Marcum Q-function

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 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 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 \nu>0. • The function \nu \mapsto Q_\nu(a,b) is log-concave on [1,\infty) for all a , b \geq 0. • The function a \mapsto 1 - Q_\nu(a,b) is log-concave on [0,\infty) for all b, \nu > 0. :: 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 ::I_{n+\frac{1}{2}}(z) = \sqrt{\frac{2z}{\pi}} \left[ g_n(z) \sinh(z) + g_{-n-1}(z) \cosh(z)\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. Recurrence relation and generating function • Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation ::Q_1(a,a) = \frac{1}{2}[1 + e^{-a^2}I_0(a^2)], :which when combined with the recursive formula gives ::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), ::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 n. • For \nu = 1/2, using the basic integral definition of generalized Marcum Q-function, we have ::Q_\nu(a,b) \sim \sum_{n=0}^\infty \psi_n, :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), ::A_n(\nu) = \frac{2^{-n}\Gamma(\frac{1}{2}+\nu+n)}{n!\Gamma(\frac{1}{2}+\nu-n)}. :The function A_n(\nu) satisfies the recursion ::A_{n+1}(\nu) = - \frac{(2n+1)^2 - 4\nu^2}{8(n+1)}A_n(\nu), :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 b>a, the first term asymptotic approximation of the generalized Marcum-Q function is :: \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 ::Q^2_\nu(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 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 ::Q_{\lfloor\nu\rfloor_{0.5}}(a,b) :However, the relative error of this bound does not tend to zero when b \to \infty. :\begin{align} Q_1(a, b)= \begin{cases} 1, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\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==