Marcum Q-Function

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

(1)

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

(2)

(Helstrom 1960, Proakis 1983).

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

A special case is given by

(3)

(Schwarz et al. 1966, eqn. A-3-2).

The Marcum -function satisfies the identity

(4)

The derivative of  with respect to  is given by

(5)

REFERENCES:

Cantrell, P. E. and Ojha, A. K. "Comparison of Generalized -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  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 -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 -Function." IEEE Trans. Info. Th. 35, 389-400, Mar. 1989.

Simon, M. K. "A New Twist on the Marcum -Function and Its Application." IEEE Commun. Lett. 2, 39-41, Feb. 1998.

Simon, M. K. "The Nuttall -Function: Its Relation to the Marcum -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 -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.