Further results on Bayesian and deterministic CRBs in the context of blind SIMO channel estimation

The performance of channel estimation is often assessed by deriving the proper Cramér-Rao Bound (CRB). However, in the blind case a special treatment is required due to the singularity of the Fisher Information Matrix (FIM). Usually a constraint is introduced to overcome the blind ambiguity and ensuing singularity. Hence, a constrained CRB has been derived in the literature since a long time ago. Although this constrained CRB has been proven to be a valid lower bound in the medium and high SNR regimes, it fails completely in the low SNR regime because unlike the MSE it does not saturate. Motivated by the shortcoming of the constrained CRB, we derive in this paper a modified constrained CRB (MCCRB) for deterministic blind channel estimation. The MCCRB is valid over the whole SNR regime. In the second part of the paper we address Bayesian blind channel estimation and explore the apparent discrepancy between channel unidentifiability with a non-singular FIM. We highlight that in the less familiar Bayesian case this relationship needs to be interpreted differently. The analytical formulas for the introduced bounds are validated by some Monte-Carlo simulations.