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 order 1 for noncentrality parameter a and argument b. Use this syntax to calculate the special case of the generalized Marcum Q-function of order m with m = 1. For the definition of the generalized Marcum Q-function, see Generalized Marcum Q-function.

Q = marcumq(a,b,m) computes the generalized Marcum Q-function of order m.

Examples

Generate and Plot Marcum $$Q$$-Function Data

Specify a range of values over which to generate data for the Marcum $$Q$$-function.

b = (0:0.1:10)';

Generate two output vectors comprising values of the Marcum $$Q$$-function of order one with noncentrality parameter values of 0 and 2.

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

Plot the resultant Marcum $$Q$$-functions.

plot(b,[Q1 Q2])

Input Arguments

a — Noncentrality parameter

nonnegative scalar | array of nonnegative numbers

Noncentrality parameter, specified as an array of nonnegative numbers.

Note

If you specify any input as an array, you must specify all other inputs as arrays of the same size or as scalars. If you specify any of the inputs as a scalar, the function expands that input to match the size of the other inputs.

Data Types: double

b — Argument of Marcum Q-function

nonnegative scalar | array of nonnegative numbers

Argument of the Marcum Q-function, specified as an array of nonnegative numbers.

Data Types: double

m — Order of generalized Marcum Q-function

positive integer | array of positive integers

Order of the generalized Marcum Q-function, specified as an array of positive integers.

Data Types: double

Output Arguments

Q — Value of Marcum Q-function

scalar in the range [0, 1] | array of numbers in the range [0, 1]

Value of the Marcum Q-function, returned as an array of numbers in the range [0, 1]. The size of this output matches the size of the input arguments.

Data Types: double

More About

Generalized Marcum Q-function

The generalized Marcum Q-function of order m is defined by

$$Q_m(a,b)=\frac{1}{a^{m-1}}{\displaystyle {\int }_b^{\infty }{x}^m\exp \left(-\frac{x^2+a^2}{2}\right)I_{m-1}(ax)dx},$$

where I_{m – 1} is the modified Bessel function of the first kind of order m – 1.

Algorithms

To compute the value of the generalized Marcum Q-function for the specified inputs, this function uses a calculation based on a method illustrated by Shnidman in [3]. This paper describes two error criteria: a relative error criterion and an absolute error criterion. This function uses the absolute error criterion.

References

[1] Cantrell, P., and A. Ojha. “Comparison of Generalized Q-function Algorithms (Corresp.).” IEEE^® Transactions on Information Theory 33, no. 4 (July 1987): 591–96. https://doi.org/10.1109/TIT.1987.1057323.

[2] Marcum, J. “A Statistical Theory of Target Detection by Pulsed Radar.” IEEE Transactions on Information Theory 6, no. 2 (April 1960): 59–267. https://doi.org/10.1109/TIT.1960.1057560.

[3] Shnidman, D.A. “The Calculation of the Probability of Detection and the Generalized Marcum Q-function,” IEEE Transactions on Information Theory 35, no. 2 (March 1989): 389–400. https://doi.org/10.1109/18.32133.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2008a