Basic simulation lab manual1

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DEPARTMENT OF ELECTRONICS AND COMMUNICATION

ENGINEERINGSiddharth 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 EXPERIMENTSS.No

Name of the Experiment

1. Basic operations on matrices.

2.Generation on various signals and Sequences (periodic and a periodic), such as unit impulse, unit step, square, saw

tooth, triangular, sinusoidal, ramp, sinc.

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

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

5. Convolution between signals and sequences

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

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

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

9. Gibbs phenomenon.

10.Finding the Fourier transform of a given signal and plotting

its magnitude and phase spectrum

11. Waveform synthesis using Laplace Transform.

Locating the zeros and poles and plotting the pole zero maps 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, kurtosis, and PSD, probability distribution function.

14. Sampling theorem verification.

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

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

17. Verification of Weiner8Khinchine relations.

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

EXP.NO: 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 clcclose all n1=-3; n2=4; n0=0; n=[n1:n2];x=[(n-n0)==0]stem(n,x)

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

[y,n]=stepseq(n0,n1stem(n,y); xlabel('n')

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

% RAMPclcclose alln=input('enter the length of ramp');t=0:n; plot(t); xlabel('t');

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

% sincx = 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’);

CONCLUSION:

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) variableA = 8; % amplitudef1 = 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');

%plot the 2 Hz sine wave scalling

t = [0:.01:1]; % independent (time) variableA = 8; % amplitudef1 = 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');ylabel('amplitude');

s3=s1/2; subplot(3ylabel('amplitude')xlabel('t'); ylabel('xlabel('t'); ylabel('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); end subplot(3,2,2) stem(t,x1);xlabel('time'); ylabel('amplitude');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);xlabel('time'); ylabel('amplitude');title('even signal'); a=h-x1; subplot(3,2,5); stem(t,a);xlabel('time'); ylabel('amplitude');

title('difference of two signal');subplot(3,2,6);stem(t,a/2);xlabel('time'); ylabel('amplitude');title('odd signal');

% energy clc;close all; clear all; x=[1,2,3]; n=3e=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);

% powerN=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 convolutionoutput:

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

CONCLUSION:

EXP.NO: 6

6. AUTO CORRELATION AND CROSS CORRELATION BETWEEN SIGNALS AND SEQUENCES.……………………………………………………………………………………………… Aim: To compute auto correlation and cross correlation between signals and sequencesEQUIPMENTS:

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 signals1. 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:

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

elsedisp('Given system is satisfy linearity property');

endsubplot(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');

elsedisp('Given system is satisfy linearity property');

endsubplot(3,1,1), stem(n,y); grid subplot(3,1,2), stem(n,yt); grid subplot(3,1,3), stem(n,d); grid

Programclc;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),grid; subplot(3,1,2),stem(yd),grid; subplot(3,1,3),stem(d),grid;

Program2: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;

CONCLUSION:

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 VerifyingIts 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);

CONCLUSION: I

EXP.NO: 9 GIBBS PHENOMENON

Aim: To verify the Gibbs Phenomenon.

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

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

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:

clcclose 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') yla

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

Exp:11LAPLACE TRNASFORMS

Aim: To perform waveform synthesis using Laplece Trnasforms of a given signal

Program for Laplace Transform:f=tsyms 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

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: 1313. 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 xx=-3:dx:3; [m,n]=size(x);

%% Defining the parameters of the pdfmu_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);gridaxis([-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);grid axis([-3 3 0 1]);title(['Gaussian Probability Distribution Function for mu_x=0 and sigma_x=', num2str(sig_x)]);title('\ite^{\omega\tau} = cos(\omega\tau) + isin(\omega\tau)')

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.

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 2wmScale 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 thereason 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*n3); 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 verifiedUsing MATLAB

EXP.No:15

REMOVAL OF NOISE BY AUTO CORRELATION/CROSS CORRELATION

Aim: removal of noise by auto correlation/cross correlation

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

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');

Result:

EXP.No:16

ION OF PERIODIC SIGNAL MASKED BY NOISE USING CORRELATION

Extraction of Periodic Signal

Masked By Noise Using Correlation

Program:

Clear all; close all; clc; n=256; k1=0:n-1;x=cos(32*pi*k1/n)+sin(48*pi*k1/n);plot(k1,x)%Module to find period of input signl k=2;xm=zeros(k,1); ym=zeros(k,1); hold onfor i=1:k

[xm(i) ym(i)]=ginput(1);

plot(xm(i), ym(i),'r*');

eriod=abs(xm(2)-xm(1)); rounded_p=round(period); m=rounded_p

% Adding noise and plotting noisy signal

y=x+randn(1,n);figure plot(k1,y)

% 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

PROGRAM:Clcclear 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.

MATLAB PROGRAM:

Clear allClcy = 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: