TABLE OF CONTENTS In this section, we will present several useful formulas typically used in digital communications system design. Characterizing the system performance typically involves evaluating integrals that contain the following forms [1 10].
The most popular integral is called the Gaussian Q function and can be written as
It has the following properties:
Another representation is given as
This is related to the error function erf( x) and complementary error function erfc( x) as shown below. Recall erf (0) = 0 and erf( ?) = 1.
Consider a normal variable y with mean m and variance ? 2; then the following relationship holds true:
A well-known upper bound is the Chernoff bound.
In addition, we have the following lower and upper bounds:
Various approximations exist for the Q- function; below we list one to assist in obtaining numerical values. The reader should consult the references for other interesting expressions.
along with the following definitions:
Often it is beneficial to have simple bounds in order to gain instant feedback and insight into performance. The Marcum Q-function is generally found in the analysis of communication systems. The generalized Marcum Q-function is defined by the following integral [1]:
with the modified Bessel function of the kth order expressed by the following integral:
The special case of the generalized Marcum Q-function is for the M = 1 case.
with the modified zero-order Bessel function.
There has been extensive work done to compare bounds for the Marcum Q-function. We...
