Cover
Half-title
Title
Copyright
Contents
Selected Applications
Preface
To the Student
Synopsis
To the Instructor
Acknowledgments
The Mathematical Core
1 Fourier’s representation for functions on R, Tp, Z and Pn
1.1 Synthesis and analysis equations
Introduction
Functions on R
Functions on Tp
Functions on Z
Functions on PN
Summary
1.2 Examples of Fourier’s representation
Introduction
The Hipparchus–Ptolemy model of planetary motion
Gauss and the orbits of the asteroids
Fourier and the .ow of heat
Fourier’s representation and LTI systems
Schoenberg’s derivation of the Tartaglia–Cardan formulas
Fourier transforms and spectroscopy
1.3 The Parse validentities and related results
The Parseval identities
The Plancherel identities
Orthogonality relations for the periodic complex exponentials
Bessel’s inequality
The Weierstrass approximation theorem
A proof of Plancherel’s identity for functions on Tp
1.4 The Fourier–Poisson cube
Introduction
Discretization by h -sampling
Periodization by p -summation
The Poisson relations
The Fourier–Poisson cube
1.5 The validity of Fourier’s representation
Introduction
Functions on PN
Absolutely summable functions on Z
Continuous piecewise smooth functions on Tp
The sawtooth singularity function on T1
The Gibbs phenomenon for w0
Piecewise smooth functions on Tp
Smooth functions on R with small regular tails
Singularity functions on R
Piecewise smooth functions on R with small regular tails
Extending the domain of validity
Further reading
Exercises
2 Convolution of functions on R, Tp, Z, and Pn
2.1 Formal definitions of…
Correlation and conjugation are closely related
2.2 Computation of fg
Direct evaluation
The sum of scaled translates
The sliding strip method
Generating functions
2.3 Mathematical properties of the convolution product
Introduction
The Fourier transform of…
Algebraic structure
Translation invariance
Differentiation of…
2.4 Examples of convolution and correlation
Convolution as smearing
Echo location
Convolution and probability
Convolution and arithmetic
Further reading
Exercises
3 Thecalculus for finding Fourier transforms of functions on R
3.1 Using the definition to find Fourier transforms
Introduction
The box function
The Heaviside step function
The truncated decaying exponential
The unit gaussian
Summary
3.2 Rules for finding Fourier transforms
Introduction
Linearity
Translation and modulation
Dilation
Inversion
Convolution and multiplication
Summary
3.3 Selected applications of the Fourier transform calculus
Evaluation of integrals and sums
Evaluation of convolution products
The Hermite functions
Smoothness and rates of decay
Further reading
Exercises
4 The calculus for finding Fourier transforms of functions on Tp, Z, and Pn
4.1 Fourier series
Introduction
Direct integration
Elementary rules
Poisson’s relation
Bernoulli functions and Eagle’s method
Laurent series
Dilation and grouping rules
4.2 Selected applications of Fourier series
Evaluation of sums and integrals
The polygon function
Rates of decay
Equidistribution of arithmetic sequences
4.3 Discrete Fourier transforms
Direct summation
Basic rules
Dilation
Poisson’s relations
4.4 Selected applications of the DFT calculus
The Euler–Maclaurin sum formula
The discrete Fresnel function
Further reading
Exercises
5 Operator identities associated with Fourier analysis
5.1 The concept of an operator identity
Introduction
Operators applied to functions on PN
Blanket hypotheses
5.2 Operators generated by powers of F
Powers of F
The even and odd projection operators
The normalized exponential transform operators
The normalized cosine transform and sine transform operators
The normalized Hartley transform operators
Connections
Tag notation
5.3 Operators related to complex conjugation
The bar and dagger operators
The real, imaginary, hermitian, and antihermitian projection operators
Symmetric functions
Symmetric operators
5.4 Fourier transforms of operators
The basic definition
Algebraic properties
5.5 Rules for Hartley transforms
5.6 Hilbert transforms
Defining relations
Operator identities
The Kramers–Kronig relations
Further reading
Exercises
6 The fast Fourier transform
6.1 Pre-FFT computation of the DFT
Introduction
Horner’s algorithm for computing the DFT
Other pre-FFT methods for computing the DFT
How big is 4N2?
The announcement of a fast algorithm for the DFT
6.2 Derivation of the FFT via DFT rules
Decimation-in-time
Decimation-in-frequency
Recursive algorithms
6.3 The bit reversal permutation
Introduction
A naive algorithm
The reverse carry algorithm
The Bracewell–Buneman algorithm
6.4 Sparse matrix factorization of F when N=2m
Introduction
The zipper identity
Exponent notation
Sparse matrix factorization of F
The action of B2m
An FFT algorithm
An alternative FFT algorithm
Precomputation of…
Application of Q4M
6.5 Sparse matrix factorization of H when N=2m
The zipper identity and factorization
Application of T4M using precomputed…
6.6 Sparse matrix factorization of F whenN=P1P2···Pm
Introduction
The zipper identity for FMP
Factorization of…
An FFT
The permutation…
The permutation…
Closely related factorizations of F,H
6.7 Kronecker product factorization of F
Introduction
The Kronecker product
Rearrangement of Kronecker products
Parallel and vector operations
Stockham’s autosort FFT
Further reading
Exercises
7 Generalized functions on R
7.1 The concept of a generalized function
Introduction
Functions and functionals
Schwartz functions
Functionals for generalized functions
7.2 Common generalized functions
Introduction
The comb III
The functions…
Summary
7.3 Manipulation of generalized functions
Introduction
The linear space G
Translate, dilate, derivative, and Fourier transform
Reflection and conjugation
Multiplication and convolution
7.4 Derivatives and simple differential equations
Differentiation rules
Derivatives of piecewise smooth functions with jumps
Solving differential equations
7.5 The Fourier transform calculus for generalized functions
Fourier transform rules
Basic Fourier transforms
Support- and bandlimited generalized functions
7.6 Limits of generalized functions
Introduction
The limit concept
Transformation of limits
7.7 Periodic generalized functions
Fourier series
The analysis equation
Convolution of p-periodic generalized functions
Discrete Fourier transforms
Connections
7.8 Alternative definitions for generalized functions
Functionals on S
Other test functions
Further reading
Exercises
Selected Applications
8 Sampling
8.1 Sampling and interpolation
Introduction
Shannon’s hypothesis
8.2 Reconstruction of f from its samples
A weakly convergent series
The cardinal series
Recovery of an alias
Fragmentation of…
8.3 Reconstruction of f from samples of a1f,a2*f,...
Filters
Samples from one filter
The Papoulis generalization
8.4 Approximation of almost bandlimited functions
Further reading
Exercises
9 Partial differential equations
9.1 Introduction
9.2 The wave equation
A physical context: Plane vibration of a taut string
The wave equation on R
The wave equation on Tp
Each point on the string vibrates with the frequency
9.3 The diffusion equation
A physical context: Heat .ow along a long rod
The diffusion equation on R
The diffusion equation on Tp
9.4 The diffraction equation
A physical context: Diffraction of a laser beam
The diffraction equation on R
The diffraction equation on Tp
9.5 Fast computation of frames for movies
Further reading
Exercises
10 Wavelets
10.1 The Haar wavelets
Interpretation of F[m,k]
Arbitrarily good approximation
Successive approximation
Coded approximation
10.2 Support-limited wavelets
The dilation equation
Smoothness constraints
Order of approximation
Orthogonality constraints
Daubechies wavelets
10.3 Analysis and synthesis with Daubechies wavelets
Coefficients for frames and details
The operators…
Samples for frames and details
The operators…
10.4 Filter banks
Introduction
Factorization of…
Fourier analysis of a filter bank
Perfect reconstruction .lter banks
Compression and reconstruction
Further reading
Exercises
11 Musical tones
11.1 Basic concepts
Introduction
Perception of pitch and loudness
Ohm’s law
Scales
Musical notation
11.2 Spectrograms
Introduction
The computation
Slowly varying frequencies
11.3 Additive synthesis of tones
Introduction
Amplitude envelopes
Synthesis of a bell tone
Synthesis of a brass tone
11.4 FM synthesis of tones
Introduction
The spectral decomposition
Dynamic spectral enrichment
11.5 Synthesis of tones from noise
Introduction
White noise
Filtered noise
11.6 Music with mathematical structure
Introduction
Transformation of frequency functions
Risset’s endless glissando
Further reading
Exercises
12 Probability
12.1 Probability density functions on R
Introduction
Generalized probability densities
12.2 Some mathematical tools
Introduction
Gaussian molli.cation and tapering
Fundamental inequalities
12.3 The characteristic function
F is bounded and continuous
Bochner’s characterization of F
Products of characteristic functions
Periodic characteristic functions
12.4 Random variables
Probability integrals
Expectation integrals
Functions of a random variable
The uncertainty relation
12.5 The central limit theorem
Sums of independent random variables
The ubiquitous bell curve
The law of errors
Further reading
Exercises
Appendices
Appendix 1 The impact of Fourier analysis
Appendix 2 Functions and their Fourier transforms
Appendix 3 The Fourier transform calculus
Appendix 4 Operators and their Fourier transforms
Appendix 5 The Whittaker–Robinson flow chart for harmonic analysis
Appendix 6 FORTRAN code for a radix 2 FFT
Appendix 7 The standard normal probability distribution
Appendix 8 Frequencies of the piano keyboard
Index