On the Error Probability of Cross-QAM With MRC Reception Over Generalized $\eta$-$\mu$ Fading Channels

The closed-form expressions for the exact average symbol error probability (SEP) and bit error probability (BEP) of arbitrary $M$-ary cross quadrature amplitude modulation (XQAM) signaling with maximal-ratio combining (MRC) diversity reception over independent but not necessarily identical $\eta$-$\mu$ fading channels are derived. The results are presented in terms of a finite (in proportion to $\sqrt{M}$) sum of Lauricella hypergeometric functions $F_{D}^{(n)}$, which can be numerically evaluated using its integral or converging series representation. For some special fading scenarios of interest, the formulas are expressed by some simpler functions, such as Gauss's single hypergeometric function $_{2}F_{1}$ and Apell's double hypergeometric function $F_{1}$, both of which are available functions in standard numerical software. For Rayleigh channels and some special cases of Nakagami-$m$ channels, the results are presented in the form of a finite sum of trigonometric functions. The analytical expressions show excellent agreement with the simulation results. Numerical evaluation with the proposed expressions reveals that cross-QAM can obtain at least 1.1-dB gain, compared with rectangular QAM when SEP $<$ 0.3.