Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers, Scientists and Mathematicians
On the monotonicity of the generalized Marcum and Nuttall Q-functions
IEEE Transactions on Information Theory
IEEE Transactions on Wireless Communications
Supertight algebraic bounds on the Gaussian Q-function
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
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We present a new family of tight lower and upper bounds on the Gaussian Q-function Q(x). It is first shown that, for any x, the integrand ϕ(θ x) of the Craig representation of Q(x) can be partitioned into a pair of complementary convex and concave segments. As a consequence of this property, integrals of ϕ(θ x) over arbitrary intervals within its convex region can be lower-bounded by Jensen's inequality and upper-bounded by Cotes' quadrature rule, with the opposite occurring for the concave region ϕ(θ x). The combination of these complementary bounds yield a complete family of both lower and upper bounds on Q(x), which are expressed in terms of elementary transcendental functions and can be made arbitrarily tight by finer segmentation. A by-product of the method is that various other functions, such as the squared Gaussian Q-function Q2(x), the 2D joint Gaussian Q-function Q(x, y, ρ), and the generalized Marcum Q-function QM(x, y), can also be both upper and lower bounded with arbitrarily tightness, which to the best of our knowledge finds no precedence in the literature. Explicit examples of the latter applications are given.