Feasibility Conditions for Interference Alignment

The degrees of freedom of MIMO interference networks with constant channel coefficients are not known in general. Determining the feasibility of a linear interference alignment solution is a key step toward solving this open problem. Our approach in this paper is to view the alignment problem as a system of bilinear equations and determine its solvability by comparing the number of equations and the number of variables. To this end, we divide interference alignment problems into two classes - proper and improper. An interference alignment problem is called proper if the number of equations does not exceed the number of variables. Otherwise, it is called improper. Examples are presented to support the intuition that for generic channel matrices, proper systems are almost surely feasible and improper systems are almost surely infeasible.

💡 Research Summary

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The paper addresses the long‑standing open problem of determining when linear interference alignment (IA) is feasible in constant‑coefficient MIMO interference networks. The authors recast the IA conditions as a system of bilinear (or “bivariate linear”) equations and propose a simple yet powerful feasibility test based on counting the number of scalar equations versus the number of scalar variables that appear in the alignment problem.

In a K‑user MIMO interference channel each transmitter i has (M_t) antennas, each receiver j has (M_r) antennas, and user i wishes to send (d_i) independent data streams. Linear IA seeks precoding matrices (V_i\in\mathbb{C}^{M_t\times d_i}) and decoding matrices (U_i\in\mathbb{C}^{M_r\times d_i}) such that for every interfering pair ((i,j), i\neq j), the interference term (U_i^{\dagger} H_{ij} V_j) is forced to zero. This condition yields (d_i d_j) scalar equations for each ordered pair ((i,j)). Summing over all ordered pairs gives the total number of equations

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