This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this purpose, we also describe two methods for inverting Laplace transforms based on the Post-Widder inversion formula. The overall procedure is illustrated by several queueing examples.

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The Fourier-series method for inverting transforms of probability distributions

Extension de la méthode des quadripôles thermiques à l'aide de transformations intégrales—calcul du transfert thermique au travers d'un défaut plan bidimensionnel

An axial dispersion model for the operation of a fixed-bed adsorber with a linear adsorption isotherm was formulated and solved analytically using the separation of variables method. The asymptotic solution for a large Peclet number and/or a small operation time was also obtained using two limiting cases. The results can be used to correlate or to predict both the mass-transfer zones and the breakthrough curves of a fixed-bed adsorber. Influence of both the Peclet number and the retardation coefficient on the operation of a fixed-bed adsorber was also studied. As an example, the axial dispersion model was applied to the experimental data of the removal of phenol from the solution in the activated carbon and the Amberlite resin XAD-4 fixed-bed adsorbers.

J.T. Hsu

Papers

Numerical inversion of certain laplace transforms by the direct application of fast fourier transform (FFT) algorithm