一、富里埃(Fourier)级数的引进 1定义:设f(x)是(-∞,+∞)上以2元为周期的函数,且f(x)在[-,]上绝对可积,称形如 a+∑( cos nx+ sin nx) 2n=1 的函数项级数为f(x)的 Fourier级数(f(x)的 Fourier展开式)

◆ Background ◆ Preliminary Concepts ◆ Sampling and the Fourier Transform of Sampled Functions ◆ The Discrete Fourier Transform of One Variable ◆ Extension to Functions of Two Variables ◆ Some Properties of the 2-D Discrete Fourier Transform ◆ The Basics of Filtering in the Frequency Domain ◆ Image Smoothing Using Frequency Domain Filters ◆ Image Sharpening Using Frequency Domain Filters ◆ Selective Filtering

§3.1 Discrete-Time Fourier Transform §3.2 DTFT Properties §3.3 Discrete Fourier Transform (DFT) §3.5 Circular Shift of a Sequence §3.6 Circular Convolution §3.7 z-transform §3.8 z-Transform Properties §3.9 Discrete Fourier Transform Computation

2.0 Introduction 2.1 Discrete-Time Signals: Sequences 2.2 Discrete-Time Systems 2.3 Linear Time-Invariant (LTI) Systems 2.4 Properties of LTI Systems 2.5 Linear Constant-Coefficient Difference Equations 2.6 Frequency-Domain Representation of Discrete-Time Signals and systems 2.7 Representation of Sequences by Fourier Transforms 2.8 Symmetry Properties of the Fourier Transform 2.9 Fourier Transform Theorems 2.10 Discrete-Time Random Signals 2.11 Summary

14-8 Definition of the Fourier transform The exponential form of the Fourier series

◆2.1 Discrete-Time Signals: Sequences ◆2.2 Discrete-Time Systems ◆2.3 Linear Time-Invariant (LTI) Systems ◆2.4 Properties of LTI Systems ◆2.5 Linear Constant-Coefficient Difference Equations ◆2.6 Frequency-Domain Representation of Discrete￾Time Signals and systems ◆2.7 Representation of Sequences by Fourier Transforms ◆2.8 Symmetry Properties of the Fourier Transform ◆2.9 Fourier Transform Theorems ◆2.10 Discrete-Time Random Signals

Dirichlet 积分 仔细观察上一节中的几幅图像后可以得到这样的直觉：对于一般 的以2π为周期的函数 f x( )，除了个别点之外（看来是不连续点），当 m → ∞ 时，它的 Fourier 级数的部分和函数序列{ m xS )( }

一、周期为 2L 的周期函数展开成 Fourier 级数 前面我们所讨论的都是以 2为周期的函数 展开成Fourier 级数，但在科技应用中所遇到的 周期函数大都是以 T 为周期，因此我们需要讨论 如何把周期为T = 2 l 的函数展开为Fourier级数 若f ( t )是以T = 2 l 为周期的函数，在[ -l , l ) 上满足Dirichlet 条件

Integral Transformation General concept of the integraltransformation Fundamental ideas of The method ofIntegral Transformation Fourier Transforms of wave problems Fourier Transforms of heat problems Fourier Transforms of steadyproblems Conclusion of this charter

Department of Electrical Engineering and Computer Science 6.003: Signals and Systems-Fall 2003 Thursday, November 13. 2003 Directions: The exam consists of 5 problems on pages 2 to 19 and additional work space on pages 20 and 21. Please make sure you have all the pages. Tables of Fourier series properties as well as CT Fourier transform and DT Fourier transform properties and pairs are supplied to you as a separate set of pages. Enter all your work and