# Verification of Linearity and time in-variance  properties of a given continuous/discrete system

Aim: To compute linearity and time in-variance properties of a given continuous /discrete system

EQUIPMENTS:

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

MATLAB Software

THEORY:

LINEARITY PROPERTY:

Any system is said to be linear if it satisfies the superposition principal

superposition principal state that Response to a weighted sum of input signal equal to the corresponding weighted sum of the outputs of the system to each of the individual input signals

X(n)-----------input signal

Y(n) --------- output signal

Y(n)=T[x(n)]

Y1(n)=T[X1(n)] : Y2(n)=T[X2(n)]

x3=[a X1(n)] +b [X2(n) ]

Y3(n) = T [x3(n)]

= T [a X1(n)] +b [X2(n) ] = a Y1(n)+ b [X2(n) ]= Z3(n)

Let a [Y1(n)]+ b [X2(n) ] =Z 3(n)

If Y3(n )- Z 3(n)=0 then the system is stable other wise it is not stable.

Program-1:

``````clc;
clear all;
close all;
n=0:40; a=2; b=1;
x1=cos(2*pi*0.1*n);
x2=cos(2*pi*0.4*n);
x=a*x1+b*x2;
y=n.*x;
y1=n.*x1;
y2=n.*x2;
yt=a*y1+b*y2;
d=y-yt;
d=round(d)
if d
disp('Given system is not satisfy linearity property');
else
disp('Given system is satisfy linearity property');
end
subplot(3,1,1), stem(n,y); grid
subplot(3,1,2), stem(n,yt); grid
subplot(3,1,3), stem(n,d); grid``````

Output:

Result: d=0 so the it displays the output is 'Given system is satisfy linearity property'

Program-2

``````clc;
clear all;
close all;
n=0:40; a=2; b=-3;
x1=cos(2*pi*0.1*n);
x2=cos(2*pi*0.4*n);
x=a*x1+b*x2;
y=x.^2;
y1=x1.^2;
y2=x2.^2;
yt=a*y1+b*y2;
d=y-yt;
d=round(d);
if d
disp('Given system is not satisfy linearity property');
else
disp('Given system is satisfy linearity property');
end
subplot(3,1,1), stem(n,y); grid
subplot(3,1,2), stem(n,yt); grid
subplot(3,1,3), stem(n,d); grid``````

output:

Result: The d is not equal to zero, so the output displays ' Given system is not satisfy linearity property'

Program-3:

Program

``````clc;
close all;
clear all;
x=input('enter the sequence');
N=length(x);
n=0:1:N-1;
y=xcorr(x,x);
subplot(3,1,1);
stem(n,x);
xlabel(' n----->');ylabel('Amplitude--->');
title('input seq');
subplot(3,1,2);
N=length(y);
n=0:1:N-1;
stem(n,y);
xlabel('n---->');ylabel('Amplitude----.');
title('autocorr seq for input');
disp('autocorr seq for input');
disp(y)
p=fft(y,N);
subplot(3,1,3);
stem(n,p);
xlabel('K----->');ylabel('Amplitude--->');
title('psd of input');
disp('the psd fun:');
disp(p)``````

LINEAR TIME INVARIENT SYSTEMS (LTI):

A system is called time invariant if its input – output characteristics do not change with time.

X(t)---- input : Y(t) ---output

X(t-T) -----delay input by T seconds : Y(t-T) ------ Delayed output by T seconds

Program-1:

``````clc;
close all;
clear all;
n=0:40;
D=10;
x=3*cos(2*pi*0.1*n)-2*cos(2*pi*0.4*n);
xd=[zeros(1,D) x];
y=n.*xd(n+D);
n1=n+D;
yd=n1.*x;
d=y-yd;
if d
disp('Given system is not satisfy time shifting property');
else
disp('Given system is satisfy time shifting property');
end
subplot(3,1,1),stem(y),grid;
subplot(3,1,2),stem(yd),grid;
subplot(3,1,3),stem(d),grid;``````

Output:

Result:  d is not equal to zero so the output displays 'Given system is not satisfy time shifting property'.

Program-2:

``````clc;
close all;
clear all;
n=0:40;
D=10;
x=3*cos(2*pi*0.1*n)-2*cos(2*pi*0.4*n);
xd=[zeros(1,D) x];
x1=xd(n+D);
y=exp(x1);
n1=n+D;
yd=exp(xd(n1));
d=y-yd;
if d
disp('Given system is not satisfy time shifting property');
else
disp('Given system is satisfy time shifting property');
end
subplot(3,1,1),stem(y),grid;
subplot(3,1,2),stem(yd),grid;
subplot(3,1,3),stem(d),grid;``````

Output: