Signal Prosessing Lab Mannual

JNU/ECE/SPL-01/EC507
1 Signal Processing Lab-I Created By Jitendra Jangid
EXPERIMENT-01
Aim:- Generation of Continuous and Discrete Elementary signals (Periodic and Non-
periodic) using mathematical expression.
Software Requirement-Matlab 7.0
Theory-
Continuous-Time and Discrete-Time Signals: A signal x(t) is a continuous-time signal if t is a
continuous variable. If t is a discrete variable, that is, x(t) is defined at discrete times, then x(t) is
a discrete-time signal.
Periodic and Non periodic Signals: A continuous-time signal x(t) is to be periodic with period T
if there is a positive nonzero value of T for which
X(t+T)=x(t) all t
Any continuous-time signal which is not periodic is called a non-periodic signal.
Periodic discrete-time signal are defined analogously. A sequence x[n] is periodic with period N
if there is a positive integer N for which
X[n+N]=x[n] all n
Coding:-
Continuous periodic time signal
time= input('Enter The time duration = ')
t=0:0.1:time ;
F=1;
y=cos(2*F*t*pi);
plot(t,y); %plot command to plot the graph%
xlabel('TIME');
ylabel('amplitude y(t)=cos(2*F*t*pi)');
title('continuous periodic signal') %title of the signal
y2=sin(2*F*t*pi);
figure;
plot(t,y2); %plot command to plot the graph%
xlabel('TIME');
ylabel('amplitude y2(t)=sin(2*F*t*pi)');
title('continuous periodic signal'); %title of the signal

Discrete periodic time signal
time= input('Enter The time duration = ')
t=0:0.1:time ;
F=1;
y=cos(2*F*t*pi);
stem(t,y); %plot command to plot the graph%
xlabel('TIME');
ylabel('amplitude y(t)=cos(2*F*t*pi)');
title('discrete periodic signal') %title of the signal
y2=sin(2*F*t*pi);
figure;
stem(t,y2); %plot command to plot the graph%
xlabel('TIME');
ylabel('amplitude y2(t)=sin(2*F*t*pi)');
title('discrete periodic signal'); %title of the signal
Non periodic continuous signal

t=0:2:50;
y=cos(t/2).*cos(t*pi/4)
plot(y)
xlabel('time')
ylabel('amplitude')
title('aperiodical continous signal ')
Non periodic Discrete signal
t=0:2:50;
y=cos(t/2).*cos(t*pi/4)
stem(y)
xlabel('time')
ylabel('amplitude')
title('aperiodical discrete signal ')

Result:- Various periodic and non-periodic signals has been studied.

EXPERIMENT-02
AIM: Generation of various continuous and discrete unit step signals.
Software Requirement- Matlab 7.0
Theory- Continuous time unit step signals is also a basic continuous time signal and is denoted
by u(t) this function is defined as.
U(t)= 1 t>0
0 t<0
Discrete time unit step signal:-
U(n) = 1 n>=0
0 n<0
Coding:-
Continuous unit step signal
t1=0:5:50
t2=-50:5:0
y1=ones(size(t1))
y2=zeros(size(t2))
t=[t2 t1]
y=[y2 y1]
plot(t,y)
xlabel('t')
title('cont. unit step')

Discrete Unit Step Signal:-
n1=0:5:30
n2=-50:5:0
y1=ones(size(n1))
y2=zeros(size(n2))
n=[n2 n1]
y=[y2 y1]
stem(n,y)
xlabel('n')
title('discrete unit step')
Result:-

EXPERIMENT NO:3
Aim: Generation of Exponential and Ramp signals in Continuous & Discrete domain.
Software used: Matlab 7.0
Theory:
Continuous-time complex exponential and sinusoidal signals:
x(t) = C*eat
where C and a are in general complex numbers.
The case a > 0 represents exponential growth. Some signals in unstable systems exhibit
exponential
growth.
The case a < 0 represents exponential decay. Some signals in stable systems exhibit exponential
decay
Ramp signal : The ramp function ( ) may be defined analytically in several ways and
definitions are:

Continuous exponential signal
n= input('Enter the value of N=');
t=0:1:n;
M=2;
y1=M*exp(t*1); %multiplication factor m =1;
subplot(211)
plot(t,y1)
xlabel('time ')
ylabel('ampitude');
title('Continuous exponential signal with m = 1 ');
subplot(212)
plot(t,y2)
xlabel('time ')
ylabel('ampitude');
title('Continuous exponential signal with m = 2');

Discrete exponent signal:
t=0:1:n;
M=2;
subplot(211)
stem(t,y1)
xlabel('time ')
ylabel('ampitude');
title('discrete exponential signal with m = 1 ');
subplot(212)
stem(t,y2)
xlabel('time ')
ylabel('ampitude');
title('discrete exponential signal with m = 2');

Continuous ramp signal
t=0:1:n;
y1=t; %multiplication factor m =1;
y2=t*2; %multiplication factor m =1;
subplot(211)
plot(t,y1)
xlabel('time ')
ylabel('ampitude');
title('continuous ramp signal with m = 1 ');
subplot(212)
plot(t,y2)
xlabel('time ')
ylabel('ampitude');
title('continuous ramp signal with m = 2');

Discrete ramp signal
t=0:1:n;
subplot(211)
stem(t,y1)
xlabel('time ')
ylabel('ampitude');
title('discrete ramp signal with m = 1 ');
subplot(212)
stem(t,y2)
xlabel('time ')
ylabel('ampitude');
title('discrete ramp signal with m = 2');
Result:

EXPERIMENT NO:4
Aim: To generate continuous and discrete convolution time signal
Theory:
Convolution theorem for continuous system:If the impulse response of an analog or continuous
time system be h(t), it means that for an input δ(t) its output is h(t). Hence for all linear system its
output for a shifted impulse δ(t-T) is h(t-T). The following are the mathematical representation
of these operations:
δ(t) h(t)
δ(t-T)  h(t-T)
x(t) δ(t-T)  x(T) h(t-T)
∫ x(T) δ(t-T) dT  ∫ x(T) h(t-T) dT
From the last two statements, we can say that an input x(T) δ(t-T) the output is x(T) h(t-T) and
for an input of continuous summation or integration of x(T) δ(t-T), the output is also continuous
summation of x(T) h(t-T). Therefore, we can write by the linearity of the system
y(t)= ∫ x(T) h(t-T) dT
The right hand side of the above equation is defined as convolution of x(t) with h(t) and now we
can write it as follow:
y(t)= convolution of x(t) and h(t)
= x(t) * h(t)
= ∫ x(T) h(t-T) dT
The symbol ‘*’ stands for convolution.
Convolution for discrete time function:
If x(n) is applied as an input to a discret time system, the response y(n) of the system is given by
y(n)= T[x(n)]=t[∑ x(k) δ(n-k)]
to study LTI systems linear convolution plays an important role.

Continuous convolution time system:
t1=[-10:1:0];
t2=[0:1:2];
t3=[2:1:10];
x=zeros(size(t1));
y=ones(size(t2));
z=zeros(size(t3));
t=[t1,t2,t3];
a=[x,y,z];
subplot(2,2 ,1);
plot(t,a,'m');
xlabel('time-->');
ylabel('u(t)-u(t-2)');
u1=[-11:1:0];
u2=[0:1:3];
u3=[3:1:11];
v1=zeros(size(u1));
v2=ones(size(u2));
v3=zeros(size(u3));
u=[u1,u2,u3];
b=[v1,v2,v3];
subplot(2,2,2);
plot(u,b,'m')
xlabel('time-->');
ylabel('u(t)-u(t-3)');
c=conv(y,v2);
subplot(2,2,3);
plot(c);
xlabel('time-->');
ylabel('(u(t)-u(t-2))*(u(t)-u(t-3))');

-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
time-->
u(t)-u(t-2)
-15 -10 -5 0 5 10 15
0
0.2
0.4
0.6
0.8
1
time-->
u(t)-u(t-3)
1 2 3 4 5 6
1
1.5
2
2.5
3
time-->
(u(t)-u(t-2))*(u(t)-u(t-3))
Result:

EXPIREMENT: 5
Aim: Adding and subtracting two given signals. (Continuous as well as Discrete signals)
Theory:
Addition of two ram signal in continuous and discrete n manner
Code:
t=0:1:n;
y3=y1+y2;
subplot(221)
plot(t,y1);
xlabel('time ')
ylabel('ampitude');
title('continuous ramp signal with m = 1 ');
subplot(222)
plot(t,y2)
xlabel('time ')
ylabel('ampitude');
title('continuous ramp signal with m =2');
subplot(223);
plot(t,y3);
xlabel('time ')
ylabel('ampitude');
title('addition of two continuous ramp signal with m = 1 and m =2');
subplot(224)
stem(t,y3)
xlabel('time ')
ylabel('ampitude');
title('addition of ramp signal in DISCRETE in time');

Addition and Subtraction of two discrete different signals
a=[0 1 1 1 2 1 ]
b=[1 0 1 1 1 2 ]
c=a+b;
d=a-b;
subplot(221)
plot(a);
xlabel('time');
ylabel('amplitude');
title('first a signal ')
subplot(222)
plot(b);
xlabel('time');
title('second b signal ')
subplot(223)
plot(c);
xlabel('time');
title('addition of a and b signal ')
subplot(224)
plot(d);

xlabel('time');
title('substraction of a and b signal ')
Addition and Subtraction of two discrete different signals
a=[0 1 1 1 2 1 ]
b=[1 0 1 1 1 2 ]
c=a+b;
d=a-b;
subplot(221)
stem(a);
xlabel('time');
title('first a signal ')
subplot(222)
stem(b);
xlabel('time');
title('second b signal ')
subplot(223)
stem(c);

xlabel('time');
title('addition of a and b signal ')
subplot(224)
stem(d);
xlabel('time');
title('substraction of a and b signal ')
Result :

EXPERIMENT: 6
Aim :To generate uniform random numbers between (0, 1).
Theory:
r = rand(n) returns an n-by-n matrix containing pseudorandom values drawn from the standard
uniform distribution on the open interval (0,1). r = rand(m,n) or r = rand([m,n]) returns an m-by-
nmatrix. r = rand(m,n,p,...) or r = rand([m,n,p,...]) returns an m-by-n-by-p-by-... array. r =
rand returns a scalar. r = rand(size(A)) returns an array the same size as A.
r = rand(..., 'double') or r = rand(..., 'single') returns an array of uniform values of the specified
class.
r = rand(n)
r = rand(m,n)
r = rand([m,n])
r = rand(m,n,p,...)
r = rand([m,n,p,...])
r = rand
r = rand(size(A))
r = rand(..., 'double')
r = rand(..., 'single')
Random number generation in continuous and discrete manner
n=input('Enter the tota number which is generated N=');
y=rand(1,n)
subplot(211)
plot(y);
xlabel('random numbers');
title('continuous plot of Random Number 0-1')
subplot(212)
stem(y)
xlabel('random numbers');
title('discrete plot of Random Number 0-1')

Result:

EXPERIMENT: 7
Aim: To generate a random binary wave.
Generation of binary wave for random signal continuous and discrete manner
n=input('Enter the tota number which is generated N=');
j=0;
y1=rand(1,n)
y=round(y1)
for i=1:n
if y(i)== 1;
j(i)=ones
else
j(i)=zeros
end
end
stem(j)
xlabel('no of random signal');
title('plot of Random in ones and zeros 0-1')

Result:

EXPERIMENT 8
Aim : To generate random sequences with arbitrary distributions, means and variances for
following :
(a) Rayleigh distribution
(b) Normal distributions: N(0,1).
Theory:
Definition The Rayleigh pdf is
The Rayleigh distribution is a special case of the Weibull distribution. If A and B are the
parameters of the Weibull distribution, then the Rayleigh distribution with parameter b is
equivalent to the Weibull distribution with parameters and B = 2.
If the component velocities of a particle in the x and y directions are two independent normal
random variables with zero means and equal variances, then the distance the particle travels per
unit time is distributed Rayleigh.
In communications theory, Nakagami distributions, Rician distributions, and Rayleigh
distributions are used to model scattered signals that reach a receiver by multiple paths.
Depending on the density of the scatter, the signal will display different fading characteristics.
Rayleigh and Nakagami distributions are used to model dense scatters, while Rician distributions
model fading with a stronger line-of-sight. Nakagami distributions can be reduced to Rayleigh
distributions, but give more control over the extent of the fading.
Parameters
The raylfit function returns the MLE of the Rayleigh parameter. This estimate is
Rayleigh distribution
Coding
x = [0:0.01:2];
p= raylpdf(x,0.5);
subplot(211)
plot(x,p)
title('Continuous Rayleigh pdf')

subplot(212)
stem(x,p)
title('Discrete Rayleigh pdf')
Normal distributions: N(0,1).
In probability theory, the normal (or Gaussian) distribution is a continuous probability
distribution, defined on the entire real line, that has a bell-shaped probability density function,
known as the Gaussian function or informally as the bell curve:[nb 1]
The parameter μ is the mean or expectation (location of the peak) and σ 2
is the variance. σ is
known as the standard deviation. The distribution with μ = 0 and σ 2
= 1 is called the standard
normal distribution or the unit normal distribution. A normal distribution is often used as a first
approximation to describe real-valued random variables that cluster around a single mean value.

Coding :
x = -5:(1/100):5;
mu = 0;
sigma = 1;
k = 1/(sigma*sqrt(2*pi));
y = k*exp(-((x-mu).^2)/(2*sigma^2));
hold off
plot(x,y,'b.-')
hold on
mu = 0.5; sigma = 1.5;
y = k*exp(-((x-mu).^2)/(2*sigma^2));
plot(x,y,'r.-')
mu = -.2; sigma = 0.5;
y = k*exp(-((x-mu).^2)/(2*sigma^2));
plot(x,y,'k.-')

EXPERIMENT 9
Aim: To plot the probability density functions. Find mean and variance for the above
distributions
Theory:
A probability density function is most commonly associated with absolutely
continuous univariate distributions. A random variable X has density f, where f is a non-
negative Lebesgue-integrable function, if:
Hence, if F is the cumulative distribution function of X, then:
and (if f is continuous at x)
Intuitively, one can think of f(x) dx as being the probability of X falling within the
infinitesimal interval [x, x + dx].
Coding : Random number distribution
data = [1 2 3 3 4]; %# Sample data
xRange = 0:10; %# Range of integers to compute a probability for
N = hist(data,xRange); %# Bin the data
plot(xRange,N./numel(data));%# Plot the probabilities for each integer
xlabel('Integer value');
ylabel('Probability');

data = [1 2 3 3 4]; %# Sample data
xRange = 0:10; %# Range of integers to compute a probability for
N = hist(data,xRange); %# Bin the data
stem(xRange,N./numel(data)); %# Plot the probabilities for each integer
xlabel('Integer value');
ylabel('Probability');

Normal distribution N( mu , sigma )
Code:
%% Explore the Normal distribution N( mu , sigma )
mu = 100; % the mean
sigma = 15; % the standard deviation
xmin = 70; % minimum x value for pdf and cdf plot
xmax = 130; % maximum x value for pdf and cdf plot
n = 100; % number of points on pdf
k = 10000; % number of random draws for histogram
% create a set of values ranging from xmin to xmax
x = linspace( xmin , xmax , n );
p = normpdf( x , mu , sigma ); % calculate the pdf
plot(x,p)
xlabel( 'x' ); ylabel( 'pdf' );
title( 'Probability Density Function' );

Code for 'Cumulative Density Function'
mu = 100; % the mean
sigma = 15; % the standard deviation
xmin = 70; % minimum x value for pdf and cdf plot
xmax = 130; % maximum x value for pdf and cdf plot
n = 100; % number of points on pdf
k = 10000; % number of random draws for histogram
% create a set of values ranging from xmin to xmax
x = linspace( xmin , xmax , n );
c = normcdf( x , mu , sigma ); % calculate the cdf
plot( x , c );
xlabel( 'x' ); ylabel( 'cdf' );
title( 'Cumulative Density Function' );

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