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

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

Publication Type | Journal Article |

Year of Publication | 2014 |

Authors | Ó. González, C. Beltrán, and I. Santamaría |

Journal | IEEE Transactions on Information Theory. |

Volume | 60 |

Issue | 3 |

Pagination | 1840-1856 |

Month Published | March |

ISSN | 0018-9448 |

Keywords | algebraic geometry, differential topology, interference alignment, MIMO interference channel, polynomial equations |

Abstract | 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. In particular, 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 complexity class. |

DOI | 10.1109/TIT.2014.2301440 |

Original Publication | On the Feasibility of Interference Alignment for the K-User MIMO Channel with Constant Coefficients |

Refereed Designation | Refereed |

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