Fiber-Optic Communications Using Nonlinear Fourier Transforms: Algorithms and a Bound

Due to the ever increasing global connectivity, the demand on the fiber-optic communication infrastructure is projected to keep increasing rapidly. A major factor currently limiting transmission capacity is the fiber nonlinearity. Some researchers have suggested the application of nonlinear Fourier transforms to exploit the fiber nonlinearity rather than ignoring or mitigating it. Nonlinear Fourier transforms allow us to solve certain nonlinear partial differential equations by transforming the complex evolution of the solution in the time-domain to a simple multiplication with a nonlinear frequency response in the nonlinear Fourier domain. This method is analogous to solving linear partial differential equations using the Fourier transform. The nonlinear Schrödinger equation is a suitable model for the propagation of light through a single-mode optical fiber. Its lossless version is solvable through a nonlinear Fourier transform. In recent years, several nonlinear Fourier transform based communication systems have been proposed. Such systems require numerical algorithms to compute the nonlinear Fourier transforms as nonlinear Fourier spectra are known analytically for only a handful of signals, and linear superposition cannot be used to compute the spectrum of a more complex signal. Computationally efficient algorithms are therefore not only essential for the real-time operation of nonlinear Fourier transform based communication systems, but are also important for their simulation. One common way to improve the spectral efficiency of a communication system is to increase the signal power in order to reduce the impact of noise. Another is to increase the signal duration in order to reduce the impact of information-free guard intervals that are inserted between transmissions to deal with the channel memory. Longer signals however require more resources to process them. The numerical problem of computing nonlinear Fourier transforms furthermore gets harder for both higher power and longer durations. Hence in the literature, we observe that the inability to perform efficient communication in these regimes is typically attributed to numerical problems of existing algorithms. In this dissertation we develop new algorithms that require shorter computation times for achieving similar accuracies as existing algorithms. Furthermore, we theoretically investigate whether some of the problems that are commonly attributed to numerical difficulties could occur in the absence of numerical effects.

The nonlinear Fourier transform for signals that decay sufficiently fast is currently the most commonly used transform in nonlinear Fourier transform based communication systems. We developed new algorithms for computing the continuous nonlinear Fourier spectrum which is one part of the nonlinear Fourier spectrum for decaying signals. We demonstrated significant improvements over existing algorithms in multiple numerical benchmarks, and implemented the algorithms in the open source software library FNFT. We also developed NFDMLab, which is a Python based open source simulation environment for nonlinear Fourier transform based communication systems that relies on FNFT. The developed forward nonlinear Fourier transform algorithms are fast higher-order methods with a complexity of O(D log²D) for computing the continuous nonlinear Fourier spectrum from D samples of a decaying signal. In the numerical benchmarks, we introduced the trade-off between accuracy and computation time as a new way to compare nonlinear Fourier transform algorithms and found that the newly proposed algorithms perform significantly better than prior work in this regard. We also provided the first counting analysis of a fast nonlinear Fourier transform algorithm.

There is also interest in using the nonlinear Fourier transform for periodic signals, as it is closer to the method used in conventional orthogonal frequency division multiplexing communication systems. The definition of the nonlinear Fourier transform for periodic signals is different from that of decaying signals. Communication systems based on nonlinear Fourier transforms for periodic signals make use of so-called finite-genus solutions of the nonlinear Schrödinger equation. Riemann theta functions are the traditional way to realize inverse nonlinear Fourier transforms that are used to synthesize finite-genus solutions. They are multi-dimensional Fourier series and their numerical computation suffers from the curse of dimensionality. This limits the genus of the signals used in the communication systems and is seen as a major bottleneck. We derived new bounds on the series truncation error and proposed two tensor-train based and a hyperbolic cross index set based algorithms for computing high-dimensional Riemann theta functions. We compared them to existing algorithms in multiple numerical benchmarks. The bounds that we derived on the truncation error of the Riemann theta functions allowed us to rule out several of the existing approaches for the high-dimension regime. We demonstrated that the algorithm based on the hyperbolic cross can compute Riemann theta functions upto 60 dimensions with moderate accuracy which is significantly higher than what was previously feasible.

We also tried to improve the performance of nonlinear Fourier transform based communication systems known as b-modulators in the highly nonlinear regime using improved numerical algorithms. When we did not see improvements, we conducted a theoretical analysis of b-modulation systems. The analysis allowed us to prove theoretically that 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. The power bound decreases with increasing signal duration which consequently decreases the signal-to-noise ratios for long signals, which explains the observed performance degradation in this regime without resorting to numerical difficulties. This result is the first of its kind as such a behaviour is not known from conventional linear systems. We also demonstrated numerically that the transmit powers achieved by an exemplary b-modulated system are close to its theoretical limits.

Fiber-optic communication systems based on nonlinear Fourier transforms have been proposed to potentially tackle fiber nonlinearity, which is a major factor currently limiting transmission capacity. Efficient numerical algorithms are essential for real-time operation as well as efficient simulations of nonlinear Fourier transform based fiber-optic communication systems. The algorithms presented in this dissertation potentially make already published nonlinear Fourier transform based communication systems more practical and also allow for development of new designs which were previously infeasible. In this dissertation furthermore a limitation on communication system design imposed by the structure of the nonlinear Fourier transform was identified. It can be used to explain the inability to perform efficient communication with long duration signals, which was previously attributed to numerical problems, and guide the design of future systems.