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1、Chapter 2Linear Time-invariant Systems Chapter 2 LTI Systems0 2 4 t1LConsider an linear time-invariant system Example 10 2 t10 1 2 t1L0 2 4 t1-1 Chapter 2 LTI Systems2.1 Discrete-time LTI Systems : The Convolution Sum卷卷积和和1. The Representation of Discrete-Time Signalsin Terms of impulsesSifting Prop
2、erty离散离散时间信号的冲激表示信号的冲激表示2. The Unit Impulse Responses单位冲激呼位冲激呼应 Chapter 2 LTI SystemsConvolution-Sum 卷卷积和和系系统在在n时辰的辰的输出包含一切出包含一切时辰辰输入脉冲的影响入脉冲的影响k时辰的脉冲在辰的脉冲在n时辰的呼辰的呼应3. 卷卷积和的和的计算算 图解法解法普通乘法普通乘法因果序列或有限因果序列或有限长度序列之度序列之间的卷的卷积 Chapter 2 LTI SystemsExample 3Determine Solution 3142hn215xn155201031426284652
3、4132210y0y1y2y3y4y5yn= 6 , 5 , 24 , 13 , 22 , 10 n=0,1,2,3,4,5 Chapter 2 LTI Systems 多多项式算法适用于有限式算法适用于有限长度序列度序列yn= 6 , 5 , 24 , 13 , 22 , 10 n=0,1,2,3,4,5利用多利用多项式算法求卷式算法求卷积和的逆运算和的逆运算知知 yn 、hn xn知知 yn 、xn hn Chapter 2 LTI SystemsExample 4Determine xn yn= 6 , 5 , 24 , 13 , 22 , 10 n=0,1,2,3,4,5y(t)0 C
4、hapter 2 LTI Systems2.2 Continuous-Time LTI Systems : The Convolution Integral 卷卷积积分分1. The Representation of Continuous-Time Signalsin Terms of impulsesSifting Property Chapter 2 LTI SystemsConvolution Integral卷卷积积分分时辰的冲激辰的冲激t 时辰的呼辰的呼应2. The Convolution Integral2.3 卷卷积的的计算算一一 由定由定义计算算积分分例例2.6 Chapt
5、er 2 LTI Systems二二 图解法解法 Chapter 2 LTI Systems例例2.7 求以下两信号的卷积求以下两信号的卷积其他其他t其他其他t解：解： Chapter 2 LTI Systems2.3 Properties of LTI SystemsLTI系系统的特性可由的特性可由单位冲激呼位冲激呼应完全描画完全描画2.3.1 Properties of Convolution Integral and Convolution Sum 1. The Commutative Property (交交换律律 Chapter 2 LTI Systems2. The Distrib
6、utive Property (分配律分配律 Chapter 2 LTI SystemsCommutative PropertyAssociative Property3. The Associative Property (结合律结合律 Chapter 2 LTI Systems4. 含有冲激的卷积含有冲激的卷积 Chapter 2 LTI Systems5. 卷积的微分、积分性质卷积的微分、积分性质微分微分积分分 Chapter 2 LTI SystemsExample otherwiseotherwiseSolution 10 1 2 3 t0 1 2 3 tIntegral Chapt
7、er 2 LTI SystemsSolution 20 1 2 3 t0 1 2 3 t Chapter 2 LTI SystemsExample Consider the convolution of the two signals(1)(1)0 t-1 0 1 t-2 -1 0 1 2 t-2-2 -1 0 1 2 t-2(-2)6 几种典型系统几种典型系统 恒等系恒等系统 微分器微分器 积分器分器 延延迟器器 累加器累加器 Chapter 2 LTI Systems2.3.4 LTI Systems with and without Memory 1. Discrete-time Sys
8、temIt is memorylessAn LTI system without memory2. Continuous-time SystemAn LTI system without memory Chapter 2 LTI Systems2.3.5 Invertibility of LTI SystemsLTI系系统的可逆性的可逆性System Inverse System identity system (恒等系统恒等系统 Chapter 2 LTI Systems2.3.6 Causality for LTI SystemsLTI系系统的因果性的因果性1. Discrete-time
9、 System与与n时辰以后输入有关时辰以后输入有关 Causal system2. Continuous-time System Causal system Chapter 2 LTI Systems2.3.7 Stability for LTI Systems (稳定性定性1. Discrete-time System Stable SystemabsolutelySummable 绝对可加绝对可加2. Continuous-time System Stable SystemabsolutelyIntegrable绝对可积绝对可积 Chapter 2 LTI Systems2.3.8 Th
10、e Unit Step Response of an LTI SystemsLTI系系统的的单位位阶跃呼呼应Discrete-time SystemContinuous-time SystemUnit Step ResponseUnit Step Response Chapter 2 LTI Systems作业：作业：2.1 2.5 2.102.7 2.11 2.12 2.22 (a) (c)2.20 2.23 2.40 2.46 2.47 Chapter 2 LTI Systems2.5Determine the value of N. Chapter 2 LTI Systems2.7 A
11、linear system S has the relationshipbetween its input and output(c) S is time-varying. Chapter 2 LTI Systems Chapter 2 LTI Systems2.10 Suppose that(a) Determine(b) If contains only three discontinuities,what is the value of a?Solution ：0 a 1 1+a t Chapter 2 LTI Systems2.12 LetShow that for Determine
12、 the value of A. Chapter 2 LTI Systems2.20. Evaluate the following integrals: Chapter 2 LTI Systems2.22(c)one period of Chapter 2 LTI Systems2.23Determine and sketch for the following value of T:(a) T=4 (b) T=2 (c) T=3/2 (d) T=1 Chapter 2 LTI Systems0 t (c) T=3/2(a) T=4-5 -4 -3 -1 0 1 3 5 t (d) T=10
13、 t -3 -1 0 1 3 t (b) T=2 Chapter 2 LTI Systems2.40 (a) an LTI system:What is the impulse response for this system? (b) Determine the response of Chapter 2 LTI Systems2.46 Consider an LTI system S and a signal andIf Determine the impulse response of S. Chapter 2 LTI Systems2.47 An LTI system with imp
14、ulse response 0 2 t In each of these cases,determinewhether or not we have enough Information to determine the output Chapter 2 LTI SystemsWe have not enough information to determine the output 2.24 (a)Find the impulse response(b)Find the response of the (c)overall system to the input Chapter 2 LTI
15、Systems例例1 知一知一LTI系统的单位系统的单位冲激呼应冲激呼应 如下图，如下图，假设输入信号为单位阶跃假设输入信号为单位阶跃信号信号 ，试求其输出，试求其输出 Chapter 2 LTI Systems例例2 知离散时间信号知离散时间信号 ，要求：，要求：1.画出画出 的波形；的波形；2.求出满足求出满足 的序列。的序列。 Chapter 2 LTI Systems图图1例例 某离散时间某离散时间LTI系统的系统框图如图系统的系统框图如图1所示，知两个子系统所示，知两个子系统的单位冲激呼应分别为的单位冲激呼应分别为假设输入信号假设输入信号 ，试求系统的输出，试求系统的输出 Chapter 2 LTI Systems
