Fourier Transforms in Radar and Signal Processing

Second Edition

David Brandwood

er I t A R T E C H H O U S E BOSTON|LONDON a r t echhouse . com

Contents

Preface xiii

Preface to the First Edition xv

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 6

1.6 Outline of the Contents 8

Reference 10

vi Fourier Transforms in Radar and Signal Processing

2 Rules and Pairs 1 |

2.1 Introduction 11

2.2 Notation 12

2.2.1 Fourier Transform and Inverse

Fourier Transform 12

2.2.2 rect and sine 13

2.2.3 (5-function and Step Function 15

2.2.4 rep and comb 17

2.2.5 Convolution 18

2.3 Rules and Pairs 20

2.4 Four Illustrations 24

2.4.1 Narrowband Waveforms 24

2.4.2 Parseval's Theorem 24

2.4.3 The Wiener-Khinchine Relation 26

2.4.4 Sum of Shifted sine Functions 26

Appendix 2A Properties of the sine Function 29

Appendix 2B Brief Derivations of the

Rules and Pairs 33

2B.1 Rules 33

2B.2 Pairs 38

3 Pulse Spectra 41

3.1 Introduction 41

3.2 Symmetrical Trapezoidal Pulse 42

3.3 Symmetrical Triangular Pulse 43

3.4 Asymmetric Trapezoidal Pulse 46

Contents vii

3.5 Asymmetrie Triangular Pulse 48

3.6 Raised Cosine Pulse 50

3.7 Rounded Pulses 53

3.8 General Rounded Trapezoidal Pulse 58

3.9 Regular Train of Identical RF Pulses 62

3.10 Carrier Gated by a Regular Pulse Train 64

3.11 Pulse Doppler Radar Target Return 65

3.12 Summary 67

4 Periodic Waveforms, Fourier Series,

and Discrete Fourier Transforms 69

4.1 Introduction 69

4.2 Power Relations for Periodic Waveforms 72

4.2.1 Energy and Power 72

4.2.2 Power in the 5-Function 72

4.2.3 General Periodic Function 74

4.2.4 Regularly Sampled Function 77

4.2.5 Note on Dimensions 78

4.3 Fourier Series of Real Functions Using

Rules and Pairs 78

4.3.1 Fourier Series Coefficients 78

4.3.2 Fourier Series of Square Wave 80

4.3.3 Fourier Series of Sawtooth 83

4.3.4 Fourier Series of Triangular Waves 85

4.3.5 Fourier Series of Rectified Sinewaves 88

VIII Fourier Transforms in Radar and Signal Processing

4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 4.4.7

4.5

Discrete Fourier Transforms General Discrete Waveform Transform of Regular Time Series Transform of Sampled Periodic Spectrum Fast Fourier Transform Examples Illustrating the FFT And DFT Matrix Representation of DFT Efficient Convolution Using the FFT

Summary

91 91 94 95 98 99

102 104

106

Appendix 4A Spectrum of Time-Limited Waveform 107

Appendix 4B Constraint on Repetition Period 108

5 Sampling Theory Ш

5.1 Introduction 111

5.2 Basic Technique 112

5.3 Wideband Sampling 113

5.4 Uniform Sampling 116 5.4.1 Minimum Sampling Rate 116

5.4.2 General Sampling Rate 117

5.5 Hilbert Sampling 120

5.6 Quadrature Sampling 122 5.6.1 Basic Analysis 122 5.6.2 General Sampling Rate 124

5.7 Low IF Analytic Signal Sampling 128

Contents ix

5.8 High IF Sampling

5.9 Summary

Appendix 5A The Hubert Transform

6 Interpolation for Delayed Waveform Time Series

131

133

134

137

6.1

6.2 6.2.1 6.2.2

6.2.3

6.2.4

6.3 6.3.1

Introduction

Spectrum Independent Interpolation

Minimum Sampling Rate Solution

Oversampling and the Spectral Gating Condition

Three Spectral Gates

Trapezoidal Gate

Trapezoidal Rounded Gate

Raised Cosine Rounded Gate

Results and Comparisons

Least Squared Error Interpolation

Method of Minimum Residual

137

138

138

142

147

147

148

151

154

158

6.3.2

6.3.3

6.4

6.4.1

Error Power

Power Spectra and Autocorrelation Functions

Rectangular Spectrum

Triangular Spectrum

Raised Cosine Spectrum

Gaussian Spectrum

Trapezoidal Spectrum

Error Power Levels

Application to Generation of Simulated Gaussian Clutter

Direct Generation of Gaussian Clutter Waveform

158

161

161

162

162

162

163

164

166

167

X Fourier Transforms in Radar and Signal Processing

6.4.2 Efficient Clutter Waveform Generation,

Using Interpolation 170

6.5 Resampling 171

6.6 Summary 172

Reference 174

7 Equalization 175

7.1 Introduction 175

7.2 Basic Approach 177

7.3 ramp and sncr Functions 181

7.4 Example of Amplitude Equalization 186

7.5 Equalization for Broadband Array Radar 188

7.6 Sum Beam Equalization 190

7.7 Difference Beam Equalization 199

7.8 Summary 214

8 Array Beamforming 217

8.1 Introduction 217

8.2 Basic Principles 218

8.3 Uniform Linear Arrays 222

8.3.1 Directional Beams 222

8.3.2 Low Sidelobe Patterns 225

8.3.3 Sector Beams 232

Contents XI

8.4 Nonuniform Linear Arrays

8.4.1 Prescribed Patterns from Nonuniform Linear Arrays

8.4.2 Sector Beams from a Nonuniform Linear Array

8.5 Summary

Final Remarks

About the Author

Index

239

239

241

248

251

253

255