Energy-Distortion Tradeoffs in Gaussian Joint Source-Channel Coding Problems

The information-theoretic notion of energy efficiency is studied in the context of various joint source-channel coding problems. The minimum transmission energy <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> ) required to communicate a source over a noisy channel so that it can be reconstructed within a target distortion <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> is analyzed. Unlike the traditional joint source-channel coding formalisms, no restrictions are imposed on the number of channel uses per source sample. For single-source memoryless point-to-point channels, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> ) is shown to be equal to the product of the minimum energy per bit <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Eb</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> of the channel and the rate-distortion function <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> ) of the source, regardless of whether channel output feedback is available at the transmitter. The primary focus is on Gaussian sources and channels affected by additive white Gaussian noise under quadratic distortion criteria, with or without perfect channel output feedback. In particular, for two correlated Gaussian sources communicated over a Gaussian multiple-access channel, inner and outer bounds on the energy-distortion region are obtained, which coincide in special cases. For symmetric channels, the difference between the upper and lower bounds on energy is shown to be at most a constant even when the lower bound goes to infinity as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> → 0. It is also shown that simple uncoded transmission schemes perform better than the separation-based schemes in many different regimes, both with and without feedback.