Spectra Lo Dev 2

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1- a) Do the following for ten realizations:i) Generate a signal y[n], N = 512 samples, using the Narrowband AR processin page 148 in thetextbook.

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ii) Estimate the parameters using the autocorrelation method.From Equation parameters of the real system:

Numerator = [ 1 0 0 0 0]

Denumerator = [1 -1.6408 2.2044 -1.4808 0.814 ]

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Then true spectrum of the system is :

Using autocorrelation method I found the estimated parameters like this :

Estimate numerator = [1.0692 0 0 0 0]

Estimate Denumerator =

[1 -1.66284812274577 2.15974478610830 -1.45767759588340 0.768503414833484]

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b) Plot all pole-zero diagrams on the same Figure.

c) Plot all spectra on the same figure.

After calculating the estimated parameters , I obtain the power spectral density for ten realizations.

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zeros-poles diagrams

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And average estimate spectrum of y[n] :

2- COVARIANCE METHODEstimated Parameters using covariance method :



[1 -1.648098544485497 2.192103123925523 -1.473035990919552 0.795286555553020]

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Estimate Power Spectral Density plots for ten different signal

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Plot of average estimate power spectral density :

3- Add white,Gaussian noise to one of the realizations such that SNR is 5 dB. Estimate theparameters using the autocorrelation method for different model order values and plot the

spectrum.

When we add noise pole-zero plot :

For model order P = 4;

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[1 -1.166358269907974 1.331382505298467 -0.642160877669500 0.388587690561766]

Estimate Numerator = [ 3.4825 0 0 0 0]

For Model order P = 6 ;

Pole-zero diagrams :

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[1 -0.955753388093975 1.076882471542758 -0.032532620043360 -0.125738012157119

0.457459488845452 0.073733723380570]

Estimate Numerator = [ 2.3689 0 0 0 0 0 0]

Real Denumerator = [1 -1.6408 2.2044 -1.4808 0.814 0 0]

Real Numerator = [ 1 0 0 0 0 0 0 ]

4- Comment on the result

Autocorrelation Method

First of all I calculated the parameters using the autocorrelation method. Autocorrelation

method is more simple method, but it produces lower resolution than the others. To understand it, I

plot the Estimate spectrum and the true spectrum on the same figure. As you see , blue one ,or Estimate

spectrum is not equal the true spectrum. It is because of low resolution of the autocorrelation method.

In this code I used biased autocorrelation method because, unbiased autocorrelation matrix is not

guaranteed to be positive definite and the variance of the spectrum estimate tends to become larger

when the matrix is ill-conditioned. Therefore, biased autocorrelation estimates are generally preferred.

When I use real-valued noise , observed this values:


[1 -1.66284812274577 2.15974478610830 -1.45767759588340 0.768503414833484]

Real Denumerator =

[1 -1.6408 2.2044 -1.4808 0.814 ]

Estimate Numerator = [ 1.0692 0 0 0 0 ]

Real Numerator = [ 1 0 0 0 0 ]

As you see, using autocorrelation method , obtain this values which are close to real values.

When I use complex noise, observed this values :

Estimate Denumerator = [ 1 + 0.00000000000000i -1.60575309677584 + 0.0198466699814833i

2.12121494934365 - 0.0257820652567119i -1.40456996565833 + 0.0260092976526211i

0.768774605203358 - 0.0123184341509201i]


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As you see, values are complex and close to real values but not equal. Resolution is low !

For better observation , I plot the true spectrum and the estimate spectrum on the same figure. From

the figure, you can understand what I mean.

Covariance Method

This method is more complex than the autocorrelation method, but it has higher resolution

which is good. Covariance method has complex calculations so it takes so much time. It is disadvantage

of the covariance method. The advantage of the covariance method over the autocorrelation method is

that no windowing of the data is required in the formation of the covariance estimates Cxx(j,k). This

method is complex because covariance matrix is hermitian symmetric, positive semi definite but not

toeplitz so we cant use Levinson algorithm. So we use cholesky decomposition which has a lot of

calculation. In these method , estimated poles are not guaranteed to be inside the unit circle,however

usually they are inside the unit circle.

When I use real-valued noise , observed this values:


[1 -1.648098544485497 2.192103123925523 -1.473035990919552 0.795286555553020]

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200Difference Between True Spectrum & Estimate Spectrum

Estimate Spectrum

True Spectrum


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Real Denumerator =

[1 -1.6408 2.2044 -1.4808 0.814 ]


Real Numerator = [ 1 0 0 0 0 ]

As you see, using covariance method , obtain this values which are close to real values.

When I use complex noise, observed this values :

Estimate Denumerator = [1+ 0.000000000000000i -1.675746488771692 - 0.224387043236265i

2.231556726273705 + 0.242639761266443i -1.497106766815033 - 0.353961312167855i

0.812295745810523 + 0.055032847788367i]


As you see, values are complex and close to real values .

I say that this method is not guarantated to be inside the unit circle. So you can see from the figure

what I mean :

Because of noise is obtained by randn(x) , each iteration it changed. Some of the iterations I observed

poles are inside the unit circle and some of the iterations they are not.

To see the difference between estimated spectrum and true spectum :

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As you see from the figure, Estimate Spectrum is almost equal to the true spectrum. This means high

resolution!

Autocorrelation method by adding noise (5dB)

Model order = P = 4 ;

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Difference between true spectrum and estimate spectrum

Estimate Spectrum

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Model order = P = 6;

When I increase the model order P , our estimate system is distorted. P = 4 is the best choice to

estimate the system. P = 4 is equal to number of parameter ( except a(0) = 1).

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CODES :

Power Spectral Density Calculatin :

function PowerSpectralDensityOfOutput = PSD(num,den,i)

% Assume we use white noise so PSD of white noise = 1

% assume Transfer function order is 4

% num : numerator

% den : denumerator

% i : figure index, which figure you want to plot

PowerSpectralDensityOfError = 1; % Error ~ N(0,1)

w = -pi:0.001:pi;z = exp(1i*w);

H = (num(1) + num(2)*z.^-1 + num(3)*z.^-2 + num(4)*z.^-3 + num(5)*z.^-4) ./(den(1) + den(2)*z.^-1 + den(3)*z.^-2 +

den(4)*z.^-3 + den(5)*z.^-4);

PowerSpectralDensityOfOutput = ((abs(H)).^2)*PowerSpectralDensityOfError;

figure(i),hold all,plot(w,PowerSpectralDensityOfOutput,'m','lineWidth',1.5); grid on;

end

Autocorrelation Method :

clear all,close all;

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% GENERATING THE TRANSFER FUNCTION IN TIME DOMAIN

% H(z) = 1/A(z);

N = 512; P = 4 ; m = 10;

w = -pi:0.001:pi;

num = [1 0 0 0 0];

den = [1 -1.6408 2.2044 -1.4808 0.8145];

% Generation of Noise

% Real NoiseWhiteNoise(1,:) = randn(1,N*m);

% Complex Noise

% WhiteNoise(1,:) = (1/sqrt(2))*randn(1,N*m) + 1i*(1/sqrt(2))*randn(1,N*m);

% FOR TEN DIFFERENT REALIZATION

PowerSpectralDensitySum = 0;

for j = 1:m

% Output Signal

y(j,:) = filter(num,den,WhiteNoise((j*N-N+1):N*j));

% FIGURE OF TEN DIFFERENT REALIZATIONS y[n] IN THE SAME PLOT

figure(2),hold all,grid on, stem(y(j,:)),title('ten different realizations of y[n]');

% AUTOCORRELATION FUNCTION

AutoCorrelation = xcorr(y(j,:),y(j,:),'biased');

% Obtaining the AutoCorrelation Matrix

Ryy = zeros(P,P);

for k = 1:(P)

Ryy(:,(k)) = (AutoCorrelation((N+1-k):(N+P-k)))';

end

ryy = (AutoCorrelation((N+1):(N+P)))';

% Estimated Parameters

EstimateParams = (inv(Ryy))*(-ryy);

EstimateParams = [ones(1);EstimateParams];

TruePowerSpectralDensity = PSD(num,den,3);

title('True PowerSpectral Density');

sum2 = 0;

for n = 1:N

sum1 = 0;

for k1 = 2:P+1

if(n+1-k1) > 0

sums = (EstimateParams(k1))*y(j,(n+1-k1));

else

sums = 0;

end

sum1 = sums + sum1;

end

sum2 = (abs(y(j,(n)) + sum1 ))^2 + sum2;

end

EstimatedVariance = (1/N)*sum2;

EstimateDenum = [EstimatedVariance; 0 ;0; 0; 0];

PowerSpectralDensityOfOutput = PSD(EstimateDenum,EstimateParams,7);

title('Estimated PowerSpectral Density');

PowerSpectralDensitySum = PowerSpectralDensityOfOutput + PowerSpectralDensitySum;

end

% Plotting the average spectrum

AverageEstimateQyy = PowerSpectralDensitySum/m;

figure(8),hold all,grid on, plot(linspace(-pi,pi,length(AverageEstimateQyy)),AverageEstimateQyy),title('Average Estimate

PowerSpectralDensity of ten different realizations');

% Pole-Zero Diagram

[zeros,poles,k1] = tf2zp(num,den);


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[EstimateZeros,EstimatePoles,k2] = tf2zp(EstimateDenum',EstimateParams');

figure(9),hold on,zplane(zeros,poles),title('zeros-poles diagrams');

figure(9),hold on,zplane(EstimateZeros,EstimatePoles);

Covariance Method :

clear all,close all;% GENERATING THE TRANSFER FUNCTION IN TIME DOMAIN

% H(z) = 1/A(z);

N = 512; P = 4 ; m = 10;

w = -pi:0.001:pi;

num = [1 0 0 0 0];

den = [1 -1.6408 2.2044 -1.4808 0.8145];


% Real Noise

% WhiteNoise(1,:) = randn(1,N*m);

% Complex Noise

WhiteNoise(1,:) = (1/sqrt(2))*randn(1,N*m) + 1i*(1/sqrt(2))*randn(1,N*m);



EstimateParams = zeros(1,P);

% EstimateParams = [ones(1),EstimateParams];

for i = 1:m

% Output Signal

y(i,:) = filter(num,den,WhiteNoise((i*N-N+1):N*i));


for j = 0:P

for k = 0:P

Cyy = 0;

for n = P:(N-1)

if(n-k)>=1 || (n-j)>=1

Cxx = y(i,n-k+1) * conj(y(i,n-j+1));

else

Cxx = 0;

end

Cyy = Cxx + Cyy;

end

CyySeq( i, j+1 , k+1 ) = Cyy / ( N - P );

end

end

C( :, :, i ) = CyySeq( i, (2:end), (2:end));

c( :, i ) = -CyySeq( i, (2:end), 1 )';

DiagonalMatrix( :, : ) = diag( diag( C( :, :, i ), 0)' );

V = chol(C(:,:,i));

U = inv(sqrt(DiagonalMatrix))*V;

LowerTriangularMatrix = (U)';

y_Matrix( :, i ) = inv( LowerTriangularMatrix( :, : ) ) * c( :, i );

EstimateParams = inv( U ) * inv( DiagonalMatrix( :, : ) ) * y_Matrix( :, i );

EstimateParams = [ones(1); EstimateParams];

% Finding of the Estimated Variance


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sum2 = 0;

for n = 1:N

sum1 = 0;

for k1 = 2:P+1

if(n+1-k1) > 0

sums = (EstimateParams(k1))*y(i,(n+1-k1));

else

sums = 0;end

sum1 = sums + sum1;

end

sum2 = (abs(y(i,(n)) + sum1 ))^2 + sum2;

end

EstimatedVariance = (1/(N-P))*sum2;




end



figure(8),grid on, plot(linspace(-pi,pi,length(AverageEstimateQyy)),AverageEstimateQyy),title('Average Estimate



% Pole-Zero Diagram





Autocorrelation method with noise (5dB):

clear all,close all;

% GENERATING THE TRANSFER FUNCTION IN TIME DOMAIN

% H(z) = 1/A(z);

N = 512; P = 4 ; m = 10;SNR = 5;

w = -pi:0.001:pi;

num = [1 zeros(1,P)];

den = [1 -1.6408 2.2044 -1.4808 0.8145 zeros(1,(P-4))];


% Real Noise

WhiteNoise(1,:) = randn(1,N*m);

% Complex Noise

% WhiteNoise(1,:) = (1/sqrt(2))*randn(1,N*m) + 1i*(1/sqrt(2))*randn(1,N*m);



for j = 1:m

% Output Signal

y(j,:) = filter(num,den,WhiteNoise((j*N-N+1):N*j));

y_with_noise(j,:) = awgn(y(j,:),SNR);


figure(2),hold all,grid on, stem(y_with_noise(j,:)),title('ten different realizations of y[n]');

% AUTOCORRELATION FUNCTION

AutoCorrelation = xcorr(y_with_noise(j,:),y_with_noise(j,:),'biased');

% Obtaining the AutoCorrelation Matrix

Ryy = zeros(P,P);

for k = 1:(P)


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Ryy(:,(k)) = (AutoCorrelation((N+1-k):(N+P-k)))';

end

ryy = (AutoCorrelation((N+1):(N+P)))';

% Estimated Parameters

EstimateParams = (inv(Ryy))*(-ryy);

EstimateParams = [ones(1);EstimateParams];


sum2 = 0;for n = 1:N

sum1 = 0;

for k1 = 2:P+1

if(n+1-k1) > 0

sums = (EstimateParams(k1))*y_with_noise(j,(n+1-k1));

else

sums = 0;

end

sum1 = sums + sum1;

end

sum2 = (abs(y_with_noise(j,(n)) + sum1 ))^2 + sum2;

end

EstimatedVariance = (1/N)*sum2;




end



figure(8),hold all,grid on, plot(linspace(-pi,pi,length(AverageEstimateQyy)),AverageEstimateQyy),title('Average Estimate


% Pole-Zero Diagram





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Spectra Lo Dev 2