  We investigate in this paper the optimal power allocation in an OFDM-SDMA
system when some users have minimum downlink transmission rate requirements. We
first solve the unconstrained power allocation problem for which we propose a
fast zero-finding technique that is guaranteed to find an optimal solution, and
an approximate solution that has lower complexity but is not guaranteed to
converge. For the more complex minimum rate constrained problem, we propose two
approximate algorithms. One is an iterative technique that finds an optimal
solution on the rate boundaries so that the solution is feasible, but not
necessarily optimal. The other is not iterative but cannot guarantee a feasible
solution. We present numerical results showing that the computation time for
the iterative heuristic is one order of magnitude faster than finding the exact
solution with a numerical solver, and the non-iterative technique is an
additional order of magnitude faster than the iterative heuristic. We also show
that in most cases, the amount of infeasibility with the non-iterative
technique is small enough that it could probably be used in practice.
