Maximum a Posteriori Probability (MAP) Joint Fine Frequency Offset and Channel Estimation for MIMO Systems With Channels of Arbitrary 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 low complexity algorithms are provided to achieve the BCRLB in almost all SNR range. Among them, the universal algorithm takes a new approach and avoids error floors of the traditional approach. We assume that the time invariant MIMO channel within a packet can have spatial correlation and nonzero 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. Within a short enough (segment of) packet where the channel and CFO are approximately time invariant, the low complexity algorithm can be employed. Similar to the belief propagation, the estimation of the previous (segment of) packet can serve as the prior knowledge for the next (segment of) packet.