小波大纲(英文)

1023080 The Outline of Wavelet Analysis Teaching
Course Code：1023080
Period：36（Prelect: 26，Experiment: 10）
Credit hour：2 credit hours
Preceding Course：Mathematics Aalysis；High Algebra Real Function analysis, Functional
Analysis.
Applying Speciality：Mathematics & Applying Mathematics
Teaching Department：the Department of mathematics
ⅰ.Property & Assignment
Wavelet Analysis is a special and elective course of mathematics. Different from Fourier analysis, the development of wavelets has been much more recent, and the subject of wavelets has become a popular tool in signal analysis and other areas of applications only within the last two decades or so. Our goal with this course is to present many of the essential ideas behind Fourier analysis and wavelets, along with some of their applications to signal analysis, to an audience of advanced undergraduate mathematics majors.
Part 1 Fourier series
1. Main content:
(1) Introduction
(2) Computation of Fourier series
(3) Convergence Theorems for Fourier Transform
2．Basic request:
Understand the trigonometric expansion of a function, be able to compute the Fourier coefficients of the continue function. Understand the convergence theorems for Fourier series. Part 2 Fourier Transform
1. Main content:
(1) Informal Development of the Fourier Transform
(2) Properties of the Fourier Transform
(3) Linear Filters
(4) The Sampling TheoremKK
(5) The Uncertainty Principle
2．Basic request:
Understand the definition and the properties of the Fourier Transform, including the convolution and the adjoint of the Fourier transform. Understand the linear filters , the sampling theorem and the uncertainty principle.
Part 3 Discrete Fourier analysis
1. Main content:
(1) The Discrete Fourier Transform
(2) Discrete Signals
2．Basic request:
Understand the discrete Fourier Transform and the discrete signal analysis. Part 4 Haar Wavelet Analysis
1. Main content:
(1) Why Wavelets
(2) Haar Wavelets
(3) Haar Decomposition and Reconstruction Algorithms
2．Basic request:
Master the basic properties of the Haar scaling function and the Haar Wavelet function. Understand the decomposition and reconstruction algorithms of Haar Wavelet.
Part 5 Multiresolution Analysis
1. Main content:
(1) The Multiresolution Framework
(2) Implementing Decomposition and Reconstruction
(3) Fourier Transform Criteria
2．Basic request:
Understand the Mallat algorithms of decomposition and reconstruction, and the processing of a signal. Know the criteria of Fourier transform.
Part 6 The Daubechies Wavelets
1. Main content:
(1) Daubechies’s Construction
(2) Classification , Moments, and Smoothness
(3) Computational Issues
(4) The Scaling Function at Dyadic Points
2．Basic request:
Understand the basic idea of Daubechies wavelet construction processing. Know the classification, aments and smoothness.
ⅲ. Assignment of the Period
ⅳ. Interpretation of the outline
During the teaching, we can change the period of teaching for taking the circumstances into consideration.
ⅴ. Teaching material and bibliography
1. Teaching material:
《A First Course in Wavelets with Fourier Analysis》， [USA]Albert Boggess, Frances J.Narcowich，Publishing House of Electronics Industry, 2002
2. Bibliography:
（1）《实用小波分析》，刘涛等编，国防工业出版社，2004年。

（2）《小波分析算法与应用》，程正兴编，西安交通大学出版社，2004年；
（3）《Ten Lectures of Wavelet》，I.Dawbechies 编, SIAM出版社，1992年；
（4）《小波变换的工程分析与应用》，杨福生编，科学出版社，2003年。

制定：函数论课程组审定人：郭运瑞
批准人：陈付贵。