Pairwise error probability is the error probability that for a transmitted signal (
X
) its corresponding but distorted version (
X
^
) will be received. This type of probability is called ″pair-wise error probability″ because the probability exists with a pair of signal vectors in a signal constellation. It's mainly used in communication systems.
In general, the received signal is a distorted version of the transmitted signal. Thus, we introduce the symbol error probability, which is the probability
P
(
e
)
that the demodulator will make a wrong estimation
(
X
^
)
of the transmitted symbol
(
X
)
based on the received symbol, which is defined as follows:
P
(
e
)
≜
1
M
∑
x
P
(
X
≠
X
^
|
X
)
where M is the size of signal constellation.
The pairwise error probability
P
(
X
→
X
^
)
is defined as the probability that, when
X
is transmitted,
X
^
is received.
P
(
e
|
X
)
can be expressed as the probability that at least one
X
^
≠
X
is closer than
X
to
Y
.
Using the upper bound to the probability of a union of events, it can be written:
P
(
e
|
X
)
≤
∑
X
^
≠
X
P
(
X
→
X
^
)
Finally:
P
(
e
)
=
1
M
∑
X
∈
S
P
(
e
|
X
)
≤
1
M
∑
X
∈
S
∑
X
^
≠
X
P
(
X
→
X
^
)
For the simple case of the additive white Gaussian noise (AWGN) channel:
Y
=
X
+
Z
,
Z
i
∼
N
(
0
,
N
0
2
I
n
)
The PEP can be computed in closed form as follows:
P
(
X
→
X
^
)
=
P
(
|
|
Y
−
X
^
|
|
2
<
|
|
Y
−
X
|
|
2
|
X
)
=
P
(
|
|
(
X
+
Z
)
−
X
^
|
|
2
<
|
|
(
X
+
Z
)
−
X
|
|
2
)
=
P
(
|
|
(
X
−
X
^
)
+
Z
|
|
2
<
|
|
Z
|
|
2
)
=
P
(
|
|
X
−
X
^
|
|
2
+
|
|
Z
|
|
2
+
2
(
Z
,
X
−
X
^
)
<
|
|
Z
|
|
2
)
=
P
(
2
(
Z
,
X
−
X
^
)
<
−
|
|
X
−
X
^
|
|
2
)
=
P
(
(
Z
,
X
−
X
^
)
<
−
|
|
X
−
X
^
|
|
2
/
2
)
(
Z
,
X
−
X
^
)
is a Gaussian random variable with mean 0 and variance
N
0
|
|
X
−
X
^
|
|
2
/
2
.
For a zero mean, variance
σ
2
=
1
Gaussian random variable:
P
(
X
>
x
)
=
Q
(
x
)
=
1
2
π
∫
x
+
∞
e
−
t
2
2
d
t
Hence,
P
(
X
→
X
^
)
=
Q
(
|
|
X
−
X
^
|
|
2
2
N
0
|
|
X
−
X
^
|
|
2
2
)
=
Q
(
|
|
X
−
X
^
|
|
2
2
.
2
N
0
|
|
X
−
X
^
|
|
2
)
=
Q
(
|
|
X
−
X
^
|
|
2
N
0
)
