Handbook of Fourier Analysis & Its Applications

Name: Handbook of Fourier Analysis & Its Applications
Brand: OUP eBook
Price: 124.99 EUR
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Robert J. Marks II(Autor*in)

OUP eBook (Verlag)

Erschienen am 8. Januar 2009

799 Seiten

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978-0-19-804430-7 (ISBN)

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Inhalt

Intro
Contents
Preface
Acronym List
Notation
1 Introduction
1.1 Ubiquitous Fourier Analysis
1.2 Jean Baptiste Joseph Fourier
1.3 This Book
1.3.1 Surveying the Contents
2 Fundamentals of Fourier Analysis
2.1 Introduction
2.2 Signal Classes
2.3 The Fourier Transform
2.3.1 The Continuous Time Fourier Transform
2.3.2 The Fourier Series
2.3.3 Relationship Between Fourier and Laplace Transforms
2.3.4 Some Continuous Time Fourier Transform Theorems
2.3.5 Some Continuous Time Fourier Transform Pairs
2.3.6 Other Properties of the Continuous Time Fourier Transform
2.4 Orthogonal Basis Functions¶
2.4.1 Parseval's Theorem for an Orthonormal Basis¶
2.4.2 Examples¶
2.5 The Discrete Time Fourier Transform
2.5.1 Relation of the DFT to the Continuous Time Fourier Transform
2.6 The Discrete Fourier Transform
2.6.1 Circular Convolution
2.6.2 Relation to the Continuous Time Fourier Transform
2.6.3 DFT Leakage
2.7 Related Transforms
2.7.1 The Cosine Transform?
2.7.2 The Sine Transform?
2.7.3 The Hartley Transform
2.7.4 The z Transforms
2.8 Exercises
2.9 Solutions for Selected Chapter 2 Exercises
3 Fourier Analysis in Systems Theory
3.1 Introduction
3.2 System Classes
3.2.1 System Types
3.2.2 Example Systems
3.3 System Characterization
3.3.1 Linear System Characterization
3.3.2 Causal Linear Systems
3.3.3 Linear Time Invariant (LTI) Systems
3.4 Amplitude Modulation
3.4.1 Coherent Demodulation
3.4.2 Envelope Demodulation
3.5 Goertzel's Algorithm for Computing the DFT
3.6 Fractional Fourier Transforms
3.6.1 Periodicity of the Fourier Transform Operator
3.6.2 Fractional Fourier Transform Criteria
3.6.3 The Weighted Fractional Fourier Transform
3.7 Approximating a Linear System with LTI Systems
3.7.1 Examples of the Piecewise Invariant Approximation
3.8 Exercises
3.9 Solutions for Selected Chapter 3 Exercises
4 Fourier Transforms in Probability, Random Variables and Stochastic Processes
4.1 Introduction
4.2 Random Variables
4.2.1 Probability Density Functions, Expectation, and Characteristic Functions
4.2.2 Example Probability Functions, Characteristic Functions, Means and Variances
4.2.3 Distributions of Sums of Random Variables
4.2.4 The Law of Large Numbers
4.2.5 The Central Limit Theorem
4.3 Uncertainty in Wave Equations
4.4 Stochastic Processes
4.4.1 First and Second Order Statistics
4.4.2 Power Spectral Density
4.4.3 Some Stationary Noise Models
4.4.4 Linear Systems with Stationary Stochastic Inputs
4.4.5 Properties of the Power Spectral Density
4.5 Exercises
4.6 Solutions for Selected Chapter 4 Exercises
5 The Sampling Theorem
5.1 Introduction
5.1.1 The Cardinal Series
5.1.2 History
5.2 Interpretation
5.3 Proofs
5.3.1 Using Comb Functions
5.3.2 Fourier Series Proof
5.3.3 Papoulis' Proof
5.4 Properties
5.4.1 Convergence
5.4.2 Trapezoidal Integration
5.4.3 The Time-Bandwidth Product
5.5 Application to Spectra Containing Distributions
5.6 Application to Bandlimited Stochastic Processes
5.7 Exercises
5.8 Solutions for Selected Chapter 5 Exercises
6 Generalizations of the Sampling Theorem
6.1 Introduction
6.2 Generalized Interpolation Functions
6.2.1 Oversampling
6.2.2 Restoration of Lost Samples
6.2.3 Criteria for Generalized Interpolation Functions
6.2.4 Reconstruction from a Filtered Signal's Samples
6.3 Papoulis' Generalization
6.3.1 Derivation
6.3.2 Interpolation Function Computation
6.3.3 Example Applications
6.4 Derivative Interpolation
6.4.1 Properties of the Derivative Kernel
6.5 A Relation Between the Taylor and Cardinal Series
6.6 Sampling Trigonometric Polynomials
6.7 Sampling Theory for Bandpass Functions
6.7.1 Heterodyned Sampling
6.7.2 Direct Bandpass Sampling
6.8 A Summary of Sampling Theorems for Directly Sampled Signals
6.9 Lagrangian Interpolation
6.10 Kramer's Generalization
6.11 Exercises
6.12 Solutions for Selected Chapter 6 Exercises
7 Noise and Error Effects
7.1 Introduction
7.2 Effects of Additive Data Noise
7.2.1 On Cardinal Series Interpolation
7.2.2 Interpolation Noise Variance for Directly Sampled Signals
7.2.3 On Papoulis' Generalization
7.2.4 On Derivative Interpolation
7.2.5 A Lower Bound on the NINV
7.3 Jitter
7.4 Filtered Cardinal Series Interpolation
7.4.1 Unbiased Interpolation from Jittered Samples
7.4.2 Effects of Jitter In Stochastic Bandlimited Signal Interpolation
7.5 Truncation Error
7.5.1 An Error Bound
7.6 Exercises
7.7 Solutions for Selected Chapter 7 Exercises
8 Multidimensional Signal Analysis
8.1 Introduction
8.2 Notation
8.3 Visualizing Higher Dimensions
8.3.1 N Dimensional Tic Tac Toe
8.3.2 Vectorization
8.4 Continuous Time Multidimensional Fourier Analysis
8.4.1 Linearity
8.4.2 The Shift Theorem
8.4.3 Multidimensional Convolution
8.4.4 Separability
8.4.5 Rotation, Scale and Transposition
8.4.6 Fourier Transformation of Circularly Symmetric Functions
8.5 Characterization of Signals from their Tomographic Projections
8.5.1 The Abel Transform and Its Inverse
8.5.2 The Central Slice Theorem
8.5.3 The Radon Transform and Its Inverse
8.6 Fourier Series
8.6.1 Multidimensional Periodicity
8.6.2 The Multidimensional Fourier Series Expansion
8.6.3 Multidimensional Discrete Fourier Transforms
8.7 Discrete Cosine Transform-Based Image Coding
8.7.1 DCT Basis Functions
8.7.2 The DCT in Image Compression
8.8 McClellan Transformation for Filter Design
8.8.1 Modular Implementation of the McClellan Transform
8.8.2 Implementation Issues
8.9 The Multidimensional Sampling Theorem
8.9.1 The Nyquist Density
8.9.2 Generalized Interpolation Functions
8.10 Restoring Lost Samples
8.10.1 Restoration Formulae
8.10.2 Noise Sensitivity
8.11 Periodic Sample Decimation and Restoration
8.11.1 Preliminaries
8.11.2 First Order Decimated Sample Restoration
8.11.3 Sampling Below the Nyquist Density
8.11.4 Higher Order Decimation
8.12 Raster Sampling
8.12.1 Bandwidth Equivalence of Line Samples
8.13 Exercises
8.14 Solutions for Selected Chapter 8 Exercises
9 Time-Frequency Representations
9.1 Introduction
9.2 Short Time Fourier Transforms and Spectrograms
9.3 Filter Banks
9.3.1 Commonly Used Windows
9.3.2 Spectrograms
9.3.3 The Mechanics of Short Time Fourier Transformation
9.3.4 Computational Architectures
9.4 Generalized Time-Frequency Representations
9.4.1 GTFR Mechanics
9.4.2 Kernel Properties
9.4.3 Marginals
9.4.4 Example GTFR's
9.5 Exercises
9.6 Solutions for Selected Chapter 9 Exercises
10 Signal Recovery
10.1 Introduction
10.2 Continuous Sampling
10.3 Interpolation From Periodic Continuous Samples
10.3.1 The Restoration Algorithm
10.3.2 Observations
10.4 Interpolation of Discrete Periodic Nonuniform Decimation
10.4.1 Problem Description
10.4.2 The Periodic Functions, ?[sub(M)](v)
10.4.3 Quadrature Version
10.5 Prolate Spheroidal Wave Functions
10.5.1 Properties
10.5.2 Application to Extrapolation
10.5.3 Application to Interval Interpolation
10.6 The Papoulis-Gerchberg Algorithm
10.6.1 The Basic Algorithm
10.6.2 Proof of the PGA using PSWF's
10.6.3 Remarks
10.7 Exercises
10.8 Solutions for Selected Chapter 10 Exercises
11 Signal and Image Synthesis: Alternating Projections Onto Convex Sets
11.1 Introduction
11.2 Geometical POCS
11.2.1 Geometrical Convex Sets
11.2.2 Projecting onto a Convex Set
11.2.3 POCS
11.3 Convex Sets of Signals
11.3.1 The Hilbert Space
11.3.2 Some Commonly Used Convex Sets of Signals
11.4 Example Applications of POCS
11.4.1 Von Neumann's Alternating Projection Theorem
11.4.2 Solution of Simultaneous Equations
11.4.3 The Papoulis-Gerchberg Algorithm
11.4.4 Howard's Minimum-Negativity-Constraint Algorithm
11.4.5 Associative Memory
11.4.6 Recovery of Lost Image Blocks
11.4.7 Subpixel Resolution
11.4.8 Reconstruction of Images from Tomographic Projections
11.4.9 Correcting Quantization Error for Oversampled Bandlimited Signals
11.4.10 Kernel Synthesis for GTFR's
11.4.11 Application to Conformal Radiotherapy
11.5 Generalizations
11.5.1 Iteration Relaxation
11.5.2 Contractive and Nonexpansive Operators
11.6 Exercises
11.7 Solutions for Selected Chapter 11 Exercises
12 Mathematical Morphology and Fourier Analysis on Time Scales
12.1 Introduction
12.2 Mathematical Morphology Fundamentals
12.2.1 Minkowski Arithmetic
12.2.2 Relation of Convolution Support to the Operation of Dilation
12.2.3 Other Morphological Operations
12.2.4 Minkowski Algebra
12.3 Fourier and Signal Analysis on Time Scales
12.3.1 Background
12.3.2 Fourier Transforms on a Time Scale
12.3.3 The Minkowski Sum of Time Scales
12.3.4 Convolution on a Time Scale
12.3.5 Additively Idempotent Time Scales
12.3.6 Discrete Convolution of AITS'
12.3.7 Multidimensional AITS Time Scales
12.4 Exercises
12.5 Solutions for Selected Chapter 12 Exercises
13 Applications
13.1 The Wave Equation, Its Fourier Solution and Harmony in Western Music
13.1.1 The Wave Equation
13.1.2 The Fourier Series Solution
13.1.3 The Fourier Series and Western Harmony
13.1.4 Pythagorean Harmony
13.1.5 Harmonics Expansions Produce Major Chords and the Major Scale
13.1.6 Fret Calibration
13.2 Fourier Transforms in Optics and Wave Propagation
13.2.1 Scalar Model for Wave Propagation
13.2.2 The Angular Spectrum
13.2.3 Rayleigh-Sommerfield Diffraction
13.2.4 A One Lens System
13.2.5 Beamforming
13.3 Heisenberg's Uncertainty Principle
13.4 Elementary Deterministic Finance
13.4.1 Some Preliminary Math
13.4.2 Compound Interest on a One Time Deposit
13.4.3 Compound Interest With Constant Periodic Deposits
13.4.4 Loan and Mortgage Payments
13.5 Exercises
13.6 Solutions for Selected Chapter 13 Exercises
14 Appendices
14.1 Schwarz's Inequality
14.2 Leibniz's Rule
14.3 A Useful Limit
14.4 Series
14.4.1 Binomial Series
14.4.2 Geometric Series
14.5 Ill-Conditioned Matrices
14.6 Other Commonly Used Random Variables
14.6.1 The Pareto Random Variable
14.6.2 The Weibull Random Variable
14.6.3 The Chi Random Variable
14.6.4 The Noncentral Chi-Squared Random Variable
14.6.5 The Half Normal Random Variable
14.6.6 The Rayleigh Random Variable
14.6.7 The Maxwell Random Variable
14.6.8 The Log Random Variable
14.6.9 The Von Mises Variable
14.6.10 The Uniform Product Variable
14.6.11 The Uniform Ratio Variable
14.6.12 The Logistic Random Variable
14.6.13 The Gibrat Random Variable
14.6.14 The F Random Variable
14.6.15 The Noncentral F Random Variable
14.6.16 The Fisher-Tippett Random Variable
14.6.17 The Gumbel Random Variable
14.6.18 The Student's t Random Variable
14.6.19 The Noncentral Student's t Random Variable
14.6.20 The Rice Random Variable
14.6.21 The Planck's Radiation Random Variable
14.6.22 The Generalized Gaussian Random Variable
14.6.23 The Generalized Cauchy Random Variable
15 References
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

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