# Mitigating the Impact of Noise on Nonlinear Frequency Division Multiplexing +
## 2 NFDM System Setup  
The NFDM setup considered in this work is sketched in the upper part of Figure 1[13; 24] and assumes that only the continuous part of the nonlinear spectrum is modulated. The encoder generates a 16-QAM signal \(s(t)\) encoding \(N_{b}\) symbols with symbol rate \(R_{s}=1/T_{s}=50\,\mathrm{GBd}\), and adds a guard time of \(N_{z}\) null symbols to account for burst transmission and group velocity dispersion (GVD). A number of techniques to map the QAM signal \(s(t)\) to the nonlinear continuous spectrum exist. In this work, unless otherwise specified, the nonlinear inverse synthesis (NIS) mapping--one of the first proposed techniques [6]--is considered. The nonlinear spectrum \(\rho(\lambda)\) is obtained by  
\[\rho(\lambda)=-S(\lambda/\pi) \tag{1}\]  
where \(S(f)\) is the ordinary Fourier transform of \(s(t)\). Other possible modulation techniques are described in Section 5. According to the NFT theory, the channel effects are fully precompensated at the transmitter (TX) by multiplying the signal \(\rho(\lambda)\) by the term \(\exp\left(4j\lambda^{2}\mathcal{L}\right)\), with \(\mathcal{L}\) being the normalized link length. While channel compensation can equivalently be done at the TX or at the receiver (RX) for standard detection (and even split between TX and RX [9]), it should be performed at the TX when using the improved detection strategies discussed in Section 4. After precompensation, the backward NFT (BNFT) is performed, the signal is reversed in time (time inversion does not affect the system performance but is considered here for the sake of simplicity), and the optical signal is obtained through a digital-to-analog converter (DAC) and launched into the fiber. At the RX, an analog-to-digital converter (ADC) provides the samples \(\tilde{\mathbf{r}}\) of the received signal, from which the transmitted information is extracted by using a given detection strategy. The standard detection strategy--referred to as forward NFT (FNFT) detection--computes the FNFT of the received samples and obtains a noisy version of the scattering data \(a(\lambda)\) and \(b(\lambda)\). The nonlinear continuous spectrum--where the information was encoded--is obtained by \(\rho(\lambda)=b(\lambda)/a(\lambda)\). However, noise (and, possibly, numerical inaccuracies) can make \(|a(\lambda)|\) very close to zero, causing \(\rho(\lambda)\) to diverge and the performance to drop. This effect can be partially mitigated with the simple expedient of replacing \(|a(\lambda)|\) with the average of the values in the adjacent \(\lambda\) when it is smaller than a certain threshold [27], here taken equal to \(10^{-1}\). After the FNFT, the detection proceeds as in a conventional receiver: the nonlinear spectrum undergoes matched filtering and sampling, and the decisions are made according to a minimum Euclidean distance criterion, usually implemented by means of multiple thresholds on the in-phase and quadrature components. Other detection strategies are discussed in Section 4.  
The physical channel is a single-mode fiber of length \(L=2000\,\mathrm{km}\), attenuation \(\alpha=0.2\,\mathrm{dB/km}\), GVD parameter \(\beta_{2}=-20.39\,\mathrm{ps^{2}/km}\), and nonlinear coefficient \(\gamma=1.22\,\mathrm{W^{-1}km^{-1}}\), along which ideal distributed amplification with spontaneous emission factor \(\eta_{\mathrm{sp}}=4\) is considered. The bandwidth of both the DAC and the ADC is \(100\,\mathrm{GHz}\).  
The system performance is reported in terms of \(Q_{\mathrm{dB}}^{2}\), where the \(Q\)-factor is evaluated from the bit error rate (BER), measured through direct error counting [13]. The performance is shown either as a function of the power \(P=E_{s}/T_{s}\), with \(E_{s}\) being the average symbol energy, or as a function of the rate efficiency \(\eta=N_{b}/(N_{b}+N_{z})\), which is changed by varying \(N_{b}\)[9; 13]. The number of null symbols is set to \(N_{z}=2000\)[13].