If, however, we intend to use the PE filter, then we have some details to attend to, and this begins to get complicated. Reviewing the Levinson recursion, we find that gaps internal to the filter tend to fill as the recursion proceeds. The filter is not as sparse as the reflection coefficients. We'll need to keep track of the nonzero filter coefficients. We need to keep track of them in order that we have a PEF that we can use in polynomial division because polynomial division is an essential part of finding missing data with the Burg method and polynomial division is a part of preconditioning the conjugate-gradient method.

A promising thought is that perhaps the Burg recursion can be run backwards. Since this would take a PEF (or its reflection coefficients) and white inputs (forward and backward prediction errors) and create a colored outputs, it seems analogous to polynomial division.

When I began multidimensional filtering studies I was ignorant of the helix and thus had not the opportunity to use the Burg or the Levinson methods. Stability was not an issue until we began to do preconditioning using polynomial division.

4/27/2000