Lower and upper bounds for the generalized Marcum and Nuttall Q-functions

Monotonicity criteria are established for the generalized Marcum Q-function Q_M (alpha, beta) and the standard Nuttall Q-function Q_M,N (alpha, beta). Specifically, we present that Q_M (alpha, beta) is monotonically increasing with regard to its order M, for all ranges of the parameters alpha, beta, whereas Q_M,N (alpha, beta) possesses analogous monotonicity behavior with respect to M + N, under the assumptions of a ges 1 and constant difference M mnplus N ges 1. For the normalized Nuttall Q-function Q_M,N (alpha, beta), we also state the same monotonicity criterion without the necessity of restricting the range of the parameter alpha. By exploiting these results, we propose closed-form upper and lower bounds for the standard and normalized Nuttall Q-functions, which for the latter case seem to be very tight. Furthermore, concerning the generalized Marcum Q-function, specific tight upper and lower bounds, that have already been proposed in the literature for the case of integer M, are appropriately utilized in order to extend its validity over real values of M. The offered theoretical results can be efficiently applied in the study of digital communications over fading channels, the capacity analysis of multiple-input multiple-output (MIMO) channels and the decoding of turbo or low-density parity-check (LDPC) codes.