广义Marcum和Nuttall Q-函数的单调性、对数性和紧界
On the monotonicity, log-concavity and tight bounds of the generalized Marcum and Nuttall Q-functions
摘要：
本文对广义Marcum和Nuttall Q-函数的单调性和对数单调性进行了全面的研究。更确切地说，首先给出了一个简单的概率方法来证明这两个函数的单调性。然后，建立了广义Marcum Q函数及其变形相对于这三个参数的对数关系。由于Nuttall Q-函数与广义Marcum Q-函数具有相似的概率解释，我们推导出了Nuttall Q-函数的对数分布。通过利用这两个函数的对数分布，我们提出了广义Marcum和Nuttall Q-函数的新的紧上下界。在大多数情况下，我们提出的界限比文献中现有的界限要紧得多。当b趋于无穷大时，我们提出的界的相对误差收敛到0。数值结果表明，在大多数情况下，所提出的边界的绝对相对误差小于5%。该界可有效地应用于认知无线电和无线传感器网络等干扰受限系统的中断概率分析、衰落信道下各种无线通信系统的误码性能研究以及差分相移键控（DPSK）信号对数似然比的提取.
英文摘要：
In this paper, we present a comprehensive study of the monotonicity and log-concavity of the generalized Marcum and Nuttall Q-functions. More precisely, a simple probabilistic method is firstly given to prove the monotonicity of these two functions. Then, the log-concavity of the generalized Marcum Q-function and its deformations is established with respect to each of the three parameters. Since the Nuttall Q-function has similar probabilistic interpretations as the generalized Marcum Q-function, we deduce the log-concavity of the Nuttall Q-function. By exploiting the log-concavity of these two functions, we propose new tight lower and upper bounds for the generalized Marcum and Nuttall Q-functions. Our proposed bounds are much tighter than the existing bounds in the literature in most of the cases. The relative errors of our proposed bounds converge to 0 as b tends to infinity. The numerical results show that the absolute relative errors of the proposed bounds are less than 5% in most of the cases. The proposed bounds can be effectively applied to the outage probability analysis of interference-limited systems such as cognitive radio and wireless sensor network, in the study of error performance of various wireless communication systems operating over fading channels and extracting the log-likelihood ratio for differential phase-shift keying (DPSK) signals.
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