Stability of filters in one and two dimensions

Linda-Jane Eaton

Filters frequently occur in signal processing and statistical time series analysis. Of particular interest is the question of stability. That is, does a filter map lp-sequences in its domain to lq-sequences, and if it does, is it continuous as an operator between subsets of lp and lq? There has been much investigation into the stability of non-adaptive one-dimensional filters when p and q are equal. We present a general theory that allows p and q to differ. In particular, we find that a filter may be (1, ∞)-stable even though it is not (p, q)-stable for other values of p and q. Although there has been some research into the stability of adaptive one-dimensional filters, no general theory has been obtained. A broad class of adaptive filters can be described by operators which are rational functions of weighted shifts. A general theory for the stability of these so called “weighted” filters is presented. The study of two-dimensional filters is also of great interest to signal analysts, especially with recent developments in image processing. Most known results focus on the (p,q)-stability of quarter-plane filters, and only for p = q = ∞. We add to these results by considering stability for other values of p and q. We also investigate the stability of filters whose domain and range is the space of sequences supported on a half-plane. As far as the author knows there has been little investigation into these questions in the past. There are also limited results in the literature regarding the stability of adaptive two-dimensional filters. We define the class of weighted two-dimensional filters, and investigate the stability of both weighted quarter-plane and weighted half-plane filters.

Stability of filters in one and two dimensions

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