# Mitigating the Impact of Noise on Nonlinear Frequency Division Multiplexing +
## 3 The Noise Problem  
Conventional systems, i.e., systems using linear modulation with no NFT, are suitably optimized for the AWGN channel. On the one hand, the use of Nyquist pulses and a QAM format at the TX ensures that a certain minimum Euclidean distance between waveforms is maintained; this distance, which is _independent of the number of transmitted symbols_ and _increases with the signal power_, determines the ultimate performance of the scheme. On the other hand, the maximum a posteriori probability (MAP) strategy is implemented at the RX by using matched filtering, sampling, and multiple thresholds to detect the closest symbol in Euclidean distance terms; this strategy ensures that the ultimate system performance is achieved. The same minimum Euclidean distance and the optimality of the detection strategy are preserved even in OFDM systems, since the IFFT and FFT operations, respectively performed at the TX and RX, are unitary transformations. Unfortunately, the fiber optic channel is not AWGN in  
Figure 1: The nonlinear frequency-division multiplexing (NFDM) system with different detection strategies. the nonlinear regime, bringing in additional issues, including signal-noise interaction and inter- and intra-channel nonlinearities. In the recent years, NFT-based systems have been proposed to overcome the limitations due to fiber nonlinearity. Due to the pioneering and complex nature of the topic, these systems have been initially modeled along the lines of the conventional ones, with no guarantee of optimality. For example, NFDM was simply developed as a nonlinear analogous of OFDM, replacing IFFT with BNFT and FFT with FNFT. Unfortunately, the BNFT and the FNFT are not unitary transformations, so that the overall system is no longer optimal in terms of minimum distance, neither at the TX nor at the RX. As far as it concerns the demodulation, the standard FNFT strategy [6] employs the FNFT followed by a conventional QAM detector, which is optimized for an AWGN channel. However, even in a back-to-back configuration, the ASE noise is AWGN in the time domain, but non-Gaussian and signal-dependent in the nonlinear frequency domain [8; 9; 19]. These modified statistics result in an unfavorable dependence of performance on the signal duration, which makes the FNFT strategy highly suboptimal, and practically limits the achievable spectral efficiency [9]. The importance of this issue is now widely acknowledged, so that some alternative detection strategies have been proposed [11; 13; 22; 23; 24; 25]. On the other hand, also the modulation is affected by a similar issue, whose importance is, however, still largely overlooked. In fact, the direct mapping of the linearly modulated signal to the nonlinear spectrum warrants a certain minimum Euclidean distance between the waveforms in the nonlinear frequency domain but not in the time-domain, where the ASE noise is injected (ASE is typically the dominating noise in long-haul optical fiber systems, but the same issue applies to all the forms of noise that are injected between the BNFT and the FNFT, including for instance transceiver noise). If the distance between waveforms is too small in the time domain, the system becomes oversensitive to noise and the performance inevitably decays, even if an optimal strategy is employed. Unfortunately, this happens for the standard NIS modulation, as shown in Figure 2a, where the (upper bound to the) squared minimum distance among transmitted waveforms in time domain (after the BNFT) is shown as a function of the power for different burst lengths. The distance is normalized by twice the value of the standard deviation \(\sigma\) of the noise generated in the considered link. For \(N_{b}=1\) the minimum distance monotonically increases with power (at least in the considered range), closely lower bounding the curve for linear modulation (valid for any \(N_{b}\)).