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# A comparison with the Wilson-Burg method

For reasons of symmetry, we can take Newton's 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
 (9)
which can easily be shown to be equivalent to the Wilson-Burg relation
 (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 Xn, 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 Xn 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.

Next: Conclusions Up: Sava & Fomel: Spectral Previous: Spectral factorization
Stanford Exploration Project
4/20/1999