Pairwise error probability

Pairwise error probability is the error probability that for a transmitted signal its corresponding but distorted version 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 that the demodulator will make a wrong estimation of the transmitted symbol based on the received symbol, which is defined as follows:
where is the size of signal constellation.
The pairwise error probability is defined as the probability that, when is transmitted, is received.
Using the upper bound to the probability of a union of events, it can be written:
Finally:

Closed form computation

For the simple case of the additive white Gaussian noise channel:
The PEP can be computed in closed form as follows:
is a Gaussian random variable with mean 0 and variance.
For a zero mean, variance Gaussian random variable:
Hence,