1. Chapter 5 Z Transform

2. z transform z transform of a sequence Two-sided z transform
One-sided z transform
If n < 0, x(n) = 0
ROC(region of convergence)
The region in the z plane in which the power series converges
(5-1)
(5-2)

3. Example 5-1
Non-causal
Fig. 5-1.
Fig. 5-1.

4. (3) Causal
(4) Causal
Fig. 5-1.
Fig. 5-1.

5. Geometric series with common ratio of
z = 2
z = 1/2
(5-3)

6. Region of convergence
Region of convergence
Fig. 5-2.

7. Example 5-2
z transform
Region of convergence
Fig. 5-3.

8. Properties of z transform
is a polynomial of z and determined from sequence,
can be reconstructed by removing in
is independent of sampling interval,
z transform of delayed signal by samples is
z transform of delayed signal
i.e.

9. becomes Discrete Fourier Transform if replacing to
let
becomes Discrete Fourier Transform
if replacing to
(5-4)

10. Table of z transform
Table 5-1.

11. Example 5-3
(1)
(2)
(3)

12. Relation between Z transform and Laplace transform
Ideally sampled function,
Laplace transform
(5-5)
(5-6)

13. z transform for sequence
The relation between z transform and Laplace transform
(5-7)
(5-8)

14. Example 5-4 

15. (3)  

16. Relation between s-plane and z-plane
Periodicity
Relation between s-plane and z-plane
(5-9)
Fig. 5-4.

17. Corresponding points between s and z planes
Left side in s plane  inside of unit circle in z plane
Right side in s plane  out of unit circle
axis in s plane  unit circle in z plane
Increased frequency in s plane such as and  mapped on same point on the unit circle in z plane

18. Corresponding points
Table 5-2. 2

19. Inverse Z transform Definition of inverse z transform
Power series expansion of
Rational function
(5-10)
(5-11)
(5-12)

20. Three methods for inverse z transform
Power series expansion
Partial fraction expansion
Residue

21. Power series expansion
Using long division
(5-13)

22. Example 5-5
Inverse z transform using long division

23. Partial fraction expansion
for ,
(5-14)
(5-15)
where is poles of ,
is coefficients for partial fraction, and
(5-16)

24. Partial fraction for N>M
(5-17)
where is calculated using long division.
(5-18)
(5-19)

25. Example 5-6
Inverse z transform
Partial fraction

27. Inverse z transform using table 5-1

28. Example 5-7
Find discrete sequence
poles

29. Partial fraction
form Eq.(5-16)

30. z transform
Inverse z transform using table 5-1

31. Example 5-8
Find discrete sequence
Partial fraction

32. Inverse z transform

33. Residue Cauchy’s theory using contour integral Calculation of residue
(5-20)
where is contour integral including all poles.
(5-21)
where ,
m is order of poles.

34. For single pole
(5-22)
Unit circle
Fig. 5-5.

35. Example 5-9 Find discrete time signal using residue If ,
Inverse z transform

36. Sum of residue

37. Example 5-10
Inverse z transform
where and as

38. Fig. 5-5.

39. n=0,

41. n>0

43. Example 5-11
Using residue
where F(z) has poles at z=0.5, and z=1.

44. Sum of residue

45. Properties of z transform
Linearity
Convolution
Differentiation
(5-23)
(5-24)
(5-25)