The document discusses z-transforms, which are a generalization of the Fourier transform used to analyze discrete-time signals and systems. It defines the z-transform, region of convergence (ROC), zeros and poles. The ROC depends on the location of poles - it is the exterior region for a right-sided sequence, interior for a left-sided sequence, and the overlap for a two-sided sequence. A system is stable if its z-transform ROC includes the unit circle, as this is where the Fourier transform is evaluated.

1. Waqas Ahmed

2. Introduction z -Transform Zeros and Poles Region of Convergence z -Transform Properties

3. Introduction

4. A generalization of Fourier transform Why generalize it? FT does not converge on all sequence Notation good for analysis Bring the power of complex variable theory deal with the discrete-time signals and systems

5. z-Transform

6. The z -transform of sequence x ( n ) is defined by Let z = e  j  . Fourier Transform

7. Fourier Transform is to evaluate z-transform on a unit circle . Re Im z = e  j  

8. Zeros and Poles

9. Give a sequence, the set of values of z for which the z -transform converges , i.e., | X ( z )|<  , is called the region of convergence . ROC is centered on origin and consists of a set of rings.

10. A stable system requires that its Fourier transform is uniformly convergent. Fact: Fourier transform is to evaluate z -transform on a unit circle. A stable system requires the ROC of z -transform to include the unit circle. Re Im 1

11. For convergence of X ( z ), we require that

12. Which one is stable? a  a Re Im 1 a  a Re Im 1

13. For convergence of X ( z ), we require that

14. Which one is stable? a  a Re Im 1 a  a Re Im 1

15. Region of Convergence

16. where P ( z ) and Q ( z ) are polynomials in z . Zeros: The values of z ’s such that X ( z ) = 0 Poles: The values of z ’s such that X ( z ) = 

17. ROC is bounded by the pole and is the exterior of a circle . Re Im a

18. ROC is bounded by the pole and is the interior of a circle . Re Im a

19. A ring or disk in the z-plane centered at the origin. The Fourier Transform of x ( n ) is converge absolutely iff the ROC includes the unit circle . The ROC cannot include any poles Finite Duration Sequences: The ROC is the entire z -plane except possibly z =0 or z =  . Right sided sequences: The ROC extends outward from the outermost finite pole in X ( z ) to z =  . Left sided sequences: The ROC extends inward from the innermost nonzero pole in X ( z ) to z =0.

20. Consider the z -transform with the pole pattern: Case 1: A right sided Sequence. Re Im a b c

21. Consider the rational z -transform with the pole pattern: Case 2: A left sided Sequence. Re Im a b c

22. Consider the rational z -transform with the pole pattern: Case 3: A two sided Sequence. Re Im a b c

23. Consider the rational z -transform with the pole pattern: Case 4: Another two sided Sequence. Re Im a b c

24. Important z -Transform Pairs

25. Sequence z -Transform ROC All z All z except 0 (if m >0) or  (if m <0)

26. z -Transform Theorems and Properties

27. Overlay of the above two ROC’s