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. We combine algebraic geometry techniques with differential topology ones and prove much stronger results than 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. The main result of the paper states that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger or equal than the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is 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. Building on these results, another contribution of the paper is to propose a simple, polynomial-complexity test of feasibility for the linear alignment problem.
In REF , generalized feasibility conditions were presented for interference alignment in MIMO channels with constant coefficients, complemented by a simple algorithm for feasibility test.
On the Feasibility of Interference Alignment for the K-User MIMO Channel with Constant Coefficients
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Computer Science