Murdoch University Research Repository

Welcome to the Murdoch University Research Repository

The Murdoch University Research Repository is an open access digital collection of research
created by Murdoch University staff, researchers and postgraduate students.

Learn more

Stability of filters in one and two dimensions

Eaton, Linda-Jane (2000) Stability of filters in one and two dimensions. PhD thesis, Murdoch University.

PDF - Whole Thesis
Available Upon Request


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.

Item Type: Thesis (PhD)
Murdoch Affiliation(s): Division of Science and Engineering
Notes: Note to the author: If you would like to make your thesis openly available on Murdoch University Library's Research Repository, please contact: Thank you.
Supervisor(s): UNSPECIFIED
Item Control Page Item Control Page