Fourier Transforms in Radar and Signal Processing (Artech House Radar Library)

212 Pages·2003·2.77 MB·English

by  David Brandwood

Fourier Transforms in Radar and Signal
Processing
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Fourier Transforms in Radar and Signal
Processing
David Brandwood
Artech House
Boston • London
www.artechhouse.com
Library of Congress Cataloging-in-Publication Data
Information for this book is on file with the Library of Congress.
British Library Cataloguing in Publication Data
Brandwood, David.
Fourier transforms in radar and signal processing. � (Artech House radar library)
1. Signal processing�Mathematics 2. Radar�Mathematics 3. Fourier
transformations I. Title
621.3’822’01515723
ISBN 1-58053-174-1
Cover design by Yekaterina Ratner
 2003 ARTECH HOUSE, INC.
685 Canton Street
Norwood, MA 02062
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International Standard Book Number: 1-58053-174-1
10 9 8 7 6 5 4 3 2 1
Contents
Preface xi
1 Introduction 1
1.1 Aim of the Work 1
1.2 Origin of the Rules and Pairs Method for
Fourier Transforms 2
1.3 Outline of the Rules and Pairs Method 3
1.4 The Fourier Transform and Generalized
Functions 4
1.5 Complex Waveforms and Spectra in Signal
Processing 7
1.6 Outline of the Contents 8
References 10
2 Rules and Pairs 11
2.1 Introduction 11
v
vi Fourier Transforms in Radar and Signal Processing
2.2 Notation 12
2.2.1 Fourier Transform and Inverse Fourier
Transform 12
2.2.2 rect and sinc 13
2.2.3 d-Function and Step Function 15
2.2.4 rep and comb 17
2.2.5 Convolution 18
2.3 Rules and Pairs 21
2.4 Three Illustrations 24
2.4.1 Narrowband Waveforms 24
2.4.2 Parseval’s Theorem 24
2.4.3 The Wiener-Khinchine Relation 26
References 27
Appendix 2A: Properties of the sinc Function 27
Appendix 2B: Brief Derivations of the Rules
and Pairs 29
3 Pulse Spectra 39
3.1 Introduction 39
3.2 Symmetrical Trapezoidal Pulse 40
3.3 Symmetrical Triangular Pulse 41
3.4 Asymmetrical Trapezoidal Pulse 44
3.5 Raised cosine Pulse 47
3.6 Rounded Pulses 49
3.7 General Rounded Trapezoidal Pulse 53
3.8 Regular Train of Identical RF Pulses 58
3.9 Carrier Gated by a Regular Pulse Train 59
Contents vii
3.10 Pulse Doppler Radar Target Return 61
3.11 Summary 62
4 Sampling Theory 65
4.1 Introduction 65
4.2 Basic Technique 66
4.3 Wideband Sampling 67
4.4 Uniform Sampling 69
4.4.1 Minimum Sampling Rate 69
4.4.2 General Sampling Rate 71
4.5 Hilbert Sampling 74
4.6 Quadrature Sampling 75
4.6.1 Basic Analysis 75
4.6.2 General Sampling Rate 78
4.7 Low IF Analytic Signal Sampling 81
4.8 High IF Sampling 84
4.9 Summary 85
References 86
Appendix 4A: The Hilbert Transform 86
5 Interpolation for Delayed Waveform Time
Series 89
5.1 Introduction 89
5.2 Spectrum Independent Interpolation 90
5.2.1 Minimum Sampling Rate Solution 90
5.2.2 Oversampling and the Spectral Gating
Condition 93
viii Fourier Transforms in Radar and Signal Processing
5.2.3 Three Spectral Gates 97
5.2.4 Results and Comparisons 105
5.3 Least Squared Error Interpolation 107
5.3.1 Method of Minimum Residual Error Power 107
5.3.2 Power Spectra and Autocorrelation Functions 111
5.3.3 Error Power Levels 114
5.4 Application to Generation of Simulated
Gaussian Clutter 114
5.4.1 Direct Generation of Gaussian Clutter
Waveform 116
5.4.2 Efficient Clutter Waveform Generation Using
Interpolation 119
5.5 Resampling 120
5.6 Summary 122
References 123
6 Equalization 125
6.1 Introduction 125
6.2 Basic Approach 126
6.3 ramp and sncr Functions 130
6.4 Simple Example of Amplitude Equalization 134
6.5 Equalization for Broadband Array Radar 135
6.6 Sum Beam Equalization 138
6.7 Difference Beam Equalization 147
6.8 Summary 158
Contents ix
7 Array Beamforming 161
7.1 Introduction 161
7.2 Basic Principles 162
7.3 Uniform Linear Arrays 164
7.3.1 Directional Beams 164
7.3.2 Low Sidelobe Patterns 167
7.3.3 Sector Beams 174
7.4 Nonuniform Linear Arrays 180
7.5 Summary 187
Final Remarks 189
About the Author 191
Index 193