Energy-Distortion Tradeoff with Multiple Sources and Feedback

Abstract The energy-distortion tradeoff for lossy transmission of sources over multi-user networks is studied. The energydistortion function E(D) is de�ned as the minimum energy required to transmit a source to the receiver within the target distortion D, when there is no restriction on the number of channel uses per source sample. For point-to-point channels, E(D) is shown to be equal to the product of the minimum energy per bit Ebmin and the rate distortion function R(D), indicating the optimality of source-channel separation in this setting. It is shown that the optimal E(D) can also be achieved by the Schalkwijk Kailath (SK) scheme, as well as separate coding, in the presence of perfect channel output feedback. Then, it is shown that the optimality of separation in terms of E(D) does not extend to multi-user networks. The scenario with two encoders observing correlated Gaussian sources in which the encoders communicate to the receiver over a Gaussian multipleaccess channel (MAC) with perfect channel output feedback is studied. First a lower bound on E(D) is provided and compared against two upper bounds achievable by separation and an uncoded SK type scheme, respectively. Even though neither of these achievable schemes meets the lower bound in general, it is shown that their energy requirements lie within a constant gap of E(D) in the low distortion regime, for which the energy requirement grows unbounded. It is shown that the SK based scheme outperforms the separation based scheme in certain scenarios, which establishes the sub-optimality of separation in this multi-user setting. I

Joint Source-Channel Coding with Time-Varying Channel and Side-Information

Transmission of a Gaussian source over a time-varying Gaussian channel is studied in the presence of time-varying correlated side information at the receiver. A block fading model is considered for both the channel and the side information, whose states are assumed to be known only at the receiver. The optimality of separate source and channel coding in terms of average end-to-end distortion is shown when the channel is static while the side information state follows a discrete or a continuous and quasiconcave distribution. When both the channel and side information states are time-varying, separate source and channel coding is suboptimal in general. A partially informed encoder lower bound is studied by providing the channel state information to the encoder. Several achievable transmission schemes are proposed based on uncoded transmission, separate source and channel coding, joint decoding as well as hybrid digital-analog transmission. Uncoded transmission is shown to be optimal for a class of continuous and quasiconcave side information state distributions, while the channel gain may have an arbitrary distribution. To the best of our knowledge, this is the first example in which the uncoded transmission achieves the optimal performance thanks to the time-varying nature of the states, while it is suboptimal in the static version of the same problem. Then, the optimal \emph{distortion exponent}, that quantifies the exponential decay rate of the expected distortion in the high SNR regime, is characterized for Nakagami distributed channel and side information states, and it is shown to be achieved by hybrid digital-analog and joint decoding schemes in certain cases, illustrating the suboptimality of pure digital or analog transmission in general.Comment: Submitted to IEEE Transactions on Information Theor