Pairwise error probability is the error probability that for a transmitted signal ( X ) its corresponding but distorted version ( 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.
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 ( X ^ ) of the transmitted symbol ( X ) based on the received symbol, which is defined as follows:
P ( e ) ≜ 1 M ∑ x P ( X ≠ X ^ | X ) where M is the size of signal constellation.
The pairwise error probability P ( X → X ^ ) is defined as the probability that, when X is transmitted, X ^ is received.
P ( e | X ) can be expressed as the probability that at least one
X ^ ≠ 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 ) ≤ ∑ X ^ ≠ X P ( X → X ^ ) Finally:
P ( e ) = 1 M ∑ X ∈ S P ( e | X ) ≤ 1 M ∑ X ∈ S ∑ X ^ ≠ X P ( X → X ^ ) For the simple case of the additive white Gaussian noise (AWGN) channel:
Y = X + Z , Z i ∼ N ( 0 , N 0 2 I n ) The PEP can be computed in closed form as follows:
P ( X → X ^ ) = P ( | | Y − X ^ | | 2 < | | Y − X | | 2 | X ) = P ( | | ( X + Z ) − X ^ | | 2 < | | ( X + Z ) − X | | 2 ) = P ( | | ( X − X ^ ) + Z | | 2 < | | Z | | 2 ) = P ( | | X − X ^ | | 2 + | | Z | | 2 + 2 ( Z , X − X ^ ) < | | Z | | 2 ) = P ( 2 ( Z , X − X ^ ) < − | | X − X ^ | | 2 ) = P ( ( Z , X − X ^ ) < − | | X − X ^ | | 2 / 2 ) ( Z , X − X ^ ) is a Gaussian random variable with mean 0 and variance N 0 | | X − X ^ | | 2 / 2 .
For a zero mean, variance σ 2 = 1 Gaussian random variable:
P ( X > x ) = Q ( x ) = 1 2 π ∫ x + ∞ e − t 2 2 d t Hence,
P ( X → X ^ ) = Q ( | | X − X ^ | | 2 2 N 0 | | X − X ^ | | 2 2 ) = Q ( | | X − X ^ | | 2 2 . 2 N 0 | | X − X ^ | | 2 ) = Q ( | | X − X ^ | | 2 N 0 ) 