Exact Symbol Error Probability of Cross-QAM in AWGN and Fading Channels

The exact symbol error probability (SEP) performance of M-ary cross quadrature amplitude modulation (QAM) in additive white Gaussian noise (AWGN) channel and fading channels, including Rayleigh, Nakagami-m, Rice, and Nakagami-q (Hoyt) channels, is analyzed. The obtained closed-form SEP expressions contain a finite (in proportion to root M) sum of single integrals with finite limits and an integrand composed of elementary (exponential, trigonometric, and/or power) functions, thus readily enabling numerical evaluation. Particularly, Gaussian Q-function is a special case of these integrals and is included in the SEP expressions. Simple and very precise approximations, which contain only Gaussian Q-function for AWGN channel and contain three terms of the single integrals mentioned above for fading channels, respectively, are also given. The analytical expressions show excellent agreement with the simulation results, and numerical evaluation with the proposed expressions reveals that cross QAM can obtain at least 1.1 dB gain compared to rectangular QAM when SEP < 0.3 in all the considered channels.