Necessary Conditions for K/2 Degrees of Freedom

Stotz et al., 2016, reported a sufficient (injectivity) condition for each user in a K-user single-antenna constant interference channel to achieve 1/2 degree of freedom. The present paper proves that this condition is necessary as well and hence provides an equivalence characterization of interference channel matrices allowing full degrees of freedom

Canonical conditions for K/2 degrees of freedom

Stotz and Bölcskei, 2015, identified an explicit condition for K/2 degrees of freedom (DoF) in constant single-antenna interference channels (ICs). This condition is expressed in terms of linear independence - over the rationals - of monomials in the off-diagonal entries of the IC matrix and is satisfied for almost all IC matrices. There is, however, a prominent class of IC matrices that admits K/2 DoF but fails to satisfy this condition. The main contribution of the present paper is a more general condition for K/2 DoF (in fact for 1/2 DoF for each user) that, inter alia, encompasses this example class. While the existing condition by Stotz and Bölcskei is of algebraic nature, the new condition is canonical in the sense of capturing the essence of interference alignment by virtue of being expressed in terms of a generic injectivity condition that guarantees separability of signal and interference