| Περίληψη: | In 2008, the commercialization by Nortel of coherent optical communications digital transceivers operating at 40 Gb/s using Differential Phase Shift Keying (DPSK) revolutionized the field of fiber-optic communications because it provided superior spectral efficiency compared to the state-of-the-art binary intensity-modulation direct-detection receivers. A precursor to this development was the introduction in the early 2000 of differential phase shift keying in conjunction with a direct-detection using a delay interferometer. The advantages of latter modulation formats were the following: a) it allowed more efficient bandwidth utilization due to its superior spectral efficiency compared to on-off keying; b) it was more resilient to fiber transmission impairments (i.e., Chromatic Dispersion (CD), Polarization Mode Dispersion (PMD), fiber nonlinearities) due to its more compact spectrum and constant envelope compared to on-off keying. Nonetheless, fiber-optic communication systems using DPSK were severely degraded by nonlinear phase noise. One way to quantify this system degradation is to accurately derive the nonlinear phase noise statistics. In previously-published articles the characteristic function of the nonlinear phase noise was calculated analytically and the Probability Density Function (PDF) of the nonlinear phase noise was calculated numerically using the Fast Fourier Transform (FFT). In this thesis, we derive an analytical expression of the PDF using the method of steepest descent in a fiber-optic communication system employing M-DPSK with direct detection. The accuracy of our approximation is by far superior to the numerical method providing great accuracy at the PDF tails from where the error probability can be determined.
Nowadays, high-speed, high-spectral efficiency, coherent optical communication systems use advanced modulation formats, such as 16 and 64 Quadrature Amplitude Modulation (QAM) in order to achieve 400 Gb/s and 1 Tb/s per channel. Major transmission impairments e.g., CD, PMD and Self-Phase Modulation (SPM) can be compensated using electronic equalization. Fiber nonlinearities such as nonlinear phase noise, and inter-channel four wave mixing are not compensated and can severely limit the maximum transmission reach.
Over the past few years, Space Division Multiplexing (SDM) utilizing multicore or Few- Mode Fibers (FMFs) have been studied intensively as a potential candidate for increasing link capacity beyond 1 Tb/s. It is anticipated that the deployment of SDM along with the Multiple-Input Multiple-Output (MIMO) nonlinear equalizers can further extend link capacity.
Digital nonlinear equalization techniques fall into two categories: a) Pre-compensation techniques at the optical transmitter that render the signal propagation more resilient to fiber nonlinearities and b) post-compensation techniques at the optical receiver, based on Digital Signal Processing (DSP), applied to the distorted signal after propagation. The latter category includes a fully numerical method called Digital Backpropagation (DBP) and a semi-analytical method based on Volterra series. These methods have been applied to the equalization of nonlinear distortion in optical super-channels either on a channel-by-channel or on a multi-channel basis. Recently-published results have revealed that a multi-channel equalization scheme, using 80 steps-per-span DBP, provides up to 3.8 dB Q^2- factor improvement after ~ 3,200 km of transmission reach. However, this impressive performance is achieved at the expense of computational complexity since many samples per symbol are required.
Considering the potential of nonlinear equalizers in the field of SDM coherent optical communication systems, previous publications have shown that linear impairments such as Differential Mode Group Delay (DMGD) and mode coupling, can be almost completely eliminated by linear MIMO-DSP techniques. Nonetheless, inter-modal nonlinearities can still limit the maximum reach in these systems. Even though nonlinear equalization techniques have been studied for dual-polarization standard single-mode fibers and six-mode fibers with two modes, to the best of our knowledge, no scalable MIMO beyond 2×2 has been presented until now for nonlinearity mitigation.
In this dissertation, we study the performance of Volterra-based equalizers when applied in channel-by-channel and in multi-channel equalization in single-mode fiber systems. In addition, we study Volterra-based MIMO nonlinear equalizers, beyond 2×2, which, to the best of our knowledge, is done for the first time.
More specifically, first, we compare, both experimentally and by simulation, the performance of a nonlinear equalizer using 3rd-order inverse Volterra series transfer function and a nonlinear equalizer based on DBP-Split-Step Fourier (SSF) method in a coherent optical communication system using 400 Gb/s coherent 16QAM Orthogonal Frequency Division Multiplexing (OFDM) super-channels. Both studies show a ~ 0.3 dB Q^2-factor improvement, compared to the linear equalization case, in the presence of inter-channel nonlinearities, generated by adjacent channels carrying either single-carrier or multi-carrier modulation formats. Then, the computational complexity of the nonlinear algorithms is estimated and is shown that, in all cases, the multi-step-per-span DBP-SSF is slightly superior to the other nonlinear equalizer but at the vast expense of computational complexity.
Second, we conduct a comparison study between the performance of a 3rd -order inverse Volterra series transfer function nonlinear equalizer and the performance of the single-step-per span and multistep-per-span DBP equalizer performing in multi-channel equalization after more than 1,000 km of a single-mode fiber link. Until now, many studies have shown that the Volterra-based nonlinear equalizers, which are essentially implemented with a single-step-per-span, are inferior compared to the single-step-per span DBP in single-channel equalization. On the contrary, we show that the inverse Volterra nonlinear equalizer performs similarly to the highly-complex DBP-SSF equalizer even with 40 samples per symbol in the case of multi-channel equalization. We demonstrate this in simulations of a 400 Gb/s, dual-polarization, 16QAM, quasi-Nyquist, multiplexed OFDM super-channel, after 1,600 km transmission distance.
Finally, we develop a MIMO 3rd-order inverse Volterra series transfer function nonlinear equalizer for a long-haul, FMF coherent optical communication system. Even though DBP technique has prevailed as the method of choice for nonlinear compensation in single-mode fiber systems, it has been shown that the single-step-per span inverse Volterra series transfer function nonlinear equalizer exhibits similar performance to much more complex multi-step-per-span DBP in multi-channel equalization. Our aim is to discover, via extensive numerical simulations, the maximum number of propagating modes for which the 3rd-order inverse Volterra series transfer function MIMO nonlinear equalizer can still provide decent compensation (i.e., at least 1 dB Q^2-factor improvement compared to linear equalization) of the nonlinear effects. Our results show that a ~ 1 dB Q^2-factor improvement can be achieved for 16QAM spatial super-channels of total capacity 6×32 Gbaud after propagation through 1,040 km of an FMF link. Our study is completed with the evaluation of the computational complexity of the nonlinear equalizers under study.
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