Power Allocation Schemes for Cooperative Communication System Using Weighted Sum Approach by Rajiv Devarajan B. Sc. Eng. (hons), University of Moratuwa, Sri Lanka, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2011 c Rajiv Devarajan 2011 Abstract This thesis investigates power allocation schemes for an amplify-and-forward dual-hop relay based cooperative communication system with perfect and imperfect channel state information (CSI). We define cost functions and propose power allocation schemes such that the cost functions are minimized. We analyze a multiuser system, where we select the best user for transmission, who incurs the least cost of transmission. In a practical system, estimated CSI is often imperfect. We assume the estimated CSI is affected by estimation errors, which are modeled as zero mean complex Gaussian. First we propose an optimization scheme where the objective is to minimize the weighted sum of source and relay powers. Then we propose a more general multi-objective optimization scheme which jointly optimizes sum power and signal-to-noise ratio (SNR). In our proposed schemes, source and relay nodes share a fixed total power, and transmission is allowed only if the minimum required SNR at the destination can be achieved with the available power budget. These schemes are analyzed under both perfect and imperfect CSI assumptions. In addition to proving the convexity of these problems, we propose analytical solutions for sum power minimization and SNR maximization schemes in the ii Abstract presence of imperfect CSI. Performance of the systems under the proposed schemes are investigated in terms of energy efficiency, throughput and outage. Simulation results show that proposed schemes reduce wastage of power by avoiding unsuccessful transmissions. iii Preface Parts of Chapter 3 have been published in and submitted to conference and journals, as mentioned below. • R. Devarajan, S. C. Jha, U. Phuyal, and V. K. Bhargava, Energy-Aware Resource Allocation for Cooperative Cellular Network Using MultiObjective Optimization Approach. Submitted to IEEE Transactions on Wireless Communications, May 2011. • R. Devarajan, S. C. Jha, U. Phuyal, and V. K. Bhargava, Energy-Aware User Selection and Power Allocation for Cooperative Communication System with Guaranteed Quality-of-Service. In Proceedings of IEEE CWIT, pages 1-5, May 2011. • R. Devarajan, A. Punchihewa, and V. K. Bhargava, Energy-Aware Power Allocation in Cooperative Communication System with Imperfect CSI. Submitted to IEEE Transactions on Communications, Nov. 2010. I am the primary author and researcher for the above mentioned manuscripts. I conducted majority of the work including but not limited to literature suriv Preface vey, identification and formulation of the problems, and proposing solutions. In addition, I implemented simulation models, analyzed results and prepared the manuscripts. Mr. S. C. Jha, Mr. U. Phuyal, Dr. A. Punchihewa, and Prof. V. K. Bhargava provided me technical guidance, and helped me in authoring the manuscripts. v Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . xii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Cooperative Communication 1.2 Literature Review 1.3 1.4 1 . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . 4 Thesis Contribution . . . . . . . . . . . . . . . . . . . . . . . 7 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . 8 vi Table of Contents 2 System and Channel Models . . . . . . . . . . . . . . . . . . . 10 2.1 System and Channel Description . . . . . . . . . . . . . . . . 10 2.2 Transmission Schemes . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Perfect CSI . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Imperfect CSI . . . . . . . . . . . . . . . . . . . . . . 13 3 Power Allocation With Perfect and Imperfect CSI . . . . . 18 3.1 Overview 3.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Power Allocation Schemes . . . . . . . . . . . . . . . . . . . . 20 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Weighted Sum Power Minimization with Perfect CSI (Scheme A) . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.2 Multi-Objective Optimization with Perfect CSI (Scheme B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3.3 Weighted Sum Power Minimization With Imperfect CSI (Scheme C) 3.3.4 . . . . . . . . . . . . . . . . . . . . . 29 Multi-Objective Optimization With Imperfect CSI (Scheme D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.5 3.4 Special Cases of Scheme D with Imperfect CSI Reduction of User Starvation 4 Numerical Results 4.1 . . . . 34 . . . . . . . . . . . . . . . . . . 41 . . . . . . . . . . . . . . . . . . . . . . . . . 43 Performance Analysis of Scheme A and Scheme B with Perfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 vii Table of Contents 4.2 Performance Analysis of Scheme C and Scheme D with Imperfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Performance Analysis of Special Cases of Scheme D with Imperfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.1 Performance of Total Power Minimization Scheme (w1D = 1, w2D = 0) 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . 63 Performance of SNR Maximization Scheme (w1D = 0, w2D = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3.3 Power Wastage Due to Ignoring the Imperfectness in CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . 74 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 viii List of Tables 4.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Jain’s Fairness Index for Different User Selection Schemes . . . 51 4.3 Jain’s Fairness Index for Different User Selection Schemes in the Presence of Channel Estimation Errors . . . . . . . . . . . 60 ix List of Figures 2.1 Three-node system model . . . . . . . . . . . . . . . . . . . . 10 3.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1 Variation of average energy efficiency per user against the endto-end SNR threshold. . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Variation of average throughput per user against the end-toend SNR threshold. . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Variation of average energy efficiency within a cluster for different number of users. Solid lines represent least cost user selection and dashed lines represent random user selection. . . 49 4.4 Variation of outage probability within a cluster for different number of users. Solid lines represent least cost user selection and dashed lines represent random user selection. . . . . . . . 50 4.5 Variation of average energy efficiency against the end-to-end SNR threshold using scheme C for (a) different weights, and (b) different estimation error variances. . . . . . . . . . . . . 54 x List of Figures 4.6 Variation of average energy efficiency against the end-to-end SNR threshold using scheme D for (a) different weights, and (b) different estimation error variances. 4.7 . . . . . . . . . . . . 56 Variation of average throughput against the end-to-end SNR threshold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.8 Variation of average energy efficiency against the number of users. Solid lines represent best user selection and dashed lines represent random user selection. . . . . . . . . . . . . . . . . . 58 4.9 Variation of average energy efficiency against the timeout period. 62 4.10 Average minimum total power versus SNR threshold for different channel estimation error variances. . . . . . . . . . . . . 64 4.11 Relative total transmission time versus SNR threshold for different channel estimation error variances. . . . . . . . . . . . . 66 4.12 Probability of transmit power shortage versus SNR threshold for different channel error variances. . . . . . . . . . . . . . . . 67 4.13 Variation of average maximum SNR with total power budget for different channel estimation error variances. . . . . . . . . 68 4.14 Probability of insufficient SNR versus SNR threshold for different channel estimation error variances. . . . . . . . . . . . . 70 4.15 Average maximum SNR versus source-relay distance for different channel estimation error variances. . . . . . . . . . . . . 71 4.16 Performance of the two schemes if estimation errors are ignored while power allocation. . . . . . . . . . . . . . . . . . . 73 xi List of Abbreviations AF Amplify-and-forward CF Compress-and-forward CSI Channel state information DF Decode-and-forward KKT Karush-Kuhn-Tucker MIMO Multiple input multiple output MMSE Minimum mean square error OFDM Orthogonal frequency division multiplexing RF Regenerate-and-forward SNR Signal-to-noise ratio ZMCG Zero mean complex Gaussian xii Acknowledgements First and foremost, I would like to express my gratitude to my supervisor, Prof. Vijay K. Bhargava, for the guidance, constructive input and support that he provided throughout my Master’s program. Furthermore, I would like to extend my utmost gratitude to my colleagues in the Information Theory and Systems laboratory for their continuous encouragement, valuable suggestions and comments. Also, I take this opportunity to thank all my friends in UBC, for making my life memorable and pleasant with their constant support. Finally, heartfelt and special thanks go to my family, who have always encouraged me and given their unconditional support throughout my career. xiii Dedication To my parents... xiv Chapter 1 Introduction 1.1 Cooperative Communication Wireless communication has become an integral part of our daily activities. For proper functioning of wireless communication systems, network operators and designers have to pay careful attention towards the issues and challenges in wireless communication systems [1]. Multiple-input multiple-output (MIMO) is a well known technique to realise an increase in spectral efficiency and link reliability in wireless communication, without additional bandwidth or transmit power [2]. Such performance improvements are achieved by the use of multiple antennas at the terminal nodes, providing spatial diversity gain [2]. However, incorporation of multiple antennas at mobile nodes may not be desirable due to increase in size of mobile devices. Cooperative communications is a technique which provides an alternative to MIMO systems to achieve spatial diversity gain [3]. In cooperative communications, geographically distributed relays are utilized to achieve spatial diversity. Relays provide multiple transmission paths from source to destination, improving transmission quality, network cover- 1 1.1. Cooperative Communication age, transmission reliability and energy efficiency [3], [4]. Relays can be fixed dedicated relays used for relaying purposes only, or relays can be other users in a multiuser network willing to cooperate by sharing their local resources [3]. Availability of alternative paths not only increases diversity, but also increases reliability. For example, if the direct path between the terminal nodes is absent due to deep fade [5], relay path can be used for transmission, increasing reliability. However, when both direct and relayed paths are available, choice of relaying largely depends on the reliability of source–relay channel [6]. In general, communication in a relay based system involves coordination and transmission phases. The coordination phase involves inter node communications to optimize performance of the subsequent transmission phase [3]. Transmission phase takes place in two time slots. In the first time slot, source node transmits to both relay and destination. In the subsequent time slot, the relay applies a cooperation technique (relaying protocol) to the received signal before retransmitting the signal to the destination. Well known cooperation techniques are amplify-and-forward (AF), decode-and-forward (DF), compress-and-forward (CF) [7], and regenerate-and-forward (RF) [8]. If a relay employs AF protocol, the relay simply amplifies the received signal and retransmits it to the destination. In this scheme, received signal is neither detected, decoded, nor compressed before retransmission [3]. In DF scheme, relay decodes the received signal, generates a new message and retransmits the newly generated signal to the destination [6]. In CF scheme 2 1.1. Cooperative Communication relay exploits the statistical dependency between the message received at the relay and destination, and compresses the received signal prior to retransmission [9]. In RF, relay retransmits a regenerated version of the detected signal to the destination [8]. In this thesis, we propose power allocation schemes for a cooperative communication system that employs AF cooperation technique. Recent tremendous growth in the usage of wireless communication has necessitated a serious attention towards energy consumption in these networks. Excessive energy consumption in such networks is a critical issue due to its adverse effects on the environment and increased operational cost [10]. As cellular system constitutes a major part of modern wireless communications, it is also responsible for a significant portion of total energy consumption of various wireless networks. Moreover, wireless data traffic is expected to increase drastically and mobile devices are anticipated to be a prime medium to access data in near future [11]. Therefore, wireless cellular system demands more attention towards energy aware system design. Green communication is a significant and growing research area, which emphasizes on incorporating energy awareness in communication systems [12]. Energy awareness in wireless networks can be achieved by application dependant relaxation of quality-of-service (QoS) and improved interference management [13]–[15]. In addition, in a cellular system, instead of deploying a single high powered source to cover the entire cell, it will be more energy efficient to deploy multiple low powered, geographically distributed relays to 3 1.2. Literature Review assist the source in transmission [13], as in cooperative communication. In the presence of multiple nodes and varying channel quality, it is important to pay attention towards allocating resources, especially in wireless networks where resources are scarce and regulated [3]. Exploiting channel state information (CSI), resources can be redistributed and optimized to achieve better system performance, in terms of throughput, bit error rate, energy-efficiency and network lifetime [7]. Providing a fair service to users in a multiuser system also depends on optimal allocation of resources. In addition, resource optimization assists in overcoming environmental and economical concerns [7]. Hence, CSI is an integral part of resource allocation in cooperative communications. CSI available at the nodes can be perfect or imperfect, and this has to be taken into account while allocating resources in order to successfully achieve the objectives. 1.2 Literature Review Most of the existing resource allocation schemes for cooperative networks are focused on throughput maximization, QoS provisioning, efficient relay selection, and relay station placement. In [5], the authors have proposed throughput maximization schemes for an AF cooperative communication system, in presence of frequency selective fading and in the absence of direct path. The authors have analyzed the system under both individual and total power constraints, and showed that 4 1.2. Literature Review maximum throughput achievable is higher in case of total power constraint due to higher degree of freedom [5]. A selective relaying scheme is proposed in [16], where the system chooses to relay only if an increase in throughput can be achieved by relaying. Otherwise, only the direct path is used for transmission. In [17], the authors investigated the allocation of power and selection of relays in an AF cooperative network. They have proposed joint schemes under the objectives of minimizing total transmit power and maximizing user rates. A multilevel relaying system with fixed subscriber stations is dealt in [18], and an algorithm is proposed to determine the minimum number of relays, optimum positions, selection and sequence of relays in order to achieve the minimum rate requirement. The authors have assumed the presence of perfect CSI in [5], [16]–[18]. A selection cooperation scheme in a frequency flat fading environment and in the presence of imperfect CSI is investigated in [19]. The authors of this paper have derived closed form expressions for the average capacity in a network with a single source, single destination, and multiple relays. In [20], the authors have investigated the outage probability of a cooperative network in the presence of channel estimation errors. They have presented a lower bound for the signal-to-noise ratio (SNR), and derived a closed-form expression for outage probability based on this bound. In [21], the authors studied the effect of channel estimation errors in a MIMO system and derived bounds for capacity. They also proposed optimal power allocation schemes in presence of channel estimation errors, when the estimation of channels is performed 5 1.2. Literature Review using minimum mean square error (MMSE) technique. In [22], the authors proposed power allocation schemes for an AF relay network with partial CSI. They allocated power such that the average maximum SNR is maximized, under two different assumptions for the availability of instantaneous channels, and statistics of the channels at the nodes. Joint optimization of the relay precoders and decoders of an AF cooperative network is investigated in [23]. The authors consider different assumptions for the availability of CSI at the destination and relay nodes. Robust relay power allocation schemes are proposed in [24], considering channel uncertainties, with individual and aggregate relay power constraints. Challenges involved in designing energy awareness in wireless mobile networks is presented in [10]. The authors highlighted two well known metrics to measure energy-efficiency, namely, energy consumed per bit (Joule/bit) and power consumed per unit area of coverage (W/m2 ). They also identified the different levels of a wireless mobile system, where there is room to improve energy-efficiency, namely, component, link, and network levels. In [14], an energy saving scheme is proposed in order to maximize the network lifetime while ensuring a predefined outage threshold. Based on the channel information, one of direct or relayed path which requires less transmission energy is selected. The authors in [15] proposed a scheme where transmission between base station and relay takes place only when a minimum SNR can be guaranteed at the destination. However, relays and users satisfying this criterion were randomly chosen from the feasible pool for transmission. 6 1.3. Thesis Contribution It is important to study the effect of power allocation and systematic user selection on energy efficiency. Optimizing energy efficiency alone may hinder the system SNR performance. Therefore, it is important to find a balance between energy efficiency and SNR. Furthermore, perfect CSI, as assumed by most of the existing schemes is difficult to achieve in practice due to highly unpredictable nature of wireless channels and the inherent error involved in channel estimation processes. Consideration of this imperfectness during resource allocation can lead to more realistic and robust schemes. On the other hand, ignoring the imperfectness in CSI may result in undesirable effects, as it will lead to wastage of resources, especially in systems where QoS guarantee is required. 1.3 Thesis Contribution In this thesis, downlink transmission of a dual hop AF relaying system is considered. The channel is assumed to suffer from frequency-flat Rayleigh fading, path loss and additive white Gaussian noise (AWGN). In our analysis, we formulate different cost functions of source and relay powers, and propose solutions to minimize the cost function, while satisfying the feasibility constraints. We assume that the source and relay powers are constrained by a sum power constraint. In addition, we incorporate a constraint on SNR, such that a minimum SNR is guaranteed. Transmission is withheld if it is impossible to guarantee the minimum required SNR with the available sum 7 1.4. Thesis Organization power budget. Then, we analyze a multiuser scenario, where a user is chosen for transmission at a given instant. Choosing a user from the feasible pool of users is performed based on their respective cost values. That is, we find the minimum cost of transmission to each user and the respective source and relay powers. Subsequently, the user that incurs the least cost is chosen for transmission. Initially, we define a cost function as a weighted sum of source and relay powers. Then, we define a cost function based on the multi-objective optimization approach, such that both power and SNR are optimized. We define the above cost functions under the assumptions of perfect and imperfect CSI. We also investigate the convexity of these problems. In addition, we provide analytical solutions for two special cases of the multi-objective optimization problem in the presence of imperfect CSI. One special case is the sum power minimization in the presence of imperfect CSI. The other special case is the SNR maximization problem. We prove the convexity of these problems and derive expressions for optimal source and relay powers. Furthermore, we evaluate the schemes through computer simulations and present graphical and numerical results, to depict the system performance. 1.4 Thesis Organization The rest of the thesis is organized as follows. In Chapter 2, underlying system model is defined, and the corresponding transmission schemes are 8 1.4. Thesis Organization introduced. Power allocation schemes in the presence of perfect and imperfect CSI, with guaranteed QoS are proposed in Chapter 3. Simulation results, which depict the performance of our proposed schemes, are presented in Chapter 4. Chapter 5 concludes the thesis and presents possible directions for future work. 9 Chapter 2 System and Channel Models 2.1 System and Channel Description We consider the downlink transmission of a three-node half duplex cooperative communication network as shown in Fig. 2.1. In this network, a source establishes a connection to the destination node with the help of an AF relay. It is assumed that the direct link between source and destination does not exist. The complete transmission takes place in two phases. In the first phase, the source transmits the signal to the relay. In the second phase, the relay amplifies the signal and retransmits the signal to the destination. This is due to either the direct link is in deep fade or the distance between source hsr Relay hrd User Source Figure 2.1: Three-node system model 10 2.2. Transmission Schemes and destination is comparably large [5]. Furthermore, it is assumed that the transmitted signal suffers from path loss, frequency-flat Rayleigh fading and AWGN. Assuming the frequency-flat Rayleigh faded channel between ¯ the effective channel coefficient h any two nodes at a unit distance to be h, ¯ + d)−θ/2 between any two nodes at a distance d can be expressed as h = h(1 [5], where θ is the path loss exponent. 2.2 2.2.1 Transmission Schemes Perfect CSI Assuming the availability of perfect CSI, signals received by the relay, ysr , and destination, yrd , in the first and second time slots are given by ysr = √ ps hsr s + nr , (2.1) √ yrd = κ1 hrd ( ps hsr s + nr ) + nd , (2.2) respectively, where ps is source power; s is the symbol transmitted by source and drawn from a unit variance constellation; hsr and hrd are the channel coefficients between source–relay and relay–destination; nr and nd are zero-mean complex Gaussian (ZMCG) noise at relay and destination with variances σn2 r and σn2 d , respectively. The signal received at relay is amplified by an amplifier with gain κ1 . As the relay uses the AF protocol, it normalizes the received signal (2.1) by a factor E{|ysr |2 } [25] and retransmits to 11 2.2. Transmission Schemes the destination with power pr . For received signal (2.1), E{|ysr |2 } can be expressed as √ √ ∗ E{ysr ysr } = E{( ps hsr s + nr )( ps h∗sr s∗ + n∗r )} = E{ps |hsr |2 |s|2 + √ ps hsr sn∗r + √ ∗ ∗ ps hsr s nr + |nr |2 }. (2.3) Considering the properties of the random variables in (2.3), the expression for the normalization factor can be simplified as E{|ysr |2 } = ps |hsr |2 + σn2 r . (2.4) Therefore, from (2.4) and considering the fact that the signal retransmitted by relay has power pr , κ1 can be expressed as √ κ1 = = pr E{|ysr |2 } pr . ps |hsr |2 + σn2 r (2.5) From (2.2), signal received by destination at the end of second time slot can be expressed as √ yrd = κ1 ps hsr hrd s + κ1 hrd nr + nd . Signal Part⇒seq (2.6) Noise Part⇒neq 12 2.2. Transmission Schemes We need to calculate the signal and noise powers at the destination in the second time slot to obtain the SNR expression. Thus, from (2.6) we can obtain signal and noise powers as psig = E{seq s∗eq } = κ21 ps |hsr |2 |hrd |2 , (2.7) pnoise = E{neq n∗eq } = κ21 |hrd |2 σn2 r + σn2 d , (2.8) respectively. Therefore, from (2.7) and (2.8) we can easily obtain the effective SNR, ρ, as psig pnoise α 1 α 2 ps pr , = α1 ps + α2 pr + 1 ρ = where α1 = |hsr |2 2 σn r and α2 = |hrd |2 . 2 σn (2.9) In the presence of perfect CSI, the through- d put of the considered system can be expressed as [5] r= 2.2.2 1 log(1 + ρ). 2 (2.10) Imperfect CSI It is assumed that both relay and destination use the MMSE based channel estimation method [26] to estimate the channels hsr and hrd , respectively. In addition, it is assumed that there exist channel estimation errors when MMSE estimation is performed for these channels. Under the MMSE estimation, the 13 2.2. Transmission Schemes channels between source and relay, and between relay and destination can be expressed as [21] ˆ sr + esr , hsr = h (2.11) ˆ rd + erd , hrd = h (2.12) ˆ sr and h ˆ rd are the MMSE estimation of the channels hsr respectively, where h and hrd , and esr and erd are the channel estimation errors of the channels hsr and hrd , respectively. Note that, the channel estimation errors and channel estimates are uncorrelated [21]. Furthermore, we assume that estimation errors esr and erd are distributed as ZMCG with variances σe2sr and σe2rd , respectively. For this system, signals received at relay, ysr , and destination, yrd , in the first and second time slots are given by ysr = √ ˆ √ ps hsr s + ps esr s + nr , yrd = κ2 √ ˆ √ ps hsr s + ps esr s + nr (2.13) ˆ rd + erd + nd , h (2.14) respectively. As mentioned in Section 2.2.1, received signal at the relay station is normalized with factor E{|ysr |2 }. For the received signal (2.13), 14 2.2. Transmission Schemes E{|ysr |2 } can be expressed as √ ˆ √ √ ˆ∗ ∗ √ ∗ ∗ ∗ } = E{( ps h ps esr s + nr )( ps h ps esr s + n∗r )} E{ysr ysr sr s + sr s + ˆ sr |2 |s|2 + ps esr h ˆ ∗ s∗ + √ps h∗ s∗ nr + ps h ˆ sr e∗ s∗ = E{ps |h sr sr sr + ps |esr |2 |s|2 + √ √ √ ps e∗sr s∗ nr ps hsr sn∗r + ps esr sn∗r + |nr |2 }. (2.15) Considering the properties of the random variables in (2.15), expression for the normalization factor can be simplified as E{|ysr |2 } = ˆ sr |2 + ps σ 2 + σ 2 . ps |h esr nr (2.16) Considering the fact that the signal transmitted by the relay has power pr , expression for κ2 can be expressed using (2.16) as √ κ2 = = pr E{|ysr |2 } pr ˆ sr |2 + σ 2 ) + σ 2 ps (|h nr esr . (2.17) SNR at the destination receiver can be expressed while considering the impact of the estimation errors as additional noise [20]. From (2.14), signal received 15 2.2. Transmission Schemes by destination at the end of second time slot can be expressed as √ ˆ ˆ yrd = κ2 ps h sr hrd s Signal Part⇒seq √ ˆ √ ˆ √ ˆ rd nr + erd nr ) + nd . + κ2 ( ps h ps hrd esr s + ps esr erd s + h sr erd s + Noise Part⇒neq (2.18) From (2.18) we can obtain signal and noise powers as ˆ sr |2 |h ˆ rd |2 , psig = E{seq s∗eq } = κ22 ps |h (2.19) pnoise = E{neq n∗eq } ˆ sr |2 σ 2 + κ2 ps |h ˆ rd |2 σ 2 = κ22 ps |h erd 2 esr ˆ rd |2 σ 2 + κ2 σ 2 σ 2 + σ 2 , + κ22 ps σe2sr σe2rd + κ22 ps |h nr 2 erd nr nd (2.20) respectively. Therefore, we can easily obtain the effective SNR, ρ, from (2.19) and (2.20) as ρˆ = = = psig pnoise ˆ sr |2 |h ˆ rd |2 |h ˆ rd |2 pr + σ 2 ˆ sr |2 σ 2 + |h ˆ rd |2 σ 2 + σ 2 σ 2 ps pr + σ 2 |h |h erd esr esr erd nr erd ps p r ˆ sr |2 ps + σ 2 + σ 2 σ 2 + σn2 d |h esr nr nd β1 β2 ps pr , (β1 + β2 + 1)ps pr + (β1 + 1)β3 ps + (β2 + 1)β4 pr + β3 β4 ··· (2.21) 16 2.2. Transmission Schemes where β1 = ˆ sr |2 |h , σe2sr β2 = ˆ rd |2 |h , σe2 β3 = rd 2 σn r , σe2sr and β4 = 2 σn d 2 σe . In the presence rd of channel estimation errors that behave as AWGN, the throughput of the considered system can be expressed as [27] rˆ = 1 log(1 + ρˆ). 2 (2.22) In addition, we select energy efficiency defined by the achievable throughput per unit power [28] as a major performance metric to evaluate the schemes in terms of energy awareness. 17 Chapter 3 Power Allocation With Perfect and Imperfect CSI 3.1 Overview In this chapter, we propose power allocation schemes for a cooperative cellular system in order to minimize the cost of transmission, assuming the availability of perfect and imperfect CSI. In the proposed schemes, we define suitable cost functions and allocate resources such that the cost functions are minimized. Cost function is first defined as the weighted sum of source and relay powers. Then we propose a more general multi-objective optimization scheme which jointly optimizes the sum of source and relay powers, and SNR. The former scheme makes the system energy efficient, while the latter scheme keeps a balance between energy efficiency and SNR. In both the schemes, QoS is guaranteed in terms of SNR at the destination. We also analyze a multiuser scenario, where the user that incurs the least cost is chosen for transmission. 18 3.2. System Description hsd hsr hrd Source Relay Cell-Edge User Relaying Region (Near-Cell-Edge) Figure 3.1: System model 3.2 System Description We consider a system as shown in Fig. 3.1. The coverage area of the source is divided into several clusters. Within every cluster, a relay is responsible for transmission to one of the users in that cluster at a given instant. Simultaneous transmissions in separate clusters are assumed to take place using channels which are orthogonal. We consider transmission within the ith cluster, which consists of a single source, single relay, i, and multiple users. Hence, when the source transmits to the destination via the relay, it becomes a three-node cooperative communication system, as in Chapter 2. The transmitted signal is assumed to suffer from path loss, frequency-flat Rayleigh fading and AWGN. In absence of the direct link between source and destination, as described in Section 2.2.1, effective SNR at the destination 19 3.3. Power Allocation Schemes receiver in the presence of perfect CSI is given by (2.9). And, as described in Section 2.2.2, effective SNR at the destination receiver in the presence of imperfect CSI is given by (2.21). In the rest of this chapter, (·)j represents a parameter for jth user served by relay i. For example, ρj represents the SNR ρ given by (2.9) for jth user within the service area of relay i. 3.3 Power Allocation Schemes We propose power allocation schemes based on the minimization of cost functions. Since the objective is to minimize cost, relay will select a user fulfilling this objective while guaranteeing the required SNR, ρmin , which is assumed to be the same for all users, without loss of generality. Furthermore, in our model, source and relay powers are constrained by a maximum power budget, pmax , i.e., ps + pr ≤ pmax . The problem of cost minimization based user selection and power allocation at the ith relay node while guaranteeing QoS can be solved in two sequential steps as explained below. First, the minimum cost for each user j served by relay i is calculated by solving the 20 3.3. Power Allocation Schemes following optimization problem, minimize psj , prj subject to Cj (psj , prj ) (3.1) psj + prj ≤ pmax , (3.2) ρj ≥ ρmin , (3.3) psj ≥ 0, prj ≥ 0, (3.4) where j ∈ {1, . . . , Mi }, Mi is the number of users served by relay i, Cj is a cost function which will be defined later and depends on psj and prj . Both psj and prj will be zero for a user who is not selected for transmission. Subsequently, the user who incurs the minimum cost among the feasible users is chosen for transmission, and user selection variable xj is updated accordingly. xj takes the value 1 if user j is selected and 0 otherwise. Selection of best user within the set of feasible users is performed as xj = 1, if argmin Ck = j and k ∈ F, k (3.5) 0, otherwise, Mi where F is the set of feasible users. Note that, xj = 1. This means j=1 that exactly one user is selected for transmission at any instant. Ties will be broken randomly. In the presence of imperfect CSI, the above problem formulation remains the same, except that constraint (3.3) will be replaced 21 3.3. Power Allocation Schemes by ρˆj ≥ ρmin . (3.6) In the rest of this chapter, we omit the subscript (·)j for brevity. Algorithmic Time Complexity of User Selection Suppose a relay is responsible for the transmission to M users on average, in a cluster. Then, the complexity of user selection algorithm within that cluster will be O(M ). This algorithmic complexity can be explained by the need to traverse M different cost values in order to choose the user with least cost for transmission, in that cluster. 3.3.1 Weighted Sum Power Minimization with Perfect CSI (Scheme A) We formulate a cost function, C A , as weighted sum of source and relay powers. Thus, weighted sum power minimization problem can be formulated as minimize ps , pr C A = w1A ps + w2A pr (3.7) subject to (3.2) − (3.4), where w1A and w2A are normalized positive weights implying the relative significance of source and relay powers and their contribution to the objective function. The weights are predefined such that w1A + w2A = 1. 22 3.3. Power Allocation Schemes Function ρ is quasiconcave in (ps , pr ), which can be verified using Bordered Hessians [29]. Bordered Hessians of function ρ are given by D1 = D2 = 0 ∂ρ ∂ps ∂ρ ∂ps ∂2ρ ∂ps ∂pr 0 ∂ρ ∂ps ∂ρ ∂pr ∂ρ ∂ps ∂2ρ ∂p2s ∂2ρ ∂ps ∂pr ∂ρ ∂pr ∂2ρ ∂pr ∂ps ∂2ρ ∂p2s =− α12 α22 p2r (α2 pr + 1)2 , (α1 ps + α2 pr + 1)4 2α13 α23 ps pr (α1 ps + 1)(α2 pr + 1) = , (α1 ps + α2 pr +)5 (3.8) (3.9) respectively. We can observe that D1 < 0 and D2 > 0 in the region of interest. Therefore, ρ is quasi-concave in (ps , pr ) [29]. Constraint (3.3) corresponds to the super-level set of ρ, which is convex [30]. In addition, C A , constraint (3.2), and constraint (3.4) are linear functions. Hence, problem (3.7) is a convex optimization problem, for which the global optimum solution can be found. The Lagrangian function for problem (3.7) can be expressed as, L(·) = w1A ps + w2A pr + λ1 (ps + pr − pmax ) + λ2 (ρmin − ρ) − λ3 ps − λ4 pr , (3.10) where λ1 , λ2 , λ3 , and λ4 are Lagrange multipliers. The KKT conditions [30] 23 3.3. Power Allocation Schemes for this problem are given by λ∗1 (p∗s + p∗r − pmax ) = 0, (3.11) λ∗2 (ρmin − ρ∗ ) = 0, (3.12) λ∗3 p∗s = 0, p∗s + p∗r ≤ pmax , ρ∗ ≥ ρmin , λ∗4 p∗r = 0, (3.13) p∗r ≥ 0, (3.14) k = 1, 2, 3, (3.15) p∗s ≥ 0, λ∗k ≥ 0, ∗ ∗ ∂L A ∗ ∗ α1 α2 pr (α2 pr + 1) − λ∗3 = 0, = w + λ − λ 1 1 2 ∂p∗s (α1 p∗s + α2 p∗r + 1)2 ∗ ∗ ∂L A ∗ ∗ α1 α2 ps (α1 ps + 1) − λ∗4 = 0, = w + λ − λ 2 1 2 ∂p∗r (α1 p∗s + α2 p∗r + 1)2 (3.16) (3.17) where (·)∗ represents value of the corresponding variable at the optimal point, (3.11)–(3.13) are complementary slackness conditions, (3.14)–(3.15) are feasibility conditions, and (3.16)–(3.17) are stationarity conditions. For successful communication from source to destination via relay, both p∗s and p∗r have to be positive. Therefore, from (3.13), λ∗3 = λ∗4 = 0. Now, from (3.16) and (3.17), λ∗2 must be non-zero since w1A + w2A = 1 and λ∗1 ≥ 0. Thus, from (3.12), ρ∗ = ρmin , i.e., constraint (3.3) becomes tight at optimal solution point of (3.7). Using this knowledge, expressions for optimal p∗s and p∗r for problem (3.7) can be obtained as p∗s = α1 α2 (w1A + λ∗1 )(w2A + λ∗1 )ρmin (ρmin + 1) ρmin + , α1 α1 α2 (w2A + λ∗1 ) (3.18) 24 3.3. Power Allocation Schemes p∗r = α1 α2 (w1A + λ∗1 )(w2A + λ∗1 )ρmin (ρmin + 1) ρmin + , α2 α1 α2 (w2A + λ∗1 ) (3.19) respectively. Now, in order to find p∗s and p∗r we need to obtain λ∗1 . We follow a similar approach as in [31] to find λ∗1 . Initially, we assume constraint (3.2) is not tight, i.e., p∗s + p∗r < pmax ; thus λ∗1 = 0 from (3.11). Subsequently, we find p∗s and p∗r from (3.18) and (3.19). Suppose, the sum of p∗s and p∗r satisfies our assumption, then we have found the optimal solution. On the other hand, if the sum is not less than pmax , we conclude that our assumption is invalid, rather (3.2) has to be active, i.e., p∗s + p∗r = pmax . Then, we find p∗s and p∗r by solving active constraints (3.2) and (3.3). If p∗s ∈ R+ and p∗r ∈ R+ , then feasible solutions exist. Otherwise, the transmission is withheld. The steps in this procedure are outlined in Algorithm 1. Note that, problem (3.7) can be simplified when w1A = w2A , which becomes an equal weight optimization. In this case, we can solve (3.7) ignoring constraint (3.2). If the calculated sum power does not satisfy (3.2), the problem is infeasible; otherwise we have the solution. Thus, the burden of finding λ∗1 will be avoided when w1A = w2A . Unless stated otherwise, we assume w1A = w2A in this chapter, which is a reasonable assumption for fixed dedicated relays. 3.3.2 Multi-Objective Optimization with Perfect CSI (Scheme B) In this section, we propose a scheme to minimize power while maximizing SNR, which are clearly conflicting objectives. Thus we use the concept of 25 3.3. Power Allocation Schemes Algorithm 1 Finding optimal source and relay powers for Scheme A Require: w1A , w2A , α1 , α2 , pmax and ρmin 1: Assume: λ∗1 = 0 2: Find p∗s and p∗r from (3.18) and (3.19) 3: if p∗s + p∗r ≤ pmax then 4: optimal p∗s and p∗r are found 5: else if p∗s + p∗r > pmax then 6: λ∗1 = 0 7: =⇒ (3.2) is active from (3.11) 8: find p∗s and p∗r from active constraints (3.2) and (3.3) 9: end if multi-objective optimization in formulating a problem which optimizes both power and SNR simultaneously [32]. The goal of this formulation is to keep a balance between the two objectives according to their relative importance. An important part of multi-objective optimization is the process of ordering the objectives in terms of preferences, which is usually done by a decision maker. Ordering of objectives can be either a priori or a posteriori of executing the optimization algorithm. In the former case, ordering is done prior to executing the optimization algorithm, which results in a single solution. In the latter case, a solution is chosen from a set of solutions resulting from the execution of the optimization algorithm. A priori use of design parameters is a well known approach to predetermine the relative importance of the objectives [32], [33]. As explained below, appropriate transformations are used in forming the global criterion function [32], which takes care of both the objective functions. The objective function for minimizing the sum of source and relay powers 26 3.3. Power Allocation Schemes can be expressed as f (ps , pr ) = ps + pr . Before forming a global criterion function, this should be transformed to ensure consistent comparison [32]. We use normalization to transform the functions. From constraints (3.2) and (3.4), we can conclude that 0 ≤ f (ps , pr ) ≤ pmax . Hence, after normalizing f (ps , pr ), transformed objective function is given by f † (ps , pr ) = (ps + pr )/pmax . Next, we formulate the objective function that represents SNR maximization. As we are interested only in the region where ρ ≥ ρmin > 0, maximizing ρ is equivalent to minimizing 1/ρ. Thus, we define the second objective function as g(ps , pr ) = 1/ρ. From constraint (3.3), it is apparent that 0 ≤ g(ps , pr ) ≤ 1/ρmin . Hence, after normalization, transformed objective function is given by g † (ps , pr ) = ρmin /ρ. Thus, multi-objective optimization problem in the presence of perfect CSI can be formulated as minimize ps , pr C B = w1B ps + pr pmax + w2B ρmin ρ (3.20) subject to (3.2) − (3.4), where w1B and w2B are the normalized non-negative weights implying the relative significance and contribution of sum power and SNR, respectively, to the objective function, and chosen such that w1B + w2B = 1. Thus, minimizing C B is equivalent to minimizing the sum power and maximizing SNR jointly, which gives a Pareto-optimal solution [32] depending on the weights. 27 3.3. Power Allocation Schemes It is clear that, when w2B = 0, C B is a linear function of ps and pr , and Scheme B will be equivalent to Scheme A for w1A = w2A . Moreover, we can show that C B is convex in (ps , pr ) for 0 < w2B ≤ 1. Convexity of a function can be proved using the positive definiteness of the second order Hessian matrix of the function [30]. The second order Hessian matrix of C B can be expressed as, H= ∂2CB ∂p2s ∂2CB ∂ps ∂pr ∂2CB ∂pr ∂ps ∂2CB ∂p2r . (3.21) According to Sylvester’s criterion, a matrix is positive definite if all of its leading principal minors are positive [34]. The first and second order leading principal minors of H are respectively given by, |H1×1 | = 2w2B ρmin (α2 pr + 1) , α1 α2 p3s pr 2 wB ρ2 (4α1 ps (α2 pr + 1) + 4α2 pr + 3) . |H2×2 | = 2 min α12 α22 p4s p4r When 0 < w2B ≤ 1, the principal minors are positive in the feasible region. Hence, from Sylvester’s criterion, H is positive definite [34]. Therefore, C B is convex in (ps , pr ) for 0 < w2B ≤ 1. Therefore, (3.20) is a convex optimization problem, and existing numerical convex problem solvers can be used to obtain the optimal solution (p∗s , p∗r ) [30]. 28 3.3. Power Allocation Schemes 3.3.3 Weighted Sum Power Minimization With Imperfect CSI (Scheme C) As in Section 3.3.1, we choose the cost function C C as weighted sum of source and relay transmit powers. Therefore, the weighted sum power minimization problem can be formulated as minimize ps , pr C C = w1C ps + w2C pr (3.22) subject to (3.2), (3.4), and (3.6). where w1C and w2C are the normalized non-negative weights. Moreover, for ρmin > 0, constraint (3.6) is equivalent to, 1/ˆ ρ ≤ 1/ρmin . (3.23) Therefore, optimization problem (3.22) can be reformulated as minimize ps , pr C C = w1C ps + w2C pr (3.24) subject to (3.2), (3.4), and (3.23). Second-order Hessian [30] of a function will be used to prove the convexity of the function 1/ˆ ρ. Function 1/ˆ ρ is said to be convex if its Hessian matrix H is positive-definite. Thus, first and second order leading principal minors 29 3.3. Power Allocation Schemes of H can be obtained as |H1×1 | = |H2×2 | = = ∂ 2 (1/ˆ ρ) 2β4 [(1 + β2 )pr + β3 ] , = ∂p2s β1 β2 p3s pr ∂ 2 (1/ˆ ρ) ∂p2s ∂ 2 (1/ˆ ρ) ∂ps ∂pr ∂ 2 (1/ˆ ρ) ∂pr ∂ps ∂ 2 (1/ˆ ρ) ∂p2r (3.25) [4(1 + β1 + β2 + β1 β2 )ps pr + 4β3 (1 + β1 )ps ··· β12 p4s +4β4 (1 + β2 )pr + 3β3 β4 ]β3 β4 , β22 p4r (3.26) respectively. It is apparent from (3.25) and (3.26) that for positive ps and pr , |H1×1 | and |H2×2 | are positive. Therefore, from Sylvester’s criterion [34], H is positive-definite. This concludes that 1/ˆ ρ is a convex function. Hence, (3.24) is a convex optimization problem, for which the global optimum solution can be found. We find the solution of (3.24) using Lagrange method and KKT conditions. The Lagrangian function for this problem can be expressed as L(·) = w1C ps + w2C pr + σ1 (ps + pr − pmax ) + σ2 1 1 − ρˆ ρmin − σ 3 ps − σ 4 pr , (3.27) where σ1 , σ2 , σ3 and σ4 are the Lagrangian multipliers. The KKT conditions 30 3.3. Power Allocation Schemes [30] for this problem are given by σ1∗ (p∗s + p∗r − pmax ) = 0, (3.28) 1 1 − ρˆ∗ ρmin = 0, (3.29) σ4∗ p∗r = 0, (3.30) p∗r ≥ 0, (3.31) l = 1, . . . , 4, (3.32) σ2∗ σ3∗ p∗s = 0, p∗s + p∗r ≤ pmax , ρˆ∗ ≥ ρmin , p∗s ≥ 0, σl∗ ≥ 0, ∗ ∗ ∂L(·) C ∗ ∗ β4 (β2 pr + pr + β3 ) = w − σ3∗ = 0, + σ − σ 1 1 2 ∗ ∂p∗s β1 β2 p∗2 p s r ∗ β (β p + p∗s + β4 ) ∂L(·) 1 s C ∗ ∗ 3 = w2 + σ 1 − σ 2 − σ4∗ = 0, ∗ ∗ ∗2 ∂pr β1 β2 ps p r (3.33) (3.34) where (3.28)–(3.30) are complementary slackness conditions, (3.31)–(3.32) are feasibility conditions, and (3.33)–(3.34) are stationarity conditions. For successful communication from source to destination via relay, both p∗s and p∗r have to be positive. Therefore, from (3.30), σ3∗ = σ4∗ = 0. Now, from (3.33) and (3.34), σ2∗ must be non-zero since σ1∗ ≥ 0. Thus, from (3.28), 1/ˆ ρ∗ = 1/ρmin , i.e., constraint (3.23) becomes tight at optimal solution point of (3.24). Thus, optimal source and relay powers can be expressed as p∗s = (β2 + 1)β4 ρmin σ1∗ + w1C + (ρmin + 1)(σ1∗ + w2C )β1 β2 β3 β4 ρmin (β1 β2 − (β1 + β2 + 1)ρmin ) σ1∗ + w1C + , (3.35) 31 3.3. Power Allocation Schemes p∗r = (β1 + 1)β3 ρmin σ1∗ + w2C + (ρmin + 1)(σ1∗ + w1C )β1 β2 β3 β4 ρmin (β1 β2 − (β1 + β2 + 1)ρmin ) σ1∗ + w2C + , (3.36) respectively. Negative values of powers are avoided by exploiting [·]+ = max{0, [·]}. Now, in order to find p∗s and p∗r we need to obtain σ1∗ . We follow a similar approach as in [31]. Initially, we assume constraint (3.2) is not tight, i.e., p∗s + p∗r < pmax ; thus σ1∗ = 0 from (3.28). Subsequently, we find p∗s and p∗r from (3.35) and (3.36). Suppose, the sum power satisfies our assumption, then we have found the optimal solution. On the other hand, if the sum is not less than pmax , we conclude that our assumption is violated, rather (3.2) has to be active, i.e., p∗s + p∗r = pmax . Then, we find p∗s and p∗r by solving active constraints (3.2) and (3.23). If p∗s ∈ R+ and p∗r ∈ R+ , then feasible solutions exist. Otherwise, (3.24) is infeasible, and no transmission will take place. The steps in this procedure are outlined in Algorithm 2. Note that problem (3.24) can be simplified when w1C = w2C . In this case, we can solve (3.24) ignoring constraint (3.2). If the calculated sum power does not satisfy (3.2), the problem is infeasible; otherwise we have the solution. Thus, the burden of finding σ1∗ can be avoided when w1C = w2C . 32 3.3. Power Allocation Schemes Algorithm 2 Finding optimal source and relay powers for Scheme C Require: w1C , w2C , β1 , β2 , β3 , β4 , pmax , and ρmin 1: Assume: σ1∗ = 0 2: Find p∗s and p∗r from (3.35) and (3.36) 3: if p∗s + p∗r ≤ pmax then 4: optimal p∗s and p∗r are found 5: else if p∗s + p∗r > pmax then 6: σ1∗ = 0 7: =⇒ (3.2) is active from (3.28) 8: find p∗s and p∗r from active constraints (3.2) and (3.23) 9: end if 3.3.4 Multi-Objective Optimization With Imperfect CSI (Scheme D) As explained in Section 3.3.2, we formulate a cost function C D , and a multiobjective optimization problem with imperfect CSI as minimize ps , pr C D = w1D ps + p r pmax + w2D ρmin ρˆ (3.37) subject to (3.2), (3.4), and (3.23). where w1D and w2D are the normalized non-negative weights. It is clear that, when w2D = 0, C D is a linear function of ps and pr , and Scheme D becomes equivalent to Scheme C with w1C = w2C . Moreover, C D is convex in (ps , pr ) for 0 < w2D ≤ 1. The Hessian matrix of C D defined in (3.37) can be expressed as H= ∂2CD ∂p2s ∂2CD ∂ps ∂pr ∂2CD ∂2CD ∂pr ∂ps ∂p2r . (3.38) 33 3.3. Power Allocation Schemes First and second order leading principal minors of H are given by, |H1×1 | = |H2×2 | = 2β4 w2D ρmin (β2 pr + pr + β3 ) , β1 β2 p3s pr (3.39) 4β3 β4 (w2D )2 ρ2min [(β1 β2 + β1 + β2 + 1)ps pr + β3 (β1 + 1)ps β12 β22 p4s p4r +β4 (β2 + 1)pr + 3β3 β4 /4] , (3.40) respectively. When 0 < w2D ≤ 1, the principal minors are positive in the feasible region. Hence, from Sylvester’s criterion, H is positive definite [34]. Therefore, C D is convex in (ps , pr ). Therefore, (3.37) is a convex optimization problem, and an existing numerical convex problem solver can be used to obtain the optimal solution (p∗s , p∗r ) [30]. Note that since (3.39) and (3.40) do ρ is also convex by substituting not depend on w1D , we can easily show that 1/ˆ w1D = 0 in C D . 3.3.5 Special Cases of Scheme D with Imperfect CSI Multi-objective optimization problem formulated in (3.37) has two special cases. The special cases occur when w1D = 1 or when w2D = 1. Problem (3.37) becomes a sum power minimization problem with imperfect CSI when w1D = 1. On the other hand, it becomes an SNR maximization problem when w2D = 1. In this section, we derive analytical solutions for source and relay powers for these special cases. 34 3.3. Power Allocation Schemes Total Power Minimization (w1D = 1, w2D = 0) When w1D = 1 and w2D = 0, problem (3.37) reduces to the minimization of total power with imperfect CSI, while achieving the minimum required SNR. This is a practical scenario where the communication system needs to optimize its transmit power in order to prolong its lifetime and limit the interference power introduced to other active users. Sum power minimization problem can be expressed as minimize ps , pr ps + pr (3.41) subject to (3.2), (3.4), and (3.23). Problem (3.41) represents a convex optimization problem. The Lagrangian function of (3.41) can be formulated as L(·) = (ps + pr ) + µ1 1 1 − ρˆ ρmin + µ2 (ps + pr − pmax ) + µ3 (−ps ) + µ4 (−pr ). (3.42) 35 3.3. Power Allocation Schemes where µ1 , µ2 , µ3 , and µ4 are non-negative Lagrange multipliers. Then, KKT conditions for optimization problem (3.41) can be expressed as µ∗1 1 1 − ρˆ ρmin = 0, (3.43) µ∗2 (p∗s + p∗r − pmax ) = 0, (3.44) µ∗3 (−p∗s ) = 0, (3.45) µ∗4 (−p∗r ) = 0, (3.46) −p∗s ≤ 0, −p∗r ≤ 0, (3.47) µ∗l ≥ 0, l = 1, . . . , 4, (3.48) 1 1 − ≤ 0, ρˆ ρmin (3.49) p∗s1 + p∗r1 − pmax ≤ 0, (3.50) ∗ ∂L(·) ∗ β4 [(1 + β2 )pr + β3 ] + µ∗2 − µ∗3 = 0, = 1 − µ 1 ∗ ∂p∗s β1 β2 p∗2 p s r ∗ β [(1 + β )p ∂L(·) 1 s + β4 ] ∗ 3 = 1 − µ + µ∗2 − µ∗4 = 0, 1 ∂p∗r β1 β2 p∗s p∗2 r (3.51) (3.52) For successful communication from source to destination via relay, both p∗s and p∗r have to be positive. Therefore, from the complementary slackness conditions (3.45) and (3.46), we can conclude that µ∗3 = µ∗4 = 0. Now, from the two stationary conditions (3.51) and (3.52), we can conclude that µ∗1 must be non-zero since µ∗2 ≥ 0, for feasibility. Thus, from the slackness condition (3.43), we can conclude that ρˆ∗ = ρmin , i.e., constraint (3.23) will be met with equality at the optimal solution point of (3.41). Thus, by solving above 36 3.3. Power Allocation Schemes KKT conditions, optimal p∗s and p∗r can be obtained as p∗s (β2 + 1)β4 ρmin + (ρmin + 1)β1 β2 β3 β4 ρmin = β1 β2 − (β1 + β2 + 1)ρmin (β1 + 1)β3 ρmin + (ρmin + 1)β1 β2 β3 β4 ρmin p∗r = β1 β2 − (β1 + β2 + 1)ρmin + , (3.53) , (3.54) + respectively. It should be noted that, constraint (3.2) may or may not be tight at the optimal point. If (3.53) and (3.54) give positive values for source and relay powers, constraint (3.2) is verified by using the sum of these two values. If constraint (3.2) is violated, it is not possible to achieve the minimum SNR with the available power budget. In such a scenario, there is no feasible solution exist for problem (3.41) for that particular channel realization. Hence, the system will withhold transmission in this situation. A step-by-step operation of this scheme is presented in Algorithm 3. Algorithm 3 Minimize sum of source and relay powers - Problem (3.41) 1: Given β1 , β2 , β3 , β4 , pmax , and ρmin 2: Calculate p∗s and p∗r from (3.53) and (3.54) 3: if p∗s + p∗r ≤ pmax then 4: Transmit with source power p∗s and relay power p∗r 5: else 6: Set p∗s = 0 and p∗r = 0 7: Transmission is withheld 8: end if 37 3.3. Power Allocation Schemes SNR Maximization (w1D = 0, w2D = 1) When w1D = 0 and w2D = 1, problem (3.37) reduces to an SNR maximization problem. This problem considers the minimum SNR required for the successful transmission while maximizing SNR. Otherwise, the resulting SNR might not be sufficient to guarantee the basic QoS, which will eventually lead to retransmissions and wastage of energy. The optimization problem for this scenario can be expressed as minimize ps , pr 1 ρˆ (3.55) subject to (3.2), (3.4), and (3.23). Optimization problem (3.55) is convex. The Lagrangian function of optimization problem (3.55) can be expressed as L(·) = 1 +∆1 (ps +pr −pmax )+∆2 (−ps )+∆3 (−pr )+∆4 ρˆ 1 1 − ρˆ ρmin , (3.56) 38 3.3. Power Allocation Schemes where ∆1 , ∆2 , ∆3 , and ∆4 are non-negative Lagrange multipliers. Then, the KKT conditions for optimization problem (3.55) can be expressed as ∆∗1 (p∗s + p∗r − pmax ) = 0, (3.57) ∆∗2 (−p∗s ) = 0, (3.58) ∆∗3 (−p∗r ) = 0, (3.59) ∆∗4 1 1 − ρˆ ρmin = 0, (3.60) −p∗s ≤ 0, −p∗r ≤ 0, (3.61) p∗s + p∗r − pmax ≤ 0, (3.62) ∆∗l ≥ 0, l = 1, . . . , 4, β4 [(1 + β2 )p∗r2 + β3 ] ∂L(·) ∗ = −(∆4 + 1) + ∆∗1 − ∆∗2 = 0, ∗ ∗2 ∗ ∂ps β1 β2 ps pr β3 [(1 + β1 )p∗s + β4 ] ∂L(·) ∗ = −(∆ + 1) + ∆∗1 − ∆∗3 = 0. 4 ∂p∗r β1 β2 p∗s p∗2 r (3.63) (3.64) (3.65) For successful communication from source to destination via relay, both p∗s and p∗r have to be positive. Therefore, from slackness conditions ∆∗2 (−p∗s ) = 0 and ∆∗3 (−p∗r ) = 0, we can conclude that ∆∗2 = ∆∗3 = 0. Now, from the two stationary conditions (3.64) and (3.65), we can conclude that ∆∗1 must be non-zero since ∆∗4 ≥ 0, for feasibility. Thus, from complementary slackness condition ∆∗1 (p∗s + p∗r − pmax ) = 0, we can conclude that p∗s + p∗r = pmax , i.e., constraint (3.2) will be met with equality at the optimal solution point of (3.55). Therefore, solving the above KKT conditions, optimal p∗s and p∗r for 39 3.3. Power Allocation Schemes optimization problem (3.55) can be obtained as p∗s = p∗r = β3 β4 [(β2 + 1)pmax + β3 ][(β1 + 1)pmax + β4 ] − β4 [(β2 + 1)pmax + β3 ] , (β1 + 1)β3 − (β2 + 1)β4 (3.66) β3 β4 [(β2 + 1)pmax + β3 ][(β1 + 1)pmax + β4 ] − β3 [(β1 + 1)pmax + β4 ] , (β1 + 1)β3 − (β2 + 1)β4 (3.67) respectively. By observation it is clear that (3.66) and (3.67) provide positive values for source and relay powers. It should be noted that, constraint (3.23) may or may not be tight at the optimal point. Therefore, constraint (3.23) is validated at a subsequent step. If constraint (3.23) is violated, it is not possible to achieve the minimum required SNR with the available power budget. This indicates that there is no feasible solution exists for optimization problem (3.55) for that particular channel realization. Therefore, the system will withhold transmission in this situation. A step-by-step operation of this scheme is presented in Algorithm 4. Algorithm 4 Maximize SNR - Problem (3.55) 1: Given β1 , β2 , β3 , β4 , pmax , and ρmin 2: Calculate p∗s and p∗r from (3.66) and (3.67) 3: Calculate ρˆ∗ from p∗s and p∗r obtained in Step 2: 4: if ρˆ∗ ≥ ρmin then 5: Transmit with source power p∗s and relay power p∗r 6: else 7: Set p∗s = 0 and p∗r = 0 8: Transmission is withheld 9: end if 40 3.4. Reduction of User Starvation 3.4 Reduction of User Starvation In our analysis of user selection, user with the least cost is always selected among feasible users for transmission. However, under certain circumstances, always selecting such user may result in long term starvation of some users leading to poor fairness performance compared to that of fixed time slot assignment or random user selection. For example, this may happen due to the variation in the relay-destination distances. Since the effect of path loss is generally higher compared to fading, users closer to the relay may be chosen for transmission most of the time. This can be overcome by adapting a scheme that takes into account user starvation while choosing users as presented in Algorithm 5. At the beginning, the algorithm checks for users which have been starved for tc or more time slots by comparing with the time slots starved, tdj , by each user. If there is at least one such user, the user with highest starvation is scheduled for transmission. Otherwise, the user which incurs the least cost is scheduled. Number of time slots starved by each user is updated accordingly. The starvation of each user is controlled by the parameter tc . Increasing tc will be beneficial in terms of reducing cost, but will adversely affect the user fairness. Hence, by carefully choosing tc , a balance between cost and fairness can be maintained. 41 3.4. Reduction of User Starvation Algorithm 5 User selection to reduce user starvation Require: tc , tdj , Cj ∀j ∈ {1, . . . Mi } 1: Initialize: xj = 0, ∀j ∈ {1, . . . , Mi } 2: k ← arg max tdj j 3: 4: 5: 6: if tdk ≥ tc then xk ← 1, tdk ← 0 else k ← arg min Cj j 7: 8: 9: 10: 11: xk ← 1, tdk ← 0 end if for j = 1 to Mi do tdj ← tdj + 1, for j = k end for 42 Chapter 4 Numerical Results In this chapter, we present selected simulation results obtained to illustrate system performance with proposed schemes. Table 4.1 gives a list of system parameters, parameter values used, and the way the parameters were generated. Parameter values and assumptions specific to different schemes are given in the respective subsections, and with figures. All simulation results were averaged over 10, 000 independent channel realizations. 4.1 Performance Analysis of Scheme A and Scheme B with Perfect CSI Selected results using Scheme A and Scheme B with perfect CSI are presented in this section. A path loss exponent of three is assumed. We assume that relay stations are placed in the midway of base station and cell boundary. As we are mainly dealing with cell-edge users, we assume that the users are uniformly distributed near the cell-edge. In such scenario, we assume that the effect of relay–destination path loss is similar for all users within a cluster. In Figs. 4.1 and 4.2, we analyze the effect of power allocation by different 43 4.1. Performance Analysis of Scheme A and Scheme B with Perfect CSI Table 4.1: Simulation Parameters Parameter pmax ρmin θ dsr drd σn2 sr σn2 rd hsr hrd esr erd ˆ sr h ˆ rd h Description and Values Power budget; unless stated otherwise pmax = 20 dB was assumed in simulations Minimum SNR threshold required at the destination; unless stated otherwise ρmin = 10 dB was assumed in simulations Path loss exponent; a shadowed urban cellular environment was assumed with either θ = 3 [1] Source–relay normalized distance; unless stated otherwise dsr = 0.5 was assumed in simulations Relay–destination normalized distance; unless stated otherwise drd = 0.5 was assumed in simulations Noise variance at relay; σn2 sr = 1 was assumed Noise variance at destination; σn2 rd = 1 was assumed Source–relay actual channel gain; assumed to undergo frequency flat Rayleigh fading, with unit variance Relay–destination actual channel gain; assumed to undergo frequency flat Rayleigh fading, with unit variance Error in source–relay channel gain estimation; ZMCG random variable, with variance as specified in specific figures Error in relay–destination channel gain estimation; ZMCG random variable, with variance as specified in specific figures Source–relay estimated channel gain Relay–destination estimated channel gain 44 4.1. Performance Analysis of Scheme A and Scheme B with Perfect CSI schemes assuming each relay selects a user randomly in its cluster. For comparison purposes, we also consider schemes where power allocation between source and relay is done such that energy efficiency is maximized with and without considering the QoS constraint (3.3). Fig. 4.1 shows the variation of average energy efficiency versus ρmin . It can be seen that energy efficiency of Scheme A is almost equal to the maximum energy efficiency achievable in presence of QoS constraint. Also, energy efficiency of Scheme B increases with the increase in the ratio w1B /w2B for a given SNR. In addition, the maximum energy efficiency achievable in the absence of constraint (3.3) serves as an upper-bound of energy efficiency. In Fig. 4.2, we show the variation of average throughput per user against ρmin . Average throughput of Scheme A and the throughput achievable by maximizing energy efficiency with constraint (3.3) are almost equal. Average throughput of Scheme B is better, and varies with the choice of weights w1B and w2B . Therefore, Scheme B provides a way to keep a balance between energy efficiency and SNR. Fig. 4.3 depicts the variation of energy efficiency against the number of users within a cluster. In Fig. 4.3(a), we compare the average energy efficiency of Scheme A using random user selection, with that using the least cost user selection (best user selection) for different number of users and different values of ρmin . It can be observed that, for a fixed ρmin , energy efficiency increases with the increase in number of users in case of least cost user selection method, while it does not change for random user selection. 45 4.1. Performance Analysis of Scheme A and Scheme B with Perfect CSI 0.7 Average Energy Efficiency per User 0.6 Maximize Energy Efficiency with QoS Maximize Energy Efficiency without QoS 0.5 Scheme A Scheme B, w1B = 0.2, w2B = 0.8 0.4 Scheme B, w1B = 0.5, w2B = 0.5 Scheme B, w1B = 0.8, w2B = 0.2 0.3 Scheme B, w1B = 0.9, w2B = 0.1 0.2 0.1 0 0 2 4 6 8 10 12 14 16 End-to-end SNR Threshold ρmin (dB) 18 20 Figure 4.1: Variation of average energy efficiency per user against the endto-end SNR threshold. 46 4.1. Performance Analysis of Scheme A and Scheme B with Perfect CSI 4 Average Throughput per User (bits/s/Hz) Maximize Energy Efficiency with QoS 3.5 Maximize Energy Efficiency without QoS 3 2.5 2 Scheme A 1.5 Scheme B, w1B = 0.2, w2B = 0.8 Scheme B, w1B = 0.5, w2B = 0.5 1 Scheme B, w1B = 0.8, w2B = 0.2 Scheme B, w1B = 0.9, w2B = 0.1 0.5 0 0 2 4 6 8 10 12 14 16 End-to-end SNR Threshold ρmin (dB) 18 20 Figure 4.2: Variation of average throughput per user against the end-to-end SNR threshold. 47 4.1. Performance Analysis of Scheme A and Scheme B with Perfect CSI Furthermore, energy efficiency decreases for high SNR requirements. Similar trends are observed for the case of Scheme B, as shown in Fig. 4.3(b). Fig. 4.4 presents the outage probability versus the number of users within a cluster. Here, outage refers to such event when ρmin cannot be achieved for the scheduled user with the available power budget. It can be seen from the graph that least cost user selection policy reduces the outage probability compared to random user selection. The outage increases with increase in ρmin , however, the performance gap is considerably high for higher ρmin . It is worthwhile to mention that the outage for Scheme B is equal to that of Scheme A. Because, outage occurs due to infeasibility of transmission, which is related to the feasibility constraints of Scheme A and Scheme B. As the set of constraints are the same for both schemes, outage events are also equal. We study the fairness offered to users by the proposed schemes. Jain’s fairness index [35] is well known to measure the fairness among users. Jain’s 2 Mi cj fairness index is given by j=0 Mi · Mi j=0 , where cj is any resource obtained by c2j jth user. We calculate the index for above schemes in terms of the timeslots assigned to each user in 1000 channel realizations, and present in Table 4.2. The system parameters are chosen as ρmin = 5 dB, pmax = 20 dB, w1A = w2A = 0.5, w1B = 0.9, and w2B = 0.1. It is obvious that random user selection offers more fairness to users compared to best user selection. However, it is also seen that the fairness of best user selection scheme is comparable to that of random user selection. As expected, a slight reduction 48 4.1. Performance Analysis of Scheme A and Scheme B with Perfect CSI 0.9 0.8 ρmin = 5 dB ρmin = 10 dB Average Energy Efficiency 0.7 ρmin = 20 dB 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 4 6 8 10 12 Number of Users 14 16 18 20 (a) Scheme A (w1A = 0.5, w2A = 0.5) 0.5 0.45 ρmin = 5 dB Average Energy Efficiency 0.4 ρmin = 10 dB ρmin = 20 dB 0.35 0.3 0.25 0.2 0.15 0.1 0.05 1 2 4 6 8 10 12 Number of Users 14 16 18 20 (b) Scheme B (w1B = 0.9, w2B = 0.1) Figure 4.3: Variation of average energy efficiency within a cluster for different number of users. Solid lines represent least cost user selection and dashed lines represent random user selection. 49 4.1. Performance Analysis of Scheme A and Scheme B with Perfect CSI 0.5 Outage Probability within a Cluster 0.45 ρmin = 5dB 0.4 ρmin = 10dB ρmin = 20dB 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1 2 4 6 8 10 12 Number of Users 14 16 18 20 Figure 4.4: Variation of outage probability within a cluster for different number of users. Solid lines represent least cost user selection and dashed lines represent random user selection. 50 4.1. Performance Analysis of Scheme A and Scheme B with Perfect CSI in fairness is evident with increase in number of users for all considered schemes. However, fairness index will increase when the number of channel realizations is increased. Table 4.2: Jain’s Fairness Index for Different User Selection Schemes Mi 2 5 10 20 50 Best User Selection Scheme A Scheme B (w1A = w2A = 0.5) (w1B = 0.9, w2B = 0.1) 0.9998 0.9998 0.9957 0.9957 0.9816 0.9816 0.9713 0.9713 0.9551 0.9551 Random User Selection 0.9999 0.9979 0.9931 0.9839 0.9598 51 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI 4.2 Performance Analysis of Scheme C and Scheme D with Imperfect CSI In this section, we present selected numerical results using Schemes C and D with imperfect CSI. Unless stated otherwise, parameters chosen and assumptions made are same as in Section 4.1 In Figs. 4.5–4.7, we show the performance results corresponding to power allocation by different schemes assuming each relay selects a user randomly from within its cluster. For comparison purposes, we also consider schemes where power allocation between source and relay is done such that energy efficiency is maximized with and without considering the QoS constraint (3.6). We present the variation of energy efficiency against end-to-end SNR threshold using Scheme C in Fig. 4.5. Fig. 4.5(a) shows the result for different relative weights of source and relay powers (i.e., w1C and w2C ). It can be seen that the energy efficiency of Scheme C is almost equal to the maximum energy efficiency achievable in presence of QoS constraint. Energy efficiency is highest when both the source and relay powers are given equal importance, i.e., w1C = w2C = 0.5. In addition, the maximum energy efficiency achievable in the absence of constraint (3.6) serves as its upper-bound. It is also observed that interchanging the values of w1C and w2C results in overlapping curves. This further implies that the contribution of both the source and relay powers towards average energy efficiency are equally important for the 52 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI chosen parameters. Fig. 4.5(b) depicts the effect of different error variances in the source– relay and relay–destination channel estimation. It is seen that when the error variances of the two hops are equal, average energy efficiency is almost equal to the case of no error in channel estimation. When the values of σe2sr and σe2rd are interchanged, the resulting energy efficiency is practically same. Thus, it can be concluded that estimation errors in the two hops equally contribute to the average energy efficiency for the chosen parameters. However, when different error variances are present in different hops, average energy efficiency decreases compared to the case of no estimation error. From both graphs in Fig. 4.5, it is observed that the average energy efficiency drops with increase in end-to-end SNR requirement in all cases, as more power is required to guarantee higher SNR. Due to the same reason, it is further observed that for very high SNR requirement, the change in error variances and choices of weights become less significant. Fig. 4.6 shows the variation of average energy efficiency versus in Scheme D. It is evident from Fig. 4.6(a) that energy efficiency of Scheme D increases with the ratio w1D /w2D for a given SNR threshold. This is because more importance is being given to minimizing power transmission with such choice of weights. In other words, energy efficiency is less when very high priority is given to SNR (w2D = 0.9). Fig. 4.6(b) shows that energy efficiency is high when there is no estimation errors. However, the energy efficiency is close to this upper bound when the error variances of both source–relay and relay– 53 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI 0.7 w 1C = 0.5, w 2C = 0.5 w 1C = 0.1, w 2C = 0.9 0.6 Average Energy Efficiency per User w 1C = 0.9, w 2C = 0.1 w 1C = 0.2, w 2C = 0.8 0.5 w 1C = 0.8, w 2C = 0.2 σe2sr = 0.010, σe2rd = 0.010 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 End-to-end SNR Threshold ρmin (dB) 18 20 (a) 0.7 σe2sr = 0, σe2rd = 0 σe2sr = 0.010, σe2rd = 0.010 0.6 Average Energy Efficiency per User σe2sr = 0.001, σe2rd = 0.008 σe2sr = 0.008, σe2rd = 0.001 0.5 σe2sr = 0.002, σe2rd = 0.001 0.4 w 1C = 0.5, w 2C = 0.5 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 End-to-end SNR Threshold ρmin (dB) 18 20 (b) Figure 4.5: Variation of average energy efficiency against the end-to-end SNR threshold using scheme C for (a) different weights, and (b) different estimation error variances. 54 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI destination channel estimation errors are same. If the error variances are not equal, then increase in one of them adversely affects the energy efficiency. This is because of the imbalance in channel estimates of two hops. In both figures, the energy efficiency declines with increase in SNR requirement for the similar reasons as explained above for Fig. 4.5. In Fig. 4.7, we show the variation of average throughput against. Average throughput of Scheme C and the throughput achievable by maximizing energy efficiency with constraint (3.6) are almost equal. Average throughput of Scheme D is better, and varies with the choice of weights w1D and w2D . It improves with decrease in the ratio w1D /w2D , which corresponds to decreasing the priority to minimizing transmit power and increasing the priority to system throughput. This further confirms that Scheme D provides a way to keep a balance between energy efficiency and SNR. Fig. 4.8 depicts the variation of energy efficiency against the number of users within a cluster for both proposed schemes. In Fig. 4.8(a), we compare the average energy efficiency of Scheme C using random user selection with that using best user selection (3.5) for different number of users and different values of ρmin . For a fixed ρmin , it is observed that energy efficiency increases with the increase in number of users in case of best user selection method, while it does not change for random user selection. Lower energy efficiency is achieved for high SNR requirements. As shown in Fig. 4.8(b), similar trends are observed for the case of Scheme D. We also study the fairness offered to users by best user selection scheme. 55 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI 0.4 w 1D = 0.5, w 2D = 0.5 w 1D = 0.1, w 2D = 0.9 0.35 Average Energy Efficiency per User w 1D = 0.9, w 2D = 0.1 w 1D = 0.2, w 2D = 0.8 0.3 w 1D = 0.8, w 2D = 0.2 0.25 σe2sr = 0.010, σe2rd = 0.010 0.2 0.15 0.1 0.05 0 0 5 10 15 End-to-end SNR Threshold ρmin (dB) 20 (a) 0.2 σe2sr = 0, σe2rd = 0 σe2sr = 0.010, σe2rd = 0.010 0.18 Average Energy Efficiency per User σe2sr = 0.001, σe2rd = 0.008 0.16 σe2sr = 0.008, σe2rd = 0.001 σe2sr = 0.002, σe2rd = 0.001 0.14 w 1D = 0.5, w 2D = 0.5 0.12 0.1 0.08 0.06 0.04 0 5 10 15 End-to-end SNR Threshold ρmin (dB) 20 (b) Figure 4.6: Variation of average energy efficiency against the end-to-end SNR threshold using scheme D for (a) different weights, and (b) different estimation error variances. 56 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI 4 Maximize Energy Efficiency with QoS Maximize Energy Efficiency without QoS Average Throughput per User (bit/s/Hz) 3.5 σe2sr = 0.010, σe2rd = 0.010 3 2.5 2 Scheme C, w1C = 0.5, w2C = 0.5 1.5 Scheme D, w1D = 0.5, w2D = 0.5 Scheme D, w1D = 0.2, w2D = 0.8 1 Scheme D, w1D = 0.8, w2D = 0.2 Scheme D, w1D = 0.1, w2D = 0.9 Scheme D, w1D = 0.9, w2D = 0.1 0.5 0 0 2 4 6 8 10 12 14 16 End-to-end SNR Threshold ρmin (dB) 18 20 Figure 4.7: Variation of average throughput against the end-to-end SNR threshold. 57 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI 0.9 0.8 Average Energy Efficiency per User ρmin = 5 dB ρmin = 10 dB 0.7 ρmin = 20 dB 0.6 σe2sr = 0.010, σe2rd = 0.010 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 12 14 Number of Users (Mi ) 16 18 20 (a) Scheme C (w1C = 0.5) 0.5 0.45 ρmin = 5 dB ρmin = 10 dB Average Energy Efficiency per User 0.4 ρmin = 20 dB 0.35 σe2sr = 0.010, σe2rd = 0.010 0.3 0.25 0.2 0.15 0.1 0.05 1 2 4 6 8 10 12 14 Number of Users (Mi ) 16 18 20 (b) Scheme D (w1D = 0.9) Figure 4.8: Variation of average energy efficiency against the number of users. Solid lines represent best user selection and dashed lines represent random user selection. 58 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI We calculate Jain’s fairness index in terms of the resource time-slot assigned to each user in 1000 trials. We evaluate the fairness of different schemes in presence of channel estimation errors for two cases of user distribution. We assume parameter values as ρmin = 5 dB, pmax = 20 dB, w1C = w2C = 0.5, w1D = 0.9, w2D = 0.1, σe2sr = σe2rd = 0.010, and tc = 2Mi . In Case I, we assume that all users within a cluster are at the same distance from the relay. In Case II, we assume that one of the users within a cluster is nearer to the relay (at 4/5th of the average distance) while others are far and at the same distance. It is evident from Table 4.3 that use of Algorithm 5 improves the fairness performance of best user selection scheme. The fairness improvement achieved by this algorithm increases with the increase in number of users. When number of users is higher, a few of them are more likely to have relatively adverse channel conditions over a long time period. Hence, Algorithm 5 is more effective in such scenario as it reduces the average starvation period. For the same reason, a slight reduction in fairness is evident with increase in number of users for all considered schemes. Having a time out period effectively improves fairness among users. However, it can have negative effect on energy efficiency of proposed schemes because it tries to provide transmission opportunity to users starving for tc time slots irrespective of the channel condition. Therefore, value of design parameter tc plays a significant role in improving the fairness among users or degrading the energy efficiency. The effect of tc on average energy efficiency is shown in Fig. 4.9, for 59 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI Table 4.3: Jain’s Fairness Index for Different User Selection Schemes in the Presence of Channel Estimation Errors Case I Mi 2 5 10 20 50 Mi 2 5 10 20 50 Scheme C Best User Algorithm 5 0.9984 0.9984 0.9966 0.9967 0.9923 0.9937 0.9707 0.9869 0.9420 0.9812 Scheme D Best User Algorithm 5 0.9984 0.9993 0.9966 0.9995 0.9923 0.9946 0.9707 0.9949 0.9420 0.9915 Case II Scheme C Scheme D Best User Algorithm 5 Best User Algorithm 5 0.8853 0.9991 0.8853 0.9993 0.7465 0.9955 0.7465 0.9989 0.7305 0.9893 0.7305 0.9973 0.6181 0.9912 0.6181 0.9965 0.5741 0.9717 0.5741 0.9875 60 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI parameter values ρmin = 5 dB, σe2sr = σe2rd = 0.01, w1C = w2C = 0.5, w1D = 0.9, and w2D = 0.1. It can be seen that when a time out period is used, the average energy efficiency increases with increase in tc . On the other hand, energy efficiency of best user selection schemes remains higher and does not depend on tc . However increase in tc usually decreases the fairness among users, as shown in Table 4.3. Therefore, a tradeoff between fairness and energy efficiency is usually required in practice. 61 4.2. Performance Analysis of Scheme C and Scheme D with Imperfect CSI 1 Mi = 20, Number of Trials = 1000 0.9 Average Energy Efficiency per User 0.8 0.7 0.6 0.5 0.4 0.3 Scheme C, Algorithm 5 0.2 Scheme C, Best Scheme D, Algorithm 5 0.1 Scheme D, Best 0 20 40 80 200 Timeout Period (tc ) 1000 Figure 4.9: Variation of average energy efficiency against the timeout period. 62 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI 4.3 Performance Analysis of Special Cases of Scheme D with Imperfect CSI Numerical results are presented to illustrate the performance of proposed power allocation schemes when w1D = 1 and w2D = 1, with different channel estimation error variances. A shadowed urban cellular radio environment with a path-loss exponent of four is considered. It is assumed that the source, relay and the destination are positioned on a straight line. 4.3.1 Performance of Total Power Minimization Scheme (w1D = 1, w2D = 0) Fig. 4.10 presents the average minimum total power required versus endto-end SNR threshold for different channel estimation error variances. It is apparent from Fig. 4.10 that the total power increases with increasing SNR for all channel estimation error variances. For a particular SNR threshold, the increase in the total power is relatively low when σe2sr = σe2rd . However, when σe2sr = σe2rd , the average total power required is comparatively high. It can be further observed that the curves overlap when the error variances of the two hops are interchanged. This shows that the channel estimation errors on both the links equally affect the average total power, for the chosen set of parameters. In Fig. 4.11, the average transmission time versus SNR is plotted. The transmission time is specified relative to the time taken in 63 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI 30 Average Minimum Total Transmit Power (dB) 25 20 15 σe2sr = 0, σe2rd = 0 10 σe2sr = 0.050, σe2rd = 0.050 σe2sr = 0.010, σe2rd = 0.001 σe2sr = 0.001, σe2rd = 0.010 5 pmax = 30 dB dsr = 0.5 0 0 2 4 σe2sr = 0.050, σe2rd = 0.001 σe2sr = 0.001, σe2rd = 0.050 6 8 10 12 14 SNR Threshold at the Destination (dB) 16 18 20 Figure 4.10: Average minimum total power versus SNR threshold for different channel estimation error variances. 64 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI the perfect CSI scenario, while considering the transmission of large fixed amount of symbols. It can be noticed from Fig. 4.11 that transmission time increases with SNR threshold in the presence of imperfect CSI. Furthermore, the transmission time is considerably high when σe2sr = σe2rd = 0.05, even though the average total power required is relatively low. This is due to the high probability of transmit power shortage. The variation of probability of power shortage with end-to-end SNR threshold is shown in Fig. 4.12. It can be seen from Fig. 4.12 that the probability of power shortage increases with SNR threshold. However, in the low SNR region the probability approaches zero for all the channel estimation error variances. The equal contribution of the two hops can be observed by the overlapping probability curves. 4.3.2 Performance of SNR Maximization Scheme (w1D = 0, w2D = 1) In Fig. 4.13, the average maximum SNR against the total power budget is plotted for various channel estimation error variances. It can be seen from Fig. 4.13 that the average maximum SNR increases with the increase of transmit power budget. However, this increment is less significant when the channel estimation error variance is high and imbalanced. Furthermore, the average maximum SNR decreases with increasing error variance for a given power budget. This reduction is high when the error variances of the two hops are imbalanced. In Fig. 4.14, we show the variation of probability of the SNR 65 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI 2.5 σe2sr = 0, σe2rd = 0 2.4 σe2sr = 0.050, σe2rd = 0.050 σe2sr = 0.010, σe2rd = 0.001 Average Relative Time Consumption 2.2 σe2sr = 0.001, σe2rd = 0.010 σe2sr = 0.050, σe2rd = 0.001 2 1.8 σe2sr = 0.001, σe2rd = 0.050 pmax = 30 dB dsr = 0.5 1.6 1.4 1.2 1 0 2 4 6 8 10 12 14 SNR Threshold at the Destination (dB) 16 18 20 Figure 4.11: Relative total transmission time versus SNR threshold for different channel estimation error variances. 66 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI 1 σe2sr = 0, σe2rd = 0 0.9 Probability P r(p∗s + p∗r > pmax ) 0.8 0.7 σe2sr = 0.050, σe2rd = 0.050 σe2sr = 0.010, σe2rd = 0.001 σe2sr = 0.001, σe2rd = 0.010 σe2sr = 0.050, σe2rd = 0.001 σe2sr = 0.001, σe2rd = 0.050 0.6 0.5 pmax = 30 dB dsr = 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 SNR Threshold at the Destination (dB) 30 35 Figure 4.12: Probability of transmit power shortage versus SNR threshold for different channel error variances. 67 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI 35 σe2sr = 0, σe2rd = 0 σe2sr = 0.010, σe2rd = 0.010 σe2sr = 0.025, σe2rd = 0.025 30 σe2sr = 0.050, σe2rd = 0.002 Average Maximum SNR (dB) σe2sr = 0.002, σe2rd = 0.050 σe2sr = 0.075, σe2rd = 0.002 25 σe2sr = 0.002, σe2rd = 0.075 ρmin = 10 dB dsr = 0.5 20 15 10 10 12 14 16 18 20 22 24 Total Transmit Power Budget(dB) 26 28 30 Figure 4.13: Variation of average maximum SNR with total power budget for different channel estimation error variances. 68 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI being lower than the SNR threshold, against the SNR threshold for different channel estimation error variances. A similar behavior as in Fig. 4.12 can be seen in this scenario as well. In addition, Fig. 4.14 reflects the probability of retransmissions and wastage of energy that can occur if careful attention is not paid towards the minimum required SNR. The average maximum SNR versus normalized distance between source and relay is plotted in Fig. 4.15. The performance of the proposed optimal power allocation scheme and the uniform power allocation scheme are compared. It can be seen from Fig. 4.15 that for all the channel estimation error variances, the proposed scheme outperforms the uniform power allocation scheme. When σe2sr = σe2rd the average maximum SNR reaches the highest value when dsr = 0.5, as the two hops are balanced [5]. The significance of source to relay hop increases with the increase of dsr . Hence, when σe2sr > σe2rd , the maximum SNR reduces with increasing dsr . A similar explanation can be given for the impact on the SNR when σe2rd > σe2sr . 4.3.3 Power Wastage Due to Ignoring the Imperfectness in CSI In Fig. 4.16, variation of total power wastage (in dB) against the end-toend SNR threshold is shown for 10,000 channel realizations. Here, power wastage is defined as the power that is used in unsuccessful transmissions due to the assumption of perfect CSI at the point of power allocation, even 69 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI 1 σe2sr = 0, σe2rd = 0 0.9 0.8 σe2sr = 0.010, σe2rd = 0.010 σe2sr = 0.025, σe2rd = 0.025 σe2sr = 0.010, σe2rd = 0.002 Probability P r(ρ < ρmin ) 0.7 0.6 σe2sr = 0.002, σe2rd = 0.010 pmax = 30 dB dsr = 0.5 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 SNR Threshold at the Destination (dB) 30 35 Figure 4.14: Probability of insufficient SNR versus SNR threshold for different channel estimation error variances. 70 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI solid line = optimal power allocation dashed line = uniform power allocation 36 ρmin = 10 dB pmax = 30 dB 34 Average Maximum SNR (dB) 32 30 28 26 24 22 σe2sr = 0, σe2rd = 0 20 σe2sr = 0.010, σe2rd = 0.010 σe2sr = 0.010, σe2rd = 0.002 18 σe2sr = 0.002, σe2rd = 0.010 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Normalized Source-Relay Distance dsr 0.8 0.9 1 Figure 4.15: Average maximum SNR versus source-relay distance for different channel estimation error variances. 71 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI though CSI is imperfect. Since we assume CSI is perfect when allocating power, although it is imperfect, power we allocate may not be sufficient to meet the required end-to-end SNR at the destination. However, transmission takes place with the allocated power. In such a scenario, the actual SNR at the destination may be less than the required minimum SNR, resulting in unsuccessful transmission. Since channel estimation errors act as additional noise, actual SNR in presence of imperfect CSI will be less than the SNR that is calculated with the assumption of perfect CSI. As it can be observed in Fig. 4.16, power wastage increases with increase in SNR threshold, in both the schemes. This is because, amount of required source and relay powers increases with the increase in minimum SNR required at the destination, regardless of the assumption on the CSI. Difference in wastage is less in sum power minimization scheme, compared to that of SNR maximization scheme. This is because, in the former scheme, we allocate the minimum power, compared to the latter scheme, where the entire power budget is utilized. On the other hand, if the imperfection in CSI is considered at the point of power allocation, there will not be any power wastage. This is because the transmission will be withheld, if the minimum required SNR cannot be guaranteed with the available power budget. Since we are concerned about energy-awareness of the system, it is mandatory to avoid such power wastage resulting from ignoring the imperfectness in CSI. 72 4.3. Performance Analysis of Special Cases of Scheme D with Imperfect CSI 65 SNR Maximization Scheme (w 1D = 0, w 2D = 1) Sum Power Minimization Scheme (w 1D = 1, w 2D = 0) Total Power Wastage (dB) 60 55 50 45 p max = 30 dB Number of Trials = 10000 σe2sr = σe2rd = 0.010 40 0 5 10 15 SNR Threshold at the Destination (dB) 20 Figure 4.16: Performance of the two schemes if estimation errors are ignored while power allocation. 73 Chapter 5 Conclusions and Future Work 5.1 Conclusions In this thesis, we proposed power allocation schemes to enhance the performance of a dual-hop, amplify-and-forward relay based cooperative communication system. In our analysis, we considered a more practical wireless channel model with channel estimation errors, and a guaranteed quality-of-service (QoS) in terms of end-to-end signal-to-noise ratio (SNR). We proposed a weighted sum power minimization scheme which minimizes the weighted sum of source and relay powers. Since this problem formulation takes care of relative importance of source and relay powers, by proper choice of the weights, we can make either source or relay more energy efficient. We also proposed a multi-objective optimization scheme which provides a trade-off between SNR and sum power. By carefully choosing the relative importance of SNR and sum power, we can achieve the required average SNR while minimizing the sum power. In addition, we proposed analytical solutions for minimizing the sum of source and relay powers, and maximizing SNR, with imperfect channel state information (CSI). Simulation results showed that the weighted 74 5.2. Future Work sum power minimization scheme enhances the performance of the system in terms of energy efficiency, with guaranteed QoS. The scheme that jointly optimizes sum power and SNR, helps to keep a balance between the same by the choice of appropriate relative importance levels. In addition, simulation results showed that power wastage can be reduced by paying careful attention to the imperfectness in CSI in a system that requires QoS guarantee. 5.2 Future Work The following list presents some of the possible extensions of our work: • We dealt with a frequency flat fading channel in our schemes. A possible extension is to consider frequency selective fading channels and exploit orthogonal frequency division multiplexing (OFDM) to overcome the issues of such a channel. This will involve power allocation across subcarriers, subcarrier pairing and modifications in QoS constraint. • Choosing best users may result in poor fairness performance. Using a timeout period will help to overcome this to some extent. However, instead of considering only the current cost of transmission, it is interesting to develop a scheme that considers the instantaneous and average costs of transmission in choosing a user, similar to the proportional fairness scheme [36]. • Another interesting topic for future work is the analysis of a system with 75 5.2. Future Work similar configuration, but with decode-and-forward relaying protocol. • We analyzed a system with a single source, single relay and multiusers within a cluster. A possible extension is considering a system with multiple relays, which involves relay selection. 76 Bibliography [1] T. Rappaport. Wireless Communications: Principles and Practice. Prentice Hall PTR, Jan. 2002. [2] H. Bolcskei. MIMO-OFDM wireless systems: Basics, perspectives, and challenges. IEEE Wireless Communications, 13(4):31 –37, Aug. 2006. [3] Y. W. Hong, W. J. Huang, F. H. Chiu, and C. C. J. Kuo. Cooperative communications in resource-constrained wireless networks. IEEE Signal Process. Mag., 24(3):47–57, May 2007. [4] E. Hossain, D. I. Kim, and V. K. Bhargava, editors. Cooperative Cellular Wireless Networks. Cambridge University Press, Mar. 2011. [5] I. Hammerstrom and A. Wittneben. On the optimal power allocation for nonregenerative OFDM relay links. In Proc. IEEE ICC, volume 10, pages 4463–4468, Jun. 2006. [6] J. N. Laneman, D. N. C. Tse, and G. W. Wornell. Cooperative diversity in wireless networks: Efficient protocols and outage behavior. IEEE Trans. Inf. Theory, 50(12):3062–3080, Dec. 2004. 77 Bibliography [7] K. J. R. Liu, Ahmed K. Sadek, W. Su, and A. Kwasinski. Cooperative Communications and Networking. Cambridge University Press, New York, NY, USA, 2009. [8] V. Mahinthan, L. Cai, J.W. Mark, and X. Shen. Partner selection based on optimal power allocation in cooperative-diversity systems. IEEE Trans. Vehicular Technology, 57(1):511 –520, Jan. 2008. [9] G. Kramer, M. Gastpar, and P. Gupta. Cooperative strategies and capacity theorems for relay networks. IEEE Trans on Inf. Theory, 51(9):3037 – 3063, Sep. 2005. [10] L.M. Correia, D. Zeller, O. Blume, D. Ferling, Y. Jading, I. G´odor, G. Auer, and L. Van Der Perre. Challenges and enabling technologies for energy aware mobile radio networks. IEEE Commun. Mag., 48(11):66– 72, Nov. 2010. [11] T. Chen, H. Kim, and Y. Yang. Energy efficiency metrics for green wireless communications. In Proc. WCSP’10, pages 1–6, Oct. 2010. [12] E. Hossain, V. K. Bhargava, and G. Fettweis, editors. Green Radio Communications Networks. under preparation. [13] C. Comaniciu, N. B. Mandayam, and H. V. Poor. Radio resource management for green wireless networks. In Proc. IEEE VTC, pages 1–5, Sep. 2009. 78 Bibliography [14] I. Krikidis, J. S. Thompson, and P. M. Grant. Cooperative relaying with feedback for lifetime maximization. In Proc. IEEE ICC, pages 1–6, May 2010. [15] V. A. Le, R. A. Pitaval, S. Blostein, T. Riihonen, and R. Wichman. Green cooperative communication using threshold-based relay selection protocols. In Proc. IEEE ICGCS, pages 521–526, Jun. 2010. [16] O. Duval, Z. Hasan, E. Hossain, F. Gagnon, and V. Bhargava. Subcarrier selection and power allocation for amplify-and-forward relaying over OFDM links. IEEE Trans. Wireless Commun., 9(4):1293 –1297, Apr. 2010. [17] K.T. Phan, D.H.N. Nguyen, and T. Le-Ngoc. Joint power allocation and relay selection in cooperative networks. In Proc. IEEE GLOBECOM’09, pages 1–5, Dec. 2009. [18] B. Lin and P. H. Ho. Dimensioning and location planning of broadband wireless networks under multi-level cooperative relaying. IEEE Trans. Wireless Commun., 8(11):5682 –5691, Nov. 2009. [19] M. Seyfi, S. Muhaidat, and Jie Liang. Capacity of selection cooperation with channel estimation errors. In Proc. 25th Biennial Symposium on Communications (QBSC), pages 361 –364, May 2010. [20] M. Seyfi, S. Muhaidat, and J. Liang. Outage probability of selection 79 Bibliography cooperation with channel estimation errors. In Proc. IEEE VTC, pages 1–5, May 2010. [21] T. Yoo and A. Goldsmith. Capacity and power allocation for fading MIMO channels with channel estimation error. IEEE Trans. Inf. Theory, 52(5):2203–2214, May 2006. [22] T. T. Pham, H. H. Nguyen, and H. D. Tuan. Power allocation in orthogonal wireless relay networks with partial channel state information. IEEE Trans. Signal Process., 58(2):869 –878, Feb. 2010. [23] Z. Yi and I. M. Kim. Joint optimization of relay-precoders and decoders with partial channel side information in cooperative networks. IEEE J. Select. Areas Commun., 25(2):447 –458, Feb. 2007. [24] T. Q. S. Quek, M. Z. Win, H. Shin, and M. Chiani. Robust power allocation for amplify-and-forward relay networks. In Proc. IEEE ICC’07, pages 957 –962, Jun. 2007. [25] Y. Zhao, R. Adve, and T.J. Lim. Improving amplify-and-forward relay networks: optimal power allocation versus selection. Wireless Communications, IEEE Transactions on, 6(8):3114 –3123, Aug. 2007. [26] F. Gao, T. Cui, and A. Nallanathan. On channel estimation and optimal training design for amplify and forward relay networks. IEEE Trans. on Wireless Commun., 7(5):1907–1916, May 2008. 80 Bibliography [27] M. Medard. The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel. IEEE Transactions on Inf. Theory, 46(3):933–946, May 2000. [28] M. Guowang, N. Himayat, and G.Y. Li. Energy-efficient link adaptation in frequency-selective channels. IEEE Trans. Commun., 58(2):545–554, Feb. 2010. [29] M. J. Osborne. Mathematical methods for economic theory: A tutorial, (§3.4). Available online at: http://www.economics.utoronto. ca/osborne/MathTutorial/QCCF.HTM, retreived 10 Oct. 2010. [30] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge Univ. Press, 2004. [31] M. O. Hasna and M. S. Alouini. Optimal power allocation for relayed transmissions over Rayleigh-fading channels. IEEE Trans. on Wireless Commun., 3(6):1999–2004, Nov. 2004. [32] R. T. Marler and J. S. Arora. Survey of multi-objective optimization methods for engineering. Struct. and Multidisciplinary Optim., 26:369– 395, Mar. 2004. [33] R. T. Marler and J. S. Arora. The weighted sum method for multiobjective optimization: New insights. Structural and Multidisciplinary Optimization, 41:853–862, Jun. 2010. 81 Bibliography [34] G. T. Gilber. Positive definite matrices and Sylvester’s criterion. Am. Math. Monthly, 98(1):44–46, Jan. 1991. [35] R. Jain, D. M. Chiu, and W. Hawe. A quantitative measure of fairness and discrimination for resource allocation in shared computer systems. DEC Research Report TR-301, Sep. 1984. [36] A. Jalali, R. Padovani, and R. Pankaj. Data throughput of CDMAHDR a high efficiency-high data rate personal communication wireless system. In Proc. 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Power allocation schemes for cooperative communication system using weighted sum approach Devarajan, Rajiv 2011
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Title | Power allocation schemes for cooperative communication system using weighted sum approach |
Creator |
Devarajan, Rajiv |
Publisher | University of British Columbia |
Date Issued | 2011 |
Description | This thesis investigates power allocation schemes for an amplify-and-forward dual-hop relay based cooperative communication system with perfect and imperfect channel state information (CSI). We define cost functions and propose power allocation schemes such that the cost functions are minimized. We analyze a multiuser system, where we select the best user for transmission, who incurs the least cost of transmission. In a practical system, estimated CSI is often imperfect. We assume the estimated CSI is affected by estimation errors, which are modeled as zero mean complex Gaussian. First we propose an optimization scheme where the objective is to minimize the weighted sum of source and relay powers. Then we propose a more general multi-objective optimization scheme which jointly optimizes sum power and signal-to-noise ratio (SNR). In our proposed schemes, source and relay nodes share a fixed total power, and transmission is allowed only if the minimum required SNR at the destination can be achieved with the available power budget. These schemes are analyzed under both perfect and imperfect CSI assumptions. In addition to proving the convexity of these problems, we propose analytical solutions for sum power minimization and SNR maximization schemes in the presence of imperfect CSI. Performance of the systems under the proposed schemes are investigated in terms of energy efficiency, throughput and outage. Simulation results show that proposed schemes reduce wastage of power by avoiding unsuccessful transmissions. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-08-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0105046 |
URI | http://hdl.handle.net/2429/36957 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2011-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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