z Transform and Inverse z & Pole Zero

8/13/2019 z Transform and Inverse z & Pole Zero

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AIM:To find Z and inverse Z transform and draw sketch pole zero plot of Z-domain transfer

function.

THEORY:

The Laplace transform and its discrete-time counterpart the z-transform are essentialmathematical tools for system design and analysis, and for monitoring the stability of a system.

A working knowledge of the z-transform is essential to the study of discretetime filters and

systems. It is through the use of these transforms that we formulate a closed-form mathematical

description of a system in the frequency domain, design the system, and then analyse the

stability, the transient response and the steady state characteristics of the system.

One of the most useful aspects of the z-transform analysis is the description of a system

in terms of the so-called poles and zeros of the system. The zeros of a transfer function H(z) arethe values of the variable z for which the transfer function (or equivalently its numerator) is

zero. Therefore the zeros are the roots of numerator polynomial . The poles of H(z) are the

values of the variable z for which H(z) is infinite. This happens when the denominator of H(z) is

zero. Therefore the poles of H(z) are the roots of the denominator polynomial.

To obtain the poles and zeros of H(z) rewrite the numerator and denominator

polynomials to avoid negative powers of the variable z as:

DHOLE PATIL COLLEGE OF ENGG. WAGHOLI , PUNE

SUBJECT: DIGITAL SIGNAL PROCESSING

Experiment No: Title: Z and inverse Z transform

Date of Performance: Class: T.E (I sem)Grade: Signature:


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Pole-Zero Description

The poles of a z-transform, H(z), are the values of z for which H(z)becomes infinity. The values of z for which H(z)becomes zero are referred to as zeros. The poles and zeros of H(z) may be real or complex. When they are complex, they occur in conjugate pairs, to ensure that the coefficients, ak

and bk, are real.

If the locations of the poles and zeros of H(z) are known, then H(z) itself can be readilyreconstructed to within a constant.

PROCEDURE:

Open MATLAB

Open new M-file Type the program Save in current directory Compile and Run the program For the output see command window\ Figure window

MATLAB CODE

%To find the inverse Z transform of

%X(Z)=z/(3z2-4z+1)

clc; %Clear the command window

%Accepting the coefficients of the equation.

b=input('Enter the numerator coefficients->')

a=input('Enter the denominator coefficients->')

%Displaying the Z transform for the given coefficients

[r,p,k] = residuez(b,a);

disp('Residues for the given Z transform will be')

disp(r)

disp('Poles for the given Z transform will be')

disp(p)


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COMMAND WINDOW OUTPUT::

Enter the numerator coefficients->[0 1]

b =

0 1

Enter the denominator coefficients->[3 -4 1]

a = 3 -4 1

Residues for the given Z transform will be

0.5000

-0.5000

Poles for the given Z transform will be

1.0000

0.3333

CODE FOR POLE ZERO

%Program to plot the pole zero diagram for

%X(Z)=(1-(1/2z))/(1+(1/z)+(2/z2)+(3/z3)

clc;

b=input('Enter the numerator coefficients->')

a=input('Enter the denominator coefficients->')

%Displaying the Z transform for the given coefficients

z=tf(b,a);

disp(z)

[z,p,k] = tf2zpk(b,a);

disp('Poles for the given Z transform will be')

disp(p)

disp('Zeros for the given Z transform will be')


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disp(z)

zplane(z,p)

xlabel('Real axis')

ylabel('Imag axis')

title('POLE ZERO PLOT')

COMMAND WINDOW OUTPUT::

Enter the numerator coefficients->[1 -2]

b =

1 -2

Enter the denominator coefficients->[1 1 2 3]

a =

1 1 2 3

tf object: 1-by-1

Poles for the given Z transform will be

0.1378 + 1.5273i

0.1378 - 1.5273i

-1.2757

Zeros for the given Z transform will be

0

0

2


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FIGURE WINDOW

CONCLUSION:

We have successfully managed to obtain the inverse Z-transform of the given Z-domain

transfer function & verify it practically.

We have also successfully plotted the pole zero diagram for the given function by

extracting the poles and zeros through the program and have verified it with the theoretical

calculations.

Documents

z Transform and Inverse z & Pole Zero