2.1.3 PPM demodulator, decoder

Typically, the individual slot statistics at the output of the detector are not available to the decoder.¹Instead, for $1 \le i \le n$, a PPM demodulator uses the $M$ slot statistics of ${\bf Y}_i$ to make an $M$-PPM symbol decision $Z_i \in \{0,\ldots, M-1\}$ by choosing the slot within each symbol that maximizes the number of detected photons, or in case of a tie, by randomly choosing a slot among those with the maximum statistic. It has recently been proven that this is the maximum likelihood rule for PPM detection when the statistics are governed by the sum of a Webb and Gaussian deviates[19]. Perhaps surprisingly, the maximum likelihood rule becomes more complicated than ``pick the largest'' when the detector output is approximated by a Gaussian distribution, in which a nonsignaling slot has mean $\mu_b$ and variance $\sigma_b^2$ and a signaling slot has mean $\mu_b + \mu_s$ and variance $\sigma_b^2 + \sigma_s^2$. We avoid this problem by not using the Gaussian approximations.