios. In most real-world applications, the target’s motion parameters such as translational velocity, translational acceleration, rotational velocity, and rotational acceleration are unknown to the radar. Furthermore, a target’s initial angular position with respect to RLOS is also an unidentified quantity. Therefore, the appropriate cross-range indexing in meters could not be possible in most cases. For this reason, the resultant ISAR image may not be displayed on the range-cross-range plane, but on the time-Doppler or range-Doppler plane.

6.1.2 Imaging Ground/Sea Targets via Aerial Radar When it comes to the aerial radar, the main application of ISAR imaging is to identify and/or classify sea or ground platforms such as tanks, ships, and vessels. Since both the target and the radar are in motion, in this case, the analysis and the processing become more complex due to the fact that the radar’s motion with respect to the target provides additional Doppler shift in the phase of the received signal. The problem of directing the radar antenna’s beam toward the target, that is, tracking, is another issue that needs to be controlled, which is not an easy task most of the time. Therefore, effective target tracking systems are essential on the radar site for a successful ISAR application. It is also worthwhile to mention that the propagation features of the air medium also play an important role in collecting a reliable received signal. The propagation characteristics of EM waves in different weather conditions (foggy, rainy, stormy, snowy, etc.) differ from the ideal case of a calm air situation. The atmospheric noise, background noise, and the radar platform’s electronic noise itself may influence the quality of the received signal as well.

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6 Range-Doppler Inverse Synthetic Aperture Radar Processing

(a) Radar

Pitch motion ISAR image (side view)

(b) Radar

Yaw motion ISAR image (Top view)

(c) Radar Roll motion

ISAR image (Front view)

Figure 6.2 Resulting 2D ISAR images for (a) pitching, (b) yawing (turning), and (c) rolling platform.

While the main source of Doppler frequency shift is due to the target’s rotational and/or tangential motion with respect to radar for aerial targets, the Doppler shift caused by the tangential motion of ground/sea targets is generally quite small since the speeds of these platforms are relatively much smaller. The major source of Doppler frequency shift for such targets is due to the targets’ rotational motion about its own axis, that is, yawing, rolling, and pitching. As demonstrated in Figure 6.2, yaw, roll, and pitch motions of the platform can produce the required angular variation during the coherent integration time (or illumination period) of the radar. A ship’s yawing, rolling, or pitching motions are usually caused by the wave motions for different sea-state conditions. In oceanography, the term sea state is commonly used to describe the general condition of the free surface on a large body of water with respect to wind waves and swell at a certain location and moment (en.wikipedia.org/wiki/Sea_state). The sea-state code index ranges from 0 (calm) to 9 (severe or phenomenal) depending on the wave height. Sea-state values of 3–4 correspond to slight to moderate sea conditions and represent wave heights from 0.5 to 2.5 m. Pitch is the rotational motion of the ship about the transverse (side-to-side) axis. Pitch motion primarily depends on the sea-state condition and the length between the perpendiculars. Longer ships tend to allow smaller pitch angles. Typical pitch

6.1 Scenarios for ISAR

angles are in the range of 1 –2 for a sea-state code of 4 and can be as large as 5 – 11 for a sea-state code of 8 (Doerry 2008). Yaw is the rotational motion of the ship about the vertical (up-down) axis and is caused by temporary bearing changes. Maximum yaw angle for a ship can be as large as tens of degrees. The period of yaw motion is generally equal to wave period (Doerry 2008). Roll is the rotational motion of the ship about the longitudinal (front-back) axis. The sea-state condition, frequency of the hitting wave, and the ship’s righting arm curve are the most important parameters that influence the maximum value of the roll angle. A typical value for the maximum roll angle is a few degrees for sea-state codes of 3–4. However, this value can jump to tens of degrees for a sea-state code of 8. While the range resolution is achieved by multifrequency sampling of the platform’s backscattered echo, the cross-range (or the Doppler frequency shift) resolution is achieved by multiangle sampling of the received signal. The corresponding 2D ISAR images for a pitching, yawing, and rolling platform are illustrated in Figure 6.2a–c, respectively: In Figure 6.2a, the platform is performing a pitch motion so that the radar collects the backscattered returns from the platform for different look angles of elevation. This elevation angle diversity provides the spatial resolution in the altitudinal direction. Also, utilizing the frequency bandwidth of the received pulses provides range resolution along the longitudinal (or the range) axis. As a result, a 2D ISAR image that shows the side view of the platform can be obtained. The case in Figure 6.2b provides a different ISAR image of the target. As the target is performing a yaw motion, radar collects echo signals from the platform for different azimuth look angles. This data setup makes it possible to resolve different points in the cross-range direction. Similar to the case in Figure 6.2a, frequency diversity of the transmitted signal makes it possible to resolve different points in the longitudinal (or range) direction so that we can obtain the 2D ISAR image as if the platform is being viewed from the top (or bottom). If the case in Figure 6.2c is considered, the roll motion of the platform makes it possible to resolve points in the altitudinal direction as similar to the case in Figure 6.2a. Since the platform is rotated by 90 in azimuth when compared with the first case, the frequency diversity provides range resolution in the beam direction of the ship platform as seen in Figure 6.2c. Therefore, the corresponding 2D ISAR image shows the front (or back) view of the target. In most real-world applications, the target’s position with respect to radar and the target’ s axial motion with respect to radar are random, as shown in Figure 6.3. If this is the situation, the target’s ISAR image is displayed on the 2D projection plane where the range axis is the RLOS axis, and the cross-range axis is the same direction as the target’s maneuver (pitch, yaw, or roll) axis, which should also be perpendicular to the range axis as demonstrated in Figure 6.3.

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6 Range-Doppler Inverse Synthetic Aperture Radar Processing

ISAR image grid Radar

Pitching target

Crossrange (or dopple r cells

Range c

ells

Range- Doppler ISAR image

Doppler

Range

Figure 6.3

6.2

The formation of the ISAR grid for a maneuvering platform.

ISAR Waveforms for Range-Doppler Processing

In real scenarios of ISAR, images of various platforms including airplanes, helicopters, ships, and tanks are usually formed by collecting the multifrequency multiaspect received signals from these targets with one of the following popular waveforms: 1) Stretch or LFM or chirp pulse train 2) SFCW pulse train These waveforms were already listed and studied in Chapter 2, Sections 2.6.5 and 2.7. Next, we are going to revisit chirp pulse train and SFCW pulse train waveforms and their uses in ISAR range-Doppler processing.

6.2.1

Chirp Pulse Train

A common radar pulse train that consists of a total of N chirp pulse waveforms is shown in Figure 6.4. Here Tp is the pulse duration or pulse length, TPRI is the pulse

6.2 ISAR Waveforms for Range-Doppler Processing Transmitted signal

Tp

TPRI

Time Pulse no.1

Pulse no.2

Pulse no. N

Received signal

Returned pulse no.1

Returned pulse no.2

Returned pulse no.N

Time

Figure 6.4 The chirp pulse train is utilized in range-Doppler processing of ISAR: (up) transmitted waveform, and (below) received waveform.

repetition interval (PRI), and T is the dwell time, often called the coherent integration time, and is determined by T = N T PRI = N PRF,

63

where PRF is the pulse repetition frequency. To avoid ambiguity in range, every returned pulse should arrive at the radar before the next pulse is transmitted. If the target is at R distant from the radar, therefore, the minimum value of the PRI should be as the following to avoid the ambiguity in range determination: T PRImin =

c , 2R

64

which means that the PRF should be always less than the maximum value: PRFmax =

2R c

65

If the target is at the range of 30 km, for example, the PRF should be less than 200 μs to avoid ambiguity in the range. The variation of the frequency within a chirp provides the necessary frequency bandwidth to resolve the points along the range dimensions. The bandwidth of a chirp pulse is selected according to the required range resolution as B=

c , 2 Δr

66

313

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6 Range-Doppler Inverse Synthetic Aperture Radar Processing

where Δr is the desired range resolution. The instantaneous frequency of a single chirp pulse waveform is given by f i = f o + K t;

0 ≤ t ≤ Tp

67

where fo is the starting frequency of the chirp and K is the chirp rate. Therefore, the chirp rate of the LFM pulse should be selected in the following way to provide the necessary bandwidth for range processing: K=

B Tp

68

Coherent integration time for the chirp pulse radar is as given in Eq. 6.3.

6.2.2

Stepped Frequency Pulse Train

SFCW pulse is one of the most frequently used radar waveforms in ISAR imaging. A detailed explanation of the SFCW is given in Chapter 2, Section 2.6.3 and its usage in range – cross-range ISAR imaging is demonstrated in Chapter 4. Here, we will show the usage of SFCW pulse train in range-Doppler ISAR imaging. In the SFCW pulse train operation, a total of M identical bursts of N pulses are generated to be transmitted toward the target as demonstrated in Figure 6.5. Each pulse in each burst is composed of a single frequency sinusoidal wave. The frequency of the first pulse is fL, and the frequencies of the subsequent pulses are incremented by Δf such that the nth pulse in any burst is given by f n = f L + n − 1 Δf

69

Therefore, the frequency of the Nth pulse in any burst is fN = fL + N −1 ≜ fH

Δf

6 10

Therefore, the frequency bandwidth of the stepped frequency pulse train is then equal to B = N Δf

6 11

The total time passed for one burst is T burst = N T PRI N , = PRF

6 12

Transmitted signal TPRI Tp TPRI

Burst no.1

(N–1)·TPRI

N·TPRI Time

Pulse no.1

Pulse no.2 (N+1)·TPRI

Burst no.2

Pulse no.N (2N–1)·TPRI

2N·TPRI Time

Pulse no. N+1

Burst no.M

Pulse no.N+2

((M-1)-N+1)-TPRI

Pulse no.2N

(M·N–1)·TPRI

M·N·TPRI Time

Burst number Pulse frequency

Pulse no. (M–1)·N+1

f1 = fL

Pulse no. (M–1)·N+2

f2 = fL + Δf

Pulse no. M·N

fN = fL + (M–1)·Δf = fH

Figure 6.5 Representation of stepped frequency transmitted signal of M bursts each having N stepped frequency waveforms.

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6 Range-Doppler Inverse Synthetic Aperture Radar Processing

and the coherent integration time or the dwell time for the M burst is equal to T = M T burst = M N T PRI MN = PRF

6 13

To avoid ambiguity in range determination, the pulse response should arrive before the next pulse is transmitted. Therefore, the maximum PRF value is the same as in the case of chirp pulse illumination and given as in Equation 6.5.

6.3

Doppler Shift’s Relation to Cross-Range

Let us start with the analysis of Doppler shift caused by target motion by assuming that the target only has a rotational motion component as illustrated in Figure 6.6. The target has a rotational motion with an angular velocity of ω. The goal here is to find the Doppler shift at the received signal caused by the rotational movement of a point scatterer, P, on the target. As illustrated in Figure 6.6, the point P(xp, yp) is located at Rp away from the rotation axis of the target. Then the tangential velocity of the point P is equal to v = Rp ω

6 14

The radial velocity of this scattering center along the radar’s line of sight direction is represented as vr and can be found by using the two similar triangles in the figure as vr =

yp v Rp Returned signal @ (fi + fDP)

Transmitted signal @ fi

6 15 ω Vr V

Scattering center P(xp, yp) yp

Rp

xp

y

x Origin (phase center)

Figure 6.6 Target’s rotational motion causes Doppler shift in the frequency of the received pulses for a point on the target.

6.3 Doppler Shift’s Relation to Cross-Range

Substituting Eq. 6.15 into 6.16 will yield yp Rp ω Rp

vr =

6 16

= yp ω, which clearly states that the “radial velocity” of a point scatterer on the target is directly related to its “cross-range value.” Now, we can easily calculate the Doppler frequency shift caused by radial velocity of the point scatterer as 2vr fi c , 2ω = yp λi

f DP =

6 17

where fi and λi are the instantaneous frequency and the corresponding wavelength of the chirp pulse waveform, respectively. Therefore, the EM wave returned from point P will have a Doppler shift with an amount of f DP as calculated in Eq. 6.17. This result clearly demonstrates that the Doppler frequency shift caused by the motion of the point scatterer is directly proportional to its cross-range position, yp. Therefore, if the returned signal from all the scattering centers is plotted in the Doppler shift domain, the resulting plot is proportional to the target’s cross-range profile. If the angular speed of the target is correctly estimated, then the cross-range profile of the target can be correctly labeled.

6.3.1 Doppler Frequency Shift Resolution For the general case, the Doppler frequency shift of a point at (x, y) on the target is equal to fD =

2ω y λi

6 18

Then, the resolution for the Doppler frequency shift, ΔfD, can be readily found as ΔfD =

2ω Δy λi

6 19

As found earlier in Chapter 4 (Equation 4.47), the cross-range resolution is given by Δy = (λ/2)/Ω where Ω is the total angular span or the total viewing angle of the target by the radar. Ω can be easily related to angular velocity, ω, and the total viewing time of the target (or the dwell time), T, as Ω=ωT

6 20

317

318

6 Range-Doppler Inverse Synthetic Aperture Radar Processing

Then the cross-range resolution becomes equal to Δy =

λi 2 ωT

6 21

Finally, ΔfD can be determined in terms of T by substituting Equation 6.21 into 6.19 as 2ω λi λi 2ωT 1 = T

ΔfD =

6 22

Therefore, the resolution for the Doppler frequency shift is the inverse of the total viewing time which is an expected result according to the Fourier theory.

6.3.2

Resolving Doppler Shift and Cross-Range

Let us consider the geometry seen in Figure 6.7 where the target is rotating with a rotational velocity of ω. The origin of the geometry is Ro away from the radar. Assuming that the target is at the far field of the radar, the phase of the returned signal from the nth scattering center located at P(xn, yn) on the target will have the form of φ = exp − j2kRn t ,

6 23

where Rn(t) is the radial distance of the point scatterer from the radar and can be written as Rn t = Ro + x n + vrn t

6 24 y ω P(xn, yn) Vr

Radar

Rp

Ro O

Figure 6.7

Geometry for Doppler processing of a rotating target.

V x

6.4 Forming the Range-Doppler Image

Here, vrn corresponds to the radial velocity of the nth point scatterer. Then the phase in Equation 6.24 can be rewritten as the following: φ = exp − j2k Ro + x n

exp − j2kvrn t

6 25

Note that the first term is constant with respect to time and only the second term is time varying. Substituting vrn = λ f Dn 2 into Eq. 6.25 will yield φ = exp − j2k Ro + x n = exp − j2k Ro + x n

exp − j2

λ f Dn 2

2π λ

t

,

6 26

exp − j2π f Dn t

where fDn is the Doppler frequency shift for the nth point scatterer. It is clear that there exists a Fourier transform (FT) relationship between time variable, t, and the Doppler frequency shift variable, fDn. Therefore, taking the inverse FT (IFT) of the received signal with respect to time, the Doppler frequency shift for the nth point scatterer can be easily resolved. As listed in Equation 6.18, cross-range is proportional to the Doppler frequency shift. If the rotational velocity ω is predicted, the cross-range dimension yn can also be correctly determined.

6.4

Forming the Range-Doppler Image

Let us assume that the target in Figure 6.7 is modeled as if it contains K point scatterers located at (xn, yn) where n runs from 1 to K. Then, the received signal can be approximated as the following: K

An exp − j2k Ro + x n

E s k, t

exp − j2π f Dn t ,

6 27

n=1

where An is the complex magnitude of the nth scattering center. Taking the origin of the target as the phase center of the geometry, the phase term exp − j2kRo can be suppressed to get K

An exp − j2π

Es f , t n=1

2f xn c

exp − j2π f Dn t

6 28

Taking the 2D IFT of the backscattered signal with respect to (2f/c) ≜ α and (t), we get K

Fα−, 1t E s f ,t

An Fα− 1 exp − j2παxn

Ft− 1 exp −j2πfD −fDn

n=1 K

=

An δ x − xn δ fD −fDn n=1

≜ ISAR x, fD 6 29

319

320

6 Range-Doppler Inverse Synthetic Aperture Radar Processing −1 where F α,t is 2D IFT operation with respect to α and t. Similar manner, F α− 1 and

F t− 1 are the 1D IFT operations with respect to α and t, respectively. This result clearly shows that the resulting 2D image data are on the range-Doppler frequency plane. As is obvious from the above analysis, the range components of the scattering centers, xns, are easily resolved as the same way that we did in the conventional ISAR imaging. The other dimension is the Doppler frequency axis that is proportional to the cross-range axis. Provided that the target’s rotational velocity is known or estimated correctly, the transformation from Doppler frequency space to crossrange space can be performed by applying the following transformation formula: y=

λc f , 2ω D

6 30

where λc is the wavelength corresponding to the center frequency. After this transformation, the cross-range components, yns, are also resolved, and the ISAR image on range cross-range plane can be succesfully generated.

6.5

ISAR Receiver

Most ISAR systems are designed for either chirp or SFCW pulse train waveforms. Some systems utilize other stretch waveforms as well (Curlander and McDonough 1991). The ISAR receiver circuitry is, therefore, designed according to the type of illumination waveform.

6.5.1

ISAR Receiver for Chirp Pulse Radar

LFM pulse train or chirp pulse train waveform is widely used in SAR and ISAR applications, thanks to its easy applicability. The general block diagram for chirp pulse ISAR receiver is shown in Figure 6.8. The receiver processes the received signal pulse-by-pulse such that range profile corresponding to each pulse is obtained. Doppler frequency shifts for each range bin are determined with the help of FT operation so that the final 2D range-Doppler image of the target is obtained. Let us analyze the ISAR receiver in Figure 6.8 in more detail. First, the chirp pulse return from the target is collected and fed to the intermediate frequency (IF) amplifier such that the signal level is amplified at the IF stage for further processing. Then, the matching filtering is applied to compress each of the incoming pulses. As demonstrated in Chapter 3, Section 3.5.3.1, the output of the matched filter (or the pulse compressor) is the compressed version of the received pulse. The result is nothing but the 1D range profile of the target for that particular pulse. At this point, N range profiles corresponding to N pulse returns are produced. Then quadrature detection (QD) follows to detect the amplitude and the phase

6.5 ISAR Receiver Received signal (N pulses) IF AMP

Pulse compressor

N range profiles

I

Sampler + A/D convertor

Quadrature detector

N range profiles, each has M range + bins

Σ

Q

+ Sampler + A/D convertor

Range alignment

1D-FT along range-bins 2D data set of N range profiles by M Doppler frequencies 2D RangeDoppler image

Figure 6.8 ISAR receiver block diagram for chirp pulse illumination.

information of the returned signal at the baseband frequencies. The details of QD will be given in Section 6.6. The entire signal processing schemes up to this point are analog. As the next step, the inphase (I) and quadrature (Q) pairs at the output of the QD are sampled and digitized by using samplers and analog-to-digital (A/D) converters such that each range profile is digitized to M range cells (or range bins). Then, the digitized range profiles of length M are put side-by-side to align the range positions such that each range cell has to correspond to the same respective range positions along the target. Otherwise, image blurring occurs due to this range walk, the movement of range positions from profile to profile. The process of range alignment will be explained in Section 6.7. After compensating the range walk in the 2D data set, 1D discrete FT (or DFT) can be applied along azimuthal time instants to transform the returns to Doppler frequency space. The resulting 2D matrix is the N-by-M range-Doppler ISAR image of the target. In Section 6.8, we will present the detailed processes and the algorithm for range-Doppler ISAR imaging.

6.5.2 ISAR Receiver for SFCW Radar SFCW signal is also one of the commonly used waveforms in radar imaging because it can provide digital data for fast processing for a reliable SAR/ISAR image. The SFCW transmitter sends out M repeated bursts of stepped frequency

321

322

6 Range-Doppler Inverse Synthetic Aperture Radar Processing Received signal (M burts of N pulses) IF AMP

I

Sampler + A/D convertor +

Quadrature detector

Σ Q

+

1D-FT along stepped frequencies

Sampler + A/D convertor

N range profiles, each has M range bins

Range alignment

2D RangeDoppler image

Figure 6.9

2D data set of N range profiles by M doppler frequencies

1D-FT along bursts

ISAR receiver block diagram for stepped frequency radar illumination.

waveforms. In each burst, a total of N stepped frequency waveforms are transmitted. A common block diagram for a SFCW-based ISAR receiver is illustrated in Figure 6.9. The receiver collects the total scattered field data that are composed of M bursts with N stepped frequency pulses. This received signal is fed to an IF amplifier such that the signal level is amplified at the IF stage for further processing. Afterward, QD is used to gather the amplitude and the phase information of the returned signal around the baseband frequencies. Then, the I and Q pairs at the output of the QD are sampled and digitized using samplers and A/D converters such that a matrix of M-by-N is constituted for M bursts that corresponds to M azimuthal time instants and N stepped frequencies. Taking 1D IFT along the stepped frequencies will yield a total of M different range profiles, each having N range bins. If the radial velocity of the target is not small and/or the PRF is not high enough (so the dwell time is long), the range profiles may not line up and alignment of range profiles may be required before processing in the azimuth direction, as will be clarified in Section 6.7. If the range arrangement is not made, the ISAR image becomes blurred due to this movement of range positions from profile to profile. For this case, range cells should be aligned for the whole 2D data set. Then, 1D IFT along bursts (or azimuthal time instants) will transform the data into the Doppler

6.6 Quadrature Detection

frequency shift domain. The resulting 2D matrix is the ISAR image of the target in range-Doppler frequency domain.

6.6

Quadrature Detection

The process of QD is commonly used in radar systems to acquire received signal phase information relative to the transmitted signal carrier. QD can also be regarded as the mixing operation that carries the received signal to the baseband to obtain the amplitude and the phase information of the received signal in the form of quadrature, that is, I and Q components. The block diagram of QD is shown in Figure 6.10. The QD receiver is usually applied after the pulse compression filter in SAR/ISAR imaging as shown in Figure 6.8. The input signal to the QD receiver is fed to the inphase (I) channel and 90 delayed with respect to the reference signal at the local oscillator (LO) or the quadrature (Q) channel. Let us assume that the transmitted signal has the form of a simple sinusoid, s t = Ai cos 2π f i t ,

6 31

where fi is the instantaneous frequency within the bandwidth of the transmitted signal. Then, the received signal from a point scatterer that is Ro away from the radar has the following form: E t = Ai cos 2π f i t −

2Ro c

,

6 32

where Ai is the backscattered amplitude. The processing in I- and Q-channels are given in detail below. Quadrature receiver I – channel LPF

Received signal

Local oscillator

~

A/D convertor

c(t)

+

Σ

π/2 filter

+ Q – channel LPF

Figure 6.10

Block diagram for quadrature detection.

A/D convertor

QR output

323

324

6 Range-Doppler Inverse Synthetic Aperture Radar Processing

6.6.1

I-Channel Processing

The received signal is multiplied with the pilot signal, c(t), that is generated by the LO: c t = Bi cos

6 33

j2π f i t

The multiplier output in the I-channel yields the following: E t c t = Ai Bi cos 2π f i t − A i Bi = 2

2Ro c

cos

2Ro cos 4π f i t − 2π f i c

j2π f i t

2Ro + cos 2π f i c

6 34

After low pass filtering, the first term at (2fi) frequency is filtered out, and the second term, that is, the time-invariant (or DC) component, will stay as s I = C i cos 2π f i

6.6.2

2Ro c

6 35

Q-Channel Processing

The Hilbert (or −π/2) filter puts a π/2 radian (or 90 ) delay to the pilot signal as c t = Bi cos

j2π f i t −

π 2

6 36

The multiplier in the Q-channel produces the following output: E t c t = Ai Bi cos 2π f i t − A i Bi 2 A i Bi = 2 =

2Ro c

cos

2Ro π − c 2 2Ro π cos 4π f i t − 2π f i − c 2 cos 4π f i t − 2π f i

j2π f i t − + cos + sin

π 2 2Ro π + c 2 2Ro − 2π f i c − 2π f i

6 37 The low pass filtering operation filters out the first term and keeps the second DC term as 2Ro c 2Ro = − C i sin 2π f i c

s Q = Ci sin

− 2π f i

6 38

6.6 Quadrature Detection

Both channels are processed with an A/D converter such that s[I] and s[Q] signals are digitized for M different frequencies (see Figure 6.10). Afterward, baseband I and Q signals are summed to give the final output as sout f t = s I + s Q 2Ro 2Ro − jC i sin 2π f i c c 2Ro − j2π f i c

= C i cos 2π f i = C i exp

6 39

The phase of this output signal has the delay of (2Ro/c) compared to the transmitted signal which obviously shows the location of the scatterer. The amplitude of this output signal is directly related to the backscattering field amplitude of the scatterer. To be able to employ digital processing of the received signal, the data should be sampled and digitized with the help of an AD converter. The range resolution has already been defined as c 6 40 Δr = 2B If the frequency bandwidth is to be digitized to a total of M discrete frequencies, then Δf =

B , M

6 41

which is also equal to c Δf = , 2Rmax

6 42

where Rmax = M Δr is the maximum range or unambiguous range extent seen by the radar. Therefore, the frequency variable fi can be replaced by the following discrete variable: f i = f o + i Δf ;

i = 0, 1, 2, …, M − 1

6 43

Here, fo denotes the initial or the start-up frequency. The output of the AD converter is the digitized version of the N range profiles, each having a total of M range cells (or range bins). Since the target is in motion in general, normally, Doppler shifts occur between the received pulses. The target may have translational motion and/or rotational motion with respect to radar. In any case, any cross-range point on the image will produce Doppler shifts along the received pulses. After digitizing the whole received signal, the data can be represented in a 2D form such that the time response of each chirp (or the range profile) is plotted in a column for every pulse received (see Figure 6.11a).

325

6 Range-Doppler Inverse Synthetic Aperture Radar Processing Received pulses Pulse 1 Pulse 2

Doppler Doppler freq. 1 freq. 2

Pulse N

Range cell 1

E11

E12

E1N

Range cell 2

E21

E22

E2N

EM1 EM2

EMN

Doppler

Doppler freq. N

1D-FT 1D-FT

Range bins /

Range cell 1

l11

l12

l1N

Range cell 2

l21

l22

l2N

lM1

lM2

lMN

Range

326

Range cells

Range cell M

Figure 6.11 signal.

6.7

1D-FT Range cell M

Formation of the range-Doppler ISAR image via digital processing of received

Range Alignment

At the end of QD and before applying the azimuth compression to each bin of the range profiles, alignment of range is necessary in most cases. This alignment is applied to compensate for the phenomenon called range walk. In the typical ISAR setup, range walk is mainly induced by the target’s radial translational motion with respect to radar. The change of range value from profile to profile results in one scatterer to “walk” among the range bins. For constant-velocity targets, the translational velocity of the target can be estimated and the alignment of range profiles can be performed accordingly. In real scenarios, however, the target’s motion can be more complex such that it may contain both radial and the tangential components of higher orders. If this is the case, the range alignment is not simple; many motion compensation algorithms have been developed to solve this problem (Kirk 1975; Chen and Andrews 1980; Wu et al. 1995; Itoh et al. 1996; Xi et al. 1999). For instance, finding a prominent scatterer in a range profile and tracking it among the other range profiles can sometimes be effective (Wang et al. 1998; Küçükkiliç 2006). Various motion compensating algorithms, including the prominent scattering technique, will be covered in Chapter 8. In some cases when the target’s velocity is low and the dwell time is short, the change in the range may stay within the range resolution such that the “walk” settles within a range cell. Therefore, no range correction is needed. In some other cases, when the integration time is sufficiently short such that the target’s motion can be approximated to a constant radial velocity, then an effective range alignment method can be applied to attain walk-free range profiles. When all the range profiles are aligned, the Doppler processing can then be reliably applied by inverse Fourier transforming the collected data for every range cell.

6.8 Defining the Range-Doppler ISAR Imaging Parameters

This operation provides the final ISAR matrix of the target in the 2D rangeDoppler frequency plane as illustrated in Figure 6.11b.

6.8 Defining the Range-Doppler ISAR Imaging Parameters Although the SFCW-based systems are easier to implement and are preferable in some ISAR imaging applications, chirp pulse-based ISAR systems work much faster than SFCW-based systems and therefore are mostly preferred when the target is moving fast, as in the case of airplanes and fighters. Furthermore, the chirp pulse systems have the property of providing good signal-to-noise ratio (SNR) at the image output as given in Chapter 3, Section 3.5.1. Therefore, it is more reliable and applicable for real-world applications when the noise is always available and unavoidable. Next, we are going to present a general approach for the implementation steps of ISAR imaging for range-Doppler ISAR processing.

6.8.1 Image Frame Dimension (Image Extends) For an ISAR application, the ultimate goal is to get an EM reflectivity of the target; therefore, the frame of the image, that is, the dimensions of the image in the range and cross-range (or Doppler) plane, should be specified to cover the whole target (see Figure 6.12). It is always safe to select the size of the image frame to be at least two to three times larger than the actual size of target’s projection on the range and cross-range frame of [Xp Yp] to avoid aliasing.

Radar

Rm

ax

Δy

Yp

Δr

Ran g

e

Figure 6.12

Ym ax Cro ss-r a (or Dop nge pler )

Rp

Selection of image frame size and resolutions in ISAR imaging.

327

328

6 Range-Doppler Inverse Synthetic Aperture Radar Processing

6.8.2

Range and Cross-Range Resolution

The range resolution should be selected according to the target’s size in the range direction. If the range resolution is selected as Δr, the number of range points along the target will have a maximum value of N = floor

Rmax , Δr

6 44

where the “floor” function rounds its argument to the lower nearest integer. Similarly, if the cross-range resolution is selected as Δy, the number of cross-range points along the target will have a maximum value of M = floor

Y max Δy

6 45

Therefore, the values of N and M should be selected such that the target features are clearly distinguished with the selected range and cross-range resolutions.

6.8.3

Frequency Bandwidth and the Center Frequency

Based on the decided value for the range resolution, the required frequency bandwidth should be B=

c 2 Δr

6 46

For fast processing of the ISAR data, the center frequency can be conveniently selected as at least 10 times the frequency bandwidth, that is, fc 10B. Otherwise, direct integration of the ISAR integral or a polar reformatting scheme should be applied, as explained in Chapter 4, Section 4.6.2.

6.8.4

Doppler Frequency Bandwidth

As given in Equation 6.19, the Doppler frequency shift resolution of a target that is rotating with an angular velocity of ω with respect to radar is ΔfD =

2fc ω Δy c

6 47

Therefore, the total Doppler frequency bandwidth can be found as BW

fD

= ΔfD M 2fc = ω Δy M c 2fc = ωY max c

6 48

6.8 Defining the Range-Doppler ISAR Imaging Parameters

6.8.5 Pulse Repetition Frequency The Doppler frequency bandwidth shown above covers all the cross-range scatterers extending over the cross-range width of Ymax. When the chirp pulse radar operation is considered, therefore, the required PRF in order to unambiguously sample the scattered field over the cross-range width of Ymax is PRF =

1 T2

2fc ωY max c

6 49

Therefore, the minimum value of this PRF is equal to PRFmin =

2fc ωY max c

6 50

or the maximum value for the PRI PRImax =

c 2 f c ωY max

6 51

On the other hand, the minimum value for the PRI is dictated by the target’s distance from the radar’s position, as the return of the pulse should arrive before the next pulse leaves the transmitter: PRImin =

2Ro c

6 52

Therefore, the PRI should be selected between these two limit values to avoid ambiguity in the cross-range dimension. When the SFCW operation is considered, a total of N pulses are to be sent for a single burst. If the pulse width of a single frequency pulse is TP, PRF 1 = N N TP 2fc ωY max c

6 53

Therefore, minimum PRF for stepped frequency radar should be PRFmin = N

2fc ωY max c

6 54

6.8.6 Coherent Integration (Dwell) Time To achieve the selected cross-range resolution, the value for the total viewing time (or the dwell time) should be equal to the following for chirp pulse illumination:

329

330

6 Range-Doppler Inverse Synthetic Aperture Radar Processing

T = N PRI c 2 f c ωY max c , = 2 f c ωΔy =N

6 55

which in turn provides a total viewing angle width of Ω = ωT

6 56

When the SFCW is considered, coherent integration time becomes equal to T = M N PRI =M N

c 2M f c ωY max

6 57

c , = 2 f c ωΔy which is identical to the result in Eq. 6.55. Therefore, SFCW systems should be N times faster than the chirp-pulse systems to have the same dwell time value.

6.8.7

Pulse Width

For the unmodulated (or the single frequency) pulse, the requirement for the minimum value for the duration or the width of the transmitted pulse is Rmax 6 58 c On the other hand, the linear frequency modulation within the chirp pulse makes it possible to have a longer pulse width, as explained in Chapter 3, Section 3.5.3.2. Therefore, the real pulse width value, Tp, is decided by considering the amount of power to be induced on the pulse. Once Tp is decided, the compression ratio, D, can be chosen according to the following condition: T pmin =

D≤

Tp T pmin

6 59

After the compression ratio is found, the chirp rate, K, can be calculated using Equation 3.47, in Chapter 3, as K=

D T 2p

6 60

so that the chirp pulse stx t

exp j2π

f ct + K

t2 2

, t ≤ Tp 2

is formed to be transmitted toward the target.

6 61

6.9 Example of Chirp Pulse-Based Range-Doppler ISAR Imaging

6.9 Example of Chirp Pulse-Based Range-Doppler ISAR Imaging This example will demonstrate the range-Doppler ISAR image of a target that is illuminated by the chirp pulse radar, as in the scenario illustrated in Figure 6.13. The radar’s location is taken as the origin and the target’s center is assumed to be located at some (xo, yo) point on the 2D coordinate system. The target reflectivity is assumed to be characterized by a number of point scatterers of equal magnitude as their locations in the 2D Cartesian coordinate system are plotted in Figure 6.14. The target platform has a constant velocity of vx along the x-direction. A supposed scenario with the corresponding parameters listed in Table 6.2 is chosen for this example. The radar parameters of chirp pulse waveform are listed in Table 6.2. Therefore, the transmitter sends out the following chirp pulses: stx t

exp j2π

f 0t + K

t2 2

m−1

T2 ≤ t ≤ m − 1

0

T2 + Tp

elsewhere, 6 62

xo Target

vx

y0

R

Radar

Figure 6.13

The scenario for range-Doppler ISAR imaging.

331

6 Range-Doppler Inverse Synthetic Aperture Radar Processing

30

20

10 y, m

332

0

–10

–20

–30

–30

–20

–10

0

10

20

30

x, m

Figure 6.14

Table 6.2

Fictitious fighter consists of perfect point scatterers of equal reflectivity.

Target simulation parameters for chirp pulse illumination.

Parameter name

Symbol

Value

Target parameters Target’s initial position in x

xo

0m

Target’s initial position in y

yo

24 km

Target’s velocity along x

vx

120 m/s

Center frequency of chirp

fc

10 GHz

Frequency bandwidth of chirp

B

2.5 GHz

Pulse duration of single pulse

TP

Radar parameters

Chirp rate

K≜

0.4 μs Tp B

6.25 e15 s/Hz

where m = 1 : Np: pulse number variable T2 = 1/PRF ≜ PRF: pulse repetition interval. The transmitted chirp pulse waveform is plotted in Figure 6.15. The received signal from a point scatterer at Ro away from the radar will be in the form of the following:

6.9 Example of Chirp Pulse-Based Range-Doppler ISAR Imaging

Transmitted signal

2 1.5 1 0.5 0 –0.5 –1 –1.5 –2 –0.2

–0.15

–0.1

–0.05

0

0.05

0.1

0.15

Time, μs

Figure 6.15

Transmitted chirp pulse waveform.

fo

srx t

exp

j2π +K

2R c t − 2R c 2

t−

0

2R ≤ t ≤ m − 1 T2 c 2R + + Tp c elsewhere,

m − 1 T2 + 2

6 63 where R=

x 0 − vx t

2

+ yo 2

1 2

6 64

As is obvious from Equation 6.64, the path of the EM wave, R is changing with time because of the motion of the target. Therefore, frequency of the received signal, srx, will have some Doppler shift components as expected. It is also important to note that there will be some additive noise due to the clutter from the scene, atmospheric effects, and from the electronic circuitry of the radar. Therefore, the total received signal at the receiver, grx(t), will be the summation of the received signal plus the additive noise as grx t = srx t + n t

6 65

333

334

6 Range-Doppler Inverse Synthetic Aperture Radar Processing

It is preferable to have as high an SNR as possible; however, this cannot be achieved most of the time. Therefore, in this example, a very bad SNR value of 0.0013 (or −28.93 dB) is considered. The additive noise signal is chosen to be white Gaussian. To be able to process the received signal digitally, the received pulses should be sampled in time. Since the bandwidth of the transmitted signal is B, this corresponds to a minimum sampling time interval of ts =

1 B

6 66

Therefore, each pulse is sampled by Nsample points of Tp ts

N sample

6 67

After this sampling process, the received data can be written as a 2D matrix of Mp times Nsample. The range compression now can be applied for each digitized pulse by applying a pulse compression procedure that includes matched filtering of each pulse return with the replica of the original pulse, as explained in detail in Chapter 3, Section 3.5.1. The matched filter response of the transmitted chirp pulse is depicted in Figure 6.16. This signal is used as the frequency-domain replica of the transmitted pulse to be used in matched filtering process. After applying the

Matched filter response

0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 –3

–2

–1

0

1

Frequency, GHz

Figure 6.16

Matched filter response of the chirp pulse.

2

3

6.9 Example of Chirp Pulse-Based Range-Doppler ISAR Imaging

0

0

0.005 –5

0.01 Azimuth time, s

0.015

–10

0.02 –15

0.025 0.03

–20

0.035 0.04

–25

0.045 50

100

150

200

250

300 dB

–30

Range bins

Figure 6.17

Range compressed data with additive random noise are present.

matched filter operation, the resultant range compressed data are plotted in Figure 6.17 where the range profile for each azimuth time instant can be easily observed at different range bins. The effect of additive noise can also be observed from the image as the noise shows up as a clutter all around the image. The range profiles for different azimuth time instants are well aligned, thanks to high PRF rate of 3000. Therefore, no range alignment procedure is needed before proceeding to the azimuth compression. If the PRF was lower, dwell time, T = Mp/PRF, will be longer, and the range profiles for different received pulses may not be lined up; therefore, range alignment procedure should be employed before applying the processing in the azimuth dimension. Finally, an IFT operation is performed along the pulse index so that the points in the cross-range dimension can be resolved as they appeared in the different Doppler frequency shift values. Therefore, the final ISAR image is obtained on the range-Doppler plane as depicted in Figure 6.18 where the point scatterers on the target are resolved well in range direction and fairly well in the cross-range direction due to some finite velocity of the target along the azimuth direction. The noise in the receiver is highly suppressed, thanks to matched filtering. Although the received noise energy was about 29 dB higher than that of the received signal energy, the noise level of the image is at least 25 dB lower than the target image points; this outcome is easily observed from Figure 6.18. The range cross-range ISAR image in Figure 6.19 can only be constructed provided that

335

Range, m

6 Range-Doppler Inverse Synthetic Aperture Radar Processing

–30

0

–20

–5

–10

–10

0

–15

10

–20

20

–25 50

Figure 6.18 present.

Range, m

336

100

150 200 Doppler index

250

300 dB

–30

Range-Doppler ISAR image of the target with additive random noise is

–30

0

–20

–5

–10

–10

0

–15

10

–20

20

–25 –80

–60

–40

–20

0

20

40

60

80

dB

–30

Cross-range, m

Figure 6.19 present.

Range cross-range ISAR image of the target with additive random noise is

the angular velocity of the target is known or estimated. Then, the transformation from Doppler frequency shift axis to cross-range axis is accomplished by the formula given in Equation 6.30.

6.10 Example of SFCW-Based Range-Doppler ISAR Imaging This example will demonstrate the range-Doppler ISAR image of a target that is illuminated by the SFCW-based radar, as in the scenario illustrated in Figure 6.20. For this scenario, the target has both radial and rotational motion components. The target is assumed to have a radial translational velocity component of vr and a radial translational acceleration of ar. Furthermore, the target is

6.10 Example of SFCW-Based Range-Doppler ISAR Imaging

Pitching with Ω

Radar LOS Radially moving with vr and ar

Figure 6.20

Table 6.3

A geometry for ISAR imaging scenario.

Target simulation parameters for SFCW illumination.

Parameter name

Symbol

Value

Target parameters Target’s initial position in range

R0

4 km

Target’s radial velocity

vr

5 m/s

Target’s radial acceleration

ar

0.04 m/s2

Target’s rotational velocity

ω

1.2 /s

Starting frequency

fo

9 GHz

Frequency bandwidth

B

125 MHz

Pulse repetition frequency

PRF

35 KHz

Number of pulses

Npulse

128

Number of bursts

MBurst

128

Radar parameters

also assumed to have a rotational velocity component of ω. Both the target and the radar parameters that are used in the simulation are listed in Table 6.3. The target is assumed to be composed of perfect point scatterers of equal magnitude as the locations of these scattering centers are shown in Figure 6.21. A Matlab code (see Matlab code 6.2 at the end of this chapter) is used to calculate the theoretical backscattered electric field from the target. White Gaussian noise was also added to the calculated field to include the effect of the noise originated by any cause. An SNR value of 3.55 (or 5.50 dB) was assumed during the simulation. First, the range profiles from the target are obtained by applying the 1D IFT operation along the frequency-diverse data. The resultant range profiles for different burst indexes are plotted in Figure 6.22. Since the range profiles seem to be aligned from burst to burst, no range alignment procedure is applied. Then, the IFT operation along the bursts makes it possible to resolve the points in the cross-range direction such that they lined up in the Doppler frequency shift axis.

337

6 Range-Doppler Inverse Synthetic Aperture Radar Processing

80 60

y, m

40 20 0 –20 –40 –60 –80

Figure 6.21

–60

–40

–20

0 x, m

20

40

60

80

Target with perfect point scatterers.

0 –2 20 –4 –6

40 Burst index

338

–8 60

–10 –12

80

–14 100

–16 –18

120 –60

–40

–20

0

20

40

60

Range, m

Figure 6.22

Range profiles of the target for different burst indexes.

dB

–20

6.11 Matlab Codes

0

20 30 40

–5

Doppler index

50 60

–10

70 80 –15 90 100 –20

110 120

–25 –60

Figure 6.23

–40

0 –20 Range, m

20

40

dB

Range-Doppler ISAR image of the target.

Therefore, the resultant image is nothing but the 2D ISAR image in the rangeDoppler plane as plotted in Figure 6.23.

6.11

Matlab Codes

Below are the Matlab source codes that were used to generate all of the Matlabproduced figures in this chapter. The codes are also provided in the CD that accompanies this book.

Matlab code 6.1 Matlab file “Figure6-14thru19.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 6.14 thru 19 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the (Continued)

339

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6 Range-Doppler Inverse Synthetic Aperture Radar Processing

% same directory: % % fighter.mat clear close all %—Radar parameters————————————————————————————————————— c = 3e8; % speed of EM wave [m/s] fc = 10e9; % Center frequency of chirp [Hz] BWf = 2.5e9; % Frequency bandwidth of chirp [Hz] T1 =.4e-6; % Pulse duration of single chirp [s] %—target parameters———————————————————————————————————— Vx = 120; % radial translational velocity of target [m/s] Xo = 0e3; % target's initial x coordinate wrt radar Yo = 24e3; % target's initial y coordinate wrt radar Xsize = 180; % target size in cross-range [m] Ysize = 60; % target size in range [m] %——set parameters for ISAR imaging—————————————————————— % range processing Ro = sqrt(Xo^2+Yo^2); % starting range distance [m] dr = c/(2*BWf ); % range resolution [m] fs = 2*BWf; % sampling frequency [Hz] M = round(T1*fs); % range samples Rmax = M*dr; % max. range extend [m] RR = -M/2*dr:dr:dr*(M/2-1); % range vector [m] Xmax = 1*Xsize; % range window in ISAR [m] Ymax = 1*Ysize; % cross-range window in ISAR [m] % Chirp processing U = Vx/Ro; % rotational velocity [rad/s] BWdop = 2*U*Ysize*fc/c; % target Doppler bandwith [Hz] (Continued)

6.11 Matlab Codes

PRFmin = BWdop; % min. PRF PRFmax = c/(2*Ro); % max. PRF N = floor(PRFmax/BWdop);% # of pulses PRF = N*BWdop; % Pulse repetition frequency [Hz] T2 = 1/PRF; % Pulse repetition interval T = T2*N; % Dwell time [s] (also = N/PRF) % cross- range processing dfdop = BWdop/N; % doppler resolution lmdc = c/fc; % wavelength at fc drc = lmdc*dfdop/2/U; % cross-range resolution RC = -N/2*drc:drc:(N/2-1)*drc; % cross-range vector %—load the coordinates of the scattering centers on the fighter———— load fighter %—Figure 6.14—————————————————————————————————————————— h = figure; plot(-Xc,Yc,'o', 'MarkerSize',6,'MarkerFaceColor',[1 0 0]);grid; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); grid minor axis([-35 35 -30 30]) xlabel('\itx, m'); ylabel('\ity, m'); %— sampling & time parameters —————————————————————————— dt = 1/fs; % sampling time interval t = -M/2*dt:dt:dt*(M/2-1); % time vector along chirp pulse XX = -Xmax/2:Xmax/(M-1):Xmax/2; F = -fs/2:fs/(length(t)-1):fs/2; % frequency vector slow_t = -M/2*dt:T2: -M/2*dt+(N-1)*T2; %— transmitted signal ————————————————————————————————— Kchirp = BWf/T1; % chirp pulse parameter s = exp(1i*2*pi*(fc*t+Kchirp/2*(t.^2)));% original (Continued)

341

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6 Range-Doppler Inverse Synthetic Aperture Radar Processing

signal sr =exp(1i*2*pi*(fc*t+Kchirp/2*(t.^2)));% replica H = conj(fft(sr)/M); % matched filter transfer function %—Figure 6.15—————————————————————————————————————————— h = figure; plot(t*1e6, s, 'k','LineWidth',0.5) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); grid minor title('\ittransmitted signal'); xlabel('\ittime, \mus') axis([min(t)*1e6 max(t)*1e6 -2 2 ]); %—Figure 6.16—————————————————————————————————————————— h = figure;plot(F*1e-9,abs(fftshift(H)), 'k','LineWidth',2) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); grid minor title('Matched filter response'); xlabel('\itfrequency, GHz') %— Received Signal —————————————————————————————————— for n=1: N Es(n,1:M) =zeros(1,M); for m =1: length(Xc); x = Xo+Xc(m)-Vx*T2*(n-1); R = sqrt((Yo+Yc(m))^2+x^2); Es(n,1:M) = Es(n,1:M)+exp(j*2*pi*(fc*(t-2*R/c) +Kchirp/2*((t-2*R/c).^2))); end % define noise noise=5*randn(1,M); NS(n,1:M)=noise; % Matched filtering EsF(n,1:M) = fft(Es(n,1:M)+noise)/M; (Continued)

6.11 Matlab Codes

ESS(n,1:M) = EsF(n,1:M).*H; ESS(n,1:M) = ifft(ESS(n,1:M)); end E_signal = sum(sum(abs(EsF.^2))); E_noise = sum(sum(abs(NS.^2))); SNR = E_signal/E_noise; SNR_db = 10*log10(SNR); %—Figure 6.17—————————————————————————————————————————— rd=30; % dynamic range of display h=figure; matplot2(1:N,slow_t,(ESS),rd); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); grid minor xlabel('\itrange bins'); ylabel('\itazimuth time, s'); %—Figure 6.18—————————————————————————————————————————— h=figure; win = hanning(N)*ones(1,M); % prepare window in crossrange direction matplot2(1:N,RC,(fftshift(fft(ESS.*win))).',rd); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\itrange, m'); xlabel('\itDoppler index'); %—Figure 6.19—————————————————————————————————————————— h=figure; (Continued)

343

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6 Range-Doppler Inverse Synthetic Aperture Radar Processing

matplot2(XX,RC,(fftshift(fft(ESS.*win))).',rd); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\itrange, m'); xlabel('\itcross-range, m'); % axis([-30 30 -90 90]) axis equal; axis tight

Matlab code 6.2 Matlab file “Figure6-21thru23.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 6-21 thru23 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % samedirectory: % % CoutUssFletcher.mat clear close all clc %—Radar parameters————————————————————————————————————— pulses = 128; % # no of pulses burst = 128; % # no of bursts c = 3.0e8; % speed of EM wave [m/s] f0 = 9e9; % Starting frequency of SFR radar system [Hz] bw = 125e6; % Frequency bandwidth [Hz] T1 = (pulses-1)/bw; % Pulse duration [s] PRF = 35e3; % Pulse repetition frequency [Hz] T2 = 1/PRF; % Pulse repetition interval [s] (Continued)

6.11 Matlab Codes

%—target parameters———————————————————————————————————— theta0 = 0; % Initial angle of target's wrt target [degree] w = 1.2; % Angular velocity [degree/s] Vr = 5.0; % radial velocity of EM wave [m/ s] ar = 0.04; % radial accelation of EM wave [m/s^2] R0 = 4e3; % target's initial distance from radar [m] dr = c/(2*bw); % range resolution [m] W = w*pi/180; % Angular velocity [rad/s] %—load the coordinates of the scattering centers on the fighter———— load CoutUssFletcher %—Figure 6.21—————————————————————————————————————————— n = 10; Xc =(xind(1:n:6142)-93.25)/1.2; Yc =-zind(1:n:6142)/1.2; h = figure; plot(Xc,-Yc,'o','MarkerSize',3,'MarkerFaceColor',[1 0 0]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); axis([-80 80 -60 90]) xlabel('X [m]'); ylabel('Y [m]'); %—Scattering centers in cylindirical coordinates———————— [theta,r] = cart2pol(Xc,Yc); Theta = theta+theta0*0.017455329; %add initial angle i = 1:pulses*burst; T = T1/2+2*R0/c+(i-1)*T2;%calculate time vector Rvr = Vr*T+(0.5*ar)*(T.^2);%Range Displacement due to radial vel. & acc. Tetw = W*T;% Rotational Displacement due to angular vel. (Continued)

345

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6 Range-Doppler Inverse Synthetic Aperture Radar Processing

i = 1:pulses; df = (i-1)*1/T1; % Frequency incrementation between pulses k = (4*pi*(f0+df ))/c; k_fac = ones(burst,1)*k; %————Calculate backscattered E-field——————————————————— Es(burst,pulses)=0.0; for scat=1:1:length(Xc) arg = (Tetw - theta(scat) ); rngterm = R0 + Rvr - r(scat)*sin(arg); range = reshape(rngterm,pulses,burst); range = range.'; phase = k_fac.* range; Ess = exp(j*phase); Es = Es+Ess; end Es = Es.'; % define noise noise=10*randn(burst,pulses); E_signal = sum(sum(abs(Es.^2))); E_noise = sum(sum(abs(noise.^2))); SNR = E_signal/E_noise SNR_db = 10*log10(SNR) Es = Es+noise.'; %—Figure 6.22—————————————————————————————————————————— % Check out the range profiles X = -dr*((pulses)/2-1):dr:dr*pulses/2;Y=X/2; RP = fft((Es.')); RP = fftshift(RP,1); h = figure; matplot2(X,1:burst,RP.',20); colormap(hot); cc = colorbar; tt = title(cc,'dB'); (Continued)

References

tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itBurst index'); %Form ISAR Image (no compansation) %—Figure 6.23—————————————————————————————————————————— ISAR = abs(fftshift(fft2((Es)))); h = figure; matplot2(X,1:burst,ISAR(:,pulses:-1:1),25); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itDoppler index'); axis([-76 50 20 128])

References Chen, C.C. and Andrews, H.C. (1980). Target-motion-induced radar imaging. IEEE Transactions on Aerospace and Electronic Systems AES-16 (1): 2–14. https://doi.org/ 10.1109/TAES.1980.308873. Curlander, J.C. and McDonough, R.N. (1991). Synthetic Aperture Radar Systems and Signal Processing. New York: Wiley. Doerry, A.W. (2008). Ship Dynamics for Maritime ISAR Imaging. Technical Report. Sandia National Laboratories. SAND2008-1020, February. https://doi.org/10.2172/ 929523en.wikipedia.org/wiki/Sea_state Itoh, T., Sueda, H., and Watanabe, Y. (1996). Motion compensation for ISAR via centroid tracking. IEEE Transactions on Aerospace and Electronic Systems 32 (3): 1191–1197. https://doi.org/10.1109/7.532283. Kirk, J.C. (1975). Motion compensation for synthetic aperture radar. IEEE Transactions on Aerospace and Electronic Systems AES-11 (3): 338–348. https://doi.org/10.1109/ TAES.1975.308083.

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Küçükkiliç, T. (2006). ISAR imaging and motion compensation. MS thesis. Middle East Technical University. Wang, Y., Ling, H., and Chen, V.C. (1998). ISAR motion compensation via adaptive joint time-frequency technique. IEEE Transactions on Aerospace and Electronic Systems 34 (2): 670–677. https://doi.org/10.1109/7.670350. Wu, H., Grenier, D., Delisle, D. et al. (1995). Translational motion compensation in ISAR image processing. IEEE Transactions on Image Processing 4 (11): 1561–1571. https://doi.org/10.1109/83.469937. Xi, L., Guosui, L., and Ni, J. (1999). Autofocusing of ISAR images based on entropy minimization. IEEE Transactions on Aerospace and Electronic Systems 35 (4): 1240– 1251. https://doi.org/10.1109/7.805442.

349

7 Scattering Center Representation of Inverse Synthetic Aperture Radar In radar imaging, the scattering center concept provides various advantages, especially when dealing with radar cross-sections of objects and synthetic aperture radar/inverse synthetic aperture radar (SAR/ISAR) imaging. After the object is illuminated by an electromagnetic (EM) wave, some locations on the object provide localized radiation/scattering energy toward the observation point. The locations of these concentrated sources of scattering energy are called scattering centers. The use of the scattering center model provides a very sparse representation of ISAR imagery such that EM scattering from complex bodies can be modeled as if it is emitting from a discrete set of points on the target (Hurst and Mittra 1987; Yu et al. 1991; Carriere and Moses 1992; Bhalla and Ling 1996; Chen and Ling 2002) as illustrated in Figure 7.1. Such representation is so powerful that it may provide many advantages, including the following: 1) A simple and sparse representation of the EM scattering and/or SAR/ISAR imagery can be obtained. 2) Since the model is sparse, the scattering data and/or SAR/ISAR image can be compressed with high data compression ratios. 3) Because of the fact that the new data set is much smaller than the original data set, the reconstruction of the scattering data and/or SAR/ISAR image can be obtained quickly. 4) The model can be used to interpolate the scattering data and/or SAR/ISAR image with infinite resolution. 5) The model can be used to extrapolate the scattering data and/or SAR/ISAR image within some finite bandwidths of frequencies and angles. 6) The model provides insights into the scattering mechanisms that may lead to understanding the cause-and-effect relationship of the scattering phenomena from the target. The scattering center representation is usually implemented through a parameterization scheme based on the point-scatterer model. Therefore, the model is first Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms: With Advanced SAR/ISAR Imaging Concepts, Algorithms, and MATLAB Codes, Second Edition. Caner Özdemir. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

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7 Scattering Center Representation of Inverse Synthetic Aperture Radar

defined and the model parameters are extracted from the scattered field data and/ or SAR/ISAR image afterward by applying an extraction algorithm. Once the model is properly extracted, the data can be reconstructed and/or interpolated/ extrapolated by applying a reconstruction routine. All these steps will be explained throughout this chapter.

7.1

Scattering/Radiation Center Model

This model is motivated by the observation that an ISAR image exhibits strong point-scatterer-like behavior as can easily be seen from the presented ISAR images in Chapters 4–6. It is well known that the backscattered signature can be modeled by a very sparse set of point radiators called scattering centers on the target (Hurst and Mittra 1987; Yu et al. 1991; Carriere and Moses 1992; Tseng and Burnside 1992; Wang and Jeng 1995; Bhalla and Ling 1996; Özdemir et al. 1998, 2000; Chen and Ling 2002). Since these scattering centers can also be regarded as the secondary point radiators, they are also called radiation centers (Özdemir et al. 1998, 2000) in some applications. When the incident field impinges on the object, surface currents flow on the surface of the object. At some regions of the target, these surface currents interfere constructively (i.e. in-phase), and the resultant scattering wave amplitude becomes greater. These dominant sources of radiation or scattering are called hot spots or scattering centers in the radar literature. Corner reflector-type structures and planar surfaces are capable of providing strong specular scattering that can be regarded as the scattering centers. In some parts of the target, however, surface currents interfere destructively (i.e. out-of-phase), and the overall scattering amplitude is decreased. Planar regions on the target (if they do not constitute any specular scattering) provide very little scattering. Those parts of target are sometimes called cold regions or ghost regions. The main idea in the scattering center representation comes from the fact that the scattered field from a target can be approximated as if it is coming from a finite number of point scatterers on the target (Figure 7.1). These point scatterers happen to be around the hot spots on the target. With this model, these scattering centers are responsible for all the scattering energy as the scattered field can be parameterized by a finite set of discrete point scatterers as N

Es k, ϕ

An exp − j2 k

rn

n=1 N

An exp − j2 k x x n + k y yn

= n=1 N

An exp − j2k cos ϕ x n + sin ϕ yn

= n=1

71

7.1 Scattering/Radiation Center Model

EM illumination

EM illumination Real scenario

Equivalent point-scatter model

Figure 7.1 Point-radiator model of the scattered field. (a) the real scenario for field scattering from a target, (b) equivalent point-scatterer model based on scattering centers.

Here, Es(k, ϕ) represents the backscattered electric field collected at different frequencies and angles. An is the complex amplitude of the nth scattering center and r n = x n x + yn y is the displacement vector of the location of the nth scattering center. Equation 7.1 suggests that the total scattered field can be written as the sum of N different point radiators at different (xn, yn) locations. This point radiator or scattering center model is more conveniently applied in the image domain rather than in the Fourier domain. Therefore, the employment of the scattering center model is more meaningful in the image domain since the ISAR image itself consists of finite number of point scatterers together with their point spread functions (PSFs). Therefore, an ISAR image is conveniently parameterized via a finite set of point radiators as N

An h x − x n , y − y n ,

ISAR x, y

72

n=1

where An is the complex amplitude, (xn, yn) is the location of the nth scattering center, and h(x, y) is called the PSF or ray spread function whose formula has already been derived in Chapter 5, Eq. 5.26 as h x, y =

exp j2k xc x

exp j2k yc y

BWky π

BWkx BWkx sin c x π π BWky sin c y , π

73

where BWkx and BWky are the finite bandwidths in kx and ky, respectively. Here, kxc and kyc are the center spatial frequencies in x and y directions. When the small bandwidth and the small-angle approximation holds true, the above PSF can be rewritten in terms of the frequency and the angle variables as

351

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7 Scattering Center Representation of Inverse Synthetic Aperture Radar

h x, y = sin c

exp j

4π f c x + ϕc y c

4fc B Ω c2

sin c

2B x c

2fc Ω y c

74

In the above equation, B and Ω are the frequency bandwidth and the look-angle span, respectively. Also, fc and ϕc are the center values for the frequency band and aspect width, respectively.

7.2

Extraction of Scattering Centers

Once the model is defined, these scattering centers can now be extracted from the image together with their corresponding PSFs. Although it is more convenient and practical to extract these point radiators or scattering/radiation centers directly in the image domain, it is also possible to convey the extraction process in the Fourier (or frequency-aspect) domain. Because the ISAR image itself is the display of these point radiators, it is easier to implement the extraction procedure in the image domain. As we shall see next, it is computationally more desirable as well.

7.2.1

Image Domain Formulation

7.2.1.1 Extraction in the Image Domain: The “CLEAN” Algorithm

There are a few methods used to extract scattering centers from the ISAR image (Bhalla and Ling 1996). Among those, the common image processing algorithm used for the extraction of scattering/radiation centers is the well-known CLEAN algorithm. CLEAN algorithm was first introduced to perform deconvolution on images created in radio astronomy (Högbom 1974; Segalovitz and Frieden 1978). It has also been successfully utilized for scattering/radiation center extraction (Tsao and Steinberg 1988; Bhalla et al. 1997; Özdemir et al. 1998) in radar imaging applications. CLEAN is a robust, iterative procedure that successively picks out the highest point in the image, assumes it is a point radiator (or a scattering center) with the corresponding amplitude, and removes its point spread response from the image. At the nth iteration of CLEAN, therefore, if An is the strength of the highest point in the image with the location of (xn, yn) the two-dimensional (2D) residual image can be written as that is to say, “2D residual image at the nth step is equal to 2D residual image at the (n − 1)th step minus the highest scattering center strength with its corresponding PSF in the residual image for the nth step” as formulated below: 2D residual image

n

=

2D residual image

n−1

− An h x − x n , y − yn

75

7.2 Extraction of Scattering Centers

The extraction process is iteratively continued until the maximum strength in the residual image reaches a user-defined threshold value. Typically, the energy in the residual image decreases quickly during the initial stages of the iteration and tapers off after reaching the noise floor. 7.2.1.1.1

Scattering Center Extraction Using CLEAN

An example for the extraction of scattering centers is demonstrated through Figures 7.2–7.5. A biplane geometry whose CAD file can be seen in Figure 7.2a has been chosen as the target for this analysis. The EM back-scattering simulation of this airplane target has been done by using the PREDICS simulator code (Özdemir et al. 2009, 2014; Kırık and Özdemir 2019) based on the simulation geometry given in Figure 7.2b. All the simulation parameters and generated ISAR imaging parameters are listed in Table 7.1. Once the backscattered electric field in collected based on the ISAR simulation geometry and parameters, the ISAR image is constructed as depicted in Figure 7.3. The perspective view of the airplane model and its projected ISAR image are given in Figure 7.3a and b, respectively. Next, the scattering center extraction procedure in Equation 7.5 is applied to the ISAR image shown in Figure 7.3. CLEAN algorithm is utilized to extract a total of 300 scattering centers by using the PSF described in Equation 7.3. The locations of first 150 extracted scattering centers are plotted in Figure 7.4a. Their sizes are shown with respect to their relative amplitudes. As seen from this figure, the dominant scattering centers are located around the noise of the airplane, landing gears, supporting struts between the wings, propellers, and tail wings. This figure clearly demonstrates the how CLEAN algorithm successfully picks and extracts the scattering centers with their PSFs from the ISAR image. The strengths of the extracted scattering centers are plotted in Figure 7.4b versus the scattering center number in the order of extraction. As seen from the figure, CLEAN algorithm converges rapidly as the energy in the residual image reduces significantly at each iteration of the extraction procedure. After extracting a sufficient number of scattering centers (about 250 in this example), the energy in the residual image starts to decrease gradually. This situation occurs when the maximum strength in the residual image reaches the noise floor of the original image. Therefore, when this level is reached, CLEAN process should be terminated. Otherwise, the extraction algorithm would try to clean the noise by the selected point-radiator model, which may continue infinitely without converging. This is because of the fact that the noise (either numerical or measurement) is not composed of point radiators. For that reason, the iterative search in CLEAN is usually halted after reaching a user-defined threshold value (which is usually the noise floor level of the image). As we have already stated, the scattering center model provides a sparse and simple representation of scattered field patterns and/or ISAR images. This feature will be

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7 Scattering Center Representation of Inverse Synthetic Aperture Radar

(a)

7.00 m

354

3.30 m

11.68 m

(b) Crossrange

y z

Airplane’s axial line

Range

x

15° Radar look direction

ϕ

Figure 7.2 The biplane model in ISAR simulation. (a) front, side and top view of the target with dimensions, (b) ISAR simulation geometry of the biplane target.

7.2 Extraction of Scattering Centers

Table 7.1 ISAR simulation parameters of biplane target for the look angle of (θ = 105 , ϕ = 0 ). EM simulation parameters

Value

Physical solver

PO + SBR + PTD

Minimum frequency

7.700 GHz

Maximum frequency

8.686 GHz

Number of frequency samples

70

Vertical look-angle (θ) [min]

105

Vertical look-angle (θ) [max]

105

Vertical look-angle (θ) [samples]

1

Horizontal look-angle (ϕ) [min]

−3.494

Horizontal look-angle (ϕ) [max]

3.441

Horizontal look-angle (ϕ) [samples]

132

Polarization

VV

Ray density

10 rays/λ

Maximum bounce

5

ISAR imaging parameters

Value

Range resolution

15 cm

Range window extend

10.5 m

Cross-range resolution

15 cm

Cross-range window extend

19.8 m

demonstrated with the help of the above numerical example. The original image size of Figure 7.3 has 280 × 528 with complex entries. Therefore, this image occupies 280 × 528 × 2 × 8 bit = 2.365 MB of disk space. However, when the scattering center model is used, one complex entry for the amplitude and two real entries for the location of each scattering center are needed. For our example of 300 scattering centers, therefore, a disk space of 300 × (2 + 2) × 8 bit = 9.6 KB is required to store all the scattering centers. So, a data compression ratio of approximately 246-to-1 is achieved. To summarize, it has been shown that a sparse model based on scattering center representation can be constructed. Once available, this model can be used to reconstruct the scattered field and the ISAR image with a very fine resolution and good fidelity, as will be shown next. 7.2.1.2

Reconstruction in the Image Domain

Once the scattering centers are extracted, the reverse process, that is, reconstruction of the ISAR image, can be readily done in real time by means of the extracted

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7 Scattering Center Representation of Inverse Synthetic Aperture Radar

(b)

Original ISAR image 0

(a) 8

Original ISAR image

–5

6 4 4 Cross-range, m

–10

2 0 8 6

4 Cr 2 os 0 sra ng –2 e, –4 m –6

0

0

–15

–2 –20

–6

,m ge

n Ra

–5

2

–4

5 –8

–25

–8 –5

5 dB

0 Range, m

–30

Figure 7.3 Original ISAR image of the biplane target displayed with CAD model file in (a) perspective view, (b) top view of range – cross range plane.

(a)

(b)

Locations of scattering centers with relative amplitudes

0.3

5 0.25

4 Amplitude, mV/m

3 Cross-range, m

356

2 1 0 –1 –2 –3

0.2 0.15 0.1 0.05

–4 –5

0 –3 –2 –1 0 1 2 Range, m

3

0

50

100

150

200

250

300

Scattering center #

Figure 7.4 (a) Locations of extracted scattering centers in range – cross-range plane displayed with respect to their relative strengths, (b) amplitudes of the extracted scattering centers.

7.2 Extraction of Scattering Centers

scattering centers. Furthermore, the Fourier domain (or frequency-aspect) data can also be reconstructed very easily with good fidelity, as we shall see next. 7.2.1.2.1

Image Reconstruction

The image can be reconstructed by putting all the extracted scattering centers back onto the ISAR image, together with their PSFs, as N

An h x − x n , y − yn

ISARr x, y =

76

n=1

This reconstruction formula is the same as the extraction formula given in Equation 7.2. Since the parameters (An, xn, yn) are already obtained after the extraction process, the reconstructed ISAR image can be obtained quickly. An example of a reconstructed ISAR image is shown in Figure 7.5 where a total of 300 extracted scattering centers are used to reconstruct the original ISAR image in Figure 7.3. Comparison of the original image in Figure 7.3 with the reconstructed image in Figure 7.5 clearly validates that almost perfect reconstruction can be attained with the help of scattering centers. This example, of course, shows the effectiveness of the scattering center model in representing the ISAR image. (b)

Reconstructed ISAR image 0

(a) 8

Reconstructed ISAR image

–5

6 4

4 Cross-range, m

–10

2 0 8 6

4 2 Cr os 0 s–2 ra ng –4 e, –6 m

5 0

–8 –5

,m ge

n Ra

2 0

–15

–2 –20

–4 –6

–25

–8 –5

0 Range, m

5 dB

–30

Figure 7.5 Reconstructed ISAR image of the biplane target based on CLEAN algorithm displayed with CAD model file in (a) perspective view, (b) top view of range – cross range plane.

357

358

7 Scattering Center Representation of Inverse Synthetic Aperture Radar

7.2.1.2.2 Field Reconstruction

The scattered electric field in frequencies and angles can be reconstructed easily by using the extracted scattering centers with the following formula: N

Esr k, ϕ =

An exp − j2 k x x n + k y yn n=1

77

N

An exp − j2k cos ϕ x n + sin ϕ yn

= n=1

One will easily notice that this formula is obtained by exploding the point-radiator model given in Equation 7.1. Therefore, the scattering centers are being used in its literal meaning such that extracted scattering centers are being used as the point radiators. Furthermore, this formula is essentially nothing but the 2D Fourier transform (FT) of Equation 7.6, as expected. Therefore, the field can be reconstructed directly using the extracted scattering centers and with the help of Equation 7.7. An example of field reconstruction is demonstrated with the help of Figures 7.6–7.11. In Figure 7.6a, original backscattered field data for the original ISAR image given in Figure 7.3 are shown. These original data are displayed on 2D multifrequency multiaspect plane. Then, by applying the expression in Equation 7.7, the reconstructed field pattern for the same frequency-angle variables is presented in Figure 7.6b. For the reconstruction of the backscattered field pattern, all of the 300 extracted scattering centers were used. Comparing both the original and the reconstructed images visually, agreement between both sets of data can be easily seen. Since the scattered field pattern is reconstructed by the analytical formula in Equation 7.7, the angular or spectral resolution can be selected at any value that offers infinite resolution. In practice, we have infinite resolution in reconstructing the scattered field values. In Figure 7.7, for example, the same scattering centers were used to reconstruct the backscattered field by sampling both the frequency and aspect variables 5 times more. Image size, therefore, was increased 25 times compared to the original image. As a result, the resolution of this new image is much finer than that of the original image. To be able to observe the success of field reconstruction using scattering centers, we will show the agreement between the original and the reconstructed fields in the one-dimensional (1D) frequency domain or aspect domain. To demonstrate the results for the same ISAR example, the reconstructed field patterns are compared with the original field patterns in Figures 7.8 and 7.9. First, the aspect field pattern comparison is done for azimuth angles varying from −3.494 to 3.441 at the frequency of 8.029 GHz. The original field data obtained by brute-force computation have 132 distinct points. After using scattering center presentation of ISAR imaging and using the formula in Equation 7.7, the field reconstruction is accomplished for the same frequency and the angle range. Since there is no

7.2 Extraction of Scattering Centers

(a)

Original back-scattered field 0 –2

150

–4 100

Angle, degree

–6 50

–8 –10

0

–12

–50

–14 –100 –16 –150 –200

–18 7.8

8

8.2

8.4

8.6

dB

–20

Frequency, GHz

(b)

Reconstructed back-scattered field 0 –2

150

–4 100 Angle, degree

–6 50

–8

0

–10 –12

–50

–14 –100 –16 –150

–18

–200

–20 7.8

8

8.2

8.4

8.6

dB

Frequency, GHz

Figure 7.6 (a) Original backscattered electric- field for the ISAR example in Figure 7.3, (b) reconstructed field patterns based on CLEAN by utilizing a total of 300 scattering centers.

359

7 Scattering Center Representation of Inverse Synthetic Aperture Radar

Reconstructed back-scattered field 0 –2

150

–4 100

Angle, degree

–6 50

–8

0

–10 –12

–50

–14 –100 –16 –150

–18 –20

–200

7.8

8

8.2

8.4

8.6

dB

Frequency, GHz

Figure 7.7 Reconstructed backscattered field patterns sampled 5 times more compared to the original image.

@ f = 8.029 GHz

3

With brute force computation With scattering centers

2.5

2 E-field, V/m

360

1.5

1

0.5

0

–3

–2

–1

0

1

2

3

ϕ, degree

Figure 7.8 Comparison of the original pattern obtained by brute-force computation to the reconstructed pattern by scattering center in the angle domain.

7.2 Extraction of Scattering Centers

@ ϕ = –1.8°

2 1.8

With brute force computation With scattering centers

1.6

E-field, V/m

1.4 1.2 1 0.8 0.6 0.4 0.2 0 7.7

7.8

7.9

8

8.1

8.2

8.3

8.4

8.5

8.6

8.7

Frequency, GHz

Figure 7.9 Comparison of the original pattern obtained by brute-force computation to the reconstructed pattern by scattering center in the frequency domain.

restriction in choosing the angular granularity in the process of reconstructing, we use 5 times more data points such that there exist a total of 660 data points in the new reconstructed field pattern, as depicted in Figure 7.8. The comparison of the original angular field data computed by brute-force calculation and the reconstructed angular field data computed by using the scattering centers clearly shows the success of the scattering center model in angular field reconstruction. This image obviously demonstrates that the scattering center model can be used to interpolate the original field data with high accuracy. This, of course, provides tremendous computation time-savings for RCS calculation or ISAR image formation with fine resolutions. The result of another comparison is plotted in Figure 7.9, where original and reconstructed field patterns in the frequency domain are shown in the same plot. This time, the radar look angle is kept constant at the azimuth angle of –1.8 and the frequency is altered from 7.7 to 8.686 GHz. Again, the reconstruction formula in Equation 7.7 is used to estimate the frequency field pattern that is 5 times more sampled than the original field pattern. The visual comparison of original and reconstructed frequency-domain electric field patterns clearly shows the fidelity of the scattering centers in estimating the field pattern. Again, the data between the original data points were successfully interpolated in a short time, thanks to the scattering center model.

361

362

7 Scattering Center Representation of Inverse Synthetic Aperture Radar

7.2.2

Fourier Domain Formulation

7.2.2.1 Extraction in the Fourier Domain

The scattering center representation can also be employed in the Fourier domain (or frequency-aspect domain). For this case, the scattering centers are to be extracted from the frequency-aspect data. This process is conceptually possible but requires more computer resources as will be explained next. The extraction in the Fourier domain can be achieved by the famous method known as “matching pursuit” (Mallat and Zhang 1993; Su et al. 2000; Bicer et al. 2018). In the extraction process, the frequency-aspect field projection of each scattering center is iteratively extracted from the 2D Fourier domain electric field data. During each iteration process, the scattering center location that gives the largest value of the following inner product is recorded: An = max

x n ,yn

< E s k, ϕ , φ x, y >

78

Here, An is the highest value of the inner product between the Fourier domain scattered electric field and the phase term φ(x, y) is equal to exp − j2 k x x n + k y yn . The inner product in the above equation is defined as the following: E s k, ϕ , φ x, y

k H ϕH

= kL

ϕL

E s k, ϕ

φ∗ x, y dϕdk

79

So, at the nth iteration of the search, (xn, yn) pair that gives the largest value of An corresponds to the location of corresponding scattering center. Once the strongest scattering center at the nth iteration is determined, that is, (An, xn, yn) values are determined, its associated scattered field is subtracted from the residual electric field as E s k, ϕ

n

= E s k, ϕ

n−1

− An φ x n , y n

7 10

This extraction process is continued until the energy in the residual field reaches a user-defined threshold value. Usually, this value is the noise floor of the collected data. The main problem in this extraction routine is that it requires an exhaustive search over 2D data for every iteration of the search process. An example of Fourier domain extraction is demonstrated via another ISAR simulation that was carried out with PREDICS software (Özdemir et al. 2009, 2014; Kırık and Özdemir 2019). For the simulation, same biplane model given in Figure 7.2a was used. The EM simulation and ISAR parameters are listed in Table 7.2. Then, 2D backscattered electric field data were collected based on the parameters listed in Table 7.2. Then, a total of 250 scattering centers are extracted from the multifrequency, multiaspect backscattered electric field data by applying the matching pursuit routine. In Figure 7.10a, amplitudes of the extracted scattering centers are given. During the Fourier domain extraction, the original data grid

7.2 Extraction of Scattering Centers

Table 7.2 ISAR simulation parameters of biplane target for the look angle of (θ = 110 , ϕ = 0 ). EM simulation parameters

Value

Physical solver

PO + SBR

Minimum frequency

8.259 GHz

Maximum frequency

8.927 GHz

Number of frequency samples

50

Vertical look-angle (θ) [min]

110

Vertical look-angle (θ) [max]

110

Vertical look-angle (θ) [samples]

1

Horizontal look-angle (ϕ) [min]

−2.498

Horizontal look-angle (ϕ) [max]

2.448

Horizontal look-angle (ϕ) [samples]

100

Polarization

VV

Ray density

10 rays/λ

Maximum bounce

5

ISAR imaging parameters

Value

Range resolution

22 cm

Range window extend

11 m

Cross-range resolution

20 cm

Cross-range window extend

20 m

is four times more sampled to better estimate the locations of the scattering centers. However, the iterative search takes more time. If the data grid is sampled more densely, the amplitudes of the scattering centers converge faster. This is, in turn, at the price of increased computation time of the extraction algorithm. The locations of extracted scattered centers are displayed in Figure 7.10b with respect to their relative amplitudes. As another evaluation for the success of the extraction process, the average energies of the original, the reconstructed and the residual image are compared by using the below formula: W=

1 M N Es MN m = 1 n = 1 m,n

2

7 11

where W is the average energy of the scattered electric field matrix Es of size M × N. For the above numerical example, original collected backscattered electric field

363

7 Scattering Center Representation of Inverse Synthetic Aperture Radar

(a)

Locations of scattering centers with relative amplitudes

(b) 2 5

1.8

4

1.6

3

1.4

Cross-range, m

Amplitude, mV/m

364

1.2 1 0.8 0.6

1 0 –1 –2 –3

0.4

–4

0.2 0

2

–5 0

50

100 150 Scattering center #

200

250

–4

–2

0 2 Range, m

4

Figure 7.10 (a) Amplitudes of the extracted scattering centers based on Fourier domain formulation, (b) Corresponding locations of the extracted scattering centers displayed with respect to their relative strengths.

matrix has an average energy of 635.92j based on the formula given in Equation 7.11. After extracting a total of 250 scattering centers from the original 2D frequency-aspect electric field, the energy left in the residual data happened to be only 3.62j which definitely shows that 99.43% of the energy from the original image has been extracted successfully thanks to Matching Pursuit algorithm whose formulation is given in Equations 7.8–7.10. 7.2.2.2 Reconstruction in the Fourier Domain

Once the scattering centers are extracted, the reconstruction either in the Fourier domain or in the image domain can be done using (An, xn, yn) values for each scattering centers. Both reconstruction routines will be given next. 7.2.2.2.1 Field Reconstruction

The scattered electric field in frequencies and angles can be reconstructed by putting back the field projections of all extracted scattering centers by using the following formula: N

Esr k, ϕ =

An exp − j2k cos ϕ x n + sin ϕ yn

7 12

n=1

Provided that the extraction has been employed such that most of the energy has been extracted from the original field data, the reconstruction field should be almost equal to the original field, that is, Esr k, ϕ Es k, ϕ An example of reconstruction field pattern is demonstrated in Figure 7.11b where the 2D

7.2 Extraction of Scattering Centers

Original back-scattered field

(a)

0

2 –5

1.5

Angle, degree

1

–10

0.5 0

–15

–0.5 –20

–1 –1.5

–25

–2 –2.5

8.3

8.4

8.5

8.6

8.7

8.8

8.9

dB

–30

Frequency, GHz Reconstructed back-scattered field

(b)

0

2 –5

1.5

Angle, degree

1

–10

0.5 0

–15

–0.5 –20

–1 –1.5

–25

–2 –2.5

8.3

8.4

8.5

8.6

8.7

8.8

8.9

dB

–30

Frequency, GHz

Figure 7.11 2D backscattered electric field data: (a) Original, (b) Reconstructed after utilizing 250 scattering centers extracted in frequency-aspect domain.

365

7 Scattering Center Representation of Inverse Synthetic Aperture Radar

Reconstructed field (x5 upsampled)

0

2 –5

1.5 1 Angle, degree

366

–10

0.5 0

–15

–0.5 –20

–1 –1.5

–25

–2 –2.5

8.3

8.4

8.5

8.6

8.7

8.8

8.9

dB

–30

Frequency, GHz

Figure 7.12 Reconstructed field pattern (5 times more sampled) gives more detailed estimate of the field pattern in between the original field data points.

frequency-aspect backscattered field is reconstructed using the 250 scattering centers for the same airplane model. The comparison of the reconstructed field in Figure 7.11b with the original field in Figure 7.11a clearly demonstrates the success of the reconstruction process. Since there is no limit in reconstructing the field at a desired granularity, the reconstruction can be employed at very fine resolutions. In Figure 7.11, the frequencyaspect backscattered data for the airplane model are reconstructed by the same 250 scattering centers with 5 times more sampled data points in each domain. Therefore, the field value in between the original data points can be estimated (or interpolated) with good fidelity. The 5 times more sampled reconstructed field data are shown in Figure 7.12. A more detailed comparison of the original frequency-aspect field data with the reconstructed one is accomplished thru the plots given in Figure 7.13. In Figure 7.13a, the comparison of the original data with the 5 times more interpolated reconstructed data along the frequencies is given for the look angle value of −1 . In a dual manner, the comparison of the original data with the 5 times more interpolated reconstructed data along the angles is provided for the frequency of 8.709 GHz. As seen from both figures, the model provided by scattering center representation successfully predicts the data in between points with good fidelity.

7.2 Extraction of Scattering Centers

(a)

@ PHI = –0.999 Deg. With brute force computation With scattering centers

E-field, V/m

6 5 4 3 2 8.3

8.4

8.5

8.6

8.7

8.8

8.9

Frequency, GHz

E-field, V/m

(b)

@ f = 8.709 GHz

10 9 8 7 6 5 4 3 2 1

With brute force computation With scattering centers

–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

ϕ, degree

Figure 7.13 Comparison of the original pattern obtained by brute-force computation to the reconstructed pattern obtained by scattering center extraction in the Fourier domain. (a) comparison in the frequency domain, (b) comparison in the azimuth angle domain.

7.2.2.2.2

Image Reconstruction

A fast and convenient way of reconstructing the ISAR image by using scattering centers extracted in the Fourier domain is by taking the 2D inverse FT (IFT) of the reconstructed field pattern. Therefore, the reconstructed ISAR image can be readily formed by ISARr x, y =

−1 2

E sr k, ϕ

7 13

where 2− 1 corresponds to 2D IFT operation. For the same biplane example, the ISAR image is reconstructed with the formula above. For comparison purposes, the original ISAR image constructed by the collected field data is also given in Figure 7.14a, whereas, the reconstructed ISAR image using the scattering centers extracted in the Fourier domain is shown in Figure 7.14b. While displaying the ISAR images, a 2D Hamming window was utilized to compress the sidelobes in the images. As can be easily observed from the original and reconstructed ISAR

367

7 Scattering Center Representation of Inverse Synthetic Aperture Radar Original ISAR image

(a) Original ISAR image

Cross-range, m

4 2 0 5 C ss ro

0

m e, ng -ra

5

–5

0 –10

–5

,m nge

Ra

0

8

–5

6

–10

4

–15

2

–20

0

–25

–2

–30

–4

–35

–6

–40

–8

–45

–10

(b)

–5

0 Range, m

5 dB

Reconstructed ISAR image Reconstructed ISAR image

4 Cross-range, m

368

2 0 5 s ro

C ,m ge

n ra

s-

0 5

–5 0 –10

–5

e, m

g Ran

–50

0

8

–5

6

–10

4

–15

2

–20

0

–25

–2

–30

–4

–35

–6

–40

–8

–45

–10

–5

0 Range, m

5 dB

–50

Figure 7.14 2D ISAR image: (a) Original, (b) Reconstructed after utilizing 250 scattering centers extracted in frequency-aspect domain.

images, almost perfect match between the two, with only small discrepancies, is achieved.

7.3

Matlab Codes

Below are the Matlab source codes that were used to generate all of the Matlabproduced figures in this Chapter 7. The codes are also provided in the CD that accompanies this book.

7.3 Matlab Codes

Matlab code 7.1 Matlab file “Figure7_3thru7_9.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 7.3 thru 7.9 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % P_Plane105.mat clear close all %________________GET THE DATA__________________________ load P_Plane105.mat c M P N

= = = =

.3; length(Freq); length(Theta); length(Phi);

% speed of light

% organize axes df = Freq(2)-Freq(1); fc = Freq(round(M/2+1)); % center freq BWf = 2*(fc-Freq(1)); dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); Phir = Phi*pi/180; dphi = pi*(aspect(2)-aspect(1))/180; % in radians Phirc = Phir(round(N/2+1)) ; BWphi = 2*(Phirc-Phir(1)); dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1);

XX = -dx*M/2:dx/4:-dx*M/2+dx/4*(4*M-1); YY = -dy*N/2:dy/4:-dy*N/2+dy/4*(4*N-1);

369

370

7 Scattering Center Representation of Inverse Synthetic Aperture Radar

K = 2*pi*Freq/c; dk = K(2)-K(1); kc = 2*pi*fc/c;

% wavenumber vector % wavenumber resolution % center wavenumber

% ISAR Es = vec2mat(Es_VV,N);

% Convert 2d DATA

ISAR = fftshift(fft2(Es)); ISAR = ISAR/M/N;

% the image

% ISAR 4x UPSAMPLED————————————————— Enew = Es; Enew(M*4,N*4) = 0; ISARnew = fftshift(fft2(Enew)); ISARnew = ISARnew/M/N; %________________2D-CLEAN____________________________ % prepare 2D sinc functions sincx = ones(1,M); sincx(1,M+1:M*4) = 0; hsncF = fft(sincx)/M; sincy=ones(1,N); sincy(1,N+1:N*4) = 0; hsncPHI = fft(sincy)/N; %initilize hh = zeros(4*M,4*N); ISARres = ISARnew.'; Amax = max(max(ISARnew)); ISARbuilt = zeros(N*4,M*4); % loop for CLEAN for nn=1:300 [A,ix] = max(max(ISARres)); [dum,iy] = max(max(ISARres.')); hsincX = shft(hsncF,ix); hsincY = shft(hsncPHI,iy); hhsinc = hsincX.'*hsincY;

7.3 Matlab Codes

ISARres = ISARres-A*hhsinc.'; SSs(nn,1:3) = [A XX(ix) YY(iy)]; II=ISARres; II(1,1) = Amax; % Image Reconstruction ISARbuilt = ISARbuilt-A*hhsinc.'; end %________________IMAGE COMPARISON______________________ %—Figure 7.3—————————————————————————————————————————— h = figure; matplot(X,Y,abs(ISARnew(4*M:-1:1,:).'),30); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; colormap(hot); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); % plot( -xyout_xout,xyout_yout,'w.'); xlabel('\itrange, m'); ylabel('\itcross-range, m'); title('Original ISAR image') axis equal; axis tight %—Figure 7.5——————————————————————————————————————————— h = figure; matplot(X,Y,abs(ISARbuilt(:,4*M:-1:1)),30); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; colormap(hot); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); % plot( -xyout_xout,xyout_yout,'w.'); xlabel('\itrange, m'); ylabel('\itcross-range, m'); title('\itReconstructed ISAR image') axis equal; axis tight

371

372

7 Scattering Center Representation of Inverse Synthetic Aperture Radar

%________________SCATTERING CENTER INFO DISPLAY________ %—Figure 7.4a—————————————————————————————————————————— h = figure; hold for m = 1:150 t = round(abs(SSs(m,1))*20/abs(SSs(1,1)))+1; plot(-SSs(m,2),SSs(m,3),'o', 'MarkerSize',t,'MarkerFaceColor',[1 0 0]); hold on end % plot(-xyout_xout,xyout_yout,'.'); axis([min(X) max(X) min(Y) max(Y)]) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itcross-range, m'); title('\itLocations of scattering centers with relative amplitudes ') grid minor

%—Figure 7.4b—————————————————————————————————————————— h = figure; plot(abs(SSs(:,1)),'square', 'MarkerSize',4,'MarkerFaceColor',[1 0 0]); grid minor set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itscattering center #'); ylabel('\itamplitude, mV/m'); %________________RECONSTRUCT THE FIELD PATTERN_________ ESR = zeros(5*M,5*N); ESr = zeros(M,N); kk = K(1): dk/5: K(1)+dk/5*(5*M-1); dkk = kk(2)-kk(1);

7.3 Matlab Codes

pp = Phir(1): dphi/5: Phir(1)+dphi/5*(5*N-1); dpp = pp(2)-pp(1);

for nn = 1:300 An = SSs(nn,1); xn = SSs(nn,2); yn = SSs(nn,3); ESR = ESR+An*exp(1i*2*xn*(kk-kk(1)).')*exp (1i*2*kc*yn*(pp-pp(1))); ESr = ESr+An*exp(1i*2*xn*(K-K(1)))*exp(1i*2*kc*yn* (Phir-Phir(1)).'); end % Elapsed time is 0.470874 seconds. %—Figure 7.6(a)———————————————————————————————————————— h = figure; matplot(Freq,Phi*180/pi,abs((Es.')),20); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; colormap(hot) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\itangle, degree'); xlabel('\itfrequency, GHz'); title('\itOriginal back-scattered field') %—Figure 7.6(b)———————————————————————————————————————— h = figure; matplot(Freq,Phi*180/pi,abs((ESr.')),20); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; colormap(hot) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\itangle, degree'); xlabel('\itfrequency, GHz'); title('\itReconstructed back-scattered field')

373

374

7 Scattering Center Representation of Inverse Synthetic Aperture Radar

%—Figure 7.7——————————————————————————————————————————— h = figure; matplot(Freq,Phi*180/pi,abs((ESR.')),20); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; colormap(hot) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\itangle, degree'); xlabel('\itfrequency, GHz'); title('\itReconstructed back-scattered field (x5 upsampled)') %—Figure 7.8——————————————————————————————————————————— nn =24; h = figure; plot(Phi,abs(Es(nn,:)),'k-.o','MarkerSize',6, 'LineWidth',2); hold; plot(pp*180/pi,abs(ESR(5*(nn-1)+1,:)),'r','LineWidth',1); hold; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\phi, degree'); ylabel('\itE-field, V/m'); tt=num2str(Freq(nn)); ZZ=['@ f = ' tt ' GHz']; axis([Phi(1) Phi(end) 0 3]); title(ZZ); drawnow; grid minor legend('\itwith brute force computation','\itwith scattering centers') %—Figure 7.9——————————————————————————————————————————— nn = 33; h = figure; plot(Freq,abs(Es(:,nn)),'k-.*','MarkerSize',8,

7.3 Matlab Codes

'LineWidth',3); hold; plot(kk*c/2/pi,abs(ESR(:,5*(nn-1)+1)),'r-', 'LineWidth',2); hold; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itfrequency, GHz'); ylabel('\itE-field, V/m'); tt=num2str(Phi(nn)); ZZ=['@ \phi = ' tt ' \circ']; title(ZZ);drawnow; grid minor legend('\itwith brute force computation','\itwith scattering centers')

375

376

7 Scattering Center Representation of Inverse Synthetic Aperture Radar

Matlab code 7.2 Matlab file ’Figure7-10thru7-14.m ’ %—————————————————————————————————————————————————————— % This code can be used to generate Figure 7.10 thru 7.14 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % BiplaneEs.mat clear close all %________________GET THE DATA__________________________ % load P_Plane105.mat load BiplaneEsT110 Es = E_VV; c M P N

= = = =

.3; length(Freq); length(Theta); length(Phi);

% speed of light

% organize axes df = Freq(2)-Freq(1); fc = Freq(round(M/2+1)); % center freq BWf = 2*(fc-Freq(1)); dx = c/2/BWf; X = -M/2*dx:dx:dx*(M/2-1); Phir = Phi*pi/180; dphi = pi*(aspect(2)-aspect(1))/180; % in radians Phirc = Phir(round(N/2+1)) ; BWphi = 2*(Phirc-Phir(1)); dy = c/2/BWphi/fc; Y = -N/2*dy:dy:dy*(N/2-1);

7.3 Matlab Codes

XX = -dx*M/2:dx/4:-dx*M/2+dx/4*(4*M-1); YY = -dy*N/2:dy/4:-dy*N/2+dy/4*(4*N-1); K = 2*pi*Freq/c; dk = K(2)-K(1); kc = 2*pi*fc/c;

% wavenumber vector % wavenumber resolution % center wavenumber

%% ___________MATCHING PURSUIT________________________ collectedData = zeros(200,3); %initilize scattering center info ES = Es; PHI = Phir; Power1 = sum(sum(Es).^2); % initial power of the data axisX = X(1):dx/4:X(end); axisY = Y(1):dy/4:Y(end); cosPhi = cos(PHI.'); sinPhi = sin(PHI.'); for NN = 1:250 % extract 300 scattering centers Amax = 0; p1Max = zeros(size(ES)); for Xn = axisX for Yn = axisY p1 = exp(-1i*2*K*(cosPhi.*Xn+sinPhi.*Yn)); A = sum(sum(ES.*p1))/(size(ES,1)*size(ES,2)); if A > Amax Amax = A; collectedData(NN,1:3) = [A Xn Yn]; p1Max = conj(p1); end end end ES = ES-(Amax.*p1Max); plot(abs(collectedData(:,1))); drawnow end %—Figure 7.10(a)——————————————————————————————————————— %—SCATTERING CENTER INFO DISPLAY—————————————————————————————

377

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7 Scattering Center Representation of Inverse Synthetic Aperture Radar

SSs = collectedData; figure; plot(abs(SSs (1:250,1)),'square','MarkerSize',4,'MarkerFaceColor',[1 0 0]); grid minor set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itScattering center #'); ylabel('\itAmplitude, mV/m'); %—Figure 7.10(b)——————————————————————————————————————— figure; hold for m=1:250 t = round(abs(SSs(m,1))*20/abs(SSs(1,1)))+1; plot(-SSs(m,2),SSs(m,3),'o','MarkerSize', t,'MarkerFaceColor',[1 0 0]); end hold grid minor axis([min(X) max(X) min(Y) max(Y)]) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itcross-range, m'); title('\itLocations of scattering centers with relative amplitudes ') %–ISAR IMAGE COMPARISON———————————————————————————————— % ISAR 4x UPSAMPLED————————————————— %—Figure 7.14(a)——————————————————————————————————————— win = hamming(M)*hamming(N).'; Enew = Es.*win; Enew(M*4,N*4) = 0; ISARorig = fftshift(fft2(Enew)); ISARorig = ISARorig/M/N; figure; matplot(X,Y,abs(ISARorig(4*M:-1:1,:).'),50);

7.3 Matlab Codes

% matplot(X,Y,abs(ISARbuilt(:,4*M:-1:1)),30); colormap(hot) cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); % line (-xyout_xout,xyout_yout,'Color','k','LineStyle','.'); xlabel('\itrange, m'); ylabel('\itcross-range, m'); title('\itOriginal ISAR image') %—————FIELD COMPARISON——————————————————— %——————Field Reconsctruction————————— Esr = zeros(M,N); for NN = 1:250 A = collectedData(NN,1); x1 = collectedData(NN,2); y1 = collectedData(NN,3); Esr = Esr + A*exp(1i*2*K*(cosPhi*x1+sinPhi*y1)); end %—Figure 7.14(b) —————————————————————————————————————— Enew = Esr.*win; Enew(M*4,N*4) = 0; ISARrec = fftshift(fft2(Enew)); ISARrec = ISARrec.'/M/N; % reconstructed ISAR image figure; matplot(X,Y,abs(ISARrec(:,4*M:-1:1)),50); colormap(hot) cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); % line (-xyout_xout,xyout_yout,'Color','k','LineStyle','.'); xlabel('\itrange, m');

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7 Scattering Center Representation of Inverse Synthetic Aperture Radar

ylabel('\itcross-range, m'); title('\itReconstructed ISAR image')

%———Figure 7.11(a)————————————————————————————————————— figure; matplot(Freq,PHI*180/pi,abs((Es.')),30); colormap(hot) cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\itangle, degree'); xlabel('\itfrequency, GHz'); title('\itOriginal back-scattered field') %—Figure 7.11(b) —————————————————————————————————————— figure; matplot(Freq,PHI*180/pi,abs((Esr.')),30); colormap(hot) cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\itangle, degree'); xlabel('\itfrequency, GHz'); title('\itReconstructed back-scattered field') %——————RECONSTRUCT THE FIELD PATTERN x5——————————— Nup = 5; Esr = zeros(M*Nup,N*Nup); k = K; kk = K(1): dk/Nup: K(1)+dk/Nup*(Nup*M-1); dkk = kk(2)-kk(1); pp = Phir(1): dphi/Nup: Phir(1)+dphi/Nup*(Nup*N-1); dpp = pp(2)-pp(1);

7.3 Matlab Codes

csP = cos(pp); snP = sin(pp); for NN = 1:250 A = collectedData(NN,1); x1 = collectedData(NN,2); y1 = collectedData(NN,3); Esr = Esr+A*exp(1i*2*kk.'*(csP.*x1+snP.*y1)); end %—Figure 7.12 ————————————————————————————————————————— figure; matplot(Freq,PHI*180/pi,abs((Esr.')),30); colormap(hot) cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\itangle, degree'); xlabel('\itfrequency, GHz'); title('Reconstructed field (x5 upsampled)') %—Figure 7.13(a)——————————————————————————————————————— nn = 31; h = figure; plot(Freq,abs(Es(:,nn)),'k-. *','MarkerSize',8,'LineWidth',2); hold on; plot(kk*c/2/pi,abs(Esr(:,Nup*(nn-1)+1)),'r','LineWidth',2); hold off; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); grid minor xlabel('\itfrequency, GHz'); ylabel('\itE-field, V/m'); tt = num2str(PHI(nn)*180/pi); ZZ = ['@ PHI = ' tt ' Deg.']; title(ZZ); drawnow;

381

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7 Scattering Center Representation of Inverse Synthetic Aperture Radar

legend('with brute force computation','with scattering centers') axis tight %—Figure 7.13(b)——————————————————————————————————————— nn = 34; h = figure; plot(PHI*180/pi,abs(Es(nn,:)),'k-.*','MarkerSize',8, 'LineWidth',2); hold on; plot(pp*180/pi,abs(Esr(Nup*(nn-1)+1,:)),'r-', 'LineWidth',2); hold off; grid minor set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\phi, degree'); ylabel('\itE-field, V/m'); tt = num2str(Freq(nn)); ZZ = ['@ f = ' tt ' GHz']; title(ZZ); drawnow; legend('with brute force computation','with scattering centers') axis tight

References Bhalla, R. and Ling, H. (1996). Three-dimensional scattering center extraction using the shooting and bouncing ray technique. IEEE Transactions on Antennas and Propagation 44 (11): 1445–1453. https://doi.org/10.1109/8.542068. Bhalla, R., Moore, J., and Ling, H. (1997). A global scattering center representation of complex targets using the shooting and bouncing ray technique. IEEE Transactions on Antennas and Propagation 45 (12): 1850–1856. https://doi.org/ 10.1109/8.650204. Bicer, M.B., Akdagli, A., and Özdemir, C. (2018). A matching-pursuit based approach for detecting and imaging breast cancer tumor. Progress in Electromagnetics Research 64: 65–76.

References

Carriere, R. and Moses, R.L. (1992). High-resolution radar target modeling using a modified Prony estimator. IEEE Transactions on Antennas and Propagation 40 (1): 13–18. https://doi.org/10.1109/8.123348. Chen, V.C. and Ling, H. (2002). Time-Frequency Transforms for Radar Imaging and Signal Analysis. Boston: Artech House. Högbom, J.A. (1974). Aperture synthesis with a non-regular distribution of interferometer baselines. Astronomy and Astrophysics Supplement 15: 417–426. Hurst, M. and Mittra, R. (1987). Scattering center analysis via Prony’s method. IEEE Transactions on Antennas and Propagation 35 (8): 986–988. https://doi.org/10.1109/ TAP.1987.1144210. Kırık, Ö. and Özdemir, C. (2019). An accurate and effective implementation of physical theory of diffraction to the shooting and bouncing ray eethod via PREDICS tool. Sigma Journal of Engineering and Natural Science 37 (4): 1153–1166. Mallat, S.G. and Zhang, Z. (1993). Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing 41 (12): 3397–3415. https://doi.org/10.1109/ 78.258082. Özdemir, C., Bhalla, R., and Ling, H. (1998). Radiation center representation of antenna synthetic aperture radar (ASAR) images. IEEE Antennas and Propagation Society International Symposium. 1998 Digest. Antennas: Gateways to the Global Network. Held in Conjunction with: USNC/URSI National Radio Science Meeting (Cat. No.98CH36). Atlanta, USA: IEEE. https://doi.org/10.1109/APS.1998.699149. Özdemir, C., Bhalla, R., and Ling, H. (2000). A radiation center representation of antenna radiation patterns on a complex platform. IEEE Transactions on Antennas and Propagation 48 (6): 992–1000. https://doi.org/10.1109/8.865235. Özdemir, C., Kirik, O., and Yilmaz, B. (2009). Sub-aperture method for the widebandwidth wide-angle inverse synthetic aperture radar imaging. 2009 International Conference on Electrical and Electronics Engineering – ELECO 2009 (5–8 November 2009), 288–292. Bursa, Turkey: IEEE. Özdemir, C., Yılmaz, B., and Kırık, Ö. (2014). pRediCS: a new GO-PO based ray launching simulator for the calculation of electromagnetic scattering and RCS from electrically large and complex structures. Turkish Journal of Electrical Engineering & Computer Sciences 22: 1255–1269. https://doi.org/10.3906/elk-1210-93. Segalovitz, A. and Frieden, B.R. (1978). A ‘CLEAN’-type deconvolution algorithm. Astronomy and Astrophysics 70 (3): 335–343. Su, T., Özdemir, C., and Ling, H. (2000). On extracting the radiation center representation of antenna radiation patterns on a complex platform. Microwave and Optical Technology Letters 26 (1): 4–7. Tsao, J. and Steinberg, B.D. (1988). Reduction of sidelobe and speckle artifacts in microwave imaging: the CLEAN technique. IEEE Transactions on Antennas and Propagation 36 (4): 543–556. https://doi.org/10.1109/8.1144.

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Tseng, N.Y. and Burnside, W.D. (1992). A very efficient RCS data compression and reconstruction technique. Technical Report No. 722780-4. Ohio State University, Electroscience Laboratory, Columbus, OH, United States. Wang, S.Y. and Jeng, S.K. (1995). Generation of point scatterer models using PTD/SBR technique. IEEE Antennas and Propagation Society International Symposium. 1995 Digest (18–23 June 1995), 1914–1917. Newport Beach, CA, USA: IEEE. https://doi. org/10.1109/APS.1995.530964. Yu, W.P., To, L.G., and Oii, K. (1991). N-Point scatterer model RCS/Glint reconstruction from high-resolution ISAR target imaging. Proceedings of End Game Measurement and Modeling Conference (January 1991), Point Mugu, California, USA, pp. 197–212.

385

8 Motion Compensation for Inverse Synthetic Aperture Radar For the operational inverse synthetic aperture radar (ISAR) situation, the target’s relative movement with respect to the radar sensor provides the angular diversity required for range-Doppler ISAR imagery as given in Chapter 6. For the groundbased ISAR systems, for example, collecting back-scattered energy from an aerial target that is moving with a constant velocity for a sufficiently long period of time can provide the necessary angular extend to form a successful ISAR image. On the other hand, real targets such as planes, ships, helicopters, and tanks do have usually complicated motion components while maneuvering. These may include translational and rotational (yaw, roll, and pitch) motion parameters such as velocity, acceleration, and jerk. Moreover, all these parameters are unknown to the radar engineer, which adds further complexities to the problem. Therefore, trying to estimate these motion parameters and also trying to invert the undesired effects of motion on the ISAR image is often called motion compensation (MOCOMP). Since the motion parameters are unknown to the radar sensor, the MOCOMP process can be regarded as a blind process and is also assumed to be one of the most challenging tasks in ISAR imaging research. The MOCOMP procedure has to be employed in all SAR and ISAR applications to obtain a clear and focused image of the scene or the target. In SAR applications, for example, the information gathered from the radar platform’s inertial measurement system, gyro, and/or global positioning system (GPS) is generally used to correct the motion effects on the phase of the received signal (Eichel et al. 1989). However, the situation is very different in ISAR applications such that all motion parameters, including velocity, acceleration, jerk, and the type of maneuver (straight motion, yaw, roll, and pitch), are not known by the radar. Therefore, these parameters must somehow be estimated and then eliminated to have a successful ISAR image of the target. If an efficient compensation routine is not applied, the resultant ISAR image is defocused and blurred in range and cross-range dimensions. Various methods Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms: With Advanced SAR/ISAR Imaging Concepts, Algorithms, and MATLAB Codes, Second Edition. Caner Özdemir. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

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from many researchers have been suggested to mitigate or eliminate these unwanted motion effects in ISAR imaging (Chen and Andrews 1980; Haywood and Evans 1989; Zhu and Wu 1991; Calloway and Donohoe 1994; Wahl 1994; Itoh et al. 1996; Wang and Bao 1997; Xi et al. 1999; Cho et al. 2000; Küçükkiliç 2006). The single-scattering referencing algorithm (Chen and Andrews 1980), the multiple-scatterer method (Zhu and Wu 1991), the centroid tracking algorithm (Itoh et al. 1996; Cho et al. 2000), the entropy minimization method (Wang and Bao 1997; Xi et al. 1999), the phase gradient autofocusing technique (Calloway and Donohoe 1994; Wahl 1994), the cross-correlation method (Haywood and Evans 1989; Küçükkiliç 2006), and the joint time-frequency (JTF) methods (Chen and Qian 1998; Wang et al. 1998; Chen and Ling 2002; Choi et al. 2003) are popular ones among numerous ISAR MOCOMP techniques.

8.1

Doppler Effect Due to Target Motion

When the scatterer on the target is moving, the Doppler shift posed by the scatterer’s line-of-sight (LOS) velocity sets “inaccurate” distance information about the position of the scatterer to the phase of the received electromagnetic (EM) wave. While the scatterer is moving fast, it may occupy several pixels in the image during the integration interval of ISAR. Therefore, the phase of the backscattered wave is altered such that the resultant ISAR image is mislocated in cross-range and defocused in both range and cross-range domains. If the scatterer is not moving fast, the ISAR image may not be blurred. However, the location of the scatterer will still not be true due to the Doppler shift imposed by the target’s movement. The effect of the target’s motion on the phase of the backscattered wave and/or to the ISAR image is investigated based on the geometry illustrated in Figure 8.1. In the general case, the target may have both radial and rotational motion during the illumination period of radar. For this reason, the point scatterer at P(x, y) on the target is assumed to have both radial and rotational motion components. According to the practical convention of radar imaging, the phase center is selected in the middle of the target and is assumed to be the origin. We would like to estimate the phase error induced due to target motion. If the target is situated at the far field of the radar, the distance of point P from the radar can be approximated as (Chen and Ling 2002) r t

R t + x cos φ t − y sin φ t

81

Here, R(t) is the target’s translational range distance from the radar and φ(t) represents the rotational angle of the target with respect to the radar LOS

8.1 Doppler Effect Due to Target Motion

y

v

x ϕ (t) u

R(t) r(t)

Radar

P(x,y)

Figure 8.1 Geometry for a moving target with respect to radar.

(RLOS) axis, u. Expanding time varying R(t) and φ(t) into their Taylor series, they can be represented to yield 1 2 at t + 2 1 φ t = φo + ωr t + αr t 2 + 2

R t = Ro + vt t +

82

Here, Ro is the initial range of the target, and vt and at are the target’s translational velocity and the acceleration, respectively. Higher-order terms starting from the target’s translational jerk follow these first three terms. Similarly, φo is the initial angle of the target with respect to the RLOS axis. ωr and αr are the angular velocity and the angular acceleration of the target, respectively. The phase of the backscattered signal from point P can be written as φ t = − 2k r t 2r t = − 2πf c

83

Therefore, the Doppler frequency shift due to motion can be calculated by taking the time derivative of this phase as 1 ∂ φt 2π ∂t 2f ∂ = − r t c ∂t 2f = − v t + at t + … c 2f + ω r + αr t + … c

fD =

84

x sin φ t + y cos φ t

Here, the first and second terms represent the radial (or translational) and the rotational Doppler frequency shifts, respectively. If the motion of the target is to be

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8 Motion Compensation for Inverse Synthetic Aperture Radar

approximately described with vt and ωr, then the translational and the rotational Doppler frequency shifts are reduced to the following terms: f trans D f rot D

−

2f vt c

2f ωr c

85 x sin φo + ωr t + y cos φo + ωr t

Therefore, the translational Doppler shift becomes directly related to the target’s translational velocity, vt. On the other hand, the rotational Doppler shift posed by the target is more complex as it depends on many parameters that can be deduced from Equation 8.5.

8.2

Standard MOCOMP Procedures

Where monostatic ISAR imaging is concerned, the received signal can be theoretically approximated as the integration of the backscattered echoes from all the scatterers inside the radar beam: ∞

st =

A x, y −∞

exp

− j4πf

r t c

86

dy dx

Here, A(x, y) is the backscattered signal intensity from any point scatterer at (x, y), and f corresponds to the frequency of the radar waveform. Substituting the range equation in Equation 8.1 into 8.6, the received signal is deduced to be equal to s t = exp

− j4πf

Rt c

∞

A x, y −∞

exp

− j4πf

f x cos φ t − y sin φ t c

dy dx 87

If the target’s initial range, Ro, and the linear translational velocity, vt, are known, the phase term prior to the above integral can be removed by multiplying Equation 8.7 by the term of “exp(j4πfR(t)/c).” This is called range tracking or the coarse MOCOMP, and it is the standard procedure for compensating the translational motion effects. After obtaining the phase-compensated backscattered signal, a Fourier transform (FT) operation can then be applied to image the backscattered signal intensity function, A(x, y) (Chen and Ling 2002). If the scatterers pass different range cells during the coherent integration time, the resultant phase-compensated image will still be defocused. Therefore, a finer compensating technique called Doppler tracking that attempts to make the Doppler shifts constant is required (Kirk 1975; Walker 1980; Wu et al. 1995). This procedure is also called the fine MOCOMP.

8.2 Standard MOCOMP Procedures

The signal processing tools used to reduce motion errors, that is, MOCOMP techniques, are usually treated in two steps: First, the effects due to translational motion components are solved. Then, the errors associated with the rotational movement of the target are dealt with.

8.2.1 Translational MOCOMP The target’s radial or translational motion is defined as the movement of the target along the range axis (or RLOS axis) of the radar. The target’s translational motion is one of the most significant components that affects image quality in the ISAR image. The main effect of the target’s translational motion is shifting the positions of the scatterers on the target along the the range axis. This is because of the fact that target’s radial distance changes for consecutive radar pulses as they are sent at different time instants while the target is moving. Therefore, the phase of the collected electric field data is misaligned along the pulses so that the Doppler frequencies that are used to estimate the exact locations of the target are spread out over a finite number of range cells. When the FT is directly applied to the collected data that contain translational motion, the location of the point scatterer is poorly estimated since the scatterer is visible for all of that finite number of range cells. Therefore, the scatterers look as if they “walk” over the range cells. The range walk phenomena can negatively affect the range resolution, range accuracy, and signal-to-noise ratio (SNR) of the resulting ISAR image. Therefore, the target’s resultant image before the compensation may be smeared in the cross-range direction and defocused in range and crossrange directions. The amount of smearing, of course, depends on the amount of target’s radial motion (or the radial velocity). Although there may be little or no smearing effects for small radial velocity values as in the case of slowly moving ship targets, the image smearing can be drastic for fast-moving targets such as fighter airplanes. Usually, an algorithm is applied to overcome the range walk issue by trying to align the range bins. The common name for keeping the scatterers in their range cells is range tracking. 8.2.1.1

Range Tracking

There are different range tracking methods that have been employed by various researchers (Chen and Andrews 1980; Xu et al. 1990; Itoh et al. 1996; Wang et al. 1998; Cho et al. 2000; Wenxian et al. 2001; Choi et al. 2003; Martorella et al. 2003; Berizzi et al. 2004; Küçükkiliç 2006; Munoz-Ferreras et al. 2006; Yu and Yang 2008). The cross-correlation method, for example, calculates the correlation coefficients between the adjacent range profiles and tries to estimate the range walk between the range profiles (Chen and Andrews 1980; Choi et al. 2003; Küçükkiliç 2006).

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Another widely applied range tracking method is called target centroid tracking (Itoh et al. 1996; Cho et al. 2000). The main idea in this method is to estimate the radial motion of the target centroid point and to compensate in such a way that the range and the Doppler shift of the target centroid are kept constant. Another famous scheme is called prominent point processing (PPP) (Wang et al. 1998; Wenxian et al. 2001; Martorella et al. 2003; Berizzi et al. 2004; Küçükkiliç 2006; Munoz-Ferreras et al. 2006; Yu and Yang 2008). The first step in the PPP algorithm is to select prominent points on the target. Assuming that the phase components of these dominant point scatterers are known, the translational motion error can be mitigated by unwrapping the higher-order phase component of the first prominent point as the second step of the algorithm. In the third step, the rotational motion error is eliminated by extracting the relationship between the rotation angle and the dwell time from the phase of the second prominent points. Finally, the rotational velocity can be estimated by measuring the phase of the third prominent points (Wang et al. 1998; Küçükkiliç 2006). 8.2.1.2 Doppler Tracking

While the range tracking procedures are capable of aligning all the scatterers at their correct range cells, the Doppler frequency shifts of these scatterers may still be varying with time in the phase of the received signal. In other words, the Doppler frequency shifts may not be constant between the scatterers. These Doppler frequency shifts can be caused by the target’s movement along the range direction and changes in the instantaneous radial velocity of the target with respect to radar (Haywood and Evans 1989). The procedure that tries to make the Doppler frequency shifts constant among the range cells is called Doppler tracking (Haywood and Evans 1989; Itoh et al. 1996; Chen and Ling 2002; Küçükkiliç 2006). Some of the popular Doppler tracking algorithms that have been widely applied are listed as follows: dominant scatterer algorithm (Steinberg 1988; Haywood and Evans 1989), sub-aperture approach (Calloway and Donohoe 1994), cross-range centroid tracking algorithm (Itoh et al. 1996), phase gradient autofocus technique (Wahl 1994), and multiple PPP technique (Werness et al. 1990; Carrara et al. 1995).

8.2.2

Rotational MOCOMP

The target’s rotational motion is defined as the movement that causes aspect change of the target from the RLOS. As thoroughly presented in Chapter 6, a small value of rotation of the target with respect to the RLOS is adequate to form the range-Doppler ISAR image. However, when the rotation of the target is not small for the coherent integration time of the radar, the Fourier-based ISAR imaging algorithm produces degradations such as blurring and smearing in the resultant ISAR image.

8.2 Standard MOCOMP Procedures

In most ISAR scenarios, the target’s motion contains not only rotational motion components but also translational motion components. Usually, the unfavorable effect caused by the translational motion is more severe than that caused by rotational motion. Therefore, the rotational MOCOMP is applied after the translational MOCOMP in the general treatment of ISAR MOCOMP. Let us consider the scenario in Figure 8.1, wherein the target has a general motion of both translational and rotational motion. Substituting Equation 8.1 into 8.3, we get the following for the phase of the received signal: 4πf R t + x cos φ t − y sin φ t 88 c To have rotationally motion-compensated ISAR image, this phase should only be a linear function of angular velocity as demonstrated in Chapter 6. With this construct, a Fourier-based ISAR imaging procedure will be able to resolve the points in the cross-range directions. However, the phase in Equation 8.3 is a nonlinear function of angular velocity and is quite complex. Let us simplify this phase by assuming that the target has only radial velocity and angular velocity such that φt = −

R t = Ro + vt t

89

φ t = φo + ωr t Then, the phase of the received signal becomes φt = −

4πf 4πf Ro + vt t − x cos φ0 + ωr t − y sin φo + ωr t c c 8 10

The first term is related to the translational motion, and the second term is responsible for the rotational motion. Without loss of generality, we can set φo = 0 to have 4πf x cos ωr t − y sin ωr t 8 11 c For sufficiently small values of angular velocity or the coherent integration time of ISAR or both, that is, the argument “ωrt” is small, then cos(ωrt) ≈ 1 and sin (ωrt) ≈ ωrt. Therefore, Φrot t = −

Φrot t

−

4πf c

x−

4πf ωr t c

y

8 12

Noting that the Doppler frequency shift is equal to fD = 2ωry/λ, this equation can then be rewritten as Φrot t

−

4πf c

x − 2π f D

t

8 13

391

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8 Motion Compensation for Inverse Synthetic Aperture Radar

This result gives exactly the same phase value as in Equation 6.26 (Chapter 6) that is necessary for an ideal range-Doppler processing. When the “ωrt” is not small, this rotational motion has to be compensated for by applying a compensating procedure to have a distortion-free ISAR image. JTF-based methods have proven to be very effective for removing the rotational motion effects from the ISAR image (Chen and Qian 1998; Wang et al. 1998; Chen and Ling 2002; Choi et al. 2003). An example based on JTF processing for compensating the rotational motion will be given in Section 8.3.3.3. Another popular method for rotational MOCOMP is PPP (Wang et al. 1998; Wenxian et al. 2001; Martorella et al. 2003; Berizzi et al. 2004; Küçükkiliç 2006; Munoz-Ferreras et al. 2006; Yu and Yang 2008).

8.3

Popular ISAR MOCOMP Techniques

As the names of MOCOMP methods have been listed in the above paragraphs, there are many algorithms studied by numerous researchers. In this section, we will present the most famous and popular MOCOMP algorithms associated with numerical examples in Matlab.

8.3.1

Cross-Correlation Method

The cross-correlation method is one of the basic and most applied range tracking algorithms. The presented algorithm here relies on the stepped-frequency continuous-wave (SFCW) radar configuration. Let us assume that radar system sends out the stepped frequency waveform of M bursts each having N pulses toward the target. The target’s translational velocity, vt, is assumed to be constant. Therefore, radar collects the two-dimensional (2D) backscattered electric field data, Es[m, n], of size (M N). Then, the phase of the mth burst and the nth pulse can be written in terms of vt as (Küçükkiliç 2006) φ Es m, n

= −

4π f n Ro − T PRI vt n − 1 + N m − 1 c

,

m

=

n

=

1, 2, …, M

, 1, 2, …, N 8 14

where fn is the stepped frequency value for the nth transmitted pulse, Ro is the initial radial location of the target from radar, and TPRI is the time lag between adjacent pulses or simply the pulse repetition interval (PRI). Similarly, the phase of the (m + 1)th burst and nth pulse is equal to φ Es m + 1 , n

= −

4π f n Ro − T PRI vt n − 1 + Nm c

8 15

8.3 Popular ISAR MOCOMP Techniques

Therefore, the phase difference between the adjacent bursts can be calculated as Δφburst-to-burst = φ Es m, n + 1 − φ E s m, n 4π f m vt T PRI M = c 4π f m = ΔRburst-to-burst , c

8 16

where ΔRburst -to- burst = vt (TPRI N) is the so-called “range walk” between the adjacent bursts. This range shift can be compensated by applying the following steps: 1) First, one-dimensional (1D) discrete FT (DFT) is applied along the pulses such that a total of M range profile vector, RPm of length N is obtained. 2) One of the range profiles is taken as the reference. In practice, the first one, RP1, is usually chosen due to the fact that its phase is usually either in advance or in lag when compared to phases of all the others. 3) The cross-correlations of the magnitudes of other (M − 1) range profiles to that of the reference range profile are calculated via computing the following crosscorrelation factor: CCRm =

IDFT DFT RPref

DFT RPm

∗

,

m = 1, 2, …, M − 1 8 17

Here, IDFT stands for the inverse DFT. Notice that each CCRm vector is also of length N. 4) The locations of the peak value for the calculated cross-correlations indicate the range shifts (or time delays) that are required to align each RP with respect to the reference range profile of RPref: K m = index max CCRm , m = 1, 2, …, M − 1

8 18

5) The resultant index vector is usually smoothed by fitting to a lower order polynomial so that the gradual change between the index vector K is almost constant: Sm = smooth K m , m = 1, 2, …, M − 1

8 19

6) Therefore, the range walk between the nth range profile, RPm, and the reference range profile, RPref, can then be approximated as ΔRn − to − ref

Sm Δr,

8 20

where Δr is the range resolution and is given by Δr = c 2B , where B is the total frequency bandwidth.

8 21

393

8 Motion Compensation for Inverse Synthetic Aperture Radar

7) As the last step, the compensating phase vector for the range profile RPm equals Δφm − to − ref =

4π f n ΔRm − to − ref , n = 1, 2, …, N c

8 22

Then, the motion-compensated range profile can be obtained by using this correcting phase as RPm = DFT Δφm − to − ref IDFT RPm

8 23

Once all M range profiles are corrected, a motion-compensated ISAR image can then be generated using conventional ISAR imaging routines.

8.3.1.1 Example for the Cross-Correlation Method

We will demonstrate the concept of range tracking by applying the crosscorrelation method over a numerical example. A hypothetical fighter, shown in Figure 8.2, composed of perfect point scatterers of equal magnitudes is chosen. The target, at an initial radial distance of Ro = 16 km, is moving toward the radar with a radial velocity of vt = 70 m/s. The target has a radial acceleration value of at = 0.1 m/s2. We also assume that target is rotating slowly with an angular velocity of φr = 0.03 rad/s. The radar sends 128 bursts, each having 128 modulated pulses. The frequency of the first pulse is fo = 10 GHz and the total frequency bandwidth is B = 128 MHz. Pulse repetition frequency (PRF) of the radar system is chosen as 20 KHz. 30 20 10 y, m

394

0 –10 –20 –30

–30

–20

–10

0

10

20

30

x, m

Figure 8.2

A fighter target composed of perfect point scatterers.

8.3 Popular ISAR MOCOMP Techniques

0 –2 20

–4 –6

Doppler index

40

–8 60

–10 –12

80

–14 100

–16 –18

120

–20 –60

–40

–20

0 20 Range, m

40

60

dB

Figure 8.3 Traditional ISAR image of the fighter target (no compensation).

First, we obtained the conventional range-Doppler ISAR image of the fighter by employing traditional ISAR imaging procedures without applying any compensation for the motion of the target. The resultant raw range-Doppler ISAR is depicted in Figure 8.3. As can be clearly seen from the figure, the effect of target’s motion is severe in the resultant ISAR image such that the image is broadly blurred in the range and Doppler domains, and the true locations of the target’s scattering centers cannot be retrieved from the image. Next, the cross-correlation method is applied to track the range and compensate for the motion of the target. First, the range profiles of the target are obtained by applying 1D inverse FT (IFT) operation to the backscattered electric field along the frequencies. After taking the first range profile, PR1, as the reference, the cross-correlation between the reference range profile and the others are calculated by using the formula in Equation 8.17. As explained in the algorithm, the index for the maximum value of the correlations indicates the time shift required to align the range profiles. After finding these indices and multiplying them with the calculated range resolution value of Δr = c/2B = 1.17 m (for this example, we get the range shifts of the range profiles with respect to PR1). These range walks and their smoothed versions with respect to the range profile index are plotted in Figure 8.4 as solid and dashed lines, respectively. While smoothing the range profile shifts to a lower order polynomial (a line for this example), Robust Lowess method is utilized (Cleveland 1979). Furthermore, the difference between the consecutive range walks is plotted in Figure 8.5a where these differences are almost constant. Then, the target’s radial translational speed can be found by dividing these range walk differences by the time differences between each burst. This calculated speed versus range profile

395

8 Motion Compensation for Inverse Synthetic Aperture Radar

80 RP shifts Smoothed RP shifts

70

60

50

40

30 20 20

Figure 8.4

40

60 80 Range profile index

100

120

Range profile shifts and their smoothened versions versus range profile index.

Range differences, m

(a) –0.3 –0.4 –0.5 –0.6 20

40

(b)

60 80 Range profile index

100

120

75 Iradial speed, m/s

396

Actual speed = 70 m/s, Est. average speed = 70.7639 m/s

70

65 20

40

60 80 Range profile index

100

120

Figure 8.5 (a) Range differences with respect range profile index, and (b) radial velocity with respect range profile index.

8.3 Popular ISAR MOCOMP Techniques

Figure 8.6 Motion-compensated ISAR image of the fighter target.

Motion compansated ISAR image 0 –2 20

–4 –6

Doppler index

40

–8 60

–10 –12

80

–14 100

–16 –18

120 –20

0 20 Range, m

dB

–20

index is plotted in Figure 8.5b. From this figure, we see that the speed is almost constant, around 70 m/s. If we take the average of this speed vector, we get an estimated average value of vest t = 70 76 m s for the target’s radial translational speed, which is very close to the actual speed of vt = 70 m/s. At the last step of the algorithm, the phase contribution caused by the target’s motion can be compensated for by multiplying the scattered field data with the below phase term as E scomp m, n = E s m, n f exp j4π n vest n−1 + N m−1 c t

,

m = 1, 2, …, M n = 1, 2, …, N

8 24

Once the phase of the collected scattered field is compensated, the ISAR image can then be obtained by applying the regular ISAR imaging procedures. Figure 8.6 shows the resultant motion-compensated ISAR image after applying the whole process explained above. The dominant motion effects of translational velocity vt = 70 m/s are successfully eliminated, and the image of the fictitious fighter is almost perfectly focused. We also notice that radial translation acceleration of at = 0.1 m/s2 and angular speed φr = 0.03 rad/s have little effect on the resultant motion-compensated image as observed from Figure 8.6. When these values are taken to be sufficiently greater, however, the image distortion/blurring is unavoidable if only the range tracking procedure is used as the compensation tool.

397

398

8 Motion Compensation for Inverse Synthetic Aperture Radar

8.3.2

Minimum Entropy Method

Another popular tool that has been used as a MOCOMP technique for removing translational motion effects in ISAR images is called the minimum entropy method. In fact, the concept of entropy is commonly utilized in engineering to measure the disorders of any system (Shannon 1951). In SAR/ISAR imagery, the entropy phenomenon is used in a similar manner to estimate the disorder in the image (Wang and Bao 1997; Xi et al. 1999; Shin and Myung 2008). In ISAR imaging, the minimum entropy method tries to estimate the motion parameters (such as velocity and acceleration) of the target. This task is accomplished by calculating the entropy of the energy in the image and minimizing this parameter by iteratively trying out the possible values of motion parameters. The details of this method will be presented next.

8.3.2.1 Definition of Entropy in ISAR Images

We assume that the target has translational motion parameters, both the translational velocity of vt and a radial acceleration of at. With this construct, the phase of the backscattered signal can be written in the following form: 4πf c

φ Es = −

Ro +

vt t +

1 2 at t 2

8 25

Here, Ro is the initial radial distance of the target from the radar. The sign of vt can be either plus or minus for an approaching or retreating target, respectively. Similarly, the sign of at can be either plus or minus for an accelerating or decelerating target, respectively. The first phase, −4πfRo/c, is constant for all time values and therefore can be suppressed for imaging purposes. With this convention, the effect of motion can then be compensated if the scattered electric field is multiplied by the following compensating phase term: S = exp

j

4πf c

vt t +

1 2 at t 2

8 26

Therefore, the goal of the algorithm is to estimate the motion parameters of vt and at to be able to successfully remove their effects from the phase of the received signal. If the ISAR image matrix is I and has M columns and N rows, then the Shannon entropy, E, is defined as (Shin and Myung 2008) M

N

E I = −

I m, n log10 I m, n m=1n=1

8 27

8.3 Popular ISAR MOCOMP Techniques

where I m, n =

I m, n M

N

8 28

I m, n m=1n=1

Here I is the normalized version of the ISAR image. The normalization is accomplished by dividing the image pixels by the total energy in the image. Once the entropy is defined for the ISAR image itself, the goal is to find the corresponding compensation vector (so the motion parameters) such that the new ISAR image has the minimum entropy (or the disorder). The process of searching for the correct values of motion parameters can be done iteratively as will be demonstrated with a numerical example next.

8.3.2.2

Example for the Minimum Entropy Method

In this example, we will demonstrate the use of the minimum entropy method for compensating the motion effects in an ISAR image. First, we use a target composed of discrete perfect point scatterers that have equal scattering amplitudes as shown in Figure 8.7. The target is moving away from the radar with a radial translational velocity of vt = 4 m/s and with a radial translational acceleration of at = 0.6 m/s2. The target’s angular velocity is chosen as ωr = 0.06 rad/s. The target’s initial radial distance from the radar is taken as Ro = 500 m. The radar sends 128 bursts, each having 128 modulated pulses. The frequency of the starting pulse is f0 = 8 GHz, and the total frequency bandwidth is B = 384 MHz. The PRF is chosen as 14.5 KHz. Without applying any compensation, the rangeDoppler ISAR image of the target is constructed with the help of conventional ISAR imaging techniques, and the corresponding range-Doppler ISAR image is obtained as shown in Figure 8.8. As is obvious from the image, the uncompensated ISAR image is highly distorted and blurred due to both the translational and the rotational motion of the target. In Figure 8.9, the spectrogram of received time pulses is plotted. This figure clearly demonstrates the progressive shift in the frequency (or in the phase) of the consecutive received time pulses. This shift occurs due to the change of target’s range distance from the radar during the integration time of the ISAR process. If a successful MOCOMP practice is applied, there is expected to be no (or minimal) range shift between consecutive time pulses. Then, the minimum entropy methodology is applied to the ISAR image data in Figure 8.8. The algorithm iteratively searches for the values of vt and at by minimizing the entropy of the compensated ISAR image of I =

−1 2

S Es ,

8 29

399

8 Motion Compensation for Inverse Synthetic Aperture Radar

20 15 10

y, m

5 0 –5 –10 –15 –20 –20

Figure 8.7

–15

–10

–5

0 x, m

5

10

15

20

A hypothetical target composed of perfect point scatterers. 0 –2

20

–4 –6

40 Doppler index

400

–8 60

–10 –12

80

–14 100

–16 –18

120 –20

Figure 8.8

–10

0 10 Range, m

20

dB

–20

Conventional ISAR image of the airplane target (no compensation).

where 2− 1 is the 2D IFT operation and S is defined as in Equation 8.26 for different values of vt and at. The estimated values for vt and at are found by iteratively searching the minimum value of the entropy as defined in Equation 8.27. Figure 8.10 shows the graph of the entropy value for different values of translational velocity and acceleration. The 2D search space makes a minimum where vest t

8.3 Popular ISAR MOCOMP Techniques

Spectrogram 0

Frequency index

50

100

150

200

250 20

40

60 80 Time pulses

100

120

Figure 8.9 Spectrogram of range cells (before compensation).

Entropy value

4 3.9 X 0.6 Y4 Z 3.662

3.8 3.7 3.6 15

1 10

5

0.5 0

Translational vel., m/s

–5

0 –10

–15

Translational acc., m/s2

Figure 8.10 Entropy plot for translational radial velocity translational radial acceleration search space.

401

8 Motion Compensation for Inverse Synthetic Aperture Radar

0 140

–2

Figure 8.11 ISAR image of the airplane after applying minimum entropy compensation.

–4 160 –6 Doppler index

402

–8

180

–10 200

–12 –14

220

–16 240

–18 –20

–10

0 10 Range, m

20

dB

–20

2 parameter becomes equal to 4 m/s and aest t parameter equals to 0.6 m/s as demonstrated in Figure 8.10. Therefore, the algorithm successfully estimates the correct values of vt and at. The effect of motion in the scattered field can then be mitigated by multiplying it with the compensating phase term as given in Equation 8.29. Consequently, the motion-compensated ISAR image is obtained as shown in Figure 8.11 by applying the regular DFT-based ISAR imaging technique. The compensated ISAR image clearly demonstrates that the unwanted effects due to target’s motion are eliminated after applying the minimum entropy methodology. The target’s scattering centers are very well displayed with good resolution. A further check is done by looking at the spectrogram of the motion-compensated received time pulses as illustrated in Figure 8.12. As is obvious from this spectrogram, the range delays between the time pulses are eliminated such that all frequency (or the phase) values of the returned pulses are aligned to each other successfully.

8.3.3

JTF-Based MOCOMP

The JTF tools have been extensively utilized in EM applications ranging from analysis and interpretation of EM signals (Moghaddar and Walton 1993; Trintinalia and Ling 1996; Özdemir and Ling 1997) to radar signature and target classification (Kim and Ling 1993; Kim et al. 2000). JTF representations, including short-time Fourier transform (STFT), Wigner–Ville distributions (WVD), continuous wavelet transform (CWT), adaptive wavelet transform (AWT), and Gabor wavelet

8.3 Popular ISAR MOCOMP Techniques

Spectrogram 0

Frequency index

50

100

150

200

250 20

Figure 8.12

40

60 80 Time pulses

100

120

Spectrogram of range cells (after compensation).

transform (GWT), have been shown to be very effective in analyzing the ISAR image of the complicated moving targets as well (Chen and Qian 1998; Xia et al. 2002), In this section, we will present the use of JTF representation in compensating the motion effects in ISAR imaging.

8.3.3.1

Received Signal from a Moving Target

In real-world scenarios, the target’s maneuver can be so complex that Doppler frequency shifts in the received signal may vary with time. If the target has complex motion such as yawing, pitching, rolling, or more generally maneuvering, regular Fourier-based MOCOMP techniques may not be sufficient to model the behavior of the motion. Therefore, the use of JTF tools may provide insights into understanding and characterizing the Doppler frequency variations such that translational and rotational motion parameters such as velocity, acceleration, and jerk can be estimated with good fidelity. Let us assume that the target has a complex maneuver that can be written as a linear combination of both the translational and rotational motion components. If R(t) is the target’s translational range distance from the radar, and φ(t) is the rotational angle of the target with respect to RLOS axis as illustrated in Figure 8.1. Expanding R(t) and φ(t) into Taylor series yields the formulation listed in Equation 8.2. We first assume that the target is modeled on point scatterers, such that there exists a total number of K point-scatterers on the target. The time-domain

403

404

8 Motion Compensation for Inverse Synthetic Aperture Radar

backscattered signal at the radar receiver can then be represented as the following sum from each scattering centers on the target as K

gt =

Ak x k , yk k=1

8 30 f − j4π o R t + x k cos φ t − yk sin φ t c

exp

Here, Ak(xk, yk) is the backscattered field amplitude from the kth point scattered. When the range profiles are concerned only, the time-domain backscattered signal at a selected range cell, x, can be written in a similar manner as follows: K

g x, t =

Ak x, yk k=1

8 31

exp

− j4π

fo R t + x cos φ t − yk sin φ t c

Here, x-axis corresponds to the range direction, and t is the coherent processing interval that can also be regarded as the pulse number. Substituting R(t) and φ(t) as listed in Eq. 8.2 into 8.31 and displaying only the leading phase terms, one can get (Chen and Ling 2002) K

g x, t =

Ak x, yk k=1

exp

− j4π

fo c

Ro + x + vt + ωr yk t 1 at − ω2r x + αr yk t 2 + … + 2 8 32

The first term in the phase is constant and can be ignored for the imaging process. To have a motion-free range-Doppler image of the target, R(t) should be fixed at Ro, and φ(t) should linearly vary with time as φ(t) = ωrt. If these ideal conditions are met, the FT operation will successfully focus the cross-range points (i.e. yks) onto their correct locations. Therefore, the MOCOMP procedure should be applied to other phase terms starting from the second order in aiming to suppress them in the phase of the received signal. 8.3.3.2 An Algorithm for JTF-Based Rotational MOCOMP

One effective solution, suggested by Chen (Chen and Qian 1998; Chen and Ling 2002), is to apply JTF tools to extract the instantaneous Doppler frequency information of the time-varying range-Doppler data such that time snapshots of the time-varying ISAR image can be constructed. The JTF-based schematic algorithm that can take the time-snapshot ISAR images of a rotating target is illustrated in Figure 8.13. The methodology can be separated into the following steps:

ωr

vt

Image at t = t1

Radar

Range cell # 1

JTF transform

Range cell # 2

JTF transform

N times

Range cell # N

N time pulses

N time pulses

1D-IFT

Cross-range Image at t = t2 M n ra ge

Range

1D-IFT

Range

P doppler frequencies

s

JTF transform

1D-IFT

M bursts

M range bins Cross-range

Scattered field

Figure 8.13

Range profiles

Schematic representation of JTF-based ISAR imaging system.

Time-range-doppler cube

Time snapshots

406

8 Motion Compensation for Inverse Synthetic Aperture Radar

1) Pulsed radar (either linear frequency modulated [LFM] or SFCW based) collects the scattered field from the target for the coherent integration time. After the received signal is digitized, let us suppose that we have a matrix size of (M N). For the SFCW radar operation, the matrix is obtained from M bursts, each having N pulses. 2) In the second step, 1D IFT operation is applied among bursts to get the 1D range profiles for N pulses. 3) Then, multiple JTF transforms are employed to the pulses for every range of cell value. Each JTF transformation operation at the single range cell yields a timeDoppler matrix that has a dimension of (M P). If the target’s rotational velocity ωr is known, the Doppler axis can be readily replaced with a cross-range axis by using the following relationship: y=

f D λc , 2ωr

8 33

where y is the cross-range variable, fD is the instantaneous Doppler frequency shift, and λc is the wavelength of center frequency. 4) After the JTF transformation operations are employed to all of the range cells, a three-dimensional (3D) time-range-Doppler (or time-range-cross-range) cube of size (M N P) is constructed. This cube has the property of providing the range-Doppler (or range-cross-range) image at a selected time instant. 5) As the final step, a total of N range-Doppler (or range-cross-range) ISAR images of the target can be obtained by taking different slices of the time-rangeDoppler (or time-range-cross-range) cube as illustrated in Figure 8.13. The resultant 2D ISAR images correspond to the time snapshots of the target while rotating.

8.3.3.3 Example for JTF-Based Rotational MOCOMP

An example for the above algorithm is demonstrated for the airplane model in Figure 8.14a. The model consists of ideal point scatterers that imitate a fighter aircraft. The simulation of the backscattered electric field is collected for a scenario such that the airplane is 16 km away from the radar and moving in the direction that makes a 30 angle with the RLOS axis. The target has a translation speed of vt = 1 m/s while rotating with an angular speed of ωr = 0.24 rad/s. A total of 128 pulses in each of 512 bursts are selected for the SFCW radar simulation of backscattered electric field. The center frequency and the frequency bandwidth are selected as fc = 3.256 GHz and B = 512 MHz, respectively. The corresponding pulse duration is then Tp =

#of pulses − 1 = 248 05 ns B

8 34

8.3 Popular ISAR MOCOMP Techniques

(a) ωr

Rotation axis

Vt

Radar

(b) 0 –60 –5

Cross range, m

–40 –10

–20 0

–15

20

–20

40 –25 60 –30 –10

0

10

dB

Range, m

Figure 8.14 (a) A target (consists of perfect point scatterers) moving with a translational speed of vt and with an angular speed of wt, (b) Traditional ISAR image of the moving target (vt = 1 m/s, ωr = 0.24 rad/s) without any compensation.

407

408

8 Motion Compensation for Inverse Synthetic Aperture Radar

The PRF is chosen as 20 KHz. So, the PRI or the time between the two consecutive bursts is then 1 = 50 μs 8 35 PRF First, the traditional ISAR image is obtained by applying 2D IFT to the backscattered field. The resulting image is plotted in Figure 8.14b, where the image suffers from blurring effect due to fast rotation rate of the target. Since the target’s angular and translational location are different for the first and the last pulse of the radar, the maneuvering effect can also be seen in the ISAR image as the aircraft’s image is smeared. This is pretty analogous to any optical imaging system: When the object point is moving fast, it occupies several pixels in the image during the time the lens stays open. Therefore, the resulting picture of the fast-moving object becomes blurred. As suggested in the above algorithm, the time-dependent ISAR images of the target can be formed, thanks to the JTF-based ISAR imaging process that can take time snapshots of the scene. Therefore, we applied the above methodology to the backscattered field data from the target while it was moving. During the implementation of the algorithm, a Gabor-wavelet transform that uses a Gaussian blur function (Gedraite and Hadad 2011) is employed as the JTF tool. At the end of the algorithm steps, a corresponding 3D time-range-cross-range cube is obtained for the simulated moving target. In Figure 8.15, time snapshots of the 3D timerange-cross-range cube are plotted for the selected nine different time instants. Each subfigure corresponds to a particular ISAR image for a particular time instant. As time progresses, the movement of the fighter’s radar image is clearly observed by looking from the first ISAR image to the last one. PRI =

8.3.4

Algorithm for JTF-Based Translational and Rotational MOCOMP

Here, we will briefly present an algorithm that compensates the motion errors in two steps. In the first step, the motion parameters are estimated using the well-known matching pursuit (MP) technique and compensate the translational motion errors. In the second step, we apply a JTF tool to compensate the rotational motion errors. Assuming that the target has both translational and rotational motion components, the time-domain backscattered signal at the radar receiver can be approximated as given in Equation 8.32. In the first step of the algorithm, translational motion parameters such as translation radial speed and translation radial acceleration are estimated and then compensated for afterward. We start the algorithm by rewriting the formula in Equation 8.32 as the following: k

g x, t =

Ak x, yk k=1 t

φ R 0 , v t , at , t

exp

− j4π

fo x c

φ ωr , αr , x, yk , t , r

8 36

8.3 Popular ISAR MOCOMP Techniques

t = 0.2848 s

t = 0.6528 s

t = 1.0208 s

–50

–50

–50

0

0

0

50

50

50

–10

0

10

–10

t = 1.3888 s

0

10

–10

t = 1.7568 s –50

–50

0

0

0

50

50

50

0

10

–10

t = 2.4928 s

0

10

–10

t = 2.8608 s –50

–50

0

0

0

50

50

50

0

10

–10

0

10

0

10

t = 3.2288 s

–50

–10

10

t = 2.1248 s

–50

–10

0

–10

0

10

Figure 8.15 2D range-cross range ISAR images for different time snapshots (images are retrieved from the 3D time-range-cross-range cube).

where φt and φr are the phase terms that contain both the translational and the rotational motion parameters as φt vt , at , t = exp φr ωr , αr , x, yk , t = exp

fo 1 Ro + vt t + at t 2 + c 2 f 1 − ω2r x + αr yk t 2 + − j4π o ωr yk t + c 2

− j4π

8 37

409

410

8 Motion Compensation for Inverse Synthetic Aperture Radar

As the first goal, translational MOCOMP has to be accomplished. We start with the algorithm by estimating the translational motion parameters with the help of an MP-type searching routine. In general, an MP algorithm predicts a suboptimal solution to the problem of a signal with known basis functions of unknown parameters. The MP algorithm is commonly used in time-frequency analysis (Mallat and Zhang 1993; Franaszczuk et al. 1998) and also in radar imaging problems (Wang et al. 1998; Su et al. 2000; Bicer et al. 2018). By utilizing the MP algorithm, the unknown translational motion parameters of vt and at are estimated with an iterative search by projecting the received signal with every motion parameter basis on a 2D space. The parameters that give the largest projection are assumed to be estimated values, as the search can be modeled as A = max

vt ,at

g x, y

φt vt , at , t

,

8 38

where the inner product is defined as g φt =

g x, t

φ t v t , at , t

∗

d at d v t

8 39

vt at est After the end of this iterative search, the parameter values of vest t and at that maximize the projection of the received signal onto 2D vt − at space are considered to be the resultant estimated values. Then, the translational MOCOMP is finalized by multiplying the original received signal with the phase correction term as

s x, y = g x, y

est φt vest t , at , t

∗

8 40

Once the translational motion errors are mitigated, only the rotational motion effects stay in the received signal. In the second part of the algorithm, therefore, the compensation for rotational motion errors is done. For this goal, the JTFT-based MOCOMP routine as explained in Section 8.3.3.2 can be applied.

8.3.4.1 A Numerical Example

We will now demonstrate an example that simulates the above algorithm. The target is assumed to consist of a set of perfect point scatterers that have equal scattering amplitudes as shown in Figure 8.7. The target’s initial location is Ro = 1.3 km away from the radar and moving with a radial translational velocity of vt = 35 m/s and with a radial translational acceleration of at = −1.9 m/s2. The target’s angular velocity of the target is ωr = 0.15 rad/s (8.5944 /s). The radar’s starting frequency is fo = 3 GHz and the total bandwidth is B = 384 MHz. The radar transmitter sends out 128 modulated pulses in each of 512 bursts. PRF is chosen to be 20 KHz. Without applying any compensation routine, the regular range-Doppler ISAR image of the target is formed using the traditional imaging methodology. The corresponding range-Doppler ISAR image is shown in Figure 8.16. As clearly seen

8.3 Popular ISAR MOCOMP Techniques

0 –2

20

–4 –6

Doppler index

40

–8 60

–10 –12

80

–14 100

–16 –18

120 –20

–10

0

10

20

–20

dB

Range, m

Figure 8.16 Conventional ISAR image of the airplane target with translational and rotational motion (no compensation). Spectrogram 0

Frequency index

0 100

–5

200

–10

300

–15

400

–20

500

–25

600

–30

700

–35

800

–40

900

–45

1000 20

Figure 8.17

40

60 80 Time pulses

100

120

–50 dB

Spectrogram of range cells (no compensation).

from the figure, the image is highly distorted, defocused, and blurred due to highvelocity values in both the translational and the rotational directions. To observe the frequency (or phase) shifts among the received time pulses, the spectrogram for consecutive time pulses is shown in Figure 8.17. This spectrogram clearly demonstrates the severe frequency shifts between the pulses. Because of the

411

8 Motion Compensation for Inverse Synthetic Aperture Radar ×104

6

X: –1.9 Y: 35 Z: 6.613e + 04

×104 6

Maximum argument

412

5

5 4

4 3

3

2 1

2 0 40 35 30 25 20

Translational velocity, m/s

–1.5

15 10

–2.5

–1

–0.5

0

0.5

–2 Translational acceleration m/s2

1

1

dB

Figure 8.18 2D Matching pursuit search space for the translational velocity and the translational acceleration.

translational acceleration, the nonlinearity of the frequency shifts is also observed. After compensating for the errors associated with target’s motion, these shifts are expected to be minimal. In the first step of the algorithm that was explained in Section 8.3.4, the translational motion parameters together with target’s initial distance Ro were estimated using an MP-type search routine. After this iterative search, the correct 2 est est values of Rest 0 = 1.3 km, vt = 35 m/s, and at = −1.9 m/s were successfully found. Figure 8.18 demonstrates the 2D search space of MP in the translational radial velocity and translational radial acceleration. As seen from the figure, the argubecomes equal to 35 m/s and ment in MP search makes a maximum when vest t 2 aest equals to −1.9 m/s . t After the translational motion parameters of the target were predicted, the translational MOCOMP was finalized by employing the formula in Equation 8.40. Then, the ISAR image corresponding to the modified electric field is plotted in Figure 8.19. This figure clearly demonstrates the success of the translational MOCOMP such that only the rotational motion-based defocusing is noted in the ISAR image. The spectrogram of the time pulses in the modified received signal is also plotted in Figure 8.20 to investigate the frequency shifts between the consecutive time pulses. As seen from this spectrogram, although severe frequency

8.3 Popular ISAR MOCOMP Techniques

0 620

–5

Doppler index

600

–10

580

–15

560

–20

540 –25 520 –20

–10

0

10

20

dB

–30

Range, m

Figure 8.19

ISAR image of the airplane target after translational motion compensation.

Spectrogram

Frequency index

0

0

100

–5

200

–10

300

–15

400

–20

500

–25

600

–30

700

–35

800

–40

900

–45

1000 20

Figure 8.20

40

60 80 Time pulses

100

120 dB

–50

Spectrogram of time pulses (after translational compensation).

shifts mainly due to the target translational velocity were mitigated, there still exists some fluctuation in the phase of the modified received signal due to rotational motion errors. In the second part of the algorithm, the errors associated with target’s rotational motion are compensated. For this goal, a Gabor-wavelet transform that uses a Gaussian blur function (Gedraite and Hadad 2011) is employed as the JTF tool

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8 Motion Compensation for Inverse Synthetic Aperture Radar

0 140 –5

Doppler index

160 –10 180 –15 200 –20 220 –25

240 –20

–10

0 10 Range, m

20

dB

–30

Figure 8.21 ISAR image of the airplane target after translational and rotational motion compensation. Spectrogram

20 Frequency index

414

40 60 80 100 120 20

40

60 80 Time pulses

100

120

Figure 8.22 Spectrogram of time pulses (after translational and rotational motion compensation).

to compensate the rotational motion effects in the ISAR image; the resultant image is given in Figure 8.21 where all the phase errors due to the translational and the rotational motion of the target were eliminated. The resultant motion-free ISAR image is very well focused and the scattering centers around the target are well localized. The final check is also performed by looking at the spectrogram of the compensated received signal as plotted in Figure 8.22 where the frequencies (or the contents of the phases) of time pulses are well aligned.

8.4 Matlab Codes

8.4

Matlab Codes

Below are the Matlab source codes that were used to generate all of the Matlabproduced figures in this chapter. The codes are also provided in the CD that accompanies this book.

Matlab code 8.1 Matlab file “Figure8-2thru8-6.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figures 8.2 – 8.6 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % Fighter.mat clear close all clc %—Radar parameters————————————————————————————————————— pulses = 128; burst = 128; c = 3.0e8; f0 = 10e9; system [Hz] bw = 128e6; T1 = (pulses-1)/bw; PRF = 20e3; T2 = 1/PRF; dr = c/(2*bw);

% % % %

# no of pulses # no of bursts speed of EM wave [m/s] Starting frequency of SFR radar

% % % % %

Frequency bandwidth [Hz] Pulse duration [s] Pulse repetition frequency [Hz] Pulse repetition interval [s] range resolution [m]

%——Target parameters——————————————————————————————————— W = 0.03;

% Angular velocity [rad/s]

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8 Motion Compensation for Inverse Synthetic Aperture Radar

Vr = 70.0; wave [m/s] ar = 0.1; R0 = 16e3; [m] theta0 = 0; [degree]

% radial translational velocity of EM % radial accelation of EM wave [m/s^2] % target's initial distance from radar % Initial angle of target's wrt target

%—Figure 8.2——————————————————————————————————————————— %load the coordinates of the scattering centers on the fighter load Fighter h = plot(-Xc,Yc,'o', 'MarkerSize',8,'MarkerFaceColor', [1 0 0]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); axis([-35 35 -30 30]) grid minor xlabel('\it x, m'); ylabel('\it y, m'); saveas(h,'Figure8_2.tif','tif'); %Scattering centers in cylindirical coordinates [theta,r]=cart2pol(Xc,Yc); theta=theta+theta0*0.017455329; %add initial angle i = 1:pulses*burst; T = T1/2+2*R0/c+(i-1)*T2;%calculate time vector Rvr = Vr*T+(0.5*ar)*(T.^2);%Range Displacement due to radial vel. and acc. Tetw = W*T;% Rotational Displacement due to angular vel. i = 1:pulses; df = (i-1)*1/T1; % Frequency incrementation between pulses k = (4*pi*(f0+df ))/c; k_fac=ones(burst,1)*k;

8.4 Matlab Codes

%Calculate backscattered E-field Es(burst,pulses)=0.0; for scat=1:1:length(Xc) arg = (Tetw - theta(scat) ); rngterm = R0 + Rvr - r(scat)*sin(arg); range = reshape(rngterm,pulses,burst); range = range.'; phase = k_fac.* range; Ess = exp(1i*phase); Es = Es + Ess; end Es = Es.'; %—Figure 8.3——————————————————————————————————————————— %Form ISAR Image (no compansation) X = -dr*((pulses)/2-1):dr:dr*pulses/2;Y=X/2; ISAR = abs(fftshift(fft2((Es)))); h=figure; matplot2(X,1:pulses,ISAR,20); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it range, m'); ylabel('\itDoppler index'); saveas(h,'Figure8_3.tif','tif'); %–Cross-Correlation Algorithm Starts here————————————— RP=(ifft(Es)).';% Form Range Profiles for l=1:burst % Cross-correlation between RPn & RPref cr(l,:) = abs(ifft(fft(abs(RP(1,:))).* fft(abs(conj (RP(l,:)))))); pk(l) = find((max(cr(l,:))== cr(l,:)));% Find max index(shift in range) end

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8 Motion Compensation for Inverse Synthetic Aperture Radar

Spk = smooth((0:pulses-1),pk,0.8,'rlowess');%smoothing the delays RangeShifts = dr*pk;% range shifts SmRangeShifts = dr*Spk;% range shifts RangeDif = SmRangeShifts(2:pulses)-SmRangeShifts(1: pulses-1);%range differences RangeDifAv = mean(RangeDif );% average range differences T_burst=T(pulses+1)-T(1); % time between the bursts Vr_Dif=(-RangeDif/T_burst); % estimated radial velocity from each RP Vr_av=(RangeDifAv /T_burst); % estimated radial velocity (average) %—Figure 8.4——————————————————————————————————————————— h = figure;plot(i,RangeShifts,'LineWidth',2);hold plot(i,SmRangeShifts,'-.k.','MarkerSize',4);hold axis tight grid minor legend('\itRP shifts','\itsmoothed RP shifts'); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange profile index'); saveas(h,'Figure8_4.tif','tif'); %—Figure 8.5——————————————————————————————————————————— h = figure; subplot(211);plot(RangeDif,'LineWidth',2); axis([1 burst -.65 -.25 ]) grid minor set(gca,'FontName', 'Arial', 'FontSize',10,'FontWeight', 'Bold'); xlabel('\itrange profile index'); ylabel('\itrange differences, m') subplot(212);plot(Vr_Dif,'LineWidth',2); axis([1 burst Vr-5 Vr+5 ])

8.4 Matlab Codes

set(gca,'FontName', 'Arial', 'FontSize',10,'FontWeight', 'Bold'); grid minor xlabel('\itrange profile index'); ylabel('\iradial speed, m/s ') text(15,74,['Actual Speed = ',num2str(Vr),' m/s, Est. average speed = ',… num2str(-Vr_av),' m/s']); saveas(h,'Figure8_5.tif','tif'); % Compansating the phase f = (f0+df );% frequency vector T = reshape(T,pulses,burst); %prepare time matrix F = f.'*ones(1,burst); %prepare frequency matrix Es_comp = Es.*exp((1i*4*pi*F/c).*(Vr_av*T));%Phase of E-field is compansated %—Figure 8.6——————————————————————————————————————————— win = hanning(pulses)*hanning(burst).'; %prepare window ISAR = abs((fft2((Es_comp.*win)))); % form the image ISAR2 = ISAR(:,28:128); ISAR2(:,102:128)=ISAR(:,1:27); h = figure; matplot2(Y,1:pulses,ISAR2,20); % motion compansated ISAR image colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it range, m'); ylabel('\itDoppler index'); title('\itMotion compansated ISAR image') saveas(h,'Figure8_6.tif','tif');

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8 Motion Compensation for Inverse Synthetic Aperture Radar

Matlab code 8.2 Matlab file “Figure8-7thru8-12.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figures 8-7 thru 8.14 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % fighter2.mat clear close all clc %—Radar parameters————————————————————————————————————— pulses = 128; burst = 128; c = 3.0e8; f0 = 8e9; system [Hz] bw = 384e6; T1 = (pulses-1)/bw; PRF = 14.5e3; T2 = 1/PRF; dr = c/(2*bw);

% % % %

# no of pulses # no of bursts speed of EM wave [m/s] Starting frequency of SFR radar

% % % % %

Frequency bandwidth [Hz] Pulse duration [s] Pulse repetition frequency [Hz] Pulse repetition interval [s] slant range resolution [m]

%—Target parameters———————————————————————————————————— W = 0.06; Vr = 4.0; of EM wave [m/s] ar = 0.6; s^2] R0 =.5e3; radar [m] theta0 = 125; target [degree]

% Angular velocity [rad/s] % radial translational velocity % radial accelation of EM wave [m/ % target's initial distance from % Initial angle of target's wrt

8.4 Matlab Codes

%—Figure 8-7——————————————————————————————————————————— %load the coordinates of the scattering centers on the fighter load fighter2 h = plot(Xc,Yc,'o', 'MarkerSize',8,'MarkerFaceColor',[1 0 0]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); axis([-20 20 -20 20]) grid minor xlabel('\it x, m'); ylabel('\it y, m'); saveas(h,'Figure8_7.tif','tif'); %Scattering centers in cylindirical coordinates [theta,r] = cart2pol(Xc,Yc); theta=theta+theta0*0.017455329; %add initial angle i = 1:pulses*burst; T = T1/2+2*R0/c+(i-1)*T2;%calculate time vector Rvr = Vr*T+(0.5*ar)*(T.^2);%Range Displacement due to radial vel. and acc. Tetw = W*T;% Rotational Displacement due to angular vel. i = 1:pulses; df = (i-1)*1/T1; % Frequency incrementation between pulses k = (4*pi*(f0+df ))/c; k_fac=ones(burst,1)*k; %Calculate backscattered E-field Es(burst,pulses)=0.0; for scat=1:1:length(Xc); arg = (Tetw - theta(scat) ); rngterm = R0 + Rvr - r(scat)*sin(arg); range = reshape(rngterm,pulses,burst);

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8 Motion Compensation for Inverse Synthetic Aperture Radar

range = range.'; phase = k_fac.* range; Ess = exp(-j*phase); Es = Es+Ess; end Es = Es.'; %—Figure 8-8 —————————————————————————————————————————— %Form ISAR Image (no compansation) X = -dr*((pulses)/2-1):dr:dr*pulses/2;Y=X/2; ISAR = abs(fftshift(fft2((Es)))); h = figure; matplot2(X,1:pulses,ISAR,20); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\itrange, m'); ylabel('\itDoppler index'); saveas(h,'Figure8-8.tif','tif'); %—Figure 8-9 —————————————————————————————————————————— % JTF Representation of range cell EsMp = reshape(Es,1,pulses*burst); % S=spectrogram(EsMp,32,30); S = spectrogram(EsMp,128,64,120); [a,b] = size(S); h = figure; matplot2((1:a),(1:b),abs(S),50); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); colormap(hot); xlabel('\ittime pulses'); ylabel('\itfrequency index'); title('\itSpectrogram');

8.4 Matlab Codes

saveas(h,'Figure8-9.tif','tif'); %Prepare time and frequency vectors f = (f0+df );% frequency vector T = reshape(T,pulses,burst); %prepare time matrix F = f.'*ones(1,burst); %prepare frequency matrix % Searching the motion parameters via min. entropy method syc=1; V = A = m = for

-15:.2:15; -0.4:.01:1; 0; Vest = V m = m+1; n = 0; for iv = A n = n+1; VI(syc,1:2) = [Vest,iv]; S = exp((1i*4*pi*F/c).*(Vest*T+(0.5*iv)*(T. ^2))); Scheck = Es.*S; ISAR = abs(fftshift(fft2((Scheck)))); SumU = sum(sum(ISAR)); I = (ISAR/SumU); Emat = I.*log10(I); EI(m,n) = -(sum(sum(Emat))); syc = syc+1; end

end [dummy,mm] = min(min(EI.')); %Find index for estimated velocity [dummy,nn] = min(min(EI)); %Find index for estimated acceleration %—Figure 8_10 ————————————————————————————————————————— h = figure;

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8 Motion Compensation for Inverse Synthetic Aperture Radar

surfc(A,V,EI); colormap(gray) set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\ittranslational vel., m/s'); xlabel('\ittranslational acc., m/s^2'); zlabel ('\itEntropy value') saveas(h,'Figure8-10.tif','tif'); % Form the mathing phase for compensation Sconj = exp((1i*4*pi*F/c).*(V(mm)*T+(0.5*A(nn)*(T. ^2)))); % Compansate S_Duz = Es.*Sconj; %—Figure 8-11 ————————————————————————————————————————— % ISAR after compensation h = figure; matplot2(X,burst,abs(fftshift(fft2(S_Duz))),20); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it range, m'); ylabel('\itDoppler index'); saveas(h,'Figure8-11.tif','tif'); %—Figure 8-12 ————————————————————————————————————————— % Check the compensation using via JFT Representation of range cells EsMp = reshape(S_Duz,1,pulses*burst); S = spectrogram(EsMp,128,64,120); [a,b] = size(S); h = figure; matplot2((1:a),(1:b),abs(S),50); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold');

8.4 Matlab Codes

colormap(hot); xlabel('\ittime pulses'); ylabel('\itfrequency index'); title('\itSpectrogram'); saveas(h,'Figure8-12.tif','tif');

Matlab code 8.3 Matlab file “Figure8-14.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 8.14 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % scat_field.mat clear close all %—Load the Scattered Field ————————————————————————————— load scat_field % Npulse = 128; burst % Nburst = 512; % f1 = 3e9; EM wave % BWf = 512e6; % T1 = (Npulse-1)/BWf; % PRF = 20e3; % PRI = 1/PRF; % W = 0.16;

% number of pulses in one % number of bursts % starting frequency for the % % % % %

bandwidth of the EM wave pulse duration Pulse Repetation Frequency Pulse Repetation Interval angular velocity [rad/s]

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8 Motion Compensation for Inverse Synthetic Aperture Radar

% Vr = 1.0; % ar = 0.0;

% radial velocity [m/s] % acceleration [m/s2]

% c = 3.0e8;

% speed of the EM wave

%—Figure 8.14a ——————————————————————————————————————— plot(-Xc,Yc,'square', 'MarkerSize',5,'MarkerFaceColor', [1 0 0]); set(gca,'FontName', 'Arial', 'FontSize',14,'FontWeight','Bold'); xlabel('\itx, m'); ylabel('\ity, m'); axis([min(-Xc)*1.1 max(-Xc)*1.1 min(Yc)*1.1 max(Yc) *1.1]) %—Figure 8.14b ——————————————————————————————————————— %—Form Classical ISAR Image ——————————————————————————— w = hanning(Npulse)*hanning(Nburst)'; Es = Es.*w; Es(Npulse*4,Nburst*4) = 0; ISAR = abs(fftshift(ifft2((Es)))); h = figure; matplot2(XX,YY,ISAR,30); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight','Bold'); xlabel('\itrange, m'); ylabel('\itcross range, m');

8.4 Matlab Codes

Matlab code 8.4 Matlab file -Figure8-15.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 8.15 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % scat_field.mat clear close all %—Load the Scattered Field ———————————————————————————— load scat_field % Npulse = 128; % Nburst = 512; % f1 = 3e9; wave % BWf = 512e6; % T1 = (Npulse-1)/BWf; % PRF = 20e3; % PRI = 1/PRF; % W = 0.16; % Vr = 1.0; % ar = 0.0; c = 3.0e8;

% number of pulses in one burst % number of bursts % starting frequency for the EM % % % % % % % %

bandwidth of the EM wave pulse duration Pulse Repetation Frequency Pulse Repetation Interval angular velocity [rad/s] radial velocity [m/s] acceleration [m/s2] speed of the EM wave

N=1; T = 2*R0/c+((1:Npulse*Nburst)-1)*PRI; % tst = PRI*Npulse; % Toplam Sinyal Tekrarlama Süresi Nt=T(1:Npulse:Npulse*Nburst); Es_IFFT = ifft(Es)'; field

% take 1D IFFT of

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8 Motion Compensation for Inverse Synthetic Aperture Radar

%——Prepare JTF filter function—————————————————————— n=0; figure for frame=90:115:1100 n=n+1; fp=145; için Kullanılan Frekans tp = ((frame-1)*tst)/2 ; frame

% select time frames % counter for plotting % Görüntünün Ortalanaması % window center for each

%——Prepare JTF filter fncn (Gabor fncn with Gaussian Blur) Alpha_p=(0.04); % Blurring coefficient for i=1:Npulse part1 = 1/sqrt(2*pi*(Alpha_p)^2); % normalization term part2 = exp(-((Nt-tp).^2)/(2*Alpha_p)); % Gaussian window part3 = exp((-1i*2*pi*fp*(Nt-tp))/N); % Harmonic function GaborWavelet(i,1:Nburst) = part1*part2.*part3;% Gabor function fp=fp+1/(Npulse*tst); end % %** Wavelet Transform St =fftshift(GaborWavelet*Es_IFFT); subplot(3,3,n);matplot2(XX,YY,(St.'),25); colormap(hot); set(gca,'FontName', 'Arial', 'FontSize',10,'FontWeight','Bold'); title(['t = ',num2str(tp),' s'],'FontAngle','Italic'); end

8.4 Matlab Codes

Matlab code 8.5 Matlab file “Figure8-16thru8-22.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figures 8.16 thru 8.22 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % Fighter3.mat clear close all clc %—Radar parameters————————————————————————————————————— pulses = 128; burst = 512; c = 3.0e8; f0 = 3e9; system [Hz] bw = 384e6; T1 = (pulses-1)/bw; PRF = 20e3; T2 = 1/PRF; W = 0.15; Vr = 35.0; wave [m/s] ar = -1.9; s^2] R0 = 1.3e3; radar [m] dr = c/(2*bw); theta0 = -30;

% % % %

# no of pulses # no of bursts speed of EM wave [m/s] Starting freq. of SFR radar

% % % % % %

Frequency bandwidth [Hz] Pulse duration [s] Pulse repetition frequency [Hz] Pulse repetition interval [s] Angular velocity [rad/s] radial translational vel. of EM

% radial accelation of EM wave [m/ % target's initial dist. from % range resolution [m] % Look angle of the target

%—Figure ———————————————————————————————————————————

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8 Motion Compensation for Inverse Synthetic Aperture Radar

%load the coordinates of the scattering centers on the fighter load Fighter3 h = plot(-Xc,Yc,'o', 'MarkerSize',8,'MarkerFaceColor', [1 0 0]); grid minor; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); axis([-20 20 -20 20]) xlabel('\it x, m'); ylabel('\it y, m'); %Scattering centers in cylindirical coordinates [theta,r] = cart2pol(Xc,Yc); theta = theta+theta0*0.017455329; %add initial angle i = 1:pulses*burst; T = T1/2+2*R0/c+(i-1)*T2;%calculate time vector Rvr = Vr*T+(0.5*ar)*(T.^2);%Range Displacement due to radial vel. & acc. Tetw = W*T;% Rotational Displacement due to angular vel. i = 1:pulses; df = (i-1)*1/T1; % Frequency incrementation between pulses k = (4*pi*(f0+df ))/c; k_fac = ones(burst,1)*k; %Calculate backscattered E-field Es(burst,pulses) = 0.0; for scat = 1:1:length(Xc) arg = (Tetw - theta(scat) ); rngterm = R0 + Rvr - r(scat)*sin(arg); range = reshape(rngterm,pulses,burst); range = range.'; phase = k_fac.* range; Ess = exp(-1i*phase); Es = Es+Ess;

8.4 Matlab Codes

end Es = Es.'; %—Figure 8.16 ————————————————————————————————————————— %Form ISAR Image (no compansation) X = -dr*((pulses)/2-1):dr:dr*pulses/2;Y=X/2; ISAR = abs((fft2((Es)))); h = figure; matplot2(X,1:pulses,ISAR,20); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it range, m'); ylabel('\itDoppler index'); %—Figure 8.17 ————————————————————————————————————————— % JTF Representation of range cell EsMp = reshape(Es,1,pulses*burst); S = spectrogram(EsMp,128,64,128); [a,b] = size(S); h = figure; matplot2((1:a),(1:b),abs(S),50); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; xlabel('\ittime pulses'); ylabel('\itfrequency index'); title('\itSpectrogram'); %Prepare time and frequency vectors f = (f0+df );% frequency vector

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8 Motion Compensation for Inverse Synthetic Aperture Radar

T = reshape(T,pulses,burst); %prepare time matrix F = f.'*ones(1,burst); %prepare frequency matrix % Searching the motion parameters via Matching Pursuit syc = 1; RR = 1e3:1e2:2e3; V = 10:40; A = -2.5:.1:1; m = 0; clear EI for Vest = V m = m+1; n = 0; for iv = A n = n+1; p = 0; for Rest = RR p = p+1; VI(syc,1:2) = [Vest,iv]; S = exp((1i*4*pi*F/c).*(Rest+Vest*T+(0.5*iv)*(T. ^2))); Scheck = Es.*S; SumU = sum(sum(Scheck)); EI(m,n,p) = abs(SumU); end end end

[dummy,pp] = max(max(max((EI)))); %Find index for estimated velocity [dummy,nn] = max(max((EI(:,:,pp)))); %Find index for estimated velocity [dummy,mm] = max(EI(:,nn,pp)); %Find index for estimated acceleration

8.4 Matlab Codes

%—Figure 8.18 ————————————————————————————————————————— figure; h = surfc(A,V,EI(:,:,pp)); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); ylabel('\ittranslational velocity, m/s'); xlabel('\ittranslational acceleration, m/s^2'); zlabel ('\itmaximum argument') % Form the mathing phase for compensation Sconj = exp((1i*4*pi*F/c).*(V(mm)*T+(0.5*A(nn)*(T. ^2)))); % Compansate S_Duz = Es.*Sconj; %—Figure 8.19 ————————————————————————————————————————— % ISAR after compensation h = figure; matplot(X,burst,abs(fftshift(fft2(S_Duz))),30); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it range, m'); ylabel('\it Doppler index'); %—Figure 8.20 ————————————————————————————————————————— % Check the compensation using via JTF Representation of range cells Sconjres = reshape(S_Duz,1,pulses*burst); S = spectrogram(Sconjres,128,64,120); [a,b] = size(S);

433

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8 Motion Compensation for Inverse Synthetic Aperture Radar

h =figure; matplot2((1:a),(1:b),abs(S),50); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it time pulses'); ylabel('\it frequency index'); title('\it Spectrogram'); %—This part for Rotational motion compensation——————————————— Ese = S_Duz; win = hamming(pulses)* hamming(burst).';% Prepare Window Esew = Ese.*win; % Apply window to the E-field Es_IFFT = (ifft(Esew)).'; % Range profiles i = 1:pulses*burst; T = T1/2+2*R0/c+(i-1)*T2; %** Apply Gaussian Blur Filter via Gabor Function N = 1; % Sampling # tst = T2*pulses; % dwell time t = T(1:pulses:pulses*burst); % time vector for bursts fp = 160; Alpha_p = 0.04; t_istenen = 100; tp = ((t_istenen-1)*tst)/2;

% Blurring coefficient % tp=1 sec, T=2.1845 sec % Center of Gaussian window

% % Gabor function and Gaussian Blur function parca1 = 1/sqrt(2*pi*(Alpha_p)^2); % normalized term parca2 = exp(-((t-tp).^2)/(2*Alpha_p)); % Gaussian

8.4 Matlab Codes

window term for i=1:pulses parca3 = exp((-1i*2*pi*fp*(t-tp))/N); % Harmonic function GaborWavelet(i,1:burst) = parca1*parca2.*parca3;% Gabor Wavelet func. fp=fp+1/(pulses*tst); end % %** Wavelet Transform St_Img = fftshift(GaborWavelet*Es_IFFT); % shift image to the center %—Figure 8.21 ————————————————————————————————————————— figure; matplot2(X,pulses,(St_Img.'),30); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; grid minor; xlabel('\it range, m'); ylabel('\itDoppler index'); %—Figure 8.22 ————————————————————————————————————————— EMp = reshape(St_Img,1,128*128); S = spectrogram(EMp,256,120); h = figure; matplot2((1:pulses),(1:pulses),abs(S.'),60); colormap(hot); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\ittime pulses');

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8 Motion Compensation for Inverse Synthetic Aperture Radar

ylabel('\itfrequency index'); title('\itSpectrogram'); grid minor

References Berizzi, F., Martorella, M., Haywood, B. et al. (2004). A survey on ISAR autofocusing techniques. 2004 International Conference on Image Processing, 2004. ICIP ’04 (4–27 October 2004), Singapore, Singapore. https://doi.org/10.1109/ICIP.2004.1418676. Bicer, M.B., Akdagli, A., and Özdemir, C. (2018). A matching-pursuit based approach for detecting and imaging breast cancer tumor. Progress in Electromagnetics Research 64: 65–76. Calloway, T.M. and Donohoe, G.W. (1994). Subaperture autofocus for synthetic aperture radar. IEEE Transactions on Aerospace and Electronic Systems 30 (2): 617– 621. https://doi.org/10.1109/7.272285. Carrara, W.G., Goodman, R.S., and Majevski, R.M. (1995). Spotlight Synthetic Aperture Radar: Signal Processing Algorithms. Boston: Artech House. Chen, C.C. and Andrews, H.C. (1980). Target-motion-induced radar imaging. IEEE Transactions on Aerospace and Electronic Systems AES-16 (1): 2–14. https://doi.org/ 10.1109/TAES.1980.308873. Chen, V.C. and Ling, H. (2002). Time-Frequency Transforms for Radar Imaging and Signal Analysis. Boston: Artech House. Chen, V.C. and Qian, S. (1998). Joint time-frequency transform for radar range-Doppler imaging. IEEE Transactions on Aerospace and Electronic Systems 34 (2): 486–499. https://doi.org/10.1109/7.670330. Cho, J.S., Kim, D.J., and Park, D.J. (2000). Robust centroid target tracker based on new distance features in cluttered image sequences. IEICE Transactions on Information and Systems E83-D (12): 2142–2151. http://hdl.handle.net/10203/75592. Choi, I.S., Cho, B.L., and Kim, H.T. (2003). ISAR motion compensation using evolutionary adaptive wavelet transform. IEE Proceedings – Radar, Sonar and Navigation 150 (4): 229–233. https://doi.org/10.1049/ip-rsn:20030639. Cleveland, W.S. (1979). Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association 74 (368): 829–836. https://doi.org/ 10.2307/2286407. Eichel, P.H., Ghiglia, D., and Jakowatz, C.V. (1989). Speckle processing method for synthetic-aperture-radar phase correction. Optics Letters 14 (1): 1–3. https://doi.org/ 10.1364/OL.14.000001.

References

Franaszczuk, P.J., Bergey, G.K., Durka, P.J., and Eisenberg, H.M. (1998). Time-frequency analysis using the matching pursuit algorithm applied to seizures originating from the mesial temporal lobe. Electroencephalography and Clinical Neurophysiology 106 (6): 513–521. https://doi.org/10.1016/S0013-4694(98) 00024-8. Gedraite, E.S. and Hadad, M. (2011). Investigation on the effect of a Gaussian Blur in image filtering and segmentation. Proceedings ELMAR-2011, Zadar, Croatia (14–16 September 2011), pp. 393–396. IEEE. Haywood, B. and Evans, R.J. (1989). Motion compensation for ISAR imaging. Australian Symposium on Signal Processing and Applications: ASSPA 89 (17–19 April 1989), Adelaide, Australia, pp. 113–117. Itoh, T., Sueda, H., and Watanabe, Y. (1996). Motion compensation for ISAR via centroid tracking. IEEE Transactions on Aerospace and Electronic Systems 32 (3): 1191–1197. https://doi.org/10.1109/7.532283. Kim, H. and Ling, H. (1993). Wavelet analysis of radar echo from finite-sized targets. IEEE Transactions on Antennas and Propagation 41 (2): 200–207. https://doi.org/ 10.1109/8.214611. Kim, K.T., Choi, I.S., and Kim, H.T. (2000). Efficient radar target classification using adaptive joint time-frequency processing. IEEE Transactions on Antennas and Propagation 48 (12): 1789–1801. https://doi.org/10.1109/8.901267. Kirk, J.C. (1975). Motion compensation for synthetic aperture radar. IEEE Transactions on Aerospace and Electronic Systems AES-11 (3): 338–348. Küçükkiliç, T. (2006). ISAR imaging and motion compensation. Master thesis. Middle East Technology University. Mallat, S.G. and Zhang, Z. (1993). Matching pursuit with time-frequency dictionaries. IEEE Transactions on Signal Processing 41 (12): 3397–3415. https://doi.org/10.1109/ 78.258082. Martorella, M., Haywood, B., Berizzi, F., and Mese, E.D. (2003). Performance analysis of an ISAR contrast-based autofocusing algorithm using real data. 2003 Proceedings of the International Conference on Radar (IEEE Cat. No.03EX695) (3–5 September 2003), Adelaide, SA, Australia. https://doi.org/10.1109/RADAR.2003.1278705. Moghaddar, A. and Walton, E.K. (1993). Time-frequency distribution analysis of scattering from waveguide cavities. IEEE Transactions on Antennas and Propagation 41 (5): 677–680. https://doi.org/10.1109/8.222287. Munoz-Ferreras, J.M., Calvo-Gallego, J., Perez-Martinez, F. et al. (2006). Motion compensation for ISAR based on the shift-and-convolution algorithm. 2006 IEEE Conference on Radar (24–27 April 2006), Verona, New York, USA. https://doi.org/ 10.1109/RADAR.2006.1631825. Özdemir, C. and Ling, H. (1997). Joint time-frequency interpretation of scattering phenomenology in dielectric – coated wires. IEEE Transactions on Antennas and Propagation 45 (8): 1259–1264. https://doi.org/10.1109/8.611245.

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Shannon, C.E. (1951). Prediction and entropy of printed English. The Bell System Technical Journal 30 (1): 50–64. https://doi.org/10.1002/j.1538-7305.1951.tb01366.x. Shin, S.Y. and Myung, N.H. (2008). The application of motion compensation of ISAR image for a moving target in radar target recognition. Microwave and Optical Technology Letters 50 (6): 1673–1678. https://doi.org/10.1002/mop.23466. Steinberg, B.D. (1988). Microwave imaging of aircraft. Proceedings of the IEEE 76 (12): 1578–1592. https://doi.org/10.1109/5.16351. Su, T., Özdemir, C., and Ling, H. (2000). On extracting the radiation center representation of antenna radiation patterns on a complex platform. Microwave and Optical Technology Letters 26 (1): 4–7. https://doi.org/10.1002/(SICI)1098-2760 (20000705)26:13.0.CO;2-2. Trintinalia, L.C. and Ling, H. (1996). Extraction of waveguide scattering features using joint time-frequency ISAR. IEEE Microwave and Guided Wave Letters 6 (1): 10–12. https://doi.org/10.1109/75.482055. Wahl, D.E. (1994). Phase gradient autofocus – a robust for high resolution SAR phase correction. IEEE Transactions on Aerospace and Electronic Systems 30 (3): 827–835. https://doi.org/10.1109/7.303752. Walker, J. (1980). Range-Doppler imaging of rotating objects. IEEE Transactions on Aerospace and Electronic Systems AES-16 (1): 23–52. https://doi.org/10.1109/ TAES.1980.308875. Wang, G.Y. and Bao, Z. (1997). The minimum entropy criterion of range alignment in ISAR motion compensation. Proceedings of Radar Systems (Conference Publication No.449) (14–16 October 1997), Edinburgh, UK. https://doi.org/10.1049/cp:19971669. Wang, Y., Ling, H., and Chen, V.C. (1998). ISAR motion compensation via adaptive joint time - frequency technique. IEEE Transactions on Aerospace and Electronic Systems 34 (2): 670–677. https://doi.org/10.1109/7.670350. Wenxian, F., Shaohong, L., and Wen, H. (2001). Motion compensation for spotlight SAR mode imaging. 2001 CIE International Conference on Radar Proceedings (Cat No.01TH8559) (15–18 October 2001), Beijing, China. https://doi.org/10.1109/ ICR.2001.984865. Werness, S.A.S., Carrara, W.G., Joyce, L.C., and Franczak, D.B. (1990). Moving target imaging algorithm for SAR data. IEEE Transactions on Aerospace and Electronic Systems 26 (1): 57–67. https://doi.org/10.1109/7.53413. Wu, H., Grenier, D., Delisle, G.Y., and Fang, D.G. (1995). Translational motion compensation in ISAR image processing. IEEE Transactions on Image Processing 4 (11): 1561–1571. https://doi.org/10.1109/83.469937. Xi, L., Guosui, L., and Ni, J. (1999). Autofocusing of ISAR image based on entropy minimization. IEEE Transactions on Aerospace and Electronic Systems 35 (4): 1240– 1252. https://doi.org/10.1109/7.805442. Xia, X.G., Wang, G., and Chen, V.C. (2002). Quantitative SNR analysis for ISAR imaging using joint time-frequency analysis – short time Fourier transform. IEEE

References

Transactions on Aerospace and Electronic Systems 38 (2): 649–659. https://doi.org/ 10.1109/TAES.2002.1008993. Xu, R., Cao, Z., and Liu, Y. (1990). Motion compensation for ISAR and noise effect. IEEE Aerospace and Electronic Systems Magazine 5 (6): 20–22. https://doi.org/ 10.1109/62.54639. Yu, J. and Yang, J. (2008). Motion compensation of ISAR imaging for high-speed moving target. 2008 IEEE International Symposium on Knowledge Acquisition and Modeling Workshop (21–22 December 2008), Wuhan, China. https://doi.org/ 10.1109/KAMW.2008.4810440. Zhu, Z.D. and Wu, X.Q. (1991). Range-Doppler imaging and multiple scatter-point location. Journal of Nanjing University of Aeronautics & Astronautics 23: 62–69.

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9 Bistatic ISAR Imaging While the monostatic configuration (that is both the transmitter and the receiver are collocated) is generally the standard approach for ISAR imaging applications, the use of bistatic ISAR (Bi-ISAR) configuration at which the transmitter and the receiver are located at different locations has been gaining more attention in regard to target recognition and classification applications (Zhu et al. 2005). The use of bistatic configuration can be attractive for many situations or practices: At the top of the list, a most likely case is when the target is moving along or near to radar line-of-sight (RLOS) direction so that the required angular look-angle width for resolving cross-range points cannot be achieved (Zhu et al. 2005; Özdemir et al. 2009). The use of Bi-ISAR imaging will solve this problem since the angular variation is assuredly obtained at the receiver site for practical usages of transmitter and receiver locations (Bhalla and Ling 1993; Ziyue et al. 2009; Drozdowicz et al. 2017). Importantly, Bi-ISAR imaging concept has the ability to be used in early warning systems and also for the detection of intrusions along maritime borders (Lazarov et al. 2014; Wang et al. 2015). The advantages of Bi-ISAR imaging in more detail will be explained next.

9.1

Why Bi-ISAR Imaging?

As given in Chapter 4, the monostatic ISAR image which can be regarded as the 2D projection of target reflectivities on the projection plane of range and cross-range domains, the resolutions in range, and cross-range domains are achieved by the frequency bandwidth of the transmitted pulse and target’s rotation with respect to RLOS direction. Therefore, it may not be possible to attain and form an ISAR image if the movement of target does not contain any rotational component with respect to its own axis. This case is illustrated in Figure 9.1a where an aircraft is flying close to the RLOS direction so that there is not enough look-angle variation of the target available to attain the required cross-range resolution to form an ISAR Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms: With Advanced SAR/ISAR Imaging Concepts, Algorithms, and MATLAB Codes, Second Edition. Caner Özdemir. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

9.1 Why Bi-ISAR Imaging?

(a)

h

t pat

Fligh

Not enough angular span! TX/RX

(b) h

t pat

Fligh

Abundant angular span! TX

RX

Figure 9.1 (a) Monostatic ISAR versus, (b) Bi-ISAR imaging configuration.

image. To overcome such scenarios, bistatic ISAR configuration can provide a solution for such blind cases of monostatic ISAR. As demonstrated in Figure 9.1b, in contrast to monostatic case, receiving antenna of the radar is put at a location different than the transmitting antenna. With this construct, the required look-angle variation over the target can be possible thanks to bistatic geometry of the radar. Therefore, bistatic ISAR technology can really provide a good solution for the geometrically adverse cases at which the target cannot provide the necessary angle variation for the monostatic ISAR. Another advantage of Bi-ISAR set-up comes from its ability to beat stealth feature of special military platforms. The low-observable platforms are being aimed to provide small values of RCS for the threatening frequencies. The low-observability is generally achieved by physical design and the paint, a.k.a. radiation-absorbent material (RAM) for the ultimate goal of providing very low monostatic RCS. In terms of physical design, open cavities are hided, S-like cavities are used for the jet inlets to avoid multiple backscattering. Furthermore, dihedral or trihedral type structures are also avoided to provide less monostatic RCS values. The main idea of the design logic is to reflect the incoming electromagnetic (EM) wave to another direction so that the platform should deliver very little backscattering energy. In

441

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9 Bistatic ISAR Imaging

terms of the paint material of stealth platforms, the most RAMs are designed for the monostatic operation of the radar. For example, the thickness of the RAM painting is decided based on the frequency of operation of the backscattering wave. To sum up, stealth technology is usually designed by considering the monostatic radar operation set-up since the possibilities are very plentiful for the bistatic case; and, therefore it is not easy to make a universal physical design and/or painting material design to mitigate the bistatic RCS value under a desired threshold value. For these reasons, bistatic ISAR technology can be a good candidate in recognition and classification of stealth platforms. This phenomenon is illustrated thru the scenarios given in Figure 9.2. In Figure 9.2a, a monostatic radar can only sense the small amount of backscattered EM energy from a low-observable aircraft. This amount of energy cannot be sufficient for detection or forming an ISAR image since the target is designed to reflect very small amount of energy along the direction of illumination. In Figure 9.2b, whereas, the bistatic radar receiver can collect the deflected energy from the Stealth aircraft in the direction of receiver’s LOS. Considering the ISAR case, the receiver will be able to see different points on the receiver as the target is moving in its path since different deflected waves will reach the bistatic ISAR receiver during the integration time of ISAR operation. All these features of bistatic configuration present significant power detection advantage when compared to monostatic operation of radar for low-observable platforms. In addition to abovementioned advantages, bistatic ISAR configuration can also provide passive ISAR (P-ISAR) imaging (Garry and Smith 2019) by exploiting illuminating sources from other transmitters of usually commercial usages. The telecommunication signals such as frequency modulated (FM) and digital television (DTV) are not designed for radar operations. However, broadcasting networks are usually composed of several transmitters at different locations and propagating at separate center frequency of operations. If a single ISAR receiver antenna can exploit the diversity of such illuminators and concurrently process the received data from these transmitting sources, a bistatic P-ISAR image can be possibly formed. Of course, bistatic P-ISAR technology offers various advantages in comparison to other radar systems including

•• •• •

providing continuous operation without being detected, difficulty of jamming, being physically smaller in size (no transmitter and related hardware), low costs of operation, maintenance, and procurement costs, and resilience to anti-radiation missiles.

Last but not the least, if the bistatic ISAR receivers are multiplied and used in a network set-up, multi-static ISAR (Mu-ISAR) configuration can be established

9.1 Why Bi-ISAR Imaging?

(a)

r te at

g in

c

s ck Ba

TX/RX

Monostatic radar can sense very little backscattering energy (not enough for detection)

(b)

Deflected scattering

TX

Bistatic radar is sensing adequate energy for detection

RX

Figure 9.2 (a) Monostatic radar can only sense backscattered wave, (b) bistatic radar has the ability to sense deflected waves.

with a small increase in the total cost. Such usage of using multiple bistatic receivers will have the advantages of

••

increasing target’s visibility in the ISAR image, opportunity of imaging some portions of the target that were shadowed for a particular look-angle or more, and

443

444

9 Bistatic ISAR Imaging

•

increased probability of identification and/or classification of the target due to inclusion of more scattering centers and features from the body of the target.

9.2

Geometry for Bi-Isar Imaging and the Algorithm

9.2.1

Bi-ISAR Imaging Algorithm for a Point Scatterer

The geometry for the bistatic ISAR imaging problem is given via a general scenario that is illustrated in Figure 9.3. The target is assumed to be a point scatterer located at position P(x0, y0) in Cartesian coordinates. The transmitter (Tx) and the receiver (Rx) are at r 1 and r 2 away from the point scatterers. The transmitted signal propagates along the wavenumber direction of k1 from transmitter to point P as shown in Figure 9.3. Then, it scatters from point P in all directions. The receiver collects the scattered wave along the wavenumber direction of k2 . The bistatic angle between these wavenumbers vectors is taken as β. As it will be explained later, a synthetic RLOS direction is introduced in the geometry of Figure 9.3. This direction is the bisector of the angle between the transmitter’s LOS and receiver’s LOS direction and makes an azimuth angle of φ with the x-axis as depicted in the figure. Based on this model and taking the origin as the phase center of the imaging geometry, the far-field scattered field from the point scatterer P at an azimuth angle φ around the receiver can be approximated as Es k, φ

Ao exp − j k 1 r 0 + k 2 r 0

ˆk 2

β/2

Rx

→ r1

Monostatic equivalent

Tx

Figure 9.3

Geometry for bistatic ISAR imaging.

P(xo, yo) → ro

β/2 φ

91

→ kˆ2 · ro

kˆ2 → r2

,

ˆk 1

x →

ˆk1 · r o

9.2 Geometry for Bi-Isar Imaging and the Algorithm

where Ao is the amplitude of the scattered electric field intensity. This equation contains summation of two-phase lags of

k1 r 0

and

k 2 r 0 . The first

phase lag is due to the fact that the transmitted wave travels an extra distance of k 1 r 0 when compared to a synthetic point at the phase center of the geometry, i.e., the origin. The same argument can be analogously made for the second phase term that is responsible for the extra travelling distance amount of k 2 r 0 when compared to the reference wave that originates from the origin. Notice that the k 1 and k 2 vectors can be written on the 2D x–y space in terms of the wave numbers in x and y directions as the followings: k 1 = k k1 =k

cos φ + β 2

x + sin φ + β 2

92

y

k 2 = k k2 =k

cos φ − β 2

x + sin φ − β 2

93

y

Here, k is the wavenumber that is equal to k = 2πf/c where f is the frequency of operation and c is the speed of light in the air. Also, x and y are the unit vectors in x and y directions, respectively. Then, the arguments in the first and the second phase terms of Equation 9.1 can be reorganized to give k1 r 0 = k =k

cos φ + β 2

x + sin φ + β 2

cos φ + β 2 x o + sin φ + β 2

y

x o x + yo y

yo 94

and k2 r 0 = k =k

cos φ − β 2

x + sin φ − β 2

cos φ − β 2 x o + sin φ − β 2

y

x o x + yo y

yo , 95

respectively. Then, Equation 9.1 can be reorganized to give the following equation: E s k,

Ao exp − jk x o cos φ + β 2 + cos φ − β 2 exp − jk yo sin φ + β 2 + sin φ − β 2

96

445

446

9 Bistatic ISAR Imaging

The trigonometric expressions in the first and second phase terms can be easily calculated by the help of Ptolemy’s sum and difference identities for cosines and sines as given in more details below: cos φ +

β 2

+ cos φ −

β 2

β β − sin φ sin 2 2 β β + sin φ sin + cos φ cos 2 2 β = 2 cos φ cos 2

= cos φ cos

97

and sin φ +

β 2

+ sin φ −

β 2

= sin φ cos

β 2

β 2

β β − cos φ sin 2 2

+ sin φ cos = 2 sin φ cos

+ cos φ sin

98

β 2

Plugging Equations 9.7 and 9.8 into 9.6, it is trivial to get Es k, φ

Ao exp

− j2k cos φ cos

β xo 2

exp

− j2k sin φ cos

β y 2 o 99

This equation provides two separate phase terms as a function of both the spatial frequency variable k and the look angle variable φ. If these phase terms are carefully examined, the Fourier relationships between (2k cos φ cos(β/2)) and x, and (2k sin φ cos(β/2)) and y can be straightforwardly noticed. Therefore, the bistatic ISAR image can be produced in range and cross-range domains by the convenience of the 2D inverse Fourier transform (IFT). For the practical usages of bistatic ISAR imaging, the famous small frequency bandwidth and narrow-angle span approximation can be applied as already explained in Section 4.5 for the monostatic ISAR imaging. Therefore, the frequency bandwidth, B, is chosen to be small compared to center frequency of operation fc, that is, B ≤ fc/10. Then the wave number in the second phase term of Equation 9.9 can be approximated as k

kc = 2π f c c

9 10

Secondly, the look-angle span, say Ω is narrow such as on the order of a few degrees but not greater than 5 –7 . Then the following approximations can be made

9.2 Geometry for Bi-Isar Imaging and the Algorithm

cos φ

1

sin φ

φ

9 11

Consequently, the scattered electric field in Equation 9.9 can then be approximated to Es k, φ

Ao exp

β xo 2

− j2k cos

2 cos Es f , φ

Ao exp

exp

− j2π

− j2π

β 2

c

2 f c cos

exp

f

− j2k c φ cos

β y 2 o

xo 9 12

β 2

c

φ

yo

In the above equation, there exist Fourier relationships between ((2f cos(β/2))/c) ≜ α and xo, and between ((2fcφ cos(β/2))/c) ≜ γ and yo. With this construct, therefore, the bistatic ISAR image on 2D x–y plane can be obtained by taking the 2D IFT of Equation 9.12 to have −1 α,γ

Es α, γ

−1 γ ∞

= Ao

E s x, y = Ao ∞

−∞

−∞

−1 β

exp − j2πα x o exp − j2πα x o

exp − j2πγ yo

exp

exp

exp − j2πγ yo j2πα x dα

j2πγ y dγ

9 13

= Ao δ x − x o , y − yo ≜ ISAR x, y Here,

−1 α,γ

stands for the 2D IFT operation with respect to α-and-γ directions, −1 γ

whereas and β− 1 are the 1D IFT operations with respect to individual α and γ directions, respectively. The result of Eq. 9.13 theoretically proves that 2D ISAR image of point P(x0, y0) manifests itself at the exact location as a 2D impulse function located at (xo, yo) with the correct EM reflectivity coefficient of Ao after applying the Fourier based above ISAR imaging integral. It is important to note that while the transmitting and receiving directions have a total angle value of “β” (also named as the bistatic angle), the ISAR image is constructed around the bisector direction of this bistatic angle. This synthetic direction around the azimuth angle of φ are usually called the bistatic aspect. This is because of the fact that the bistatic ISAR imaging geometry delivers an ISAR image as if it has been constructed for the look direction at this aspect value of a synthetic

447

448

9 Bistatic ISAR Imaging

monostatic radar equivalent as demonstrated in Figure 9.3. As we shall see in the upcoming sections, the bistatic angle seriously affects the range and cross-range resolutions of the bistatic ISAR image and also observed bistatic RCS of the target.

9.2.2

Bistatic ISAR Imaging Algorithm for a Target

The ISAR formulation in the previous section is formulated for a single point scatterer that was assumed to be located on the target. Based on the scattering center model of the ISAR imaging, it is a common practice to approximate the scattered electric field from a target as the summation of scattering from a finite number of single point scatterers, called scattering centers on the target. Therefore, Eq. 9.1 can be generalized to give K

Es k, φ

Ai exp − j k 1i r 0i + k 2i r 0i

9 14

i=1

Here, K represents the total number of effective bistatic scattering centers and Ai stands for the complex bistatic scattered field amplitude for the ith scattering center. k 1i and k 2i are the wavenumber vectors along the directions from transmitter to ith scattering center and from ith scattering center to receiver, respectively. r 0 is the displacement vector from origin to the ith scattering center location at (xi, yi). Following the very similar formulation steps from Equation 9.2 to 9.13, it is trivial to obtain the following 2D ISAR image of the target by the following 2D inverse Fourier integral of the collected 2D scattered field data as ∞

ISAR x, y =

−∞ ∞

Es k, φ

exp

j2παx exp

j2πγy dαdγ

K

−∞ i=1

Ai exp − j k 1i r 0i + k 2i r 0i

exp j2π αx + γy dαdγ ∞

K

=

Ai i=1

−∞

exp − j k 1i r 0i + k 2i r 0i

exp j2π αx + γy dαdγ ∞

K

=

Ai i=1

−∞

exp

j2πα x − x i

exp

j2πγ y − yi dαdγ

K

Ai δ x − x i , y − yi

= i=1

9 15

9.3 Resolutions in Bistatic ISAR

Thus, the resultant ISAR image is composed of summation of K scattering centers with their EM bistatic reflectivity coefficients of Ais. Obviously, the limits of the integral in Equation 9.15 have to be finite in practice due to the fact that the field data can be collected within a finite bandwidth and finite aspect width. Therefore, the practical ISAR image response distorts from the impulse function to the sinc function for a finite frequency bandwidth of B and a finite look-angular width of Ω. As a result, Equation 9.15 converts to K

ISAR x, y =

Ai sinc

1 span α

x − xi

sinc

1 span γ

Ai sinc

2 cos β B c

x − xi

sinc

2 f c cos β Ω c

i=1 K

= i=1

y − yi y − yi 9 16

9.3

Resolutions in Bistatic ISAR

Range and the cross-range resolutions in bistatic ISAR imaging can be obtained by the help of the formula given in Eq. 9.16.

9.3.1 Range Resolution The first sinc term in Eq. 9.16 can be used to determine the resolution metric for the range. The −4 dB width of this sinc term can be calculated by taking the inverse of the argument in front of (x − xi) expression as Δx =

c 2B cos β 2

9 17

As similar to monostatic ISAR, wider frequency bandwidth provides sharper resolution along the range direction. On the other hand, there is an extra term of “cos (β/2)” that is present in the denominator of the resolution formula for the bistatic ISAR case. This offers that the range resolution varies for different values of bistatic angle. The best bistatic range resolution can be Δx = c/2B for β = 0 that represents exactly the monostatic ISAR case. It is obvious that β is always greater than 0 for any bistatic case; therefore, the range resolution in bistatic ISAR is always worse than that of monostatic ISAR case. For example, while a 1 GHz bandwidth of transmitted signal is providing a range resolution of 15 cm for the monostatic ISAR, the resolution becomes 17.32 cm for the bistatic angle of 60 for the bistatic ISAR. To give a more insight into the range resolution of bistatic ISAR, range resolution figures for different values of bistatic

449

450

9 Bistatic ISAR Imaging

Table 9.1 Bistatic range resolution outcome for different values of bistatic angle. Bistatic angle, β ( )

Frequency bandwidth, B (GHz)

Monostatic range resolution, Δx (cm)

Bistatic range resolution, Δx (cm)

15

1

15.00

15.13

30

1

15.00

15.53

45

1

15.00

16.24

60

1

15.00

17.32

90

1

15.00

21.21

120

1

15.00

30.00

141

1

15.00

45.00

145

1

15.00

49.88

151.05

1

15.00

60.00

156.93

1

15.00

75.00

160

1

15.00

86.38

175

1

15.00

343.88

180

1

15.00

Infinity

angle are given in Table 9.1 for the fixed transmitted signal’s bandwidth of 1 GHz. Table 9.1 clearly states that the bistatic range resolution is decreasing slowly up to bistatic angle of 90 . It becomes twice times worse than that of monostatic case for the bistatic angle of 120 . When the bistatic angle is chosen as 141 , the range resolution becomes three times coarser than the monostatic range resolution. When the bistatic angle reaches to 180 , the bistatic resolution becomes infinity, and therefore, the ISAR imaging cannot be possible as expected since the transmitter directly looks towards the receiver and mesmerize it. To sum up, it seems practical to use bistatic angles up to 145 –150 to keep a reasonable range resolution performance in the reconstructed bistatic ISAR image.

9.3.2

Cross-Range Resolution

The second sinc term in Equation 9.16 can be used to retrieve the formula for the cross-range resolution. The −4 dB width of this sinc term, i.e. the resolution, can be found by taking the reverse of the argument in front of (y − yi) term as c 2 f c Ωcos β 2 λc = 2Ωcos β 2

Δy =

9 18

9.3 Resolutions in Bistatic ISAR

Table 9.2

Bistatic cross-range resolution outcome for different values of bistatic angle.

Bistatic angle, β( )

Frequency of operation, fc (GHz)

Data collection angular width, Ω( )

Monostatic crossrange resolution, Δy (cm)

Bistatic crossrange resolution, Δy (cm)

15

10

5.73

15.00

15.13

30

10

5.73

15.00

15.53

45

10

5.73

15.00

16.24

60

10

5.73

15.00

17.32

90

10

5.73

15.00

21.21

120

10

5.73

15.00

30.00

141

10

5.73

15.00

45.00

145

10

5.73

15.00

49.88

151.05

10

5.73

15.00

60.00

156.93

10

5.73

15.00

75.00

160

10

5.73

15.00

86.38

175

10

5.73

15.00

343.88

180

10

5.73

15.00

Infinity

where λc corresponds to the wavelength for the center frequency, fc. Equation 9.18 suggests the higher the angle span width the better the bistatic resolution in the cross-range direction as in the case monostatic ISAR. Again, bistatic cross-range resolution has also the “cosβ/2” term present in the denominator of the formula which will worsen the cross-range resolution metric for wider values of aspect angle. To give an example, a monostatic ISAR radar operating at 10 GHz provides a cross-range resolution value of 15 cm for a data collection angular span of 5.73 according to the formula given in Equation 4.47. If bistatic set-up is used, this cross-range resolution becomes coarser with the increasing value of the bistatic angle as listed in Table 9.2. The resolution becomes twice, three times, four times, and five times than that of monostatic case for the bistatic angle values of 120 , 141 , 151.05 , and 156.93 , respectively. Again, for an operational values of cross-range resolution metrics, bistatic configuration with bistatic angle less than 120 is reasonable. Even, bistatic angle values up to 145 –150 are still workable. For wider angles, the bistatic ISAR imaging will highly suffer from the deteriorated resolution metrics both in range and cross-range directions.

9.3.3 Range and Cross-Range Extends Once the resolutions in range and cross-range domains are determined as in Equations 9.17 and 9.18, it is trivial to find the range and cross-range spatial extends, i.e.

451

452

9 Bistatic ISAR Imaging

the bistatic ISAR image frame size by using the sampling points in frequency and aspect domains. If the frequency bandwidth is sampled by Nx times and the angular width is sampled by Ny times, corresponding image domain extends are given as X max = N x Δx =

Nx c 2B cos β 2

Y max = N y Δy =

9 19

N y λc 2Ωcos β 2

This results clearly show that bistatic ISAR window size is always bigger than that of monostatic case because of “cos(β/2)” term in the denominators of Xmax and Ymax. For the same examples given above, if the bistatic angle is 90 and the 2D frequency-aspect data are collected over 256 sampling points in each domain, the bistatic ISAR image size becomes 41.6 m × 41.6 m by that is to be compared with the monostatic ISAR window size of 38.4 m × 38.4 m for the same radar parameters.

9.4

Design Procedure for Bi-ISAR Imaging

The basic algorithm for designing a Bi-ISAR image is given thru the flowchart in Figure 9.4. The steps of the algorithm are briefly given below: Step 1: One of the important decisions of ISAR imaging is choosing the image window extends, i.e. the size of the ISAR image frame. Similar to monostatic ISAR, it is mandatory to judge the ISAR image size by considering the target size. For a successful ISAR image, the range window extends Xmax and cross-range window extend Ymax should be sufficiently greater than the target’s physical dimensions along these dimensions. It is important to note that the ISAR image is the 2D projection of the target’s 3D shape onto range cross-range according to the look-angle of the radar. Step 2: Another important selection is the range resolution Δx and cross-range resolution Δy. These parameters are so critical that the selection of how many physical details of the target is going to be seen in the image is determined by these parameters. As similar to monostatic case, the number of sampling points in the range, Nx, and cross-range, Ny, are determined via the same equations:

9.4 Design Procedure for Bi-ISAR Imaging

Figure 9.4 Flowchart for the Bi-ISAR imaging algorithm.

Start

Image size: Xmax, Ymax Resolutions: Δx, Δy

Nx = Ny =

Δf = Δϕ =

Xmax Δx Ymax Δy

c/2 Xmax cos(β/2) λc/2 Ymax cos(β/2)

B = Nx Δf Ω = Ny Δϕ

Collect Es(f,ϕ) for the determined frequencies and angles

Take 2D IFT of Es(f,ϕ) ISAR(x, y) Stop

453

454

9 Bistatic ISAR Imaging

X max Δx Y max Ny = Δy

Nx =

9 20

Another important role of these parameters is that they also equal to the number of the sampling point in frequency and aspect as Nf = Nx and Nφ = Ny, respectively. Step 3: In this step, we determine, the frequency bandwidth, center frequency of operation, and aspect span width based on the parameters that were decided in the first two steps of the algorithm. According to the Fourier theory, the expressions in Equations 9.17 and 9.18 suggest the following frequency and look-angle resolutions, namely Δf and Δφ, respectively as c 2 X max cos β 2 λc 2 Δφ = Y max cos β 2 Δf =

9 21

Then, the frequency bandwidth, B, and the angle span width, Ω, can be found via Nx c 2X max cos β 2 N y λc Δφ = 2Y max cos β 2

B = N f Δf = Ω = Nφ

9 22

Once the frequency bandwidth, B, is calculated, the center frequency of operation can be chosen to be fc 10 B if the small bandwidth narrowaspect ISAR imaging can be employed. Then, the bistatic scattering should also be collected within a small aspect span of Ω, preferable not more than 7 or 8 . Step 4: Now, the 2D bistatic electric field data, Es(f, φ), can be collected using the following frequency and aspect vectors

f = φ=

Nf Δf 2 Nφ Δφ φc − 2 fc −

Nf − 1 Δf 2 Nφ φc − − 1 Δφ 2 fc −

fc φc

fc + φc +

Nf − 1 Δf 2 Nφ − 1 Δφ 2 9 23

Step 5: As the final step of the algorithm, useful 2D IFT operation can be applied to get the bistatic 2D ISAR image of Es(x, y).

9.5 Bi-Isar Imaging Examples

9.5

Bi-Isar Imaging Examples

9.5.1 Bi-ISAR Design Example #1 In this example, an airplane model that was assumed to be consist of perfect point scatterer as shown in Figure 9.5a is considered. The model has dimensions of 36 and 26.7 m in x and y directions, respectively. The target is thought to be situated at 45 km away from the transmitter and the receiver is also at the far field of the target as the distance between them is 55 km. Both the transmitter and the receiver are directed to the target from the front (nose-on) and side (broadside) as depicted in Figure 9.5a. The bistatic angle between the transmitter and the receiver is 90 , respectively as illustrated in Figure 9.5a. The bistatic is radar is assumed to be operating at the center frequency of 11.5 GHz with a bandwidth of 1 GHz. It is further considered that the receiver can observe the target for some period of time such that the corresponding total look-angle span is 7 (0.1222 rad). Based on the above configuration and radar parameters, following design procedure is applied to form the ISAR image of the target. i)

First image window size is determined such that the airplane model has to fall inside the ISAR image frame. Therefore, an image size of (Xmax, Ymax) = 50 m × 50 m was chosen. ii) Based on the frequency bandwidth of the radar and the look-angle span of the target, the range and cross-range resolutions can be found using Equations 9.17 and 9.18 as Δx =

c 3 108 = 9 2B cos β 2 2 10 cos 45

Δy =

c 3 108 = 9 2 f c Ωcos β 2 2 11 5 10 0 1222 cos 45

= 21 21 cm

9 24

= 15 10 cm 9 25

iii) Then, total number of range bins, Nx, and total number of cross-range bins, Ny, can be practically found by X max 50 = round = 236 0 2121 Δx Y max 50 = = round = 331 Δy 0 1510

Nx = Ny

9 26

Nx and Ny also represent the image pixel numbers along range and cross-range directions, respectively. If no extrapolation scheme (such as zero padding) is

455

9 Bistatic ISAR Imaging

(a) 20 10

TX

Y, m

0 –10 –20 –30 –40 –40

–30

–20

–10 X, m

0

10

20

RX

(b) 0

–25 –20

–5

–15

Moonstatic equivalent

Cross-range, m

456

–10

–10 45°

–5

–15

0 –20

5 10

–25

15 –30

20 –20

–10

0

10

20

dB

–35

Range, m

Figure 9.5 (a) Bi-ISAR imaging geometry for an airplane model, (b) constructed BiISAR image.

9.5 Bi-Isar Imaging Examples

applied, the ISAR image would have a total count of 236 × 331 = 78116 pixels (or ~78 kilopixels). iv) The calculated numbers in Equation 9.26 also represent the total number of sampling points in frequency and aspect domains, respectively. Therefore, the frequency sampling spacing, Δf, and aspect sampling spacing,Δφ, can be calculated as given below: B B 5 109 4 24 MHz = = Nf Nx 236 Ω Ω 7 = Δφ = 0 02 = Nφ Ny 331

Δf =

9 27

v) Now, the design is almost complete and the rest of the algorithm is to collect the 2D frequency-aspect bistatic scattered data for the calculated frequencies and aspects with so that 2D multi-frequency multi-aspect scattered electric field matrix of Es(f, φ) is gathered. vi) At the final step of the algorithm, 2D IFT is applied to this data to have the final constructed bistatic ISAR image as given in Figure 9.5b. While constructing the image, 2D Hamming window is applied for sidelobe suppression and four-times zero-padding procedure is also employed for interpolation purposes. Therefore, the constructed image is about 1.3 Megapixels. Looking at the bistatic ISAR image in the figure, one can easily observe that point scatterers in the airplane model are correctly pinpointed and mapped at their correct locations and true amplitudes. Furthermore, the ISAR image is rotated exactly 45 that is half of the bistatic angle of 90 as expected. This phenomenon was previously explained in detail in Section 9.2.1. Therefore, the final image has an orientation as if it has been constructed with a synthetic monostatic radar looking towards the target from the halfway of the bistatic angle between the transmitter and the receiver as expected.

9.5.2 Bi-ISAR Design Example #2 In this example, another airplane model that was supposed to be consists of perfect point scatterers in Figure 9.6 is considered. The model has an extend of 29 m in x direction and 20 m in y direction. The target is assumed to be 37 km away from the transmitter. The receiver is also at the far field of the target with a range distance of 43 km. The bistatic radar is assumed to be operating between 11 and 12 GHz. It is further considered that the receiver can observe the target for some period of time such that the corresponding total look-angle span is 7.2 . For this example, the effect of bistatic angle on the ISAR image is investigated especially in terms of resolutions in range and cross-range domains. For this

457

9 Bistatic ISAR Imaging

15 10 5 y, m

458

0 –5 –10 –15 –20

Figure 9.6

–15

–10

–5

0 x, m

5

10

15

20

Fighter aircraft model for the Bi-ISAR imaging example #2.

purpose, the bistatic angle is varied from 20 to 179 for a total of 10 different angle values as listed in the first column of Table 9.3. Based on the above transmitter-target-receiver configurations and presumed radar parameters, the following design steps are applied to form the ISAR images of the target for different bistatic angle values. i) The size of the ISAR image window is selected such that the airplane model has to be within the ISAR image frame. Therefore, an image size of (Xmax, Ymax) = 36 m × 36 m was chosen. ii) Based on the frequency bandwidth of the radar and the look-angle span of the target, the range and cross-range resolutions can be found using Equations 9.17 and 9.18 as c 3 108 = 9 28 Δx = 9 2B cos β 2 2 10 cos β 2 Δy =

c 3 108 = 9 2 f c Ωcos β 2 2 11 5 10 0 1222 cos β 2

9 29

The theoretical range and cross-range resolution values for the simulated angles can be seen from the second and the third columns of Table 9.3, respectively. It is seen from the list that the calculated resolution values become worse as the angle between the transmitter and the receiver rises. iii) Next, total number of range bins, Nx, and total number of cross-range bins, Ny, can be readily found using Equation 9.20. As it can be observed from

9.5 Bi-Isar Imaging Examples

Table 9.3

Calculated simulation parameters with respect to bistatic angle.

Bistatic angle, β( )

Range resolution, Δx (cm)

Crossrange resolution, Δy (cm)

# of range bins, Nx

# of crossrange bins, Ny

Frequency sampling spacing, Δf (MHz)

Aspect sampling spacing, Δφ ( )

20

15.23

10.54

236

342

4.24

0.021

40

15.96

11.10

226

326

4.42

0.022

60

17.32

11.99

208

300

4.81

0.024

80

19.58

13.55

184

266

5.43

0.027

100

23.34

16.15

154

223

6.49

0.032

120

30.00

20.76

120

173

8.33

0.042

140

43.86

30.35

82

119

12.20

0.061

160

86.38

59.77

42

60

23.81

0.120

170

1.721

1.191

21

30

47.62

0.240

175

3.439

2.380

10

15

100.0

0.480

the table, these numbers decrease due to worsened resolution values for increased values of bistatic angle resulting less pixels in the ISAR image. If no extrapolation is employed, the ISAR image size is 236 × 342 for β = 20 ; whereas it is 42 × 60 for β = 160 , for example. iv) These calculated Nx and Ny also determine the total number of sampling points in frequency and aspect domains, respectively. So, it is trivial to calculate the frequency sampling spacing, Δf, and aspect sampling spacing, Δφ, by using Equation 9.21 so that the results are listed in the last two columns of Table 9.3. v) Then, the bistatic scattered signal is collected with the calculated parameters that are found and listed in previous steps. Therefore, the 2D frequency aspect bistatic scattered data Es(f, φ) are gathered for all bistatic angles listed in Table 9.3. vi) Lastly, 2D Bi-ISAR images are formed using the 2D IFT formula given in Equation 9.15. The constructed 2D Bi-ISAR images corresponding various bistatic angles ranging from 20 to 175 are given in Figure 9.7a through 9.7j, respectively. During the processing, 2D Hamming window and fourtimes zero-padding procedures were applied for sidelobe reduction and interpolation purposes, respectively. Several observations can be made from these Bi-ISAR images in Figure 9.7:

•

As expected, the Bi-ISAR images were rotated with exactly the half of the bistatic angle values due to the reason that was explained in Section 9.2.1.

459

9 Bistatic ISAR Imaging

(a)

Bistatic angle: 20°

0

–15 –5

–10 Cross-range, m

–5

–10

0 –15

5 10

–20

15 –15

–10

(b)

–5

0 5 Range, m

10

15

dB

–25

Bistatic angle: 40° 0

–15 –5

–10 Cross-range, m

460

–5

–10

0 –15

5 10

–20

15 –15

–10

–5

5 0 Range, m

10

15

dB

–25

Figure 9.7 Bi-ISAR image of the aircraft model for bistatic angle of (a) 20 , (b) 40 , (c) 60 , (d) 80 , (e) 100 , (f ) 120 , (g) 140 , (h) 160 , (i) 170 , and (j) 175 .

9.5 Bi-Isar Imaging Examples

(c)

Bistatic angle: 60° 0

–15 –5

–10 Cross-range, m

–5

–10

0 –15

5 10

–20

15 –15

–10

(d)

–5

0 5 Range, m

10

15

dB

–25

Bistatic angle: 80° 0

–15 –5

Cross-range, m

–10 –5

–10

0 –15

5 10

–20

15 –15

–10

Figure 9.7 (Continued)

–5

5 0 Range, m

10

15

dB

–25

461

9 Bistatic ISAR Imaging

(e) Bistatic angle: 100°

0

–15 –5

Cross-range, m

–10 –5

–10

0 –15

5 10

–20

15 –15

–10

(f)

–5

0 5 Range, m

10

15

dB

Bistatic angle: 120°

–25

0

–15 –5

–10 Cross-range, m

462

–5

–10

0 –15

5 10

–20

15 –15

Figure 9.7

–10

(Continued)

–5

5 0 Range, m

10

15

dB

–25

9.5 Bi-Isar Imaging Examples

(g)

Bistatic angle: 140° 0 –15 –5

Cross-range, m

–10 –5

–10 0 –15

5 10

–20

15 –15

–10

(h)

–5

0 5 Range, m

10

15

dB

Bistatic angle: 160°

–25

0

–15 –5

Cross-range, m

–10 –5

–10 0 –15

5 10

–20

15 –15

–10

Figure 9.7 (Continued)

–5

0 5 Range, m

10

15

dB

–25

463

9 Bistatic ISAR Imaging

(i) Bistatic angle: 170°

0

–15 –5

Cross-range, m

–10 –5

–10 0 –15

5 10

–20

15 –25 –15

–10

(j)

–5

0 5 Range, m

10

15

dB

Bistatic angle: 175°

0

–15

–10

–5

–5 Cross-range, m

464

–10 0 –15

5

10

–20

15 –25 –15

Figure 9.7

–10

(Continued)

–5

0 Range, m

5

10

dB

9.6 Mu-ISAR Imaging

• •

One can easily observe the resolution degradation as the bistatic angle between the transmitter and the receiver increases from 20 to 175 in Figure 9.7a–j, respectively. This result is in accordance with the theoretical resolution values listed in Table 9.3. This study suggests that the Bi-ISAR image construction with good fidelity is possible for bistatic angles up to 150 . Beyond that, the ISAR image greatly suffers from bad resolution performance in both directions.

9.6

Mu-ISAR Imaging

Multi-static radar system can be constructed by using multiple number of transmitters and receivers each situated at different spatial locations. The main idea behind of using multi-receivers in ISAR imaging is to increase the visibility of the target on the radar screen. An image fusion algorithm is then used to construct the final image on the same coordinate system. Various multi-static geometries can be obtained by utilizing different multiple transmit and receive configurations. Basically, a total of three different multi-static configuration can be possible as follows:

• • •

One-transmitter multi-receiver configuration: In this geometry, there is only one transmitting antenna and a number of receiving antennas are situated at different spatial locations as illustrated in Figure 9.8a. This set-up provides the cheapest solution among others since there is single power transmitting site. Multi-transmitter one-receiver configuration: In this approach, bistatic link is being formed via a number of active transmitters and with a single receiver as shown in Figure 9.8b. Since the EM propagation path is reversed in this set-up when compared to the previous geometry, the same data-set could be collected at the receiving site provided that a multiplexing (either in time or in frequency) is employed. This multi-static configuration may require more financial resources since it contains several active transmitting units. Multi-transmitter multi-receiver configuration: For this case, illumination of the target is being provided thru several transmitters at different locations as depicted in Figure 9.8c. Furthermore, multi-receivers are collecting the bistatic scattered field from the target. This geometry can also be thought as multipleinput multiple-output (MIMO) system that is used to improve the spatial resolution (Pastina et al. 2010; Zhu et al. 2017). This type of configuration improves the signal-to-noise ratio (SNR) and therefore, increases the probability of detection of the target at the price of more complex data and signal processing steps. If there are M transmitting and N receiving sites are at present, various MIMO

465

466

9 Bistatic ISAR Imaging

(a)

Earth Receiver #N

Transmitter

Receiver #2 Receiver #1

(b)

Earth Transmitter #N

Receiver

Transmitter #2 Transmitter #1

(c)

Earth Transmitter #1

Receiver #2

Transmitter #M Receiver #N

Receiver #1 Transmitter #2

Figure 9.8 Various geometries for Mu-ISAR imaging configurations: (a) single-transmitter multi-receiver, (b) multi-transmitter single-receiver, and (c) multi-transmitter multi-receiver.

9.6 Mu-ISAR Imaging

bistatic links can be configured by selecting a total of P active transmitters and K active receivers where 2 ≤ P ≤ M and 2 ≤ K ≤ N.

9.6.1 Challenges in Mu-ISAR Imaging While multi-static configuration provides some advantages as listed above, there are some difficulties that need to be overcome while utilizing it:

• •

• • •

Since the receivers are at the different spatial locations, the collected dataset from each receiver has to be transferred to a common location. Since the total bistatic range distances are different for each transmitter-receiver duo, associated arrival times of echoes will not be the same. This will lead blurring and defocusing effects at the final reconstructed image. Therefore, a synchronization routine should also be employed to coherently form the image. Another challenge can be attributed to the computational hardness of transforming each single bistatic image into a common coordinate system. It should be noticed that each ISAR image is formed on the 2D plane that contains the vectors from transmitter to target and target to receiver. In fact, the Bi-ISAR image is the projection of every scattering centers on this plane. On the other hand, each transmitter-receiver duo offers different ISAR image plane unless all the transmitters and receivers are actually on the same plane. Therefore, the projection of each scattering centers of the target will not be on the same 2D plane; therefore, coordinate transformations need to be carefully employed to have a correctly mapped final multi-static ISAR image. It is obvious that target’s relative motion along yaw, pitch, and roll directions with respect to different transmitters and/or receivers differ. Therefore, the cross-range resolutions in each Bi-ISAR image will also vary which will lead to resolution mismatches in the final ISAR image so that blurring and defocusing phenomena arise. It is also demonstrated in Section 9.2 that the range and cross-range resolutions in bistatic ISAR image are also related to the bistatic angle between the transmitter and the receiver. Since each bistatic duo in a multi-static scenario presents different bistatic angle, its bistatic ISAR image’s resolutions will be different. This phenomenon also offers another mismatch between the resolutions of each Bi-ISAR image that results in further blurring and defocusing effects in the final merged ISAR image. It is also no doubt that the type of scattering from target’s various parts can vary with respect to bistatic angle. For example, a geometry on the target may offer single bounce type scattering in some bistatic angles while its scattering characteristics can be changed to multiple bounce type scattering features. It is also known from Section 4.5.3 that multi-bounce mechanisms in ISAR image are

467

468

9 Bistatic ISAR Imaging

delayed in range direction. Since the final multi-static image is formed for a selected look-direction by transforming each Bi-ISAR images to this particular coordinates, the delayed image signatures of multi-bounces mechanisms will not be aligned to the range direction of this look-angle and therefore, they are wrongly interpreted. Similar arguments can also be made for other scattering types such as edge scatterings and volume scatterings.

9.6.2

Mu-ISAR Imaging Example

Based on the scenario that is given in Figure 9.9, a Mu-ISAR imaging example is constructed in Matlab simulation environment. As it can be observed from the multi-static scene in the figure, there is one transmitting antenna that is R0 = 54 km away from the target and there are three receivers with their target distances of R1 = 33 km, R2 = 28 km and R3 = 49 km away, respectively. The target, the transmitter, and the receivers are assumed to lie on the same plane. The target is fighter aircraft that is modeled with perfect point scatterers as depicted in Figure 9.10. The aircraft target model has a length of 14.84 m and has a wing extend length of 10.20 m. In the multi-static simulation, following parameters are considered. The bistatic angles between the transmitter and the three receivers are 36 , 68 , and 135 , respectively as illustrated in Figure 9.9. The Mu-ISAR system has the center frequency of operation of 9.9 GHz with a frequency bandwidth of 1.6 GHz. It is also assumed that the target’s movement provides an angular span of 4.3 with respect to transmitter and the receivers. Since the resolutions in range and cross-range directions are bistatic angle dependent (see Equation 9.19), the different resolution metrics can be obtained at the receiver sites as listed in Table 9.4. Obviously, the

36°

68°

R0

R2

135°

R3

R1 TX RX1

Figure 9.9

RX2

Multi-static ISAR scenario with single transmitter and three receivers.

RX3

9.6 Mu-ISAR Imaging

8 6 4

y, m

2 0 –2 –4 –6 –8 –10

–5

Figure 9.10

0 x, m

5

10

Fighter craft modeled with perfect point scatterers.

range and cross-range resolutions become worse as the bistatic angle increases. Still, both resolutions are less than 25 cm even if the bistatic angle is 135 for the third receiver. Such a resolution performance is still good enough to resolve major scattering centers on an aircraft model. The simulated bistatic electric field data Esj k, φ (j = 1, 2 and 3 for Rx #1, Rx #2 and Rx #3, respectively) have been obtained with the help of a simulation script that was coded in Matlab. Since the target distances from the transmitter and receivers can be instantly measured from the delay time of received pulses, the phases contributions that are responsible for the bistatic travel mechanisms from transmitter-to-target and target-to-receivers can be compensated as follows: s Ej

k, φ =

Esj

k, φ

exp jk R0 + R j

for j =

1

, Rx#1

2

, Rx#2

3

, Rx#3 9 30

s

Here, E j k, φ are the phase compensated bistatic electric field data such that the s

phase center of each E j k, φ is set to center of target for its monostatic equivalent configuration as seen from the half of the bistatic angle. Afterward, standard bistatic ISAR imaging steps given in the flowchart of Figure 9.4 are employed based on the simulation parameters that are listed in Table 9.4, the individual Bi-ISAR images are formed as given in Figure 9.11. The Bi-ISAR images formed

469

Table 9.4 Parameters for Mu-ISAR imaging example. Input parameters Center frequency, fc

Frequency bandwidth, B

Angular span width, Ω

Target distances

9.9 GHz

1.6 GHz

4.3

R0 R1 R2 R3

= = = =

54 33 28 49

km km km km

Bistatic angles

ISAR image size

βtx-rx1 = 36 βtx-rx2 = 68 βtx-rx3 = 135

20 m × 20 m

Calculated parameters Range resolution

Cross-range resolution

# of sampling points in frequency

# of sampling points in aspect

Δxrx1 = 9.86 cm Δxrx2 = 11.31 cm Δxrx3 = 24.50 cm

Δyrx1 = 10.61 cm Δyrx2 = 12.17 cm Δyrx3 = 26.38 cm

Nf, rx1 = 203 Nf, rx2 = 177 Nf, rx3 = 82

Na, rx1 = 188 Na, rx2 = 164 Na, rx3 = 76

9.6 Mu-ISAR Imaging

(a) Bistatic angle: 36°

Bistatic angle: 68°

Bistatic angle: 135°

Combined multi-static ISAR image

(b)

0

–8

Cross-range, m

–6

–5

–4 –2

–10

0 2

–15

4 6

–20

8 10 –10

–25 –5

0 Range, m

5

dB

Figure 9.11 Bi-ISAR images of aircraft model obtained at (a) Rx #1, (b) Rx #2, and (c) Rx #3. (d) Constructed Mu-ISAR image (bottom) from the Bi-ISAR images.

by the first, second, and third receiver can be seen in the upper left, upper middle, and upper right images, respectively. As expected, the resultant Bi-ISAR images are obtained as if seen from their monostatic equivalents with corresponding half-angle values of 18 , 34 , and 67.5 for the Rx #1, Rx #2, and Rx #3, respectively. As expected, one can easily observe the resolution degradation both in range and cross-range domains when progressing from the left image to right image as the value of the bistatic angle increases. Lastly, all three Bi-ISAR images are combined together to yield a multi-static image. Since the bistatic angles are found since the antennas were assumed to be tracked to the target, each bistatic ISAR image can be transformed to a preferable coordinate system by rotating each image accordingly.

471

472

9 Bistatic ISAR Imaging

For the above example, Bi-ISAR images in the upper part of Figure 9.11 were rotated by half of bistatic angles such that all of them were lined along the same range direction. During the transformation, a bilinear interpolation scheme was utilized for the new grid of Mu-ISAR image. Then, all aligned images are summed up to form the final Mu-ISAR image that is presented in the lower part of Figure 9.11. As seen from this image, scattering centers that constitute the target outline is now more pronounced that may rise to increased probability of true classification of the target.

9.7

Matlab Codes

Below are the Matlab source codes that were used to generate all of the Matlabproduced figures in this chapter. The codes are also provided inside the CD of this book.

Matlab code 9.1 Matlab file “Figure9-5.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 9.5 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % Airplane.mat % AirplaneEs2.mat % matplot2.m clear; c=3e8;

% speed of light

%%————————Input Parameters——————————————————————————— % Transmitter and Receiver locations beta = 90*pi/180; % bistatic angle [rad] theta0 = 90*pi/180; % angle of Rx w.r.t target center [rad]

9.7 Matlab Codes

R0 = 45e3; R1 = 55e3; R = R0+R1;

% TX distance from target center [m] % RX distances from target center [m] % Total travel distance of EM wave[m]

% Image extent Dmax = 25;

% max. dimension = 2*Dmax

% Min. and max. frequencies [Hz] Fmin = 11e9; Fmax = 12e9; Fc = (Fmin+Fmax)/2; % Angular span-extend from center BWthd = 3.5; % In degrees: from -BWthd to +BWthd %%————————Pre-calculations for ISAR———————————————————— % Frequency bandwidth [Hz] BWf = Fmax-Fmin; % Angle spanwidth [rad] BWth = BWthd*pi/180; % Resolutions and corresponding f and theta vectors dx = c/(2*BWf*cos(beta/2)) ; % range resolution dy = c/(2*Fc*cos(beta/2)*(2*BWth));% cross-range resolution Nf = round((2*Dmax)/dx); Nth = round((2*Dmax)/dy);

% NUM of freq. points % NUM of cross-range points

df = BWf/(Nf ); % frequency resolution f = Fmin:df:Fmax-df; % frequency vector k = 2*pi*f/c; % wavenumber vector dth = 2*BWth/(Nth); % angle resolution theta = theta0+dth*(-Nth/2:Nth/2-1);% angle vector % Image resolutions and range and x-range vectors X = -dx*Nf/2:dx:dx*(Nf/2-1); % form range vector Y = -dy*Nth/2:dy:dy*(Nth/2-1); % form cross-range vector

473

474

9 Bistatic ISAR Imaging

% Load target scattering centers load Airplane figure; plot(-xx,yy,'o','MarkerSize',4,'MarkerFaceColor',[1 0 0]); grid on; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); axis([-45 25 -45 25]); xlabel('X [m]'); ylabel('Y [m]'); % Prepare for other signal processing steps [phi,rho] = cart2pol(xx,yy); l=length(xx); % Load Bistatic Scattered E-field load AirplaneEs2 %————POST PROCESSING OF ISAR————————————————————————— kk = repmat(k',1,Nth); Es = Es.*exp(1i*kk*2*R); % compansate extra phase % windowing w = hamming(Nf )*hamming(Nth).'; Ess = Es.*w; % zero padding by 4 times Enew = Ess; Enew(Nf*4,Nth*4)=0; % ISAR image formatiom ISARnew = fftshift(fft2(Enew.')); ISARnew = ISARnew/Nf/Nth; figure; matplot2(-X,Y,ISARnew,35); axis equal axis tight colormap(hot);

9.7 Matlab Codes

cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it range, m'); ylabel('\it cross-range, m');

Matlab code 9.2 Matlab file “Figure9-6and7.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 9.6 and 9.7 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the same % directory: % % Fighter1.mat % Fighter20Es2.mat, Fighter40Es2.mat, Fighter60Es2. mat % Fighter80Es2.mat, Fighter100Es2.mat, Fighter120Es2. mat % Fighter140Es2.mat, Fighter160Es2.mat, Fighter170Es2.mat % Fighter175Es2.mat % matplot2.m clear ; c=3e8; % speed of light %% Input Paramaters % Image extent Dmax = 18; % max. dimension = 2*Dmax %Load target scattering centers load Fighter1 h = figure(1); plot(-xx,yy,'o','MarkerSize',4,'MarkerFaceColor',[1 0 0]);

475

476

9 Bistatic ISAR Imaging

set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); grid minor axis([-Dmax Dmax -Dmax Dmax]); axis equal xlabel('\it x, m'); ylabel('\it y, m'); % saveas(h,'Figure9_5a.png','png'); % Frequencies Fmin = 11e9; Fmax = 12e9; Fc = (Fmin+Fmax)/2; % Angular span-extend from center BWthd = 3.6; % In degrees: from -BWthd to +BWthd theta0 = 90*pi/180;% R0 = 37e3; % R1 = 43e3; % R = R0+R1; %

angle of Rx w.r.t target center TX distance from target center [m] RX distances from target center [m] Total travel distance of EM wave[m]

% Different bistatic angles betaAll = [ 20 40 60 80 100 120 140 160 170 175]*pi/180; p = length(betaAll); figure for n=1:p % loop for bistatic angles beta = betaAll(n); % single bistatic angle % Pre calculations OF ISAR % Frequency bandwidth BWf = (Fmax-Fmin); % Angle spanwidth (radians) BWth = BWthd*pi/180;

% radians

% resolutions and corresponding f and theta vectors dx = c/(2*BWf*cos(beta/2)) ; % range resolution dy = c/(2*Fc*cos(beta/2)*(2*BWth)) ; % cross-range

9.7 Matlab Codes

resolution Nf = round((2*Dmax)/dx); points due to max range Nth = round((2*Dmax)/dy); range points due to max cr-range df = BWf/(Nf ); resolution f = Fmin:df:Fmax-df; k = 2*pi*f/c; vector dth = 2*BWth/(Nth); theta = theta0+dth*(-Nth/2:Nth/2-1);

% NUM of freq % NUM of cross-

% frequency % frequency vector % wavenumber % angle resolution % angle vector

%Image resolutions and range and x-range vectors X=-dx*Nf/2:dx:dx*(Nf/2-1); % range vector Y=-dy*Nth/2:dy:dy*(Nth/2-1); % cross-range vector %Prepare for other signal processing steps [phi,rho] = cart2pol(xx,yy); l=length(xx); %————————————— load Bistatic Scattered E-field S = num2str(beta*180/pi); Stitle = ['Fighter' S 'Es2']; load(Stitle) kk = repmat(k',1,Nth); Es = Es.*exp(1i*kk*2*R);

% compansate extra phase

%————POST PROCESSING OF ISAR————————————————————————— %windowing; w = hanning(Nf )*hanning(Nth).'; Ess = Es.*w; %zero padding with 4 times Enew = Ess; Enew(Nf*4,Nth*4)=0;

477

478

9 Bistatic ISAR Imaging

% ISAR image formatiom ISARnew = fftshift(fft2(Enew.')); ISARnew = ISARnew/Nf/Nth; %ISARnew(1,1)=2.62 h2 = figure(2); matplot2(X,-Y,ISARnew.',25); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; axis equal; axis tight set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it range, m'); ylabel('\it cross-range, m'); S = num2str(beta*180/pi); Stitle = ['\it Bistatic angle: ' S '^{\circ}']; title(Stitle) ; drawnow pause%(.1) end

Matlab code 9.3 Matlab file “Figure9-10and11.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 9.10 and 9.11 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % % %

F35.mat F35_36Es2.mat, F35_68Es2.mat, F35_135Es2.mat DATA1.mat, DATA2.mat, DATA3.mat matplot2.m

clear;

9.7 Matlab Codes

c=3e8;

% speed of light

%% Input Parameters theta0 = 90*pi/180;% R0 = 12e3; % R1 = 13e3; % R = R0+R1; % % Image extent Dmax = 10;

angle of Rx w.r.t target center TX distance from target center [m] RX distances from target center [m] Total travel distance of EM wave[m]

% max. dimension = 2*Dmax

% Frequencies [Hz] Fmin = 9.1e9; Fmax = 10.7e9; Fc = (Fmin+Fmax)/2; % Frequency bandwidth [Hz] BWf = Fmax-Fmin; % Angle spanwidth BWthd = 4.3;

% In degrees: from -BWthd to +BWthd

% Angle spanwidth [rad] BWth = BWthd*pi/180; %—————Load target scattering centers————————————— load F35 xx = -Xind; yy = Yind; figure; plot(-xx,yy,'ko','MarkerSize',2,'MarkerFaceColor',[0 0 0]); set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); axis([-Dmax Dmax -8 8]); grid minor axis equal xlabel('\it x, m'); ylabel('\it y, m');

479

480

9 Bistatic ISAR Imaging

%Prepare for other signal processing steps [phi,rho] = cart2pol(xx,yy); l=length(xx); %%———————Pre calculations OF ISAR—————————————— % Different bistatic angles betaAll = [36 68 135]*pi/180; p = length(betaAll); for n=1:p % loop for bistatic angles beta = betaAll(n); % single bistatic angle % resolutions and corresponding f and theta vectors dx = c/(2*BWf*cos(beta/2)); % range resolution dy = c/(2*Fc*cos(beta/2)*(2*BWth)) ; % cr-range resolution Nf = round((2*Dmax)/dx); Nth = round((2*Dmax)/dy); points

% NUM of freq. points % NUM of cross-range

df = BWf/(Nf ); % frequency resolution f = Fmin:df:Fmax-df; % frequency vector k = 2*pi*f/c; % wavenumber vector dth = 2*BWth/(Nth); % angle resolution theta = theta0+dth*(-Nth/2:Nth/2-1);% angle vector %Image resolutions and range and x-range vectors X=-dx*Nf/2:dx:dx*(Nf/2-1); % range vector Y=-dy*Nth/2:dy:dy*(Nth/2-1); % cross-range vector %load Bistatic Scattered E-field S = num2str(beta*180/pi); Stitle = ['F35_' S 'Es2']; load(Stitle) kk = repmat(k',1,Nth);

9.7 Matlab Codes

Es = Es.*exp(1i*kk*2*R);

% phase compensation

%————————POST PROCESSING OF ISAR ——————————————— w = hanning(Nf )*hanning(Nth).'; Ess = Es.*w; % Ess = Es.*1; %zero padding with 4 times Enew = Ess; Enew(Nf*4,Nth*4)=0; % ISAR image formatiom ISARnew = fftshift(fft2(Enew.')); ISARnew = ISARnew/Nf/Nth; figure; matplot2(X,-Y,ISARnew.',25); colormap(hot) axis equal; axis tight set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it range, m'); ylabel('\it cross-range, m'); S = num2str(beta*180/pi); Stitle = ['\it Bistatic angle: ' S '^{\circ}']; title(Stitle) ; drawnow end %% ——— MULTISTATIC IMAGE ——————————— % Load previously stored ISAR images load DATA36; ISAR1 = ISARnew.'; X1 = X; Y1 = Y; load DATA68; ISAR2 = ISARnew.'; X2 = X; Y2 = Y; load DATA135; ISAR3 = ISARnew.'; X3 = X; Y3 = Y; [Xq,Yq] = meshgrid(-Dmax:2*Dmax/999:Dmax); % Rotate bistatic images % 1st image rotation ISAR1m = imrotate(ISAR1, -36/2,'bilinear','crop');

481

482

9 Bistatic ISAR Imaging

[a,b] = size(ISAR1m); [X1m,Y1m] = meshgrid(linspace(min(X1),max(X1),a), linspace(min(Y1),max(Y1),b)); ISAR1n = interp2(X1m,Y1m,ISAR1m.',Xq,Yq); % 2nd image rotation ISAR2m = imrotate(ISAR2, -68/2,'bilinear','crop'); [a,b] = size(ISAR2m); [X2m,Y2m] = meshgrid(linspace(min(X2),max(X2),a), linspace(min(Y2),max(Y2),b)); ISAR2n = interp2(X2m,Y2m,ISAR2m.',Xq,Yq); % 3nd image rotation ISAR3m = imrotate(ISAR3, -135/2,'bilinear','crop'); [a,b] = size(ISAR3m); [X3m,Y3m] = meshgrid(linspace(min(X3),max(X3),a), linspace(min(Y3),max(Y3),b)); ISAR3n = interp2(X3m,Y3m,ISAR3m.',Xq,Yq); Final_Image = ISAR1n + ISAR2n + ISAR3n; % Combined ISAR image figure; matplot2(X,-Y,Final_Image.',25); colormap(hot); cc = colorbar; tt = title(cc,'dB'); tt.Position = [ 8 -15 0 ]; set(gca,'FontName', 'Arial', 'FontSize',12,'FontWeight', 'Bold'); xlabel('\it range, m'); ylabel('\it cross-range, m'); title('\itCombined Multi-static ISAR Image'); axis tight; axis equal; grid off

References

References Bhalla, R. and Ling, H. (1993). ISAR image formation using bistatic data computed from the shooting and bouncing ray technique. Journal of Electromagnetic Waves and Applications 7 (9): 1271–1287. https://doi.org/. https://doi.org/10.1163/ 156939393X00255. Drozdowicz, J., Samczynski, P., and Baczyk, M. K. (2017). Three-dimensional imaging of a rotating airborne target using bistatic inverse synthetic aperture radar. 2017 Signal Processing Symposium (SPSympo), Jachranka, Poland (12–14 September 2017). https://doi.org/10.1109/SPS.2017.8053643. Garry, J.L. and Smith, G.E. (2019). Passive ISAR part I: framework and considerations. IET Radar, Sonar & Navigation 13 (2): 169–180. http://dx.doi.org/. https://doi.org/ 10.1049/iet-rsn.2018.5233. Lazarov, A., Kabakchiev, H., and Kostadinov, T. (2014). DVB-T bistatic forward scattering inverse synthetic aperture radar imaging. 2014 15th International Radar Symposium (IRS), Gdansk, Poland (16–18 June 2014). https://doi.org/10.1109/ IRS.2014.6869261. Özdemir, C., Demirci, S., Yilmaz, B., Ak, C., and Yigit, E. (2009). A new and practical formulation of bistatic inverse synthetic aperture radar imaging and verification of the formulation using numerical examples. 2009 International Conference on Electrical and Electronics Engineering – ELECO 2009, Bursa, Turkey (5–8 November 2009), pp. II-161–II-164. Pastina, D., Bucciarelli, M., and Lombardo, P. (2010). Multistatic and MIMO distributed ISAR for enhanced cross-range resolution of rotating targets. IEEE Transactions on Geoscience and Remote Sensing 48 (8): 3300–3317. Wang, Y., Chen, D., and Liu, Z. (2015). A correction algorithm of migration through Doppler resolution cell for bistatic inverse synthetic aperture radar. 2015 8th International Congress on Image and Signal Processing (CISP), Shenyang, China (14–16 October 2015), pp. 1515–1519. https://doi.org/10.1109/CISP.2015.7408124. Zhu, Z., Zhang, Y., and Tang, Z. (2005). Bistatic inverse synthetic aperture radar imaging. IEEE International Radar Conference, 2005, Arlington, VA, USA (9–12 May 2005). https://doi.org/10.1109/RADAR.2005.1435850. Zhu, R., Zhou, J., Jiang, G., and Fu, Q. (2017). Range migration algorithm for near-field MIMO-SAR imaging. IEEE Geoscience and Remote Sensing Letters 14 (12): 2280– 2284. https://doi.org/10.1109/LGRS.2017.2761838. Ziyue, T., Zhenbo, Z., Lixiao, Z., and Shaoying, S. (2009). Research on imaging of ship target based on bistatic ISAR. 2009 2nd Asian-Pacific Conference on Synthetic Aperture Radar, Xian, Shanxi, China (26–30 October 2009). https://doi.org/10.1109/ APSAR.2009.5374261.

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10 Polarimetric ISAR Imaging Inverse synthetic aperture radar (ISAR) imaging has been generally utilized for target detection, automatic target classification (ATC), and automatic target recognition (ATR) applications (Musman et al. 1996; Kim et al. 2005; Baumgartner 2018). In most ISAR systems, the radar’s antenna is usually polarized with a specific linear polarization, either vertical or horizontal. As given in more detail in Chapter 2, Section 2.3, the radar cross section (RCS) of an object is also dependent on the polarization of the incoming wave. Therefore, some features of the target cannot be extracted via the selected polarization. In fact, as we shall see next, some structures either manmade or natural are sensitive to some polarization and not others. Therefore, using different polarization of the radar’s antenna will definitely yield more features of the target to be sensed by the receiver such that the probability of classifying and recognizing the target will eventually increase. Also, the ATC and ATR procedures in traditional single-polarization ISAR imaging are usually be done by assessing the amplitude-ISAR image of the target without exploiting its phase information. With the use of other polarization in ISAR images will offer the opportunity of using the phase information of each single-polarization ISAR image for further signal and image processing tools to improve the ATC and ATR applications. All these features of polarimetric usage of ISAR imaging will be explored within this chapter.

10.1

Polarization of an Electromagnetic Wave

Polarization of an electromagnetic (EM) wave can be formally defined as the locus of the electric field vector perpendicular to the direction of propagation (Boerner et al. 1992). In other words, the orientation and shape of the pattern traced by the tip electric field vector determine the polarization type and the sensitivity of the

Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms: With Advanced SAR/ISAR Imaging Concepts, Algorithms, and MATLAB Codes, Second Edition. Caner Özdemir. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

10.1 Polarization of an Electromagnetic Wave

EM wave. As more detailed information of the EM, wave polarization can be found in any antenna theory book such as (Balanis 1989), we will just briefly review the types and sensitivities of polarization since our main focus is just the usage of polarization in radar imaging.

10.1.1

Polarization Type

For the general case, the tip of the electric field traces the form of an ellipse that is called the polarization ellipse (Boerner et al. 1992; Volakis 2007) as illustrated in Figure 10.1. This fictitious ellipse is orthogonal to the EM wave’s direction of propagation and can be represented by two perpendicular axes of x and y, or horizontal (H) and vertical (V). The magnitudes and phases of the tip of the electric field vector in H and V directions define the type of the polarization of the EM wave, namely linear, circular, or elliptical. When the phases of the tip of the electric field in H and V direction are the same or opposite phases, i.e. 0 or 180 (regardless of the amplitudes), this type of EM wave is called linearly polarized (LP). The polarization vector’s tilt angle is defined according to the polarization vector’s amplitudes of H and V directions. If they are equal, for instance, the tilt angle becomes 45 . When there is no component in H direction, the tilt angle becomes 90 , and the EM wave is named as vertically polarized. Analogously, if there exists no component in V direction, the tilt angle turns to 0 , and the EM wave is termed horizontally polarized. When there exists a 90 phase difference between the H and V components of the tip of the electric field, the field is either circularly or elliptically polarized. If the amplitudes of these two components are equal, the field is called circularly polarized (CP), if not, it is named elliptically polarized (EP). If the tip of electric field y(V)

u uv u min

Figure 10.1

Tilt angle: Ψ

x

u ma

uh

Polarization ellipse of an EM wave.

x(H)

485

486

10 Polarimetric ISAR Imaging

vector is represented as u, it can be written as the summation of two perpendicular vector components of ux (or uh) and uy (or uv) as illustrated in Figure 10.1 provided that EM wave is propagating in z-direction. u = ux ax + uy ay = uh ah + uv av

10 1

Here, ax (ah ) and ay (av ) are the unit vectors in horizontal and vertical directions, respectively. In an alternative representation, u can be written in terms of two circular polarization unit vectors of aL = 12 ah − jav and aR = 12 ah + jav that correspond to left-handed and right-handed unit vectors, respectively as u = u L a L + u R aR

10 2

Here, uL = uL exp jδL and uR = uR exp jδR are the complex left-handed and right-handed components of the tip of electric field vector u . As another parameter, the ratio of the major axis to the minor axis of polarization ellipse is referred to as the axial ratio (AR): AR =

umax , umin

1 ≤ AR ≤ ∞

10 3

As demonstrated in Figure 10.1, the tilt angle is used to identify the spatial orientation of the polarization ellipse and also equal to Ψ=

1 δL − δR 2

10 4

Linear, circular and elliptical polarizations can be practically defined by the help of above representation with the assessments below:

• • •

When uh and uv are in the same phase, i.e. δL = δR, then the polarization ellipse collapses a single line that represents a linearly polarized (LP) wave. When |uh| = |uv| and (δL − δR) is the odd multiples of 90 , the polarization ellipse turns to a circle, so the wave is called circularly polarized (CP). For other cases, the EM wave is generally elliptically polarized (EP) provided that (δL − δR) is not integer multiplies of 180 for which the wave becomes LP.

10.1.2

Polarization Sensitivity

Another important parameter for a CP or EP wave is the polarization sensitivity that defines the rotation of tip of electric field vector u either in clockwise or in counter-clockwise direction.

10.1 Polarization of an Electromagnetic Wave

• •

Polarization sensitivity of an EP wave can be decided by the following arguments: – If uL > uR, the wave is left-hand elliptically polarized (LHEP). – If uL < uR, the wave is right-hand elliptically polarized (RHEP). Polarization sensitivity of a CP wave can be decided by the following arguments: – If uL 0 and uR = 0, the wave is left-hand circular polarized (LHCP). – If uL = 0 and uR 0, the wave is right-hand circular polarized (RHCP).

10.1.3

Polarization in Radar Systems

For LP radar systems (either H or V), the convention is determined by the transmit and receive polarizations by a pair of symbols with the following four possible channels:

•• ••

HH – for horizontal transmit and horizontal receive, VV – for vertical transmit and vertical receive, HV – for horizontal transmit and vertical receive, and VH – for vertical transmit and horizontal receive.

•• ••

LL – for circularly left transmit and circularly left receive, LR – for circularly left transmit and circularly right receive, RL – for circularly right transmit and circularly left receive, and RR – for circularly right transmit and circularly right receive.

The basic system architecture model for a fully polarized monostatic radar transceiver is depicted in Figure 10.2. As it can be seen from the figure, a single antenna can be used for both transmit and receive the EM wave by the help of circulators. Dual polarized antenna has the feature of transmitting and receiving EM waves in two-orthogonal polarizations. The received signals in each channel are enhanced using a low-noise amplifier (LNA) to be fed to signal processor. Then, a fully polarimetric (HH, HV, VH, and VV) scattered field can be measured such that polarization scattering matrix of [S] can be constructed (Raney et al. 2011). The same convention can also be used for ISAR systems either monostatic or bistatic. There are also radar systems that use CP for the transmitting and receiving antennas. The polarization of the waves can, then, be represented by using circular left (L) and circular right (R) polarizations as follows:

487

488

10 Polarimetric ISAR Imaging

Transmitter Timing and control

90°

V LNA

H

LNA

V

Receiver

H Dualpolarized antenna

H– channel

V– channel

H– processor

V– processor

H HH

V H

Figure 10.2

10.2

V HV VH

SHH

SHV

SVH

SVV

VV

The basic system architecture for a LP polarized radar transceiver.

Polarization Scattering Matrix

The backscattering from a target measured at far-field can be completely described by the 2 × 2 polarization scattering matrix [S] as follows: Er =

exp − jkR S Et R

10 5

Here, E r and E t represent the complex received (or scattered) and transmitted (or incident) electric field vectors. The propagation term exp(−jkR)/R, where k is the wavenumber and R is the total distance of the EM wave traveled, is not a target related

10.2 Polarization Scattering Matrix

parameter, and therefore, normalized out. The elements of [S] are the complex scattering amplitudes Sij = Sij exp jϕij which are dependent only on the target characteristics, for a selected frequency and particular viewing geometry of fixed azimuth angle ϕ, elevation angle θ. The diagonal and off-diagonal elements of scattering matrix are called as co-polarized (co-pol) and cross-polarized (cross-pol) scattering terms, respectively. In monostatic configurations, [S] becomes symmetric, that is SHV = SVH for all reciprocal scattering media. In the general case, [S] can be measured in any orthogonal basis such as LP or CP. If the scattering matrix is measured in LP basis (i.e. H and V ), for instance, then it can be represented as the following (Demirci et al. 2020): ErH ErV

=

Et exp − jkR S LP H R E tV

10 6

where [S]LP is LP-based scattering matrix and given by S LP =

SHH

SHV

SVH

SVV

10 7

Here, E rH and ErV stands for the received horizontal and received vertical electric fields, respectively. In a dual manner, E tH and EtV are the transmitted horizontal and transmitted vertical electric fields, respectively. In fact, the scattering matrix can be measured in any basis of orthogonal polarizations without losing any polarization information. It is important to note that the information content of scattering matrix is independent of the polarization basis used for its measurement, i.e. changing the polarization of the incident wave yields alteration in the scattered wave, but physical scattering information contained within [S] remains the same. The transformation from LP to CP basis can be readily obtained as S CP = U S LP U T ,

10 8

where [U] is unitary matrix, that is, [U][U∗]T = [I] and given by U =

1 1 2 1

j −j

10 9

In the above expressions, ∗ and T are for complex conjugate and matrix trans1 0 pose operations, respectively and [I] is the identity matrix given by [I]= . 0 1 Therefore, the elements of [S]CP can be get in terms of elements of [S]LP by applying the transformation in Eq. 10.8 as

489

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10 Polarimetric ISAR Imaging

SLL

SLR

SRL

SRR

=

SHH − SVV SHV + SVH +j 2 2 SHH + SVV SHV − SVH +j 2 2

SHH + SVV SHV − SVH −j 2 2 SHH − SVV SHV + SVH −j 2 2 10 10

If the medium is reciprocal, the cross-pol parameters becomes equal, that is SHV = SVH. Therefore, this equation simplifies to SLL

SLR

SRL

SRR

=

SHH − SVV + jSHV 2 SHH + SVV 2

SHH + SVV 2 SHH − SVV − jSHV 2

10 11

The elements of the scattering matrix can be used to characterize particular features of simple and complex objects. To better comprehend how the S-matrix elements are unique in identifying various canonical objects, scattering matrices of some special targets are given in Table 10.1 for LP and CP bases (Lee and Pottier 2009). For the horizontal pole (or wire), for instance, only the first entry of the Smatrix of LP basis (i.e. SHH element) flashes while other entries are zero. For the vertical dihedral, the vertical incoming wave bounces back without changing its polarization whereas horizontal incoming wave has to have 180 out of phase while leaving the structure. Therefore, their S-matrix entries are 1 and −1, respectively as listed in Table 10.1. For a pole structure of 45 inclination, all entries of the S-matrix provide identical coefficients of 0.5 as shown in Table 10.1 that is also in accordance with the physical expectation.

10.2.1

Relation to RCS

As the RCS, σ of a target is defined earlier in the second chapter via the Equation 2.11 that relates the incident (or transmitted) electric field,Eitto the scattered (or reflected) electric field E sr as σ tr = lim R

∞

4πR2

E sr

2

Eit

2

10 12

Here, subscripts t and r represent the polarization type of transmitted and received signals, respectively. For instance, if the transmitted signal is H-polarized and the received one is V-polarized, the term σHV represents the cross-pol RCS of the target. Or, if the transmitter and the receiver are left-hand circular polarized, Eq. 10.12 turns to a co-polar RCS value of σLL.

10.2 Polarization Scattering Matrix

Table 10.1

Scattering matrices for various objects. Linear-H/V basis

Circular-R/L basis

S HH

S HV

S LL

S LR

Target

S VH

S VV

S RL

S RR

Horizontal pole

1 0

Vertical pole

0 0

1 1 1 2 1 1

0 0

1 2

0 1 45 inclined pole Horizontal dihedral

−1

1 1

1 j 2 1

1

0

1

0

0

−1

0

1

0 Odd-bounce targets (flat plates, trihedrals, and curved surfaces)

1

1

1 1 2 1

−1 0

Vertical dihedral

−1 1 −j

−1

0

0

−1

1

1 0

0

1

0 1

1

0

To derive the relationship between the scattering matrix [S] and the RCS, one can easily start with plugging Equation 10.5 into 10.12 as σtr = lim

4πR

∞

R

= 4π lim R

= 4π Str

∞ 2

2

Str

exp − jkR R

E it

Str Eit

2

2

10 13

2

where Str corresponds to the any element in the polarization scattering matrix. If Str is written in terms of the RCS of the target, it is trivial to get the following: Str =

1 2 π

σ tr

10 14

Therefore, it is important to note that the polarization scattering coefficients are proportional to the square root of the RCS value of the target of same polarization.

10.2.2

Polarization Characteristics of the Scattered Wave

To better interpret the polarization scattering matrix, it is needed to understand the polarization characteristics of the EM wave scattered from a target or a scene.

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10 Polarimetric ISAR Imaging

The behavior of scattering (or back-scattering) of EM wave varies significantly for different targets and different terrains. For basic shapes of canonical targets, the characteristics of scattered EM wave were already mentioned in Chapter 2, Sections 2.1 and 2.2 with associated illustrations given in Figure 2.1. Most of canonical objects such as poles, dipoles, plates, dihedrals, and trihedrals are named as deterministic or coherent targets since they produce coherent radar echoes. Coherent targets completely produce polarized scattered waves, that is the scattered EM field is at a specific polarization. As illustrated in Figure 10.3a, man-made structures such as houses, buildings, and machines generally give rise to polarized scattered waves since they usually consist of canonical substructures such as plates, cubes, and other regularly shaped surfaces. For the case of radar targets that have many substructures that produce numerous scattering phenomena, they are usually named as distributed or extended targets. For instance, millions of raindrops illuminated by a radar pulse can be regarded as a distributed target. Almost all natural objects including trees, grass, leaves, and vegetation fall into this category. Such extended targets tend to produce depolarized scattered waves, i.e. the scattered EM wave is diffracted to various directions with different orientations such that it is in the form of different polarization than that of incident wave as illustrated in Figure 10.3b. In high-frequency approximation of EM propagation, the backscattered signal from a coherent target is dominated by local scattering centers on the target. If the target structure is complex such as airplanes and ships, the scattering mechanisms occurring around these target regions tend to be highly complicated due to coherent addition of contributions from single, double, triple, and higher-order bounce scattering mechanisms. To reveal the complexity within the scattered signal and classify the scattering characteristics, target decomposition techniques can (a)

(b) ret Pola urn rize ed d wa ve

re Pol tu ar rn iz ed ed wa ve

492

G d en ret epo eral ur lar ly ne ize dw d av e

Deterministic scattering

Random scattering

Figure 10.3 Scattering characteristics of EM wave from (a) an urban area, and (b) a vegetated area.

10.2 Polarization Scattering Matrix

be utilized to decouple this overlapping of various scattering phenomena (Duquenoy et al. 2006, 2010, Dallmann 2017, Dallmann and Heberling 2017, Martorella et al. 2008, 2011). In particular, coherent decompositions utilize directly the polarization scattering matrix of [S] and express it with a combination of simpler responses for an easier interpretation that we shall investigate next.

10.2.3

Polarimetric Decompositions of EM Wave Scattering

The main objective of applying coherent decompositions is to express the polarization scattering matrix [S] with basis scattering responses of known canonical structures. The basics of polarimetric decomposition theory were first presented by Huynen (1970), afterward, other researchers have proposed various decomposition techniques for coherent and/or distributed targets (Duquenoy et al. 2006, 2010; Martorella et al. 2008, 2011; Dallmann 2017; Dallmann and Heberling 2017). So, as in the case of radar imaging, target decomposition techniques aim to express the total scattering from the target as a sum of independent elementary scattering mechanisms to associate the physical phenomena beyond each scattering component. Based on coherent (deterministic) and incoherent (distributed or random) categorization of targets based on their EM scattering characteristics, target decomposition techniques have also been categorized as coherent target decompositions (CTDs) and incoherent target decompositions (ITDs), respectively (Verma 2012). The CTDs are utilized to characterize the completely polarized scattering waves from the deterministic targets for which the polarimetric information can be entirely defined by the polarization scattering matrix of [S]. For instance, sphere, dipole, diplane, and helix are considered to be the canonical targets. As obvious, dipole and diplane are designated as a function of the orientation angle with respect to the radar line-of-sight (RLOS) direction. With the help of CTDs, measured [S] can be decomposed into sphere, dipole, diplane, and helix components. By doing so, it becomes possible to resolve different types of scatterers within the scattered data. Most common and popular CTDs have been named by their inventors, Pauli (Martorella et al. 2008, 2011), Cameron (Duquenoy et al. 2006, 2010), and Krogager et al. (1997). It has been demonstrated by various researchers that CTDs can be successfully utilized for various realistic targets including, simple reflectors, benchmark objects, and scale models of vehicles such as a tank and a boat (Huynen 1970; Krogager et al. 1997; Duquenoy et al. 2006, 2010; Martorella et al. 2008, 2011; Verma 2012; Dallmann 2017; Dallmann and Heberling 2017). ITDs rely on the use of covariance or coherency matrix for describing the complex scattering features of natural, distributed objects. The very basic ITD is so-called Freeman decomposition that models the covariance matrix as the contribution of three scattering mechanisms, namely

493

494

10 Polarimetric ISAR Imaging

i) Volume scattering where a canopy scatterer is modeled as a cloud of randomly oriented dipoles, ii) Double-bounce scattering modeled by a pair of orthogonal surfaces, and iii) Surface or single-bounce scattering modeled by a first-order Bragg scattering from a moderate rough surface (Richards 2009). Other commonly used ITDs are Huynen decomposition (Huynen 1970), Barnes decomposition (Holm and Barnes 1988), and Eigenvector–Eigenvalue based decomposition (Cloude 2010). For the case of SAR imaging, the scattering features of the terrain together with man-made objects become really important and crucial for characterizing the targets and/or the terrain. Therefore, both CTDs and ITDs are being utilized for SAR applications ranging from mapping of soil surface roughness to agricultural crop monitoring (Brown et al. 2003; Morrison and Williams 2005; Lim and Koo 2008; Wang et al. 2014, 2015). For the ISAR point of view, the targets to be imaged are usually the man-made objects such as fighters, tanks, boats, helicopters, drones, UAVs, etc. For this reason, the main body and the sub-structures of a typical ISAR target are composed of canonical objects that provide deterministic or coherent target features. Therefore, the scattered data from a fully polarimetric ISAR (Pol-ISAR) arrangement can practically provide the measurement of the magnitude and phase of the reflected wave for an orthogonal set of polarization states by utilizing any type of CTD. For instance, Krogager’s technique is based on three-component decomposition of the complex Sinclair matrix associated with canonical scattering mechanisms (Krogager et al. 1997). In an alternative method, Cameron has performed a factorization of the measured scattering matrix based on two fundamental properties of radar targets: reciprocity and symmetry (Cameron et al. 1996; Dallmann 2017). Among CTDs, the most basic and the efficient one is so-called the Pauli decomposition scheme that is successfully employed in imaging applications such as SAR urban monitoring. We choose the Pauli decomposition technique as a sample tool for Pol-ISAR imaging as we shall investigate throughout this chapter.

10.2.4

The Pauli Decomposition

10.2.4.1 Description of Pauli Decomposition

The Pauli decomposition represents the polarimetric scattering matrix of [S] into so-called the Pauli basis matrices of ΨP. The usual arrangement of ΨP is based on conventional orthogonal linear (H and V ) basis; hence, the Pauli bases of [S]a, [S]b, [S]c, and [S]d are constituted by the following 2 × 2 matrices for HH, HV, VV, and VH polarizations, respectively as

10.2 Polarization Scattering Matrix

1 0 Sa=

2 0 1

Sb=

,

0 1 Sc=

2 1 0

Sd=

,

1

0

0

−1

0

−1

1

0

2 10 15

2

Noticing that the reciprocity is valid for the monostatic arrangement of ISAR (that is, SHV = SVH), the explicit expression of ΨP can then be expressed with the three orthogonal 2 × 2 matrices as the following: ΨP =

S a, S b, S c

=

2

1

0

0

1

, 2

1

0

0

−1

, 2

0

1

1

0

10 16

With this construct, the measured [S] can then be expressed in terms of Pauli decomposition bases with the following expression: S =

SHH SHV

SHV SVV

= α S a + β S b + γ S c,

10 17

where SHH + SVV 2 SHH − SVV β= 2 γ = 2SHV

α=

10 18

are the complex coefficients that correspond to the weights of the associated basis matrix. From Eq. 10.18, the span (or the total power) of [S] can also be described in terms of these coefficients as the following: span S

10.2.4.2

= SHH 2 + SVV 2 + 2 SHV = α2+ β2+ γ2

2

10 19

Interpretation of Pauli Decomposition

The phenomenological interpretation of the Pauli decomposition can be made by evaluating the meaning of the basis matrices. The first matrix [S]a with SHH = SVV and SHV = SVH = 0 can be interpreted as the scattering matrix of a “single-bounce” or “odd-bounce” scatterer caused probably

495

496

10 Polarimetric ISAR Imaging

by spheres, flat surfaces, and trihedral corner reflectors (TCRs). So, the weighting α (or |α|2) represents the amount of the contributions from such scatterers in the measured electric field (or power) data. The second matrix [S]b involves 180 phase difference between the co-polarized terms (of HH and VV ) and corresponds to the scattering matrix of a dihedral along the RLOS direction. This indicates “double-bounce” or “even-bounce” scattering similar to the one that can be observed from dihedral corner reflectors (DCRs). The third matrix [S]c corresponds to the scattering matrix of a diplane rotated 45 about the RLOS direction. Noting that this matrix is defined with respect to LP basis, it represents a scatterer that transforms the incident polarization into its orthogonal state. This term is sometimes associated with “volume scattering” such as from vegetation, forest canopy, or from even-bounce objects with 45 orientation. The weightings of |β|2 and |γ|2 represent the scattered power amount from “evenbounce” objects and from 45 rotated diplane structures within the scattered EM signal, respectively. 10.2.4.3 Polarimetric Image Representation Using Pauli Decomposition

The polarimetric information within the scattered EM signal is displayed with the help of single radar image with RGB (red–green–blue) color-coding based on Pauli decomposition basis. Therefore, the polarization matrix, [S] is represented by the combination of |α|2, |β|2, and |γ|2 in the resultant RGB image (or so-called the Pauli image) to interpret the odd-, double-, and 45 oriented (or vegetation) scattering mechanisms, respectively. The most used color-coding convention for the Pauli images are as follows: Red Green Blue

α 2 or SHH γ

2

β

2

2

or 2 SHV or SVV

2

10 20

2

Corresponding Pauli RGB color-palette is provided in Figure 10.4 in which the scattering mechanisms that correspond to three orthogonal Pauli bases of the first, second, and third components, i.e. odd-bounce, double-bounce, and random scatterings are labeled accordingly. If a single radar image cell contains only one of these three basic mechanisms, it is displayed with its corresponding color value in the radar image. If there exists a dihedral structure, for instance, then the local image signature is displayed in red color. If there are more than one scattering mechanism going around the image pixel location, then the corresponding mixtures of colors are displayed according to the power levels of each orthogonal scattering bases. For example, if there are both single and double bounces available from a sub-structure of the target, combination of blue and red, i.e., magenta color signature is shown around the corresponding image pixels.

10.4 ISAR Imaging with Full Polarization

Red Magenta

Yellow

White Green

Cyan

Blue

Blue : Odd-bounces (flat and curved surfaces, trihedrals) Red : Double-bounces (target-ground reflection, dihedrals) Green : Incoherent scattering (volume scattering, random scattering, tree-canopies)

Figure 10.4 Pauli RBG color palette with scattering explanations. (For whole assessment of the color figures, please see the electronic version of the book.)

10.3

Why Polarimetric ISAR Imaging?

The conventional radar systems utilize the frequency, amplitude, and also relative and absolute phase information of the received signal to resolve physical features of scatterers and the background environment. In high-resolution polarimetric radar imaging such as ISAR, on the other hand, the ultimate objective is to bring out the whole vector nature of the target or scene, i.e. in addition to amplitude, frequency, phase, and also polarization state information of the transmitted and received wave is incorporated into signal and image processing. By doing so, valuable information for the features and characterization of the target or the background environment can be attained. Full polarimetric analysis of the ISAR data, then, can be utilized for improved identification and classification techniques including ACR and ATR.

10.4

ISAR Imaging with Full Polarization

10.4.1

ISAR Data in LP Basis

The conventional ISAR arrangement sweeps a band of frequencies and a swath of angles to coherently collect and measure the backscattered EM waveform for a single polarization, usually, H or V. As explained in chapter 4 in detail, the use of twodimensional (2D) Fourier transform (FT) operation converts the frequency-aspect data onto 2D real spatial plane of range and cross-range axes. The 2D ISAR image

497

498

10 Polarimetric ISAR Imaging

displays the target’s EM reflectivity where each resolution cell depicts the single polarization response of the target in the scene. With the employment of a fully polarimetric radar as in the case of Figure 10.2, it is possible to represent the target response in an ISAR resolution cell with the polarization scattering matrix, [S] as given in Eq. 10.7. If the fully polarimetric backscattered data are collected with LP antennas of H and V polarizations, [S] matrix is obtained for every frequencies and aspect of the collected data set providing all channels of the polarimetry. Although the entries of [S] are shown in LP bases, it can be represented in any orthogonal bases with the physical scattering information within [S] is preserved since the scattering feature of the target is unique. If the Pauli decomposition scheme is chosen, the [S] matrix is represented as given in Eq. 10.17 with the Pauli bases described in Equation 10.20. If the Krogager or the Cameron bases are chosen, the bases function should be altered, but the physical polarization scattering information within [S] remains unchanged.

10.4.2

ISAR Data in CP Basis

If left-hand and right-hand circularly polarized antennas are being used for transmit and receive, then polarization scattering matrix [S] can be expressed in terms of L and R circular bases by employing the transformation formula given in Equation 10.11. Polarimetric measurement via L/R bases provides different insights of the backscattered signal. For instance, cross-pol (either |SLR| or |SRL|) represents single and odd-bounce scattering phenomena whereas co-pol (|SLL| and |SRR|) characterizes double and even-number of bounce scattering phenomena for a CP wave (Demirci et al. 2020). These features can be readily observed from the scattering matrices of various objects listed for LP and CP bases in Table 10.1. For odd-bounce targets of flat plates and trihedrals, for instance, 0 1 SLL SLR = . Therefore, circular cross-pol components are the classifySRL SRR 1 0 ing metrics, whereas circular co-pol components are zero. For the LP excitation, on SHH SHV 1 0 the other hand, the scattering matrix is = which means that SVH SVV 0 1 linear co-pol components are the identifying polarizations while linear cross-pol components become zero. Therefore, the response of LP and CP basis representations can be completely different such as being the reverse for various targets. For the dihedral target that is put vertically, the circular L/R basis scattering −1 0 SLL SLR = that clearly reveal that circular matrix becomes S = SRL SRR 0 −1 co-pol components light up for even-bounce targets while the circular cross-pol

10.5 Polarimetric ISAR Images

components give null responses. Similar decisive conclusions can be done for determining the different characteristics of various targets listed in Table 10.1 and also for other similar canonical objects. Noticing that the ISAR imaging procedure is nothing but spatially spotting the EM reflectivity of the target associated with its corresponding scattering amplitudes, having the full-polarimetric response of the target will certainly lead to characterize the features of the target together with its substructures with proper polarimetric interpretation.

10.5

Polarimetric ISAR Images

In forming fully Pol-ISAR images of a target, standard imaging algorithms are applied for the selected single-polarization channel such that polarization images of ISARHH, ISARHV, ISARVH, and ISARVV are obtained for LP transmit and receive antennas based on the radar set-up is given in Figure 10.2. If desired, CP-ISAR images can then also formed by applying the transformation formula given in Equation 10.11 or directly formed by using CP transmit and receive antennas. The Pol-ISAR imaging concept will be presented by various examples given below.

10.5.1 10.5.1.1

Pol-ISAR Image of a Benchmark Target The “SLICY” Target

The concept of Pol-ISAR imaging is first demonstrated via a benchmark object whose scattering features can be easily foreseen. For this purpose, the well-known radar benchmark target of SLICY whose CAD view can be seen in Figure 10.5 is chosen. SLICY has been a common test-object that has been frequently utilized to validate the radar imaging algorithms in the literature (Özdemir and Ling 1999; Coburn et al. 2005; Glentis et al. 2013). This model of SLICY has extensions of 7.56, 8.50, and 5.24 m in x, y, and z directions, respectively. SLICY is an attractive test object since it contains several distinct scattering shapes, such as a short open cylinder (C1), a tall closed cylinder (C2), a trihedral (T), two step-like dihedrals (D1 and D2), a quarter cylinder surface (S), and a flat plate (P) has like all of these substructures are labeled in Figure 10.5 accordingly. These substructures of SLICY have the valuable feature of providing various coherent scattering mechanisms including single-, double-, triple-, multiplebounces, specular, and surface reflections. 10.5.1.2

Fully Polarimetric EM Simulation of SLICY

The EM backscattering simulation of SLICY has been carried out by the commercial RCS simulator tool of PREDICS (Özdemir et al. 2014; Kırık and Özdemir 2019)

499

500

10 Polarimetric ISAR Imaging

z

C1 C2 D1

T S

θ = 90° D2 P

ok ar lo Rad n ctio dire

x

y

Figure 10.5 SLICY target in ISAR simulation for the look-aspect direction of object of (θi = 90 , ϕi = 0 ).

for the all the LP polarimetric signatures, i.e. HH, HV, VH, and VV with using PO (physical optics) + SBR (shooting and bouncing ray) + PTD (physical theory of diffraction) solver. All the simulation parameters and yielded ISAR imaging parameters are listed in Table 10.2. 10.5.1.3 LP Pol-ISAR Images of SLICY

The simulation results of PREDICS yields full-polarimetric backscattered electric field values for the frequencies and angles listed in Table 10.2. For each polarization channel, standard ISAR imaging procedures that are explained in detail in Chapter 4 are applied to from the ISAR images in HH, HV, VH, and VV polarizations as presented in Figure 10.6a–d, respectively. The CAD view of SLICY is also projected onto ISAR images for image referencing purposes. LP images in Figure 10.6 show that all the noteworthy scattering energies are shown up only in co-pol (|SHH| and |SVV|) images, whereas the scattering energies in cross-pol (|SHV| and |SVH|) images are stayed quite low. In fact, they are at least 40 dB lower than the co-pol ones. As briefly explained in Section 10.2.2 that the man-made objects of canonical shapes support coherent scattering and produce deterministic radar echoes. In fact, SLICY target is a good example of this and is composed of many canonical structures as explained in Section 10.5.1.1. Therefore, most parts of SLICY support co-pol mechanisms (or coherent) and the only structures that can produce some cross-pol mechanisms (or incoherent) are the cylinders (C1 and C2) and the inclined surface (S) on account of diffractions. These phenomena can be clearly seen in Figure 10.6. For this particular look direction of

10.5 Polarimetric ISAR Images

Table 10.2

Full-polarimetric ISAR simulation parameters of SLICY.

EM simulation parameters

Value

Physical solver

PO + SBR + PTD

Frequency [min]

11.531 GHz

Frequency [max]

12.459 GHz

Number of frequency samples

100

Vertical look-angle (θ) [min]

90

Vertical look-angle (θ) [max]

90

Number of vertical look-angle (θ) samples

1

Horizontal look-angle (ϕ) [min]

−2.238

Horizontal look-angle (ϕ) [max]

2.193

Number of horizontal look-angle (ϕ) samples

100

Polarization

HH, HV, VH, VV

Ray density

10 rays/λ

Maximum bounce

10

ISAR imaging parameters Range resolution

16 cm

Range window extend

16 m

Cross-range resolution

16 cm

Cross-range window extend

16 m

θi = 90 and ϕi = 0 , the main scattering mechanisms are illustrated in Figure 10.7 as most of them (P, D1, D2, C1, and C2) experience single bounces, whereas only the trihedral (T) behaves like a horizontal dihedral and therefore produces doublebounces. Of course, all these reflections are of deterministic types; therefore, they all showed up in both co-pol (|SHH| and |SVV|) images, but not in cross-pol (|SHV| and |SVH|) images. Since the incident wave is coming perpendicular to the substructures of plane (P), dihedrals (D1 and D2), there happens specular reflections from these surfaces such that high RCS levels around 20 dBsm are observed from Figure 10.6a and d. We also observe other coherent single surface reflections from open cylinder (C1) and closed cylinder (C2); therefore, their ISAR signatures also appear in co-pol (|SHH| and |SVV|) images of Figure 10.6a and d, respectively. It is also noteworthy to mention that C1 and C2 also support some weak incoherent echoes (of diffracted fields) due to their circular structures along azimuth which can be readily observed from the in cross-pol (|SHV| and |SVH|) images of from Figure 10.6b and c, respectively. The RCS levels of these echoes are around

501

10 Polarimetric ISAR Imaging

(a)

(b)

∣SHH∣

∣SHV∣

20 –6

–2

5

0

0

2

–5 –10

4

10

–2

5

0

0

2

–5 –10

4

–15

6

15

–4

10 Cross-range, m

Cross-range, m

20 –6

15

–4

–15

6

–20 –6

–4

–2

0

2

4

6

–20 –6

dBsm

–4

–2

Range, m

(c)

0

2

4

6

–6

∣SVV∣ 20

–6

15

–4

5

0

0

2

–5 –10

4

–15

6

Cross-range, m

–2

15

–4

10

10

–2

5

0

0

2

–5 –10

4

–15

6

–20 –6

–4

–2

0

2

dBsm

Range, m

(d)

∣SVH∣ 20

Cross-range, m

502

4

6

dBsm

Range, m

–20 –6

–4

–2

0

2

4

6

dBsm

Range, m

Figure 10.6 LP-ISAR images of the “SLICY” target for the radar look direction of (θi = 90 , ϕi = 0 ): (a) HH, (b) HV, (c) VH, and (d) VV.

−5 dBsm which is 20–25 dB lower than the coherent ones. The curved surface (S) region between the steps of SLICY does not support adequate back-scattered energy to appear within the dynamic range of the display for this particular look-angle and therefore not enough energy contribution can be seen within the polarization images for this particular structure. 10.5.1.4 CP Pol-ISAR Images of SLICY

Although LP ISAR imagery presents a tool for classification between the deterministic (coherent) and random (incoherent) radar echoes, they do not offer a tool to discriminate between odd- and double-bounce scattering. As mentioned in Section 10.4.2, CP representation with L/R basis has the attractive feature of judging odd and double bounce mechanisms that can be used to extract more features

10.5 Polarimetric ISAR Images

Odd bounce C1

Double bounce

T

D1 S P

C2

D2

Figure 10.7 Illustration of various scattering mechanisms from the “SLICY” target for the radar look direction of (θi = 90 , ϕi = 0 )

from the ISAR images. For this goal, the CP-based ISAR images of LL, LR, RL, and RR have been formed by applying the LP to CP transformation formula in Equation 10.11 to form the corresponding CP images of SLICY in Figure 10.8a–d, respectively. As illustrated in Figure 10.7, the canonical substructures of SLICY mostly provide single bounce radar echoes, namely front plane (P), dihedrals (D1 and D2), and cylinders (C1 and C2) for the incident wave angle of (θi = 90 , ϕi = 0 ). Therefore, all these signatures are being seen in cross-pol (|SLR| and |SRL|) CP images of Figure 10.8b and c, respectively. The only double mechanism is from the trihedral (T) that becomes a dihedral for the radar look aspect of (θi = 90 , ϕi = 0 ); therefore, its signature shows up in the co-pol (|SLL| and |SRR|) CP images of Figure 10.8a and d, respectively. This is a very good example and evidence of how CP basis representation provides distinct classification of the single- and double-bounce echoes.

10.5.1.5

Pauli Decomposition Image of SLICY

While each of ISAR images for LP and CP channels provides significant scattering characteristics as mentioned above, the ISAR image constructed after the polarization decomposition techniques (such as the Pauli, Krogager, and Cameron) have the advantageous feature of displaying the scattering mechanism discrimination in a single ISAR image. Figure 10.9, for instance, shows the reconstructed Pauli ISAR image obtained by applying the decompositions given in Eqs. 10.17 and 10.18, and also using image representation based on the RGB color-coding convention in Eq. 10.20. Thanks to the Pauli image, one can easily distinguish different scattering mechanisms from each other and based on the Pauli RBG color palette that is provided in Figure 10.4.

503

10 Polarimetric ISAR Imaging

(a)

(b)

∣SLL∣

∣SLR∣

20 –6

–2

5

0

0

2

–5 –10

4

10

–2

5

0

0

2

–5 –10

4

–15

6

15

–4

10 Cross-range, m

Cross-range, m

20 –6

15

–4

–15

6

–20 –6

–4

–2

0

2

4

6

–20 –6

dBsm

–4

–2

Range, m

(c)

0

2

4

6

–6

∣SRR∣ 20

–6

15

–4

5

0

0

2

–5 –10

4

–15

6

Cross-range, m

–2

15

–4

10

10

–2

5

0

0

2

–5 –10

4

–15

6

–20 –6

–4

–2

0

2

Range, m

dBsm

Range, m

(d)

∣SRL∣ 20

Cross-range, m

504

4

6

dBsm

–20 –6

–4

–2

0

2

4

6

dBsm

Range, m

Figure 10.8 CP-ISAR images of the “SLICY” target for the radar look direction of (θi = 90 , ϕi = 0 ): (a) LL, (b) LR, (c) RL, and (d) RR.

Using Pauli image of SLICY in Figure 10.9, now, it is really much easier to classify the radar echoes from it as follows: For the coherent scattering mechanisms, the trihedral (T) supports a double-bounce mechanism for the look angle direction of (θi = 90 , ϕi = 0 ); therefore, it showed up in the Pauli image as a signature of red color (labeled as “t”). The flat surface (P), dihedrals (D1 and D2), and cylinders (C1 and C2) experience single-bounce mechanisms of specular reflections, so, they all produce signatures of blue color with labels “p”, “d1”, “d2”, “c1”, and “c2”, respectively. For the incoherent scattering mechanisms such as the ones deflected random echoes from the cylinders produce green color signatures as they are observed as crescent-like signatures (labeled as “i1” and “i2”) around the single-bounce scattering spot of blue color and propagate away from the radar. Although these green color incoherent signatures are weaker compared to the

10.5 Polarimetric ISAR Images

Pauli image

d1

–6

c1 i1

Cross-range, m

–4

–2

0 p

t

2

4

6

i2

c2 d2 –6

–4

–2

0

2

4

6

Range, m

Figure 10.9 Pauli image of SLICY for the radar look direction of (θi = 90 , ϕi = 0 ). (For the whole assessment of the color figures, please see the electronic version of the book)

coherent ones, they are still apparent in the Pauli image providing a lot of insights about the various scattering features off the target. (For whole assessment of the color figures, please see the electronic version of the book.) For a quantitative examination of the echo signatures of sub-structures on SLICY, the scattering and RCS matrices are formed from the LP and CP ISAR images in Figures 10.6 and 10.8, respectively. The echo signatures are extracted by using a rectangular window centered at the canonical object’s peak value and covering the whole ISAR echo signature. Table 10.3 lists the average values of scattering and RCS matrix entries of SLICY’s substructures for LP and CP bases for the conducted simulation study. The canonical objects, its acting structures, extracted [S] and RCS matrices are given throughout the columns from left to right. The obtained complex scattering [S] matrix entries are in accordance with the theoretical ones given in Table 10.1: For instance, trihedral (T) object has the LP co-polar scattering matrix entries of SHH = 12.3ej54 and SVV = 12.3e−j125 that clearly shows that |SHH| |SVV| with almost 180 phase difference between HH and VV channels. Furthermore, the magnitudes of the cross-pol entries,

505

Table 10.3 [S ] and RCS matrices of SLICY substructures in LP and CP bases for the radar look angle of (θi = 90 , ϕi = 0 ). LP basis

CP basis RCS (dBsm)

Scattering matrix

Object (label)

Structure (scattering mechanism)

Short cylinder (C1)

Cylinder surface (single-bounce)

0 05e

j14

2 9e

Tall cylinder (C2)

Cylinder surface (single-bounce)

5 6e

j120

0 02e − j52

0 02e − j70

Trihedral (T)

Dihedral (double-bounce)

Step-like dihedrals (D1 and D2)

Flat surface (single-bounce)

Flat plate (P)

Flat surface (single-bounce)

2 1 10 − 7 e

Scattering matrix

RCS (dBsm)

S HH

S HV

σ HH

σ HV

S LL

S LR

σ LL

σ LR

S VH

S VV

σ VH

σ VV

S RL

S RR

σ RL

σ RR

2 9e

j73

j14

0 05e

j73

92

− 26

0 05e

j76

− 26

92

3 05e

j73

0 02e − j135

15

− 34

j120

− 34

15

7 9 10 − 6 e − j147

21 8

2 4 10 − 5 e − j39

12 3e − j125

6 1e − j127 9 1 10 − 7 e − j14

12 3e

9 6e

5 6e j54

5 7e

j165

− 102

12 2e

j98

− 92

21 8

0 05e − j37

9 6 10 − 7 e − j14

15 7

− 120

0 01e

6 1e − j127

− 121

15 7

6 1e

19 6

− 126

0 03e

− 133

19 6

9 6e − j141

5 10 − 7 e

j128 j147

9 6e

j179

j128

j82

j142

j119

− 26

95

95

− 26

j165

− 34

15 1

j99

15 1

− 34

0 05e − j37

21 7

− 26

− 26

21 7

j142

− 40

15 7

0 01e j82

15 7

− 40

3 05e

j73

0 05e − j90 5 7e

0 02e 12 2e 6 1e

j98

9 6e − j141 0 03e

j119

− 30 4

19 6

19 6

− 30 4

10.5 Polarimetric ISAR Images

i.e. |SHV| and |SVH|, are negligible when compared to co-polar entries of [S]; thereby indicating a double-bounce scattering from a horizontal dihedral (see Table 10.1) which is expected for the angle of incidence. The same deduction can also be made if the LP basis RCS matrix is considered: The co-pol RCS values are about the same and at least 110 dB greater than that of cross-pol RCS values and thus again indicating a deterministic scattering. If CP based [S] and RCS matrices in Table 10.3 are considered for the same trihedral (T) object. Again, co-polar [S] entries of SLL = SRR = 12.2ej98 provide identical values that are much higher in amplitude than the cross-pol entries, i.e., |SLR| = |SRL| = 0.05e−j37 . This interpretation is in very good accordance with the theoretical expectation that is given in Table 10.1. For the CP basis RCS matrix, the situation is very similar to LP basis RCS matrix as in parallel with the theoretical expectations. The co-polar RCS entries of σ LL = σ RR = 21 dBsm that is almost 50 dB higher than cross-pol entries of σ LR = σ RL = − 26 dBsm which is indicating a coherent scattering. The obtained scattering matrix elements of other canonical structures are again in good agreement with the theoretical expectations listed in Table 10.1 and left to the reader for further investigation and interpretation.

10.5.2 10.5.2.1

Pol-ISAR Image of a Complex Target The “Military Tank” Target

Pol-ISAR imaging concept is also demonstrated by a much more complex target that is a tank model whose CAD file can be seen in Figure 10.10. The dimensions of the tank model are indicated in the figure. The tank target has been meshed with

3.242 m

3.358 m

9.751 m

Figure 10.10

Military Tank model: side, front, and top views with dimensions.

507

508

10 Polarimetric ISAR Imaging

a total of 362,026 small triangular patches,therefore, a very detailed model is available for the EM simulation. With such a complex substructure, this model serves as a perfect target for extracting scattering characteristics from it. 10.5.2.2 Fully Polarimetric EM Simulation of “Tank” Target

The EM monostatic backscattering, RCS, and ISAR simulation of tank model have been carried out by the PREDICS simulator (Özdemir et al. 2014; Kırık and Özdemir 2019) again for calculating all the LP polarimetric signatures, i.e. HH, HV, VH, and VV by the help of PO + SBR + PTD solver. For the ISAR simulation, the radar look direction to the tank model was (θi = 75 , ϕi = 30 ) as illustrated in Figure 10.11. All the simulation parameters and produced ISAR imaging parameters are listed in Table 10.4. 10.5.2.3 LP Pol-ISAR Images of “Tank” Target

The EM calculation using PREDICS simulator has produced the full-polarimetric backscattered electric field values for the frequencies and angles listed in Table 10.4. Again, standard ISAR imaging procedures that are given in detail in Chapter 4 are applied for each polarization channel to form the ISAR images in HH, HV, VH, and VV polarizations as presented in Figure 10.12a–d, respectively. The silhouette of the tank model is projected onto polarization ISAR images for referencing purposes. Constructed LP-ISAR images in Figure 10.12 present many scattering signatures in all polarizations as clearly seen from the images. As expected, co-pol (|SHH| and z

θ = 75°

ϕ=

30°

x y ok r lo da tion a R irec d

Figure 10.11 Military Tank target in ISAR simulation for the look-aspect direction of object of (θi = 75 , ϕi = 30 )

10.5 Polarimetric ISAR Images

Table 10.4

Full-polarimetric ISAR simulation parameters of Tank model.

EM simulation parameters

Value

Physical solver

PO + SBR + PTD

Frequency [min]

11.200 GHz

Frequency [max]

12.788 GHz

Number of frequency samples

128

Vertical look-angle (θ) [min]

75

Vertical look-angle (θ) [max]

75

Number of vertical look-angle (θ) samples

1

Horizontal look-angle (ϕ) [min]

26.180

Horizontal look-angle (ϕ) [max]

33.760

Number of horizontal look-angle (ϕ) samples

128

Polarization

HH, HV, VH, VV

Ray density

10 rays/λ

Maximum bounce

10

ISAR imaging parameters Range resolution

9.375 cm

Range window extend

12 m

Cross-range resolution

9.375 cm

Cross-range window extend

12 m

|SVV|) ISAR images (see Figures 10.12a and d, respectively) exhibit more stronger scattering energies compared to cross-pol (|SHV| and |SVH|) ISAR images (see Figures 10.12b and c, respectively). It is no doubt that the tank model was constructed numerous canonical objects that obviously support coherent scattering mechanisms. Therefore, observing highest radar echoes in HH and VV polarizations is fairly normal. On the other hand, there happen to be noticeable crosspolarization scattering centers of the tank platform that are about 10–15 dB lower in magnitude than co-polar scattering centers. This is due to the fact that there exist so many small and detailed structures on the model that the incoming wave deflects to various directions after hitting these structures so that cross-pol echoes are also generated besides the strong co-pol ones. Since the target is illuminated obliquely from the left side, most of the scattering centers are accumulated on this side as expected. The bogie wheels on the left track provide powerful backscattered energy such that all of seven bogie wheels can be visually distinguished from each other especially in co-pol LP ISAR images. Two turret storage bins toward the back of the tank also give very high scattering centers owing to their rectangular cavity-type

509

10 Polarimetric ISAR Imaging

(a)

(b)

∣sHH∣

∣sHV∣ –10

–10 –5 –15

–15

–20

–20

–25

0

–30

Cross-range, m

Cross-range, m

–5

–25

0

–30 –35

–35 –40

5 –5

0

–40

5 –5

5 dBsm

(c)

0

5 dBsm

Range, m

Range, m

(d)

∣sVH∣

∣sVV∣ –10

–10 –5

–5 –15

–15

–20

–20

–25

0

–30

Cross-range, m

Cross-range, m

510

–25

0

–30 –35

–35 –40

5 –5

0

–40

5

5 dBsm

Range, m

–5

0

5 dBsm

Range, m

Figure 10.12 LP-ISAR images of the “Military Tank” target for the radar look direction of (θi = 75 , ϕi = 30 ): (a) HH, (b) HV, (c) VH, and (d) VV.

shapes. It can also be seen many hot spots on and around turret especially toward main gun. For instance, the cupola hatch can be easily distinguished in co-pol LP ISAR images as the scattering centers form almost a circle along the perimeter of the cupola hatch.

10.5.2.4 CP Pol-ISAR Images of “Tank” Target

To be able to investigate the incoherent radar echoes together with the coherent ones, the CP-based ISAR images of the tank model is given in Figure 10.13 by the applying the LP to CP transformation formula given in Eq. 10.11. Therefore, LL, LR, RL, and RR ISAR images of tank target reconstructed as presented in Figure 10.13a–d, respectively.

10.5 Polarimetric ISAR Images

Interpretations of the CP ISAR images provide various classifying properties of the backscattering mechanisms from the tank target. Overall, the double bounce energies happen to be higher than odd-bounce ones for the radar illumination direction of (θi = 75 , ϕi = 30 ). This can be easily realized by comparing the co-pol LL and RR ISAR images (see Figure 10.13a and d) to the cross-pol LR and RL ISAR images (see Figure 10.13b and c). There are many classifying properties that can be sensed by these CP ISAR images. For example, the CP scattering centers of two turret storage bins show up in co-pol (|SLL| and |SRR|) ISAR images which means that these structures mainly provide double bounces for the particular look-angle. Cross-pol (|SLR| and |SRL|) ISAR images provide much weaker energies for the same objects; therefore, odd-bounce responses (mainly triple

(a)

(b)

∣sLL∣

∣sLR∣ –10

–10 –5 –15

–15

–20

–20

–25

0

–30

Cross-range, m

Cross-range, m

–5

–25

0

–30 –35

–35 –40

5 –5

0

–40

5 –5

5 dBsm

(c)

0

(d)

∣sRL∣

∣sRR∣ –10

–10 –5

–5 –15

–15

–20

–20

–25

0

–30

Cross-range, m

Cross-range, m

5 dBsm

Range, m

Range, m

–25

0

–30 –35

–35 –40

5 –5

0 Range, m

5 dBsm

–40

5 –5

0

5 dBsm

Range, m

Figure 10.13 CP-ISAR images of the “Military Tank” target for the radar look direction of (θi = 75 , ϕi = 30 ): (a) LL, (b) LR, (c) RL, and (d) RR.

511

512

10 Polarimetric ISAR Imaging

for this case) are much weaker. Another strong double-bounce scattering occurs from the left-front corner of the tank target which can be easily observed again in co-pol (|SLL| and |SRR|) ISAR images, but not in cross-pol (|SLR| and |SRL|) ISAR images. The bogie wheels principally support double-bounce mechanisms; so, the scattering centers on them shine out in co-pol CP ISAR images. On the other hand, wave interaction between bogies and tracks also generates multiple reflections such that weaker odd-bounce mechanisms also show themselves up in crosspol CP ISAR images in Figure 10.13. 10.5.2.5 Pauli Decomposition Image of “Tank” Target

While LP ISAR images show the back-scattering echoes in each singular channel that can be used to detect coherent scattering features from the target and CP ISAR images can provide further information such as odd- and double-bounce mechanisms together with incoherent scattering features from the target. To be able to make a judgment for all these different scattering features, all four images of LP and/or CP based ISAR images should be accounted. Pauli image (like the other decomposition images such as Krogager and Cameron), on the other hand, has the very practical and attractive feature of displaying all these valuable scattering features in a single image with the help of a color-coding convention. For the tank model given in Figure 10.10, the Pauli image is formed by employing the decomposition formulae given in Equations 10.17 and 10.18, and also using the RGB color-coding convention image representation that is defined in Equation 10.20. The resultant Pauli ISAR image based on RBG colors is presented in Figure 10.14. It is quite trivial for anyone to notice that this Pauli image clearly present many different scattering mechanisms with various color coding with different tones. A detailed investigation of this image is required to interpret and classify the scattering features from the tank target as we shall do next. To aid the interpretation, the major and interesting scattering mechanisms in various color coding in the Pauli image of Figure 10.14 are labeled from “a” to “i” for better localization of the scattering features on the tank target. Below are the brief explanations for the radar signatures (with corresponding labels) seen in the Pauli image: a) These scatterings are from the main gun of the tank. Since they come up in blue-color in the Pauli image, they represent single-bounce mechanism as demonstrated in Figure 10.15a. In fact, the left one that is much weaker corresponds to the back-scattered wave from the noise of the main gun while the stronger blue echo on the right side presents a single-bounce reflection from the cylindrical joint part of the main gun. b) This double-bounce scattering corresponds to the left-front side of the tank as illustrated in Figure 10.15a. Since it is a double-bounce mechanism, it showed up in the Pauli decomposition image in red color.

10.5 Polarimetric ISAR Images

Pauli decomposition ISAR image –3 i

a –2

g h

Cross-range, m

–1 0 b 1

c 2 f

3 4

e

d –4

–3

–2

–1

0 1 Range, m

2

3

4

Figure 10.14 Pauli images of the “Military Tank” target for the radar look direction of (θi = 75 , ϕi = 30 ). (For the whole assessment of the color figures, please see the electronic version of the book)

c) As demonstrated in Figure 10.15b, two of turret storage bins at the end of turret provides double-bounce echoes since they behave like dihedrals for the direction of incoming EM wave. For this reason, the back-scattering from them appear in red color in the Pauli image of Figure 10.14. The farther storage bin has a larger dihedral structure; therefore, its echo is stronger and larger than that of the other storage bin that is nearer to the radar. d) When the Pauli ISAR responses of bogie wheels are carefully examined, one can clearly observe blue, red, green, yellow, purple, violet, and even white signatures on and around seven bogie wheels. As some of them illustrated in Figure 10.15c, there are innumerous echoing responses from simple ones such as single-bounce and double-bounce to much more complex ones such as multi-bounce backscattering mechanisms. While odd-bounce backscattering mechanisms are providing blue color signatures, double-bounce ones show up in red color. It is also observed many green color signatures that correspond to incoherent scattering mechanisms of deflected waves after many multi-bounces. (e and f ) While these very complex scattering mechanisms occur on top of each other, we see other tones of red, blue, and green colors. This can be seen toward the left-back side of the tank in the Pauli image on which one can notice the

513

10 Polarimetric ISAR Imaging

(a)

Single

Double

(b)

Double

Double

(c)

Single

Multiple Double Multiple

Single

Double

ng le

Tri p

le

(d)

Multiple

Si

514

Figure 10.15 Various scattering mechanism from different parts of Military Tank target: (a) front and main gun, (b) backside of turret, (c) bogie wheels, and (d) top side of turret.

10.7 Matlab Codes

color mixture that ranges from green to yellow, and yellow to red, and then red to violet (they are labeled as “e” in Figure 10.14). Even these three distinctive mechanisms of odd-bounces, double-bounces, random, and/or volume scattering occurs, then all R, G, and B colors sum up to give white color in the Pauli image signature labeled as “f ” in Figure 10.14. (g and h) On top of turret, again there happen to be many distinctive radar echoes that come up in the Pauli image of Figure 10.14. Some of these various echoes are illustrated in in Figure 10.15d where a few single, double, triple, and multibounce scattering mechanisms are shown. While the odd-bounce echoes are indicated with “g” that provide blue-color signatures, the double-bounce ones are labeled with “h” which come up in red-colored responses. i) Since the top part of turret is composed of abundant small substructures of various shapes, it is expected some of the incoming wave’s energy will deflect in different directions with changing its polarization. In fact, the green-colored radar echoes in Figure 10.14 basically present incoherent scattering mechanisms as expected.

10.6

Feature Extraction from Polarimetric Images

Once the scattering-type classification is attained by the help of Pol-ISAR images by the help of polarization scattering matrix and the polarization decomposition techniques, the knowledge of different scattering mechanisms can be used to extract features of target’s ISAR image for developing ATR algorithms. One of the most basic and practical techniques that has been used in extracting features from Pol-ISAR images is the CLEAN algorithm (Martorella et al. 2008) that was detailly explained in Chapter 7. This technique relies on standard procedure of scattering center extraction together with its spread response from the image based on polarimetric assessment of the scattering characteristics of this particular scattering center. The other methods rely on using artificial neural networks (ANNs) and Eigenvector Decomposition techniques that can be reached from the literature (Martorella et al. 2009; Paladini et al. 2011; Bai et al. 2019).

10.7

Matlab Codes

Below are the Matlab source codes that were used to generate all of the Matlabproduced figures in this chapter. The codes are also provided inside the CD of this book.

515

516

10 Polarimetric ISAR Imaging

Matlab code 10.1 Matlab file “Figure10-6.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 10.6 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % M_F12_90_0_with_wind.mat clear ; close all; clc; %% load SLICY ISAR data @TH=90, PHI=0 load M_F12_90_0_with_wind image_data = SHH SVV SHV SVH

= = = =

M_F12_90_0_with_wind;

squeeze(image_data(:,:,1)); squeeze(image_data(:,:,4)); squeeze(image_data(:,:,2)); squeeze(image_data(:,:,3));

SHH_dB SVV_dB SHV_dB SVH_dB

= = = =

20*log10(abs(SHH(:))); 20*log10(abs(SVV(:))); 20*log10(abs(SHV(:))); 20*log10(abs(SVH(:)));

max_dB_CO = max(max([SHH_dB SVV_dB])); max_dB_X = max(max([SHV_dB SVH_dB])); N = size(image_data, 1); P = size(image_data, 2); %% DISPLAY DYN_RANGE=45; % Dynamic range of the image h = figure; for kk = 4:-1:1

10.7 Matlab Codes

image = abs(squeeze(image_data(:,:,kk))); image_dB=20*log10(image); imagesc(x_image,-y_image, image_dB, [max_dB_CO-DYN_RANGE max_dB_CO]); colormap(hot) set(gca,'FontName', 'Arial', 'FontSize',10,'FontWeight', 'Bold'); set(gca,'YDir','reverse'), set(gca,'XDir','normal') xlabel('\itrange (m)'); ylabel('\itcross range (m)'); % POL-ISAR images in LP switch kk case 1 title ('\it|S_H_H|', case 2 title ('\it|S_H_V|', case 3 title ('\it|S_V_H|', otherwise title ('\it|S_V_V|', end c = colorbar; tt = title(c,'dBsm'); tt.Position = [8 -15 0]; pause(2)

'FontSize',10) 'FontSize',10) 'FontSize',10) 'FontSize',10)

end

Matlab code 10.2 Matlab file “Figure10-8.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 10.8 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % M_F12_90_0_with_wind.mat

517

518

10 Polarimetric ISAR Imaging

clear all; close all; clc; clear ; close all; clc; %% load SLICY ISAR data @TH=90, PHI=0 load M_F12_90_0_with_wind image_data =

M_F12_90_0_with_wind;

N = size(image_data, 1); P = size(image_data, 2); % Extract Polarimetric Channels SHH = image_data(:,:,1); SHV = image_data(:,:,2); SVH = image_data(:,:,3); SVV = image_data(:,:,4); M_tar_lps = zeros(2,2,N,P); M_tar_lps(1,1,:,:) M_tar_lps(1,2,:,:) M_tar_lps(2,1,:,:) M_tar_lps(2,2,:,:)

= = = =

SHH; SHV; SVH; SVV;

%% Circular polarization conversion matrices RLC = 1/sqrt(2)*[1 -1i; 1 1i]; TLC = 1/sqrt(2)*[1 1i; 1 -1i]; inv_TLC = inv(TLC); M_tar_cps = zeros(2,2,N,P); % Convert to CP for i = 1:P for j = 1:N M_tar_cps(:,:,j,i) = RLC*M_tar_lps(:,:,j,i)

10.7 Matlab Codes

*inv_TLC; end end M_tar_cp = zeros(N,P,4); M_tar_cp(:,:,1) = M_tar_cps(1,1,:,:); M_tar_cp(:,:,2) = M_tar_cps(1,2,:,:); M_tar_cp(:,:,3) = M_tar_cps(2,1,:,:); M_tar_cp(:,:,4) = M_tar_cps(2,2,:,:); SRR SLL SRL SLR

= = = =

squeeze(M_tar_cp(:,:,1)); squeeze(M_tar_cp(:,:,4)); squeeze(M_tar_cp(:,:,2)); squeeze(M_tar_cp(:,:,3));

SRR_dB SLL_dB SRL_dB SLR_dB

= = = =

20*log10(abs(SRR(:))); 20*log10(abs(SLL(:))); 20*log10(abs(SRL(:))); 20*log10(abs(SLR(:)));

max_dB_CO = max(max([SRR_dB SLL_dB])); max_dB_X = max(max([SRL_dB SLR_dB])); clear SHH SHV SVH SVV M_tar_cps image_data %% DISPLAY DYN_RANGE=45;% Dynamic range of the image for kk = 4:-1:1 image = abs(squeeze(M_tar_cp(:,:,kk))); image_dB=20*log10(image); imagesc(x_image,-y_image,image_dB, [max_dB_CODYN_RANGE max_dB_CO]); colormap(hot) set(gca,'FontName', 'Arial', 'FontSize',10,'FontWeight', 'Bold'); set(gca,'YDir','reverse'), set(gca,'XDir','normal') xlabel('\itrange (m)'); ylabel('\itcross range (m)'); % POL-ISAR images in CP switch kk

519

520

10 Polarimetric ISAR Imaging

case 1 title case 2 title case 3 title otherwise title

('\it|S_R_R|', 'FontSize',10) ('\it|S_R_L|', 'FontSize',10) ('\it|S_L_R|', 'FontSize',10)

('\it|S_L_L|', 'FontSize',10) end c = colorbar; tt = title(c,'dBsm'); tt.Position = [ 8 -15 0 ]; pause(2) end

Matlab code 10.3 Matlab file “Figure10-9.m” %—————————————————————————————————————————————————————— % This code can be used to generate Figure 10.9 %—————————————————————————————————————————————————————— % This file requires the following files to be present in the % same directory: % % M_F12_90_0_with_wind.mat % newjet.mat % imlocalbrighten % imcomplement % imreducehaze % linspecer clear ; close all; clc; load newjet cmap = newjet;

% A specific colormap

%% load SLICY ISAR data @TH=90, PHI=0 load M_F12_90_0_with_wind

10.7 Matlab Codes

image_data =

M_F12_90_0_with_wind;

N = size(image_data, 1); P = size(image_data, 2); % Extract Polarimetric Channels SHH = image_data(:,:,1); SHV = image_data(:,:,2); SVH = image_data(:,:,3); SVV = image_data(:,:,4); max_val = max(abs(image_data(:))); % Normalize SHH = image_data(:,:,1)/max_val; SHV = image_data(:,:,2)/max_val; SVH = image_data(:,:,3)/max_val; SVV = image_data(:,:,4)/max_val; %% PAULI IMAGING SXX = SHV; Channnel

% Form average Cross-pol

% Form Pauli scattering vectors A = (SHH+SVV)./sqrt(2); % Isotropic odd bounce scatterer (SHH = SVV, SVH=SHV=0), Spheres, Flat plates, TCRs B = (SHH-SVV)./sqrt(2); % Isotropic even bounce scatterer (SHH = -SVV, SHV=SVH=0), Dihedral corner reflectors C = sqrt(2)*SXX; % Isotropic even bounce scatterer with a relative orientation of pi/4 w.r.t horizontal [SPAN] = deal(zeros(N,P)); for mm=1:N for nn=1:P SPAN(mm,nn) = abs(A(mm,nn))^2+abs(B(mm,nn))^2 +abs(C(mm,nn))^2; end end

521

522

10 Polarimetric ISAR Imaging

ind=find(SPAN==0); SPAN(ind)=min(min(SPAN)); SPAN_dB = 10*log10(abs(SPAN)); % DISPLAY PAULI Sall = zeros(N, P, 3); Sall(:,:,1) = B; Sall(:,:,2) = C; Sall(:,:,3) = A; Is = abs(Sall); %% DISPLAY %%% Extract the individual red, green, and blue color channels. dyn_range = 50; redChannel = Is(:, :, 1); greenChannel = Is(:, :, 2); blueChannel = Is(:, :, 3); mask = SPAN_dB