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.
In exactly the same setting as the present paper, @cite_8 have independently proposed a similar achievable strategy for critical @math satisfying both and @math . @cite_8 is limited to critical values of @math and contains no converse arguments beyond the equation counting bound of @cite_0 and @cite_20 .
@cite_0 @cite_20 @cite_8
1525037223 1971328044 2949418501
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. Consider a K-user flat fading MIMO interference channel where the kth transmitter (or receiver) is equipped with Mk (respectively Nk) antennas. If an exponential (in K) number of generic channel extensions are used either across time or frequency, Cadambe and Jafar [1] showed that the total achievable degrees of freedom (DoF) can be maximized via interference alignment, resulting in a total DoF that grows linearly with A even if Mk and Nk are bounded. In this work we consider the case where no channel extension is allowed, and establish 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 Mk = M, Nk = N, 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). We also show that this bound is tight when the number of antennas at each transceiver is divisible by d, the number of data streams per user. In this paper, the 3-user multiple-input multiple-output Gaussian interference channel with M antennas at each transmitter and N antennas at each receiver is considered. It is assumed that the channel coefficients are constant and known to all transmitters and receivers. A novel scheme is presented that spans a new achievable degrees of freedom region. For some values of M and N, the proposed scheme achieve higher number of DoF than are currently achievable, while for other values it meets the best known upperbound. Simulation results are presented showing the superior performance of the proposed schemes to earlier approaches.