Pairwise error probability

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

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

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

where M is the size of signal constellation.

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

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

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

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

Finally:

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

Closed form computation

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

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

The PEP can be computed in closed form as follows:

${\begin{aligned}P(X\to {\widehat {X}})&=\mathbb {P} (||Y-{\widehat {X}}||^{2}<||Y-X||^{2}|X)\\&=\mathbb {P} (||(X+Z)-{\widehat {X}}||^{2}<||(X+Z)-X||^{2})\\&=\mathbb {P} (||(X-{\widehat {X}})+Z||^{2}<||Z||^{2})\\&=\mathbb {P} (||X-{\widehat {X}}||^{2}+||Z||^{2}+2(Z,X-{\widehat {X}})<||Z||^{2})\\&=\mathbb {P} (2(Z,X-{\widehat {X}})<-||X-{\widehat {X}}||^{2})\\&=\mathbb {P} ((Z,X-{\widehat {X}})<-||X-{\widehat {X}}||^{2}/2)\end{aligned}}$

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

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

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

Hence,

${\begin{aligned}P(X\to {\widehat {X}})&=Q{\bigg (}{\tfrac {\tfrac {||X-{\widehat {X}}||^{2}}{2}}{\sqrt {\tfrac {N_{0}||X-{\widehat {X}}||^{2}}{2}}}}{\bigg )}=Q{\bigg (}{\tfrac {||X-{\widehat {X}}||^{2}}{2}}.{\sqrt {\tfrac {2}{N_{0}||X-{\widehat {X}}||^{2}}}}{\bigg )}\\&=Q{\bigg (}{\tfrac {||X-{\widehat {X}}||}{\sqrt {2N_{0}}}}{\bigg )}\end{aligned}}$

See also

• Signal processing
• Telecommunication
• Electrical engineering
• Random variable

Further reading

• Simon, Marvin K.; Alouini, Mohamed-Slim (2005). Digital Communication over Fading Channels (2. ed.). Hoboken: John Wiley & Sons. ISBN 0471715239.