Minimum Energy Analysis for Robust Gaussian Joint Source-Channel Coding with a Distortion-Noise Profile

Minimum energy required to achieve a distortion-noise profile, i.e., a function indicating the maximum allowed distortion value for each channel noise level, is studied for transmission of Gaussian sources over Gaussian channels. A general coding scheme is proposed to upper bound the minimum energy needed for any distortion-noise profile. Conversely, utilizing a family of lower bounds originally derived for broadcast channels with power constraints, the minimum required energy is lower bounded for any profile. As examples, linear, exponential, square-law, and staircase profiles are studied. It is shown that for the linear profile, as well as for any concave profile, uncoded transmission is optimal. As a negative result, it is shown that exponential profiles are not achievable with finite energy. For square-law and staircase profiles, upper bounds and lower bounds are derived and the gaps between them are studied.