4.1 Bit error rate vs. background level

We used Eq. (13) to determine the lowest bit error rate theoretically possible for PPM signaling using the Nd:YAG laser and SLiK APD. The capacity was determined by numerically evaluating Eq. (5) and plugging into Eq.s (1); substitution into (13) gives the bound on bit error rate. Fig.s 3-5 indicate the bounds for $M=256$,

Figure 3: Comparison of RS performance to Shannon limit. Parameters: $M=256$, $\bar n_s=100$, $T_s=31.25$ns, and the SLiK APD detector. ($T_d$ is an irrelevant parameter.)

Figure 4: Comparison of RS performance to Shannon limit. Parameters: $M=64$, $\bar n_s=100$, $T_s=31.25$ns, and the SLiK APD detector. ($T_d$ is an irrelevant parameter.)

Figure 5: Shannon limit on BER of 2-PPM. Parameters: $M=2$, $\bar n_s=100$, $T_s=31.25$ns, and the SLiK APD detector. ($T_d$ is an irrelevant parameter.)

$M=64$, and $M=2$, respectively. As can be seen, when operating at a BER of $10^{-6}$, the use of rate 7/8 codes promises the ability to withstand background levels over 40dB stronger than an uncoded system. Rate 7/8 Reed-Solomon (RS) codes operate within 3.5dB of the limit for rate 7/8 codes. In an uncoded system with $M=256$ we must have $\bar n_b \le 0.001$ in order to achieve a BER of $10^{-6}$; with a RS(255,224) code we required $\bar n_b \le 7.1$; and capacity implies $\bar n_b \le 16.0$. Note in Table 2 that when $M=64$, a RS code is further from capacity than when $M=256$.

Table 1: This is a test.
Table 2: Maximum background light that can be handled while operating with a coded BER of $10^{-6}$. The table indicates that codes operating at the Shannon limit can withstand 2.3 to 7.6dB higher levels of background light, compared to RS codes. Parameters: $M=256,64,2$, $R_c = 7/8$ or $1/2$, $\bar n_s=100,T_s=31.25$ ns, SLiK detector.

$M$	$R_c$	$\bar n_b$, Maximum	$\bar n_b$, RS coding	Difference (dB)
256	7/8	16.0	7.1	3.5
64	7/8	29.3	5.1	7.6
2	7/8	115	-	-
256	1/2	37.8	22.5	2.3
64	1/2	69.9	30.5	3.6
2	1/2	475	-	-