Causal with causal inverse

Theoretically, a PEF is a causal filter with a causal inverse. This adds confidence to the likelihood that deconvolution of natural processes with a PEF might get the correct phase spectrum as well as the correct amplitude spectrum. Naturally, the PEF does not give the correct phase to an ``all-pass'' filter. That is a filter with a phase shift but a constant amplitude spectrum. (I think most migration operators are in this category.)

Theoretically we should be able to use a PEF in either convolution or polynomial division. There are some dangers though, mainly connected with dealing with data in small windows. Truncation phenomena might give us PEF estimates that are causal, but whose inverse is not, so they cannot be used in polynomial division. This is a lengthy topic in the classic literature. This old, fascinating subject is examined in my books, FGDP and PVI. A classic solution is one by John Parker Burg. We should revisit the Burg method in light of the helix.