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Error performance is considered as one of the most important performance measures, and deriving the closed-form expressions for efficient modulation techniques over generalized fading channels is important for future cellular systems. In this paper, the performance of a dual-hop amplify-and-forward multi-relay system with best relay selection is analyzed over independent and non-identically distributed (i.n.i.d.) Nakagami-m fading links with both integer and non-integer fading parameters. The impact of practical constraints of imperfect channel state information (CSI) and non-linear power amplifier (NLPA) at each of the relays are considered. Closed-form expressions for the outage probability are derived for both integer and non-integer fading parameters, and asymptotic analysis on the outage probability is performed to obtain the diversity order of the considered multi-relay system. Based on the cumulative distribution function approach, average symbol error rate (ASER) expressions for general order hexagonal QAM, general order rectangular QAM, and 32-cross QAM schemes are also derived. The comparative analysis of ASER for various QAM schemes with different constellations is also illustrated. Furthermore, the impact of the number of relays, fading parameter, channel estimation error, and non-linear distortion on the system performance is also highlighted. Finally, the derived analytical results are validated through Monte-Carlo simulations. INDEX TERMS Nakagami-m, multi-relay, imperfect CSI, non-linear power amplifier (NLPA), hexagonal QAM (HQAM), rectangular QAM (RQAM), cross QAM (XQAM).

Communication system's performance is sensitive to bandwidth, power, cost etc. There have been various solutions to improve the performance, out of them, one of the fundamental solutions over the years is design of optimum modulation schemes. As the research on beyond 5G heats up, we survey and explore power and bandwidth efficient modulation schemes for the next generation communication systems in details. In the existing literature, initially square quadrature amplitude modulation (SQAM) was considered. However, only square constellations are not sufficient for varying channel conditions and rate requirements, thus, efficient odd power of 2 constellations were introduced. For odd power of 2 constellations, rectangular QAM (RQAM) is most commonly used. However, RQAM is not a good choice and modified cross QAM (XQAM) constellation is preferred which provides improved power efficiency over RQAM due to its energy efficient two dimensional (2D) structure. The increasing demand for high data-rates has further encouraged research towards more compact 2D constellations which leads to hexagonal lattice structure based hexagonal QAM (HQAM) constellations. In this work, various QAM constellations are discussed and detailed study of star QAM, XQAM, and HQAM is presented. Generation, peak and average energies, peak-to-average-power ratio, symbol-error-rate, decision boundaries, bit mapping, Gray code penalty, and bit-error-rate of star QAM, XQAM, and HQAM constellations for different constellation orders are presented. Finally, a comparative study of various QAM constellations is presented which justifies the supremacy of HQAM over other QAM constellations for various wireless communication systems and a potential modulation scheme for future standards.

In this work, the performance of a dual-hop variable gain amplify-and-forward (AF) mixed radio frequency (RF)/free space optics (FSO) system is analyzed in details. As variable gain AF relay network is considered, processing delay results in outdated channel state information (CSI) during amplification at the relay. For general applicability, the RF link is modeled with the generalized Nakagami-m fading. The FSO link is modeled with the Gamma-Gamma distribution which is affected with the atmospheric turbulence and pointing error impairments, and both the intensity modulation with direct detection (IM/DD) and heterodyne detection are employed at the FSO receiver. In this context, analytical expressions of outage probability and ergodic capacity are derived in terms of Meijer-G function and extended generalized bivariate Meijer-G function. For diversity order, asymptotic outage probability expression is also derived. Further, cumulative distribution function based generalized average symbol error rate expressions for various quadrature amplitude modulation (QAM) schemes such as hexagonal QAM, rectangular QAM, and cross QAM are derived in terms of Meijer-G function. Furthermore, a detailed comparative study of various modulation schemes is presented and the impact of pointing error, atmospheric turbulence, outdated CSI, and Nakagami-m parameter are highlighted on the system performance. Finally, all the analytical results are verified through Monte-Carlo simulations.