In this paper an original interacting particle system approach is developed for studying Markov chains in rare event regimes. The proposed particle system is theoretically studied through a genealogical tree interpretation of Feynman–Kac path measures. The algorithmic implementation of the particle system is presented. An estimator for the probability of occurrence of a rare event is proposed and its variance is computed, which allows to compare and to optimize different versions of the algorithm. Applications and numerical implementations are discussed. First, we apply the particle system technique to a toy model (a Gaussian random walk), which permits to illustrate the theoretical predictions. Second, we address a physically relevant problem consisting in the estimation of the outage probability due to polarization-mode dispersion in optical fibers.

/pdf/genealogical-particle-analysis-of-rare-events-1gfl4bkhdu.pdf

Genealogical particle analysis of rare events

Polarization-related impairments have become a critical issue for high-data-rate optical systems, particularly when considering polarization-mode dispersion (PMD) and polarization-dependent loss (PDL). In general, polarization effects are stochastic processes and can occur on short or long time scales. Therefore, control and monitoring of these effects may be crucial in any systems-level mitigation. This tutorial will highlight the following key issues: PMD and PDL, monitoring schemes, emulation of proper statistics, interaction, and applications of polarization scrambling.

Monitoring and control of polarization-related impairments in optical fiber systems

The basic theory of importance sampling (IS) as relevant to polarization-mode dispersion (PMD) in optical fibers is discussed, and its application to Monte Carlo (MC) simulations of PMD-induced transmission impairments is demonstrated. The use of IS allows rare PMD events to be simulated much more efficiently than with standard MC methods. As a consequence, methods employing IS provide natural and effective tools to assess PMD-induced impairments and outages in optical transmission systems at realistic probability levels.

/pdf/importance-sampling-for-polarization-mode-dispersion-3daeb9xvhm.pdf

Importance sampling for polarization-mode dispersion: techniques and applications

A simulation method that targets all possible combinations of first- and second-order polarization-mode dispersion (PMD) is described. Use of this method in importance-sampled Monte Carlo simulations yields a more comprehensive determination of PMD-induced system penalties than first-order biasing alone and significantly speeds up the calculation of outage probabilities, particularly when PMD compensation is employed. The technique is demonstrated by using it to calculate the probability distribution function (pdf) of second-order PMD and the joint pdf of the magnitude of first- and second-order PMD.

/pdf/multiple-importance-sampling-for-first-and-second-order-2ykgzr6eld.pdf

Multiple importance sampling for first- and second-order polarization-mode dispersion

We demonstrate a practical polarization-mode-dispersion (PMD) emulator using programmable differential-group-delay (DGD) elements. The output PMD statistics of the emulator can be chosen by varying the average of the Maxwellian DGD distribution applied to each element. The emulator exhibits good stability and repeatability in a laboratory environment. In addition, we demonstrate how this emulator may be used to experimentally employ the powerful technique of importance sampling to quickly generate extremely low probability events. This technique is used to measure the Q-factor degradation due to both average and rare PMD values in a 10-Gb/s transmission system.

Polarization-mode-dispersion emulator using variable differential-Group-delay (DGD) elements and its use for experimental importance sampling

A method and a system to produce, either in numerical simulations or in experiments, specified amounts of first, second and higher order PMD in a controlled manner, in particular large amounts. Parameters can be adjusted to obtain specific ranges of first, second and higher order PMD, and importance sampling can be used to determine the probability that the resulting PMD events can be obtained in realistic situations. Individual results obtained using specific parameter values can be combined to produce even larger ranges of PMD.

Method and system for the controlled production of polarization mode dispersion

We have demonstrated a simulation method that can efficiently generate not only large first- and second-order PMD, but also combinations where one of the two is large and the other is small. Furthermore, the use of importance sampling allows the accurate determination of the associated probability distribution, even for very rare events.

Multiple importance sampling for first- and second-order PMD

https://ieeexplore.ieee.org/iel5/8042/22213/01036424.pdf

Multiple importance sampling fur flrst- and secondorder PMD

We have shown analytically and numerically that, when a variable-delay polarisation mode dispersion (PMD) compensator is used, the residual pulse broadening (or its minimum value) is proportional to second-order PMD, even in cases where the PMD correlation bandwidth exceeds the pulse bandwidth. Since second-order PMD is one of the main factors contributing to the pulse broadening after PMD compensation, numerical investigations of PMD compensators should take into account a wide range of second-order PMD values.

S.L. Fogal

Papers

Multiple importance sampling for first- and second-order polarization-mode dispersion

Method and system for the controlled production of polarization mode dispersion

Multiple importance sampling for first- and second-order PMD

Multiple importance sampling fur flrst- and secondorder PMD

Importance-sampled pulse broadening statistics before and after PMD compensation