Marcum Q-function explained

In statistics, the generalized Marcum Q-function of order

\nu is defined as Q _{\nu} (a,b) = (1)/(a ^{\nu-1}) \int ^{inf ty} _{b} x ^{\nu} \exp \left( - (x ^{2} + a ^{2})/(2) \right) I _{\nu-1} (ax) dx

where

b \geq 0 and a, \nu > 0 and I _{\nu-1} is the modified Bessel function of first kind of order \nu -1 . If b > 0 , the integral converges for any \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 \nu = 1 , and hence named after, by Jess Marcum for pulsed radars.^[1]

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 asQ _{\nu} (a,b) = 1 - (1)/(a ^{\nu-1}) \int ^{b} _{0} x ^{\nu} \exp \left( - (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^{[2] [3]} Q _{n(a,b)} = \left\{ \begin{array}{lr} H _{n(a,b)} & a < b, \\ (1)/(2) + H _{n(a,a)} & a = b, \\ 1 + H _{n(a,b)} & a > b, \end{array} \right.

where

H _{n(a,b)} = (\zeta ^{1-n})/(\exp\left( - 2 \pi) (a ^{2+b} ^{2})/(2) \right) \int ^{2 \pi} _{0} (\cos(n -1 )\theta - \zeta \cos n\theta)/(1-2 \zeta\cos\theta + \zeta ^{2}) \exp(ab\cos\theta) d \theta,

and the ratio

\zeta = a/b is a constant.

For any real

\nu > 0 , such finite trigonometric integral is given by^[4] Q _{\nu(a,b)} = \left\{ \begin{array}{lr} H _{\nu(a,b)} + C _{\nu(a,b)} & a < b, \\ (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)} = (\sin(\nu\pi))/(\exp\left( - \pi) (a ^{2+b} ^{2})/(2) \right) \int ^{1} _{0} ((x/\zeta) ^{\nu-1})/(\zeta + x) \exp\left[ - (ab)/(\left(x + 2) (1)/(x) \right) \right] 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 . ^[5]

• The function

\nu \mapsto Q _{\nu(a,b)} is log-concave on [ 1 , inf ty) for all a , b \geq 0 . ^[5]

• The function

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

• The function

a \mapsto 1 - Q _{\nu(a,b)} is log-concave on [ 0 , inf ty) for all b, \nu > 0 . ^[5]

Series representation

• The generalized Marcum Q function of order

\nu > 0 can be represented using incomplete Gamma function as^[7] Q _{\nu} (a,b) = 1 - e ^{- a ^{2/2}} \sum ^{inf ty} _{k =0} (1)/(k!) (\gamma(\nu + k, (b ^{2})/() 2))/(\Gamma(\nu + k)) \left( (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^[7] Q _{\nu} (a,b) = 1 - e ^{- a ^{2/2}} \sum ^{inf ty} _{k =0} ( -1 ) ^{k} (L ^{(\nu -1 )} _{k} ( (a ^{2})/(2) ))/(\left( \Gamma(\nu + k +1 )) (b ^{2})/(2) \right) ^{k+\nu} ,

where

L ^{(\alpha)} _{k} ( ⋅ ) 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^[4] Q _{\nu} (a,b) = e ^{- (a ^{2} + b ^{2)/2}} \sum ^{inf ty} _{\alpha =1- \nu} \left( (a)/(b) \right) ^{\alpha} I _{-\alpha} (ab), 1 - Q _{\nu(a,b)} = e ^{- (a ^{2} + b ^{2)/2}} \sum ^{inf ty} _{\alpha = \nu} \left( (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 Q _{n+1/2} (a,b) = (1)/(2) \left[ erfc \left( (b - a)/(\sqrt{ 2) }\right) + \mathrm\left(\frac\right) \right] + e^ \sum_^n \left(\frac\right)^ I_(ab),

where

erfc ( ⋅ ) is the complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as^[4] I _{\pm(n+0.5)} (z) = (1)/(\sqrt{\pi) } \sum_^n \frac \left[\frac{(-1)^k e^z \mp (-1)^n e^{-z}}{(2z)^{k+0.5}} \right],

where

n is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have^[4] Q _{n+1/2} (a,b) = Q(b - a) + Q(b + a) + (1)/(b\sqrt{ 2 \pi) } \sum_^ \left(\frac\right)^i \sum_^ \frac \left[\frac{(-1)^k e^{-(a-b)^2/2} + (-1)^i e^{-(a+b)^2/2}}{(2ab)^k} \right],

for non-negative integers

n , where Q( ⋅ ) 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:^[8] I _{n + (1)/(2)} (z) = \sqrt{ (2 z)/(\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) = ( 2 n +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^[9]

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

• The above formula is easily generalized as

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

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 = inf ty , we obtain the Neumann series representation of the generalized Marcum Q-function.

• The related three-term recurrence relation is given by

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

c _{\nu(a,b)} = \left( (b)/(a) \right) (I _{\nu(ab)})/(I _{\nu+1} (ab)) .

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

(a ^{2})/(2) Q _{\nu+2} (a,b) = \left( (a ^{2})/(2) - \nu\right) Q _{\nu+1} (a,b) + \left( (b ^{2})/(2) + \nu\right)Q _{\nu} (a,b) - (b ^{2})/(2) Q _{\nu-1} (a,b).

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

Q _{\nu+1} (a,b) = Q _{\nu(a,b)} + (1)/(a) (\partial)/(\partial a) Q _{\nu(a,b),} Q _{\nu-1} (a,b) = Q _{\nu(a,b)} + (1)/(b) (\partial)/(\partial b) Q _{\nu(a,b).}

• The ordinary generating function of

Q _{\nu(a,b)} for integral \nu is \sum ^{inf ty} _{n =-inf ty} t ^{n} Q _{n(a,b)} = e ^{- (a ^{2+b} ^{2)/2}} (t)/(1- t) e ^{(b ^{2} t + a ^{2/t)/2}} ,

where

|t|< 1 .

Symmetry relation

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

\nu = n Q _{n(a,b)} + Q _{n(b,a)} = 1 + e ^{- (a ^{2+b} ^{2)/2}} \left[ I _{0(ab)} + \sum ^{n -1} _{k =1} (a ^{2k} + b ^{2k})/((ab) ^{k}) I _{k(ab)} \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).}

Special values

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

Q _{\nu(0,0)} = 1 , Q _{\nu(a,0)} = 1 , Q _{\nu(a,+infty)} = 0 , Q _{\nu(0,b)} = (\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^[10] Q _{1(a,a)} = (1)/(2) [ 1 + e ^{- a ^{2}} I ^{2 )],} _{0 (a}

which when combined with the recursive formula gives

Q _{n(a,a)} = (1)/(2) [ 1 + e ^{- a ^{2}} I ^{2 )]} _{0 (a} + e ^{- a ^{2}} \sum ^{n -1} _{k =1} I ^{2 ),} _{k(a} Q _{-n} (a,a) = (1)/(2) [ 1 + e ^{- a ^{2}} I ^{2 )]} _{0 (a} - e ^{- a ^{2}} \sum ^{n} _{k =1} I ^{2 ),} _{k(a}

for any non-negative integer

n .

• For

\nu = 1 / 2 , using the basic integral definition of generalized Marcum Q-function, we have^{[9] [10]} Q _{1/2} (a,b) = (1)/(2) \left[ erfc \left( (b - a)/(\sqrt{ 2) }\right) + \mathrm\left(\frac\right) \right].

• For

\nu =3 / 2 , we haveQ _{3/2} (a,b) = Q _{1/2} (a,b) + \sqrt{ (2)/(\pi) } \, \frac e^.

• For

\nu = 5 / 2 we have Q _{5/2} (a,b) = Q _{3/2} (a,b) + \sqrt{ (2)/(\pi) } \, \frac e^.

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^[11] Q _{\nu(a,b)} \sim \sum ^{inf ty} _{n =0} \psi _{n,}

where

\psi _{n} is given by\psi _{n} = (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[ ((b - a) ^{2})/(2 ab) \right] ^{n - (1)/(2)} \Gamma\left( (1)/(2) - n, ((b - a) ^{2})/(2) \right), A _{n(\nu)} = (2 ^{- n} \Gamma( (1)/(2) + \nu + n))/(n!\Gamma( (1)/(+ \nu - n) 2)) .

The function

A _{n(\nu)} satisfies the recursion A _{n+1} (\nu) = - (( 2 n +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} = (\sqrt{ 2 \pi ab)/() } \mathrm\left(\frac\right).

Hence, assuming

b>a , the first term asymptotic approximation of the generalized Marcum-Q function is^[11] Q _{\nu(a,b)} \sim \psi _{0} = \left( (b)/(a) \right) ^{\nu - (1)/(2)} Q(b - a),

where

Q( ⋅ ) 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^[11] Q _{\nu(a,b)} \sim 1- \psi _{0} =1- \left( (b)/(a) \right) ^{\nu - (1)/(2)} Q(a - b).

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^{[12] [13]} (\partial)/(\partial a) Q _{\nu(a,b)} = a \left[Q _{\nu+1} (a,b) - Q _{\nu} (a,b)\right] = a \left( (b)/(a) \right) ^{\nu} e ^{- (a ^{2+b} ^{2)/2}} I _{\nu} (ab), (\partial)/(\partial b) Q _{\nu(a,b)} = b \left[Q _{\nu-1} (a,b) - Q _{\nu} (a,b)\right] = - b \left( (b)/(a) \right) ^{\nu-1} e ^{- (a ^{2+b} ^{2)/2}} I _{\nu-1} (ab).

We can relate the two partial derivatives as

(1)/(a) (\partial)/(\partial a) Q _{\nu(a,b)} + (1)/(b) (\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^[10] (\partial ^{n})/(\partial a ^{n}) Q _{\nu(a,b)} = n! ( - a) ^{n} \sum ^{[n/ 2 ]} _{k =0} (( -2 a ^{2)} ^{-k})/(k!(n -2 k)!) \sum ^{n - k} _{p =0} ( -1 ) ^{p} \binom{n - k}{p} Q _{\nu+p} (a,b), (\partial ^{n})/(\partial b ^{n}) Q _{\nu(a,b)} = (n! a ^{1-\nu})/(2 ^{n} b ^{n-\nu+1}) e ^{- (a ^{2+b} ^{2)/2}} \sum ^{n} _{k = [n/ 2 ]} (( -2 b ^{2)} ^{k})/((n - k)!( 2 k - n)!) \sum ^{k -1} _{p =0} \binom{k -1 }{p} \left( - (a)/(b) \right) ^{p} I _{\nu-p-1} (ab).

Inequalities

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

Q ^{2} _{\nu(a,b)} > (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^{[14] [9] [15]} 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

b \to inf ty .^[5] For integral values of \nu = n , this bound reduces to 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

\nu = n is obtained by taking the arithmetic mean of the upper and lower bound^[15] Q _{n(a,b)} ≈ (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 , inf ty) as^[5] Q _{\nu _{1}} (a,b) ^{\nu _{2} - v} Q _{\nu _{2}} (a,b) ^{v - \nu _{1}} < Q _{\nu(a,b)} < (Q _{\nu _{2}} (a,b) ^{\nu _{2} - \nu +1})/(Q _{\nu _{2} +1} (a,b) ^{\nu _{2} - \nu}) ,

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 inf ty .^[5] For integral values of \nu = n , this bound reduces to \sqrt{Q _{n} (a,b) Q _{n} (a,b)} < Q _{n(a,b)} < Q _{n} (a,b) \sqrt{ (Q _{n} (a,b))/(Q _{n} (a,b)) }.

Cauchy-Schwarz bound

Using the trigonometric integral representation for integer valued

\nu = n , the following Cauchy-Schwarz bound can be obtained^[3] e ^{- b ^{2/2}} \leq Q _{n(a,b)} \leq \exp\left[ - (1)/(2) (b ^{2} + a ^{2)\right]} \sqrt{I _{0(2ab)}} \sqrt{ (2 n -1)/(2) + (\zeta ^{2(1-n)})/(2 ( 1- \zeta ^{2)}) }, \qquad \zeta < 1,1 - Q _{n(a,b)} \leq \exp\left[ - (1)/(2) (b ^{2+a} ^{2)\right]} \sqrt{I _{0(2ab)}} \sqrt{ (\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^{[16] [3]} e ^{- (b + a) ^{2/2}} \leq Q _{n(a,b)} \leq e ^{- (b - a) ^{2/2}} + (\zeta ^{1-n} -1)/(\pi( 1- \zeta)) \left[e ^{- (b - a) ^{2/2}} - e ^{- (b + a) ^{2/2}} \right], \zeta < 1 , Q _{n(a,b)} \geq 1 - (1)/(2) \left[e ^{- (a - b) ^{2/2}} - e ^{- (a + b) ^{2/2}} \right], \zeta > 1 .

When

n =1 , the bound simplifies to givee ^{- (b + a) ^{2/2}} \leq Q _{1(a,b)} \leq e ^{- (b - a) ^{2/2}} , \zeta < 1 , 1 - (1)/(2) \left[e ^{- (a - b) ^{2/2}} - e ^{- (a + b) ^{2/2}} \right] \leq Q _{1(a,b),} \zeta > 1 .

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

e ^{- b ^{2/2}} \leq Q _{n(a,b)} \leq (1)/(\sqrt{ 2) (2 n -1)/(2) + (\zeta ^{2(1-n)})/(2 ( 1- \zeta ^{2)}) } \left[e^{-(b-a)^2/2} + e^{-(b+a)^2/2} \right], \qquad \zeta < 1Q _{n(a,b)} \geq 1 - (1)/(\sqrt{ 2) (\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^{[16] [3]} ( 1-2λ ) ^{-n} \exp \left( -λ b ^{2} + (λ n a ^{2})/(1-2λ) \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

( 0 < λ < 1 / 2 ) has optimum value λ _{0} ofλ _{0} = (1)/(2) \left( 1 - (n)/(b ^{2}) - (n)/(b ^{2}) \sqrt{ 1 + ((ab) ^{2})/(n) }\right).

Semi-linear approximation

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

\begin{align} Q _{1(a,} b) = \begin{cases} 1 , ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ if ~ b < c _{1} \ - \beta _{0} e ^{- (1)/(\left(a ^{2 \right)} _{0 \right)} 2)} I _{0\left(a\beta} _{0\right)\left(b-\beta} _{0\right)+Q} _{1\left(a,\beta} _{0\right),} ~~~~~ if ~ c _{1} \leq b \leq c _{2} \\ 0 , ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ if ~ b> c _{2 \end{cases} \end{align}} where\begin{align} \beta _{0} = (a + \sqrt{a ^{2+2})/() },\end\begin{align} c _{1(a)} = max ( 0 ,\beta _{0+ (Q _{1\left(a,\beta} _{0\right)-1})/(\beta _{0} e ^{- (1)/(\left(a ^{2 \right)} _{0 \right)} 2)} I _{0\left(a\beta} _{0\right)})} ) , \end{align} and\begin{align} c _{2(a)} = \beta _{0+ (Q _{1\left(a,\beta} _{0\right)})/(\beta _{0} e ^{- (1)/(\left(a ^{2 \right)} _{0 \right)} 2)} 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^[18]

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:^[18] P _{N(X,Y)} = \sum ^{inf ty} _{k =0} e ^{-NX} ((NX) ^{k})/(k!) \sum ^{N -1+ k} _{m =0} e ^{-Y} (Y ^{m})/(m!) ,

form two:^[18]

P _{N(X,Y)} = \sum ^{N -1} _{m =0} e ^{-Y} (Y ^{m})/(m!) + \sum ^{inf ty} _{m = N} e ^{-Y} (Y ^{m})/(m!) \left( 1 - \sum ^{m - N} _{k =0} e ^{-NX} ((NX) ^{k})/(k!) \right),

form three:^[18]

1 - P _{N(X,Y)} = \sum ^{inf ty} _{m = N} e ^{-Y} (Y ^{m})/(m!) \sum ^{m - N} _{k =0} e ^{-NX} ((NX) ^{k})/(k!) ,

form four:^[18]

1 - P _{N(X,Y)} = \sum ^{inf ty} _{k =0} e ^{-NX} ((NX) ^{k})/(k!) \left( 1 - \sum ^{N -1+ k} _{m =0} e ^{-Y} (Y ^{m})/(m!) \right),

and form five:^[18]

1 - P _{N(X,Y)} = e ^{-(NX+Y)} \sum ^{inf ty} _{r = N} \left( (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

X \sim Exp ( λ ) is a exponential distribution with rate parameter λ , then its cdf is given by F _{X(x)} = 1 - Q _{1\left(0,\sqrt{2} λ x}\right)

• If

X \sim Erlang (k, λ ) is a Erlang distribution with shape parameter k and rate parameter λ , then its cdf is given by F _{X(x)} = 1 - Q _{k\left(0,\sqrt{2} λ 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 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 Weibull (k, λ ) is a Weibull distribution with shape parameters k and scale parameter λ , then its cdf is given by F _{X(x)} = 1 - Q _{1} \left( 0 , \sqrt{ 2 } \left( (x)/(λ) \right) ^{(k)/(2)} \right)

• If

X \sim 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 _{(d)/(p)} \left( 0 , \sqrt{ 2 } \left( (x)/(a) \right) ^{(p)/(2)} \right)

• If

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

• If

X \sim Rayleigh (\sigma) is a Rayleigh distribution with parameter \sigma , then its cdf is given by F _{X(x)} = 1 - Q _{1 \left( 0 , (x)/(\sigma)} \right)

• If

X \sim 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 , (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 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{ (2 m)/(\Omega) }x\right)

• If

X \sim 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( (\nu)/(\sigma)} , (x)/(\sigma) \right)

• If

X \sim \chi _{k(λ)} is a non-central chi distribution with non-centrality parameter λ and k degrees of freedom, then its cdf is given by F _{X(x)} = 1 - Q _{k/2} ( λ ,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,
6. Y. Sun and A. Baricz (2008). Inequalities for the generalized Marcum Q-function. Applied Mathematics and Computation 203(2008) 134-141.
7. 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.
8. M. Abramowitz and I.A. Stegun (1972). Formula 10.2.12, Modified Spherical Bessel Functions, Handbook of Mathematical functions, p. 443
9. 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,
10. Y.A. Brychkov (2012). On some properties of the Marcum Q function. Integral Transforms and Special Functions 23(3), 177-182.
11. N.M. Temme (1993). Asymptotic and numerical aspects of the noncentral chi-square distribution. Computers Math. Applic., 25(5), 55-63.
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,
• 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