广义Marcum和Nuttall Q-函数的单调性
On the Monotonicity of the Generalized Marcum and Nuttall Q-Functions
摘要：
建立了广义Marcum Q函数$\emph{Q}_{M}$、标准Nuttall Q函数$\emph{Q}_{M,N}$和归一化Nuttall Q函数$\mathcal{Q}_{M,N}$关于它们的实阶指数M、N的单调性准则。此外，对于M，N为0.5的奇数倍和$M\geq N$的情况，导出了标准Nuttall Q函数和归一化Nuttall Q函数的计算公式.利用这些结果，新的上下界$\emph{Q}_{M,N}$和$\mathcal{Q}_{M,N}$提出。此外，具体的紧的上限和下限为$\emph{Q}_{M}$，以前在文献中报道，扩展为真正的M值。所提供的理论结果可有效地应用于衰落信道下数字通信的研究、多输入多输出系统的信息论分析以及概率论中随机过程的描述等。
英文摘要：
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.
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