Monotonicity criteria are established for the generalized Marcum Q-function, $\emph{Q}_{M}$, the standard Nuttall Q-function, $\emph{Q}_{M,N}$, and the normalized Nuttall Q-function, $\mathcal{Q}_{M,N}$, with respect to their real order indices M,N. Besides, closed-form expressions are derived for the computation of the standard and normalized Nuttall Q-functions for the case when M,N are odd multiples of 0.5 and $M\geq N$. By exploiting these results, novel upper and lower bounds for $\emph{Q}_{M,N}$ and $\mathcal{Q}_{M,N}$ are proposed. Furthermore, specific tight upper and lower bounds for $\emph{Q}_{M}$, previously reported in the literature, are extended for real values of M. The offered theoretical results can be efficiently applied in the study of digital communications over fading channels, in the information-theoretic analysis of multiple-input multiple-output systems and in the description of stochastic processes in probability theory, among others.
Deep Dive into On the Monotonicity of the Generalized Marcum and Nuttall Q-Functions.
Monotonicity criteria are established for the generalized Marcum Q-function, $\emph{Q}_{M}$, the standard Nuttall Q-function, $\emph{Q}_{M,N}$, and the normalized Nuttall Q-function, $\mathcal{Q}_{M,N}$, with respect to their real order indices M,N. Besides, closed-form expressions are derived for the computation of the standard and normalized Nuttall Q-functions for the case when M,N are odd multiples of 0.5 and $M\geq N$. By exploiting these results, novel upper and lower bounds for $\emph{Q}_{M,N}$ and $\mathcal{Q}_{M,N}$ are proposed. Furthermore, specific tight upper and lower bounds for $\emph{Q}_{M}$, previously reported in the literature, are extended for real values of M. The offered theoretical results can be efficiently applied in the study of digital communications over fading channels, in the information-theoretic analysis of multiple-input multiple-output systems and in the description of stochastic processes in probability theory, among others.
A N extended version of the (standard) Marcum Q- function, Q(α, β) = ∞ β xe -x 2 +α 2 2 I 0 (αx)dx, where α, β ≥ 0, originally appeared in [1, Appendix, eq. ( 16)], defines the standard 1 Nuttall Q-function [2, eq. ( 86)], given by the integral representation
where the order indices are generally reals with values M ≥ 0 and N > -1, I N is the N th order modified Bessel function of the first kind [3, eq. (9.6.
3)] and α, β are real parameters with α > 0, β ≥ 0. It is worth mentioning here, that the negative values of N , defined above, have not been of interest in any practical applications so far. However, the extension of the Nuttall Q-function to negative values of N has been introduced here in order to facilitate more effectively the relation of this function to the more common generalized Marcum Q-function, as will be shown in the sequel. An alternative version of Q M,N (α, β) is the normalized Nuttall Q-function, Q M,N (α, β), which constitutes a normalization of the former with respect to the parameter α, defined simply by the relation
Typical applications involving the standard and normalized Nuttall Q-functions include: (a) the error probability performance of noncoherent digital communication over Nakagami fading channels with interference [4], (b) the outage probability of wireless communication systems where the Nakagami/Rician faded desired signals are subject to independent and identically distributed (i.i.d.) Rician/Nakagami faded interferers, respectively, under the assumptions of minimum interference and signal power constraints [4]- [7], (c) the performance analysis and capacity statistics of uncoded multipleinput multiple-output (MIMO) systems operating over Rician fading channels [8]- [10], and (d) the extraction of the required log-likelihood ratio for the decoding of differential phase-shift keying (DPSK) signals employing turbo or low-density paritycheck (LDPC) codes [11].
Since both types of the Nuttall Q-function are not considered to be tabulated functions, their computation involved in the aforementioned applications was handled considering the two distinct cases of M + N being either odd or even, in order to express them in terms of more common functions. The possibility of doing such when M + N is odd was suggested in [2], requiring particular combination of the two recursive relations [2, eqs. (87), (88)]. However, the explicit solution was derived only in [4, eq. (13)] entirely in terms of the Marcum Q-function and a finite weighted sum of modified Bessel functions of the first kind. Having all the above in mind, along with the fact that the calculation of Q(α, β) itself requires numerical integration, the issue of the efficient computation of (1) and ( 2) still remains open.
The generalized Marcum Q-function [12] of positive real order M , is defined by the integral [13, eq. ( 1)]
where α, β are non-negative real parameters2 . For M = 1, it reduces to the popular standard (or first-order) Marcum Qfunction, Q 1 (α, β) (or Q(α, β)), while for general M it is related to the normalized Nuttall Q-function according to [14, eq. (4.105)]
An identical function to the generalized Marcum Q is the probability of detection3 [1, eq. ( 49)], which has a long history in radar communications and particularly in the study of target detection by pulsed radar with single or multiple observations [1], [16]- [18]. Additionally, Q M (α, β) is strongly associated with: (a) the error probability performance of noncoherent and differentially coherent modulations over generalized fading channels [14], [19]- [23], (b) the signal energy detection of a primary user over a multipath channel [24], [25], and finally (c) the information-theoretic study of MIMO systems [26]. Aside from these applications, the generalized Marcum Qfunction presents a variety of interesting probabilistic interpretations. Most indicatively, for integer M , it is the complementary cumulative distribution function (CCDF) of a noncentral chi-square (χ 2 ) random variable with 2M degrees of freedom (DOF) [27, eq. (2.45)]. This relationship was extended in [28] to work for the case of odd DOF as well, through a generalization of the noncentral χ 2 CCDF. Similar relations can be found in the literature involving the generalized Rician [29, (2.1-145)], the generalized Rayleigh [30, pp. 1] (for α = 0) and the bivariate Rayleigh [31, Appendix A], [32] (for M = 1) CCDF’s. Finally, in a recent work [33], a new association has been derived between the generalized Marcum Q-function and a probabilistic comparison of two independent Poisson random variables.
More than thirty algorithms have been proposed in the literature for the numerical computation of the standard and generalized Marcum Q-functions, among them power series expansions [34]- [36], approximations and asymptotic expressions [37]- [40], and Neumann series expansions [41]- [43]. However, the above representations may not always provide sufficient information about the relat
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