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1、1,Chapter 2,Linear Time-invariant Systems,2,Consider an linear time-invariant system,Example 1,3,2.1 Discrete-time LTI Systems : The Convolution Sum （卷积和）,1. The Representation of Discrete-Time Signals in Terms of impulses,Sifting Property,离散时间信号的冲激表示,2. The Unit Impulse Responses 单位冲激响应,4,Convoluti
2、on-Sum （卷积和）,系统在n时刻的输出包含所有时刻输入脉冲的影响,k时刻的脉冲在n时刻的响应,3. 卷积和的计算, 图解法,普通乘法因果序列或有限长度序列之间的卷积,5,Example 3,Determine,Solution,3 1 4 2 hn,2 1 5 xn,15 5 20 10,3 1 4 2,6 5 24 13 22 10,yn= 6 , 5 , 24 , 13 , 22 , 10 n=0,1,2,3,4,5,6, 多项式算法（适用于有限长度序列）,yn= 6 , 5 , 24 , 13 , 22 , 10 n=0,1,2,3,4,5,利用多项式算法求卷积和的逆运算,已知 y
3、n 、hn xn 已知 yn 、xn hn,7,Example 4,Determine xn,yn= 6 , 5 , 24 , 13 , 22 , 10 n=0,1,2,3,4,5,8,2.2 Continuous-Time LTI Systems : The Convolution Integral （卷积积分）,1. The Representation of Continuous-Time Signals in Terms of impulses,Sifting Property,Convolution Integral 卷积积分,时刻的冲激,t 时刻的响应,2. The Convolu
4、tion Integral,9,2.3 卷积的计算,一 由定义计算积分,二 图解法,10,11,12,2.3 Properties of LTI Systems,LTI系统的特性可由单位冲激响应完全描述,2.3.1 Properties of Convolution Integral and Convolution Sum,1. The Commutative Property (交换律）,13,2. The Distributive Property (分配律）,14,3. The Associative Property (结合律）,15,4. 含有冲激的卷积,16,5. 卷积的微分、积分
5、性质,17,Example,Solution 1,18,Solution 2,19,Example Consider the convolution of the two signals,20,6 几种典型系统, 恒等系统, 微分器, 积分器, 延迟器, 累加器,21,2.3.4 LTI Systems with and without Memory,1. Discrete-time System,It is memoryless,An LTI system without memory,2. Continuous-time System,An LTI system without memor
6、y,22,2.3.5 Invertibility of LTI Systems LTI系统的可逆性,23,2.3.6 Causality for LTI Systems LTI系统的因果性,1. Discrete-time System,Causal system,2. Continuous-time System,24,2.3.7 Stability for LTI Systems (稳定性）,1. Discrete-time System,absolutely Summable 绝对可加,2. Continuous-time System,absolutely Integrable 绝对可
7、积,25,2.3.8 The Unit Step Response of an LTI Systems LTI系统的单位阶跃响应,Discrete-time System,Continuous-time System,Unit Step Response,Unit Step Response,26,作业：,2.1 2.5 2.10 2.7 2.11 2.12 2.22 (a) (c) 2.20 2.23 2.40 2.46 2.47,27,28,(c) S is time-varying.,29,30,Solution ：,31,32,2.20. Evaluate the following integrals:,33,34,35,36,37,38,We have not enough information to determine the output,39,Find the impulse response Find the response of the overall system to the input,40,例1 已知一LTI系统的单位,冲激响应 如图所示，,若输入信号为单位阶跃,信号 ，试求其输出,41,例2 已知离散时间信号 ，要求：,1.画出 的波形；,2.求出满足 的序列。,42,