Signals and Systems Lecture 28

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1 Signals and Systems Lecture 28
Properties of z-Transform ROC of z-Transform Inverse z-Transform

2 Chapter 10 The Z-Transform
§10.5 Properties of the z-Transform § Linearity

3 Chapter 10 The Z-Transform
§ Time Shifting Consider z=0 or z=∞

4 Chapter 10 The Z-Transform
Example 1 Nst order zero Zeros: z=0 Poles:

5 Chapter 10 The Z-Transform
§ Scaling in the z-Domain Poles of : Poles of : Specially ,

6 Chapter 10 The Z-Transform

7 Chapter 10 The Z-Transform
§ Time Reversal Poles of : Poles of :

8 Chapter 10 The Z-Transform
§ Time Expansion

9 Chapter 10 The Z-Transform
§ Conjugation If is real , 实序列 的复极点共轭成对出现。

10 Chapter 10 The Z-Transform
§ The Convolution Property

11 Chapter 10 The Z-Transform
Example 10.15 Example 10.16 Consider a summation Example

12 Chapter 10 The Z-Transform
§ Differentiation in the z-Domain Example 10.17

13 Chapter 10 The Z-Transform
§ The Initial-Value Theorem If then If then Example Determine the initial-value

14 Chapter 10 The Z-Transform
§ The Final-Value Theorem 因果序列 如果 的极点均在单位圆内（允许在z=1有一个一阶极点） Example 终值不存在。 终值不存在。

15 Chapter 10 The Z-Transform
§10.7 Analysis and Characterization of LTI Systems using z-Transforms § Causality A discrete-time system is causal including infinity. If is rational function, 系统因果 分子阶数不大于分母阶数

16 Chapter 10 The Z-Transform
Example 10.20 This system is not causal. Example 10.21 It is a causal system.

17 Chapter 10 The Z-Transform
§ Stability A stable system A discrete-time system is stable Example 10.21 The system is causal but not stable. The system is not causal but stable. The system is anticausal and not stable.

18 Chapter 10 The Z-Transform
如果 为有理函数, 系统因果、稳定 的极点均在单位圆内 Example 10.24

19 ROC Chapter 10 The Z-Transform
§ Linear Constant-Coefficient Difference Equations ROC

20 Chapter 10 The Z-Transform
Example Consider an causal system for which the input and output satisfy the linear constant-coefficient equation Determine the unit impulse response Determine the unit step response Determine the unit impulse response of another system which satisfy the following linear constant -coefficient equation.

21 Chapter 10 The Z-Transform
例 已知一因果LTI系统的单位阶跃响应 ，当输入 为 时，其零状态响应 ，求输入

22 Chapter 10 The Z-Transform
Example 10.26 Suppose that we are given the following information about an LTI system: 2. If ，then the output is Determine the system function for this system, and deduce the causality and stability of this system. Write the difference equation characterizes the system.

23 Chapter 10 The Z-Transform

24 Chapter 10 The Z-Transform
Example 10.27 一具有有理系统函数 的因果、稳定系统， 在 有一极点，在单位圆上某处有一零点， 其余零极点未知，试判断下列说法是否正确。 收敛。 2. 对某一ω值有

25 Chapter 10 The Z-Transform
为有限长序列 为实信号。 无法判断。 单位脉冲响应 是一因果、稳定系统的

26 Chapter 10 The Z-Transform
§10.8 System Function Algebra and Block Diagram Representations Three basic operations 1. Addition 2. Multiplication by a coefficient 3. Delay Z-1

27 Chapter 10 The Z-Transform
Example 10.30 Consider the causal LTI system -1/4 1/8

28 S2 Chapter 10 The Z-Transform Example 10.31
Consider the causal LTI system 1. 直接模拟 S2 -1/4 1/8 -7/4 -1/2

29 Chapter 10 The Z-Transform
-7/4 1 Z-1 -1/2 1/8 -1/4 公共点

30 Chapter 10 The Z-Transform
2. 级联模拟 1 1 1 Z-1 Z-1 1 1 -1/2 1/4 1/4 -2

31 Chapter 10 The Z-Transform
2. 并联模拟 1 1 3/5 4 -14/3 1 1

32 Problem Set P