A Brief History of Shot Noise

John A. Gubner

The most basic shot-noise statistics, namely the mean and variance, were reported by Campbell in 1909 [1]-[2]. Shot noise was also investigated by Schottky in his 1918 paper on spontaneous current fluctuations in electric conductors [14]. In 1944-45, Rice [12] gave an extensive analysis of shot noise when the underlying Poisson process has a constant intensity. In particular, he showed that as the intensity tends to infinity, the probability distribution of the shot noise tends to a normal distribution. In 1971 Papoulis [11], considering underlying Poisson processes with time-varying intensity, gave numerical bounds on the difference between the true shot-noise distribution and the Gaussian approximation. The first nonasymptotic results concerning the cumulative distribution of shot noise appeared in the 1960 paper by Gilbert and Pollak [4]. For an underlying Poisson process with constant intensity, they derived an integral equation satisfied by the shot-noise distribution. They were able to solve this integral equation for some special cases of the shot-noise impulse response. Progress on the numerical computation of the density of shot noise was reported by Richter and Smits [13] in 1974. Their approach was to approximate the characteristic function by piecewise polynomial segments, which could then be inverse Fourier transformed in closed form. In 1975 Foschini et al. [3] gave a detailed analysis of the detection of a shot-noise signal in the presence of additive white Gaussian noise. They employed several approximations in order to obtain manageable expressions for the likelihood function, from which they could gain insight into the general structure of the optimum detector. They did not analyze the effects of their approximations or attempt to analyze the performance of their approximately optimal detector. The 1976 paper of Mazo and Salz [9] analyzed the performance of integrate-and-dump filters. They also obtained exact formulas for the probability distribution of the gains in physical avalanche diodes. Further work on computing the density of shot noise appeared in the 1978 paper by Yue et al. [15]. Their approach was to approximate the shot-noise density by a weighted sum of normal densities. Of particular interest to us is the 1984 paper of Morris [10]. Although his imaging model assumes that the photon locations are known, he processes them with a linear filter, and so ends up analyzing a filtered Poisson process. He is fortunate that in his example the shot-noise characteristic function is available in closed form. In 1988 Kadota [7] reported approximately optimum detection of deterministic signals in Gaussian and compound Poisson noise. His model included a noise process that consisted of samples of a Gaussian process where the sample times were Poisson distributed. In 1990, Lowen and Teich [8] considered shot-noise processes with a power-law impulse response to obtain 1/f noise. They assumed that the underlying Poisson process had a constant intensity, and they showed that such shot-noise random variables need not converge to a normal law as the intensity increases. In 1991 Hero [6] approximated the likelihood ratio for observing a shot-noise process in the presence of additive white Gaussian noise. His approximation became more accurate as impulse response of the shot noise became more narrow and more closely resembled the underlying point process. Most recently, Helstrom and Ho [5] have successfully applied numerical contour integration to the inversion of the shot-noise characteristic function to obtain the shot-noise cumulative distribution in the case when additive Gaussian noise is present.

For references to my own work on shot noise, see my publications.

References

  1. N. Campbell, ``The study of discontinuous phenomena,'' Proc. Cambr. Phil. Soc., vol. 15, pp. 117-136, 1909.
  2. N. Campbell, ``Discontinuities in light emission,'' Proc. Cambr. Phil. Soc., vol. 15, pp. 310-328, 1909.
  3. G. J. Foschini, R. D. Gitlin, and J. Salz, ``Optimum direct detection for digital fiber optic communication systems,'' Bell Syst. Tech. J., vol. 54, no. 8, pp. 1389-1430, Oct. 1975.
  4. E. N. Gilbert and H. O. Pollak, ``Amplitude distribution of shot noise,'' Bell Syst. Tech. J., vol. 39, pp. 333-350, Mar. 1960.
  5. C. W. Helstrom and C.-L. Ho, ``Analysis of avalanche diode receivers by saddlepoint integration,'' IEEE Trans. Commun., vol. 40, no. 8, pp. 1327-1338, Aug. 1992.
  6. A. O. Hero, ``Timing estimation for a filtered Poisson process in Gaussian noise,'' IEEE Trans. Inform. Theory, vol. 37, no. 1, pp. 92-106, Jan. 1991.
  7. T. T. Kadota, ``Approximately optimum detection of deterministic signals in Gaussian and compound Poisson noise,'' IEEE Trans. Inform. Theory, vol. 34, no. 6, pp. 1517-1527, Nov. 1988.
  8. S. B. Lowen and M. C. Teich, ``Power-law shot noise,'' IEEE Trans. Inform. Theory, vol. 36, no. 6, pp. 1302-1318, Nov. 1990.
  9. J. E. Mazo and J. Salz, ``On optical data communication via direct detection of light pulses,'' Bell Syst. Tech. J., vol. 55, no. 3, pp. 347-369, Mar. 1976.
  10. G. M. Morris, ``Scene matching using photon-limited images,'' J. Opt. Soc. Amer. A, vol. 1, no. 5, pp. 482-488, May 1984.
  11. A. Papoulis, ``High density shot noise and Gaussianity,'' J. Appl. Prob., vol. 8, no. 1, 118-127, Mar. 1971.
  12. S. O. Rice, ``Mathematical analysis of random noise,'' Bell Syst. Tech. J., vol. 23, pp. 282-332, July 1944; vol. 24, pp. 46-156, Jan. 1945. [Reprinted in Selected Papers on Noise and Stochastic Processes, N. Wax, Ed. New York: Dover, 1954, pp. 133-294].
  13. W. J. Richter Jr. and T. I. Smits, ``Numerical evaluation of Rice's integral representation of the probability density function for Poisson impulse noise,'' J. Acoust. Soc. Amer., vol. 56, pp. 481-496, 1974.
  14. W. Schottky, ``Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern,'' Annalen der Physik, vol. 57, pp. 541-567, 1918.
  15. O.-C. Yue, R. Lugannani, and S. O. Rice, ``Series approximations for the amplitude distribution and density of shot processes,'' IEEE Trans. Commun. vol. COM-26, no. 1, pp. 45-54, Jan. 1978.
Sep 27 1996
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