BER computation of 4/M-QAM hierarchical constellations

• Published in: IEEE transactions on broadcasting Vol. 47; no. 3; pp. 228 - 239
• Main Authors: Vitthaladevuni, P.K., Alouini, M.-S.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2001; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Additive white noise; Applied sciences; Approximation; AWGN; Bit error rate; Broadcasting. Videocommunications. Audiovisual; Carriers; Channels; Constellations; Digital multimedia broadcasting; Digital video broadcasting; Exact sciences and technology; Fading; Genetic expression; Mathematical analysis; Mathematical models; Messages; Multimedia systems; Noise; Protection; Quadrature amplitude modulation; Studies; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Television; Transmission and modulation (techniques and equipments); Fading channels; Performance evaluation; Quadrature amplitude modulation; Multimedia systems; Bit error rate; Wireless telecommunication; Carrier to noise ratio; Numerical method; Digital transmission; AWGN channels; Rayleigh fading; Hierarchical system; Digital video broadcasting
• ISSN: 0018-9316; 1557-9611
• DOI: 10.1109/11.969372

Hierarchical constellations offer a different degree of protection to the transmitted messages according to their relative importance. As such they found interesting application in digital video broadcasting systems as well as wireless multimedia services. Although a great deal of attention has been devoted in the 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. Indeed the only available expressions "leading-term" approximate BER expressions for 4/16-QAM and 4/64-QAM. 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.