digital signal processing laboratory

8/12/2019 Digital Signal Processing Laboratory

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DIGITAL SIGNAL PROCESSING LABORATORY

Sub Code : 10ECL57

A LIST OF EXPERIMENTS USING MATLAB/SCILAB/OCTAVE/WAB

1. Verification of sampling theorem.

2. Impulse response of a given system

3. Linear convolution of two given sequences.

4. Circular convolution of two given sequences

5. Autocorrelation of a given sequence and verification of its properties.

6. Cross correlation of given sequences and verification of its properties.

7. Solving a given difference equation.

8. Computation of N point DFT of a given sequence and to plot magnitude

and phase spectrum.

9. Linear convolution of two sequences using DFT and IDFT.

10. Circular convolution of two given sequences using DFT and IDFT

11. Design and implementation of FIR filter to meet given specifications.

12. Design and implementation of IIR filter to meet given specifications.

B. LIST OF EXPERIMENTS USING DSP PROCESSOR

(Note: Experiments no: 1, 2, 3 & 7 may be performed on CCS)

1. Linear convolution of two given sequences.

2. Circular convolution of two given sequences.

3. Computation of N- Point DFT of a given sequence

4. Realization of an FIR filter (any type) to meet given specifications .The

input can be a signal from function generator / speech signal .

5. Audio applications such as to plot time and frequency (Spectrum)

display of Microphone output plus a cosine using DSP. Read a wav file

and match with their respective spectrograms

6. Noise: Add noise above 3kHz and then remove; Interference suppression

using 400 Hz tone.


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7. Impulse response of first order and second order system

REFERENCE BOOKS:

1. Sanjeet Mitra, Digital signal processing using MATLAB, TMH,2001

2. J.G.Proakis & Ingale, Digital signal processing using MATLAB,.MGH, 2000

3. B.Venkataramani and Bhaskar, Digital signal processors,TMH,2002

Experiment 1Verification of Sampling theorem

Aim :To verify sampling theorem.

Theory :

Sampling

The process of obtaining the signal values from the continuous signal at regular tim

intervals is known as sampling. The sampling interval is denoted as Ts and its reciproca

the sampling frequency or sample-rate is denoted as fs, where fs = 1/Ts. The result of th

process is just a sequence of numbers called a discrete time signal and is denoted as x[

where n is the index used to refer to the samples.

Interpolation

We can reconstruct the original continuous signal from its samples by using Interpolatio

Interpolation is the process of guessing signal values at arbitrary instants of time, whic

fall - in general - in between the actual samples. Thereby interpolation creates a continuou

time signal and can be seen as an inverse process to sampling.

Ideally, the continuous signal obtained from interpolation should be equal to the origin

continuous signal. One of the very popular interpolation methods is linear interpolation -


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reconstruct a signal value we simply connect the values at our sampling instants wit

straight lines.

sophisticated interpolation methods such as sinc-interpolation will yield smoother an

more faithful reconstructions than the simple linear interpolation.

Sampling Theorem:

Sampling theorem states that Exact reconstruction of a continuous time signal from its

samples is possible, if the signal is band-limited and the sampling frequency is greater tha

twice the signal bandwidth.

While sampling a signal the sampling frequencyfsmust be greater than twice the

highest frequency componentfmin the signal. iefs>=2fm.This is the Nyquist Sampling

Theorem.

When the above condition is not met, i.e, If the signal is sampled at a frequenc

lesser than twice the highest frequency present in the signal and reconstruct the signal an

way, an odd phenomenon called Aliasing occurs.

The following figure shows a 100Hz signal and is sampled at a sampling rate

80Hz .When the signal is reconstructed from the samples it results in a different sign

with frequency of 20 Hz.Entirely different from the original 100Hz signal !!

Procedure:1)Generate a Sinusoidal Wave of given frequency f

2)Sample the given signal with sampling rate fs.

3)Plot the discrete time signal.


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4)Reconstruct signal by using linear interpolation (plotting the samples).

5)Repeat the steps 1 to 4 for different sampling frequencies.

6) In step 4 use sinc interpolation to reconstruct the signal repeat the procedur

1 to 5.

Program 1.

%**********************************************************************************%To verify smpling theorem

% using linear interpolation Reconstruction.

% This m-file illustrates the effects of aliasing in time domain.

%********************************************************************************************

% Clears Command window/Close all figures/Clear all variables

clc;

close all;

clear all;

%******************************Generate Analog Signal *****************************

t=0:0.001:.3;

xt=sin(2*pi*30*t);

plot(t,xt);

xlabel('t --->');ylabel(' x(t) ----->');title('Analog signal')

disp('Observe analog Signal . To Sample the signal press ENTER')

%*****************************************Sampling**********************************

fs =input('Enter sampling frequency : ')

ts=1/fs;

n=0:ts:.3;

xn=sin(2*pi*30*n);

% ***************************************Plot the discrete sequence**************

hold

stem(n,xn,'r', 'filled')

xlabel('n --->');ylabel(' x(n) ----->');title('Analog signal in BLUE, Smpled Signal in RED')


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disp('TO RECONSTRUCT THE SIGNAL FOM SAMPLES PRESS ENTER ')

pause

% ***************************************Reconstruction of signal**************

plot(n,xn,'g')

title('Analog signal in BLUE , smpled signal in RED ,Reconstructed signal in GREEN')

Sample run Out put :

Observe analog Signal . To Sample the signal press ENTER

Enter sampling frequency : 65

fs =

65

Current plot held

TO RECONSTRUCT THE SIGNAL FOM SAMPLES PRESS ENTER

Compare the reconstructed signal with the original signal.

Program 2.%*************************************************************************************************


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%To verify smpling theorem

% Reconstruction using sinc interpolation.

% This m-file illustrates the effects of aliasing in time domain.

%**************************************************************************************************

Clears Command window/Close all figures/Clear all variables

clc;

close all;

clear all;

%******************************Generate Analog Signal****************************************

t=0:0.001:.3;

xt=sin(2*pi*30*t);

plot(t,xt);

xlabel('t --->');ylabel(' x(t) ----->');title('Analog signal')

disp('Observe analog Signal . To Sample the signal press ENTER')

%*****************************************Sampling******************************************

fs =input( 'Enter sampling frequency')

ts=1/fs;

n=0:ts:.3;

xn=sin(2*pi*30*n);

% ***************************************Plot the discrete sequence**********************

hold

stem(n,xn,'r', 'filled')

xlabel('n --->');ylabel(' x(n) ----->');title('Analog signal in BLUE, Smpled Signal in RED')

disp('TO RECONSTRUCT THE SIGNAL FOM SAMPLES PRESS ENTER ')

pause

% ***************************************Reconstruction of signal**********************

ni=0:0.001:.3;

xr=interp1(n,xn,ni,'sinc');

plot(ni,xr,'g')


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title('Analog signal in BLUE , smpled signal in RED ,Reconstructed signal in GREEN')

Sample run:

Observe analog Signal . To Sample the signal press ENTER

Enter sampling frequency : 100

fs =

100

Current plot held

TO RECONSTRUCT THE SIGNAL FOM SAMPLES PRESS ENTER

Observe that reconstructed signal is almost matching with the original signal.

Exercise :1.Generate the sinusoidal signals with following frequencies and verify sampling theorem.

i) 100Hz ii)3000 Hz iii) 10,000Hz iv) 1 MHz

2.Generate the following signal and verify sampling theorem.

x(t) =2sin(40t) + 3cos(1000t)

3.Generate the following signal and verify sampling theorem.

x(t) =2sin(40t) cos(1000t)

Experiment 2 Impulse response of a given system

Aim:To find the impulse response h(n) of a system

i) The difference equation characterizing the system.

ii) The system function H(Z) is given.


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Theory:

The response of a discrete-time system to a unit sample sequence[n]is called theunit

sample responseor, simply, theimpulse response, and denoted ash[n].

The impulse response can be used to represent the relation between input and the output

of the system.

A discrete-time system is called afinite impulse response(FIR) system if its impulse

responseh[n] is of finite length. Otherwise, it is aninfinite impulse response(IIR) system.

If a linear, time-invariant discrete-time system is characterized by an impulse responseh[n

The responsey[n] to a input excitationx[n] is given by

=

=k

knhkxny ][][][

and is referred to as convolution.

It is clear that the convolution process in the Z-domain entails the multiplication of two

polynomials.)().()(][*][][ ZHZXZYnhnxny ==

)(

)()(

ZX

ZYZH =

When the system function H(Z) is known the impulse response can be computed by finding

the inverse Z transform of )(ZH . ))((][1 ZHZnh =

The responsey[n] to a input excitationx[n] can also be given by using a difference equation

== +=M

k

N

k 0

k

1

k k)-x(nbk)-y(nay(n) OR

y(n)= b0x(n) + b1x(n-1) + b2x(n-2) +.+ b3x(n-M) - a1y(n-1) - a2y(n-2) -.- a3y(n-N)

Let's take theztransform of both sides; The terms in Y (Z) may be grouped together on

theleft-hand side to get


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Y(Z)= b0X(Z) + b1Z-1X(Z) + b2Z

-2X(Z) +.+ b3Z-MX(Z)

- a1Z-1Y(Z) - a2Z

-2Y(Z) -.- a3Z-NY(Z)

Factoring out the common terms Y(Z) and X(Z) and rearranging

)(.....1

.....

)(

)(2

2

1

1

2

2

1

10ZH

ZaZaZa

ZbZbZbb

ZX

ZY

N

N

M

M=

++++

++++=

The impulse response can be computed by finding the inverse Z transform of )(ZH .

Program:

The difference equation that characterizes the system is given. Compute impulse response

of the system.

y[n]0.4y[n1]+0.75y[n2] = 2.2403x[n]+2.4908x[n1]+2.2403x[n2]

% Compute the impulse response y

close all, clear all

N = 40;

num = [2.2403 2.4908 2.2403]; % coefficients of x[n-k] or numerator of H(Z)

den = [1 -0.4 0.75]; % coefficients of y[n-k] or denomenator of H(Z)

y = impz(num,den,N);

% Plot the impulse response

stem(y);

xlabel('Time index n'); ylabel('Amplitude');

title('Impulse Response'); grid;

Experiment 3 Linear convolution of two given sequencesAIM:To compute the linear convolution of

i) two causal sequences.

ii) Noncausal sequences.


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The linear, time-invariant discrete-time system is characterized by a signal called th

IMPULSE RESPONSE denoted by h[n]. The procedure of finding the output of such an LTI

system for a given input can be described by a mathematical operation called

CONVOLUTION.

The responsey[n] to a input excitationx[n] is given by the superposition of weighted,

shifted impulse responses. Mathematically,

=

=k

knhkxny ][][][

Procedure:

a.Causal sequences

i. Read the input sequenceii. convolve

iii.Plot input and out put signal.

b. Non Causal sequences

For Non-causal sequences using of function conv may not be appropriate since the indexes

may not match.

In Convolution

Beginning index of the convolved sequence =

Index of the first element of the sequence x(n) + index of the first element of the sequence

y(n)

Ending index of the convolved sequence =

Index of the Last element of the sequence x(n) + index of the Last element of the sequence

y(n)

So we modify the program for convolution to perform convolution of Non-causal sequences

as given below.

Accept the Sequences x and h

Accept the index vectors for x and h

Compute Starting and Ending indexes for y


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Compute convolution using function conv.

Plot computed indexes for y Vs y

Program 1

%linear convolution

clc;

close all;

clear all;

x=input('Enter the input sequence x[n] with origin at 0: ');

h=input ('Enter the impulse response h[n] of the system with origin at 0: ');

y=conv(x,h)

subplot(311)

n1=0:length(x)-1;

stem(n1,x, 'filled');

title ('Plot of input sequence x[n]');

n2=0:length(h)-1;

subplot(312)

stem(n2,h,'filled');

title ('Plot of output responce h[n]');

n3=0:length(y)-1;

subplot(313)

stem(n3,y,'filled');

title('Plot of output response y[n]');

xlabel('n---->');ylabel('Amplitude-->')

Sample run

Enter the input sequence x[n] with origin at 0: [1 2 2 1]


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Enter the impulse response h[n] of the system with origin at 0: [1 2 3 4]

y =

1 4 9 15 16 11 4

Program 2:

% to compute the convolution of two sided sequences (Causal Sequences).

clc;

x=input ('enter the first sequence: ');

h=input ('enter the second sequence: ');

nx=input('index vector for first sequence: ');

nh=input('index vector for second sequence: ');

nyb=nx(1)+nh(1);

nye=nx(length(x))+nh(length(h));

ny=[nyb:nye];

y=conv(x,h)

subplot(3,1,1);stem(nx,x)

ylabel('x----->')

subplot(3,1,2);stem(nh,h)

ylabel('h----->')

subplot(3,1,3);stem(ny,y)

xlabel('n--->');ylabel('y----->')

%************ ***************END***********************************

Sample Run:

enter the first sequence: [1 2 3 4 5]

enter the second sequence: [3 4 3]

index vector for first sequence: [-1:3]

index vector for second sequence: [-1:1]

y =


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3 10 20 30 40 32 15

Exercise :

ii) Write and use the function file given below to compute convolution of non causal

sequences.

Function conv_m.m :

Note :Save this Function as a separate fileconv_m.m

%************************* Function Begins Here***********************

% modified convolution routine for Convolution of two sequences

function[y,ny]=conv_m(x,nx,h,nh);

nyb=nx(1)+nh(1);

nye=nx(length(x))+nh(length(h));

ny=[nyb:nye];

y=conv(x,h)

%******************************END ofFunction****************************************

Experiment 4 Cross Correlation

% cross correlation

% Exp a% x ! ["# 11# $# 0# -1# # 2]

% nx![-"&"]

x!inp't(Enter te inp't se'ence x[n]!)nx !inp't(Enter +n,ex ector for x[n])

[y#ny] ! si.sift(x#nx#2)

/ ! ran,n(1#len.t(y))n/ ! ny % .enerate /(n)

[y#ny] ! si.a,,(y#ny#/#n/) % obtain y(n) ! x(n-2) /(n)

nyb!nx(1)ny(1)

nye!nx(len.t(x))ny(len.t(y))nrxy!nyb&nye

rxy!xcorr(x#y)

s'bplot(2#1#1)ste(nrxy#rxy)axis([-##-30#230])xlabel(la. ariable l)

ylabel(rxy)title(4rosscorrelation& noise se'ence)

*****************Supporting Functions********************

sigadd.m

f'nction [y#n] ! si.a,,(x1#n1#x2#n2)

% ipleents y(n) ! x1(n)x2(n)% -----------------------------


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% [y#n] ! si.a,,(x1#n1#x2#n2)% y ! s' se'ence oer n# /ic incl',es n1 an, n2

% x1 ! first se'ence oer n1

% x2 ! secon, se'ence oer n2 (n2 can be ,ifferent fro n1)%

n ! in(in(n1)#in(n2))&ax(ax(n1)#ax(n2)) % ,'ration of y(n)

y1 ! 5eros(1#len.t(n)) y2 ! y1 % initiali5ationy1(fin,((n6!in(n1))7(n8!ax(n1))!!1))!x1 % x1 /it ,'ration of y

y2(fin,((n6!in(n2))7(n8!ax(n2))!!1))!x2 % x2 /it ,'ration of y

y ! y1y2 % se'ence a,,ition

sigfold.m

f'nction [y#n] ! si.fol,(x#n)% ipleents y(n) ! x(-n)

% -----------------------

% [y#n] ! si.fol,(x#n)%

y ! fliplr(x) n ! -fliplr(n)

sigshift.m

f'nction [y#n] ! si.sift(x##n0)

% ipleents y(n) ! x(n-n0)% -------------------------

% [y#n] ! si.sift(x##n0)

%n ! n0 y ! x

Experiment 5 AutoCorrelation

clcclear all

close all

,isp(9'tocorrelation &),isp(Exaple +np'ts )

,isp( x ! ["# 11# $# 0# -1# # 2] )

,isp(nx![-"&"])x!inp't(Enter te inp't se'ence x[n]!)

nx ! inp't(Enter +n,ex ector nx for x[n]!)

nyb!2*nx(1)nye!2*nx(len.t(x))nrxy!nyb&nye

rxy!xcorr(x)

s'bplot(2#1#1)ste(nrxy#rxy)axis([-##-30#230])xlabel(la. ariable l)

ylabel(rxy)title(9'tocorrelation& noise se'ence)

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