Improper Complex-Valued Multiple-Model Adaptive Estimation

Motivated by the problem of estimating the discrete and continuous states of an improper complex-valued stochastic hybrid system, the paper proposes a class of widely linear (augmented) multiple model adaptive estimation algorithms, referred to as the C/MMAE. We show that for an improper complex-valued signal, pseudo-covariance of the innovation sequence is not zero and, therefore, carries useful statistical information regarding the unknown behaviour mode of the hybrid system. A new Bayesian law is, therefore, derived as a function of the pseudo-covariance of the innovation sequence and used to compute the probability that a hypothesized model is in effect at a certain time. We show that the C/MMAE, which uses the new Bayesian law and utilizes the complete second-order statistical characterization of the complex-valued innovation sequence, convergencies faster than its counterpart, which only uses the conventional covariance of the innovation sequence. In order to simplify the computational complexity, we develop two circularized versions of the C/MMAE using a preprocessing step, referred to as the circularizing filter (CF). The CF is incorporated to convert the improper observations/innovations into the proper ones in order to reduce the computational complexity of the hypothesis testing step. Finally, an interacting version of the C/MMAE, referred to as C/IMM, is developed for improper complex-valued systems with Markovian switching coefficients. Simulation results indicate that the proposed hybrid estimators provide improved performance and convergence properties over their traditional counterparts.