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The design of the precoder the maximizes the mutual information in linearvector Gaussian channels with an arbitrary input distribution is studied.Precisely, the precoder optimal left singular vectors and singular values arederived. The characterization of the right singular vectors is left, ingeneral, as an open problem whose computational complexity is then studied inthree cases: Gaussian signaling, low SNR, and high SNR. For the Gaussiansignaling case and the low SNR regime, the dependence of the mutual informationon the right singular vectors vanishes, making the optimal precoder designproblem easy to solve. In the high SNR regime, however, the dependence on theright singular vectors cannot be avoided and we show the difficulty ofcomputing the optimal precoder through an NP-hardness analysis.

On optimal precoding in linear vector Gaussian channels with arbitrary input distribution