Energy-distortion tradeoff with multiple sources and feedback

The energy-distortion tradeoff for lossy transmission of sources over multi-user networks is studied. The energy-distortion function E(D) is defined 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 E_bmin 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 multiple-access 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.