ankit sharma(dsp)

8/6/2019 Ankit sharma(dsp)

http://slidepdf.com/reader/full/ankit-sharmadsp 1/33

Digital

Signal

Processing

BY:- Ankit Sharma

Roll No:- 0657013108

BTECH(IT)



Experiment No. 1

Aim: To study MATLAB and various commands and file types associated with it and henceappreciate the use of MATLAB in signal processing.

THEORY

MATLAB (short for Matrix Laboratory) is a special purpose computer program organized to

perform engineering and scientific calculations.

MATLAB has many advantages compared to conventional computer languages for technical

problem solving .Among them are:

1) Ease of use:->Programs may be easily written and modified with the built in integrateddevelopment environment and debugged with the MATLAB debugger. Because the language is

so easy to use, it is ideal for the rapid prototyping of new programs.

2) Platform independence:->MATLAB is supported on many different computer systems ,

providing a large measure of platform independence. As a result, programs written in MATLAB

can migrate to new platforms when the needs of the user change.

3) Predefined functions:->MATLAB comes complete with an extensive library of predefined

functions that provide tested and pre-packaged solutions to many basic technical tasks.

4) Device independent plotting:->The plots and images can be displaced on any graphical outputdevice supported by the computer on which MATLAB is running. This capability makes

MATLAB an outstanding tool for visualizing technical data.

5) Graphical user interface:->MATLAB includes tools that allow a programmer to interactively

construct a graphical user interface for program.

6) MATLAB compiler:-> MATLAB¶s flexibility and platform independence is achieved by

compiling MATLAB programs into a device independent p-code and then interpreting the p-

code at run time.

Commands & Essential functions in MATLAB:

y clc- clear the command window only.

y clear-removes all variables from workspace.

y clear all- removes all variables, globals & functions.



y Ls-listing of cursor directory.

y Pwd- present, working, directory.

y Load-load data from a file

y «-continues a MATLAB statement in the following line.

y Date-current date

y Title-adds a title to a plot

y Save-save data from workspace in a file.

y Print-prints a figure window.

y Max(x)-returns the maximum value in vector x, and optionally the location of that value.

y Min(x)-returns the minimum value in vector x, and optionally the location of that value.

y Log(x)-calculates natural logarithm of x.

y Help«.(command name)-give brief description of asked command.

y doc«.(command name)-full description.

y Whos-list all variables in current workspace,size.edit-matlab editor «.etc.

y edit-matlab editor

y Size(a)-return size of matrix a.

y Length(a)-maximum dimension of matrix.

y A¶-to change column vector to row.

y Logspace-generates logarithmic generated spaced vectors.

y Find(a)-returns indexes (row and column no. of non-zero elements).

y Disp-accepts an array arguments and displays the value .

y Abs(x)-calculates the absolute value of x.

y Ans-default variable used to store the result of expression not assigned to another

variable.

y Acos(x)-calculates inverse cosine of x.

y Asin(x)- calculates inverse sine of x.

y Atan(x)- calculates inverse tangent of x.

y Ceil(x)-rounds x to the nearest integer towards positive infinity.

y Char-converts a matrix of nos. into a character string.

y Clock-current time

y Cos(x)-calculates cosine of x.

y Eps-represents machine precision.

y Inf-represents machine infinity.

y Mod(n,m)-remainder or modulo function

y Sqrt-calculates square of a no.



Experiment No. 2

Aim:Write a program to develop elementary signal function modules (m-files) for unit sample,

unit step, exponential and unit ramp sequences.

THEORY

Exponential Function:

A function f(x)=ex

is defined as:

0<f(x)�1; for -��x�0

1�f(x)<�; for 0�x��

Unit impulse Function:

Unit impulse is a signal that is 0 everywhere except at n=0 where it is 1. In discrete time domain

the unit impulse signal is defined as:

�(n) = 1; n=0

= 0; elsewhere

Unit Step Function:

The integral of the impulse function is a unit step signal. In discrete time unit step signal is

defined as:

U(n) = 1; n>=0

= 0; n<0

Ramp signal:

Ramp signal increases linearly with sample number n. It is defined as

R(n) = n; n>=0

= 0; n<0



DESCRIPTION OF MATLAB FUNCTION USED:

Exp:

The exp function is an elementary function that operates element-wise on arrays. Its domain

includes complex numbers.

Y = exp(X) returns the exponential for each element of X.

Subplot:

subplot divides the current figure into rectangular panes that are numbered rowwise. Each pane

contains an axes. Subsequent plots are output to the current pane.

subplot(m,n,p) creates an axes in the pth pane of a figure divided into an m-by-n matrix of

rectangular panes.

The new axes becomes the current axes. If p is a vector, it specifies an axes having a position

that covers all the subplot positions listed in p.

Stem:

A two-dimensional stem plot displays data as lines extending from a baseline along the x-axis. Acircle (the default) or other marker whose y-position represents the data value terminates each

stem.

stem(X,Y) plots X versus the columns of Y. X and Y must be vectors or matrices of the same

size. Additionally, X can be a row or a column vector and Y a matrix with length(X) rows.

Xlabel:

xlabel('string') labels the x-axis of the current axes.

Ylabel:

ylabel('string') labels the y-axis of the current axes.



PROGRAM:

%TO PLOT EXPONENTIAL SIGNAL

t=0:7;

a=1;

y2=exp(a*t);

subplot(2,2,1);

stem(t,y2);

ylabel('amplitude-->');

xlabel('n-->')

%TO PLOT UNIT SAMPLE SIGNAL

clc;

clear all;

clear all;

t=-2:1:2;

y=[zeros(1,2),ones(1,1),zeros(1,2)];

subplot(2,2,2);

stem(t,y);


xlabel('n-->');



%TO PLOT RAMP SIGNAL

n1=7;

t=0:n1;

subplot(2,2,3);

stem(t,t);


xlabel('(c)n-->');

%TO PLOT UNIT STEP SIGNAL

n=7;

t=0:1:n-1;

y1=ones(1,n);

subplot(2,2,4);

stem(t,y1);


xlabel('n-->');



OUTPUT:



Experiment No. 3

Aim:Write a program to develop program modules based on operation on sequences like signal

shifting,signal folding,signal addition and signal multiplication.

THEORY Signal processing is group of basic operations applied to an input signal resulting in another

signal as the output. There are various operations which can be performed on a signal.

Signal addition:

{x1(n)}+{x2(n)}={x1(n)+x2(n)}

Summation: Sum all the values of the sequence to obtain the value of the output at a given

time.

Signal multiplication:

{x1(n)}*{x2(n)}={x1(n)*x2(n)}

Multiplication: Multiply x1(n) by x2(n) to obtain the value of the output sequence.

Scaling of signal:

S*{x(n)}={s*x(n)}

Shifting of signal:

The shift signal takes the input sequence and shifts the values by an increment of the independent

variable. The shifting may delay or advance the sequences in time.

Y(n)={x(n-k)}



Shifting: Shift x(-k) by n to the right(left) if n is positive(negative) to obtain x(n-k).

Folding of signal:

Y(n)=x(-n)

Here x(n) is the input and y(n) is the output.

Folding: Fold x(n) about n=0 to obtain x(-n).


Exp:




Subplot:




rectangular panes.



Stem:

A two-dimensional stem plot displays data as lines extending from a baseline along the x-axis. A

circle (the default) or other marker whose y-position represents the data value terminates each

stem.





Xlabel:


Ylabel:


Sin:

The sin function operates element-wise on arrays. The function's domains and ranges include

complex values. All angles are in radians.

Y = sin(X) returns the circular sine of the elements of X.

Cos:

The cos function operates element-wise on arrays. The function's domains and ranges include


Y = cos(X) returns the circular cosine for each element of X.

Max:

C = max(A) returns the largest elements along different dimensions of an array.

Min:

C = min(A) returns the smallest elements along different dimensions of an array.



Zeros:

Creates an array of all zeros.

B = zeros(m,n) or B = zeros([m n]) returns an m-by-n matrix of zeros.

Fliplr:

B = fliplr(A) returns A with columns flipped in the left-right direction, that is, about a vertical

axis.

Disp:

disp(X) displays an array, without printing the array name. If X contains a text string, the stringis displayed.

PROGRAM:

%% Signal addition

n1=[-5:10];

n2=[-2:11];

x1=sin(n1);

x2=cos(n2);

subplot(2,2,1);stem(x1,n1);

subplot(2,2,2);stem(x2,n2);

n=min(min(n1),min(n2)): max(max(n1),max(n2));

y1=zeros(1,length(n));

y2=y1;

y2(find((n>=min(n2))&(n<=max(n2))==1))=x2;


y=y1+y2;



subplot(2,2,3);

stem(y,n);

OUTPUT:



%% Signal multiplication

n1=[-5:10];

n2=[-2:11];

x1=sin(n1);

x2=cos(n2);

n=min(min(n1),min(n2)): max(max(n1),max(n2));

y1=zeros(1,length(n));

y2=y1;



y=y1.*y2;

stem(n,y);

OUTPUT:



%% Signal Shifting

a=8;

n1=[3:18];

x1=n1;

subplot(1,2,1);stem(n1,x1);

n=a+n1;

y=x1;

subplot(1,2,2);stem(n,y);

OUTPUT:



%% Signal Folding

x=[-2:3];

n1=-2;

n2=3;

n=[n1:n2];

m=-fliplr(n);

z=fliplr(x);

disp(n);

disp(x);

disp(m);

disp(z);

subplot(2,1,1);

stem(n,x);

title('given signal');

subplot(2,1,2);

stem(m,z);

title('folded signal');



OUTPUT:



Experiment No. 4

Aim:Write a program to generate the given sequences and decompose them into their even and

odd parts.

a) x(n)=0.9^n cos(0.2n+/3) ; 0�n�20

b) x(n)=10 sin(0.008 n^2+) ;0�n�100 and is a random sequence.

THEORY

Even and odd signals bear some important symmetry properties. Under reversal of independent

variable, these signals either remain the same (even signal) or get reflected or flipped (odd

signal) about the horizontal axis.

Equations or definitions (5.1) and (5.2) mathematically express these properties for both

continuous and discrete time cases.

Even Signals: x(t) = x(±t), x[n] = x[±n] (5.1)

Odd Signals: x(t) = ±x(±t), x[n] = ±x[±n] (5.2)

It is possible to decompose any signal into its even and odd parts,

xe(t)=1/2(x(t)+x(-t))

xo(t) =1/2(x(t)-x(-t))

x(t)=xe(t)+xo(t)


Exp:






Subplot:



subplot(m,n,p) creates an axes in the pth pane of a figure divided into an m-by-n matrix of rectangular panes.



Stem:



stem.



Xlabel:


Ylabel:


Clf:

clf deletes from the current figure all graphics objects whose handles are not hidden (i.e., their

HandleVisibility property is set to on).

Figure:

figure(h) does one of two things, depending on whether or not a figure with handle h exists. If h

is the handle to an existing figure, figure(h) makes the figure identified by h the current figure,



makes it visible, and raises it above all other figures on the screen. The current figure is the target

for graphics output. If h is not the handle to an existing figure, but is an integer, figure(h) creates

a figure and assigns it the handle h. figure(h) where h is not the handle to a figure, and is not an

integer, is an error.

Cos:

The cos function operates element-wise on arrays. The function's domains and ranges include


Y = cos(X) returns the circular cosine for each element of X.

Fliplr:

B = fliplr(A) returns A with columns flipped in the left-right direction, that is, about a vertical

axis.

a)

PROGRAM:

n=[0:20];

x=(0.9.^n).*cos((0.2.*n.*pi)+(pi/3));

[xe,xo,m]=evenodd(x,n);

figure(1),clf;

subplot(2,2,1);stem(n,x);

title('Main');

subplot(2,2,2);stem(m,xo);

title('ODD');

subplot(2,2,3);stem(m,xe);

title('EVEN');



FUNCTION:

function[xo,xe,m]=evenodd(x,n)

if any(imag(x)~=0)

error('X is not a real sequence');

end

m=-fliplr(n);

m1=min([m,n]); m2=max([m,n]);

m=m1:m2;

nm=n(1)-m(1); n1=1:length(n);

x1=zeros(1,length(m));

x1(n1+nm)=x; x=x1;

xe=0.5*(x+fliplr(x));

xo=0.5*(x-fliplr(x));

OUTPUT:



b)

PROGRAM:

n=[0:100];

x=10.*sin(0.008.*pi.*(n.^2)+randn(1,101));

[xe,xo,m]=evenodd(x,n);

figure(1),clf;

subplot(2,2,1);stem(n,x);

title('Main');

subplot(2,2,2);stem(m,xo);

title('ODD');

subplot(2,2,3);stem(m,xe);

title('EVEN');

function:

function[xo,xe,m]=evenodd(x,n)

if any(imag(x)~=0)

error('X is not a real sequence');

end

m=-fliplr(n);

m1=min([m,n]); m2=max([m,n]);

m=m1:m2;

nm=n(1)-m(1); n1=1:length(n);

x1=zeros(1,length(m));

x1(n1+nm)=x; x=x1;

xe=0.5*(x+fliplr(x));xo=0.5*(x-fliplr(x));



OUTPUT:



Experiment No. 6

Aim: Write a program to show the application of cross correlation in comparing the signals.

THEORY

In signal processing, cross-correlation is a measure of similarity of two waveforms as a function

of a time-lag applied to one of them. This is also known as a sliding dot product or inner-

product . It is commonly used to search a long duration signal for a shorter, known feature. It also

has applications in pattern recognition, single particle analysis, electron tomographic

averaging, cryptanalysis, and neurophysiology.

For continuous functions, f and g , the cross-correlation is defined as:

where f * denotes the complex conjugate of f .

Similarly, for discrete functions, the cross-correlation is defined as:


Exp:




Subplot:






rectangular panes.



Stem:



stem.



Xlabel:


Ylabel:


Xcorr:

c = xcorr(x,y) returns the cross-correlation sequence in a length 2*N-1 vector, where x and y are

length N vectors (N>1).

If x and y are not the same length, the shorter vector is zero-padded to the length of the longer

vector. By default, xcorr computes raw correlations with no normalization.



PROGRAM:

x=[1,2,3,4];

y=[4,3,2,1];

nx=[0:8];

[c,lags]=xcorr(x,y);

subplot(1,1,1);

stem(lags,c);

xlabel('n');

ylabel('a');

OUTPUT:



Experiment No. 8

Aim: Write a program to compute the partial expansion of a given transfer function in z-domain

and compute its poles and zeros. Also plot poles and zeros , and determine the ROC of z-

transform.

THEORY

Impulse Invariant Transformation

In this technique the desired impulse response of the digital filter is obtained by uniformly

sampling the impulse response of the analog filter.

H(s) = (s+1)/(s2+5s+6)

H(z) = (1-0.8966z-1

)/(1-1.5595 z-1

+0.66065 z-1

)

Bilinear Transformation

In this technique the desired impulse response of the digital filter is obtained uniformly sampling

the impulse response of the analog filter.

H(s) = (s+1)/ (s2+5s+6)

Taking T=1sec.

H(z)=(0.15-0.1z-1

-0.05z-2

)/(1+0.2z-1

)

Region of convergence

The region of convergence (ROC) is the set of points in the complex plane for which the Z-

transform summation converges.




Zplane:

This function displays the poles and zeros of quantized filters, as well as the poles and zeros of

the associated unquantized reference filter.

Residue:

[r,p,k] = residue(b,a) finds the residues, poles, and direct terms of a partial fraction expansion of

the ratio of two polynomials, b(z) and a(z). Vectors b and a specify the coefficients of the

polynomials of the discrete-time system b(z)/a(z) in descending powers of z

PROGRAM:

>> n=[2 16 44 56 32]

n =

2 16 44 56 32

>> d=[3 3 -15 18 -12]

d =

3 3 -15 18 -12

>> [r,p,k]=residue(n,d)

r =

-0.0177

9.4914

-3.0702 + 2.3398i

-3.0702 - 2.3398i

p =

-3.2361



1.2361

0.5000 + 0.8660i

0.5000 - 0.8660i

k =

-2.6667

>> zplane(n,d)

OUTPUT:



Experiment No. 7

Aim: To compute the circular convolution of two sequences using MATLAB functions andusing the fold, shift and multiply procedure, then compare the result with that obtained using

DFT & IDFT.

THEORY

Let x1(n) and x2(n) are finite duration sequences both of length N with DFTs X1(k) and

X2(k).Now we find a sequence x3(n) for which the DFT is X3(k), where

X3(K)=X1(K)*X2(K)

We know from earlier computations that

N-1

X3(n)= x1(m)x2(n-m)N

m=0

This equation represents circular convolution of x1(n) and x2(n) which is represented as

X3( n)=x1(n) x x2(n)

We find that DFT of:

DFT[x1(n) x2(n)]=X1(K)*X2(K)

PROGRAM:

%% circular convolution

clc;

clear all;

close all;

N

N



n=50;

x1=1:n;

figure,stem(x1);

title('first seq');

x2=n:-1:1;

figure,stem(x2);

title('second seq');

x3=cconv(x1,x2,n);

figure,stem(x3);

title('Circular convolution using conv');

x4=fliplr(x1);

x4=circshift(x4,[0,1]);

for i=1:n

x5(i)=sum(x2.*circshift(x4,[0,i-1]));

end

figure,stem(x5);

title(' circular convolution using shift for n multiply procedure');

x6=fft(x1).*fft(x2);

x6==ifft(x6);



figure,stem(x6);

title('fourier transform');

OUTPUT:

0 5 10 15 20 25 30 35 40 45 50 0

5

10

15

20

25

30

35

40

45

50 first seq

0 5 10 15 20 25 30 35 40 45 50 0

5

10

15

20

25

30

35

40

45

50 second seq

ankit sharma(dsp)

Documents