DEGREE PROGRAMME IN WIRELESS COMMUNICATIONS ENGINEERING

MASTER’S THESIS

PERFORMANCE ANALYSIS OF MIMODUAL-HOP AF RELAY NETWORKS OVER

ASYMMETRIC FADING CHANNELS

Author Praneeth Jayasinghe

Supervisor Prof. Markku Juntti

Second Supervisor Prof. Matti Latva-aho

Technical Advisor L.K. Saliya Jayasinghe

May, 2014

Jayasinghe P. (2014) Performance Analysis of MIMO Dual-hop AF Relay Networksover Asymmetric Fading Channels. University of Oulu, Department of Communica-tions Engineering, Master’s Degree Program in Wireless Communications Enginee-ring. Master’s thesis, 77 p.

ABSTRACT

We analyze the performance of dual-hop multiple-input multiple-output (MIMO)amplify-and-forward (AF) relay systems by considering the source-to-relay andrelay-to-destination channels undergo Rayleigh and Rician fading, respectively.Several MIMO techniques and practical relaying scenarios are considered to in-vestigate the effect of such asymmetric fading on the MIMO AF relaying systems.

First, we investigate the performance of the optimal single stream beamformingon non-coherent AF MIMO relaying. We use tools of finite-dimensional randommatrix theory to statistically characterize the instantaneous signal-to-noise ratio(SNR). The closed-form expressions of the cumulative distribution function, pro-bability density function, and moments of SNR are derived and used to analyzethe performance of the system with outage probability, bit error rate (BER), andergodic capacity. Numerical simulations are carried out to investigate the effectsof the Rician factor, rank of the line-of-sight (LoS) component, and the number ofantennas at the nodes on the system performance. Additionally, the performanceis compared with orthogonal space-time block-coding (OSTBC) based AF MIMOsystem.

Next, we investigate relay selection schemes for non-coherent dual-hop AF re-laying with OSTBC over asymmetric fading channels. We propose two relay se-lection methods as optimal and sub-optimal schemes. The performance of pro-posed schemes are discussed with respect to the outage probability, BER and theergodic capacity.

Finally, we study the effect of co-channel interference (CCI) and feedback delayon the multi-antenna AF relaying over asymmetric fading channels. Here, trans-mit beamforming vector is calculated using outdated channel state informationdue to the feedback delay from relay-to-source, and the relay node experienceCCI due to frequency reuse in the cellular network. The performance is investi-gated using the outage probability, BER and ergodic capacity to analyze the effectof the Rician factor, CCI, feedback delay and number of antennas.

All these discussions are useful to evaluate the performance of AF MIMO sys-tems in asymmetric fading channels. Our analysis suggests that having good LoScomponent increases the performance of the system for multiple-input-single-output (MISO) and single-input-multiple-output (SIMO) scenarios of relay-destinationchannel. Having good scattering component increases the performance for MIMOcases.

Keywords: Amplify-and-forward relaying, bit error rate, cumulative distributionfunction, multiple-input multiple-output, moments, optimal beamforming, pro-bability density function, Rayleigh fading, Rician fading.

TABLE OF CONTENTS

ABSTRACT

TABLE OF CONTENTS

FOREWORD

LIST OF ABBREVIATIONS AND SYMBOLS

1. INTRODUCTION 9

2. BACKGROUND REVIEW 132.1. Cooperative Communication . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1. Decode-and-Forward Relaying . . . . . . . . . . . . . . . . . 132.1.2. Amplify-and-Forward Relaying . . . . . . . . . . . . . . . . 14

2.2. MIMO Communications . . . . . . . . . . . . . . . . . . . . . . . . 152.3. MIMO Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4. Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5. Relay Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6. CCI and Feedback delay . . . . . . . . . . . . . . . . . . . . . . . . 19

3. OPTIMAL BEAMFORMING 213.1. System and Channel Model . . . . . . . . . . . . . . . . . . . . . . . 213.2. Statistics of the SNR . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1. Cumulative Distribution Function . . . . . . . . . . . . . . . 233.2.1.1. Non-i.i.d. Rician Fading . . . . . . . . . . . . . . . 233.2.1.2. Low-rank Rician Fading . . . . . . . . . . . . . . . 253.2.1.3. i.i.d. rician . . . . . . . . . . . . . . . . . . . . . . 263.2.1.4. Rayleigh Fading . . . . . . . . . . . . . . . . . . . 27

3.2.2. Probability Density Function . . . . . . . . . . . . . . . . . . 283.2.3. Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3. Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.1. Outage Probability . . . . . . . . . . . . . . . . . . . . . . . 313.3.2. High SNR Analysis . . . . . . . . . . . . . . . . . . . . . . . 323.3.3. Effect of the Relay Saturation . . . . . . . . . . . . . . . . . 333.3.4. Symbol Error Rate . . . . . . . . . . . . . . . . . . . . . . . 353.3.5. Ergodic Capacity . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4. RELAY SELECTION WITH OSTBC 414.1. System and Channel Models . . . . . . . . . . . . . . . . . . . . . . 414.2. Statistics of the Relay Selection Schemes . . . . . . . . . . . . . . . 42

4.2.1. Optimal Relay Selection Scheme . . . . . . . . . . . . . . . . 424.2.2. Sub-optimal Relay Selection Scheme . . . . . . . . . . . . . 43

4.3. Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5. EFFECT OF CCI AND FEEDBACK DELAY 495.1. System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2. Statistics of the SNR . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3. Performance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3.1. Outage Probability . . . . . . . . . . . . . . . . . . . . . . . 515.3.2. Symbol Error Rate . . . . . . . . . . . . . . . . . . . . . . . 515.3.3. Ergodic Capacity . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3.3.1. Upper bound . . . . . . . . . . . . . . . . . . . . . 525.3.3.2. Lower bound . . . . . . . . . . . . . . . . . . . . . 53

5.4. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 535.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6. SUMMARY AND CONCLUSIONS 57

7. REFERENCES 59

8. APPENDICES 64A. Derivation of the c.d.f. of the ξmax . . . . . . . . . . . . . . . . . . . 64

A.1. Case 1: NS ≥ q . . . . . . . . . . . . . . . . . . . . . . . . . 64A.2. Case 2: NS < q . . . . . . . . . . . . . . . . . . . . . . . . . 66A.3. For j ≤ q −NS in (106) . . . . . . . . . . . . . . . . . . . . 67A.4. For j > q −NS in (106) . . . . . . . . . . . . . . . . . . . . 67

B. Derivation of the c.d.f. of ξmax for the Low-rank Rician, i.i.d. Rician . 67B.1. Low-rank Rician . . . . . . . . . . . . . . . . . . . . . . . . 67B.2. Case 1: NS ≥ q . . . . . . . . . . . . . . . . . . . . . . . . . 68B.3. Case 2: NS < q . . . . . . . . . . . . . . . . . . . . . . . . 68B.4. i.i.d Rician . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

C. Derivation of the p.d.f. of the ξmax . . . . . . . . . . . . . . . . . . . 69C.1. Validity of the p.d.f. expression for q = 1 . . . . . . . . . . . 70

D. Derivation of Moments of the ξmax . . . . . . . . . . . . . . . . . . . 71D.1. Case min(NR, ND) = 1 . . . . . . . . . . . . . . . . . . . . 71D.2. Case NS = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 71

E. Derivation of the Asymptotic Expansion of the Outage Probability . . 71E.1. Asymptotic expansion for q = 1 . . . . . . . . . . . . . . . . 71E.2. Asymptotic expansion for NS = 1 . . . . . . . . . . . . . . . 72

F. Derivation of the Asymtotic Ergodic Capacity . . . . . . . . . . . . . 73G. Derivation of the c.d.f. and p.d.f. for sub-optimal relay selection scheme 74

G.1. c.d.f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74G.2. p.d.f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

H. Derivation of the c.d.f. of γ and γu . . . . . . . . . . . . . . . . . . . 75H.1. Derivation of the c.d.f. of the γ . . . . . . . . . . . . . . . . . 75H.2. Derivation of the c.d.f. of the γu . . . . . . . . . . . . . . . . 75

I. Ergodic Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

FOREWORD

This Master thesis is focused on performance analysis of MIMO dual-hop AF relaynetworks over asymmetric fading channels. I would like to gratefully acknowledgethe Centre for wireless communication (CWC), Department of communications En-gineering (DCE), University of Oulu for providing financial support for this researchwork.

I would like to express my deepest thank to my supervisor, Prof. Markku Juntti, forhis great supervision throughout my study period. His door was always opened mefor discuss. Without his full supervision and inspiration, this research would not havebeen accomplished successfully. My special thank goes to My elder brother Mr. L.K. Saliya Jayasinghe for always there to motivate and to give valuable thoughts to mystudies whenever needed. Without his guidance and inspiration, this research wouldnot have been accomplished successfully.

I would like to express my deep thank to Prof. Matti Latva-aho for his guidance, pre-cious suggestions and comments during overall research period. Also, many thanks forgiving me opportunity to work in CWC to obtain financial assistance for my researchwork.

A special thank goes to my research partner Mr. Antti Roivainen and to my dearfriends Mr. Ishwore and Mr. Jigo for your lovely memories and companion we hadduring my stay in Oulu.

Finally, my deepest thank and appreciations go to my beloved parents for their loveand encouragement for my studies. I also thank to my family members for their loveand support.

LIST OF ABBREVIATIONS AND SYMBOLS

AF Amplify and ForwardBER Bit Error RateBPSK Binary Phase Shift KeyingCCI Co-channel interferencec.d.f. cumulative distribution functionCSI Channel State InformationDF Decode and Forwardiid Independent and identically distributedMIMO Multiple Input Multiple OutputMISO Multiple Input Single OutputMRC Maximum ratio combiningMRT Maximum ratio transmissionLoS Line of SightOSTBC Orthogonal Space Time Block Codingp.d.f. Probability Density FunctionSER Symbol Error RateSIMO Single Input Multiple OutputSINR Signal to Interference and Noise RatioSISO Single Input Single OutputSNR Signal-to-Noise RatioTB Transmit beamforming

a/an Fixed gaina(θr,i) Array response vectorb/bn Positive real numberhij Channel coefficients between i node to j nodehir/αir ith Interferer-relay channelhjd jth Interferer-relay channelhsd Source-Destination channeln0 White gaussian noisep/pn Maximum dimension of a matrixq/qn Minimum dimension of a matrixr Received signals min(NS, q)si ith symbol of M iid signalsw max(NS, q)x Modulated symbolx Estimate of xxi/xj Signal from Interfererysr Received signal at the Relayn Noise vectorn1 Noise vector at the sourcenR/nnk Noise at the Relay

nd/nk/nD Noise at the Destinationvmax Maximum eigen vectorwt Transmit beamforming vectorwr Receive beamforming vectorx Transmitted signal vectorxk Transmitted signal vector at kth symbol periody/yk/ynk Received signal vectorC Complex numberB(., .) Beta functionG Gain at the Relay

Gm,np,,q

(x |a1,...,ap

b1,...,bq

)Meijer’s G function

K Rician factorKv(·) modified Bessel function of the second kindM No of iid symbols at OSTBC matrixNS No of anntennas at sourceNR No of anntennas at RelayND No of anntennas at DestinationNT No of symbol periodsPi Transmit power of the ith interfererPR Transmit power at the relayPS Transmit power at the sourcePmax Maximum transmit power at the RelayQ(.) Gaussian Q-functionR Code rateRn No of antennas at the nth RelayU(., ., .) Confluent hypergeometric functionWa,b(.) Whittaker function.CN Complex noiseG Constant Gain MatrixH Channel matrixhsr/h1/H1/HSn Source to Relay channelhrd/h2/H2/HnD Relay to Destination channelh2/H2/HnD LoS component of Relay to Destination channelh2/H2/HnD Scattered component of Relay to Destination channelI0(.) Zeroth order modified Bessel function of the first kindIi Identity matrix with dimention iX OSTBC matrix

α Constantβi Complex amplitude of the ith pathη/ηn Coefficient of the line of sight componentγ Signal to noise ratioγ0/γ Average SNRγmax Maximum instantaneous SNR

γsub Instantaneous SNR of suboptimal pathγR Instantaneous input power at the Relayγth Threshold SNRλi/λ Ordered eigen valuesψ(.) Eular’s psi functionρ Average transmit powerσ/σn Coefficient of the scattered componentξmax Maximum eigen valueΓ(·) Gamma functionΛ Diagonal matrix with ordered eigen values

min(α, β) Minimum between α and βmax(α, β) Maximum between α and β(p)l Pochhammer symbol||.||F Frobenius norm2F1(., .; .; .) Gauss hypergeometric function(·)−1 Inverse of a Matrix(·)T Transpose of a Matrix(·)H Hermitian of a MatrixE{.} Expectation of a random variable⊗ Kronecker operator

1. INTRODUCTION

Modern wireless communication systems push for high data rates, reliable communica-tions, coverage enhancements, and less power requirements. Multiple-input multiple-output (MIMO) relaying can be identified as a candidate for meeting these challenges.MIMO technique provides higher spectral efficiency and improves the reliability ofthe communication systems [1, 2]. Cooperative relay communication enhances thethroughput and extends the coverage area [3, 4, 5, 6, 7]. This also reduces the needto use high transmitter power, which in turn results in reduced interference to othernodes. Both techniques can also be used to achieve spatial diversity. Recent studieshave increased interest on the MIMO relaying that can optimally utilize key resourcesof wireless fading, and achieve the benefits of both techniques [8].

MIMO relaying is mainly based on the relaying protocols such as amplify-and-forward (AF) and decode-and-forward (DF). In the AF protocol, the relay node for-wards the amplified version of the received signal to the destination node, and providessignificant gains with using less complicated processing. On the other hand, the DFprotocol based relay node decodes the received signal and re-encodes, modulates andsends to the destination. Several researchers showed that the AF protocol can be imple-mented in practice [9, 10, 11]. In [10], the authors implemented an AF relaying systemand compared it to the direct transmission. The performance evaluation shows a clearadvantage of using AF relays, revealing a significant bit error rate (BER) improvementunder realistic wireless conditions. Both the AF and DF systems were implemented in[11], and compared in terms of implementation loss and the complexity. They showedthat the AF protocol is less complex and has lower implementation loss (i.e., perfor-mance is very similar to the theoretical studies). These findings further support the ideathat an AF relaying protocol requires less complicated processing. However, many an-alytical studies on AF MIMO relaying assume that the instantaneous channel stateinformation (CSI) of source-relay and relay-destination channels are available at therelay node [12, 13], where the relay node has to consume a lot of resources to estimateCSI. This is contradictory to the reason of using AF MIMO relaying as a less com-plex relaying protocol. As an alternative solution, more realistic and less complex AFscheme called ’fixed-gain’ AF relaying is studied in [14, 15, 16, 17, 18], which applieda fixed gain at the relay node.

Most of the studies on fixed-gain AF MIMO relaying have been carried out us-ing tools of large random-matrix theory. They are mainly focused on the asymptoticnetwork capacity and fundamental information-theoretic performance limits of largenetworks [19, 20, 21, 22]. In [19], the authors discussed both variable and fixed-gainAF MIMO systems to investigate the asymptotic capacity in large relay networks. An-other study on a large scale multi-level AF relay network has been analyzed in [20].More realistic scenario of AF relaying with multi user interference is discussed in [21],and they found expressions for ergodic capacity and derived some bounds. Jin et al.[22], analytically characterized the ergodic capacity for fixed-gain AF MIMO dual-hopsystem.

Recent studies in [23, 24, 25, 26, 27, 28, 29] investigate analytical performances onAF MIMO dual-hop systems with the use of finite dimensional random matrix theory.These systems are mainly based on orthogonal space time coding (OSTBC) and opti-mal beamforming. OSTBC based AF MIMO relaying systems can be used to improve

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the link reliability when the CSI is not available at the transmitter. Dharmawansa etal. [23] studied the performance of the OSTBC transmission for fixed-gain AF MIMOdual-hop system when the channels undergo Rayleigh fading. They have used mo-ment generating function to discuss the performance with different metrics. Song etal. [24] investigated the diversity order in an OSTBC based MIMO system. MIMObeamforming can also use to mitigate the fading effect through diversity by using CSIknowledge. Performance analysis on beamforming based AF MIMO systems are car-ried out in [26, 27, 28]. Louie et al.[26] studied the performance of such system withantenna correlation in terms of outage probability and symbol error rates. Da Costaet al. [27] studied performance of AF MIMO beamfoming over Nakagami-m fadingchannels. Furthermore, Min et al. [28] studied the outage probability of such AFMIMO dual-hop system with optimal beamforming. However, most of these stud-ies assume CSI at the transmitter and relay node. Also, these studies are carried outassuming a single antenna at the relay node.

A fixed-gain AF MIMO dual-hop system with optimal beamforming is discussedin [29], where the beamforming vectors are computed at the destination, and transmitbeamforming vector is sent back to the transmitter via a feedback link. An optimal sin-gle stream beamforming at the transmitter side is considered in their work. However,this work is limited to the Rayleigh fading scenarios, and which is usually not practicalin actual relay deployments. It is more appropriate to consider a situation where therelay is chosen such that the relay-to-destination channel has a line-of-sight (LoS) path[30, 31, 25]. This makes the relay-destination channel to experience Rician fading dueto the dominant LoS component. Even though Rician fading is very common in manypractical situations, it is not discussed enough due to the complexity associated withRician fading. Jayasinghe et al. [25] carried out performance analysis on fixed-gainAF MIMO dual hop system with OSTBC, and assumed asymmetric fading channelsbetween the source-to-relay and relay-to-destination. Performance analysis on optimalbeamforming based fixed-gain AF MIMO dual hop system over asymmetric fadingchannels will be useful to find performance differences with OSTBC based fixed-gainAF MIMO systems, and which in turn provides the pathway to select the best optionfor any practical deployment.

Cooperative communication can also be used to overcome adverse channel fadingin wireless environments. The use of multiple relays in such fading scenario helps toprovide a reliable transmission for the source to the destination, where these adversechannel effects can be mitigated using relay selection strategies such as selecting thebest relay. Various studies on the relay selection have been carried out to increase thediversity order, reduce adverse channel effects, and to overcome the half-duplex loss ofthe use of relay [32, 8, 33]. In [32], a relay selection scheme is proposed to maximizethe SNR at the destination. In [8, 12], the authors show that the diversity gains can beachieved in the order of the number of relays assisting the communication. This alsocompensates for the half-duplex loss. Most of these relay selection works are basedon relaying protocols such as AF and DF [33, 12, 34, 35]. Several studies address theimperfect CSI in relay selection schemes [36, 37]. Additionally, the authors in [38]investigate the impact of using outdated CSI on the relay selection.

Dharmawansa et al.[23] carried out analysis on non-coherent AF MIMO dual hopsystem with OSTBC. MIMO AF relaying with OSTBC gives considerable improve-ments for link reliability in the system. These systems provide the same diversity

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order as maximal-ratio receiver combining [2, 39, 40]. Relay selection schemes withnon-coherent AF MIMO OSTBC relaying can enhance the system performances fur-ther without extra signal processing capabilities at relay nodes. To have a clear un-derstanding of such systems with several relay nodes, these scenarios require furtherinvestigations.

AF relaying protocol has been the most attractive relaying protocol among the re-search community, which forwards amplified version of the received signal at the relaynode. AF relaying can be categorized into variable gain relaying [12, 13, 28] or fixedgain relaying [15, 14, 25] based on the availability of CSI at the relay node. Most ofthe research on variable gain AF relaying have been carried out assuming that the CSIof source-to-relay and relay-to-destination channels are available at the relay terminal.MIMO beamforming can be used to mitigate the severe effects of fading by exploitingchannel knowledge at both the transmitter and receiver. In particular, transmit beam-forming (TB) can be used at the transmitter, and maximum ratio combining (MRC) atthe receiver. Dual hop MIMO AF relaying systems employing TB/MRC provide sig-nificant performance gains as in [41, 42]. However, TB requires CSI at the transmitter,and it is not perfect for many practical scenarios. Also, the relay based communicationsystems are required to investigate for interferences at the nodes, such as co-channelinterference (CCI) at the relay and the destination. These practical aspects are studiedwith dual hop MIMO AF systems in [43, 44, 45]. In [43], the authors investigate theoutage probability in the presence of CCI. In [44], the authors analyze the effect offeedback delay on the outage probability and average SER. Huang et al. [45] derivesome capacity bounds of the MIMO AF relaying system in the presence of both CCIand feedback delay. However, these studies are limited to Rayleigh fading scenarios,and which is not practical in actual relay deployment.

Motivated with the above considerations, in this thesis, we analyze the performanceof different type of MIMO dual hop system over asymmetric fading channels. In par-ticular, Optimal beamforming AF MIMO relaying, Relay Selection on dual hop AFMIMO with OSTBC, Effect of CCI and feedback delay on the multi-antenna AF re-laying. Asymmetric fading of the dual hop system is considered as the source-relayand the relay-destination channels undergoes Rayleigh and Rician fading respectively.The rest of the thesis structure is as follows. Chapter 2 provides basic literature review.

In Chapter 3, we analyze the performance of an optimal beamforming scheme forfixed-gain AF MIMO dual-hop system in a situation where source-relay and relay-destination channels undergo Rayleigh and Rician fading respectively. In deep fadingscenarios, the source-to-destination communication requires higher reliability, whichis possible with a single stream transmission and a higher diversity order. However,the rate can be improved by using multiple stream transmission in a reliable fadingscenario. In this study, we focus on providing a higher diversity order to the sys-tem by having the single stream transmission. Both transmit and receive beamform-ing vectors are obtained to maximize the instantaneous signal-to-noise ratio (SNR) atthe destination. We use tools of finite dimensional random-matrix theory with differ-ent Rician fading scenarios to analyze the system. Here, we consider cases of relay-destination channel to undergo independent non identical Rician fading, where meanis non-identical (non-i.i.d Rician), low-rank Rician fading, where mean (LoS compo-nent) has a lower rank, and independent identical Rician fading (i.i.d Rician). Newstatistical results of the instantaneous SNR at the destination are derived in terms of

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the cumulative distribution function (c.d.f.), probability density function (p.d.f.), andmoments. Then, we use these statistical expressions to derive equations for outageprobability, symbol error rate (SER) and ergodic capacity. Diversity orders are derivedfor simplified scenarios in the high SNR analysis. These performance metrics are usedto evaluate the performance of optimal beamforming AF MIMO system with differentantenna configurations, Rician factors, different Rician fading scenarios. Additionally,the saturation loss at the relay is quantified in terms of the outage probability. Finally,the system is compared with the OSTBC based AF MIMO system.

In Chapter 4, we investigate optimal relay selection schemes for orthogonal space-time block coded multiple-input multiple-output system with non-coherent amplify-and-forward relays, where channel state information is not available at the source andrelays. The source-relay and relay-destination channels undergo Rayleigh and Ricianfading, respectively. Two possible relay selection schemes are proposed, and both arestatistically characterized by deriving an exact closed form expression for the cumula-tive distribution function and probability density function of the instantaneous SNR atthe destination. In the first relay selection method, maximizing instantaneous SNR atthe destination is considered to select the best relay. In the second scheme, maximiz-ing relay-destination channel is considered. The derived statistical results are used toanalyse the performance of the system with outage probability, average bit error rateand ergodic capacity. Finally, we compare both relay selection schemes with respectto the relay pool size and Rician factor.

In Chapter 5, we analyze the performance of of dual-hop multiple antenna AF re-laying systems using TB/MRC when the relay is subjected to multiple interferers andthe source-relay and relay-destination channels undergo Rayleigh and Rician fadingrespectively. We derive set of new statistical results to the instantaneous signal-to-interference plus noise ratio (SINR) at the destination. Then, apply these statisticalresults to study the performance of beamforming systems in terms of three key per-formance metrics, i.e., outage probability, BER and ergodic capacity. Finally, the per-formance metrics are used to investigate the impact of key system parameters such asRician factor, CCI, feedback delay and number of antennas. Summary of the thesis ispresented in Chapter 6.

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2. BACKGROUND REVIEW

2.1. Cooperative Communication

In wireless communication, users are suffers from channel fading, that the signal atten-uation varies significantly over the transmission. This can be overcome using properdiversity methods such as spatial, temporal, and frequency diversity. Cooperative com-munication introduces new diversity method called cooperative diversity. Cooperativediversity achieves with the cooperation of distributed relay nodes that help to establisha communication link between source node and destination node. Several theoreticalstudies have been carried out on cooperative communication and those studies suggestthat the cooperative communication required less transmit power at the source and therelay. This makes cooperative communication is interference less and power efficientcommunication method. Also, cooperative communication enhances the coverage andthe throughput of the system. [3, 4, 5, 6, 7].

In this thesis work, we study dual hop transmission, which is a three-node net-work consisting of a source, a destination, and a relay. Figure 1 illustrates simpleschematic of dual hop communication system. Here, fading channels source-relay,relay-destination and source-destination are denoted as hsr,hrd and hsd, respectively.For this system, received signal at the relay node can be obtained as [13]

ysr =√PShsrx+ nr, (1)

where PS is the average transmit power at the source, x is the signal transmitted at thesource and nr is the noise at the relay.

Figure 1: Dual hop relay communication.

Dual hop transmission system can be classified into two main categories dependshow the signal processing is done to received signal ysr at the relay node. Those areregenerative (decode-and-forward) and non-regenerative (amplify-and-forward) sys-tems.

2.1.1. Decode-and-Forward Relaying

DF relaying is an example for regenerative system. DF based relay node decodesthe received signal and re-encodes, modulates and sends to the destination. Figure 2

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illustrates simple schematic of the DF relying system [7]. The main advantage of DFmethod is that, it eliminates the noise at relay node with the expense of processingdelay due to modulation/demodulation and encoding/decoding.

In the DF system, the relay demodulates the received signal ysr to estimate x. Thenthe estimated signal x is forward to the destination in order to complete the transmis-sion. This signal estimation can be carried out in symbol by symbol or for the entirecodeword by cosidering the required system performance and the complexity at therelay. Recieved signal yrd at the destination can be obtained as,

yrd =√PRhrdx+ nd, (2)

where PR is the average transmit power at the relay and nd is the noise at the destina-tion.

Figure 2: Decode and forward relaying system.

2.1.2. Amplify-and-Forward Relaying

In the AF protocol, the relay node forwards the amplified version of the received signalto the destination node, and provides significant gains with using less complicatedprocessing. AF protocol can be implemented in practice and it is less complex and haslower implementation loss.

Simple AF relaying system is illustrated in Figure 3 [7]. In the AF system, receivedsignal ysr at the relay is subject to the amplification factor G at the relay node beforeforward it to the destination. Then received signal yrd at the destination can be obtainedas

yrd = G√PRhrdysr + nr. (3)

Depending on the availability of instantaneous CSI at the relay node, the AF relayingcan be categorized into two schemes. Those are variable gain AF relaying and fixedgain AF relaying. In variable gain relaying, it amplifies the received signal at the relaynode based on the instantaneous CSI. On the other hand, fixed gain is applied in AFfixed gain method considering the average behavior of the channel.

• Fixed gain AF relay : The amplification factorG is obtained according to averagechannel status as

15

Figure 3: Amplify and forward relaying system.

G =

√PR

E (|ysr|2). (4)

• Variable gain AF relay :The amplification factor G is obtained according to in-stantaneous CSI as

G =

√PR|ysr|2

. (5)

2.2. MIMO Communications

A MIMO system uses multiple antennas at both the transmitter and receiver to im-prove the communication system performance by use of diversity and multiplexingtechniques. MIMO system provides higher spectral efficiency, improves the reliability,fading mitigation and improved resistance to interference [1, 2].

Figure 4: MIMO system.

16

There are three main MIMO techniques have been proposed in the literature, suchas precoding, spatial multiplexing, and diversity coding. Precoding is a technique thatuse the knowledge of CSI at the transmitter and the receiver to design precoder formulti-stream beamforming. In spatial multiplexing, a high rate signal is split into eachtransmit antenna with different low rate date streams and every stream use the samefrequency band. In a situation where CSI is not available at the transmitter, diversitycoding can be used to achieve better diversity gain similar to MRC system. In diversitycoding method, signal is transmitted by applying space-time coding at the transmitter.

Basic MIMO system illustrates in Figure 4. This MIMO system consist with ntransmit antennas and m received antennas. Channel between ith receive antenna andjth transmit antenna is denote as hij . Therefore received signal can be modeled as

y = Hx + n, (6)

where y is the received signal vector, x transmitted signal vector, n is the noise vectorand H is the channel matrix with (i, j)th component is hij .

Beamforming uses precoding technique and multiple antennas for directional signaltransmission and reception [46]. This directionality of the transmission is obtainedby multiplying the transmit/receive signal with precoding vector in order to obtainconstructive interference in the relevant direction and destructive interference in otherdirections. Beamforming methods can be applied at both the transmitter and the re-ceiver. Also, beamforming significantly reduces the interference and improves systemcapacity.

Maximal ratio transmission (MRT) is the beamforming technique that can achieveboth diversity and the array gains with transmit beamforming. MRC is the optimalcombining method, where the signals from the received antenna elements are com-bined in the way that the instantaneous SNR is maximized. MRT with MRC providesreference for the optimum performance that the system may obtain using both transmitand receive diversity.

In single stream beamforming, same signal is transmit through each of the transmitantennas after precoding with transmit beamforming vector. Then receive beamform-ing vector is design in the way that end-to-end SNR is maximized at the receiver input.Received signal of single stream beamforming can be modeled as,

y = Hwtx+ n, (7)

where wt is the transmit beamforming vector and x is the transmit signal from all theantennas. At the receiver, receive combining vector wr is applied to y. This can beexpressed as,

y = wHr Hwtx+ wH

r n. (8)

Alamouti proposed a new way of transmit diversity scheme with the use of twotransmit antennas when CSI is not available at the transmitter [2]. This achieved bytransmitting a pair of symbols using two antennas at first and then transmits the trans-formed version of the symbols. This Alomouti scheme is led to the progress of space-time block coding technique. In OSTBC, the data transmitted with orthogonal coding,such that multiple copies of the data transmit across multiple antennas. This improvesthe reliability of data transmission. Transmitting multiple copies of data, increases thechance of correctly decode the received signal using the redundantly received data.

17

This OSTBC coding exploits the independent fading in the multiple antennas to im-prove the diversity gain.

At the transmitter, OSTBC encoding done with N i.i.d. symbols s1, s2, .., sN aremapped to a row orthogonal matrix X ∈ Cm×NT , where entries of X obtained by linearcombinations of s1, s2, .., sN and their conjugates [47]. Also, NT is the number ofsymbol periods used to send a code word. Therefore, the code rate is R = N/NT . Letxk be the transmitted signal during the kth symbol period. We take X = (x1, ..., xNT ).During the kth symbol period, we have the received signal at the destination as

yk = Hxk + nk, k = 1, 2, .., NT , (9)

where nk is the noise vector at the destination.

2.3. MIMO Relaying

MIMO relaying is an interesting research direction that can optimally utilize key re-sources of wireless fading, and achieve the benefits of both MIMO and cooperativecommunication. We investigate different types of MIMO relaying systems in thisthesis work, such as optimal single stream beamforming based AF relaying, OSTBCbased AF relaying and TB/MRC based AF relaying.

Figure 5: MIMO cooperative communication system.

In dual hop optimal single stream beamforming method, optimal transmit beam-forming vector and optimal combining vector are designed in order to maximize end-to-end instantaneous SNR by use of CSI knowledge of the source-relay and relay-destination channels. The beamforming vectors are computed at the destination, andtransmit beamforming vector is sent back to the transmitter via a feedback link. In deepfading scenarios, the source-to-destination communication requires higher reliability,which is possible with a single stream transmission and a higher diversity order. Moredetailed analysis carried out in Chapter 3.

OSTBC based AF MIMO relaying systems can be used to improve the link reliabil-ity when the CSI is not available at the transmitter. In this case, it is important to havemultiple antennas at the transmitter to apply OSTBC and receiver (relay, destination)

18

can be either single antenna or multiple antennas. In Chapter 4, we discuss further onOSTBC based dual hop systems.

Another dual hop system with MIMO beamforming is investigated in Chapter 5. Inthis MIMO relaying system, TB is used at the transmitter considering the source-relayCSI, and MRC at the receiver considering the relay-destination CSI.

2.4. Fading Channels

To Analytically investigate MIMO dual hop systems, It is important understand sta-tistical behavior of the fading channels. The magnitude of the fading is random in awireless channel. This is because of the multipath and the random location of objectsin the environment [48]. Two fading models, that very much popular in the researchcommunity are Rayleigh and Rician fading. This thesis work considers asymmetricfading channels, such as source-relay channel undergoes Rayleigh fading and relay-destination channel undergoes Rician fading.

Rayleigh fading is caused by multipath reception with no LoS component. Thismodel is suited when the signal received from the large number of reflected waves andscattered waves. Rayleigh fading is exponentially distributed with respect to SNR and,p.d.f. of Rayleigh fading channel is given by [49]

pγ(γ) =1

γ0

exp

(− γ

γ0

): γ ≥ 0 , (10)

where γ0 is the average SNR.When the received signal is consists of multipath components and considerable LoS

component, the fading can be modeled as Rician fading. Probability distribution of theRician fading is given by [49]

Pr(r) =r

σexp

(−r

2 + s2

2σ

)I0

(rsσ2

): r ≥ 0 , (11)

where σ is the variance of, s represents the amplitude of the line-of-sight path compo-nent and I0(.) is the zeroth order modified Bessel function of the first kind.

2.5. Relay Selection

The use of multiple relays in an adverse fading environment helps to provide a reliabletransmission for the source to the destination, where these adverse channel effects canbe mitigated using relay selection strategies such as selecting the best relay. The relayselection can be use to increase the diversity order, reduce adverse channel effects, andto overcome the half duplex loss of the use of relay. Various relay selection schemesproposed in the literature such as best relay selection, nearest neighbor selection, bestharmonic mean selection and best worse channel selection [12]. For those relay selec-tion schemes, it is assumed that all CSI is available at the destination and relay nodehas its own CSI.

Figure 6 illustrates possible relaying paths in dashed lines and the best path in solidline. By using suitable relay selection scheme, diversity gains can be achieved in the

19

Figure 6: Cooperative communications system with multiple relays.

order of the number of relays assisting the communication. We consider two relayselection schemes in Chapter 4 based on best relay selection and nearest neighborselection.

• Best relay selection : The selection method is following a SNR policy in thesense that the selected relay achieves a maximum instantaneous SNR.

• Nearest neighbor selection : Selects the relay, which is nearest to the base station.This nearest relay can find using the best channel between source-relay or relay-destination.

2.6. CCI and Feedback delay

In cellular networks, frequency reuse introduces CCI to wireless communication sys-tem. In more realistic analysis, it is important to consider CCI effect on relay basedcommunication system. This CCI issue is experienced in both the relay and the des-tination. Also, in practice CSI is not perfect. Therefore, it is important to study thesystems with imperfect CSI.

In dual hop AF relaying system received signal at the relay ysr with presence of CCIis given by

ysr =√PShsrx+

N1∑i=1

√Pihirxi + nr, (12)

20

where Pi is ith interferer’s signal power, hir is the ith interferer-relay channel and xiis the interference signal. Similarly, the received signal at the destination yrd withpresence of CCI is given by

yrd = G√PRhrdysr +

N2∑j=1

√Pihjdxj + nd, (13)

where Pj is jth interferer’s signal power, hjd is the jth interferer-destination channeland xj is the interference signal.

Imperct CSI is a performance limiting factor in wireless communication system,which is occurred due to the feedback delay between relay-source and destination-relay. Imperfect CSI due to feedback delay can be modeled using Jake’s autocorrela-tion model. As an example the source-relay channel hsr(t) with feedback delay τ canbe modeled as [50]

hrs(t− τ) =√ρhsr(t) +

√1− ρn0, (14)

where n0 is the white gaussian noise, and ρ is the correlation coefficient between hsr(t−τ) and hsr(t).

21

3. OPTIMAL BEAMFORMING

3.1. System and Channel Model

We consider fixed-gain AF MIMO dual hop system as shown in the Figure 7. Thesource, the relay, and the destination have NS, NR and ND antennas respectively. Thesource transmits to the relay during the first time slot, and the relay node transmits theamplified version of received data to the destination during the second timeslot. Weassume that there is no direct link between the source and the destination. The source-relay channel is assumed to undergo Rayleigh fading, and the relay-destination channelis subject to Rician fading. We denote H1 ∈ CNR×NS and H2 ∈ CND×NR as MIMOchannel matrices from the source-to-relay and the relay-to-destination, respectively.Entries of H1 are assumed to be ∼ CN (0,1). H2 is modeled as

H2 = ηH2 + σH2, (15)

where H2 is the LoS component, and H2 is the scattered component. H2 consistsof non-identical complex elements having unit magnitude, entries of H2 are assumedto be ∼ CN (0,1). In non-i.i.d. Rician fading, H2 is full rank matrix that consistsof complex elements having unit magnitude. For low-rank Rician fading, H2 is anarbitrary rank matrix that consists of complex elements having unit magnitude. H2

has rank one for i.i.d. Rician fading scenario. We also consider both η, σ to satisfyη2 + σ2 = 1 in all these scenarios.

Figure 7: AF MIMO dual hop system with NS source antennas, NR relay antennas andND destination antennas.

We consider optimal beamforming vectors at the source and the destination. Thesource node transmits beamformed version of modulated symbol x after being beam-formed by a transmit beamforming vector wt ∈ CNS , where E{|x|2} = ρ, and||wt||2 = 1. The relay amplifies the received signal by a constant gain matrixG =

√aINR , where a = b

NR(1+ρ)with b is a positive real number. This selection

of a satisfies the average total power constraint E{||GH1wtx + GnR||2 ≤ b}, wherenR ∼ CN (0, INR) is noise vector at the relay. A situation where the total power ex-ceeds the maximum available power at the relay node is discussed in a later section.

22

The destination detects the signal after using a receive beamforming vector wr ∈ CND ,where ||wr||2 = 1. The received signal r at the destination is given as

r =√awH

r H2H1wtx+√awH

r H2nR + wHr nD, (16)

where nD ∼ CN (0, IND) is noise vector at the destination. The instantaneous SNR atthe destination is given by

γ =aρwH

r H2H1wtwHt HH

1 HH2 wr

wHr (aH2HH

2 + IND)wr. (17)

Here, the SNR γ can be maximized by optimal beamforming design. We assumethat the channel state information (CSI) is only available at the destination, and thedestination computes the optimal beamforming vectors wt and wr. Then, the optimaltransmit beamforming vector is sent back to the transmitter via a dedicated feedbacklink [29]. The optimal receive beamforming vector wr that maximize the SNR is givenby

wr = (aH2HH2 + IND)−1H2H1wt. (18)

Then, substituting (18) into (17) gives

γ = aρwHt HH

1 HH2 (aH2H

H2 + IND)−1H2H1wt. (19)

Next, we consider eigenvalues of HH1 HH

2 (aH2HH2 + IND)−1H2H1 as (ξ1, ξ2, .., ξNS).

The maximum eigenvalue is denoted as ξmax, which is given by

ξmax = max(ξ1, ξ2, .., ξNS). (20)

The eigenvector associated with ξmax is vmax ∈ CNS . Therefore,

HH1 HH

2 (aH2HH2 + IND)−1H2H1vmax = ξmaxvmax. (21)

It is evident that wt = vmax provides the maximum instantaneous SNR γ in (19), andγ is simplified into

γ = aρξmax. (22)

To gain more insight into the system performance, we need to analyze the statisticalproperties of the SNR in (22). For the sake of convenience we use the followingnotations; p = max(ND, NR), q = min(ND, NR), w = max(NS, q), s = min(NS, q).

3.2. Statistics of the SNR

Statistical parameters of the SNR γ directly depend on the statistical results of themaximum eigenvalue ξmax. Here, we derive new analytical expressions for the c.d.f.,p.d.f., and moments of the ξmax using the tools of finite dimensional random matrixtheory. These results are used to find the statistical parameters of the SNR γ. We startwith a scenario where the relay-destination channel undergoes non-i.i.d Rician fading.This simplifies the derivations for other scenarios like low-rank Rician, i.i.d. Rician,and Rayleigh fading.

23

3.2.1. Cumulative Distribution Function

The following theorems are used to find the c.d.f. of the SNR γ.

3.2.1.1. Non-i.i.d. Rician Fading

Here, the relay-destination channel undergoes non-i.i.d. Rician fading, hence the LoScomponent H2 is full rank.

Theorem 1: The c.d.f. of the maximum eigenvalue ξmax of HH1 HH

2 (aH2HH2 +

IND)−1H2H1 is given by

Fξmax(x) =c(−1)NS(q−s)

det(V)∏s

i=1 Γ(NS − i+ 1)det(I(x)), (23)

where I(x) is a q × q matrix with (i, j)th element is given by

(I(x))ij =

∞∑l=0

λli(−1)q−NS−j

(t)ll!σ(2p+2l−q+1)

( q−j∑k=0

(q−jk

)akσ2wjΓ(wj)

)j ≤ q − s

∞∑l=0

λli(w−j)!(t)ll!σ(2p+2l−q+1)

( sj∑k=0

(sjk

)akσ2uΓ(u)

−w−j∑m=0

m+sj∑r=0

2(m+sjr )arxme−ax(xσ2)

v2

m!Kv

(2√

xσ2

) )j > q − s,

(24)

where 0 < λ1 < λ2 < .. < λq < ∞ are non-zero ordered eigenvalues of|η|2|σ|2 H2H

H

2 , det(V) is a q × q vandermonde matrix whose determinant is given by

det(V) = det(λq−ji ) =∏q

l<k(λl − λk), (t)l is the Pochhammer symbol which is givenby (t)l = Γ(t+l)

Γ(t)and Kv(.) is the modified Bessel function of the second kind, , Γ(.) is

the gamma function, sj = j+s−q−1, v = p+ l+r−m−sj, u = p+ l+k−sj, wj =p+ j + l + k − q, and the constant c is given as

c =e−Tr(Λ)

(Γ(t))q, (25)

where Λ = diag(λ1, λ2, ...., λq).Proof : See Appendix A.Now, the c.d.f. of the SNR γ can be obtained as

Fγ(γ) = Fξmax

(γ

aρ

). (26)

Here, the c.d.f. of the SNR γ is not directly dependent on the entries of channelmatrices. It depends on the eigenvalues of |η|

2

σ2 H2H2H

. Also, the infinite summation in(24) is converging rapidly, and can obtain accurate values by truncating the series withsufficient depth. Table I shows the convergence of the infinite summation for the an-tenna configuration (3,2,5) with ρ = 10 dB and b = 1. The scenarios with low Rician

24

Table 1: Convergence of the c.d.f. with no of terms l in the infinite summation forantenna configuration (3,2,5)

l γ=10dB, K=-10dB γ=10dB, K=0dB γ=10dB, K=10dB10 0.665274800319336 0.355952041857909 9.57498590406942×10−31

20 0.665274831431821 0.659020512821794 2.89446439264914×10−22

50 0.665274831431821 0.660140383027912 4.03509585927486×10−8

100 0.665274831431821 0.660140383027912 0.392343964457507150 0.665274831431821 0.660140383027912 0.633807625339485200 0.665274831431821 0.660140383027912 0.633807828259881250 0.665274831431821 0.660140383027912 0.633807828259881

factor requires a smaller number of terms to converge, whereas higher Rician factorscenarios require the larger numbers of terms. We obtained the accuracy up to 15thdecimal point by truncating the infinite summation term l = 20, l = 50 and l = 200for Rician factors K = −10 dB, K = 0 dB and K = 10 dB respectively. This newc.d.f. expression is used to evaluate performance metrics of the system in later sec-tions. Next, the most important and common MIMO configurations are discussed asspecial cases of the general solution (23).

Corollary 1 : The c.d.f. of the ξmax for min(NR, ND) = 1 is given by

Fξmax(x) = 1−∞∑l=0

NS−1∑m=0

m∑r=0

2λle−λxm+ v2

(mr

)are−axKv

(2√

xσ2

)Γ(p)(p)ll!m!σ2(m−r+ v

2 ), (27)

where v = p+ l −m+ r, u = p+ l, λ is the eigenvalue of |η|2

|σ|2 H2HH

2 which is given

by |η|2

|σ|2pq.Proof : The proof is straightforward by substituting q = 1 to (23). This result is usefulto analyze the cases with the single relay antenna or the single destination antenna.

Corollary 2: The c.d.f. of the ξmax for NS = 1 is given by

Fξmax(x)=c(−1)q−1

det(V)

q∑d=1

( ∞∑l=0

λld(σ2p+2lΓ(p+ l)−2e−ax(xσ2)

p+l2 Kp+l

(2√

xσ2

) )(t)ll!σ(2p+2l−q+1)

)Ad,

(28)where Ad is the (d, q)th cofactor of a q × q matrix with (i, j)th entry is given by

(Ad)ij =∞∑l=0

λli(−1)q−NS−j

(t)ll!σ(2p+2l−q+1)

( q−j∑k=0

(q − jk

)akσ2wjΓ(wj)

), (29)

where wj = p+ j + l + k − q.Proof : The proof is straightforward by substituting NS = 1 to (23) and applying theLaplace expansion along the last column of the determinants. These expressions alsoconsist of infinite summations. However, they converge rapidly.

Figure 8 illustrates both analytical and simulation curves of the c.d.f. of the SNRγ for different antenna configurations. These antenna configurations are selected to

25

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CDF

Fγ(γ)

SNR γ

Analytical (3,2,5) K=10 dB

Analytical (3,2,5) K= -10 dB

Analytical (3,2,5) K= 0 dB

Analytical (2,5,1) K=10 dB

Analytical (2,4,6) K=10 dB

Analytical (1,5,2) K=10 dB

Simulation

K increasing

(2,4,6)

(2,5,1)

(1,5,2)

(3,2,5)

Figure 8: c.d.f of the γ for different antenna configurations (NS, NR, ND), and withdifferent values of Rician factor K = −10, 0, 10 dB. ρ = 10 dB, and b = 1. Montecarlo simulation with 106 iterations.

match the casesNS ≥ q, NS < q , q = 1 andNS = 1. The caseNS ≥ q (NS = 3, NR =

2, ND = 5) is considered with different Rician factors K(= |η|2σ2 ) to investigate the

effect of Rician fading. For all other cases, Rician factor is considered to be K = 10dB. Here, the LoS component H2 is generated according to [51] with full rank, andentries are non-identical complex elements with unit magnitude. We also considerρ = 10 dB, and b = 1. Derived closed form results are used to plot the analyticalcurves and those are matched with simulated curves with greater accuracy. The c.d.f.of the SNR move towards to high SNR region when the number of antennas is high.That can be observed with the c.d.f. curves that are obtained for the cases (3,2,5) and(2,4,6). Additionally, channels with high Rician factors seem to have better SNR formost of time than channels with low Rician factor.

3.2.1.2. Low-rank Rician Fading

Here, we consider the scenario where the relay-destination channel undergoes low-rank Rician fading, where LoS component H2 is not full rank. We consider arbitraryrank m (0 ≤ m < q) for H2. Then |η|

2

σ2 H2H2H

has m non-zero eigenvalues.

26

Corollary 3: The c.d.f. of the ξmax is given by,

Fξmax(x)low =clow(−1)NS(q−s)+q(q−1)/2

det(Vm)(λq+1−mλq+2−m...λq)q−mdet(Ilow(x)), (30)

where Ilow(x) is a q × q matrix with (i, j)th element is given by

(Ilow(x))ij =

(−1)q−NS−j

σ(2p−q+2i−1)

( q−j∑k=0

(q−jk

)akσ2wijΓ(wij)

)j ≤ q − s, i ≤ q −m

∞∑l=0

λli(−1)q−NS−j

(t)ll!σ(2p+2l−q+1)

( q−j∑k=0

(q−jk

)akσ2wjΓ(wj)

)j ≤ q − s, i > q −m

(w−j)!σ(2p−q+2i−1)

( sj∑k=0

(sjk

)akσ2uijΓ(uij)−

w−j∑m=0

m+sj∑r=0

2(m+sjr

)ar x

m

m!e−ax(xσ2)

vij2 Kvij

(2√

xσ2

) )j > q − s, i ≤ q −m

∞∑l=0

λli(w−j)!(t)ll!σ(2p+2l−q+1)

( sj∑k=0

(sjk

)akσ2uΓ(u)−

w−j∑m=0

m+sj∑r=0

2(m+sjr

)ar x

m

m!e−ax(xσ2)

v2Kv

(2√

xσ2

) )j > q − s, i > q −m ,

(31)

where 0 < λq−m+1 < λq−m+2 < .. < λq < ∞ are non-zero ordered eigenvalues of|η|2σ2 H2H2

H, Vm is m×m vandermonde matrix with determinant given by det(Vm) =∏q

q−m<l<k(λk−λl), sj = j+s−q−1, v = p+ l+r−m−sj, u = p+ l+k−sj, wj =p+j+l+k−q, uij = p+k−sj+i−1, vij = p+r−m−sj+i−1, wij = p+k+j+i−q−1,and the constant clow is given by

clow =e−Tr(Λm)

Γ(t)m∏q−m

z=1 Γ(p− q + z)∏q−1

r=m Γ(q − r)∏s

i=1 Γ(NS − i+ 1), (32)

where Λm = diag(λq−m+1, λq−m+2, ...., λq).Proof : See Appendix B.

3.2.1.3. i.i.d. rician

Here, we consider the relay-destination channel undergoes i.i.d Rician fading, whereLoS component H2 has one eigenvalue.

27

Corollary 4: The c.d.f. of the ξmax is given by

Fξmax(x)iid =e−λq(−1)NS(q−s)+q(q−1)/2

Γ(t)λq−1q

∏q−1m=1 Γ(p− q +m)Γ(q −m)

∏si=1 Γ(NS − i+ 1)

det(Iiid(x)),

(33)

where Iiid(x) is a q × q matrix with (i, j)th element given by

(Iiid(x))ij =

(−1)q−NS−j

σ(2p−q+2i−1)

( q−j∑k=0

(q−jk

)akσ2wijΓ(wij)

)j ≤ q − s, i ≤ q − 1

∞∑l=0

λli(−1)q−NS−j

(t)ll!σ(2p+2l−q+1)

( q−j∑k=0

(q−jk

)akσ2wjΓ(wj)

)j ≤ q − s, i = q

(w−j)!σ(2p−q+2i−1)

( sj∑k=0

(sjk

)akσ2uijΓ(uij)

−w−j∑m=0

m+sj∑r=0

2(m+sjr

)ar x

m

m!e−ax(xσ2)

vij2 Kvij

(2√

xσ2

) )q − s < j, i ≤ q − 1

∞∑l=0

λli(w−j)!(t)ll!σ(2p+2l−q+1)

( sj∑k=0

(sjk

)akσ2uΓ(u)

−w−j∑m=0

m+sj∑r=0

2(m+sjr

)ar x

m

m!e−ax(xσ2)

v2Kv

(2√

xσ2

) )q − s < j, i = q

(34)

where sj = j+s−q−1, v = p+l+r−m−sj , u = p+l+k−sj , wj = p+j+l+k−q,uij = p+ k − sj + i− 1, vij = p+ r −m− sj + i− 1, wij = p+ k + j + i− q − 1.Proof : See Appendix B.

3.2.1.4. Rayleigh Fading

Here, LoS component H2 becomes zero.Corollary 5: The c.d.f. of the ξmax is given by

Fξmax(x)R =(−1)NS(q−s)∏q

m=1 Γ(p− i+ 1)Γ(q − i+ 1)det(IR(x)), (35)

28

where IR(x) is a q × q matrix with (i, j)th element given by

(I(x)R)ij =

(−1)q−NS−jq−j∑k=0

(q−jk

)akΓ(wij) j ≤ q − s

sj∑k=0

(sjk

)akΓ(uij)−

w−j∑m=0

m+sj∑r=0

2(m+sjr

)ar x

m

m!e−ax(x)

vij2 Kvij (2

√x)

j > q − s,(36)

where sj = s+ j− q− 1, uij = p+ k− sj + i− 1, vij = p+ r−m− sj + i− 1, wij =p+ k + j + i− q − 1.Proof : The proof is similar to i.i.d. Rician scenario with λq goes to zero.

Analitical expressions for the cases of low-rank Rician, and i.i.d. Rician also consistof infinite summations. However, truncating those with sufficient depth gives accuratevalues. These expressions are used to analyze the system in terms of outage probability.During next sections, we consider non-i.i.d. Rician fading as the default Rician fadingscenario for the relay-destination channel. Expansions for all other scenarios can besimilarly done as in the c.d.f.

3.2.2. Probability Density Function

Here, we derive new expression for p.d.f. of the SNR γ. This is useful to discuss theSER and ergodic capacity. The p.d.f. of the SNR γ can be obtained as

fγ(γ) =fξmax

(γaρ

)aρ

, (37)

where following theorem presents the p.d.f. of the ξmax, when the relay-destinationchannel undergoes non-i.i.d. Rician fading.

Theorem 2: The p.d.f. of the maximum eigenvalue ξmax of HH1 HH

2 (aH2HH2 +

IND)−1H2H1 is given by

fξmax(x) =c(−1)NS(q−s)

det(V)∏s

i=1 Γ(NS − i+ 1)

q∑l=q−s+1

det(Il(x)), (38)

where Il(x) is a q × q matrix with (i, j)th element is given by

(Il(x))ij =

(I(x))ij j 6= l

∞∑l=0

w−j∑m=0

m+sj∑r=0

2λli(w−j)!(m+sjr )are−axxm+ v

2

(t)ll!m!σ(2p+2l−q+1−v)

×( (a−

(mx

))Kv

(2√

xσ2

)+√

1xσ2Kv−1

(2√

xσ2

) )j = l,

(39)where v = p+ l + r −m− sj, u = p+ l + k − sj .Proof : See Appendix C.

29

Next, we derive expression for the p.d.f. of ξmax for special cases which can be usefulin many situations.

Corollary 6: The p.d.f. of the ξmax for min(NR, ND) = 1 is given by

fξmax(x) =∞∑l=0

NS−1∑m=0

m∑r=0

2λle−λ−ax(mr

)arx

v2

+m

Γ(p)(p)ll!m!σ2(m−r+ v2 )

×

((a−m

x

)Kv

(2

√x

σ2

)+

√1

xσ2Kv−1

(2

√x

σ2

)), (40)

where v = p+ l + r −m.Proof : The proof is straightforward by substituting q = 1 into (38).

Corollary 7: The p.d.f. of the ξmax for NS = 1 is given by

fξmax(x) =c(−1)q−1

det(V)

q∑d=1

(∞∑l=0

2λlde−ax(xσ2)

p+l2

(t)ll!σ(2p+2l−q+1)

×(aKp+l

(2

√x

σ2

)+

√1

xσ2Kp+l−1

(2

√x

σ2

))Ad,

(41)

Proof : The proof is straightforward by taking first derivative of (28).Here, the expressions (39)-(41) have infinite summations. However, truncating at

finite length gives accurate results.Figure 9 compares the p.d.f. of the SNR γ with different antennas configurations

and Rician factors (K = −10, 0, 10 dB). Simulation curves of the p.d.f. are obtainedusing normalized histogram method. We used 106 random channel realizations and 0.1bin size in the simulation. All other parameters are used similar to the Figure 8. In allcases, the analytical and simulation curves are matched with reasonable accuracy. Thisconfirms our derivations for all scenarios. The p.d.f. move towards high SNR whenthe number of antennas are higher at all nodes. A similar behavior can be observedwith the Rician factor as in the Figure 8.

3.2.3. Moments

Moments of the SNR γ are another useful statistical property that can use to measurethe performance of the system in many aspects. Here, we present expressions for mo-ments of the SNR γ by deriving moments of the ξmax. Deriving an analytical expressionfor the general case is mathematically hard. However, we can derive solutions for thecases min(NR, ND) = 1 and NS = 1.

30

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

PDF

Fγ(γ

)

SNR γ

Analytical (3,2,5) K=10dB

Analytical (3,2,5) K=-10dB

Analytical (3,2,5) K=0dB

Analytical (2,5,1)

Analytical (2,4,6)

Analytical (1,5,2)

Simulation

(2,5,1)

(1,5,2)

K increasing

(3,2,5)

(2,4,6)

Figure 9: p.d.f of the γ for different antenna configurations (NS, NR, ND), and with dif-ferent values of Rician factor K = −10, 0, 10 dB. ρ = 10 dB, and b = 1. Normalizedhistogram with 106 iterations and 0.1 bin size.

Theorem 3 : The nth moment of ξmax for min(NR, ND) = 1 is given by

E{ξnmax} =∞∑l=0

NS−1∑m=0

m∑r=0

λle−λ(mr

)Γ(n+m+ v + 1)

Γ(p)(p)ll!m!an+m−rσ2m−2r

× Γ(n+m+ 1)

(U(n+m+ 1, 1− v, 1

aσ2

)

+U(n+m+ 1, 2− v, 1

aσ2

)(aσ2)(n+m+ v)

−m× U

(m+ n, 1− v, 1

aσ2

)(n+m+ v)(m+ n)

), (42)

where U(., ., .) is the confluent hypergeometric function.Proof : See Appendix D.

Theorem 4 : The nth moment of ξmax for NS = 1 is given by

E{ξnmax} =c(−1)q−1

det(V)

q∑d=1

(∞∑l=0

λldΓ(n+ t+ l + 1)Γ(n+ 1)

(t)ll!σ(−q+1)an

×

(U(n+ 1, 1− t− l, 1

aσ2

)+U(n+ 1, 2− t− l, 1

aσ2

)(aσ2)(n+ t+ l)

))Ad, (43)

31

where Ad is the (d, q)th cofactor of a q × q matrix as given in (29).Proof : See Appendix D.Then, analytical expressions for moments of the SNR γ can be obtained from followingrelationship.

E{γn} = anρnE{ξnmax}. (44)

3.3. Performance Analysis

In this section, we evaluate the performance of the system using statistical propertiesderived in the previous section. The effect of Rician factor, the rank of the Ricianfading channel, and the antenna configurations are investigated by considering threeimportant performance measures, i.e., outage probability, average SER and ergodiccapacity. The diversity order and the relay saturation effects are discussed in terms ofoutage probability. We also compare the system performance with an OSTBC basedAF MIMO dual hop system [25].

3.3.1. Outage Probability

The outage probability is an important quality of service measure, defined as the prob-ability that the instantaneous SNR falls below a pre-defined threshold γth, and can beexpressed as

Pout(γth) = Pr(γ ≤ γth) = Fξmax

(γth

aρ

). (45)

Figure 10 shows the outage probability versus average transmit SNR ρ for differentRician fading environments such as non-i.i.d Rician, low-rank Rician, i.i.d. Rician andRayleigh. Rician factor variations (K = η2

σ2 ) are also considered for all Rician fadingscenarios. Both analytical and simulation curves are obtained for K = −10, 0, 10 dBwith the antenna configuration of (3, 5, 3). We use (45) to plot the analytical curves ofthe outage probability. Simulation curves are obtained by generating entries of H1, H2

as circular symmetric Gaussian variables with unit variance, and the LoS componentis generated according to [51],

H2 =m∑i=1

βia(θr,i)a(θt,i)T (46)

where a(θr,i) is the array response vector and a(θt,i) is the array steering vector for ithdominant path. βi is the complex amplitude of the ith path. The rank of the matrixdepends on the number of dominant paths and for full rank m = q. We also assumeγth = 10 dB and b = ρ. Non-i.i.d. Rician fading (rank 3 on H2) has better outageperformance than the i.i.d. Rician (rank 1) and low-rank Rician (rank 2) scenarios forall Rician factors. The performance difference is higher when the Rician factor is high.However, the outage performance is decreasing with the Rician factor for all cases ofRician fading. This means that the Rayleigh fading has better outage performancethan the rest of the cases, i.e., having a good scattering component is more beneficialin MIMO scenarios.

32

2 4 6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

100

OutageProbabilityP

out

SNR (dB)

Analytical Rank 3, K=10 dBAnalytical Rank 3, K=0 dBAnalytical Rank 3, K=-10 dBAnalytical Rank 2, K=10 dBAnalytical Rank 2, K=0 dBAnalytical Rank 2, K=-10 dBAnalytical Rank 1, K=10 dBAnalytical Rank 1, K=0 dBAnalytical Rank 1, K=-10 dBAnalytical RayleighSimulation

K = 0 dBK = 0 dB

K = 10 dBK = -10 dB

Rank increasing

Figure 10: Illustration on the effect of Rician fading on the outage probability vs av-erage SNR ρ when NS = 3, NR = 5, ND = 3. γth = 10 dB, b = ρ. Monte carlosimulation with 106 iterations.

3.3.2. High SNR Analysis

We consider asymptotic analysis for the outage probability to have further understand-ing on the system performance at the high SNR. Here, we derive diversity order ofthe system for some special cases (min(NR, ND) = 1 and NS = 1) of antenna con-figurations. Analytical derivation of the asymptotic outage probability for the generalantenna configuration is mathematically hard.

Theorem 5 : For min(NR, ND) = 1, asymptotic expression of the outage probabil-ity is given by

Pout(γth) =

∑∞l=0

∑NSr=0

λle−λ(NSr )arΓ(p+l+r−NS)

Γ(p)(p)ll!NS!σ2(NS−r)

(γthaρ

)NS

for NS < p

e−λ ln(aργth

)Γ(p)(NS)!σ2NS

(γthaρ

)NS

for NS = p

e−λΓ(NS−p)Γ(p+1)(NS−1)!σ2p

(γthaρ

)pfor NS > p

(47)

Proof : See Appendix E.

33

Theorem 6 : For NS = 1, asymptotic expression of the outage probability is givenby

Pout(γth) =c det(Φ)

Γ(q + 1) det(V)

(γth

aρ

)q, (48)

where Φ is a q × q matrix with (i, j)th element given by

(Φ)ij =∞∑l=0

λliΓ(p+ l − j)(t)ll!

(1

σ2+ a(p+ l − j)

). (49)

Proof : See Appendix E.

Diversity order for the q = 1 case equals to min(NS, p), andNS = 1 case equals to q.These match full diversity order in such situations. We use the derived analytical resultsto confirm with the numerical simulation. Figure 11 shows the outage probabilitycurves for the special cases q = 1 and NS = 1. Curves for analytical, simulationand asymptotic outage probabilities are obtained for Rician factors K = 0 and 10dB. Similar simulation parameters are used as in the previous cases. Analytical andsimulation curves have an accurate match while the asymptotic curves converge withboth in the high SNR values. Interestingly, when q = 1 the outage performance isimproves with the Rician factor. This means that the Rician fading provides betterperformance in MISO (multiple-input single-output) or SIMO (single-output multiple-input) scenarios of the relay-destination channel. This is opposite to the observationwe had for the cases with q 6= 1.

3.3.3. Effect of the Relay Saturation

Here, we study the relay saturation effect on the outage probability. The fixed-gainAF relay can go into saturation when the amplified signal at the relay exceed the max-imum available transmit power of the relay. This is a common situation in slightlyfaded source-relay scenarios. However, this occurs only seldomly in severely fadedenvironments. The following requirement is needed to avoid such situation. i.e.,

a‖H1wt‖2ρ+ a‖nR‖2 ≤ Pmax, (50)

where Pmax is the maximum transmit power at the relay. In the previous sections, weselected a = b

NR(1+ρ)and b is the average transmit power at the relay. The constraint

(50) may not be satisfied for all channel realizations. Then the relay can go into sat-uration and there can be a loss in the performance. To evaluate the loss, we selectPmax = ρ, which guarantees the AF relay to have the similar transmit power as thesource. Now, a is selected according to [6],

a =

ρ

NR(1+ρ)γR ≤ NR(1 + ρ)

ργR

γR > NR(1 + ρ) ,

(51)

34

5 10 15 20 2510

−3

10−2

10−1

100

OutageProbabilityP

out

SNR (dB)

Analytical (2,5,1) K=10 dB

Analytical (2,5,1) K= 0 dB

Asymptotic (2,5,1) K=10 dB

Asymptotic (2,5,1) K=0 dB

Analytical (1,5,2) K=10 dB

Analytical (1,5,2) K= 0 dB

Asymptotic (1,5,2) K=10 dB

Asymptotic (1,5,2) K=0 dB

Simulation

K increasing

(2,5,1)

K increasing

(1,5,2)

Figure 11: Asymptotic analysis of outage probability with the Rician factor for antennaconfigurations (2,5,1) and ((1,5,2). γth = 10 dB, b = ρ. Monte carlo simulation with106 iterations.

where γR is the instantaneous input power at the relay node. i.e.,

γR = ‖H1wt‖2ρ+ ‖nR‖2. (52)

Here, a is selected such a way that the transmit power at the relay is limited to ρ.Signals with low receive power are subject to fixed-gain according to the case 1 in(51). Signals with high received power are subject to a variable gain. The variablegain is adjusted according to instantaneous received power as case 2 in (51). Figure12 illustrates the outage performance for the antenna configuration (2,2,3) with differ-ent Rician factors and threshold SNR γth = 10 dB. Simulations are carried out usingMonte Carlo method with 106 iterations. Simulation results reveal that outage perfor-mance is reduced due to the relay saturation. The relay saturation loss is higher at thelow SNR regions. A similar loss can be observed for all the Rician factors. However,the saturation effect is negligible at the high SNR regions. Additionally, the saturationloss can be minimized in all regions by selecting a higher Pmax.

35

0 2 4 6 8 10 12 14 16 18 2010

−4

10−3

10−2

10−1

100

OutageProbabilityP

out

SNR (dB)

Analitical K=10 dB No saturation

Simulation K=10 dB No saturation

Simulation K=10 dB With saturation

Analitical K=0 dB No saturation

Simulation K=0 dB No saturation

Simulation K=0 dB With saturation

Analitical K=-10 dB No saturation

Simulation K=-10 dB No saturation

Simulation K=-10 dB With saturation

K increasing

Figure 12: Effect on the relay saturation on Outage probability with the Rician factorfor antenna configurations (2,2,3). γth = 10 dB. Monte Carlo simulation with 106

iterations.

3.3.4. Symbol Error Rate

Average SER is another important performance metric to characterize the performanceof a communication system. The common formula to find the average SER for manymodulation schemes is given by

SER = Eγ{a1Q(√

2a2γ)}, (53)

where Q(.) is the Gaussian Q-function, and a1, a2 are modulation specific constants(for BPSK case a1 = 1, a2 = 1) [52]. Alternatively, we can find the average SERusing the c.d.f. of SNR as [53]

SER =a1√a2

2√π

∫ ∞0

e−a2γ

√γFγ(γ)dγ. (54)

Solving the integral for general c.d.f. expression (23) is quite complicated. How-ever, mathematical tool such as Matlab and Mathematica can be easily used to findthe analytical results. Here, we only solve the integration for two special casesmin(NR, NS) = 1 and NS = 1.

36

Theorem 7 : For min(NR, NS) = 1, the average SER is given by

SER=a1

2−∞∑l=0

NS−1∑m=0

m∑r=0

λle−λ(mr

)ara1√a2

Γ(p)(p)ll!m!σ2(m−r) (55)

×

(Γ(m+ v + 1

2)Γ(m+ 1

2)U(m+ 1

2, 1− v, 1

( 1ρ

+a2)σ2

)2√π(aρ)m(1

ρ+ a2)m+ 1

2

).

Proof : Proof is straightforward by substituting (27) into (54), and use the integralrelationship in (131) .

Theorem 8 : For NS = 1, the average SER is given by

SER =c(−1)q−1a1

√a2

2√π det(V)

q∑d=1

( ∞∑l=0

λld(t)ll!σ(−q+1)

(√ π

a2

Γ(p+ l)

−(Γ(p+ l + 1

2)Γ(1

2)U(1

2, 1− p− l, 1

(a+a2aρ)σ2 )

(1ρ

+ a2)12

))Ad.

(56)

Proof : Proof is straight forward and omitted.Figure 13 shows average BER versus average transmit SNR ρ for BPSK modulationscheme. Similar simulation parameters are assumed as in the previous cases with b =ρ. The average BER performance is considered with different Rician factors for threeantenna configurations. Analytical and simulation curves match with a reasonableaccuracy which confirms the validity of derived results. The average BER performanceis better in low Rician factors for (1,5,2) and (3,2,5) antenna configurations. This issimilar to the behavior we observed in outage performances for the case with q 6= 1. Inantenna configuration (2,5,1), the average BER performance increases with the Ricianfactor. These observations confirm that having good scattering component of the relay-destination channel benefits the MIMO beamforming system performance. MISO orSIMO scenarios are benefited by good LoS component. In Figure 14, an OSTBC basedAF MIMO system [25] is also considered comparing system performance. AlamoutiOSTBC is used with code rate one to make a fair comparison. For all Rician factors,the optimal single stream beamforming based MIMO dual hop system performs betterthan OSTBC based AF MIMO system.

3.3.5. Ergodic Capacity

The ergodic capacity C(ρ), is evaluated using

C(ρ) =1

2E{log2(1 + γ)}

=1

2

∫ ∞0

log2(1 + γ)fγ(γ)dγ =1

2

∫ ∞0

log2(1 + aρx)fξmax(x)dx.(57)

Solving the integral (43) for ergodic capacity is complicated. Mathematical tools canbe used to find the numerical results. However, we can use moments of ξmax to obtain

37

0 2 4 6 8 10 12 14 1610

−4

10−3

10−2

10−1

100

BER

SNR (dB)

Analytical K=10 dB

Analytical K=0 dB

Analytical K=-10 dB

Simulation

K Increasing

(2,5,1)

K Increasing

K Increasing

(3,2,5)

(1,5,2)

Figure 13: Effect of Rician factor and antenna configuration on the BER vs averageSNR. BPSK modulation is used with b = ρ. Monte carlo simulation with 106 iterations.

0 2 4 6 8 10 12 14 1610

−6

10−5

10−4

10−3

10−2

10−1

BER

SNR (dB)

Analytical K=10 dB

Analytical K=0 dB

Analytical K=-10 dB

Simulation

(2,4,2)OSTBC

K increasing

K increasing

(2,4,2)Beamforming

Figure 14: Difference between OSTBC and beamforming based AF MIMO system.Alamouti OSTBC, BPSK modulation and b = ρ is used with antenna configuration(2,4,2). Monte carlo simulation with 106 iterations.

38

an upper bound and an approximation of the ergodic capacity. The upper bound Cup(ρ)can be obtained by using the Jensen’s inequality. Thus,

Cup(ρ) =1

2log2(1 + aρE{ξmax}). (58)

The second order approximation Capp(ρ) for the ergodic capacity can be obtained ac-cording to [54]

Capp(ρ) =1

2log2(1 + aρE{ξmax})−

(aρ)2(E{ξ2max} − E{ξmax}2)

4 ln 2(1 + aρE{ξmax})2. (59)

Theorems 3 and 4 can be used to evaluate Cup(ρ) and Capp(ρ) analytically for the casesq = 1 and NS = 1. To get more insight on ergodic capacity behavior, we derive anexpression for the asymptotic ergodic capacity considering the special case q = 1.

Theorem 9 : For q = 1, the analytical expression for asymptotic ergodic capacitybecomes

Chigh(ρ) =1

2

(ρ |dB

3dB− L∞

)+ o(1), (60)

where L∞ is given by

L∞ = log2

1

a+∞∑0

λle−λG1,33,2

[aσ2 |1−p−l,1,11,0

]ln 2 Γ(p)l!(p)l

−∞∑0

λle−λ(ψ(p+ l) + ln σ2)

l! ln 2− 1

ln 2ψ(NS), (61)

and Gm,np,,q

(x |a1,...,ap

b1,...,bq

)is the Meijer’s G function according to [55 Eq. 8.4.6.5], ψ(.) is

the Eular’s psi function [56 Eq. 8.360].Proof : See Appendix FFigure 15 illustrates the ergodic capacity versus average transmit SNR ρ. Analyticalresults are obtained with mathematical tools, and similar simulation parameters areassumed as in the previous cases. Ergodic capacity increases with SNR ρ, and similarbehavior can be observed for different Rician factors for the cases q = 1 and q 6= 1.

Figure 16 illustrates Cup(ρ), Capp(ρ) and Chigh(ρ) versus the average transmit SNR ρ.The results are obtained for two special cases NS = 1 (antenna configuration (1,4,2)and (1,4,4)) and q = 1 (antenna configuration (2,4,1) and (4,4,1)). The curves forCapp(ρ) and the simulated ergodic capacity are matched with a high accuracy. This re-flects the fact that the Capp(ρ) provides a good approximation for the ergodic capacity.Also, the upper bound of the ergodic capacity Cup(ρ) always has some margin withthe simulated curves. For the antenna configurations (2,4,1) and (4,4,1), the asymp-totic curves Chigh(ρ) are obtained with SNR ρ. These asymptotic curves provide goodestimates of the ergodic capacity when the SNR is higher than 15 dB.

39

0 2 4 6 8 10 12 14 160

0.5

1

1.5

2

2.5

3

3.5

Ergodic

Capacity

SNR (dB)

Analytical K=10 dB

Analytical K=0 dB

Analytical K=-10 dB

Simulation

K increasing

(3,2,5)

K increasing

K increasing

(2,3,5)

K increasing(2,5,1)

(1,5,2)

Figure 15: Effect of Rician factor and antenna configuration on the ergodic capacity vsaverage SNR. b = ρ. Monte carlo simulation with 106 iterations.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

4

Ergodic

Capacity

SNR (dB)

Monte Carlo Simulation

Capacity Upper Bound

Capacity Aproximation

Asymptotic Capacity

(1,4,2)

(2,4,1)

(4,4,1)

(1,4,4)

Figure 16: Ergodic capacity behavior vs SNR for different antenna configurations(1,4,2), (1,4,4), (2,4,1) and (4,4,1) with Rician factor K = 10 dB and b = ρ. Montecarlo simulation with 106 iterations.

40

3.4. Conclusion

We analyze an optimal single stream beamforming for a fixed-gain dual-hop AFMIMO relay system with Rayleigh fading in the source-relay channel and Rician fad-ing in the relay-destination channel. The optimal transmit and receive beamformingvectors are designed to maximize the instantaneous SNR at the destination. New ex-pressions for the c.d.f, p.d.f, and moments of the instantaneous SNR at the destinationare derived using finite dimensional random matrix theory. These statistical proper-ties are used to analyze the system performance with outage probability, average SERand ergodic capacity. New analytical results for outage probability and average SERare derived for special cases of antenna configurations. The system performances arediscussed with Rician factor, rank of the Rician channel, and with different antennaconfigurations. Numerical analysis suggest that having good LoS component increasesthe performance of the system for MISO and SIMO scenarios of the relay-destinationchannel. Having good scattering component increases the performance for MIMOcases. Additionally, results reveal that the average BER performance of the optimalbeamforming is superior to the OSTBC based AF MIMO system.

41

4.RELAYSELECTIONWITHOSTBC

4.1.SystemandChannelModels

WeconsideracooperativecommunicationsystemmodelasinFigure17,whereasource-to-destinationcommunicationispossiblewithasinglerelayoutofatotalofNrelaynodes.Allrelaysperformnon-coherentAFrelaying.Thesourceandthedesti-nationnodeshaveNS,andNDantennas,respectively.RelaynodenhasRn

NS

HS1

H1D

ND

Source

DestinationRelay 1

Relay n

Relay N

HSn

HSN

HnD

HND

antennas.

Figure17:AFMIMOsystemwithNrelays

Duringthefirsttimeslot,thesourcetransmitstorelaynodes.Allrelaynodesam-plifytheirreceivedsignal,subjecttoanaveragepowerconstraint,andtransmittothedestinationinthesecondtimeslot.Generalnon-coherentAFrelayingassumptionismadeas[23,25],whereinstantaneousCSIisavailableonlyatthedestination.Wealsoassumethatthereisnodirectlinkbetweenthesourceanddestination.Thesource-relaychannelsareassumedtoundergoRayleighfadingandrelay-destinationchannelsaresubjecttonon-i.i.d.Ricianfading.Non-i.i.dRicianisreferredasindependentnonidenticalRicianfading,wheremeanisnon-identical.Letusdenotethechannelmatricesbetweenthesource-to-nthrelayandnthrelay-to-destinationasHSn∈C

Rn×NSandHnD∈CND×Rn,respectively.EntriesofHSn

areassumedtobe∼CN(0,1). WemodelHnDwithaspecularcomponent(LoS)

HnDandvariablecomponent(scattered)HnDasHnD=ηnHnD+σnHnD[25].HnDconsistsofcomplexelementshavingunitmagnitude,entriesofHnDareassumedtobe∼CN(0,1).Wealsoconsiderbothηn,σntosatisfyη

2n+σ

2n=1inourdiscussion.

SourceemploysOSTBCencoding[25,57,58],andM i.i.d.symbolss1,s2,..,sMaremappedtoaroworthogonalmatrixX∈CNS×NT,whereentriesofXobtainedbylinearcombinationsofs1,s2,..,sM andtheirconjugates.NTisthenumberofsymbolperiodsusedtosendacodeword.Therefore,thecoderateisR=M/NT.TheexactconstructionmethodonthespecificOSTBCisgivenin[47].Denotexkbethetransmittedsignalduringthekthsymbolperiod(i.e.,thekthcolumnofX).Here,we

42

take X = (x1, ...,xNT ) where xi ∈ CNS×1 and ρ as the total transmit power across allNS source antennas E{‖xk‖2} = ρ. During the kth symbol period, the received signalat the destination through the nth relay, ynk is given by

ynk = anHnDHSnxk + anHnDnnk + nk, k = 1, .., NT (62)

where relay gain an =√

bnRn(1+ρ)

and bn is a positive real number, nnk ∼ CN (0, IRn)

and nk ∼ CN (0, IND) are noise vectors at the nth relay and destination nodes, respec-tively. Prior to decoding symbols, we apply the noise whitening filter (a2

nHnDHHnD +

IND)−12 to ynk. Finally, after linear OSTBC processing, we can obtain the instanta-

neous SNR through the n th relay as

γn=αγa2nTr

(HH

SnHHnD(a2

nHnDHHnD+IND)−1HnDHSn

), (63)

where α = 1/RNS, γ is the average transmit SNR which equals ρ due to unit noisevariance. Since instantaneous SNR during the kth symbol period is independent ofsymbol period, we can denote the instantaneous SNR through nth relay as γn.

Here, we assume CSI is not available at the relay nodes. However, CSI is avail-able at the destination node. Channel products between source-to-relay and relay-to-destination, i.e. a2

nHnDHSn (∀n), and relay-to-destination channels HnD (∀n) are suf-ficient to find these instantaneous SNR values. Suitable channel estimation techniquescan be used to find these[59].

We investigate two relay selection schemes at the destination. In the first relay selec-tion scheme, the instantaneous SNR received at the destination node is used to selectthe best relay, where the selection can be done in an optimal manner. The secondscheme is based on the best relay-to-destination channel, which is sub-optimal andCSI of relay-to-destination channels HnD (∀n) are sufficient to select the relay node.The second scheme can be simple to implement.

In next sections, we discuss statistical behavior of these schemes, and focus on theperformance difference. We also use the following notations: pn = max(ND, Rn),qn = min(ND, Rn).

4.2. Statistics of the Relay Selection Schemes

Here, we statistically characterize the instantaneous SNR at the destination consideringboth relay selection methods.

4.2.1. Optimal Relay Selection Scheme

The selection method is following a SNR policy in the sense that the selected relayachieves a maximum instantaneous SNR. Relay nodes may consist of different gains,antenna configurations and amplifying powers. Additionally, the LoS components be-tween relay nodes and destination nodes are different from each other. This selectionis optimum from the SNR perspective for such scenarios.

A relay node is selected such that the instantaneous SNR is maximized, i.e,

n∗opt = arg maxn∈{1,...,N}

γn. (64)

43

We denote γmax as the instantaneous SNR at the selected relay node n∗opt. The statisticalparameters of γmax are obtained in terms of the c.d.f. and p.d.f. These parameters areused in the sequel to investigate the performance of this scheme.

Since γmax is selected as the maximum of the N independent SNRs, The c.d.f. ofγmax can be written in terms of Fγn(γ) as

Fγmax(γ) = Pr(γmax ≤ γ) = Pr(max(γ1, γ2, .., γN) ≤ γ)

= Pr((γ1 ≤ γ) ∩ ..(γN ≤ γ)) =N∏n=1

Fγn(γ),(65)

where Fγn(γ) is the c.d.f. of the SNR received through relay n. Closed form solutionof Fγn(γ) for the case qn = 1 is given by [25]

Fγn(γn) = 1−2e−(λn+ γnαγ )

Γ(p)

NS−1∑k=0

k∑i=0

∞∑l=0

(ki

)γknλ

ln

kαkγka2in σ

2in

×

(√γn

a2nσ

2nαγ

)pn+l−iKpn+l−i

(2√

γna2nσ

2nαγ

)(pn)lΓ(l + 1)

, (66)

where λn is non-zero eigenvalue of |ηn|2

|σn|2 HnDHnDH

, Kv(z) is the modified Besselfunction of the second kind and of order v, Γ(.) is the gamma function, (pn)l is thePochhammer symbol which is given by (pn)l = Γ(pn+l)

Γ(pn).

Therefore, we can find the closed form solution of the Fγmax(γ) by substituting (66)into (65).

Next, we find the the p.d.f. of γmax in terms of Fγn(γ) and fγn(γ) as

fγmax(γ) =d

dγ

N∏n=1

Fγn(γ)

=N∑t=1

fγt(γ)N∏

n=1,n 6=t

Fγn(γ),

(67)

where fγn(γ) is the p.d.f. of the SNR received through relay n. The exact probabilitydistribution of the SNR γn for qn = 1 is given by[25]

fγn(γn) =2γNS−1

n e(−γnαγ )e−λn

Γ(NS)Γ(p)(αγσ2na

2n)NS

NS∑j=0

∞∑l=0

(NSj

)a2jn σ

2jn λ

ln

(p)lΓ(l + 1)

×(

γna2nσ

2nαγ

)rn/2Krn

(2

√γn

a2nσ

2nαγ

), (68)

where rn = pn + l+ j−NS. Then we can find the closed form solution of the fγmax(γ)by substituting (66) and (68) into (67) .

4.2.2. Sub-optimal Relay Selection Scheme

Here, we consider less complicated relay selection scheme which uses instantaneouschannel HnD between the relay and destination as the selection criteria. This is possible

44

since we assume CSI of HnD (∀n) at the destination node. Then, the relay node, whichhas maximum Tr(HnDHH

nD) is selected in this method, i.e.,

n∗sub = arg maxn∈{1,...,N}

Tr(HnDHHnD). (69)

This selection method is easy to implement with partial CSI knowledge. We denoteγsub as the instantaneous SNR at the destination when using the selected relay noden∗sub.

Derivations of the c.d.f. and p.d.f. for general antenna configurations is mathemat-ically hard. Here, we consider qn = 1 with similar parameters of pn, an, ηn, and σn(∀n) to derive the exact expressions of the c.d.f. and p.d.f.

Theorem 1 : The c.d.f. of SNR γsub can be obtain as,

Fγsub(γ) =1−NS−1∑k=0

k∑i=0

∞∑l=0

2N(ki

)γke−λ−

γαγ λl

Γ(p)(p)ll!k!(αγ)k(aσ)2i

N−1∑m=0

(N − 1

m

)(−1)me−mλ

×

(∞∑l1=0

∞∑l2=0

...

∞∑lm=0

l1+p−1∑r1=0

l1+p−1∑r2=0

...

lm+p−1∑rm=0

λl1+l2+...+lm

l1!l2!...lm!r1!r2!...rm!(70)

×(

γ

(m+ 1)αγ(aσ)2

)0.5(p+l−i+∑mj=1 rj)

K(p+l−i+∑mj=1 rj)

(2

√(m+ 1)γ

αγ(aσ)2

)).

Proof : See Appendix G.

Theorem 2 : The p.d.f. of SNR γsub can be obtained as,

fγsub(γ) =

NS−1∑k=0

k∑i=0

∞∑l=0

2N(ki

)γke−λ−

γαγ λl

Γ(p)(p)ll!k!(αγ)k(aσ)2i

N−1∑m=0

(N − 1

m

)(−1)me−mλ

×

(∞∑l1=0

∞∑l2=0

...∞∑

lm=0

l1+p−1∑r1=0

l1+p−1∑r2=0

...

lm+p−1∑rm=0

λl1+l2+...+lm(

γ(m+1)αγ(aσ)2

)0.5(p+l−i+∑mj=1 rj)

l1!l2!...lm!r1!r2!...rm!

×

((1

αγ− k

x

)K(p+l−i+

∑mj=1 rj)

(2

√(m+ 1)γ

αγ(aσ)2

)

+

√(m+ 1)

αγ(aσ)2γK(p+l−i−1+

∑mj=1 rj)

(2

√(m+ 1)γ

αγ(aσ)2

))). (71)

Proof : See Appendix G.Next, we use these statistical results to study the system performance.

4.3. Performance Analysis

We use derived closed form expressions of the c.d.f. and p.d.f. for both relay selectionschemes to investigate the performance of the system with different system perfor-mance metrics. We consider the outage probability, symbol error rate and ergodic

45

capacity for this purpose. The outage probability, Pout, defined as the probability thatthe instantaneous end-to-end SNR falls below a certain threshold SNR, γth, is given by

Pout = Pr{γ ≤ γth} =

∫ γth

0

fγ(γ)dγ = Fγ(γth). (72)

The exact expressions obtain for the c.d.f. in the both selection schemes can be usedto obtain exact outage probability by substituting desired threshold SNR γth.

In addition to the outage probability, average symbol error rate (PS) is another im-portant metric for characterizing the performance of a communication system, whichis defined as

PS = E{Q(√γ)} =

∫ ∞0

Q(√γ)fγ(γ)dγ. (73)

where Q(.) is the Gaussian Q-function.Finally, the ergodic capacity, C, can be evaluated using

C =1

2E{log(1 + γ)} =

1

2

∫ ∞0

log(1 + γ)fγ(γ)dγ, (74)

where the 12

factor accounts for orthogonal transmission over the two links. TheseSER and ergodic capacity can be evaluated analytically using mathematical tools suchas Mathematica and Matlab.

4.4. Numerical Examples

Performance of relay selection schemes are investigated by changing system parame-ters such as, the size of the relay pool, and Rician factors (Kn = η2

n/σ2n) of the relay-

destination paths. Simulation curves are obtained by generating entries of HSn, HnD

as circular symmetric Gaussian variables with unit variance. LoS components are gen-erated according to [51], where entries are non-identical complex elements with a unitmagnitude. The analytical expressions in (66), (68), (70), and (71) have infinite sum-mations, but those converge rapidly and we can obtain sufficiently accurate results forthe outage probability, ergodic capacity and symbol error rate by truncating these sum-mations. Here, infinite summations are truncated after the first 20 terms. We considerthe Alamouti coding as OSTBC scheme, which gives code rate R = 1. The averageSNR is considered as γ = 10 dB, and relay gain parameter bn = 1 for all relays.

Figure 18 shows the outage probability versus threshold SNR for different relay poolsizes. In a given pool size, a similar antenna configuration of (NS = 2, Rn = 4, ND =1) is assumed. All relay-destination channels behave independently with the Ricianfactor of Kn = 0 dB (∀n). As expected, both analytical and simulated curves arematched with reasonable accuracy. The outage probability decreases when the relaypool size increases. Also, the optimal relay selection approach provides significantperformance improvement compare to the suboptimal approach. Therefore, we canuse a higher number of relay nodes to assist source to destination and take advantageof diversity gain.

The outage probability variation with threshold SNR for different Rician factors areshown in Figure 19. We investigate the cases of Kn = −10, 0, 10 dB with the relay

46

−5 0 5 10

10−1

100

OutageProbabilityP

out

Threshold SNR γth (dB)

Analytical N=2,Optimal scheme

Analytical N=3,Optimal scheme

Analytical N=4,Optimal scheme

Analytical N=2,Suboptimal

Analytical N=3,Suboptimal

Analytical N=4,Suboptimal

Simulation

N increasing

N increasing

Figure 18: Illustration on the effect of the relay pool size N on the outage probabilityfor antenna configuration (2,4,1) with Kn = 0 dB (∀n).

pool size N = 2, and antenna configuration (2, 4, 1). In the optimal relay selec-tion method, the outage probability improves with Rician factors. Therefore, havinghigher LoS components, improves the performance of such a relay selection strategy.A similar behavior can be observed when increasing the size of the relay pool. On theother hand, the sub-optimal relay selection approach does not provide significant per-formance gains for the higher Rician factor, and the outage probability curves behavesimilarly.

The average bit error rate for binary phase shift keying with average SNR is investi-gated in Figure 20. The BER for the optimal relay selection scheme is considered forthree Rician factors with N = 2, where the performance improves with the Rician fac-tor similar to the observation in the outage scenario. Moreover, the BER performancewith respect to the relay pool size N is obtained for Rician factor K = 0 dB. When Nincreases, there is a significant performance improvement in the optimal relay selec-tion scheme. In sub-optimal scheme, the performance improvement is comparativelylesser than in the optimal case.

Finally, Figure 21 illustrates the ergodic capacity gap between two proposed schemes.Here, the capacity gap studied by changing the relay pool sizeN and Rician factorKn.It can be observed that the ergodic capacity gap increases significantly with the sizeof relay pool and Rician factor. This verifies that the optimal scheme provides betterresults in both high and low SNR.

47

−5 0 5 10

10−1

100

OutageProbabilityP

out

Threshold SNR γth (dB)

K=10 dB,Optimal scheme

K=0 dB,Optimal scheme

K=-10 dB,Optimal scheme

K=10 dB,Suboptimal

K=0 dB,Suboptimal

K=-10 dB,Suboptimal

Simulation

K increasing

Figure 19: Illustration on the effect of the Rician factor K on the outage probabilityfor antenna configuration (2,4,1) with N = 2.

0 2 4 6 8 10 1210

−3

10−2

10−1

Bit

ErrorRate

PS

Average SNR (dB)

K=0 dB,Optimal scheme

K=10 dB,Optimal scheme

K=-10 dB,Optimal scheme

N=2,K=0 dB

N=3,K=0 dB

N=4,K=0 dB

Simulation

N increasing

K increasing

N increasing

Sub-optimal scheme

Optimal scheme

Figure 20: BER for BPSK vs average SNR γ. Curves are obtained for the differentRician factors and relay pool sizes N with antenna configuration (2,4,1).

48

0 2 4 6 8 10 120.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Cop−

Csub

Average SNR (dB)

N=2,K=0 dBN=2,K=10 dBN=2,K=-10 dBN=3,K=0 dBN=4,K=0 dBSimulation

N increasing K increasing

Figure 21: Ergodic capacity gap vs average SNR γ. Curves are obtained for the differ-ent relay pool sizes N with antenna configuration (2,4,1) and K = 0 dB.

4.5. Conclusion

We propose possible relay selection schemes for non-coherent AF MIMO OSTBC sys-tem under asymmetric fading channels. Proposed relay selection schemes are based onmaximum instantaneous SNR at the destination and the best channel between relaysand the destination. CSI is assumed as unknown at source and relay nodes, and we de-rive expressions for the c.d.f. and p.d.f. of instantaneous SNR for both schemes. Theseexpressions are used to analyse the system performance with the outage probability,BER, and ergodic capacity. The maximum SNR based relay selection scheme (optimalscheme) improves performances significantly with the size of the relay pool. Havinghigher LoS component between the relay and destination channel also provides bettersystem performance. The relay selection based on the best relay-destination channel(sub-optimal) always gives lesser performance compared first method. However, wecan still have higher system performance with the larger relay pool size.

49

5. EFFECT OF CCI AND FEEDBACK DELAY

5.1. System model

We consider dual-hop AF relaying with multiple antennas at the source and destination.The number of antennas at the source, relay, and destination nodes are NS , 1 andND, respectively. The source-relay channel undergoes Rayleigh fading, and the relay-destination channel undergoes independent non identical Rician fading, where meanis non-identical (non-i.i.d Rician). Relay operates on half-duplexing, and the sourceto destination transmission completes in two time slots. We assume that there is afeedback delay of the relay-source channel, and CCI is experienced at the relay node,not at the destination.

In the first time slot, the source transmits x(t) ∈ C beamformed by transmit beam-forming vector wt(t) ∈ CNS×1. wt(t) is evaluated using outdated CSI information.We express received signal at the relay r(t) ∈ C as

r(t) =√PSh†1(t)wt(t)x(t) +

NI∑i=1

√Piαir(t)xi(t), (75)

where PS is the transmit power at the source, h1 ∈ CNS×1 is the relay-to-source chan-nel. Entries of h1 are assumed to be ∼ CN (0,1), Pi is the transmit power of the ithinterference channel, αir(t) is the interfering channel coefficients ∼ CN (0, 1) of theith interference channel, and xi(t) is the ith interference symbol. Noise at the relaynode is neglected.

The TB vector is obtained as wt(t) = h1(t − τ)/‖h1(t − τ)‖F , where τ is thefeedback delay. Here, the h1(t− τ) is obtained as in [50],

h1(t− τ) =√ρh1(t) +

√1− ρn1(t), (76)

where n1 ∼ CN (0, INS) is the noise vector at the source, and ρ is the correlationcoefficient between h1(t − τ) and h1(t), which is obtained by Jakes’s autocorrelationmodel [60].

In the second time slot, the relay forwards the amplified version of the receivedsignal. Here, the relay gain a is considered as

a =

(PS | h†1(t)wt(t) |2 +

NI∑i=1

Pi | αir(t) |2)−1/2

. (77)

The destination node performs MRC, and the received signal at the destination y(t) ∈C is given by,

y(t) =√PRaw

†r(t)h2(t)r(t) + w†r(t)nD, (78)

where PR is the transmit power at the relay, wr(t) ∈ CND×1 is the MRC vector whichis defined as wr(t) = h2(t)/‖h2(t)‖F , nD ∼ CN (0, IND) is the noise vector at thedestination, and h2 ∈ CND×1 is the relay-destination channel, which is

h2 = ηh2 + σh2, (79)

50

where h2 ∈ CND×1 is the LoS component, and h2 ∈ CND×1 is the scattered compo-nent. h2 consist of non identical complex elements having unit magnitude, entries ofh2 are assumed to be ∼ CN (0,1). We also consider both η, σ to satisfy η2 + σ2 = 1in the analysis.

Here, the end-to-end SINR at the destination can be expressed as

γ =γ1γ2

γ1 + (γ2 + 1)γ3

, (80)

where γ1 = PS | h†1(t)wt(t) |2, γ2 = PR‖h2(t)‖F , γ3 =∑NI

i=1 Pi | αir(t) |2.

5.2. Statistics of the SNR

We derive c.d.f. of the SINR γ which is important to obtain exact results to outageprobability, BER and ergodic capacity.Theorem 1: The c.d.f. of the end-to-end SINR Fγ(x) is given by,

Fγ(x) = 1−NS−1∑m=0

NS−m−1∑k=0

∞∑l=0

η(A1)∑i=1

φi(A1)∑j=1

k∑n=0

l+ND−1∑r=0

(k

n

)(l +ND − 1

r

)βmk!

×(

1

γ1

)ke−λ− x

γ2 λlχi,j(A1)

Γ(ND)l!(ND)lγl+ND2

µ−j〈i〉(j − 1)!

(1 + x)k+r−n

2 xl+ND−1+n+k−r2

(γ2

γ1

)n+r+1−k2

×Γ(n+ r + j + 1)Γ(k + j)√1

γ1γ2

ex(x+1)γ1γ2

/2

(1

µ〈i〉+ xγ1

)(1

µ〈i〉+

x

γ1

)−0.5(k+n+r+2j)

× W−0.5(k+n+r+2j),0.5(n+r+1−k)

(x(x+ 1)

γ1γ2

/(1

µ〈i〉+

x

γ1

)).

(81)

Here, Γ(.) is the Gamma function, γ1 = PS , βm =(NS−1m

)ρ2(NS−m−1)(1− ρ2)m, γ2 =

PR(1+K)

, K = |η|2|σ|2 is the Rician factor, λ is the eigenvalue of |η|

2

|σ|2 h2h†2 which is given

by |η|2|σ|2ND, (ND)l is the Pochhammer symbol which is given by (ND)l = Γ(ND+l)

Γ(ND),

A1 = diag(µ1, µ2, ..., µNI ) and µ〈i〉 = Pi is the average power of the ith interference,η(A1) is the number of distinct diagonal elements of A1, µ〈1〉 > µ〈2〉 > ... > µ〈η(A1)〉are the distinct diagonal elements in decreasing order, φi(A1) is the multiplicity ofµ〈i〉 and χi,j(A1) is the (i, j) th characteristics coefficient of A1[61], Wa,b(.) is theWhittaker function.

Proof : See Appendix H

The c.d.f. of the SINR depends on the correlation coefficient ρ, Rician factor K,thenumber of antennas, and transmit powers at source and relay. More importantly, thec.d.f. is independent of the exact values of h2. It depends only on the eigenvalue λ of|η|2|σ|2 h2h

†2. The infinite summation gives accurate answers with sufficient truncation of

the series. Next, we derive a upper bounded c.d.f. of the SINR, which can be used toapproximate BER.

51

Theorem 2: The c.d.f. of the upper bounded end-to-end SINR is given by ,

Fγu(γ)=1−NS−1∑m=0

NS−m−1∑k=0

η(A1)∑i=1

φi(A1)∑j=1

∞∑l=0

l+ND−1∑r=0

βmχi,j(A1)Γ(k + j)λle−λ− γ

γ2 γk+r

k!(j − 1)!µj〈i〉l!r!γk1γ

r2

(γγ1

+ 1µ〈i〉

)k+j.

(82)

Proof : See Appendix H

5.3. Performance analysis

5.3.1. Outage Probability

Outage probability is a good measure to analyze the system performance, which isdefined as the probability that the end-to-end SINR γ falls below a predefined thresholdγth.

Pout(γth) = Pr(γ ≤ γth) = Fγ(γth). (83)

By substituting threshold SINR γth into (81) , we can obtain the closed form expres-sion for the outage probability.

5.3.2. Symbol Error Rate

Symbol error rate is another important metric to characterize the performance of acommunication system. The common formula to find the average SER for many mod-ulation schemes is given by,

PE = Eγ{aQ(√

2bγ)}, (84)

where Q(.) is the Gaussian Q-function, and a, b are modulation specific constant (ForBPSK case a = 1, b = 1) [52]. Alternatively, we can find the average SER using thec.d.f. of SINR as [53].

PE =a√b

2√π

∫ ∞0

e−bγ√γFγ(γ)dγ. (85)

Finding the closed form solution for exact SER is mathematically hard. Therefore, weuse numerical integration tools in mathematica to obtain analytical results. However,we can obtain a closed form expression of BER for upper bounded SINR as

PEu =a

2− a√b

2√π

NS−1∑m=0

NS−m−1∑k=0

η(A1)∑i=1

φi(A1)∑j=1

∞∑l=0

l+ND−1∑r=0

×βmχi,j(A1)Γ(k + j)λle−λΓ

(k + r + 1

2

)γr+ 1

21

k!(j − 1)!l!r!γr2µr+ 1

2

〈i〉

× U(k + r +

1

2, r +

3

2− j, γ1

µ〈i〉

(b+

1

γ2

)).

(86)

52

The proof is possible applying c.d.f. Fγu(γ) in (85), solving the integral by using [56Eq. 3.383.6].

5.3.3. Ergodic Capacity

Finally, the ergodic capacity, C, is evaluated using

C =1

2ln 2

∫ ∞0

1− Fγ(γ)

1 + γdγ. (87)

To obtain the exact analytical results, we used numerical integration in Mathematica.We can derive following upper and lower bounds of ergodic capacity to get furtherunderstanding of the system.

5.3.3.1. Upper bound

Upper bound of the ergodic capacity is given as

Cu =1

2ln 2

NS−1∑m=0

NS−m−1∑k=0

η(A1)∑i=1

φi(A1)∑j=1

βmχi,j(A1)Γ(k + j)µk〈i〉

k!(j − 1)!γk1B(k + 1, j)

× 2F1

(j + k, k + 1; j + k + 1; 1−

µ〈i〉γ1

)+∞∑l=0

λle−λG1,33,2

[γ2|

1−ND−l,1,11,0

]Γ(ND)l!(ND)l2ln 2

− 1

2log2(1 + eα1−α3 + eα2),

(88)

where αi(= E(ln γi) for i = 1, 2, 3) are given in (169) - (171), B(., .) denotes theBeta function [56 Eq. 8.380] and 2F1(., .; .; .) denotes the Gauss hypergeometricfunction [56 Eq. 9.100], Gm,n

p,,q

(x |a1,...,ap

b1,...,bq

)is the Meijers G function according to [55

Eq. 8.4.6.5], ψ(.) is the Eular psi function [56 Eq. 8.360].Proof : See Appendix I

53

5.3.3.2. Lower bound

Lower bound of the ergodic capacity Cl can be obtained as

Cl =1

2ln 2

NS−1∑m=0

NS−m−1∑k=0

η(A1)∑i=1

φi(A1)∑j=1

βmχi,j(A1)Γ(k + j)µk〈i〉

k!(j − 1)!γk1B(k + 1, j)

× 2F1

(j + k, k + 1; j + k + 1; 1−

µ〈i〉γ1

)+

1

2ln 2

η(A1)∑i=1

φi(A1)∑j=1

χi,j(A1)

×(ψ(j) + ln µ〈i〉) +∞∑l=0

λle−λ

Γ(ND)l!(ND)l2ln 2G1,3

3,2

[γ2|

1−ND−l,1,11,0

]−1

2log2

(γ1

NS−1∑m=0

βm(NS −m) +

(1 +

∞∑l=0

λle−λγ2 (ND + l)

l!

)η(A1)∑i=1

×φi(A1)∑j=1

χi,j(A1)jµ〈i〉

).

(89)

Proof : See Appendix I

5.4. Numerical Examples

Here, we use the analytical and simulation results to investigate the system behaviorto find the effect of Rician factor K, correlation coefficient ρ, number of antennas, andtransmit powers.

In Figure 22, both analytical and simulation curves for the c.d.f. of γ are shown fordifferent Rician factors (K = −10, 0, 10 dB). Analytical curves are obtained using(81), where the infinite series is truncated after first 50 terms. Here, we consider thecorrelation coefficient ρ = 0.5, PS = PR = 10 dB, and three interference channels(NI = 3) with Pi = 0 dB. The analytical and simulation curves have an accuratematch, and the Rician factor has a positive impact on the end-to-end SINR.

Figure 23 shows the outage probability versus average transmit power PS for differ-ent antenna configurations, Rician factors, and correlation coefficients. We considerthreshold SINR γth = 10 dB, PR = 10 dB, and three interference channels (NI = 3)with Pi = 0 dB. From the outage curves that are obtained for different ρ, it can beseen that outage performance improves with the value of ρ. The outage performanceimproves also with the number of transmit and receive antennas. The Rician fadinghas a constructive effect on the SINR, high Rician factor provide better outage per-formance. Additionally, we can observe that the outage curves are converged at highaverage transmit power PS . The curves are started to converge regardless the value ofρ. This is due to, as γ1 →∞, the end-to-end SINR γ ' γ2.

Figure 24 illustrates the BER versus transmit power PS for different parametersas in the earlier case. We consider transmitting power at the relay PR = PS , andNI = 3 with Pi = 0 dB. Analytical curves for exact BER (PE) are obtained using(85) applying numerical integration tools in mathematica. Analytical curves for BER

54

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CDF

Fγ(γ

)

SNR γ (dB)

Analytical K=0 dB

Analytical K=10 dB

Analytical K=−10 dB

Simulation

K increasing

Figure 22: c.d.f. of γ for the antenna configuration NS = 2, ND = 2 with differentvalues of Rician factor K = 0,−10, 10 dB and correlation coefficient ρ = 0.5 andPS = PR = 10 dB, Pi = 0 dB.

0 2 4 6 8 10 12 14 16 18 2010

−2

10−1

100

OutageProbabilityPout

PS (dB)

Analytical ρ = 0.5(black)

Analytical ρ = 0.8(red)Analytical K= 0 dBAnalytical K=10 dBAnalytical K=-10 dBSimulation

NS = ND = 2

NS = ND = 4

K increasing

ρ increasing K=0 dB

K=-10 dB

K=10 dB

Figure 23: Outage probability of γ for the γth = 10 dB with different values of Ricianfactor K = 0,−10, 10 dB and PR = 10 dB, Pi = 0 dB.

55

0 1 2 3 4 5 6 7 8 9 10

10−2

10−1

BER

PS (dB)

Analytical ρ = 0.8(black)

Analytical ρ = 0.5(red)Analytical K= 0 dBAnalytical K=10 dBAnalytical K=-10 dBSimulation

NS = ND = 4

NS = ND = 2,PEu

PEu

PE

K increasing

ρ increasing

Figure 24: BER versus the transmit power PS with different values of Rician factorK = 0,−10, 10 dB and PR = PS, Pir = 0 dB.

with upper bounded SINR (PEu) are obtained from (86). From the BER curves thatare obtained for NS = ND = 4, we can observe that PEu always lies below PEHowever, PEu provides alternative measure for BER. The BER performance improveswith Rician factor, correlation coefficient, and number of antennas similar to Figure23. Finally, Figure 25 illustrates the ergodic capacity versus transmit power (PS),and similar simulation parameters are used as in the Figure 24. Analytical curvesfor the upper and lower bounded ergodic capacity are obtained from (88) and (89),respectively. Exact ergodic capacity curves are obtained using (87) applying numericalintegration tools in mathematica. For NS = ND = 4, it can be observed that theexact and bounded results matched with reasonable accuracy, and this provide a goodestimate for the performance range of the system. The ergodic capacity improves withthe Rician factor. Higher correlation coefficient ρ also provides better ergodic capacity.However, significant improvements can be achieved through the increasing number ofantennas at the nodes.

5.5. Conclusion

We analyze multi-antenna AF relaying system that employs TB/MRC over asymmetricfading channels in the presence of CCI and feedback delay. Exact closed form expres-sions for the c.d.f. of SINR and c.d.f. of the upper bounded SINR are derived. Thesestatistical expressions are used to obtain the closed form expressions for the outageprobability, BER of the upper bounded SINR, lower and upper bounds of the ergodic

56

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

Ergodic

Capacity

PS (dB)

ExactUpper boundLower boundK=0 dBK=10 dBK=-10 dBSimulation

K increasing

NS = ND = 2

NS = ND = 4ρ = 0.8

ρ = 0.8

ρ = 0.5

Figure 25: Ergodic capacity versus the transmit power PS with different values ofRician factor K = 0,−10, 10 dB and PR = PS, Pir = 0dB.

capacity. Numerical results for these performance metrics confirm these derivations,where we have an exact match between simulation and analytical results. The perfor-mance of the system is dependent on the feedback delay, CCI, number of antennas,and Rician factor. Higher correlation between estimated channel and actual channelprovides better performance in all cases. Having good LoS component improves theperformance of the system, and it is significant when the number of antennas small.However, the number of antennas at nodes improves the performance compared torest.

57

6. SUMMARY AND CONCLUSIONS

In this thesis, the performance of MIMO dual hop AF relay networks over asymmet-ric fading channels was studied. We considered three different system models withdifferent MIMO techniques and practical aspects.

We analyzed the performance of an optimal single stream beamforming scheme fora MIMO relay network with dual-hop fixed-gain AF relaying. The source-relay andrelay-destination channels undergo Rayleigh and Rician fading, respectively. DifferentRician fading scenarios are considered for relay destination channel, depending on therank of the Rician channel matrix. The channel state information is only available atthe destination, and the destination computes the optimal transmit and receive beam-forming vectors to maximize the SNR. The optimal transmit beamforming vector issent back to the transmitter via a dedicated feedback link. We derive new analyticalexpressions for the c.d.f., p.d.f., and moments to statistically characterize the prop-erties of the instantaneous SNR. These statistical properties are used to analyze thesystem performance in terms of the outage probability, BER, and the ergodic capac-ity. The performance analysis investigates the effects of the Rician factor, rank of theLoS component, and number of antennas at the nodes on the system performance.Outage probability of the system investigated for the rank of the LoS component andhigh SNR behavior. Also, relay saturation effect studied by using outage probability.Symbol error rate of the system investigated for different MIMO antenna configura-tions and compared performance with OSTBC based AF MIMO system. The resultsreveal that the optimal single stream beamforming system provides better performancethan an OSTBC. Ergodic capacity of the system investigated for different MIMO an-tenna configurations and Rician factors. Also, approximated results and high SNRresults obtained for ergodic capacity. Numerical analysis suggest that having goodLoS component increases the performance of the system for MISO and SIMO sce-narios of relay-destination channel. Having good scattering component increases theperformance for MIMO cases. Also, the number of antennas at the source, relay andthe destination nodes improve the system performance. Diversity gain of the systemimproves with the number of antennas at each of the node.

Then, we investigated optimal relay selection schemes for OSTBC MIMO systemwith non-coherent AF relays, where channel state information is not available at thesource and relays. The source-relay and relay-destination channels undergo Rayleighand Rician fading, respectively. Two possible relay selection schemes are proposed,and both are statistically characterized by deriving an exact closed form expression forthe c.d.f., p.d.f. of the instantaneous SNR at the destination. In the first relay selec-tion method, maximizing instantaneous SNR at the destination is considered to selectthe best relay (optimal scheme). In the second scheme, maximizing relay-destinationchannel is considered (suboptimal scheme). For the optimal relay selection, the des-tination node decide best relay comparing instantaneous SNR of through each relay.This requires CSI knowledge of all the fading channels at the destination. However,suboptimal selection scheme requires only relay-destination CSI knowledge at the des-tination. The derived statistical results are used to analyze the performance of the sys-tem with outage probability, average bit error rate and ergodic capacity. Then, compareboth relay selection schemes with respect to the relay pool size and Rician factor. Theoptimal relay selection scheme improves performances significantly with the size of

58

the relay pool. Having higher LoS component between the relay and destination chan-nel also provides better system performance. The sub-optimal relay selection schemealways gives lesser performance compared first method. However, we can still havehigher system performance with the larger relay pool size.

Finally, we investigated the performance of dual hop multiple antenna AF relay-ing system of TB and MRC considering feedback delay and CCI at the relay node.Source-relay and relay-destination channels experience Rayleigh and Rician fadingrespectively. TB vector is calculated using outdated channel state information dueto the feedback delay from relay-to-source, and the relay node experience CCI dueto frequency reuse in the cellular network. We derive new closed form expressionsfor the exact cumulative distribution function of the end-to-end signal-to-interferenceplus-noise ratio to statistically characterized the system. We also derived closed formexpression for the outage probability, bit error rate and ergodic capacity. The systemperformance is investigated using the derived performance metrics to analyze the ef-fect of Rician fading, CCI, feedback delay and number of antennas. Higher correlationbetween estimated channel and actual channel provides better performance in all cases.Having good LoS component improves the performance of the system, and it is signif-icant when the number of antennas small. However, the number of antennas at nodesimproves the performance compared to rest.

Studies of this thesis can be further extend for different system models. We canextend this for multi hop relay networks, multiple users systems, full duplex systems,different relaying protocol such as DF. Also, we can extend this work analysing difer-ent asymetric fading scenarioes, considering different fading models.

59

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64

8. APPENDICES

A. Derivation of the c.d.f. of the ξmax

We use eigenvalue decomposition of HH2

(aH2H

H2 + IND

)−1 H2 as

HH2

(aH2HH

2 + IND)−1 H2 = UHΞU, (90)

where U is unitary matrix, Ξ = diag(β1, ..., βq) with βi = σ2φiaσ2φi+1

, in which φi is the

eigenvalues of 1σ2 H2HH

2 . Then, we can express HH1 HH

2

(aH2HH

2 + IND)−1 H2H1 as

HH1 HH

2

(aH2HH

2 + IND)−1 H2H1 = H

H

1 ΞH1, (91)

where H1 = UH1 ∼ CN q,NS(0, Iq ⊗ INS).Maximum eigenvalue ξmax of HH

1 HH2

(aH2HH

2 + IND)−1 H2H1 has the similar statis-

tical properties as the maximum eigenvalue of HH

1 ΞH1. Therefore, we use maximumeigenvalue of H

H

1 ΞH1, and consider two cases of NS ≥ q and NS < q to derive c.d.f.of the ξmax.

A.1. Case 1: NS ≥ q

The conditional c.d.f. of the maximum eigenvalue ξmax on Ξ is given by [62]

Fξmax(x | Ξ) =det(Ψ1(x))

det(V1)

q∏i=1

Γ(NS − i+ 1)

, (92)

where V1 is a q × q matrix, with the determinant of

det(V1) =

(q∏i=1

βNSi

) ∏1<l<k<q

(1

βk− 1

βl

)=

(q∏i=1

βNSi

) ∏1<l<k<q

(βl − βkβkβl

),

(93)and Ψ1(x) is a q × q matrix with entries given by

(Ψ1(x))ij = βNS−i+1j γ

(NS − i+ 1,

x

βj

), (94)

where γ(n, x) is the lower incomplete gamma function.We also know that the joint eigenvalue distribution of 1

σ2 H2HH2 is given by [15]

fφ1,..,φq(φ1, .., φq) =c

det(V)det( 0F1(t;φiλj)) det(W)

q∏k=1

φp−qk e−φk , (95)

where 0 < φ1 < φ2 < ... < φq < ∞, 0F1(·; ·) is the generalized hypergeometricfunction, 0 < λ1 < λ2 < .. < λq <∞ are non-zero ordered eigenvalues of |η|

2

σ2 H2H2H

,

65

V is a q × q vandermonde matrix with det(V) = det(λq−ji ) =∏q

l<k(λl − λk), W isq × q vandermonde matrix with its determinant given by det(W) = det(φq−ji ) =∏q

l<k(φl − φk), and

c =e−Tr(Λ)

(Γ(t))q, (96)

where Λ = diag(λ1, λ2, ...., λq).A variable transformation φi = βi

σ2(1−aβi) gives the distribution of β1, β2, .., βq as

fβ(β1, ..., βq) = fφ

(β1

σ2(1− aβ1), ....,

βqσ2(1− aβq)

)| J |, (97)

where | J |=∏q

i=11

σ2(1−aβi)2 . The joint distribution given in (97) is used to obtain theunconditional c.d.f. as

Fξmax(x) =c

det(V)∏q

i=1 Γ(NS − i+ 1)det(I(x)), (98)

where

det(I(x)) =

∫..

∫D

det(Ψ1(x))

det(V1)det

(0F1

(t;

βiλjσ2(1− aβi)

))(99)

× det

((βi

σ2(1− aβi)

)q−j) q∏k=1

(βk

σ2(1−aβk)

)p−qe− βkσ2(1−aβk)

σ2(1− aβk)2dβ1..dβq,

where D = {0 ≤ β1, .. ≤ βq <1a}.

We can further simplify (99) into

det(I(x)) =

∫..

∫D

det(Ψ1(x) det

(0F1

(t;

βiλjσ2(1− aβi)

))(100)

×∏

1<l<k<q

βkβlσ2(1− aβk)(1− aβl)

q∏k=1

βp−q−NSk e− βkσ2(1−aβk)

σ2(p−q+1)(1− aβk)p−q+2dβ1..dβq.

By [63 Corollary 2], we can solve the multiple integral and obtain

det(I(x)) = (101)

det

(∫ 1a

0

γ

(NS − j + 1,

x

y

)0F1

(t;

yλiσ2(1− ay)

)yp−je

− y

σ2(1−ay)

σ(2p−q+1)(1− ay)p+1dy

).

Next, we use series expansion of 0F1

(t; yλiσ2(1−ay)

)=∑∞

l=0

(yλi

σ2(1−ay)

)l(t)ll!

, and use the

property of lower incomplete gamma function γ(1+n, x) = n!(1− e−x

(∑nm=0

xm

m!

))[56 8.352]. Equation (101) becomes

det(I(x)) = det

(∞∑l=0

λli(NS − j)!(t)ll!σ(2p+2l−q+1)

(∫ 1a

0

yp−j+le− y

σ2(1−ay)

(1− ay)p+l+1dy

−NS−j∑m=0

xm

m!

∫ 1a

0

yp−j+l−me−xy− y

σ2(1−ay)

(1− ay)p+l+1dy))

.

(102)

66

A variable transformation is required as z = y1−ay , and (102) becomes

det(I(x)) = det

(∞∑l=0

λli(NS − j)!(t)ll!σ(2p+2l−q+1)

(∫ ∞0

zp−j+le−zσ2 (1 + az)j−1dz

−NS−j∑m=0

xm

m!e−ax

∫ ∞0

zp−j+l−me−xz− zσ2 (1 + az)j+m−1dz

)).

(103)

Finally, using [56 3.381.4 and 3.471.9]

det(I(x)) = det

(∞∑l=0

λli(NS − j)!(t)ll!σ(2p+2l−q+1)

(j−1∑k=0

(j − 1

k

)akσ2uΓ(u)

−NS−j∑m=0

m+j−1∑r=0

2

(m+ j − 1

r

)arxm

m!e−ax(xσ2)

v2Kv

(2

√x

σ2

))),

(104)

where v = p+ l + r −m+ 1− j, u = p+ l + k + 1− j.

A.2. Case 2: NS < q

Here, the conditional c.d.f. of the maximum eigenvalue ξmax on Ξ is given by [62]

Fξmax(x | Ξ) =(−1)NS(q−NS)det(Ψ2(x))

det(V1)

NS∏i=1

Γ(NS − i+ 1)

, (105)

where elements of Ψ2(x) is given by

(Ψ2(x))ij =

(−1βj

)q−NS−ifor i ≤ q −NS

βq−i+1j γ

(q − i+ 1, x

βj

)for i > q −NS,

(106)

and det(V1) is defined as in the case 1. Unconditional c.d.f. can be written as

Fξmax(x) =c(−1)NS(q−NS)

det(V)∏NS

i=1 Γ(NS − i+ 1)det I(x), (107)

where

det(I(x)) =

∫..

∫D

det(Ψ2(x)) det

(0F1

(t;

βiλjσ2(1− aβi)

))

×q∏

k=1

βp−NS−1k e

− βkσ2(1−aβk)

σ(2p−q+1)(1− aβk)p+1dβ1..dβq. (108)

Multiple integrals can be reduced by applying [63 Corollary 2] to (108)

det(I(x)) = det

(∫ 1a

0

(Ψ2(x))ji 0F1

(t;

yλiσ2(1− ay)

)yp−NS−1e

− y

σ2(1−ay)

σ(2p−q+1)(1− ay)p+1dy

).

(109)

67

A.3. For j ≤ q −NS in (106)

det(I(x)) = det

(∫ 1a

0

(−1)q−NS−j 0F1

(t;

yλiσ2(1− ay)

)yp+j−q−1e

− y

σ2(1−ay)

σ(2p−q+1)(1− ay)p+1dy

).

(110)A variable change is considered as z = y

1−ay , and using [56 3.381.4] we can obtain

det(I(x)) = det

(∞∑l=0

λli(−1)q−NS−j

(t)ll!σ(2p+2l−q+1)

(q−j∑k=0

(q − jk

)akσ2wjΓ(wj)

)), (111)

where wj = p+ j + l + k − q.

A.4. For j > q −NS in (106)

The result is similar to case 1 (104).

det(I(x)) = det

(∞∑l=0

λli(q − j)!(t)ll!σ(2p+2l−q+1)

( sj∑k=0

(sjk

)akσ2uΓ(u)

−q−j∑m=0

m+sj∑r=0

2

(m+ sjr

)arxm

m!e−ax(xσ2)

v2Kv

(2

√x

σ2

))),

(112)

where sj = NS+j−q−1, u = p+k+l−sj, v = p+r+l−m−sj . All these scenariosare considered together, and the closed form solution for Fξmax(x) can be mentioned asin Theorem 1.

B. Derivation of the c.d.f. of ξmax for the Low-rank Rician, i.i.d. Rician

B.1. Low-rank Rician

Here, we derive the c.d.f. for low-rank Rician fading using the c.d.f. of non.i.i.d. Ricianfading scenario. |η|2

σ2 H2H2H

have only m positive eigenvalues, where 0 < m < q.Eigenvalues λ1, λ2, .., λq−m are considered as zero, and 0 < λq−m+1 < λq−m+2 < .. <

λq < ∞ are non-zero ordered eigenvalues of |η|2

σ2 H2H2H

. The c.d.f. can be obtainedby taking the limit of (23)

Fξmax(x)low = limλ1,λ2,..,λq−m→0

Fξmax(x), (113)

and this gives 0/0 for the limit. Therefore, we use l’Hopital rule to evaluate the limit.This can be done by taking (i − 1)th derivative with respect to λi for elements in ithcolumn/ row of I(x) and c ,V as in [15]. Then the limit is evaluated as λi goes to zerofor i = 1, .., q −m.

68

B.2. Case 1: NS ≥ q

Here, we use intermediate result (101) to apply l’Hopital rule. This gives

limλi→0

di−1

dλi−1i

(I(x))i,j = limλi→0

(∫ 1a

0

Γ(t)γ(NS − j + 1, x

y

)Γ(p−q+i)

×0F1

(t+ i− 1; yλi

σ2(1−ay)

)yp−j+i−1e

− y

σ2(1−ay)

σ(2p−q+2i−1)(1− ay)p+idy

),

=Γ(t)

Γ(p−q+i)

(∫ 1a

0

γ(NS − j + 1, x

y

)yp−j+i−1e

− y

σ2(1−ay)

σ(2p−q+2i−1)(1− ay)p+idy

).

(114)

B.3. Case 2: NS < q

Here, we have two separate cases j ≤ q −Ns and j > q −NS .

• For j ≤ q −Ns we use intermediate result in (111) to apply l’Hopital rule. Thisgives

limλi→0

di−1

dλi−1i

(I(x))i,j = limλi→0

(∫ 1a

0

Γ(t)(−1)q−NS−j

Γ(p−q+i)

×0F1

(t+ i− 1; yλi

σ2(1−ay)

)yp+j−q+i−2e

− y

σ2(1−ay)

σ(2p−q+2i−1)(1− ay)p+idy

)

=Γ(t)

Γ(p−q+i)

(∫ 1a

0

(−1)q−NS−jyp+j−q+i−2e

− y

σ2(1−ay)

σ(2p−q+2i−1)(1− ay)p+idy

).

(115)

• For j > q −NS , the solution is similar to (114).

Exact expressions for the integrals in (114) and (115) are evaluated using the variabletransformation z = y

1−ay and with [56 3.381.4 and 3.471.9].In both of these cases, det(V) becomes

det(V) = (−1)q(q−1)/2(λq+1−mλq+2−m...λq)q−m

q−1∏i=m

Γ(q − i) det(Vm). (116)

where Vm is m×m vandermonde matrix with its determinant given by

det(Vm) =

q∏q−m<l<k

(λk − λl). (117)

69

Also, the constant c now becomes clow = e−Tr(Λm)

Γ(t)q.

After some algebraic manipulations we can obtain the result in corollary 3. Here, wedenote the resultant determinant as det(Ilow(x)). Entries of 1st to (q −m)th rows aregiven by solutions obtained for integrals in (114) and (115), and entries of (q−m+1)th

to qth rows are same as in (24) .

B.4. i.i.d Rician

In the i.i.d. Rician scenario, entries of H2 are identical. Therefore, only one eigenvalueis non-zero in |η|

2

σ2 H2H2H

. We consider λq as the non-zero eigenvalue. Thus taking thelimit of c.d.f. in non-i.i.d. Rician we have,

Fξmax(x)iid = limλ1,λ2,..,λq−1→0

Fξmax(x). (118)

Here, we can obtain the solution by applying m = 1. Resultant determinant is denotedas det(Iiid(s)). The column entries of det(Iiid(s)) are the same as the entries of thedet(Ilow(s)) with m = 1. The det(V) becomes

det(V) = (−1)q(q−1)/2λq−1q

q−1∏i=1

Γ(q − i). (119)

This can be further simplified to obtain (33).

C. Derivation of the p.d.f. of the ξmax

We know by definition p.d.f. can be obtained as fξmax(x) = ddxFξmax(x), and this gives

fξmax(x) =c(−1)NS(q−s)

det(V)∏s

i=1 Γ(NS − i+ 1)

d

dxdet(I(x)). (120)

ddx

det(I(x)) can be obtained as

d

dxdet I(x)) =

q∑l=1

det(Il(x)), (121)

where Il(x) is q × q matrix with entries given by

(Il(x))ij =

(I(x))ij for j 6= l

ddx

(I(x))ij for j = l,

(122)

For the case j ≤ q − s, ddx

(I(x))ij is equal to zero. Therefore, for 1 ≤ l ≤ q − s,det(Il(x)) is equal to zero. However, when j > q − s, we can obtain the derivative as

d

dx(Il(x))ij =

∞∑l=0

w−j∑m=0

m+sj∑r=0

2λli(w − j)!(m+sjr

)are−axxm+ v

2

(t)ll!m!σ(2p+2l−q+1−v)

×

((a−

(mx

))Kv

(2

√x

σ2

)+

√1

xσ2Kv−1

(2

√x

σ2

)). (123)

70

Here, we used following property of the BesselK function.

d

dxxvKv(x) = −xvKv−1(x) (124)

Finally, the expression for fξmax(x) is mentioned as in the theorem 2 by combiningabove results.

C.1. Validity of the p.d.f. expression for q = 1

To give more insight, we prove the validity of the p.d.f. expression for q = 1 (40),i.e., proving the integral from 0 → ∞ is equal to unity and p.d.f. is a non-negativefunction. Integral can be simplified as∫ ∞

0

fξmax(x)dx =

∫ ∞0

∞∑l=0

NS−1∑m=0

m∑r=0

2λle−λ−ax(mr

)arx

v2

+m

Γ(p)(p)ll!m!σ2(m−r+ v2 )

×

((a−m

x

)Kv

(2

√x

σ2

)+

√1

xσ2Kv−1

(2

√x

σ2

))dx

=∞∑l=0

NS−1∑m=0

m∑r=0

2λle−λ(mr

)ar

Γ(p)(p)ll!m!σ2(m−r+ v2 )

[− e−axx

v2

+mKv

(2

√x

σ2

)]∞0.

(125)

By using properties of BesselK function,

[e−axx

v2

+mKv

(2

√x

σ2

)]∞0

=

−1

2Γ(v)σv for m = 0

0 for m 6= 0.

(126)

Then the integral simplifies into∫ ∞0

fξmax(x)dx =∞∑l=0

2λle−λ

Γ(p)(p)ll!σ2( v2 )

1

2Γ(v)σv

=∞∑l=0

λle−λΓ(p+ l)

Γ(p)(p)ll!= e−λ

∞∑l=0

λl

l!= 1. (127)

This satisfies the first condition.To prove p.d.f. is a non-negative function, we simplify the nested summation term

in (40). Then the p.d.f. is given by

fξmax(x) =∞∑l=0

NS∑r=0

2λle−λ−ax(NSr

)arxNS−1

Γ(p)(p)ll!(NS − 1)!σ2(NS−r)

( xσ2

)v/2Kv

(√2x

σ2

), (128)

where v = p + l + r − NS . For every x ∈ [0,∞], product of exponential function,BesselK function and polynomial component in (128) provide positive values. There-fore, this is non-negative, and the p.d.f. expression is valid.

For general case, the proof is mathematically complex. However, simulation curvesof the p.d.f. have an exact match with analytical results. This justifies the validity ofthe analytical p.d.f. expression.

71

D. Derivation of Moments of the ξmax

In general, the nth moment can be obtained as

E{ξnmax} =

∫ ∞0

xnfξmax(x)dx. (129)

D.1. Case min(NR, ND) = 1

Here, we use p.d.f. given in Corollary 6 (40), and substitute that to (129). This gives

E{ξnmax} =

∫ ∞0

∞∑l=0

NS−1∑m=0

m∑r=0

2λle−λ−ax(mr

)arx

v2

+n+m

Γ(p)(p)ll!m!σ2(m−r+ v2 )

((a−m

x

)Kv

(2

√x

σ2

)

+

√1

xσ2Kv−1

(2

√x

σ2

))dx. (130)

By using below relationship we can solve the above integral to get the result in (42).∫ ∞0

xµe−mxKv

(2√βx)dx =

Γ(µ+ v2

+ 1)Γ(µ− v2

+ 1)

2mµ− v2

+1

× β−v2 U

(µ− v

2+ 1, 1− v, β

m

)(131)

D.2. Case NS = 1

Here, we use Corollary 7 in (41). This gives

E{ξnmax} =

∫ ∞0

c(−1)q−1

det(V)

q∑d=1

Ad

(∞∑l=0

2λlde−axσp+l

(t)ll!σ(2p+2l−q+1)

(ax

2n+p+l2 Kp+l

(2

√x

σ2

)+ x

2n+p+l−12

√1

σ2Kp+l−1

(2

√x

σ2

))dx (132)

This also can be simplified by using (131) to get the result in (43).

E. Derivation of the Asymptotic Expansion of the Outage Probability

Near zero behaviour of c.d.f. Fξmax(x) is required to obtain the asymptotic expansionfor the outage probability.

E.1. Asymptotic expansion for q = 1

Here, the near zero behaviour of the conditional c.d.f. Fξmax(x | Ξ) can be obtainedfrom (92) as,

Fξmax(x | Ξ) ≈ e−xβ

xNS

βNSNS!. (133)

72

The above conditional c.d.f. is used with (97) (for q = 1) to find the unconditionalc.d.f. of ξmax as

Fξmax(x) ≈∫ 1

a

0

e−xβ

xNS

βNSNS!fφ

(β

σ2(1− aβ)

)1

σ2(1− aβ)2dβ. (134)

After expanding generalized hypergeometric function with infinite series expansion,

Fξmax(x) ≈∞∑l=0

λle−λxNS

Γ(p)(p)ll!NS!σ(2p+2l)

∫ 1a

0

βp−1+l−NSe− xβ− β

σ2(1−aβ)

(1− aβ)p+l+1dβ. (135)

Next, a variable transformation is required as y = β1−aβ . After some algebraic manip-

ulations and using [56 Eq. 3.471.9 ], we can obtain

Fξmax(x) ≈∞∑l=0

NS∑r=0

2λle−λ−ax(NSr

)arxNS

Γ(p)(p)ll!NS!σ2(NS−r)

( xσ2

)v/2Kv

(√x

σ2

), (136)

where v = p+ l+ r−NS . The near zero behaviour of the BesselK function is knownas

Kv(x) ≈

−γc − log x

2for v = 0

12(| v | −1)!

(x2

)−|v| for v ∈ Z and v 6= 0,

(137)

where γc is the Euler-Mascheroni constant. From (136) and (137), we can obtain theminimum exponent of the x and non-zero coefficient. This depends on the p S NS ,and asymptotic expansions of outage probability can be obtained as in (47).

E.2. Asymptotic expansion for NS = 1

ForNS = 1, the maximum eigenvalue of HH

1 ΞH1 is equivalent to the maximum eigen-value of h

H

1 Ξh1, where h1 ∈ CN q×1 and Ξ is given as in Appendix A. Now, the p.d.f.fξmax(x | Ξ) (p.d.f. of maximum eigenvalue ξmax conditioned on Ξ) is the sum of qindependent chi-square random variables. Near zero behavior of this p.d.f. can beobtained as

fξmax(x | Ξ) ≈ 1

Γ(q)

q∏i=1

β−1i xq−1. (138)

Now, the conditional c.d.f. is Fξmax(x | Ξ) ≈ 1Γ(q+1)

∏qi=1 β

−1i xq, and this is used with

(97) to find unconditional c.d.f. of ξmax

Fξmax(x) ≈ cxq

Γ(q + 1) det(V)

∫..

∫D

det

(0F1

(t;

βiλjσ2(1− aβi)

))

× det

((βi

σ2(1− aβi)

)q−j) q∏k=1

(βk

σ2(1−aβk)

)p−qe− βkσ2(1−aβk)

βkσ2(1− aβk)2dβ1..dβq.

(139)

After applying [63 Corollary 2], and performing some manipulations using [56 3.351.3],we can obtain the asymptotic expression of the outage probability as in (48).

73

F. Derivation of the Asymtotic Ergodic Capacity

In the high SNR region, the ergodic capacity can be characterized using high SNRslope S∞ and power offset L∞ as in [64],

Chigh(ρ) = S∞

(ρ |dB

3dB− L∞

)+ o(1), (140)

where S∞ = limρ→∞C(ρ)

log2(ρ)and L∞ = limρ→∞

(log2(ρ) − C(ρ)

S∞

). We can easily

derive that S∞ is equal to 12, and power offset L∞ becomes,

L∞ = log2(1

a)− E{log2(ξmax)}. (141)

For q = 1, the maximum eigenvalue ξmax can alternatively expressed as [29],

ξmax =hH1 h1hH2 h2

ahH2 h2 + 1, (142)

where h1 ∈ CNS×1 and h2 ∈ Cp×1. Therefore,

E{log2(ξmax)} =E{log2(hH1 h1)}+ E{log2(hH2 h2)} − E{log2(ahH2 h2 + 1)}, (143)

The p.d.f. of hH1 h1 is given by,

fγ1(y) =yNS−1

Γ(NS)e−y. (144)

From (95), the p.d.f. of hH2 h2 is given as,

fγ2(y) =e−λyp−1σ2p

Γ(p)det(

0F1

(t;yλ

σ2

))e−

y

σ2 . (145)

Next, we use (144) and (145) and with [56 4.352.1]. This gives

E{log2 hH1 h1} =1

ln 2ψ(NS), (146)

E{log2 hH2 h2} =∞∑0

λle−λ(ψ(p+ l) + lnσ2)

l! ln 2. (147)

Using the properties of Meijer’sG function ln(1+x) = G1,22,2

[x |1,11,0

]and [56 7.813.1],

we have

E{log2 ahH2 h2 + 1} =∞∑0

λle−λG1,33,2

[aσ2 |1−p−l,1,11,0

]ln 2 Γ(p)l!(p)l

. (148)

This completes the derivation of L∞. Therefore, the asymptotic ergodic capacity canbe expressed as in (61).

74

G. Derivation of the c.d.f. and p.d.f. for sub-optimal relay selection scheme

G.1. c.d.f.

We know Tr(HnDHHnD) =

∑qni=1 σ

2nφni, where φni is nonzero ordered eigenvalues (0 <

φn1 < φn2 < ... < ∞) of 1σ2nHnDHH

nD. Here, we consider qn = 1, pn = p, ηn = η andσn = σ, hence denote φni = φn. The PDF of φn can be written as [25]

fφn(φn) =e−λ

Γ(p)0F1(p;φnλ) φp−1

n e−φn . (149)

where λ = |η|2p|σ|2 , 0F1(p;φnλ) is the generalized hypergeometric function [56]. Then,

we can write,

maxn∈{1,...,N}

Tr(HnDHHnD) =⇒ max

n∈{1,...,N}φn =⇒ φmax. (150)

The PDF of the φmax can be derived as

fφmax(φ) =Ne−λ

Γ(p)0F1(p;φλ) φp−1e−φ

(1−

∞∑l=0

l+p−1∑r=0

λle−λ

l!r!e−φφr

)N−1

. (151)

Next, we use moment generating function (m.g.f) as in [23] to derive the CDF of theγsub. The following expression for Mγsub|φmax(s) in [23 Eq. 30] can be used with q = 1,

Mγsub|φmax(s) =

(1

1 + sαγa2σ2φmax1+a2σ2φmax

)NS

. (152)

We take conditional CDF of the SNR as given by [23]

Fγsub|φmax(γ) = L−1

{Mγ|φmax(s)

s

}= 1− e−

γαγ

(1+ 1

a2σ2φmax

) NS−1∑k=0

γk

k!αkγk

(1 +

1

a2σ2φmax

)k, (153)

where L−1 the inverse Laplace transform operator. The CDF of the SNR can be ob-tained by using (151) and (153) as,

Fγsub(γ) =

∫ ∞0

Fγ|φmax(γ)fφmax(φ) dφ. (154)

After some algebraic manipulations, and using [56 Eq. 3.478.4], we can obtain (70).

G.2. p.d.f.

The PDF of the sub-optimal scheme can be derived by taking the derivative of (70)with respect to γ and applying following property of the BesselK function.

d

dxxvKv(x) = −xvKv−1(x). (155)

75

H. Derivation of the c.d.f. of γ and γu

H.1. Derivation of the c.d.f. of the γ

From the SINR γ in (80), we can obtain the CDF as in [43],

Fγ(x) = 1−∫ ∞

0

∫ ∞0

(1− Fγ1(θ))fγ2(y + x)fγ3(z)dydz, (156)

where θ = x(y + x+ 1)z/y.The relay-to-source channel h1 undergoes Rayleigh fading, therefore, the the CDF

of γ1 can be obtained as [44],

Fγ1(x) = 1−NS−1∑m=0

NS−m−1∑k=0

βmk!

(x

γ1

)ke− xγ1 . (157)

The relay-to-destination channel h2 undergoes Rician fading, and the probability den-sity function (PDF) of γ2 is given by [25],

fγ2(y) =e−λyND−1

Γ(ND)γND2

0F1

(ND;

yλ

γ2

)e− yγ2 , (158)

where 0F1(.; .) is the generalized hypergeometric function defined in [56].The PDF of γ3 is given by [61],

fγ3(z) =

η(A1)∑i=1

φi(A1)∑j=1

χi,j(A1)µ−j〈i〉z

j−1

(j − 1)!e− zµ〈i〉 . (159)

By substituting (157),(158),(159) into (156), we have

Fγ(x) =1−NS−1∑m=0

NS−m−1∑k=0

∞∑l=0

η(A1)∑i=1

φi(A1)∑j=1

βmxk

k!γk1

e−λ− x

γ2 λl

Γ(ND)l!(ND)lγl+ND2

χi,j(A1)µ−j〈i〉

(j − 1)!

×∫ ∞

0

∫ ∞0

(y + x+ 1)k(x+ y)l+Nr−1

yke−x(x+1)z/γ1

y− yγ2 dy zk+j−1e

− zµ〈i〉− xzγ1 dz

(160)

We solve the integrations by using [56 Eq. 3.471.9] and [56 Eq. 6.643.3]. Finally, wecan obtain the CDF as in (81).

H.2. Derivation of the c.d.f. of the γu

The SINR expression in (80) can be reformulated as,

γ =

γ1

γ3γ2

γ1

γ3+ 1 + γ2

. (161)

76

The SINR γ is upper bounded by γu = min(γ1

γ3, γ2). Here, the CDF of γ1

γ3can be

obtained as,

F γ1γ3

(γ) = 1−NS−1∑m=0

NS−m−1∑k=0

η(A1)∑i=1

φi(A1)∑j=1

βmk!χi,j(A1)

×µ−j〈i〉Γ(k + j)

(j − 1)!

(γ

γ1

)k (γ

γ1

+1

µ〈i〉

)(−k−j)

,

(162)

and the CDF of γ2 can be obtained as in [25],

Fγ2(γ) = 1−∞∑l=0

l+ND−1∑r=0

λle−λ

l!r!e− γγ2

(γ

γ2

)r. (163)

Finally, the CDF of the γu can obtain using the relationship,

Fγu(γ) =1− (1− F γ1γ3

(γ))(1− Fγ2(γ)), (164)

and the exact solution can be obtained as (82).

I. Ergodic Capacity

The upper bound of the ergodic capacity Cu can be evaluated using following expres-sion [65],

Cu =1

2E{log2(1 +

γ1

γ3

)}+1

2E{log2(1 + γ2)} − 1

2log2

(1 + eE{lnγ1}−E{lnγ3} + eE{lnγ2}

).

(165)

Using (162), we have

E{log2(1+γ1

γ3

)}=1

ln 2

NS−1∑m=0

NS−m−1∑k=0

η(A1)∑i=1

φi(A1)∑j=1

βmk!

χi,j(A1)Γ(k + j)µk〈i〉

(j − 1)!γk1B(k + 1, j)

× 2F1

(j + k, k + 1; j + k + 1;1−

µ〈i〉γ1

).

(166)

Then, we use following property of Meijers G function [55 Eq. 8.4.6.5] to evaluateE{log2(1 + γ2)}.

ln(1 + x) = G1,22,2

[x|1,11,0

]. (167)

Further manipulations using [56 Eq. 7.813.1] and (163) gives

E{log2(1 + γ2)} =∞∑l=0

λle−λγl2G1,33,2

[γ2|

1−ND−l,1,11,0

]Γ(ND)l!(ND)lln 2

. (168)

77

We can simplify other components of (165) as

E{ln γ1} =

NS−1∑m=0

βm(ψ(NS −m) + ln γ1), (169)

E{ln γ2} =∞∑l=0

λle−λ(ψ(ND + l) + ln γ2)

l!, (170)

E{ln γ3} =

η(A1)∑i=1

φi(A1)∑j=1

χi,j(A1)(ψ(j) + ln µ〈i〉). (171)

By substituting these, we can obtain (88).The lower bound of the ergodic capacity can be evaluated using following expession

[65],

Cl =1

2E{log2(γ1 + γ3)}+

1

2E{log2(1 + γ2)} − 1

2log2 (E{ γ1}+ E{γ2γ3}+ E{γ3})

(172)

Here, we can find the terms as E{log2(γ1 + γ3)} = E{log2(1 + γ1

γ3) + E{log2(γ3),

E{γ1} = γ1

∑NS−1m=0 βm(NS − m), E{γ2} =

∑∞l=0

λle−λγ2 (ND+l)l!

, and E{γ3)} =∑η(A1)i=1

∑φi(A1)j=1 χi,j(A1)jµ〈i〉. Finally, we can obtain the (89).