Minimum distortion transmission of Gaussian sources over fading channels

We consider the problem of reliably communicating the output of a Gaussian source across average power constrained multi-antenna channels. We consider both fading and constant gain channels, and assume that in the former case the receiver has perfect channel state information. We are interested in designing simple memoryless encoding and decoding strategies to minimize the mean-square error between the transmitted and reconstructed vectors. We find the optimal encoder and decoder pair when the encoder is restricted to the class of linear transformations on the input and evaluate its performance. We also derive the theoretically achievable optimal performance. For the constant gain channel, we find that linear coding can be optimal under certain conditions. For the Rayleigh fading channel, we prove that the performance of the linear encoder/decoder pair in the low SNR regime is the same as the optimal to the first order. We prove the same approximate optimality result for the Rician channel in some special cases.