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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C_{J}</tex> bit/J as compared to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> bit/s, to emphasize the energy efficiency and to deemphasize the speed of the telemetry. The energy channel capacity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C_{J}</tex> , for any waveform channel with well-defined capacity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> bit/s at signal power <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> watts, is defined as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C/S</tex> bit/J. It is shown that for coherent binary frequency shift keying (BFSK) and waveform channels, the supremum of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C_{J}</tex> over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> is approached as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> approaches zero. For the Gaussian channel the best coding uses narrow bands with the highest S/N.