Basic Siumlation Lab - 12-13, I Semester- SRAWAN

7/29/2019 Basic Siumlation Lab - 12-13, I Semester- SRAWAN

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Basic Simulation lab

AURORAS ENGINEERING COLLEGEBHONGIR, NALGONDA DIST. 508116.

Lab manual of

BASIC SIMULATION LAB

2nd

Year 1st

Semester of ECE

(As per R09 Academic Regulation)

DEPARTMENT OF

ELECTRONICS AND COMMUNICATION ENGINEERING


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PREFACE

Research and development over the past 30 years has lead to significant advances in the field of DSP.

Covering the basics of linear continuous time and discrete time systems and their applications along

with MATLAB programs is the need of the day.

This lab is suitable for subjects like S&S, DSP. The various concepts of the subject are arranged logically

and explained in a simple user friendly language like MATLAB.

Working with MAT LAB will enable the students to enhance their concepts in S&S, its applications. This

lab totally consists of basics of signals and systems that is signals and their operations, Fourier

transforms and how to do the operations on matrices. Also contains about the systems, convolution and

correlation and how to remove the noise using correlation.

Sampling theorem and wiener khintchine theorem can be understood by using this lab. MATLAB

software which is 20075B version is user friendly with enhanced feature and improved tools related to

DSP and compatible to interface DSP processors.


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LAB CODE

1. Students should report to the concerned labs as per the time table schedule.

2. Students who turn up late to the labs will in no case be permitted to perform the experiment

scheduled for the day.

3. After completion of the experiment, certification of the concerned staff in- charge in the

observation book is necessary.

4. Students should bring a note book of about 100 pages and should enter the readings/

observations into the note book while performing the experiment.

5. The record of observations along with the detailed experimental procedure of the experiment

performed in the immediate last session should be submitted and certified by the staff member

in-charge.

6. Not more than three students in a group are permitted to perform the experiment on a setup.

7. The group-wise division made in the beginning should be adhered to, and no mix up of student

among different groups will be permitted later.

8. The components required pertaining to the experiment should be collected from stores in-

charge after duly filling in the requisition form.

9. When the experiment is completed, students should disconnect the setup made by them, and

should return all the components/instruments taken for the purpose.

10. Any damage of the equipment or burn-out of components will be viewed seriously either

by putting penalty or by dismissing the total group of students from the lab for the

semester/year.

11. Students should be present in the labs for the total scheduled duration.

12. Students are required to prepare thoroughly to perform the experiment before coming to

Laboratory.

13. Procedure sheets/data sheets provided to the students groups should be maintained neatly andto be returned after the experiment.


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INDEXSl NO. Name of the Experiment

1 BASIC MATHEMATICAL OPERATIONS ON MATRICES.

2 GENERATION OF VARIOUS SIGNALS

3 OPERATIONS ON SIGNALS

4 SYMMETRY OF A SIGNAL

5 CONVOLUTION OF SIGNALS & SEQUENCES

6 CORRELATION OF SIGNALS

7 LINEARITY AND TIME INVARIENCE PROPERTIES

8 RESPONSE OF LTI SYSTEMS

9 GIBBS PHENOMENON

10 FOURIER TRANSFORM

11 WAVEFORM SYNTHESIS USING LAPLACE TRANFORM

12 POLE-ZERO MAP

13 GENERATION OF GAUSSIONNOISE14 SAMPLING THEOREM VERIFICATION

15 NOISE REMOVAL USING CORRELATION

16 PERIODIC SIGNAL EXTRACTION USING CORRELATION

17 RAISED COSINE FILTER

18 WIENERKHINTCHINE RELATION


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1. BASIC MATHEMATICAL OPERATIONS ON MATRICESAim: Write a MATLAB program to generate matrix and perform basic operations on matrices.

Program:

A=[1 2 3 4;3 4 7 6;5 6 7 8;2 5 4 1] % defining matrix AB=[3 4 5 6;5 6 7 8;1 2 3 4;4 5 6 7] % defining matrix Bdisp('-->size of matrix A is')C=size(A) % displays size of matrixdisp('-->rank of matrix A is')D=rank(A) % displays rank of matrixdisp('-->trace of matrix A is')E=trace(A) % displays trace of matrixdisp('-->addition of matrices A&B is')F=A+B % Addition of two maticesdisp('-->subtraction of matrices A&B is')

G=A-B % subtraction of two maticesdisp('-->multiplication of matrices A&B is')H=A*B % multiplication of matricesdisp('-->element by element multiplication of matrices A&B is')I=A.*B % element by element multiplicationdisp('-->determinant of matrix A is')J=det(A) % determinant of Adisp('-->inverse of matrix A is')K=inv(A) % inverse of AL=K*A % multiplication of A with inv(A) I matrixdisp('-->transpose of matrix A is')

M=transpose(A) % transpose of Adisp('-->scalar addition of matrix A is')N=2+A % scalar additiondisp('-->scalar subtraction of matrix A is')O=2-A % scalar subtractiondisp('-->Identity matrix is')P=eye(3) % Identity matrixdisp('-->Magic matrix is')Q=magic(3) % sum of rows and columns are equaldisp('-->sub matrices of A are')R=A(2:3,2:4) % submatrix of A

S=A(1:4,3:4)A(:,3)=[] % empty matrix

Output:A =

1 2 3 4

3 4 7 65 6 7 8

2 5 4 1


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B =

3 4 5 6

5 6 7 8

1 2 3 44 5 6 7

-->size of matrix A isC =

4 4

-->rank of matrix A isD =

4

-->trace of matrix A is

E =13

-->addition of matrices A&B is

F =4 6 8 10

8 10 14 14

6 8 10 12

6 10 10 8-->subtraction of matrices A&B is

G =

-2 -2 -2 -2-2 -2 0 -2

4 4 4 4

-2 0 -2 -6

-->multiplication of matrices A&B isH =

32 42 52 62

60 80 100 12084 110 136 162

39 51 63 75

-->element by element multiplication of matrices A&B isI =

3 8 15 24

15 24 49 48

5 12 21 328 25 24 7

-->determinant of matrix A is

J = -80.0000

-->inverse of matrix A isK =

-0.9000 0.1000 0.4000 -0.2000

0.4750 -0.4000 0.0250 0.3000-0.2500 0.5000 -0.2500 0.0000

0.4250 -0.2000 0.0750 -0.1000


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L =

1.0000 -0.0000 0 -0.0000

0 1.0000 -0.0000 -0.0000

0.0000 0.0000 1.0000 0.00000.0000 -0.0000 -0.0000 1.0000

-->transpose of matrix A is

M =

1 3 5 22 4 6 5

3 7 7 4

4 6 8 1

-->scalar addition of matrix A is

N =3 4 5 6

5 6 9 8

7 8 9 10

4 7 6 3

-->scalar subtraction of matrix A is

O =1 0 -1 -2

-1 -2 -5 -4

-3 -4 -5 -6

0 -3 -2 1-->Identity matrix is

P =

1 0 00 1 0

0 0 1

-->Magic matrix isQ =

8 1 6

3 5 7

4 9 2-->sub matrices of A are

R =

4 7 6

6 7 8S =

3 4

7 67 8

4 1


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A =

1 2 4

3 4 6

5 6 82 5 1

Result: In this experiment basic operations on matrices using MATLAB have been performed.


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2. GENERATION OF VARIOUS SIGNALS AND SEQUENCES

Aim: Write a MATLAB program to generate different types of signals and sequences such as

unit impulse, unit step, sawtooth, triangular, square, sinusoidal, ramp and sinc functions.

A)Unit impulse and unit step signal and sequence

Program:

n=input('enter the value n')t=-n:0.02:n;x=(t==0);subplot(3,1,1)plot(t,x)xlabel('time');ylabel('amplitude')title('impulse signal')y=(t>=0);

subplot(3,1,2)plot(t,y)xlabel('time');ylabel('amplitude')title('unit step signal')subplot(3,1,3)stem(t,y)xlabel('time');ylabel('amplitude')title('unit step sequence')

Output:

enter the value n 4

Impulse, step signal and sequence


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B) Sawtooth and Triangular signalProgram:

n=input('enter the value n');fs=100;t=-n:1/fs:n;x=sawtooth(2*pi*5*t);figuresubplot(2,1,1)plot(t,x)xlabel('time');ylabel('amplitude')title('sawtooth signal')subplot(2,1,2)stem(t,x)xlabel('time');ylabel('amplitude')title('sawtooth sequence')y=sawtooth(2*pi*5*t,0.5);

figuresubplot(2,1,1)plot(t,y)xlabel('time');ylabel('amplitude')title('triangular signal')subplot(2,1,2)stem(t,y)xlabel('time');ylabel('amplitude')title('triangular sequence')

Output:

enter the value n 1 Sawtooth


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Triangular

C) Square and ramp signal and sequence.Program:

n=input('enter the value n');fs=100;t=0:1/fs:n;x=square(2*pi*5*t);

figuresubplot(2,1,1)plot(t,x)xlabel('time');ylabel('amplitude')title('square signal')subplot(2,1,2)stem(t,x)xlabel('time');ylabel('amplitude')title('square wave sequence')y=t;figure

subplot(2,1,1)plot(t,y)xlabel('time');ylabel('amplitude')title('ramp signal')subplot(2,1,2)stem(t,y)xlabel('time');ylabel('amplitude')title('ramp sequence')


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Output:enter the value n 2

Square wave

Ramp signal


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D) Sinusoidal and sinc functionsProgram:

fs=100;t=-1:1/fs:1;x=sin(2*pi*5*t);figuresubplot(2,1,1)plot(t,x)gridxlabel('time');ylabel('amplitude')title('sin signal')subplot(2,1,2)stem(t,x)gridxlabel('time');ylabel('amplitude')

title('sin wave sequence')y=sinc(10*t);figuresubplot(2,1,1)plot(t,y)gridxlabel('time');ylabel('amplitude')title('sinc signal')subplot(2,1,2)stem(t,y)grid

xlabel('time');ylabel('amplitude')title('sinc sequence')

Output: Sine signal


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Sinc signal

E) Sine wave with harmonics and noiseProgram:

fs=1000;ts=1/fs;t=0:ts:1;

x=sin(2*pi*10*t)+sin(2*pi*20*t);subplot(2,1,1);plot(t,x);title('sine with two harmonics');xlabel('time');ylabel('amplitude');grid;x1=x+2*randn(size(t));subplot(2,1,2);plot(t,x1);title('sine with noise');xlabel('time');ylabel('amplitude');grid;


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Output:Sine with harmonics and sine wave with noise

Result: In this experiment various signals have been generated using MATLAB.


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3. OPERATIONS ON SIGNALS AND SEQUENCES

Aim: Write a MATLAB program to study the operations of the signals and sequences such as

addition, multiplication, scaling, shifting, folding and computation of energy and average power.

Program:

A) Operation on independent variablex(t)= 5 + t -5


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subplot(2,1,2)plot(t,y5)grid;xlabel('time');ylabel('amplitude')title('time scaled signal')

Calling function

function x=y(t)x1=t+5;x2=11+4*t;x3=24-9*t;x4=t-6;x=x1.*(-5*t&t


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Time shifted and time scaled signal

Time shifted and scaled signal


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B) Operation on dependent variableProgram:

clear alltmin=-15;tmax=15;t=tmin:0.1:tmax;y0=3*y(t);y1=3*y(t+4); % time shiftingy2=y0+y1; % additiony3=y0.*y1; % multiplicationy4=2*y0; % scalar multiplicationy5=y0/2; % scalar divisionfiguresubplot(2,1,1)plot(t,y0)grid;xlabel('time');ylabel('amplitude')

title('actual signal')subplot(2,1,2)plot(t,y1)grid;xlabel('time');ylabel('amplitude')title('time shifted signal')figuresubplot(2,1,1)plot(t,y2)grid;xlabel('time');ylabel('amplitude')title('addition of two signals')subplot(2,1,2)

plot(t,y3)grid;xlabel('time');ylabel('amplitude')title('multiplication of two signals')figuresubplot(2,1,1)plot(t,y4)grid;xlabel('time');ylabel('amplitude')title('amplitude scaled signal')subplot(2,1,2)plot(t,y5)grid;xlabel('time');ylabel('amplitude')

title('amplitude scaled signal')


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Output:

Actual and time shifted signal

Addition and multiplication of two signals


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Amplitude scaled signals


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C) Computation of energy of the a periodic signalProgram:

t=0:1:50;x=(1/2).^t;plot(t,x)grid;xlabel('time');ylabel('amplitude')title('actual signal')disp('the calculated energy of the signal is')E=sum(abs(x).^2)disp('the theoritical energy of the signal is')E_theory=4/3

Output:

the calculated energy of the signal is

E = 1.3333

the theoritical energy of the signal is

E_theory =

1.3333

Signal


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D) Computation of power of the periodic signalProgram:

N=input('enter the value N');t=-N:0.0001:N;x=cos(2*pi*50*t);y=cos(2*pi*50*t).^2;plot(t,x)axis([-0.2 0.2 -2 2])grid;xlabel('time');ylabel('amplitude')title('actual signal')disp('the calculated power of the signal is')P=sum(abs(y).^2)/length(x)disp('the theoritical power of the signal is')E_theory=3/8Output:

enter the value N 5the calculated power of the signal isP =

0.3750

the theoritical power of the signal is

E_theory =0.3750

Signal

Result:In this experiment the various operations on signals have been performed and computation of

energy and power of various signals have been calculated using MATLAB software.


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4. FINDING EVEN AND ODD PARTS OF THE SIGNAL

Aim: Write a MATLAB program to find the even and odd parts of the signal/sequence and real

and imaginary part of the signal.

A)Even and odd parts of the signal

Program:t=0:0.005:4;x=sin(2*pi*5*t)+cos(2*pi*5*t);subplot(221)plot(t,x)gridxlabel('time');ylabel('amplitude')title('original signal')y=sin(-t)+cos(-t);subplot(222)plot(t,y)gridxlabel('time');ylabel('amplitude')title('folded signal')z=(x+y)/2;subplot(223)plot(t,z)gridxlabel('time');ylabel('amplitude')title('even part of the signal')subplot(224)

p=(x-y)/2;plot(t,p)gridxlabel('time');ylabel('amplitude')title('odd part of the signal')q=z+p;figureplot(t,q)gridxlabel('time');ylabel('amplitude')title('original signal')


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Output:

Even and odd part of the signal

Reconstructed signal from even and odd parts


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B) Real and imaginary part of the signal.

Program:

x=input('enter complex number')i=imag(x)r=real(x)disp('imaginary part is')disp(i)disp('real part is')disp(r)

Output:enter complex number[1+2i 2+3i 4+5i 5 6+7i]

imaginary part is

2 3 5 0 7

real part is

1 2 4 5 6

Result: In this experiment even and odd parts of the signals and sequence and real and imaginary

part of the signal have been calculated using MATLAB software.


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5. CONVOLUTION OF SIGNALS AND SEQUENCES

Aim: Write a MATLAB program to plot the convolution of signals and sequences.

A) For signalsProgram:

t=0:0.01:10;x=cos(2*pi*5*t);h=sin(2*pi*2*t);y=conv(x,h);subplot(3,1,1)plot(x)xlabel('time');ylabel('amplitude');

title('signal1')subplot(3,1,2)plot(h)xlabel('time');ylabel('amplitude');title('signal2')subplot(3,1,3)plot(y)xlabel('time');ylabel('amplitude');title('convolution of two sequences')

Output:

Convolution of two signals


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B) For sequencesProgram:

x=input('enter the first sequence');h=input('enter the second sequence');n1=length(x);n2=length(h);y=conv(x,h);subplot(3,1,1)t1=0:n1-1;stem(t1,x)xlabel('time');ylabel('amplitude');title('first sequence')subplot(3,1,2)t2=0:n2-1;stem(t2,h)

xlabel('time');ylabel('amplitude');title('second sequence')subplot(3,1,3)t3=0:n1+n2-2;stem(t3,y)xlabel('time');ylabel('amplitude');title('convolution of two sequences')

Output:enter the first sequence[1 2 3 5]

enter the second sequence[4 5 7 6]

convolution of two sequences4 13 29 55 58 53 30

Convolution of two sequences

Result: In this experiment convolution of signals and sequences are generated using MATLAB.


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6. AUTO CORRELATION AND CROSS CORRELATION BETWEEN SIGNALS AND

SEQUENCES

Aim: Write a MATLAB program to find the auto correlation and cross correlation of signals andsequences.

A) Auto correlationx=input('enter the sequence');D=input('type the delay');xd=[zeros(1,D) x];[r,lag]=xcorr(x,xd);stem(lag,r)title('cross correlation of x and dealayed x');xlabel('lag index')ylabel('autocorrelation')

Output:

enter the sequence[1 2 3 -2 -1]

type the delay3cross correlation of x and dealayed x

Columns 1 through 6

-1.0000 -4.0000 -4.0000 4.0000 19.0000 4.0000

Columns 7 through 12-4.0000 -4.0000 -1.0000 0.0000 0.0000 0.0000

Columns 13 through 15

0 -0.0000 -0.0000Auto correlation of two sequences


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B) Cross correlationProgram:

x=input('enter the first sequence');y=input('enter the second sequence');[r,lag]=xcorr(x,y);disp('cross correlation of x and y')disp(r)stem(lag,r)title('cross correlation of x and y');xlabel('lag index')ylabel('cross correlation')

Output:enter the first sequence[1 2 3 4]

enter the second sequence[4 5 6 7]

cross correlation of x and yColumns 1 through 6

7.0000 20.0000 38.0000 60.0000 47.0000 32.0000

Column 716.0000

Cross correlation of two sequnences

Result: In this experiment correlation of various signals has been performed using MATLAB.


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7. VERIFICATION OF LINEARITY AND TIME INVARIANCE PROPERTIES OF A

GIVEN CONTINUOUS/DISCRETE SYSTEMS

Aim: Write a MATLAB program to verify linearity and time invariance properties of a given

continuous/discrete systems.

A)Linear system

Program:

n=0:40; %veriying upto 40 samplesa=2;b=1; %multiplication constantsx1=cos(2*pi*0.1*n); %first inputx2=cos(2*pi*0.4*n); %second inputx=a*x1+b*x2; %single outputy=n.*x; %output upto 40 samplesy1=n.*x1; %first input outputy2=n.*x2; %second input outputyt=a*y1+b*y2; %addition first output,second outputd=y-yt;d=round(d);if d==0

disp('given system is satisfy linearity property')else

disp('given system is not satisfy linearityproperty')endsubplot(3,1,1)

stem(n,y)grid;xlabel('time');ylabel('amplitude')title('output of the system with added inputs')subplot(3,1,2)stem(n,yt)grid;xlabel('time');ylabel('amplitude')title('output of the system with weighted added inputs')subplot(3,1,3)stem(n,d)grid;xlabel('time');ylabel('amplitude')title('differece')


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Output:

given system is satisfy linearity property

B)Nonlinear system

Program:

n=0:40; %veriying upto 40 samplesa=2;b=-3; %multiplication constantsx1=cos(2*pi*0.1*n); %first inputx2=cos(2*pi*0.4*n); %second inputx=a*x1+b*x2; %single outputy=n.*x.^2; %output upto 40 samplesy1=n.*x1.^2; %first input outputy2=n.*x2.^2; %second input output

yt=a*y1+b*y2; %addition first output,second outputd=y-yt;d=round(d);if (d==0)

disp('given system is satisy linearity property')else

disp('given system is not satisy linearity property')endsubplot(3,1,1)


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stem(n,y)grid;xlabel('time');ylabel('amplitude')title('output of the system with added inputs')subplot(3,1,2)stem(n,yt)

grid;xlabel('time');ylabel('amplitude')title('output of the system with weighted added inputs')subplot(3,1,3)stem(n,d)grid;xlabel('time');ylabel('amplitude')title('differece')

Output:

Result: In this experiment linearity and time invariance properties of given system has been

verified using MATLAB.


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8. COMPUTATION OF UNIT SAMPLE, UNIT STEP, SINUSOIDAL RESPONSES OF

THE GIVEN LTI SYSTEM

Aim: Write a MATLAB program to compute unit sample, unit step, sinusoidal responses of the

given LTI system using MATLAB.

Program:

b=input('enter the numerator coefficients');a=input('enter the numerator coefficients');t=0:0.1:10;x1=sin(2*pi*2*t);h=impulse(b,a);s=step(b,a);si=filter(b,a,x1);subplot(3,1,1)plot(h)grid;xlabel('time');ylabel('amplitude');title('impulse response')subplot(3,1,2)plot(s)grid;xlabel('time');ylabel('amplitude');title('step response')subplot(3,1,3)plot(t,si)grid;xlabel('time');ylabel('amplitude');title('sinusoidal response')

Output:

enter the numerator coefficients[1]

enter the numerator coefficients[1 0.2 0.1]

Transfer function:

1

-----------------

s^2 + 0.2 s + 0.1


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Impulse , step, sinusoidal response of a system

Result: In this experiment unit impulse, unit step, sinusoidal responses of given LTI system havebeen calculated using MATLAB software.


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9. GIBBS PHENOMENON

Aim: Write a MATLAB program to verify the Gibbs phenomenon.

Program:

t=-1:0.01:1;y=sin(2*pi*5*t);subplot(5,1,1)plot(t,y)grid;xlabel('time');ylabel('amplitude')title('Gibbs phenomenon')h=2;for k=3:2:9

y=y+sin(k*2*pi*5*t)/k;subplot(5,1,h)plot(t,y)

grid;xlabel('time');ylabel('amplitude');h=h+1;

end

Output:

Gibbs phenomenon

Result: In this experiment Gibbs phenomenon has been verified using MATLAB.


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10. FOURIER TRANSFORM OF A GIVEN SIGNAL

Aim: Write a MATLAB program to find the Fourier transform of a signal and plot its magnitude

and phase spectrum.

Program:

Fs=100;t=0:1/Fs:10;x=sin(2*pi*20*t)+cos(2*pi*40*t);N=512;X=fft(x,N);f=Fs*(0:N-1)/N;Xm=abs(X);Xp=angle(X)*180/pi;subplot(3,1,1)plot(t,x)axis([0 1 -1 1])

grid;xlabel('time');ylabel('amplitude');title('signal')subplot(3,1,2)plot(f,Xm)grid;xlabel('frequency');ylabel('magnitude');title('magnitude spectrum')subplot(3,1,3)plot(f,Xp)grid;xlabel('frequency');ylabel('magnitude');title('phase spectrum')

Output: Signal, its magnitude and phase spectrum

Result: The Fourier transform of the given signal is found and its magnitude and phase

spectrums are plotted using MATLAB.


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11. WAVEFORM SYNTHESIS USING LAPLACE TRANSORM

Aim: To perform waveform synthesis using Laplace transform of a given signal. A) Laplace transform

Program:syms f t;

f=t;L=laplace(f);disp('laplace transform is')disp(L)

Output:laplace transform is

1/s^2

B) Laplace transformProgram:syms t w sf1=sin(w*t);f2=sin(w*(t+1));L1=laplace(f1);L2=laplace(f2);disp('laplace transorm first signal is')disp(L1)disp('laplace transorm second signal is')disp(L2)

Output:

laplace transorm first signal isw/(s^2 + w^2)

laplace transorm second signal is(w*cos(w) + s*sin(w))/(s^2 + w^2)

C) Inverse Laplace transformProgram:syms F sF=24/(s*(s+8));I=ilaplace(F);disp('inverse laplace transform is')

disp(I)

Output:inverse laplace transform is

3 - 3/exp(8*t)


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D) Signal synthesize using Laplace transformProgram:

Output:

Result: In this experiment the signals are synthesized using Laplace transform using MATLAB.


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12. LOCATING POLES AND ZEROS

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

the given transfer function using MATLAB.A) S-plane

Prgram:

b=input('enter numerator coeficients');a=input('enter numerator coeficients');H=tf(b,a)[p,z]=pzmap(H);pzmap(H);h=impulse(H);figureplot(h)gridxlabel('time');ylabel('amplitude');title('impulse response')disp('zeros are at')disp(z)disp('poles are at')disp(p)if max(real(p))>0

disp('all poles do not lie in the let half of S-plane')disp('so the system is unstable')

elsedisp('all poles lie in the let half of S-plane')disp('so the system is stable')

end

Output:

enter numerator coeficients[1 2 3 4]enter numerator coeficients[4 5 6 7]

Transfer function:

s^3 + 2 s^2 + 3 s + 4

-----------------------4 s^3 + 5 s^2 + 6 s + 7

zeros are at

-1.6506-0.1747 + 1.5469i

-0.1747 - 1.5469i

poles are at

-1.2078

-0.0211 + 1.2035i-0.0211 - 1.2035i


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all poles lie in the let half of S-plane

so the system is stable

Pole zero map

Impulse response


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B) Z-planeProgram:

b=input('enter numerator coeficients');a=input('enter numerator coeficients');

H=filt(b,a)z=zero(H)[r,p,k]=residuez(b,a)radpole=abs(p);[p,z]=pzmap(H);zplane(b,a);title('pole zero map of given system in Z-plane')figureimpz(b,a)disp('zeros are at')disp(z)disp('poles are at')disp(p)if max(radpole)>=1

disp('all poles do not lie within the unit circle')disp('so the system is unstable')

elsedisp('all poles lie within the unit circle')disp('so the system is stable')

end

Output:enter numerator coeficients[1 2 3 4]

enter numerator coeficients[4 5 7 6]

Transfer function:

1 + 2 z^-1 + 3 z^-2 + 4 z^-3----------------------------

4 + 5 z^-1 + 7 z^-2 + 6 z^-3

Sampling time: unspecifiedz =

-1.6506

-0.1747 + 1.5469i

-0.1747 - 1.5469i

r =

-0.0972 + 0.0242i-0.0972 - 0.0242i

-0.2222


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p =

-0.1250 + 1.2183i

-0.1250 - 1.2183i

-1.0000k =

0.6667zeros are at-1.6506

-0.1747 + 1.5469i

-0.1747 - 1.5469ipoles are at

-0.1250 + 1.2183i

-0.1250 - 1.2183i

-1.0000all poles do not lie within the unit circle

so the system is unstable

Pole zero map


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Impulse response

Result: In this experiment the zeros and poles are located and pole zero maps are plotted in s-

plane and z-plane for given transfer function using MATLAB.


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13. GAUSSIAN NOISE

Aim: Write a MATLAB program to generate gaussian noise and computes its mean, mean

square value, skew, kurtosis, power spectral distribution, probability distribution function.

Program:

x1=randn(1,5000);x2=randn(1,5000);x3=rand(1,5000);x4=rand(1,5000);xmean=mean(x1);xmsq=sum(x1.^2/length(x1));xstandard=std(x1);xvar=var(x1);xskew=skewness(x1);xkurt=kurtosis(x1);

disp('mean of gaussian noise is');disp(xmean);disp('mean squre value of gaussian noise is');disp(xmsq);disp('standard deviation of gaussian noiseis');disp(xstandard);disp('variance of gaussian noise is');disp(xvar);disp('skew of gaussian noise is');disp(xskew);disp('kurtosis of gaussian noise is');disp(xkurt);figurehist(x1)title('Gaussian(random) distribution')figure

hist(x3)title('uniform distribution')figuresubplot(2,1,1)plot(x1,x2,'.');title('scatter plot of normal(gaussian)distribution randomnumbers')subplot(2,1,2)plot(x3,x4,'.');title('scatter plot of uniform distribution randomnumbers')

Output:

mean of gaussian noise is

-0.0145

mean squre value of gaussian noise is

0.9667


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standard deviation of gaussian noise is

0.9832

variance of gaussian noise is

0.9667

skew of gaussian noise is

-0.0243

kurtosis of gaussian noise is

3.0355

Distribution of Gaussian noise

Uniform distribution


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14. SAMPLING THEOREM VERIFICATION

Aim: Write a MATLAB program to verify the Sampling theorem.

Program:

clc;clear all;t=-5:0.0001:5;f1=5;f2=20;x=cos(2*pi*f1*t)+cos(2*pi*f2*t);figure(1)plot(t,x)axis([-0.4 0.4 -2 2])grid;xlabel('time');ylabel('amplitude')title('continuous time signal');

%%%%% case 1 %%%%%%%%fs1=1.4*f2;ts1=1/fs1;n1=-0.4:ts1:0.4;xs1=cos(2*pi*f1*n1)+cos(2*pi*f2*n1);figure(2)stem(n1,xs1)axis([-0.4 0.4 -2 2])hold on;plot(t,x,'r');hold off

grid;xlabel('time sample');ylabel('Amplitude');title('Discrete time signal sampling rate fs


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figure(4)stem(n3,xs3)axis([-0.4 0.4 -2 2])hold on;plot(t,x,'r');

hold offgrid;xlabel('time sample');ylabel('Amplitude');title('Discrete time signal sampling rate fs>2*fmax');

Output:

Continuous time signal

Sampling at fs


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Sampling at fs=2*fmax

Sampling at fs>2*fmax

Result: In this experiment sampling theorem has been verified using MATLAB.


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15. REMOVAL OF NOISE BY AUTO CORRELATION/CROSS CORRELATION

Aim: Write a MATLAB program to remove noise by auto correlation/cross correlation.

A) Using auto correlationProgram:

t=0:0.01:4;signal=sin(2*pi*10*t);noise=randn(size(t));noisysignal=signal+noise;sc=xcorr(signal,signal);nc=xcorr(noise,noise);snc=xcorr(noisysignal,noisysignal);asnc=sc+nc;figure

subplot(2,1,1)plot(signal)grid;xlabel('time');ylabel('amplitude');title('signal')subplot(2,1,2)plot(noisysignal)grid;xlabel('time');ylabel('amplitude');title('noise added signal')figuresubplot(2,1,1)plot(sc)

grid;xlabel('time');ylabel('amplitude');title('correlation of signal')subplot(2,1,2)plot(nc)grid;xlabel('time');ylabel('amplitude');title('correlation of noise signal')figuresubplot(2,1,1)plot(snc)grid;xlabel('time');ylabel('amplitude');title('correlation of noisy signal')

subplot(2,1,2)plot(asnc)grid;xlabel('time');ylabel('amplitude');title('addition of correlation of signal and noise')


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Output:Signal and noise added signal

Correlation of signal and correlation of noise signal


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Correlation o noisy signal and addition of correlation of signal and noise

B) Using cross correlationProgram:t=0:0.01:1;signal=sin(2*pi*10*t);

testsignal=cos(2*pi*10*t); %testing signl with samefrequency of signalnoise=randn(size(t));noisysignal=signal+noise;sc=xcorr(signal,testsignal);nc=xcorr(noise,testsignal);snc=xcorr(noisysignal,testsignal);asnc=sc+nc;figuresubplot(2,1,1)plot(signal)

grid;xlabel('time');ylabel('amplitude');title('signal')subplot(2,1,2)plot(testsignal)grid;xlabel('time');ylabel('amplitude');title('test signal')figuresubplot(2,1,1)plot(noisysignal)


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grid;xlabel('time');ylabel('amplitude');title('noisy signal')subplot(2,1,2)plot(sc)grid;xlabel('time');ylabel('amplitude');

title('correlation of signal&test signal')figuresubplot(3,1,1)plot(nc)grid;xlabel('time');ylabel('amplitude');title('correlation of noise&test signal')subplot(3,1,2)plot(snc)grid;xlabel('time');ylabel('amplitude');title('correlation of noisysignal and testsignal')subplot(3,1,3)plot(asnc)grid;xlabel('time');ylabel('amplitude');title('addtion of correlations sc & nc')

Output:Two periodic signals (with same period)


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Noisy signal and correlation of signal and test signal

Correlation of noise and test signal, Correlation of noisy signal and test signal, addition of sc and nc

Result: In this experiment the detecting of noise by auto correlation/cross correlation has beenverified using MATLAB.


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16. Extraction of periodic signal masked by noise using Correlation

Aim: Write a MATLAB program to extract periodic signal masked by noise using correlation.

Program:

t=-2:0.001:2;signal=sin(2*pi*t);impulse=ones(1,20);%testing signal with same frequencyof signalnoise=randn(size(t));noisysignal=signal+noise;si=xcorr(signal,impulse);ni=xcorr(noise,impulse);nsic=xcorr(noisysignal,impulse);figuresubplot(2,1,1)

plot(t,signal)grid;xlabel('time');ylabel('amplitude');title('signal')subplot(2,1,2)stem(impulse)grid;xlabel('time');ylabel('amplitude');title('test signal')figuresubplot(2,1,1)plot(noise)grid;xlabel('time');ylabel('amplitude');

title('noisy signal')subplot(2,1,2)plot(noisysignal)grid;xlabel('time');ylabel('amplitude');title('correlation of signal&test signal')figuresubplot(3,1,1)plot(si)grid;xlabel('time');ylabel('amplitude');title('correlation of noise&test signal')subplot(3,1,2)

plot(ni)grid;xlabel('time');ylabel('amplitude');title('correlation of noisysignal and testsignal')subplot(3,1,3)plot(nsic)grid;xlabel('time');ylabel('amplitude');title('addtion of correlations sc & nc')


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Output:

Actual signal and periodic impulse signal

Noisy signal and Correlation of noise and impulse


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Correlation of noise and impulse, Correlation of noisy signal and impulse, addition of sc and nc

Result: In this experiment the removal of noise by correlation has been verified using

MATLAB.


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17. VERIFICATION OF WIENER-KHINCHINE RELATION

Aim: Verification of Wiener-Khinchine relation using MATLAB.

Program:

Fs=100;t=0:1/Fs:10;x=sin(2*pi*15*t)+sin(2*pi*30*t);N=512;X=fft(x,N);f=Fs*(0:N-1)/N;power=X.*conj(X)/N;rxx=xcorr(x,x);Sxx=fft(rxx,512);subplot(2,1,1)plot(f,power)

title('power spectrum using fourier transform')grid;xlabel('frequency');ylabel('Power')subplot(2,1,2)plot(f,abs(Sxx))title('power spectrum using fourier transform of ACF')grid;xlabel('frequency');ylabel('Power')

Output:

Result: In this experimentthe Wiener-Khinchine relation has been verified using MATLAB.


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18. CHECKING A RANDOM PROCESS FOR STATIONARITY IN WIDE SENSE

Aim: Write a MATLAB program to check a random process for stationarity in wide sense.

Program:

Output:

Result: In this experiment the random process for stationarity in wide sense has been verified

using MATLAB.

Basic Siumlation Lab - 12-13, I Semester- SRAWAN

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