Chapter 5 Z Transform

2/45  Z transform –Representation, analysis, and design of discrete signal –Similar to Laplace transform –Conversion of digital signal into frequency domain 1. Introduction

3/45  z transform –Two-sided z transform –One-sided z transform If n < 0, x(n) = 0 2. z transform (5-1) (5-2)

4/45  Example 5-1 (1)Non-causal (2)Non-causal Fig. 5-1.

5/45  Example 5-1 (3) Causal (4) Geometric series with common ratio of Fig. 5-1.

6/45 Convergence of series Region of convergence Fig Region of convergence (5-3)

7/45 Z=2 Z=1/2

8/45  Example 5-2 –z transform Region of convergence Fig. 5-3.

9/45  Characteristic of z transform (1) is a polynomial equation of z and determined from samples, (2) can be reconstructed by removing in (3) is independent to sampling interval, (4)z transform of delayed signal by samples is z transform of delayed signal Expression of difference equation

10/45 (5) Same to discrete Fourier transform by replacing to (5-4)

11/45  Table of z transform Table 5-1.

12/45  Example 5-3 (1) (2) (3)

13/45  Ideally sampled function, –Laplace transform 3. Relation between Z transform and Laplace transform (5-5) (5-6)

14/45 –z transform –Relation (5-7) (5-8)

15/45  Example 5-4 (1)   (2)  

16/45 (3)  

17/45 –Periodicity –s-plane and z-plane Fig (5-9)

18/45  Corresponding points (1)Left side plane in s plane  inside of unit circle in z plane (2)Right side plane in s plane  out of unit circle (3) axis in s plane  unit circle in z plane (4)Increased frequency in s plane  mapped on the unit circle in z plane

19/45 –Corresponding points

20/45  Definition of inverse z transform –Power series of 4. Inverse Z transform (5-10) (5-11) (5-12)

21/45 –Three methods to obtain inverse z transform Power series expansion Partial fraction expansion Residue

22/45 –Power series expansion Long division (5-13)

23/45  Example 5-5 –Inverse z transform using long division

24/45 –Partial fraction expansion where is poles of, is coefficients for partial fraction, and

25/45 Partial fraction for N>M where is calculated using long division. (5-17) (5-18) (5-19)

26/45  Example 5-6 –Inverse z transform

27/45

28/45 Inverse z transform using table 5-1

29/45  Example 5-7 –Inverse z transform poles

30/45 Partial fraction

31/45 z transform Inverse z transform

32/45  Example 5-8 –Inverse z transform

33/45 Inverse z transform

34/45 –Residue Cauchy’s theory using contour integral Calculation of residue where contour integral including all poles. where, m is order of poles. (5-20) (5-21)

35/45 For single pole Unit circle Fig (5-22)

36/45  Example 5-9 –Find discrete time signal If, Inverse z transform

37/45 Sum of residue

38/45  Example 5-10 –Inverse z transform where and as

39/45 Fig. 5-5.

40/45 n=0,

41/45 n>0

42/45  Example 5-11 where F(z) has poles at z=0.5, and z=1.

43/45 Sum of residue

44/45  linearity  Convolution  Differentiation 5. Characteristic of z transform (5-23) (5-24) (5-25)