A system theoretical view on nonlinear fiber propagation

In communication theory, discrete-time end-to-end channel models play a fundamental role in developing advanced transmission and equalization schemes. Most notable, the discrete-time linear, dispersive channel with additive white Gaussian noise (AWGN) is often used to model point-to-point transmission scenarios. In the last decades, numerous transmission methods for such linear channels have emerged and are now applied in many digital transmission standards. With the advent of high-speed CMOS technology, those schemes have also been adopted in applications for fiber-optical transmission with digital-coherent reception. Many of the applied techniques (e.g., coded modulation, signal shaping, and equalization) are still designed for linear channels whereas the fiber-optical channel is inherently nonlinear. A channel model which obtains the (discrete-time) output symbol sequence from a given (discrete-time) input symbol sequence by an explicit input/output relation is highly desirable to make further advances in developing strategies optimized for fiber-optical transmission. In the past two decades, considerable effort was spent developing channel models for fiber-optical transmission with good trade-offs between computational complexity and numerical accuracy. Most of the early work is, however, concerned with the phenomenology in the optical domain alone, i.e., both the source and the effect of fiber nonlinearity is studied in the continuous-time, optical domain not considering the transmitter and/or receiver front-ends. Here, one promising strategy is, e.g., based on the so-called perturbative approach where non-linear effects are considered as small perturbations to the optical signal. By now, the optical community is faced with a vast number of models based on the perturbation premise, each model with its own assumptions, simplifications, and objectives. The connection between already existing models using complementary views (e.g., one in time-, the other in frequency-domain) is often unclear. A detailed and rigorous derivation for some prominent models is still pending. E.g., the transition from the original continuous-time to the more relevant discrete-time end-to-end model lacks a comprehensive, system-theoretic analysis. Hence, in this dissertation, nonlinear fiber propagation is assessed from a systems-theoretic point of view with applications for communication systems. Based on the theory of nonlinear systems, the present work aims to connect the dots between various, existing channel models, unifying and comparing the different approaches. To that end, the perturbation approach in continuous-time is revisited with a special emphasis on the dual representations of nonlinear systems in time and frequency. From that, a discrete-time end-to-end fiber-optical channel model is derived which includes the transmit-side pulse shaping, the receive-side matched filtering, and T-spaced sampling. As before, two complementary representations of the now time-discretized end-to-end model are present—one in (discrete) time domain, the other in 1/T-periodic continuous-frequency domain. The time-domain formulation coincides with the well-known pulse-collision picture. The novel frequency-domain picture incorporates the sampling operation via an aliased and hence 1/T-periodic formulation of the nonlinear system. This gives rise to an alternative perspective on the end-to-end input/output relation between the spectrum of the discrete-time transmit symbol sequence and the spectrum of the receive symbol sequence. Both views can be extended from a regular, i.e., solely additive model, to a combined regular-logarithmic model to take the multiplicative nature of certain distortions into consideration. A novel algorithmic implementation of the discrete and periodic frequency-domain model is presented. The derived end-to-end model requires only a single computational step and shows good agreement in the mean-squared error sense compared to the oversampled and inherently sequential split-step Fourier method.