1、1,Chapter 2,Linear Time-invariant Systems,2,Consider a linear time-invariant system,Example 1 an LTI system,3,2.1 Discrete-time LTI Systems : The Convolution Sum （卷积和）,2.1.1 The Representation of Discrete-Time Signals in Terms of impulses,Sifting Property,离散时间信号的冲激表示,4,2.1.2 The Discrete-Time Unit I
2、mpulse Responses and the Convolution-Sum Representation of LTI Systems,The Unit Impulse Responses 单位冲激响应,Let,Let,5,2. Convolution-Sum （卷积和）,Time Invariant,Scaling,Additivity,Convolution-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,6,3. 卷积和的计算, 图解法,例2.3,图解法步骤：, 求乘积, 对每一个n求和,7,Example 2.4,Determine
3、the output signal,Solution,(a) n0,(b) 0n4,8,(d),(e),9,Summarizing ,we obtain,Ly=11,Lx=5 Lh=7,Ly=Lx+Lh-1,10,不带进位的普通乘法适用于因果序列或有限长度序列之间的卷积,11,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=0,1,2,3,4,5,12, 多项式算法（适用于有限长度序列）,yn= 6 , 5 ,
4、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 yn 、hn xn 已知 yn 、xn hn,13,Example 4,Determine xn,yn= 6 , 5 , 24 , 13 , 22 , 10 n=0,1,2,3,4,5,14,2.2 Continuous-Time LTI Systems : The Convolution Integral（卷积积分）,2.2.1 The Representation of Continuous-Time Signals in Terms of impulses,Sifting Propert
5、y,For example:,15,According to Sampling Property,2.2.1 The Continuous-Time Unit Impulse Response and the Convolution Integral Representation of LTI Systems,Time Invariant,Scaling,16,Convolution Integral 卷积积分,时刻的冲激,t 时刻的响应,2.3 卷积的计算,一 由定义计算积分,17,二 图解法,18,19,2.3 Properties of LTI Systems,LTI系统的特性可由单位冲
6、激响应完全描述,Example 2.9, LTI system,输入输出关系是唯一的,20, Nonlinear System,非线性系统无法用单位冲激响应完全描述,本课程主要研究线性时不变（LTI）系统,21,2.3.1 Properties of Convolution Integral and Convolution Sum,1. The Commutative Property (交换律）,22,2. The Distributive Property (分配律）,23,3. The Associative Property (结合律）,交换积分次序,24,25,4. 含有冲激的卷积,
7、26,5. 卷积的微分、积分性质, 微分, 积分,27, 推广,n=1 m=-1,n0 微分 n0 积分,28,29,Example,Solution 1,30,31,Example Consider the convolution of the two signals,32,6 几种典型系统, 恒等系统, 微分器, 积分器, 延迟器, 累加器,33,2.3.4 LTI Systems with and without Memory,1. Discrete-time System,It is memoryless,An LTI system without memory,2. Continuo
8、us-time System,An LTI system without memory,34,2.3.5 Invertibility of LTI Systems LTI系统的可逆性,35,2.3.6 Causality for LTI Systems LTI系统的因果性,1. Discrete-time System,Causal system,2. Continuous-time System,36,Consider a linear system,For any time t0,2.3.7 Stability for LTI Systems (稳定性）,1. Discrete-time
9、System,37,2. Continuous-time System,absolutely Summable绝对可加,The system is stable.,absolutely Integrable 绝对可积,The system is not stable.,38,2.3.8 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 StepResponse,Unit StepResponse,39,2.4 Singularity Fun
10、ctions (奇异函数）,2.4.1 The Unit Impulse as an Idealized Short Pulse,40,Example 2.16,41,Example 2.17,42,2.4.2 Defining the Unit Impulse through Convolution 单位冲激的卷积定义,For any x(t),For any normal ,which is continuous at time t=0.,Let t=0,Sifting Property,The unit impulse has unit area,43,2.4.3 Unit Double
11、ts and Other Singularity Functions 单位冲激偶和其它奇异函数,1.,In general ,44,45,3. Properties of Unit Doublets （冲激偶的性质）,2. Defining,For any,46,In general,Derivatives of different orders of the unit impulse单位冲激的各阶积分,47,2. 5 Causal LTI Systems described by Differential and Difference Equations 用微分方程和差分方程描述的因果LTI
12、系统,2. 5.1 Linear Constant-coefficient Differential Equations 线性常系数微分方程,一 经典解法,1.Homogeneous solution 齐次解,特征方程,48, 特征方程有N个不同的单根, 特征方程有1个 r 阶重根 ，其余N-r个根各不相同,Homogeneous Solution Natural Response齐次解 自然响应,2. Particular SolutionForced Response特解 受迫响应,的函数形式取决于输入的函数形式,49,不是特征单根 是特征单根,或,50,齐次解：,特解：,全响应：,51,
13、 若初始条件不为零，,若初始条件不跃变，,代入,全响应,N阶系统，初始松弛：,初始松弛条件下：,52,二 零输入、零状态解法,全响应：,1.零输入响应 ：,具有齐次解的形式；,假定特征方程含有 N个不同的单根,2.零状态响应 ：,53,由初始状态 唯一决定,由零初始状态 及输入共同决定,由初始状态 及输入决定,函数形式由 输入信号决定,1. 零输入零状态解法,2. 经典解法,三 两种响应分解形式的关系,54,Zero-input response,Zero-stateresponse,3. Full response,55,2.5.2 Linear Constant-coefficient D
14、ifference Equations 线性常系数差分方程,Nth order,Recursive Solution 递推算法,Recursive Equation 递归方程,56,Particularly N=0,Nonrecursive Equation 非递归方程,Unit impulse response,Finite impulse response (FIR) system,Example 2.15,57,58,59,2. 差分方程的经典解法,Full Homogeneous Particular, Homogeneous solution,Eigenequation 特征方程,(
15、a) 特征方程有N个不同的单根,(b) 特征方程有1个 r 阶重根1其余N-r 个根各不相同,60, Particular solution (Forced Response),不是特征单根 是特征单根,或,61,3. 零输入、零状态解法,Full Zero-input Zero-state, 零输入响应 ：,具有齐次解的形式；, 零状态响应 ：,假定特征方程含有 N个不同的单根,62,由初始状态 唯一决定,由零初始状态 及输入共同决定,由初始状态 及输入决定,函数形式由 输入信号决定,4. 两种响应分解形式的关系,63,Solution 1,Homogeneous Solution,Part
16、icularSolution,Full Solution,64,Initial State:,Full Solution,Solution 2,Zero-input Solution,65,Zero-state Solution,Full Solution,Zero-input response,Zero-state response,Natural response,Forced response,66,2. 5.3 Block Diagram Representations of First-Order System 一阶系统的方框图表示,1. Discrete-time systems,Addition,Multiplication by a coefficient,Delay,Recursive Equation,67,2. Continuous-time systems,Addition,Multiplication by a coefficient,Integrator,68,Initial Condition,69,作业： 2.22 (a) (c) 2.20 2.23 2.40 2.46 2.47,70,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,
