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Design and analysis of high-performance cooperative relay networks
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Design and analysis of high-performance cooperative relay networks
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DESIGN AND ANALYSIS OF HIGH-PERFORMANCE COOPERATIVE RELAY NETWORKS by Wan-Jen Huang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2008 Copyright 2008 Wan-Jen Huang Dedication This dissertation is dedicated to my beloved family. ii Acknowledgement January 2008 is definitely a memorable month for me. It is because that I passed my defense exam and finished my PhD dissertation eventually!! Along the way toward com- pleting my degree, I am very lucky to have various supports and encouragement from many people around me. First of all, I would like to express the most sincere thankful- nesstomyadvisor, Prof. JayKuo, forhisguidanceintheseyears. EventhoughIsuffered difficulties finding my research topic again and again, he still kept supporting me to con- quer the barrier. There is one more person plays an important role in my dissertation– Prof. Peter Hong is a great teacher as well as a helpfulfriendforme. Hispositive person- ality and rigorous attitude toward research inspire me a lot. Besides, I truly appreciate Prof. Xiaoli Yu for her kind help and encouragement, and Prof. Michael Neely for his helpful suggestions on my dissertation. I also sincerely thank Prof. Fengzhu Sun and Prof. Keith Chugg for their generous help on joining the committee of my qualifying exam and defense exam. Duringtheseyears, Iamsogladtomeetmanygreat lab-matesinourgroup,including Fu-Hsuan, Roger, Ronald, Layla, Feng-Tsun, Hsao-Hsuan, Lawrence, Simon,Usman..etc. Thank you, buddies, for the wonderful memories in USC. I especially thank Fu-Husan for his patient mentoring and sharing his experiences. I am also grateful to my friends, iii Grace, Shao-Chen, Chao, Yi-Ping, Yueh-Cheng, Wan-Jhu, and Margaret. With you, my life in LA is more joyful and colorful. Finally, Iwould like to thank my beloved family with the highest gratefulness. Thank you, dad and mom, for your huge support and consideration at all times. Thank you, Wei-Feng!! You always stand by me, and gave me more courage to overcome those frustrations. I am so lucky to have your accompany along the road pursuing our dreams. iv Table of Contents Dedication ii Acknowledgement iii List Of Figures viii Abstract xi Chapter 1: Introduction 1 1.1 Significance of the Research . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Review of Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Contribution of the Research . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 2: Background Review 10 2.1 Fundamental of Cooperative Communications . . . . . . . . . . . . . . . . 10 2.2 Power allocation in resource-constrained cooperative networks . . . . . . . 13 2.2.1 Three-Node Relay Networks . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1.1 Case I: Nodes with Full CSI . . . . . . . . . . . . . . . . 14 2.2.1.2 Case II: Nodes with Partial CSI . . . . . . . . . . . . . . 18 2.2.2 Dual-Hop Relay Networks . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2.1 Case I: Full CSI at Relay and Destination Nodes . . . . . 20 2.2.2.2 Case II: Channel Gain Known to Relays and Full CSI at Destination . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.2.3 Case III: Partial CSI at Relays and Full CSI at Destination 25 2.2.3 Multi-Hop Relay Networks . . . . . . . . . . . . . . . . . . . . . . 26 Chapter3: ComparisonofPowerControlSchemesforEnergy-Constrained Relay Networks 28 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Power Control Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.1 Three Basic Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.2 Performance Comparisons of three power control schemes . . . . . 34 3.4 Power control with consideration of limited battery energy . . . . . . . . . 37 v 3.4.1 Power Saving Strategy for Relay Nodes . . . . . . . . . . . . . . . 38 3.4.2 Performance Comparison with Energy Constraint . . . . . . . . . . 39 3.5 Optimality of the opportunistic scheme . . . . . . . . . . . . . . . . . . . 44 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.7 Appendix : Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . 48 Chapter 4: Lifetime Maximization for Amplify-and-Forward Cooperative Networks 50 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Network Lifetime Maximization . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Discrete Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.5 Markov Analysis of Network Lifetime . . . . . . . . . . . . . . . . . . . . . 63 4.5.1 Performance Analysis of the Proposed Strategies . . . . . . . . . . 65 4.5.2 Optimal Selection Strategy with Global CSI . . . . . . . . . . . . . 67 4.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.6.1 Continuous Power Allocation . . . . . . . . . . . . . . . . . . . . . 68 4.6.2 Discrete Power Allocation and Peak Power Constraint . . . . . . . 73 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Chapter5: Relay-AssistedDecorrelatingMultiuserDetector(RAD-MUD) for Cooperative CDMA Networks 78 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2.1 Phase I : Direct Transmission Phase . . . . . . . . . . . . . . . . . 82 5.2.2 Phase II : Cooperative Transmission Phase . . . . . . . . . . . . . 83 5.3 Relay-Assisted Decorrelating Multiuser Detector (RAD-MUD) . . . . . . 85 5.4 MMSE Multiuser Receivers and Signal Combining . . . . . . . . . . . . . 88 5.4.1 MMSE Multiuser Detection at Relays . . . . . . . . . . . . . . . . 89 5.4.2 MMSE Signal Combining at Destination . . . . . . . . . . . . . . . 90 5.5 Cooperative Transmission Strategies . . . . . . . . . . . . . . . . . . . . . 95 5.5.1 Transmit Beamforming . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5.1.1 Beamforming for Relay-Destination (R-D) Channels . . . 96 5.5.1.2 BeamformingforSource-Relay-Destination(S-R-D)Chan- nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5.2 Selective Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.5.2.1 Threshold Selection Strategy . . . . . . . . . . . . . . . . 97 5.5.2.2 Best Selection Strategy . . . . . . . . . . . . . . . . . . . 99 5.5.3 Distributed Space-Time Coding (DSTC) . . . . . . . . . . . . . . . 99 5.6 Numerical Simulation and Discussion . . . . . . . . . . . . . . . . . . . . . 100 5.6.1 Cases Without Direct Links . . . . . . . . . . . . . . . . . . . . . . 100 5.6.2 Cases With Direct Links Combined. . . . . . . . . . . . . . . . . . 106 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.8 Appendix: Derivation of Beamforming Coefficients in (5.25) . . . . . . . . 112 5.9 Appendix: Diversity of Best Selection Strategy . . . . . . . . . . . . . . . 113 vi Chapter 6: Relay-Assisted Multiuser Detection for Amplify-and-Forward Asynchronous Cooperative Uplink Networks 117 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3 Relay-Assisted Decorrelation at Relays . . . . . . . . . . . . . . . . . . . . 123 6.4 MMSE Multiuser Detection at Receivers . . . . . . . . . . . . . . . . . . . 126 6.4.1 MMSE Estimation at Relays . . . . . . . . . . . . . . . . . . . . . 126 6.4.2 MMSE Multiuser Detection at Destination . . . . . . . . . . . . . 128 6.5 Cooperative Transmission Strategies . . . . . . . . . . . . . . . . . . . . . 131 6.5.1 Transmit Beamforming . . . . . . . . . . . . . . . . . . . . . . . . 132 6.5.2 Selective Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.5.3 Distributed Space Time Coding . . . . . . . . . . . . . . . . . . . . 133 6.6 Performance Comparison and Computer Simulations . . . . . . . . . . . . 135 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Chapter 7: Conclusion and Future Work 141 7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Bibliography 145 vii List Of Figures 2.1 A three-node cooperative network model. . . . . . . . . . . . . . . . . . . 10 2.2 A dual-hop cooperative network. . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Comparison of outage probabilities of AF and DF schemes in the three- node network with equal and optimal power allocation. . . . . . . . . . . 18 3.1 A system model of the proposed cooperative relay network. . . . . . . . . 30 3.2 Comparison of the BER performance as a function of the relay node num- ber with different power control schemes. . . . . . . . . . . . . . . . . . . 35 3.3 Comparison of the BER performance as a function of the total transmit power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Comparison of the averaged transmit power of the opportunistic scheme with power saving as a function of the relay node number. . . . . . . . . . 41 3.5 The averaged relay lifetime of the opportunistic scheme with power saving as a function of the relay node number. . . . . . . . . . . . . . . . . . . . 42 3.6 ComparisonoftheaveragedBERperformanceandrelaylifetimeforvarious power control schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 Long-term BER performance for different power control schemes. . . . . . 44 3.8 The averaged transmission power versus the number of relay nodes.. . . . 47 4.1 The system model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 The state transition diagram of an energy-consuming process. . . . . . . . 64 4.3 The averaged network lifetime versus the initial energy of each relay for cooperative network with six relays. . . . . . . . . . . . . . . . . . . . . . 69 viii 4.4 The averaged transmission power versus the number of relays for a fixed aggregate initial energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.5 The averaged wasted energy versus the number of relays for a fixed aggre- gate initial energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.6 The averaged network lifetime versus the number of relays for a fixed ag- gregate initial energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.7 The averaged network lifetime of selective strategies with discrete power allocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.8 The averaged network lifetime of Strategies MTP, MEI and MOP. . . . . 74 4.9 Lifetime achieved with strategy MEI for different L. . . . . . . . . . . . . 75 4.10 Average lifetime of a network with non-identical channel statistics. . . . 76 5.1 Illustration of a cooperative CDMA uplink system with K sources and L relays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 The block diagram of the RAD-MUD system. . . . . . . . . . . . . . . . . 87 5.3 BER performance comparison for different precoding and MUD strategies at relays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.4 BER performance comparison for systems employing transmit beamform- ing without MMSE combining at the destination. . . . . . . . . . . . . . . 103 5.5 BER performance comparison for systems employing selective relaying without MMSE combining at the destination. . . . . . . . . . . . . . . . . 104 5.6 BERperformancecomparisonforsystemsemployingDSTCwithoutMMSE combining at the destination. . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.7 BER performance comparison for systems employing transmit beamform- ing and performing MMSE combining at the destination. . . . . . . . . . 107 5.8 BERperformancecomparisonforsystemsemployingselective relayingand performing MMSE combining at the destination. . . . . . . . . . . . . . . 108 5.9 BER performance comparison for systems employing DSTC performing and MMSE combining at the destination. . . . . . . . . . . . . . . . . . . 109 ix 5.10 BERcomparisonofthecase withtwo usersandtwo relayswithcorrelation coefficients ρ=0.75 and 0.25 . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.11 TheBERperformanceasafunctionoftotal transmitpower forK =8 and L =2 (circles), 4 (triangles) and 8(stars). . . . . . . . . . . . . . . . . . . 111 5.12 PerformancecomparisonbetweenMMSEjointdecodingin(??)(thedashed line) and the alternative low-complexity MMSE combining in (??) (the solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1 The system model of a cooperative network with multiple users. . . . . . 121 6.2 The block diagram of RAD-MUD with zero-forcing precoding at relays. . 125 6.3 The BER performance comparison for the system using equally weighing factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.4 The BER performance comparison for the system using transmit beam- forming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.5 The BER performance comparison for the system using selective relaying. 139 6.6 The BER performance comparison for the system using DSTC. . . . . . . 140 x Abstract Cooperative communications have attracted a considerable amount of attention in recent years by exploiting the spatial diversity gain inherent in a multi-user wireless system withouttheneedofmultipleantennasateachnode. Inthisdissertation,weconsiderways to enhance the performance of energy-constrained cooperative networks in two aspects. First, we devise power allocation strategies to maximize the lifetime of the amplify- and-forward (AF) cooperative network. Based on selective relaying, we propose three strategies that take into account the local channel state information (CSI) and the local residual energy information (REI) at each relay to prolong the network lifetime. With a finite number of power levels, the energy dissipation process can be modeled as a finite-state Markov chain and the optimal lifetime maximization strategy can be derived using dynamic programming. We demonstrate that the network lifetime can be extended considerably by exploiting both CSIandREI via numerical simulation. The performance of the proposed strategies that utilize only local CSI and REI is shown to be comparable to that of the optimal strategy that demands global CSI and REI. Second, we examine the uplink of a synchronous cooperative CDMA network, where users cooperate by relaying each other’s messages to the base station. When spreading waveforms are not orthogonal, multiple access interference (MAI) exists at relays and the xi destination, causing the cooperative diversity gain to diminish. To address this problem, we propose the relay-assisted decorrelating multiuser detector (RAD-MUD) to separate interfering signals at the destination with the help of precoding at relays along with pre- whitening at the destination. Three cooperative transmission strategies are considered on top of RAD-MUD; namely, transmit beamforming, selective relaying and distributed space-time coding. Since the reliability of each source-relay and/or relay-destination links is different, relay transmissions are weighted accordingly to further combat MAI. The RAD-MUD system is extended to cooperative networks, where asynchronism of transmissions among the source nodes and relay nodes is taken into consideration. The advantages of RAD-MUD over existing cooperative MUD schemes are verified by com- puter simulation. xii Chapter 1 Introduction 1.1 Significance of the Research The multiple-input multiple-output (MIMO) techniques have been proposed to exploit spatial and temporal diversities to combat fading in a wireless environment [17]. How- ever, emerging wireless applications such as sensor and wireless mesh networks have an increasing demand for small and low cost devices that are densely deployed over a wide area. In this case, the use of cooperation among simple and constrained users for wireless messagetransmissionbecomesanattractive alternative. Cooperativecommunications, as firstproposedin[38,49,51,52],exploitthespatialdiversityinherentinmultiusersystems by allowing users with diverse channel qualities to cooperate and relay each other’s mes- sagestowards thedestination. Sinceeach transmittedmessageispassedthroughmultiple independent relay paths, the probability that the message fails to reach the destination is significantly reduced. 1 In energy-constrained wireless systems such as sensor networks, the replacement of batteries couldbeexpensiveoreven prohibitive. Thus, theproblemoflifetime maximiza- tion becomes increasingly important. For this reason, it is often desirable to exploit the knowledge of the channel state information (CSI) to redistribute the resource and/or the traffic load to improve communication efficiency. Based on different network topologies and cooperation methods, optimal resource allocation policies have been derived under various performance criteria and system constraints [23], where significant gains in per- formance have been observed. However, most of existing work has focusedon minimizing thetransmissionpowertomeettheQoSconstraintatthedestinationwithoutconsidering the residual battery energy at each node. Without balanced energy usage among users, the network may become non-functional even when some users still have a large amount of battery energy remaining. Hence, one open research problem is how to maximize the lifetime of cooperative relay networks by considering CSI as well as the residual energy information (REI) at intermediate nodes. Thespatial diversity gainof acooperative systemhasbeenstudiedextensively forthe case with a single source user or multiple source users where each transmits through an orthogonal channel [19, 38, 40]. However, the orthogonal assumption is difficult to meet in practice. It is not obvious whether cooperation is still advantageous in the presence of multiple access interference (MAI). To answer this question, we investigate a cooperative CDMA uplink system where users cooperate by relaying each other’s messages to the base station and relays are allowed to forward the message from more than one source to the base station. Spreading codes used to transmit and forward source signals may not be perfectly orthogonal. To mitigate MAI, multiuser detection (MUD) techniques 2 [60], which were studied for conventional non-cooperative uplink systems in the past, are adopted at both relays and the base station. To take advantage of cooperative networks, one key idea is to design an appropriate precoder at relays to mitigate MAI using CSI and correlation properties among spreading codes. 1.2 Review of Previous Work Several cooperation strategies with different relaying techniques [38], including amplify- and-forward(AF) anddecode-and-forward(DF), have beenstudiedto acquirethespatial diversity gain in wireless networks. To deal with the fading effect and enhance power efficiency, user cooperation has been introduced in wireless sensor networks (WSNs) [53, 50]. That is, a sensor node located between the transmit node and the receive node is chosen as a relay node that cooperates with the transmit node in packet forwarding. A multi-hop AF cooperative relay scheme was developed and analyzed in [53], where a significant gain in the network lifetime due to node cooperation was demonstrated. Besides, the gain increases with the node density. For a given outage probability, the energy consumption of AF and DF cooperative relay schemes was compared with that of a direct transmission scheme in [50]. For a Rayleigh fading channel model, it was shown that cooperative relaying is more energy efficient when the distance between nodes is large. Consider the scenario that a group of users serve as relays that forward messages from a specific source node to the destination at each time instance. These relays form a distributed antenna array to achieve the diversity gain similar to a multiple-input 3 multiple-output(MIMO)system. Techniquesusedinmultipleantennasystemshavebeen examined in cooperative networks to improve bandwidth efficiency, such as beamforming [1, 18], space-time coding [32, 39] and selective relaying [6, 8, 67]. Among them, selective relaying (also referred to as cooperative relaying) is attractive due to its simplicity and ease of implementation. Only the relay with the best channel condition is chosen to transmit in selective relaying. It is proven to achieve full diversity and be optimal under different optimizing criteria, e.g. [8, 23, 67]. Furthermore, it only demands local CSI at each relay, and can be conducted in a distributed manner [6]. Lifetime maximization has been considered in wireless cooperative networks by sev- eral researchers in the past. With the statistical knowledge of CSI, a suboptimal power allocation algorithm wasproposedin[22]to maximize theaverage network lifetime which is defined as the minimum lifetime among all nodes in a network. To improve the device lifetime, relays can be deployed and dedicated to message-forwarding in a cooperative network. A recursive algorithm to determine the position of relays to maximize the net- work lifetime was studied in [22]. To maximize the lifetime of wireless sensor networks with quasi-static channel fading, one may exploit the instantaneous CSI knowledge and the residual energy information (REI) of sensor nodes e.g. [10, 11, 13]. Sensor node scheduling based on the max-min residual energy [11] or the maximum energy efficiency index [10] (i.e., the ratio between the residual energy and the transmit power) results in longer network lifetime as compared with that using REI or CSI alone. The scheduling to maximize network lifetime can be conducted in a distributed manner by opportunistic carrier sensing. 4 Withtheglobal knowledge ofCSIandREI,theevolution ofuser’sresidualenergycan be modeled as a finite state Markov chain, where each state records the REI of all users [13]. The problem of maximizing the network lifetime can be modeled as a stochastic shortest path problem. However, the computational complexity grows dramatically with the number of power levels, the number of relays and the initial battery energy values. An upper bound of the maximum expected lifetime was derived in [13] at much lower complexity, and the scheduling strategy using the energy efficiency index was shown to have an expected lifetime comparable with the upper bound. MUD techniques have been developed to mitigate MAI in CDMA uplink systems. Several famous MUD schemes include maximum likelihood (ML) MUD [59, 60], decor- relating MUD [43, 44] and minimum-mean-square-error (MMSE) MUD [45, 34]. MUD is examined in the context of cooperative CDMA networks in recent years. Sets of two userswere treated asa cooperating pairin [9, 58], whereMUD is usedat relays to decode messages from a single source of interest. With full CSI knowledge, user cooperation is usefulinreducingthe biterrorrate (BER) whenthe MMSElinear detector [9]is adopted at the relay. In a more practical setting where each relay has only partial CSI and the spreading waveform of its partner while the base station has CSI of connected links, blind MUD can be deployed at both relays and the base station as proposedin [58]. This scheme achieves the maximal diversity gain for blind detection with a large sample size. 5 1.3 Contribution of the Research Several important contributions have been made in this research. They are highlighted below. • Three power allocation schemes for a space-time coded AF relay system are com- pared, wherepowerallocation dependson thelinkquality ofwireless channels. Our main focus is their impact on the power consumption of relay nodes. Among these three schemes, the opportunistic scheme aided by SNR-constrained power reduc- tion gives the best performance in terms of both the BER and network lifetime. Moreover, the energy efficiency of the opportunistic scheme (which is also referred to the selective relaying) is demonstrated. Basically, it demands the minimal total relay power subject to the SNR requirement at the destination. • To maximize network lifetime, three power allocation strategies that exploit the channel state information (CSI), the residual battery energy and the QoS require- ment are proposed under the setting of AF selective relaying. It is shown that the network lifetime can be extended considerably by taking all these three factors into account. • We provide a practical way to measure the lifetime of cooperative networks. In the sensor network literature, the network lifetime is mostly defined as the duration of time for which all sensors are active. This definition may not be suitable for cooperative networks since the operability of the system is not governed by the life/deathofasingledevice. InChapter3,wemeasuretheoperabilityofthenetwork astheaveragedend-to-endoutageprobabilityfromuserstotheirdestination. Then, 6 even though the death of a node due to energy depletion will result in a loss in diversity, the desired QoS may be maintained for other nodes. • The average network lifetime of proposed strategies are analyzed with respect to discrete power allocation. By modeling the residual energy of each node as states of a Markov Chain, the averaged network lifetime is equivalent to the average time from the initial state to an absorption state for which the outage probability is no longer achievable. As compared with the optimal strategy obtained from solving a equivalent stochastic shortest path problem [13], which requires global CSI and REI, the performance loss of our proposed strategies that demand only local CSI and REI is negligible. • WeinvestigatetheuseofMUDandusercooperationintheuplinkofansynchronous CDMA network, where each relay is allowed to cooperate with multiple users si- multaneously and messages received from multiple sources are jointly processed at each relay. The MMSE MUD is applied at relays to mitigate MAI and detected symbols at relays are retransmitted after passing through the proposed precoder. To enhance the overall detection performance, signals transmitted by both source nodes and relays are combined at the destination using the MMSE criterion. • We propose relay-assisted decorrelating MUD (RAD-MUD) in cooperative uplink systems, where relay nodes perform one half of decorrelating operations while the destination performstheotherhalfofdecorrelating operations. Unlikeconventional zero-forcing (ZF)precoding[61]orthedecorrelating MUD [43], wheredecorrelating operations are conducted only at the transmitter or only at the receiver, there is 7 neither power expansion at transmitters nor noise amplification at the receiver in RAD-MUD. • Three cooperative transmission strategies are adopted on top of RAD-MUD. They are transmit beamforming, selective relaying and distributed space-time coding. Since the channel on each source-relay path is different, relay transmissions are weighted in our schemes to combat MAI. The advantages of RAD-MUD over ZF precoding and other existing cooperative MUD schemes are shown via computer simulations. • We examine asynchronous transmissions among source nodes and relay nodes since synchronism is difficult to achieve at all relay nodes. With asynchronous trans- mission, the inter-symbol interference (ISI) tends to degrade the detection per- formance. We extend RAD-MUD to the case of asynchronous transmissions and show that RAD-MUD with user cooperation outperforms synchronous directional transmission significantly. 1.4 Organization of the Dissertation The rest of the dissertation is organized as follows. The concept of user cooperation and power allocation strategies based on different cooperation strategies and network topolo- gies are reviewed in Chapter 2. Three power control schemes for AF space-time coded relay channel and the SNR-constrained power reduction are presented and compared in Chapter 3. Then, the optimality of selective relaying is shown in Sec. 3.5. With selective relaying, three relay strategies to maximize the network lifetime are proposed, compared 8 and analyzed in Chapter 4. In the uplink of a synchronous cooperative network, the use of cooperative transmissions on top of RAD-MUD is studied in Chapter 5. With asyn- chronous transmissions among sources and relays, RAD-MUD together with cooperative strategies is investigated in Chapter 6. Finally, concluding remarks are drawn and the future work is mentioned in Chapter 7. 9 Chapter 2 Background Review 2.1 Fundamental of Cooperative Communications Cooperative communications [51, 52, 38, 49] provide spatial diversity gains through the cooperationofmultipletransmittingterminals. Withcooperation, theusersthatmomen- tarily experience a deep fade in its link towards the destination can utilize the quality channels provided by its partners to achieve the desired Quality of Service (QoS). In this case, the probability that the terminals will simultaneously experience a bad channel will be significantly reduced and, thus, the energy or bandwidthresources required to achieve the desired QoS can be reduced. User 1 User 2 Destination Figure 2.1: A three-node cooperative network model. 10 In Fig. 2.1, we show a canonical example where two users cooperate in transmitting their messages to the destination. In the first transmission phase, the source (either user 1 or user 2) transmits its own message to the destination while the partner receives the message simultaneously with no extra cost due to the broadcast nature of wireless trans- mission. In the second phase, the users, serving as the partner to the source, relays the information that it received in the previous phase to the destination, where optimal com- bining is then performed for detection. The relaying strategies studied in the literature [51, 38] are mostly variants of Decode-and-Forward (DF) or Amplify-and-Forward (AF) methods, while other strategies such as Compress-and-Forward, [37] Selective Relaying, Incremental Relaying [38] or Coded Cooperations[31] have been proposed as well. In- stead of transmitting its own messages, each user expends part of its resources to relay its partners messages, thus, exploring the spatial diversity to enhance the users average performance, especially in the high SNR regime. The cooperation schemes can be easily extended to a large network where each user may have more than one cooperating partner. As shown in Fig. 2.2, each user utilizes the cooperating terminals to form a distributed antenna array which achieves a diversity gain that is proportionalto the numberof relays [39]; hence, for a fixedQoS requirement, such as the received SNR or the bit error rate, the total transmit power is inversely pro- portional to the number of relays [35]. Many cooperative strategies have been studied in the literature for different network models and system assumptions. We can categorize these work using the following few properties [c.f. Fig. 2.2]: (1) with/without direct transmission from the source; (2) with/without feedback from the receiver; and (3) mul- tiple access strategies, e.g. TDMA, FDMA or CDMA. Due to the difficulty of network 11 h S1 h 1D h 2D h ND h S2 h SN Source (S) Relay 1(R 1 ) Relay 2(R 2 ) Relay N(R N ) Destination(D) . . . . . . . . . . . . . . . X 0 Z X 1 X 2 X N U 1 =f 1 (X 1 ) U N =f N (X N ) U 2 =f 2 (X 2 ) (feedback) Figure 2.2: A dual-hop cooperative network. synchronization, several asynchronous cooperation schemes [49, 62] have been proposed as well. Foranetworkdeployedoverawidearea, thecooperationschemecanalsobeextended to include multi-hop transmissions by concatenating multiple layers of the cooperating relays shown in Figs. 2.1 and 2.2. When using cooperative multi-hop transmissions, the energy from both the source and the multiple relaying nodes are combined at the receivers, thus, reducing the overall transmit power as the network size increases [36, 24]. However, extending the cooperation over multiple hops necessarily complicates the problem and raises several challenges in the system design, such as routing, medium access control, channel estimation and the available diversity gains, which are subject to future investigation. In the following section, we show that, by providing the transmitters and relays with the channel state information (CSI), we are able to optimally allocate the energy or bandwidth resources and maximize the efficiency of the resource expenditure for each particular strategy. 12 2.2 Power allocation in resource-constrained cooperative networks An extensive review of power allocation methods under different network topologies, multiple access channels, cooperation methods and CSI assumptions discussed in the literature is given in this section. We first study the three-node topology as shown in Fig. 2.1, then the dual-hop topology as shown in Fig. 2.2 and finally a general multi-hop topology. When the CSI is not known to the transmitter, the spatial diversity gain is achieved by allowing users to have a fair share of each others’ resources. With the CSI knowledge, significant improvements in terms of BER, outage probability or capacity can be attained by applying optimal power allocation among cooperating nodes. 2.2.1 Three-Node Relay Networks Considerthe three-node relay network shown inFig. 2.1. Without loss of generality, user 1 is the source node (S) that intends to transmit a message to the destination (D) while user 2 serves as the relay node (R). In the first step, source S transmits symbol X S to both R and D. The received signals can be expressed as X R =h SR ·X S +W R and X D1 =h SD ·X S +W D1 , respectively, whereh SR andh SD arethechannelcoefficientsfortheS-RandtheS-Dlinks, and W R and W D1 denote the additive channel noise. In the second step, R transmits 13 symbol U = f(X R ) as a function of the received signal X R , while source S is silent. Consequently, the signal received at D can be written as X D2 =h RD ·U +W D2 =h RD ·f(X R )+W D2 , whereh RD isthechannelcoefficientbetweentheR-DpairandW D2 istheadditivechannel noise. In the following discussion, W R , W D1 and W D2 are assumed to be i.i.d. circularly symmetricadditivewhiteGaussiannoisewithvarianceN 0 =1. Thetransmittedmessages X S andU havethevariancesP S andP R ,respectively, whichrepresentsthepoweremitted by each node. The main objective is to determine the optimal allocation of P S and P R to maximize theQoSperformanceatD,subjecttothetotal powerconstraintP S +P R ≤P 0 . The optimal power allocation scheme depends on specific QoS measures such as the outage probability, capacity, SNR and BER. We consider cases with full and partial CSI separately. 2.2.1.1 Case I: Nodes with Full CSI When full CSI is available to S, R and D (i.e., complex coefficients h SR , h SD , h RD are known),thepoweremittedbyeachnodecanberedistributedtocompensatefornon-ideal channels. This problem has been studied for both DF and AF cooperation schemes and their solutions depend on whether the direct S-D link is taken into account (i.e., X D1 is combinedwithX D2 insignaldetection). IfbothX D1 andX D2 arecombinedfordetection 14 at the destination, it is referred to as the case with diversity. If only X D2 is considered, it is the case without diversity, which reduces to a simple multi-hop relay problem. We first examine the DF scheme that maximizes the channel capacity. If there is no direct link between S and D, it is equivalent to the conventional dual-hop transmission withoutdiversity. Itisobviousthatthecapacityoftherelaypathisequaltotheminimum of the S-R link capacity and the R-D link capacity. Thus, the optimal power allocation becomes a standard max-min problem [48], i.e., C DF,w/o diversity = max {P S ,P R } min 1 2 log(1+|h SR | 2 P S ), 1 2 log(1+|h RD | 2 P R ) . The solution must yield an equal capacity (or SNR) for both links, i.e., log 1+P S |h SR | 2 =log 1+P R |h RD | 2 . Hence, we have P S =P 0 |h RD | 2 |h SR | 2 +|h RD | 2 and P R =P 0 |h SR | 2 |h SR | 2 +|h RD | 2 [48]. If there is a direct link between S and D, more power should be allocated to S since its transmission contributes to the direct path as well as the relay path. If the direct channel has better quality than the S-R link or the R-D link, it is natural to allocate all power to S alone. An interesting scenario to justify the DF scheme is considered in [38], where R retransmits only when it correctly decodes the message and D is said to receive a message successfully only when its transmissions through both paths are successful. 15 Under such a scenario, the overall capacity is bounded by the channel capacity of S-R link. Thus, the power allocation problem can be formulated as C DF,diversity = max {P S ,P R } min 1 2 log(1+|h SR | 2 P S ), 1 2 log(1+|h SD | 2 P S +|h RD | 2 P R ) , and the capacity is maximized with [48] P S =P 0 |h RD | 2 |h SR | 2 +|h RD | 2 −|h SD | 2 and P R =P 0 |h SR | 2 −|h SD | 2 |h SR | 2 +|h RD | 2 −|h SD | 2 . As expected, more power is allocated to S as compared to the case without diversity. The optimal power allocation of the AF scheme with respect to the end-to-end capac- itycanbederivedsimilarly. IntheAFscheme,Rdoesnotdecodethemessagebutsimply retransmits an amplified version of the received signal. Since the signal transmitted byR will contain an amplified version of the noise along the S-R link, both the noise variance, N 0 , and the total power, P 0 , play a role in power allocation. Specifically, for the case without diversity, the ratio between P S and P R becomes [33] P S P R = s |h RD | 2 P 0 +N 0 |h SR | 2 P 0 +N 0 . With diversity, the power allocation problem exists only when the S-R link and the R-D link are sufficiently good when compared with the S-D link. To be more specific, when min(|h SR | 2 ,|h RD | 2 )<|h SD | 2 , one should simply allocate all the power toS. When power allocation is needed, a similar dependence on P 0 and N 0 is observed. For example, when 16 |h SR |≈|h RD | and are both sufficiently larger than |h SD |, the ratio between P S and P R can be approximated as [33] P S P R ≈ |h SR | 2 |h RD | 2 P 0 +|h RD | 2 |h SD | 2 P 0 +|h SD | 2 N 0 |h SR | 2 |h RD | 2 P 0 −|h SR | 2 |h SD | 2 P 0 −|h SD | 2 N 0 . Example of Case I Considera three-node network whose relay nodeislocated inthe middleofS andD, and its distance to both nodes is d=1. All nodes have full knowledge of channel coefficients h SR , h RD and h SD , which are i.i.d. circularly symmetric Gaussian random variables with zero mean and variances σ 2 SR = 1, σ 2 RD = 1 and σ 2 SD = 1/2 α , where α = 3 is the path loss coefficient. We would like to achieve rate R = 1 at D under the total power constraint P S +P R = P 0 . An outage occurs when the channel capacity cannot achieve the transmission rate R. The outage probabilities of AF and DF schemes with diversity are compared in Fig. 2.3. For each scheme, we plot results obtained from equal and optimal power allocation methods. Although the DF scheme achieves higher capacity when averaged over channel realizations, it does not provide a good diversity gain since the transmission depends on successfuldecodingatR. Therefore,theoutageprobabilityoftheAFschemeoutperforms thatoftheDFschemewhentheSNRvalueissufficientlyhigh. Besides,theoptimalpower 17 0 5 10 15 20 25 30 35 40 10 −4 10 −3 10 −2 10 −1 10 0 Total Power Constraint P 0 (dB) Outage Probability Direct DF. equal power alloc AF. equal power alloc DF. opt power alloc AF. opt power alloc Figure 2.3: Comparison of outage probabilities of AF and DF schemes in the three-node network with equal and optimal power allocation. allocation scheme has a SNR gain of approximately 3dB over the equal power allocation method. 2.2.1.2 Case II: Nodes with Partial CSI It is often difficult to have full CSI in a highly dynamic environment as described above, since all nodes have to track the channel status continuously. To address this issue, power allocation strategies based on partial CSI have been developed. For example, a power allocation strategy for the DF scheme was developed in [63] based on the averaged channel gains, i.e., E[|h SR | 2 ] and E[|h RD | 2 ], which are easier to obtain in practice. The strategy proposed in [63] minimizes an upper bound of the symbol error rate (SER) for M-ary modulations, e.g. M-QAM or M-PSK, which is shown to be near optimal at high 18 SNR regimes. When diversity combining is performed at the destination, D, the power allocation ratio is found to be [63] P S P R = 1+ p 1+CE[|h RD | 2 ]/E[|h SR | 2 ] 2 >1, where C is a positive constant that depends on the specific modulation used. It is worthwhile to point out that more power is allocated to S since it contributes to both the direct and relay paths. Interestingly, the channel gain of the S-D link plays no role in the above power allocation scheme. Furthermore, ifE[|h SR | 2 ]≪E[|h RD | 2 ], all power should be allocated to S since R would not be able to decode messages reliably. On the other hand, if E[|h SR | 2 ] ≫ E[|h RD | 2 ], the power should be equally distributed betweenS andR. UndersimilarCSIassumptions,thepowerallocationfortheAFscheme was derived in [20, 14] with respect to the outage probability. 2.2.2 Dual-Hop Relay Networks Thepowerallocationproblemhasbeenstudiedextensivelyforthedual-hoprelaynetwork shown in Fig 2.2. In this case, power allocation becomes much more interesting due to the increased degree of freedom as a result of more relay nodes. As shown in Fig 2.2, let us consider N relay nodes, denoted by R k , k = 1,··· ,N, and let h Sk and h kD denote the complex channel coefficients from the source S to the relay R k and from R k to destination node D, respectively. A two-stage cooperation is adopted. That is, S 19 broadcastsitsmessageinthefirststageandthesetofrelays{R k ,k =1,··· ,N}transmits simultaneously in the second stage. The transmit powers of S and R k are denoted by P S and P k , respectively. The total power constraint is imposed on the summation of relay powers, i.e., P N k=1 P k ≤ P R . Since power allocation among P S and P R can be determined using techniques derived in the previous subsection, we focus on the power allocation among relay nodes in this subsection. 2.2.2.1 Case I: Full CSI at Relay and Destination Nodes The system of multiple relay nodes in Fig. 2.2 can be viewed as a virtual antenna array thattransmitsnoisyversionsofthesourcemessages. WhenfullCSIisknownattherelays, a precoding technique similar to that in MIMO systems can be used to compensate for both the channel gain and the phase rotation experienced by the relays to achieve better detection performance. The optimal solution depends on the orthogonality of the relay channels as discussed below. For orthogonal relaying channels, D receives N copies of the source symbol from the relay nodes with no interference among each other. With knowledge of the exact channel coefficients, the N symbols can be combined coherently at D to increase the received SNR. With the AF scheme, the capacity of the parallel relay channel can befoundas [46] C AF,orthogonal = 1 2 log 1+ N X k=1 |h Sk | 2 |h kD | 2 P S P k |h Sk | 2 P S +|h kD | 2 P k +1 ! , 20 and the capacity-maximizing power allocation strategy results in the following water- filling solution [46] P k = |h Sk | 2 √ γ k 1 √ η − 1 √ γ k + , where(a) + =max(a,0)andγ k = |h Sk | 2 |h kD | 2 P S |h Sk | 2 +1 . ThevalueP S P k γ k isthepowerofthesignal component of the output of relay node R k and the Lagrange multiplier, η, is chosen to meet the total power constraint of the relay nodes. Note that relay node R k is allowed to transmit if and only if γ k >η. Power allocation for the DF scheme with orthogonal relay channels was derived in [39] to maximize the capacity. Consider a set of relay nodes, denoted byR D , that is able to correctly decode the messages transmitted by S. That is, for all k∈R D , the desired transmission rate is smaller than the capacity of the S-R k link. These relays decode and forward the messages to D, acting as multiple antennas on a single terminal. In the wideband or the low SNR regime [47], the capacity can be approximated by C DF,orthogonal ≈ 1 2 X k∈D P k |h kD | 2 , if P k |h kD | 2 ≪ 1. Thus, it is converted to an equivalent problemthat maximizes the sumof the SNR values from the set, R D , of decodable relay nodes. The solution to the above optimization problem is to choose the relay node among R D with the best channel towards D and allocate all the power to that node. This means that the selective relaying scheme is optimal for the DF scheme with orthogonal relay channels. 21 Let us now consider the case of non-orthogonal channels. If the signals forwarded by the relay nodes arrive simultaneously at D, the received signal at D can be written as Z = N X k=1 h kD U k +W D , where W D is the AWGN with unit variance and N is the total number of relay nodes in the network. The transmitted symbol U k = f(X k ) at relay node R k is a function of received signal X k and the specific cooperation scheme. When the CSI is not known to relay nodes, signals arriving at D may be mixed constructively or destructively due to different carrier phase shifts at D. On the other hand, if both the amplitude and phase information of all channels are known to the relay nodes, the phase shift effect can be compensated and the signals can be added coherently at D with a beamforming technique. For the AF scheme over non-orthogonal channels, relays can be viewed as multiple antennas with complex gains applied to the output of each antenna, i.e., the transmitted symbolcanbewrittenasU k =w AF k X k . WhenfullCSIis available at therelays, theopti- mal beamforming factors were derived in [18] to optimize the received SNR. Specifically, the gain applied at R k is equal to w AF k =λ AF |h Sk ||h kD | 1+P S |h Sk | 2 +P k |h kD | 2 · h ∗ Sk |h Sk | · h ∗ kD |h kD | , 22 whereλ AF isaconstantusedtomeetthetotalpowerconstraint,i.e., P N k=1 |w AF k | 2 (P S |h Sk | 2 + 1) =P R , and the transmit power allocated to node R k is equal to P k =λ 2 AF |h Sk | 2 |h kD | 2 (P S |h Sk | 2 +1) (1+P S |h Sk | 2 +P k |h kD | 2 ) 2 . The beamforming factor for the AF scheme was also derived in [35] using the minimum mean square error (MMSE) criterion. Please note that the phase rotation along the S-R k link and the R k -D link must be compensated by w AF k in the AF scheme. However, for the DF scheme, only the phase rotation along the R k -D link have to be compensated since the decoding at the relay eliminates the effect of phase rotation along the S-R k link. The beamforming factors must take into account decoding errors at relay nodes as proposed in [1]. When the BPSK modulation is used, the optimal beamforming factor of R k that maximizes the SNR at D becomes w DF k =λ DF (1−2p e k )h ∗ kD 1+4P R |h kD | 2 p e k (1−p e k ) , where p e k = Q( p 2P S |h Sk | 2 ) is the error probability at node R k for BPSK. As p e k ap- proaches 0.5, the power allocated to R k goes to zero. Similarly, λ DF is chosen to satisfy the total power constraint. 23 2.2.2.2 CaseII:ChannelGainKnown to Relays andFull CSIatDestination When the phase information is not available to the relays, it is difficult to compute the beamforming gain accurately and a noncoherent combination of signals may result in random constructive or destructive interference at D. To avoid the random interference among different relay nodes, we may allocate all power to one relay as proposed in [6, 67, 7]. It was shown in [7] that this selective relaying strategy is optimal in minimizing the outage probability for the DF space-time-encoded scheme under the total power constraint. Specifically, power P R should be allocated to the node with k ∗ DF =argmax k min{P S |h Sk | 2 ,P R |h kD | 2 }. Selective relaying was also proposed for the AF scheme in [8]. This strategy achieves a diversity orderofN while the equal powerdistributionmethod providesno diversity gain if space-time codes are not used. It was shown in [67] that selective relaying achieves better throughput than the case of orthogonal channels even with the optimal power allocation over sub-bands since the latter scheme requires N times the bandwidth. With selective relaying, power allocation strategies can be derived to maximize the lifetime of a wireless sensor network, which is the longest time that the system remains operational. The power allocation strategy that maximizes the capacity or SER may not extend the network lifetime since these objective functions do not take the residual battery energy of each relay node into account. To extend the network lifetime, the 24 selection strategy k ∗ = argmax k e k /P k was used in [10, 30], where e k is the residual battery energy at relay R k . With this strategy, the network lifetime can be extended by approximately 6% as compared to the case that does not consider residual energy. 2.2.2.3 Case III: Partial CSI at Relays and Full CSI at Destination The power allocation strategies presented above were shown to offer significant perfor- mance gains under the total power constraint. However, it is often difficult to obtain the instantaneous CSIforall links ofthe systemin practice. Thisproblemismade even more challenging when the number of users increases. To address this issue, power allocation strategies with less stringent assumptions on the CSI have been proposed. Specifically, a power allocation strategy for the DF space-time-encoded scheme was derived in [42] by assuming that relay R k knows only the instantaneous channel gain of the S-R k link, i.e., |h Sk | 2 , and the average channel gain of the R k -D link, i.e., E[|h kD | 2 ]. Then, a near optimal solution that minimizes the outage probability by selecting a set of relays and allocating them with an equal share of power was developed. The set of selected relays is B={k:P S |h Sk | 2 >η,P R E[|h kD | 2 ]>η} [ {k:P S |h Sk | 2 >η,E[|h kD | 2 ]>E[|h jD | 2 ],∀j6=k}, where η is the SNR needed for reliable decoding. With the same amount of channel information, the optimal power allocation strategy for the AF scheme was derived in [66]. In this case, since instantaneous values of h kD , for all k, are not known to relays, they cannot compensate for the phase rotation properly. 25 Then, the best selection leads to the optimal power allocation strategy. Specifically, it is optimal [66] to select the node with the highest SNR value that is averaged over the channel gain between the relay and the destination. A near optimal solution is obtained by approximating the SNR with the first-order Taylor’s expansion so that k ∗ =argmax k P S P R |h Sk | 2 E{|h kD | 2 } 1+P S |h Sk | 2 +P R E{|h kD | 2 } . 2.2.3 Multi-Hop Relay Networks The cooperative transmission system can be extended to a multi-hop scenario by con- catenating multiples of the three-node or the dual-hop networks. Instead of restricting to the two-hop cooperation, the signals from M hops away can be combined to enhance thedetection at thedestination. Inconventional multi-hopsystems, manyofthe received signals that contain insufficient energy for reliable detection are discarded, e.g. signals from distant transmitters. On the contrary, with cooperation, the receiver may combine signals transmitted via different relays, regardless of the signal strength, to enhance the performance. No signals are discarded which leads to a significant gain in energy effi- ciency. This advantage and the respective power allocation strategies have been studied in [55, 65, 49, 24]. The challenge lies in the fact that, since a network could be deployed over a large area, we can no longer assume that the signals from all users arrive at the receiver simultaneously. Instead, we should view this system as the transmission of a source signal through a multi-path fading channel generated by the asynchronous relays, 26 which can be resolved by the RAKE receiver or some equalization technique, as treated in [49, 24, 62]. The complexity of the optimal power allocation increases exponentially with the number of nodes in the network [24]. To control the complexity, scalable yet suboptimal solutions were proposed and significant energy saving can still be observed. 27 Chapter 3 Comparison of Power Control Schemes for Energy-Constrained Relay Networks 3.1 Introduction In resource-constrained wireless systems such as sensor networks, it is often desirable to exploit the knowledge of the channel state information (CSI) and perform optimal power allocation for cooperative relays to minimize the energy consumption or prolong the network lifetime. Several power allocation strategies were proposed based on different cooperation strategies and network topologies [23]. Most of the existing works focus on maximizing the information rate or the instantaneous capacity with the perfect CSI [46, 19]. Here, instead of maximizing the achievable rate, we target at the saving of the aggregate transmit power since power consumption is a more critical issue in most wireless sensor network (WSN) applications. In this chapter, we consider two adjacent cluster heads in a WSN whose data trans- mission is aided by neighboring sensor nodes, which act as relay nodes to provide the 28 spatial diversity gain and save the path loss between cluster heads. Distributed space- timeblockcoding(DSTC)[32]. andpowerallocation schemesareadoptedbytheserelays. Atfirst,threepowerallocationsarecomparedwhichdependonthelinkqualityofwireless channels. Our main focus is their impact on the power consumption of relay nodes. This is because that sensor nodes have a more stringent battery resource as compared with that of cluster heads in heterogeneous WSNs. The three power control schemes under our study are stated below[25]. • Scheme 1. The transmission power of a relay node is proportional to the SNRvalue of the single-relay path. • Scheme 2. Relay nodes with their path SNR lower than a threshold are dropped and others transmit with the same power. • Scheme 3. Only the best node is allowed to connect using all available power. Scheme 3 is also called the selective relaying (or opportunistic relaying) [6], which exploits the spatial diversity most effectively and maximizes the averaged system SNR. However,inasensornetworkwithslowlyvaryinglinkconditions,relaynodeswithabetter channelconditionwillsufferfromenergydepletionmuchearlier. Toaddressthisproblem, we propose an SNR-constrained power reduction algorithm that reduces the transmit power when the system SNR meets a target value. The performance of the opportunistic scheme aided by SNR-constrained power reduction is analyzed in Sec. 3.4.2. It is shown bysimulationthattherelaylifetimecanbeprolongedsignificantlyatthecostofaslightly 29 h S1 h 1D h 2D h ND h S2 h SN Source Relay 1 Relay 2 Relay N Destination . . . . . . . . . . . . . . . x r 1 r 2 r N t 1 t 2 t N z Figure 3.1: A system model of the proposed cooperative relay network. increased error rate. Among various opportunistic schemes, the optimal power allocation strategy is obtained by minimizing the total relay power subject to the SNR requirement at the destination. This scheme is proven to achieve the full diversity, and it is optimal with respect to several different optimizing criteria, e.g. [23, 8, 67]. Furthermore, it only demands local CSI at each relay, and can be conducted in a distributed manner [6]. 3.2 System Model ConsideranetworkwhereN+1nodescooperateintransmittingmessagesto thedestina- tion. At any time instance, we have one useract as the source andthe remainingN users serve as cooperative partners that relay the source message to the destination as shown in Fig. 3.1. In this chapter, we consider the case where the source is fixed throughout the whole transmission process. The cooperation takes on two phases of transmission. In the first phase, the source sends messagex of dimension T×1 with zero-mean and covariance matrix E{x † x} =I T 30 to relay nodes, whereI T is a T×T identity matrix. The signal received at the k-th relay is r k = p P S h Sk x+v k , k =1,2,...,N, (3.1) where h Sk is the channel coefficient from the source to the k-th relay, v k is the additive white Gaussian noise (AWGN) at the k-th relay with E{v k v † j } = I T · δ kj (δ kj is the Kronecker delta and † is the conjugate transpose.), and P S is the transmit power of the source. Channel coefficients, h Sk , 1 ≤ k ≤ N, are assumed to be independent and circularly symmetric complex Gaussian random variables withCN(0,σ 2 Sk ). In the second phase, each relay transmits an amplified version of the received signal to the destination using the DSTC proposed in [32]. Specifically, the signal transmitted by the k-th node is expressed as t k = s P k P S |h Sk | 2 +1 A k r k , (3.2) whereP k isthe transmitpower of thek-th relay andA k istheT×T space-time encoding matrix that is chosen randomly at relay k. Let A k , for k = 1,··· ,N, be unitary and 31 i.i.d. isotropically randomwithzero mean [32]. It was shownthat, forT >N, the DSTC achieves full diversity at high SNR. Then, the signal received at the destination becomes z = N X k=1 h kD t k +w, (3.3) = N X k=1 s P S P k P S |h Sk | 2 +1 h Sk h kD A k x+ N X k=1 s P k P S |h Sk | 2 +1 h kD A k v k +w (3.4) where h kD is the channel coefficient from the k-th relay to the destination and w is the AWGN at the destination with E{ww † } = I. Again, h kD is assumed to be i.i.d. circularly symmetric with distributionCN(0,σ 2 kD ). The maximum likelihood (ML) detection scheme is performedat the destination with full knowledge of the CSI, i.e., h Sk and h kD for all k, and the ST coding matrices A k . b x =argmin s i kz− N X k=1 s P S P k P S |h Sk | 2 +1 h Sk h kD A k s i k 2 . (3.5) Notethatwearenotconcernedwiththecombiningofsignalstransmittedfromthesource at the destination but focus on the power allocation over relays in this work. 3.3 Power Control Schemes 3.3.1 Three Basic Schemes In this section, we consider the power control scheme with total power constraint, i.e., P N k=1 P k = P R . The transmit power of each relay is controlled by the destination using 32 variouspowercontrolstrategiesundertheassumptionthatthefeedbackchannelisreliable through channel coding. The SNR at the destination averaged over random choices of coding matrices is given by SNR = P N k=1 P S P k P S |h Sk | 2 +1 |h Sk h kD | 2 1+ P N k=1 P k |h kD | 2 P S |h Sk | 2 +1 , (3.6) = N X k=1 α k ρ k , (3.7) where ρ k is the SNR value that all relay power P R is allocated to the k th relay node. That is, ρ k ≡ P S P R |h Sk h kD | 2 1+P S |h Sk | 2 +P R |h kD | 2 , k =1,2,··· ,N. (3.8) The weight in (3.7) is related to power allocation {P k } via α k ∼ P k P S |h Sk | 2 +1 (1+P S |h Sk | 2 +P R |h kD | 2 ), (3.9) subject to P N k=1 α k =1. In this section, we would like to compare the following three power control schemes by taking the path SNR value, ρ k , into account in different ways. • Scheme 1: assignment in proportion with path SNR The transmitted power assigned to each relay node is proportional to ρ k in (3.8). • Scheme 2: assignment by dropping paths of low path SNR We drop paths whose path SNR values are lower than a threshold, and assign all 33 active nodes with equal power. The diversity gain of employing the DSTC in this scheme depends on the number of active nodes. • Scheme 3: opportunistic assignment All power is assigned to the single relay node that has the largest path SNR. Besides,weusetheequalpowerschemei.e.,P k =P R /N),astheperformancebenchmark. From the linearity between system SNR and path SNR, we can conclude that the system SNR in equation (3.7) attains its maximum when opportunistic assignment is applied. Actually, the equal power scheme and the opportunistic scheme are two extreme cases of Scheme 2 by choosing the lowest and the highest threshold values, respectively. 3.3.2 Performance Comparisons of three power control schemes We compare the performance of various power control schemes described above in the following. Here, the limited battery power at the relays is not considered ,i.e., all nodes still have sufficient battery energy. The comparison is on the bit rate error performance under different relay schemes. When the total transmit power (i.e., P S +P R ) is fixed, it is optimal for the equal powerscheme bysetting P S =P R ifh Sk andh kD are i.i.d. [32]. To fairly compareit with three power control schemes described in Sec. 3.3.1, we also choose P S =P R =P 0 in our simulations. Fig. 3.2 compares the bit error rate (BER) as a function of the number of relay nodes, N, for different power control schemes. The BER is obtained from Monte Carlo simulations. Source symbols are modulated by QPSK and transmitted with block 34 size T =N. The total transmit power is 15 dB, and the thresh old of the second scheme is set to the averaged path SNRover all relay nodes, i.e., nodeswith thepath SNRbelow the averaged value are dropped. The bit error rate (BER) for direct transmission with the same transmit power in the same fading channel is about 0.14. We see that the BER performance is significant improved by AF relaying and it decreases with the number of relay nodes. Theopportunisticscheme (scheme3)withN =3performsaswellasscheme 1 with N = 8 and scheme 2 with N = 5. This can be explained by the fact that scheme 3 exploits the spacial diversity most effectively and is most suitable for a high density WSN with full CSI. 1 2 3 4 5 6 7 8 9 10 10 −4 10 −3 10 −2 10 −1 Number of relay nodes (R) Bit Error Rate equal power scheme 1 scheme 2 scheme 3 Figure 3.2: Comparison of the BER performance as a function of the relay node number with different power control schemes. 35 InFig. 3.3, we show the BER performanceversus the total transmit power fora relay system with N =6 relay nodes . The complex unitary coding matrixA k is isotropically random. The threshold of Scheme 2 is the same as that in Fig. 3.2. We see that the three power control schemes outperforms the equal power scheme by 3dB, 4dB and 8dB, respectively. Whenapplyingtheopportunisticpower control scheme (Scheme 3), it isnot necessary to encode retransmit signals with DSTC since only one node relays signal at a time. Therefore, the opportunistic power control scheme always provides the best choice if the instantaneous channel status is known and the lifetime energy of relay nodes is not of concern. Another issue is the amount of feedback information needed in each power control scheme. For Scheme 1, the size of feedback messages depends on the resolution of the power level. For Scheme 2, only N bits are needed to notify which nodes are on and off. In our simulations, Scheme 2 slightly outperforms Scheme 1 by simply choosing the thresholdastheaveragedpathSNRwhilethepowercontrolmessageforScheme2ismuch simpler. For the opportunistic scheme (Scheme 3), only log 2 N is needed to specify the bestnode. Thus,itgives thebestperformancebyconsideringboththeerrorperformance and the feedback message amount. 36 0 2 4 6 8 10 12 14 16 18 20 22 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 2P 0 (dB) Bit Error Rate equal power Scheme 1 Scheme 2 Scheme 3 Figure 3.3: Comparison of the BER performance as a function of the total transmit power. 3.4 Power control with consideration of limited battery energy If the battery energy of each relays is limited, the number of active relay nodes decreases with time and it influences the cooperative diversity gain. Therefore, it is important to manage the power consumption of relay nodes effectively. If the channel condition varies slowly in a wireless sensor network, the node with the largest path SNR will be the only relay node for a long while in the opportunistic scheme. Consequently, it suffers from power depletion faster than other nodes. Due to the decreased number of active relay nodes, the diversity gain of the opportunistic scheme decreases and the error rate increases with time. On the other hand, all relay nodes share the power usage equally at 37 all time in the equal power scheme, all links last for a longer periodat the price of having a lower SNR value at the receive end. 3.4.1 Power Saving Strategy for Relay Nodes To obtain a good balance between the diversity gain and fairness of battery usage, we propose a power saving strategy that minimizes the aggregate transmitted power of relay nodes subject to a target SNR constraint. Let γ be the target system SNR and κ the power reduction ratio. When the system SNR exceeds γ, we can adjust the transmit power of relay nodes to be P (s) k =κP k , (3.10) where κ = γ P N k=1 P k |h kD | 2 P S |h Sk | 2 +1 (P S |h Sk | 2 −γ) . (3.11) The value of κ is obtained by substituting P k in (3.6) with P (s) k in (3.10). When the denominator of (3.11) is negative, it means that the propagated error at relays is amplified such that system SNR fails to achieve the target value γ no matter how large the relay power is. When the target SNR γ is achieved, 0≤ κ≤ 1. We see from (3.11) that a large value of the channel strength between relays and the receiver, indicated by 38 |h kD | 2 , is helpful to decrease factor κ. However, when the channel strength between the transmitter and relays|h Sk | 2 increases, the value of P S |h Sk | 2 −γ P S |h Sk | 2 +1 also increases, and it will be saturated at 1. Therefore, the benefit of power saving for a large value of|h Sk | 2 is limited and the reason is that the forwarding power is normalized at the relay nodes. The choice of γ depends on the desired error probability. For example, if the signal is modulated by QPSK and the number of relay nodes is large, then the bit error rate of the received symbols is roughly proportional to Q( √ γ) [21]. 3.4.2 Performance Comparison with Energy Constraint In this section, we consider the constraint of limited battery power on relay nodes and study their effect on the system. We compare the averaged transmit and relay lifetime of each scheme. Since the opportunistic scheme has severe unbalanced power usage among relay nodes (especially in a slowly fading environment), we are more interested in the effect of the SNR-constrained power saving strategy on this scheme. The transmit power of the best relay node after power saving with target SNR γ is P opp,γ =min k γ |h kD | 2 P S |h Sk | 2 +1 P S |h Sk | 2 −γ . (3.12) 39 Weseefrom(3.12)thatalargervalueoftherelay-receiverchannelgain|h kD |ismorehelp- fulinreducingthepowerconsumption, sincetheamplification causedbythetransmitter- relay channel gain |h Sk | is normalized before retransmission at relay nodes. Under Rayleigh fading environments, the distribution function of the transmit power of the opportunistic scheme P opp,γ with target SNR γ is f opp,γ (v) =N Z ∞ v f P,γ (u)du N−1 f P,γ (v), (3.13) where f P,γ is the distribution of the transmit power for a single relay node to achieve target SNR γ and of the following form: f P,γ (v) = γ v 2 e − γ(P S +v) P S v ( 2γ(1+γ) P S K 0 ( s 4γ(1+γ) P S v ) + s 4γ(1+γ) P S v K 1 ( s 4γ(1+γ) P S v )). (3.14) whereK ν (·) isthe modifiedBessel functionof the secondkindandorderν. Theaveraged transmit power after power saving with target SNR γ becomes P opp,γ = Z P S 0 uf opp,γ (u)du+P S Z ∞ P S f opp,γ (u)du. (3.15) Wedefinetherelaylifetimetobethedurationthatallrelaynodesarestillactive. After applying the power saving strategy to the opportunistic scheme, the averaged transmit power and the averaged relay lifetime versus the number of relay nodes, N, are shown in 40 Fig. 3.4andFig. 3.5, respectively. Itisassumedinthesimulationthatthechannelsuffers fromslowlyfadingwithf d T b =5,whereT b isthedurationofrelaynodesbeingactivewith energy consumption P S . Without the power saving strategy, the total transmit power is fixed at 15dB as shown in Fig. 3.4. With the power saving strategy, the average transmit power becomes smaller for a larger value in the relay node number (N) and/or a smaller target SNR value (γ). As shown in Fig. 3.5, the relay lifetime is significantly prolonged especially for a smaller target SNR value and the performance degradation fromγ=12dB to γ=9dB on the BER performance is acceptable as shown in Fig. 3.6. 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 12 14 16 18 20 Number of relay nodes (R) Averaged transmitted power scheme 3 scheme 3, γ =9dB scheme 3, γ =10dB scheme 3, γ =11dB scheme 3, γ =12dB Figure 3.4: Comparison of the averaged transmit power of the opportunistic scheme with power saving as a function of the relay node number. In Fig. 3.6, we compare the averaged relay lifetime and the BER performance for the equal power scheme, the three power control schemes, and the opportunistic schemes 41 1 2 3 4 5 6 7 8 9 10 0 100 200 300 400 500 600 700 800 Number of relay nodes (R) averaged relay lifetime(units) scheme 3 scheme 3, γ =9dB scheme 3, γ =10dB scheme 3, γ =11dB scheme 3, γ =12dB Figure 3.5: The averaged relay lifetime of the opportunistic scheme with power saving as a function of the relay node number. with power saving. The number of relay nodes, N, is 6 and the total transmit power is 15dB. Without power saving, the equal power scheme has the longest relay lifetime while the opportunistic scheme has the shortest. We see clearly a tradeoff between the BER performanceandtherelay lifetime. However, wecanget abettertradeoff bychoosing the opportunistic scheme with the power saving strategy. For γ = 9dB, the relay lifetime is significantly prolonged (by 84.38%) while the BER performance is still very good. When the target SNR is set to 12dB, we can get almost identical BER performance as the opportunistic scheme while the relay lifetime is prolonged by 6.25%. In other words, the powersaving strategy can improvethe opportunisticschemebyreaching abettertradeoff between signal quality and relay lifetime. 42 equal powerscheme 1 scheme 2 scheme 3 scheme 3 scheme 3 Bit error rate Relay lifetime 0.0335 0.0107 0.0062 0.0018 0.0034 0.0018 302 277.5 267.5 224 413 238 γ = 9 dB γ = 12 dB Figure 3.6: Comparison of the averaged BER performance and relay lifetime for various power control schemes. We compare the long term BER performance of four power control schemes in Fig. 3.7. The simulation environment is the same as that in Fig. 3.4. After the death of the firstrelay node, the performanceof the opportunisticscheme beginsto degrade gradually due to decrease in diversity. At the 300th time step, relay nodes with the equal power scheme suffer power outage simultaneously. From the beginning up to this point, we see that all power control schemes have better BER performance than the equal power scheme. For the opportunistic scheme with power saving and target SNR equal to 12 dB, the time of the death of the first relay nodes is slightly extended. By setting the target SNR at 9dB, the long-term performance clearly outperforms other power control 43 schemes in terms of lifetime and BER. The price paid is small performance degradation as compared with Scheme 3. 0 50 100 150 200 250 300 350 400 450 500 10 −4 10 −3 10 −2 10 −1 10 0 Time (unit) Bir Error Rate Scheme 1 Scheme 2 Equal Power Scheme 3 Scheme 3; γ =9dB Scheme 3; γ =12dB Figure 3.7: Long-term BER performance for different power control schemes. 3.5 Optimality of the opportunistic scheme The simulation results shows that the opportunistic scheme has the maximal lifetime compared with other power control strategies. In this section, we would like to show the optimality of the opportunistic scheme by deriving the optimal power allocation that minimizesthesumoftheinstantaneoustransmissionpowerssubjecttotheaveraged SNR requirement at the destination based on the system model in Sec. 3.2 [30]. 44 The optimization problem can be formulated as min N X k=1 P k subject to (i) SNR≥γ and (ii) P k ≥0,∀k, (3.16) where γ is the target destination SNR. From (3.6), constraint (i) can be written in linear form as N X k=1 P k |h kD | 2 1− γ+1 P S |h Sk | 2 +1 ≥γ. (3.17) WhenP S |h Sk | 2 <γ, thecontribution ofrelayk tothesummationin(3.17) isnegative and thereby should be allocated with zero power, i.e., P k = 0. This corresponds to the case where the S-k link is so noisy that the amplification at the relay cannot provide any advantage. These relays are ineligible for transmission. The complement set R S ={k :P S |h Sk | 2 ≥γ} forms the set of reliable relays. If all channels from the source to relays experience deep fading simultaneously so that the target SNR is not achievable, R S will be an empty set and will result in an outage. The solution to (3.16) is described by the following proposition with its proof given in the appendix. Proposition 1: For nonemptyR S , the optimal power allocation solution to (3.16) is P k = γ |h kD | 2 P S |h Sk | 2 +1 P S |h Sk | 2 −γ , k =k ∗ , 0, otherwise, (3.18) 45 where k ∗ =arg min k∈R S γ |h kD | 2 P S |h Sk | 2 +1 P S |h Sk | 2 −γ . (3.19) We note that the minimum transmission power needed for the k-th relay to achieve the target SNR, γ, with no cooperation is γ |h kD | 2 P S |h Sk | 2 +1 P S |h Sk | 2 −γ ,w k , which is called the weight for k∈R S . Following from the proposition, the optimal power allocation scheme is in form of the selective relaying that chooses the relay with the minimum required transmission power, i.e., minimum weight w k , to relay the message. Besides, when the setR S is nonempty, k ∗ = arg min k∈R S γ |h kD | 2 P S |h Sk | 2 +1 P S |h Sk | 2 −γ = arg min 1≤k≤N P S P R |h Sk h kD | 2 1+P S |h Sk | 2 +P R |h kD | 2 , (3.20) i.e., the optimal solution induces to the opportunistic scheme with the power reduction in section 3.4.1. We show in Fig. 3.8 the averaged total transmission power required to achieve the target SNR as the number of relays increases. In this experiment, P S = 12dB, γ =8dB, and σ 2 Sk = σ 2 kD = 1 for all k. We set 1 unit of power to be the transmit power required to achieve SNR = 0dB at the receiver with channel gain equal to one. We compare 46 2 4 6 8 10 12 14 16 0 5 10 15 20 25 Number of relay nodes (N) Averaged sum of transmit power (dB) Equal Power over R S Optimal Power Allocation Figure 3.8: The averaged transmission power versus the number of relay nodes. the performance between the optimal power allocation and the case with an equal power distribution among users inR S . Since the diversity increases with N, the total power for both cases decreases as N increases. In fact, the gain achieved by power allocation also increases since a larger degree of freedom is available for power distribution. A gain of approximately 5dB is observed for 16 relays. Opportunistic relaying has been studied under different scenarios and optimizing cri- teria[6,8,67]. Ithasbeenshowntoachievefulldiversityevenwithoutcentralizedcontrol. For example, with the contention-based method proposed in [6], it can be implemented in a distributed manner since weights w k can be computed with the knowledge of local CSI at each relay. 47 3.6 Conclusion Three power control schemes for the amplify-and forward(AF) space-time coded relay channel were studied and compared in this chapter. It was shown that the relay system benefitssignificantlyfromthepowercontrolschemes. Amongallschemes,theopportunis- tic scheme performs the best without the consideration of the relay lifetime. For relay nodes with finite battery lifetime a SNR-constrained power saving strategy was proposed to avoid the early power outage for the opportunistic scheme under a slow fading chan- nel. Consequently, the opportunistic power control scheme with power saving reaches the best balance between link reliability and relay lifetime. It was also shown that the opportunistic power control scheme that chooses only the relay with the best composite channel requires the minimum energy consumption to achieve the SNR requirement at the destination. This implies that no DSTC is needed when the CSI is known at relays since only a single relay is used at any time instance. 3.7 Appendix : Proof of Proposition 1 Since the objective function and constraints in Eq. (3.16) are linear of{P k }, the optimal power allocation is a linear programming problem. To begin with, let us re-write the constraint of the SNR requirement in Eq. (3.17) as N X k=1 P k |h kD | 2 γ P S |h Sk | 2 −γ P S |h Sk | 2 +1 = N X k=1 P k 1 w k ≥1. (3.21) 48 Since the optimal power allocation has to satisfy two constraints, the solution lies in the feasible regionA where all linear constraints are satisfied A= N \ k=1 {P k ≥0} \ { N X k=1 P k w k ≥1}. Note that the feasible regionA is a polyhedron surroundedby several hyperplanes which indicate equalities of constraints. The extreme points of the region occur at the intersec- tion of these hyperplanes. Thus, an extreme point is located at the P k -axis if w k ≥ 0, i.e., k∈R S , and the number of extreme points of the feasible regionA is|R S |. The set of extreme points is{w k 1 k :∀k∈R S }. On the other hand,A is empty if and only if the values of{w k } are all negative, i.e., R S =∅ and no optimal power allocation exists. The objective of the optimization is to find an optimal solution with the minimal total transmission power, P P k , within regionA. According to the fundamental theorem of linear programming, if there is a bounded optimal solution of the linear programming problem, there exists an extreme point of the feasible region which is optimal. Since all extreme nodes are located on the axes, the optimal solution naturally turns out to be selective relaying and the selected optimal relay node is k ∗ =arg min k∈R S w k , (3.22) which leads to the results of the proposition. 49 Chapter 4 Lifetime Maximization for Amplify-and-Forward Cooperative Networks 4.1 Introduction Cooperation communications [51, 38, 49] allow users in the system to cooperate in relay- ingeachother’smessagestothedestination. Withthespatialdiversitygains, cooperative systems have theability to reducethe total energyrequiredto meet the quality-of-service (QoS) requirement at the destination. Several power allocation strategies were proposed based on different cooperation strategies and network topologies [23]. However, the even- tual goal of power allocation should be to prolong the lifetime of users operating with a limited amount of battery energy. In contrast with the single-transmitter-single-receiver system, minimizing the instantaneous power consumption in each transmission may not maximize the network lifetime since the lifetime depends not only on the short term 50 power consumptionbut the residual energy information (REI) of the usersas well. With- out balanced energy usage among users, the network may become non-functional even when many users still have a large amount of battery energy remaining. ItwasshowninSec.3.5 thatselective relaying(i.e., theopportunisticschemeinChap- ter3) is optimal intermsofminimizingthe total instantaneous transmissionpowerinthe AF cooperative network with distributed space-time codes(DSTC) [32]. In other words, the optimal power control policy in this case should allocate power only to the node with the best channel. This implies that DSTC is not needed if instantaneous CSI is known at relays. The selective relaying scheme, also referred to as opportunistic relaying, is also known to achieve full diversity [67]. Since the implementation of selective relaying demands only local channel state information (CSI) at relays, it can be conducted in a distributed manner with the method described in [6]. Basedonselectiverelaying,weproposethreenetworklifetimemaximizationstrategies, where the lifetime is defined as the duration in which the destination SNR requirement can be met with a certain probability, i.e., the outage probability must be lower than a predetermined value. Lifetime maximization has been considered in sensor networks, e.g. [10, 11, 13, 12], but most work along this line defines the network lifetime as the duration in which all users in the network are alive. Such a definition is not proper in our current context since the loss of one node may not disable a cooperative network. Withproperpowerallocation, userswithlowerresidualenergycanbeaidedbyuserswith 51 higher residual energy. In this sense, cooperation not only provides diversity for effective utilization of channels, but for residual energy levels as well. To implement cooperative systems with low-cost high-speed digital circuits, we ana- lyze the network lifetime performance for devices with only a finite set of power levels. The evolution of the users’ residual energies can be modeled as a finite state Markov chain, where each state records the REI of all users [13, 12]. In this case, the averaged network lifetime is equivalent to the average time to absorption in the Markov chain [26], where the absorption states correspond to the non-operable REI values. Furthermore, with global knowledge of CSI and REI, the problem of maximizing the network lifetime can be modeled as a stochastic shortest path problem [13]. As compared with the op- timal strategy that requires global CSI and REI, the performance loss of our proposed strategies that demand only local CSI and REI is negligible[29]. 4.2 System Model Consider a network where N +1 nodes cooperate to transmit messages from the source to the destination. At any instant in time, we have one user act as the source and the remainingN usersserve ascooperative partnersthatrelay themessage tothedestination asshowninFig. 4.1. Weconsiderthecasewherethesourceisfixedthroughoutthewhole transmission process. 52 h S1 h 1D h 2D h ND h S2 h SN Source Relay 1 Relay 2 Relay N Destination . . . . . . . . . . . . . . . x r 1 r 2 r N t 1 t 2 t N z Figure 4.1: The system model. The cooperation takes on two phases of transmission. In the first phase, the source sends signal x with varianceE|x| 2 =1 to the relay nodes. The received signal at the i-th relay is r i = p P S h Si x+v i , i=1,2,...,N, (4.1) where h Si is the channel coefficient from the source to the i-th relay node, v i is i.i.d. additive white Gaussiannoise (AWGN) at thei-th relay withunitvariance, andP S is the transmit power of the source. The channel coefficients h Si , 1≤ i≤ N, are independent complex and circularly symmetric Gaussian random variables; namely, h Si ∼CN(0,σ 2 Si ). Inthischapter, itisassumedthechannelcoefficients variesblock-by-block independently. Let us consider the selective cooperation method using AF schemes that only one relay node is chosen to transmit. If thek ∗ -th relay nodeis selected, it forwards the signal with power P k ∗ in the second phase. The signal received at the destination is z = s P S P k ∗ P S |h Sk ∗| 2 +1 h Sk ∗h k ∗ D x+ s P k ∗ P S |h Sk ∗| 2 +1 h k ∗ D v k ∗ +v D , (4.2) 53 whereh k ∗ D ∼CN(0,σ 2 k ∗ D )is thechannel coefficient fromtherelay R k ∗ to thedestination and v D is AWGN at the destination with unit variance. Let {h Si ,∀i} and {h jD ,∀j} be independent random variables. Assume the amplitude of local channel gains are known at each relay, i.e., |h sk | and |h kD | are known at the relay R k , and the global CSI is known at the destination. As shown in Sec.3.5, the relay node k ∗ shall be selected from R S = {k : P S |h Sk | 2 ≥ γ} which contains the set of reliable relays able to achieve the target SNR. Let the transmit power for the relay k ∈R S to achieve a target SNR γ be w k which is given w k = γ |h kD | 2 P S |h Sk | 2 +1 P S |h Sk | 2 −γ . (4.3) Through this chapter, the selected relay R k ∗ transmits with power P k ∗ =w k ∗. 4.3 Network Lifetime Maximization One important goal of power allocation in wireless networks is to prolong the lifetime of battery-powered devices. With each user operating under an individual battery con- straint, the lifetime of a cooperative system will depend on the power distribution over relays. In other words, it cannot be maximized by simply minimizing the sum of the transmission powers as done in Sec.3.5. Most previous work [11, 13, 12, 25](including the chapter 3 in this thesis proposal) views the network lifetime as the duration in which all users in the network remain active. In other words, the network is dead if one of the users is depleted with battery 54 energy. However, thisdefinitiondoesnotfullycharacterizetheoperabilityoftheproposed cooperative systemsincetheenergydepletionofasinglenodeonlydecreasesthediversity available to the system. The QoS may still be achieved cooperatively with the remaining relays. In this section, we define the network lifetime to be the duration in which the SNR requirement at the destination is achieved with a certain probability. Based on the selective relaying scheme described in Sec. 3.5, we discuss the lifetime performance of several relay selection strategies. Let e k [m] be the residual energy of relay k after the m-th message is transmitted. When relay k is chosen to transmit, the outage will occur if the SNR at the relay is lower than γ, i.e., k / ∈R S , or if the residual energy is insufficient to reach the target SNR. The outage probability of the k-th relay after the m-th time slot is a function of the residual energy e k [m], i.e., P out (e k [m]) = Pr{k / ∈R S or w k >e k [m]} = Pr min P S e k [m]|h Sk h kD | 2 1+P S |h Sk | 2 +e k [m]|h kD | 2 ,P S |h Sk | 2 <γ = Pr P S e k [m]|h Sk h kD | 2 1+P S |h Sk | 2 +e k [m]|h kD | 2 <γ . (4.4) For Rayleigh fading channels,|h Sk | 2 and|h kD | 2 are exponentially distributed with mean σ 2 Sk and σ 2 kD , respectively. Then, it can be shown explicitly that P out (e k [m]) =1−F k (e k [m]), (4.5) 55 where F k (e k [m]) =e −( γ P S σ 2 Sk + γ e k [m]σ 2 kD ) s 4γ(γ +1) P S e k [m]σ 2 Sk σ 2 kD K 1 s 4γ(γ +1) P S e k [m]σ 2 Sk σ 2 kD ! (4.6) is the distribution function of w k , i.e., the probability that the minimum required trans- mission power of the k-th relay is smaller than e k [m], and K 1 (·) is the modified Bessel function of the second kind of order one. Let e[m] = [e 1 [m],··· ,e N [m]], where e[0] gives the initial energy distribution at the relays. It is assumed that each transmission lasts for one time unit and the transmission power is equal to the amount of energy consumed during the interval. With cooperation, the outage probability at time m is given by P out (e[m]) = N Y k=1 P out (e k [m]). The network lifetime is defined mathematically as L(e[0]) =min m {m:P out (e[m])>η} where threshold η is the maximum tolerable outage probability. It is clear that the network lifetime is determined by the initial energy distribution e[0] and parameters η and γ. 56 By the strong law of large numbers (SLLN), the averaged network lifetime can be expressed as [11] E[L(e[0])] = P N k=1 e k [0]−E w E r , (4.7) where E w is the averaged total residual energy at all relays when the network dies (i.e., the wasted energy) and E r is the averaged total energy consumed by all relays during each transmission. To maximize the network lifetime, the relay selection scheme must minimizetheaveraged energyconsumptionaswellasthewastedenergy, whichareclosely coupled. We see from Fig. 3.8 that the minimum averaged transmission power decreases with the number of relays due to the increased spatial diversity. Therefore, to maximize network lifetime, it is desirable to maintain the maximum number of active relays to minimize the rate of energy consumption. Thus, both REI and CSI must be considered in the lifetime maximization policy. Three relay selection methods that exploit local instantaneous CSI and REI, i.e., parameters w k [m] and e k [m] for relay k, to maximize the averaged lifetime in (4.7) are considered. In these schemes, only nodes that belong to set R E =R S ∩{k :e k [m]≥w k [m]} are eligible to relay the m-th message. When R E is empty, no relay is selected and an outage occurs. As long as the outage probability of the network satisfies the QoS requirement,thenetworkremainsactiveeventhoughthecurrenttransmissionfails. Since 57 these methods demand only the instantaneous information, index m is omitted in the following discussion. These three methods along with the minimum power solution in Sec. 3.5 are listed as follows. (I) The minimum transmission power strategy (MTP) Choose the node that requires the minimum transmission power, which is the method described in Sec.3.5, i.e., k ∗ MTP =arg min k∈R E w k . (II) The maximum residual energy strategy (MRE) Choose the node with the largest residual energy after retransmitting the current message [11], i.e., k ∗ MRE =arg max k∈R E e k −w k . The goal is to maintain the maximum diversity gain by preventing any node from depleting itsenergyearlier thanothers. Thismethodbalances theenergyconsump- tion across relays. 58 (III) The maximum energy-efficiency index strategy (MEI) The energy efficiency index is defined as the ratio between e k and w k , i.e., ρ k = e k w k [10]. Thestrategy selects nodek ∗ MEI withthe maximal energy efficiency index, i.e., k ∗ MEI =arg max k∈R E e k w k . In other words, we choose the node whose transmission consumes the least portion of its current residual energy. (IV) The minimum outage probability strategy (MOP) To reduce the increasing rate of the outage probability, we choose the relay that has the minimum outage probability after the current message is transmitted, i.e., k ∗ MOP = arg min k∈R E P out (e−w k 1 k ) = arg min k∈R E P out (e−w k 1 k ) P out (e) = arg min k∈R E P out (e k −w k ) P out (e k ) , (4.8) where1 k is an N×1 vector with the k-th element equal to 1 and zero everywhere else. Although both strategies MRE and MEI consider CSI and REI in a simple and in- tuitive way, the latter actually achieves better performance in terms of network lifetime 59 maximization. This is due to that, when P k e k [0] ≫E w , maximizing the averaged life- timeisequaltomaximizingtheratiobetweentheresidualenergyandthetransmitpower, i.e., the energy efficiency index, as shown in (4.7). Moreover, when the two strategies select different relays, i.e., k ∗ MRE 6=k ∗ MEI , we have e k ∗ MRE −w k ∗ MRE > e k ∗ MEI −w k ∗ MEI , e k ∗ MRE /w k ∗ MRE < e k ∗ MEI /w k ∗ MEI . Thus, we have w k ∗ MEI <w k ∗ MRE . When the node selection disagrees, strategy MEI always selects a relay that has lower transmission power and, thus, is more energy efficient. Strategy MOP attempts to minimize the outage probability based on the local infor- mation. Whene k issufficiently large, thevalue ofδ k isvery small andthe Bessel function in (4.5) can be approximated by K 1 (δ k )≈δ −1 k . In this case, strategy MOP reduces to k ∗ outage =argmax k∈S D (e k −w k ) e k w k , (4.9) which is a combination of strategies MRE and MEI. Since these strategies depend only on local REI and CSI at each relay, they can be implemented in a distributed manner as proposed in [6]. The performance of these methods are evaluated in Sec. 4.6 by computer simulation. 60 4.4 Discrete Power Allocation Due to the practical hardware constraint, most low cost mobile devices can emit signals thattakeononlyafinitesetofpowerlevels. Besides, themaximumtransmissionpoweris constrained by the linear operating range of the power amplifier. Suppose that users are limited to L discrete power levels 0 < ε 1 < ε 2 <··· < ε L <∞. When relay k is chosen to transmit, the transmission power allocated to the k-th relay is the minimum power level needed to meet the target destination SNR. When the battery energy is sufficient, the discrete power level assigned to the k-th relay is w k,d = min 1≤i≤L {ε i :ε i ≥w k }, if k∈R S and w k ≤ε L , 0, otherwise. (4.10) For relay k to be eligible for retransmission, its residual energy must exceed the discrete power level w k,d . In this case, we define the new eligible set of relays as ˆ R E =R S ∩{k:w k ≤ε L }∩{k :e k ≥w k,d }. When ˆ R E is empty, the SNR requirement is not achievable and an outage occurs. The discrete power allocation scheme is obtained by quantizing continuous power w k into discrete levels. Generally speaking, the scheme consumes more energy in each transmission due to the quantization effect. The probability that relay k is not eligible dueto a badchannel is 1−F k (ε L ) whereF k (·) is the distributionfunction ofw k as shown 61 in (4.6). When all relays are active, the minimum outage probability is achieved if the maximum power level ε L satisfies N Y k=1 1−F k (ε L )<η. (4.11) By considering the limitation on the residual battery energy at each node, the outage probability is given by P out,d (e) = N Y k=1 1−F k (⌊e k ⌋), (4.12) where ⌊e k ⌋ = max{ε i : ε i ≤ e k ,i = 1,2,··· ,L} is the maximum transmission power emitted by the k-th relay. The selective relaying schemes described in Sec. 4.3 are modified for the discrete case as follows. Strategy II is omitted since it is shown to result in bad performance later. (I’) Discrete MTP strategy: k ∗ MTP =argmin k∈ ˆ R E w k,d . (III’) Discrete MEI strategy: k ∗ MEI =argmax k∈ ˆ R E e k w k,d . (IV’) Discrete MOP strategy: We apply strategy MOP in Sec. 4.3 by choosing k ∗ MOP =arg min k∈ ˆ R E P out (e k −w k,d ) P out (e k ) . (4.13) We may have more than one optimal relay for selection in the discrete case. If this hap- pens,wemayselectonefromthesetofoptimalrelaysrandomlywithanequalprobability. 62 4.5 Markov Analysis of Network Lifetime With finite initial battery energy and discrete power levels, the set containing all possible values of residual energye[m] is finite. Since the power emitted duringeach transmission dependsonlyoncurrentREIandCSIintheproposedstrategies, andbyassumingthatthe channels are independent during each transmission, the evolution of the residual energy levels{e[m]} ∞ m=0 can bemodeled as a finite-state Markov chain. Thenetwork lifetime for selective relaying strategies is derived analytically by Markov analysis. The state space of the Markov chain,S, is the set of all possible residual energy levels, i.e., S = ( e:e k =e k [0]− L X l=1 υ l ε l ≥0,∀υ l ∈N∪{0},∀k ) , (4.14) where e k [0] is the initial battery energy of the k-th relay. A state transition occurs af- ter each transmission, where the transition probability is characterized by the current energy state, the selection strategy and the strategy-dependent probability of the trans- mitted power levels. The states where the destination QoS cannot be met form a set of terminating states S T , i.e., S T ={e∈S :P out,d (e)>η}. (4.15) Let E[L(e)] be the averaged residual lifetime starting from state e. By definition, E[L(e)] = 0 for e ∈ S T . The Markov chain is shown in Fig. 4.2. For a state e / ∈ S T , 63 e 0 e 1 . . . . . . L(e 0 ) L(e 1 ) L(e 2 ) L(e n )=0 L(e n+1 )=0 Pr(e 0 e 1 ) e 2 Pr(e 1 e 2 ) e n e n+1 Pr(e 0 e 2 ) Pr(e 1 e 3 ) . . . . . . . . . . . . P out,d (e 0 ) P out,d (e 1 ) P out,d (e 2 ) S T Figure 4.2: The state transition diagram of an energy-consuming process. the averaged network lifetime is equal to the averaged number of transitions from state e to any one of the terminating states. Note that when an outage occurs, no energy is consumed and a self-transition takes place at non-terminating states, i.e., for e / ∈ S T . When a transition occurs frome / ∈S T toe ′ , a unit time passes and the averaged residual lifetime is E[L(e ′ )]. Then, the averaged network lifetime of energy state e / ∈S T is given by E[L(e)] = X 0≤e ′ ≤e Pr{e→e ′ }(1+E[L(e ′ )]), (4.16) where Pr{e→e ′ } is the transition probability from state e to state e ′ . Since the prob- ability of self-transition at state e is equal to outage probability P out,d (e), the averaged network lifetime in (4.16) becomes 1 E[L(e)] = 1 1−P out,d (e) 1+ X 0≤e ′ <e Pr{e→e ′ }E[L(e ′ )] . (4.17) 1 Here, we define vectors u≥v if u k ≥ v k ,∀k, and define u >v if u≥v and max k {u k −v k } > 0. 64 The averaged lifetime of a network with initial energy e 0 is E[L(e 0 )], which can be evaluated recursively from terminating states. If the channels of all relays are i.i.d., the averaged residual lifetime is invariant to the ordering of residual energies in the state vector e. This property can be utilized to reduce the computational complexity when evaluating (4.17). 4.5.1 Performance Analysis of the Proposed Strategies Sinceatmostonerelay ischosen duringeach transmissioninthesestrategies, thenumber of valid transitions for each state is at most N × L. For e > e ′ ≥ 0, the transition probability from state e / ∈S T to e ′ is greater than zero only if e−e ′ =ε j 1 k , ∀1≤k≤N, 1≤j≤L. The transition probability is then determined by the probability that relay k is selected to transmit for a given strategy and that the required transmission power is ε j . For the discrete MTP strategy, an eligible relay node k is selected to transmit with power ε j only if other relays are not eligible or their required power levels no less than ε j , i.e., w m,d =0, w m,d >e m , or w m,d ≥ε j ,∀m6=k. 65 If more than one eligible relay requires the same minimal transmission power ε j , one of them is selected at random with an equal probability. For a valid transition frome to e ′ ande−e ′ =ε j 1 k , the transition probability is Pr (I ′ ) (e→e ′ )=Pr{w k,d =ε j } Y m6=k Pr{w m,d >min(ε j ,e m ) or w m,d =0} + 1 2 X n6=k:en≥ε j Pr{w n,d =ε j } Y m6=k,n Pr{w m,d >min(ε j ,e m ) or w m,d =0}+··· , (4.18) where Pr{w k,d =ε j }=F k (ε j )−F k (ε j−1 ). Similarly, for the discrete MEI strategy, the k-th relay node is selected to transmit with power ε j only if the energy efficiency indices of other eligible relays are no greater thane k /ε j . Foravalidtransitionfrometoe ′ ande−e ′ =ε j 1 k ,thetransitionprobability is given by Pr (III ′ ) (e→e ′ )=Pr{w k,d =ε j } Y m6=k Pr{ e m w m,d < e k ε j or w m,d =0} + 1 2 X n6=k Pr{ e n w n,d = e k ε j } Y m6=k,n Pr{ e m w m,d < e k ε j or w m,d =0}+··· . (4.19) For the discrete MOP strategy, the k-th relay node is selected to transmit with power ε j only if the ratio of P out (e m −w m,d ) to P out (e m ) of any other eligible relay m is not less 66 than P out (e k −ε j )/P out (e k ). For a valid transition from e to e ′ and e−e ′ = ε j 1 k , the transition probability is Pr (IV ′ ) (e→e ′ ) =Pr{w k,d =ε j } Y m6=k Pr{ P out (e m −w m,d ) P out (e m ) > P out (e k −ε j ) P out (e k ) or m / ∈ ˆ R E } + 1 2 X n6=k Pr{ P out (e n −w n,d ) P out (e n ) = P out (e k −ε j ) P out (e k ) } Y m6=k,n Pr{ P out (e m −w m,d ) P out (e m ) > P out (e k −ε j ) P out (e k ) orm/ ∈ ˆ R E }+··· . (4.20) 4.5.2 Optimal Selection Strategy with Global CSI For a given set of power levels and initial battery energy distributions, the optimal net- work lifetime can be computed using techniques similar to the stochastic shortest path problem [13]. Specifically, we construct a path for each state that goes through the maxi- mum averaged number of transitions before entering a terminating state. The maximum averaged network lifetime is obtained uniquely by Bellman’s equation [5], E[L optimal (e)]= 1 1−P out,d (e) 1+ X {u>0:max k e k −u k ≥0} Pr{w d =u}max k∈ ˆ R E E[L optimal (e−u k 1 k )] , (4.21) where u = (u 1 ,··· ,u N ) is a set of discrete transmission powers with elements u k ∈ {0,ε 1 ,··· ,ε L }. For a given set of power levels {ε i } L i=1 , the optimal network lifetime E[L optimal (e[0])] is obtained backwards from terminating states. With global knowledge of REI and CSI at all relays, the strategy that maximizes the averaged network lifetime is given by k ∗ optimal =arg max k∈ ˆ R E E[L optimal (e−w k,d 1 k )]. (4.22) 67 Although this strategy maximizes the averaged network lifetime, it is difficult to imple- ment in practice since it requires global CSI and REI, and that E[L optimal (e)] must be computed in advance. The computational complexity grows dramatically with the num- ber of power levels, the number of relays and the initial battery energy values [13]. The performanceof thisoptimal scheme serves primarilyas anupperboundto other selection strategies. 4.6 Simulation Results We compare the averaged network lifetime of several relay strategies discussed earlier by computer simulation in this section. In the simulation, the transmission power of the source and the target SNR are chosen to be P S =12dB and γ =8 dB, respectively. The threshold for the outage probability is η = 0.1. Channel coefficients h Sk and h kD are i.i.d. complex Gaussian variables with unit variance and varies independently in each transmission. 4.6.1 Continuous Power Allocation First, we compare the network lifetime of strategies (I)–(IV) with no constraints on the adjustable power levels. We begin with the same initial energy level at each relay, i.e., e k [0] =E 0 for all k. For simplicity, we choose E 0 to be an integer multiple of P S ranging from250P S to450P S . Notethatthevariationintheaveragedtransmissionpowerandthe wasted energy with respect to the initial energy is almost negligible for a fixed number of relays. For a network of N =6 relays, the averaged network lifetime of different schemes is compared in Fig. 4.3. We see from Eq. (4.7) that the averaged network lifetime is 68 a linear function of the total initial energy and the slope is determined by the averaged transmission power of each corresponding strategy. We see that MTP, MEI and MOP all have similar slopes indicating similar averaged transmission power. Although MRE has a much lower amount of wasted energy, its network lifetime is reduced since it tends to choose a node with higher transmission power. Since the initial energy in this example is much larger than the wasted energy, MEI provides good performance since (4.7) is likely to be maximized by the energy efficiency index. We also show that the strategy in (4.9) closely approximates the MOP, wheretheoutage probability is usedasthe selection criterion. SincethenetworklifetimeisdefinedintermsoftheoutageprobabilityandMOP performs step-by-step optimization, MOP does not yield the globally optimal solution in principle. 240 260 280 300 320 340 360 380 400 420 440 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 4200 Inital Battery energy E 0 (× P S ) Averaged Network Lifetime MTP MRE MEI MOP Approx. MOP Figure 4.3: The averaged network lifetime versus the initial energy of each relay for cooperative network with six relays. 69 8 10 12 14 16 18 20 22 6 6.5 7 7.5 8 Number of relay nodes (N) Averaged Transmit power (dB) MTP MRE MEI MOP Figure 4.4: The averaged transmission power versus the number of relays for a fixed aggregate initial energy. We compare the averaged wasted energy, transmission power and lifetime of the four strategies for different values of N in Figs. 4.4-4.6. The total initial relay energy is 60P S , which is equally distributed among all relays. In this case, the initial energy is small and the wasted energy in (4.7) cannot be neglected. This is especially true for a large value of N. As the number of relays increases, the averaged energy consumption during the lifetime decreases because of the spatial diversity gain. We show the averaged power per transmission in Fig. 4.4, which is taken over the duration of its lifetime. We see that MEI and MOP demand almost the same transmission power as MTP, which minimizes the power consumed in each transmission. Rather than paying the price for balancing the energy distribution, MOP even requires slightly less power due to the preservation of spatial diversity. Fig. 4.5 shows that the wasted energy decreases for a small value 70 9 12 15 18 21 100 105 110 115 120 125 130 135 140 Number of relay nodes (N) Averaged wasted energy MTP MRE MEI MOP Figure 4.5: The averaged wasted energy versus the numberof relays for a fixed aggregate initial energy. of N and increases for a large number of relays. When the residual energy of each node drops below the averaged transmission power, it is less likely for the relay to forward one more message to the destination successfully. Since the decreasing rate of averaged transmissionpowerissmallerthan1/N forlargeN,thewastedenergygraduallyincreases withthe numberof relays. By considering REI,MRE, MEIandMOP reduce the amount of residual energy considerably as compared to MTP. MRE has the least wasted energy for a small numberof relays, while MOP achieves the least wasted energy for large values ofN. Itisinteresting that whenN isverylarge, MTPhasalmost thesame orlesswasted energyeventhoughitdoesnottake REIintoaccount. AsshowninFig. 4.6, theresultant averaged network lifetime increases with the number of relays. Since the wasted energy cannot be neglected in this case, the strategy based on the energy efficiency index can 71 9 12 15 18 21 120 130 140 150 160 170 180 190 200 210 220 Number of relay nodes (N) Averaged Network Lifetime MTP MRE MEI MOP Figure4.6: Theaveragednetworklifetimeversusthenumberofrelaysforafixedaggregate initial energy. no longer be used to approximate the optimum strategy. For a small value of N, MEI performsaswellastheonethatminimizestheoutageprobability, whichisthebestinthis scenario. However, it gradually degrades when N increases. That is, it moves closer to that of MTP. MOP achieves the best performance when the initial energy is comparable to the wasted energy. In this case, only a few transmissions can occur before the network dies so that the performance of the step-by-step maximization is close to that of the globally optimal solution. Overall, MEI that chooses the relay with the largest energy efficiency index e k /w k is a good one since it offers a relatively long lifetime in both Figs. 4.3 and 4.6. 72 4.6.2 Discrete Power Allocation and Peak Power Constraint We compare the averaged lifetime of a cooperative network with 3 relays for different discrete power allocation schemes. In Fig. 4.7, we compare the averaged lifetime of discrete MTP, MEI and MOP along with the optimal performance obtained from Eq. (4.21), where e k [0] =E 0 for all k and that the number of discrete power levels is L=10. The maximum power is ε L = 82.25 and the discrete power levels are set as ε i = i× ε L L . The averaged network lifetime is computed by the Markov Chain analysis as described in Sec. 4.5. We see that the averaged transmission powers for MEI and MOP are almost the same as the optimal strategy. 150 200 250 300 350 400 450 15 20 25 30 35 40 45 50 55 Inital Battery energy E 0 Averaged Network Lifetime Discrete MTP Discrete MEI Discrete MOP Discrete Optimal Figure 4.7: The averaged network lifetime of selective strategies with discrete power allocation. In Fig. 4.8, we show the averaged lifetime of three power allocation strategies for the continuous-power case (dashed line), the discrete-power case with L = 10 (solid) 73 82.25 164.5 246.75 329 411.25 493.5 0 10 20 30 40 50 60 70 80 90 100 Inital Battery energy E 0 Averaged Network Lifetime MTP MEI MOP Continuous Discrete, L=10 Discrete, L=5 Figure 4.8: The averaged network lifetime of Strategies MTP, MEI and MOP. and L = 5 (dash-dot), where the discrete power levels and the maximum power are the same as Fig. 4.7. The lines marked by triangles, stars and circles indicate the performance of discrete MTP, MEI and MOP strategies, respectively. To compare with the continuous power allocation schemes fairly, we introduce the same maximal power constraint ε max =82.25 for the continuous power case. Due to the quantization effect, we see from Fig. 4.8 that discrete power allocation loses by approximately 22% and 39% for L = 10 and L = 5, respectively. Note also that both MEI and MOP outperform MTP, especially in discrete power allocation. Although discrete MTP minimizes the energy consumption of each transmission, MEI and MOP achieve a balanced usage of battery energy at the relays. Using the energy efficiency index as the selection criterion, we compare in Fig. 4.9 the lifetime of the continuous-power case and cases of discrete power levels with L = 74 100 150 200 250 300 350 400 450 500 0 10 20 30 40 50 60 70 80 90 Initial Energy E 0 Averaged Network Lifetime Contiuous Disc. L=100 Disc. L=80 Disc. L=60 Disc. L=40 Disc. L=20 Disc. L=10 Disc. L= 5 Figure 4.9: Lifetime achieved with strategy MEI for different L. 5,10,20,40,60,80,100 for a network with three cooperative relays. Other system param- eters are the same as before. The results are obtained with Monte Carlo simulations. As shownin Fig. 4.9, the loss of discrete power levels decreases rapidly asL increases from 5 to 40. A loss of roughly 7.5% is still observed for L=40. Increasing the power level after thispoint doesnot providesignificant improvement. Thus, witha suitable discrete power level, the design of a good selection strategy is actually more important than increasing the power levels with complex hardware. Finally, in Fig.4.10, we compare the average lifetime of a cooperative network where the three relays are at different distances to the destination. Specifically, let the source, the three relays and the destination be arranged in order on a line with equal distance, where the distance between source and destination d SD is 2. The transmit power of the source is P S = 15dB, and the value of target SNR and the desired outage probability 75 100 150 200 250 300 350 400 450 500 550 600 0 10 20 30 40 50 60 70 80 Inital Battery energy E 0 Average Network Lifetime MTP MEI MOP L=10 L=5 Figure 4.10: Average lifetime of a network with non-identical channel statistics. are the same as Fig.4.8. The channel coefficients h Sk and h kD are independent complex Gaussian distributed with zero mean and variances d −2 Sk and d −2 kD respectively. In Fig. 4.10, we show the average lifetime of the three power allocation strategies for the case with discrete power levels with L = 10 (solid) and with L = 5 (dash-dot). Also, we set ε max = 100. When MTP is applied, the relay in the middle of the line is most likely to be chosen since it often achieves the target SNR with lower transmit power. In this case, the relay is more likely to run out of battery energy and, thus, reduces the network lifetime. On the other hand, the average lifetimes of MEI with L=5 and L=10 are respectively 12% and 16% longer than the lifetimes of MTP strategy. MOP does not perform the best since it minimizing outage probability step-by-step, which does not guarantee to minimize the outage performance globally. It shows that the MEI achieves 76 the best balance between minimizing energy consumption and preserving residual energy when the channel statistics are not identical. 4.7 Conclusion In this chapter, several lifetime maximization strategies were proposed accordingly based on the selective relaying approach. The proposed strategies take both the CSI and REI into account with either continuous or discrete power allocation levels. For the discrete case, the averaged network lifetime of the proposed strategies were derived via a finite- state Markov chain analysis. The proposed strategies that consider local REI and CSI jointly were shown to be nearly optimal with significantly lower complexity. 77 Chapter 5 Relay-Assisted Decorrelating Multiuser Detector (RAD-MUD) for Cooperative CDMA Networks 5.1 Introduction In this chapter, we examine a cooperative CDMA system where the spreading codes of source users are not necessarily orthogonal and relays are allowed to forward messages from more than one source. In this case, the signal detection performance may degrade due to multiple-access interference (MAI). MAI is especially severe when the target user is distant from the receiver, i.e., the near-far effect. It is not obvious whether the ad- vantages of cooperation still remain in the presence of MAI. Multi-user detection (MUD) techniques [60] have beenproposedto mitigate the MAI effect in non-cooperative CDMA networks. The main contribution of this chapter is to exploit the use of MUD at both relays and the destination to mitigate MAI in a multi-user environment and demonstrate that cooperation is still advantageous in such a context. 78 There are several well known MUD techniques such as the maximum likelihood (ML) detector, the decorrelating detector [43], the minimum-mean-square-error (MMSE) de- tector [45], the decision feedback detector [15] and the successive or parallel interference cancellation schemes (SIC or PIC) [3]. The ML detector minimizes the error probabil- ity but has complexity that increases exponentially with the number of users. To lower the complexity, linear decorrelating and MMSE receivers, which require only polynomial complexity, were proposed. However, the reduced computational complexity comes at the cost of higher bit-error-rates (BER). In particular, the decorrelating receiver elimi- nates MAI butmay lead to noise amplification whenspreadingcodes are non-orthogonal. The MMSE receiver controls noise amplification up to a certain degree but results in higher residual MAI. Nonlinear decision-feedback and interference cancelation schemes offer good performance but experience large latency and error propagation. Most existing work on cooperative communications assumes that there is only one sourceinthenetworkwithallotherusersservingasrelaysortherearemultiplesourcesbut each transmits over an orthogonal channel (which implies the availability of orthogonal spreading codes). In practical systems, the requirement for orthogonality is difficult to satisfyand,thus,MAIcannotbeignored. Morerecently, MUDhasbeenstudiedin[9,58] for pair-wise cooperative systems, where each user is grouped with another user into a cooperative pair and is only allowed to forward messages transmitted by its dedicated partner. In other words, each relay helps forward the message from only one source and utilizes MUD to suppress signals transmitted by all other sources. Consequently, the degree of freedom provided by multiple relays and relays’ capability to process sources’ messages jointly are not fully exploited. 79 In this work, we consider a different scenario where each relay may cooperate with multipleuserssimultaneously. Messages received frommultiplesourcesaredecodedusing theMMSEmultiuserdetector(MMSE-MUD) atrelaysandjointlyprocessedbeforebeing retransmittedtothebasestation. Byexploitingrelay’scapabilitytopreprocessmessages, we propose the relay-assisted decorrelating multiuser detector (RAD-MUD) to separate (or to decorrelate) multiple access interfering signals at the destination. Although decor- relation of signals can also be achieved with the zero-forcing (ZF) precoder [61] or the decorrelating MUD [43], these decorrelating operations are performed entirely at either the transmitter or the receiver, resulting in power expansion at the transmitter or noise amplification at the receiver. Unlike ZF precoding or decorrelating MUD, RAD-MUD performsonehalfofthedecorrelatingoperationatrelaysandtheotherhalfatthedestina- tion. This unique feature of RAD-MUD avoids power expansion and noise amplification and, therefore, results in better BER performance as compared with existing cooperative MUD schemes. Three cooperative transmission schemes are considered on top of RAD- MUD here; namely, transmit beamforming, selective relaying and distributed space-time coding (DSTC). Moreover, since the fading and MAI on each source-relay path (or each relay-destination path) is different, relay transmissions are weighted in our schemes to combat MAI furthermore. 5.2 System Model Consider a cooperative network with K users, denoted by S 1 ,S 2 ,··· ,S K , serving as sources and L users, denoted by R 1 ,R 2 ,··· ,R L , serving as relays that forward messages 80 from sources to the destination as shown in Fig. 5.1. Each user is assigned a unique spreading code that is non-orthogonal but linearly independent of each other. The sys- temperformstwo phasesoftransmission. InPhaseI,sourcessendtheirmessagesdirectly to the destination using their respective spreading codes. These transmissions are over- heard and decoded by relays in Phase II, and retransmitted to the destination using the same set of spreading codes. Transmissionsfrom all users are assumed to be synchronous such that transmitted symbols arrive at receivers simultaneously. This simplifying as- sumption allows us to focus on the benefits of relaying in a multiuser system and bounds on achievable performance in a more practical setting. S 1 S 2 S K R 1 R 2 R 3 R L D .... .... ... h S1R1 h S2R1 h SKR1 h R1D h S1RL h S2RL h SKRL h R2D h RLD Phase I Phase II Figure 5.1: Illustration of a cooperative CDMA uplink system with K sources and L relays. 81 5.2.1 Phase I : Direct Transmission Phase In Phase I, each source transmits a message with M data symbols to the destination, denoted by D. Let x[m] = [x 1 [m],··· ,x K [m]] T be the BPSK data symbols transmitted by sources S 1 ,··· ,S K during the m-th symbol period. Thus, x k [m]∈{−1,1}. We have E[x[m]x[ℓ] T ] = I K×K , if m=ℓ, 0 K×K , otherwise, where I K×K is the K×K identity matrix and 0 K×K the K×K matrix with zeros in all elements. Let P S k be the power transmitted by source S k and s k (t) be the spreading waveformofS k . Underthesynchronousassumption,thesignalreceivedatthedestination during Phase I is y I (t)= M X m=1 K X k=1 h S k D p P S k x k [m]s k (t−mT s )+v I (t), (5.1) where T s is the symbol period, h S k D is the complex channel coefficient from S k to D, and v I (t) is the additive white Gaussian noise (AWGN). We consider a block fading environment where channel coefficients remain constant over the M-symbol block and are independent and identically distributed (i.i.d.) from block to block. The channel coefficient, h S k D , is assumed to be circularly symmetric complex Gaussian with zero- mean and variance σ 2 S k D , i.e., h S k D ∼CN(0,σ 2 S k D ), and independent among sources. If N is the spreading gain, the spreading waveform for S k can be expressed as s k (t) = 1 √ N N X n=1 c k [n]ϕ(t−nT c ), k =1,··· ,K, (5.2) 82 where c k [n] is the n-th element of the±1 spreading sequence assigned to S k , and ϕ(t) is the normalized chip waveform with unit energy and duration T c =T s /N. The received signal, y I (t), at destination D is passed through a matched filterbank (MFB) with spreading waveforms s 1 (t),··· ,s K (t), which are assumed to be known at both relays and the destination. LetR be the correlation matrix of the spreading wave- forms with its (i,j)-th element equal to [R] i,j = Z Ts 0 s i (t)s j (t)dt. The signal obtained at the MFB output during the m-th symbol period in Phase I is given by y I [m]=RH SD x[m]+v I [m], m=1,··· ,M, (5.3) whereH SD =diag( p P S 1 h S 1 D , p P S 2 h S 2 D ,··· , p P S K h S K D )andv I [m]istheAWGN with distribution CN(0 K×1 ,σ 2 v R). The MFB outputs obtained over the entire symbol block are denoted byY I =[y I [1],··· ,y I [M]]. 5.2.2 Phase II : Cooperative Transmission Phase Due to the broadcast nature of wireless channels, signals emitted by the sources in Phase I are also received at relays R 1 , R 2 ,···, R L . Similar to (5.3), the signal received at R l is passed through an MFB and the output in the m-th symbol period is u l [m] =RH SR l x[m]+n l [m], m=1,··· ,M, (5.4) 83 where H SR l = diag( p P S 1 h S 1 R l ,··· , p P S K h S K R l ) and n l [m] is the AWGN at relay R l with covariance matrix σ 2 n R, and where h S k R l ∼CN(0,σ 2 S k R l ) is the channel coefficient between S k andR l , which isassumed to beindependentfor each source-relay link. Based on the MFB output, data symbols are detected at each relay using the MMSE multiuser detector as described in Sec. 5.4. The block of symbols detected at R l are denoted by ˆ X l =[ˆ x l [1],··· ,ˆ x l [M]], where ˆ x l [m] is the detection made at relay R l on transmitted symbol vector x[m]. Depending on the cooperative strategy, detected symbols ˆ X l may be re-encoded into theK×M symbolmatrixT l =f( ˆ X l )=[t l [1],··· ,t l [M]],wheret l [m]=[t l,1 [m],··· ,t l,K [m]] T is the symbol vector transmitted by R l during the m-th symbol period. When beam- forming or selective relaying is considered [c.f. Sec. 5.3], t l [m] depends only on detected symbols in the m-th time period, i.e., ˆ x l [m]. When space-time codes are considered, t l [m] depends on entire matrix ˆ X l . During the m-th symbol period, t l,k [m] will be trans- mitted using the spreading waveform of source S k and the signal received at destination D during Phase II is given by y II (t)= M X m=1 K X k=1 L X l=1 h R l D t l,k [m]s k (t−mT s )+v II (t), (5.5) where h R l D is the channel coefficient between R l and D and v II (t) is the AWGN. The power transmitted by R l in the m-th symbol period is given by P R l , E Z ∞ −∞ K X k=1 t l,k [m]s k (t−mT) 2 dt = E[t l [m] H Rt l [m]], (5.6) 84 whichisfixedoverallsymbolperiodsandmustsatisfythefollowingtotalpowerconstraint L X l=1 P R l ≤P R . (5.7) Similarly, in the m-th symbol period, the signal at D is passed through an MFB to yield the output y II [m]= L X l=1 h R l D Rt l [m]+v II [m], m=1,··· ,M, (5.8) where h R l D ∼CN(0,σ 2 R l D ) and v II ∼CN(0 K×1 ,σ 2 v R). The channel coefficients of each source-relay, relay-destination, and source-destination links are assumed to be indepen- dent of each other and i.i.d. over each block. The MFB output over the entire symbol block isY II =[y II [1],··· ,y II [M]]. Signals received in Phase I and II are then combined at the destination to further increase the diversity gain. 5.3 Relay-AssistedDecorrelatingMultiuserDetector(RAD- MUD) As shown in (5.3) and (5.8), signals obtained at the MFB output are subject to MAI if spreading waveforms are non-orthogonal. In this case, MUD can be employed at both relays and the destination to mitigate MAI. For the non-cooperative CDMA system, the decorrelating MUD was proposed in [43] to eliminate MAI by multiplying the MFB 85 outputwiththeinverse ofthecorrelation matrix. Thatis, forthesignalreceived inPhase II, as given by (5.8), the decorrelator output at D is equal to R −1 y II [m] = L X l=1 h R l D t l [m]+R −1 v II [m]. In this case, the k-th term of the decorrelator output depends only on symbols trans- mitted with spreading code s k (t), i.e., the k-th term in each of vectors t 1 [m],··· ,t L [m]. Although this method eliminates MAI, the noise variance may increase due to the corre- lation among spreading codes. This is observed from the noise covariance matrix which is given by E[R −1 v II [m](R −1 v II [m]) H ] =σ 2 v R −1 . To address this issue, we proposed the relay-assisted decorrelating multiuser detector (RAD-MUD) which, with the help of precoding at relays, allows us to decorrelate signals at destination D without noise en- hancement. The block diagram of RAD-MUD is shown in Fig. 5.2. Suppose that relays (i.e., R l , l =1,···,L) have knowledge of the spreadingcodes of all sources. Each relay, say R l , first encodesthedetected symbolmatrix, ˆ X l , intoaK×M matrixg l ( ˆ X l ), whereg l (·)depends only on the specific cooperative transmission strategy employed, as discussed in Sec. 5.5. For example, with beamforming, we have g l ( ˆ X l )=W l ˆ X l , whereW l is a diagonal matrix consisting of beamforming coefficients. The output of the cooperative operation g l ( ˆ X l ) is then precoded by matrixL −H , whereL is the Cholesky decomposition ofR such that R=LL H , whereL is a K×K lower triangular matrix. The symbol matrix transmitted by R l is given by T l =f( ˆ X l ) =L −H g l ( ˆ X l ). (5.9) 86 The mapping g l (·) must be chosen to satisfy the total power constraint in (5.6) and (5.7) such that L X l=1 E[t l [m] H Rt l [m]]= L X l=1 E[g l ( ˆ X l )[m] H g l ( ˆ X l )[m]]≤P R , where g l ( ˆ X l )[m] is the m-th column of g l ( ˆ X l ). It is worthwhile to point out that, unlike the ZF precoding scheme in [61], the precoding employed in RAD-MUD as given in (5.9) does not result in power expansion since the transmitted power depends only on the cooperative transmission strategy but not the correlation of spreading codes. MUD U 1 L −H X 1 T 1 L −1 MMSE Combining g 1 (X 1 ) Y II Y II Z D Y I MUD U 2 L −H X 2 T 2 g 2 (X 2 ) MUD U L L −H X L T L g L (X L ) . . . . . . . . . . . . V II Figure 5.2: The block diagram of the RAD-MUD system. With precodingat relays, theMFB outputatD canbeobtainedbysubstituting(5.9) into (5.8). It is given by Y II = L X l=1 h R l D RL −H g l ( ˆ X l )+V II , (5.10) 87 where V II = [v II [1],··· ,v II [M]]. The received signal in (5.10) is then pre-multiplied withL −1 to yield ˘ Y II = L X l=1 h R l D g l ( ˆ X l )+ ˘ V II , (5.11) where ˘ Y II =L −1 Y II =[˘ y II [1],··· ,˘ y II [M]] and ˘ V II =L −1 V II =[˘ v II [1],··· ,˘ v II [M]]. It is worthwhile to point out that the noise term, ˘ v II [k]=L −1 v II [k], has covariance matrix E[˘ v II [k]˘ v II [k] H ]=σ 2 v I K×K and, thus, L −1 can be viewed as a whitening filter. To con- clude, with precoding at relays and pre-whitening at D, signals transmitted by different spreading codes are decorrelated at the destination without noise amplification or power expansion and, thus, constructing K orthogonal channels between the relays and the destination. 5.4 MMSE Multiuser Receivers and Signal Combining Although transmissions from relays to the destination can be decorrelated with RAD- MUD, signals received by relays from sources are still subject to MAI. In the following, we describe the use of MMSE-MUD at relays 1 and MMSE signal combining methods at D to improve system performance. 1 MMSE receivers are chosen here simply as a representative scheme. RAD-MUD is not reliant on any particular detection method at relays. 88 5.4.1 MMSE Multiuser Detection at Relays AsdescribedinSec. 5.2, each relay receives anaggregate signalfromthesourcesinPhase I and passes it through an MFB to obtainu l [m], which is given in (5.4). At relay R l , we apply MMSE-MUD tou l [m] and obtain the following detected symbol vector ˆ x l [m] =sgn(ℜ{z l [m]}) =sgn(ℜ{C l u l [m]}), (5.12) where z l [m]=C l u l [m] is the MMSE estimate of x[m] at relay R l and C l is the K×K matrix chosen to minimize the mean square error (MSE); namely, E[|C l u l [m]−x[m]| 2 ]. Here,ℜ{a} denotes the real part of a and, for real b, sgn[b] =1 if b≥0 and−1 if b<0. When spreading codes are linearly independent and correlation matrix R is of full rank, the MMSE solution can be computed as C l =E[x[m]u l [m] H ]E[u l [m]u l [m] H ] −1 =H H SR l R(RH SR l H H SR l R+σ 2 n R) −1 , (5.13) where index m is omitted in C l since symbols x[m] are assumed to be i.i.d. with respect to m so that C l is constant over time. Let C l = [c (l) 1 ,c (l) 2 ,··· ,c (l) K ] T , where c (l) k =[c (l) k,1 ,··· ,c (l) k,K ] T for all k. The detected symbol corresponding to S k is ˆ x l,k [m]= 89 sgn(ℜ{z l,k [m]})=sgn[ℜ{(c (l) k ) T u l [m]}]. The BER of S k ’s transmitted symbol at R l is given by α l,k = 1 2 K−1 X x i [m]∈{±1}∀i6=k x k [m]=1 Pr(ℜ{z l,k [m]}<0|x k [m],x i [m],∀i6=k) = 1 2 K−1 X x i [m]∈{±1}∀i6=k x k [m]=1 Q v u u t 2ℜ{(c (l) k ) T RH SR l x[m]} 2 σ 2 n (c (l) k ) T R(c (l) k ) ∗ , (5.14) where Q(x) = R ∞ x 1 √ 2π e − t 2 2 dt. When the number of interfering users is large, we can approximate the MAI con- tribution by a Gaussian model and the BER of the message transmitted by S k can be expressed as α l,k ∼ =Q v u u t 2[(c (l) k ) T RH SR l ] 2 k E[|z l,k [m]| 2 ]−[(c (l) k ) T RH SR l ] 2 k =Q s 2[Γ l ] k,k 1−[Γ l ] k,k ! , (5.15) where Γ l =H H SR l (H SR l H H SR l +σ 2 n R −1 ) −1 H SR l , (5.16) and [B] i,j is the (i,j)-th element of matrix B. 5.4.2 MMSE Signal Combining at Destination Based on the system model in Sec. 5.2, D receives two sets of signals: one directly from sources in Phase I and the other from relays in Phase II as given in (5.3) and (5.8), respectively. When the precoding strategy at relays and channel coefficients of all links are known at D, signals from both direct and cooperative paths can be combined at D 90 to improve the detection performance. For example, sources that suffer from the near-far effect at certain relays may be detected reliably by others. Therefore, diversity gains can be enhanced by properly weighting signals received along each path. SupposethatT l =L −H g l ( ˆ X l )istransmittedbyR l andg l ( ˆ X l )takes onthelinearform W l ˆ X l (e.g. inthe case of beamformingandselective relaying [c.f. Sec. 5.5]), whereW l = diag(w l,1 ,··· ,w l,K ) is a diagonal weighting matrix whose elements are determinedby the transmission strategy and the total power constraint. To analyze errors in the detected symbol matrix, we define a random diagonal matrix Θ l [m]=diag(θ l,1 [m],··· ,θ l,K [m]), where θ l,k [m] ∈ {±1} is a Bernoulli random variable with Pr(θ l,k [m]=−1)= α l,k and Pr(θ l,k [m]=1)= 1−α l,k . More specifically, we set θ l,k [m]=1 if the detection of S k ’s m-th symbol at R l is correct and θ l,k [m] =−1 if it is incorrect (α l,k is equivalent to the BER). Then, the detected symbol matrix can be written as ˆ X l =[ˆ x l [1],··· ,ˆ x l [M]], where ˆ x[m]=Θ l [m]x l [m], m=1,2,...,M. (5.17) Random variables θ l,k [m] are i.i.d. over time with mean E[θ l,k [m]] =1−2α l,k . Signals received at D in both phases can be expressed jointly as Y D =[Y T I ˘ Y T II ] T =[y D [1],··· ,y D [M]], where y D [m]= y I [m] ˘ y II [m] = RH SD P L l=1 h R l D W l Θ l [m] x[m]+ v I [m] ˘ v II [m] (5.18) 91 is the signal received in both phases during the m-th transmission period. Based on the signal given in (5.18), we can compute the MMSE multiuser detector as ˆ x D [m]=sgn(ℜ{z D [m]}) =sgn(ℜ{C D y D [m]}), (5.19) whereC D =E[x[m]y H D [m]]E[y D [m]y H D [m]] −1 . Alternatively, to reduce the detection complexity, we firsttake the MMSE estimate of each source’s symbol based on the signal received from the direct path, i.e., y I [m]. The MMSE estimate of each source obtained from the direct path is then combined with the corresponding decorrelated signal from the cooperative path using the MMSE criterion. Specifically, D first computes the MMSE estimate ofx[m] based ony I [m] as z I [m] = E[x[m]y I [m] H ]E[y I [m]y I [m] H ] −1 y I [m] = H H SD R(RH SD H H SD R+σ 2 v R) −1 y I [m]. (5.20) The k-th element of z I [m] is the MMSE estimate of symbol x k [m] and given by z I,k [m] =[Γ D ] k,k x k [m]+ξ k [m], (5.21) whereΓ D =H H SD (H SD H H SD +σ 2 v R −1 ) −1 H SD andξ k [m]isthecombinedMAI-plus-Gaussian- noiseterm. Byexploitingthefactthatx k [m]isindependentofξ k [m]andE[z I [m]z I [m] H ]= Γ D , one can show that ξ k [m] has zero mean and variance [Γ D ] k,k −[Γ D ] 2 k,k . 92 The estimate of S k ’s symbol received from both direct and cooperative paths can be expressed as y (k) D [m] = z I,k [m] ˘ y II,k [m] = [Γ D ] k,k P L l=1 h R l D w l,k θ l,k [m] x k [m]+ ξ k [m] ˘ v II,k [m] = h (k) D x k [m]+v (k) D , (5.22) where h (k) D = h [Γ D ] k,k , P L l=1 h R l D w l,k θ l,k i T and v (k) D = [ξ k [m], ˘ v II,k [m]] T . The MMSE estimate of x k [m] is then computed based on the signal in (5.22), which is given by z k [m] =C ′ D y (k) D with C ′ D =E h x k [m](y (k) D [m]) H i E h y (k) D [m](y (k) D [m]) H i −1 . (5.23) Note that E h x k [m](y (k) D [m]) H i = " [Γ D ] k,k , L X l=1 h ∗ R l D w ∗ l,k (1−2α l,k ) # and E h y (k) D [m](y (k) D [m]) H i = [Γ D ] k,k [Γ D ] k,k P L l=1 h ∗ R l D w ∗ l,k (1−2α l,k ) [Γ D ] k,k P L l=1 h R l D w l,k (1−2α l,k ) PP L l,l ′ =1 h R l D h ∗ R l ′D w l,k w ∗ l ′ ,k E[θ l,k [m]θ l ′ ,k [m]]+σ 2 v The detection made on S k ’s transmitted symbol is given by ˆ x D,k [m]=sgn(ℜ{z k [m]}). This method yields little performance loss as compared with the previous method, which will be shown by computer simulation in Sec. 5.6. 93 With RAD-MUD, it appears at the first sight that the choice of spreading wave- forms does not have direct influence on symbol detection in Phase II. However, higher correlations between spreading waveforms result in higher decoding BERs at relays and, consequently, errors at the destination as well. The overall performance can be improved by choosing weighting factors so as to take detection reliability at relays into account as discussed in Sec. 5.5. Given a set of channel realizations H={h S k D ,h S k R l ,h R l D ,for k=1,··· ,K, and l=1,··· ,L} and correlation matrix R, we can compute the BER of S k ’s decoded symbols at D, which is denoted by BER k|H,R . As shown in (5.18), the detection performance depends on specific error patterns at relays, i.e., θ k ,[θ 1,k ,···,θ L,k ]. The conditional BER for decoding S k ’s messages is BER k|H,R = X θ k ∈{±1} L BER k|H,R,θ k ·Pr(θ k |H,R), (5.24) whereBER k|H,R,θ k istheBERconditionedonH,R,θ k andPr(θ k |H,R)istheprobability of detection errors θ k given H andR. By treating the interference as a Gaussian random variable, conditional BER can be computed from (5.22) as BER k|H,R,θ k =Pr −ℜ{C ′ D h (k) D }+ℜ{C ′ D v (k) D [m]}>0 x k [m] =−1,H,R,θ k =Q q SINR D,k (H,R,θ k ) ·1 {ℜ{C ′ D h (k) D }≥0} + 1−Q q SINR D,k (H,R,θ k ) ·1 {ℜ{C ′ D h (k) D }<0} , 94 where 1 {B} represents the indicator function, which is equal to 1 if the statement B is true and 0, otherwise. The signal-to-interference-plus-noise ratio becomes SINR D,k (H,R,θ k ) = (ℜ{C ′ D h (k) D }) 2 1 2 E h C ′ D v (k) D [m](C ′ D v (k) D [m]) H i = n [Γ] k,k 1−[Γ] k,k h σ 2 v +4 P l |h R l D w l,k | 2 (α l,k −α 2 l,k ) i +ℜ n P l P l ′h R l D h ∗ R l ′D w l,k w ∗ l,k (1−2α l ′ ,k )θ l,k oo 2 1 2 n [Γ] k,k 1−[Γ] k,k h σ 2 v +4 P l |h R l D w l,k | 2 (α l,k −α 2 l,k ) i +σ 2 v | P l h R l D w l,k (1−2α l,k )| 2 o . The above derivation holds for all cooperative transmission strategies if g l ( ˆ X l ) = W l ˆ X l . 5.5 Cooperative Transmission Strategies In RAD-MUD, relay precoding allows us to construct K orthogonal channels between relays and the destination, similar to that in [39]. The message of each source is then transmitted through orthogonal channels with the help of a distributed antenna array formed by relays. In this case, cooperative signal processing techniques studied in the literature canbereadily appliedto thissystem. Based on theavailability of channel state information (CSI), we discuss and compare the performance of three cooperative trans- mission strategies: (a) transmit beamforming; (b) selective relaying; and (c) distributed space-time coding. For transmit beamforming and selective relaying schemes, symbols transmitted by R l can be expressed in linear form as g l ( ˆ X l )=W l ˆ X l . To maintain fair- ness, we assume that the same total relay power is used to retransmit detected symbols corresponding to each source, i.e., P L l=1 |w l,k | 2 =P R /K, for all k. 95 5.5.1 Transmit Beamforming When full CSI is available to all relays, transmit beamforming can be applied, treating relaysasmultipleantennas, tocompensateforthephasedifferenceoneachrelaypathand obtain coherent addition of signals at D. Here, we consider two beamforming methods. 5.5.1.1 Beamforming for Relay-Destination (R-D) Channels In this scheme, beamforming coefficients at each relay are chosen based on values of the relay-destination (R-D) channels. This is the solution that maximizes the signal-to- noise ratio at the destination when antennas are co-located at a single terminal, or, when errorsat relays arenegligible. For transmitbeamforming,we writeg l ( ˆ X l )=W l ˆ X l where W l =diag(w l,1 ,··· ,w l,K ) contains beamforming coefficients for each source at R l on the diagonal. Specifically, we have w l,k =β RD h ∗ R l D , ∀ l,k, whereβ RD = q P R /(K P L l=1 |h R l D | 2 ) is chosen to meet the power constraint P L l=1 |w l,k | 2 = P R /K. 5.5.1.2 Beamforming for Source-Relay-Destination (S-R-D) Channels If errors at relays are not negligible, one must consider the detection reliability at each relay in deriving beamforming coefficients. Let w k =[w 1,k ,··· ,w L,k ] T be beamforming coefficientsusedtoretransmitS k ’sdetectedsymbolsatrelaysR 1 ,··· ,R L . Byconsidering 96 the reliability of relay detection, we choose w k to minimize the MMSE at the destina- tion for signals corresponding to S k in Phase II. Specifically, with derivations given in Appendix??, we obtain optimal coefficients as w k =argmin w k min c k E[|c k ˘ y II,k [m]−x k [m]| 2 ] =β k,SRD Φ k + Kσ 2 v P R I L×L −1 p k ,(5.25) wherep k =[h R 1 D (1−2α 1,k ),··· ,h R L D (1−2α L,k )] H andΦ k isanL×Lmatrixwithelements [Φ k ] l,l ′ = |h R l D | 2 , if l =l ′ , h ∗ R l D h R ′ l D E[θ l,k θ l ′ ,k ], if l6=l ′ , (5.26) where β k,SRD = v u u t P R /K p H k Φ k + Kσ 2 v P R I L×L −2 p k is chosen to meet the power constraint P L l=1 |w l,k | 2 = P R /K. Note that, in p k , smaller weights are given to relays with larger decoding errors and/or less reliable R-D channels. 5.5.2 Selective Relaying When only partial CSI is available at relays, we consider two selective relaying strategies as detailed below. 5.5.2.1 Threshold Selection Strategy Consider the case where each relay only has the knowledge of the local S-R and R- D channel coefficients. In this strategy, a relay decides independently whether it will forward a particular source’s message using a locally selected threshold. Suppose that 97 SINR S k R l is the signal-to-MAI-plus-noise ratio for S k ’s transmitted signal measured at R l and SNR R l D is the signal-to-noise ratio at the destination for the signal transmitted by R l . Then, we get γ l,k =min{SINR S k R l ,SNR R l ,D } =min ( [Γ l ] k,k 1−[Γ l ] k,k , β 2 Th,k |h R l D | 2 σ 2 v ) , where SINR S k R l and Γ l are given in (5.15), R l relays the message transmitted by S k if and only ifγ l,k is no less than threshold γ T . That is, w l,k =β Th,k ifγ l,k ≥γ T . Otherwise, w l,k = 0. The rationale behind this selection criterion is that, since the BER of S k ’s symbol at the destination is dominated by the maximum error probability between the S-R link and the R-D link, we allow R l to relay the symbol only if the maximum error probabilityofthetwo linksaresufficientlyreliable, i.e., whentheminimumSNRorSINR exceeds a certain threshold. Since only local channel coefficients are known at each relay, β Th,k is chosen to satisfy theaverage powerconstraint P L l=1 E[|w l,k | 2 ]=P R /K. Thresholdγ T ischosennumerically to minimize the average BER at the destination in Sec. 5.6. Furthermore, the threshold selection strategy can be combined with the beamforming R-D strategy. In this case, only the SINR on the S-R link is used for comparison since full CSI of the R-D channel is available. Specifically, we set w l,k = β Th−BF h ∗ R l D if SINR S k R l ≥ γ T , and w l,k = 0, otherwise. This method can be viewed as a compromise between Beamforming R-D and Beamforming S-R-D in terms of complexity and performance. 98 5.5.2.2 Best Selection Strategy Whenglobal CSIisavailable butthephasecoherent transmissionisnotachievable dueto imperfectsynchronization, wecanemploythebestselection strategywhereonlyonerelay is chosen to transmit for each source. In this case, we set w k,l =β Th,k if γ l,k ≥γ l ′ ,k ,∀ l ′ , and w i,k = 0, otherwise, where β Th,k = p P R /K. When relays have only local CSI, the selection can be conducted at the destination in a centralized manner or, distributedly, usingtheopportunisticcarriersensingmethodproposedin[6]. Thebestselectionstrategy has been studied extensively in the literature and shown to achieve full diversity in the absence of MAI. In Sec. 5.9, we show that the full diversity can also be achieved by RAD-MUD even in the presence of MAI. 5.5.3 Distributed Space-Time Coding (DSTC) Whenno CSIis available at relays, we canemploy distributedspace-time coding (DSTC) [32, 39, 54] to exploit the diversity in a cooperative system. In particular, we adopt the linear dispersive space time code [32], where the space time code is obtained by multiplying the symbol block with a random unitary matrix generated independently at each relay. Suppose that DSTC is applied to M consecutive symbols at each relay, where M is chosen to be greater than L in order to achieve full diversity [32]. LetA l be the unitary and isotropic random matrix of dimension M ×M used to encode detected symbols from each source. The transmitted symbol matrix is then given by T l =L −H W l ˆ X l A l , l =1,··· ,L. (5.27) 99 Since no CSI is available at relays, all diagonal elements of the weighting matrix are set to the same value. At the destination, the MMSE estimate obtained on the direct path and the signal received from the orthogonal cooperative path for each source is combined together to perform the maximum likelihood (ML) detection. (The residual MAI of the MMSE estimate is treated as Gaussian when applying ML in Sec. 5.6.) Note that the MMSE detector is not used here since it does not achieve diversity in this case [c.f. [41]]. Furthermore, if only local CSI of S-R links is available at relays, we can combine the thresholdselectionstrategywithDSTCsothatw l,k =β DSTC ifandonlyifSINR S k R l >γ T . Similarly, β DSTC is chosen to satisfy the average power constraint as described in Sec. 5.5.2. 5.6 Numerical Simulation and Discussion The performance of RAD-MUD with various cooperation methods discussed in Sec. 5.5 is studied by computer simulation in this section. In Sec. 5.6.1, the signal received from the first phase is so weak that signal detection at the destination relies on the signal re- transmitted by relays. In Sec. 5.6.2, the signal from sources has the same received power as the signal transmitted from relays, and the MMSE combining scheme as described in Sec.5.4.2 is performed at the destination. 5.6.1 Cases Without Direct Links In this section, we consider a cooperative network with K =8 sources and L= 8 relays. Channel coefficients h S k R l and h R l D are assumed to be i.i.d. with distributionCN(0,1) forallkandl. Channelcoefficientsh S k D areassumedi.i.d. withdistributionCN(0,1/2 2 ). 100 This corresponds to the case where relays are located in the middle of sources and the destination. In this subsection, the signal received from the direct path is not considered, andthediversity gain comes fromtheS-R-Dpathonly. Each sourcetransmitswithequal power P s and the sum of the transmit power of all sources is equal to that of all relays, i.e., KP s =P R =P. The spreading waveforms are generated randomly with non-singular correlation matrices, and the spreading gain is N=8. 8 16 24 32 40 10 −3 10 −2 10 −1 Total Transmit Power P(dB) Bit Error Rate ZF Precoding, Equal [Cao&Vojcic ’04] Cooperative MMSE−MUD,Equal RAD−MUD, Equal Figure 5.3: BER performance comparison for different precoding and MUD strategies at relays. In Fig. 5.3, we compare the proposed RAD-MUD with three different transmission schemes: (i)Zero-Forcingprecoding[61]; (ii)Cao&VojcicCooperativeMUD[9]; and(iii) CooperativeMMSE-MUDwithoutprecodingatrelays(CooperativeMMSE-MUD).Inall these schemes, MMSE-MUD is used at relays to decode the messages from all sources. 101 At the destination, the MMSE multiuser detector is derived under the assumption that θ l,k is independent of relays for simplicity, which is suboptimal detection. In scheme (i), each relay performsZF precoding (as proposedin [61]) before messages are retransmitted to the destination. That is, the symbol transmitted by R l is T l = β zf R −1 g l ( ˆ X l )whereβ zf ischosentosatisfythetotalpowerconstraintin(5.7). Inscheme (ii), each relay forwardsonlythe message ofone source (i.e., its dedicated partner) while, in scheme (iii), each relay decodes-and-forwards messages from all sources, similar to RAD-MUD,butdonotperformprecodingattherelays,i.e.,T l =g l ( ˆ X l ). Thecooperative source-relay pairs for scheme (ii) are chosen randomly in experiments. We assume that all relays transmit with equal power and the weighting factors of all relayed symbols are identicalforscheme(iii)andRAD-MUD,i.e.,g l ( ˆ X l )=W l ˆ X l ,whereW l = p P/KLI K×K , ∀ l. It is worthwhile to point out that the system withZF precodingat the relays achieves the same performance as the system that employs decorrelating MUD at the destination without precoding at relays. Therefore, the latter is not shown redundantlyin the figure. Although signals received at the destination are decoupled in both ZF precoding and RAD-MUD, we see that RAD-MUD outperforms ZF precoding by 3 dB since no power expansion occurs at relays. For Cao& Vojcic’s scheme(scheme (i)), no decorrelating op- eration is performed at relays, which is a special case of scheme (iii) with one non-zero element in W l . As compared with schemes (i) and (iii), it is observed that scheme (iii) withequalweightsprovidesahigherdiversityorderbecausedetectedsymbolswithhigher error rates may be compensated by symbols detected from other relays. We also see that 102 RAD-MUD outperformsschemes(ii) and(iii) since signalsreceived atthe destination are free of MAI. 8 16 24 32 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Total Transmit Power P(dB) Bit Error Rate Direct Transmission Cooperative MMSE−MUD, BF R−D Cooperative MMSE−MUD, BF R−D w. Threshold SEL Cooperative MMSE−MUD, BF S−R−D RAD−MUD, BF R−D RAD−MUD. BF R−D w. Threshold SEL RAD−MUD. BF S−R−D Figure 5.4: BER performance comparison for systems employing transmit beamforming without MMSE combining at the destination. In Figs. 5.4, 5.5 and 5.6, we show the performance of RAD-MUD with different cooperative transmis- sion strategies at relays as described in Sec. 5.5 but not combining the signal received in the first phase at the destination. Since the cooperative MMSE MUD of scheme (iii) outperforms schemes (i) and (ii) as shown in Fig. 5.3 and has a structure that incorpo- rates the cooperative transmission strategies (i.e., each relay in scheme (iii) is allowed to decode-and-forward messages from all sources), we can compare RAD-MUD with the Cooperative MMSE-MUD only in the following experiments. The performance of the direct transmission with a total transmit power set to 2P is plotted in all figures as a 103 reference. The BER performance as a function of the total transmit power P is shown in Figs. 5.4, 5.5 and 5.6 for transmit beamforming, selective relaying and DSTC, respec- tively. These methods are utilized to achieve cooperative diversity gains besides reducing MAI with MUD. We see that RAD-MUD outperforms cooperative MMSE-MUD in all cases as detailed below. To be concise, the word beamforming is abbreviated as “BF” and selective as “SEL” in the legends. 8 16 24 32 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Total Transmit Power P(dB) Bit Error Rate Direct Transmission Cooperative MMSE−MUD, Threshold SEL Cooperative MMSE−MUD, Best SEL RAD−MUD, Threshold SEL RAD−MUD, Best SEL Figure 5.5: BER performance comparison for systems employing selective relaying with- out MMSE combining at the destination. TheBERcurvesaveragedoverallsourcesforthecasesemployingtransmitbeamform- ing at relays are compared in Fig. 5.4. For RAD-MUD and cooperative MMSE-MUD schemes, we see that diversity is not well achieved if decoding errors at relays are not considered even though signals from all relays are coherently combined at the destina- tion (i.e., the case of beamforming R-D). By considering global channel information, the 104 beamformingS-R-D strategy providesastonishing improvement ontheBER performance and the diversity order. The improvement is more obvious if the beamforming S-R-D is used on top of RAD-MUD, which outperforms the case with beamforming S-R-D on top of cooperative MMSE MUD by 7.5 dB at BER= 10 −4 . The strategy with beamform- ing R-D and threshold selection provides a compromise between the required channel information and BER performance. In Fig. 5.5, we compare the average BER as a function of the transmit power for the case of threshold selection and best selection. The threshold for the threshold selec- tion strategy is obtained numerically. Without MMSE combining, the diversity order of selective relaying strategies is roughly the same for RAD-MUD and cooperative MMSE MUD. However, RAD-MUD outperforms by 3-4 dB due to MAI mitigation. 8 16 24 32 10 −4 10 −3 10 −2 10 −1 10 0 Bit Error Rate Direct Transmission Cooperative MMSE−MUD, DSTC Cooperative MMSE−MUD, DSTC w. Threshold SEL RAD−MUD, DSTC RAD−MUD, DSTC w. Threshold SEL Figure 5.6: BER performance comparison for systems employing DSTC without MMSE combining at the destination. 105 The BER performance with the DSTC strategy is compared in Fig. 5.6. Without exploiting any CSI at relays, RAD-MUD associated with DSTC outperforms cooperative MMSE MUD by 5dB. If the SINR of S-R links is used in threshold selection, an obvious diversity gain can be observed in RAD-MUD, which outperforms cooperative MMSE MUD by 8 dB. 5.6.2 Cases With Direct Links Combined In this section, it is assumed that channel coefficients h S k R l are i.i.d. for all k and l with distributionCN(0,1) and that all sources and relays have equal distance to D such that h S k D ,h R l D ∼CN(0,1/16), for all k and l. This corresponds to the case where sources and relays are located in the vicinity of each other and are all sufficiently far from the destination such that the distances to the destination can be considered as equal. Each source transmits with equal power P s and the transmit power sum of sources is equal to the transmit power sum of relays, i.e., KP s = P R = P. The spreading waveforms are generated randomly with non-singular correlation matrices. That is, received signals in both phases have comparable influence in signal detection after performing MMSE combining as described in Sec.5.4.2. The variance of the AWGN is equal to 1 for all receivers. ThespreadinggainisN =8. Atthedestination, theMMSEmultiuserdetector isderivedunderthesimplifyingassumptionthatθ l,k isindependentofrelays,whichresults in suboptimal detection. For a cooperative network with K =8 sources and L=8 relays, we first compare the performanceofRAD-MUDwiththedifferentcooperativetransmissionstrategiesatrelays after MMSE combining at the destination. Similar to Sec.5.6.1, we compare RAD-MUD 106 with the cooperative MMSE-MUD in the following experiments. The performance of the direct transmission with the total transmit power equal to 2P is shown as a benchmark. Signals received at the destination in both phases are combined using the low complexity MMSE combining method given in (5.23). 8 16 24 32 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 Total Transmit Power P(dB) Bit Error Rate Direct Transmission Cooperative MMSE−MUD, BF R−D Cooperative MMSE−MUD, BF w.SEL Cooperative MMSE−MUD, BF S−R−D RAD−MUD, BF R−D RAD−MUD, BF w.SEL RAD−MUD, BF S−R−D Figure 5.7: BER performance comparison for systems employing transmit beamforming and performing MMSE combining at the destination. The BER performance as a function of the total transmit power P is shown in Figs. 5.7, 5.8 and 5.9 for transmit beamforming, selective relaying and DSTC, respectively. We see that RAD-MUD outperforms the cooperative MMSE-MUD in all cases. The improvementwithRAD-MUDismostpronouncedforcooperative transmissionstrategies that yield large cooperative diversity gains since MAI may dominate the BER in these cases and it becomes increasingly important to mitigate MAI effectively. 107 10 15 20 25 30 35 40 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Total Transmit Power P (dB) Bit Error Rate Direct Transmission Cooperative MMSE−MUD, Threshold SEL Cooperative MMSE−MUD, Best SEL RAD−MUD, Threshold SEL RAD−MUD, Best SEL Figure 5.8: BER performance comparison for systems employing selective relaying and performing MMSE combining at the destination. InFig. 5.7,wecomparetheperformanceofCooperativeMMSE-MUDandRAD-MUD forthreebeamformingschemes: beamformingS-R,beamformingS-R-Dandbeamforming withthresholdselection. Weseethat,byweightingrelaysymbolswithdetectionreliability atrelaysinbeamformingS-R-D,wecanimprovetheBERperformanceoverbeamforming R-D significantly. The beamforming with threshold selection serves as a compromise between the above two schemes since it allows a relay to forward the symbol only when thedetectionisreliable. TheimprovementwithRAD-MUDisthelargestforbeamforming S-R-D since this scheme yields the largest cooperative gain. In Fig. 5.8, the threshold selection and the best selection strategies are compared, where γ T , in the threshold selection strategy, is obtained numerically to minimize the average of BER at D. We see that best selection outperforms threshold selection since 108 8 16 24 32 40 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Total Transmit Power P(dB) Bit Error Rate Direct Transmission Cooperative MMSE−MUD, DSTC RAD−MUD, DSTC Figure 5.9: BER performance comparison for systems employing DSTC performing and MMSE combining at the destination. it utilizes global CSI. More interestingly, the increase in the diversity order for the best selection scheme is larger for RAD-MUD than the cooperative MMSE-MUD since the performance is less restricted by MAI. When DSTC is applied, RAD-MUD outperforms cooperative MMSE-MUD by 7 dB when BER=10 −3 as shown in Fig. 5.9. We consider the case with K =L=2 and compare the BER performance forρ=0.25 and 0.75 in Fig. 5.10, where ρ= R s 1 (t)s 2 (t)dt is the correlation between two spreading codes. The direct transmission, equal gain and best selection methods are adopted in both cooperative MMSE-MUD and RAD-MUD. As shown in the figure, RAD-MUD effectively combats MAI when the correlation between two spreading codes is high, i.e., ρ=0.75. When ρ=0.25, MAI is small and RAD-MUD and cooperative MMSE-MUD have comparable performance. 109 10 15 20 25 10 −3 10 −2 10 −1 Total Transmit Power P(dB) Bit Error Rate Direct Transmission, r=0.75 Direct Transmission, r=0.25 Cooperative MMSE−MUD, Equal, r=0.75 Cooperative MMSE−MUD, Equal, r=0.25 Cooperative MMSE−MUD, Best SEL, r=0.75 Cooperative MMSE−MUD, Best SEL, r=0.25 RAD−MUD, Equal, r=0.75 RAD−MUD, Equal, r=0.25 RAD−MUD, Best SEL, r=0.75 RAD−MUD, Best SEL, r=0.25 Figure 5.10: BER comparison of the case with two users and two relays with correlation coefficients ρ=0.75 and 0.25 In Fig. 5.11, the BER of the best selection scheme is shown for K=8 and L=2,4,8. We see that the increase in diversity is more evident for RAD-MUD as the number of relays increases. When BER=10 −4 , we see 3.5 dB improvement as L increases from 2 to 8 with cooperative MMSE-MUD, and 7 dB improvement with RAD-MUD. To show the effectiveness of the low complexity MMSE combining method in (5.23), we compare the BER performance of the optimal MMSE combining (dashed-line) and the alternative low-complexity method (solid-line) in Fig. 5.12 for the equal gain, selective relaying, and beamforming strategies. As shown in the figure, the alternative method yields little performance loss while demanding much lower decoding complexity. 110 8 16 24 32 40 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Total Transmit Power P (dB) Bit Error Rate Direct Transmission Cooperative MMSE−MUD, L=2 Cooperative MMSE−MUD, L=4 Cooperative MMSE−MUD, L=8 RAD−MUD, L=2 RAD−MUD, L=4 RAD−MUD, L=8 Figure 5.11: The BER performance as a function of total transmit power for K =8 and L=2 (circles), 4 (triangles) and 8(stars). 8 16 24 32 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Total Transmit Power P (dB) Bit Error Rate Equal Threshold SEL Best SEL BF R−D BF w. SEL BF S−R−D Figure5.12: PerformancecomparisonbetweenMMSEjointdecodingin(5.19)(thedashed line) and the alternative low-complexity MMSE combining in (5.23) (the solid line). 111 5.7 Conclusion Therelay-assisted decorrelating multiuserdetector (RAD-MUD) scheme wasproposedto decouple users’ signals at the base-station without noise amplification. This is achieved by precoding transmitted messages at the relays along with pre-whitening of the received signals at the destination. Three cooperative transmission schemes were studied on top of the RAD-MUD system. They are transmit beamforming, selective relaying, and dis- tributed space time coding. The performance of the proposed RAD-MUD scheme was studied and shown to outperform cooperative MUD without relay-assisted decorrelation with computer simulation in representative test cases. 5.8 Appendix: Derivation of Beamforming Coefficients in (5.25) Here, we derive the beamforming coefficients in (5.25). Given ˘ y II,k [m], the MMSE esti- mate of symbol x k [m] is computed and the corresponding MSE is equal to min c k E[|c k ˘ y II,k [m]−x k [m]| 2 ]=E|x k [m]| 2 − |E[x k [m]˘ y ∗ II,k [m]]| 2 E|˘ y ∗ II,k [m]| 2 =1− |p H k w k | 2 w H k Φ k w k +σ 2 v , wherep k andΦ k are defined as in Sec. 5.5.1. Under the power constraint|w k | 2 =P R /K, ∀ k, beamforming coefficients are given by w k =arg min w:|w| 2 = P R K 1− |p H k w| 2 w H (Φ k + Kσ 2 v P R I)w . (5.28) 112 Then, by definingu= Φ k + Kσ 2 v P R I 1 2 w, the optimization problem can be expressed as u k =argmin u 1− p H k (Φ k + Kσ 2 v P R I) −1 2 u |u| 2 . (5.29) By the Cauchy-Schwarz inequality, the minimum of MSE is attained whenu k is given by u k =β k,SRD Φ k + Kσ 2 v P R I −1 2 p k , where β k,SRD is a non-zero arbitrary constant. It follows that w k =β k,SRD Φ k + Kσ 2 v P R I L×L −1 p k and, to satisfy the power constraint, we set β k,SRD = v u u t P R /K p H k Φ k + Kσ 2 v P R I L×L −1 p k . 5.9 Appendix: Diversity of Best Selection Strategy ThebestselectionstrategyinRAD-MUDachievesfulldiversityasshowninthefollowing. Without loss of generality, source S 1 is the user of interest while R l ∗ is the selected relay. For simplicity, we choose P S =P R /K=P and σ 2 S k R l =σ 2 R l D =σ 2 v =σ 2 n =1, for all k and l. In the meanwhile, we consider the case where [R] i,j =ρ<1, for i6=j and [R] i,i =1, for all i (i.e., the set of spreading codes is generated using the shifted versions of m-sequences 113 [56]). Without considering the direct transmission, the outage probability[38] of user 1 is given by P out,1 =Pr 1 2 min{log(1+SINR S 1 R l ∗ ),log(1+SNR R l ∗D )}<λ =Pr{γ l ∗ ,1 <γ}, (5.30) where λ is the target rate and γ=2 2λ −1. With best selection, we choose relay R l ∗ such that γ l ∗ ,1 =max l γ l,1 and the outage probability can be written as P out,1 =Pr{max l γ l,1 <γ}=Pr{γ l,1 <γ,∀ l=1,2,··· ,L}. (5.31) Since γ l,1 =min [Γ l ] 1,1 1−[Γ l ] 1,1 ,P|h R l D | 2 , forΓ l in (5.16), we see that the value of γ l,1 depends only on h S 1 R l ,··· ,h S K R l and h R l D . Therefore, γ 1,1 ,γ 2,1 ,··· ,γ L,1 are independent and the outage probability can be written as P out,1 = L Y l=1 Pr(γ l,1 <γ) = L Y l=1 [1−Pr(γ l,1 ≥γ)] = L Y l=1 1−Pr min [Γ l ] 1,1 1−[Γ l ] 1,1 ,P|h R l D | 2 ≥γ = L Y l=1 1−Pr [Γ l ] 1,1 1−[Γ l ] 1,1 ≥γ Pr P|h R l D | 2 ≥γ . (5.32) Note that, from (5.16) and the fact that σ 2 n =1, we have Γ l =H H SR l (H SR l H H SR l +σ 2 n R −1 ) −1 H SR l =(I+H −1 SR l R −1 H −H SR l ) −1 . 114 Then, we can show that I−Γ l (a) = H −1 SR l (R+H −H SR l H −1 SR l ) −1 H −H SR l (b) = H −1 SR l h ρe K e T K +(1−ρ)I K×K +H −H SR l H −1 SR l i −1 H −H SR l , where (a) follows from the matrix inversion lemma, which states that (A+BCB H ) −1 =A −1 −A −1 B(C −1 +B H A −1 B) −1 B H A −1 , by taking A=I, B=H −1 SR l , C=R −1 and (b) follows by rewriting the correlation matrix as R=ρe K e T K +(1−ρ)I K×K , where e K is a K×1 all-1 vector. By applying the matrix inversion lemma again, with A=(1−ρ)I K×K +H −H SR l H −1 SR l (which is diagonal), B=e K and C=ρ, we have (R+H −H SR l H −1 SR l ) −1 =diag(κ l,1 ,···,κ l,K )− [κ l,1 ,···,κ l,K ] T [κ l,1 ,···,κ l,K ] ρ −1 + P K k=1 κ l,k (5.33) where κ l,k = P|h S k R l | 2 1+(1−ρ)P|h S k R l | 2 . Thus, it follows that the (1,1) entry ofI−Γ l is [I−Γ l ] 1,1 = 1 P|h S 1 R l | 2 κ l,1 − κ 2 l,1 ρ −1 + P K k=1 κ l,k ! . Moreover, since [Γ l ] 1,1 [I−Γ l ] 1,1 = 1 [I−Γ l ] 1,1 −1, we have Pr [Γ l ] 1,1 1−[Γ l ] 1,1 ≥γ =Pr P|h S 1 R l | 2 1−ρ+ 1 V MAI ≥γ , (5.34) 115 where V MAI = 1 ρ + K X k=2 P|h S k R l | 2 1+P|h S k R l | 2 (1−ρ) . Note thatV MAI depends on the the channel from sources S 2 , S 3 , ..., S K to R l . Assume thatthecorrelationcoefficientρisstrictlylessthanone. AthighSNR(i.e., P≫1),V MAI can be approximated as a deterministic variable V MAI ≈ 1 ρ + K−1 1−ρ and the probability in (5.34) can be written as Pr [Γ l ] 1,1 1−[Γ l ] 1,1 ≥γ ≈exp − γ P 1+ρ(K−2) 1+ρ(K−2)−ρ 2 (K−1) . (5.35) Finally, by substituting (5.35) into (5.32) and from the fact that Pr{P|h R l D | 2 ≥ γ} = exp −γ/P , h R l D ∼CN(0,1), we have P out,1 ≈ L Y l=1 1−exp − γ P 1+ρ(K−2) 1+ρ(K−2)−ρ 2 (K−1) exp(− γ P ) ≈ γ P 2(1−ρ)+ρ(2−ρ)(K−1) (1−ρ)[1+(K−1)ρ] L = exp −Lln P γ (1−ρ)[1+(K−1)ρ] 2(1−ρ)+ρ(2−ρ)(K−1) = exp −L ln P 2 2R −1 +lnC(ρ,K) , whereC(ρ,K)= (1−ρ)[1+(K−1)ρ] 2(1−ρ)+ρ(2−ρ)(K−1) ∈[0,1)representstheperformancedegradationcaused byMAI.Hence,wehaveshownthatthebestselectionstrategyachievesthediversityorder equal to the number of relays L even in the presence of MAI. 116 Chapter 6 Relay-Assisted Multiuser Detection for Amplify-and-Forward Asynchronous Cooperative Uplink Networks 6.1 Introduction Consider the uplink of a cooperative CDMA system where users cooperate by relaying each other’s messages to the base station. This is an application scenario of cooperative communications studied in [23, 38, 51, 52]. At each time instance, some users serve as source nodes while others serve as relay nodes. Under this context, most existing work studies cooperative strategies where there is one single source node or there are multiple source nodes and each transmits over orthogonal channel [18, 39, 40]. However, the requirement of orthogonal channels is difficult to achieve in practice and the multiple access interference (MAI) among channels may not be ignored. Recently, cooperative uplinksystemswhereuserstransmitdatausingnon-orthogonalspreadingwaveformswere studied in [9, 27, 28, 58]. With the multiuser detection (MUD) [60] employed at relays, the advantage of user cooperation is observed in the presence of MAI. Nevertheless, in 117 all previous work (including Chapter 5), transmissions among all sources and relays are assumed to be synchronous, which is an idealistic assumption. Since the transmission delays between all source nodes and relays are different, it is difficult or prohibitive to achieve the synchronous assumption. In this chapter, we want to relax the synchronous requirement of transmissions from all sources to each relay, and study the diversity gain in such an environment. In asynchronous code division multiple access (CDMA) uplink systems, MUD has been studied extensively to mitigate MAI and inter-symbol-interference (ISI). The opti- mum detection was proposed in [59] to minimize the error probability, but its complexity is exponentially proportional to the number of users. To lower the complexity, several suboptimum linear multiuser detectors were proposed, such as the decorrelating detector [44], theminimummean-squareerror(MMSE)detector [34,64]andthedecisionfeedback detector [16]. The decorrelating MUD effectively mitigates the interference, but may re- sult innoise amplification. Thelinear MMSE MUD cannot eliminate all interference, but controlstheresidualnoiseandinterferencetoacertaindegree. Thedetectionperformance of the decision feedback detector can be improved with the feedback of correct decisions, but may suffer from error propagation with error decisions at low signal-to-noise ratio (SNR). The outage performance of asynchronous cooperative uplinks was studied in [57], wherethedecode-and-forwardrelayingtechniquesandspace-timecoding[39]wereapplied at relays. Under the assumption that transmissions among nodes are asynchronous, loss of spectral efficiency due to non-orthogonal transmissions and asynchronism has been 118 examined. However, practical multiuser detection methods in the asynchronous CDMA uplink system was not addressed in [57]. In this chapter, we consider cooperative networks with asynchronous transmissions between sources and relays. Instead of decode-and-forward relaying, we consider the amplify-and-forward relay method to prevent errors caused by the hard decision made at relays. The transmission between relays and the destination is assumed to be syn- chronous since it is affordable to achieve synchronous with position information. Under the synchronous assumption, transmitted symbols from all relays can be decoupled with relay-assisted decorrelating MUD (RAD-MUD) as proposed in Chapter 5 [27, 28], which performs one half of decorrelating operations at relays and the other half decorrelating operations at the destination. RAD-MUD is able to construct parallel channels between relays and the destination to prevent additional MAI. Moreover, with optimal weight- ing or selection at relays, estimated symbols suffer from severe interference at certain relays may be suppressed while being enhanced at other relays where the interference is less severe. Hence, the cooperative diversity is used to combat fading and mitigate MAI. With the channel information available at relays, we study three cooperative relay- ing schemes on top of RAD-MUD: (1) transmit beamforming, (2) selective relaying and (3) distributed space-time coding (DSTC). The diversity gain with user cooperation in asynchronous CDMA uplink systems is compared with that in synchronous uplink with- out cooperation in Sec. 6.6. Moreover, it is shown that RAD-MUD can mitigate MAI effectively especially with cooperative relaying schemes. 119 6.2 System Model Consider a cooperative network where users cooperate by relaying each other’s message. As shown in Fig.6.1, there are K source users, denoted by S = {S 1 ,··· ,S K }, and L cooperating users, denoted by R ={R 1 ,··· ,R L }, which serve as relay nodes at a time instance. The cooperative transmission takes two phases. In the first phase, each source node, say node S k , transmits M data symbols x k [m],m = 1,··· ,M using dedicated spreading waveforms with transmission power P S k . The signal transmitted by the source node can be expressed by u k (t)= M X m=1 p P S k x k [m]s k (t−mT s −τ (k) l ), k =1,2,··· ,K, where each data symbol x k [m] ∈ M is identical and independently distributed (i.i.d.) with unit power and M is the signal constellation to modulate the data. Let N be the spreading gain, the spreading waveforms assigned to source node S k can be expressed as s k (t) = 1 √ N N X n=1 c k [n]ϕ(t−nT c ), k =1,··· ,K, where c k [n] is the n-th element of the±1 spreading sequence assigned to S k , and ϕ(t) is the normalized chip waveform with unit energy and duration T c =T s /N. Itisassumedinmostpreviouswork, e.g.,[9,27,28,58], thatsignals transmittedfrom source nodes arrive at all relays simultaneously. However, since path delays from source users to all relays are different, synchronism at all relays is difficult to achieve. Here, 120 S 1 S 2 S K R 1 R 2 R 3 R L D .... .... ... h S1R 1 h S2R 1 h SKR 1 h R1D h S1R L h S2R L h SKR L h R2D h RLD Figure 6.1: The system model of a cooperative network with multiple users. we consider asynchronous transmission among sources and relays. The signal received at relay R l is given by y l (t) = K X k=1 M X m=1 p P S k h S k R l x k [m]s k (t−mT s −τ (k) l )+n l (t), (6.1) where h S k R l ∈CN(0,σ 2 S k R l ) and τ (k) l ∈ [0,T s ) are the channel coefficient and the trans- mission delay from source S k to relay node R l , and n l (t) is the AWGN with power σ 2 n . It is assumed that the channel state and transmission delays τ (1) l ,··· ,τ (K) l are known at relay R l . At relay node R l , the received signal y l (t) is first passed through a match 121 filter bank (MFB) corresponding to s k (t−τ (k) l ),k = 1,··· ,K. During the m-th symbol period, the output of the MFB is given by y l [m] = R l [0]H SR l x[m]+R l [1]H SR l x[m−1]+R l [−1]H SR l x[m+1]+n l [m](6.2) = (R l [m]H SR l )∗x[m]+n l [m], (6.3) wherex[m] =[x 1 [m],··· ,x K [m]],H SR l =diag( p P S 1 h S 1 R l ,··· , p P S K h S K R l ),andR l [m](m = −1,0,1) is the correlation matrix of asynchronous waveforms with [R l [m]] k,k ′ = Z s k ′(t−τ (k ′ ) l )s k (t+mT s −τ (k) l )dt. The noise vector at the MFB output is denoted by n l [m] with the power spectral den- sity(PSD) function equal to S n l (ω) =σ 2 n R l (ω) =σ 2 n (R l [−1]e jω +R l [0]+R l [1]e −jω ). In the second phase, the MFB outputs at R l , i.e.,Y l =[y l [1],y l [2],··· ,y l [M]] is pre- codedasaK×M matrixT l =[t l [1],t l [2],··· ,t l [M]],wheret l [m] =[t l,1 [m],··· ,t l,K [m]] T . The proposed precoding procedure will be described in Sec. 6.3. Precoded symbolst l [m] are re-transmitted by relay R l using the set of spreading waveforms {s 1 (t),··· ,s K (t)} during the m-th symbol period. More specifically, the signal transmitted by node R l in the second phase is given by u l (t) = M X m=1 K X k=1 t l,k [m]s k (t−mT s ). 122 It is assumed that transmission delays from relays to the destination can be compen- sated with the position information so that signals arrive at the destination simultane- ously. In the second phase, the signal received at the destination is given by y D (t) = M X m=1 K X k=1 L X l=1 h R l D t l,k [m]s k (t−mT s )+v D (t), where h R l D is the complex channel coefficient from R l to D, and v D (t) is the AWGN with variance σ 2 v . Channel coefficient h R l D is assumed to be i.i.d. circularly symmetric complex Gaussian with zero mean and variance σ 2 R l D . The received signal in the second phase is then passed through the MFB corresponding to s k (t−mT S ), k = 1,2,··· ,M. During the m-th symbol period in the second phase, the MFB output is given by y D [m]= L X l=1 h R l D Rt l [m]+v D [m], (6.4) where R is the correlation matrix of spreading waveforms with the (k,k ′ )-th element [R] k,k ′ = R Ts 0 s k (t)s k ′(t)dt, and v D [m] is the Gaussian noise vector with distribution CN(0 K×1 ,σ 2 v R). Signal detection will be discussed in Sec.6.4. 6.3 Relay-Assisted Decorrelation at Relays The received signal at each relay contains the multiple access interference (MAI) and white noise as given in (6.3). To mitigate MAI propagated from relays, the received signal at the l-th relayy l [m] is first passed through a linear minimum mean square error (MMSE) estimator to obtain an estimate of data symbols. Let z l [m] be the MMSE 123 estimate ofx[m] atR l , which will bedescribedinSec. 6.4. Insteadof using harddecision for data symbol detection, estimates of symbols are weighted and re-transmitted directly. The amplify-forward relaying method avoids detection errors at relays. The block diagram of the zero-forcing precoding scheme is shown in Fig. 6.2. We use Z l =[z l [1],··· ,z l [M]] to denote the block of M consecutive estimates of data symbols at the l-th relay. As shown in Fig. 6.2, the estimate of symbolsZ l is first mapped to g l (Z l ). Since the MMSE estimate of each data symbol is forwarded without hard decision, the residual multiple access interference (MAI) and Gaussian noise observed at relays are forwardedas well. To leverage the spatial diversity in multiusernetworks, theestimate of symbolswithsmallerresidualinterferenceplusnoiseshallbeenhanced,whiletheestimate of symbols with larger residual interference plus noise should be suppressed. With the residual SINR and the CSI of R-D links, mapping functions {g 1 (·),g 2 (·),··· ,g L (·)} are determined by the cooperative strategy employed at relays, e.g. transmit beamforming, selective relaying and distributed space time coding, which will be described in Sec. 6.5. Being similar to RAD-MUD proposed in Chapter 5, one half of decorrelating oper- ations is performed at relays by multiplying g l (Z l ) with L −H at the last stage of the precoder, where L is the Cholesky factorization of R subject to R = LL H . Thus, the symbols re-transmitted by the l-th relay T l =[t l [1],t l [2],··· ,t l [M]] is given by T l =L −H g l (Z l ). (6.5) 124 y 1 [i] L -H L -1 y 2 [i] L -H L -H . . . . . . . . . R 1 R 2 R L y L [i] y II [i] v D [i] y II [i] g 1 ( . ) g 2 ( . ) g L (.) t 1 [i] t 2 [i] t L [i] MUD MMSE Estimation MMSE Estimation MMSE Estimation Figure 6.2: The block diagram of RAD-MUD with zero-forcing precoding at relays. Mapping g l (·) must satisfy the aggregate relay transmit power constraint, i.e., L X l=1 E Z ∞ ∞ K X k=1 t l,k [m]s k (t−mT s ) 2 dt = L X l=1 E t l [m] H Rt l [m] = L X l=1 E g l (Z l )[m] H g l (Z l )[m] ≤P R , (6.6) where g l (Z l )[m] is the i-th column of g l (Z l ). Withprecodingatrelays,theMFBoutputatthedestinationin(6.4)canbere-written as y D [m] = L X l=1 h R l D RL −H g l (Z l )[m]+v D [m]. (6.7) The MFB output in (6.7) is first pre-multiplied withL −1 , which results in ˘ y D [m]=L −1 y D [m]= L X l=1 h R l D g l (Z l )[m]+˘ v D [m]. (6.8) 125 Note that the resultant noise vector ˘ v D [m] =L −1 v D [m] has covariance matrix σ 2 v I K×K . That is, the multiplication of L −1 carries out one half of decorrelating operation as well as noise whitening at the destination. With the relay assisted decorrelation, symbols in ˘ y D are decoupled without power expansion at the relays or noise amplification at the destination. They only depend on g l (Z l ). 6.4 MMSE Multiuser Detection at Receivers Although the transmission over different spreading waveforms can be decorrelated for linksfromrelaystothedestination,theoveralldetectionperformancereliesontheresidual SINR of symbol estimates at relays. In this section, symbol estimation at relays and MMSE multiuser detection (MMSE-MUD) at the destination will be described. 6.4.1 MMSE Estimation at Relays We examine the MMSE estimation performed at relays 1 . LetC l (ω) = P C l [m]e −jmω be the transfer function of the MMSE estimator at relay R l . Based on (6.3), the output of the estimator can be expressed as z l [m]=C[m]∗y l [m]=(C l [m]∗R l [m]H SR l )∗x[m]+C l [m]∗n l [m], l=1,2,··· ,L. (6.9) Estimator C l (ω) is chosen to minimize the mean square error (MSE) between z l [m] andx[m], i.e.,E |C l [m]∗y l [m]−x[m]| 2 . TheMMSE estimator canbeobtained bythe 1 It is worthwhile to mention that the MMSE estimation is just an example and other estimation methods can be used for cooperative transmission on top of RAD-MUD as well. 126 orthogonality principle, i.e., E (C l [m]∗y l [m]−x[m])y H l [m ′ ] = 0, for any integer m ′ , and it follows that C l [m]∗R y l [m] =R xy l [m], l =1,2,··· ,L, (6.10) whereR y l [m] =E[y l [m ′ ]y l [m ′ −m] H ]istheautocorrelationfunctionofy l [m]andR xy l [m] = E[x[m ′ ]y l [m ′ −m] H ] is the cross-correlation function betweenx[m] andy l [m]. Then, the transfer function of the MMSE estimator C l (ω) is obtained by taking the discrete time Fourier transform on both sides of (6.10) as C l (ω) =S xy l (ω)S −1 y l (ω), l =1,2,··· ,L, (6.11) where S xy l (ω) = F{R xy l [m]} =(R l (ω)H SR l ) H S y l (ω) = F{R y l [m]} =R l (ω)H SR l H H SR l R H l (ω)+σ 2 n R l (ω). With estimators in (6.11), the MMSE estimate at R l can be expressed by z l [m]=Γ l [m]∗x[m]+C l [m]∗n l [m]=Γ l [0]x[m]+ X m ′ 6=0 Γ l [m ′ ]x[m−m ′ ]+C l [m]∗n l [m], (6.12) 127 whereΓ l [m]=C l [m]∗R l [m]H SR l has the following transfer function Γ l (ω) = C l (ω)R l (ω)H SR l = H H SR l R H l (ω) R l (ω)H SR l H H SR l R H l (ω)+σ 2 n R l (ω) −1 R l (ω)H SR l . The correlation matrix ofz l [m] can be computed as E z l [m]z l [m] H = 1 2π Z π −π C l (ω)S y l (ω)C H l (ω) dω= 1 2π Z π −π Γ l (ω) dω=Γ l [0]. To analyze the SINR of each estimate, z l,k [m] in (6.12) can be re-written as z l,k [m] =[Γ l [0]] k,k x k [m]+ξ l,k [m], (6.13) where [B] i,j is the (i,j)-th element in matrixB and ξ l,k [m] consists of the residual MAI, inter-symbol interference (ISI) and Gaussian noise. Since ξ l,k [m] is independent of x k [m], we have E[|ξ l,k [m]| 2 ] =E[|z l,k [m]| 2 ]−[Γ l [0]] 2 k,k . Thus, the SINR of the MMSE estimate z l,k [m] is given by SINR S k R l = [Γ l [0]] 2 k,k [Γ l [0]] k,k −[Γ l [0]] 2 k,k = [Γ l [0]] k,k 1−[Γ l [0]] k,k . 6.4.2 MMSE Multiuser Detection at Destination It is assumed that the precoding strategy at relays and all channel coefficients and the correlation matrix among spreading waveforms are known at the destination. We use the MMSE linear estimator to detect data symbolsat the destination. Supposethat g l (Z l )= 128 W l Z l (e.g. in the case of beamforming and selective relaying [c.f. Sec. 6.5]), where W l =diag(w l,1 ,··· ,w l,K ) is a diagonal weighting matrix whose elements are determined by the transmission strategy subject to the total power constraint 2 . Based on (6.8) and (6.12), the signal received at the destination can be written as ˘ y D [m] = L X l=1 h R l D W l z l [m]+ ˘ v D [m] = L X l=1 h R l D W l Γ l [m]∗x[m]+ L X l=1 h R l D W l C l [m]∗n l [m]+˘ v D [m]. (6.14) LetC D (ω)= P C D [m]e −jmω be the transfer function of the MMSE linear estimator applied on ˘ y D [m]. By the orthogonality principle,C D (ω) can be found as C D (ω)=F{E x[m ′ ]˘ y D [m ′ −m] H }F{E ˘ y D [m ′ ]˘ y D [m ′ −m] H } −1 = L X l=1 h R l D W l Γ l (ω) ! H L X l=1 L X l ′ =1 h R l D h ∗ R ′ l D W l S z l z l ′ (ω)W H l ′ +σ 2 v I K×K ! −1 , (6.15) where S z l z l ′ (ω) =F{E[z l [m ′ ]z l ′[m ′ −m]]} = Γ l (ω), l =l ′ . Γ l (ω)Γ l ′(ω), otherwise. The MMSE estimate of x[m] at the destination is then given by z D [m]=C D [m]∗˘ y D [m]=Γ D [m]∗x[m]+C D [m]∗ L X l=1 h R l D W l C l [m]∗n l [m]+˘ v D [m] ! , 2 For the case of distributed space time coding, the maximum likelihood detection shall be used at the destination as described in Sec. 6.5.3 129 whereΓ D [m]=C D [m]∗ P L l=1 h R l D W l Γ l [m] has the following transfer function Γ D (ω)= L X l=1 h R l D W l Γ l (ω) ! H L X l=1 L X l ′ =1 h R l D h ∗ R ′ l D W l S z l z l ′ (ω)W H l ′ +σ 2 v I K×K ! −1 L X l=1 h R l D W l Γ l (ω) ! . It is worthwhile to mention that the autocorrelation function ofz D [m] is equal toΓ D [m]. The mean-squared-error of estimation z D [m] is given by MSE=trace E (x[m]−z D [m])(x[m]−z D [m]) H =trace(I K×K −Γ D [0]), (6.16) where Γ D [0] = 1 2π Z π −π Γ D (ω) dω. Finally, data symbols are detected using hard decision on z D [m]. For instance, if symbols are modulated by QPSK, they are detected via ˆ x[m] = 1 √ 2 (sgn(ℜ{z D [m]})+j∗sgn(ℑ{z D [m]})). By the Gaussian approximation, the bit error rate (BER) of S k ’s data symbols with QPSK modulation is given by BER(x k )=Q q SINR(z D,k ) =Q s [Γ D [0]] k,k 1−[Γ D [0]] k,k ! . (6.17) The bit error rate of the system with different cooperative strategies will be numerically compared in Sec. 6.6. 130 6.5 Cooperative Transmission Strategies With RAD-MUD, relays constructK orthogonal channelsto forwardMMSEestimates of source’s symbols. Thisis similar to the single-source case where relays from a distributed antennaarraytoincreasethediversityoftransmissionfromeachsourcetothedestination. Each symbol’s estimate contains the desired data symbol, as well as MAI, ISI and the Gaussian noise, which should be considered in mapping function g l (·). Based on CSI and correlation properties available at relays, three cooperative strategies, i.e., transmit beamforming, selective relaying and distributed space time coding (DSTC), are used to construct mapping functions g 1 (·),··· ,g L (·) exploit the spatial diversity in the multiuser system. For transmit beamforming and selective relaying strategies, the mapping function at R l can be expressed as g l (Z l ) =W l Z l , where W l = diag(w l,1 ,w l,2 ,··· ,w l,k ) is a K×K diagonal weighting matrix. Theweighting factorshave tomeet thetotal powerconstraint in (6.6), i.e., L X l=1 trace E g l (Z l )[m]g l (Z l )[m] H = L X l=1 trace W l E z l [m]z l [m] H W H l = L X l=1 K X k=1 |w l,k | 2 [Γ l [0]] k,k ≤P R , (6.18) where the transmission power of the estimate of x k at R l is|w l,k | 2 [Γ l [0]] k,k . For fairness, weighting factors for each source’s symbols are subject to the following individual power constraint: L X l=1 |w l,k | 2 [Γ l [0]] k,k ≤P R /K, k =1,2,··· ,K. (6.19) 131 6.5.1 Transmit Beamforming When global CSI and correlation matrices are fully known, we consider transmit beam- formingthatoptimizestheweightingfactorsuchthattheSINRofreceivedsymbols ˘ y D [m] is maximized. Based on (6.14) and (6.13), the k-th symbol in ˘ y D [m] can be expressed as ˘ y D,k [m] = L X l=1 h R l D w l,k z l,k [m]+˘ v D,k [m] = L X l=1 h R l D w l,k [Γ l [0]] k,k x k [m]+ L X l=1 h R l D w l,k ξ l,k [m]+˘ v D,k [m]. Since ξ l,k [m] and ˘ v D,k [m] are independent of desired symbol x k [m], the SINR of ˘ y D,k [m] is given by SINR(˘ y D,k [m]) = P L l=1 h R l D w l,k [Γ l [0]] k,k 2 E|˘ y D,k [m]| 2 − P L l=1 h R l D w l,k [Γ l [0]] k,k 2 , (6.20) where E|˘ y D,k [m]| 2 =σ 2 v + L X l=1 L X l ′ =1 h l,D w l,D h ∗ l ′ ,D w ∗ l ′ ,D E[z l,k [m]z l ′ ,k [m]] andE[z l,k [m]z l ′ ,k [m]] = 1 2π R π −π S z l z l ′ (ω)dω. ForeachsourceS k ,thecorrespondingweight- ing factors, denoted by w k = [w 1,k ,w 2,k ,··· ,w L,k ] T , to maximize the SINR in (6.20) is given by w k =β BF Λ −1 k Σ k −h H RD h H RD Λ k + Kσ 2 v P R I L×L −1 h H RD , ,k =1,2,··· ,K, (6.21) 132 where h RD = [h R 1 D ,h R 2 D ,··· ,h R L D ], Λ k = diag([Γ 1 [0]] k,k ,[Γ 2 [0]] k,k ,··· ,[Γ L [0]] k,k ), and Σ k is an L×L matrix with elements [Σ k ] i,j = h j,D h ∗ i,D E[z j,k [m]z i,k [m]]. Constant β BF is chosen to meet the power constraint in (6.19). 6.5.2 Selective Relaying In only the channel gain is available yet without phase-coherent transmission, we may use the selective relaying strategy. That is, only one relay is selected to serve a specific source node. For source S k , a relay is selected if it can attain the maximal SINRin (6.20) subject to the transmit power constraint. Thus, for each source S k , w l,k = s P R K[Γ l [0]] k,k , only if l = arg max l ′ =1,···,L |w l ′ ,k | 2 |h R l ′D | 2 [Γ l [0]] 2 k,k |w l ′ ,k | 2 |h R l ′D | 2 ([Γ l ′[0]] k,k −[Γ l [0]] 2 k,k )+σ 2 v = arg max l ′ =1,···,L (P R /K)|h R l D | 2 [Γ l [0]] k,k (P R /K)|h R l D | 2 (1−[Γ l [0]] k,k )+σ 2 v ; (6.22) otherwise, w l,k = 0. When the relay has the knowledge of local CSI only, relay selection can be achieved by the opportunistic carrier sensing method proposed in [6]. 6.5.3 Distributed Space Time Coding If there is no channel information available at relays, we can simply employ DSTC to achieve the spatial diversity [39, 32]. Here, we consider space-time linear block codes with block length M, which has to be greater than L to attain full diversity [32]. Let 133 A l is an M×M unitary space time coding matrix randomly generated at relays. When DSTC is applied, g l (Z l ) = W l Z l A l , where W l is a diagonal weighting matrix. Then, transmitted symbols at R l are given by T l =L −H W l Z l A l , l =1,2,··· ,L. EachrowinZ l containsM consecutiveestimatesofS k ’ssymbols. Withmappingfunction g l (Z l ) at R l , the k-th row ofZ l is encoded withA l inthe time domain and then weighted byw l,k . Since matrixA l is unitary, the aggregate transmit power over M symbol periods remains the same for any unitaryA l . Thus, with DSTC, weighting factorsW 1 ,··· ,W L are set to meet the power constraint in (6.19). Weighting matrixW l can be designed to enhance the SINR value of the MMSE estimate at R l . If the global information of S-R linksisnotavailable,theweightingfactorcanbesimplysettow l,k = p (P R /KL[Γ l [0]] k,k ), for all k and l. At the destination, received symbols in M consecutive symbol periods ˘ Y D = [˘ y D [1], ˘ y D [2], ···, ˘ y D [M]] can be expressed as ˘ Y D = L X l=1 h R l D W l Z l A l + ˘ V D , where ˘ V D =[˘ v D [1],˘ v D [2],··· ,˘ v D [M]]. The optimal maximum likelihood (ML) solution of all data symbols has a complexity of O(|M| M×K ), where |M| is the size of symbol constellation. To reduce the complexity, we adopt a sub-optimal solution under the assumption that the interference-plus-noise term ξ l,k [m] in each estimate z l,k [m] [c.g. 134 (6.13)] is approximated by a Gaussian random variable and the correlation between each other is small enough to be neglected. Thus, symbols of each source can be detected separately by [ˆ x k [1],ˆ x k [2],··· ,ˆ x k [M]]=arg min b∈M 1×M ˘ Y D (k)−b L X l=1 h R l D w l,k [Γ l [0]] k,k A l ! 2 , (6.23) where ˘ Y D (k) is the k-th row of ˘ Y D . The performance of suboptimal ML detection will be examined in Sec. 6.6. 6.6 Performance Comparison and Computer Simulations We compare the performance of cooperative strategies on top of RAD-MUD with com- puter simulation in this section. We consider a cooperative network consisting of K = 8 source nodes and L=8 relays. In these experiments, channel coefficients h S k R l and h R l D are assumed to be i.i.d. for all k and l with distribution CN(0,1). This corresponds to the case that the distance between all sources and all relays and the distance from all relays to the destination are roughly equal. Transmission delay τ (k) l is i.i.d. for all l and k andtakes values from{ nTc 4 ,n=0,1,··· ,4N−1} withequal probability. Thevariances ofAWGN atall relays andthedestinationareset tounity, i.e., σ 2 n =σ 2 v =1. Each source transmits with equal power P S and the total transmit power of all sources is equal to the total transmit power of all relays, i.e., KP S = P R = P. Spreading waveforms are gen- erated randomly with non-singular correlation matrices with spreading gain N = 8. For comparison, the performanceofdirect transmission(i.e., without cooperation) withtotal transmit power 2P is plotted for benchmarking. The channel coefficient from sources to 135 the destination, h S k D , is assumed i.i.d. with distribution CN(0,1/2 2 ) and transmission between sources and the destination is assumed to be synchronous. We compare the following two precoding schemes at relays with RAD-MUD. 1. Zero-forcing precoding Each relay performs zero-forcing [61] before re-transmitting symbols, i.e., T l = β zf R −1 g l (Z l ), where β zf is chosen to meet the total transmit power constraint. 2. Cooperative MMSE-MUD without precoding EachrelaytransmitsthemappedestimateT l =β c g l (Z l )directly,whereβ c ischosen to meet the total power constraint. For the above two schemes, the linear MMSE estimation scheme presented in Sec. 6.4 is usedat all relays andthedestination. All source’s dataaremodulatedasQPSKsymbols. In Figs. 6.3–6.5, BERs of each source’s symbols are evaluated by (6.17) and averaged over 10000 channel realizations. In Fig. 6.3, we compared three schemes under the condition that the transmit power of each estimated symbol at relays is the same, i.e., w l,k = p P/(KL[Γ l [0]] k,k ), for all l and k. We see that the multiuser uplink system outperforms the synchronous direct transmission scheme although the S-R link is asynchronous and no specific cooperative strategy is applied. Clearly, the cooperative diversity is advantageous in the multiuser uplink system even with asynchronous S-R links. Besides, RAD-MUD is better than the other two precoding schemes by 2.5-3 dB since MAI caused by the transmission over R-D links is eliminated without transmit power expansion or receive noise amplification. 136 8 16 24 32 40 10 −3 10 −2 10 −1 10 0 Total Transmit Power P(dB) Bit Error Rate Direct Transmission Zero−Forcing, Equal MMSE cooperation, Equal RAD−MUD, Equal Figure 6.3: The BER performance comparison for the system using equally weighing factors. With cooperative strategies, the advantage of RAD-MUD over the other precoding schemes is more pronounced since MAI can be greatly reduced by RAD-MUD. We com- pare the BER performances of three schemes with transmit beamforming in Fig. 6.4. With global CSI, the BER performance with multiuser cooperation is greatly improved. Among the tree schemes, zero-forcing precoding performs the worse at low SNR because of power expansion at the transmitter. At high SNR, MAI caused by non-orthogonal transmission of R-D links is coheretly combined at the destination, which dominates the performance of cooperative MMSE-MUD scheme. With transmit beamforming, RAD- MUD outperforms the other two schemes by 8-10 dB at BER=10 −5 . We compare the BER performance of three selective relaying schemes in Fig. 6.5. The influence of MAI in selective relaying is less than that in transmit beamforming. 137 8 16 24 32 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Total Transmit Power P(dB) Bit Error Rate Direct Transmission Zero−Forcing, Beamforming MMSE cooperation, Beamforming RAD−MUD, Beamforming Figure 6.4: The BER performance comparison for the system using transmit beamform- ing. The cooperative MMSE scheme outperforms the zero-forcing scheme by 1.5 dB. The proposedRAD-MUD outperformsthe other precoding schemes with selective relaying by 5.5-7 dB. We compare the BER performance of three schemes with DSTC in Fig. 6.6. The averaged BER is obtained from the Monte Carlo simulation over 15000 random chan- nel and en-coding matrix realizations. We employ the linear MMSE estimator of fixed length N C l = 7 at relays [64]. We adopt the equal weight for DTSC. In all schemes, the suboptimal ML detection scheme as given in (6.23) is employed at the destination. For cooperative MMSE-MUD, theMFB outputisfirstmultipliedbyaone-tap matrixMMSE equalizer before sub-optimal ML detection. The proposed RAD-MUD outperforms other 138 8 16 24 32 10 −4 10 −3 10 −2 10 −1 10 0 Total Transmit Power P(dB) Bit Error Rate Direct Transmission Zero−Forcing, Selective MMSE cooperation, Selective RAD−MUD, Selective Figure 6.5: The BER performance comparison for the system using selective relaying. two schemes by 2-4 dB. As compared with RAD-MUD with equal weights in Fig. 6.3, DSTC has 15 dB performance gain at BER = 0.005 without requiring further channel information at relays. 6.7 Conclusion The use of cooperative relaying in a multi-user cooperative CDMA network was stud- ied, where the asynchronism of transmission between sources and relays was considered. The RAD-MUD technique is employed to perform one half of decorrelating at relays and the other half at the desitnation in order to decouple estimated symbols sent by all relays. RAD-MUD was used in association with three cooperation methods; namely, beamforming, selective relaying anddistributedspace-time coding(DSTC). It was shown 139 8 16 24 32 40 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Total Transmit Power P (dB) Bit Error Rate Direct Transmission Zero−Forcing, DSTC Cooperation MMSE−MUD, DSTC RAD−MUD, DSTC Figure 6.6: The BER performance comparison for the system using DSTC. that the use of RAD-MUD is advantageous to mitigate MAI even without any cooper- ative strategy applied. The effect of RAD-MUD is however more pronounced when the cooperative strategy is employed. With interfering source users, RAD-MUD was shown to outperform cooperative MMSE-MUD in preserving the diversity order. RAD-MUD is also better than zero-forcing precoding in combating noise amplification. 140 Chapter 7 Conclusion and Future Work 7.1 Conclusion Research work presented in this dissertation is briefly summarized below. First, we pro- posedandcomparedthreepowercontrolschemesfortheamplify-andforward(AF)space- time coded relay channel in Chapter 3. It was shown that the relay system benefits significantly from the power control schemes that are determined by the composite chan- nel strength of each relay path. By considering the finite battery lifetime at relays, a SNR-constrained power saving strategy was proposed to avoid the early power outage of the opportunistic scheme under a slow fading channel. Then, it was shown that the opportunistic scheme aided by the power saving strategy gives the best balance between link reliability and relay lifetime. At the end of the chapter, it was proved that, for the AF space-time codedrelay channels, the opportunisticscheme demandsminimumenergy consumption to achieve the SNR requirement at the destination. This implies that no DSTC is needed when the CSI is known at relays since only one single relay is used at any time instance. 141 Based on the selective relaying scheme where only one relay is selected to transmit at any time, power allocation strategies that incorporate REI and CSI to maximize the network lifetime were proposed and studied in Chapter 4. With a finite level of transmit power, the expected network lifetime of proposed strategies was derived via finite-state Markovchainanalysis. Itwasprovedanalytically thatthenetworklifetimeofcooperative networksisnolongermaximizedbytheminimumpowersolution(i.e.,theMTPstrategy). Furthermore, it was shown by computer simulation that the MEI strategy gives the longest average lifetime when the battery energy is sufficient while the MOP strategy gives better performance when the initial battery energy is comparable with the wasted energy. As compared with the optimal network lifetime by solving the stochastic shortest path problem, strategies that consider local REI and CSI jointly were shown to be nearly optimal with significantly lower complexity. The use of multi-user detection (MUD) in a cooperative CDMA network was inves- tigated for the uplink in synchronous CDMA systems in Chapter 5. We proposed the relay-assisted decorrelating multiuser detector (RAD-MUD) to decouple users’ signals at the base-station without noise amplification. This is achieved by precoding transmitted messages at relays along with pre-whitening of received signals at the destination. Three cooperative strategies; namely, beamforming, selective relaying and distributed space- time coding (DSTC), were studied on top of RAD-MUD. By computer simulation, we show that RAD-MUD systems outperform other cooperative MUD schemes. The perfor- mance gain of RAD-MUD is more pronounced in associated with cooperative strategies which exploit global CSI, which demonstrates that RAD-MUD is advantageous in MAI mitigation. 142 In Chapter 6, the use of RAD-MUD in association with user cooperation was in- vestigated in the cooperative uplink network under a more practical assumption, i.e. transmissions from all source nodes to each relay are asynchronous. At relay nodes, the amplify-and-forwardrelaying scheme isusedtoavoid detection errorscausedbyMAIand ISI. It is observed from computer simulation that the use of RAD-MUD is advantageous in mitigating MAI even without any cooperative strategy applied. With cooperative strategies, such as transmit beamforming, selective relaying and DSTC, the performance gain of RAD-MUD is much more obvious. This demonstrates that RAD-MUD can ex- ploit the multiuser diversity in a cooperative CDMA uplink system under the practical asynchronous assumption. 7.2 Future Work Studies on the use of MUD in asynchronous cooperative CDMA networks as reported in Chapter 6 can be continued along several directions. The optimization of the precoder at relay nodes was not done with that of the detector at the base station jointly due to the high complexity of joint optimization. However, a recursive algorithm to optimize the precoder and the detecor jointly should be interesting and worth further investiga- tion. The upper bound on the performance of the cooperative uplink system is another interesting topic. Theoretical performance analysis of asynchronous cooperative uplink systemswith a specificcooperative strategy ontop of RAD-MUD isstill lacking, e.g., the outage behavior and the BER performance (in terms of the ratio of the source number 143 and the relay number). To reduce the system complexity, some alternative scheme such as group detection and interference cancelation can be considered as well. Network coding [2] has been proposed to increase network capacity in recent years. The common underlying principle of cooperative communications and network coding is the provision of multiple paths by neighboring nodes distributed in the network. Redun- dantpacketsforwardedthroughmultiplepathsalsoprovidecertainprotectionfromtrans- mission failures. For a large scale network, adaptive network-coded cooperation (ANCC) was proposed in [4] to enhance spectral efficiency by coupling the network-on-graph with the code-on-graph (i.e., LDPC codes), and shown to outperform the conventional fixed cooperative strategy (e.g. DF) significantly. A joint network-channel coding based on LDPC was developed for a two-user uplink channel cooperating with one relay node, and a significant gain over the non-cooperating case was demonstrated. Most previous work was studied for some extreme cases with either a huge amount of relays or a single re- lay. With a limited number of intermediate relays, the use of distributed network coding and space-time coded cooperation may provide a feasible way to achieve diversity and, therefore, spectral efficiency. This could be another interesting future research topic. 144 Bibliography [1] M.M. AbdallahandH.C.Papadopoulos, “Beamforming algorithms fordecode-and- forward relaying in wireless networks,” in Proc. Conference on Information Sciences and Systems (CISS’05), 2005. [2] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inform. Theory, vol. 46, no. 4, pp. 1214–1216, July 2000. [3] J. G. Andrews, “Interference cancellation for cellular systems: A contemporary overview,” IEEE Wireless Commun. Mag., vol. 12, no. 2, pp. 19–29, Apr. 2005. [4] X. Bao and J. 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Abstract (if available)
Abstract
Cooperative communications have aroused considerable attention in recent years since it exploits the spatial diversity gains inherent in multi-user wireless systems without the need of multiple antennas at each node. In this dissertation, we consider to enhance the performance of energy-constrained ooperative networks in two aspects.
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Huang, Wan-Jen
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Core Title
Design and analysis of high-performance cooperative relay networks
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Viterbi School of Engineering
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Doctor of Philosophy
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Electrical Engineering (Multimedia and Creative Technology)
Publication Date
04/18/2008
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01/23/2008
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University of Southern California
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coopertive networks,lifetime maximization,multiuser detection,OAI-PMH Harvest,power allocation
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English
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Kuo, C.-C. Jay (
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