Abstract
The problem of optimal smoothing for noncausal two-dimensional (2-D) systems is formulated and solved based on the 2-D descriptor (generalized state-space) representation.
The problem is to obtain the linear least square estimate of the random signal corrupted by observation noise. The random signal to be estimated is the output of a noncausal 2-D shaping filter driven by white noise.
It has been shown by the authors that a noncausal 2-D system whose two-sided (z, s)-transform is rational can be represented by a 2-D descriptor system.
In the paper, after formulating the best estimation problem by describing the random signal as the output of a 2-D descriptor system driven by white noise, we obtain a set of linear equations of the best estimate.
Next, we convert the set of linear equations into the linear equations over a commutative algebra of two-sided Z-transforms.
We show that under some assumptions on the 2-D descriptor realization of the random signal shaping filter, the above equations can be solved iteratively for the best estimate of that signal.
Finally, a numerical example and simulation results for design of optimal smoothers based on the proposed method are presented.
