Robust deconvolution for ARMAX models with Gaussian uncertainties

In this paper we propose a robust deconvolution filter design that optimises a functional motivated by the a posteriori probability of the signals to be estimated. The problem is formulated in the framework of uncertain linear systems represented by discrete-time input–output ARMAX models, where the uncertainty is modelled as the realisation of a stochastic process with known statistics. The design is based on the use of a horizon of measurements in such a way that, for FIR systems, the functional to be optimised coincides with the one that maximises the a posteriori probability (MAP); and for ARMAX systems, the functional converges to the MAP functional as the length of the horizon is increased. The goal is to estimate signals with Gaussian or truncated Gaussian probability density functions based on measurements correlated with them. The robust design shows a very significant improvement, in a probabilistic sense for different systems, of the relative standard deviation of the estimation error when compared with the nominal model filter design.