Consider a MIMO interference channel whereby each transmitter and receiverare equipped with multiple antennas. The basic problem is to design optimallinear transceivers (or beamformers) that can maximize system throughput. Therecent work [1] suggests that optimal beamformers should maximize the totaldegrees of freedom and achieve interference alignment in high SNR. In thispaper we first consider the interference alignment problem in spatial domainand prove that the problem of maximizing the total degrees of freedom for agiven MIMO interference channel is NP-hard. Furthermore, we show that evenchecking the achievability of a given tuple of degrees of freedom for allreceivers is NP-hard when each receiver is equipped with at least threeantennas. Interestingly, the same problem becomes polynomial time solvable wheneach transmit/receive node is equipped with no more than two antennas. Finally,we propose a distributed algorithm for transmit covariance matrix design, whileassuming each receiver uses a linear MMSE beamformer. The simulation resultsshow that the proposed algorithm outperforms the existing interferencealignment algorithms in terms of system throughput.
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