# A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
## 5 Applications of NFT and INFT on an Optical Communication System
The data are encoded only on \(b(\lambda,0)\) as part of the nonlinear spectrum with the following transformation:  
\[b(\lambda,0)=AS(\lambda,0) \tag{66}\]  
where:  
\[A=\frac{\sqrt{1-e^{-|S(\lambda,0)|^{2}}}}{\sqrt{\left|S(\lambda,0)\right|^{2}}} \tag{67}\]  
The continuous scattering data \(a(\lambda,0)\) are related to \(b(\lambda,0)\) as:  
\[a(\lambda,0)=\sqrt{1-|b(\lambda,0)|^{2}}e^{j\mathcal{H}\left(\frac{\log\left(1 -|\lambda(\lambda,0)|^{2}\right)}{2}\right)} \tag{68}\]  
where \(\mathcal{H}(\cdot)\) denotes the Hilbert transform. The reflection coefficient will be denoted as \(\eta_{c,b}(\lambda,0)\). The details of b-modulation are already discussed in several other papers [30; 51; 52]. We conducted different simulation in MATLAB to transmit the three modulated signals in an optical fiber medium of total length 1600 km. The fiber was divided into 20 spans of 80 km each, where at the end of each span, white noise originating from distributed Raman amplification was added, with a noise figure of 4 dB. The optical fiber dispersion parameter is \(\beta_{2}=-20.4\) ps\({}^{2}\)/km, and the nonlinear coefficient is \(\gamma\) = 1.3 W\({}^{-1}\) km\({}^{-1}\). At the receiver side, we are going to measure the performance using the Q-factor in dB found directly from the constellation affected by the white noise, as in [53]:  
Figure 5:0  
For historical reasons, the performance in optical systems are usually measured in terms of the quality Q-factor, given as: \(Q=\sqrt{2}erfc^{-1}(2 BER)\), where \(erfc^{-1}\) is the inverse complementary error function, and \(BER\) is the bit error rate. For a fixed \(BER=10^{-3}\), Q-factor is equal to 9.8 dB. Systems with a Q-factor of less than 9.8 dB will fail to work, as the number of errors is very large. We can think of this value as a lower bound for our system to work. The reason for why we are using Equation (69) as a quality parameter is because the number of transmitted symbols in our numerical simulation is only 1024, and the minimum \(BER\) that we can achieve is only \(4.8\times 10^{-4}\). To fully investigate the system, even for lower \(BER\), we use Equation (69). Because the oversampling factor used in this work is 40, we cannot increase the number of symbols to achieve a lower \(BER\), as the computation cost will be very high. The local oscillator at the receiver side has no frequency drift and has a linewidth of 100 kHz. The total number of symbols transmitted was \(2^{10}\). We added 25% of overhead to satisfy Step 1 of the direct NFT algorithm. An oversampling factor of 40 was selected for the simulations. The system performance for the three different modulations applying our algorithm are shown in Figures 6 and 7. In Figure 6, we show the Q-factor versus the average power in the input of the optical fiber for all three modulations. We observe that the b-modulation performs better compared to the other two modulations, which on the other hand, show very similar performances. The maximum value of the Q-factor, Q = 25.9 dB, is achieved when the average power is equal to \(-1\) dBm, for the b-modulation shown in the blue line in Figure 5:1 Compared to the other two modulations, the Q-factor is increased by 2 dB. If the average power is less than \(-1\) dBm, the performance is deteriorating as the Q-factor is decreasing. The reason behind this is the noisy channel assumed in the system, which is added by the distributed optical Raman amplifiers assumed over the optical links. If the average power is greater than \(-1\) dBm, then the Q-factor is also decreasing due to the nonlinear effects and the numerical errors from many approximation values in our method. Figure 5:2 Q-factor versus input average power for all three modulations for a 50 Gbaud QPSK modulation over 1600 km of optical fiber. In blue line is shown the b-modulation; in black line, we show the modulation with a normalized reflection coefficient \(\eta_{C,n}\), given in Equation (64), and in red line, the normal modulation with reflection coefficient \(\eta_{C,n}\), given in Equation (63).