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Gaussian MIMO channel under total transmit and multiple interference powerconstraints (TPC and IPCs) is considered. A closed-form solution for itsoptimal transmit covariance matrix is obtained in the general case (up to dualvariables). A number of more explicit closed-form solutions are obtained insome special cases, including full-rank and rank-1 (beamforming) solutions,which differ significantly from the well-known water-filling solutions (e.g.signaling on the channel eigenmodes is not optimal anymore and the capacity canbe zero for non-zero transmit power). A whitening filter is shown to be animportant part of optimal precoding under interference constraints. Capacityscaling with transmit power is studied: its qualitative behaviour is determinedby a natural linear-algebraic structure induced by MIMO channels of multipleusers. A simple rank condition is given to characterize the cases wherespectrum sharing is possible. An interplay between the TPC and IPCs isinvestigated, including the transition from power-limited tointerference-limited regimes. A number of unusual properties of an optimalcovariance matrix under IPCs are pointed out and a bound on its rank isestablished. Partial null forming known in the adaptive antenna arrayliterature is shown to be optimal from the information-theoretic perspective aswell in some cases.

On The Capacity of Gaussian MIMO Channels Under Interference Constraints (full version)