VERIFICATION OF BOTH LINEAR AND TIME INVARIANT SYSTEMS

LINEARITY PROPERTY:

If any system is claimed to be linear if it satisfies the superposition principal. Superposition principal state that Response to a weighted sum of input adequate to the corresponding weighted sum of the outputs of the system to every of the individual input signals.
If x(n) may be a input and y(n) may be a output then

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

 y₁(n) =T[x₁(n)] and 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 y₂(n)

Program:

clc;

close all;

clear all;

% Verification of Linearity of a given System.

% a) y(n)=nx(n) b) y=x^2(n)

n=0:40;

a1=input(‘enter the scaling factor a1=’);

a2=input(‘enter the scaling factor a2=’);

x1=cos(2*pi*0.1*n);

x2=cos(2*pi*0.4*n);

x3=a1*x1+a2*x2;

%y(n)=n.x(n);

y1=n.*x1;

y2=n.*x2;

y3=n.*x3;

yt=a1*y1+a2*y2;

yt=round(yt);

y3=round(y3);

if y3==yt

 disp(‘given system [y(n)=n.x(n)]is Linear’);

else

 disp(‘given system [y(n)=n.x(n)]is non Linear’);

end

%y(n)=x(n).^2

x1=[1 2 3 4 5];

x2=[1 4 7 6 4];

x3=a1*x1+a2*x2;

y1=x1.^2;

y2=x2.^2;

y3=x3.^2;

yt=a1*y1+a2*y2;

if y3==yt

 disp(‘given system [y(n)=x(n).^2 ]is Linear’);

else

 disp(‘given system is [y(n)=x(n).^2 ]non Linear’);

end

Output will obtained as

Enter the scaling factor a1=3

Enter the scaling factor a2=5

Given system [y(n) =n.x(n)]is Linear

Given system is [y(n) =x(n).^2 ]non Linear

Time-Invariant System:

A system is named time-invariant if its input-output characteristics don’t change with time. X(t) is the input and Y(t) will be the output, X(t-k) is a delayed form of input by k seconds, Y(t-k) is delayed form of output by k seconds.

 If Y(t)=T[X(t)] while Y(t-k)=T[X(t-k)] then system is time invariant system.

clc;

close all;

clear all;

% Verification of your Time Invariance of a Discrete System

% a)y=x^2(n) b) y(n)=nx(n)

clc;

clear all;

close all;

n=1:9;

x(n)=[2 1 4 3 6 5 8 7 9];

d=3; % some time delay

xd=[zeros(1,d),x(n)];%x(n-k)

y(n)=x(n).^2;

yd=[zeros(1,d),y];%y(n-k)

disp(‘transformation of delay signal yd:’);

disp(yd)

dy=xd.^2; % T[x(n-k)]

disp(‘delay of transformation signal dy:’);

disp(dy)

if dy==yd

 disp(‘given system [y(n)=x(n).^2 ]is time invariant’);

else

 disp(‘given system is [y(n)=x(n).^2 ]not time invariant’);

end

y=n.*x;

yd=[zeros(1,d),y(n)];

disp(‘transformation of delay signal yd:’);disp(yd);

n1=1:length(xd);

dy=n1.*xd;

disp(‘delay of transformation signal dy:’);disp(dy);

if yd==dy

 disp(‘given system [y(n)=nx(n)]is a time invariant’);

else

 disp(‘given system [y(n)=nx(n)]not a time invariant’);

 end

Output:

Transformation of delay signal yd:

     0     0     0     4     1    16     9    36    25    64    49    81

Delay of transformation signal dy:

     0     0     0     4     1    16     9    36    25    64    49    81

Given system [y(n)=x(n).^2 ]is time invariant

Transformation of delay signal yd:

     0     0     0     2     2    12    12    30    30    56    56    81

Delay of transformation signal dy:

     0     0     0     8     5    24    21    48    45    80    77   108

Given system [y(n)=nx(n)]not a time invariant