BER Upper Bound Expressions in Coded Two-Transmission Schemes With Arbitrarily Spaced Signal Constellations

The main contribution of this letter is the derivation of an upper bound BER expression, as a function of distances between signal points, for arbitrary constellations, in a generic, two-transmission scheme such as relaying, HARQ, or CoMP. The approach utilizes the product-state matrix, and thus the arbitrarily chosen constellations together with the encoders do not need to satisfy the quasi-regularity property that includes geometrical uniformity and symmetry. The channel fading is modeled using a Nakagami-
m
distribution, both with and without correlation between the two transmissions. We also allow for different path loss at each transmission. The results are valid for general coded schemes as long as a transfer function expression can be derived (convolutional codes and trellis coded modulation). The upper bounds are very accurate for BER values lower than
$10^{-2}$
for any chosen constellation.