Chapter 5 Z Transform

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)

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

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

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

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

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

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.

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

Table of z transform Table 5-1.

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

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

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

Example 5-4 

(3)  

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

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

Corresponding points Table 5-2. 2

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

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

Power series expansion Using long division (5-13)

Example 5-5 Inverse z transform using long division

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

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

Example 5-6 Inverse z transform Partial fraction

Inverse z transform using table 5-1

Example 5-7 Find discrete sequence poles

Partial fraction form Eq.(5-16)

z transform Inverse z transform using table 5-1

Example 5-8 Find discrete sequence Partial fraction

Inverse z transform

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.

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

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

Sum of residue

Example 5-10 Inverse z transform where and as

Fig. 5-5.

n=0,

n>0

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

Sum of residue

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