LIST OF EXPERIMENTS Experiment Name Page No. · 8. Computation of unit sample, unit step and...

Department of ECE Basic Simulation Lab Anurag College of Engineering

1

LIST OF EXPERIMENTS

Experiment Name Page No.

1. Basic operations on matrices. 2

2. Generation on various signals and Sequences (periodic and aperiodic),

such as unit impulse, unit step, square, sawtooth, triangular, sinusoidal, ramp, sinc 10

3. Operations on signals and sequences such as addition, multiplication, scaling,

shifting, folding computation of energy and average power. 20

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

of signal. 29

5. Convolution between signals and sequence 33

6. Auto correlation and cross correlation between signals and sequences 37

7. Verification of linearity and time invariance properties of a given

continuous /discrete systems 43

8. Computation of unit sample, unit step and sinusoidal response of the

given LTI system and verifying its physical Realizability and stability properties. 47

9. Gibbs phenomenon 51

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

and phase spectrum. 52

11. Waveform synthesis using Laplace Transform 54

12. Locating the zeros and poles and plotting the pole zero maps in s-plane

and z-plane for the given transfer function. 57

13. Generation of Gaussian Noise(real and complex), computation of its

mean, M.S. Value and its skew kurtosis, and PSD, probability distribution function 59

14. Sampling theorem verification 62

15. Removal of noise by auto correlation/cross correlation 65

16. Extraction of periodic signal masked by noise using correlation. 71

17. Verification of Weiner-Khinchine relations. 74

18. Checking a random process for stationarity in wide sense. 76


2

Experiment No:1

Basic operations on matrices

Aim: To perform basic operation on matrices Using MATLAB

1. Addition of matrices

2. Subtraction of matrices

3. Multiplication of Matrices

4. Element to element matrix multiplication

5. Element to element matrix division

6. Transpose of a matrix

7. Identity matrix

8. Rank of matrix

Requirements: A system with windows (XP/NT/2000/7)

MAT LAB Software.

Procedure: 1. Open matlab icon on the desktop

2. Open new editor window(ctrl+N)

3. Type the program and save with “.m” extension

4. Run the program

5. Observe the output in command window/figure window

Program: %1. Addition of matrices

clc;

clear all;

close all;

a=[2 3 5; 7 2 5];


3

b=[- 8 5 2; 1 0 5];

c=a+b;

disp(‘addition of following 2 matrices’);

disp(a);

disp(‘and’);

disp(b);

disp(‘is’);

disp(c);

Output: addition of following 2 matrices

2 3 5

7 2 5

and

-8 5 2

1 0 5

is

-6 8 7

8 2 10

% 2. Subtraction of matrices

clc;

clear all;

close all;

a=[2 3 5; 7 2 5];

b=[- 8 5 2; 1 0 5];

c=a-b;

disp('addition of following 2 matrices');


4

disp(a);

disp('and');

disp(b);

disp('is');

disp(c);

Output: addition of following 2 matrices

2 3 5

7 2 5

and

-8 5 2

1 0 5

is

10 -2 3

6 2 0

% 3. Multiplication of matrices

clc;

clear all;

close all;

a=[2 3 ; 7 2 ];

b=[- 8 5 ; 1 5];

c=a*b;

disp('multiplication of following 2 matrices');

disp(a);


5

disp('and');

disp(b);

disp('is');

disp(c);

Output: multiplication of following 2 matrices

2 3

7 2

and

-8 5

1 5

is

-13 25

-54 45

% 4.Elememt to element matrix multiplication

clc;

clear all;

close all;

a=[2 3 ; 7 2 ];

b=[- 8 5 ; 1 5];

c=a.*b;

disp('element multiplication of following 2 matrices');

disp(a);


6

disp('and');

disp(b);

disp('is');

disp(c);

Output: element multiplication of following 2 matrices

2 3

7 2

and

-8 5

1 5

is

-16 15

7 10

% 5 .Element to element matrix division

clc;

clear all;

close all;

a=[2 3 ; 7 2 ];

b=[- 8 5 ; 1 5];

c=a./b;

disp('element division of following 2 matrices');

disp(a);


7

disp('and');

disp(b);

disp('is');

disp(c);

Output: element division of following 2 matrices

2 3

7 2

and

-8 5

1 5

is

-0.2500 0.6000

7.0000 0.4000

% 6. Transpose of a matrix

clc;

clear all;

close all;

a=[2 3 ; 7 2 ];

c=transpose(a);

disp('transpose of');

disp(a);

disp('is');

disp(c);


8

Output: transpose of

2 3

7 2

is

2 7

3 2

% 7. Identity matrix

clc;

clear all;

close all;

c=eye(3);

disp('identity matrix is');

disp(c);

Output: identity matrix is

1 0 0

0 1 0

0 0 1

% 8. Rank of matrix

clc;

clear all;

close all;

a=[2 3 ; 7 2 ];

c=rank(a);

disp('rank of');

disp(a);

disp('is');

disp(c);


9

Output: rank of

2 3

7 2

is

2

Result: Basic operations on matrices i.e addition, subtraction, multiplication, transpose, identity,

rank of matrices has been performed using matlab.


10

Experiment No: 2

Generation on various signals and Sequences

Aim: To generate the various signals 1. Unit step signal

2. Unit ramp signal

3. Unit impulse signal

4. sinusoidal signal

5. Square wave

6. Triangular wave

7. sawtooth wave

8. sinc signal

Using matlab.


MAT LAB Software.




4. Run the program



11

Program: clc;

clear all;

close all;

t=-10:0.001:10;

a=1.*(t>=0)+0.*(t<0);

subplot(2,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('continuous time unit step signal');

n=-10:1:10;

b=1.*(n>=0)+0.*(n<0);

subplot(2,1,2);

stem(n,b)

xlabel('n');

ylabel('x(n)');

title(' unit step sequence');


12

2. Unit ramp signal

clc;

clear all;

close all;

t=-10:0.001:10;

a=t.*(t>=0)+0.*(t<0);

subplot(2,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('continuous time unit ramp signal');

n=-10:1:10;

b=n.*(n>=0)+0.*(n<0);

subplot(2,1,2);

stem(n,b)

xlabel('n');

ylabel('x(n)');

title(' unit ramp sequence');

-10 -8 -6 -4 -2 0 2 4 6 8 100

5

10

t

x(t)

continuous time unit ramp signal

-10 -8 -6 -4 -2 0 2 4 6 8 100

5

10

n

x(n)

unit ramp sequence


13

3. Unit impulse signal

clc;

clear all;

close all;

t=-10:0.001:10;

a=1.*(t==0)+0.*(t<0)+0.*(t>0);

subplot(2,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('continuous time unit impulse signal');

n=-10:1:10;

b=1.*(n==0)+0.*(n<0)+0.*(n>0);

subplot(2,1,2);

stem(n,b)

xlabel('n');

ylabel('x(n)');

title(' unit impulse sequence');

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.5

1

t

x(t)

continuous time unit impulse signal

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.5

1

n

x(n)

unit impulse sequence


14

4. sinusoidal signal

clc;

clear all;

close all;

t=-10:0.001:10;

a=sin(2*pi*(1/20)*t);

subplot(2,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('continuous time sinusoidal signal');

n=-10:1:10;

b=sin(2*pi*(1/20)*n);

subplot(2,1,2);

stem(n,b)

xlabel('n');

ylabel('x(n)');

title(' sinusoidal sequence');

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

t

x(t)

continuous time sinusoidal signal

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

n

x(n)

sinusoidal sequence


15

5. Square wave

clc;

clear all;

close all;

t=-10:0.001:10;

a=square(2*pi*(1/7)*t);

subplot(2,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('continuous time square wave');

n=-10:1:10;

b=square(2*pi*(1/7)*n);

subplot(2,1,2);

stem(n,b)

xlabel('n');

ylabel('x(n)');

title(' square wave sequence');

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

t

x(t)

continuous time square wave

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

n

x(n)

square wave sequence


16

6. Triangular wave

clc;

clear all;

close all;

t=-10:0.001:10;

a=sawtooth(2*pi*(1/7)*t,0.5);

subplot(2,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('continuous time triangular wave');

n=-10:1:10;

b=sawtooth(2*pi*(1/7)*n,0.5);

subplot(2,1,2);

stem(n,b)

xlabel('n');

ylabel('x(n)');

title(' triangular wave sequence');

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

t

x(t)

continuous time triangular wave

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

n

x(n)

triangular wave sequence


17

7. sawtooth wave

clc;

clear all;

close all;

t=-10:0.001:10;

a=sawtooth(2*pi*(1/7)*t);

subplot(2,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('continuous time sawtooth wave');

n=-10:1:10;

b=sawtooth(2*pi*(1/7)*n);

subplot(2,1,2);

stem(n,b)

xlabel('n');

ylabel('x(n)');

title(' sawtooth sequence');

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

t

x(t)

continuous time sawtooth wave

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

n

x(n)

sawtooth sequence


18

8. Sinc signal

clc;

clear all;

close all;

t=-10:0.001:10;

a=sinc(2*pi*(1/7)*t);

subplot(2,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('continuous time sinc wave');

n=-10:1:10;

b=sinc(2*pi*(1/7)*n);

subplot(2,1,2);

stem(n,b)

xlabel('n');

ylabel('x(n)');

title(' sinc sequence');

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.5

0

0.5

1

t

x(t)

continuous time sinc wave

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.5

0

0.5

1

n

x(n)

sinc sequence


19

Result: Basic signals unit step, unit ramp, impulse, sinusoidal, square, triangular, sawtooth waves

has been generated using matlab.


20

Experiment No:3

Operations on signals and sequences

Aim: To perform operations on signals Using MATLAB

1. Time shiting

2. Time scaling

3. Time reversal

4. Amplitude scaling

5. Signal addition

6. Signal multiplication


MAT LAB Software.




4. Run the program


Program:

1. Time shiting clc;

clear all;

close all;

t=-10:0.001:10;


subplot(3,1,1);

plot(t,a);


21

xlabel('t');

ylabel('x(t)');


subplot(3,1,2);

plot(t-2,a);

xlabel('t');

ylabel('x(t-2)');

title('left shifted sawtooth signal');

subplot(3,1,3);

plot(t+2,a);

xlabel('t');

ylabel('x(t+2)');

title('right shifted sawtooth signal');

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

t

x(t

)


-12 -10 -8 -6 -4 -2 0 2 4 6 8-1

0

1

t

x(t

-2)

left shifted sawtooth signal

-8 -6 -4 -2 0 2 4 6 8 10 12-1

0

1

t

x(t

+2)

right shifted sawtooth signal


22

2. Time scaling

clc;

clear all;

close all;

t=-10:0.001:10;


subplot(3,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');


subplot(3,1,2);

plot(t*2,a);

xlabel('t');

ylabel('x(t-2)');

title('scaled sawtooth signal');

subplot(3,1,3);

plot(t*(1/2),a);

xlabel('t');

ylabel('x(t*2)');

title('scaled sawtooth signal');


23

3. Time reversal

clc;

clear all;

close all;

t=-10:0.001:10;


subplot(2,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');


subplot(2,1,2);

plot(-t,a);

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

t

x(t

)


-20 -15 -10 -5 0 5 10 15 20-1

0

1

t

x(t

*2)

scaled sawtooth signal

-5 -4 -3 -2 -1 0 1 2 3 4 5-1

0

1

t

x(t

(1/2

)

scaled sawtooth signal


24

xlabel('t');

ylabel('x(-t)');

title('time reversal signal');

4. Amplitude Scaling:

clc;

clear all;

close all;

t=-10:0.001:10;


b=2*a;

c=(1/2)*a;

subplot(3,1,1);

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

t

x(t

)


-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

t

x(-

t)

time reversal signal


25

plot(t,a);

xlabel('t');

ylabel('x(t)');


subplot(3,1,2);

plot(t,b);

xlabel('t');

ylabel('2x(t)');

title('amplitude scaled sawtooth signal');

subplot(3,1,3);

plot(t,c);

xlabel('t');

ylabel('1/2x(t)');

title('amplitude scaled sawtooth signal');

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

t

x(t

)


-10 -8 -6 -4 -2 0 2 4 6 8 10-2

0

2

t

2x(t

)

amplitude scaled sawtooth signal

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.5

0

0.5

t

1/2

x(t

)

amplitude scaled sawtooth signal


26

5. Signal addition:

clc;

clear all;

close all;

t=-10:0.001:10;

a=2*sawtooth(2*pi*(1/5)*t);

b=2*sin(2*pi*(1/5)*t);

c=a+b;

subplot(3,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');


subplot(3,1,2);

plot(t,b);

xlabel('t');

ylabel('y(t)');

title('continuous time sine wave');

subplot(3,1,3);

plot(t,c);

xlabel('t');

ylabel('x(t)+y(t)');

title('addition');


27

6. Signal multiplication:

clc;

clear all;

close all;

t=-10:0.001:10;

a=2*sawtooth(2*pi*(1/5)*t);

b=2*sin(2*pi*(1/5)*t);

c=a.*b;

subplot(3,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');


-10 -8 -6 -4 -2 0 2 4 6 8 10-2

0

2

t

x(t

)


-10 -8 -6 -4 -2 0 2 4 6 8 10-2

0

2

t

y(t

)

continuous time sin wave

-10 -8 -6 -4 -2 0 2 4 6 8 10-5

0

5

t

x(t

)+y(t

)

addition


28

subplot(3,1,2);

plot(t,b);

xlabel('t');

ylabel('y(t)');

title('continuous time sin wave');

subplot(3,1,3);

plot(t,c);

xlabel('t');

ylabel('x(t)*y(t)');

title('multiplication');

Result: Basic operations time shifting, time scaling, time reversal, amplitude scaling, signal

addition, multiplication has been performed using matlab.

-10 -8 -6 -4 -2 0 2 4 6 8 10-2

0

2

t

x(t

)


-10 -8 -6 -4 -2 0 2 4 6 8 10-2

0

2

t

y(t

)

continuous time sin wave

-10 -8 -6 -4 -2 0 2 4 6 8 10-5

0

5

t

x(t

)+y(t

)

multiplication


29

Experiment No: 4

Finding the even and odd parts of signal/sequence and real and

imaginary parts of signal

Aim: To find Finding the even and odd parts of signal/sequence and real and imaginary parts of

signal.


MAT LAB Software.




4. Run the program


Program:

Even and odd parts of a signal:

clc;

clear all;

close all;

t=-10:0.001:10;

a=sin(t);

b=sin(-t);

c=(1/2)*(a+b);

d=(1/2)*(a-b);

subplot(3,1,1);

plot(t,a);

xlabel('t');


30

ylabel('x(t)');

title('continuous time sin wave');

subplot(3,1,2);

plot(t,c);

xlabel('t');

ylabel('xe(t)');

title('even parts');

subplot(3,1,3);

plot(t,d);

xlabel('t');

ylabel('xo(t)');

title('odd parts');

-10 -8 -6 -4 -2 0 2 4 6 8 100

1

2

t

x(t

)


-10 -8 -6 -4 -2 0 2 4 6 8 100

1

2

t

xe(t

)

even parts

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

t

xo(t

)

odd parts


31

Real and Imaginary parts:

clc;

clear all;

close all;

t=-10:0.001:10;

a=exp(j*t);

c=real(a);

d=imag(a);

subplot(3,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('complex exponential wave');

subplot(3,1,2);

plot(t,c);

xlabel('t');

ylabel('xe(t)');

title('Real parts');

subplot(3,1,3);

plot(t,d);

xlabel('t');

ylabel('xo(t)');

title('Imaginary parts');


32

Result: Even parts and odd parts of a signal and Real and imaginary parts of a signal has been

found.

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

t

x(t

)

complex exponential wave

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

t

xe(t

)

Real parts

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

t

xo(t

)

Imaginary parts


33

Experiment No: 5

Convolution between signals and sequences

Aim: To observe convolution between signals and sequences


MAT LAB Software.




4. Run the program


Program: Convolution of signals

clc;

clear all;

close all;

t=-10:0.001:10;

a=2.*(t>=0)+0.*(t<0);

c=sin(t);

d=conv(a,c);

subplot(3,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('step signal');

subplot(3,1,2);


34

plot(t,c);

xlabel('t');

ylabel('y(t)');

title('sine wave');

subplot(3,1,3);

plot(d);

xlabel('t');

ylabel('z(t)');

title('convolution');

-10 -8 -6 -4 -2 0 2 4 6 8 100

1

2

t

x(t

)

step signal

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

t

y(t

)

sine wave

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

-5000

0

5000

t

z(t

)

convolution


35

Convolution of sequences:

clc;

clear all;

close all;

t1=-1:1:2;

t2=0:1:2;

t3=-1:1:4;

a=[2 5 9 7 ];

c=[7 5 4];

d=conv(a,c);

subplot(3,1,1);

stem(t1,a);

xlabel('t');

ylabel('x(t)');

title('sequence1');

subplot(3,1,2);

stem(t2,c);

xlabel('t');

ylabel('y(t)');

title('sequence 2');

subplot(3,1,3);

stem(t3,d);

xlabel('t');

ylabel('z(t)');

title('convolution');


36

Result: Convolution of signals and sequences has been observed using matlab.

-1 -0.5 0 0.5 1 1.5 20

5

10

t

x(t

)

sequence1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

t

y(t

)

sequence 2

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

100

200

t

z(t

)

convolution


37

Experiment No: 6

Auto correlation and cross correlation

Aim: To observe cross correlation and auto correlation between signals and sequences


MAT LAB Software.




4. Run the program


Program: Cross correlation of signals

clc;

clear all;

close all;

t=-10:0.001:10;

a=2.*(t>=0)+0.*(t<0);

c=sin(t);

d=xcorr(a,c);

subplot(3,1,1);

plot(t,a);

xlabel('t');

ylabel('x(t)');

title('step signal');

subplot(3,1,2);


38

plot(t,c);

xlabel('t');

ylabel('y(t)');

title('sine wave');

subplot(3,1,3);

plot(d);

xlabel('t');

ylabel('z(t)');

title('cross correlation');

-10 -8 -6 -4 -2 0 2 4 6 8 100

1

2

t

x(t

)

step signal

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

0

1

t

y(t

)

sine wave

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

-5000

0

5000

t

z(t

)

cross correlation


39

Auto Correlation of signal

clc;

clear all;

close all;

t=-10:0.001:10;

c=sin(t);

d=xcorr(c,c);

subplot(2,1,1);

plot(t,c);

xlabel('t');

ylabel('x(t)');

title('sine signal');

subplot(2,1,2);

plot(t,d);

xlabel('t');

ylabel('y(t)');

title('sine wave');

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1

t

x(t)

sine signal

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

-1

-0.5

0

0.5

1x 10

4

t

y(t)

sine wave


40

Cross correlation of sequences:

clc;

clear all;

close all;

t1=-1:1:2;

t2=0:1:2;

t3=-1:1:5;

a=[2 5 9 7 ];

c=[7 5 4];

d=xcorr(a,c);

subplot(3,1,1);

stem(t1,a);

xlabel('t');

ylabel('x(t)');

title('sequence1');

subplot(3,1,2);

stem(t2,c);

xlabel('t');

ylabel('y(t)');

title('sequence 2');

subplot(3,1,3);

stem(t3,d);

xlabel('t');

ylabel('z(t)');

title('cross correlation');


41



Auto correlation of sequences:

clc;

clear all;

close all;

t1=-1:1:2;

a=[2 5 9 7 ];

d=xcorr(a,a);

subplot(2,1,1);

stem(t1,a);

xlabel('t');

-1 -0.5 0 0.5 1 1.5 20

5

10

t

x(t

)

sequence1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

t

y(t

)

sequence 2

-1 0 1 2 3 4 50

100

200

t

z(t

)

cross correlation


42

ylabel('x(t)');

title('sequence1');

subplot(2,1,2);

stem(d);

xlabel('t');

ylabel('y(t)');

title('Auto correlation');

Result: Cross correlation and auto correlation of signals and sequences has been observed.

-1 -0.5 0 0.5 1 1.5 20

5

10

t

x(t

)

sequence1

1 2 3 4 5 6 70

50

100

150

200

t

y(t

)

Auto correlation


43

Experiment No:7

Verification of linearity and time invariance

Aim: To Verify linearity and time invariance properties of given system.


MAT LAB Software.




4. Run the program


Program: Linearity1

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;


44

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);

subplot(3,1,2),

stem(n,yt);

subplot(3,1,3);

stem(n,d);

Output: Given system is satisfy linearity property

0 5 10 15 20 25 30 35 40-100

0

100

200

0 5 10 15 20 25 30 35 40-100

0

100

200

0 5 10 15 20 25 30 35 40-1

0

1


45

Time Invariance:

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);


46

grid;

Output: Given system is not satisfy time shifting property

Result: Linearity and time invariance properties of a system have been observed using matlab.

0 5 10 15 20 25 30 35 40 45-200

0

200

0 5 10 15 20 25 30 35 40 45-400

-200

0

200

0 5 10 15 20 25 30 35 40 45-200

0

200


47

Experiment No:8

Computation of unit sample, unit step and sinusoidal response

Aim: To Compute unit sample, unit step and sinusoidal response of the given LTI system and

verifying its physical Realizability and stability properties.


MAT LAB Software.




4. Run the program


Program: Impulse response:

clc;

clear all;

close all;

b=[1 0 0];

a=[1,-0.5,.9];

h=tf(b,a)

impulse(h);

0 50 100 150 200 250

-12

-10

-8

-6

-4

-2

0

2

4

6x 10

26 Impulse Response

Time (seconds)

Ampli

tude


48

Step Response:

clc;

clear all;

close all;

b=[1 2 3];

a=[1 5 3];

h=tf(b,a)

step(h);

0 2 4 6 8 10 120.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Step Response

Time (seconds)

Am

plit

ude


49

Sinusoidal response:

clc;

clear all;

close all;

t=0:0.01:10;

x=sin(t);

b=[1 2 3];

a=[1 5 3];

h=tf(b,a)

lsim(h,x,t);

Result: Step response, Impulse response and sinusoidal response have been observed using matlab.

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Linear Simulation Results

Time (seconds)

Am

plit

ude


50

Experiment No: 9

Gibbs phenomenon

Aim: To observe the gibbs phenomenon and compare with square wave.


MAT LAB Software.




4. Run the program


Program: Gibbs phenomenon

clc;

clear all

close all

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;

for k=3:2:9

y=y+sin(k*t)/k;


51

subplot(5,1,h);

plot(t,y);

xlabel('k');


h=h+1;

end

Result:Gibbs phenomenon has been observed for 3 to 9 sinusoidal waves.

0 5 10 15 20 25 30-101

k

am

plitu

de gibbs phenomenon

0 5 10 15 20 25 30-101

k

am

plitu

de

0 5 10 15 20 25 30-101

k

am

plitu

de

0 5 10 15 20 25 30-101

k

am

plitu

de

0 5 10 15 20 25 30-101

k

am

plitu

de


52

Experiment No:10

Fourier transform

Aim: To Find the Fourier transform of a given signal and plotting its magnitude and phase

spectrum.


MAT LAB Software.




4. Run the program


Program:

clc

close all

clear 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)')


53

subplot(2,1,2)

plot(phase);

grid

xlabel('k')

ylabel('degrees')

Result: Fourier transform of a given signal and its magnitude and phase spectrum have been

plotted.

1 2 3 4 5 6 7 80

1

2

3

4

k

X(K

)

1 2 3 4 5 6 7 8-100

-50

0

50

100

k

degre

es


54

Experiment No:11

Waveform synthesis using Laplace Transform

Aim: To Find the Laplace transform of a given signal


MAT LAB Software.




4. Run the program


Program:

clc;

clear all;

syms f t;

f=t;

laplace(f) %laplace transform

%f(s)=16/s(s+8) invese Laplce Transform

syms F s

F=24/(s*(s+8));

ilaplace(F)

Output: ans = 1/s^2

ans = 3 - 3*exp(-8*t)


55

clc;

clear all;

close all;

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);


56

f=f1+f2+f3;

figure;

plot(t,f);

laplace(f);

Result: Laplace transform and wave form synthesis have been observed.

0 50

1

2

3

4

5

0 50

0.5

1

1.5

2

0 5 100

1

2

3

4

5

0 5 10-8

-6

-4

-2

0

2

0 5 10-8

-6

-4

-2

0

0 5 100

1

2

3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


57

Experiment No:12

Locating the zeros and poles

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

given transfer function.


MAT LAB Software.




4. Run the program


Program:

clc;

clear all;

close all;

b=[1 0];

a=[1,-0.5];

h=tf(b,a)

figure;

pzmap(h);

figure;

zplane(b,a);


58

Result: Poles and zeros of a system plotted on s-plane and z-plane.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Pole-Zero Map

Real Axis (seconds-1)

Imagin

ary

Axis

(seconds-1

)

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Part

Imagin

ary

Part


59

Experiment No:13

Generation of Gaussian Noise

Aim: To generate Gaussian noise using matlab


MAT LAB Software.




4. Run the program


Program:

clc;

clear all;

close all;

dx=0.01;

x=-3:dx:3;

[m,n]=size(x);

mu_x=0;

sig_x=0.1;

a=1/(sqrt(2*pi)*sig_x);

for j=1:n

px1(j)=a*exp([-((x(j)-mu_x)/sig_x)^2]/2);

end

cum_Px(1)=0;

for j=2:n


60

cum_Px(j)=cum_Px(j-1)+dx*px1(j);

end

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);

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')

-3 -2 -1 0 1 2 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Gaussian pdf for mux=0 and sigma

x=0.1

--> x

--> p

df


61

Result: Gaussian noise has been generated using matlab

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1e = cos() + isin()

--> x

-->

PD

F


62

Experiment No:14

Sampling theorem

Aim: To verify sampling theorem(over sampling, under sampling and critical sampling) using

matlab


MAT LAB Software.




4. Run the program


Program:

clc;

clear all;

close all;

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;


63

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;


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;


64

n3=-20:1:20;


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;

Result: Sampling theorem(over sampling, under sampling and critical sampling) have been

verified using matlab.

-10 -5 0 5 10-1

-0.5

0

0.5

1

time

x(t

)

continous time signal

-4 -2 0 2 4-1

-0.5

0

0.5

1

time

x(n

)

discrete time signal with fs<2fm

-5 0 5-1

-0.5

0

0.5

1

time

x(n

)

discrete time signal with fs=2fm

-20 -10 0 10 20-1

-0.5

0

0.5

1

time

x(n

)

discrete time signal with fs>2fm


65

Experiment No:15

Removal of noise

Aim: To remove noise by auto correlation / cross correlation using matlab.


MAT LAB Software.




4. Run the program


Program: Removal of noise by auto correlation

clc;

clear all;

t=0:0.1:pi*4;

s=sin(t);

k=2;

subplot(6,1,1)

plot(s);

title('signal s');

xlabel('t');


n = randn([1 126]);

f=s+n;

subplot(6,1,2)

plot(f);


66

title('signal f=s+n');

xlabel('t');


as=xcorr(s,s);

subplot(6,1,3)

plot(as);

title('auto correlation of s');

xlabel('t');


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);


67

title('addition of as+an');

xlabel('t');


Removal of noise by cross correlation:

clc;

clear all;

t=0:0.1:pi*4;

s=sin(t);

0 20 40 60 80 100 120 140-1

01

signal s

t

am

plitude

0 20 40 60 80 100 120 140-505

signal f=s+n

t

am

plitude

0 50 100 150 200 250 300-100

0100

auto correlation of s

t

am

plitude

0 50 100 150 200 250 300-200

0200

auto correlation of n

t

am

plitude

0 50 100 150 200 250 300-200

0200

auto correlation of f

t

am

plitude

0 50 100 150 200 250 300-200

0200

addition of as+an

t

am

plitude


68

k=2;

subplot(7,1,1)

plot(s);

title('signal s');

xlabel('t');


c=cos(t);

subplot(7,1,2)

plot(c);

title('signal c');

xlabel('t');


n = randn([1 126]);

f=s+n;

subplot(7,1,3)

plot(f);

title('signal f=s+n');

xlabel('t');


asc=xcorr(s,c);

subplot(7,1,4)

plot(asc);


69

title(' correlation of s and c');

xlabel('t');


anc=xcorr(n,c);

subplot(7,1,5)

plot(anc);

title(' correlation of n and c');

xlabel('t');


cfc=xcorr(f,c);

subplot(7,1,6)

plot(cfc);

title(' correlation of f and c');

xlabel('t');


hh=asc+anc;

subplot(7,1,7)

plot(hh);

title('addition of sc+nc');

xlabel('t');



70

Result: Noise have been removed by auto correlation and cross correlation using matlab.

0 20 40 60 80 100 120 140-1

0

1signal s

t

am

plitu

de

0 20 40 60 80 100 120 140-1

0

1signal c

t

am

plitu

de

0 20 40 60 80 100 120 140-5

0

5signal f=s+n

t

am

plitu

de

0 50 100 150 200 250 300-100

0

100 correlation of s and c

t

am

plitu

de

0 50 100 150 200 250 300-20

0

20 correlation of n and c

t

am

plitu

de

0 50 100 150 200 250 300-100

0

100 correlation of f and c

t

am

plitu

de

0 50 100 150 200 250 300-100

0

100addition of sc+nc

t

am

plitu

de


71

Experiment No:16

Extraction of periodic signal

Aim: To extract periodic signal masked by noise using correlation using matlab.


MAT LAB Software.




4. Run the program


Program:

clc

clear all;

close all;

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 on

for i=1:k


72

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

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

end

period=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));


73

Result: Periodic signal have been extracted by correlation using matlab.

0 50 100 150 200 250 300-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300-4

-3

-2

-1

0

1

2

3

4


74

Experiment No:17

Weiner-Khinchine relations

Aim: To verify weiner-khinchine relations using matlab.


MAT LAB Software.




4. Run the program


Program:

clc

clear all;

close 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);


75

subplot(3,2,4);

plot(fw);

fw2=(abs(fw)).^2;

subplot(3,2,5);

plot(fw2);

Result: Weiner-khinchine relations using matlab have been verified.

0 10 20 30 40 50 60 70-1

-0.5

0

0.5

1

0 20 40 60 80 100 120 140-40

-20

0

20

40

0 20 40 60 80 100 120 1400

500

1000

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-40

-20

0

20

40

0 10 20 30 40 50 60 700

500

1000


76

Experiment No:18

CHECKING A RANDOM PROCESS FOR STATIONARITY IN WIDE

SENSE

Aim: To check random process for stationary in wide sense using matlab.


MAT LAB Software.




4. Run the program


Program:

clc;

clear all;

close all;

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));


77

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;

solve(c);

disp('values are constants, it does not dependent on time, so it is wide sence

stationary ')

Output: ans = 0

-2

values are constants, it does not dependent on time, so it is wide sence stationary

Result: Random process for stationary in wide sense stationary using matlab.


78

1. Define Signal

2. Define determistic and Random Signal

3. Define Delta Function

4. Define Periodic and a periodic Signal

5. Define Symetric and Anti-Symmetric Signals

6. Define Continuous and Discrete Time Signals

7. What are the Different types of representation of discrete time signals

8. What are the Different types of representation of discrete time signals

9. What are the Different types of Operation performed on signals

10. What is System

11. What is Causal Signal

12. What are the Different types of Systems

13. What is Linear System

14. What is Time Invariant System

15. What is Static and Dynamic System

16. What is Even Signal

17. What is Odd Signal

18. Define the Properties of Impulse Signal

19. What is Causality Condition of the Signal

20. What is Condition for System Stability

21. Define Convolution

22. Define Properties of Convolution

23. What is the Sufficient condition for the existence of F.T

24. Define the F.T of a signal

25. State Paeseval’s energy theorem for a periodic signal

26. Define sampling Theorem

27. What is Aliasing Effect

28. what is Under sampling

29. What is Over sampling


79

30. Define Correlation

31. Define Auto-Correlation

32. Define Cross-Correlation

33. Define Convolution

34. Define Properties of Convolution

35. What is the Difference Between Convolution& Correlation

36. What are Dirchlet Condition

37. Define Fourier Series

38. What is Half Wave Symmetry

39. What are the properties of Continuous-Time Fourier Series

40. Define Laplace-Transform

41. What is the Condition for Convergence of the Laplace Transform

42. What is the Region of Convergence(ROC)

43. State the Shifting property of L.T

44. State convolution Property of L.T

45. Define Transfer Function

46. Define Pole-Zeros of the Transfer Function

47. What is the Relationship between L.T & F.T &Z.T

48. Fined the Z.T of a Impulse and step

49. What are the Different Methods of evaluating inverse z-T

50. what are the ROC properties of a Z.T

51. Define Initial Value Theorem of a Z.T

52. Define Final Value Theorem of a Z.T

53. Define Nyquist Rate

54. Define the condition for distortionless transmission through the system

55. What is signal band width

56. What is system band width

57. What is the relationship between rise time and band width

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