dsp lab manual gec dahod 1

Department of

Electronics & Communication Engineering

Laboratory Manual Digital Signal Processing

B. E. SEMESTER: VII (Subject Code: 171003)

(GUJARAT TECHNOLOGICAL UNIVERSITY) Term Date: 1-7-2013 to 30-10-2013

Prepared by: Prof. S. B. Prajapati

Dept. of Electronics and Comm. GEC, Dahod | Digital Signal Processing Lab Manual 2

Government Engineering College, Dahod

Certificate

This is to certify that Mr./Ms........

of Electronics and Communication department, semester. Enrollment

No. has satisfactorily completed his/her laboratory work in the

course .......................... course code ... for the term

ending in ..2013.

Date of Submission

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INDEX

Sr. No.

Title of the Experiment Date Page Sign Marks

1 Familiarization with MATLAB software and general functions.

2

Write a MATLAB program to develop some elementary continues time (CT) signals a. Sinusoidal b. Complex waveform c. Unit Impulse d. Unit Step e. Unit Ramp f. Exponential g. Noise

3 Write a MATLAB program to find the sum of sinusoidal signals and understanding of the concept of harmonics.

4

Write a MATLAB program to develop some elementary sequences or discrete time (DT) signals a. Sinusoidal b. Complex waveform c. Unit Impulse d. Unit Step e. Unit Ramp f. Exponential g. Noise

5 Write a MATLAB program to verify the Sampling theorem.

6

Write a MATLAB program to find the impulse response of a LTI system defined by a difference equation: y(n) + 0.7y(n-1) 0.45y(n-2) 0.6y(n-3) = 0.8x(n) 0.44x(n-1) + 0.36x(n-2) + 0.2x(n-3)

7 Write a MATLAB program to plot the Frequency response of a given LTI sequence: y(n) 5 y(n1) = x(n) + 4 x(n1)

8 Write a MATLAB program to compute linear convolution of two given sequences.

9 Write a MATLAB program to find the Discrete Time Fourier Transform DTFT of the given sequence.


10

write a MATLAB program to plot magnitude response and phase response of digital FIR filter using Rectangular window

a. Low pass filter b. High pass filter c. Bandpass filter d. Bandstop filter

Hanning Window and Hamming Window Design


Experiment No: 1

FAMILIARISATION WITH MATLAB

Aim: To familiarise with MATLAB software and general functions. MATLAB: MATLAB is a software package for high performance numerical computation and visualization provides an interactive environment with hundreds of built in functions for technical computation, graphics and animation. The MATLAB name stands for MATrix Laboratory produced by Mathworks Inc., USA. At its core ,MATLAB is essentially a set (a toolbox) of routines (called m files or mex files) that sit on your computer and a window that allows you to create new variables with names (e.g. voltage and time) and process those variables with any of those routines (e.g. plot voltage against time, find the largest voltage, etc). It also allows you to put a list of your processing requests together in a file and save that combined list with a name so that you can run all of those commands in the same order at some later time. Furthermore, it allows you to run such lists of commands such that you pass in data and/or get data back out (i.e. the list of commands is like a function in most programming languages). Once you save a function, it becomes part of your toolbox (i.e. it now looks to you as if it were part of the basic toolbox that you started with). For those with computer programming backgrounds: Note that MATLAB runs as an interpretive language (like the old BASIC). That is, it does not need to be compiled. It simply reads through each line of the function, executes it, and then goes on to the next line. (In practice, a form of compilation occurs when you first run a function, so that it can run faster the next time you run it.) MATLAB Windows: MATLAB works with through three basic windows. Command Window: This is the main window. It is characterized by MATLAB command prompt >> when you launch the application program MATLAB puts you in this window all commands including those for user-written programs ,are typed in this window at the MATLAB prompt. Graphics window: the output of all graphics commands typed in the command window are flushed to the graphics or figure window, a separate gray window with white background colour the user can create as many windows as the system memory will allow. Edit window: This is where you write edit, create and save your own programs in files called M files. Input-output: MATLAB supports interactive computation taking the input from the screen and flushing, the output to the screen. In addition it can read input files and write output files. Data Type: The fundamental data type in MATLAB is the array. It encompasses several distinct data objects- integers, real numbers, matrices, character strings, structures and cells. There is no need to declare variables as real or complex, MATLAB automatically sets the variable to be real.


Dimensioning: Dimensioning is automatic in MATLAB. No dimension statements are required for vectors or arrays. We can find the dimensions of an existing matrix or a vector with the size and length commands. Where to work in MATLAB?

All programs and commands can be entered either in the

a) Command window.

b) As an M file using Matlab editor.

Note: Save all M files in the folder 'work' in the current directory. Otherwise you have to locate

the file during compiling. Typing quit in the command prompt >> quit, will close MATLAB

Development Environment.

Basic commands in Matlab

1. T = 0:1:10

This instruction indicates a vector T which as initial value 0 and final value 10 with an

increment of 1

Therefore T = [0 1 2 3 4 5 6 7 8 9 10]

2. F= 20:1:100

Therefore F = [20 21 22 23 24 100]

3. T= 0: 1/pi: 1

Therefore T= [0, 0.3183, 0.6366, 0.9549]

4. zeros (1, 3)

The above instruction creates a vector of one row and three columns whose values are zero

Output= [0 0 0]

5. zeros (2,4)

Output = 0 0 0 0 0 0 0 0

6. ones (5,2)

The above instruction creates a vector of five rows and two columns Output = 1 1 1 1 1 1 1 1 1 1 7. a = [ 1 2 3] b = [4 5 6] a.*b = [4 10 18] 8. If C= [2 2 2] b.*C results in [8 10 12]


9. plot (t, x) If x = [6 7 8 9] t = [1 2 3 4] This instruction will display a figure window which indicates the plot of x versus t.

10. stem (t,x) :- This instruction will display a figure window as shown

11. Subplot: This function divides the figure window into rows and columns.

Subplot (2 2 1) divides the figure window into 2 rows and 2 columns 1 represent number

of the figure

(2,2,1)

(2,2,2)

(2,2,3)

(2,2,4)


12. Conv Syntax: w = conv(u,v) Description: w = conv(u,v) convolves vectors u and v. Algebraically, convolution is the same operation as multiplying the polynomials whose coefficients are the elements of u and v. 13. Disp Syntax: disp(X) Description: disp(X) displays an array, without printing the array name. If X contains a text string, the string is displayed. Another way to display an array on the screen is to type its name, but this prints a leading "X=," which is not always desirable. Note that disp does not display empty arrays. 14. xlabel Syntax: xlabel('string') Description: xlabel('string') labels the x-axis of the current axes. 15. ylabel Syntax : ylabel('string') Description: ylabel('string') labels the y-axis of the current axes. 16. Title Syntax : title('string') Description: title('string') outputs the string at the top and in the centre of the current axes. 17. grid on Syntax : grid on Description: grid on adds major grid lines to the current axes. 18. FFT FFT(X) is the discrete Fast Fourier transform (FFT) of vector X. For matrices, the FFT operation is applied to each column. For N-D arrays, the FFT operation operates on the first non-singleton dimension. FFT(X,N) is the N-point FFT, padded with zeros if X has less than N points and truncated if it has more. 19. ABS Absolute value. ABS(X) is the absolute value of the elements of X. When X is complex, ABS(X) is the complex modulus (magnitude) of the elements of X. 20. ANGLE Phase angle. ANGLE(H) returns the phase angles, in radians, of a matrix with complex elements.


21. INTERP Resample data at a higher rate using low-pass interpolation. Y = INTERP(X,L) re-samples the sequence in vector X at L times the original sample rate. The resulting re-sampled vector Y is L times longer, LENGTH(Y) = L*LENGTH(X). 22. DECIMATE Resample data at a lower rate after low-pass filtering. Y = DECIMATE(X,M) re-samples the sequence in vector X at 1/M times the original sample rate. The resulting re-sampled vector Y is M times shorter, i.e., LENGTH(Y) = CEIL(LENGTH(X)/M). By default, DECIMATE filters the data with an 8th order Chebyshev Type I low-pass filter with cut-off frequency .8*(Fs/2)/R, before re-sampling. Some important commands in MATLAB Help List topics on which help is available Help command name Provides help on the topic selected Demo Runs the demo program Who Lists variables currently in the workspace Whos Lists variables currently in the workspace with their size Clear Clears the workspace, all the variables are removed Clear x,y,z Clears only variables x,y,z Quit Quits MATLAB


Experiment No: 2

GENERATION OF ELEMENTARY CONTINUES TIME SIGNALS

Aim : Write a MATLAB program to develop some elementary continues time (CT) signals

a. Sinusoidal

b. Square

c. Complex waveform

d. Unit Step

e. Unit Ramp

f. Exponential

g. Noise

MATLAB Code :

a. Sinusoidal Waveform clc; clear all; close all; t = 0 : 0.001 : 5; f = input ('Enter the value of frequency'); a = input ('Enter the value of amplitude'); subplot (2,1,1); y =a*sin(2*pi*f*t); plot (t,y,'r'); xlabel ('time'); ylabel ('amplitude'); title ('sine wave') grid on;

Output:

Enter the value of frequency 2 Enter the value of amplitude 1


b. Square Waveform clc; clear all; close all; t = 0 : 0.001 : 5; f = input ('Enter the value of frequency'); a = input ('Enter the value of amplitude'); subplot (2,1,1); y =a*square(2*pi*f*t); plot (t,y,'r'); xlabel ('time'); ylabel ('amplitude'); title ('Square wave') grid on;

Output :

Enter the value of frequency 1 Enter the value of amplitude 1

c. Complex Waveform clc; clear all; close all; t = 0 : 0.001 : 5; f = input ('Enter the value of frequency'); a = input ('Enter the value of amplitude'); subplot (2,1,1); y = [exp((-0.2+2*i)*t)]; plot (t,y,'r'); xlabel ('time'); ylabel ('amplitude'); title ('Complex wave') grid on;


Output : Enter the value of frequency 2 Enter the value of amplitude 1

d. Unit Step Waveform clc; clear all; close all; t = 0 : 0.001 : 5; a = input ('Enter the value of amplitude'); subplot(2,1,1) plot (t,a,'r'); xlabel ('time'); ylabel ('amplitude'); title ( 'Complex Wave') grid on;

Output: Enter the value of amplitude 1


e. Unit Ramp Waveform clc; clear all; close all; t = 0 : 0.001 : 5; m = input ('Enter the value of slope'); subplot (2,1,1); y = m*t; plot (t,y,'r'); xlabel ('time'); ylabel ('amplitude'); title ('Ramp wave') grid on;

Output: Enter the value of slope 1

f. Exponential Waveform clc; clear all; close all; a=0.5; t=0:0.1:10; disp('EXPONENTIAL DECAYING SIGNAL'); x=a.^t; subplot(2,1,1); plot(t,x); xlabel('Time'); ylabel('Amplitude'); title('Exponential Decaying Signal Response'); grid on;


disp('EXPONENTIAL GROWING SIGNAL'); x=a.^-t; subplot(2,1,2); plot(t,x); xlabel('Time'); ylabel('Amplitude'); title('Exponential Growing Signal Response'); grid on; Output: EXPONENTIAL DECAYING SIGNAL

EXPONENTIAL GROWING SIGNAL

g. Noise Signal

clc; clear all; close all;

t=0:0.001:50; % generate the uncorrupted signal s=2*t.*(0.9.^t); subplot(2,1,1); plot(t,s); title('Uncorrupted sequence'); xlabel('Time index'); ylabel('Amplitude');


% generate the noise signal subplot(2,1,2); x=rand(50,1); plot(x); title('Noise'); xlabel('Time index'); ylabel('Amplitude');

Output:

Conclusion:


Experiment No: 3

CONCEPT OF HARMONICS

Aim: Write a MATLAB program to find the sum of sinusoidal signals and understand the concept of harmonics. MATLAB Code :

clc; clear all; close all; t=0:.01:2*pi; %generation of sine signals% y1=sin(t); y2=sin(3*t)/3; y3=sin(5*t)/5; y4=sin(7*t)/7; y5=sin(9*t)/9; %y = sin(t) + sin(3*t)/3 + sin(5*t)/5 + sin(7*t)/7 + sin(9*t)/9; % y = y1+y2+y3+y4+y5; plot (t,y,t,y1,t,y2,t,y3,t,y4,t,y5); legend ('y','y1','y2','y3','y4','y5'); title ('generation of sum of sinusoidal signals'); grid on; ylabel ('---> Amplitude'); xlabel ('---> t'); Output:


Exercise:

1. Plot the synthesized waveform using the summation of even component of the series.

2. Plot the synthesized triangular waveform.

Conclusion:


Experiment No: 4

GENERATION OF ELEMENTARY DISCRETE TIME SIGNALS

Aim : Write a MATLAB program to develop some elementary discrete time (DT) signals

a. Sinusoidal

b. Square

c. Complex waveform

d. Unit Impulse

e. Unit Step

f. Unit Ramp

g. Exponential

MATLAB Code :

a. Sinusoidal DT signal

clc; clear all; close all; N = input('Enter Number of Samples : '); n = 0:0.1:N; x = sin(n); stem (n,x); xlabel ('Time'); ylabel ('Amplitude'); title ('Discrete Time Sine Signal'); grid on;

Output:

Enter Number of Samples : 15


b. Square DT signal

clc; clear all; close all; N = input('Enter the number of Samples:); n = 0:0.1:N; s = square(2*n); stem (n,s); xlabel ('time'); ylabel ('amplitude'); title ('square wave') grid on; Output: Enter the number of Samples : 10

c. Complex DT signal

clc; clear all; close all; n = 0 : 0.1 : 15; subplot (2,1,1); y = [exp((-0.2+2*i)*n)]; stem (n,y,'r'); xlabel ('time'); ylabel ('amplitude'); title ('Discrete Time Complex wave') grid on;


d. Unit Impulse DT signal clc; clear all; close all; N = input('Enter the Number of Samples: '); n = -N:1:N; x = [zeros(1,N) 1 zeros(1,N)]; subplot (2,1,1); stem (n,x,'r'); xlabel ('Time'); ylabel ('Amplitude'); title ('Impulse Response'); grid on; Output: Enter the number of Samples : 7

e. Unit Step DT signal

clc; clear all; close all; N = input(' Enter the Number of Samples : '); n = -N:1:N; x = [zeros(1,N) 1 ones(1,N)]; subplot (2,1,1); stem (n,x,'r'); xlabel ('Time'); ylabel ('Amplitude'); title ('Unit Step Response'); grid on; Output: Enter the number of Samples : 10


f. Unit Ramp DT signal clc; clear all; close all; N = input('Enter Number of Samples : '); a = input('Enter Amplitude : '); n = 0:1:N; x = a*n; subplot (2,1,1); stem (n,x,'r'); xlabel ('Time'); ylabel ('Amplitude'); title ('Unit Ramp Response'); grid on; Output: Enter Number of Samples : 50 Enter Amplitude : 1

g. Exponential DT signal clc; clear all; close all; N = input('Enter Number of Samples : '); % EXPONENTIAL DECAYING SIGNAL a = 0.5; n = 0:.1:N; x = a.^n; subplot (2,1,1); stem (n,x,'r'); xlabel ('Time'); ylabel ('Amplitude'); title ('Exponential Decaying Signal Response'); grid on; %EXPONENTIAL GROWING SIGNAL subplot (2,1,2); x = a.^-n; stem (n,x,'r'); xlabel ('Time'); ylabel ('Amplitude'); title ('Exponential Growing Signal Response'); grid on;


Output: Enter Number of Samples : 5


Experiment No: 5

SAMPLING THEOREM

Aim: Write a MATLAB program to verify the Sampling theorem.

MATLAB Code:

clear all; close all; clc; tf = 0.05; t = 0 : 0.00005 : tf; f = input ('Enter the analog frequency,f = : '); xt = cos (2*pi*f*t); % Under Sampling Sampling Frequency less than Nyquist rate => fs < 2f fs1 = 1.3*f; n1 = 0:1/fs1:tf; xn = cos(2*pi*f*n1); subplot (3,1,1); plot (t,xt,'b',n1,xn,'r*-'); title ('Undersampling plot'); xlabel ('time'); ylabel ('Amplitude'); % Nyquist Sampling Sampling Frequency fs = 2f fs2 = 2*f; n2 = 0 : 1/fs2 : tf; xn = cos(2*pi*f*n2); subplot (3,1,2); plot (t,xt,'b',n2,xn,'r*-'); title ('Nyquist plot'); xlabel ('time'); ylabel ('Amplitude'); % Over Sampling Sampling Frequency greater than Nyquist rate => fs > 2f fs3 = 6*f; n3 = 0 : 1/fs3 : tf; xn = cos(2*pi*f*n3); subplot (3,1,3); plot (t,xt,'b',n3,xn,'r*-'); title ('Oversampling plot'); xlabel ('time'); ylabel ('Amplitude');


Output : Enter the analog frequency,f = : 200

Conclusion:


Experiment No: 6

IMPULSE RESPONSE OF THE LTI SYSTEM

Aim: Write a MATLAB program to find the impulse response of a LTI system defined by a difference equation: y(n) + 0.7y(n-1) 0.45y(n-2) 0.6y(n-3) = 0.8x(n) 0.44x(n-1) + 0.36x(n-2) + 0.2x(n-3)

MATLAB Code:

clc; clear all; close all; N = input ('Enter the required length of impulse response N = '); n = 0 : N-1; b = input ('Enter the co-efficients of x(n), b = '); a = input ('Enter the co=efficients of y(n), a = '); % Impulse response of the LTI system given by the difference equation x = [1, zeros(1,N-1)]; y = impz (b,a,N); subplot (2,1,1); stem (n,y,r); xlabel ('time'); ylabel ('amplitude'); title ('IMPULSE RESPONSE'); grid on; % Pole Zero distribution of the LTI system given by the difference equation subplot (2,1,2); zplane (b,a); xlabel ('Real part'); ylabel ('Imaginary part'); title ('Poles and Zeros of H[z] in Z-plane'); grid on;


Output : Enter the required length of impulse response N = 40

Enter the co-efficients of x(n), b = [0.8 -0.44 0.36 0.02] Enter the co=efficients of y(n), a = [1 0.7 -0.45 -0.6]

Exercise:

Find out the impulse response of the following difference equation and plot the pole-zero

response of the same.

1. 0.25y(n-3) + 0.7y(n-2)+y(n) = 3x(n) 0.9x(n-1) + 1.3x(n-2)

2. y(n) y(n-1) = x(n) x(n-1)


Conclusion:


Experiment No: 7

FREQUENCY RESPONSE OF THE LTI SYSTEM

Aim: Write a MATLAB program to plot the Frequency response of a given LTI sequence:

y(n) 5 y(n1) = x(n) + 4 x(n1)

MATLAB Code: b = [1, 4]; %Numerator coefficients a = [1, -5]; %Denominator coefficients w = -2*pi: pi/256: 2*pi; % Frequency Response of the LTI system [h] = freqz(b, a, w); subplot(2, 1, 1), plot(w, abs(h)); xlabel ('Frequency \omega'), ylabel ('Magnitude'); grid on; % Phase Response of the LTI system subplot(2, 1, 2), plot(w, angle(h)); xlabel('Frequency \omega'), ylabel('Phase - Radians'); grid on; Output:


Exercise:

Find out the frequency response of the following difference equations.

1. 0.25y(n-3) + 0.7y(n-2)+y(n) = 3x(n) 0.9x(n-1) + 1.3x(n-2)

2. y(n) y(n-1) = x(n) x(n-1)

Conclusion:


Experiment No: 8

LINEAR CONVOLUTION

Aim: Write a MATLAB program to compute linear convolution of two given sequences.

MATLAB Code:

a. Linear convolution computation (Program 1)

clear all; close all; clc;

x1 = input ('enter the first sequence '); subplot (2,2,1); stem (x1,'r'); ylabel ('amplitude'); title ('plot of the first sequence'); grid on;

x2 = input ('enter 2nd sequence '); subplot (2,2,2); stem (x2, 'r'); ylabel ('amplitude'); title ('plot of 2nd sequence'); grid on; f = conv (x1,x2);

disp ('output of linear conv is'); disp (f); subplot (2,2,3); stem (f,'r'); xlabel ('time index n'); ylabel ('amplitude f'); title('linear conv of sequence'); grid on;

Output:

enter the first sequence [1 2 3 4] enter 2nd sequence [2 -1 2 3] output of linear conv is

2 3 6 12 8 17 12


b. Linear convolution computation (Program 2)

clc; clear all; close all; a=input('Enter the starting point of x[n]='); b=input('Enter the starting point of h[n]='); x=input('Enter the co-efficients of x[n]='); h=input('Enter the co-efficients of h[n]='); y=conv(x,h); subplot(3,1,1); p=a:(a+length(x)-1); stem(p,x,'r'); grid on; xlabel('Time'); ylabel('Amplitude'); title('INPUT x(n)'); subplot(3,1,2); q=b:(b+length(h)-1); stem(q,h,'r'); grid on;


xlabel('Time'); ylabel('Amplitude'); title('IMPULSE RESPONSE h(n)'); subplot(3,1,3); n=a+b:length(y)+a+b-1; stem(n,y,'r'); grid on; disp(y) xlabel('Time'); ylabel('Amplitude'); title('Convolution x(n)* h(n)');

Output: Enter the starting point of x[n]=-2 Enter the starting point of h[n]=1 Enter the co-efficients of x[n]=[1 2 3 4 5] Enter the co-efficients of h[n]=[-1 -2 1 1 2] -1 -4 -6 -7 -6 1 15 13 10

Exercise: Find out the Convolution of other two series and find the starting point of the result.


Conclusion:


Experiment No: 9

DISCRETE TIME FOURIER TRANSFORM - DTFT

Aim: Write a MATLAB program to find the Discrete Time Fourier Transform (DTFT) of the given sequence.

MATLAB Program:

% Evaluation of the DTFT clf; % Compute the frequency samples of the DTFT w = -4*pi : 8*pi/511 : 4*pi; num = [2 1]; den = [1 -0.6];

h = freqz(num, den, w);

% Plot the DTFT subplot(4,1,1) plot(w,real(h)); grid on; title('Real part of H(e^{j\omega})') xlabel('\omega /\pi'); ylabel('Amplitude'); subplot(4,1,2) plot(w,imag(h));grid title('Imaginary part of H(e^{j\omega})') xlabel('\omega /\pi'); ylabel('Amplitude'); subplot(4,1,3) plot(w,abs(h));grid title('Magnitude Spectrum |H(e^{j\omega})|') xlabel('\omega /\pi'); ylabel('Amplitude'); subplot(4,1,4) plot(w,angle(h));grid title('Phase Spectrum arg[H(e^{j\omega})]') xlabel('\omega /\pi'); ylabel('Phase, radians');


Output:

Exercise: Find the Fourier transform of the ( ) (

) ( )


Conclusion:


Experiment No: 10

FIR FILTER DESIGN

Aim: Write a MATLAB program to plot magnitude response and phase response of digital FIR Filter using Rectangular window

a. Low pass Filter

b. High pass Filter

c. Bandpass Filter

d. Bandstop Filter

MATLAB Program:

a. LOW PASS FIR Filter Designing


N=input('Enter the value of N:'); wc=input('Enter cutoff frequency:');

h=fir1(N,wc/pi,rectwin(N+1)); freqz(h); Output: Enter the value of N: 5 Enter cutoff frequency: 0.5*pi


b. HIGH PASS FIR Filter Designing



h=fir1(N,wc/pi, high,rectwin(N+1)); freqz(h); Output: Enter the value of N: 28 Enter cutoff frequency: 0.5*pi

c. BAND PASS FIR Filter Designing



h=fir1(N,wc/pi,rectwin(N+1)); freqz(h); Output: Enter the value of N: 28 Enter cutoff frequency: [0.3*pi 0.7*pi]


d. BAND STOP FIR Filter Designing



h=fir1(N,wc/pi, stop,rectwin(N+1)); freqz(h); Output: Enter the value of N: 28 Enter cutoff frequency: [0.2*pi 0.7*pi]


Hanning Window:

1. h=fir1(N,wc/pi,hanning(N+1)); - Low Pass FIR Filter 2. h=fir1(N,wc/pi, high,hanning(N+1)); - High Pass FIR Filter

3. h=fir1(N,wc/pi,hanning(N+1)); - Band Pass FIR Filter 4. h=fir1(N,wc/pi, stop, hanning(N+1)); - Band Stop FIR Filter

Hamming Window:

1. h=fir1(N,wc/pi,hamming(N+1)); - Low Pass FIR Filter 2. h=fir1(N,wc/pi, high,hamming(N+1)); - High Pass FIR Filter

3. h=fir1(N,wc/pi,hamming(N+1)); - Band Pass FIR Filter 4. h=fir1(N,wc/pi, stop, hamming(N+1)); - Band Stop FIR Filter

DepartmentofElectronics & Communication EngineeringB. E. SEMESTER: VII

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