Locating the zeros and poles and plotting the pole zero maps in z-plane for the given transfer function 

Aim: To locating the zeros and poles and plotting the pole zero maps in z-plane for the given transfer function

EQUIPMENTS:

PC with windows (95/98/XP/NT/2000).

MATLAB Software

Theory: 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.

Bilateral Z-transform

The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the function X(z) defined as

Unilateral Z-transform

Alternatively, in cases where x[n] is defined only for n = 0, the single-sided or unilateral Z-transform is defined as

In signal processing, this definition is used when the signal is causal.

The roots of the equation P(z) = 0 correspond to the 'zeros' of X(z)

The roots of the equation Q(z) = 0 correspond to the 'poles' of X(z)

The ROC of the Z-transform depends on the convergence of the

Program-1:   plotting poles and zeros on z-plane

clc;
close all
clear all;
%b= input('enter the numarator cofficients')
%a= input('enter the dinomi cofficients')
b=[1 2 3 4]
a=[1 2 1 1 ]
zplane(b,a);

Output:

Applications :Z-Transform is used to find the system responses

Program-2:   plotting poles and zeros on z-plane     

clc;  
clear all; close all;
num=input('enter the numerator polynomial vector \n');                  %[1 0 0]  
den=input('enter the denominator polynomial vector \n');                 %[1 1 0.16]
H=filt(num,den)
z=zero(H);
disp('the zeros are at ');
disp(z);
[r p k]=residuez(num,den);
disp('the poles are at ');
disp(p);
zplane(num,den);
title('Pole-Zero map in the Z-plane');
if max(abs(p))>=1
   disp('all the poles do not lie with in the unit circle');
   disp('hence the system is not stable');
else
   disp('all the poles lie with in the unit circle');
   disp('hence the system is stable');
end;

OUTPUT:-

Enter the numerator polynomial vector  

[1 0 0]

Enter the denominator polynomial vector  

[1 1 0.16]

Transfer function:

        1

--------------------

1 + z^-1 + 0.16 z^-2

The zeros are at  

    0

    0

The poles are at  

  -0.8000

  -0.2000

All the poles lie with in the unit circle

Hence the system is stable  

Result: In this experiment the zeros and poles and plotting the pole zero maps in s-plane and z-plane for the given transfer function using MATLAB.

Viva Questions:

1.what are the ROC properties of a Z.T

Ans: a1 x1[n] + a2 x2[n] = a1 X1(z) + a2 X2(z)

2.Define Initial Value Theorem of a Z.T

Ans:

if x(n] is casual

3. Define Final Value Theorem of a Z.T

Ans: 

Only if poles of (z-1)X(z)  are inside the unit circle

4. Define the condition for distortion-less transmission through the system

Ans:   Transmission is said to be distortion less if the input and output have identical wave shapes within a multiplicative constant.A delayed output that retains the input waveform is also considered distortion less.Thus in distortion-less transmission, the input x(t) and output y(t) satisfy the condition:y(t) = Kx(t - t) where t is the delay time and k is a constant