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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.

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Presentation on theme: "Chapter 10 The Z-Transform Complex Frequency Domain (Z-Domain) Analysis of LTI System ● Representation of Aperiodic Signals ● Response of LTI System to."— Presentation transcript:

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 §5 §10

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

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

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

5 B. Understanding of The Transform Pairs ﹡ Inverse Transform

6 Frequency

7 ﹡ The Transform ﹡ Similarity :

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

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

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

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

12 ① if, Unit Circle ROC Unit Circle

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

14 ROC for Left-sided Right-sided Integral ROC for Integral

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

16 C. ROC : ROC 双边信号 环形收敛域 或无收敛域

17 双边信号 环形收敛域 双边信号 无收敛域

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

19 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 ﹡ left-sided signal ﹡ Two-sided signal ﹡ right-sided signal ROC: Bounded by poles ﹡ ﹡

23 10.4 Geometric Evaluation of The Fourier Transform From The Zero-Pole Plot Geometric Evaluation of Z-Transform A. The Method 零点距离积 极点距离积 零点相位和 极点相位和 zero pole

24 B. Example

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

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

27

28 Scaling in the Z-Domain Scaling 平移 外扩 or 内收

29 Time Reversal 1/R

30 Time Expansion Where , if n is a multiple of k , else (时域扩展) integer k=3 -4k -3k -2k -k 0 k 2k 3k 4k 补零

31 Conjugation For real signal :

32 The Convolution Property Differentiation in Z- Domain

33 Differentiation Differentiation , Linear Important : useful in Inverse Z-Transform

34 Linearity, Time-scaling

35 The Initial-value Theorem Table 10.1 include all properties 10.6 Some Common Z-Transform Pairs Table 10.2 For causal , we have ( 检验变换的正确性 ) ,

36 10.3 Inverse Z-Transform ① Contour Integral 围线积分 ROC Integral line: ② Partial-Fraction Expansion 部分分式展开 for any kind of for rational

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

38 2. Idea ① ② Get by Formula in Appendix (Partial-Fraction Expansion) ③ ④ 一阶极点二阶极点一阶极点 ROC

39 B. Examples ① ②

40 ③ for ROC: 左 右 右 ② 左

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 or Transfer Function System Function

42 B. Explanation of 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 (类似于 ) or 函数集 的选择 其中: 相频特性(给定 ) 幅频特性(给定 )

43 Integral Line

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

45 Coefficient of right-side of Equ. Coefficient of left-side of Equ.

46 System Performance vs. A. Causality vs. CausalityROC: Causality Rational ROC: 2. including Cross outer most pole ① ② ①② 2. Including 1. exterior outside of a circle

47 B. Stability vs. Stability Fourier Transform ROC StableUnstable

48 Unstable, noncausal Stable, noncausal Unstable, causal

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

50 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 ③响应合成

51 2. Steps ① ③ ② Key : (类似于 域分析) ① ③① ③ (For zero-state response)

52 选择合成 的函数集 ① : 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 ② : 规定了每个函数集的幅相调整方法

53 4. Example , :求 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 ① ② causal For causal signal

57 For non-causal signal 单边化 non-causal 2 1 ① ②

58 Properties of Unilateral Z-Transform Table 10.3 (Compared to Table 10.1) Difference A. Roc: B. Time Reversal: Don’t exist C. Convolution:

59 D. Time Shifting:

60 Solving Difference Equation Using the Unilateral Z-Transform Causal LTI System input state Zero input response Zero state response,, Full Response

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

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


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