Pairwise error probability

Probability theory
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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.^[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 $P(e)$ 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

References

1 2 Stüber, Gordon L. Principles of mobile communication (3rd ed.). New York: Springer. p. 281. ISBN 1461403642.

Further reading

Prasad, 5th IEEE International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC '94) The Hague, the Netherlands, September 18–22, 1994 ; ICCC Regional Meeting on Wireless Computer Networks (WCN), the Hague, the Netherlands, September 21–23, 1994 ; edited by Jos H. Weber, Jens C. Arnbak, and Ramjee (1994). Wireless networks : catching the mobile future : proceedings. Amsterdam: IOS Press. pp. 564–575. ISBN 9051991932.
Simon, Marvin K.; Alouini, Mohamed-Slim (2005). Digital Communication over Fading Channels (2. ed.). Hoboken: John Wiley & Sons. ISBN 0471715239.

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