Thursday, November 13, 2003 Directions: The exam consists of 6 problems on pages 2 to 19 and 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 tables are supplied to you at the end of this booklet. Enter all your work and your answers directly in the spaces provided on the printed pages of this book- let. Please make sure your name is on all sheets. You may use bluebooks for scratch work, but we will not grade them at all. all sketches must be adequately

2.1 The Continuous-Time Fourier Transform 2.2 The Discrete-Time Fourier transform 2.3 the discrete-time periodic sequences in the frequency-domain 2.4 Relationship of DTFT and FT 2.5 Z-Transform

其它展开 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=2l的函数展开为 Fourier级数 若f(t)是以T=2l为周期的函数,在[-l,l) 上满足 Dirichlet条件

其它展开 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=2l的函数展开为 Fourier级数 若f(t)是以T=2l为周期的函数,在[-l,l) 上满足 Dirichlet条件

前面关于 Fourier 级数的论述都是对周期函数而言的，那么对于 非周期函数，又该如何处理呢？ 在(−,+) 上可积的非周期函数 f (x)可以看成是周期函数的极限 情况，处理思路是这样的：

Fourier级数的分析性质 为简单起见,假定f(x)的周期为2π。 首先,利用 Riemann引理可以直接得出 定理16.3.1设f(x)在[-上可积或绝对可积,则对于f(x)的 Fourier系数an与b

离散 Fourier 变换 人们刚开始利用无线电技术传输信号时，是将连续信号进行某种 调制处理后直接传送的（图 16.5.1），本质上传送的还是连续信号（也 叫模拟信号）。这样的传输方式抗干扰能力差，失真严重，尤其是经 过长距离传送或多级传递后，信号可能面目全非，质量自然难尽人意

◆2.1 Discrete-Time Signals: Sequences ◆High and Low Frequencies in Discrete-time signal ◆2.2 Discrete-Time Systems ◆Memoryless (memory); Linear; Time-Invariant; Causality; Stability(BIBO) ◆2.3 Linear Time-Invariant (LTI) Systems ◆LTI Systems:Convolution( 系统适用吗?) ◆2.4 Properties of LTI Systems ◆Stability and Causality of LTI systems;FIR and IIR systems ◆2.5 Linear Constant-Coefficient Difference Equations ◆The output for a given input is not uniquely specified. Auxiliary conditions are required; initial-rest conditions ◆2.6 Frequency-Domain Representation of Discrete￾Time Signals and systems ◆ Eigenfunction and Eigenvalue for LTI 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

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