4.2.3 Case 3: Weak Signal, Strong Background

In this weak signal, strong background case, we let $\bar n_s = 10$, $\bar n_b = 100$ and $G=30$. Here, $\rho_0 =0.0995$, $\rho_+=1.76$, $\Delta=6.08$, and $\beta_0=0.566$, 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. 4. Here, the SNR $\rho_0$ plays the only non-negligible role, with $\rho_+$, $\Delta$, and $\beta_0$ more than two orders of magnitude behind. This implies that when the signal strength is low, capacity is almost completely a function of one SNR parameter, and is very sensitive to the precise signal level.

Indeed, we see in Fig. 7 that ${{\bar n}_s\over C} \vert{\partial C\over\partial {\bar n}_s}\vert$ and ${\eta\over C} \vert{\partial C\over\partial \eta}\vert$ are much higher than in cases 1 and 2. In fact, they are greater than one, meaning a more than one-for-one return on investment is possible. Capacity sensitivity with respect to other physical parameters are similar to cases 1 and 2.

Figure 6: Capacity sensitivity of $2^k$-PPM, with respect to fundamental parameters, case 3.

Figure 7: Capacity sensitivity of $2^k$-PPM with respect to physical parameters, case 3.