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 SNR 2 + o(SNR). These constellations also attain the minimum bit-energy-to-noise-variance ratio of -1.59 dB at vanishing SNR. A natural question concerns the extent to which this universality extends to other common channel models. We consider here the discrete-time Poisson channel, frequently used to represent optical communication channels, and quantity the gap between the channel capacity and the mutual information for fixed signal constellations. As a by-product of our analysis, we also determine the asymptotic form of the capacity at vanishing signal energy. Consider a memoryless channel with input X and output Y given by the sum Y = S(X) + Z (1) of a noise Z and a signal component S(X), itself a function of the input X . The input X is a non-negative real number (i. e. it has units of energy), drawn from a unit-energy set X according to a probability distribution P (x). We let S(X) be distributed according to a Poisson distribution of parameter εsX , where εs is an average signal energy. The output components S(X), Z , and Y are nonnegative integers. A. Martinez is with Centrum Wiskunde & Informatica, The Netherlands. e-mail: [email protected]. January 4, 2014 DRAFT 2 We study three channel models: noiseless, with Z = 0; additive Poisson noise, where Z follows a Poisson distribution of mean εn > 0; and additive geometric noise, with Z distributed according to a geometric distribution of mean εn > 0. With additive Poisson noise the channel transition probability, denoted by Q(y|x), is given by Q(y|x) = esn (εsx + εn) y y! , (2) where εn ≥ 0. For the channel with geometric noise, we have Q(y|x) = y ∑ l=0 e 1 + εn ( εn 1 + εn y (