Pairwise error probability

Part of a series on statistics
Probability theory
Probability Axioms Determinism System Indeterminism Randomness
Probability space Sample space Event Collectively exhaustive events Elementary event Mutual exclusivity Outcome Singleton Experiment Bernoulli trial Probability distribution Bernoulli distribution Binomial distribution Exponential distribution Normal distribution Pareto distribution Poisson distribution Probability measure Random variable Bernoulli process Continuous or discrete Expected value Variance Markov chain Observed value Random walk Stochastic process
Complementary event Joint probability Marginal probability Conditional probability
Independence Conditional independence Law of total probability Law of large numbers Bayes' theorem Boole's inequality
Venn diagram Tree diagram
v t e

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]

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:

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

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,

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

• Signal processing
• Telecommunication
• Electrical engineering
• Random variable

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