Data Transmission Based on Exact Inverse Periodic Nonlinear Fourier Transform, Part I: Theory

The nonlinear Fourier transform (NFT) decomposes waveforms propagating through optical fiber into nonlinear degrees of freedom, which are preserved during transmission. By encoding information on the nonlinear spectrum, a transmission scheme inherently compatible with the nonlinear fiber is obtained. Despite potential advantages, the periodic NFT (PNFT) has been studied less compared to its counterpart based on vanishing boundary conditions, due to the mathematical complexity of the inverse transform. In this article we extract the theory of the algebro-geometric integration method underlying the inverse PNFT from the literature, and tailor it to the communication problem. We provide a complete algorithm to compute the inverse PNFT. As an application, we employ the algorithm to design a novel modulation scheme called nonlinear frequency amplitude modulation, where four different nonlinear frequencies are modulated independently. Finally we provide two further modulation schemes that may be considered in future research. The algorithm is further applied in Part II of this article to the design of a PNFT-based communication experiment.