信号教学课件(华中科技大学).ppt
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1、CHAPTER 2 LINEAR TIME- INVARIANT SYSTEMS 2.0 INTRODUCTION Representation of signals as linear combination of delayed impulses. Convolution sum(卷积和) or convolution integral(卷积 积分) representation of LTI systems. Impulse response and systems properties Solutions to linear constant-coefficient differenc
2、e and differential equations (线性常系数差分或微分方程). 2.1 DISCRETE-TIME SYSTEMS: THE CONVOLUTION SUM Derivation steps: Step 1: Representing discrete-time signals in terms of unit samples: Step 2: Defining Unit sample response hn : response of the LTI system to the unit sample n. n hn Step 3: Writing any arbi
3、trary input xn as: Step 4: By taking use of linearity and time-invariance, we can get the response yn to xn which is the weighted linear combination of delayed unit sample responses as following: The Convolution Sum Representation of LTI Systems convolution sum or superposition sum : Convolution ope
4、ration symbol: LTI system is completely characterized by its response to the unit sample -hn . Example 2.1 n xn 0 1 2 n hn -2 0 2 (a) Consider a LTI system with unit sample response hn and input xn, as illustrated in Figure (a). Calculate the convolution sum of these two sequences graphically. k xk
5、0 1 2 k h-k -2 0 2 (b) h-2h2 1 1 0.5 k xk 0 1 2 k hn-k n-2 n+2 If n4, Graph of yn in Example 2.1 From Example 2.1, we can draw the following table: Thus, we obtain a method for the computation of convolution sum, that is suitable for two short sequences. xn = 1,1,10hn = 0.5, 1, 0.5, 1, 0.5-2 xn*hn =
6、 0.5,1.5,2,2.5,2,1.5,0.5-2 0.5 1.5 2 2.5 2 1.5 0.5 0.5 1 0.5 1 0.5 xn h-2 h-1 h0 h1 h2 hn x0 x1 x2 : x0h-2 x0h-1 x0h0 x0h1 x0h2 0 0 x1h-2 x1h-1 x1h0 x1h1 x1h2 0 0 x2h-2 x2h-1 x2h0 x2h1 x2h2 0 0 0 0 0 0 0 0 0 y- 2 y- 1 y0 y1 y3y2y4 0.5 1 0.5 1 0.5 1 1 1 0.5 1 0.5 1 0.5 + + 0.5 1 0.5 1 0.5 Example 2.2
7、 Consider an input xn and a unit sample response hn given by Determine and plot the output Using the geometric sum formula to evaluate the equation, we have n 2 1 yn Graph of yn in Example 2.2 2.2 CONTINUOUS-TIME LTI SYSTEMS: THE CONVOLUTION INTEGRAL The Representation of Continuous-Time Signals in
8、Terms of Impulses: t -02 k x(t) Staircase approximation to a continuous-time signal x(t) x(-2 ) t -2- x(0) 0 t 2 x() t -0 x(-) t Mathematical representation for the rectangular pulses as , the summation approaches an integral and is the unit impulse function Compared with the Sampling property of th
9、e unit impulse: Give the as the response of a continuous-time LTI system to the input , then the response of the system to pulse is Thus, the response to is As , in addition, the summing becomes an integral. Therefore, convolution integral or superposition integral : unit impulse response h(t) : the
10、 response to the input . (单单位冲激响应应) Convolution integral symbol: A continuous-time LTI system is completely characterized by its unit impulse response h(t) . Example 2.3 Consider the convolution of the following two signals, which are depicted in (a): 0 T 1 x(t) t 0 2T 2T h(t) t (a) 2T h(t) t-2T t 0
11、 T 1 x() For t 0 Interval 1. For t 0, there is no overlap between the nonzero portions of and , and consequently, From the definition of the convolution integral of two continuous- time signals, 2Th(t) t-2T 0 t T 1 x() For 0 t T . Thus, for 0 t T . Interval 2. For 0 t T, 2T h(t) t-2T T t 1 x() For T
12、 t 2T Thus, for T t T but t-2T 0, i.e. T t 0, but t-2T T, i.e. 2T t 3T Thus, for 2T t 3T . For 2T t T, or equivalently, t 3T, there is no overlap between the nonzero portions of and hence, x() 2T h(t) 0 T t-2T t 1 For t 3T Summarizing, 2.3 PROPERTIES OF CONVOLUTION OPERATION 2.3.1 The Commutative Pr
13、operty(交换律) in discrete time : in continuous time: 2.3.2 The Distributive Property (分配律) in discrete time : in continuous time: y(t) y2(t) y1(t) h2(t) x(t) h1(t) x(t) h1(t)+h2(t) y(t) Two equivalent systems: having same impulse responses 2.3.3 The Associative Property (结合律) in discrete time : in con
14、tinuous time: xnh1nynh2n xnhn=h1n*h2nyn xnhn=h2n*h1nynxnh2nynh1n Four equivalent systems 2.3.4 Convolving with Impulse 2.3.5 Differentiation and Integration of Convolution Integral Combining the two properties, we have 2.3.6 First Difference and Accumulation of Convolution Sum 2.4.1 LTI Systems with
15、 and without Memory for a discrete-time LTI system without memory: for a continuous-time LTI system without memory: 2.4.2 Invertibility of LTI Systems The impulse responses of a system and its inverse system satisfy the following condition: in discrete-time : in continuous-time: Since 2.4 PROPERTIES
16、 OF LTI SYSTEMS 2.4.3 Causality for LTI Systems for a causal discrete-time LTI system: for a causal continuous-time LTI system: 2.4.4 Stability for LTI Systems for a stable discrete-time LTI system: for a stable continuous-time LTI system: absolutely summable absolutely integrable Suppose Proof: The
17、n If Then Therefore, the absolutely summable is a sufficient condition to guarantee the stability of a discrete-time LTI system. To show that the absolutely summable is also a necessary condition for the stability of a discrete-time LTI system, Let where, is conjugate . Then, xn is bounded by 1, tha
18、t is However, IfThen 2.5 The Unit Step Response(单位阶跃响应) of an LTI System The unit step response, sn or s(t), is the output of an LTI system when input xn=un or x(t)=u(t). The unit step response of a discrete-time LTI system is the running sum of its unit sample response: The unit sample response of
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