INVERSE Z TRANSFORM

4. INVERSE Z TRANSFORM



AIM: To develop a program for Computing Inverse Z-Transform

EQUIPMENTS: MATLAB 7.5

Learning Objectives: To make the students familiar with concept of inverse Z-transform
with the use of MATLAB.

THEORY:
Description: In mathematics and signal processing, the Z-transform converts a discrete
time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-
domain representation. The Z-transform, like many other integral transforms, can be defined as
either a one-sided or two-sided transform.
The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the function X(z)
defined as
.
Alternatively, in cases where x[n] is defined only for n = 0, the single-sided or unilateral
Z-transform is defined as
Description: X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \
Description: X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=0}^{\infty} x[n] z^{-n} \


In signal processing, this definition is used when the signal is causal.
Rational Z-transform to partial fraction form:
Consider the transfer function in the rational form i-e;
18z3
G(z)= ------------------
18z3+3z2-4z-1
We can evaluate the partial fraction form of the above system using matlab command. The
partial fraction form be,
G(z)= 0.36__ + __0.24__ + _0.4____
1 – 0.5z-1 1+0.33 z-1 (1+0.33 z-1)
Matlab command that converts rational z-transform in to partial fraction form is
‘residuez’.
If you want to see the poles and zeros in a zplane. This function displays the poles and zeros
of discrete-time systems. Use the under given matlab command
zplane(b,a)

ALGORITHM:


1. Write the poles and zeros of the input sequence.
2. Returned vector R contains the residues, Column vector contains P contains the pole
locations. And row vector contains the direct terms.



PROGRAM CODE:
%program to perform Inverse Z-Transform
b=[1,0.4*sqrt(2)];
a=[1,-0.8*sqrt(2),0.64];
[R,P,C]=residuez(b,a);
R
P
C
Zplane(b,a);