# Mitigating the Impact of Noise on Nonlinear Frequency Division Multiplexing +
## 5 Improved Modulations
### The Power-Tilt Strategy  
The power-tilt strategy is a simple and effective approach--proposed here for the first time--to improve the performance of NFDM, when NIS is employed. Based on the observations made about Figures 2b and 4a, the idea is that of decreasing the amplitude of the first symbols modulating \(s(t)\)while increasing that of the last ones, trying to equalize the power along the transmitted waveform. This should cause the last symbols to have a larger impact on the waveform, while reducing the impact of the first symbols, yielding an overall improvement of the minimum distance. This manuscript considers only a linear power tilt, characterized by the tilt strength \(0\leq\delta<1\), but more involved tilt strategies may be investigated. Let \(\tilde{x}_{1},\ldots,\tilde{x}_{N_{b}}\) be the symbols to be transmitted, drawn from a regular QAM alphabet. The signal \(s(t)\) is modulated by the tilted symbols \(x_{k}=a_{k}\tilde{x}_{k}\) for \(k=1,\ldots,N_{b}\), with  
\[a_{k}=\sqrt{1-\delta+2\delta(k-1)/(N_{b}-1)}. \tag{6}\]  
If \(\delta=0\), no tilt is applied. It can easily be verified that the mean energy per symbol remains unchanged when \(\delta\) is varied. The power-tilt strategy applied to NIS is sketched in Figure 3d. The applied tilt is then accounted for at the receiver side. The power-tilt strategy is illustrated in Figure 5a-c, by comparing the effect of a 0.9 tilt with the case where no tilt is used. Figure 5a,b show that, when applying the 0.9 tilt, the energy of the first symbols (and of the corresponding pulses of the modulated signal) is reduced, whereas that of the last ones is increased. Figure 5c reports the corresponding optical waveforms after the BNFT, showing how the tilt strategy partly compensates for the power decay of the optical signal with time. Figure 6a reports the total number of bit errors accumulated after the transmission of 100 different sequences of \(N_{b}=1024\) symbols as a function of the symbol position in the sequence, considering the NFDM system described in Section 2 with the DF-BNFT detection, a power \(P=-7\) dBm, and different tilt strengths. With no tilt (\(\delta=0\)), the number of errors increases from left to right, with most of the errors occurring on the rightmost symbols. On the other hand, when increasing the tilt strength (\(\delta>0\)), more energy is allocated on the last symbols and less energy on the first ones, counterbalancing the previous effect--i.e., reducing the error probability on the last symbols and increasing that on the first symbols. This improves the overall performance up to an optimal tilt value (\(\delta=0.9\) in this example). Figure 5: Illustration of the power-tilt strategy for a signal with 16 symbols drawn from a 16 QAM constellation and a launch power \(P=7\) dBm. Blue lines and symbols refer to a power tilt \(\delta=0.9\), whereas red lines and symbols refer to the case without tilt (\(\delta=0\)). (**a**) Energy of the symbols; (**b**) QAM signal \(s(t)\) before the BNFT; (**c**) optical signal \(q(t)\) after the backward nonlinear Fourier transform (BNFT). If the power tilt is further increased, e.g., to \(\delta=0.99\), the mean energy of the first symbols becomes too low (zero, in the limit \(\delta=1\)) and the corresponding error probability too high, worsening again the overall performance. The optimal tilt is the one that minimizes the average error probability, best balancing the errors occurring on the first and last symbols. Therefore, it is expected to depend on the launch power, burst length, and modulation format. Next, Figure 6b shows the number of bit errors accumulated in the same conditions as Figure 6a, but with a fixed tilt \(\delta=0.9\) and different launch powers. When the power is very low (linear regime) the tilt causes a lot of errors on the first symbols, as in this case the optical signal is weak and very similar to the QAM signal \(s(t)\), so that the power tilt further decreases the energy of the first symbols and increases the corresponding number of errors.