  A scenario involving a source, a channel, and a destination, where the
destination is interested in {\em both} reliably reconstructing the message
transmitted by the source and estimating with a fidelity criterion the state of
the channel, is considered. The source knows the channel statistics, but is
oblivious to the actual channel state realization. Herein it is established
that a distortion constraint for channel state estimation can be reduced to an
additional cost constraint on the source input distribution, in the limit of
large coding block length. A newly defined capacity-distortion function thus
characterizes the fundamental tradeoff between transmission rate and state
estimation distortion. It is also shown that non-coherent communication coupled
with channel state estimation conditioned on treating the decoded message as
training symbols achieves the capacity-distortion function. Among the various
examples considered, the capacity-distortion function for a memoryless Rayleigh
fading channel is characterized to within 1.443 bits at high signal-to-noise
ratio. The constrained channel coding approach is also extended to multiple
access channels, leading to a coupled cost constraint on the input
distributions for the transmitting sources.
