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Marcum Q-Function

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A function that arises in performance analysis of partially coherent, differentially coherent, and noncoherent communications. The generalized Marcum Q-function is defined by

Q_M(alpha,beta)=1/(alpha^(M-1))int_beta^inftyx^Me^(-(x^2+alpha^2)/2)I_(M-1)(alphax)dx,
(1)

where I_n(x) is a modified Bessel function of the first kind. It also has the series form

Q_M(alpha,beta)=e^(-(alpha^2+beta^2)/2)sum_(k=1-M)^infty(alpha/beta)^kI_k(alphabeta)
(2)

(Helstrom 1960, Proakis 1983).

The Marcum Q-function is implemented in the Wolfram Language as MarcumQ[M, a, b].

A special case is given by

Q_1(alpha,alpha)=1/2[1+e^(-alpha^2)I_0(alpha^2)]
(3)

(Schwarz et al. 1966, eqn. A-3-2).

The Marcum Q-function satisfies the identity

Q_1(alpha,beta)+Q_1(beta,alpha)=1+e^(-(alpha^2+beta^2)/2)I_0(alphabeta).
(4)

The derivative of Q_M(alpha,beta) with respect to b is given by

(partialQ_M)/(partialbeta)=-(beta^M)/(alpha^(M-1))e^(-(alpha^2+beta^2)/2)I_(M-1)(alphabeta).
(5)

See also

Nuttall Q-Function

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References

Cantrell, P. E. and Ojha, A. K. "Comparison of Generalized Q-Function Algorithms." IEEE Trans. Info. Th. 33, 591-596, Jul. 1987.Helstrom, C. W. Statistical Theory of Signal Detection. New York: Pergamon, 1960.Marcum, J. I. Table of Q Functions. U.S. Air Force RAND Research Memorandum M-339. Santa Monica, CA: Rand Corporation, Jan.  1, 1950.Parl, S. "A New Method of Calculating the Generalized Q Function." IEEE Trans. Info. Th. 26, 121-124, Jan. 1980.Peterson, R. L.; Ziemer, R. E.; and Borth, D. E. "Numerical Approximations for the Chi-Squared Probability Distribution and Marcum's Q-Function." Appendix F in Introduction to Spread-Spectrum Communications. Englewood Cliffs, NJ: Prentice-Hall, pp. 672-681, 1995.Proakis, J. Digital Communications. New York: McGraw-Hill, 1983.Schwartz, M.; Bennett, W. R.; and Stein, S. Communication Systems and Techniques. New York: McGraw-Hill, 1966.Shnidman, D. A. "The Calculation of the Probability of Detection and the Generalized Marcum Q-Function." IEEE Trans. Info. Th. 35, 389-400, Mar. 1989.Simon, M. K. "A New Twist on the Marcum Q-Function and Its Application." IEEE Commun. Lett. 2, 39-41, Feb. 1998.Simon, M. K. "The Nuttall Q-Function: Its Relation to the Marcum Q-Function and Its Application in Digital Communication Performance Evaluation." IEEE Trans. Commun. 50, 1712-1715, 2002.Simon, M. K. and Alouini, M.-S. "Exponential-Type Bounds on the Generalized Marcum Q-function with Application to Error Probability Analysis over Fading Channels." IEEE Trans. Commun. 48, 359-366, Mar. 2000.Simon, M. K. and Alouini, M.-S. Digital Communication over Fading Channels: A Unified Approach to Performance Analysis. New York: Wiley, 2000.Simon, M. K.; Hinedi, S. M.; and Lindsey, W. C. Digital Communication Techniques: Signal Design and Detection. Englewood Cliffs, NJ: Prentice-Hall, 1965.Stein, S. "Unified Analysis of Certain Coherent and Noncoherent Binary Communication Systems." IEEE Trans. Information Th. 10, 43-51, Jan. 1964.Van Trees, H. L. Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, pp. 394-395 and 411, 1968.

Referenced on Wolfram|Alpha

Marcum Q-Function

Cite this as:

Weisstein, Eric W. "Marcum Q-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MarcumQ-Function.html

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