Channel and frequency offset estimation is a classic topic with a large body of prior work using mainly maximum likelihood (ML) approach together with Cramer-Rao Lower bounds (CRLB) analysis. We provide the maximum a posteriori (MAP) estimation solution which is particularly useful for for tracking where previous estimation can be used as prior knowledge. Unlike the ML cases, the corresponding Bayesian Cramer-Rao Lower bound (BCRLB) shows clear relation with parameters and a low complexity algorithm achieves the BCRLB in almost all SNR range. We allow the time invariant channel within a packet to have arbitrary correlation and mean. The estimation is based on pilot/training 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 belief propagation, the estimation of the previous (segment of) packet can serve as the prior knowledge for the next (segment of) packet.
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