Computation of unit sample, unit step and sinusoidal response of  the given LTI system and verifying  its physical reliability and stability properties 

Aim: To Unit Step And Sinusoidal Response Of The Given LTI System And Verifying Its Physical Realizability And Stability Properties.

EQUIPMENT:

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

MATLAB Software
 

Theory:

A discrete time system performs an operation on an input signal based on predefined criteria to produce a modified output signal. The input signal x(n) is the system excitation, and y(n) is the system response. The transform operation is shown as,

If the input to the system is unit impulse i.e. x(n) = d(n) then the output of the system is known as impulse response denoted by h(n) where,

h(n) = T[d(n)]

we know that any arbitrary sequence x(n) can be represented as a weighted sum of discrete impulses. Now the system response is given by,

For linear system (1) reduces to

%given difference equation y(n)-y(n-1)+.9y(n-2)=x(n);

Program - 1:   Calculate and plot the impulse response and step response

b=[1];

a=[1,-1,.9];

x=impseq(0,-20,120);

n = [-20:120];

h=filter(b,a,x);

subplot(3,1,1);stem(n,h);

title('impulse response');

xlabel('n');ylabel('h(n)');

stepseq(0,-20,120);

s=filter(b,a,x);

s=filter(b,a,x);

subplot(3,1,2);

stem(n,s);

title('step response');

xlabel('n');ylabel('s(n)')

t=0:0.1:2*pi;

x1=sin(t);

%impseq(0,-20,120);

n = [-20:120];

h=filter(b,a,x1);

subplot(3,1,3);stem(h);

title('sin response');

xlabel('n');ylabel('h(n)');

figure;

zplane(b,a);

Output

Program - 2:  Computation of Unit Sample response :

%This program finds the unit sample response of the given discrete  system  

a=input('enter the coefficient vector of input  starting from the coefficient of x(n) term')
b=input('enter the  coefficient vector of  output starting from the coefficient of y(n) term')
n1=input('enter the lower limit of the range of impulse response')
n2=input('enter the upper limit of the range of impulse response')
n=[n1:n2];
x=zeros(1,length(n));
i=find(n==0);
x(i)=1;
h=filter(a,b,x);
stem(n,h)
xlabel('time')
ylabel('amplitude')
grid
title('Unit Sample response of the given discrete system y(n)-y(n-1)+0.9y(n-2)=x(n)')

RESULT:

enter the coefficient vector of input  starting from the coefficient of x(n) term 1

a =  1

enter the  coefficient vector of  output starting from the coefficient of y(n) term  [1 -1 0.9]

b = 1.0000   -1.0000    0.9000

enter the lower limit of the range of impulse response-50

n1 =  -50

enter the upper limit of the range of impulse response50

n2 =  50

Output:

Program - 3:  Computation of Unit Step response 

%This program finds the unit step response of the given discrete  system

a=input('enter the coefficient vector of input  starting from the coefficient of x(n) term')
b=input('enter the  coefficient vector of  output starting from the coefficient of y(n) term')
n1=input('enter the lower limit of the range of impulse response')
n2=input('enter the upper limit of the range of impulse response')
n=[n1:n2];
x=zeros(1,length(n));
i=find(n==0);
x(i:length(x))=1;
h=filter(a,b,x);
stem(n,h)
xlabel('time')
ylabel('amplitude')
grid
title('Unit Step response of the given discrete system y(n)-y(n-1)+0.9y(n-2)=x(n)')

RESULT:

enter the coefficient vector of input  starting from the coefficient of x(n) term 1  

a =  1

enter the  coefficient vector of  output starting from the coefficient of y(n) term  [1 -1 0.9]

b = 1.0000   -1.0000    0.9000

enter the lower limit of the range of impulse response-50

n1 =  -50

enter the upper limit of the range of impulse response50

n2 =  50

Output:

Program - 4: Computation of Sinusoidal response 

%This program finds the Sinusoidal response of the  given discrete  system

a=input('enter the coefficient vector of input  starting from the coefficient of x(n) term')
b=input('enter the  coefficient vector of  output starting from the coefficient of y(n) term')
f=input('enter the sampling frequency')
T=1/f;
t=0:T/f:T;
x=sin(2*pi*f*t);
h=filter(a,b,x);
subplot(2,1,1)
stem(t,x)
xlabel('time')
ylabel('amplitude')
grid
title('Sinusoidal input(f=50) for the  discrete system y(n)-y(n-1)+0.9y(n-2)=x(n)')
subplot(2,1,2)
stem(t,h)
xlabel('time')
ylabel('amplitude')
grid
title('Sinusoidal(f=50) response of  the discrete system y(n)-y(n-1)+0.9y(n-2)=x(n)')

RESULT:

enter the coefficient vector of input  starting from the coefficient of x(n) term 1  

a =  1

enter the  coefficient vector of  output starting from the coefficient of y(n) term  [1 -1 0.9]

b = 1.0000   -1.0000    0.9000

enter the sampling frequency100

f = 100

Output:

 VERIFYING ITS PHYSICAL RELIABILITY AND STABILITY PROPERTIES:

a=input('enter the coefficients of numerator in the order of decreasing order of the variable z')
b=input('enter the coefficients of denominator in the order of decreasing order of the variable z')
p=roots(a);
q=roots(b);
i=find(abs(q)<1);
R1=input('enter the lower bound of ROC')
R2=input('enter the upper bound of ROC')
if length(p)<=length(q) &amp; R2==inf
   disp('The system is causal')
else
   disp('The system is not causal')
end
if R1<1&amp;R2>1 | length(i)==length(q)
   disp('The system is stable')
else
   disp('The system is unstable')
end

OUTPUT:

1.H(z)=z/(3z2-4z+1) ROC |z|>1

enter the coefficients of numerator in the order of decreasing order of the variable z

a =1     0

enter the coefficients of denominator in the order of decreasing order of the variable z  

b=3    -4     1

enter the lower bound of ROC

R1 =1

enter the upper bound of ROC

R2 =Inf

The system is causal. The system is unstable

2.H(z)=z/(3z2-4z+1) ROC 1/3<|z|<1

enter the coefficients of numerator in the order of decreasing order of the variable z

a =1     0

enter the coefficients of denominator in the order of decreasing order of the variable z  

b=3    -4     1

enter the lower bound of ROC

R1 =1/3

enter the upper bound of ROC

R2 =1

The system is not causal. The system is unstable

3.H(z)= (z2-1.5z)/[z2-(5/6)z+1/6] ROC |z|>0.5

enter the coefficients of numerator in the order of decreasing order of the variable z

a =1.0000   -1.5000         0

enter the coefficients of denominator in the order of decreasing order of the variable z

b =1.0000   -0.8333    0.1667

enter the lower bound of ROC

R1 =0.5000

enter the upper bound of ROC

R2 =inf.  

The system is causal. The system is stable

4.H(z)= (z2-1.5z)/(z2-(5/6)z+1/6) ROC 1/3< |z|<0.5

enter the coefficients of numerator in the order of decreasing order of the variable z

a =1.0000   -1.5000         0

enter the coefficients of denominator in the order of decreasing order of the variable z

b =1.0000   -0.8333    0.1667

enter the lower bound of ROC

R1 =1/3

enter the upper bound of ROC

R2 =0.5.  

The system is not causal. The system is stable

LOCATING THE POLES AND ZEROS IN S-PLANE AND Z-PLANE:  

Z-transforms

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  

PROGRAM:-   ZEROS AND POLES IN S- PLANE    

clc;  
clear all;  
close all;
num=input('enter the numerator polynomial vector\n');    % [1 -2 1]
den=input('enter the denominator polynomial vector\n');    % [1 6 11 6]
H=tf(num,den)
[p z]=pzmap(H);
disp('zeros are at ');
disp(z);
disp('poles are at ');
disp(p);
pzmap(H);
if max(real(p))>=0
   disp(' All the poles do not lie in the left half of S-plane ');
   disp(' the given LTI systen is not a stable system ');
else
   disp('All the poles lie in the left half of S-plane ');
   disp(' the given LTI systen is a stable system ');  
end;

OUTPUT:-

Enter the numerator polynomial vector

[1 -2 1]

Enter the denominator polynomial vector

[1 6 11 6]

Transfer function:

   s^2 - 2 s + 1

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

s^3 + 6 s^2 + 11 s + 6

Zeros are at  

    1

    1

Poles are at  

  -3.0000

  -2.0000

  -1.0000

All the poles lie in the left half of S-plane  

The given LTI system is a stable system  

Result: In this experiment computation of unit sample, unit step and sinusoidal response of the given lti system and verifying its physical reliability and stability properties Using MATLAB.

Viva Questions:

1. What is Even Signal

Ans: If x(t)= x(-t) then x(t) is Even signal.

2. What is Odd Signal

Ans: If x(t)= -x(-t) then x(t) is Odd signal.

3. State the difference between a Signal and Sequence?

Ans: Signal is a function varies with time, sequence consisting of number samples.

4. What is Static and Dynamic System

Ans: Dynamic system output -constantly changing and dynamic system carry past and present inputs to get output,

Static system carry only present input to generate output