New Bounds for the Generalized Marcum $Q$-Function
New Bounds for the Generalized Marcum $Q$-Function
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DOI：
10.1109/tit.2009.2021370
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发表时间：
2009-07
影响因子：
2.5
通讯作者：
Á. Baricz;Yin Sun
中科院分区：
文献类型：
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作者：
Á. Baricz;Yin Sun
本文研究广义Marcum Q函数Q (a, b)，其中a， b = 0, b = 0。我们的目标是扩展Corazza和Ferrari的结果(IEEE Trans。为了推导出一些新的紧下界和上界，引证了广义Marcum q -函数，vol. 48, pp. 3003-3008, 2002。在我们的证明中，关键的工具是涉及第一类修正贝塞尔函数和一些经典不等式，即Cauchy-Buniakowski-Schwarz和Chebyshev积分不等式的某些函数的单调性。对于大b，这些边界是非常紧的，也就是说，随着b的增加，边界的相对误差收敛于零。理论分析和数值结果都证明了边界的严密性。
In this paper, we study the generalized Marcum Q-function Q nu(a, b), where a, nu > 0 and b ges 0. Our aim is to extend the results of Corazza and Ferrari (IEEE Trans. Inf. Theory, vol. 48, pp. 3003-3008, 2002) to the generalized Marcum Q-function in order to deduce some new tight lower and upper bounds. The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind and some classical inequalities, i.e., the Cauchy-Buniakowski-Schwarz and Chebyshev integral inequalities. These bounds are shown to be very tight for large b, i.e., the relative errors of our bounds converge to zero as b increases. Both theoretical analysis and numerical results are provided to show the tightness of our bounds.