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

3. Introduction

5. z-Transform

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

8. Zeros and Poles

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

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