On the Number of Interference Alignment Solutions for the K-User MIMO Channel with Constant Coefficients

In this paper, we study the number of different interference alignment (IA) solutions in a K-user multiple-input multiple-output (MIMO) interference channel, when the alignment is performed via beamforming and no symbol extensions are allowed. We focus on the case where the number of IA equations matches the number of variables. In this situation, the number of IA solutions is finite and constant for any channel realization out of a zero-measure set and, as we prove in the paper, it is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of alignment solutions is the scaled average of the determinant of a certain Hermitian matrix related to the geometry of the problem. Interestingly, while the value of this determinant at an arbitrary point can be used to check the feasibility of the IA problem, its average (properly scaled) gives the number of solutions. For single-beam systems the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented. Nonetheless, our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas and streams per user.Comment: Submitted to IEEE Transactions on Information Theor

Interference alignment using finite and dependent channel extensions: the single beam case

Vector space interference alignment (IA) is known to achieve high degrees of freedom (DoF) with infinite independent channel extensions, but its performance is largely unknown for a finite number of possibly dependent channel extensions. In this paper, we consider a $K$-user $M_t \times M_r$ MIMO interference channel (IC) with arbitrary number of channel extensions $T$ and arbitrary channel diversity order $L$ (i.e., each channel matrix is a generic linear combination of $L$ fixed basis matrices). We study the maximum DoF achievable via vector space IA in the single beam case (i.e. each user sends one data stream). We prove that the total number of users $K$ that can communicate interference-free using linear transceivers is upper bounded by $NL+N^2/4$, where $N = \min\{M_tT, M_rT \}$. An immediate consequence of this upper bound is that for a SISO IC the DoF in the single beam case is no more than $\min\left\{\sqrt{ 5K/4}, L + T/4\right\}$. When the channel extensions are independent, i.e. $L$ achieves the maximum $M_r M_t T$, we show that this maximum DoF lies in $[M_r+M_t-1, M_r+M_t]$ regardless of $T$. Unlike the well-studied constant MIMO IC case, the main difficulty is how to deal with a hybrid system of equations (zero-forcing condition) and inequalities (full rank condition). Our approach combines algebraic tools that deal with equations with an induction analysis that indirectly considers the inequalities.Comment: 43 pages. Revised version; title changed. A shorter version (without proofs for simple cases) accepted by IEEE Trans. on Info. Theor