Rational modeling and linear prediction of random fields
Krogmeier, James Vincent
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Permalink
https://hdl.handle.net/2142/23374
Description
Title
Rational modeling and linear prediction of random fields
Author(s)
Krogmeier, James Vincent
Issue Date
1990
Doctoral Committee Chair(s)
Arun, K.S.
Department of Study
Electrical and Computer Engineering
Discipline
Electrical and Computer Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Electronics and Electrical
Language
eng
Abstract
An approach to the two-dimensional spectrum estimation problem is proposed that is based upon modeling a random field as the output of a rational linear system driven by the innovations of the field. A variety of linear prediction problems, each depending upon the definition of past, may be formulated for random fields; consequently, there are an equal number of innovations representation models. For the rational modeling application, a good innovations representation model should provide a finite parametrization of the spectrum estimation problem; therefore, the model should be a rational linear system when the spectrum of the random field is itself rational.
In this dissertation, a non-causal linear interpolation problem is posed and solved by generalizing some results of one-dimensional linear interpolation theory. Spectral conditions are found for the regularity and determinism of a random field with respect to two-dimensional interpolation. The form of the non-causal innovations representation filter and the spectrum of the non-causal innovations are derived. It is shown that the interpolation problem admits a unique Wold decomposition. These results are compared to earlier work on random field linear prediction using causal definitions of past. It is shown that only the interpolation problem gives rise to a rational innovations representation filter for all fields with rational spectra.
In order to develop an estimation procedure for the parameters of the non-causal innovations representation filter, a theory of generalized Hankel forms is given for non-causal two-dimensional linear systems. A Kronecker theorem is proven for these forms relating their rank and null vectors to the rationality of the non-causal system, its minimal order, and its transfer function. This theory is used in an algorithm for rational spectrum estimation.
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