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. Now you can solve many of these problems with the integration-free approach to carrying out Fourier transforms (and Fourier series) presented in this book.
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.
Author(s): David Brandwood
Series: Artech House radar library
Publisher: Artech House
Year: 2003
Language: English
Pages: 212
City: Boston
TeamLiB......Page 0
Cover......Page 1
Contents......Page 6
Preface......Page 12
1.1 Aim of the Work......Page 14
1.2 Origin of the Rules and Pairs Method for Fourier Transforms......Page 15
1.3 Outline of the Rules and Pairs Method......Page 16
1.4 The Fourier Transform and Generalized Functions......Page 17
1.5 Complex Waveforms and Spectra in Signal Processing......Page 20
1.6 Outline of the Contents......Page 21
2.1 Introduction......Page 24
2.2.1 Fourier Transform and Inverse Fourier Transform......Page 25
2.2.2 rect and sinc......Page 26
2.2.3 Function and Step Function......Page 28
2.2.4 rep and comb......Page 30
2.2.5 Convolution......Page 31
2.3 Rules and Pairs......Page 34
2.4.1 Parseval’s Theorem......Page 37
2.4.3 The Wiener- Khinchine Relation......Page 39
Appendix 2A: Properties of the sinc Function......Page 40
Appendix 2B: Brief Derivations of the Rules and Pairs......Page 42
3.1 Introduction......Page 52
3.2 Symmetrical Trapezoidal Pulse......Page 53
3.3 Symmetrical Triangular Pulse......Page 54
3.4 Asymmetrical Trapezoidal Pulse......Page 57
3.5 Raised cosine Pulse......Page 60
3.6 Rounded Pulses......Page 62
3.7 General Rounded Trapezoidal Pulse......Page 66
3.8 Regular Train of Identical RF Pulses......Page 71
3.9 Carrier Gated by a Regular Pulse Train......Page 72
3.10 Pulse Doppler Radar Target Return......Page 74
3.11 Summary......Page 75
4.1 Introduction......Page 78
4.2 Basic Technique......Page 79
4.3 Wideband Sampling......Page 80
4.4.1 Minimum Sampling Rate......Page 82
4.4.2 General Sampling Rate......Page 84
4.5 Hilbert Sampling......Page 87
4.6.1 Basic Analysis......Page 88
4.6.2 General Sampling Rate......Page 91
4.7 Low IF Analytic Signal Sampling......Page 94
4.8 High IF Sampling......Page 97
4.9 Summary......Page 98
Appendix 4A: The Hilbert Transform......Page 99
5.1 Introduction......Page 102
5.2.1 Minimum Sampling Rate Solution......Page 103
5.2.2 Oversampling and the Spectral Gating Condition......Page 106
5.2.3 Three Spectral Gates......Page 110
5.2.4 Results and Comparisons......Page 118
5.3.1 Method of Minimum Residual Error Power......Page 120
5.3.2 Power Spectra and Autocorrelation Functions......Page 124
5.4 Application to Generation of Simulated Gaussian Clutter......Page 127
5.4.1 Direct Generation of Gaussian Clutter Waveform......Page 129
5.4.2 Efficient Clutter Waveform Generation Using Interpolation......Page 132
5.5 Resampling......Page 133
5.6 Summary......Page 135
6.1 Introduction......Page 138
6.2 Basic Approach......Page 139
6.3 ramp and sncr Functions......Page 143
6.4 Simple Example of Amplitude Equalization......Page 147
6.5 Equalization for Broadband Array Radar......Page 148
6.6 Sum Beam Equalization......Page 151
6.7 Difference Beam Equalization......Page 160
6.8 Summary......Page 171
7.1 Introduction......Page 174
7.2 Basic Principles......Page 175
7.3.1 Directional Beams......Page 177
7.3.2 Low Sidelobe Patterns......Page 180
7.3.3 Sector Beams......Page 187
7.4 Nonuniform Linear Arrays......Page 193
7.5 Summary......Page 200
Final Remarks......Page 202
About the Author......Page 204
Index......Page 206
