Nonlinear Fourier Transforms (NFTs) are generalizations of the conventional Fourier transform that allow to solve specific nonlinear evolution equations in a way that is analogous to how Fourier solved the heat equation. The evolution of the nonlinear Fourier spectrum is, exactly like in the linear case, much simpler than the evolution of the original signal. The celebrated fast Fourier transform (FFT) algorithms are linear approximations of the Fourier transform. For smooth signals the result of the FFT convergences very quickly to the true Fourier transform. Unfortunately, unlike for the FFT, the accuracy of the FNFTs remains (at most) second-order even when the signal is smooth. In this work we developed an NFT algorithm that requires only O(D log^2 D) floating point operations, but achieves a sixth-order [O(D ^−6 )] error decay.
FNFT is a software library for the numerical computation of (inverse) nonlinear Fourier transforms, which are also known as (inverse) scattering transforms. The focus of the library is on fast algorithms, but it also contains non-fast methods. FNFT is written in C and comes with a MATLAB interface. A Python interface is available separately.
Starting to work in the area of NFDM currently requires a significant investment of time. Even though a software library for numerical computation of (inverse) NFTs are accessible, setting up a complete NFDM transceiver requires a lot more work. NFDMLab is an open source software environment that simulates fiber-optic data transmissions using nonlinear Fourier transforms. Our open source software environment NFDMLab allows to investigate many different variants of NFDM in a simple and intuitive way, even if there is no prior experience with NFDM. NFDMLab can be controlled using a simple and intuitive graphical environment built on the celebrated Jupyter notebook technology. The simulation lets users choose different constellation formats, modulation techniques, carrier shapes, and fiber and amplifier models in an interactive way. It provides visualizations for the whole transceiver chain and provides standard quality measures such as the uncoded bit error ratio.
The performance of various nonlinear frequency division multiplexed (NFDM) fiber-optic transmission systems has been observed to decrease with increasing signal duration. For a class of NFDM systems known as b-modulators, we show that the nonlinear bandwidth, signal duration, and power are coupled when singularities in the nonlinear spectrum are avoided. When the nonlinear bandwidth is fixed, the coupling results in an upper bound on the transmit power that decreases with increasing signal duration. Signal-to-noise ratios are consequently expected to decrease, which can help explain drops in performance observed in practice. Furthermore, we show that there is often a finite bound on the transmit power of b-modulators even if spectral singularities are allowed.