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1、Chapter 2Linear Time-invariant Systems Chapter 2 LTI Systems10 2 4 t1LConsider an linear time-invariant system Example 10 2 t10 1 2 t1L0 2 4 t1-1 Chapter 2 LTI Systems22.1 Discrete-time LTI Systems:The Convolution Sum（卷积和）（卷积和）1.The Representation of Discrete-Time Signalsin Terms of impulsesSifting
2、Property离散时间信号的冲激表示离散时间信号的冲激表示2.The Unit Impulse Responses单位冲激响应单位冲激响应 Chapter 2 LTI Systems3Convolution-Sum（卷积和）（卷积和）系统在系统在n时刻的输出包含所有时刻输入脉冲的影响时刻的输出包含所有时刻输入脉冲的影响k时刻的脉冲在时刻的脉冲在n时刻的响应时刻的响应3.卷积和的计算卷积和的计算 图解法图解法普通乘法普通乘法因果序列或有限长度序列之间的卷积因果序列或有限长度序列之间的卷积 Chapter 2 LTI Systems4Example 3Determine Solution 314
3、2hn215xn1552010314262846524132210y0y1y2y3y4y5yn=6,5,24,13,22,10 n=0,1,2,3,4,5 Chapter 2 LTI Systems5 多项式算法（适用于有限长度序列）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利用多项式算法求卷积和的逆运算已知已知 yn 、hn xn已知已知 yn 、xn hn Chapter 2 LTI Systems6Example 4Determine xn yn=6,5,24,13,22,10 n=0,1,2,3,4,5
4、y(t)0 Chapter 2 LTI Systems72.2 Continuous-Time LTI Systems:The Convolution Integral （卷积积分）（卷积积分）1.The Representation of Continuous-Time Signalsin Terms of impulsesSifting Property Chapter 2 LTI SystemsConvolution Integral卷积积分卷积积分时刻的冲激时刻的冲激t 时刻的响应时刻的响应2.The Convolution Integral82.3 卷积的计算卷积的计算一一 由定义计
5、算积分由定义计算积分例例2.6 Chapter 2 LTI Systems二二 图解法图解法9 Chapter 2 LTI Systems例例2.7 求下列两信号的卷积求下列两信号的卷积其余其余t其余其余t解：解：10 Chapter 2 LTI Systems112.3 Properties of LTI SystemsLTI系统的特性可由单位冲激响应完全描述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）交换律）Chapt
6、er 2 LTI Systems122.The Distributive Property (分配律）分配律）Chapter 2 LTI Systems13Commutative PropertyAssociative Property3.The Associative Property (结合律）结合律）Chapter 2 LTI Systems144.含有冲激的卷积含有冲激的卷积 Chapter 2 LTI Systems155.卷积的微分、积分性质卷积的微分、积分性质微分微分积分积分 Chapter 2 LTI Systems16Example otherwiseotherwiseSol
7、ution 10 1 2 3 t0 1 2 3 tIntegral Chapter 2 LTI Systems17Solution 20 1 2 3 t0 1 2 3 t Chapter 2 LTI Systems18Example 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)196 几种典型系统几种典型系统 恒等系统恒等系统 微分器微分器 积分器积分器 延迟器延迟器 累加器累加器 Chapter 2 LTI Systems202.3.4 LTI Sy
8、stems with and without Memory 1.Discrete-time SystemIt is memorylessAn LTI system without memory2.Continuous-time SystemAn LTI system without memory Chapter 2 LTI Systems212.3.5 Invertibility of LTI SystemsLTI系统的可逆性系统的可逆性System Inverse System identity system(恒等系统）恒等系统）Chapter 2 LTI Systems222.3.6 Ca
9、usality for LTI SystemsLTI系统的因果性系统的因果性1.Discrete-time System与与n时刻以后输入有关时刻以后输入有关 Causal system2.Continuous-time System Causal system Chapter 2 LTI Systems232.3.7 Stability for LTI Systems (稳定性）稳定性）1.Discrete-time System Stable SystemabsolutelySummable 绝对可加绝对可加2.Continuous-time System Stable Systemabs
10、olutelyIntegrable绝对可积绝对可积 Chapter 2 LTI Systems242.3.8 The Unit Step Response of an LTI SystemsLTI系统的单位阶跃响应系统的单位阶跃响应Discrete-time SystemContinuous-time SystemUnit Step ResponseUnit Step Response Chapter 2 LTI Systems25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 S
11、ystems262.5Determine the value of N.Chapter 2 LTI Systems272.7 A linear system S has the relationshipbetween its input and output(c)S is time-varying.Chapter 2 LTI Systems28 Chapter 2 LTI Systems292.10 Suppose that(a)Determine(b)If contains only three discontinuities,what is the value of a?Solution：
12、0 a 1 1+a t Chapter 2 LTI Systems302.12 LetShow that for Determine the value of A.Chapter 2 LTI Systems312.20.Evaluate the following integrals:Chapter 2 LTI Systems322.22(c)one period of Chapter 2 LTI Systems332.23Determine and sketch for the following value of T:(a)T=4 (b)T=2 (c)T=3/2 (d)T=1 Chapte
13、r 2 LTI Systems340 t (c)T=3/2(a)T=4-5 -4 -3 -1 0 1 3 5 t (d)T=10 t -3 -1 0 1 3 t (b)T=2 Chapter 2 LTI Systems352.40 (a)an LTI system:What is the impulse response for this system?(b)Determine the response of Chapter 2 LTI Systems362.46 Consider an LTI system S and a signal andIf Determine the impulse
14、 response of S.Chapter 2 LTI Systems372.47 An LTI system with impulse 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 382.24 (a)Find the impulse response(b)Fin
15、d the response of the(c)overall system to the input Chapter 2 LTI Systems39例例1 已知一已知一LTI系统的单位系统的单位冲激响应冲激响应 如图所示，如图所示，若输入信号为单位阶跃若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 Systems40例例2 已知离散时间信号已知离散时间信号 ，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 Systems41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，已知两个子系统所示，已知两个子系统的单位冲激响应分别为的单位冲激响应分别为若输入信号若输入信号 ，试求系统的输出，试求系统的输出 Chapter 2 LTI Systems42