Optimal Power Allocation for Downstream xDSL With Per-Modem Total Power Constraints: Broadcast Channel Optimal Spectrum Balancing (BC-OSB)

The duality between multiple-input multiple-output (MIMO) multiple-access channels (MAC) and MIMO broadcast channels (BCs) has been established under a total power constraint. The same set of rates for MAC can be achieved in BC exploiting the MAC-BC duality formulas while preserving the total power constraint. In this paper, we describe the BC optimal power allocation applying this duality in a downstream x-digital subscriber lines (xDSLs) context under a total power constraint for all modems over all tones. Then, a new algorithm called BC-optimal spectrum balancing (BC-OSB) is devised for a more realistic power allocation under per-modem total power constraints. The capacity region of the primal BC problem under per-modem total power constraints is found by the dual optimization problem for the BC under per-modem total power constraints which can be rewritten as a dual optimization problem in the MAC by means of a precoder matrix based on the Lagrange multipliers. We show that the duality gap between the two problems is zero. The multiuser power allocation problem has been solved for interference channels and MAC using the OSB algorithm. In this paper we solve the problem of multiuser power allocation for the BC case using the OSB algorithm as well and we derive a computational efficient algorithm that will be referred to as BC-OSB. Simulation results are provided for two VDSL2 scenarios: the first one with differential-mode (DM) transmission only and the second one with both DM and phantom-mode (PM) transmissions.