Joint Source–Channel Codes for MIMO Block-Fading Channels

We consider transmission of a continuous amplitude source over an $L$-block Rayleigh-fading $M_t \times M_r$ multiple-input multiple-output (MIMO) channel when the channel state information is only available at the receiver. Since the channel is not ergodic, Shannon's source–channel separation theorem becomes obsolete and the optimal performance requires a joint source–channel approach. Our goal is to minimize the expected end-to-end distortion, particularly in the high signal-to-noise ratio (SNR) regime. The figure of merit is the distortion exponent, defined as the exponential decay rate of the expected distortion with increasing SNR. We provide an upper bound and lower bounds for the distortion exponent with respect to the bandwidth ratio among the channel and source bandwidths. For the lower bounds, we analyze three different strategies based on layered source coding concatenated with progressive superposition or hybrid digital/analog transmission. In each case, by adjusting the system parameters we optimize the distortion exponent as a function of the bandwidth ratio. We prove that the distortion exponent upper bound can be achieved when the channel has only one degree of freedom, that is $L=1$, and $\min\{M_t,M_r\}=1$ . When we have more degrees of freedom, our achievable distortion exponents meet the upper bound for only certain ranges of the bandwidth ratio. We demonstrate that our results, which were derived for a complex Gaussian source, can be extended to more general source distributions as well.