Homotopy continuation for vector space interference alignment in MIMO X networks

2016 24th European Signal Processing Conference (EUSIPCO)

2016

In this paper we study the degrees of freedom (DoF) achieved by interference alignment (IA) for cellular networks in reverse time division duplex (R-TDD) mode, a new configuration associated to heterogeneous networks. We derive a necessary feasibility condition for interference alignment in the multi-cell R-TDD scenario, which is then specialized to the particular case of symmetric demands and antenna distribution. We show that, for those symmetric networks for which the properness condition holds with equality, R-TDD does not improve the DoF performance of conventional synchronous TDD systems. Nevertheless, our simulation results indicate that, in more asymmetric scenarios, significant DoF benefits can be achieved by applying the R-TDD approach.

Section: Simulation Resultsmentioning

confidence: 99%

On the spatial degrees of freedom benefits of reverse TDD in multicell MIMO networks

Fanjul

2016 24th European Signal Processing Conference (EUSIPCO)

2016

“…In this paper we propose an algorithm, which extends and generalizes our previous work in [1], to compute interference alignment solutions for general asymmetric X networks. It is based on homotopy continuation, a numerical method which is frequently used to solve multivariate systems of nonlinear equations [30]- [33].…”

Section: A Summary Of Contributionsmentioning

confidence: 99%

“…• In comparison to [1], which was proposed only for generic channels, in this work we explicitly enforce the rank constraints on the direct channels by representing them as an additional set of bilinear equations. In this way, the new algorithm is able to find alignment precoders and decoders also in scenarios with symbol extensions, which result in non-generic (structured) MIMO channels.…”

Section: A Summary Of Contributionsmentioning

confidence: 99%

See 1 more Smart Citation

Homotopy Continuation for Spatial Interference Alignment in Arbitrary MIMO X Networks

Fanjul

González

IEEE Trans. Signal Process.

et al. 2017

Self Cite

In this paper we propose an algorithm to design interference alignment (IA) precoding and decoding matrices for arbitrary MIMO X networks. The proposed algorithm is rooted in the homotopy continuation techniques commonly used to solve systems of nonlinear equations. Homotopy methods find the solution of a target system by smoothly deforming the solution of a start system which can be trivially solved. Unlike previously proposed IA algorithms, the homotopy continuation technique allows us to solve the IA problem for both unstructured (i.e., generic) and structured channels such as those that arise when time or frequency symbol extensions are jointly employed with the spatial dimension. To this end, we consider an extended system of bilinear equations that include the standard alignment equations to cancel the interference, and a new set of bilinear equations that preserve the desired dimensionality of the signal spaces at the intended receivers. We propose a simple method to obtain the start system by randomly choosing a set of precoders and decoders, and then finding a set of channels satisfying the system equations, which is a linear problem. Once the start system is available, standard prediction and correction techniques are applied to track the solution all the way to the target system. We analyze the convergence of the proposed algorithm and prove that, for many feasible systems and a sufficiently small continuation parameter, the algorithm converges with probability one to a perfect IA solution. The simulation results show that the proposed algorithm is able to consistently find solutions achieving the maximum number of degrees of freedom (DoF) in a variety of MIMO X networks with or without symbol extensions. Further, the algorithm provides insights into the feasibility of IA in MIMO X networks for which theoretical results are scarce.

“…For the sake of completeness, channel realization does not exist. Still, one may try to compute them by repeatedly running some iterative algorithm such as the ones in [34], [35] or [36] from different initialization points. 4 This idea is illustrated in Figure 2, where the sum-rate performance associated to 973 different solutions is shown.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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

González

Beltrán

IEEE Trans. Inform. Theory

2015

Self Cite

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...