Channel capacity in bits per joule

Underwater acoustic telemetry has a total input energy constraint, since the energy is stored in the transmitter's batteries. This work is primarily rephrasing the work in channel capacity in terms of C₉ bit/J as compared to C bit/s, to emphasize the energy efficiency and to deemphasize the speed of the telemetry. The energy channel capacity C₉, for any waveform channel with well-defined capacity C bit/s at signal power S watts, is defined as C/S bit/J. It is shown that for coherent binary frequency shift keying (BFSK) and waveform channels, the supremum of C₉ over S is approached as S approaches zero. For the Gaussian channel the best coding uses narrow bands with the highest S/N.