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Structured codes in network information theory
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Structured codes in network information theory
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STRUCTURED CODES IN NETWORK INFORMATION THEORY by Song-Nam Hong A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2014 Copyright 2014 Song-Nam Hong Dedication This dissertation is dedicated to my dear wife, Taeeun and lovely daughter, Chloe Dahyun, with great appreciation and happiness. ii Acknowledgements First, I would like to thank my advisor, Professor Giuseppe Caire. For me, he is an ideal advisor and I feel extremely fortunate to be able to work with him. He is passionate, energetic, rigorous, and highly supportive. He gave me the freedom in choosing what I wanted to work on, and always encouraged me to pursue further into the topics. He also tried his best to help me get around the many diculties I faced. He taught me the im- portance of a good example and to always keep searching for something bigger. Without his great encouragement and inspiration, this research work would not be possible. I am also very grateful for all the great guidance from my qualifying exam and de- fense committee members: Professor Andrea Molisch, Professor Urbashi Mitra, Professor Michael Neely, Professor Jason Fulman, and Professor Susan Montgomery. Their insight- ful comments and invaluable advice made this work technically sound and meaningful. I was so fortunate to have great opportunities to discuss with Professor Sholomo Shamai, Professor Bobak Nazer, and Dr. Dariush Divsalar. I want to express my special thanks to Professor Dong-Joon Shin for his endless support. Without him, I would not have a great chance to study at USC. I want to say thank you to all my groupmates, Huh Hoon, Ozgun Bursalioglu, An- suman Adhikary, Arash Saber Tehrani, Vasillis Ntranos, Ryan Rogalin, Myngyue Ji, Dilip Bethanabhotla, and Karthikeyan Shanmugam. It is you who made my life at USC much more memorable and be lled with joy. I remember all the wonderful moments I had with you. Thanks also to Hassan Ghozlan, Feng Hao, Junyang Shen, Seun Sangodoyin, Joongheon Kim and all the other members of Communication Science Institute (CSI) for many inspirational discussions inside and outside of research. iii Finally, I would like to thank my parents and family for their endless support, sacrice, and love. Without my dearest wife, Taeeun, I could not have done anything. Her full faith and love has been a source of power for me and I cannot express in words how grateful I am to her and also to her parents. My beloved daughter, Chloe Dahyun, I wish she will know someday that she is joy in Los Angeles. iv Table of Contents Dedication ii Acknowledgements iii List of Figures viii Abstract xi Chapter 1 Introduction 1 1.1 Cooperative Distributed Antenna Systems . . . . . . . . . . . . . . . . . . 3 1.2 Full-Duplex Relaying with Half-Duplex Relays . . . . . . . . . . . . . . . 7 1.3 Some Two-User Gaussian Networks with Cognition, Coordination, and Two Hops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter 2 Cooperative Distributed Antenna Systems 20 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Nested Lattice Codes . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.3 Compute and Forward . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Compute and Forward for the DAS Uplink . . . . . . . . . . . . . . . . . 25 2.3 Reverse Compute and Forward for the DAS Downlink . . . . . . . . . . . 29 2.4 Low-Complexity Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4.1 QCoF and LQF for the DAS Uplink . . . . . . . . . . . . . . . . . 37 2.4.2 RQCoF for the DAS Downlink . . . . . . . . . . . . . . . . . . . . 39 2.4.3 Gaussian Approximation . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Comparison with Known Schemes on the Wyner Model . . . . . . . . . . 43 2.5.1 Review of some Classical Coding Strategies . . . . . . . . . . . . . 43 2.5.1.1 Quantize reMap and Forward (QMF) . . . . . . . . . . . 43 2.5.1.2 Decode and Forward (DF) . . . . . . . . . . . . . . . . . 44 2.5.1.3 Compressed Dirty Paper Coding (CDPC) . . . . . . . . . 45 2.5.1.4 Compressed Zero Forcing Beamforming (CZFB) . . . . . 46 2.5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.6 Antenna and User Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.6.1 Antenna Selection for the DAS Uplink . . . . . . . . . . . . . . . . 50 2.6.1.1 AT selection for CoF and QCoF . . . . . . . . . . . . . . 51 v 2.6.1.2 AT selection for LQF . . . . . . . . . . . . . . . . . . . . 53 2.6.2 User Selection for the DAS Downlink . . . . . . . . . . . . . . . . . 53 2.6.3 Comparison on the Bernoulli-Gaussian Model . . . . . . . . . . . . 54 2.7 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.7.1 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.7.2 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter 3 Full-Duplex Relaying with Half-Duplex Relays 62 3.1 System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.1 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 Achievable Rates of Virtual Full-Duplex Relay Channel . . . . . . . . . . 68 3.3 Proof of Theorem 3.1: Achievable Coding Schemes . . . . . . . . . . . . . 75 3.3.1 DF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.2 AF with Successive Decoding . . . . . . . . . . . . . . . . . . . . . 76 3.3.3 QMF with Successive Decoding . . . . . . . . . . . . . . . . . . . . 78 3.3.4 CoF with Forward Substitution . . . . . . . . . . . . . . . . . . . . 85 3.4 Multihop Virtual Full-Duplex Relay Channel . . . . . . . . . . . . . . . . 88 3.4.1 AF with Successive Decoding . . . . . . . . . . . . . . . . . . . . . 95 3.4.2 QMF with Successive Decoding . . . . . . . . . . . . . . . . . . . . 97 3.4.3 CoF with Forward Substitution . . . . . . . . . . . . . . . . . . . . 103 Chapter 4 Some Two-User Gaussian Networks with Cognition, Coordination, and Two Hops 110 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1.1 Compute-and-Forward and Integer-Forcing . . . . . . . . . . . . . 111 4.2 Network-Coded Cognitive Interference Channel . . . . . . . . . . . . . . . 114 4.2.1 Capacity Region for nite-eld Network-Coded CIC . . . . . . . . 115 4.2.2 Scaled Precoded CoF . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.2.3 An achievable rate region for the Gaussian Network-Coded CIC . . 122 4.2.4 Generalized Degrees of Freedom . . . . . . . . . . . . . . . . . . . 127 4.3 Two-User MIMO IC: Coordination, Cognition, Two-Hop . . . . . . . . . . 130 4.3.1 CoF Framework based on Channel Integer Alignment . . . . . . . 134 4.3.2 Linear Precoding over deterministic networks . . . . . . . . . . . . 138 4.3.2.1 Network-Coded ICC . . . . . . . . . . . . . . . . . . . . . 138 4.3.2.2 2 2 2 IC . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.3.3 Network-Coded CIC . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4 Improving the sum rates using successive cancellation . . . . . . . . . . . 142 4.5 Optimization of achievable rates . . . . . . . . . . . . . . . . . . . . . . . 148 4.5.1 Finite SNR Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.6.1 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.6.1.1 Achievable scheme . . . . . . . . . . . . . . . . . . . . . . 154 4.6.1.2 Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.6.2 Proof of Corollary 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.6.3 Proof of Theorem 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . 159 vi Chapter 5 Conclusions 162 References 166 vii List of Figures 1.1 Cooperative Distributed Antenna System with 5 UTs and 4 ATs (e.g., K = 5 and L = 4), and digital backhaul links of rate R 0 . . . . . . . . . . . 4 1.2 Two-hop relay network with full-duplex relay. . . . . . . . . . . . . . . . . 9 1.3 Tow-hop relay network with virtual full-duplex relay. The 2R + denotes the inter-relay interference level. Black-solid lines are active for every even time slot and red-dashed lines are active for every odd time slot. . . . . . 9 1.4 In the classical CIC, the data router sends the one of information messages to the non-cognitive transmitter. In the Network-Coded CIC, the data router forwards \mixed data" to the non-cognitive transmitter. . . . . . . 14 1.5 The generalized degrees-of-freedom (GDoF) of the two-user Gaussian Network- Coded CIC. For the interference regimes with 1=2, the gap between the Network-Coded CIC and CIC becomes arbitrarily large as SNR and INR goes to innity. This shows that mixed-data at the non-cognitive transmitter can provide the unbounded capacity gain at high SNR. . . . . 17 2.1 DAS Uplink Architecture using Compute and Forward: L = 4 and K = 4. 26 2.2 DAS Downlink Architecture Using Reverse Compute and Forward: L = 4 and K = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 Implementation of the modulo operation (analog component-wise saw- tooth transformation) followed by the scalar quantization functionQ (=p)Z[j] () function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4 SNR = 25dB and L =1. Achievable rates per user as a function of R 0 , for the DAS uplink in the Wyner model case with inter-cell interference parameter = 0:7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.5 SNR = 25dB and L = 10. Achievable sum rates as a function of R 0 , for the DAS downlink in the Wyner model case with inter-cell interference parameter = 0:7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.6 DAS uplink withK = 5,L = 25 andR 0 = 6 bit/channel use: average sum rate vs. SNR on the Bernoulli-Gaussian model with = 0:5. . . . . . . . . 54 viii 2.7 DAS downlink with K = 25, L = 5 and R 0 = 6 bit/channel use: average sum rate vs. SNR on the Bernoulli-Gaussian model with = 0:5. . . . . . 55 2.8 DAS uplink with K = 5 and L = 50, Bernoulli-Gaussian model with = 0:5: Colors represent the relative gain of CoF versus QF (e.g., ratio of sum rates R CoF =R QF ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.9 DAS downlink with K = 50 and L = 5, Bernoulli-Gaussian model with = 0:5: Colors represent the relative gain of RCoF versus CDPC (e.g., ratio of sum rates R RCoF =R CDPC ). . . . . . . . . . . . . . . . . . . . . . . . 58 2.10 DAS uplink with SNR = 25 dB, K = 5 and L = 50: achievable sum rates as a function of R 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.11 DAS downlink with SNR = 25 dB, K = 50 and L = 5: achievable sum rates as a function of R 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1 Multihop virtual full-duplex relay channels when K = 5 (i.e., 6-hop net- work). Black-solid lines are active for every even time slot and red-dashed lines are active for every odd time slot. . . . . . . . . . . . . . . . . . . . . 62 3.2 Four possible cuts for two-hop relay channel with half-duplex relays. . . . 65 3.3 Simplied channel model in case of eliminating the inter-relay interference at destination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 SNR = 15. Achievable rates of various coding schemes as a function of the inter-relay interference level . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5 SNR = 30 dB. Achievable rates of various coding schemes as a function of the inter-relay interference level . . . . . . . . . . . . . . . . . . . . . . . 74 3.6 Simplied model for QMF. . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.7 DPC scheme for 3-hop virtual full-duplex relay channel. . . . . . . . . . . 89 3.8 Condensed network of (K + 1)-hop virtual full duplex relay channel. . . . 90 3.9 SNR = 20 dB. Achievable ergodic rates of various coding schemes averaging over 2 Unif(0:9; 1:1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.10 SNR = 20 dB. Achievable ergodic rates of various coding schemes averaging over 2 Unif(0:5; 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.11 Noise accumulation of multihop AF scheme. . . . . . . . . . . . . . . . . . 95 3.12 Time expanded 3-hop network. The ` k;t denotes the relay k's message at time slot t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.13 Equivalent model of QMF scheme for (K + 1)-hop network. . . . . . . . . 98 ix 3.14 Message ow over (K + 1)-hop virtual full-duplex relay channel where the coecients of black line and red line are 1 and q, respectively. . . . . . . . 103 4.1 Distributed zero-forcing precoding for nite-eld Network-Coded CIC. Dif- ferently from RLNC, the cognitive transmitter carefully chooses the coef- cients of linear combination according to the channel coecients q ij 's. . . 115 4.2 Encoding and decoding structures of the proposed achievability scheme. Transmitter 1 uses the DPC to cancel the interference at its intended receiver 1, and also performs precoding over nite-eld to eliminate the interference at receiver 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3 Average sum rate for Gaussian Network-Coded CIC with i.i.d. channel coecientsCN (0; 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.4 Two-User Gaussian networks with coordination, cognition, and two hops. 129 4.5 A deterministic noiseless 2 2 2 nite-eld IC. . . . . . . . . . . . . . . 139 4.6 Performance comparison of PCoF with CIA and time-sharing with respect to ergodic symmetric sum rates for 2 2 2 MIMO interference channel with M = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.7 Performance comparison of PCoF with CIA and time-sharing with respect to ergodic symmetric sum rates for MIMO interference coordination channel.151 x Abstract Claude Shannon showed that unstructured random codes are shown to be optimal in various single user channels. In the past forty years, one of the major trusts of information theory has been extended this theory to the multi-terminal settings (known as network information theory). In this dissertation, we make progress on understanding the role of structured codes in several network settings. In the rst part of this dissertation, we present a novel coding scheme nicknamed Reverse Compute and Forward for broadcast layered relay networks, where source wishes to transmit independent messages to the corresponding destinations with the aid of in- termediate relays. This information theoretic model can capture the one of promising future wireless network architectures known as cooperative distributed antenna systems or cloud radio access networks. In the proposed scheme, each destination reliably decodes a linear combination of relays' messages (over suitable nite eld). This is enabled by exploiting the algebraic closure properties of lattice codes. Then, the end-to-end inter- ference (over the nite eld) is completely eliminated by zero-forcing precoding at the source, namely, the decoded linear combination at each destination is nothing but its own desired message. We further show that the proposed scheme outperforms the state of the art information theoretic scheme called Compressed Dirty Paper Coding, when the source-relay link capacity is nite. In addition, we introduce a \virtual" full-duplex relay channel for which each relay stage in a multi-hop relaying network is formed by at least two relays, used alternatively in transmission and reception modes, such that while one relay transmits its signal to the next stage, the other relay receives a signal from the previous stage. With such a pipelined scheme, the source is active and sends a new information message in each time xi slot as if full-duplex relays are employed. For such channel, we show that structured code can almost achieve the upper bound when the channel gains have controlled uctuations not larger than 3 dB, yielding a rate that does not depend on the number of relaying stages. This has not been obtained by other schemes (based on random codes) since their rates degrade linearly or logarithmically with the number of stages. Hence, those schemes are very far from the optimality in particular when multihop transmission is considered. Finally, we study a number of two-user interference networks with multiple-antenna transmitters/receivers (MIMO), transmitter side information (cognition) in the form of linear combinations (over an appropriate nite eld) of the information messages, and two-hop relaying. It is shown that in a cognitive Gaussian interference channel, if one node has a rank decient linear combination of two messages, this can yield degrees of freedom (DoF) and generalized DoF (GDoF) gains on the wireless segment, even though the coecients of the linear combination are chosen at random and a priori (independent of the channel realization). This is the rst result, as far as we know, that network coding in the wired part of the network is shown to yield DoF gain on the wireless part of the network and shows that structured codes can be used jointly with other structured network coding techniques, such as linear network coding, even beyond the \physical layer network coding" ideas. Also, we characterize the symmetric sum rate of the two-user MIMO IC with coordination, cognition, and two-hops and provide nite signal-to-noise ratio results (not just DoF) which show that the proposed structured codes are competitive against the state of the art interference avoidance based on orthogonal access, for standard randomly generated Rayleigh fading channels. xii Chapter 1 Introduction Classic information theory has generally relied on random coding arguments to charac- terize the fundamental limits of communication. For example, the capacity achieving code for Additive White Gaussian Noise (AWGN) point-to-point channel is constructed by in- dependent and identically distributed codewords from the power constraint shpere [CT12]. Notice that a particular algebraic structure is not imposed on the codes. In his seminal works, Shannon showed that the random coding argument is sucient to achieve the capacities in all single user channels. Furthermore, the same type of random codes can also be used to achieve the capacity of regions of some multi-terminal settings, including multiple access channel and special case of broadcast channel [Sha01]. In the late six- ties, structured codes were studied and it was found that they can signicantly reduce the encoding/decoding complexity but their performances were, at best, the same with the optimal random codes. Consequently, it is tempting to believe that random codes are sucient to characterize the fundamental limits of communication systems and the structured codes are not needed in classical information theory. On the other hand, the advantage of structured codes was rst demonstrated by Ko- rner and Marton. They showed that purely random code constructions are not always sucient while structured random codes may be required on the proof of achievability. They studied the distributed source coding problem when the destination wishes to re- cover a function of the source messages [KM79]. To be specic, source 1 observes a 1 binary source U 1 and source 2 observes another binary source U 2 , and decoder desires to reconstruct the mod-2 sum of the observations as U 1 U 2 . When the sources are correlated, it was shown that recovering the mod-2 sum of the observations is more ef- cient than recovering the individual observations, and the optimal rate region is larger than the Slepian-Wolf region. Moreover, the unstructured random coding arguments were insucient here, and structured linear codes were used instead. Linear codes have the property that the sum (over the proper nite eld) of two codewords is again a codeword, which plays an important role to send the mod-2 sum of the source observations in this distributed setting. Lattice codes form the real (or complex) counterpart of linear codes and extend the linearity property from a nite eld into the real (or complex) eld [Ban99]. Hence, they form an important class of structured codes and can be used in many wireless network settings. There exist a large body of work on lattice codes and their applications in com- munications. We cannot do justice to all this work here and point an interested reader to an excellent survey by Zamir [Zam09]. In particular for AWGN networks, nested lattice codes have been shown to outperform the unstructured random codes in several scenar- ios. As an example, Philosof, Zamir, Erez, and Khisti demonstrated that lattice codes enable distributed dirty paper coding for Gaussian multiple access channels [PZEK11]. The subsequent work by Sanderovich, Peleg, and Shamai exploited the algebraic struc- ture of lattice codes to derive better scaling laws for decentralized processing in cellular networks [SPS08]. In [SPS08], Narayanan, Wilson, and Sprintson developed a nested lattice strategy for the two-way relay channel. Also, Nam, Chung, and Lee introduced an asymmetric power constraints for the two-way relay channel and Gaussian multiple access channels [NCL08, NCL11]. Regarding the interference alignments, lattice codes can be used to realize these gains at nite SNR. Bresler, Parekh, and Tse employed lattice codes to approximate the capacity of the many-to-one and one-to-many interference channels to within a constant number of bits [BPT10]. This scheme was extended to bursty inter- ference channels in [KPV09]. For symmetric interference channels, Sridharan, Jafarian, 2 Vishwanath, Jafar, and Shamai developed a layered lattice strategy in [SJV + 08]. Re- cently, Ordentlich, Erez, and Nazer developed a new framework for lattice interference alignment [OEN12a], based on compute and forward framework [NG11]. Krithivasan and Pradhan have employed nested lattice codes for distributed compression of linear functions of jointly Gaussian sources [KP07]. Wagner derived an outer bound for the Gaussian case in [Wag11]. He and Yener showed that lattices are useful for informa- tion theoretic secrecy [HY09]. Finally, recent work by Zhan, Nazer, Erez, and Gastpar demonstrated that structured codes can be used to improver receiver design for MIMO channels [GZE + 10] and its dual approach was presented in [HC12] by Hong and Caire. The main goal of this dissertation is to demonstrate the advantage of structured codes in the following network models: 1) Cooperative distributed antenna systems (a.k.a, Cloud radio access networks); 2) Virtual full-duplex relay networks; 3) Some two-user interference networks with coordination, cognition, and two hop. In the following sections we provide an overview and a summary of our main contributions for each scenario. 1.1 Cooperative Distributed Antenna Systems A cloud base station is a Distributed Antenna System (DAS) formed by a number of simple antenna terminals (ATs) [SW11], spatially distributed over a certain area, and connected to a central processor (CP) via wired backhaul [Fla11,LSZ + 10,MBG11]. Cloud base station architectures dier by the type of processing made at the ATs and at the CP, and by the type of wired backhaul. At one extreme of this range of possibilities, the ATs perform just analog ltering and (possibly) frequency conversion, the wired links are analog (e.g., radio over ber [NNM + 04]), and the CP performs demodulation to baseband, A/D and D/A conversion, joint decoding (uplink) and joint pre-coding (downlink). At the other extreme we have \small cell" architectures where the ATs 3 AT 1 AT 2 AT3 AT4 UT 1 UT 2 UT 3 UT 4 UT 5 Figure 1.1: Cooperative Distributed Antenna System with 5 UTs and 4 ATs (e.g., K = 5 and L = 4), and digital backhaul links of rate R 0 . perform encoding/decoding, the wired links send data packets, and the CP performs high- level functions, such as scheduling, link-layer error control, and macro-diversity packet selection. In this work we focus on an intermediate DAS architecture where the ATs perform partial decoding (uplink) or precoding (downlink) and the backhaul is formed by digital links of xed rateR 0 . In this case, the DAS uplink is an instance of a multi-source single destination layered relay network where the rst layer is formed by the user terminals (UTs), the second layer is formed by the ATs and the third layer contains just the CP (see Fig. 1.1). The corresponding DAS downlink is an instance of a broadcast layered relay network with independent messages. In our model, analog forwarding from ATs to CP (uplink) or from CP to ATs (down- link) is not possible. Hence, some form of quantization and forwarding is needed. A general approach to the uplink is based on the Quantize reMap and Forward (QMF) paradigm of [ADT11] (extended in [LKEGC11] where it is referred to as Noisy Network Coding). In this case, the ATs perform vector quantization of their received signals at 4 some rate R 0 R 0 . They map the blocks of nR 0 quantization bits into binary words of length nR 0 by using some randomized hashing function (notice that this corresponds to binning if R 0 > R 0 ), and let the CP perform joint decoding of all UTs' messages based on the observation of all the (hashed) quantization bits. 1 It is known in [ADT11] that QMF achieves a rate region within a bounded gap from the cut-set outer bound [CT12], where the bound depends only on the network size and on R 0 , but it is independent of the channel coecients and of the operating SNR. For the broadcast-relay downlink, a general coding strategy has been proposed in [KRV12] based on a combination of Marton coding for the general broadcast channel [NGD + 75] and a coding scheme for determin- istic linear relay networks, \lifted" to the Gaussian case. Specializing the above general coding schemes to the DAS considered here, for the uplink we obtain the scheme based on quantization, binning and joint decoding of [SSPS09], and for the downlink we obtain the Compressed Dirty Paper Coding (CDPC) scheme of [SSS + 09b]. From an implementation viewpoint, both QMF and CDPC are not practical, the former requiring vector quantiza- tion at the ATs and joint decoding of all UT messages based on the hashed quantization bits at the CP, and the latter requiring Dirty Paper Coding and vector quantization at the CP. A lower complexity alternative strategy for general relay networks was proposed in [NG11] and goes under the name of Compute and Forward (CoF). CoF makes use of lattice codes, such that each relay can reliably decode a linear combination with integer coecients of the interfering codewords. Thanks to the fact that lattices are modules over the ring of integers, this linear combination translates directly into a linear combination of the information messages dened over a suitable nite eld. CoF can be immediately used for the DAS uplink. The performance of CoF was examined in [NSGS09] for the DAS uplink in the case of the overly simplistic Wyner model [Wyn94]. It was shown that 1 The information-theoretic vector quantization of [ADT11], [LKEGC11] can be replaced by scalar quantization with a xed-gap performance degradation [OD13]. 5 CoF yields competitive performance with respect to QMF for practically realistic values of SNR. This work contributes to the subject in the following ways: 1) for the DAS uplink, we consider the CoF approach and examine the corresponding system optimization at nite SNR for a general channel model including fading and shadowing (i.e., beyond the nice and regular structure of the Wyner model); 2) For the downlink, we propose a novel precoding scheme nicknamed Reverse Compute and Forward (RCoF); 3) For both uplink and downlink, we present low-complexity versions of CoF and RCoF based on standard scalar quantization at the receivers. These schemes are motivated by the observation that the main bottleneck of a digital receiver is the Analog to Digital Conversion (ADC), which is costly, power-hungry, and does not scale with Moore's law. Rather the number of bit per second produced by an ADC is roughly a constant that depends on the power consumption [W + 99,SPM09]. Therefore, it makes sense to consider the ADC as part of the channel. The proposed schemes, nicknamed Quantized CoF (QCoF) and Quantized RCoF (RQCoF), lead to discrete-input discrete-output symmetric memoryless channel models naturally matched to standard single-user linear coding. In fact, QCoF and RQCoF can be easily implemented usingq-ary Low-Density Parity-Check (LDPC) codes [HC11,TNBH12,FSK10] with q =p 2 and p prime, yielding essentially linear complexity in the code block length and polynomial complexity in the system size (minimum between number of ATs and UTs). The two major impairments that deteriorate the performance of DAS with CoF/RCoF are the non-integer penalty (i.e., the residual self-interference due to the fact that the channel coecients take on non-integer values in practice) and the rank-deciency of the resulting system matrix over the q-ary nite eld. In fact, the wireless channel is characterized by fading and shadowing. Hence, the channel matrix from ATs to UTs does not have any particularly nice structure, in contrast to the Wyner model case, where the channel matrix is tri-diagonal [NSGS09]. Thus, in a realistic setting, the system matrix resulting from CoF/RCoF may be rank decient. This is especially relevant when 6 the size q of the nite eld is small (e.g., it is constrained by the resolution of the A/D and D/A conversion). The proposed system optimization counters the above two problems by considering power allocation, network decomposition and antenna selection at the receivers (ATs selection in the uplink and UTs selection in the downlink). We show that in most practical cases the AT and UT selection problems can be optimally solved by a simple greedy algorithm. Numerical results show that, in realistic networks with fading and shadowing, the proposed optimization algorithms are very eective and essentially eliminate the problem of system matrix rank deciency, even for small eld size q. 1.2 Full-Duplex Relaying with Half-Duplex Relays In the evolution of wireless networks from voice-centric to data-centric networks, the throughput of cell-edge users is becoming a signicant system bottleneck. This prob- lem is further exacerbated in systems operating at higher frequencies (mm-waves [DH07, RGBD + 13, RSM + 13]), due to the fact that at those frequencies the pathloss exponent is large [RGBD + 13, Yan04]. In these cases, the use of relays represents a promising technique in order to extend network coverage, combat shadowing eects, and improve network throughput [DWV08,SYF02,ID08,LSZ + 10]. In addition, multi-hop relaying can be instrumental to implement a wireless backhaul able to overcome non-line of sight prop- agation, providing a cost-eective and rapidly deployable alternative to the conventional backbone wired network. The use of relays was rst standardized in IEEE802.16j [PH09]. Later, also LTE- advanced considered various relay strategies in order to meet target throughput and coverage requirements [SWAYKT + 09, PDF + 08]. In these practical systems, relays op- erate in a half-duplex mode due to the non-trivial implementation problems related to transmitting and receiving in the same frequency band and during the same time slot [PH09, SWAYKT + 09, PDF + 08]. Since a half-duplex relay can forward a message 7 from source to destination over two time slots, it makes an inecient use of the radio channel resource. Alternatively, relays can operate in full-duplex mode, transmitting data while receiving new data to be forwarded in the next time slot. Yet, the implementa- tion of full-duplex relays is quite demanding in practice, due to the signicant amount of self-interference between transmitting and receiving RF chains (see Fig. 1.2). For example, WiFi signals are transmitted at 20 dBm average power and the noise oor is around90 dBm. Thus, the self-interference has to be canceled by 110 dB to reduce it to the noise oor. Otherwise, any residual interference treated as noise would degrade the performance. Although, ideally, the self-interference can be perfectly removed from the received signal since it is perfectly known by the relay, in practice this is not possible since the large power imbalance between transmit and received signal saturates completely the receiver RF chain (in particular, the dynamic range of the Analog-to-Digital Conversion (ADC)) such that digital interference cancellation in the receiver baseband is not possible. Recent works [BPM07,DS10,CJS + 10,BMK13,DDS12] have shown the practical feasi- bility of full-duplex relays by suppressing the impact of self-interference in a mixed analog- digital fashion. These architectures are based on some form of analog self-interference cancellation, in order to prevent the receiver ADC from being saturated by the transmit- ter power, followed by digital self-interference cancellation in the baseband domain. In some of these schemes, the self-interference cancellation in the analog domain is obtained by transmitting with multiple antennas, such that the transmit signal superimposes in phase opposition and therefore cancels at the receiving antennas. A more recent alterna- tive [BMK13] makes use of a single antenna, and of a signal splitter called \circulator" that connects the transmitter chain to the antenna and the antenna to the receiver chain, while providing sucient isolation between the transmitter port and the receiver port. Building on the idea of using multiple antennas to cope with the isolation of the re- ceiver from the transmitter, we may consider a \distributed version" of such approach where the antennas belong to physically separated nodes. This has the advantage that 8 se lf -inte r f e r enc e Figure 1.2: Two-hop relay network with full-duplex relay. Virtual Full-Duplex Relay Figure 1.3: Tow-hop relay network with virtual full-duplex relay. The 2 R + denotes the inter-relay interference level. Black-solid lines are active for every even time slot and red-dashed lines are active for every odd time slot. each of such nodes operates in conventional half-duplex mode. Furthermore, by allow- ing a large physical separation between the nodes, the problem of receiver saturation is eliminated. In this paper, we study such a \virtual" full-duplex relay scheme formed by two-half duplex relays (see Fig. 1.3). As argued above, this can be seen (to some extent) as the distributed version of full-duplex proposals based on multiple antennas. At each time slot, one of relays (in receive mode) receives a new data slot from the source while other relay (in transmit mode) forwards the processed data slot (obtained in the previous time interval) to the destination. The role of the relays is swapped at each time interval. This relaying operation is known as \successive relaying" [OS04, RGK08, BMK10]. In this way, the source can send a new message to the destination at every time slot as if full-duplex relay was employed. It is interesting to notice that the network topology is identical to the well-known diamond relay network, with the addition of one interfering link between the two relays. The main performance bottleneck of successive relaying is 9 the so-called inter-relay interference, corresponding to the self-interference in full-duplex relays. For the given successive relaying operation, an upper bound on the achievable rate is easily obtained as C(SNR) = log(1 +SNR). This will be referred to as the successive upper bound. In this work, we examine several information theoretic coding schemes and their achievable rates for successive relaying. For non-interfering relays (i.e., inter-relay interference link = 0), it was shown in [BMK10] that the Decode-and-Forward (DF) strategy is optimal. The non-interfering model can capture some practical scenarios for which relays are located far from each other or xed infrastructure relays are deployed with high-directional antennas [MK10,DN10,PWH09,IKS12]. In case of interfering relay (i.e., > 0), Dirty-Paper Coding (DPC) is optimal, i.e., achieves the performance of ideal full-duplex relay [CCL07,RGK08]. Since the source has non-causal information on relay's transmit signal and inter-relay interference channel , it can completely eliminate the \known" interference at intended receiver, using DPC. Therefore, for the 2-hop network with a single relay stage, the performance of ideal full-duplex relay is achievable by using practical half-duplex relays based on successive relaying and DPC. A natural question aries: can we achieve the performance of ideal full-duplex relay by using half-duplex relays for a multihop network, with multiple relay stages? We rst show that DPC is no longer applicable for a K-stage relay network K 2. Hence, we consider several alternative coding strategies that cancel the inter-relay interference at either the relay or the destination. We rst focus on 2-hop networks to explain these coding schemes and compare them with DPC. In particular, we consider the DF strategy [CG79, KGG05], where inter-relay interference is removed at the relays by joint decoding. A second approach consists of letting the destination remove interference. This is because, with the pipelined transmission, the destination also \knows" the already causally decoded interference. However, due to the transmission power-constraint and the capacity limit of the relay-to-destination link, the relay needs to perform some form of processing on its received signal. In particular, we consider: (i) the relay forwards a scaled 10 version of its received signal to destination (i.e., Amplify and Forward (AF) [LTW04]); (ii) the relay quantizes its received signal, random bins the quantization bits and forward the (digitally encoded) quantization bit index to the destination (i.e., Quantize reMap and Forward (QMF) [ADT11], also known as Noisy Network Coding (NNC) [LKEGC11]); (iii) the relay forwards a noiseless linear combination of the incoming messages over an appropriate nite eld (i.e., Compute and Forward (CoF) [NG11]). For the cases of (i) and (ii), the destination eliminates the known interference signal in the signal domain (before decoding). In the case of (iii), the destination cancels the interference in the message domain (after decoding). For a 2-hop network, we show that QMF and CoF achieve the optimal performance (i.e., DPC rate) within 1 bit. Also, CoF with power allocation may outperform QMF if the inter-relay interference level is large enough (i.e., 2 0:5). Then, we generalize those coding schemes to multihop virtual full-duplex relay channel described in Fig. 3.1, and derive their achievable rates. Since this model is a special case of a single-source single destination (non-layered) network, QMF in [ADT11] and NNC in [LKEGC11] can be applied to this model. By setting the quantization distortion levels to be at background noise level, QMF and NNC achieve the capacity within a constant (with respect to SNR and ) gap that scales linearly with the number of nodes in the network. For the multihop model considered in this paper, we provide an improvement result by using the principle of QMF (or NNC) and optimizing the quantization levels. The resulting scheme achieves a gap that scales logarithmically with the number of nodes. Also, the proposed QMF scheme is a special case of \short-message" NNC [HK13] and has lower decoding complexity by using successive decoding instead of joint simultaneous decoding as in [ADT11,LKEGC11]. In addition, we also show that CoF can achieve the upper bound within 0.5 bits if the inter-relay interference level tends to an integer. We also derive an upper bound that is independent of K and coincides with the successive upper bound log(1 +SNR). For more general cases, CoF (including power allocation) can achieve the upper bound within about 1.5 bits and outperform the DF, AF, and 11 QMF, having a gap that increases with K. Therefore, we can approximately achieve the performance of ideal full-duplex relay for multihop channel. 1.3 Some Two-User Gaussian Networks with Cognition, Coordination, and Two Hops Interference is one of the fundamental aspects of wireless communication networks. Al- though the full characterization of the Interference Channel (IC) capacity is elusive, much progress has been made in recent years. The capacity region of the two-user Gaussian IC was characterized within 1 bit, by using superposition coding with an appropriate power allocation of the private and common message codewords, and by providing a new upper bounding technique [ETW08]. Degrees of Freedom (DoF) results are obtained under the assumption of full channel knowledge for the two-user multiple input multiple output (MIMO) IC with arbitrary number of antennas at each node in [JF07], for the K user IC with time-varying or frequency-selective channels in [CJ08], for the K-user IC with constant channel coecients in [MOgMaK09], and for theK-user IC with multiple anten- nas [GJ10]. Also, Generalized DoF (GDoF) results are found for the two-user MIMO IC in [KV12] and for symmetricK user IC in [JV10]. Further results for various interference networks can be found in [Jaf11]. In many practical communication systems, transmitters or receivers are not isolated. For example, in cellular systems the base stations are connected through a wired back- haul network through which information messages and some form of channel state in- formation or coordination can be shared [Fla11, LSZ + 10, MBG11, HC12, HC13a]. In wired networks, routing is generally optimal only for the single-source single-destination case [FF56]. In the more general case of multiple sources and multiple destinations (multi- source multicasting), linear network coding is known to achieve the min-cut max- ow bound [ACLY00]. In practice, random linear network coding is of particular interest for its simplicity. In this case, intermediate nodes forward liner combinations of the incoming 12 messages by randomly and independently choosing the coecients from an appropriate nite-eld [HMK + 06]. Going back to the cellular systems case, if random linear network coding is used in the backhaul network, the base stations obtain linear combination of the messages instead of individual messages. If the backhaul link serving a given base station has capacity large enough, the rank (per unit time) of such linear combinations is equal to the number of independent messages (per unit time), so that the base station knows all the messages. In contrast, if the backhaul link is a capacity bottleneck, the rank (per unit time) is less than the number of independent messages (per unit time). In this case, the base station has access to \mixed data", i.e., rank-decient linear combinations of the messages. We refer to this model as the Network-Coded Cognitive IC (CIC). An example of Network-Coded CIC is shown in Fig. 1.4, including a cellular BS and a home BS (e.g., a femtocell access point). The cellular BS is connected to the data router, which generates both messages, via a high capacity link supporting rate 2R 0 . The home BS is connected to the same data router via lower capacity link supporting only rate R 0 . In this case, the data router sends two information messages to the cellular BS (equivalently, a rank-2 linear combination thereof) and a rank-1 linear combination of the messages to the home BS. In the case of routing, this linear combination has coecients 0 and 1, reducing to the classical CIC, which has been extensively investigated in the literature [MYK07,WVA07,JV09]. In particular, the Gaussian CIC capacity region was approximately characterized within 1.87 bits in [RTD10]. If general network coding is used instead of routing, the rank-1 linear combination has generally non-zero coecients and therefore contains mixed data. Notice that in the model of Fig. 1.4 mixed data can be provided without violating the backhaul capacity constraint of R 0 . At this point, a natural question arises: Does mixed data at the \non-cognitive" transmitter provide a capacity increase \for free" for the Network-Coded CIC over the conventional CIC? Proceeding along this line, we observe that a basic level of multicell cooperation named interference coordination has been investigated and it is currently considered in industry for its practical aspects. A simple two-user model for interference coordination consists 13 Data Router Cognitive transmitter Non-cognitive transmitter Figure 1.4: In the classical CIC, the data router sends the one of information messages to the non-cognitive transmitter. In the Network-Coded CIC, the data router forwards \mixed data" to the non-cognitive transmitter. of a data router, two M-antenna base stations, and two M-antenna user receivers. The wired backhaul links from the router to the base stations have the same capacity equal to R 0 . The data router has no knowledge on channel state information (CSI), due to the separation between physical layer and network layer. However, the base stations have full CSI of both the direct and interfering links, obtained from the users through feedback channels. Such CSI knowledge allows the base stations to coordinate in their sharing strategies such as power allocation and beamforming directions [GHH + 10]. From an information theoretic viewpoint, this model is a two-user MIMO IC. Assuming that network coding is used in the backhaul, the data router can deliver linear combinations of the messages instead of individual messages, at the same cost in terms of backhaul capacity constraint. Hence, the model becomes a two-user MIMO IC with mixed data at both transmitters, and shall be referred to as the Network-Coded Interference Coordina- tion Channel (ICC). In this paper, we address the following question: Does mixed data at the transmitters provide a capacity increase \for free" for the Network-Coded ICC over conventional interference coordination? Finally, building on the insight gained in the above Network-Coded cognitive net- works, we study the 2 2 2 MIMO IC, consisting of two transmitters (sources), two relays, and two receivers (destinations), where nodes have M multiple antennas. This 14 model is non-cognitive but has some commonality with the previous models in the sense that it consists of two cascaded two-user MIMO ICs where the relay can have access to mixed messages if proper alignment and coding over the nite-elds is used in the rst hop. The 2 2 2 Gaussian IC has received much attention recently, being one of the fundamental building blocks to characterize two- ow networks [SA11]. One natural ap- proach is to consider this model as a cascade of two ICs. In [SSS + 09b], the authors apply the Han-Kobayashi scheme [TSK81] for the rst hop to split each message into private and common parts. Relays can cooperate using the shred information (i.e., common mes- sages) for the second hop, in order to enhance the data rates. This approach is known to be highly suboptimal at high SNR, since two-user IC can only achieve 1 DoF. In [CJ09] it was shown that 4 3 DoF is achievable by viewing each hop as an X-channel. This is accomplished using the interference alignment scheme for each hop. More recently, the optimal DoF was obtained in [GJW + 12] using a new scheme called aligned interference neutralization, which appropriately combines interference alignment and interference neu- tralization. Also, the KKK Gaussian IC was recently studied in [SA12], where it is shown that the KK MIMO cut-set upper bound (equal to K) can be eectively achieved using aligned network diagonalization. This work contributes in the following subjects: Network-Coded CIC: single antenna case: We characterize the capacity region of a nite-eld Network-Coded CIC using distributed zero-forcing precoding. We no- tice that this region is equivalent to that of a nite-eld vector broadcast channel. This shows that in this case partial cooperation yields the same performance of full coopera- tion, as long as the non-cognitive transmitter knows the mixed message rather than its own individual message only. Thus, we conclude that mixed data at the non-cognitive transmitter can increase capacity. It is worthwhile noticing that the nite-eld model is itself meaningful in practical wireless communication systems, by the observation that the main bottleneck of a digital receiver is the Analog to Digital Conversion (ADC), which is costly, power-hungry, and does not scale with Moore's law. Rather, the number of 15 bit per second produced by ADC is roughly a constant that depends on the power con- sumption [W + 99,SPM09]. Therefore, it makes sense to consider the ADC as part of the channel. This, together with the algebraic structure induced by lattice coding, produces a nite-eld model as shown by the authors in [HC13a, HC11]. Motivated by this rst successful result, we present a novel scheme nicknamed Precoded Compute-and-Forward (PCoF) for Gaussian Network-Coded CIC. CoF makes use of nested lattice codes, such that each receiver can reliably decode a linear combination with integer coecient of the interfering codewords [NG11]. Thanks to the fact that lattice are modules over the ring of integers, this linear combination translates directly into a linear combination of the information messages dened over a suitable nite-eld. For brevity, we refer to this fact as \lattice linearity" in the following. Finally, the interference in the nite-eld domain is completely eliminated by distributed zero forcing precoding (over nite-eld). This scheme can be thought of as a distributed approach of Reverse CoF (RCoF), proposed by the authors in [HC12,HC13a] for the downlink of distributed antenna systems. Another novel contribution of this work is the characterization of the sum GDoF (see [ETW08] and denition in (4.51)) of the Gaussian Network-Coded CIC. This is obtained by combining an improved achievability result that makes use of both Dirty-Paper Coding (DPC) and PCoF, with a new outer bound on the sum rate. As a consequence of the GDoF analysis, we show that mixed data can provide an unbounded capacity gain with respect to conventional cognitive interference channels. As shown in Fig. 1.5, the sum GDoF of the Network-Coded CIC is larger than the sum GDoF of the standard IC when = logINR logSNR , the ratio of the interference power over the direct link power (expressed in dB), is larger than 1=3, and it is larger than the sum GDoF of the standard CIC when > 1=2. In contrast, for < 1=2 it is better not to mix the data on the side information link to the non-cognitive transmitter (i.e., use routing in the backhaul link). It is also interesting to notice that for = 1 the use of mixed data provides the same two degrees of freedom of full cooperation (two-users vector broadcast channel), mimicking the result of the nite-eld case. 16 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3 3.5 Figure 1.5: The generalized degrees-of-freedom (GDoF) of the two-user Gaussian Network-Coded CIC. For the interference regimes with 1=2, the gap between the Network-Coded CIC and CIC becomes arbitrarily large as SNR and INR goes to innity. This shows that mixed-data at the non-cognitive transmitter can provide the unbounded capacity gain at high SNR. 17 Network-Coded MIMO IC: As anticipated before, we have considered three com- munication models having the two-user MIMO IC as a building block: Network-Coded ICC (representative of a cellular system downlink with interference coordination), Network- Coded CIC with MIMO generalization, and 2 2 2 IC [GJW + 12, SA11]. In all these models, we assume that all nodes haveM transmit/receive antennas. Our coding scheme is based on the extension of the PCoF idea to the MIMO case. This scheme consists of two phases: 1) Using the CoF framework in order to transform the two-user MIMO IC into a deterministic nite-eld IC; 2) Using linear precoding on the nite-eld domain in order to eliminate interference. The main performance bottleneck of CoF consists of the non- integer penalty, which ultimately limits the performance of CoF at high SNR [NP12]. To overcome this bottleneck, we employ Channel Integer Alignment (CIA) in order to create an \aligned" channel matrix for which exact integer forcing is possible. We de- rive achievable symmetric sum rate results for all three channel modes and prove that PCoF with CIA can achieve sum DoF equal to 2M 1 in all cases. In particular, for the Network-Coded CIC, we prove that the optimal 2M sum DoF is achieved by appro- priately combining DPC and PCoF as in the scalar case. Beyond the DoF results, we further employ the lattice codes algebraic structure in order to obtain good performance at nite SNRs. We use the integer-forcing receiver (IFR) approach of [ZNEG10] and integer-forcing beamforming (IFB), proposed by the authors in [HC12, HC13a], in order to minimize the power penalty at the transmitters. We provide numerical results showing that PCoF with CIA outperforms time-sharing even at reasonably moderate SNR, with increasing performance gain as SNR increases. 1.4 Dissertation Outline The remainder of this dissertation is organized as follows. In Chapter 2, we consider the application of CoF to the DAS uplink and introduce the novel concept of network decomposition to improve the CoF sum rate. Also, we consider 18 the DAS downlink and presents the RCoF scheme. For both schemes, we introduce the low-complexity \quantized" versions of CoF and RCoF. Focusing on the symmetric Wyner model, we presents a simple power allocation strategy to alleviate the impact of non-integer penalty. In the case of a realistic DAS channel model including fading, shadowing, and pathloss, a low-complexity greedy algorithm for ATs selection (uplink) and UTs selection (downlink) is presented. In Chapter 3, we examine several information theoretic coding schemes and compare them with respect to their achievable rates. Some detail explanation of their encoding and decoding schemes are provided and their achievable rates are obtained. We generalize successive relaying to multihop virtual full-duplex relay channel and derive the achiev- able rates of various information theoretic coding schemes. Also, their performances are compared analytically and numerically. In Chapter 4, we characterize the capacity region of nite-eld Network-Coded CIC and present PCoF, as a natural extension of nite eld scheme, for the Gaussian counter- part. Further, we derive an achievable rate region of the Gaussian Network-Coded CIC, by appropriately combining PCoF and DPC, and characterize the sum GDoF. Also, we characterize an achievable symmetric sum rate for the two-user MIMO ICs under investi- gation, and derive the DoF of these channels. The um rates are improved using successive cancellation in terms of CoF. We optimize the (symmetric) sum rate and provide some numerical results, showing good performance at intermediate and \practical" values of SNR. The dissertation is concluded in Chapter 5 with the nal remarks on the contributions of this dissertation. 19 Chapter 2 Cooperative Distributed Antenna Systems In this chapter we study a distributed antenna system where L antenna terminals (ATs) are connected to a Central Processor (CP) via digital error-free links of nite ca- pacity R 0 , and serve K user terminals (UTs). For the uplink, we apply the \Compute and Forward" (CoF) approach and examine the corresponding system optimization at nite SNR. For the downlink, we propose a novel precoding scheme nicknamed \Re- verse Compute and Forward" (RCoF). In both cases, we present low-complexity versions of CoF and RCoF based on standard scalar quantization at the receivers, that lead to discrete-input discrete-output symmetric memoryless channel models for which near- optimal performance can be achieved by standard single-user linear coding. The proposed uplink and downlink system optimization focuses specially on the ATs and UTs selection problem. In both cases, for a given set of transmitters, the goal consists of selecting a subset of the receivers such that the corresponding system matrix has full rank and the sum rate is maximized. We present low-complexity ATs and UTs selection schemes and demonstrate, through Monte Carlo simulation in a realistic environment with fading and shadowing, that the proposed schemes essentially eliminate the problem of rank deciency of the system matrix and greatly mitigate the non-integer penalty aecting CoF/RCoF at high SNR. Comparison with other state-of-the art information theoretic schemes, such as Quantize reMap and Forward for the uplink and Compressed Dirty Paper Coding for the 20 downlink, show competitive performance of the proposed approaches with signicantly lower complexity 2.1 Preliminaries In this section we provide some basic denitions and results that will be extensively used in the sequel. Table 2.1 shows the acronyms used throughout the chapter. Table 2.1: Acronyms used in the paper CoF Compute and Forward QCoF Quantized Compute and Forward DF Decode and Forward QMF Quantize reMap and Forward QF Quantize and Forward LQF Lattice Quantize and Forward RCoF Reverse Compute and Forward RQCoF Reverse Quantized Compute and Forward CDPC Compressed Dirty Paper Coding CZFB Compressed Zero Forcing Beamforming 2.1.1 Network Model We consider a DAS with L ATs and K UTs, each of which is equipped with a single antenna. The ATs are connected to the CP via digital backhaul links of rate R 0 (see Fig. 1.1). A block ofn channel uses of the discrete-time complex baseband uplink channel is described by Y =HX +Z; (2.1) where we use \underline" to denote matrices whose horizontal dimension (column index) denotes \time" and vertical dimension (row index) runs across the antennas (UTs or ATs), the matrices X = 2 6 6 6 6 4 x 1 . . . x K 3 7 7 7 7 5 and Y = 2 6 6 6 6 4 y 1 . . . y L 3 7 7 7 7 5 21 contain, arranged by rows, the UT codewords x k 2 C 1n and the AT channel output vectorsy ` 2C 1n , fork = 1;:::;K, and` = 1;:::;L, respectively. The matrixZ contains i.i.d. Gaussian noise samplesCN (0; 1), and the matrix H = [h 1 ;:::;h L ] T 2 C LK contains the channel coecients, assumed to be constant over the whole block of length n and known to all nodes. Similarly, a block of n channel uses of the discrete-time complex baseband downlink channel is described by ~ Y = ~ H ~ X + ~ Z, where we use \tilde" to denote downlink variables, ~ X2 C Ln contains the AT codewords, ~ Y; ~ Z2 C Kn contain the channel output and Gaussian noise at the UT receivers, and ~ H = [ ~ h 1 ;:::; ~ h K ] T 2 C KL is the downlink channel matrix. Since ATs and UTs are separated in space and powered independently, we assume a symmetric per-antenna power constraint for both the uplink and the downlink, given by 1 n E[kx k k 2 ]SNR for all k and by 1 n E[k~ x ` k 2 ]SNR for all `, respectively. 2.1.2 Nested Lattice Codes LetZ[j] be the ring of Gaussian integers and p be a Gaussian prime. 1 Let denote the addition overF p 2, and letg :F p 2!C be the natural mapping ofF p 2 ontofa +jb :a;b2 Z p gC. We recall the nested lattice code construction given in [NG11]. Let =f = zT : z2Z n [j]g be a lattice in C n , with full-rank generator matrix T2C nn . LetC =fc = wG :w2F r p 2 g denote a linear code overF p 2 with block length n and dimension r, with generator matrix G. The lattice 1 is dened through \construction A" (see [EZ04] and references therein) as 1 =p 1 g(C)T + ; (2.2) 1 Gaussian integer is called a Gaussian prime if it is a prime inZ[j]. A Gaussian prime a +jb satises exactly one of the following conditions [Sti03]: 1)jaj =jbj = 1; 2) one of a;b is zero and the other is a prime number inZ of the form 4n + 3 (with n a nonnegative integer); 3) both of a;b are nonzero and a 2 +b 2 is a prime number inZ of the form 4n + 1. In this paper, p is assumed to be a prime number congruent to 3 modulo 4, which is an integer Gaussian prime according to condition 2). 22 where g(C) is the image ofC under the mapping g (applied component-wise). It follows that 1 p 1 is a chain of nested lattices, such thatj 1 =j =p 2r andjp 1 = 1 j = p 2(nr) . For a lattice and r2C n , we dene the lattice quantizer Q (r) = arg min 2 kr k 2 , the Voronoi regionV = fr 2 C n : Q (r) = 0g and [r] mod = rQ (r). For and 1 given above, we dene the lattice code L = 1 \V with rate R = 1 n logjLj = 2r n logp. Construction A provides a natural labeling of the codewords ofL by the information messages w2 F r p 2 . Notice that the set p 1 g(C)T is a system of coset representatives of the cosets of in 1 . Hence, the natural labeling function f :F r 2 !L is dened by f(w) =p 1 g(wG)T mod . 2.1.3 Compute and Forward We recall here the CoF scheme of [NG11]. Consider theK-user Gaussian multiple access channel (G-MAC) dened by y = K X k=1 h k x k +z; (2.3) where h = [h 1 ;:::;h K ] T , and the elements of z are i.i.d. CN (0; 1). All users make use of the same nested lattice codebookL = 1 \V , where has second moment 2 = 1 nVol(V) R V krk 2 dr = SNR. Each user k encodes its information message w k 2F r p 2 into the corresponding codeword t k = f(w k ) and produces its channel input according to x k = [t k +d k ] mod ; (2.4) where the dithering sequences d k are mutually independent across the users, uniformly distributed overV , and known to the receiver. Notice that, as in many other applica- tions of nested lattice coding and lattice decoding (e.g., [ZSE02,EZ04,DEGC03]), random dithering is instrumental for the information theoretic proofs, but a deterministic dither- ing sequence that scrambles the input and makes it zero-mean and uniform over the shaping region can be eectively used in practice, without need of common randomness. 23 The decoder's goal is to recover a linear combination v = [ P K k=1 a k t k ] mod with integer coecient vector a = [a 1 ;:::;a K ] T 2 Z K [j]. Since 1 is a Z[j]-module (closed under linear combinations with Gaussian integer coecients), then v2L. Letting ^ v be the decoded codeword (for some decoding function which in general depends on h and a), we say that a computation rateR is achievable for this setting if there exist sequences of lattice codesL of rate R and increasing block length n, such that the decoding error probability satises lim n!1 P(^ v6=v) = 0. In the scheme of [NG11], the receiver computes ^ y = " y K X k=1 a k d k # mod (2.5) = [v +z e (h;a;)] mod ; (2.6) where z e (h;a;) = K X k=1 (h k a k )x k +z (2.7) denotes the eective noise, including the non-integer self-interference (due to the fact that h k = 2Z[j] in general) and the additive Gaussian noise term. From [NG11], the density of the non-i.i.d. vector z e (h;a;) can be upper bounded (up to a constant scaling) by the density of an i.i.d. zero-mean Gaussian vector whose variance approaches 2 z e (h;a;) =khak 2 +kk 2 as n!1. The scaling, dither removal, and modulo- operation in (2.6) is referred to as the CoF receiver mapping in the following. By minimizing the eective noise variance of z e (h;a;) with respect to , we obtain 2 (h;a) = min 2 z e (h;a;) =SNR kak 2 SNRkh H ak 2 1 +SNRkhk 2 (a) = a H (SNR 1 I +hh H ) 1 a; (2.8) 24 where (a) follows from the Matrix Inversion Lemma [Har08, Thm 18.2.8]. Since is uniquely determined byh anda as = SNRh H a 1+SNRkhk 2 , it will be omitted in the following, for the sake of notation simplicity. That is, the eective noise in (2.7) with this specic choice of is denoted by z e (h;a) in the following. From [NG11], we know that by applying lattice decoding to ^ y given in (2.6) the following computation rate is achievable: R(h;a;SNR) = log + SNR a H (SNR 1 I +hh H ) 1 a ; (2.9) where log + (x), maxflog(x); 0g. The computation rate R(h;a;SNR) can be maximized by minimizing 2 (h;a) with respect to a. The quadratic form (2.8) is positive denite for any SNR <1, since the matrix (SNR 1 I +hh H ) 1 has eigenvalues i = 8 > < > : SNR=(1 +khk 2 SNR) i = 1 SNR i> 1: (2.10) By Cholesky decomposition, there exists a lower triangular matrixL such that 2 (h;a) = kL H ak 2 . It follows that the problem of minimizing 2 (h;a) overa2Z K [j] is equivalent to nding the "shortest lattice point" of the K-dimensional lattice generated by L H . This can be eciently obtained using the complex LLL algorithm [LLL82, Nap96] possibly followed by Phost or Schnorr-Euchner enumeration (see [DEGC03]) of the non-zero lattice points in a sphere centered at the origin, with radius equal to the shortest vector found by complex LLL. Algorithm 1 summarizes the procedures used in this paper to nd the optimal integer coecient vector a2Z K [j]. 2.2 Compute and Forward for the DAS Uplink In this section we review the DAS uplink based on CoF in [NSGS09,HC11] (see Fig. 2.1) and provide an algorithm to eciently compute sum rates for a general network using the 25 Algorithm 2.1 Find the optimal integer coecients 1. Take F =L H 2. Find the reduced basis matrix F red , using the (complex) LLL algorithm 3. Take the column of F red with minimum Euclidean norm, call it b ? 4. Let r =kb ? k + for some very small > 0 5. Use Phost or Schnorr-Euchner enumeration with F red to nd all lattice points in the sphere centered at 0, with radius r. Notice that this algorithm will nd for sure the point 0 (discarded), the point b ? , and possibly some shorter non-zero points. CP Messages recovered by Gaussian Elimination if is full rank AT 1 AT 2 AT 3 AT 4 UT 1 UT 2 UT 3 UT 4 Figure 2.1: DAS Uplink Architecture using Compute and Forward: L = 4 and K = 4. 26 idea of network decomposition. For simplicity of exposition, we restrict to consider the same numberK =L of UTs and ATs. The notation, however, applies also to the case of K < L addressed in Section 2.6, when considering AT selection. The UTs make use of the same lattice codeL of rate R, and produce their channel input x k , k = 1;:::;K, ac- cording to (2.4). Each AT` decodes the codeword linear combinationv ` = h P K k=1 a `;k t k i mod , for a target integer vectora ` = (a `;1 ;:::;a `;K ) T 2Z K [j] determined according to Algorithm 1, independently of the other ATs ifRR(h ` ;a ` ;SNR). Lettingu ` =f 1 (v ` ) denote the information message corresponding to the target decoded codeword v ` , the code linearity overF p 2 and theZ[j]-module structure of 1 yield u ` = K M k=1 q `;k w k ; (2.11) where q `;k =g 1 ([a `;k ] mod pZ[j]). After decoding, each AT ` forwards the correspond- ing information message ^ u ` to the CP via wired links of xed R 0 . This can be done if R R 0 . The CP collects all the messages ^ u ` for ` = 1;:::;L and forms the system of linear equations overF p 2 2 6 6 6 6 4 ^ u 1 . . . ^ u L 3 7 7 7 7 5 =Q 2 6 6 6 6 4 ^ w 1 . . . ^ w K 3 7 7 7 7 5 ; (2.12) where we dene A = [a 1 ;:::;a L ] T and the system matrix Q = [q 1 ;:::;q L ] T =g 1 ([A] mod pZ[j]): Provided that Q has rank K over F p 2, the CP obtains the decoded messagesf ^ w k g by Gaussian elimination. Assuming this full-rank condition and R < R(h ` ;a ` ;SNR) for all 27 ` = 1;:::;L, the error probabilityP( ^ w k 6=w k for somek) can be made arbitrarily small for suciently large n. The resulting achievable rate per user is given by [NSGS09]: R = minfR 0 ; min ` fR(h ` ;a ` ;SNR)gg: (2.13) Remark 1 Since each AT` determines its integer coecient vector a ` in a decentralized way, by applying Algorithm 1 independently of the other ATs' channel coecients, the resulting system matrix Q may be rank-decient. If K < L, requiring that all ATs can decode reliably is unnecessarily restrictive: it is sucient to select a subset of K ATs which can decode reliably and whose coecients form a full-rank system matrix. This selection problem will be addressed in Section 2.6. Although the elements of H are non-zero, the corresponding Q may include zeros, since some elements of the vectors a ` may be zero modulo pZ[j]. Following the [NG11, Thm 5], the sum rate of CoF-based DAS can be improved by taking zero integer- coecients into account: R sum (H;A) = K X k=1 R k ; where R k denotes the rate of UT k with R k = min `:a `k 6=0 R(h ` ;a ` ;SNR): Because of the presence of zero elements, the system matrixQ may be put in block diago- nal form by column and row permutations. If the permuted system matrix hasS diagonal blocks, the corresponding network graph decomposes into S independent subnetworks (i.e., network decomposition) and CoF can be applied separately to each subnetwork such that taking the minimum of the computation rates over the subnetworks is not needed. Hence, the sum rate is given by the sum (over the subnetworks) of the sum rates of each network component. In turns, the common UT rate of each indecomposable subnetwork 28 takes on the form (2.13). For given Q, the disjoint subnetwork components can be found eciently using depth-rst or breadth-rst search [AMO93]. This also essentially reduces the computation complexity of Gaussian elimination, which is performed independently for each subnetwork. We assume that, up to a suitable permutation of rows and columns, Q can be put in block diagonal form with diagonal blocks Q(A s ;U s ) for s = 1;:::;S, where we use the following notation: for a matrix Q with rows index set [1 : L] and column index set [1 :K], Q(A;U) denotes the submatrix obtained by selecting the rows inA [1 :L] and the columns inU [1 :K]. The following results are immediate: Lemma 2.1 IfQ is a full-rankKK matrix, the diagonal blocksQ(A s ;U s ) are full-rank square matrices for every s. Corollary 2.1 CoF with network decomposition, applied to a DAS uplink with channel matrix H = [h 1 ;:::;h K ] T 2C KK , achieves the sum rate R CoF (H;A) = S X s=1 jA s j min R 0 ; min k2As fR(h k ;a k ;SNR)g ; where A = [a 1 ;:::;a K ] T is the matrix of CoF integer coecients, and where the system matrixQ =g 1 ([A] mod pZ[j]) has full rankK overF p 2 and can be put in block diagonal form by rows and columns permutations, with diagonal blocks Q(A s ;U s ) for s = 1;:::;S. 2.3 Reverse Compute and Forward for the DAS Downlink In this section we propose a novel downlink precoding scheme nicknamed \Reverse" CoF (RCoF). Again, we restrict to the case K = L although the notation applies to the case of K > L, treated in Section 2.6. In a DAS downlink, the role of the ATs and UTs can be reversed with respect to the uplink. Each UT can reliably decode an integer linear combination of the lattice codewords sent by the ATs. However, the UTs 29 CP Precoding over finite-field UT 1 AT 1 AT 2 AT 3 AT 4 UT 2 UT 3 UT 4 Figure 2.2: DAS Downlink Architecture Using Reverse Compute and Forward: L = 4 and K = 4. cannot share the decoded codewords as in the uplink, since they have no backhaul links. Instead, the \interference" in the nite eld domain can be totally eliminated by zero- forcing precoding (over the nite eld) at the CP. RCoF has a distinctive advantage with respect to its CoF counterpart viewed before: since each UT sees only its own lattice codeword plus the eective noise, each message is rate-constrained by the computation rate of its own intended receiver, and not by the minimum of all computation rates across all receivers, as in the uplink case. In order to achieve dierent coding rates while preserving the lattice Z[j]-module structure, we use a family of nested lattices L 1 , obtained by a nested construction A as described in [NG11, Sect. IV.A]. In particular, we let ` =p 1 g(C ` )T + with =Z n [j]T and withC ` denoting the linear code over F p 2 generated by the rst r ` rows of a common generator matrix G, with r L r L1 r 1 . The corresponding nested lattice codes are given by L ` = ` \V , and have rate R ` = 2r ` n logp. We let ~ A = [~ a 1 ;:::; ~ a K ] T , where ~ a k 2Z L [j] denotes the integer coecient vector used at UT k for the modulo- receiver mapping (see (2.6)), and we let ~ Q = g 1 ([ ~ A] mod pZ[j]) denote the downlink system matrix, assumed to have rank L. Then, RCoF scheme proceeds as follows (see Fig. 2.2): 30 The CP sendsL independent messages toL UTs (ifK >L, then a subset ofL UTs is selected, as explained in Section 2.6). We letk ` denote the UT destination of the `-th message, encoded byL ` at rate R ` . The CP forms the messages ~ w ` 2 F r 1 p 2 by appending r 1 r ` zeros to each `-th information message of r ` symbols, so that all messages have the same length r 1 . The CP produces the precoded messages 2 6 6 6 6 4 ~ 1 . . . ~ L 3 7 7 7 7 5 = ~ Q 1 2 6 6 6 6 4 ~ w 1 . . . ~ w L 3 7 7 7 7 5 : (2.14) (notice: ifK >L then ~ Q is replaced by theLL submatrix ~ Q(fk 1 ;:::;k L g; [1 :L])). The CP forwards the precoded message ~ ` to AT ` for all ` = 1;:::;L, via the digital backhaul link. AT ` locally produces the lattice codeword ` = f(~ ` )2L 1 (the densest lattice code) and transmits the corresponding channel input ~ x ` according to (2.4). Because of linearity, the precoding and the encoding over the nite eld commute. Therefore, we can write [~ T 1 ;:::; ~ T L ] T = B[ ~ t T 1 ;:::; ~ t T L ] T mod , where ~ t ` = f( ~ w ` ) and B = g( ~ Q 1 ). 31 Each UT k ` applied the CoF receiver mapping as in (2.6), with integer coecient vector ~ a k ` and scaling factor k ` , yielding ^ ~ y k ` = 2 6 6 6 6 6 6 6 4 ~ a T k ` 2 6 6 6 6 6 6 6 4 ~ 1 . . . ~ L 3 7 7 7 7 7 7 7 5 + ~ z e ( ~ h k ` ; ~ a k ` ) 3 7 7 7 7 7 7 7 5 mod = 2 6 6 6 6 4 ~ a T k ` B 2 6 6 6 6 4 ~ t 1 . . . ~ t L 3 7 7 7 7 5 + ~ z e ( ~ h k ` ; ~ a k ` ) 3 7 7 7 7 5 mod (a) = 2 6 6 6 6 4 h ~ a T k ` B i mod pZ[j] 2 6 6 6 6 4 ~ t 1 . . . ~ t L 3 7 7 7 7 5 + ~ z e ( ~ h k ` ; ~ a k ` ) 3 7 7 7 7 5 mod (b) = h ~ t ` + ~ z e ( ~ h k ` ; ~ a k ` ) i mod ; (2.15) where (a) is due to the fact that [pt] mod =0 for any codewordt2 ` , and (b) follows from the following result: Lemma 2.2 Let ~ Q = g 1 ([ ~ A] mod pZ[j]). Assuming ~ Q invertible over F p 2, if B = g( ~ Q 1 ), then we have: [ ~ AB] mod pZ[j] =I: Proof Using [ ~ A] mod pZ[j] =g( ~ Q), we have: [ ~ AB] mod pZ[j] = [([ ~ A] mod pZ[j])B] mod pZ[j] = [g( ~ Q)g( ~ Q 1 )] mod pZ[j] = [g( ~ Q ~ Q 1 )] mod pZ[j] = I: 32 From (2.15) we have that RCoF induces a point-to-point channel at each desired UT k ` , where the integer-valued interference is eliminated by precoding, and the remaining eective noise is due to the non-integer residual interference and to the Gaussian channel noise. The scaling coecient k ` and the integer coecient vector ~ a k ` are optimized independently by each UT using (2.8) and Algorithm 1. It follows that the desired message ~ w ` can be recovered with arbitrarily small probability of error if R ` R( ~ h k ` ; ~ a k ` ;SNR), where the latter takes on the form given in (2.9). Including the fact that the precoded messages can be sent from the CP to the ATs if R ` R 0 , we arrive at: Theorem 2.1 RCoF applied to a DAS downlink with channel matrix ~ H = [ ~ h 1 ;:::; ~ h L ] T 2 C LL achieves the sum rate R RCoF ( ~ H; ~ A) = L X `=1 minfR 0 ;R( ~ h ` ; ~ a ` ;SNR)g: Remark 2 When the channel matrix ~ H has the property that each row` is a permutation of the rst row (e.g., in the case ~ H is circulant, as in the Wyner model [Wyn94]), each UT has the same computation rate and hence a single lattice codeL =L 1 = =L L is sucient. 2.4 Low-Complexity Schemes This section considers low-complexity versions of the schemes of Sections 2.2 and 2.3, using one-dimensional lattices and scalar quantization. Our approach is suited to the practically relevant case where the receivers are equipped with ADCs of xed nite res- olution, such that scalar quantization is included as an unavoidable part of the channel model. In this case, CoF and RCoF, as well as QMF and CDPC, are not possible since 33 lattice quantization (or any form of high-dimensional vector quantization) requires to have access to the unquantized (soft) signal samples. The quantized versions of CoF and RCoF follow as special cases, by choosing the generator matrix of the shaping lattice to be T = I, with = p 6SNR in order to satisfy the per-antenna power constraint with equality. The resulting lattice code is L = 1 \V with = Z n [j] and 1 = (=p)g(C) + , for a linear codeC over F p 2 of rate R = 2r n logp. Furthermore, we introduce a scalar quantization stage as part of each receiver. This is dened by the function Q (=p)Z[j] (), applied component-wise. Since is the n-dimensional complex cubic lattice, the modulo- operations in CoF/RCoF are also performed component-wise. Hence, we can restrict to a symbol-by-symbol channel model instead of considering n-vectors as before. Consider the same G-MAC setting of Section 4.1.1. Given the information message w k 2 F r p 2 , encoder k produces the codeword c k = w k G and the corresponding lattice codeword t k =f(w k ) = (=p)g(c k ) mod . The i-th component of its channel input x k is given by x k;i = [t k;i +d k;i ] mod Z[j]; (2.16) where the dithering samples d k;i are i.i.d. across users and time dimensions, and uni- formly distributed over the square region [0;) +j[0;). As already noticed, random dithers can be replaced in practice by deterministic signals [ZSE02, EZ04, DEGC03]. The received signal is given by (2.3). The receiver selects the integer coecient vec- tor a = (a 1 ;:::;a K ) T 2Z K [j] and produces the sequence u2F n p 2 with components u i = g 1 p " Q (=p)Z[j] y i K X k=1 a k d k;i !# mod Z[j] !! (2.17) = g 1 " Q Z[j] p K X `=1 a k t k;i + i (h;a;) !!# mod pZ[j] ! ; (2.18) 34 scalar quantizer sawtooth transformation Figure 2.3: Implementation of the modulo operation (analog component-wise sawtooth transformation) followed by the scalar quantization function Q (=p)Z[j] () function. for i = 1;:::;n, where i (h;a;) = K X k=1 (h k a k )x k;i +z i : (2.19) Since p t k;i 2 Z[j] by construction, and using the obvious identity Q Z[j] (v +) = v + Q Z[j] () with v2Z[j] and 2C, we arrive at u = K M k=1 q k c k (h;a;); (2.20) whereq k =g 1 ([a k ] mod pZ[j]) and where the components of the discrete additive noise (h;a;) are given by i (h;a;) =g 1 ([Q Z[j] ((p=) i (h;a;))] mod pZ[j]). This shows that the concatenation of the lattice encoders, the G-MAC and the receiver mapping (2.17) reduces to an equivalent discrete linear additive-noise Finite-Field MAC (FF-MAC) given by (2.20). Notice that the eective noise (h;a;) in the induced FF-MAC is i.i.d., dierently from the eective noise in (2.7) induced when the coarse lattice is good for mean square error (MSE) quantization. Remark 3 Notice thatu is obtained from the channel outputy by component-wise analog operations (scaling by and translation by P K k=1 a k d k ), scalar quantization and modulo reduction. In fact, the scalar quantization and the modulo lattice operations commute, 35 i.e., the modulo operation can be performed directly on the analog signals by wrapping the complex plane into the Voronoi region ofZ[j], and then the scalar quantizerQ (=p)Z[j] () can be applied to the wrapped samples. This corresponds to the analog sawtooth trans- formation, followed by scalar quantization, applied to the real and imaginary parts of the complex baseband signal, as shown in Fig. 2.3. The marginal pmf of i (h;a;) can be calculated numerically. We have noticed that a Gaussian approximation for the same variance (i.e., (p=) i (h;a;)CN (0; 2 )) yields very accurate approximated results, even for small number of transmitters. However, the goodness of such approximation depends, in general, on the specic channel coecients and operating SNR. In Section 2.4.3, we obtain an easy way to calculate the pmf of the eective noise component i (h;a;) based on such Gaussian approximation. The optimal choice of a and for the discrete channel (2.20) consists of minimizing the entropy of the discrete additive noise H( i (h;a;)). The folded Gaussian density was studied in [FTC00,Fis05] and the was numerically optimized with respect to minimizing the entropy and MSE in [Fis05]. In our case, we need to jointly optimize the integer vector a and the coecient , which does not lead to an elegant numerical method. Instead, we resort to the minimization of the unquantized eective noise variance 2 , which leads to (2.8), by choosing = SNRh H a 1+SNRkhk 2 and using the integer search of Algorithm 1. We assume that and a are determined in this way, independently, by each receiver, and again omit from the notation. In the following, we will present coding schemes for the induced FF-MAC in (2.20) and for the corresponding Finite-Field Broadcast Channel (FF-BC) resulting from the downlink, by exchanging the roles of ATs and UTs. We follow the notation used in Sections 2.2 and 2.3 and let Q = g 1 ([A] mod pZ[j]) and ~ Q = g 1 ([ ~ A] mod pZ[j]) denote the system matrix for the uplink and for the downlink, respectively. 36 2.4.1 QCoF and LQF for the DAS Uplink In this section we present two schemes referred to as Quantized CoF (QCoF) and Lattice Quantize and Forward (LQF), which dier by the processing at the ATs. QCoF is a low-complexity quantized version of CoF. The quantized channel output at AT ` is given by u ` =v ` (h ` ;a ` ); (2.21) where, by linearity,v ` = L K k=1 q `;k c k is a codeword ofC. This is a point-to-point channel with discrete additive noise over F p 2. AT ` can successfully decode v ` if R 2 logp H((h ` ;a ` )). This is an immediate consequence of the well-known fact that linear codes achieve the capacity of symmetric discrete memoryless channels [Dob63]. If R R 0 , each AT ` can forward the decoded message linear nite-eld combination to the CP, so that the original UT messages can be obtained by Gaussian elimination (see Section 2.2). With the same notation of Theorem 2.1, including network decomposition which applies verbatim here, we have: Theorem 2.2 QCoF with network decomposition, applied to a DAS uplink with channel matrix H = [h 1 ;:::;h K ] T 2C KK , achieves the sum rate R QCoF (H;A) = S X s=1 jA s j minfR 0 ; min k2As f2 logpH((h k ;a k ))gg: Next, we consider the LQF scheme, which may provide an attractive alternative in the case 2 logp R 0 , i.e., when R 0 is large and a small value of p is imposed by the ADC complexity and/or power consumption constraints. In LQF, the UTs encode their in- formation messages by using independently generated, not nested, random linear codes fC k g overF q , in order to allow for dierent coding ratesfR k g. In this case, the ne lat- tice for UT k is k = (=p)g(C k ) +Z n [j] and the symbol by symbol quantization maps the channel into an additive MAC channel over F q , with discrete additive noise. Hence, 37 independently generated random linear codes are optimal for this channel (this is easily seen form the fact that the channel is additive over the nite eld). In LQF, the ATs forward its quantized channel observations directly to the CP without local decoding. Hence, LQF can be seen as a special case of QMF without binning. From (2.21), the CP sees a FF-MAC with L-dimensional output: 2 6 6 6 6 4 u 1 . . . u L 3 7 7 7 7 5 =Q 2 6 6 6 6 4 c 1 . . . c K 3 7 7 7 7 5 2 6 6 6 6 4 (h 1 ;a 1 ) . . . (h L ;a L ) 3 7 7 7 7 5 : (2.22) The following result provides an achievable sum rate of LQF subject to the constraint 2 logpR 0 . Theorem 2.3 Consider the FF-MAC, dened by Q2F KK p 2 as in (2.22). If Q has rank K, the following sum rate is achievable by linear coding R FF-MAC = 2K logp K X k=1 H((h k ;a k )): (2.23) Proof: See Section 2.7.1. The relative merit of QCoF and LQF depends onR 0 ,p, and on the actual realization of the channel matrixH. In symmetric channel cases (i.e., Wyner model [Wyn94]), where the ATs have the same computation rate, QCoF beats LQF by makingp suciently large. On the other hand, if the modulation orderp is predetermined as in a conventional wireless communication system, and this is relatively small with respect to R 0 , LQF outperforms QCoF by breaking the limitation of the minimum computation rate over the ATs. When R 0 is given as system parameter andp can be chosen as design parameter, one can either choose a smallp and apply LQF or choose a largerp and apply QCoF. Notice that while QCoF requires decoding at each AT, LQF is suitable for very simple or \dumb" ATs, but requires joint decoding or successive decoding for the underlying FF-MAC at the CP. 38 2.4.2 RQCoF for the DAS Downlink Exchanging the roles of ATs and UTs and using (2.20), the DAS downlink with quanti- zation at the receivers is turned into the FF-BC 2 6 6 6 6 4 ~ u 1 . . . ~ u K 3 7 7 7 7 5 = ~ Q 2 6 6 6 6 4 ~ c 1 . . . ~ c L 3 7 7 7 7 5 2 6 6 6 6 4 ( ~ h 1 ; ~ a 1 ) . . . ( ~ h K ; ~ a K ) 3 7 7 7 7 5 : (2.24) The following result yields that simple matrix inversion over F p 2 can achieve the ca- pacity of this FF-BC. Intuitively, this is because there is no additional power cost with Zero-Forcing Beamforming (ZFB) in the nite-eld domain (unlike ZFB in the complex domain). Theorem 2.4 Consider the FF-BC in (2.24) for K = L. If ~ Q has rank L, the sum capacity is C FF-BC = 2L logp L X `=1 H(( ~ h ` ; ~ a ` )): (2.25) and it can be achieved by linear coding. Proof See Section 2.7.2. Motivated by Theorem 2.4, we present the RQCoF scheme using nite-eld matrix inver- sion precoding at the CP. As for RCoF, we useL nested linear codesC L C 1 where C ` is generated by the rst r ` rows of a common generator matrix G, with r L r 1 and has rate R ` = 2r ` n logp. Let k ` denote the UT destination of the `-th message, en- coded byC ` . The CP precodes the zero-padded information messagesf ~ w ` :` = 1;:::;Lg as in (2.14) and sends the precoded message ~ ` to AT ` for all ` = 1;:::;L, via the 39 digital backhaul link. AT ` generates the codeword ~ c ` = ~ ` G2C 1 , and the correspond- ing transmitted signal ~ x ` according to (2.16), with ~ t ` = f(~ ` ) Each UT k ` produces its quantized output according to the scalar mapping (2.17) and obtains: ~ u k ` = 0 B B B B @ ~ q T k ` 2 6 6 6 6 4 ~ c 1 . . . ~ c L 3 7 7 7 7 5 1 C C C C A ( ~ h k ` ; ~ a k ` ) (2.26) = 0 B B B B @ ~ q T k ` ~ Q 1 2 6 6 6 6 4 ~ w 1 G . . . ~ w L G 3 7 7 7 7 5 1 C C C C A ( ~ h k ` ; ~ a k ` ) = ~ v ` ( ~ h k ` ; ~ a k ` ) (2.27) where ~ v ` = ~ w ` G is a codeword ofC ` . Thus, UT k ` can recover its desired message if R ` 2 logpH(( ~ h k ` ; ~ a k ` )). Summarizing, we have: Theorem 2.5 RQCoF applied to a DAS downlink with channel matrix ~ H = [ ~ h 1 ;:::; ~ h L ] T 2 C LL achieves the sum rate R RQCoF ( ~ H; ~ A) = L X `=1 min n R 0 ; 2 logpH(( ~ h ` ; ~ a ` )) o : Remark 4 The complexity of RCoF is almost identical to that of CDPC since both imple- mentations utilize the same building blocks; they require nested lattice coding with shaping lattice of high dimension and apply modulo- operations at both the transmitter and the receiver. In RQCoF, we employ one-dimensional cubic lattice =Z[j] to make the modulo- operation of manageable complexity at the expense of the shaping loss. The corresponding scheme to DPC is known to be Tomlinson-Harashima Precoding (THP) in [Tom71,HM72], applied to multiuser MIMO precoding in [FWLH02,WFVH04,BTC06, CS04]. However, it is important to notice that in RQCoF the CP forwards only (digital) 40 message indices to the ATs, while in THP the CP needs high-dimensional vector quan- tization in order to compress the THP precoded signal to each AT. If scalar rather than vector quantization is used in quantized THP, an additional penalty is incurred due to the use of suboptimal quantization. Unfortunately, the latter is dicult to take into account in closed form and a precise comparison of RQCoF with scalar-quantized THP is beyond the scope of the present work. 2.4.3 Gaussian Approximation Let" = (p=)Ref i (h;a;)gCN (0; 2 " ) with 2 " = 2 =2. We consider the distribution of the discrete random variable =Q Z ("). The pmf of i (h;a;) is obtained by considering i.i.d. real and imaginary parts, both distributed as . Dene the function (x) , P "> (2x 1) 2 P "> (2x + 1) 2 (2.28) = G t (2x 1) 2 " G t (2x + 1) 2 " ; where G t (z) = 1 p 2 Z 1 z exp t 2 2 dt (2.29) is the Gaussian tail function. Recall that g maps theZ p =f0; 1;:::;p 1g into the set of integersf0; 1;:::;p 1gR. We dene an intervalI(x) by I(x), [x 0:5;x + 0:5]: (2.30) The pmf of can be computed as P( =),P 0 @ "2 [ m2Z I(g() +pm) 1 A : (2.31) 41 For any 1 ; 2 6= 0 satisfyingg( 1 )+g( 2 ) =p, we haveP( = 1 ) =P( = 2 ), which can be immediately proved using the symmetry of Gaussian distribution (about the origin): P 0 @ "2 [ m2Z I(g( 1 ) +pm) 1 A = P 0 @ "2 [ m2Z + [f0g I(g( 1 ) +pm) 1 A +P 0 @ "2 [ m2Z [f0g I(g( 1 )p +pm) 1 A = P 0 @ "2 [ m2Z + [f0g I(g( 1 ) +pm) 1 A +P 0 @ "2 [ m2Z + [f0g I(pg( 1 ) +pm) 1 A = P 0 @ "2 [ m2Z + [f0g I(g( 1 ) +pm) 1 A +P 0 @ "2 [ m2Z + [f0g I(g( 2 ) +pm) 1 A ; whereZ + andZ denote the positive and negative integers, respectively. Thus, we only need to nd the pmf of with p1 2 and other probabilities are directly obtained by symmetry. Using (2.32) for 6= 0, we can quickly compute the pmf of using (x) dened in (2.28): P( = 0) = (0) + 2 1 X m=1 (g() +pm) (2.32) P( =) = 1 X m=0 (g() +pm) + (pg() +pm): (2.33) In fact, (x) is monotonically decreasing function on x and in general, quickly converges to 0 asx increases. Therefore, we only need a nite number of summations in (2.32) and (2.33) and we observed that it is enough to sum over m = 0; 1; 2 in all numerical results presented in this chapter. 42 2.5 Comparison with Known Schemes on the Wyner Model In order to obtain clean performance comparisons with other state-of-the art information theoretic coding strategies, we consider the symmetric Wyner model [Wyn94], which has been used in several other works for its simplicity and analytic tractability. In particular, we consider comparisons with Quantize reMap and Forward (QMF) and Decode and Forward (DF) for the DAS uplink, and Compressed Dirty Paper Coding (CDPC) and Compressed Zero-Forcing Beamforming (CZFB) for the DAS downlink. In the symmetric Wyner model with L ATs and L UTs, the received signal at the `-th receiver (AT for the uplink or UT for the downlink) is given by y ` =x ` + (x `1 +x `+1 ) +z ` ; (2.34) where 2 (0; 1] quanties the strength of inter-cell interference and z ` has i.i.d. compo- nentsCN (0; 1). 2.5.1 Review of some Classical Coding Strategies 2.5.1.1 Quantize reMap and Forward (QMF) Each AT performs vector quantization of its received signal at some rate R 0 R 0 and maps the blocks of nR 0 quantization bits into binary words of length nR 0 by using a hashing function (binning). The CP performs joint decoding of all UTs' messages based on the observation of all the (hashed) quantization bits. Using random coding with Gaussian codes and random binning, [SSPS09, Prop IV.1] proves the following achievable rate of QMF: R QMF = max 0r min S[1:L] n jSj(R 0 r) (2.35) + log det I +SNR (1 2 r )H(S c ; [1 :L])H(S c ; [1 :L]) H oo : (2.36) 43 As R 0 ! 1, R QMF tends to the sum rate of the underlying multi-antenna G-MAC channel with L users and one L-antenna receiver. For SNR!1 and xed R 0 , then R QMF !LR 0 [SSPS09,NSGS09]. While for a general channel matrix computing (2.36) is non-trivial, a remarkable result of [SSPS09] is that for the Wyner model in the limit of L!1 the QMF rate per user can be simplied to R QMF, per-user =F (r ); (2.37) where F (r) = Z 1 0 log 1 +SNR 1 2 r (1 + 2 cos(2)) 2 d; and wherer is the solution of the equationF (r) =R 0 r. A simplied version of QMF does not use binning, and simply forwards to the CP the quantization bits collected at the ATs. We refer to this scheme as Quantize and Forward (QF), without the re-mapping. In this case, the quantization rate is R 0 =R 0 . From [SSS09a], the achievable sum rate of QF is given by R QF = log det I +SNRDHH H ; (2.38) where D = diag(1=(1 +D ` ) :` = 1;:::;L) and D ` = (1 +SNRkh ` k 2 )=(2 R 0 1) denotes the variance of quantization noise at AT `. 2.5.1.2 Decode and Forward (DF) In the Wyner model, each AT ` sees the three-input G-MAC formed by UTs ` 1, ` and ` + 1. In this scheme, each AT decodes a single message such that the sum rate across all distinct messages is maximized. Since in the symmetric Wyner model 1, the optimal decoding strategy is such that each AT ` decodes the corresponding UT ` 44 message. Imposing either to treat interference as noise, or to decode all messages at each AT, yields [SSPS09]: R 1 = log 1 + SNR 1 + 2 2 SNR R 2 = min 1 2 log(1 + 2 2 SNR); 1 3 log(1 + (1 + 2 2 )SNR) R DF = L minfmax(R 1 ;R 2 );R 0 g: This scheme has no joint-processing gain. However, when R 0 is suciently small com- pared to the rates achievable over the wireless channel, or when is very small, this scheme can be optimal [SSPS09,NSGS09]. In fact, DF is what is implemented today in a network of small cells, where each AT operates as a stand-alone base station, and the de- coded packets are sent to a common node that may use packet selection macro-diversity, in the case some of the base stations fail to decode. Therefore, it is useful to compare with DF to quantify the potential gains of other schemes with respect to current technology. 2.5.1.3 Compressed Dirty Paper Coding (CDPC) We focus now on the DAS downlink. In CDPC the CP performs joint DPC under per- antenna power constraint and sends the compressed (or quantized) DPC codewords to the corresponding ATs via wired links. As a consequence, the ATs also transmit quantization noise. Let ~ v ` be the DPC-encoded signal to be transmitted by AT` and let _ v ` denote its quantized version. Dene 2 ` = 1 n E[k~ v ` k 2 ] and _ 2 ` = 1 n E[k _ v ` k 2 ]. From the standard rate distortion theory, an achievable quantization distortion D ` is given by R(D ` ) = min P ^ V ` jV ` :E[kV ` ^ V ` k 2 ]D ` I(V ` ; ^ V ` )I(V ` ; _ V ` ) = log(1 + 2 ` =D ` ); 45 where the upper bound follows from the choice _ V ` =V ` + _ Z ` with _ Z ` CN (0;D ` ) andV ` with variance 2 ` . Letting R 0 = log(1 + 2 ` =D ` ) and solving for D ` we obtain D ` = 2 ` 2 R 0 1 : (2.39) Using the fact that _ 2 ` = 2 ` +D ` , the per-antenna power constraint _ 2 ` SNR imposed at each AT ` yields 2 ` SNR 2 R 0 1 2 R 0 for` = 1;:::;L: (2.40) Using (2.40) in (2.39), we obtainD ` =SNR 2 R 0 for` = 1;:::;L. At the`-th UT receiver, the variance of the eective noise is given by ~ 2 ` = 1 +k ~ h ` k 2 SNR 2 R 0 : (2.41) Then, an achievable sum rate of CDPC is equal to the sum capacity of the resulting vector BC with the above modications (i.e., per-antenna power constraint and noise variance). This can be computed using the ecient algorithm given in [HPC09], based on Lagrangian duality. Further, the closed form rate-expression was provided in [SSS + 09b] for the so- called soft-hando Wyner model, a simplied variant of the Wyner model where each receiver has only one interfering signal from its left neighboring cell. While CDPC is expected to be near optimal for large R 0 , it is generally suboptimal at nite (possibly small) R 0 . 2.5.1.4 Compressed Zero Forcing Beamforming (CZFB) CP performs precoding with the inverse channel matrix B = ~ H 1 and sends the com- pressed ZFB signals to the corresponding ATs via wired links. As in CDPC, the ATs forward also quantization noise, such that the variance of eective noise at the `-th UT is given again by (2.41). The transmit power constraint (2.40) holds verbatim. Because 46 of the non-unitary precoding, the useful signal power is given by SNR 2 R 0 1 2 R 0kb ` k 2 where b T ` is the `-th row of the precoding matrix B. It follows that CZFB achieves the sum rate R CZFB = L X `=1 log 1 + SNR=kb ` k 2 1 + (1 +k ~ hk 2 SNR)=(2 R 0 1) : 2.5.2 Numerical Results Thanks to the banded structure of the Wyner model channel matrix, the resulting system matrix of CoF (resp., RCoF) is guaranteed to have rankL although every AT (resp., UT) determines its integer coecient vector in a distributed way. In addition, the non-integer penalty which may be relevant for specic values of can be mitigated by using a power allocation (PA) strategy, in order to create more favorable channel coecients for the integer conversion at each receiver. In [NSGS09] a further improved strategy is proposed based on superposition coding, where the user messages are split into two layers, and one layer is treated as noise while the other is treated by CoF. Here we focus on simple power allocation, since it is practical and captures a signicant fraction of the gains achieved with superposition coding. The power allocation strategy works as follows: odd-numbered UTs (resp., ATs) transmit at power P and even-numbered UTs (resp., ATs) transmit at power (2)P , for 2 [0; 1]. The role of odd- and even-numbered UTs (or ATs) is alternately reversed in successive time slots, such that each UT (resp., AT) satises its individual power constraint on average. Accordingly, the eective coecients of the channel for odd-numbered and even-numbered relays are h o = [ p 2; p ; p 2] and h e = [ p ; p 2; p ]. For given , the parameter 2 [0; 1] can be optimized to make the eective channels better suited for the integer approximation in the CoF receiver mapping. We have two computation rates, R(h o ;a o ) and R(h e ;a e ), at the odd and even numbered receivers. The achievable symmetric rate of CoF (or RCoF) with power allocation is given by minfR 0 ;R(h o ;a o ;SNR);R(h e ;a e ;SNR)g. Notice that the odd- and even-numbered relays can optimize their own equation coecients independently, but the optimization with respect to is common to all, and the computation rate is the minimum 47 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 R 0 (bits per channel use) Rates per user (bits per channel use) Upper Bound QMF QF DF CoF CoF with PA QCoF with PA Figure 2.4: SNR = 25dB and L =1. Achievable rates per user as a function of R 0 , for the DAS uplink in the Wyner model case with inter-cell interference parameter = 0:7. computation rate over all the relays, since the same lattice codeL is used across all users. In Fig. 2.4, we show the performance of various relaying strategies for the DAS uplink with SNR = 25 dB, as a function of backhaul rate R 0 . L =1 is assumed in order to use the simple rate expression of QMF in (2.37). Fig. 2.4 shows that the power allocation strategy signicantly reduces the integer approximation penalty and almost achieves the cut-set outer bound (i.e., capacity) forR 0 7 bits where the cut-set outer bound is given in [NSGS09] as R = minfC MIMO ;R 0 g; and C MIMO = Z 1 0 log(1 +SNR(1 + 2 cos 2) 2 )d: Not surprisingly, QCoF withp = 251 only pays the shaping penalty with respect to CoF, i.e., it approaches the performance of the corresponding high-dimensional scheme within 0:5 bit per complex dimension. 48 We observe a similar trend for the downlink schemes, shown in Fig. 2.5. In this case, the achievable sum rate of RCoF with power allocation is given by R sum = 1 2 (minfR 0 ;R(h o ;a o ;SNR)g + minfR 0 ;R(h e ;a e ;SNR)g); (2.42) where the average between odd and even numbered UTs is due to the fact that in RCoF we can use two dierent lattice codes and therefore the rates are not constrained to be all equal. RCoF outperforms CDPC for R 0 6:5 bits per channel use. Remark 5 As shown in Fig. 2.4, there is a crossing point such that for R 0 below this point CoF outperforms QMF and for R 0 above this point QMF outperforms CoF. This crossing point depends on SNR. In particular, as SNR increases the value of R 0 at the crossing point also increases. This behavior is explained by observing that CoF suers from the local decoding constraints for lower SNR, and QMF can enjoy the coherence gain when the quantization noise is small enough (i.e., R 0 is large). A similar trend can be observed for RCoF and CDPC in the downlink. Interestingly, we observe that the fully practical and easily implementable quantized schemes QCoF and RQCoF can outperform other conventional practical schemes such as DF and CZFB, respectively. Further, we notice that the proposed schemes can be signicantly improved by mitigating the impact of the non-integer penalty. In this model, power allocation is eective thanks to the system symmetric structure. However, it is not clear how to extend the power allocation approach in the general case of a wireless network whose channel matrix is the result of fading, shadowing and pathloss, and therefore it does not enjoy any special easily parameterized structure. In the next section we address such case and show that multiuser diversity (i.e., AT/UT selection) can greatly improve the performance of the basic schemes. 49 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 R 0 (bits per channel use) Sum Rates (bits per channel use) CDPC CZFB RCoF RCoF with PA RQCoF with PA Figure 2.5: SNR = 25dB and L = 10. Achievable sum rates as a function of R 0 , for the DAS downlink in the Wyner model case with inter-cell interference parameter = 0:7. 2.6 Antenna and User Selection Since the proposed schemes require an equal number of ATs and UTs active at each given time, in a general DAS with K UTs and L ATs the system must select which terminals are active in every scheduling slot. We dene the \active" set of UTsU [1 : K] as the subset of UTs that are actually scheduled for transmission (resp., reception) on the current uplink (resp., downlink) slot, comprising n channel uses. Similarly, the \active" set of ATsA [1 :L] is dened as the subset of ATs that are used for reception (resp., transmission) on the current uplink (resp., downlink) slot. 2.6.1 Antenna Selection for the DAS Uplink We assume that the active set of UTs is xed a priori. Without loss of generality, we can xU = [1 : K] and assume K < L. Our goal is to select a subsetA [1 : L] of ATs of cardinality K. Recall that every AT chooses the integer combination coecients, and therefore its vector q ` , using Algorithm 1 in order to maximize its own computation rate 50 R ` =R(h ` ;a ` ;SNR). The CP knowsfq ` ;R ` :`2 [1 :L]g. The CP aims at maximizing the sum rate such that the resulting system matrix is full-rank, by selecting a subset of ATs for the given UT active setU. 2.6.1.1 AT selection for CoF and QCoF From Theorem 2.1, the AT selection problem consists of ndingA solution of: max A[1:L] S(A) X s=1 jA s j minfR 0 ; minfR ` :`2A s gg (2.43) subject to Rank(Q(A;U)) =jUj; (2.44) where S(A) indicates the number of disjoint subnetworks with respect to Q(A;U). This problem has no particularly nice structure and the optimal solution is obtained, in general, by exhaustive search over alljUjjUj submatrices ofQ([1 :L];U). Yet, we notice that if an optimal solutionA ? does not decompose (i.e.,S(A ? ) = 1), the simple greedy Algorithm 2 given below nds it (see Lemma 2.3). Namely, there exists a low-complexity algorithm to nd an optimal AT selection for dense networks whose system matrix Q([1 : L];U) cannot be decomposed in block-diagonal form. In general, we may have several disjoint subnetworks, each of which does not decom- pose further, even when removing some ATs. Then, we can perform antenna selection by using Algorithm 2 on each subnetwork component. If the optimum solution of each subnetwork component does not involve further network decomposition, by Lemma 2.3 we are guaranteed to arrive at an optimal global solution. This generally suboptimal (but ecient) approach can be summarized as For given Q = Q([1 : L];U), perform network decomposition using depth-rst or breadth-rst search [AMO93], yielding disjoint subnetworks Q(A s ;U s ) for s = 1;:::;S. 51 For each subnetworkQ(A s ;U s ), run Algorithm 2 and nd a good selectionA ? s A s withjA ? s j =jU s j. Finally, obtain the set of active ATs,A ? =[ S s=1 A ? s , such thatjA ? j =jUj. Algorithm 2.2 The Greedy Algorithm Input: (Q,fw ` : ` = 1;:::;mg) where Q is a full-rank mn matrix with m > n and w ` = minfR 0 ;R ` g Output:S [1 :m] withjSj =n 1. Sort [1 :m] such that w 1 w 2 w m 2. Initially, ` = 1 andS =; 3. If Rank(Q(S[f`g; [1 :n]))> Rank(Q(S; [1 :n])), thenS S[f`g 4. Set ` =` + 1 5. Repeat 3)-4) untiljSj =n We have: Lemma 2.3 If Rank(Q) =n, Algorithm 2 nds a solution to the problem max S[1:m] minfw ` :`2Sg (2.45) subject to Rank(Q(S; [1 :n])) =n: (2.46) Proof Let ^ Q be the row-permuted matrix of Q according to the decreasing ordering of the weights w ` . The problem is then reduced to nding the minimum row index ` y such that ^ Q([1 :` y ]; [1 :n]) has rank n. This is precisely what Algorithm 2 does. An immediate corollary of Lemma 2.3 is that, if one disregards network decomposition, then Algorithm 2 nds the maximum computation rate over the AT selection. In fact, it is sucient to use Algorithm 2 with m =L, n =K, and input Q =Q([1 :L];U) and w ` = minfR 0 ;R ` g for ` = 1;:::;L. 52 2.6.1.2 AT selection for LQF From Theorem 2.3, the AT selection problem consists of ndingA solution of: max A[1:L] X `2A minfR 0 ;R ` g (2.47) subject to Rank(Q(A;U)) =jUj; (2.48) where we let R ` = 2 logpH((h ` ;a ` )) (see Section 2.4.1). This problem consists of the maximization of a linear function subject to a matroid constraint, where the ma- troidM = ( ;I) is dened by the ground set = [1 : L] and by the collection of independent setsI =fA : Q(A;U) has linearly independent rowsg. Rado and Ed- monds [Rad57, Edm71] proved that a greedy algorithm nds an optimal solution. In this case, such algorithm coincides with Algorithm 2 with input Q = Q([1 : L];U) and w ` = minfR 0 ;R ` g. 2.6.2 User Selection for the DAS Downlink In this case we assume that the set of ATsA = [1 : L] is xed and K > L. Hence, we wish to select a subsetU [1 :K] of cardinalityL such that the resulting system matrix has rank L and the DAS downlink sum rate is maximized. The CP has knowledge of the downlink system matrix ~ Q([1 :K];A) = [~ q 1 ;:::; ~ q K ] T and the set of individual user computation rates, ~ R k = R( ~ h k ; ~ a k ;SNR) for RCoF, or ~ R k = 2 logpH(( ~ h k ; ~ a k )) for RQCoF (see Theorem 2.4). The UT selection problem consists of ndingU solution of: max U[1:K] X k2U minfR 0 ; ~ R k g (2.49) subject to Rank( ~ Q(U;A)) =jAj: (2.50) As noticed before, this can be regarded as the maximization of a linear function over a matroid constraint. Therefore, Algorithm 2 with input Q = ~ Q([1 : K];A) and w k = minfR 0 ; ~ R k g provides an optimal solution. 53 0 5 10 15 20 25 0 5 10 15 20 25 SNR [dB] Sum Rates (bits per channel use) CoF, Random Selection (5 ATs) CoF, Random Selection (15 ATs) CoF, Random Selection (25 ATs) CoF, Greedy Selection (5 ATs) DF, Optimal Selection (5 ATs) Random Selection Figure 2.6: DAS uplink with K = 5, L = 25 and R 0 = 6 bit/channel use: average sum rate vs. SNR on the Bernoulli-Gaussian model with = 0:5. 2.6.3 Comparison on the Bernoulli-Gaussian Model We consider a DAS with channel matrix with i.i.d. elements [H] `;k = h `;k `;k , where h `;k CN (0; 1) and `;k is a Bernoulli random variable with P( `;k = 1) = . This model captures the presence of Rayleigh fading and some extreme form of path-blocking shadowing, and it is appropriate for a DAS deployed in buildings, or dense urban en- vironments where the ATs are not mounted on tall towers, in contrast to conventional macro-cellular systems. For the downlink results, we assume a channel matrix ~ H with the same statistics. We compute the ergodic sum rates by Monte Carlo averaging with respect to the channel matrix. If the resulting system matrix, after AT (resp., UT) selection is rank decient, then the achieved instantaneous sum rate is zero, for that specic realization. Hence, rank deciency can be regarded as a sort of \information outage" event. With the path gain coecients and noise variance normalization adopted here, the SNR coincides with the individual nodes power constraint. 54 5 10 15 20 25 30 0 5 10 15 20 25 30 SNR [dB] Sum−Rates (bits per channel use) RCoF, Greedy Selection (5 UTs) RQCoF (p=7), Greedy Selection (5 UTs) RCoF, Random Selection (5 UTs) RQCoF (p=7), Random Selection (5 UTs) Figure 2.7: DAS downlink withK = 25,L = 5 andR 0 = 6 bit/channel use: average sum rate vs. SNR on the Bernoulli-Gaussian model with = 0:5. Fig. 2.6 shows the average sum rate for a DAS uplink withK = 5 UTs,L = 25 ATs and channel blocking probability = 0:5. This result clearly show that the proposed \greedy" AT selection scheme yields a large improvement over random selection of a xed number of ATs, and essentially eliminates the problem of system matrix rank deciency, provided thatLK. The curves denoted as \Random Selection" indicate the case where a xed number L 0 < L of ATs is randomly selected with uniform probability, independent of channel realizations. In random selection, we considered three cases asL 0 = 5; 15, and 25, in order to see the tradeo between the rank-deciency and the minimum of computation rates. For example, the rank-deciency problem can be alleviated by increasing L 0 while the common user rate (i.e., the minimum computation rate) is generally decreasing with L 0 . For L 0 = 25 the DAS uses all the available ATs all the time, yet its performance is much worse than selecting 5 ATs out of 25 according to the proposed selection scheme. In Fig. 2.6, the performance of CoF with Random Selection of 5 ATs is equivalent to the case with K =L = 5, by symmetry of the user channel statistics. In this case, CoF 55 does not provide a satisfactory performance, dierently from the result of well-structured Wyner models considered in the previous section and in [NSGS09]. Fig. 2.6 provides also a comparison with the DF performance. In this case, DF is directly related to the conventional approach of non-cooperating ATs, each of which acts as a separated base station. Following the system architecture in Remark 1, in a DF scheme the ATs decode one of the messages in a decentralized way. DF can be optimized over the allocation of messages to the ATs, by nding 5 ATs that can decode the 5 user messages in order to maximize the sum rate. This is equivalent to restricting the system matrix of CoF to be a 5 5 permutation matrix (i.e., an integer matrix with a single 1 in each row and column and all zero elsewhere), representing the association of user messages to the ATs. Fig. 2.7 shows a similar trend for the DAS downlink. Here, random selection indicates that 5 UTs are chosen at random out of the 25 UTs. We notice that the sum rate vs. SNR curves for both greedy and random UT selection have the same slope, indicating that the rank-deciency problem is not signicant in both cases. However, greedy selection achieves a very evident multiuser diversity gain over random selection. This is not only due to selecting channel vectors with large gains, as in conventional multiuser diversity, but also to the fact that the greedy selection is able to choose channels that are adapted to the RCoF strategy, i.e., whose coecients are well approximated by integers (up to a common scaling factor). It is also interesting to notice that RQCoF with greedy selection does not suer from the rank-deciency of the system matrix even for p as small as 7, in the example. This is indicated by the fact that the sum rate gap between RQCoF and RCoF is essentially equal to the shaping loss (0.5 bits per user). We compared the proposed schemes with QF (uplink) and CDPC (downlink) over the Bernoulli-Gaussian model. Recall that QF is a special case of QMF without binning, whose achievable sum rate is given in (2.38). In QF, more observations (i.e., more active ATs) generally improve the sum rate and thus AT selection is not needed for the sake of 56 5 10 15 20 25 1 2 3 4 5 6 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Figure 2.8: DAS uplink with K = 5 andL = 50, Bernoulli-Gaussian model with = 0:5: Colors represent the relative gain of CoF versus QF (e.g., ratio of sum rates R CoF =R QF ). maximizing the sum rate. Yet, for a fair comparison with the same total backhaul capac- ity, we considered a greedy search that selects L 0 =K <L active ATs, by maximizing at each step the achievable sum rate. From Fig. 2.8, we observe that CoF outperforms QF when R 0 is small relatively to the channel SNR. In this regime, the quantization noise dominates with respect to the non-integer penalty. Instead, whenR 0 increases, eventually QF outperforms CoF. Fig. 2.9 presents a comparison between RCoF and CDPC, leading to similar conclusions for the DAS downlink. Next, we examine the performance of the proposed low-complexity schemes QCoF, LQF, and RQCoF, by focusing on a small cell network scenario, where ATs and UTs are close to each other. This is re ected by considering a xed and relatively large SNR value (SNR = 25 dB in our simulation), and comparing performances versusR 0 , which becomes the main system bottleneck. Fig. 2.10 shows that QCoF and LQF are competitive with respect to the performance of QF. Furthermore, an additional remarkable feature of the lattice-based schemes is that they can substantially reduce the channel state information overhead. When QCoF (or LQF) withp = 7 is used, each AT` only requires 2K log(7) 57 5 10 15 20 25 1 2 3 4 5 6 0.7 0.8 0.9 1 1.1 1.2 Figure 2.9: DAS downlink with K = 50 and L = 5, Bernoulli-Gaussian model with = 0:5: Colors represent the relative gain of RCoF versus CDPC (e.g., ratio of sum rates R RCoF =R CDPC ). 28 (with K = 5) bits of feedback per scheduling slot in order to forward the integer combination coecients (i.e., q ` = (q `;1 ;:::;q `;K )) to the CP. In Fig. 2.11, RQCoF withp = 17 can achieve the same spectral eciency of CDPC for R 0 5 bits and outperforms CZFB in the range ofR 0 6 bits. For CZFB, we made use of the standard greedy user selection approach [DS05,YG06] to nd a subset of K 0 <K (with K 0 = 5 in our simulation) active UTs. As expected, QCoF (resp., RQCoF) can achieve the performance of QF (resp., CDPC) when the wired backhaul rate R 0 is not over-dimensioned with respect to the capacity of the wireless channel. These observations point out that the proposed schemes are suitable for low-complexity implementation of cooperative home networks where small home-based access points are connected to the CP via digital subscriber line (DSL). 58 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 5 10 15 20 25 30 R 0 (bits per channel use) Sum−Rates (bits per channel use) QF QCoF (p=7) LQF (p=3) LQF (p=5) LQF (p=7) Figure 2.10: DAS uplink with SNR = 25 dB,K = 5 andL = 50: achievable sum rates as a function of R 0 . 1 2 3 4 5 6 7 0 5 10 15 20 25 30 35 R 0 (bits per channel use) Sum−Rates (bits per channel use) CDPC CZFB RQCoF (p=17) Figure 2.11: DAS downlink with SNR = 25 dB,K = 50 andL = 5: achievable sum rates as a function of R 0 . 59 2.7 Proofs 2.7.1 Proof of Theorem 2.3 Consider the FF-MAC dened by y = Qx where x = (x 1 ;:::;x K ) T 2 F K p 2 and y = (y 1 ;:::;y K ) T 2 F K p 2 . The capacity region is the union of the rate regions dened by [EGK11] X k2S R k I (fx k :k2S);yjfx k :2S c g;q); 8 S [1 :K]; (2.51) over all pmfs P x;q = P q Q K k=1 P x k jq . Since for any xed such pmf the region (2.51) is a polymatroid, the maximum sum rate achieved on the dominant face P K k=1 R k =I(x;yjq). Since the expectation of the maxima is larger or equal to the maximum of the expectation, we have that P L k=1 R k max Px;q I(x;yjq). Finally, since q!x!y, we have: I(x;yjq)I(x;q;y) =I(x;y) +I(q;yjx) =I(x;y); (2.52) showing that time-sharing is not needed for the maximum sum rate. Since Q is full rank, uniform i.i.d. inputsF p 2 achieve I(x;y) = H(y)H(yjx) =H(y)H() (2.53) 2K logpH(): (2.54) Finally, since P K k=1 H( k ) H( 1 ;:::; K ), we conclude that the sum rate in (2.23) is achievable. 2.7.2 Proof of Theorem 2.4 We consider the FF-BC dened by y = ~ Qx , where x = (x 1 ;:::;x L ) T and y = (y 1 ;:::;y L ) T . SinceQ is invertible, lettingx = ~ Q 1 v forv2F L p 2 yields the orthogonal BC y =v. The achievable sum rate for this decoupled channel is obviously given by the 60 sum of of the capacities of each individual additive-noise nite-eld channel, irrespectively of the statistical dependence across the noise components. Each `-th channel capacity is achieved by using independently generated random linear codes over F p 2 [Dob63]. It follows that the sum rate (2.25) is achievable. In order to show that this is in fact the sum- capacity of the FF-BC, we notice that a trivial upper-bound on the broadcast capacity region is given by [EGK11]: R ` max Px I(x;y ` ) for ` = 1;:::;L: (2.55) This is the capacity of the single-user channel with transition probability P y ` jx . Due to the additive noise nature of the channel, we haveI(x;y ` ) =H(y ` )H( ` ). Furthermore, H(y ` ) 2 logp and this upper bound is achieved by letting x uniformly distributed over F L p 2 . Summing over ` we nd that the upper bound on the sum capacity coincides with (2.25). 61 Chapter 3 Full-Duplex Relaying with Half-Duplex Relays In this chapter we consider \virtual" full-duplex relaying by means of half-duplex relays. In this conguration, each relay stage in a multi-hop relaying network is formed by at least two relays, used alternatively in transmit and receive modes, such that while one relay transmits its signal to the next stage, the other relay receives a signal from the previous stage. With such a pipelined scheme, the source is active and sends a new information message in each time slot. We consider the achievable rates for dierent coding schemes and compare them with a cut-set upper bound, which is tight in certain conditions. In particular, we show that both lattice-based Compute and Forward (CoF) and Quantize reMap and Forward (QMF) yield attractive performance and can be easily implemented. In particular, QMF in this context does not require \long" messages and joint (non-unique) decoding, if the quantization mean-square distortion at the relays is chosen appropriately. Also, in the multi-hop case the gap of QMF from the cut-set upper Figure 3.1: Multihop virtual full-duplex relay channels whenK = 5 (i.e., 6-hop network). Black-solid lines are active for every even time slot and red-dashed lines are active for every odd time slot. 62 bound grows logarithmically with the number of stages, and not linearly as in the case of \noise level" quantization. Furthermore, we show that CoF is particularly attractive in the case of multi-hop relaying, when the channel gains have uctuations not larger than 3dB, yielding a rate that does not depend on the number of relaying stages. In particular, we argue that such architecture may be useful for a wireless backhaul with line-of-sight propagation between the relays. 3.1 System Models In this paper, encoding/decoding operations are performed over time slots consisting of n channel uses of a discrete-time Gaussian channel. Also, successive relaying is assumed such that, at each time slot t, the source transmits a new message w t 2f1;:::; 2 nR g to one of the relays and the destination decodes a new message w t1 from the other relay (see Fig. 1.3). The role of relays 1 and 2 is alternatively reversed in successive time slots. During N + 1 time slots, the destination decodes N messagesfw t :t = 1;:::;Ng. Hence, the achievable rate is given by N N+1 R. By letting N !1, we can achieve the rateR, provided that the message error probability vanishes withn. As in standard relay channels (see for example [CG79, KGG05, ADT11, LKEGC11, HK13]), we take rst the limit for n!1 and then for N!1, and focus on the achievability of rate R. A block of n channel uses of the discrete-time channel is described by For odd t, y R 2 [t] = x S [t] + x R 1 [t] +z R 2 [t] (3.1) y D [t] = x R 1 [t] +z D [t] (3.2) 63 For even t, y R 1 [t] = x S [t] + x R 2 [t] +z R 1 [t] (3.3) y D [t] = x R 2 [t] +z D [t] (3.4) where 2R + denotes the inter-relay interference level. Here, x S [t]2C 1n and x R k [t]2 C 1n denote the transmit signals at source and relay k, respectively. Also, y D [t]2C 1n and y R k [t]2 C 1n denote the received signals at destination and relay k, respectively. For simplicity of notation, we will drop the relay index k in the rest of paper since it is implicitly identied by time index t. Also, it is assumed that the channel coecients are time-invariant and known to all nodes. 3.1.1 Upper Bound Fixing the relay operation to be successive relaying, an upper bound onR is immediately obtained by considering the cut between source and relay (or between relay and desti- nation). We call this successive upper bound, and yields R C(SNR). This bound is achievable by DPC [CCL07,RGK08]. In this section we prove that, even without xing a priory the relay operation, C(SNR) is an upper bound on the achievable rate if SNR 1 and if the relay state (transmit/receive) is independent of the messages. Throughout the paper, it is assumed thatSNR 1, such thatR upper =C(SNR) is a general upper bound. Since successive relaying with DPC achieves this bound, the capacity of this channel is equal to C(SNR). In order to prove this result, we start from the upper bound on a general half-duplex relay network derived in [KSA03], by introducing the concept of state. In [KSA03], it is assumed that the sequence of network state is known to all nodes at each time and is predened. Thus, the state is independent of the information messages. Here, the state of network is a partitioning of its nodes into three disjoint setsI (idle),T (transmitters) andR (receivers), such that S 2 T [I and D 2 R[I, and there is no link that 64 (a) cut 1: state 4 is meaningless (b) cut 2: state 1 is meaningless (c) cut 3: state 2 is meaningless (d) cut 4: state 3 is meaningless Figure 3.2: Four possible cuts for two-hop relay channel with half-duplex relays. 65 arrives at a transmitter or idle node. Lett m denote the fraction of time during which the network operates in state m2f1; 2;:::;Mg. In our model, there are only four states: (i) T =fS; R 1 g,R =fD; R 2 g,I =;; (ii)T =fS; R 2 g,R =fD; R 1 g,I =;; (iii)T =fSg, R =fR 1 ; R 2 g,I =fDg; (iv)T =fR 1 ; R 2 g,R =fDg,I =fSg. Notice that successive relaying only consists of two states (i) and (ii), i.e., t 1 = t 2 = 1=2 and t 3 = t 4 = 0. From [KSA03], we can derive the upper bound of our channel such as R upper = max t 1 ;t 2 ;t 3 ;t 4 minfI 1 ;I 2 ;I 3 ;I 4 g (3.5) subject to t 1 +t 2 +t 3 +t 4 = 1 t 1 ;t 2 ;t 3 ;t 4 0; where I 1 = t 1 C (SNR) +t 2 C (SNR) +t 3 C (2SNR) I 2 = t 2 C (1 + (1 + ) 2 )SNR +SNR 2 +t 3 C (SNR) +t 4 C (SNR) I 3 = t 1 C (1 + (1 + ) 2 )SNR +SNR 2 +t 3 C (SNR) +t 4 C (SNR) I 4 = t 1 C (SNR) +t 2 C (SNR) +t 4 C (4SNR); whereI 1 ;I 2 ;I 3 , andI 4 correspond to the four possible cuts (see Fig. 3.2). Then, we have: Lemma 3.1 If SNR 1, the solution of the optimization problem (3.5) is given by t 1 =t 2 = 1=2 and t 3 =t 4 = 0. Proof Let s = t 1 +t 2 . For any given s, the choice of t 1 = t 2 maximizes the objective function since I 1 and I 4 only depend on s, and minfI 2 ;I 3 g is maximized when t 1 = t 2 . For given s with t 1 =t 2 , dene R(s) = max t 3 ;t 4 :t 3 +t 4 =1s minfI 1 ;I 2 ;I 3 ;I 4 g: (3.6) 66 The proof follows by showing that R(1) R(s) for any 0 s < 1. Suppose that s is strictly less than 1, i.e., t 3 +t 4 = 1s > 0. In this case, we can observe that minfI 1 ;I 4 g is maximized when t 3 = (1s)C(4SNR)=(C(2SNR) +C(4SNR)) and t 4 = (1s)C(2SNR)=(C(2SNR) +C(4SNR)). For given s, we have: R(s) = max t 3 ;t 4 :t 3 +t 4 =1s minfI 1 ;I 2 ;I 3 ;I 4 g minfI 1 ;I 4 g with t 3 and t 4 (3.7) sC(SNR) + (1s) C(2SNR)C(4SNR) C(2SNR) +C(4SNR) (3.8) (a) sC(SNR) + 1s 2 C(3SNR); (3.9) where (a) is from Lemma 3.2, provided below. Using the (3.9), we can show that R(1)R(s) C(SNR)sC(SNR) 1s 2 C(3SNR) (3.10) = (1s) 2 (2C(SNR)C(3SNR)) (3.11) = (1s) 2 log (1 +SNR) 2 1 + 3SNR (3.12) (a) 0; (3.13) where (a) is due to the fact that SNR 2 SNR under the assumption of SNR 1. Lemma 3.2 The following inequality is hold: C(2SNR)C(4SNR) (C(2SNR) +C(4SNR))C(3SNR): (3.14) Proof Using the concavity of the logarithm, we have that C(3SNR) 1 2 (C(2SNR) + C(4SNR)). Then, we have: (C(2SNR) +C(4SNR))C(3SNR) 1 2 (C(2SNR) +C(4SNR)) 2 (3.15) C(2SNR)C(4SNR): (3.16) 67 Encoder Decoder si de- i nf or m ati on Figure 3.3: Simplied channel model in case of eliminating the inter-relay interference at destination. 3.2 Achievable Rates of Virtual Full-Duplex Relay Channel For a 2-hop virtual full-duplex relay channel, we examine the performances of various coding schemes that are categorized into three approaches: 1) Coping with interference at the source: since the source has non-causal information on the relay's transmit signal, it can completely eliminate the \known" interference at the other relay, using DPC; 2) Coping with interference at the relays: the receiving relay can decode the source message either by treating the inter-relay interference as noise or by using joint decoding, depending on interference level; 3) Coping with interference at the destination: since the inter-relay interference is also a \known" signal at the destination, the latter can use it as side information, i.e., the destination can cancel the inter-relay interference. In the third case, our goal is to design encoding/decoding functions in Fig. 3.3 that eciently uses the destination side information. The high-level description is as follows: The relay encoder produces a noisy (or noiseless) function of the two incoming signals such that x =L(x S (w);x R ). The destination decoder recovers the desired messagew by using a noisy observation ofL(x S (w);x R ) and the side-information x R . Encoding functions can be constructed by using various relaying strategies such as AF, QMF, and CoF. In AF, the relay simply operates as a repeater and transmits a power 68 scaled version of its received signal to the destination. Hence, the relay's transmit- ted signal can be regarded as a noisy linear combination of two incoming signals, i.e., L(x S (w);x R ) = (x S (w) +x R +z R ) for some power-scaling constant . In QMF, the relay performs vector quantization of its received signal at some rate R 0 C(SNR). Then, it maps the resulting block of nR 0 quantization bits into a binary word of length nC(SNR) by using some randomized hashing function (notice that this corresponds to binning if R 0 > C(SNR)). Finally, the relay forwards the the binary word (bin index) to the destination. In this case, the bin index encodes a noisy linear combination of the incoming signals. Finally, CoF makes use of lattice codes, such that relay can reliable decode an integer linear combination of the interfering lattice codewords. Thanks to the fact that lattices are modules over the ring of integers, this linear combination translates directly into a linear combination of the information messages dened over a suitable nite eld [NG11, HC13b]. Namely, relay forwards a noiseless linear combination of the messages (over a suitable nite-eld) to the destination. As we shall see, the decoding function at the destination consists of \successive decoding" for AF and QMF, and \for- ward substitution" for CoF. The detailed encoding/decoding procedures will be explained in Section 3.3. With these schemes, we have: Theorem 3.1 For the 2-hop virtual full-duplex relay channel, DPC, DF, AF, QMF, and CoF can achieve the following rates: R DPC = log(1 +SNR) R DF = min log(1 +SNR); max log 1 + SNR 1 + 2 SNR ; 1 2 log(1 + (1 + 2 )SNR) R AF = log 1 + SNR 2 (1 +SNR) (1 + (1 + 2 )SNR)(1 + 2SNR) R QMF = log 1 + SNR 2 1 + 2SNR R CoF = min log + SNR b H (SNR 1 I +hh H ) 1 b ; log(1 + 2 2 SNR) ; 69 for some b6= 02Z 2 [j], = ( 1 ; 2 ) withj i j 1, where h = [ 1 ; 2 ] T . Proof See Section 3.3. Remark 6 The CoF rate in Theorem 3.1 can be maximized by optimizing the power allocation (PA) parameter = ( 1 ; 2 ) and the integer coecients b where 1 and 2 represent the power back-o values at the source and the transmitting relay, respectively. Since the role of the transmitting relay alternates with time t, the same constant 2 is applied to both relays. In fact, the optimization of the PA parameter does not lend itself to a closed form solution and requires, in general, an exhaustive search. For the sake of analytical tractability, we consider three possible PA strategies: (i) = (1; 1) (No PA); (ii) = ( =d e; 1); (iii) = (1;b c= ), wherebxc is the largest integer x anddxe is the smallest integer x. The goal of PA strategies (ii) and (iii) is to mitigate the non-integer penalty, which ultimately limits the performance of CoF especially in high SNR [NW12]. The PA strategy (ii) is chosen ifd e b c and vice versa. In this paper, we will use the notation R CoF to represent the CoF rate without PA and R CoFP with PA (i.e., maximum rate of PA strategies (ii) and (iii)). For given , the CoF rate can be maximized by minimizing the b H (SNR 1 I +hh H ) 1 b with respect to an integer vector b2 Z 2 [j]. It was shown in [HC13b] that this is equivalent to a \shortest lattice point" problem, that can be eciently obtained using the complex LLL algorithm, possibly followed by Phost or Schnorr-Euchner enumeration (see Algorithm 1 in [HC13b]). Corollary 3.1 The CoF rate with PA satises the lower bound R CoFP log 1 1 +d e 2 + 2 max SNR ; (3.17) where max = max d e ; b c : (3.18) Proof See Section 4.1.1. 70 Remark 7 From Lemma 3.3 in Section 3.3.3, the achievable rate of QMF with noise- level quantization (as in the general strategy of [ADT11]) is R QMFN = log(1 +SNR) 1: (3.19) This rate is within 0.5 bits of the QMF rate achieved by optimal quantization, and this gap vanishes as SNR grows. Nevertheless, noise-level quantization requires joint decoding of message and quantization index while in the case of optimal quantization a much simpler successive decoding strategy, as in classical \Wyner-Ziv" compress and forward relaying [CG79], turns out to be sucient. In the following, we will compare the performances of coding schemes in terms of their achievable rates. Corollary 3.2 QMF achieves the performance of DPC (i.e., the capacity) within 1 bit: R DPC R QMF 1: (3.20) Proof R DPC R QMF = log(1 +SNR) log 1 + SNR 2 1 + 2SNR = log (1 +SNR)(1 + 2SNR) (1 +SNR) 2 = log 1 + SNR 1 +SNR log 2 = 1: Corollary 3.3 AF achieves the performance of QMF within log(1 + 2 ) bits: R QMF R AF log(1 + 2 ): (3.21) 71 Proof Letting A =SNR 2 =(1 + 2SNR), we have: R QMF R AF = log(1 +A) log 1 + A(1 +SNR) 1 + (1 + 2 )SNR = log (1 +A)(1 + (1 + 2 )SNR) 1 + (1 + 2 )SNR +A(1 +SNR) = log 1 + 2 ASNR 1 + (1 + 2 )SNR +A(1 +SNR) log(1 + 2 ): Corollary 3.4 In high SNR (i.e., SNR 1) and strong interference ( 2 0:5), CoF with PA can outperform QMF: R CoFP R QMF : (3.22) Proof Under the high SNR condition, we only need to show that 2 max 1 2 . When < 1, we have that 2 max = 2 =d e 2 = 2 . Since it is assumed that 2 0:5, we show that 2 max 1 2 . For the case of 1, we consider the two cases: (a) d e b c : Since 2 d eb c, we have: 2 max = 2 d e 2 b c d e 1 2 : (3.23) (b) d e b c : Since 2 d eb c, we have: 2 max = b c 2 2 b c d e 1 2 : (3.24) Here, we used the fact that b c d e 1 2 since 1. 72 0.5 1 1.5 2 0 1 2 3 4 5 6 Inter ï relay interference levels Achievable Rates DPC DF CoF CoF w/ PA QMF AF Figure 3.4: SNR = 15. Achievable rates of various coding schemes as a function of the inter-relay interference level . 73 0.5 1 1.5 2 2 3 4 5 6 7 8 9 10 11 Inter−relay interference levels Achievable Rates DPC DF CoF CoF w/ PA QMF AF Figure 3.5: SNR = 30 dB. Achievable rates of various coding schemes as a function of the inter-relay interference level . 74 In order to conrm our analytical results, we numerically evaluate the achievable rates of all the considered coding schemes for dierent values of 2R + and SNRs. Figs. 3.4 and 3.5 show that numerical results are well matched to the analytical results in the above corollaries. Also, the performance of CoF is uctuated as SNR increases due to the impact of non-integer penalty. It is remarkable that PA strategy dramatically improves the performance of CoF, especially in high SNR (see Fig. 3.5). 3.3 Proof of Theorem 3.1: Achievable Coding Schemes We prove Theorem 3.1 by considering separately the schemes based on DF, AF, QMF, and CoF in the following subsections. 3.3.1 DF Each relay treats the other relay's signal as interference and decodes a source message. Depending on the inter-relay interference level , the relay decodes the source message either by treating interference as noise or by joint decoding. Since the interference is completely eliminated at relay, the destination can recover a desired message if R C(SNR). This scheme yields the achievable rate: R DF = min max log 1 + SNR 1 + 2 SNR ; 1 2 log(1 + (1 + 2 )SNR) ;C(SNR) ; (3.25) where the rst and second terms are achievable rates obtained by treating interference as noise and joint (unique) decoding, respectively. Notice that the same achievable rate region is obtained by simultaneous non-unique decoding [BGK12], i.e., the achievable rate region of simultaneous non-unique decoding is the union of the regions of treating interference as noise and joint (unique) decoding. 75 3.3.2 AF with Successive Decoding In this scheme, the relay's operation consists of forwarding a scaled version of the received signal to the destination. At each time slot t + 1, the relay transmits the received signal during slot t with power scaling : x R [t + 1] =y R [t] =(x S [t] + x R [t] +z R [t]); (3.26) where is chosen to satisfy the power constraint equal to SNR: = s SNR 1 + (1 + 2 )SNR : (3.27) The destination observes y D [t + 1] = x R [t + 1] +z D [t + 1] (3.28) = (x S [t] + x R [t] +z R [t]) +z D [t + 1] for t = 1;:::;N; (3.29) and uses successive decoding as follows: The destination can decode message w 1 from the received signal y D [2] =x S [1] + z R [1] +z D [2] if R log 1 + 2 SNR 1 + 2 : (3.30) In order to decode message w 2 , the destination rst cancels the \known" interfer- ence signal x S [1] (obtained from the decoded message w 1 ) from the observation y D [3], obtaining: y D [3] 2 x S [1] = x S [2] + x R [2] +z R [2] +z D [3] 2 x S [1] (3.31) = x S [2] + 2 z R [1] +z R [2] +z D [3]; (3.32) 76 where recall that x R [t] =(x S [t 1] + x R [t 1] +z R [t 1]) and x R [1] = 0. From (3.32), message w 2 can be decoded if R log 1 + 2 SNR 1 + 2 + 4 2 : (3.33) For t = 3; 4; 5;:::, the destination proceeds to decode message w t by canceling the \known" interference signals obtained from the decoded messagesfw 1 ;:::;w t1 g. It is easy to generalize (3.32) to y D [t] t2 X `=1 ( ) t`1 x S [`] =x S [t 1] +z e [t]; (3.34) where the eective noise of resulting point-to-point channel is given by z e [t] = t1 X `=1 ( ) t`1 z R [`] +z D [t]: (3.35) Then, the eective noise variance is given by 2 e [t] = 1 + 2 t1 X `=1 (( ) 2 ) t`1 : (3.36) We notice that 2 e [t] is an increasing function on t and it is upper bounded by lim t!1 2 e [t] = 1 + 2 1 ( ) 2 ; (3.37) since we have = s 2 SNR 1 + (1 + 2 )SNR < 1: (3.38) Based on (3.37), destination can decode all source messages if R log 1 + 2 SNR 1 + 2 =(1 ( ) 2 ) : (3.39) 77 By plugging the in (3.27) into (3.39), the achievable rate of AF is obtained as R AF = log 1 + SNR 2 (1 +SNR) (1 + (1 + 2 )SNR)(1 + 2SNR) : (3.40) 3.3.3 QMF with Successive Decoding We rst derive an achievable rate of QMF as a function of the quantization rate R 0 (associated with quantization distortion level 2 q ). Then, we optimize this parameter to maximize the achievable rate. The QMF scheme is described as follows. For eacht, the relays make use of a quantization codebookf _ y R;t (1);:::; _ y R;t (2 nR 0 )g of block length n, generated at random with i.i.d. components according to the distribution of _ Y R =Y R + _ Z, whereY R CN (0; (1+ 2 )SNR+1) and _ ZCN (0; 2 q ) are independent. The quantization codebook is partitioned into 2 n(R 0 C(SNR)+) bins, by random assignment, such that each bin has size 2 n(C(SNR)) , for some > 0. At time slot t, the relay in receive mode observes: y R [t] =x S [t] + x R [t] +z R [t]: (3.41) and quantizes as t = Q t (y R [t]), whereQ t : C n ! f1;:::; 2 nR 0 g is a suitable quantization function based on the codebookf _ y R;t (1);:::; _ y R;t (2 nR 0 )g. Let ` t 2f1;:::; 2 n(C(SNR)) g denote the index of the bin containing the quantiza- tion codeword _ y R;t ( t ). Then, the relay encodes ` t into its downstream codeword x R [t + 1], and transmits it to the destination in slot t + 1. The destination applies joint typical decoding using the side information x R [t] (al- ready decoded codeword at the previous slot) and the bin-index ` t , in order to decodew t . Notice that for all> 0 and suciently largen, the probability of error incurred in decoding ` t can be made as small as desired. 78 Lemma 3.3 For any given quantization level 2 q , the above QMF scheme achieves the rate R = min log 1 + SNR 1 + 2 q ; log(1 +SNR) log 1 + 1 2 q : (3.42) Proof We derive an achievable rate of QMF for given quantization quadratic distortion 2 q . The problem reduces to considering the simplied model shown in Fig. 3.6. In the model, we let x R denote the realization of an i.i.d. random vector 1 X n R inde- pendent of the source information messageW and with componentsP X R (some known probability distribution). As a matter of fact, this is the codeword sent by the other relay and interfering at the input of the receiving relay. Since this is not a random vector but a codeword out of the relay codebook, one may wonder if treating it as a random i.i.d. vector is rigorous. Indeed, because of the random codebook generation and the random mapping of the bin index onto the relay codewords, following the rigorous argument given in [ADT11, LKEGC11], we know that this is indeed the case. This argument is not re- peated here for the sake of brevity, and since it is by now well-known. In addition, in the special case of Wyner-Ziv quantization, we know that the Wyner-Ziv rate distortion func- tion in the Gaussian-Quadratic case is achievable even for an arbitrary realization of the additive interference/side information. This follows from universal structured schemes based on nested lattices and minimum distance lattice quantization and decoding (see for example [ZSE02]), replacing the usual typicality arguments valid for i.i.d. interfer- ence/side information. Codebook Generation: Fix > 0, > 0 and 0 > 0. Randomly and independently generate 2 nR codewordsx S (w) of lengthn indexed by w2f1;:::; 2 nR g with i.i.d. componentsP X S , such thatE[jX S j 2 ] =SNR=(1 + 0 ). 1 We use the following notation convention: vectors of length n over C are denoted by underlined boldface small case letters (e.g., x). Such vectors maybe realization of random vectors, denoted by X n = (X1;:::;Xn). When a random vector X n is i.i.d., we denote by the same capital letter X the random variable such that XiX for all i = 1;:::;n. 79 Randomly and independently generate 2 nR codewords x(w) of length n indexed by w2f1;:::; 2 n(C(SNR)) g with i.i.d. components P X , such that E[jXj 2 ] = SNR=(1 + 0 ). DeneY R =X S +X R +Z R , whereZ R CN (0; 1), and independently generate 2 nR 0 codewords _ y R () of length n, indexed by 2f1;:::; 2 nR 0 g, with i.i.d. components _ Y R , where _ Y R =Y R +Z q ; (3.43) with Z q CN (0; 2 q ). The quantization codewords are randomly and independently assigned with uniform probability to 2 n(C(SNR=(1+ 0 ))) bins, for some > 0. We denote the `-th bin by B ` with `2f1;:::; 2 n(C(SNR=(1+ 0 ))) g. Source and relaying operation: The source transmits message w2f1;:::; 2 nR g by sending the codeword x S (w). If x S (w) does not satises the transmit power constraint, the all-zero vector is transmitted. For all 0 > 0 and suciently large n the probability of violating the transmit power constraint can be made arbitrarily small, and we shall not consider this even further for the sake of brevity. The relay in receiving mode observes y R =x S (w) +x R +z R , and nds such that (y R ; _ y R ())2T (n) (Y R ; _ Y R ), where the latter denotes the jointly -typical set for P Y R ; _ Y R dened as above. If no quantization codeword satises the joint typicality condition, the relay chooses = 1. The relay nds the bit index` such that _ y R ()2B ` , and transmits the downstream codeword x(`) to the destination. The same consideration made before about the transmit power constraint applies here. The destination observes y D =x(`) +z D , and knows the side information x R from previous decoding steps. 80 Decoding at the destination: Since the coding rate of the relay is strictly less than C(SNR=(1 + 0 )), the destination can decode the bin-index ` from its own received signal with vanishing probability of error. Then, it performs joint typical decoding to nd ^ w using the bin-index ` and the known signal x R , i.e., it nd a unique message ^ w2f1;:::; 2 nR g such that x S ( ^ w); _ y R ( 0 );x R 2T (n) (X S ; _ Y R ;X R ) for some _ y R ( 0 )2B ` : (3.44) AnalysisofProbabilityofError: By the standard random coding symmetrization argument [CT12], we can assume that the transmitted source message isw = 1 and relay selected bin index is ` = 1. Furthermore, from the covering lemma in [EGK11], we have that the probability of quantization error P (Y n R ; _ Y n R ()) = 2T (n) (Y R ; _ Y R ) 8 = 1;:::; 2 nR 0 can be made as small as desired if we choose R 0 = I(Y R ; _ Y R ) +. We make this choice and implicit assume that all the error events below are intersected with the quantization success event n (Y n R ; _ Y n R ())2T (n) (Y R ; _ Y R ) for some = 1;:::; 2 nR 0 o : At this point, we analyze the average probability of error at the destination, averaged also over the random coding ensemble and over the random realization of the interference/side information X n R . We consider the events: E 1 : n X n S (1); _ Y n R ( 0 );X n R = 2T (n) (X S ; _ Y R ;X R ) for some _ Y n R ( 0 )2B 1 o E 2 : n X n S (w6= 1); _ Y n R ( 0 );X n R 2T (n) (X S ; _ Y R ;X R ) for some _ Y n R ( 0 )2B 1 o : 81 Quantization Random Binning (side information) Joint Typical Decoding Figure 3.6: Simplied model for QMF. For suciently large n, P (E 1 ) by [CT12, Lemma 10.6.1]. Using the union bound, we have: P(E 2 ) 2 nR P X n S (2); _ Y n R ( 0 );X n R 2T (n) (X S ; _ Y R ;X R ) for some _ Y n R ( 0 )2B 1 :(3.45) The eventE 2 can be divided into two disjoint error events according to quantization sequence: 1. E 21 : n X n S (2); _ Y n R ();x n R 2T (n) (X S ; _ Y R ;X R ) o (i.e., true quantized sequence) 2. E 22 : n X n S (2); _ Y n R ( 0 );x n R 2T (n) (X S ; _ Y R ;X R ) for some _ Y n R ( 0 )2B 1 o with 0 6=. The probability ofE 21 can be upper bounded as P(E 21 ) = P X n S (2); _ Y n R ();X n R 2T (n) (X; _ Y R ;X R ) P X n R = 2T (n) (X R ) + +P n X n S (2); _ Y n R ();X n R 2T (n) (X; _ Y R ;X R ) o \ n X n R 2T (n) (X R ) o X x R 2T (n) (X R ) P X n R (x R ) X (x S ; _ y R )2T (n) (X S ; _ Y R jx R ) P X n S (x S )P _ Y n R jX n R ( _ y R jx R ) = X (x n S (2); _ y n R )2T (n) (X S ; _ Y R jx n R ) P(x n S )P( _ y n R jx n R ) 2 nh(X S ; _ Y R jX R ) 2 nh(X S ) 2 nh( _ Y R jX R ) : (3.46) 82 Letting e Y n denote a random vector distributed as _ Y n R but independent ofY n R (and there- fore of X n S (w) for all w and of X n R ), the probability ofE 22 can be upper bounded as P(E 22 ) = P X n S (2); _ Y n R ( 0 );x n R 2T (n) (X S ; _ Y R ;X R ) for some _ Y n R ( 0 )2B 1 jB 1 jP n X n S (2); e Y n ;X n R 2T (n) (X S ; _ Y R ;X R ) o jB 1 j2 nh(X S ; _ Y R jX R ) 2 nh(X S ) 2 nh( _ Y R ) = jB 1 j2 nh( _ Y R jX S ;X R ) 2 nh( _ Y R ) ; (3.47) where we used the fact that h(X S ; _ Y R jX R ) =h( _ Y R jX S ;X R ) +h(X S ). Using (3.46) and (3.47) in the union bound (3.45) and the fact that jB 1 j : = 2 n(I(Y R ; _ Y R )C(SNR=(1+ 0 ))+2) ; we nd thatP(E 2 ) vanishes as n!1 under the following conditions: From (3.46): R < h(X S ) +h( _ Y R jX R )h(X S ; _ Y R jX R ) (3.48) = h( _ Y R jX R )h( _ Y R jX S ;X R ) (3.49) = I(X S ; _ Y R jX R ) = log 1 + SNR=(1 + 0 ) 1 + 2 q ; (3.50) where the last equality follows by choosing X S CN (0;SNR=(1 + 0 )). From (3.47): R < h( _ Y R )h( _ Y R jX S ;X R )I(Y R ; _ Y R ) +C(SNR=(1 + 0 )) 2 (3.51) = C(SNR=(1 + 0 ))I(Y R ; _ Y R jX S ;X R ) 2 (3.52) = log(1 +SNR=(1 + 0 )) log 1 + 1 2 q 2; (3.53) with again the same choice X S CN (0;SNR=(1 + 0 )). 83 From (3.50) and (3.53), since ; 0 and are arbitrary, we conclude that any R satisfying R< min log 1 + SNR 1 + 2 q ; log(1 +SNR) log 1 + 1 2 q is achievable. From (3.42), we observe that the rst rate constraint is a decreasing function of 2 q and the second rate constraint is an increasing function of 2 q . Hence, the optimal value of 2 q is obtained by solving: 1 + SNR 1 + 2 q = 2 q (1 +SNR) 1 + 2 q : (3.54) This yields 2 q;opt = 1 +SNR SNR : (3.55) Remarkably, the optimal quantization distortion level (3.55) depends only on SNR but is independent of the inter-relay interference level . Also, for high-SNR, the optimal quantization distortion converges to noise-level quantization (i.e., 2 q = 1). Finally, the resulting rate achievable by QMF is given by: R QMF = log 1 + SNR 2 1 + 2SNR : (3.56) Remark 8 We observe that the optimal quantization level in (3.55) coincides with the Wyner-Ziv distortion [CT12, Thm 6] used in classical compress and forward relaying, where the relay quantizes its received signal so that, using side-information x R [t], the des- tination can uniquely recover the quantization sequence _ y R;t ( t ). That is, the quantization level 2 q is chosen to satisfy the condition I(Y R ; _ Y R jX R ) =C(SNR): (3.57) 84 Table 3.1: Compute-and-Forward with Forward Substitution. time slot 1 time slot 2 time slot 3 time slot 4 X S x S (w 1 ) x S (w 2 ) x S (w 3 ) x S (w 4 ) Y R 1 u 2 =q 1 w 2 +q 2 u 1 u 4 =q 1 w 4 +q 2 u 3 X R 1 x R (u 2 ) Y R 2 u 1 =w 1 u 3 =q 1 w 3 +q 2 u 2 X R 2 x R (u 1 ) x R (u 3 ) Y D ^ w 1 =u 1 ^ w 2 =q 1 1 u 2 q 1 1 q 2 u 1 ^ w 3 =q 1 1 u 3 q 1 1 q 2 u 2 Then, destination can avoid the complexity of joint typical decoding and just use classical successive decoding: The destination rst decodes the relay's message (i.e., bin-index) ` t ; Using the bin-index ` t and side-information x R [t], the destination nds the quanti- zation codeword _ y R;t ( t ); Then, it cancels the \known" inter-relay interference x R [t] such as _ y R;t ( t ) x R [t] =x S [t] +z R [t] + _ z[t]; (3.58) where _ z[t] represent the \quantization noise". This can be interpreted and used as the output of a virtual point-to-point channel, from which the destination can decode the desired message. 3.3.4 CoF with Forward Substitution CoF applied to the virtual full-duplex relay channel is summarized in Table 3.1. At even time slots, relay 1 decodes a linear combination of two messages sent by the source and relay 2, and in the next time slot, the decoded linear combination is re-encoded and transmitted to destination. At odd time slot, the role of relays 1 and 2 are reversed. 85 This scheme can be regarded as a generalization of DF in the sense that it reduces to (a special case of) DF by setting the coecient of the linear combination equal to [1; 0] T , i.e., zero coecient to the inter-relay interference message. The major impairment that deteriorates the performance of CoF is the non-integer penalty (i.e., the residual self- interference due to the fact that the channel coecients take on non-integer values), which ultimately limits the performance of CoF at high SNR [NW12]. In our model, the non-integer penalty which may be relevant for specic values of , can be mitigated by using power allocation in order to create more favorable channel coecients for the integer conversion at each receiver [HC13b]. In this way, the source transmits at power 2 1 SNR and the transmitting relay at power 2 2 SNR, where 1 and 2 are chosen such that j 1 j;j 2 j 1, in order to satisfy the transmit power constraint. By including the power allocation into channel coecients, the eective channel vector is given byh = [ 1 ; 2 ] T . Letb = [b 1 ;b 2 ] T 2Z[j] 2 . Also, we letq ` =g 1 ([b ` ] mod pZ[j]) for` = 1; 2. The receiver's goal consists of decoding a the message combination u t = q 1 w t +q 2 u t1 where u t1 denotes the relay's message. In this scheme, all messages are dened over an appropriate nite-eld. From [NG11, Thm 5], the relay can reliably decode the linear combination u t =q 1 w t +q 2 u t1 if R log + SNR b H (SNR 1 I +hh H ) 1 b : (3.59) During the next time slot, the decoded linear combination can be reliably transmitted to destination if R log 1 + 2 2 SNR : (3.60) After N + 1 time slots, the destination can observe the noiseless linear combinations fu t = q 1 w t +q 2 u t1 : t = 2;:::;N + 1g with u 1 = w 1 . Using forward substitution, desired source messages can be recovered as: ^ w t =q 1 1 (u t q 2 u t1 ); t = 2;:::;N (3.61) 86 with initial value u 1 =w 1 . It is perhaps interesting to notice that this scheme does not suer from catastrophic error propagation: if a message u t is erroneously decoded, it will aect at most two decoded source messages. From (3.59) and (3.60), the achievable rate of CoF is obtained by R CoF = min log + SNR b H (SNR 1 I +hh H ) 1 b ; log(1 + 2 2 SNR) ; (3.62) for some b6= 02Z 2 [j] and = ( 1 ; 2 )2R 2 + . As mentioned in Remark 6, instead of trying to exhaustively optimize with respect to the PA parameter , we have considered only the two choices = ( =d e; 1) and = (1;b c= ). Both PA strategies satisfy the constraint ofj 1 j;j 2 j 1. PAStrategy1) Relay decodes a linear combination of lattice codewords with integer coecientsb = [1;d e] T . From [NG11, Thm 1], the variance of eective noise is given by 2 e () =SNR d e 1 2 +j d ej 2 ! +jj 2 ; (3.63) for some 2C. Also, we can optimize to minimize the above variance and get: opt = SNR( =[ ] + [ ]) 1 +SNR( =[ ]) 2 +SNR 2 : (3.64) By letting = opt in (3.63), we have: 2 e ( opt ) = ( =d e + d e) 2 SNR 2 + (1 +d e 2 )SNR (1 + (( =d e) 2 + 2 )SNR) 2 ; (3.65) which yields a rate-constraint: R = log SNR 2 e ( opt ) = log 1 1 +d e 2 + 2 d e 2 SNR : (3.66) 87 PA Strategy 2) Relay decodes a linear combination with integer coecients b = [1;b c] T . Similarly, the variance of eective noise is given by 2 e () =SNR j 1j 2 +jb cb cj 2 +jj 2 ; (3.67) and yields the rate-constraint: R log 1 1 +b c 2 +SNR : (3.68) Due to the change of relay's transmission power, we have the following rate-constraint obtained from relay-to-destination transmission: R log(1 + 2 2 SNR) = log 1 + b c 2 2 SNR (3.69) Therefore, an achievable rate of CoF with the second PA strategy is given by R = log 1 1 +b c 2 + b c 2 2 SNR : (3.70) By taking the maximum rate over the two PA strategies, the following rate is achiev- able: R CoFP = log 1 1 +d e 2 + 2 max SNR ; (3.71) where max = max d e ; b c : (3.72) This proves Corollary 3.1. 3.4 Multihop Virtual Full-Duplex Relay Channel In this section we generalize the results of Section 3.2 to the case of a (K + 1)-hop virtual full-duplex relay network comprising K relay layers (see Fig. 3.1). It is assumed that all 88 ``known" interference is canceled by DPC unknown at R 1 S S S D R 1 R 2 R 2 R 3 R 4 R 1 R 4 R 1 cannot perform DPC time slot 1 time slot 3 time slot 2 R 2 R 3 Figure 3.7: DPC scheme for 3-hop virtual full-duplex relay channel. inter-relay interference levels are identical and equal to 2R + (i.e., symmetric channel model). We start by considering an upper bound on capacity: Lemma 3.4 IfSNR 1, the capacity of the (K +1)-hop virtual full-duplex relay network shown in Fig. 3.1 is upper bounded by R (K) upper = log(1 +SNR): (3.73) Proof The proof is obtained by induction. From Lemma 3.1, we have that the bound holds for a 2-hop network (i.e.,K = 1). Assume that it also holds for theK-hop network and consider the (K+1)-hop network. From the hypothesis assumption, the capacity from source to a relay in the last hop is bounded by log(1 +SNR). Then, we can consider the condensed 2-hop network consisting of source, two relays in the last hop, and destination, as illustrated in Fig. 3.8. Since the resulting model is equivalent to the case of K = 1, the upper bound of this model is equal to log(1 +SNR). 89 Figure 3.8: Condensed network of (K + 1)-hop virtual full duplex relay channel. Regarding the achievable schemes, we show that DPC is no longer applicable for K > 2. For instance, consider the 3-hop network in Fig. 3.7. At time slot 3, relay 1 wants to cancel the inter-relay interference sent from relay 3, using DPC. However, it is not possible since relay 1 does not receive relay 4's message w 1 during the previous time slots. On the other hand, other coding schemes in Section 3.3 can be applied to the (K + 1)-hop network and their achievable rates are derived in Sections 3.4.1, 3.4.2 and 3.4.3. These achievable rates are summarized in Theorem 3.2 and their performance degradation with respect to K is given in Corollary 3.5. Theorem 3.2 For a symmetric (K + 1)-hop virtual full-duplex relay network as shown in Fig. 3.1, the following rates are achievable: R (K) DF = R (1) DF (3.74) R (K) AF = log 1 + 1 +SNR 1 + (1 + 2 )SNR K SNR K+1 (1 +SNR) K+1 SNR K+1 ! (3.75) R (K) QMF = log 1 + SNR K+1 (1 +SNR) K+1 SNR K+1 (3.76) R (K) CoF = R (1) CoF (3.77) R (K) CoFP = log(SNR) +K log( 2 max ); (3.78) where max = maxf =d e;b c= g. Proof See Sections 3.4.1, 3.4.2 and 3.4.3. 90 Corollary 3.5 With high-SNR condition (i.e., SNR 1), the performance degradations according to the number of relay stages K are given by R (1) DF R (K) DF = 0 (3.79) R (1) AF R (K) AF = (K 1) log(1 + 2 ) (3.80) R (1) QMF R (K) QMF = log K + 1 2 (3.81) R (1) CoF R (K) CoF = 0 (3.82) R (1) CoFP R (K) CoFP = (K 1) log(1= 2 max ): (3.83) Proof See Sections 3.4.1, 3.4.2 and 3.4.3. Corollary 3.6 When the inter-relay interference level is equal to direct channel gain (i.e., = 1), CoF achieves the upper bound within 0:5 bit. Proof In this case, the achievable rate of CoF is given by R CoF = log 1 2 +SNR ; (3.84) with integer coecients b = [1; 1] T . From the upper bound in Lemma 3.4, we have: R upper R CoF = log(1 +SNR) log(1=2 +SNR) 0:5; (3.85) since it is assumed that SNR 1. Corollary 3.6 shows that CoF is almost optimal for multihop virtual full-duplex relay channel provided that the inter-relay interference and the direct channel gains are bal- anced. This result does not capture the impact of non-integer penalty, which may greatly degrade the performance of CoF. In order to demonstrate the actual performance of CoF in this setting, we considered Monte Carlo averaging over the inter-relay interference level . The corresponding results are plotted in Figs. 3.9 and 3.10. When 2 is close to 1 (i.e., 91 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 Number of relay stages (K) Achievable Rates Upper Bound DF AF CoF CoF w/ PA QMF Figure 3.9: SNR = 20 dB. Achievable ergodic rates of various coding schemes averaging over 2 Unif(0:9; 1:1). 2 Unif(0:9; 1:1)), CoF almost achieves the upper bound and generally outperforms the other coding schemes, especially when K increases. Fig. 3.10 shows that even if 2 is not always close to 1, CoF gives the best performance for suciently large number of relay stages (in this case, K > 3). Also, for K 3, CoF with PA outperforms the other schemes. Therefore, CoF (with or without PA) appears to be a strong candidate for the practical implementation of multihop virtual full-duplex relay networks, especially when the relative power of the interfering and direct links can be tuned by node placement and line of sight propagation, making the channel coecients essentially deterministic. 92 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 Number of relay stages (K) Achievable Rates Upper Bound DF AF CoF CoF w/ PA QMF Figure 3.10: SNR = 20 dB. Achievable ergodic rates of various coding schemes averaging over 2 Unif(0:5; 1). 93 Remark 9 The multihop virtual full-duplex relay channel is a special case of a general multiple multicast relay network studied in [ADT11] and later in [LKEGC11] for a larger class of relay networks. In [LKEGC11], NNC consists of message repetition encoding (i.e., one long message with repetitive encoding), signal quantization at relay, and simul- taneous joint typical decoding on the received signals from all the blocks without explicitly decoding quantization indices. Recently, Short-Message NNC (SNNC) has been proposed in [HK13], which overcomes the long delay of NNC, by transmitting many short messages in blocks rather than using one long message with repetitive encoding. By setting the quan- tization distortion levels to be at the background noise level, NNC (or QMF) achieves the capacity within a constant gap where the gap scales linearly with the number of nodes in the network, but it is independent of SNR. For the channel considered in this paper, we provide an improved result as shown in Corollary 3.5 by using optimal quantization at the relays, where the gap scales logarithmically with the number of relay stages (K). Fur- ther, we have a lower decoding complexity at destination than NNC, for which successive decoding is used instead of joint simultaneous decoding. To be specic, the destination successively decodes all relays' messages (i.e., quantization indices) and then, the source message. Notice that all relays' messages are explicitly decoded, dierently from NNC, and are used as side-information at the next time slot, which makes it possible to employ Wyner-Ziv quantization. Thanks to Wyner-Ziv quantization, the destination can avoid the complexity of joint decoding in order to decode each relay's message. In our scheme, the destination rst nds an unique quantized sequence using side-information and the bin index, and then decodes the message from the quantized observation (see Section 3.4.2). For comparison, we also derive the achievable rate of QMF with noise-level quantization. In this case, the destination must perform joint decoding in order to decode the relays' messages. We notice that the joint decoding here is separately performed for each relay 94 Figure 3.11: Noise accumulation of multihop AF scheme. message while it is done over entire network information in [ADT11, LKEGC11]. From Lemma 3.5, the achievable rate of QMF with noise-level quantization is given by R (K) QMFN = log(1 +SNR)K: (3.86) By comparing (3.86) with (4.74) we see that Wyner-Ziv quantization provides a substan- tial gain over noise-level quantization, having a larger gap as K grows. 3.4.1 AF with Successive Decoding In order to compute an achievable rate, we need to compute the variance of eective noise and the desired signal power at the destination. As shown in Section 3.3.2, the signal power is reduced by 2 for each transmission. Hence, we have: SNR e = 2K SNR; (3.87) where the scaling constant is given in (3.27) as = s SNR 1 + (1 + 2 )SNR : (3.88) 95 Next, we will derive the variance of eective noise at destination for K + 1 hops. Let 2 e;K [t] denote the variance of eective noise at the destination. Dene the stationary limit 2 e;K = lim t!1 2 e;K [t]. In Section 3.3.2, we found that 2 e;1 = 1 + 2 1 2 2 : (3.89) For the sake of notation simplicity, we letr = 2 1 2 2 = SNR 1+SNR < 1. Considering the noise accumulation scheme of Fig. 3.11, and letting zCN (0; 2 z ) the eective noise at the receiver input of relay 1 of stage K, and z D CN (0; 1) the thermal noise at the input of the destination receiver, the eective noise at the destination receiver for slot times t and t + 1 are given by z (K) e [t] = z +z D (3.90) z (K) e [t + 1] = (z (K1) e [t] + z) +z D : (3.91) Squaring and taking expectation of both sides of the above equations, and taking the limit for t!1, we nd: 2 e;K = 2 2 z + 1 (3.92) 2 e;K = 2 ( 2 e;K1 + 2 2 2 z ) + 1: (3.93) Notice that a relay in the stageK can be considered as a destination of aK-hop network, due to the symmetric structure of network. Using (3.92) and (3.93), we can solve for 2 z and obtain: 2 z = 2 e;K1 1 2 2 : (3.94) Replacing in (3.92), we nd a recursion for the eective noise variance: 2 e;K = 2 2 e;K1 1 2 2 + 1 =r 2 e;K1 + 1; (3.95) 96 which yields 2 e;K =r K1 2 e;1 + K1 X i=1 r i1 : (3.96) Using (3.89), i.e., 2 e;1 = 1 +r, we nally arrive at: 2 e;K = K X i=0 r i = 1r K+1 1r = (1 +SNR) 1 SNR 1 +SNR K+1 ! : (3.97) The achievable rate of AF for the (K + 1)-hop network is eventually obtained as R (K) AF = log 1 + SNR K+1 (1 +SNR) K (1 + (1 + 2 )SNR) K (1 +SNR) K+1 SNR K+1 ! : (3.98) For high SNR, we have: R (1) AF R (K) AF = (K 1) log (1 + (1 + 2 )SNR) SNR + log (1 +SNR)((1 +SNR) K+1 SNR K+1 ) (1 + 2SNR)(1 +SNR) K : Since the second term is larger than or equal to zero, we have the lower bound on the performance degradation: R (1) AF R (K) AF (K 1) log 1 + (1 +j j 2 )SNR SNR : (3.99) In the high-SNR, the gap is approximated by (K 1) log(1 +j j 2 ). 3.4.2 QMF with Successive Decoding In this section we prove the achievable rate expression for QMF for the (K + 1)-hop net- work, and show that it degrades logarithmically on the number of relay stagesK. Fig. 3.12 shows the time expanded network for 3-hop virtual full-duplex relay network. Notice that the destination knows the inter-relay interferences as side information since this is com- pletely determined by the previously decoded relays' messages (i.e., bin indices). Focusing 97 source destination 2-stage relays 1-stage relays known interferences Figure 3.12: Time expanded 3-hop network. The ` k;t denotes the relay k's message at time slot t. side-information side-information known interferences (a) Channel model for decoding source message 3 (b) Equivalent model for (K+1)-hop network Figure 3.13: Equivalent model of QMF scheme for (K + 1)-hop network. 98 on decodingw 3 , we can introduce the simplied channel model as illustrated in Fig. 3.13 (a). Also, this model applies to decoding of any source message w t for t 3. Hence, we can drop the time index in the simplied model and derive an achievable rate of QMF for the (K + 1)-hop network based on the equivalent model of Fig. 3.13 (b). We follow the notations in Fig. 3.13 (b) for the \known" interferences and additive Gaussian noises. Letx R (` k ) denote the transmit signal of relayk with message` k , fork = 1;:::;K. Also, let R k denote the message rate of relay k (i.e., ` k 2f1;:::; 2 nR k g. Letting y k denote the received signal at relay k, we have: y k =x R (` k1 ) +x k +z R k for k = 1;:::;K; (3.100) where z k consists of an i.i.d. complex Gaussian random variable with zero mean and variance 1. Also, the received signal at destination is given by y D =x R (` K ) +z D ; (3.101) where z D consists of an i.i.d. complex Gaussian random variable with zero mean and variance 1. The procedures of encoding and decoding are as follows. Encoding: Source transmits x S (w) to relay 1. Relay k quantizes its received signal y k =x R (` k1 ) +x k +z R k into a quantization codeword _ y k . The quantization codebooks are constructed as for the case ofK = 1, previously treated. The relay nds an bin index ` k 2f1;:::; 2 nR k g such that the corresponding bin contains the quantization codeword _ y k , it encodes the bin index as x R (` k ), and 99 transmits to the next stage relayk + 1. Here, Wyner-Ziv quantization is used, such that the quantization distortion level is chosen by imposing R k =I(Y k ; _ Y k jX k ) = log 1 +SNR + 2 q;k 2 q;k ! : (3.102) Decoding at destination: From the received signal y D , the destination can decode the bin-index ` K if R K C(SNR): (3.103) Using the decoded bin-index ` K and side-information x K , it can nd an unique quantization codeword _ y K = x R (` K1 ) +x K +z R K + _ z K , from which the known interference x K can be canceled, obtaining _ y K x K =x R (` K1 ) +z R K + _ z K : (3.104) Then, the destination can decode the bin index ` K1 if R K1 log 1 + SNR 1 + 2 q;K ! : (3.105) By repeating the above procedure until all the relay bin indices have been decoded, the destination obtains the observation _ y 1 =x S (w) +z R 1 + _ z 1 : (3.106) Finally, destination can decode the source message w if R log 1 + SNR 1 + 2 q;1 ! : (3.107) 100 In order to derive an achievable rate, we need to compute 2 q;1 by considering all rate constraints. First of all, from the rate-constraint in (3.103), we have: log 1 +SNR + 2 q;K 2 q;K ! = log(1 +SNR)) 2 q;K = 1 +SNR SNR : (3.108) Also, from the rate-constraint in (3.105), we have: log 1 +SNR + 2 q;K1 2 q;K1 ! = log 1 + SNR 1 + 2 q;K ! ; (3.109) which yields 2 q;K1 = 1 +SNR SNR (1 + 2 q;K ): (3.110) In general, we have the following relation: 2 q;k1 = 1 +SNR SNR (1 + 2 q;k ) for k =K;K 1;:::; 1; (3.111) with initial value 2 q;K = (1 +SNR)=(SNR). Using this relation, we nd q;1 = K X i=1 1 +SNR SNR i = (1 +SNR) K+1 (1 +SNR)SNR K SNR K : (3.112) Then, an achievable rate of QMF of rthe (K + 1)-hop network is given by R (K) QMF = log 1 + SNR 1 + 2 q;1 ! = log 1 + SNR K+1 (1 +SNR) K+1 SNR K+1 : (3.113) Finally, we compute the performance degradation of QMF according to the number of hops K: R (1) QMF R (K) QMF = log (1 +SNR) K+1 SNR K+1 (1 +SNR) K1 (1 + 2SNR) (3.114) = log P K i=0 (1 +SNR) Ki SNR i (1 +SNR) K1 (1 + 2SNR) ! : (3.115) 101 In the high SNR regime, the above gap is approximated by R (1) QMF R (K) QMF = log K + 1 2 : (3.116) Next, we consider the performance of QMF with noise-level quantization (i.e., 2 q;k = 1 for k = 1;:::;K). This is given by the following lemma. Lemma 3.5 The QMF with noise-level quantization ( 2 q;k = 1 for k = 1;:::;K) can achieve the following rate: R (K) QMFN = log(1 +SNR)K: (3.117) Proof The decoding procedure is similar to what seen above, i.e., the destination rst decodes the relays' messages ` K ;` K1 ;:::;` 1 in the order, and the source message w. Only dierence is that the destination must perform joint typical set decoding in order to decode the messages ` k with side information x k and the previously decoded relay's message (i.e., the bin index)` k+1 . Then, we can immediately achievable rateR k to decode message ` k from Lemma 3.3. Yet, we have a minor change in the second rate-constraint which becomes R k+1 1 since, in our case, the number of bins is changed from 2 nC(SNR) to 2 nR k+1 . In case of noise-level quantization (i.e., 2 q = 1), the second rate constraint is less than the rst rate constraint. Therefore, the index ` k can be successfully decoded if R k R k+1 1: (3.118) From this, we have the following relations: R k+1 R k = 1 for k = 0;:::;K 1; (3.119) 102 D S Figure 3.14: Message ow over (K + 1)-hop virtual full-duplex relay channel where the coecients of black line and red line are 1 and q, respectively. where R k j k=0 =R denotes the source message rate. With the initial value R K = log(1 + SNR) and using telescoping sum, we arrive at: R = log(1 +SNR)K: (3.120) 3.4.3 CoF with Forward Substitution For the case of CoF with no PA, the achievable rate is independent of the number of relay stages K since the scheme does not propagate noise and does not attenuate the signal, and the network is symmetric. In this section we focus on the application of the PA schemes of Section 4.1.1 to the case of multihop networks. Consider the (K + 1)- hop network with K + 1 transmitters (i.e., one source and K relays) for every time slot. In our achievability scheme, the PA parameters are constant over the time slots but dierent nodes have dierent parameters. We let = ( 1 ;:::; K+1 ), where 1 is the PA parameter for the source and k is the corresponding parameter for the transmitting relay in stagesk = 2;:::;K + 1. As done in Section 4.1.1, we consider two PA strategies: 1) k = d e K+1k for k = 1;:::;K + 1; 2) k = b c k1 for k = 1;:::;K + 1. In the following example, we motivate why the PA parameters depend on the stage index k. 103 Example 1 Consider a 3-hop network with relay indexing as in Fig. 3.7, and PA strategy 2) in the odd time slot, i.e., when source, relay 1, and relay 3 are in transmit mode. As done in Section 4.1.1, we can choose 1 = 1 and 2 = b c , which produces the integer- valued eective channel [1;b c] T for the multiple access channel (MAC) at relay 2. By including the transmission power change of relay 1, the channel coecients of the MAC at relay 4 are given by h b c ; i T . Then, we can choose 3 = b c 2 2 in order to make the eective channel b c [1;b c] T integer-valued up to a common non-integer factor, which can be undone at the receiver of relay 4 (at the cost of some noise power enhancement). Notice that the computation rate of the second MAC is lower than that of the rst MAC, since b c 1. Hence, the performance of CoF with PA degrades on K. The achievable rates of CoF with PA strategies are derived as follows. PA Strategy 1) By including the impact of power allocation in the channel, the eective channel at the k-th MAC is given by h k d d k ; k d e k1 i T = k1 d d k1 h d d ; i T . We observe that the channel gains decrease with k since d d 1. As done in Section 4.1.1, we can choose the integer coecientsb = [1;d e] T (same for allk) such that the variance of eective noise is given by 2 e;k () =SNR k d e k 1 2 + k d e k1 d e ! : (3.121) Following the procedures in (3.63) and (3.64), we have: R log 1 1 +d e 2k + 2k d e 2k SNR : (3.122) Since the rate-constraint is a non-increasing function ofk, the most stringent computation rate constraint is given by k =K and, accordingly, we have: R log 1 1 +d e 2K + 2K d e 2K SNR : (3.123) 104 PAStrategy2) Similarly, the eective channel at thek-th MAC is h b c k1 k1 ; b c k k1 i T = b c k1 k1 [1;b c] T . Following a similar procedure as before, the rate constraints with integer coecients b = [1;b c] T are given by: R log 1 1 +b c 2k +SNR : (3.124) The most stringent rate constraint is given by: R log 1 + b c 2K 2K SNR : (3.125) From (3.122) and (3.125), we have R CoFP = log (SNR) +K log( 2 max ); (3.126) where max = maxf =d e;b c= g. Next, we illustrate the forward substitution that the destination node can use in order to recover the desired messages from observed linear combinationsfu ` : ` = 1;:::;tg. Without loss of generality, we assume that each relay decodes a linear combination of incoming message with coecients (1;q), using CoF. Let u (K) t denote the linear combi- nation available at the destination at time slot t for the (K + 1)-hop network. Since the destination begins to receive a signal after K time slots, we have u (K) t = 0 for tK: (3.127) We also have that u (K) K+1 = w 1 since the rst signal is not interfered. In case of K = 1, we can easily compute the following relation: u (1) t+1 = t X `=1 q t` w ` : (3.128) 105 At time slott + 1, the above equation has only one unknownw t since the destination has been already decodedfw ` :` = 1;:::;t 1g during the previous time slots. Thus, it can recover the desired message w t such as w t = u (1) t+1 t1 X `=1 q t` w ` (3.129) = u (1) t+1 qu (1) t : (3.130) Yet, an extension to a general K is not straightforward. Using the symmetric structure of network (see Fig. 3.14), we can derive the following relation: u (K) t+1 qu (K) t =u (K1) t : (3.131) Here, we used the fact that the relay in the last hop can be considered as the destination of aK-hop network. Using (3.131) and (3.128), we obtain a linear equation to recursively recover the desired messages. For example, when K = 3, we have: A =u (3) t+3 qu (3) t+2 = u (2) t+2 (3.132) B = (u (3) t+2 qu (3) t+1 ) = u (2) t+1 ; (3.133) from which we obtain: AqB =u (3) t+3 2qu (3) t+2 +q 2 u (3) t+1 =u (1) t+1 = t X `=1 q t` w ` : (3.134) At time slot t + 3, the destination can decode w t using previously decoded messages fw ` :` = 1;:::;t 1g and observationsfu ` :` = 1;:::;t + 3g such as w t =u (3) t+3 2qu (3) t+2 +q 2 u (3) t+1 t1 X `=1 q t` w ` : (3.135) 106 From (3.135), it seems that this scheme suers from catastrophic error-propagation: if we make a wrong decision in somew ` , this will aect all the subsequent messages. However, this impact can be avoided by obtainingw t as a function of a sliding window of theK +1 observationsfu ` :` =t;:::;t +Kg as follows. By substituting t intot 1 in (3.134), we have: u (3) t+2 2qu (3) t+1 +q 2 u (3) t = t1 X `=1 q t1` w ` ; (3.136) from which we obtain: t1 X `=1 q t` w ` =q(u (3) t+2 2qu (3) t+1 +q 2 u (3) t ): (3.137) By replacing the last term in (3.135) by (3.137), we have: w t = u (3) t+3 2qu (3) t+2 +q 2 u (3) t+1 q(u (3) t+2 2qu (3) t+1 +q 2 u (3) t ) (3.138) = u (3) t+3 3qu (3) t+2 + 3q 2 u (3) t+1 q 3 u (3) t : (3.139) Using (3.139) instead of (3.135) to decode w t , we can signicantly reduce the impact of error-propagation: if a message u t is erroneously decoded, it will aect at most four decoded source messages (i.e., in general, K + 1 decoded source messages). The general result is given by: Lemma 3.6 For the (K + 1)-hop network with CoF, the following relation holds: K X `=1 (q) `1 0 B @ K 1 ` 1 1 C Au (K) t`+K+1 = t X `=1 q t` w ` : (3.140) Hence, the destination can decode the desired message w t at time slot t +K by ways of w t = K+1 X `=1 (q) `1 0 B @ K ` 1 1 C Au (K) t`+K+1 : (3.141) 107 Proof The result is proved by induction. By (3.128), the result holds for K = 1. As- suming that it holds K 1, we show that it also holds for K + 1: t X `=1 q t` w ` (a) = K X `=1 (q) `1 0 B @ K 1 ` 1 1 C Au (K) t`+K+1 (a) = K X `=1 (q) `1 0 B @ K 1 ` 1 1 C A (u (K+1) t`+K+2 qu (K+1) t`+K+1 ) = K X `=1 (q) `1 0 B @ K 1 ` 1 1 C Au (K+1) t`+K+2 + K X `=1 (q) ` 0 B @ K 1 ` 1 1 C Au (K+1) t`+K+1 (c) = K+1 X `=1 (q) `1 0 B @ K 1 ` 1 1 C Au (K+1) t`+K+2 + K+1 X `=1 (q) `1 0 B @ K 1 ` 2 1 C Au (K+1) t`+K+2 (d) = K+1 X `=1 (q) `1 0 B @ K ` 1 1 C Au (K+1) t`+K+2 ; (3.142) where (a) is from the hypothesis assumption, (b) is from (3.131), (c) and (d) are due to the fact that 0 B @ K 1 K 1 C A = 0; 0 B @ K 1 1 1 C A = 0; and 0 B @ K 1 ` 1 1 C A + 0 B @ K 1 ` 2 1 C A = 0 B @ K ` 1 1 C A: (3.143) 108 Also, from (3.142), we have: w t = K+1 X `=1 (q) `1 0 B @ K ` 1 1 C Au (K+1) t`+K+2 t1 X `=1 q t` w ` (3.144) (a) = K+1 X `=1 (q) `1 0 B @ 0 B @ K ` 1 1 C A + 0 B @ K ` 2 1 C A 1 C Au (K+1) t`+K+2 + (q) K+1 u (K+1) t (3.145) (b) = K+1 X `=1 (q) `1 0 B @ K + 1 ` 1 1 C Au (K+1) t`+K+2 + (q) K+1 u (K+1) t (3.146) = K+2 X `=1 (q) `1 0 B @ K + 1 ` 1 1 C Au (K+1) t`+K+2 ; (3.147) where (a) is due to the fact that t1 X `=1 q t` w ` =q 0 B @ K+1 X `=1 (q) `1 0 B @ K ` 1 1 C Au (K+1) t`+K+1 1 C A; (3.148) and (b) is from (3.143). 109 Chapter 4 Some Two-User Gaussian Networks with Cognition, Coordination, and Two Hops In this chapter we study a number of two-user interference networks with multiple- antenna transmitters/receivers (MIMO), transmitter side information in the form of linear combinations (over an appropriate nite-eld) of the information messages, and two-hop relaying. We start with a Cognitive Interference Channel (CIC) where one of the trans- mitters (non-cognitive) has knowledge of a rank-1 linear combination of the two infor- mation messages, while the other transmitter (cognitive) has access to a rank-2 linear combination of the same messages. This is referred to as the Network-Coded CIC, since such linear combination may be the result of some random linear network coding scheme implemented in the backbone wired network. For such channel we develop an achiev- able region based on a few novel concepts: Precoded Compute and Forward (PCoF) with Channel Integer Alignment (CIA), combined with standard Dirty-Paper Coding. We also develop a capacity region outer bound and nd the sum symmetric Generalized Degrees of Freedom (GDoF) of the Network-Coded CIC. Through the GDoF characterization, we show that knowing \mixed data" (linear combinations of the information messages) provides an unbounded spectral eciency gain over the classical CIC counterpart, if the ratio (in dB) of signal-to-noise (SNR) to interference-to-noise (INR) is larger than certain threshold. Then, we consider a Gaussian relay network having the two-user MIMO IC as the main building block. We use PCoF with CIA to convert the MIMO IC into a 110 deterministic nite-eld IC. Then, we use a linear precoding scheme over the nite-eld to eliminate interference in the nite-eld domain. Using this unied approach, we char- acterize the symmetric sum rate of the two-user MIMO IC with coordination, cognition, and two-hops. We also provide nite-SNR results (not just degrees of freedom) which show that the proposed coding schemes are competitive against state of the art interfer- ence avoidance based on orthogonal access, for standard randomly generated Rayleigh fading channels. 4.1 Preliminaries In this section we provide some basic denitions and background results that will be extensively used in the sequel. 4.1.1 Compute-and-Forward and Integer-Forcing We recall here the CoF scheme of [NG11] applied to a particular case of Gaussian MIMO channel with joint processing of the receiver antennas and independent lattice coding at each transmit antenna. Our reference model is given by Y =HCX +Z; (4.1) where H2 C MM , C2 Z[j] MS , X k 2 C Sn , and where Z contains i.i.d. Gaussian noise samplesCN (0; 1). Here, S denotes the number of independent and indepen- dently encoded information streams (messages) sent by a virtual \super-user" collecting all channel inputs. For k = 1;:::;S, each k-th independent message w k 2F r q is encoded input the codeword t k =f(w k ) of the same lattice codeL of rate R and mapped to the channel input sequence x k = [t k +d k ] mod ; (4.2) 111 where the dithering sequencesfd k g are mutually independent, uniformly distributed over V , and known to the receiver. 1 The encoded sequencesfx k g are arranged by rows into the transmit signal matrixX. We also dene the matrixT of dimensionsSn containing ft k g arranged by rows, and the dithering matrix D with rowsfd k g. Channel matrices in the formHC as in (4.1) will appear several times in this paper as a consequence of channel integer alignment, explicitly designed such that [CT] mod has lattice codewords arranged by rows. 2 The decoder's goal is to recover L M integer linear combinations of the S lattice codewords, given by the rows s ` of the matrix S = [B H CT] mod , for some integer matrix B2Z[j] ML . Letting b ` denote the `-th column of B, the receiver computes ^ y ` = h H ` Yb H ` CD i mod = [b H ` CT + H ` (HCX +Z)b H ` C(T +D)] mod = [s ` +z e (HC;b ` ; ` )] mod ; (4.3) where ` 2C M1 andz e (HC;b ` ; ` ) is the`-th eective noise sequence, distributed as: H ` Hb H ` CD | {z } non-integer penalty + H ` Z |{z} Gaussian noise : (4.4) Choosing H ` =b H ` H 1 , the variance of the eective noise is given by 2 e;` =k(H 1 ) H b ` k 2 : (4.5) This choice is referred to in [ZNEG10] as the exact Integer Forcing Receiver (IFR). In this way, the non-integer penalty of CoF is completely eliminated. More in general, the 1 The dithering sequences in this paper have these properties, and this fact will be understood in the sequel even though not explicitly stated. 2 The modulo reduction applied to matrices is intended row by row. 112 decoding performance can be improved especially at low SNR by minimizing the eective noise variance with respect to ` for given b ` . This yields 2 e;` = b H ` C(SNR 1 I +C H H H HC) 1 C H b ` : (4.6) Sinceb ` andC are integer-valued,s ` = [b H ` CT] mod is a codeword ofL. From [NG11], we know that by applying lattice decoding to ^ y ` given in (4.3) there exist sequences of lattice codesL of rate R and increasing block length n such that s ` can be decoded successfully with arbitrarily high probability as n!1, provided that 3 R< log + SNR 2 e;` ! ; (4.7) where the expression in the right-hand side of (4.7) is the computation rate for the modulo- additive noise channel (4.3) with given SNR and eective noise variance. All the L linear combinations can be reliably decoded if R min ` ( log + SNR 2 e;` !) : (4.8) The requirement that all inputs are encoded with the same lattice code with rate con- strained by the worst computation rate over the desired linear combinations (columns of B) can be relaxed by allowing the inputs to be encoded at dierent rates, using a family of nested lattice codes, and using successive cancellation with respect to CoF, according to the scheme proposed and analyzed in [OEN12b]. The application of this idea to the networks treated in this paper is examined in Section 4.4. 3 We dene log + (x), maxflog(x); 0g. 113 Using the linearity of lattice encoding, 4 the correspondingL linear combinations over F q for the messages are given by U = g 1 ([B H ] mod pZ[j])g 1 ([C] mod pZ[j])W = [B H ] q [C] q W; (4.9) where we use the notation [B H ] q ,g 1 ([B H ] mod pZ[j]). 4.2 Network-Coded Cognitive Interference Channel A two-user Gaussian Network-Coded CIC consists of a Gaussian interference channel where transmitter 1 (the cognitive transmitter) knows both user 1 and user 2 information messages (or, equivalently, two independent linear combinations thereof) and transmitter 2 (the non-cognitive transmitter) only knows only one linear combination of the messages. Without loss of generality, we assume that transmitter 1 knows (w 1 ;w 2 ), and transmitter 2 has w 1 w 2 , where w k 2 F r q denotes the information message desired at receiver k, at rate R k bit/symbol, for k = 1; 2. We assume that if R 1 6= R 2 then the lowest rate message is zero-padded such that both messages have a common length, given by r = maxfnR 1 ;nR 2 g, wheren denotes the coding block length. A block ofn channel uses of the discrete-time complex baseband two-user IC is described by y 1 = h 11 x 1 +h 12 x 2 +z 1 (4.10) y 2 = h 21 x 1 +h 22 x 2 +z 2 ; (4.11) where z k 2 C n1 contains i.i.d. Gaussian noise samplesCN (0; 1) and h ij 2 C de- notes the channel coecients, assumed to be constant over the whole block of length n and known to all nodes. Also, we have a common per-user power constraint, given by 4 Lattice encoding linearity refers to the isomorphism betweenF r q andL induced by the natural labeling f dened before. 114 Figure 4.1: Distributed zero-forcing precoding for nite-eld Network-Coded CIC. Dif- ferently from RLNC, the cognitive transmitter carefully chooses the coecients of linear combination according to the channel coecients q ij 's. 1 n E[kx k k 2 ] SNR, for k = 1; 2. Each receiver k observes the channel output y k and produces an estimate ^ w k of the desired message w k . A rate pair (R 1 ;R 2 ) is achievable if there exists a family of codes satisfying the power constraint, such that the average decoding error probability satises lim n!1 P( ^ w k 6=w k ) = 0, for both k = 1; 2. 4.2.1 Capacity Region for nite-eld Network-Coded CIC In order to build an intuition for Gaussian channel, we consider the corresponding nite- eld model and show that distributed zero-forcing precoding achieves the capacity of nite- eld Network-Coded CIC. A block ofn channel uses of the discrete-time nite-eld IC is described by 2 6 4 y 1 y 2 3 7 5 =Q 2 6 4 x 1 x 2 3 7 5 2 6 4 1 2 3 7 5; (4.12) where k 2F n q contains i.i.d. additive noise samples, and q ij 2F q is the (i;j)-th element of Q, denoting the channel coecients from transmitter j to receiver i, assumed to be constant over the whole block of length n and known to all nodes. Theorem 4.1 If det(Q) 6= 0 and q 11 ;q 21 6= 0, the capacity region of the nite-eld Network-Coded CIC is the set of all rate pairs (R 1 ;R 2 ) such that R k logqH( k ) for k = 1; 2: (4.13) 115 Proof We rst derive a simple upper bound by assuming the full transmitter cooperation. In this case, this model reduces to the nite-eld vector broadcast channel. A trivial upper-bound on the broadcast capacity region is given by [EGK11]: R k max P X 1 ;X 2 I(X 1 ;X 2 ;Y k ) for k = 1; 2: (4.14) Due to the additive noise nature of the channel, we have I(X 1 ;X 2 ;Y k ) =H(Y k )H( k ). Furthermore, H(Y k ) logq and this upper bound is achieved by letting (X 1 ;X 2 ) Uniform overF 2 q . This bound coincides with (4.13). Next, we derive an achievable rate using distributed zero-forcing precoding technique. Without loss of generality, it is assumed that H( 1 )H( 2 ). We use two nested linear codesC 2 C 1 whereC k has rate R k = r k n logq. Let w 1 and w 2 be the zero-padded information messages to common length r 1 . The detailed procedures of distributed zero forcing technique is as follows (see Fig. 4.1). Transmitter 1 produces the codewords c 1 = w 1 G and c 2 = w 2 G, and transmits the precoded codeword x 1 =m 1 c 1 m 2 c 2 , for some coecients m 1 ;m 2 2F q . Transmitter 2 produces the codeword c 1 c 2 = (w 1 w 2 )G and transmits the precoded codeword x 2 =m 3 (c 1 c 2 ) with coecient m 3 2F q . Receiver 1 observes: y 1 = q 11 x 1 q 12 x 2 1 = 11 c 1 12 c 2 1 ; (4.15) where 11 = (q 11 m 1 q 12 m 3 ) and 12 = (q 11 m 2 q 12 m 3 ). Receiver 2 observes: y 2 = q 21 x 1 q 22 x 2 2 = 22 c 2 21 c 1 2 ; (4.16) where 22 = (q 21 m 2 q 22 m 3 ) and 21 = (q 21 m 1 q 22 m 3 ). 116 The goal is to nd a precoding vector m = (m 1 ;m 2 ;m 3 ) T to cancel interference at both receivers (i.e., such that 12 = 21 = 0), while preserving the desired codewords (i.e., such that 11 ; 22 6= 0). Equivalently, we want to nd a non-zero vector m to satisfy the following conditions: Condition 1 (canceling the interferences) Cm =0; (4.17) where C = 2 6 4 0 q 11 q 12 q 21 0 q 22 3 7 5: (4.18) Condition 2 (preserving the desired signals) det(QM) = 11 22 6= 0; (4.19) where M = 2 6 4 m 1 m 2 m 3 m 3 3 7 5: (4.20) Since Rank(C) 2, there exist non-zero vectors m 2 Null(C) that satises Condition 1. Since Q has full rank, Condition 2 is equivalent to requiring that M has rank 2, i.e., that m 3 (m 1 m 2 )6= 0. In short, we have to nd the conditions for which a vector m in the null-space of C satises m 3 6= 0 and m 1 6= m 2 . Assuming q 11 6= 0 and q 21 6= 0, we have that Cm =0 yields m 2 = q 12 q 11 m 3 ; m 1 = q 22 q 21 m 3 : Using this in the expression of M, we nd that det(M)6= 0 if we choose m 3 6= 0 and if q 12 q 11 + q 22 q 21 = q 12 q 21 +q 11 q 22 q 11 q 21 = det(Q) q 11 q 21 6= 0: 117 By assumption, the above condition is always true, therefore we conclude that a vector m satisfying Conditions 1 and 2 can always be found. In this case, the precoded channel decouples into two parallel additive noise channels y 1 = 11 c 1 1 (4.21) y 2 = 22 c 2 2 ; (4.22) for which rates R k logqH( k ) are clearly achievable by linear coding [Dob63]. Remark 10 The capacity region of nite-eld Network-Coded CIC under the assump- tions of Theorem 4.1 is equivalent to the capacity region of the corresponding nite-eld vector broadcast channel. In other words, partial network-coded cooperation and full co- operation yield the same performance. Remark 11 It is interesting to notice that if q 11 = 0 and det(Q)6= 0, then q 21 ;q 12 6= 0. This implies thatm in the null space ofC takes on the form (0;m 2 ; 0) T for somem 2 6= 0. If q 21 = 0 and det(Q)6= 0, then q 11 ;q 22 6= 0. This implies that m in the null space of C takes on the form (m 1 ; 0; 0) T for somem 1 6= 0. In both cases, det(M) = 0 and interference cannot be removed without eliminating the useful signal at one of the two receivers. The observation in the above remark is strengthened by the following unfeasibility result: Lemma 4.1 Perfect channel decoupling is not possible if the conditions of Theorem 4.1 do not hold. Proof We will show that if the conditions of Theorem 4.1 are not satised, then it is not possible to achieve the sum rate of two perfectly decoupled channels, i.e., the sum rate 118 is lower than 2 logq (H( 1 ) +H( 2 )). We employ the upper bounds derived in Section 4.6.1 such as minfR 1 ;R 2 g minfI(X 1 ;Y 1 jX 2 );I(X 1 ;Y 2 jX 2 )g (4.23) maxfR 1 ;R 2 g maxfI(X 1 ;X 2 ;Y 1 );I(X 1 ;X 2 ;Y 2 )g (4.24) = logq minfH( 1 );H( 2 )g: (4.25) Notice that the sum rate is equal to minfR 1 ;R 2 g + maxfR 1 ;R 2 g. When q 11 = 0, the receiver 1 observes the Y 1 = q 12 X 2 1 . Then, we have that minfR 1 ;R 2 g = 0 since I(X 1 ;Y 1 jX 2 ) = H(Y 1 jX 2 )H(Y 1 jX 1 ;X 2 ) = 0. Using (4.23) and (4.25), we have that R 1 +R 2 logq minfH( 1 );H( 2 )g. Similarly when q 21 = 0, i.e, Y 2 = q 22 X 2 2 , the minfR 1 ;R 2 g = 0 is also zero because of I(X 1 ;Y 2 jX 2 ) = 0. Thus, we have that R 1 +R 2 logq minfH( 1 );H( 2 )g. In both cases, the sum rates are strictly less than 2 logq (H( 1 ) +H( 2 )). 4.2.2 Scaled Precoded CoF Motivated by the above result, we present a novel scheme named Precoded Compute- and-Forward (PCoF) for the Gaussian Network-Coded CIC. Using CoF decoding, each receiver can reliably decode an integer linear combination of the lattice codewords sent by transmitters. Then, the \interference" in the nite-eld domain can be completely eliminated by distributed zero-forcing precoding, provided that the conditions of Theorem 4.1 are satised. In order to achieve dierent coding rates while preserving the lattice Z[j]-module structure, we use a family of nested lattices 2 1 , where k = p 1 g(C k )T + with =Z n [j]T and whereC k denotes the linear code overF q generated by the rst r k rows of a generator matrix G, with r 2 r 1 . The corresponding nested lattice codes are given byL k = k \V , and have rate R k = r k n logq. We let B = [b 1 ;b 2 ]2Z[j] 22 , where b k denotes the integer coecients vector used at receiver k for 119 the modulo- receiver mapping (see (4.3)), and we let Q = [B H ] q 2F 22 q . For the time being, it is assumed that det(Q);q 11 ;q 21 6= 0 overF q . PCoF proceeds as follows: Transmitters 1 and 2 produce the precoded messages: u 1 = m 1 w 1 m 2 w 2 (4.26) u 2 = m 3 (w 1 w 2 ); (4.27) respectively, where m = (m 1 ;m 2 ;m 3 ) is a non-zero vectorm2 Null(C) whereC is related to Q as dened in (4.18). Each transmitter k produces the lattice codeword v k = f(u k )2L 1 (the densest lattice code) and transmits the channel inputs x k = [v k +d k ] mod , where d k are dithering sequences. By lattice linearity we have: v 1 = g(m 1 )t 1 +g(m 2 )t 2 mod (4.28) v 2 = g(m 3 )t 1 +g(m 3 )t 2 mod ; (4.29) where t k = f(w k ). As in the proof of Theorem 4.1, we choose the precoding vector m = (m 1 ;m 2 ;m 3 ) to satisfy Condition 2, such that QM = diag( 11 ; 22 ) for some 11 ; 22 6= 0; (4.30) where M is related to m as dened in (4.20). 120 Each receiverk applied the CoF receiver mapping (4.3) with integer coecients vector b k and (scalar) scaling factor k , yielding ^ y k = 2 6 4b H k 2 6 4 v 1 v 2 3 7 5 +z e (h k ;b k ; k ) 3 7 5 mod (4.31) = 2 6 4b H k g(M) 2 6 4 t 1 t 2 3 7 5 +z e (h k ;b k ; k ) 3 7 5 mod (4.32) (a) = 2 6 4([b H k g(M)] mod pZ[j]) 2 6 4 t 1 t 2 3 7 5 +z e (h k ;b k ; k ) 3 7 5 mod (4.33) (b) = [g( kk )t k +z e (h k ;b k ; k )] mod ; (4.34) where h k = [h k1 ;h k2 ], where (a) follows from the fact that [pt] mod = 0 for any codeword t2L k , and where (b) is due to the following result: Lemma 4.2 Let Q = [B H ] q . If QM = diag( 11 ; 22 ) over F q , then [B H g(M)] mod pZ[j] = diag(g( 11 );g( 22 )): (4.35) Proof Using [B H ] mod pZ[j] =g(Q), we have: [B H g(M)] mod pZ[j] = [([B H ] mod pZ[j])g(M)] mod pZ[j] = [g(Q)g(M)] mod pZ[j] = [g (QM)] mod pZ[j] = [g (diag( 11 ; 22 ))] mod pZ[j] = diag(g( 11 );g( 22 )): 121 From the results summarized in Section 4.1.1, we know that lattice decoding applied to the observation ^ y k at each receiver k can reliably decode the desired message if R k log + SNR b H k (SNR 1 I +h k h H k ) 1 b k ! : This rate can be improved if each transmitter k scales its signal by some factor k 2P, whereP =f2 C :jj 1g denotes the unit disk in C. This choice of k guarantees that the power constraint is satised at each transmitter. The eective channel matrix induced by this scaling is given by ~ H( 1 ; 2 ) = 2 6 4 1 h 11 2 h 12 1 h 21 2 h 22 3 7 5: (4.36) We have: Theorem 4.2 Scaled PCoF applied to Gaussian Network-Coded CIC with H = [h ij ]2 C 22 achieves the rate pairs (R 1 ;R 2 ) such that R k log + SNR b H k (SNR 1 I + ~ h k ~ h H k ) 1 b k ! ; for any full rank integer matrix B = [b 1 ;b 2 ] with b 11 ;b 21 6= 0 mod pZ[j] and 1 ; 2 2P, where ~ h k = [ 1 h k1 ; 2 h k2 ]. 4.2.3 An achievable rate region for the Gaussian Network-Coded CIC It was shown in Section 4.2 that distributed zero-forcing precoding is optimal for nite- eld Network-Coded CIC. In the Gaussian case, however, the channel coecients are not integers and hence Scaled PCoF may not be optimal due to the non-integer penalty. Using the fact that transmitter 1 has non-causal information of message 2, we can completely eliminate the interference of signal from transmitter 2 at receiver 1 by using DPC [Cos83]. Also, we can remove the non-integer penalty at the receiver 2 by using Scaled PCoF with 122 precoding known interference Figure 4.2: Encoding and decoding structures of the proposed achievability scheme. Transmitter 1 uses the DPC to cancel the interference at its intended receiver 1, and also performs precoding over nite-eld to eliminate the interference at receiver 2. a careful choice of the scaling factor of transmitter 2. In other words, while Scaled PCoF cannot simultaneously remove the non-integer penalty at both receivers, it can completely eliminate it at receiver 2, while interference at receiver 1 is handled by DPC precoding. We let b = [b 1 ;b 2 ]2Z[j] 2 denote the integer coecients vector used at receiver 2 for the CoF receiver mapping (4.3), and we letq k = [b k ] q . Again, it is assumed thatq 1 ;q 2 6= 0 overF q . The proposed achievability scheme proceeds as follows (see Fig. 4.2): Transmitter 2 produces the lattice codeword v 2 = f(w 1 w 2 ) and produces the channel input with power scaling factor 2P: x 0 2 =x 2 ; (4.37) where x 2 = [v 2 +d 2 ] mod . Transmitter 1 produces the precoded message mw 1 where m2F q is given by q 1 mq 2 = 0)m = (q 1 ) 1 (q 2 ); (4.38) where (q 1 ) 1 denotes the multiplicative inverse ofq 1 and (q 2 ) denotes the additive inverse ofq 2 . Then, it uses DPC for the known interference signalh 12 x 0 2 and forms: x 1 = [v 1 1 (h 12 =h 11 )x 0 2 +d 1 ] mod ; (4.39) 123 where v 1 =f(mw 1 ). By lattice linearity we have: v 1 = g(m)t 1 mod (4.40) v 2 = t 1 +t 2 mod ; (4.41) where t 1 =f(w 1 ) and t 2 =f(w 2 ). Receivers 1 and 2 observe the y 1 and y 2 such as y 1 = h 11 x 1 +h 12 x 2 +z 1 (4.42) y 2 = h 21 x 1 +h 22 x 2 +z 2 : (4.43) Receiver 1 performs the in ated modulo-lattice mapping as ^ y 1 = [ 1 y 1 =h 11 d 1 ] mod . This results in the mod- additive noise channel given by: ^ y 1 = [( 1 =h 11 )[h 11 x 1 +h 12 x 0 2 +z 1 ]d 1 ] mod = [v 1 v 1 + 1 x 1 + 1 (h 12 =h 11 )x 0 2 + ( 1 =h 11 )z 1 d 1 ] mod = [v 1 (1 1 )u 1 + ( 1 =h 11 )z 1 ] mod ; where u 1 is uniformly distributed onV and is independent of z 1 and v 1 by the inde- pendence and uniformity of dithering and by the Crypto Lemma. From standard DPC results [ZSE02], choosing 1 = 1;MMSE = SNRjh 11 j 2 1 +SNRjh 11 j 2 ; (4.44) the coding rate R 1 is achievable if R 1 log(1 +jh 11 j 2 SNR): (4.45) 124 Letting ~ h() = [h 21 ; ~ h 22 ] with ~ h 22 =h 22 1;MMSE h 12 h 21 =h 11 , receiver 2 applies the CoF receiver mapping (4.3) with integer coecientsb and scaling factor 2 =b 1 =h 21 , yielding ^ y 2 = [ 2 y 2 b 1 d 1 b 2 d 2 ] mod = [b 1 v 1 +b 2 v 2 + 2 (h 21 x 1 +h 22 x 0 2 +z 2 )b 1 [v 1 +d 1 ]b 2 [v 2 +d 2 ]] mod = [b 1 v 1 +b 2 v 2 + ( 2 h 21 b 1 )[v 1 +d 1 ] + ( 2 ~ h 22 b 2 )x 2 + 2 h 21 + 2 z 2 ] mod (a) = 2 6 4b T 2 6 4 v 1 v 2 3 7 5 + (b 1 ~ h 22 =h 21 b 2 )u 2 + (b 1 =h 21 )z 2 3 7 5 mod = h 0 B @b T 2 6 4 g(m) 0 1 1 3 7 5 mod pZ[j] 1 C A 2 6 4 t 1 t 2 3 7 5 +z e ( ~ h();b) i mod (b) = [([b 2 ] mod pZ[j])t 2 +z e ( ~ h();b)] mod ; where u 2 is uniformly distributed onV and is independent of v 1 , v 2 , and z 2 by the independence and uniformity of dithering and by the Crypto Lemma, = Q (v 1 1;MMSE (h 12 =h 11 )x 2 +d 1 ), (a) is due to the fact that 2 h 21 =b 1 2 , and (b) follows from the fact that m is chosen to satisfy the (4.38), i.e., b 1 g(m) +b 2 mod pZ[j] = 0. Furthermore, we dene z e ( ~ h();b) = (b 1 ~ h 22 =h 21 b 2 )u 2 + (b 1 =h 21 )z 2 : (4.46) Receiver 2 decodes t 2 by applying lattice decoding to ^ y 2 . This yields the achievable region given by: Theorem 4.3 Scaled PCoF and DPC applied to Gaussian Network-Coded CIC withH = [h ij ]2C 22 achieves the rate pairs (R 1 ;R 2 ) such that R 1 log(1 +jh 11 j 2 SNR) (4.47) R 2 log + SNR 2 e () ; (4.48) 125 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 SNR [dB] Average sum rates (bits per channel use) Full Cooperation (Vector BC) Scaled PCoF and DPC Scaled PCoF Figure 4.3: Average sum rate for Gaussian Network-Coded CIC with i.i.d. channel coecientsCN (0; 1). for any b2Z[j] 2 with b 1 ;b 2 6= 0 mod pZ[j] and any 2P, where 2 e () = b 1 ~ h 22 h 21 b 2 2 SNR + b 1 h 21 2 : Example 2 We evaluate the performance of proposed schemes with respect to their av- erage achievable sum rate, where averaging is with respect to the channel realizations with i.i.d. coecients h ij CN (0; 1). Also, we considered the performance of full-cooperation (i.e., vector broadcast channel with sum-power constraint (see for example [Yu06] for an ecient algorithm to compute the vector broadcast channel sum-capacity). In Fig. 4.3, Scaled PCoF shows the satisfactory performance in the moderate SNRs (i.e., SNR < 20 dB). Yet, this scheme suers from the non-integer penalty at high SNRs. Remarkably, Scaled PCoF with DPC (and optimization with respect to the scaling factor in Theorem 4.3) performs within a constant gap with respect to full-cooperation at any SNR. 126 4.2.4 Generalized Degrees of Freedom In the high SNR regime, a useful proxy for the performance of wireless networks is pro- vided by the Generalized Degrees-of-Freedom (GDoFs), which characterize the capacity pre-log factor in dierent relative scaling regimes of the channel coecients, as SNR grows to innity [ETW08]. In this section we study the symmetric GDoFs. In particular, we consider the following channel model: y 1 = h 11 p SNRx 1 +h 12 p INRx 2 +z 1 (4.49) y 2 = h 21 p INRx 1 +h 22 p SNRx 2 +z 2 ; (4.50) where h ij 2C are bounded non-zero constants independent of SNR;INR, z k is the i.i.d. Gaussian noiseCN (0; 1), and 1 n E[kx k k 2 ] 1 fork = 1; 2. The channel is parameterized by SNR and INR, both growing to innity such that INR = SNR as SNR!1, where 0 denes the relative strength of the direct and interference paths. LettingC sum (SNR;) denote the sum capacity for givenSNR and, the sum symmetric GDoF is dened by d sum () = lim SNR!1 C sum (SNR;) logSNR : (4.51) The main result of this section is given by: Theorem 4.4 For the Gaussian Network-Coded CIC, the sum symmetric GDoF is given by d sum () = 1 +: (4.52) Proof See Section 4.6.1. 127 In order to demonstrate the benet gain of the mixed message at the non-cognitive transmitter, we compare the sum GDoF of Gaussian IC and Gaussian CIC. The GDoF of Gaussian IC is computed in [ETW08], and it is given by d sum () = 8 > > > > > > > > > > < > > > > > > > > > > : 2(1); 0< 1 2 2; 1 2 < 2 3 2; 2 3 < 1 ; 1< 2 2; 2: (4.53) Also, from the constant gap result in [RTD10], we can immediately compute the sum GDoF of Gaussian CIC as d sum () = 8 > < > : 2; 1 ; > 1: (4.54) The sum symmetric GDoF of these three channel models are shown in Fig. 1.5. It is also immediate to observe that the sum GDoF of full-cooperation is given by d sum () = 2 maxf1;g: (4.55) In this case, the upper bound can be obtained from the 22 MIMO capacity with full CSI, and an easily analyzable achievable scheme consists of employing simple linear precoding given by BH 1 , where B = 2 6 4 1 0 0 1 3 7 5 for 1 and B = 2 6 4 0 1 1 0 3 7 5 for > 1: (4.56) Observing that (4.52) and (4.55) coincide for = 1, we conclude that the sum DoF of the Network-Coded CIC coincides with the sum DoF of full-cooperation, while the sum 128 (a) Network-Coded ICC (b) Network-Coded CIC (c) IC No CSIT Figure 4.4: Two-User Gaussian networks with coordination, cognition, and two hops. GDoF is strictly worse than full cooperation when 6= 1. Furthermore, the network- coded cognition yields higher sum GDoFs than the conventional cognition when 1=2 and higher sum GDoFs than the standard IC when 1=3. Remark 12 Apparently, having a rank-1 linear combination of both messages at trans- mitter 2 instead of just message 2 hurts for small (weak interference) and it is helpful in the intermediate to strong interference regime. Obviously, in a system where the backhaul network is rate-constrained but can be optimized with respect to the employed network code used, one would dispatch to the non-cognitive transmitter its own message only if the wireless segment operates in the regime of weak interference, and a linear combina- tion of the two messages if it operates in the medium or strong interference regimes, thus obtaining the upper envelope of the conventional and Network-Coded CIC sum GDoF. 129 4.3 Two-User MIMO IC: Coordination, Cognition, Two- Hop In this section, we study three communication channels (see Fig. 4.4) with the two-user MIMO IC as a building block, namely, Network-Coded ICC (representative of a cellular system downlink with interference coordination), Network-Coded CIC (the MIMO gen- eralization of the model of Section 4.2), and 2 2 2 IC (a canonical two- ows two-hop network that has attracted considerable attention in recent literature [GJW + 12,SA11]). In all these models, we assume that all nodes have M transmit/receive antennas. Let w k;` 2F r q denote the independent messages intended for destination k, for k = 1; 2. For simplicity of exposition, we dene the message matrix W k with rows w k;1 ;:::;w k;M , where w k;` can be all-zero vectors for ` > S k if user k has S k independent information messages. In the Network-Coded ICC, the source has no knowledge of the CSI and can deliver xed (i.e., not dependent on the wireless channel matrices) linear combinations of the information messages to each transmitter, such that each transmitter k knows M linear combinations as S k1 W 1 S k2 W 2 , for suitable integer matrices S ki . In the wireless channel, a block of n channel uses of the discrete-time complex baseband MIMO IC is described by 2 6 4 Y 1 Y 2 3 7 5 = 2 6 4 F 11 F 12 F 21 F 22 3 7 5 2 6 4 X 1 X 2 3 7 5 + 2 6 4 Z 1 Z 2 3 7 5; (4.57) where the matrices X k and Y k contain, arranged by rows, the channel input sequences x k;` 2C 1n , the channel output sequences y k;` 2C 1n , and where F jk 2C MM denotes the channel matrix between transmitter k and receiver j. The Network-Coded CIC has the wireless channel component given in (4.57), but in this case the two transmitters have dierent knowledge on the messages. In particular, transmitter 1 (the cognitive transmit- ter) knows both messagesW 1 ;W 2 and transmitter 2 (the non-cognitive transmitter) only knows linear combinations S 21 W 1 S 22 W 2 , where the rank of the linear combinations 130 is not sucient to recover the individual messages. Finally, we consider the 2 2 2 IC, as shown in Fig. 4.4 (c), where each transmitter k (referred to as \source" in this relay setting) has a message for its intended destination k, fork = 1; 2. In this model, the rst hop is also described by (4.57) and in the second hop a block of n channel uses of the discrete-time complex MIMO IC is described by 2 6 4 Y 3 Y 4 3 7 5 = 2 6 4 F 33 F 34 F 43 F 44 3 7 5 2 6 4 X 3 X 4 3 7 5 + 2 6 4 Z 3 Z 4 3 7 5: (4.58) where we denote the two transmitter-receiver pairs in the second hop by k = 3; 4, and whereZ k contains i.i.d. Gaussian noise samplesCN (0; 1). We assume that the elements ofF jk are drawn i.i.d. according to a continuous distribution (i.e., Gaussian distribution). The channel matrices are assumed to be constant over the whole block of length n and known to all nodes, and we consider a total power constraint equal to P sum at each transmitter (both sources and relays). Before stating the main results of this section, it is useful to introduce the following notation. With reference to Section 4.1.1, for a set of modulo- additive noise channel of the type (4.3), induced by nested lattice coding, by the channel matrix H k C k and by the integer combining matrix B k with columns b k;` , for k = 1; 2 and ` = 1;:::;L k , for some integer L k , we dene R comp (H k C k ;B k ;SNR) = min `=1;:::;L k ( log + SNR 2 e;k;` !) ; (4.59) where 2 e;k;` = b H k;` C k (SNR 1 I +C H k H H k H k C k ) 1 C H k b k;` : (4.60) 131 Also, we dene the constant matrices: C 12 = 2 6 4 0 1(M1) I (M1)(M1) 3 7 5; (4.61) C 22 = 2 6 4 I (M1)(M1) 0 1(M1) 3 7 5: (4.62) With this notation, we have: Theorem 4.5 For the Network-Coded ICC and Network-Coded CIC, PCoF with CIA can achieve the symmetric sum rate of (2M 1)R with all messages of the same rate given by R = min k=1;2 fR comp (H k C k ;B k ;SNR)g; (4.63) for any full-rank integer matrices A 1 2 Z[j] MM ;A 2 2 Z[j] (M1)(M1) and B 1 ;B 2 2 Z[j] MM , and any alignment precoding matrices V k satisfying the alignment conditions in (4.72), where H k = F k1 V 1 ; C k = A 1 C k2 A 2 (4.64) SNR = min k=1;2 P sum tr(V k A k A H k V H k ) : (4.65) Theorem 4.6 For the 222 IC, PCoF with CIA can achieve the symmetric sum rate of (2M 1)R with all messages of the same rate given by R = min k=1;2 fR comp (H k C k ;B k ;SNR)g; min k=3;4 fR comp (H k C k ;B k ;SNR 0 )g ; (4.66) 132 for any full-rank integer matrices A 1 ;A 3 2 Z[j] MM ;A 2 ;A 4 2 Z[j] (M1)(M1) and B k 2 Z[j] MM ;k = 1;:::; 4, and any alignment precoding matrices V k satisfying the alignment conditions in (4.72), where H k = F k1 V 1 ; C k = A 1 C k2 A 2 ; k = 1; 2 (4.67) H k = F k3 V 3 ; C k = A 3 C (k2)2 A 4 ; k = 3; 4 (4.68) SNR = min k=1;2 P sum tr(V k A k A H k V H k ) (4.69) SNR 0 = min k=3;4 P sum tr(V k A k A H k V H k ) : (4.70) The next result shows that the per-message rateR grows as logSNR whenP sum !1, thus obtaining an achievable sum DoF result for all the above channel models: Corollary 4.1 PCoF with CIA achieves sum DoF equal to (2M 1) for the Network- Coded ICC, Network-Coded CIC, and 2 2 2 IC, when all nodes have M multiple antennas. Proof See Section 4.6.2. For the Network-Coded CIC, we can improve the DoF by appropriately combining the DPC and PCoF as done in Section 4.2.3 for single antenna case. Exploiting this idea, we obtain: Theorem 4.7 For the Network-Coded CIC, the sum DoF is equal to M when all nodes have M multiple antennas. Proof See Section 4.6.3. The proofs of Theorems 4.5 and 4.6 are provided in Sections 4.3.1 and 4.3.2. Our achievable scheme is based on the extension of the PCoF approach to the MIMO case. 133 This scheme consists of two phases: 1) Using the CoF framework, we transform the two-user MIMO IC into a deterministic nite-eld IC. 2) A linear precoding scheme is used over nite-eld to eliminate the interferences (see Figs. 4.5). The main performance bottleneck of CoF consists of the non-integer penalty, which ultimately limits the per- formance of CoF at high SNR [NP12]. To overcome this bottleneck, we employ CIA in order to create an \aligned" channel matrix for which exact integer forcing is possible, similarly to what done in Section 4.1.1. Specically, we use alignment precoding matrices V 1 and V 2 at the two transmitters such that F k1 V 1 F k2 V 2 =H k C k ; (4.71) where H k 2 C MM and C k 2 Z[j] M2M . Linear precoding over the complex eld may produce a power-penalty due to the non-unitary nature of the alignment matrices, and this can degrade the performance at nite SNR. In order to counter this eect, we use Integer Forcing Beamforming (IFB) [HC12]. The main idea is that V k can be pre- multiplied (from the left) by some appropriately chosen full-rank integer matrix A k since its eect can be undone by precoding over F q , using [A k ] q . Then, we can optimize the integer matrix in order to minimize the power penalty of alignment. The optimization of alignment and IFB in order to obtain good nite SNR performance is postponed to Section 4.5. The details of the coding scheme are given in the following sections. 4.3.1 CoF Framework based on Channel Integer Alignment In this section we show how to turn any two-user MIMO IC into a deterministic nite- eld IC using the CoF framework. Consider the MIMO IC in (4.57). For k = 1; 2, let W T k = S k1 W 1 S k2 W 2 denote the network coded messages at transmitter k. We let w T k;` 2 F r q denote the `-th row of W T k , and we let W T 1 ;W 1 have dimension Mr and W T 2 ;W 2 have dimension (M 1)r. The precoding matrices S k1 ;S k2 over F q will be determined in Section 4.3.2. We let V 1 = [v 1;1 ;:::;v 1;M ]2 C MM and V 2 = 134 [v 2;1 ;:::;v 2;M1 ]2C M(M1) denote the precoding matrices used at transmitters 1 and 2, respectively, chosen to satisfy the alignment conditions F 11 v 1;`+1 = F 12 v 2;` F 21 v 1;` = F 22 v 2;` ; (4.72) for ` = 1;:::;M 1. The feasibility of conditions (4.72) is shown in [GJW + 12] for any integer M 2, almost surely with respect to the continuously distributed channel matricesfF jk g. Let A 1 2Z[j] MM and A 2 2Z[j] (M1)(M1) denote full rank integer matrices (the optimization of which in order to minimize the transmit power penalty is discussed in Section 4.5). The transmitters make use of the same lattice codeL of rateR, where is chosen such that 2 =SNR. Then, CoF based on CIA proceeds as follows. Encoding: Each transmitter k precodes its messages overF q as W 0 T k = [A k ] 1 q W T k ; k = 1; 2: (4.73) Then, the precoded messages (rows of W 0 T k ) are encoded using the nested lattice codes ast 0 k;` =f(w 0 T k;` ). Finally, the channel input sequences are given by the rows of X 00 k =V k A k X 0 k ; (4.74) where X 0 k has rows x 0 k;` = [t 0 k;` +d k;` ] mod . Due to the sum-power constraint equal to P sum at each transmitter, the second moment of coarse lattice (i.e., SNR) must satisfy SNR tr(V k A k A H k V H k )P sum for k = 1; 2: (4.75) 135 Thus, we can choose: SNR = min P sum tr(V k A k A H k V H k ) : k = 1; 2 : (4.76) Decoding: Receiver 1 observes: Y 1 = F 11 X 00 1 +F 12 X 00 2 +Z 1 (4.77) (a) = F 11 V 1 | {z } ,H 1 [ I MM C 12 ] 2 6 4 A 1 X 0 1 A 2 X 0 2 3 7 5 +Z 1 (4.78) = H 1 C 1 2 6 4 X 0 1 X 0 2 3 7 5 +Z 1 ; (4.79) where (a) follows from the fact that the precoding vectors satisfy the alignment conditions in (4.72) and C 1 = [ A 1 C 12 A 2 ]. Similarly, receiver 2 observes the aligned signals: Y 2 = F 21 X 1 +F 22 X 2 +Z 2 (4.80) = F 21 V 1 | {z } ,H 2 [ I MM C 22 ] 2 6 4 A 1 X 0 1 A 2 X 0 2 3 7 5 +Z 2 (4.81) = H 2 C 2 2 6 4 X 0 1 X 0 2 3 7 5 +Z 2 ; (4.82) where C 2 = [ A 1 C 22 A 2 ]. 136 Notice that the channel matrices in (4.79) and (4.82) follow the particular form in (4.1). Following the CoF framework in (4.8) and (4.9), if RR comp (H k C k ;B k ;SNR), receiver k can decode the M linear combinations with full-rank integer coecients matrix B k : U k = [B H k ] q [C k ] q 2 6 4 W 0 T 1 W 0 T 2 3 7 5 (4.83) = [B H k ] q [A 1 ] q [C k2 ] q [A 2 ] q 2 6 4 W 0 T 1 W 0 T 2 3 7 5 (4.84) (a) = [B H k ] q I MM [C k2 ] q 2 6 4 W T 1 W T 2 3 7 5; (4.85) where (a) is due to the precoding overF q in (4.73). Let ^ W T 1 = [B H 1 ] 1 q U 1 and ^ W T 2 denote the rstM 1 rows of [B H 2 ] 1 q U 2 . The mapping betweenfW T 1 ;W T 2 g andf ^ W T 1 ; ^ W T 2 g denes a deterministic nite-eld IC given by: 2 6 4 ^ W T 1 ^ W T 2 3 7 5 =Q sys 2 6 4 W T 1 W T 2 3 7 5; (4.86) where the system matrix is dened by Q sys = 2 6 4 I MM Q 12 Q 21 I (M1)(M1) 3 7 5; (4.87) and where Q 12 = 2 6 4 0 1(M1) I (M1)(M1) 3 7 5; Q 21 = [ I (M1)(M1) 0 (M1)1 ]: (4.88) Notice that the system matrix is xed and independent of the channel matrices, since it is determined only by the alignment conditions. 137 4.3.2 Linear Precoding over deterministic networks In this section we determine linear precoding schemes to eliminate the interferences in the nite-eld domain. Recall that transmitter 2 sends only M 1 messages in order to use CIA. Accordingly, Q sys in (4.86) has dimension (2M 1) (2M 1). 4.3.2.1 Network-Coded ICC In this model, the source can deliver linear combinations of information messages with coecients Q 1 sys . We have: Lemma 4.3 The system matrix Q sys dened in (4.87) is full-rank over F q . Proof The determinant of Q sys is given by det(Q sys ) = det(I MM )det(I (M1)(M1) (Q 21 Q 12 )) (4.89) = det(I (M1)(M1) (Q 21 Q 12 )) = 1; (4.90) sinceI (M1)(M1) (Q 21 Q 12 ) is a lower triangular matrix with unit diagonal elements. Such precoding yields immediately ^ W T k =W k fork = 1; 2. This proves Theorem 4.5 for the Network-Coded ICC. 4.3.2.2 2 2 2 IC We use the CoF framework based on CIA illustrated in Section 4.3.1 in order to turn each hop (i.e., a two-user MIMO IC) into a deterministic nite-eld IC dened by Q sys in (4.87). At the two sources, no precoding is used such that W T k = W k , for k = 1; 2. Hence, the deterministic nite-eld IC corresponding to the rst-hop of the 2 2 2 IC network has outputs ^ W T 1 ; ^ W T 2 related to W 1 and W 2 by (4.86). 138 Source 1 Source 2 "not used" "not used" precoding Relay 2 Relay1 Destination1 Destination 2 Figure 4.5: A deterministic noiseless 2 2 2 nite-eld IC. Relays 1 and 2 perform precoding of the decoded linear combination messages such as W T 3 =M 1 ^ W T 1 and W T 4 =M 2 ^ W T 2 , where the precoding matrices M 1 and M 2 are dened in Lemma 4.4. Operating in a similar way as for the rst hop, the second hop deterministic nite-eld IC is given by 2 6 4 ^ W T 3 ^ W T 4 3 7 5 = Q sys 2 6 4 W T 3 W T 4 3 7 5: (4.91) Concatenating the two hops, the end-to-end nite-eld deterministic network is described by 2 6 4 ^ W T 3 ^ W T 4 3 7 5 =Q sys 2 6 4 M 1 0 0 M 2 3 7 5Q sys 2 6 4 W 1 W 2 3 7 5: (4.92) Lemma 4.4 shows that the decoded linear combinations are equal to its desired messages at destination 1 and are equal to the messages with a change of sign (multiplication by 1 in the nite-eld) at destination 2 (see Fig. 4.5). This proved Theorem 4.6. Lemma 4.4 Choosing precoding matrices M 1 and M 2 as M 1 = (I MM (Q 12 Q 21 )) 1 (4.93) M 2 = (I (M1)(M1) (Q 21 Q 12 )) 1 ; (4.94) 139 the end-to-end system matrix becomes a diagonal matrix: Q sys 2 6 4 M 1 0 0 M 2 3 7 5Q sys = 2 6 4 M 1 Q 12 M 2 Q 21 M 1 Q 12 Q 12 M 2 Q 21 M 1 M 2 Q 21 Q 21 M 1 Q 12 M 2 3 7 5 (4.95) = 2 6 4 I MM 0 0 I (M1)(M1) 3 7 5: (4.96) Proof From the Matrix Inversion Lemma [Har08, Thm 18.2.8], we can rewrite M 1 and M 2 as M 1 = I MM Q 12 (I (M1)(M1) (Q 21 Q 12 )) 1 Q 21 (4.97) M 2 = (I (M1)(M1) Q 21 (I MM (Q 12 Q 21 )) 1 Q 12 ): (4.98) Canceling the interferences: M 1 Q 12 Q 12 M 2 = (I MM (Q 12 Q 21 )) 1 Q 12 Q 12 ((I (M1)(M1) (Q 21 Q 12 )) 1 ) = Q 12 Q 12 (I (M1)(M1) (Q 21 Q 12 )) 1 Q 21 Q 12 Q 12 ((I (M1)(M1) (Q 21 Q 12 )) 1 ) = Q 12 (Q 12 )(I (M1)(M1) (Q 21 Q 12 )) 1 ((Q 21 Q 12 )I (M1)(M1) ) = 0 M(M1) : Q 21 M 1 M 2 Q 21 = Q 21 Q 21 Q 12 (I (M1)(M1) (Q 21 Q 12 )) 1 Q 21 ((I (M1)(M1) (Q 21 Q 12 )) 1 Q 21 ) = Q 21 (I (M1)(M1) (Q 21 Q 12 ))(I (M1)(M1) (Q 21 Q 12 )) 1 (Q 21 ) = 0 (M1)M : 140 Preserving the desired signals: M 1 Q 12 M 2 Q 21 = I MM Q 12 (I (M1)(M1) (Q 21 Q 12 )) 1 Q 21 (Q 12 (I (M1)(M1) (Q 21 Q 12 )) 1 Q 21 ) =I MM Q 21 M 1 Q 12 M 2 = Q 21 (I MM (Q 12 Q 21 )) 1 Q 12 (I (M1)(M1) ) (Q 21 (I MM (Q 12 Q 21 )) 1 Q 12 ) =I (M1)(M1) : This completes the proof. 4.3.3 Network-Coded CIC In this case we assume that transmitter 1 knows both messages W 1 and W 2 , and transmitter 2 only knows M 1 linear combinations S 1 W 0 1 S 2 W 2 , where S 1 ;S 2 2 F (M1)(M1) q are full-rank matrices and where W 0 1 = Q 21 W 1 contains the rst M 1 rows ofW 1 (see the denition ofQ 21 in (4.88)). In fact, we may assume that transmitter 2 also knows the interference-free messagew 1;M (the last row ofW 1 ) but this is not used in our scheme. Transmitters 1 and 2 perform the precoding (over F q ) in the following way: W T 1 = M 1 (W 1 Q 12 S 1 1 S 2 W 2 ) (4.99) W T 2 = M 2 S 1 1 (S 1 W 0 1 S 2 W 2 ); (4.100) 141 where M 1 and M 2 are dened in Lemma 4.4. From (4.86), we have: 2 6 4 ^ W T 1 ^ W T 2 3 7 5 = Q sys 2 6 4 W T 1 W T 2 3 7 5 (4.101) = Q sys 2 6 4 M 1 0 0 M 2 3 7 5 2 6 4 I MM Q 12 S 1 1 S 2 Q 21 S 1 1 S 2 3 7 5 2 6 4 W 1 W 2 3 7 5 (4.102) = Q sys 2 6 4 M 1 0 0 M 2 3 7 5Q sys 2 6 4 I MM 0 0 S 1 1 S 2 3 7 5 2 6 4 W 1 W 2 3 7 5 (4.103) (a) = 2 6 4 I MM 0 0 S 1 1 S 2 3 7 5 2 6 4 W 1 W 2 3 7 5; (4.104) where (a) follows from Lemma 4.4. This shows that the decoded linear combinations are equal to its desired messages at receiver 1 and are equal to the messages with multipli- cation by full-rank matrix (S 1 1 S 2 ) 1 (in the nite-eld domain) at receiver 2. Based on this, Theorem 4.5 is proved for the Network-Coded CIC. 4.4 Improving the sum rates using successive cancellation In this section we improve the sum rate in Theorem 4.5 by using CoF with successive cancellation. We focus on Network-Coded ICC to explain the proposed scheme. As shown before, precoding of information messages (over F q ) can eliminate interference from the other transmitter so that each receiver observes full-rank integer linear combinations of its own intended lattice codewords in the corresponding MIMO modulo channel. Once the network is reduced to two decoupled MIMO modulo channels, each receiver can perform successive cancellation following the idea rst proposed in [OEN12b]. In this way, each message can be recovered reliably at rate equal to the computation rate of the corresponding equation, without being constrained by the equal rate requirement (minimum of the computation rates of all equations). 142 In order to achieve the dierent coding rates while preserving the latticeZ[j]-module structure, we use a family of nested lattice codes 2M1 1 , obtained by a nested construction A as described in [NG11, Sect. IV.A]. In particular, we let ` = p 1 g(C ` )T + with = Z n [j]T and withC ` denoting the linear code over F q generated by the rst r ` rows of a common generator matrix G, with r 2M1 r 1 . The corresponding nested lattice codes are given byL ` = ` \V for ` = 1;:::; 2M 1. Let w k;` 2F r 1 q be the zero-padded message to the common length r 1 . Encoding follows the same procedure outlined in Section 4.3. Namely, we let 2 6 4 W T 1 W T 2 3 7 5 =Q 1 sys 2 6 4 W 1 W 2 3 7 5; (4.105) in order to eliminate interference. Recall that each transmitter k precodes its messages over F q as in (4.73) where the integer matrix A k is used for IFB with the purpose of minimizing the power penalty (see later). Then, the precoded messages are encoded using the densest lattice codeL 1 ast 0 `;k =f(w 0 T k;` ). Finally, the channel input sequences are given by the rows of X 00 k dened in (4.74). Let t k;` = f(w k;` ) denote the lattice codeword corresponding to information message w k;` . Using lattice linearity, we can express the precoding in the complex (lattice) domain as: 2 6 4 T 0 1 T 0 2 3 7 5 = 2 6 4 A 1 1 0 0 A 1 2 3 7 5g(Q 1 sys ) 2 6 4 T 1 T 2 3 7 5 mod : (4.106) From (4.79) and (4.82), each receiver k observes the integer aligned signals: Y k = H k C k 2 6 4 X 0 1 X 0 2 3 7 5 +Z k (4.107) = H k I MM C k2 2 6 4 A 1 0 0 A 2 3 7 5 2 6 4 X 0 1 X 0 2 3 7 5 +Z k : (4.108) 143 The modulo vector channel after applying the CoF receiver mapping (4.3) with integer coecients matrix B k seen at each receiver k = 1; 2 is given as follows: At receiver 1 we have: ^ Y 1 = 2 6 4B H 1 I MM C 12 2 6 4 A 1 0 0 A 2 3 7 5 2 6 4 T 0 1 T 0 2 3 7 5 +Z e (H 1 C 1 ;B 1 ) 3 7 5 mod (a) = 2 6 4B H 1 I MM C 12 g(Q 1 sys ) 2 6 4 T 1 T 2 3 7 5 +Z e (H 1 C 1 ;B 1 ) 3 7 5 mod (b) = h B H 1 T 1 +Z e (H 1 C 1 ;B 1 ) i mod ; (4.109) where (a) follows the (4.106), (b) is due to the fact that I MM C 12 q Q 1 sys = I MM 0 MM1 : (4.110) and where Z e (H 1 C 1 ;B 1 ) denotes the Mn matrix of eective noises with rows z e (H 1 C 1 ;b 1;` ; 1;` ), and the projection vector 1;` is determined as a function of H 1 C 1 ;b 1;` as said in Section 4.1.1. Similarly, at receiver 2 we have: ^ Y 2 = 2 6 4B H 2 I MM C 22 2 6 4 A 1 0 0 A 2 3 7 5 2 6 4 T 0 1 T 0 2 3 7 5 +Z e (H 2 C 2 ;B 2 ) 3 7 5 mod = 2 6 4B H 2 I MM C 22 g(Q 1 sys ) 2 6 4 T 1 T 2 3 7 5 +Z e (H 2 C 2 ;B 2 ) 3 7 5 (a) = 2 6 4B H 2 2 6 4 T 2 t 3 7 5 +Z e (H 2 C 2 ;B 2 ) 3 7 5 mod : (4.111) 144 wheret denotes some linear combination of lattice codewords, irrelevant for receiver 2, (a) follows from the fact that I MM C 22 q Q 1 sys = 2 6 4 0 M1M1 I (M1)(M1) ? 3 7 5; (4.112) and where? denotes some non-zero vector inF 1(2M1) q . In (4.111),Z e (H 2 C 2 ;B 2 ) is dened similarly to Z e (H 1 C 1 ;B 1 ). Receiver 2 can recover its M 1 messages as long as it has M 1 full-rank linear combinations of its own messages. In order to remove the unintended messages collected in t, we choose B 2 in the form: B H 2 = ~ B H 2 0 ; (4.113) where ~ B 2 2 Z[j] (M1)(M1) is full-rank. Then, the rst M 1 observations of receiver 2 is given by ^ Y 0 2 = h ~ B H 2 T 2 +Z e (H 2 C 2 ;B 2 ) i mod : (4.114) From (4.109) and (4.114), we have that each receiver obtains a full-rank interference-free MIMO integer valued modulo channel with eective additive noise. At this point, each receiver can perform successive cancellation [OEN12b], thus relaxing the mini- mum common computation rate constraint. Focusing on receiver 1, we illustrate the successive cancellation procedure with given integer matrix B 1 and computation rates flog + (SNR= 2 e;1;` ) : ` = 1;:::;Mg. The same procedure can be straightforwardly ap- plied to receiver 2, given the formal equivalence of (4.109) and (4.114). Without loss of generality, assume that 2 e;1;1 2 e;1;M : (4.115) Letting R 1;` denote the rate of `-th message of user 1, we have R 1;` = r j 1;` for some j 1;` 2f1;:::; 2M 1g, i.e., the `-th message of user 1 is encoded using nested lattice 145 codesL j 1;` . For the time being, we assume that R 1;1 R 1;2 R 1;M (the ordering will be determined later on, according to column permutation of B 1 that is required for successive cancellation). Receiver 1 can reliably decode s 1 = [b H 1;1 T 1 ] mod as long as R 1;1 log + SNR 2 e;1;1 ! : (4.116) Then, it proceeds to decode s 2 = [b H 1;2 T 1 ] mod . Using the previously decoded s 1 , it can perform the cancellation: [^ y 2 +e 21 s 1 ] mod = [s 2 +e 21 s 1 +z e (H 1 C 1 ;b 1;2 ; 1;2 )] mod (4.117) = [~ s 2 +z e (H 1 C 1 ;b 1;2 ; 1;2 )] mod : (4.118) Here, e 21 2 Z[j] is chosen so that [(b 1;2 (1) +e 21 b 1;1 (1))] mod pZ[j] = 0 where b(j) denotes the j-th element of vector b. Then ~ s 2 does not include t 1 and hence receiver 1 can reliably decode ~ s 2 as long as R 1;2 log + SNR 2 e;1;2 ! : (4.119) Now, receiver 1 can obtain s 2 such as s 2 = [~ s 2 e 21 s 1 ] mod . Receiver 1 can decode the remaining linear combinations s ` for ` 3 in a similar manner. Namely, before decodings ` , receiver 1 adds h P `1 j=1 e `j s j i mod (i.e., an integer valued linear combina- tions of previously decoded s j 's). Here the coecients e `j are chosen so that the impact of t 1 ;:::;t `1 is canceled out from s ` . Assuming that such coecients exist, receiver 1 can decode ~ s ` = h s ` + P `1 j=1 e `j s j i mod as long as R 1;` is less than the corresponding computation rate of the `-th equation. 146 From [OEN12b, Lemma 2], such cancellation coecients exist for at least one column permutation vector 1 of B 1 . Accordingly, all M linear combinations can be decoded as long as R 1; 1 (`) log + SNR 2 e;1;` ! for ` = 1;:::;M: (4.120) Therefore, the sum rate P M `=1 log + SNR 2 e;1;` is achievable. Similarly, there exists at least one column permutation vector 2 of B 2 for which all M 1 linear combinations at receiver 2 can be decoded as long as R 2; 2 (`) log + SNR 2 e;2;` ! for ` = 1;:::;M 1; (4.121) where we letR 2;` =r j 2;` , for some index mappingj 2;` 2f1;:::; 2M 1g, denote the rate of `-th message of user 2. The exactly same procedure can be applied to the Network- Coded CIC. The successive cancellation replaces the sum-rate formula in Theorem 4.5 as follows: Theorem 4.8 For the Network-Coded ICC and Network-Coded CIC, PCoF with CIA can achieve sum rate as R sum = M X `=1 log + SNR 2 e;1;` ! + M1 X `=1 log + SNR 2 e;2;` ! ; (4.122) for any full rank integer matricesA 1 2Z[j] MM ;A 2 2Z[j] (M1)(M1) ,B 1 2Z[j] MM ; ~ B 2 2Z[j] (M1)(M1) , and any alignment precoding matricesV k to satisfy the alignment conditions in (4.72), where B H 2 = ~ B H 2 0 (4.123) H k = F k1 V 1 ;C k = A 1 C k2 A 2 ; k = 1; 2 (4.124) SNR = min k=1;2 P sum V k A k A H k V H k ; (4.125) 147 and where 2 e;k;` = b H k;` C k (SNR 1 I +C H k H H k H k C k ) 1 C H k b k;` ; k = 1; 2: 4.5 Optimization of achievable rates In this section we optimize the integer matrices A k and B k in Theorems 4.5-4.8 by assuming that the precoding matrices V k are given. The dimensions of A k and B k can be either MM or (M 1) (M 1), depending on k. Since this does not change the optimization problem, we will drop the index k and just consider dimension M. The power-penalty optimization with respect to A takes on the form: arg min tr VAA H V H = M X `=1 kVa ` k 2 subject to A is full rank overZ[j]; (4.126) where a ` denotes the `-th column of A. Also, the minimization of the eective noise variance with respect to B takes on the form: arg min max ` kLb ` k 2 subject to B is full rank overZ[j]; (4.127) whereL denotes a square-root factor of (SNR 1 I+H H H) 1 ,H denotes an aligned channel matrix and b ` denotes the `-th column of B. We notice that problem (4.126) (resp., (4.127)) is equivalent to nding a reduced basis for the lattice generated by V (resp., L). In particular, the reduced basis takes on the form VU where U is a unimodular matrix over Z[j]. Hence, choosing A = U yields the minimum power-penalty subject to the full rank condition in (4.126). In practice we 148 used the (complex) LLL algorithm [Nap96] 5 , with renement of the LLL reduced basis approximation by Phost or Schnorr-Euchner lattice search [DEGC03]. We let u ` denote the `-th column of U. Phost or Schnorr-Euchner enumeration generates all non-zero lattice points in a sphere centered at the origin, with radius equal to d = max ` kVu ` k 2 . This radius guarantees the existence of solutions because of having the trivial solution fu 1 ;:::;u M g. Dene the set of integer vectors such that the corresponding lattice points are in a sphere with radius d by (A;W) =f(a ` ;w ` ) :w ` =kVa ` k 2 dg: (4.128) The following lemma shows that the greedy algorithm (Algorithm 1) nds a solution (i.e., M linearly independent integer vectors) to the problem (4.126) (or (4.127)). Lemma 4.5 For given (A;W) dened in (4.128), Algorithm 1 nds a solution to the following two problems: min SA X `2S w ` (4.129) subject to fa ` :`2Sg are linearly independent (4.130) jSj =M: (4.131) and min SA maxfw ` :`2Sg (4.132) subject to fa ` :`2Sg are linearly independent (4.133) jSj =M: (4.134) Proof The rst problem consists of the minimization of linear function subject to a ma- troid constraint, where the matroidM( ;I) is dened by the ground set = [1 :jAj] and 5 We can also use the HKZ and Minkowski lattice basis reduction algorithm (see [SHV12] for details). 149 by the collection of independent setsI =fS :fa ` :`2Sg are linearly independentg. Rado and Edmonds [Rad57,Edm71] proved that a greedy algorithm (Algorithm 1) nds an optimal solution. In case of the second problem, we provide a simple proof as follows. Suppose that the indices of elements inA are rearranged according to the increasing ordering of the weights w ` . The problem is then reduced to nding the minimum index ` y such thatfa 1 ;:::;a ` yg includes the M linearly independent vectors. This is precisely what Algorithm 1 does. Algorithm 4.1 The Greedy Algorithm Input: (A; ) =f(a ` ;w ` ) :a ` 2Z[j] M1 ;w ` 2Z + g Output:SA withjSj =M 1. Rearrange the indices of vectors inA such that w 1 w 2 w jAj 2. Initially, ` = 1 andS = 3. If Rank(S[f`g)> Rank(S) thenS S[f`g 4. Set ` =` + 1 5. Repeat 3)-4) untiljSj =M 4.5.1 Finite SNR Results We evaluate the performance of PCoF with CIA in terms of its average achievable sum rates. We computed the ergodic sum rates by Monte Carlo averaging with respect to the channel realizations with i.i.d. elementsCN (0; 1). Recall that we consider a total power constraint equal toP sum at each transmitter (both sources and relays). We rst consider the symmetric sum rates in Theorem 4.6 for 2 2 2 Gaussian IC. For comparison, we considered the performance of time-sharing where IFR is used for eachMM MIMO IC. We used the IFR since it is known to almost achieve the performance of joint maximum 150 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 P sum /M [dB] Average symmetric sum rates (bits per channel use) Time Sharing PCoF with CIA (w/o opt) PCoF with CIA 3 DoF 2 DoF Figure 4.6: Performance comparison of PCoF with CIA and time-sharing with respect to ergodic symmetric sum rates for 2 2 2 MIMO interference channel with M = 2. 0 10 20 30 40 50 60 70 0 20 40 60 80 100 120 P sum /M [dB] Average sum rates (bits per channel use) Time Sharing (M=2) PCoF with CIA (M=2) Time Sharing (M=3) PCoF with CIA (M=3) Figure 4.7: Performance comparison of PCoF with CIA and time-sharing with respect to ergodic symmetric sum rates for MIMO interference coordination channel. 151 likelihood receiver [ZNEG10] and has a similar complexity with PCoF. In this case, an achievable symmetric sum rate of time-sharing is obtained as R = minfR comp (F 11 ;B 1 ;SNR);R comp (F 33 ;B 2 ;SNR)g; (4.135) for any full-rank matrices B 1 ;B 2 2Z[j] MM , where SNR =P sum . Here, we used 2P sum for power constraint since with time-sharing each transmitter is active on only half of the time slots. For PCoF with CIA, we need to nd precoding matrices for satisfying the alignment condition in (4.72). For M = 2, the conditions are given by F 11 v 1;2 =F 12 v 2;1 and F 21 v 1;1 =F 22 v 2;1 : For the simulation, we used the following precoding matrices to satisfy the above condi- tions: V 1 = F 1 21 F 22 1 F 1 11 F 12 1 and v 2;1 =1: (4.136) Also, the same construction is used for the second hop. We used the complex LLL algorithm to optimize integer matrices, yielding lower bound on achievable rate in Theo- rem 4.6. Since source 1 (or relay 1) transmits one more stream than source 2 (or relay 2), the former always requires higher transmission power. In order to eciently satisfy the average power-constraint, the role of sources 1 and 2 (equivalently, relays 1 and 2) is al- ternatively reversed in successive time slots. In Fig. 4.6, we observe that PCoF with CIA can have the SNR gain about 5 dB by optimizing the integer matrices for IFR and IFB, comparing with simply using identity matrices. Also, PCoF with CIA provides a higher sum rate than time-sharing ifSNR 15 dB, and its gain over time-sharing increases with SNR, showing that in this case the DoF result matters also at nite SNR. In addition, we evaluate the performance of the Network-Coded ICC with respect to sum rates. The achievable sum rate is given in Theorem 4.8. For comparison, we 152 considered the performance of time-sharing where the achievable sum rate is equal to the capacity of the individual (interference free) MIMO channel with full CSI at both transmitter and receiver, given by R sum = M X `=1 log(1 +P ` ` (F H kk F kk )); (4.137) whereP ` is obtained via water-lling over the eigenvalues ofF H kk F kk , denoted by ` (F H kk F kk ), such as P ` = 1 ` (F H kk F kk ) + ; (4.138) with is chosen to satisfy the total power constraint P M `=1 P ` = 2P sum . Again, with time-sharing the per-slot power constraint is 2P sum . For PCoF with CIA, we used the same construction method in (4.136) forM = 2. Also, we considered the case withM = 3 and in this case, the alignment conditions are given by F 11 v 1;2 = F 12 v 2;1 F 11 v 1;3 = F 12 v 2;2 F 21 v 1;1 = F 22 v 2;1 F 21 v 1;2 = F 22 v 2;2 : For the simulation, we used the following precoding matrices to satisfy the above condi- tions: V 1 = F 1 21 F 22 F 1 12 F 11 1 1 F 1 11 F 12 F 1 22 F 21 1 (4.139) V 2 = F 1 12 F 11 1 F 1 22 F 21 1 : (4.140) Fig. 4.7 shows that PCoF with CIA provides a higher sum rate than time-sharing, having a larger gap as SNR increases. 153 4.6 Proofs 4.6.1 Proof of Theorem 4.4 4.6.1.1 Achievable scheme We use the achievable rates given in Theorem 4.3. It is immediately shown that the achievable GDoF of message 1 (cognitive user), obtained by d 1 () = lim SNR!1 log(1 +jh 11 j 2 SNR) logSNR = 1: (4.141) In this proof, we show that message 2 (non-cognitive user) achieves GDoF equal to , by carefully choosing the power scaling factor 2P. The eective channel for Scaled PCoF is given by ~ h() = [h 21 p INR;(h 22 p SNR 1;MMSE (h 12 h 21 =h 11 )SNR 1 2 )] and can be rewritten as ~ h() =SNR =2 [h 21 ; ~ h 22 ]; (4.142) where ~ h 22 =h 22 SNR (1)=2 hSNR (1)=2 andh = 1;MMSE (h 12 h 21 =h 11 ). Here, we choose = ? ,h 21 =( ~ h 22 ), where 1 is an integer with =djh 21 = ~ h 22 je2Z + . This produces a kind of \aligned" channel: ~ h =SNR =2 [h 21 ;h 21 = ]: (4.143) Letting b 1 = , and b 2 = 1, the eective noise in (4.46) is obtained by z e ( ~ h;b) = h 21 SNR =2 z 2 : (4.144) 154 This shows that non-integer penalty is completely eliminated. Also, we can use the zero forcing precoding over F q since the chosen integer coecients b 1 = and b 2 = 1 are non-zero. From this, we have the lower bound on the achievable rate of Scaled PCoF: max R 2 ()R 2 ( ) = log(jh 21 j 2 SNR) 2 log( ): (4.145) The lower and upper bounds on is given by 1 1 + h 21 h 22 SNR (1)=2 hSNR (1)=2 ; (4.146) where converges to a const as SNR!1. Finally, the achievable GDoF of the non- cognitive transmitter is derived as d 2 () lim SNR;INR!1 R 2 ( ) logSNR =: (4.147) From (4.141) and (4.147), the achievable sum GDoF is 1 +. 4.6.1.2 Converse For given ratesR 1 andR 2 , we deneR min = minfR 1 ;R 2 g andR 4 = maxfR 1 ;R 2 gR min . If R 1 > R 2 then W 1 = (M 1 ;M 4 ) and W 2 = (M 2 ;0). In the reverse case, we have that W 1 = (M 1 ;0) and W 2 = (M 2 ;M 4 ). In both cases, the non-cognitive transmitter knows the linear combination,W 1 W 2 = (M 1 M 2 ;M 4 ). From the well-known Crypto Lemma, theM 1 M 2 is mutually statistically independent ofM 1 , as well asM 1 M 2 is mutually 155 statistically independent of M 2 . In this proof, we derive the upper bounds on R min and R max =R min +R 4 . First, we derive the upper bound on the minimum rate R min : nR min = H(M 1 ) =H(M 1 jM 1 M 2 ;M 4 ) = H(M 1 jM 1 M 2 ;M 4 )H(M 1 jY n 1 ;M 1 M 2 ;M 4 ) +H(M 1 jY n 1 ;M 1 M 2 ;M 4 ) (a) I(M 1 ;Y n 1 jM 1 M 2 ;M 4 ) +n n = h(Y n 1 jM 1 M 2 ;M )h(Y n 1 jM 1 M 2 ;M ;M 1 ) +n n (b) = h(Y n 1 jX n 2 ;M 1 M 2 ;M )h(Y n 1 jX n 1 ;X n 2 ;M 1 M 2 ;M ;M 1 ) +n n h(Y n 1 jX n 2 )h(Y n 1 jX n 1 ;X n 2 ;M 1 M 2 ;M ;M 1 ) +n n (c) = h(Y n 1 jX n 2 )h(Y n 1 jX n 1 ;X n 2 ) +n n = I(X n 1 ;Y n 1 jX n 2 ) +n n n log(1 +jh 11 j 2 SNR) +n n ; where (a) follows from the Fano's inequality and data processing inequality as H(M 1 jY n 1 ;M 1 M 2 ;M 4 )H(M 1 jY n 1 )H(M 1 j ^ M 1 )n n ; (4.148) (b) follows from the fact that encoder 2 has (M 1 M 2 ;M ), therefore for any coding scheme X n 2 is a function of (M 1 M 2 ;M ), and (c) follows from the fact that there is a Markov chain (M 1 ;M 2 ;M 4 )! (X n 1 ;X n 2 )!Y n 1 : 156 In the same manner, we get: nR min = H(M 2 ) =H(M 2 jM 1 M 2 ;M 4 ) = H(M 2 jM 1 M 2 ;M 4 )H(M 2 jY n 2 ;M 1 M 2 ;M 4 ) +H(M 2 jY n 2 ;M 1 M 2 ;M 4 ) I(M 2 ;Y n 2 jM 1 M 2 ;M 4 ) +n n h(Y n 2 jX n 2 )h(Y n 2 jX n 1 ;X n 2 ) +n n = I(X n 1 ;Y n 2 jX n 2 ) +n n n log(1 +jh 21 j 2 INR) +n n : From (4.148) and (4.149), we have R min minflog(1 +jh 11 j 2 SNR); log(1 +jh 21 j 2 INR)g: (4.149) An obvious upper bound on R 1 and R 2 are given by nR 1 I(X n 1 ;X n 2 ;Y n k ) +n n (4.150) n log(1 +jh 11 j 2 SNR +jh 12 j 2 INR) +n n (4.151) nR 2 I(X n 1 ;X n 2 ;Y n 2 ) +n n (4.152) n log(1 +jh 21 j 2 INR +jh 22 j 2 SNR) +n n : (4.153) Since R min +R 4 = maxfR 1 ;R 2 g, we have: R max maxflog(1 +jh 11 j 2 SNR +jh 12 j 2 INR); log(1 +jh 21 j 2 INR +jh 22 j 2 SNR)g: (4.154) 157 Using (4.149), (4.154), and INR = SNR , we have the upper bounds in the asymptotic case: lim SNR!1 R min logSNR + R max logSNR minf1;g + maxf1;g; (4.155) yielding the upper bound on the sum symmetric GDoF as d sum = lim SNR!1 R min +R max logSNR 1 +: (4.156) This completes the proof. 4.6.2 Proof of Corollary 4.1 For the DoF proof, we assume that P sum goes to innity and equivalently, SNR goes to innity. In this proof, we will show that the individual messages rate R (equal for all messages) grows as logSNR, i.e., lim SNR!1 R logSNR = 1: (4.157) Assuming that we use exact IFR (see Section 4.1.1, eq. (4.5)), we have: R comp (H k C k ;B k ;SNR) = log(SNR) max ` log k(H 1 k ) H b k;` k 2 : (4.158) This denitely shows that lim SNR!1 Rcomp(H k C k ;B k ;SNR) logSNR = 1. Accordingly, R grows as logSNR. However, H k must be full-rank in order to allow exact IFR. Since H k =F k1 V 1 fork = 1; 2 andH k =F k3 V 3 fork = 3; 4, we need to show thatV 1 andV 3 are full rank. For the alignment, we use the following construction method proposed in [GJW + 12]: v 1;`+1 = (F 1 11 F 12 F 1 22 F 21 ) ` v 1;1 (4.159) v 2;` = (F 1 22 F 21 F 1 11 F 12 ) `1 F 1 22 F 21 v 1;1 ; (4.160) 158 for ` = 1;:::;M 1. Once v 1;1 is determined, other vectors are completely determined by the above equations. As argued in [GJW + 12], since the channel matrices are drawn form a continuous distribution then F, (F 1 11 F 12 F 1 22 F 21 ) has all distinct eigenvalues almost surely. From [HJ12, Thm 1.3.9], F is diagonalizable such as F =E 2 6 6 6 6 4 1 . . . M 3 7 7 7 7 5 E 1 ; where thei-th column ofE is an eigenvector ofF associated with i . Choosingv 1;1 =E1, the alignment precoding matrix V 1 can be rewritten as V 1 =E 2 6 6 6 6 4 1 1 M1 1 . . . . . . . . . . . . 1 M M1 M 3 7 7 7 7 5 | {z } =J ; (4.161) where J denotes the Vandermonde matrix. Therefore, the determinant of the V 1 is computed by det(V 1 ) = det(E)det(J) (4.162) = det(E) Y 1ijM ( j i )6= 0: (4.163) This shows that V 1 is full rank . With the same procedure, we can show that V 3 is full rank. 4.6.3 Proof of Theorem 4.7 We prove that PCoF with CIA and DPC can achieve the optimal 2M DoF. This scheme can be regarded as MIMO extension of Scaled PCoF and DPC proposed in Section 4.2.3. Consider the MIMO IC in (4.57). Letfw k;` : ` = 1;:::;Mg denote the independent 159 messages to be intended for receiver k, for k = 1; 2. Without loss of generality, it is assume that transmitter 2 knows the W 1 W 2 . Our achievable scheme proceeds as follows. Encoding: Transmitter 1 independently produces theM lattice codewordst 2;` =f(w 1;` w 2;` ) for ` = 1;:::;M and transmits the channel input: X 0 2 =VX 2 ; (4.164) where X 2 = [T 2 +D 2 ] mod . Transmitter 2 performs the DPC using the known interference signal F 12 X 0 2 to get: X 1 = T 1 F 1 11 F 12 X 0 2 +D 1 mod ; (4.165) where T 1 = f(W 1 ) denotes the lattice codewords corresponding to precoded messagesW 1 . Decoding: Receiver 1 performs the modulo-lattice mapping as ^ Y 1 = [F 1 11 Y 1 D 1 ] mod that yields: ^ Y 1 = [T 1 T 1 +X 1 +F 1 11 F 12 X 0 2 D 1 +F 1 11 Z 1 ] mod = [T 1 +X 1 [T 1 F 1 11 F 12 X 0 2 +D 1 ] mod +F 1 11 Z 1 ] mod = [T 1 +F 1 11 Z 1 ] mod : This shows that receiver 1 has interference-free channel, thus achieving theM DoF. 160 Receiver 2 applies the CoF receiver mapping (4.3) with integer coecients B H = I I and scaling factor F 1 21 , yielding ^ Y 2 = F 1 21 Y 2 D 1 D 2 = T 1 +T 2 + ((F 1 21 F 22 F 1 11 F 12 )VI)X 2 +F 1 21 Z 2 mod = 2 6 4B H 2 6 4 T 1 T 2 3 7 5 + (IB 1 )[T 1 +D 1 ] + (F 1 21 ~ F 22 VB 2 )X 2 +F 1 21 Z 2 3 7 5 mod (a) = [T 1 +T 2 +F 1 21 Z 2 ] mod = [f(W 1 ) +f(W 1 ) +f(W 2 ) +F 1 21 Z 2 ] mod (b) = [f(W 2 ) +F 1 21 Z 2 ] mod ; where (a) is due to the fact that the precoding matrix is chosen as V = (F 1 21 F 22 F 1 11 F 12 ) 1 and (b) follows the precoding over F q using the coecient (1) at the cognitive transmitter . Then, receiver 2 can achieve the M DoF. 161 Chapter 5 Conclusions In this thesis, we demonstrated that structured codes have a crucial role in network information theory. First, we studied a cooperative distributed antenna system (a.k.a., cloud radio access network). For the DAS uplink, we proposed system optimization based on network decomposition and on greedy selection of the antenna terminals for a given set of desired active user terminals. For the DAS downlink, we proposed a novel precoding scheme referred to as Reverse Compute and Forward (RCoF). This scheme reverse the role of ATs and UTs with respect to the CoF for the uplink, and uses linear precoding over the nite eld domain in order to eliminate multiuser interference. In this case, we considered system optimization consisting of selection a subset of UTs for a given set of active ATs. It turns out that in this case the problem can be formulated as the maximization of a linear function subject to a matroid constraint, for which a simple greedy procedure is known to be optimal. We also considered strategies that incorporate the presence of a ADC at the receiver as an unavoidable part of the channel model. In this case, we can design lattice based strategies that explicitly take into account the presence of the nite resolution scalar quantization at the receivers. In particular, this leads to very simple single-user linear coding schemes over F q with q = p 2 , and p a prime. Our own results in [HC11] and others' results in [TNBH12,FSK10] show that it is possible to approach the theoretical performance of random coding usingq-ary LDPC codes with linear complexity in the code block length and polynomial complexity in the number of network nodes. We 162 provided extensive comparison of the proposed lattice-based strategies with information- theoretic strategies for the DAS uplink and downlink, namely QMF and CDPC, known to be near-optimal. We observed that the proposed strategies achieve similar and sometimes better performance in certain relevant regimes, while providing a clear path for practical implementation, while the information-theoretic terms of comparisons are notoriously dicult to be implemented in practice. As a matter of fact, today's technology relies on the widely suboptimal decode and forward (DF) scheme for the uplink, or on the compressed linear beamforming approach for the downlink, which are easily outperformed by the proposed schemes with similar, if not better, complexity. We wish to point out that the proposed schemes are competitive when the wired backhaul rateR 0 is a limiting factor of the overall system sum rate. For example, in a typical home Wireless Local Area Network setting, the rates supported by the wireless segment are of the order of 10 to 50 Mbits/s, while typical DSL connection between the wireless router and the DSL central oce (playing the role of the CP in our scenario) has rates between 1 and 10 Mb/s. In this case, the schemes proposed in this paper can provide a viable and practical approach to uplink and downlink centralized processing at manageable complexity. Second, we considered \virtual" full-duplex relaying by means of half-duplex relays. This scheme can be seen as an information theoretic version of several full-duplex relay proposals implemented in hardware, by using two or more antennas in the same node. While the use of two or more antennas in full-duplex hardware is motivated by the neces- sity of creating sucient attenuation of the self-interference in the RF (analog) domain, such that the transmit signal does not saturate the receiver ADC, in our setting we as- sume that such attenuation is always large enough due to the fact that the two antennas at the two half-duplex relays forming one full-duplex relaying stage are physically sepa- rated. In contrast, while self-interference cancellation in the same full duplex device is not subject to a power constraint and can always be done, at least in principles, in the separated \virtual" scheme the inter-relay interference must be handled by appropriate coding and decoding techniques, subject to the transmit power constraint of each node. 163 In this work, we have considered several previously proposed techniques and have char- acterized their performance in this specic context. In particular, we obtained simple cut-set upper bounds for both the 2-hop and the multi-hop relay networks. This bound is tight and is achieved by DPC cancellation from the source for SNR 1. We showed that both lattice-based Compute and Forward (CoF) and Quantize reMap and Forward (QMF) yield attractive performance and can be easily implemented in the 2-hop net- work. In particular, QMF in this context does not require \long" messages and joint (non-unique) decoding, if the quantization mean-square distortion at the relays is chosen appropriately. In the multi-hop case, the gap of QMF from the cut-set upper bound grows logarithmically with the number of stages, and not linearly as in the case of \noise level" quantization. Furthermore, we have shown that CoF is particularly attractive in the case of multi-hop relaying, when the channel gains have controlled uctuations not larger than 3dB, yielding a rate that essentially does not depend on the number of relaying stages. We would like to conclude with an observation of possible practical interest. A widely accepted and on-going trend in the next generation of wireless networks (generally re- ferred to as 5G) considers the use of higher and higher frequency bands (mm-waves). At these frequencies, attenuation and non-line of sight propagation represent a signicant impairment for coverage. Hence, a dense deployment of small cells is envisaged, to han- dle low-mobility and high capacity trac. While blanketing a large area with tiny cells operating at high frequencies (e.g., 20 to 60 GHz [RGBD + 13, RSM + 13]) will certainly yield very large area spectral eciency, the cost of providing wired backhaul links to such a dense deployment may be prohibitive, especially in areas where ubiquitous ber is not already deployed. In this case, wireless backhaul is a cost-eective attractive option. We believe that the multi-hop virtual relaying network studied here can be applied, as a guiding principle, to the implementation of a wireless backhaul formed by multiple virtual full duplex relaying stages operating in line of sight to each other, such that the channel coecients can be accurately learned and the link attestations can be balanced such that the CoF scheme becomes very ecient (e.g., as in Fig. 3.9). 164 Lastly, we studied a number of two-user interference networks with coordination, cognition, and two hops. For the interference network with cognition (called network- coded cognitive interference channel), we developed an achievable region based on a few novel concepts: Precoded Compute and Forward (PCoF) with Channel Integer Align- ment (CIA), combined with standard Dirty-Paper Coding. We also developed a capacity region outer bound and found the sum symmetric Generalized Degrees of Freedom. Via the GDoF characterization, we showed that knowing \mixed data" together with exploit- ing structured codes provides an unbounded gain over the classical counterpart. Then, we considered a Gaussian relay network having the two-user MIMO IC as the main building block. We used PCoF with CIA to convert the MIMO IC into a deterministic nite-eld IC. Then, we employed a linear precoding scheme over the underlying nite-eld to elimi- nate interference in the nite-eld domain. Using this unied approach, we characterized the symmetric sum rate of the two-user MIMO IC with coordination, cognition, and two hops. We also provided nite-SNR results (not just degrees of freedom) which show that the proposed coding schemes are competitive against the state-of-the art interference avoidance based on orthogonal access, for standard randomly generated Rayleigh fading channels. 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Hong, Song-Nam
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Structured codes in network information theory
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Viterbi School of Engineering
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Doctor of Philosophy
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Electrical Engineering
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07/15/2014
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07/14/2014
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coding theory
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network information theory
structured codes
wireless networks