ANZIAM  J.  44 (2003), 485-500
An optimal linear filter for random signals with realisations in a separable Hilbert space

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


Abstract
Let $u$ be a random signal with realisations in an infinite-dimensional vector space $X$ and $v$ an associated observable random signal with realisations in a finite-dimensional subspace $Y \subseteq X$. We seek a pointwise-best estimate of $u$ using a bounded linear filter on the observed data vector $v$. When $x$ is a finite-dimensional Euclidean space and the covariance matrix for $v$ is nonsingular, it is known that the best estimate $\hat{u}$ 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 Hilbert-Schmidt operators. An extended example is given.
Download the article in PDF format (size 117 Kb)
TeXAdel Scientific Publishing ©  Australian MS