Marcum Q-function

In statistics, the generalized Marcum Q-function of order ${\displaystyle \nu }$ is defined as

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

where ${\displaystyle b\geq 0}$ and ${\displaystyle a,\nu >0}$ and ${\displaystyle I_{\nu -1}}$ is the modified Bessel function of first kind of order ${\displaystyle \nu -1}$. If ${\displaystyle b>0}$, the integral converges for any ${\displaystyle \nu }$. The Marcum Q-function occurs as a complementary cumulative distribution function for noncentral chi, noncentral chi-squared, and Rice distributions. In engineering, this function appears in the study of radar systems, communication systems, queueing system, and signal processing. This function was first studied for ${\displaystyle \nu =1}$, and hence named after, by Jess Marcum for pulsed radars.^[1]

Properties

Finite integral representation

Using the fact that ${\displaystyle Q_{\nu }(a,0)=1}$, the generalized Marcum Q-function can alternatively be defined as a finite integral as

${\displaystyle Q_{\nu }(a,b)=1-{\frac {1}{a^{\nu -1}}}\int _{0}^{b}x^{\nu }\exp \left(-{\frac {x^{2}+a^{2}}{2}}\right)I_{\nu -1}(ax)\,dx.}$

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 ${\displaystyle \nu =n}$, such a representation is given by the trigonometric integral^[2][3]

${\displaystyle Q_{n}(a,b)=\left\{{\begin{array}{lr}H_{n}(a,b)&a<b,\\{\frac {1}{2}}+H_{n}(a,a)&a=b,\\1+H_{n}(a,b)&a>b,\end{array}}\right.}$

where

${\displaystyle H_{n}(a,b)={\frac {\zeta ^{1-n}}{2\pi }}\exp \left(-{\frac {a^{2}+b^{2}}{2}}\right)\int _{0}^{2\pi }{\frac {\cos(n-1)\theta -\zeta \cos n\theta }{1-2\zeta \cos \theta +\zeta ^{2}}}\exp(ab\cos \theta )\mathrm {d} \theta ,}$

and the ratio ${\displaystyle \zeta =a/b}$ is a constant.

For any real ${\displaystyle \nu >0}$, such finite trigonometric integral is given by^[4]

${\displaystyle Q_{\nu }(a,b)=\left\{{\begin{array}{lr}H_{\nu }(a,b)+C_{\nu }(a,b)&a<b,\\{\frac {1}{2}}+H_{\nu }(a,a)+C_{\nu }(a,b)&a=b,\\1+H_{\nu }(a,b)+C_{\nu }(a,b)&a>b,\end{array}}\right.}$

where ${\displaystyle H_{n}(a,b)}$ is as defined before, ${\displaystyle \zeta =a/b}$, and the additional correction term is given by

${\displaystyle C_{\nu }(a,b)={\frac {\sin(\nu \pi )}{\pi }}\exp \left(-{\frac {a^{2}+b^{2}}{2}}\right)\int _{0}^{1}{\frac {(x/\zeta )^{\nu -1}}{\zeta +x}}\exp \left[-{\frac {ab}{2}}\left(x+{\frac {1}{x}}\right)\right]\mathrm {d} x.}$

For integer values of ${\displaystyle \nu }$, the correction term ${\displaystyle C_{\nu }(a,b)}$ tend to vanish.

Monotonicity and log-concavity

• The generalized Marcum Q-function ${\displaystyle Q_{\nu }(a,b)}$ is strictly increasing in ${\displaystyle \nu }$ and ${\displaystyle a}$ for all ${\displaystyle a\geq 0}$ and ${\displaystyle b,\nu >0}$, and is strictly decreasing in ${\displaystyle b}$ for all ${\displaystyle a,b\geq 0}$ and ${\displaystyle \nu >0.}$^[5]

• The function ${\displaystyle \nu \mapsto Q_{\nu }(a,b)}$ is log-concave on ${\displaystyle [1,\infty )}$ for all ${\displaystyle a,b\geq 0.}$^[5]

• The function ${\displaystyle b\mapsto Q_{\nu }(a,b)}$ is strictly log-concave on ${\displaystyle (0,\infty )}$ for all ${\displaystyle a\geq 0}$ and ${\displaystyle \nu >1}$, which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.^[6]

• The function ${\displaystyle a\mapsto 1-Q_{\nu }(a,b)}$ is log-concave on ${\displaystyle [0,\infty )}$ for all ${\displaystyle b,\nu >0.}$^[5]

Series representation

• The generalized Marcum Q function of order ${\displaystyle \nu >0}$ can be represented using incomplete Gamma function as^[7][8][9]

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

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

• The generalized Marcum Q function of order ${\displaystyle \nu >0}$ can also be represented using generalized Laguerre polynomials as^[9]

${\displaystyle Q_{\nu }(a,b)=1-e^{-a^{2}/2}\sum _{k=0}^{\infty }(-1)^{k}{\frac {L_{k}^{(\nu -1)}({\frac {a^{2}}{2}})}{\Gamma (\nu +k+1)}}\left({\frac {b^{2}}{2}}\right)^{k+\nu },}$

where ${\displaystyle L_{k}^{(\alpha )}(\cdot )}$ is the generalized Laguerre polynomial of degree ${\displaystyle k}$ and of order ${\displaystyle \alpha }$.

• The generalized Marcum Q-function of order ${\displaystyle \nu >0}$ can also be represented as Neumann series expansions^[4][8]

${\displaystyle Q_{\nu }(a,b)=e^{-(a^{2}+b^{2})/2}\sum _{\alpha =1-\nu }^{\infty }\left({\frac {a}{b}}\right)^{\alpha }I_{-\alpha }(ab),}$

${\displaystyle 1-Q_{\nu }(a,b)=e^{-(a^{2}+b^{2})/2}\sum _{\alpha =\nu }^{\infty }\left({\frac {b}{a}}\right)^{\alpha }I_{\alpha }(ab),}$

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

• For non-negative half-integer values ${\displaystyle \nu =n+1/2}$, we have a closed form expression for the generalized Marcum Q-function as^[8][10]

${\displaystyle Q_{n+1/2}(a,b)={\frac {1}{2}}\left[\mathrm {erfc} \left({\frac {b-a}{\sqrt {2}}}\right)+\mathrm {erfc} \left({\frac {b+a}{\sqrt {2}}}\right)\right]+e^{-(a^{2}+b^{2})/2}\sum _{k=1}^{n}\left({\frac {b}{a}}\right)^{k-1/2}I_{k-1/2}(ab),}$

where ${\displaystyle \mathrm {erfc} (\cdot )}$ is the complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as^[4]

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

where ${\displaystyle n}$ is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have^[4]

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

for non-negative integers ${\displaystyle n}$, where ${\displaystyle Q(\cdot )}$ 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:^[11]

${\displaystyle I_{n+{\frac {1}{2}}}(z)={\sqrt {\frac {2z}{\pi }}}\left[g_{n}(z)\sinh(z)+g_{-n-1}(z)\cosh(z)\right],}$

where ${\displaystyle g_{0}(z)=z^{-1}}$, ${\displaystyle g_{1}(z)=-z^{-2}}$, and ${\displaystyle g_{n-1}(z)-g_{n+1}(z)=(2n+1)z^{-1}g_{n}(z)}$ for any integer value of ${\displaystyle n}$.

Recurrence relation and generating function

• Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation^[8][10]

${\displaystyle Q_{\nu +1}(a,b)-Q_{\nu }(a,b)=\left({\frac {b}{a}}\right)^{\nu }e^{-(a^{2}+b^{2})/2}I_{\nu }(ab).}$

• The above formula is easily generalized as^[10]

${\displaystyle Q_{\nu -n}(a,b)=Q_{\nu }(a,b)-\left({\frac {b}{a}}\right)^{\nu }e^{-(a^{2}+b^{2})/2}\sum _{k=1}^{n}\left({\frac {a}{b}}\right)^{k}I_{\nu -k}(ab),}$

${\displaystyle Q_{\nu +n}(a,b)=Q_{\nu }(a,b)+\left({\frac {b}{a}}\right)^{\nu }e^{-(a^{2}+b^{2})/2}\sum _{k=0}^{n-1}\left({\frac {b}{a}}\right)^{k}I_{\nu +k}(ab),}$

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

• The related three-term recurrence relation is given by^[7]

${\displaystyle Q_{\nu +1}(a,b)-(1+c_{\nu }(a,b))Q_{\nu }(a,b)+c_{\nu }(a,b)Q_{\nu -1}(a,b)=0,}$

where

${\displaystyle c_{\nu }(a,b)=\left({\frac {b}{a}}\right){\frac {I_{\nu }(ab)}{I_{\nu +1}(ab)}}.}$

We can eliminate the occurrence of the Bessel function to give the third order recurrence relation^[7]

${\displaystyle {\frac {a^{2}}{2}}Q_{\nu +2}(a,b)=\left({\frac {a^{2}}{2}}-\nu \right)Q_{\nu +1}(a,b)+\left({\frac {b^{2}}{2}}+\nu \right)Q_{\nu }(a,b)-{\frac {b^{2}}{2}}Q_{\nu -1}(a,b).}$

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

${\displaystyle Q_{\nu +1}(a,b)=Q_{\nu }(a,b)+{\frac {1}{a}}{\frac {\partial }{\partial a}}Q_{\nu }(a,b),}$

${\displaystyle Q_{\nu -1}(a,b)=Q_{\nu }(a,b)+{\frac {1}{b}}{\frac {\partial }{\partial b}}Q_{\nu }(a,b).}$

• The ordinary generating function of ${\displaystyle Q_{\nu }(a,b)}$ for integral ${\displaystyle \nu }$ is^[10]

${\displaystyle \sum _{n=-\infty }^{\infty }t^{n}Q_{n}(a,b)=e^{-(a^{2}+b^{2})/2}{\frac {t}{1-t}}e^{(b^{2}t+a^{2}/t)/2},}$

where ${\displaystyle |t|<1.}$

Symmetry relation

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

${\displaystyle Q_{n}(a,b)+Q_{n}(b,a)=1+e^{-(a^{2}+b^{2})/2}\left[I_{0}(ab)+\sum _{k=1}^{n-1}{\frac {a^{2k}+b^{2k}}{(ab)^{k}}}I_{k}(ab)\right].}$

In particular, for ${\displaystyle n=1}$ we have

${\displaystyle Q_{1}(a,b)+Q_{1}(b,a)=1+e^{-(a^{2}+b^{2})/2}I_{0}(ab).}$

Special values

Some specific values of Marcum-Q function are^[6]

• ${\displaystyle Q_{\nu }(0,0)=1,}$
• ${\displaystyle Q_{\nu }(a,0)=1,}$
• ${\displaystyle Q_{\nu }(a,+\infty )=0,}$
• ${\displaystyle Q_{\nu }(0,b)={\frac {\Gamma (\nu ,b^{2}/2)}{\Gamma (\nu )}},}$
• ${\displaystyle Q_{\nu }(+\infty ,b)=1,}$
• ${\displaystyle Q_{\infty }(a,b)=1,}$
• For ${\displaystyle a=b}$, by subtracting the two forms of Neumann series representations, we have^[10]

${\displaystyle Q_{1}(a,a)={\frac {1}{2}}[1+e^{-a^{2}}I_{0}(a^{2})],}$

which when combined with the recursive formula gives

${\displaystyle Q_{n}(a,a)={\frac {1}{2}}[1+e^{-a^{2}}I_{0}(a^{2})]+e^{-a^{2}}\sum _{k=1}^{n-1}I_{k}(a^{2}),}$

${\displaystyle Q_{-n}(a,a)={\frac {1}{2}}[1+e^{-a^{2}}I_{0}(a^{2})]-e^{-a^{2}}\sum _{k=1}^{n}I_{k}(a^{2}),}$

for any non-negative integer ${\displaystyle n}$.

• For ${\displaystyle \nu =1/2}$, using the basic integral definition of generalized Marcum Q-function, we have^[8][10]

${\displaystyle Q_{1/2}(a,b)={\frac {1}{2}}\left[\mathrm {erfc} \left({\frac {b-a}{\sqrt {2}}}\right)+\mathrm {erfc} \left({\frac {b+a}{\sqrt {2}}}\right)\right].}$

• For ${\displaystyle \nu =3/2}$, we have

${\displaystyle Q_{3/2}(a,b)=Q_{1/2}(a,b)+{\sqrt {\frac {2}{\pi }}}\,{\frac {\sinh(ab)}{a}}e^{-(a^{2}+b^{2})/2}.}$

• For ${\displaystyle \nu =5/2}$ we have

${\displaystyle Q_{5/2}(a,b)=Q_{3/2}(a,b)+{\sqrt {\frac {2}{\pi }}}\,{\frac {ab\cosh(ab)-\sinh(ab)}{a^{3}}}e^{-(a^{2}+b^{2})/2}.}$

Asymptotic forms

• Assuming ${\displaystyle \nu }$ to be fixed and ${\displaystyle ab}$ large, let ${\displaystyle \zeta =a/b>0}$, then the generalized Marcum-Q function has the following asymptotic form^[7]

${\displaystyle Q_{\nu }(a,b)\sim \sum _{n=0}^{\infty }\psi _{n},}$

where ${\displaystyle \psi _{n}}$ is given by

${\displaystyle \psi _{n}={\frac {1}{2\zeta ^{\nu }{\sqrt {2\pi }}}}(-1)^{n}\left[A_{n}(\nu -1)-\zeta A_{n}(\nu )\right]\phi _{n}.}$

The functions ${\displaystyle \phi _{n}}$ and ${\displaystyle A_{n}}$ are given by

${\displaystyle \phi _{n}=\left[{\frac {(b-a)^{2}}{2ab}}\right]^{n-{\frac {1}{2}}}\Gamma \left({\frac {1}{2}}-n,{\frac {(b-a)^{2}}{2}}\right),}$

${\displaystyle A_{n}(\nu )={\frac {2^{-n}\Gamma ({\frac {1}{2}}+\nu +n)}{n!\Gamma ({\frac {1}{2}}+\nu -n)}}.}$

The function ${\displaystyle A_{n}(\nu )}$ satisfies the recursion

${\displaystyle A_{n+1}(\nu )=-{\frac {(2n+1)^{2}-4\nu ^{2}}{8(n+1)}}A_{n}(\nu ),}$

for ${\displaystyle n\geq 0}$ and ${\displaystyle A_{0}(\nu )=1.}$

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

${\displaystyle \phi _{0}={\frac {\sqrt {2\pi ab}}{b-a}}\mathrm {erfc} \left({\frac {b-a}{\sqrt {2}}}\right).}$

Hence, assuming ${\displaystyle b>a}$, the first term asymptotic approximation of the generalized Marcum-Q function is^[7]

${\displaystyle Q_{\nu }(a,b)\sim \psi _{0}=\left({\frac {b}{a}}\right)^{\nu -{\frac {1}{2}}}Q(b-a),}$

where ${\displaystyle Q(\cdot )}$ is the Gaussian Q-function. Here ${\displaystyle Q_{\nu }(a,b)\sim 0.5}$ as ${\displaystyle a\uparrow b.}$

For the case when ${\displaystyle a>b}$, we have^[7]

${\displaystyle Q_{\nu }(a,b)\sim 1-\psi _{0}=1-\left({\frac {b}{a}}\right)^{\nu -{\frac {1}{2}}}Q(a-b).}$

Here too ${\displaystyle Q_{\nu }(a,b)\sim 0.5}$ as ${\displaystyle a\downarrow b.}$

Differentiation

• The partial derivative of ${\displaystyle Q_{\nu }(a,b)}$ with respect to ${\displaystyle a}$ and ${\displaystyle b}$ is given by^[12][13]

${\displaystyle {\frac {\partial }{\partial a}}Q_{\nu }(a,b)=a\left[Q_{\nu +1}(a,b)-Q_{\nu }(a,b)\right]=a\left({\frac {b}{a}}\right)^{\nu }e^{-(a^{2}+b^{2})/2}I_{\nu }(ab),}$

${\displaystyle {\frac {\partial }{\partial b}}Q_{\nu }(a,b)=b\left[Q_{\nu -1}(a,b)-Q_{\nu }(a,b)\right]=-b\left({\frac {b}{a}}\right)^{\nu -1}e^{-(a^{2}+b^{2})/2}I_{\nu -1}(ab).}$

We can relate the two partial derivatives as

${\displaystyle {\frac {1}{a}}{\frac {\partial }{\partial a}}Q_{\nu }(a,b)+{\frac {1}{b}}{\frac {\partial }{\partial b}}Q_{\nu +1}(a,b)=0.}$

• The n-th partial derivative of ${\displaystyle Q_{\nu }(a,b)}$ with respect to its arguments is given by^[10]

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

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

Inequalities

• The generalized Marcum-Q function satisfies a Turán-type inequality^[5]

${\displaystyle Q_{\nu }^{2}(a,b)>{\frac {Q_{\nu -1}(a,b)+Q_{\nu +1}(a,b)}{2}}>Q_{\nu -1}(a,b)Q_{\nu +1}(a,b)}$

for all ${\displaystyle a\geq b>0}$ and ${\displaystyle \nu >1}$.

Bounds

Based on monotonicity and log-concavity

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

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

${\displaystyle \lfloor x\rfloor _{0.5}=\lfloor x-0.5\rfloor +0.5}$

${\displaystyle \lceil x\rceil _{0.5}=\lceil x+0.5\rceil -0.5}$

where ${\displaystyle \lfloor x\rfloor }$ and ${\displaystyle \lceil x\rceil }$ denote the integer floor and ceiling functions.

• The monotonicity of the function ${\displaystyle \nu \mapsto Q_{\nu }(a,b)}$ for all ${\displaystyle a\geq 0}$ and ${\displaystyle b>0}$ gives us the following simple bound^[14][8][15]

${\displaystyle Q_{\lfloor \nu \rfloor _{0.5}}(a,b)<Q_{\nu }(a,b)<Q_{\lceil \nu \rceil _{0.5}}(a,b).}$

However, the relative error of this bound does not tend to zero when ${\displaystyle b\to \infty }$.^[5] For integral values of ${\displaystyle \nu =n}$, this bound reduces to

${\displaystyle Q_{n-0.5}(a,b)<Q_{n}(a,b)<Q_{n+0.5}(a,b).}$

A very good approximation of the generalized Marcum Q-function for integer valued ${\displaystyle \nu =n}$ is obtained by taking the arithmetic mean of the upper and lower bound^[15]

${\displaystyle Q_{n}(a,b)\approx {\frac {Q_{n-0.5}(a,b)+Q_{n+0.5}(a,b)}{2}}.}$

• A tighter bound can be obtained by exploiting the log-concavity of ${\displaystyle \nu \mapsto Q_{\nu }(a,b)}$ on ${\displaystyle [1,\infty )}$ as^[5]

${\displaystyle Q_{\nu _{1}}(a,b)^{\nu _{2}-v}Q_{\nu _{2}}(a,b)^{v-\nu _{1}}<Q_{\nu }(a,b)<{\frac {Q_{\nu _{2}}(a,b)^{\nu _{2}-\nu +1}}{Q_{\nu _{2}+1}(a,b)^{\nu _{2}-\nu }}},}$

where ${\displaystyle \nu _{1}=\lfloor \nu \rfloor _{0.5}}$ and ${\displaystyle \nu _{2}=\lceil \nu \rceil _{0.5}}$ for ${\displaystyle \nu \geq 1.5}$. The tightness of this bound improves as either ${\displaystyle a}$ or ${\displaystyle \nu }$ increases. The relative error of this bound converges to 0 as ${\displaystyle b\to \infty }$.^[5] For integral values of ${\displaystyle \nu =n}$, this bound reduces to

${\displaystyle {\sqrt {Q_{n-0.5}(a,b)Q_{n+0.5}(a,b)}}<Q_{n}(a,b)<Q_{n+0.5}(a,b){\sqrt {\frac {Q_{n+0.5}(a,b)}{Q_{n+1.5}(a,b)}}}.}$

Cauchy-Schwarz bound

Using the trigonometric integral representation for integer valued ${\displaystyle \nu =n}$, the following Cauchy-Schwarz bound can be obtained^[3]

${\displaystyle e^{-b^{2}/2}\leq Q_{n}(a,b)\leq \exp \left[-{\frac {1}{2}}(b^{2}+a^{2})\right]{\sqrt {I_{0}(2ab)}}{\sqrt {{\frac {2n-1}{2}}+{\frac {\zeta ^{2(1-n)}}{2(1-\zeta ^{2})}}}},\qquad \zeta <1,}$

${\displaystyle 1-Q_{n}(a,b)\leq \exp \left[-{\frac {1}{2}}(b^{2}+a^{2})\right]{\sqrt {I_{0}(2ab)}}{\sqrt {\frac {\zeta ^{2(1-n)}}{2(\zeta ^{2}-1)}}},\qquad \zeta >1,}$

where ${\displaystyle \zeta =a/b>0}$.

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 ${\displaystyle \zeta =a/b>0}$, one such bound for integer valued ${\displaystyle \nu =n}$ is given as^[16][3]

${\displaystyle e^{-(b+a)^{2}/2}\leq Q_{n}(a,b)\leq e^{-(b-a)^{2}/2}+{\frac {\zeta ^{1-n}-1}{\pi (1-\zeta )}}\left[e^{-(b-a)^{2}/2}-e^{-(b+a)^{2}/2}\right],\qquad \zeta <1,}$

${\displaystyle Q_{n}(a,b)\geq 1-{\frac {1}{2}}\left[e^{-(a-b)^{2}/2}-e^{-(a+b)^{2}/2}\right],\qquad \zeta >1.}$

When ${\displaystyle n=1}$, the bound simplifies to give

${\displaystyle e^{-(b+a)^{2}/2}\leq Q_{1}(a,b)\leq e^{-(b-a)^{2}/2},\qquad \zeta <1,}$

${\displaystyle 1-{\frac {1}{2}}\left[e^{-(a-b)^{2}/2}-e^{-(a+b)^{2}/2}\right]\leq Q_{1}(a,b),\qquad \zeta >1.}$

Another such bound obtained via Cauchy-Schwarz inequality is given as^[3]

${\displaystyle e^{-b^{2}/2}\leq Q_{n}(a,b)\leq {\frac {1}{2}}{\sqrt {{\frac {2n-1}{2}}+{\frac {\zeta ^{2(1-n)}}{2(1-\zeta ^{2})}}}}\left[e^{-(b-a)^{2}/2}+e^{-(b+a)^{2}/2}\right],\qquad \zeta <1}$

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

Chernoff-type bound

Chernoff-type bounds for the generalized Marcum Q-function, where ${\displaystyle \nu =n}$ is an integer, is given by^[16][3]

${\displaystyle (1-2\lambda )^{-n}\exp \left(-\lambda b^{2}+{\frac {\lambda na^{2}}{1-2\lambda }}\right)\geq \left\{{\begin{array}{lr}Q_{n}(a,b),&b^{2}>n(a^{2}+2)\\1-Q_{n}(a,b),&b^{2}<n(a^{2}+2)\end{array}}\right.}$

where the Chernoff parameter ${\displaystyle (0<\lambda <1/2)}$ has optimum value ${\displaystyle \lambda _{0}}$ of

${\displaystyle \lambda _{0}={\frac {1}{2}}\left(1-{\frac {n}{b^{2}}}-{\frac {n}{b^{2}}}{\sqrt {1+{\frac {(ab)^{2}}{n}}}}\right).}$

Semi-linear approximation

The first-order Marcum-Q function can be semi-linearly approximated by ^[17]

${\displaystyle {\begin{aligned}Q_{1}(a,b)={\begin{cases}1,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathrm {if} ~b<c_{1}\\-\beta _{0}e^{-{\frac {1}{2}}\left(a^{2}+\left(\beta _{0}\right)^{2}\right)}I_{0}\left(a\beta _{0}\right)\left(b-\beta _{0}\right)+Q_{1}\left(a,\beta _{0}\right),~~~~~\mathrm {if} ~c_{1}\leq b\leq c_{2}\\0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathrm {if} ~b>c_{2}\end{cases}}\end{aligned}}}$

where

${\displaystyle {\begin{aligned}\beta _{0}={\frac {a+{\sqrt {a^{2}+2}}}{2}},\end{aligned}}}$

${\displaystyle {\begin{aligned}c_{1}(a)=\max {\Bigg (}0,\beta _{0}+{\frac {Q_{1}\left(a,\beta _{0}\right)-1}{\beta _{0}e^{-{\frac {1}{2}}\left(a^{2}+\left(\beta _{0}\right)^{2}\right)}I_{0}\left(a\beta _{0}\right)}}{\Bigg )},\end{aligned}}}$

and

${\displaystyle {\begin{aligned}c_{2}(a)=\beta _{0}+{\frac {Q_{1}\left(a,\beta _{0}\right)}{\beta _{0}e^{-{\frac {1}{2}}\left(a^{2}+\left(\beta _{0}\right)^{2}\right)}I_{0}\left(a\beta _{0}\right)}}.\end{aligned}}}$

Equivalent forms for efficient computation

It is convenient to re-express the Marcum Q-function as^[18]

${\displaystyle P_{N}(X,Y)=Q_{N}({\sqrt {2NX}},{\sqrt {2Y}}).}$

The ${\displaystyle P_{N}(X,Y)}$ can be interpreted as the detection probability of ${\displaystyle N}$ incoherently integrated received signal samples of constant received signal-to-noise ratio, ${\displaystyle X}$, with a normalized detection threshold ${\displaystyle Y}$. In this equivalent form of Marcum Q-function, for given ${\displaystyle a}$ and ${\displaystyle b}$, we have ${\displaystyle X=a^{2}/2N}$ and ${\displaystyle Y=b^{2}/2}$. Many expressions exist that can represent ${\displaystyle P_{N}(X,Y)}$. However, the five most reliable, accurate, and efficient ones for numerical computation are given below. They are form one:^[18]

${\displaystyle P_{N}(X,Y)=\sum _{k=0}^{\infty }e^{-NX}{\frac {(NX)^{k}}{k!}}\sum _{m=0}^{N-1+k}e^{-Y}{\frac {Y^{m}}{m!}},}$

form two:^[18]

${\displaystyle P_{N}(X,Y)=\sum _{m=0}^{N-1}e^{-Y}{\frac {Y^{m}}{m!}}+\sum _{m=N}^{\infty }e^{-Y}{\frac {Y^{m}}{m!}}\left(1-\sum _{k=0}^{m-N}e^{-NX}{\frac {(NX)^{k}}{k!}}\right),}$

form three:^[18]

${\displaystyle 1-P_{N}(X,Y)=\sum _{m=N}^{\infty }e^{-Y}{\frac {Y^{m}}{m!}}\sum _{k=0}^{m-N}e^{-NX}{\frac {(NX)^{k}}{k!}},}$

form four:^[18]

${\displaystyle 1-P_{N}(X,Y)=\sum _{k=0}^{\infty }e^{-NX}{\frac {(NX)^{k}}{k!}}\left(1-\sum _{m=0}^{N-1+k}e^{-Y}{\frac {Y^{m}}{m!}}\right),}$

and form five:^[18]

${\displaystyle 1-P_{N}(X,Y)=e^{-(NX+Y)}\sum _{r=N}^{\infty }\left({\frac {Y}{NX}}\right)^{r/2}I_{r}(2{\sqrt {NXY}}).}$

Among these five form, the second form is the most robust.^[18]

Applications

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

• If ${\displaystyle X\sim \mathrm {Exp} (\lambda )}$ is a exponential distribution with rate parameter ${\displaystyle \lambda }$, then its cdf is given by ${\displaystyle F_{X}(x)=1-Q_{1}\left(0,{\sqrt {2\lambda x}}\right)}$

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

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

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

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

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

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

• If ${\displaystyle X\sim \mathrm {Rayleigh} (\sigma )}$ is a Rayleigh distribution with parameter ${\displaystyle \sigma }$, then its cdf is given by ${\displaystyle F_{X}(x)=1-Q_{1}\left(0,{\frac {x}{\sigma }}\right)}$

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

• If ${\displaystyle X\sim \chi _{k}}$ is a chi distribution with ${\displaystyle k}$ degrees of freedom, then its cdf is given by ${\displaystyle F_{X}(x)=1-Q_{k/2}(0,x)}$

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

• If ${\displaystyle X\sim \mathrm {Rice} (\nu ,\sigma )}$ is a Rice distribution with parameters ${\displaystyle \nu }$ and ${\displaystyle \sigma }$, then its cdf is given by ${\displaystyle F_{X}(x)=1-Q_{1}\left({\frac {\nu }{\sigma }},{\frac {x}{\sigma }}\right)}$

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

Footnotes

1. J.I. Marcum (1960). A statistical theory of target detection by pulsed radar: mathematical appendix, IRE Trans. Inform. Theory, vol. 6, 59-267.
2. 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.
3. 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.
4. A. 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
5. 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
6. Y. Sun and A. Baricz (2008). Inequalities for the generalized Marcum Q-function. Applied Mathematics and Computation 203(2008) 134-141.
7. N.M. Temme (1993). Asymptotic and numerical aspects of the noncentral chi-square distribution. Computers Math. Applic., 25(5), 55-63.
8. A. 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
9. S. Andras, A. Baricz, and Y. Sun (2011) The Generalized Marcum Q-function: An Orthogonal Polynomial Approach. Acta Univ. Sapientiae Mathematica, 3(1), 60-76.
10. Y.A. Brychkov (2012). On some properties of the Marcum Q function. Integral Transforms and Special Functions 23(3), 177-182.
11. M. Abramowitz and I.A. Stegun (1972). Formula 10.2.12, Modified Spherical Bessel Functions, Handbook of Mathematical functions, p. 443
12. W.K. Pratt (1968). Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(7), 1220-1221.
13. R. Esposito (1968). Comment on Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(12), 2195-2195.
14. 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.
15. 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.
16. M.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.
17. 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.
18. D.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.
• Nuttall, Albert H. (1975): 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.