A Novel Algorithm to Evaluate Definite Integrals by the Gauss-Legendre Integration Rule Based on the Stochastic Arithmetic: Application in the Model of Osmosis System

Finding the optimal iteration of Gaussian quadrature rule is one of the important problems in the computational methods. In this study, we apply the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library to find the optimal iteration and optimal approximation of the Gauss-Legendre integration rule (G-LIR). A theorem is proved to show the validation of the presented method based on the concept of the common significant digits. Applying this method, an improper integral in the solution of the model of the osmosis system is evaluated and the optimal results are obtained. Moreover, the accuracy of method is demonstrated by evaluating other definite integrals. The results of examples illustrate the importance of using the stochastic arithmetic in discrete case in comparison with the common computer arithmetic.

Influence of phase noise and interchannel interference on the performance of optical heterodyne FSK receiver

The moments approach is generally considered a systematic way to perform the analysis of coherent optical systems. Exact moments of the filtered signal corrupted by phase noise enable the construction of the Gaussian quadrature rule, which may be used to calculate the system error probability. In this paper, we extend the method to a wider class of systems to include the cases where the interchannel interference may be significant. We derive the essential equations in the matrix form and compare the moments approach with numerical simulation and Fokker-Planck approach. To illustrate the results, we apply the moment's method to two-channel optical heterodyne FSK system with dual-filter receiver structure, and evaluate the required channel spacing to have less than 1 dB penalty due to crosstalk.