The Reliability Function of Variable-Length Lossy Joint Source-Channel Coding with Feedback

We consider transmission of discrete memoryless sources (DMSes) across discrete memoryless channels (DMCs) using variable-length lossy source-channel codes with feedback. The reliability function (optimum error exponent) is shown to be equal to $\max \{0, B(1-R(D)/C)\}$ , where $R(D)$ is the rate-distortion function of the source, $B$ is the maximum relative entropy between output distributions of the DMC, and $C$ is the Shannon capacity of the channel. We show that in this asymptotic regime, separate source-channel coding is, in fact, optimal.