On the Monotonicity of the Generalized Marcum and Nuttall ${Q}$ -Functions

Monotonicity criteria are established for the generalized Marcum <i>Q</i> -function, <i>QM</i>(alpha,beta), the standard Nuttall <i>Q</i>-function, <i>QM</i>,<i>N</i>(alpha,beta) , and the normalized Nuttall <i>Q</i>-function, <i>QM</i>,<i>N</i>(alpha,beta), with respect to their real order indices <i>M</i>,<i>N</i>. Besides, closed-form expressions are derived for the computation of the standard and normalized Nuttall <i>Q</i>-functions for the case when <i>M</i>,<i>N</i> are odd multiples of 0.5 and <i>M</i> ges <i>N</i>. By exploiting these results, novel upper and lower bounds for <i>QM</i>,<i>N</i>(alpha,beta) and <i>QM</i>,<i>N</i>(alpha,beta) are proposed. Furthermore, specific tight upper and lower bounds for <i>QM</i>(alpha,beta), previously reported in the literature, are extended for real values of <i>M</i>. 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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