4.2.4 Case 4: Weak Signal, Weak Background

In case 4, we let ${\bar n}_s = 10$, ${\bar n}_b = 1$, and $G=140$. It follows that $\rho_0 =3.58$, $\rho_+=1.28$, $\Delta=2.93$, $\beta_0=0.284$, and the Jacobian is given in Appendix B.

A full simulation was run at this operating point. The capacity sensitivity with respect to the fundamental parameters are shown in Fig. 8. As in all previous cases, the SNR parameter $\rho_0$ has the greatest effect on capacity. In this case, however, the skewness difference $\Delta$ has a greater effect at higher PPM orders than the excess SNR $\rho_+$. The sensitivity with respect to both $\Delta$ and $\rho_+$ are significantly higher than in the previous cases, because when both signal and background are weak, the difference of the variances or skewnesses in the signal and nonsignal slots becomes relatively more important in distinguishing signals. As in all previous cases, the fraction of the signal that is Webb distributed, $\beta_0$, plays a minor role at this operating point.

The capacity sensitivity with respect to the physical parameters is shown in Fig. 9 As in case 3, ${\bar n}_s$ and $\eta$ are the critical physical parameters. The background intensity ${\bar n}_b$ is even less influential than in cases 1-3. Reducing the background intensity incident on the detector would not be even as effective as, e.g., reducing the effective noise temperature of the detector.

Figure 8: Capacity sensitivity of $2^k$-PPM, with respect to fundamental parameters, case 4.

Figure 9: Capacity sensitivity of $2^k$-PPM with respect to physical parameters, case 4.