On the optimal power allocation for additive color noise parallel channels with limited access constraint

In this paper, we consider an (N,K)-limited access system consisting of N parallel additive noise channels with spatial dependency, where the receiver starts to decode the information being transmitted when at least K out of N channel outputs are received. We investigate the optimal power allocation that maximizes the minimum mutual information among all possible cases of partial reception. A universal guideline is then obtained for a group of permutation-invariant channels, in which the system mutual information remains unchanged when permuting the parameters that characterize the partial reception and signal-to-noise power ratio (SNR) of channels, that a channel with less noise power should have larger SNR. When all N channels belong to a permutation-invariant group, we also have that the optimal power allocation problem can be transformed to an equivalent problem for K parallel channels without limited access constraint via a water-filling noise-power-redistribution process. The merit of this transformation can be more evidently seen when the channel input-noise pairs are reduced to be spatially independent with distributions scaled from a common random vector, for which the optimal power allocation solution can be simply obtained by a two-phase water-filling process.