Subspace alignment chains and the degrees of freedom of the three-user MIMO interference channel

Abstract

We show that the 3-user M T × M R MIMO interference channel where each transmitter is equipped with M T antennas and each receiver is equipped with M R antennas has d(M, N ) = min M 2−1/κ , N 2+1/κ degrees of freedom (DoF) normalized by time, frequency, and space dimensions, whereWhile the DoF outer bound of d(M, N ) is established for every M T , M R value, the achievability of d(M, N ) DoF is established in general subject to a normalization with respect to spatial-extensions, i.e., the scaling of the number of antennas at all nodes. Specifically, we show that qd(M, N ) DoF are achievable for the 3-user qM T × qM R MIMO interference channel, for some positive integer q which may be seen as a spatial-extension factor. q is the scaling factor needed to make the value qd(M, N ) an integer. Given spatial-extensions, the achievability relies only on linear beamforming based interference alignment schemes and requires neither channel extensions nor channel variations in time or frequency. In the absence of spatial extensions, it is shown through examples how essentially the same interference alignment scheme may be applied over time-extensions over either constant or time-varying channels. The central new insight to emerge from this work is the notion of subspace alignment chains as DoF bottlenecks. The subspace alignment chains are instrumental both in identifying the extra dimensions to be provided by a genie to a receiver for the DoF outer bound, as well as in the construction of the optimal interference alignment schemes.The DoF value d(M, N ) is a piecewise linear function of M, N , with either M or N being the bottleneck within each linear segment while the other value contains some redundancy, i.e., it can be reduced without reducing the DoF. The corner points of these piecewise linear segments correspond to two sets, A = {1/2, 2/3, 3/4, · · · } and B = {1/3, 3/5, 5/7, · · · }. The set A contains all those values of M/N and only those values of M/N for which there is redundancy in both M and N , i.e., either can be reduced without reducing DoF. The set B contains all those values of M/N and only those values of M/N for which there is no redundancy in either M or N , i.e., neither can be reduced without reducing DoF. Because A and B represent settings with maximum and minimum redundancy, essentially they are the basis for the DoF outer bounds and inner bounds, respectively.Our Our results show that M/N ∈ A are the only values for which there is no DoF benefit of joint processing among co-located antennas at the transmitters or receivers. This may also be seen as a consequence of the maximum redundancy in the M/N ∈ A settings.

Degrees of freedom achieved using subspace alignment chains for three-cell networks

“…While the optimal DoF of single-input single-output (SISO) wireless networks and some multiple-input multiple-output (MIMO) wireless networks [3], [4], [7] are established using the asymptotic alignment scheme of [1] or the rational dimensions framework of [2], linear beamforming schemes with finite symbol extensions play an important role in establishing the optimal DoF of several MIMO networks [8]. In particular, linear beamforming strategies for interference alignment (IA) are used to establish the optimal DoF of networks such as the three-user MIMO interference channel (IC) [8], twocell cellular networks [9], [10] etc. In this work, we focus on the achievable DoF using linear beamforming for three-cell MIMO cellular networks.…”

“…This paper provides one such approach for a class of three-cell networks by extending the notion of SACs introduced in [8] for the three-user IC. A SAC identifies a sequence of transmit subspaces at different transmitters such that each transmit subspace in the sequence aligns its interference with the preceding and succeeding transmit subspaces in the chain.…”

Beamformer design for interference alignment using reweighted frobenius norm minimization

“…While not all proper systems are feasible [10], improper systems have been shown to be almost surely infeasible [14]. In this paper, we only consider proper systems that are known to be feasible.…”

“…While several iterative algorithms have been proposed to design aligned beamformers [3]- [9], they are not always guaranteed to converge to a set of aligned beamformers that achieve the requisite number of DoF. Third, constructive approaches to design aligned beamformers such as subspace alignment chains [10] are not yet available for a broad class of (G, K, M, N ) networks. Thus, iterative algorithms for designing aligned beamformers are very much of interest to the research community.…”

Degrees of Freedom of the Interference Channel with a Cognitive Helper

“…The notion of DoF, also known as the number of independent signaling dimensions, is of great significance for understanding the high signal-to-noise power ratio (SNR) behavior of the capacity. By revealing the most essential aspects of the communication problem, DoF investigations have generated many fundamental ideas such as interference alignment [11], [6], deterministic channel models [10], rational dimensions [15], [7], aligned interference neutralization [9], subspace alignment chains [3], and genie chains [4]. While DoF characterizations have recently been obtained for a wide variety of wireless networks, in this letter we are primarily interested in an interference network with a cognitive helper, a setup which has not been considered before.…”