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A two-way relay channel with independent parallel Gaussian channels between the relay and the two terminals is considered. Focusing on the decode-and-forward protocol, the second phase of the communication, in which the relay broadcasts the two messages to their respective receivers, is studied. Precisely, the problem of computing the power allocation among the parallel channels that maximizes the weighted sum rate assuming arbitrarily distributed channel inputs (such as m-QAM) is stated and shown to be convex. A numerical algorithm is provided to solve the problem for the general case and, for the particular cases of high and low power regimes, expressions for the optimal power allocation are derived in closed form.
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