Feasibility Conditions for Interference Alignment

The degrees of freedom (DoF) of K-user MIMO interference networks with constant channel coefficients are not known in general. Determining the feasibility of a linear interference alignment is a key step toward solving this open problem. Our approach in this paper is to view the alignment problem for interference networks as a multivariate polynomial system and determine its solvability by comparing the number of equations and the number of variables. Consequently, we divide the interference networks into two classes proper and improper, where interference alignment is and is not achievable, respectively. An interference network is called proper if the cardinality of every subset of equations in the corresponding polynomial system is less than or equal to the number of variables involved in that subset of equations. Otherwise, it is called improper. Our intuition in this paper is that for general channel matrices, proper systems are almost surely feasible and improper systems are almost surely infeasible. We prove the direct link between proper (improper) and feasible (infeasible) systems for some important cases, thus significantly strengthening our intuition. Numerical simulation results also support our intuition.