FIM Regularity for Gaussian Semi-Blind MIMO FIR Channel Estimation

We consider the problem of semi-blind (SB) channel estimation for multiple-input multiple-output (MIMO) finite impulse response (FIR) channels. We motivate a Fisher information matrix (FIM) based analysis of the semi-blind estimation problem and demonstrate that the semi-blind FIM can be written as the sum of a blind FlM J^B and training symbol FTM J^t. We show that the blind FIM J^B is rank deficient and establish the minimum number of training symbols necessary to achieve regularity (full-rank) of the FIM for identifiability. We also illustrate that an SB scheme can potentially be very efficient compared to an exclusively training based scheme since it estimates very few un-constrained parameters. It is then demonstrated that the rank deficiency of the FIM for an irreducible MIMO FIR channel arises because of a unitary (rotation) matrix indeterminacy. Based on this analysis a semi-blind algorithm is proposed for MIMO FIR channel estimation