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.{{cite book|last=Stüber|first=Gordon L.|title=Principles of mobile communication|date=8 September 2011 |publisher=Springer|location=New York|isbn=978-1461403647|pages=281|edition=3rd}} 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 {{math|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:

: $P(e|X)\le\sum_{\widehat{X}\neq X} P(X \to \widehat{X})$

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:

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

$(Z,X-\widehat{X})$ 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,

: $\begin{align}
P(X \to \widehat{X}) & =Q \bigg(\tfrac{\tfrac$

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{align}

Pairwise error probability

Expansion of the definition

Closed form computation

See also

References

Further reading