The Pennsylvania State University the Graduate School

The Pennsylvania State University The Graduate School

OUTAGE PROBABILITY AND TRANSMITTER POWER

ALLOCATION FOR NC-BASED COOPERATIVE NETWORKS

OVER NAKAGAMI-M FADING CHANNEL

A Thesis in Electrical Engineering by Yi-Lin Sung

c 2014 Yi-Lin Sung

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Master of Science

May 2014 The thesis of Yi-Lin Sung was reviewed and approved∗ by the following:

Sedig S. Agili Associate Professor, Department Head of Electrical Engineering Program Coordinator, Master of Science in Electrical Engineering Thesis Co-Advisor

Aldo W. Morales Professor of Electrical Engineering Thesis Co-Advisor

Robert Gray Associate Professor of Electrical Engineering

∗Signatures are on ﬁle in the Graduate School. Abstract

In previous work, we proposed a modiﬁed network coding (NC) cooperation system based on complementary code (CC)-code division multiple access (CDMA) technique. This technique not only consumes fewer time slots in either synchronous or asynchronous transmission but also alleviates multi-path interference. In that system, a multiplier is used at the base station where the received signal is multiplied with redundant signals coming from the relays, rather than performing a binary sum operation in the media access control (MAC) layer, which is normally used in traditional network coding. In this thesis, a further investigation is car- ried out in the proposed system in terms of outage probability and transmitter power allocation. Outage probability was obtained by setting some conditions on a modiﬁed mutual information equation. In addition, using the max-min optimization method, the optimal power allocation was obtained. Simulation results show that the proposed NC-based cooperative system reduces the overall system outage probability and provides a better transmitter power allocation as compared to the traditional cooperative communication system. These results are obtained by using a Nakagami-m fading channel.

Index Terms–Bit error rate, network coding, complementary code-code division multiple access (CC-CDMA), cooperative, relay, diversity, outage probability, capacity, optimal power.

iii Table of Contents

List of Figures vii

List of Tables x

List of Abbreviations and Symbols xi

Acknowledgments xv

Chapter 1 Introduction 1 1.1 Introduction ...... 1 1.2 Outage performance ...... 1 1.2.1 Outage probability ...... 1 1.2.2 Optimal power allocation ...... 2 1.2.3 System outage probability introduction ...... 2 1.3 The system framework ...... 2 1.3.1 Cooperation ...... 3 1.3.2 Network coding ...... 4 1.3.3 Multiplier ...... 4 1.4 Benefits of the proposed system ...... 5 1.4.1 TDMA ...... 6 1.4.2 CDMA ...... 6 1.4.3 Spectrum efficiency ...... 7 1.4.4 Benefits of complementary codes ...... 7

Chapter 2 Error rate probability 10 2.1 System model ...... 10

iv 2.1.1 Signals received at relay nodes ...... 11 2.1.2 Signals received at the destination ...... 14 2.2 DIVERSITY ANALYSIS ...... 15 2.2.1 Sub-optimal weights ...... 17 2.2.2 Probability of error at destination ...... 18 2.3 Numerical analyses ...... 20 2.3.1 NC-based cooperative by using a multiplier ...... 20 2.3.2 Multiplier network coding vs. ML distributed phase steering 21 2.3.3 Multiplier network coding vs. Network-Coded Cooperative Diversity (NCCD) ...... 23 2.3.4 Multiplier network coding vs. ML detection network coding 23 2.4 Summary of the advantages of using a multiplier network coding . . 24

Chapter 3 Outage probability 31 3.1 Outage probability ...... 31 3.2 Numerical analyses ...... 37 3.2.1 Outage probability for NC-based cooperative by using a multiplier ...... 37 3.2.2 Multiplier network coding vs. Two users cooperative network coding ...... 37 3.2.3 Multiplier network coding vs. Network-coding-based hybrid AF and DF system ...... 38 3.2.4 Multiplier network coding vs. Distributed orthogonal Space- Time block codes ...... 38 3.3 Summary of the advantages of using a CC-CDMA cooperative network coding ...... 40

Chapter 4 Optimal power allocation 53 4.1 Optimal power allocation ...... 53 4.1.1 Standard max-min problem ...... 53 4.1.2 Lagrangian problem ...... 56 4.2 Summary of the advantages of using optimal power allocation . . . 58 4.3 Numerical analyses ...... 59

Chapter 5 Conclusions 71 5.1 Conclusions ...... 71 5.2 Future work ...... 72

v Appendix A Received signal at relay rj or destination d 73

Appendix B Received noise at relay rj or destination d 76

Appendix C Maximum output SNR analysis 78

Appendix D Error probability closed form solution for a special case and diversity gain analysis 82

Bibliography 86

vi List of Figures

1.1 Traditional cooperative communication scheme. Mobile user 2 conveys user 1’s signal to the base station because of its better communication channel quality...... 3 1.2 An example of a network-coding-based cooperative communication with two mobile users, one relay station, and a receiver...... 4 1.3 Comparison between the traditional XOR operation and a multiplier using BPSK signals...... 5

2.1 Source t1 transmitter block diagram...... 10 2.2 Transmitter-to-relay proposed method using CC-CDMA...... 11 2.3 Received signals at the base station. Note that the boxed part of this block diagram will be depicted in detail in Fig. 2.4...... 15 2.4 Detail diagram of the shaded area in Fig. 2.3 at the destination

for DF case.y ˜t2,d is multiplied byy ˜r1,d in order to be a redundant

signal for diversity achievement withy ˜t1,d...... 16 2.5 A general case of a NC-based cooperative communication system with K sources and Z relays...... 20

2.6 Error probability when changing mt1 value with 1, 2, and 3, under

the conditions of setting mt2 = 1, mr1 = 1...... 25

2.7 Error probability when changing mt1 value with 1, 2, and 3, under

the conditions of setting mt2 = 1, mr1 = 1, mr2 = 1...... 26

2.8 Error probability when changing mt1 value with 1, 2, and 3, under

the conditions of setting mt2 = 1, mr1 = 1, mr2 = 1, mr3 = 1. . . . . 27

2.9 Error probability when changing mt1 value with 1, 2, and 3, under

the conditions of setting mt2 = 1, mr1 = 1, mr2 = 1, mr3 = 1,

mr4 =1...... 28

2.10 Error probability when ﬁxing all mti and mrj values with 1, under the conditions of changing the number of relays...... 29 2.11 BPSK error probability performance...... 30

vii 3.1 Illustration of a cooperative relay network with N parallel potential relays [35]...... 32 3.2 General case: K source nodes with Z relay nodes. Users in this case cannot be relay...... 33 3.3 Outage probability vs. SNR with R = 0.5 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m...... 40 3.4 Outage probability vs. SNR with R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m...... 41 3.5 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 42 3.6 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 43 3.7 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 44 3.8 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 45 3.9 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 46 3.10 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 47 3.11 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 48 3.12 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 49 3.13 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 50 3.14 Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 51 3.15 Outage probability comparisons...... 52

4.1 Two phases of DF cooperative strategy...... 54 4.2 Illustration of water ﬁlling for power allocation case...... 57 4.3 Optimal power vs. equal power with R = 0.5 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m...... 59 4.4 Optimal power vs. equal power with R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m...... 60 4.5 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 61 4.6 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 62

viii 4.7 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 63 4.8 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 64 4.9 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 65 4.10 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 66 4.11 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 67 4.12 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 68 4.13 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 69 4.14 Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. . 70

ix List of Tables

1.1 Chart of multiplier and binary sum XOR...... 5

2.1 BER comparisons vary from SNR 5 to 20 dB...... 21

3.1 Outage performance comparisons vary from SNR 5 to 20 dB. . . . . 41

x List of Abbreviations and Symbols

2GS econd Generation

3GT hird Generation

ACFA uto Correlation Function

AFA mplify and Forward

BERB it Error Rate

BPSKB inary Phase Shift Keying

BSB ase Station

CC-CDMAC omplementary Code - Code Division Multiple Access

CCFC ross Correlation Function

DFD ecode and Forward

MAIM ultiple Aaccess Interferece

MIM ultipath Interference

MLM aximum Likelihood

MGFM oment Gernerating Function

MRCM aximum Ratio Combining

NCN etwork Coding

PDFP robability Density Function

xi PGP roccessing Gain

SNRS ignal Noise Ratio

TDMAT ime Division Multiple Access

cti,m(t) spreading signal or spreading sequence at transmitter i

crj ,m(t) spreading signal or spreading sequence at relay node j

fm central frequency

hti,rj (t) channel response via i-th transmitter and j-th relay

nm(t) m-th subcarrier noise K the number of users which the complementary code can support

Z the number of relay nodes which the complementary code can support

Q(x) Q-function

sti (t) transmitting signal at transmitter i

srj (t) transmitting signal at relay node j

wt wideband noise

wti,d weight of ti to d

wrj ,d weight of rj to d

xti (t) local information at transmitter i

xrj (t) local information at relay node j

yti,rj received signals at rj from ti

yti,d received signals at d from ti

yrj ,d received signals at d from rj

y˜d,sum total signals multiplied by the corresponding weight received at d

y˜t1,r1 decision variable at r1 from t1

xii αti,rj ,l(t) Nakagami-m distributed random variable

αti,d,l(t) Nakagami-m distributed random variable

αrj ,d,l(t) Nakagami-m distributed random variable

θti,rj ,l(t) phase of i-th transmitter to z-th relay

θti,d,l(t) phase of i-th transmitter to d

θrj ,d,l(t) phase of j-th relay to d

λti,rj ,l(t) delay via i-th transmitter and j-th relay δ Dirac-delta function

(DF ) Γti,rj useful signals via ti and rj in DF case

Γ(DF ) useful signals via t and d in DF case ti,d i

Γ(DF ) useful signals via r and d in DF case rj ,d j

(DF ) ξti,rj noise via ti and rj in DF case

ξ(DF ) noise via t and d in DF case ti,d i

ξ(DF ) noise via r and d in DF case rj ,d j

the conditional expected value ofy ˜(t1) for relay one case E d,sum ¯ the conditional expected value ofy ˜(t1) for relay two case E d,sum ˜ the conditional expected value ofy ˜(t1) for relay three case E d,sum ˘ the conditional expected value ofy ˜(t1) for relay four case E d,sum ˆ the conditional expected value ofy ˜(t1) for general case E d,sum the conditional variance ofy ˜(t1) for relay one case V d,sum ¯ the conditional variance ofy ˜(t1) for relay two case V d,sum ˜ the conditional variance ofy ˜(t1) for relay three case V d,sum

xiii ˘ the conditional variance ofy ˜(t1) for relay four case V d,sum ˆ the conditional variance ofy ˜(t1) for general case V d,sum

γb,1r SNR at d for relay one case

γb,2r SNR at d for relay two case

γb,3r SNR at d for relay three case

γb,4r SNR at d for relay four case

Pout outage probability Ψ( , ) the lower incomplete gamma function · · Lagrangian L λ Lagrangian multiplier

xiv Acknowledgments

I would like to express the deepest appreciation to my committee chairs Dr. Aldo W. Morales and Dr. Sedig S. Agili, who have the attitude and the substance of a genius: they continually and convincingly conveyed a spirit of adventure in regard to research and an excitement in regard to teaching. Without their guidance and persistent help this thesis would not have been possible. I would like to thank my committee member, Dr. Robert Gray for giving me such important and helpful suggestions for further improvement in this thesis, In addition, a thank you to Dr. Jerry F. Shoup, who introduced me to science, and whose enthusiasm for the electrical engineering ﬁeld had lasting eﬀect.

xv Dedication

I lovingly dedicate this thesis to my parents, who supported me each step of the way.

xvi Chapter 1

Introduction

1.1 Introduction

In this thesis, the outage probability performance and optimal power allocation, of a network coding cooperative communication system based on complementary code CDMA (NC-CC-CDMA), is investigated. In addition, our previous work in NC-CC-CDMA is further validated with several new MATLAB simulations.

1.2 Outage performance

1.2.1 Outage probability

There are two typical ways to improve and measure a communication system performance, which are bit error rate (BER: P (e)) and outage probability (Pout). In this thesis, we focus on reducing Pout to improve our system performance. The outage performance method is based on channel capacity and mutual information [1]. Outage occurs when the channel capacity is less than the transmission rate R, i.e., log(1 + SNR h 2) < R. Hence, the main idea of the performance improvement | | is increasing the channel capacity, which can be achieved by several transmission techniques such as CDMA [3]-[7], cooperative communication [8]-[10] and network coding [11] (see also Section 1.4). The essential ideas of these transmission techniques will be introduced in the following sections. The outage probability performance comparisons with others’ works are also depicted in Chapter 3. 2

1.2.2 Optimal power allocation

Optimization methods are categorized into two parts: standard max-min problem [12] (see Section 4.1.1) and ”Lagrangian problem [13]” (see Section 4.1.2). The optimization is always the main concept to achieve the optimal situation. In other words, we not only pursue the best system performance but also provide an optimal power allocation for mobile users. Optimal power allocation can meet the limited radio resources requirements such as bandwidth and transmission power. The main objective of this research is to ﬁnd solutions that can improve the channel capacity and utilization of the radio resources that are available to the service providers. We will show with numerical results in Chapter 4, that the outage probability can be further reduced with optimal power allocation.

1.2.3 System outage probability introduction

The outage probability performance analysis is performed on NC-CC-CDMA that was proposed in our previous work [2]. The NC-CC-CDMA system will be brieﬂy introduced in the following section. Due to the cooperative nature of the system and its diversity, the outage probability can be dramatically reduced; while the network coding part improves the system’s channel capacity

1.3 The system framework

In this Section, we briefly review the network-coding-based cooperation from our previous work [2], and introduce the main idea in our system model. In Chapter 2, mathematical analyses for the model will be presented. In NC-based cooperative communication system, network coding is used as the second transmission phase into the traditional cooperative networks [15]-[25]. The network coding method improved the cooperation system capacity as compared to the conventional one, which is a significant enhancement for outage probability performance analysis. In addition, the traditional NC-based cooperative communication system achieved diversity, which is an important characteristic for system reliability. In [2], we offered an alternative solution for diversity by introducing a 3 multiplier at the base station (BS) as well as using CC-CDMA in the cooperation system. This method, NC-based CC-CDMA, provided a better BER performance as compared to the traditional cooperative communication in addition to saving time slots and channel bandwidth. Moreover, the system’s complexity was also reduced because there was no maximum likelihood detection used at the base station (no MAC layer decoding).

1.3.1 Cooperation

User 2 User 2's signal

User 1's signal User 1's signal

User 1's signal User 1 Base station

User 1's signal (strong) User 2's signal (strong)

User 2 transmits user 1's signal (strong)

User 1's signal (weak)

Figure 1.1. Traditional cooperative communication scheme. Mobile user 2 conveys user 1’s signal to the base station because of its better communication channel quality.

In this subsection, we brieﬂy review how cooperative networks work. As shown in Fig. 1.1, cooperation is simply a process that mobile user 1 asks relay(s) – mobile user 2 – for help to convey their signals to the BS. The reason we use cooperation is because the communication link can have several barriers (see Fig. 1.1) on the light of sight (LOS). For example, the channel between mobile user 1 and the BS has couple of buildings, which can be regarded as barriers. Thus, this route may weaken the transmitted signal. Taking into account the uncontrollable conditions of the wireless transmission, another mobile user, who has a clear communicating 4 channel with the BS, can receive and retransmit signals. This simple idea improves the overall performance of the wireless communication system; however, it taxes the limited power resources of mobile users.

1.3.2 Network coding

st1

ar1 = at1 ⊕ at 2 User 1 Sr1 st1 Relay station

st 2

Base station

User 2 S ,S : Analog signals st 2 ti rj ati ,arj :Digital signals

Figure 1.2. An example of a network-coding-based cooperative communication with two mobile users, one relay station, and a receiver.

In the traditional network coding mechanism, two or more digital signals are XOR-ed (in binary) at the MAC layer at the relay(s) and the BS (see Fig. 1.2 ). After passing through the cooperative user’s channel, the received messages, which have been XOR-ed at the relays, will be XOR-ed with the original signal at the BS to recover the mobile user’s signal. This idea is widely used in a cellular system as shown in Fig. 1.2, where, for simplicity, one relay with two mobile users and a receiver is depicted in [2].

1.3.3 Multiplier

The traditional network coding [11] uses an XOR operation throughout the communication link, as shown in Fig. 1.2. Our scheme used a multiplier at the BS that matches the operation of the traditional XOR, as shown in Table 1.1 and Fig. 1.3. However, the main advantage of using the multiplier is its use in the received signal, at the BS, thereby avoiding one extra decoding step in the protocol stack. 5

Multiplier (be- XOR gate (af- fore the deci- ter the decision sion device) device)

s1 s2 s1 s2 + ×+ = + + ⊕+ = + + × = + ⊕ = × −+ = − ⊕ −+ = − − × =− + − ⊕ =− + − × − − ⊕ −

Table 1.1. Chart of multiplier and binary sum XOR.

1 0 1 0 + - + - ⊕ 0 1 1 0 ⊗ + - - + 1 1 0 0 + + - - Traditional Multiplier XOR operator

Figure 1.3. Comparison between the traditional XOR operation and a multiplier using BPSK signals.

In order to show that the multiplier can accomplish the same function as the XOR operation, BPSK signals on the quadrant that are basically separated by positive and negative axes can be considered. The results of multiplication of s1 and s2 analog signals give either a positive or negative phase, which has equivalent results as the XORTraditional logic XOR gateoperator (see Table 1.1). In theMultiplier decode-and-forward (DF) case, decisions on the signals (s1 and s2) are made after despreading and demodulation to obtain the original analog signals from the transmitters. Therefore, if the channel conditions are such that the signal phases do not severely change, we can safely replace the XOR operation by a multiplier in the case of analog signals.

1.4 Beneﬁts of the proposed system

In this Section, we briefly introduce the major differences between TDMA and CDMA and their spectral efficiency properties [4]-[7]. 6

1.4.1 TDMA

1. Advantages of TDMA [7]: There are plenty of advantages of TDMA in cellular communication systems.

It can easily adapt to data transmission as well as voice communication. • It is capable of carrying 64 kbps to 120 Mbps of data rates. This • allows the operator to do services like fax, voice band data, and short message service (SMS) as well as bandwidth-intensive application such as multimedia and videoconferencing. TDMA separates users by time slots to ensure there will be no interfer- • ence with transmissions. It provides users with an extended battery life, because it transmits • only a portion of the time during conversations. It is the most cost-eﬀective technology for upgrading analog to digital. • 2. Disadvantages of TDMA [7]:

Each user has a predefined time slot. When a user is moving from one • cell to another, if all time slots in this cell are fully loaded, the user might be disconnected. It is subjected to multipath distortion. A message sent from a tower to • a receiver might come from any one of multi-directions. It might have bounced off a couple of different buildings before arriving.

1.4.2 CDMA

1. Advantages of CDMA [7]:

Failures happen only when the mobile device is at least twice as far • from the base station. Hence, it is used in the rural areas where GSM (TDMA) cannot cover. 7

It has a high spectral capacity that accommodates more users per MHz • of bandwidth.

2. Disadvantages of CDMA [7]:

The major problem in CDMA technology is channel pollution, i.e., sig- • nals from too many cells exist in the subscriber’s mobile device but none of them are dominant. Another drawback in this technology when compared to TDMA is the • lack of international roaming capabilities.

1.4.3 Spectrum eﬃciency

Spectrum efficiency is defined as the maximum number of travel channels per MHz per cell. The spectrum efficiency depends upon the average bit error rate (BER) that is sufficient for a service quality. In a TDMA system, the channel capacity is fixed to a limited number of time slots and new users cannot be added when each of these slots is loaded. Due to GSM’s better error management and frequency hopping, the interference of the co-channel is reduced. This allows the frequency to be reused without ruining quality of service (QoS) [7]. CDMA spectral efficiency can be improved by radio resource management techniques, resulting in a higher number of simultaneous calls, and higher data rates can be achieved without adding more radio spectrum or more base station sites.

1.4.4 Beneﬁts of complementary codes

CC-CDMA was introduced in [3] and [4] and offers several advantages. For example, the codes are perfectly orthogonal ones in a sense that they offer zero cross-correlation used between them in both synchronous and asynchronous transmission channels. Therefore, a CC-CDMA system can achieve multi-access interference (MAI)-free operation for both uplink and downlink transmissions. As it was mentioned in [3] ”The joint effect of ideal cross-correlation functions and ideal auto- correlation functions makes a CDMA system using them virtually interference-free, to make a truly noise-limited CDMA system!” 8

There are many diﬀerent ways to characterize the CDMA codes, but nothing can be more intuitive and eﬀective than the auto-correlation function (ACF) and cross-correlation function (CCF) [4]. Following the statement as mentioned above, the new proposed CDMA system [4] (CC-CDMA) can be concluded in three main advantage points:

1. The significant property attribute of CC-CDMA system is that the ”offset stacked” spreading modulation method is particularly beneficial for multi- rate data transmission in multimedia services.

2. This offset stacking spreading technique can show data transmission rate by simply shifting more than one chip (at most L chips) between two neighboring offset stacked bits. When L chips are shifted between two consecutive bits, the proposed system reduces to a traditional CDMA system, yielding the lowest data rate1 while the highest data rate is achieved if only one chip is shifted between two neighboring offset stacking bits.

3. By doing these steps, the highest spreading eﬃciency equivalent to one can be achieved, which implies that every chip is capable of carrying one bit information.

Thus, the proposed system – CC-CDMA – is capable of delivering a higher bandwidth eﬃciency than a traditional CDMA system under the same processing gain. This thesis is organized as follows: in Chapter 1, we introduce the main top- ics of this thesis: outage probability and optimal power allocation. The outage performance system’s background is also described in Section 1.3. In Section 1.3, a brief review of network-coding-based cooperation is presented and the modiﬁed system model for a NC-based cooperative system is proposed. The outage and error probability’s system model is analyzed in detail in Chapter 2, Section 2.1, under the framework of CC-CDMA NC-based cooperative system. Diversity analysis is presented in Section 2.2. Numerical simulations using MATLAB and Nakagami- m fading channel are provided in Section 2.3. In Chapter 3, outage probability is derived and simulation results, from the derived outage probability equations,

1Note that the spreading efficiency (SE) of all conventional CDMA-based mobile communication systems, such as IS-95, CDMA2000, and W-CDMA, are equal to 1/N, which is far less than 1, which implies that these systems can not offer a better bandwidth efficiency. 9 are presented and discussed. Similarly, in Chapter 4, optimal power allocation expression is derived, simulated and discussed. The numerical results are shown by MATLAB figures. Conclusions follow in Chapter 5. Supporting equations are included in Appendices A-C [2] and Appendix D, which is an extension analysis for the diversity gain in this thesis. Chapter 2

Error rate probability

2.1 System model

In this Section, we will review the mathematical basis of the NC-based cooperative system [2] introduced in Section 1.3 where the decode-and-forward [9] mode is used. In the following subsections, the proposed method [2] was further analyzed by using maximum ratio combining (MRC) that leaded to error probability calculation. ct() rM1 , 2/ETbbcos(2 ft M ) ctrM, () 2/cos(2)E Tft ˆ Note that the network coding is only used at the base station. 1 bb M

s ()t   t1 ,1

ctt ,1 () 2/cos(2)E Tft 1 bb 1 s ()t b att () t1 t1 1 ()M Polar NRZ  ytd1,   s ()t t1 ,2 ct() t1 ,2 2/cos(2)EbbTft 2    y (1) s ()t td2,   tM1 ,

ct() tM1 , 2/cos(2)EbbTft M

Figure 2.1. Source t1 transmitter block diagram.

()M ytd2,

(1) yrd1,

()M yrd1,

123 3 4

yˆ 11

stt ,1 ()   1

2E / T cos(2 f t ) ct() bb 1 t1 ,1    st() tM1 , ct() 2Eb / T b cos(2 f M t ) Source 1 tM1 ,

at() st() t1 Spreading t Polar 1 & NRZ Modulation

Nakagami-m fading Relay 1 channel

yt() y nTb at() tr, Despreading t1 11 tr11, Decision Polar & device NRZ Demodulation

Spreading & yt(1) () y(1) Modulation tr11, tr, Bandpass filter (nT 1) b 11 ( )dt   nT f11 W  f  f  W b

2 /Tb cos(2 f1 t ) ct() t1 ,1  yt()M () y()M tr, Bandpass filter 11 (nT 1) b tr11, ( )dt   nT fMM W  f  f  W b 2 /T cos(2 f t ) bMct() tM1 ,

Figure 2.2. Transmitter-to-relay proposed method using CC-CDMA.

2.1.1 Signals received at relay nodes

According to [3], the complementary code (CC)-CDMA transmitted signal from st1 , as shown in Fig. 2.1 and Fig. 2.2, can be expressed as:

M 2E s (t) = b c (t) cos(2πf t)x (t), (2.1) t1 T t1,m m t1 m=1 r b X where ∞ x (t) = a (n)g(t nT ), a (n) +1, 1 , (2.2) t1 t1 − b t1 ∈ { − } n=−∞ X 12 and c (t) is the m-th element spreading sequence [3] assigned to s , 1 m M, t1,m t1 ≤ ≤ ct1,m is the normalized chip waveform with unit energy.

1 1 ct ,m(t) , , nTb t (n + 1)Tb. (2.3) 1 ∈ √M −√M ≤ ≤ and g(t nT ) is the bit pulse waveform function. The rectangle function g(x) − b is a function that is 0 outside the interval [0,Tb/2] and unity inside it. It is also called the gate function, pulse function, or window function. An L-tap transversal ﬁlter is used to model the multipath Nakagami-m fading channel. The complex low-pass impulse response of the channel for the ti-th space [2] can be expressed by L −jθ h (t) = α e ti,rj δ(t λ ), (2.4) ti,rj ti,rj − ti,rj Xl=1 where αti,rj , θti,rj , and λti,rj are the ti-th space and rj-th path’s envelope, phase, and delay, respectively. δ(t) is a Dirac-delta function; αti,rj is a Nagakami-m distributed random variable. Since the CC-CDMA system consists of M subcarriers, therefore, there are M channels between source and relay nodes. Then, the complete impulse response can be expressed as a matrix

h = [h(1) (t), , h(m) (t), , h(M) (t)]. (2.5) ti,rj ,l ti,rj ,l ··· ti,rj ,l ··· ti,rj ,l

Therefore, signals received at the relay node, after transmission through a slow multipath Nakagami-m fading channel, can be written as:

M L ζ (t) = h(m) (t) ? s (t) ti,rj ti,rj ,l ti,rj ,m m=1 X Xl=0 M L (m) (m) −jθ (m) = α e ti,rj ,l s (t λ ), (2.6) ti,rj ,l ti,rj ,m − ti,rj ,l m=1 X Xl=0 where ’?’ represents the convolution operation. The transmitted signals, after the convolution with the channels and without channel gain and phase, can be written as:

2E s (t λ(m) ) = b c (t λ(m) ) cos 2πf (t λ(m) ) ti,rj ,m ti,rj ,l T ti,m ti,rj ,l m ti,rj ,l − r b − − 13

x (t λ(m) ), (2.7) × ti,rj − ti,rj ,l where λ(m) represents the m-subcarrier delay between t to r channel. Conse- ti,rj ,l i j quently, the received signals yti,rj at rj from ti are:

yti,rj (t) = ζti,rj (t) + w(t)

M L (m) (m) −jθ (m) = α e ti,rj ,l s (t λ ) + w(t), (2.8) ti,rj ,l ti,rj ,m − ti,rj ,l m=1 X Xl=0 where w(t) is the wideband noise. By adding up the multipath signals, after despreading and demodulation, at the relay node as shown in Fig. 2.2,

(1) (m) (M) y˜ti,rj (t),..., y˜ti,rj (t),..., y˜ti,rj (t),

(m) wherey ˜ti,rj (t) is the m-th subcarrier signal from transmitter ti to relay node rj

Tb 2 y˜(m) (t) = y(m) (t) cos(2πf t)c (t)dt, (2.9) ti,rj ti,rj T m ti,m Z0 r b

Similarly, the direct signal from a transmitter ti to the destination can be expressed as: Tb 2 y˜(m)(t) = y(m)(t) cos(2πf t)c (t)dt, (2.10) ti,d ti,d T m ti,m Z0 r b where L (m) (m) (m) −jθ (m) y (t) = α e ti,rj ,l s (t λ ) + n (t), (2.11) ti,rj ti,rj ,l ti,rj ,m − ti,rj ,l m Xl=0 and L (m) (m) (m) −jθ (m) y (t) = α e ti,d,l s (t λ ) + n (t), (2.12) ti,d ti,d,l ti,d,m − ti,d,l m Xl=0 where λ(m) represents a m-th subcarrier time delay between source t to relay r ti,rj ,l i j of the Nagakami-m channel. Hence, the detected signals, using CC-CDMA, at the relays can be written as:

M (m) y˜ti,rj = y˜ti,rj (t) m=1 X 14

M Tb 2 = y(m) (t) cos(2πf t)c (t)dt ti,rj T m t1,m m=1 0 r b X Z M Tb L (m) (m) −jθ (m) = α e ti,rj ,l s (t λ ) + n (t) ti,rj ,l ti,rj ,m − ti,rj ,l m m=1 0 X Z Xl=0 2 cos(2πf t)c (t)dt × T m ti,m r b M L Tb (m) 2 (m) −jθ (m) = α e ti,rj ,l s (t λ ) ti,rj ,l ti,rj ,m ti,rj ,l Tb − m=1 l=0 0 r X X Z cos(2πf t)c (t)dt × m ti,m M Tb 2 + n (t) cos(2πf t)c (t) T m m ti,m m=1 Z0 r b (DFX) (DF ) = Γti,rj + ξti,rj

0 −jθti,rj ,0 (DF ) = Ebati (n )αti,rj ,0e + ξti,rj , (2.13) p where DF represents decode-and-forward. Similarly, the detected signals at the base station are given by,

y˜ = Γ(DF ) + ξ(DF ) ti,d ti,d ti,d 0 −jθ (DF ) = E a (n )α e ti,d,0 + ξ . (2.14) b ti ti,d,0 ti,d p Where Γ(DF ) and ξ(DF ) are signal and noise, respectively and are discussed in detail ti,d ti,d in Appendices A and B.

2.1.2 Signals received at the destination

Similar to the analysis shown above, the signals received at the destination from the relays can be written as:

M (m) 0 −jθ (DF ) y˜ = y˜ (t) = E a (n )α e rj ,d,0 + ξ . (2.15) rj ,d rj ,d b rj rj ,d,0 rj ,d m=1 X p 15

2.2 DIVERSITY ANALYSIS

For diversity analysis, we will use the proposed method, which is shown in Fig. 2.3. The details of the boxed area in Fig. 2.3 is shown in Fig. 2.4. Note that in the proposed method, a multiplier is used before the decision device whereas in previous method the XOR operation is used after the decision device, therefore reducing system’s complexity and reducing probability of error.

Detecting Signal signals from t1 Despreading y from t1 to d t1,d Decision Device & sum (Threshold) Demodulation

Signal from t2 Despreading yt ,d to d 2 & Demodulation

yt ,d × yr ,d Signal multiplier 2 1 from r1 Despreading to d & y Demodulation r1,d

y analog signal t1,d The main works at the base station. Signals are sent y analog signal from source 1, 2, and relay, where relay’s signal t2 ,d contains source 1 and 2's information. y analog signal r1,d y y analog signal t2 ,d × r1,d

Detected t1's signal (DIGITAL)

Base station

Figure 2.3. Received signals at the base station. Note that the boxed part of this block diagram will be depicted in detail in Fig. 2.4.

In the following analysis, we focus on the diversity properties of received signal from user 1, t1, at the destination. Before diversity analysis, two issues need to be tackled, optimal weights and probability of error by ﬁrst calculating the output ofy ˜ y˜ which is a redundant signal containing t information.y ˜ will t2,d × r1,d 1 t1,d be multiplied by an optimal weight wt1,d, andy ˜t2,d will be multiplied byy ˜r1,d and 16

y(1) td1 ,

. ()t1 nT y y b td1 , d, sum Decision .   device . w td1 , ()M ytd, 1 ()t nT y 2 b d, sum Decision (1)  y device td2 , . w y rd1 , td2 , .  . w td2 , y()M td2 ,

(1) y w Redundant signal rd1 , rd1 , . y rd1 , .  .

y()M rd1 ,

Figure 2.4. Detail diagram of the shaded area in Fig. 2.3 at the destination for DF case. y˜t2,d is multiplied byy ˜r1,d in order to be a redundant signal for diversity achievement withy ˜t1,d.

another optimal weight wr1,d, (see Fig. 2.4) as shown below:

y˜(t1) = w y˜ + w y˜ y˜ . (2.16) d,sum t1,d t1,d r1,d t2,d × r1,d

The expected value and variance ofy ˜t1,d are shown in Appendix C. 17

2.2.1 Sub-optimal weights

Let ”wt1,d” be the sub-optimal weight from transmitter one to the destination and ”wr1,d” be the optimal weight from relay one to the destination. Following the results described in the Appendix C, Eq. (C.14) to (C.18), the sub-optimal weights are given by:

√ ∗ Ψ Υ w = q q t1,d N0 2 N0 2 2 Φ 2 +Ψ Υ q  N0 (2.17) ∗ Φ 2 w = √ q  r1,d 2 N0 2 Υ Φ 2 +Ψ Υ  The output signal to noise ratio is found in the Appendix C and is given by:

w2 Ψ2 + 2w w ΨΦ + w2 Φ2 (SNR) = t1,d t1,d r1,d r1,d , (2.18) o w2 N0 + w2 Υ t1,d 2 r1,d

By substituting Eq. (2.17) (sub-optimal weights) into Eq. (2.18) (output SNR equation), the maximum output signal-to-noise ratio is given by:

q √ N0 2 Ψ Υ Φ 2 q q Ψ + √ q Φ N0 2 N0 2 2 N0 2 2 Φ 2 +Ψ Υ Υ Φ 2 +Ψ Υ (SNR)o = √ 2 q N 2 Φ 0 Ψ Υ N0 + 2 Υ q N q N 2 √ q N 0 Φ2 0 +Ψ2Υ Υ Φ2 0 +Ψ2Υ 2 2 2 q √ 2 N 2 2 Φ 0 1 Ψ Υ √ 2 N q + 2 0 2 N0 Υ Φ 2 +Ψ Υ = 2 √ 2 q N 2 Φ 0 1 Ψ Υ N0 √ 2 N q + Υ Φ2 0 +Ψ2Υ N0 2 Υ 2 2 q 2 √ 2 N0 Ψ2 Υ Φ 2 q + √ N0 Υ 2 ! = , (2.19) 2 2 N0 Ψ Υ + Φ 2 where Ψ, Υ, and Φ are deﬁned in the Appendix C. 18

2.2.2 Probability of error at destination

Considering a BPSK CC-CDMA-based transmission technique, and following Fig. 2.3, the probability of error of the received signal, after the decision device at the base station, from transmitter t1 is analyzed (extension to other received signals can be similarly done). Assuming that signals are broadcasted from all transmitters and relays, and noting that the output ofy ˜ y˜ is a redundant signal containing t2,d× r1,d t1 information, the error probability is given by:

∞ ∞ ∞ P (e) = P (e αt ,d,0,αr ,d,0)pα (αt ,d,0)pα (αt ,d,0) | i 1 t1,d,0 1 t2,d,0 2 Z0 Z0 Z0 p (α )dα dα dα , (2.20) αr1,d,0 r1,d,0 t1,d,0 t2,d,0 r1,d,0 where the conditional error probability in Eq. (2.20), given the channel responses to transmitter ti and relay r1 depicted by αti,d,0 and αr1,d,0, respectively, is as follows:

∞ P (e α ,α ) = p(˜y(t1) a = 1, α ,α )dy˜(t1) , (2.21) | ti,d,0 r1,d,0 d,sum| t1 − ti,d,0 r1,d,0 d,sum Z0

(t1) and the conditional probability distribution of the decision variabley ˜d,sum(t), given also the channel responses, is:

2 (t1) − y˜d,sum+E /2V (t1) e p(˜y at = 1, αt ,d,0,αr ,d,0) = , (2.22) d,sum| 1 − i 1 √2π V where and are shown in the Appendix C. The Nakagami-m PDF is in essence E V a central chi-square distribution and is given by:

m 2m−1 2m α 2 p (α) = e−mα , α 0. (2.23) α Γ(m) ≥

Where m is the Nakagami-m fading parameter, which ranges from 0.5 to . Com- ∞ bining Eqs. (2.20) and (2.23), provides the following error probability:

∞ ∞ ∞ mt1 2mt1 −1 2 2m α 2 t1 t1,d,0 −mt α P (e) = Q E e 1 t1,d,0 Γ(m ) Z0 Z0 Z0 r V t1 19

mt2 2mt2 −1 mr1 2mr1 −1 2m α 2 2m α 2 t2 t2,d,0 −mt α r1 r1,d,0 −mr α e 2 t2,d,0 e 1 r1,d,0 × Γ(mt2 ) Γ(mr1 ) dα dα dα , (2.24) × t1,d,0 t2,d,0 r1,d,0 where the Q-function in Eq. (2.24) is given by:

1 π/2 x2 Q(x) = exp dφ, x 0. (2.25) π − 2 sin2 φ ≥ Z0 Therefore, the error probability can be rewritten as:

π/2 ∞ ∞ ∞ 2 1 γb,1r P (e) = exp E(1R) π − sin2 φ Z0 Z0 Z0 Z0 V(1R) mt1 2mt1 −1 mt2 2mt2 −1 2m α 2 2m α 2 t1 t1,d,0 −mt α t2 t2,d,0 −mt α e 1 t1,d,0 e 2 t2,d,0 × Γ(mt1 ) Γ(mt2 )

mr1 2mr1 −1 2m α 2 r1 r1,d,0 −mr α 1 r1,d,0 e dαt1,d,0dαt2,d,0dαr1,d,0dφ. (2.26) × Γ(mr1 )

The above integrations do not have closed form solutions1; therefore, a numerical integration, in MATLAB, is performed on Eq. (2.26). The system’s performance can improve dramatically when the number of relays increases, as discussed in the next sections. For instance, Eq. (2.26) can be extended from a single relay equation to a more general form including several relays (see Fig. 2.5). For simplicity, Eq. (2.27) shows the probability of error using a four-relay NC-based CC-CDMA cooperative communication system:

π/2 ∞ ∞ ∞ ∞ ∞ ∞ 2 1 γb,1r P (e) = exp E(4R) π − sin2 φ Z0 Z0 Z0 Z0 Z0 Z0 Z0 V(4R) mt1 2mt1 −1 mt2 2mt2 −1 2m α 2 2m α 2 t1 t1,d,0 −mt α t2 t2,d,0 −mt α e 1 t1,d,0 e 2 t2,d,0 × Γ(mt1 ) Γ(mt2 )

mr1 2mr1 −1 mr2 2mr2 −1 2m α 2 2m α 2 r1 r1,d,0 −mr α r2 r2,d,0 −mr α e 1 r1,d,0 e 2 r2,d,0 × Γ(mr1 ) Γ(mr2 )

mr3 2mr3 −1 mr4 2mr4 −1 2m α 2 2m α 2 r3 r3,d,0 −mr α r4 r4,d,0 −mr α e 3 r3,d,0 e 4 r4,d,0 × Γ(mr3 ) Γ(mr4 )

dαt1,d,0dαt2,d,0dαr1,d,0dαr2,d,0dαr3,d,0dαr4,d,0dφ. (2.27)

1For a special case, we have a closed form as shown in subsection Appendix D. 20

r 1 h rd1 , htr, 11 h trK , 1 r2 h h rd2 , tr12, t1 h htr, td1 , K 2 d htr, h 11Z  tdK ,

tK h trKZ, 1 r h Z 1 rdZ 1 , h h tr1 , Z trK , Z hrd, rZ Z

y ytd, Received y rd1 , 1 rd2 , signals d y rdZ1 , yrd, y Z tdK ,

Figure 2.5. A general case of a NC-based cooperative communication system with K sources and Z relays.

2.3 Numerical analyses

2.3.1 NC-based cooperative by using a multiplier

In this Section, we analyze the proposed system’s performance using the probability of error described in Eq. (2.26), where P (e) is plotted against the SNR denoted by γb,1r, in Figs. 2.6, 2.7, and 2.8 under diﬀerent Nagakami-m parameters. In these

ﬁgures, mti represents the Nagakami-m parameter fading channel from transmitter ti to relay rj and the base station d; mrj represents the Nagakami-m parameter fading channel from relay rj to the base station d. Notice that the system’s performance dramatically improves when the mt1 value increases (see Figs. 2.6 - 2.8).

In Fig. 2.6, a single relay NC-based cooperative communication system, mt2 and mrj ’s Nakagami-m parameters are ﬁxed at 1, while mt1 changes from 1, 2, and 3 (note that when m = , the channel has no fading). Similarly, in Figs. 2.7 and ∞ 21

SNR 5 dB SNR 10.5 dB SNR 15 dB SNR = 20 dB ≈ ≈ ≈ The proposed system 2.39e-02 2.846e-03 3.808e-04 2.18e-05 Distributed phase steering (0◦) [21] 1.727e-01 9.725e-02 5.629e-02 2.799e-02 Distributed phase steering (90◦) [21] 1.041e-01 3.916e-02 1.6e-02 4.926e-03 NCCD [22] 8.025e-02 7.109e-03 1.023e-03 9.059e-05 ML detection [23] 5.19e-02 6.768e-03 7.17e-04 6.95e-05 Table 2.1. BER comparisons vary from SNR 5 to 20 dB.

2.8, mt2 and mrj ’s Nakagami-m parameters are ﬁxed at 1, the system performance

improves when mt1 value increases. In Fig. 2.10, we categorize multi-relay system performance into a graph to

show diversity property. When mti and mrj ’s Nakagami-m parameters are ﬁxed at 1, system performance improves when the number of relays increase.

2.3.2 Multiplier network coding vs. ML distributed phase steering

Considering the worst case (θ = 0◦) and the best case (θ = 90◦) [21] in Rayleigh

channel fading, which is equivalent to Nakagami-m with mt1 = 1, comparisons are made with a similar scheme [21], using decode-and-forward with maximum likelihood detection method. The proposed system performs better as depicted in Fig. 2.11. In addition, in [21], it is assumed a distributed phase precoding, which is not the case in the proposed method. A BPSK constellation, i.e., x 1 , j ∈ {± } the received waveform is 2 h gives four possibilities for any fixed set of j=1 ± j fading coefficients. The distributedP phase steering with the maximum likelihood (ML) detection [21], when θ = 0◦, the received signals map onto two different quaternary pulse amplitude modulation constellations (4-PAM); while θ = 90◦, the constellation becomes a 4-QAM. By the BER analysis in Fig. 2.11, when SNR = 20 dB, their performance improves to BER = 2.799 10−2 when θ = 0◦, × which is the worst case; and the performance improves to BER = 4.926 10−3 × when θ = 90◦, which is the best case while the proposed method has a better performance, P (e) = 2.18 10−5. However, this performance is achieved using × multi-relays with a simple multiplier at the destination, while [21] uses a more advanced demodulator at the MAC layer. This comparison shows that there is a 22 trade-off between performance and receiver complexity. The BER performance in Fig. 2.11 [21] is shown as:

1. For the best case (θ = 90◦):

¯b 1 γ Pθ= π = 1 , (2.28) 2 2 − 2 + γ r

1 where γ = 2 is the average received SNR. σn 2. For the worst case (θ = 0◦):

1 √γ 1 γ arctan √1 + γ P¯b + + θ=0 ≈ 4 − 4√4 + γ 24 + 12γ 24(1 + γ)1.5 γ 3 + −24(1 + γ)(2 + γ) 24 + 16γ √ 4 3γ arctan 1 + 3 γ 3γ + 2(3 + 4qγ)1.5 − 4(3 + 4γ)(3 + 2γ) √ 2+5γ 1 √2γ arctan √ + 2(1+3γ) 12(1 + 2γ)(4 + 9γ) − 12(2 + 5γ)1.5 γ(1 + 3γ) 3 + + 6(2 + 5γ)(4 + 17γ + 18γ2) 8(3 + 8γ)(2 + 6γ) √ 3+10γ √3γ arctan √ 3γ(1 + 4γ) 3(1+4γ) + − (3 + 10γ)1.5 2(3 + 10γ)(6 + 34γ + 48γ2) √ √2γ arctan √ 2+5γ 2(1+γ) 9 − 12(2 + 5γ)1.5 − 16(3 + 8γ)(3 + γ) √3(3+10γ) γ(1 + γ) √3γ arctan + (3+4γ) + 6(2 + 5γ)(4 + 9γ + 2γ2) − 2(3 + 10γ)1.5 3γ(3 + 4γ) 1 + , (2.29) 4(3 + 10γ)(9 + 27γ + 8γ2) − 12(1 + 2γ)(4 + γ)

1 where γ = 2 is the average received SNR. σn 23

2.3.3 Multiplier network coding vs. Network-Coded Co- operative Diversity (NCCD)

Considering the same fading channel (Rayleigh which is equivalent to Nakagami-m with mt1 = 1), comparisons are made with a similar scheme [22], using decode- and-forward method. The proposed system performs better as depicted in Fig.

2.11. Note that [22] uses only only transmitter (Ns = 1) and one relay, which typically oﬀers better performance than using two transmitters and one relay as in the proposed scheme. Consider the cooperation scheme with a single relay with DF [22], when SNR = 20 dB, their performance improves to BER = 4.52 10−5 × when N = 1, and BER = 9.059 10−5 when N = 2, while the proposed method s × s has a better performance, P (e) = 2.18 10−5 with two transmitters (N = 2) × s and a single relay. Note that [22] always uses cooperative maximum ratio combining (C-MRC) as the proposed system does; however, they use a more advanced demodulator at the MAC layer. This comparison shows that there is a trade-oﬀ between performance and receiver complexity. The BER performance in Fig. 2.11 [22] is shown as:

1 Ns 1 Ns 1 1 P i C1 + C2 + , (2.30) s,X $ γ¯ X γ¯ X γ¯ γ¯ fi i=1 gi j=1 fi R ! X Xj=6 i

√ where BPSK . C1 and C2 are 45+ 5 and 3 , respectively. X ∈ { } X X 160 16

2.3.4 Multiplier network coding vs. ML detection network coding

Considering the same fading channel (Rayleigh which is equivalent to Nakagami-m with mt1 = 1), comparisons are made with a similar scheme [23], using decode- and-forward with maximum likelihood detection method. The proposed system performs better or equivalent to as depicted in Fig. 2.11. In addition, in [23], it is assumed an error/delay-free control channel, which is not the case in the proposed method. With the DF maximum likelihood detection [23], when SNR = 20 dB, their performance improves to BER = 3.35 10−5 for one user and one × 24 relay 2, and BER = 6.95 10−5 for two users and one relay; while the proposed × method has a better performance, P (e) = 2.18 10−5, still with two transmitters. × However, this performance is achieved using multi-relay with a simple multiplier at the destination, while [23] uses a more advanced demodulator at the MAC layer. This comparison shows that there is a trade-oﬀ between performance and receiver complexity. The BER performance in Fig. 2.11 is shown as:

rD,Ns+1 = hRDxˆC + nD,Ns+1 h Ns = RD xˆ + n √N i D,Ns+1 s i=1 X h Ns = RD (x + e ) + n √N i i D,Ns+1 s i=1 X h Ns h Ns = RD x + RD e + n . (2.31) √N i √N i D,Ns+1 s i=1 s i=1 X X where Ns represents the number of users.

2.4 Summary of the advantages of using a multiplier network coding

Comparisons between a multiplier XOR and a conventional XOR operation are made as follows:

1. The ﬁrst advantage of using a multiplier XOR is that considering maximum ratio combining (MRC) at the destination d for both multiplier and conventional XOR operation, the sum of weighted signals at d from the proposed system are much stronger than from the traditional XOR system. At the destination in the proposed system, each received relay signal is multiplied

by received signal from t2 to generate a t1-like signal, which is considered as a

copy for t1 signal to achieve diversity performance. The detailed mathematics

2 Note that [23] uses only one transmitter (Ns = 1) and one relay, which typically oﬀers better performance than using two transmitters and one relay as in the proposed scheme. 25

DF cases for m =1, m =2, m =3 t1 t1 t1 0 10

−1 10

−2 10

−3 10

−4 10

−5 10 Error probability (Pe)

−6 10 (2−Tx, 1−Relay, m =1) t1

−7 (2−Tx, 1−Relay, m =2) 10 t1 (2−Tx, 1−Relay, m =3) t1 −8 10 0 2 4 6 8 10 12 14 16 18 20 SNR(dB)

Figure 2.6. Error probability when changing mt1 value with 1, 2, and 3, under the conditions of setting mt2 = 1, mr1 = 1.

for the multiplier XOR is shown as:

4 (t1) y˜d,sum = wt1,dy˜t1,d +y ˜t2,d wrj ,dy˜rj ,d j=1 X (t1) (t1) =y ˜s +y ˜n , (2.32)

where the valued signal term is:

(t1) 0 −jθt1,d,0 y˜s = wt1,d Ebat1 (n )αt1,d,0e 4 p 0 −jθrj ,d,0 +˜yt2,d Eb wrj ,darj (n )αrj ,d,0e j=1 p X 26

DF cases for m =1, m =2, m =3 t1 t1 t1 0 10

−1 10

−2 10

−3 10

−4 10

−5 10

Error probability (Pe) −6 10

−7 (2−Tx, 2−Relay, m =1) 10 t1

(2−Tx, 2−Relay, m =2) −8 t1 10 (2−Tx, 2−Relay, m =3) t1 −9 10 0 2 4 6 8 10 12 14 16 18 20 SNR(dB)

Figure 2.7. Error probability when changing mt1 value with 1, 2, and 3, under the conditions of setting mt2 = 1, mr1 = 1, mr2 = 1.

0 −jθt1,d,0 0 −jθt2,d,0 = wt1,d Ebat1 (n )αt1,d,0e + Ebat2 (n )αt2,d,0e 4 p 0 −jθ p E w a (n )α e rj ,d,0 . (2.33) × b rj ,d rj rj ,d,0 j=1 p X 0 Note that arj ,d(n ) is a conventional XOR signal at each relay node, which is a a = a . Following the digital signals simpliﬁcation in Eq. (2.33), t1 ⊕ t2 rj the channel weight for a multiplier XOR operation depends on each t r 2 × j channel, so it can generate a stronger output SNR (see Eq. (2.19)) by larger weights (see Eq. (2.17)); while a conventional XOR can be regarded as an

equal gain channel because t1 signals is recovered by t2 and rj’s signals after 27

DF cases for m =1, m =2, m =3 t1 t1 t1 0 10

−1 10

−2 10

−3 10

−4 10

−5 10

Error probability (Pe) −6 10

−7 (2−Tx, 3−Relay, m =1) 10 t1 (2−Tx, 3−Relay, m =2) t1 −8 10 (2−Tx, 3−Relay, m =3) t1 −9 10 0 2 4 6 8 10 12 14 16 18 20 SNR(dB)

Figure 2.8. Error probability when changing mt1 value with 1, 2, and 3, under the conditions of setting mt2 = 1, mr1 = 1, mr2 = 1, mr3 = 1.

decoding, so the weights are exactly the same. In other words, the output SNR cannot be greater than the output SNR in the multiplier XOR method. Therefore, BER performance comparison results in Fig. 2.11 can clearly show that system performance for a multiplier XOR is better.

2. A multiplier operation is simpler than using any signal detection method like maximum likelihood (ML) detection [23]-[21], or multiuser detection (MUD) [19] method. The error rate probability in the numerical analysis clearly shows that using a simple multiplier method cannot only improve system performance, but also reduce the system’s complexity. 28

DF cases for m =1, m =2, m =3 t1 t1 t1 0 10

−1 10

−2 10

−3 10

−4 10

−5 10

Error probability (Pe) −6 10 (2−Tx, 4−Relay, m =1) t1

−7 10 (2−Tx, 4−Relay, m =2) t1

−8 10 (2−Tx, 4−Relay, m =3) t1

−9 10 0 2 4 6 8 10 12 14 16 18 20 SNR(dB)

Figure 2.9. Error probability when changing mt1 value with 1, 2, and 3, under the conditions of setting mt2 = 1, mr1 = 1, mr2 = 1, mr3 = 1, mr4 = 1. 29

DF cases for m =1 t1 0 10

−1 10

−2 10

−3 10

−4 10 Error probability (Pe)

−5 10

(2−Tx, 1−Relay)

−6 10 (2−Tx, 2−Relay) (2−Tx, 3−Relay) (2−Tx, 4−Relay) −7 10 0 2 4 6 8 10 12 14 16 18 20 SNR(dB)

Figure 2.10. Error probability when ﬁxing all mti and mrj values with 1, under the conditions of changing the number of relays. 30

Comparison of the proposed method vs. other methods 0 10

−1 10

−2 10

−3 10 Error probability (Pe) Proposed method with 2−Tx, 1−Relay, m=1

DF ML method with 2−Tx, 1−Relay, m=1

−4 0 10 distributed phase method (=90 ), 2−Tx, 1−Relay, m=1

0 distributed phase method (=0 ), 2−Tx, 1−Relay, m=1

NCCD with 2−Users (Tx or Relay), m=1 −5 10 0 2 4 6 8 10 12 14 16 18 20 SNR(dB)

Figure 2.11. BPSK error probability performance. Chapter 3

Outage probability

In this Chapter, we introduce a outage probability analysis to determine the system’s performance. In Section 3.1, we build up channel mutual information and channel capacity equations, respectively. Following the channel capacity equation, we can derive a closed form outage probability based on Nakagami-m fading channel. Next, in Section 3.2, we show the outage performance in our work and three signiﬁcant comparisons with results of other researchers.

3.1 Outage probability

Outage happens when the signals cannot be reliably decoded at the destination. From an information theoretic point of view, given the information rate R and the channel gain h, outage occurs if the channel capacity is less than R, i.e., log(1 + SNR h 2) < R. Therefore, outage probability under the channel gain h can | | be expressed as a function of the transmission rate as given below:

P (R) Pr(log(1 + SNR h 2) < R) out , | | = Pr(SNR h 2 < 2R 1). (3.1) | | −

We expect that our system shall have a better outage probability performance because of CC-CDMA and network-coding-based system. As we mentioned in the introduction, network coding helps characteristics of the proposed system save bandwidth. Therefore, the greater the channel bandwidth, the better the outage 32

performance.LU et al.: OUTAGE PROBABILITY OF COOPERATIVE RELAY NETWORKS IN TWO-WAVE WITH DIFFUSE POWER FADING CHANNELS 43

Phase 1 Phase 2 ℎ 2 as the fading power variable between 𝑢 and 𝑣. Based ∣ 𝑢,𝑣∣ on (1), the PDF of 𝑋𝑢,𝑣 can be expressed as r1 r1 2𝐿 𝑥+𝜅𝑙 m relays 1 𝑎˜𝑙 𝑢,𝑣 √𝜅𝑙𝑢,𝑣 𝑥 ⋯ 𝑝 (𝑥)= 𝑒− 2𝜎2 𝐼 , (3) N relays 𝑋𝑢,𝑣 2 0 2 rn rn 2𝜎 2 𝜎 ∑𝑙=1 ( ) hr ,d ⋯ 1 hs,r ⋯⋯ (𝑙 1)/2 𝜋 1 r 𝑙 2 N r where 𝜅𝑙𝑢,𝑣 = 𝐾𝑢,𝑣 1+( 1) Δ𝑢,𝑣 cos ⌊ − ⌋ 2𝜎 , h N 2𝐿 1 s,rn hr ,d − − n 𝑎˜ = 𝑎 ,and 𝑥 denotes the greatest integer less hs,r 𝑙 (𝑙+1)/2 ( ) s N d s d than or⌊ equal to⌋ . Hereafter,⌊ ⌋ to alleviate the notation, we hs,d 𝑥 source destination source destination ˆ ˆ define 𝐾0 ≜ 𝐾𝑠,𝑑, Δ0 ≜ Δ𝑠,𝑑, 𝐾𝑛 ≜ 𝐾𝑠,𝑟𝑛 , Δ𝑛 ≜ Δ𝑠,𝑟𝑛 , ˇ ˇ 𝐾𝑛 ≜ 𝐾𝑟𝑛,𝑑, Δ𝑛 ≜ Δ𝑟𝑛,𝑑, 𝜅0,𝑙 ≜ 𝜅𝑙𝑠,𝑑 , 𝜅ˆ𝑛,𝑙 ≜ 𝜅𝑙𝑠,𝑟𝑛 , , ,and . Figure 3.1.Fig. 1.Illustration Illustration of of a cooperative a cooperative relay relay network network with with N𝑁 parallel potential potential relays𝜅ˇ𝑛,𝑙 ≜ 𝜅𝑙𝑟𝑛,𝑑 𝑋0 ≜ 𝑋𝑠,𝑑 𝑋𝑟𝑛 ≜ 𝑋𝑟𝑛,𝑑 [35]. relays. III. OUTAGE PROBABILITY ANALYSIS In [8], [32], and [36], Laneman et al. proposed the maximum average mutual in- parallel potential relays 𝑟 𝑁 , as depicted in Fig. 1. In A. Exact Outage Probability formation [1] for repetition-coded decode-and-forward𝑛 𝑛=1 as shown in Fig. 3.1, which this network, the adaptive{ DF} relaying is performed in two is given by, We first focus our attention on the exact outage probability consecutive phases. In the first phase, the source broadcasts of adaptive DF relaying in TWDP fading channels. The outage its signal to the destination and 𝑁 relays in one time slot. 1 2 probability, 𝑃out,isdefined as the probability that the channel IDF = min log(1 + SNR αs,r ), Let 𝑠,𝑟𝑛 denote the2 instantaneous{ mutual| | information of the average mutual information lies beneath a given rate 𝑅.In ℐ 2 2 channel between 𝑠log(1and +𝑟𝑛 SNR,andα 𝑅 denote+ SNR theα predetermined) , (3.2) ℐ | s,d| | r,d| } this network, the outage probability is expressed as transmission rate. If 𝑠,𝑟 𝑅,the𝑛th relay has the ability ℐ 𝑛 ≥ 𝑁 as a functionto decode of the the fading signal random correctly variables. and thusIn addition, belonging outage to the event active for spectral 𝑃out = Pr( <𝑅 𝑚)Pr( 𝑚), (4) efficiencyrelayR was set given𝑚 by=I 𝑟𝑛<: R 𝑠,𝑟and𝑛 is equivalent𝑅, 𝑛 =1 to, 2 the, event,,𝑁.We ℐ ∣𝒟 𝒟 𝒟 DF{ ℐ ≥ ⋅⋅⋅ } 𝑚=0 assume that consists of 𝑚 active relays, i.e. = 𝑚. ∑ 𝒟𝑚 ∣𝒟𝑚∣ In the second phase, the active relays forward22R the1 signals to the where can be calculated as [3, eq. (8)] min α 2, α 2 + α 2 < − . (3.3) ℐ destination in 𝑁{|consecutives,r| | s,d| time| r,d slots.| } Finally,SNR the destination 1 combines the direct signals from the source and the indirect = log 1+𝛾 𝑋 + 𝑋 , (5) Therefore, the overall outage probability can be written as: ℐ 𝑁 +1 2 0 𝑟𝑛 signals through the relays according to MRC. [ ( 𝑟 )] 𝑛∑∈𝒟𝑚 For this network, we consider a practical assumption that P out(SNR,R) := Pr[I < R] with 𝛾 denoting the transmitted SNR. Here, we note that for allDF the links experienceDF independent and non-identical TWDP 𝑁 22R 1 22R 1 agiven𝑚,thereare 𝑚 = 𝑚 possible choices of the active fading. We denote= Pr ℎ𝑢,𝑣α 2as< the− channel+ Pr fadingα 2 coefficient− 𝒩 | s,r| SNR | s,r| ≥ SNR relay set 𝑚. As such, we rewrite (4) as the summation over between 𝑢 and 𝑣,where 𝑢 𝑠,𝑛 𝑟 , 𝑣 𝑟𝑛,𝑑 ,and 𝒟 ( ) ∈{ } 2R ∈{ } all possible active relay sets as 𝑢 = 𝑣. Although it is intractable2 to derive2 2 an exact1 closed-form Pr αs,d + αr,d < − . (3.4) ∕ × | | | | SNR 𝑁 𝑚 PDF expression for the amplitude of TWDP fading coefficient, 𝒩 𝑃 = Pr ( <𝑅 )Pr( ) , (6) Durgin et al. presented a family of approximated PDF results out ℐ ∣𝒟𝑚,𝑞 𝒟𝑚,𝑞 𝑚=0 𝑞=1 in [10]. Using these results, the PDF of the amplitude of ℎ𝑢,𝑣 ∑ ∑ is approximated as where 𝑚,𝑞 is a set of the 𝑞th possible choice of 𝑚 relays 𝒟 𝑁 2 2 𝐿 from 𝑟𝑛 𝑛=1. 𝑟 𝑟 +2𝜎 𝐾𝑢,𝑣 𝑟 { } − 2𝜎2 (1) We then proceed to evaluate the conditional probability 𝑝 ℎ𝑢,𝑣 (𝑟)= 2 𝑒 𝑎𝑖𝐷 ; 𝐾𝑢,𝑣,𝛼𝑖𝑢,𝑣 , ∣ ∣ 𝜎 𝜎 Pr ( <𝑅 ) and the probability Pr ( ) in (6). Firstly, 𝑖=1 ( ) 𝑚,𝑞 𝑚,𝑞 ∑ usingℐ (5), we∣𝒟 re-express Pr ( <𝑅 𝒟) as where 𝑟 is the fading amplitude, 2𝜎2 is the average ℐ ∣𝒟𝑚,𝑞 power of the diffuse waves, 2 2 2 𝐾𝑢,𝑣 =(𝑉1𝑢,𝑣 + 𝑉2𝑢,𝑣 )/2𝜎 Pr ( <𝑅 𝑚,𝑞)=Pr(𝐻(˜𝜒𝑚,𝑞) <𝜂) , (7) is the ratio of the total specular power to diffuse ℐ ∣𝒟 (𝑁+1)𝑅 where 𝜂 = 2 1 /𝛾, 𝜒˜𝑚,𝑞 = 𝑋0,𝜒𝑚,𝑞 with waves, 𝛼𝑖𝑢,𝑣 =Δ𝑢,𝑣 cos [(𝑖 1)𝜋/(2𝐿 1)], Δ𝑢,𝑣 = 2 2 − − − { } is the relative strength of the two 𝜒𝑚,𝑞 = 𝑋𝑟𝑛 𝑟𝑛 𝑚,𝑞 ,and𝐻(𝜒) is a function to calculate 2𝑉1𝑢,𝑣 𝑉2𝑢,𝑣 /(𝑉1𝑢,𝑣 + 𝑉2𝑢,𝑣 ) the summation{ (} over∈𝒟 all the) elements in the set 𝜒. Secondly, specular components, 𝑉1𝑢,𝑣 and 𝑉2𝑢,𝑣 are the voltages of the two specular waves, 𝐿 𝐾𝑢,𝑣Δ𝑢,𝑣/2 is the order of the PDF, based on the assumption that all links are mutually indepen- ≥ 𝐿 the first five values of are given in Table II of [10], dent, we re-express Pr ( 𝑚,𝑞) as 𝑎𝑖 𝑖=1 𝒟 and { } Pr ( 𝑚,𝑞)= Pr (𝑋𝑠,𝑟𝑛 𝜂) 1 𝑦𝑧 𝒟 ≥ 𝐷 (𝑥; 𝑦,𝑧)= 𝑒 𝐼 𝑥 2𝑦(1 𝑧) 𝑟𝑛 𝑚,𝑞 2 0 − ∈𝒟∏ 1 ( ) Pr (𝑋𝑠,𝑟𝑛 <𝜂) , (8) 𝑦𝑧 √ × + 𝑒− 𝐼0 𝑥 2𝑦(1 + 𝑧) (2) 𝑟 𝑐 2 𝑛∈𝒟∏𝑚,𝑞 ( ) with 𝐼 ( ) denoting the zeroth-order modi√ fied Bessel function where 𝑐 is the complement of . From (7) and (8), 0 ⋅ 𝒟𝑚,𝑞 𝒟𝑚,𝑞 of the first kind. When 𝐾𝑢,𝑣 =0, TWDP fading reduces to we find that it is essential to determine the CDF of 𝐻(˜𝜒𝑚,𝑞) Rayleigh fading, and when 𝐾 =0and Δ =0,TWDP and 𝑋 . Given that 𝐻(˜𝜒 ) is the summation of 𝑚 +1 𝑢,𝑣 ∕ 𝑢,𝑣 𝑠,𝑟𝑛 𝑚,𝑞 fading reduces to Rician fading. We further denote 𝑋𝑢,𝑣 = independent TWDP fading power variables, we present the 33

r 1 h rd1 , htr, 11 h trK , 1 r2 h h rd2 , tr12, t1 h htr, td1 , K 2 d htr, h 11Z  tdK ,

tK h trKZ, 1 r h Z 1 rdZ 1 , h h tr1 , Z trK , Z hrd, rZ Z

y ytd, Received y rd1 , 1 rd2 , signals d y rdZ1 , yrd, y Z tdK ,

Figure 3.2. General case: K source nodes with Z relay nodes. Users in this case cannot be relay.

Based on the above method in [8], the mutual information in our work can be shown:

1 2 K I = log 1 + SNR α 2 DF 2 S ti,rj | ti,r| i=1 X 1 2 K 2 Z + log 1 + SNR α 2 + SNR α 2 , (3.5) 2 S ti,d| ti,d| S rj ,d| rj ,d| i=1 j=0 X X where K represents the number of users, Z represents the number of relay nodes, and S represents the spreading sequence. Therefore, the outage probability can be written as:

out PDF (SNR,R) := Pr[IDF < R] K (22R 1)S = Pr α 2 < − | ti,rj | 2SNR i=1 ti,rj X 34

K (22R 1)S + Pr α 2 − | ti,rj | ≥ 2SNR i=1 ti,rj X 1 K 1 Z 22R 1 Pr α 2 + α 2 < − , (3.6) × S | ti,d| S | rj ,d| 2SNR i=1 n=0 ti,d X X where SNRs are deﬁned as:

SNR at relay j from t can be written as: • i

Pti,rj (SNR)ti,rj = , (3.7) σn

SNR at d from t can be written as: • i

Pti,d (SNR)ti,d = , (3.8) σd

SNR at d from relay j can be written as: •

Prj ,d (SNR)rj ,d = . (3.9) σd

Knowing that h is a Nakagami-m distribution with parameters m, Ω, and | i,j| θ. θ is assumed to be uniformly distributed over (0, 2π]. Hence, the PDF of h | i,j| is given by, m 2 m 2m−1 − m x2 f (x) = x e Ω U(x). (3.10) |hi,j | Γ(m) Ω Each of the channel gain, i.e. h 2 follows a Gamma distribution with parameters | i,j| α = m > 0 and β = Ω /m > 0. Therefore, the PDF of h 2 is given as: i i i i i | i,j|

xαi−1e−x/βi 2 f|hi,j | (x) = αi U(x). (3.11) Γ(αi)βi

For example, when K = 2 users, Z = 1 relay node, and the spreading sequence is S = 4, the mutual information is given by,

1 2 K I = log 1 + SNR α 2 DF 2 4 ti,rj | ti,r| i=1 X 35

1 2 K 2 Z + log 1 + SNR α 2 + SNR α 2 . (3.12) 2 4 ti,d| ti,d| 4 rj ,d| r,d| i=1 j=0 X X

Assuming that SNRti,d = SNRrj ,d, the overall outage probability can be written as:

2R out 2 2 1 P (SNR,R) = Pr α < − DF | ti,rj | SNR ti,rj 22R 1 + Pr α 2 − | ti,rj | ≥ SNR ti,rj 3(22R 1) Pr α 2 + α 2 < − . (3.13) × | ti,d| | rj ,d| 2SNR ti,d Let h 2 = h 2 = h 2 = X be independent Gamma variates with parameter | s,r| | s,d| | r,d| i α and β, therefore, the PDF of Ys,r = Xs1,r + Xs2,r can be expressed as:

y2α−1e−y/β f (y = x + x ) = f (x ) f (x ) = U(y). (3.14) Y s1,r s2,r X 1 ∗ X 2 Γ(2α)β2α

Thus, Ys,d;r,d = Xs1,d + Xs2,d + Xr,d can be expressed as:

y3α−1e−y/β f (y) = f (x ) f (x ) f (x ) = U(y). (3.15) Y X 1 ∗ X 2 ∗ X 3 Γ(3α)β3α

According to [34],

u xv−1e−µxdx = µ−vΨ(v, µu), for v > 0 , (3.16) <{ } Z0 where Ψ( , ) is the lower incomplete gamma function deﬁned as: · · x Ψ(κ, x) = e−ttκ−1dt. (3.17) Z0 Therefore,

β2α x F (x) = Ψ 2α, , (3.18) Γ(2α)β2α β where F (x) = x f (y)dy. Hence, the identity Ψ(α, x) = α−1xα F (α; 1 + α; x) 0 Y 1 1 − R 36

[34]. Therefore,

x x 2α x Ψ 2α, = (2α)−1 F 2α; 1 + 2α; . (3.19) β β 1 1 −β Finally, Eq. (3.13) can be written as

x1 y2α−1e−y/β L pout(SNR,R) = dy DF Γ(2α)β2α Z0 x1 y2α−1e−y/β L x2 y3α−1e−y/β + 1 dy dy − Γ(2α)β2α Γ(3α)β3α 0 0 Z Z L x2α(2α)−1 x = 1 F 2α; 1 + 2α; 1 Γ(2α)β2α 1 1 − β ! L x2α(2α)−1 x + 1 1 F 2α; 1 + 2α; 1 − Γ(2α)β2α 1 1 − β ! x3α(3α)−1 x 2 F 3α; 1 + 3α; 2 , (3.20) × Γ(3α)β3α 1 1 − β where L = 1, 2, 3, 4 represent the number of transmission paths between users and relays. Since x = γ = (22R 1)/SNR and x = γ = (22R 1)/SNR , so 1 1 − ti,rj 2 2 − ti,d that Eq. (3.20) can be rewritten as:

L γ2α(2α)−1 γ pout(SNR,R) = 1 F 2α; 1 + 2α; 1 DF Γ(2α)β2α 1 1 − β ! L γ2α(2α)−1 γ + 1 1 F 2α; 1 + 2α; 1 − Γ(2α)β2α 1 1 − β ! γ3α(3α)−1 γ 2 F 3α; 1 + 3α; 2 . (3.21) × Γ(3α)β3α 1 1 − β 37

3.2 Numerical analyses

3.2.1 Outage probability for NC-based cooperative by using a multiplier

In our work, under the same consideration and fading channel, the outage probability results of the proposed system (Figs. 3.3-3.14) are better than those in [38], [40], and [39]. The detailed of the performance analysis are shown in the following subsections, which are based on Fig. 3.15.

3.2.2 Multiplier network coding vs. Two users cooperative network coding

Considering the same fading channel (Rayleigh which is equivalent to Nakagami-m with mt1 = 1), the reference network coding cooperative system uses users’ cooperation – all sources can be transmitters and relays, while the proposed system re- stricts sources to be only transmitters but cannot be relays. The proposed system performs better as depicted in Fig. 3.15. When the spectral efficiency is fixed at R = 0.5 [38] at SNR = 18.95 dB, their performance improves to P = 6.201 10−5, out × while the proposed method has a better performance, P = 2.93 10−9. How- out × ever, this performance is achieved using multiple relays with CC-CDMA transmission technique at the destination (see Fig. 3.15), while [38] uses TDMA, yet, it consumes more time and affect the system’s capacity even though TDMA can facilitate orthogonal transmission. For completeness, the mutual information and outage probability in [38] are shown as:

2 2 I = log 1 + γ h 2 + log 1 + γ h 2 + γ h 2 , (3.22) NC 3 | u1,d | 3 | u1,d | | u2,d | and

P [R] = P [INC < R] 2 2 = P log 1 + γ h 2 + log 1 + γ h 2 + γ h 2 . (3.23) 3 | u1,d | 3 | u1,d | | u2,d | 38

3.2.3 Multiplier network coding vs. Network-coding-based hybrid AF and DF system

Considering the same fading channel, the proposed system performs better as depicted in Fig. 3.15. When the spectral eﬃciency is ﬁxed at R = 0.5 [39] at SNR = 25 dB, their performance improves to P = 2.987 10−9 in case2, and out × P = 1.495 10−9 in case4, while the proposed method has a better performance, out × even when SNR = 20 dB, P = 1.111 10−9. However, this performance is out × achieved using multi-relays with CC-CDMA transmission technique at the destination Fig. 3.15. The reference network coding system [39] proposed a hybrid Amplify-and-Forward and Decode-and-Forward for the network coding system. They show that the proposed scheme outperforms conventional protocols on outage probability and prove that it has full diversity at relatively high-SNR regime. The outage probability for the best two cases out of four in [39] are shown as:

P Case2 = P r(γ < 22R 1) P r(γ < 22R 1) out 1,2 − · 2,1 − γ γ P r γ + 1,2 2,d < 22R 1 , (3.24) · 1,d γ + γ + 1 − 1,2 2,d and

P Case4 = P r(γ < 22R 1) P r(γ > 22R 1) out 1,2 − · 2,1 − P r(γ2 < 22R 1) P r(γ < 22R 1) · AF − · 2,d − 2R 2R +P r(γ2,d > 2 1)P r(γ1,d < 2 1) , (3.25) − − where γ2 = γ + γ1,2γ2,d . AF 1,d γ1,2γ2,d+1

3.2.4 Multiplier network coding vs. Distributed orthogonal Space-Time block codes

Considering the same fading channel, the reference cooperative system uses distributed extended orthogonal space-time-block code (D-EO-STBC). The proposed system performs better as depicted in Fig. 3.15. When the spectral eﬃciency is

ﬁxed at R = 0.5 [40] at SNR = 20 dB, their performance improves to Pout = 39

5.824 10−2 for one relay scheme, and P = 2.643 10−3 for two relays scheme; × out × while the proposed method has a better performance, P = 1.668 10−5 for only out × one relay system. However, this performance is achieved using multi-relays with CC-CDMA transmission technique at the destination Fig. 3.15, while [40] uses time slots. It requires two time slots to complete the transmission between the source and destination and achieve a minimum outage probability threshold. The outage probabilities for one relay and two relays in [40] are shown as:

Pout(γ) = FΓ1 (γ) − γ = 1 e λ1 , (3.26) − and

Pout(γth) = FΓ2 (γth)

− γ − γ 1 λ2 1 λ1 λ 1 e λ 1 e 1 − ! − 2 − ! = 1 1 . (3.27) λ1 − λ2 40

Outage probability vs. SNR (1 Relay) 0 10

−1 10

−2 10

−3 10

−4 10

−5 10

−6 10 Outage probability (R = 0.5) m = 0.5 −7 10 m = 1

−8 10 m = 1.5 m = 2 −9 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.3. Outage probability vs. SNR with R = 0.5 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m.

3.3 Summary of the advantages of using a CC- CDMA cooperative network coding

Comparisons between a CC-CDMA and a TDMA cooperative network coding system are made as follows:

1. As we mentioned at subsection 1.4.3 in Chapter 1, CDMA provides a better spectrum eﬃciency than TDMA. The results are shown in outage probability performance in Fig. 3.15. The proposed system uses CDMA rather than TDMA, which uses time slots for transmission while providing orthogonal 41

SNR 6.3 dB SNR 10.5 dB SNR 16.8 dB SNR 20 dB ≈ ≈ ≈ ≈ The proposed system 3.067e-04 6.824e-06 2.037e-08 1.11e-09 2-Users coop-NC [38] 1.814e-02 2.857e-03 1.627e-04 4.28e-05 NC Hybird AF-DF (case 1) [39] 8.051e-04 6.023e-05 6.863e-07 1.124e-07 NC Hybird AF-DF (case 4) [39] 0.461e-04 3.181e-05 3.473e-07 5.659e-08 D-EO-STBC (1-Relay) [40] 7.538e-01 4.123e-01 1.168e-01 5.824e-02 D-EO-STBC (2-Relay [40] 4.547e-01 1.318e-01 1.06e-02 2.643e-03 Table 3.1. Outage performance comparisons vary from SNR 5 to 20 dB.

Outage probability vs. SNR (1 Relay) 0 10

−1 10

−2 10

−3 10

−4 10

−5 10

−6

Outage probability (R=1) 10 m = 0.5

−7 10 m = 1

−8 m = 1.5 10 m = 2 −9 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.4. Outage probability vs. SNR with R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m.

transmission, i.e., multi-users interference-free communication.

2. Due to the channel pollution, the disadvantage of CDMA technology, we introduce CC-CDMA into our NC-based cooperation system to eﬃciently 42

Outage probability vs. SNR (2 Relays) 0 10

−5 10

−10 10

Outage probability (R = 0.5) m = 0.5 −15 10 m = 1

m = 1.5

m = 2 −20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.5. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m.

eliminates multiple access interference (MAI). CC-CDMA technology not only improves outage capacity, but also provides a higher bandwidth eﬃcient than a traditional CDMA system. Fig. 3.15 shows that the proposed system outperforms a conventional TDMA system and a modiﬁed TDMA system – hybrid AF system [39]. 43

Outage probability vs. SNR (2 Relays) 0 10

−2 10

−4 10

−6 10

−8 10

−10 10

−12 10 Outage probability (R = 1) −14 10 m = 0.5

−16 10 m = 1

m = 1.5 −18 10 m = 2 −20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.6. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 44

Outage probability vs. SNR (3 Relays) 0 10 m = 0.5

m = 1

−5 m = 1.5 10 m = 2

−10 10 Outage probability (R = 0.5) −15 10

−20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.7. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 45

Outage probability vs. SNR (3 Relays) 0 10 m = 0.5 −2 10 m = 1

−4 10 m = 1.5

m = 2 −6 10

−8 10

−10 10

−12 10 Outage probability (R = 1) −14 10

−16 10

−18 10

−20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.8. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 46

Outage probability vs. SNR (4 Relays) 0 10 m = 0.5 −2 10 m = 1

−4 10 m = 1.5

−6 m = 2 10

−8 10

−10 10

−12 10

Outage probability (R = 0.5) −14 10

−16 10

−18 10

−20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.9. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 47

Outage probability vs. SNR (4 Relays) 0 10 m = 0.5 −2 10 m = 1

−4 10 m = 1.5

−6 m = 2 10

−8 10

−10 10

−12 10 Outage probability (R = 1) −14 10

−16 10

−18 10

−20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.10. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 48

Outage probability vs. SNR (m=1) 0 10

1−Relay

−2 10 2−Relay

3−Relay −4 10 4−Relay

−6 10

−8 10

−10 10 Outage probability (R = 0.5)

−12 10

−14 10

−16 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.11. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 49

Outage probability vs. SNR (m=1) 0 10

−2 10

−4 10

−6 10

−8 10

−10 10 Outage probability (R=1)

−12 1−Relay 10 2−Relay

−14 10 3−Relay

4−Relay −16 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.12. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 50

Outage probability vs. SNR (m=0.5) 0 10

−2 10

−4 10

−6 10

−8 10 Outage probability (R = 0.5) −10 10 1−Relay

2−Relay

−12 10 3−Relay

4−Relay −14 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.13. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 51

Outage probability vs. SNR (m=0.5) 0 10

−2 10

−4 10

−6 10

−8 10 Outage probability (R=1)

−10 10 1−Relay

2−Relay

−12 10 3−Relay

4−Relay

−14 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.14. Outage probability vs. SNR with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 52

Outage probability vs. SNR (1 Relay, m=1) 0 10

−1 10

−2 10

−3 10

−4 10

−5 10

−6 10 The proposed system Outage probability (R = 0.5) 2 Users cooperation −7 10 NC HAD Case 4 NC HAD Case 2 −8 10 1−Relay STD, m=1 2−Relay STD, m=1 −9 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 3.15. Outage probability comparisons. Chapter 4

Optimal power allocation

In this Chapter, we introduce two optimization solutions for the outage probability and the optimal power allocation outage performance. Due to the limited radio resources such as bandwidth and transmission power, the optimal power allocation strategy references can reduce outage probability while optimizing power allocation. In Section 4.1.1, we use a standard maximum-minimum method to discuss the optimization solution approaches. Next, in Section 4.1.2, we use a Lagrangian multiplier method to solve the optimization problem.

4.1 Optimal power allocation

4.1.1 Standard max-min problem

First of all, we can use a capacity maximization problem to examine the optimal power allocation of the proposed system, as a special case (See. Fig. 4.1) with the DF cooperative strategy. The capacity of the source to relay link is given [12]:

1 2 K C = log 1 + SNR α 2 SR 2 S ti,rj | ti,r| i=1 X 1 2 K P = log 1 + ti α 2 , (4.1) 2 S σ2 | ti,r| i=1 r X 54

�! Phase 2

Phase 1 � �

�!

Figure 4.1. Two phases of DF cooperative strategy. where K represents the number of users, Z represents the number of relay nodes, and S represents the spreading sequence. In the second phase, the destination combines the signals from both source and relay systems by using maximum ratio combining (MRC), so that the capacity can be written as:

1 2 K 2 Z C = 1 + SNR α 2 + SNR α 2 SR;RD 2 S ti,d| ti,d| S rj ,d| rj ,d| i=1 j=0 X X 1 2 K P 2 Z P = 1 + ti α 2 + rj α 2 . (4.2) 2 S σ2 | ti,d| S σ2 | rj ,d| i=1 d j=0 d X X The channel capacity in the DF cooperative framework is limited by the minimum of these two phases (K = 2, Z = 1, and S = 4):

1 2 2P C = min log 1 + ti α 2 , DF 2 4 σ2 | ti,r| ( r 1 2 2P 2 P 1 + ti α 2 + rj α 2 . (4.3) 2 4 σ2 | ti,d| 4 σ2 | rj ,d| d d ) 55

Hence, the optimization problem turns into a standard max-min problem [12] as shown:

1 P maximize min log 1 + ti α 2 , 2 ti,r Pt ,Pr ,∀i,j 2 σ | | i j ( r 1 P 1 P 1 + ti α 2 + rj α 2 2 σ2 | ti,d| 2 σ2 | rj ,d| d d ) K Z subject to P + P P , and P ,P 0, i, j. ti rj ≤ sum ti,d rj ,d ≥ ∀ i=1 j=1 X X If α 2 α 2 and α 2 α 2, the capacity is maximized with equal | rj ,d| ≥ | ti,d| | ti,r| ≥ | ti,d| capacity for the two phases, i.e.,

1 P 1 P 1 P log 1 + ti α 2 = 1 + ti α 2 + rj α 2 . (4.4) 2 σ2 | ti,r| 2 σ2 | ti,d| 2 σ2 | rj ,d| r d d Given the total power constraint 2P + P = 3P P . Therefore, we can show ti rj ≤ sum that the optimal power allocation is:

2 |αrj ,d| 2 ∗ 3P σd P = 2 , (4.5) ti |α |2 |α |2 |αr ,d| 2 ti,r ti,d + j σ2 σ2 σ2 r − d d and 2 2 |αti,r| |αti,d| σ2 σ2 ∗ r − d P = 3P 2 . (4.6) rj |α |2 |α |2 |αr ,d| ti,r ti,d + j σ2 σ2 σ2 r − d d Following Eqs. (4.5) and (4.6), we can see that there is more power allocated to source ti than to relay rj. Using the optimal power, Eqs. (4.5) and (4.6), we can rewrite the outage probability (Eq. (3.13)) as:

2R 2R out 2 2 1 2 2 1 P (SNR,R) = Pr α < − + Pr α − DF | ti,rj | SNR | ti,rj | ≥ SNR ti,rj ti,rj 1 Pr SNR α 2 + SNR α 2 < 22R 1 × ti,d| ti,d| 2 rj ,d| rj ,d| − 56

22R 1 22R 1 = Pr α 2 < − + Pr α 2 − | ti,rj | P ∗/σ2 | ti,rj | ≥ P ∗/σ2 ti d ti d P ∗ P ∗ Pr ti α 2 + rj α 2 < 22R 1 . (4.7) × σ2 | ti,d| 2σ2 | rj ,d| − d d

Assuming that αti,d = αrj ,d for simplicity, hence, the optimal power allocation outage probability can be rewritten as:

Huang proposed that the optimal power allocation [13] can be found by maximizing the achievable rate (or, equivalently, the eﬀective SNR) at the destination. Given the total power constraint K P + Z P P , the problem can be i=1 ti,d j=1 rj ,d ≤ sum formulated equivalently as aP linear programmingP given as follows:

K Z 2 2 2 2 maximize log 1 + SNRti,d αti,d + SNRrj ,d αr,d Pt ,d,Pr ,d,∀i,j S | | S | | i j i=1 j=0 X X K Z subject to P + P P , and P ,P 0, i, j. ti,d rj ,d ≤ sum ti,d rj ,d ≥ ∀ i=1 j=1 X X

2 Where S = 4 represents the spreading sequence, SNRti,d = Pti,d/σd, and SNRrj ,d = 2 Prj ,d/σd. For example, when K = 2 (source), Z = 1 (relay), using the Lagrangian function to ﬁnd the optimal solution gives,

2 2 2 2P α 2 Pr ,d αr ,d (2P + P , λ) = log 1 + ti,d| td,d| + j | j | L ti,d rj ,d 4 σ2 4 σ2 d d λ 2P + P P . (4.9) − ti,d rj ,d − sum 57

For the special case, when Pti,d = Prj ,d and αti,d = αrj ,d, Eq. (4.9) can be simpliﬁed as: P α 2 (3P , λ) = log 1 + ti,d| ti,d| λ(3P P ). (4.10) L ti,d 2σ2 − ti,d − sum d

Taking the ﬁrst derivative with respect to Pti,d, thus,

∂ (3P , λ) P α 2 −1 α 2 L ti,d = log e 1 + ti,d| ti,d| | ti,d| 3λ 0. (4.11) ∂P 2 · 2σ2 · 2σ2 − ≤ ti,d d d

Therefore,2.3 Capacity following of Wireless the Channels Karush-Kuhn-Tucker (KKT) conditions [44], the optimal57

Fig. 2.19 Illustration of waterﬁlling solution in (2.90). Figure 4.2. Illustration of water ﬁlling for power allocation case. transmission power P ∗ is as shown: ti,d of (P1, ,PM ,λ) with respective to Pk is given by L ··· 2 log2 e 2σd 3λ |α |2 otherwise1 P ∗ = − ti,d 2 − 2 (4.12) ∂ (P1, ,PMt,λi,d ) Pk μk 2 μk  |αs,d| L ··· = (log02 e) 1+ if| 2 | η.| 2| λ. (2.89) ∂P · σ σ2 · σ − k  w d ≥ w 1 2σ2 + Following the KKT conditions,=  η the optimald transmission. power is (4.13) 3 − α 2 | ti,d| 2 2 0, if (log2 e) μk /σw λ, log2 Pe = 2 | | ≤ , Where η = λ kis a constantlog2 e chosenσw to satisfy the sum power constraint, and 2 , otherwise. , λ μk (x)+ = max(x, 0). With the− optimal| | pre-coder and the optimal decoder, the σ2 + = η w ,k=1, 2, ,M, (2.90) − μ 2 ··· | k|

where η =log2 e/λ is a constant set to meet the sum power constraint in (2.87) and (x)+ =max(x, 0). The value of η can be found by the water-ﬁlling algorithm, as illustrated in Fig. 2.19, and the optimal solution in (2.90) is referred to as the water-ﬁlling solution. With the optimal pre-coder and the optimal decoder, the channel capacity is given by

2 η μk C = log2 | 2 | . (2.91) σw k:Pk >0 Consider the fading channel for example again, if the transmitter has knowledge of the instantaneous channel state H, the covariance Rs of channel input symbols can always be adapted accordingly to achieve the channel capacity. Therefore, the ergodic capacity of a MIMO system with full CSI at the transmitter is given by 58 channel capacity is given by,

1 η α 2 C = log | ti,d| . (4.14) 2 σ2 d 4.2 Summary of the advantages of using optimal power allocation

Comparisons between optimal power allocation and equal power strategies give us the following:

1. Optimal power allocation can meet the limited radio resources requirements such as bandwidth and transmission power. The main objective of this research is to ﬁnd solutions that can improve the channel capacity and utilization of the radio resources that are available to the service providers.

2. As depicted in Eqs. (4.5) and (4.6), we can see that there is more power allocated to sources than to relays. Due to this optimization property, Eq. (4.8) shows a optimization solution for the outage probability (See Figs. 4.3- 4.14). The numerical results prove that a optimal power allocation strategy maximizes the channel capacity; thus, its outage performance outperforms the equal power outage probability. 59

4.3 Numerical analyses

Optimal power vs. Equal power (1 Relay) 0 10

−2 10

−4 10

−6 10

−8 Optimal power (m=0.5)

Outage probability (R = 0.5) 10 Equal power (m=0.5) Optimal power (m=1) Equal power (m=1) −10 10 Optimal power (m=1.5) Equal power (m=1.5) Optimal power (m=2) Equal power (m=2) −12 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.3. Optimal power vs. equal power with R = 0.5 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 60

Optimal power vs. Equal power (1 Relay) 0 10

−2 10

−4 10

−6 10

−8 Outage probability (R = 1) 10 Optimal power (m=0.5) Equal power (m=0.5) Optimal power (m=1) Equal power (m=1) −10 10 Optimal power (m=1.5) Equal power (m=1.5) Optimal power (m=2) Equal power (m=2) −12 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.4. Optimal power vs. equal power with R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 61

Optimal power vs. Equal power (2 Relays) 0 10 Optimal power (m=0.5)

−2 Equal power (m=0.5) 10 Optimal power (m=1) Equal power (m=1) −4 10 Optimal power (m=1.5) Equal power (m=1.5) −6 Optimal power (m=2) 10 Equal power (m=2)

−8 10

−10 10

−12 10

Outage probability (R = 0.5) −14 10

−16 10

−18 10

−20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.5. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 62

Optimal power vs. Equal power (2 Relays) 0 10

−2 10

−4 10

−6 10

−8 10

−10 10

−12 10 Outage probability (R = 1) −14 Optimal power (m=0.5) 10 Equal power (m=0.5) Optimal power (m=1) −16 10 Equal power (m=1) Optimal power (m=1.5)

−18 Equal power (m=1.5) 10 Optimal power (m=2) Equal power (m=2) −20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.6. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 63

Optimal power vs. Equal power (3 Relays) 0 10 Optimal power (m=0.5) Equal power (m=0.5) Optimal power (m=1) −5 10 Equal power (m=1) Optimal power (m=1.5) Equal power (m=1.5) Optimal power (m=2) −10 10 Equal power (m=2)

−15 10

−20

Outage probability (R = 0.5) 10

−25 10

−30 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.7. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 64

Optimal power vs. Equal power (3 Relays) 0 10

−5 10

−10 10

−15 10

−20 Outage probability (R = 1) 10 Optimal power (m=0.5) Equal power (m=0.5) Optimal power (m=1) Equal power (m=1) −25 10 Optimal power (m=1.5) Equal power (m=1.5) Optimal power (m=2) Equal power (m=2) −30 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.8. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 65

Optimal power vs. Equal power (4 Relays) 0 10 Optimal power (m=0.5) Equal power (m=0.5) Optimal power (m=1) −5 10 Equal power (m=1) Optimal power (m=1.5) Equal power (m=1.5) Optimal power (m=2) −10 10 Equal power (m=2)

−15 10

−20

Outage probability (R = 0.5) 10

−25 10

−30 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.9. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 66

−15 10

−20 Outage probability (R = 1) 10

−25 10

−30 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.10. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 67

Optimal power vs. Equal power (m=1) 0 10 Optimal power (m=1)

−2 10 Equal power (m=1) Equal power (m=1) −4 Optimal power (m=1) 10 Equal power (m=1)

−6 Optimal power (m=1) 10 Equal power (m=1)

−8 Optimal power (m=1) 10

−10 10

−12 10

Outage probability (R = 0.5) −14 10

−16 10

−18 10

−20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.11. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 68

Optimal power vs. Equal power (m=1) 0 10

−2 10

−4 10

−6 10

−8 10

−10 10

−12 10 Optimal power (1−Relay) Outage probability (R = 1) −14 10 Equal power (1−Relay) Equal power (2−Relay) −16 Optimal power (2−Relay) 10 Equal power (3−Relay)

−18 Optimal power (3−Relay) 10 Equal power (4−Relay) Optimal power (4−Relay) −20 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.12. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 69

Optimal power vs. Equal power (m=0.5) 0 10

−2 10

−4 10

−6 10

−8 10 Optimal power (1−relay) Equal power (1−relay) Outage probability (R = 0.5) −10 10 Equal power (2−relay) Optimal power (2−relay) Equal power (3−relay) −12 10 Optimal power (3−relay) Equal power (4−relay) Optimal power (4−relay) −14 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.13. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. 70

Optimal power vs. Equal power (m=0.5) 0 10

−2 10

−4 10

−6 10

−8 Optimal power (1−Relay) Outage probability (R = 1) 10 Equal power (1−Relay) Equal power (2−Relay) Optimal power (2−Relay) −10 10 Equal power (3−Relay) Optimal power (3−Relay) Equal power (4−Relay) Optimal power (4−Relay) −12 10 0 2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 4.14. Optimal power vs. equal power with R = 0.5 and R = 1 bit/sec/Hz, while the Gamma distribution with parameters α = m, β = Ω/m. Chapter 5

Conclusions

5.1 Conclusions

In our previous wok [2], we proposed a modified method of combining CC-CDMA cooperative communication system and network coding, hence minimizing the time slots problems and alleviating multi-path interference. We also introduced a multiplier operation at the destination, instead of using the traditional XOR operation. In Chapter 2, we refined the analysis of CC-CDMA by providing new simulation results showing that the modified system can improve diversity, i.e. reliability, without deterioration of the average probability of error. In Chapter 3, the outage probability of the proposed CC-CDMA system is analyzed. Simulation results shows that the proposed system outperforms other similar systems. In Chapter 4, we further prove that by using an optimization method, it is possible to maximize the system performance, i.e., minimize the outage problem. Consequently, the proposed system can improve communication between mobile consumer electronic devices. Based on the analysis presented and comparisons made with other similar systems, we can conclude that the proposed method performs better not only in reducing time slots, but also channel capacity, without an advanced demodulator or the assumption of an error/delay-free control channel. Moreover, the more relays used in the two-transmitters NC-based cooperative system the better the performance of detecting signals. It confirms the assumption that the redundant signals can improve the overall diversity and hence a system’s performance. 72

5.2 Future work

In our current research, we have used maximum ratio combining (MRC) with decode-and-forward (DF) strategy; however, there are still a couple of other coding strategies that we have not discussed yet, such as amplify-and-forward, selection relay, repetition relay, etc. This could be included in further research. The diversity-multiplexing-tradeoﬀ is also an important system performance approach to determine the cooperative system diversity gain, which can be also analyzed in future work. Following Chapters 2 and 3, error probability and outage probability can approach a high SNR regime to get a diversity order. Finally, further analysis using other technique such as a Monte-Carlo simulation, could be used to show that the numerical analyses are accurately calculated so that the BER [45] and outage probability performance [46]-[47] results are reliable. Appendix A

Received signal at relay rj or destination d

In our previous work [2], the detected signal shown in Eq. (2.14) are explained in detail below:

M L Tb (m) (DF ) 2 (m) −jθ (m) Γ = α e ti,rj ,l s (t λ ) cos(2πf t)c (t)dt ti,rj ti,rj ,l ti,rj ,m ti,rj ,l m ti,m Tb − m=1 l=0 0 r X X Z M L Tb (m) 2 (m) −jθ 2Eb (m) = α e ti,rj ,l c (t λ ) ti,rj ,l ti,m ti,rj ,l Tb Tb − m=1 l=0 0 r r X X Z cos 2πf (t λ(m) ) x (t λ(m) ) cos(2πf t)c (t) dt, (A.1) × m − ti,rj ,l ti,rj − ti,rj ,l m ti,m cos A cos B | {z } where the cosine| product{z in Eq. (A.1)} can be simpliﬁed as shown:

1 cos A cos B = cos 2πf (t λ(m) ) + 2πf t 2 m − ti,rj ,l m + cos 2πf (t λ(m) ) 2πf t m − ti,rj ,l − m 1 = cos 4πf t 2πf λ(m) + cos 2πf λ(m) . (A.2) 2 m − m ti,rj ,l m ti,rj ,l Because cos 4πf t 2πf λ(m) is ﬁltered out by a low pass ﬁlter since the center m − m ti,rj ,l frequency 4πf t is two times higher than 2πf λ(m) . Therefore, Eq. (A.1) can m m ti,rj ,l 74 be rewritten as:

M L Tb (m) (DF ) 1 (m) −jθ (m) (m) Γ E α e ti,rj ,l c (t λ )x (t λ ) ti,rj b ti,rj ,l t1,m ti,rj ,l ti,rj ti,rj ,l ≈ Tb − − m=1 l=0 0 X X Z p cos(2πf λ(m) )c (t)dt × m ti,rj ,l ti,m M Tb (m) 1 (m) −jθ (m) (m) = E α e ti,rj ,0 c (t λ )x (t λ ) T b ti,rj ,0 ti,m − ti,rj ,0 ti,rj − ti,rj ,0 m=1 0 b X Z p M L Tb (m) (m) 1 (m) −jθ cos(2πf λ )c (t)dt + E α e ti,rj ,l m ti,rj ,0 ti,m b ti,ri,l × Tb m=1 l=1 0 X X Z p c (t λ(m) )x (t λ(m) ) cos(2πf λ(m) )c (t)dt. (A.3) × ti,m − ti,rj ,l ti,rj − ti,rj ,l m ti,rj ,l ti,m

Deﬁne λ(m) = ε(m) T , where the range of ε(m) is: ti,rj ,l ti,rj ,l c ti,rj ,l

0, for l = 0 ε(m) = (A.4) ti,rj ,l ν, ν 1, 2, ,N, , for l = 0  ∈ { ··· ···} 6 Assume the channels are fully correlated, in other words the signal bandwidth is narrower than the coherence channel’s bandwidth, thus:

α(1) = α(2) = = α(M) = α , (A.5) ti,rj ,l ti,rj ,l ··· ti,rj ,l ti,rj ,l and θ(1) = θ(2) = = θ(M) = θ , (A.6) ti,rj ,l ti,rj ,l ··· ti,rj ,l ti,rj ,l and λ(1) = λ(2) = = λ(M) = λ . (A.7) ti,rj ,l ti,rj ,l ··· ti,rj ,l ti,rj ,l Now deﬁne: Tb 1 c2(t)dt = . (A.8) M Z0 Using the property of complementary code, which are Eqs. (A.5), (A.6), and (A.7), (DF ) thus, the ﬁrst part of Γti,rj can be rewritten as:

M Tb (m) (m) −jθ (m) Part I. = α e ti,rj ,0 c (t λ )c (t)dt ti,rj ,0 ti,m − ti,rj ,0 ti,m m=1 0 X Z 75

M Tb −jθ (m) = α e ti,rj ,0 c (t λ )c (t)dt ti,rj ,0 ti,m − ti,rj ,0 ti,m m=1 0 X Z −jθti,rj ,0 (m) αt ,r ,0e Tb, for λ = 0 = i j ti,rj ,0 , (A.9) 0, for λ(m) = 0  ti,rj ,l 6

(DF ) and the second part of Γti,rj is as shown:

M L Tb (m) (m) −jθ (m) Part II. = α e ti,rj ,l c (t λ )c (t)dt ti,rj ,l ti,m − ti,rj ,l ti,m m=1 0 X Xl=1 Z M L Tb −jθ (m) = α e ti,rj ,l c (t λ )c (t)dt ti,rj ,l ti,m − ti,rj ,l ti,m m=1 0 X Xl=1 Z = 0. (A.10)

Consequently, by Eq. (A.9) and Eq. (A.10), hence, Eq. (A.3) can be rewritten as:

M Tb (DF ) √Eb −jθ (m) (m) Γ α e ti,rj ,0 c (t λ ) x (t λ ) ti,rj ≈ T ti,rj ,0 ti,m − ti,rj ,0 ti,rj − ti,rj ,0 b m=1 0 Z ∞ X P ati (n)g(t−nTb) n=−∞ | {z } cos(2πf λ(m) )c (t)dt × m ti,rj ,0 ti,m M Tb √Eb −jθ (m) = α e ti,rj ,0 c (t λ )c (t)dt T ti,rj ,0 ti,m − ti,rj ,0 ti,m b m=1 Z0 X ∞ a (n0)g(t n0T ) + a (n)g(t nT ) cos(2πf λ(m) ) . × ti − b ti − b m ti,rj ,0 n=−∞ ! nX=6 n0 out of the window | {z } (A.11)

(m) If λti,rj ,0 = 0, thus Eq. (A.11) can be rewritten as:

(DF ) 0 −jθti,rj ,0 Γti,rj = Ebati (n )αti,rj ,0e . (A.12) p Similarly, (DF ) 0 −jθ Γ = E a (n )α e ti,d,0 . (A.13) ti,d b ti ti,d,0 p Appendix B

Received noise at relay rj or destination d

In our previous work [2], consider that

M Tb 2 ξ(DF ) = c(t)n(t)cos(2πf t)dt, (B.1) T m m=1 0 r b X Z where n(t) is a zero mean, variance N0/2 Gaussian procedure, c(t) is a spreading code. (DF ) Tb 2 Let ξm = c(t)n(t) cos(2πf t)dt, thus, 0 Tb m R q Tb 2 2 E[ξ2 ] = E c(t)n(t) cos(2πf t)dt m T m Z0 r b Tb 2 Tb 2 = E c(t)n(t) cos(2πf t)dt c(t0)n(t) cos(2πf t0)dt0 T m T m Z0 r b Z0 r b Tb Tb 2 = E c(t)c(t0)n(t)n(t0) cos(2πf t) cos(2πf t0)dtdt0 T m m Z0 Z0 b Tb Tb 2 = E[n(t)n(t0)]c(t)c(t0) cos(2πf t) cos(2πf t0)dtdt0 T m m Z0 Z0 b Tb Tb 2 N = 0 δ(t t0)c(t)c(t0) cos(2πf t) cos(2πf t0)dtdt0 T 2 − m m Z0 Z0 b Tb N = 0 c2(t0) cos2(2πf t0)dt0 T m Z0 b 77

Tb N 1 1 = 0 c2(t0) + cos(4πf t0) dt0 T 2 2 m Z0 b N = 0 . (B.2) 2M

(DF ) Therefore, ξm is a zero mean, variance N0/2M Gaussian distribution. Since

M (DF ) (DF ) ξ = ξm , (B.3) m=1 X and ξ(DF ), ξ(DF ), , ξ(DF ) are i.i.d., thus, 1 2 ··· M M M N N E[ξ2] = E[ξ2 ] = 0 = 0 . (B.4) m 2M 2 m=1 m=1 X X (DF ) therefore, ξ is a zero mean, variance N0/2 Gaussian distribution. Appendix C

Maximum output SNR analysis

In our previous work [2], following by Eq. (2.16) in Section 2.2, a decision variable

(t1) valuey ˜d,sum can be written as:

y˜(t1) = w y˜ + w [˜y y˜ ] d,sum t1,d t1,d r1,d t2,d × r1,d 0 −jθ (DF ) = w E a (n )α e t1,d,0 + w ξ t1,d b t1 t1,d,0 t1,d t1,d

0 0 −j(θr1,d,0+θt2,d,0) +wr1p,d Ebar1 (n )at2 (n )αr1,d,0αt2,d,0e

0 −jθ (DF ) 0 −jθ (DF ) + E ha (n )α e r1,d,0 ξ + E a (n )α e t2,d,0 ξ b r1 r1,d,0 t2,d b t2 t2,d,0 r1,d +ξp(DF )ξ(DF ) p r1,d t2,d i 0 −jθt1,d,0 = Eb wt1,dat1 (n )αt1,d,0e p 0 0 −j(θr1,d,0+θt2,d,0) +wr1,d Ebar1 (n )at2 (n )αr1,d,0αt2,d,0e p (DF ) 0 −jθ (DF ) + w ξ + w E a (n )α e r1,d,0 ξ t1,d t1,d r1,d b r1 r1,d,0 t2,d p 0 −jθ (DF ) (DF ) (DF ) +w E a (n )α e t2,d,0 ξ + w ξ ξ r1,d b t2 t2,d,0 r1,d r1,d r1,d t2,d (t1) p(t1) =y ˜s +y ˜n , (C.1)

(t1) (t1) wherey ˜s andy ˜n represent signal part and noise part, respectively. If the channel responses αti,d,0 and αrj ,d,0 are ﬁxed, for transmitters i = 1, 2, and relay j = 1. 79

(t1) The conditional expected value ofy ˜d,sum is given by,

Exp = E y˜(t1) α ,α cond d,sum| tk,d,0 r1,d,0 0 −jθt1,d,0 = Eb wt1,dat1 (n )αt1,d,0e

0 0 −j(θr1,d,0+θt2,d,0) +pwr1,d Ebar1 (n )at2 (n )αr1,d,0αt2,d,0e

(t1) =y ˜s . p (C.2)

We can also simplify Eq. (C.2) as follows:

Exp = E = , (C.3) cond E(1r) b E p where

0 −jθ = w a (n )α e t1,d,0 E(1r) t1,d t1 t1,d,0 0 0 −j(θr1,d,0+θt2,d,0) +wr1,d Ebar1 (n )at2 (n )αr1,d,0αt2,d,0e . (C.4) p By letting

0 −jθt1,d,0 Ebat1 (n )αt1,d,0e = Ψ, (C.5) and p

0 0 −j(θr1,d,0+θt2,d,0) Ebar1 (n )at2 (n )αr1,d,0αt2,d,0e = Φ, (C.6) so, Eq. (C.2) can be rewritten as:

(t1) y˜s = wt1,dΨ + wr1,dΦ. (C.7)

Similarly, consider the same conditions as in Eq. (C.2), then the conditional vari-

(t1) ance ofy ˜d,sum is given by,

Var = V ar y˜(t1) α ,α cond d,sum| tk,d,0 r1,d,0 2 N0 2 2 0 2 −2jθ N0 = w + w E a (n)α e r1,d,0 t1,d 2 r1,d b r1 r1,d,0 2 2 2 0 2 −2jθ N0 N0 +E a (n )α e t2,d,0 + b t2 t2,d,0 2 4 2 2 = E y˜(t1) α ,α = y˜(t1) . (C.8) n | tk,d,0 r1,d,0 n 80

Also, we can simplify Eq. (C.8) as follows:

N Var = 0 = , (C.9) cond V(1r) 2 V where

2 2 2 0 2 −2jθ 2 0 2 −2jθ = w + w E a (n )α e r1,d,0 + E a (n )α e t2,d,0 V(1r) t1,d r1,d b r1 r1,d,0 b t2 t2,d,0 N + 0 . (C.10) 2

2 2 0 2 −2jθ N0 2 0 2 −2jθ N0 N0 By letting E a (n )α e r1,d,0 +E a (n )α e t2,d,0 + = Υ, there- b r1 r1,d,0 2 b t2 t2,d,0 2 4 fore, Eq. (C.8) can be rewritten as:

2 N y˜(t1) = w2 0 + w2 Υ. (C.11) n t1,d 2 r1,d The output signal-to-noise ratio (SNR)o can be written as:

y˜(t1) 2 y˜(t1) 2 (SNR) = s = s . (C.12) o (t1) 2 w2 N0 + w2 Υ y˜n t1,d 2 r1,d By Eqs. (C.7) and (C.11), thus, (C.12) can be rewritten as:

w Ψ + w Φ 2 (SNR) = t1,d r1,d o w2 N0 + w2 Υ t1,d 2 r1,d w2 Ψ2 + 2w w ΨΦ + w2 Φ2 = t1,d t1,d r1,d r1,d . (C.13) w2 N0 + w2 Υ t1,d 2 r1,d

Let f(wt1,d, wr1,d) = (SNR)o, hence, after performing partial derivative of f(wt1,d, wr1,d) and let it equal to 0 to obtain the maximum SNR.

∂f(w , w ) t1,d r1,d = 0, (C.14) ∂wt1,d and

∂f(w , w ) t1,d r1,d = 0. (C.15) ∂wr1,d 81

By Eqs. (C.14) and (C.15) yield

2w Ψ2 + 2w ΨΦ w2 N0 + w2 Υ t1,d r1,d t1,d 2 r1,d 2 " wt1,dΨ + wr1,dΦ wt1,dN0 # − 2 = 0, (C.16) w2 N0 + w2 Υ t1,d 2 r1,d and

2w ΨΦ + 2w Φ2 w2 N0 + w2 Υ t1,d r1,d t1,d 2 r1,d 2 " wt1,dΨ + wr1,dΦ 2wr1,dΥ # − 2 = 0. (C.17) w2 N0 + w2 Υ t1,d 2 r1,d By simplifying Eq. (C.16) and (C.17) yields,

w (w Φ + w Ψ)( N w Φ + 2w ΥΨ) = 0, (C.18) r1,d r1,d t1,d − 0 t1,d r1,d and

w (w Φ + w Ψ)(N w Φ 2w ΥΨ) = 0. (C.19) t1,d r1,d t1,d 0 t1,d − r1,d Appendix D

Error probability closed form solution for a special case and diversity gain analysis

In this Section, an extension diversity gain analysis from our previous work [2] is performed, using similar set up as Section 2.2. For this analysis, the moment- generating function (MGF) of a Nakagami-m fading channel is needed which is deﬁned as:

sγb,1R Mγb,1R (s) = e ∞

= exp(sγ¯b,1R)pγb,1R (γb,1R)dγb,1R −∞ Z −m sγ¯ = 1 b,1R , (D.1) − m ! where γb,1R is deﬁned in Eq. (2.19) andγ ¯b,1R represents the average SNR; while

sγb,1R sγb,1R < e > denotes the expectation value of e and pγb,1R is the probability density function (pdf) of a Nakagami-m fading channel, which is:

m m−1 m γb,1R mγb,1R pγb,1R = exp . (D.2) Γ(m)¯γb,1R − γ¯b,1R ! 83

For independent random variables X and Y , the moment-generating function sat- isﬁes:

s(x+y) Mx+y(s) = e = esxesy = esx e sy

= Mx( s)My (s). (D.3)

Using the above results and when mt1 = mt2 = mr1 (i.e., two transmitters and single relay case), Eq. (2.27) can be written as:

π/2 ∞ 2 1 γb,1R P (e) = exp E(1R) (p )3dγ dφ π − sin2 φ γb,1R b,1R Z0 Z0 (1R) V −3m 1 π/2 2 = M E(1R) dφ π γb,1R − sin2 φ Z0 V(1R) ! −3m 1 π/2 2 γ¯ = 1 + E(1R) b,1R dφ. (D.4) π sin2 φ m Z0 V(1R) ! Eq. (D.4) can be approximated to:

−3m 1 γ¯c,1r (1 +γ ¯c,1r) Γ(3m + 2 ) 1 1 P (e) 2F1 1, 3m + ; 3m + 1; , ≈ 1 +γ ¯c,1r 2√π Γ(3m + 1) 2 1 +γ ¯c,1r r (D.5) 2 E(1R) whereγ ¯c,1r = γ¯b,1R, and 2F1( , ; ; ) [43] is the Gauss hypergeometric func- mV(1R) · · · · tion. By deﬁnition,

1 ∞ (1) m + 1 xi F 1, m + ; m + 1; x = i 2 i , (D.6) 2 1 2 (m + 1) i! i=0 i X where 1 x = , (D.7) 1 +γ ¯c,t1,r1 and (a) = Γ(a + i)/Γ(a) = a(a + 1) (a + i 1). (D.8) i ··· −

From (D.8), it is seen that (1)i = i!, (m + 1)i = (m + i)!/m! and (m + 1/2)i = 84

(1/2)m+i/(1/2)m. Thus, (D.6) can be rewritten as

1 ∞ 1 m! F 1, m + ; m + 1; x = 2 m+i xi 2 1 2 1 (m + i)! i=0 2 m X ∞ 1 m!x−m = 2 m+i xm+i 1 (m + i)! 2 m i=0 X m! ∞ 1 xi m−1 1 xi = x−m . (D.9) 1 2 i! − 2 i! 2 m i=0 i i=0 i X X For the case where we have L links (from transmitters to destination and relays to destination) and when all the Nakagami-m links’ parameters are equal, then we can rewrite Eq. (D.4) as:

π/2 ∞ 2 γ 1 ((L−2)R) b,(L−2)r L P (e) = exp E (p ) dγ − dφ π − sin2 φ γb,(L−2)r b,(L 2)r Z0 Z0 ((L−2)R) V −Lm 1 π/2 2 E((L−2)R) = Mγb,(L−2)r 2 dφ π − − sin φ Z0 V((L 2)R) ! −Lm 1 π/2 2 γ¯ E((L−2)R) b,(L−2)r = 1 + 2 dφ π − sin φ m Z0 V((L 2)R) ! −Lm 1 γ¯c,(L−2)r (1 +γ ¯c,(L−2)r) Γ(Lm + 2 ) ≈ s1 +γ ¯c,(L−2)r 2√π Γ(Lm + 1) 1 1 F 1, Lm + ; Lm + 1; , L > 0 (D.10) ×2 1 2 1 +γ ¯ c,(L−2)r

2 E((L−2)R) whereγ ¯c,(L−2)r = γ¯b,(L−2)r. mV((L−2)R) The deﬁnition of diversity gain is given by [33],

log P (SNR) D = lim e . (D.11) − SNR→∞ log SNR

For example, when using two transmitters, L 2 = Z (where Z is the number − of relays used in the scheme), and under the Nakagami-m distribution when m = 1 (Rayleigh distribution), the diversity gain in the proposed system, using Eq. 85

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