Efficiency of Optimized Pulse Position Modulation with Noisy Direct Detection

We analyze theoretically the impact of background counts on the efficiency of optical communication in the photonstarved regime using the pulse position modulation (PPM) format with direct detection. Degradation of the photon information efficiency is studied in the case when the background count rate is at most comparable with the rate of photodetection events generated by the incoming optical signal. The PPM symbol length is optimized under the constraint of a fixed average spectral power density using an analytical approximation. The results are compared with generalized on-off keying (OOK) optimized over the a priori probability distribution for the input binary alphabet. The generalized OOK scheme can be viewed as a relaxation of the PPM scheme by removing the requirement that a light pulse must occur exactly once in each fixed-length frame of time bins that constitute the PPM symbol. It is shown that the asymptotic scaling of the photon information efficiency does not change qualitatively in this scenario.