In this paper, we have considered the optimization of the M-ary quadrature amplitude modulation (MQAM) constellation size to minimize the bit energy consumption under average bit error rate (BER) constraints. In the computation of the energy expenditure, the circuit, transmission, and retransmission energies are taken into account. A combined log-normal shadowing and Rayleigh fading model is employed to model the wireless channel. The link reliabilities and retransmission probabilities are determined through the outage probabilities under log-normal shadowing effects. Both single-hop and multi-hop transmissions are considered. Through numerical results, the optimal constellation sizes are identified. Several interesting observations with practical implications are made. It is seen that while large constellations are preferred at small transmission distances, constellation size should be decreased as the distance increases. Similar trends are observed in both fixed and variable transmit power scenarios. We have noted that variable power schemes can attain higher energy-efficiencies. The analysis of energy-efficient modulation design is also conducted in multi-hop linear networks. In this case, the modulation size and routing paths are jointly optimized, and the analysis of both the bit energy and delay experienced in the linear network is provided.
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