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Distributed interference management in large wireless networks
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Distributed interference management in large wireless networks
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Distributed Interference Management in Large Wireless Networks Thesis by Vasilis Ntranos In Partial Fulfillment of the Requirements for the degree of DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) UNIVERSITY OF SOUTHERN CALIFORNIA Los Angeles, California August 2015 (Defended April 8, 2015) Copyright 2015 Vasilis Ntranos The dissertation of Vasilis Ntranos, titled “Distributed Interference Management in Large Wireless Networks,” is approved: Prof. Giuseppe Caire, (Chair) EE Dept. date: Prof. Salman Avestimehr, EE Dept. date: Prof. Jason Fulman, Math Dept. date: University of Southern California ii to my family iii CEn oÚda, íti oÎdàn oÚda. Swkr' athc iv Acknowledgments First of all, I would like to deeply thank my advisor, Prof. Giuseppe Caire, for providing me with the guidance, inspiration and support to complete this thesis. His unique way of bringing the elegance of pure theory together with the art of engineering has shaped me as a researcher and influenced me as a thinker. One of the most important things I have learned while working with him over these years is to aim high and never give up; even from the very beginning, Giuseppe has taught me to look at open problems – not as brick walls in my academic path – but as new opportunities to challenge my understanding. I would also like to thank Dr. Mohammad Ali Maddah-Ali whose contribution in this thesis has been unparalleled not only on a technical, but also on a personal level. I started working with Mohammad as a summer intern at Bell labs but it’s only fair to say that over the last two years he has not only been my mentor but also a co-advisor on this thesis. His direct and insightful way of explaining things have made our countless meetings unforgettable. Mohammad, thank you again for your support and for being always there. A special thanks goes to Bobak Nazer and Viveck Cadambe, and to all my other collaborators, Alex Dimakis, Nikos Armenatzoglou, Dimitris Papadias and Fernando Rosas, with whom I have explored very different but equally exciting research paths outside the topics of this thesis. Thank you all for expanding my research horizons and for making my academic journey a bit more adventurous, interesting and fun. v I would also like to express my gratitude to my undergraduate thesis advisor, Prof. Nikos Sidiropoulos, whose guidance and support during my undergrad years, have made this academic journey possible. Kyrie Niko, thank you for introducing me to the fascinating world of research and for planting the right seeds for my future academic steps. A big acknowledgement goes to my friends Thimio, Stavro, Chari and Thano who believed in me from the very beginning and have been by my side since my undergrad years in Crete. The good times we had in LA with my USC friends Dimitri, Stelio, George and Antoni have made these past five years an amazing and unforgettable experience. I would like, hence, to take this chance and thank you all for your constant support and for making the road to my Ph.D even more enjoyable and fun. Finally, a big thank you goes to Jim, Georgia and Melina for their welcoming hospitality and for making Los Angeles feel like home. The greatest acknowledgment of all goes to my family and especially my mother IoannaandmybrotherAchilleswhohavealwaysbeenthereformeandhaveoffered me unconditional love and support throughout my life. Without your continuous encouragement and solid guidance, this thesis would not have been possible. Last but not least, I want to thank my partner, Hara, for the tremendous love, patience and understanding she has shown me over these years. She has been by my side through all the ups and downs that come along with a Ph.D thesis. Her love, advice and constant support lies behind every sentence and every equation you will come across in this dissertation. vi Abstract This thesis makes progress towards the fundamental understanding of distributed interference management in large wireless networks and provides practical and valuable system design guidelines for a much more efficient and scalable wireless network architecture. Practical interference alignment for cellular networks: Interference alignment promises that, in Gaussian interference channels, each link cansupporthalfofadegreeoffreedom(DoF)perpairoftransmit-receiveantennas. However, in general, this result requires to precode the data bearing signals over a signal space of asymptotically large diversity, e.g., over an infinite number of dimensions for time-frequency varying fading channels, or over an infinite number of rationally independent signal levels, in the case of time-frequency invariant chan- nels. In this thesis we consider hexagonal network topologies (e.g, sectored cellular networks, omnidirectional small cells) in which receivers observe interference from all of their neighbors and show that the promised optimal DoFs can be achieved with linear interference alignment in one-shot (i.e., over a single time-frequency slot). To enable these results, we propose a non-iterative, local message-passing cooperation scheme in which base-stations process and share information with their neighbors to contribute in a global, network-wide interference cancellation. Throughout this thesis we design practical interference alignment schemes for large cellular networks and provide the necessary theoretical tools for their analysis. vii Fundamental limits of distributed interference management: A fundamental question that naturally arises from any cooperative communication scenario is “How much backhaul capacity is required in order to achieve a given communication rate?” or, equivalently “What is the best communication rate that one can achieve for a given constraint on the total backhaul capacity?”. In the second part of this thesis we make significant progress towards answering this question, from an information theoretic perspective. We consider an interference channel model in whichK receivers cooperatively attempt to decode their intended messages locally by processing and sharing information through limited capacity backhaul links. In contrast to distributed antenna architectures that have been proposed in the literature, where data processing is utterly performed in a central- izedfashion, ourmodelaimstocapturetheessenceofdecentralized(overthecloud) processing, allowing for a more general class of interference management strategies. Focusing on the three-user case, we characterize the fundamental tradeoff between the achievable communication rates and the corresponding backhaul cooperation rate, in terms of degrees of freedom (DoF). Surprisingly, we show that the optimum communication-cooperation tradeoff remains the same when we move from two- user to three-user interference channels. In the absence of cooperation, this is due to interference alignment, which keeps the fraction of communication dimensions wasted for interference unchanged. When backhaul cooperation is available, we develop a new idea that we call cooperation alignment, which guarantees that the cooperation overhead also remains the same as we increase the number of users. viii Contents Acknowledgments v Abstract vii List of Figures xii 1 Introduction 1 1.1 Interference Alignment . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Practical Challenges . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Cooperation in Wireless Networks . . . . . . . . . . . . . . . . . . . 5 1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Cellular Interference Alignment 10 2.1 Cellular Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Interference Graph . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Network Interference Cancellation . . . . . . . . . . . . . . . 16 2.1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Networks with No Intra-cell Interference . . . . . . . . . . . . . . . 19 2.2.1 Achievability . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Converse (Proof of Theorem 2.2) . . . . . . . . . . . . . . . 24 2.3 Networks with Intra-Cell Interference . . . . . . . . . . . . . . . . . 27 2.3.1 Joint intra-cell processing scheme . . . . . . . . . . . . . . . 28 2.4 Topological Robustness . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 The Compound Cellular Network . . . . . . . . . . . . . . . 32 2.4.2 Topologically Robust Scheme . . . . . . . . . . . . . . . . . 34 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ix 3 Omni-Directional Antennas and Asymmetric Configurations 44 3.1 Omni-directional Cellular Model . . . . . . . . . . . . . . . . . . . . 45 3.1.1 Interference Graph . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.2 Network Interference Cancellation . . . . . . . . . . . . . . . 48 3.2 Cellular Networks with M = 2,N = 2 . . . . . . . . . . . . . . . . . 49 3.2.1 Achievability (Proof of Theorem 3.1) . . . . . . . . . . . . . 50 3.2.2 Converse (Proof of Theorem 3.2) . . . . . . . . . . . . . . . 56 3.3 Cellular Networks with Asymmetric Antenna Configurations . . . . 63 3.3.1 Achievability for M = 2, N = 3. (Proof of Thm 3.3) . . . . . 64 3.3.2 Achievability for M = 2, N = 4. (Proof of Thm 3.4) . . . . . 66 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4 Towards a general framework for Cellular IA 71 4.1 Cellular Interference Alignment for the Downlink . . . . . . . . . . 72 4.1.1 Successive encoding based on DPC . . . . . . . . . . . . . . 72 4.1.2 Uplink-Downlink Duality . . . . . . . . . . . . . . . . . . . . 75 4.2 Arbitrary Local Interference Topologies . . . . . . . . . . . . . . . . 79 4.2.1 Directed Interference and Acyclic Decomposition . . . . . . 79 4.2.2 Single-antenna networks and acyclic scheduling . . . . . . . 82 4.2.3 Multiple antenna networks and topology projections . . . . . 83 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5 Fundamental limits of Distributed Interference Management 89 5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.1.1 Distributed Cooperation Channel Model . . . . . . . . . . . 91 5.1.2 Achievable Rates, Capacity, and Degrees of Freedom . . . . 92 5.1.3 Example: Centralized processing . . . . . . . . . . . . . . . 94 5.2 Communication vs Cooperation Tradeoff . . . . . . . . . . . . . . . 94 5.2.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 Cooperation Alignment . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.3.1 The achievability of DoF(1) = 1 . . . . . . . . . . . . . . . . 98 5.3.2 DoF-Backhaul Upper Bound . . . . . . . . . . . . . . . . . 104 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 A Appendix for Chapter 2 107 A.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 107 A.1.1 M is even . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 A.1.2 M is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 A.2 Proof of Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.2.1 The cardinality ofV (r) . . . . . . . . . . . . . . . . . . . . . 113 A.2.2 The cardinality ofT (r) . . . . . . . . . . . . . . . . . . . . . 114 A.2.3 Proof of|V (r) ex | =O q |V (r) | . . . . . . . . . . . . . . . . . 116 x A.3 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 116 A.4 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.5 Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 121 B Appendix for Chapter 3 123 B.1 Proof of Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 B.2 Proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 B.3 Proof of Lemma 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 B.4 Proof of Lemma 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 B.4.1 The cardinality ofV (r) . . . . . . . . . . . . . . . . . . . . . 127 B.4.2 The cardinality ofT (r) . . . . . . . . . . . . . . . . . . . . . 128 B.4.3 The cardinality ofV (r) ex . . . . . . . . . . . . . . . . . . . . . 129 B.5 The Linear Programming Converse for M×M Cellular Systems . . 131 Bibliography 134 xi List of Figures 1.1 InterferenceAlignmentforthe 3-user 2×2MIMOInterferenceChan- nel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 The cellular network topology and the corresponding interference graph: we consider the uplink of a MIMO cellular network with 120 o sector receivers as depicted in Fig. 2.1a. Each receiver is interested in decoding the message of the mobile terminal associated with it and observes all other transmissions as interference. In our cellular model we assume four dominant sources of interference for each sector shown as orange arrows originating from its closest out-of- cell transmitters. Interference between the sectors of the same cell is depicted with black arrows. Fig. 2.1b shows the corresponding interference graph by taking into account all interfering links in a given cellular network, in which vertices represent transmit-receive pairs within sectors and edges indicate interfering neighbors. The dashed black edges in the above graph correspond to interference between sectors of the same cell that we are going to ignore until Section 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 The Eisenstein integers Z(ω) on the complex plane. . . . . . . . . 15 2.3 The directed interference graphG π ∗(V,E π ∗) after network interfer- encecancellationaccordingtothe“left-to-right, top-down”decoding order π ∗ . The transmitter of a sector associated with node i causes interference only to its neighboring sector receivers j with j < i. Orange arrows indicate out-of-cell interference while black dashed arrows correspond to interference from within each cell. . . . . . . 17 xii 2.4 Out-of-Cell Interference in the neighborhood of a blue node. Sectors are labeled here with letters to avoid confusion with the underlying decoding order (cf. Fig. 2.3). . . . . . . . . . . . . . . . . . . . . . 22 2.5 Interference Alignment Scheme. The alignment conditions are de- picted here with arrows connecting interfering streams that have to be aligned. The direction of the arrows show the corresponding beamforming dependencies (e.g., the arrow labeled with the number 1 requires that v a is chosen as a function of v b ). . . . . . . . . . . . 23 2.6 The set of triangles [u,v,w]∈T forG(V,E). All the circle nodes belong toV in and participate in exactly one triangle and the setV ex contains the remaining (colored) nodes on the boundary of the graph. 25 2.7 Intra-Cell Interference Elimination. The sectors of each cell can jointly process their received signals and successively decode their desired messages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.8 The achievable scheme of Section 2.3.1 is not topologically robust. Sector b will not able to decode its own desired message if Sector a does not observe interference from its neighbors. . . . . . . . . . . 31 2.9 The decoding order π s . The highlighted sector pairs decode their messages simultaneously. . . . . . . . . . . . . . . . . . . . . . . . 35 2.10 Robust decoding for sectors b and c. . . . . . . . . . . . . . . . . . 36 2.11 Robust decoding for sector a. . . . . . . . . . . . . . . . . . . . . . 38 2.12 Compound channel-state configurations for sectors a, b, c and d. Case 1 captures all parameter configurations α, in which there is no interference between sectors c and d, and Case 2 corresponds to configurations with α ad = 1. In Cases 3 and 4, we assume that [α cd ,α ad ] = [1, 0] and distinguish between configurations in which sector a observes at most one, or two interfering signals. . . . . . . 39 3.1 The cellular network topology and the corresponding dominant in- terferencegraph. Thedesireduplinkchannelsfrommobileterminals to their associated base-stations are depicted with black arrows in each cell. Each base-station observes six dominant interfering sig- nals from its six adjacent cells (red arrows) and weaker interference from outer cells (orange arrows) located at distance two or more. The interference graph by construction captures only the dominant sources of interference for each cell; the vertices represent transmit- receive pairs and edges indicate interfering neighbors. . . . . . . . . 46 3.2 The Eisenstein integers Z(ω) on the complex plane. . . . . . . . . 47 xiii 3.3 The directed interference graph ˚ G π ∗(V,E π ∗) after network interfer- encecancellationaccordingtothe“left-to-right, top-down”decoding order π ∗ . The transmitter of a cell associated with node i causes interference only to its neighboring base-station receiversj withj <i. 48 3.4 The interference sub-graph ˆ ˚ G π ∗(V\V 0 ,E π ∗\E 0 ) represented on the complex plane. The transparent vertices correspond to inactive cells v ∈ V 0 with labels φ(v) ∈ Λ 0 and the transparent edges correspond to their adjacent edgesE 0 . The edges in the cluster S(ω) are highlighted in a tilted rectangle that is centered at the pointz =ω. Notice that all the interfering edges in the above graph can be partitioned into smaller (isomorphic) sets by translatingS(z) over all z∈ Λ 0 +ω (black vertices). . . . . . . . . . . . . . . . . . 51 3.5 The interference edges in the clusterS(a). The transmit beamform- ing vectors v a , v b and v c (gray nodes) are uniquely associated with the edges inS(a) and have to be designed such that interference is aligned at receivers a, d and e. . . . . . . . . . . . . . . . . . . . . . 53 3.6 The set of triangles [a,b,c]∈T for ˚ G(V,E). All the circle nodes belong toV in and participate in exactly three triangles (n v = 3). The setV ex contains the colored nodes on the boundary for which n v < 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.7 The minimum of the piecewise linear function 1 3 · max i {s i −λ·g i }. 63 3.8 Interference alignment solution when M = 2, and N = 3. . . . . . . 65 3.9 The directed interference graph ˚ G(V,E) divided in horizontal stripes. 66 3.10 Interference alignment solution in a single stripe when M = 2, and N = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.11 The interference clusters A and B. . . . . . . . . . . . . . . . . . . . 68 4.1 Successive decoding in the uplink versus successive encoding in the downlink. In both cases, base-station 1 (BS1) will use the backhaul to give the corresponding information to base-station 2 (BS2). In the uplink, BS2 can use ˆ W 1 to reconstruct the corresponding signal and subtract the interference coming from user 1. In the downlink, BS2 can use Q(x 1 ) and DPC to avoid interference from BS1. . . . . 73 4.2 Uplink and Downlink with reverse decoding/encoding orders,π and π. Afterthecorrespondingnetwork-wideinterferencecancellationin bothcases,theremaininginterferencechannelgainsforthedownlink are reciprocal to the ones obtained in the uplink, and are given by H uv =H H vu ,∀[v,u]∈E π . . . . . . . . . . . . . . . . . . . . . . . . . 77 xiv 4.3 Locallyconnectednetworkmodelandthecorrespondinginterference graph: This figure shows the interference topology between four transmit-receive pairs in the uplink. Green arrows represent the desired links and red arrows indicate interference. The dashed lines showthelocalbackhaulconnectionsbetweenthecorrespondingbase stations (BS). Notice that there is no backhaul connection between BS1 and BS4 since the corresponding transmit-receive pairs do not interferewitheachother. Theresulting(directed)interferencegraph is shown in the right hand side of the figure. . . . . . . . . . . . . 80 4.4 The acyclic chromatic number ofD vs the chromatic number of its symmetric graphD. Notice that in the first caseχ A (D) =χ(D) but D6≡D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 The asymmetric Wyner model. . . . . . . . . . . . . . . . . . . . . . 82 4.6 An interference graph (left) and its bipartite alignment graph (right). 83 4.7 Example: An interference graph (left), its bipartite alignment graph (middle) and the corresponding tree with root node 1. . . . . . . . . 86 4.8 The interference graph (left) of a 4-user 2× 2 IC. The dashed and solid edges illustrate the acyclic decomposition of the graph intoD N (dashed lines) andD IA (solid lines). Notice that since the bipartite alignment graph ofD IA (right) has only one cycle, Theorem 4.3 can be applied in this case to achieve 4 total DoF. . . . . . . . . . . . . 87 5.1 Channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 A simple scheme to achieve 1 DoF per user with α = 2 (K−1) K . . . . 94 5.3 Achievable DoF vsα: The dashed line corresponds to the achievable tradeoffforK-userchannelsbycentralizedprocessing(Section5.1.3) and the yellow region shows the corresponding gap from theK-user outer bound of Theorem 5.1. In Theorem 5.2 we show that coop- eration alignment is able to close this gap and achieve the optimal communication vs cooperation tradeoff for K = 3. . . . . . . . . . . 96 A.1 The beamforming choices (a), (b) and (c). In this example, all the nodes with dashed outline have chosen their beamforming vectors at random. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 A.2 Theset oftriangles [u,v,w]∈T for ˆ G V, ˆ E A ∗ . Allthecirclenodes belong toV in and participate in one triangle and the setV ex contains the colored nodes on the boundary. . . . . . . . . . . . . . . . . . . 122 xv CHAPTER1 Introduction Interference is one of the major limiting factors in the performance of today’s wireless networks. When there is no interference a single user can enjoy the entire systems spectrum and communicate with its intended receiver at a rate that scales withthelogarithmoftheavailablepower[Sha48]. Whenweaddasecondtransmit- receive pair in the network – that is competing over the same channel resources – a natural approach is to split the available spectrum into two non-overlapping parts, and allocate the first half of it to user one and the other half to user two. In that way, both users can simultaneously communicate, however, at a rate that will be 1/2 times their interference-free rate. Following the same principle, in networks with K transmit-receive pairs, every user will only be able to communicate at a fraction 1/K of its corresponding interference-free rate. Broadly speaking, this is how we deal with interference in todays wireless systems; frequency reuse, time- division-multiple access (TDMA) and other conventional interference mitigation techniques are all based on this simple intuitive approach. Looking into the future, however, the wireless communication industry is expecting an exponential growth in demands for cellular data traffic [Cis15]. Conventional techniques, such as time/frequency orthogonalization, waste valuable channel resources in order to avoid concurrent interfering transmissions, and will 1 Chapter 1 Introduction soon not be able to keep up with our rapidly increasing bandwidth demands. Clearly, one of the most important technical challenges in the design of next generation wireless systems will be the implementation of more sophisticated interference management schemes that are able to break this orthogonalization barrier and achieve a spectral efficiency that goes beyond 1/K. 1.1 Interference Alignment Recent advances in information theory [CJ08,MGMAK14,NGJV12] have shown that transmission schemes based on interference alignment [MAMK08,CJ08] are able to provide significant gains compared to conventional approaches. In many communication settings, techniques based on interference alignment are able to achieve the theoretically optimal scaling of the user rates under the presence of interference. In particular, the results of [MAMK08,CJ08] on the K-user interfer- ence channel, showed that it is possible for all users to simultaneously communicate over the same channel resources and still achieve 1/2 their interference-free rates. This is an order-of-K improvement over the conventional 1/K, which shows that wireless networks are not inherently limited by interference and, contrary to the conventional belief, their sum capacity can scale linearly with the number of users. The basic idea of interference alignment is simple enough, yet very powerful: “From each receiver’s perspective, all interference should look as if it originated from a single transmitter”. This effect can be achieved in a K-user interference channel, as long as all the transmitted signals are carefully designed to “align” their aggregate effect on each receiver’s signal space. An illustrating example is given below for the 3-user 2× 2 interference channel. Assume that each user i wants to transmit a single stream 1 x i to its intended receiver using a two-dimensional beamforming vectorv i . Thejth receiver’s two-dimensional observation is given by y j =H j1 v 1 x 1 +H j2 v 2 x 2 +H j3 v 3 x 3 +z j , j∈{1, 2, 3}, 1 For simplicity we omit the time dimension and focus on the transmission of a single symbol. 2 Chapter 1 Introduction whereH ji denotes the 2×2 channel gain matrix between transmitteri and receiver j, which is assumed to be chosen at random from a continuous distribution and remains fixed throughout the entire communication. Notice that the interference at each receiver will in general arrive along two vectors (e.g., H 12 v 2 and H 13 v 3 for receiver 1) in a two-dimensional observation space. However, if we choose v 2 and v 3 such that H 12 v 2 and H 13 v 3 are co-linear, then all interference at receiver 1 will be confined in a one-dimensional subspace, along the span ofH 12 v 2 . That is, interference will be aligned, and from receiver’s 1 perspective, it would seem as if only a single user interferes with its desired signal. Hence, choosing v 1 , v 2 and v 3 such that the same condition is simultaneously satisfied for all receivers, would yield the interference alignment solution shown in Fig. 1.1. Interference occupies half of the dimensions in each receiver’s observation space and the other half can be used for interference-free communication. user 1 user 2 user 3 v 1 v 2 v 3 H 11 v 1 H 22 v 2 H 33 v 3 Interference Alignment | 3-users Figure 1.1: Interference Alignment for the 3-user 2× 2 MIMO Interference Channel. One might argue that as we increase the number of transmit-receive pairs in the network the number of interference alignment conditions that need to be satisfied will be more than the available number of variables in the beamforming vectors. In fact a straightforward equation counting will show that even with one extra transmit-receive pair (fourth user), interference alignment in the above setting would be impossible. Surprisingly, Cadambe and Jafar [CJ08] were able 3 Chapter 1 Introduction to circumvent this problem by showing that alignment can be accomplished asymptotically, by a precoding over many parallel channels obtained from different frequencies or time slots. 1.1.1 Practical Challenges While transmission schemes based on interference alignment promise significant gains compared to conventional interference mitigation techniques, the extent to which such gains can be realized in practice has been so far limited. A crucial observation is that the topology of the network can significantly affect the deployment of interference alignment. As we have seen, interference alignment can be easily applied to a three-user interference channel, where each transmitter and receiver are equipped with two antennas. However, adding just one more user radically changes the nature of the problem. The known solutions for four-user interference channels, rely on asymptotic expansion of the signaling and forming exponentially-many 2 data streams, each carrying a vanishingly-small data rate [MAMK08, CJ08, MGMAK14, SY13]. This asymptotic interference alignment approach requires exponentially-large delay or exponentially-accurate channel estimation, which makes it unfavorable in practice. On the other hand, there are some results showing that without such exponential expansion, the DoF gain of any linear interference alignment scheme in fully connected networks vanishes [YGJK10,RLL12,BCT13]. Overcoming the exponential expansion of decoding delay or channel state resolution is one of the main (conceptual) fundamental challenges of interference alignment applied to practical scenarios, particularly in cellular systems. There are several approaches that try to resolve this problem: • Clustering: In this approach, the network is split into smaller sub-networks, where non-asymptotic interference alignment is feasible within each cluster (see for example [SHT11]). The problem with this approach is that the remaining interference, between the clusters, will eliminate the potential gain of interference alignment in large networks. 2 In fact, this grows exponentially with the square of the number of users. 4 Chapter 1 Introduction • Relaying: It is shown that in some scenarios, using relays in the network can help to align interference without expansion [NMK10,NLL10]. The major bottleneck of this approach is that the number of antennas in each relay is required to grow with the size of the network. • Feedback: Using output feedback has been shown to achieve the full DoFs without asymptotic expansion [NCNC13, PSD12,GKV14]. However, the current solutions do not scale with the number of users and therefore cannot be used in large networks. While all of the above approaches deserve more exploration and are currently the subject of extensive research, in this thesis we pursue an alternative solution which is particularly suitable for wireless cellular systems. Our approach relies on two key properties specific of cellular networks, namely, local base station cooperation via backhaul and the locality of interference. One of the main contributions of this thesis is to use backhaul collaboration in order to change the effective interference pattern in the network, such that practical interference alignment is possible. The proposed communication schemes that will be discussed in the following chapters can be applied in large cellular networks and is able to achieve the optimal DoFs without asymptotic symbol expansion. 1.2 Cooperation in Wireless Networks Cooperation of multiple base-stations with some form of data or signal sharing throughthewiredbackhaulnetworkisasubjectofintenseresearchbothintermsof information theoretic fundamental limits and in terms of practical signaling/coding schemes. Most works have focused on the setting where a central processor is connected to all the cooperating base-stations through orthogonal noiseless links of given capacity [SSPS09a,SSPSS09,HC13b]. In the uplink, all base stations can share their (quantized) received signal samples over the backhaul of the network, concentrate all these observations in a common central processor, and jointly decode the corresponding user messages. Similarly, in the downlink, all user messages can be shared across the entire network, so that base stations can cooperatively transmit the messages to the corresponding users by joint downlink precoding. This technique, which relies on full cooperation of the base stations, is 5 Chapter 1 Introduction referred to as "Network MIMO" in the literature [KFV06,FKV06,GHH + 10], and effectively reduces the system to a (network-wide) multiple-antenna multiaccess channel for the uplink, or to a multiple-antenna broadcast channel [CS03,WSS06] for the downlink. Clustered Cooperation In an effort to reduce the significant backhaul load requirements of Network MIMO, clustered cooperation has been proposed, where for the downlink [ZCA + 09, SW11, LLSW12, GAV11, GAV12, GV11] the message of each base station is shared within a local cooperation cluster of base stations, and for the uplink [LSW07,Ven07,SSPS09b,LLSW12,KZS13] the sampled (or quantized) received signal of each base station is shared within a cluster of base stations. In [SW11,LLSW12,LSW07] cooperation is defined in terms of sharing messages (for the downlink) or received signals (for the uplink) in a window of neighboring cells, while [GAV11] a generalized notion of message sharing for the downlink where any message can be shared by arbitrary M cells. This approach is further generalizedin[GAV12,GV11], whereanaveragebackhaulloadconstraintisdefined in terms of the average number of shared message per cell. A very large body of works on the finite SNR performance analysis of applied Network MIMO schemes based on local cooperation is also available in the literature, such as the large- system analysis in [HMK + 11,HTC12,HCPR12,ZH12], the uplink and downlink nested lattice structured coding techniques in [HC13a], and the simulation studies in [HTH + 09,MF09]). Process-and-Share A fundamentally different form of collaboration among base stations – that is the primary focus of of this thesis – is what we call process and share. In this class of cooperation for the uplink, base stations process all the signals that they have collected so far, including, (i) the signal sampled from its receiver antennas, and (ii) the signals it received from other base stations through the backhaul, and then form a “backhaul message” that can be shared with other base stations in a cluster. Similarly, for the downlink, each base station shares over the backhaul, a signal which is the result of jointly processing (i) its own signal, and (ii) what 6 Chapter 1 Introduction it received from other base stations over backhaul. In other words, under this class of backhaul collaboration schemes, base stations collaborate by sharing a function of all the information they have gathered up to that point, instead of simply forwarding raw data. The choice of this function is a design parameter and the corresponding signal processing can be either linear or non-linear. This approach has been used in [WT11a,WT11b,WPVW14,BMK10] to approximately characterize the capacity of two-user interference channels with limited transmitter (or receiver) collaboration, and in [SSPS09b] to derive achievable rates for local cooperation in linear Wyner-type cellular networks. 1.3 Dissertation Outline In Chapter 2, we focus on the uplink of a symmetric cellular system, where each cell is split into three sectors with orthogonal intra-sector multiple access and consider a “local” interference model, in which interference is due to transmitters in neighboring cells only. We propose a non-iterative local cooperation scheme where base stations pass to their neighbors their decoded messages such that interference from already decoded messages can be canceled. Therefore, for a given decoding order, the interference between sectors is described by a directed locally connected graph. The problem consists of maximizing the per-sector DoFs over all possible decoding orders and precoding schemes. In particular, we provide a decoding order and a one-shot interference alignment scheme able to achieve optimal per-sector DoFs, up to an additive gap due to boundary effects, that vanishes as the size of the network becomes large. Then, we extend our treatment by considering the case of inter-sector interference with joint processing of the three sector at each cell site. Finally, in order to avoid signaling schemes relying on the strength of interference, we further introduce the notion of topologically robust schemes, which are able to guaranteeaminimumrate(orDoFs)irrespectivelyofthestrengthoftheinterfering links. Towards this end, we design a different decoding order and alignment scheme which is topologically robust and still achieves the same optimum DoFs. In Chapter 3, we extend our results to networks with omni-directional cells. Thissettingyieldsaninterferencegraphtopologythatissignificantlymoreinvolved than the sectored case studied in Chapter 2, both from a theoretical and practical 7 Chapter 1 Introduction perspective. Throughout this chapter we consider the most popular wireless network configurations where users have 2 antennas, and base-stations have 2, 3 or 4 antennas. In particular, we provide linear one-shot IA schemes for the 2× 2, 2× 3 and 2× 4 cases and show the achievability of 3/4, 1 and 7/6 DoFs per user respectively. A major challenge in this setting is to provide tight upper bounds on the achievable DoFs. In general, DoFs converses under linear schemes require the solution of a non-linear discrete optimization problem over a number of variables thatgrowswiththesizeofthenetwork. Inthischapter, weproposeanewapproach to transform this challenging optimization problem to a tractable linear program (LP) with significantly fewer variables. This approach, which is of independent theoretical interest, is used in this chapter to show that the achievable 3/4 DoFs per user are indeed optimal for a large (extended) cellular network with 2×2 links. In Chapter 4 we generalize our cellular interference alignment framework in twodirections. First, inSection4.1, wefocusonthedownlinkofthecellularmodels thatwehavebeenstudiedinChapters2and3. Morespecifically, weconsideradual process-and-share framework for the cellular downlink based on dirty paper coding (DPC) that utilizes the local backhaul connections between base stations to enable transmitter cooperation and provide the same interference alignment gains. We propose a network-wide successive encoding scheme for the downlink in which base stations share their (quantized) DPC precoded signals over the backhaul in order to successively help their neighbors’ encoding process. This form of cooperation can be seen as the dual of the decoded message passing scheme for the uplink and is able to provide an uplink-downlink DoF duality result for cellular interference alignment: Any linear IA scheme that can achieved DoFs per user in the uplink of acellularsystem, canbetranslatedunderourframeworktoaduallinearIAscheme that can achieve the same DoFs in the downlink. Then, in Section 4.2, we consider general interference graph topologies and develop results that can be applied to a large class of partially connected interference networks. In particular, we study the gains of cooperation through network interference cancellation (or network-DPC forthedownlink)insingleantennanetworksfromagraphtheoreticperspectiveand identify a key structural property of multiple antenna networks that characterizes a large class of topologies for which practical one-shot IA schemes can achieve M/2 DoF per user. 8 Chapter 1 Cellular Interference Alignment Finally, in Chapter 5, inspired by the effectiveness of the decoded message- passing architecture within the cellular interference alignment framework, we aim to study the information theoretic limits of distributed interference management under the most general form of process-and-share cooperation. In particular, we propose a channel model in which receivers are able to exchange any function of the information that is currently available to them, through wired backhaul links of finite capacity. Focusing on the three-user interference channel, we characterize the fundamental tradeoff between the achievable communication rates and the corresponding backhaul cooperation overhead that is required under any cooperation scheme, in terms of degrees of freedom (DoF). Surprisingly, we show that the optimum communication-cooperation tradeoff remains the same when we move from two-user to three-user interference channels. It is well known that, in the absence of cooperation, this is due to interference alignment, which keeps the fraction of communication dimensions wasted for interference unchanged. When backhaul cooperation is available, we develop a new idea that we call cooperation alignment, which guarantees that the cooperation overhead also remains the same as we increase the number of users. The achievable scheme on the transmitters side is based on splitting each data stream into many sub-streams, each carrying a small fraction of the total DoF. At the receivers side, we consider an iterative process- and-share approach in which receivers decode, recombine and share carefully chosen sub-streams over backhaul to distributedly eliminate interference across the network. This iterative approach – that can be seen as the information theoretic limit of network interference cancellation – is shown to be asymptotically optimal and is able to outperform cooperation schemes relying in centralized decoding. 9 CHAPTER2 Cellular Interference Alignment The main goal of this chapter is to show that a practical communication scheme is able to achieve to optimal gains of interference alignment in the uplink of cellular systems with minimal cooperation. To enable interference alignment without any expansion, we consider here a non-linear process-and-share cooperation framework that is motivated by schemes embraced in practice (see [BKKR11] for an example). Whilethegeneralprocess and share frameworkencompassesmanypossiblevariants (e.g., including (i) compressed version of all available signals, (ii) entire messages, (iii) part of the messages, (iv) some combination of the message over a lattices, soft-information decoding in the form of log-likelihood ratios, as in certain iterative decoding schemes), in this work we consider a successive decoding scheme where each base station processes the signal received from its own antennas, as well as the decoded messages received through the backhaul links from (some of) its neighboring base stations which have already decoded their messages. In turns, the base station shares its decoded message locally, over the backhaul, to neighboring base stations which haven’t yet decoded their messages, in order to help their decoding process. We show that this local and one directional (non-iterative) data exchange — restricted only to decoded messages — is enough to reduce the uplink 10 Chapter 2 Cellular Interference Alignment of a sectored cellular network to a topology in which half of DoFs per transmit- receive antenna can be achieved by linear interference alignment schemes without requiring time-frequency expansion or lattice alignment. The proposed algorithm takes advantage of the partial connectivity of extended cellular networks 1 and has several desired properties that are necessary in practical applications: • Scalability: The overall performance of the scheme materializes irrespectively of the size of the cellular network, i.e., when the number of transmit-receive pairs becomes arbitrarily large. • Locality: The transmission scheme operates under local information ex- change, and exploits the distributed nature of the cellular network. • Spectral Efficiency: The scheme achieves high spectral-efficiency by allowing more (interference-free) parallel transmissions to take place within the same spectrum. The ideaof combining interference alignment withdecoded message sharing has been studied in [GPK09] for small network configurations (e.g, with three active receivers in the uplink). In contrast, here we focus on large cellular networks in whichinterferencealignmentwithoutasymptoticsymbolexpansionisnotknownto be feasible. We emphasize that locally sharing decoded information messages over the backhaul and restricting to single-user decoding (or joint three-sector decoding in the same cell) can be easily implemented within the current technology. In general, in coordinated cell processing strategies, there is always the risk that the signaling scheme relies on the strength of interference in order to achieve reliable communication. However, practical systems are not designed to guarantee that strength. On the contrary, current system deployment is geared to making interfering links as weak as possible. Hence, a scheme that relies on “strong interference” links would fail if applied to a system which was designed according to the current guidelines. In order to address this issue, we introduce the concept of topological robustness, where the goal is to design communication schemes that can maintain a minimum rate (or DoFs) no matter if the interference links are 1 Following [XK04,LT05,OJTL10] we refer to an “extended” network as a network with a fixed spatial density of cells and increasing total coverage area, in contrast to a “dense” network where the total coverage area is fixed and the cell density increases. 11 Chapter 2 Cellular Interference Alignment strong or weak. In particular, we show that such schemes exists in our framework and prove their optimality using a compound network formulation. This chapter is organized as follows. First, in Section 2.1 we describe the cellular model that we consider in this work and give a formal problem statement. Then, in Section 2.2 we state our results for networks with no intra-cell interference and give the corresponding achievability and converse theorems. In Section 2.3 we extend our model to incorporate both out-of-cell and intra-cell interference and, finally, in Section 2.4 we focus on the design and optimality of topologically robust transmission schemes. 2.1 Cellular Model Consider a large multiple-input multiple-output (MIMO) cellular network with three sectors per cell. As in current 4G cellular systems [STB09], orthogonal intra- sector multiple access is used in the uplink, such that, without loss of generality, we can consider a single user per sector, as shown in Fig. 2.1. Within each sector, the receiver is interested in decoding the uplink message of the user associated with it and observes all other transmissions as interference. We consider here a symmetric configuration in which all transmitters and receivers in the network are equipped with M antennas each, and assume frequency-flat channel gains that remain constant throughout the entire communication. Becauseofshadowingeffectsanddistance-dependentpathloss, thatareinherent to wireless communications [Mol05], we assume that the interference seen at each receiver is generated locally, by transmitters located in neighboring sectors. 2 Let S be the sector index set and letN (i) denote the set of the interfering neighbors of the ith sector. The received signals in our model can be written as y i =H ii x i + X j∈N (i) H ij x j +z i , i∈S (2.1) 2 In practice, the aggregate effect of non-neighboring transmitters is treated as noise, and contributes to the “noise floor” of the system. In [GNAJ13], necessary and sufficient conditions on the channel gain coefficients of a Gaussian K-user interference channels are found such that “treating interference as noise” (TIN) is approximately optimal in the sense that, subject to these conditions, the TIN-achievable region is within an SNR-independent gap of the capacity region. 12 Chapter 2 Cellular Interference Alignment 4 (a) Cellular network (b) Interference graph Fig. 1: The cellular network topology and the corresponding interference graph: we consider the uplink of a MIMO cellular network with 120 o sector receivers as depicted in Fig. 1a. Each receiver is interested in decoding the message of the mobile terminal associated with it and observes all other transmissions as interference. In our cellular model we assume four dominant sources of interference for each sector shown as orange arrows originating from its closest out-of-cell transmitters. Interference between the sectors of the same cell is depicted with black arrows. Fig. 1b shows the corresponding interference graph by taking into account all interfering links in a given cellular network, in which vertices represent transmit-receive pairs within sectors and edges indicate interfering neighbors. The dashed black edges in the above graph correspond to interference between sectors of the same cell that we are going to ignore until Section IV. Because of shadowing effects and distance-dependent pathloss, that are inherent to wireless communications [36], we assume that the interference seen at each receiver is generated locally, by transmitters located in neighboring sectors. 3 LetS be the sector index set and letN(i) denote the set of the interfering neighbors of the ith sector. The received signals in our model can be written as y i =H ii x i + X j2 N(i) H ij x j +z i ,i2 S (1) 3 In practice, the aggregate effect of non-neighboring transmitters contributes to the “noise floor” of the system. In [37], necessary and sufficient conditions on the channel gain coefficients of a GaussianK-user interference channels are found such that “treating interference as noise” (TIN) is approximately optimal in the sense that, subject to these conditions, the TIN-achievable region is within an SNR-independent gap of the capacity region. Figure 2.1: The cellular network topology and the corresponding interference graph: we consider the uplink of a MIMO cellular network with 120 o sector receivers as depicted in Fig. 2.1a. Each receiver is interested in decoding the message of the mobile terminal associated with it and observes all other transmissions as interference. In our cellular model we assume four dominant sources of interference for each sector shown as orange arrows originating from its closest out-of-cell transmitters. Interference between the sectors of the same cell is depicted with black arrows. Fig. 2.1b shows the corresponding interference graph by taking into account all interfering links in a given cellular network, in which vertices represent transmit-receive pairs within sectors and edges indicate interfering neighbors. The dashed black edges in the above graph correspond to interference between sectors of the same cell that we are going to ignore until Section 2.3. where H ij is the M × M matrix of channel gains between the transmitter (user terminal) associated with sector j and the receiver of sector i and x i are the corresponding transmitted signals satisfying the average power constraint E h ||x i || 2 i ≤P. In this chapter, we will consider two interference models based on the choice of the setsN (i), i∈S. In the first part, we will assume that the sectors located in the same cell do not interfere with each other and focus only on interference generated by nearby out-of-cell transmitters. This assumption can be motivated by taking into account the physical orientation and radiation patterns of the antennas used in sectored cellular systems, where the interference power from users in different sectors of the same cell should be much less than the interference power observed 13 Chapter 2 Cellular Interference Alignment from out-of-cell users located in the sector’s line of sight. Then, in Section 2.3, we aregoingtoliftthisassumptionandconsiderthecasewheresectorreceiversobserve both out-of-cell and intra-cell interference. This extension takes into account the fact that users near the sector boundary may produce significant interference to the neighboring sector in the same cell, due to possibly non-ideal sectored antenna radiation patterns. 2.1.1 Interference Graph A useful representation of our cellular model can be given by the corresponding interference graphG(V,E) shown in Fig. 2.1b. In this graph, vertices represent transmit-receive pairs within each sector and edges indicate interfering neighboring links: the transmitter associated with a node u ∈ V causes interference to all receivers associated with nodes v∈V if there is an edge (u,v)∈E. Notice that the interference graph is undirected and hence interference between sectors in our model goes in both directions. More formally, we can define the interference graphG(V,E) as follows. First, we are going to define the setV through a one-to-one mapping between the vertices of the graph and a set of complex numbers that we will refer to as node labels. The real and imaginary parts of these labels can be interpreted as the coordinates of the corresponding nodes embedded on the complex plane in a way that resembles the specific sector layout of our cellular system. A natural choice for this labeling is the set of the Eisenstein integersZ(ω) that exhibits the hexagonal lattice structure shown in Fig. 2.2. Definition 2.1 (Eisenstein integers). The set of Eisenstein integers, denoted as Z(ω), is formed by all complex numbers of the formz =a +bω, wherea,b∈Z and ω = 1 2 (−1 +i √ 3). DefineB r ,{z∈C:|Re(z)|≤r,|Im(z)|≤ √ 3r 2 } and let φ :V→Z(ω)∩B r be a one-to-one mapping between the elements ofV and the set of bounded Eisenstein integers given by Z(ω)∩B r . For any v∈V we say that φ(v) is the label of the corresponding vertex in the interference graph. Correspondingly, the set of vertices V is given by V = n φ −1 (z) :z∈Z(ω)∩B r o . (2.2) 14 Chapter 2 Cellular Interference Alignment Re Im ! 1 3+2! (!) Figure 2.2: The Eisenstein integers Z(ω) on the complex plane. We now explicitly describe the set of edgesE in the interference graph in terms of the function φ. Consider the set of three segments in C Δ(z) ={(z,z +ω), (z,z +ω + 1), (z +ω,z +ω + 1)} and define the set D, [ a,b∈Z: [a+b]mod 36=0 Δ(a +bω) (2.3) to be the union of Δ(a +bω) over alla,b∈Z such that [a +b] mod 36= 0. Observe that the segments in Δ(z) form a triangle with vertices in the Eisenstein integers z,z +ω andz +ω +1, as shown in Fig. 2.2. The functionf(a+bω), [a +b] mod 3 partitions the hexagonal lattice Z(ω) into three cosets. In particular, all points z such that f(z) = 0 form a sublattice Λ 0 of Z(ω), and the points z for which f(z) = 1 and f(z) = 2 corresponds to its cosets Λ 0 + 1 and Λ 0 − 1. In Fig. 2.2, the points of Λ 0 , Λ 0 + 1 and Λ 0 − 1 are shown with squares, circles and diamonds, respectively. Without loss of generality, we assume that for allz∈ Λ 0 the segments in Δ(z)correspondtolinksbetweenthethreesectorsofthesamecell. Hence, under the assumption that such sectors do not interfere, we exclude the corresponding {Δ(z) : z∈ Λ 0 } in the definition ofD in Eq.2.3. Eventually, the set of edgesE representing out-of-cell interference is given by E ={(u,v) :u,v∈V and (φ(u),φ(v))∈D}. (2.4) 15 Chapter 2 Cellular Interference Alignment Definition 2.2 (Interference Graph). The out-of-cell interference graphG(V,E) is an undirected graph defined by the set of verticesV given in Eq.2.2 and the corresponding set of edgesE given in Eq.2.4. The graph vertices represent transmit- receive pairs in our cellular model and edges indicate interfering neighbors. 2.1.2 Network Interference Cancellation We further consider a message-passing network architecture for our cellular system, in which sector receivers communicate locally in order to exchange decoded messages. Any receiver that has already decoded its own user’s message can use the backhaul of the network and pass it as side information to one or more of its neighbors. In turn, the neighboring sectors can use the received decoded messages in order to reconstruct the corresponding interfering signals and subtract them from their observation. It is important to note that this scheme only requires sharing (decoded) information messages between sector receivers and does not require sharing the baseband signal samples, which is much more demanding for the backbone network. Theaboveoperationeffectivelycancelsinterferenceinonedirection: alldecoded messages propagate through the backhaul of the network, successively eliminating certain interfering links between neighboring sectors according to a specified decoding order. Fig. 2.3 illustrates the above network interference cancellation process in our cellular graph model assuming a “left-to-right, top-down” decoding order. Notice that edges are now directed in order to indicate the interference flow over the network. For example, if an undirected edge (u,v) exists inE and, under this message-passing architecture, node v decodes its message before node u and passes it to node u through the backhaul, then the resulting interference graph will contain the directed link [u,v], indicating that the interference is from node (sector) u node (sector) v only. A decoding order π can be specified by defining a partial order “≺ π ” over the set of verticesV in our interference graph. Then, the message of the user associated with vertex v∈V will be decoded before the one associated with vertex u∈V if v≺ π u. In principle, we can choose any decoding order that partially orders the setV and hence π can be treated as an optimization parameter in our model. 16 Chapter 2 Cellular Interference Alignment 2 1 5 7 11 12 14 15 17 19 18 21 24 25 26 28 31 32 34 3 6 10 13 16 20 23 27 30 33 9 8 29 22 4 Figure 2.3: The directed interference graphG π ∗(V,E π ∗) after network interference can- cellation according to the “left-to-right, top-down”decodingorderπ ∗ . Thetransmitterof a sector associated with nodei causes interference only to its neighboring sector receivers j with j <i. Orange arrows indicate out-of-cell interference while black dashed arrows correspond to interference from within each cell. Definition 2.3 (Directed Interference GraphG π ). For a given partial order “≺ π ” onV, the directed interference graph is defined asG π (V,E π ) whereE π is a set of ordered pairs [u,v] given byE π ={[u,v] : (u,v)∈E and v≺ π u}. Next, we formally specify the “left-to-right, top-down” decoding order π ∗ that has been chosen in Fig. 2.3. As we will show in the following section, this decoding order is indeed optimum and can lead to the maximum possible DoF per user in large cellular networks. Definition 2.4 (The Decoding Order π ∗ ). The “left-to-right, top-down” decoding order π ∗ is defined by 3 v≺ π ∗ u⇔ Im (φ(v))> Im (φ(u)), or Im (φ(v)) = Im (φ(u))and Re (φ(v))< Re (φ(u)), for all u,v∈V. 3 Notice that π ∗ is a total order of the setV. 17 Chapter 2 Cellular Interference Alignment 2.1.3 Problem Statement Here we define the achievable rate region of the cellular systemG(V,E) under the network interference cancellation framework introduced in the previous subsection. Definition 2.5 (Achievable Rates). The rates{R v , v∈V}∈R (π) are achievable inG(V,E) under the network interference cancellation framework with decoding order π if, for any > 0 and sufficiently large n, there exist a length-n coding scheme defined by: • The message setsW v ={1, 2,..., 2 nRv }, v∈V. • The encoding functions f (n,π) v :W v →C M×n , v∈V, such that X n v =f (n,π) v (W v ), 1 n E h ||X i || 2 F i ≤P, • The decoding functions g (n,π) v :C M×n × Y u≺πv,(u,v)∈E W u →W v , that give ˆ W v ,g (n,π) v (Y n v ,{ ˆ W u :u≺ π v, (u,v)∈E}), such that the corresponding probability of error given by P (n,π) e , max v n P { ˆ W v 6=W v } o , is less that. The achievable rate regionR is defined as the union over all possible decoding orders π given byR = S π R (π) . Our main goal is to design efficient communication schemes for the cellular networks that fall within the above network interference cancellation framework. As a first-order approximation of a scheme’s efficiency, we will consider here the achievable DoFs that can be broadly defined as the number of point-to-point interference-free channels that can be created between transmit-receive pairs in the network. More specifically, for the achievable schemes, we are going to limit ourselves to linear beamforming strategies over multiple antennas assuming constant (frequency-flat) channel gains without allowing symbol extensions. We refer to 18 Chapter 2 Cellular Interference Alignment such schemes as “one-shot”, indicating that precoding is achieved over a single time-frequencyslot(symbol-by-symbol). Ourgoalittomaximize, overalldecoding orders π, the average (per sector) achievable DoFs d G,π , 1 |V| X v∈V d v , (2.5) whereG(V,E) is the interference graph defined in Section 2.1.1 andd v denotes the DoFs achieved by the transmit-receive pair associated with the node v∈V, where d v = lim P→∞ R v (P ) log(P ) , and{R v (P ),v∈V}∈R (π) is an achievable rate vector for a decoding orderπ and transmit power constraint P. 2.2 Networks with No Intra-cell Interference Here we state our main results for the case where there is no interference between the sectors of the same cell. It is worth pointing out that in this section we do not assume any form of collaboration between sector receivers other than the message passingschemedescribedinSection2.1.2. Themainresultsofthissectionaregiven by the following achievability and converse theorems. For the sake of clarity and in order to build intuition on the achievability coding scheme we treat in detail the case of two-antenna terminals (M = 2) in Section 2.2.1 and provide the complete proof in Appendix A.1. The proof of the converse theorem is given in Section 2.2.2. Theorem 2.1. For a sectored cellular system G(V,E) in which transmitters and receivers are equipped with M antennas each, there exist a one-shot linear beamforming scheme that achieves the average (per sector) DoFs d G,π ∗ = M 2 , M is even M 2 − 1 6 , M is odd (2.6) under the network interference cancellation framework with decoding order π ∗ . 19 Chapter 2 Cellular Interference Alignment Remark 2.1. Notice that d G,π ∗ is not exactly M/2 for odd values of M. This is because we have insisted on one-shot schemes. By precoding over two time- frequency varying slots it is not difficult to show that M/2 DoFs per sector are indeed achievable also for odd M. Theorem 2.2. For a sectored cellular systemG(V,E) without intra-cell interfer- ence, in which transmitters and receivers are equipped with M antennas each, for any network interference cancellation decoding order π, the average (per sector) DoFs d G,π are upper bounded by M/2 +O (1/ √ |V|). The above theorems yield a tight DoFs result for large extended cellular networks, for which|V| → ∞. The termO (1/ √ |V|) comes from the fact that sectors on the boundary observe less interference, and therefore can achieve higher DoFs. However, the number of sectors on the boundary is small compared to the total number or sectors|V|, and therefore, their effect vanishes as the size of the network increases. Remark 2.2. We should point out, that our converse holds for a general class of achievability schemes (formally described in Section 2.1.3) and is not necessarily restricted to one-shot linear beamforming/alignment strategies. In short, this class of solutions requires each receiver to first decode its own message entirely and then share it with its neighbors. Notice that this form of collaboration allows for transmission schemes with any symbol expansion across time, but it rules out strategies in which messages are partially decoded and shared between receivers. 2.2.1 Achievability For the purpose of illustrating our main ideas, we will consider here the case where sector receivers and mobile terminal transmitters are equipped with M = 2 antennas and describe the linear beamforming scheme that is able to achieve one DoF per link for the entire network. Consider the directed interference graph G π ∗(V,E π ∗) shown in Fig. 2.3 and assume that all user terminals v∈V are simultaneously transmitting their signals x v to their corresponding receivers. Recall that each sector receiver that is able to decode its own message, is also able to pass it as side information to its neighbors, effectively eliminating interference in that direction. Hence, following the “left-to- right, top-down” decoding orderπ ∗ introduced in Section 2.1.2, the sector receiver 20 Chapter 2 Cellular Interference Alignment associatedwiththenodeu∈V isabletoeliminateinterferencefromallneighboring sectors v≺ π ∗ u and attempt to decode its own message from the two-dimensional received signal observation y u given by y u =H uu x u + X v:[v,u]∈E π ∗ H uv x v +z u . (2.7) Our goal is to design the transmitted signals x v such that all interference observed in y u is aligned in one dimension for every sector receiver u in our cellular system. Let u u and v u denote the 2-dimensional receive and transmit beamforming vectors associated with node u ∈ V and assume that every user terminal in the network has encoded its message in the corresponding codeword. Although codewords span many slots (in time), we focus here on a single slot and denote the corresponding coded symbol of user u by s u . Then, the vector transmitted by user u is given by x u = v u s u and each receiver can project its observation y u along u u to obtain ˆ y u =u H u H uu v u s u + X v:[v,u]∈E π ∗ u H u H uv v v s v + ˆ z u . We will show next that it is possible to designu u andv u across the entire network G π ∗(V,E π ∗) such that the following interference alignment conditions are satisfied: u H u H uu v u 6= 0, ∀u∈V and (2.8) u H u H uv v v = 0, ∀[v,u]∈E π ∗. (2.9) Hence, each receiver in the network can decode its own desired symbol s u from an interference-free channel observation of the form ˆ y u = ˆ h u s u + ˆ z u (2.10) where ˆ h u =u H u H uu v u and ˆ z u =u H u z u . Inordertodescribethealignmentprecodingscheme, wewillpartitionthenodes inG π ∗(V,E π ∗) into three sets based on their interference in-degree, defined as the number of incoming interfering links. Notice that in Fig. 2.3 all the square nodes observeatmostthreeincominginterferinglinks, whilethein-degreesofall diamond 21 Chapter 2 Cellular Interference Alignment v c v a v b v d d c b e H ea H eb H ad H ab H ac H dc a d c b e a Figure 2.4: Out-of-Cell Interference in the neighborhood of a blue node. Sectors are labeled here with letters to avoid confusion with the underlying decoding order (cf. Fig. 2.3). and circle nodes are at most two and one respectively. LetV square ={v : φ(v)∈ Λ 0 },V circle ={v :φ(v)∈ Λ 0 +1} andV diamond ={v :φ(v)∈ Λ 0 −1} denote the sets of square, diamond, and circle nodes respectively, as introduced in Section 2.1.1. First we are going to propose an interference alignment solution for a small part ofthe network thatwewillrefer toas the neighborhood ofa squarenode, denotedas S(u), u∈V square , and then explain how this solution can be extended and applied in the entire network. Fig. 2.4 shows the interfering links and transmit-receive pairs that belong to the neighborhoodS(a). Intheaboveneighborhood, thegoalistodesignthe 2-dimensionalbeamforming vectors v a , v b , v c and v d such that all interference occupies a single dimension in everyreceiver. Wewillhencerequirethatspan(H ea v a ) = span(H eb v b )forreceivere and span(H ab v b ) = span(H ac v c ) = span(H ad v d ) for receiver a. These interference alignment conditions can be satisfied if we choose: v a . = H −1 ea H eb v b (2.11) v b . = H −1 ab H ac v c (2.12) v c . = H −1 ac H ad v d , (2.13) where v . = u is a shorthand notation for v∈ span(u). Notice that in the above solution the beamforming vectorsv a ,v b andv c depend on the chosen direction for v d . This is a key observation in order to embed the above beamforming strategy in the entire network. 22 Chapter 2 Cellular Interference Alignment b c a 1 2 3 1 2 3 1 2 3 Figure 2.5: Interference Alignment Scheme. The alignment conditions are depicted here with arrows connecting interfering streams that have to be aligned. The direction of the arrows show the corresponding beamforming dependencies (e.g., the arrow labeled with the number 1 requires that v a is chosen as a function of v b ). All the transmitters associated with square nodes a∈V square can choose their beamforming vectorsv a such that the first alignment condition Eq.2.11 is satisfied in every neighborhoodS(a). This beamforming choice is shown in Fig. 2.5 with an arrow labeled with the number 1, connecting the two interfering links that have to be aligned. The direction of the arrow indicates that v a has been chosen as a function ofv b . In a similar fashion, following the arrows labeled with the number 2, every circle nodeb∈V circle can beamform to satisfy the second alignment condition Eq.2.12 by choosing v b as a function of v c . Now, in order to ensure that the third condition Eq.2.13 is also satisfied in every neighborhood we can choose the beamformingvectorsofdiamondnodesc∈V diamond accordingtothearrowslabeled with the number 3, as shown in Fig 2.5. Notice that, from each neighborhood’s perspective, v c is chosen as a function of an arbitrary vector v d that has in turn been chosen to satisfy an alignment condition in a different neighborhood. Following this procedure, all the transmit- ters are able design their beamforming vectors sequentially, as functions of their neighbors’ choices, starting from the boundary of the network. It is not hard to verify that with the above beamforming strategy, every receiver in the network will observe all interference aligned in one dimension that can subsequently be zero-forced in order to obtain an observation in the form of Eq.2.10. In that way, under the network interference cancellation framework with decoding order π ∗ , 23 Chapter 2 Cellular Interference Alignment all transmit-receive pairs inG π ∗(V,E π ∗) can successively create a one-dimensional interference-free channel for communication and hence achieve d v = 1,∀v∈V. 2.2.2 Converse (Proof of Theorem 2.2) In the previous section we described a linear one-shot beamforming scheme that can be applied inG(V,E) whenM = 2, and achieved v = 1, for allv∈V, following the “left-to-right, top-down” decoding order π ∗ . Here, we are going to show that the above DoFs are almost optimal for our cellular network in the sense that for any decoding orderπ, the average achievable (per sector) DoFsd G,π are upper bounded by M/2 +O (1/ √ |V|). In the following lemma, we show that under the class of schemes described in Section 2.1.3, any two interfering sectors inG(V,E) cannot achieve more than M DoF. Intuitively, we can argue that the effect of network interference cancellation between any two interfering sectors inG(V,E) is at most able to induce aZ-channel betweenthesesectors, whichcorrespondstoadirectedinterferingedgeinG π (V,E π ). Lemma 2.1 (DoF achievability conditions). For any network interference cancel- lation decoding order π, the achievable DoF inG(V,E) must satisfy: 0≤d v ≤M, ∀v∈V, (2.14) d u +d v ≤M, ∀(u,v)∈E. (2.15) Proof. Consider any two sectorsu,v∈V with (u,v)∈E and an arbitrary decoding orderπ. Without loss of generality we will assume thatu≺ π v. Let ˆ W b (j),{ ˆ W i : i≺ π j, (i,j)∈E} denote the decoded messages available through the backhaul to receiver j∈V. Under the class of cooperation schemes defined in Section 2.1.3, the decoding functions of receivers u and v are given by ˆ W u =g (n,π) u (Y n u , ˆ W b (u)) and ˆ W v =g (n,π) v (Y n v , ˆ W b (v)). Notice that, since u ≺ π v, we have that ˆ W u ∈ ˆ W b (v) and ˆ W v / ∈ ˆ W b (u). Assuming that a genie is able to give to both receivers all the messages in the network except W u and W v , and following standard arguments (e.g, as in [CJ08, Lemma 1]), we can upper bound the achievable degrees of freedom d u and d v under our framework by considering a two user MIMOZ-channel where no further 24 Chapter 2 Cellular Interference Alignment cooperationisallowedbetweenthereceivers. Hence, applyingtheZ-channelbound for every pair of neighboring vertices in in the network, we obtain the necessary DoF achievability conditions Eq.2.14 and Eq.2.15 as stated by the lemma. Now, given the result of Lemma 2.1, we can obtain an upper bound on the average achievable DoFs inG(V,E), by considering the optimization problem Q 1 (G) : maximize {dv :v∈V} 1 |V| X v∈V d v subject to: Eq.2.14,Eq.2.15. In particular, we will derive an upper bound ˆ d G for the optimal value of Q 1 (G), such that ˆ d G ≥ opt(Q 1 (G)), and show that ˆ d G = 1 +O (1/ √ |V|). Notice that due to Lemma 2.1, the corresponding bound will hold for all possible network interference cancellation schemes and decoding orders. As a first step, we are going to rewrite the sum in the objective of Q 1 (G) as a sum over connected vertex triplets [u,v,w] that we are going to call the triangles T of our graph. u v w Figure 2.6: The set of triangles [u,v,w]∈T forG(V,E). All the circle nodes belong toV in and participate in exactly one triangle and the setV ex contains the remaining (colored) nodes on the boundary of the graph. 25 Chapter 2 Cellular Interference Alignment In order to formally describe the set of trianglesT inG(V,E), we consider the set of ordered Eisenstein integer triplets P ={[z,z +ω,z +ω + 1] :z∈ Λ 0 − 1}. Recall from Section 2.1.1 that when z∈Z(ω)∈ Λ 0 − 1, the points z, z +ω and z +ω + 1 form the line segments Δ(z)⊆D and the corresponding graph vertices φ −1 (z), φ −1 (z +ω) and φ −1 (z +ω + 1) form a connected triangle inG(V,E). We can hence define the set of vertex triangles as T ={[u,v,w] : [φ(u),φ(v),φ(w)]∈P,u,v,w∈V}. (2.16) The above definition is illustrated in Fig. 2.6 in which shaded triangles connect the corresponding vertex triplets [u,v,w] ∈ T . Notice that apart from some vertices on the external boundary of the graph, all other nodes participate in exactlyonetriangleinT andhencewecanrewritethesumintheobjectivefunction of Q 1 (G) as a sum overT instead ofV. LetV ex denote the set of external vertices that lie on the outside boundary of our graph and do not participate in any triangle inT and letV in ,V\V ex denote the internal nodes respectively. It is not hard to see that P v∈V in d v = P [i,j,k]∈T (d i +d j +d k ), and therefore the objective function of Q 1 (G) can be written as 1 |V| X v∈V d v = 1 |V| X [i,j,k]∈T (d i +d j +d k ) + X u∈Vex d u . (2.17) Now, taking into account the constraints Eq.2.14 and Eq.2.15, we can upper bound Eq.2.17 as follows. Since d v ≤ M, ∀v ∈ V, we can upper bound the second term in the right-hand side of Eq.2.17 by M|V ex |. Further, sinced u +d v ≤ M, ∀(u,v)∈E, we have that 2(d i +d j +d k )≤ 3M, and hence we can upper bound the first term by 3|T| M 2 =|V in | M 2 ≤|V| M 2 . Therefore, the optimal value of Q 1 (G) is bounded by opt(Q 1 (G))≤ M 2 + M|V ex | |V| . (2.18) 26 Chapter 2 Cellular Interference Alignment Finally, we can use the following lemma to show that M|V ex |/|V| =O (1/ √ |V|) and hence conclude that the average (per sector) achievable DoF inG(V,E) are indeed bounded by 1 |V| X v∈V d v ≤ M 2 +O (1/ √ |V|), as stated by Theorem 2.2. Lemma 2.2. By construction, the interference graph G(V,E) satisfies |V ex | = O q |V| . Proof. The proof of this lemma follows directly from the construction of the graph and the choice of the setT shown in Fig. 2.6. In order to bound the cardinality of V ex , we can argue that the number of vertices on the boundary ofG(V,E) will scale proportionally to the circumference of a square while the total number of vertices will scale proportionally to the corresponding area. Hence, as|V| increases, the number of external vertices|V ex | is bounded byO q |V| . For a more detailed proof, we refer the reader to Appendix A.2. 2.3 Networks with Intra-Cell Interference In this section we extend our cellular model to incorporate both out-of-cell and intra-cell interference. Namely we will assume here that a sector receiver observes interference not only from its out-of-cell neighbors but also from the other transmitters located within the same cell. These intra-cell interfering links are shown as black arrows in Fig. 2.1a and correspond to the dashed edges in the interference graph shown in Fig. 2.1b. The interference graph, denoted here as ˆ G V, ˆ E , is the same as the graph defined in Section 2.1.1 with the only difference that the set ˆ E now includes both out-of-cell and intra-cell interference edges. Similarly we can define the directed interference graph ˆ G π V, ˆ E π for any network interference cancellation decoding order π. We will see next that these additional interfering links in ˆ E do not affect the achievable DoFs in our cellular system as long as we allow the sectors of each cell to 27 Chapter 2 Cellular Interference Alignment jointly process their received signals. 4 Again, we state here our main achievability result and focus on the case where M = 2 in Section 2.3.1, while the full proof is postponed to Appendix A.3. Theorem 2.3. For a sectored cellular system ˆ G V, ˆ E in which transmitters and receivers are equipped with M antennas each, there exists a one-shot linear beamforming scheme that achieves the average (per sector) DoFs d ˆ G,π ∗ = M 2 , M is even M 2 − 1 6 , M is odd (2.19) under the network interference cancellation framework with decoding orderπ ∗ , and with joint processing within the sectors of each cell. 2.3.1 Joint intra-cell processing scheme Consider the beamforming scheme described in Section 2.2.1 for M = 2 and focus on the cell{a,b,c} shown in Fig. 2.7. Without loss of generality, we will describe here how to jointly process the received observations iny a ,y b andy c such that all intra-cell interference can be eliminated and show that under the network interference cancellation framework and the beamforming choices of Section 2.2.1, every sector receiver in the network is able to decode its own desired message. a d c b Figure 2.7: Intra-Cell Interference Elimination. The sectors of each cell can jointly process their received signals and successively decode their desired messages. 4 It is interesting to notice that joint sector processing at the same cell base station site is implemented in current technology. 28 Chapter 2 Cellular Interference Alignment According to the “left-to-right, top-down” decoding orderπ ∗ , at the time when sector a attempts to decode, all the interfering links from transmitters located “above and to the left” of a have already been eliminated. As we can also see in Fig. 2.7, sector a will observe intra-cell interference from sectors b and c, and out-of-cell interference from sectord. Hence, the received signal available to sector a is given by y a =H aa v a s a + X u∈{b,c,d} H au v u s u +z a . (2.20) At the same time, the receivers b and c will be observing interference from all their neighboring sectors that have not decoded their messages yet. Notice however that with the specific beamforming choices described in Section 2.2.1, all interference that comes from sectors whose messages will be decoded after sectors b and c according to π ∗ , occupy a single dimension in each receiver and can hence be zero-forced. It is only the transmitter associated with sector d that is going to cause interference after the projection. Therefore, the corresponding observations from sectors b and c that are available when receiver a attempts to decode are given by u H b y b =u H b H bb v b s b + X u∈{a,c} u H b H bu v u s u +u H b z b , (2.21) u H c y c =u H c H cc v c s c + X u∈{a,b,d} u H c H cu v u s u +u H c z c . (2.22) We will see next that the cell with sectors{a,b,c} can jointly process the above observations such that all the corresponding sector receivers will be able to decode their desired messages in the order s a ,s b ,s c specified by π ∗ . Indeed, if we let s = [s a ,s b ,s c ,s d ] T , the observations Eq.2.20, Eq.2.21 and Eq.2.22 can be written in vector form as ˜ y = ˜ Hs +˜ z (2.23) where ˜ y = [y a ,u H b y b ,u H c y c ] T , ˜ z = [z a ,u H b z b ,u H c z c ] T 29 Chapter 2 Cellular Interference Alignment and ˜ H = H aa v a H ab v b H ac v c H ad v d u H b H ba v a u H b H bb v b u H b H bc v c 0 u H c H ca v a u H c H cb v b u H c H cc v c u H c H cd v d . Now, assuming that the channel matrices in our cellular network are chosen at random from an absolutely continuous joint probability distribution, we can show that ˜ H∈ C 4×4 is full rank with probability one. One can check that the beamforming vectors u i and v j do not depend on the above channel realizations (since by construction are based on expressions that only involve out-of-cell interference), and therefore the elements of ˜ H satisfy P h det( ˜ H)6= 0 i = 1. Hence, the given cell can always decode the corresponding messages from˜ y in the required order, as soon as the observations Eq.2.20, Eq.2.21 and Eq.2.22 become available to sectors a, b and c. In order to state the decoding process more explicitly, consider Q = [q 1 ,q 2 ,q 3 ,q 4 ] to be the unitary matrix obtained by the QL-decomposition of ˜ H such that Q H ˜ H = ` 11 0 0 0 ` 21 ` 22 0 0 ` 31 ` 32 ` 33 0 ` 41 ` 42 ` 43 ` 44 . The sector receivers a, b and c can first project ˜ y along q 1 , q 2 and q 3 in order to obtain their corresponding observations in the form y 0 a =` 11 s a +z 0 1 (2.24) y 0 b =` 21 s a +` 22 s b +z 0 2 (2.25) y 0 c =` 31 s a +` 32 s b +` 33 s c +z 0 3 , (2.26) and then successively decode their desired messages s a , s b ands c according to the specified order. In general, the above observations can be generated for every cell in the network just before their first sector receiver attempts to decode. Therefore, following the “left-to-right, top-down” decoding order π ∗ , all the sectors in ˆ G π ∗ V, ˆ E π ∗ can 30 Chapter 2 Cellular Interference Alignment decode their desired messages using the above procedure and hence the average (per sector) DoFs d ˆ G,π ∗ = 1 are achievable. 2.4 Topological Robustness In this section we introduce the concept of topological robustness for interference networks. Broadly speaking, an achievable scheme is said to be robust with respect to a network topology if its performance does not depend on the existence (or strength) of interference. This is a very important property to take into account if we want to apply a communication scheme in practice. Cellular systems are in principle designed such that most interfering links are weak and hence any scheme that solely depends on the existence (or strength) of interference will fail whenever the corresponding links are missing (or weak). Consider for example the achievable scheme described in Section 2.3.1 and assume that for a given channel realization all interference observed at receivera is zero. In this network instance, depicted in Fig. 2.8, the equivalent channel matrix a d c b Figure 2.8: The achievable scheme of Section 2.3.1 is not topologically robust. Sectorb will not able to decode its own desired message if Sector a does not observe interference from its neighbors. in the joint receiver observation for the cell{a,b,c}, given by ˜ H 0 = H aa v a 0 2×1 0 2×1 0 2×1 0 u H b H bb v b u H b H bc v c 0 0 u H c H cb v b u H c H cc v c u H c H cd v d , 31 Chapter 2 Cellular Interference Alignment is rank-deficient and hence the desired transmitted messages cannot be resolved from Eq.2.23. Even though sector a can always decode its own message, its observation cannot help sectors b and c eliminate the remaining interference, and therefore sector b cannot decode its message. It is not surprising that the above scheme fails in this case; the receiver has been designed to rely on a specific interference topology (cf. Fig. 2.7) in order obtain the required linearly-independent observations. Whenever the corresponding links are missing, the decoding process fails and therefore the scheme proposed in Section 2.3.1 cannot be considered topologically robust. In practice, interference will never be exactly zero as in the previous example. However, any communication scheme that critically depends on sufficiently strong interfering links (e.g. , such that the corresponding messages can be decoded and interference can be canceled) will suffer from significant noise enhancement in the decoding process whenever the corresponding channel gains are below a certain threshold. In this case the corresponding receiver will not be able to decode within the operating SNR range of the network and the weaker interference links will become the bottleneck in its performance. Under this framework, one could consider all possible channel realizations and design a family of transmission schemes, each one specifically optimized for the corresponding interference topology. Even though this is a tractable approach for small networks, it becomes more challenging as the size of the network increases. Here, we take a unified approach and propose a topologically robust transmission scheme for large cellular systems that is able to maintain the same performance for all network configurations, no matter if the interference links are strong or weak. 2.4.1 The Compound Cellular Network In order to formally capture the concept of topological robustness in our cellular model, we will consider here a compound scenario in which any subset of the interfering links could be potentially missing from the network. More precisely, we focus on the sectored cellular system ˆ G(V, ˆ E) defined in Section 2.3 and we assume that every directed edge [v,u]∈ ˆ E is associated with a binary channel- state parameter α uv ∈{0, 1} that determines whether the corresponding link will exist in the network or not. 32 Chapter 2 Cellular Interference Alignment The compound channel matrices are generated in the form of α uv ·H uv ,∀[v,u]∈ ˆ E, as a function of the channel-state configuration A, n α uv ∈{0, 1} : [v,u]∈ ˆ E o , (2.27) and the compound cellular network is defined over all possible choices ofA ∈ {0, 1} 2| ˆ E| . We assume that the channel-state configuration A is known to all receivers but is a priori unavailable 5 to the transmitters in the above compound network, in the sense that the interference alignment precoding scheme (although a function of the channel matricesH uv and of the interference graph ˆ G(V, ˆ E)) must be designed irrespectively ofA. A topologically robust transmission scheme is required to maintain the same performance for all channel-state parametersA∈{0, 1} 2| ˆ E| . Let ˆ G(V, ˆ E A) be the interference graph generated in the above compound network when the channel- state is A and let d ˆ G (A) denote the average (per sector) DoFs achievable in ˆ G(V, ˆ E A). Definition 2.6. A communication scheme designed for a sectored cellular system ˆ G(V, ˆ E) is said to be topologically robust with robustness level d > 0, if it can achieve d ˆ G (A)≥d, for all channel-state configurationsA∈{0, 1} 2| ˆ E| . As we have seen before, the achievable scheme described in Section 2.3.1 is not topologically robust according to the above definition: even though it can achieve d ˆ G (A) = 1, when α uv = 1,∀(u,v)∈ ˆ E, there exists a configurationA 0 , shown in Fig. 2.8, in which the decoding process fails. Underthisframework,weareinterestedinthedesignofcommunicationschemes that maximize the compound DoFs, d C , min A∈{0,1} | ˆ E| d ˆ G (A). (2.28) Notice that a topologically robust scheme with robustness level d achieves (by definition) the compound DoFs d C =d. 5 It is important to note that in the original set-up, the channel state information is available at the transmitters, and therefore obviously the channel is not compound. However, this rather artificial compound model allows us to design a unified achievable scheme, which works, independent of the strength of interference links. 33 Chapter 2 Cellular Interference Alignment The following theorems show the existence of topologically robust schemes that achieve the optimum compound DoFs performance, which coincides with the optimum DoFs performance in the non-compound setting with intra-cell interference given in Section 2.3.1. Theorem 2.4. For a compound sectored cellular system n ˆ G V, ˆ E A :A∈{0, 1} 2| ˆ E| o , in which transmitters and receivers are equipped with M antennas each, there exists a one-shot linear beamforming scheme that achieves the average (per sector) compound DoFs d ∗ C = M 2 , M is even M 2 − 1 6 , M is odd (2.29) under the network interference cancellation framework, assuming local receiver cooperation within each cell. Theorem 2.5. The compound DoFs d C of a sectored cellular system n ˆ G V, ˆ E A :A∈{0, 1} 2| ˆ E| o achievable under the network interference cancellation framework are bounded by M/2 +O (1/ √ |V|). As before, we discuss in detail the caseM = 2 and sketch the proof of Theorem 2.4 in the general case in Appendix A.4. Theorem 2.5 is proved in Appendix A.5. 2.4.2 Topologically Robust Scheme In this section, we focus on the case where M = 2 and describe a topologically robust transmission scheme for ˆ G(V, ˆ E) that is able to achieve d C = 1. We will consider a scheme very similar to the one described in Section 2.3.1. We will use the same beamforming strategy, but consider a new decoding order that is able to guarantee topological robustness. In the terminology to follow, we distinguish between primary and secondary sectors in our network according to their relative position within each cell. We say that a sector v∈V is primary if v∈V circle (i.e., it is located in the upper-left corner of a cell) and secondary otherwise. We consider here a new decoding order under the network interference cancellation framework in which cells decode their messages in diagonal groups, 34 Chapter 2 Cellular Interference Alignment decoding order D 1 D 2 D 3 ... ⇡ s Figure 2.9: The decoding orderπ s . The highlighted sector pairs decode their messages simultaneously. starting from the upper-left corner of the network. Within each group, the cells first decode their primary messages (i.e, the ones associated with primary sectors) following a top-down decoding order and then proceed to their secondary messages which are decoded in the opposite direction. This process leads to the “curly-S” decoding order shown in Fig. 2.9 and will be denoted here as π s . An important property of the above decoding order is that it maintains, under network interference cancellation, the same out-of-cell interfering link directions as the “left-to-right, top-down” decoding order π ∗ . We have that E πs ,{[u,v] : (u,v)∈E and v≺ πs u} =E π ∗ and hence the beamforming scheme designed forE π ∗ (Section 2.2.1) can be directly applied in this case and satisfy the out-of-cell alignment conditions u H u H uv v v = 0,∀[v,u]∈E πs . (2.30) Recall the example shown in Fig. 2.8 and assume that receiver a has already decoded its own message. With the previous decoding order, π ∗ , the receivers b andc were unable to jointly decode their messages due to the existing interference 35 Chapter 2 Cellular Interference Alignment from sector d. With the new decoding order however, this is no longer an issue. According toπ s , the receiver in sectord will be decoded before sectorsb andc, and hence its message will be available to the corresponding receivers for interference cancellation. a b c d messages decoded before (b), (c) Figure 2.10: Robust decoding for sectors b and c. The network instance described above is depicted in Fig. 2.10. Under the network interference cancellation framework with decoding order π s , the receiver observations at the time when sectors b and c attempt to decode are given by y b =H bb v b s b +H bc v c s c + X v:[v,b]∈Eπs H bv v v s v +z b , and y c =H cc v c s c +H cb v b s b + X v:[v,c]∈Eπs H cv v v s v +z c , where H uv =α uv ·H uv are the compound channel matrices with state parameters α uv ∈{0, 1}. Notice that the secondary sectorsb andc, no longer need the primary observation from sectora in order to decode their messages. From Eq.2.30 we have that u H b X v:[v,b]∈Eπs H bv v v s v = 0 and u H c X v:[v,c]∈Eπs H cv v v s v = 0, 36 Chapter 2 Cellular Interference Alignment for all compound channel states and hence the corresponding observations can be written in vector form as u H b y b u H c y c = u H b H bb v b u H b H bc v c u H c H cb v b u H c H cc v c | {z } , ˜ H(α bc ,α cb ) s b s c + ˜ z. (2.31) The equivalent channel matrix ˜ H(α bc ,α cb )∈C 2×2 given in the above observa- tion depends on the compound channel-state parameters α bc ,α cb ∈{0, 1}, which determine whether sectors b and c interfere with each other or not. We can see that the resulting channel matrices, ˜ H(0, 0) = u H b H bb v b 0 0 u H c H cc v c , ˜ H(1, 0) = u H b H bb v b u H b H bc v c 0 u H c H cc v c , ˜ H(0, 1) = u H b H bb v b 0 u H c H cb v b u H c H cc v c , ˜ H(1, 1) = u H b H bb v b u H b H bc v c u H c H cb v b u H c H cc v c , are all full-rank, and hence the receivers in sectorsb andc are always able to decode their messages, irrespective of the compound channel-state parameters. Similarly, we can show that all transmitted messages in ˆ G V, ˆ E A associated with secondary sectors, can be successfully decoded according to π s , for all compound channel-state configurationsA∈{0, 1} 2| ˆ E| . It remains to argue that primary sectors are also able to decode their messages in the above compound network and hence show that the average (per sector) degrees of freedom d ∗ C = 1 are achievable. Consider the cell{a,b,c} shown in Fig. 2.11 just before its primary sector receivera attempts to decode. According toπ s , the available receiver observations in this cell are given by y a =H aa v a s a + X v∈{b,c,d} H av v v s v +z a , u H b y b =u H b H bb v b s b + X v∈{a,c} u H b H bv v v s v +u H b z b , u H c y c =u H c H cc v c s c + X v∈{a,b,d} u H c H cv v v s v +u H c z c , 37 Chapter 2 Cellular Interference Alignment a b c d messages decoded before (a) Figure 2.11: Robust decoding for sector a. where all interference coming from sectors v≺ πs a has already been eliminated. The above observations can be written in vector form (cf. Eq. 2.23) as, ˜ y = ˜ H(α) ˜ s + ˜ z, (2.32) where the matrix ˜ H(α) depends on the channel-state parameters α , [α ab ,α ac ,α ad ,α ba ,α bc ,α ca ,α cb ,α cd ]∈{0, 1} 8 and is given by ˜ H(α) = H aa v a H ab v b H ac v c H ad v d u H b H ba v a u H b H bb v b u H b H bc v c 0 u H c H ca v a u H c H cb v b u H c H cc v c u H c H cd v d . Notice that ˜ H(α) has the same structure as the matrix ˜ H we considered in Section 2.2.1, and as we have already seen in the example of Fig. 2.8, there exist channel-state configurations (e.g,α = [0, 0, 0, 0, 1, 0, 1, 1]), for which ˜ H(α) becomes rank-deficient. However, this is not necessarily a problem here, since we are only interested in decoding the primary sector’s message s a . In this case, we just have to guarantee that the following condition, [1, 0, 0, 0]∈ rowspan ˜ H(α) , (2.33) 38 Chapter 2 Cellular Interference Alignment holds for every channel-state configuration α. Of course, when α is the all- ones vector, the matrix ˜ H(α) ∈ C 4×4 is full-rank and the above condition is automatically satisfied. In order to show that the primary sector’s message can always be decoded and that Eq.2.33 holds for all α∈{0, 1} 8 , we will consider here the following cases: 1. α cd = 0, for all [α ab ,α ac ,α ad ,α ba ,α bc ,α ca ,α cb ]∈{0, 1} 7 . 2. α ad = 1, for all [α ab ,α ac ,α ba ,α bc ,α ca ,α cb ,α cd ]∈{0, 1} 7 . 3. [α cd ,α ad ] = [1, 0], for all [α ab ,α ac ,α ad ,α ba ,α bc ,α ca ,α cb ,α cd ] withα ab ·α ac = 0. 4. [α cd ,α ad ] = [1, 0], for all [α ab ,α ac ,α ad ,α ba ,α bc ,α ca ,α cb ,α cd ] withα ab ·α ac = 1. Notice that these four cases (illustrated in Fig. 2.12) cover all possible compound channel-state configurations for the interfering links between the sectors a,b,c, andd. Beforeproceedingtoexaminethesecasesseparately, wegivealemma that will be repeatedly used. a b c d a b c d a b c d a b c d Case 1 Case 2 Case 3 Case 4 ↵ cd =0 ↵ ad =1 ↵ ab ·↵ ac =0 ↵ ab ·↵ ac =1 ↵ cd =1 ↵ ad =0 ↵ cd =1 ↵ ad =0 , , Figure 2.12: Compound channel-state configurations for sectors a, b, c and d. Case 1 captures all parameter configurationsα, in which there is no interference between sectors c and d, and Case 2 corresponds to configurations with α ad = 1. In Cases 3 and 4, we assume that [α cd ,α ad ] = [1, 0] and distinguish between configurations in which sector a observes at most one, or two interfering signals. 39 Chapter 2 Cellular Interference Alignment Lemma 2.3. Let H be an n×n matrix whose elements are chosen at random from an absolutely continuous joint probability distribution. For any binary matrix A∈{0, 1} n×n with diagonal elementsa ii = 1,i = 1,...,n, the rank of the Hadamard (pointwise) product (A◦H) is equal to n with probability one. Proof. LetG = (A◦H)anddefinethemultivariatepolynomialQ(h 1,1 ,h 1,2 ,...,h n,n ) as being equal to det(G). Using the Leibnitz formula for the determinant we have that Q(h 1,1 ,h 1,2 ,...,h n,n ) = X σ∈Sn sgn(σ) n Y i=1 G i,σ(i) (2.34) = X σ∈Sn sgn(σ) n Y i=1 a i,σ(i) n Y i=1 h i,σ(i) (2.35) = n Y i=1 h i,i + X σ∈Sn\{σ ∗ } sgn(σ) n Y i=1 a i,σ(i) n Y i=1 h i,σ(i) (2.36) and hence Q(h 1,1 ,h 1,2 ,...,h n,n )6≡ 0, for all A with a i,i = 1. Further, assuming that h i,j are chosen at random from an absolutely continuous joint probability distribution, we have that P[Q(h 1,1 ,h 1,2 ,...,h n,n )6= 0] = 1, and therefore the matrix G = (A◦H) is full-rank with probability one. Case 1: When α cd = 0, the receiver a can first zero-force the interference from sector d and obtain u H a y a =u H a H aa v a s a + X v∈{b,c} u H a H av v v s v +u H a z a . Then it can use the projected observations from sectors b and c, which are given in this case by u H b y b =u H b H bb v b s b + X v∈{a,c} u H b H bv v v s v +u H b z b , u H c y c =u H c H cc v c s c + X v∈{a,b} u H c H cv v v s v +u H c z c , 40 Chapter 2 Cellular Interference Alignment in order to create a three-dimensional vector observation of the form u H b y b u H b y b u H c y c = u H a H aa v a u H a H ab v b u H a H ac v c u H b H ba v a u H b H bb v b u H b H bc v c u H c H ca v a u H c H cb v b u H c H cc v c | {z } , ˜ H(α ab ,αac,α ba ,α bc ,αca,α cb ) s a s b s c + ˜ z. Notice that ˜ H(α ab ,α ac ,α ba ,α bc ,α ca ,α cb ) can be written as the pointwise product (A◦H) = 1 α ab α ac α ba 1 α bc α ca α cb 1 ◦ u H a H aa v a u H a H ab v b u H a H ac v c u H b H ba v a u H b H bb v b u H b H bc v c u H c H ca v a u H c H cb v b u H c H cc v c , whereH andA satisfy the conditions of Lemma 2.3, and hence it is full-rank for all channel-state parameters [α ab ,α ac ,α ba ,α bc ,α ca ,α cb ]. We can therefore argue that receiver a is always able in this case to decode its desired message from the above joint observation. Case 2: When α ad = 1, the equivalent channel matrix ˜ H(α) is going to be full-rank for every choice of [α ab ,α ac ,α ba ,α bc ,α ca ,α cb ,α cd ]∈{0, 1} 7 and hence s a can be decoded directly from Eq.2.32. In order to show this we will first write the matrix ˜ H(α) in its product form (A◦ ˜ H), where A = 1 α ab α ac 1 1 α ab α ac 1 α ba 1 α bc 0 α ca α cb 1 α cd , and consider a permutation matrix P σ that reorders the rows of A according to σ(1) = 4, σ(2) = 1, σ(3) = 2, σ(4) = 3. We have that rank(A◦ ˜ H) = rank P σ (A◦ ˜ H) = rank(P σ A◦P σ ˜ H) 41 Chapter 2 Cellular Interference Alignment and since [P σ A] ii = 1, ∀i, we can use Lemma 2.3 to show that the above matrix is indeed full-rank for any choice of channel-state parameters [α ab ,α ac ,α ba ,α bc ,α ca ,α cb ,α cd ]∈{0, 1} 7 . Case 3: Whenα ad = 0 and [α ab ,α ac ] = [0, 0], receivera observes no interference and can directly decode its own message. Now, when [α ab ,α ac ]∈{[0, 1], [1, 0]}, the receiver a has only one interfering link which can always be zero-forced from its two-dimensional observation y a . Without loss of generality, assume that α ac = 1 and choose u a ∈C 2 such that u H a H ac v c = 0. Then, u H a y a =u H a H aa v a s a +u H a z a , and since u H a H aa v a 6= 0 with probability one, the message s a can be decoded in this case as well. Case 4: In this case, the equivalent channel matrix ˜ H(α) can be written as (A◦ ˜ H), with A = 1 1 1 0 1 1 1 0 α ba 1 α bc 0 α ca α cb 1 1 , and, as in Case 2, we can use Lemma 3 to show that it is full-rank, by swapping the second and third rows of A. Hence, s a can be decoded from Eq.2.32 for all [α ba ,α bc ,α ca ,α cb ]∈{0, 1} 4 . 2.5 Summary In this chapter we have shown that the promised DoFs gain of interference alignment can be achieved in cellular networks with straightforward one-shot alignment precoding, without requiring symbol extensions over very large number of time-frequency dimensions, or infinite resolution of “rationally independent” signal levels. In particular, we have shown schemes that achieve 1/2 DoFs per antenna in the uplink of a cellular system with three sectors per cell and one active user per sector, where both the user transmitter and the sector receiver have M antennas. Our result applies immediately to the case of M even, while it requires extension over two time/frequency varying slots for M odd. The application of 42 Chapter 2 Cellular Interference Alignment interference alignment to large cellular networks is enabled by a simple form of local base station cooperation, where base stations are ordered, and each base base station passes to its successive neighbors its own decoded message. Furthermore, for the case where there is (possibly) interference between sectors of the same cell, weconsideredaschemethatexploitsjointprocessing(infact, successivedecodingis sufficient) of the three sectors in the same cell and achieves the same optimal DoFs. For this scenario, we have also defined the notion of “topological robustness”, as the ability to achieve fixed average DoFs irrespectively of the presence/absence of the interfering links. In particular, we have shown that topologically robust one- shot linear schemes exist, achieving the same optimal DoFs of the original network where all links are present. 43 CHAPTER3 Omni-Directional Antennas and Asymmetric Configurations In the previous chapter we introduced a novel framework for the uplink of sectored cellular networks in which interference alignment can achieve the promised optimal DoFsbylinearprecodinginone-shot(i.e., byprecodingoverasingletime-frequency slot). As we have seen, a distributed backhaul network architecture – in which nearby receivers can exchange already decoded messages – is enough to reduce the uplink of a sectored cellular network to a topology in which the optimal degrees of freedomcanbeachievedbylinearinterferencealignmentschemeswithoutrequiring time-frequency expansion or lattice alignment. In this chapter we follow a similar approach and focus on the uplink of non- sectored cellular systems where base-stations are equipped with omni-directional antennas. The motivation behind this cellular model is that omni-directional base-stations are typically used in dense small-cell deployments [ACD + 12] where inter-cell interference from neighboring cells is a major impairment. Compared to the sectored case , the wireless system scenario considered here is much more challenging, both from a theoretical and an engineering perspective, since base- station receivers operate in a richer interference environment: In Chapter 2, each sector receiver observed four dominant interfering links from its neighboring sector transmitters due to the deployment of directional antennas, whereas here it is 44 Chapter 3 Omni-directional Antennas and Asymmetric Configurations natural to assume that every base-station receiver in the network will observe significant interference from all six surrounding neighboring cells. Under this framework, we focus on cellular system scenarios in which user terminals are equipped with M = 2 transmit antennas and provide one-shot linear interference alignment schemes for the practically relevant cases in which base-stations are equipped with N = 2, 3 and 4 receive antennas. Further, we propose a new converse technique for cellular networks with M×M links (i.e., transmitters and receivers have the same number M of antennas) based on the DoFs feasibility inequalities for linear interference alignment introduced in [RLL12], [BCT13]. Direct application of these inequalities in our setting yields a challenging (non- linear, discrete optimization) problem over a number of variables that grows with the number of transmit-receive pairs in the network. To overcome this difficulty, we exploit the topology of the interference graph and relax the corresponding optimization problem into a linear program (LP) with a small number of variables that does not depend on the size of the network. Using this approach, we are able to provide a tight outer bound on the achievable DoFs in the 2× 2 case and show that our one-shot IA scheme achieves the optimal DoFs in a large (extended) cellular network. This chapter is organized as follows. First, in Section 3.1 we revisit the cellular network model introduced in the previous chapter and provide the corresponding definitions for the omni-directional case. Then, in Section 3.2 we state our results for 2× 2 cellular networks and give the corresponding achievability and converse theorems. Finally, in Section 3.3 we consider networks with asymmetric antenna configurations and provide the corresponding interference alignment schemes for 2× 3 and 2× 4 systems. 3.1 Omni-directional Cellular Model As depicted in Figure 3.1a, we focus here on a large MIMO cellular network with K base-stations, each one serving a user’s mobile terminal. In contrast to the sectoredcellularmodelthatweconsideredinChapter2, here, weassumethatbase- stations are equipped with omni-directional antennas and serve one user in each cell (instead of three sectors serving three users per cell in the sectored case). In line with our previous assumptions, within each cell, the base-station receivers are 45 Chapter 3 Omni-directional Antennas and Asymmetric Configurations 3 (a) Cellular network (b) Interference graph Fig. 1: The cellular network topology and the corresponding dominant interference graph. The desired uplink channels from mobile terminals to their associated base-stations are depicted with black arrows in each cell. Each base-station observes six dominant interfering signals from its six adjacent cells (red arrows) and weaker interference from outer cells (orange arrows) located at distance two or more. The interference graph by construction captures only the dominant sources of interference for each cell; the vertices represent transmit-receive pairs and edges indicate interfering neighbors. Taking into account path loss and shadowing effects that are inherent to wireless transmissions, we assume that all interference in our cellular model is generated locally between transmitters and receivers of neighboring cells. As depicted in Fig. 1a, each base-station receiver in our model will observe six dominant interfering signals from all six adjacent cells and weaker interference from transmitters located at distance two or more. In the context of this paper, we will not explicitly consider interference from non-neighboring transmitters, assuming that their aggregate effect contributes to the “noise floor” of the system 3 . Let S be the index set of all cells in the network and let N(i) denote the six interfering neighbors of the cell i2 S. Within our framework, the received observation of the ith base-station can be written as y i =H ii x i + X j2 N(i) H ij x j +z i (1) where H ij is the N⇥ M matrix of channel gains between the transmitter associated with cell j and the receiver of cell i and x i are the corresponding transmitted signals satisfying the average power constraint E ⇥ ||x i || 2 ⇤ P. 3 In [23], necessary and sufficient conditions on the channel gain coefficients of a GaussianK-user interference channels are found such that “treating interference as noise” (TIN) is approximately optimal in the sense that, subject to these conditions, the TIN-achievable region is within an SNR-independent gap of the capacity region. Figure3.1: The cellular network topology and the corresponding dominant interference graph. The desired uplink channels from mobile terminals to their associated base- stations are depicted with black arrows in each cell. Each base-station observes six dominant interfering signals from its six adjacent cells (red arrows) and weaker interference from outer cells (orange arrows) located at distance two or more. The interference graph by construction captures only the dominant sources of interference for each cell; the vertices represent transmit-receive pairs and edges indicate interfering neighbors. only interested in decoding the uplink transmissions of the user that is associated with them and observe all other simultaneous transmissions as interference. In this chapter we will further focus on asymmetric antenna configurations where the base-station receivers and the mobile terminal transmitters are equipped with N and M antennas respectively. Morespecifically, lettingS betheindexsetofallcellsinthenetworkandletting N (i) denote the six interfering neighbors of the celli∈S, the received observation of the ith base-station can be written as y i =H ii x i + X j∈N (i) H ij x j +z i (3.1) whereH ij is theN×M matrix of channel gains between the transmitter associated with cell j and the receiver of cell i that are assumed, as before, to be generic and remain emain constant throughout the entire communication, and x i are 46 Chapter 3 Omni-directional Antennas and Asymmetric Configurations the corresponding transmitted signals satisfying the average power constraint E h ||x i || 2 i ≤P. 3.1.1 Interference Graph The interference graph that corresponds to the omni-directional cellular model is shown in Fig. 3.1b. To avoid confusion with our previous notation, throughout this chapter, we will denote the omni-directional interference graph as ˚ G(V,E) to indicate that we are referring to the omni-directional model where vertices and edges refer to the hexagonal cell structure shown in Fig. 3.1b. For completeness we redefine here the omni-directional cellular model with respect to the Eisenstein integer labels. Re Im ! 1 3+2! (!) Figure 3.2: The Eisenstein integers Z(ω) on the complex plane. DefineB r ,{z∈C:|Re(z)|≤r,|Im(z)|≤ √ 3r 2 } and let φ :V→Z(ω)∩B r be a one-to-one mapping between the elements ofV and the set of bounded Eisenstein integers given by Z(ω)∩B r . For any v∈V we say that φ(v) is the unique label of the corresponding node in our graph. Correspondingly, the set of verticesV is given by V = n φ −1 (z) :z∈Z(ω)∩B r o . (3.2) In order to explicitly describe the set of edgesE in terms of the function φ, we define the set D, [ z∈Z(ω) Δ(z), (3.3) 47 Chapter 3 Omni-directional Antennas and Asymmetric Configurations where Δ(z) ={(z,z +ω), (z,z +ω + 1), (z +ω,z +ω + 1)} (3.4) is the set of the three line segments in C (shown in Fig. 3.2) that form a triangle with verticesz,z +ω andz +ω + 1. The set of edgesE in our graph can hence be given by E ={(u,v) :u,v∈V and (φ(u),φ(v))∈D}. (3.5) Definition 3.1 (Omni-directional Interference Graph). The interference graph ˚ G(V,E) is an undirected graph defined by the set of verticesV given in Eq.3.2 and the corresponding set of edgesE given in Eq.3.5. The graph vertices represent transmit-receive pairs in our cellular model and edges indicate interfering neighbors. 3.1.2 Network Interference Cancellation As in Chapter 2, we will consider here the same message-passing network architecture for our cellular system, in which base-station receivers communicate locally in order to exchange decoded messages. Fig. 3.3 illustrates the above network interference cancellation process in our cellular graph model assuming a “left-to-right, top-down” decoding order. And all the corresponding definitions follow directly for Section 2.1.2 2 1 3 4 5 6 7 8 9 11 10 12 13 14 15 16 17 19 18 20 21 22 23 24 25 26 28 27 29 30 31 32 33 34 Figure 3.3: The directed interference graph ˚ G π ∗(V,E π ∗) after network interference can- cellation according to the “left-to-right, top-down” decoding order π ∗ . The transmitter of a cell associated with node i causes interference only to its neighboring base-station receivers j with j <i. 48 Chapter 3 Omni-directional Antennas and Asymmetric Configurations Definition 3.2 (Directed Interference Graph ˚ G π ). For a given partial order “≺ π ” onV, the directed interference graph is defined as ˚ G π (V,E π ) whereE π is a set of ordered pairs [u,v] given byE π ={[u,v] : (u,v)∈E and v≺ π u}. Problem Statement Our objective is to maximize, over all decoding orders π, the average (per cell) achievable DoFs d ˚ G,π , 1 |V| X v∈V d v , (3.6) where ˚ G(V,E) is the interference graph defined in Section 3.1.1 andd v denotes the DoFs achieved by the transmit-receive pair associated with the node v∈V, where d v = lim P→∞ R v (P ) log(P ) , and R v (P ) is the achievable rate in cell v∈V under the per-user transmit power constraint P. 3.2 Cellular Networks with M = 2,N = 2 Theorem 3.1 (Achievability). For a 2× 2 cellular system ˚ G(V,E), there exist a one-shot linear beamforming scheme that achieves the average (per cell) DoFs, d ˚ G,π ∗ = 3/4, under the network interference cancellation framework with decoding order π ∗ . Theorem 3.2 (Converse). For a 2× 2 cellular system ˚ G(V,E) the average (per cell) DoFs d ˚ G,π that can be achieved by any one-shot linear beamforming scheme, for any network interference cancellation decoding order π, are bounded by d ˚ G,π ≤ 3/4 +O (1/ √ |V|). Remark 3.1. The above results can be directly translated to achievability and converse theorems for sectored cellular systems with intra-cell interference [NMAC15a]. In Chapter 2, we studied a wireless network scenario in which co- located sectors (i.e., the sectors of the same cell) are able to jointly process their received observations and showed that for M×M links the optimal M/2 DoFs (per user) are achievable. An interesting observation is that the cellular model 49 Chapter 3 Omni-directional Antennas and Asymmetric Configurations considered here leads to an interference graph that is similar (isomorphic) to the one considered in the above case. The fundamental difference here is that we do not allow any receivers to jointly process their signals. Therefore, it is possible to restate the results of Theorems 3.1 and 3.2 for the sectored case and assess the gain of joint cell processing in such systems. WhenM = 2,N = 2, we can see that jointly processing the received signals within each cell yields 33% gain in terms of the average achievable (per user) DoFs, compared to single-user decoding. 3.2.1 Achievability (Proof of Theorem 3.1) Consider the interference graph ˚ G(V,E) introduced in Section 3.1.1 and assume that all user terminals v∈V whose labels φ(v) belong to the sub-lattice Λ 0 , 2·Z(ω) (3.7) are turned off, while the remaining user terminals with φ(v)∈ Z(ω)\ Λ 0 are all simultaneously transmitting their signals x v to their corresponding receivers. Let V 0 ,{v∈V :φ(v)∈ Λ 0 } denote the set of inactive vertices and letE 0 ,{(u,v)∈ E :φ(u) or φ(v)∈ Λ 0 } be the set of edges that are adjacent toV 0 . Recall that under our framework, each base-station receiver that is able to decode its own message, is also able to pass it as side information to its neighbors, effectively eliminating interference in that direction. Hence, following the “left- to-right, top-down” decoding order π ∗ introduced in Section 3.1.2, the receivers associated with the active nodesu∈V\V 0 are able to eliminate interference from all neighboring cells v≺ π ∗ u and attempt to decode their own message from the two-dimensional received signal observation y u given by y u =H uu x u + X v:[v,u]∈E π ∗\E 0 H uv x v +z u . (3.8) Our goal is to design the transmitted signals x v such that all interference observed in y u is aligned in one dimension for all u∈V\V 0 . In that way, we can show that the DoFs d v = 1, v∈V\V 0 0, v∈V 0 50 Chapter 3 Omni-directional Antennas and Asymmetric Configurations are achievable in ˚ G(V,E) and hence d ˚ G,π ∗ = 1 |V| X v∈V d v = |V\V 0 | |V| = 1− |V 0 | |V| . Later, we will see that Λ 0 has been chosen so that|V 0 |≤ 1 4 |V|, and hence obtain that the average (per cell) DoFs d ˚ G,π ∗ = 3/4 are indeed achievable under our framework. The setting described above is illustrated in Fig. 3.4, where transparent vertices correspond to inactive cells v∈V 0 and transparent edges correspond toE 0 . The resultinginterferencegraph, denotedhereas ˆ ˚ G π ∗ (V\V 0 ,E π ∗\E 0 ), isasub-graphof ˚ G π ∗(V,E π ∗) and represents the corresponding interference between active transmit- receive pairs in the network. Re Im S(!) Figure3.4: The interference sub-graph ˆ ˚ G π ∗(V\V 0 ,E π ∗\E 0 ) represented on the complex plane. The transparent vertices correspond to inactive cellsv∈V 0 with labelsφ(v)∈ Λ 0 and the transparent edges correspond to their adjacent edgesE 0 . The edges in the cluster S(ω) are highlighted in a tilted rectangle that is centered at the pointz =ω. Notice that all the interfering edges in the above graph can be partitioned into smaller (isomorphic) sets by translatingS(z) over all z∈ Λ 0 +ω (black vertices). Let u u and v u denote the 2-dimensional receive and transmit beamforming vectors associated with the active nodes u∈V\V 0 and assume that the active 51 Chapter 3 Omni-directional Antennas and Asymmetric Configurations user terminals in the network have encoded their messages in the corresponding codewords. Although codewords span many slots (in time), we focus here on a single slot and denote the corresponding coded symbol of user u by s u . Then, the vector transmitted by user u is given by x u = v u s u and each receiver can project its observation y u along u u to obtain ˆ y u =u H u H uu v u s u + X v:[v,u]∈E π ∗\E 0 u H u H uv v v s v + ˆ z u . We will show next that it is possible to design u u and v u across the entire network such that the following interference alignment conditions are satisfied: u H u H uu v u 6= 0, ∀u∈V\V 0 and (3.9) u H u H uv v v = 0, ∀[v,u]∈E π ∗\E 0 . (3.10) Hence, every active receiver in the network can decode its own desired symbol s u from an interference-free channel observation of the form ˆ y u = ˆ h u s u + ˆ z u (3.11) where ˆ h u =u H u H uu v u and ˆ z u =u H u z u . In order to describe the alignment precoding scheme, we will partition the interfering edges in ˆ ˚ G π ∗(V\V 0 ,E π ∗\E 0 ) into smaller sets that we will refer to as the interference clusters, given by S(z), n [z,z− 1], [z,z + 1 +ω], (3.12) [z + 1,z], [z + 1,z + 1 +ω], (3.13) [z− 1−ω,z− 1], [z− 1−ω,z] o , (3.14) for z∈ Λ 0 +ω. To illustrate the above definition, in Fig. 3.4, the interference clusterS(ω) is highlighted in a tilted rectangle that is centered at z =ω and the 52 Chapter 3 Omni-directional Antennas and Asymmetric Configurations points z∈ Λ 0 +ω are shown in black color. It is not hard to verify that the set of edges can be written as E π ∗\E 0 = [ z∈Λ 0 +ω {[u,v] : [φ(u),φ(v)]∈S(z),u,v∈V} (3.15) = [ {v∈V:φ(v)∈Λ 0 +ω} S(v) (3.16) whereS(v), n [u,w] : [φ(u),φ(w)]∈S φ −1 (v) ,u,w∈V o andS(v) T S(v 0 ) =∅, for all v6=v 0 with φ(v),φ(v 0 )∈ Λ 0 +ω. Hence, as we can also see in Fig. 3.4, the setE π ∗\E 0 can be partitioned into smaller interference clusters by translating the highlighted tilted rectangle over all black vertices. Apart from a few exceptions (due to the finite boundary of the network), the resulting interference clustersS(v) will be isomorphic and exhibit the same structure as the one shown in Fig. 3.5. In the following, we will describe the interference alignment solution that lies at thecoreofourachievabilitytheorem,namelytheprecodingschemethatcanachieve d v = 1forallcellsv inagivencluster. Then, wewillseethattheabovesolutioncan be readily extended to the entire network through Eq.3.16 and therefore show that the required interference alignment conditions Eq.3.9 and Eq.3.10 are satisfied in ˆ ˚ G π ∗(V\V 0 ,E π ∗\E 0 ). v c v a v b a c e b d a b c d e H ea v a H da v a H ab v b H db v b H ac v c H ec v c Figure 3.5: The interference edges in the clusterS(a). The transmit beamforming vectors v a , v b and v c (gray nodes) are uniquely associated with the edges inS(a) and have to be designed such that interference is aligned at receivers a, d and e. In the above cluster, the goal is to design the 2-dimensional beamforming vectors v a , v b , and v c such that all interference occupies a single dimension in every receiver. We will hence require that span(H ab v b ) = span(H ac v c ) for receiver 53 Chapter 3 Omni-directional Antennas and Asymmetric Configurations a, span(H db v b ) = span(H da v a ) for receiver d, and span(H ea v a ) = span(H ec v c ) for receiver e. These alignment conditions can be written as, v a . = H −1 da H db v b (3.17) v b . = H −1 ab H ac v c (3.18) v c . = H −1 ec H ea v a , (3.19) where v . =u is a shorthand notation for v∈ span(u), and are satisfied as long as v a . =H −1 da H db H −1 ab H ac H −1 ec H ea v a . (3.20) Therefore, if we choose v a to be an eigenvector of the above matrix and set v b . = H −1 ab H ac H −1 ec H ea v a and (3.21) v c . = H −1 ec H ea v a , (3.22) allinterferenceobservedatthereceiversa,dandewillbealignedinonedimension, and can hence be zero-forced by the corresponding receiver projections u a , u d and u e . Assuming that the channel matrices are drawn from a continuous non- degenerate distribution, the projected useful signal coefficients will be non-zero with probability one and therefore, the desired messages s a , s d and s e can be successfully decoded from the interference-free observationsu H a y a ,u H d y d andu H e y e . It is important to note that the transmit beamforming choices v a , v b , and v c as well as the receiver projections u a , u d , and u e are uniquely associated with the interference edges in the above cluster and do not participate in any other interference alignment conditions in the network. Therefore, the eigenvector solution Eq.3.20-Eq.3.22 can be applied locally for all interference clustersS(v) in the network in order to choose the appropriate beamforming vectors v v and u v for all active cells v∈V\V 0 . Since the interference alignment conditions Eq.3.9 and Eq.3.10 can be satisfied across the entire network, the DoFs d v = 1, v∈V\V 0 0, v∈V 0 54 Chapter 3 Omni-directional Antennas and Asymmetric Configurations are achievable in ˚ G(V,E) and hence d ˚ G,π ∗ = 1 |V| X v∈V d v = |V\V 0 | |V| = 1− |V 0 | |V| . (3.23) Lemma 3.1. The number of inactive cells|V 0 | can always be chosen in ˚ G(V,E) to be less than or equal to|V|/4. Proof. Recall that the total number of cells is|V| =|Z(ω)∩B r | and the number of inactive cells is|V 0 | =|Λ 0 ∩B r |, whereB r ,{z∈C :|Re(z)|≤r,|Im(z)|≤ √ 3r 2 } is defined in Section 3.1.1 and Λ 0 , 2·Z(ω) is chosen in Eq.3.7. Let R(r) be the ratio |V 0 | |V| = |Λ 0 ∩Br| |Z(ω)∩Br| as a function of the network size parameter r. For large cellular networks we can already see that lim r→∞ R(r) = Vol (Z(ω)) Vol (2·Z(ω)) = 1 4 . We want to show however that R(r)≤ 1 4 , for all r≥ 1. Notice that the points in Z(ω)∩B r can be partitioned into 2r + 1 horizontal lines. When r is odd, the set Z(ω)∩B r has r + 1 lines with 2r points and r lines with 2r + 1 points. Therefore the total number of points can be calculated as|Z(ω)∩B r | = (r+1)2r+r(2r+1) = 4r 2 +3r. Now, allthepointsthatcorrespondtother+1linescontainoddmultiples ofω and hence cannot be in Λ 0 . From the remainingr lines, r+1 2 lines haver points in Λ 0 and r−1 2 lines haver + 1 points in Λ 0 . The total number of points in Λ 0 ∩B r is therefore given by|Λ 0 ∩B r | = r+1 2 r + r−1 2 (r + 1) = r 2 + (r− 1)/2, and the corresponding ratio can be calculated as R(r) = r 2 + (r− 1)/2 4r 2 + 3r ≤ r + 1/2 4r + 2 = 1 4 . For the case where r is even the same result can be obtained if we choose Λ 0 = 2·Z(ω) +ω. We omit the details here for brevity. From the above lemma and the result obtained in Eq.3.23, we conclude that the average (per cell) DoFs d ˚ G,π ∗ = 3/4, are indeed achievable in ˚ G(V,E) under the network interference cancellation framework with decoding order π ∗ , and the proof of Theorem 3.1 is completed. 55 Chapter 3 Omni-directional Antennas and Asymmetric Configurations 3.2.2 Converse (Proof of Theorem 3.2) We begin by stating a useful lemma that will help us bound the total degrees of freedom achievable in our setting. Lemma 3.2 (Linear IA Feasibility [RLL12], [BCT13]). Under the network interference cancellation framework, the degrees of freedom achievable by linear schemes in anM×M flat-fading MIMO cellular network with directed interference graph ˚ G π (V,E π ), must satisfy d v ∈{0,...,M},∀v∈V (3.24) d u +d v ≤M,∀[u,v]∈E π (3.25) 2 X v∈V (M−d v )d v ≥ X (u,v)∈Eπ d u d v (3.26) for any decoding orderπ, assuming all channel gains are chosen from a continuous distribution. The above inequalities are necessary conditions for the achievability of any degrees of freedom{d v ,v∈V} in ˚ G π (V,E π ). We are going to use these conditions here to obtain an upper bound on the average degrees of freedom achievable in our network by considering the optimization problem Q 1 ( ˚ G π ) : maximize {dv,v∈V} 1 |V| X v∈V d v subject to: Eq.3.24, Eq.3.25, Eq.3.26. As we can see, Q 1 ( ˚ G π ) is a very difficult problem to solve in its current form: Not only it involves non-linear integer constrains but also it is defined over many variables (since we want to consider large networks). In the following, we are going to transform (and relax) this problem to a tractable linear program (LP) with significantly fewer variables that do not scale with the size of the network. Our approach can be summarized in three steps: • First, we rewrite the objective and the constraints in Q 1 ( ˚ G π ) as sums over connected vertex triplets [a,b,c] that we are going to call the trianglesT of our graph. 56 Chapter 3 Omni-directional Antennas and Asymmetric Configurations • Then, we identify the set of distinct triangle-DoF configurations that are allowed in ˚ G(V,E) and formulate an LP relaxation over their relative frequencies (i.e., the number of times each triangle-DoF configuration occurs in the graph). • Finally we solve the resulting linear program using duality theory and show that d ˚ G,π ≤ opt(Q 1 ( ˚ G π ))≤ 3/4 +O (1/ √ |V|), for all decoding orders π. u v w Figure 3.6: The set of triangles [a,b,c]∈T for ˚ G(V,E). All the circle nodes belong to V in and participate in exactly three triangles (n v = 3). The setV ex contains the colored nodes on the boundary for which n v < 3. In order to formally describe the set of trianglesT in ˚ G(V,E), we consider the set of ordered Eisenstein integer triplets P ={[z,z +ω,z +ω + 1] :z∈Z(ω)}. Recall from Section 3.1.1 that the points z, z +ω and z +ω + 1 form the line segments Δ(z)⊆D and that the corresponding graph vertices φ −1 (z), φ −1 (z +ω) and φ −1 (z +ω + 1) form a connected triangle in ˚ G(V,E). The setT can hence be defined as T ,{[a,b,c] : [φ(a),φ(b),φ(c)]∈P,a,b,c∈V}. (3.27) 57 Chapter 3 Omni-directional Antennas and Asymmetric Configurations The above definition is illustrated in Fig. 3.6 in which shaded triangles connect the corresponding vertex triplets [a,b,c]∈T . Notice that apart from the vertices on the external boundary of the graph, all other nodes participate in exactly three triangles inT . This observation will be particularly useful in rewriting the sum in the objective function of Q 1 ( ˚ G π ) as a sum overT instead ofV. Let n v , X [a,b,c]∈T 1 v∈{a,b,c} (3.28) denote the number of triangles [a,b,c]∈T that include a given vertex v∈V. As we have seen,n v can only take values in{0, 1, 2, 3} for anyv∈V. More specifically n v = 3 for all internal vertices in ˚ G(V,E), while n v < 3 only for external vertices that lie on the outside boundary of the graph. We define the set of internal and external vertices as V in = {v∈V :n v = 3}, and (3.29) V ex = {v∈V :n v < 3}, (3.30) and in Fig. 3.6 we show the above distinction by coloring all graph vertices v that belong to the setV ex ⊆V. Lemma 3.3. The degrees of freedom{d v ,v∈V}, satisfy 1 |V| X v∈V d v ≤ 1 3|V| X [a,b,c]∈T s(d a ,d b ,d c ) + M|V ex | |V| , (3.31) where s(d a ,d b ,d c ),d a +d b +d c . (3.32) Proof. See Appendix B.1 Lemma 3.4. Any degrees of freedom{d v ,v∈V} that satisfy Eq.3.26 also satisfy: X [a,b,c]∈T g(d a ,d b ,d c )≤ 3M 2 2 |V ex |, (3.33) 58 Chapter 3 Omni-directional Antennas and Asymmetric Configurations where g(d a ,d b ,d c ), (d a +d b ) 2 + (d a +d c ) 2 + (d b +d c ) 2 +d a d b +d a d c +d b d c − 2M(d a +d b +d c ). (3.34) Proof. See Appendix B.2 Therefore, we arrive at Q 2 ( ˚ G π ) : maximize {dv,v∈V} 1 3|V| X [a,b,c]∈T s(d a ,d b ,d c ) + M|V ex | |V| (3.35) subject to: d v ∈{0,...,M},∀v∈V (3.36) d u +d v ≤M,∀[u,v]∈E π (3.37) X [a,b,c]∈T g(d a ,d b ,d c )≤ 3M 2 2 |V ex |, (3.38) whose optimal value satisfies opt(Q 2 ( ˚ G π ))≥ opt(Q 1 ( ˚ G π )), i.e., the solution of Q 2 provides an outer bound for the achievable DoFs in the cellular network. A key observation is that the functions s(d a ,d b ,d c ) and g(d a ,d b ,d c ) can only take specific values for each triangle due to the constraints Eq.3.36 and Eq.3.37 and are invariant under permutations of their arguments. If we letD denote the set of all distinct triangle-DoF configurations given by D, [i,j,k] : i≤j≤k∈{0,...,M} i +j≤M j +k≤M k +i≤M (3.39) and define x (i,j,k) , 1 |T| X [a,b,c]∈T 1 (d a ,d b ,d c ) = (i,j,k) , (3.40) 59 Chapter 3 Omni-directional Antennas and Asymmetric Configurations to be the fraction of triangles [a,b,c]∈T with the specific DoF configuration [i,j,k]∈D, we can write the corresponding sums in Eq.3.35 and Eq.3.38 as X [a,b,c]∈T s(d a ,d b ,d c ) =|T| X [i,j,k]∈D s(i,j,k)·x (i,j,k) (3.41) and X [a,b,c]∈T g(d a ,d b ,d c ) =|T| X [i,j,k]∈D g(i,j,k)·x (i,j,k) . (3.42) Relaxing the integrality constraints in Eq.3.36 and letting x (i,j,k) ∈ R, we can reduce the optimization problem Q 2 ( ˚ G π ) to a linear program (LP) over the variables x (i,j,k) , [i,j,k]∈D. The goal is to identify the relative frequencies x (i,j,k) of the triangle-DoF configurations that, under the (relaxed) feasibility constraints, maximize the (upper-bounded) degrees of freedom in ˚ G. The corresponding linear program is given by LP( ˚ G π ) : maximize n x (i,j,k) ∈R, [i,j,k]∈D o |T| 3|V| X [i,j,k]∈D s(i,j,k)·x (i,j,k) + M|V ex | |V| (3.43) subject to: X [i,j,k]∈D g(i,j,k)·x (i,j,k) ≤ 3M 2 |V ex | 2|T| (3.44) X [i,j,k]∈D x (i,j,k) = 1 (3.45) x (i,j,k) ≥ 0,∀[i,j,k]∈D (3.46) For the rest of this section we will focus on M = 2 and show that the solution to the above linear program is upper bounded by 3/4 +O (1/ √ |V|). For additional results and the extension of the corresponding bounds to the general M×M case we refer the reader to Appendix B.5. When M = 2, the feasible triangle-DoF configurations are given by D ={[0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, 1], [1, 1, 1]}. 60 Chapter 3 Omni-directional Antennas and Asymmetric Configurations If we define the 5-dimensional vectors s, h s(0, 0, 0), s(0, 0, 1), s(0, 0, 2), s(0, 1, 1), s(1, 1, 1) i T = h 0, 1, 2, 2, 3 i T (3.47) g, h g(0, 0, 0), g(0, 0, 1), g(0, 0, 2), g(0, 1, 1), g(1, 1, 1) i T = h 0,−2, 0,−1, 3 i T (3.48) and let x , h x (0,0,0) , x (0,0,1) , x (0,0,2) , x (0,1,1) , x (1,1,1) i T , we arrive at the linear program: LP 2 ( ˚ G π ) : maximize { x∈R 5 } |T| 3|V| s T x + 2 |V ex | |V| (3.49) subject to: g T x≤ 6 |V ex | |T| , 1 T x = 1, x≥ 0. (3.50) To obtain some intuition for the above optimization problem, notice that in Fig. 3.6, the trianglesT in ˚ G(V,E) are almost as many as the verticesV and therefore we can argue that |T| 3|V| → 1 3 as|V|→∞. On the other hand, the number ofexternalvertices|V ex |ismuchsmallerthat|V|andwehavethat lim |V|→∞ |V ex |/|V| = 0. Consequently, one should expect that in large cellular systems, the average achievable DoFs are upper bounded by the solution of the linear program: maximize { x∈R 5 } 1 3 s T x, subject to: g T x≤ 0, 1 T x = 1, x≥ 0. (3.51) Quiteremarkably, thesolutionoftheaboveLPisgivenbytherelativefrequency vector x ∗ = h x (0,0,0) , x (0,0,1) , x (0,0,2) , x (0,1,1) , x (1,1,1) i T = h 0, 0, 0, 3/4, 1/4 i , and its optimal value is 1 3 s T x ∗ = 3/4. It is interesting to note that the optimal relative frequencies x (0,1,1) = 3/4 and x (1,1,1) = 1/4 given in the above solution are exactly the same as the ones used in our achievability scheme in Fig. 3.4. In the rest of this section we are going to use LP duality theory to formally show that the optimal value of LP 2 ( ˚ G π ) in Eq.3.49-Eq.3.50 is indeed bounded by 3/4 +O (1/ √ |V|) and hence complete the proof of Theorem 3.2. 61 Chapter 3 Omni-directional Antennas and Asymmetric Configurations Lemma 3.5. Let opt(s,g,α,β,γ) denote the optimal value of the linear program given by maximize {x∈R n } α·s T x +β (3.52) subject to: g T x≤γ, 1 T x = 1, x≥ 0. (3.53) where α,β,γ≥ 0 and s,g∈R n . Then, for any λ≥ 0, we have that opt(s,g,α,β,γ)≤α· max i {s i −λ·g i } +λ·αγ +β. Proof. See Appendix B.3. Using the result of the above lemma, we can show that the solution of the linear program LP 2 ( ˚ G π ) is upper bounded by opt(LP 2 ( ˚ G π ))≤ |T| 3|V| · max i {s i −λ·g i } + 2(λ + 1) |V ex | |V| , (3.54) for anyλ≥ 0, by identifyingα = |T| 3|V| ,β = 2 |Vex| |V| andγ = 6 |Vex| |T| . In order to obtain a tight bound for opt(LP 2 ( ˚ G π )) we are going to choose λ ∗ = arg min λ≥0 1 3 · max i {s i −λ·g i } (3.55) The function 1 3 ·max i {s i −λ·g i } = max{1/3+2λ/3, 2/3, 2/3+λ/3, 1−λ} is a convex piecewise linear function inλ (shown in Fig. 3.7) and one can verify that λ ∗ = 1/4 and 1 3 max i {s i −λ ∗ ·g i } = 3/4. Substituting back in Eq.3.54 we obtain opt(LP 2 ( ˚ G π ))≤ |T| 3|V| · max i {s i −λ ∗ ·g i } + 2(λ ∗ + 1) |V ex | |V| (3.56) = 3 4 · |T| |V| + 5 2 · |V ex | |V| . (3.57) 62 Chapter 3 Omni-directional Antennas and Asymmetric Configurations 1 1 1/4 1/2 2/3 1/3 1/3+2 /3 1 2/3+ /3 1 3 ·min 0 max i {s i ·g i }=3/4 3/4 Figure 3.7: The minimum of the piecewise linear function 1 3 · max i {s i −λ·g i }. Lemma 3.6. By construction, the interference graph ˚ G(V,E) satisfies |T|≤|V|, and (3.58) |V ex | =O q |V| . (3.59) Proof. See Appendix B.4. Therefore, from Eq.3.57 and Lemma 3.6 we obtain opt(LP 2 ( ˚ G π ))≤ 3 4 +O 1/ q |V| , (3.60) which concludes the proof of Theorem 3.2. 3.3 Cellular Networks with Asymmetric Antenna Configurations Theorem 3.3. For a 2× 3 cellular system ˚ G(V,E), there exist a one-shot linear beamforming scheme that is able to achieve the average (per cell) DoFs, d ˚ G,π ∗ = 1, under the network interference cancellation framework with decoding order π ∗ . 63 Chapter 3 Omni-directional Antennas and Asymmetric Configurations Theorem 3.4. For a 2× 4 cellular system ˚ G(V,E), there exist a one-shot linear beamforming scheme that is able to achieve the average (per cell) DoFs, d ˚ G,π ∗ = 7/6, under the network interference cancellation framework with decoding orderπ ∗ . 3.3.1 Achievability for M = 2, N = 3. (Proof of Thm 3.3) Here, wewillfocusonthecasewheremobileterminaltransmittersandbase-station receivers are equipped with M = 2 and N = 3 antennas and describe the linear beamforming scheme that is able to achieve one DoF per cell for the entire network. Consider the directed interference graph ˚ G π ∗(V,E π ∗) shown in Fig. 3.8 and assume that all user terminals v∈V are simultaneously transmitting their signals x v =v v s v to their corresponding receivers. Following the “left-to-right, top-down” decoding order π ∗ introduced in Section 3.1.2, each base-station will attempt to decode its own desired message from the three-dimensional receiver observation y u =H uu v u s u + X v:[v,u]∈E π ∗ H uv v v s v +z u , (3.61) by projecting alongu u ∈C 3×1 . As before, the goal is to designv u ∈C 2×1 andu u ∈ C 3×1 for all u∈V such that all interference is zero forced and the corresponding messages can be decoded from the projected observationsu H u y u =u H u H uu v u s u + ˆ z u . In order to show achievability, we will first focus on the cellsa,b,c andd shown inFig.3.8andthendescribehowthecorrespondingsolutioncanbeextendedacross the entire network. The receiver a first projects its observation onto the two dimensional subspace orthogonal to H ab v b , effectively zero-forcing interference from transmitter b. The corresponding two-dimensional projected signal at receiver a is given by P ⊥ ab y a =P ⊥ ab H aa v a s a +P ⊥ ab H ac v c s c +P ⊥ ab H ad v d s d | {z } interference +ˆ z a , (3.62) 64 Chapter 3 Omni-directional Antennas and Asymmetric Configurations where P ⊥ ab is a 2× 3 matrix that satisfies P ⊥ ab H ab v b = 0. Further, transmitter d can choose its beamforming vector as a function of v c , P ⊥ ab H ad v d . =P ⊥ ab H ac v c ⇔ (3.63) v d . = (P ⊥ ab H ad ) −1 P ⊥ ab H ac v c (3.64) such that all interference in the above observation is aligned in one dimension, and can hence be zero-forced by projecting along the two-dimensional vector ˜ u a . Overall, the three-dimensional receiver projection is given by u a = (P ⊥ ab ) H ˜ u a . (3.65) The two key steps in the above scheme are depicted in Fig. 3.8. Receiver a projects its observation on a two dimensional subspace to zero-force (ZF) interferencefromcellb, andtransmitterdchoosesv d inordertoaligntheremaining interference (IA) with the cell c. ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF ZF IA IA IA IA IA IA IA IA IA IA IA IA IA IA IA IA IA IA IA IA IA IA IA a b c d Figure 3.8: Interference alignment solution when M = 2, and N = 3. Exactly the same procedure can be followed to yield an interference alignment solution across the entire network. Starting from the boundary, the receiversu∈V can zero-force interference coming from the cell v with φ(v) = φ(u) + 1 (right neighbor) by projecting onto P ⊥ uv , and the transmitters v ∈ V can align their transmitted signals with the transmitteru withφ(u) =φ(v)− 1 (left neighbor) by choosing v v as a function of v u . 65 Chapter 3 Omni-directional Antennas and Asymmetric Configurations 3.3.2 Achievability for M = 2, N = 4. (Proof of Thm 3.4) In order to describe the interference alignment scheme for this case, we will first partition the directed interference graph ˚ G π ∗(V,E π ∗) into horizontal stripes as shown in Fig. 3.9. Each stripeS k contains three consecutive horizontal lines of nodes that we will refer to as top, middle and bottom according to their relative position within each stripe. S k S k 1 S k 2 Figure 3.9: The directed interference graph ˚ G(V,E) divided in horizontal stripes. In the following we will describe a transmission scheme in which (at least) half of the nodes located in the middle line of each stripe (black nodes) are able to achieve d v = 2 while all the remaining nodes in the network (white nodes) are able to achieved v = 1. LetV 1 andV 2 denote the corresponding subsets of vertices v∈V for which d v = 1 and d v = 2 respectively. SinceV =V 1 ∪V 2 , the average (per cell) DoFs that will be achievable in ˚ G(V,E) with the above configuration can be written as d ˚ G,π ∗ = 1 |V| X v∈V d v = |V 1 | + 2|V 2 | |V| = 1 + |V 2 | |V| , (3.66) 66 Chapter 3 Omni-directional Antennas and Asymmetric Configurations and sinceV 2 can be chosen so that|V 2 |≥ 1 6 |V|, 1 we obtain that d ˚ G,π ∗ ≥ 7/6. First, notice that all the receivers associated with nodes in the bottom line of each stripeS k observe two interfering links coming from the transmitters in the top line ofS k−1 and one interfering link coming from their adjacent (right) neighbor in S k , showninFig.3.9withdashedgrayarrows. AccordingtoourDoFconfiguration, these interfering links will occupy three out of the total four dimensions (N = 4) in each receivers’ subspace and can therefore be zero-forced in order to achieve d v = 1. This important observation allows us to decouple the beamforming choices betweendifferentstripesand proposeaninterferencealignmentsolution forasingle stripe that can be replicated across the entire network. Fig. 3.10 shows the single stripe of the network that we will focus on for the rest of this section. The nodes are now shown with different shapes (diamonds, squares and circles) to illustrate their distinct roles in the achievability scheme that follows. Further, notice that the interfering edges in the bottom line of this stripe are not shown here since all interference in the corresponding nodes has already been zero forced (to decouple the consecutive stripes of the network). a c b d e f cluster A cluster B a c b d cluster A e f cluster B S2 C 4⇥ 3 S2 C 4⇥ 3 Figure3.10: Interference alignment solution in a single stripe whenM = 2, andN = 4. Assume that all the black nodes (d v = 2) have chosen their two-dimensional beamforming subspaces at random and consider the interference cluster A, shown in Fig. 3.10. Notice that receiverd observes three interference links (one from each cell in the cluster) and receiver b observers four interference streams in total (one from cell a and three streams coming from the transmitters outside cluster A). 1 Notice that the setV 2 contains at least half of the nodes in every third line of the graph in Fig. 3.9. This construction can be generalized for any interference graph ˚ G(V,E) so that |V 2 |/|V|≥ 1/6. 67 Chapter 3 Omni-directional Antennas and Asymmetric Configurations We will first show that all interference observed at receiver b can be aligned in a three dimensional subspace by appropriately choosing the beamforming vector of transmitter a. Let S ∈ C 4×3 denote the three-dimensional (out-of-cluster) interference observed at receiver b (shown in Fig. 3.11) and let u b ∈ C 4×1 be a vector in the left nullspace of S (i.e., such that u H b S = 0). The receiver b can project its observation along u b to zero-force the out-of-cluster interference and obtain u H b y b = (u H b H bb v b )s b + (u H b H ba v a )s a + ˆ z. (3.67) Choosingv a ∈C 2×1 to be orthogonal toH H ba u b ∈C 2×1 yieldsu H b H ba v a = 0 and the receiver b can decode its message from the above interference-free observation. a c b d e f cluster A cluster B a c b d cluster A e f cluster B S2 C 4⇥ 3 S2 C 4⇥ 3 Figure 3.11: The interference clusters A and B. Our next objective, after choosing v a , is to choose v b and v c such that all interference is aligned in a two-dimensional subspace at receiver d. The corresponding observation can be written as y d =H dd V d s d +H da v a s a +H db v b s b +H dc v c s c | {z } interference +z (3.68) =H dd V d s d +G˜ s +z, (3.69) where G, [H da v a H db v b H dc v c ]∈C 4×3 and ˜ s, [s a s b s c ] T . Lemma 3.7. For any v a 6= 0, there exist v b , v c ∈ C 2×1 with||v b || ≤ 1 and ||v c ||≤ 1 such thatrank(G) = 2 with probability one, assuming thatH da ,H db , and H dc ∈C 4×2 are chosen at random from a continuous (non-degenerate) distribution. Proof. Consider the matrix F = [H da v a H db H dc ] ∈ C 4×5 and let x = [x 1 x 2 x 3 x 4 x 5 ] be a vector in the nullspace of F such that Fx = 0. Since 68 Chapter 3 Omni-directional Antennas and Asymmetric Configurations F is generic, i.e., the corresponding matrices have been chosen from a non- degenerate distribution, we have that x i 6= 0, ∀i, with probability one. Let ˜ v b = [−x 2 /x 1 −x 3 /x 1 ] T , ˜ v c = [−x 4 /x 1 −x 5 /x 1 ] T , γ = min n 1 ||˜ v b || , 1 ||˜ vc|| o and setv b =γ· ˜ v b andv c =γ· ˜ v c . Then one can check thatH db v b +H dc v c =γ·H da v a , and therefore show that rank [H da v a H db v b H dc v c ] =rank(G) = 2. From the above lemma we can see that receiver d will observe interference aligned in two dimensions that can be zero-forced by projecting y d along U d ∈ C 4×2 . The corresponding symbols s d can then be decoded from the interference- free observation U H d y d =U H d H dd V d s d + ˆ z. (3.70) Up to this point we have shown that all the cells in cluster A (nodes a, b, c and d) are able to decode their desired messages. Going back to Fig. 3.10, we can see that we can follow exactly the same beamforming procedure for the adjacent cluster of cells that is highlighted in a dash rectangle. In a similar manner we can show that all the cells in the middle and bottom lines of the corresponding stripe can successfully decode their desired messages. Toconcludeourproofwehavetoshowthatallthecellsassociatedwithdiamond nodes (top line of the stripe) can also decode their corresponding messages. Towards this end, consider cluster B shown in Fig. 3.10 and notice that receiver f observes one interfering stream from transmitter e and three interfering streams from (out-of-cluster) transmitters d and b. Since V d and v b have already been chosen (in cluster A), our only option is to design v e such that H fe v e ∈ span [H fd V d H fb v b ] | {z } ,S∈C 4×3 . (3.71) As we have seen before (for receiver b), this is possible by choosing u f in the left nullspaceofSandv e tobeorthogonaltoH H fe u f . Inasimilarfashion, wecanextend the above beamforming choices for all diamond nodes so that the corresponding receivers will observe all interference aligned in a three-dimensional subspace that can be zero-forced allowing them to successfully decode their desired symbols and achieve d v = 1. 69 Chapter 3 Omni-directional Antennas and Asymmetric Configurations 3.4 Summary In this chapter we have extended the Cellular IA framework to the case of omni- directional base-stations with user terminals equipped with M = 2 transmit antennas and we provided one-shot linear interference alignment schemes for the practically relevant cases in which base-stations have N = 2, 3 and 4 receive antennas. Omni-directional cells yield an interference graph topology that is significantly more involved than the sectored case previously studied in Chapter 2. Nevertheless, omni-directional antennas are commonly used in small and densely deployed cells, which is an on-going technology trend, and therefore deserve to be studied. For the case of 2×2 links, we have shown that 3/4 DoFs per user (i.e., per cell) are achievable in one-shot (without symbol extensions) by linear interference alignmentprecoding. Moreover, wehaveproventhatinthiscasetheachievable 3/4 DoFs per user are asymptotically optimal for a large extended network, where the boundary effect of the cells at the edge of the network vanishes. For the practically relevant cases of 2× 3 and 2× 4 links, we have provided explicit one-shot linear IA constructions to achieve 1 and 7/6 DoFs per user respectively. Our converse for the 2× 2 case is based on the bounds on the degrees of freedom feasible by one-shot precoding given in [RLL12], [BCT13]. Applying the corresponding bounds in extended networks results in a challenging non-linear optimization problem over a number of variables that grows with the number of transmit-receive pairs in the network. Taking into account specific structural properties of the cellular interference graph, we reformulated (relaxed) the above optimization problem into a tractable linear program with a small number of variables that does not depend on the size of the network and provided a tight converse bound for the 2× 2 case using duality theory. This approach can be generalized to the symmetric M× M case, and is a technique of independent interest. Overall, our work points out the importance of exploiting the network topology in extended (large) network models, and local decoded message sharing through backhaul links, which is not a demanding task for the backhaul (e.g., it can be implemented by IP tunneling over a shared Fiber-Optical infrastructure) and, yet,canprovideunboundedcapacitygainswithrespecttoconventionalapproaches, when combined with carefully designed IA precoding schemes. 70 CHAPTER4 Towards a general framework for Cellular IA In this chapter we extend the cellular interference alignment framework in two directions. First we consider the downlink of the cellular models that we have introduced in chapters 2 and 3. More specifically, in Section 4.1, we provide the dual framework for the downlink of cellular networks with the same backhaul architecture, and show that for every “one-shot” IA scheme that can achieved DoFs per user in the uplink, there exists a dual “one-shot” IA scheme that can achieve the same DoFs in the downlink. To enable cellular interference alignment for the downlink, base-stations will now use the same local backhaul links to exchange quantized versions of the dirty-paper precoded signals instead of user messages. Then, in Section 4.2, we consider general interference graph topologies and develop results that can be applied to a large class of partially connected interference networks. In particular, we characterize the gains of cooperation through network interference cancellation (or network-DPC for the downlink) in single antenna networks from a graph theoretic perspective and identify a key structural property of multiple antenna networks that characterizes a large class of topologies for which practical one-shot IA schemes can achieve M/2 DoF per user. 71 Chapter 4 Towards a general framework for Cellular IA 4.1 CellularInterferenceAlignmentfortheDownlink In Chapters 2 and 3 we have shown that “one-shot” interference alignment schemes can achieve the optimal degrees of freedom in the uplink of a cellular network topology in which base-stations (receivers) can exchange decoded messages locally over the backhaul links. A natural question that comes to mind is whether similar results can be obtained for the downlink of such networks. In the downlink, the local backhaul connections between base-stations can be used to enable transmitter cooperation, as opposed to receiver cooperation in the uplink. Inthissection, weconsideranovelsuccessive encoding scheme forthedownlink, based on DPC, that is able to create the same directed interference graph as the one we created in the uplink with network interference cancellation; base- stations will first quantize and then share their DPC precoded signals with their neighbors, who can in turn successively encode their messages using DPC to avoid the known interference. Within this framework, we will show that any DoFs that are achievable by linear one-shot interference alignment in the uplink of a cellular system with a given decoding order π, are also achievable in the downlink of this network with the same linear IA precoding scheme, as long as the corresponding encoding order π (under which base-stations encode, quantize and share their DPC signals) is reversed. 4.1.1 Successive encoding based on DPC Let us first focus on two neighboring base-stations of the cellular network that are connected through a limited capacity backhaul link and describe how they can successively encode their messages using DPC such that interference is pre- canceled in one direction. We will consider here for simplicity the case where both transmitters and receivers are equipped with a single antenna in order to outline the main idea behind our successive encoding scheme. The received signals observed at the mobile users associated with base-station 1 (BS1) and base-station 2 (BS2) are given by y 1 = ˜ h 11 x 1 + ˜ h 12 x 2 +z 1 , y 2 = ˜ h 22 x 2 + ˜ h 21 x 1 +z 2 , (4.1) 72 Chapter 4 Towards a general framework for Cellular IA W1 x 1 W2 x 2 quantizer encoder ... Q(x2) backhaul quantizer backhaul Q(x1) encoder Dirty Paper Coding ˆ W2 ˆ W1 Q(x1) Downlink decoder ... backhaul decoder y2 =h22x2 +h21(x1 Q(x1))+z2 y1=h11x1+h12x2+z1 ˆ W2 ˆ W1 backhaul Interference Cancellation ˆ W1 Uplink BS1 BS2 BS1 BS2 Figure 4.1: Successive decoding in the uplink versus successive encoding in the downlink. In both cases, base-station 1 (BS1) will use the backhaul to give the corresponding information to base-station 2 (BS2). In the uplink, BS2 can use ˆ W 1 to reconstruct the corresponding signal and subtract the interference coming from user 1. In the downlink, BS2 can use Q(x 1 ) and DPC to avoid interference from BS1. where x 1 , x 2 are the transmitted signals (represented as vectors with block length n) of BS1 and BS2, satisfying the average power constraint 1 n E[x H i x i ]≤P,i = 1, 2 and z i is i.i.d Gaussian noise with unit variance. We will assume without loss of generality that the BS1 has already encoded its message using DPC, to eliminate some other interfering links in the network, and focus on BS2. Fig. 4.1 shows the successive encoding scheme for the downlink (in the setting we consider here) in comparison to the successive decoding scheme that we used in the previous sections for the uplink. In the downlink, BS1 will first quantize its transmitted DPC signal x 1 to obtain Q(x 1 ). Since x 1 is Gaussian i.i.d 1 with average power P, its quantized version Q(x 1 ) will also be Gaussian i.i.d, and can be represented at rate R(D) = log(P/D) with average distortion (quantization noise variance) given by 1 n E [||x 1 −Q(x 1 )|| 2 ]≤ D. In order to keep the quantization noise at the system’s noise level, we will set the distortionD = 1. Now, assuming that the backhaul rate between BS1 and BS2 is at least log(P ), we can let BS1 give its quantized DPC 1 It is well-known that the DPC precoded signal can be treated as Gaussian iid. This follows from the fact that the random variable X forming the auxiliary variable U =X +αS in Costa’s coding scheme [Cos83] is Gaussian and independent of the interference S, and from standard strong typicality arguments (see e.g., the appendix of [WNC10]). Also, if the universal lattice precoding scheme of [ZSE02] is used instead of Costa’s scheme, it is well-known that the dithered modulo lattice precoded signal is Gaussian i.i.d. in the limit of large dimension for a sequence of shaping-good lattices. 73 Chapter 4 Towards a general framework for Cellular IA signalQ(x 1 ) to BS2. As a result, BS2 will know the quantized part of interference Q(x 1 ) coming from BS1 and can use it to successively encode its own DPC signal as follows. The observed signal at the intended receiver of BS2 can be written as y 2 = ˜ h 22 x 2 + ˜ h 21 x 1 +z 2 = ˜ h 22 x 2 + ˜ h 21 Q(x 1 ) + ˜ h 21 (x 1 −Q(x 1 )) | {z } quantization noise +z 2 = ˜ h 22 x 2 + ˜ h 21 Q(x 1 ) +z Q +z 2 , (4.2) where z Q , ˜ h 21 (x 1 − Q(x 1 )) denotes the effective i.i.d Gaussian noise with variance| ˜ h 21 | 2 due to quantization. Notice that since the quantization noise z Q is independent of x 1 , the above observation can be written in the standard form: y 2 = ˜ h 22 x 2 +s + ˜ z, (4.3) where s = ˜ h 21 ·Q(x 1 ) is the known interference at BS2 and ˜ z = z Q +z 2 is the effective Gaussian noise with variance 1 +| ˜ h 21 | 2 . Using DPC at BS2 to avoid the known interference s, we can obtain an achievable rate at user 2 given by R 2 = log 1 + | ˜ h 22 | 2 P 1 +| ˜ h 21 | 2 ! , (4.4) which has the same pre-log factor (equal to 1 DoF) as if interference was not present, due to the fact that the quantization noise variance is constant and not a function of P. Therefore, at high SNR, we can see that this successive encoding scheme for the downlink has exactly the same network interference cancellation properties as the successive decoded message passing scheme that we have used for the uplink; In the following subsection we will use this scheme to enable directed interference cancellation across the entire network and obtain the corresponding uplink-downlink duality result for the one-shot DoFs achievable by Cellular IA. Remark 4.1. It is worth pointing out that sharing the quantized DPC signals is fundamental for this scheme to be embedded in the context of a larger cellular network. To enable network interference cancellation, one could be tempted to use an approach in which base-stations share user messages (as in [SW11,LLSW12,GAV11,GAV12,GV11,LSW07]) instead of quantized codewords 74 Chapter 4 Towards a general framework for Cellular IA over the backhaul. However, we can see that in that case, interference would propagate through the cellular system – from neighbor-to-neighbor, along the network interference cancellation paths – and subsequent base-stations would observe interfering signals that are functions of all their predecessors’ messages in the encoding order. 4.1.2 Uplink-Downlink Duality A first step towards our uplink-downlink DoF duality result will be to show that the encoding scheme based on DPC that we introduced in the previous section is indeed able to successively remove directed interfering links in the downlink, across the entire networkG(V,E), according to a given (predefined) encoding order π. For simplicity, in order to illustrate the main ideas behind our scheme, we will only consider here cellular networks without intra-cell interference. It can be seen however that a similar approach can be applied to networks with intra-cell interference, aslongasthecollocatedsectortransmittersareabletojointlyprecode their signals. Let V v ∈ C M×dv ,U v ∈ C M×dv denote the transmit and receive beamforming matrices associated with each cell v∈V, where M is the number of the available transmit/receive antennas and d v is the number of transmitted signals (of block lengthn) in cellv∈V denoted byX v ∈C dv×n . As in the previous sections,H uv ∈ C M×M denote the (constant, flat-fading) channel gains between the transmitter of cell v∈V and the receiver of the cell u∈V, that are chosen at random from a continuous distribution and are identically zero for all (u,v) / ∈E,u6=v. Lemma 4.1. The effective channel between the downlink transmit-receive pair associated with cell u∈V after DPC is given by U H u Y u =U H u H uu V u X u + X v:u≺πv U H u H uv V v X v +U H u ˜ Z u , (4.5) where the columns of ˜ Z u ∈C M×n are i.i.d zero-mean Gaussian noise vectors with covariance C z , 1 n E[ ˜ Z H u ˜ Z u ] = I + X v:u πv H uv V v V H v H H uv . (4.6) 75 Chapter 4 Towards a general framework for Cellular IA Proof. The projected received signal at cell v∈V is given by U H u Y u =U H u H uu V u X u + X v∈N (u) U H u H uv V v X v +U H u Z u , whereN (u)denotesthesetofalltheinterferingneighborsofu∈V. Thissetcanbe partitioned according to the encoding order π into two sets,{v∈N (u) : u≺ π v} and{v∈N (u) : u π v}. All the transmitters v that belong to the set that has already encoded their messages (i.e, u π v) will quantize their DPC signals X v into Q(X v ) at rate d v · log(P ) and give them to base-station u through the backhaul. The received signal can hence be written as U H u Y u = U H u H uu V u X u + X v:u≺πv U H u H uv V v X v + X v:u πv U H u H uv V v ·Q(X v ) | {z } known interference +U H u X v:u πv H uv V v (X v −Q(X v )) | {z } quantization noise +U H u Z u . (4.7) If we let ˜ Z u ,Z u + X v:u πv H uv V v (X v −Q(X v )) and encodeX u using DPC to avoid the known interference we obtain the effective channel given by Eq.4.5 with (column-wise) i.i.d Gaussian noise whose covariance is given by Eq.4.6. Remark 4.2. Although in the above scheme base-stations share quantized DPC signals instead of messages, the rate required for the backhaul links in the downlink is the same (in the leading order of P) as the rate required for corresponding the local message-passing scheme in the uplink. This follows from the fact that DPC in this setting is used on top of the linear precoding scheme over the antennas: If we let ˜ H uv ,U H u H uv V v ∈C du×dv ,∀(u,v)∈E, then from each encoder’s perspective, DPC is performed over a d u ×d u equivalent MIMO channel: ˜ Y u = ˜ H uu X u + ˜ S + ˜ Z u , 76 Chapter 4 Towards a general framework for Cellular IA with d u -dimensional interference ˜ S = P v:u πv ˜ H uv ·Q(X v ). The interference ˜ S is therefore known to the encoder, as long as the corresponding base-station is able to get thed v -dimensionalQ(X v ) at rate log(P ) per dimension over the local backhaul links. This is exactly the same backhaul rate scaling required for exchanging messages and hence both the downlink and the uplink schemes can operate under the same backhaul network infrastructure. a b c a b c ⇡ Uplink decoding order ¯ ⇡ Downlink encoding order a b c a b c H ab H ac H bc 1 2 3 y 2 y 3 1 2 3 W 2 W 3 ↵ DoF 3/2 3 0 4/3 d(↵ )=3/2+↵ ·9/8 H H ab H H ac H H bc Figure 4.2: Uplink and Downlink with reverse decoding/encoding orders, π and π. After the corresponding network-wide interference cancellation in both cases, the remaining interference channel gains for the downlink are reciprocal to the ones obtained in the uplink, and are given by H uv =H H vu ,∀[v,u]∈E π . Theorem4.1 (Uplink-DownlinkDuality). Any degrees of freedom{d v ,v∈V} that are achievable in the uplink of the cellular networkG(V,E) in one-shot by linear IA with beamforming matrices{V v ∈ C M×dv ,v∈V} and{U u ∈ C M×dv ,u∈V}, under the network interference cancellation framework with decoding order π, are also achievable in the downlink of the same cellular networkG(V,E) under the successive DPC framework with the reverse encoding order π and beamforming matrices given by{V v =U v ∈C M×dv ,v∈V} and{U u =V u ∈C M×du ,u∈V}. Proof. Let the partial order “≺ π ”, defined on the setV, be the inverse of “≺ π ” such that u≺ π v⇔v≺ π u,∀u,v∈V, (4.8) andconsiderthecorrespondingdirectedinterferencegraphsG π (V,E π )fortheuplink andG π (V,E π ) for the downlink. Since the degrees of freedom{d v ,v ∈V} are 77 Chapter 4 Towards a general framework for Cellular IA achievableintheuplinkwecanarguethatthecorrespondingbeamformingmatrices chosen for the uplink,{V v ∈C M×dv ,v∈V} and{U u ∈C N×dv ,u∈V}, satisfy: U H u H uv V v = 0,∀[v,u]∈E π , and (4.9) rank U H v H vv V v =d v ,∀v∈V. (4.10) As illustrated in Fig. 4.2, for every directed edge [v,u] ∈E π there exists a directed edge [u,v]∈E π and the corresponding channels are reciprocal to each other. That is, the downlink channel matrices denoted byH vu ∈C M×M , [u,v]∈E π are given by H vu =H H uv , (4.11) whereH uv ∈C M×M are the corresponding uplink channel matrices associated with opposite edges [v,u]∈E π . Now, we can rewrite Eq.4.9 as follows. U H u H uv V v = 0,∀[u,v]∈E π ⇔U H v H uv V u = 0,∀[u,v]∈E π (4.12) ⇔V H u H H uv U v = 0,∀[u,v]∈E π (4.13) ⇔V H u H vu U v = 0,∀[u,v]∈E π , (4.14) whereEq.4.12followsfromthefactthatπ andπ satisfyEq.4.8, Eq.4.12isobtained by transposing all equations, and Eq.4.14 by substituting the downlink channel matrices from Eq.4.11. It has become clear now from Eq.4.14 that if we choose the downlink transmit beamforming matrices V v ∈ C M×dv to be the corresponding uplink receive beamforming matrices U v ∈ C M×dv and vice versa (i.e., U u = V u ), the following IA conditions are satisfied in the downlink: U H u H uv V v = 0,∀[v,u]∈E π , and (4.15) rank U H v H vv V v =d v ,∀v∈V. (4.16) Now from Lemma 1 we have that the signal observation for every receiveru∈V is given by U H u Y u =U H u H uu V u X u + X v:u≺πv U H u H uv V v X v +U H u ˜ Z u , (4.17) 78 Chapter 4 Towards a general framework for Cellular IA and from Eq.4.15 we can see that P v:u≺πv U H u H uv V v X v = 0, which in turn yields U H u Y u =U H u H uu V u X u +U H u ˜ Z u . (4.18) From Eq.4.16 and since the noise variance does not scale with the transmit power P, we can argue that every transmit-receive pair u∈V in the downlink of the cellular networkG(V,E) will achieve d u degrees of freedom and we conclude the proof. 4.2 Arbitrary Local Interference Topologies Here we consider a general interference graph model for wireless networks that can be used to study arbitrary local interference topologies with local cooperation through network interference cancellation [BKKR11]. Examples that can be put in thisframeworkincludeallthepreviouslystudiedhexagonalcellularnetworkmodels that have been introduced in Chapters 2 and 3. In the following, we will only focus on the uplink of locally connected networks, since – as we have seen before – all the corresponding DoF results can be easily translated for the downlink as well, through the dual Network-DPC architecture presented in Chapter 4.1. The channel model that we will use here is a natural extension of the channel model presented in the previous chapters. The interfering transmit receive pairs are specified by an (arbitrary) interference graph and the corresponding channel matrices assumed to be constant and generic (i.e., drawn from a continuous, non-degenerate distribution). Furthermore, for the backhaul links we will only consider nearest neighbor connectivity, assuming that when two transmit-receive pairs are close enough to interfere they are also close enough to cooperate over the backhaul. An example of this locally connected network model and the corresponding backhaul connectivity is illustrated in Fig. 4.3. 4.2.1 Directed Interference and Acyclic Decomposition To avoid ambiguity, in the remaining of this chapter we will only consider directed graphs. A directed graphD(V,E) is defined by the set of verticesV and a set of ordered pairsE⊆{[u,v] :u,v∈V} that represent the directed edges inD. 79 Chapter 4 Towards a general framework for Cellular IA 1 2 3 4 2 4 3 1 1 4 2 3 Figure4.3: Locallyconnectednetworkmodelandthecorrespondinginterferencegraph: This figure shows the interference topology between four transmit-receive pairs in the uplink. Green arrows represent the desired links and red arrows indicate interference. The dashed lines show the local backhaul connections between the corresponding base stations (BS). Notice that there is no backhaul connection between BS1 and BS4 since the corresponding transmit-receive pairs do not interfere with each other. The resulting (directed) interference graph is shown in the right hand side of the figure. Any directed graphD(V,E) can be decomposed into two acyclic graphs as follows. For any ordering π :V→{1, 2,...,|V|} of the vertices, we first partition the edgesE into two setsE π andE π given by E π ={[u,v]∈E :π(u)<π(v)}, and E π ={[u,v]∈E :π(u)>π(v)}. Then it is easy to see (by the definition of π) that the graphsD π (V,E π ) and D π (V,E π ) are acyclic and sinceE =E π ∪E π we can writeD(V,E) =D(V,E π ∪E π ). We denote the above decomposition as D =D π ⊕D π , Acyclic chromatic number Before we proceed we will first provide some useful definitions and a lemma. 80 Chapter 4 Towards a general framework for Cellular IA Definition 4.1 (Chromatic Number). The chromatic number of an undirected graphG(V,E), denoted by χ(G), is defined as the minimum positive integer r > 0 such that there exists a decomposition of the verticesV into r disjoint sets, each of which induces a disconnected subgraph ofG. Definition 4.2 (Acyclic Chromatic Number). The acyclic chromatic number of a directed graphD(V,E), denoted by χ A (D), is defined as the minimum positive integerr> 0 such that there exists a decomposition of the verticesV intor disjoint sets, each of which induces an acyclic subgraph ofD. Definition 4.3 (Symmetric graphG). The symmetric graphD(V,E) of directed graphD(V,E) is given by including inE all the directed edgesE as well as their inverse directions, i.e.,E ={[u,v] : [u,v] or [v,u]∈E}. Remark 4.3. Note that the symmetric graph is directed but equivalent to an undirected graph with the same edges. Lemma 4.2. For all directed graphsD(V,E) we have that χ A (D)≤ χ(D), with equality ifD≡D. However, χ A (D) =χ(D) does not implyD≡D. Proof. This lemma follows directly from definitions 4.1, 4.2 and 4.3. See Fig. 4.4 for an illustrating example. A (G)=2 (G)=2 (G)=3 A (G)=1 G!G G!G Figure 4.4: The acyclic chromatic number of D vs the chromatic number of its symmetric graphD. Notice that in the first case χ A (D) =χ(D) butD6≡D. 81 Chapter 4 Towards a general framework for Cellular IA 4.2.2 Single-antenna networks and acyclic scheduling Lemma 4.3. If the interference graphD(V,E) of a network with|V| = K is acyclic, then K DoF are achievable inD by network interference cancellation. Proof. Consider the decoding order π for which u≺ π v⇔ [u,v]∈E. Since the interference graphD(V,E) is acyclic, one can see thatE π ={[u,v]∈E :v≺ π u}≡ ∅ and hence after network interference cancellation, all the vertices in the graph D(V,E π ) will be disconnected. Therefore, K DoF can be achieved. Example: Asymmetric Wyner Model. As an example for Lemma 4.3 we can consider the linear asymmetric channel model shown in Fig. 4.5. Using the decoding order π ={1, 2,..., 8} we can see that all interference can be cancelled and hence we can achieve a total of 8 DoF in this network. In contrast, without network interference cancellation one can only achieve 4 DoF (by the conventional independent set scheduling). Example | Asymmetric Wyner Model y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 W 1 W 2 W 3 W 4 W 5 W 6 W 7 W 8 •Uplink: Figure 4.5: The asymmetric Wyner model. Theorem 4.2. For any interference graph D(V,E), with |V| = K, network interference cancellation can achieve K χ A (D) DoF. Proof. The achievability proof follows directly from Lemma 4.3 by scheduling the χ A (D) independent directed acyclic subgraphs ofD. Remark 4.4. Asonewouldexpect, theDoFachievableundernetworkinterference cancellation are greater than (or equal to) the DoF achieved by the conventional independent set scheduling approach, since χ A (D)≤χ(D) (Lemma 4.2). 82 Chapter 4 Towards a general framework for Cellular IA 4.2.3 Multiple antenna networks and topology projections The main idea that we propose in this section is to use multiple antennas and align interference in a specific pattern such that after a zero-forcing projection the resulting interference topology will be modified to our favor. For ease of exposition, here we will only consider the case where all transmitters and receivers in the network are equipped with M = 2 antennas. Definition 4.4 (Bipartite alignment graph). Given a directed interference graph D(V,E) we construct the corresponding bipartite alignment graph as follows. First we make a copyU of the vertex setV and we let f :U →V be a one-to-one mapping betweenU andV. Then we draw an edge (v,u) between v∈V and u∈U if and only if [v,f(u)] is a directed edge in the original graphD. 5 4 3 1 2 4 3 5 1 2 5 3 4 1 2 Figure 4.6: An interference graph (left) and its bipartite alignment graph (right). Lemma 4.4. Consider a directed interference graph D(V,E), |V| = K, of a network in which transmitters and receivers are equipped with M = 2 antennas. If the bipartite alignment graph ofD is acyclic or has disjoint cycles, then a linear “one-shot” interference alignment scheme can achieve K DoF. Proof. To prove this it suffices to show that, under the conditions stated in the lemma, all interference can be aligned in a single dimension at every receiverv∈V inthenetworkD(V,E). Withoutlossofgeneralitywewillassumethatthebipartite alignment graphB ofD(V,E) is connected and we will distinguish between the two 83 Chapter 4 Towards a general framework for Cellular IA casesstatedinthelemma: FirstwearegoingtoconsiderthecasewhereB isacyclic and then focus on the case whereB contains one cycle 2 . Recall from definition 4.4 that the bipartite alignment graph contains two types of nodes denoted byV, which is the set of nodes inD(V,E), andU which is a copy ofV. LetB(V,U,A) denote the bipartite alignment graph whereA denotes the corresponding set of edges. By the construction ofB(V,U,A) one can think of the nodesV as the transmitters, the nodesU as the receivers and the edges A as the interfering links that have to be aligned. Letting v v denote the two- dimensional beamforming vectors of the transmitter nodes v∈V, and u u be the two-dimensional zero-forcing vectors of the receiver nodes, it suffices to show that u T u H uv v v = 0,∀(u,v)∈A. (4.19) Case 1: If the bipartite alignment graphB(V,U,A) is acyclic then it can be represented by a tree. Notice that from the bipartite property ofB, all the edges inA will connect one node fromV to one node fromU. Take r∈V to be an arbitrary root node and construct the corresponding tree, denoted byT (V∪U,A) (e.g., by breadth-first search inB starting from r∈V). Further, let depth(x,T ) denote the distance of a node x from the root of the treeT with the convention that depth(r,T ) = 0. Since the tree was constructed from a bipartite graph, it will have the property that x∈ V if and only if depth(x,T ) is even and x∈ U if and only if depth(x,T ) is odd (See Fig. 4.7 for an illustrating example of this construction). In order to satisfy Eq.4.19 we can choose the beamforming vectors v v , for all v∈V as follows. For any node v∈V\{r} let u∈U be its parent inT , i.e., (u,v)∈A and depth(v,T ) = depth(u,T ) + 1. Similarly, let w ∈ V be the parent of u. For example, in Fig. 4.7, if v∈V is the blue node with the number 2 then u is the green node number 4 and w is the blue node number 1. With this notation in place, the beamforming vectorv v , for anyv∈V can be chosen as a function ofv w given by v v . =H −1 uv H uw v w . 2 Note that since we have assumed (without loss of generality) thatB is connected the above two cases cover the conditions stated by the lemma:B will either be acyclic or contain one cycle. 84 Chapter 4 Towards a general framework for Cellular IA Notice that in this case, the receiver u∈ U will observe the interference from v aligned with the observed interference from w. Following the same procedure for all nodes v∈V\{r} of the treeT , it is not hard to see that all interference will be aligned and every beamforming vector will be a function of the beamforming vectorv r of the root noder∈V. Choosing the direction ofv r arbitrarily andu u be the corresponding zero-forcing vectors that project the received signals orthogonal to interference we can see that K DoF can be achieved in this case. Case 2: Assume now that bipartite alignment graphB(V,U,A) contains a cycle of length L and let v 1 −u 2 −...−v L−1 −u L −v 1 denote the nodes in the cycle. Without loss of generality we can choose the edge (u L ,v 1 )∈A (indicated by a dashed red line in the example of Fig. 4.7) and remove it from the bipartite graph in order to make it acyclic. Hence, we can construct the corresponding tree T (V∪U,A\{(u L ,v 1 )}), as we did before, starting from an arbitrary root node r∈V. Following the same procedure as in Case 1, we can choose all beamforming vectors as a (linear) function of the beamforming vector v r of the root noder∈V which in turn will satisfy all interference alignment constraints – except the one that corresponds to the edge (u L ,v 1 )∈A. Letv 1 =Av r andv L−1 =Bv r , whereA andB are the corresponding matrices that are obtained by applying the procedure of Case 1. In order to satisfy this last constraint, the beamforming vectors of the transmitters v 1 and v L−1 must be aligned at the receiver u L , which means that v 1 . = H −1 u L v 1 H u L v L−1 v v L−1 ⇔ (4.20) Av r . = H −1 u L v 1 H u L v L−1 Bv r ⇔ (4.21) v r . = A −1 H −1 u L v 1 H u L v L−1 Bv r (4.22) Therefore, if we choose the beamforming vector ofv r of the root noder∈V to be the eigenvector of the matrixA −1 H −1 u L v 1 H u L v L−1 B, the above equation (Eq.4.22) will be satisfied without affecting any other alignment condition in the network. Choosing the receive vectors u u , for all u ∈ U to project the received signals orthogonal to the aligned interference at every receiver we can see thatK DoF can be achieved in this case as well and conclude the proof of the lemma. 85 Chapter 4 Towards a general framework for Cellular IA 5 4 3 1 2 4 3 5 1 2 7 6 6 7 5 4 3 1 2 7 6 5 4 3 1 2 4 3 5 1 2 7 6 6 7 Interference Graph Bipartite Graph Tree Structure Figure 4.7: Example: An interference graph (left), its bipartite alignment graph (middle) and the corresponding tree with root node 1. Theorem 4.3. For any interference graphD(V,E),|V| =K, of a network, where transmitters and receivers are equipped with M = 2 antennas, if there exists an decoding order that induces a decomposition ofD =D N ⊕D IA such thatD IA satisfies the conditions of Lemma 4.4, then K DoF are achievable by linear interference alignment under the network interference cancellation framework. Proof. The proof of this theorem follows immediately by applying the achievable scheme of Lemma 4.4 onD IA , projecting out the aligned interference and applying Lemma 4.3 to the resulting (one-dimensional) acyclic networkD N . Examples: • The 4-user 2× 2 (fully-connected) IC: As illustrated in Fig. 4.8 the interference graph of a four user interference channel can be decomposed (by acyclic decomposition, using any decoding order) intoD N (dashed lines) andD IA (solid lines). Now, sinceD IA has a bipartite alignment graph with exactly one cycle, we can directly apply Theorem 4.3 to show that 1 DoF per user is achievable by network interference cancellation in this setting. • Cell IA with sectors: As we have seen in Chapter 2, the sectored cellular model for networks without intra-cell interference can be decomposed (using the left-to-right top-down decoding order) into two acyclic graphsG N and 86 Chapter 4 Towards a general framework for Cellular IA 4 3 1 2 4 3 1 2 4 3 1 2 Figure 4.8: The interference graph (left) of a 4-user 2× 2 IC. The dashed and solid edges illustrate the acyclic decomposition of the graph intoD N (dashed lines) andD IA (solid lines). Notice that since the bipartite alignment graph ofD IA (right) has only one cycle, Theorem 4.3 can be applied in this case to achieve 4 total DoF. G π ∗. It is not hard to show that the corresponding bipartite alignment graph forG π ∗ is acyclic, and, hence, Theorem 4.3 can be used for the 2× 2 case to show that every transmit receive pair in the network can achieve 1 DoF. • Cell IA omnidirectional: The omnidirectional cellular model that was introduced in Chapter 3 does not immediately fall within this framework. One can see that any decoding order π will generate a directed interference graphG π whosebipartitealignmentgraphhasoverlappingcycles. Inthiscase Theorem 4.3 cannot be directly applied and one would expect that 1 DoF per cell is not achievable in this case. Indeed, as we formally proved with a linear programming converse in Chapter 3, the average (per user) DoF in this case are upper bounded by 3/4 +O(1/ √ K). Theorem 4.3 however gives a very interesting interpretation of our achievability results for this cellular network model; turning off a minimal subset of cells in the network yields a sub- network whose bipartite alignment graph has only disjoint cycles. Applying Theorem 4.3 to the sub-graph shown in Fig. 3.4 (that has been obtained by carefully choosing to turn off a set of K/4 cells), guarantees the existence of a linear interference alignment solution that achieves 1 DoF in every active cell, which in turn shows the achievability of 3/4 average DoF per cell. 87 Chapter 4 Towards a general framework for Cellular IA 4.3 Summary In this chapter we have shown that cellular interference alignment schemes can be applied to a general class of communication settings with local backhaul connectivity. We considered a dual process-and-share framework for the downlink based on dirty paper coding (DPC) that utilizes the local backhaul connections between base stations to enable transmitter cooperation and provide the same interference alignment gains. More specifically, we proposed a network-wide successive encoding scheme for the downlink in which base stations share their (quantized) DPC precoded signals over the backhaul in order to successively help their neighbors’ encoding process. This form of cooperation can be seen as the dual of the decoded message passing scheme for the uplink and is able to provide an uplink-downlink DoF duality result for cellular interference alignment: Any linear IA scheme that can achieve d DoFs per user in the uplink of a cellular system, can be translated under our framework to a dual linear IA scheme that can achieve the same DoFs in the downlink. Then we extended our results towards arbitrary network topologies and identified a key structural property that allows for the practical “one-shot” IA communication schemes under network interference cancellation (or network DPC). In particular we focused on directed interference graphsD(V,E) with 2× 2 links and showed that if the bipartite alignment graph of any acyclic subgraph, that is induced by a setS ⊆V and a decoding order π, does not contain overlapping cycles, then|S|/|V| DoF per user are achievable inD. Interestingly, this characterization includes the previously studied hexagonal cellular models in Chapters 2 and 3, and provides a graph theoretic perspective on the corresponding achievability proofs. 88 CHAPTER5 Fundamental limits of Distributed Interference Management Consider a wireless network in which K transmitters want to convey independent information messages to their intended receivers throughK interfering links and a wired backhaul network that consists of noiseless links through which all receivers can interact with each other and cooperate in the decoding process. A fundamental question that naturally arises from this cooperative communication setup is “How much backhaul capacity is required in order to achieve a given communication rate?” or, equivalently “What is the best communication rate that one can achieve for a given constraint on the total backhaul capacity?” For the two-user case, the above communication vs cooperation tradeoff has been characterized within a constant gap in [WT11a]. From a degrees of freedom (DoF) perspective, if the average (per user) rate scales as R = DoF· log(P ) + o(log(P )) and the average (per user) backhaul cooperation load scales as L = α·log(P )+o(log(P )), the results of [WT11a] can be used to show that the optimal communication vs cooperation tradeoff for the two-user interference channel is given by DoF ∗ (α) = min{1, 1+α 2 }. This is a very intuitive result in terms of the achievable DoF; when α = 0, one can achieve DoF(0) = 1/2 by orthogonal user scheduling and when α = 1, DoF(1) = 1 can easily be achieved by exchanging the user’s (appropriately quantized) received observations over the backhaul. However, 89 Chapter 5 Fundamental limits of Distributed Interference Management we can immediately see that following the same approach for the K-user case is not optimal in general. To begin with, it is well-known that transmission schemes based on interference alignment [MAMK08,CJ08,MGMAK14] are still able to achieve DoF(0) = 1/2 no matter how many users are interfering in the network. The fundamental question that we aim to answer in this work is whether the same holds for all values of α≥ 0; or, to put it in other words, whether the entire communication vs cooperation tradeoff remains unaffected by the presence of more than two interfering links. Surprisingly, our results show that this is indeed the case for the three-user interference channel. In order to do this, in this chapter, we develop the new idea of cooperation alignment that has the same effect on the backhaul load as interference alignment has on the degrees of freedom; from each receiver’s perspective, it appears as if a single user jointly processes the observations of the entire network and only shares the necessary information over the backhaul. The main contributions of the work presented in this chapter are as follows. • We propose an information theoretic channel model that reveals the funda- mental challenges of decentralized (over the cloud) backhaul cooperation in wireless networks, • We characterize the optimum communication vs cooperation tradeoff for three-user interference channels, and, • We show that a new form of backhaul alignment – that we term cooperation alignment – is required to achieve it. Related Work: Several results that have developed and used techniques that are closely related to our achievable schemes, can be found in [NNW13,NW12, SA14,NG11,MGMAK14]. In particular, [NNW13,NW12] proposed lattice coding schemes for compute-and-forward [NG11] based on techniques developed for interference alignment [MGMAK14,NGJV12] to show that K relays can reliably decode a (jointly) invertible function of the K interfering messages sent by the transmitters and achieve a computation rate with K degrees of freedom. More recently, [SA14] focused on a K×K×K (two-hop) wireless relay network and used similar techniques to design a novel aligned network diagonalization scheme that is able to distributedly invert the corresponding decoded functions of the K 90 Chapter 5 Fundamental limits of Distributed Interference Management transmitted messages, over the second wireless channel from the K relays to the K receivers. This chapter is organized as follows. In Section 5.1 we provide the basic definitions and formally describe the proposed channel model. In Section 5.2 we outline our main results for the communication vs cooperation tradeoff and in Sections 5.3 and 5.3.2 we give the corresponding proofs. 5.1 Problem Statement 5.1.1 Distributed Cooperation Channel Model The channel model considered in this chapter is illustrated in Fig. 5.1. Each transmitter i ∈ {1,...,K} has a message W i (intended for receiver i) which is encoded into a block length-n codeword [x i (t)] n t=1 satisfying the average power constraint 1 n P n t=1 |x i (t)| 2 ≤ P. The received signal at the ith receiver at time t = 1,...,n is given by y i (t) = K X j=1 h ij x j (t) +z i (t), (5.1) where h ij ∈C is the (complex) channel gain between the jth transmitter and the ith receiver, andz i (t) is the additive circularly-symmetric complex Gaussian noise observed at receiver i with zero mean and unit variance. Encoder 1 Encoder 2 Encoder K ... W 1 W 2 W K ˆ W K ˆ W 1 ˆ W 2 ... Decoder 1 x 1 x 2 x K y K y 1 y 2 z 2 z 1 z K h ij Decoder 2 Decoder K Backhaul Network [m 12 ,m 13 ,...,m 1K ] [m 21 ,m 23 ,...,m 2K ] [m K1 ,m K2 ,...,m K,K 1 ] [m 1K ,m 2K ,...,m K 1,K ] [m 21 ,m 31 ,...,m K1 ] [m 12 ,m 32 ,...,m K2 ] ... Figure 5.1: Channel model 91 Chapter 5 Fundamental limits of Distributed Interference Management The decoders are able to collaborate over the backhaul in order to produce their estimates, ˆ W i , i∈{1,...,K}. We assume that the backhaul network consists of directed noiseless links [i, ˆ i], between every pair of decodersi6= ˆ i∈{1,...,K}, and the rate from decoder i to decoder ˆ i is denoted by R [i, ˆ i] b . The backhaul message from decoderi to decoder ˆ i, that passes through the link [i, ˆ i] at timet, is denoted bym i→ ˆ i (t) and is given as a function of all the previously received signals [y i (τ)] t−1 τ=1 atreceiveriandall thepreviouslyreceivedmessages [m `→i (τ)] t−1 τ=1 fromall decoders `∈{1,...,K},`6=i. The rate of each backhaul linkR [i, ˆ i] b is therefore determined by the average joint entropy of the messages [m i→ ˆ i (τ)] n τ=1 that pass through it and is given by R [i, ˆ i] b = 1 n H [m i→ ˆ i (τ)] n τ=1 . (5.2) Animportantquantitythatwillbeusedintherestofthischapteristheaverage (per user) backhaul cooperation rate given by R b , 1 K K X i=1 X ˆ i6=i R [i, ˆ i] b . (5.3) 5.1.2 Achievable Rates, Capacity, and Degrees of Freedom The rate tuple (R 1 ,R 2 ,...,R K ) is called achievable under an average backhaul cooperation rate constraint R b ≤ L, if for any > 0 and sufficiently large n, there exist a length-n coding scheme defined by: • The message setsW i ={1, 2,..., 2 nR i }, i = 1,...,K. • The encoding functions f i :W i →C n , i = 1,...,K. • The backhaul relaying functions g [t] i ˆ i that generate m i→ ˆ i (t) such that m i→ ˆ i (t) =g [t] i ˆ i [y i (τ)] t τ=1 ,M [t−1] i ∈B t i ˆ i , where M [t] i ,{[m `→i (τ)] t τ=1 ,`∈{1,...,K},`6=i} is the collection of all the backhaul messages m `→i (τ) received at decoder i up to time t, andB t i ˆ i is a finite set that denotes the message alphabet used for the backhaul link [i, ˆ i] at time t. 92 Chapter 5 Fundamental limits of Distributed Interference Management • The decoding functions η i :C n × Y `={1,...,K},t={1,...,n} B t `i →W i , that give ˆ W i ,η (n) i [y i (τ)] n τ=1 , h M [τ] i i n−1 τ=1 , such that the corresponding probability of error given by P (n) e , P ∪ i∈{1,...,K} { ˆ W i 6=W i } is less than, and the average backhaul cooperation rate satisfies the backhaul load constraint R b = 1 K K X i=1 X ˆ i6=i 1 n H [m i→ ˆ i (τ)] n τ=1 ≤L. Definition5.1 (CapacityRegion). The capacityregionC L is defined as the closure of the set of all the rate tuples (R 1 ,R 2 ,...,R K ) that are achievable with an average backhaul cooperation rate R b ≤L. Remark 5.1. The region C 0 coincides with the capacity region of the K-user Gaussian interference channel (no cooperation) and the regionC ∞ with the capacity region of the K-user Gaussian MIMO multiple access channel with K receive antennas (full cooperation). For an achievable scheme, we are interested in characterizing the tradeoff between the average (per user) backhaul cooperation load given by α, lim P→∞ L(P ) log(P ) , (5.4) and the average (per user) achievable degrees of freedom (DoF) given by DoF(α), lim inf P→∞ 1 K log(P ) K X k=1 R K . (5.5) The average DoF (per user) of the channel is denoted by DoF ∗ (α) and defined as the supremum of DoF(α). Remark5.2. Notice that whenα = 0, the average degrees of freedomDoF(0) = 1/2 can be achieved (without any cooperation) by interference alignment. On the other hand, whenα =∞ (full cooperation) the average degrees of freedom is DoF(∞) = 1 can be achieved by jointly decoding the K received observations. 93 Chapter 5 Fundamental limits of Distributed Interference Management 5.1.3 Example: Centralized processing Under this framework, we can designate a specific receiver, say receiver 1, to take the role of the centralized processor and let all the other receivers quantize (within a constant distortion) and forward their observations to it. Now receiver 1 can jointly process all the observations to decode both its own message and the other the K− 1 messages and subsequently forward the K− 1 decoded messages back to their intended receivers. As we can see this scheme is able to achieve the full DoF of 1 with backhaul cooperation load α = 2 (K−1) K . If we time-share between this scheme and the asymptotic interference alignment scheme that can achieve DoF(0) = 1/2, we can obtain the boundary shown with the dashed line in Fig. 5.3. Encoder 1 Encoder 2 Encoder K ... W 1 W 2 W K ... x 1 x 2 x K y K y 1 y 2 z 2 z 1 z K h ij Quantizer 2 Quantizer K ˆ W K ˆ W 1 ˆ W 2 central processor Decoder 2 Decoder K ... Q(y K ) Q(y 2 ) Backhaul cooperation H 1 ... Decoder 1 Figure 5.2: A simple scheme to achieve 1 DoF per user with α = 2 (K−1) K 5.2 Communication vs Cooperation Tradeoff 5.2.1 Main Results Our main results on the communication vs cooperation tradeoff for the channel model introduced in the previous section, are described in the following theorems. 94 Chapter 5 Fundamental limits of Distributed Interference Management Theorem 5.1 (Upper Bound). In the K-user interference channel with average backhaul load α, we have DoF ∗ (α)≤ min n 1, 1+α 2 o . This outer-bound is derived based on considering every pair of links in the network, and developing a bound on communication versus cooperation tradeoff between these two, while the remaining links are effectively eliminated from the system (by a genie giving their messages to both receivers). We refer the reader to Section 5.3.2 for the detailed proof. Remark 5.3. Notice that Theorem 5.1 shows that for the K-user interference channel with average backhaul load α = lim P→∞ L(P ) log(P ) = 0, the per user DoF are bounded by DoF ∗ (0)≤ 1/2. This matches the well known DoF outer bound for the K-user interference channel (without cooperation), and implicitly shows that even when cooperation rates are allowed to scale as L(P ) =o(log(P )), there is no cooperation scheme that can increase the DoF achievable by interference alignment. Theorem 5.2 (Achievability). In the 3-user interference channel with average backhaul load α, we have DoF ∗ (α)≥ min 1, 1 +α 2 . (5.6) Theorems 5.1 and 5.2 characterize DoF ∗ (α) for the three-user interference channel as DoF ∗ (α) = min n 1, 1+α 2 o . Recall that, as observed in the chapter’s introduction, for two-user interference channels, the optimum DoF per user is again DoF ∗ (α) = min n 1, 1+α 2 o , and it is achievable by time-sharing orthogonal access (for α = 0) and quantize and forward (α = 1). The question is why for the three-user case, we are able to achieve the same tradeoff. For α = 0, the answer is well-understood. We can achieve 1/2 DoF per user with interference alignment. However, it seems surprising that the DoF per user is the same for other values of α. This result basically says that not only in the wireless channel, the extra link does not affect the DoF per user (due to interference alignment), but also in the backhaul, the load of collaboration per user does not scale with the number of users. The reason is that in the backhaul, we implement another form of alignment, which we call cooperation alignment, that is able to hide the additional collaboration load that is due to the extra link. 95 Chapter 5 Fundamental limits of Distributed Interference Management ↵ average (per user) backhaul load average (per user) DoF 1 1 2 1 (K 1) K 2 DoF(↵ ) DoF ⇤ (↵ )=min ⇢ 1, 1+↵ 2 Figure5.3: AchievableDoFvsα: Thedashedlinecorrespondstotheachievabletradeoff forK-user channels by centralized processing (Section 5.1.3) and the yellow region shows the corresponding gap from the K-user outer bound of Theorem 5.1. In Theorem 5.2 we show that cooperation alignment is able to close this gap and achieve the optimal communication vs cooperation tradeoff for K = 3. Illustrating Example: To describe the main idea behind cooperation alignment that we will later use in our achievability proof, we consider here a specific, yet illustrating, three-user interference channel with h 31 =γh 21 and h 33 =γh 23 , γ∈C, given by y 1 = h 11 x 1 +h 12 x 2 + h 13 x 3 +z 1 , y 2 = h 21 x 1 +h 22 x 2 + h 23 x 3 +z 2 , y 3 =γh 21 x 1 +h 32 x 2 +γh 23 x 3 +z 3 . This example is constructed such that the channel coefficients of x 1 and x 3 are aligned at receivers two and three, which allows us to implement and explain the cooperation alignment scheme in a simple way. Let us assume that each transmitter uses a Gaussian codebook, carrying one DoF. We aim to show that each receiver is able to decode its own message, with 96 Chapter 5 Fundamental limits of Distributed Interference Management backhaul load of α = 1. In the above example, receiver 3 can first form the backhaul messagem 3→2 as the quantized version ofy 3 (with a constant distortion) and send it to receiver 2. Since x 1 and x 3 are aligned in m 3→2 and y 2 , receiver 2 is able to decode x 2 at full rate 1 by subtracting m 3→2 from γy 2 . Notice that at this point receiver 2 can also extract the term h 21 x 1 +h 23 x 3 from its observation. In order to help receiver 1 decode x 1 , receiver 2 can now combine x 2 and h 21 x 1 + h 23 x 3 into a single message m 2→1 as the quantized version of h 12 x 2 + h 13 h 23 (h 21 x 1 + h 23 x 3 ). This combination is formed such thatx 2 andx 3 iny 1 andm 2→1 are aligned. Sharingm 2→1 over the backhaul will therefore help receiver 1 decodex 1 at full rate and subsequently extract the interfering term h 12 x 2 +h 13 x 3 . In a similar fashion, receiver 1 can recombine the newly available terms x 1 and h 12 x 2 + h 13 x 3 into the message m 1→3 as the quantized version of γh 21 x 1 + h 32 h 12 (h 12 x 2 +h 13 x 3 ) to help receiver 3 decodex 3 as well. This cooperative process, in which receivers iteratively decode desired messages, recombine interfering terms and share aligned backhaul messages that help another receiver decode, is what we refer to as cooperation alignment. In the above example, we could start the iteration because the combination of the signals observed at receiver 3 (γh 21 x 1 +γh 23 x 3 ) was already aligned with the interference observed at receiver 2. Therefore, by quantizing and forwarding y 3 to receiver 2, the latter could immediately eliminate its interference and decode x 2 with 1 DoF. The major challenge in the three-user IC with generic channel coefficients is that it would have been impossible to find a starting point for the above process since the corresponding channel coefficients are distinct in all receivers with probability 1. However – as we will see in the next section– we are able to create this form of alignment asymptotically by splitting the data streams into many sub-streams, each carrying a small fraction of the total DoF. The iterative approach is then started from a vanishing fraction of sub-streams that do not have any interference. 1 Here we assume that a Lattice vector quantizer with dither is used, thus quantization can be modeled with an additive independent quantization noise. 97 Chapter 5 Fundamental limits of Distributed Interference Management 5.3 Cooperation Alignment 5.3.1 The achievability of DoF(1) = 1 Proof of Theorem 5.2: We first define some short-hand notations. For a natural number N ∈ N, let s ij ∈{1,...,N}, i,j∈{1, 2, 3}, and s, [s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,s 31 ,s 32 ,s 33 ]. Then, clearlys∈S N ,{1,..,N} 9 . In addition, fors∈S N , we define the monomial ν s , as ν s = Q i,j h s ij ij . Furthermore, for a positive real number Q, we define Z Q , Z∩ [−Q,Q], i.e. the integer numbers between−Q and Q. To form the transmit signals, we use the methodology proposed in [MGMAK14] for real interference alignment, and its extention for complex channels in [MA10]. In addition, here, we follow the modulation technique used in [NW12] to achieve DoF of K for compute and forward (but with different encoding and decoding schemes). At transmitter one, the message W 1 is split into N 9 sub-messages, for some N∈N. Each sub-message is then coded, with a rate that will be specified later, and modulated over the integer constellation Z Q , Z∩ [−Q,Q] to form a sub-stream. Each sub-stream of message W 1 is indexed by a unique s∈S N and denoted by{a s (t)} n t=1 . The transmitter one at time t sends a weighted linear combination of sub-streams a s (t), and scaled by Γ, as x 1 (t) = Γ· P s∈S N ν s a s (t). (5.7) Recallthatν s = Q i,j h s ij ij . Thescalingfactor Γguaranteesthatthepowerconstraint is satisfied and will be determined later. We apply the same scheme at transmitters two and three to respectively form sub-streams{b s (t)} n t=1 and{c s (t)} n t=1 , s∈S N , and transmit signals x 2 (t) = Γ· P s∈S N ν s b s (t) and x 3 (t) = Γ· P s∈S N ν s c s (t). For simplicity of exposition, in the rest of the proof we drop the time indext, unless it 98 Chapter 5 Fundamental limits of Distributed Interference Management is required for clarification. At time t, the corresponding received observations for i∈{1, 2, 3} are given by y i = Γ· P s∈S N+1 ν s ·r i,s +z i , (5.8) where r 1,s , a (s 11 −1),s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,s 31 ,s 32 ,s 33 +b s 11 ,(s 12 −1),s 13 ,s 21 ,s 22 ,s 23 ,s 31 ,s 32 ,s 33 +c s 11 ,s 12 ,(s 13 −1),s 21 ,s 22 ,s 23 ,s 31 ,s 32 ,s 33 , (5.9) r 2,s , a s 11 ,s 12 ,s 13 ,(s 21 −1),s 22 ,s 23 ,s 31 ,s 32 ,s 33 +b s 11 ,s 12 ,s 13 ,s 21 ,(s 22 −1),s 23 ,s 31 ,s 32 ,s 33 +c s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,(s 23 −1),s 31 ,s 32 ,s 33 , (5.10) r 3,s , a s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(s 31 −1),s 32 ,s 33 +b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,s 31 ,(s 32 −1),s 33 +c s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,s 31 ,s 32 ,(s 33 −1) . (5.11) Inaddition, forsimplicity, inEq.5.8, Eq.5.9, Eq.5.10, andEq.5.11, foranys / ∈S N , we assume that a s = b s = c s = 0, and we follow this assumption throughout the proof. We further note that r i,s ∈ Z 3Q for i∈{1, 2, 3} and s∈S N+1 . If we let I = (N + 1) 9 , Q = 1 3 P (1−ε)/(I+1+2ε) , and Γ = c 1 P (I−1+4ε)/(2(I+1+2ε)) , for some positiveconstantc 1 andε, wecanshowthatthepowerconstantateachtransmitter is satisfied [MA10]. At time t, in each receiver i, we apply Maximum Likelihood (ML) detection to estimate r i,s (t), for all s∈S N+1 from the received signal y i (t). Notice that the above detection could be erroneous for some receivers. For now, let us focus on a specific time t, where r i,s (t), s∈S N+1 are correctly decoded for all receivers and we will discuss the probability of error and its effect on the achievable rates later. In the rest of this proof we aim to show that the receivers one, two, and three can respectively resolve desired symbols a s , b s and c s for all s∈S N based on the already individually detected sums r i,s by successively exchanging and processing 99 Chapter 5 Fundamental limits of Distributed Interference Management information over the backhaul. For convenience, in the notation to follow, we will denote addition in the sub-message vector indices s∈S N with corresponding superscripts; e.g, a (s 11 −1),s 12 ,...,s 33 in Eq.5.9 will be written as a s −1 11 ,s 12 ,...,s 33 . To start unraveling r i,s for all s∈S N+1 , i∈{1, 2, 3}, and resolve the desired symbolsateachreceiver, westartfromtheboundariesasfollows. Noticethatforall s 0 ∈{s∈S N+1 :s 31 =N + 1}, we have that r 3,s 0 = a s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 , and hence receiver 3 has resolved some of the symbols of receiver 1 without interference. Hence receiver 3 can give these symbols to receiver 1 directly over the backhaul. The result of this message passing step is that receiver 1 knows its desired symbols a s , for all s∈S N with s 31 = N. The corresponding backhaul rate that has been used for this step is given by N 8 log(2Q + 1). Now receiver 1 can subtract a s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 from its corresponding observations in Eq.5.9 and obtain the interference terms b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 +c s 11 ,s +1 12 ,s −1 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 , (5.12) for all s∈S N with s 31 = N. In order to help receiver 2, receiver 1 will form the sums b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 +c s 11 ,s +1 12 ,s −1 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 +a s 11 ,s +1 12 ,s −1 13 ,s −1 21 ,s 22 ,s +1 23 ,N,s 32 ,s 33 , (5.13) by adding the symbols a s 11 ,s +1 12 ,s −1 13 ,s −1 21 ,s 22 ,s +1 23 ,N,s 32 ,s 33 to the interference terms in Eq.5.12, and give them to receiver 2 over the backhaul. From Eq.5.10 we can see that receiver 2 has already detected the sums a s 11 ,s +1 12 ,s −1 13 ,s −1 21 ,s 22 ,s +1 23 ,N,s 32 ,s 33 +b s 11 ,s +1 12 ,s −1 13 ,s 21 ,s −1 22 ,s +1 23 ,N,s 32 ,s 33 +c s 11 ,s +1 12 ,s −1 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 (5.14) and hence subtracting them from Eq.5.13 will create the terms b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 −b s 11 ,s +1 12 ,s −1 13 ,s 21 ,s −1 22 ,s +1 23 ,N,s 32 ,s 33 , (5.15) 100 Chapter 5 Fundamental limits of Distributed Interference Management from which receiver 2 can successively resolve its desired symbols b s for alls∈S N with s 31 =N. The backhaul load for this step is equal to N 8 log(6Q + 1). Now we can already see that since receiver 2 knows b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 , it can extract from Eq.5.10 the interference terms c s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 +a s 11 ,s 12 ,s 13 ,s −1 21 ,s 22 ,s +1 23 ,N,s 32 ,s 33 , (5.16) for all s ∈ S N with s 31 = N. Giving the above terms to receiver 3 over the backhaul, will help it to decode c s for all s∈S N with s 31 =N, because receiver 3 already hasa s for alls∈S N withs 31 =N. Since the number of symbols that have been exchanged in Eq.5.16 is N 8 , the backhaul rate used for this step is equal to N 8 log(4Q + 1). Knowing c s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s 32 ,s 33 , receiver 3 can extract from Eq.5.11 the interference terms a s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−1),s 32 ,s 33 +b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s −1 32 ,s 33 (5.17) and subsequently create the terms a s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−1),s 32 ,s 33 +b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s −1 32 ,s 33 +c s 11 ,s +1 12 ,s −1 13 ,s 21 ,s 22 ,s 23 ,N,s −1 32 ,s 33 (5.18) by adding c s 11 ,s +1 12 ,s −1 13 ,s 21 ,s 22 ,s 23 ,N,s −1 32 ,s 33 to Eq.5.17 . Giving Eq.5.18 to receiver 1 will help it to resolve the symbols a s for all s ∈ S N with s 31 = N− 1, because receiver 1 already knows the interfering sums b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,N,s −1 32 ,s 33 + c s 11 ,s +1 12 ,s −1 13 ,s 21 ,s 22 ,s 23 ,N,s −1 32 ,s 33 from Eq.5.12. Similarly, as in Eq.5.13 and Eq.5.15, receiver 1 will help receiver 2 to resolve b s for all s∈S N with s 31 = N− 1. Now that receiver 2 knowsb s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−1),s 32 ,s 33 , it can extract from Eq.5.10 the interference terms c s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−1),s 32 ,s 33 +a s 11 ,s 12 ,s 13 ,s −1 21 ,s 22 ,s +1 23 ,(N−1),s 32 ,s 33 101 Chapter 5 Fundamental limits of Distributed Interference Management and create the terms c s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−1),s 32 ,s 33 +a s 11 ,s 12 ,s 13 ,s −1 21 ,s 22 ,s +1 23 ,(N−1),s 32 ,s 33 +b s 11 ,s 12 ,s 13 ,s −1 21 ,s 22 ,s +1 23 ,N,s −1 32 ,s 33 (5.19) by adding the symbols b s 11 ,s 12 ,s 13 ,s −1 21 ,s 22 ,s +1 23 ,N,s −1 32 ,s 33 (that are already known to receiver 2) in order to match the interference terms in Eq.5.17 that are known to receiver 3. Now from Eq.5.19 and Eq.5.17 receiver 3 can extract c s for all s∈S N with s 31 =N− 1 and create as in Eq.5.18 the terms a s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−2),s 32 ,s 33 +b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−1),s −1 32 ,s 33 +c s 11 ,s +1 12 ,s −1 13 ,s 21 ,s 22 ,s 23 ,(N−1),s −1 32 ,s 33 (5.20) to help receiver 1 resolve a s for all s∈S N with s 31 =N− 2. Following the same pattern we can see that the backhaul communication will requireN rounds, where in each roundr∈{0,...,N− 1}, the receivers can resolve their desired symbols a s , b s and c s for all s∈S N with s 31 =N−r. In round r we have the following message passing steps: • Receiver 3 gives to receiver 1 M [r] 3→1 = n b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−r+1),s −1 32 ,s 33 (5.21) +c s 11 ,s +1 12 ,s −1 13 ,s 21 ,s 22 ,s 23 ,(N−r+1),s −1 32 ,s 33 +a s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−r),s 32 ,s 33 |s ij ∈{1,...,N} o , • Receiver 1 gives to receiver 2 M [r] 1→2 = n a s 11 ,s +1 12 ,s −1 13 ,s −1 21 ,s 22 ,s +1 23 ,(N−r),s 32 ,s 33 (5.22) +c s 11 ,s +1 12 ,s −1 13 ,s 21 ,s 22 ,s 23 ,(N−r),s 32 ,s 33 +b s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−r),s 32 ,s 33 |s ij ∈{1,...,N} o , 102 Chapter 5 Fundamental limits of Distributed Interference Management • Receiver 2 gives to receiver 3 M [r] 2→3 = n a s 11 ,s 12 ,s 13 ,s −1 21 ,s 22 ,s +1 23 ,(N−1),s 32 ,s 33 (5.23) +b s 11 ,s 12 ,s 13 ,s −1 21 ,s 22 ,s +1 23 ,N,s −1 32 ,s 33 +c s 11 ,s 12 ,s 13 ,s 21 ,s 22 ,s 23 ,(N−1),s 32 ,s 33 |s ij ∈{1,...,N} o . The above set of symbolsM [r] i→j are carefully created in each roundr based on the availableobservationsandsymbolsatreceiveriinthepreviousroundssuchthatthe interfering terms match exactly the interference terms observed at receiver j. The recipient ofM [r] i→j is therefore enabled to extract its desired symbol by subtracting the sum of the interfering symbols without knowing their individual values. The total number of symbols that have been exchanged over the backhaul in the above scheme is given by #msg = N−1 X r=0 M [r] 1→2 + M [r] 2→3 + M [r] 3→1 (5.24) where each symbol is in Z 3Q . Therefore, the average (per user) backhaul rate that has been used can be calculated as R b ≤ #msg·log(6Q+1) 3 . Since M [r] 1→2 = M [r] 2→3 = M [r] 3→1 = N 8 , for all r = 0, 1,...,N− 1, we have that #msg = 3N 9 . Then lim P→∞ R b log(P ) ≤ N 9 1−ε (N+1) 9 +1+2ε , which is arbitrary close to one, given large enough N and small enough ε. For each time slott such that the ML detection ofr i,s (t),∀s∈S N+1 at all three receivers is performed without an error, the above unraveling process guarantees that receivers one, two, and three will be able to obtain respectively the correct a s (t), b s (t), and c s (t), for all s∈S N . On the other hand, for the time slots t such that error occurs in the ML detection at any of the receivers, the above unraveling process will inevitably deliver incorrect symbols. Using Fano’s inequality and the union bound, we can show that at each time t, the probability of error in detecting r i,s (t), for some s ∈ S N+1 and some i ∈ {1, 2, 3}, is upper-bounded by p e , 3(N + 1) 9 exp(−c 2 P ε/2 ) for some constant c 2 [MA10]. Thus, each sub- stream can carry the rate of at least (1−p e ) log(2Q + 1)− 1, which has DoF of 1−ε (N+1) 9 +1+2ε . Therefore, the DoF of each message can be at least N 9 1−ε (N+1) 9 +1+2ε , which is arbitrary close to one, given large enough N and small enough ε. 103 Chapter 5 Fundamental limits of Distributed Interference Management 5.3.2 DoF-Backhaul Upper Bound Proof of Theorem 5.1: First we are going to bound the rates R 1 +R 2 by following an approach similar to the two-user bound developed in [WT11a]. Consider a genie that gives y n [2:K] , [y 2 (τ),y 3 (τ),...,y K (τ)] n τ=1 andx n 2 , [x 2 (τ)] n τ=1 to receiver one, and the messages W [3:K] , [W 3 ,W 4 ,...,W K ] to both receivers one and two, as side information. Starting from Fano’s inequality we have that n(R 1 +R 2 − n ) = I W 1 ;y n 1 ,M [n] 1 +I W 2 ;y n 2 ,M [n] 2 (a) ≤ I W 1 ;y n 1 ,M [n] 1 W [3:K] +I W 2 ;y n 2 ,M [n] 2 W [3:K] (b) ≤ I x n 1 ;y n 1 ,M [n] 1 W [3:K] +I x n 2 ;y n 2 ,M [n] 2 W [3:K] (c) ≤ I x n 1 ;y n 1 ,M [n] 1 ,x n 2 ,y n [2:K] W [3:K] +I x n 2 ;y n 2 ,M [n] 2 W [3:K] (d) = I x n 1 ;y n [1:K] x n 2 W [3:K] +I x n 2 ;y n 2 ,M [n] 2 W [3:K] (e) ≤ I x n 1 ;y n [1:K] x n 2 W [3:K] +I x n 2 ;y n 2 W [3:K] +n P i6=2 R [i,2] b , where (a) follows from the genie giving the messages W [3:K] to both receivers, (b) form the data processing inequality, (c) from the genie giving y n [2:K] and x n 2 to receiver 1, (d) from the fact that x n 1 and x n 2 are independent and that M [n] 1 is a function of y n [1:K] and (e) from the chain rule and the fact that I x n 2 ;M [n] 2 y n 2 ,W [3:K] ≤H(M [n] 2 )≤ X i6=2 H([m i2 (τ)] n τ=1 ) =n P i6=2 R [i,2] b . The remaining two terms in the RHS of (e) can be further bounded as 104 Chapter 5 Fundamental limits of Distributed Interference Management I x n 1 ;y n [1:K] x n 2 W [3:K] +I x n 2 ;y n 2 W [3:K] = h y n 1 ,y n [3:K] y n 2 x n 2 W [3:K] −h z n [1:K] +h y n 2 W [3:K] = h h 11 x n 1 +z n 1 ,h 31 x n 1 +z n 3 ,...,h K1 x n 1 +z n K h 21 x n 1 +z n 2 +h h 21 x n 1 +h 22 x n 2 +z n 2 −h z n [1:K] (a 0 ) ≤ X i6=2 h h i1 x n 1 +z n i h 21 x n 1 +z n 2 +h h 21 x n 1 +h 22 x n 2 +z n 2 −h z n [1:K] (b 0 ) = X i6=2 h h i1 x n 1 +z n i −h i1 h −1 21 (h 21 x n 1 +z n 2 ) h 21 x n 1 +z n 2 +h h 21 x n 1 +h 22 x n 2 +z n 2 −h z n [1:K] ≤ X i6=2 h z n i −h i1 h −1 21 z n 2 +h h 21 x n 1 +h 22 x n 2 +z n 2 −h z n [1:K] (c 0 ) ≤ n X i6=2 log(1 +|h i1 | 2 /|h 21 | 2 ) +n log(1 +P (|h 21 | 2 +|h 22 | 2 )), where (a 0 ) follows from the chain rule and the fact that conditioning reduces entropy, (b 0 ) follows from the translational invariance property, and (c 0 ) from the factthattheGaussiandistributionmaximizesentropyforagivenvariance. Putting everything together, we conclude that R 1 +R 2 ≤ log(1 +P (|h 21 | 2 +|h 22 | 2 )) + X i6=2 R [i,2] b +o(log(P )). In a similar way, we can obtain bounds of the same form for the pairsR 2 +R 3 , R 3 +R 4 , up to R K−1 +R K andR K +R 1 which we can add together to show that 2 K X k=1 R k ≤ K+1 X `=2 log(1 +P (|h `,`−1 | 2 +|h `` | 2 )) +KR b +o(log(P )). Since R b ≤ L(P ) for any achievable scheme, dividing by log(P ) and taking the limit asP→∞ yields 2DoF(α)≤ 1 +α as required. Further, by considering each 105 Chapter 5 Fundamental limits of Distributed Interference Management user separately (single user bound) we can trivially obtain that DoF(α)≤ 1, and hence conclude that DoF(α)≤ min 1, 1 +α 2 , α≥ 0 as stated by Theorem 5.1. 5.4 Summary In this chapter, in order to study the fundamental limits of cooperation in wireless networks, we considered an interference channel model in which K receivers cooperatively attempt to decode their intended messages locally by processing and sharing information through limited capacity backhaul links. In contrast to distributed antenna architectures that have been proposed in the literature, where data processing is utterly performed in a centralized fashion, the model that we considered here aims to capture the essence of decentralized (over the cloud) processing, allowing for a more general class of interference management strategies. Focusingonthethree-usercase, wecharacterizedthefundamentaltradeoffbetween the achievable communication rates and the corresponding backhaul cooperation rate, in terms of degrees of freedom (DoF). Surprisingly, we showed that the optimum communication-cooperation tradeoff remains the same when we move from two-user to three-user interference channels. In the absence of cooperation, this is due to interference alignment, which keeps the fraction of communication dimensions wasted for interference unchanged. When backhaul cooperation is available, we developed a new technique that we called cooperation alignment, which guarantees that the average (per user) backhaul load remains the same as we increase the number of users. 106 APPENDIXA Appendix for Chapter 2 A.1 Proof of Theorem 2.1 Consider the directed interference graph G π ∗ V,E π ∗ and let V v ,U v ∈ C M×dv denote the transmit and receive beamforming matrices associated with each node v∈V. We will show here that it is possible to choose d v , V v and U v for every v∈V such that the following conditions are satisfied. U H u H uv V v = 0,∀[v,u]∈E π ∗ (A.1) rank U H v H vv V v =d v ,∀v∈V, and (A.2) 1 |V| X v∈V d v ≥ M 2 , M is even M 2 − 1 6 , M is odd. (A.3) Recallthedefinitionsoff(·)andφ(·)thataregiveninSection2.1.1andconsider the sets V k ={v∈V :f(φ(v)) =k}, k = 0, 1, 2. (A.4) 107 Appendix A Note that the setsV 0 ,V 1 andV 2 , are the same as the setsV square ,V circle andV diamond given in Section 2.2.1. The above setsV k satisfy V = 2 [ k=0 V k and V i ∩V j =∅, i6=j and hence form a partition ofV. An important observation is that, according toE π ∗, every receiver associated with a node u∈V 1 has at most one interfering transmitter. More specifically, for every u∈V 1 there exist [v,u]∈E π ∗ if and only if there exist v∈V 2 with φ(v) =φ(u)− 1−ω. Similarly, the receivers associated with the nodes u∈V 2 have at most two interfering transmitters and hence we can argue that for every u∈V 2 there exist [v,u]∈E π ∗ if and only if there existv 0 ∈V 0 withφ(v 0 ) =φ(u) + 1 orv 1 ∈V 1 with φ(v 1 ) =φ(u)−ω. Finally, every receiver associated with a node u∈V 0 observes at most three interferers and we have that for every u∈V 0 there exist [v,u]∈E π ∗ if and only if there exist v 1 ∈V 1 with φ(v 1 ) = φ(u) + 1 or v 2 ∈V 2 with φ(v 2 ) = φ(u)−ω or v 0 1 ∈V 1 with φ(v 0 1 ) =φ(u)− 1−ω. For any full column rank matrixA∈C m×n withm>n, we letP ⊥ A ∈C m×(m−n) be a basis for the nullspace of A H , such that (P ⊥ A ) H A = 0. A.1.1 M is even Let d v = M 2 , for all v∈V and consider the beamforming choices for the following node configurations (also shown in Fig. A.1). (a) For all v 0 ∈V 0 such that there exist v 1 ∈V 1 and u∈V 2 with φ(v 1 ) = φ(v 0 )− 1−ω and φ(u) =φ(v 0 )− 1, set V v 0 =H −1 uv 0 H uv 1 V v 1 /||H −1 uv 0 H uv 1 V v 1 ||. Otherwise choose V v 0 ∈C M×dv at random. 108 Appendix A (b) Forallv 1 ∈V 1 suchthatthereexistsv 2 ∈V 2 andu∈V 0 withφ(v 2 ) =φ(v 1 )+1 and φ(u) =φ(v 1 ) + 1 +ω, set V v 1 =H −1 uv 1 H uv 2 V v 2 /||H −1 uv 1 H uv 2 V v 2 ||. Otherwise choose V v 1 ∈C M×dv at random. (c) For all v 2 ∈V 2 such that there exists v 1 ∈V 1 and u∈V 0 with φ(v 1 ) = φ(v 2 ) + 1 +ω and φ(u) =φ(v 2 ) +ω, set V v 2 =H −1 uv 2 H uv 1 V v 1 /||H −1 uv 2 H uv 1 V v 1 ||. Otherwise choose V v 2 ∈C M×dv at random. (d) For all u∈V such that there exists an edge [v,u]∈E π ∗ for some v∈V, set U u =P ⊥ HuvVv . Otherwise choose U u ∈C M×dv at random. v 0 v 1 u v 1 v 2 u v 1 u (a) (b) (c) v 2 Figure A.1: The beamforming choices (a), (b) and (c). In this example, all the nodes with dashed outline have chosen their beamforming vectors at random. 109 Appendix A Notice that the conditions (A.2) and (A.3) are automatically satisfied (with probability one) since d v = M 2 ,∀v ∈V and H uv are chosen at random from a continuous distribution. We are going to show next that the conditions (A.1) are also satisfied for all [v,u]∈E π ∗. Consider the sets: E (k) π ∗ ={[v,u]∈E π ∗ :u∈V k }. As we have seen, every receiver associated with u∈V 1 observers at most one interfering transmitter and hence U H u H uv V v = (P ⊥ HuvVv ) H H uv V v = 0 for all [v,u] ∈ E (1) π ∗ . For every receiver u ∈ V 2 there exist at most two interfering transmitters given by v 0 ∈ V 0 , v 1 ∈ V 1 with φ(v 0 ) = φ(u) + 1 and φ(v 1 ) = φ(u)−ω. Notice that in this case φ(v 1 ) = φ(v 0 )− 1−ω and according to (a), H uv 0 V v 0 = H uv 1 V v 1 /||H −1 uv 0 H uv 1 V v 1 ||∈ span(H uv 1 V v 1 ). Hence we can also argue that U H u H uv V v = 0 for all [v,u]∈E (2) π ∗ . Now consider the setE (0) π ∗ . In a similar fashion, we can see that according to the beamforming choices (b) and (c), all interference observed by receivers u∈V 0 aligns in M/2 dimensions. That is for everyu∈V 0 that observes interference from the transmitters v 1 ∈V 1 ,v 2 ∈V 2 and v 0 1 ∈V 1 withφ(v 1 ) =φ(u) + 1 ,φ(v 2 ) =φ(u)−ω andφ(v 0 1 ) =φ(u)− 1−ω we have that span(H uv 1 V v 1 ) = span(H uv 0 1 V v 0 1 ) = span(H uv 2 V v 2 ) and hence U H u H uv V v = 0 for all [v,u]∈E (0) π ∗ as well. Since by definitionE (0) π ∗ ∪E (1) π ∗ ∪E (2) π ∗ =E π ∗ we conclude that the conditions (A.1) are satisfied for all [v,u]∈E π ∗. A.1.2 M is odd Let ˜ d v = M−1 2 ,∀v∈V and consider the beamforming matrices ˜ U v ∈C M× M+1 2 and ˜ V v ∈ C M× M−1 2 given by (a), (b), (c) and (d). Following the same arguments as before we can see that if we use the above beamforming subspaces for transmission, every receiveru∈V will observe interference aligned in M−1 2 dimensions and hence we could directly achieve 1 |V| P v∈V ˜ d v = M−1 2 . Notice however that in this case, any receiver that zero-forces M−1 2 out of M dimensions can in principle support one extra dimension for transmission since M− M−1 2 = ˜ d v + 1. Furthermore, any receiver that uses only ˜ d v = M−1 2 dimensions for desired symbols can zero-force the remaining M+1 2 dimensions and can hence tolerate one additional interfering stream from its neighbors. 110 Appendix A LetV ∗ ⊆V beasetofnodessuchthatthefollowingtwoconditionsaresatisfied: (u,v) / ∈E,∀u,v∈V ∗ (A.5) {v∈V ∗ : [v,u]∈E π ∗} ≤ 1,∀u / ∈V ∗ . (A.6) The first condition requires thatV ∗ is an independent set inG(V,E) and the second one states that for every u / ∈V ∗ there is at most one v∈V ∗ such that [v,u]∈E π ∗. Consider the following beamforming choices given in terms of ˜ V v and ˜ U v : • For all v∈V ∗ set V v = [ ˜ V v ,v v ]/||[ ˜ V v ,v v ]||, for some v v / ∈ span( ˜ V v ) and let U H v = ˜ U H v . • For all u / ∈V ∗ set V u = ˜ V u . If there exists v∈V ∗ such that [v,u]∈E π ∗ set and U H u =P ⊥ ˜ U H u Huvvv ˜ U H u . Otherwise set U H u = ˜ U H u . We have thatd v = ˜ d v +1 for allv∈V ∗ andd v = ˜ d v for allv / ∈V ∗ . We are going to show next that with the above beamforming choices the interference alignment conditions (A.1) and (A.2) are satisfied and hence the average (per sector) DoFs 1 |V| X v∈V d v = M− 1 2 + |V ∗ | |V| (A.7) are achievable. Then we are going to show that it is always possible to find a set V ∗ ⊆V that satisfies the properties (A.5) and (A.6) with|V ∗ |≥ |V| 3 and hence show that 1 |V| P v∈V d v ≥ M 2 − 1 6 as required by (A.3). First notice that the conditions (A.2) are automatically satisfied (with proba- bility one) since all the channel matrices H uv have been chosen at random from a continuous distribution. In order to show that the zero-forcing conditions (A.1) are also satisfied, consider the sets V (0) ∗ ,{u∈V\V ∗ : {v∈V ∗ : [v,u]∈E π ∗} = 0}, V (1) ∗ ,{u∈V\V ∗ : {v∈V ∗ : [v,u]∈E π ∗} = 1}. Notice that according to (A.6), the setsV ∗ , V (0) ∗ andV (1) ∗ form a partition of V. According to (A.5), every receiver associated with u∈V ∗ will only observe 111 Appendix A interference from transmitters v / ∈ V ∗ and since U u = ˜ U u for all u ∈ V ∗ and V v = ˜ V v for all v / ∈V ∗ , we have that U H u H uv V v = ˜ U H u H uv ˜ V v = 0,∀[u,v]∈ E π ∗,u∈V ∗ . Similarly, U H u H uv V v = ˜ U H u H uv ˜ V v = 0,∀[u,v]∈E π ∗,u∈V (0) ∗ . Now, consider the receivers associated with the nodes u∈V (1) ∗ and let v 0 ∈{v∈V ∗ : [v,u]∈E π ∗}. By construction, we have that span(H uv V v )⊆ span(H uv 0 V v 0 ) = span([H uv 0 ˜ V v 0 ,H uv 0 v v 0 ]) for all v∈V such that [v,u]∈E π ∗ and since U H u H uv 0 V v 0 =[ ˜ U H u H uv 0 ˜ V v 0 ,U H u H uv 0 v v 0 ] = =[P ⊥ ˜ U H u Huv 0 vv 0 ˜ U H u H uv 0 ˜ V v 0 ,P ⊥ ˜ U H u Huv 0 vv 0 ˜ U H u H uv 0 v v 0 ] = 0, we get U H u H uv V v = 0,∀[u,v]∈E π ∗,u∈V (1) ∗ . Putting everything together, since the setsV ∗ ,V (0) ∗ andV (1) ∗ form a partition ofV, we can argue that U H u H uv V v = 0 for all [u,v]∈E π ∗ and hence show that the conditions (A.1) are satisfied. For the last part of the proof consider the setsV k given in (A.4) and recall that they form a partition ofV. First notice that since|V| =|V 0 | +|V 1 | +|V 2 |, there must exist some k ∗ ∈{0, 1, 2} such that|V k ∗|≥ |V| 3 . By symmetry, we have that |V 1 | =|V 2 | since f(z) = 1⇔ f(−z) = 2,∀z∈ Z(ω), and hence we can assume without loss of generality that k ∗ is either 0 or 2. Furthermore the setV k ∗ will satisfy (A.5) since for every (u,v)∈E we can write φ(u) = φ(v) +δ, for some δ∈{±1,±ω,±(ω + 1)} and hence f(φ(u))6= f(φ(v)),∀(u,v)∈E. Finally, recall that 1) for every u∈V 1 there exist [v,u]∈E π ∗ if and only if there exist v ∈V 2 with φ(v) = φ(u)− 1−ω, 2) for every u∈V 2 there exist [v,u]∈E π ∗ if and only if there existv 0 ∈V 0 withφ(v 0 ) =φ(u) + 1 orv 1 ∈V 1 with φ(v 1 ) =φ(u)−ω and 3) for everyu∈V 0 there exist [v,u]∈E π ∗ if and only if there exist v 1 ∈V 1 with φ(v 1 ) = φ(u) + 1 or v 2 ∈V 2 with φ(v 2 ) = φ(u)−ω or v 0 1 ∈V 1 with φ(v 0 1 ) = φ(u)− 1−ω. Therefore, {v∈V k ∗ : [v,u]∈E π ∗} ≤ 1,∀u∈V and hence the setV k ∗, k ∗ ∈{0, 2} will also satisfy (A.6). In order to complete the proof we setV ∗ = V k ∗ and obtain 1 |V| P v∈V d v = M−1 2 + |V k ∗| |V| ≥ M 2 − 1 6 as required. 112 Appendix A A.2 Proof of Lemma 2.2 Recall that the set of verticesV of the graphG(V,E) is defined in terms of a parameter r≥ 1 as V = n φ −1 (z) :z∈Z(ω)∩B r o , where B r , ( z∈C :|Re(z)|≤r,|Im(z)|≤ √ 3r 2 ) . Since the size of the graph depends on the choice of r, we will consider here the sequence of graphs G (r) (V (r) ,E (r) ), indexed by r ∈ Z + and provide the corresponding results in terms of the above parameter. A.2.1 The cardinality ofV (r) By definition|V (r) | =|Z(ω)∩B r |. Hence, our goal is to count the number of Eisenstein integers that belong to the setZ(ω)∩B r . We define the sets L(k) = ( z∈Z(ω)∩B r :|Im(z)| = √ 3k 2 ) (A.8) for all k∈{−r,..., 0,...,r}. Notice that the sets L(k) contain all the Eisenstein integersthatlieonthesamehorizontallineonthecomplexplaneandhence S k L(k) forms a partition of the setZ(ω)∩B r . Therefore, |Z(ω)∩B r | = r X k=−r |L(k)|. A key observation coming from the triangular structure of Z(ω) is that |L(k)| = |L(0)|, k is even |L(1)|, k is odd. Hence, we can write |Z(ω)∩B r | =K [r] even |L(0)| +K [r] odd |L(1)|. 113 Appendix A where K [r] even ,K [r] odd denote the cardinalities of even and odd integers in {−r,..., 0,...,r}. Ifr is even thenK [r] even =r + 1 andK [r] odd =r, whereas ifr is odd thenK [r] even =r andK [r] odd =r + 1. Since|L(0)| = 2r + 1 and|L(1)| = 2r for allr≥ 1 we have that |V (r) | =|Z(ω)∩B r | = 4r 2 + 3r + 1, r is even 4r 2 + 3r, r is odd. (A.9) A.2.2 The cardinality ofT (r) We will associate here each ordered vertex triplet [u,v,w]∈T (r) with its leading vertex u∈V (r) in a one-to-one fashion and define the set A (r) ={φ(u)∈Z(ω)∩B r : [u,v,w]∈T (r) }. In order to determine the cardinality ofT (r) , it suffices to count the number of Eisenstein integers that belong to the setA (r) , since|T (r) | =|A (r) | by definition. Consider the sets S(k) =A (r) ∩L(k) for all k∈{−r,..., 0,...,r− 1}. The set S(k) contains all the Eisenstein integers thatareassociatedwithaleadingvertexofatriangleandlieonthesamehorizontal line L(k). As before, S k S(k) forms a partition ofA (r) and hence |A (r) | = r−1 X k=−r |S(k)|. Intuitively|S(k)| counts the number of triangles that are formed between the lines L(k) and L(k + 1) and hence the total number of triangles can be obtained by adding all|S(k)| up to k =r− 1. It is not hard to verify that |S(k)| = |S(0)|, k is even |S(1)|, k is odd. 114 Appendix A for all r≥ 2 and hence |A (r) | =K [r] even |S(0)| +K [r] odd |S(1)| where K [r] even ,K [r] odd denote the cardinalities of even and odd integers in {−r,..., 0,...,r− 1}. We have that K [r] even =K [r] odd =r and hence |A (r) | =r (|S(0)| +|S(1)|). It follows from the definitions ofT (r) ,A (r) and S(0) that z∈S(0)⇔ z∈L(0),f(z) = 1 and z +ω,z +ω + 1∈L(1). WecanarguehencethatthesetS(0)hencecontainstheintegersa∈{−r+1,...,r− 1} for which [a]mod3 = 1. Similarly, z∈S(1)⇔ z∈L(1),f(z) = 1 and z +ω,z +ω + 1∈L(2). And hence the set S(1) contains the Eisenstein integers z = a +ω, for all a∈ {−r + 1,r} that satisfy [a + 1]mod3 = 1. It follows that |S(0)| = 2 r− 1 3 and |S(1)| = r− 1 3 + r 3 . We can hence conclude that |T (r) | =r 3 r− 1 3 + r 3 ≥ 4r r− 1 3 ≥ 4r r− 1 3 . (A.10) 115 Appendix A A.2.3 Proof of|V (r) ex | =O q |V (r) | By definition, we have that|V (r) ex | =|V (r) |− 3|T (r) |. From (A.9) and (A.10) we have that|V (r) |≤ 4r 2 + 3r + 1 and|T (r) |≥ 4r(r− 1)/3 and hence we can upper bound|V (r) ex | as |V (r) ex |≤ 4r 2 + 3r + 1− 4r 2 + 4r = 7r + 1. (A.11) Since q |V (r) |≥ 2r for all r≥ 1, we have that |V (r) ex |≤ 7r + 1≤ 8r≤ 4 q |V (r) |,∀r≥ 1, and hence|V (r) ex | =O q |V (r) | . A.3 Proof of Theorem 2.3 Consider the setV as defined as in Section 2.1.1 and let D in , [ a,b∈Z: [a+b]mod 3=0 Δ(a +bω), (A.12) and D out , [ a,b∈Z: [a+b]mod 36=0 Δ(a +bω) (A.13) where Δ(z) ={(z,z +ω), (z,z +ω + 1), (z +ω,z +ω + 1)}. The set of out-of-cell edges can be defined as E out ={(u,v) :u,v∈V and (φ(u),φ(v))∈D out }, (A.14) and the set of intra-cell edges as E in ={(u,v) :u,v∈V and (φ(u),φ(v))∈D in }. (A.15) 116 Appendix A The interference graph in this case is given by ˆ G V, ˆ E , where ˆ E =E out ∪E in . We further define the sets C(z) ={z,z + 1,z−ω,z−ω− 1}∩B r for all z ∈ Z(ω) such that f(z) = 1. Notice that if|C(z)| = 4, the set C(z) corresponds to the labels of four vertices a,b,c,d∈V with φ(a) =z, φ(b) =z + 1, φ(c) =z−ω, φ(d) =z−ω− 1. Moreover, the vertices{a,b,c} are connected in ˆ G V, ˆ E only with edges inE in and hence correspond to sectors of the same cell (cf. Fig. 2.7). FirstwearegoingtoshowthatwiththebeamformingchoicesgiveninAppendix A.1, the above cell{a,b,c} can jointly decode its corresponding messages according to the decoding orderπ ∗ . Notice that at the time when receivera wants to decode, all the sectors that correspond to vertices v∈V : v≺ π ∗ a have already decoded their messages and no longer cause interference to their neighbors. Hence, the received signal for a sector associated with u∈V can be written as y u =H uu V u s u + X (u,v)∈ ˆ E: a≺ π ∗v H uv V v s v +z u . (A.16) The interfering transmitters for receiver a are given by{v : (a,v)∈ ˆ E,a≺ π ∗ v} ={b,c,d}. In order to identify the interfering transmitters for receivers b and c notice that for any u∈{b,c} the set{v : (u,v)∈ ˆ E,a≺ π ∗ v} can be written as {v : (u,v)∈E in } [ {v : (u,v)∈ ˆ E out ,a≺ π ∗ v}. For receiver b the set{v : (b,v)∈ ˆ E out ,a≺ π ∗ v} ={v : [v,b]∈ ˆ E out π ∗} and for receiver c we have that{v : (c,v)∈ ˆ E out ,a≺ π ∗ v} ={d}∪{v : [v,c]∈ ˆ E out π ∗}. 117 Appendix A Putting everything together, the interfering transmitters for receivers b and c are given by {v : (b,v)∈ ˆ E,a≺ π ∗ v} ={a,c}∪{v : [v,b]∈ ˆ E out π ∗}, and {v : (c,v)∈ ˆ E,a≺ π ∗ v} ={a,c,d}∪{v : [v,c]∈ ˆ E out π ∗}. A key observation is that according to (A.1) and the achievability scheme of Appendix A.1 all the interference from the transmitters in{v : [v,b]∈ ˆ E out π ∗} and {v : [v,c]∈ ˆ E out π ∗} can be zero-forced at receivers b and c by projecting along U H b and U H c respectively. The corresponding observations are given by U H b y b =U H b H bb V b s b + X v∈{a,c} U H b H bv V v s v +z b , (A.17) U H c y c =U H c H cc V c s c + X v∈{a,c,d} U H c H c V v s v +z c . (A.18) We are going to show next that it is possible for the cell{a,b,c} to jointly decode the desired messages s a , s b and s c from the received signals y a , U H b y b and U H c y c . Let ˜ s = s a s b s c s d , ˜ y = y a U H b y b U H c y c , ˜ z = z a U H b z b U H c z c , and ˜ H = H aa V a H ab V b H ac V c H ad V d U H b H ba V a U H b H bb V b U H b H bc V c 0 d b ×d d U H c H ca V a u H c H cb V b U H c H cc V c U H c H cd V d , 118 Appendix A such that the available observations in the cell{a,b,c} can be written in vector form as ˜ y = ˜ H˜ s +˜ z. (A.19) Lemma A.1. If the channel gains H uv ∈C M×M , (u,v)∈ ˆ E are chosen at random from an absolutely continuous joint probability distribution, the matrix ˜ H has full column rank with probability one. Proof. The matrix ˜ H has M +d b +d c rows and d a +d b +d c +d d columns. Since d a + d d ≤ M for all M (odd or even), we have to show that Pr h rank( ˜ H) = d a +d b +d c +d d i = 1. Recall that the beamforming matricesV v ,U u do not depend on the channel realizations H ij used in the definition of the above matrix. Hence, assumingthatallchannelgainsarechosenatrandomfromanabsolutelycontinuous joint probability distribution, we can argue that the joint distribution of the non- zero entries in ˜ H (given by the projectionsU H u H ij V v ) is also absolutely continuous. LetF =P ˜ Hbethematrixobtainedbyrearrangingtherowsof ˜ Hsuchthatallzero entries are in the upper-right corner and consider ˜ F to be the square sub-matrix defined by the d a +d b +d c +d d bottom rows of F. We can see that the diagonal elements of ˜ F are going to be non-zero, and hence, we can show by Lemma 2.3 that the rank of ˜ F is full with probability one. Therefore, ˜ H will always have d a +d b +d c +d d linearly independent rows andPr h rank( ˜ H) =d a +d b +d c +d d i = 1. In view of the above lemma, the vector observation in (A.19) can be used to decode the symbols in ˜ s and hence the cell{a,b,c} is able to recover the desired messages s a , s b and s c . Applying the above procedure successively, according to the decoding orderπ ∗ , we can argue that that all the cells{a,b,c}⊆V whose labels correspond to a set C(z) with|C(z)| = 4, can decode their desired messages using the beamforming choices of Appendix A.1. In order to conclude the proof it remains to consider all the degenerate cases for cells that lie on the boundary of ˆ G V, ˆ E and correspond toC(z) with|C(z)|≤ 3. When C(z) = 1, there is only out-of-cell interference and hence the scheme works as described in Appendix A.1. This is also the case when|C(z)| = 2 and φ(d) = 119 Appendix A z− 1−ω∈ C(z). If|C(z)| = 2 and φ(d) = z− 1−ω / ∈ C(z) the two sectors u,v of the given cell can zero-force all out-of-cell interference and use their vector observation U H u y u U H v y v = U H v H vv V v U H v H vu V u U H u H uv V v U H u H uu V u s u s v + ˜ z to jointly decode the desired messages s u and s v . In a similar fashion, all the cells {a,b,c} that correspond to a setC(z) with|C(z)| = 3 andφ(d) =z−1−ω / ∈C(z) candecodetheirmessagesusingtheprojectedobservationsU H a y a ,U H b y b andU H c y c . Finally, the only possible cell configuration with|C(z)| = 3 and φ(d) = z− 1− ω∈ C(z), is{a,c} with φ(a) = z and φ(c) = z−ω. The corresponding vector observation is given by y a U H c y c = H aa V a H ac V c H ad V d U H c H ca V a U H c H cc V c U H c H cd V d s a s c s d + ˜ z. Notice that the above channel matrix hasM +d c rows andd a +d c +d d columns. According to the beamforming choices of Appendix A.1, we have thatd a +d d ≤M for allM and hence we can argue as before that the above matrix has full column rank with probability one. Therefore, the receiversa andc can jointly decode their desired messages in this case as well. A.4 Proof of Theorem 2.4 Theproofcanbeobtainedasastraightforwardgeneralizationoftheproofdescribed in Section 2.4.2 using the beamforming design of Theorem 2.3 given in Appendix A.3. Applying Lemma 2.3, we can show that primary and secondary sectors are always able to decode their messages from the available observations y a U H b y b U H c y c = H aa V a H ab V b H ac V c H ad V d U H b H ba V a U H b H bb V b U H b H bc V c 0 d b ×d d U H c H ca V a U H c H cb V b U H c H cc V c U H c H cd V d s a s b s c s d + ˜ z, (A.20) 120 Appendix A and U H b y b U H c y c = U H b H bb V b U H b H bc V c U H c H cb V b U H c H cc V c s b s c + ˜ z, (A.21) for all channel-state configurations given in Section 2.4.2. We omit the details here for brevity. A.5 Proof of Theorem 2.5 Here, we are going to follow an approach similar to the one in Section 2.2.2 and show that for any decoding order π, any linear scheme for the system n ˆ G V, ˆ E A :A∈{0, 1} 2| ˆ E| o achievescompoundDoFsd C upperboundedbyM/2+ O (1/ √ |V|), as stated in Theorem 2.5. First we will upper bound d C by conditioning on a specific channel-state configurationA ∗ shown in Fig. A.2. We have that d C = min A∈{0,1} 2| ˆ E| d ˆ G (A)≤d ˆ G (A ∗ ), (A.22) whereA ∗ is given by setting α ij = 1 for all edges [i,j]∈ ˆ E that belong to the trianglesT and α ij = 0 otherwise. Notice, that the channel state configurationA ∗ and the setT has been chosen here such that for all [u,v,w]∈T , the sectors associated with the nodes u, v, w belong to different cells and hence cannot be jointly decoded. Therefore the results of Lemma 2.1 and Lemma 2.2 can be applied in this case as well and hence we can directly obtain that d C =M/2 +O (1/ √ |V|). 121 Appendix A u v w Figure A.2: The set of triangles [u,v,w]∈T for ˆ G V, ˆ E A ∗ . All the circle nodes belong toV in and participate in one triangle and the setV ex contains the colored nodes on the boundary. 122 APPENDIXB Appendix for Chapter 3 B.1 Proof of Lemma 3.3 From the definition of n v in (3.28), we have that X [a,b,c]∈T (d a +d b +d c ) = X v∈V n v d v . (B.1) Splitting the sum in terms ofV in andV ex we get X v∈V n v d v = X v∈V in n v d v + X u∈Vex n u d u (B.2) = 3 X v∈V d v + X v∈V in (n v − 3)d v + X u∈Vex (n u − 3)d u (B.3) = 3 X v∈V d v + X v∈Vex (n v − 3)d v , (B.4) where the last step follows from the fact that n v = 3 for all v∈V in . Rearranging the terms and dividing by 3 gives X v∈V d v = X [a,b,c]∈T d a +d b +d c 3 ! + X u∈Vex 1− n u 3 d u , (B.5) 123 Appendix B and hence , we can rewrite the average DoFs as 1 |V| X v∈V d v = 1 3|V| X [a,b,c]∈T (d a +d b +d c ) + C ex |V| , (B.6) where C ex = X v∈Vex 1− n v 3 d v . Since 0≤n v < 3 and 0≤d v ≤M,∀v∈V we have that C ex ≤ X v∈Vex d v ≤M|V ex |, (B.7) and hence 1 |V| X v∈V d v ≤ 1 3|V| X [a,b,c]∈T (d a +d b +d c ) + M|V ex | |V| . (B.8) B.2 Proof of Lemma 3.4 First, observe that each triangle [a,b,c]∈T uniquely covers the three correspond- ing edges inE π , and since d v ≥ 0 for all v∈V we can lower bound the right-hand side of (3.26) by X [u,v]∈Eπ d u d v ≥ X [a,b,c]∈T (d a d b +d a d c +d b d c ). Then, using (B.5), we can rewrite the left-hand side of (3.26) as 2 X v∈V (M−d v )d v = X [a,b,c]∈T 2 3 (M−d a )d a + (M−d b )d b + (M−d c )d c + 2 X v∈Vex 1− n v 3 (M−d v )d v = 1 3 X [a,b,c]∈T 2M(d a +d b +d c )− 2(d 2 a +d 2 b +d 2 c ) + 2 3 X v∈Vex (3−n v ) (M−d v )d v . 124 Appendix B Putting everything together, we can argue that any degrees of freedom{d v ,v∈V} that satisfy (3.26) must also satisfy: X [a,b,c]∈T 2(d 2 a +d 2 b +d 2 c ) + 3 (d a d b +d a d c +d b d c )− 2M(d a +d b +d c ) ≤D ex ⇔ X [a,b,c]∈T (d a +d b ) 2 + (d a +d c ) 2 + (d b +d c ) 2 +d a d b +d a d c +d b d c − 2M(d a +d b +d c ) | {z } ,g(da,d b ,dc) ≤D ex , (B.9) where D ex = 2 X v∈Vex (3−n v ) (M−d v )d v . Since 0≤n v < 3 and 0≤d v ≤M,∀v∈V we have that D ex ≤ X v∈Vex 6(M−d v )d v ≤ 3M 2 2 |V ex |, and hence X [a,b,c]∈T g(d a ,d b ,d c )≤ 3M 2 2 |V ex |. (B.10) B.3 Proof of Lemma 3.5 Consider the optimization problem LP (s,g,α,β,γ) given by maximize {x∈R n } α·s T x +β (B.11) subject to: g T x≤γ, 1 T x = 1, x≥ 0, (B.12) where α,β,γ≥ 0 and s,g∈R n , and let opt(s,g,α,β,γ) denote its optimal value. The Langrangian associated with LP (s,g,α,β,γ) is given by L(x,λ,μ,ν) =α·s T x +β−λ(g T x−γ)−μ(1 T x− 1) +ν T x (B.13) = (αs−λg−μ1 +ν) T x +λγ +μ +β (B.14) 125 Appendix B where λ, μ∈ R and ν ∈ R n are the Lagrange multipliers associated with the constraints g T x≤ γ, 1 T x = 1 and x≥ 0. From duality theory we have that the Lagrange dual function defined as h(λ,μ,ν), sup x∈R n L(x,λ,μ,ν) (B.15) satisfies opt(s,g,α,β,γ)≤h(λ,μ,ν) (B.16) for any λ≥ 0, ν≥ 0 and μ∈R. From (B.14) and (B.15) we obtain that h(λ,μ,ν) = λγ +μ +β when αs−λg−μ1 +ν = 0 ∞ otherwise. (B.17) Since ν≥ 0 we have that αs−λg−μ1 +ν = 0⇔ μ1≥ αs−λg and therefore setting μ ∗ = max i {αs i −λg i } we get opt(s,g,α,β,γ)≤h(λ,μ ∗ ,ν) = max i {αs i −λg i } +λγ +β, (B.18) for all λ≥ 0. Rewriting the above bound as α· max i {s i − ˜ λ·g i } + ˜ λαγ +β where ˜ λ,λ/α≥ 0 we obtain the desired result. B.4 Proof of Lemma 3.6 Recall that the set of verticesV inG(V,E) is defined in terms of a parameterr≥ 1 as V = n φ −1 (z) :z∈Z(ω)∩B r o , where B r , ( z∈C :|Re(z)|≤r,|Im(z)|≤ √ 3r 2 ) . Since the size of the graph depends on the choice of r, we will consider here the sequence of graphs G (r) (V (r) ,E (r) ), indexed by r ∈ Z + and provide the corresponding results in terms of the above parameter. 126 Appendix B B.4.1 The cardinality ofV (r) By definition|V (r) | =|Z(ω)∩B r |. Hence, our goal is to count the number of Eisenstein integers that belong to the setZ(ω)∩B r . We define the sets L(k) = ( z∈Z(ω)∩B r :|Im(z)| = √ 3k 2 ) (B.19) for all k∈{−r,..., 0,...,r}. Notice that the sets L(k) contain all the Eisenstein integersthatlieonthesamehorizontallineonthecomplexplaneandhence S k L(k) forms a partition of the setZ(ω)∩B r . Therefore, |Z(ω)∩B r | = r X k=−r |L(k)|. A key observation coming from the triangular structure of Z(ω) is that |L(k)| = |L(0)|, k is even |L(1)|, k is odd. Hence, we can write |Z(ω)∩B r | =K [r] even |L(0)| +K [r] odd |L(1)|. where K [r] even ,K [r] odd denote the cardinalities of even and odd integers in {−r,..., 0,...,r}. Ifr is even thenK [r] even =r +1 andK [r] even =r, whereas ifr is odd thenK [r] even =r andK [r] even =r + 1. Since|L(0)| = 2r + 1 and|L(1)| = 2r for allr≥ 1 we have that |V (r) | =|Z(ω)∩B r | = 4r 2 + 3r + 1, r is even 4r 2 + 3r, r is odd. (B.20) 127 Appendix B B.4.2 The cardinality ofT (r) We will associate here each ordered vertex triplet [a,b,c]∈T (r) with its leading vertex a∈V (r) in a one-to-one fashion and define the set A (r) ={φ −1 (u)∈Z(ω)∩B r : [a,b,c]∈T (r) }. In order to determine the cardinality ofT (r) , it suffices to count the number of Eisenstein integers that belong to the setA (r) , since|T (r) | =|A (r) | by definition. Consider the sets S(k) =A (r) ∩L(k) for all k∈{−r,..., 0,...,r− 1}. The set S(k) contains all the Eisenstein integers thatareassociatedwithaleadingvertexofatriangleandlieonthesamehorizontal line L(k). As before, S k S(k) forms a partition ofA (r) and hence |A (r) | = r−1 X k=−r |S(k)|. Intuitively|S(k)| counts the number of triangles that are formed between the lines L(k) and L(k + 1) and hence the total number of triangles can be obtained by adding all|S(k)| up to k =r− 1. It is not hard to verify that |S(k)| = |S(0)|, k is even |S(1)|, k is odd, for all r≥ 2 and hence |A (r) | =K [r] even |S(0)| +K [r] odd |S(1)| where K [r] even ,K [r] odd denote the cardinalities of even and odd integers in {−r,..., 0,...,r− 1}. We have that K [r] even =K [r] odd =r and hence |A (r) | =r (|S(0)| +|S(1)|). 128 Appendix B It follows from the definitions ofT (r) ,A (r) and S(0) that z∈S(0)⇔z∈L(0) andz +ω,z +ω + 1∈L(1). WecanarguehencethatthesetS(0)hencecontainstheintegersa∈{−r+1,...,r− 1}. Similarly, z∈S(1)⇔z∈L(1) andz +ω,z +ω + 1∈L(2), and hence the set S(1) contains the Eisenstein integers z = a +ω, for all a∈ {−r + 1,...,r}. It follows that|S(0)| = 2r− 1 and|S(1)| = 2r and therefore |T (r) | =|A (r) | = 4r 2 −r. (B.21) Comparing (B.21) with (B.20) gives|T (r) |≤|V (r) | as stated in the first part of the lemma. B.4.3 The cardinality ofV (r) ex We will upper bound|V (r) ex | as follows. From Lemma B.5 we have that X u∈V (r) ex 1− n u 3 x u = X v∈V (r) x v − X [i,j,k]∈T (r) x i +x j +x k 3 , for any{x v :v∈V (r) }. Setting x v = 1,∀v∈V (r) , we obtain |V (r) ex |− X u∈Vex n u 3 =|V (r) |−|T (r) |. Since n v ≤ 2 for all v∈V (r) ex we have that X u∈Vex n u 3 ≤ 2 3 |V (r) ex |, and hence |V (r) ex |≤ 3 |V (r) |−|T (r) | . (B.22) 129 Appendix B From (B.20) and (B.22) we obtain |V (r) ex | ≤ 3 4r 2 + 3r + 1− 4r 2 +r = 12r + 3. (B.23) From (B.20) it follows that q |V (r) |≥ 2r for all r≥ 1. From (B.23) we have that |V (r) ex |≤ 12r + 3≤ 9 q |V (r) |,∀r≥ 1, and hence|V (r) ex | =O q |V (r) | . 130 Appendix B B.5 The Linear Programming Converse for M×M Cellular Systems Recall from Section 3.2.2 that the following linear program can be used to upper bound the average (per cell) DoFs achievable inG(V,E) for any decoding order π. LP(G π ) : maximize n x (i,j,k) ∈R, [i,j,k]∈D o |T| 3|V| X [i,j,k]∈D s(i,j,k)·x (i,j,k) + M|V ex | |V| subject to: X [i,j,k]∈D g(i,j,k)·x (i,j,k) ≤ 3M 2 |V ex | 2|T| X [i,j,k]∈D x (i,j,k) = 1 x (i,j,k) ≥ 0,∀[i,j,k]∈D Using the result of Lemma 3.5, we can show that the solution of the linear program LP(G π ) is upper bounded by opt(LP(G π ))≤ |T| 3|V| · max [i,j,k]∈D ( s(i,j,k)− g(i,j,k) 2M ) + 5M 4 · |V ex | |V| , (B.24) by identifying α = |T| 3|V| , β = M|Vex| |V| , γ = 3M 2 |Vex| 2|T| and setting λ = 1 2M . Further, using Lemma 3.6 we obtain opt(LP(G π ))≤ 1 3 · max [i,j,k]∈T D ( s(i,j,k)− g(i,j,k) 2M ) +O 1 q |V| . (B.25) 131 Putting everything together and defining the function f M (i,j,k), 1 3 s(i,j,k)− g(i,j,k) 2M ! (B.26) = 1 3 2(i +j +k)− (i +j) 2 +ij + (i +k) 2 +ik + (j +k) 2 +jk 2M ! , (B.27) we arrive at the following theorem that upper bounds the average (per cell) DoFs in our framework. Theorem B.1. For a cellular systemG(V,E) in which transmitters and receivers are equipped with M antennas each and for any network interference cancellation decoding order π, the average (per cell) DoFs that can be achieved by any one-shot linear beamforming scheme are bounded by d G,π ≤ max [i,j,k]∈D f M (i,j,k) +O (1/ √ |V|), whereD is the set of all distinct triangle-DoF configurations given in (3.39) and f M (i,j,k) is given by (B.27). Corollary B.1. We have that max [i,j,k]∈T D f M (i,j,k)≤ 2M 5 , for allM. Proof. The function f M (i,j,k) is concave with global maximum max [i,j,k]∈R 3 f M (i,j,k) = 2M/5 at [i,j,k] = h 2M 5 , 2M 5 , 2M 5 i . Since max [i,j,k]∈T D f M (i,j,k)≤ max [i,j,k]∈R 3 f M (i,j,k) for all M, the corollary follows. 132 Table B.1: The setsT D and corresponding values off M (i,j,k) forM = 2, 3 and 4. The asterisks indicate the configurations [i,j,k]∈D that attain the maximum of f M (i,j,k). M = 2 D f M (i,j,k) [0, 0, 0] 0 [0, 0, 1] 1/2 [0, 0, 2] 2/3 [0,1,1] 3/4 ∗ [1,1,1] 3/4 ∗ M = 3 D f M (i,j,k) [0, 0, 0] 0 [0, 0, 1] 5/9 [0, 0, 2] 8/9 [0, 0, 3] 1 [0, 1, 1] 17/18 [0, 1, 2] 10/9 [1,1,1] 7/6 ∗ [1,1,2] 7/6 ∗ M = 4 D f M (i,j,k) [0, 0, 0] 0 [0, 0, 1] 7/12 [0, 0, 2] 1 [0, 0, 3] 5/4 [0, 0, 4] 4/3 [0, 1, 1] 25/24 [0, 1, 2] 4/3 [0, 1, 3] 35/24 [0, 2, 2] 3/2 [1, 1, 1] 11/8 [1, 1, 2] 37/24 [1, 1, 3] 37/24 [1,2,2] 19/12 ∗ [2, 2, 2] 3/2 133 Bibliography [ACD + 12] Jeffrey G Andrews, Holger Claussen, Mischa Dohler, Sundeep Rangan, and Mark C Reed. Femtocells: Past, present, and future. IEEE J. on Selected. Areas in Commun. (JSAC), 30(3):497–508, April 2012. [BCT13] G. Bresler, D. Cartwright, and D. Tse. 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Ntranos, Vasilis
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Distributed interference management in large wireless networks
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Viterbi School of Engineering
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Electrical Engineering
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06/22/2015
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04/08/2015
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base station cooperation,cellular systems,compute and forward,cooperative communication,distributed interference management,interference alignment,interference cancellation,interference channel,OAI-PMH Harvest
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Caire, Giuseppe (
committee chair
), Avestimehr, Salman (
committee member
), Fulman, Jason (
committee member
)
Creator Email
ntranos@usc.edu,vntranos@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-575678
Unique identifier
UC11302127
Identifier
etd-NtranosVas-3497.pdf (filename),usctheses-c3-575678 (legacy record id)
Legacy Identifier
etd-NtranosVas-3497.pdf
Dmrecord
575678
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Ntranos, Vasilis
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
base station cooperation
cellular systems
compute and forward
cooperative communication
distributed interference management
interference alignment
interference cancellation
interference channel