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[−π,π]上可积或绝对可积，则对于 f x( )的 Fourier 系数an与bn，有

§7.1 Fourier变换的概念与性质 §7.2离散 Fourier变换 §7.3 Fourier变换的应用

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

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

◆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

◆2.1 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

§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

◆ 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

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