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Chapter 10 The Z-Transform

§10. 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. §5 Frequency Domain Analysis. 1. 2. 3. √. Frequency analysis. ﹡ Condition:. Frequency analysis.

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Chapter 10 The Z-Transform

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  1. §10 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

  2. §5 Frequency Domain Analysis 1 2 3 √ Frequency analysis ﹡ Condition: Frequency analysis ﹡ Problems: 10.0 Introduction

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

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

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

  6. Frequency Frequency

  7. ﹡ Similarity : ﹡ Similarity : ﹡The Transform

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

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

  10. Condition=ROC ① if , Unit Circle ROC ROC 10.2 The Region of Convergence of The Z-Transform 10.2.1 The ROC. <Examples 10.1>

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

  12. Unit Circle ROC ① if , Unit Circle < Examples10.2>

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

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

  15. A. ROC : B. Poles ROC 10.2.3. General Rule for ROC 右边信号 双边信号 左边信号

  16. : ROC C. ROC 双边信号 环形收敛域 或无收敛域

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

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

  19. pole zero <Example 10.6>

  20. 收敛域向外 右边信号 E.

  21. 收敛域向内 左边信号 F.

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

  23. zero pole 零点距离积 极点距离积 零点相位和 极点相位和 10.4 Geometric Evaluation of The Fourier Transform From The Zero-Pole Plot 10.4.1 Geometric Evaluation of Z-Transform A. The Method

  24. B. Example

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

  26. 10.5.1 Linearity 10.5.2 Time shifting shift （ 可能加入或去掉） 10.5 Properties of Z-Transform

  27. <Proof> <Example>

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

  29. 1/R 10.5.4 Time Reversal <Proof>

  30. Where integer ，if n is a multiple of k ， else k=3 -4 -3 -2 -1 0 1 2 3 4 -4k -3k -2k -k 0 k 2k 3k 4k 补零 （时域扩展） 10.5.5 Time Expansion

  31. For real signal : 10.5.6 Conjugation

  32. 10.5.7 The Convolution Property 10.5.8 Differentiation in Z- Domain

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

  34. Linearity, Time-scaling <Example>

  35. For causal ， 10.5.9 The Initial-value Theorem (检验变换的正确性) ，we have 10.5.10 Table 10.1 include all properties 10.6 Some Common Z-Transform Pairs Table 10.2

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

  37. A. Partial-Fraction Expansion for Rational 1. Basic Z-Transform Pairs (10.5.8 example)

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

  39. B. Examples ① ②

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

  41. 10.7 Analysis and Configuration of LTI systems using Z-Transform 10.7.1 System Function of LTI System : A. Response of LTI System to ,where System Function System Function or Transfer Function

  42. 幅频特性（给定 ） 相频特性（给定 ） 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

  43. <Example> 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. <Example>

  46. 10.7.2 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 ROC: ② 2. Including Rational ① ②

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

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