Cauchy–Schwarz bound on the generalized Marcum Q‐function with applications

Abstract

The Cauchy-Schwarz bounding technique is used to derive useful bounds on the generalized Marcum Q-function and its complement. Three new exponential-type bounds on Q M (˛,ˇ) are derived, and these are found to be tight and useful for a number of applications of interest. One such example is the derivation of an upper bound on the average symbol error rate probability for noncoherent and differentially coherent communication systems over generalized fading channels. It is shown that these exponential-type bounds are considerably tighter than the Chernoff bound (Rappaport SS. IEEE Trans. on Information Theory 1971; 17: 497-498) counterpart. Numerical results also reveal that the tightness of one of the exponential-type bounds is comparable to the bound obtained in Simon and Alouini (IEEE Trans. on Communications 2000: 359-366), while another is found to be superior than that obtained in Simon and Alouini over a wide range of arguments.

Computing and Bounding the Generalized Marcum Q-Function via a Geometric Approach

“…We also prove that Q m+0.5 (a, b) and Q m−0.5 (a, b) are, respectively, upper and lower bounds on Q m (a, b), irrespective of whether n is even or odd. Thus, for the case of even n, or equivalently integer m, Q m (a, b) can be bounded by Q m+0.5 (a, b) and Q m−0.5 (a, b) which can be evaluated using our new representation for odd n. Our numerical results show that these new bounds are very tight, much tighter than the existing bounds available in the literature [5], [6] in most cases, and their average also approximates Q m (a, b) very well.…”

Tight bounds for the generalized Marcum Q-function

“…Since, the precise computation of the Marcum Q -function, generalized Marcum Q -function, respectively is quite difficult, in the last few decades many engineers, statisticians and mathematicians established approximation formulas and bounds for the function b → Q ν (a, b). For more details on approximations, lower and upper bounds we refer to the most recent papers [2,5,7,9,12,13,20,21,27,28] and to the references therein. In this field ν is the number of independent samples of the output of a square-law detector, and hence in most of the papers the authors deduce lower and upper bounds for the generalized Marcum Q -function with order ν integer.…”

Generic Exponential Bounds on the Generalized Marcum Q-Function via the Geometric Approach

“…We now show some numerical results, and compare the tightness of our new bounds on Q m (a, b) in Section II with the tightness of the existing exponential bounds in [5], [6].…”

“…(18)]. Our new bounds include the generic exponential bounds GUBm1-KL in (5) and GLBm1-KL in (6). For simplicity, we just choose equispaced points for the parameter θ i in GUBm1-KL and GLBm1-KL, i.e., θ i = iπ/N .…”

“…To further simplify analytical results, some simple exponential bounds, which involve only the exponential function, have been given in [5], [6]. Some I kbounds, which involve I 0 (·) or {I k (·)} m−1 k=0 , have also been given in [6], [7], but these I k -bounds are too complicated to use in further analytical manipulations of Q m (a, b), such as in averaging Q m (a, b) over a fading distribution. All these existing results were obtained by using a mathematical approach.…”