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A Student’s Guide to Fourier Transforms
with applications in physics and engineering
Second Edition
J. F. JAMES
Honorary Research Fellow,
The University of Glasgow
و المحتوى كما يلي :
Contents
Preface to the first edition
Preface to the second edition
1 Physics and Fourier transforms
1.1 The qualitative approach
1.2 Fourier series
1.3 The amplitudes of the harmonics
1.4 Fourier transforms
1.5 Conjugate variables
1.6 Graphical representations
1.7 Useful functions
1.8 Worked examples
2 Useful properties and theorems
2.1 The Dirichlet conditions
2.2 Theorems
2.3 Convolutions and the convolution theorem
2.4 The algebra of convolutions
2.5 Other theorems
2.6 Aliasing
2.7 Worked examples
3 Applications 1: Fraunhofer diffraction
3.1 Fraunhofer diffraction
3.2 Examples
3.3 Polar diagrams
3.4 Phase and coherence
3.5 Exercises
4 Applications 2: signal analysis and communication theory
4.1 Communication channels
4.2 Noise
4.3 Filters
4.4 The matched filter theorem
page vil
IX
11248
10
1 I
1 1
18
20
20
21
23
29
30
33
35
38
38
42
52
53
57
58
58
60
61
62
vvi Contents
4.5 Modulations
4.6 Multiplex transmission along a channel
4.7 The passage of some signals through simple filters
4.8 The Gibbs phenomenon
5 Applications 3: spectroscopy and spectral line shapes
5.1 Interference spectrometry
5.2 The shapes of spectrum lines
6 Two-dimensional Fourier transforms
6.1 Cartesian coordinates
6.2 Polar coordinates
6.3 Theorems
6.4 Examples of two-dimensional Fourier transforms
with circular symmetry
6.5 Applications
6.6 Solutions without circular symmetry
7 Multi-dimensional Fourier transforms
7.1 The Dirac wall
7.2 Computerized axial tomography
7.3 A ‘spike’ or ‘nail’
7.4 The Dirac fence
7.5 The ‘bed of nails’
7.6 Parallel plane delta-functions
7.7 Point arrays
7.8 Lattices
8 The formal complex Fourier transform
9 Discrete and digital Fourier transforms
9.1 History
9.2 The discrete Fourier transform
9.3 The matrix form of the DFT
9.4 The BASIC FFT routine
Appendix
Bibliography
63
69
69
70
76
76
81
86
86
87
88
89
90
92
94
94
97
101
103
104
106
106
107
109
116
116
117
118
122
126
131134 Index
Dirac comb (cant.)
delta function 15, 20, 101
FT of 16
fence 103
wall 94, 97
spike 101
point arrays 106
Dirichlet conditions 20, 58, 109
discrete Fourier transform 116 et seq
matrix form 118
distributive rule 25
Doppler broadening 83
Gaussian profile 13, 27, 83
ghosts, Rowland 68
Gibbs phenomenon 70
graphical representation 11, 112, 113
Hankel transforms 87, 91, 128
harmonics 2, 4, 8
amplitude of 4
harmonic integrator 72
Hartley-Shannon theorem 65
Hermitian functions 115
Heaviside step 69, 71, 126
history, of discrete transforms 116
Huygens’ principle 38
wavelets 50
electric charge, accelerated 81
error, periodic in grating ruling 66
exponential decay 14
exponentials, complex 7 impulse response 24
integrator, harmonic 72
intensity
of a wave 41 et seq
in single-slit diffraction 42
in a diffraction grating 46
interference spectrometry 76
interferogram 78
interferometer, Michelson 77
interpolary function theory 32,
34 et seq
interpolation 111
theorem 34
interval, sampling 32
instrumental function 24
inverse transform 9
inversion formulae 7
Fabry-Perot etalon 53, 72, 84
fast Fourier transform 116 et seq
BASIC routine for 122
filters 61
matched, theorem 62
folding frequency 32 et seq
Fourier coefficients 7, 129
inversion theorem 9
pairs 9
series 2, 17, 128, 129
Fourier transforms 1, 9
digital 116 et seq
matrix form 118
formal complex 109
modular 1, 109
phase 109
power 109
sine & cosine 8, 112
two-dimensional 86 et seq
multi-dimensional 94 et seq
Fraunhofer diffraction theory 38
et seq
two-dimensional 90 et seq
frequency
angular 10
fundamental 2
modulation 65
spectrum 33, 64
functions
aperture 40
circ 89
disk 89
Gaussian 13
sawtooth 18, 35, 37
top-hat 11
fundamental 2
FWHM (Fullwidth at half maximum) 13,
14, 15, 47, 83
Jacobi expansion 66, 128
jinc-function 89
Johnson noise 60
lifetime of an excited state 83
Lorentz profile 14, 82, 83, 84
matched filter theorem 62
Maxwellian velocity distribution 83
Michelson harmonic integrator 72
Michelson interferometer 77
Miller indices 108
modular tranfroms 109
modulating signal 63
modulation
amplitude 64
frequency 65
index 66
pulse height 68
pulse width 68
pulse position 68
multiplex advantage 79Index 135
multiplex transmission 69
multiplexing
time- 69
frequency 69
sine-function 12, 13, 45, 46, 79
-defined 12
slice theorem 99
spectral power density (SPD) 11, 23, 59, 60,
62, 78, 81, 82
spectrometer, perfect 23
spectrum
energy 10
lines, shapes of 81
power 10
spike 101
square-wave 5, 17
Stratton, harmonic integrator 72
superposition of planes 97
symmetric parts 109 et seq
symmetry 109
anti- 109, 114
nail 101
noise 60 et seq
Johnson 60
semi-conductor 61
white 60
photon shot- 61
Nyquist frequency 32
oblique incidence 41
orthogonality
of sines and cosines 4
of Bessel functions 128
overtones 1
temperature broadening 83
theorems
addition 22
cardinal, of interpolary function theory 32
convolution 26
convolution derivative 31
derivative 30
interpolation 34
inversion 9
matched filter 62
Parseval’s 32, 127
power 32
Rayleigh’s 32, 88, 127
sampling 32 et seq, 79
similarity 35
shift 16, 22
tomography, computer axial 97
top-hat function 11
transition probability 83
triangle function 28
twiddle factors 120
Parseval’s theorem 32, 127
periodic errors 66 et seq
phase 6
and coherence 53
-angle 7, 54
-change 38, 52
-delay 52
-difference 7, 40, 55
transform 109
point-spread function 24
polar coordinates 87
diagrams 52
power spectrum 10, 11
theorem 32
projection function 97
slice theorem 99
pulse train, passage through a
filter 72
Radon transform 97, 99
Rayleigh criterion 92
Rayleigh’s theorem 32, 88, variables
abstract 9
conjugate 10
physical 10
visibility, of fringes 55
Voigt profile 84
separation of components 84
voltage step, passage through a filter 73
127
reciprocal lattice 108
rect function 11
resolution, grating 46
Rowland ghosts 68
sampling 69, 78
theorem 32 et seq, 79
saw-tooth wave 18, 35
serial link 69
shah (=III)-function 17
shift theorem 16, 22
signal analysis 58
signal/noise ratio 62
similarity theorem 35
wavenumber 76
Weierstrass’ function 20
Whitaker’s interpolary function theory 34
Wiener-Khinchine theorem 29, 59, 60, 84
Yagi aerial 53
Young’s slits 43, 55
Index
addition theorem 22, 58
Airy disc 24, 89, 91
aliasing 33
amplitude 6
of the harmonics 4
diffracted 40
modulation 64
in Fourier transforms 80
analytic expansion 5
analytic signal 40, 61
wave vector 54
angular frequency 10
angular measure 10
annulus 89
antenna theory 52
antisymmetric 19, 109 et seq
aperture function 40
grating 45 et seq, 66
apodisation 48, 80
apodising mask 48 et seq
Argand plane 11, 74
associative rule 25
autocorrelation theorem 28
cardinal theorem, interpolation 32
Cartesian coordinates 86
Cauchy’s integral formula 74
circular symmetry 87, 129
coherence 53
partial 55
communication channel 58
commutative rule 25
complex exponentials 7
computerized axial tomography 97
conjugate variables 10
Connes’ apodising function 80
convolution 17, 23
of two Gaussians 27
theorem 26
corollary 110
derivative theorem 31
convolutions 23 et seq
algebra of 25, 29
examples of 26
damped oscillator 81
deconvolution 47
De l’Hopital’s rule 6
De Moivre’s theorem 7, 129
delta function 15
derivative theorem 30
diffraction
Fraunhofer 38 et seq
grating 45
intensity distribution 46
resolution 46
single slit 42
three slit 44
two slit 44
dipole radiation 81, 83
Dirac comb 17, 32, 33, 36, 37, 45, 51, 66, 84,
bandwidth, channel 65
BASIC program for FFT 122
baud-rate 65
Beer’s law 100
Bessel functions 66, 87
integral expansion 128
Jacobi expansion 128
Bessel’s equation 129
bit-reversed order 120
blaze angle 51, 52
blaze wavelength 51
blazing of diffraction gratings
52
box-car function 11
Breit-Wigner formula 83
117
bed of nails 104 et seq
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