Preface to the first edition ................................... ix
Preface to the second edition .................................. xi
Preface to the third edition ................................. xiii
1 Physics and Fourier transforms ............................... 1
1.1 The qualitative approach ................................ 1
1.2 Fourier series .......................................... 2
1.3 The amplitudes of the harmonics ......................... 4
1.4 Fourier transforms ...................................... 8
1.5 Conjugate variables .................................... 10
1.6 Graphical representations .............................. 11
1.7 Useful functions ....................................... 11
1.8 Worked examples ........................................ 18
2 Useful properties and theorems .............................. 20
2.1 The Dirichlet conditions ............................... 20
2.2 Theorems ............................................... 22
2.3 Convolutions and the convolution theorem ............... 22
2.4 The algebra of convolutions ............................ 30
2.5 Other theorems ......................................... 31
2.6 Aliasing ............................................... 34
2.7 Worked examples ........................................ 36
3 Applications 1: Fraunhofer diffraction ...................... 40
3.1 Fraunhofer diffraction ................................. 40
3.2 Examples ............................................... 44
3.3 Babinet's principle .................................... 54
3.4 Dipole arrays .......................................... 55
3.5 Polar diagrams ......................................... 58
3.6 Phase and coherence .................................... 58
3.7 Fringe visibility ...................................... 60
3.8 The Michelson stellar interferometer ................... 61
3.9 The van Cittert-Zernike theorem ........................ 64
4 Applications 2: signal analysis and communication theory .... 66
4.1 Communication channels ................................. 66
4.2 Noise .................................................. 68
4.3 Filters ................................................ 69
4.4 The matched filter theorem ............................. 70
4.5 Modulations ............................................ 71
4.6 Multiplex transmission along a channel ................. 77
4.7 The passage of some signals through simple filters ..... 77
4.8 The Gibbs phenomenon ................................... 81
5 Applications 3: interference spectroscopy and spectral
line shapes ................................................. 86
5.1 Interference spectrometry .............................. 86
5.2 The Michelson multiplex spectrometer ................... 86
5.3 The shapes of spectrum lines ........................... 91
6 Two-dimensional Fourier transforms .......................... 97
6.1 Cartesian coordinates .................................. 97
6.2 Polar coordinates ...................................... 98
6.3 Theorems ............................................... 99
6.4 Examples of two-dimensional Fourier transforms with
circular symmetry ..................................... 100
6.5 Applications .......................................... 101
6.6 Solutions without circular symmetry ................... 103
7 Multi-dimensional Fourier transforms ....................... 105
7.1 The Dirac wall ........................................ 105
7.2 Computerized axial tomography ......................... 108
7.3 A 'spike' or 'nail' ................................... 112
7.4 The Dirac fence ....................................... 114
7.5 The 'bed of nails' .................................... 115
7.6 Parallel-plane delta-functions ........................ 116
7.7 Point arrays .......................................... 118
7.8 Lattices .............................................. 119
8 The formal complex Fourier transform ....................... 120
9 Discrete and digital Fourier transforms .................... 127
9.1 History ............................................... 127
9.2 The discrete Fourier transform ........................ 128
9.3 The matrix form of the DFT ............................ 129
9.4 A BASIC FFT routine ................................... 133
Appendix ...................................................... 137
Bibliography .................................................. 141
Index ......................................................... 143
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