Evaluation of Gram-Charlier coefficients

A formulation of the Gram-Charlier coefficients for the expansion of a frequency distribution of data in terms of Hermite polynomials is presented. The formulation has been found to be particularly amenable to computer programming.     

A Convenient Multivariate Gram-Charlier Type A Series

It is often necessary to approximate the probability density function of a random variable from given statistical moments. The Gram-Charlier Type A series is one well known method for such representations. In this note, the Gram-Charlier Type A series is generalized to the multidimensional case.     

Generalization of the Gram-Charlier/Edgeworth Series and Application to Time-Frequency Analysis

We derive a general equation relating probability densities and as special cases we the obtain Gram-Charlier and Edgeworth series. This allows us to generalize these methods and clarify a number of issues pertaining to both probability theory and time-frequency analysis. In particular we show how the Gram-Charlier and Edgeworth series are related to the kernel method of time-frequency analysis. The approach allows us to construct densities that satisfy given constraints such as joint moments or conditional moments. Also, we show that the kernel has to be signal dependent and that to obtain a proper distribution it should be the ratio of two characteristic functions.     

Fiber Optics Receiver Error Rate Prediction Using the Gram-Charlier Series

This paper applies the Gram-Charlier series method to the caculation of error probabilties in digital optical receivers. This method allows the calculation of "exact" error probabilities including the effects of avalanche noise, thermal noise, and arbitrary posidetection processing filter. The predictions of this method are compared with those of a simple Gaussian approximation and with the Chernoff bounds. Finally, the effects of modal noise are included in the theory, and some specific cases are explored numerically.     

Approximations to Error Functions

This paper addresses approximations to error functions and points out three representative approximations, each with its own merits. Cody's approximation is the most computationally intensive of the three, it is not overly so, and there is no arguing over its accuracy. The other two approximations are much simpler computationally, and they both yield accuracies that would be considered more than sufficient in most practical situations. Absolute relative error provides an effective measure of goodness, and, for approximations to the Q-function, it also places a loose bound on the absolute error in the approximation. Cody's approximation is an effective surrogate for the true error function; the values provided by that approximation match the actual values of the error function to within roughly the precision of double-precision floating point arithmetic.     

A recursive method for computing the coefficients of the Gram-Charlier series

The Gram-Charlier series is a known tool for approximating a probability density function when the moments or the cumulants of a random variable are known. A recursive procedure is presented which is well suited for the numerical computation of the coefficients of the series.     

Simple Approximations for Token Rings

Recently, a conservation law for the waiting times at a token ring has been derived. This conservation law is used here to develop simple approximations to the waiting times at individual queues on a token ring. Although exact algorithms are available, they may require significant computation time. In contrast, the approximation developed here is easily calculated on a pocket calculator, and is more accurate than earlier approximations.     

Chromatic Derivatives and Local Approximations

We present a detailed motivation for the notions of chromatic derivatives and chromatic expansions. Chromatic derivatives are special, numerically robust linear differential operators; chromatic expansions are the associated local expansions, which possess the best features of both the Taylor and the Nyquist expansions. We give a simplified treatment of some of the basic properties of chromatic derivatives and chromatic expansions which are relevant for applications. We also consider some signal spaces with a scalar product defined by a Cesaro-type sum of products of chromatic derivatives, as well as an approximation of such a scalar product which is relevant for signal processing. We also introduce a new kind of local approximations based on trigonometric functions.     

Accurate evaluation of bit-error rates of optical communication systems using the Gram-Charlier series

The probability densities and cumulative distribution functions of decision statistics of optical communications systems are expanded as a Gram-Charlier (G-C) series, leading to arbitrarily accurate systematic evaluation of bit-error rates (BERs) and optimal decision thresholds of optical communication systems. The method displays negligible computational complexity and is applicable whenever the moment or cumulant generating functions of the decision statistics are analytically available. We applied the technique to a birth-and-death Markovian model of a direct-detection receiver with optical preamplifier in a two-level amplitude-shift keying system. The modal expansion series rapidly converged, whereas the alternative saddlepoint approximation method predicted a BER which deviated by 7% from the G-C result.     

A Gram-Charlier series method of calculating the probability of error in lightwave transmission systems

A Gram-Charlier series method has previously been applied to the calculation of the conditional probability of error, for a fixed data sequence, in the presence of intersymbol interference, detector multiplication noise, shot noise, and thermal noise. The probability of error is obtained by averaging the conditional probability of error over all possible data sequences. In this paper it is shown how the computational efficiency of the Gram-Charlier series method can be improved by calculating the probability of error without the need for an exhaustive averaging procedure.     

Approximations to inverse error functions

We have presented four approximations to the inverse cumulative distribution function of a standard normal random variable. The first two were meant purely as a tool to motivate the other approximations, although either could be useful when extreme accuracy is not required and one doesn't venture too far into the tails of the distribution. There are many other such simple approximations to F_X ^-1, suitable for use on a hand calculator.     

Approximations for the renewal function

This paper describes an accurate, computable approximation for evaluating the renewal function (RF). The method uses Pade approximants to compute the RF near the origin and switches to the asymptotic values farther from the origin. There is a polynomial switch-over function in terms of the coefficient of variation of the distribution, enabling one to determine a priori if the asymptotic value can be used instead of computing the Pade approximant. The results are tested with the truncated Gaussian distribution. The method yields a set of approximants to the RF that are re-usable, and can be used to compute the derivative and the integral of the RF. Results for the RF are within 1% of the optimal solution for most coefficients of variation     

Approximations for dielectric or plasma scatterers

Two approximate methods have been developed for calculating the radar cross sections of nonconducting bodies. The superposition approximation used to obtain the radar cross sections of dielectric clad bodies requires that the scattering properties of the component parts be known. The modified geometrical optics method, which yields the radar cross sections of dielectric bodies, requires no such initial information. These methods can be applied to some bodies with shapes that cannot be treated by means of the classical boundary-value solutions. Furthermore, the computations required are usually considerably simpler than for the boundary-value solutions even when the latter are obtainable.     

Diffusion Approximations for Time-Dependent Queueing Systems

This paper deals with queueing systems which reveal timedependent behavior. The goal of the paper is to present a set of results which can be applied to the analysis of computer-communication systems. Some new results on the time-dependent analysis of theGI/GI/1queue and of general queueing networks are given. All these results are based on a transient diffusion model of a single server queue. A parametric-decomposition method of the time-dependent analysis of queueing networks is presented. The method permits the analysis, of the networks where the arrival, service processes and also the routing matrix are time-dependent. It makes possible the analysis of various aspects of computer-communication systems, especially the adaptive routings, the strategies of flow and congestion control, the influence of short overload peaks and other time-varying phenomena in such systems. To illustrate these applications, a simple example of the analysis of the centralized adaptive routing in packet-switching networks is presented.     

Approximations for the cylindrical wire kernel

The authors introduce several new approximations to the cylindrical wire kernel. These approximations are derived from a recently found exact series solution for the cylindrical wire kernel. It is also demonstrated that the well-known reduced kernel and far-zone approximations may be obtained as special limiting cases of this series     

Approximations for characteristics of nomadic communications

In this paper we consider a mobile communication network in which users are moving along a road. It is desirable that each point of the road is in the range of at least one base‐station such that each user can communicate with a base‐station at any time. We model such a communication network with a Boolean model which we intersect with a line, the road on which the users are moving. Applying point process techniques we derive approximation formulae for the distribution of the covered and uncovered parts of the road. Furthermore, we examine for a rather general model the probability that exactly K users are communicating with a base station. Numerical examples illustrate how accurate the approximations are. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximations for lognormally fading optical signals

A lognormally distributed random variable can be used to approximate accurately a noncentral chi-square random variable, This letter uses this result to approximate the photoelectron count distribution of a lognormally fading optical signal.     

Recursive methods for numerical computation of the coefficients of a generalization of the Gram-Charlier series

Recursive methods well suited for the numerical computation of the coefficients appearing in the approximation of a probability density function (pdf) with the aid of a known and well behaved reference pdf and its successive derivatives are presented in this letter. The use of the normal law as a reference law is examined.     

Generalized Gram-Charlier series with application to the sum of log-normal variates (Corresp.)

A generalized Gram-Charlier series, applicable to non-Gaussian problems, is developed. Expressions are given for the first six error coefficients. The high inherent accuracy of the series is demonstrated by development of the expansion for the sum of independent, identically distributed log-normal variates.     

Approximations to hybrid mode slot line behaviour

New closed form formulas are presented for the dispersion characteristic (?'/ ?) and characteristic impedance (Z0) of single slot line. ?'/? and Z0 are given as functions of the slot width w over substrate thickness H ratio and the ratio H/? where ? is the free-space wavelength. For the ranges of variable considered ?'/? values agree with S. B. Cohn's and Mariani et al. results to within 5% and Z0 agrees to within 14.5% or better.