Recall erf (0) 0 and .
erfc (x)
2 1p
# 3
`
x
exp(t2) dt 1 erf (x)
erf (x)
2 1
p
# 3
x
0
exp(t2) dt
(x 0) Q(x)
2
# erfca x 12b
1
2
# c1 erf a x 12bd
erf (`) 1
(x 0) Q(x)
1p 3
p/2
0
exp c
x2
2 # sin 2u ddu
Q(x) 1 Q(x)
Q(0)
1
2
lim
xS`
Q(x) 1
lim
xS`
Q(x) 0
Q(x) 3
`
x
1
2p
# e
y2
2 dy
507
Consider a normal variable y with mean m and variance ; 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.
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