Cryptography Reference
In-Depth Information
observation does not fall between two thresholds.
TYPE 1: Probabilities that the observation is higher or is lower than a threshold
= r
A
j
1)
A
Pe
(
M−
1)
=(
M
−
1)
A
|
A
j
=(
M
−
2)
A
2
|
=Pr
r<
(
M
1)
A
−
A
j
=(
M
−
= r
A
j
1)
A
Pe
−
(
M−
1)
=
−
(
M
−
1)
A
|
A
j
=
−
(
M
−
2)
A
2
|
=Pr
r>
1)
A
−
(
M
−
A
j
=
−
(
M
−
TYPE 2: Probabilities that the observation does not fall between two thresholds
Pe
2
j−
1
−M
=Pr
A
j
M
)
A
=(2
j
−
1
−
M
)
A
|
A
j
=(2
j
−
1
−
M
)
A
2
M
)
A
2
|
Pr
(2
j
Pe
2
j−
1
−M
=1
−
−
2
−
<r<
(2
j
−
}
Observation
r
is Gaussian conditionally to a realization of the amplitude
A
j
,
with mean
A
j
=(2
j
−
1
−
M
)
A
A
j
T/
2
and variance
N
0
/
2
. The conditional probabilities have
the expressions:
±
A
2
T
2
N
0
1
2
erfc
Pe
(
M−
1)
=
Pe
−
(
M−
1)
=
A
2
T
2
N
0
where the complementary error function is always defined by:
Pe
(2
j−
1
−M
)
=
erfc
+
∞
2
√
π
u
2
)
du
erfc
(
x
)=1
−
erf
(
x
)=
exp(
−
x
To calculate the mean error probability on the groups of data, we have two
conditional probabilities of type 1, and
(
M
−
2)
conditional probabilities of type
2.
A
2
T
2
N
0
Pes
=
M
−
1
erfc
M
(
M
2
−
1)
3
Introducing the average energy
E
s
=
A
2
T
2
received by group of data, the
mean error probability is again equal to:
erfc
Pes
=
M
−
1
3
M
2
E
s
N
0
M
−
1
