# On the Number of Interference Alignment Solutions for the K-User MIMO Channel with Constant Coefficients

Title | On the Number of Interference Alignment Solutions for the K-User MIMO Channel with Constant Coefficients |

Publication Type | Journal Article |

Year of Publication | 2015 |

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

Journal | IEEE Transaction on Information Theory |

Volume | 61 |

Issue | 11 |

Pagination | 6028-6048 |

Month Published | November |

Abstract | In this paper, we study the number of different interference alignment (IA) solutions that exists for the K-user multiple-input multiple-output (MIMO) interference channels with constant coefficients, when the alignment is performed via beamforming and without symbol extensions. When counting the number of IA solutions for a given problem, the most interesting case happens when the number of equations and variables of the polynomial system of equations are the same and the system is feasible. In this situation, the number of IA solutions is finite and, as we show in this paper, is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of solutions is the scaled average over a subset of the solution variety (formed by all triplets of channels, precoders and decoders satisfying the IA polynomial equations) 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, the average of the determinant (properly scaled) gives us the number of solutions. Our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas and streams per user. |

DOI | 10.1109/TIT.2015.2482493 |

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