On non-stationary convolution and inverse convolution

James Rickett

james@sep.stanford.edu

ABSTRACT

Recursive inverse filtering with non-stationary filters is becoming a useful tool in a range of applications, from multi-dimensional inverse problems to wave extrapolation. I formulate causal non-stationary convolution and combination and their adjoints in such a way that it is apparent that the corresponding non-stationary recursive filters are true inverse processes. Stationary recursive inverse-filtering is stable if, and only if, the filter is minimum-phase. I show that recursive inverse-filtering with a filter-bank consisting of minimum-phase two-point filters is also unconditionally stable. However, I demonstrate that, for a more general set of minimum-phase filters, stability of non-stationary recursive inverse-filtering is not guaranteed.