Class of tight bounds on the Q-function with closed-form upper bound on relative error

摘要

In this paper, we propose a novel class of parametric bounds on the Q-function, which are lower bounds for 1 <= a x > x(t) = (a (a-1) / (3-a))(1/2), and upper bound for a = 3. We prove that the lower and upper bounds on the Q-function can have the same analytical form that is asymptotically equal, which is a unique feature of our class of tight bounds. For the novel class of bounds and for each particular bound from this class, we derive the beneficial closed-form expression for the upper bound on the relative error. By comparing the bound tightness for moderate and large argument values not only numerically, but also analytically, we demonstrate that our bounds are tighter compared with the previously reported bounds of similar analytical form complexity.