  We consider transmission of stationary and ergodic sources over non-ergodic
composite channels with channel state information at the receiver (CSIR).
Previously we introduced alternate capacity definitions to Shannon capacity,
including the capacity versus outage and the expected capacity. These
generalized definitions relax the constraint of Shannon capacity that all
transmitted information must be decoded at the receiver. In this work alternate
end-to-end distortion metrics such as the distortion versus outage and the
expected distortion are introduced to relax the constraint that a single
distortion level has to be maintained for all channel states. For transmission
of stationary and ergodic sources over stationary and ergodic channels, the
classical Shannon separation theorem enables separate design of source and
channel codes and guarantees optimal performance. For generalized communication
systems, we show that different end-to-end distortion metrics lead to different
conclusions about separation optimality even for the same source and channel
models.
