  Channel and frequency offset estimation is a classic topic with a large body
of prior work using mainly maximum likelihood (ML) approach together with
Cram\'er-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 Cram\'er-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.
