MAP Joint Frequency and Channel Estimation for MIMO Systems with Spatial Correlation

Carrier frequency offset (CFO) and channel estimation is a classic topic with a large body of prior work using the maximum likelihood (ML) approach together with the Cramér-Rao lower bound (CRLB) analysis. We give the maximum a posteriori probability (MAP) estimation solution which is particularly useful for tracking. Unlike the ML cases, the corresponding Bayesian CRLB (BCRLB) shows a clear relation with parameters and a low complexity algorithm achieves the BCRLB in almost all SNR range. We allow the time invariant MIMO channel within a packet to have arbitrary spatial correlation and mean. The estimation is based on pilot signals. An unexpected result is that the joint MAP estimation is equivalent to an individual MAP estimation of the frequency offset first, again different from the ML results. We provide insight on the pilot/training signal design based on the BCRLB. Unlike past algorithms that trade performance and/or complexity for the accommodation of time varying channels, the MAP solution provides a different route for dealing with time variation.