Diophantine Equations for k-Step
Ahead Prediction and Fixed Lag Smoothing:
Recursive and Analytical Solutions
Glasgow, Scotland, April 15-16, 1999.
Diophantine equations constitute an important conceptual
tool for solving problems which are formulated in the
polynomial systems framework. Besides spectral
factorization, Diophantine equations appear in most
H2-optimal (MSE) solutions. It is therefore essential that the
numerical procedures, used to solve these equations,
provide adequate performance.
Although reliable and efficient algorithms exist,
it is of great interest to take advantage of situations
where analytical and recursive solutions can be obtained.
For signals which can be modelled as multi-variable
ARMA-processes, contaminated by either white or
coloured noise, we present a set of predictors, filters
and fixed-lag smoothers which can be obtained
with a minimal use of numerical procedures.
If the noise is coloured, the Diophantine equation,
associated with the signal estimation problem, has to be
solved numerically for one value of the prediction horizon,
or the smoothing lag. Once the solution is obtained, the
whole set of predictors and smoothers, up to a predefined
prediction horizon or smoothing lag, can be obtained by
analytical expressions, based on the previously obtained
If the noise is white, no numerical solution at all will be required.
PhD Thesis by Lars Lindbom,
deriving these results
and applying them to the design of adaptive filters.
Paper on design of adaptation
algorithms, with a derivation and motivation of the equations
that were presented here.
Book Chapter on H2-design
of model-based nominal and robust filters.