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

Electrical engineers working in radar, sonar, and signal processing use Fourier transform relationships everyday on the job. This book presents a coordinated system for performing Fourier transforms on a variety of functions. It illustrates how to apply Fourier transforms to many specific examples in radar, signal processing, and antenna design. Fourier Transforms in Radar and Signal Processing Second Edition 2 Contents 6 Preface 14 Preface to the First Edition 16 1 Introduction 18 1.1 Aim of the Work 18 1.2 Origin of the Rules-and-Pairs Method for Fourier Transforms 19 1.3 Outline of the Rules-and-Pairs Method 20 1.4 The Fourier Transform and Generalized Functions 21 1.5 Complex Waveforms and Spectra in Signal Processing 23 1.6 Outline of the Contents 25 References 27 2 Rules and Pairs 28 2.1 Introduction 28 2.2 Notation 29 2.2.1 Fourier Transform and Inverse Fourier Transform 29 2.2.2 rect and sinc 30 2.2.3 d-function and Step Function 32 2.2.4 rep and comb 34 2.2.5 Convolution 35 2.3 Rules and Pairs 37 2.4 Four Illustrations 41 2.4.1 Narrowband Waveforms 41 2.4.2 Parseval’s Theorem 41 2.4.3 The Wiener-Khinchine Relation 43 2.4.4 Sum of Shifted sinc Functions 43 Appendix 2B: Brief Derivations of the Rules and Pairs 50 2B.1 Rules 50 2B.2 Pairs 54 3 Pulse Spectra 58 3.1 Introduction 58 3.2 Symmetrical Trapezoidal Pulse 59 3.3 Symmetrical Triangular Pulse 60 3.4 Asymmetric Trapezoidal Pulse 63 3.5 Asymmetric Triangular Pulse 65 3.6 Raised Cosine Pulse 67 3.7 Rounded Pulses 70 3.8 General Rounded Trapezoidal Pulse 75 3.9 Regular Train of Identical RF Pulses 79 3.10 Carrier Gated by a Regular Pulse Train 81 3.11 Pulse Doppler Radar Target Return 82 3.12 Summary 84 4 Periodic Waveforms, Fourier Series,and Discrete Fourier Transforms 86 4.1 Introduction 86 4.2 Power Relations for Periodic Waveforms 89 4.2.1 Energy and Power 89 4.2.2 Power in the d -Function 89 4.2.3 General Periodic Function 91 4.2.4 Regularly Sampled Function 94 4.2.5 Note on Dimensions 95 4.3 Fourier Series of Real Functions Using Rules and Pairs 95 4.3.1 Fourier Series Coefficients 95 4.3.2 Fourier Series of Square Wave 97 4.3.3 Fourier Series of Sawtooth 100 4.3.4 Fourier Series of Triangular Waves 102 4.3.5 Fourier Series of Rectified Sinewaves 105 4.4 Discrete Fourier Transforms 108 4.4.1 General Discrete Waveform 108 4.4.2 Transform of Regular Time Series 111 4.4.3 Transform of Sampled Periodic Spectrum 112 4.4.4 Fast Fourier Transform 115 4.4.5 Examples Illustrating the FFT and DFT 116 4.4.6 Matrix Representation of DFT 119 4.4.7 Efficient Convolution Using the FFT 121 4.5 Summary 123 Appendix 4A: Spectrum of Time-Limited Waveform 124 Appendix 4B: Constraint on Repetition Period 125 5 Sampling Theory 128 5.1 Introduction 128 5.2 Basic Technique 129 5.3 Wideband Sampling 130 5.4 Uniform Sampling 133 5.4.1 Minimum Sampling Rate 133 5.4.2 General Sampling Rate 134 5.5 Hilbert Sampling 137 5.6 Quadrature Sampling 139 5.6.1 Basic Analysis 139 5.6.2 General Sampling Rate 141 5.7 Low IF Analytic Signal Sampling 145 5.8 High IF Sampling 148 5.9 Summary 150 References 151 Appendix 5A: The Hilbert Transform 151 6 Interpolation for Delayed WaveformTime Series 154 6.1 Introduction 154 6.2 Spectrum Independent Interpolation 155 6.2.1 Minimum Sampling Rate Solution 155 6.2.2 Oversampling and the Spectral Gating Condition 159 6.2.3 Three Spectral Gates 164 6.2.4 Results and Comparisons 171 6.3 Least Squared Error Interpolation 175 6.3.1 Method of Minimum Residual Error Power 175 6.3.2 Power Spectra and Autocorrelation Functions 178 6.3.3 Error Power Levels 181 6.4 Application to Generation of Simulated Gaussian Clutter 183 6.4.1 Direct Generation of Gaussian Clutter Waveform 184 6.4.2 Efficient Clutter Waveform Generation, Using Interpolation 187 6.5 Resampling 188 6.6 Summary 189 References 191 7 Equalization 192 7.1 Introduction 192 7.2 Basic Approach 194 7.3 ramp and sncr Functions 198 7.4 Example of Amplitude Equalization 203 7.5 Equalization for Broadband Array Radar 205 7.6 Sum Beam Equalization 207 7.7 Difference Beam Equalization 216 7.8 Summary 231 8 Array Beamforming 234 8.1 Introduction 234 8.2 Basic Principles 235 8.3 Uniform Linear Arrays 239 8.3.1 Directional Beams 239 8.3.2 Low Sidelobe Patterns 242 8.3.3 Sector Beams 249 8.4 Nonuniform Linear Arrays 256 8.4.1 Prescribed Patterns from Nonuniform Linear Arrays 256 8.4.2 Sector Beams from a Nonuniform Linear Array 258 8.5 Summary 265 Final Remarks 268 About the Author 270 Index 272 Fourier Transforms Are Used Widely, And Are Of Particular Value In The Analysis Of Single Functions And Combinations Of Functions Found In Radar And Signal Processing. Still, Many Problems That Could Have Been Tackled By Using Fourier Transforms May Have Gone Unsolved Because They Require Integration That Is Difficult And Tedious. This Newly Revised And Expanded Edition Of A Classic Artech House Book Provides You With An Up-to-date, Coordinated System For Performing Fourier Transforms On A Wide Variety Of Functions. Along Numerous Updates Throughout The Book, The Second Edition Includes A Critical New Chapter On Periodic Waveforms A Topic Not Covered In Any Other Book And Detailed Coverage Of Asymmetric Triangular Pulse. By Building Upon Woodward's Well Known Rules And Pairs Method And Related Concepts And Procedures, This Book Establishes A Unified System That Makes Implicit The Integration Required For Performing Fourier Transforms On A Wide Variety Of Functions. It Details How Complex Functions Can Be Broken Down To Their Constituent Parts For Analysis. You Can Now Concentrate On Functional Relationships Instead Of Getting Bogged Down In The Details Of Integration. This Approach To Implementing Fourier Transforms Is Illustrated With Many Specific Examples From Digital Signal Processing As Well As Radar And Antenna Operation. Dvd-rom Included! Contains Matlab Programs That Implement Many Of The Results Presented In The Book. Accompanying DVD-ROM contains MATLAB programs that implement many of the results presented in this book