# Pairwise error probability

Pairwise error probability is the error probability that for a transmitted signal (${\displaystyle X}$) its corresponding but distorted version (${\displaystyle {\widehat {X}}}$) 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.[1] It's mainly used in communication systems.[1]

## 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 ${\displaystyle P(e)}$ that the demodulator will make a wrong estimation ${\displaystyle ({\widehat {X}})}$ of the transmitted symbol ${\displaystyle (X)}$ based on the received symbol, which is defined as follows:

${\displaystyle P(e)\triangleq {\frac {1}{M}}\sum _{x}\mathbb {P} (X\neq {\widehat {X}}|X)}$

where M is the size of signal constellation.

The pairwise error probability ${\displaystyle P(X\to {\widehat {X}})}$ is defined as the probability that, when ${\displaystyle X}$ is transmitted, ${\displaystyle {\widehat {X}}}$ is received.

${\displaystyle P(e|X)}$ can be expressed as the probability that at least one ${\displaystyle {\widehat {X}}\neq X}$ is closer than ${\displaystyle X}$ to ${\displaystyle Y}$.

Using the upper bound to the probability of a union of events, it can be written:

${\displaystyle P(e|X)\leq \sum _{{\widehat {X}}\neq X}P(X\to {\widehat {X}})}$

Finally:

${\displaystyle P(e)={\tfrac {1}{M}}\sum _{X\in S}P(e|X)\leq {\tfrac {1}{M}}\sum _{X\in S}\sum _{{\widehat {X}}\neq X}P(X\to {\widehat {X}})}$

## Closed form computation

For the simple case of the additive white Gaussian noise (AWGN) channel:

${\displaystyle Y=X+Z,Z_{i}\sim {\mathcal {N}}(0,{\tfrac {N_{0}}{2}}I_{n})\,\!}$

The PEP can be computed in closed form as follows:

{\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}}}

${\displaystyle (Z,X-{\widehat {X}})}$ is a Gaussian random variable with mean 0 and variance ${\displaystyle N_{0}||X-{\widehat {X}}||^{2}/2}$.

For a zero mean, variance ${\displaystyle \sigma ^{2}=1}$ Gaussian random variable:

${\displaystyle P(X>x)=Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{+\infty }e^{-}{\tfrac {t^{2}}{2}}dt}$

Hence,

{\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}}}