optimal signal processing exam

8/2/2019 Optimal Signal Processing Exam

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Sivaramakrishnan, Aishwarya, 19871120-8663

Solutions to Exam

Optimal Signal Processing, ET 2410

Sivaramakrishnan,Aishwarya,19871120-8663

[email protected]
mailto:[email protected]:[email protected]:[email protected]


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Task 1

Task 1-ALeast Square Polynomial Curve Fitting is a theorem which tells about,

Least Squares Method that used to give the best fit for the curve, Polynomial in this LS sense is used to minimize the error between the observed data

points to the estimated curve.

Here, we use least square polynomial for the given N data pairs ,Thus, the equation is given by,

where, the output depends on the input parameter and the white guassian noise with varianceNow, the least square polynomial of degree k is given by,

that can fit the N data pairs of x and y by solving the linear system given by,

x . P = y

This can then be converted into vandermonde equation and then, this can be solved by

multiplying

on both the sides of the system,

Now, we know that, This is the equation to derive polynomial for the least squares.


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The error is given by the difference between the measured value with the estimated value as mentioned.Least Mean Square Error is given by, Thus, now we can substitute the value of here, we get,

This is how the variance is derived for the noise signal v, because for the unknown Guassiannoise, mean is zero and the variance is given as . So least mean square is equal to variance.Task 1-B

Defining and deriving the expression for the polynomial constant p and the variance of the

guassian noise(v).

Matlab Coding:

%Function defining the Least Square Polynomial Curve Fitting%

%%p= polynomial coeffecient of least square sense and s - guassian noise vectorfunction [p s]=LSPCF(x,y,K)x=x(:);y=y(:); %Defining the Value of (xn,yn) data points

N=length(x); %no. of samples of xn

J=N-K; %defining subscript

%Vandermonde Function is used to implement the vandermonde matrix for the polynomial

coeffecient p

vand=vander(x);

%Here J is expected to be real valued positive integer or logicals

X=vand(:,J:N);

%expression for the polynomial coeffecient p

a=inv(X.'*X);

b=(a*X.');p=b*y;

%Estimated Curve Function


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y1=X*p;

%Error Function is the difference betw. the measured data points to the

%estimated curvee=y-y1;

s=sum(e.^2)/J;

Table 1 for task coding 1

Task 1-C:

The given lines are coded and the o/p with K=5 as the coefficient, the output is generated.

The values of P and S are then determined with the x and y input.

P=

-0.00078393346777

0.09872199258599

0.00933460439849-0.98931337214534

1.98587916101385

-2.01305977294848

s =

0.0191

clc

clear allload lspcf-polyval-3

[p, s] = LSPCF (x, y, 5); % function call of least square polynomial curve fitting% figure

%given datasN = size(x); % it collects the row and column of x in a row

n = sqrt(s)*randn(N); %Random noise given as n

y1 = polyval(p,x)+n;

figureplot(y)

title('Comparison of Original Data Pairs vs Estimated data curve')hold on

plot(y1,'R')

xlabel('samples n');

ylabel('Data Value');


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legend('sample data pairs','Estimated Data Curve',2);

Table 2 for task coding 1-b

Figure 1-c Least Square polynomial curve fitting

The curve seems to be fitting enough when the coefficient K=5. With the increase in K, there is

little a difference in o/p until K=15, but after then again the curve loses track and does not fit to

the points. But, K=5 seems to be appropriate for the given signal vectors x and y.


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Task 2:

Speech Coding is the process of compressing the speech data available in the digital

audio signal. Speech signal that is understood by the human auditory system is only transmitted

by preserving intelligibility and pleasantness. The speech signal is temporally compressed into

low bits/per second and then reconstructed back by preserving most of the speech content.

Task 2-A:

Now, in this task, the signal transmitted is expected to be compressed when passing

through the channel with limited bandwidth but without missing its quality in the receiver end.

Here, speechcoding.mat is given as the input containing the values of speech signals, sample data

rate and the error.

Now, we derive a function called stepup to be carried out in the speech coding. It steps up

the data with every corresponding value of the column vector.

Here, in the speech coding, c1 and e1 column vectors are utilized to develop the code, steppingup the reflection coefficient, filtering and then reconstructing back at the receiver end without

any too much loss in quality and intelligibility. Plotting the reconstructed signal gives us a clear

picture.

function a = stepup(gamma)gamma(:);

a = gamma(1);

for n=2:length(gamma)m=length(a);

b=[a' 0];

c=gamma(n);d=-a(m:-1:1)';

a = b + c*[d 1];

a = a(:);end

clear allclear all

close all

clcload speechcoding% loading the speechsignal[Z1,Z2] = size(c1);%row and column of c1

for i = 1:Z2

% Picking column vector to find a.J=2:Z1;gama = c1(J,i);

% Finding stepping up gama to find a


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Table 3 for task coding 2-a

Figure 2-a Speechcoding

a = stepup(gama);

PredEF = [1 -a']; % Predictor error Filter

A = sqrt(c1(1,i));

e = e1(:,i);

% Passing the Error Signal from inverse of H(z)x(:,i) = filter(A,PredEF,e);end

L = length(e1);

% signal Reconstruction.

y = reconstruction(x,L);plot(1:length(y),abs(fft(y)))

xlabel('Time in Seconds');

ylabel('Sample amplitude');

title('SPEECH CODING');


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Task 2-B

Since, e2 is not available, a command called awgn is used in order to get a new white guassian

noise signal.

clear allclose all

clc

load speechcoding% loading the speechsignal

[Z1,Z2] = size(c1);%row and column of c1for i = 1:Z2% Picking column vector to find a.

J=2:Z1;

gama = c1(J,i);% Finding stepping up gama to find a

a = stepup(gama);

PredEF = [1 -a']; % Predictor error FilterA = sqrt(c1(1,i));

e = awgn(PredEF,0.9);%white guassian noise% Passing the Error Signal from inverse of H(z)x(:,i) = filter(A,PredEF,e);

end

L = length(e);% signal Reconstruction.

y = reconstruction(x,L);

plot(1:length(y),abs(fft(y)), 'r')xlabel('Time in Seconds');

ylabel('Sample amplitude');

title('SPEECH CODING');



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Figure 2-b Speechcoding with white guassian noise.

Task 3:

Beamforming is a process of collecting all the signals together in an array from the sensor, filtereach signal and give out together as one single sensitive signal out which can be received by a

loud speaker etc.at the receiver end. It involves space and time.

Task 3-A

This task ask us to define a function calledfilterbf of the filtered input and the input weights.

The beamformer filter is used to filter the input signal. Here, the column vector is considered forthe filtering operation and also for the input signal. Each column vector of H is considered

everytime using the for loop to filter the corresponding column vector of X. Thus, the sum of allthe filtered signal are given as the output of the beamformer.

function y=filterbf(H,X)

%initializing the o/p signaly=0;

% No. of Rows and Columns in the filtered signal;


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hk=size(H);

% Column Vector of the Filtered Signal is only considered

colvector=hk(2);

%To Generate Filter of the Input Signal but the Filtering Signal for p=1:colvector

a=H(:,p);b=X(:,p);y=filter(a,1,b)+y;

end


Task 3-B

The cross correlation vector of the spatiotemporal domain is given by,

So this can be written in the vector form when length of the vector is from 1 to k.

The vector length or the dimension is considered to be K*L. So this length of vector is the

stacked vector of the spatiotemporal domain.

Similarly, the autocorrelation of the spatiotemporal domain is given by,

Since its the multiplication vector of the input signal with its same transpose vector, the matrix

form is defined as below,

Here, for the matrix, the vector length is considered to be K*L by K*L. So, this length of the

matrix is the stacked matrix of the spatiotemporal domain.


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Task 3-C

stcorrmtx

Spatio temporal correlation matrix X .

Each columns -sensor signal no. of columns - no. of sensors. L- coefficient Rxx=Autocorrelation Matrix

%function defining the stacked correlation matrixfunction Rxx=stcorrmtx(X,L)

%corresponding row nd column of X

[R,C]=size(X);

Rxxmain=[];%autocorrelation matrix initialized to zero

for p=1:C

x=convmtx(X(:,p),L);%convoluting X with Lx1=x'*x;Rxxmain=[Rxxmain; x1];%matrix formation

endRxx=Rxxmain;

Table 5 for task coding 3-c

stcorrvec

Spatio-temporal Correlation vector d with matrix X

Each columns -sensor signal no. of columns - no. of sensors. L- coefficient Rdx=Cross-correlation VEctor

%Function determining the stacked correlation vector

function rdX=stcorrvec(d,X,L)[rows,columns]=size(X)% rows nd columns takes the corresponding row nd column of X

for i=1:columns %initializationx1=convmtx(X(:,i),L);%function call of convolution matrix

d(length(x1),1)=0;%defining d at length y1 as zerordx1(:,i)=x1.'*d;%cross correlation vector

end


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rdX=rdx1(:);%all the rows and columns of rdx

Table 6 for task coding 3-c

Task 3-D

Wiener beamformer filter helps in filtering out the unwanted portions.

Here, Wiener Beamformer Filter Coefficient is derived as h.

Further described in the script.

%function defining the wiener filter of beamformerfunction h=wienerbf(X,d,L)

rdX=stcorrvec(d,X,L);%corrlation vctor function caaling

Rxx=stcorrmtx(X,L);%correlation matrix function call

[R,C]=size(X);

for p=1:Ca=p*L;

%correlation process

autocorr=Rxx(a-L+1:a,1:L);crosscorr=rdX(a-L+1:a,1);

%optimal filter of beamformer

wopt=Rxx\rdX;

%normalizing the optimal filter coeffecient

W=norm(wopt);h(:,p)=wopt/W;

end

h=h./norm(h);

Table 7 for task coding 3-D

Task 3-E

The file beamforming gives the input signal containing the required values of d,Fs.

close allclear all

clc

load beamforming%loading the given signal

L=25;%taps%wiener beamform filter function call

H=wienerbf(X,d,L);% H=[1:4;4:7];


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%values defined in the source

F=(0:200)/201*Fs/2;

A=(-90:90);

G=beampattern(H,d,Fs,F,A);%beampattern function callimagesc(A,F,G);%image with scaling full range image data

xlabel('Direction (degrees)');ylabel('frequency (Hz)');

Table 8 for task coding 3-E

Figure 3e-a. Beam pattern after wiener filtering in space.

It is observed that the direction extends from -90 to 90 in the upper frequency region and it slows

down and gets attenuated at the bottom frequencies. In the bottom, it is observed that the

direction of angles -20 to 20 only shows high gain and in the other angles, the beamformation is

out of phase. From 0 to 1500KHz, the beam pattern seems to have a high gain.


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Task3-F

This is the function to define the signal to noise interference ratio. It is expected to keep this ratio

high where the signal power is high while the noise power is low.

S-Input Signal N-Noise Signal Each column of S and N- Sensor Signal

%function defining the signal to noise and interference ratio beamformer

function h=snirbf(S,N,L)

%rows determine the sensor signal and the columns determine the total no.of sensors[sensorsig,senscount]=size(S);

%correlation matrix function call

Rss=stcorrmtx(S,L);%for signalRnn=stcorrmtx(N,L);% for noise

for p=1:senscount

a=p*L;

Rssj=Rss(a-L+1:a,1:L);

Rnnj=Rnn(a-L+1:a,1:L);

%v determines the eigen vector and lambda being the eigen value

[v1 lamda]=eig(Rssj,Rnnj);h=v1(:,end);%entire eigen vector

endh=h./norm(h);%normalizing the filter

Table 9 for task coding 3-F

Task 3-G

The file beamforming gives the input signal containing the required values of d,Fs,S,N.

close all

clear allclc

load beamforming%loading the given signal


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Table 10 for task coding 3-G

Figure 3g-a. Beam pattern using signal and noise ratio interference.

From the figure, it is deduced that the beam formation happened in the very central part in the

entire direction from -90 to 90. Even when the angles exceed,there is no change in the

beampattrn which tells that the gain is extremely strong in the direction( at all the angles) but

% values are given as mentioned

L=25;%taps

H=snirbf(S,N,L);%sig. to noise nd interf. filter

% H=[1:4;4:7];

F=(0:200)/201*Fs/2;A=(-90:90);G=beampattern(H,d,Fs,F,A);%function call of beam pattern

imagesc(A,F,G);

xlabel('Direction (degrees)');

ylabel('frequency (Hz)');


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only at the central frequency. Then, it is attenuated and is slowly faded out at the other remaining

part of frequency, out of (1950 -2050 KHz) for Fs-8KHz.

Task 3-H

The beamformer filter is designed to have the linear constrained minimum variance. This is

developed in order to minimize the output power of the beamformer.

The beamformer filter is defined by,

Where w is the normalised frequency as stated.

Thus, the fourier transform can be rewritten as,

We can very well understand from the constraint that it minimizes the output power, it seems to

be very much similar to the Minimum Variance Spectrum Estimation.

It is given that the minimum delay is given by,

As the sensors are placed at a certain distance in space,the delay filter is been designed which is

a pure delay,

So the total beamformer filter including the delay is given by,

Using vector stacking, the total transfer function is rewritten as,,

L being the length of the filter,

The o/p power in discrete form can be written as,

P =


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Then, this equation can be written as,

hThe minimization problem is given by, (This derivation is similar to the LECTURE 7 for

minimum variance spectrum estimation)

The constraint that is subject to a certain frequency and at certain direction shall be passed

through the beamformer undistorted which makes the total beamformer filter shall be unity.

) which means, Constrained optimization using Lagrange Multiplier,

J Because, the lagrange multiplier converts the constrained problem into non-constrained problem.

So the cost function J(h) results as above.

Choose h and to minimize J,

So, acc. to constraint, , being unknown.Therefore,

Inserting this into the optimal filter equation,

This is the optimal filter coefficient of a beamformer in estimating the power at frequency

and

at the direction ofundistorted.This function lcmvbfdefines the Linear Constrained Minimum Variance Beamformer which hassome specific frequency at specific direction in which a signal is passed, undistorted.

%% Function to determine the Linear constrained Minimum Variance Beamformer

function h=lcmvbf(K,L,t0,f0,Fs,d)


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Table 11 for task coding 3-H

Task 3-I

LCMV Beamformer is designed with the given file as the input signal.

Rxx=eye(L); %identity matrix of L

for k=0:K-1%initialization

for l=0:L-1%initialization

C=344;%speed of soundmindelay=(sin(t0*pi/180)*d*Fs)/(C); %relative minimum delay of signal

w=f0/Fs;%omegae=exp(-j*2*pi*l*w);Dfilter=exp(-j*2*pi*w*k*mindelay); %Delayed filter

e1(l+1,k+1)=Dfilter*e;

end

e2=e1(:,k+1);h(:,k+1)=(inv(Rxx)*e2)/(e2'*inv(Rxx)*e2); %optimal filter coefficients of the LCMV

beamformer filter

end


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Table 12 for task coding 3-I

close allclear all

clc

load beamforming

L=25;

K=4;

% values are given as mentioned% h=[1:4;4:7];%tried with this values

F=(0:200)/201*Fs/2;%frequency range from 0 to Fs/2

A=(-90:90);%direction from -90 to 90

C=344;%speed of soundt0=45;

f0=2000;

H=lcmvbf(K,L,t0,f0,Fs,d);% function call of LCMV beamformer filterfigure

G=beampattern(H,d,Fs,F,A);%function call of forming a beam pattern

% imagesc(A,F,G)plot(A,G)%plot against direction and gain

xlabel('Direction (degrees)');

ylabel('gain (dB)');figure

G=beampattern(H,d,Fs,F,45);

% imagesc(A,F,G)% wi=linspace(0,pi,Fs);% tried with line spacing

plot(F,(G))%plot against frequency and gain with constant direction of 45 degree

xlabel('frequency (Hz)');

ylabel('gain (dB)');


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Figure 3i-a. Beam pattern using LCMV with direction between -90 to 90.

Figure 3i-b. Beam pattern using LCMV with direction of 45 degrees and varying frequency.


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From the figure a and b, it is deduced that both of them satisfy the optimization criterion and

design constraint. its is seen that, the gain is strong and high at the direction of -20 to 100 and

more. In the lesser angles than -20, the beam pattern starts attenuating. Also, the beam pattern

shows a sinc with high main lobe at 45 degrees at 2000Hz without any distorted beamformation.

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