Energy Efficiency and Performance Analysis of Multihop Wireless Communication over Nakagami-m Fading Channels

Transcription

1 Energy Efficiency and Performance Analysis of Multihop Wireless Communication over Nakagami-m Fading Channels Thesis by Itsikiantsoa Randrianantenaina In Partial Fulfillment of the Requirements For the Degree of Master of Science King Abdullah University of Science and Technology, Thuwal, Kingdom of Saudi Arabia (April, 2015)

2 The thesis of Itsikiantsoa Randrianantenaina is approved by the examination committee Committee Chairperson: Dr. Mohamed-Slim Alouini Committee Member: Dr. Tareq AlNaffouri Committee Member: Dr. George Turkiyyah 2

3 Copyright 2015 Itsikiantsoa Randrianantenaina All Rights Reserved 3

4 4 ABSTRACT Energy Efficiency and Performance Analysis of Multihop Wireless Communication over Nakagami-m Fading Channels Itsikiantsoa Randrianantenaina It has been proven in the literature that the concept of multihop relaying (where the source communicates with the destination via many intermediate nodes) can improve wireless communications from multiple aspects such as broadening coverage, overcoming shadowing, reducing the transmit power, and increasing the capacity of the network at a low additional cost. On the other hand, the problem of energy efficiency is currently one of the biggest challenges for green radio communications. In fact, there has been a massive increase in the number of cellular network subscribers during the last two decades. Therefore, there has been an exponential expansion of the cellular network market, followed by an increase in the number of base stations that have costly energy requirements. In addition, for battery-powered devices, transmit and circuit energy consumption has to be minimized for better battery lifetime and performance. In this work, the performance of multihop relaying communication over Nakagamim fading is investigated without diversity combining as well as with. Closed form expression of the average ergodic capacity is given. Then, an expression of the outage probability is obtained using the inverse of Laplace transform and the average bit error rate is bounded using the Moment Generating Function approach. The energy

5 efficiency is analyzed using the "consumption factor" as a metric, and it is derived in closed form for Amplify-and-Forward, and compared to the energy efficiency if Decode-and-Forward. Then, based on the obtained expressions, a power allocation strategy maximizing the consumption factor is proposed. Simulations and numerical results are presented to show the accuracy of the analysis, and the performance of the energy-aware power allocation scheme is compared, in terms of energy efficiency, as well as other common performance metrics, with other power allocation schemes from the literature. 5

6 6 ACKNOWLEDGEMENTS I would like to express my gratitude to my advisor Professor Mohamed-Slim Alouini (Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology), and Professor Mustapha Benjillali (Institut National des Postes et Télécommunications, Rabat, Morocco). They were available, generous in support, assistance and suggestions throughout this project.

7 7 TABLE OF CONTENTS Examination Committee Approval 2 Copyright 3 Abstract 4 Acknowledgements 6 List of Abbreviations 9 List of Symbols 10 List of Figures 11 1 Introduction and Basic Notions General Introduction Channel Model Pathloss Multipath Propagation Green communications Energy efficiency metrics Fundamenal trade-offs Multihop communications Motivations Forwarding strategies Diversity Combining Motivations Diversity Techniques Combining Techniques Multihop Relayed Communication Without Diversity Combining System model

8 2.2 Ergodic capacity Outage probability Bit-error-rate Energy efficiency and power allocation strategy Energy-aware power allocation Low complexity suboptimal power allocation Conclusion Multihop Relayed Communication With Diversity Combining System model Ergodic capacity Outage probability Bit-error-rate Conclusion Numerical and Simulation Results Ergodic Capacity Bit-Error-Rate Outage Probability Consumption Factor Concluding Remarks Summary Ongoing and Future Research Work References 55 Appendices 59 8

9 9 LIST OF ABBREVIATIONS AF AWGN Amplify and Forward Additive White Gaussian Noise CDF CF Cumulative Distribution Function Consumption Factor DE DF Deployment efficiency Decode and Forward ECR EE EGC ETSI Energy Consumption Rate Energy efficiency Equal Gain Combining European Telecommunications Standards Institute LoS Line of Sight MGF MRC Moment Generating Function Maximal Ratio Combining OSI Open Systems Interconnection PDF Probability Distribution Function QoS Quality of Service SE SNR Spectrum efficiency Signal to Noise Ratio

10 10 LIST OF SYMBOLS F X (.) CDF of a random variable X E[.] The expectation of a random variable L 1 (.) Inverse Laplace transform operator the PDF L(.) Laplace transform operator M X (.) MGF of a random variable X f X (.) PDF of a random variable X Pr[X] Probability of an event X

11 11 LIST OF FIGURES 2.1 The adopted multihop system model without diversity combining The adopted multihop system model with diversity combining Variation of end-to-end ergodic capacity with the number of hops Variation of the end-to-end ergodic capacity with Ω DT e2e Variation of C with the a fixed SNR for the equivalent DT Comparison of Chernoff bound with the approximation for Q-function Variation of P out with the number hop Variation of C with the SNR for the equivalent DT Variation of CF with the number of hops for AF Variation of CF with the number of hops for DF Variation of CF according to the total transmit power budget for AF Variation of CF according to the total transmit power budget for DF Tradeoff between the C AF (Capacity) and the CF AF Tradeoff between the C DF (Capacity) and the CF DF Effect of dissimilar fading on CF

12 12 Chapter 1 Introduction and Basic Notions 1.1 General Introduction Traditionally, cellular network subscribers are served directly by their nearest base station, those base stations are fixed and are wire-connected to a public telecommunications infrastructure. However, due to a better understanding of the wireless channel and the evolution of digital communications hardware, there has been an evolutionary improvement in the quality of services offered to the consumer. And some of the constraints of the cellular network are removed progressively. One of the new proposed concepts is the multihop communication [1, 2], where the source communicates with the destination via one or many intermediate nodes. Those intermediate nodes could be small-antennas designed only for this purpose or users mobile devices. On the other hand, during the last two decades, there was a huge expansion of cellular network market with an exponential increase of the subscribers number. This is partially due to the ceaseless introduction of new attractive devices, the mobile streaming video, the online game development, and the success of social networking. According to the Gartner forecast, mobile devices surpassed the personal computer as the most common web access device from 2013, and by 2015, over 80 percent of

13 Chapter 1. Introduction the handsets sold in mature markets will be smartphones. Moreover, this growth is expected to persist in the future with the introduction of the 4G and 5G mobile technologies. With this growth comes the inevitable increase in the number of base station sites [3], each of them consumes averagely about 24 megawatt per year. Consequently, cellular networks energy consumption is currently a serious concern. From the users side, for battery-powered devices, transmit and circuit energy consumption has to be minimized for better battery lifetime and performance. In addition, public concern for possible impacts of these communication systems has emerged in terms of greenhouse gases emission and electromagnetic radiation. Therefore, the problem of energy efficiency is one of the current biggest challenges towards green radio communications [4]. This thesis conducts a comprehensive analytical framework of the exact end-toend performance and energy efficiency of multihop communication with and without diversity combining, for an arbitrary number of hops between the source and the destination, considering a generalized fading distribution that is Nakagami-m. Multihop Amplify-and-Forward (AF) relay network performance is compared to the performance of a single-hop system, and the case with diversity combining is compared to the simple relaying. In order to do so, the most common performance assessment criteria in literature are derived and analyzed: the ergodic capacity, the outage probability and average Bit Error Rate (BER). Then, the energy efficiency is studied, using the Consumption Factor (CF) as metric. This thesis report is organized as follows. The rest of this chapter introduces basic notions about the concepts, model and metrics employed/studied through this project. Chapter 2 focuses on the case of multihop communication without diversity combining. The performance metrics and the CF are derived, then an energy-aware node power allocation is proposed. In Chapter 3, the case of multihop communication with diversity combining using 13

14 Chapter 1. Introduction Maximal Ratio Combining (MRC) is studied and the same performance metrics as for Chapter 2 are derived. The analytic results are verified with simulations and numerical results in Chapter 4, different comparisons are presented. And the gain in Energy Efficiency (EE) using the proposed energy-aware power allocation is highlighted. 1.2 Channel Model Using wireless radio channel to reliably transmit data with high-speed is challenging as it is susceptible to random noises, interferences and change over time in unpredictable ways. The received signal power varies because of distance (path loss), large obstacles in the surrounding environment (shadowing), and multipath due to scattering Pathloss The pathloss models the dissipation of the power radiated caused only by distance, thus, it has the same at a given transmit receive distance. It is defined as the ratio of transmitted power and received power. In this project, the effect of pathloss is captured by the factor d ν multiplying the transmitted power to get the received power. d is the distance between the transmitter and receiver, and ν is a constant called the pathloss exponent, varying from 2 to 8, depending on the environment Multipath Propagation The transmitted signal may have multiple copies through different paths to reach the receiver, the shortest path is the Line of Sight (LoS). This is called the multipath effect, each signal copy experiences different attenuation coefficient, delay and phase shift, and the received signal is the sum of those different copies. The result can be 14

15 Chapter 1. Introduction either constructive or destructive, i.e. it can amplify the signal power seen at the receiver if the signals are in phase, or attenuate it otherwise. Nakagami-m is one of the distributions used to model the randomness of multipath propagation. This distribution was developed based on the empirical results. It covers a wide types of fading as its parameters can be adjusted to fit a variety of empirical measurements. The Nakagami-m distribution has two parameters, the shape parameter m and the spread parameter Ω, and the expression of the corresponding PDF of the channel amplitude α is f α (α) = 2 ( m ) mα 2m 1 e m Ω α2. (1.1) Γ(m) Ω The shape parameter m corresponds to the degrees of freedom related to the number of added Gaussian random variables. If m = 1, Rayleigh fading is recovered. This is another type of fading distribution, modeling the case without dominant component (no LoS). The PDF of the channel amplitude in that case is f α (α) = 2α λ e α2 λ, where α 0 and λ = E[α 2 ], (1.2) and consequently, the power follows an exponential distribution. The range 0.5 m < 1 is used to model fading channels that are more severe than Rayleigh fading. And m > 1 is reserved for the case where the fluctuation of the signal amplitude is reduced compared to the Rayleigh case. Note that, when the channel amplitude is Nakagami-m distributed, the power of the received signal (γ) follows a Gamma distribution Gamma(m, Ω), where m is the shape parameter and Ω the mean, having PDF: f γ (γ) = 1 ( m ) mγ exp( m 1 mγ ) Γ(m) Ω Ω (1.3) 15

16 Chapter 1. Introduction 1.3 Green communications Different schemes and solutions have already been proposed in order to decrease the energy consumption of the wireless network (or, equivalently, increase the EE). It has been shown that one promising technique to achieve this goal is multihop relaying [5]. Several investigations have been already conducted in that direction. By way of illustration, the authors of [6] presented several route selection methods in multihop communication and evaluated their performance in terms of Spectrum Efficiency (SE) and EE. The instantaneous trade-off between the total energy consumption per bit transmitted and the end-to-end rate under spatial reuse in wireless multihop networks was developed and analyzed in [7]. For given relays positions, it was shown that the total energy consumption-per-bit is minimized by optimally selecting the endto-end rates. The authors of [8] investigated the basic trade-offs between energy consumption, hop distance and robustness against fading. They expressed the energy consumption rate as a function of the source destination distance and derived optimal hop distances Energy efficiency metrics In order to compare the greenness of one system to another, metrics of EE are required. Carbon footprints or CO 2 emission are not sufficient to measure the greenness in telecommunication networks since there are other motivations to reduce the energy consumption such as the cost, better practical usage (user battery), electromagnetic radiation, etc.. Therefore, some standardization organizations or researchers defined metrics to quantify and compare the greenness of communication systems, those standard metrics are more convenient for a long term research goals of reducing the energy consumption. They can be classified according to the concerned part of the network and/or different level of possible intervention. For our interest, i.e. at network level, 16

17 Chapter 1. Introduction the EE can be defined in different ways. For any cellular network, EE could be defined as the covered area or number of served subscribers per unit of power consumed by the BS site. If any technique of power saving such as sleeping mode is applied, then the EE efficiency is measured by efficiency of the applied technique, that is the ratio of the saved energy to the consumed energy without power saving technique. In the next chapters, the CF is used to measure the energy efficiency of the network system. CF is defined as the maximum achievable rate (given by Shannon s capacity) per unit of power consumed to transmit (considering the transmit power itself and the circuit powers) Fundamenal trade-offs This subsection discusses the fundamental issues towards green wireless communications, known as trade-offs between the energy efficiency and other network parameters that has to be taken account. Energy saving within a wireless network is limited by other concerns such as the equipment and installation costs, the available resources, and the performance of the system. There are four fundamental trade-offs in green communication [9]. First, the trade-off between EE and Deployment Efficiency (DE), DE is the cost of the network equipment, the installation and maintenance of these equipment, per throughput). To decrease this cost, engineers try to limit the number of cells covering a given area. This leads to a raise of transmit power to maintain the minimum acceptable SNR, and therefore a growth of the consumed energy. However, in the practical case, considering the static power and the circuit power, an optimal point (cell size) may exist depending on each scenario of deployment. Another compromise has to be made between the bandwidth and the consumed power, since those are the two most important, but limited, resources in wireless communications. From Shannon capacity formula, for a point-to-point communication, the power consumption is a decreasing function of the bandwidth. Therefore, 17

18 Chapter 1. Introduction the minimum power consumption is achieved with an infinite bandwidth and vice versa. However, further study showed that the relation between the consumed power and the bandwidth becomes more complicated in practical, and that using the whole available bandwidth may be not the most energy efficient. The third trade-off is between the EE and the transmission delay or the service latency. This requires further attention as the delay affects directly the user, and it is one of the Quality of Service (QoS) parameters. Another important trade-off is between the EE and the SE, SE is the achieved throughput per unit of bandwidth. SE and EE are related by Shannon s formula of the achievable data rate as follows R = Blog 2 ( 1+ P N 0 ), (1.4) where R is the achievable data rate for a given bandwidth B, transmit power P and noise power N 0. Defining SE as η SE = R, and EE as η B EE = R, the expression of SE P as a function of EE from (1.4) is η EE = η SE (2 η SE 1)N0. (1.5) This tradeoff between EE and SE is studied in this project for the multihop communication, and it is shown that an optimal operating point for both exists, considering not only the transmit power but also the power consumed in circuits. 1.4 Multihop communications Motivations Multihop relaying communication is one of the techniques that can improve wireless channel by alleviating the effects of shadowing and fading [10, 11]. It has gained the 18

19 Chapter 1. Introduction attention of researchers because of its advantages compared to the traditional direct transmission in terms of deployment, connectivity, adaptability and capacity, only with a low additional cost [12]. In case of bad channel conditions (in terms of fading, shadowing, and path loss), serving distant users becomes difficult and expensive in transmit power, therefore breaking this long distance into several range transmissions with significantly reduced transmit power is one of the solutions to mitigate wireless channel impairments [10, 11, 13] and to save energy [14]. Moreover, relaying communications can also ensure broader coverage [15] and can increase the capacity of the network at a low additional cost [12]. The multihop communications could be applied to cellular networks, ad-hoc, vehicular transmission and hybrid networks in order to increase coverage, throughput, and capacity. Moreover, in some practical scenarios, the deployment of traditional fixed infrastructures are not feasible, and in those cases relaying offers more flexibility in time and reduces the location constraints. Relaying can help to improve the coverage of existing network as well, by extending the coverage at the periphery of the cells or closing the internal gaps [15]. The performance of multihop relaying communication in terms of capacity, outage probability, error rate and energy efficiency has been studied through many research works, either for the case of two hops [10,15 17], or for the generalized multihop case. For instance, the energy consumption for a BER-constrained system is analyzed and optimized in [18], and [19] give an expression of the average symbol error probabilities of multihop transmission with AF relays. [13] analyzes the performance of multihop relaying communication over Rayleigh Fading Forwarding strategies There are two main forwarding strategies in wireless multihop communication, Amplify and Forward (AF) and Decode and Forward (DF). Those strategies define the actions that are performed by the nodes between the source and the destination. 19

20 Chapter 1. Introduction Amplify-and-Forward The relay node amplifies the received signal with a gain G without decoding and retransmits the output of the amplifier to the next node. It is a layer-1-relaying in OSI model, in that case the noise is amplified with the useful signal [11]. This strategy is also known as analog relaying. The expression of the end-to-end SNR in this case depends on the gain G. One possibility is taking the gain at the i-th relay to be G i = ε i α i 1 2 ε i 1 +N 0, (1.6) where ε i is the maximum energy that could be transmitted at the i th relay, α i 1 is the channel amplitude of the previous hop. In that case, it can be shown that the instantaneous end-to-end SNR is γ AF e2e,1 = [ N ) 1 (1+ 1] 1γi, (1.7) where, γ i is the SNR at the i-th relay. Another possibility is using G i = ε i α i 1 2 ε i 1, (1.8) in this case, the end-to-end SNR is γ AF e2e,2 = [ N ] 1 1. (1.9) It can be shown that γe2e,2 AF is a lower bound of γaf e2e,1, and this bound is tight for high SNR. γ i 20

21 Chapter 1. Introduction Decode-and-Forward For DF, each relay digitally decodes and re-encodes the received signal from the preceding terminal before retransmitting it to the next node. Therefore, from a decoding point of view, the equivalent end-to-end SNR is the minimum of SNRs of individual hops, i.e., γ DF e2e = min...n γ i. (1.10) 1.5 Diversity Combining Motivations Diversity is one efficient way to alleviate the bad effects of fading channels since it protects the signal against deep fades. The idea is to create multiple channels or branches that have uncorrelated fading, and receive multiple independent replicas of the same signal through those different channels. Therefore, the probability that all the received signals are simultaneously in deep fading is very low. Diversity can be used either to mitigate the effect of fading (named in that case microdiversity), or to mitigate the effect of shadowing (macrodiversity) Diversity Techniques There are several methods to create those independent realizations of the channel to get diversity in a wireless system such as time diversity (same information sent over the same channel at different times), frequency diversity (same information sent at different carrier frequencies separated by at least the coherence bandwidth), or space diversity. This last one exploits the fact that two antennas separated by several wavelengths will not generally experience fades at the same time. This assumption is reasonably satisfied in relaying communication as the relays contributing to the 21

22 Chapter 1. Introduction multihop communications are widely separated. Therefore, spatial diversity can be achieved and exploited with multihop communications by combining the signals received from different nodes in the relay chain. In fact, using diversity combining with multihop relaying attracted the interest of researchers. [20] considers the case where every node combines the signals received from all previous relays in addition to the signal from the source, and analyzes the SER. Then, the relay decodes the signal before forwarding it to the next hops, the authors analyse the symbol-error-rate for MPSK and QAM and propose an optimal power allocation scheme. Moreover, the authors of [13] study and compare four schemes in terms of bit-error-rate (BER) over Rayleigh fading channels: AF/DF, with/without diversity Combining Techniques The combining technique defines the process used at the receiver to exploit the independent copies of the signal received. The receiver can selects the signal with highest instantaneous SNR (selection combining) or select the first branch with a SNR higher than a predetermined SNR (threshold combining)..., however the optimal combining technique is the MRC. In MRC, every branch is first co-phased (i.e. multiplied by a coefficient that cancels the phase) then summed with optimal weighting to maximize the combined output SNR. Intuitively, branches with a high SNR should be weighted more than the ones with a low SNR, and it can be proven that the optimal weight for a given branch is its SNR. In that case, the output SNR is the sum of the SNR of all combined branches. 22

23 23 Chapter 2 Multihop Relayed Communication Without Diversity Combining In this chapter, the case of multihop communication without diversity combining is considered. Performance metrics are derived (capacity in Section 2.2, outage probability in Section 2.3 and BER in Section 2.4). Based on the expression of CF, an-energy aware power allocation is proposed. 2.1 System model h 1 h R 1 R 2 h 3 2 R 3 R 3 R N+1 d 1 d 2 d 3 D Figure 2.1: The adopted multihop system model without diversity combining, with N 1 AF relays. We consider a source R 1 and a destination R N+1 communicating through (N 1) AF half-duplex relays as shown in Fig The distance between R 1 andr N+1 is fixed and it is denoted by D. Each node uses only the information received from its immediate predecessor. The i-th hop link is of length d i, has ν i as pathloss exponent, m i as shape parameter of fading, and h i as instantaneous channel coefficient. The h i s are considered to be independent and not necessarily identically distributed Nakagami-

24 Chapter 2. Multihop Relaying m i fading. Therefore, the SNR follows a Gamma distribution Gamma(m i,ω i ). We denote by γ i = h i 2 Pi t and Ω N 0 (d i ) ν i = E[γ i ] = E[ h i 2] Pi t the instantaneous and average i N 0 (d i ) ν i SNRs of the i th hop, respectively. All channel coefficients are assumed to be perfectly known at the receiving node, as channel estimation issues are out of scope here. Throughout the analysis, and without any loss of generality, fading parameters m i are assumed to be integer, and the noise over all channels is zero-mean AWGN with the same variance N 0. Each node is in one of two possible states: transmission or reception. The power consumed by node R i during a transmission phase is given by P t i /ε+pct i, where P t i is the transmit power used by R i, ε is the power amplifier s efficiency (ε ]0,1]), and P ct i is the circuit power consumption (PC) in transmission mode. Similarly, we denote by P cr i and the PCs of R i during the reception mode. At the beginning of the communication, the source node sends the first packet, and the other nodes receive then relay to the next node. Therefore, it takes N slots before all the nodes are operational. But, subsequently, and since the source is not idle while other nodes are relaying, the destination receives one packet every time slot. 2.2 Ergodic capacity The ergodic capacity defines the maximum data rate that can be sent over the channel with asymptotically small error probability. Its instantaneous value is given by Shannon s well-known formula [21], for a unity of bandwidth C = 1 ln(2) ln(1+γ e2e), (2.1) 24

25 Chapter 2. Multihop Relaying where γ e2e is the instantaneous SNR. Assuming that the gain is the one given in (1.6), the instantaneous end-to-end capacity is [ C = 1 ln(2) ln N ( ) ] 1 γi , = 1 ln(2) ln 1+ γ i N γ i N (γ i +1) N N = 1 ln(2) ln (γ i +1) N N (γ i +1) γ i γ i = 1 ln(2) ln(1 f N), (2.2) where f N = N γ i. Using Taylor series expansion N (1+γ i ) ln(1 x) = j=1 x j, for x < 1, (2.3) j we get C = 1 ln(2) j=1 f j N j. (2.4) The average capacity is then obtained by averaging the expression of the instantaneous capacity over the joint PDF of all γ i s. Since γ i s are assumed to be independent, their joint PDF is just the product of their individual PDF f γi (γ i ), therefore, 25

26 Chapter 2. Multihop Relaying the average ergodic capacity can be written as follows, C = 1 ln(2) = 1 ln(2) = 1 ln(2) 0 j=1 j= j 1 j 0 0 N... j=1 ( 1 N j 0 ( γi 0 ) j γ i N f γi (γ i )dγ i 1+γ i ) j f (γ γi i)dγ i ( γi N 1+γ i 1+γ i ) j f γi (γ i)dγ i. (2.5) Since γ i is following a Gamma distribution with shape parameter m i and mean Ω i, using the expression of the PDF of Gamma distribution given in in (1.3), we get C = 1 ln(2) j=1 1 j N ( γi 0 1+γ i ) j 1 Γ(m i ) ( mi Ω i ) mi γ exp( mi 1 m ) iγ i dγ i. (2.6) Ω i Using [22, Eq.(13.2.5)], we have + ( γi 0 1+γ i ) j γ m i 1 i ) mi e m iγ i mi Ω i dγ i =( Γ(j +m i ) U( j +m i,1+m i, m ) i. (2.7) Ω i Ω i Therefore, C= 1 1 ln(2) j j=1 N ( Γ(j +m i ) U j+m i,1+m i, m ) i. (2.8) Γ(m i ) Ω i Note that the infinite sum in (2.8) converges quickly for j Outage probability An outage occurs when the the end-to-end SNR falls below a certain threshold, and the receiver is not anymore able to decode the signal. This threshold is chosen to guarantee a certain quality of service which essentially depends on the type of modulation employed and the type of application supported. The probability of the outage is another performance criteria that can be used to characterize a wireless communication 26

27 Chapter 2. Multihop Relaying system. Relative to a threshold SNR γ th, it is defined as P out = Pr[γ e2e γ th ] = [ 1 = Pr 1 ] γ e2e γ th = 1 F 1 γ e2e ( 1 γ th γth 0 f γe2e (γ e2e )dγ e2e ). (2.9) We know that for X 0, x F X (x) = f X (t)dt 0 ( x = L (L 1 0 )) f X (t)dt (2.10) Using one of the properties of the Laplace transform [23, Eq. ( )], we get ( ) F X (x) = L 1 MX (s) (2.11) s x Applying this result to (2.10), we obtain P out = 1 L 1 ( M 1 γ e2e (s) s ) 1 γ th. (2.12) To get a tractable expression of the outage probability we exploit the fact that the inverse of the end-to-end SNR is approximately the sum of the inverse of each hop SNR (1.9) therefore, 1 γ e2e = M 1 γ e2e (s) = N N 1 γ i, (2.13) M 1 γ i (s), (2.14) 27

28 Chapter 2. Multihop Relaying where every M 1 γ i (s) is obtained in closed-form as follows, M 1 (s) = 1 γ i Γ(m i ) = 1 Γ(m i ) ( mi Ω i ( mi Ω i ) mi ( γ mi 1 exp m ) iγ i 0 Ω i ) mi γ mi 1 exp 0 ( m iγ i Ω i s γ i ) exp ( sγi dγ i, ). (2.15) Using [23, Eq. ( )], we get M 1 γ i (s) = 2 Γ(m i ) ( mi s Ω i ) mi /2K mi ( 2 mi s Ω i ). (2.16) 2.4 Bit-error-rate Generally, the expression of error rate is complex as the symbol error rate depends on the modulation scheme, and the bit error rate on the symbol error rate and the mapping of symbols to bits. However, it has been shown in [24] that the instantaneous BER or SER can be expressed or approximated with a general form, for any modulation scheme, that is ( ) BER(γ e2e ) αq βγe2e, (2.17) α and β are constants, their values depend on the modulation scheme used. For BPSK: BER(γ e2e ) = Q( γ e2e ). (2.18) Most of the analysis of the BER in the literature are conducted by approximating/upperbounding the Q-function with the Chernoff bound. Here, we use the approximation of the complementary error function that is given in [25] to obtain a better approxi- 28

29 Chapter 2. Multihop Relaying mation of the Q-function: Q(x) 1 1 2x 12 e x e 3 (2.19) Therefore, average BER over the fading channel can be approximated as BER α α 0 (e βx 2 ( Mγe2e( β 2 ) e 4 2βx 3 ) + M γ e2e ( 2β 3 ) 4 f γe2e (x)dx, ). (2.20) The expression of M γe2e (s) from M 1 γ e2e (s) can be obtained using one of the results of [19], that is and we get M γe2e (s) = M X (s) = 0 0 ( ) J 0 2 su u M 1 (u)du, (2.21) X N+1 ( ) J 0 2 su u M 1 (u) γ l l=2 N k=1,k l M 1 γ k (u)du. (2.22) M 1 γ l (s) is given in (2.17). 2.5 Energy efficiency and power allocation strategy In order to analyze how energy efficient is the multihop relayed communication system, we use the CF (introduced in [26]) as metric of EE. In order to use CF, we first define the total consumed power for an end-to-end transmission. P AF tot = 1 ε N P t i +P c +NP AF c, (2.23) 29

30 Chapter 2. Multihop Relaying where P c represents all the circuit powers (during transmission, reception and idle modes) from R 1 to R N+1, i.e., P c = N Pct i + N+1 i=2 Pcr i. Pc AF is the additional power consumed to amplify the received signal before forwarding at every node. We obtain the expression of the average CF using the expression of the average capacity derived in Sec. 2.2, CF = c N j=1 1 j N ( Γ(j +m i ) U j+m i,1+m i, m ) i, (2.24) Γ(m i ) Ω i with c N = [ 1 ln(2) ε 1 N Pt i +P c +NPc AF ]. (2.25) Energy-aware power allocation In this subsection, we derive an energy-efficient, CF-optimal, transmit power allocation strategy for the analyzed multihop relaying setup. Given a total power constraint P tot, the optimization problem can be formulated as follows min P t i s.t. CF(P t 1,Pt 2,...,Pt N ) N P t i P max, (2.26) and the Lagrangian of the problem is given by L(P t,µ) = CF(P t 1,Pt 2,...,Pt N )+µ ( N P t i P max ), (2.27) 30

31 Chapter 2. Multihop Relaying where µ is the Lagrange multiplier corresponding to the constraint. The Karush- Kuhn-Tucker (KKT) conditions can be expressed as CF +µ = 0, i = 1,...,N Pi t N Pi t P max. (2.28) Note that solving the equations system in (2.29) using Newton s method is quite complex as the expressions of the first and second derivatives of CF are not straightforward. Alternatively, in this work, we adopt the Automatic Differentiation (AD) 1 to compute the gradient of the objective function, then it is passed to MATLAB s fmincon 2 as one of the parameters, while using the interior-point algorithm to further accelerate the calculation time. Note that, in addition to the increasing computation complexity (with increasing N), the optimization problem in (2.27) has to be solved by a central unit aware of the channel statistics of all hops, which then broadcasts the optimal transmit powers to the relaying nodes. To avoid this, we further propose a low complexity decentralized algorithm yielding close-to-optimal transmit powers Low complexity suboptimal power allocation The idea is to solve the optimization sequentially, assuming at every node that all following links has the same statistics. The first node solves the optimization problem (2.27), assuming that the rest of the links have the same channel statistics as the first hop. Therefore, the optimization can be done with only one variable. The calculated 1 For MATLAB, AD was implemented by R. D. Neidinger in 2008 through the valder class which implements AD by operator overloading: it computes the first order derivative or multivariable gradient vectors starting with a known simple valder and propagating it through elementary functions and operators. 2 MATLAB s fmincon is a powerful method for solving constrained optimization problems. However, it is not fast enough for a considerable number of hops in our context; especially for the AF case. 31

32 Chapter 2. Multihop Relaying optimal value is the operating transmit power of the first node. Then, the first node transmits its corresponding term in the expression of CF (calculated with the optimal calculated transmit power and the channel statistic) to the next node. Once the second node receives the information from the first, it formulates a new optimization problem using the information obtained from the first node and assuming that all the following hops have the same channel statistics as the second hop. The process continues until the last node. Practically, at the n th node R n, first, the following optimization problem is solved max x CF(x) s.t. (N n+1)x P max P max,n 1. (2.29) where n 1 P max,n 1 = P t i. (2.30) The expression of CF(x) is given by CF(x) = c N (x) J j=1 ( ( )) 1 j T Γ(j +m n ) n 1(j) U j+m n,1+m n, m N n+1 nn 0 (d n ) ν Γ(m n ) E [ h n 2]. x (2.31) where we consider only the first J terms for the infinite sum in (2.25), and T n 1 (j) = n 1 Γ(j +m i ) Γ(m i ) ( U j+m i,1+m i, m ) i, (2.32) Ω i c N (x) = ( 1 ε B/ ln(2) ) (N n+1)x+ n 1 P t i +P c +NP AF c. (2.33) After solving the optimization problem in (2.30), the computed optimal value of x is the operating transmit power P t n for the nth node. This node transmits to the next node P max,n = P max,n 1 +P t n and T n(j) for j = 1,2,...,J. Note that, recursively, we 32

33 Chapter 2. Multihop Relaying can write T n (j) = T n 1 (j) Γ(j +m n) Γ(m n ) ( U j+m n,1+m n, m ) n. (2.34) Ω n The performance of this algorithm is shown and compared to the performance of the optimal allocation in Chapter Conclusion This chapter gives analytic expressions of performances metrics and CF. The ergodic capacity is derived in closed form as function of the network and channel parameters. An expression of the outage probability is provided with the help of the inverse Laplace transform, and the BER is derived in a generalized form valid for every modulation scheme, using the MGF approach. Then, a closed form expression of the CF is given, and based on this expression a power allocation technique optimizing the CF with a total transmit power constraint is proposed, followed by the description of a simplified algorithm to perform the allocation. The analytic expressions and the performance of the proposed algorithm are verified/analyzed with numerical and simulation results presented in Chapter 4. 33

34 34 Chapter 3 Multihop Relayed Communication With Diversity Combining In this chapter, the case of AF multihop relayed communication with diversity combining is analyzed. The system model is detailed in Section 3.1. Then, we derive a closed-form expression of the ergodic capacity in Section 3.2, an expression of the outage probability in Section 3.3, and the expression BER is derived in Section System model We consider that each node combine the information received from all of its predecessor using MRC before amplifying and forwarding. All links are considered to be independent and not necessarily identically distributed Nakagami-m fading. The link between the k-th and hop j-th nodes is of length d k,j, has ν k,j as pathloss exponent, (m k,j,ω k,j ) as shape and spread parameter of the fading, and h k,j as instantaneous channel coefficient. Throughout the analysis, fading parameters m k,j are assumed to be integer, and the noise over all channels is zero-mean additive white gaussian (AWGN) with the same variance N 0. We denote by γ k,j = h k,j 2 Pk,j t and N 0 (d k,j ) ν k,j Ω k,j = E[γ k,j ] the instantaneous and average SNRs of the link from the k-th to j-th

35 Chapter 3. MRC Multihop Relaying γ 1,j γ 1,N+1 γ 1,N+1 γ 1,k γ k,j γ j,n+1 R 1 R k R j R N+1 γ 1,2 γ k 1,k γ k,k+1 γ j 1,j γ j,j+1 γ N,N+1 Figure 3.1: The adopted multihop system model with diversity combining, with N 1 AF relays. nodes, respectively. γ c j is the combined SNR at the j-th node. j 1 γj c = γ k,j. (3.1) k=1 The PDF of γ c j is given in [27] m k,j j 1 f γ c j (x) = c j k=1 q=1 ρ (m k,j q) k,j x q 1 e xm k,j Ω k,j (m k,j q)!(q 1)!, (3.2) where and ρ (m k,j q) k,j j 1 c j = = dm k,j q ds m k,j q l=1 [ j 1 l k ( ml,j Ω l,j ) ml,j, (3.3) ( ) ] ml,j ml,j +s. (3.4) Ω l,j m s= k,j Ω k,j We denoted by γ D e2e the new end-to-end SNR when diversity combining is performed. We assume that an appropriate gain is used to amplify the signal at every node in order to maintain the expressions of the instantaneous end-to-end SNR for AF at the receiver. The derivation of this gain is beyond the scope of this work. 3.2 Ergodic capacity Let us assume that there exists an optimal gain that every node can use in order to keep the same form of the expression of the end-to-end SNR as in the previous 35

36 Chapter 3. MRC Multihop Relaying chapter i.e [ N+1 ( ) 1 γ γe2e D c = i +1 1]. (3.5) i=2 This assumption is taken in order to get a tractable analytic expression of the capacity and the other performance metrics. In that case, the same techniques used for the derivation of the average ergodic capacity in Sec. 2.2 can be applied to get the expression of the instantaneous capacity γ c i : C D = 1 ln(2) ( ) f D j N, (3.6) j j=1 where f D N = N+1 γi c i=2 N+1 (3.7) (1+γi c) i=2 The average capacity is then obtained by averaging the expression of the instantaneous capacity over the joint PDF of all γ c i s. Since γ k,j s are independent from each other for all k s and all j s, γ c i are independent from each other as well, and their joint PDF is just the product of their individual PDF. Therefore, the average ergodic 36

37 Chapter 3. MRC Multihop Relaying capacity can be written as follows C D = 1 ln(2) = 1 ln(2) = 1 ln(2) (a) = 1 ln(2) (b) = 1 ln(2) 0 j=1 j=1 j=1 j= j 1 j 0 0 N+1 i=2 N+1 1 j i=2 N+1 1 j i=2... j=1 0 0 ( N+1 1 j i=2 N+1 i=2 ( γ c i 1+γ c i i 1 m k,i c i k=1 q=1 i 1 m k,i c i k=1 q=1 γ c i 1+γ c i ( γ c i 1+γ c i ) jn+1 i=2 ) j f γ c i (γc i )dγc i f γ c i (γ c i )dγc i ) j f γc i (γc i)dγ c i ρ (m k,i q) k,i (m k,i q)!(q 1)! 0 ( x j+q 1 (x+1) exp m ) k,ix dx j Ω k,i ρ (m k,i q) ( k,i Γ(j +q) (m k,i q)!(q 1)! U j +q,1+q, m ) k,i. (3.8) Ω k,i (a) is obtained by substituting the PDF of γ c i by its expression given in (3.2) and we get (b) solving the integral using (2.7). 3.3 Outage probability The outage probability relative to a threshold SNR γ th is : P D out = Pr[ γ D e2e γ th]. (3.9) Using the same technique as in (2.3) [ 1 P D out = Pr 1 ] γe2e D γ th M 1 (s) = 1 L 1 γ e2e D (3.10) s 1 γ th 37

38 Chapter 3. MRC Multihop Relaying Since then 1 γ D e2e = N+1 i=2 i=2 1, (3.11) γ c i N+1 M 1 (s) = M γ e2e D 1 (s), (3.12) γ i c where every M 1 (s) is calculated can be obtained in closed form as follows, γ i c M 1 (s) = γ i c j 1 m k,j c j 0 k=1 q=1 m k,j j 1 = c j k=1 q=1 ρ (m k,j q) ( k,j (m k,j q)!(q 1)! xq 1 exp xm ) ( k,j exp s ) dx, Ω k,j x ρ (m k,j q) k,j (m k,j q)!(q 1)! 0 ( x q 1 exp xm k,j s ) dx, Ω k,j x (3.13) Using [23, Eq. ( )], we get M 1 γ c i m k,i i 1 (s) = c i k=1 q=1 2ρ (m k,i q) k,i (m k,i q)!(q 1)! ( Ωk,i s m k,i )q 2 Kq ( 2 mk,i s Ω k,i ). (3.14) 3.4 Bit-error-rate Following the same reasoning as in Sec. 2.4, the expression of the instantaneous BER for multihop communication with diversity, for any modulation scheme can be expressed or approximated as ( ( 1 BER D α 12 exp γd e2e )+ 14 ( )) 2 exp 2γD e2e 3 (3.15) 38

39 Chapter 3. MRC Multihop Relaying Therefore, average BER over the fading channel can be approximated as BER D α α (e βx 2 2βx 3 ) e 4 ( Mγ D (β) e2e 2 + M γ D (2β) e2e f γ D e2e (x)dx, ). (3.16) Using the result from [19], we get the expression of M γe2e (s) M D γ e2e (s) = 0 N+1 ( ) J 0 2 su l=2 u M 1 γ c l (u) N+1 k=2,k l M 1 (u)du. (3.17) γ k c The expression of M 1 γ c l is the one given in (3.14). 3.5 Conclusion Performance metrics are derived for multihop communications with diversity combining in this chapter. The spatial diversity is achieved when every node considers the signal received from all of its previous nodes. The expressions are checked with Monte Carlo simulations in Chapter 4, and compared to the case without diversity in order to see if whether it is advantageous or not. 39

40 40 Chapter 4 Numerical and Simulation Results In this section, numerical and simulation results are presented to prove the accuracy of the analysis. For different channel conditions, quantified by the value of the fading distribution shape parameter m, three transmission schemes are compared: the direct transmission (DT), the multihop relayed communication without diversity combining (noted WMRC) and the multihop relayed communication with diversity combining (noted MRC in the figures). The variation of every performance metric is presented according to two of the system parameters: the number of hops between the first transmitter and the last receive, then the total transmit power 1 equally distributed among the nodes. Monte Carlo simulations over 50,000 iterations are conducted to verify every derived analytical expression. Unless otherwise specified, the network parameters are selected as follows, the total distance between the source and the destination and the bandwidth are normalized to unity, and the relays are positioned uniformly between the source and the destination. The pathloss exponent is the same for all links, ν i = ν = 2.5, and the expected value of the squared channel gain is equal to 1 for all links, i.e. for the case without diversity combining E[ h i 2 ] = 1 for all i and for the case with diversity combining E[ h k,j 2 ] = 1 for all (k,j) unless otherwise specified (for the CF analysis). The value of the fading distribution shape parameter m is constant for all hops in a given 1 This transmit power is obtained from the end-to-end SNR of the equivalent DT of Ω DT e2e

41 Chapter 4. Numerical and Simulation Results scenario Ergodic Capacity Fig. 4.1 captures the variation of the end-to-end ergodic capacity with the increase of number of hops between the source and the destination. Two cases are considered for WMRC: in the first case (left figures), the cancellation of the interference from other nodes is assumed to be perfect, i.e the signal received form the immediate predecessor is interference-free. In that case, all the nodes are allowed to transmit using the same spectrum range. The second case (right figure) assumes that no interference cancellation technique is applied even though, the interference from the other relays might be strong. Therefore, every node has to transmit in different range of spectrum, and the transmission consumes N times more bandwidth. And consequently, the capacity is scaled by 1 / N. For the case with MRC, since every node receives from all of its previous relays have to use different spectrum in order to distinguish the signal received from a node to another, hence the capacity is always scaled by 1 / N, unlike the direct transmission with one unit of bandwidth. With perfect interference cancellation, it can be seen that, in general the capacity increases with the number of hops. However, in case of bad channel conditions (m = 1), the simple relaying is not advantageous compared to the DT, unless number of hops is relatively high. The gain from DT to simple relaying enhances as the channel condition improves. On the other hand, if every relay uses different range of spectrum from all the other relays, then performing multihop communication is not advantageous in terms of data rate. The performance of the multihop relaying with MRC in terms of data rate is not satisfactory, mainly because of the scaling with the bandwidth consumed.%. 2 In the figures, the black color corresponds to the case m = 1, blue is for m = 2 and red for m = 3. 41

42 Chapter 4. Numerical and Simulation Results Erodic Capacity WMRC WMRC-MC DT MRC-MC MRC Erodic Capacity DT MRC-MC MRC WMRC WMRC-MC N N Figure 4.1: Variation of end-to-end ergodic capacity with the number of hops for different fading parameters. Ω DT e2e = 5 db. One the left, perfect interference cancellation for WMRC, and on the right, without interference cancellation. The variation of the ergodic capacity according to Ω DT e2e is shown in fig. 4.2 for a fixed number of hops N = 4. As expected, for all cases, the capacity is an increasing function of Ω DT e2e. However, again, when the channels conditions are not favorable and Ω DT e2e is relatively low, then it is preferable to operate in direct transmission. Ergodic capacity WMRC WMRC-MC DT MRC-MC MRC Ergodic capacity DT MRC-MC MRC WMRC WMRC-MC Ω DT e2e Ω DT e2e Figure 4.2: Variation of the end-to-end ergodic capacity with the the end-to-end SNR for the equivalent DT of hops for different fading parameters. N = 4. One the left, perfect interference cancellation for WMRC, and on the right, without interference cancellation. 4.2 Bit-Error-Rate The variation of the BER with Ω DT e2e is presented in fig

43 Chapter 4. Numerical and Simulation Results Average BER DT WMRC WMRC-MC MRC MRC-MC Ω DT e2e Figure 4.3: Variation of C with the a fixed SNR for the equivalent DT for different fading parameters. N = 2. Fig. 4.4 exhibits the accuracy of the approximation of the Q-function using the sum of two exponentials given in (2.20) compared to the Chernoff bound. It can be seen that in our case the Chernoff bound does not provide a tight upper-bound of the average BER. 43

44 10 0 Chernoff Bound Chapter 4. Numerical and Simulation Results Approximation (2.20) Monte Carlo 10 1 Average BER Ω DT e2e Figure 4.4: The average BER obtained using the Chernoff bound of Q-function compared to the one obtained using the approximation given in (2.20). 4.3 Outage Probability It can be seen in figs.4.5 and 4.6, that the outage probability is a decreasing function of the number of hops and the transmit power at every node. Unlike the ergodic capacity, multihop communication is at least as good as the DT in terms of outage, even when the channel conditions are not promising (m = 1), however the gain from relaying improves with the enhancement of the channel. On the other hand, the amount of gain obtained from MRC is the same for a given number of hop independently from the channel condition. This gain can go up to 95%. 44