bsl lab manual

8/4/2019 BSL Lab Manual

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>> X=det(A)

X =

0

Generation of various signals

Matlab code for the generation of Unit Step Sequencen = input('enter the n value');

t = 0:1:n-1;

y1= ones(1,n);subplot(2,2,2);

stem(t,y1);

ylabel('amp');xlabel('(b)n');

Output:

Enter the n value 10

Matlab code for the generation of Unit impulse Sequence


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clc; %clear the data

t = -2:1:2;

y=[zeros(1,2),ones(1,1),zeros(1,2)]; %Create an array of all zeros and onessubplot(2,2,1);

stem(t,y); %function displays two-dimensional discrete sequence data.

ylabel('amp'); %label the Yaxesxlabel('(a)n'); %label the Xaxes

1a) Matlab code to generate sinusoidal(continous , discrete)

clc;clear all;close all;

t=0:0.01:1;

x=sin(2*pi*2*t);subplot(2,1,1);

plot(t,x);

xlabel('--> time');ylabel('--> magnitude');

title(' continuoes time cose wave ');

grid;

n=0:0.05:1;x=sin(2*pi*2*n);

subplot(2,1,2);

stem(n,x,'r');xlabel('--> time');

ylabel('--> magnitude');

title('discrete time cose wave');grid;


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Matlab code to generate cosine(continous , discrete)

clc;

clear all;

close all;t=0:0.01:1;

x=cos(2*pi*2*t);subplot(2,1,1);plot(t,x);

xlabel('--> time');

ylabel('--> magnitude');title(' continuoes time cose wave ');

grid;

n=0:0.05:1;x=cos(2*pi*2*n);

subplot(2,1,2);

stem(n,x,'r');


title('discrete time cose wave');

grid;


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Matlab code to generate sawtooth signal(continous, discrete)

clc;clear all;

close all;

t=-1:0.001:1;x=sawtooth(2*pi*2*t);

subplot(2,1,1);

plot(t,x);


title(' continuoes time sawtooth wave ');

grid;

n=-1:0.05:1;x=sawtooth(2*pi*2*n);

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

xlabel('--> time');

ylabel('--> magnitude');title('discrete time sawtooth wave');

grid;


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Matlab code to generate triangular waveform

fs = 10000; %Set the sampling frequency fs as 10000.

t = 0 : 1/fs : 1.5; %Generate a vector of 15001 samples for t with a

value between 0 & 1.5

%with an increment of 0.0001x = sawtooth (2 * pi * 50 * t , 0.5);%Generate a Triangular wave of

frequency 50 Hz, by calling the

%MATLAB built-in function sawtooth with 0.5 as the

duty cycle parameter.

plot(t,x); axis ( [ 0 0.1 -1 1 ] ); %Plot the graph. Adjust the X and Y

coordinates to view a few cycles of the triangular wave.

xlabel ('Time Index t (sec.)');

ylabel ('Amplitude');%Label the X and Y axes

title ('Triangular Wave Signal Sequence'); %Add an appropriate title to thegraph


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clc;

clear all;

close all;t=0:0.001:1;

x=square(2*pi*2*t);

subplot(2,1,1);plot(t,x);

xlabel('--> time');


title(' continuoes time square wave ');axis([0 1 -2 2]);

grid;

n=0:0.1:1;x=square(2*pi*2*n);

subplot(2,1,2);

stem(n,x);xlabel('--> time');


title('discrete time square wave');grid;


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APERIODIC SIGNALS

clc;

clear all;close all;t=-5:0.01:5;

%Unit impulse

x1=1;

x2=0;

x=x1.*(t==0)+x2.*(t~=0);subplot(3,3,1);

plot(t,x,'r');

xlabel('->t');

ylabel('->x(t)');axis([-5 5 -1 2]);

title('Unit impulse signal');

grid;

%unit step

x=x1.*(t>=0)+x2.*(t


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plot(t,x);

xlabel('t');

ylabel('x(t)');axis([-5 5 -1 2]);

title('unit step signal');

grid;

%ramp signal

x1=t;x2=0;

x=x1.*(t>=0)+x2.*(t=0)+x2.*(t0)+x2.*(t==0)+x3.*(t


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subplot(3,3,6);

plot(t,x);

xlabel('t');ylabel('x(t)');

axis([-5 5 -2 2]);

grid;title('signum signal');

%gaussian pulse

x=exp(-t.^2);subplot(3,3,7);

plot(t,x);

xlabel('t');

ylabel('x(t)');grid;

title('gaussian pulse');

% decreasing exponential signal

x=exp(-0.2*t);subplot(3,3,8);

plot(t,x);xlabel('t');

ylabel('x(t)');

title('decreasing exponential signal');

grid;

% increasing exponential signal

x=exp(0.2*t);

subplot(3,3,9);plot(t,x);

xlabel('t');

ylabel('x(t)');title('increasing exponential signal');

grid;


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3.Operations on signals

3a) MATLAB CODE FOR SIGNAL ADDITION

clc; % Clear Command Window

clear all ; %Clears all the previously defined variables

t = 0 : 0.01 : 30; %Generate a vector of 3001 samples for t with a value%between 0 & 30 with an increment of 0.01

x1 = sin( 2 * pi * 1/3 * t ); %Generate the sinusoidal signal of frequency 1/3 Hz

x2 = sin( 2 * pi * 1/5 * t ); % Generate the sinusoidal signal of frequency 1/5 Hzy = x1 + x2; % Perform sample by sample vector addition on x1 and x2

subplot( 3,1,1 ); plot( t , x1 ); %Divide the graph space into three parts.

grid; %Plot signal 1 in the first part with grid.xlabel( ' Time Index t (sec.) ' );

ylabel( ' x1(t) ' ); %Label the X and Y axes

title( ' Signal 1 : Sine Wave of Frequency 1/3 Hz ' ); %Add a title to the graph

subplot( 3,1,2 ); plot( t , x2 );


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grid; %Plot the graph of signal 2 in the second part with grid.

xlabel( ' Time Index t (sec.) ' );

ylabel( ' x2(t) ' ); %Label the X and Y axestitle( ' Signal 2 : Sine Wave of Frequency 1/5 Hz ' ); %Add a title to the graph

subplot( 3,1,3 ); plot( t , y );

grid; % Plot the graph of resultant signal after summation, in the third part with grid.xlabel( ' Time Index t (sec.) ' );

ylabel( ' y(t) = x1(t) + x2(t) ' ); %Label the X and Y axes

title( ' Resultant Signal : Signal 1 + Signal 2 ' ); %Add A title to the graph

3b MATLAB CODE FOR SIGNAL MULTIPLICATION

clc; clear all; % Clear Command Window and all the previously defined variables

t = 0 : 0.01 : 30; % Generate a vector of 3001 samples for t with a%value between 0 & 30 with an increment of 0.01

x1 = sin( 2 * pi * 1/3 * t ); %Generate the sinusoidal signal of frequency 1/3 Hz

x2 = sin( 2 * pi * 1/5 * t ); %Generate the sinusoidal signal of frequency 1/5 Hzy = x1 .* x2; %Perform sample by sample multiplication on x1 and x2.

subplot( 3,1,1 ); plot( t , x1 ); %Divide the graph space into three parts.grid; % Plot signal 1 in the first part, with grid.xlabel( ' Time t (sec.) ' );

ylabel( ' x1(t) ' ); %Label the X and Y axes

title( ' Signal 1 : Sine Wave of Frequency 1/3 Hz ' ); %Add an appropriate title to the graph

subplot( 3,1,2 ); plot( t , x2 );grid; %Plot the graph of signal 2 in the second part.Include the grid.

xlabel( ' Time Index t (sec.) ' ); ylabel( ' x2(t) ' ); %Label the X and Y axes


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title( ' Signal 2 : Sine Wave of Frequency 1/5 Hz ' ); %Add an appropriate title to the graph

subplot( 3,1,3 ); plot( t , y );

grid; %Plot the graph of resultant signal after summation, in the thirdpart.

xlabel( ' Time Index t (sec.) ' );

ylabel( ' y(t) = x1(t) .* x2(t) ' ); %Label the X and Y axestitle( ' Resultant Signal : Dot Product of Signal 1 and Signal 2 ' );

3c) MATLAB CODE FOR SIGNAL SCALING

clc; % Clear Command Windowclear all; % Clear all the previously defined variablesN = input ( ' Type the desired length of the signal '); %Input a number for N, thelength of the signalt = 0 : 0.01 : N-1; %Generate a vector t of N samplesx = sin( 2 * pi * 1/5 * t ); % Generate a sine wave for exampleA = input ( ' Please input a SCALE FACTOR(>1 or a +ve fraction) for A ');%Input a No. greater than unity or a positive fraction, using which the givensignal is to be scaledy = A * sin( 2 * pi * 1/5 * t ); %Scale the signal by the scalefactor A

subplot( 2,1,1 ); plot( t , x ); % Divide the graph space into twoparts, Plot x in first part.axis ( [ 0 N-1 -A A ] ); grid; %Aadjust the axes appropriatelyand use grid.xlabel( ' Time Index t (sec.) ' );ylabel( ' x(t) ' ); %Label the X and Y axestitle( ' Signal 1 : Sine Wave of Frequency 1/5 Hz ' ); %Add a title to the graph


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subplot( 2,1,2 ); plot( t , y ); % Plot the scaled signal y in secondpart.axis ( [ 0 N-1 -A A ] ); grid; %adjust the axes appropriately anduse grid.xlabel( ' Time Index t (sec.) ' );ylabel( ' y(t) ' ); %Label the X and Y axestitle( ' Signal 2 : Scaled Version of Signal 1 ' );%Add a title to the graph

output:Type the desired length of the signal 10Please input a SCALE FACTOR(>1 or a +ve fraction) for A 2

3d) MATLAB CODE FOR SIGNAL SHIFTINGclear all; clc ; %Clear Command Window and all the previously defined variablesx = input( ' Type the samples of a signal ' ); %Generate any arbitrary signal vector x by

%typing the samples

n1 = input( ' Type the time origin at which the first signal sample is ' );

%Considering the X-axis range from n1 to n2 for the signal x, enter the value ofn1, where thefirst signal sample begins.

N = length(x); %Find the length of the signal sequence

n2 = n1 + N -1; %Find the time index of the last sample of the signal xn = n1 : n2; %Set the range for X-axis from n1 to n2

disp( ' The program will now ask for the amount of shift ' );

%Display on screen the statement in parenthesisdisp( ' Enter a POSITIVE number for delay & NEGATIVE number for Advancement ' );

%Display on screen the statement in parenthesis

disp( ' Press any key to continue ' ); pause; %Display on screen the statement in parenthesis

%and wait until the above two statements are read.d = input( ' Enter the desired amount of shift of the signal ' ); %Input the value of time shift

nn = n + d; xs = x; %Shift the time index by the amount of shift and leave the original

%signal as it is by just assigning it to a different variable.


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subplot(2,1,1); stem( n , x ); %Plot the original signal as stems in the First part of the graphic.

xlabel( ' Time Index n (sec.) ' );

ylabel( ' x(n) ' ); %Label the X and Y axestitle( ' Original Signal ' ); %Add a title to the graph

Subplot(2,1,2); stem( nn , x ); %Plot the shifted signal in the second part of the

graphic.xlabel( ' Time Index n (sec.) ' ); ylabel( ' xs(n) ' ); %Label the X and Y axes

title( ' Time Shifted Signal ' ); %Add a title to the graph

output:

Type the samples of a signal 7

Type the time origin at which the first signal sample is 0

The program will now ask for the amount of shiftEnter a POSITIVE number for delay & NEGATIVE number for Advancement

Press any key to continue

Enter the desired amount of shift of the signal 4

3e) MATLAB CODE FOR SIGNAL FOLDING

clc; % Clear Command Window

clear all ; %Clear all the previously defined variables

t = 0 : 0.1 : 5; %Generate a vector n using a step size of 0.1

x = 0.5 * t; %Generate a ramp as an example signallx = length(x); %Find the length of the signal sequence

nx = 0 : lx-1; %Set a range for positive X-axisxf = fliplr( x ); %Perform the folding operation of signal x by using the built-in function

fliplr

nf = -fliplr( nx ); %Fold the vector nx to get a mirror image vector nf.%This will set the range of negative X-axis for the folded signal

subplot(2,1,1); % Divide the graph space into two parts


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stem( nx , x ); % Plot the original signal in the first part as stems

xlabel( ' nx ' );

ylabel( ' x(nx) ' ); %Label the X and Y axestitle('Original Signal' ); %Add a title to the graph

subplot(2,1,2);

stem( nf , xf ); %Plot the folded signal in the second part.xlabel( ' nf ' );

ylabel( ' xf(nf) ' ); %Label the X and Y axes

title( ' Folded Signal ' ); %Add a title to the graph

The

3f) MATLAB CODE FOR COMPUTATION OF ENERGY

n = 0:1:20; %Create a vector n of 101 samples

x = (1/2) .^ n; %Create the signal x(n) = (1/2)n u(n)

stem(n,x); axis([0 25 0 1]); %Plot the signal as stems and adjust the axes appropriately.disp( ' The calculated Energy E of the signal is '); %Display the message in brackets on screen

E = sum(abs(x).^2) %Calculate the energy of the signaldisp( ' The Theoretical Energy of the signal is ' ); %Display the message in brackets on screen

E_Theory = 4/3 ; %Calculate the theoretical value of the energy of the signal

Output:The calculated Energy E of the signal is

E =

1.3333


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3g) MATLAB CODE FOR COMPUTATION OF POWER

N = input( 'Type a value for N ');% Type a value for N, which is the one sided length of the

signal.t = -N : .0001 : N; %Create a vector t that will act as a time index

x = cos (2 * pi * 50 * t ) .^ 2; %Generate the signal x = cos 2 (2 f t)

disp( ' The calculated Power P of the signal is ' ) %Display the message on the screen

P = sum(abs(x).^2)/length(x) %Calculate the power of the signal x anddisplay its value on screen.

plot(t,x); axis ([0 0.1 0 1]); %Plot the signal by adjusting the axes appropriately.disp( ' The Theoretical Power of the signal is ' ) %Display the message in brackets on screen

P_Theory = 3/8 %Show the theoretical value of the power of the signal on screen.

Output:

Type a value for N 20

The calculated Power P of the signal is

P =

0.3750The Theoretical Power of the signal is

P_Theory =


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0.3750

4. Finding the even and odd parts of a signal or sequence and real and imaginary

parts of a signal4a) MATLAB CODE TO FIND WHETHER A SIGNAL IS EVEN OR ODD AND FIND

ITS EVEN AND ODD PARTS:

clc;clearall;close all;tmin=-4;tmax=4; dt=0.1;

t=tmin:dt:tmax;

%x1=exp(2*t);%x2=exp(-2*t);

x1=sin(2*pi*0.2*t);

x2=sin(-(2*pi*0.2*t));

%x1=cos2*pi*0.2*t);

%x2=cos(-(2*pi*0.2*t));if(x2==x1)

disp(' The signal is even signal ');

elseif(x2==(-x1))

disp(' The signal odd signal ');else

disp('The signal is neither even nor odd signal');

endend

xe=(x1+x2)/2;

xo=(x1-x2)/2;ymin=min([min(x1),min(x2),min(xe),min(xo)]);


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ymax=max([max(x1),max(x2),max(xe),max(xo)]);

subplot(2,2,1); plot(t,x1); axis([tmin tmax ymin ymax]);

xlabel('t'); ylabel('x1(t)'); title('signal x(t)');grid;

subplot(2,2,2); plot(t,x2); axis([tmin tmax ymin ymax]);

xlabel('t'); ylabel('x2(t)'); title('signal x(-t)');grid;

subplot(2,2,3); plot(t,xe); axis([tmin tmax ymin ymax]);

xlabel('t'); ylabel('xe(t)'); title('even part of x(t)');grid;

subplot(2,2,4); plot(t,xo); axis([tmin tmax ymin ymax]);

xlabel('t'); ylabel('xo(t)'); title('odd part of x(t)');

grid;

4b) MATLAB CODE TO FIND THE REAL AND IMAGINARY PARTS OF A SIGNAL

clc;

clearall;


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x = [0, 2+j*4, -3+j*2, 5-j*1, -2-j*4, -j*3, 0] %Create the complex signal

%vector x

n = -3 : 3; %Set a range for X-axis using vector nxc = conj( x ); %Compute the complex conjugate of x

xc_folded = fliplr(xc); %Fold the signal xc using the built-in %function 'fliplr'

xc_even = 0.5 * [x + xc_folded]; %Compute the even part of xcxc_odd = 0.5 * [x - xc_folded]; %Compute the odd part of xc

figure; %Open a new figure window (Figure 1)

subplot(2,1,1) ; stem(n , real(x) ) %Plot the real part of complex signal x in the 1st half ofFigure1.

title('Real part of Complex Signal x(n)')

xlabel ( ' n ' );

ylabel ('Magnitude of Real [x(n)]');

subplot(2,1,2); stem( n , imag(x) ) % Plot the imaginary part of complex

%signal x in the 2nd half of Figure 1.

title('Imaginary part of Complex Signal x(n)')%Add A title to the graphxlabel ( ' n ' );

ylabel ('Magnitude of Imag[ x(n)]');figure; %Open a new figure window (Fig 2)

subplot(2,1,1) ; stem(n , real(xc_even)); %Plot the real part of even signal

%xc_even in the 1st half of Fig 2.

title('Real part of even signal xc(n)');xlabel ( ' n ' );

ylabel ('Magnitude of Real (xc-even)');

subplot(2,1,2) ; stem( n , imag(xc_even) ) %Plot the imaginary part of even%signal xc_even in the 2nd half of %Figure 2.

title('Imaginary part of even signal xc(n)')

xlabel ( ' n ' );ylabel ('Magnitude of Imag (xc-even)');

figure; %Open a new figure window (Fig3)

subplot(2,1,1) ; stem( n , real(xc_odd)) %Plot the real part of odd signal%xc_odd in the 1st half of Fig 3.

title('Real part of odd signal xc(n)')

xlabel ( ' n ' );

ylabel ('Magnitude of Real (xc-odd)');subplot(2,1,2);stem(n,imag(xc_odd)) %Plot the imaginary part of odd signal

% xc_odd in the 2nd half of Fig 3.

title('Imaginary part of odd signal xc(n)')xlabel ( ' n ' );

ylabel ('Magnitude of Imag (xc-odd)');


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5.Convolution between signals

MATLAB Code Explanation

x = input (' type the samples of input signal x(n) '); %Enter the values of Input Signal x(n)

nx1 = input( ' type the time origin of the first sample of x(n) ' );%Enter the time origin of the firstsample of x(n)

NX = length(x); % Find the length of the signal x(n)

nx2 = nx1 + NX -1; %Find the upper limit of X-axis range

nx = nx1 : nx2; %Set X-axis range for the graph of x(n)figure;

stem(nx,x) % Open a new figure window (Figure 1) and plot a stem graph of x(n)

title (' Input Signal x(n) '); %Add an appropriate title to the graphxlabel ('Time Index n');

ylabel ( ' x(n) ' ); %Label the X and Y axes

h = input (' type the samples of Impulse response signal h(n) '); %Enter the values of Impulseresponse Signal h(n)

nh1 = input( ' type the time origin of the first sample of h(n) ' ); %Enter the time origin of the

first sample of h(n)NH = length(h); %Find the length of the signal h(n)


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nh2 = nh1 + NH -1; %Find the upper limit of X-axis range

nh = nh1 : nh2; %Set X-axis range for the graph of h(n)

figure; stem(nh,h) %Open a new figure window (Figure 2) and plot a stem graph of h(n)title (' Impulse Response Signal h(n) ');%Add an appropriate title to the graph

xlabel ('Time Index n');

ylabel ( ' h(n) ' ); %Label the X and Y axesy = conv ( x , h ); %Compute the LTI system output y(n) using the built-in MATLAB

function 'conv'

NY = NX + NH -1; %Find the length of the signal y(n)ny1 = nx1 + nh1; %Find the time origin at which the first signal sample of y(n) is available

ny = ny1 : NY-abs(ny1)-1; %Set X-axis range for the graph of y(n)

figure; stem(ny,y) % Open a new figure window (Figure 3) and plot a stem graph of y(n)

title (' Output Signal y(n) = x(n)*h(n)'); %Add an appropriate title to the graphxlabel ('Time Index n'); ylabel ( ' y(n) ' ); %Label the X and Y axes

disp ('Time Index of x(n) = '); disp(nx); %Display on screen the message in parentheses and

the time index of x(n)

disp ('Input Signal Sequence x(n) = ');disp(x);

disp ('Time Index of h(n) = ');disp(nh);

disp ('Impulse Response Sequence h(n) = ');

disp(h);

disp ('Time Index of y(n) = ');disp(ny);

disp ('output Signal Sequence y(n)= ');

disp(y);

Output:


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7 a)MATLAB code for verification of linearity property of a discrete time system

x1=input(type the samples of x1);%enter the values of first input signal x1.x2=input(type the samples of x2); % enter the values of second input signal x2

if(length(x1)~= length(x2) )% check if the lengths of two vectors x1 and x2 are same

disp(ERROR :Lengths of x1&x2 are different); %If the sizes are different display the errormessage

return; %get out of the program, as you cannot proceed further, and go to the command promptend;

h=input(type the samples of h); % enter the samples of impulse response h of the LTI systemN=length(x1)+length(h)-1; %find the length of the o/p sequence

disp(length of the output signal will be );

disp(N);a1=input(the scale factor a1 is); %Enter the scale factor a1 of the path of x1

a2=input(the scale factor a2 is); %Enter the scale factor a2 of the path of x2

x=a1*x1+a2*x2; % find the intermediate o/p x of system 1 as a1x1 + a2 x 2

y01=conv(x,h); %find the final o/p of system 1 as the convolution of x and h.

y1=conv(x1,h); %find the 1st

intermediate o/p y1 of system 1 as the convolution of x1 andh.

y1s=a1*y1; %scale y1 with the scaling factor a1

y2=conv(x2,h); %find the 1st intermediate o/p y1 of system 2 as the convolution of x1 andh.

y2s=a2*y2; %scale y2 with the scaling factor a2


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y02=y1s+y2s; % find the final o/p of system 2 by adding the two intermediate outputs.

disp(input signal x1 is);

disp(x1);

disp(input signal x2 is);disp(x2);

disp(Output sequence y01 is);

disp(y01);disp(Output sequence y02 is);

disp(y02);

if(y01==y02)

disp(y01=y02.Hence the LTI system is LINEAR);end;

output:

type the samples of x1[2 5 3 7 -1]

type the samples of x2[-4 6 -2 3 1]

type the samples of h[1 -3 4 -2 7 2 -1 3]

length of the output signal will be

12

the scale factor a1 is3

the scale factor a2 is2input signal x1 is

2 5 3 7 -1

input signal x2 is

-4 6 -2 3 1

Output sequence y01 is-2 33 -84 124 -130 286 33 168 123 -14 82

-3

Output sequence y02 is


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-2 33 -84 124 -130 286 33 168 123 -14 82

-3

y01=y02.Hence the LTI system is LINEAR>>

7 b)MATLAB code for verification of time invariance property of a discrete time system

x=input(type the samples of x(n)); %enter the values of the input signal x(n)

h=input(type the samples of h(n)); %enter the values of the impulse response h(n) of system H

y=conv(x,h); %find the output signal y(n) as the convolution of x(n)and h(n)disp(enter a POSITIVE number for delay);

d=input(desired delay of the signal is); %enter the desired shift by which the signal x(n) needs

to be delayedxd=[zeros(1,d),x]; %generate the delayed input signal xd(n) by padding x(n) with d zeros

nxd=0:length(xd)-1; %set the x-axis range for input signals graphyd=conv(xd,h); %convolve the delayed input signal xd(n) with impulse response h(n) to produce

a new output signalyd(n)nyd=0:length(yd)-1; %set the x-axis range for output signals graph

disp(original input signal x(n) is);

disp(x);disp(delayed input signal xd(n) is);

disp(xd);

disp(original output signal y(n) is);disp(y);

disp(delayed output signal y(n) is);

disp(yd);xp=[x,zeros(1,d)]; %append d zeros to vector x so as to plot x with xd on the same graphfigure;

subplot(2,1,1);

stem(nxd,xp);grid;

xlabel(time index n);

ylabel(x(n));title(original input signal x(n));

subplot(2,1,2);

stem(nxd,xd);

grid;xlabel(time index n);

ylabel(xd(n));

title(delayed input signal xd(n));yp=[y,zeros(1,d)]; %append d zeros to vector y so as to plot y with yd on the same graph

figure

subplot(2,1,1);stem(nyd,yp);


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grid;


ylabel(y(n));title(original output signal y(n));

subplot(2,1,2);

stem(nyd,yd);grid;


ylabel(yd(n));title(delayed output signal yd(n));

type the samples of x(n)[0 1 1 1 2 2 2]

type the samples of h(n)[1 3 5 2 1]

enter a POSITIVE number for delaydesired delay of the signal is4

original input signal x(n) is

0 1 1 1 2 2 2

delayed input signal xd(n) is

0 0 0 0 0 1 1 1 2 2 2

original output signal y(n) is

0 1 4 9 12 16 21 21 16 6 2

delayed output signal y(n) is

Columns 1 through 14

0 0 0 0 0 1 4 9 12 16 21 21 16 6

Column 15

2


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

and verifying its physical realizability and stability properties

8a)MATLAB code for Unit Impulse Response of LTI System

num = input ('type the numerator vector ');%Enter the fficients of the Numerator polynomial');den = input ('type the denominator vector '); %Enter the coefficients of the Denominator

polynomial

N = input ('type the desired length of the output sequence N '); %Enter the number of outputsamples that you wish to see in a graph

n = 0 : N-1;% Set a range for X-axis

imp = [ 1 zeros(1, N-1) ]; %Generate the Unit Impulse signal of N samples

h = filter ( num, den, imp );% Obtain the output of the system, using the builtof h(n)

disp('the impulse response of lti system is');disp(h);

stem(n,h);xlabel ('time index n');

ylabel ('h(n)');

title ('Impulse Response of LTI system');

outputAll zero system:


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All pole system:


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Pole zero system :


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8b)MATLAB code for Unit Step Response of LTI System

num = input ('type the numerator vector ');%Enter the coefficients of the Numerator polynomial

den = input ('type the denominator vector ');%Enter the coefficients of the Denominator

polynomialN = input ('type the desired length of the output sequence N ');%Enter the number of output

samples that you wish to see in a graph

n = 0 : 1 : N-1; %Set a range for X-axisu = ones (1, N); %Generate the Unit Step signal of N samples

s = filter ( num, den, u );% Obtain the output s(n) of the system,using the built-in MATLABfunction 'filter'

disp('The step response of LTI system is'); disp(s); %Display the message in parentheses and thevalues of s(n)

stem(n,s) %Plot a stem graph of s(n)

xlabel ('time index n');ylabel ('s(n)');

title ('Step Response of LTI system');


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Pole-zero system:


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8c)MATLAB code for Frequency Response of LTI System

num = input ('type the numerator vector '); %Enter the coefficients of the Numerator polynomial

den = input ('type the denominator vector '); %Enter the coefficients of the Denominatorpolynomial

N = input ('number of frequency points '); %Enter the number of frequency points at which the

frequency response need to be calculatedw = 0 : pi / N : pi; %Set a range for X-axis from 0 to ?

H = freqz(num, den, w); %Calculate the frequency response H(ejw) of the transfer function H(z)

for w ranging from 0 to ?figure; %Open a new figure window (Figure 1) for plotting the real and imaginary parts of H

subplot(2 , 1 , 1 );

plot( w/pi , real(H) ); %Divide figure window into two portions and plot the real part of H in the

1st portionxlabel( ' omega \pi' );

ylabel(' Amplitude ' )

title ( 'Real part' );

subplot( 2 , 1 , 2 );plot( w/pi , imag(H) );%Plot the imaginary part of H in the 2nd portion of figure window

xlabel( ' \omega / \pi' );ylabel(' Amplitude ' )

title ( 'Imaginary part' );

figure;% Open a new figure window (Figure 2) for plotting the magnitude and phase of H

subplot( 2 , 1 , 1 );plot( w/pi , abs(H) );% plot the magnitude part of H in the 1st portion

xlabel(' omega \pi');

ylabel(' Magnitude')title ('Magnitude Spectrum');

subplot( 2 , 1 , 2 );

plot( w/pi , angle(H) ); %Plot the phase part of H in the 2nd portion of figure windowxlabel(' omega \pi');

ylabel(' Phase(radians) ');

title ('Phase ');

output :


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8d)MATLAB code for finding the Stability of LTI System

num=input(type the numerator);


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den = input (' type the denominator vector ');

[z,p,k] = tf2zp(num,den); %Obtain the location of zeros, poles and the gain constant using the

built-in function 'tf2zp'disp ('Gain constant is '); disp(k); %Display the message in parentheses and the value of gain

constant k

disp (' Zeros are at '); disp(z); %Display the message in parentheses and the location of zerosdisp ('radius of Zeros ') ;

radzero =abs(z)%Display the message in parentheses and the radius of zeros

disp ('Poles are at '); disp(p); %Display the message in parentheses and the location of polesdisp ('radius of Poles ') ; radpole = abs(p) %Display the message in parentheses and the radius of

poles

if max(radpole) >= 1 %If at least one pole is outside the unit circle, do the following

disp (' ALL the POLES do not lie within the Unit Circle '); %Display the message in parenthesesdisp (' Oooooops..The given LTI system is NOT a stable system '); %Display the message in

parentheses

else %If all the poles lie within the unit circle, do the following

disp (' ALL the POLES lie WITHIN the Unit Circle ');disp (' The given LTI system is a REALIZABLE and STABLE system ');

end; %End of IF statementzplane(num,den) %Plot the pole zero map of the given system in Z-plane, using the built-in

function 'zplane'

title ( ' Pole-Zero Map of the LTI system ' );

Output:


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9.Gibbs phenomenon

Matlab code for observing gibbs phenomenon


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clc; clear all; %Clear C ommand Window and all the previously defined

variables

N = input( 'type the total number of harmonics' ); %Input the number of sine wave%harmonics that you wish to add to generate the synthesized signal

t = 0 : 0.001 : 1; %Generate a vector of 1001 samples for t with values between 0 & 1 andwith %an increment of 0.001y = square( 2 * pi * t ); %Generate the square wave signal of period T=1, using the built-in

%function 'square'

plot( t , y , 'r' , 'linewidth' , 2 ) %Plot the original square wave signal y(t) with a red line and%thickness of 2 units, for comparison with the synthesized square wave signal

axis( [ 0 1 -1.5 1.5 ] ) Adjust the scale of X and Y axes to have a clear view of signals

hold; %Hold the original square wave plot so as to superimpose the synthesized square%wave on to it for comparison

sq = zeros( size(t) ); %Generate a vector of 1001 zeros to initialize the sq vector

for n = 1 : 2 : N %Generate odd values for n beginning with 1 and with an

%increment of 2, in each for loopsq = sq + (4 / (pi * n) * sin( 2 * pi * n * t)); %Generate the synthesized square

%wave based on equation (10.9)

end; %End of 'For' loopplot( t , sq ) %Plot the graph of synthesized square wave signal .

grid; %Include grid lines in the graph

xlabel( 't' ); ylabel( 'sq(t)' ); %Label the X and Y axestitle('Synthesized Square Wave Using Fourier Series'); %Add an appropriate title to the

graph

output:

type the total number of harmonics5

Current plot held

type the total number of harmonics50


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Current plot held

10. Finding the fourier transform of a given signal and plotting magnitude and

phase spectrum

Matlab code for computing the DFT

clc; clear all; %Clear Command Window and all the previously defined variables

f = 100; %Fix the frequency of the Cosine wave at 100 Hz.

Fs = 1000; % Fix the Sampling frequency at 1 KHz, following the Nyquist criterion.Ts = 1/Fs; % Find the sampling interval

N = 1024; % Let the length of the DFT sequence N be 1024.

n = [0 : N-1]* Ts; %Set the scale of X-axis for plotting the example signal.x = 0.8 * cos( 2 * pi * f * n); % Generate the Cosine wave of frequency 100 Hz

figure; % Open a new figure window (Fig. 1) to plot the Cosine signal x(n).

plot( n , x ); grid; % Plot x(n) and include grid.

axis( [ 0 0.05 -1 1 ] ); % Adjust the scale of X and Y axes to have a clear view of the signaltitle(' Cosine Signal of Frequency f'); % Add an appropriate title to the graph

xlabel('time n (sec.)'); ylabel('x(n)'); ````````% Label the X and Y axes

Xk = fft( x , N ); % Compute the N-point DFT using the built-in function 'fft'k = 0 : N-1; % Set a range for the frequency index variable k

figure; % Open a new figure window 2 to plot the magnitude and phase spectra of x(n)

Xmag = abs(Xk); %Compute the magnitude of X(k)subplot(2,1,1); % Divide the Fig. 2 graphic space into two portions

plot(k,Xmag); , plot the magnitude of X(k) in the 1st portion.

title(' Magnitude of Fourier Transform'); %Add an appropriate title to the graph

xlabel('frequency index k');

ylabel('Magnitude'); %Label the X and Y axessubplot(2,1,2);

plot(k,angle(Xk)); %Plot the phase angle of X(k) in the 2nd portion of Fig.2.grid;

title(' Phase of Fourier Transform'); %Add an appropriate title to the graph

xlabel('frequency index k');ylabel('Phase');


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12. Locating zeros and poles, and plotting the pole-zero maps in S-plane and

Z-plane

12a) MATLAB Code for drawing the pole zero map in S-domainclc; clear all;

num = input ( 'type the numerator polynomial vector ' );

%Enter the coefficients of numerator polynomial N(s) in square bracketsden = input( ' type the denominator polynomial vector ' );

%Enter the coefficients of denominator polynomial D(s) in square brackets

H = tf( num , den ) %Find the transfer function H(s)[ p , z ] = pzmap( H ); %Find the locations of poles and zeros. p and z will be column

vectors.

disp (' zeros are at ');disp( z ) %Display the message in parentheses and the location of zeros

disp ('poles are at '); disp( p ) %Display the message in parentheses and the location of poles

figure; %Open a new figure window (Fig.1) for plotting the pole zero map

pzmap( H ) %Plot the pole zero map

[ r, p, k ] = residue( num , den );% Find the PFE coefficients (residues) r,%pole locations p and the gain constant k of H(s)

disp ('PFE coefficients '); disp( r ); %Display the message in parentheses and%the residues r

disp ('Gain constant is '); disp( k );% Display the message in parentheses and

%the gain constant kif max(real(p)) >= 1 %If the real part of any of the poles is greater than or equal to 1,

do the%following

disp (' All poles DO NOT LIE in the Left Half of S-Plane ');%Display the message in parentheses

disp (' Oooooops..The given LTI system is NOT a stable system ');

Else %If the real part of any of the poles is less than 1, do the followingdisp (' ALL the POLES lie in the Left Half of S-Plane ');

disp (' The given LTI system is a STABLE system ');end; %End of the IF statement

figure; %Open a new figure window (Fig.2) for plotting the impulse response h(t)

t = 0 : 0.1 : 5; %Set a time range for X-axish = impulse( H , t ); %Find the impulse response h(t), using the built-in function

'impulse'

plot( t , h ) %Plot the impulse response h(t) versus time t

xlabel('t'); ylabel('h(t)'); %Label the X and Y axestitle ( ' Impulse Response of the LTI system ' );

output:

simple distinct poles:type the numerator polynomial vector [1 -2 1]

type the denominator polynomial vector [1 6 11 6]

Transfer function:s^2 - 2 s + 1


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----------------------

s^3 + 6 s^2 + 11 s + 6

zeros are at

1

1poles are at

-3.0000

-2.0000-1.0000

PFE coefficients

8.0000

-9.00002.0000

Gain constant is

ALL the POLES lie in the Left Half of S-Plane

The given LTI system is a STABLE system


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Complex conjugate poles:

type the numerator polynomial vector [0 1 2 1]


Transfer function:s^2 + 2 s + 1

---------------------s^3 + 2 s^2 + 4 s + 8

zeros are at

-1

-1poles are at

-2.0000

0.0000 + 2.0000i0.0000 - 2.0000i

PFE coefficients0.12500.4375 - 0.0625i

0.4375 + 0.0625i

Gain constant isAll poles DO NOT LIE in the Left Half of S-Plane

Oooooops..The given LTI system is NOT a stable system

Multiple Order Poles:


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type the numerator polynomial vector [1]


Transfer function:1

-----------

s^3 + 2 s^2

zeros are at

poles are at0

0

-2

PFE coefficients0.2500

-0.2500

0.5000

Gain constant isAll poles DO NOT LIE in the Left Half of S-Plane



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12b) MATLAB Code for drawing the pole zero map in Z-domain

clc; clear all; %Clear Command Window and all the previously defined variablesnum = input (' type the numerator vector '); %Enter the coefficients of numerator

%polynomial N(z) in square brackets

den = input (' type the denominator vector ');%Enter the coefficients of denominator polynomial D(z) in square

brackets

H = filt(num , den) %Find the transfer function H(z) using the%built-in function 'filt'

z = zero(H); %Find the locations of zeros. z will be a column vector.

disp (' zeros are at '); disp(z) %Display the message in parentheses and the

%location of zerosdisp ('radius of Zeros ') ; radzero = abs(z)

%Display the message in parentheses and the radius of zeros

[r,p,k] = residuez(num,den); %Find the PFE coefficients (residues) r, pole%locations p and the gain constant k of H(z)

disp ('poles are at '); disp(p) %Display the message in parentheses and the

%location of polesdisp ('radius of poles ') ; radpole = abs(p)

%Display the message in parentheses and the radius of poles

disp ('PFE coefficients '); disp(r); %Display the message in parentheses and %residues r

disp ('Gain constant is '); disp(k); %Display the message in parentheses and

%gain constant k

figure; %Open a new figure window (Fig.1) for plotting the pole zero map


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zplane(num,den) %Plot the pole zero map in z-plane

title ( ' Pole-Zero Map of the LTI system in Z-Plane' );

if max(radpole) >= 1 %If the magnitude of any of the poles is

%greater than or equal to 1, do the following

disp (' ALL the POLES do not lie within the Unit Circle ');

disp (' Oooooops..The given LTI system is NOT a stable system ');

else %If the magnitude of any of the poles is less than 1, do the following

disp (' ALL the POLES lie WITHIN the Unit Circle ');

disp (' The given LTI system is a REALIZABLE and STABLE system ');

end; %End of the IF statement

figure; %Open a new figure window (Fig.2) for plotting the impulse response h(n)

impz(num,den) %Plot the time domain impulse response h(n) of the LTI system H(z)

output:simple and distinct poles:type the numerator vector [1 -1 1]

type the denominator vector [1 1 0.16]Transfer function:

1 - z^-1 + z^-2

--------------------1 + z^-1 + 0.16 z^-2

Sampling time: unspecified

zeros are at0.5000 + 0.8660i0.5000 - 0.8660i

radius of Zeros

radzero =

1.0000

1.0000poles are at

-0.8000

-0.2000

radius of polesradpole =

0.80000.2000

PFE coefficients

5.0833-10.3333


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Gain constant is

6.2500

ALL the POLES lie WITHIN the Unit CircleThe given LTI system is a REALIZABLE and STABLE system


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Complex and conjugate poles:

type the numerator vector [1 1]type the denominator vector [1 -1 0.5]

Transfer function:1 + z^-1

-------------------

1 - z^-1 + 0.5 z^-2Sampling time: unspecified

zeros are at

0

-1radius of Zeros

radzero =

0

1poles are at

0.5000 + 0.5000i0.5000 - 0.5000i

radius of poles

radpole =

0.7071

0.7071PFE coefficients

0.5000 - 1.5000i

0.5000 + 1.5000iGain constant is

ALL the POLES lie WITHIN the Unit Circle

The given LTI system is a REALIZABLE and STABLE system


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Multiple order poles:

type the numerator vector [1]

type the denominator vector [1 0.5 -1.25 0.375]Transfer function:

1

-------------------------------------1 + 0.5 z^-1 - 1.25 z^-2 + 0.375 z^-3

Sampling time: unspecified

zeros are at0

0

0

radius of Zerosradzero =

0

00

poles are at-1.5000

0.5000 + 0.0000i

0.5000 - 0.0000i

radius of polesradpole =

1.50000.5000

0.5000

PFE coefficients0.5625

0.1875 + 0.0000i

0.2500 - 0.0000iGain constant is

ALL the POLES do not lie within the Unit Circle



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13.Generation of Gaussian noise(real and complex), computation its mean,

mean square values, Skew, Kurtosis, PSD, Probability Distribution function

clc; clear all; % Clear Command Window and all the previously defined variables


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x1 = randn(1,5000); %Generate first set of 5000 samples of Gaussian distributed random

numbers

x2 = randn(1,5000); %Generate second set of 5000 samples of Gaussian distributed randomnumbers

figure; plot( x1 , x2 , ' . ' ) %Plot the joint distribution of both the sets using 'dots' so as to get a

scatter plot in a new figure windowtitle('Scatter Plot of Gaussian Distributed Random Numbers'); %Add an appropriate title to

thegraph

x1 = rand(1,5000);% Generate first set of 5000 samples of uniformly distributed randomnumbersx2 = rand(1,5000); %Generate second set of 5000samples of uniformly distributed random

numbersfigure; plot( x1 , x2 , ' . ' ) Plot the joint distribution of boththe sets using 'dots' so as to

get a scatter plot in a new figure window

title('Scatter Plot of Uniform Distributed Random Numbers');%Add an appropriate title to thegraph

x3 = rand(1,100000);% Generate one lakh samples ofuniformly distributed random numbers

figure; subplot(2,1,1); hist(x3)% Plot a histogram graph of x3 in the1st portion of a new figure

windowtitle('Uniform Distribution');%Add an appropriate title to the graph

y = randn(1,100000);% Generate one lakh samples of Gaussian distributed randomnumbers TextBook : Basic Simulation Lab with MATLAB

subplot(2,1,2); hist(y)% Plot a histogram graph of y in the2nd portion of the figure window

title('Gaussian Distribution');%Add an appropriate title to thegraph

ymu = mean(y) %Find the mean value of yymsq = sum(y .^ 2 ) / length(y)% Find the mean square value of y

ysigma = std(y) %Find the standard deviation of y

yvar = var(y)% Find the variance value of yyskew = skewness(y) %Find the skew value of y

ykurt = kurtosis(y) %Find the kurtosis value of y

output:

ymu =

0.0038

ymsq =

1.0043

ysigma =

1.0022

yvar =

1.0043

yskew =

-0.0178

ykurt =

3.0212


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14. Matlab code for verification of Sampling Theoremclc;clear allt=-5:0.0001:5;


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F1=3;F2=23;x=cos(2*pi*F1*t)+cos(2*pi*F2*t);figure(1);plot(t,x);axis([-0.4 0.4 -2 2]);

xlabel('timet(sec)');ylabel('x(t)');title('continous signal:x(t)=cos(2piF1t)+cos(2piF2t)');%case 1Fs1=1.4*F2;ts1=1/Fs1;n1=-0.4:ts1:0.4;xs1=cos(2*pi*F1*n1)+cos(2*pi*F2*n1);figure(2);stem(n1,xs1);hold on;plot(t,x,'r');

axis([-0.4 0.4 -2 2]);hold off;xlabel('time sample(n)');ylabel('amplitude');title('discrete time signal');legend('Fs


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xlabel('time sample(n)');ylabel('amplitude');title('discrete time signal');legend('Fs>2Fmax');


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17.Impulse response of a raised cosine filterMatlab code for plotting Impulse Response of RCFclc;clear all;

t=linspace(-5,5,1000);beta1=0.2;beta2=0.4;beta3=0.6;

Ts=1;h=(sin(pi*t/Ts))./(pi*t/Ts);h1=(cos(pi*beta1*t/Ts))./(1-(2*beta1*t/Ts).^2);h2=(cos(pi*beta2*t/Ts))./(1-(2*beta2*t/Ts).^2);h3=(cos(pi*beta3*t/Ts))./(1-(2*beta3*t/Ts).^2);ht=h.*h1;plot(t,ht);hold on;

ht=h.*h2;plot(t,ht,'r');ht=h.*h3;plot(t,ht,'g');xlabel('timet(sec)');ylabel('h(t)');grid;legend('beta-->0.2','0.4','0.6');hold off;

Matlab code for Cross Correlation of x(n) and Delayed x(n)


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x= [1 2 3 -2 -1];% Create a sample signal vector x(n)D = input( ' type the delay D ' );%Enter the desired amount of delay Dxd = [ zeros(1,D) x ]; %Create a delayed signal xd(n) with a delay D[ r , lag ] = xcorr( x , xd ); %Compute the cross correlation of x(n)and xd(n) usingthe built-in function'xcorr'stem( lag , r ) %Plot a stem graph of cross correlated output versus the lag

title( ' Cross Correlation of x(n) andDelayed x(n) ' );%Add appropriate title to thegraphxlabel( ' Lag Index l ' );ylabel( ' Rxy( l ) ' );%Label the X and Y axesoutput:type the delay D 5

Matlab code for Time Delay Estimation using CrossCorrelation

x= [1 2 3 -2 -1];% Create a sample signal vector x(n)

D = input('type the round tripdelay D ');%Enter the desired amount of delay Dalpha= input('type the attenuation factor ');%Enter a value (


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if obj == 1 %If the object is present run the next two statements to attenuate the delayed signal

and add noise to it

xa= alpha * xd; %Attenuate the strength of xd(n) by timesto account for loss of amplitudey = xa + w; %Corrupt the attenuated signal xa(n) byadding random noise

else %If the object is not present, then thereceived signal will have noise only. Runthe next

statement.y = w; %Equate the received signal to noise

end; %End of IF statement.figure; Open a new figure window (Fig. 1)

xp = [x zeros(1,D)]; %Append 'D' zeros to x(n) so as to plot it onthe same scale as that of y(n)subplot(2,1,1); stem(nxd,xp); %Divide Fig.1 window into two parts, plot astem graph of x(n) in

the 1st part

xlabel ('n '); ylabel ('x(n)'); %Label the X and Y axes

title(' Transmitted Signal x(n)'); %Add appropriate title to the graphNY = length(y); %Find the length of the received signal

ny = 0 : NY-1; %Set a range for X-axis of the plot of y(n)

subplot(2,1,2); stem(ny,y); %Plot a stem graph of y(n) in the 2nd part ofFig. 1 window

xlabel ('n '); ylabel ('y(n)'); %Label the X and Y axestitle(' Received Signal y(n)');%Add appropriate title to the graph

[r,lag] = xcorr(x,y); %Compute the cross correlation of x(n) and y(n) using the built-in function'xcorr'

figure; %Open a new figure window (Fig. 2)

stem(lag,r); %Plot a stem graph of cross correlatedoutput versus the lag

xlabel ('Lag '); ylabel ('Rxy(l)');% Label the X and Y axestitle ('Time Delay Estimation Using Cross Correlation ');%Add appropriate title to the graph

output:

type the round tripdelay D 4type the attenuation factor 0.5

type 1 for presence of object or a 0 for absence 1


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clc; clear all; %Clear Command Window and all the previously defined variables

M = 256; %Set an upper limit for X-axis

n = 0 : M-1; %Set the range for X-axis from 0 to 255

x = cos(16*pi*n/M) + sin(32*pi*n/M); % Generate the periodic signal x(n)snr = input('Type the desired SNR '); %Enter a value for the desired SNR


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px = var(x) %Find the power of x(n)

w = sqrt(12) * (rand( 1 , M )-0.5); %Generate the zero mean unity power white random noise

signalan = sqrt(px * (10 ^ ((-1*snr)/10))) %Find the scale factor by which to scale w(n) so as to get

the desired SNR

w = w .* an; %Scale w(n) by the scaling factor 'an'pn = var(w) %Find the power of noise

SNRdb = 10 * log10(px/pn) %Find the actual SNR

y = x + w; %Generate noisy signal y(n) by corrupting x(n) with w(n)N = M /8; %Find the common period of x(n)

L = floor( M / N ); %Find L, the number of complete cycles of x(n) in y(n)

d = zeros( 1 , M );% Generate a row vector 'd' of M zeros for storing the impulse train

for i = 1 : M %Begin a 'for' loopif rem(i-1,N) == 0 %If the remainder of (i-1) / N is zero i.e.at i = 0, 32, 64 etc. do the following

d(i) = 1; %Generate a unit impulse at that value of i in the vector

end; %End of 'IF' loop

end; %End of 'FOR' loopCyd = ifft( fft(y,M).*fft(d,M)) / M; %Find the circular cross correlation of the impulse train y(n)

and d(n) using 'fft' and 'ifft' commandsr = Cyd * (M/L); %Multiply the result by the scale factor M/L as per (17.15)

figure(1); %Open a new figure window (Fig.1) for plotting x(n)

plot(n,x,'b'); axis([1 80 -3 3]) %Plot x(n) in Fig.1 in blue color and adjust the scale of X and Y

axes for a clear view of few cycles of x(n)xlabel( 'n' ); ylabel( 'x(n)' ); %Label the X and Y axes

title('Periodic Signal x(n)') %Add an appropriate title to the graph

figure(2); %%Open a new figure window (Fig.2)subplot(2,1,1); plot(n,y,'r'); % Divide Fig.2 into two parts, plot the noisy signal y(n) in red color

in the 1stpart.

grid; axis([1 96 -3 3]); %Include grid and adjust the scale of X and Y axes for a clear view offew cycles of y(n)

xlabel( 'n' ); ylabel( 'y(n)' ); %Label the X and Y axes

title('Noisy Signal y(n)') %%Add an appropriate title to the graphsubplot(2,1,2);stem(n,d);% Plot a stem graph of Impulse train d(n) in the 2ndpart of Fig.2.

grid;axis([1 96 -0.5 1.5]); %Include grid and adjust the scale of X and Y axes for a clear view of

few cycles of d(n)

xlabel( 'n' ); ylabel( 'd(n)' ); %Label the X and Y axestitle('Impulse Train d(n)') %Add an appropriate title to the graph

figure(3); %Open a new figure window (Fig.3) for plotting the extracted periodic signal r(n)

plot(n,r,'k'); axis([1 80 -3 3])% Plot r(n) in black color and adjust the scale of axes for a clearview of few cycles of r(n)

xlabel( 'n' ); ylabel( 'r(n)' ); %Label the X and Y axes

title('Extracted Periodic Signal r(n)') %Add an appropriate title to the graphfigure(4); %Open a new figure window (Fig.4) for plotting r(n) in comparison to x(n)

plot(n,x,'b'); hold on;

axis([1 80 -3 3]) %Plot x(n) in blue color, hold the graph so as to superimpose the plot of r(n) on

the same graph and adjust the scale of axes


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plot(n,r,'r:'); hold off;

axis([1 80 -3 3]) %Plot r(n) using red dotted line, release the graph and adjust the scale of axes

legend('x(n)','r(n)') %Add an appropriate legend to the graphoutput:

Type the desired SNR 5

px =

1.0039

an =

0.5634

pn =

0.3095

SNRdb =

5.1105


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clc; clear all; %Clear Command Window and all the previously defined variablesN = 100; %Fix the length of the signals as 100

n = 0:N-1; %Set a range for time on the X-axis

dsnr = input('type desired SNR in dB'); %Enter the desired Signal to Noise Ratio in decibels (dB)x = sqrt(2) * sin((pi/5)*n); %Generate an example sine wave signal of period 10 and unity power

figure(1); stem(n,x);grid% Open a new figure window(Fig.1), Plot a stem graph of x(n) and

includegrid

axis([ 0 50 -1.5 1.5]) %Adjust the scale of X and Y axes for a clear view of few cycles of

x(n)

xlabel( ' n ' ); ylabel( 'x(n)' ); %Label the X and Y axestitle('Sinusoidal Signal x(n)'); %Add an appropriate title to the graph

px = var(x) %Find the power px of x(n) and display it on screen

an = sqrt(px * (10^(-1 * dsnr/10)));% Find the scale factor by which to scale w(n) so as to get thedesired SNR

w = sqrt(12) * (rand(1, N )-0.5); %Generate 100 samples of zero mean unit y power random

signal w(n)

w = w * an;% Scale w(n) by the scaling factor vnpn = var(w) %Find the power of noise signal w(n) and display it on screen

disp('The calculated SNR ');% Display a message on screen

SNRdb = 10*log10(px / pn)% Find the SNR in dB and display on screenfigure(3); stem(n,w);

grid; %Open a new figure window(Fig.3),

%Plot a stem graph of w(n) and include grid


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axis([0 50 min(w) max(w)])% Adjust the scale of X and Y axes for a clear view of few cycles

of w(n)

xlabel('n'); ylabel('w(n)'); %Label the X and Y axestitle('Random Noise Signal w(n)');% Add an appropriate title to the graph

y = x + w; %Corrupt x(n) by adding noise w(n)

figure(6); %Open a new figure window (Fig.6)subplot(2,1,1);

stem(n,y);

grid ;%Divide the Fig.6 into two parts, plot y(n) as a stem graph in the 1st part, include grid

axis( [ 0 50 min(y) max(y) ] ) %Adjust the scale of X and Y axes for a

%clear view of few cycles of y(n)

xlabel( ' n ' );ylabel( 'y(n)= x(n) + w(n)' );%Label the X and Y axes

title('Sinusoidal Signal Corrupted with Random Noise')%Add an appropriate title to the graph

[ryy,lag] = xcorr(y,y,'unbiased'); %Find the normalized autocorrelation of y(n)

subplot(2,1,2);stem(lag,ryy);

grid; %plot Ryy(l) as a stem graph in the 2nd part, include gridaxis([0 50 -1.5 1.5]) %Adjust the scale of X and Y axes for a clear view of few cycles of y(n)

xlabel( 'Lag Index l' ); ylabel( 'R_y_y(l)' );%Label the X and Y axes

title('Autocorrelation Signal R_y_y(l)') %Add an appropriate title to the graph

[rxx,lag] = xcorr(x,x); %Find the autocorrelation Rxx(l) of x(n)figure(2);

stem(lag,rxx);

grid;% Open a new figure window(Fig.2), Plot a stem graph of Rxx(l) and include gridaxis([-20 20 min(rxx) max(rxx)]) %Adjust the scale of X and Y axes for a clear view of few

cycles of Rxx(l)

xlabel('Lag Index l');ylabel('R_x_x(l)');%Label the X and Y axes

title('Autocorrelation Signal R_x_x(l)') %Add an appropriate title to the graph

[rxw,lag] = xcorr(x,w); %Find the cross correlation Rxw(l) between x(n) and w(n)figure(5);

stem(lag,rxw);

grid;% Open a new figure window(Fig.5), Plot a stem graph of Rxw(l) and include grid

axis([-20 20 min(rxw) max(rxw)]) %Adjust the scale of X and Y axes for a clear view of fewcycles of Rxw(l)

xlabel('Lag Index l');

ylabel('R_x_w(l)');%Label the X and Y axestitle('Cross Correlation Between x(n) and w(n) ')%Add an appropriate title to the graph

[rww,lag] = xcorr(w,w); %Find the autocorrelation Rww(l) of w(n)

figure(4);stem(lag,rww);grid; %Open a new figure window(Fig.4), %Plot a stem graph of Rww(l) and include grid

axis([-20 20 min(rww) max(rww)]) %Adjust the scale of X and Y axes for a clear view of few

cycles of Rww(l)

xlabel('Lag Index l');


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ylabel('R_w_w(l)');%Label the X and Y axes

title('Autocorrelation Signal R_w_w(l)') %Add an appropriate title to the graph

bsl lab manual

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