SIGNAL AND SYSTEM

LABORATORY MANUAL

Course No: PCCECE44L

B.Tech. 4th Semester

DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING

Institute of Technology Kashmir University

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LABORATORY INSTRUCTIONS

1. While entering the Laboratory, the students should follow the dress code. (Wear shoes

and White apron,Female Students should tie their hair back).

2. The students should bring their observation book, record, calculator, necessary stationery

items and graph sheets if any for the lab classes without which the students will not be allowed

for doing the experiment.

3. All the Equipment and components should be handled with utmost care. Any breakage or

damage will becharged.

4. If any damage or breakage is noticed, it should be reported to the concerned in charge

immediately.

5. The theoretical calculations and the updated register values should be noted down in the

observation book and should be corrected by the lab in-charge on the same day of the

laboratory session.

6. Each experiment should be written in the record note book only after getting signature from

the lab in-charge in the observation notebook.

7. Record book must be submitted in the successive lab session after completion of experiment.

8. 100% attendance should be maintained for the laboratory classes.

‘

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Precautions.

1. Check the connections before giving the supply.

2. Observations should be done carefully.

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INDEX

S.No. Name of the Experiment Page

No.
1 Write a program to generate various Signals and Sequences: Periodic and 15-21
Aperiodic, Unit Impulse, Unit Step, Square, Saw tooth, Triangular,
Sinusoidal, Ramp, Sinc function.
2 Perform operations on Signals and Sequences: Addition, Multiplication, 22-27
Scaling, Shifting, Folding, Computation of Energy and Average Power.
3 Write a program to find the trigonometric & exponential Fourier series 28-31
coefficients of a rectangular periodic signal. Reconstruct the signal by
combining the Fourier series coefficients with appropriate weightages-
Plot the discrete spectrum of the signal.
4 Write a program to find Fourier transform of a given signal. Plot its 32-37
amplitude and phase spectrum.
5 Write a program to convolution two discrete time sequences. Plot all the 38-40
sequences.
6 Write a program to find auto correlation and cross correlation of given 41-43
sequences
7 Write a program to verify Linearity and Time Invariance properties of a 44-46
given Continuous/Discrete System.
8 Write a program to generate discrete time sequence by sampling a 47-48
continuous time signal. Show that with sampling rates less than
Nyquist rate, aliasing occurs while reconstructing the signal.
9 Write a program to find magnitude and phase response of first order 49-50
low pass and high pass filter. Plot the responses in logarithmic scale.
10 Write a program to find response of a low pass filter and high pass 51
filter, when a speech signal is passed through these filters.
11 Write a program to generate Complex Gaussian noise and find its mean, 52-53
variance, Probability Density Function (PDF) and Power Spectral
Density (PSD).
12 Generate a Random data (with bipolar) for a given data rate (say 54
10kbps). Plot the same for a time period of 0.2 sec.
13 To plot pole-zero diagram in S-plane/Z-plane of given signal/sequence and 55-57
verify its stability.

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MATLAB INTRODUCTION
MATLAB, which stands for Matrix Laboratory, is a state-of-the-art mathematical software
package, which is used extensively in both academia and industry. It is an interactive program
for numerical computation and data visualization, which along with its programming capabilities
provides a very useful tool for almost all areas of science and engineering. Unlike other
mathematical packages, such as MAPLE or MATHEMATICA, MATLAB cannot perform
symbolic manipulations without the use of additional Toolboxes. It remains however, one of the
leading software packages for numerical computation.

As you might guess from its name, MATLAB deals mainly with matrices. A scalar is a 1-by-1
matrix and a row vector of length say 5, is a 1-by-5 matrix. One of the many advantages of
MATLAB is the natural notation used. It looks a lot like the notation that you encounter in a
linear algebra. This makes the use of the program especially easy and it is what makes MATLAB
a natural choice for numerical computations. The purpose of this experiment is to familiarize
MATLAB, by introducing the basic features and commands of the program.

MATLAB is case-sensitive, which means that a + B is not the same as a + b. The

MATLAB prompt (») in command window is where the commands are entered.

Operators:

1. + addition

2. -subtraction

3. * multiplication

4. ^ power

5. ' transpose

6. \ left division

7. / right division

Remember that the multiplication, power and division operators can be used in conjunction with
a period to specify an element-wise operation.

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Built in Functions:

1. Scalar Functions:

Certain MATLAB functions are essentially used on scalars, but operate element-wise when

applied to a matrix (or vector). They are summarized below.

1. sin -trigonometric sine

2. cos -trigonometric cosine

3. tan -trigonometric tangent

4. asin -trigonometric inverse sine (arcsine)

5. acos -trigonometric inverse cosine (arccosine)

6. atan -trigonometric inverse tangent (arctangent)

7. exp -exponential

8. log -natural logarithm

9. abs -absolute value 10. sqrt -square root

11. rem -remainder 12. round -round towards nearest integer

13. floor -round towards negative infinity 14. ceil -round towards positive infinity

2. Vector Functions:

Other MATLAB functions operate essentially on vectors returning a scalar value. Some of

these functions are given below.

1. max largest component : get the row in which the maximum element lies

2. min smallest component

3. length length of a vector

4. sort sort in ascending order

5. sum sum of elements

6. prod product of elements

7. median median value

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8. mean mean value std standard deviation

3. Matrix Functions:

Much of MATLAB’s power comes from its matrix functions. These can be further separated into
two sub-categories. The first one consists of convenient matrix building functions, some of
which are given below.

1. eye -identity matrix

2. zeros -matrix of zeros

3. ones -matrix of ones

4. diag -extract diagonal of a matrix or create diagonal matrices

5. triu -upper triangular part of a matrix

6. tril -lower triangular part of a matrix

7. rand -randomly generated

matrix eg:

diag([0.9092;0.5163;0.2661])

ans = 0.9092

00

0 0.5163 0

0 0 0.2661

Commands in the second sub-category of matrix functions are

1. size size of a matrix

2. det determinant of a square matrix

3. inv inverse of a matrix

4. rank rank of a matrix

5. rref reduced row echelon form

6. eig eigenvalues and eigenvectors

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1. Write a program to generate various Signals and Sequences: Periodic and

Aperiodic, Unit Impulse, Unit Step, Square, Saw tooth, Triangular,
Sinusoidal, Ramp, Sinc function.

Aim: Generate various signals and sequences (Periodic and aperiodic ), such as Unit Impulse, Unit
Step, Square, Saw tooth, Triangular, Sinusoidal, Ramp, Sinc.
Software Required: Matlab software 7.0 version and above
Theory: If the amplitude of the signal is defined at every instant of time then it is called continuous
time signal. If the amplitude of the signal is defined at only at some instants of time then it is
called discrete time signal. If the signal repeats itself at regular intervals then it is called periodic
signal. Otherwise they are called aperiodic signals.
EX: ramp,Impulse,unit step, sinc- Aperiodic signals square,sawtooth,triangular sinusoidal –
periodic signals.
» plot(x,y)

It is good practice to label the axis on a graph and if applicable indicate what each axis
represents. This can be done with the xlabel and ylabel commands.

» xlabel('x')

» ylabel('y=cos(x)')

Inside parentheses, and enclosed within single quotes, we type the text that we wish to be
displayed along the x and y axis, respectively. We could even put a title on top using

» title('Graph of cosine from -pi to pi')

» plot (x,y,’g’)

Where the third argument indicating the color, appears within single quotes. We could get a
dashed line instead of a solid one by typing

» plot (x,y,’--’)

or even a combination of line type and color, say a blue dotted line by typing

» plot (x,y,’b:’)

We can get both graphs on the same axis, distinguished by their line type, using

» plot(x,y,'r--',x,z,'b:')

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When multiple curves appear on the same axis, it is a good idea to create a legend to label and
distinguish them. The command legend does exactly this.

» legend ('cos(x)','sin(x)')

The text that appears within single quotes as input to this command, represents the legend
labels. We must be consistent with the ordering of the two curves, so since in the plot command
we asked for cosine to be plotted before sine, we must do the same here.

At any point during a MATLAB session, you can obtain a hard copy of the current plot
by either issuing the command print at the MATLAB prompt, or by using the command menus
on the plot window. In addition, MATLAB plots can by copied and pasted (as pictures) in your
favorite word processor (such as Microsoft Word). This can be achieved using the Edit menu on
the figure window. Another nice feature that can be used in conjunction with plot is the
command grid, which places grid lines to the current axis (just like you have on graphing paper).
Type help grid for more information.

The 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 2*pi and height

The sinc function has a value of 1 when x is equal to zero, and a value of for all other
elements of x.
% Generation of signals and sequences
clc;
clear all;
close all;
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%generation of unit impulse signal
t1=-1:0.01:1
y1=(t1==0);
subplot(2,2,1);
plot(t1,y1);
xlabel('time');
ylabel('amplitude');
title('unit impulse signal');
%generation of impulse sequence
subplot(2,2,2);

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stem(t1,y1);
xlabel('n');
ylabel('amplitude');
title('unit impulse sequence');
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%generation of unit step signal
t2=-10:1:10;
y2=(t2>=0);
subplot(2,2,3);
plot(t2,y2);
xlabel('time');
ylabel('amplitude');
title('unit step signal');
%generation of unit step sequence
subplot(2,2,4);
stem(t2,y2);
xlabel('n');
ylabel('amplitude');
title('unit step sequence');
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%generation of square wave signal
t=0:0.002:0.1;
y3=square(2*pi*50*t);
figure;
subplot(2,2,1);
plot(t,y3);
axis([0 0.1 -2 2]);
xlabel('time');
ylabel('amplitude');
title('square wave signal');
%generation of square wave sequence
subplot(2,2,2);
stem(t,y3);
axis([0 0.1 -2 2]);
xlabel('n');
ylabel('amplitude');
title('square wave sequence');
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%generation of sawtooth signal
y4=sawtooth(2*pi*50*t);
subplot(2,2,3);
plot(t,y4);
axis([0 0.1 -2 2]);
xlabel('time');
ylabel('amplitude');
title('sawtooth wave signal');
%generation of sawtooth sequence

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subplot(2,2,4);
stem(t,y4);
axis([0 0.1 -2 2]);
xlabel('n');
ylabel('amplitude');
title('sawtooth wave sequence');
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%generation of triangular wave signal
y5=sawtooth(2*pi*50*t,.5);
figure;
subplot(2,2,1);
plot(t,y5);
axis([0 0.1 -2 2]);
xlabel('time');
ylabel('amplitude');
title(' triangular wave signal');
%generation of triangular wave sequence
subplot(2,2,2);
stem(t,y5);
axis([0 0.1 -2 2]);
xlabel('n');
ylabel('amplitude');
title('triangular wave sequence');
%generation of sinsoidal wave signal
y6=sin(2*pi*40*t);
subplot(2,2,3);
plot(t,y6);
axis([0 0.1 -2 2]);
xlabel('time');
ylabel('amplitude');
title(' sinsoidal wave signal');
%generation of sin wave sequence
subplot(2,2,4);
stem(t,y6);
axis([0 0.1 -2 2]);
xlabel('n');
ylabel('amplitude');
title('sin wave sequence');
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%generation of ramp signal
y7=t;
figure;
subplot(2,2,1);
plot(t,y7);
xlabel('time');
ylabel('amplitude');
title('ramp signal');

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%generation of ramp sequence

subplot(2,2,2);
stem(t,y7);
xlabel('n');
ylabel('amplitude');
title('ramp sequence');
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%generation of sinc signal
t3=linspace(-5,5);
y8=sinc(t3);
subplot(2,2,3);
plot(t3,y8);
xlabel('time');
ylabel('amplitude');
title(' sinc signal');
%generation of sinc sequence
subplot(2,2,4);
stem(y8);
xlabel('n');
ylabel('amplitude');
title('sinc sequence');

Result: Various signals & sequences generated using Matlab software.

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Results and discussion : Periodic and Aperiodic, Unit Impulse, Unit Step, Square, Saw
tooth, Triangular, Sinusoidal, Ramp, Sinc function are generated and plotted.

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2. Perform operations on Signals and Sequences: Addition,

Multiplication, Scaling, Shifting, Folding, Computation of Energy and
Average Power.
Aim: To performs functions on signals and sequences such as addition, multiplication, scaling,
shifting, folding, computation of energy and average power.

Software Required: Matlab software 7.0 version and above

Theory:Signal Addition

Any two signals can be added to form a third signal,

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

Multiplication:
Multiplication of two signals can be obtained by multiplying their values at every instant. z
z(t) = x (t) y (t)
Time reversal/Folding:
Time reversal of a signal x(t) can be obtained by folding the signal about
t=0. Y(t)=y(-t)
Signal Amplification/Scaling : Y(n)=ax(n) if a < 1 attenuation
a >1 amplification
Time shifting: The time shifting of x(n) obtained by delay or advance the signal in time by
using y(n)=x(n+k)
If k is a positive number, y(n) shifted to the right i e the shifting delays the signal
If k is a negative number, y(n ) it gets shifted left. Signal Shifting advances the signal

Program :

clc;
clear all;
close all;
% generating two input signals
t=0:.01:1;
x1=sin(2*pi*4*t);
x2=sin(2*pi*8*t);
subplot(2,2,1);
plot(t,x1);
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xlabel('time');
ylabel('amplitude');
title('input signal 1');
subplot(2,2,2);
plot(t,x2);
xlabel('time');
ylabel('amplitude');
title('input signal 2');
% addition of signals
y1=x1+x2;
subplot(2,2,3);
plot(t,y1);
xlabel('time');
ylabel('amplitude');
title('addition of two signals');
% multiplication of signals
y2=x1.*x2;
subplot(2,2,4);
plot(t,y2);
xlabel('time');
ylabel('amplitude');
title('multiplication of two signals');
% scaling of a signal1
A=2;
y3=A*x1;
figure;
subplot(2,2,1);
plot(t,x1);
xlabel('time');
ylabel('amplitude');
title('input signal')
subplot(2,2,2);
plot(t,y3);
xlabel('time');
ylabel('amplitude');
title('amplified input signal');
% folding of a signal1
h=length(x1);
nx=0:h-1;
subplot(2,2,3);
plot(nx,x1);
xlabel('nx');
ylabel('amplitude');
title('input signal')
y4=fliplr(x1);
nf=-fliplr(nx);
subplot(2,2,4);

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plot(nf,y4);
xlabel('nf');
ylabel('amplitude');
title('folded signal');
%shifting of a signal 1
figure;
subplot(3,1,1);
plot(t,x1);
xlabel('time t');
ylabel('amplitude');
title('input signal');
subplot(3,1,2);
plot(t+2,x1);
xlabel('t+2');
ylabel('amplitude');
title('right shifted signal');
subplot(3,1,3);
plot(t-2,x1);
xlabel('t-2');
ylabel('amplitude');
title('left shifted signal');
%operations on sequences
n1=1:1:9;
s1=[1 2 3 0 5 8 0 2 4];
figure;
subplot(2,2,1);
stem(n1,s1);
xlabel('n1');
ylabel('amplitude');
title('input sequence 1');
s2=[1 1 2 4 6 0 5 3 6];
subplot(2,2,2);
stem(n1,s2);
xlabel('n2');
ylabel('amplitude');
title('input sequence 2 ');
% addition of sequences
s3=s1+s2;
subplot(2,2,3);
stem(n1,s3);
xlabel('n1');
ylabel('amplitude');
title('sum of two sequences');
% multiplication of sequences
s4=s1.*s2;
subplot(2,2,4);
stem(n1,s4);

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xlabel('n1');
ylabel('amplitude');
title('product of two sequences');
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
% program for energy of a sequence
z1=input('enter the input sequence');
e1=sum(abs(z1).^2);
disp('energy of given sequence is');e1
% program for energy of a signal
t=0:pi:10*pi;
z2=cos(2*pi*50*t).^2;
e2=sum(abs(z2).^2);
disp('energy of given signal is');e2
% program for power of a sequence
p1= (sum(abs(z1).^2))/length(z1);
disp('power of given sequence is');p1
% program for power of a signal
p2=(sum(abs(z2).^2))/length(z2);
disp('power of given signal is');

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Results and discussions

Output:
enter the input sequence[1 3 2 4 1]
energy of given sequence is
e1 = 31
energy of given signal is
e2 = 4.0388
power of given sequence is
p1 = 6.2000
power of given signal is
p2 = 0.3672
Result: Various operations on signals and sequences are performed.

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3. Write a program to find the trigonometric & exponential Fourier series

coefficients of a rectangular periodic signal. Reconstruct the signal by
combining the Fourier series coefficients with appropriate weightages - Plot
the discrete spectrum of the signal.
Aim: To find the trigonometric & exponential Fourier series coefficients of a rectangular
periodic signal. Reconstruct the signal by combining the Fourier series coefficients with
appropriate weightages - Plot the discrete spectrum of the signal.

Software Required: Matlab software 7.0 version and above

Theory: to compute the trigonometric fourier series coefficients of a periodic square wave time
signal that has a value of 2 from time 0 to 3 and a value of -12 from time 3 to 6. It then repeats
itself. I am trying to calculate in MATLAB the fourier series coefficients of this time signal and
am having trouble on where to begin.

The equation is x(t) = a0 + sum(bk*cos(2*pi*f*k*t)+ck*sin(2*pi*f*k*t))

The sum is obviously from k=1 to k=infinity. a0, bk, and ck are the coefficients
Program:
% Description: This M-file plots the truncated Fourier Series
% representation of a square wave as well as its
% amplitude and phase spectrum.

clear; % clear all variables

clf; % clear all figures

N = 11; % summation limit (use N odd)

wo = pi; % fundamental frequency (rad/s)
c0 = 0; % dc bias
t = -3:0.01:3; % declare time values

figure(1) % put first two plots on figure 1

% Compute yce, the Fourier Series in complex exponential form

yce = c0*ones(size(t)); % initialize yce to c0

for n = -N:2:N, % loop over series index n (odd)

cn = 2/(j*n*wo); % Fourier Series Coefficient
yce = yce + real(cn*exp(j*n*wo*t)); % Fourier Series computation
end

subplot(2,1,1)
plot([-3 -2 -2 -1 -1 0 0 1 1 2 2 3],... % plot original y(t)

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[-1 -1 1 1 -1 -1 1 1 -1 -1 1 1], ':');

hold;
plot(t,yce); % plot truncated exponential FS
xlabel('t (seconds)'); ylabel('y(t)');
ttle = ['EE341.01: Truncated Exponential Fourier Series with N = ',...
num2str(N)];
title(ttle);
hold;

% Compute yt, the Fourier Series in trigonometric form

yt = c0*ones(size(t)); % initialize yt to c0

for n = 1:2:N, % loop over series index n (odd)

cn = 2/(j*n*wo); % Fourier Series Coefficient
yt = yt + 2*abs(cn)*cos(n*wo*t+angle(cn)); % Fourier Series computation
end

subplot(2,1,2)
plot([-3 -2 -2 -1 -1 0 0 1 1 2 2 3],... % plot original y(t)
[-1 -1 1 1 -1 -1 1 1 -1 -1 1 1], ':');
hold; % plot truncated trigonometric FS
plot(t,yt);
xlabel('t (seconds)'); ylabel('y(t)');
ttle = ['EE341.01: Truncated Trigonometric Fourier Series with N = ',...
num2str(N)];
title(ttle);
hold;

% Draw the amplitude spectrum from exponential Fourier Series

figure(2) % put next plots on figure 2

subplot(2,1,1)
stem(0,c0); % plot c0 at nwo = 0

hold;
for n = -N:2:N, % loop over series index n
cn = 2/(j*n*wo); % Fourier Series Coefficient
stem(n*wo,abs(cn)) % plot |cn| vs nwo
end
for n = -N+1:2:N-1, % loop over even series index n
cn = 0; % Fourier Series Coefficient
stem(n*wo,abs(cn)); % plot |cn| vs nwo
end

xlabel('w (rad/s)')
ylabel('|cn|')
ttle = ['EE341.01: Amplitude Spectrum with N = ',num2str(N)];
title(ttle);
grid;
hold;

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% Draw the phase spectrum from exponential Fourier Series

subplot(2,1,2)
stem(0,angle(c0)*180/pi); % plot angle of c0 at nwo = 0

hold;
for n = -N:2:N, % loop over odd series index n
cn = 2/(j*n*wo); % Fourier Series Coefficient
stem(n*wo,angle(cn)*180/pi); % plot |cn| vs nwo
end
for n = -N+1:2:N-1, % loop over even series index n
cn = 0; % Fourier Series Coefficient
stem(n*wo,angle(cn)*180/pi); % plot |cn| vs nwo
end
xlabel('w (rad/s)')
ylabel('angle(cn) (degrees)')
ttle = ['EE341.01: Phase Spectrum with N = ',num2str(N)];
title(ttle);
grid;
hold;

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Results and discussions

Result: Trigonometric & exponential Fourier series coefficients of a rectangular periodic signals
are plotted.

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4. Write a program to find Fourier transform of a given signal. Plot

its amplitude and phase spectrum.

Aim: To find the Fourier Transform of a given signal and plotting its magnitude and phase
spectrum.
Software Required: Matlab software 7.0 version and above
Theory:
Fourier Transform:
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 Fourier Transform of signal f(t) is defined as

Programs with symbolic matlab tool

Compute the Fourier transform of common inputs. By default, the transform is in terms of w.

Function Input and Output

Rectangular pulse syms a b t

f = rectangularPulse(a,b,t);
f_FT = fourier(f)
f_FT =
- (sin(a*w) + cos(a*w)*1i)/w + (sin(b*w) + cos(b*w)*1i)/w

Unit impulse (Dirac delta) f = dirac(t);

f_FT =
fourier(f) f_FT
=
1
Absolute value f = a*abs(t);

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Function Input and Output

f_FT =
fourier(f) f_FT
=
-(2*a)/w^2
Step (Heaviside) f = heaviside(t);
f_FT = fourier(f)
f_FT =
pi*dirac(w) - 1i/w

Constant f = a;
f_FT = fourier(a)
f_FT =
pi*dirac(1, w)*2i

Cosine f = a*cos(b*t);
f_FT =
fourier(f) f_FT
=
pi*a*(dirac(b + w) + dirac(b - w))
Sine f = a*sin(b*t);
f_FT =
fourier(f) f_FT
=
pi*a*(dirac(b + w) - dirac(b - w))*1i
Sign f = sign(t);
f_FT = fourier(f)
f_FT =
-2i/w

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Triangle syms c
f = triangularPulse(a,b,c,t);
f_FT = fourier(f)
f_FT =

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Function Input and Output

-(a*exp(-b*w*1i) - b*exp(-a*w*1i) - a*exp(-c*w*1i) + ...

c*exp(-a*w*1i) + b*exp(-c*w*1i) - c*exp(-b*w*1i))/ ...
(w^2*(a - b)*(b - c))

Right-sided exponential Also calculate transform with condition a > 0. Clear assumptions.
f = exp(-t*abs(a))*heaviside(t);
f_FT = fourier(f)

assume(a > 0)
f_FT_condition = fourier(f)
assume(a,'clear')
f_FT =
1/(abs(a) + w*1i) - (sign(abs(a))/2 - 1/2)*fourier(exp(-t*abs(a)),t,w)

f_FT_condition =
1/(a + w*1i)

Double-sided exponential Assume a > 0. Clear assumptions.

assume(a > 0)
f = exp(-a*t^2);
f_FT =
fourier(f)
assume(a,'clear')
f_FT =
(pi^(1/2)*exp(-w^2/(4*a)))/a^(1/2)
Gaussian Assume b and c are real. Simplify result and clear
assumptions. assume([b c],'real')
f = a*exp(-(t-b)^2/(2*c^2));
f_FT = fourier(f)

f_FT_simplify = simplify(f_FT)
assume([b c],'clear')

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Function Input and Output

f_FT =
(a*pi^(1/2)*exp(- (c^2*(w + (b*1i)/c^2)^2)/2 - b^2/(2*c^2)))/ ...
(1/(2*c^2))^(1/2)

f_FT_simplify =
2^(1/2)*a*pi^(1/2)*exp(-(w*(w*c^2 + b*2i))/2)*abs(c)

Bessel of first kind with nu Simplify the result.

=1 syms x
f = besselj(1,x);
f_FT = fourier(f);
f_FT = simplify(f_FT)
f_FT =
(2*w*(heaviside(w - 1)*1i - heaviside(w + 1)*1i))/(1 - w^2)^(1/2)

Specify Independent Variable and Transformation Variable

Compute the Fourier transform of exp(-t^2-x^2). By default, symvar determines the independent
variable, and w is the transformation variable. Here, symvar chooses x.

syms t x

f = exp(-t^2-x^2);

fourier(f)

ans =

pi^(1/2)*exp(- t^2 - w^2/4)

Specify the transformation variable as y. If you specify only one variable, that variable is the
transformation variable. symvar still determines the independent variable.

syms y

fourier(f,y)

ans =

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pi^(1/2)*exp(- t^2 - y^2/4)

Specify both the independent and transformation variables as t and y in the second and third
arguments, respectively.

fourier(f,t,y)

ans =
pi^(1/2)*exp(- x^2 - y^2/4)
Fourier Transforms Involving Dirac and Heaviside Functions
Compute the following Fourier transforms. The results are in terms of the Dirac and Heaviside
functions.
syms t w
fourier(t^3, t, w)
ans =
-pi*dirac(3, w)*2i
syms t0
fourier(heaviside(t - t0),t,w)
ans =
exp(-t0*w*1i)*(pi*dirac(w) - 1i/w)
Program:
clc;
clear all;
close all;
fs=1000;
N=1024; % length of fft sequence
t=[0:N-1]*(1/fs);
% input signal
x=0.8*cos(2*pi*100*t);
subplot(3,1,1);
plot(t,x);
axis([0 0.05 -1 1]);
grid;
xlabel('t');
ylabel('amplitude');
title('input signal');
% Fourier transform of given signal
x1=fft(x);
% magnitude spectrum
k=0:N-1;
Xmag=abs(x1);
subplot(3,1,2);
plot(k,Xmag);
grid;

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xlabel('t');
ylabel('amplitude');
title('magnitude of fft signal')
%phase spectrum
Xphase=angle(x1);
subplot(3,1,3);
plot(k,Xphase);
grid;
xlabel('t');
ylabel('angle');
title('phase of fft signal');

Result: Magnitude and phase spectrum of FFT of a given signal is plotted.

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5. Write a program to convolution two discrete time sequences.

Plot all the sequences
Aim: Write the program for convolution between two signals and also between two
sequences.
Software Required: Matlab software 7.0 version and above
Theory:
Convolution involves the following operations.
1. Folding
2. Multiplication
3. Addition
4. Shifting

Convolution is an integral concatenation of two signals. It is used for the determination of the
output signal of a linear time-invariant system by convolving the input signal with the impulse
response of the system. Note that convolving two signals is equivalent to multiplying the Fourier
transform of the two signals.

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

x[n]= Input signal Samples
h[ n-k]= Impulse response co-efficient.
y[ n]= 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;
%program for convolution of two sequences
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 sequence')

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subplot(3,1,2);
stem(h);
xlabel('n');
ylabel('h(n)');
title('impulse response sequence')
subplot(3,1,3);
stem(y);
xlabel('n');
ylabel('y(n)');
title('linear convolution')
disp('linear convolution y=');
disp(y)
%program for signal convolution
t=0:0.1:10;
x1=sin(2*pi*t);
h1=cos(2*pi*t);
y1=conv(x1,h1);
figure;
subplot(3,1,1);
plot(x1);
xlabel('t');
ylabel('x(t)');
title('input
signal')
subplot(3,1,2);
plot(h1);
xlabel('t');
ylabel('h(t)');
title('impulse response')
subplot(3,1,3);
plot(y1);
xlabel('n');
ylabel('y(n)');
title('linear
convolution');

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Results and discussions

Output:
enter input sequence: [1 3 4 5]
enter impulse response: [2 1 4]
linear convolution y= {2 7 15 26 21 20 }

Result: convolution between signals and sequences is computed.

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6. Write a program to find auto correlation and cross

correlation of given sequences.
Aim: To compute Auto correlation and Cross correlation between signals and sequences.
Software Required: Matlab software 7.0 version and above
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
Program:
clc;
close
all; clear
all;
% two input sequences
x=input('enter input sequence');
h=input('enter the impulse suquence');
subplot(2,2,1);
stem(x);
xlabel('n');
ylabel('x(n)');
title('input sequence');
subplot(2,2,2);
stem(h);
xlabel('n');
ylabel('h(n)');
title('impulse sequence');
% cross correlation between two
y=xcorr(x,h);
subplot(2,2,3);
stem(y);
xlabel('n');
ylabel('y(n)');
title(' cross correlation between two sequences ');
% auto correlation of input sequence
z=xcorr(x,x);
subplot(2,2,4);
stem(z);
xlabel('n');
ylabel('z(n)');
title('auto correlation of input sequence');
% cross correlation between two signals
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% generating two input signals

t=0:0.2:10;
x1=3*exp(-2*t);
h1=exp(t);
figure;
subplot(2,2,1);
plot(t,x1);
xlabel('t');
ylabel('x1(t)');
title('input signal');
subplot(2,2,2);
plot(t,h1);
xlabel('t');
ylabel('h1(t)');
title('impulse signal');
% cross
correlation
subplot(2,2,3);
z1=xcorr(x1,h1);
plot(z1);
xlabel('t');
ylabel('z1(t)');
title('cross correlation
');
% auto
correlation
subplot(2,2,4);
z2=xcorr(x1,x1);
plot(z2);
xlabel('t');
ylabel('z2(t)');
title('auto correlation
');

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Results and discussions

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Output: enter input sequence [1 2 5 7]

enter the impulse sequence [2 6 0 5 3]

Result: Auto correlation and Cross correlation between signals and sequences is computed.

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7 Write a program to verify Linearity and Time Invariant properties of a
given Continuous/Discrete System.
Aim: Verify the Linearity of a given Discrete System.
Software Required: Matlab software 7.0 version and above
Theory:
Any system is said to be linear if it satisfies the superposition principal. Superposition principal
state that Response to a weighted sum of input signal equal to the corresponding weighted sum
of the outputs of the system to each of the individual input signals. If x(n) is a input signal and
y(n) is a output signal then y(n)=T[x(n)]
y1(n)=T[x1(n)] and y2(n)=T[x2(n)]
x3=[a*x1(n) +b *x2(n) ]
Y3(n)= T [x3(n)]
T [a*x1(n)+b*x2(n) ] = a y1(n)+ b y2(n)
Program (A) :
% Verification of Linearity of a given System.
% a) y(n)=nx(n) b) y=x^2(n)
clc;
clear all;
close all;
n=0:40;
a1=input('enter the scaling factor a1=');
a2=input('enter the scaling factor a2=');

x1=cos(2*pi*0.1*n);
x2=cos(2*pi*0.4*n);
x3=a1*x1+a2*x2;
%y(n)=n.x(n);
y1=n.*x1;
y2=n.*x2;
y3=n.*x3;
yt=a1*y1+a2*y2;
yt=round(yt);
y3=round(y3);
if y3==yt
disp('given system [y(n)=n.x(n)]is Linear');
else
disp('given system [y(n)=n.x(n)]is non Linear');
end
%y(n)=x(n).^2
x1=[1 2 3 4 5];
x2=[1 4 7 6 4];
x3=a1*x1+a2*x2;
y1=x1.^2;
y2=x2.^2;
y3=x3.^2;
yt=a1*y1+a2*y2;
if y3==yt

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disp('given system [y(n)=x(n).^2 ]is Linear');

else
disp('given system is [y(n)=x(n).^2 ]non Linear');
end
Result: The Linearity of a given Discrete System is verified.
Output:
enter the scaling factor a1=3
enter the scaling factor a2=5
given system [y(n)=n.x(n)]is Linear
given system is [y(n)=x(n).^2 ]non Linear
Program (B) :

% Verification of Time Invariance of a Discrete System

% a)y=x^2(n) b) y(n)=nx(n)
clc;
clear all;
close all;
n=1:9;
x(n)=[1 2 3 4 5 6 7 8 9];
d=3; % time delay
xd=[zeros(1,d),x(n)];%x(n-k)
y(n)=x(n).^2;
yd=[zeros(1,d),y];%y(n-k)
disp('transformation of delay signal yd:');disp(yd)
dy=xd.^2; % T[x(n-k)]
disp('delay of transformation signal dy:');disp(dy)
if dy==yd
disp('given system [y(n)=x(n).^2 ]is time invariant');
else
disp('given system is [y(n)=x(n).^2 ]not time invariant');
end

y=n.*x;
yd=[zeros(1,d),y(n)];
disp('transformation of delay signal yd:');disp(yd);
n1=1:length(xd);
dy=n1.*xd;
disp('delay of transformation signal dy:');disp(dy);
if yd==dy
disp('given system [y(n)=nx(n)]is a time invariant');
else
disp('given system [y(n)=nx(n)]not a time invariant');
end

Output:

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transformation of delay signal yd:

0 0 0 1 4 9 16 25 36 49 64 81
delay of transformation signal dy:
0 0 0 1 4 9 16 25 36 49 64 81
given system [y(n)=x(n).^2 ]is time invariant
transformation of delay signal yd:
0 0 0 1 4 9 16 25 36 49 64 81
delay of transformation signal dy:
0 0 0 4 10 18 28 40 54 70 88 108
given system [y(n)=n x(n)]not a time invariant

Result: The Time Invariance of a given Discrete System is verified.

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8. Write a program to generate discrete time sequence by sampling a

continuous time signal. Show that with sampling rates less than Nyquist rate,
aliasing occurs while reconstructing the signal.
Aim: Verify the sampling theorem.
Software Required: Matlab software 7.0 version and above
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."

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

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;
n1=-4:1:4;

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fs1=1.6*fm;
fs2=2*fm;
fs3=8*fm;
x1=cos(2*pi*fm/fs1*n1);
subplot(2,2,2);
stem(n1,x1);
xlabel('time');
ylabel('x(n)');
title('discrete time signal with fs<2fm');
hold on;
subplot(2,2,2);
plot(n1,x1);
grid;
n2=-5:1:5;
x2=cos(2*pi*fm/fs2*n2);
subplot(2,2,3);
stem(n2,x2);
xlabel('time');
ylabel('x(n)');
title('discrete time signal with fs=2fm');
hold on;
subplot(2,2,3);
plot(n2,x2)
grid;
n3=-20:1:20;
x3=cos(2*pi*fm/fs3*n3);
subplot(2,2,4);
stem(n3,x3);
xlabel('time');
ylabel('x(n)');
title('discrete time signal with fs>2fm')
hold on;
subplot(2,2,4);
plot(n3,x3)
grid;

OUTPUT :

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Result: Sampling theorem is verified.

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9. Write a program to find magnitude and phase response of first order low
pass and high pass filter. Plot the responses in logarithmic scale.
Aim: To find magnitude and phase response of first order low pass and high pass filter. Plot the
responses in logarithmic scale.

Software Required: Matlab software 7.0 version and above

Theory: To get the phase response in the time domain you need to estimate the delay between
the input and the output at the test frequency and then convert to phase. For sure, taking the angle
of the RMS ratio will not yield that delay. If you really want to estimate the delay in the time
domain, there is a function called find delay that may be of use.
However, the alternative approach is to work in the frequency domain, i.e., use the ratio
of the frequency response of the output to the frequency response of the input to directly estimate
the magnitude and phase response of the system. The code that follows shows how to do that for
a simple low pass filter. Keep in mind, that any approach you use may begin to suffere as you
test frequency gets close to the Nyquist frequency. You can experiment with this code to see how
close you can get before this simple approach begins to break down.

Program :

f=1:15;
% second order filter with 10 Hz pass band
H = @(f,s) 1./(1./(2*pi*f).^2*s.^2 + 2./(2*pi*f)*s + 1);
H10 = @(s) H(10,s);
% magnitude estimation
magn = abs(H10(1j*2*pi*f));
% phase estimation
phase = rad2deg( angle(H10(1j*2*pi*f)) );
figure;
subplot(2,1,1);
semilogx(f,mag2db(magn));
title('Frequency response of the filter using abs(Out/Inp)')
xlabel('Frequency');
ylabel ('Change in output/input P-P' );
subplot(2,1,2);
semilogx(f,phase);
title('Phase response of the filter using angle(Out/Inp)')
xlabel('Frequency');
ylabel ('phase' );
% check against inbuilt functions
H_tf = @(f) tf(1,[1./(2*pi*f).^2 2./(2*pi*f) 1]);
figure;
bodeplot(H_tf(10),2*pi*f)

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Result: The magnitude and phase response of LPF plotted.

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10 Write a program to find response of a low pass filter and high

pass filter, when a speech signal is passed through these filters.
Aim: To find response of a low pass filter and high pass filter, when a speech signal is passed
through these filters.

Software Required: Matlab software 7.0 version and above

Theory:

y = lowpass(x,wpass) filters the input signal x using a lowpass filter with normalized passband
frequency wpass in units of π rad/sample. lowpass uses a minimum-order filter with a stopband
attenuation of 60 dB and compensates for the delay introduced by the filter. If x is a matrix, the
function filters each column independently.

y = lowpass(x,fpass,fs) specifies that x has been sampled at a rate of fs hertz. fpass is the
passband frequency of the filter in hertz.

y = lowpass(xt,fpass) lowpass-filters the data in timetable xt using a filter with a passband

frequency of fpass hertz. The function independently filters all variables in the timetable and all
columns inside each variable.

y = lowpass( ,Name,Value) specifies additional options for any of the previous syntaxes using
name-value pair arguments. You can change the stopband attenuation, the transition band
steepness, and the type of impulse response of the filter.

[y,d] = lowpass( ) also returns the digitalFilter object d used to filter the input.

lowpass( ) with no output arguments plots the input signal and overlays the filtered signal.

Program :
% Read standard sample tune that ships with MATLAB.
[dataIn, Fs] = audioread('guitartune.wav');
% Filter the signal
fc = 800; % Make higher to hear higher frequencies.
% Design a Butterworth filter.
[b, a] = butter(6,fc/(Fs/2));
freqz(b,a)
% Apply the Butterworth filter.
filteredSignal = filter(b, a, dataIn);
% Play the sound.
player = audioplayer(filteredSignal, Fs);
play(player);

Result: Response of a low pass filter and high pass filter, when a speech signal studied.

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11 Write a program to generate Complex Gaussian noise and find its

mean, variance, Probability Density Function (PDF) and Power Spectral
Density (PSD).

Aim: Write the program for generation of Gaussian noise and computation of its mean, mean
square value, standard deviation, variance, and skewness.
Software Required: Matlab software 7.0 version and above
Theory:
Gaussian noise is statistical noise that has a probability density function (abbreviated
pdf) of the normal distribution (also known as Gaussian distribution). In other words, the values
the noise can take on are Gaussian-distributed. It is most commonly used as additive white noise
to yield additive white Gaussian noise (AWGN).Gaussian noise is properly defined as the noise
with a Gaussian amplitude distribution. says nothing of the correlation of the noise in time or of
the spectral density of the noise. Labeling Gaussian noise as 'white' describes the correlation of
the noise. It is necessary to use the term "white Gaussian noise" to be correct. Gaussian noise is
sometimes misunderstood to be white Gaussian noise, but this is not the case.
Program:
clc;
clear all;
close all;
%generates a set of 2000 samples of Gaussian distributed random numbers
x=randn(1,2000);
%plot the joint distribution of both the sets using dot.
subplot(211)
plot(x,'.');
title('scatter plot of gaussian distributed random numbers');
ymu=mean(x)
ymsq=sum(x.^2)/length(x)
ysigma=std(x)
yvar=var(x)
yskew=skewness(x)
p=normpdf(x,ymu,ysigma);
subplot(212);
stem(x,p);
title(' gaussian distribution');
Output:
ymu = 0.0403
ymsq = 0.9727
ysigma = 0.9859
yvar = 0.9720
yskew = 0.0049

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OUTPUT :

Result: Gaussian noise and its characteristics are studied.

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12. Generate a Random data (with bipolar) for a given data rate
(say 10kbps). Plot the same for a time period of 0.2 sec.
Aim: To Generate a Random data (with bipolar) for a given data rate (say 10kbps). Plot the same
for a time period of 0.2 sec.

Software Required: Matlab software 7.0 version and above

Theory:

X = rand returns a single uniformly distributed random number in the interval

(0,1). X = rand(n) returns an n-by-n matrix of random numbers.

X = rand(sz1,...,szN) returns an sz1-by-...-by-szN array of random numbers

where sz1,...,szN indicate the size of each dimension. For example, rand(3,4) returns a 3-by-4
matrix.

X = rand(sz) returns an array of random numbers where size vector sz specifies size(X). For
example, rand([3 4]) returns a 3-by-4 matrix.

X = rand( ,typename) returns an array of random numbers of data type typename.

The typename input can be either 'single' or 'double'. You can use any of the input arguments in
the previous syntaxes.

Program :
clc;
clear all;
a=-3
b=3
x=rand
c=a+(b-a)*x
y=c^2
z=y
for i=1:1000
x1=rand c1=a+
(b-a)*x1
y1=c1^2

if y1<z
z=y1
else
z;
end
end

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