Capacity of Pulse-Position Modulation (PPM) on Gaussian and Webb Channels

This article computes the capacity of various idealized soft-decision channels modeling an optical channel using an avalanche photodiode detector (APD) and pulse-position modulation (PPM). The capacity of this optical channel depends in a complicated way on the physical parameters of the APD and the constraints imposed by the PPM orthogonal signaling set. This article attempts to identify and separate the effects of several fundamental parameters on the capacity of the APD-detected optical PPM channel. First, an overall signal-to-noise ratio (SNR) parameter is de ned such that the capacity as a function of a bit-normalized version of this SNR drops precipitously toward zero at quasi-brick-wall limits on bit SNR that are numerically the same as the well-understood brick-wall limits for the standard additive white Gaussian noise (AWGN) channel. A second parameter is used to quantify the effects on capacity of one unique facet of the optical PPM channel (as compared with the standard AWGN channel) that causes the noise variance to be higher in signal slots than in nonsignal slots. This nonuniform noise variance yields interesting capacity effects even when the channel model is AWGN. A third parameter is used to measure the effects on capacity of the difference between an AWGN model and a non-Gaussian model proposed by Webb (see reference in [2]) for approximating the statistics of the APD-detected optical channel. Finally, a fourth parameter is used to quantify the blending of a Webb model with a pure AWGN model to account for thermal noise. Numerical results show that the capacity of M-ary orthogonal signaling on the Webb channel exhibits the same brick-wall Shannon limit, (M ln 2)=(M 1), as on the AWGN channel ( 1:59 dB for large M). Results also compare the capacity obtained by hard- and soft-output channels and indicate that soft-output channels o er a 3-dB advantage.