1 Low-Signal-Energy Asymptotics of Capacity and Mutual Information for the Discrete-Time Poisson Channel

The first terms of the low-signal-energy asymptotics for the mutual information in the discrete-time Poisson channel are derived and compared to an asymptotic expression of the capacity. In the presence of non-zero additive noise (either Poisson or geometric), the mutual information is concave at zero signal-energy and the minimum energy per bit is not attained at zero capacity. Fixed signal constellations which scale with the signal energy do not attain the minimum energy per bit. The minimum energy per bit is zero when additive Poisson noise is present and εn log 2 when additive geometric noise of mean εn is present. I. MOTIVATION AND NOTATION In the complex-valued Gaussian channel with signal-to-noise ratio SNR the mutual information of very general constellations (e. g. zero-mean with uncorrelated real and imaginary parts each of energy 1 2 [1]) has the same low-SNR asymptotics as the channel capacity, namely SNR − 1 2 SNR2 + o(SNR 2). These constellations also attain the minimum bit-energy-to-noise-variance ratio of-1.59 dB at vanishing SNR. A natural question concerns th