A theoretical analysis of photon-arrival jitter in an optical pulse-position-modulation (PPM) communication channel has been performed, and now constitutes the basis of a methodology for designing receivers to compensate so that errors attributable to photon-arrival jitter would be minimized or nearly minimized. Photon-arrival jitter is an uncertainty in the estimated time of arrival of a photon relative to the boundaries of a PPM time slot. Photon-arrival jitter is attributable to two main causes: (1) receiver synchronization error [error in the receiver operation of partitioning time into PPM slots] and (2) random delay between the time of arrival of a photon at a detector and the generation, by the detector circuitry, of a pulse in response to the photon. For channels with sufficiently long time slots, photon-arrival jitter is negligible. However, as durations of PPM time slots are reduced in efforts to increase throughputs of optical PPM communication channels, photon-arrival jitter becomes a significant source of error, leading to significant degradation of performance if not taken into account in design.
For the purpose of the analysis, a receiver was assumed to operate in a photon-starved regime, in which photon counts follow a Poisson distribution. The analysis included derivation of exact equations for symbol likelihoods in the presence of photon-arrival jitter. These equations describe what is well known in the art as a matched filter for a channel containing Gaussian noise. These equations would yield an optimum receiver if they could be implemented in practice.
Because the exact equations may be too complex to implement in practice, approximations that would yield suboptimal receivers were also derived. One approximation is based on the assumption that the jitter in the arrival of each photon is independent. Another approximation is based on the assumption that only photon counts over finite time bins are available. Yet another approximation is based on the counts-over-finite-time-bins assumption with the additional assumption that the counts follow a Poisson distribution. For jitter with a standard deviation of 0.28 of a slot, computational-simulation tests have shown that receivers designed to compensate using the exact or approximate equations would exhibit error-rate reductions, relative to receiver designs based on neglect of photon-arrival jitter, equivalent to power increases of the order of 1 dB (see figure).