# Book:Ronald N. Bracewell/The Fourier Transform and its Applications/Second Edition

## Ronald N. Bracewell: The Fourier Transform and its Applications

Published $\text {1978}$, McGraw-Hill

ISBN 07-066196-0.

### Contents

Preface to the Second Edition
Preface to the First Edition
Chapter 1 Introduction
Chapter 2 Groundwork
The Fourier transform and Fourier's integral theorem
Conditions for the existence of Fourier transforms
Transforms in the limit
Oddness and evenness
Significance of oddness and evenness
Complex conjugates
Cosine and sine transforms
Interpretation of the formulas
Problems
Bibliography
Chapter 3 Convolution
Examples of convolution
Serial products
Inversion of serial multiplication / The serial product in matrix notation / Sequences as vectors
The autocorrelation function
Pentagram notation for cross correlation
The energy spectrum
Appendix
Problems
Chapter 4 Notation for Some Useful Functions
Rectangle function of unit height and base, $\map \Pi x$
The triangle function of unit height and area, $\map \Lambda x$
Various exponentials and Gaussian and Rayleigh curves
Heaviside's unit step function, $\map H x$
The sign function, $\map \sgn x$
The filtering or interpolating function, $\map {\operatorname {sinc} } x$
Pictorial representation
Summary of special symbols
Chapter 5 The Impulse Symbol
The sifting property
The sampling or replicating symbol $\map {\operatorname {III} } x$
The even and odd impulse pairs, $\map {\operatorname {II} } x$ and $\map {\operatorname {I_I} } x$
Derivatives of the impulse symbol
Null functions
Some functions in two and more dimensions
The concept of generalized function
Particularly well-behaved functions / Regular sequences / Generalized functions / Algebra of generalized functions / Differentiation of ordinary functions
Problems
Chapter 6 The Basic Theorems
A few transforms for illustration
Similarity theorem
Shift theorem
Modulation theorem
Convolution theorem
Rayleigh's theorem
Power theorem
Autocorrelation theorem
Derivative theorem
Derivative of a convolution integral
The transform of a generalized function
Proofs of theorems
Addition theorem / Similarity and shift theorems / Derivative theorem / Power theorem
Summary of theorems
Problems
Chapter 7 Doing Transforms
Integration in closed form
Numerical Fourier transformation
Generation of transforms by theorems
Application of the derivative theorem to segmented functions
Chapter 8 The Two Domains
Definite integral
The first moment
Centroid
Moment of inertia (second moment)
Moments
Mean-square abscissa
Variance
Smoothness and compactness
Smoothness under convolution
Asymptotic behavior
Equivalent width
Autocorrelation width
Mean-square widths
Some inequalities
Upper limits to ordinate and slope / Schwarz's inequality
The uncertainty relation
Proof of uncertainty relation / Example of uncertainty relation
The finite difference
Running means
Central-limit theorem
Summary of correspondences in the two domains
Problems
Chapter 9 Electrical Waveforms, Spectra, and Filters
Electrical waveforms and spectra
Filters
Interpretation of theorems
Similarity theorem / Addition theorem / Shift theorem / Modulation theorem / Converse of modulation theorem
Linearity and time invariance
Problems
Chapter 10 Sampling and Series
Sampling theorem
Interpolation
Rectangular filtering
Undersampling
Ordinate and slope sampling
Interlaced sampling
Sampling in the presence of noise
Fourier series
Gibbs phenomenon / Finite Fourier transforms / Fourier coefficients
The shah symbol is its own Fourier transform
Problems
Chapter 11 The Laplace Transform
Convergence of the Laplace integral
Theorems for the Laplace transform
Transient-response problems
Laplace transform pairs
Natural behavior
Impulse response and transfer function
Initial-value problems
Setting out initial-value problems
Switching problems
Problems
Chapter 12 Relatives of the Fourier Transform
The two-dimensional Fourier transform
Two-dimensional convolution
The Hankel transform
Fourier kernels
The three-dimensional Fourier transform
The Hankel transform in $n$ dimensions
The Mellin transform
The $z$ transform
The Abel transform
The Hilbert transform
The analytic signal / Instantaneous frequency and envelope / Causality
Problems
Chapter 13 Antennas
One-dimensional apertures
Analogy with waveforms and spectra
Beam width and aperture width
Beam swinging
Arrays of arrays
Interferometers
Physical aspects of the angular spectrum
Two-dimensional theory
Problems
Chapter 14 Television Image Formation
The convolution relation
Test procedure by response to point source
Testing by frequency response
Equalization
Edge response
Raster sampling
Problems
Chapter 15 Convolution in Statistics
Distribution of a sum
Consequences of the convolution relation
The characteristic function
The truncated exponential distribution
The Poisson distribution
Problems
Chapter 16 Noise Waveforms
Discrete representation by random digits
Filtering a random input: effect on amplitude distribution
Digression on independence / The convolution relation
Effect on autocorrelation
Effect on spectrum
Spectrum of random input / The output spectrum
Some noise records
Envelope of bandpass noise
Detection of a noise waveform
Measurement of noise power
Problems
Chapter 17 Heat Conduction and Diffusion
One-dimensional diffusion
Gaussian diffusion from a point
Diffusion of a spatial sinusoid
Sinusoidal time variation
Problems
Chapter 18 The Discrete Fourier Transform
The discrete transform formula
Cyclic convolution
Examples of discrete Fourier transforms
Reciprocal property
Oddness and evenness
Examples with special symmetry
Complex conjugates
Reversal property
Shift theorem
Convolution theorem
Product theorem
Cross-correlation
Autocorrelation
Sum of sequence
First value
Generalized Parseval-Rayleigh theorem
Packing theorem
Similarity theorem
The fast Fourier transform
Practical considerations
Is the discrete Fourier transform correct?
Applications of the FFT
Two-dimensional data
Power spectra
Chapter 19 Pictorial Dictionary of Fourier Transforms
Chapter 20 Supplementary Problems
Chapter 21 Tables
Index

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