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1、Chapter 4The Continuous-Time Fourier Transform Chapter 4 Fourier Transformfactor Synthesis equationAnalysis equationFourier Transform PairSpectrum频谱 of 4.1 Representation of Aperiodic Signals :The Continuous-Time Fourier TransformThe Fourier coefficients of are proportional to samples of the Fourier
2、 transform of one period of Consider a periodic signal Defining Chapter 4 Fourier Transform Chapter 4 Fourier Transform4.1.3 Fourier Transforms of Typical SignalsFF Chapter 4 Fourier Transform4.2 The Fourier Transforms for Periodic Signals Chapter 4 Fourier TransformMore generally Periodic square wa
3、ve Chapter 4 Fourier Transform -0 0 0 20Periodic impulses train Chapter 4 Fourier Transform Chapter 4 Fourier Transform4.3 Properties of the Fourier Transforms 4.3.1 Linearity Chapter 4 Fourier Transform4.3.2 Time Shifting Example 4.90 1 2 3 4 t Chapter 4 Fourier Transform-2 -1 0 1 2 t Example4.3.3
4、Conjugation and Conjugate Symmetry Chapter 4 Fourier Transformreal evenreal evenreal oddPurely imaginary odd Chapter 4 Fourier Transform4.3.4 Differentiation and Integration1. Differentiation Chapter 4 Fourier Transform2. Integration-2 0 2 t1-1-2 0 2 t (1)(-2)(1)Example Chapter 4 Fourier Transform4.
5、3.5 Time and Frequency ScalingLet Chapter 4 Fourier TransformExample 交换交换4.3.6 Duality对偶性偶性Example Chapter 4 Fourier TransformFFFExample 4.13 Chapter 4 Fourier TransformExample 4.144.3.7 Differentiation in Frequency Domain Chapter 4 Fourier TransformExample 1 More generally Chapter 4 Fourier Transfo
6、rmExample 2-2 -1 0 1 2 t -2 -1 0 1 2 t -2 -1 0 1 2 t Chapter 4 Fourier TransformExample 3 Chapter 4 Fourier Transform4.3.8 Frequency Shifting Chapter 4 Fourier Transform4.3.9 Parsevals RelationSolution of Infinite Differentiation Chapter 4 Fourier TransformExample 1 FExample 2 Chapter 4 Fourier Tran
7、sformExample 3 Chapter 4 Fourier TransformExample 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 Transform4.4 The Convolution Property Chapter 4 Fourier TransformThe usefulness of the Convolution Pr
8、opertyExample 4.19Solution Chapter 4 Fourier TransformExample 4.20FF Chapter 4 Fourier Transform4.5 The Multiplication property (modulation property)调制特性制特性 Chapter 4 Fourier TransformExample 4.23 Determine the Fourier transform of the signal Chapter 4 Fourier TransformExampleIn Figure (a) ,a system
9、 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 Transform-7W -5W -3W 0 3W 5W 7W -5W -3W 0 3W 5W -8W -6W -2W 0 2W 6W 8W -2W 0 2W Chapter 4 Fourier Transform4.6 Frequency-domain analysis of LTI s
10、ystemsLTI系系统的的频域分析域分析1. Stable System Chapter 4 Fourier Transform2. Linear Constant-Coefficients Differential EquationsFTA ratio of polynomials in Chapter 4 Fourier TransformExample 4.25Consider a stable LTI system Chapter 4 Fourier TransformExample 4.25Consider the LTI system with input Chapter 4 F
11、ourier TransformExample Consider an LTI system Chapter 4 Fourier TransformHomework: 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 TransformProblems 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.