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
(2):

(4) |

If we now divide (4) by we obtain

(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*).

7/5/1998