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

Chapter 10 The Z-Transform §10.5 Properties of the z-Transform §10.5.1 Linearity

Chapter 10 The Z-Transform §10.5.2 Time Shifting Consider z=0 or z=∞

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

Chapter 10 The Z-Transform §10.5.3 Scaling in the z-Domain Poles of : Poles of : Specially ,

Chapter 10 The Z-Transform

Chapter 10 The Z-Transform §10.5.4 Time Reversal Poles of : Poles of :

Chapter 10 The Z-Transform §10.5.5 Time Expansion

Chapter 10 The Z-Transform §10.5.6 Conjugation If is real , 实序列 的复极点共轭成对出现。

Chapter 10 The Z-Transform §10.5.7 The Convolution Property

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

Chapter 10 The Z-Transform §10.5.8 Differentiation in the z-Domain Example 10.17

Chapter 10 The Z-Transform §10.5.9 The Initial-Value Theorem If then If then Example Determine the initial-value

Chapter 10 The Z-Transform §10.5.10 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。

Chapter 10 The Z-Transform §10.7 Analysis and Characterization of LTI Systems using z-Transforms §10.7.1 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

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

Chapter 10 The Z-Transform §10.7.2 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.

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

ROC Chapter 10 The Z-Transform §10.7.3 Linear Constant-Coefficient Difference Equations ROC

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.

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

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.

Chapter 10 The Z-Transform

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。 1. 收敛。 2. 对某一ω值有

Chapter 10 The Z-Transform 3. 为有限长序列 4. 为实信号。 无法判断。 单位脉冲响应 5. 是一因果、稳定系统的

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

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

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

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

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

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

Problem Set P804 10.30