Computation of Shot-Noise Probability Distributions and Densities

The computation of the cumulative distribution (cdf), the complementary cdf (ccdf), and the density of certain shot-noise random variables is discussed. After subtracting off a few terms that can be computed in closed form, what remains can be approximated by a general method for approximating samples of a cdf or ccdf by summing a Fourier series whose coefficients are modulated samples of their characteristic function. To approximate the density, a spline is fit to the cdf samples and then differentiated. When the density has corners, it is important that the spline have coincident knots at these locations. For shot-noise densities, these locations are easily identified.

[1]  Tommy Elfving,et al.  Interpolation and approximation by monotone cubic splines , 1991 .

[2]  H. Pollak,et al.  Amplitude distribution of shot noise , 1960 .

[3]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[4]  J. E. Mazo,et al.  On optical data communication via direct detection of light pulses , 1976, The Bell System Technical Journal.

[5]  R. E. Edwards,et al.  Fourier series : a modern introduction , 1982 .

[6]  A. Papoulis.  HIGH DENSITY SHOT NOISE AND GAUSSIANITY , 1971 .

[7]  T. Kawata.  Fourier analysis in probability theory , 1972 .

[8]  Malvin Carl Teich,et al.  Power-law shot noise , 1990, IEEE Trans. Inf. Theory.

[9]  I. Rubin,et al.  Random point processes , 1977, Proceedings of the IEEE.

[10]  George Michael Morris.  Scene matching using photon-limited images , 1984 .

[11]  J. Joseph,et al.  Fourier Series , 2018, Encyclopedia of GIS.

[12]  T. I. Smits,et al.  Numerical Evaluation of Rice's Integral Representation of the Probability Density Function for Poisson Impulsive Noise , 1971 .

[13]  Carl W. Helstrom,et al.  Analysis of avalanche diode receivers by saddlepoint integration , 1992, IEEE Trans. Commun..

[14]  S. Rice.  Mathematical analysis of random noise , 1944 .

[15]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[16]  Norman C. Beaulieu,et al.  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 , 1990, IEEE Trans. Commun..

[17]  Donald L. Snyder,et al.  Random Point Processes in Time and Space , 1991 .

[18]  On-Ching Yue,et al.  Series Approximations for the Amplitude Distribution and Density of Shot Processes , 1978, IEEE Trans. Commun..