This thesis aims to theoretically study a modern linear transceiver design strategy, namely interference alignment, in wireless networks. We consider an interference channel whereby each transmitter and receiver are equipped with multiple antennas. The basic problem is to design optimal linear transceivers (or beamformers) that can maximize the system throughput. The recent work [1] suggests that optimal beamformers should maximize the total degrees of freedom through the interference alignment equations. In this thesis, we first state the interference alignment equations and study the computational complexity of solving these equations. In particular, we prove that the problem of maximizing the total degrees of freedom for a given interference channel is NP-hard. Moreover, it is shown that even checking the achievability of a given tuple of degrees of freedom is NP-hard when each receiver is equipped with at least three antennas. Interestingly, the same problem becomes polynomial time solvable when each transmit/receive node is equipped with no more than two antennas. The second part of this thesis answers an open theoretical question about interference alignment on generic channels: What degrees of freedom tuples (d1, d2, ..., dK) are achievable through linear interference alignment for generic channels? We partially answer this question by establishing a general condition that must be satisfied by any degrees of freedom tuple (d1, d2, ..., dK) achievable through linear interference alignment. For a symmetric system with dk = d for all k, this condition implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than K(M + N)/(K + 1), where M and N are the number of transmit and receive antennas, respectively. We also show that this bound is tight when the number of antennas at each transceiver is divisible by the number of data streams.
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