# PAM-\(N\)--Fundamental Limits in Chromatic Dispersive-Uncompensated Channels
## 7 BER Analysis
Analysis results for the attenuation penalty vs. fiber length (km) for 1 st, 2 nd, and 3 rd order SG filters with a 3 dB BW at 40 GHz and 80 GHz are shown in Figure 11a,b, respectively. The dashed curve stands for Equation (34) (blue curve in Figure 7). Figure 10: Bit error rate (BER) curves vs. optical input power for different fiber lengths (km) using a 2 nd order SG filter with 3 dB BW at 40 GHz. Figure 9: PAM-4 short-reach transmission. The purpose of the comparison to the wide (80 GHz) filter's performance is to emphasize the effect of the narrow one. The penalty was evaluated at a BER of 1e-3 and plotted vs. the fiber length. Therefore, the penalty is with respect to the same system but without the fiber (which is already at a 14 dB loss). The simulation results show that beyond 3.3 km there is a drastic decline in the BER performance. Referring to a BER value of 0.5, this result is consistent with the fundamental limit of 3.77 km (see Table 1) and with the theory of Equation (34). In the PAM-8 case presented in Figure 12, declines in the BER performance occur for distances larger than 2.3 km, which is again consistent with the fundamental limit of 2.62 km. Moreover, as can be seen from these data, a narrower spectral bandwidth is associated with a greater dependence on the SG order. This conduct is consistent with the following reasoning. Using Equation (37), the SG equivalent of Equation (28) can be written as follows:  
\[m=-\frac{\ln\left[-2\frac{\ln\left(\pi/4\right)}{\ln 2}\right]}{2\ln(2 BW/ \Delta)}\cong\frac{0.18}{\ln(2 BW/\Delta)} \tag{38}\]  
Figure 11: Attenuation penalty vs. fiber length (km) of PAM-4 with SG filter orders of 1, 2, 3, and theory (Equation (34)) with 3 dB BW at (**a**) 40 GHz and (**b**) 80 GHz. Figure 12: Penalty vs. fiber length (km) of PAM-8 with SG filter orders of 1, 2, and 3 (in legend) with 3 dB BW at (**a**) 40 GHz and (**b**) 80 GHz. In the presence of noise, for a given channel's noise level and a given BER value, the power penalty (in dB) from a fiber's length \(z_{0}\) to \(z\) can be approximated by the simple expression \(26.44N\beta_{2}^{2}B^{4}\big{(}z^{2}-z_{0}^{2}\big{)}\). To further validate the analytical analysis, simulations of PAM-4 and PAM-8 short-reach transmissions were introduced in Figures 10-12. The simulation results showed dispersion limitations that were consistent with the fundamental dispersion limitation presented in Table 1, and the spectral bandwidth in the SG filter order behavior (Section 6) was consistent with the analytical solution derived in Equation (32). Moreover, the power penalty showed high agreement with the analytical analysis of additive noise and dispersion presented in Section 6. Conceptualization, E.G. and G.K.; methodology, E.G. and G.K.; software, E.G. and G.K.; validation, E.G. and G.K.; formal analysis, E.G. and G.K.; investigation, E.G. and G.K.; resources, E.G. and G.K.; data curation, E.G. and G.K.; writing--original draft preparation, E.G. and G.K.; writing--review and editing, E.G. and G.K.; visualization, E.G. and G.K.; supervision, E.G. and G.K.; project administration, E.G. and G.K.; funding acquisition, E.G. and G.K.. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The authors declare no conflict of interest.