# Mitigating the Impact of Noise on Nonlinear Frequency Division Multiplexing +
## 1 Introduction  
Fiber nonlinearity can be regarded not only as a detrimental effect, but also as an opportunity for all-optical signal processing and regeneration and for the establishment of favorable conditions for signal propagation. Key properties of the fiber propagation equation are its integrability via the _nonlinear Fourier transform_ (NFT) and the existence of soliton solutions [1]. In the context of fiber communications, these properties have fostered the dream of a sort of \"transparent\" propagation regime, virtually immune to both dispersion and nonlinearity. Soliton systems, where information is carried by solitary waves propagating unchanged through the fiber, have been deeply studied in the eighties and nineties of the last century [2]. More recently, also thanks to great progress in DSP technology, the same dream has been revived and widened by the proposal of a _nonlinear frequency-division multiplexing_ (NFDM) scheme, which, similarly to the well known orthogonal frequency-division multiplexing (OFDM) for linear channels, uses the NFT to encode information on the _nonlinear spectrum_ of the optical signal, relying on its substantial invariance during propagation [3; 4; 5]. In recent years, numerous theoretical and numerical investigations and experimentaldemonstrations of NFDM have been reported [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18], though, possibly due to the pioneering nature of the topic, a clear advantage over conventional systems has yet to be demonstrated. A fundamental difference of NFDM with respect to OFDM is that the NFT, unlike the ordinary Fourier transform, is not a unitary transformation, so that the statistics of the amplified spontaneous emission (ASE) noise, modeled as additive white Gaussian noise (AWGN) in time domain, are not preserved in the nonlinear frequency domain [3; 8; 19; 20; 21]. These modified statistics result in an unfavorable dependence of performance on the signal duration, which practically limits the achievable spectral efficiency [9]. A possible explanation of this negative result comes from the use of detection strategies borrowed from conventional communication systems, and optimized for the AWGN channel rather than for the actual noise statistics. Indeed, a number of _improved_ detection strategies have been proposed to address this issue [11; 13; 22; 23; 24; 25] but, unfortunately, without fully solving it. Another possibility is that the linear modulations (e.g., quadrature amplitude modulation (QAM)) that are employed to encode the information on the nonlinear spectrum, while assuring a certain _minimum Euclidean distance_ between the waveforms in the nonlinear frequency domain, resulting in a too-small distance in the time domain, making the system oversensitive to the ASE noise. In this paper, after briefly introducing NFDM systems, we discuss the noise problem in their earliest implementations, highlighting why both the standard modulation and demodulation steps are not optimal [26]. In the light of the above, we discuss some improved modulation and detection strategies, introducing for the first time a power-tilt strategy to mitigate some of the aforementioned issues. The various strategies are then compared in terms of performance through numerical simulations. While highlighting the significant improvements obtained by properly designing the NFDM system, we remark the unfavorable behavior with respect to conventional (i.e., with no NFT) systems, which causes a performance decay of NFDM systems at high spectral efficiency. The analysis is performed in a simple ideal scenario, assuming ideal distributed amplification along the fiber, modulating only the continuous spectrum of a single polarization, and favoring accuracy over complexity in the implementation of the NFT algorithms, so that numerical issues with the latter are avoided.