On the degrees of freedom achievable through interference alignment in a MIMO interference channel

Abstract

Consider a K-user flat fading MIMO interference channel where the k-th transmitter (or receiver) is equipped with M k (respectively N k ) antennas. If a large number of statistically independent channel extensions are allowed either across time or frequency, the recent work [1] suggests that the total achievable degrees of freedom (DoF) can be maximized via interference alignment, resulting in a total DoF that grows linearly with K even if M k and N k are bounded. In this work we consider the case where no channel extension is allowed, and establish a general condition that must be satisfied by any degrees of freedom tuple achievable through linear interference alignment. When M k = M and N k = N for all k, this condition implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than M + N − 1. If, in addition, all users have the same DoF d = 1, then this upper bound on the total DoF is actually tight for almost all MIMO interference channels. †

Relay-aided interference alignment for multiple partially connected subnetworks

“…shall hold to align these interferences at d 4 , namely v (2) and v (3) shall be parallel. Let this be denoted by v (2) ∥ v (3) .…”

“…Let this be denoted by v (2) ∥ v (3) . In other words, the variables of subnetwork 1 shall be chosen to satisfy v (2) ∥ v (3) in addition to its intra-subnetwork interferencenulling conditions. Example 2: In the same network, d 3 receives nonnegligible inter-subnetwork interference from s 7 .…”

“…Example 2: In the same network, d 3 receives nonnegligible inter-subnetwork interference from s 7 . Therefore, the variables of subnetwork 1 and subnetwork 3 shall be chosen to satisfy v (7) ⊥ u (3) in addition to the intra-subnetwork interference-nulling conditions in each individual subnetwork so that the inter-subnetwork interference could be nullified at d 3 .…”

“…However, the study of feasibility conditions shows that the goal of "everyone gets half the cake" is difficult to reach in constant channels, especially when there are lots of users. In constant multiple-input-multiple-output (MIMO) interference networks for instance, the number of antennas required at every user increases linearly with the total number of users [3]- [5]. In contrast to this, IA aided by half-duplex relays guarantees the achievement of 1/2 DoF per user even if each user only has a single antenna, but requires lots of relays or relay antennas depending on the network size [6]- [8].…”

On the Number of Interference Alignment Solutions for the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-User MIMO Channel With Constant Coefficients

“…In plain words, these are MIMO interference networks for which the number of variables is larger than the number of equations in the polynomial system. On the other hand, systems with less variables than equations are always infeasible [13], [14], [17]. Herein we will focus on the case in between, where the dimensions of V and H are exactly the same (identical number of variables and equations), and consequently, the number of IA solutions is finite (it may be even zero) and constant out of a zero measure set of H as also proved in [17].…”