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An optimal linear filter for random signals with realisations in a separable Hilbert space

Published online by Cambridge University Press:  17 February 2009

P. G. Howlett ,
C. E. M. Pearce  and
A. P. Torokhti
P. G. Howlett
Affiliation:
Centre for Industrial and Applicable Mathematics, University of South Australia.
C. E. M. Pearce
Affiliation:
School of Applied Mathematics, The University of Adelaide, Adelaide, SA 5005, Australia; e-mail: cpearce@maths.adelaide.edu.au.
A. P. Torokhti
Affiliation:
Centre for Industrial and Applicable Mathematics, University of South Australia and School of Applied Mathematics, The University of Adelaide.
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Abstract

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Let u be a random signal with realisations in an infinite-dimensional vector space X and υ an associated observable random signal with realisations in a finite-dimensional subspace Y ⊆ X. We seek a pointwise-best estimate of u using a bounded linear filter on the observed data vector υ. When x is a finite-dimensional Euclidean space and the covariance matrix for υ is nonsingular, it is known that the best estimate û of u is given by a standard matrix expression prescribing a linear mean-square filter. For the infinite-dimensional Hilbert space problem we show that the matrix expression must be replaced by an analogous but more general expression using bounded linear operators. The extension procedure depends directly on the theory of the Bochner integral and on the construction of appropriate HilbertSchmidt operators. An extended example is given.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 4 , April 2003 , pp. 485 - 500
Copyright
Copyright © Australian Mathematical Society 2003

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