A strong converse for joint source-channel coding

We consider a discrete memoryless joint source-channel setting. In this setting, if a source sequence is reconstructed with distortion below some threshold, we declare a success event. We prove that for any joint source-channel scheme, if this threshold lower (better) than the optimum average distortion, then the success probability approaches zero as the block length increases. Furthermore, we show that the probability has an exponential behavior, and evaluate the optimal exponent. Surprisingly, the best exponential behavior is attainable by a separation-based scheme.