Polynomial division

Kolmogorov cross-spectral factorization, therefore, provides a tool to factor the helical 1-D filter of length 2N_x + 1 into minimum-phase causal (and maximum-phase anti-causal) filters of length N_x +1. Deconvolution with minimum-phase filters is unconditionally stable. However, inverse-filtering with the entire filters would be an expensive operation. Fortunately, filter coefficients drop away rapidly from either end. In practice, small-valued coefficients can be safely discarded, without violating the minimum-phase requirement; so for a given grid-size, the cost of the matrix inversion scales linearly with the size of the grid.

The unitary form of equation () can be maintained by factoring the right-hand-side matrix, ${\bf A}$ in equation (), with Kolmogorov before applying it to ${\bf q}_z$.

$$\begin{eqnarray} {\bf L} {\bf L}^T \; {\bf q}_{z+\Delta z} & = & \left({\bf L} ... ... {\bf L}^T}{\left( {\bf L} {\bf L}^T \right)^\ast} \; {\bf q}_{z}\end{eqnarray}$$ (44) (45)

Chapter  and Appendix  extend the concept of recursive inverse filtering to handle non-stationarity. There are pitfalls associated with this process, however; consequently, in this Chapter I limit the examples to the constant velocity case.