What Is Pairwise Error Probability?

Pairwise error probability (PEP) is a fundamental metric in digital communications theory that measures the probability of a receiver incorrectly deciding in favor of one transmitted signal when a different signal was actually sent. Formally, the PEP between two codewords or signal points X and X' is the probability that the maximum-likelihood (or minimum-distance) decoder selects X' when X was transmitted. This pairwise quantity serves as a building block for analyzing the total error performance of communication systems: by bounding the overall error probability as a sum of PEP contributions across all codeword pairs (the union bound), engineers can derive design criteria for codes that minimize errors under additive noise and fading conditions.

PEP analysis is particularly central to the design and evaluation of space-time codes for multiple-antenna (MIMO) wireless systems, where the interaction between transmit diversity, channel fading, and code structure is captured precisely through the pairwise error probability framework. The technique draws on probability theory, linear algebra, and information theory, and forms a bridge between abstract code properties and measurable system performance.

Definition and Computation

For a system transmitting over an additive white Gaussian noise (AWGN) channel, the PEP between two signal points can be computed in closed form using the Q-function of the Euclidean distance between the signals scaled by the channel signal-to-noise ratio (SNR). In fading channels, the computation is more involved because the channel coefficients are random, requiring expectation of the conditional PEP over the fading distribution. For Rayleigh fading channels, this expectation can often be expressed in closed form through the moment generating function (MGF) approach, which yields tractable expressions for systems with BPSK and QPSK modulation. The IEEE paper on exact pairwise error probability of space-time codes derives closed-form PEP expressions for spatially and temporally correlated Rayleigh fading, providing tools applicable to practical MIMO configurations.

Union Bound and Performance Analysis

Because exhaustive computation of all codeword-pair error events is generally impractical, the union bound approximates the word error probability as the sum over all pairwise error probabilities. At high SNR, this bound becomes tight and reveals which codeword pairs dominate error performance. In fading channels, the PEP decreases with SNR at a rate determined by two key parameters: the diversity order (the exponent on SNR in the probability's inverse relationship, determining the slope of the error curve on a log-log scale) and the coding gain (a multiplicative factor that shifts the curve left or right). Analysis using pairwise error bounds is used extensively in spatial interference cancellation systems, as surveyed in IEEE research on spatial interference cancellation and PEP analysis, which examines how antenna array processing affects the PEP for competing streams.

Space-Time Codes and the Diversity-Multiplexing Tradeoff

In MIMO systems, space-time codes transmit structured signals across multiple antennas to exploit spatial diversity and improve error probability. The design criteria for such codes are derived directly from PEP analysis: the rank criterion requires that the difference matrix between any two distinct codewords have full rank, and the determinant criterion requires maximizing the minimum determinant of that matrix over all codeword pairs. The Alamouti scheme, the simplest full-rate orthogonal space-time block code for two transmit antennas, achieves second-order diversity as confirmed by its PEP decay at SNR^(-2). Extensions to distributed relay networks introduce additional complications: the Springer analysis of distributed space-time coding using the Alamouti scheme in wireless relay networks derives PEP expressions accounting for both hop-by-hop fading and inter-relay interference.

Applications

Pairwise error probability has applications in a range of fields, including:

• MIMO and massive MIMO wireless system design and code optimization
• Space-time block and trellis code performance evaluation
• Cooperative and relay communication network analysis
• Coded modulation scheme comparison under fading channel models
• Iterative detection and decoding convergence analysis