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:


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

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}$$