Pairwise error probability is the error probability that for a transmitted signal (
X
{\displaystyle X}
distorted version (
X
^
{\displaystyle {\widehat {X}}}
Expansion of the definition
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
)
{\displaystyle P(e)}
demodulator will make a wrong estimation
(
X
^
)
{\displaystyle ({\widehat {X}})}
(
X
)
{\displaystyle (X)}
P
(
e
)
≜
1
M
∑
x
P
(
X
≠
X
^
|
X
)
{\displaystyle P(e)\triangleq {\frac {1}{M}}\sum _{x}\mathbb {P} (X\neq {\widehat {X}}|X)}
where M is the size of signal constellation.
The pairwise error probability
P
(
X
→
X
^
)
{\displaystyle P(X\to {\widehat {X}})}
probability that, when
X
{\displaystyle X}
X
^
{\displaystyle {\widehat {X}}}
P
(
e
|
X
)
{\displaystyle P(e|X)}
X
^
≠
X
{\displaystyle {\widehat {X}}\neq X}
X
{\displaystyle X}
Y
{\displaystyle 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
^
)
{\displaystyle P(e|X)\leq \sum _{{\widehat {X}}\neq X}P(X\to {\widehat {X}})}
Finally:
P
(
e
)
=
1
M
∑
X
∈
S
P
(
e
|
X
)
≤
1
M
∑
X
∈
S
∑
X
^
≠
X
P
(
X
→
X
^
)
{\displaystyle P(e)={\tfrac {1}{M}}\sum _{X\in S}P(e|X)\leq {\tfrac {1}{M}}\sum _{X\in S}\sum _{{\widehat {X}}\neq X}P(X\to {\widehat {X}})}
Closed form computation
For the simple case of the additive white Gaussian noise (AWGN) channel:
Y
=
X
+
Z
,
Z
i
∼
N
(
0
,
N
0
2
I
n
)
{\displaystyle Y=X+Z,Z_{i}\sim {\mathcal {N}}(0,{\tfrac {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
)
{\displaystyle {\begin{aligned}P(X\to {\widehat {X}})&=\mathbb {P} (||Y-{\widehat {X}}||^{2}<||Y-X||^{2}|X)\\&=\mathbb {P} (||(X+Z)-{\widehat {X}}||^{2}<||(X+Z)-X||^{2})\\&=\mathbb {P} (||(X-{\widehat {X}})+Z||^{2}<||Z||^{2})\\&=\mathbb {P} (||X-{\widehat {X}}||^{2}+||Z||^{2}+2(Z,X-{\widehat {X}})<||Z||^{2})\\&=\mathbb {P} (2(Z,X-{\widehat {X}})<-||X-{\widehat {X}}||^{2})\\&=\mathbb {P} ((Z,X-{\widehat {X}})<-||X-{\widehat {X}}||^{2}/2)\end{aligned}}}
(
Z
,
X
−
X
^
)
{\displaystyle (Z,X-{\widehat {X}})}
random variable with mean 0 and variance
N
0
|
|
X
−
X
^
|
|
2
/
2
{\displaystyle N_{0}||X-{\widehat {X}}||^{2}/2}
For a zero mean, variance
σ
2
=
1
{\displaystyle \sigma ^{2}=1}
P
(
X
>
x
)
=
Q
(
x
)
=
1
2
π
∫
x
+
∞
e
−
t
2
2
d
t
{\displaystyle P(X>x)=Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{+\infty }e^{-}{\tfrac {t^{2}}{2}}dt}
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
)
{\displaystyle {\begin{aligned}P(X\to {\widehat {X}})&=Q{\bigg (}{\tfrac {\tfrac {||X-{\widehat {X}}||^{2}}{2}}{\sqrt {\tfrac {N_{0}||X-{\widehat {X}}||^{2}}{2}}}}{\bigg )}=Q{\bigg (}{\tfrac {||X-{\widehat {X}}||^{2}}{2}}.{\sqrt {\tfrac {2}{N_{0}||X-{\widehat {X}}||^{2}}}}{\bigg )}\\&=Q{\bigg (}{\tfrac {||X-{\widehat {X}}||}{\sqrt {2N_{0}}}}{\bigg )}\end{aligned}}}
See also
References
^ Stüber, Gordon L. (8 September 2011). Principles of mobile communication (3rd ed.). New York: Springer. p. 281. ISBN 978-1461403647
Further reading
Simon, Marvin K.; Alouini, Mohamed-Slim (2005). Digital Communication over Fading Channels (2. ed.). Hoboken: John Wiley & Sons. ISBN 0471715239 Categories :
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↑
) its corresponding but 
) 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.
that the 
of the transmitted symbol 
based on the received symbol, which is defined as follows:
is defined as the 
can be expressed as the probability that at least one 
is closer than 
.
is a Gaussian 
.
Gaussian random variable:
