A Lower Bound on the Expected Distortion of Joint Source-Channel Coding

We consider the classic joint source-channel coding problem of transmitting a memoryless source over a memoryless channel. The focus of this work is on the long-standing open problem of finding the rate of convergence of the smallest attainable expected distortion to its asymptotic value, as a function of the blocklength n. Our main result is that in general the convergence rate is not faster than n(-1/2). In particular, we show that for the problem of transmitting i.i.d uniform bits over a binary symmetric channels with Hamming distortion, the smallest attainable distortion (bit error rate) is at least Omega(n(-1/2)) above the asymptotic value, if the "bandwidth expansion ratio" is above 1.