The stability of stationary recursive inverse filtering depends on the phase of the causal filter: if (and only if) the filter is minimum phase, then its inverse filter is stable. This raises the question: is non-stationary inverse filtering stable if all filters contained in the filter-bank are minimum-phase?
For the case of inverse filtering with a two-point filter (N_{a}=2), equation (9) reduces to x_{0}=y_{0}, and the following formula for k>0:
x_{k} = y_{k} - a_{1,(k-1)} x_{k-1}. | (13) |
(14) |
(15) |
There is a larger class () of stable non-stationary recursive filters that can be obtained by repeatedly multiplying stable bidiagonal matrices. However, given a general non-stationary filter matrix, there is no straightforward way to determine whether it is a member of this stable class. In fact, it is relatively easy to find an example filter-bank consisting of minimum-phase individual filters whose recursive output is unbounded for finite input. Consider the filter-bank, , consisting of minimum-phase filters,
(16) |
one
Figure 1 Impulsive input (a) and response (b) to non-stationary filtering with filter-bank defined in equation (16). |
The instability stems from the fact that as N increases, so does the number of boundaries between different filters. Such rapid non-stationary filter variations, as in the example above, are pathological in the context of seismic applications, where filters are typically quasi-stationary. For these applications instability is rarely observed; however, we must be aware that we are not dealing with an unconditionally stable operator, and instability may rear its ugly head from time-to-time.