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

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

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

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

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.This work has been supported by the Ministerio de Economía y Competitividad (MINECO) of Spain under grant TEC2013-47141- C4-R (RACHEL project) and FPI grant BES-2014-069786

Power minimization in multi-tier networks with flexible duplexing

In this paper we present an algorithm to minimize transmit power in multiple-input multiple-output (MIMO) heterogeneous networks (HetNets) with flexible duplexing, a promising strategy that allows the coexistence of uplink and downlink cells within the same time and frequency resource block. First, the proposed algorithm minimizes transmit power for a given uplink/downlink (UL/DL) combination, and afterwards, the optimal solution out of the explored UL/DL combinations is selected. To reduce the computational cost of exploring all the UL/DL settings, we propose a hierarchical switching (HS) approach that considers a reduced subset of transmit directions. By means of Monte Carlo simulations, we show that the proposed technique provides significant power savings with respect to a conventional time-division duplex (TDD) scheme.This work has been supported by the MINECO of Spain and AEI/FEDER funds from the EU, under grant TEC2016-75067-C4-4-R (CARMEN project) 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

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

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

The Fifth NASA/DOD Controls-Structures Interaction Technology Conference, part 2

This publication is a compilation of the papers presented at the Fifth NASA/DoD Controls-Structures Interaction (CSI) Technology Conference held in Lake Tahoe, Nevada, March 3-5, 1992. The conference, which was jointly sponsored by the NASA Office of Aeronautics and Space Technology and the Department of Defense, was organized by the NASA Langley Research Center. The purpose of this conference was to report to industry, academia, and government agencies on the current status of controls-structures interaction technology. The agenda covered ground testing, integrated design, analysis, flight experiments and concepts