Q(a,b) 3
`
b
x # expa
x2 a2
2 b # I0(ax) dx
Ik(x)
1
2p
# 3
p
p
Aj # ejuBk # ex# sin u du
QM(a,b)
1
aM1
# 3
`
b
xM # exp cax2 a2
2 bd # IM1(ax) dx
f 0.5307027145
d 0.7265760135
c 0.7107068705
b 0.142248368
a 0.127414796
x
1
1 0.231641888 # y
(y 0) Q(y) > (ax bx2 cx3 dx4 fx5) # expay2
2 b
Q(x)
1 22px
# expax2
2 b
Q(x)
1 22px
# a1
1
x2b # expax2
2 b
Q(x)
1
2
# expa
x2
2 b
P(y z) QQz m
s R s2
508 APPENDIX A
with the modified zero-order Bessel function.
There has been extensive work done to compare bounds for the Marcum Q-function. We provide
a summary of the work presented in the reference section. We will first present the upper
bound, assuming (b a) and using the following Bessel relationship I0(ax) exp(ax) in the region
of x 0 [2]:
Next we will present the lower bound, assuming (b a) and using the following Bessel relationship
in the region of x b [2]:
For b a, the following holds true:
For large argument values, the following can be used for approximations:
erfc(x) >
exp(x2) 2px
I0(x) >
exp(x) 22px
Q(a,b) 1
1
2
# cexpa
(b a)2
2 b expa
(b a)2
2 bd
Q(a,b) exp c
(b a)2
2 d
Q(a,b) expa
a2 b2
2 b # I0(ab)
Q(a,b)
b
b a
# exp c
(b a)2
2 d
Q(a,b)
I0(ab) # b
exp(ab)
# Ap
2
# erfcab a 22 b
I0(x)
I0(b)
exp(b)
# exp(x)
x
Q(a,b) exp c
(b a)2
2 d
Q(a,b) expa
a2 b2
2 b # I0(ab) a # Ap
8
# erfcab a 22 b
Q(a,b)
b
b a
# exp c
(b a)2
2 d
Q(a,b)
I0(ab)
exp(ab)
# eexp c
(b a)2
2 d a # Ap
2
# erfcab a 22 b f
I0(x)
1p
# 3
p
0
exp(x # cos u)du
USEFUL FORMULAS 509
Some other useful variations to the above Marcum Q-function are given below.
Pages:
814
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