Data Transmission based on Exact Inverse Periodic Nonlinear Fourier Transform, Part I: Theory
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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 paper 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 fully automatized 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 obtained algorithm is applied in Part II of this paper in the design of a PNFT-based communication experiment.
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