Using the fact that , the generalized Marcum Q-function can alternatively be defined as a finite integral as
However, it is preferable to have an integral representation of the Marcum Q-function such that (i) the limits of the integral are independent of the arguments of the function, (ii) and that the limits are finite, (iii) and that the integrand is a Gaussian function of these arguments. For positive integer values of , such a representation is given by the trigonometric integral[2][3]
where
and the ratio is a constant.
For any real , such finite trigonometric integral is given by[4]
where is as defined before, , and the additional correction term is given by
For integer values of , the correction term tend to vanish.
Monotonicity and log-concavity
The generalized Marcum Q-function is strictly increasing in and for all and , and is strictly decreasing in for all and [5]
The function is strictly log-concave on for all and , which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.[6]
where is the generalized Laguerre polynomial of degree and of order .
The generalized Marcum Q-function of order can also be represented as Neumann series expansions[4][8]
where the summations are in increments of one. Note that when assumes an integer value, we have .
For non-negative half-integer values , we have a closed form expression for the generalized Marcum Q-function as[8][10]
where is the
complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as[4]
where is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have[4]
for non-negative integers , where is the
Gaussian Q-function. Alternatively, we can also more compactly express the Bessel functions with half-integer as sum of hyperbolic sine and cosine functions:[11]
where , , and for any integer value of .
Recurrence relation and generating function
Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation[8][10]
for positive integer . The former recurrence can be used to formally define the generalized Marcum Q-function for negative . Taking and for , we obtain the Neumann series representation of the generalized Marcum Q-function.
The related three-term recurrence relation is given by[7]
where
We can eliminate the occurrence of the Bessel function to give the third order recurrence relation[7]
Another recurrence relationship, relating it with its derivatives, is given by
The ordinary generating function of for integral is[10]
where
Symmetry relation
Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral
Various upper and lower bounds of generalized Marcum-Q function can be obtained using monotonicity and log-concavity of the function and the fact that we have closed form expression for when is half-integer valued.
Let and denote the pair of half-integer rounding operators that map a real to its nearest left and right half-odd integer, respectively, according to the relations
where and denote the integer floor and ceiling functions.
The monotonicity of the function for all and gives us the following simple bound[14][8][15]
However, the relative error of this bound does not tend to zero when .[5] For integral values of , this bound reduces to
A very good approximation of the generalized Marcum Q-function for integer valued is obtained by taking the arithmetic mean of the upper and lower bound[15]
A tighter bound can be obtained by exploiting the log-concavity of on as[5]
where and for . The tightness of this bound improves as either or increases. The relative error of this bound converges to 0 as .[5] For integral values of , this bound reduces to
Cauchy-Schwarz bound
Using the trigonometric integral representation for integer valued , the following Cauchy-Schwarz bound can be obtained[3]
where .
Exponential-type bounds
For analytical purpose, it is often useful to have bounds in simple exponential form, even though they may not be the tightest bounds achievable. Letting , one such bound for integer valued is given as[16][3]
When , the bound simplifies to give
Another such bound obtained via Cauchy-Schwarz inequality is given as[3]
Chernoff-type bound
Chernoff-type bounds for the generalized Marcum Q-function, where is an integer, is given by[16][3]
where the Chernoff parameter has optimum value of
Semi-linear approximation
The first-order Marcum-Q function can be semi-linearly approximated by [17]
where
and
Equivalent forms for efficient computation
It is convenient to re-express the Marcum Q-function as[18]
The can be interpreted as the detection probability of incoherently integrated received signal samples of constant received signal-to-noise ratio, , with a normalized detection threshold . In this equivalent form of Marcum Q-function, for given and , we have and . Many expressions exist that can represent . However, the five most reliable, accurate, and efficient ones for numerical computation are given below. They are form one:[18]
If is a
chi distribution with degrees of freedom, then its cdf is given by
If is a
Nakagami distribution with as shape parameter and as spread parameter, then its cdf is given by
If is a
Rice distribution with parameters and , then its cdf is given by
If is a
non-central chi distribution with non-centrality parameter and degrees of freedom, then its cdf is given by
Footnotes
^J.I. Marcum (1960). A statistical theory of target detection by pulsed radar: mathematical appendix, IRE Trans. Inform. Theory, vol. 6, 59-267.
^M.K. Simon and M.-S. Alouini (1998). A Unified Approach to the Performance of Digital Communication over Generalized Fading Channels, Proceedings of the IEEE, 86(9), 1860-1877.
^
abcdeA. Annamalai and C. Tellambura (2001). Cauchy-Schwarz bound on the generalized Marcum-Q function with applications, Wireless Communications and Mobile Computing, 1(2), 243-253.
^
abcdA. Annamalai and C. Tellambura (2008). A Simple Exponential Integral Representation of the Generalized Marcum Q-Function QM(a,b) for Real-Order M with Applications. 2008 IEEE Military Communications Conference, San Diego, CA, USA
^
abcdefgY. Sun, A. Baricz, and S. Zhou (2010). On the Monotonicity, Log-Concavity, and Tight Bounds of the Generalized Marcum and Nuttall Q-Functions. IEEE Transactions on Information Theory, 56(3), 1166–1186,
ISSN0018-9448
^
abY. Sun and A. Baricz (2008). Inequalities for the generalized Marcum Q-function. Applied Mathematics and Computation 203(2008) 134-141.
^
abcdefN.M. Temme (1993). Asymptotic and numerical aspects of the noncentral chi-square distribution. Computers Math. Applic., 25(5), 55-63.
^
abcdefA. Annamalai, C. Tellambura and John Matyjas (2009). "A New Twist on the Generalized Marcum Q-Function QM(a, b) with Fractional-Order M and its Applications". 2009 6th IEEE Consumer Communications and Networking Conference, 1–5,
ISBN978-1-4244-2308-8
^
abS. Andras, A. Baricz, and Y. Sun (2011) The Generalized Marcum Q-function: An Orthogonal Polynomial Approach. Acta Univ. Sapientiae Mathematica, 3(1), 60-76.
^
abcdefgY.A. Brychkov (2012). On some properties of the Marcum Q function. Integral Transforms and Special Functions 23(3), 177-182.
^W.K. Pratt (1968). Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(7), 1220-1221.
^R. Esposito (1968). Comment on Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(12), 2195-2195.
^V.M. Kapinas, S.K. Mihos, G.K. Karagiannidis (2009). On the Monotonicity of the Generalized Marcum and Nuttal Q-Functions. IEEE Transactions on Information Theory, 55(8), 3701-3710.
^
abR. Li, P.Y. Kam, and H. Fu (2010). New Representations and Bounds for the Generalized Marcum Q-Function via a Geometric Approach, and an Application. IEEE Trans. Commun., 58(1), 157-169.
^
abM.K. Simon and M.-S. Alouini (2000). Exponential-Type Bounds on the Generalized Marcum Q-Function with Application to Error Probability Analysis over Fading Channels. IEEE Trans. Commun. 48(3), 359-366.
^ H. Guo, B. Makki, M. -S. Alouini and T. Svensson, "A Semi-Linear Approximation of the First-Order Marcum Q-Function With Application to Predictor Antenna Systems," in IEEE Open Journal of the Communications Society, vol. 2, pp. 273-286, 2021, doi: 10.1109/OJCOMS.2021.3056393.
^
abcdefgD.A. Shnidman (1989). The Calculation of the Probability of Detection and the Generalized Marcum Q-Function. IEEE Transactions on Information Theory, 35(2), 389-400.
References
Marcum, J. I. (1950) "Table of Q Functions". U.S. Air Force RAND Research Memorandum M-339. Santa Monica, CA: Rand Corporation, Jan. 1, 1950.
Shnidman, David A. (1989): The Calculation of the Probability of Detection and the Generalized Marcum Q-Function,IEEE Transactions on Information Theory, 35(2), 389-400.
Weisstein, Eric W. Marcum Q-Function. From MathWorld—A Wolfram Web Resource.
[1]
Using the fact that , the generalized Marcum Q-function can alternatively be defined as a finite integral as
However, it is preferable to have an integral representation of the Marcum Q-function such that (i) the limits of the integral are independent of the arguments of the function, (ii) and that the limits are finite, (iii) and that the integrand is a Gaussian function of these arguments. For positive integer values of , such a representation is given by the trigonometric integral[2][3]
where
and the ratio is a constant.
For any real , such finite trigonometric integral is given by[4]
where is as defined before, , and the additional correction term is given by
For integer values of , the correction term tend to vanish.
Monotonicity and log-concavity
The generalized Marcum Q-function is strictly increasing in and for all and , and is strictly decreasing in for all and [5]
The function is strictly log-concave on for all and , which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.[6]
where is the generalized Laguerre polynomial of degree and of order .
The generalized Marcum Q-function of order can also be represented as Neumann series expansions[4][8]
where the summations are in increments of one. Note that when assumes an integer value, we have .
For non-negative half-integer values , we have a closed form expression for the generalized Marcum Q-function as[8][10]
where is the
complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as[4]
where is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have[4]
for non-negative integers , where is the
Gaussian Q-function. Alternatively, we can also more compactly express the Bessel functions with half-integer as sum of hyperbolic sine and cosine functions:[11]
where , , and for any integer value of .
Recurrence relation and generating function
Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation[8][10]
for positive integer . The former recurrence can be used to formally define the generalized Marcum Q-function for negative . Taking and for , we obtain the Neumann series representation of the generalized Marcum Q-function.
The related three-term recurrence relation is given by[7]
where
We can eliminate the occurrence of the Bessel function to give the third order recurrence relation[7]
Another recurrence relationship, relating it with its derivatives, is given by
The ordinary generating function of for integral is[10]
where
Symmetry relation
Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral
Various upper and lower bounds of generalized Marcum-Q function can be obtained using monotonicity and log-concavity of the function and the fact that we have closed form expression for when is half-integer valued.
Let and denote the pair of half-integer rounding operators that map a real to its nearest left and right half-odd integer, respectively, according to the relations
where and denote the integer floor and ceiling functions.
The monotonicity of the function for all and gives us the following simple bound[14][8][15]
However, the relative error of this bound does not tend to zero when .[5] For integral values of , this bound reduces to
A very good approximation of the generalized Marcum Q-function for integer valued is obtained by taking the arithmetic mean of the upper and lower bound[15]
A tighter bound can be obtained by exploiting the log-concavity of on as[5]
where and for . The tightness of this bound improves as either or increases. The relative error of this bound converges to 0 as .[5] For integral values of , this bound reduces to
Cauchy-Schwarz bound
Using the trigonometric integral representation for integer valued , the following Cauchy-Schwarz bound can be obtained[3]
where .
Exponential-type bounds
For analytical purpose, it is often useful to have bounds in simple exponential form, even though they may not be the tightest bounds achievable. Letting , one such bound for integer valued is given as[16][3]
When , the bound simplifies to give
Another such bound obtained via Cauchy-Schwarz inequality is given as[3]
Chernoff-type bound
Chernoff-type bounds for the generalized Marcum Q-function, where is an integer, is given by[16][3]
where the Chernoff parameter has optimum value of
Semi-linear approximation
The first-order Marcum-Q function can be semi-linearly approximated by [17]
where
and
Equivalent forms for efficient computation
It is convenient to re-express the Marcum Q-function as[18]
The can be interpreted as the detection probability of incoherently integrated received signal samples of constant received signal-to-noise ratio, , with a normalized detection threshold . In this equivalent form of Marcum Q-function, for given and , we have and . Many expressions exist that can represent . However, the five most reliable, accurate, and efficient ones for numerical computation are given below. They are form one:[18]
If is a
chi distribution with degrees of freedom, then its cdf is given by
If is a
Nakagami distribution with as shape parameter and as spread parameter, then its cdf is given by
If is a
Rice distribution with parameters and , then its cdf is given by
If is a
non-central chi distribution with non-centrality parameter and degrees of freedom, then its cdf is given by
Footnotes
^J.I. Marcum (1960). A statistical theory of target detection by pulsed radar: mathematical appendix, IRE Trans. Inform. Theory, vol. 6, 59-267.
^M.K. Simon and M.-S. Alouini (1998). A Unified Approach to the Performance of Digital Communication over Generalized Fading Channels, Proceedings of the IEEE, 86(9), 1860-1877.
^
abcdeA. Annamalai and C. Tellambura (2001). Cauchy-Schwarz bound on the generalized Marcum-Q function with applications, Wireless Communications and Mobile Computing, 1(2), 243-253.
^
abcdA. Annamalai and C. Tellambura (2008). A Simple Exponential Integral Representation of the Generalized Marcum Q-Function QM(a,b) for Real-Order M with Applications. 2008 IEEE Military Communications Conference, San Diego, CA, USA
^
abcdefgY. Sun, A. Baricz, and S. Zhou (2010). On the Monotonicity, Log-Concavity, and Tight Bounds of the Generalized Marcum and Nuttall Q-Functions. IEEE Transactions on Information Theory, 56(3), 1166–1186,
ISSN0018-9448
^
abY. Sun and A. Baricz (2008). Inequalities for the generalized Marcum Q-function. Applied Mathematics and Computation 203(2008) 134-141.
^
abcdefN.M. Temme (1993). Asymptotic and numerical aspects of the noncentral chi-square distribution. Computers Math. Applic., 25(5), 55-63.
^
abcdefA. Annamalai, C. Tellambura and John Matyjas (2009). "A New Twist on the Generalized Marcum Q-Function QM(a, b) with Fractional-Order M and its Applications". 2009 6th IEEE Consumer Communications and Networking Conference, 1–5,
ISBN978-1-4244-2308-8
^
abS. Andras, A. Baricz, and Y. Sun (2011) The Generalized Marcum Q-function: An Orthogonal Polynomial Approach. Acta Univ. Sapientiae Mathematica, 3(1), 60-76.
^
abcdefgY.A. Brychkov (2012). On some properties of the Marcum Q function. Integral Transforms and Special Functions 23(3), 177-182.
^W.K. Pratt (1968). Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(7), 1220-1221.
^R. Esposito (1968). Comment on Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(12), 2195-2195.
^V.M. Kapinas, S.K. Mihos, G.K. Karagiannidis (2009). On the Monotonicity of the Generalized Marcum and Nuttal Q-Functions. IEEE Transactions on Information Theory, 55(8), 3701-3710.
^
abR. Li, P.Y. Kam, and H. Fu (2010). New Representations and Bounds for the Generalized Marcum Q-Function via a Geometric Approach, and an Application. IEEE Trans. Commun., 58(1), 157-169.
^
abM.K. Simon and M.-S. Alouini (2000). Exponential-Type Bounds on the Generalized Marcum Q-Function with Application to Error Probability Analysis over Fading Channels. IEEE Trans. Commun. 48(3), 359-366.
^ H. Guo, B. Makki, M. -S. Alouini and T. Svensson, "A Semi-Linear Approximation of the First-Order Marcum Q-Function With Application to Predictor Antenna Systems," in IEEE Open Journal of the Communications Society, vol. 2, pp. 273-286, 2021, doi: 10.1109/OJCOMS.2021.3056393.
^
abcdefgD.A. Shnidman (1989). The Calculation of the Probability of Detection and the Generalized Marcum Q-Function. IEEE Transactions on Information Theory, 35(2), 389-400.
References
Marcum, J. I. (1950) "Table of Q Functions". U.S. Air Force RAND Research Memorandum M-339. Santa Monica, CA: Rand Corporation, Jan. 1, 1950.
Shnidman, David A. (1989): The Calculation of the Probability of Detection and the Generalized Marcum Q-Function,IEEE Transactions on Information Theory, 35(2), 389-400.
Weisstein, Eric W. Marcum Q-Function. From MathWorld—A Wolfram Web Resource.
[1]