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...

We study two distinct, but overlapping, networks that operate at the same time, space, and frequency. The first network consists of n randomly distributed primary users, which form either an ad hoc network, or an infrastructuresupported ad hoc network with l additional base stations. The second network consists of m randomly distributed, ad hoc secondary users or cognitive users. The primary users have priority access to the spectrum and do not need to change their communication protocol in the presence of secondary users. The secondary users, however, need to adjust their protocol based on knowledge about the locations of the primary nodes to bring little loss to the primary network's throughput. By introducing preservation regions around primary receivers and avoidance regions around primary base stations, we propose two modified multihop routing protocols for the cognitive users. Base on percolation theory, we show that when the secondary network is denser than the primary network, both networks can simultaneously achieve the same throughput scaling law as a stand-alone network. Furthermore, the primary network throughput is subject to only a vanishingly fractional loss. Specifically, for the ad hoc and the infrastructure-supported primary models, the primary network achieves sum throughputs of order n 1/2 and max{n 1/2 , l}, respectively. For both primary network models, for any δ > 0, the secondary network can achieve sum throughput of order m 1/2−δ with an arbitrarily small fraction of outage. Thus, almost all secondary source-destination pairs can communicate at a rate of order m −1/2−δ .

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This paper analyzes the impact and benefits of infrastructure support in improving the throughput scaling in networks of $n$ randomly located wireless nodes. The infrastructure uses multi-antenna base stations (BSs), in which the number of BSs and the number of antennas at each BS can scale at arbitrary rates relative to $n$. Under the model, capacity scaling laws are analyzed for both dense and extended networks. Two BS-based routing schemes are first introduced in this study: an infrastructure-supported single-hop (ISH) routing protocol with multiple-access uplink and broadcast downlink and an infrastructure-supported multi-hop (IMH) routing protocol. Then, their achievable throughput scalings are analyzed. These schemes are compared against two conventional schemes without BSs: the multi-hop (MH) transmission and hierarchical cooperation (HC) schemes. It is shown that a linear throughput scaling is achieved in dense networks, as in the case without help of BSs. In contrast, the proposed BS-based routing schemes can, under realistic network conditions, improve the throughput scaling significantly in extended networks. The gain comes from the following advantages of these BS-based protocols. First, more nodes can transmit simultaneously in the proposed scheme than in the MH scheme if the number of BSs and the number of antennas are large enough. Second, by improving the long-distance signal-to-noise ratio (SNR), the received signal power can be larger than that of the HC, enabling a better throughput scaling under extended networks. Furthermore, by deriving the corresponding information-theoretic cut-set upper bounds, it is shown under extended networks that a combination of four schemes IMH, ISH, MH, and HC is order-optimal in all operating regimes.Comment: 26 pages, 10 figures, 1 table, Under revision for IEEE Transactions on Information Theor