1. 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

2. √ 10.0 Introduction §5 Frequency Domain Analysis Frequency analysis2 3
﹡ Condition:
Frequency analysis √
﹡ Problems:
Frequency analysis

3. ﹡Cause: Basic signal:
represent
﹡Measure: Basic signal: √
represent

4. 10.1 The Z-Transform Pair A. The Transform Pair Under Condition
we have ② ① 反 正
z-plane
Integral line

5. B. Understanding of The Transform Pairs
﹡Inverse Transform

6. Frequency
Frequency

7. ﹡The Transform
﹡ Similarity :
﹡ Similarity :

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

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

10. 10.2 The Region of Convergence of The Z-Transform
The ROC.
<Examples 10.1>
Condition=ROC
① if ,
Unit Circle ROC
ROC

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

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

13. 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

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

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

16. C.
ROC
: ROC
双边信号
环形收敛域
或无收敛域

17. <Example 10.7>
双边信号
环形收敛域
双边信号
无收敛域

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

19. <Example 10.6 >
pole
zero

20. E.
右边信号
收敛域向外

21. F.
左边信号
收敛域向内

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

23. 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

24. B. Example

25. 10.4.2 Geometric Evaluation of Fourier Transform
A. The Method
as above,
if Unit Circle ROC,
Let
B. Example

26. 10.5 Properties of Z-Transform
Linearity
Time shifting
shift
（ 可能加入或去掉）

27. <Proof>
<Example>

28. 10.5.3 Scaling in the Z-Domain外扩 or 内收
Scaling in the Z-Domain
<Proof>
Scaling 平移

29. Time Reversal
<Proof>
1/R

30. 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

31. Conjugation
For real signal :

32. 10.5.7 The Convolution Property
Differentiation in Z- Domain

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

34. <Example>
Linearity, Time-scaling

35. 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

36. 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

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

38. 2. Idea
一阶极点
二阶极点
一阶极点 ① ③
② Get by Formula in Appendix (Partial-Fraction Expansion) ④
ROC

39. B. Examples ① ②

40. ② 左 ③
for ROC: 右 左
for ROC: 右
for ROC:

41. 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

42. 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
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 ）

43. <Example>
Integral Line

44. 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

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

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

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

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

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

50. 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

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

52. 3. Role of LTI System explained by Z-Domain Analysis
幅度调整
相位调整
调整幅度
调整相位 :
选择合成 的函数集
② :
规定了每个函数集的幅相调整方法

53. 4. Example ， ：求
<Solution> ① ③ ②

54. 10.8 System Function Block Program of LTI System

55. 10.9 The Unilateral Laplace Transform
Definition
i.e. √
单边化

56. <10.32>
causal ①
单边化
For causal signal ②

57. <10.33>
non-causal ①
单边化
For non-causal signal ② 1 2

58. 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:

59. D. Time Shifting:

60. 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

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

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