Abstract-We studied the efficiency of different implementations of the split-step Fourier method for solving the nonlinear Schrödinger equation that employ different step-size selection criteria. We compared the performance of the different implementations for a variety of pulse formats and systems, including higher order solitons, collisions of soliton pulses, a single-channel periodically stationary dispersion-managed soliton system, and chirped return to zero systems with single and multiple channels. We introduce a globally third-order accurate split-step scheme, in which a bound on the local error is used to select the step size. In many cases, this method is the most efficient when compared with commonly used step-size selection criteria, and it is robust for a wide range of systems providing a system-independent rule for choosing the step sizes. We find that a step-size selection method based on limiting the nonlinear phase rotation of each step is not efficient for many optical-fiber transmission systems, although it works well for solitons. We also tested a method that uses a logarithmic step-size distribution to bound the amount of spurious four-wave mixing. This method is as efficient as other second-order schemes in the single-channel dispersion-managed soliton system, while it is not efficient in other cases including multichannel simulations. We find that in most cases, the simple approach in which the step size is held constant is the least efficient of all the methods. Finally, we implemented a method in which the step size is inversely proportional to the largest group velocity difference between channels. This scheme performs best in multichannel optical communications systems for the values of accuracy typically required in most transmission simulations.

In the first part of this paper the theory of differential characters is developed completely from a de Rham -Federer viewpoint. Characters are defined as equivalence classes of special currents, called sparks, which appear naturally in the theory of singular connections. As in de Rham -Federer cohomology, there are many different spaces of currents which yield the character groups. The fundamental exact sequences in the theory are easily derived from methods of geometric measure theory. A multiplication of de Rham-Federer characters is defined using transversality results for flat and rectifiable currents established in the appendix. It is shown that there is a natural equivalence of ring functors from de Rham -Federer characters to the classical Cheeger-Simons characters given, as in de Rham cohomology, via integration. This discussion rounds out the approach to differential character theory introduced by Gillet-Soulé and Harris. The groups of differential characters have an obvious topology and natural smooth Pontrjagin duals (introduced here). It is shown that the

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Abstract-The authors have derived a receiver model that provides an explicit relationship between the factor and the optical signal-to-noise ratio (OSNR) in optical fiber communication systems for arbitrary pulse shapes, realistic receiver filters, and arbitrarily polarized noise. It is shown how the system performance depends on both the degree of polarization of the noise and the angle between the Stokes' vectors of the signal and the noise. The results demonstrate that the relationship between the OSNR and the factor is not unique when the noise is partially polarized. This paper defines the enhancement factor and three other parameters that explicitly quantify the relative performance of different modulation formats in a receiver. The theoretical and experimental results show that the performance of the return-to-zero format is less sensitive to variations in the receiver characteristics than is the performance of the nonreturn-to-zero format. Finally, a validation of the formula is presented for computing the factor from the OSNR and the Stokes vectors of the signal and the noise by comparison with both experiments and Monte Carlo simulations.

Abstract-We evaluate the error-correcting performance of a low-density parity-check (LDPC) code in an AWGN channel using a novel dual adaptive importance sampling (DAIS) technique based on multicanonical Monte Carlo (MMC) simulations, that allows us to calculate bit error rates as low as 10 −19 for a (96, 50) LDPC code without a priori knowledge of how to bias. Our results agree very well with standard MC simulations, as well as the union bound for the code.