Marcum Q-function

Properties

Finite integral representation

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

: 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 \nu = n, such a representation is given by the trigonometric integral

: Q_n(a,b) = \left{ \begin{array}{lr} H_n(a,b) & a \frac{1}{2} + H_n(a,a) & a=b, \ 1 + H_n(a,b) & a b, \end{array} \right.

where

: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 \zeta = a/b is a constant.

For any real \nu 0, such finite trigonometric integral is given by

: Q_\nu(a,b) = \left{ \begin{array}{lr} H_\nu(a,b) + C_\nu(a,b) & a \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, \zeta = a/b, and the additional correction term is given by

: 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 \nu, the correction term C_\nu(a,b) tend to vanish.

Monotonicity and log-concavity

• The generalized Marcum Q-function Q_\nu(a,b) is strictly increasing in \nu and a for all a \geq 0 and b, \nu 0, and is strictly decreasing in b for all a, b \geq 0 and \nu0.
• The function \nu \mapsto Q_\nu(a,b) is log-concave on 1,\infty) for all a , b \geq 0.
• The function b \mapsto Q_\nu(a,b) is strictly log-concave on (0,\infty) for all a \geq 0 and \nu 1, which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.
• The function a \mapsto 1 - Q_\nu(a,b) is log-concave on [0,\infty) for all b, \nu 0.

Series representation

• The generalized Marcum Q function of order \nu 0 can be represented using incomplete Gamma function as

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

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

:: 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 L_k^{(\alpha)}(\cdot) is the generalized Laguerre polynomial of degree k and of order \alpha.

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

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

:: 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 \alpha assumes an integer value, we have I_{\alpha}(ab) = I_{-\alpha}(ab).

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

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

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

:for non-negative integers n, where 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:

: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.

Recurrence relation and generating function

• Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation

:: 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

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

• The related three-term recurrence relation is given by

:where

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

::\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

• The ordinary generating function of Q_\nu(a,b) for integral \nu is

::\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 |t|

Symmetry relation

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

:In particular, for n = 1 we have

Special values

Some specific values of Marcum-Q function are

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

:which when combined with the recursive formula gives

:for any non-negative integer n.

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

:: 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 \nu=3/2, we have

• For \nu = 5/2 we have

Asymptotic forms

• Assuming \nu to be fixed and ab large, let \zeta = a/b 0, then the generalized Marcum-Q function has the following asymptotic form

:where \psi_n is given by

::\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 \phi_n and A_n are given by

::\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),

:The function A_n(\nu) satisfies the recursion

:for n \geq 0 and A_0(\nu)=1.

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

::\phi_0 = \frac{\sqrt{2 \pi ab}}{b-a} \mathrm{erfc}\left(\frac{b-a}{\sqrt{2}}\right).

:Hence, assuming ba, the first term asymptotic approximation of the generalized Marcum-Q function is

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

:For the case when a b, we have

:Here too Q_\nu(a,b) \sim 0.5 as a \downarrow b.

Differentiation

• The partial derivative of Q_\nu(a,b) with respect to a and b is given by

:: \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), :: \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

::\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 Q_\nu(a,b) with respect to its arguments is given by

:: \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),
:: \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

:for all a \geq b 0 and \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 \nu \mapsto Q_\nu(a,b) and the fact that we have closed form expression for Q_\nu(a,b) when \nu is half-integer valued.

Let \lfloor x \rfloor_{0.5} and \lceil x \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

:\lfloor x \rfloor_{0.5} = \lfloor x - 0.5 \rfloor + 0.5 : \lceil x \rceil_{0.5} = \lceil x + 0.5 \rceil - 0.5

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

• The monotonicity of the function \nu \mapsto Q_\nu(a,b) for all a \geq 0 and b 0 gives us the following simple bound

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

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

:: 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 \nu \mapsto Q_\nu(a,b) on [1,\infty) as

:where \nu_1 = \lfloor\nu\rfloor_{0.5} and \nu_2 = \lceil\nu\rceil_{0.5} for \nu \geq 1.5. The tightness of this bound improves as either a or \nu increases. The relative error of this bound converges to 0 as b \to \infty. For integral values of \nu = n, this bound reduces to

::\sqrt{Q_{n - 0.5}(a,b) Q_{n + 0.5}(a,b)}

Cauchy-Schwarz bound

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

: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 - 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 \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 \zeta = a/b 0, one such bound for integer valued \nu = n is given as

: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 :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 n=1, the bound simplifies to give

:e^{-(b+a)^2/2} \leq Q_1(a,b) \leq e^{-(b-a)^2/2}, \qquad \zeta :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

: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 :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 \nu = n is an integer, is given by

:(1-2\lambda)^{-n} \exp \left(-\lambda b^2 + \frac{\lambda n a^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 \end{array} \right.

where the Chernoff parameter (0 has optimum value \lambda_0 of

:\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

:\begin{align} Q_1(a, b)= \begin{cases} 1, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathrm{if}~ b -\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}

Equivalent forms for efficient computation

It is convenient to re-express the Marcum Q-function as

: P_N(X,Y) = Q_N(\sqrt{2NX},\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) = \sum_{k=0}^\infty e^{-NX} \frac{(NX)^k}{k!} \sum_{m=0}^{N-1+k} e^{-Y} \frac{Y^m}{m!},

form two:

: 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:

: 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:

: 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:

: 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.

Applications

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

• If X \sim \mathrm{Exp}(\lambda) is an exponential distribution with rate parameter \lambda, then its cdf is given by F_X(x) = 1 - Q_1\left(0,\sqrt{2 \lambda x}\right)
• If X \sim \mathrm{Erlang}(k,\lambda) is a Erlang distribution with shape parameter k and rate parameter \lambda, then its cdf is given by F_X(x) = 1 - Q_k\left(0,\sqrt{2 \lambda x}\right)
• If X \sim \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,\sqrt{x})
• If X \sim \mathrm{Gamma}(\alpha,\beta) is a gamma distribution with shape parameter \alpha and rate parameter \beta, then its cdf is given by F_X(x) = 1 - Q_{\alpha}(0,\sqrt{2 \beta x})
• If X \sim \mathrm{Weibull}(k,\lambda) is a Weibull distribution with shape parameters k and scale parameter \lambda, then its cdf is given by F_X(x) = 1 - Q_1 \left( 0, \sqrt{2} \left(\frac{x}{\lambda}\right)^{\frac{k}{2}} \right)
• If X \sim \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_{\frac{d}{p}} \left( 0, \sqrt{2} \left(\frac{x}{a}\right)^{\frac{p}{2}} \right)
• If X \sim \chi^2_k(\lambda) is a non-central chi-squared distribution with non-centrality parameter \lambda and k degrees of freedom, then its cdf is given by F_X(x) = 1 - Q_{k/2}(\sqrt{\lambda},\sqrt{x})
• If X \sim \mathrm{Rayleigh}(\sigma) is a Rayleigh distribution with parameter \sigma, then its cdf is given by F_X(x) = 1 - Q_1\left(0,\frac{x}{\sigma}\right)
• If X \sim \mathrm{Maxwell}(\sigma) is a Maxwell–Boltzmann distribution with parameter \sigma, then its cdf is given by F_X(x) = 1 - Q_{3/2}\left(0,\frac{x}{\sigma}\right)
• If X \sim \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 \sim \mathrm{Nakagami}(m,\Omega) is a Nakagami distribution with m as shape parameter and \Omega as spread parameter, then its cdf is given by F_X(x) = 1 - Q_{m}\left(0,\sqrt{\frac{2m}{\Omega}}x\right)
• If X \sim \mathrm{Rice}(\nu,\sigma) is a Rice distribution with parameters \nu and \sigma, then its cdf is given by F_X(x) = 1 - Q_1\left(\frac{\nu}{\sigma},\frac{x}{\sigma}\right)
• If X \sim \chi_k(\lambda) is a non-central chi distribution with non-centrality parameter \lambda and k degrees of freedom, then its cdf is given by F_X(x) = 1 - Q_{k/2}(\lambda,x)

Footnotes

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 QM Function, IEEE Transactions on Information Theory, 21(1), 95–96,
• 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. https://mathworld.wolfram.com/MarcumQ-Function.html

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