A Student’s Guide to Fourier Transforms with Applications in Physics and Engineering – J. M. James – 2nd Edition

Preface to the first edition
Preface to the second edition
Preface to the third edition
1.1 The qualitative approach
1.2 Fourier series
1.3 The amplitudes of the harmonics
1.4 Fourier transforms
1.5 Conjugate variables
1.7.1 The `top-hat' function
1.7.2 The sinc-function
1.7.3 The Gaussian function
1.7.4 The exponential decay
1.7.5 The Dirac `delta-function'
1.7.7 The Dirac comb
1.8 Worked examples
2.1 The Dirichlet conditions
2.3 Convolutions and the convolution theorem
2.3.1 The convolution theorem
2.3.2 Examples of convolutions
2.3.3 The autocorrelation theorem
2.4.1 Convolution of two delta-functions
2.5.1 The derivative theorem
2.5.3 Parseval's theorem
2.5.4 The sampling theorem
2.6 Aliasing
2.6.1 The interpolation theorem
2.7.1 An arithmetical result using Parseval's theorem
2.7.2 Alternating pulse-heights
2.7.4 Convolution with a sinusoid
3.1 Fraunhofer diffraction
3.2.1 Single-slit diffraction, normal incidence
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)
3.2.5 The transmission diffraction grating
3.2.6 Apertures with phase-changes instead of amplitude changes
3.2.7 Diffraction at an aperture with a prism
3.2.8 The blazed diffraction grating
3.3 Babinet's principle
3.4 Dipole arrays
3.6 Phase and coherence
3.7 Fringe visibility
3.8 The Michelson stellar interferometer
3.9 The van Cittert-Zernike theorem
4.1 Communication channels
4.1.1 The Wiener-Khinchine theorem
4.2 Noise
4.3 Filters
4.4 The matched filter theorem
4.5 Modulations
4.7.1 The Heaviside step-function
4.7.2 The passage of a voltage step through a `perfect' low-pass filter
4.8 The Gibbs phenomenon
4.8.1 The passage of a train of pulses through a low-pass filter
4.8.2 Passage of a voltage step through a simple high-pass filter
5.2 The Michelson multiplex spectrometer
5.2.1 The theory of the Michelson-Fourier spectrometer
5.3 The shapes of spectrum lines
6.1 Cartesian coordinates
6.2 Polar coordinates
6.3 Theorems
6.4 Examples of two-dimensional Fourier transforms with circular symmetry
6.5.1 Fraunhofer diffraction by a rectangular slot
6.5.2 Fraunhofer diffraction by a circular aperture
6.6 Solutions without circular symmetry
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
9.1 History
9.2 The discrete Fourier transform
9.3 The matrix form of the DFT
9.3.1 Two-dimensional FFTs
9.4.1 A routine for 1024 complex numbers
A.1 Parseval's theorem and Rayleigh's theorem
The Jacobi expansion
The Hankel transform
A.3 Conversion of Fourier-series coefficients into complex exponential form
Bibliography
Index