Signals and Systems Lecture 27 The z-Transform ROC of z-Transform Inverse z-Transform

Chapter 10 The Z-Transform

Chapter 10 The Z-Transform unit circle Z-plane

Chapter 10 The Z-Transform Example 10.1 Example 10.2

Chapter 10 The Z-Transform Example 10.3

Chapter 10 The Z-Transform §10.2 The Region of Convergence for the Z-Transform Property 1: The ROC of X(z) consists of a ring in the z-plane centered about the origin. Property 2: The ROC does not contain any poles.

Chapter 10 The Z-Transform Property 3: If is of finite duration, then the ROC is the entire z-plane, except possibly z=0 and/or z=∞. positive powers of z negative powers of z

Chapter 10 The Z-Transform Example 10.5 Example 10.6 (N-1)st order pole Zeros:

Chapter 10 The Z-Transform Property 4: If is right sided, If is right sided, Furthermore, if If is not causal, positive powers of z

Chapter 10 The Z-Transform Example

Chapter 10 The Z-Transform Property 5: If is left sided, If is left sided, Furthermore, if If is not anticausal, negative powers of z

Chapter 10 The Z-Transform Example

Chapter 10 The Z-Transform Property 6: If is two sided, The ROC of X(z) is Example 10.7 X(z) does not exist.

Chapter 10 The Z-Transform Example is right sided is two sided is left sided

Chapter 10 The Z-Transform Basic Z-Transform pairs:

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Chapter 10 The Z-Transform §10.3 The Inverse Z-Transform 1. Partial-Fraction Expansion Example 10.9 Determine for all possible ROC.

Chapter 10 The Z-Transform 2. Power-Series Expansion (幂级数展开法） Power-Series of z-1 Example 10.12

Chapter 10 The Z-Transform Example 10.13 Consider

Chapter 10 The Z-Transform Example 10.14 Consider

Chapter 10 The Z-Transform Example 10.14 Consider the z-transform

Chapter 10 The Z-Transform §10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot Pole vector: Zero vector:

Chapter 10 The Z-Transform Example Consider a first-order system

Chapter 10 The Z-Transform §10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot Pole vector: Zero vector:

Chapter 10 The Z-Transform Example Consider a first-order system

Problem Set P801 10.22 (a) (b) 10.23 (a) (b)