Spectra factorization

Suppose now that we want to apply the same procedure to obtain the factors of spectral functions of (Z-transforms). Let S(Z) be the auto-correlation that we seek to factor into causal and anticausal parts. Burg (1998, personal communication) recognized that we can use the Newton method to factor by writing an equation equivalent to ():  

If we now divide () by we obtain   

Equation () leads to the Wilson-Burg algorithm:

1.

Compute the right side of () by polynomial division forwards and backwards and then add 1.

2.

Abandon negative lags, to only keep the positive powers of the Z polynomial, and also keep half of the zero lag. Now you have A_t+1(Z)/A_t(Z).

3.

Multiply out (convolve) the denominator A_t(Z). Now we have the desired result A_t+1(Z).

Iterate as long as you wish.