Water-filling is the limiting case of a general capacity maximization principle

The optimal power allocation for Gaussian vector channels subject to sum power constraints is achieved by the well known water-filling principle. In this correspondence, we show that the discontinuous water filling solution is obtained as the limiting case of p-norm bounds on the power covariance matrix as p tends to one. Directional derivatives are the main vehicle leading to this result. An easy graphical representation of the solution is derived by the level crossing points of simple power functions, which in the limit p = 1 gives a nice dual view of the classical representation.