Bs Lab Manual

Basic Simulation Lab

Department of Electronics and Communication Engineering, TITS, Hyderabad

Laboratory manual

for

BASIC SIMULATION LAB

II B. Tech I Semester

By

Mr. J. Sunil Kumar

&

Mrs. P. Saritha

Department of Electronics & Communication Engineering

Turbomachinery Institute of Technology & Sciences

(Approved by AICTE & Affiliated to JNTUH)

Indresam(v), Patancheru(M), Medak(Dist)



Turbomachinery Institute of Technology & Sciences

Certificate

This is to certify that Mr. / Ms. ………………………………….. RollNo……………..… of I/II/III/IV

B.Tech I / II Semester of …………….……………………..…………branch has completed the laboratory work

satisfactorily in …………………..…….……..... Lab for the academic year 20 … to 20 …as prescribed in the

curriculum.

Place: …………….….

Date: ……………..….

Lab In charge Head of the Department Principal



INSTRUCTIONS FOR STUDENTS

Students shall read the points given below for understanding the theoretical concepts and practical applications.

1. Listen carefully to the lecture given by teacher about importance of subject, curriculum philosophy, Learning

structure, skills to be developed, information about equipment, instruments, procedure, method of continuous

assessment, tentative plan of work in laboratory and total amount of work to be done in a semester.

2. Students shall undergo study visit of the laboratory for types of equipment, instruments and material to be used,

before performing experiments.

3. Read the write up of each experiment to be performed, a day in advance.

4. Organize the work in the group and make a record of all observations.

5. Understand the purpose of experiment and its practical implications.

6. Student should not hesitate to ask any difficulty faced during conduct of practical / exercise.

7. Student shall develop maintenance skills as expected by the industries.

8. Student should develop the habit of pocket discussion / group discussion related to the experiments/ exercises so

that exchanges of knowledge / skills could take place.

9. Student should develop habit to submit the practical, exercise continuously and progressively on the scheduled dates

and should get the assessment done.

10. Student shall attempt to develop related hands - on - skills and gain confidence.

11. Student shall focus on development of skills rather than theoretical or codified knowledge.

12. Student shall visit the nearby workshops, workstation, industries, laboratories, technical exhibitions trade fair etc.

even not included in the Lab Manual. In short, students should have exposure to the area of work right in the student

hood.

13. Student shall develop the habit of evolving more ideas, innovations, skills etc. those included in the scope of the

manual.

14. Student shall refer to technical magazines, proceedings of the Seminars, refer websites related to the scope of the

subjects and update their knowledge and skills..

15. The student shall study all the questions given in the laboratory manual and practice to write the answers to these

questions.



INDEX

S.no Name of the Experiment Page

No

Date of

Perform

ance

Date

of

Submi

ssion

Assessmen

t marks

(Max 10)

Sign.

of

Facult

y

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11

12

13

14

15

16



List of Experiments:

1. Basic Operations on Matrices.

2. Generation of Various Signals and Sequences (Periodic and Aperiodic), such as Unit impulse, unit

step, square, saw tooth, triangular, sinusoidal, ramp, sinc.

3. Observations 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 parts of signal.

5. Convolution between signals and sequences.

6. Autocorrelation and cross correlation between signals and sequences.

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

8. Computation of unit sample, unit step and sinusoidal responses of the given LTI system and

verifying its physical realizability 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 Tronsform.

12. Locating the zeros and poles and plotting the pole-zero maps in S plane and Z-plane for the given

transfer function.

13. 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 autocorrelation / cross correlation.

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

17. Verification of winer-khinchine relations.

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



Exp. No: 1 Date:

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

Theory:

MATLAB treats all variables as matrices. For our purposes a matrix can be thought of as an array, in fact,

that is how it is stored.

• Vectors are special forms of matrices and contain only one row OR one column.

• Scalars are matrices with only one row AND one column.A matrix with only one row AND one

column is a scalar. A scalar can be reated in MATLAB as follows:

» a_value=23

a_value =23

• A matrix with only one row is called a row vector. A row vector can be created in MATLAB as

follows :

» rowvec = [12 , 14 , 63]

rowvec = 12 14 63

• A matrix with only one column is called a column vector. A column vector can be created in

MATLAB as follows:

» colvec = [13 ; 45 ; -2]

colvec =

13

45

-2

• A matrix can be created in MATLAB as follows:

» matrix = [1 , 2 , 3 ; 4 , 5 ,6 ; 7 , 8 , 9]

matrix =

1 2 3

4 5 6

7 8 9

Extracting a Sub-Matrix



A portion of a matrix can be extracted and stored in a smaller matrix by specifying the names of both

matrices and the rows and columns to extract. The syntax is:

sub_matrix = matrix ( r1 : r2 , c1 : c2 ) ;

Where r1 and r2 specify the beginning and ending rows and c1 and c2 specify the beginning and ending

columns to be extracted to make the new matrix.

• A column vector can beextracted from a matrix.

• As an example we create a matrix below:

» matrix=[1,2,3;4,5,6;7,8,9]

matrix =

1 2 3

4 5 6

7 8 9

Here we extract column 2 of the matrix and make a column vector:

» col_two=matrix( : , 2)

col_two =

2 5 8

• A row vector can be extracted from a matrix.

As an example we create a matrix below:

» matrix=[1,2,3;4,5,6;7,8,9]

matrix =

1 2 3

4 5 6

7 8 9

• Here we extract row 2 of the matrix and make a row vector. Note that the 2:2 specifies the second

row and the 1:3 specifies which columns of the row.

» rowvec=matrix(2 : 2 , 1 :3)

rowvec =4 5 6

» a=3;

» b=[1, 2, 3;4, 5, 6]

b =



1 2 3

4 5 6

» c= b+a % Add a to each element of b

c =

4 5 6

7 8 9

• Scalar - Matrix Subtraction

» a=3;

» b=[1, 2, 3;4, 5, 6]

b =

1 2 3

4 5 6

» c = b - a %Subtract a from each element of b

c =

-2 -1 0

1 2 3

• Scalar - Matrix Multiplication

» a=3;

» b=[1, 2, 3; 4, 5, 6]

b =

1 2 3

4 5 6

» c = a * b % Multiply each element of b by a

c =

3 6 9

12 15 18

• Scalar - Matrix Division

» a=3;

» b=[1, 2, 3; 4, 5, 6]



b =

1 2 3

4 5 6

» c = b / a % Divide each element of b by a

c =

0.3333 0.6667 1.0000

1.3333 1.6667 2.0000

a = [1 2 3 4 6 4 3 4 5]

a =

1 2 3 4 6 4 3 4 5

b = a + 2

b =

3 4 5 6 8 6 5 6 7

A = [1 2 0; 2 5 -1; 4 10 -1]

A =

1 2 0

2 5 -1

4 10 -1

B = A'

B =

1 2 4

2 5 10

0 -1 -1

C = A * B

C =

5 12 24

12 30 59



24 59 117

Instead of doing a matrix multiply, we can multiply the corresponding elements of two matrices or

vectors using the .* operator.

C = A .* B

C =

1 4 0

4 25 -10

0 -10 1

Let's find the inverse of a matrix …

X = inv(A)

X =

5 2 -2

-2 -1 1

0 -2 1

and then illustrate the fact that a matrix times its inverse is the identity matrix.

I = inv(A) * A

I =

1 0 0

0 1 0

0 0 1

to obtain eigenvalues

eig(A)

ans =

3.7321

0.2679

1.0000

as the singular value decomposition.

svd(A)



ans =

12.3171

0.5149

0.1577

CONCLUSION: Inthis experiment basic operations on matrices Using MATLAB have been demonstrated.

3. Perform following operations on any two matrices

A+B

A-B

A*B

A.*B

A/B

A./B

A\B

A.\B

A^B,A.^B,A',A.

4. Enter the following matrix at the command window and perform the following

A= [3 4 5 1

5 6 7 2

7 8 9 3]

i. All the elements of all rows but first column



ii. All the elements of first row but all columns

iii. Element in second row and third column

iv. Reshape this matrix to (4×3) matrix

v. Determine inverse , Determinant ,rank, size, transpose, eigen values, of A

vii. Multiply A with another Matrix B

viii. perform arry multiplication of A and B

5. x= [2 3 4] ; y=[2 5 1] perform all logical operations (AND, OR, NOT, EXOR)

6. Perform following operations on any two matrices

A+B, A-B, A*B, A.*B, A/B, A./B, A^B, A.^B, A',A



EXP.NO: 2 Date:

GENERATION ON 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.


MATLAB Software

Theory :

Unit Impulse: a) Continuous signal:

And

Also called unit impulse function. The value of delta function can also be defined in the sense of

generalized function

φ(t): Test Function

b) Unit Sample sequence: δ(n)= 1, n=0

0, n≠0

i.e

1)( =∫∞

∞−dttδ

=∞

≠=

0

00)(

t

ttδ



Matlab program:

%unit impulse generation

clc

close all

n1=-3;

n2=4;

n0=0;

n=[n1:n2];

x=[(n-n0)==0]

stem(n,x)

2) Unit Step Function u(t):

b) Unit Step Sequence u(n): )= 1, n ≥ 0

0, n < 0

% unit step generation

n1=-4;

<

>=

00

01)(

t

ttu

∫∫∞∞

∞−=

0)()()( dttdtttu φφ



n2=5;

n0=0;

[y,n]=stepseq(n0,n1,n2);

stem(n,y);

xlabel('n')

ylabel('amplitude');

title('unit step');

Square waves: Like sine waves, square waves are described in terms of period, frequency and amplitude:

Peak amplitude, Vp , and peak-to-peak amplitude, Vpp , are measured as you might expect. However,

the rms amplitude, Vrms , is greater than that of a sine wave. Remember that the rms amplitude is the DC

voltage which will deliver the same power as the signal. If a square wave supply is connected across a lamp,



the current flows first one way and then the other. The current switches direction but its magnitude remains

the same. In other words, the square wave delivers its maximum power throughout the cycle so that Vrms is

equal to Vp . (If this is confusing, don't worry, the rms amplitude of a square wave is not something you need

to think about very often.)

Although a square wave may change very rapidly from its minimum to maximum voltage, this

change cannot be instantaneous. The rise time of the signal is defined as the time taken for the voltage to

change from 10% to 90% of its maximum value. Rise times are usually very short, with durations measured

in nanoseconds (1 ns = 10-9

s), or microseconds (1 µs = 10-6

s), as indicated in the graph

% square 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])



title('Square Periodic Wave');

subplot(2,2,3);

stem(t,x2);

axis([0 0.1 -1.2 1.2])





Sawtooth:

The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is named a sawtooth based

on its resemblance to the teeth on the blade of a saw. The convention is that a sawtooth wave ramps upward

and then sharply drops. However, there are also sawtooth waves in which the wave ramps downward and

then sharply rises. The latter type of sawtooth wave is called a 'reverse sawtooth wave' or 'inverse sawtooth

wave'. As audio signals, the two orientations of sawtooth wave sound identical. The piecewise linear function

based on the floor function of time t, is an example of a sawtooth wave with period 1.

A more general form, in the range −1 to 1, and with period a. This sawtooth function has the same phase as

the sine function. A sawtooth wave's sound is harsh and clear and its spectrum contains both even and odd

harmonics of the fundamental frequency. Because it contains all the integer harmonics, it is one of the best

waveforms to use for synthesizing musical sounds, particularly bowed string instruments like violins and

cellos, using subtractive synthesis.



Applications

The sawtooth and square waves are the most common starting points used to create sounds with

subtractive analog and virtual analog music synthesizers. The sawtooth wave is the form of the vertical and

horizontal deflection signals used to generate a raster on CRT-based television or monitor screens.

Oscilloscopes also use a sawtooth wave for their horizontal deflection, though they typically use electrostatic

deflection. On the wave's "ramp", the magnetic field produced by the deflection yoke drags the electron beam

across the face of the CRT, creating a scan line. On the wave's "cliff", the magnetic field suddenly collapses,

causing the electron beam to return to its resting position as quickly as possible. The voltage applied to the

deflection yoke is adjusted by various means (transformers, capacitors, center-tapped windings) so that the

half-way voltage on the sawtooth's cliff is at the zero mark, meaning that a negative voltage will cause

deflection in one direction, and a positive voltage deflection in the other; thus, a center-mounted deflection

yoke can use the whole screen area to depict a trace. Frequency is 15.734 kHz on NTSC, 15.625 kHz for

PAL and SECAM)

% 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');

Triangle wave

A triangle wave is a non-sinusoidal waveform named for its triangular shape. A band limited triangle

wave pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz

(A2).Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll

off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed

to just the inverse).It is possible to approximate a triangle wave with additive synthesis by adding odd

harmonics of the fundamental, multiplying every (4n−1)th harmonic by −1 (or changing its phase by π), and

rolling off the harmonics by the inverse square of their relative frequency to the fundamental. This infinite

Fourier series converges to the triangle wave:



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 Generation

The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs

often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields.

Its most basic form as a function of time (t) is:

where:

• A, the amplitude, is the peak deviation of the function from its center position.

• ω, the angular frequency, specifies how many oscillations occur in a unit time interval, in radians per

second

• φ, the phase, specifies where in its cycle the oscillation begins at t = 0.

A sampled sinusoid may be written as:

where f is the signal frequency, fs is the sampling frequency, θ is the phase and A is the amplitude of the

signal. The program and its output is shown below: Note that there are 64 samples with sampling frequency

of 8000Hz or sampling time of 0.125 mS (i.e. 1/8000). Hence the record length of the signal is

64x0.125=8mS. There are exactly 8 cycles of sinewave, indicating that the period of one cycle is 1mS which

means that the signal frequency is 1KHz.



% 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');


% RAMP

clc

close all



n=input('enter the length of ramp');

t=0:n;

plot(t);

xlabel('t');


title ('ramp')

Sinc function:

The sinc function computes the mathematical sinc function for an input vector or matrix x. Viewed as a

function of time, or space, the sinc function is the inverse Fourier transform of the rectangular pulse in

frequency centered at zero of width 2π and height 1. The following equation defines the sinc function:

The sinc function has a value of 1 whenx is equal to zero, and a value of

% 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’);

Conclusion: In this experiment various signals have been generated Using MATLAB

Exercise Questions: Generate following signals using MATLAB

1. x(t)=e-t

2. x(t)= t 2 / 2

3. Generate rectangular pulse function

4. Generate signum sunction sinc(t)= 1 t > 0

0 t=0

-1 t<0

5. Generate complex exponential signal x(t)= e st for different values of σ and Ω



Exp. No: 3 Date:

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.


MATLAB Software

Theory :

Basic Operation on Signals:

Time shifting: y(t)=x(t-T)The effect that a time shift has on the appearance of a signal If T is a positive

number, the time shifted signal, x (t -T ) gets shifted to the right, otherwise it gets shifted left.

Signal Shifting and Delay:

. Shifting : y(n)=x(n-k) ; m=n-k; y=x;

Time reversal: Y(t)=y(-t) Time reversal _ips the signal about t = 0 as seen in

Figure 1.

Signal Addition and Subtraction:

Addition: any two signals can be added to form a third signal,

z (t) = x (t) + y (t)

Signal Amplification/Atténuation:



Multiplication/Division: of two signals, their product is also a signal.

z (t) = x (t) y (t)

folding: y(n)=x(-n) ; y=fliplr(x); n=-fliplr(n);

%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);



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) variable

A = 8; % amplitude



subplot(3,2,1)

plot(s1);

xlabel('t');


s2=2*s1;

subplot(3,2,2)

plot(s2);



xlabel('t');


s3=s1/2;

subplot(3,2,3)

plot(s3);

xlabel('t');


subplot(3,2,4)

stem(s1);

xlabel('t');


s2=2*s1;

subplot(3,2,5)

stem(s2);

xlabel('t');


s3=s1/2;

subplot(3,2,6)

stem(s3);

xlabel('t');




Exercise 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: In this experiment the various operations on signals have been performed. Using MATLAB

has been demonstrated.



Exp. No : 4 Date:

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.


MATLAB Software

Theory : Even and Odd Signal

One of characteristics of signal is symmetry that may be useful for signal analysis. Even signals are

symmetric around vertical axis, and Odd signals are symmetric about origin.

Even Signal: A signal is referred to as an even if it is identical to its time-reversed counterparts; x(t) = x(-t).

Odd Signal: A signal is odd if x(t) = -x(-t). An odd signal must be 0 at t=0, in other words, odd signal

passes the origin. Using the definition of even and odd signal, any signal may be decomposed into a sum of

its even part, xe(t), and its odd part, xo(t), as follows:

It is an important fact because it is relative concept of Fourier series. In Fourier series, a periodic signal can

be broken into a sum of sine and cosine signals. Notice that sine function is odd signal and cosine function is

even signal.

%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);

xlabel('time');


title('folded signal');

z=h+x1

subplot(3,2,3);

stem(t,z);


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 and Power Signal: A signal can be categorized into energy signal or power signal: An energy

signal has a finite energy, 0 < E < ∞. In other words, energy signals have values only in the limited time

duration. For example, a signal having only one square pulse is energy signal. A signal that decays

exponentially has finite energy, so, it is also an energy signal. The power of an energy signal is 0, because of

dividing finite energy by infinite time (or length).

On the contrary, the power signal is not limited in time. It always exists from beginning to end and it never

ends. For example, sine wave in infinite length is power signal. Since the energy of a power signal is infinite,

it has no meaning to us. Thus, we use power (energy per given time) for power signal, because the power of

power signal is finite, 0 < P < ∞.

% 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: In this experiment even and odd parts of various signals and energy and power of signals have

been calculated Using MATLAB

Exercise Questions: find even and odd parts of the following signals

1. x(n)= e(-0.1+j0.3)n

,- 10≤n≤10

2. Calculate the following signals energies

3. x(t)=u(t)-u(t-15)

4. cos(10∏t)u(t)u(2-t-5)

5. Calculate the following signals energies

a. x(t)= e

j(2t+∏/4)

b. x(t)=cos(t)

c. x(t)=cos(∏/4) n



EXP.NO: 5 Date:

LINEAR CONVOLUTION

Aim: To find the out put with linear convolution operation Using MATLAB Software.


MATLAB Software

Theory:

Linear Convolution involves the following operations.

1. Folding

2. Multiplication

3. Addition

4. Shifting

These operations can be represented by a Mathematical Expression as follows:

x[ ]= Input signal Samples

h[ ]= Impulse response co-efficient.

y[ ]= Convolution output.

n = No. of Input samples

h = No. of Impulse response co-efficient.

Example : X(n)=1 2 -1 0 1, h(n)= 1,2,3,-1

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



Conclusion: In this experiment convolution of various signals have been performed Using MATLAB

Applications:

Convolution is used to obtain the response of an LTI system to an arbitrary input signal.It is used to find the

filter response and finds application in speech processing and radar signal processing.

Exercise 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: 6 Date:

6. AUTO CORRELATION AND CROSS CORRELATION BETWEEN

SIGNALS AND SEQUENCES.

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


MATLAB Software

Theory:

Correlations of sequences:

It is a measure of the degree to which two sequences are similar. Given two real-valued sequences x(n)

and y(n) of finite energy,

Convolution involves the following operations.

1. Shifting

2. Multiplication

3. Addition

These operations can be represented by a Mathematical Expression as follows:

Crosscorrelation

The index l is called the shift or lag parameter

Autocorrelation

The special case: y(n)=x(n)

% Cross Correlation

clc;

close all;

clear all;

x=input('enter input sequence');

h=input('enter the impulse suquence');

∑+∞

−∞=

−=n

yx lnynxlr )()()(,

∑+∞

−∞=

−=n

xx lnxnxlr )()()(,



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)');


subplot(3,1,2);

stem(y);



xlabel('n');

ylabel('y(n)');


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 has 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




1. X(n)=[1 -1 4 ], h(n) = [ -1 2 -3 1]


x1(t)=e-3t

u(t)-u(t-2) , x2(t)= e -3t

for 0 ≤ t ≤ 2



EXP.NO: 7 Date:

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

Theory:

Linearity Property

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);

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)

Linear Time Invariant Systems(LTI):



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);

subplot(3,1,3);

stem(d);



Conclusion: In this experiment Linearity and Time invariance property of given system has bees verified

performed Using MATLAB

Applications: It is used to measure the degree to which the two signals are similar and it is also used for

radar detection by estimating the time delay. It is also used in Digital communication defence applications

and sound navigation


1. X(n)=[1 -1 4 ], h(n) = [ -1 2 -3 1]


x1(t)=e-3t

u(t)-u(t-2) , x2(t)= e -3t

for 0 ≤ t ≤ 2



EXP.NO:8 Date:

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

Reliability And Stability Properties.


MATLAB Software

A discrete time system performs an operation on an input signal based on predefined criteria to

produce a modified output signal. The input signal x(n) is the system excitation, and y(n) is the system

response. The transform operation is shown as,

If the input to the system is unit impulse i.e. x(n) = δ(n) then the output of the system is known as impulse

response denoted by h(n) where,

h(n) = T[δ(n)]

we know that any arbitrary sequence x(n) can be represented as a weighted sum of discrete impulses. Now

the system response is given by,

For linear system (1) reduces to

%given difference equation y(n)-y(n-1)+.9y(n-2)=x(n);



%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');


=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');




figure;

zplane(b,a);



Conclusion: In this experiment computation of unit sample, unit step and sinusoidal response of the given lti

system and verifying its physical realizability and stability properties Using MATLAB



EXP.NO: 9 Date:

GIBBS PHENOMENA

Aim: To verify the Gibbs Phenomenon.


MATLAB Software

Theory:

The Gibbs phenomenon, the Fourier series of a piecewise continuously differentiable periodic function

behaves at a discontinuity. the n the approximated function shows amounts of ripples at the points of

discontinuity. This is known as the Gibbs Phenomena. Partial sum of the Fourier series has large oscillations

near the jump, which might increase the maximum of the partial sum above that of the function itself. The

overshoot does not die out as the frequency increases, but approaches a finite limit

The Gibbs phenomenon involves both the fact that Fourier sums overshoot at a jump discontinuity, and that

this overshoot does not die out as the frequency increases

Gibbs Phenomina Program :

t=0:0.1:(pi*8);

y=sin(t);

subplot(5,1,1);

plot(t,y);

xlabel('k');


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');


h=h+1;



end

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



EXP.NO: 10. Date:

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


MATLAB Software

Fourier Transform Theorems:

We may use Fourier series to motivate the Fourier transform as follows. Suppose that ƒ is a function which is

zero outside of some interval [−L/2, L/2]. Then for any T ≥ L we may expand ƒ in a Fourier series on the

interval [−T/2,T/2], where the "amount" of the wave e2πinx/T

in the Fourier series of ƒ is given by

By definition

Program:

Aim: To compute N-point FFT


MATLAB Software

Theory:

DFT of a sequence X[K] = [ ] N

KnjN

K

enx

∏−−

=∑

21

0

Where N= Length of sequence.

K= Frequency Coefficient.

n = Samples in time domain.

FFT : -Fast Fourier transformer .



There are Two methods.

1. Decimation in time (DIT FFT).

2. Decimation in Frequency (DIF FFT).

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')

Applications:

The no of multiplications in DFT = N2.

The no of Additions in DFT = N(N-1).



For FFT.

The no of multiplication = N/2 log 2N.

The no of additions = N log2 N.

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 Date:

LAPLACE TRNASFORMS

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

Bilateral Laplace transform :

When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is

normally intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform or

two-sided Laplace transform by extending the limits of integration to be the entire real axis. If that is done

the common unilateral transform simply becomes a special case of the bilateral transform where the

definition of the function being transformed is multiplied by the Heaviside step function.

The bilateral Laplace transform is defined as follows:

Inverse Laplace transform

The inverse Laplace transform is given by the following complex integral, which is known by various names

(the Bromwich integral, the Fourier-Mellin integral, and Mellin's inverse formula):

Program for Laplace Transform:

f=t

syms f t;

f=t;

laplace(f)

Program for inverse 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) %inverse LT poles and zeros



Signal syntheses using Laplace Transform:

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

Applications of laplace transforms:

1. Derive the circuit (differential) equations in the time domain, then transform these ODEs to the s-

domain;

2. Transform the circuit to the s-domain, then derive the circuit equations in the s-domain (using the

concept of "impedance").

The main idea behind the Laplace Transformation is that we can solve an equation (or system of equations)

containing differential and integral terms by transforming the equation in "t-space" to one in "s-space". This

makes the problem much easier to solve



EXP.NO: 12 Date:

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:

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

MATLAB Software

Z-transforms

the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers,

into a complex frequency-domain representation.The Z-transform, like many other integral transforms, can

be defined as either a one-sided or two-sided transform.

Bilateral Z-transform

The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the function X(z) defined as

Unilateral Z-transform

Alternatively, in cases where x[n] is defined only for n ≥ 0, the single-sided or unilateral Z-transform is

defined as

In signal processing, this definition is used when the signal is causal.

The roots of the equation P(z) = 0 correspond to the 'zeros' of X(z)

The roots of the equation Q(z) = 0 correspond to the 'poles' of X(z)

The ROC of the Z-transform depends on the convergence of the



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);

Applications :Z-Transform is used to find the system responses

Conclusion: In this experiment the zeros and poles and plotting the pole zero maps in s-plane and z-plane for

the given transfer function using MATLAB.



EXP.NO: 13 Date:

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);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 Date:

14. SAMPLING THEOREM VERIFICATION

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


MATLAB Software

Theory:

Sampling Theorem:

A band limited signal can be reconstructed exactly if it is sampled at a rate atleast twice the maximum

frequency component in it." Figure 1 shows a signal g(t) that is bandlimited.

Figure 1: Spectrum of bandlimited signal g(t)

The maximum frequency component of g(t) is fm. To recover the signal g(t) exactly from its samples it has

to be sampled at a rate fs ≥ 2fm. The minimum required sampling rate fs = 2fm is called ' Nyquist rate

Proof: Let g(t) be a bandlimited signal whose bandwidth is fm (wm = 2πfm).



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 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 the bandwidth of the signal.

Oversampling: In practice signal are oversampled, where fs is sampling frequency 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;


subplot(2,2,3);

stem(n2,x2);


title('discrete time signal with fs=2fm')

hold on

subplot(2,2,3);

plot(n2,x2)

grid;

n3=-20:1:20;


subplot(2,2,4);

stem(n3,x3);


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



EXP.No:15 Date:

REMOVAL OF NOISE BY AUTO CORRELATION/CROSS CORRELATION

Aim: removal of noise by auto correlation/cross correlation


MATLAB Software

Theory:

Detection of a periodic signal masked by random noise is of greate importance .The noise signal

encountered in practice is a signal with random amplitude variations. A signal is uncorrelated with any

periodic signal. If s(t) is a periodic signal and n(t) is a noise signal then

T/2

Lim 1/T ∫ S(t)n(t-T) dt=0 for all T

T--∞ -T/2

Qsn(T)= cross correlation function of s(t) and n(t) Then Qsn(T)=0

Detection by Auto-Correlation:

S(t)=Periodic Signal mixed with a noise signal n(t).Then f(t) is [s(t ) + n(t) ]

Let Qff(T) =Auto Correlation Function of f(t)

Qss(t) = Auto Correlation Function of S(t)

Qnn(T) = Auto Correlation Function of n(t)

T/2

Qff(T)= Lim 1/T ∫ f(t)f(t-T) dt

T--∞ -T/2

T/2

= Lim 1/T ∫ [s(t)+n(t)][s(t-T)+n(t-T)] dt

T--∞ -T/2

=Qss(T)+Qnn(T)+Qsn(T)+Qns(T)

The periodic signal s(t) and noise signal n(t) are uncorrelated

Qsn(t)=Qns(t)=0 ;Then Qff(t)=Qss(t)+Qnn(t)

The Auto correlation function of a periodic signal is periodic of the same frequency and the Auto correlation

function of a non periodic signal is tends to zero for large value of T since s(t) is a periodic signal and n(t) is



non periodic signal so Qss(T) is a periodic where as aQnn(T) becomes small for large values of T Therefore

for sufficiently large values of T Qff(T) is equal to Qss(T).

Detection by Cross Correlation:

f(t)=s(t)+n(t)

c(t)=Locally generated signal with same frequencyas that of S(t)

T/2

Qfc (t) = Lim 1/T ∫ [s(t)+n(t)] [ c(t-T)] dt

T--∞ -T/2

= Qsc(T)+Qnc(T)

C(t) is periodic function and uncorrelated with the random noise signal n(t). Hence Qnc(T0=0) Therefore

Qfc(T)=Qsc(T)

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');


n = randn([1 126]);

f=s+n;

subplot(6,1,2)

plot(f);

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);

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');

Conclusion: In this experiment the removal of noise by auto correlation/cross correlation have been verified

using MATLAB



EXP.No:16 Date:

EXTRACTION OF PERIODIC SIGNAL MASKED BY NOISE USING CORRELATION

Extraction of Periodic Signal Masked By Noise Using Correlation

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 on

for i=1:k

[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));



Applications:

The theorem is useful for analyzing linear time-invariant systems, LTI systems, when the inputs and outputs

are not square integrable, so their Fourier transforms do not exist. A corollary is that the Fourier transform of

the autocorrelation function of the output of an LTI system is equal to the product of the Fourier transform of

the autocorrelation function of the input of the system times the squared magnitude of the Fourier transform

of the system impulse response. This works even when the Fourier transforms of the input and output signals

do not exist because these signals are not square integrable, so the system inputs and outputs cannot be

directly related by the Fourier transform of the impulse response. Since the Fourier transform of the

autocorrelation function of a signal is the power spectrum of the signal, this corollary is equivalent to saying



that the power spectrum of the output is equal to the power spectrum of the input times the power transfer

function. This corollary is used in the parametric method for power spectrum estimation.

Conclusion: In this experiment the Weiner-Khinchine Relation has been verified using MATLAB.



Exp. No:17 Date:

VERIFICATION OF WIENER–KHINCHIN RELATION Aim: Verification of wiener–khinchine relation


MATLAB Software

Theory:

The Wiener–Khinchin theorem (also known as the Wiener–Khintchine theorem and sometimes as the

Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem) states that the power

spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding

autocorrelation function.

Continuous case:

Where

is the autocorrelation function defined in terms of statistical expectation, and where is the power spectral

density of the function . Note that the autocorrelation function is defined in terms of the expected value of a

product, and that the Fourier transform of does not exist in general, because stationary random functions are

not square integrable.

The asterisk denotes complex conjugate, and can be omitted if the random process is real-valued.

Discrete case:

Where and where is the power spectral density of the function with

discrete values Being a sampled and discrete-time sequence, the spectral density is periodic in the frequency

domain.



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);



Exp No.18. Date:

CHECKING A RANDOM PROCESS FOR STATIONARITY IN WIDE SENSE.

Aim: Checking a random process for stationarity in wide sense.


MATLAB Software

Theory: A stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a

stochastic process whose joint probability distribution does not change when shifted in time or space. As a

result, parameters such as the mean and variance, if they exist, also do not change over time or position..

Definition

Formally, let Xt be a stochastic process and let represent the cumulative

distribution function of the joint distribution of Xt at times t1…..tk. Then, Xt is said to be stationary if, for all

k, for all τ, and for all t1…..tk

Weak or wide-sense stationarity

A weaker form of stationarity commonly employed in signal processing is known as weak-sense

stationarity, wide-sense stationarity (WSS) or covariance stationarity. WSS random processes only

require that 1st and 2nd moments do not vary with respect to time. Any strictly stationary process which has

a mean and a covariance is also WSS.

So, a continuous-time random process x(t) which is WSS has the following restrictions on its mean function

and autocorrelation function

The first property implies that the mean function mx(t) must be constant. The second property implies that the

correlation function depends only on the difference between t1 and t2 and only needs to be indexed by one

variable rather than two variables. Thus, instead of writing,

we usually abbreviate the notation and write



This also implies that the auto covariance depends only on τ = t1 − t2, since

When processing WSS random signals with linear, time-invariant (LTI) filters, it is helpful to think of the

correlation function as a linear operator. Since it is a circulant operator (depends only on the difference

between the two arguments), its eigenfunctions are the Fourier complex exponentials. Additionally, since the

eigenfunctions of LTI operators are also complex exponentials, LTI processing of WSS random signals is

highly tractable—all computations can be performed in the frequency domain. Thus, the WSS assumption is

widely employed in signal processing algorithms.

Applicatons: Stationarity is used as a tool in time series analysis, where the raw data are often transformed

to become stationary, for example, economic data are often seasonal and/or dependent on the price level.

Processes are described as trend stationary if they are a linear combination of a stationary process and one or

more processes exhibiting a trend. Transforming these data to leave a stationary data set for analysis is

referred to as de-trending

Stationary and Non Stationary Random Process:

A random X(t) is stationary if its statistical properties are unchanged by a time shift in the time origin.When

the auto-Correlation function Rx(t,t+T) of the random X(t) varies with time difference T and the mean value

of the random variable X(t1) is independent of the choice of t1,then X(t) is said to be stationary in the wide-

sense or wide-sense stationary . So a continous- Time random process X(t) which is WSS has the following

properties

1) E[X(t)]=µX(t)= µX(t+T)

2) The Autocorrelation function is written as a function of T that is

3) RX(t, t+T)=Rx(T)

If the statistical properties like mean value or moments depends on time then the random process is said

to be non-stationary.

When dealing with two random process X(t) and Y(t), we say that they are jointly wide-sense stationary

if each process is stationary in the wide-sense.

Rxy(t,t+T)=E[X(t)Y(t+T)]=Rxy(T).

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

Viva-voice:

1. Define Signal

2. Define deterministic and Random Signal

3. Define Delta Function

4. What is Signal Modeling

5. Define Periodic and a periodic Signal

6. Define Symmetric and Anti-Symmetric Signals

7. Define Continuous and 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 Signum function?

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

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 L.T

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. Explain Time-Shifting property of a Z.T

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

52. Define Initial Value Theorem of a Z.T

53. Define Final Value Theorem of a Z.T

54. Define Sampling Theorem

55. Define Nyquist Rate

56. Define Energy of a Signal

57. Define Power of a signal

58. Define Gibbs Phenomena

59. Define the condition for distortionless transmission through the system

60. What is signal band width

61. What is system band width

62. What is Paley-Winer criterion?

63. Derive relationship between rise time and band width.

64. State the relation ship between PSD and ACF?

65. What is the integration of ramp signal?

66. Difference between vectors and signals?

67. What is the important of dot product over cross product in signals?

68. Define Hilbert transform?



69. Relation ship between FT and ZT?

70. What are the different properties of signals?

71. What are the different properties of systems?

72. what LTI system?

73. What is time variant and time in variant with examples?

74. Define inevitability?

75. Define stable and un stable?

76. what is the condition for WSS random process?

77. Define random variable with examples?

78. Define random process with examples?

79. Difference between ACF and CCF?

80. what are the different noises?

LAB PRACTICE QUESTIONS

Write and simulate a Matlab function to generate a Unit parabola

Write and simulate a Matlab function to generate a Unit cubic

Write and simulate a Matlab for triangle wave, sine wave of 3vpp and frequency 1000hz

Generate Matlab program to exp(-t)cos(2*pi*4*t)

Generate Matlab program to t*(cos2*pi*4*t)

Generate Matlab program to t *exp(-t)

Generate Matlab program to step function

Generate a Exponential sequence multiplied with ‘cos’ function

Generate a sin function output with exponential sequences.

Program to plot u(t-4), u(t+4),δ(t+5), δ(t+6),π((t-1)/2),u(-t+3),u(2t+4)

Create the function x=y(t) such that

y(t)=t+5; -5<t<-2;

11+4t; -2<t<1;

24-19t; 1<t<3

t-6; 3<t<6

hence plot y(t), y(t+4), 2y(t-3), y(2t), y(2t-3), y(t/2).

Genarate the following using MATLAB

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

x(t)=3r(t-1)

x(t)=U(n+2-u(n-3)

x(n)=x1(n)+x2(n)where x1(n)=1,3,2,1,x2(n)=1,-2,3,2

x(t)=r(t)-2r(t-1)+r(t-2)

x(n)=2δ(n+2)-2δ(n-4), -5≤ n ≥5.



Calculate the following signals energies

i. x(t)=u(t)-u(t-15)

ii .cos(10∏t)u(t)u(2-t-5)

Calculate the following signals energies

d. x(t)= e

j(2t+∏/4)

e. x(t)=cos(t)

f. x(t)=cos(∏/4) n

Write a program to accept a finite duration causal sequence x[n] and plot the sequence and it’s odd and

even parts.

Find the even and odd parts of the signal x(t)=sint+cost+sintcost and plot the individual signal using

MATLAB

Write a Mat lab program to convolve the signals x(t) = π[(t-1)/2] and h(t) = e-0.5t

u(t) plot the two signals

x(t) and h(t) along with the result of the convolution, y(t).

Write a Mat lab program using the conv function to obtain the Hilbert transforms of the Following signals

a) A Square pulse; b) A Triangle;

Simulate a matlab program on autocorrelation and PSD if x = [1 2 3 4 5 ]

Simulate a matlab program on autocorrelation and PSD if x = [1 1 0 1 ]

x (n)=[1 -1 4 ], h(n) = [ -1 2 -3 1] find y(n), where x(n) is input, h(n) is impulse response, and y(n) is the

output of the system.

Perform convolution between the. Two periodic sequences

x1(t)=e-3t

u(t)-u(t-2) , x2(t)= e -3t

for 0 ≤ t ≤ 2

x(n)=[1 -1 4 ], h(n) = [ -1 2 -3 1] find y(n) where x(n) is input, h(n) is impulse response, and y(n) is the

output of the system.

Check the linearity property of the systems y(t)=2x(t)+3 and y(t)=x2(t)

Check the stability of the systems y(t)= t e-at

u(t) and y(t)=t eatu(t)

Check the time invariant property of the systems y(t)=tx(t) and y(t)= x(4t)

Let the impulse response of an LTI system be given by h(t)=u(t-1)-u(t-4) find output of system if input

x(t)=u(t)+u(t-1)-2u(t-2)

Write and simulate a program on impulse response if y(n) = x(n) + x(n-1) + x(n-2)

Write and simulate a program on unit step sequence if y(n) = u(n) + (n-2) + u(n-4)

Write and simulate a program on impulse response if y(t)= cos(t) + sin(t)

Write and simulate an impulse response of y(n) = 2x(n -1) – 2x(n – 2) + 4y(n – 1)

Write and simulate an impulse response of y(n) = x(n) – x(n -1) – x(n-2)



Write a program for fourier transform of periodic signal

Write a program of FT of cos(t)

x(t)= 3e-tu(t) be the input to a system with impulse response h(t)= 2 e

-2tu(t) find y(t) and its fourier transform.

Given the following polynomials in S, find the inverse Laplace transforms, y1 (t), y2 (t), y3 (t), and y4 (t). in

Symbolic Mat lab Dirac(n,t) denotes the nth derivative of the unit impulse Function. X1(s) = s3 + 6s2 +11s

+6, X2(s) = [s2 + 4]2 and X3(s) = s2 + 4s + 4

a) Y1(s) = X1(s)/ X2(s), b) Y2(s) = X1(s)/ X3(s), c) Y3(s) =1/ X1(s) X2(s).

Use the MATLAB command pzpole to plot the poles and zeros of

H(s)= (s3+1)/(s

4+2s

2+1)

Find the laplace transform of delta function, unit function, ramp function, shifted unit function, shifted unit,

shifted ramp functions,

Find the Laplace transform of ‘sinwt’ and ‘coswt’

Find the inverse laplace transform of F=24/(s*(s+8));

Use the MATLAB command roots to determine the poles and zeros of the system

H(s)=(s2+2)/(s

3++2s

2-s+1)

H(s)=(s3+1)/(s

4++2s

2+1)

Sketch the magnitude response of the LTI system having transfer function

H(s)=(s-0.5)/((s+0.1-5j)(s+0.1+5j))

Write a program to simulate parseval’s relation

Bs Lab Manual

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