傅立叶分析和应用Fourier analysis and applications : filtering, numerical computation, wavelets
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作 者：Claude Gasquet，Robert D. Ryan 著
分类号：
ISBN：9780387984858
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简介
This applied mathematic text focuses on Fourier analysis, filters and signal analysis. Scientists and engineers are confronted by the necessity of using classical mathematics such as Fourier transforms, convolution, distribution and more recently wavelet analysis in all areas of modelling. The object of this book is two-fold - on the one hand to convey to the mathematical reader a rigorous presentation and exploration of the important applications of analysis leading to numerical calculations and on the other hand to convey to the physics reader a body of theory in which the well-known formulae find their justification. The reader will find the basic study of fundamental notions such as Lebesgue integration and theory of distribution and these permit the establishment of the following areas: Fourier analysis and convolution Filters and signal analysis time-frequency analysis (gabor transforms and wavelets) The book is aimed at engineers and scientists and contains a large number of exercises as well as selected worked out solutions. The words `Translated by Robert D Ryan' should be included in ALL promotion material regarding the book.
目录
Translator's Preface v (2)
Preface to the French Edition vii
Chapter Ⅰ Signals and Systems
Lesson 1 Signals and Systems
1.1 General considerations
1.2 Some elementary signals
1.3 Examples of systems
Lesson 2 Filters and Transfer Functions
2.1 Algebraic properties of systems
2.2 Continuity of a system
2.3 The filter and its transfer function
2.4 A standard analog filter: the RC cell
2.5 A first-order discrete filter
Chapter Ⅱ Periodic Signals
Lesson 3 Trigonometric Signals
3.1 Trigonometric polynomials
3.2 Representation in since and consincs
3.3 Orthogonality
3.4 Exercises
Lesson 4 Periodic Signals and Fourier Series
4.1 The space L(2)(p)(0, a)
4.2 The idea of approximation
4.3 Convergence of the approximation
4.4 Fourier coefficients of real, odd, and even functions
4.5 Formulary
4.6 Exercises
Lesson 5 Pointwise Representation
5.1 The Riemann-Lebesgue theorem
5.2 Pointwise convergence?
5.3 Uniform convergence of Fourier series
5.4 Exercises
Lesson 6 Expanding a Function in an Orthogonal Basis
6.1 Fourier series expansions on a bounded interval
6.2 Expansion of a function in an orthogonal basis
6.3 Exercises
Lesson 7 Frequencies, Spectra, and Scales
7.1 Frequencies and spectra
7.2 Variations on the scale
7.3 Exercises
Chapter Ⅲ The Discrete Fourier Transform and Numerical Computations
Lesson 8 The Discrete Fourier Transform
8.1 Computing the Fourier coefficients
8.2 Some properties of the discrete Fourier transform
8.3 The Fourier transform of real data
8.4 A relation between the exact and approximate Fourier coefficients
8.5 Exercises
Lesson 9 A Famous, Lightning-Fast Algorithm
9.1 The Cooley-Tukey algorithm
9.2 Evaluating the cost of the algorithm
9.3 The mirror permutation
9.4 A recursive program
9.5 Exercises
Lesson 10 Using a FFT for Numerical Computations
10.1 Computing a periodic convolution
10.2 Nonperiodic convolution
10.3 Computations on high-order polynomials
10.4 Polynomial interpolation and the chebyshev basis
10.5 Exercises
Chapter Ⅳ The Lebesgue Integral
Chapter Ⅴ Spaces
Chapter Ⅵ Convolution and the Fourier Transform of Functions
Chapter Ⅶ Analog Filters
Chapter Ⅷ Distributions
Chapter Ⅹ Filters and Distributions
Chapter Ⅸ Convolution and the Fourier Transform of Distributions
Chapter Ⅺ Sampling and Discrete Filters
Chapter Ⅻ Current Trends: Time-Frequency Analysis
References
Index
Preface to the French Edition vii
Chapter Ⅰ Signals and Systems
Lesson 1 Signals and Systems
1.1 General considerations
1.2 Some elementary signals
1.3 Examples of systems
Lesson 2 Filters and Transfer Functions
2.1 Algebraic properties of systems
2.2 Continuity of a system
2.3 The filter and its transfer function
2.4 A standard analog filter: the RC cell
2.5 A first-order discrete filter
Chapter Ⅱ Periodic Signals
Lesson 3 Trigonometric Signals
3.1 Trigonometric polynomials
3.2 Representation in since and consincs
3.3 Orthogonality
3.4 Exercises
Lesson 4 Periodic Signals and Fourier Series
4.1 The space L(2)(p)(0, a)
4.2 The idea of approximation
4.3 Convergence of the approximation
4.4 Fourier coefficients of real, odd, and even functions
4.5 Formulary
4.6 Exercises
Lesson 5 Pointwise Representation
5.1 The Riemann-Lebesgue theorem
5.2 Pointwise convergence?
5.3 Uniform convergence of Fourier series
5.4 Exercises
Lesson 6 Expanding a Function in an Orthogonal Basis
6.1 Fourier series expansions on a bounded interval
6.2 Expansion of a function in an orthogonal basis
6.3 Exercises
Lesson 7 Frequencies, Spectra, and Scales
7.1 Frequencies and spectra
7.2 Variations on the scale
7.3 Exercises
Chapter Ⅲ The Discrete Fourier Transform and Numerical Computations
Lesson 8 The Discrete Fourier Transform
8.1 Computing the Fourier coefficients
8.2 Some properties of the discrete Fourier transform
8.3 The Fourier transform of real data
8.4 A relation between the exact and approximate Fourier coefficients
8.5 Exercises
Lesson 9 A Famous, Lightning-Fast Algorithm
9.1 The Cooley-Tukey algorithm
9.2 Evaluating the cost of the algorithm
9.3 The mirror permutation
9.4 A recursive program
9.5 Exercises
Lesson 10 Using a FFT for Numerical Computations
10.1 Computing a periodic convolution
10.2 Nonperiodic convolution
10.3 Computations on high-order polynomials
10.4 Polynomial interpolation and the chebyshev basis
10.5 Exercises
Chapter Ⅳ The Lebesgue Integral
Chapter Ⅴ Spaces
Chapter Ⅵ Convolution and the Fourier Transform of Functions
Chapter Ⅶ Analog Filters
Chapter Ⅷ Distributions
Chapter Ⅹ Filters and Distributions
Chapter Ⅸ Convolution and the Fourier Transform of Distributions
Chapter Ⅺ Sampling and Discrete Filters
Chapter Ⅻ Current Trends: Time-Frequency Analysis
References
Index
傅立叶分析和应用Fourier analysis and applications : filtering, numerical computation, wavelets
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