A Student's Guide to Fourier Transforms

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to the first edition......Page 11
Preface to the second edition......Page 13
Preface to the third edition......Page 15
1.1 The qualitative approach......Page 17
1.2 Fourier series......Page 18
1.3 The amplitudes of the harmonics......Page 20
1.4 Fourier transforms......Page 24
1.5 Conjugate variables......Page 26
1.7.1 The `top-hat\' function......Page 27
1.7.2 The sinc-function......Page 28
1.7.3 The Gaussian function......Page 29
1.7.4 The exponential decay......Page 30
1.7.5 The Dirac `delta-function\'......Page 31
1.7.7 The Dirac comb......Page 33
1.8 Worked examples......Page 34
2.1 The Dirichlet conditions......Page 36
2.3 Convolutions and the convolution theorem......Page 38
2.3.1 The convolution theorem......Page 42
2.3.2 Examples of convolutions......Page 43
2.3.3 The autocorrelation theorem......Page 45
2.4.1 Convolution of two delta-functions......Page 46
2.5.1 The derivative theorem......Page 47
2.5.3 Parseval\'s theorem......Page 48
2.5.4 The sampling theorem......Page 49
2.6 Aliasing......Page 50
2.6.1 The interpolation theorem......Page 51
2.7.1 An arithmetical result using Parseval\'s theorem......Page 52
2.7.2 Alternating pulse-heights......Page 53
2.7.4 Convolution with a sinusoid......Page 54
3.1 Fraunhofer diffraction......Page 56
3.2.1 Single-slit diffraction, normal incidence......Page 60
3.2.3 Two slits, each of width a, with centres separated by a distance b (Young\'s slits, Fresnel\'s biprism, Lloyd\'s mirror, Rayleigh\'s refractometer, Billet\'s split-lens)......Page 61
3.2.5 The transmission diffraction grating......Page 62
3.2.6 Apertures with phase-changes instead of amplitude changes......Page 67
3.2.7 Diffraction at an aperture with a prism......Page 68
3.2.8 The blazed diffraction grating......Page 69
3.3 Babinet\'s principle......Page 70
3.4 Dipole arrays......Page 71
3.6 Phase and coherence......Page 74
3.7 Fringe visibility......Page 76
3.8 The Michelson stellar interferometer......Page 77
3.9 The van Cittert-Zernike theorem......Page 80
4.1 Communication channels......Page 82
4.1.1 The Wiener-Khinchine theorem......Page 83
4.2 Noise......Page 84
4.3 Filters......Page 85
4.4 The matched filter theorem......Page 86
4.5 Modulations......Page 87
4.7.1 The Heaviside step-function......Page 93
4.7.2 The passage of a voltage step through a `perfect\' low-pass filter......Page 96
4.8 The Gibbs phenomenon......Page 97
4.8.1 The passage of a train of pulses through a low-pass filter......Page 98
4.8.2 Passage of a voltage step through a simple high-pass filter......Page 99
5.2 The Michelson multiplex spectrometer......Page 102
5.2.1 The theory of the Michelson-Fourier spectrometer......Page 104
5.3 The shapes of spectrum lines......Page 107
6.1 Cartesian coordinates......Page 113
6.2 Polar coordinates......Page 114
6.3 Theorems......Page 115
6.4 Examples of two-dimensional Fourier transforms with circular symmetry......Page 116
6.5.1 Fraunhofer diffraction by a rectangular slot......Page 117
6.5.2 Fraunhofer diffraction by a circular aperture......Page 118
6.6 Solutions without circular symmetry......Page 119
7.1 The Dirac wall......Page 121
7.2 Computerized axial tomography......Page 124
7.3 A `spike\' or `nail\'......Page 128
7.4 The Dirac fence......Page 130
7.5 The `bed of nails\'......Page 131
7.6 Parallel-plane delta-functions......Page 132
7.7 Point arrays......Page 134
7.8 Lattices......Page 135
8 The formal complex Fourier transform......Page 136
9.1 History......Page 143
9.2 The discrete Fourier transform......Page 144
9.3 The matrix form of the DFT......Page 145
9.3.1 Two-dimensional FFTs......Page 148
9.4.1 A routine for 1024 complex numbers......Page 149
A.1 Parseval\'s theorem and Rayleigh\'s theorem......Page 153
The Jacobi expansion......Page 154
The Hankel transform......Page 155
A.3 Conversion of Fourier-series coefficients into complex exponential form......Page 156
Bibliography......Page 157
Index......Page 159