Spectra factorization

Suppose now that we want to apply the same procedure to obtain the factors of spectral functions of $Z=e^{i\omega \Delta t}$(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 $F(Z)=A(Z)\bar A(1/Z)$ by writing an equation equivalent to (2):

 

$$S(Z)-A_t(Z)\bar A_t(1/Z)= A_t( Z)[\bar A_{t+1}(1/Z)-\bar A_t(1/Z)]+ \bar A_t(1/Z)[ A_{t+1}( Z)- A_t( Z)]$$ (4)

If we now divide (4) by $\bar A_t(1/Z) A_t(Z)$ we obtain   

$${\bar A_{t+1}(1/Z) \over \bar A_t(1/Z)} \ +\ {A_{t+1}(Z) \over A_t(Z)} = 1 \ +\ {S(Z) \over \bar A_t(1/Z)\ A_t(Z)}$$ (5)

Equation (5) leads to the Wilson-Burg algorithm:

1.

Compute the right side of (5) 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.