# The Fourier-series method for inverting transforms of probability distributions

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## Abstract

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.

### Keywords

Computational probability numerical inversion of transforms characteristic functions Laplace transforms generating functions Fourier transforms cumulative distribution functions calculating tail probabilities numerical integration Fourier series Poisson summation formula the Fourier-series method the Gaver-Stehfest method## Preview

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### References

- [1]J. Abate and H. Dubner, A new method for generating power series expansions of functions, SIAM J. Numer. Anal. 5 (1968) 102–112.Google Scholar
- [2]J. Abate and W. Whitt, Transient behavior of regulated Brownian motion I: starting at the origin, Adv. Appl. Prob. 19 (1987) 560–598.Google Scholar
- [3]J. Abate and W. Whitt, Transient behavior of the
*M/M*/1 queue via Laplace transforms, Adv. Appl. Prob. 20 (1988) 145–178.Google Scholar - [4]J. Abate and W. Whitt, Approximations for the
*M/M*/1 busy-period distribution, in:*Queueing Theory and its Applications, Liber Amicorum for J.W. Cohen*, eds. O.J. Boxma and R. Syski (North-Holland, Amsterdam, 1988) pp. 149–191.Google Scholar - [5]J. Abate and W. Whitt, Simple spectral representations for the M/M/1 queue, Queueing Systems 3 (1988) 321–346.Google Scholar
- [6]J. Abate and W. Whitt, Numerical inversion of Laplace transforms of probability distributions, AT&T Bell Laboratories, Murray Hill, NJ (1991).Google Scholar
- [7]J. Abate and W. Whitt, Numerical inversion of probability generating functions, AT&T Bell Laboratories, Murray Hill, NJ (1991).Google Scholar
- [8]M. Abramowitz and I.A. Stegun,
*Handbook of Mathematical Functions*(National Bureau of Standards, Washington, DC, 1972).Google Scholar - [9]N.C. Beaulieu, An infinite series for the computation of the complementary probability distribution function of a sum of independent random variables and its application to the sum of Rayleigh random variables, IEEE Trans. Commun. COM-38 (1990) 1463–1474.Google Scholar
- [10]R. Bellman, R.E. Kalaba and J. Lockett,
*Numerical Inversion of the Laplace Transform. Application to Biology, Economics, Engineering and Physics*(American Elsevier, New York, 1966).Google Scholar - [11]B.C. Berndt,
*Ramanujan's Notebooks*, Part II (Springer, New York, 1989).Google Scholar - [12]D. Bertsimas and D. Nakazato, Transient and busy period analysis for the
*GI/G*/1 queue; the method of stages, Queueing Systems 10 (1992) 153–184.Google Scholar - [13]H. Bohman, A method to calculate the distribution function when the characteristic function is known, Ark. Mat. 4 (1960) 99–157.Google Scholar
- [14]H. Bohman, A method to calculate the distribution function when the characteristic function is known, BIT 10 (1970) 237–242.Google Scholar
- [15]H. Bohman, From characteristic function to distribution function via Fourier analysis, BIT 12 (1972) 279–283.Google Scholar
- [16]E.O. Brigham and R.E. Conley, Evaluation of cumulative probability distribution functions: improved numerical methods, IEEE Proc. 58 (1970) 1367–1368.Google Scholar
- [17]A.S. Carasso, Infinitely divisible pulses, continuous deconvolution, and the characterization of linear time invariant systems, SIAM J. Appl. Math. 47 (1987) 892–927.Google Scholar
- [18]H.S. Carslaw,
*Introduction to the Theory of Fourier's Series and Integrals*, 3rd ed. (Dover, New York, 1930).Google Scholar - [19]J.K. Cavers, On the fast Fourier transform inversion of probability generating functions, J. Inst. Math. Appl. 22 (1978) 275–282.Google Scholar
- [20]D.C. Champeney,
*A Handbook of Fourier Theorems*(Cambridge University Press, New York, 1987).Google Scholar - [21]K.L. Chung,
*A Course in Probability Theory*, 2nd ed. (Academic Press, New York, 1974).Google Scholar - [22]J.W. Cooley, P.A.W. Lewis and P.D. Welch, Application of the fast Fourier transform to the computation of Fourier integrals, Fourier series, and convolution integrals, IEEE Trans. AU-15 (1967) 79–84.Google Scholar
- [23]J.W. Cooley, P.A.W. Lewis and P.D. Welch, Historical notes on the fast Fourier transform, Proc. IEEE 55 (1967) 1675–1677.Google Scholar
- [24]J.W. Cooley, P.A.W. Lewis and P.D. Welch, The fast Fourier transform algorithm: programming considerations of sine, cosine and Laplace transforms, J. Sound Vib. 12 (1970) 315–337.Google Scholar
- [25]J.W. Cooley and J.W. Tukey, An algorithm for the machine computation of complex Fourier series, Math. Comp. 19 (1965) 297–301.Google Scholar
- [26]D.R. Cox,
*Renewal Theory*(Methuen, London, 1962).Google Scholar - [27]K.S. Crump, Numerical inversion of Laplace transforms using a Fourier-series approximation, J. ACM 23 (1976) 89–96.Google Scholar
- [28]J.N. Daigle, Queue length distributions from probability generating functions via discrete Fourier transforms, Oper. Res. Lett. 8 (1989) 229–236.Google Scholar
- [29]B. Davies and B.L. Martin, Numerical inversion of Laplace transforms: a critical evaluation and review of methods, J. Comp. Phys. 33 (1970) 1–32.Google Scholar
- [30]R.B. Davies, Numerical inversion of a characteristic function, Biometrika 60 (1973) 415–417.Google Scholar
- [31]R.B. Davies, The distribution of a linear combination of
*X*^{2}random variables, Appl. Stat. 29 (1980) 323–333.Google Scholar - [32]P.J. Davis and P. Rabinowitz,
*Methods of Numerical Integration*, 2nd ed. (Academic Press, New York, 1984).Google Scholar - [33]M.A.B. Deakin, Euler's version of the Laplace transform, Amer. Math. Monthly 87 (1980) 264–269.Google Scholar
- [34]G. de Balbine and J. Franklin, The calculation of Fourier integrals, Math. Comp. 20 (1966) 570–589.Google Scholar
- [35]F.R. de Hoog, J.H. Knight and A.N. Stokes, An improved method for numerical inversion of Laplace transforms, SIAM J. Sci. Stat. Comput. 3 (1982) 357–366.Google Scholar
- [36]G. Doetsch,
*Introduction to the Theory and Application of the Laplace Transformation*(Springer, New York, 1974).Google Scholar - [37]B.T. Doshi, Analysis of clocked schedules — high priority tasks, AT&T Tech. J. 64 (1985) 633–659.Google Scholar
- [38]B.T. Doshi and J. Kaufman, Sojourn times in an
*M/G*/1 queue with Bernoulli feedback, in:*Queueing Theory and Its Applications, Liber Amicorum for J.W. Cohen*, eds. O.J. Boxma and R. Syski (North-Holland, Amsterdam, 1988).Google Scholar - [39]H. Dubner, Partitions approximated by finite cosine series, Math. Computation, to appear.Google Scholar
- [40]H. Dubner and J. Abate, Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform, J. ACM 15 (1968) 115–123.Google Scholar
- [41]F. Durbin, Numerical inversion of Laplace transforms: An efficient improvement to Dubner and Abate's method, Comput. J. 17 (1974) 371–376.Google Scholar
- [42]W. Feller,
*An Introduction to Probability Theory and its Applications*, Vol. I, 3rd ed. (Wiley, New York, 1968).Google Scholar - [43]W. Feller,
*An Introduction to Probability Theory and its Applications*, Vol. II, 2nd ed. (Wiley, New York, 1971).Google Scholar - [44]H.E. Fettis, Numerical calculation of certain definite integrals by Poisson's summation formula, Math. Tables Other Aids Comput. 9 (1955) 85–92.Google Scholar
- [45]B. Fornberg, Numerical differentiation of analytic functions, ACM Trans. Math. Software 7 (1981) 512–526.Google Scholar
- [46]J. Foster and F.B. Richards, The Gibbs phenomenon for piecewise-linear approximations, Amer. Math. Monthly 98 (1991) 47–49.Google Scholar
- [47]B.S. Garbow, G. Giunta, J.N. Lyness and A. Murli, Algorithm 662, A FORTRAN software package for the numerical inversion of the Laplace transform based on Weeks' method, ACM Trans. Meth. Software 14 (1988) 171–176.Google Scholar
- [48]W. Gautschi, On the condition of a matrix arising in the numerical inversion of the Laplace transform, Math. Comput. 23 (1969) 109–118.Google Scholar
- [49]D.P. Gaver, Jr., Observing stochastic processes and approximate transform inversion, Oper. Res. 14 (1966) 444–459.Google Scholar
- [50]D.P. Gaver, Jr., Diffusion approximations and models for certain congestion problems, J. Appl. Prob. 5 (1968) 607–623.Google Scholar
- [51]J. Gil-Palaez, Note on the inversion theorem, Biometrika 38 (1951) 481–482.Google Scholar
- [52]I.J. Good, Analogs of Poisson's sum formula, Amer. Math. Monthly 69 (1962) 259–266.Google Scholar
- [53]F.J. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform, Proc. IEEE 66 (1978) 51–83.Google Scholar
- [54]P.G. Harrison, Laplace transform inversion and passage-time distributions in Markov processes, J. Appl. Prob. 27 (1990) 74–87.Google Scholar
- [55]H. Heffes and D.M. Lucantoni, A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance, IEEE J. Sel. Areas Commun. SAC-4 (1986) 856–868.Google Scholar
- [56]D.P. Heyman, Mathematical models of database degradation, ACM Trans. Database Sys. 7 (1982) 615–631.Google Scholar
- [57]G. Honig and U. Hirdes, A method for the numerical inversion of Laplace transforms, J. Comput. Appl. Math. 10 (1984) 113–129.Google Scholar
- [58]T. Hosono, Numerical inversion of Laplace transform, J. Inst. Elec. Eng. Jpn. 54–A64 (1979) 494 (in Japanese).Google Scholar
- [59]T. Hosono, Numerical inversion of Laplace transform and some applications to wave optics, Radio Sci. 16 (1981) 1015–1019.Google Scholar
- [60]T. Hosono,
*Fast Inversion of Laplace Transform by BASIC*(Kyoritsu Publishers, Japan, 1984; in Japanese).Google Scholar - [61]T. Hosono, Numerical algorithm for Taylor series expansion, Electronics and Communications in Japan 69 (1986) 10–18.Google Scholar
- [62]T. Hosono, K. Yuda and A. Itoh, Analysis of transient response of electromagnetic waves scattered by a perfectly conducting sphere. The case of back- and forward-scattering, Electronics and Communications in Japan 71 (1988) 74–86.Google Scholar
- [63]J.T. Hsu and J.S. Dranoff, Numerical inversion of certain Laplace transforms by the direct application of fast Fourier transform (FFT) algorithm, Comput. Chem. Engng. 11 (1987) 101–110.Google Scholar
- [64]S. Ichikawa and A. Kishima, Application of Fourier-series technique to inverse Laplace transform (Part I), Mem. Fac. Eng. Kyoto U. 34 (1972) 53–67.Google Scholar
- [65]S. Ichikawa and A. Kishima, Application of Fourier-series technique to inverse Laplace transform (Part II), Mem. Fac. Eng. Kyoto U. 35 (1973) 393–400.Google Scholar
- [66]D.L. Jagerman, An inversion technique for the Laplace transform with applications, Bell Sys. Tech. J. 57 (1978) 669–710.Google Scholar
- [67]D.L. Jagerman, An inversion technique for the Laplace transform, Bell Sys. Tech. J. 61 (1982) 1995–2002.Google Scholar
- [68]D.L. Jagerman, MATHCALC, AT&T Bell Laboratories, Holmdel, NJ (1987).Google Scholar
- [69]D.L. Jagerman, The approximation sequence of the Laplace transform, AT&T Bell Laboratories, Holmdel, NJ (1989).Google Scholar
- [70]R. Johnsonbaugh, Summing an alternating series, Amer. Math. Monthly 86 (1979) 637–648.Google Scholar
- [71]J. Keilson, Exponential spectra as a tool for the study of single-server systems, J. Appl. Prob. 15 (1978) 162–170.Google Scholar
- [72]D.G. Kendall, A summation formula for finite trigonometric integrals, Quart. J. Math. 13 (1942) 172–184.Google Scholar
- [73]J.E. Kiefer and G.H. Weiss, A comparison of two methods for accelerating the convergence of Fourier-series, Comput Math. Appl. 7 (1981) 527–535.Google Scholar
- [74]Y. Kida, UBASIC Version 8.12, Faculty of Science, Kanazawa University, 1-1 Marunouchi, Kanazawa 920, Japan (1990).Google Scholar
- [75]L. Kleinrock,
*Queueing Systems, Vol. 1: Theory*(Wiley, New York, 1975).Google Scholar - [76]H. Kobayashi,
*Modeling and Analysis*(Addison-Wesley, Reading, MA, 1978).Google Scholar - [77]S. Koizumi, A new method of evaluation of the Heaviside operational expression by Fourier series, Phil. Mag. 19 (1935) 1061–1076.Google Scholar
- [78]V.I. Krylov and N.S. Skoblya,
*A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation*(Mir Publ., Moscow, 1977).Google Scholar - [79]Y.K. Kwok and D. Barthez, An algorithm for the numerical inversion of Laplace transforms, Inverse Problems 5 (1989) 1089–1095.Google Scholar
- [80]E. Lukacs,
*Characteristic Functions*, 2nd ed. (Hafner, New York, 1970).Google Scholar - [81]Y.L. Luke, Simple formulas for the evaluation of some higher transcendental functions, J. Math. Phys. 34 (1955) 298–307.Google Scholar
- [82]J.N. Lyness, Differentiation formulas for analytic functions, Math. Comp. 22 (1968) 352–356.Google Scholar
- [83]J.N. Lyness and G. Giunta, A modification of the Weeks method for numerical inversion of the Laplace transform, Math. Comp. 47 (1986) 313–322.Google Scholar
- [84]J.N. Lyness and C.B. Moler, Numerical differentation of analytic functions, SIAM J. Numer. Anal. 4 (1967) 202–210.Google Scholar
- [85]W.F. Magnus, F. Oberhettinger and R.P. Soni,
*Formulas and Theorems for the Special Functions of Mathematical Physics*(Springer, New York, 1966).Google Scholar - [86]M.R. Middleton, Transient effects in
*M/G*/1 queues, Ph.D. dissertation, Stanford University (1979).Google Scholar - [87]P.L. Mills, Numerical inversion of z-transforms with application to polymer kinetics, Comp. Chem. 2 (1987) 137–151.Google Scholar
- [88]A. Murli and M. Rizzardi, Algorithm 682, Talbot's method for the Laplacc inversion problem, ACM Trans. Math. Software 16 (1990) 158–168.Google Scholar
- [89]N. Mullineux and J.R. Reed, Numerical inversion of integral transforms, Comput. Math. Appl. 3 (1977) 299–306.Google Scholar
- [90]R.E. Nancc, U.N. Bhat and B.G. Claybrook, Busy period analysis of a time sharing system: transform inversion, J. ACM 19 (1972) 453–463.Google Scholar
- [91]I.P. Natanson,
*Constructive Function Theory, Vol. I, Uniform Approximation*(F. Ungar, New York, 1964).Google Scholar - [92]W.D. Neumann, UBASIC: A public-domain BASIC for mathematics, Notices Amer. Math. Soc. 36 (1989) 557–559.Google Scholar
- [93]A.H. Nuttall, Numerical evaluation of cumulative probability distribution functions directly from characteristic functions, IEEE Proc. 57 (1969) 2071–2072.Google Scholar
- [94]F. Oberhettinger,
*Fourier Transforms of Distributions and Their Inverses*(Academic, New York, 1973).Google Scholar - [95]W.C. Obi, LAPLACE — A performance analysis library (PAL) module, AT&T Bell Laboratories, Holmdel, NJ (1987).Google Scholar
- [96]R. Piessens, A bibliography on numerical inversion of the Laplace transform and its applications, J. Comput. Appl. Math. 1 (1975) 115–128.Google Scholar
- [97]R. Piessens, and N.D.P. Dang, A bibliography on numerical inversion of the Laplace transform and its applications: A supplement, J. Comput. Appl. Math. 2 (1976) 225–228.Google Scholar
- [98]R. Piessens and R. Huysmans, Algorithm 619. Automatic numerical inversion of the Laplace transforms, ACM Trans. Math. Softw. 10 (1984) 348–353.Google Scholar
- [99]L.K. Platzman, J.C. Ammons and J.J. Bartholdi, III, A simple and efficient algorithm to compute tail probabilities from transforms, Oper. Res. 36 (1988) 137–144.Google Scholar
- [100]S.D. Poisson, Mémoire sur le Calcul Numérique des Integrales Défines, Mem. Acad. Sci. Inst. France 6 (1823) 571–602.Google Scholar
- [101]E.L. Post, Generalized differentiation, Trans. Amer. Math. Soc. 32 (1930) 723–781.Google Scholar
- [102]L.R. Rabiner and B. Gold,
*Theory and Application of Digital Signal Processing*(Prentice-Hall, Englewood Cliffs, NJ, 1975).Google Scholar - [103]A.A.G. Requicha, Direct computation of distribution functions from characteristic functions using the fast Fourier transform, IEEE Proc. 58 (1970) 1154–1155.Google Scholar
- [104]J. Riordan,
*Stochastic Service Systems*(Wiley, New York, 1962).Google Scholar - [105]S.O. Rice, Efficient evaluation of integrals of analytic functions by the trapezoidal rule, Bell Sys. Tech. J. 52 (1973) 707–722.Google Scholar
- [106]S.O. Rice, Numerical evaluation of integrals with infinite limits and oscillating integrands, Bell. Sys. Tech. J. 54 (1975) 155–164.Google Scholar
- [107]S. Ross,
*Stochastic Processes*(Wiley, New York, 1983).Google Scholar - [108]B. Schorr, Numerical inversion of a class of characteristic functions, BIT 15 (1975) 94–102.Google Scholar
- [109]M. Silverberg, Improving the efficiency of Laplace-transform Inversion for network analysis, Electronics Lett. 6 (1970) 105–106.Google Scholar
- [110]R.M. Simon, M.T. Stroot and G.H. Weiss, Numerical inversion of Laplace transforms with applications to percentage labeled experiments, Comput. Biomed. Res. 6 (1972) 596–607.Google Scholar
- [111]W.L. Smith, On the distribution of queueing times, Proc. Camb. Phil. Soc. 49 (1953) 449–461.Google Scholar
- [112]W. Squire, The numerical treatment of Laplace transforms: the Koizumi inversion method, Int. J. Num. Meth. Eng. 20 (1984) 1697–1702.Google Scholar
- [113]H. Stehfest, Algorithm 368. Numerical inversion of Laplace transforms, Commun. ACM 13 (1970) 479–49 (erratum 13, 624).Google Scholar
- [114]F. Stenger, (1981) Numerical methods based on Whittaker cardinal, or sine functions, SIAM Rev. 23 (1981) 165–224.Google Scholar
- [115]D. Stoyan,
*Comparison Methods for Queues and Other Stochastic Models*(Wiley, Chichester, 1983).Google Scholar - [116]A. Talbot, The accurate numerical inversion of Laplace transforms, J. Inst. Math. Appl. 23 (1979) 97–120.Google Scholar
- [117]D. ter Haar, An easy approximate method of determining the relaxation spectrum of a viscoelastic material, J. Polymer Sci. 6 (1951) 247–250.Google Scholar
- [118]H.C. Tijms,
*Stochastic Modelling and Analysis: A Computational Approach*(Wiley, Chichester, 1986).Google Scholar - [119]G.P. Tolstov,
*Fourier Series*(Dover, New York, 1976).Google Scholar - [120]B. Van Der Pol and H. Bremmer,
*Operational Calculus*(Cambridge Press, 1955; reprinted, Chelsea, New York, 1987).Google Scholar - [121]J.M. Varah, Pitfalls in the numerical solution of linear ill-posed problems, SIAM J. Sci. Stat. Comput. 4 (1983) 164–176.Google Scholar
- [122]F. Veillon, Une nouvelle méthode de calcul de la transformée inverse d'une fonction au sens de Laplace et de la deconvolution de deux fonctions, R.A.I.R.O. 6 (1972) 91–98.Google Scholar
- [123]F. Veillon, Algorithm 486. Numerical inversion of Laplace transform, Commun. AGM 17 (1974) 587–589.Google Scholar
- [124]W.T. Weeks, Numerical inversion of Laplace transforms using Laguerre functions, J. ACM 13 (1966) 419–426.Google Scholar
- [125]D.V. Widder, The inversion of the Laplace integral and the related moment problem, Trans. Amer. Math. Soc. 36 (1934) 107–200.Google Scholar