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Fundamentals of two user-centric architectures for 5G: device-to-device communication and cache-aided interference management

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Fundamentals of two user-centric architectures for 5G: device-to-device communication and cache-aided interference management

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Content Fundamentals of Two User-Centric Architectures for 5G: Device-to-Device Communication and Cache-Aided Interference Management Author: Navid Naderializadeh A Dissertation Submitted to the Faculty of the USC Graduate School In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Committee Members: Prof. Salman Avestimehr (Chair) Prof. Andreas F. Molisch Prof. Meisam Razaviyayn Department of Electrical Engineering University of Southern California Los Angeles, CA 90089 December 2016 Abstract We focus on the fundamental limits of two possible architectures in fifth-generation mobile networks (5G) that bring connectivity and contents closer to the users in the network. In particular, we study the problem of interference management in wireless device-to-device (D2D) communication systems and cache-aided wireless networks. For D2D networks, we first focus on the so-called “low-interference” regime; i.e., the regime where the level of interference in the network is so low that it can be safely “ignored” . In particular, we present a general condition for the K-user fully-connected fully-asymmetric Gaussian interference channel, under which the simple scheme of treating interference as noise (TIN) is shown to be not only optimal for the entire GDoF region, but also within a constant gap of the entire capacity region. Based on this condition, we then propose a new spectrum sharing mechanism for D2D networks, named information-theoretic link scheduling (in short, ITLinQ), which identifies the subsets of users in the network that satisfy the TIN optimality condition and schedules the links in such subsets to transmit data at the same time. We characterize the guaranteed fraction of the capacity region that ITLinQ is able to achieve in a specific network setting. Furthermore, we introduce a distributed version of the ITLinQ scheme and show that it outperforms the state-of-the-art by more than a 100% in terms of sum-throughput, while keeping the complexity at the same level. For the case of cache-aided wireless networks, we consider a system comprising a library of files (e.g., movies) and a wireless network with an arbitrary number of transmitters and an arbitrary number of receivers, where each transmitter and each receiver is equipped with a local cache of certain size. Each receiver will ask for one of the files in the library, which needs to be delivered. In this setting, we characterize the one-shot sum degrees-of- freedom (sum-DoF) of the networks to within a factor of 2 for all system parameters. Our result shows that the one-shot sum-DoF scales linearly with the aggregate cache size in the network. We propose an achievable scheme that exploits the redundancy of the contents at transmitters’ caches to cooperatively zero-force some outgoing interference and availability of the unintended content at receivers’ caches to cancel (subtract) some of the incoming interference. We develop a particular pattern for cache placement that maximizes the overall gains of cache-aided transmit and receive interference cancellations. For the converse, we present an integer optimization problem which minimizes the number of communication blocks needed to deliver any set of requested files to the receivers. We then provide a lower bound on the value of this optimization problem, hence leading to an upper bound on the linear one-shot sum-DoF of the network, which is within a factor of 2 of the achievable sum-DoF. We finally extend the aforementioned result to the wireless cellular networks with caches at both base stations and receivers, in which each receiver is only able to receive signals from its three neighboring base stations due to the presence of path loss and fading. We characterize the degrees-of-freedom (DoF) per cell to within additive and multiplicative i Abstract gaps of 2 for all system parameters, under one-shot linear schemes. Our result indicates that the one-shot linear DoF per cell scales linearly with the total amount of cache that is available in the cell. To establish the result, we propose a randomized cache placement for the receivers and a delivery scheme which utilizes the overlap of contents at the base station caches to zero-force part of their outgoing interference and also uses the cache contents of the receivers to create coded multicasting opportunities, so that the receivers are able to eliminate the remaining interference due to undesired packets. We also provide a converse argument which shows that the achievable one-shot linear DoF per cell of our scheme is within constant additive and multiplicative gaps of 2 of its optimum value. ii Acknowledgements Over the course of my doctoral studies, I had the privilege of having the amazing support of many individuals, without which my PhD would not have been possible. First and foremost, I would like to thank my advisor, Prof. Salman Avestimehr. It has been an absolute honor to work with him. His diligent guidance, tenacity and decent knowledge have paved the way toward the accomplishments that I have had during the past years. I would also like to thank the members of my qualifying exam and dissertation committee members, Prof. Andreas Molisch, Prof. Keith Chugg, Prof. Bhaskar Krishnamachari, Prof. Yan Liu and Prof. Meisam Razaviyayn, whose insightful comments have greatly helped improve the quality of this dissertation. I spent Summer 2015 as an intern at Bell Labs, working with Dr. Reinaldo Valenzuela and Dr. Mohammad Ali Maddah-Ali. I am really grateful to them for this valuable opportunity that also served as a starting point for my further ongoing research collaboration with Dr. Maddah-Ali. Thanks to all our brilliant group members and visitors at USC, Mehrdad Kiamari, Songze Li, David Kao, Aly El Gamal, Eyal En Gad, Heecheol Yang, and Qian Yu. I really enjoyed working and having rewarding discussions with them. Also, thanks to the several smart colleagues at the USC Communication Sciences Institute, Vinod Kristem, Daoud Burghal, Arash Tehrani, Mingyue Ji, Hassan Ghozlan, Hao Feng, Ahmad Fallahpour, Amirhossein Mohajerin Ariaei, Morteza Ziyadi, Sajjad Beygi, and Sina Aboutorabi. It has been a great experience being a member of CSI alongside with them. I would also like to thank the CSI staff, Susan Wiedem, Gerrielyn Ramos and Corine Wong, who have greatly helped me during my years at USC. My sincere gratitude goes to my wonderful friends that have helped me feel home in Los An- geles: Seyed Mohammad Asghari, Mehdi Jafarnia, Aria Samiei, Sepideh Kiani, Mona Shar- ifi, Mohammad Noormohammadpour, Ehsan Ebrahimzadeh, Pardis Tavakolian, Shahrzad Gholami, Aslan Etminan, Keyvan Noury and Mehrdad Showkatbakhsh. They have been with me through the ups and downs of my years here and made me enjoy every moment of my life more than I could imagine. Special thanks to my friends, Amirkhosro Gouran, Shervin Minaee, Hamidreza Aghasi, Morteza Hashemi and Alireza Sheikhattar, whose remote support have made my graduate studies run smoothly. Last but not least, I would like to express my deepest gratitude to my lovely family, my parents and my brother, for their relentless love and support. This dissertation is dedicated to them. iii Contents Abstract i Acknowledgements iii Contents iv List of Figures vii 1 Introduction 1 2 On the Optimality of Treating Interference as Noise 5 2.1 System Model and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Generalized Degrees of Freedom . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Capacity Region within a Constant Gap . . . . . . . . . . . . . . . . 9 2.1.3 Achievable Rate Region of TIN Scheme . . . . . . . . . . . . . . . . 10 2.2 Condition for Optimality of TIN . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Polyhedral Relaxation of TIN . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Dual Characterization of Polyhedral TIN Region via Potential Functions 14 2.2.3 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Constant Gap to Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 The General Achievable GDoF Region of TIN . . . . . . . . . . . . . . . . . 22 2.5 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Spectrum Sharing in D2D Networks: Information-Theoretic Link Schedul- ing (ITLinQ) 28 3.1 Description and Analysis of the Information-Theoretic Link Scheduling Scheme 31 3.1.1 Description of ITIS and ITLinQ . . . . . . . . . . . . . . . . . . . . 31 3.1.2 Capacity Analysis of the ITLinQ Scheme . . . . . . . . . . . . . . . 35 3.1.2.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . 38 3.1.2.2 Impact of Rayleigh Fading on the Capacity Analysis . . . . 42 3.2 A Distributed Method for Implemenation of ITLinQ . . . . . . . . . . . . . 46 3.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.1 Performance of Distributed ITLinQ under the Model of Section 3.1.2 49 iv Contents 3.3.2 Performance Comparison of the distributed ITLinQ and FlashLinQ 50 4 Fundamental Limits of Cache-Aided Interference Management 55 4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.1 Problem Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.2 Detailed Problem Description . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Main Result and its Implications . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 Achievable Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3.1 Description of the Achievable Scheme via an Example . . . . . . . . 65 4.3.2 Description of the General Achievable Scheme . . . . . . . . . . . . . 68 4.3.2.1 Prefetching Phase . . . . . . . . . . . . . . . . . . . . . . . 68 4.3.2.2 Delivery Phase . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.3 Analysis of the Sum-DoF of the Proposed Achievable Scheme . . . . 75 4.4 Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4.1 Conversion to a Virtual MISO Interference Channel . . . . . . . . . 77 4.4.2 Integer Program Formulation . . . . . . . . . . . . . . . . . . . . . . 79 4.4.3 Relaxing Worst-Case Demands to Average Demands and Optimizing over Caching Realizations . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4.4 Lower Bound on the Number of Communication Blocks . . . . . . . 81 5 Cache-Aided Interference Management in Wireless Cellular Networks 83 5.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Achievable Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3.1 3μ T = 1 (No BS collaboration) . . . . . . . . . . . . . . . . . . . . . 92 5.3.2 3μ T = 2 (Partial BS collaboration) . . . . . . . . . . . . . . . . . . . 94 5.3.3 3μ T = 3 (Full BS collaboration) . . . . . . . . . . . . . . . . . . . . . 97 5.4 Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.4.1 Conversion to a Virtual MISO Interference Channel . . . . . . . . . 98 5.4.2 Integer Program Formulation . . . . . . . . . . . . . . . . . . . . . . 101 5.4.3 Relaxing Worst-Case Demands to Average Demands and Optimizing over Caching Realizations . . . . . . . . . . . . . . . . . . . . . . . . 101 5.4.4 Lower Bound on the Number of Communication Blocks . . . . . . . 102 6 Concluding Remarks and Future Directions 104 Appendix A Replacing α ij < 0 with α ij = 0 in (2.2) 108 Appendix B Characterization of The Polyhedral TIN RegionP 114 Appendix C Proof of Theorem 2.3 116 Appendix D Proof of Theorem 2.5 118 v Contents Appendix E Proof of Lemma 4.1 121 Appendix F Proof of Lemma 4.4 123 Bibliography 127 vi List of Figures 2.1 The GDoF “W” curve for the two-user symmetric Gaussian interference chan- nel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 (a) A 3-user interference channel, where the value on each link is equal to its channel strength level, and (b) The GDoF region of this network, which is a convex polyhedron and can be achieved by TIN. . . . . . . . . . . . . . . . 12 2.3 (a) The directed graph D in which the green, blue and red arcs belong to A 1 ,A 2 andA 3 , respectively. For simplicity, only some parts of the edges are shown in this figure. (b) The corresponding directed graph D for Example 2.1. 15 2.4 A 3-user cyclic interference channel, where the strength levels for each link is shown in the figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 The TIN region of the network in Figure 2.4, which is the union of the yellow region (P ∅ ) and the blue region (P {3} ). . . . . . . . . . . . . . . . . . . . . . 25 2.6 (a) A 10-user interference channel where the black circle, green circles, red triangles, and blue crosses represent the whole cell area, the coverage area of base stations, base stations (transmitters), and receivers, respectively. The coverage radius of each transmitter is taken to ber = 100m. (b) Effect of the coverage radius and the number of users on the probability that the sufficient condition (2.9) is satisfied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1 A wireless network composed of n source-destination pairs, where the green and red lines represent the direct and cross channel gains, respectively. . . . 32 3.2 A deterministic view of the optimality condition for treating interference as noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Cumulative distribution function (CDF) of the actual gap of the achievable rate region by TIN with respect to the capacity region for networks with 8 links satisfying condition (3.1) and its comparison with the worst-case gap of log 3n = log 24≈ 4.58. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Comparison of the guaranteed achievable fraction of capacity region by the ITLinQ scheme in different regimes with TDMA and independent set schedul- ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Average achievable fraction of the capacity region by ITLinQ scheme with and without fading and comparison with time-sharing. . . . . . . . . . . . . 44 3.6 Comparison of the average gap to the achievable fraction of the capacity region by ITLinQ with and without fading. . . . . . . . . . . . . . . . . . . 45 vii List of Figures 3.7 Performance comparison of distributed ITLinQ with time-sharing under the model of Section 3.1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.8 (a) Comparison of the sum-rate performance of distributed ITLinQ, Flash- LinQ and the no-scheduling case, and (b) Comparison of the cumulative distribution function of the average link rate achieved by distributed ITLinQ and FlashLinQ in a network of 1024 links. . . . . . . . . . . . . . . . . . . . 51 3.9 An example of the case where distributed ITLinQ might be unfair. . . . . . 52 3.10 Comparison of the average link rate CDF of distributed ITLinQ, fair ITLinQ and FlashLinQ for a network with 1024 links. . . . . . . . . . . . . . . . . . 54 3.11 Comparison of the sum-rate achievable by fair ITLinQ and FlashLinQ. . . . 54 4.1 Wireless network with K T transmitters and K R receivers, where each trans- mitter and each receiver caches up to M T F packets and M R F packets, re- spectively, from a library of N files, each composed of F packets. . . . . . . 58 4.2 Delivery phase for the example in Section 4.3.1 for respective requests of files A, B and C by receivers Rx 1 , Rx 2 and Rx 3 , whereL(α,β) denotes some linear combination of α andβ. In every step, each pair of transmitters collaborate to zero-force the interference due to a specific subfile at a certain undesired receiver. Moreover, each receiver also uses its cache contents to cancel the interference due to the other interfering packet. Therefore, the communication is interference-free in all 6 steps. . . . . . . . . . . . . . . . 67 4.3 More detailed description of the linear encoding and decoding schemes used in the delivery phase step in Figure 4.2–(a). In this step, Tx 1 and Tx 2 zero-force A 12,2 at Rx 3 , Tx 1 and Tx 3 zero-forceC 13,1 at Rx 2 , and Tx 2 and Tx 3 zero-force B 23,3 at Rx 1 . Moreover, Rx 1 , Rx 2 and Rx 3 can cancel the interference due to C 13,1 , A 12,2 , and B 23,3 , respectively, since they already have each respective subfile in their own cache. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.1 A cellular network model with K R = 4. . . . . . . . . . . . . . . . . . . . . 86 5.2 Base station prefetching pattern. . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 The delivery activation pattern for the case of 3μ T = 1. . . . . . . . . . . . 91 5.4 Two steps of delivery in Example 5.1 in which the base station delivers (a) subfilesA 1,2 andB 1,1 to receivers Rx 1 and Rx 2 , and (b) subfilesA 1,23 ,B 1,13 , and C 1,12 to receivers Rx 1 , Rx 2 , and Rx 3 simultaneously. . . . . . . . . . . . 93 5.5 The delivery activation pattern for the case of 3μ T = 2. . . . . . . . . . . . 95 C.1 The m-user cyclic interference channel. . . . . . . . . . . . . . . . . . . . . . 117 viii To My Dearest Father, Mother and Brother. ix Chapter 1 Introduction The advent of new devices, applications and demands for wireless communication systems is dramatically altering the shape of today’s wireless networks. In order to address all these demands, future wireless networks and in particular the fifth-generation mobile networks (5G) need scalable approaches that keep the users at the center of the network in order to bring connectivity, contents and computing closer to the end users [1]. There are two specific architectures that can help with the aforementioned user-centric design paradigms, which are drawing a significant amount of attention from both theoretical and practical points of view; device-to-device (D2D) communication and cache-aided wireless networks. Device-to-device communication among mobile users is being considered as a potential key component of next-generation wireless communication systems. The D2D communi- cation functionality can enable various applications and services (see, e.g. [2, 3]), such as proximity-based applications involving discovering and communicating with nearby devices (e.g., Internet of Things (IoT)). International Data Corporation (IDC) predicts that the installed base of the IoT will grow to 212 billion “things” by the end of 2020 [4]. This shows that D2D communication can play a fundamental role in the wireless industry in the coming years. Such a type of communication can also enable higher data rates and system capacity by leveraging the underlying peer-to-peer wireless network that can be created via local communication among the users (see, e.g. [5–9]). This will reshape the conventional wire- less networks with uplink and downlink communications in which the connectivity revolves around the base stations [10]. 1 Chapter 1. Introduction On the other hand, recent trends show that the networks are shifting toward a “content- centric” design, in which the major part of the traffic is due to requests of some popular contents (such as videos) which are generated well-ahead of the communication time [11]. This suggests that it would be advantageous to bring the contents closer to the users [12–14]. Therefore, the memories which are nowadays available at all the devices throughout wireless networks can be used for caching (parts of) popular contents before communication. In both of the aforementioned scenarios, as the network grows in size and complexity, the problem of spectrum sharing and interference management gains in more importance. The interference management approaches for D2D networks should require low complexity, low coordination among the nodes and also simple physical layer schemes, mainly due to the inherent distributed nature of these networks. Moreover, cache memories of both trans- mitters and receivers can be helpful in managing the interference in wireless networks. In particular, caches at the transmitter side can be used to induce collaboration among trans- mitters in order to zero-force their outgoing interference to unintended receivers, while the receivers can also use their cache contents to potentially remove the incoming interference due to undesired packets. The question, however, is: If we have a limited amount of memory as a resource in a wireless network, where should this memory be installed? Should it be installed at the transmitters or at the receivers? In this dissertation, we focus on fundamental limits of interference management and spec- trum sharing in D2D and cache-aided wireless networks. In the D2D setting, we first char- acterize a condition under which treating interference as noise is information-theoretically optimal (to within a constant gap). We show that in a K-user interference channel, if for each user the desired signal strength is no less than the sum of the strengths of the strongest interference from this user and the strongest interference to this user (all values in dB scale), then the simple scheme of using point to point Gaussian codebooks with appropriate power levels at each transmitter and treating interference as noise at every receiver (in short, TIN scheme) achieves all points in the capacity region to within a constant gap. The generalized degrees of freedom (GDoF) region under this condition is a polyhedron, which is shown to be fully achieved by the same scheme, without the need for time-sharing. The results are proved by first deriving a polyhedral relaxation of the GDoF region achieved by TIN, then providing a dual characterization of this polyhedral region via the use of potential functions, and finally proving the optimality of this region in the desired regime. 2 Chapter 1. Introduction We then consider the problem of spectrum sharing in D2D communication systems. In- spired by the optimality condition for treating interference as noise, we define a new con- cept of “information-theoretic independent sets” (ITIS), which indicates the sets of links for which simultaneous communication and treating the interference from each other as noise is information-theoretically optimal (to within a constant gap). Based on this concept, we develop a new spectrum sharing mechanism, called “information-theoretic link scheduling” (in short, ITLinQ), which at each time schedules those links that form an ITIS. We first provide a performance guarantee for ITLinQ by characterizing the fraction of the capacity region that it can achieve in a network with sources and destinations located randomly within a fixed area. Furthermore, we demonstrate how ITLinQ can be implemented in a distributed manner, using an initial 2-phase signaling mechanism which provides the re- quired channel state information at all the links. Through numerical analysis, we show that distributed ITLinQ can outperform similar state-of-the-art spectrum sharing mechanisms, such as FlashLinQ [15], by more than a 100% of sum-rate gain, while keeping the complexity at the same level. Finally, we discuss a variation of the distributed ITLinQ scheme which can also guarantee fairness among the links in the network and numerically evaluate its performance. Moreover, in cache-aided wireless networks, we consider a system comprising a library of N files (e.g., movies) and a wireless network withK T transmitters, each equipped with a local cache of size of M T files, and K R receivers, each equipped with a local cache of size of M R files. Each receiver will ask for one of theN files in the library, which needs to be delivered. The objective is to design the cache placement (without prior knowledge of receivers’ future requests) and the communication scheme to maximize the throughput of the delivery. In this setting, we show that the sum degrees-of-freedom (sum-DoF) of min{ K T M T +K R M R N ,K R } is achievable, and this is within a factor of 2 of the optimum, under one-shot linear schemes. This result shows that (i) the one-shot sum-DoF scales linearly with the aggregate cache size in the network (i.e., the cumulative memory available at all nodes), (ii) the transmitters’ and receivers’ caches contribute equally in the one-shot sum-DoF, and (iii) caching can offer a throughput gain that scales linearly with the size of the network. To prove the result, we propose an achievable scheme that exploits the redundancy of the content at transmitters’ caches to cooperatively zero-force some outgoing interference and availability of the unintended content at receivers’ caches to cancel (subtract) some of the incoming interference. We develop a particular pattern for cache placement that maximizes the overall gains of cache-aided transmit and receive interference cancellations. 3 Chapter 1. Introduction For the converse, we present an integer optimization problem which minimizes the number of communication blocks needed to deliver any set of requested files to the receivers. We then provide a lower bound on the value of this optimization problem, hence leading to an upper bound on the linear one-shot sum-DoF of the network, which is within a factor of 2 of the achievable sum-DoF. We finally extend the aforementioned result to the wireless cellular networks with caches at both base stations and receivers, in which each receiver is only able to receive signals from its three neighboring base stations due to the presence of path loss and fading. We characterize the degrees-of-freedom (DoF) per cell to within additive and multiplicative gaps of 2 for all system parameters, under one-shot linear schemes. Our result indicates that the one-shot linear DoF per cell scales linearly with the total amount of cache that is available in the cell, which is the sum of the cache at the central base station and all the receivers within the cell. To establish the result, we propose a randomized cache placement that is decentralized and a delivery scheme which, on one hand, utilizes the overlap of contents at the base station caches to zero-force part of their outgoing interference, and on the other hand, uses the cache contents of the receivers to create coded multicasting opportunities, so that the receivers are able to eliminate the remaining interference due to undesired packets. We also provide a converse argument which shows that the achievable one-shot linear DoF per cell of our scheme is within constant additive and multiplicative gaps of 2 of its optimum value. 4 Chapter 2 On the Optimality of Treating Interference as Noise Treating interference as noise (TIN) when it is sufficiently weak is one of the key principles of interference management. As a robust principle that is also known to be optimal under certain conditions, TIN is interesting both from practical and theoretical perspectives. From a practical perspective, TIN is attractive for its low complexity and robustness to channel uncertainty. TIN involves the use of only point-to-point channel codes, that are well understood, quite practical, and near-optimal in their ability to deal with unstructured noise. Further, since it requires only a coarse knowledge of the signal to interference and noise power ratio (SINR) at the transmitters, the overhead associated with acquiring channel state information at the transmitters (CSIT) is minimal for the TIN scheme. The practical appeal of the TIN scheme has motivated several studies of the achievable rate region of TIN in the literature. However, as noted by e.g., [18, 19], in spite of the simplicity of TIN, the structure of the TIN rate region is non-trivial — it involves the optimization of the power levels at the transmitters, and is generally non-convex by itself, i.e., if time-sharing is not involved. From a theoretical perspective, it is the optimality of TIN that has attracted the most attention. It is shown in [20–22] that in a so-called “noisy interference” regime, TIN achieves This chapter is mainly taken from [16,17], coauthored by the author of this dissertation. 5 Chapter 2. On the Optimality of Treating Interference as Noise the sum capacity of the interference channel. An extension of the noisy interference regime is obtained for multiple-input multiple-output (MIMO) Gaussian interference channels in [23]. In terms of generalized degrees-of-freedom (GDoF), the well-known “W” curve [24] in Fig. 2.1 demonstrates that for the two-user symmetric Gaussian interference channel, when the strengths of both direct channels are assumed as SNR and the interference channels are not stronger than √ SNR, TIN achieves the symmetric GDoF for each user. This result is generalized to K-user fully-connected symmetric Gaussian interference channels in [25] and to cyclic asymmetric Gaussian interference channels in [26]. However, not much is known about the regime where TIN is GDoF-optimal for the general fully-connected, fully- asymmetric K-user Gaussian interference channel. 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 TIN is GDoF optimal Figure 2.1: The GDoF “W” curve for the two-user symmetric Gaussian interference channel. In this chapter we present a general condition for theK-user fully-connected fully-asymmetric Gaussian interference channel, under which TIN is shown to be not only optimal for the en- tire GDoF region, but also within a constant gap of the entire capacity region. The general condition is stated in words as follows: If for each user, the desired signal strength is no less than the sum of the strengths of the strongest interference from this user and the strongest interference to this user (all values in dB scale), then TIN is GDoF optimal. As an example, consider a K-user interference channel, where all the desired links have strength SNR and the interference links, each of which may have a different strength, are not stronger than √ SNR. In other words, the ratio of signal strength to the desired signal 6 Chapter 2. On the Optimality of Treating Interference as Noise strength (each in dB scale) for each interferer is 0.5 or less and for the desired signal is 1. Then it follows, from the result shown in this chapter, that the GDoF-optimal scheme is TIN with appropriate power allocation at each transmitter. Our proof of the optimality of TIN in the aforementioned regime consists of three steps. First, we introduce a relaxed version of TIN, called polyhedral TIN. Second, we show that, quite interestingly, the achievable GDoF region through polyhedral TIN, namely the poly- hedral TIN region, can be characterized by checking the existence of a potential function for an induced fully-connected directed graph, with nodes representing the source-destination pairs in the original interference channel (with the addition of a “ground” node), and a specific assignment of lengths to each arc. Using this equivalence and a potential theorem, we derive a dual characterization of the polyhedral TIN region. The significance of our dual characterization is the elimination of power allocation variables in the characteriza- tion. Finally, we prove the outer bounds to establish the optimality of polyhedral TIN in the regime of interest. Since TIN performs no worse than polyhedral TIN, this proves that TIN is optimal in that regime in the GDoF sense. Moreover, following the proof for GDoF-optimality of TIN, we show that in the same regime, TIN can also achieve the whole capacity region of theK-user interference channel to within a constant gap. Finally, we show that for general channel gain values in aK-user interference channel, the achievable GDoF region of TIN, namely TIN region, is composed of the union of 2 K polyhedra. However, for the regime of interest, one polyhedron subsumes all the others, hence the TIN region reduces to a single polyhedron, which is the polyhedral TIN region. 2.1 System Model and Preliminaries As our starting point, consider the canonical model of a fully-asymmetric K-user wireless interference channel, with the input-output relationship Y k (t) = K X i=1 h ki ˜ X i (t) +Z k (t), ∀k∈ [K],{1, 2,...,K}, (2.1) where at each time index t, ˜ X i (t) is the transmitted symbol of transmitter i, Y k (t) is the received signal of receiver k, h ki is the complex channel gain value from transmitter i to receiver k, and Z k (t)∼CN (0, 1) is the additive white Gaussian noise (AWGN) at receiver 7 Chapter 2. On the Optimality of Treating Interference as Noise k. All the symbols are complex. Each transmitter i is subject to the power constraint E[| ˜ X i (t)| 2 ]≤P i . We will translate the standard channel model (2.1) into an equivalent normalized form that is more conducive for GDoF studies. We define the signal-to-noise ratio (SNR) of user i and interference-to-noise ratio (INR) of transmitter i at receiver k as follows 1 . SNR i , max(1,|h ii | 2 P i ), INR ki , max(1,|h ki | 2 P i ), i6=k, ∀i,k∈ [K]. (2.2) As in [24], for the GDoF metric, we preserve the ratios of different signal strengths in dB scale as all SNR’s approach infinity. To this end, taking P > 1 as a nominal power value, we define α ii , log SNR i logP , α ki , log INR ki logP , i6=k, ∀i,k∈ [K], (2.3) implying that for each useri, SNR i =P α ii and for any two distinct usersi,k, INR ki =P α ki . Now according to (2.2) and (2.3), we can represent the original channel model in (2.1) in the following form, Y k (t) = K X i=1 √ P α ki e jθ ki X i (t) +Z k (t), ∀k∈ [K]. (2.4) In this equivalent channel model, X i (t) = ˜ X i (t)/ √ P i is the transmit symbol of transmitter i, and the power constraint for each transmitter is normalized to unity; i.e., E[|X i (t)| 2 ]≤ 1, ∀i∈ [K]. The transmit power in the original channel model is absorbed in the channel coefficients, so that √ P α ki and θ ki are the magnitude and the phase, respectively, of the channel between transmitteri and receiverk,∀i,k∈ [K]. We will call the exponentα ki the channel strength level of the link between transmitter i and receiver k. In the rest of the chapter, unless otherwise stated, we will consider the equivalent channel model in (2.4). Since this is a K-user interference channel, transmitter i has message W i intended for receiveri, and the messagesW i are independent,∀i∈ [K]. We denote the size of the message set of useri by|W i |. For codewords spanningn channel uses, the ratesR i = log|W i | n ,i∈ [K], are achievable if the probability of error at all the receivers can be made arbitrarily small as n approaches infinity. The channel capacity regionC is the closure of the set of all achievable 1 It is not difficult to verify that assigning a value of 1 to SNR’s and INR’s that are less than 1, or equivalently, assigning a 0 value to αij that might otherwise be negative, is only a matter of convenience, and has no impact on the GDoF or the constant gap result. The details are deferred to Appendix A. 8 Chapter 2. On the Optimality of Treating Interference as Noise rate tuples. Collecting the channel strength levels and phases in the sets α,{α ki }, θ,{θ ki }, ∀i,k∈ [K], (2.5) the capacity region is a function of α,θ,P , and is denoted asC(P,α,θ). 2.1.1 Generalized Degrees of Freedom The GDoF region of the K-user interference channel as represented in (2.4) is defined in (2.6). D(α,θ), n (d 1 ,d 2 ,...,d K ) : ∃(R 1 (P ),R 2 (P ),...,R K (P ))∈C(P,α,θ) as a function of P , such that d i = lim P→∞ R i (P ) logP , ∀i∈ [K] o . (2.6) In general, the channel capacity (GDoF) region of complex Gaussian interference channel may depend on both the channel strength levelsα, and the channel phasesθ. However, the capacity (GDoF) inner and outer bounds that we present in this chapter depend only on the channel strength levels α. As such, our results hold regardless of whether or not the channel phase information is available to the transmitters. 2.1.2 Capacity Region within a Constant Gap Following the same definition as in [24] and [26], an achievable rate region is said to be within x (x≥ 0) bits of the capacity region if for any rate tuple (R 1 ,R 2 ,...,R K ) on the boundary of the achievable rate region, the rate tuple (R 1 +x,R 2 +x,...,R K +x) is outside the channel capacity region. Equivalently, ((R 1 −x) + , (R 2 −x) + ,..., (R K −x) + ) is in the achievable region for any rate tuple (R 1 ,R 2 ,...,R K ) in the capacity region, where for a real number β, (β) + denotes max{0,β}. 9 Chapter 2. On the Optimality of Treating Interference as Noise 2.1.3 Achievable Rate Region of TIN Scheme In the TIN scheme, transmitter i uses a transmit power of P r i , r i ≤ 0, 2 and each receiver treats all the incoming interference as noise, so that the SINR at receiver i is given by SINR i = P α ii ×P r i 1 + P j6=i P α ij ×P r j . This implies that the rate achieved by user i through TIN is equal to R i = log(1 + SINR i ) = log 1 + P α ii +r i 1 + P j6=i P α ij +r j ! , (2.7) and therefore, the GDoF achieved by user i equals d i = max{0,α ii +r i − max{0, max j:j6=i (α ij +r j )}}. (2.8) The achievable GDoF region through TIN, which we denote byP ∗ , is the set of allK-tuples (d 1 ,d 2 ,...,d K ) for which there existr i ’s,r i ≤ 0,i∈ [K], such that (2.8) holds for alli∈ [K]. 2.2 Condition for Optimality of TIN The main result of this section is the following theorem, which introduces a condition under which TIN is GDoF-optimal. Theorem 2.1. In a K-user interference channel, where the channel strength level from transmitter i to receiver j is equal to α ji ,∀i,j∈ [K], if the following condition is satisfied α ii ≥ max j:j6=i {α ji } + max k:k6=i {α ik }, ∀i,j,k∈ [K], (2.9) then power control and treating interference as noise can achieve the whole GDoF region. Moreover, the GDoF region is the set of all K-tuples (d 1 ,d 2 ,...,d K ) satisfying 0≤d i ≤α ii , ∀i∈ [K] (2.10) 2 Recall that to obtain the equivalent channel model (2.4), in fact the transmit power of each user is absorbed into the channel coefficients. As P goes to infinity, although it appears that the transmit power of user i vanishes as an artifact of the GDoF framework, its actual SNR approaches infinity when the user achieves a positive GDoF value. 10 Chapter 2. On the Optimality of Treating Interference as Noise m X j=1 d i j ≤ m X j=1 (α i j i j −α i j−1 i j ), ∀(i 1 ,i 2 ,...,i m )∈ Π K , ∀m∈{2, 3,...,K}, (2.11) where Π K is the set of all possible cyclic sequences 3 of all subsets of [K] with cardinality no less than 2, and the modulo-m arithmetic is implicitly used on the user indices, e.g., i m =i 0 . Remark 2.1. Condition (2.9) can be stated in words as — for each user the desired signal strength is no less than the sum of the strengths of the strongest interference from this user and the strongest interference to this user (all values in dB scale). Theorem 2.1 claims that under this condition, TIN is GDoF-optimal. Remark 2.2. Both condition (2.9) and the GDoF region specified by (2.10)-(2.11) display a natural duality in the sense that they are both unchanged if the roles of the transmitters and receivers are switched, i.e., if allα ij values are switched withα ji values. In other words, for the same channel strengths, if we consider the reciprocal network (in the same sense as a multiple access channel being the reciprocal of a broadcast channel), then again under condition (2.9), TIN is GDoF-optimal, and the GDoF region is the same as in the original network. Such a duality holds also for the entire TIN regionP ∗ (defined as the set of all K-tuples (d 1 ,d 2 ,...,d K ) for which there exist r i ’s, r i ≤ 0, i∈ [K], such that (2.8) holds for all i∈ [K]), and a similar duality relationship for the symmetric rate has been observed in [27]. Example 2.1. To interpret the results in Theorem 2.1, we derive and plot the GDoF region for a 3-user network in which condition (2.9) is satisfied. Consider the 3-user network in Fig. 2.2a. In this network, the channel strength level between transmitter i and receiver j, α ji , is shown on the corresponding link,∀i,j∈{1, 2, 3}. For the case of K = 3, Π K = {(1, 2), (1, 3), (2, 3), (1, 2, 3), (1, 3, 2)}. According to Theorem 2.1, the GDoF region is the set of all (d 1 ,d 2 ,d 3 ) satisfying 0≤d 1 ≤ 2 0≤d 2 ≤ 1 0≤d 3 ≤ 1.5 d 1 +d 2 ≤ 2.3 d 1 +d 3 ≤ 2.4 3 Each cyclic sequence in ΠK is essentially a cyclically ordered subset of user indices, without repetitions. In ΠK , there exist P K m=2 K m (m− 1)! cyclic sequences. 11 Chapter 2. On the Optimality of Treating Interference as Noise d 2 +d 3 ≤ 1.5 d 1 +d 2 +d 3 ≤ 3.7 d 1 +d 2 +d 3 ≤ 2.5, which is depicted in Fig. 2.2b. Recall that condition (2.9) is satisfied in the network of Fig. 2.2a for all users i∈{1, 2, 3}. Therefore, Theorem 2.1 implies that TIN achieves the entire GDoF region of this network. 4 2 1 0.5 0.1 1 0.5 1.5 0.2 0.5 (a) (b) Figure 2.2: (a) A 3-user interference channel, where the value on each link is equal to its channel strength level, and (b) The GDoF region of this network, which is a convex polyhedron and can be achieved by TIN. We prove Theorem 2.1 through the following steps. We first introduce a relaxation of the TIN scheme called polyhedral TIN, and show that the achievable GDoF region by poly- hedral TIN is in fact a polyhedral region. We study the polyhedral TIN region in some detail to understand its structure. In particular, we show that the polyhedral TIN re- gion can be characterized by checking the existence of a potential function for an induced fully-connected directed graph, with nodes representing the source-destination pairs in the original interference channel (with the addition of a “ground” node) and a specific assign- ment of lengths to the arcs of the graph. Afterwards, we derive a dual characterization of the polyhedral TIN region and prove the outer bounds to establish the optimality of polyhedral TIN, hence TIN, whenever condition (2.9) is satisfied. 12 Chapter 2. On the Optimality of Treating Interference as Noise 2.2.1 Polyhedral Relaxation of TIN In the first step toward proving Theorem 2.1, we introduce a polyhedral version of the TIN scheme. Ignoring the first max{0,...} term in (2.8) changes the scheme to a relaxed version, which we call the polyhedral TIN scheme. With this modification, the achievable GDoF region via polyhedral TIN denoted byP will be the set of all K-tuples (d 1 ,d 2 ,...,d K ) for which there exist r i ’s, i∈ [K], such that r i ≤ 0, ∀i∈ [K] (2.12) d i ≥ 0, ∀i∈ [K] (2.13) d i =α ii +r i − max{0, max j:j6=i (α ij +r j )},∀i∈ [K]. (2.14) In the polyhedral TIN scheme, we require that the right hand side of (2.14) is non-negative for all users. Otherwise, the obtained K-tuple is not a valid GDoF tuple and hence its corresponding power exponents r i ’s are not acceptable for the polyhedral TIN scheme. Recall that in the original TIN scheme, for all r i ≤ 0, i∈{1, 2,..,K}, we always obtain a valid GDoF tuple according to (2.8). Therefore, this modification actually puts more constraints on the power exponents r i ’s besides the constraints of r i ≤ 0. In general, this can only shrink the achievable GDoF region of TIN. Thus it is true thatP⊆P ∗ . Example 2.2. Consider a 2-user interference channel with α ij = 1, ∀i,j ∈{1, 2}. In the polyhedral TIN scheme, since the right hand side of (14) is non-negative for all users i∈{1, 2}, we have 1 +r 1 − max{0, 1 +r 2 }≥ 0 1 +r 2 − max{0, 1 +r 1 }≥ 0. Combining with the constraints ofr i ≤ 0, it is easy to verify that the valid power exponents r i ’s for the polyhedral TIN scheme satisfy r 1 = r 2 and r 1 ,r 2 ∈ [−1, 0], under which the polyhedral TIN regionP only consists of the single point (0,0). While in the original TIN scheme, according to (2.8), the achievable GDoF regionP ∗ is the union of two line segments, i.e.,P ∗ ={(d 1 ,d 2 ) : 0≤d 1 ≤ 1,d 2 = 0}∪{(d 1 ,d 2 ) :d 1 = 0, 0≤d 2 ≤ 1}. Obviously, in this exampleP⊂P ∗ . 4 However, as we will show in the following, under condition (2.9), the above relaxation incurs no loss. In other words, when condition (2.9) is satisfied, the TIN regionP ∗ is 13 Chapter 2. On the Optimality of Treating Interference as Noise equal to the polyhedral TIN regionP. From (2.12)-(2.14), the polyhedral TIN regionP can be characterized by a number of linear inequalities, which as we will see, significantly contributes to understanding the TIN regionP ∗ . In fact, it is easy to prove thatP is the set of all K-tuples (d 1 ,d 2 ,...,d K ) for which there exist r i ’s, i∈ [K], such that the inequalities (2.15)-(2.18) are satisfied. The details are relegated to Appendix B. r i ≤ 0, ∀i∈ [K] (2.15) d i ≥ 0, ∀i∈ [K] (2.16) d i ≤α ii +r i ⇔r i ≥d i −α ii , ∀i∈ [K] (2.17) d i ≤α ii +r i − (α ij +r j )⇔r i −r j ≥α ij + (d i −α ii ),∀i,j∈ [K],i6=j. (2.18) As we will show, after the elimination of the variables r i ,i∈ [K], the regionP can be fully characterized by (2.10)-(2.11). Moreover, as demonstrated in Example 2.1, this region is a polyhedron, which is why the scheme is called polyhedral TIN. 2.2.2 DualCharacterizationofPolyhedralTINRegionviaPotentialFunc- tions Equipped with the aforementioned description of polyhedral TIN, we now characterize the polyhedral TIN regionP for general channel strength levels. As mentioned in (2.15)-(2.18), for a GDoF tuple (d 1 ,d 2 ,...,d K )∈ R K + (the non-negative orthant of the K-dimensional Euclidean space), it is in the regionP if and only if there exist r i ’s, i∈ [K], satisfying r i ≤ 0, ∀i∈ [K] (2.19) r i ≥d i −α ii , ∀i∈ [K] (2.20) r i −r j ≥α ij + (d i −α ii ),∀i,j∈ [K],i6=j. (2.21) Now, we define a directed graph D = (V,A), as shown in Fig. 2.3a, where V ={v 1 ,...,v K ,u} A =A 1 ∪A 2 ∪A 3 A 1 ={(v i ,v j ) :i,j∈ [K],i6=j} A 2 ={(v i ,u) :i∈ [K]} A 3 ={(u,v i ) :i∈ [K]}, 14 Chapter 2. On the Optimality of Treating Interference as Noise v 1 v i v K v j v 2 v K-1 u (a) v 1 v 3 v 2 u (b) Figure 2.3: (a) The directed graphD in which the green, blue and red arcs belong toA 1 ,A 2 and A 3 , respectively. For simplicity, only some parts of the edges are shown in this figure. (b) The corresponding directed graph D for Example 2.1. and we assign a length l(a) to every arc a∈A as follows. l(v i ,v j ) =α ii −d i −α ij l(v i ,u) =α ii −d i l(u,v i ) = 0. As an example, the corresponding directed graph D for Example 2.1 is drawn in Fig. 2.3b. Evidently, this is a fully-connected directed graph, in which the length of each arc depends on the channel strength levels and the GDoFs we intend to achieve. This careful assignment of the lengths to the arcs of this graph allows us to use the following lemma. Lemma 2.1. IfP denotes the polyhedral TIN region of aK-user interference channel, then for a GDoF tuple (d 1 ,d 2 ,...,d K )∈ R K + , it is in the regionP if and only if there exists a valid potential function for the graph D. Proof. By definition [28], a function p :V →R is called a potential if for every two nodes a,b∈ V such that (a,b)∈ A, l(a,b)≥ p(b)−p(a). These inequalities only depend on the difference between potential function values. Therefore, without loss of generality, if there exists a valid potential function for the graph, we can make one node, say node u, ground; i.e., p(u) = 0. Letting r i :=p(v i ), the potential function values should satisfy the following 15 Chapter 2. On the Optimality of Treating Interference as Noise conditions. α ii −d i −α ij ≥r j −r i , ∀i,j∈ [K],i6=j (2.22) α ii −d i ≥−r i , ∀i∈ [K] (2.23) 0≥r i , ∀i∈ [K]. (2.24) The above inequalities exactly match the ones in (2.19)-(2.21). This completes the proof. Next, we invoke the potential theorem of [28], re-stated below, to complete the characteri- zation of the polyhedral TIN region,P. Potential Theorem [Theorem 8.2 of [28]]: There exists a potential function for a directed graph D if and only if each directed circuit in D has non-negative length. Combining Lemma 2.1 and the potential theorem, we conclude that for a GDoF tuple (d 1 ,d 2 ,...,d K )∈ R K + , it is in the polyhedral TIN regionP if and only if each directed circuit in the graph D has a non-negative length. Therefore, it just remains to interpret the conditions of non-negative length for the circuits. We can categorize the circuits of D in three classes: • Circuits in the form of (u,v i ,u),∀i∈ [K]. For these circuits, we have α ii −d i ≥ 0⇔d i ≤α ii . (2.25) • Circuits in the form of (v i 0 ,v i 1 ,...,v im ), where i 0 = i m ,∀(i 1 ,i 2 ,...,i m )∈ Π K , ∀m∈ {2, 3,...,K}; i.e., the circuits which do not include node u. For these circuits, the non-negative length condition will be m−1 X j=0 (α i j i j −d i j −α i j i j+1 )≥ 0 ⇔ m−1 X j=0 d i j ≤ m−1 X j=0 (α i j i j −α i j i j+1 ) (a) ⇔ m X j=1 d i j ≤ m X j=1 (α i j i j −α i j−1 i j ). (2.26) 16 Chapter 2. On the Optimality of Treating Interference as Noise where in step (a) we just reorder the terms in the right hand side and use the fact that i m =i 0 . • Circuits in the form of (u,v i 1 ,...,v im ,u),∀(i 1 ,i 2 ,...,i m )∈ Π K , ∀m∈{2, 3,...,K}. For these circuits, the following inequality should hold. m−1 X j=1 (α i j i j −d i j −α i j i j+1 ) + (α imim −d im )≥ 0. (2.27) Since α imi 1 ≥ 0, we have α imim −d im ≥ α imim −d im −α imi 1 . Therefore, given the conditions in (2.26), the conditions in this class of circuits are redundant. Consequently, we will end up with conditions (2.25)-(2.26). Finally adding the non-negativity constraint on d i ’s, we obtain (2.10)-(2.11). This directly leads us to the following theorem which characterizes the polyhedral TIN regionP for general channel strength levels. Theorem 2.2. The GDoF region achieved through polyhedral TIN, denoted byP, is the set of all K-tuples (d 1 ,d 2 ,...,d K ) satisfying 0≤d i ≤α ii , ∀i∈ [K] m X j=1 d i j ≤ m X j=1 (α i j i j −α i j−1 i j ), ∀(i 1 ,i 2 ,...,i m )∈ Π K , ∀m∈{2, 3,...,K}, where Π K is the set of all possible cyclic sequences of all subsets of [K] with cardinality no less than 2, and the modulo-m arithmetic is implicitly used on the user indices, e.g., i m =i 0 . Now, we are at a stage to complete the proof of Theorem 2.1. 2.2.3 Proof of Theorem 2.1 Finally, to prove Theorem 2.1, we will show that under condition (2.9), the polyhedral TIN regionP coincides with the GDoF region outer bound, therefore establishing the optimality of TIN under (2.9) and proving Theorem 2.1. Note that from Theorem 2.2, the region (2.10)-(2.11) is exactly equal to the polyhedral TIN regionP and therefore this GDoF region can be achieved by TIN. Therefore, we only need to prove the outer bounds on the GDoF region. To this end, we first present the following theorem. 17 Chapter 2. On the Optimality of Treating Interference as Noise Theorem 2.3. For the K-user interference channel with channel input-output relationship in (2.1), the capacity region is included in the set of rate tuples (R 1 ,R 2 ,...,R K ) such that R i ≤ log(1 +|h ii | 2 P i ), ∀i∈ [K] (2.28) m X j=1 R i j ≤ m X j=1 log 1 +|h i j i j+1 | 2 P i j+1 + |h i j i j | 2 P i j 1 +|h i j−1 i j | 2 P i j ! , ∀(i 1 ,i 2 ,...,i m )∈ Π K , ∀m∈{2, 3,...,K}, (2.29) where Π K is the set of all possible cyclic sequences of all subsets of [K] with cardinality no less than 2, and the modulo-m arithmetic is implicitly used on the user indices, e.g., i m =i 0 . The proof of Theorem 2.3 follows [24, 26] and is relegated to Appendix C. Equipped with Theorem 2.3, we can now complete the converse of Theorem 2.1 through the following corollary. Corollary 2.1. For theK-user interference channel with channel input-output relationship in (2.4), when condition (2.9) is satisfied, its GDoF region is included in the set of GDoF tuples (d 1 ,d 2 ,...,d K ) such that d i ≤α ii , ∀i∈ [K] (2.30) m X j=1 d i j ≤ m X j=1 (α i j i j −α i j−1 i j ),∀(i 1 ,i 2 ,...,i m )∈ Π K , ∀m∈{2, 3,...,K}, (2.31) where Π K is the set of all possible cyclic sequences of all subsets of [K] with cardinality no less than 2, and the modulo-m arithmetic is implicitly used on the user indices, e.g., i m =i 0 . Proof of Corollary 2.1. The individual bounds in (2.30) follow directly from the inequalities in (2.28). In fact, from (2.28) we have d i = lim P→∞ R i logP ≤ lim P→∞ log(1 +P α ii ) logP =α ii , for any i∈ [K]. Also, the cyclic outer bounds in (2.31) follow from the outer bounds in (2.29). In fact, for any cycle (i 1 ,i 2 ,...,i m )∈ Π K we have 18 Chapter 2. On the Optimality of Treating Interference as Noise m X j=1 d i j = lim P→∞ P m j=1 R i j logP ≤ lim P→∞ P m j=1 log 1 +P α i j i j+1 + P α i j i j 1+P α i j−1 i j logP = m X j=1 max{0,α i j i j+1 ,α i j i j −α i j−1 i j } = m X j=1 (α i j i j −α i j−1 i j ), where the last equality is due to condition (2.9). This completes the proof. Note that as we explained before, when condition (2.9) is satisfied, the TIN regionP ∗ is equal to the polyhedral TIN regionP, which is a convex polyhedron as shown in Theorem 2.2. This means that in this regime, time-sharing cannot help enlarge the achievable GDoF region via TIN. 2.3 Constant Gap to Capacity In this section, we show that when condition (2.9) holds, so that TIN is GDoF-optimal, we can apply the insight gained in the GDoF study to prove that TIN can also achieve the whole channel capacity region to within a constant gap at any finite SNR. The main result of this section is mentioned in the following theorem. Theorem 2.4. In a K-user interference channel, where the channel strength level between transmitter i and receiver j is α ji , if condition (2.9) holds, then TIN can achieve to within log 2 (3K) bits of the capacity region. Proof. (Converse) Recall that using Theorem 2.3, we obtain the following outer bounds. R i ≤ log 2 (1 +P α ii ), ∀i∈ [K] m X j=1 R i j ≤ m X j=1 log 2 1 +P α i j i j+1 + P α i j i j 1 +P α i j−1 i j , ∀(i 1 ,i 2 ,...,i m )∈ Π K , ∀m∈{2, 3,...,K}. 19 Chapter 2. On the Optimality of Treating Interference as Noise Since P > 1, it follows that R i ≤ log 2 (1 +P α ii )≤α ii log 2 P + 1, ∀i∈ [K], (2.32) m X j=1 R i j ≤ m X j=1 log 2 1 +P α i j i j+1 + P α i j i j 1 +P α i j−1 i j < m X j=1 log 2 1 +P α i j i j+1 + P α i j i j P α i j−1 i j = m X j=1 log 2 P α i j−1 i j +P α i j i j+1 +α i j−1 i j +P α i j i j P α i j−1 i j ≤ m X j=1 log 2 3P α i j i j P α i j−1 i j = m X j=1 [(α i j i j −α i j−1 i j ) log 2 P + log 2 3], (2.33) for all cycles (i 1 ,i 2 ,...,i m )∈ Π K ,∀m∈{2, 3,...,K}. (Achievability) Consider the power control and TIN scheme, where the power allocated to each transmitter is equal to P r i (r i ≤ 0,∀i∈ [K]), and the achievable rate for each user is R i,TIN = log 2 1 + P r i +α ii 1 + P j6=i P r j +α ij . (2.34) From the proof of Theorem 2.1, we know that under condition (2.9), if d i ’s satisfy (2.10) and (2.11), then there exist r i ’s such that r i +α ii − max j6=i {0,r j +α ij } =d i , ∀i,j∈ [K], (2.35) r i ≤ 0, ∀i∈ [K]. (2.36) Therefore, we can write R i,TIN = log 2 1 + P r i +α ii 1 + P j6=i P r j +α ij ≥ log 2 P r i +α ii P 0 + P j6=i P r j +α ij ≥ log 2 P r i +α ii KP r i +α ii −d i =d i log 2 P + log 2 1 K . (2.37) 20 Chapter 2. On the Optimality of Treating Interference as Noise In other words, whend i ’s satisfy (2.10) and (2.11), the rates in (2.37) are always achievable by TIN,∀i∈ [K]. Thus it is not hard to obtain that the achievable rate region by TIN includes the rate tuples (R 1,TIN ,R 2,TIN ,...,R K,TIN ) satisfying 0≤R i,TIN ≤ max 0,α ii log 2 P + log 2 1 K , ∀i∈ [K], (2.38) m X j=1 R i j ,TIN ≤ max 0, m X j=1 (α i j i j −α i j−1 i j ) log 2 P + log 2 1 K (2.39) for all cycles (i 1 ,i 2 ,...,i m )∈ Π K ,∀m∈{2, 3,...,K}. Comparing (2.32)-(2.33) with (2.38)-(2.39), we can characterize the approximate channel capacity to within a constant gap, which is only dependent on the number of users K. We can show that TIN achieves to within log 2 (3K) bits of the capacity region. To this end, we need to show that each of the rate constraints in (2.38) and (2.39) is within log 2 (3K) bits of its corresponding outer bound in (2.32) and (2.33); i.e., the following inequalities always hold 4 , σ R i < log 2 (3K), ∀i∈ [K] (2.40) σ P m j=1 R i j ≤m log 2 (3K), ∀(i 1 ,i 2 ,...,i m )∈ Π K , ∀m∈{2, 3,...,K}, (2.41) where σ (.) denotes the difference between the achievable rate in (2.38) and (2.39) and its corresponding outer bound in (2.32) and (2.33). For σ R i , we consider the following two cases, • α ii log 2 P + log 2 1 K ≤ 0: In this case, we obtain σ R i =α ii log 2 P + 1≤ 1 + log 2 K < log 2 (3K). • α ii log 2 P + log 2 1 K > 0: In this case, we have σ R i = α ii log 2 P + 1 − α ii log 2 P + log 2 1 K = 1 + log 2 K < log 2 (3K). Similarly, for σ P m j=1 R i j , we consider the following two cases, 4 Notice that since in the second line of (2.33) there exists a “<”, “≤” is fine for the inequality in (2.41). 21 Chapter 2. On the Optimality of Treating Interference as Noise • P m j=1 (α i j i j −α i j−1 i j ) log 2 P + log 2 1 K ≤ 0: In this case, we obtain σ P m j=1 R i j = m X j=1 (α i j i j −α i j−1 i j ) log 2 P + log 2 3 ≤ m X j=1 [log 2 3 + log 2 K] =m log 2 (3K). • P m j=1 (α i j i j −α i j−1 i j ) log 2 P + log 2 1 K > 0: In this case, we have σ P m j=1 R i j = m X j=1 (α i j i j −α i j−1 i j ) log 2 P + log 2 3 − m X j=1 (α i j i j −α i j−1 i j ) log 2 P + log 2 1 K = m X j=1 [log 2 3 + log 2 K] =m log 2 (3K). Combining the above results, we complete the proof. 2.4 The General Achievable GDoF Region of TIN In this section, we remove the constraint (2.9) on the channel gains, and investigate the achievable GDoF region by TIN for K-user interference channels with general channel strength levels. As we show, the TIN regionP ∗ is equal to the union of multiple poly- hedra, each of which is in the form of the polyhedral TIN region of a subset of the users of the network. Remarkably, the TIN region is almost the same as the polyhedral TIN region in the sense that the measure of the difference of the two sets is zero in R K . We have shown that when (2.9) holds, the original TIN regionP ∗ is equal to the polyhedral TIN regionP. Now, the natural question to ask is what the TIN regionP ∗ is for K-user interference channels with general channel strength levels. The following theorem settles this issue. Theorem 2.5. In a K-user interference channel, where the channel strength level from transmitteri to receiverj is equal toα ji , the achievable GDoF region through power control 22 Chapter 2. On the Optimality of Treating Interference as Noise and treating interference as noise, denoted byP ∗ , is equal to P ∗ = [ S⊆[K] P S , (2.42) whereP S ,S⊆ [K], is defined as P S = (d 1 ,d 2 ,...,d K ) : d i = 0,∀i∈S, 0≤d j ≤α jj ,∀j∈S c , m X j=1 d i j ≤ m X j=1 (α i j i j −α i j−1 i j ),∀(i 1 ,i 2 ,...,i m )∈ Π S c , and Π S c is the set of all possible cyclic sequences of all subsets ofS c with cardinality no less than 2. The proof is given in Appendix D. In words, the TIN regionP ∗ is the union of 2 K polyhedral TIN regionsP S , each of which corresponds to the case where the users inS are made silent. Note thatP φ is actually the polyhedral TIN regionP defined in Theorem 2.2. Except for the polyhedral TIN regionP, all the otherP S ’s have zero volume in R K since in each of them the users inS always have zero GDoF value. Therefore, the TIN regionP ∗ is almost the same as the polyhedral TIN regionP in the sense that the measure of the difference of the two sets is zero inR K . Furthermore, as opposed to the polyhedral TIN regionP, the TIN regionP ∗ may not be convex in general, and if time-sharing is allowed alongside with TIN, the achievable region may become substantially larger. Therefore, the above theorem also reveals that when the sufficient condition (2.9) is violated, time-sharing may help enlarge the achievable GDoF region of TIN. Example 2.3. Consider the 3-user cyclic interference channel shown in Fig. 2.4. Notice that for user 3 the sufficient condition (2.9) does not hold. First, if all the users are active, we can get the polyhedral TIN region as follows. P ∅ = (d 1 ,d 2 ,d 3 ) : 0≤d i ≤ 1, ∀i∈{1, 2, 3}, d 1 +d 2 ≤ 1.9, d 2 +d 3 ≤ 1.4, d 1 +d 3 ≤ 1.1, d 1 +d 2 +d 3 ≤ 1.4 , (2.43) which is in fact the polyhedral TIN regionP we defined earlier. 23 Chapter 2. On the Optimality of Treating Interference as Noise 1 1 0.9 0.6 1 0.1 Figure 2.4: A 3-user cyclic interference channel, where the strength levels for each link is shown in the figure. Then, consider the cases in which only one of the three users is made silent and hence has zero GDoF, and the other two users are active. In such cases, we only need to consider the Z-channel between the remaining two users, implying that P {1} = (d 1 ,d 2 ,d 3 ) : d 1 = 0, 0≤d 2 ≤ 1, 0≤d 3 ≤ 1,d 2 +d 3 ≤ 1.4 P {2} = (d 1 ,d 2 ,d 3 ) : d 2 = 0, 0≤d 1 ≤ 1, 0≤d 3 ≤ 1,d 1 +d 3 ≤ 1.1 P {3} = (d 1 ,d 2 ,d 3 ) : d 3 = 0, 0≤d 1 ≤ 1, 0≤d 2 ≤ 1,d 1 +d 2 ≤ 1.9 . It is easy to verify that P {1} ⊆P ∅ , P {2} ⊆P ∅ , but P {3} 6⊆P ∅ . For instance, the GDoF tuple (1, 0.9, 0)∈P {3} is not in the GDoF regionP ∅ since it violates the cycle bound d 1 +d 2 +d 3 ≤ 1.4. Next, consider the cases in which two users are made silent. P {2,3} = (d 1 ,d 2 ,d 3 ) : 0≤d 1 ≤ 1, d 2 =d 3 = 0 P {1,3} = (d 1 ,d 2 ,d 3 ) : 0≤d 2 ≤ 1, d 1 =d 3 = 0 P {1,2} = (d 1 ,d 2 ,d 3 ) : 0≤d 3 ≤ 1, d 1 =d 2 = 0 , 24 Chapter 2. On the Optimality of Treating Interference as Noise and it can be verified that P {2,3} ⊆P ∅ , P {1,3} ⊆P ∅ , P {1,2} ⊆P ∅ . Finally, we have P {1,2,3} = (d 1 ,d 2 ,d 3 ) : d 1 =d 2 =d 3 = 0 ⊆P ∅ . Therefore, the TIN region is equal to P ∗ = [ S⊆{1,2,3} P S =P ∅ ∪P {3} . (2.44) This region is illustrated in Fig. 2.5, where the yellow region corresponds toP ∅ and the blue region corresponds toP {3} . Note that since for user 3, the sufficient condition (2.9) is violated, the polyhedral TIN regionP =P ∅ is not the whole GDoF region for this 3- user cyclic channel. Moreover, as Fig. 2.5 shows, the regionP ∗ is not convex. Therefore, time-sharing betweenP ∅ andP {3} can help enlarge the achievable GDoF region via TIN. 4 Figure 2.5: The TIN region of the network in Figure 2.4, which is the union of the yellow region (P ∅ ) and the blue region (P {3} ). 25 Chapter 2. On the Optimality of Treating Interference as Noise 2.5 Numerical Analysis In this section, we numerically compute the probability that the sufficient condition (2.9) is satisfied in a typical wireless scenario. We consider a circular cell with a radius of 1 km and placeK base stations (transmitters) randomly and uniformly over the cell area. Each base station is assumed to have a coverage radius of r. In order to create a K-user interference channel with strong enough direct links, we consider K mobile receivers such that the i-th mobile receiver is located randomly and uniformly inside the coverage area of the i-th base station, i∈ [K]. A realization of such a network scenario is depicted in Fig. 2.6a. T 1 R 1 T 2 R 2 T 3 R 3 T 4 R 4 T 5 R 5 T 6 R 6 T 7 R 7 T 8 R 8 T 9 R 9 T 10 R 10 (a) 20 40 60 80 100 120 140 160 180 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r (radius) Probability that condition (9) is satisfied K=5 K=10 K=15 (b) Figure 2.6: (a) A 10-user interference channel where the black circle, green circles, red trian- gles, and blue crosses represent the whole cell area, the coverage area of base stations, base stations (transmitters), and receivers, respectively. The coverage radius of each transmitter is taken to be r = 100m. (b) Effect of the coverage radius and the number of users on the probability that the sufficient condition (2.9) is satisfied. For the channel gain values, we make use of the Erceg model [29], operating at a frequency of 2GHz and using the terrain category of hilly/light tree density. Taking the noise floor as -110 dBm, we choose the transmit power of all the base stations such that the expected value of the SNR at the boundary of their coverage area is 0 dB. Then, we randomly locate the base stations and mobile receivers according to the coverage radius r. Fig. 2.6b demonstrates the result of our numerical analysis. As illustrated in this plot, the probability that the sufficient condition (2.9) for the GDoF- optimality of TIN is satisfied decreases as the density of the network increases, either by 26 Chapter 2. On the Optimality of Treating Interference as Noise increasing the number of users or by increasing the coverage radius of each base station. However, as a typical scenario, it is noteworthy that for the case of a 10-user interference channel with the coverage radius of 100m for each base station, the sufficient condition (2.9) is satisfied half the times. This means that with a probability of 50%, TIN is GDoF-optimal and can also achieve the whole capacity region of the network to within a constant gap. It therefore implies that the sufficient condition (2.9) can be actually satisfied in practice with a reasonably high probability, enabling the results in this chapter on the optimality of treating interference as noise to be put into use in practice. 27 Chapter 3 Spectrum Sharing in D2D Networks: Information-Theoretic Link Scheduling (ITLinQ) As mentioned in Chapter 1, device-to-device (D2D) communication introduces new chal- lenges for interference management and spectrum sharing. In particular, due to the spe- cific characteristics of these networks, neither fully coordinated synchronous cellular-type approaches that rely on advanced physical layer designs, nor fully distributed and asyn- chronous WiFi-type mechanisms (such as CSMA/CA) are adequate. The downside of the first type of interference management mechanisms is that they need levels of centraliza- tion, coordination, and information at the mobile nodes that are difficult to accomplish in practice. On the other hand, the problem with the second type of approaches is that their performance degrades significantly as the number of links grows. These issues have motivated a more recent approach that is based on a minimal level of coordination among the links which also maintains its promising performance for large numbers of links. This scheme, called FlashLinQ [15], is a distributed scheduling scheme which demonstrates considerable improvement over pure CSMA/CA. In a system of multiple source-destination pairs (links), this scheduling algorithm first orders the links according This chapter is mainly taken from [30–32], coauthored by the author of this dissertation. 28 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) to a randomly selected priority list. Then, starting from the higher-order links, each link is scheduled if it does not cause and does not receive “much” interference from the already scheduled links. The level of acceptable interference is determined based on the observed signal-to-interference ratio (SIR) at all the previously scheduled links and also the current link. FlashLinQ has also been implemented and experimented in practice and shown to demonstrate promising performance compared to previous scheduling schemes. FlashLinQ scheduling can also be viewed as a refinement of the conventional independent set scheduling which is based on using a conflict graph to model the interference among the links (see, e.g. [33–37] and the protocol model in [38]). In the independent set scheduling approach, two links (source-destination pairs) are considered to be mutually non-interfering, hence able to transmit data at the same time, if the interference that they cause on each others’ destinations is below a certain threshold. The drawback of this scheme is that this threshold is set at a fixed value (often at noise level) which does not capture the effect of the number of links, their density inside the cell area, etc. More importantly, the scheme does not consider the signal-to-noise ratio (SNR) level that each link itself can achieve and only takes the interference levels into account. FlashLinQ, however, overcomes this problem by comparing the direct signal power level that each link gets with its incoming interference power level. Also, in the FlashLinQ scheduling algorithm, if a link does not cause/receive much interference to/from higher-priority links, but does not get a high direct signal power itself, it gets silent and “yields” such that lower-priority links have the opportunity to contribute more to the overall sum-throughput of the network. Hence, both FlashLinQ and independent set scheduling approaches aim at finding subsets of links in which the interference among them is at a “sufficiently” low level, so that their simultaneous transmissions are not detrimental to each other. This gives rise to a nat- ural question: what would be a theoretically-justified way of creating such subsets, and determining whether the interference among them is at a “sufficiently” low level? In this chapter, we propose an answer to this question. We define a new concept of information-theoretic independent sets (ITIS), which indicates the sets of links for which simultaneous communication and treating the interference from each other as noise is information-theoretically optimal (to within a constant gap). In other words, a subset of links forms an ITIS if by a simple scheme of using point-to-point Gaussian codebooks with appropriate power levels at each transmitter and treating interference as noise at every receiver we can achieve the entire information-theoretic capacity region of that subset of 29 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) links (to within a constant gap). We use the optimality condition for treating interference as noise, developed in Chapter 2 to provide a description of ITIS based on the channel gains among the links in the network. In fact, as we will see later, a subset of links is defined to create an ITIS if for any link in the subset, the SNR level is no less than the sum of its strongest incoming interference-to-noise ratio (INR) and its strongest outgoing INR (all measured in dB scale). It is important to note that this condition is quite different from that of FlashLinQ and independent set scheduling which only rely on thresholds on SIR and INR values to identify the subsets of links with “sufficiently” low levels of interference. Furthermore, we propose our new spectrum sharing mechanism, named information-theoretic link scheduling (in short, ITLinQ), which schedules the links in an information-theoretic in- dependent set to transmit data at the same time. We characterize the guaranteed fraction of the capacity region that ITLinQ is able to achieve in a specific network setting. In par- ticular, we consider a set of n source-destination pairs, where the source nodes are spread randomly and uniformly over a circular cell of fixed radius and each destination node is located within a distance r n ∝ n −β of its corresponding source node. For the channel gains, we only consider the path-loss model. In such a setting, we show that the criteria for defining information-theoretic independent sets transforms the network into a random geometric graph which enables us to characterize the fraction of capacity region that can be achieved by the ITLinQ scheme. In fact, depending on the value of β, we identify three regimes in each of which ITLinQ can achieve a fraction λ of the capacity region within a gap of k almost-surely: • For 0<β < 1, λ = Θ n β−1 and k =O log 3n n 1−β . 1 • For β = 1, λ = ln(lnn) lnn and k =O log(lnn) + ln(lnn) lnn . • For β > 1, λ = Θ(1) and k =O(log 3n). This shows a considerable improvement over the fraction of the capacity region that the conventional independent set scheduling can achieve, which is 1 n (derived via numerical analysis). We will also show that the network model in which each destination gets asso- ciated with the closest source to itself is a subclass of the above model for any β < 1 2 , and 1 For two functions f(n) and g(n) defined on the set of positive integers Z+, f(n) =O(g(n)) if and only if there exists a positive real number a and a positive integer n0 such that for all n>n0,|f(n)|≤a|g(n)|. Also, f(n) = Θ(g(n)) if and only if there exist two positive real numbers a1 and a2 and a positive integer n0 such that for all n>n0, a1|g(n)|≤|f(n)|≤a2|g(n)|. 30 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) therefore we can asymptotically achieve a fraction Θ( √ n) of the capacity region in this case almost-surely. Afterwards, we will focus on the challenge of distributed implementation of the ITLinQ scheme. We will develop a distributed spectrum sharing scheme based on ITLinQ, whose complexity is comparable to the FlashLinQ algorithm. The conditions that need to be sat- isfied for the sources and destinations in this scheme are based on the sufficient conditions for the optimality of treating interference as noise (as derived in Chapter 2), hence pro- viding a strong theoretical backbone for the algorithm. We will numerically evaluate the performance of our distributed scheme and compare it with FlashLinQ in an outdoor set- ting with 8 - 4096 links of random lengths spread uniformly at random in a square cell. We observe that the sum-rate achieved by the distributed ITLinQ scheme improves over that of FlashLinQ by more than 100%, while keeping the complexity basically at the same level. Finally, we introduce a slight variation to the distributed ITLinQ scheme so as to consider fairness among the links in the network and show that our fair ITLinQ scheme achieves almost the same tail distribution for the link rates as in FlashLinQ, while demonstrating over 50% sum-rate gain. 3.1 Description and Analysis of the Information-Theoretic Link Scheduling Scheme In this section, we introduce our scheduling scheme, which we call “information-theoretic link scheduling” (in short, “ITLinQ”). We start by defining the notion of “information- theoretic independent set” (in short, “ITIS”) and then move forward to describe the ITLinQ scheme. Afterwards, we will consider a specific network setting and in that setting, we will characterize the fraction of capacity region that ITLinQ is able to achieve to within a gap. 3.1.1 Description of ITIS and ITLinQ We consider a wireless network composed of n sources{S i } n i=1 and n destinations{D i } n i=1 in which each source aims to communicate a message to its corresponding destination. All the links (i.e., source-destination pairs) are considered to share the same spectrum, which gives rise to interference among all the transmissions. We assume that all the nodes (i.e., all the sources and the destinations) know how many links exist in the network and they 31 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) also agree on a specific ordering of the links, where by ordering we mean a labeling of the links from 1 ton. Furthermore, we assume that the nodes are synchronous; i.e., there exists a common clock among them. The physical-layer model of the network is considered to be the AWGN model in which each source S i intends to send a message W i to its corresponding destination D i , and does so by encoding its message to a codeword X l i of length l and transmitting it within l time slots. There is a power constraint ofE h 1 l kX l i k 2 i ≤P on the transmit vectors. The received signal vector of destination j over the l time slots will be equal to Y l j = n X i=1 h ji X l i +Z l j , whereh ji denotes the channel gain between source i and destination j, andZ l j denotes the additive white Gaussian noise vector at destinationj with distributionCN (0,NI l ), I l being thel×l identity matrix. An example of such a network configuration is illustrated in Figure 3.1. S 1 S i S n S j D i D j D n D 1 h ii h jj h ji h ij Figure 3.1: A wireless network composed ofn source-destination pairs, where the green and red lines represent the direct and cross channel gains, respectively. We assume that at each destination, all the incoming interference is treated as noise. There- fore, each source-destination pair S i −D i can achieve the rate ofR i = log(1+SINR i ), where SINR i , P|h ii | 2 P j6=i P|h ij | 2 +N denotes the signal-to-interference-plus-noise ratio at destination i. In general, treating interference as noise (TIN) is known to be suboptimal for the general interference channel and numerous more sophisticated physical-layer coding schemes (such as message splitting and successive interference cancellation [24,39], interference alignment [40,41], and structured coding [42–44]) have been proposed in order to improve it. However, as we showed in Chapter 2, if condition (2.9) is satisfied, the TIN is information-theoretically 32 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) optimal (to within a constant gap). It is easy to verify that for the network model mentioned in this section, this condition is equivalent to SNR i ≥ max j6=i INR ij max k6=i INR ki , ∀i = 1,...,n, (3.1) where SNR i , P|h ii | 2 N and INR ij , P|h ij | 2 N denote the signal-to-noise ratio of link i and the interference-to-noise ratio of source j at destination i, respectively. Remark 3.1. This result can intuitively be explained under the deterministic channel model of [45] as follows. Consider the deterministic model for link i as shown in Figure 3.2. In this figure, each little circle represents a signal level. The transmit and received signal T i : Signal levels of S i causing interference at other links R i : Signal levels of D i receiving interference from other links SNR i (dB) S i D i Figure 3.2: A deterministic view of the optimality condition for treating interference as noise. levels are sorted from MSB to LSB from top to bottom at the source and the destination respectively. The channel gain between source i and destination j in the deterministic model, denoted by n ji indicates how many of the first MSB transmitted signal levels of source i are received at destination node j. Now, let us consider all transmit signal levels of source i that are interfering to destinations j6= i in the network. The total number of them is max j6=i n ji , and they are depicted by the the setT i in Figure 3.2. Similarly, we can consider all received signal levels of destinationi that are receiving interference from sources j6=i in the network. The total number of them is max j6=i n ij , and they are depicted by the the setR i in Figure 3.2. Now, by taking logarithm of both sides of inequality (1) in Theorem 1, the condition in (1) can be described as|T i | +|R i |≤ log SNR i =n ii , which means that there is no connection (or coupling) between the signal levels of source i that are causing interference and the received signal levels of destination i that are getting interference from all other nodes in the network. 33 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) Remark 3.2. The gap of log 3n mentioned in Theorem 2.4 is in fact the worst-case gap between the achievable rate region by TIN and the outer bound. We have numerically evaluated the actual gap for the case of randomly-generated networks with 8 links which satisfy condition (3.1) and the result is illustrated in Figure 3.3 in the form of the cumulative distribution function (CDF). As the figure suggests, the actual gap to the capacity region 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 Gap to the capacity region (Bits) Probability Worst−case gap Figure 3.3: Cumulative distribution function (CDF) of the actual gap of the achievable rate region by TIN with respect to the capacity region for networks with 8 links satisfying condition (3.1) and its comparison with the worst-case gap of log 3n = log 24≈ 4.58. achievable by TIN is much smaller than the worst-case gap of log 3n with a high probability. Therefore, if we consider any subset of the source-destination pairs in a wireless network and show that condition (3.1) is satisfied in that subset, then we know that TIN is information- theoretically optimal in that subset of the links (to within a constant gap). This implies that the interference is at such a low level in this subnetwork that makes it suitable to call such a subset an “information-theoretic independent subset” . More formally, we have the following definition. Definition 3.1 (ITIS). In a wireless network ofn links, a subset of the linksS⊆{1,...,n} is called an information-theoretic independent set (in short, ITIS) if for any link i∈S, SNR i ≥ max j∈S\{i} INR ij max k∈S\{i} INR ki . (3.2) As it is clear, the difference between such a concept and the regular notion of an independent set lies in the fact that in the latter case, the interference between any pair of links should 34 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) be below a certain threshold (e.g., noise level), whereas in the former case, the interference between all of the links is at such a low level (determined by condition (3.1)) that makes it (to within a constant gap) information-theoretically optimal to treat all the interference as noise. Based on the concept of ITIS, we define our scheduling scheme as follows. Definition 3.2 (ITLinQ). The information-theoretic link scheduling (in short, ITLinQ) scheme is a spectrum sharing mechanism which at each time, schedules the sources in an information-theoretic independent set (ITIS) to transmit simultaneously. Moreover, all the destinations will treat their incoming interference as noise. Remark 3.3. In order to gain more intuition about the information theoretic independent sets, one can consider a simple sufficient condition for the scheduling condition in (3.2). It is easy to verify that a subset of linksS form an ITIS if for any link i∈S, INR ij ≤ p SNR i , INR ji ≤ p SNR i , ∀j∈S\{i}, which is the same as the condition for the optimality of TIN for a network with only two links which is mentioned in [24]. In fact, this condition compares the ratio between the INR and SNR values in dB scale with a fixed threshold of 1 2 . This is the main distinction of this condition compared to the conditions used in FlashLinQ, in which the difference between the INR and SNR values in dB scale is compared with a fixed threshold. We will use this sufficient condition later in the chapter for both the capacity analysis and the distributed implementation of the ITLinQ scheme. In Section 3.2, we will show how to implement the ITLinQ scheme in a distributed way. However, for now, we will focus on characterizing the fraction of the capacity region that ITLinQ is able to achieve in a specific network setting. 3.1.2 Capacity Analysis of the ITLinQ Scheme In this section, we analyze the fraction of the capacity region that the ITLinQ scheme can achieve to within a gap in a network with a large number of links. We consider a network in which the sources are placed uniformly and independently inside a circle of radius R. After placing the source nodes, each destination node D i (associated with the source node S i ) is assumed to be located within a distance r n = r 0 n −β of S i , where r 0 is a fixed distance and β is a positive exponent. Note that we do not assume any particular 35 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) distribution for the placement of the destination nodes as long as each of them lies within a circle of radius r 0 n −β around its corresponding source node. This represents a dense network in which the intended destinations are typically located closer to their sources, which is a characteristic of D2D networks. Moreover, in this section, we assume that each channel gain is a deterministic function of the distance between its corresponding source and destination. In fact, we consider the path-loss model for the channel gains in which the squared magnitude of the channel gain at distance r is equal to g 0 r −α , where g 0 ∈R is a fixed real number and α denotes the path-loss exponent. In Section 3.1.2.2, we will study the case in which Rayleigh fading is included in the channel model as well. For such a network and channel model, we have the following theorem (which will be proved later in this section) that presents a guarantee on the fraction of the capacity region that can be achieved by the ITLinQ scheme. Theorem 3.1. For sufficiently large number of links (n→∞) in the above model, the ITLinQ scheme can almost-surely achieve a fraction λ of the capacity region within a gap of k bits 2 , where                λ = 2πR 2 √ 3γ 2 n β−1 , k≤ 2πR 2 √ 3γ 2 log 3n n 1−β if 0<β < 1 λ = ln(lnn) lnn , k≤ log(lnn) + (log 3) ln(lnn) lnn if β = 1 λ = 1 j 1 β−1 + 1 2 k +1 , k≤ log 3n j 1 β−1 + 1 2 k +1 if β > 1, in which γ = 2α q P N g 0 r α 0 is a constant independent of n. Remark 3.4. The achievable fraction of the capacity region expressed in Theorem 3.1 is only a lower bound on the fraction of the capacity region that ITLinQ is able to achieve, and therefore, ITLinQ can guarantee the achievability of this fraction of the capacity region (to within a gap of k bits). Figure 3.4 illustrates the impact of the maximum source-destination distance decreasing rate on the fraction of the capacity region that can be achieved by ITLinQ. 3 If the maximum 2 This implies that for any rate tuple (R1,...,Rn) in the capacity region of the network, ITLinQ is (almost- surely) able to achieve the rate tuple (λR1−k,...,λRn−k). 3 Since the focus of the comparison is on the order of the fractions achievable by ITLinQ in different regimes ofβ, the parameters are chosen such that the constant 2 √ 3πR 2 3γ 2 in Theorem 3.1 is equal to 1. Hence, any choice of the parameters for which 2 √ 3πR 2 3γ 2 = 1 is a valid choice. 36 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) source-destination distance is proportional to n −β such that 0 < β < 1, then the ITLinQ scheme is capable of asymptotically achieving a fraction proportional to 1 n 1−β of the capac- ity region, within a vanishing gap. However, if the maximum source-destination distance scales as n −1 , then the achievable fraction of the capacity region decreases as ln(lnn) lnn which declines much slower than the previous case. In this case, the gap increases very slowly with respect to n. Finally, in the case that the maximum distance between each source and its corresponding destination scales faster than n −1 , we can achieve at least a constant frac- tion of the capacity region for asymptotically large number of links which is a considerable improvement, whereas the gap is increasing with the number of links. This matches the natural intuition that the closer the destinations are located to their corresponding sources, the more the signal-to-interference-plus-noise ratio and the higher the fraction of the ca- pacity that can be achieved by the ITLinQ scheme. Also, as a baseline, we have included the fraction of the capacity region that TDMA and independent set scheduling can achieve, which is 1 n for both schemes. 4 40 60 80 100 120 140 160 180 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Number of links (n) Achievable fraction of the capacity region TDMA and Independent Set Scheduling ITLinQ, β=0.5 ITLinQ, β=1 ITLinQ, β=2 Figure 3.4: Comparison of the guaranteed achievable fraction of capacity region by the ITLinQ scheme in different regimes with TDMA and independent set scheduling. Remark 3.5. As an immediate application of Theorem 3.1, one can consider the model in which all then source and then destination nodes are located uniformly and independently within a circular area of radius R, and each destination gets associated with its closest source. For such a model, it is straightforward to show that ITLinQ can almost-surely achieve a fractionλ = 2 √ 3πR 2 3γ 2 n β−1 of the capacity region to within a gap ofk≤ 2 √ 3πR 2 3γ 2 log 3n n 1−β for any β < 1 2 , when n→∞. 4 The achievable fraction of the capacity region by independent set scheduling was derived through nu- merical analysis. 37 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) 3.1.2.1 Proof of Theorem 3.1 In order to characterize the fraction of the capacity region that ITLinQ is able to achieve and prove Theorem 3.1, we seek to find the minimum number of information-theoretic independent sets which cover all the links and we will then do time-sharing among these subsets. More precisely, if we denote the set of all the information-theoretic independent subsets of a network composed of n source-destination pairs byS n , then we are interested in the minimum-cardinality subset ofS n whose members cover all the links; i.e., their union is equal to the set of all the links{1,...,n}. Denote such a subset byS ∗ n and let κ n =|S ∗ n |. We will show that time-sharing among theseκ n information-theoretic independent sets can achieve the fractions of the capacity region mentioned in Theorem 3.1. As the first step of the proof, we characterize the achievable fraction of the capacity region by the ITLinQ scheme and its gap with respect to the random variable κ n in the following lemma. Lemma 3.1. The ITLinQ scheme can achieve a fraction 1 κn of the capacity region of a network composed of n source-destination pairs to within a gap of log 3n κn . Proof. Consider any rate tuple (R 1 ,...,R n ) inside the capacity region of the network and consider any ITISU ∈S ∗ n . From the result in Theorem 2.4, since TIN is information- theoretically optimal inU (to within a constant gap), the rate tuple ( ¯ R 1,U ,..., ¯ R n,U ) is achievable in the 1 κn fraction of time which is allocated toU, where ¯ R i,U =      R i − log 3|U| i∈U 0 i / ∈U. Therefore, the rate achieved by any link i∈{1,...,n} in the network through the ITLinQ scheme, denoted by R i,ITLinQ , can be lower bounded as R i,ITLinQ = 1 κ n X U∈S ∗ n ¯ R i,U = 1 κ n X U∈S ∗ n :i∈U (R i − log 3|U|) ≥ 1 κ n (R i − log 3n) (3.3) = 1 κ n R i − log 3n κ n , 38 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) where (3.3) follows from the fact that the subsets inS ∗ n cover all the links{1,...,n} and that for everyU∈S ∗ n , we have|U|≤n. This completes the proof. Therefore, to find an achievable fraction of the capacity region by ITLinQ, we need to find an upper bound on κ n , that is the minimum number of information-theoretic independent subsets which cover all of the links. One way to find such an upper bound is to restrict the TIN-optimality condition in (3.1). In other words, we need to find another condition that implies condition (3.1), but is more restricted and more tractable than (3.1). Imposing such a restricted sufficient condition will reduce the number of information-theoretic independent subsets, hence leading to an upper bound on κ n . To this end, we present Lemma 3.2. In the following, we denote the distance between source i and destination j by d S i D j and the distance between sources i and j by d S i S j ,∀i,j. Lemma 3.2. If in a network of n source-destination pairs within the framework of the model in Section 3.1.2, the distance between S i and S j satisfies d S i S j > γn −β/2 +r 0 n −β , then max((INR ji ) 2 , (INR ij ) 2 )< min(SNR i , SNR j ). Proof. Based on the model considered in Section 3.1.2, we know that d S i D i ≤ r 0 n −β and d S j D j ≤r 0 n −β . Moreover, from the triangle inequality, we will haved S i D j ≥d S i S j −d S j D j > γn −β/2 . Similarly, we have d S j D i >γn −β/2 . Therefore, we can get SNR i = P N g 0 d S i D i −α ≥ P N g 0 r 0 n −β −α = P N g 0 r 0 −α n αβ , (3.4) and INR ji = P N g 0 d S i D j −α < P N g 0 γn −β/2 −α = P N g 0 γ −α n αβ/2 . (3.5) Combining (3.4) and (3.5), we will have (INR ji ) 2 < P N g 0 2 γ −2α n αβ = P N g 0 r 0 −α n αβ ≤ SNR i , (3.6) 39 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) and likewise, we can show that (INR ij ) 2 < SNR i . (3.7) Combining (3.6) with (3.7) yields max((INR ji ) 2 , (INR ij ) 2 )< SNR i . By symmetry, we will also have max((INR ji ) 2 , (INR ij ) 2 )< SNR j . This completes the proof. Consequently, Lemma 3.2 implies that there exists a threshold distance of d th,n =γn −β/2 + r 0 n −β such that if the distance between two sources is greater than this threshold, the corresponding pair of links are considered to be information-theoretically independent; i.e., the interference they cause on each other is at a sufficiently low level that it is information- theoretically optimal to treat it as noise (to within a constant gap). Therefore, given an network of n source-destination pairs with nodes spread as mentioned in the model in the beginning of Section 3.1.2, we can build a corresponding undirected graph G n = (V n ,E n ) where V n ={1,...,n} is the set of vertices and (i,j)∈E n if and only if d S i S j ≤ d th,n ; i.e., two nodes are connected together if and only if the distance between their sources is no larger than the threshold distance d th,n . We call the resultant graph G n the information-theoretic conflict graph of the original network. Clearly, this graph is a random geometric graph [46]. To return to our original problem, note that we needed to find an upper bound on κ n . The following lemma provides such an upper bound. Lemma 3.3. κ n ≤χ(G n ), where χ(.) denotes the chromatic number. Proof. The chromatic number of G n is the smallest number of colors that can be assigned to all of the nodes of G n such that no two adjacent nodes have the same color. Therefore, considering the subsets of the links which receive the same color, χ(G n ) is the minimum number of subsets of the links which cover all the links and each of which consist of links whose sources have distance larger than d th,n . From Lemma 3.2, it is easy to show that if for three distinct links i,j,k, all the pairwise source distances are larger than d th,n , then we will have that all the squared INR’s within the subnetwork consisting of links{i,j,k} are less than all the SNR’s. Extending this argument, we can see that all the independent subsets of G n automatically satisfy the TIN-optimality condition of (3.1) and hence are also information-theoretic independent subsets. Therefore,κ n , which denotes the minimum 40 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) number of information-theoretic independent subsets that cover all the links, can be no more than χ(G n ), the chromatic number of G n . Thus the final step is to characterize the asymptotic distribution of χ(G n ). In this step, we will use parts (i), (ii) and (iv) of Theorem 1.1 in [46] which we bring here for the sake of completeness. Consider a positive integer d and a normk.k on R d . Suppose we have n points x 1 ,...,x n inR d and a threshold distance r, where lim n→∞ r = 0. Then, the random geometric graph G n is defined as a graph with vertex set{1,...,n} in which vertices i and j are adjacent if and only if thekx i −x j k≤r. Defining f(n)g(n) and f(n)∼g(n) to be equivalent to lim n→∞ f(n) g(n) = 0 and lim n→∞ f(n) g(n) = 1, respectively, we have the following theorem. Theorem 3.2 ( [46]). For the random geometric graph G n , the following hold. (i) If nr d ≤n −α for some fixed α> 0, then P χ(G n )∈ lnn ln(nr d ) + 1 2 , lnn ln(nr d ) + 1 2 + 1 for all but finitely many n = 1. (ii) If n − nr d lnn for all > 0, then χ(G n )∼ lnn/ ln lnn nr d a.s. (iii) If nr d lnn (but still r→ 0), then χ(G n )∼ vol(B) 2 d δ σnr d a.s., where B is the unit ball in R d , σ is the “maximum density” of the distribution of nodes in R d and δ is the “packing density”, defined in [46]. Using Theorem 3.2, the following lemma characterizes the asymptotic behavior of the chro- matic number of the information-theoretic conflict graph G n . Lemma 3.4. For the information-theoretic conflict graph G n , χ(G n ) exhibits the following behavior as n→∞: • If 0<β < 1, then χ(Gn) n 1−β a.s. −→ √ 3 2πR 2 γ 2 . 41 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) • If β = 1, then χ(Gn) lnn/ln(lnn) a.s. −→ 1. • If β > 1, then P χ(G n )−→ j 1 β−1 + 1 2 k or χ(G n )−→ j 1 β−1 + 1 2 k + 1 = 1. Proof. Since the information-theoretic conflict graph G n is a random geometric graph with threshold distanced th,n =γn −β/2 +r 0 n −β and the nodes are distributed inR 2 , we can make use of Theorem 3.2. We will have the following cases: • If 0<β < 1, then nd 2 th,n =γ 2 n 1−β +r 2 0 n 1−2β lnn, and therefore we can use part (iii) of Theorem 3.2. Note that the dominant term in γ 2 n 1−β +r 2 0 n 1−2β is the first term, since β > 0. Also, as mentioned in [46], for the case of Euclidean norm in R 2 , we haveδ = π 2 √ 3 and vol(B) =π. Also, since the distribution of the nodes is uniform on a circle of radius R, we have σ = 1 πR 2 . Therefore, we can get χ(Gn) n 1−β a.s. −→ √ 3 2πR 2 γ 2 . • If β = 1, then nd 2 th,n = γ 2 +r 2 0 n 1−2β which converges to a constant asymptotically, since 1− 2β < 0. This enables us to use part (ii) of Theorem 3.2, since n − γ 2 +r 2 0 n 1−2β lnn for all > 0. Therefore, we have χ(Gn) lnn/ln(lnn) a.s. −→ 1. • If β > 1, then nd 2 th,n =γ 2 n −(β−1) +r 2 0 n −(2β−1) , where 2β− 1>β− 1> 0. Thus, we can make use of part (i) of Theorem 3.2 to get P χ(G n )−→ j 1 β−1 + 1 2 k or χ(G n )−→ j 1 β−1 + 1 2 k + 1 = 1. The proof of Theorem 3.1 then follows immediately from Lemmas 3.1, 3.3 and 3.4 and also the fact that the continuous function f(x) = 1 x preserves almost-sure convergence (continuous mapping theorem [47]). 3.1.2.2 Impact of Rayleigh Fading on the Capacity Analysis One of the most important phenomena in wireless networks is the concept of channel fading. Even though fading seems to be a detrimental aspect of wireless networks, it can also be helpful if it is viewed in a careful way. Probably the most well-known example for this is 42 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) receive diversity at multi-antenna receivers, where we can make use of independently faded signals to combine them in the best way, leading to an improvement in the received SNR. Hence, it would be interesting to figure out how fading can affect the results we derived so far on the fraction of the capacity region that ITLinQ can achieve. In this section, we focus on this problem, considering the same model for the spatial location of the nodes as in Section 3.1.2 with the difference that here, we consider the squared magnitude of the channel gain at distancer to beg 0 r −α whereg 0 represents the Rayleigh fade of the channel modeled as an exponential random variable with normalized mean of 1. We consider a slow fading scenario (i.e., block fading), where the rate of change of the channel characteristics is much smaller than the rate of change of the transmitted signal. Hence, the channel fade g 0 remains fixed during the transmission within each block of communication (which corresponds to a scheduling phase of ITLinQ) and changes i.i.d from one block to the next. The definition of information-theoretic independent set (ITIS) still remains the same as before, i.e., within each block of communication with revisited channel gain values (modeled as g 0 r −α ), a subsetS⊆{1,...,n} in which for any link i∈S condition (3.2) is satisfied is an ITIS in that block. Obviously, introducing Rayleigh fading into the channel model adds another source of randomness in the analysis of the fraction of the capacity region achieved by ITLinQ, which is due to the dependence of ITIS’s on the random fade of the channels. However, we can still make use of Lemma 3.1 to characterize the fraction of the capacity region that ITLinQ can achieve in each block of communication when Rayleigh fading is also included in the channel model. In this case, the faded interference may no longer be Gaussian, but we can circumvent this issue due to the recent result in [48]. In [48], the authors show that in a multi-user network, Gaussian noise is the worst-case additive noise in the sense that any rate tuple that can be achieved under the assumption of Gaussian noise can also be achieved under non-Gaussian additive noise of the same variance. Therefore, by treating the aggregate (non-Gaussian) noise plus faded interference at each destination as a Gaussian noise, we achieve a lower bound on the achievable rate of ITLinQ. As a result, Lemma 3.1 would still hold in a fading scenario, meaning that in each block of communication ITLinQ can achieve a fraction 1 κn of the capacity region to within a gap of log 3n κn , where now κ n , the minimum number of ITIS’s whose union contains all the links, depends both on the spatial location of the links and the realization of the fading. Characterizing the distribution of κ n in the fading scenario (even in the asymptote of n→∞) is quite challenging, hence we will use numerical evaluation in the rest of this 43 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) section to analyze the average fraction of the capacity that ITLinQ achieves in the fading scenario (i.e.,E h 1 κn i ). We consider the same network model of Section 3.1.2 for the placement of the nodes (in which source nodes are distributed uniformly within a circle of radiusR and each destination node is located within a distance r 0 n −β of its corresponding source node) and we evaluate the average fraction of the capacity region that ITLinQ is able to achieve (to within a gap) for both cases of with and without Rayleigh fading. The result is illustrated is Figure 3.5. 5 6 7 8 9 10 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Average achievable fraction of capacity region Number of links (n) β=0.5, Path Loss β=0.5, Path Loss+Fading β=1, Path Loss β=1, Path Loss+Fading β=2, Path Loss β=2, Path Loss+Fading Time Sharing Figure 3.5: Average achievable fraction of the capacity region by ITLinQ scheme with and without fading and comparison with time-sharing. By considering Figure 3.5, we can now compare the average fraction of the capacity region that ITLinQ achieves in the fading and non-fading scenario (both with the same average channel gains). We interestingly note that there is an improvement in the case where Rayleigh fading is also included (in particular, for β ≤ 1). The intuition behind this improvement can be explained as follows. Consider the ITIS condition (3.2) rewritten in the following form g ii d −α S i D i ≥ P N max j∈S\{i} g ij d −α S j D i max k∈S\{i} g ki d −α S i D k , (3.8) where∀i,j, g ij is the exponential fading random variable of the channel between source j and destination i. For fixed power and noise levels and spatial distribution of the nodes in the network, condition (3.8) reveals the opportunity that fading is providing in this case. In fact, there are specific locations of nodes in the network for which condition (3.8) cannot be satisfied in a deterministic path-loss setting. However, our numerical results show that the 44 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) randomness due to the inclusion of Rayleigh fading can help this condition to be satisfied for more subsets of the links, resulting in an improvement in the achievable fraction of the capacity region. This can, therefore, be viewed as another case where fading is helpful in terms of the system performance. For the case of β > 1, since the destination nodes get very close to their corresponding source nodes, interference is already at a very low level and therefore fading cannot be of much help and may even degrade the performance by a small amount, as depicted in Figure 3.5. Finally, using Lemma 1, we can also quantify the gap to the fraction of the capacity region that ITLinQ is able to achieve in each block of communication to be log 3n κn . For the above network model, we numerically evaluate and plot the average gap (i.e., E h log 3n κn i )in Figure 3.6. Similar to the non-fading scenario of Section 3.1.2, we note that, for the case of β≤ 1, 5 6 7 8 9 10 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Number of links (n) Average gap to the achievable fraction of capacity region (Bits) β=0.5, Path Loss β=0.5, Path Loss+Fading β=1, Path Loss β=1, Path Loss+Fading β=2, Path Loss β=2, Path Loss+Fading Figure 3.6: Comparison of the average gap to the achievable fraction of the capacity region by ITLinQ with and without fading. the average gap again does not scale with the size of the network and always remains less than 1.2 bits, independent of the number of links. However, for the case of β > 1 where a constant fraction of the capacity region can be achieved asymptotically, the gap increases with the number of links and Theorem 3.1 predicts that the increase is logarithmic with respect to n. 45 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) 3.2 A Distributed Method for Implemenation of ITLinQ In this section, we present a distributed algorithm for putting the ITLinQ scheme into practice in real-world networks. The algorithm is inspired by the FlashLinQ distributed algorithm [15] and its complexity is exactly at the same level as the FlashLinQ algorithm. However, as we will demonstrate through numerical analysis in Section 3.3.2, it significantly outperforms FlashLinQ in a certain network scenario. As mentioned in Section 3.1.1, we consider wireless networks consisting ofn source-destination pairs. In each execution of the algorithm, to address the issue of fairness among the links, we first permute the links randomly and reindex them from 1 to n based on the realization of the random permutation, as also done in [15]. This new indexing of the links corresponds to a priority order of the links: linki has higher priority than linkj ifi<j,∀i,j∈{1,...,n}. Then, link 1 is always scheduled to transmit at the current time frame and for the remain- ing links, each link is scheduled if it does not cause and receive “too much” interference to and from the higher priority links. The conditions for defining the level of “too much” interference for link j∈{2,...n} are as follows, where η is a design parameter: • At D j , the following conditions must be satisfied: INR ji ≤ SNR η j , ∀i<j, (3.9) which imply that D j does not receive too much interference from higher-priority links. • At S j , the following conditions must be satisfied: INR ij ≤ SNR η j , ∀i<j, (3.10) which imply that S j does not cause too much interference at higher-priority links. As it is clear, there are two major differences here with respect to the FlashLinQ scheduling conditions: The first difference is that instead of considering the raw fraction SIR = SNR INR , here we are considering an exponent for the SNR term, which is completely inspired by the condition for the optimality of TIN (3.1). The second difference is that in condition (3.10), the outgoing interference of each link is compared to its own SNR rather than other links’ SNR’s. This is also inspired by the TIN-optimality condition (3.1). 46 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) In fact, if the parameter η is set to η = 0.5, then conditions (3.9) and (3.10) imply that the TIN-optimality condition (3.1) is satisfied at link j. This means that link j can safely be added to the information-theoretic independent subset of higher priority links and get scheduled to transmit in the current time frame. This algorithm, therefore, seeks to find the largest possible distributed information-theoretic independent subset based on the priority ordering of the links. However, it is clear that selectingη = 0.5 might be too pessimistic and restrictive, and may prevent some links which cause and receive low levels of interference from being scheduled. Therefore, we will leave this variable as a design parameter, and as we will see in Section 3.3.2, tuning this parameter can indeed improve the achievable sum-rate by this scheduling algorithm. The remaining question is: How can the sources and destinations check whether their pertinent conditions are satisfied? This can be done by a simple signaling mechanism which is inspired by the FlashLinQ algorithm [15] and is a two-phase process, in each of which we assume that each link uses its own frequency band and transmissions are interference-free: • In the first phase, all the sources transmit signals at their full power P . The des- tinations will receive their own desired signals and also all the interfering signals in separate frequency bands. Then, the destinations estimate their received SNR’s and INR’s and check if their desired conditions (3.9) are satisfied. This phase is the same as that of FlashLinQ [15]. • In the second phase, contrary to the “inverse power echo” mentioned in the FlashLinQ algorithm [15], the destinations also transmit signals at the same power level P of the sources. Similar to the first phase, in this phase all the sources can estimate the value of their desired SNR’s and INR’s in order to verify the validity of condition (3.10). Remark 3.6. Clearly, power control at the transmitters may lead to an improvement in the performance of the scheme. However, due to the complication in implementing power control among the links in a distributed way, we disregard it in our scheme and use full power at all the transmitters. See, e.g., [49] on power control algorithms in D2D underlaid cellular networks. As it is obvious, the complexity of our distributed signaling mechanism is completely com- parable to that of the FlashLinQ algorithm. 47 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) 3.3 Numerical Analysis In this section, we numerically analyze the performance of distributed ITLinQ in two distinct settings. First, in Section 3.3.1 we assess the performance of distributed ITLinQ under the model developed in Section 3.1.2. Furthermore, in Section 3.3.2 we compare the performance of distributed ITLinQ with FlashLinQ in a model similar to the one considered in [15]. For concreteness, the algorithm used in this section for the performance evaluation of dis- tributed ITLinQ is illustrated in pseudo-code format in Algorithm 3.1. Here, we assume that through multiple iterations of the training mechanism introduced in Section 3.2, each link is aware of the values of its own SNR and all its incoming and outgoing INR’s and it is also aware of the active higher-order links. We assume that the knowledge of the active higher-order links is also available to each link in implementing the FlashLinQ scheme. For Algorithm 3.1 Implementation of Distributed ITLinQ 1: initialize active(1) = 1, active(j) = 0, ∀j = 2,...,n; 2: for j = 2,...,n 3: S j ={i :i≤j and active(i) = 1} 4: if INR ji ≤MSNR η j at D j ,∀i∈S j 5: flag D j =1; 6: endif 7: if INR ij ≤MSNR η j at S j ,∀i∈S j 8: flag S j =1; 9: endif 10: active(j) =flag D j . flag S j ; 11: end 12: return active the implementation of ITLinQ distributively, we also consider a second tuning parameter M which adds more flexibility to our scheme. This parameter can in general be tuned to optimize the performance of the algorithm in any network setting. For the results of this section, We will set M to be equal to 25 dB. Algorithm 3.1 returns a vector active of length n which specifies whether or not each link should be scheduled. In particular, for any j∈{1,...,n}, link j is scheduled if and only if active(j) = 1. 48 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) 3.3.1 Performance of Distributed ITLinQ under the Model of Section 3.1.2 In this section, we analyze the performance of distributed ITLinQ under the model of Section 3.1.2. In Section 3.1.2 we characterized the fraction of the capacity region that ITLinQ can achieve (to within a gap) with full knowledge of all the channel gains for an asymptotically large number of links. One may wonder how well ITLinQ can perform under this model for finite number of links where each link only has a local knowledge of the channel gain values. This provides the motivation for studying distributed ITLinQ in this setting. For the sake of numerical analysis in this section, we assume that the sources are distributed uniformly within a circle of radius R = 10km and each destination is assumed to be uni- formly distributed in a circle of radius r n = r 0 n −β around its corresponding source node where r 0 is chosen to be equal to 1km. The squared channel gain at distance r is taken to be equal to r −2.5 (the constant g 0 is assumed to be normalized to 1). The transmit power is taken to be 10 dBm and the additive white Gaussian noise variance at the destination nodes is set to -110 dBm. We have evaluated the sum-rate achievable by distributed ITLinQ in this setting with the value of η set to 0.5 and β taking values 0.5, 1 and 2. The result is illustrated in Figure 3.7. For the sake of comparison, we have also plotted the average sum-rate that can be 10 20 30 40 50 60 70 80 90 100 0 1000 2000 3000 4000 5000 6000 7000 Number of links (n) Average achievable sum−rate (Bits/Sec/Hz) Distributed ITLinQ, β=0.5 Distributed ITLinQ, β=1 Distributed ITLinQ, β=2 TDMA, β=0.5 TDMA, β=1 TDMA, β=2 Figure 3.7: Performance comparison of distributed ITLinQ with time-sharing under the model of Section 3.1.2. achieved by time-sharing among the links. As the figure demonstrates, there is a huge 49 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) sum-rate improvement by using distributed ITLinQ over time-sharing. Moreover, while our theoretical analysis in Theorem 3.1 characterizes the fraction of the capacity region achievable by fully-centralized ITLinQ, what we observe in Figure 3.7 is that the network can still enjoy the significant sum-rate improvement that a distributed implementation of ITLinQ can provide. 3.3.2 Performance Comparison of the distributed ITLinQ and FlashLinQ In this section, we will illustrate the performance of our distributed algorithm and compare it with FlashLinQ through numerical analysis. We drop n links randomly in a 1km× 1km square. The length of each link, which is the distance between its corresponding source and destination, is taken to be a uniform random variable in the interval [2, 65m]. As in [15], we use the carrier frequency of 2.4 GHz and a bandwidth of 5 MHz. The noise power spectral density is considered to be -184 dBm/Hz. The transmit power is set to 20 dBm. Moreover, the channel follows the LoS model in ITU-1411. In particular, if the base station antenna height is denoted by h b , the mobile station antenna height is denoted by h m and the transmission wavelength is denoted byλ, then the transmission loss (in dB) at distance d is taken to be equal to L =L bp + 6 +      20 log 10 d R bp if d≤R bp 40 log 10 d R bp if d>R bp , where R bp = 4h b hm λ denotes the breakpoint distance and L bp = 20 log 10 λ 2 8πh b hm denotes the basic transmission loss at the break point. As in [15], we assume all the antenna heights to be equal to 1.5m, alongside with a log-normal shadowing with standard deviation of 10 dB. The antenna gain per device is taken to be -2.5 dB and the noise figure is assumed to be 7 dB. Figure 3.8a demonstrates the sum-rate achievable by the distributed ITLinQ scheme for different values of η and its comparison to FlashLinQ. The implementation of FlashLinQ follows the same steps as in [15] and in particular, the threshold values γ TX and γ RX are taken to be equal to 9 dB. As the figure illustrates, tuning the parameter η can lead to considerable gains over FlashLinQ. For the case of η = 0.5, in which conditions (3.9) and (3.10) are sufficient for the optimality of TIN (to within a constant gap), distributed ITLinQ exhibits over 28% gain compared to FlashLinQ for 4096 links. Interestingly, setting η = 0.7 50 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Number of links (n) Average achievable sum−rate (Bits/Sec/Hz) Distributed ITLinQ, η=0.5 Distributed ITLinQ, η=0.6 Distributed ITLinQ, η=0.7 Distributed ITLinQ, η=1 FlashLinQ No Scheduling (a) 0 200 400 600 800 1000 1200 1400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sum−rate (Bits/Sec/Hz) Probability Distributed ITLinQ, η=0.5 Distributed ITLinQ, η=0.6 Distributed ITLinQ, η=0.7 Distributed ITLinQ, η=1 FlashLinQ (b) Figure 3.8: (a) Comparison of the sum-rate performance of distributed ITLinQ, FlashLinQ and the no-scheduling case, and (b) Comparison of the cumulative distribution function of the average link rate achieved by distributed ITLinQ and FlashLinQ in a network of 1024 links. results in more than 110% gain over FlashLinQ for 4096 links. However, as we increase η to 1, more and more links get scheduled which results in a degradation in the overall performance. As a baseline, the achievable sum-rate when there is no scheduling (i.e., all the links operate simultaneously) is also plotted in Figure 3.8a. Moreover, in the same setting, we also study the cumulative distribution function (CDF) of the sum-rate in a network of 1024 links. The result is depicted in Figure 3.8b. Again, the 51 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) same trend occurs in this plot, showing that distributed ITLinQ, especially for the value of η = 0.7, can result in considerable uniform gain compared to the sum-rate achievable by FlashLinQ. For instance, with 50% probability, the sum-rate achieved by FlashLinQ is less than 540 bits/sec/Hz while with the same probability, the sum-rate achieved by distributed ITLinQ is less than 928 bits/sec/Hz. Another natural aspect of distributed scheduling schemes that is of considerable importance is the issue of fairness among the links. In particular, the scheduling scheme should take care of all links fairly, regardless of them being strong or weak. It can be seen that the distributed ITLinQ scheme favors strong links more than weak links. To highlight this issue, a network with two links AB and CD is shown in Figure 3.9. In this figure, link AB is a A B C D Figure 3.9: An example of the case where distributed ITLinQ might be unfair. low-SNR link and link CD is a high-SNR link. Moreover, Destination node B suffers from strong interference due to the source node C. To see why ITLinQ may be unfair in such a scenario regardless of the priority of the links, consider the following two cases: • If link AB has a lower priority than link CD, link CD is first scheduled. Then, destination B checks its scheduling condition INR BC ≤ MSNR η BA and with a high probability may find that it is not satisfied (since the interference from C is strong compared to the signal power received from A). This will prevent link AB from being scheduled. • If link AB has a higher priority than link CD, it will be scheduled first. Then, since both destination node D is receiving a low amount of interference from A (compared to the signal power from D) and source node C is causing a low amount of interference at B (compared to the signal power it delivers to D), link CD will also get scheduled and hurts the transmission of link AB. Therefore, in both cases, the low-SNR link AB will not get a high rate, if any. This motivates a modification of the distributed ITLinQ scheme to account for this issue. To 52 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) this end, we present a fair version of distributed ITLinQ as follows. Inspired by the example in Figure 3.9, in the fair ITLinQ algorithm, the high-SNR links should get scheduled in a more restrictive way. This can be done by decreasing the parameters η (and M) in the scheduling condition for the outgoing interference of high-SNR links. In general, η (and M) need to be a descending function of SNR. However, one simple solution would be to choose a threshold SNR th such that if the SNR of a link is higher than this threshold, η and M are altered to decreased values ¯ η and ¯ M. The pseudo-code for the fair ITLinQ scheme is presented in Algorithm 3.2. To assess the performance of fair ITLinQ in terms Algorithm 3.2 Fair ITLinQ 1: initialize active(1) = 1, active(j) = 0, ∀j = 2,...,n; 2: for j = 2,...,n 3: S j ={i :i≤j and active(i) = 1} 4: if INR ji ≤MSNR η j at D j ,∀i∈S j 5: flag D j =1; 6: endif 7: if SNR j ≤ SNR th 8: if INR ij ≤MSNR η j at S j ,∀i∈S j 9: flag S j =1; 10: endif 11: else 12: if INR ij ≤ ¯ MSNR ¯ η j at S j ,∀i∈S j 13: flag S j =1; 14: endif 15: endif 16: active(j) =flag D j . flag S j ; 17: end 18: return active of fairness, we have numerically evaluated the CDF of the link rates (averaged over both priorities and locations) for a network with 1024 links under the same model as the one mentioned in the beginning of this Section. The threshold value for high-SNR is chosen to be SNR th = 110 dB and the modified parameters are set to ¯ η = 0.6 and ¯ M = 20 dB. Figure 3.10 compares the CDF of the average link rate by distributed ITLinQ (with η = 0.7), fair ITLinQ and FlashLinQ. As the figure illustrates, fair ITLinQ can improve the tail distribution of distributed ITLinQ and perform as well as FlashLinQ in terms of fairness. This certainly does not come for free and in fact, there is a trade-off between fairness and the achievable sum-rate. The sum-rate achieved by fair ITLinQ is compared with FlashLinQ in Figure 3.11. As the figure illustrates, for the case of 4096 links, the sum-rate gain of fair 53 Chapter 3. Spectrum Sharing via Information-Theoretic Link Scheduling (ITLinQ) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Link rate (Bits/Sec/Hz) Probability Distributed ITLinQ, η=0.7 Fair ITLinQ FlashLinQ Figure 3.10: Comparison of the average link rate CDF of distributed ITLinQ, fair ITLinQ and FlashLinQ for a network with 1024 links. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 200 400 600 800 1000 1200 1400 Number of links (n) Average achievable sum−rate (Bits/Sec/Hz) Fair ITLinQ FlashLinQ Figure 3.11: Comparison of the sum-rate achievable by fair ITLinQ and FlashLinQ. ITLinQ over FlashLinQ is more than 50%. For more information regarding the software implementation of ITLinQ, the reader is referred to [50]. 54 Chapter 4 Fundamental Limits of Cache-Aided Interference Management Over the last decade, video delivery has emerged as the main driving factor of the wireless traffic. In this context, there is often a large library of pre-recorded content (e.g. movies), out of which, users may request to receive a specific file. One way to reduce the burden of this traffic is to employ memories distributed across the networks and closer to the end users to prefetch some of the popular content. This can help system to deliver the content with higher throughput and less delay. As a result, there have been significant interests in both academia and industry in char- acterizing the impact of caching on the performance of communication networks (see, e.g. [53–65]). In particular, in a network with only one transmitter broadcasting to several receivers, it was shown in [54] that local delivery attains only a small fraction of the gain that caching can offer, and by designing a particular pattern in cache placement at the users and exploiting coding in delivery, a significantly larger global throughput gain can be achieved, which is a function of the entire cache throughout the network. This also demonstrates that the gain of caching scales with the size of the network. As a follow-up, This chapter is mainly taken from [51,52], coauthored by the author of this dissertation. 55 Chapter 4. Fundamental Limits of Cache-Aided Interference Management this work has been extended to the case of multiple transmitters in [55], where it was shown that the gain of caching can be improved if several transmitters have access to the entire library of files. Caching at the transmitters was also considered in [56, 57] and used to in- duce collaboration between transmitters in the network. It is also shown in [59] that caches at the transmitters can improve load balancing and increase the opportunities for interfer- ence alignment. More recently, the authors in [60] evaluated the performance of cellular networks with edge caching via a hypergraph coloring problem. Furthermore, in [61], the authors studied the problem of maximizing the delivery rate of a fog radio access network for arbitrary prefetching strategies. In this chapter, we consider a general network setting with caches at both transmitters and receivers, and demonstrate how one can utilize caches at both transmitters and receivers to manage the interference and enhance the system performance in the physical layer. In particular, we consider a library ofN files and a wireless network withK T transmitters and K R receivers, in which each transmitter and each receiver is equipped with a cache memory of a certain size. In particular, each transmitter and each receiver can cache up to M T and M R files, respectively. The system operates in two phases. The first phase is called the prefetching phase, where each cache is populated up to its limited size from the content of the library. This phase is followed by a delivery phase, where each user reveals its request for a file in the library. The transmitters then need to deliver the requested files to the receivers. Note that in the prefetching phase, the system is still unaware of the files that the receivers will request in the delivery phase. The goal is to design the cache contents in the prefetching phase and communication scheme in the delivery phase to achieve the maximum throughput for arbitrary set of requested files. Due to their practical appeal, in this work we focus on one-shot linear delivery strategies. Interestingly, many of the previous works on caching have relied on one-shot schemes for content delivery (see, e.g. [56,60]). Our main result in this chapter is the characterization of the one-shot linear sum degrees- of-freedom (sum-DoF) of the network, i.e., number of the receivers that can be served interference-free simultaneously, within a factor of 2 for all system parameters. In fact, we show that the one-shot linear sum-DoF of min n K T M T +K R M R N ,K R o is achievable, and this is within a factor of 2 of the optimum. This result shows that the one-shot linear sum- DoF of the network grows linearly with the aggregate cache size in the network (i.e., the cumulative memory available at all nodes). It also implies that caches at the transmitters’ side are equally valuable as the caches on the receivers’ side in the one-shot linear sum-DoF 56 Chapter 4. Fundamental Limits of Cache-Aided Interference Management of the network. Our result, therefore, establishes a fundamental limit on the performance of one-shot delivery schemes for cache-aided interference management. To achieve the aforementioned sum-DoF, we propose a particular pattern in cache placement so that each piece of each file in the library is available in the caches of K T M T N transmitters and K R M R N receivers. Once caching is done this way, we can show that for delivering any set of requested contents to the receivers, min n K T M T +K R M R N ,K R o of the receivers can be served at each time, interference-free. This gain is achieved by simultaneously exploiting the opportunity of collaborative interference cancellation (i.e., zero-forcing) at the transmitters’ side and opportunity of eliminating known interference contributions at the receivers’ side. The first opportunity is created by caching the pieces of each file at several transmitters. The second opportunity is available since pieces of a file requested by one user has been cached at some other receivers, and thus do not impose interference at those receivers effectively. Our proposed cache placement pattern maximizes the overall gain achieved by these opportunities for any arbitrary set of receiver requests and this gain can be achieved even with a simple one-shot linear delivery scheme. Moreover, we demonstrate that our achievable sum-DoF is within a factor of 2 of the optimal sum-DoF for one-shot linear schemes. To prove the outer bound, we take a four-step approach in order to lower bound the number of communication blocks needed to deliver any set of requested files to the receivers. First, we show that the network can be converted to a virtual MISO interference channel in each block of communication. Using this conversion, we next write an integer optimization problem for the minimum number of communication blocks needed to deliver a fixed set of requests for a given caching realization. We then show how we can focus on average demands instead of the worst-case demands to derive an outer optimization problem on the number of communication blocks optimized over the caching realizations. Finally, we present a lower bound on the value of the aforementioned optimization problem, which leads to the desired upper bound on the one-shot linear sum- DoF of the network. This result illustrates that in this setting, caches at transmitters’ side are equally valuable as caches at receivers’ side. It also shows that caching offers a throughput gain that scales linearly with the size of the network. The rest of the chapter is organized as follows. We present the problem formulation in Section 4.1. We state the main result in Section 4.2. We prove the achievability of our main result in Section 4.3 and the converse in Section 4.4. 57 Chapter 4. Fundamental Limits of Cache-Aided Interference Management 4.1 Problem Formulation In this section, we first provide a high-level description of the problem setting and the main parameters in the system model, and then we present a detailed description of the problem formulation. 4.1.1 Problem Overview Consider a wireless network, as illustrated in Figure 4.1, with K T transmitters and K R receivers, and also a library ofN files, each of which containsF packets, where each packet is a vector of B bits. Each node in the network is equipped with a local cache memory of a certain size that can be used to cache contents arbitrarily from the library before the receivers reveal their requests and communication begins. In particular, each transmitter and each receiver is equipped with a cache of size M T F and M R F packets, respectively. Tx1 Tx2 TxKT Rx1 Rx2 RxKR N files F packets b b b Library Transmitter Receiver packets MRF QKR packets MRF Q2 Q1 MRF packets P1 MTF packets P2 PKT caches caches MTF packets MTF packets Figure 4.1: Wireless network with K T transmitters and K R receivers, where each trans- mitter and each receiver caches up to M T F packets andM R F packets, respectively, from a library of N files, each composed of F packets. We assume that the system operates in two phases, namely the prefetching phase and the delivery phase. In the prefetching phase, each node can cache contents arbitrarily from the library subject to its cache size constraint. In particular, each transmitter selects up to M T F packets out of the entire library to store in its cache, and each receiver selects up to M R F packets out of the entire library to store in its cache. In the delivery phase, each receiver requests an arbitrary file from the library. Since each receiver may have cached 58 Chapter 4. Fundamental Limits of Cache-Aided Interference Management parts of its desired file in the prefetching phase, the transmitters need to deliver the rest of the requested packets to the receivers over the wireless channel. We assume that at each time, the transmitters employ a one-shot linear scheme, where a subset of requested packets are selected to be delivered interference-free to a corresponding subset of receivers. Each transmitter transmits a linear combination of the subset of the selected packets which it has cached in the prefetching phase. The interference is cancelled with the aid of cached contents as follows. Since each requested packet may be cached at multiple transmitters, the transmitters can collaborate in order to zero-force the outgoing interference of that packet at some of the unintended receivers. Moreover, the receivers can also use their cached packets as side information to eliminate the remaining incoming interference from to undesired packets. Our objective is to design a cache placement scheme and a delivery scheme which maximize the number of packets that can be delivered at each time interference-free. In this setting, we define the one-shot linear sum-degrees of freedom as the ratio of the number of delivered packets over the number of blocks needed for communicating those packets for any set of receiver demands. Finally, we define the one-shot linear sum-DoF of the network, denoted by DoF ∗ L,sum (N,M T ,M R ), as the maximum achievable one-shot linear sum-DoF over all caching realizations. 4.1.2 Detailed Problem Description We consider a discrete-time additive white Gaussian noise channel, as illustrated in Figure 4.1, withK T transmitters denoted by{Tx i } K T i=1 andK R receivers denoted by{Rx i } K R i=1 . The communication at time t over this channel is modeled by Y j (t) = K T X i=1 h ji X i (t) +Z j (t), (4.1) where X i (t)∈ C denotes the signal transmitted by Tx i ,i∈ [K T ],{1,...,K T } and Y j (t) denotes the receive signal by Rx j ,j∈ [K R ]. Moreover,h ji ∈C denotes the channel gain from Tx i to Rx j , assumed to stay fixed over the course of communication, and Z j (t) denotes the additive white Gaussian noise at Rx j at time slot t, distributed asCN (0, 1). The transmit signal at Tx i ,i∈ [K T ],{1,...,K T } is subject to the power constraintE |X i (t)| 2 ≤P . 59 Chapter 4. Fundamental Limits of Cache-Aided Interference Management We assume that each receiver will request an arbitrary file out of a library of N files {W n } N n=1 , which should be delivered by the transmitters. Each file W n in the library contains F packets{w n,f } F f=1 , where each packet is a vector of B bits; i.e., w n,f ∈ F B 2 . Furthermore, we assume that each node in the network is equipped with a cache memory of a certain size that can be used to cache arbitrary contents from the library before the receivers reveal their requests and communication begins. In particular, each transmitter and each receiver is equipped with a cache of size M T F and M R F packets, respectively. We assume that the network operates in two phases, namely the prefetching phase and the delivery phase, which are described in more detail as follows. Prefetching Phase: In this phase, each node can store an arbitrary subset of the packets from the files in the library up to its cache size. In particular, each transmitter Tx i chooses a subsetP i of theNF packets in the library, where|P i |≤M T F , to store in its cache. Likewise, each receiver Rx i stores a subsetQ i of the packets in the library, where|Q i |≤ M R F . Caching is done at the level of whole packets and we do not allow breaking the packets into smaller subpackets. Also, this phase takes place unaware of the receivers’ future requests. Delivery Phase: In this phase, each receiver Rx j ,j∈ [K R ], reveals its request for an ar- bitrary file W d j from the library for some d j ∈ [N]. We let d = [d 1 ... d K R ] T denote the demand vector. Depending on the demand vector d and the cache contents, each receiver has already cached some packets of its desired file and there is no need to deliver them. The transmitters will be responsible for delivering the rest of the requested packets to the receivers. In order to make sure that any piece of content in the library is stored at the cache of at least one transmitter in the network, we assume that the transmitter cache size satisfies K T M T ≥N. Each transmitter first employs a random Gaussian coding scheme ψ : F B 2 → C ˜ B of rate logP + o(logP ) to encode each of its cached packets into a coded packet composed of ˜ B complex symbols, so that each coded packet carries one degree-of-freedom (DoF). We denote the coded version of each packet w n,f in the library by ˜ w n,f ,ψ(w n,f ). Afterwards, the communication takes place over H blocks, each of length ˜ B time slots. In each block m∈ [H], the goal is to deliver a subset of the requested packets, denoted byD m , to a subset of receivers, denoted byR m , such that each packet inD m is intended to exactly one of the receivers inR m . In addition, the set of transmitted packets in all blocks and the cache 60 Chapter 4. Fundamental Limits of Cache-Aided Interference Management contents of the receivers should satisfy {w d j ,f } F f=1 ⊂ H [ m=1 D m ! ∪Q j , ∀j∈ [K R ], (4.2) which implies that for any receiver Rx j ,j∈ [K R ], each of its requested packets should be either transmitted in one of the blocks or already stored in its own cache. In each blockm∈ [H], we assume a one-shot linear scheme where each transmitter transmits an arbitrary linear combination of a subset of the coded packets inD m that it has cached. Particularly, Tx i ,i∈ [K T ] transmits x i [m]∈C ˜ B , where x i [m] = X (n,f): w n,f ∈P i ∩Dm v i,n,f [m] ˜ w n,f , (4.3) and v i,n,f [m]’s denote the complex beamforming coefficients that Tx i uses to linearly com- bine its coded packets in block m. On the receivers’ side, the received signal of each receiver Rx j ∈R m in block m, denoted by y j [m]∈C ˜ B , can be written as y j [m] = K T X i=1 h ji x i [m] + z j [m], (4.4) where z j [m]∈ C ˜ B denotes the noise vector at Rx j in block m. Then, receiver Rx j will use the contents of its cache to cancel (subtract out) the interference of some of undesired packets inD m , if they exist in its cache. In particular, each receiver Rx j ∈R m , forms a linear combinationL j,m , as L j,m (y j [m], ˜ Q j ) (4.5) to recover ˜ w d j ,f ∈D m , where ˜ Q j denotes the set of coded packets cached at receiver Rx j . The communication in block m∈ H to transmit the packets inD m is successful, if there exist linear combinations (4.3) at the transmitters’ side and (4.5) at receivers’ side, such that for all Rx j ∈R m , L j,m (y j [m], ˜ Q j ) = ˜ w d j ,f + z j [m]. (4.6) 61 Chapter 4. Fundamental Limits of Cache-Aided Interference Management The channel created in (4.6) is a point-to-point channel, whose capacity is logP +o(logP ). Hence, since each coded packet ˜ w d j ,f is coded with rate logP +o(logP ), it can be de- coded with vanishing error probability as B increases. We assume that the communication continues forH blocks until all the desired packets are successfully delivered to all receivers. Since each packet carries one degree-of-freedom, the one-shot linear sum-degrees-of-freedom (sum-DoF) of|D m | is achievable in each block m∈ [H]. This implies that throughout the H blocks of communication, the one-shot linear sum-DoF of S H m=1 Dm H is achievable. Therefore, for a given caching realization, we define the one-shot linear sum-DoF to be maximum achievable one-shot linear sum-DoF for the worst case demands; i.e., DoF {P i } K T i=1 ,{Q i } K R i=1 L,sum = inf d sup H,{Dm} H m=1 H S m=1 D m H . (4.7) This leads us to the definition of the one-shot linear sum-DoF of the network as follows. Definition 4.1. For a network with a libraryN files, each containingF packets, and cache size of M T and M R files at each transmitter and receiver, respectively, we define the one- shot linear sum-DoF of the network as the maximum achievable one-shot linear sum-DoF over all caching realizations; i.e., DoF ∗ L,sum (N,M T ,M R ) = sup {P i } K T i=1 ,{Q i } K R i=1 DoF {P i } K T i=1 ,{Q i } K R i=1 L,sum (4.8) s.t. |P i |≤M T F, ∀i∈ [K T ] (4.9) |Q i |≤M R F, ∀i∈ [K R ], (4.10) where DoF {P i } K T i=1 ,{Q i } K R i=1 L,sum is defined in (4.7). 4.2 Main Result and its Implications In this section, we present our main result on the one-shot linear sum-DoF of the network and its implications. Theorem 4.1. For a network with a library ofN files, each containingF packets, and cache size of M T and M R files at each transmitter and each receiver, respectively, the one-shot 62 Chapter 4. Fundamental Limits of Cache-Aided Interference Management linear sum-DoF of the network, as defined in Definition 4.1, satisfies min K T M T +K R M R N ,K R ≤ DoF ∗ L,sum (N,M T ,M R )≤ min 2 K T M T +K R M R N ,K R , (4.11) for sufficiently large F . In the following, we highlight the implications of Theorem 4.1 and its connections to some prior works: 1. (Within a factor of 2 characterization) The upper bound in (4.11) is within a factor of 2 of the lower bound in (4.11). Therefore, Theorem 4.1 characterizes the one-shot linear sum-DoF of a cache-aided wireless network to within a factor of 2, for all system parameters. 2. (Aggregate cache size matters) The one-shot linear sum-DoF characterized in Theorem 4.1 is proportional to the aggregate cache size that is available throughout the network, even-though these caches are isolated. 3. (Equal contribution of transmitter and receiver caches) Perhaps interestingly, the caches at both sides of the network, i.e., the transmitters’ side and the receivers’ side, are equally valuable in the achievable one-shot linear sum-DoF of the network. Note that in practice, size of each transmitter’s cache, M T , could be large. However, the number of transmitters (e.g., base stations)K T is often small. On the other hand, size of the cache M R at the receivers (e.g., cellphones) is small, whereas the number of receivers K R is large. Therefore K T M T could be comparable with K R M R . Our result in Theorem 4.1 shows that neither caches at the transmitters nor caches at the receivers should be ignored. 4. (Linear scaling of DoF with network size) LettingK T =K R =K, we observe that the one-shot linear sum-DoF scales linearly with the number of users in a fully-connected interference channel. Note that without caches, the one-shot linear sum-DoF of a fully-connected interference channel is bounded by 2, as shown in [66]. Hence, caching enables linear growth of the DoF without the need for more complex physical layer schemes. 5. (Role of transmitter and receiver caches) As we will show in Section 4.3, in (4.11), K T M T N represents the contribution of collaborative zero-forcing at the transmitters’ 63 Chapter 4. Fundamental Limits of Cache-Aided Interference Management side, and K R M R N represents the gain of canceling the known interference at the re- ceivers’ side. 6. (Connection to single-server coded caching [54]) A special case of our network model is the case with a single transmitter, which was previously considered in [54]. In this case, it can be shown that a sum-DoF of min n 1 + K R M R N ,K R o is achievable, which is equivalent to the global caching gain introduced in [54], indicating the number of receivers in the network that can be served simultaneously, interference-free. Hence, our result subsumes the result of [54] by generalizing it for the case of multiple trans- mitters. 7. (Connection to multi-server coded caching [55]) Another special case of our network model is the case where each transmitter has space to cache the entire library; i.e., M T = N. This case was previously considered in [55] and it can be verified that in this case, a sum-DoF of min n K T + K R M R N ,K R o is achievable. Hence, our result can also be viewed as a generalization of the result in [55] where the cache size of each transmitter may be arbitrarily smaller than the entire library size. Remark 4.1. In practice, the files in the library have nonuniform demands and some of them are more popular than the rest. In this case, our algorithm can be used to cache and deliver the N most popular files. If a user requests one of the remaining less popular files, it can be directly served by a central base station. The parameter N can be tuned, based on the popularity pattern of the contents, in order to attain the best average performance. Example 4.1. As an illustrative example, consider a cellular network with 5 base stations as transmitters, each with a 10 TB memory and 100 cellphones as receivers, each with a 32 GB memory. Moreover, consider a library of the 1000 most popular movie titles on Netflix, each with size of 5 GB. Then, Theorem 4.1 implies that at each time, around 11 cellphones can be served simultaneously interference-free, no matter what their demands are, in contrast to the naive time-sharing scheme, where at each time only 1 cellphone can be served. The rest of the chapter is devoted to the proof of Theorem 4.1. In particular, we illustrate the achievable scheme in Section 4.3 and we present the converse argument in Section 4.4. 64 Chapter 4. Fundamental Limits of Cache-Aided Interference Management 4.3 Achievable Scheme In this section, we prove the achievability of Theorem 4.1 by presenting an achievable scheme which utilizes the caches at the transmitters and receivers efficiently to exploit the zero- forcing and interference cancellation opportunities at the transmitters’ and receivers’ sides, respectively. In particular, we introduce a prefetching strategy which maximizes the gains attained by the aforementioned opportunities in the delivery phase, no matter what the receiver demands are. We first explain our achievable scheme through a simple, illustrative example and then proceed to mention our general achievable scheme. 4.3.1 Description of the Achievable Scheme via an Example Consider a system withK T = 3 transmitters andK R = 3 receivers, where each transmitter has space to cache M T = 2 files and each receiver has space to cache M R = 1 file. The library has N = 3 files W 1 =A, W 2 =B, and W 3 =C, each consisting of F packets. In the following, we will describe the prefetching and delivery phases in detail. Prefetching Phase: In this phase, each fileW n ,n∈ [3] in the library is broken into 3 2 3 1 = 9 disjoint subfiles W n,T,R for anyT ⊆ [K T ] = [3] andR⊆ [K R ] = [3] such that|T| = 2 and |R| = 1, where each subfile consists of F/9 packets. Each subfile W n,T,R is then stored at the caches of the two transmitters inT and the single receiver inR. For example, file A is broken into 9 subfiles as follows: A 12,1 ,A 12,2 ,A 12,3 ,A 13,1 ,A 13,2 ,A 13,3 ,A 23,1 ,A 23,2 ,A 23,3 , whereA 12,1 is stored at transmitters Tx 1 and Tx 2 as well as receiver Rx 1 ,A 12,2 is stored at transmitters Tx 1 and Tx 2 as well as receiver Rx 2 , etc. We do the same partitioning for files B and C, as well. It is easy to verify that each transmitter caches 6 subfiles of each file, hence the total size of its cached content is 3∗ (6∗F/9) = 2F packets which satisfies its memory constraint. Also, each receiver caches 3 subfiles of each file and its total cached content has size 3∗(3∗F/9) =F packets, hence satisfying its memory constraint. Note that in this phase, we are unaware of receivers’ future requests. 65 Chapter 4. Fundamental Limits of Cache-Aided Interference Management Delivery Phase: In this phase, each receiver reveals its request for a file in the library. Without loss of generality, assume that receivers Rx 1 , Rx 2 and Rx 3 request files W d 1 =A, W d 2 =B and W d 3 =C, respectively. Note that each receiver has already stored 3 subfiles of its desired file in its own cache, and therefore the transmitters need to deliver the 6 remaining subfiles of each requested file. In particular, the following 18 subfiles need to be delivered by the transmitters to the requesting receivers: A 12,2 ,A 12,3 ,A 13,2 ,A 13,3 ,A 23,2 ,A 23,3 to receiver Rx 1 , B 23,3 ,B 13,1 ,B 12,3 ,B 23,1 ,B 13,3 ,B 12,1 to receiver Rx 2 , (4.12) C 13,1 ,C 23,2 ,C 23,1 ,C 12,2 ,C 12,1 ,C 13,2 to receiver Rx 3 . We now show that we can break the 18 subfiles in (4.12) into 6 sets, each containing 3 subfiles, such that the subfiles in each set can be delivered simultaneously to the receivers, interference-free. Such a partitioning is illustrated through the 6 steps in Figure 4.2, where each step takes F 9 blocks. In each step, 3 subfiles are delivered to all the receivers simul- taneously, while all the inter-user interference can be eliminated. For example, in the first step, as in Figure 4.2–(a), subfiles A 12,2 ,B 23,3 ,C 13,1 are respectively delivered to receivers Rx 1 , Rx 2 , and Rx 3 at the same time. In Figure 4.3, we show in detail how the interference is cancelled in this step. The transmit signals of transmitters Tx 1 , Tx 2 and Tx 3 can be respectively written as X 1 =−h 32 ˜ A 12,2 +h 23 ˜ C 13,1 , X 2 = h 31 ˜ A 12,2 −h 13 ˜ B 23,3 , X 3 =−h 21 ˜ C 13,1 +h 12 ˜ B 23,3 , where for any subfile W n,T,R , ˜ W n,T,R denotes its coded version. For simplicity, in this example, we ignore the power constraint at the transmitters. On the other hand, the received signals by receivers Rx 1 , Rx 2 and Rx 3 can be respectively written as Y 1 = (h 12 h 31 −h 11 h 32 ) ˜ A 12,2 + (h 11 h 23 −h 13 h 21 ) ˜ C 13,1 +Z 1 , Y 2 = (h 23 h 12 −h 22 h 13 ) ˜ B 23,3 + (h 22 h 31 −h 21 h 32 ) ˜ A 12,2 +Z 2 , Y 3 = (h 31 h 23 −h 33 h 21 ) ˜ C 13,1 + (h 33 h 12 −h 32 h 13 ) ˜ B 23,3 +Z 3 . Now, note that receivers Rx 1 , Rx 2 and Rx 3 can cancel the interference due to C 13,1 , A 12,2 , 66 Chapter 4. Fundamental Limits of Cache-Aided Interference Management and B 23,3 , respectively, since they already have each respective subfile in their own cache. Therefore, all the interference in the network can be effectively eliminated and the receivers will be able to decode their desired subfiles. Likewise, one can verify that all the receivers can receive their desired subfiles interference-free in all the 6 steps of communication depicted in Figure 4.2. (a) (b) X3 =L( ˜ B13,1, ˜ C23,2) X2 =L( ˜ A12,3, ˜ C23,2) X1 =L( ˜ A12,3, ˜ B13,1) C23,2 B13,1 A12,3 (f) X3 =L( ˜ A23,3, ˜ C13,2) X2 =L( ˜ A23,3, ˜ B12,1) X1 =L( ˜ B12,1, ˜ C13,2) C13,2 B12,1 A23,3 A23,2 B13,3 C12,1 (e) A13,3 B23,1 C12,2 X1 =L( ˜ A13,3, ˜ C12,2) X2 =L( ˜ B23,1, ˜ C12,2) X3 =L( ˜ A13,3, ˜ B23,1) (d) (c) C23,1 B12,3 A13,2 X1 =L( ˜ A13,2, ˜ B12,3) X2 =L( ˜ B12,3, ˜ C23,1) X3 =L( ˜ A13,2, ˜ C23,1) X3 =L( ˜ A23,2, ˜ B13,3) X2 =L( ˜ A23,2, ˜ C12,1) X1 =L( ˜ B13,3, ˜ C12,1) X1 =L( ˜ A12,2, ˜ C13,1) X2 =L( ˜ A12,2, ˜ B23,3) X3 =L( ˜ B23,3, ˜ C13,1) Rx3 Rx2 Rx1 Rx3 Rx2 Rx1 Rx3 Rx2 Rx1 Tx3 Tx2 Tx1 Tx3 Tx2 Tx1 Tx3 Tx2 Tx1 Tx3 Tx2 Tx1 Tx3 Tx2 Tx1 Tx3 Tx2 Tx1 Rx3 Rx2 Rx1 Rx3 Rx2 Rx1 Rx3 Rx2 Rx1 A12,2 Decoder C13,1 B23,3 Decoder A12,2 C13,1 Decoder B23,3 Decoder C23,1 Decoder A13,2 Decoder B12,3 Decoder C12,1 Decoder A23,2 Decoder B13,3 Decoder B13,1 Decoder C23,2 Decoder A12,3 Decoder B23,1 Decoder C12,2 Decoder A13,3 Decoder B12,1 Decoder C13,2 Decoder A23,3 Figure 4.2: Delivery phase for the example in Section 4.3.1 for respective requests of filesA, B andC by receivers Rx 1 , Rx 2 and Rx 3 , whereL(α,β) denotes some linear combination of α and β. In every step, each pair of transmitters collaborate to zero-force the interference due to a specific subfile at a certain undesired receiver. Moreover, each receiver also uses its cache contents to cancel the interference due to the other interfering packet. Therefore, the communication is interference-free in all 6 steps. A12,2 Rx3 Rx2 Rx1 Tx3 Tx2 Tx1 h11 h21 X1 = (−h32× ˜ A12,2) + ( h23× ˜C13,1) X2 = ( h31× ˜ A12,2) + (−h13× ˜B23,3) ZF at Rx3 ZF at Rx2 ZF at Rx1 Decoder C13,1 B23,3 Decoder A12,2 C13,1 Decoder B23,3 X3 = (−h21× ˜C13,1) + ( h12× ˜B23,3) Figure 4.3: More detailed description of the linear encoding and decoding schemes used in the delivery phase step in Figure 4.2–(a). In this step, Tx 1 and Tx 2 zero-forceA 12,2 at Rx 3 , Tx 1 and Tx 3 zero-force C 13,1 at Rx 2 , and Tx 2 and Tx 3 zero-force B 23,3 at Rx 1 . Moreover, Rx 1 , Rx 2 and Rx 3 can cancel the interference due to C 13,1 , A 12,2 , and B 23,3 , respectively, since they already have each respective subfile in their own cache. 67 Chapter 4. Fundamental Limits of Cache-Aided Interference Management Consequently, the 18 subfiles in (4.12), each of which consists of F/9 packets, are delivered to the receivers in 6 steps, each consisting of F/9 blocks. Note that our particular file splitting pattern in the prefetching phase and the particular scheduling pattern in the delivery phase allows us to maximally exploit the two gains of zero-forcing the outgoing interference on the transmitters’ side and canceling the known interference on the receivers’ side, no matter what the receiver demands are in the delivery phase. Therefore, the sum- DoF of 18∗F/9 6∗F/9 = 3 = min n K T M T +K R M R N ,K R o is achievable in this network. 4.3.2 Description of the General Achievable Scheme Our general achievable scheme is given in Algorithm 4.1. In this algorithm, we use the notation t T , K T M T N , t R , K R M R N , (4.13) and for now, we assume that t T and t R are integers. Recall that in the example in Section 4.3.1, t T = 2 and t R = 1. In the following, we will describe the prefetching and delivery phases in more detail. 4.3.2.1 Prefetching Phase For any file W n in the library, n∈ [N], we partition it into K T t T K R t R disjoint subfiles of equal sizes 1 , denoted by W n = n W n,T,R o T⊆[K T ]:|T|=t T R⊆[K R ]:|R|=t R . (4.14) Based on the above partitioning, in the prefetching phase, each transmitter Tx i stores a subsetP i of the packets in the library as described below. P i ={W n,T,R :i∈T}. (4.15) 1 Due to the assumption that F is sufficiently large, we can assume that it is an integer multiple of K T t T K R t R . 68 Chapter 4. Fundamental Limits of Cache-Aided Interference Management Algorithm 4.1 Achievable scheme for Theorem 4.1 Prefetching Phase: 1: for n = 1,...,N 2: Partition W n into K T t T K R t R disjoint subfiles{W n,T,R } T⊆[K T ],|T|=t T ,R⊆[K R ],|R|=t R of equal sizes. 3: end 4: for i = 1,...,K T 5: Tx i caches all W n,T,R for which i∈T . 6: end 7: for j = 1,...,K R 8: Rx j caches all W n,T,R for which j∈R. 9: end Delivery Phase: 10: for j∈ [K R ] 11: forT ⊆ [K T ] s.t.|T| =t T 12: forR⊆ [K R ]\{j} s.t.|R| =t R 13: partition W d j ,T,R to t R ![K R −(t R +1)]! [K R −(t R +t T )]! disjoint subfiles n W d j ,T,π,π 0 o π∈Π R π 0 ∈Π [K R ]\(R∪{j}),t T −1 of equal sizes. 14: end 15: end 16: end 17: forT ⊆ [K T ] s.t.|T| =t T 18: forR⊆ [K R ] s.t.|R| =t T +t R 19: for π∈ Π circ R 20: Each transmitter Tx i transmits a linear combination of the coded subfiles as in X i =L i,T,π n ˜ W d π(l) ,T⊕ K T (l−1),π[l+1:l+t R ],π[l+t R +1:l+t R +t T −1] : l∈ [t T +t R ],i∈T⊕ K T (l− 1) o ! using the linear combinations shown in Lemma 4.2 such that the subfiles n W d π(l) ,T⊕ K T (l−1),π[l+1:l+t R ],π[l+t R +1:l+t R +t T −1] :l∈ [t T +t R ] o are simultaneously delivered to the receivers inR interference-free. 21: end 22: end 23: end 69 Chapter 4. Fundamental Limits of Cache-Aided Interference Management Illustration 4.1. For instance, in the example network considered in Section 4.3.1, trans- mitter Tx 3 stores the following subset of packets in its cache. P 3 ={W 1,13,1 ,W 1,13,2 ,W 1,13,3 ,W 1,23,1 ,W 1,23,2 ,W 1,23,3 , W 2,13,1 ,W 2,13,2 ,W 2,13,3 ,W 2,23,1 ,W 2,23,2 ,W 2,23,3 , W 3,13,1 ,W 3,13,2 ,W 3,13,3 ,W 3,23,1 ,W 3,23,2 ,W 3,23,3 } ={A 13,1 ,A 13,2 ,A 13,3 ,A 23,1 ,A 23,2 ,A 23,3 , B 13,1 ,B 13,2 ,B 13,3 ,B 23,1 ,B 23,2 ,B 23,3 , C 13,1 ,C 13,2 ,C 13,3 ,C 23,1 ,C 23,2 ,C 23,3 }. Based on the above caching strategy, we can verify that the total number of packets cached by transmitter Tx i equals N K T − 1 t T − 1 ! K R t R ! F K T t T K R t R =NF t T K T =M T F packets, hence satisfying its memory size constraint, where K T −1 t T −1 is the number of subsetsT ⊆ [K T ] of size t T which include the transmitter index i. Likewise, in the prefetching phase, each receiver Rx j stores a subsetQ j of the packets in the library as described below. Q j ={W n,T,R :j∈R}. (4.16) Illustration 4.2. For instance, in the example network considered in Section 4.3.1, receiver Rx 2 stores the following subset of packets in its cache. Q 2 ={W 1,12,2 ,W 1,13,2 ,W 1,23,2 ,W 2,12,2 ,W 2,13,2 ,W 2,23,2 ,W 3,12,2 ,W 3,13,2 ,W 3,23,2 } ={A 12,2 ,A 13,2 ,A 23,2 ,B 12,2 ,B 13,2 ,B 23,2 ,C 12,2 ,C 13,2 ,C 23,2 }. This suggests that the total number of packets cached by receiver Rx j is equal to N K T t T ! K R − 1 t R − 1 ! F K T t T K R t R =NF t R K R =M R F packets, which also satisfies its memory size constraint. 70 Chapter 4. Fundamental Limits of Cache-Aided Interference Management 4.3.2.2 Delivery Phase In this section, we first describe the delivery phase for the case where t T +t R ≤ K R , so that the first term in the lower bound in (4.11) is dominant. We will later show how to deal with the case where t T +t R >K R . In the delivery phase, the receiver requests are revealed, and in particular, each receiver Rx j ,j∈ [K R ] requests a file W d j from the library and the transmitters need to deliver the subfiles in {W d j ,T,R :j / ∈R} to receiver Rx j ; i.e., the subfiles of fileW d j which have not been already stored in the cache of receiver Rx j . In the following, our goal is to show that the set of packets which need to be delivered to the receivers can be partitioned into subsets of size t T +t R such that the packets in each subset can be scheduled together. To this end, we need to further break each subfile to smaller subfiles. In particular, for any j∈ [K R ],T ⊆ [K T ] s.t. |T| = t T ,R⊆ [K R ]\{j} s.t.|R| =t R , we partition W d j ,T,R to t R ![K R −(t R +1)]! [K R −(t R +t T )]! smaller disjoint subfiles of equal sizes denoted by W d j ,T,R = n W d j ,T,π,π 0 o π∈Π R π 0 ∈Π [K R ]\(R∪{j}),t T −1 , (4.17) where for a setS, Π S denotes the set of permutations ofS, and for anyt∈{1,...,|S|}, Π S,t denotes the set of all permutations of all subsets ofS of size t; i.e., Π S,t = [ A⊆S,|A|=t Π A . Remark 4.2. Note that in the example setting discussed in Section 4.3.1, t R ![K R −(t R +1)]! [K R −(t R +t T )]! = 1, which implies that further partitioning of the subfiles is not needed. The advantage of further breakdown of the subfiles in (4.17) is that we can now partition the set of the subfiles which need to be delivered to the receivers into certain subsets of size t T +t R such that each subfile W d j ,T,π,π 0 intended for receiver Rx j is zero-forced at the receivers with indices in π 0 . Moreover, since this subfile is also already cached at the 71 Chapter 4. Fundamental Limits of Cache-Aided Interference Management receivers with indices in π, the communication will be interference-free for each set of the t T +t R subfiles. We show how to do such a partitioning in Lemma 4.1. In this lemma, we use the following notation: For a setR, we let Π circ R denote the set of (|R|− 1)! circular permutations ofR. 2 Moreover, for a setS, a permutationπ∈ Π S and two integersi,j satisfyingj≥i, we define π[i :j] as π[i :j] = [π(i⊕ |S| 0) π(i⊕ |S| 1) π(i⊕ |S| 2) ... π(i⊕ |S| (j−i))], where for an integer m, i⊕ m j is defined as i⊕ m j = 1 + (i +j− 1 mod m). (4.18) Finally, for a setT and an integer j, we letT⊕ m j denote entry-wise addition of elements ofT with j modulo m, as defined in (4.18). Lemma 4.1. Given the prefetching phase in Section 4.3.2.1, for any receivers’ demand vector d, the set of subfiles which need to be delivered to the receivers can be partitioned into disjoint subsets of size t T +t R as [ T⊆[K T ]:|T|=t T R⊆[K R ]:|R|=t T +t R π∈Π circ R n W d π(l) ,T⊕ K T (l−1),π[l+1:l+t R ],π[l+t R +1:l+t R +t T −1] :l∈ [t T +t R ] o . (4.19) Proof. See Appendix E. Illustration 4.3. For the example network mentioned in Section 4.3.1, the set of 18 subfiles which need to be delivered to the receivers, as in (4.12), can be partitioned to the following 6 sets. {A 12,2 ,B 23,3 ,C 13,1 }∪{A 12,3 ,B 13,1 ,C 23,2 }∪{A 13,2 ,B 12,3 ,C 23,1 } ∪{A 13,3 ,B 23,1 ,C 12,2 }∪{A 23,2 ,B 13,3 ,C 12,1 }∪{A 23,3 ,B 12,1 ,C 13,2 }. (4.20) 2 A circular permutation of a setR is a way of arranging the elements ofR around a fixed circle. The number of distinct circular permutations of a setR is equal to (|R|− 1)!. For example, ifR ={1, 2, 3}, then Π circ R ={[1, 2, 3], [1, 3, 2]}. 72 Chapter 4. Fundamental Limits of Cache-Aided Interference Management Based on the partitioning of the small subfiles that need to be delivered to the receivers in Lemma 4.1, we will have K T t T K R t T +t R (t T +t R − 1)! steps of communication, where at each step, specific setsT andR and a permutationπ are fixed as in (4.19), and each transmitter Tx i will transmit a linear combination of the coded subfiles for whichi∈T⊕ K T (l−1); i.e., X i =L i,T,π n ˜ W d π(l) ,T⊕ K T (l−1),π[l+1:l+t R ],π[l+t R +1:l+t R +t T −1] : l∈ [t T +t R ],i∈T⊕ K T (l− 1) o ! , (4.21) where for any subfileW d j ,T,π,π 0, ˜ W d j ,T,π,π 0 denotes the corresponding coded subfile contain- ing PHY coded symbols, andL i,T,π (.) represents the linear combination that transmitter Tx i chooses for sending the subfiles in (4.21). We will next show that under such a delivery scheme, there always exists a choice of linear combinations at the transmitters so that at each step, the communication will be interference-free and all the t T +t R receivers inR can decode their desired packets, as we also showed in the example setting in Section 4.3.1. Lemma 4.2. For any subset of t T transmittersT ⊆ [K T ], any subset of t T +t R receivers R ⊆ [K R ], and any circular permutation π ∈ Π circ R , there exists a choice of the linear combinations{L i,T,π (.)} K T i=1 in (4.21) such that the set of t T +t R subfiles in n W d π(l) ,T⊕ K T (l−1),π[l+1:l+t R ],π[l+t R +1:l+t R +t T −1] :l∈ [t T +t R ] o , (4.22) can be delivered simultaneously and interference-free by the transmitters in S l∈[t T +t R ] T⊕ K T (l− 1) to the receivers inR. Proof. For ease of notation and without loss of generality, assume T ={1,...,t T },T⊕ K T (l− 1) ={l,...,t T +l},R ={1,...,t T +t R },π = [1,...,t T +t R ]. First, we need to determine the subset of the subfiles which is available at each transmitter. It is easy to verify that 73 Chapter 4. Fundamental Limits of Cache-Aided Interference Management • If i∈{1,...,t T − 1}, then transmitter Tx i has subfiles n ˜ W d π(l) ,T⊕ K T (l−1),π[l+1:l+t R ],π[l+t R +1:l+t R +t T −1] :l∈{1,...,i} o ; (4.23) • If i∈{t T ,...,t T +t R }, then transmitter Tx i has subfiles n ˜ W d π(l) ,T⊕ K T (l−1),π[l+1:l+t R ],π[l+t R +1:l+t R +t T −1] :l∈{i−t T + 1,...,i} o ; (4.24) • and if i∈{t T +t R + 1,..., 2t T +t R − 1}, then transmitter Tx i has subfiles n ˜ W d π(l) ,T⊕ K T (l−1),π[l+1:l+t R ],π[l+t R +1:l+t R +t T −1] :l∈{i−t T + 1,...,t T +t R } o . (4.25) Since each transmitter sends a linear combination of the subfiles that it has, the transmit signal of transmitter Tx i can be written as X i =                      i P l=1 v i,l ˜ W d l ,{l,...,t T +l},{l+1,...,l+t R },{l+t R +1,...,l+t R +t T −1} , if i∈{1,...,t T − 1} i P l=i−t T +1 v i,l ˜ W d l ,{l,...,t T +l},{l+1,...,l+t R },{l+t R +1,...,l+t R +t T −1} , if i∈{t T ,...,t T +t R } t T +t R P l=i−t T +1 v i,l ˜ W d l ,{l,...,t T +l},{l+1,...,l+t R },{l+t R +1,...,l+t R +t T −1} , if i∈{t T +t R + 1,..., 2t T +t R − 1} . (4.26) This implies that the received signal at receiver Rx j ,j∈{1,...,t T +t R } can be written as Y j = 2t T +t R −1 X i=1 h ji X i +Z j (4.27) = t T +j X i=j h ji v i,j ˜ W d j ,{j,...,t T +j},{j+1,...,j+t R },{j+t R +1,...,j+t R +t T −1} + j+t T −1 X l=j+1 t T +l X i=l h ji v i,l ˜ W d l ,{l,...,t T +l},{l+1,...,l+t R },{l+t R +1,...,l+t R +t T −1} + j−1 X l=j−t R t T +l X i=l h ji v i,l ˜ W d l ,{l,...,t T +l},{l+1,...,l+t R },{l+t R +1,...,l+t R +t T −1} +Z j . (4.28) 74 Chapter 4. Fundamental Limits of Cache-Aided Interference Management Now, note that in (4.28), the first term corresponds to the desired subfile of receiver Rx j , while the second and third terms correspond to the undesired subfiles whose interference needs to be canceled at this receiver. However, note that the subfiles in the third term are already cached at receiver Rx j and hence it is able to cancel their incoming interference. Hence, in order for all receivers Rx j ,j∈{1,...,t T +t R } to receive their subfiles interference- free, there should exist a choice of linear combination coefficients{v i,l } such that t T +j X i=j h ji v i,j = 1,∀j∈{1,...,t T +t R } (4.29) t T +l X i=l h ji v i,l = 0,∀j∈{1,...,t T +t R },∀l∈{j + 1,...,j +t T − 1}. (4.30) Equations (4.29)-(4.30) introduce a system of t T (t T +t R ) linear equations. On the other hand, the number of variables{v i,l } is also equal to t T (t T +t R ). This indicates that there always exists a choice of linear combination coefficients{v i,l } such that (4.29)-(4.30) are satisfied. Finally, note that by scaling all the transmit signals by a large enough factor, the power constraint at all the transmitters can also be satisfied. Hence the proof is complete. Remark 4.3. As mentioned in Section 4.1, we assume that the channel gains remain constant over the course of communication. However, for the delivery scheme presented in the proof of Lemma 4.2, this assumption can be relaxed, since we only need the channel gains to remain unchanged for each block of communication and they can be allowed to vary among different blocks. Remark 4.4. In the delivery scheme presented in the proof of Lemma 4.2, we only used zero-forcing at the transmitters in order to cancel their outgoing interference, which is DoF- optimal. However, in general one can use any scheme that exploits the collaboration among the transmitters in order to optimize the actual rates in the finite-SNR regime (such as the schemes suited for the MIMO broadcast channels [67]). 4.3.3 Analysis of the Sum-DoF of the Proposed Achievable Scheme As a result of Lemmas 4.1 and 4.2, it is clear that for any set of receiver demands in the delivery phase, we can schedule all the requested subfiles in groups of size t T +t R . Now, if t T and/or t R are not integers, we can split the memories and the files proportionally 75 Chapter 4. Fundamental Limits of Cache-Aided Interference Management so that for each new partition, the aforementioned scheme can be applied for updated t T and t R which are integers. Hence, combining the schemes over different partitions allows us to serve t T +t R simultaneously, interference-free, for any values of t T and t R such that t T +t R ≤K R . 3 Finally, ift T +t R >K R , then since we cannot serve more thanK R receivers, we can neglect some of the caches at either the transmitters’ side or the receivers’ side and use a fraction of the caches with new sizes N K T ≤M 0 T ≤M T andM 0 R ≤M R so that K T M 0 T +K R M 0 R N =K R . We can then use Algorithm 4.1 to serve all theK R receivers simultaneously without interference. As we showed in Section 4.3.2.1, our prefetching phase respects the cache size constraint of all the transmitters and receivers. Moreover, given our prefetching phase, each receiver Rx j caches K T t T K R −1 t R −1 F ( K T t T )( K R t R ) = M R N F packets of each file in the library. Hence, for each set of requested files by the receivers, a total of K R 1− M R N F packets need to be delivered by the transmitters to the receivers. Therefore, based on the delivery phase mentioned in Section 4.3.2, the number of blocks required to deliver all theK R 1− M R N F packets to the receivers is equal to K R 1− M R N F min{t T +t R ,K R } . This suggests that for any set of receiver demands, sum-DoF of K R 1− M R N F K R 1− M R N F min{t T +t R ,K R } = min{t T +t R ,K R } = min K T M T +K R M R N ,K R is achievable, hence completing the proof of achievability of Theorem 4.1. 4.4 Converse In this section, we prove the converse of Theorem 4.1. In particular, we show that the lower bound on the one-shot linear sum-DoF in (4.11) is within a factor of 2 of the optimal one- shot linear sum-DoF. In order to prove the converse, we take four steps as detailed in the following sections. First, we demonstrate how in each block of communication, the network can be converted into a virtual MISO interference channel. Second, we use this conversion to write an integer optimization problem for the minimum number of communication blocks 3 In [54], this method is referred to as memory-sharing, which resembles time-sharing in network infor- mation theory. 76 Chapter 4. Fundamental Limits of Cache-Aided Interference Management needed to deliver a set of receiver demands for a given caching realization. Third, we show how we can focus on average demands instead of the worst-case demands to derive an outer optimization problem on the number of communication blocks optimized over the caching realizations. Finally, we present a lower bound on the value of the aforementioned outer optimization problem, which leads to the desired upper bound on the one-shot linear sum-DoF of the network. 4.4.1 Conversion to a Virtual MISO Interference Channel Consider any caching realization {P i } K T i=1 ,{Q i } K R i=1 and any demand vector d. As discussed in Section 4.1, in each communication blocks a subset of requested packets are selected to be sent to a corresponding subset of distinct receivers. Now, we can state the following lemma, which bounds the number of packets that can be scheduled together in a single communication block using a one-shot linear scheme. Lemma 4.3. Consider a single communication block where a set{w n l ,f l } L l=1 of L packets are scheduled to be transmitted together to L distinct receivers. In order for each receiver to successfully decode its desired packet, the number of these concurrently-scheduled packets should be bounded by L≤ min l∈[L] |T l | +|R l |, (4.31) where for any l∈ [L],T l andR l denote the set of transmitters and receivers which have cached the packet w n l ,f l , respectively. Proof. For ease of notation and without loss of generality, suppose that in the considered block, L packets{w 1,1 ,..., w L,1 } are scheduled to be sent to L receivers{Rx 1 ,..., Rx L }, respectively. Each transmitter Tx i ,i∈ [K T ] will transmit x i = X l:i∈T l v i,l,1 ˜ w l,1 , (4.32) where we have dropped the dependency on the block index, since we are focusing on a single block. On the other hand, the received signal of receiver Rx j ,j∈ [L] can be written as y j = K T X i=1 h ji x i + z j (4.33) 77 Chapter 4. Fundamental Limits of Cache-Aided Interference Management = K T X i=1 h ji X l:i∈T l v i,l,1 ˜ w l,1 + z j (4.34) = L X l=1 X i∈T l h ji v i,l,1 ˜ w l,1 + z j . (4.35) Therefore, (4.35) implies that we can effectively convert the network into a new MISO in- terference channel with L virtual transmitters{ c Tx l } L l=1 , where c Tx l is equipped with|T l | antennas, and L single-antenna receivers{Rx j } L j=1 , in which each virtual transmitter c Tx l intends to send the coded packet ˜ w l,1 to receiver Rx l . Each antenna in the new network corresponds to a transmitter in the original network. Hence, the channel vectors are corre- lated in the new network. In fact, as (4.35) suggests, all the antennas corresponding to the same transmitter in the original network have the same channel gain vectors to the receivers in the new network. In the constructed MISO interference channel, we take a similar approach as in [66] in order to bound the one-shot linear sum-DoF of the network. Each virtual transmitter c Tx l in the constructed MISO network will select a beamforming vector v l ∈ C |T l |×1 (which consists of the coefficients chosen by the original transmitters corresponding to its antennas) to transmit its desired symbol. Denoting the channel gain vector between transmitter c Tx l and receiver Rx j as h jl ∈C |T l |×1 , the decodability conditions can be written as h T jl v l = 0, ∀l6=j s.t. j / ∈R l (4.36) h T jj v j 6= 0, ∀j∈ [L]. (4.37) Now, each of the vectors v l , l∈ [L] can be written as v l =q l P l   1 ¯ v l   , (4.38) where q l is a non-zero scalar, P l is a|T l |×|T l | permutation matrix and ¯ v l is a vector of size (|T l |− 1)× 1. Also, for any two distinct pairs l6=j, the channel gain vector h jl can be permuted as ¯ h jl = P −1 l h jl , and we can partition ¯ h jl as ¯ h jl =   ¯ h (1) jl ¯ h (2) jl   , (4.39) 78 Chapter 4. Fundamental Limits of Cache-Aided Interference Management where ¯ h (1) jl is a scalar and ¯ h (2) jl is of size (|T l |− 1)× 1. Therefore, the nulling condition in (4.36) can be rewritten as h ¯ h (1) jl ¯ h (2)T jl i   1 ¯ v l   = 0⇔ ¯ h (1) jl + ¯ h (2)T jl ¯ v l = 0. (4.40) Now, since the packet sent by the virtual transmitter c Tx l is available in the caches of at most|R l | receivers in the network, the interference of each transmitter should be nulled at least atL−|R l |− 1 unintended receivers. This implies that the free beamforming variables at transmitter l, i.e., ¯ v l , should satisfy at least L−|R l |− 1 linear equations in the form of (4.40). This is not possible unless the number of equations is no greater than the number of variables, or L−|R l |− 1≤|T l |− 1⇒L≤|T l | +|R l |. (4.41) Since the above inequality holds for all l∈ [L], the proof is complete. 4.4.2 Integer Program Formulation Equipped with Lemma 4.3, we define a set of packetsD m selected to be transmitted at block m to be feasible if its size satisfies condition (4.31) in Lemma 4.3. We can then write the following integer program (P1) to minimize the number of required communication blocks for any given caching realization and set of receiver demands: min H (P1-1) s.t. H [ m=1 D m = K R [ j=1 W d j \Q j (P1-2) D m is feasible, ∀m∈ [H], (P1-3) where (P1-2) states that all the demanded packets that are not cached at the requesting receivers need to be delivered by the transmitters over the H blocks of communication. 79 Chapter 4. Fundamental Limits of Cache-Aided Interference Management 4.4.3 Relaxing Worst-Case Demands to Average Demands and Optimiz- ing over Caching Realizations We can now write an optimization problem to minimize the number of communication blocks required for delivering the worst-case demands optimized over the caching realizations. However, before that, we need to introduce some notation. Given any caching realization {P i } K T i=1 ,{Q i } K R i=1 , we can break each file W n ,n∈ [N], in the library into (2 K T −1)(2 K R ) subfiles{W n,T,R } T⊆ ∅ [K T ],R⊆[K R ] , whereW n,T,R denotes the subfile of W n exclusively stored in the caches of the transmitters inT and receivers inR, and we use the shorthand notationT ⊆ ∅ [K T ] to denoteT ⊆ [K T ],T 6=∅. We definea n,T,R as the number of packets in W n,T,R . Denoting the answer to the optimization problem (P1) byH ∗ {P i } K T i=1 ,{Q i } K R i=1 , d , the be- low optimization problem yields the number of communication blocks required for delivering the worst-case demands, minimized over all caching realizations: min {P i } K T i=1 ,{Q i } K R i=1 max d H ∗ {P i } K T i=1 ,{Q i } K R i=1 , d (P2-1) s.t. X T⊆ ∅ [K T ] X R⊆[K R ] a n,T,R =F, ∀n∈ [N] (P2-2) N X n=1 X R⊆[K R ] X T⊆[K T ]: i∈T a n,T,R ≤M T F,∀i∈ [K T ] (P2-3) N X n=1 X T⊆ ∅ [K T ] X R⊆[K R ]: j∈R a n,T,R ≤M R F,∀j∈ [K R ] (P2-4) a n,T,R ≥ 0,∀n∈ [N],∀T ⊆ ∅ [K T ],∀R⊆ [K R ]. (P2-5) To lower bound the value of the above optimization problem, we can write the following optimization problem, which yields the number of communication blocks averaged over all the π(N,K R ) = N! (N−K R )! permutations of distinct receiver demands, denoted byP N,K R : min {P i } K T i=1 ,{Q i } K R i=1 1 π(N,K R ) X d∈P N,K R H ∗ {P i } K T i=1 ,{Q i } K R i=1 , d (P3-1) s.t. X T⊆ ∅ [K T ] X R⊆[K R ] a n,T,R =F, ∀n∈ [N] (P3-2) 80 Chapter 4. Fundamental Limits of Cache-Aided Interference Management N X n=1 X R⊆[K R ] X T⊆[K T ]: i∈T a n,T,R ≤M T F,∀i∈ [K T ] (P3-3) N X n=1 X T⊆ ∅ [K T ] X R⊆[K R ]: j∈R a n,T,R ≤M R F,∀j∈ [K R ] (P3-4) a n,T,R ≥ 0,∀n∈ [N],∀T ⊆ ∅ [K T ],∀R⊆ [K R ]. (P3-5) 4.4.4 Lower Bound on the Number of Communication Blocks Having the optimization problem in (P3), we now present the following lemma which pro- vides a lower bound on the value of (P3). Lemma 4.4. The value of the optimization problem (P3) is bounded from below by K R NF 1− M R N 2 K T M T +K R M R . Proof. See Appendix F. Since the total number of packets delivered over the channel is K R 1− M R N F in the optimization problem (P3), Lemma 4.4 immediately yields the following upper bound on the one-shot linear sum-DoF: DoF ∗ L,sum (N,M T ,M R )≤ K R 1− M R N F K R NF 1− M R N 2 K T M T +K R M R = K T M T +K R M R N−M R . Combining the above bound with the trivial bound on the one-shot linear sum-DoF which is the number of receivers, K R , we have DoF ∗ L,sum (N,M T ,M R )≤ min K T M T +K R M R N−M R ,K R . (4.42) Now, consider the following two cases: 81 Chapter 4. Fundamental Limits of Cache-Aided Interference Management • M R ≤ N 2 : In this case, (4.42) implies that DoF ∗ L,sum (N,M T ,M R )≤ min ( K T M T +K R M R N− N 2 ,K R ) ≤ min 2 K T M T +K R M R N ,K R . • M R > N 2 : In this case, (4.11) implies that one-shot linear sum-DoF of DoF L,sum (N,M T ,M R )> min ( K T M T +K R N 2 N ,K R ) > K R 2 , can be achieved, while the upper bound in (4.42) implies that DoF ∗ L,sum (N,M T ,M R )≤K R . Therefore, in both cases, the inner bound in (4.11) is within a factor of 2 of the outer bound in (4.11), which completes the proof of the converse of Theorem 4.1. 82 Chapter 5 Cache-Aided Interference Management in Wireless Cellular Networks In Chapter 4, we considered a fully-connected interference channel with caches at both transmitters’ side and receivers’ side and characterized the sum degrees-of-freedom (sum- DoF) to within a constant factor under one-shot linear schemes. In this chapter, we consider a wireless cellular network with caches at both base stations and receivers, and aim to understand the fundamental limits that caching can help to increase the system capacity through the metric of DoF per cell. In particular, we consider a cellular network with C cells, where each cell contains a central base station and K R receivers located around its corners. Due to path-loss and fading effects, we assume that each receiver is only able to receive signals from its three neighboring base stations and does not receive signals (hence, no interference) from the remaining base stations. We further assume that there is a library of N files of the same size out of which the receivers will request their desired contents. Before the requests are revealed, each node in the network can prefetch some parts of the library in its own cache. In particular, each base station and each receiver is assumed to be equipped with a cache of sizeμ T N files andμ R N files, respectively. After the base stations This chapter is mainly taken from [68], coauthored by the author of this dissertation. 83 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks and the receivers fill up their caches, each receiver reveals its request for a file in the library, and the base stations need to deliver the requested contents to the receivers. Note that compared to the setting in Chapter 4, the aforementioned setting is different in two key aspects. First, this setting incurs a cellular topology in which each receiver is only connected to three neighboring base stations. This is in stark contrast with the settings con- sidered in Chapter 4 in which every receiver can receive signals from every transmitter in the network. Second, we focus on a scenario in which the receiver locations are unknown during the prefetching phase. This is particularly important in cellular networks due to the mobil- ity of the users [69], as the prefetching phase occurs long before the recievers reveal their requests, whereas in the settings considered in Chapter 4, a centralized prefetching scheme has been used to cache the contents at the receivers. Therefore, this chapter generalizes the results of Chapter 4 by posing two new challenges, namely a partially-connecgted topology, in which not all the base station-receiver links can be utilized to deliver the requested con- tents and also an unknown topology, in which receiver locations are unknown during the prefetching phase, and hence the prefetching scheme cannot depend on the topology of the network. In this chapter, we characterize the one-shot linear DoF that caching can achieve per cell to within additive and multiplicative gaps of 2. We introduce prefetching and delivery mechanisms and evaluate their achievable DoF per cell, and we also develop a converse argument to show that our achievable DoF per cell is within additive and multiplicative gaps of 2 of the optimum under the restriction of one-shot linear schemes. Our result shows that the gain of caching is proportional to the entire cache inside each cell, which linearly increases with the number of receivers and also the size of caches at the base stations and the receivers. Our achievable scheme makes use of a random prefetching scheme at the receivers which is independent of the number, location and identity of the users. In this scheme, similar to the one proposed in [70], each receiver caches a random fraction of size μ R of all the files in the library. The base stations, on the other hand, utilize the reuse pattern of clusters of 3 cells to prefetch the contents from the library, such that each piece of content is available at 3μ T base stations in each cluster of 3 adjacent cells. The delivery phase is also based on the base station cache sizes. For small cache sizes, where there is no collaboration, only one-third of the base stations are activated, while for larger cache sizes, two-thirds or all the base stations can be activated simultaneously. The active base stations at each step 84 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks make use of the overlaps in their cache contents to zero-force part of the interference at the surrounding, undesired receivers. The receivers will also utilize their cache contents to cancel the interference from part of their surrounding active base stations. Furthermore, we show that the DoF per cell achievable by our scheme is within additive and multiplicative gaps of 2 of its optimal value for all system parameters under the restriction of one-shot linear schemes. Our proof of converse consists of multiple steps. We first show that through conversion to a virtual MISO inteference channel, the one-shot linear DoF of the cellular network can be upper bounded by a fully-connected wireless network of the same size, in which the number of packets that can be delievered simultaneously to the receivers is bounded by the number of transmitters and receivers that have cached those packets. We then cast the converse as a scheduling problem in the form of an integer program. We will then relax the worst-case demands to average demands and bound the value of the final optimization problem using similar techniques as in Section 4.4, namely LP relaxation and Cauchy-Schwartz inequality. The rest of the chapter is organized as follows. In Section 5.1 we describe the system model. We present our main result and its implications in Section 5.2. We prove the achievability and converse of our main result in Sections 5.3 and 5.4, respectively. 5.1 System Model Consider a cellular network, as illustrated in Figure 5.1, consisting of C hexagonal cells, where each celli∈ [C] contains a central base station (BS) denoted by Tx i andK R receivers randomly located at the 6 cell corners. As for the physical layer, we assume that all the receivers located at the intersection of each three cells can only receive signals from the three neighboring base stations and do not receive signals from the rest of the base stations due to path-loss and fading effects. In particular, at each time slot t, the received signal of each receiver Rx j in the network, j∈ [CK R ],{1, 2,...,CK R }, is given by Y j (t) = X i∈N j h ji X i (t) +Z j (t), (5.1) whereN j denotes the set of 3 base stations whose signals can be received by Rx j , h ji ∈C denotes the channel gain from Tx i to Rx j , X i (t)∈ C denotes the transmit signal of Tx i at time slot t, and Z j (t) denotes the additive white Gaussian noise at Rx j at time slot t, 85 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks distributed asCN (0, 1). The transmit signal at base station Tx i ,i∈ [C] is subject to the power constraintE |X i (t)| 2 ≤P . Figure 5.1: A cellular network model with K R = 4. We assume that each receiver will request an arbitrary file out of a library of N files {W n } N n=1 , which should be delivered by the base stations. Each file W n in the library contains F packets{w n,f } F f=1 , where each packet is a vector of B bits; i.e., w n,f ∈ F B 2 . Furthermore, we assume that each node in the network is equipped with a cache memory of certain size that can be used to cache arbitrary contents from the library before the receivers reveal their requests and communication begins. In particular, each base station and each receiver is equipped with a cache of size μ T NF and μ R NF packets, respectively. We call μ T and μ R the fractional cache sizes of the base stations and the receivers, respectively. We assume that the network operates in two phases, namely the prefetching phase and the delivery phase, which are described as follows. Prefetching Phase: In this phase, each node can store an arbitrary subset of the packets from the files in the library up to its cache size. In particular, each base station Tx i chooses a subsetP i of the NF packets in the library, where|P i |≤ μ T NF , to store in its cache. Likewise, each receiver Rx j stores a subsetQ j of the packets in the library, where |Q j |≤μ R NF . Caching is done at the level of whole packets and we do not allow breaking the packets into smaller subpackets. Also, this phase takes place unaware of the receivers’ future requests. 86 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks Delivery Phase: In this phase, each receiver Rx j in the network reveals its request for an arbitrary file W d j from the library for some d j ∈ [N]. We let d = [d 1 ... d CK R ] T denote the vector of demands for all the receivers. Depending on the demand vector d and the cache contents, each receiver has already cached some packets of its desired file and there is no need to deliver them. The base stations will be responsible for delivering the rest of the requested packets to the receivers. In order to make sure that any piece of content in the library is stored at the cache of at least one base station in the neighborhood of each receiver, we assume that the base station cache size satisfies 3μ T ≥ 1. The details of the delivery phase are similar to the one mentioned in Section 4.1. To make the chapter self-contained, we highlight the main steps in the following. • Coding the packets: Each base station first employs a random Gaussian coding scheme ψ :F B 2 →C ˜ B of rate logP +o(logP ) to encode each of its cached packets into a coded packet composed of ˜ B complex symbols, so that each coded packet carries one degree- of-freedom (DoF). We denote the coded version of each packet w n,f in the library by ˜ w n,f ,ψ(w n,f ). • Block communication: The communication takes place over H blocks, each of length ˜ B time slots. In each block m∈ [H], the goal is to deliver a subset of the requested packets, denoted byD m , to a subset of receivers, denoted byR m , such that each packet inD m is intended to exactly one of the receivers inR m . In addition, the set of transmitted packets in all blocks and the cache contents of the receivers should satisfy {w d j ,f } F f=1 ⊂ H [ m=1 D m ! ∪Q j , ∀j∈ [CK R ], (5.2) which implies that for any receiver Rx j , each of its requested packets should be either transmitted in one of the blocks or already stored in its own cache. • One-shot linear scheme — encoding: In each block m∈ [H], we assume a one-shot linear scheme where each base station transmits an arbitrary linear combination of a subset of the coded packets inD m that it has cached. Particularly, Tx i ,i∈ [C] transmits x i [m]∈C ˜ B , where x i [m] = X (n,f): w n,f ∈P i ∩Dm v i,n,f [m] ˜ w n,f , (5.3) 87 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks and v i,n,f [m]’s denote the complex beamforming coefficients that Tx i uses to linearly combine its coded packets in block m. • One-shot linear scheme — decoding: On the receivers’ side, the received signal of each receiver Rx j ∈R m in block m, denoted by y j [m]∈C ˜ B , can be written as y j [m] = X i∈N j h ji x i [m] + z j [m], (5.4) where z j [m]∈ C ˜ B denotes the noise vector at Rx j in block m. Then, receiver Rx j will use the contents of its cache to cancel (subtract out) the interference of some of undesired packets inD m , if they exist in its cache. In particular, each receiver Rx j ∈R m , forms a linear combinationL j,m , as L j,m (y j [m], ˜ Q j ) (5.5) to recover ˜ w d j ,f ∈D m , where ˜ Q j denotes the set of coded packets cached at receiver Rx j . The communication in block m∈ H to transmit the packets inD m is successful, if there exist linear combinations (5.3) at the base stations and (5.5) at receivers, such that for all Rx j ∈R m , L j,m (y j [m], ˜ Q j ) = ˜ w d j ,f + z j [m]. (5.6) The channel created in (5.6) is a point-to-point channel, whose capacity is logP + o(logP ). Hence, since each coded packet ˜ w d j ,f is coded with rate logP +o(logP ), it can be decoded with vanishing error probability as B increases. We assume that the communication continues for H blocks until all the desired packets are successfully delivered to all receivers. • Degrees-of-freedom: Since each packet carries one degree-of-freedom, the one-shot linear sum-degrees-of-freedom (sum-DoF) of|D m | is achievable in each block m∈ [H] for sufficiently large file size F . This implies that throughout the H blocks of communication, the one-shot linear sum-DoF of S H m=1 Dm H is achievable. Therefore, for a given caching realization, we define the one-shot linear sum-DoF to be maximum 88 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks achievable one-shot linear sum-DoF for the worst case demands; i.e., LDoF {P i } C i=1 ,{Q j } CK R j=1 sum = inf d sup H,{Dm} H m=1 H S m=1 D m H . (5.7) We further define the one-shot linear sum-DoF of the network as the maximum achiev- able one-shot linear sum-DoF over all caching realizations; i.e., LDoF ∗ sum (C) = sup {P i } C i=1 ,{Q j } CK R j=1 LDoF {P i } C i=1 ,{Q j } CK R j=1 sum (5.8) s.t. |P i |≤μ T NF, ∀i∈ [C] (5.9) |Q j |≤μ R NF, ∀j∈ [CK R ]. (5.10) Finally, we define the one-shot linear DoF per cell as LDoF ∗ = lim C→∞ lim N→∞ LDoF ∗ sum (C) C . (5.11) 5.2 Main Result In this section, we present our main result and its implications. Theorem 5.1. For a cellular network with fractional cache sizes ofμ T andμ R at each base station and each receiver, respectively, the one-shot linear DoF per cell satisfies min{μ T +K R μ R − 2,K R }≤ LDoF ∗ ≤ min{2(μ T +K R μ R ),K R }. (5.12) Remark 5.1. Note that the upper bound in (5.12) is within additive and multiplicative factors of 2 of the lower bound in (5.12). Therefore, Theorem 5.1 characterizes the one- shot linear DoF per cell of a cache-aided wireless cellular network to within additive and multiplicative gaps of 2 for all system parameters. Remark 5.2. The one-shot linear DoF per cell characterized in Theorem 5.1 scales linearly with the aggregate cache size that is available inside each cell, despite the fact that these caches are isolated. Moreover, the caches at both sides of the network, i.e., the base stations and the receivers, are equally valuable in the achievable one-shot linear DoF per cell. In 89 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks particular, the term μ T is the contribution of base station caches which are used to zero- force part of the interference at undesired receivers, and the term K R μ R is the contribution of receiver caches which are used as side information to eliminate part of the incoming interference due to undesired packets. Our result also demonstrates that for a given cache size, the one-shot linear DoF per cell scales linearly with the number of receivers in the cell. Hence, caching has the potential to provide a scalable solution for wireless cellular networks. Remark 5.3. The phenomenon mentioned in Remark 5.2 was also observed for the case of fully-connected wireless networks in Chapter 4, in which the one shot linear sum-DoF in the entire network was proportional to the size of the entire caches at all the transmitters and receivers. Therefore, Theorem 5.1 suggests that the partial connectivity and the fact that cache placement should be done without the knowledge of user locations during the delivery phase incurs no loss in the cellular settings compared to the fully-connected case in Chapter 4. This can be intuitively explained as follows. In the fully-connected case, transmit zero- forcing and receiver interference cancellation could be used across the entire network. On the contrary, in the cellular case, the partial connectivity does not allow delivery from every base station to every receiver. However, the missing links in the network also reduce the amount of interference that needs to be canceled at the receivers. Interestingly, Theorem 5.1 implies that these two effects cancel each other almost completely. Moreover, using a randomized prefetching scheme due to the unknown location of the receivers during the prefetching phase will only result in a negligible loss that vanishes as the number of receivers grows. We will prove Theorem 5.1 by providing an achievable scheme and an outer bound argument in Sections 5.3 and 5.4, respectively. 5.3 Achievable Scheme In this section, we prove the achievability part of Theorem 5.1 by introducing an achievable scheme which is decentralized in its prefetching phase and characterize its DoF per cell. For prefetching the contents on the receivers’ side, since their location is unknown during the prefetching phase, we take a randomized approach similar to [70] in which each receiver stores μ R F packets of each file in the library uniformly at random. 90 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks As the locations of the base stations are fixed, the prefetching at the base stations is done according to the reuse pattern with clusters of size 3 as in Figure 5.2, and the base stations are labeled accordingly so that the base stations with the same label have identical cache contents. 2 3 1 2 1 3 2 2 1 1 2 1 2 1 1 2 1 2 1 2 1 3 2 3 2 1 2 1 2 3 1 2 3 2 3 2 3 1 2 3 2 3 1 2 3 1 1 3 2 3 1 1 1 1 1 1 1 1 1 1 1 1 Figure 5.2: Base station prefetching pattern. Figure 5.3: The delivery activation pattern for the case of 3μ T = 1. The details of base station prefetching phase and also the delivery phase depend on the size of caches at the base stations and we need to consider the following three cases. 1 1 For the case where 3μT is not an integer, we can use the memory sharing technique [51,54] in order to achieve a convex combination of the DoF per cell achieved with base station fractional cache sizes μ 0 T and μ 00 T such that 3μ 0 T =b3μTc and 3μ 00 T =d3μTe. 91 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks 5.3.1 3μ T = 1 (No BS collaboration) In this case, we partition each file W n ,n ∈ [N] in the library into 3 disjoint subfiles {W n,1 ,W n,2 ,W n,3 } of the same size. Each transmitter with index t∈ [3] will then store all the subfiles in{W n,t :n∈ [N]}. Due to the aforementioned prefetching strategy, each base station caches 1 3 of the entire library and no cooperation can exist among the base stations with different labels. Hence, at each time, we only turn on the base stations with the same label and turn off the rest. For instance, one step of the delivery scheme is shown in Figure 5.3. We then do time sharing to activate the base stations with different labels. At each step, each active base station will serve all the users around its six corners. Therefore, it is clear that using such an activation pattern of the cells, there is no interference between active cells. Let us denote the indices of the active cells as{1, 2,..., C 3 } and denote the number of receivers around the corners of the i th active cell as K i . Since there are CK R receivers in the entire network, it is true that P C 3 i=1 K i = CK R . By the law of large numbers, the number of packets of each file that are cached exactly at each subset of s receivers is approximately equal to q s (1−q) K i −s F, (5.13) where q =μ R is the probability of each packet being cached at each receiver. Now, in the i th active cell, using a scheme similar to the one in [70], the base station needs to find the packets that are cached at equal number of receivers out of the set ofK i receivers around itself, and deliver them together. In particular, all the packets that are cached at s receivers can be delivered in groups of s + 1 to the receivers. To recapitulate the scheme, consider the following example. Example 5.1. Consider one of the active cells in Figure 5.3 with label t = 1 and assume that there are a total of 3 receivers around the cell corners. For the ease of notation, let us denote these receivers by Rx 1 , Rx 2 , and Rx 3 . Suppose that during the delivery phase, these receivers request files A, B, and C, respectively. As mentioned in the base station prefetching phase, each base station will only have access to one-third of each file. Let A 1 , B 1 , and C 1 denote the parts of the requested files that is available at the considered base station. Each of these parts can be partitioned to 2 3 = 8 disjoint subfiles based on the 92 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks receivers that have cached those subfiles. For example, we can partition A 1 to 8 subfiles as {A 1,∅ ,A 1,1 ,A 1,2 ,A 1,3 ,A 1,12 ,A 1,13 ,A 1,23 ,A 1,123 }, (5.14) where for anyR⊆{1, 2, 3},A 1,R denotes the subfile ofA 1 cached exclusively at the receivers inR. Note that due to (5.13), each subfile A 1,R includes approximately q |R| (1−q) 3−|R|F 3 packets. We can also do the same partitioning for B 1 and C 1 . Now, note that receiver Rx 1 has already cached the 4 subfiles{A 1,1 ,A 1,12 ,A 1,13 ,A 1,123 } and therefore, the base station needs to deliver the remaining 4 subfiles of A 1 to this receiver. Similarly, receiver Rx 2 and Rx 3 also need 4 subfiles of B 1 and C 1 , respectively. In order to deliver these subfiles, the base station uses the cache contents of the recievers to create coded multicasting opportunities. Two example steps of the delivery are shown in Figure 5.4, where for any subfileW n,t,R , ˜ W n,t,R denotes the corresponding coded subfile containing PHY coded symbols. In Figure 5.4–a, subfiles A 1,2 and B 1,1 are delivered to receivers Rx 1 and Rx 2 simultaneously, while in Figure 5.4–b, subfiles A 1,23 ,B 1,13 , andC 1,12 are delivered to receivers Rx 1 , Rx 2 , and Rx 3 simultaneously. In both cases, the receivers are able to Rx 1 Rx 2 Rx 3 (a) Rx 1 Rx 2 Rx 3 (b) A 1,2 B 1,1 A 1,23 C 1,12 A 1,23 B 1,13 B 1,13 C 1,12 A 1,23 +B 1,13 +C 1,12 A 1,2 +B 1,1 Figure 5.4: Two steps of delivery in Example 5.1 in which the base station delivers (a) subfiles A 1,2 and B 1,1 to receivers Rx 1 and Rx 2 , and (b) subfiles A 1,23 , B 1,13 , and C 1,12 to receivers Rx 1 , Rx 2 , and Rx 3 simultaneously. recover their desired subfiles using cache contents to cancel the interference due to other subfiles. Algorithm 5.1 illustrates the delivery scheme employed by the i th active base station with label t∈ [3] for the demand vector [d 1 ... d K i ]. In this algorithm, for any n∈ [N] and R⊆ [K i ], W n,t,R denotes the subfile of W n,t exclusively cached at the receivers inR and ˜ W n,t,R denotes the corresponding coded subfile containing PHY coded symbols.. 93 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks Algorithm 5.1 Delivery phase for the i th base station with label t∈ [3] , for the case 3μ T = 1 1: for s =K i ,K i − 1,..., 1 2: forR⊆ [K i ] s.t.|R| =s 3: The base station transmits P k∈R ˜ W d k ,t,R\{k} 4: end 5: end This implies that the following sum-DoF is achievable in the entire network: LDoF ∗ sum (C)≥ C 3 X i=1 1 1−q K i −1 X s=0 K i − 1 s ! q s (1−q) K i −s (s + 1) = C 3 X i=1 K i −1 X s=0 K i − 1 s ! q s (1−q) K i −1−s (s + 1) = C 3 + C 3 X i=1 K i −1 X s=0 s K i − 1 s ! q s (1−q) K i −1−s (5.15) = C 3 + C 3 X i=1 (K i − 1)q (5.16) = C 3 + CK R − C 3 μ R =Cμ T +C K R − 1 3 μ R ≥C μ T +K R μ R − 1 3 , where (5.16) follows from the fact that the inner summation in (5.15) is the expectation of a binomial random variable with parameters (K i − 1,q). This suggests that in this case, the achievable DoF per cell is lower bounded by LDoF ∗ = lim C→∞ lim N→∞ LDoF ∗ sum (C) C ≥μ T +K R μ R − 1 3 . (5.17) 5.3.2 3μ T = 2 (Partial BS collaboration) In this case, we partition each file W n ,n ∈ [N] in the library into 3 disjoint subfiles {W n,12 ,W n,13 ,W n,23 } of the same size. Each base station with label t∈ [3] will then store 94 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks all the subfiles in{W n,T :t∈T,n∈ [N]}. The aforementioned prefetching scheme is such that each piece of content is available at 2 base stations. Hence, at each step, one-third of the cells corresponding to a single label is turned off, and the rest of the base stations will collaborate to deliver the requested contents to the receivers. An example of such an activation pattern is depicted in Figure 5.5. Figure 5.5: The delivery activation pattern for the case of 3μ T = 2. Using this activation pattern, all the packets cached exclusively at subsets of s receivers can be delivered in groups of size min{s + 2C 3 ,CK R }. This is because each packet is cached at two-third of the base stations, and hence we have 2C 3 active base stations at each step. If the network was fully connected (i.e., if all the receivers could receive signals from all the base stations), using the scheme in Chapter 4, it is clear that all such packets can be delivered in groups of size min{s + 2C 3 ,CK R }. This is also true for the case of partially- connected cellular networks under study, since in this case, we can activate at least one receiver in each active cell, so that the number of transmit beamforming variables is enough to provide zero-forcing opportunities by the base stations (corresponding to the 2C 3 term in s + 2C 3 ) alongside with the receiver cancellation opportunities (corresponding to the s term in s + 2C 3 ). The detailed delivery scheme is demonstrated in Algorithm 5.2 for the demand vector [d 1 ... d CK R ]. In this algorithm, for any n∈ [N] andR⊆ [CK R ], W n,T,R denotes the subfile of W n,T exclusively cached at the receivers inR and ˜ W n,T,R denotes the corresponding coded subfile containing PHY coded symbols. Moreover, for a setR, we 95 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks let Π circ R denote the set of (|R|− 1)! circular permutations ofR. 2 Finally, for a setS, a permutation π∈ Π S and two integers i,j satisfying j≥i, we define π[i :j] as π[i :j] = [π(i⊕ |S| 0) π(i⊕ |S| 1) π(i⊕ |S| 2) ... π(i⊕ |S| (j−i))], where for an integer m, i⊕ m j is defined as i⊕ m j = 1 + (i +j− 1 mod m). (5.18) Algorithm 5.2 Delivery phase for base stations with labelsT ⊂ [3],|T| = 2 , for the case 3μ T = 2 1: for s =CK R − 2C 3 ,CK R − 2C 3 − 1,..., 0 2: for j∈ [CK R ] 3: forR⊆ [CK R ]\{j} s.t.|R| =s 4: LetR 0 ={R 0 ⊆ [CK R ]\ (R∪{j}) :|R 0 | = 2C 3 − 1 andR 0 ∪R∪{j} contains at least one receiver in each active cell}. 5: Let Π 0 = S R 0 ∈R 0 Π R 0 . 6: partition W d j ,T,R to s!|Π 0 | disjoint subfiles n W d j ,T,π,π 0 o π∈Π R π 0 ∈Π 0 of equal sizes. 7: end 8: end 9: forR⊆ [CK R ] s.t.|R| =s+ 2C 3 andR contains at least one receiver in each active cell 10: for π∈ Π circ R 11: Each base station Tx i transmits a linear combination of the coded subfiles as in X i =L i,T,π n ˜ W d π(l) ,T,π[l+1:l+s],π[l+s+1:l+s+ 2C 3 −1] :l∈ [s + 2C 3 ] o such that the subfiles n W d π(l) ,T,π[l+1:l+s],π[l+s+1:l+s+ 2C 3 −1] :l∈ [s + 2C 3 ] o are simultaneously delivered to the receivers inR interference-free. 12: end 13: end 14: end 2 A circular permutation of a setR is a way of arranging the elements ofR around a fixed circle. The number of distinct circular permutations of a setR is equal to (|R|− 1)!. For example, ifR ={1, 2, 3}, then Π circ R ={[1, 2, 3], [1, 3, 2]}. 96 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks Consequently, our delivery scheme can achieve the following DoF per cell: LDoF ∗ ≥ lim C→∞ 1 C · 1 1−q CK R −1 X s=0 CK R − 1 s ! q s (1−q) CK R −s min s + 2C 3 ,CK R = lim C→∞ 1 C CK R −1 X s=0 CK R − 1 s ! q s (1−q) CK R −1−s min s + 2C 3 ,CK R = lim C→∞ 1 C CK R −1 X s=0 CK R − 1 s ! q s (1−q) CK R −1−s s + 2C 3 − lim C→∞ 1 C CK R −1 X s=CK R − 2C 3 +1 CK R − 1 s ! q s (1−q) CK R −1−s s + 2C 3 −CK R = lim C→∞ μ T + K R − 1 C μ R − lim C→∞ 1 C CK R −1 X s=CK R − 2C 3 +1 CK R − 1 s ! q s (1−q) CK R −1−s s + 2C 3 −CK R ≥μ T +K R μ R − lim C→∞ 2 3 CK R −1 X s=CK R − 2C 3 +1 CK R − 1 s ! q s (1−q) CK R −1−s =μ T +K R μ R − 2 3 lim C→∞ Pr X≥CK R − 2C 3 + 1 (5.19) ≥μ T +K R μ R − 2 3 lim C→∞ E[X] CK R − 2C 3 + 1 (5.20) =μ T +K R μ R − 2 3 lim C→∞ (CK R − 1)q CK R − 2C 3 + 1 ≥μ T +K R μ R − 2 3 K R K R − 2 3 ≥μ T +K R μ R − 2, where in (5.19), X∼B(CK R − 1,q), and (5.20) follows from Markov’s inequality. 5.3.3 3μ T = 3 (Full BS collaboration) In this case, sinceμ T = 1, each base station stores the whole library in its cache. Therefore, all the base stations can be activated together in order to deliver all the packets. Considering subsets of receivers of sizes, all the packets cached ats receivers can be delivered in groups of size min{s+C,CK R }. Hence, following the same lines as the previous case, we can show 97 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks that if K R ≥ 2 the following DoF per cell is achievable: LDoF ∗ ≥μ T +K R μ R − 2. (5.21) 5.4 Converse In this section, we prove the converse of Theorem 5.1. The converse essentially follows the same lines as in Section 4.4, consisting of four steps. The main difference with the converse in Section 4.4 is in the first step, where we show that in each block of communication, the network can be converted into a virtual MISO interference channel. The next steps, i.e., writing an integer optimization problem for the minimum number of communication blocks needed to deliver a set of receiver demands for a given caching realization, switching the focus to average demands instead of the worst-case demands, and deriving a lower bound on the value of the aforementioned optimization problem follow the same lines as Section 4.4 and we briefly mention their main ingredients here for the sake of completeness. 5.4.1 Conversion to a Virtual MISO Interference Channel Consider any caching realization {P i } C i=1 ,{Q j } CK R j=1 and any demand vector d. As dis- cussed in Section 5.1, in each communication blocks a subset of requested packets are selected to be sent to a corresponding subset of distinct receivers. Now, we can state the following lemma, which bounds the number of packets that can be scheduled together in a single communication block using a one-shot linear scheme. Lemma 5.1. Consider a single communication block where a set{w n l ,f l } L l=1 of L packets are scheduled to be transmitted together to L distinct receivers. In order for each receiver to successfully decode its desired packet, the number of these concurrently-scheduled packets should be bounded by L≤ min l∈[L] |T l | +|R l |, (5.22) where for any l∈ [L],T l andR l denote the set of base stations and receivers which have cached the packet w n l ,f l , respectively. 98 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks Proof. For ease of notation and without loss of generality, suppose that in the considered block, L packets{w 1,1 ,..., w L,1 } are scheduled to be sent to L receivers{Rx 1 ,..., Rx L }, respectively. Each base station Tx i ,i∈ [C] will transmit x i = X l:i∈T l v i,l ˜ w l,1 . (5.23) On the other hand, the received signal of receiver Rx j ,j∈ [L] can be written as y j = X i∈N j h ji x i + z j (5.24) = X i∈N j h ji X l:i∈T l v i,l ˜ w l,1 + z j (5.25) = L X l=1 X i∈T l ∩N j h ji v i,l ˜ w l,1 + z j , (5.26) This implies that we can effectively convert the network into a new MISO interference channel with L virtual transmitters{ c Tx l } L l=1 , where c Tx l is equipped with|T l | antennas, and L single-antenna receivers{Rx j } L j=1 , in which each virtual transmitter c Tx l intends to send the coded packet ˜ w l,1 to receiver Rx l . Each antenna in the new network corresponds to a transmitter in the original network. This implies that the network will not be fully- connected, and each receiver is connected only to at most 3 antennas at each transmitter. In the constructed MISO interference channel, we take a similar approach as in [66] in order to bound the one-shot linear sum-DoF of the network. Each virtual transmitter c Tx l will select a beamforming vector v l ∈ C |T l |×1 (which consists of the coefficients chosen by the original transmitters corresponding to its antennas) to transmit its desired symbol. Denoting the channel gain vector between transmitter c Tx l and receiver Rx j by h jl ∈C |T l |×1 , the decodability conditions can be written as h T jl v l = 0, ∀l6=j s.t. j / ∈R l (5.27) h T jj v j 6= 0, ∀j∈ [L]. (5.28) 99 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks Note that the channel gain vectors h jl will be sparse, only containing at most 3 non-zero elements corresponding toT l ∩N j . Now, each of the vectors v l , l∈ [L] can be written as v l =q l P l   1 ¯ v l   , (5.29) where q l is a non-zero scalar, P l is a|T l |×|T l | permutation matrix and ¯ v l is a vector of size (|T l |− 1)× 1. Also, for any two distinct pairs l6=j, the channel gain vector h jl can be permuted as ¯ h jl = P −1 l h jl , and we can partition ¯ h jl as ¯ h jl =   ¯ h (1) jl ¯ h (2) jl   , (5.30) where ¯ h (1) jl is a scalar and ¯ h (2) jl is of size (|T l |− 1)× 1. Therefore, the nulling condition in (5.27) can be rewritten as h ¯ h (1) jl ¯ h (2)T jl i   1 ¯ v l   = 0⇔ ¯ h (1) jl + ¯ h (2)T jl ¯ v l = 0. (5.31) Now, note that by the assumption that the set of 3 neighboring base stations of each receiver need to have cached the entire library, the packet sent by the virtual transmitter c Tx l should be cached at least by one base station around any receiver in the network. This means that the virtual transmitter c Tx l is connected to all theL receivers in the constructed MISO network. Now, since the packet sent by the virtual transmitter c Tx l is available in the caches of|R l | receivers in the network, the interference of each transmitter should be nulled at least at L−|R l |− 1 unintended receivers. On the other hand, for each receiver Rx j , the number of free variables in (5.31) is equal to|T l ∩N j |− 1. Therefore, the nulling equations are not satisfied unless the number of equations is no greater than the number of variables, or L−|R l |− 1≤ [ j/ ∈R l (T l ∩N j ) − 1≤|T l |− 1⇒L≤|T l | +|R l |. (5.32) Since the above inequality holds for all l∈ [L], the proof is complete. 100 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks 5.4.2 Integer Program Formulation Equipped with Lemma 5.1, we define a set of packetsD m selected to be transmitted at block m to be feasible if its size satisfies condition (5.22) in Lemma 5.1. We can then write the following integer program (P4) to minimize the number of required communication blocks for any given caching realization and set of receiver demands: min H (P4-1) s.t. H [ m=1 D m = CK R [ j=1 W d j \Q j (P4-2) D m is feasible, ∀m∈ [H], (P4-3) where (P4-2) states that all the demanded packets that are not cached at the requesting receivers need to be delivered by the base stations over the H blocks of communication. 5.4.3 Relaxing Worst-Case Demands to Average Demands and Optimiz- ing over Caching Realizations We can now write an optimization problem to minimize the number of communication blocks required for delivering the worst-case demands optimized over the caching realizations. However, before that, we need to introduce some notation. Given any caching realization {P i } C i=1 ,{Q j } CK R j=1 , we can break each file W n ,n∈ [N], in the library into (2 C −1)(2 CK R ) subfiles{W n,T,R } T⊆ ∅ [C],R⊆[CK R ] , whereW n,T,R denotes the subfile of W n exclusively stored in the caches of the base stations inT and receivers inR, and we use the shorthand notationT ⊆ ∅ [C] to denoteT ⊆ [C],T 6=∅. We define a n,T,R as the number of packets in W n,T,R . To lower bound the number of communication blocks needed to deliver the worst-case de- mands, we can instead focus on the average number of blocks needed for delivering distinct receiver demands. Therefore, denoting the answer to the optimization problem (P4) by H ∗ {P i } C i=1 ,{Q j } CK R j=1 , d , the below optimization problem yields the number of communi- cation blocks averaged over all theπ(N,CK R ) = N! (N−CK R )! permutations of distinct receiver 101 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks demands, denoted byP N,CK R : min {P i } C i=1 ,{Q j } CK R j=1 1 π(N,CK R ) X d∈P N,CK R H ∗ {P i } C i=1 ,{Q j } CK R j=1 , d (P5-1) s.t. X T⊆ ∅ [C] X R⊆[CK R ] a n,T,R =F, ∀n∈ [N] (P5-2) N X n=1 X R⊆[CK R ] X T⊆[C]: i∈T a n,T,R ≤μ T NF,∀i∈ [C] (P5-3) N X n=1 X T⊆ ∅ [C] X R⊆[CK R ]: j∈R a n,T,R ≤μ R NF,∀j∈ [CK R ] (P5-4) a n,T,R ≥ 0,∀n∈ [N],∀T ⊆ ∅ [C],∀R⊆ [CK R ]. (P5-5) 5.4.4 Lower Bound on the Number of Communication Blocks Having the optimization problem in (P5), we now present the following lemma which pro- vides a lower bound on the value of (P5). Lemma 5.2. The value of the optimization problem (P5) is bounded from below by CK R F (1−μ R ) 2 Cμ T +CK R μ R . Proof. The proof follows the same lines as Appendix F. Since the total number of packets delivered over the channel is CK R (1−μ R )F in the optimization problem (P5), Lemma 5.2 immediately yields the following upper bound on the one-shot linear sum-DoF: LDoF ∗ sum (C)≤ CK R (1−μ R )F CK R F (1−μ R ) 2 Cμ T +CK R μ R = Cμ T +CK R μ R 1−μ R , which implies that LDoF ∗ ≤ μ T +K R μ R 1−μ R . 102 Chapter 5. Cache-Aided Interference Management in Wireless Cellular Networks Combining the above bound with the trivial bound on the one-shot linear DoF per cell which is the number of receivers, K R , we have LDoF ∗ ≤ min μ T +K R μ R 1−μ R ,K R . (5.33) Now, consider the following two cases: • μ R ≤ 1 2 : In this case, (5.33) implies that LDoF ∗ ≤ min ( μ T +K R μ R 1− 1 2 ,K R ) = min{2(μ T +K R μ R ),K R }. • μ R > 1 2 : In this case, (5.12) implies that one-shot linear per cell DoF of LDoF ∗ > min μ T + K R 2 − 2,K R > K R 2 − 2, can be achieved, while the upper bound in (5.33) implies that LDoF ∗ ≤K R . Therefore, in both cases, the inner bound in (5.12) is within additive and multiplicative gaps of 2 of the outer bound in (5.12), which completes the proof of the converse of Theorem 5.1. 103 Chapter 6 Concluding Remarks and Future Directions In this dissertation, we focused on two user-centric architectures for 5G, namely device- to-device communication and cache-aided wireless networks, which bring connectivity and contents closer to the users. In both settings, we studied the fundamental limits of spectrum sharing and interference management through the lens of information theory. We first introduced a condition for fully-connected fully-asymmetric K-user interference channels under which power control at the transmitters and treating interference as noise at the receivers, in short, the TIN scheme, was proven to be GDoF-optimal. The GDoF re- gion under this condition was shown to be a polyhedron. The analysis was also generalized to show that under the same condition, TIN can achieve the whole capacity region of the network to within a constant gap that only depends on the number of users K. Further- more, the achievable GDoF region by TIN for general values of channel gains in a K-user interference channel was also characterized fully. Based on the aforementioned result, we presented a new spectrum sharing scheme, called information-theoretic link scheduling (ITLinQ), in order to manage the interference in wire- less D2D networks. At each time, the scheme schedules a subset of links which satisfy the TIN optimality condition. We presented a performance guarantee of the ITLinQ scheme by characterizing the fraction of the capacity region that it is able to achieve in a specific network setting. Moreover, we developed a distributed way of implementing the ITLinQ scheme and showed, via numerical analysis, that it yields considerable gains over FlashLinQ, 104 Chapter 6. Concluding Remarks and Future Directions a similar recently-proposed scheduling algorithm. We also showed how to address the issue of fairness among the links in the network by introducing a fair version of the distributed ITLinQ scheme. In the second part of the dissertation, we considered a wireless network setting with arbitrary numbers of transmitters and receivers, where all transmitters and receivers in the network are equipped with cache memories of specific sizes. We characterized the one-shot linear sum-DoF of the network to within a gap of 2. In particular, we showed that the one-shot linear sum-DoF of the network is proportional to the aggregate cache size in the network, even though the cache of each node is isolated from all the other nodes. We presented an achievable scheme which loads the caches carefully in order to maximize the opportunity for zero-forcing the outgoing interference from the transmitters and interference cancellation due to previously-cached content at the receivers. We also demonstrated that the achievable one-shot linear sum-DoF of our scheme is within a multiplicative factor of 2 of the optimal one-shot linear sum-DoF by bounding the number of communication blocks required to deliver any set of requested files to the receivers using an integer programming approach. Finally, as an extension of the aforementioned result, we considered cache-aided cellular networks in which each cell has a central base station and multiple receivers around its corners and all the base stations and receivers are equipped with cache memories. We characterized the one-shot linear DoF per cell to within additive and multiplicative gaps of 2 for all system parameters. Our result demonstrated that the partial connectivity of the network and also the unknown locations of the receivers during the prefetching phase has almost no impact on the performance of the system compared to the centralized fully- connected case. We provided a randomized prefetching scheme at the receivers and a delivery scheme which uses the caches at the base stations and receivers in order to zero- force the outgoing interference from the base stations and cancel the incoming interference at the receivers, respectively. We also provided an outer bound which shows that as the number of receivers per cell grows to infinity, the DoF per cell achievable by our introduced scheme is within a multiplicative factor of 2 of the optimal DoF per cell under one-shot linear schemes. The results presented in this dissertation reveal numerous research directions worth pursu- ing. First, it would be interesting to determine whether condition (2.9) is also necessary for TIN to be GDoF-optimal. For the 2-user case, it can be shown that except for a set of channel gain values of measure zero, the condition is also necessary for the GDoF-optimality 105 Chapter 6. Concluding Remarks and Future Directions of TIN. However, going beyond two users makes the problem more challenging. We sus- pect that again outside a set of channel gains of measure zero, condition (2.9) is necessary for TIN to be GDoF-optimal. However, the necessity of this condition for the networks comprising more than 2 users remains open. Second, one can think about generalizing the ITLinQ scheme to multihop D2D networks. Especially, the result in [71] shows that coupling between interference management and relaying strategies can provide significant gains. Hence, it would be worth figuring out the performance improvement that ITLinQ is able to guarantee in a multihop setting. Third, it would be interesting to investigate the impact of more advanced interference management techniques, such as successive interference cancellation, on ITLinQ. For example, recent results in [72–76] demonstrate that by a careful use of repetition coding at the transmitters and temporal interference neutralization at the receivers, one can achieve spectral efficiency gains that are considerably beyond the common interference avoidance approach. Thus, an interesting future direction can be to characterize the impact of structured repetition coding and temporal interference neutralization on ITLinQ. A fourth direction would be to combine caching with more sophisticated interference man- agement schemes. We have reported some initial restuls in [77] where we used the replication in the cache contents at the transmitters in order to improve the system performance using ITLinQ. We derived an achievable fraction of the users in the network all of which can be served simultaneously by the so-called cached ITLinQ scheme. This is done through a greedy source-user association policy called the “greedy closest-source policy” . Our result shows that such an association policy can provide an order of magnitude gain in spectral reuse over conventional schemes. We have also numerically demonstrated that using ITLinQ in a network with users caching the files can provide order of magnitude throughput gains over the state-of-the-art cluster-based delivery scheme proposed in [58], for both small and large library sizes. It would be interesting to study the role of transmitter and receiver caches illustrated in this work in improving the achievable system throughput that more sophisticated delivery schemes such as ITLinQ can provide. Another important extension would be to consider cache-aided cellular networks where the receivers may also receive interference from base stations farther than its immediate neighboring ones. Developing algorithms for interference mitigation in such a scenario is a problem worth pursuing. Last but not least, we studied the impact of caching in cellular networks from a DoF perspective. It would be interesting to analyze the actual rate 106 Chapter 6. Concluding Remarks and Future Directions improvement that caching can provide in other network scenarios such as device-to-device (D2D) networks (see, e.g., [58,77]). 107 Appendix A Replacing α ij < 0 with α ij = 0 in (2.2) We refer to the channel with potentially negative α ij ’s (i,j∈ [K]) as the original channel, and the channel with all negative α ij ’s replaced with zeros as the modified channel. To prove the claim that replacing α ij < 0 with α ij = 0 does not impact the GDoF or the constant gap result, we go through the following steps: • First, we show that the capacity region of the original channel is within a constant gap per user to that of the modified channel, which also shows that the two channels have the same GDoF region. The proof requires two directions, namely C original ⊆C modified + constant, and C modified ⊆C original + constant. The channel input-output relationship for the original channel is given by ¯ Y k (t) = K X i=1 √ P ¯ α ki e jθ ki X i (t) + ¯ Z k (t), ∀k∈ [K], 108 Appendix A. Replacing α ij < 0 with α ij = 0 in (2.2) where ¯ Z k (t) ∼ CN (0, 1) and certain ¯ α ki ’s might be negative. Define α ij , ¯ α + ij , ∀i,j ∈ [K], where for any real number β, β + stands for max{0,β}. The received signal of user k, k∈ [K], in the modified channel is Y k (t) = K X i=1 √ P α ki e jθ ki X i (t) +Z k (t) = K X i=1 q P ¯ α + ki e jθ ki X i (t) +Z k (t) = X i∈N k e jθ ki X i (t) + X i/ ∈N k √ P ¯ α ki e jθ ki X i (t) +Z k (t), where Z k (t)∼CN (0, 1) is independent of ¯ Z k (t),N k is the set of transmitter indices whose link to receiverk is with negative channel strength level in the original channel; i.e., N k = i∈ [K] : ¯ α ki < 0 . First, we proveC original ⊆C modified + constant. DefineW,{W 1 ,W 2 ,...,W K }, and let ˆ Y k (t) = ¯ Y k (t)−Y k (t) = X i∈N k ( √ P ¯ α ki − 1)e jθ ki X i (t) + ¯ Z k (t)−Z k (t). Then, we have I(W k ; ¯ Y n k )≤I(W k ;Y n k , ˆ Y n k ) =I(W k ;Y n k ) +I(W k ; ˆ Y n k |Y n k ) =I(W k ;Y n k ) +h( ˆ Y n k |Y n k )−h( ˆ Y n k |Y n k ,W k ) (a) ≤ I(W k ;Y n k ) +h( ˆ Y n k )−h( ˆ Y n k |Y n k ,W) =I(W k ;Y n k ) +h( ˆ Y n k )−h( ¯ Z n k −Z n k |W,Z n k ) (b) ≤ I(W k ;Y n k ) + n X t=1 h( ˆ Y k (t))−h( ¯ Z n k ) (c) ≤ I(W k ;Y n k ) +n log[2πe(K + 2)]−n log(2πe) =I(W k ;Y n k ) +n log(K + 2), where step (a) follows the facts that dropping conditioning does not reduce entropy (for the second term) and adding conditioning does not increase entropy (for the third 109 Appendix A. Replacing α ij < 0 with α ij = 0 in (2.2) term), step (b) follows the chain rule and the fact that dropping conditioning does not reduce entropy, and step (c) holds since|N k |≤ K and Gaussian distribution maximizes differential entropy under a given variance constraint. This implies that C original ⊆C modified + constant. Similarly, we can prove the other directionC modified ⊆C original + constant, I(W k ;Y n k )≤I(W k ; ¯ Y n k , ˆ Y n k ) =I(W k ; ¯ Y n k ) +I(W k ; ˆ Y n k | ¯ Y n k ) =I(W k ; ¯ Y n k ) +h( ˆ Y n k | ¯ Y n k )−h( ˆ Y n k | ¯ Y n k ,W k ) ≤I(W k ; ¯ Y n k ) +h( ˆ Y n k )−h( ˆ Y n k | ¯ Y n k ,W) =I(W k ; ¯ Y n k ) +h( ˆ Y n k )−h( ¯ Z n k −Z n k |W, ¯ Z n k ) ≤I(W k ; ¯ Y n k ) + n X t=1 h( ˆ Y k (t))−h(Z n k ) ≤I(W k ; ¯ Y n k ) +n log(K + 2). • Next, we prove that, regardless of whether or not TIN is GDoF-optimal, the original and modified channels always have the same achievable GDoF region through TIN P ∗ . To this end, we only need to show that using the same transmit power vector (P r 1 ,P r 2 ,...,P r K ) in the modified and original channels, useri∈ [K] in both channels achieves the same GDoF value by treating interference as noise. Recall that in the modified channel, when each transmitter i uses a transmit power of P r i , r i ≤ 0 and each receiver treats all the incoming interference as noise, the rate achieved by user i is R i = log 1 + P α ii +r i 1 + P j6=i P α ij +r j ! , and the achievable GDoF by user i through TIN equals d i = max{0,α ii +r i − max{0, max j:j6=i (α ij +r j )}}. (A.1) Now consider the original channel. Similarly, applying the same transmit power P r i to each transmitteri and treating interference as noise at each receiver, the achievable 110 Appendix A. Replacing α ij < 0 with α ij = 0 in (2.2) rate of user i is ¯ R i = log 1 + P ¯ α ii +r i 1 + P j6=i P ¯ α ij +r j ! . In the original channel, denote the set of user indices whose direct link is with negative channel strength level asU. For all the users i∈U, it is easy to verify that the achievable GDoF through TIN is ¯ d i = 0, (A.2) while for the users i / ∈U, we have ¯ d i = max{0, ¯ α ii +r i − max{0, max j:j6=i (¯ α ij +r j )}} (d) = max{0, ¯ α + ii +r i − max{0, max j:j6=i (¯ α ij +r j )}} (e) = max{0, ¯ α + ii +r i − max{0, max j:j6=i (¯ α + ij +r j )}} = max{0,α ii +r i − max{0, max j:j6=i (α ij +r j )}}, (A.3) where step (d) follows from the fact that ¯ α + ii = ¯ α ii for users i / ∈U, and step (e) holds since when ¯ α ij < 0, we have ¯ α ij +r j < 0, ¯ α + ij +r j ≤ 0, and replacing the former with the latter does not impact the final result. Combining (A.2) and (A.3), we obtain that for user i∈ [K] ¯ d i = max{0,α ii +r i − max{0, max j:j6=i (α ij +r j )}}. (A.4) Comparing (A.1) with (A.4), we establish that the original and modified channels have the same TIN GDoF regionP ∗ . • Finally, we show that for the original channel, TIN achieves the capacity region to within log 2 (3K) bits, when the following condition holds ¯ α + ii ≥ max j:j6=i {¯ α + ji } + max k:k6=i {¯ α + ik }, ∀i∈ [K]. (A.5) 111 Appendix A. Replacing α ij < 0 with α ij = 0 in (2.2) We start with the converse. For the original channel, when condition (A.5) holds, based on Theorem 2.3, we have R i ≤ log 2 (1 +P ¯ α ii )≤ log 2 (1 +P ¯ α + ii ) ≤ ¯ α + ii log 2 P + 1 =α ii log 2 P + 1, ∀i∈ [K] (A.6) m X j=1 R i j ≤ m X j=1 log 2 1 +P ¯ α i j i j+1 + P ¯ α i j i j 1 +P ¯ α i j−1 i j ! ≤ m X j=1 log 2   1 +P ¯ α i j i j+1 + P ¯ α + i j i j 1 +P ¯ α i j−1 i j   = m X j=1 log 2   1 +P ¯ α i j i j+1 + P ¯ α + i j i j P 0 +P ¯ α i j−1 i j   < m X j=1 log 2   1 +P ¯ α i j i j+1 + P ¯ α + i j i j P ¯ α + i j−1 i j   = m X j=1 log 2   P ¯ α + i j−1 i j +P ¯ α i j i j+1 +¯ α + i j−1 i j +P ¯ α + i j i j P ¯ α + i j−1 i j   ≤ m X j=1 log 2   3P ¯ α + i j i j P ¯ α + i j−1 i j   = m X j=1 [(¯ α + i j i j − ¯ α + i j−1 i j ) log 2 P + log 2 3] = m X j=1 [(α i j i j −α i j−1 i j ) log 2 P + log 2 3], (A.7) for all cycles (i 1 ,i 2 ,...,i m )∈ Π K ,∀m∈{2, 3,...,K}. Comparing (2.32) and (2.33) (the outer bounds of the modified channel in Section 2.3) with (A.6) and (A.7), one can find that the modified and original channels have exactly the same outer bounds. Then, consider the achievability. For the modified channel, denote the achievable rate region through TIN under condition (2.9) (the TIN-optimality condition in Theorem 2.1) asR TIN . In the modified channel, for any rate tuple R TIN = (R 1 ,R 2 ,...,R K ) in the achievable TIN regionR TIN , we have a corresponding transmit power vector P TIN = (P r 1 ,P r 2 ,...,P r K ). We denote the set of the transmit power vectors for all the rate tuples inR TIN asP TIN . In the original channel, applying the same set of transmit power vectorsP TIN for transmitters and treating interference as noise at each receiver, 112 Appendix A. Replacing α ij < 0 with α ij = 0 in (2.2) one can obtain an achievable TIN region ¯ R TIN such that (i) any user k / ∈U achieves a rate no less than that in the modified channel when the same transmit power vector is utilized, as for that user the interfering links in the original channel are no stronger than those in the modified channel, which indicates that for users k / ∈U the constant gap cannot increase in the original channel; (ii) the constant gap for any user k∈U is at most 1 bit, since according to (A.6) the achievable rate of that user is upper bounded by 1 bit. Therefore, combining with the constant gap result for the modified channel (see Theorem 2.4), we establish that for the original channel, when condition (A.5) is satisfied, TIN achieves to within log 2 (3K) bits of the capacity region. Combining the above steps, we prove that assigning a 0 value to negativeα ij ,∀i,j∈ [K], has no impact on the GDoF or the constant gap results presented in Chapter 2 (i.e., Theorems 2.1, 2.4 and 2.5). 113 Appendix B Characterization of The Polyhedral TIN RegionP To prove the equivalence of the two representations (2.12)-(2.14) and (2.15)-(2.18), we first define that ¯ P is the set of all K-tuples (d 1 ,d 2 ,...,d K ) for which there exist r i ’s, i∈ [K], such that (2.15)-(2.18) hold, and then show thatP = ¯ P. Obviously, it is true thatP⊆ ¯ P. This is because if (d 1 ,d 2 ,...,d K )∈P, then there existr i ’s such that (2.12)-(2.14) are satisfied, which in turn implies that (2.15)-(2.18) are satisfied. This leads to the fact that (d 1 ,d 2 ,...,d K )∈ ¯ P. Also, it is easily verified that ¯ P⊆P. This is because when (d 1 ,d 2 ,...,d K )∈ ¯ P, there exist r i ’s such that (2.15)-(2.18) are satisfied, implying that d i ≤α ii +r i − max{0, max j:j6=i (α ij +r j )} ⇔r i ≥d i −α ii + max{0, max j:j6=i (α ij +r j )}, ∀i∈ [K]. (B.1) Now, to prove (d 1 ,d 2 ,...,d K )∈P, we need to show that when (2.15)-(2.18) hold, there exist ¯ r i ’s such that ¯ r i ≤ 0, ∀i∈ [K], (B.2) ¯ r i =d i −α ii + max{0, max j:j6=i (α ij + ¯ r j )}, ∀i∈ [K]. (B.3) 114 Appendix B. Characterization of The Polyhedral TIN RegionP Denote r = (r 1 ,r 2 ,...,r K ) and ¯ r = (¯ r 1 , ¯ r 2 ,..., ¯ r K ). Writing (B.2) and (B.3) in the vector form, we have ¯ r≤ 0, (B.4) ¯ r =f(¯ r). (B.5) Then it is easy to verify the following conditions are satisfied: (i)f is a continuous increasing function; (ii) From (B.1), the setS ¯ r ={¯ r : ¯ r≥f(¯ r)} is nonempty; (iii) The setS ¯ r is bounded from below. Then, according to the fixed point theorem in [78] (Proposition 6 in [78]), we know that for each r∈S ¯ r , there exists a ¯ r≤ r such that ¯ r =f(¯ r). Combining this result with (2.15), we know when (2.15)-(2.18) hold, there exist such ¯ r i ’s satisfying (B.2) and (B.3), which implies that (d 1 ,d 2 ,...,d K )∈P. Therefore, we establish thatP = ¯ P. 115 Appendix C Proof of Theorem 2.3 First, each individual bound in (2.28) is simply the cut-set upper bound for user i∈ [K]. Next, we consider the cyclic bound (2.29), where the modulo-m arithmetic is implicitly used on the user indices, e.g.,i m =i 0 . For any cyclic sequence (i 1 ,i 2 ,...,i m )∈ Π K , we start with the fully connected K-user interference channel with input-output relationship (2.1), and go through the following steps: • Eliminate all the users i∈ [K]\{i 1 ,i 2 ,...,i m } and their desired messages; • Remove all the interfering links but the links from transmitter i j to receiver i j−1 , ∀j∈{1, 2,...,m}. We end up with the m-user cyclic interference channel as depicted in Fig. C.1. The above two steps cannot hurt the rates of the remaining messages. Therefore, the sum rate of users i∈{i 1 ,i 2 ,...,i m } in the original K-user interference channel is upper bounded by that of the m-user cyclic interference channel. Define S i j (t) =h i j−1 i j ˜ X i j (t) +Z i j−1 (t), ∀j∈{1, 2,...,m}. Then for receiver i j , we provide S n i j through a genie. From Fano’s inequality, we have n(R i j −) ≤I(W i j ;Y n i j ,S n i j ) =h(Y n i j ,S n i j )−h(Y n i j ,S n i j |W i j ) 116 Appendix C. Proof of Theorem 2.3 …… …… ࢀ ࢏ ૚ ࢀ ࢏ ૛ ࢀ ࢏ ૜ ࢀ ࢏ ࢓ ࡾ ࢏ ૚ ࡾ ࢏ ૛ ࡾ ࢏ ૜ ࡾ ࢏ ࢓ Figure C.1: The m-user cyclic interference channel. =h(S n i j ) +h(Y n i j |S n i j )−h(S n i j |W i j )−h(Y n i j |S n i j ,W i j ) =h(S n i j ) +h(Y n i j |S n i j )−h(Z n i j−1 )−h(S n i j+1 ). Taking the sum of n(R i j −) for all j∈{1, 2,...,m}, we obtain n m X j=1 (R i j −)≤ m X j=1 h h(Y n i j |S n i j )−h(Z n i j ) i ≤ n X t=1 m X j=1 h h(Y i j (t)|S i j (t))−h(Z i j (t)) i , where the last inequality follows chain rule and the fact that dropping conditioning does not reduce entropy. Finally, using the fact that the circularly symmetric Gaussian distribution maximizes conditional differential entropy under a given covariance constraint, we end up with the desired outer bound m X j=1 R i j ≤ m X j=1 log 1 +|h i j i j+1 | 2 P i j+1 + |h i j i j | 2 P i j 1 +|h i j−1 i j | 2 P i j . 117 Appendix D Proof of Theorem 2.5 We prove the theorem in two steps. • Step 1: S S⊆[K] P S ⊆P ∗ . It suffices to show that for allS⊆ [K],P S ⊆P ∗ ; i.e., the regionP S can be achieved through TIN. Note that ifS =∅, thenP S =P ∅ =P⊆P ∗ . Now, ifS6=∅, then to make the users inS silent, we setr i =−∞,∀i∈S. This forces d i = 0,∀i∈S. Then, for the remaining users, i.e., the users inS c , we use polyhedral TIN. Therefore, the polyhedral TIN region where all the users inS are removed from the network, can be achieved. This region is in factP S , and hence,P S ⊆P ∗ . • Step 2: P ∗ ⊆ S S⊆[K] P S . To prove this, we first define the sets ˜ P S asP S restricted to strictly positive GDoF’s for users inS c ; i.e., ˜ P S ={(d 1 ,d 2 ,...,d K )∈P S :d i > 0,∀i∈S c }, for anyS⊆ [K]. It is obvious that ˜ P S ⊆P S and therefore, [ S⊆[K] ˜ P S ⊆ [ S⊆[K] P S . (D.1) Now, we will prove that P ∗ ⊆ S S⊆[K] ˜ P S . Assume there exists a GDoF point (d 1 ,d 2 ,...,d K ) lying outside all of the sets ˜ P S . Such a point should satisfy at least one of the following conditions: 118 Appendix D. Proof of Theorem 2.5 – d i < 0 or d i >α ii for some user i∈ [K]. In this case, it is trivial that the GDoF point is not achievable by TIN. – P m j=1 d i j > P m j=1 (α i j i j −α i j−1 i j ) for some cyclic sequence (i 1 ,i 2 ,...,i m )∈ Π K such that d i j > 0,∀j∈{1, 2,...,m}. Note the modulo-m arithmetic is implicitly used on the user indices, e.g., i m = i 0 . In this case, we show that this GDoF point cannot belong toP ∗ by contradiction. Assume otherwise; i.e., suppose (d 1 ,d 2 ,...,d K ) is achievable by TIN. For all j∈{1, 2,...,m}, since it is assumed that d i j > 0, there exist r i j ’s such that d i j =r i j +α i j i j − max{0, max i k 6=i j (r i k +α i j i k )}. Therefore, we will have m X j=1 r i j +α i j i j − max{0, max i k 6=i j (r i k +α i j i k )}> m X j=1 (α i j i j −α i j−1 i j ) ⇒ m X j=1 r i j +α i j−1 i j − max{0, max i k 6=i j (r i k +α i j i k )}> 0. (D.2) On the other hand, for all j∈{1, 2,...,m} we have max{0, max i k 6=i j (r i k +α i j i k )}≥r i j+1 +α i j i j+1 , which in turn implies that m X j=1 r i j +α i j−1 i j − max{0, max i k 6=i j (r i k +α i j i k )} ≤ m X j=1 r i j +α i j−1 i j − (r i j+1 +α i j i j+1 ) = m X j=1 (r i j −r i j+1 ) + m X j=1 (α i j−1 i j −α i j i j+1 ) = 0. But considering (D.2), this is a contradiction. Therefore in this case, the GDoF point is not achievable by TIN, too. This implies thatP ∗ ⊆ S S⊆[K] ˜ P S , which combined with (D.1) yieldsP ∗ ⊆ S S⊆[K] P S . 119 Appendix D. Proof of Theorem 2.5 Combining steps 1 and 2 leads to (2.42), therefore completing the proof. 120 Appendix E Proof of Lemma 4.1 For anyT ⊆ [K T ] s.t.|T| =t T and for anyl∈ [t T +t R ], it is clear that the setT⊕ K T (l−1) is of size t T . Also, for anyR⊆ [K R ] s.t.|R| =t T +t R , and for any permutation π∈ Π circ R , the vectorπ[l + 1 :l +t R ] is of sizet R and the vectorπ[l +t R + 1 :l +t R +t T − 1] is of size t T − 1. Furthermore, note that W d π(l) ,T⊕ K T (l−1),π[l+1:l+t R ],π[l+t R +1:l+t R +t T −1] is a subfile of the file W d π(l) requested by receiver Rx π(l) . However, since π(l) / ∈ π[l + 1 : l +t R ], receiver Rx π(l) has not stored the packets in this subfile in its cache and therefore, this subfile needs to be delivered to this receiver. Finally, each set inside the union in (4.19) is composed of t T +t R subfiles. The number of such sets is equal to K T t T ! K R t T +t R ! (t T +t R − 1)!. (E.1) Hence, the total number of subfiles in (4.19) is equal to K T t T ! K R t T +t R ! (t T +t R − 1)!(t T +t R ) = K T t T ! K R t T +t R ! (t T +t R )!. (E.2) On the other hand, each receiver Rx j has already cached K T t T K R −1 t R −1 subfiles as in (4.16) in its cache, and needs the rest of the subfiles of its requested file, i.e., K T t T K R −1 t R subfiles, where each subfile is further partitioned into t R ![K R −(t R +1)]! [K R −(t R +t T )]! smaller subfiles. Hence, the 121 Appendix E. Proof of Lemma 4.1 total number of small subfiles that need to be delivered to all the receivers is equal to K R " K T t T ! K R − 1 t R !# t R ![K R − (t R + 1)]! [K R − (t R +t T )]! = K T t T ! K R t T +t R ! (t T +t R )!, (E.3) which equals the total number of small subfiles in (4.19), calculated in (E.2). Consequently, the set of requested subfiles which are not cached at the corresponding receivers can be partitioned as in (4.19), hence the proof is complete. 122 Appendix F Proof of Lemma 4.4 According to the constraint (4.31), each of the packets of order s, which are available at s nodes, either on the transmitter side or the receiver side, can be scheduled with at most s− 1 packets of the same order. Therefore, for any given caching realization and set of demands, we have the lower bound H ∗ {P i } K T i=1 ,{Q i } K R i=1 , d ≥ K T +K R X s=K R K R X j=1 X T⊆[K T ]: |T|∈[s] X R⊆[K R ]: |R|=s−|S T | j/ ∈R a d j ,T,R K R + K R −1 X s=1 K R X j=1 X T⊆[K T ]: |T|∈[s] X R⊆[K R ]: |R|=s−|S T | j/ ∈R a d j ,T,R s ≥ K T +K R X s=1 K R X j=1 X T⊆[K T ]: |T|∈[s] X R⊆[K R ]: |R|=s−|S T | j/ ∈R a d j ,T,R s . (F.1) Now, denoting the objective function in (P3-1) by ¯ H {P i } K T i=1 ,{Q i } K R i=1 , we have ¯ H {P i } K T i=1 ,{Q i } K R i=1 ≥ 1 π(N,K R ) K T +K R X s=1 1 s K R X j=1 X T⊆[K T ]: |T|∈[s] X R⊆[K R ]: |R|=s−|S T | j/ ∈R π(N− 1,K R − 1) N X n=1 a n,T,R = 1 N K T +K R X s=1 1 s K R X j=1 X T⊆[K T ]: |T|∈[s] X R⊆[K R ]: |R|=s−|S T | j/ ∈R N X n=1 a n,T,R 123 Appendix F. Proof of Lemma 4.4 = 1 N K T X r=1 K R X r 0 =0 1 r +r 0 K R X j=1 X T⊆[K T ]: |T|=r X R⊆[K R ]: |R|=r 0 j/ ∈R N X n=1 a n,T,R = 1 N K T X r=1 K R X r 0 =0 K R −r 0 r +r 0 X T⊆[K T ]: |T|=r X R⊆[K R ]: |R|=r 0 N X n=1 a n,T,R = 1 N K T X r=1 K R −1 X r 0 =0 b r,r 0 r +r 0 , (F.2) where for any r∈ [K T ] and r 0 ∈ [K R − 1]∪{0}, we define b r,r 0, K R X j=1 X T⊆[K T ]: |T|=r X R⊆[K R ]: |R|=r 0 j/ ∈R N X n=1 a n,T,R = (K R −r 0 ) X T⊆[K T ]: |T|=r X R⊆[K R ]: |R|=r 0 N X n=1 a n,T,R . (F.3) Moreover, adding the constraint in (P3-3) over all transmitters yields K T M T F≥ K T X i=1 N X n=1 X R⊆[K R ] X T⊆[K T ]: i∈T a n,T,R (F.4) = N X n=1 X R⊆[K R ] K T X i=1 X T⊆[K T ]: i∈T a n,T,R (F.5) = N X n=1 X R⊆[K R ] K T X r=1 r X T⊆[K T ]: |T|=r a n,T,R . (F.6) Likewise, adding the constraint in (P3-4) over all receivers yields K R M R F≥ K R X j=1 N X n=1 X T⊆ ∅ [K T ] X R⊆[K R ]: j∈R a n,T,R (F.7) = N X n=1 X T⊆ ∅ [K T ] K R X j=1 X R⊆[K R ]: j∈R a n,T,R (F.8) = N X n=1 X T⊆ ∅ [K T ] K R X r 0 =0 r 0 X R⊆[K R ]: |R|=r 0 a n,T,R , (F.9) 124 Appendix F. Proof of Lemma 4.4 and from (F.6) and (F.9), we have (K T M T +K R M R )F≥ N X n=1      X R⊆[K R ] K T X r=1 r X T⊆[K T ]: |T|=r a n,T,R + X T⊆[K T ] K R X r 0 =0 r 0 X R⊆[K R ]: |R|=r 0 a n,T,R      (F.10) = K T X r=1 K R X r 0 =0 (r +r 0 ) X T⊆[K T ]: |T|=r X R⊆[K R ]: |R|=r 0 N X n=1 a n,T,R (F.11) ≥ K T X r=1 K R −1 X r 0 =0 r +r 0 K R −r 0 b r,r 0. (F.12) Now, using the Cauchy-Schwarz inequality, we can write K R −1 X r 0 =0 b r,r 0≤ v u u t K R −1 X r 0 =0 r +r 0 K R −r 0 b r,r 0 v u u t K R −1 X r 0 =0 K R −r 0 r +r 0 b r,r 0. (F.13) Summing the above inequality over r yields K T X r=1 K R −1 X r 0 =0 b r,r 0≤ K T X r=1   v u u t K R −1 X r 0 =0 r +r 0 K R −r 0 b r,r 0 v u u t K R −1 X r 0 =0 K R −r 0 r +r 0 b r,r 0   (F.14) ≤ v u u t K T X r=1 K R −1 X r 0 =0 r +r 0 K R −r 0 b r,r 0 v u u t K T X r=1 K R −1 X r 0 =0 K R −r 0 r +r 0 b r,r 0 (F.15) ≤ q (K T M T +K R M R )F v u u t K T X r=1 K R −1 X r 0 =0 K R −r 0 r +r 0 b r,r 0, (F.16) where in (F.15) we have invoked the Cauchy-Schwarz inequality again and (F.16) follows from (F.12). On the other hand, we have K T X r=1 K R −1 X r 0 =0 b r,r 0 = K T X r=1 K R −1 X r 0 =0 K R X j=1 X T⊆[K T ]: |T|=r X R⊆[K R ]: |R|=r 0 j/ ∈R N X n=1 a n,T,R (F.17) 125 Appendix F. Proof of Lemma 4.4 = K T X r=1 K R X r 0 =0 K R X j=1              X T⊆[K T ]: |T|=r X R⊆[K R ]: |R|=r 0 N X n=1 a n,T,R      −         X T⊆[K T ]: |T|=r X R⊆[K R ]: |R|=r 0 j∈R N X n=1 a n,T,R                 (F.18) =K R   N X n=1 X T⊆ ∅ [K T ] X R⊆[K R ] a n,T,R   − K R X j=1 N X n=1 X T⊆ ∅ [K T ] X R⊆[K R ]: j∈R a n,T,R (F.19) ≥K R (N−M R )F, (F.20) where the inequality is due to (P3-2) and (F.7). Therefore, we can continue (F.2) to bound the objective function in (P3-1) as ¯ H {P i } K T i=1 ,{Q i } K R i=1 ≥ 1 N K T X r=1 K R −1 X r 0 =0 b r,r 0 r +r 0 (F.21) ≥ 1 K R N K T X r=1 K R −1 X r 0 =0 (K R −r 0 )b r,r 0 r +r 0 (F.22) ≥ 1 K R NF (K T M T +K R M R )   K T X r=1 K R −1 X r 0 =0 b r,r 0   2 (F.23) ≥ 1 K R NF (K T M T +K R M R ) K R (N−M R )F 2 (F.24) = K R NF 1− M R N 2 K T M T +K R M R , (F.25) where (F.23) and (F.24) follow from (F.16) and (F.20), respectively. 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Abstract (if available)

Abstract We focus on the fundamental limits of two possible architectures in fifth-generation mobile networks (5G) that bring connectivity and contents closer to the users in the network. In particular, we study the problem of interference management in wireless device-to-device (D2D) communication systems and cache-aided wireless networks. For D2D networks, we first focus on the so-called ""low-interference"" regime

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Asset Metadata

Creator Naderializadeh, Navid (author)

Core Title Fundamentals of two user-centric architectures for 5G: device-to-device communication and cache-aided interference management

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 11/15/2016

Defense Date 10/12/2016

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag 5G,caching,device-to-device communication,interference management,ITLinQ,OAI-PMH Harvest,wireless networks

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Avestimehr, Amir Salman (committee chair), Molisch, Andreas (committee member), Razaviyayn, Meisam (committee member)

Creator Email naderial@usc.edu,navid.naderi11@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-673567

Unique identifier UC11336617

Identifier etd-Naderializ-4920.pdf (filename),usctheses-c16-673567 (legacy record id)

Legacy Identifier etd-Naderializ-4920-0.pdf

Dmrecord 673567

Document Type Dissertation

Rights Naderializadeh, Navid

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