Performance of the Optimum Receiver for Pulse-Position Modulation Signals With Avalanche Photodiode Statistics

Statistical modeling of the avalanche photodiode electron output is discussed. The exact McIntyre{Conradi density function for the output electron count is reviewed, along with the Webb and Gaussian approximations. In addition, the maximum-likelihood decision statistic for detection of pulse-position-modulation (PPM) signals is derived using the more accurate Webb density rather than the Poisson or Gaussian approximations that have been used previously. It is shown that for Webb-distributed output electrons, the maximum-likelihood rule is to choose the PPM word corresponding to the slot with the maximum electron count. Error probabilities for 256-ary PPM signals are calculated using the maximum-electron-count decision rule in the absence of thermal noise. The detection of weak optical signals is hampered by the presence of additive thermal noise, generated by resistors in the electrical circuit, in the measurement. This problem can be overcome by the use of photomultiplier tubes (PMTs) or avalanche photodiodes (APDs), both of which amplify the electrical current generated by absorbed photons. PMTs have very high gain and, hence, can be used to detect individual photons. However, the spectral response of these devices is limited to visible wavelengths. APDs provide less gain but remain useful over a wider range of wavelengths for overcoming thermal noise.