Documentation

marcumq

Generalized Marcum Q function

Syntax

Q = marcumq(a,b)
Q = marcumq(a,b,m)

Description

Q = marcumq(a,b) computes the Marcum Q function of a and b, defined by

$Q\left(a,b\right)={\underset{b}{\overset{\infty }{\int }}x\mathrm{exp}\left(-\frac{{x}^{2}+{a}^{2}}{2}\right)}^{}{I}_{0}\left(ax\right)dx$

where a and b are nonnegative real numbers. In this expression, I0 is the modified Bessel function of the first kind of zero order.

Q = marcumq(a,b,m) computes the generalized Marcum Q, defined by

${Q}_{}\left(a,b\right)=\frac{1}{{a}^{m-1}}\underset{b}{\overset{\infty }{\int }}{x}^{m}\mathrm{exp}\left(-\frac{{x}^{2}+{a}^{2}}{2}\right){I}_{m-1}\left(ax\right)dx$

where a and b are nonnegative real numbers, and m is a positive integer. In this expression, Im-1 is the modified Bessel function of the first kind of order m-1.

If any of the inputs is a scalar, it is expanded to the size of the other inputs.

Examples

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This example shows how to use the marcumq function.

Create an input vector, x.

x = (0:0.1:10)';

Generate two output vectors for a=0 and a=2.

Q1 = marcumq(0,x);
Q2 = marcumq(2,x);

Plot the resultant Marcum Q functions.

plot(x,[Q1 Q2]) References

 Cantrell, P. E., and A. K. Ojha, “Comparison of Generalized Q-Function Algorithms,” IEEE Transactions on Information Theory, Vol. IT-33, July, 1987, pp. 591–596.

 Marcum, J. I., “A Statistical Theory of Target Detection by Pulsed Radar: Mathematical Appendix,” RAND Corporation, Santa Monica, CA, Research Memorandum RM-753, July 1, 1948. Reprinted in IRE Transactions on Information Theory, Vol. IT-6, April, 1960, pp. 59–267.

 Shnidman, D. A., “The Calculation of the Probability of Detection and the Generalized Marcum Q-Function,” IEEE Transactions on Information Theory, Vol. IT-35, March, 1989, pp. 389–400.