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.
Almost all other work has focused on various heuristic algorithms, mainly iterative in nature (see @cite_17 , @cite_12 , and @cite_11 ). Some have proofs of convergence, but performance guarantees are not available. @cite_11 , @cite_5 study a refined version of the single-transmit dimension problem, where for the case that alignment is possible (as mentioned above, feasibility of alignment is known for the single-transmit case @math ) they attempt to choose a good solution among the many possible solutions. Papailiopoulos and Dimakis @cite_3 relax the problem of maximizing degrees of freedom to that of a constrained rank minimization and propose an iterative algorithm.
@cite_17 @cite_3 @cite_5 @cite_12 @cite_11
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We consider the joint optimization of beamformers and linear receivers in a MIMO interference network. Each transmitter transmits a single beam corresponding to a rank-one precoder. When the number of users K is greater than the number of antennas at each terminal N, the maximum degrees of freedom is achieved via spatial interference alignment. Interference alignment is feasible for up to K = 2N−1 users, in which case there is a finite number of solutions to the alignment conditions. This number of solutions increases rapidly with N, and the solutions depend only on the cross-channel coefficients (i. e., they are independent of the direct channels). To maximize the achievable sum rate at high SNRs we therefore wish to select an aligned solution which is best matched to the direct channels. We evaluate the performance of this scheme for large K and N, assuming that the solution is the best out of a random subset of aligned solutions. We then compare numerically this performance with the performance of previously proposed numerical (e. g., forward-backward) techniques for optimizing beams, and a new technique which tracks the local optimum as the SNR is incrementally increased, similar to a homotopy method for improving convergence properties. We observe that the incremental technique typically achieves better performance than the previously proposed methods. In this paper an improved iterative algorithm is proposed for distributed transmit beamforming to achieve carrier phase alignment at the receiver. The transmitters adjust their phase offsets based on one-bit feedback from the receiver in each time slot. The proposed algorithm has an improvement in the convergence speed of phase alignment compared to a previously proposed algorithm in the literature by exploiting one-bit feedback information more efficiently. Simulation results show that the improved algorithm on average has a 20 faster convergence speed. The minimum number of time slots required to achieve specified beamforming gains and the corresponding perturbation steps are obtained by Monte Carlo simulations. 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. We show that the maximization of the sum degrees-of-freedom for the static flat-fading multiple-input multiple-output (MIMO) interference channel is equivalent to a rank constrained rank minimization problem (RCRM), when the signal spaces span all available dimensions. The rank minimization corresponds to maximizing interference alignment (IA) so that interference spans the lowest dimensional subspace possible. The rank constraints account for the useful signal spaces spanning all available spatial dimensions. That way, we reformulate all IA requirements to requirements involving ranks. Then, we present a convex relaxation of the RCRM problem inspired by recent results in compressed sensing and low-rank matrix completion theory that rely on approximating rank with the nuclear norm. We show that the convex envelope of the sum of ranks of the interference matrices is the normalized sum of their corresponding nuclear norms and introduce tractable constraints that are asymptotically equivalent to the rank constraints for the initial problem. We also show that our heuristic relaxation can be tuned for the multi-cell interference channel. Furthermore, we experimentally show that in many cases the proposed algorithm attains perfect interference alignment and in some cases outperforms previous approaches for finding precoding and zero-forcing matrices for interference alignment.