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

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## Ronald N. Bracewell:

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

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

- ISBN 07-066196-0

### Subject Matter

### 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**Addition 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**Radius of gyration**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**Addition theorem**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*

## Further Editions

## Source work progress

- 1978: Ronald N. Bracewell:
*The Fourier Transform and its Applications*(2nd ed.) ... (previous) ... (next): Chapter $2$: Groundwork: The Fourier transform and Fourier's integral theorem