Homotopy continuation for vector space interference alignment in MIMO X networks

In this paper we propose an algorithm to design interference alignment (IA) precoding and decoding matrices for MIMO X networks (XN). 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 known solutions of a start system which can be trivially solved. The key observation leading to a simple start system is realizing that the inverse IA problem, i.e., finding the channels that satisfy the IA conditions given a set of precoders and decoders, is linear and, therefore, a convenient trivial system. Once the start system has been solved, standard prediction and correction techniques are applied to track the solution all the way to the target system. Our results show that the proposed algorithm is able to consistently find solutions achieving the maximum number of degrees of freedom (DoF) whereas alternating minimization techniques, which typically work well for the interference channel (IC), repeatedly fail for the XN. Further, the algorithm provides insights into the feasibility of alignment in MIMO X networks for which theoretical results are scarce.This work was supported by the Spanish Government, Ministerio de Ciencia e Innovación (MICINN), under project COSIMA (TEC2010-19545-C04-03), project COMONSENS (CSD2008-00010, CONSOLIDER-INGENIO 2010 Program) and FPU grant AP2009-1105

Homotopy continuation for spatial interference alignment in arbitrary MIMO X networks

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 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.This work has been supported by the Ministerio de Economía y Competitividad (MINECO) of Spain, under grants TEC2013-47141-C4-R (RACHEL), TEC2016-75067-C4-4-R (CARMEN), MTM2014-57590-P, and FPI grant BES-2014-069786

Interference alignment for the MIMO interference channel

We study vector space interference alignment for the MIMO interference channel with no time or frequency diversity, and no symbol extensions. We prove both necessary and sufficient conditions for alignment. In particular, we characterize the feasibility of alignment for the symmetric three-user channel where all users transmit along d dimensions, all transmitters have M antennas and all receivers have N antennas, as well as feasibility of alignment for the fully symmetric (M=N) channel with an arbitrary number of users. An implication of our results is that the total degrees of freedom available in a K-user interference channel, using only spatial diversity from the multiple antennas, is at most 2. This is in sharp contrast to the K/2 degrees of freedom shown to be possible by Cadambe and Jafar with arbitrarily large time or frequency diversity. Moving beyond the question of feasibility, we additionally discuss computation of the number of solutions using Schubert calculus in cases where there are a finite number of solutions.Comment: 16 pages, 7 figures, final submitted versio

Statistical analysis of single-beam interference alignment schemes

In this work, we derive analytical approximate expressions for the user rates achievable by interference alignment (IA) algorithms in single-beam multiple-input multiple-output (MIMO) networks for a fixed channel realization. Unlike previous works that perform a large-system analysis in which the number of users, antennas, or streams is required to tend to infinity, in this paper we only require that the number of different IA solutions (precoders and decoders) for the given scenario is sufficiently high, which typically happens even for moderate-size feasible networks. Based on the assumption that the IA beamformers for a given channel realization are random vectors isotropically distributed on the complex unit sphere, we characterize the user rates by averaging over the (possible finite) set of IA solutions. Some simulation results show the accuracy of the proposed rate expressions.This work was supported by the Ministerio de Economía y Competitividad (MINECO), Spain, under project RACHEL (TEC2013-47141-C4-3-R) and FPI grant BES-2014-069786

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 this 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.The work of Ó. González and I. Santamaría was supported by MICINN (Spanish Ministry for Science and Innovation) under grants TEC2013-47141-C4-3-R (RACHEL), TEC2010-19545-C04-03 (COSIMA), CONSOLIDER-INGENIO 2010 CSD2008-00010 (COMONSENS) and FPU grant AP2009-1105. Carlos Beltrán was partially supported by MICINN grant MTM2010-16051

Experimental evaluation of flexible duplexing in multi-tier MIMO networks

In this paper, we present an experimental evaluation of the performance benefits provided by flexible duplexing, an access technique that allows uplink and downlink cells to coexist within the same time-frequency resource blocks. In order to replicate a wireless multi-tier network composed of 1 macro-cell and 2 small cells, a measurement campaign has been conducted using an indoor wireless testbed comprised of a total of 6 multiple-input multiple-output (MIMO) software-defined radio (SDR) devices. Since each cell has a single active user, each uplink/downlink configuration can be identified with a different interference channel, over which interference alignment (IA) is used as an inter-cell interference management technique and compared to other existing methods. The obtained results show that flexible duplexing clearly outperforms the conventional time-division duplex (TDD) access approach, where all cells operate synchronized either in uplink or dowlink mode. Additionally, interference alignment consistently provides better results in most of the interference regimes when compared to minimum means quare error (MMSE)-based schemes. The impact of channel estimate quality on the different communication strategies is also studied. It is worth highlighting that the presented over-the-air (OTA) experiments represent the first implementation of IA with real-time precoding and decoding.The work of Jacobo Fanjul, Jesús Ibáñez and Ignacio Santamaria has been supported by the Ministerio de Economía, Industria y Competitividad (MINECO) of Spain, and AEI/FEDER funds of the E.U., under grant TEC2016-75067-C4-4-R (CARMEN), grant PID2019-104958RB-C43 (ADELE), and FPI grant BES-2014-069786. The work of José A. García-Naya has been funded by the Xunta de Galicia (ED431G2019/01), the Agencia Estatal de Investigación of Spain (TEC2016-75067-C4-1-R, RED2018-102668-T), and ERDF funds of the E.U. (AEI/FEDER, UE)

Aeronautical engineering: A continuing bibliography with indexes (supplement 272)

This bibliography lists 719 reports, articles, and other documents introduced into the NASA scientific and technical information system in November, 1991. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Technology for large space systems: A bibliography with indexes (supplement 22)

This bibliography lists 1077 reports, articles, and other documents introduced into the NASA Scientific and Technical Information System between July 1, 1989 and December 31, 1989. Its purpose is to provide helpful information to the researcher or manager engaged in the development of technologies related to large space systems. Subject areas include mission and program definition, design techniques, structural and thermal analysis, structural dynamics and control systems, electronics, advanced materials, assembly concepts, and propulsion

Large space structures and systems in the space station era: A bibliography with indexes (supplement 04)

Bibliographies and abstracts are listed for 1211 reports, articles, and other documents introduced into the NASA scientific and technical information system between 1 Jul. and 30 Dec. 1991. Its purpose is to provide helpful information to the researcher, manager, and designer in technology development and mission design according to system, interactive analysis and design, structural concepts and control systems, electronics, advanced materials, assembly concepts, propulsion, and solar power satellite systems