Marcum Q-function

Abstract: Some integrals are presented that can be expressed in terms of the Q_M function, which is defined as \begin{equation} Q_M(a,b) = \int_b^{\infty} dx x(x/a)^{M-1} \exp (- \frac{x^2 + a^2}{2}) I_{M-1}(ax), \end{equation} where I_{M-1} is the modified Bessel function of order M-1 . Some integrals of the Q_M function are also evaluated.

Abstract: A highly reliable, accurate, and efficient method of calculating the probability of detection, P/sub N/(X,Y), for N incoherently integrated samples, where X is the constant received signal-to-noise ratio of a single pulse and Y is the normalized threshold level, is presented. The useful range of parameters easily exceeds most needs. On a VAX/11 computer with double precision calculations, better than 13-place absolute accuracy is normally achieved. There is a gradual loss of accuracy with increasing parameter values. For example, for N=10/sup 9/, and with both NX and Y near 10/sup 7/, the accuracy can drop to ten places. The function P/sub N/(X,Y) can be equated to the generalized Marcum Q-function, Q/sub m/( alpha , beta ). The corresponding limits on alpha and beta are roughly 4500 for the 13-place accuracy and 60000 for ultimate (INTEGER*4) limit. >

Abstract: 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\ura ?. 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.

Abstract: Strict upper and lower bounds of exponential-type are derived for the generalized (mth order) Marcum Q-function which enable simple evaluation of a tight upper bound on the average bit-error probability performance of a wide class of noncoherent and differentially coherent communication systems operating over generalized fading channels. For the case of frequency selective fading with arbitrary statistics per independent fading path, the resulting upper hound on performance is expressed in the form of a product of moment generating functions of the instantaneous power random variables that characterize these paths.