Nonlinear frequency division multiplexing with b-modulation Shifting the energy barrier

: The recently proposed b-modulation method for nonlinear Fourier transform-based fiber-optic transmission offers explicit control over the duration of the generated pulses and therewith solves a longstanding practical problem. The currently used b-modulation however suffers from a fundamental energy barrier. There is a limit to the energy of the pulses, in normalized units, that can be generated. In this paper, we discuss how the energy barrier can be shifted by proper design of the carrier waveform and the modulation alphabet. In an experiment, it is found that the improved b-modulator achieves both a higher Q -factor and a further reach than a comparable conventional b-modulator. Furthermore, it performs significantly better than conventional approaches that modulate the reflection coefficient.

control over the duration of the pulse. One solution to this problem is to use the NFT for periodic signals instead of the more common NFT for vanishing signals [13][14][15]. The transmitter only transmits one period of the generated signal plus a cyclic prefix in this scenario, similar to conventional OFDM. The challenge of the periodic NFT approach is that the NFT for periodic signals is mathematically more complicated. It is in particular not straight-forward to enforce a desired period. So far, only relatively simple systems with a few degrees of freedom have been demonstrated. An alternative solution is our proposed bmodulation method [16], which is based on the NFT for vanishing signals and can generate pulses of a finite, pre-specified duration in a simple way. This method was adopted in an experimental demonstration of 100 Gbps b-modulated nonlinear frequency division multiplexed (NFDM) transmission using 132 subcarriers [17]. A dual polarization NFDM transmission achieving a record net data rate of 400 Gbps based on b-modulation was demonstrated in [18].
In this paper, we extend our previous work [16] and add several modifications to the original b-modulation scheme, including flat top carriers and constellation shaping. The modified method is numerally studied to show the advantages of limited signal time duration compared with conventional modulation of the continuous spectrum (q -modulation). In a back-to-back (B2B) scenario, simulation results show that signal-noise interactions through NFT-processing can be significantly reduced for improved b-modulated signals. Based on the results, we experimentally compare the modified b-modulator techniques with conventional b-and  q -modulation schemes for a 14.4 Gbps 16QAM NFDM transmission over 640km standard single-mode fiber (SSMF). The proposed b-modulation scheme demonstrates a Qfactor gain of ~1.2 dB and nonlinear tolerance (launched power) gain of ~4 dB over a conventional FDM system. The results serve as another step forward in designing high performance NDFM signaling techniques for nonlinear transmission systems.

Basics of the nonlinear Fourier transform
The NFT of a signal ( ) q t , which in our context is either the (normalized) input to or the (normalized) output of a single-mode fiber with anomalous dispersion, is defined in a twostep procedure. First, consider the Zakharov-Shabat problem (see, e.g., [8] where λ is parameter. The NFT of ( ) q t has two parts defined in terms of the limits The first part is the continuous spectrum where the eigenvalues k λ are the solutions to ( ) 0 a λ = in 0 λ > I , and the residues are given by The main advantage of the NFT is that it simplifies the nonlinear Schrödinger equation which models the evolution of the complex envelope ( , ) u z t at location z and at retarded can be reconstructed from the NFT of the fiber output ( , ) u z t using the relations

Conventional modulation methods for the continuous spectrum
We first aim to embed data in the continuous spectrum of the fiber input. The discrete spectrum is not used and chosen to be empty. Several methods have been proposed to modulate a block of symbols N s , , N s − … ∈ , where  is a finite modulation alphabet, into the continuous spectrum. Let ( ) ψ ξ denote a carrier waveform, A 0 > a power control factor, and s 0 ξ > a shift. Most modulation methods (e.g., [3,19]) for  ( ) q ξ take one of two forms, where the power control factor A 0 > is a constant and

The original b-modulation method
The conventional modulation methods for the continuous spectrum offer no control over the duration of the fiber input and suffer from poor utilization of the temporal domain. Motivated by a classic result for the NFT with respect to the Korteweg-de Vries equation [24], it was [16]. The modulation scheme in [16] It was observed that the generated fiber-input ( ) q t would be time-limited with if the carrier waveform ( ) ψ ξ was bandlimited in the sense that its conventional inverse T T ∉ − . Note that this condition is a continuous-time version of the realizability conditions derived in the context of codirectional coupler design [22]: in the absence of eigenvalues, the discrete-time version of ( ) q t is zero outside a given range if and only if the Fourier series coefficients of the discrete-time version of ( ) b ξ are zero outside a related range. Also note that it is essential that the power scaling factor is a constant w.r.t. to the nonlinear spectral parameter ξ . Except in very specific special cases, a ξ -dependent power control factor will lead to a time-domain signal that is no longer time-limited even if the carrier wave fulfills the condition mentioned above. The original b-modulation scheme in [16] is one of the first NFT-based modulation method that offers explicit control over the duration of the generated fiber inputs. It has been demonstrated experimentally in [17,23], where the carrier waveform was a sinc pulse. We also remark that it was recently proposed [21] to embed information in the analytic extension ( ) b ξ of  ( ) q ξ , but the methods in [21] do not lead to time-limited signals.

The energy barrier
The energy of the generated fiber-input is known to satisfy [8] ( ) ( ) On the other hand, it is also known [8] that a valid ( ) b ξ satisfies ( ) It was observed in [16] that even if the power control factor A is driven towards the limit imposed by the condition ( ) ( ) The maximum energy we can achieve by adjusting the power control factor A in this case thus is We call MCE[ψ] the maximum carrier energy of the carrier waveform ( ) ψ ξ . The MCE can be both finite or infinite, depending on the carrier waveform. Consider, e.g., The MCE of this carrier waveform is finite for any value of n , The carrier waveforms in [16] were impulse responses of raised cosines, and it can be checked numerically that their MCE is indeed finite as well. Interestingly, this is not true for all carrier waveforms. The MCE of a rectangular carrier waveform, is actually infinite, Rectangular carriers however defeat the purpose of b-modulation -their inverse Fourier transform ( ) Ψ τ is not compactly supported, so that the duration of the generated pulses is not finite anymore. The same discussion applies if a root raised cosine is chosen as the carrier q ξ as in [18]. In both cases, the energy barrier is defeated, but the signals are no longer of finite duration because ( ) b ξ is not bandlimited.
In contrast, the energy barrier is not that relevant for conventional modulation of the continuous spectrum, i.e., the modulation of  ( ) As soon as the absolute value of the carrier waveform can be lower bounded by some rectangle, the energy will go to infinity for A → ∞ . The same holds for the modulation of  2 ( ) q ξ , which has been especially designed to enable explicit control the pulse energy.

Carrier waveform
In light of the discussion in the previous section, we find that the carrier waveform should at least fulfill the following two conditions: ( ) ψ ξ should have a i) compactly supported ( ) Ψ τ to ensure finite pulse durations; and ii) large enough MCE to enable sufficiently high signal energies.
The sinc and raised cosine carriers used for b-modulation so far satisfy these conditions, but there is nevertheless an issue with them that has not been obvious so far since we focused on the single carrier case until now. Ideally, the maximum energy we can achieve with a multicarrier system containing 2 1 To address all three conditions i)-iii), we propose to use Fourier-transformed flat top windows (e.g., [20]) as carrier waveforms since they are bandlimited, concentrated in the ξ domain, and approximate a rectangle at their center (which has infinite MCE). In our experiments and simulations, we used the carrier that corresponds to the flat-top window which was designed using the "Program 1" Matlab script in [20] with inputs that put equal weight on the perfect flatness of the carrier around zero and the decay of the sidelobes (i.e.,  Table 1.
Both are shown as Fig. 1 for the duration 4.5 T = .

Constellation shaping
The energy of the fiber input in the original b-modulation method ( ) as in (4) was adjusted through the power control factor A 0 > . The energy of a single carrier that has been modulated with a symbol n s ∈  is, as above, To avoid the generation of disproportionately weak carriers, we propose to abandon the power control factor A 0 > and use a reshaped version of the given modulation alphabet In The reshaped modulation alphabet is chosen as where m 1, , .

M = …
The term in the middle of this equation is monotonously increasing in m γ , while the right-hand side is known and independent of m γ . Therefore, we could determine the m γ using the bisection method; the integral was computed numerically. Note that our choice of m γ ensures that the energy ratios of the modulated carriers match the energy ratios of their symbols with respect to the original modulation alphabet, i.e., 2 In other words, the generation of disproportionately weak carriers is avoided. Another advantage is that the average modulated carrier energy matches the desired average energy,

Simulation examples
In this subsection, we numerically investigate the performance of the improved b-modulator with that of several other methods in a back-to-back (B2B) scenario. The original modulation alphabet is a 16-QAM. It is shown together with its shaped version, for a desired carrier energy d E 4 = , in Fig. 2(a). The fiber inputs generated by the original b-modulator (i.e., We emphasize that in our setup both b-modulators, original and improved, use the same flat-top carrier. The reason for not using a sinc or the impulse response of a raised cosine was that, with such carriers, the original b-modulation scheme was not able to match the energies of the improved b-modulator. The advantage of this choice is that we can isolate the effect of constellation shaping in our investigations. Some example pulses are shown in Fig. 2(b). The symbol duration for the b-modulators is 4.5 T = . For the  q -modulation methods, much larger windows are used to generate the initial signals and then truncated to 4 . The truncation error made will be much lower than for conventional  q -modulation because the tail is rapidly, and not slowly, decaying to zero. To corroborate this claim, we show the 99.9% durations and bandwidths of each ( ) q t modulated with randomly chosen blocks of symbols by the four methods in Fig. 2(c). It can be seen that the 99.9% durations of the pulses generated by the b-modulators are consistently lower than that for the conventional modulation methods. Figure 2(d) shows the (conventional) Fourier transforms of the four fiber input types, which are all very similar.  cessing, we num generated by malized SNR ( tigated the per he same flat-top 2 = before tran methods and t rage energy d E n in Fig. 3(a). A chemes and the on with the lin tion scheme. S ing in this exa f reshaped con e b-modulation imilar phenom spectrum in [ hat the resulti anation is that noises (AWGN ear Fourier do ear Fourier coe cking. In additi various averag wn in Fig. 3

Experime
We also cond experimental symbols were e and in the lite arises whether t these benefits e.g., [8]) that