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

Subject Matter

 * Fourier Analysis

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



Source work progress
* : Chapter $2$: Groundwork: The Fourier transform and Fourier's integral theorem