Optimum Power Allocation for Parallel Gaussian Channels with Arbitrary Input Distributions

The sum mutual information of independent parallel Gaussian- noise channels is maximized, under an aggregate power constraint, by independent Gaussian inputs whose power is allocated according to the waterfilling policy. In practice, discrete signalling constellations with limited peak-to-average ratios (m-PSK, m-QAM, etc) are used in lieu of the ideal Gaussian signals. This paper gives the power allocation policy that maximizes the mutual information over parallel channels with arbitrary input constellations. The solution is a generalized form of waterfilling that we refer to as mercury/waterfilling. The relationship between mutual information of Gaussian channels and nonlinear minimum mean-square error proves key to solving the power allocation problem.