A unified coding scheme for hybrid transmission of Gaussian source over Gaussian channel

We show that when transmitting a Gaussian source over an average-power-constrained Gaussian channel with mismatched bandwidth, there exists an uncountable set of hybrid digital analog schemes which are optimal under the minimum mean squared error criterion. This generalizes the recent result by Bross et al. that for the bandwidth matched case, there exists an uncountable set of optimal schemes, with the uncoded transmission and the separation approach being the two extremes. The proposed schemes for bandwidth expansion and compression both require proper combination of various components such as power splitting, bandwidth splitting, rate splitting, Wyner-Ziv coding and dirty-paper coding. This set of schemes includes all the existing known optimal schemes as special cases. We show that this set of schemes can be applied to the broadcast scenario with three receivers, when the receiver with median channel SNR achieves optimal distortion, and it offers distortion tradeoff between of the good receiver and bad receiver. Interestingly, though continuous, this tradeoff curve is in fact concave, implying that its performance is worse than a direct time-sharing approach in this three user scenario. We further show even in a more general broadcast setting with a continuum of receivers the time sharing scheme is better than any given scheme in this set; somewhat surprisingly, there exists a unique time-sharing ratio for this to hold.