Degrees of Freedom in Vector Interference Channels

Abstract

This paper continues the Wu-Shamai-Verdu program [3] on characterizing the\ndegrees of freedom (DoF) of interference channels (ICs) through Renyi\ninformation dimension. Specifically, we find a single-letter formula for the\nDoF of vector ICs, encompassing multiple-input multiple-output (MIMO) ICs,\ntime- and/or frequency-selective ICs, and combinations thereof, as well as\nscalar ICs as considered in [3]. The DoF-formula we obtain lower-bounds the DoF\nof all channels--with respect to the choice of the channel matrix--and\nupper-bounds the DoF of almost all channels. It applies to a large class of\nnoise distributions, and its proof is based on an extension of a result by\nGuionnet and Shlyakthenko [3] to the vector case in combination with the Ruzsa\ntriangle inequality for differential entropy introduced by Kontoyiannis and\nMadiman [4]. As in scalar ICs, achieving full DoF requires the use of singular\ninput distributions. Strikingly, in the vector case it suffices to enforce\nsingularity on the joint distribution of each individual transmit vector. This\ncan be realized through signaling in subspaces of the ambient signal space,\nwhich is in accordance with the idea of interference alignment, and, most\nimportantly, allows the scalar entries of the transmit vectors to have\nnon-singular distributions. The DoF-formula for vector ICs we obtain enables a\nunified treatment of "classical" interference alignment a la Cadambe and Jafar\n[5], and Maddah-Ali et al. [6], and the number-theoretic schemes proposed in\n[7], [8]. Moreover, it allows to calculate the DoF achieved by new signaling\nschemes for vector ICs. We furthermore recover the result by Cadambe and Jafar\non the non-separability of parallel ICs [9] and we show that almost all\nparallel ICs are separable in terms of DoF. Finally, our results apply to\ncomplex vector ICs, thereby extending the main findings of [2] to the complex\ncase.\n