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The Z-Transform: Properties, Inverse Transform, and Examples

Learn about the definition, properties of ROC, inverse z-transform, and z-transform properties with examples in Chapter 3. Explore methods like inspection, partial fraction expansion, and power series. Understand the relation between H(z) and frequency response using B=[2,1,0,-1] and A=[1,0,0.5,0,-1]. Discover the characteristics of different sequences like causal, stable, two-sided, and more in the z-transform domain. Solve exercises 3.37 and 3.38 without contour integral method.

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The Z-Transform: Properties, Inverse Transform, and Examples

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  1. Chapter 3 the z-Transform 3.1 definition 3.2 properties of ROC 3.3 the inverse z-transform 3.4 z-transform properties

  2. 3.1 definition Figure 3.2

  3. the condition for convergence：

  4. ROC takes the poles as its boundary EXAMPLE:

  5. 3.2 properties of ROC

  6. 3.3 the inverse z-transform 1.inspection method 2.partial fraction expansion 3.power series expansion

  7. EXAMPLE: EXAMPLE:

  8. 3.4 z-transform properties

  9. 7.x[n] is causal

  10. EXAMPLE:

  11. Relation between H(z) and frequency responce: B=[2,1,0,-1] A=[1,0,0.5,0,-1] freqz(B,A)

  12. summary: 3.1 the z-transform 3.2 properties of ROC right-sides sequence: inside the circle left-sides sequence: outside the circle finite-duration sequence: the entire z-plane two-sided sequence: a ring causal sequence: including infinite stable sequence: including the unit circle 3.3 the inverse z-transform: inspection method partial fraction expansion power series expansion 3.4 z-transform properties

  13. Keys and difficulties： ROC; the convolution property; the relationship among system function, the impulse response, frequency response and difference equation; exercises: 3.37 3.38 （these can be solved without contour integral method）

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