The Stability of Digital Filters

A fundamental problem in the analysis of seismic traces is increasing the resolution of overlapping events. One method of attack is the use of so-called "inverse" digital filters. An inverse digital filter frequently has the undesirable characteristic that its memory function grows without limit with increasing time. Such inverse filters are called "unstable." In the present treatment, filter stability is investigated in some detail. In particular, a digital filter of the "minimum-delay" type is shown to have an inverse that consists only of a stable memory function. On the other hand, a filter of the "maximum-delay" type has an inverse that consists only of a stable anticipation function. A filter which is neither minimum-delay nor maximum-delay, but which is of the "mixed-delay" type, has a stable inverse, this inverse being made up of a memory component and an anticipation component. Filters with nonvanishing anticipation components are physically nonrealizable only if one wishes them to work in real time, that is, filters functioning in real time cannot operate on future values of the input. This difficulty may be overcome by making a complete and permanent record of the entire data sequence to be analyzed prior to any filtering operation. In this manner, filters need only operate on already available input values. Any filter design allowing one to delay the output until the entire input has been recorded will thus satisfy our requirements.