4.2.1 Case 1: Strong Signal, Strong Background, Optimized Gain

Here, we consider an EG&G SliK APD detector with physical parameters $\eta=0.38$, $F=2.42572$, $I_b=40$fA, $I_s=2.00$nA, $T=300$K, and $R_L=179.7$k$\Omega$. It has been shown [6,7] that for this set of parameters, $G=65$ is the optimum gain for hard-decision detection of 256-PPM. This is also a good estimate of the gain which maximizes capacity on the soft-decision channel, which turned out to be $G=59$. We use a Q-switched Nd:YAG laser modulated with a slot width of $T_s=31.25$ns. A high signal strength ${\bar n}_s=100$ is incident on the detector, and a high background level ${\bar n}_b=100$ is also present, which corresponds to reception on a clear, sunny day.

Plugging these parameters into Eq.s (13-16), it follows that $\rho_0 =13.7$, $\rho_+=15.7$, $\Delta=45.3$, and $\beta_0=0.873$. The likelihood function $L(v)$ computed for these parameters is shown in Fig. 1.

Figure 1: Likelihood function $L(v)$ over the range stored in the look-up table, case 1.

Using the finite differences method described above, the partial derivative of capacity with respect to each fundamental parameter, i.e., the components of the gradient, were computed. These components, when normalized as in Eq. (17), give the capacity sensitivity with respect to each of the fundamental parameters, which is shown in Fig. 2. Note that by far, the SNR parameter $\rho_0$ has the greatest effect on capacity, followed by the excess SNR parameter $\rho_+$. The blending fraction $\beta_0$ and skewness difference $\Delta$ play a lesser role.

The Jacobian was evaluated (see numerical value in Appendix B) and used to determine the $\partial C\over\partial x$ for each physical parameter $x$. The capacity sensitivity with respect to the physical parameters is shown in Fig. 3.

Figure 2: Capacity sensitivity of $2^k$-PPM, with respect to fundamental parameters, case 1.

Figure 3: Capacity sensitivity of $2^k$-PPM with respect to physical parameters, case 1.

As is no surprise, the signal intensity ${\bar n}_s$ and background intensity ${\bar n}_b$ are two of the most important physical parameters. Curiously, capacity sensitivity w.r.t. ${\bar n}_s$ is approximately twice the sensitivity w.r.t. ${\bar n}_b$. This contrasts with the usual AWGN channel, where signal and noise affect capacity in precisely equal amounts, i.e., only through their ratio. Also, note that sensitivity of ${\bar n}_s$ is the sum of the sensitivities of ${\bar n}_b$ and $\eta$. This is a consequence of the fact that the number of absorbed photons is proportional to $\eta$.

The most capacity-sensitive detector parameter is the quantum efficiency $\eta$, which at the operating point shown is even more influential than the background intensity. Capacity is also sensitive to the excess noise ratio $F$. The slot width $T_s$, noise temperature $T$, and load resistance $R_L$ play lesser roles, which are nearly equal because of their occurence together in Eq. (13) and (16). Capacity sensitivity with respect to $I_b$ and $I_s$ are both more than two orders of magnitude lower than the other parameters. This is because $I_s$ and $I_b$ contribute only neglibly to $\rho_0$ and $\beta_0$ at this operating point compared to the other physical parameters in Eq. (13) and Eq. (16). Since the gain has been optimized, capacity is not sensitive to the gain.