1 Lecture 5 Properties of LTI Systems The Solution of LCCDE

2 Chapter 2 LTI Systems §2.3 Properties of LTI Systems An LTI system is completely characterized by its impulse response h(t)/h[n] Example 2.9 ① LTI system ② Nonlinear System ③ Time-variant System

3 Chapter 2 LTI Systems §2.3.1 Properties of Convolution Integral and Convolution Sum 1. The Commutative Property ( 交换律）

4 Chapter 2 LTI Systems 2. The Distributive Property ( 分配律）

5 Chapter 2 LTI Systems 3. The Associative Property ( 结合律） Commutative Property Associative Property

6 Chapter 2 LTI Systems 4. 含有冲激的卷积 ① ②

7 Chapter 2 LTI Systems 5. 卷积的微分、积分性质 ① 微分性质 ② 积分性质 ③ 推广形式 n>0 微分 n<0 积分

8 Chapter 2 LTI Systems 特殊地 n=1 m=-1 Example otherwise Consider the convolution of the two signals

9 Example Consider the convolution of the two signals Chapter 2 LTI Systems

10 Chapter 2 LTI Systems 6 几种典型系统 ① 恒等系统 ② 微分器 ③ 积分器 ④ 延迟器 ⑤ 累加器

11 Chapter 2 LTI Systems §2.3.4 LTI Systems with and without Memory 1. Discrete-time System An LTI system without memory 2. Continuous-time System An LTI system without memory

12 Chapter 2 LTI Systems §2.3.5 Invertibility of LTI Systems LTI 系统的可逆性 System Inverse System identity system ( 恒等系统）

13 Chapter 2 LTI Systems §2.3.6 Causality for LTI Systems LTI 系统的因果性 1. Discrete-time System Causal system 2. Continuous-time System Causal system

14 Chapter 2 LTI Systems Consider a LTI system Causal Initial Rest ( 初始松弛） For any time t 0 The system is initial rest. ( 初始松弛）

15 Chapter 2 LTI Systems §2.3.7 Stability for LTI Systems ( 稳定性） 1. Discrete-time System Stable System Absolutely summable （绝对可加） 2. Continuous-time System Stable System Absolutely integrable （绝对可积 ）

16 Chapter 2 LTI Systems §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

17 Define Chapter 2 LTI Systems §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.

18 Chapter 2 LTI Systems §2.4.3 Unit Doublets and Other Singularity Functions 单位冲激偶和其它奇异函数 1. In general, 2. For any

19 Chapter 2 LTI Systems 3. Properties of Unit Doublets （冲激偶的性质） ① ② ③ 4.Derivatives of different orders of the unit impulse 单位冲激的各阶积分

20 Chapter 2 LTI Systems §2. 5 Causal LTI Systems described by Differential and Difference Equations § Linear Constant-coefficient Differential Equations 一 经典解法 Homogeneous solution Natural Response Particular Solution Forced Response

21 Chapter 2 LTI Systems 输入特解 C （常数） B （常数） α 不是特征单根 α 是特征单根 或

22 Chapter 2 LTI Systems Example 2.14 (a) The system is initial rest Determine the output

23 Chapter 2 LTI Systems 由初始状态 唯一决定 零输入响应 由零初始状态 及输入共同决定 零状态响应 由初始状态 及输入决定 自然响应 函数形式由 输入信号决定 受迫响应 零输入零 状态解法 经典解法 三 两种响应分解形式的关系 二 零输入、零状态解法

24 Chapter 2 LTI Systems 1.Zero-input response 2.Zero-state response 3. Full response Example 2.14 Determine the output

25 Chapter 2 LTI Systems §2.5.2 Linear Constant-coefficient Difference Equations 线性常系数差分方程 Nth order 1.Recursive Solution （递推算法） Example 2.15 (a) The system is initial rest, and the input Determine the output

26 Chapter 2 LTI Systems 2. Typical solution Homogeneous solution Natural Response Particular Solution Forced Response

27 Chapter 2 LTI Systems ② Particular solution (Forced Response) 输入特解 α 不是特征单根 α 是特征单根 或

28 Chapter 2 LTI Systems 3. 零输入、零状态解法 Zero-input Response 4. 两种响应分解形式的关系 Zero-state Response

29 Chapter 2 LTI Systems Example Consider an LTI system: Determine the full response Solution 1 Homogeneous Solution Particular Solution Full Solution

30 Chapter 2 LTI Systems § Block Diagram Representations of First-Order System 一阶系统的方框图表示 1. Discrete-time systems Three basic operations Addition Multiplication by a coefficient Delay D Recursive Equation D Initial rest :

31 Chapter 2 LTI Systems 2. Continuous-time systems Three basic operations Addition Multiplication by a coefficient Integrator ∫

32 Chapter 2 LTI Systems ∫ Initial Condition

33 图解法步骤： ㈠ 反折 ㈡ 平移 ㈢ 求乘积 ㈣ 对每一个 n 求和

34 Chapter 2 LTI Systems

35 卷积的微分、积分性质 Chapter 2 LTI Systems Properties of LTI systems An LTI system without memory Invertibility of LTI Systems

36 Chapter 2 LTI Systems Discrete-time System Continuous-time System Causal system Causal system Stable System Stable System

37 Chapter 2 LTI Systems Defining the Unit Impulse through Convolution

38 Homogeneous Solution Natural Response Particular Solution Forced Response

39 Readlist Signals and Systems –3.0~3.3

40 Problem Set 2.40,2.46,2.47(a)(c)(e)