This paper studies vector space interference alignment for the three-user MIMO interference channel with no time or frequency diversity. The main result is a characterization of the feasibility of interference alignment in the symmetric case where all transmitters have M antennas and all receivers have N antennas. If N >= M and all users desire d transmit dimensions, then alignment is feasible if and only if (2r+1)d = N. It turns out that, just as for the 3-user parallel interference channel BT09 , the length of alignment paths captures the essence of the problem. In fact, for each feasible value of M and N the maximum alignment path length dictates both the converse and achievability arguments. One of the implications of our feasibility criterion is that simply counting equations and comparing to the number of variables does not predict feasibility. Instead, a more careful investigation of the geometry of the alignment problem is required. The necessary condition obtained by counting equations is implied by our new feasibility criterion.
@cite_16 apply alignment on the signal scale to the @math -user @math MIMO IC. The converse arguments in that paper are obtained by forming a two-user interference channel with two users transmitting and decoding jointly; thus they obtain the inequality @math corresponding to @math in of the present paper.
@cite_16
1525037223 2139960897 2111785671 2028676993
Determining the feasibility conditions for vector space interference alignment in the K-user MIMO interference channel with constant channel coefficients has attracted much recent attention yet remains unsolved. The main result of this paper is restricted to the symmetric square case where all transmitters and receivers have N antennas, and each user desires d transmit dimensions. We prove that alignment is possible if and only if the number of antennas satisfies N>= d(K+1) 2. We also show a necessary condition for feasibility of alignment with arbitrary system parameters. An algebraic geometry approach is central to the results. Using interference alignment, it has been shown that the number of degrees of freedom in the interference channel scales linearly with the number of users. Unfortunately, closed-form solutions for interference alignment over constant-coefficient channels with more than 3 users are difficult to derive. This paper proposes an algorithm for interference alignment in the MIMO interference channel with an arbitrary number of users, antennas, or spatial streams. The algorithm is an alternating minimization over the precoding matrices at the transmitters and the interference subspaces at the receivers, and is proven to converge. Numerical results show how the algorithm is useful for simulation and can give insight into the limitations of interference alignment. We consider interference alignment in the partially connected K-user MIMO interference channel (IC). Conversely to the fully-connected case, we show that interference alignment can be achievable for an arbitrary number of users K in the network, while the per-user signaling dimension remains fixed, provided that the number of interference links per user is bounded. For this class of channels, which we denote by L-interfering K-user MIMO IC, we provide a criterion applicable to symmetric systems for the system of IA equations to be proper, according to the framework introduced earlier by Properness is a necessary condition for IA to be feasible. Interestingly, this criterion is independent from the number of users K. Furthermore, we propose an iterative algorithm to solve the alignment problem for this class of channels. Consider a MIMO 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 system throughput. The recent work [13] suggests that optimal beamformers should maximize the total degrees of freedom and achieve interference alignment in high SNR. In this paper we first consider the interference alignment problem in spatial domain and prove that the problem of maximizing the total degrees of freedom for a given MIMO interference channel is NP-hard. Furthermore, we show that even checking the achievability of a given tuple of degrees of freedom for all receivers is NP-hard when each receiver is equipped with at least three antennas. Moreover, in case where each transmitter and receiver use at most two antennas, the same problem is polynomial time solvable. Finally, we propose a distributed algorithm for transmit covariance matrix design, while assuming each receiver uses a linear MMSE beamformer. The simulation results show that the proposed algorithm outperforms the existing interference alignment algorithms in terms of system throughput.