# Mitigating the Impact of Noise on Nonlinear Frequency Division Multiplexing +
## 3 The Noise Problem
On the other hand, for \(N_{b}>1\), the minimum distance reaches a maximum and then decreases--the longer the burst length \(N_{b}\), the lower the peak value of the minimum distance. The behavior of the minimum distance obtained with NFDM modulation is quite counterintuitive and totally different from that obtained with linear modulation, where, by contrast it grows monotonically with power and independently of the length of the transmitted sequence. This peculiar behavior highlights an intrinsic and critical limitation of NFDM modulation, which can in fact achieve a reasonable performance only with short bursts, i.e., for low spectral efficiencies, regardless of the adopted detection strategy. Figure 2a reports the actual minimum distance only for the short bursts with \(N_{b}=1,2,3\). For longer bursts, since the exhaustive search of the minimum distance among all the possible waveforms, becomes too computationally expensive, the search was limited only to a subset of possible waveforms, hence obtaining an upper bound--in these cases, the actual minimum distance might indeed be even smaller, but not larger. An example of the upper bounding procedure for \(N_{b}=512\) is shown in Figure 2b, which reports the squared minimum distance found between waveforms when considering a fixed number of realizations and changing only some specific symbols. The figure reports, with the label \"several symbols change\", the squared minimum distance found among 150 random sequences. Furthermore, the figure shows the squared minimum distance found, among 150 random sequences, by changing just one symbol. This is done considering the sequences obtained by testing all the 16 constellation symbols in a fixed position: the change in the first position,  
last position, and random position are indicated as \"first symbol\", \"last symbol\", and \"random symbol\", respectively. As expected, the distance is larger when changing a random number of symbols in a random position, and smaller when changing only one symbol. In particular, when the first symbol is changed, the minimum distance is small at low power and increases at high power, approaching the curve with several changes. Conversely, when only the last symbol is changed, the smallest minimum distance is obtained. In this case, the minimum distance reaches a maximum and then decreases. This happens because, according to the causality property of the NFT (see next section), the first symbol influences all the signal, while the last symbol only the tail, this effect being negligible at low power and strong at high power. According to this analysis, the best lower bound is obtained when changing only the last symbol of the sequence. This is, in fact, the bound considered in Figure 2a for \(N_{b}>3\). Finally, Figure 2b suggests that NFDM modulation induces an uneven behavior along the time-domain waveform--variations of the first symbols of the sequence cause larger changes of the waveform than variations of the last ones--revealing possible ways to improve the overall performance, as it will be shown in Section 5.3. In conclusion, while a clear advantage of NFDM systems over conventional ones has not been yet demonstrated, NFDM can still be improved. Aiming at an optimal system, both the modulation and the detection steps should be modified and tailored for the NFDM paradigm. Possible improvements are discussed in the next sections.