Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

B.19 Q-FUNCTION AND ERROR FUNCTIONS

Definition: Q-Function The Q-function is defined as an area under the standard Gaussian pdf:

(B.101)

for with , , Q(0) = 0.5, and

(B.102)

The Q-function is used to express a definite integral of a Gaussian pdf for which no closed form exists. The cdf of the standard Gaussian random variable can be written as

(B.103)

A plot of the Q-function is shown in Figure B.16(a). (Note that the Q-function is not a special case of the Marcum Q-function described later.) Since a closed form does not exist for Q(x), some tabulated results are provided in Table B.4. However, it is often convenient to have a closed-form approximation for Q(x). It is relatively straightforward to show that for , Q(x) is upper bounded as follows:

Figure B.16 Functions based on the Gaussian ...