Blind and semiblind channel estimation is a topic that enjoyed explosive developments throughout the nineties, and then came to a standstill, probably because of perceived unsatisfactory performance. Blind channel estimation techniques were developed and usually evaluated for a given channel realization, i.e. with a deterministic channel model. Such blind channel estimates, especially those based on subspaces in the data, are often only partial and ill-conditioned. On the other hand, in wireless communications the channel is typically modeled as Rayleigh fading, i.e. with a Gaussian (prior) distribution expressing variances of and correlations between channel coefficients. In recent years, such prior information on the channel has started to get exploited in pilot-based channel estimation, since often the pure pilot-based (deterministic) channel estimate is of limited quality due to limited pilots. In this paper we explore a Bayesian approach to (semi-)blind channel estimation, exploiting a priori information on fading channels. In the case of deterministic unknown input symbols, it suffices to augment the classical blind (quadratic) channel criterion with a quadratic criterion reflecting the Rayleigh fading prior. In the case of a Gaussian symbol model the blind criterion is more involved. The joint ML/MAP estimation of channels, deterministic unknown symbols, and channel profile parameters can be conveniently carried out using Variational Bayesian techniques. Variational Bayesian techniques correspond to alternating maximization of a likelihood w.r.t. subsets of parameters, but taking into account the estimation errors on the other parameters. To simplify exposition, we elaborate the details for the case of MIMO OFDM systems.
