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Publication# MIMO Compute-and-Forward

Michael Christoph Gastpar

*Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa, *2009

Article de conférence

Article de conférence

Résumé

In many network communication scenarios, a relay in the network may only need to recover and retransmit an equation of the transmitted messages. In previous work, it has been shown that if each transmitter employs the same lattice code, the interference structure of the channel can be exploited to recover an equation much more efficiently than possible with standard multiple-access strategies. Here, we generalize this compute-and-forward framework to the multiple antenna setting. Our results show that it is often beneficial to use extra antennas at the receiver to rotate the channel coefficients towards the nearest integer vector instead of separating out the transmitted signals. We also demonstrate that in contrast to classical strategies, the multiplexing gain of compute-and-forward increases if the transmitters have channel state information. Finally, we apply our scheme to the two way relay network and observe performance gains over traditional strategies.

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The main goal in network information theory is to identify fundamental limits of communication over networks, and design solutions which perform close to such limits. After several decades of effort, many important problems still do not have a characterization of achievable performance in terms of a finite dimensional description. Given this discouraging state of affairs, a natural question to ask is whether there are systematic approaches to make progress on these open questions. Recently, there has been significant progress on several open questions by seeking a (provably) approximate characterization for these open questions. The main goal of approximation in network information theory is to obtain a universal approximation gap between the achievable and the optimal performance. This approach consists of four ingredients: simplify the model, obtain optimal solution for the simplified model, translate this optimal scheme and outer bounds back to the original model, and finally bound the gap between what can be achieved using the obtained technique and the outer bound. Using such an approach, recent progress has been made in several problems such as the Gaussian interference channel, Gaussian relay networks, etc. In this thesis, we demonstrate that this approach is not only successful in problems of transmission over noisy networks, but gives the first approximation for a network data compression problem. We use this methodology to (approximately) resolve problems that have been open for several decades. Not only do we give theoretical characterization, but we also develop new coding schemes that are required to satisfy this approximate optimality property. These ideas could give insights into efficient design of future network communication systems. This thesis is split into two main parts. The first part deals with the approximation in lossy network data compression. Here, a lossy data compression problem is approximated by a lossless counterpart problem, where all the bits in the binary expansion of the source above the required distortion have to be losslessly delivered to the destination. In particular, we study the multiple description (MD) problem, based on the multi-level diversity (MLD) coding problem. The symmetric version of the MLD problem is well-studied, and we can directly use it to approximate the symmetric MD problem. We formulate the asymmetric multi-level diversity problem, and solve it for three-description case. The optimal solution for this problem, which will be later used to approximate the asymmetric multiple description problem, is based on jointly compressing of independent sources. In both symmetric and asymmetric cases, we derive inner and outer bounds for the achievable rate region, which together with the gap analysis, provide an approximate solution for the problem. In particular, we resolve the symmetric Gaussian MD problem, which has been open for three decades, to within 1 bit. In the second part, we initiate a study of a Gaussian relay-interference network, in which relay (helper) nodes are to facilitate competing information flows over a wireless network. We focus on a two-stage relay-interference network where there are weak cross-links, causing the networks to behave like a chain of Z Gaussian channels. For these Gaussian ZZ and ZS networks, we establish an approximate characterization of the rate region. The outer bounds to the capacity region are established using genie-aided techniques that yield bounds sharper than the traditional cut-set outer bounds. For the inner bound of the ZZ network, we propose a new interference management scheme, termed interference neutralization, which is implemented using structured lattice codes. This technique allows for over-the-air interference removal, without the transmitters having complete access to the interfering signals. We use insights gained from an exact characterization of the corresponding linear deterministic version of the problem, in order to study the Gaussian network. We resolve the Gaussian relay-interference network to within 2 bits. The new interference management technique (interference neutralization) shows the use of structured lattice codes in the problem. We also consider communication from a source to a destination over a wireless network with the help of a set of authenticated relays, and presence of an adversarial jammer who wishes to disturb communication. We focus on a special diamond network, and show that use of interference suppression (nulling) is crucial to approach the capacity of the network. The exact capacity characterization for the deterministic network, along with an approximate characterization (to within 4 bits) for the Gaussian network is provided. The common theme that binds the diverse network communication problems in this thesis is that of approximate characterization, when exact resolutions are difficult. The approach of focusing on the deterministic/lossless problems underlying the noisy/lossy network communication problems has allowed us to develop new techniques to study these questions. These new techniques might be of independent interest in other network information theory problems.

Complex networks exist for a number of purposes. The neural, metabolic and food networks ensure our survival, while the social, economic, transportation and communication networks allow us to prosper. Independently of the purposes and particularities of the physical embodiment of the networks, one of their fundamental functions is the delivery of information from one part of the network to another. Gossip and diseases diffuse in the social networks, electrochemical signals propagate in the neural networks and data packets travel in the Internet. Engineering networks for robust information flows is a challenging task. First, the mechanism through which the network forms and changes its topology needs to be defined. Second, within a given topology, the information must be routed to the appropriate recipients. Third, both the network formation and the routing mechanisms need to be robust against a wide spectrum of failures and adversaries. Fourth, the network formation, routing and failure recovery must operate under the resource constraints, either intrinsic or extrinsic to the network. Finally, the autonomously operating parts of the network must be incentivized to contribute their resources to facilitate the information flows. This thesis tackles the above challenges within the context of several types of networks: 1) peer-to-peer overlays – computers interconnected over the Internet to form an overlay in which participants provide various services to one another, 2) mobile ad-hoc networks – mobile nodes distributed in physical space communicating wirelessly with the goal of delivering data from one part of the network to another, 3) file-sharing networks – networks whose participants interconnect over the Internet to exchange files, 4) social networks – humans disseminating and consuming information through the network of social relationships. The thesis makes several contributions. Firstly, we propose a general algorithm, which given a set of nodes embedded in an arbitrary metric space, interconnects them into a network that efficiently routes information. We apply the algorithm to the peer-to-peer overlays and experimentally demonstrate its high performance, scalability as well as resilience to continuous peer arrivals and departures. We then shift our focus to the problem of the reliability of routing in the peer-to-peer overlays. Each overlay peer has limited resources and when they are exhausted this ultimately leads to delayed or lost overlay messages. All the solutions addressing this problem rely on message redundancy, which significantly increases the resource costs of fault-tolerance. We propose a bandwidth-efficient single-path Forward Feedback Protocol (FFP) for overlay message routing in which successfully delivered messages are followed by a feedback signal to reinforce the routing paths. Internet testbed evaluation shows that FFP uses 2-5 times less network bandwidth than the existing protocols relying on message redundancy, while achieving comparable fault-tolerance levels under a variety of failure scenarios. While the Forward Feedback Protocol is robust to message loss and delays, it is vulnerable to malicious message injection. We address this and other security problems by proposing Castor, a variant of FFP for mobile ad-hoc networks (MANETs). In Castor, we use the same general mechanism as in FFP; each time a message is routed, the routing path is either enforced or weakened by the feedback signal depending on whether the routing succeeded or not. However, unlike FFP, Castor employs cryptographic mechanisms for ensuring the integrity and authenticity of the messages. We compare Castor to four other MANET routing protocols. Despite Castor's simplicity, it achieves up to 40% higher packet delivery rates than the other protocols and recovers at least twice as fast as the other protocols in a wide range of attacks and failure scenarios. Both of our protocols, FFP and Castor, rely on simple signaling to improve the routing robustness in peer-to-peer and mobile ad-hoc networks. Given the success of the signaling mechanism in shaping the information flows in these two types of networks, we examine if signaling plays a similar crucial role in the on-line social networks. We characterize the propagation of URLs in the social network of Twitter. The data analysis uncovers several statistical regularities in the user activity, the social graph, the structure of the URL cascades as well as the communication and signaling dynamics. Based on these results, we propose a propagation model that accurately predicts which users are likely to mention which URLs. We outline a number of applications where the social network information flow modelling would be crucial: content ranking and filtering, viral marketing and spam detection. Finally, we consider the problem of freeriding in peer-to-peer file-sharing applications, when users can download data from others, but never reciprocate by uploading. To address the problem, we propose a variant of the BitTorrent system in which two peers are only allowed to connect if their owners know one another in the real world. When the users know which other users their BitTorrent client connects to, they are more likely to cooperate. The social network becomes the content distribution network and the freeriding problem is solved by leveraging the social norms and reciprocity to stabilize cooperation rather than relying on technological means. Our extensive simulation shows that the social network topology is an efficient and scalable content distribution medium, while at the same time provides robustness to freeriding.

The focus of this thesis is on the study of decentralized wireless multi-hop networks. We are particularly interested in establishing bounds on the traffic-carrying capabilities of wireless ad hoc networks and conditions on the scalability of such networks with node mobility. This theoretical investigation brings forward challenges on the design of such networks. This leads to a second part of this thesis that considers the feasibility and the design of physical layer architectures and schemes for decentralized wireless multi-hop networks. In the first part of this thesis, bounds on the capacity of wireless ad hoc networks with two types of non-uniform traffic patterns are established. We focus on the impact of traffic patterns where local communications predominate and show the improvement in terms of per user-capacity over ad hoc networks with unbounded average communication distances. We then study the capacity of hybrid wireless networks, where long-distance relaying is performed by a fixed overlay network of base-stations. We investigate the scaling of capacity versus the number of nodes and the density of base-stations in the area of the network. It is shown that the gain in performance is mainly due to the reduction in the mean number of hops from source to destination. Then, we investigate the impact of mobility on the ad hoc network capacity. We propose a set of necessary and sufficient conditions under which the long-term averaged throughput in an ad hoc network can remain constant as the number of nodes increases. The main idea is to use a connectivity graph that does not represent the actual physical network, but rather the available communication resources. This graph also allows to translate the problem of maximizing the throughput in ad hoc networks to the multi-commodity flow problem and directly apply related results. In contrast to these macroscopic studies, in the second part we focus on a microscopic analysis of ad hoc wireless networks. We are interested in characterizing the performance of decentralized multiple-access and retransmission schemes for multi-hop wireless networks with the goal of drawing conclusions on cross-layer design. We investigate different transmission strategies in order to assess the tradeoff between spatial density of communications and the range of each transmission. We present tools for characterizing the spatial throughput as a function of topological parameters (e.g node population density) and system parameters (propagation, bandwith etc). The results of this work also show that coding and retransmissions provide means of reliable communication coupled with a completely decentralized multiple-access strategy. Finally, an efficient protocol for the delay-limited fading Automatic Retransmission reQuest (ARQ) single relay channel is considered for cooperative communications. The proposed protocol exploits two kinds of diversity: (i) space diversity available through the cooperative (relay) terminal, which retransmits the source's signals, (ii) ARQ diversity obtained by leveraging the retransmission delay to enhance the reliability. The performance characterization is in terms of the achievable diversity, multiplexing gain and delay tradeoff for a high signal-to-noise ratio (SNR) regime. Then, by letting the source's power level vary over the retransmission rounds, we show the benefits of power control on the diversity.