A recursive algorithm for the exact BER computation of generalized hierarchical QAM 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 were published on the BER performance of hierarchical QAM constellations. In this correspondence, we obtain exact and generic expressions in M for the BER of the generalized hierarchical M-PAM (pulse amplitude modulation) constellations over additive white Gaussian noise (AWGN) and fading channels. We also show how these expressions can be extended to generalized hierarchical M-QAM constellations (square and rectangular). 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.