The Fourier Transform and Its Applications, Ronald N. Bracewell, 3rd Edition, McGraw-Hill, 2000
Preface XVll
1 Introduction 1
2 Groundwork 5
The Fourier Transform and Fourier's Integral Theorem 5
Conditions for the Existence of Fourier Transforms 8
Transforms in the Limit 10
Oddness and Evenness 11
Significance of Oddness and Evenness 13
Complex Conjugates 14
Cosine and Sine Transforms 16
Interpretation of the Formulas 18
3 Convolution 24
Examples of Convolution 27
Serial Products 30
Inversion of serial multiplication / The serial product in matrix notation /
Sequences as vectors
Convolution by Computer 39
The Autocorrelation Function and Pentagram Notation 40
The Triple Correlation 45
The Cross Correlation 46
The Energy Spectrum 47
4 Notation for Some Useful Functions 55
Rectangle Function of Unit Height and Base, n(x) 55
Triangle Function of Unit Height and Area, A(x) 57
Various Exponentials and Gaussian and Rayleigh Curves 57
Heaviside's Unit Step Function, H(x) 61
The Sign Function, sgn x 65
The Filtering or Interpolating Function, sine x 65
Pictorial Representation 68
Summary of Special Symbols 71
5 The Impulse Symbol 74
The Sifting Property 78
The Sampling or Replicating Symbol III(x) 81
The Even and Odd Impulse Pairs n(x) and II (x) 84
Derivatives of the hnpulse Symbol 85
Null Functions 87
Some Functions in Two or More Dimensions 89
The Concept of Generalized Function 92
Particularly well-behaved functions I Regular sequences I Generalized functions I
Algebra of generaliud functions / Differentiation of ordinary functions
6 The Basic Theorems 105
A Few Transforms for Illustration 105
Similarity Theorem 108
Addition Theorem 110
Shift Theorem 111
Modulation Theorem 113
Convolution Theorem 115
Rayleigh's Theorem 119
Power Theorem 120
Autocorrelation Theorem 122
Derivative Theorem 124
Derivative of a Convolution Integral 126
The Transform of a Generalized Function 127
Proofs of Theorems 128
Similarity and shift theorems / Derivative theorem / Power theorem
Summary of Theorems 129
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