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Chapter 10 The z-Transform

Chapter 10 The z-Transform. 崔琳莉. The Z-Transform. Another transform, any end? Motivation : analogous to LT in CT So, the definition and properties of the z-Transform closely parallel those of the Laplace Transform Pay attention to the relation and difference between LT and zT.

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Chapter 10 The z-Transform

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  1. Chapter 10 The z-Transform 崔琳莉

  2. The Z-Transform • Another transform, any end? • Motivation: analogous to LT in CT • So, the definition and properties of the z-Transform closely parallel those of the Laplace Transform • Pay attention to the relation and differencebetween LT and zT

  3. In this chapter, we will learn… • The z-Transform (bilateral and unilateral) and the inverse z-transform • The Region of Convergence for the zT • Properties of the z-Transform • The Pole-Zero plot • Analysis LTI system using zT • Block Diagram

  4. x[n]=zn y[n]=H(z)zn h[n] 10.1 The Z-Transform • Review (assuming it converges)

  5. x[n]=zn y[n]=H(z)zn h[n] When z=ejw (unit magnitude), H(ejw) corresponds to the FT of h[n] When z=rejw (complex), H(z) corresponds to the zT of h(t) Definition: The (Bilateral) Z-Transform

  6. Im Unit circle z-plane w 1 Re • Relationship between DTFT and ZT (Similar to FT and LT)

  7. x[n] r-n • Region of Convergence ( ROC ) ROC: Range of |z| for X(z) to converge

  8. Relationship between LT and ZT If and sampled signal (1) suppose (2) We can get (1) = (2) Then, s-plane vs.z-plane ??

  9. Example 10.1

  10. Im Im Z-plane Z-plane a Re -1 1 Re -1 1 Specially, unit circle unit circle The ROC of signals in example 10.1 (right-sided signal)

  11. include |z|=1stable ROC outside |z|=1unstable

  12. include |z|=1stable ROC outside |z|=1unstable

  13. Example 10.2

  14. Im a - Re 1 1 The ROC of Example 10.2 (a left-sided signal)

  15. ROC include |z|=1 stable inside |z|=1 unstable

  16. Example The ROC is the entire z-plane .

  17. Example 10.3

  18. Example 10.4

  19. Rational z-Transforms • x[n] = linear combination of exponentials for n > 0 and for n < 0 X(z) is rational polynomials in z — characterized (except for a prefactor) by its poles and zeros — sometimes, it is convenient for X(z) to be expressed in terms of polynomials in z-1

  20. Im Re a -1 b 10.2 The Region of Convergence for z-Transform Property1: The ROC of X(z) consists of a ring in the z-plane centered about the origin. Property2: the ROC does not contain any poles. (equivalent to a vertical strip in the s-plane) (same as in LT)

  21. Property3: If x[n] is of finite duration, then the ROC is the entire z-plane ,except possibly z=0 and/or z= Example

  22. Property4: If x[n] is right sided, and if the circle |z|=r0 is in the ROC,then all finite values of z for which |z|>r0will also in the ROC. Z-plane Prove The ROC of a right-sided signal

  23. converges slower than

  24. Im Z-plane r Re b 0 Property5: If x[n] is left sided, and if the circle |z|=r0is in the ROC,then all values of z for which0<|z|<r0will also in the ROC. The ROC of a left-sided signal

  25. Property6: If x[n] is two sided, and if the circle |z|=r0is in the ROC,then the ROC will consist of a ring in the z-plane that includes the circle|z|=r0 .Normally, r1<|z|<r2. (r1<r2)

  26. Quiz: What types of signals do the following ROC correspond to? right-sided left-sided two-sided

  27. Example 10.6 Then zeros: or ROC:

  28. Im unit circle a Re (N - 1) st order poles: z=0 (N-1)st order Note: z=a is cancelled.

  29. Example 10.7

  30. Clearly, ROC does notexist if b > 1 ⇒ Noz-transform for b|n|.

  31. Im Unit circle z-plane x x x x Re Property7: If the z transform X(z) of x[n] is rational, then its ROC is bounded by poles or extends to infinity. In addition,no poles of X(z) are contained in the ROC.

  32. Property8: If the z transform X(z) of x[n] is rational — if x[n] is right sided，then the ROC is the region in the z-plane outside the outmost pole i.e., outside the circle of radius equal to the largest magnitude of the poles of X(z). —Furthermore,if x[n] is causal,then the ROC also includes z=∞.

  33. Property9: If the Z transform X(z) of x[n] is rational, — if x[n] is left sided, then the ROC is the region in the z-plane inside the innermost nonzero pole i.e., inside the circle of radius equal to the smallest magnitude of the poles of X(z) other than any at z=0 and extending inward to and possibly including z=0. — in particular. if x[n] is anticausal,then the ROC also includes z=0.

  34. Im Unit circle z-plane Re Example 10.8 How may possible ROCs in this Figure?

  35. Im Im Im Unit circle Unit circle Unit circle z-plane z-plane z-plane Re Re Re There are 3 possible ROCs shown in Figure before. ROC1 right sided ROC3 left sided ROC2 two sided Homework: 10.2 10.3 10.6 10.7

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