(9) |

(10) |

In an analogous way, we can take the general relation from Table 3

and convert it to We can then consider a symmetrical relation where on the left side we insert the anticausal part of the spectrum, and obtain Finally, we can sum the preceding two equations and get(11) |

Equation (11) represents our general formula for
spectral factorization. If we consider the particular case when *G* is
*X*_{n}, we obtain equation (10), which we have shown
to be equivalent to the Wilson-Burg formula.

From the computational standpoint, our equation is more expensive than
the Wilson-Burg because it requires two more convolutions on the
numerator of the right-hand side. However, our equation offers more
flexibility in the convergence rate. If we try to achieve a quick
convergence, we can take *G* to be *X*_{n} and get the Wilson-Burg
equation. On the other hand, if we worry about the stability,
especially when some of the
roots of the auto-correlation function are close to the unit circle,
and we fear losing the minimum-phase property of the factors,
we can take *G* to be some damping function, more tolerant of
numerical errors.

Moreover, by using the Equation (11), we can achieve
fast convergence in cases when the auto-correlations we are
factorizing have a very similar form, for example, in nonstationary
filtering. In such cases, the solution at the preceding step can be
used as the *G* function in the new factorization. Since *G* is already
very close to the solution, the convergence is likely to occur quite
fast.

4/20/1999