Autoregressive all-pole model parameters — modified covariance method
a = armcov(x,p)
[a,e] = armcov(x,p)
a = armcov(x,p) uses the modified covariance method to fit a pth-order autoregressive (AR) model to the input signal, x, which is assumed to be the output of an AR system driven by white noise. This method minimizes the forward and backward prediction errors in the least-squares sense. The output array, a, contains the normalized estimates of the AR system parameters, A(z), in descending powers of z. a has p + 1 columns. If x is a vector, then a is a row vector. If a is a matrix, then the coefficients along the nth row of a model the nth column of x.
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 modified covariance 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 = armcov(y,4)
arcoeffs = 1.0000 -2.7741 3.8404 -2.6841 0.9360
Generate 50 realizations of the process, changing each time the variance of the input noise. Compare the modified-covariance-estimated variances to the actual values.
nrealiz = 50; noisestdz = rand(1,nrealiz)+0.5; randnoise = randn(1024,nrealiz); for k = 1:nrealiz y = filter(1,A,noisestdz(k) * randnoise(:,k)); [arcoeffs,noisevar(k)] = armcov(y,4); end plot(noisestdz.^2,noisevar,'*') title('Noise Variance') xlabel('Input') ylabel('Estimated')
Repeat the procedure using armcov's multichannel syntax.
realiz = bsxfun(@times,noisestdz,randnoise); Y = filter(1,A,realiz); [coeffs,variances] = armcov(Y,4); hold on plot(noisestdz.^2,variances,'o') q = legend('Single channel loop','Multichannel'); q.Location = 'best';
Let y(n) be a wide-sense stationary random process obtained by filtering a white noise input with variance e with the system function A(z). If Py(ejω) is the power spectral density of y(n), then
Because the method characterizes the input data using an all-pole model, the correct choice of the model order, p, is important.