Pairwise error probability

Pairwise error probability is the error probability that for a transmitted signal ($X$) its corresponding but distorted version ($\hat{X}$) 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 $(\hat{X})$ of the transmitted symbol $(X)$ based on the received symbol, which is defined as follows:

$P(e)\triangleq \frac{1}{M}\sum _x\mathbb{P}(X\ne \hat{X}|X)$

where M is the size of signal constellation.

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

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

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

$P(e|X)\le \sum _{\hat{X}\ne X}P(X\to \hat{X})$

Finally:

$P(e)=\frac{1}{M}\sum _{X\in S}P(e|X)\le \frac{1}{M}\sum _{X\in S}\sum _{\hat{X}\ne X}P(X\to \hat{X})$

Closed form computation

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

$Y=X+Z,Z_i\sim 𝒩(0,\frac{N_0}{2}{I}_n)$

The PEP can be computed in closed form as follows:

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

$(Z,X-\hat{X})$ is a Gaussian random variable with mean 0 and variance $N_0||X-\hat{X}|{|}^2/2$.

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

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

Hence,

$\begin{array}{cc}P(X\to \hat{X})& =Q\left(\frac{\frac{||X-\hat{X}|{|}^2}{2}}{\sqrt{\frac{N_0||X-\hat{X}|{|}^2}{2}}}\right)=Q(\frac{||X-\hat{X}|{|}^2}{2}.\sqrt{\frac{2}{N_0||X-\hat{X}|{|}^2}})\\ & =Q\left(\frac{||X-\hat{X}||}{\sqrt{2N_0}}\right)\end{array}$

See also

• Signal processing
• Telecommunication
• Electrical engineering
• Random variable