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:

Result: d=0 so output displays 'Given system is satisfy time shifting property'.
CONCLUSION: In this experiment Linearity and Time in-variance property of given system has bees verified performed Using MATLAB
Applications:it is used to measure the degree to which the two signals are similar and it is also used for radar detection by estimating the time delay.it is also used in Digital communication defense applications and sound navigation
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CreatedDec 08, 2019
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UpdatedMar 01, 2020
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Views409
Introduction
Basic operations on Matrices
Genaration of various signals and sequences
Operations on signals and sequences
Finding the even and odd parts of signal/sequence and real and imaginary parts of signal
Verification of Linearity and time in-variance properties of a given continuous/discrete system
Linear Convolution
Auto correlation and cross correlation between signals and sequences
Computation of unit sample, unit step and sinusoidal response of the given LTI system and verifying its physical reliability and stability properties
GIBBS phenomenon
Sampling theorem verification
Finding the Fourier transform of a given signal and plotting its magnitude and phase spectrum
Laplace Transforms
Locating the zeros and poles and plotting the pole zero maps in z-plane for the given transfer function
Gaussian Noise
Verification of Wiener- Khinchin relation
Removal of noise by auto-correlation/cross-correlation
Extraction of Periodic signal masked by noise using correlation.
Checking a Random process for stationarity in wide sense
To find a mean and variance of a discrete random variable
To find a moment generating function of a discrete random variable
Computation of Energy of sinusoidal signal
Computation of energy of rectangular pulse
Computation of Average Power
Waveform Synthesis
Find and plot the cumulative distribution and probability density functions of a random variable
Verification of central limit theorem