BER computation of 4/M-QAM hierarchical constellations

Hierarchical constellations offer a different degree of protection to the transmitted messages according to their relative importance. As such they find interesting application in digital video broadcasting systems as well as wireless multimedia services. Although a great deal of attention has been devoted in the recent literature to the study of the bit error rate (BER) performance of uniform quadrature amplitude modulation (QAM) constellations, very few results have been published on the BER performance of hierarchical QAM constellations. Indeed the only available expressions are "leading-term" approximate BER expressions for 4/16-QAM and 4/64-QAM. In this paper, we obtain exact and generic expressions in M for the BER of the 4/M-QAM (square and rectangular) constellations over additive white Gaussian noise (AWGN) and fading channels. For the AWGN case, these expressions are in the form of a weighted sum of complementary error functions and are solely dependent on the constellation size M, the carrier-to-noise ratio, and a constellation parameter which controls the relative message importance. Because of their generic nature, these new expressions readily allow numerical evaluation for various cases of practical interest. In particular, numerical results show that the leading-term approximation gives significantly optimistic BER values at low carrier-to-noise ratio (CNR) in particular over Rayleigh fading channels but is quite accurate in the high CNR region.