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This paper continues the Wu-Shamai-Verdu&#x0301; program on characterizing the degrees of freedom (DoF) of interference channels (ICs) through Re&#x0301;nyi information dimension. Specifically, we find a single-letter formula for the DoF of vector ICs, encompassing multiple-input multiple-output ICs, time- and/or frequency-selective ICs, and combinations thereof, as well as scalar ICs as considered by Wu et al., 2015. The DoF-formula we obtain lower-bounds the DoF of all channels-with respect to the choice of the channel matrix-and upper-bounds the DoF of almost all channels. It applies to a large class of noise distributions, and its proof is based on an extension of a result by Guionnet and Shlyakthenko, 2007, to the vector case in combination with the Ruzsa triangle inequality for differential entropy introduced by Kontoyiannis and Madiman, 2015. As in scalar ICs, achieving full DoF requires the use of singular input distributions. Strikingly, in the vector case, it suffices to enforce singularity on the joint distribution of each transmit vector. This can be realized through signaling in subspaces of the ambient signal space, which is in accordance with the idea of interference alignment, and, most importantly, allows the scalar entries of the transmit vectors to have non-singular distributions. The DoF-formula for vector ICs we obtain enables a unified treatment of classical interference alignment a&#x0300; la Cadambe and Jafar, 2008, and Maddah-Ali et al., 2008, and the number-theoretic schemes proposed by Motahari et al., 2014, and by Etkin and Ordentlich, 2009. Moreover, it allows to calculate the DoF achieved by new signaling schemes for vector ICs. We furthermore recover the result by Cadambe and Jafar on the non-separability of parallel ICs, 2009, and we show that almost all parallel ICs are separable in terms of DoF. Finally, our results apply to complex vector ICs, thereby extending the main findings of Wu et al., 2015 to the complex case.

Degrees of Freedom in Vector Interference Channels