Arbitrarily Tight Upper and Lower Bounds on the Gaussian Q-Function and Related Functions

—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 ...

• Type: Conference
• Year: 2009
• Where: ICC
• Authors: Giuseppe Thadeu Freitas de Abreu
• Keywords: Communications, Complementary Convex, Gaussian Q-function, ICC 2009, Upper Bounds