DEGREE PROGRAMME IN WIRELESS COMMUNICATIONS ENGINEERING MASTER S THESIS

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1 DEGREE PROGRAMME IN WIRELESS COMMUNICATIONS ENGINEERING MASTER S THESIS PERFORMANCE ANALYSIS OF MIMO DUAL-HOP AF RELAY NETWORKS OVER ASYMMETRIC FADING CHANNELS Author Supervisor Second Supervisor Technical Advisor Praneeth Jayasinghe Prof. Markku Juntti Prof. Matti Latva-aho L.K. Saliya Jayasinghe May, 214

2 Jayasinghe P. 214 Performance Analysis of MIMO Dual-hop AF Relay Networks over Asymmetric Fading Channels. University of Oulu, Department of Communications Engineering, Master s Degree Program in Wireless Communications Engineering. 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 and relay-to-destination channels undergo Rayleigh and Rician fading, respectively. Several MIMO techniques and practical relaying scenarios are considered to investigate the effect of such asymmetric fading on the MIMO AF relaying systems. First, we investigate the performance of the optimal single stream beamforming on non-coherent AF MIMO relaying. We use tools of finite-dimensional random matrix theory to statistically characterize the instantaneous signal-to-noise ratio SNR. The closed-form expressions of the cumulative distribution function, probability density function, and moments of SNR are derived and used to analyze the performance of the system with outage probability, bit error rate BER, and ergodic capacity. Numerical simulations are carried out to investigate the effects of the Rician factor, rank of the line-of-sight LoS component, and the number of antennas at the nodes on the system performance. Additionally, the performance is compared with orthogonal space-time block-coding OSTBC based AF MIMO system. Next, we investigate relay selection schemes for non-coherent dual-hop AF relaying with OSTBC over asymmetric fading channels. We propose two relay selection methods as optimal and sub-optimal schemes. The performance of proposed schemes are discussed with respect to the outage probability, BER and the ergodic capacity. Finally, we study the effect of co-channel interference CCI and feedback delay on the multi-antenna AF relaying over asymmetric fading channels. Here, transmit beamforming vector is calculated using outdated channel state information due to the feedback delay from relay-to-source, and the relay node experience CCI due to frequency reuse in the cellular network. The performance is investigated using the outage probability, BER and ergodic capacity to analyze the effect of the Rician factor, CCI, feedback delay and number of antennas. All these discussions are useful to evaluate the performance of AF MIMO systems in asymmetric fading channels. Our analysis suggests that having good LoS component increases the performance of the system for multiple-input-singleoutput MISO and single-input-multiple-output SIMO scenarios of relay-destination channel. Having good scattering component increases the performance for MIMO cases. Keywords: Amplify-and-forward relaying, bit error rate, cumulative distribution function, multiple-input multiple-output, moments, optimal beamforming, probability density function, Rayleigh fading, Rician fading.

3 TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS FOREWORD LIST OF ABBREVIATIONS AND SYMBOLS 1. INTRODUCTION 9 2. BACKGROUND REVIEW Cooperative Communication Decode-and-Forward Relaying Amplify-and-Forward Relaying MIMO Communications MIMO Relaying Fading Channels Relay Selection CCI and Feedback delay OPTIMAL BEAMFORMING System and Channel Model Statistics of the SNR Cumulative Distribution Function Non-i.i.d. Rician Fading Low-rank Rician Fading i.i.d. rician Rayleigh Fading Probability Density Function Moments Performance Analysis Outage Probability High SNR Analysis Effect of the Relay Saturation Symbol Error Rate Ergodic Capacity Conclusion RELAY SELECTION WITH OSTBC System and Channel Models Statistics of the Relay Selection Schemes Optimal Relay Selection Scheme Sub-optimal Relay Selection Scheme Performance Analysis Numerical Examples

4 4.5. Conclusion EFFECT OF CCI AND FEEDBACK DELAY System model Statistics of the SNR Performance analysis Outage Probability Symbol Error Rate Ergodic Capacity Upper bound Lower bound Numerical Examples Conclusion SUMMARY AND CONCLUSIONS REFERENCES APPENDICES 64 A. Derivation of the c.d.f. of the ξ max A.1. Case 1: N S q A.2. Case 2: N S < q A.3. For j q N S in A.4. For j > q N S in B. Derivation of the c.d.f. of ξ max for the Low-rank Rician, i.i.d. Rician. 67 B.1. Low-rank Rician B.2. Case 1: N S q B.3. Case 2: N S < q B.4. i.i.d Rician C. Derivation of the p.d.f. of the ξ max C.1. Validity of the p.d.f. expression for q = D. Derivation of Moments of the ξ max D.1. Case minn R, N D = D.2. Case N S = E. Derivation of the Asymptotic Expansion of the Outage Probability.. 71 E.1. Asymptotic expansion for q = E.2. Asymptotic expansion for N S = F. Derivation of the Asymtotic Ergodic Capacity G. Derivation of the c.d.f. and p.d.f. for sub-optimal relay selection scheme 74 G.1. c.d.f G.2. p.d.f H. Derivation of the c.d.f. of γ and γ u H.1. Derivation of the c.d.f. of the γ H.2. Derivation of the c.d.f. of the γ u I. Ergodic Capacity

5 FOREWORD This Master thesis is focused on performance analysis of MIMO dual-hop AF relay networks over asymmetric fading channels. I would like to gratefully acknowledge the Centre for wireless communication CWC, Department of communications Engineering DCE, University of Oulu for providing financial support for this research work. I would like to express my deepest thank to my supervisor, Prof. Markku Juntti, for his great supervision throughout my study period. His door was always opened me for discuss. Without his full supervision and inspiration, this research would not have been 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 my studies whenever needed. Without his guidance and inspiration, this research would not have been accomplished successfully. I would like to express my deep thank to Prof. Matti Latva-aho for his guidance, precious suggestions and comments during overall research period. Also, many thanks for giving me opportunity to work in CWC to obtain financial assistance for my research work. A special thank goes to my research partner Mr. Antti Roivainen and to my dear friends Mr. Ishwore and Mr. Jigo for your lovely memories and companion we had during my stay in Oulu. Finally, my deepest thank and appreciations go to my beloved parents for their love and encouragement for my studies. I also thank to my family members for their love and support.

6 LIST OF ABBREVIATIONS AND SYMBOLS AF BER BPSK CCI c.d.f. CSI DF iid MIMO MISO MRC MRT LoS OSTBC p.d.f. SER SIMO SINR SISO SNR TB Amplify and Forward Bit Error Rate Binary Phase Shift Keying Co-channel interference cumulative distribution function Channel State Information Decode and Forward Independent and identically distributed Multiple Input Multiple Output Multiple Input Single Output Maximum ratio combining Maximum ratio transmission Line of Sight Orthogonal Space Time Block Coding Probability Density Function Symbol Error Rate Single Input Multiple Output Signal to Interference and Noise Ratio Single Input Single Output Signal-to-Noise Ratio Transmit beamforming a/a n Fixed gain aθ r,i Array response vector b/b n Positive real number h ij Channel coefficients between i node to j node h ir /α ir ith Interferer-relay channel h jd jth Interferer-relay channel h sd Source-Destination channel n White gaussian noise p/p n Maximum dimension of a matrix q/q n Minimum dimension of a matrix r Received signal s minn S, q s i ith symbol of M iid signals w maxn S, q x Modulated symbol ˆx Estimate of x x i /x j Signal from Interferer y sr Received signal at the Relay n Noise vector n 1 Noise vector at the source n R /n nk Noise at the Relay

7 n d /n k /n D v max w t w r x x k y/y k /y nk C B.,. G G m,n p,,q K K v M N S N R N D N T P i P R P S P max Q. R R n x a 1,...,a p b 1,...,b q U.,.,. W a,b. CN G H h sr /h 1 /H 1 /H Sn h rd /h 2 /H 2 /H nd h 2 /H 2 /H nd h 2 / H 2 / H nd I. I i X Noise at the Destination Maximum eigen vector Transmit beamforming vector Receive beamforming vector Transmitted signal vector Transmitted signal vector at kth symbol period Received signal vector Complex number Beta function Gain at the Relay Meijer s G function Rician factor modified Bessel function of the second kind No of iid symbols at OSTBC matrix No of anntennas at source No of anntennas at Relay No of anntennas at Destination No of symbol periods Transmit power of the ith interferer Transmit power at the relay Transmit power at the source Maximum transmit power at the Relay Gaussian Q-function Code rate No of antennas at the nth Relay Confluent hypergeometric function Whittaker function. Complex noise Constant Gain Matrix Channel matrix Source to Relay channel Relay to Destination channel LoS component of Relay to Destination channel Scattered component of Relay to Destination channel Zeroth order modified Bessel function of the first kind Identity matrix with dimention i OSTBC matrix α β i η/η n γ γ / γ γ max Constant Complex amplitude of the ith path Coefficient of the line of sight component Signal to noise ratio Average SNR Maximum instantaneous SNR

8 γ sub γ R γ th λ i /λ ψ. ρ σ/σ n ξ max Γ Λ Instantaneous SNR of suboptimal path Instantaneous input power at the Relay Threshold SNR Ordered eigen values Eular s psi function Average transmit power Coefficient of the scattered component Maximum eigen value Gamma function Diagonal matrix with ordered eigen values minα, β Minimum between α and β maxα, β Maximum between α and β p l Pochhammer symbol. F Frobenius norm 2F 1.,.;.;. Gauss hypergeometric function 1 Inverse of a Matrix T Transpose of a Matrix H Hermitian of a Matrix E{.} Expectation of a random variable Kronecker operator

9 1. INTRODUCTION Modern wireless communication systems push for high data rates, reliable communications, coverage enhancements, and less power requirements. Multiple-input multipleoutput MIMO relaying can be identified as a candidate for meeting these challenges. MIMO technique provides higher spectral efficiency and improves the reliability of the communication systems [1, 2]. Cooperative relay communication enhances the throughput and extends the coverage area [3, 4, 5, 6, 7]. This also reduces the need to use high transmitter power, which in turn results in reduced interference to other nodes. Both techniques can also be used to achieve spatial diversity. Recent studies have increased interest on the MIMO relaying that can optimally utilize key resources of wireless fading, and achieve the benefits of both techniques [8]. MIMO relaying is mainly based on the relaying protocols such as amplify-andforward AF and decode-and-forward DF. In the AF protocol, the relay node forwards the amplified version of the received signal to the destination node, and provides significant gains with using less complicated processing. On the other hand, the DF protocol based relay node decodes the received signal and re-encodes, modulates and sends to the destination. Several researchers showed that the AF protocol can be implemented in practice [9, 1, 11]. In [1], the authors implemented an AF relaying system and compared it to the direct transmission. The performance evaluation shows a clear advantage of using AF relays, revealing a significant bit error rate BER improvement under realistic wireless conditions. Both the AF and DF systems were implemented in [11], and compared in terms of implementation loss and the complexity. They showed that the AF protocol is less complex and has lower implementation loss i.e., performance is very similar to the theoretical studies. These findings further support the idea that an AF relaying protocol requires less complicated processing. However, many analytical studies on AF MIMO relaying assume that the instantaneous channel state information CSI of source-relay and relay-destination channels are available at the relay node [12, 13], where the relay node has to consume a lot of resources to estimate CSI. This is contradictory to the reason of using AF MIMO relaying as a less complex relaying protocol. As an alternative solution, more realistic and less complex AF scheme called fixed-gain AF relaying is studied in [14, 15, 16, 17, 18], which applied a fixed gain at the relay node. Most of the studies on fixed-gain AF MIMO relaying have been carried out using tools of large random-matrix theory. They are mainly focused on the asymptotic network capacity and fundamental information-theoretic performance limits of large networks [19, 2, 21, 22]. In [19], the authors discussed both variable and fixed-gain AF MIMO systems to investigate the asymptotic capacity in large relay networks. Another study on a large scale multi-level AF relay network has been analyzed in [2]. 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-hop system. Recent studies in [23, 24, 25, 26, 27, 28, 29] investigate analytical performances on AF 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 optimal beamforming. OSTBC based AF MIMO relaying systems can be used to improve

10 the link reliability when the CSI is not available at the transmitter. Dharmawansa et al. [23] studied the performance of the OSTBC transmission for fixed-gain AF MIMO dual-hop system when the channels undergo Rayleigh fading. They have used moment generating function to discuss the performance with different metrics. Song et al. [24] investigated the diversity order in an OSTBC based MIMO system. MIMO beamforming can also use to mitigate the fading effect through diversity by using CSI knowledge. Performance analysis on beamforming based AF MIMO systems are carried out in [26, 27, 28]. Louie et al.[26] studied the performance of such system with antenna correlation in terms of outage probability and symbol error rates. Da Costa et al. [27] studied performance of AF MIMO beamfoming over Nakagami-m fading channels. Furthermore, Min et al. [28] studied the outage probability of such AF MIMO dual-hop system with optimal beamforming. However, most of these studies assume CSI at the transmitter and relay node. Also, these studies are carried out assuming a single antenna at the relay node. A fixed-gain AF MIMO dual-hop system with optimal beamforming is discussed in [29], where the beamforming vectors are computed at the destination, and transmit beamforming vector is sent back to the transmitter via a feedback link. An optimal single 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 practical in actual relay deployments. It is more appropriate to consider a situation where the relay is chosen such that the relay-to-destination channel has a line-of-sight LoS path [3, 31, 25]. This makes the relay-destination channel to experience Rician fading due to the dominant LoS component. Even though Rician fading is very common in many practical situations, it is not discussed enough due to the complexity associated with Rician fading. Jayasinghe et al. [25] carried out performance analysis on fixed-gain AF MIMO dual hop system with OSTBC, and assumed asymmetric fading channels between the source-to-relay and relay-to-destination. Performance analysis on optimal beamforming based fixed-gain AF MIMO dual hop system over asymmetric fading channels will be useful to find performance differences with OSTBC based fixed-gain AF MIMO systems, and which in turn provides the pathway to select the best option for any practical deployment. Cooperative communication can also be used to overcome adverse channel fading in wireless environments. The use of multiple relays in such fading scenario helps to provide a reliable transmission for the source to the destination, where these adverse channel effects can be mitigated using relay selection strategies such as selecting the best relay. Various studies on the relay selection have been carried out to increase the diversity order, reduce adverse channel effects, and to overcome the half-duplex loss of the use of relay [32, 8, 33]. In [32], a relay selection scheme is proposed to maximize the SNR at the destination. In [8, 12], the authors show that the diversity gains can be achieved in the order of the number of relays assisting the communication. This also compensates for the half-duplex loss. Most of these relay selection works are based on relaying protocols such as AF and DF [33, 12, 34, 35]. Several studies address the imperfect 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 hop system with OSTBC. MIMO AF relaying with OSTBC gives considerable improvements for link reliability in the system. These systems provide the same diversity 1

11 order as maximal-ratio receiver combining [2, 39, 4]. Relay selection schemes with non-coherent AF MIMO OSTBC relaying can enhance the system performances further without extra signal processing capabilities at relay nodes. To have a clear understanding of such systems with several relay nodes, these scenarios require further investigations. AF relaying protocol has been the most attractive relaying protocol among the research community, which forwards amplified version of the received signal at the relay node. AF relaying can be categorized into variable gain relaying [12, 13, 28] or fixed gain relaying [15, 14, 25] based on the availability of CSI at the relay node. Most of the research on variable gain AF relaying have been carried out assuming that the CSI of 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 exploiting channel knowledge at both the transmitter and receiver. In particular, transmit beamforming TB can be used at the transmitter, and maximum ratio combining MRC at the receiver. Dual hop MIMO AF relaying systems employing TB/MRC provide significant 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 communication systems are required to investigate for interferences at the nodes, such as co-channel interference CCI at the relay and the destination. These practical aspects are studied with dual hop MIMO AF systems in [43, 44, 45]. In [43], the authors investigate the outage probability in the presence of CCI. In [44], the authors analyze the effect of feedback delay on the outage probability and average SER. Huang et al. [45] derive some capacity bounds of the MIMO AF relaying system in the presence of both CCI and 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 performance of different type of MIMO dual hop system over asymmetric fading channels. In particular, Optimal beamforming AF MIMO relaying, Relay Selection on dual hop AF MIMO with OSTBC, Effect of CCI and feedback delay on the multi-antenna AF relaying. Asymmetric fading of the dual hop system is considered as the source-relay and 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 for fixed-gain AF MIMO dual-hop system in a situation where source-relay and relaydestination channels undergo Rayleigh and Rician fading respectively. In deep fading scenarios, the source-to-destination communication requires higher reliability, which is 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 fading scenario. In this study, we focus on providing a higher diversity order to the system by having the single stream transmission. Both transmit and receive beamforming vectors are obtained to maximize the instantaneous signal-to-noise ratio SNR at the destination. We use tools of finite dimensional random-matrix theory with different Rician fading scenarios to analyze the system. Here, we consider cases of relaydestination channel to undergo independent non identical Rician fading, where mean is non-identical non-i.i.d Rician, low-rank Rician fading, where mean LoS component has a lower rank, and independent identical Rician fading i.i.d Rician. New statistical results of the instantaneous SNR at the destination are derived in terms of 11

12 the cumulative distribution function c.d.f., probability density function p.d.f., and moments. Then, we use these statistical expressions to derive equations for outage probability, symbol error rate SER and ergodic capacity. Diversity orders are derived for simplified scenarios in the high SNR analysis. These performance metrics are used to evaluate the performance of optimal beamforming AF MIMO system with different antenna 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 spacetime block coded multiple-input multiple-output system with non-coherent amplifyand-forward relays, where channel state information is not available at the source and relays. The source-relay and relay-destination channels undergo Rayleigh and Rician fading, respectively. Two possible relay selection schemes are proposed, and both are statistically characterized by deriving an exact closed form expression for the cumulative distribution function and probability density function of the instantaneous SNR at the destination. In the first relay selection method, maximizing instantaneous SNR at the destination is considered to select the best relay. In the second scheme, maximizing relay-destination channel is considered. The derived statistical results are used to analyse the performance of the system with outage probability, average bit error rate and ergodic capacity. Finally, we compare both relay selection schemes with respect to the relay pool size and Rician factor. In Chapter 5, we analyze the performance of of dual-hop multiple antenna AF relaying systems using TB/MRC when the relay is subjected to multiple interferers and the source-relay and relay-destination channels undergo Rayleigh and Rician fading respectively. We derive set of new statistical results to the instantaneous signal-tointerference plus noise ratio SINR at the destination. Then, apply these statistical results to study the performance of beamforming systems in terms of three key performance metrics, i.e., outage probability, BER and ergodic capacity. Finally, the performance metrics are used to investigate the impact of key system parameters such as Rician factor, CCI, feedback delay and number of antennas. Summary of the thesis is presented in Chapter 6. 12

13 13 2. BACKGROUND REVIEW 2.1. Cooperative Communication In wireless communication, users are suffers from channel fading, that the signal attenuation varies significantly over the transmission. This can be overcome using proper diversity methods such as spatial, temporal, and frequency diversity. Cooperative communication introduces new diversity method called cooperative diversity. Cooperative diversity achieves with the cooperation of distributed relay nodes that help to establish a communication link between source node and destination node. Several theoretical studies have been carried out on cooperative communication and those studies suggest that the cooperative communication required less transmit power at the source and the relay. This makes cooperative communication is interference less and power efficient communication method. Also, cooperative communication enhances the coverage and the throughput of the system. [3, 4, 5, 6, 7]. In this thesis work, we study dual hop transmission, which is a three-node network consisting of a source, a destination, and a relay. Figure 1 illustrates simple schematic of dual hop communication system. Here, fading channels source-relay, relay-destination and source-destination are denoted as h sr,h rd and h sd, respectively. For this system, received signal at the relay node can be obtained as [13] y sr = P S h sr x + n r, 1 where P S is the average transmit power at the source, x is the signal transmitted at the source and n r is the noise at the relay. Figure 1: Dual hop relay communication. Dual hop transmission system can be classified into two main categories depends how the signal processing is done to received signal y sr at the relay node. Those are regenerative decode-and-forward and non-regenerative amplify-and-forward systems Decode-and-Forward Relaying DF relaying is an example for regenerative system. DF based relay node decodes the received signal and re-encodes, modulates and sends to the destination. Figure 2

14 14 illustrates simple schematic of the DF relying system [7]. The main advantage of DF method is that, it eliminates the noise at relay node with the expense of processing delay due to modulation/demodulation and encoding/decoding. In the DF system, the relay demodulates the received signal y sr to estimate x. Then the estimated signal ˆx is forward to the destination in order to complete the transmission. This signal estimation can be carried out in symbol by symbol or for the entire codeword by cosidering the required system performance and the complexity at the relay. Recieved signal y rd at the destination can be obtained as, y rd = P R h rdˆx + n d, 2 where P R is the average transmit power at the relay and n d is the noise at the destination. Figure 2: Decode and forward relaying system Amplify-and-Forward Relaying In the AF protocol, the relay node forwards the amplified version of the received signal to the destination node, and provides significant gains with using less complicated processing. AF protocol can be implemented in practice and it is less complex and has lower implementation loss. Simple AF relaying system is illustrated in Figure 3 [7]. In the AF system, received signal y sr at the relay is subject to the amplification factor G at the relay node before forward it to the destination. Then received signal y rd at the destination can be obtained as y rd = G P R h rd y sr + n r. 3 Depending on the availability of instantaneous CSI at the relay node, the AF relaying can be categorized into two schemes. Those are variable gain AF relaying and fixed gain AF relaying. In variable gain relaying, it amplifies the received signal at the relay node based on the instantaneous CSI. On the other hand, fixed gain is applied in AF fixed gain method considering the average behavior of the channel. Fixed gain AF relay : The amplification factor G is obtained according to average channel status as

15 15 Figure 3: Amplify and forward relaying system. P R G = E y sr 2. 4 Variable gain AF relay :The amplification factor G is obtained according to instantaneous CSI as P R G = y sr MIMO Communications A MIMO system uses multiple antennas at both the transmitter and receiver to improve the communication system performance by use of diversity and multiplexing techniques. MIMO system provides higher spectral efficiency, improves the reliability, fading mitigation and improved resistance to interference [1, 2]. Figure 4: MIMO system.

16 16 There are three main MIMO techniques have been proposed in the literature, such as precoding, spatial multiplexing, and diversity coding. Precoding is a technique that use the knowledge of CSI at the transmitter and the receiver to design precoder for multi-stream beamforming. In spatial multiplexing, a high rate signal is split into each transmit antenna with different low rate date streams and every stream use the same frequency band. In a situation where CSI is not available at the transmitter, diversity coding can be used to achieve better diversity gain similar to MRC system. In diversity coding method, signal is transmitted by applying space-time coding at the transmitter. Basic MIMO system illustrates in Figure 4. This MIMO system consist with n transmit antennas and m received antennas. Channel between ith receive antenna and jth transmit antenna is denote as h ij. 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 vector and H is the channel matrix with i, jth component is h ij. Beamforming uses precoding technique and multiple antennas for directional signal transmission and reception [46]. This directionality of the transmission is obtained by multiplying the transmit/receive signal with precoding vector in order to obtain constructive interference in the relevant direction and destructive interference in other directions. Beamforming methods can be applied at both the transmitter and the receiver. Also, beamforming significantly reduces the interference and improves system capacity. Maximal ratio transmission MRT is the beamforming technique that can achieve both diversity and the array gains with transmit beamforming. MRC is the optimal combining method, where the signals from the received antenna elements are combined in the way that the instantaneous SNR is maximized. MRT with MRC provides reference for the optimum performance that the system may obtain using both transmit and receive diversity. In single stream beamforming, same signal is transmit through each of the transmit antennas after precoding with transmit beamforming vector. Then receive beamforming 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 = Hw t x + n, 7 where w t is the transmit beamforming vector and x is the transmit signal from all the antennas. At the receiver, receive combining vector w r is applied to y. This can be expressed as, ŷ = w H r Hw t x + w H r n. 8 Alamouti proposed a new way of transmit diversity scheme with the use of two transmit antennas when CSI is not available at the transmitter [2]. This achieved by transmitting a pair of symbols using two antennas at first and then transmits the transformed version of the symbols. This Alomouti scheme is led to the progress of spacetime block coding technique. In OSTBC, the data transmitted with orthogonal coding, such that multiple copies of the data transmit across multiple antennas. This improves the reliability of data transmission. Transmitting multiple copies of data, increases the chance of correctly decode the received signal using the redundantly received data.

17 17 This OSTBC coding exploits the independent fading in the multiple antennas to improve the diversity gain. At the transmitter, OSTBC encoding done with N i.i.d. symbols s 1, s 2,.., s N are mapped to a row orthogonal matrix X C m N T, where entries of X obtained by linear combinations of s 1, s 2,.., s N and their conjugates [47]. Also, N T is the number of symbol periods used to send a code word. Therefore, the code rate is R = N/N T. Let x k be the transmitted signal during the kth symbol period. We take X = x 1,..., x NT. During the kth symbol period, we have the received signal at the destination as where n k is the noise vector at the destination. y k = Hx k + n k, k = 1, 2,.., N T, MIMO Relaying MIMO relaying is an interesting research direction that can optimally utilize key resources of wireless fading, and achieve the benefits of both MIMO and cooperative communication. We investigate different types of MIMO relaying systems in this thesis work, such as optimal single stream beamforming based AF relaying, OSTBC based AF relaying and TB/MRC based AF relaying. Figure 5: MIMO cooperative communication system. In dual hop optimal single stream beamforming method, optimal transmit beamforming vector and optimal combining vector are designed in order to maximize endto-end instantaneous SNR by use of CSI knowledge of the source-relay and relaydestination channels. The beamforming vectors are computed at the destination, and transmit beamforming vector is sent back to the transmitter via a feedback link. In deep fading scenarios, the source-to-destination communication requires higher reliability, which is possible with a single stream transmission and a higher diversity order. More detailed analysis carried out in Chapter 3. OSTBC based AF MIMO relaying systems can be used to improve the link reliability when the CSI is not available at the transmitter. In this case, it is important to have multiple antennas at the transmitter to apply OSTBC and receiver relay, destination

18 18 can be either single antenna or multiple antennas. In Chapter 4, we discuss further on OSTBC based dual hop systems. Another dual hop system with MIMO beamforming is investigated in Chapter 5. In this MIMO relaying system, TB is used at the transmitter considering the source-relay CSI, and MRC at the receiver considering the relay-destination CSI Fading Channels To Analytically investigate MIMO dual hop systems, It is important understand statistical behavior of the fading channels. The magnitude of the fading is random in a wireless channel. This is because of the multipath and the random location of objects in the environment [48]. Two fading models, that very much popular in the research community are Rayleigh and Rician fading. This thesis work considers asymmetric fading channels, such as source-relay channel undergoes Rayleigh fading and relaydestination channel undergoes Rician fading. Rayleigh fading is caused by multipath reception with no LoS component. This model is suited when the signal received from the large number of reflected waves and scattered waves. Rayleigh fading is exponentially distributed with respect to SNR and, p.d.f. of Rayleigh fading channel is given by [49] p γ γ = 1 exp γγ : γ, 1 γ where γ 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 the Rician fading is given by [49] P r r = r σ exp r2 + s 2 2σ I rs σ 2 : r, 11 where σ is the variance of, s represents the amplitude of the line-of-sight path component and I. is the zeroth order modified Bessel function of the first kind Relay Selection The use of multiple relays in an adverse fading environment helps to provide a reliable transmission for the source to the destination, where these adverse channel effects can be mitigated using relay selection strategies such as selecting the best relay. The relay selection can be use to increase the diversity order, reduce adverse channel effects, and to overcome the half duplex loss of the use of relay. Various relay selection schemes proposed in the literature such as best relay selection, nearest neighbor selection, best harmonic mean selection and best worse channel selection [12]. For those relay selection schemes, it is assumed that all CSI is available at the destination and relay node has its own CSI. Figure 6 illustrates possible relaying paths in dashed lines and the best path in solid line. By using suitable relay selection scheme, diversity gains can be achieved in the

19 19 Figure 6: Cooperative communications system with multiple relays. order of the number of relays assisting the communication. We consider two relay selection schemes in Chapter 4 based on best relay selection and nearest neighbor selection. Best relay selection : The selection method is following a SNR policy in the sense 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 relaydestination CCI and Feedback delay In cellular networks, frequency reuse introduces CCI to wireless communication system. In more realistic analysis, it is important to consider CCI effect on relay based communication system. This CCI issue is experienced in both the relay and the destination. Also, in practice CSI is not perfect. Therefore, it is important to study the systems with imperfect CSI. In dual hop AF relaying system received signal at the relay y sr with presence of CCI is given by y sr = N 1 P S h sr x + Pi h ir x i + n r, 12 i=1

20 2 where P i is ith interferer s signal power, h ir is the ith interferer-relay channel and x i is the interference signal. Similarly, the received signal at the destination y rd with presence of CCI is given by y rd = G N 2 P R h rd y sr + Pi h jd x j + n d, 13 where P j is jth interferer s signal power, h jd is the jth interferer-destination channel and x j 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 destinationrelay. Imperfect CSI due to feedback delay can be modeled using Jake s autocorrelation model. As an example the source-relay channel h sr t with feedback delay τ can be modeled as [5] h rs t τ = ρh sr t + 1 ρn, 14 where n is the white gaussian noise, and ρ is the correlation coefficient between h sr t τ and h sr t. j=1

21 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. The source, the relay, and the destination have N S, N R and N D antennas respectively. The source transmits to the relay during the first time slot, and the relay node transmits the amplified version of received data to the destination during the second timeslot. We assume that there is no direct link between the source and the destination. The sourcerelay channel is assumed to undergo Rayleigh fading, and the relay-destination channel is subject to Rician fading. We denote H 1 C N R N S and H 2 C N D N R as MIMO channel matrices from the source-to-relay and the relay-to-destination, respectively. Entries of H 1 are assumed to be CN,1. H 2 is modeled as H 2 = ηh 2 + σ H 2, 15 where H 2 is the LoS component, and H 2 is the scattered component. H 2 consists of non-identical complex elements having unit magnitude, entries of H 2 are assumed to be CN,1. In non-i.i.d. Rician fading, H 2 is full rank matrix that consists of complex elements having unit magnitude. For low-rank Rician fading, H 2 is an arbitrary rank matrix that consists of complex elements having unit magnitude. H 2 has rank one for i.i.d. Rician fading scenario. We also consider both η, σ to satisfy η 2 + σ 2 = 1 in all these scenarios. Figure 7: AF MIMO dual hop system with N S source antennas, N R relay antennas and N D destination antennas. We consider optimal beamforming vectors at the source and the destination. The source node transmits beamformed version of modulated symbol x after being beamformed by a transmit beamforming vector w t C N S, where E{ x 2 } = ρ, and w t 2 = 1. The relay amplifies the received signal by a constant gain matrix G = b ai NR, where a = N R with b is a positive real number. This selection 1+ρ of a satisfies the average total power constraint E{ GH 1 w t x + Gn R 2 b}, where n R CN, I NR is noise vector at the relay. A situation where the total power exceeds the maximum available power at the relay node is discussed in a later section.

22 22 The destination detects the signal after using a receive beamforming vector w r C N D, where w r 2 = 1. The received signal r at the destination is given as r = aw H r H 2 H 1 w t x + aw H r H 2 n R + w H r n D, 16 where n D CN, I ND is noise vector at the destination. The instantaneous SNR at the destination is given by γ = aρwh r H 2 H 1 w t wt H H H 1 H H 2 w r. 17 wr H ah 2 H H 2 + I ND w r Here, the SNR γ can be maximized by optimal beamforming design. We assume that the channel state information CSI is only available at the destination, and the destination computes the optimal beamforming vectors w t and w r. Then, the optimal transmit beamforming vector is sent back to the transmitter via a dedicated feedback link [29]. The optimal receive beamforming vector w r that maximize the SNR is given by w r = ah 2 H H 2 + I ND 1 H 2 H 1 w t. 18 Then, substituting 18 into 17 gives γ = aρw H t H H 1 H H 2 ah 2 H H 2 + I ND 1 H 2 H 1 w t. 19 Next, we consider eigenvalues of H H 1 H H 2 ah 2 H H 2 + I ND 1 H 2 H 1 as ξ 1, ξ 2,.., ξ NS. The maximum eigenvalue is denoted as ξ max, which is given by ξ max = maxξ 1, ξ 2,.., ξ NS. 2 The eigenvector associated with ξ max is v max C N S. Therefore, H H 1 H H 2 ah 2 H H 2 + I ND 1 H 2 H 1 v max = ξ max v max. 21 It is evident that w t = v max provides the maximum instantaneous SNR γ in 19, and γ is simplified into γ = aρξ max. 22 To gain more insight into the system performance, we need to analyze the statistical properties of the SNR in 22. For the sake of convenience we use the following notations; p = maxn D, N R, q = minn D, N R, w = maxn S, q, s = minn S, q Statistics of the SNR Statistical parameters of the SNR γ directly depend on the statistical results of the maximum eigenvalue ξ max. Here, we derive new analytical expressions for the c.d.f., p.d.f., and moments of the ξ max using the tools of finite dimensional random matrix theory. These results are used to find the statistical parameters of the SNR γ. We start with 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 Cumulative Distribution Function The following theorems are used to find the c.d.f. of the SNR γ Non-i.i.d. Rician Fading Here, the relay-destination channel undergoes non-i.i.d. Rician fading, hence the LoS component H 2 is full rank. Theorem 1: The c.d.f. of the maximum eigenvalue ξ max of H H 1 H H 2 ah 2 H H 2 + I ND 1 H 2 H 1 is given by F ξmax x = c 1 N Sq s detv s i=1 ΓN detix, 23 S i + 1 where Ix is a q q matrix with i, j th element is given by q j λ l i 1q N S j q j t l l!σ 2p+2l q+1 k a k σ 2w j Γw j Ix ij = k= λ l i w j! sj t l l!σ 2p+2l q+1 w j m= m+s j r= k= sj k a k σ 2u Γu j q s 2 m+s j r a r x m e ax xσ 2 v 2 m! K v 2 x σ 2 j > q s, 24 where < λ 1 < λ 2 <.. < λ q < are non-zero ordered eigenvalues of η 2 H σ 2 2 H H 2, detv is a q q vandermonde matrix whose determinant is given by detv = detλ q j i = q l<k λ l λ k, t l is the Pochhammer symbol which is given by t l = Γt+l and K Γt v. is the modified Bessel function of the second kind,, Γ. is the gamma function, s j = j +s q 1, v = p+l+r m s j, u = p+l+k s j, w j = p + j + l + k q, and the constant c is given as where Λ = diagλ 1, λ 2,..., λ q. Proof : See Appendix A. Now, the c.d.f. of the SNR γ can be obtained as c = e TrΛ Γt q, 25 F γ γ = F ξmax γ aρ. 26 Here, the c.d.f. of the SNR γ is not directly dependent on the entries of channel matrices. It depends on the eigenvalues of η 2 H H σ 2 2 H 2. Also, the infinite summation in 24 is converging rapidly, and can obtain accurate values by truncating the series with sufficient depth. Table I shows the convergence of the infinite summation for the antenna configuration 3,2,5 with ρ = 1 db and b = 1. The scenarios with low Rician

24 24 Table 1: Convergence of the c.d.f. with no of terms l in the infinite summation for antenna configuration 3,2,5 l γ=1db, K=-1dB γ=1db, K=dB γ=1db, K=1dB factor requires a smaller number of terms to converge, whereas higher Rician factor scenarios require the larger numbers of terms. We obtained the accuracy up to 15th decimal point by truncating the infinite summation term l = 2, l = 5 and l = 2 for Rician factors K = 1 db, K = db and K = 1 db respectively. This new c.d.f. expression is used to evaluate performance metrics of the system in later sections. Next, the most important and common MIMO configurations are discussed as special cases of the general solution 23. Corollary 1 : The c.d.f. of the ξ max for minn R, N D = 1 is given by F ξmax x = 1 N S 1 m= m 2λ l e λ x m+ v m 2 r a r e ax K v 2 x σ 2, 27 Γpp l l!m!σ 2m r+ v 2 r= where v = p + l m + r, u = p + l, λ is the eigenvalue of η 2 H σ 2 2 H H 2 which is given by η 2 pq. σ 2 Proof : The proof is straightforward by substituting q = 1 to 23. This result is useful to analyze the cases with the single relay antenna or the single destination antenna. Corollary 2: The c.d.f. of the ξ max for N S = 1 is given by F ξmax x= c 1q 1 detv q d=1 λ l d σ 2p+2l Γp + l 2e ax xσ 2 p+l 2 K p+l 2 x σ 2 A t l l!σ 2p+2l q+1 d, where A d is the d, qth cofactor of a q q matrix with i, jth entry is given by A d ij = λ l i 1 q N S j q j t l l!σ 2p+2l q+1 k= q j k 28 a k σ 2w j Γw j, 29 where w j = p + j + l + k q. Proof : The proof is straightforward by substituting N S = 1 to 23 and applying the Laplace expansion along the last column of the determinants. These expressions also consist of infinite summations. However, they converge rapidly. Figure 8 illustrates both analytical and simulation curves of the c.d.f. of the SNR γ for different antenna configurations. These antenna configurations are selected to

25 ,5, ,4,6 3,2,5 CDF Fγγ ,5,2 K increasing SNR γ Analytical 3,2,5 K=1 db Analytical 3,2,5 K= -1 db Analytical 3,2,5 K= db Analytical 2,5,1 K=1 db Analytical 2,4,6 K=1 db Analytical 1,5,2 K=1 db Simulation Figure 8: c.d.f of the γ for different antenna configurations N S, N R, N D, and with different values of Rician factor K = 1,, 1 db. ρ = 1 db, and b = 1. Monte carlo simulation with 1 6 iterations. match the cases N S q, N S < q, q = 1 and N S = 1. The case N S q N S = 3, N R = 2, N D = 5 is considered with different Rician factors K= η 2 to investigate the σ 2 effect of Rician fading. For all other cases, Rician factor is considered to be K = 1 db. Here, the LoS component H 2 is generated according to [51] with full rank, and entries are non-identical complex elements with unit magnitude. We also consider ρ = 1 db, and b = 1. Derived closed form results are used to plot the analytical curves 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 for most of time than channels with low Rician factor Low-rank Rician Fading Here, we consider the scenario where the relay-destination channel undergoes lowrank Rician fading, where LoS component H 2 is not full rank. We consider arbitrary rank m m < q for H 2. Then η 2 σ 2 H 2 H 2 H has m non-zero eigenvalues.

26 26 Corollary 3: The c.d.f. of the ξ max is given by, c low 1 N Sq s+qq 1/2 F ξmax x low = detv m λ q+1 m λ q+2 m...λ q deti lowx, 3 q m where I low x is a q q matrix with i, j th element is given by q j k a k σ 2w ij Γw ij q j 1 q N S j σ 2p q+2i 1 k= q j λ l i 1q N S j t l l!σ 2p+2l q+1 j q s, i q m k= q j k a k σ 2w j Γw j j q s, i > q m I low x ij = sj w j! σ 2p q+2i 1 w j m= k= m+s j r= sj k a k σ 2u ij Γu ij 2 m+s j r a r x m m! e ax xσ 2 vij 2 K vij 2 x σ 2 j > q s, i q m 31 λ l i w j! sj t l l!σ 2p+2l q+1 w j m= m+s j r= k= 2 m+s j r sj k a k σ 2u Γu a r x m m! e ax xσ 2 v 2 K v 2 x σ 2 j > q s, i > q m, where < λ q m+1 < λ q m+2 <.. < λ q < are non-zero ordered eigenvalues of η 2 σ 2 H 2 H 2 H, Vm is m m vandermonde matrix with determinant given by detv m = q q m<l<k λ k λ l, s j = j +s q 1, v = p+l +r m s j, u = p+l +k s j, w j = p+j+l+k q, u ij = p+k s j +i 1, v ij = p+r m s j +i 1, w ij = p+k+j+i q 1, and the constant c low is given by c low = e TrΛm Γt m q m z=1 Γp q + z q 1 r=m Γq r s i=1 ΓN S i + 1, 32 where Λ m = diagλ q m+1, λ q m+2,..., λ q. Proof : See Appendix B i.i.d. rician Here, we consider the relay-destination channel undergoes i.i.d Rician fading, where LoS component H 2 has one eigenvalue.