Aligned Interference Neutralization and the Degrees of Freedom of the 2$\,\times \,$2$\,\times \,$2 Interference Channel

Abstract

We show that the 2 × 2 × 2 interference network, i.e., the multihop interference network formed by concatenation of two 2-user interference channels achieves the min-cut outer bound value of 2 DoF, for almost all values of channel coefficients, for both time-varying or fixed channel coefficients. The key to this result is a new idea, called aligned interference neutralization, that provides a way to align interference terms over each hop in a manner that allows them to be cancelled over the air at the last hop. Interference Alignment ApproachAny approach that treats either hop (or both hops) of the 2 × 2 × 2 IC as an interference channel can only achieve a maximum of 1 DoF, because of the bottleneck created by the 2 user interference channel which has only 1 DoF [35]. Interestingly, the interference channel approach is highly suboptimal at high SNR. This is because the interference channel approach precludes interference alignment.Interference alignment refers to a consolidation of undesired signals into smaller dimensions so that the signaling dimensions available for desired signals at each receiver are maximized. Interference alignment was observed first by Birk and Kol [36] for the index coding problem, and then by Maddah-Ali et. al. for the X channel in [37], followed by Weingarten et. al. for the compound vector broadcast channel in [38]. The idea was crystallized as a general concept in [2,3] by Jafar and Shamai, and Cadambe and Jafar, respectively, and has since been applied in increasingly sophisticated forms [13,39,40,41,42,20,5,43,44,45] across a variety of communication networks -both wired and wireless -often leading to surprising new insights.2 Unlike the interference channel approach which can achieve no more than 1 DoF, Cadambe and Jafar show in [4] that the 2 × 2 × 2 IC can achieve 4 3 DoF almost surely. This is accomplished by a decode and forward approach that treats each hop as an X channel. Specifically, each transmitter divides its message into two independent parts, one intended for each relay. This creates a total of 4 messages over the first hop, one from each source to each relay node, i.e., the 2 × 2 X channel setting. After decoding the messages from each transmitter, each relay has a message for each destination node, which places the second hop into the X channel setting as well. It is known that the 2 × 2 X channel with single antenna nodes has 4 3 DoF. The result was shown first by Jafar and Shamai in [2] under the assumption that the channel coefficients are time-varying. By using a combination of linear beamforming, symbol extensions and asymmetric complex signaling, Cadambe et. al. showed in [40] that 4 3 DoF are achievable on the 2 × 2 X channel even if the channels are held constant for almost all values of channel coefficients. Motahari et. al. [46] proposed the framework of rational dimensions which allows 4 3 DoF to be achieved almost surely even if the channels are fixed and restricted to real values. Thus, regardless of whether the channels are time-varying or constan...

Aligned interference neutralization and the degrees of freedom of the 2×2×2 interference channel with interfering relays

“…Since the min-cut DoF outer bound is 2, we only need to provide an achievable scheme to achieve 2 DoF for this channel. Recall that for the 2 × 2 × 2 IC where two relays do not interfere with each other, 2 DoF can be achieved using aligned interference neutralization [7]. As we can see, the only difference between the 2 × 2 × 2 IC studied in [7] and the channel studied in this paper is the additional assumption that two relays are interfering with each other.…”

“…While the DoF for the general cases are not known, some progress has been made for some special cases. In [7], it is shown that the 2×2×2 interference network, i.e., the 2-hop interference network formed by concatenation of two 2-user interference channels, achieves the cut-set outer bound value of 2 DoF for almost all channel coefficients. Although the 2 × 2 × 2 setting seems special, the new idea of aligned interference neutralization that emerged out of the study of this channel turns out to be important to achieve the optimal DoF of a variety of multihop multiple unicast networks.…”

“…It is shown that such networks can only have 1, 2 or 3/2 DoF. Remarkably, a class of 2-unicast layered networks can be condensed to the 2 × 2 × 2 setting where aligned interference neutralization scheme introduced in [7] is utilized. Moreover, this setting turns out to be the only setting where interference alignment is essential.…”

“…And we will show in this paper that the outer bound can be achieved for almost all channel coefficients. The achievable scheme is based on aligned interference neutralization designed for the 2 × 2 × 2 IC in [7] as well as exploiting memory at sources and relays which helps to cancel interference between relays.…”

Research on the communication method of mobile network shadow fading based on interference alignment algorithm

“…Relay and interference alignment algorithm, can greatly improve the system capacity. Interference alignment algorithm for 2L + 1X relay network [46] proposed an alignment of interference and mechanisms. The basic idea is aligned each hop in the network interference, and interference cancellation at the last hop.…”