On the Number of Interference Alignment Solutions for the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-User MIMO Channel With Constant Coefficients

Abstract

In this paper, we study the number of different interference alignment (IA) solutions in a Kuser 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. Ó. González and I. Santamaria are with the Communications Engineering Editorial Area Communications I. INTRODUCTION Interference alignment (IA) has received a lot of attention in recent years as a key technique to achieve the maximum degrees of freedom (DoF) of wireless networks in the presence of interference. Originally proposed in [1], [2], the basic idea of IA consists of designing the transmitted signals in such a way that the interference at each receiver falls within a lower-dimensional subspace, therefore leaving a subspace free of interference for the desired signal [3]. This idea has been applied in different forms (e.g., ergodic interference alignment [4], signal space alignment [1], or signal scale alignment [5], [6]), and adapted to various wireless networks such as interference networks [1], X channels [2], downlink broadcast channels in cellular communications [7] and, more recently, to two-hop relay-aided networks in the form of interference neutralization [8].In this paper we consider the linear IA problem (i.e., signal space alignment by means of linear beamforming) for the K-user multiple-input multiple-output (MIMO) interference channel with constant channel coefficients. Moreover, the MIMO channels are considered to be generic, without any particular structure, which happens, for instance, when the channel matrices have independent entries drawn from a continuous distribution. This setup has also been the preferred option for recent experimental studies on IA [9], [10], [11]. The feasibility of linear IA for MIMO interference networks, which amounts to study the solvability of a set of polynomial equations, has been an active research topic during the last years [12], [13], [14],[15], [16]. Combining algebraic geometry tools with differential topology, it has been rece...

Finding the number of feasible solutions for linear interference alignment problems

“…The main results of the paper are Theorems 1 and 2 below, which give integral expressions for the number of IA solutions when the system is tightly feasible (s = 0): this number is denoted as (π −1 1 (H 0 )) and is the same for all channel realizations out of some zero-measure set. Detailed proofs of the theorems can be found in [16].…”

“…As it can be observed, the estimate of the integral formula in Theorem 2 converges much faster than that of Theorem 1, thus allowing us to get smaller relative errors. More examples can be found in [16].…”

Low-Overhead Feedback Topology Design for the K-User MIMO Interference Alignment

“…Although six feedback structures have already been provided based on specific closed-form IA solutions in the field of feedback topology construction, it does not mean that the potential advantage of feedback topologies in the saving of CSI overhead has been completely revealed. In other words, it is possible to design a new feedback structure with less CSI overhead by finding an appropriate closed-form IA solution different from those used in the existing six feedback topologies, because multiple IA solutions exist in the K-user MIMO interference channel [15][16]. For this reason, in this paper, we concentrate on the CSI overhead reduction from the perspective of feedback topology design based on closed-form IA solution.…”

Statistical analysis of single-beam interference alignment schemes

“…In this work, we derive analytical approximate expressions for the per-user rates achievable by IA algorithms in singlebeam MIMO networks for a fixed channel realization. Instead of performing a large-system analysis in which the number of users, antennas, or streams is required to grow, we only require that the number of different IA solutions for the given scenario is sufficiently high, which typically happens for moderate size networks [12], [13].…”