A Feasibility Test for Linear Interference Alignment in MIMO Channels with Constant Coefficients

In this paper, we consider the feasibility of linear interference alignment (IA) for multiple-input multiple-output (MIMO) channels with constant coefficients for any number of users, antennas and streams per user; and propose a polynomial-time test for this problem. Combining algebraic geometry techniques with differential topology ones, we first prove a result that generalizes those previously published on this topic. Specifically, we consider the input set (complex projective space of MIMO interference channels), the output set (precoder and decoder Grassmannians) and the solution set (channels, decoders and precoders satisfying the IA polynomial equations), not only as algebraic sets but also as smooth compact manifolds. Using this mathematical framework, we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective. If that mapping is not surjective, then the solution variety projects into the input space in a singular way and the projection is a zero-measure set. This result naturally yields a simple feasibility test, which amounts to checking the rank of a matrix. We also provide an exact arithmetic version of the test, which proves that testing the feasibility of IA for generic MIMO channels belongs to the bounded-error probabilistic polynomial (BPP) complexity class.Comment: To be published in IEEE Transactions on Information Theor

Generalized Interference Alignment --- Part I: Theoretical Framework

Interference alignment (IA) has attracted enormous research interest as it achieves optimal capacity scaling with respect to signal to noise ratio on interference networks. IA has also recently emerged as an effective tool in engineering interference for secrecy protection on wireless wiretap networks. However, despite the numerous works dedicated to IA, two of its fundamental issues, i.e., feasibility conditions and transceiver design, are not completely addressed in the literature. In this two part paper, a generalised interference alignment (GIA) technique is proposed to enhance the IA's capability in secrecy protection. A theoretical framework is established to analyze the two fundamental issues of GIA in Part I and then the performance of GIA in large-scale stochastic networks is characterized to illustrate how GIA benefits secrecy protection in Part II. The theoretical framework for GIA adopts methodologies from algebraic geometry, determines the necessary and sufficient feasibility conditions of GIA, and generates a set of algorithms that can solve the GIA problem. This framework sets up a foundation for the development and implementation of GIA.Comment: Minor Revision at IEEE Transactions on Signal Processin

Interference Alignment for Cognitive Radio Communications and Networks: A Survey

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).Interference alignment (IA) is an innovative wireless transmission strategy that has shown to be a promising technique for achieving optimal capacity scaling of a multiuser interference channel at asymptotically high-signal-to-noise ratio (SNR). Transmitters exploit the availability of multiple signaling dimensions in order to align their mutual interference at the receivers. Most of the research has focused on developing algorithms for determining alignment solutions as well as proving interference alignment’s theoretical ability to achieve the maximum degrees of freedom in a wireless network. Cognitive radio, on the other hand, is a technique used to improve the utilization of the radio spectrum by opportunistically sensing and accessing unused licensed frequency spectrum, without causing harmful interference to the licensed users. With the increased deployment of wireless services, the possibility of detecting unused frequency spectrum becomes diminished. Thus, the concept of introducing interference alignment in cognitive radio has become a very attractive proposition. This paper provides a survey of the implementation of IA in cognitive radio under the main research paradigms, along with a summary and analysis of results under each system model.Peer reviewe

On the Feasibility of Linear Interference Alignment for MIMO Interference Broadcast Channels with Constant Coefficients

In this paper, we analyze the feasibility of linear interference alignment (IA) for multi-input-multi-output (MIMO) interference broadcast channel (MIMO-IBC) with constant coefficients. We pose and prove the necessary conditions of linear IA feasibility for general MIMO-IBC. Except for the proper condition, we find another necessary condition to ensure a kind of irreducible interference to be eliminated. We then prove the necessary and sufficient conditions for a special class of MIMO-IBC, where the numbers of antennas are divisible by the number of data streams per user. Since finding an invertible Jacobian matrix is crucial for the sufficiency proof, we first analyze the impact of sparse structure and repeated structure of the Jacobian matrix. Considering that for the MIMO-IBC the sub-matrices of the Jacobian matrix corresponding to the transmit and receive matrices have different repeated structure, we find an invertible Jacobian matrix by constructing the two sub-matrices separately. We show that for the MIMO-IBC where each user has one desired data stream, a proper system is feasible. For symmetric MIMO-IBC, we provide proper but infeasible region of antenna configurations by analyzing the difference between the necessary conditions and the sufficient conditions of linear IA feasibility.Comment: 14 pages, 3 figures, accepted by IEEE Trans. on Signal Processin