Prediction Polynomial

This example shows how to obtain the prediction polynomial from an autocorrelation sequence. The example also shows that the resulting prediction polynomial has an inverse that produces a stable all-pole filter. You can use the all-pole filter to filter a wide-sense stationary white noise sequence to produce a wide-sense stationary autoregressive process.

Create an autocorrelation sequence defined by

r(k)=(24/5)2|k|(27/10)3|k|k=0,1,2

k = 0:2;
rk = (24/5)*2.^(-k)-(27/10)*3.^(-k);

Use ac2poly to obtain the prediction polynomial of order 2.

A = ac2poly(rk);

The prediction polynomial of order 2 is

A(z)=15/6z1+1/6z2

Examine the pole-zero plot of the FIR filter to see that the zeros are inside the unit circle.

zplane(A,1)

The inverse all-pole filter is stable with poles inside the unit circle.

zplane(1,A)

Use the all-pole filter to produce a realization of a wide-sense stationary AR(2) process from a white noise sequence. Set the random number generator to the default settings for reproducible results.

rng default;
x = randn(1000,1);
y = filter(1,A,x);

Compute the sample autocorrelation of the AR(2) realization and show that the sample autocorrelation is close to the true autocorrelation.

[xc,lags] = xcorr(y,2,'biased');
[xc(3:end) rk']
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