Conclusions

There are three important points to draw from this Appendix. Firstly, I have formulated 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. If you think of causal non-stationary filtering as a lower triangular matrix, then recursive inverse filtering applies the inverse matrix.

The second important point is that recursive inverse-filtering with a filter-bank consisting of minimum-phase two-point filters is unconditionally stable, and as such it is totally safe to apply in any circumstance.

However, the final point is that for a more general set of minimum-phase filters, stability of non-stationary recursive inverse-filtering is not guaranteed: use with care.