Mercury/waterfilling: optimum power allocation with arbitrary input constellations

For parallel independent Gaussian-noise channels with an aggregate power constraint, independent Gaussian inputs whose powers are allocated according to the waterfilling policy maximize the sum mutual information. In practice, however, discrete signalling constellations such as m-PSK or m-QAM are used in lieu of the ideal Gaussian signals. This paper gives the power allocation policy, referred to as mercury/waterfilling, that maximizes the sum mutual information over parallel channels with arbitrary input constellations