Use a vector of polynomial coefficients to generate an AR(4) process by filtering 1024 samples of white noise. Reset the random number generator for reproducible results. Use the Yule-Walker method to estimate the coefficients.

rng default

A = [1 -2.7607 3.8106 -2.6535 0.9238];

y = filter(1,A,0.2*randn(1024,1));

arcoeffs = aryule(y,4)

arcoeffs = 1×5

    1.0000   -2.7262    3.7296   -2.5753    0.8927

Generate 50 realizations of the process, changing each time the variance of the input noise. Compare the Yule-Walker-estimated variances to the actual values.

nrealiz = 50;

noisestdz = rand(1,nrealiz)+0.5;

randnoise = randn(1024,nrealiz);
noisevar = zeros(1,nrealiz);

for k = 1:nrealiz
    y = filter(1,A,noisestdz(k) * randnoise(:,k));
    [arcoeffs,noisevar(k)] = aryule(y,4);
end

plot(noisestdz.^2,noisevar,'*')
title('Noise Variance')
xlabel('Input')
ylabel('Estimated')

Repeat the procedure using the function's multichannel syntax.

Y = filter(1,A,noisestdz.*randnoise);

[coeffs,variances] = aryule(Y,4);

hold on
plot(noisestdz.^2,variances,'o')
hold off
legend('Single channel loop','Multichannel','Location','best')

Use a vector of polynomial coefficients to generate an AR(2) process by filtering 1024 samples of white noise. Reset the random number generator for reproducible results.

rng default

y = filter(1,[1 -0.75 0.5],0.2*randn(1024,1));

Use the Yule-Walker method to fit an AR(10) model to the process. Output and plot the reflection coefficients. Only the first two coefficients lie outside the 95% confidence bounds, indicating that an AR(10) model significantly overestimates the time dependence in the data. See AR Order Selection with Partial Autocorrelation Sequence for more details.

[ar,nvar,rc] = aryule(y,10);

stem(rc)
xlim([0 11])
conf95 = sqrt(2)*erfinv(0.95)/sqrt(1024);
[X,Y] = ndgrid(xlim,conf95*[-1 1]);
hold on
plot(X,Y,'--r')
hold off
title('Reflection Coefficients')