Cryptography Reference
In-Depth Information
Under this hypothesis, the error probability
Peb
(
ρ
)
is equal to :
Peb
(
ρ
)=
1
2
erfc
√
ρ
(2.175)
with:
L
E
b
N
0
(
α
i
)
2
ρ
=
n
=1
By averaging the probability
Peb
(
ρ
)
on the different realizations of the random
variable
ρ
, we obtain the bit error probability in presence of diversity of order
L
:
∞
Peb
=
Peb
(
ρ
)
p
(
ρ
)
dρ
0
The random variable
ρ
follows a
χ
2
law with probability density:
1
ρ
m
1)!
m
L
ρ
L−
1
exp
p
(
ρ
)=
−
(2.176)
(
L
−
where
m
is equal to:
N
0
E
(
α
j
)
2
After integration, the bit error probability is equal to:
Peb
=
1
E
b
m
=
L L−
1
L
1+
η
2
n
−
η
−
1+
n
n
(2.177)
2
n
=0
with:
L
=
E
b
/LN
0
1+
E
b
/LN
0
(
L
1+
n
)!
n
!(
L
−
−
1+
n
n
η
=
and
1)!
where
E
b
is the average total energy used to transmit an information bit (
E
b
=
LE
b
).
For a high signal to noise ratio, an approximation of the bit error probability
Peb
is given by:
−
2
L
L
E
b
LN
0
1
4
E
b
/LN
0
−
1
Peb
≈
pour
>>
1
(2.178)
L
In the presence of diversity, the bit error probability
Peb
decreases following the
inverse of the signal to noise ratio to the power of
L
.