Region of Convergence (ROC) in the z-plane

region of convergence (ROC) in the z-plane

1. ROC of Z-transform PREPARED BY JAMI VENKATA SUMAN ECE DEPARTMENT GMRIT

2. The z-transform and the DTFT The z-transform is a function of the complex z variable Convenient to describe on the complex z-plane If we plot z= e j  for =0 to 2 we get the unit circle Re Im Unit Circle  r=1 2 2  2

3. Stability and the ROC For a > : If the ROC is outside the unit circle, the signal is unstable. If the ROC includes the unit circle, the signal is stable. 3

4. For a < : If the ROC is outside the unit circle, the signal is unstable. If the ROC includes the unit circle, the signal is stable. 4

5. Example : If: The z -Transform is the same, but the region of convergence is different. 5

6. For : If the ROC includes the unit circle, the signal is stable. If the ROC includes the unit circle, the signal is unstable. 6

7. Two-Sided Exponential Sequence Re Im o o x x 7

8. Properties of the ROC 8

9. 1. The ROC is an annular ring in the z -plane centered about the origin (which is equivalent to a vertical strip in the s-plane ). 2. DTFT exists if and only if the ROC includes the unit circle 3. The ROC does not contain any poles (similar to the Laplace transform). 4. If x [ n ] is of finite duration, then the ROC is the entire z -plane except possibly z = and/or z =  : 5. If x [ n ] is a right-sided sequence, and if | z | = r is in the ROC, then all finite values of z for which | z | > r are also in the ROC . 6. If x [ n ] is a left-sided sequence, and if | z | = r is in the ROC, then all finite values of z for which | z | < r are also in the ROC . 9

10. 7. If x [ n ] is a two-sided sequence, and if | z | = r is in the ROC, then the ROC consists of a ring in the z-plane including | z | = r . Example: right-sided left-sided two-sided 10

11. 8. The ROC must be a connected region 9. If the z-transform of x(n) is rational then its ROC is bounded by poles or extended to infinity. 10. If the z-transform of x(n) is rational and right sided then ROC is z-plane outside the outermost pole. 11. If the z-transform of x(n) is rational and left sided then ROC is the region in z-plane inside the innermost pole. 11

12. Z-Transform Region of Convergence |z|<a b<|z| b<|z|<a all |z| 12

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