Preface ........................................................ XI
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 Reference ........................ 8
2 Rules and Pairs ............................................. 10
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 δ-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
3.5 Asymmetric 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 (δ-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
4.4 Discrete Fourier Transforms ............................ 91
4.4.1 General Discrete Waveform ....................... 91
4.4.2 Transform of Regular Time Series ................ 94
4.4.3 Transform of Sampled Periodic Spectrum .......... 95
4.4.4 Fast Fourier Transform .......................... 98
4.4.5 Examples Illustrating the FFT And DFT ........... 99
4.4.6 Matrix Representation of DFT ................... 102
4.4.7 Efficient Convolution Using the FFT ............ 104
4.5 Summary ............................................... 106
Appendix 4A Spectrum of Time-Limited Waveform ............. 197
Appendix 4B Constraint on Repetition Period ............... 108
5 Sampling Theory ............................................ 111
5.1 Introduction .......................................... 111
5.2 Basic Technique ....................................... 112
5.3 Wideband Sampling ..................................... 113
5.4 Uniform Sampling ...................................... 115
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
6 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
8.4 Nonuniform Linear Arrays .............................. 239
8.4.1 Prescribed Patterns from Nonuniform Linear
Arrays ......................................... 239
8.4.2 Sector Beams from a Nonuniform Linear Array .... 241
8.5 Summary ............................................... 248
Final Remarks ................................................. 251
About the Author .............................................. 253
Index ......................................................... 255
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