Application to the Poisson channel

In the case of PPM on a Poisson channel,

$$p_0(k) = \frac{n_b^k e^{-n_b}}{k!}$$ (7)

$$p_1(k) = \frac{(n_s+n_b)^k e^{-(n_s+n_b)}}{k!},$$ (8)

where $n_b$ represents the average number of background counts detected in a slot, and $n_s$ represents the average number of signal counts detected in a signal slot. Figure 1 shows the symbol error

Figure 1: The symbol error rate of 64-PPM on a Poisson channel, with $n_b = 1$, as computed using (1), (3), and (6).

rate as a function of $n_s$, when $n_b = 1$ and $M=64$. Using (1) or (2), the error rate computation became inaccurate whenever the true error rate was below 0.01. This is because the square-bracket term in (1) evaluated to zero (e.g., $(1+10^{-17})^{64}-1$ is evaluated as zero) for significant terms of the sum. Using (3), the computation becomes inaccurate for error rates below approximately $10^{-15}$. Using (6), the error rate could be accurately computed for for error rates down to $10^{-323}$, which is the limit of representable floating point numbers in the IEEE 754 double precision format.