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1、Chapter 4The Continuous-Time Fourier Transform1 Chapter 4 Fourier Transformfactor Synthesis equationAnalysis equationFourier Transform PairSpectrum（频谱）频谱） of 4.1 Representation of Aperiodic Signals :The Continuous-Time Fourier Transform2The Fourier coefficients of are proportional to samples of the
2、Fourier transform of one period of Consider a periodic signal Defining Chapter 4 Fourier Transform3 Chapter 4 Fourier Transform4.1.3 Fourier Transforms of Typical SignalsFF4 Chapter 4 Fourier Transform54.2 The Fourier Transforms for Periodic Signals Chapter 4 Fourier Transform6More generally Periodi
3、c square wave Chapter 4 Fourier Transform7 -0 0 0 20Periodic impulses train Chapter 4 Fourier Transform8 Chapter 4 Fourier Transform4.3 Properties of the Fourier Transforms 4.3.1 Linearity Chapter 4 Fourier Transform4.3.2 Time Shifting 9Example 4.90 1 2 3 4 t Chapter 4 Fourier Transform-2 -1 0 1 2 t
4、 Example104.3.3 Conjugation and Conjugate Symmetry Chapter 4 Fourier Transformreal evenreal evenreal oddPurely imaginary odd11 Chapter 4 Fourier Transform124.3.4 Differentiation and Integration1. Differentiation Chapter 4 Fourier Transform2. Integration13-2 0 2 t1-1-2 0 2 t (1)(-2)(1)Example Chapter
5、 4 Fourier Transform144.3.5 Time and Frequency ScalingLet Chapter 4 Fourier TransformExample 15交换交换4.3.6 Duality（对偶性）对偶性）Example Chapter 4 Fourier Transform16FFFExample 4.13 Chapter 4 Fourier Transform17Example 4.144.3.7 Differentiation in Frequency Domain Chapter 4 Fourier Transform18Example 1 More
6、 generally Chapter 4 Fourier Transform19Example 2-2 -1 0 1 2 t -2 -1 0 1 2 t -2 -1 0 1 2 t Chapter 4 Fourier Transform20Example 3 Chapter 4 Fourier Transform214.3.8 Frequency Shifting Chapter 4 Fourier Transform4.3.9 Parsevals Relation22Solution of Infinite Differentiation Chapter 4 Fourier Transfor
7、m23Example 1 FExample 2 Chapter 4 Fourier Transform24Example 3 Chapter 4 Fourier Transform25Example 4 Consider a signal with Fourier Transform Evaluate the following frequency-domain expressions:-1 0 1 2 3 t is real even function. Chapter 4 Fourier Transform264.4 The Convolution Property Chapter 4 F
8、ourier Transform27The usefulness of the Convolution PropertyExample 4.19Solution Chapter 4 Fourier Transform28Example 4.20FF Chapter 4 Fourier Transform294.5 The Multiplication property (modulation property)（调制特性）调制特性） Chapter 4 Fourier Transform30Example 4.23 Determine the Fourier transform of the
9、signal Chapter 4 Fourier Transform31ExampleIn Figure (a) ,a system is shown with input and outputThe input signal has the Fourier transform shown in Figure (b)Determine and sketch . Figure (a)Figure (b) Chapter 4 Fourier Transform32-7W -5W -3W 0 3W 5W 7W -5W -3W 0 3W 5W -8W -6W -2W 0 2W 6W 8W -2W 0
10、2W Chapter 4 Fourier Transform334.6 Frequency-domain analysis of LTI systemsLTI系统的频域分析系统的频域分析1. Stable System Chapter 4 Fourier Transform342. Linear Constant-Coefficients Differential EquationsFTA ratio of polynomials in Chapter 4 Fourier Transform35Example 4.25Consider a stable LTI system Chapter 4
11、 Fourier Transform36Example 4.25Consider the LTI system with input Chapter 4 Fourier Transform37Example Consider an LTI system Chapter 4 Fourier Transform38Homework: 4.3 4.4 4.10 4.11 4.14 4.15 4.24 4.25 4.32 4.35 4.36 4.37* 4.43* Chapter 4 Fourier Transform39Problems for Fourier Analysis Consider an LTI system with unit impulse response 2. If the input is determine the output 1. Determine the frequency response of the whole system. 40
