Appendix B: Gaussian Q Function

B.1 Gaussian Q‐Function

The Gaussian Q‐function, which is defined by

represents the area under the tail (between x and ∞) of a zero‐mean and unit variance Gaussian pdf fZ(z) (see Figure B.1). Since the area under a pdf is equal to unity, Q(−∞) = 1 and Q(∞) = 0. Owing to the symmetry of the Gaussian pdf with respect to the origin, Q(0) = 1/2 and

(B.2)images
Graph depicting Gaussian Q-Function as the Q(x) curve (solid line) and the Gaussian normal Pdf as the fx(x) curve (dash line) with zero-mean and unity variance.

Figure B.1 Gaussian Q‐Function and the Gaussian (Normal) Pdf with Zero‐Mean and Unity Variance.

The Craig’s definition of the Q‐function [1] can easily be derived from (B.1) as follows:

If we make a transformation of Cartesian coordinates to polar coordinates as images and inserting images into (B.3) (see Figure B.2), one gets

(B.4)images

Figure B.2 Coordinate Transformation For Deriving the Craig’s Formula For ...

Get Digital Communications now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.