The exact transition probability and bit error probability of two-dimensional signaling

A new signal-space partitioning method is proposed for the calculation of transition probabilities of arbitrary two-dimensional (2-D) signaling with polygonal decision regions. This enables straightforward formulation of exact bit error probability, symbol error probability, mutual information (MI), and error probabilities of individual bits in the constellation. Previously published symbol error rate (SER) results for additive white Gaussian noise (AWGN) and various fading channels, diversity reception, and imperfect channel estimation (ICE) can all be used in conjunction with the new signal-space partitioning method to obtain the desired performance measures. Numerical results of the bit error rates (BERs) of several 8-ary and 16-ary constellations in AWGN and in Rayleigh, Ricean, and Nakagami fading channels with diversity reception are presented. Pilot-symbol-assisted 2-D signaling is also studied. The proposed new approach is both exact and numerically efficient. It provides new insights into the effect of bit mapping, constellation parameter optimizations, unequal error protection (UEP) at different bit positions, and performance comparisons of different constellations.