4.3 Optimization of PPM order

Fig. 4 begs the question of what PPM order optimizes the data rate. For nighttime reception in which $\bar n_b \ll 1$, the optimal PPM order is near $M=2048$. This closely follows the discussion in Section 2.2.2 regarding the errorless channel. For daytime reception in which $\bar n_b \approx 100$, we can see from Fig. 4 that the optimal PPM order is well under 256. To be more precise, the order of PPM that maximizes capacity in bits per second can be seen directly from a plot of capacity versus $M$. This is shown in Fig. 5, and the optimal PPM orders for various values of $\bar n_b$ are summarized in Table 2.

Figure 5: Capacity of $M$-PPM on an optical channel, with $P_b = 10^{-6}$, $\bar n_s=100$, $\bar n_b\in\{0.1,1,10,50,100\}$, $T_s=31.25$ ns, $T_d = 432000$ ns, and the SLiK APD detector.

Table 2: Optimal PPM orders $M$ when $P_b = 10^{-6}$, $\bar n_s=100$, $\bar n_b\in\{0.1,1,10,50,100\}$, $T_s=31.25$ ns, $T_d = 432000$ ns, and the SLiK APD detector.

$\bar n_b$	Optimal $M$
0.1	2036
1	1815
10	634
50	52
100	18

This suggests use of a multiple PPM order communications system. During nighttime reception it should use $M$ on the order of thousands, and during daytime reception it should use $M$ on the order of dozens. Unoptimized PPM orders can be costly. As can be seen from Fig. 5, using $M=2036$ during the day would be disastrous for the data rate. Using $M=18$ at night reduces capacity by over half.