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# Chapter 10 The Z-Transform

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Chapter 10 The Z-Transform
Complex Frequency Domain (Z-Domain) Analysis of LTI System ● Representation of Aperiodic Signals ● Response of LTI System to Aperiodic Signals §10 §5

√ 10.0 Introduction §5 Frequency Domain Analysis Frequency analysis
2 3 ﹡ Condition: Frequency analysis ﹡ Problems: Frequency analysis

﹡Cause: Basic signal: represent ﹡Measure: Basic signal: represent

10.1 The Z-Transform Pair A. The Transform Pair Under Condition
we have z-plane Integral line

B. Understanding of The Transform Pairs
﹡Inverse Transform

Frequency Frequency

﹡The Transform ﹡ Similarity : ﹡ Similarity :

C. The Convergence Region of the Z-Transform
ROC generally : ROC Integral line 点： ：基本信号

D. Relations Between Z-Transform and Discrete-Time Fourier Transform
if ROC Let ROC Z-Transform on Unit Circle ＝ Discrete-Time Fourier Transform

10.2 The Region of Convergence of The Z-Transform
The ROC. <Examples 10.1> Condition=ROC ① if , Unit Circle ROC ROC

② if , Unit Circle ROC Unit Circle 一般：右边信号 收敛域向外

< Examples 10.2> ① if , Unit Circle ROC Unit Circle

then the Unit Circle ROC
② if , then the Unit Circle ROC Unit Circle 一般：左边信号 收敛域向内

ROC for Integral Left-sided Right-sided ROC for Integral
< > <10.1> <10.2> Integral ROC for Integral Right-sided Left-sided ROC for

General Rule for ROC 右边信号 A. ROC : 双边信号 B. Poles ROC 左边信号

C. ROC : ROC 双边信号 环形收敛域 或无收敛域

<Example 10.7> 双边信号 环形收敛域 双边信号 无收敛域

possibly except D. is finite duration ROC: entire Z-plane, Pole at
Poles at “环形” “向内” “向外”

<Example 10.6 > pole zero

E. 右边信号 收敛域向外

F. 左边信号 收敛域向内

G. Rational ROC: Bounded by poles ﹡left-sided signal ﹡ Two-sided signal ﹡right-sided signal

10.4 Geometric Evaluation of The Fourier Transform From The Zero-Pole Plot
Geometric Evaluation of Z-Transform A. The Method zero pole 零点距离积 极点距离积 零点相位和 极点相位和

B. Example

10.4.2 Geometric Evaluation of Fourier Transform
A. The Method as above, if Unit Circle ROC, Let B. Example

10.5 Properties of Z-Transform
Linearity Time shifting shift （ 可能加入或去掉）

<Proof> <Example>

10.5.3 Scaling in the Z-Domain

Time Reversal <Proof> 1/R

10.5.5 Time Expansion （时域扩展） Where ，if n is a multiple of k 补零 integer
， else integer k=3 -4k -3k -2k -k k 2k 3k 4k 补零

Conjugation For real signal :

10.5.7 The Convolution Property
Differentiation in Z- Domain

Important : useful in Inverse Z-Transform
<Example> Differentiation Differentiation，Linear Important : useful in Inverse Z-Transform

<Example> Linearity, Time-scaling

10.6 Some Common Z-Transform Pairs
The Initial-value Theorem (检验变换的正确性) For causal ，we have Table include all properties 10.6 Some Common Z-Transform Pairs Table 10.2

10.3 Inverse Z-Transform ① Contour Integral 围线积分 for any kind of
ROC Integral line: for any kind of ② Partial-Fraction Expansion 部分分式展开 for rational

A. Partial-Fraction Expansion for Rational
1. Basic Z-Transform Pairs ( example)

2. Idea 一阶极点 二阶极点 一阶极点 ② Get by Formula in Appendix (Partial-Fraction Expansion) ROC

B. Examples

for ROC: for ROC: for ROC:

10.7 Analysis and Configuration of LTI systems using Z-Transform
System Function of LTI System : A. Response of LTI System to ,where System Function System Function or Transfer Function

B. Explanation of （类似于 ） 对各衰减因子各频率的衰减复正弦信号的幅度调整和相位调整作用 其中： or 函数集 的选择 幅频特性（给定 ） 相频特性（给定 ）

<Example> Integral Line

2. From the Linear-Coefficient Different Equation of LTI System
C. The Method to Obtain 1. From : 2. From the Linear-Coefficient Different Equation of LTI System , Linearity, Time-Shifting

<Example> Coefficient of right-side of Equ.
Coefficient of left-side of Equ. <Example>

1. exterior outside of a circle Causality ROC: 2. including
System Performance vs. A. Causality vs. 1. exterior outside of a circle Causality ROC: 2. including Cross outer most pole Causality 1. exterior outside of a circle ROC: 2. Including Rational

B. Stability vs. Stability Fourier Transform ROC ROC Stable Unstable

<Example> Unstable, causal Stable, noncausal Unstable, noncausal

C. Stable & Causal System ~
Causality All poles lies inside unit circle Exterior to the circle Acrossing outer most pole Rational Stability

10.7.3 Z-Domain Analysis of LTI System
1. Idea : Basic relation between input and output : Relation between any input and output ①信号分解 ②已知输入输出 ③响应合成

(For zero-state response)
2. Steps (For zero-state response) Key : ① ③ （类似于 域分析）

3. Role of LTI System explained by Z-Domain Analysis

4. Example ：求 <Solution>

10.8 System Function Block Program of LTI System

10.9 The Unilateral Laplace Transform
Definition i.e. 单边化

<10.32> causal 单边化 For causal signal

<10.33> non-causal 单边化 For non-causal signal 1 2

10.9.2 Properties of Unilateral Z-Transform
Table (Compared to Table 10.1) Difference A. Roc: B. Time Reversal: Don’t exist C. Convolution:

D. Time Shifting:

10.9.3 Solving Difference Equation Using the Unilateral Z-Transform
<10.37> Causal LTI System , , state input Full Response Zero state response Zero input response

Causal ① If (Full Response) ROC ② If (zero-state) Causal ROC

* Alternative Way of Solving Zero-State Response: when
Causal →ROC 实际未说明初始状态都是零状态

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