  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 algebrogeometric 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 paper to the design of a PNFT-based communication experiment.
