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SIGNAL PROCESSING LAB M Tech (PTPG), SSP, I SEM

1. GENERATIONOF BASIC SIGNALS

( unit impulse, exponential, unit step, ramp )

AIM : To generate the Basic Signals like unit impulse, step, ramp, and exponential.

SOFTWARE REQUIRED : MAT LAB 7.0

PROGRAM DESCRIPTION : In this program basic signals like unit impulse, unit step,

ramp, and exponential signals are to be generated based on the following definitions.

1. Unit impulse signal: This signal is having unit amplitude only at origin and else

where the amplitude zero. It is denoted by (t).

2. Unit step signal: This signal is having unit amplitude for all the positive values of

time axis and else where the amplitude zero. It is denoted by u (t). The sequence is

denoted by u[n].

3. Ramp signal: This signal is having amplitude values linearly increasing same as

the time axis values from origin to all the positive values and else where theamplitude zero. It is denoted by r(t). The sequence is denoted by r[n].

4. Exponential signal: This signal is having the amplitudes exponentially increasing

with time. This is a plot ofeat.Where e denotes exponential function, a is aninteger and t is the time scale.

PROGRAM :

%Generation of various signals: impulse, exponential, step, ramp.

clc;

clear all;

clear all;

%impulse sequence

t=-2:1:2;

y1=[zeros(1,2) 1 zeros(1,2)];

subplot (2,2,1);

stem(t,y1);

grid

title ('Impulse Response');

xlabel ('no. of samples :n-->');

ylabel ('--> Amplitude');

%Exponential Sequence

n=input('enter the length of Exponential Sequence');

t=0:1:n;

a=input('Enter "a" value');

y2=exp(a*t);

subplot(2,2,2);

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stem(t,y2);

grid

title ('Exponential Sequence');



%Step Sequence

s=input ('enter the length of step sequence');

t=-s:1:s;

y3=[zeros(1,s) ones(1,1) ones(1,s)];

subplot(2,2,3);

stem(t,y3);

grid

title ('Step Sequence');



%Ramp Sequence

x=input ('enter the length of Ramp sequence');

t=0:x;subplot(2,2,4);

stem(t,t);

grid

title ('Ramp Response');



OUPUTS:

enter the length of Exponential Sequence: 5

Enter "a" value :2

enter the length of step sequence: 5

enter the length of Ramp sequence: 6

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GRAPHS :

-2 -1 0 1 20

0.5

1Impulse Response

no. of samples :n-->

-->

Amplitude

0 2 4 60

1

2

3x 10

4Exponential Sequence


-->

Amplitude

-5 0 50

0.5

1Step Sequence


-->

Amplitud

e

0 2 4 60

2

4

6Ramp Response


-->

Amplitud

e

INFERENCE : In the generation of the basic signals we have defined the signals alongtime and amplitude axes for unit impulse, step, ramp and exponential signals and plotted

the graphs. For different values the graphs are plotted.

RESULT : The basic signals like unit impulse , unit step, ramp and exponential signals

are generated and the waveforms are plotted.

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2. GENERATION OF BASIC SIGNALS

( Sinusoidal signal )

AIM : To Generate sinusoidal signal of frequency f Hz and phase phi


PROGRAM DESCRIPTION : A sinusoidal signal is a continuous signal whoseamplitude varies between two fixed values on the positive and negative amplitude axes.

This is a plotted as a function of frequency, time and phase for different values. This signal

is represented as y(t)=A sin ( 2ft+ ). Where A is amplitude, f is the frequency, t is time

and is phase.

PROGRAM :

% Generation of sinusoidal signal of f Hz and phase phi

clc;

clear all;

close all;

A=input('Enter the amplitude of the sinusoidal signal= ');

f = input('Enter the frequency of the sinusoid sequence f = ');

phi = input('Enter the phase of the sinusoid sequence phi = ');

fs = input('Enter the sampling frequency fs = ');

T = input('Enter the duration of the sequence T = ');

dt = 1/fs;

t = 0:dt:T;

y = A*sin(2*pi*f*t + phi);

plot(t,y);

title('Sinusoidal signal','fontsize',12);

xlabel('t -->');

ylabel('Amplitude -->');

OUTPUTS:Enter the amplitude of the sinusoidal signal= 3

Enter the frequency of the sinusoid sequence f = 4

Enter the phase of the sinusoid sequence phi = 10

Enter the sampling frequency fs = 1000

Enter the duration of the sequence T = 2

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GRAPHS :

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3

-2

-1

0

1

2

3Sinusoidal signal

t -->

Amplitude-->

INFERENCE : The frequency f indicates the no. of cycles in one second i.e., in this

program 4 cycles in one second as f = 4. Similarly duration indicates the total signal which

is to be displayed, here duration is 2, means from 0 to 2 sec. the signal is plotted.

RESULT : For a particular frequency f Hz and phase phi, the sinusoidal signal is generated

and plotted.

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3. BASIC OPERATIONS ON SIGNALS

( Shifting, Folding, Multiplication and Addition)

AIM : To perform different operations like shifting, addition, folding, multiplication and

shifting of basic signals.

SOFTWARE REQUIRED : MAT LAB 7.0PROGRAM DESCRIPTION : In this program the basic operations like shifting, addition,

folding, multiplication and shifting is operated on basic signals like impulse, step, ramp,

exponential.

PROGRAM :

% Basic operations on impulse signal: shifting operationclc;

clear all;

clear all;

%impulse sequence

t=-3:1:3;

y1=[zeros(1,3) 1 zeros(1,3)];

subplot (3,1,1);

stem(t,y1);

grid;


xlabel ('time -->');


%impulse sequence shifted right by one location

t=-3:1:3;

y2=[zeros(1,4) 1 zeros(1,2)];

subplot (3,1,2);

stem(t,y2);

grid;title ('Impulse Response shifted right by one location');

xlabel ('time-->');

ylabel ('Amplitude-->');

%impulse sequence shifted left by two location

t=-3:1:3;

y3=[zeros(1,1) 1 zeros(1,5)];

subplot (3,1,3);

stem(t,y3);

grid;

title ('Impulse Response shifted left by two locations');

xlabel ('time-->');

ylabel ('Amplitude-->');

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GRAPHS:

-3 -2 -1 0 1 2 30

0.5

1Impulse Response

time -->

-->

Am

plitude

-3 -2 -1 0 1 2 30

0.5

1Impulse Response shifted right by one location

time-->

Amplitude-->

-3 -2 -1 0 1 2 30

0.5

1Impulse Response shifted left by two locations

time-->

Amplitude-->

%Basic operations on step signal: folding operationclc;clear all;

clear all;

%Step Sequence


t=-s:1:s;

y1=[zeros(1,s) ones(1,1) ones(1,s)];

subplot(1,2,1);

stem(t,y1);

grid




%Step Sequence after folding

y2=[ones(1,s) ones(1,1) zeros(1,s)];

subplot(1,2,2);

stem(t,y2);

grid

title ('Step Sequence after folding');



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OUPUTS:enter the length of step sequence: 5

GRAPHS:

-5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Step Sequence

time -->

-->

Amplitude

-5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Step Sequence after folding

time -->

-->

Amplitude

%Basic operations on signals: folding operation on ramp signalclc;

clear all;

clear all;

%Ramp Sequence

x=input('enter the length of the ramp sequence:')

t=0:xsubplot(2,1,1)

stem(t,t)

grid

title ('Ramp Response');



%Ramp Sequence after folding

t=-x:0

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y=-t

subplot(2,1,2)

stem(t,y)

grid

title ('Ramp Response after folding');



OUTPUTS:enter the length of the ramp sequence:5

x = 5t = 0 1 2 3 4 5

t = -5 -4 -3 -2 -1 0

y = 5 4 3 2 1 0

GRAPHS:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

Ramp Response

time -->

-->

Amplitude

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 00

2

4

6Ramp Response after folding

time -->

-->

Amplitude

%Basic operations on signals: folding operation on exponential signal

clc;clear all;

clear all;

%Exponential Sequence

n=input('enter the length of Exponential Sequence:');

t=0:1:n;

a=input('Enter "a" value:');

y=exp(a*t);

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subplot(2,1,1);

stem(t,y);

grid




%Exponential Sequence after folding

subplot(2,1,2)

stem(-t,y)

grid




OUTPUTS:enter the length of Exponential Sequence: 5

Enter "a" value: 0.8

GRAPHS:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40


time -->

-->

Amplitude

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 00

20

40


time -->

-->

Amplitude

INFERENCE : In this program the shifting operation is performed n impulase signal,folding operation is performed on step, ramp, and exponential signals.

RESULT : Different operations like shifting, addition, folding, multiplication and shifting

of basic signals like impulse, step, ramp, and exponential is performed and the graphs areplotted.

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4. FINDING THE FFT OF DIFFERENT SIGNALS

AIM : To find the FFT of different signals like impulse, step, ramp and exponential.


PROGRAM DESCRIPTION : In this program using the command FFT for impulse, step,

ramp and exponential sequences the FFT is generated. In the process of finding the FFT the

length of the FFT is taken as N. The FFT consists of two parts: MAGNITUDE PLOT andPHASE PLOT. The magnitude plot is the absolute value of magnitude versus the samples

and the phase plot is the phase angle versus the samples.

PROGRAM :

% FFT of the impulse sequence : magnitude and phase responseclc;

clear all;

close all;%impulse sequence

t=-2:1:2;

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

subplot (3,1,1);

stem(t,y);

grid;

input('y=');

disp(y);




xn=y;

N=input('enter the length of the FFT sequence: ');xk=fft(xn,N);

magxk=abs(xk);

angxk=angle(xk);

k=0:N-1;

subplot(3,1,2);

stem(k,magxk);

grid;

xlabel('k');

ylabel('|x(k)|');

subplot(3,1,3);

stem(k,angxk);

disp(xk);

grid;

xlabel('k');ylabel('arg(x(k))');

OUTPUTS:

y= 0 0 1 0 0

enter the length of the FFT sequence: 10

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1.0000 0.3090 - 0.9511i -0.8090 - 0.5878i -0.8090 + 0.5878i 0.3090 + 0.9511i

1.0000 0.3090 - 0.9511i -0.8090 - 0.5878i -0.8090 + 0.5878i 0.3090 + 0.9511i

GRAPHS:

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1ImpulseResponse

time-->

-->Amplitude

0 1 2 3 4 5 6 7 8 90

0.5

1

k

|x(k)|

0 1 2 3 4 5 6 7 8 9-5

0

5

k

arg(x(k))

% FFT of the step sequence : magnitude and phase responseclc;

clear all;close all;

%Step Sequence


t=-s:1:s;

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

subplot(3,1,1);

stem(t,y);

grid

input('y=');

disp(y);




xn=y;N=input('enter the length of the FFT sequence: ');

xk=fft(xn,N);

magxk=abs(xk);

angxk=angle(xk);

k=0:N-1;

subplot(3,1,2);

stem(k,magxk);

grid

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xlabel('k');

ylabel('|x(k)|');

subplot(3,1,3);

stem(k,angxk);

disp(xk);

grid

xlabel('k');

ylabel('arg(x(k))');

OUTPUTS:enter the length of step sequence: 5

y= 0 0 0 0 0 1 1 1 1 1 1


5.0000 -1.0000 + 3.0777i 0 -1.0000 + 0.7265i 0 -1.0000 0-1.0000 - 0.7265i 0 -1.0000 - 3.0777i

GRAPHS:

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1Step Sequence

time -->

-->

Amplitude

0 1 2 3 4 5 6 7 8 90

5

k

|x

(k)|

0 1 2 3 4 5 6 7 8 9-5

0

5

k

arg(x(k))

% FFT of the Ramp sequence: magnitude and phase response

clc;

clear all;

close all;

%Ramp Sequence

s=input ('enter the length of Ramp sequence: ');

t=0:s;

y=t

subplot(3,1,1);

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stem(t,y);

grid

input('y=');

disp(y);

title ('ramp Sequence');



xn=y;

N=input('enter the legth of the FFT sequence: ');

xk=fft(xn,N);

magxk=abs(xk);

angxk=angle(xk);

k=0:N-1;

subplot(3,1,2);

stem(k,magxk);

grid

xlabel('k');

ylabel('|x(k)|');

subplot(3,1,3);

stem(k,angxk);

disp(xk);grid

xlabel('k');


OUTPUTS:enter the length of Ramp sequence: 5

y = 0 1 2 3 4 5


15.0000 -7.7361 - 7.6942i 2.5000 + 3.4410i -3.2639 - 1.8164i 2.5000 + 0.8123i

-3.0000 2.5000 - 0.8123i -3.2639 + 1.8164i 2.5000 - 3.4410i -7.7361 + 7.6942i

GRAPHS:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5rampSequence

time-->

-->Amplitud

0 1 2 3 4 5 6 7 8 90

10

20

k

|x(k)|

0 1 2 3 4 5 6 7 8 9-5

0

5

k

arg(x(k))

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% FFT of the Exponential Sequence : magnitude and phase response

clc;

clear all;

close all;

%exponential sequence

n=input('enter the length of exponential sequence: ');t=0:1:n;

a=input('enter "a" value: ');

y=exp(a*t);

input('y=')

disp(y);

subplot(3,1,1);

stem(t,y);

grid;

title('exponential response');

xlabel('time');

ylabel('amplitude');

disp(y);

xn=y;

N=input('enter the length of the FFT sequence: ');

xk=fft(xn,N);

magxk=abs(xk);

angxk=angle(xk);

k=0:N-1;

subplot(3,1,2);

stem(k,magxk);

grid;

xlabel('k');

ylabel('|x(k)|');

subplot(3,1,3);

stem(k,angxk);

grid;

disp(xk);xlabel('k');


OUTPUTS:enter the length of exponential sequence: 5

enter "a" value: 0.8

y= 1.0000 2.2255 4.9530 11.0232 24.5325 54.5982


98.3324 -73.5207 -30.9223i 50.9418 +24.7831i -41.7941 -16.0579i38.8873 + 7.3387i -37.3613 38.8873 - 7.3387i -41.7941 +16.0579i50.9418 -24.7831i -73.5207 +30.9223i

GRAPHS :

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

50

100exponential response

time

amplitude

0 1 2 3 4 5 6 7 8 90

50

100

k

|x(k)|

0 1 2 3 4 5 6 7 8 9-5

0

5

k

arg(x(k))

INFERENCE : The FFT for impulse, step, ramp and exponential sequences is generatedusing the FFT command. The magnitude plot is the absolute value of magnitude versus the

samples and the phase plot is the phase angle versus the samples is plotted for different

signals for different values. This program is very simple and requires defining the signaland finding FFT and plotting.

RESULT : The FFT of different signals like impulse, step, ramp and exponential is found

and the magnitude and phase plots of the same is plotted.

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5. PROGRAM TO VERIFY DECIMATION AND INTERPOLATION

AIM : To verify Decimation and Interpolation of a given Sequences


PROGRAM DESCRIPTION : The sampling rate alteration that is employed to generate

a new sequence with a sampling rate higher or lower than that of a given sequence. Thus, if

x[n] is a sequence with a sampling rate of FT Hz and it is used to generate another sequencey[n] with a desired sampling rate of FT

' Hz, then the sampling rate alteration ratio is given

by FT'/ FT = R.

If R > 1, the process is called interpolation and results in a sequence with a higher

sampling rate. If R < 1, the sampling rate is decreased by a process called decimation and it

results in a sequence with a lower sampling rate.

The decimation and interpolation can be carried out by using the pre-defined commandsdecimate and interp respectively.

PROGRAMS:

% DECIMATION

clc;

clear all;

close all;

disp('Let us take a sinusoidal sequence which has to be decimated: ');

fm=input('Enter the signal frequency fm: ');

fs=input('Enter the sampling frequnecy fs: ');T=input('Enter the duration of the signal in seconds T: ');

dt=1/fs;

t=dt:dt:T;

M=length(t);

m=cos(2*pi*fm*t);

r=input('Enter the factor by which the sampling frequency has to be

reduced r: ');

md=decimate(m,r);

figure(1);

subplot(3,1,1);

plot(t,m);

grid;

xlabel('t-->');

ylabel('Amplitude-->');

title('Sinusoidal signal before sampling');

subplot(3,1,2);

stem(m);

grid;

xlabel('n-->');

ylabel('Amplitudes of m -->');

title('Sinusoidal signal after sampling before decimation');

subplot(3,1,3);

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stem(md);

grid;

title('Sinusoidal after decimation');

xlabel('n/r-->');

ylabel('Amplitude of md-->');

Output:Enter the signal frequency fm: 2Enter the sampling frequnecy fs: 100

Enter the duration of the signal in seconds T: 1

Enter the factor by which the sampling frequency has to be reduced r: 2

GRAPHS:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

t-->

Amplitude--> Sinusoidal signal beforesampling

0 10 20 30 40 50 60 70 80 90 100-1

0

1

n-->Amplitudesof

m-->

Sinusoidal signal after samplingbeforedecimation

0 5 10 15 20 25 30 35 40 45 50-1

0

1Sinusoidal after decimation

n/r-->Amplitudeofmd-->

% INTERPOLATION

clc;

clear all;

close all;

disp('Let us take a sinusoidal sequence which has to be interpolated: ');

fm=input('Enter the signal frequency fm: ');

fs=input('Enetr the sampling frequnecy fs: ');

T=input('Enter the duration of the signal in seconds T: ');

dt=1/fs;

t=dt:dt:T;

M=length(t);

m=cos(2*pi*fm*t);

r=input('Enter the factor by which the sampling frequency has to beincreased r: ');

md=interp(m,r);

figure(1);

subplot(3,1,1);

plot(t,m);

grid;

xlabel('t-->');

ylabel('Amplitude-->');

title('Sinusoidal signal before sampling');

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subplot(3,1,2);

stem(m);

grid;

xlabel('n-->');

ylabel('Amplitudes of m -->');

title('Sinusoidal signal after sampling before interpolation');

subplot(3,1,3);

stem(md);

grid;

title('Sinusoidal after interpolation');

xlabel('n x r-->');

ylabel('Amplitude of md-->');

Outputs:Enter the signal frequency fm: 2

Enetr the sampling frequnecy fs: 100

Enter the duration of the signal in seconds T: 1

Enter the factor by which the sampling frequency has to be increased r: 2

GRAPHS:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

t-->

Amplitude-->

Sinusoidal signal before sampling

0 10 20 30 40 50 60 70 80 90 100-1

0

1

n-->Amplitudesofm

-->

Sinusoidal signal after sampling before interpolation

0 20 40 60 80 100 120 140 160 180 200-2

0

2Sinusoidal after interpolation

n x r-->Amplitudeofmd-->

INFERENCE : The only constraint about the program is that the factors of decimation or

interpolation should be an integers. If we want to change the sampling frequency by afactor which is not an integer it can be done by using the command resample by which we

can change the sampling rate by a factor I / D. For this we have to interpolate by an integer

factor I and then decimate by an integer factor D

RESULT : The Decimation and Interpolation of given sequences is verified and graphs

are plotted.

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6. PROGRAM TO CONVERT CD DATA INTO DVD DATA

AIM : To convert the CD data into DVD data.


PROGRAM DESCRIPTION : In this program the CD data is converted into DVD data

by using the command resample. Here the CD data is of the sampling frequency 44.1

kHz and the DVD data is of the sampling frequency 96.0 kHz. This is also one kind ofsampling rate conversion by a factor. In this program first we generate the CD signal and

from that we generate the decimated version by downsampling it and upsampling results in

DVD data in a sampled version. Then plot the CD data and DVD data signals.

PROGRAM :

clc;

clear all;

close all;fm=input('enter the signal frequency fm:');

fs=input('enter the sampling frequency fs:');

T=input('enter the duration of the signal in seconds:');

dt=1/fs;

t=0:pi/100:pi

m=sin(2*pi*fm*t);

subplot(411);

plot(m);

xlabel('time-->');

ylabel('amplitude-->');

subplot(412);

stem(m);

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

y=resample(m,96,44);

subplot(413)

stem(y);

xlabel('time-->');


subplot(414)

plot(y);

xlabel('time-->');


OUTPUTS:

enter the signal frequency fm:2enter the sampling frequency fs:100

enter the duration of the signal in seconds:1

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GRAPHS :

0 20 40 60 80 100 120

-1

0

1

time-->amplitude-->

0 20 40 60 80 100 120-1

0

1

time-->amplitude-->

0 50 100 150 200 250-2

0

2

time-->amplitude-->

0 50 100 150 200 250-2

0

2

time-->amplitude-->

INFERENCE : This program is similar to resampling of the signal by a factor of

96000/41000. Since CD data is of the frequency 44.1 kHz and DVD data is of 96 kHz. We

have to use the resample command only to carryout this experiment. Another way of doingthis is to Interpolate the given signal by a factor of 96000 and then decimate by 44100.

RESULT : Hence the CD data is converted into DVD data and the graphs are plotted.

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7. GENERATION OF DUAL TONE MULTIPLE FREQUENCY (DTMF) SIGNALS

AIM: To generation of dual tone multiple frequency ( DTMF ) signals.

SOFTWARE REQUIRED: MAT LAB 7.0

PROGRAM DESCRIPTION: In this program the Dual tone multiple frequency signal

which is used in telephone systems is generated. A DTMF signal consists of a sum of two

tones, with frequencies taken from two mutually exclusive groups of preassignedfrequencies. Each pair of such tones represents a unique number or a symbol. Decoding of

a DTMF signal thus involves identifying the two tones in that signal and determining their

corresponding number or symbol.

PROGRAM:

clc;

clear all;

close all;

number=input('enter a phone number with no spaces:','s');

fs=8192;

T=0.5;

x=2*pi*[697 770 852 941];

y=2*pi*[1209 1336 1477 1602];

t=[0:1/fs:T]';

tx=[sin(x(1)*t),sin(x(2)*t),sin(x(3)*t),sin(x(4)*t)]/2;

ty=[sin(y(1)*t),sin(y(2)*t),sin(y(3)*t),sin(y(4)*t)]/2;

for k=1:length(number)

switch number(k)

case'1'

tone=tx(:,1)+ty(:,1);

sound(tone);

plot(tone);

case'2'

tone=tx(:,1)+ty(:,2);

sound(tone);

plot(tone);

case'3'

tone=tx(:,1)+ty(:,3);

sound(tone);

plot(tone);

case'A'

tone=tx(:,1)+ty(:,4);

sound(tone);

plot(tone);

case'4'

tone=tx(:,2)+ty(:,1);

sound(tone);

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plot(tone);

case'5'

tone=tx(:,2)+ty(:,2);

sound(tone);

plot(tone);

case'6'

tone=tx(:,2)+ty(:,3);

sound(tone);

plot(tone);

case'B'

tone=tx(:,2)+ty(:,4);

sound(tone);

plot(tone);

case'7'

tone=tx(:,3)+ty(:,1);

sound(tone);

plot(tone);

case'8'

tone=tx(:,3)+ty(:,2);

sound(tone);

plot(tone);case'9'

tone=tx(:,3)+ty(:,3);

sound(tone);

plot(tone);

case'C'

tone=tx(:,3)+ty(:,4);

sound(tone);

plot(tone);

case'#'

tone=tx(:,4)+ty(:,1);

sound(tone);

plot(tone);

case'0'

tone=tx(:,4)+ty(:,2);

sound(tone);

plot(tone);

case'*'

tone=tx(:,4)+ty(:,3);

sound(tone);

plot(tone);

case'D'

tone=tx(:,4)+ty(:,4);sound(tone);

plot(tone);

otherwise

disp('invalid number');

end;

pause(0.75);

end;

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OUTPUT:enter a phone number with no spaces: 963

GRAPHS:

0 500 1000 1500 2000 2500 3000 3500 4000 4500-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000 4500-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000 4500-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

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INFERENCE: In this program the two frequencies which are going to produce the dual

tone when we press any button, that is why it is called Dual Tone Multiple Frequency

(DTMF). The only constraint is if the number we pressed is not there in the list then it willgenerate the output as invalid number. But here the frequencies are fixed. If we want to

generate different tone we have to change the frequencies every time.

RESULT: Hence Dual Tone Multiple Frequency (DTMF) signals are generated and the

output is verified.

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8. THE AUTOCORRELATION OF THE GIVEN SEQUENCE

AIM : To find the autocorrelation of the given sequence


PROGRAM DESCRIPTION : In this program, the autocorrelation of sequence x[n]

is computed by folding the sequence with respect to real axis and then taking the

convolution of the folded sequence with the original sequence. It is generally denoted

by

rxx ( l) = x(n)x(n-l) where l =..-2,-1,0,1,2,..n = -

PROGRAM :

clc;

clear all;

close all;

a=input('Enter the first sequence: ');

subplot(211);

stem(a);

grid;

xlabel('time samples-->');

ylabel('amplitudes-->');

title('given sequence');

n1=length(a);

n2=n1-1;

n3=n1+n2;

b=[zeros(1,n2) a zeros(1,n2)];

for i=1:n3

j=n3c=[zeros(1,i-1) a zeros(1,j-i)];

d=transpose(c);

e(i)=b*d;

disp('The auto correlation is: ');e

subplot(212)

stem(e);

grid;

xlabel('time samples-->');

ylabel('amplitudes-->');

title('The auto correlation of the given sequence');

end;

OUTPUTS:Enter the first sequence: [2 3 4 5]

The auto correlation is:

e = 10 23 38 54 38 23 10

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GRAPHS :

1 1.5 2 2.5 3 3.5 40

2

4

6

time samples-->

amplitudes-->

given sequence

1 2 3 4 5 6 70

20

40

60

time samples-->

amplitude

s-->

The auto correlation of the given sequence

INFERENCE : In this program first we have to append the zeros before and after the

sequence which is equal to length of the sequence minus one. Then we can find the

autocorrelation of that sequence by shifting the sequence every time by multiplying withthe same sequence. Hence it results in the autocorrelation of the sequence.

RESULT : Hence the autocorrelation of the given sequence is calculated and plotted.

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9. POWER SPECTRAL DENSITY USING SQUARE MAGNITUDE AND

AUTOCORRELATION

AIM : To compute the Power Spectral Density of given signals using square magnitude

and autocorrelation


PROGRAM DESCRIPTION : The stationary random processes do not have finite energyand hence do not processes a Fourier Transform. Such signals have finite average power

and hence are characterized by a Power Density Spectrum or Power Spectral Density

(PSD). In this program we have used the property of PSD that it can be calculated from its

autocorrelation function by taking the time average. In the Square magnitude method aftertaking the FFT by taking the absolute value we get the PSD.

PROGRAM :

clc;

clear all;

close all;

f1=input('Enter the frequency of first sequence in Hz: ');

f2=input('Enter the frequency of the second sequence in Hz: ');

fs=input('Enter the sampling frequency in Hz: ');

t=0:1/fs:1

x=2*sin(2*pi*f1*t)+3*sin(2*pi*f2*t)+rand(size(t));

px1=abs(fft(x).^2)

px2=abs(fft(xcorr(x),length(t)));

subplot(211)plot(t*fs,10*log10(px1));%square magnitude

grid;

xlabel('Freq.in Hz-->');

ylabel('Magnitude in dB-->');

title('PSD using square magnitude method');

subplot(212)

plot(t*fs,10*log10(px2));%autocorrelation

grid;

xlabel('Freq.in Hz-->');

ylabel('Magnitude in dB-->');

title('PSD using auto correlation method');

OUTPUTS:Enter the frequency of first sequence in Hz: 200

Enter the frequency of the second sequence in Hz: 400

Enter the sampling frequency in Hz: 1000

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GRAPHS :

0 100 200 300 400 500 600 700 800 900 1000-50

0

50

100

Freq.in Hz-->

MagnitudeindB

-->

PSD using square magnitude method

0 100 200 300 400 500 600 700 800 900 100030

40

50

60

Freq.in Hz-->

MagnitudeindB-->

PSD using auto correlation method

INFERENCE : The only constraint is to take the two frequencies must be two time lesserthan the sampling frequencies.

RESULT : Hence the PSD is computed using the square magnitude and Autocorrealtionand graph is plotted.

10. POWER SPECTRAL DENSITY USING PERIODOGRAM

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AIM : To compute the Power Spectral Density of given signals using Periodoram




(PSD). The property of PSD that it can be calculated from its autocorrelation function bytaking the time average. In the Square magnitude method after taking the FFT by taking the

absolute value we get the PSD. Here we used the command periodogram to estimate the

PSD of the given signal of 200 Hz frequency.

PROGRAM :clc;

clear all;

close all;

Fs=1000;t=0:1/Fs:0.3

x=cos(2*pi*t*200)+0.1*randn(size(t));

periodogram(x,[ ],'twosided',512,Fs);

GRAPHS:

0 100 200 300 400 500 600 700 800 900-70

-60

-50

-40

-30

-20

-10

Frequency (Hz)

Power/frequency(dB/

Hz)

Periodogram Power Spectral Density Estimate

INFERENCE : In the computation of the PSD here we are using the command

periodogram so it very simple to generate the PSD of the given signal and the output is

verified and can be compared with the other PSD generation methods.

RESULT : Hence the PSD is computed using the periodogram and graph is plotted.

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11. POWER SPECTRAL DENSITY ESTIMATION USING PERIODOGRAM AND

MODIFEIED PERIODOGRAM

AIM : To compute the Power Spectral Density of given signals using Periodoram and

Modified periodogram




(PSD). The property of PSD that it can be calculated from its autocorrelation function by

taking the time average. In the Square magnitude method after taking the FFT by taking theabsolute value we get the PSD. The same can be calculated using periododram and

modified periodogram methods.

PROGRAM :

%% clearing screen

clc;

close all;

clear all;

%% psd calculation using fft

t = 0:1023;

x = sin(2*pi*50/1000*t)+sin(2*pi*120/1000*t);

y = x+randn(size(t));

N=length(y);

f1=1000*(0:N/2-1)/N;

p=abs(fft(y).^2)/N;

subplot(221);

plot(f1,10*log10(p(1:512)));

grid;

xlabel('frequency-->');

ylabel('Power in dB-->');

title(' PSD estimation using fft')

%% psd calculation using periodogram

[Pxx,w]=periodogram(y);

subplot(222);

plot(1000*w/(2*pi),10*log10(Pxx));

grid;



title('PSD using periodogram');

%% psd calculation using modified periodogramw=hanning(N);

[Pxx,w]=periodogram(y,w);

subplot(223);

plot(1000*w/(2*pi),10*log10(Pxx));

grid;



title('Modified periodogram');

%% psd calculation using auto correlation and fft

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z=xcorr(y);

m=length(z);

g1=abs(fft(z));

n2=1000*(0:(m/2)-1)/m;

subplot(224);

plot(n2,10*log10(g1(1:1023)));

grid;



title('PSD estimate by auto correlation and fft')

GRAPHS :

0 200 400 600-40

-20

0

20

40

frequency-->

PowerindB

-->

PSD est imation using fft

0 200 400 600-40

-20

0

20

frequency-->

PowerindB

-->

PSD using periodogram

0 200 400 600-60

-40

-20

0

20

frequency-->

Powerind

B-->

Modified periodogram

0 200 400 600-20

0

20

40

60

frequency-->

Powerind

B-->

PSD estimate by auto correlation and fft

INFERENCE : In the computation of the PSD here we are using the commandperiodogram and modified periodogram is estimated by using the hanning window. So it

very simple to generate the PSD of the given signal and the output is verified and it is

compared with the other PSD generation methods.

RESULT : Hence the PSD is estimated and compared with different methods.

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12. POWER SPECTRAL DENSITY ESTIMATION USING BARLETT METHOD

AIM : To estimate the Power Spectral Density using Barlett method


PROGRAM DESCRIPTION : The PSD can be calculated from the autocorrelation

function of a signal by taking its the Fourier transform, called the periodogram. In general

the variance of the estimate Pxx ( f ) does not decay to zero as N tends to infinity. Thus

The periodogram is not a consistent estimate of the true power density spectrumi.e., it does not converge to the true power density spectrum. Thus the estimated spectrum

suffers from the smoothing effects and the leakage embodied in the Barlett window. Thesmoothing and leakage ultimately limit our ability to resolve closely spaced spectra. The

methods Barlett along with Blackman Tukey and Welch are classical methods and make no

assumption about how the data were generated and hence called nonparametric methods.

In this method for reducing the variance in the periodogram involves three steps:1. First the N-point sequence is subdivided into K nonoverlapping segments, where

segment has length M.2. For each segment we compute the periodogram.

3. Finally, we average the periodograms for the K segments to obtain the Barlett

power spectrum estimate.

Thus Barlett methodis called as Averaging Periodograms.

PROGRAM:

clc;

close all;clear all;

t=0:1023

x=sin(2*pi*50/1000*t)+sin(2*pi*120/1000*t);

y=x+randn(size(t));

N=length(y);

k=4;

M=N/k;

x1=y(1:M);

x2=y(M+1:2*M);

x3=y(2*M+1:3*M);

x4=y(3*M+1:4*M);

px41=(1/M)*((abs(fft(x1))).^2);

px42=(1/M)*((abs(fft(x2))).^2);px43=(1/M)*((abs(fft(x3))).^2);

px44=(1/M)*((abs(fft(x4))).^2);

px5=(px41+px42+px43+px44)/k;

n1=1000*(0:M/2)/M;

plot(n1,px5(1:M/2+1));

grid;

xlabel('Normalized frequency in Hz -->');

ylabel('-->Power Spectrum in dB');

title('PSD ESTIMATION USING BARLETT METHOD');

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GRAPH:

0 50 100 150 200 250 300 350 400 450 5000

10

20

30

40

50

60

Normalized frequency in Hz -->

-->PowerSpectrumi

ndB

PSD ESTIMATION USING BARLETT METHOD

INFERENCE : Estimation of Power Spectral Density using Barlett method is observed

from the graph.

RESULT : Hence the PSD is estimated using Barlett method.

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13. POWER SPECTRAL DENSITY ESTIMATION USING WELCH METHOD

AIM : To estimate the Power Spectral Density using Welch method









Welch made two modifications to the Barlett method.1. First, he allowed the data sequence to overlap.

2. The second modification made by Welch to the Barlett method is to window thedata segments prior to computing the periogram.

Thus Welch methodis called asModified Periodogram.

PROGRAM:

clc;

close all;

clear all;

fs=1024;

f1=200;

f2=400;

M=128;

t=0:1/fs:1;

x=sin(2*pi*f1*t)+sin(2*pi*f2*t)+rand(size(t));

L=length(x);

K=L/M;

wi=hann(M+1);

m=0;

su=[];

for i=1:M/2:L-M+1

y=wi'.*x(i:M+i);

w2=abs(fft(y).^2);su=[su;w2];

end;

su1=sum(su);

su2=sum(wi.^2);

w1=su1/su2;

w12=10*log(w1);

fs1=(fs/K)+2;

t1=((1:fs1)/fs1);

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plot(t1,w12);

grid;

xlabel('Normalized frequency -->');


title('Welch method of Power Spectrum Estimation');

GRAPH:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10

0

10

20

30

40

50

60

Normalized frequency -->

-->PowerSpectrumi

ndB

Welch method of Power Spectrum Estimation

INFERENCE : Estimation of Power Spectral Density using Welch method is observedfrom the graph.

RESULT : Hence the PSD is estimated using Welch method.

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14. PSD ESTIMATION USING BLACKMAN AND TUKEY METHOD

AIM : To estimate the Power Spectral Density using Blackman and Tukey method.









Blackman and Tukey proposed and analyzed the method in which the sampleautocorrelation sequence is windowed first and then Fourier transformed to yield the

estimate of the power spectrum.

PROGRAM:

clc;

close all;

clear all;

fs=1024;

f1=200;

f2=400;

t=0:1/fs:1;

x=sin(2*pi*f1*t)+sin(2*pi*f2*t)+rand(size(t));

L=length(x);

y=xcorr(x);

wi=hann(length(y));

y1=wi'.*y;

i=length(y);

b=abs(fft(xcorr(y1)))/L;

i1=length(b);

t1=(1:i1)/i1;

b1=10*log(b);

plot(t1,b1);

grid;

xlabel('Normalized frequency -->');ylabel('-->Power Spectrum in dB');

title('Blackman and Tukey method of Power Spectrum Estimation');

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GRAPH:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-50

0

50

100

150

200


-->P

owerSpectrumi

ndB

Blackman and Tukey method of Power Spectrum Estimation

INFERENCE : Estimation of Power Spectral Density using Blackman and Tukey method

is observed from the graph.

RESULT : Hence the PSD is estimated using Blackman and Tukey method.

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15. POWER SPECTRUM ESTIMATION USING YULE-WALKER METHOD

AIM : To estimate the Power Spectral Density using Yule-Walker method.


PROGRAM DESCRIPTION : The Power Spectral Density can be calculated by Barlett

along with Blackman Tukey and Welch are the classical methods and make no assumption

about how the data were generated and hence called nonparametric methods. Thenonparametric power spectrum estimation methods require the availability of long data

records in order to obtain the necessary frequency resolution required in many applications.

The limitations of nonparametric methods:1. The inherent assumption that the auto correlation estimate is zero for m >= N.

2. Another inherent assumption is that in the periodogram estimate the data are

periodic with period N.

Neither one of these assumptions is realistic.

The modeling approach eliminates the need for window functions and the assumption thatthe auto correlation is zero for |m| >= N. As a consequence, Parametric ( model-based )power spectrum estimation methods avoid the problem of leakage and provide better

resolution than do the FFT based, nonparametric methods. The Yule-Walker and Burg

methods are calledParametric methods of Power Spectrum Estimation.

Yule-Walker method: The Yule-walker method estimates simply the auto correlation

from the data and use the estimates to solve for the AR model parameters.The result is astable AR model.

PROGRAM:

clc;

close all;

clear all;

a=[1 -2.2137 2.9408 -2.1697 0.9609]; % AR filter coefficients

randn('state',1);

x=filter(1,a,randn(256,1)); % AR system output

pyulear(x,4); % Fourth-order estimate



title('Yule-Walker method of Power Spectrum Estimation');

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GRAPH:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-30

-20

-10

0

10

20

30


-->P

owerSpectrumi

ndB

Yule-Walker method of Power Spectrum Estimation

INFERENCE : Estimation of Power Spectral Density using Yule-Walker method is

observed from the graph.

RESULT : Hence the PSD is estimated using Yule-Walker method.

16. POWER SPECTRUM ESTIMATION USING BURG METHOD

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AIM : To estimate the Power Spectral Density using Burg method.


PROGRAM DESCRIPTION : The Power Spectral Density can be calculated by Barlettalong with Blackman Tukey and Welch are the classical methods and make no assumption

about how the data were generated and hence called nonparametric methods. The

nonparametric power spectrum estimation methods require the availability of long datarecords in order to obtain the necessary frequency resolution required in many applications.

The limitations of nonparametric methods:

1. The inherent assumption that the auto correlation estimate is zero for m >= N.2. Another inherent assumption is that in the periodogram estimate the data are

periodic with period N.

Neither one of these assumptions is realistic.

The modeling approach eliminates the need for window functions and the assumption that

the auto correlation is zero for |m| >= N. As a consequence, Parametric ( model-based )power spectrum estimation methods avoid the problem of leakage and provide better

resolution than do the FFT based, nonparametric methods. The Yule-Walker and Burgmethods are calledParametric methods of Power Spectrum Estimation.

Burg method: The method devised by Burg for estimating the AR parameters can be

viewed as an order recursive least squares lattice method, based on the minimization of the

forward and backward errors in linear predictors, with the constraint that the ARparameters satisfy the Levinson-Durbin recursion.

PROGRAM:

clc;

close all;

clear all;

a=[1 -2.2137 2.9408 -2.1697 0.9609]; % AR filter coefficients

randn('state',1);

x=filter(1,a,randn(256,1)); % AR system output

pburg(x,4); % Fourth-order estimate



title('Burg method of Power Spectrum Estimation');

GRAPH:

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-30

-20

-10

0

10

20

30


-->PowerSpectrumi

ndB

Burg method of Power Spectrum Estimation

INFERENCE : Estimation of Power Spectral Density using Burg method is observed from

the graph.

RESULT : Hence the PSD is estimated using Burg method.

Dsp Record

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