Marcum Q-function

Using the fact that $Q\_\backslash nu\; (a,0)\; =\; 1$, the generalized Marcum Q-function can alternatively be defined as a finite integral as

: $Q\_\backslash nu\; (a,b)\; =\; 1\; -\; \backslash frac\{1\}\{a^\{\backslash nu-1\}\}\; \backslash int\_0^b\; x^\backslash nu\; \backslash exp\; \backslash left(\; -\backslash frac\{x^2\; +\; a^2\}\{2\}\; \backslash right)\; I\_\{\backslash nu-1\}(ax)\; \backslash ,\; dx.$

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 $\backslash nu\; =\; n$, such a representation is given by the trigonometric integralM.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.A. 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.

:$$

Q_n(a,b) = \left\{

\begin{array}{lr}

H_n(a,b) & a < b, \\

\frac{1}{2} + H_n(a,a) & a=b, \\

1 + H_n(a,b) & a > b,

\end{array}

\right.

where

:$H\_n(a,b)\; =\; \backslash frac\{\backslash zeta^\{1-n\}\}\{2\backslash pi\}\; \backslash exp\backslash left(-\backslash frac\{a^2+b^2\}\{2\}\backslash right)\; \backslash int\_0^\{2\backslash pi\}\; \backslash frac\{\backslash cos(n-1)\backslash theta\; -\; \backslash zeta\; \backslash cos\; n\backslash theta\}\{1-2\backslash zeta\backslash cos\backslash theta\; +\; \backslash zeta^2\}\; \backslash exp(ab\backslash cos\backslash theta)\; \backslash mathrm\{d\}\; \backslash theta,$

and the ratio $\backslash zeta\; =\; a/b$ is a constant.

For any real $\backslash nu\; >\; 0$, such finite trigonometric integral is given byA. Annamalai and C. Tellambura (2008). A Simple Exponential Integral Representation of the Generalized Marcum Q-Function Q_M(a,b) for Real-Order M with Applications. 2008 IEEE Military Communications Conference, San Diego, CA, USA

:$$

Q_\nu(a,b) = \left\{

\begin{array}{lr}

H_\nu(a,b) + C_\nu(a,b) & a < b, \\

\frac{1}{2} + H_\nu(a,a) + C_\nu(a,b) & a=b, \\

1 + H_\nu(a,b) + C_\nu(a,b) & a > b,

\end{array}

\right.

where $H\_n(a,b)$ is as defined before, $\backslash zeta\; =\; a/b$, and the additional correction term is given by

:$C\_\backslash nu(a,b)\; =\; \backslash frac\{\backslash sin(\backslash nu\backslash pi)\}\{\backslash pi\}\; \backslash exp\backslash left(-\backslash frac\{a^2+b^2\}\{2\}\backslash right)\; \backslash int\_0^1\; \backslash frac\{(x/\backslash zeta)^\{\backslash nu-1\}\}\{\backslash zeta+x\}\; \backslash exp\backslash left[\; -\backslash frac\{ab\}\{2\}\backslash left(x+\backslash frac\{1\}\{x\}\backslash right)\; \backslash right]\; \backslash mathrm\{d\}x.$

For integer values of $\backslash nu$, the correction term $C\_\backslash nu(a,b)$ tend to vanish.

• The generalized Marcum Q-function $Q\_\backslash nu(a,b)$ is strictly increasing in $\backslash nu$ and $a$ for all $a\; \backslash geq\; 0$ and $b,\; \backslash nu\; >\; 0$, and is strictly decreasing in $b$ for all $a,\; b\; \backslash geq\; 0$ and $\backslash nu>0.$Y. 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, {{ISSN|0018-9448}}

• The function $\backslash nu\; \backslash mapsto\; Q\_\backslash nu(a,b)$ is log-concave on $[1,\backslash infty)$ for all $a\; ,\; b\; \backslash geq\; 0.$

• The function $b\; \backslash mapsto\; Q\_\backslash nu(a,b)$ is strictly log-concave on $(0,\backslash infty)$ for all $a\; \backslash geq\; 0$ and $\backslash nu\; >\; 1$, which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.Y. Sun and A. Baricz (2008). Inequalities for the generalized Marcum Q-function. Applied Mathematics and Computation 203(2008) 134-141.

• The function $a\; \backslash mapsto\; 1\; -\; Q\_\backslash nu(a,b)$ is log-concave on $[0,\backslash infty)$ for all $b,\; \backslash nu\; >\; 0.$

• The generalized Marcum Q function of order $\backslash nu\; >\; 0$ can be represented using incomplete Gamma function asS. Andras, A. Baricz, and Y. Sun (2011) The Generalized Marcum Q-function: An Orthogonal Polynomial Approach. Acta Univ. Sapientiae Mathematica, 3(1), 60-76.

:: $Q\_\backslash nu\; (a,b)\; =\; 1\; -\; e^\{-a^2/2\}\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1\}\{k!\}\; \backslash frac\{\backslash gamma(\backslash nu+k,\backslash frac\{b^2\}\{2\})\}\{\backslash Gamma(\backslash nu+k)\}\; \backslash left(\; \backslash frac\{a^2\}\{2\}\; \backslash right)^k,$

:where $\backslash gamma(s,x)$ is the lower incomplete Gamma function. This is usually called the canonical representation of the $\backslash nu$-th order generalized Marcum Q-function.

• The generalized Marcum Q function of order $\backslash nu\; >\; 0$ can also be represented using generalized Laguerre polynomials as

::$Q\_\{\backslash nu\}(a,b)\; =\; 1\; -\; e^\{-a^2/2\}\; \backslash sum\_\{k=0\}^\backslash infty\; (-1)^k\; \backslash frac\{L\_k^\{(\backslash nu-1)\}(\backslash frac\{a^2\}\{2\})\}\{\backslash Gamma(\backslash nu+k+1)\}\; \backslash left(\backslash frac\{b^2\}\{2\}\backslash right)^\{k+\backslash nu\},$

:where $L\_k^\{(\backslash alpha)\}(\backslash cdot)$ is the generalized Laguerre polynomial of degree $k$ and of order $\backslash alpha$.

• The generalized Marcum Q-function of order $\backslash nu\; >\; 0$ can also be represented as Neumann series expansions

:: $Q\_\backslash nu\; (a,b)\; =\; e^\{-(a^2\; +\; b^2)/2\}\; \backslash sum\_\{\backslash alpha=1-\backslash nu\}^\backslash infty\; \backslash left(\; \backslash frac\{a\}\{b\}\backslash right)^\backslash alpha\; I\_\{-\backslash alpha\}(ab),$

:: $1\; -\; Q\_\backslash nu(a,b)\; =\; e^\{-(a^2\; +\; b^2)/2\}\; \backslash sum\_\{\backslash alpha=\backslash nu\}^\backslash infty\; \backslash left(\; \backslash frac\{b\}\{a\}\backslash right)^\backslash alpha\; I\_\{\backslash alpha\}(ab),$

:where the summations are in increments of one. Note that when $\backslash alpha$ assumes an integer value, we have $I\_\{\backslash alpha\}(ab)\; =\; I\_\{-\backslash alpha\}(ab)$.

• For non-negative half-integer values $\backslash nu\; =\; n\; +\; 1/2$, we have a closed form expression for the generalized Marcum Q-function as

::$Q\_\{n+1/2\}(a,b)\; =\; \backslash frac\{1\}\{2\}\backslash left[\; \backslash mathrm\{erfc\}\backslash left(\backslash frac\{b-a\}\{\backslash sqrt\{2\}\}\backslash right)\; +\; \backslash mathrm\{erfc\}\backslash left(\backslash frac\{b+a\}\{\backslash sqrt\{2\}\}\backslash right)\; \backslash right]\; +\; e^\{-(a^2\; +\; b^2)/2\}\; \backslash sum\_\{k=1\}^n\; \backslash left(\backslash frac\{b\}\{a\}\backslash right)^\{k-1/2\}\; I\_\{k-1/2\}(ab),$

:where $\backslash mathrm\{erfc\}(\backslash cdot)$ is the complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as

::$I\_\{\backslash pm(n+0.5)\}(z)\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash sum\_\{k=0\}^n\; \backslash frac\{(n+k)!\}\{k!(n-k)!\}\; \backslash left[\; \backslash frac\{(-1)^k\; e^z\; \backslash mp\; (-1)^n\; e^\{-z\}\}\{(2z)^\{k+0.5\}\}\; \backslash right],$

:where $n$ is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have

::$Q\_\{n+1/2\}(a,b)\; =\; Q(b-a)\; +\; Q(b+a)\; +\; \backslash frac\{1\}\{b\backslash sqrt\{2\backslash pi\}\}\; \backslash sum\_\{i=1\}^\{n\}\; \backslash left(\backslash frac\{b\}\{a\}\backslash right)^i\; \backslash sum\_\{k=0\}^\{i-1\}\; \backslash frac\{(i+k-1)!\}\{k!(i-k-1)!\}\; \backslash left[\; \backslash frac\{(-1)^k\; e^\{-(a-b)^2/2\}\; +\; (-1)^i\; e^\{-(a+b)^2/2\}\}\{(2ab)^k\}\; \backslash right],$

:for non-negative integers $n$, where $Q(\backslash cdot)$ 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:M. Abramowitz and I.A. Stegun (1972). [https://personal.math.ubc.ca/~cbm/aands/page_443.htm Formula 10.2.12, Modified Spherical Bessel Functions], Handbook of Mathematical functions, p. 443

::$I\_\{n+\backslash frac\{1\}\{2\}\}(z)\; =\; \backslash sqrt\{\backslash frac\{2z\}\{\backslash pi\}\}\; \backslash left[\; g\_n(z)\; \backslash sinh(z)\; +\; g\_\{-n-1\}(z)\; \backslash cosh(z)\backslash right],$

:where $g\_0(z)\; =\; z^\{-1\}$, $g\_1(z)\; =\; -z^\{-2\}$, and $g\_\{n-1\}(z)\; -\; g\_\{n+1\}(z)\; =\; (2n+1)\; z^\{-1\}\; g\_n(z)$ for any integer value of $n$.

• Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relationA. Annamalai, C. Tellambura and John Matyjas (2009). "A New Twist on the Generalized Marcum Q-Function Q_M(a, b) with Fractional-Order M and its Applications". 2009 6th IEEE Consumer Communications and Networking Conference, 1–5, {{ISBN|978-1-4244-2308-8}}

:: $Q\_\{\backslash nu+1\}(a,b)\; -\; Q\_\backslash nu(a,b)\; =\; \backslash left(\; \backslash frac\{b\}\{a\}\; \backslash right)^\{\backslash nu\}\; e^\{-(a^2\; +\; b^2)/2\}\; I\_\{\backslash nu\}(ab).$

• The above formula is easily generalized as

::$Q\_\{\backslash nu-n\}(a,b)\; =\; Q\_\backslash nu(a,b)\; -\; \backslash left(\backslash frac\{b\}\{a\}\backslash right)^\backslash nu\; e^\{-(a^2+b^2)/2\}\backslash sum\_\{k=1\}^n\; \backslash left(\backslash frac\{a\}\{b\}\backslash right)^k\; I\_\{\backslash nu-k\}(ab),$

::$Q\_\{\backslash nu+n\}(a,b)\; =\; Q\_\backslash nu(a,b)\; +\; \backslash left(\backslash frac\{b\}\{a\}\backslash right)^\backslash nu\; e^\{-(a^2+b^2)/2\}\backslash sum\_\{k=0\}^\{n-1\}\; \backslash left(\backslash frac\{b\}\{a\}\backslash right)^k\; I\_\{\backslash nu+k\}(ab),$

:for positive integer $n$. The former recurrence can be used to formally define the generalized Marcum Q-function for negative $\backslash nu$. Taking $Q\_\backslash infty(a,b)=1$ and $Q\_\{-\backslash infty\}(a,b)=0$ for $n\; =\; \backslash infty$, we obtain the Neumann series representation of the generalized Marcum Q-function.

• The related three-term recurrence relation is given by

::$Q\_\{\backslash nu+1\}(a,b)\; -\; (1+c\_\backslash nu(a,b))Q\_\backslash nu(a,b)\; +\; c\_\backslash nu(a,b)\; Q\_\{\backslash nu-1\}(a,b)\; =\; 0,$

:where

::$c\_\backslash nu(a,b)\; =\; \backslash left(\backslash frac\{b\}\{a\}\backslash right)\; \backslash frac\{I\_\backslash nu(ab)\}\{I\_\{\backslash nu+1\}(ab)\}.$

:We can eliminate the occurrence of the Bessel function to give the third order recurrence relation

::$\backslash frac\{a^2\}\{2\}\; Q\_\{\backslash nu+2\}(a,b)\; =\; \backslash left(\backslash frac\{a^2\}\{2\}\; -\; \backslash nu\backslash right)\; Q\_\{\backslash nu+1\}(a,b)\; +\; \backslash left(\backslash frac\{b^2\}\{2\}\; +\; \backslash nu\backslash right)Q\_\{\backslash nu\}(a,b)\; -\; \backslash frac\{b^2\}\{2\}\; Q\_\{\backslash nu-1\}(a,b).$

• Another recurrence relationship, relating it with its derivatives, is given by

::$Q\_\{\backslash nu+1\}(a,b)\; =\; Q\_\backslash nu(a,b)\; +\; \backslash frac\{1\}\{a\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; a\}\; Q\_\backslash nu(a,b),$

::$Q\_\{\backslash nu-1\}(a,b)\; =\; Q\_\backslash nu(a,b)\; +\; \backslash frac\{1\}\{b\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; b\}\; Q\_\backslash nu(a,b).$

• The ordinary generating function of $Q\_\backslash nu(a,b)$ for integral $\backslash nu$ is

::$\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; t^n\; Q\_n(a,b)\; =\; e^\{-(a^2+b^2)/2\}\; \backslash frac\{t\}\{1-t\}\; e^\{(b^2\; t\; +\; a^2/t)/2\},$

:where

• Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral $\backslash nu\; =\; n$

::$Q\_n(a,b)\; +\; Q\_n(b,a)\; =\; 1\; +\; e^\{-(a^2+b^2)/2\}\; \backslash left[\; I\_0(ab)\; +\; \backslash sum\_\{k=1\}^\{n-1\}\; \backslash frac\{a^\{2k\}\; +\; b^\{2k\}\}\{(ab)^k\}\; I\_k(ab)\; \backslash right].$

:In particular, for $n\; =\; 1$ we have

::$Q\_1(a,b)\; +\; Q\_1(b,a)\; =\; 1\; +\; e^\{-(a^2+b^2)/2\}\; I\_0(ab).$

Some specific values of Marcum-Q function are

• $Q\_\backslash nu(0,0)\; =\; 1,$
• $Q\_\backslash nu(a,0)\; =\; 1,$
• $Q\_\backslash nu(a,+\backslash infty)\; =\; 0,$
• $Q\_\backslash nu(0,b)\; =\; \backslash frac\{\backslash Gamma(\backslash nu,b^2/2)\}\{\backslash Gamma(\backslash nu)\},$
• $Q\_\backslash nu(+\backslash infty,b)\; =\; 1,$
• $Q\_\backslash infty(a,b)\; =\; 1,$
• For $a=b$, by subtracting the two forms of Neumann series representations, we haveY.A. Brychkov (2012). On some properties of the Marcum Q function. Integral Transforms and Special Functions 23(3), 177-182.

::$Q\_1(a,a)\; =\; \backslash frac\{1\}\{2\}[1\; +\; e^\{-a^2\}I\_0(a^2)],$

:which when combined with the recursive formula gives

::$Q\_n(a,a)\; =\; \backslash frac\{1\}\{2\}[1\; +\; e^\{-a^2\}I\_0(a^2)]\; +\; e^\{-a^2\}\; \backslash sum\_\{k=1\}^\{n-1\}\; I\_k(a^2),$

::$Q\_\{-n\}(a,a)\; =\; \backslash frac\{1\}\{2\}[1\; +\; e^\{-a^2\}I\_0(a^2)]\; -\; e^\{-a^2\}\; \backslash sum\_\{k=1\}^\{n\}\; I\_k(a^2),$

:for any non-negative integer $n$.

• For $\backslash nu\; =\; 1/2$, using the basic integral definition of generalized Marcum Q-function, we have

::$Q\_\{1/2\}(a,b)\; =\; \backslash frac\{1\}\{2\}\backslash left[\; \backslash mathrm\{erfc\}\backslash left(\backslash frac\{b-a\}\{\backslash sqrt\{2\}\}\backslash right)\; +\; \backslash mathrm\{erfc\}\backslash left(\backslash frac\{b+a\}\{\backslash sqrt\{2\}\}\backslash right)\; \backslash right].$

• For $\backslash nu=3/2$, we have

::$Q\_\{3/2\}(a,b)\; =\; Q\_\{1/2\}(a,b)\; +\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\; \backslash ,\; \backslash frac\{\backslash sinh(ab)\}\{a\}\; e^\{-(a^2\; +\; b^2)/2\}.$

• For $\backslash nu\; =\; 5/2$ we have

::$Q\_\{5/2\}(a,b)\; =\; Q\_\{3/2\}(a,b)\; +\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\; \backslash ,\; \backslash frac\{ab\; \backslash cosh\; (ab)\; -\; \backslash sinh\; (ab)\; \}\{a^3\}\; e^\{-(a^2\; +\; b^2)/2\}.$

• Assuming $\backslash nu$ to be fixed and $ab$ large, let $\backslash zeta\; =\; a/b\; >\; 0$, then the generalized Marcum-Q function has the following asymptotic formN.M. Temme (1993). Asymptotic and numerical aspects of the noncentral chi-square distribution. Computers Math. Applic., 25(5), 55-63.

::$Q\_\backslash nu(a,b)\; \backslash sim\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash psi\_n,$

:where $\backslash psi\_n$ is given by

::$\backslash psi\_n\; =\; \backslash frac\{1\}\{2\backslash zeta^\backslash nu\; \backslash sqrt\{2\backslash pi\}\}\; (-1)^n\; \backslash left[\; A\_n(\backslash nu-1)\; -\; \backslash zeta\; A\_n(\backslash nu)\; \backslash right]\; \backslash phi\_n.$

:The functions $\backslash phi\_n$ and $A\_n$ are given by

::$\backslash phi\_n\; =\; \backslash left[\; \backslash frac\{(b-a)^2\}\{2ab\}\; \backslash right]^\{n-\backslash frac\{1\}\{2\}\}\; \backslash Gamma\backslash left(\backslash frac\{1\}\{2\}\; -\; n,\; \backslash frac\{(b-a)^2\}\{2\}\backslash right),$

::$A\_n(\backslash nu)\; =\; \backslash frac\{2^\{-n\}\backslash Gamma(\backslash frac\{1\}\{2\}+\backslash nu+n)\}\{n!\backslash Gamma(\backslash frac\{1\}\{2\}+\backslash nu-n)\}.$

:The function $A\_n(\backslash nu)$ satisfies the recursion

::$A\_\{n+1\}(\backslash nu)\; =\; -\; \backslash frac\{(2n+1)^2\; -\; 4\backslash nu^2\}\{8(n+1)\}A\_n(\backslash nu),$

:for $n\; \backslash geq\; 0$ and $A\_0(\backslash nu)=1.$

• In the first term of the above asymptotic approximation, we have

::$\backslash phi\_0\; =\; \backslash frac\{\backslash sqrt\{2\; \backslash pi\; ab\}\}\{b-a\}\; \backslash mathrm\{erfc\}\backslash left(\backslash frac\{b-a\}\{\backslash sqrt\{2\}\}\backslash right).$

:Hence, assuming $b>a$, the first term asymptotic approximation of the generalized Marcum-Q function is

::$Q\_\backslash nu(a,b)\; \backslash sim\; \backslash psi\_0\; =\; \backslash left(\backslash frac\{b\}\{a\}\backslash right)^\{\backslash nu-\backslash frac\{1\}\{2\}\}\; Q(b-a),$

:where $Q(\backslash cdot)$ is the Gaussian Q-function. Here $Q\_\backslash nu(a,b)\; \backslash sim\; 0.5$ as $a\; \backslash uparrow\; b.$

:For the case when $a\; >\; b$, we have

::$Q\_\backslash nu(a,b)\; \backslash sim\; 1-\backslash psi\_0\; =\; 1-\backslash left(\backslash frac\{b\}\{a\}\backslash right)^\{\backslash nu-\backslash frac\{1\}\{2\}\}\; Q(a-b).$

:Here too $Q\_\backslash nu(a,b)\; \backslash sim\; 0.5$ as $a\; \backslash downarrow\; b.$

• The partial derivative of $Q\_\backslash nu(a,b)$ with respect to $a$ and $b$ is given byW.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.

::$\backslash frac\{\backslash partial\}\{\backslash partial\; a\}\; Q\_\backslash nu(a,b)\; =\; a\; \backslash left[Q\_\{\backslash nu+1\}(a,b)\; -\; Q\_\{\backslash nu\}(a,b)\backslash right]\; =\; a\; \backslash left(\backslash frac\{b\}\{a\}\backslash right)^\{\backslash nu\}\; e^\{-(a^2+b^2)/2\}\; I\_\{\backslash nu\}(ab),$

::$\backslash frac\{\backslash partial\}\{\backslash partial\; b\}\; Q\_\backslash nu(a,b)\; =\; b\; \backslash left[Q\_\{\backslash nu-1\}(a,b)\; -\; Q\_\{\backslash nu\}(a,b)\backslash right]\; =\; -\; b\; \backslash left(\backslash frac\{b\}\{a\}\backslash right)^\{\backslash nu-1\}\; e^\{-(a^2+b^2)/2\}\; I\_\{\backslash nu-1\}(ab).$

:We can relate the two partial derivatives as

::$\backslash frac\{1\}\{a\}\backslash frac\{\backslash partial\}\{\backslash partial\; a\}\; Q\_\backslash nu(a,b)\; +\; \backslash frac\{1\}\{b\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; b\}\; Q\_\{\backslash nu+1\}(a,b)\; =\; 0.$

• The n-th partial derivative of $Q\_\backslash nu(a,b)$ with respect to its arguments is given by

::$\backslash frac\{\backslash partial^n\}\{\backslash partial\; a^n\}\; Q\_\backslash nu(a,b)\; =\; n!\; (-a)^n\; \backslash sum\_\{k=0\}^\{[n/2]\}\; \backslash frac\{(-2a^2)^\{-k\}\}\{k!(n-2k)!\}\; \backslash sum\_\{p=0\}^\{n-k\}\; (-1)^p\; \backslash binom\{n-k\}\{p\}\; Q\_\{\backslash nu+p\}(a,b),$

::$\backslash frac\{\backslash partial^n\}\{\backslash partial\; b^n\}\; Q\_\backslash nu(a,b)\; =\; \backslash frac\{n!\; a^\{1-\backslash nu\}\}\{2^n\; b^\{n-\backslash nu+1\}\}\; e^\{-(a^2+b^2)/2\}\; \backslash sum\_\{k=[n/2]\}^n\; \backslash frac\{(-2b^2)^k\}\{(n-k)!(2k-n)!\}\; \backslash sum\_\{p=0\}^\{k-1\}\; \backslash binom\{k-1\}\{p\}\; \backslash left(-\backslash frac\{a\}\{b\}\backslash right)^p\; I\_\{\backslash nu-p-1\}(ab).$

• The generalized Marcum-Q function satisfies a Turán-type inequality

::$Q^2\_\backslash nu(a,b)\; >\; \backslash frac\{Q\_\{\backslash nu-1\}(a,b)\; +\; Q\_\{\backslash nu+1\}(a,b)\}\{2\}\; >\; Q\_\{\backslash nu-1\}(a,b)\; Q\_\{\backslash nu+1\}(a,b)$

:for all $a\; \backslash geq\; b\; >\; 0$ and $\backslash nu\; >\; 1$.

Various upper and lower bounds of generalized Marcum-Q function can be obtained using monotonicity and log-concavity of the function $\backslash nu\; \backslash mapsto\; Q\_\backslash nu(a,b)$ and the fact that we have closed form expression for $Q\_\backslash nu(a,b)$ when $\backslash nu$ is half-integer valued.

Let $\backslash lfloor\; x\; \backslash rfloor\_\{0.5\}$ and $\backslash lceil\; x\; \backslash rceil\_\{0.5\}$ denote the pair of half-integer rounding operators that map a real $x$ to its nearest left and right half-odd integer, respectively, according to the relations

:$\backslash lfloor\; x\; \backslash rfloor\_\{0.5\}\; =\; \backslash lfloor\; x\; -\; 0.5\; \backslash rfloor\; +\; 0.5$

:$\backslash lceil\; x\; \backslash rceil\_\{0.5\}\; =\; \backslash lceil\; x\; +\; 0.5\; \backslash rceil\; -\; 0.5$

where $\backslash lfloor\; x\; \backslash rfloor$ and $\backslash lceil\; x\; \backslash rceil$ denote the integer floor and ceiling functions.

• The monotonicity of the function $\backslash nu\; \backslash mapsto\; Q\_\backslash nu(a,b)$ for all $a\; \backslash geq\; 0$ and $b\; >\; 0$ gives us the following simple boundV.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.R. 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.

::

:However, the relative error of this bound does not tend to zero when $b\; \backslash to\; \backslash infty$. For integral values of $\backslash nu\; =\; n$, this bound reduces to

::

:A very good approximation of the generalized Marcum Q-function for integer valued $\backslash nu\; =\; n$ is obtained by taking the arithmetic mean of the upper and lower bound

::$Q\_n(a,b)\; \backslash approx\; \backslash frac\{Q\_\{n-0.5\}(a,b)\; +\; Q\_\{n+0.5\}(a,b)\}\{2\}.$

• A tighter bound can be obtained by exploiting the log-concavity of $\backslash nu\; \backslash mapsto\; Q\_\backslash nu(a,b)$ on $[1,\backslash infty)$ as

::

:where $\backslash nu\_1\; =\; \backslash lfloor\backslash nu\backslash rfloor\_\{0.5\}$ and $\backslash nu\_2\; =\; \backslash lceil\backslash nu\backslash rceil\_\{0.5\}$ for $\backslash nu\; \backslash geq\; 1.5$. The tightness of this bound improves as either $a$ or $\backslash nu$ increases. The relative error of this bound converges to 0 as $b\; \backslash to\; \backslash infty$. For integral values of $\backslash nu\; =\; n$, this bound reduces to

::

Using the trigonometric integral representation for integer valued $\backslash nu=n$, the following Cauchy-Schwarz bound can be obtained

:

:$1\; -\; Q\_n(a,b)\; \backslash leq\; \backslash exp\backslash left[-\backslash frac\{1\}\{2\}(b^2+a^2)\backslash right]\; \backslash sqrt\{I\_0(2ab)\}\; \backslash sqrt\{\backslash frac\{\backslash zeta^\{2(1-n)\}\}\{2(\backslash zeta^2-1)\}\},\; \backslash qquad\; \backslash zeta\; >\; 1,$

where $\backslash zeta\; =\; a/b\; >0$.

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 $\backslash zeta\; =\; a/b\; >0$, one such bound for integer valued $\backslash nu\; =\; n$ is given asM.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.

:

:$Q\_n(a,b)\; \backslash geq\; 1\; -\; \backslash frac\{1\}\{2\}\backslash left[e^\{-(a-b)^2/2\}\; -\; e^\{-(a+b)^2/2\}\; \backslash right],\; \backslash qquad\; \backslash zeta\; >\; 1.$

When $n=1$, the bound simplifies to give

:

:$1\; -\; \backslash frac\{1\}\{2\}\backslash left[e^\{-(a-b)^2/2\}\; -\; e^\{-(a+b)^2/2\}\; \backslash right]\; \backslash leq\; Q\_1(a,b),\; \backslash qquad\; \backslash zeta\; >\; 1.$

Another such bound obtained via Cauchy-Schwarz inequality is given as

:

:$Q\_n(a,b)\; \backslash geq\; 1\; -\; \backslash frac\{1\}\{2\}\; \backslash sqrt\{\backslash frac\{\backslash zeta^\{2(1-n)\}\}\{2(\backslash zeta^2-1)\}\}\; \backslash left[\; e^\{-(b-a)^2/2\}\; +\; e^\{-(b+a)^2/2\}\; \backslash right],\; \backslash qquad\; \backslash zeta\; >\; 1.$

Chernoff-type bounds for the generalized Marcum Q-function, where $\backslash nu\; =\; n$ is an integer, is given by

:$(1-2\backslash lambda)^\{-n\}\; \backslash exp\; \backslash left(-\backslash lambda\; b^2\; +\; \backslash frac\{\backslash lambda\; n\; a^2\}\{1\; -\; 2\backslash lambda\}\; \backslash right)\; \backslash geq\; \backslash left\backslash \{$

\begin{array}{lr}

Q_n(a,b), & b^2 > n(a^2+2) \\

1 - Q_n(a,b), & b^2 < n(a^2+2)

\end{array}

\right.

where the Chernoff parameter has optimum value $\backslash lambda\_0$ of

:$\backslash lambda\_0\; =\; \backslash frac\{1\}\{2\}\backslash left(1\; -\; \backslash frac\{n\}\{b^2\}\; -\; \backslash frac\{n\}\{b^2\}\; \backslash sqrt\{1\; +\; \backslash frac\{(ab)^2\}\{n\}\}\backslash right).$

The first-order Marcum-Q function can be semi-linearly approximated by 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.

:$\backslash begin\{align\}$

Q_1(a, b)=

\begin{cases}

1, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathrm{if}~ b < c_1 \\

-\beta_0 e^{-\frac{1}{2}\left(a^2+\left(\beta_0\right)^2\right)}I_0\left(a\beta_0\right)\left(b-\beta_0\right)+Q_1\left(a,\beta_0\right), ~~~~~\mathrm{if}~ c_1 \leq b \leq c_2 \\

0, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathrm{if}~ b> c_2

\end{cases}

\end{align}

where

:$$

\begin{align}

\beta_0 = \frac{a+\sqrt{a^2+2}}{2},

\end{align}

:$$

\begin{align}

c_1(a) = \max\Bigg(0,\beta_0+\frac{Q_1\left(a,\beta_0\right)-1}{\beta_0 e^{-\frac{1}{2}\left(a^2+\left(\beta_0\right)^2\right)}I_0\left(a\beta_0\right)}\Bigg),

\end{align}

and

:$$

\begin{align}

c_2(a) = \beta_0+\frac{Q_1\left(a,\beta_0\right)}{\beta_0 e^{-\frac{1}{2}\left(a^2+\left(\beta_0\right)^2\right)}I_0\left(a\beta_0\right)}.

\end{align}

It is convenient to re-express the Marcum Q-function asD.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.

: $P\_N(X,Y)\; =\; Q\_N(\backslash sqrt\{2NX\},\backslash sqrt\{2Y\}).$

The $P\_N(X,Y)$ can be interpreted as the detection probability of $N$ incoherently integrated received signal samples of constant received signal-to-noise ratio, $X$, with a normalized detection threshold $Y$. In this equivalent form of Marcum Q-function, for given $a$ and $b$, we have $X\; =\; a^2/2N$ and $Y\; =\; b^2/2$. Many expressions exist that can represent $P\_N(X,Y)$. However, the five most reliable, accurate, and efficient ones for numerical computation are given below. They are form one:

: $P\_N(X,Y)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; e^\{-NX\}\; \backslash frac\{(NX)^k\}\{k!\}\; \backslash sum\_\{m=0\}^\{N-1+k\}\; e^\{-Y\}\; \backslash frac\{Y^m\}\{m!\},$

form two:

: $P\_N(X,Y)\; =\; \backslash sum\_\{m=0\}^\{N-1\}\; e^\{-Y\}\; \backslash frac\{Y^m\}\{m!\}\; +\; \backslash sum\_\{m=N\}^\backslash infty\; e^\{-Y\}\; \backslash frac\{Y^m\}\{m!\}\; \backslash left(\; 1\; -\; \backslash sum\_\{k=0\}^\{m-N\}\; e^\{-NX\}\; \backslash frac\{(NX)^k\}\{k!\}\; \backslash right),$

form three:

: $1\; -\; P\_N(X,Y)\; =\; \backslash sum\_\{m=N\}^\backslash infty\; e^\{-Y\}\; \backslash frac\{Y^m\}\{m!\}\; \backslash sum\_\{k=0\}^\{m-N\}\; e^\{-NX\}\; \backslash frac\{(NX)^k\}\{k!\},$

form four:

: $1\; -\; P\_N(X,Y)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; e^\{-NX\}\; \backslash frac\{(NX)^k\}\{k!\}\; \backslash left(\; 1\; -\; \backslash sum\_\{m=0\}^\{N-1+k\}\; e^\{-Y\}\; \backslash frac\{Y^m\}\{m!\}\; \backslash right),$

and form five:

: $1\; -\; P\_N(X,Y)\; =\; e^\{-(NX+Y)\}\; \backslash sum\_\{r=N\}^\backslash infty\; \backslash left(\backslash frac\{Y\}\{NX\}\backslash right)^\{r/2\}\; I\_r(2\backslash sqrt\{NXY\}).$

Among these five form, the second form is the most robust.

The generalized Marcum Q-function can be used to represent the cumulative distribution function (cdf) of many random variables:

• If $X\; \backslash sim\; \backslash mathrm\{Exp\}(\backslash lambda)$ is an exponential distribution with rate parameter $\backslash lambda$, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_1\backslash left(0,\backslash sqrt\{2\; \backslash lambda\; x\}\backslash right)$

• If $X\; \backslash sim\; \backslash mathrm\{Erlang\}(k,\backslash lambda)$ is a Erlang distribution with shape parameter $k$ and rate parameter $\backslash lambda$, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_k\backslash left(0,\backslash sqrt\{2\; \backslash lambda\; x\}\backslash right)$

• If $X\; \backslash sim\; \backslash chi^2\_k$ is a chi-squared distribution with $k$ degrees of freedom, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_\{k/2\}(0,\backslash sqrt\{x\})$

• If $X\; \backslash sim\; \backslash mathrm\{Gamma\}(\backslash alpha,\backslash beta)$ is a gamma distribution with shape parameter $\backslash alpha$ and rate parameter $\backslash beta$, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_\{\backslash alpha\}(0,\backslash sqrt\{2\; \backslash beta\; x\})$

• If $X\; \backslash sim\; \backslash mathrm\{Weibull\}(k,\backslash lambda)$ is a Weibull distribution with shape parameters $k$ and scale parameter $\backslash lambda$, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_1\; \backslash left(\; 0,\; \backslash sqrt\{2\}\; \backslash left(\backslash frac\{x\}\{\backslash lambda\}\backslash right)^\{\backslash frac\{k\}\{2\}\}\; \backslash right)$

• If $X\; \backslash sim\; \backslash mathrm\{GG\}(a,d,p)$ is a generalized gamma distribution with parameters $a,\; d,\; p$, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_\{\backslash frac\{d\}\{p\}\}\; \backslash left(\; 0,\; \backslash sqrt\{2\}\; \backslash left(\backslash frac\{x\}\{a\}\backslash right)^\{\backslash frac\{p\}\{2\}\}\; \backslash right)$

• If $X\; \backslash sim\; \backslash chi^2\_k(\backslash lambda)$ is a non-central chi-squared distribution with non-centrality parameter $\backslash lambda$ and $k$ degrees of freedom, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_\{k/2\}(\backslash sqrt\{\backslash lambda\},\backslash sqrt\{x\})$

• If $X\; \backslash sim\; \backslash mathrm\{Rayleigh\}(\backslash sigma)$ is a Rayleigh distribution with parameter $\backslash sigma$, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_1\backslash left(0,\backslash frac\{x\}\{\backslash sigma\}\backslash right)$

• If $X\; \backslash sim\; \backslash mathrm\{Maxwell\}(\backslash sigma)$ is a Maxwell–Boltzmann distribution with parameter $\backslash sigma$, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_\{3/2\}\backslash left(0,\backslash frac\{x\}\{\backslash sigma\}\backslash right)$

• If $X\; \backslash sim\; \backslash chi\_k$ is a chi distribution with $k$ degrees of freedom, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_\{k/2\}(0,x)$

• If $X\; \backslash sim\; \backslash mathrm\{Nakagami\}(m,\backslash Omega)$ is a Nakagami distribution with $m$ as shape parameter and $\backslash Omega$ as spread parameter, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_\{m\}\backslash left(0,\backslash sqrt\{\backslash frac\{2m\}\{\backslash Omega\}\}x\backslash right)$

• If $X\; \backslash sim\; \backslash mathrm\{Rice\}(\backslash nu,\backslash sigma)$ is a Rice distribution with parameters $\backslash nu$ and $\backslash sigma$, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_1\backslash left(\backslash frac\{\backslash nu\}\{\backslash sigma\},\backslash frac\{x\}\{\backslash sigma\}\backslash right)$

• If $X\; \backslash sim\; \backslash chi\_k(\backslash lambda)$ is a non-central chi distribution with non-centrality parameter $\backslash lambda$ and $k$ degrees of freedom, then its cdf is given by $F\_X(x)\; =\; 1\; -\; Q\_\{k/2\}(\backslash lambda,x)$

• 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.
• Nuttall, Albert H. (1975): [https://dx.doi.org/10.1109/TIT.1975.1055327 Some Integrals Involving the Q_M Function], IEEE Transactions on Information Theory, 21(1), 95–96, {{ISSN|0018-9448}}
• 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. [http://mathworld.wolfram.com/MarcumQ-Function.html]

Category:Functions related to probability distributions