Deconvolution with Maximum Entropy Solution to Determine Local Extinction Coefficient and Local Volume Concentration Values from Laser Diffraction Data

The traditional use of the laser diffraction technique provides line‐of‐sight liquid spray drop‐size distribution. However, deconvolution of the measurements can be performed for axisymmetric spray in order to determine local spray characteristics. In a previous publication, a new deconvolution technique making use of the maximum entropy principle was established and applied to determine the local drop‐size distributions. The entire approach was experimentally validated. In this work, the technique is employed to determine local extinction coefficient values. As in the previous investigation, the measurement procedure consists of scanning a laser beam through the spray cross‐section from the center to the edge of the spray. By use of the transmittance theory, the local extinction coefficients allow the local volume concentrations to be calculated. This theory introduces the mean scattering coefficient. The results show that this coefficient must be determined as a function of the Sauter mean diameter in order to avoid overestimation of the volume concentration. Although no proper validation is presented, the coherence of the overall approach is discussed in detail and solutions for improving the spatial resolution are presented. Finally, the local volume concentrations are combined with the local drop‐size distribution to provide local volume‐weighted, drop‐size distributions. These distributions provide information on the localization of the drops according to their diameter as well as on the spatial liquid distribution. This work illustrates applications and performances of laser diffraction technique that are rarely used.     

The Residual ISI for Which the Convolutional Noise Probability Density Function Associated with the Blind Adaptive Deconvolution Problem Turns Approximately Gaussian

In a blind adaptive deconvolution problem, the convolutional noise observed at the output of the deconvolution process, in addition to the required source signal, is—according to the literature—assumed to be a Gaussian process when the deconvolution process (the blind adaptive equalizer) is deep in its convergence state. Namely, when the convolutional noise sequence or, equivalently, the residual inter-symbol interference (ISI) is considered small. Up to now, no closed-form approximated expression is given for the residual ISI, where the Gaussian model can be used to describe the convolutional noise probability density function (pdf). In this paper, we use the Maximum Entropy density technique, Lagrange’s Integral method, and quasi-moment truncation technique to obtain an approximated closed-form equation for the residual ISI where the Gaussian model can be used to approximately describe the convolutional noise pdf. We will show, based on this approximated closed-form equation for the residual ISI, that the Gaussian model can be used to approximately describe the convolutional noise pdf just before the equalizer has converged, even at a residual ISI level where the “eye diagram” is still very closed, namely, where the residual ISI can not be considered as small.     

Numerical Algorithms for Estimating Probability Density Function Based on the Maximum Entropy Principle and Fup Basis Functions

Estimation of the probability density function from the statistical power moments presents a challenging nonlinear numerical problem posed by unbalanced nonlinearities, numerical instability and a lack of convergence, especially for larger numbers of moments. Despite many numerical improvements over the past two decades, the classical moment problem of maximum entropy (MaxEnt) is still a very demanding numerical and statistical task. Among others, it was presented how Fup basis functions with compact support can significantly improve the convergence properties of the mentioned nonlinear algorithm, but still, there is a lot of obstacles to an efficient pdf solution in different applied examples. Therefore, besides the mentioned classical nonlinear Algorithm 1, in this paper, we present a linear approximation of the MaxEnt moment problem as Algorithm 2 using exponential Fup basis functions. Algorithm 2 solves the linear problem, satisfying only the proposed moments, using an optimal exponential tension parameter that maximizes Shannon entropy. Algorithm 2 is very efficient for larger numbers of moments and especially for skewed pdfs. Since both Algorithms have pros and cons, a hybrid strategy is proposed to combine their best approximation properties.     

Electrical Power Diversification: An Approach Based on the Method of Maximum Entropy in the Mean

Electrical energy is generated in different ways, each located at some specific geographical area, and with different impact on the environment. Different sectors require heterogeneous rates of energy delivery, due to economic requirements. An important problem to solve is to determine how much energy must be sent from each supplier to satisfy each demand. Besides, the energy distribution process may have to satisfy ecological, technological, or economic cost constraints.     

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution, which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

Calibration Curve with Improved Limit of Detection for Cadmium in Soil: An Approach to Minimize the Matrix Effect in Laser-Induced Breakdown Spectroscopic Analysis

The present article describes a working or combined calibration curve in laser-induced breakdown spectroscopic analysis, which is the cumulative result of the calibration curves obtained from neutral and singly ionized atomic emission spectral lines. This working calibration curve reduces the effect of change in matrix between different zone soils and certified soil samples because it includes both the species' (neutral and singly ionized) concentration of the element of interest. The limit of detection using a working calibration curve is found better as compared to its constituent calibration curves (i.e., individual calibration curves). The quantitative results obtained using the working calibration curve is in better agreement with the result of inductively coupled plasma–atomic emission spectroscopy as compared to the result obtained using its constituent calibration curves.