Basic simulation lab manual1

DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
Siddharth Institute of Engineering and Technology
(Affiliated to J.N.T.UNIVERSITY, ANANTAPUR)
Narayanavanam, puttur, AP.

II YEAR BTECH I SEMESTER

BASIC SIMULATION LAB MANUAL

PREPARED BY:
VERIFIED BY:

LIST OF EXPERIMENTS
S.No Name of the Experiment
Basic operations on matrices.
1.
Generation on various signals and Sequences (periodic and a
periodic), such as unit impulse, unit step, square, saw tooth,
2.
triangular, sinusoidal, ramp, sinc.

Operations on signals and sequences such as addition,
multiplication, scaling, shifting, folding, computation of
3.
energy and average power.

Finding the even and odd parts of signal/sequence and real
4. and imaginary part of signal.

5. Convolution between signals and sequences
Auto correlation and cross correlation between signals and
6. sequences.

Verification of linearity and time invariance properties of a
7. given continuous /discrete system.

Computation of unit sample, unit step and sinusoidal
response of the given LTI system and verifying its physical
8.
Reliability and stability properties.

Gibbs phenomenon.
9.

Finding the Fourier transform of a given signal and plotting
10.
its magnitude and phase spectrum

11. Waveform synthesis using Laplace Transform.

Locating the zeros and poles and plotting the pole zero maps
12. in s8plane and z8plane for the given transfer function.

Generation of Gaussian Noise (real and
complex),computation of its mean, M.S. Value and its skew,
13.
kurtosis, and PSD, probability distribution function.

14. Sampling theorem verification.

15. Removal of noise by auto correlation/cross correlation.

Extraction of periodic signal masked by noise using
16.
correlation.

17. Verification of Weiner8Khinchine relations.

18. Checking a random process for stationary in wide sense.

1

BASIC OPERATIONS ON MATRICES

Aim: To generate matrix and perform basic operation on matrices Using
MATLAB Software.
EQUIPMENTS:
PC with windows (95/98/XP/NT/2000).
MATLAB Software

CONCLUSION:

EXP.NO: 2

GENERATION OF VARIOUS SIGNALS AND SEQUENCES (PERIODIC
AND APERIODIC), SUCH AS UNIT IMPULSE, UNIT STEP,
SQUARE, SAWTOOTH, TRIANGULAR, SINUSOIDAL, RAMP,
SINC.

Aim: To generate different types of signals Using MATLAB Software.
EQUIPMENTS:
PC with windows
(95/98/XP/NT/2000).
MATLAB Software

Matlab program:

%unit impulse
generation clc
close all
n1=-3;
n2=4;
n0=0;
n=[n1:n
2];
x=[(n-n0)==0]
stem(n,x)

% unit step
generation n1=-4;
n2=5;
n0=0;

[y,n]=stepseq(n0,n1,n2);
stem(n,y); xlabel('n') ylabel('amplitude'); title('unit step');

% square wave wave
generator fs = 1000;
t = 0:1/fs:1.5;
x1 = sawtooth(2*pi*50*t); x2 =
square(2*pi*50*t);
subplot(2,2,1),plot(t,x1), axis([0 0.2 -1.2
1.2])
xlabel('Time (sec)');ylabel('Amplitude'); title('Sawtooth Periodic Wave')
subplot(2,2,2),plot(t,x2), axis([0 0.2 -1.2 1.2])
xlabel('Time (sec)');ylabel('Amplitude'); title('Square Periodic Wave');
subplot(2,2,3),stem(t,x2), axis([0 0.1 -1.2 1.2])
xlabel('Time (sec)');ylabel('Amplitude');

% sawtooth wave
generator fs = 10000;
t = 0:1/fs:1.5;
x = sawtooth(2*pi*50*t);
subplot(1,2,1);
plot(t,x), axis([0 0.2 -1
1]);
xlabel('t'),ylabel('x(t)')
title('sawtooth signal');
N=2; fs = 500;n =
0:1/fs:2; x =
sawtooth(2*pi*50*n);
subplot(1,2,2);

stem(n,x), axis([0 0.2 -1
1]);
xlabel('n'),ylabel('x(n)')
title('sawtooth
sequence');

To generate a trianguular pulse
A=2; t = 0:0.0005:1;
x=A*sawtooth(2*pi*5*t,0.25); %5 Hertz wave with duty cycle 25%
plot(t,x);
grid
axis([0 1 -3 3]);

%%To generate a trianguular
pulse fs = 10000;t = -1:1/fs:1;
x1 = tripuls(t,20e-3); x2 = rectpuls(t,20e-3);
subplot(211),plot(t,x1), axis([-0.1 0.1 -0.2 1.2])
xlabel('Time (sec)');ylabel('Amplitude'); title('Triangular Aperiodic Pulse')
subplot(212),plot(t,x2), axis([-0.1 0.1 -0.2 1.2])

xlabel('Time (sec)');ylabel('Amplitude'); title('Rectangular Aperiodic Pulse')
set(gcf,'Color',[1 1 1]),

%%To generate a rectangular pulse
t=-5:0.01:5;
pulse = rectpuls(t,2); %pulse of width 2 time units
plot(t,pulse)
axis([-5 5 -1 2]);
grid

% sinusoidal signal
N=64; % Define Number of samples
n=0:N-1; % Define vector
n=0,1,2,3,...62,63 f=1000; % Define
the frequency
fs=8000; % Define the sampling
frequency x=sin(2*pi*(f/fs)*n); %
Generate x(t) plot(n,x); % Plot x(t) vs.
t
title('Sinewave [f=1KHz,
fs=8KHz]'); xlabel('Sample
Number'); ylabel('Amplitude');

% RAMP
clc
close all
n=input('enter the length of ramp');
t=0:n; plot(t); xlabel('t');

ylabel('amplitude');
title ('ramp')

% sinc
x = linspace(-5,5); y =
sinc(x);
subplot(1,2,1);plot(x,y
) xlabel(‘time’);
ylabel(‘amplitude’);
title(‘sinc function’);
subplot(1,2,2);stem(x,
y); xlabel(‘time’);
ylabel(‘amplitude’);
title(‘sinc function’);

EXP.NO: 3

OPERATIONS ON SIGNALS AND SEQUENCES SUCH AS ADDITION,
MULTIPLICATION, SCALING, SHIFTING, FOLDING,
COMPUTATION OF ENERGY AND AVERAGE POWER

Aim: To perform arithmetic operations different types of signals Using
MATLAB Software.
EQUIPMENTS:
PC with windows
(95/98/XP/NT/2000).
MATLAB Softwar
%plot the 2 Hz sine wave in the top panel
t = [0:.01:1]; % independent (time) variable
A = 8; % amplitude
f1 = 2; % create a 2 Hz sine wave lasting
1 sec s1 = A*sin(2*pi*f1*t);
f2 = 6; % create a 4 Hz sine wave lasting
1 sec s2 = A*sin(2*pi*f2*t);
figure subplot(4,1,1) plot(t, s1)
title('1 Hz sine wave')
ylabel('Amplitude')
%plot the 4 Hz sine wave in the middle panel subplot(4,1,2)
plot(t, s2)
title('2 Hz sine wave')
ylabel('Amplitude')
%plot the summed sine waves in the bottom panel subplot(4,1,3)
plot(t, s1+s2) title('Summed sine waves') ylabel('Amplitude') xlabel('Time (s)')
xmult=s1.*s2;
subplot(4,1,4); plot(xmult); title('multiplication'); ylabel('Amplitude') xlabel('Time (s)')

%signal folding clc; clear all t=0:0.1:10; x=0.5*t; lx=length(x); nx=0:lx-1;
xf=fliplr(x);
nf=-fliplr(nx); subplot(2,1,1); stem(nx,x); xlabel('nx'); ylabel('x(nx)');
title('original signal'); subplot(2,1,2); stem(nf,xf); xlabel('nf'); ylabel('xf(nf)');
title('folded signal');

23

%plot the 2 Hz sine wave scalling

t = [0:.01:1]; % independent (time) variable
A = 8; % amplitude
f1 = 2; % create a 2 Hz sine wave
lasting 1 sec s1 = A*sin(2*pi*f1*t);
subplot(3,2,1) plot(s1); xlabel('t');
ylabel('amplitude'); s2=2*s1; subplot(3,2,2) plot(s2);
xlabel('t');

s3=s1/2; subplot(3,2,3) plot(s3); xlabel('t');
ylabel('amplitude'); subplot(3,2,4) stem(s1);
xlabel('t'); ylabel('amplitude'); s2=2*s1; subplot(3,2,5) stem(s2);
xlabel('t'); ylabel('amplitude'); s3=s1/2; subplot(3,2,6) stem(s3);
xlabel('t');
ylabel('amplitude
');

Excersize questions: Sketch the following questions using MATLAB

1. x(t)= u(-t+1)

2. x(t)=3r(t-1)
3. x(t)=U(n+2-u(n-3)
4. x(n)=x1(n)+x2(n)where x1(n)={1,3,2,1},x2(n)={1,-2,3,2}
5. x(t)=r(t)-2r(t-1)+r(t-2)
6. x(n)=2δ(n+2)-2δ(n-4), -5≤ n ≥5.
7. X(n)={1,2,3,4,5,6,7,6,5,4,2,1} determine and plot the following
sequence a. x1(n)=2x(n-5-3x(n+4))
b. x2(n)=x(3-n)+x(n)x(n-
2)

CONCLUSION: Inthis experiment the various oprations on signals
have been performedUsing MATLAB have been demonstrated.

EXP.NO: 4

FINDING THE EVEN AND ODD PARTS OF SIGNAL/SEQUENCE AND
REAL AND IMAGINARY PART OF SIGNAL

Aim: program for finding even and odd parts of signals Using MATLAB
Software.
EQUIPMENTS:
PC with windows
(95/98/XP/NT/2000). MATLAB
Software

%even and odd signals program:

t=-4:1:4;
h=[ 2 1 1 2 0 1 2 2 3 ];
subplot(3,2,1)
stem(t,h);
xlabel('time'); ylabel('amplitude');
title('signal');
n=9;

for i=1:9 x1(i)=h(n); n=n-1;
end subplot(3,2,2)
stem(t,x1);
title('folded
signal'); z=h+x1
subplot(3,2,3);
stem(t,z);
xlabel('time');
ylabel('amplitude'); title('sum
of two signal'); subplot(3,2,4);
stem(t,z/2);
title('even
signal'); a=h-
x1;
subplot(3,2,5);
stem(t,a);
title('difference of two signal');
subplot(3,2,6);
stem(t,a/2);
title('odd signal');

% energy clc;
close all; clear all; x=[1,2,3]; n=3
e=0;
for i=1:n;
e=e+(x(i).*x(i));
end

% energy clc;
close all; clear all; N=2 x=ones(1,N) for i=1:N
y(i)=(1/3)^i.*x(i);
end n=N;
e=0;
for i=1:n;
e=e+(y(i).*y(i));
end

%
power
clc;
close all;
clear all;
N=2
x=ones(1,
N) for
i=1:N
y(i)=(1/3)^i.*x(i);
end
n=
N;
e=0
;
for i=1:n;
e=e+(y(i).*y(i))
;
end
p=e/(2*N+
1);

% power
N=input('type a value for
N'); t=-N:0.0001:N;
x=cos(2*pi*50*t).^2;
disp('the calculated power p of the
signal is'); P=sum(abs(x).^2)/length(x)
plot(t,x);
axis([0 0.1 0 1]);
disp('the theoretical power of the
signal is'); P_theory=3/8

CONCLUSION:

EXP.NO:
5 LINEAR CONVOLUTION

Aim: To find the out put with linear convolution operation Using MATLAB
Software.
EQUIPMENTS:
PC with windows
(95/98/XP/NT/2000). MATLAB
Software

Program:
clc;
close all;
clear all;
x=input('enter input
sequence'); h=input('enter
impulse response');
y=conv(x,h);
subplot(3,1,1);
stem(x);
xlabel('n');ylabel('x(n)'
); title('input signal')
subplot(3,1,2);
stem(h);
xlabel('n');ylabel('h(n)'
); title('impulse
response')
subplot(3,1,3);

stem(y);
xlabel('n');ylabel('y(n)')
; title('linear
convolution')
disp('The resultant signal is');
disp(y)

linear convolution
output:
enter input sequence[1 4 3
2] enter impulse response[1
0 2 1] The resultant signal
is
1 4 5 11 10 7 2

EXP.NO: 6

6. AUTO CORRELATION AND CROSS CORRELATION BETWEEN
SIGNALS AND SEQUENCES.
………………………………………………………………………………………………
Aim: To compute auto correlation and cross correlation between signals and
sequences
EQUIPMENTS:
PC with windows
(95/98/XP/NT/2000). MATLAB
Software

% Cross
Correlation clc;
close all;
clear all;
x=input('enter input sequence');
h=input('enter the impulse suquence');
subplot(3,1,
1); stem(x);
xlabel('n');
ylabel('x(n)');
title('input signal');
subplot(3,1,
2); stem(h);
xlabel('n');
ylabel('h(n)');
title('impulse
signal');
y=xcorr(x,h);
subplot(3,1,3);
stem(y);
xlabel('n');
ylabel('y(n)');
disp('the resultant signal is');
disp(y);
title('correlation signal');

% auto
correlation clc;
close all;
clear all;
x = [1,2,3,4,5]; y = [4,1,5,2,6];
subplot(3,1,
1); stem(x);
xlabel('n');
ylabel('x(n)');
title('input
signal');
subplot(3,1,2);
stem(y);
xlabel('n');
ylabel('y(n)');
title('input
signal');
z=xcorr(x,x);
subplot(3,1,3);
stem(z);
xlabel('n');
ylabel('z(n)');
title('resultant signal signal');

CONCLUSION: In this experiment correlation of various signals
have been 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, defence applications and sound
navigation

Excersize questions: perform convolution between the following signals
1. X(n)=[1 -1 4 ], h(n) = [ -1 2 -3 1]
2. perform convolution between the. Two periodic
sequences x1(t)=e-3t{u(t)-u(t-2)} , x2(t)= e -3t
for 0 ≤ t ≤ 2

EXP.NO: 7

VERIFICATION OF LINEARITY AND TIME INVARIANCE PROPERTIES OF A
GIVEN CONTINUOUS /DISCRETE SYSTEM.

Aim: To compute linearity and time invariance properties of a given continuous
/discrete system

EQUIPMENTS:

MATLAB Software

Program1:
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

Program2:

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

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)

Program1:
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),gri
d;
subplot(3,1,2),stem(yd),g
rid;
subplot(3,1,3),stem(d),gri
d;

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),gri
d;
subplot(3,1,2),stem(yd),g
rid;
subplot(3,1,3),stem(d),gri
d;

EXP.NO:8

COMPUTATION OF UNIT SAMPLE, UNIT STEP AND SINUSOIDAL
RESPONSE OF THE GIVEN LTI SYSTEM AND VERIFYING ITS
PHYSICAL REALIZABILITY AND STABILITY PROPERTIES.

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

EQUIPMENTS:
PC with windows
(95/98/XP/NT/2000).
MATLAB
Software

%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);

EXP.NO: 9 GIBBS PHENOMENON

Aim: To verify the Gibbs Phenomenon.

EQUIPMENTS:
MATLAB Software

Gibbs Phenomina Program :

t=0:0.1:(pi*8); y=sin(t); subplot(5,1,1); plot(t,y); xlabel('k');
ylabel('amplitude'); title('gibbs phenomenon'); h=2;
%k=3;
for k=3:2:9 y=y+sin(k*t)/k; subplot(5,1,h);
plot(t,y); xlabel('k'); ylabel('amplitude'); h=h+1;
end

CONCLUSION: In this experiment Gibbs phenomenon have been
demonstrated Using MATLAB

EXP.NO: 10.

FINDING THE FOURIER TRANSFORM OF A GIVEN SIGNAL AND
PLOTTING ITS MAGNITUDE AND PHASE SPECTRUM

Aim: to find the fourier transform of a given signal and plotting its
magnitude and phase spectrum

EQUIPMENTS:
MATLAB Software

EQUIPMENTS:

MATLAB Software

Program:
clc;
close all;
clear all;
x=input('enter the sequence'); N=length(x);
n=0:1:N-1; y=fft(x,N) subplot(2,1,1); stem(n,x);
title('input sequence'); xlabel('time index n----->'); ylabel('amplitude x[n]-
---> '); subplot(2,1,2);
stem(n,y);
title('output sequence');
xlabel(' Frequency index K---->');
ylabel('amplitude X[k]------>');

FFT magnitude and Phase plot:

clc
close all x=[1,1,1,1,zeros(1,4)]; N=8;
X=fft(x,N); magX=abs(X),phase=angle(X)*180/pi; subplot(2,1,1)
plot(magX); grid xlabel('k')
ylabel('X(K)') subplot(2,1,2) plot(phase);

grid xlabel('k') ylabel('degrees')

CONCLUSION: In this experiment the fourier transform of a given signal
and plotting its magnitude and phase spectrum have been demonstrated
using matlab

Exp:11
LAPLACE TRNASFORMS

Aim: To perform waveform synthesis using Laplece Trnasforms of a given
signal
Program for Laplace Transform:
f=t
syms f t; f=t; laplace(f)
Program for nverse Laplace Transform
f(s)=24/s(s+8) invese LT
syms F s F=24/(s*(s+8)); ilaplace(F)
y(s)=24/s(s+8) invese LT poles and zeros

Signal synthese using Laplace Tnasform:
clear all clc t=0:1:5 s=(t);
subplot(2,3,1) plot(t,s); u=ones(1,6) subplot(2,3,2) plot(t,u); f1=t.*u;
subplot(2,3,3) plot(f1);
s2=-2*(t-1); subplot(2,3,4); plot(s2);
u1=[0 1 1 1 1 1]; f2=-2*(t-1).*u1; subplot(2,3,5); plot(f2);
u2=[0 0 1 1 1 1]; f3=(t-2).*u2; subplot(2,3,6); plot(f3); f=f1+f2+f3; figure;
plot(t,f);
% n=exp(-t);
% n=uint8(n);
% f=uint8(f);
% R = int(f,n,0,6)
laplace(f);

CONCLUSION: In this experiment the Triangular signal synthesised using
Laplece Trnasforms using MATLAB

EXP.NO: 12

LOCATING THE ZEROS AND POLES AND PLOTTING THE POLE ZERO
MAPS IN S-PLANE AND Z-PLANE FOR THE GIVEN TRANSFER FUNCTION.

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

EQUIPMENTS:
MATLAB Software

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

Result: :

EXP.NO: 13
13. Gaussian noise

%Estimation of Gaussian density and Distribution Functions

%% Closing and Clearing
all clc;
clear all;
close all;

%% Defining the range for the Random
variable dx=0.01; %delta x
x=-3:dx:3; [m,n]=size(x);

%% Defining the parameters of the pdf
mu_x=0; % mu_x=input('Enter the value of mean');
sig_x=0.1; % sig_x=input('Enter the value of varience');

%% Computing the probability density
function px1=[];
a=1/(sqrt(2*pi)*sig_x);
for j=1:n
px1(j)=a*exp([-((x(j)-mu_x)/sig_x)^2]/2);
end

%% Computing the cumulative distribution
function cum_Px(1)=0;
for j=2:n
cum_Px(j)=cum_Px(j-1)+dx*px1(j);
end

%% Plotting the
results figure(1)
plot(x,px1);grid
axis([-3 3 0 1]);
title(['Gaussian pdf for mu_x=0 and sigma_x=', num2str(sig_x)]);
xlabel('--> x')
ylabel('--> pdf')
figure(2)
plot(x,cum_Px);gri
d axis([-3 3 0 1]);
title(['Gaussian Probability Distribution Function for mu_x=0 and
sigma_x=', num2str(sig_x)]);
title('ite^{omegatau} = cos(omegatau) + isin(omegatau)')
xlabel('--> x')
ylabel('--> PDF')

EXP.NO: 14
14. Sampling theorem verification

Aim: To detect the edge for single observed image using sobel edge detection
and canny edge detection.

EQUIPMENTS:
MATLAB Software

Figure 2: (a) Original signal g(t) (b) Spectrum G(w)
δ (t) is the sampling signal with fs = 1/T > 2fm.

Figure 3: (a) sampling signal δ (t) ) (b) Spectrum δ (w)

Let gs(t) be the sampled signal. Its Fourier Transform Gs(w) isgiven by

Figure 4: (a) sampled signal gs(t) (b) Spectrum Gs(w)

To recover the original signal G(w):
1. Filter with a Gate function, H2wm(w) of width 2wm
Scale it by T.

Figure 5: Recovery of signal by filtering with a fiter of width 2wm

Aliasing
{ Aliasing is a phenomenon where the high frequency components of the
sampled signal interfere with each other because of inadequate sampling
ws < 2wm.

Figure 6: Aliasing due to inadequate sampling

Aliasing leads to distortion in recovered signal. This is the
reason why sampling frequency should be atleast twice thebandwidth of
the signal. Oversampling
{ In practice signal are oversampled, where fs is
signi_cantly higher than Nyquist rate to avoid
aliasing.

Figure 7: Oversampled signal-avoids aliasing t=-10:.01:10;
T=4; fm=1/T; x=cos(2*pi*fm*t); subplot(2,2,1); plot(t,x);
xlabel('time');ylabel('x(t)') title('continous time signal') grid;
n1=-4:1:4 fs1=1.6*fm; fs2=2*fm; fs3=8*fm;
x1=cos(2*pi*fm/fs1*n1); subplot(2,2,2); stem(n1,x1); xlabel('time');ylabel('x(n)')
title('discrete time signal with fs<2fm')
hold on subplot(2,2,2); plot(n1,x1) grid;
n2=-5:1:5; x2=cos(2*pi*fm/fs2*n2); subplot(2,2,3); stem(n2,x2);
xlabel('time');ylabel('x(n)')
title('discrete time signal with fs=2fm')
hold on

subplot(2,2,3); plot(n2,x2) grid;
n3=-20:1:20;

x3=cos(2*pi*fm/fs3*n
3); subplot(2,2,4);
stem(n3,x3);
xlabel('time');ylabel('x(
n)')
title('discrete time signal with fs>2fm')
hold on
subplot(2,2
,4);
plot(n3,x3)
grid;

CONCLUSION: In this experiment the sampling theorem have been verified
Using MATLAB

REMOVAL OF NOISE BY AUTO CORRELATION/CROSS
CORRELATION

Aim: removal of noise by auto correlation/cross correlation

EQUIPMENTS:
MATLAB Software

a)auto correlation clear all
clc t=0:0.1:pi*4; s=sin(t);
k=2; subplot(6,1,1) plot(s); title('signal s'); xlabel('t');
ylabel('amplitude'); n = randn([1 126]); f=s+n; subplot(6,1,2) plot(f);
title('signal f=s+n'); xlabel('t'); ylabel('amplitude'); as=xcorr(s,s); subplot(6,1,3)
plot(as);
title('auto correlation of s'); xlabel('t'); ylabel('amplitude'); an=xcorr(n,n);
subplot(6,1,4)
plot(an);

title('auto correlation of
n'); xlabel('t');
cff=xcorr(f,f);
subplot(6,1,5)
plot(cff);
title('auto correlation of
f'); xlabel('t');
hh=as+an;
subplot(6,1,6)
plot(hh);
title('addition of
as+an'); xlabel('t');

B)CROSS CORRELATION :

clear all clc
t=0:0.1:pi*4;
s=sin(t);
k=2;
%sk=sin(t+k);

subplot(7,1,1)
plot(s);
title('signal s');xlabel('t');ylabel('amplitude');
c=cos(t); subplot(7,1,2) plot(c);
title('signal c');xlabel('t');ylabel('amplitude');
n = randn([1 126]); f=s+n; subplot(7,1,3) plot(f);
title('signal f=s+n');xlabel('t');ylabel('amplitude');
asc=xcorr(s,c); subplot(7,1,4) plot(asc);
title('auto correlation of s and c');xlabel('t');ylabel('amplitude');
anc=xcorr(n,c); subplot(7,1,5) plot(anc);
title('auto correlation of n and c');xlabel('t');ylabel('amplitude');
cfc=xcorr(f,c); subplot(7,1,6) plot(cfc);
title('auto correlation of f and c');xlabel('t');ylabel('amplitude');
hh=asc+anc; subplot(7,1,7) plot(hh);
title('addition of asc+anc');xlabel('t');ylabel('amplitude');

76

EXP.No:16
Program:
EXTRACTION OF
Clear all; close all; clc; n=256; k1=0:n-1;
P
x=cos(32*pi*k1/n)+sin(48*pi*k1/n);
E
plot(k1,x)
R
%Module to find period of input signl k=2;
I
xm=zeros(k,1); ym=zeros(k,1); hold on
O
for i=1:k
D
[xm(i) ym(i)]=ginput(1);
I
plot(xm(i), ym(i),'r*');
C
end
period=abs(xm(2)-xm(1)); rounded_p=round(period);
S
m=rounded_p
I
% Adding noise and plotting noisy signal
G
N
A
L

M
A
y=x+randn(1,n);
S
figure plot(k1,y)
K
E
D

B
Y

N
O
I
S
E

U
S
I
N
G

C
O
R
R
E
L
A
T
I
O
N
Extraction of
Periodic Signal
Masked By Noise
Using
Correlation

% To generate impulse train with the period as that of input signal
d=zeros(1,n);
for i=1:n

if (rem(i-1,m)==0)

d(i)=1;

end end
%Correlating noisy signal and impulse train cir=cxcorr1(y,d);
%plotting the original and reconstructed signal m1=0:n/4;
figure

Plot (m1,x(m1+1),'r',m1,m*cir(m1+1));

CONCLUSION: In this experiment the Weiner-Khinchine Relation have
been verified using MATLAB.

EXP.No:17
VERIFICATION OF WIENER–KHINCHIN RELATION

AIM: Verification of wiener–khinchine relation

EQUIPMENTS:
MATLAB Software

PROGRAM:
Clc
clear all;
t=0:0.1:2*pi; x=sin(2*t); subplot(3,2,1); plot(x); au=xcorr(x,x); subplot(3,2,2);
plot(au); v=fft(au); subplot(3,2,3); plot(abs(v)); fw=fft(x); subplot(3,2,4);
plot(fw);
fw2=(abs(fw)).^2;
subplot(3,2,5); plot(fw2);

Result:

EXP18.

CHECKING A RANDOM PROCESS FOR STATIONARITY IN WIDE SENSE.

AIM: Checking a random process for stationary in wide sense.

EQUIPMENTS:
MATLAB Software

MATLAB PROGRAM:

Clear all
Clc
y = randn([1 40]) my=round(mean(y));
z=randn([1 40]) mz=round(mean(z)); vy=round(var(y)); vz=round(var(z));
t = sym('t','real'); h0=3; x=y.*sin(h0*t)+z.*cos(h0*t); mx=round(mean(x));
k=2;
xk=y.*sin(h0*(t+k))+z.*cos(h0*(t+k));
x1=sin(h0*t)*sin(h0*(t+k));
x2=cos(h0*t)*cos(h0*(t+k)); c=vy*x1+vz*x1;
% if we solve “c=2*sin (3*t)*sin (3*t+6)" we get c=2cos (6)
% which is a costant does not depent on variable’t’
% so it is wide sence stationary

Result:

Basic simulation lab manual1

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