On the monotonicity, log-concavity, and tight bounds of the generalized marcum and nuttall Q-Functions

Yin Sun, Árpád Baricz, Shidong Zhou

Research output: Article

53 Citations (Scopus)


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 first 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 μ ∞. 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.

Original languageEnglish
Article number5429099
Pages (from-to)1166-1186
Number of pages21
JournalIEEE Transactions on Information Theory
Issue number3
Publication statusPublished - márc. 1 2010


ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

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