Introduction to Random Signals and Noise

Appendix F

The Q(.) and erfc Functions

The Q function is defined as

The function is used to evaluate the error probability of transmission systems that are disturbed by additive Gaussian noise. Some textbooks use a different function for that purpose, namely the complementary error function, abbreviated as erfc. This latter function is defined as

From Equations (F.1) and (F.2) it follows that the Q function is related to the erfc function as follows:

The integral in these equations cannot be solved analytically. A simple and accurate expression (error less than 0.27 %) is given by

Most modern mathematical software packages such as Matlab, Maple and Mathematica comprise the erfc function as a standard function. Both functions are presented graphically in Figures F.1 and F.2.

Figure F.1

Figure F.2

Introduction to Random Signals and Noise W. van Etten © 2005 John Wiley & Sons, Ltd