Energy-efficient wireless communication is important for wireless devices with a limited battery life and cannot be recharged. In this study, a bit allocation algorithm to minimise the total energy consumption for transmitting a bit successfully is proposed for a two-way orthogonal frequency-division multiplexing relay system, whilst considering the constraints of quality-of-service and total transmit power. Unlike existing bit allocation schemes, which maximise the energy efficiency (EE) by measuring ‘bits-per-Joule’ with fixed bidirectional total bit rates constraint and no power limitation, their scheme adapts the bidirectional total bit rates and their allocation on each subcarrier with a total transmit power constraint. To do so, they propose an idea to decompose the optimisation problem. The problem is solved in two general steps. The first step allocates the bit rates on each subcarrier when the total bit rate of each user is fixed. In the second step, the Lagrangian multipliers are used as the optimisation variants, and the dimension of the variant optimisation is reduced from 2N to 2, where N is the number of subcarriers. They also prove that the optimal point is on the bounds of the feasible region, thus the optimal solution could be searched through the bounds.ud
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