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; y2=M*exp(t*2); %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:

n= input('Enter the value of N='); t=0:1:n; M=2; y1=M*exp(t*1); %multiplication factor m =1; y2=M*exp(t*2); %multiplication factor m =1; 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

n= input('Enter the value of N='); 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

n= input('Enter the value of N='); t=0:1:n; y1=t; %multiplication factor m =1; y2=t*2; %multiplication factor m =1; 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 5 100

0.2

0.4

0.6

0.8

1

time-->

u(t

)-u(t

-2)

-15 -10 -5 0 5 10 150

0.2

0.4

0.6

0.8

1

time-->

u(t

)-u(t

-3)

1 2 3 4 5 61

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:

n= input('Enter the value of N='); t=0:1:n; y1=t; %multiplication factor m =1; y2=t*2; %multiplication factor m =2; 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'); ylabel('amplitude'); title('second b signal ') subplot(223) plot(c); xlabel('time'); ylabel('amplitude'); title('addition of a and b signal ') subplot(224) plot(d);



xlabel('time'); ylabel('amplitude'); 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'); ylabel('amplitude'); title('first a signal ') subplot(222) stem(b); xlabel('time'); ylabel('amplitude'); title('second b signal ') subplot(223) stem(c);



xlabel('time'); ylabel('amplitude'); title('addition of a and b signal ') subplot(224) stem(d); xlabel('time'); ylabel('amplitude'); 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'); ylabel('amplitude'); title('continuous plot of Random Number 0-1') subplot(212) stem(y) xlabel('random numbers'); ylabel('amplitude'); 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'); ylabel('amplitude'); 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; k = 1/(sigma*sqrt(2*pi)); y = k*exp(-((x-mu).^2)/(2*sigma^2)); plot(x,y,'r.-') mu = -.2; sigma = 0.5; k = 1/(sigma*sqrt(2*pi)); 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' );

signal prosessing lab mannual

Education