# PAM-\(N\)--Fundamental Limits in Chromatic Dispersive-Uncompensated Channels
## 7 BER Analysis  
The maximum transmission distance Equation (22) was determined for an ideal bare system. On the one hand, it did not incorporate passive or active dispersion compensation elements (to keep the system affordable), and on the other hand, it did not take into account degrading effects, such as signal chirp, polarization effects, jitter, spectral width, nonlinear effects, and noise (to find the fundamental system's limitations). The analysis in Section 6 did incorporate noise but used ideal filters. In this section, we validate the analytical results of the previous sections using a real parameters simulation. We investigate the channel's BER and its power penalty as a function of the fiber's length and filter's bandwidth. The consistency of these realistic results with the analytical evaluations presented in Equations (22) and (34) is investigated. To simulate the channel using commercial software for BER analysis, we take advantage of the equivalence between the smooth rectangular spectral filter (Equation (11)) and super Gaussian (SG) filters. Specifically, the following SG spectral filter:  
\[H(f)=\exp\left[-\frac{1}{2}\ln 2\left(\frac{2f}{FWHM}\right)^{2m}\right] \tag{36}\]  
Figure 7: Power penalty as a function of the fiber’s length. The solid curve stands for Equation (34) and the dashed curve stands for the approximation of Equation (35). The data (\(N=4\)) was transmitted at a rate of \(B=53\) Gbaud in the C-band. where _FWHM_ is the filter's spectral intensity full-width-at-half-maximum, and \(m\) is the SG's order, which can replace with great accuracy the smooth rectangular filter of Equation (11) if:  
\[m=\frac{\sqrt{\pi}}{\ln 2}\frac{\Delta}{\delta}\text{ and }FWHM=\Delta\cdot 2^{-1/2m}= \Delta\cdot 2^{-\delta\ln 2/2\Delta\sqrt{\pi}} \tag{37}\]  
Figure 8 illustrates the similarity between the smooth rectangular filter (Equation (11)) and the SG (Equation (30)) filter, provided the parameters for Equation (31) are chosen. The figure presents high agreement between the two. To study the BER performance of a realistic optical link, a simulation was conducted of a PAM-4 transmission using a combination of Matlab and VPI software tools over a standard single-mode fiber (ITU-T Rec. G.652). This system is presented in Figure 9. The electrical domain modulating signal of 53 Gbaud PAM-4 was generated by the digital-to-analog converter (DAC) modeled by Matlab and introduced an electrical output swing of 0.8 Vpp. This output signal was injected into the VPI simulation software, where the electrical DAC output signal passed through an SG LPF and was converted into the optical domain by a Mach-Zehnder modulator (MZM) with V\(\pi\) of 4 V. After this, the signal was transmitted over a short fiber (<4 km), converted to the electrical domain by a photodiode with referred input noise of 20\(p\frac{A}{\sqrt{Hz}}\), and passed through a 50 GHz (at 3 dB) LPF. The simulation's output consisted of samples of the receiver output, which in turn were loaded back to the BER processing. The LPF in the simulation is a combination of the detectors limited bandwidth and any additional filter, electrical, or digital associated with the receiver side. Figure 8: Comparison between smooth rectangular (SR) filter (solid curve) and its equivalent super Gaussian (SG) filter (dashed curve). Figure 10 presents nine BER vs. optical attenuation curves for different fiber lengths (km) using a 2 nd order SG filter with a 3 dB bandwidth (BW) at 40 GHz (half FWHM of the filter intensity). It was found that the maximum fiber length that enabled a pre-FEC BER value of \(1\times 10^{-3}\) was 3.2 km. This value was shorter than the theoretical value of 3.77 km obtained above. This mismatch was associated with some differences between the two analyses (realistic simulation vs. theoretical). First, the theoretical analysis was evaluated for a BER value of 0.5, and second, in the theoretical analysis, ideal (i.e., noiseless) detectors were considered, while in the more realistic simulation, additive (thermal) noise was introduced.