Fundamental Limits of Non-Coherent Interference Alignment via Matroid Theory

We consider the problem of non-coherent interference alignment, in which the goal is to align the signals of multiple interfering transmitters at a single receiver where the transmitters are not aware of the channel state information. We cast this problem as a problem of determining rank loss conditions for a column concatenation of full-rank matrices, such that each row of the composing matrices is scaled by a random coefficient. We determine necessary and sufficient conditions for the design of each matrix, such that the random ensemble will almost surely lose rank by a certain amount. The result is proved by converting the problem to determining rank loss conditions for the union of some specific matroids, and then using tools from matroid and graph theories to derive the necessary and sufficient conditions. As an application, we discuss how this result can be applied to the problem of topological interference management, and characterize the linear symmetric degrees of freedom for a class of network topologies.