State estimation of stochastic systems with switching measurements: a polynomial approach

The state estimation problem is here investigated for a class of stochastic linear switching-output systems, in which the output matrix switches in a finite set of possible values according to a not directly measured discrete Markov sequence. This note presents a real-time algorithm, based on the optimal polynomial filtering approach, which achieves the simultaneous estimation of both the continuous system state and the switching parameter. The state and observation noises do not need to be Gaussian. It is shown that the optimal filter of degree one (best affine filter) does not solve the parameter estimation problem, due to a structural first-order unobservability property, and therefore the use of higher-order filters becomes necessary. As an application of the proposed filter, the problem of the online simultaneous estimation of the transmitted signal and of the impulse response samples of a multipath fast-fading digital communication channel is considered in this paper. Differently from other approaches, the polynomial filter solves the problem without the use of training sequences (preambles) in the transmitted data, so that the information flow through the channel is not interrupted.