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Signals and Systems Lecture 27
The z-Transform ROC of z-Transform Inverse z-Transform
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Chapter 10 The Z-Transform
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Chapter 10 The Z-Transform
unit circle Z-plane
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Chapter 10 The Z-Transform
Example 10.1 Example 10.2
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Chapter 10 The Z-Transform
Example 10.3
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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.
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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
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Chapter 10 The Z-Transform
Example 10.5 Example 10.6 (N-1)st order pole Zeros:
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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
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Chapter 10 The Z-Transform
Example
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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
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Chapter 10 The Z-Transform
Example
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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.
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Chapter 10 The Z-Transform
Example is right sided is two sided is left sided
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Chapter 10 The Z-Transform
Basic Z-Transform pairs:
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More
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and More
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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.
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Chapter 10 The Z-Transform
2. Power-Series Expansion (幂级数展开法） Power-Series of z-1 Example 10.12
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Chapter 10 The Z-Transform
Example Consider
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Chapter 10 The Z-Transform
Example Consider
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Chapter 10 The Z-Transform
Example 10.14 Consider the z-transform
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Chapter 10 The Z-Transform
§10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot Pole vector: Zero vector:
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Chapter 10 The Z-Transform
Example Consider a first-order system
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Chapter 10 The Z-Transform
§10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot Pole vector: Zero vector:
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Chapter 10 The Z-Transform
Example Consider a first-order system
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Problem Set P (a) (b) 10.23 (a) (b)
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