  Recently, the duality between Multiple Input Multiple Output (MIMO) Multiple
Access Channels (MAC) and MIMO Broadcast Channels (BC) 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 allo- cation
applying this duality in a downstream x-Digital Subscriber Lines (xDSL) 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
multi-user power allocation problem has been solved for interference channels
and MAC using the OSB algorithm. In this paper we solve the problem of
multi-user 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.
