Abstract
Probability generating functions of the number of photoelectrons emitted by a detector during a given interval can often be relatively easily calculated from a physical model of the emissive processes, but inverting them analytically to obtain the distributions of the number of so-called “photocounts” is usually complicated or impossible. Techniques for computing those distributions can be based on numerical evaluation of the inversion integral along a suitably chosen contour passing through a saddlepoint of the integrand and lying as close as possible to a path of steepest descent of that integrand. The method can be extended to include the effects of additive Gaussian noise, intersymbol interference, and postdetector filtering.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Debye, P., “Näherungsformeln für die Zylinderfunktionen für grosse Werte des Arguments and unbeschränkt veränderliche Werte des Index,” Math. Annalen, vol. 67, 535–558 (1909).
Riemann, B., “Sullo svolgimento del quoziente di due serie ipergeometriche in frazione continua infinita,” (Oct. 1863). In Collected Works of Bernhard Riemann, H. Weber, Ed., 424–430, New York: Dover Publications Inc. (1953).
Budden, K. G., The Propagation of Radio Waves, Cambridge: Cambridge University Press (1985).
Daniels, H. G., “Saddle point approximations in statistics,” Ann. Math. Stat., vol. 25, 631–650 (1954).
Mazo, J. E., Salz, J., “Probability of error for quadratic detectors,” Bell System Tech. J., vol. 44, 2165–2186 (Nov. 1965).
Helstrom, C. W., “Approximate evaluation of detection probabilities in radar and optical communications,” IEEE Trans. on Aerospace & Electronic Systems, vol. AES-14, 630–640 (July, 1978).
Fry, Thornton, C., Probability and Its Engineering Uses. 2nd ed., 257–264, Princeton, N.J.: D. Van Nostrand Company (1965).
Rice, S. O., “Uniform asymptotic expansions for saddle point integrals — Application to a probability distribution occurring in noise theory,” Bell Sys. Tech. J., vol. 47, 1971–2013 (Nov. 1968).
Lugannani, R., Rice, S. O., “Saddlepoint approximation for the distribution of the sum of independent random variables,” Adv. Appl. Prob., vol. 12, 475–490 (1980).
Carrier, G. F., Krook, M., Pearson, C. E., Functions of a Complex Variable, 257–264, New York: McGraw-Hill Book Co. (1966).
Rice, S. O., “Distribution of quadratic forms in normal random variables — Evaluation by numerical integration,” SIAM J. Sci. Stat. Comput., vol. 1, 438–448 (Dec. 1980).
Lugannani, R., Rice, S. O., “Distribution of the ratio of quadratic forms in normal variables — Numerical methods,” SIAM J. Sci. Stat. Comput., vol. 5, 476–488 (June, 1984).
Helstrom, C. W., Ritcey, J. A., “Evaluating radar detection probabilities by steepest descent integration,” IEEE Trans. on Aerospace & Electronic Systems, vol. AES-20, 624–634 (Sept. 1984).
Helstrom, C. W., “Computation of photoelectron counting distributions by numerical contour integration,” J. Opt. Soc. Am., vol. A2, 674–682 (May, 1985).
Helstrom, C. W., “Computation of photoelectron counting distributions by numerical contour integration. II. Light with rational spectral density,” J. Opt. Soc. Am., vol. A4, 1245–1255 (July, 1987).
Gagliardi, R. M., Karp, S., Optical Communications, 67–70, New York: J. Wiley & Sons (1976).
Personick, S. D., “New results on avalanche multiplication statistics with applications to optical detection,” Bell Sys. Tech. J., vol. 50, 167–189 (Jan. 1971).
Schwartz, C., “Numerical integration of analytic functions,” J. Computational Physics, vol. 4, 19–29, 1969.
Rice, S. O., “Efficient evaluation of integrals' of analytic functions by the trapezoidal rule,” Bell System Tech. J., vol. 52, 707–722 (May-June, 1973).
Helstrom, C. W., “Output distributions of electrons in a photomultiplier,” J. Appl. Phys., vol. 55, 2786–2792 (Apr. 1, 1984).
Helstrom, C. W., “Computation of output electron distributions in avalanche photodiodes,” IEEE Trans. on Electron Devices, vol. ED-31, 955–958 (July, 1984).
Helstrom, C. W., Rice, S. O., “Computation of counting distributions arising from a single-stage multiplicative process,” J. Computational Physics, vol. 54, 289–324 (May, 1984).
Helstrom, C. W., “Computing the performance of optical receivers with avalanche diode detectors,” IEEE Trans. on Communications, vol. COM-36, 61–66 (Jan. 1988).
Helstrom, C. W., Ho, C.-L., “Analysis of avalanche diode receivers by saddlepoint integration,” submitted to IEEE Trans. on Communications.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag
About this paper
Cite this paper
Helstrom, C.W. (1991). Calculating photocount distributions by saddlepoint methods. In: Bendjaballah, C., Hirota, O., Reynaud, S. (eds) Quantum Aspects of Optical Communications. Lecture Notes in Physics, vol 378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-53862-3_179
Download citation
DOI: https://doi.org/10.1007/3-540-53862-3_179
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-53862-2
Online ISBN: 978-3-540-46366-5
eBook Packages: Springer Book Archive