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Beamforming schemes for next generation wireless communication systems Liang, Yangwen 2011-10-13

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BEAMFORMING SCHEMES FOR NEXT GENERATION WIRELESS COMMUNICATION SYSTEMS by Yangwen Liang M.A.Sc., The University of British Columbia, 2006 B.Eng., McMaster University, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2011 c© Yangwen Liang, 2011Abstract Multiple¸input multiple¸output (MIMO) and relaying are two promising techniques which will be employed in next generation wireless communication systems. Transmit beamform- ing (BF) and receive combining are simple yet popular methods for performance enhance- ment for MIMO and/or relaying. This thesis investigates several BF schemes for MIMO and relaying systems. For systems combining MIMO and orthogonal frequency division multiplexing (MIMO¸ OFDM) technology, we propose a novel time¸domain BF (TD¸BF) scheme which uses cyclic BF ¿lters (C¸BFFs). Both perfect and partial channel state information at the transmitter are considered. The C¸BFFs are optimized for maximum average mutual in- formation per sub¸carrier and minimum average uncoded bit error rate. We show that TD¸BF has a more favorable performance/feedback rate trade¸o¾ than previously pro- posed frequency¸domain BF schemes. Secondly, BF for one¸way cooperative networks with multiple multi¸antenna amplify¸ and¸forward relays in frequency¸nonselective channels is considered. The source BF vector and the amplify¸and¸forward BF matrices at the relays are optimized for maximization of the signal¸to¸interference¸plus¸noise ratio (SINR) at the destination under three di¾erent power constraints. We show the bene¿ts of having multiple antennas at the source and/or multiple multi¸antenna relays. Subsequently, we investigate ¿lter¸and¸forward BF (FF¸BF) for one¸way relay net- works in frequency¸selective channels. For the processing at the destination, we investi- iiAbstract gate two di¾erent cases: a simple slicer, and a linear equalizer (LE) or a decision¸feedback equalizer (DFE). For both cases, we optimize the FF¸BF matrix ¿lters at the relays for maximization of the SINR under a transmit power constraint, and for the ¿rst case we con- sider additionally optimization of the FF¸BF matrix ¿lters for minimization of the total transmit power under a quality of service constraint. Leveraging results from one¸way relaying, we also investigate FF¸BF for two¸way relay networks. For the simple slicer case, we show that the optimization problems are convex. For the LE/DFE case, we establish an upper and an achievable lower bound for an SINR max¸min problem. iiiPreface Chapters 2¸5 are based on work conducted at UBC by myself under the supervision of Professor Robert Schober. In addition, the research in Chapters 2, 4, and 5 was performed in collaboration with Professor Wolfgang Gerstacker from the University of Erlangen¸ Nuremberg. For the work in Chapters 4 and 5, I also collaborated with Dr. Aissa Ikhlef, a postdoctoral fellow in Professor Robert Schober's group at the University of British Columbia. For all chapters, I conducted the paper survey on related topics, formulated the problems, proposed problem solutions, and performed the analysis and the simulations of the considered communication systems. Professor Robert Schober, Professor Wolfgang Gerstacker, and Dr. Aissa Ikhlef provided valuable feedback on my manuscript drafts. Three papers related to Chapter 2 have been published: • Y. Liang, R. Schober, and W. Gerstacker, ³Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems with Finite Rate Feedback´. IEEE Transactions on Com- munications, 57(9): 2828¸2838, Sept. 2009. • Y. Liang, R. Schober, and W. Gerstacker, ³Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems´. In Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM), Washington DC, USA, Nov. 2007. • Y. Liang, R. Schober, and W. Gerstacker, ³Minimum BER Transmit Beamforming for MIMO¸OFDM Systems with Finite Rate Feedback´. In Proceedings of the IEEE International Conference on Communications (ICC), Beijing, China, May 2008. ivPreface Four papers related to Chapter 3 have been published: • Y. Liang and R. Schober, ³Cooperative Amplify¸and¸Forward Beamforming with Multiple Multi¸Antenna Relays´. IEEE Transactions on Communications, 59(9): 2605¸2615, Sept. 2011. • Y. Liang and R. Schober, ³Cooperative Amplify¸and¸Forward Beamforming for OFDM Systems with Multiple Relays´. In Proceedings of the IEEE International Conference on Communications (ICC), Dresden, Germany, Jun. 2009. • Y. Liang and R. Schober, ³Cooperative Amplify¸and¸Forward Beamforming with Multi¸Antenna Source and Relays´ (invited paper). In Proceedings of the Third International Workshop on Computational Advances in Multi-Sensor Adaptive Pro- cessing (CAMSAP), Aruba, Dec. 2009. • Y. Liang and R. Schober, ³Amplify¸and¸Forward Multi¸Antenna Beamforming with Joint Source¸Relay Power Constraint´. In Proceedings of the IEEE Vehicular Tech- nology Conference (VTC), Ottawa, Canada, Sept. 2010. Three papers related to Chapter 4 have been published: • Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, ³Cooperative Filter¸and¸Forward Beamforming with Equalization for Frequency¸Selective Channels´. IEEE Trans. on Wireless Commun., 10(1): 228¸239, Jan. 2011. • Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, ³Filter¸and¸Forward Beamform- ing with Multiple Multi¸Antenna Relays for Frequency¸Selective Channels´ (invited paper). In Proceedings of the International ICST Conference on Communications and Networking in China (Chinacom), Aug. 2010. vPreface • Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, ³Cooperative Filter¸and¸Forward Beamforming with Equalization for Frequency¸Selective Channels´. In Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM), Miami, USA, Dec. 2010. Two papers related to Chapter 5 have been accepted/published, and one paper is under preparation for submission: • Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, ³Two¸Way Filter¸and¸Forward Beamforming for Frequency¸Selective Channels´. Accepted by the IEEE Transac- tions on Wireless Communications, Oct. 2011. • Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, ³Cooperative Two¸Way Filter¸ and¸Forward Beamforming for Frequency¸Selective Channels´. In Proceedings of the IEEE International Conference on Communications (ICC), Kyoto, Japan, Jan. 2011. • Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, ³Two¸Way Filter¸and¸Forward Beamforming with Multiple Multi¸Antenna Relays for Frequency¸Selective Chan- nels´. In preparation, Oct. 2011. viTable of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Beamforming for MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Cooperative Relay Network . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Two¸Way Cooperative Relay Network . . . . . . . . . . . . . . . . . . . . 7 1.4 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 10 viiTable of Contents 2 Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems . . 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Transmitter Processing for TD¸BF . . . . . . . . . . . . . . . . . . 16 2.2.2 MIMO Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.3 Receiver Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.4 Feedback Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Maximum AMI Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Formulation of the Optimization Problem . . . . . . . . . . . . . . 20 2.3.2 Solution of the Optimization Problem for Lg = Nc . . . . . . . . . 21 2.3.3 Solution of the Optimization Problem for Lg < Nc . . . . . . . . . 23 2.4 Minimum BER Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1 Formulation of the Optimization Problems . . . . . . . . . . . . . 27 2.4.2 Solution of the Optimization Problems for Lg = Nc . . . . . . . . . 28 2.4.3 Solution of the Optimization Problems for Lg < Nc . . . . . . . . . 29 2.5 Finite¸Rate Feedback and Comparison . . . . . . . . . . . . . . . . . . . . 32 2.5.1 Finite¸Rate Feedback Case . . . . . . . . . . . . . . . . . . . . . . 32 2.5.2 Comparison with FD¸BF . . . . . . . . . . . . . . . . . . . . . . . 33 2.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.2 Maximum AMI Criterion . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.3 Minimum BER Criterion . . . . . . . . . . . . . . . . . . . . . . . 39 2.6.4 Comparison of Maximum AMI and Minimum BER Criteria . . . . 44 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 viiiTable of Contents 3 Cooperative AF¸BF with Multiple Multi¸Antenna Relays . . . . . . . 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 System Model and Optimization Problem . . . . . . . . . . . . . . . . . . 49 3.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.2 Formulation of the Optimization Problem . . . . . . . . . . . . . . 51 3.3 Optimal AF¸BF Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.1 AF¸BF with Individual Power Constraints for Relays . . . . . . . 53 3.3.2 AF¸BF with Joint Power Constraint for Relays . . . . . . . . . . . 57 3.3.3 AF¸BF with Joint Power Constraint for Source and Relays . . . . 58 3.3.4 Comparison of the Solutions for the Di¾erent Constraints . . . . . 59 3.4 Optimal BF Vector at the Source . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.1 AF¸BF with Individual Power Constraints for Relays . . . . . . . 60 3.4.2 AF¸BF with Joint Power Constraint for Relays . . . . . . . . . . . 62 3.4.3 AF¸BF with Joint Power Constraint for Source and Relays . . . . 65 3.4.4 Comparison of the Solutions and CSI Feedback Requirements . . . 67 3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.1 Comparison of Source BF Vector Optimization Methods . . . . . 69 3.5.2 Impact of Network Parameters on Performance . . . . . . . . . . . 74 3.5.3 Impact of Power Constraints on Performance . . . . . . . . . . . . 76 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 Cooperative FF¸BF with Multiple Multi¸Antenna Relays . . . . . . . 81 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.1 FF¸BF at Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.2 Processing at Destination . . . . . . . . . . . . . . . . . . . . . . . 86 ixTable of Contents 4.2.3 Feedback Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3 FIR FF¸BF without Equalization . . . . . . . . . . . . . . . . . . . . . . 87 4.3.1 SINR Maximization Under Relay Power Constraint . . . . . . . . . 90 4.3.2 Relay Power Minimization Under SINR Constraint . . . . . . . . . 91 4.3.3 SINR Maximization Under Source¸Relay Power Constraint . . . . 92 4.3.4 Source¸Relay Power Minimization Under SINR Constraint . . . . 93 4.4 FF¸BF with Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.1 Optimal IIR FF¸BF with Equalization . . . . . . . . . . . . . . . 94 4.4.2 Optimal FIR FF¸BF with Equalization . . . . . . . . . . . . . . . 104 4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.5.1 FF¸BF without Equalization . . . . . . . . . . . . . . . . . . . . . 110 4.5.2 FF¸BF with Equalization . . . . . . . . . . . . . . . . . . . . . . . 115 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5 Two¸Way FF¸BF with Multiple Single Antenna Relays . . . . . . . . . 125 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.2.1 FF¸BF at Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.2.2 Transceiver Processing . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.3 FIR FF¸BF without Equalization . . . . . . . . . . . . . . . . . . . . . . . 130 5.3.1 Max¸min Criterion Under Relay Power Constraint . . . . . . . . . 133 5.3.2 Relay Power Minimization Under SINR Constraints . . . . . . . . 135 5.4 FF¸BF with Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.4.1 Optimal IIR FF¸BF with Equalization . . . . . . . . . . . . . . . . 137 5.4.2 FIR FF¸BF Filter Optimization . . . . . . . . . . . . . . . . . . . 143 5.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 xTable of Contents 5.5.1 Relay Power Minimization for FF¸BF without Equalization . . . . 145 5.5.2 Max¸min SINR Optimization for FF¸BF with and without Equal- ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.5.3 Max¸min SINR vs. Minimum Sum MSE Optimization for FF¸BF with ZF¸LE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.5.4 Impact of Number of Relays NR . . . . . . . . . . . . . . . . . . . 151 5.5.5 BER Performance for Fading Channels . . . . . . . . . . . . . . . . 153 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6 Summary of Thesis and Future Research Topics . . . . . . . . . . . . . . 157 6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2.1 Two¸way Relaying with Multiple Multi¸antenna Relays . . . . . . 160 6.2.2 Cooperative Communications for Multi¸user Systems . . . . . . . . 161 6.2.3 Synchronization for Cooperative Communications . . . . . . . . . . 161 6.2.4 Cooperative Communications for Cognitive Radio . . . . . . . . . 162 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 xiList of Tables 2.1 Calculation of the optimum C¸BFFs g for the maximum AMI and the mini- mum average BER criterion using a GA, respectively. Termination constant  has a small value (e.g.  = 10−4). i denotes the iteration and δi is the adaptation step size necessary for the GA. . . . . . . . . . . . . . . . . . . 26 2.2 Feedback Requirements for TD¸BF, ideal FD¸BF, and FD¸BF with modi- ¿ed spherical (MS), Grassmannian (GS), and geodesic (GD) interpolation. 34 3.1 Gradient algorithm for calculation of source BF vector gˆ for individual and joint relay power constraints. The de¿nitions of gˆ and the gradient gradk depend on the power constraint, cf. Section 3.4. Termination constant  has a small value (e.g.  = 10−5). k denotes the iteration index and ak is the adaptation step size chosen through a backtracking line search [1]. . . . . . 62 3.2 Gradient algorithm for calculation of source BF vector g and power allo- cation for joint source¸relay power constraint. Termination constant  has a small value (e.g.  = 10−5). k denotes the iteration index and ak is the adaptation step size chosen through a backtracking line search [1]. . . . . . 67 xiiList of Tables 4.1 Numerical algorithm for ¿nding the optimum power allocation p(f) for IIR FF¸BF ¿lters at the relays. X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver, respectively. Termination constant  and frequency spacing ∆f have small values (e.g.  = 10−5, ∆f = 10−5). i denotes the iteration index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2 Gradient algorithm (GA) for calculation of near¸optimal FIR FF¸BF ¿lter vector a. Termination constant  has a small value (e.g.  = 10−5). i denotes the iteration index and δi is the adaptation step size chosen through a backtracking line search [1]. . . . . . . . . . . . . . . . . . . . . . . . . . 108 xiiiList of Figures 1.1 Di¾erent relaying protocols. (a) decode¸and¸forward (DF) relaying pro- tocol, and (b) amplify¸and¸forward (AF) protocol. xs: signal broadcasted from the source, xr: signal transmitted from the relay, xˆs: regenerated signal after decoding, TSx: time slot x, and a: scaling factor. . . . . . . . . . . . 5 1.2 Di¾erent two¸way relaying protocols. (a) Bidirectional one¸way relaying protocol, (b) Time Division Broadcast (TDBC) protocol, and (c) Multiple Access Broadcast (MABC) protocol. s1: signal transmitted from transceiver 1 (TC1), s2: signal transmitted from transceiver 2 (TC2), and f(s1, s2): processed version of the received signals. . . . . . . . . . . . . . . . . . . . 8 2.1 MIMO¸OFDM system with TD¸BF. P/S: Parallel¸to¸serial conversion. S/P: Serial¸to¸parallel conversion. CE: Channel estimation. . . . . . . . . . . . 17 2.2 AMI of TD¸BF (AMI criterion), MS¸FD¸BF [2], and GD¸FD¸BF [3] with perfect CSI. NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. For comparison the AMIs for ideal FD¸BF and single¸input single¸output (SISO) transmission (NT = 1, NR = 1) are also shown. . . . . . . . . . . . 37 2.3 AMI of TD¸BF (AMI criterion) vs. number of feedback bits B per channel update. NT = 2, NR = 1, Nc = 512, Es/N0 = 10 dB, and IEEE 802.11n Channel Model B. For comparison the AMIs for GD¸FD¸BF with codebooks from [4] are also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 xivList of Figures 2.4 BER of coded MIMO¸OFDM system with TD¸BF (AMI criterion), MS¸ FD¸BF [2], and GD¸FD¸BF [3]. Perfect CSI and ¿nite¸rate feedback, NT = 2, NR = 1, Nc = 512, Rc = 1/2, and IEEE 802.11n Channel Model B. For comparison the BERs for ideal FD¸BF and SISO transmission (NT = 1, NR = 1) are also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5 Average BER of uncoded MIMO¸OFDM system with TD¸BF. Minimum average BER criterion (solid lines) and max¸min criterion (dashed lines), perfect CSI, NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. For comparison the BERs for ideal FD¸BF and SISO transmission (NT = 1, NR = 1) are also shown. . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 Average BER of uncoded MIMO¸OFDM system with TD¸BF (average BER criterion) vs. number of feedback bits B per channel update. GA was used for C¸BFF optimization. NT = 2, NR = 1, Nc = 512, Eb/N0 = 10 dB, and IEEE 802.11n Channel Model B. . . . . . . . . . . . . . . . . . . . . . . . 42 2.7 Average BER of uncoded and coded MIMO¸OFDM system with TD¸BF (average BER criterion). GA was used for C¸BFF optimization and Lg = 2 is valid for all curves shown. Perfect CSI (bold lines) and ¿nite¸rate feedback channel, NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. 43 2.8 Average BER of uncoded and coded MIMO¸OFDM system employing TD¸ BF with perfect CSI. Average BER criterion (dashed lines) and AMI crite- rion (solid lines), NT = 3, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 xvList of Figures 3.1 Cooperative network with one multi¸antenna source, multiple multi¸antenna relays, and one single¸antenna destination. gi, 1 ≤ i ≤ NT , denotes the ith element of source BF vector g. ni,µ, 1 ≤ µ ≤Mi, is the µth element of noise vector n1,i at relay i, 1 ≤ i ≤ NR. . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Locations of source, destination, and relays in simulation. . . . . . . . . . . 70 3.3 CDF of the instantaneous SINR for AF¸BF with joint relay power constraint (PC) and one relay located at (a) and (e), respectively. Results for di¾erent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A path¸loss exponent of 3, NT = 2, and d = 1 are assumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4 CDF of the instantaneous SINR for AF¸BF with joint source¸relay power constraint (PC) and NR relays. Results for di¾erent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A path¸loss exponent of 3, NT = 2, and d = 1 are assumed. The relays are located at (a) and (e) for NR = 2, (a)¸(e) for NR = 5, and (a)¸(e) with 2 relays at each location for NR = 10. . . . . . . . . . . . . . . 72 3.5 CDF of the instantaneous SINR for AF¸BF with individual relay power con- straints (PCs) and NR = 5 single¸antenna relays at locations (a)¸(e). Re- sults for di¾erent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A path¸loss exponent of 3 and d = 1 are assumed. . . . . . . . . . . . . . . . . . . . . . . . . . . 73 xviList of Figures 3.6 Average SINR vs. distance d for AF¸BF with joint relay power constraint (PC) and di¾erent numbers of antennas NT at the source. A path¸loss exponent of 3 is assumed. For comparison the SNR without relaying for a source transmit power of P = 2 and the SINR for relay selection are also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.7 Average SINR vs. distance d for AF¸BF with individual relay power con- straints (PCs) and di¾erent numbers of antennas NT at the source. A path¸ loss exponent of 3 is assumed. For comparison the SNR without relaying for a source transmit power of P = 2 and the SINR for relay selection are also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.8 Average SINR vs. distance d for AF¸BF with joint source¸relay power con- straint (PC) and di¾erent numbers of relays and di¾erent numbers of relay antennas. A path¸loss exponent of 3 is assumed. For comparison the SNR without relaying for a source transmit power of P = 2 and the SINR for relay selection are also shown. . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.9 Average mutual information (AMI) in (bits/s/Hz) vs. distance d for two di¾erent network setups and di¾erent power constraints (PCs). The relays are in locations (a) and (e) for NR = 2 and (a)¸(e) for NR = 5. The proposed gradient methods are used for computation of the source BF vector g. A path¸loss exponent of 4 is assumed. For comparison the average mutual information without relaying for a source transmit power of P = 2 and the average mutual information for relay selection are also shown. . . . . . . . 78 xviiList of Figures 3.10 Average BER vs. σ2d/σ2n for two di¾erent network setups and di¾erent power constraints. The relays are in locations (a) and (e) for NR = 2 and (a)¸(e) for NR = 5. The proposed gradient methods are used for computation of the source BF vector g. A path¸loss exponent of 3 and d = 1 are assumed. AF¸BF: 16¸QAM. Direct transmission: QPSK, source transmit power P = 2. 79 4.1 Cooperative network with one single¸antenna source, multiple multi¸antenna relay nodes, and one single¸antenna destination. EQ is the equalizer at the destination. sˆ[k] are estimated symbols after the equalizer or slicer. . . . . 85 4.2 Locations of source, destination, and relays in simulation. . . . . . . . . . . 110 4.3 Average SINR vs. decision delay k0 for FIR FF¸BF without equalization (EQ) at the destination. The FF¸BF ¿lters were optimized for SINR max- imization under joint relay power constraint. Exponentially decaying chan- nel power delay pro¿le with Lg = Lh = 5, d = 1, NR = 5, Mz = 1, z ∈ {1, 2, . . . , 5} and γg = γh = 10 dB. . . . . . . . . . . . . . . . . . . . . 111 4.4 Average SINR vs. distance d for FIR FF¸BF without equalization (EQ) at the destination. The FF¸BF matrix ¿lters were optimized for a joint relay power constraint. Exponentially decaying channel power delay pro¿le with σt = 2 and Lg = Lh = 5, and γg = γh = 10 dB. Results for direct transmission with transmit power P = 2 at the source are also included. . . 112 4.5 Average SINR vs. distance d for FIR FF¸BF without equalization (EQ) at the destination. The FF¸BF matrix ¿lters were optimized for a joint source¸ relay power constraint. Exponentially decaying channel power delay pro¿le with σt = 2 and Lg = Lh = 5, and γg = γh = 10 dB. Results for direct transmission with transmit power P = 2 at the source are also included. . . 113 xviiiList of Figures 4.6 Total average source and relay transmit power vs. required SINR γ for FIR FF¸BF without equalization (EQ) at the destination for relay power min- imization and joint source¸relay power minimization. Exponentially de- caying power delay pro¿le with σt = 2 and Lg = Lh = 5, d = 1, and γg = γh = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.7 Feasibility probability vs. required SINR γ for FIR FF¸BF without equaliza- tion (EQ) at the destination for relay power minimization and joint source¸ relay power minimization. Exponentially decaying power delay pro¿le with σt = 2 and Lg = Lh = 5, d = 1, and γg = γh = 10 dB. . . . . . . . . . . . . 115 4.8 SINR vs. iteration number i of GA given in Table 4.2 for FIR FF¸BF with MMSE¸DFE at the destination. γg = γh = 10 dB, Lg = Lh = 5, and g¯1,z[k] = ¯h1,z[k] = 1/√5, 0 ≤ k < 5, 1 ≤ z ≤ 5. NR = 5 relays with Mz = 1, 1 ≤ z ≤ NR, at locations (a)¸(e), respectively. For comparison the SINR for IIR FF¸BF is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.9 Frequency responses of IIR FF¸BF ¿lters for γg = γh = 10 dB, NR = 1 single antenna relay, Lg = Lh = 2, and g¯1,1[k] = ¯h1,1[k] = 1/√2, k ∈ {0, 1}. For comparison the frequency response of the test channel is also shown. . 118 4.10 Frequency responses of IIR FF¸BF ¿lter and FIR FF¸BF ¿lters of various lengths for MMSE¸DFE at the receiver. All channel parameters are identical to those in Fig. 4.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 xixList of Figures 4.11 Average SINR vs. distance d for FF¸BF with MMSE¸LE, MMSE¸DFE, and an MF receiver at the destination. NR = 2 relays with M1 = 2 and M2 = 3, exponentially decaying power delay pro¿le with σt = 2 and Lg = Lh = 5, and γg = γh = 10 dB. For comparison the SINRs of FF¸BF without (w/o) equalization (EQ) at the destination and without relaying are also shown, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.12 Average SINR vs. decay parameter σt for FF¸BF with MMSE¸LE, MMSE¸ DFE, and an MF receiver at the destination. NR = 2 relays with M1 = 2 and M2 = 3, distance d = 1, exponentially decaying power delay pro¿le with Lg = Lh = 5, and γg = γh = 10 dB. For comparison the SINRs of FF¸BF without (w/o) equalization (EQ) at the destination and without relaying are also shown, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.13 Average BER of BPSK vs. transmit SNR γ for FF¸BF with MMSE¸LE, MMSE¸DFE, and an MF receiver at the destination. Exponentially decay- ing power delay pro¿le with σt = 2 and Lg = Lh = 5. For comparison the BER of FF¸BF without (w/o) equalization (EQ) at the destination is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.1 Cooperative two¸way network with two transceiver nodes and NR relay nodes. EQ is the equalizer at the transceivers. sˆ1[k] and sˆ2[k] are estimated received symbols at TC2 and TC1, respectively. . . . . . . . . . . . . . . . 127 5.2 Locations of TC1, TC2, and the relays in the simulations. . . . . . . . . . 146 xxList of Figures 5.3 Total average relay transmit power vs. required SINRs γ1 and γ2 for FIR FF¸BF without equalization at the transceivers. The FF¸BF ¿lters were optimized for minimization of the relay transmit power. Exponentially de- caying power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.4 Feasibility probability vs. required SINRs γ1 and γ2 for FIR FF¸BF without equalization at the transceivers. The FF¸BF ¿lters were optimized for min- imization of the relay transmit power. Exponentially decaying power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. . . 147 5.5 Average worst¸case SINR at the transceivers vs. distance d for FF¸BF with/without equalization at the transceivers. The FF¸BF ¿lters were op- timized for maximization of the minimum transceiver SINR. Exponentially decaying channel power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . 148 5.6 Average SINR at transceivers vs. distance d for FF¸BF with/without equal- ization (EQ) at the transceivers. The FF¸BF ¿lters were optimized for maximization of the minimum transceiver SINR. Exponentially decaying channel power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.7 Average SINR at transceivers vs. distance d for FF¸BF with ZF¸LE at the transceivers. Exponentially decaying channel power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. . . . . . . . . . 151 xxiList of Figures 5.8 Average SINR vs. number of relays NR for FF¸BF with MMSE¸DFE, ZF¸ LE, MF, and slicer (no equalizer) receivers at the transceivers. Exponentially decaying power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, and γg = γh = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.9 Average BER of BPSK vs. transmit SINR γ for FF¸BF with MMSE¸DFE, MF, and slicer receiver at the transceivers. BERs for FIR FF¸BF with EQ and IIR FF¸BF with MMSE¸DFE were generated using the FF¸BF ¿lters from the achievable lower bound of the max¸min criterion. Exponentially decaying power delay pro¿le with σt = 2, Lg = Lh = 5, NR = 5, and d = 1. 154 5.10 Average BER of BPSK vs. transmit SINR γ for FF¸BF with ZF¸LE and MF receiver at the transceivers. For the min¸max criterion, BERs were generated using the FF¸BF ¿lters from the achievable lower bound. Expo- nentially decaying power delay pro¿le with σt = 2, Lg = Lh = 5, NR = 5, and d = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 xxiiList of Abbreviations 3GPP 3rd Generation Partnership Project 4G The Fourth Generation AF Amplify¸and¸Forward AMI Average Mutual Information AWGN Additive White Gaussian Noise BER Bit Error Rate BF Beamforming BICM Bit Interleaved Coded Modulation BPSK Binary Phase Shift Keying C¸BFF Cyclic Beamforming Filter CC Convolutional Code CDF Cumulative Distribution Function CIR Channel Impulse Response CP Cyclic Pre¿x CSI Channel State Information CSIT CSI at the Transmitter DF Decode¸and¸Forward DFE Decision¸Feedback Equalizer DFT Discrete Fourier Transform DSTC Distributed Space¸Time Coding xxiiiList of Abbreviations EDGE Enhanced Data Rates for GSM Evolution EQ Equalization FCC Federal Communications Commission FEC Forward Error Correction FF¸BF Filter¸and¸Forward Beamforming FD Frequency Domain FFT Fast Fourier Transform FIR Finite Impulse Response GA Gradient Algorithm GD Geodesic GS Grassmannian GSM Global System for Mobile Communications GVQ Global Vector Quantization IDFT Inverse Discrete Fourier Transform IEEE Institute of Electrical and Electronic Engineers IFFT Inverse Fast Fourier Transform i.i.d. Independent and Identically Distributed IIR In¿nite Impulse Response ISI Inter¸Symbol Interference LE Linear Equalizer LTE (3GPP) Long Term Evolution MABC Multiple Access Broadcast MF Matched Filter MIMO Multiple¸Input Multiple¸Output MLSE Maximum Likelihood Sequence Estimation xxivList of Abbreviations MMSE Minimum Mean Square Error MS Modi¿ed Spherical MSE Mean Square Error NP¸hard Non¸Deterministic Polynomial¸Time Hard OFDM Orthogonal Frequency Division Multiplexing OFDMA Orthogonal Frequency Division Multiple Access PC Power Constraint QAM Quadrature Amplitude Modulation QOQC Quadratic Objective Quadratic Constraint QoS Quality of Service QPSK Quaternary Phase Shift Keying SDP Semide¿nite Programming SINR Signal¸to¸Interference¸plus¸Noise Ratio SISO Single¸Input Single¸Output SIMO Single¸Input Multiple¸Output SOCP Second¸Order Cone Programming SM Spatial Multiplexing SNR Signal¸to¸Noise Ratio STBC Space¸Time Block Code STTC Space¸Time Trellis Code TC Transceiver TD Time Domain TDBC Time Division Broadcast WiMAX Worldwide Interoperability for Microwave Access WLAN Wireless Local Area Network xxvList of Abbreviations ZF Zero Forcing xxviNotation (·)T Transpose (·)H Hermitian transpose (·)∗ Complex conjugate 0X All¸zero column vector of length X IX X ×X identity matrix [X]ij Element of matrix X in row i and column j det(·) Matrix determinant diag{x1, . . . xN} Diagonal matrix with x1, . . . , xN on the main diagonal diag{X1, . . . , XN} Block diagonal matrix with X1, . . ., XN on the main diagonal <{·} Real part of a complex number ={·} Imaginary part of a complex number | · | Absolute value of a complex number ‖ · ‖2 Euclidean norm ‖ · ‖F Frobenius norm λmax (·) Maximum eigenvalue of a matrix λi(·) ith eigenvector of a matrix vec(·) Vectorization of a matrix (stacking the columns of · on top of each other) trace(·) Trace of a matrix x+ max(0, x) ⊗ Kronecker product xxviiNotation ⊕ Kronecker sum ∗ Discrete¸time convolution E{·} Statistical expectation F{x[k]} Fourier transform of discrete¸time signal x[k], i.e. ∞∑ k=−∞ x[k]e−j2pifk Q(·) Gaussian Q¸function [5] A  B A−B is positive semide¿nite xxviiiAcknowledgments It is a long journey for each Ph.D. degree. I am lucky that I am not walking all the way alone by myself. Without support from a large number of people, this work would have not been done. First, I would like to express my deep and sincere gratitude to my advisor, Professor Robert Schober, for his support and invaluable advice during my Ph.D. study. As a pro- found and distinguished professor, Dr. Schober sets an example of being a great researcher. The knowledge and training I got from him bene¿t me then, now and forever. I am much indebted for his patience and encouragement over the years. Without his support and guidance, this thesis would not be possible. I also would like to express my gratitude to Professor Wolfgang Gerstacker and Dr. Aissa Ikhlef for many stimulating and helpful discussions on the work we have done together. Also, I greatly thank the members of my doctoral committee, Dr. Vijay Bhargava, Dr. Vikram Krishnamurthy, Dr. Lutz Lampe, Dr. Victor Leung, and Dr. Vincent Wong for the time and e¾ort in evaluating my work and providing valuable feedback and suggestions. Many thanks go to the members of the Communications Theory Group for all the fruitful discussions, group meetings, and feedback at many points. Last but not least, I owe my deepest gratitude to my lovely wife, Yifan Tian, for her love, understanding, patience and encouragement. I thank my parents, Guosong Liang and Qinjun Li, from the bottom of my heart for their love and inspration from thousands of miles away. Special thanks are owed to my parents¸in¸law, Wendong Tian and Zhichun xxixAcknowledgments Dong, for their understanding and encouragement. Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), the University of British Columbia, the Killam Trusts, IEEE Canada, the IEEE Communications Society, and the China Scholarship Council (CSC) is gratefully acknowledged. xxxDedication To My Family xxxiChapter 1 Introduction Higher data rates, more reliable communication, and higher number of users are the main driving forces for physical layer advancement for next generation wireless communication systems. Multiple¸input¸multiple¸output (MIMO) schemes, orthogonal frequency division multiplexing (OFDM), and relaying schemes are some of the enabling techniques to achieve all the aforementioned objectives. Hence, we will provide a brief overview of some related techniques in this chapter. This chapter is organized as follows. In Section 1.1, we brieÀy review beamforming (BF) for MIMO wireless systems. In Section 1.2, we discuss one¸way relaying protocols, and in Section 1.3, we introduce two¸way relaying protocols. We brieÀy outline the contributions made in this thesis in Section 1.4, and the thesis organization is given in Section 1.5. 1.1 Beamforming for MIMO Systems In the past two decades, the application of multiple antennas at both the transmitter and the receiver has attracted considerable interest within both academia and industry as a means of providing signi¿cant performance gains over conventional single antenna based solutions [6]¸[8]. These MIMO systems enable a spatial diversity gain, a spatial multiplex- ing gain, or both, leading to high performance next generation wireless communication systems. Spatial diversity is achieved by sending the data signal over multiple independent fading 1Chapter 1. Introduction paths in space (di¾erent transmit antennas) and by utilizing appropriate combining tech- niques at the receiver. Several schemes have been proposed to exploit the spatial diversity gain. For example, space¸time block codes (STBC) [9, 10] and space¸time trellis codes (STTC) [11] are well¸known transmit diversity techniques, which lead to improved link re- liability. Spatial multiplexing techniques yield a system capacity increase by transmitting independent and separately encoded data streams from the multiple transmit antennas in parallel over the spatial channels. The maximum number of data streams the system can support is limited by the minimum of the number of transmit and the number of receive antennas. Several schemes have been proposed to exploit the spatial multiplexing gain. Examples include the vertical Bell Labs layered space¸time (V¸BLAST) [12] and the diagonal Bell Labs layered space¸time (D¸BLAST) [13] schemes. The aforementioned techniques to achieve a spatial diversity gain or a spatial multiplex- ing gain are based on the so¸called open¸loop con¿guration, where only the receiver has knowledge of the communication channel. Recent research shows that system performance can be further enhanced by so¸called closed¸loop MIMO techniques, where the transmitter also knows the channel. By exploiting channel state information (CSI) at the transmitter, transmit BF and receiver combining can be used to exploit the spatial diversity gain o¾ered by MIMO systems to mitigate the e¾ects of fading in wireless communications, cf. e.g. [14] and reference therein. In practical systems, ideal BF is not possible since the amount of information that can be fed back from the receiver to the transmitter is limited. Therefore, BF for quantized CSI and ¿nite¸rate feedback channels has recently received considerable attention [15]¸[19]. To avoid complex equalization at the receiver, MIMO is often combined with OFDM which converts broadband frequency¸selective channels into a number of parallel narrow- band frequency¸Àat channels [20]. In such a MIMO¸OFDM system, spatial multiplexing, 2Chapter 1. Introduction space¸time coding, and other signal processing algorithms are usually employed in order to approach the MIMO channel capacity. MIMO¸OFDM has been adopted in various recent standards such as IEEE 802.11 (WLAN) [21], IEEE 802.16 (Worldwide Interoperability for Microwave Access (WiMAX) standard) [22], and Long Term Evolution (LTE) [23]. Trans- mit BF techniques proposed for narrowband channels can be easily extended to broadband MIMO¸OFDM systems by applying independent BF in each sub¸carrier [24, 25]. However, the obvious drawback of this approach is that the amount of CSI data that has to be fed back from the receiver to the transmitter is prohibitively large for practical OFDM systems with moderate¸to¸large number of sub¸carriers Nc (e.g. Nc ≥ 64). Since the fading gains as well as the corresponding BF vectors are correlated across OFDM sub¸carriers, in [2] it was proposed to reduce the amount of feedback by only feeding back the BF vectors for a small number of sub¸carriers. The remaining BF vectors are obtained by modi¿ed spherical interpolation. This approach signi¿cantly reduces the required amount of feed- back at the expense of some loss in performance. The required number of feedback bits of this frequency¸domain BF (FD¸BF) scheme can be further reduced by post¸processing of the feedback bits [26] and/or by adopting improved interpolator designs such as Grass- mannian interpolators [27] or geodesic interpolators [3]. However, fundamentally for all of these FD¸BF schemes the required amount of feedback to achieve a certain performance is tied to the number of OFDM sub¸carriers. This may be problematic in OFDM systems with a large number of sub¸carriers and stringent limits on the a¾ordable amount of feed- back. Therefore, this motivates us to propose a novel time¸domain (TD) approach to BF in MIMO¸OFDM systems in Chapter 2. 3Chapter 1. Introduction 1.2 Cooperative Relay Network Multiple¸antenna processing is a promising approach to improve the capacity and reliability of next generation communication systems as pointed out in the previous section. However, such technique requires that multiple antennas are separated by at least one¸half of the wavelength of the transmitted signal from each other to obtain low correlation between the spatial channels. This requirement fundamentally limits the possibility of having multiple antennas on small communication devices. It has been recently shown that the performance of a wireless communication network can also be enhanced by relaying, which leads to an improved network coverage, throughput, and transmission reliability [28]¸[31]. Indeed, relay networks can mimic MIMO systems and introduce spatial diversity in a distributed fashion. As a result, cooperative and relay communications have been one of the most widely studied topics in communications over the past few years. The two most important relay protocols in the literature are amplify¸and¸forward (AF) relaying and decode¸and¸forward (DF) relaying [30, 31]. An illustration of the half¸duplex AF and DF protocols is given in Fig. 1.1. In both protocols, cooperative transmissions are initiated by having the source broadcasts its signal to both the relays and the destination. If the AF protocol is employed, each relay performs linear processing on the received signal and forwards the resulting signal directly to the destination without performing decoding. On the other hand, if the DF protocol is employed, each relay will decode and regenerate a new signal to the destination in the subsequent time slot. Thereby, AF relaying is generally believed to be less complex. At the destination, for AF relaying, signals from both the source and the relays are combined to provide better detection performance. If no direct link between the source and the destination is available, only signals from the relays will be used for detection. AF spatial multiplexing (AF¸SM) relaying for single¸relay networks with multiple an- 4Chapter 1. Introduction Relay DestinationSource Relay DestinationSource (a) (b) TS1 TS2 Relay DestinationSource Relay DestinationSource xs xr = xˆs xs xs xs xr = axs Figure 1.1: Di¾erent relaying protocols. (a) decode¸and¸forward (DF) relaying protocol, and (b) amplify¸and¸forward (AF) protocol. xs: signal broadcasted from the source, xr: signal transmitted from the relay, xˆs: regenerated signal after decoding, TSx: time slot x, and a: scaling factor. tennas at the source, the relay, and the destination was discussed in [32]¸[34]. However, in downlink transmission the destination node can often support only a single antenna. In this case, BF is an eÁcient and popular approach to exploit the spatial diversity o¾ered by the channel. AF¸BF for wireless relay networks was considered in [35]¸[44] and ref- erences therein. In particular, AF¸BF for networks with one single¸antenna source and multiple single¸antenna relays was considered in [35, 36, 40, 41] and [39, 42] under a joint power constraint for all relays and individual relay power constraints, respectively. Since both the source and the relays were assumed to have only one antenna, respectively, the resulting signal¸to¸interference¸noise ratio (SINR) maximization problem at the destina- tion involved only the optimization of one scalar BF gain for each relay. In contrast, in [37, 38], AF¸BF for a network with a single relay and multiple antennas at the relay and the source was investigated and closed¸form solutions for the BF vector at the source and 5Chapter 1. Introduction the AF¸BF matrix at the relay were provided. Furthermore, in [43, 44], the performance of AF¸BF with multiple antennas at the source and one single¸antenna relay was investi- gated. However, in practice, a relay network may comprise multiple relays and both the relays and the source may have multiple antennas. The extension of the results provided in those aforementioned paper to this general case is not straightforward. This problem will be discussed in details in Chapter 3. The combination of relaying and OFDM has also attracted a lot of attention recently. Relaying for wideband OFDM¸based cooperative networks is investigated in [45]¸[48]. In [45], a relay network with one transmitter¸receiver pair and a single AF relay is considered and all three nodes are equipped with multiple antennas. A power allocation scheme which maximizes the instantaneous rate of the network is proposed for this scenario. Linear ¿ltering for relay networks with one relay node was introduced in [46]. In [49], a time¸ domain AF¸BF scheme for cooperative OFDM networks with multiple relays is proposed. As mentioned above, BF for cooperative networks with single¸carrier transmission over frequency¸nonselective channels and multi¸carrier transmission over frequency¸selective channels has been studied extensively in the literature. In contrast, the literature on BF (and other relay processing techniques as well) for single¸carrier transmission over frequency¸selective channels is very sparse. Nevertheless, wireless channels are typically frequency selective and multi¸carrier modulation is not applicable in still evolving legacy systems such as Global System for Mobile Communications (GSM) and Enhanced Data Rates for GSM Evolution (EDGE) whose standard is still being further extended, and wireless sensor networks, for which the cost and power consumption of the highly lin- ear power ampli¿ers required for OFDM may be prohibitive. To compensate for the ef- fect of frequency¸selective channels, ¿lter¸and¸forward (FF) beamforming (FF¸BF) for frequency¸selective channels is proposed in [50]. However, in [50], only a simple slicer was 6Chapter 1. Introduction employed at the destination requiring the FF¸BF ¿lters at the relays to equalize both the source¸relay and the relay¸destination channels. This motivates us to consider the case if a simple linear equalization (LE) or decision¸feedback equalization (DFE) is performed at the destination in Chapter 3. In recent years, cooperative communication and relay technologies have gradually made their way into wireless standards, such as IEEE 802.16j [51] (an amendment to IEEE 802.16e mobile WiMAX standard), and LTE¸Advanced [52]. The goal of utilizing cooper- ative communications in both standards is to increase the data rates available to cell¸edge users and to increase coverage at a given data rate. 1.3 Two¸Way Cooperative Relay Network Most of the published results on distributed beamforming consider a one¸way relaying pro- tocol where the relays cooperate with each other to deliver the signals transmitted from a source (or several sources) to a destination (or several destinations). In two¸way relay- ing, the relays cooperate with each other to establish reliable bidirectional communication between two transceivers [53, 54]. The capacity and achievable rate region for two¸way relaying protocols have been studied in [55, 56] and references therein. The choice between one¸way and two¸way relaying mainly depends on the application. One¸way relaying is of interest for unidirectional communication, whereas two¸way relaying is preferable for bidirectional communication. Various protocols for two¸way relaying exist in the literature. The most common two¸ way relaying protocols are the bidirectional one¸way relaying protocol, the time division broadcast (TDBC) protocol, and the multiple access broadcast (MABC) protocol. To achieve bidirectional communication between two transceivers, a straightforward approach is to employ two successive one¸way relaying operations, as shown in Fig. 1.2 (a). However, 7Chapter 1. Introduction XOR(s1, s2) TS1 TS3 TS4 TS2 Relay TC2TC1 (a) (b) s1 s1 s2 s2 (c) TC2RelayTC1 s1 s2 RelayTC1 TC2 s1 s2 f(s1, s2) f(s1, s2) XOR(s1, s2) Figure 1.2: Di¾erent two¸way relaying protocols. (a) Bidirectional one¸way relaying pro- tocol, (b) Time Division Broadcast (TDBC) protocol, and (c) Multiple Access Broadcast (MABC) protocol. s1: signal transmitted from transceiver 1 (TC1), s2: signal transmitted from transceiver 2 (TC2), and f(s1, s2): processed version of the received signals. this protocol requires four time slots to accomplish the exchange of signals between the two transceivers, which is costly from a bandwidth eÁciency point of view. A single¸relay TDBC protocol was introduced in [53] where a network coding based method was used to reduce the required number of time slots from four to three as shown in Fig. 1.2 (b). In the ¿rst two time slots, the transceivers transmit the signals to the relays and during the third time slot the relays broadcast the XOR version of the decoded signals. As a result, each transceiver can retrieve its signal of interest by performing an XOR operation on its transmitted signal and its received signal. Intuitively, TDBC is better than the ¿rst protocol in terms of bandwidth eÁciency. A detail comparison of the bidirectional one¸way relaying protocol and the single¸relay TDBC protocol is given in [54]. The third protocol is called MABC, cf. e.g. [57, 58] and references therein. When direction link between the two transceivers does not exist, MABC is considered the most bandwidth eÁcient of the three protocols [56, 58]. In this protocol, the transceivers simultaneously send the signals to the relays during the ¿rst time slot and the relays broadcast a processed version of the received signals during the second time slot, as shown in Fig. 1.2 (c). Recently, a few papers have studied the beamforming problem in two¸way relay networks. In [57, 59, 8Chapter 1. Introduction 60], a two¸way relay network with single multi¸antenna relays is considered, whereas [58] considered a network with multiple single antenna relays. However, all the aforementioned papers consider frequency¸nonselective channels. This motivates us to investigate two¸way relaying schemes for frequency¸selective channels in Chapter 5. 1.4 Contributions of the Thesis This thesis considers BF schemes for performance enhancement that may ¿nd application in several current or upcoming wireless communication standards. The main contributions of this thesis are as follows. 1. We propose a novel time¸domain approach to BF in MIMO¸OFDM systems. The proposed time¸domain BF scheme employs cyclic BF ¿lters. Simulation results con- ¿rm the excellent performance of the proposed scheme and show that time¸domain BF has a more favorable performance/feedback rate trade¸o¾ than previously pro- posed frequency¸domain BF schemes. 2. We propose BF schemes for cooperative networks with one multi¸antenna source, multiple multi¸antenna AF relays, and one single¸antenna destination. For a given BF vector at the source, we ¿nd the optimal AF¸BF matrices at the relays for each of the three considered power constraints, namely individual relay power constraint, joint relay power constraint and joint source¸relay power constraint. Several numer- ical methods for ¿nding the optimal source BF vectors are also proposed. 3. We investigate FF¸BF for one¸way relay networks employing single¸carrier trans- mission over frequency¸selective channels. We consider two cases for the receive pro- cessing at the destinations: (1) a slicer and (2) LE/DFE. For both cases, we optimize the FF¸BF ¿lters for maximization of the SINR under a transmit power constraint. 9Chapter 1. Introduction In addition, for case (1), we also optimize the FF¸BF ¿lters for minimization of the transmit power under a QoS constraint, respectively. We ¿nd closed¸form/near¸ optimal solutions for the IIR and FIR FF¸BF matrix ¿lters at the relays. 4. Drawing from the ¿ndings on one¸way relaying, we investigate FF¸BF for two¸way relay networks with multiple single¸antenna relays. We consider two cases for the receive processing at the transceivers: (1) a slicer and (2) LE/DFE. For both cases, we optimize the FF¸BF ¿lters at the relays for an SINR balancing objective under a relay transmit power constraint. Additionally, for case (1) we also consider the optimization of the FF¸BF ¿lters for minimization of the total transmit power subject to two QoS constraints to guarantee a certain level of performance. For case (1), we show that the optimization problems are convex. For case (2), we provide an upper bound and an achievable lower bound for the optimization problem, and our results show that the gap between both bounds is small. 1.5 Organization of the Thesis In the following, we provide a brief overview of the remainder of this thesis. In Chapter 2, we propose a novel single¸data stream, time¸domain BF scheme for MIMO¸OFDM systems which uses cyclic BF ¿lters (C¸BFFs). Assuming perfect CSI at the transmitter, the C¸BFFs are optimized for two di¾erent criteria, namely, maximum average mutual information (AMI) per sub¸carrier and minimum average uncoded bit error rate (BER). If the C¸BFF length Lg is equal to the number of sub¸carriers Nc, closed¸form solutions to both optimization problems exist. For the practically relevant case Lg < Nc we present numerical methods for calculation of the optimum C¸BFFs for both criteria. Using a global vector quantization (GVQ) approach, the C¸BFFs are quantized 10Chapter 1. Introduction for practical ¿nite¸rate feedback channels. Simulation results for typical IEEE 802.11n channels con¿rm the excellent performance of the proposed scheme and show that TD¸ BF has a more favorable performance/feedback rate trade¸o¾ than previously proposed FD¸BF schemes. In Chapter 3, we consider BF for cooperative networks with one multi¸antenna source, multiple multi¸antenna AF relays, and one single¸antenna destination. The source BF vector and the AF¸BF matrices at the relays are optimized for maximization of the SINR at the destination under three di¾erent power constraints. In particular, we consider indi- vidual relay power constraints, a joint relay power constraint, and a joint power constraint for the source and the relays. We solve the associated optimization problems in two stages. In the ¿rst stage, we ¿nd the optimal AF¸BF matrices for a given BF vector at the source. For the cases of individual and joint relay power constraints, closed¸form solutions for the AF¸BF matrices are provided, respectively. Furthermore, for the case of a joint source¸ relay power constraint, the direction of the AF¸BF matrices is derived in closed form and an eÁcient numerical algorithm for the power allocation between the source and the relays is provided. In the second stage, the optimal source BF vectors are computed. Thereby, we show that for the joint relay and the joint source¸relay power constraints, the resulting problem can be transformed into a non¸convex polynomial programming problem which allows for an exact solution for small scale networks. For large scale networks and net- works with individual relay power constraints, we propose eÁcient suboptimal optimization methods for the source BF vector. Simulation results show the bene¿ts of having multiple antennas at the source and/or multiple multi¸antenna relays and illustrate the performance di¾erences introduced by the three di¾erent power constraints. In Chapter 4, we investigate FF¸BF for relay networks employing single¸carrier trans- mission over frequency¸selective channels. In contrast to prior work, which concentrated 11Chapter 1. Introduction on multiple single¸antenna relay nodes, we consider networks employing multiple multi¸ antenna relay nodes. For the processing at the destination, we investigate two di¾erent cases: (1) a simple slicer without equalization and (2) a LE or a DFE. For both cases, we optimize the FF¸BF matrix ¿lters at the relays for maximization of the SINR under a transmit power constraint, and for the ¿rst case we consider additionally optimization of the FF¸BF matrix ¿lters for minimization of the total transmit power under a quality of service (QoS) constraint. For the ¿rst case, we obtain closed¸form solutions for the optimal FIR FF¸BF matrix ¿lters, whereas for the second case, we provide the optimal solution for IIR FF¸BF matrix ¿lters, and an eÁcient gradient algorithm for recursive calculation of near¸optimal FIR FF¸BF matrix ¿lters. Our simulation results reveal that for a given total number of antennas in the network, a small number of multiple¸antenna relays can achieve signi¿cant performance gains over a large number single¸antenna relays. In Chapter 5, we consider FF¸BF for two¸way relay networks employing single¸carrier transmission over frequency¸selective channels. We adopt the MABC protocol for two¸way relaying with single¸antenna relays is assumed. Similar to the one¸way relaying with FF¸ BF, the relay nodes ¿lter the received signal using FIR or IIR ¿lters. For the processing at the transceivers, we investigate two di¾erent cases: (1) a simple slicer without equalization and (2) LE/DFE. For the ¿rst case, we optimize FIR FF¸BF ¿lters, respectively, for max- imization of the minimum transceiver SINR subject to a relay transmit power constraint and for minimization of the total relay transmit power subject to two QoS constraints to guarantee a certain level of performance. We show that both problems can be transformed into a convex second¸order cone programming (SOCP) problem, which can be eÁciently solved using standard tools. For the second case, we optimize IIR and FIR FF¸BF ¿lters for max¸min optimization of the SINR, and for transceivers with zero¸forcing LE, also for minimization of the sum MSE at the equalizer outputs of both transceivers. Leveraging 12Chapter 1. Introduction results from FF¸BF for one¸way relaying, we establish an upper and an achievable lower bound for the max¸min problem and an exact solution for the sum MSE problem. Since the gap between the upper and the lower bound for the max¸min problem is small, a close¸to¸ optimal solution is obtained. Our simulation results reveal that the performance of FF¸BF without equalization at the transceivers crucially depends on the slicer decision delay and transceivers with slicers can closely approach the performance of transceivers with equal- izers provided that the FF¸BF ¿lters are suÁciently long and a suÁcient number of relays is deployed. Finally, Chapter 6 summarizes the contributions of this thesis and outlines areas of future research. 13Chapter 2 Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 2.1 Introduction As pointed out in Chapter 1, transmit BF and receiver combining are simple yet eÁcient techniques for exploiting the bene¿ts of MIMO¸OFDM systems [14]. Several FD¸BF with CSI feedback reduction schemes have been proposed in recent publications, e.g. [2, 3, 26, 27]. With the observation that the fading gains as well as the corresponding BF vectors are correlated across OFDM sub¸carriers, [2] proposed to reduce the amount of feedback by only feeding back the BF vectors for a small number of so¸called pilot sub¸ carriers. The remaining BF vectors are obtained by modi¿ed spherical interpolation. This approach signi¿cantly reduces the required amount of feedback at the expense of some loss in performance. The required number of feedback bits of this FD¸BF scheme can be further reduced by post¸processing of the feedback bits [26] and/or by adopting improved interpolator designs such as Grassmannian interpolators [27] or geodesic interpolators [3]. However, fundamentally for all of these FD¸BF schemes the required amount of feedback to achieve a certain performance is tied to the number of OFDM sub¸carriers. In this chapter, we propose a novel TD approach to BF in MIMO¸OFDM systems 1 . 1 In this chapter, we only consider single¸stream BF which is sometimes also referred to as maximal¸ ratio transmission. We note, however, that the concept of TD¸BF can also be extended to multi¸stream BF which is also referred to as spatial multiplexing. 14Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems The motivation for considering a TD approach is that the fading correlations in the FD, which are exploited for interpolation in [2, 3, 27], have their origin in the TD. These correlations are due to the fact that the number of sub¸carriers is typically much larger than the number of non¸zero channel impulse response (CIR) coeÁcients. Therefore, tackling the problem directly in the TD is a natural choice. The proposed TD¸BF scheme employs C¸BFFs of length Lg ≤ Nc. The C¸BFFs are optimized for maximization of the AMI and for minimization of the average uncoded BER, respectively. While other C¸BFF optimization criteria are certainly possible (e.g., maximum cut¸o¾ rate, minimum coded BER), the adopted criteria can be considered as extreme cases in the sense that they cater to systems using very powerful (ideally capacity¸achieving) forward error correction (FEC) coding (AMI criterion) and systems with weak or no FEC coding (uncoded BER criterion), respectively. For perfect CSI both criteria lead to (di¾erent) nonlinear eigenvalue problems for the C¸BFF coeÁcient vectors, and we show that closed¸form solutions to both problems exist for Lg = Nc. However, for the practically more interesting case of Lg < Nc, a closed¸form solution does not exist for either problem, and we provide eÁcient numerical methods for calculation of the C¸BFFs. Furthermore, for the case of a ¿nite¸rate feedback channel we draw from the ¿ndings in [61, 62] and propose a global vector quantization (GVQ) algorithm for maximum AMI and minimum BER codebook design, respectively. This chapter also provides a detailed comparison between TD¸BF and FD¸BF [2, 3, 27]. We note that TD pre¸processing for MIMO¸OFDM has been considered in di¾erent contexts before. For example, TD¸BF schemes with one scalar BF weight per antenna (as opposed to C¸BFFs) have been proposed to reduce the number of inverse discrete Fourier transforms (IDFTs) required at the transmitter of MIMO¸OFDM systems, cf. e.g. [63] and references therein. Similarly, cyclic delay diversity, which is a simple form of space¸time coding, cf. e.g. [64, 65], may be viewed as a TD MIMO¸OFDM pre¸processing technique. 15Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems Furthermore, BF with linear BFFs has been considered for single¸carrier transmission over frequency¸selective channels and DFE at the receiver in [62]. However, the concept of employing C¸BFFs for (limited feedback) BF in MIMO¸OFDM systems is novel and has not been considered before. The remainder of this chapter is organized as follows. In Section 2.2, the considered system model is presented. The optimization of the C¸BFFs for maximization of the AMI and minimization of the average BER is discussed in Sections 2.3 and 2.4, respectively. In Section 2.5, a GVQ algorithm for ¿nite¸rate feedback TD¸BF and a detailed comparison between TD¸BF and FD¸BF are presented. Simulation results are provided in Section 2.6, and some conclusions are drawn in Section 2.7. 2.2 System Model We consider a MIMO¸OFDM system with NT transmit antennas, NR receive antennas, and Nc OFDM sub¸carriers. The block diagram of the discrete¸time overall transmission system in equivalent complex baseband representation is shown in Fig. 2.1. In the next four subsections, we introduce the models for the transmitter, the channel, the receiver, and the feedback channel. 2.2.1 Transmitter Processing for TD¸BF The modulated symbols D[n], 0 ≤ n < Nc, are taken from a scalar symbol alphabet A and have variance σ2D = E{|D[n]|2} = 1. The transmit symbol vector x , [x[0] x[1] . . . x[Nc − 1]]T after the IDFT operation can be represented as x , WD, (2.1) 16Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems D[0] Y [0] Y [Nc − 1] Remove CP n1[k] nNR[k] Add CP & P/S DFT & DFT & CE CE C∗1[0] C∗1 [Nc − 1] C∗NR[0] C∗NR[Nc − 1] D[Nc − 1] Feedback Channel S/P Remove CP S/P & & Add CP & P/S IDFT C-BFF C-BFF Figure 2.1: MIMO¸OFDM system with TD¸BF. P/S: Parallel¸to¸serial conversion. S/P: Serial¸to¸parallel conversion. CE: Channel estimation. where D , [D[0] D[1] . . . D[Nc− 1]]T and W is the unitary IDFT matrix [66], i.e., x[k] = 1√ Nc ∑Nc−1 n=0 D[n]ej2pink/Nc . At transmit antenna nt sequence x[k] is ¿ltered with a C¸BFF with impulse response gnt [k], 0 ≤ k < Lg, 1 ≤ nt ≤ NT , of length Lg ≤ Nc. The resulting OFDM symbol after cyclic ¿ltering is given by snt = ¯Gntx, (2.2) where ¯Gnt is an Nc × Nc column¸circulant matrix with ¿rst column [gTnt 0TNc−Lg ]T , gnt , [gnt[0] gnt[1] . . . gnt [Lg − 1]]T . We note that in practice the cyclic ¿ltering in (2.2) can be implemented using the following three simple steps: 1. Add a cyclic pre¿x (CP) of length Lg−1 to x to generate x¯ , [x[Nc−Lg+1] . . . x[Nc− 1] xT ]T . 2. Pass the elements of x¯ through a linear ¿lter with coeÁcients gnt [k], 0 ≤ k < Lg, to generate s¯nt , [s¯nt[0] s¯nt[1] . . . s¯nt [Nc+Lg−2]]T , where s¯nt [k] = ∑Lg−1κ=0 gnt [κ]x¯[k−κ] and x¯[k], 0 ≤ k < Nc + Lg − 1, are the elements of x¯ and x¯[k] = 0 for k < 0. 17Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 3. Remove the CP from s¯nt to obtain snt = [s¯nt [Lg − 1] . . . s¯nt [Nc + Lg − 2]]T . After cyclic ¿ltering a CP is added to snt . We assume that the CP length is not smaller than L − 1, where L is the length of the CIR. We note that due to the cyclic structure of ¯Gnt , TD¸BF does not a¾ect the length requirements of the CP, i.e., the required CP length for TD¸BF is identical to that for single¸antenna transmission. 2.2.2 MIMO Channel We model the wireless channel as a frequency¸selective and spatially correlated MIMO channel. The spatial correlations may be introduced by insuÁcient antenna spacing. The channel between transmit antenna nt and receive antenna nr is characterized by its impulse response hntnr [l], 0 ≤ l < L. Note that the impulse response coeÁcients for a given transmit/receive antenna pair are also generally mutually correlated due to transmit and receive ¿ltering. As is typically done in the BF literature, e.g. [15]¸[19], [2, 3, 27], we assume that the transmitted data is organized in frames. The channel remains constant during each frame but changes randomly between frames (block fading model). 2.2.3 Receiver Processing TD¸BF does not a¾ect the processing at the receiver, i.e., standard OFDM receiver pro- cessing is applied. After CP removal the discrete¸time received signal at receive antenna nr, 1 ≤ nr ≤ NR, can be modeled as rnr = NT∑ nt=1 ¯Hntnr ¯Gntx + nnr , (2.3) where ¯Hntnr is an Nc×Nc column¸circulant matrix with ¿rst column [hntnr[0] . . . hntnr [L− 1] 0TNc−L]T and nnr is an additive white Gaussian noise (AWGN) vector whose entries 18Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems nnr [k], 0 ≤ k < Nc, are independent and identically distributed (i.i.d.) with zero mean and variance σ2n. After DFT we obtain at antenna nr Rnr = W Hrnr = NT∑ nt=1 HntnrGntD + Nnr , (2.4) where Hntnr , W H ¯HntnrW = diag{Hntnr[0] . . . Hntnr [Nc − 1]}, Gnt , W H ¯GntW = diag{Gnt[0] . . . Gnt [Nc−1]}, and Nnr , W Hnnr = [Nnr[0] . . . Nnr [Nc− 1]]T . The Nnr [n], 0 ≤ n < Nc, are i.i.d. AWGN samples with variance σ2n. The FD channel gains Hntnr [n] and the C¸BFF gains Gnt [n] are given by Hntnr [n] , L−1∑ l=0 hntnr [l]e−j2pinl/Nc , (2.5) Gnt [n] , Lg−1∑ l=0 gnt [l]e−j2pinl/Nc . (2.6) Considering now the nth sub¸carrier and assuming an NR¸dimensional receive combining vector C[n] , [C1[n] . . . CNR [n]]T , with (2.4) the combined received signal can be expressed as Y [n] = CH [n]H [n]G[n]D[n] + CH [n]N [n], 0 ≤ n < Nc, (2.7) where NR×NT matrix H [n] contains Hntnr [n] in row nr and column nt, G[n] , [G1[n] . . . GNT [n]]T , and N [n] , [N1[n] . . . NNR [n]]T . In this chapter, we assume that the receiver has perfect knowledge of H [n], 0 ≤ n < Nc. In this case, the combining vector C[n] that maximizes the SNR of Y [n] is given by C[n] = H [n]G[n] (maximal¸ratio combining). 19Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 2.2.4 Feedback Channel We assume that a feedback channel from the receiver to the transmitter is available, cf. Fig. 2.1. In the idealized case, where the feedback channel has in¿nite capacity, the receiver sends the unquantized C¸BFF vector g, g , [gT1 . . . gTNT ]T , to the trans- mitter (perfect CSI case). In the more realistic case, where the feedback channel can only support the transmission of B bits per channel update, the receiver and the trans- mitter have to agree on a pre¸designed C¸BFF vector codebook G , {gˆ1, gˆ2, . . . , gˆN} of size N = 2B, where gˆn is an NT Lg¸dimensional vector. For a given channel vector h , [h11[0] h11[1] . . . h11[L − 1] h21[0] . . . hNT NR [L − 1]]T the receiver determines the ad- dress n of the codeword (C¸BFF vector) gˆn ∈ G, 1 ≤ n ≤ N , which maximizes the prescribed optimality criterion (maximum AMI or minimum BER). Subsequently, index n is sent to the transmitter which then utilizes g = gˆn for BF. Similar to [2, 3, 27] we assume that the feedback channel is error¸free and has zero delay. 2.3 Maximum AMI Criterion In this section, we optimize the C¸BFFs for maximization of the AMI per sub¸carrier. After rigorously formulating the optimization problem, we present a closed¸form solution for Lg = Nc and numerical methods for computation of the optimum C¸BFFs for Lg < Nc. 2.3.1 Formulation of the Optimization Problem Assuming i.i.d. Gaussian input symbols D[·], the mutual information (in bit/s/Hz) of the nth sub¸carrier is given by [66] C[n] = log2 (1 + SNR[n]) . (2.8) 20Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems For maximal¸ratio combining the SNR of the nth sub¸carrier can be obtained from (2.7) as SNR[n] = 1 σ2n GH [n]HH [n]H [n]G[n]. (2.9) We note that G[n] can be expressed as G[n] = F [n]g, (2.10) where the ntth row of NT × NT Lg matrix F [n] is given by [0T(nt−1)Lg fT [n] 0T(NT −nt)Lg ], 1 ≤ nt ≤ NT , with f [n] , [1 e−j2pin/Nc . . . e−j2pi(Lg−1)n/Nc ]T . Therefore, the AMI per sub¸ carrier depends on g and is given by C = 1Nc ∑Nc−1 n=0 C[n]. The optimization problem can now be formulated as max g Nc−1∑ n=0 C[n] (2.11) s.t. gHg = 1, (2.12) where (2.12) is a transmit power constraint. 2.3.2 Solution of the Optimization Problem for Lg = Nc Although in practice Lg  Nc is desirable to minimize the amount of feedback, it is insight- ful to ¿rst consider Lg = Nc since in this case a closed¸form solution to the optimization problem in (2.11), (2.12) exists. In addition, the solution for Lg = Nc serves as a per- formance upper bound for the practically relevant case Lg < Nc. For Lg = Nc matrix F , [F T [0] . . . F T [Nc− 1]]T is invertible, and for a given G , [GT [0] . . . GT [Nc− 1]]T the 21Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems C¸BFF vector g can be obtained from g = F −1G, (2.13) cf. (2.10). This means (2.11) and (2.12) are equivalent to max G Nc−1∑ n=0 log2 ( 1 + 1 σ2n GH [n]HH [n]H [n]G[n] ) (2.14) s.t. GHG = Nc. (2.15) The solution to this equivalent problem can be obtained as G[n] = α[n]Emax[n], 0 ≤ n < Nc, (2.16) where Emax[n] is that eigenvector of matrix HH [n]H [n] which corresponds to the maximum eigenvalue λmax[n], and α[n] is obtained from α[n] = √ Ncσ2n ( 1 λ − 1 Ncλmax[n] )+ , (2.17) where x+ , max(0, x) and λ is the solution to the water¿lling equation σ2n Nc−1∑ n=0 ( 1 λ − 1 Ncλmax[n] )+ = 1. (2.18) Once G[n], 0 ≤ n < Nc, has been calculated, the optimum g can be obtained from (2.13). Therefore, in this case, TD¸BF is equivalent to ideal FD¸BF with water¿lling which is not surprising since for Lg = Nc there are as many degrees of freedom in the TD as there are in the FD. 22Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 2.3.3 Solution of the Optimization Problem for Lg < Nc For Lg < Nc the NT Nc × NT Lg matrix F is not invertible, i.e., (2.11) and (2.14) are not equivalent anymore 2 . For convenience we rewrite (2.11), (2.12) as max g Nc−1∑ n=0 log2 ( 1 + 1 σ2n gHM [n]g ) (2.19) s.t. gHg = 1 (2.20) with NT Lg × NT Lg matrix M [n] , F H [n]HH [n]H [n]F [n]. Unfortunately, the objective function in (2.19) is not a concave function, i.e., (2.19), (2.20) is not a convex optimization problem. In fact, (2.19) and (2.20) are equivalent to the maximization of a product of Rayleigh coeÁcients ˜L(g) , Nc−1∏ n=0 gH ( σ2nINT Lg + M [n] ) g gHg , (2.21) which is a well¸known diÁcult mathematical problem that is not well understood for Nc > 1, cf. e.g. [67, 68]. In the remainder of this subsection, we will ¿rst consider a relaxation of (2.19), (2.20) to ¿nd a suboptimum solution and then provide a numerical algorithm for calculation of the optimum C¸BFF vector. 1) Relaxation of the Optimization Problem: A popular approach for solving non¸convex optimization problems is to transform the original non¸convex problem into a convex one by relaxing the constraints [1]. This leads in general to a suboptimum (but often close¸ to¸optimum) solution for the original problem. For the problem at hand we may de¿ne a 2 Note that pseudo inverse can be used as an alternative way to ¿nd the optimal g in this case. However, we found that the resulting performance is not comparable with the performance from the algorithm introduced in this section. 23Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems matrix S , ggH and rewrite (2.19), (2.20) as max S Nc−1∑ n=0 log2 det ( INR + 1 σ2n H [n]F [n]SF H [n]HH [n] ) (2.22) s.t. trace{S} ≤ 1, (2.23) S  0, (2.24) rank{S} = 1, (2.25) where S  0 means that S is a positive¸semide¿nite matrix. It is easy to show that equality is satis¿ed in (2.23) when S is optimal. The equivalent optimization problem in (2.22)¸(2.25) is still non¸convex due to the rank condition in (2.25) but can be relaxed to a convex problem by dropping this rank condition. The resulting relaxed problem is a convex semide¿nite programming (SDP) problem which can be solved with standard algorithms, cf. [1]. If the S found by this procedure has rank one, the corresponding g is also the solution to the original, non¸convex problem. On the other hand, if the optimum S does not have rank one, the eigenvector of S corresponding to its maximum eigenvalue can be used as (suboptimum) approximate solution to the original non¸convex problem. Unfortunately, the amount of time to solve the relaxed optimization problem strongly depends on Nc, and for medium numbers of sub¸carriers (e.g. Nc ≥ 64) standard optimiza- tion software (e.g. ³yalmip´ and ³SeDuMi´) takes a very long time to ¿nd the optimum S. Therefore, this relaxation approach is most useful for the practically less relevant case when the number of sub¸carriers is small (e.g. Nc < 64). 2) Gradient Algorithm: The Lagrangian of (2.19), (2.20) can be formulated as L(g) = Nc−1∑ n=0 log2 ( 1 + 1 σ2n gHM [n]g ) − µgHg, (2.26) where µ denotes the Lagrange multiplier. The optimum C¸BFF vector has to ful¿ll 24Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems ∂L(g)/∂g∗ = 0NT Lg , which leads to the non¸linear eigenvalue problem [ Nc−1∑ n=0 M [n] σ2n + gHM [n]g ] g = µg. (2.27) For very low SNRs (i.e., σ2n →∞) the optimum C¸BFF vector can be obtained from (2.27) as the unit¸norm eigenvector of ∑Nc−1 n=0 M [n] which corresponds to the maximum eigen- value of that matrix, i.e., a closed¸form solution exists for this special case. Unfortunately, the low SNR solution for g does not yield a good performance for ¿nite, practically rele- vant SNRs. Therefore, we provide in Table 2.1 a gradient algorithm (GA) for optimization problem (2.19), (2.20). Since the considered problem (2.19), (2.20) is not a convex opti- mization problem, we cannot guarantee that the GA will converge to the globally optimum solution. However, if the step size δi is chosen appropriately, the GA will converge to a local optimum, cf. e.g. [69] for guidelines on the choice of step sizes for GAs. To which local optimum the GA converges, generally depends on the initial vector g0. For the problem at hand, our simulations have shown that the choice of the initial vector g0 is not critical and the GA always achieved very similar AMI values for di¾erent random g0. Furthermore, for those cases where the relaxation method discussed in 1) found the solution to the original problem (2.19), (2.20), i.e., S had rank one, the solution found with the GA achieved the same AMI. We note that the speed of convergence of the GA depends on the adaptation step size δi. For the results shown in Section 2.6, we have adopted the backtracking line search procedure outlined in [69, p. 41], which optimizes the step size δi in each iteration. Thereby, starting from an initial value δi = ¯δ > 0 the step size is gradually reduced as δi ← ρ δi with contraction factor ρ ∈ (0, 1) until the so¸called Armijo condition with constant c is ful¿lled [69, p. 41]. We found that for the problem at hand, the GA in Table 2.1 with backtracking line search (c = 0.49, ρ = 0.9, and ¯δ = 1) typically terminates after around 100 iterations 25Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems Table 2.1: Calculation of the optimum C¸BFFs g for the maximum AMI and the minimum average BER criterion using a GA, respectively. Termination constant  has a small value (e.g.  = 10−4). i denotes the iteration and δi is the adaptation step size necessary for the GA. 1 Let i = 0 and initialize the C¸BFF vector with some g0 ful¿lling gH0 g0 = 1. 2 Update the C¸BFF vector: AMI: g˜i+1 = gi + δi [ Nc−1∑ n=0 M [n] σ2n + gHi M [n]gi ] gi BER: g˜i+1 = gi + δi [ Nc−1∑ n=0 exp ( − c2 σ2n gHi M [n]gi ) M [n] ] gi 3 Normalize the C¸BFF: gi+1 = g˜i+1√ g˜Hi+1g˜i+1 4 If 1− |gHi+1gi| < , goto Step 5, otherwise increment i→ i + 1 and goto Step 2. 5 gi+1 is the desired C¸BFF vector. if the termination constant (de¿ned in Table 2.1) is set to  = 10−4. However, in practice, the speed of convergence of the GA is not critical, since in the realistic ¿nite¸rate feedback case, the GA is only used to ¿nd the C¸BFF codebook, which is done o¾¸line. 2.4 Minimum BER Criterion The main criterion considered for C¸BFF optimization in this section is the BER averaged over all sub¸carriers. However, we will also consider the minimization of the maximum sub¸ carrier BER for optimization of the C¸BFFs. Besides the additional insight that this second BER criterion o¾ers, it also provides a useful starting point for numerical computation of the minimum average BER C¸BFF ¿lters, cf. Section 2.4.3. 26Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 2.4.1 Formulation of the Optimization Problems While closed¸form expressions for the BER or/and symbol error rate exist for most regular signal constellations such as M¸ary quadrature amplitude modulation (M¸QAM) and M¸ary phase¸shift keying (M¸PSK), these expressions are quite involved which is not desirable for C¸BFF optimization. Therefore, we adopt here the simple yet accurate BER approximations from [70], which allow us to express the approximate BER of the nth sub¸carrier as BER[n] ≈ c1 exp (−c2 SNR[n]) , (2.28) where the nth sub¸carrier SNR is de¿ned in (2.9) and c1 and c2 are modulation dependent constants. For example, for square M¸QAM we have c1 = 0.2 and c2 , 3/[2(M − 1)] [70]. Throughout this chapter we assume that all sub¸carriers use the same modulation scheme. 1) Average BER Criterion: The (approximate) average BER is given by BER = 1 Nc ∑Nc−1 n=0 BER[n]. Consequently, the minimum average BER optimization problem can be formulated as min g Nc−1∑ n=0 BER[n] (2.29) s.t. gHg = 1. (2.30) 2) Max¸Min Criterion: Since the exponential function is monotonic, we observe from (2.28) that minimizing the maximum sub¸carrier BER is equivalent to maximizing the minimum sub¸carrier SNR. The resulting max¸min problem becomes max g min ∀n SNR[n] (2.31) s.t. gHg = 1. (2.32) 27Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems Since for high SNR, the maximum sub¸carrier BER dominates the average BER, we expect that in this case both optimization criteria lead to similar performances. 2.4.2 Solution of the Optimization Problems for Lg = Nc For the solution of the optimization problem we exploit again the fact that for Lg = Nc matrix F is invertible, i.e., for a given G the C¸BFF vector g can be obtained from (2.13). 1) Average BER Criterion: Eq. (2.13) implies that (2.29) and (2.30) are equivalent to min G Nc−1∑ n=0 exp ( − c2 σ2n GH [n]HH [n]H [n]G[n] ) (2.33) s.t. GHG = Nc. (2.34) Formulating (2.33) and (2.34) as a Lagrangian, it can be shown that the optimum G[n] is again proportional to Emax[n], i.e., (2.16) is still valid. However, now α[n] in (2.16) is given by α[n] = √ σ2n c2λmax[n] [ ln ( λmax[n] λ )]+ , (2.35) where λ is the solution to the water¿lling problem σ2n c2Nc Nc−1∑ n=0 [ ln (λmax[n]/λ) λmax[n] ]+ = 1. (2.36) For high SNR, i.e., σ2n  1, λmax[n] > λ, 0 ≤ n < Nc, holds and the sub¸carrier BER can be calculated as BER[n] = c1λ/λmax[n], where λ = exp([∑Nc−1n=0 (ln(λmax[n])/λmax[n])− c2Nc/σ2n]/[∑Nc−1n=0 1/λmax[n]]), cf. (2.9), (2.28), (2.35), and (2.36). This means for high SNR the sub¸carrier BER is inversely proportional to the maximum sub¸carrier eigenvalue λmax[n]. 28Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 2) Max¸Min Criterion: Exploiting (2.13) also for the max¸min criterion, it can be shown that the optimum solution has again the general form given by (2.16) with α[n] = ( λmax[n] Nc Nc−1∑ n=0 1 λmax[n] ) − 1 2 . (2.37) This means that for the max¸min criterion and Lg = Nc all sub¸carrier SNRs are equal to SNR[n] = Nc/(σ2n∑Nc−1n=0 1/λmax[n]). Therefore, in contrast to the minimum average BER solution, for the max¸min solution all sub¸carriers have identical BERs. 2.4.3 Solution of the Optimization Problems for Lg < Nc Since F is not invertible for Lg < Nc, we present alternative approaches for solving the BER optimization problems in this subsection. 1) Average BER Criterion: For convenience we rewrite (2.29), (2.30) as min g Nc−1∑ n=0 exp ( − c2 σ2n gHM [n]g ) (2.38) s.t. gHg = 1, (2.39) where M [n] was de¿ned in Section 2.3. Unfortunately, the objective function in (2.38) is not a convex function, i.e., (2.38), (2.39) is not a convex optimization problem. Therefore, similar to Section 2.3.3, we ¿rst pursue a relaxation approach to ¿nd a suboptimum solution 29Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems to the problem. In particular, letting again S = ggH we can rewrite (2.38), (2.39) as min S Nc−1∑ n=0 exp ( − c2 σ2n trace ( H [n]F [n]SF H [n]HH [n])) (2.40) s.t. trace{S} ≤ 1, (2.41) S  0, (2.42) rank{S} = 1. (2.43) The equivalent optimization problem (2.40)¸(2.43) is still non¸convex due to the rank condition in (2.43) but can be relaxed to a convex SDP problem by dropping this rank condition. The resulting convex problem has similar properties as the relaxed convex problem in the AMI case. In particular, a (possibly suboptimum) solution to the original minimum BER problem is given by that eigenvector of the optimum S which corresponds to its maximum eigenvalue. Furthermore, the complexity of the relaxed problem again strongly depends on Nc, and becomes prohibitive for a moderate number of sub¸carriers (e.g. Nc ≥ 64). 2) Max¸Min Criterion: For the max¸min criterion, we may rewrite (2.31), (2.32) as max g min ∀n gHM [n]g (2.44) s.t. gHg = 1, (2.45) 30Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems which constitutes a quadratic objective quadratic constraint (QOQC) NP¸hard problem [71]. This problem can be restated in equivalent form as [71] max t (2.46) s.t. trace{S} ≤ 1, (2.47) trace{M [n]S} ≥ t, ∀n, (2.48) S  0, (2.49) rank{S} = 1. (2.50) By dropping the rank condition (2.50) the optimization problem (2.46)¸(2.50) can be re- laxed to an SDP problem. Unlike the SDP problems for the maximum AMI and the minimum average BER criteria, the complexity of the SDP problem (2.46)¸(2.49) is domi- nated by Lg and not by Nc. Since we are mainly interested in the case where Lg  Nc, the relaxed problem for the max¸min criterion can be solved even for large Nc (e.g. Nc ≥ 256) using standard software (e.g. ³SeDuMi´). 3) Gradient Algorithm: Unfortunately, for both relaxed optimization problems pre- sented in this section the resulting S has a high rank most of the time, and the dominant eigenvector of S is a suboptimum solution which may entail a signi¿cant performance degradation. However, a GA may be used to recursively improve the initial C¸BFF vector found through relaxation. In Table 2.1, we provide the GA for the average BER criterion since this is our primary BER¸related criterion. However, if the average BER SDP problem (2.40)¸(2.42) cannot be solved since the number of sub¸carriers Nc is too large, we use the solution found for the max¸min SDP problem (2.46)¸(2.49) for initialization of the GA. In this context, we note that the initial vector g0 seems to have a larger impact on the quality of the solution found by the GA for the minimum BER criterion than for the max- 31Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems imum AMI criterion discussed in Section 2.3. Nevertheless, for the speed of convergence of the GA for the minimum BER criterion, similar statements hold as for the GA for the maximum AMI criterion. 2.5 Finite¸Rate Feedback and Comparison In this section, we brieÀy discuss codebook design for ¿nite¸rate feedback channels based on the GVQ algorithm in [62]. Furthermore, we also compare TD¸BF with interpolation¸ based FD¸BF [2, 3, 27]. 2.5.1 Finite¸Rate Feedback Case Vector quantization can be used to design a codebook G of size N for the ¿nite¸rate feedback channel case, cf. Section 2.2.4. Here, we adopt the GVQ algorithm introduced in [62]. For this purpose a set H , {h1, h2, . . . , hT} of T channel vectors hn is generated. Thereby, the NT NRL¸dimensional vector hn contains the CIR coeÁcients of all NT NR CIRs of the nth MIMO channel realization. For each of these channel realizations the corresponding C¸BFF vector g = g¯n is generated using the GA for the maximum AMI criterion or the GA for the minimum BER criterion, cf. Table 2.1, yielding the set GT , {g¯1, g¯2, . . . , g¯T}. The vector quantizer can then be represented as a function Q: GT → G. Ideally, this function is optimized for minimization of the mean quantization error MQE , 1 T T∑ i=1 d(Q(g¯i), g¯i), (2.51) 32Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems where d(gˆm, g¯i) denotes the distortion caused by quantizing g¯i ∈ GT to gˆm ∈ G. The distortion measure depends on the optimization criterion and is given by d(gˆm, g¯i) , − Nc−1∑ n=0 log2 ( 1 + SNR(gˆm,hi)[n] ) (2.52) and d(gˆm, g¯i) , Nc−1∑ n=0 exp ( −c2 SNR(gˆm,hi)[n] ) (2.53) for the maximum AMI and the minimum BER criterion, respectively. Here, SNR(gˆm,hi)[n] is de¿ned in (2.9) and the subscripts indicate that G[n] and H [n] have to be calculated for gˆm and hi, respectively. With this de¿nition for the distortion measure the GVQ algorithm given in [62, Section IV] can be straightforwardly applied to ¿nd G. We omit here further details and refer the interested reader to [61, 62] and references therein. Once the o¾¸line optimization of the codebook is completed, G is conveyed to the transmitter and the receiver. For a given channel realization h the receiver selects that C¸BFF gˆm ∈ G which minimizes the distortion measure (2.52) [AMI criterion] or (2.53) [BER criterion] and feeds back the corresponding index to the transmitter. 2.5.2 Comparison with FD¸BF We compare TD¸BF with FD¸BF in terms of feedback requirements and computational complexity. 1) Feedback Requirements: The required number of complex feedback symbols S for TD¸BF, interpolation¸based FD¸BF with modi¿ed spherical (MS¸FD¸BF) [2], Grassman- nian (GS¸FD¸BF) [27], and geodesic (GD¸FD¸BF) [3] interpolation, and ideal FD¸BF are summarized in Table 2.2, where K denotes the cluster size in interpolation¸based FD¸BF [2], i.e., Nc/K is the number of sub¸carriers for which CSI is assumed to be available at the 33Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems Table 2.2: Feedback Requirements for TD¸BF, ideal FD¸BF, and FD¸BF with modi¿ed spherical (MS), Grassmannian (GS), and geodesic (GD) interpolation. BF Scheme Number of Complex Feedback Symbols per Frame Ideal FD¸BF S = NcNT MS¸FD¸BF [2] S = NcK (NT + 1) GS¸FD¸BF [27] and GD¸FD¸BF [3] S = NcK NT Proposed TD¸BF S = NT Lg transmitter. We will use S to compare the feedback requirements of TD¸BF and FD¸BF in Section 2.6. 2) Computational Complexity: The calculation of the C¸BFFs and the GVQ¸based codebook design for the proposed TD¸BF scheme are more involved than the calculation of the BF weights and the codebook design method adopted in [2, 3, 27] for FD¸BF, respectively. However, in practice, codebook design is done very infrequently. In fact, if the statistical properties of the MIMO channel do not change (as is typically the case in downlink scenarios), the codebook has to be designed only once. Therefore, in practice, the computational e¾ort for C¸BFF calculation and codebook design can be neglected. The interpolation of BF weights in FD¸BF has to be done in every frame. The interpolation complexity is generally proportional to Nc but strongly depends on the interpolator used. For example, modi¿ed spherical interpolation requires a grid search whereas Grassmannian and geodesic interpolation do not. Assuming a codebook of size N selecting the beamformer index at the receiver requires evaluation of N and NNc/K distortion measures for TD¸BF and interpolation¸based FD¸BF, respectively. However, a fair quantitative comparison of the associated complexities is diÁcult since the required N to achieve a similar performance may be very di¾erent in both cases. Similar to [63] we assume that the inverse IDFTs and the BF itself dominate the com- plexities of TD¸BF and FD¸BF. As is customary in the literature, we adopt the required 34Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems number of complex multiplications as measure for complexity and assume that the (I)DFT is implemented as a (inverse) fast Fourier transform ((I)FFT). Following [2] we assume that one (I)FFT operation requires Nc log2(Nc)/2 complex multiplications. Therefore, since FD¸BF requires NT IFFT operations and NT Nc complex multiplications for BF, a total of MFD = NT Nc 2 log2(Nc) + NT Nc (2.54) complex multiplications are obtained. In contrast, assuming a straightforward TD imple- mentation of convolution, MTD = Nc 2 log2(Nc) + LgNT Nc (2.55) complex multiplications are required for TD¸BF. A comparison of MFD and MTD shows that the complexity of TD¸BF is lower than that of FD¸BF if Lg < NT − 1 2NT log2(Nc) + 1. (2.56) For example, assuming Nc = 512 sub¸carriers and NT = 2, 3 ≤ NT < 9, and NT ≥ 9 TD¸ BF requires a lower complexity than FD¸BF for Lg ≤ 3, Lg ≤ 4, and Lg ≤ 5, respectively. Our results in Section 2.6 show that generally a high performance can be achieved with these small values of Lg. 2.6 Simulation Results In this section, we present simulation results for the AMI and the BER of MIMO¸OFDM with TD¸BF. Besides the uncoded BER, we also consider the BER of a coded system employing the popular bit interleaved coded modulation (BICM) concept, since the com- 35Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems bination of BICM and OFDM has been adopted in various recent standards, cf. e.g. [20]. However, ¿rst we brieÀy discuss the parameters used in our simulations. 2.6.1 Simulation Parameters Throughout this section we consider a MIMO¸OFDM system with NT = 2 or NT = 3 transmit antennas, NR = 1 receive antenna, and Nc = 512 OFDM sub¸carriers. If BICM is employed, the data bits are encoded with the quasi¸standard (171, 133)8 convolutional code of rate Rc = 1/2, possibly punctured, interleaved, and Gray mapped to the data symbols D[·] [20, 25]. At the receiver standard Viterbi soft decoding is applied. For all BER results 16¸QAM was used. For practical relevance we adopted for our simulations the IEEE 802.11n Channel Model B with L = 9 assuming a carrier frequency of 2.5 GHz and a transmit antenna spacing of λ0/2, where λ0 is the wavelength [72]. All simulation results were averaged over 100,000 independent channel realizations. For Lg < Nc the C¸BFF vectors were calculated with the algorithms given in Table 2.1. The all¸ones vector and the solution of the relaxed max¸min problem were used for initialization of the GAs for the maximum AMI and the minimum BER criterion, respectively. For Lg = Nc (equivalent to ideal FD¸BF) the closed¸form solutions for the C¸BFF provided in Sections 2.3.2 and 2.4.2 were used. For the ¿nite¸rate feedback case the C¸BFF vector codebook was generated with the GVQ algorithm discussed in Section 2.5.1 based on a training set of T = 1000 independent channel realizations. 2.6.2 Maximum AMI Criterion We ¿rst consider TD¸BF with AMI¸optimized C¸BFFs and compare its performance with that of MS¸FD¸BF [2] and GD¸FD¸BF [3], respectively. We note that in [2] an AMI criterion is used for interpolator optimization, whereas the interpolator optimization in [3] 36Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 0 2 4 6 8 10 12 14 1.5 2 2.5 3 3.5 4 4.5 5 ideal FD−BF TD−BF (Lg = 1, S = 2) TD−BF (Lg = 2, S = 4) TD−BF (Lg = 4, S = 8) MS−FD−BF (K = 256, S = 6) MS−FD−BF (K = 64, S = 24) GD−FD−BF (K = 256, S = 4) GD−FD−BF (K = 64, S = 16) NT=1, NR=1 AM I (bit/s/Hz ) Es/N0 [dB] Figure 2.2: AMI of TD¸BF (AMI criterion), MS¸FD¸BF [2], and GD¸FD¸BF [3] with perfect CSI. NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. For com- parison the AMIs for ideal FD¸BF and single¸input single¸output (SISO) transmission (NT = 1, NR = 1) are also shown. is not directly tied to the AMI or BER. Throughout this subsection NT = 2 is valid. Fig. 2.2 shows the AMI per sub¸carrier vs. Es/N0 (Es: energy per received symbol, N0: power spectral density of underlying continuous¸time passband noise process) for the proposed TD¸BF, MS¸FD¸BF, and GD¸FD¸BF for the case of perfect CSI at the transmitter. To facilitate a fair comparison between TD¸BF with C¸BFFs of length Lg and FD¸BF with cluster size K, we have included in the legend of Fig. 2.2 the respective required number of complex feedback symbols S, cf. Table 2.2. As can be observed, TD¸ 37Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 0 1 2 3 4 5 6 7 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 TD−BF (Finite−Rate Feedback, Lg = 1) TD−BF (Perfect CSI, Lg = 1) TD−BF (Finite−Rate Feedback, Lg = 2) TD−BF (Perfect CSI, Lg = 2) TD−BF (Finite−Rate Feedback, Lg = 3) TD−BF (Perfect CSI, Lg = 3) GD−FD−BF (Finite−Rate Feedback, K = 512) GD−FD−BF (Finite−Rate Feedback, K = 256) B AM I (bit/s/Hz ) Figure 2.3: AMI of TD¸BF (AMI criterion) vs. number of feedback bits B per channel update. NT = 2, NR = 1, Nc = 512, Es/N0 = 10 dB, and IEEE 802.11n Channel Model B. For comparison the AMIs for GD¸FD¸BF with codebooks from [4] are also shown. BF provides a better performance/feedback trade¸o¾ than interpolation¸based FD¸BF. For example, TD¸BF with S = 2 (Lg = 1) outperforms MS¸FD¸BF and GD¸FD¸BF with S = 6 (K = 256) and S = 4 (K = 256), respectively. MS¸FD¸BF with S = 24 (K = 64) is necessary to outperform TD¸BF with S = 8 (Lg = 4) which performs only less than 0.5 dB worse than ideal FD¸BF. In Fig. 2.3, we consider the AMI of TD¸BF with ¿nite¸rate feedback channel as a 38Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems function of the number of feedback bits B (solid lines) for an SNR of Es/N0 = 10 dB. For comparison, Fig. 2.3 also contains the AMI for TD¸BF with perfect CSI (dashed lines) and the AMIs for GD¸FD¸BF with the best known codebooks from [4] and K = 512 and K = 256. For B = 0 the codebook has just one entry and no feedback is required. As can be observed from Fig. 2.3, ¿nite¸rate feedback TD¸BF approaches the performance of the perfect CSI case as B increases. Furthermore, as expected, the number of feedback bits required to approach the perfect CSI case increases with increasing Lg. The performance of the GD¸FD¸BF scheme is signi¿cantly worse than that of the TD¸BF scheme for the same number of feedback bits. From further simulations we have observed that GD¸FD¸ BF requires more than B = 80 feedback bits to achieve the same performance as TD¸BF with 7 feedback bits and Lg = 3. Fig. 2.4 shows the BERs of a coded MIMO¸OFDM system (Rc = 1/2) employing TD¸ BF, MS¸FD¸BF, and GD¸FD¸BF vs. Eb/N0, where Eb denotes the average energy per information bit. Both perfect CSI and ¿nite¸rate feedback are considered. With perfect CSI at the transmitter, at a BER of 10−4 the performance of TD¸BF with S = 6 is about 0.8 dB and 0.77 dB worse than that of MS¸FD¸BF with S = 48 and GD¸FD¸BF with S = 64, respectively. However, in case of ¿nite¸rate feedback the performance of TD¸BF with B = 7 is slightly better than that of GD¸FD¸BF with B = 64 and MS¸FD¸BF with B = 80, where we adopted the codebooks from [4] for GD¸FD¸BF and MS¸FD¸BF, respectively. 2.6.3 Minimum BER Criterion Now, we shift our attention to TD¸BF with BER¸optimized C¸BFFs. NT = 2 is still valid. Assuming perfect CSI we show in Fig. 2.5 the average BERs for the average BER criterion and the max¸min criterion, respectively. As expected, for Lg = Nc (ideal FD¸ 39Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 0 2 4 6 8 10 12 10−5 10−4 10−3 10−2 10−1 Ideal FD−BF TD−BF (Lg = 3, S = 6) MS−FD−BF (K = 32, S = 48) GD−FD−BF (K = 16, S = 64) TD−BF (Lg=3, B=7, Finite−Rate Feedback) MS−FD−BF (K=32, B=80, Finite−Rate Feedback) GD−FD−BF (K=16, B=64, Finite−Rate Feedback) NT=1, NR=1 BE R Eb/N0 [dB] Figure 2.4: BER of coded MIMO¸OFDM system with TD¸BF (AMI criterion), MS¸FD¸ BF [2], and GD¸FD¸BF [3]. Perfect CSI and ¿nite¸rate feedback, NT = 2, NR = 1, Nc = 512, Rc = 1/2, and IEEE 802.11n Channel Model B. For comparison the BERs for ideal FD¸BF and SISO transmission (NT = 1, NR = 1) are also shown. BF) the average BER criterion leads to a lower average BER than the max¸min criterion. However, the di¾erence between both criteria is less than 1 dB at BER = 10−3. For Lg = 1 and Lg = 5 we show the average BER obtained for the relaxed max¸min criterion. As can be observed the performance is quite poor in this case and a comparison with single¸ antenna transmission (NT = 1) suggests that the diversity o¾ered by the second antenna 40Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 0 5 10 15 20 25 30 10−4 10−3 10−2 10−1 Ideal FD−BF, (Average BER Criterion) Ideal FD−BF (Max−Min Criterion) TD−BF, (Perfect CSI, GA, Lg=1) TD−BF, (Perfect CSI, GA, Lg=2) TD−BF, (Perfect CSI, GA, Lg=3) TD−BF, (Perfect CSI, GA, Lg=4) TD−BF, (Perfect CSI, GA, Lg=5) TD−BF, (Max−Min Criterion, Lg=1) TD−BF, (Max−Min Criterion, Lg=5) NT = 1, NR = 1 BE R Eb/N0 [dB] Figure 2.5: Average BER of uncoded MIMO¸OFDM system with TD¸BF. Minimum average BER criterion (solid lines) and max¸min criterion (dashed lines), perfect CSI, NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. For comparison the BERs for ideal FD¸BF and SISO transmission (NT = 1, NR = 1) are also shown. is not exploited. However, Fig. 2.5 clearly shows that this diversity can be exploited if the GA is used to improve the relaxed max¸min solution. In this case, the BER approaches the BER of the limiting Lg = Nc case as Lg increases. For example, for Lg = 5 the performance loss compared to Lg = Nc = 512 is less than 1.5 dB at BER = 10−3. In Fig. 2.6, we investigate the e¾ect of a ¿nite¸rate feedback channel on the average BER. In particular, we show the average BER as a function of the number of feedback bits B 41Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 0 1 2 3 4 5 6 7 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 Finite−Rate Feedback, Lg = 1 Perfect CSI, Lg = 1 Finite−Rate Feedback, Lg = 2 Perfect CSI, Lg = 2 Finite−Rate Feedback, Lg = 3 Perfect CSI, Lg = 3 BE R B Figure 2.6: Average BER of uncoded MIMO¸OFDM system with TD¸BF (average BER criterion) vs. number of feedback bits B per channel update. GA was used for C¸BFF optimization. NT = 2, NR = 1, Nc = 512, Eb/N0 = 10 dB, and IEEE 802.11n Channel Model B. (solid lines) for an SNR of Eb/N0 = 10 dB. For comparison, Fig. 2.6 also contains the BERs for perfect CSI (dashed lines). As can be observed, ¿nite¸rate feedback BF approaches the performance of the perfect CSI case as B increases. Furthermore, as expected, the number of feedback bits required to approach the perfect CSI case increases with increasing Lg. Therefore, smaller Lg are preferable if only few feedback bits can be a¾orded. In Fig. 2.7 we show the average BER for uncoded and coded (Rc = 1/2) transmission 42Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 0 5 10 15 20 25 30 10−4 10−3 10−2 10−1 TD−BF, (Perfect CSI, Lg = 2) TD−BF, (Lg = 2, B = 0) TD−BF, (Lg = 2, B = 1) TD−BF, (Lg = 2, B = 3) TD−BF, (Lg = 2, B = 7) uncoded BE R Eb/N0 [dB] coded (Rc = 1/2) Figure 2.7: Average BER of uncoded and coded MIMO¸OFDM system with TD¸BF (average BER criterion). GA was used for C¸BFF optimization and Lg = 2 is valid for all curves shown. Perfect CSI (bold lines) and ¿nite¸rate feedback channel, NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. with ¿nite¸rate feedback TD¸BF and TD¸BF with perfect CSI, respectively. C¸BFFs of length Lg = 2 were used in all cases and the C¸BFF vector codebook was optimized for Eb/N0 = 10 dB. Interestingly, for coded transmission signi¿cantly fewer feedback bits are required to approach the performance of the perfect CSI case than for uncoded transmis- sion. For example, for BER = 10−4 and B = 3 feedback bits the performance loss compared to perfect CSI is 0.45 dB and 3.8 dB for coded and uncoded transmission, respectively. 43Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems 0 2 4 6 8 10 12 14 16 18 20 10−5 10−4 10−3 10−2 10−1 AMI Criterion, Lg = 2 Average BER Criterion, Lg = 2 AMI Criterion, Lg = 4 Average BER Criterion, Lg = 4 AMI Criterion, Lg = Nc Average BER Criterion, Lg = Nc BE R coded (Rc = 3/4) coded (Rc = 1/2) uncoded Eb/N0 [dB] Figure 2.8: Average BER of uncoded and coded MIMO¸OFDM system employing TD¸BF with perfect CSI. Average BER criterion (dashed lines) and AMI criterion (solid lines), NT = 3, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. 2.6.4 Comparison of Maximum AMI and Minimum BER Criteria In Fig. 2.8, we compare the average BERs of uncoded and coded MIMO¸OFDM systems employing minimum average BER (dashed lines) and maximum AMI (solid lines) TD¸BF, respectively. We assume perfect CSI, NT = 3, Lg = 2, 4, and Nc (ideal FD¸BF). As one would expect, for uncoded transmission the minimum average BER criterion yields a 44Chapter 2. Time¸Domain Transmit Beamforming for MIMO¸OFDM Systems signi¿cantly better performance than the maximum AMI criterion. However, although the employed convolutional codes are by no means capacity achieving, for the coded case the maximum AMI criterion yields a lower BER than the minimum average BER criterion. 2.7 Conclusions In this chapter, we have proposed a novel TD approach to BF in MIMO¸OFDM systems. The C¸BFFs have been optimized for maximization of the AMI and minimization of the BER, respectively, and eÁcient algorithms for recursive calculation of the optimum C¸BFFs have been provided for both criteria. In contrast to existing FD¸BF schemes, for TD¸BF the number of complex feedback symbols to be conveyed to the transmitter is independent from the number of OFDM sub¸carriers. For the case of a ¿nite¸rate feedback channel a GVQ algorithm has been introduced for codebook design. Simulation results for the IEEE 802.11n Channel Model B have con¿rmed the excellent performance of TD¸BF and have shown that TD¸BF achieves a more favorable performance/feedback rate trade¸o¾ than FD¸BF. 45Chapter 3 Cooperative Amplify¸and¸Forward Beamforming with Multiple Multi¸Antenna Relays 3.1 Introduction In the previous chapter, we have introduced a novel TD¸BF scheme for direct point¸to¸ point transmission. Starting from this chapter, we consider BF schemes for cooperative relay networks. Since the AF protocol is generally believed to be less complex than the DF protocol, we will consider AF in all the remaining chapters. Recently, AF¸BF for wireless relay networks was considered in [35]¸[44] and [73]. AF¸ BF for networks with one single¸antenna source and multiple single¸antenna relays was considered in [39, 42] for individual relay power constraints, [35, 36, 40, 41] for a joint power constraint for all relays, and [73] for and a joint power constraint for the source and all relays, respectively. Since both the source and the relays were assumed to have only one antenna, respectively, the resulting SINR maximization problem at the destination involved only the optimization of one scalar BF gain for each relay. In contrast, in [37, 38], AF¸BF for a network with a single relay and multiple antennas at the relay and the source was investigated and closed¸form solutions for the BF vector at the source and the AF¸BF 46Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays matrix at the relay were provided. Furthermore, in [43, 44], the performance of AF¸BF with multiple antennas at the source and one single¸antenna relay was investigated. We note that in practice a relay network may comprise multiple relays and both the relays and the source may have multiple antennas. The extension of the results in the aforementioned papers to this general case is not straightforward as it results in complex non¸convex optimization problems for the AF¸BF matrices at the relays and the BF vector at the source. We note that multiple multi¸antenna relays were considered in [74]. However, in [74], DF relaying was assumed and the source had only a single antenna. In this chapter, we consider AF¸BF for networks with one multi¸antenna source (e.g. a base station), multiple multi¸antenna relays, and one single¸antenna destination (e.g. a mobile phone). The SINR at the destination is adopted as performance criterion and the BF vector at the source and the AF¸BF matrices at the relays are optimized under three di¾erent power constraints. In particular, we consider the cases of individual relay power constraints, a joint power constraint for all relays, and a joint source¸relay power constraint. This chapter makes the following contributions: • For a given BF vector at the source, we ¿nd the optimal AF¸BF matrices at the relays for each of the three considered power constraints. In particular, we provide closed¸form solutions for the AF¸BF matrices for the individual and joint relay power constraints, respectively. For the joint source¸relay power constraint, we derive the direction of the AF¸BF matrices in closed form and provide a simple numerical method for ¿nding the optimal power allocation for the source and the relays. In case of a single relay, this power allocation is given in closed form. • For the joint relay and the joint source¸relay power constraints, we show that the optimization problem for the source BF vector can be converted into a polynomial programming problem. Although this problem is non¸convex, it can be eÁciently 47Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays solved with the GloptiPoly or SOSTOOLS software tools [75, 76] for small scale networks (e.g. two antennas at the source and two relays with arbitrary numbers of antennas). For large scale networks and networks with individual relay power constraints, we provide eÁcient suboptimal methods for computation of the optimal source BF vector. • To implement the proposed AF¸BF scheme, the source node has to acquire the channel state information of all source¸relay channels and the Euclidean norm of each relay¸destination channel vector for computation of the optimal source BF vector. In contrast, for all considered power constraints, the relays have to know only their own source¸relay and relay¸destination channels if the source feeds back one complex scalar to each relay (individual power constraints), one complex scalar to all relays (joint relay power constraint), or one complex and one real scalar to all relays (joint source¸relay power constraint). • Our simulation results con¿rm that the proposed suboptimal optimization methods for the source BF vector achieve a close¸to¸optimal performance. Furthermore, our results show that increasing the number of antennas at the source is highly bene¿cial if the source¸relay channels have a lower SNR than the relay¸destination channels. In contrast, increasing the number of relays or the number of relay antennas is always bene¿cial. The remainder of this chapter is organized as follows. In Section 3.2, the considered system model is presented and the proposed optimization problem is rigorously formulated. The optimization of the AF¸BF matrices for maximization of the SINR for a given BF vector at the source is discussed in Section 3.3. In Section 3.4, the optimization of the source BF vector is investigated. Simulation results are provided in Section 3.5, and some conclusions are drawn in Section 3.6. 48Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays d n2 g1 gNT r nNR,1 BNR B1H1 HNR fNR f 1 n1,1 n1,M1 nNR,MNR Figure 3.1: Cooperative network with one multi¸antenna source, multiple multi¸antenna relays, and one single¸antenna destination. gi, 1 ≤ i ≤ NT , denotes the ith element of source BF vector g. ni,µ, 1 ≤ µ ≤ Mi, is the µth element of noise vector n1,i at relay i, 1 ≤ i ≤ NR. 3.2 System Model and Optimization Problem We consider the downlink of a relay network with one source node, NR relays, and one destination node. A block diagram of the discrete¸time overall transmission system in equivalent complex baseband representation is shown in Fig. 3.1. We assume that NT , Mi, and one antennas are available at the source (e.g. base station or access point), relay i, 1 ≤ i ≤ NR, and the destination (e.g. mobile phone), respectively. As usual, transmission is organized in two intervals. In the ¿rst transmission interval, the source node sends a data packet to the relays, which forward this packet to the destination node in the second transmission interval. We assume that there is no direct link between the source node and the destination node. 49Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 3.2.1 System Model In the ¿rst transmission interval, the source transmits the elements of NT¸dimensional vector, x = g d, (3.1) over its NT antennas, where g denotes the NT¸dimensional BF vector, and d is the modu- lated symbol taken from a scalar symbol alphabet A with variance σ2d , E{|d|2} = 1. The signal received at the Mi antennas of relay i, 1 ≤ i ≤ NR, can be modeled as qi = H ix + n1,i, (3.2) where [H i]µν , 1 ≤ µ ≤ Mi, 1 ≤ ν ≤ NT , is the channel gain between antenna ν of the source and antenna µ of relay i, and the elements of vector n1,i represent AWGN with variance σ21. In the second transmission interval, relay i transmits the µth element of vector si = Biqi (3.3) over antenna µ, 1 ≤ µ ≤Mi, where Bi is an Mi ×Mi AF¸BF matrix. The received signal at the destination node is given by r = NR∑ i=1 fTi si + n2, (3.4) where the µth element of Mi¸dimensional vector f i is the channel gain between antenna µ, 1 ≤ µ ≤ Mi, of relay i and the destination node, and n2 is AWGN with variance σ22. 50Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays Combining (3.1)¸(3.4), the received signal at the destination node can be expressed as r = NR∑ i=1 fTi BiH ig d + NR∑ i=1 fTi Bin1,i + n2 = fT BDHg d + fT BDn1 + n2 , (3.5) with relay¸destination channel vector f , [fT1 . . . fNR ]T , AF¸BF block diagonal matrix BD , diag{B1, . . . , BNR}, ( ∑NR i=1 Mi) × NT source¸relay channel matrix H , [HT1 . . . HTNR ]T , and relay noise vector n1 , [nT1,1 . . . nT1,NR ]T . 3.2.2 Formulation of the Optimization Problem From (3.5) the SINR at the destination node can be obtained as SINR = |f T BDHg|2 ‖fT BD‖22 σ21 + σ22 . (3.6) The design problem considered in this chapter is the optimization of the BF vector g at the source and the AF¸BF matrices Bi, 1 ≤ i ≤ NR, at the relays for maximization of the SINR at the destination node while constraining the power emitted by the source and the relays. Formally, the resulting optimization problem can be formulated as follows: max g, Bi, 1≤i≤NR SINR (3.7a) s.t. Power Constraints (3.7b) For the power constraints, we consider three di¾erent scenarios: Constraint I (Individual Power Constraints for Relays): If the transmit power of the 51Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays source and each relay is limited, the power constraints are given by ‖g‖22 ≤ P1, (3.8a) ‖BiH ig‖22 + σ21‖Bi‖2F ≤ P2,i, 1 ≤ i ≤ NR, (3.8b) where P1 and P2,i denote the maximum transmit powers of the source and relay i, respec- tively. Constraint II (Joint Power Constraint for Relays): As an alternative to the individual relay power constraint, we may impose a joint relay power constraint resulting in ‖g‖22 ≤ P1, (3.9a) NR∑ i=1 (‖BiH ig‖22 + σ21‖Bi‖2F)≤ P2 , (3.9b) where P1 and P2 denote the maximum transmit powers of the source and all relays, re- spectively. Constraint III (Joint Power Constraint for Source and Relays): Finally, we may impose a joint power constraint on the source and the relays, which leads to ‖g‖22 + NR∑ i=1 (‖BiH ig‖22 + σ21‖Bi‖2F)≤ P, (3.10) where P is the maximum total transmit power. Since Constraint I is more restrictive than Constraint II and Constraint II is more restrictive than Constraint III, we expect Constraint I to result in the lowest SINR in (3.7a) and Constraint III in the highest SINR among the three sets of constraints if the maximum overall power budget is the same, i.e., P = P1 + P2 and P2 = ∑NR i=1 P2,i. In the next two sections, we will solve problem (3.7) for the three di¾erent constraints 52Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays in (3.8)¸(3.10). 3.3 Optimal AF¸BF Matrices It is convenient to solve problem (3.7) in two steps. In Subsections 3.3.1¸3.3.3, we determine the optimal AF¸BF matrices Bi, 1 ≤ i ≤ NR, for a given BF vector g at the source under the three considered power constraints. The obtained solutions are compared in Subsection 3.3.4. The optimization of the BF vector will be tackled in Section 3.4. For the following, it is convenient to de¿ne vector ui , H ig, 1 ≤ i ≤ NR. 3.3.1 AF¸BF with Individual Power Constraints for Relays Combining (3.7) and (3.8) we obtain the optimization problem max Bi,1≤i≤NR ∣∣∣ ∑NR i=1 fTi Biui ∣∣∣ 2 σ21 ∑NR i=1 fTi BiBHi f ∗i + σ22 (3.11a) s.t. uHi BHi Biui + σ21||Bi||2F ≤ P2,i, 1 ≤ i ≤ NR, (3.11b) where we have ignored the source power constraint (3.8a) since g is assumed to be ¿xed. Next, we introduce the de¿nitions wi , u∗i ⊗ f ∗i , bi , vec{Bi}, T i , IMi ⊗ fTi , and Qi , uTi ⊗ IMi . With these de¿nitions, we can rewrite problem (3.11) in equivalent form as max bi,1≤i≤NR ∣∣∣ ∑NR i=1 w H i bi ∣∣∣ 2 σ21 ∑NR i=1 b H i T Hi T ibi + σ22 (3.12a) s.t. bHi ( QHi Qi + σ21IM2i ) bi ≤ P2,i, 1 ≤ i ≤ NR. (3.12b) 53Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays For the next step, we introduce matrix J i, which is obtained from matrix QHi Qi+σ21IM2i , JHi J i via Cholesky decomposition, and vector yi , J ibi. Vector yi can be represented as yi , √ ˜P2,i xi, where ˜P2,i = ||yi||22 and xi is a unit norm vector. Now, we can restate problem (3.12) as max ˜P2,i,xi,1≤i≤NR ∣∣∣∣ ∑NR i=1 √ ˜P2,iwHi J−1i xi ∣∣∣∣ 2 ∑NR i=1 x H i ( σ21 ˜P2,iJ−Hi T Hi T iJ−1i + σ22 NR IM2i ) xi (3.13a) s.t. ||xi||22 = 1, ˜P2,i ≤ P2,i, 1 ≤ i ≤ NR. (3.13b) Assuming that the powers ˜P2,i, 1 ≤ i ≤ NR, are ¿xed, we can ¿nd direction vectors xi, 1 ≤ i ≤ NR, that maximize (3.13a) by di¾erentiating the objective function with respect to xi and by accounting for the constraint ||x||22 = 1 by using Lagrange multipliers. After some algebraic manipulations, this leads to xi = αi ( J−Hi T Hi T iJ−1i + βiIM2i ) −1 J−Hi wi , (3.14) where αi and βi are complex and positive real constants, respectively, whose exact value is not important for the ¿nal result as will be shown in the following. In particular, using the de¿nitions of J i, T i, and wi in (3.14), we obtain xi = αiJ i ( T Hi T i + βi(QHi Qi + σ21IM2i ) ) −1 wi = αiJ i ( (IMi ⊗ fTi )H(IMi ⊗ fTi ) + βi(uTi ⊗ IMi)H(uTi ⊗ IMi) + σ21βiIM2i ) −1 wi = αiJ i ((IMi ⊗ f ∗i fTi ) +(βi(u∗i uTi + σ21IMi)⊗ IMi))−1 (u∗i ⊗ f ∗i ) , (3.15) where we have used the identity (A ⊗B)(C ⊗D) = AC ⊗BD [77]. xi can be further simpli¿ed by introducing the Kronecker sum (A⊗IM)+ (IM ⊗B) = A⊕B in (3.15) and 54Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays exploiting the relation [78] (M ⊕N )−1 = M∑ i=1 M∑ j=1 (mi ⊗ nj)(m¯i ⊗ n¯j)H λi(M ) + λj(N ) , (3.16) where mi, ni, m¯i, and n¯i denote the eigenvectors of M ×M matrices M , N , MH , and NH , respectively. This leads to xi = αiJ i (βi(u∗i uTi + σ21IMi)⊕ f ∗i fTi )−1 (u∗i ⊗ f ∗i ) = αiJ i ( u∗i||ui||2 ⊗ f∗i||f i||2 )( u∗i||ui||2 ⊗ f∗i||f i||2 )H ||f i||22 + βi(||ui||22 + σ21) (u ∗ i ⊗ f ∗i ) = αi ||f i||22 + βi(||ui||22 + σ21) J i(u ∗ i ⊗ f ∗i ) . (3.17) Exploiting (3.17) along with bi = √ ˜P2,iJ−1i xi, we obtain for the AF¸BF matrix Bi the expression Bi = ci f ∗i uHi , 1 ≤ i ≤ NR , (3.18) where complex scalar ci has to be optimized taking into account the per¸relay power constraint. Eq. (3.18) reveals that under a per¸relay power constraint eigenbeamforming with respect to the source¸relay and the relay¸source channel is optimal. For the special case where the source and all relays have only a single antenna, i.e., f i and ui are scalars, this result has already been derived in [39]. Substituting (3.18) into problem (3.11), it is obvious that all ci have to have the same phase θ to achieve the maximum SINR, i.e., ci = |ci|ejθ. The resulting optimization 55Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays problem is given by max |ci| (∑NR i=1 |ci|‖f i‖2‖ui‖2 )2 σ21 ∑NR i=1 |ci|2‖f i‖22 + σ22 (3.19a) s.t. |ci| ≤ √ P2,i ‖ui‖22 + σ21 , 1 ≤ i ≤ NR. (3.19b) Problem (3.19) is equivalent to the power allocation problem for relaying with multiple single¸antenna relays, which was solved in [39]. For completeness, we provide the solution here using the notation of this chapter. De¿ne φi = ‖ui‖2 √‖ui‖22 + σ21√ P2,i‖f i‖2 (3.20) and sort φi in descending order φτ1 ≥ φτ2 ≥ · · · ≥ φτNR , where (τ1, . . . , τNR) is an ordering of (1, . . . , NR). The optimal solution to problem (3.19) is given by [39] ci =    √ P2,i ‖ui‖22+σ21 e jθ , i = τ1, . . . , τj, κj ‖ui‖2 ‖f i‖2 e jθ , i = τj+1, . . . , τNR , (3.21) where κj , σ22 + σ21 ∑j m=1 P2,τm‖fτm‖22 ‖uτm‖22+σ21 σ22 ∑j m=1 √ P2,τm‖fτm‖2‖uτm‖2√ ‖uτm‖22+σ21 (3.22) and j is the smallest index such that κj < φ−1τj+1 . For a given source BF vector g, (3.18) and (3.21) fully specify the optimal AF¸BF matrices for multiple multi¸antenna relays with individual relay power constraints. 56Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 3.3.2 AF¸BF with Joint Power Constraint for Relays Considering (3.13) and taking into account the di¾erences between constraints (3.8) and (3.9), the optimization problem for the joint relay power constraint can be rewritten as max ˜P2,y yHJ−HwwHJ−1y yH ( σ21J−HT HTJ−1 + σ22/ ˜P2IM ) y (3.23a) s.t. ||y||22 ≤ P2 , (3.23b) where ||y||22 = ˜P2 ≤ P2, y , Jb, b = [bT1 . . . bTNR ]T , J , diag{J1, . . . , JNR}, T , diag{T 1, . . . , T NR}, and M , ∑NR i=1 M2i . We observe from (3.23a) that the maximum is achieved for ˜P2 = P2, i.e., the inequality in (3.23b) can be replaced by an equality. Thus, problem (3.23) reduces to a generalized eigenvalue problem. Consequently, the solution to problem (3.23) is given by [77] y = c ( σ21J−HT HTJ−1 + σ22 P2 IM ) −1 J−Hw , (3.24) where c is a complex scaling factor. Using similar operations as in (3.15)¸(3.17) and b = J−1y we obtain for the optimal BF matrix for AF relays with a joint power constraint Bi = c si f ∗i uHi , 1 ≤ i ≤ NR, (3.25) where si , P2 P2‖f i‖22σ21 + ‖ui‖22σ22 + σ21σ22 , 1 ≤ i ≤ NR, (3.26) c , ( NR∑ i=1 P2‖f i‖22‖ui‖22 (‖ui‖22 + σ21) (P2‖f i‖22σ21 + ‖ui‖22σ22 + σ21σ22)2 ) −1/2 ejθ (3.27) 57Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays with arbitrary phase θ. We note that the proposed solution for the AF¸BF matrix includes the result in [35] as a special case if the source and all relays have only a single antenna. 3.3.3 AF¸BF with Joint Power Constraint for Source and Relays For the joint source¸relay power constraint, problem (3.7a), (3.10) can be rewritten as max P1,g,Bi,1≤i≤NR |fT BDu|2 ‖fT BD‖22σ21 + σ22 (3.28a) s.t. ‖g‖22 ≤ P1 (3.28b) NR∑ i=1 ‖Bigi‖22 + σ21‖Bi‖2F ≤ P − P1 . (3.28c) For given P1 and g, problem (3.28) is equivalent to the joint relay power constraint problem considered in Section 3.3.2. Thus, the optimal AF¸BF matrix is given by (3.25)¸(3.27) if we let P2 = P − P1. Using this result in (3.28) and assuming the direction of g is ¿xed, the optimization problem reduces to a power allocation problem between the source and the relays, i.e., max P1 NR∑ i=1 P1 (P − P1) Γ1,iΓ2,i P1Γ1,i + (P − P1) Γ2,i + 1 (3.29a) s.t. 0 ≤ P1 ≤ P , (3.29b) where we have introduced the equivalent source¸relay SNR Γ1,i , ||ui||2/(σ21||g||2) and the equivalent relay¸destination SNR Γ2,i , ||f i||2/σ22. It is easy to show that the second derivative of the objective function (SINR) in (3.29a) with respect to P1 is always negative: ∂2SINR ∂P 21 = − NR∑ i=1 2Γ1,iΓ2,i (PΓ1,i + 1) (PΓ2,i + 1) [P1Γ1,i + (P − P1)Γ2,i + 1]3 < 0 , when 0 ≤ P1 ≤ P. (3.30) 58Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays Therefore, the objective function is concave and the optimum power allocation can be obtained with a simple bisectional search method based on [1] ∂SINR ∂P1 = NR∑ i=1 Γ1,iΓ2,i [−P 21 (Γ1,i − Γ2,i)− 2 (PΓ2,i + 1) P1 + P (PΓ2,i + 1)] [P1Γ1,i + (P − P1)Γ2,i + 1]2 = 0 . (3.31) For the special case when there is only one relay in the cooperative network, a closed¸form solution for the optimal P1 is obtained as P1 =    √(PΓ1,1+1)(PΓ2,1+1)+(PΓ2,1+1) Γ2,1−Γ1,1 , if Γ1,1 < Γ2,1, P/2 , if Γ1,1 = Γ2,1,√(PΓ1,1+1)(PΓ2,1+1)−(PΓ2,1+1) Γ1,1−Γ2,1 , if Γ1,1 > Γ2,1. (3.32) Eq. (3.32) shows that the optimal power allocation tries to balance the received SNRs of the source¸relay and the relay¸destination channels by allocating more power to the weaker channel. This result is intuitively pleasing since the performance of two¸hop links is limited by the SNR of the weaker link. 3.3.4 Comparison of the Solutions for the Di¾erent Constraints A comparison of (3.18) and (3.25) shows that the optimal AF¸BF matrices for all power constraints can be expressed as Bi = cisif ∗i uHi , 1 ≤ i ≤ NR, where si = 1, 1 ≤ i ≤ NR, and ci = c, 1 ≤ i ≤ NR, for individual relay power constraints and joint relay/joint relay¸ source power constraints, respectively. The structure of the optimal Bi reveals that for all three power constraints, eigenbeamforming with respect to the source¸relay and the relay¸destination channels is optimal. We note that although this result may have been intuitively expected, it was not obvious from (3.7). It is also interesting to observe that while for the joint relay and the joint source¸relay power constraints the relays and the 59Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays source always utilize the full available transmit power, some relays may not utilize the maximum available power if individual relay power constraints are imposed on the relays, cf. (3.21). 3.4 Optimal BF Vector at the Source We ¿rst note that for the case of NT = 1 source antenna, g/||g||2 = 1 is optimal and the optimal AF¸BF matrices obtained in Section 3.3 constitute the solution to problem (3.7). In Subsections 3.4.1¸3.4.3, we propose optimal and suboptimal solutions for the BF vector g for the case NT > 1 assuming that the optimal AF¸BF matrices obtained in Subsections 3.3.1¸3.3.3 are adopted at the relays, respectively. In Subsection 3.4.4, we discuss the feedback requirements of the proposed AF¸BF scheme. 3.4.1 AF¸BF with Individual Power Constraints for Relays The degree to which the optimization problem for g can be solved largely depends on the underlying power constraints. Thereby, individual power constraints for the relays lead to the most diÁcult and least tractable problem. Considering (3.19) and using ui = H ig, the optimal g is the solution to the following optimization problem max g SINR(g) = (∑NR i=1 |ci(g)|‖f i‖2‖H ig‖2 )2 σ21 ∑NR i=1 |ci(g)|2‖f i‖22 + σ22 (3.33a) s.t. ||g||22 = P1, (3.33b) where we have made the dependence of ci on g explicit, cf. (3.21). Since SINR(g) depends on g in a complicated manner, it does not seem possible to obtain the globally optimal solution to problem (3.33). Hence, we propose two suboptimal methods for optimization 60Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays of g. 1) Ad hoc Method: One suboptimal solution is to perform eigenbeamforming at the source with respect to the average source¸relay channel. This means we choose g as the dominant eigenvector of matrix ∑NR i=1 H H i H i and normalize it to ||g||22 = P1. 2) Gradient Method: The solution obtained with the ad hoc method can be improved using a gradient method. We note, however, that since problem (3.33) is not convex, the gradient method may not achieve the globally optimal solution. Since the derivative of SINR(g) in (3.33a) with respect to g is cumbersome, we express the SINR as a function of g¯ , [<{g}T ={g}T ]T and use a gradient estimate given by [79] ∇g¯SINR(g¯) = 12δ [ (SINR(g¯ + δe1)− SINR(g¯ − δe1)) . . . (SINR(g¯ + δe2NT )− SINR(g¯ − δe2NT )) ]T (3.34) where δ is a small positive constant and ei has a one in position i, 1 ≤ i ≤ 2NT , and zeros in all other positions (ith unit norm vector). This leads to the gradient algorithm given in Table 3.1, where gˆk , g¯k and gradk , ∇g¯SNR(gˆk). The gradient algorithm is guaranteed to ¿nd a locally optimal solution that is not worse than the solution obtained with the ad hoc method, which is used for initialization, cf. Table 3.1. We note that for computation of the gradient estimate in (3.34), constants ci, 1 ≤ i ≤ NR, have to be computed for all 4NT vectors g¯k ± δei, 1 ≤ i ≤ 2NT , using (3.19). 61Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays Table 3.1: Gradient algorithm for calculation of source BF vector gˆ for individual and joint relay power constraints. The de¿nitions of gˆ and the gradient gradk depend on the power constraint, cf. Section 3.4. Termination constant  has a small value (e.g.  = 10−5). k denotes the iteration index and ak is the adaptation step size chosen through a backtracking line search [1]. 1 Let k = 0 and initialize vector gˆ0 with solution of ad hoc method 2 Update the BF vector: g˜k+1 = gˆk + ak gradk 3 Rescale the BF vector: gˆk+1 = √ P1g˜k+1/||g˜k+1||2 4 If 1− |gˆHk+1gˆk|/P1 < , goto Step 5, otherwise increment counter k and goto Step 2 5 gˆk+1 is the desired BF vector 3.4.2 AF¸BF with Joint Power Constraint for Relays In this case, applying (3.25)¸(3.27) in (3.7) and (3.9), we obtain max g NR∑ i=1 P2‖f i‖22‖H ig‖22 P2‖f i‖22σ21 + ‖H ig‖22σ22 + σ21σ22 (3.35a) s.t. ||g||22 ≤ P1 . (3.35b) For the following it is convenient to rewrite the objective function in (3.35a) as SINR = P2 σ22 NR∑ i=1 ‖f i‖22 − NR∑ i=1 ei gHAig , (3.36) where ei , σ21 P2σ22 ‖f i‖22(P2‖f i‖22 + σ22) and Ai , σ22HHi H i + σ21 P1 (P2‖f i‖22 + σ22)INT are independent of g. The ¿rst term in (3.36) is the SINR achieved with beamforming in point¸to¸point transmission without relaying where all the relay antennas are located at one transmitter. Thus, the second term in (3.36) may be interpreted as the penalty incurred because the considered system uses AF¸BF with distributed relays and not BF for the relay¸destination channel with co¸located antennas. Consequently, maximization problem 62Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays (3.35) is equivalent to the following minimization problem min g NR∑ i=1 ei gHAig (3.37a) s.t. ||g||22 ≤ P1 . (3.37b) For the special case of NR = 1 relay having M1 antennas it is obvious from (3.37) that the optimal g is simply the dominant eigenvector of matrix HH1 H1. This result is not new and has already been mentioned in [38]. However, here we are interested in the more diÁcult case of multiple relays, for which a solution has not been provided before. We note that for NR > 1 (3.37) is a diÁcult non¸convex optimization problem. In the following, we provide the optimal and three suboptimal solutions to problem (3.37) which di¾er in their complexity and performance. 1) Transformation Method: Problem (3.37) can be transformed into the following poly- nomial programming problem min g¯, ti, 1≤i≤NR NR∑ i=1 ti (3.38a) s.t. tig¯T  <{Ai} −={Ai} ={Ai} <{Ai}   g¯ ≥ ei, 1 ≤ i ≤ NR (3.38b) g¯T g¯ ≤ P1 , (3.38c) where g¯ = [<{g}T ={g}T ]T . Although the polynomial programming problem in (3.38) is still non¸convex, for small NT and small NR (e.g., NT = 2 and NR = 2), it can be solved by using the GloptiPoly or SOSTOOLS software [75, 76]. In this case, we can indeed obtain the globally optimal solution to the AF¸BF problem. However, for large NT and NR ¿nding the globally optimal solution with the aforementioned software tools does not 63Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays seem feasible. Thus, it is desirable to provide suboptimal methods for optimization of g having a lower complexity than the transformation method. 2) Ad hoc Method: Assuming that the relay¸destination channels have a much higher SNR than the source¸relay channels, i.e., P2 σ22 ||f i||22  1σ21 ||H ig||22, 1 ≤ i ≤ NR, it is easy to see from (3.35a) that the optimal BF vector g is the dominant eigenvector of matrix∑NR i=1 H H i H i normalized to ||g||22 = P1. This dominant eigenvector can also be considered as an ad hoc solution to the problem if the underlying condition on the SNRs of the subchannels is not ful¿lled. We note that for the case where the relay¸destination channels have a much lower SNR than the source¸relay channels, the objective function in (3.35a) becomes independent of g and optimization of the BF vector at the source is not necessary. 3) Gradient Method: Similar to the case of individual relay power constraints, we may use a gradient algorithm to improve the solution obtained with the ad hoc method. The corresponding algorithm is again given in Table 3.1 with gˆk , gk and gradk ,[∑NR i=1 eiAi/(gˆHk Aigˆk)2 ] gˆk. The gradient method will ¿nd that local optimum of the objective function which is closest to the solution provided by the ad hoc method. Since problem (3.37) is not convex, there is no guarantee that this local optimum coincides with the global optimum. Nevertheless, our simulation results in Section 3.5 suggest that the solution found with the gradient method achieves a performance comparable to that of the global optimum. 4) Relaxation Method: Considering (3.36) a ³good" suboptimal strategy to achieving a high SINR is to maximize the minimum value of gHAig, 1 ≤ i ≤ NR. This results in a new (relaxed) optimization problem: max g min i, 1≤i≤NR 1 ei gHAig (3.39a) s.t. ||g||22 ≤ P1 . (3.39b) 64Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays The max¸min problem in (3.39) can be easily relaxed to a semide¿nite programming (SDP) problem and eÁciently solved using SeDuMi in Matlab [71]. 3.4.3 AF¸BF with Joint Power Constraint for Source and Relays In this case, the optimal source BF vector g and power P1 that maximize the SINR in (3.28a) have to be found. This leads to the following problem: max g NR∑ i=1 (P − P1)‖f i‖22‖H ig‖22 (P − P1)‖f i‖22σ21 + ‖H ig‖22σ22 + σ21σ22 (3.40a) s.t. ||g||22 ≤ P1 (3.40b) 0 ≤ P1 ≤ P . (3.40c) Clearly, this non¸convex problem is in general more diÁcult than the problem with the joint relay power constraint considered in Section 3.4.2. Nevertheless, we will show in the following that similar approaches as in Section 3.4.2 can also be applied to problem (3.40). For the special case of NR = 1 relay, it can be observed from (3.40a) that the optimal direction g/||g||2 of the source BF vector is given by the dominant eigenvector of matrix HH1 H1. The corresponding optimal power P1 is given by (3.32), where Γ1,1 = λmax(HH1 H1)/σ21 and Γ2,1 = ||f 1||22/σ22. For the general case of NR > 1, a closed¸form solution cannot be found. Nevertheless, in the following, we provide the globally optimal and two suboptimal solutions to problem (3.40). 1) Transformation Method: Problem (3.40) can be transformed into the following poly- 65Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays nomial programming problem max g¯, P1, ti ,1≤i≤NR NR∑ i=1 ti (3.41a) s.t. ((P − P1)‖f i‖22 − tiσ22)¯gT <{H i} −={H i} ={H i} <{H i}   g¯ ≥ ti σ21 ((P − P1)‖f i‖22 − σ22), 1 ≤ i ≤ NR (3.41b) g¯T g¯ ≤ P1 (3.41c) 0 ≤ P1 ≤ P , (3.41d) where again g¯ = [<{g}T ={g}T ]T is used. Compared to problem (3.38), problem (3.41) has one additional optimization variable (P1) and one additional constraint. Despite its non¸convexity, for small scale networks (e.g. NT = 2 and NR = 2), the globally optimal solution for problem (3.41) can be readily obtained using the GloptiPoly or SOSTOOLS software [75, 76]. For large scale networks, we turn again to suboptimal solutions to reduce complexity. 2) Ad hoc Method: As an ad hoc solution, we may adopt the dominant eigenvector of∑NR i=1 H H i H i for the direction of the BF vector g, i.e., for g/||g||2. The optimal power allocation for this direction can be found with (3.31). 3) Gradient Method: For both small¸scale and large¸scale networks the solution found with the ad hoc method can be improved with a gradient algorithm. In each iteration, the gradient algorithm ¿rst improves the direction of the BF vector and subsequently computes the power allocation for the new BF vector. The gradient algorithm is given in detail in Table 3.2. 66Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays Table 3.2: Gradient algorithm for calculation of source BF vector g and power allocation for joint source¸relay power constraint. Termination constant  has a small value (e.g.  = 10−5). k denotes the iteration index and ak is the adaptation step size chosen through a backtracking line search [1]. 1 Let k = 0 and initialize g0, P1,0, and P2,0 = P − P1,0 with the solution obtained with the ad hoc method and calculate the corresponding SNR0 (objective function in (3.40a)) 2 Update the BF vector: g˜k+1 = gk + ak [∑NR i=1 eiAi/(gHk Aigk)2 ] gk 3 Rescale the BF vector: gk+1 = √ P1,kg˜k+1/||g˜k+1||2 4 Find the optimal power allocation P1,k+1 and P2,k+1 = P − P1,k+1 for gk+1 based on (3.31) using the bisectional search method and compute the corresponding SNRk+1 5 If |SNRk+1 − SNRk| < , goto Step 6, otherwise increment counter k and goto Step 2 6 gk+1 and P1,k+1 are the desired BF vector and power, respectively 3.4.4 Comparison of the Solutions and CSI Feedback Requirements Optimality: Our results in Sections 3.4.1¸3.4.3 show that for the special case of NR = 1 relay, the optimal source BF vector g can be found in closed form for all three constraints (note that for NR = 1 the individual power constraint is identical to the joint power constraint for the relays). In contrast for NR > 1 numerical methods have to be used to obtain g. While the globally optimal solution can be found in principle for the joint relay and the joint source and relay power constraints, this does not seem possible for the individual relay power constraints. Feedback Requirements: We ¿rst consider the feedback necessary for computation of the source BF vector g. We assume that in a ¿rst training phase the relays and the destination transmit suitable pilot symbols such that the source can estimate all source¸ 67Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays relay channels H i, 1 ≤ i ≤ NR, and each relay can estimate its own relay¸destination channel f i. Subsequently, relay i feeds back real number ||f i||22 to the source. With the knowledge of H i and ||f i||22, 1 ≤ i ≤ NR, the source can compute the optimal BF vector g for all three considered power constraints. Now, we consider the feedback required for computation of the optimal BF matrices at the relays. We ¿rst recall from Section 3.3 that for all considered constraints the AF¸BF matrix can be expressed as Bi = cisif ∗i uHi , where ci depends on the channel gains of all source¸relay and all relay¸destination links and si depends on the source¸relay and relay¸ destination channels of relay i only. The speci¿c values of ci and si depend on the power constraint. We assume that after it has obtained the optimal BF vector g, the source transmits in a second training phase pilot symbols such that each relay can estimate its (e¾ective) source¸relay channel ui = H ig. Thus, relay i knows f i and ui and can compute si, while the source can compute ci. The additional feedback requirements depend on the particular form of ci and are slightly di¾erent for the three considered power constraints. For the individual relay power constraints, ci depends on i and the source has to feedback one complex number ci to each relay, cf. (3.18). For the joint relay power constraint, ci = c, 1 ≤ i ≤ NR, and the source has to broadcast only one complex number c to all relays, cf. (3.27). For the joint source¸relay power constraint, the source has to broadcast complex constant c and the power P2 (which a¾ects si in this case) to all relays. Overall the feedback requirements for the proposed AF¸BF scheme are considered to be moderate. In particular, we note that the source may need the CSI of all links in the network also for other purposes such as cross¸layer resource allocation. 68Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 3.5 Simulation Results In this section, we present simulation results for the SINR, the mutual information, and the BER of a cooperative network with AF¸BF. For all mutual information and SINR results presented in this section we assume a cooperative network with σ21 = σ 2 2 = 0.1. For the individual relay power constraints, the joint relay power constraint, and the joint source¸relay power constraint, we use (P1 = 1, P2,i = 1/NR, 1 ≤ i ≤ NR), (P1 = 1, P2 = 1), and P = 2, respectively. The locations of the source, the destination, and the relays are shown in Fig. 3.2, where the numbers on top and beside the arrows indicate the normalized distance between the nodes. Potential relay locations are marked by (a)¸(e). The normalized distance between the source and the destination is equal to 2 and the normalized horizontal distance between the source and the potential relay locations is d. The fading gains of all links are modeled as independent, identically distributed Rayleigh fading. Furthermore, a path¸loss exponent of 3 is assumed and all results were averaged over 100, 000 independent realizations of the fading channels unless speci¿ed otherwise. The optimal BF vectors at the source and the optimal AF¸BF matrices at the relays were obtained with the algorithms introduced in Sections 3.3 and 3.4. For a fair evaluation of the gain achievable with multi¸relay BF, we compare the perfor- mance of the proposed schemes with relay selection [80], which has a lower implementation complexity. For relay selection, we compute the optimal source BF vector and the opti- mal AF¸BF matrix for each relay in the network, and select subsequently the relay which achieves the highest SINR for transmission. 3.5.1 Comparison of Source BF Vector Optimization Methods First, we compare the performance of the proposed suboptimal source BF vector optimiza- tion methods. For this purpose, we show in Figs. 3.3¸3.5 cumulative distribution functions 69Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays (c) (b) (d) (e) 2 − dd source destination 1/4 1/4 1/4 1/4 (a) Figure 3.2: Locations of source, destination, and relays in simulation. (CDFs) of the achieved SINR, i.e., the probability that the achieved SINR is smaller than the SINR value on the x¸axis. Since the optimal source beamforming vectors can be com- puted with the proposed transformation methods only for joint relay and joint source¸relay power constraints and NT = 2 and NR = 2, we also consider a gradient method with mul- tiple random initializations. In particular, we run the gradient algorithms in Sections 3.4.2 and 3.4.3 for 100 random initializations and for the solution of the ad hoc method. Subse- quently, we select the beamforming vector which yields the highest SINR among the 101 obtained solutions. Results for the gradient method with random initialization are shown in Figs. 3.4 and 3.5. In Fig. 3.3, we compare the performances of the di¾erent source BF vector optimization methods proposed for the joint relay power constraint. There are NT = 2 antennas at the source and one relay at locations (a) and (e), respectively. For the relays we consider the cases M1 = M2 = 1 and M1 = 2, M2 = 3, respectively. As can be observed, for both considered numbers of relay antennas the gradient method closely approaches the global optimal solution, which was found with the transformation method. The loss in performance su¾ered by the relaxation method and the ad hoc method is larger for M1 = M2 = 1 than for M1 = 2, M2 = 3. Relay selection su¾ers from a signi¿cant loss in 70Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   Global Optimum Gradient Method Ad hoc Method Relaxation Method Relay Selection (Per−Relay PC) CD F M1 = M2 = 1 SINR [dB] M1 = 2, M2 = 3 Figure 3.3: CDF of the instantaneous SINR for AF¸BF with joint relay power constraint (PC) and one relay located at (a) and (e), respectively. Results for di¾erent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A path¸loss exponent of 3, NT = 2, and d = 1 are assumed. performance since it cannot exploit the BF gain across the relays. In Fig. 3.4, we compare the performance of the proposed source BF vector optimization techniques for the joint source¸relay power constraint for NT = 2 antennas at the source and NR single¸antenna relays for d = 1. For NR = 2 the gradient algorithm achieves practically the same performance as the optimal transformation method, which becomes too complex for NR = 5 and NR = 10. For NR = 5 and NR = 10, it can be observed 71Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 0 2 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   Global Optimum Gradient Method w/ Random Initialization Gradient Method Ad hoc Method Relay Selection (Joint Source−Relay PC) NR = 10 CD F NR = 2 NR = 5 SINR [dB] Figure 3.4: CDF of the instantaneous SINR for AF¸BF with joint source¸relay power constraint (PC) and NR relays. Results for di¾erent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A path¸loss exponent of 3, NT = 2, and d = 1 are assumed. The relays are located at (a) and (e) for NR = 2, (a)¸(e) for NR = 5, and (a)¸(e) with 2 relays at each location for NR = 10. that additional random initializations cannot signi¿cantly improve the performance of the gradient method, which suggests that the gradient method initialized with the solution of the ad hoc method is close¸to¸optimal also for large numbers of relays. The performance gap between the gradient method and the ad hoc method is practically independent of the number of relays. In contrast, the performance loss su¾ered by relay selection increases 72Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 5 6 7 8 9 10 11 12 13 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   Gradient Method Gradient Method w/ Random Initialization Ad hoc Method Relay Selection (Per−Relay PC) CD F SINR [dB] NT = 2 NT = 5 NT = 10 Figure 3.5: CDF of the instantaneous SINR for AF¸BF with individual relay power con- straints (PCs) and NR = 5 single¸antenna relays at locations (a)¸(e). Results for di¾erent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A path¸loss exponent of 3 and d = 1 are assumed. with increasing numbers of relays. In Fig. 3.5, we consider the case of individual relay power constraints and show the CDFs achieved with the di¾erent source BF vector optimization methods for NR = 5 single¸ antenna relays located at positions (a)¸(e) in Fig. 3.2 for d = 1. For the gradient method, the performance gain achievable with additional random initializations is negligible even for NT = 10 source antennas. However, the performance loss su¾ered by the ad hoc method 73Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays compared to the gradient method increases with increasing number of source antennas. For example, for a CDF value of 0.5, the performance di¾erence between both schemes is 0.5 dB and 1.1 dB for NT = 2 and NT = 10, respectively. 3.5.2 Impact of Network Parameters on Performance Figs. 3.6 and 3.7 show the average SINR vs. distance d for AF¸BF with di¾erent numbers of transmit antennas for joint relay and individual relay power constraints, respectively. We assume NR = 2 relays with one relay located in (a) and (e), respectively. For both consid- ered constraints multi¸relay AF¸BF enables considerable performance gains compared to relay selection and direct transmission. Direct transmission is preferable only if the relay¸ destination SNR is poor because the relays are located close to the source (small d). The performance loss su¾ered by relay selection is between 1 and 2 dB. Increasing the number of source antennas is bene¿cial for both constraints unless the relays are located close to the source. In the latter case, the relay¸destination channel is the performance bottleneck and more source antennas cannot improve performance. If only NT = 1 source antenna is available, BF is not used at the source (i.e., g/||g||2 = 1). For NT = 2 and NT = 5 source antennas the gradient methods achieve the highest SINRs in both ¿gures. Fig. 3.6 shows that while the max¸min relaxation method outperforms the ad hoc method for small d, the ad hoc method is preferable for large d (e.g. d ≥ 1.4 for NT = 5). In the latter case, the SINR of the source¸relay channels is much lower than that of the relay¸destination channels and the ad hoc method becomes optimal, cf. Section 3.4.2. Next we investigate the impact of the number of relays and the number of relay an- tennas. In Fig. 3.8, we show the average SINR vs. distance d for AF¸BF with NT = 5 source antennas for the joint source¸relay power constraint. For the case with two relays (in positions (a) and (e)) increasing the number of relay antennas from M1 = M2 = 1 to 74Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −4 −2 0 2 4 6 8 10 12   NT = 1 BF NT = 2 (no relay) BF NT = 5 (no relay) AF−BF NT = 1, M1 = 2, M2 = 3 AF−BF NT = 2, M1 = 2, M2 = 3 AF−BF NT = 5, M1 = 2, M2 = 3 Gradient Method Ad hoc Method Relaxation Relay Selection (Per−Relay PC) d A v erag e SIN R [dB ] Figure 3.6: Average SINR vs. distance d for AF¸BF with joint relay power constraint (PC) and di¾erent numbers of antennas NT at the source. A path¸loss exponent of 3 is assumed. For comparison the SNR without relaying for a source transmit power of P = 2 and the SINR for relay selection are also shown. M1 = 2, M2 = 3 signi¿cantly improves performance. Furthermore, Fig. 3.8 shows that it is preferable to have the 5 relay antennas located in just two relays rather than having them distributed over ¿ve relays. This can be explained by the fact that in the former case the AF¸BF matrices have 9 + 4 = 13 elements that can be optimized whereas in the latter case they have only 5× 1 = 5 elements. Similar to Fig. 3.3, we observe from Fig. 3.8 that the gradient algorithm yields larger gains over the ad hoc method for single¸antenna 75Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −4 −2 0 2 4 6 8 10 12   NT = 1 BF NT = 2 (no relay) BF NT = 5 (no relay) AF−BF NT = 1, M1 = 2, M2 = 3 AF−BF NT = 2, M1 = 2, M2 = 3 AF−BF NT = 5, M1 = 2, M2 = 3 Gradient Method Ad hoc Method Relay Selection (Per−Relay PC) d A v erag e SIN R [dB ] Figure 3.7: Average SINR vs. distance d for AF¸BF with individual relay power constraints (PCs) and di¾erent numbers of antennas NT at the source. A path¸loss exponent of 3 is assumed. For comparison the SNR without relaying for a source transmit power of P = 2 and the SINR for relay selection are also shown. relays than for multi¸antenna relays. 3.5.3 Impact of Power Constraints on Performance In Fig. 3.9, we compare the average mutual information of AF¸BF for the three considered power constraints and di¾erent network setups. For NT = 2 and NT = 5 source antennas the respective gradient methods were used to ¿nd the optimal source BF vector. If the 76Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 3 4 5 6 7 8 9 10 11 12   BF NT = 5 (no relay) AF−BF NT = 5, M1 = 1, M2 = 1 AF−BF NT = 5, M1 = 2, M2 = 3 AF−BF NT = 5, M1 = M2 = M3 = M4 = M5 = 1 Gradient Method Ad hoc Method Relay Selection (Joint Source−Relay PC) d A v erag e SIN R [dB ] Figure 3.8: Average SINR vs. distance d for AF¸BF with joint source¸relay power con- straint (PC) and di¾erent numbers of relays and di¾erent numbers of relay antennas. A path¸loss exponent of 3 is assumed. For comparison the SNR without relaying for a source transmit power of P = 2 and the SINR for relay selection are also shown. relays are located in the middle between the source and the destination (i.e., d ≈ 1) all three constraints result in a comparable performance. Furthermore, because of the symmetry of the considered setups, the performance di¾erence between the joint relay power constraint and the individual relay power constraints is comparatively small. In contrast, the joint source¸relay power constraint can yield signi¿cant performance gains if the relays are close to the source or close to the destination, respectively, by Àexibly allocating more or less 77Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 1.5 2   Joint Source−Relay PC Joint Relay PC Individual Relay PC Relay Selection (Per−Relay PC) Relay Selection (Joint Source−Relay PC) NT = 5, M1 = 2, M2 = 3 NT = 2, M1 = M2 = M3 = M4 = M5 = 1 BF with NT = 5 (no relay) BF with NT = 2 (no relay) d AM I (bit/s/Hz ) Figure 3.9: Average mutual information (AMI) in (bits/s/Hz) vs. distance d for two di¾erent network setups and di¾erent power constraints (PCs). The relays are in locations (a) and (e) for NR = 2 and (a)¸(e) for NR = 5. The proposed gradient methods are used for computation of the source BF vector g. A path¸loss exponent of 4 is assumed. For comparison the average mutual information without relaying for a source transmit power of P = 2 and the average mutual information for relay selection are also shown. power to the source. Fig. 3.10 shows the BER of 16¸ary quadrature amplitude modulation (16¸QAM) for the three considered power constraints. For comparison we also show the BER for direct trans- mission with quaternary phase¸shift keying (QPSK), i.e., the data rates for transmission with and without relaying are identical. Fig. 3.10 clearly shows that for the same number 78Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays 0 2 4 6 8 10 12 14 16 10−4 10−3 10−2 10−1   Joint Source−Relay PC Joint Relay PC Individual Relay PC Relay Selection (Per−Relay PC) Relay Selection (Joint Source−Relay PC) NT = 5, M1 = 2, M2 = 3 NT = 2, M1 = M2 = M3 = M4 = M5 = 1 QPSK BF with NT = 5 (no relay) QPSK BF with NT = 2 (no relay) σ2d/σ21 BE R Figure 3.10: Average BER vs. σ2d/σ2n for two di¾erent network setups and di¾erent power constraints. The relays are in locations (a) and (e) for NR = 2 and (a)¸(e) for NR = 5. The proposed gradient methods are used for computation of the source BF vector g. A path¸loss exponent of 3 and d = 1 are assumed. AF¸BF: 16¸QAM. Direct transmission: QPSK, source transmit power P = 2. of source transmit antennas, AF¸BF yields signi¿cant performance gains in termsof the achievable BER compared to direct transmission and relay selection. Thereby, the achiev- able BER is the lower, the less restrictive the power constraints are, i.e., for a given SINR, the joint source¸relay power constraint yields a lower BER than the joint relay power con- straint and the joint relay power constraint yields a lower BER than the individual relay 79Chapter 3. Cooperative AF¸BF with Multiple Multi¸Antenna Relays power constraints. 3.6 Conclusions In this chapter, we have considered AF¸BF for cooperative networks with one multi¸ antenna source, multiple multi¸antenna relays, and one single¸antenna destination for three di¾erent power constraints. The obtained solutions show that while the source node requires the CSI of all channels in the network to compute the optimal BF vector, the relays only have to know their own source¸relay and relay¸destination channels for im- plementation of the optimal AF¸BF matrices if the source can provide a small amount of feedback to each relay. For a given BF vector at the source, we have fully character- ized the optimal AF¸BF matrices for all three constraints. Furthermore, for small scale networks with joint relay or joint source¸relay power constraints the optimal source BF vector can be found using polynomial programming. For large scale networks and networks with individual relay power constraints eÁcient suboptimal ad hoc and gradient methods for optimization of the source BF vectors have been provided. Simulation results con¿rm the close¸to¸optimal performance of the proposed gradient methods and show that the relative performance of the three considered power constraints signi¿cantly depends on the network topology. Furthermore, our results show that increasing the number of antennas at the source is particularly bene¿cial if the relays are located far away from the source. In contrast, increasing the number of antennas at the relays or the number of relays is always bene¿cial regardless of the location of the relays. 80Chapter 4 Cooperative Filter¸and¸Forward Beamforming for Frequency¸Selective Channels with Multiple Multi¸Antenna Relays 4.1 Introduction In the previous chapter, we have investigated BF for cooperative networks in frequency¸ nonselective channels. Starting from this chapter, we will focus on BF schemes for coop- erative networks in frequency¸selective channels. Particularly, in this chapter, we consider one¸way cooperative networks with one single¸antenna source, one single¸antenna desti- nation, and multiple multi¸antenna relay nodes. We assume single¸carrier transmission and frequency¸selective channels. Relaying schemes for single¸carrier transmission over frequency¸selective channels have received little attention in the literature so far with [50, 81] being two notable exceptions. Speci¿cally, a cooperative ¿lter¸and¸forward (FF) BF technique was proposed and opti- mized under the assumptions that (1) there is no direct link between the source and the destination, (2) an equalizer is not available at the destination, and (3) full CSI of all links 81Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays is available [50]. We note that FF relaying for frequency¸Àat channels was also considered in [46]. For the frequency¸selective case, distributed space¸time block coding at the relays and equalization at the destination has been proposed in [81]. Distributed space¸time coding does not require full CSI but has a worse performance than FF¸BF. In this chapter, we investigate cooperative FF¸BF for frequency¸selective channels for the case where the destination node has either (1) a simple slicer without equalization or (2) enough processing power to perform low¸complexity equalization such as LE or DFE. Similar to [50] we assume that the central node, which computes the optimal FF¸ BF ¿lters, has full CSI of all links. However, unlike [50], our model also includes multiple multi¸antenna relays and equalization at the destination. This chapter makes the following contributions: • For the simple slicer case, we optimize the FF¸BF ¿lters for maximization of the SINR under a transmit power constraint and for minimization of the transmit power under a QoS constraint, respectively. For both optimization criteria we ¿nd a closed¸form solution for the optimal FIR FF¸BF matrix ¿lters at the relays. • For the LE/DFE case, we assume FIR and IIR ¿lters at the relays. We optimize FF¸BF for maximization of the SINR at the output of LE and DFE as well as an idealized matched ¿lter (MF) receiver ignoring any inter¸symbol interference (ISI) in the ¿lter output. The latter provides a natural performance upper bound for any equalization scheme [5] and allows us to bound possible performance gains achiev- able with more complex equalization schemes such as maximum likelihood sequence estimation (MLSE). • For IIR FF¸BF with equalization, we show that the frequency response vector of the optimal FF¸BF ¿lters can be decomposed into a unit¸norm direction vector and a scalar power allocation factor across frequencies. We provide a uni¿ed closed¸form 82Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays solution for the direction vector valid for all three considered equalization receiver structures and an eÁcient numerical method with guaranteed convergence for the power allocation. • For FIR FF¸BF with equalization, we show that the FF¸BF ¿lter optimization prob- lem is related to a diÁcult mathematical problem for which an exact solution in closed form does not seem to exist. Therefore, we provide an eÁcient numerical method for recursive calculation of the optimum FIR FF¸BF ¿lters. • Our simulation results show that (1) the performance of FF¸BF without equalization at the destination crucially depends on the slicer decision delay, (2) with the same FF¸BF ¿lter length, the addition of simple LE and DFE equalizers at the destination node yields large performance gains compared to FF¸BF with a slicer, (3) if long FIR FF¸BF ¿lters are employed, the simple slicer receiver with optimized decision delay closely approaches the same performance as equalizers, (4) relatively short FIR FF¸BF ¿lters with equalization suÁce to closely approach the performance of IIR FF¸BF ¿lters, (5) the gap between FF¸BF with LE and DFE, respectively, and the MF receiver is small implying that little can be gained by adopting more complex equalization schemes, and (6) if the total number of antennas at the relays is the same, it is preferable to have fewer relays with multiple antennas rather than more relays with less antennas each. The remainder of this chapter is organized as follows. In Section 4.2, the adopted system model is presented. The optimization of FIR FF¸BF ¿lters when the destination employs only a simple slicer is discussed in Section 4.3, and the case where the destination employs LE/DFE is considered in Section 4.4. Simulation results are provided in Section 4.5, and some conclusions are drawn in Section 4.6. 83Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 4.2 System Model We consider a relay network with one single¸antenna source node, NR multi¸antenna relays, and one single¸antenna destination node. A block diagram of the discrete¸time overall transmission system in equivalent complex baseband representation is shown in Fig. 4.1. As usual, transmission is organized in two intervals. In the ¿rst interval, the source node transmits a data packet which is received by the relays. In the second interval, the relays ¿lter the received packet and forward it to the destination node. We assume that there is no direct link between the source and the destination node (FF¸BF for LE/DFE with direct link has been considered in our journal paper [82]). At the destination, the data packets received during the second interval are processed and detected. In Fig. 4.1, the discrete¸time CIRs between the source and the ith antenna of the zth relay, gi,z[k], 0 ≤ k ≤ Lg − 1, and between the ith antenna of relay z and the destination, hi,z[k], 0 ≤ k ≤ Lh − 1, contain the combined e¾ects of transmit pulse shaping, the continuous¸time channel, receive ¿ltering, and sampling. Here, Lg and Lh denote the lengths of the source¸relay and the relay¸destination CIRs, respectively. Furthermore, we assume that relay z has Mz antennas and de¿ne hz[k] , [h1,z[k] . . . hMz ,z[k]]T and gz[k] , [g1,z[k] . . . gMz ,z[k]]T . In the following, we describe the processing performed at the relays and the destination in detail. 4.2.1 FF¸BF at Relays The signal received at the ith antenna, i = 1, . . . ,Mz, of the zth relay, z = 1, . . . , NR, during the ¿rst time interval is given by yi,z[k] = gi,z[k] ∗ s[k] + ni,z[k] , (4.1) 84Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays n0[k] s[k] sˆ[k]EQ/Slicer nMz,z[k] hz[k] n1,z[k] gz[k] gNR[k] hNR[k] Relay NR Az[k] Relay z Relay 1 g1[k] h1[k] Figure 4.1: Cooperative network with one single¸antenna source, multiple multi¸antenna relay nodes, and one single¸antenna destination. EQ is the equalizer at the destination. sˆ[k] are estimated symbols after the equalizer or slicer. where s[k] are i.i.d. symbols taken from a scalar symbol alphabet A such as PSK or QAM with variance σ2s , E{|s[k]|2}, and ni,z[k] denotes the AWGN at the ith receive antenna of the zth relay with variance σ2n , E{|ni,z[k]|2}. The FF¸BF matrix ¿lter impulse response coeÁcients of relay z are denoted by Mz×Mz matrix Az[k], −ql ≤ k ≤ qu, with elements aji,z[k] on row j and column i. For IIR FF¸BF matrix ¿lters ql → ∞ and qu → ∞ and for FIR FF¸BF ¿lters ql = 0 and qu = La − 1, where La is the FIR FF¸BF matrix ¿lter length. The signal transmitted by the jth antenna, j = 1, . . . ,Mz, of the zth relay, z = 1, . . . , NR, during the second time interval can be expressed as tj,z[k] = Mz∑ i=1 aji,z[k] ∗ yi,z[k] = Mz∑ i=1 aji,z[k] ∗ gi,z[k] ∗ s[k]+ Mz∑ i=1 aji,z[k] ∗ ni,z[k] . (4.2) 85Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 4.2.2 Processing at Destination Since there is no direct link between the source and the destination, the signal received at the destination is given by r[k] = NR∑ z=1 Mz∑ j=1 hj,z[k] ∗ tj,z[k] + n0[k] = heq[k] ∗ s[k] + n[k] , (4.3) where n0[k] is AWGN with variance σ2v , E{|n0[k]|2}. The equivalent CIR heq[k] between source and destination and the e¾ective noise n[k] are given by heq[k] , NR∑ z=1 Mz∑ j=1 hj,z[k] ∗ Mz∑ i=1 aji,z[k] ∗ gi,z[k] , (4.4) and n[k] , NR∑ z=1 Mz∑ j=1 hj,z[k] ∗ Mz∑ i=1 aji,z[k] ∗ ni,z[k] + n0[k] , (4.5) respectively. Note that n[k] is colored noise because of the ¿ltering of ni,z[k] by hz[k] and Az[k]. Eq. (4.3) shows that a cooperative relay network with FF¸BF can be modeled as an equivalent SISO system. Therefore, as long as the destination knows the statistics of the colored noise n[k], at the destination the same equalization, channel estimation, and channel tracking techniques as for point¸to¸point single¸antenna transmission can be used. Here, we consider two cases: (1) The destination makes a decision based on r[k] without equalization. (2) The destination ¿rst equalizes r[k] using LE or DFE optimized under zero¸forcing (ZF) and minimum mean¸squared error (MMSE) criteria before making a decision [5]. The optimization of the corresponding FF¸BF matrix ¿lters will be discussed in Sections 4.3 and 4.4, respectively. 86Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 4.2.3 Feedback Channel We assume that the destination estimates the relay¸destination CIRs hi,z[k], 0 ≤ k ≤ Lh − 1, 1 ≤ i ≤ NR and 1 ≤ z ≤ NR, during a training phase. Similarly, relay i estimate its own source¸relay CIR gi,z[k], 0 ≤ k ≤ Lg − 1, and forwards the estimate to the destination node. Alternatively, the destination may directly estimate the combined CIR of the source¸relay and relay¸destination channels, hi,z[k]∗gi,z[k] if relay i retransmits the training signal received from the source. The destination can then extract gi,z[k] from hi,z[k] ∗ gi,z[k] and hi,z[k] via deconvolution. Subsequently, the destination node computes the FF¸BF ¿lters using the CSI of all links and feeds back the ¿lter coeÁcients to the relays. Throughout this chapter we assume that the CSI and the feedback channel are perfect, which implies that the nodes in the network have limited mobility, and thus, all channels are slowly fading. We note that similar assumptions regarding the availability of CSI and the feedback channel are typically made in the distributed BF literature for both frequency¸Àat and frequency¸selective channels, cf. e.g. [35, 39, 42, 44, 50]. 4.3 FIR FF¸BF without Equalization In this section, we consider the case where the destination node cannot a¾ord an equalizer due to size and/or power limitations. Therefore, we assume that a simple slicer is employed at the destination throughout this section. In the following, we will optimize FIR FF¸BF matrix ¿lters for maximization of the SINR at the slicer output under a power constraint and for minimization of the transmit power under a QoS constraint, respectively. We note that the results for multi¸antenna relays in Sections 4.3.1 and 4.3.2 are extensions of the results for single¸antenna relays given in [50]. Joint source¸relay power constraints as considered in Sections 4.3.3 and 4.3.4 were not discussed in [50]. Also, for relaying with 87Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays single antenna relays a decision delay k0 was not considered in [50], i.e., k0 = 0. However, as will be shown in Section 4.5, the proper choice of decision delay k0 is important for system performance. The equivalent CIR heq , [heq[0] heq[1] . . . heq[La +Lg +Lh− 3]]T between source and destination in Eq. (4.4) can be rewritten as heq = NR∑ z=1 Hz ¯Gza¯z , HGDa, (4.6) where H , [H1 . . . HNR ], GD , diag { ¯G1, . . . , ¯GNR } , and a , [aT1 . . . aTNR ]T with (La + Lg + Lh − 2) × (La + Lg − 1)Mz matrix Hz , [H1,z H2,z . . . HMz ,z], (La + Lg − 1)Mz × M2z La matrix ¯Gz , IMz ⊗ [ ¯G1,z . . . ¯GMz ,z], and M2z La × 1 vector az , [aT11,z aT12,z . . . aT1Mz ,z aT21,z . . . aTMzMz ,z]T with aij,z , [aij,z[0] aij,z[1] . . . aij,z[La − 1]]T . Moreover, (La +Lg−1)×La matrix ¯Gi,z and (La +Lg +Lh−2)×(La +Lg−1) matrix H i,z are column circular matrices with [gi,z[0] . . . gi,z[Lg − 1] 0TLa−1]T and [hi,z[0] . . . hi,z[Lh − 1] 0TLa+Lg−2]T in the ¿rst columns, respectively. Matrix H can be separated into one vector hk0 and one sub¸matrix Hk0 , i.e., length (La + Lg − 1)∑NRz=1 Mz vector hTk0 is the row k0 of Hk0 , and Hk0 , [H]ij, i ∈ {1, . . . , k0 − 1, k0 + 1, . . . (La + Lg + Lh − 2)}, j ∈ {1, . . . , (La + Lg − 1)∑NRz=1 Mz}. Therefore, the ¿rst term in (4.3) can be decomposed into a signal part and an ISI part heq[k] ∗ s[k] = heq[k0]s[k − k0] + La+Lg+Lh−3∑ l=1, l 6=k0 heq[l]s[k − l] = hTk0GDas[k − k0] ︸ ︷︷ ︸ desired signal + sT [k]Hk0GDa ︸ ︷︷ ︸ ISI (4.7) with s[k] = [s[k] . . . s[k− k0 + 1] s[k− k0 − 1] . . . s1[k− (La + Lg + Lh − 3)]]T , and k0 is the slicer decision delay at the destination. Therefore, the power of the desired signal and 88Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays the ISI can be obtained as E {∣∣hTk0GDas[k − k0]∣∣2} = σ2saHGHD h∗k0hTk0GDa (4.8) and E {∣∣sT [k]Hk0GDa∣∣2} = σ2saHGHDHHk0Hk0GDa , (4.9) respectively. Similarly, n[k] in (4.5) can be rewritten as n[k] = NR∑ z=1 nz[k] ˘Hza¯z + n0[k] , N [k] ˘Ha + n0[k] (4.10) with length ∑NR z=1(La + Lh − 1)Mz row vector n[k] , [nT1 [k] . . . nTNR [k]]T and ∑NR z=1(La + Lh − 1)Mz ×∑NRz=1 M2z La matrix ˘H , diag { ˘H1, . . . , ˘HNR } . Moreover, nz[k] , [nT1,z[k] . . . nTMz ,z[k]]T with ni,z[k] , [ni,z[k] . . . ni,z[k − (La + Lh − 2)]]T , and ˘Hz , [IMz ⊗ ¯H1,z . . . IMz ⊗ ¯HMz ,z], where (La + Lh − 1) × La matrix ¯H i,z is column circular matrix with [hi,z[0] . . . hi,z[Lh − 1] 0TLa−1]T in the ¿rst column. The noise power can be obtained as E{|n[k]|2} = σ2naH ˘HH ˘Ha + σ2v . (4.11) From (4.8), (4.9), and (4.11), the SINR at the destination can be obtained as SINR (a) = E {∣∣hTk0GDas[k − k0]∣∣2} E {|sT [k]Hk0GDa|2}+ E{|n[k]|2} = aHW 1a aHW 2a + aHW 3a + 1 (4.12) with W 1 , σ2sGHD h∗k0hTk0GD/σ2v , W 2 , σ2sGHDHHk0Hk0GD/σ2v , and W 3 , σ2n ˘HH ˘H/σ2v . 89Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays From (4.2), the total transmit power, PR(a), of the relays can be obtained as PR(a) = NR∑ z=1 Mz∑ j=1 E {|tj,z[k]|2}= aHDa , (4.13) with D , σ2sGHDGD + σ2nILa ∑NRz=1 M2z . In the following, we will formulate various FF¸BF ¿lter optimization problems based on (4.12) and (4.13). 4.3.1 SINR Maximization Under Relay Power Constraint First, we consider the optimization of the FF¸BF matrix ¿lters for maximization of the SINR subject to maximum relay power P [50]. In comparison to [50], we consider a more general case where relays have multiple antennas, and the resulting optimization problem is more involved. Accordingly, the optimization problem can be formulated as max a SINR (a) (4.14a) s.t. aHDa ≤ P . (4.14b) By letting w , D1/2a, where D1/2 is the Cholesky decomposition of D, the optimization problem in (4.14) can be reformulated as a generalized eigenvalue problem. The optimum w can be obtained as wopt = √ Pu { Q−11 D−H/2W 1D−1/2 } = √ PQ−11 D−H/2GHD h∗k0√ hTk0GDD−1/2Q−21 D−H/2GHD h∗k0 , (4.15) 90Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays where Q1 , D−H/2 (W 2 + W 3) D−1/2 + 1P ILa ∑NRz=1 M2z and u{X} is the principle eigen- vector of matrix X. Therefore, the maximum SINR can be obtained as SINRmax = σ2s σ2v hTk0GD ( W 2 + W 3 + 1 P D ) −1 GHD h∗k0 , (4.16) and the corresponding optimum FF¸BF matrix ¿lter in vector form is given as aopt = √ P ( W 2 + W 3 + 1P D ) −1 GHD h∗k0√ hTk0GDD−1/2Q−21 D−H/2GHD h∗k0 . (4.17) 4.3.2 Relay Power Minimization Under SINR Constraint Here, we optimize the FF¸BF matrix ¿lters for minimization of the relay transmit power, PR(a), subject to an SINR constraint [50]. Again, we extend the results from [50] for single¸ antenna relays to multiple¸antenna relays. The optimization problem can be formulated as min a PR(a) = aHDa (4.18a) s.t. aHW 1a aHW 2a + aHW 3a + 1 ≥ γ , (4.18b) where γ is the QoS requirement (minimal required SINR) at the destination. We let w = D1/2a again and note that the above problem is infeasible when Q2 , D−H/2 ( W 1−γW 2 −γW 3 ) D−1/2, is negative semide¿nite. If the problem is feasible, the optimum FF¸BF matrix ¿lter can be obtained as aopt = ( γ λmax{Q2} )1/2 D−1/2u{Q2} (4.19) 91Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays and the corresponding minimum relay power is Pmin = γ λmax{Q2} . (4.20) 4.3.3 SINR Maximization Under Source¸Relay Power Constraint Compared to the case with separate power constraints for the source and the relays, which was considered in Section 4.3.1, additional performance gains are possible with a joint source¸relay transmit power constraint. We note that the joint source¸relay transmit power constraint cases in Subsections 4.3.3 and 4.3.4 were not considered in [50]. The corresponding optimization problem can be formulated as max a, σ2s aHW 1a aHW 2a + aHW 3a + 1 (4.21a) s.t. aHDa + σ2s ≤ P (4.21b) The optimal solution can be found with a divide¸and¸conquer method. In particular, if we assume that σ2s is ¿xed, problem (4.21) is identical to problem (4.14). The optimum FF¸BF matrix ¿lter is obtained as aopt = √ P − σ2s ( W 2(σ2s) + W 3 + D(σ2s)P−σ2s ) −1 GHD h∗k0√ hTk0GDD−1/2(σ2s)Q−21 (σ2s)D−H/2(σ2s)GHD h∗k0 , (4.22) and the corresponding maximum SINR is given by SINRmax(σ2s) = σ 2 s σ2v hTk0GD ( W 2(σ2s) + W 3 + D(σ 2 s) P − σ2s ) −1 GHD h∗k0 (4.23) 92Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays where Q1(σ2s) , D−H/2(σ2s) ( W 2(σ2s) + W 3 ) D−1/2(σ2s) + 1P − σ2s ILa ∑NR z=1 M2z . (4.24) Note that D, W 1, and W 2 de¿ned earlier depend on σ2s in this case. The remaining problem is to ¿nd the optimal σ2s such that SINRmax(σ2s) is maximized, i.e. max σ2s , 0≤σ2s≤P SINRmax(σ2s) . (4.25) Problem (4.25) can be easily solved by a grid search or other numerical methods given in [1]. 4.3.4 Source¸Relay Power Minimization Under SINR Constraint In this case, the goal is to minimize the joint source¸relay transmit power subject to a destination SINR constraint. The optimization problem can be formulated as min a, σ2s aHDa + σ2s (4.26a) s.t. aHW 1a aHW 2a + aHW 3a + 1 ≥ γ (4.26b) Again, we assume that σ2s is ¿xed, and the resulting problem is identical to problem (4.18). If the problem is feasible, the optimum FF¸BF matrix ¿lter is given by aopt = ( γ λmax{Q2(σ2s)} )1/2 D−1/2(σ2s)u{Q2(σ2s)} , (4.27) 93Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays where λmax{·} denotes maximum eigenvalue of a matrix, and the corresponding minimum joint source¸relay transmit power is Pmin = γ λmax{Q2(σ2s)} + σ 2 s (4.28) with Q2(σ2s) , D−H/2(σ2s) (W 1(σ2s)− γW 2(σ2s)− γW 3) ×D−1/2(σ2s). The remaining op- timization problem is min σ2s γ λmax{Q2(σ2s)} + σ 2 s (4.29a) s.t. λmax{Q2(σ2s)} > 0 . (4.29b) Note that λmax{Q2(σ2s)} = 0 when σ2s = 0. Therefore, σ2s = 0 has been ignored in problem (4.29) due to the fact that (4.29b) is satis¿ed only if σ2s > 0. Problem (4.29) can be easily solved by numerical methods given in [1]. 4.4 FF¸BF with Equalization Throughout this section we assume that the destination node employs LE or DFE with IIR equalization ¿lters. In a practical implementation, FIR equalization ¿lters are used, of course. However, suÁciently long FIR ¿lters will approach the performance of IIR ¿lters arbitrarily close. Assuming IIR equalization ¿lters has the advantage that relatively simple and elegant expressions for the SINR at the equalizer output exist [83, 84]. 4.4.1 Optimal IIR FF¸BF with Equalization In order to exploit the SINR expressions in [83, 84], we ¿rst have to whiten the noise impairing the signal received at the destination. The power spectral density of n[k] in 94Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays (4.5) can be obtained as Φn(f) = σ2n NR∑ z=1 Mz∑ i=1 ∣∣∣∣∣ Mz∑ j=1 Hj,z(f)Aji,z(f) ∣∣∣∣∣ 2 + σ2v = σ 2 na H(f)Γ(f)a(f) + σ2v (4.30) with ∑NR z=1 M2z× ∑NR z=1 M2z square matrix Γ(f) , diag {Γ1(f), . . . , ΓNR(f)}, where Γz(f) ,( h∗z(f)hTz (f) ) ⊗ IMz and hz(f) , [H1,z(f), . . . , HMz ,z(f)]T . The frequency response of the relay¸destination channel corresponding to the jth antenna of the zth relay is given by Hj,z(f) , F{hj,z[k]}. The frequency responses of the FF¸BF matrix ¿lters are collected in vector a(f) , [aT1 (f) . . . aTNR(f)]T with az(f) , [A11,z(f) A12,z(f) . . . AMzMz ,z(f)]T , where Aji,z(f) , F{aji,z[k]} denotes the frequency response of the FF¸BF matrix ¿lter at relay z corresponding to the ith receive antenna and the jth transmit antenna. The whitening ¿lter W (f) for n[k] can be easily obtained as W (f) = σ2n NR∑ z=1 Mz∑ i=1 ∣∣∣∣∣ Mz∑ j=1 Hj,z(f)Aji,z(f) ∣∣∣∣∣ 2 + σ2v −1/2 = ( σ2na H(f)Γ(f)a(f) + σ2v ) −1/2 , (4.31) and we denote the output of the whitening ¿lter by r′[k]. Taking into account the whitening, the frequency response of the equivalent overall channel can be obtained as H ′eq(f) , W (f)F{heq[k]} = ( σ2na H(f)Γ(f)a(f) + σ2v ) −1/2 qT (f)a(f) (4.32) with q(f) , [qT1 (f) . . . qTNR(f)]T , qz(f) , hz(f) ⊗ gz(f), gz(f) , [G1,z(f) G2,z(f) . . . GMz ,z(f)]T , Gi,z(f) , F{gi,z[k]}, and hz(f) , [H1,z(f) H2,z(f) . . . HMz ,z(f)]T . The power spectral density of the noise component, n′[k], of r′[k] is Φn′(f) = 1. In the remainder of this section, we formulate and solve the IIR FF¸BF ¿lter opti- 95Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays mization problems for LE, DFE, and an idealized matched ¿lter (MF) receiver in a uni¿ed manner. After introducing Z(a(f)) , |H ′eq(f)|2 = a H(f)q∗(f)qT (f)a(f) σ2na H(f)Γ(f)a(f) + σ2v , (4.33) we can express the SINRs at the outputs of a decision feedback and a linear equalizer as [83, 84] SINRDFE(a(f)) = σ2s exp    1/2∫ −1/2 ln (Z(a(f)) + ξ) df    − χ (4.34) and SINRLE(a(f)) = σ2s   1/2∫ −1/2 (Z(a(f)) + ξ)−1 df   −1 − χ , (4.35) respectively. In (4.34) and (4.35), we have χ = 0, ξ = 0 and χ = 1, ξ = 1/σ2s if the equalization ¿lters are optimized based on a ZF and an MMSE criterion, respectively. Similarly, if only a single, isolated symbol s[k] is transmitted, the SINR at the output of an MF is given by [5] SINRMF(a(f)) = σ2s 1/2∫ −1/2 Z(a(f)) df. (4.36) It is not diÁcult to show that regardless of how the FF¸BF ¿lter frequency responses a(f) are chosen, we always have [84] SINRMF(a(f)) ≥ SINRDFE(a(f)) ≥ SINRLE(a(f)). (4.37) Thus, the MF receiver constitutes a performance upper bound for DFE and LE with 96Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays continuous transmission of symbols s[k]. In fact, it can be shown that the MF receiver provides a performance upper bound for any realizable equalization structure including optimal MLSE [5]. Note, however, that the MF receiver generally has a poor performance for continuous symbol transmission since it does not combat ISI. In this section, our goal is to optimize the FF¸BF matrix ¿lters for maximization of the SINRs at the output of the considered equalizers. To make the problem well de¿ned, we constrain the relay transmit power, PR, which is given by PR = NR∑ z=1 Mz∑ j=1 1/2∫ −1/2 Φtj,z(f)df = 1/2∫ −1/2 aH(f)D(f)a(f)df (4.38) where Φtj,z(f) , σ2s | ∑Mz i=1 Aji,z(f)Gi,z(f)|2 + σ2n ∑Mz i=1 |Aji,z(f)|2, z = 1, . . . , NR, j = 1, . . . , Mz, is the power spectral density of the transmit signal tj,z[k] at the jth antenna of the zth relay, D(f) , σ2sGH(f)G(f) + σ2nI∑NR z=1 M2z , G(f) , diag {G1(f), . . . , GNR(f)}, and Gz(f) , IMz ⊗ gTz (f). Formally, the IIR FF¸BF ¿lter optimization problem can now be stated as max a(f) SINRX(a(f)) (4.39a) s.t. 1/2∫ −1/2 aH(f)D(f)a(f) df ≤ P, (4.39b) where P denotes the maximum relay transmit power, and X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver, respectively. It is convenient to introduce vector v(f) , D1/2(f)a(f), which can be expressed as v(f) = √p(f)u(f) without loss of generality, where p(f) denotes the power of v(f) and u(f) has unit norm, ‖u(f)‖2 = 1. Furthermore, 97Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays we introduce ¯Z(v(f)) = ¯Z(√p(f)u(f)) , Z(a(f)), which is given by ¯Z(v(f)) = a H(f)q∗(f)qT (f)a(f) σ2na H(f)Γ(f)a(f) + σ2v = uH(f)J(f)u(f) uH(f)X(f)u(f) (4.40) with rank one, positive semi¸de¿nite matrix J(f) = p(f)D−1/2(f)q∗(f)qT (f)D−1/2(f) (4.41) and full rank, positive de¿nite matrix X(f) = σ2np(f)D−1/2(f)Γ(f)D−1/2(f) + σ2vINR . (4.42) Introducing SINRX(v(f)) = SINRX (√ p(f)u(f) ) , SINRX(a(f)), we can restate prob- lem (4.39) in equivalent form as max p(f),u(f) SINRX (√ p(f)u(f) ) (4.43a) s.t. 1/2∫ −1/2 p(f) df ≤ P (4.43b) p(f) ≥ 0. (4.43c) In the following, we provide a uni¿ed solution to problem (4.43) valid for all three considered equalization schemes. 1) Optimum IIR FF¸BF Filters : We observe from (4.43) that the constraints of the considered optimization problem do not depend on u(f). Thus, without loss of gen- erality, we can ¿nd the globally optimal solution of problem (4.43) by ¿rst maximizing the SINR with respect to u(f) for a given power allocation p(f) and by subsequently 98Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays optimizing the resulting SINR expression with respect to p(f). Furthermore, for all three considered receiver structures, the SINR SINRX(v(f)) is monotonically increasing in ¯Z (√ p(f)u(f) ) . Thus, for any given power allocation p(f), we can maximize the SINR SINRX(v(f)) by maximizing ¯Z (√ p(f)u(f) ) with respect to u(f) for all frequencies f . Hence, the optimal FF¸BF direction vector, uopt(f), can be found from the following optimization problem max u(f) ¯Z (√ p(f)u ) = uH(f)J(f)u(f) uH(f)X(f)u(f) . (4.44) Problem (4.44) is a generalized eigenvalue problem for which a closed¸form solution exists since matrix J(f) has rank one and matrix X(f) has full rank. The solution of problem (4.44) can be easily obtained as uopt(f) = c(f)X−1(f)D−1/2(f)q∗(f) , (4.45) where c(f) is a real¸valued scaling factor which is given by c(f) = 1√ qT (f)D−1/2(f)X−2(f)D−1/2(f)q∗(f) . (4.46) The maximum ¯Z (√ p(f)u(f) ) achievable with uopt(f) is ¯Z (√ p(f)uopt(f) ) = p(f)qT (f)D−1/2(f)X−1(f)D−1/2(f)q∗(f) = p(f)qT (f)(σ2np(f)Γ(f) + σ2vD(f))−1 q∗(f) . (4.47) Now, we can express the optimal FF¸BF ¿lter frequency response vector (for a given 99Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays power allocation), aopt(f), as aopt(f) = √ p(f)c(f)(σ2np(f)Γ(f) + σ2vD(f))−1 q∗(f) . (4.48) From (4.48), the optimum individual FF¸BF ¿lter of relay z, aoptz (f), can be simpli- ¿ed as aoptz (f) = √ p(f)c(f) ( σ2np(f)Γz(f) + σ2vσ2sGHz (f)Gz(f) + σ2nσ2vIM2z ) −1 q∗z(f) = √ p(f)c(f) ( σ2np(f) [ h∗z(f)hTz (f) ] ⊕ [ σ2vσ 2 sg ∗ z(f)gTz (f) + σ2nσ2vIMz ])−1 × (h∗z(f)⊗ g∗z(f)) (4.49) = √ p(f)c(f) (h∗z(f)⊗ g∗z(f)) σ2np(f)‖hz(f)‖2 + σ2vσ2s‖gz(f)‖2 + σ2nσ2v . (4.50) The transformation from (4.49) to (4.50) is accomplished by exploiting the relation [78] (M ⊕N )−1 = N∑ i=1 N∑ j=1 (mi ⊗ nj) (m¯i ⊗ n¯j)H λi(M ) + λj(N) , (4.51) where mi, ni, m¯i, and n¯i denote the eigenvectors of N ×N matrices M , N , MH , and NH , respectively. Therefore, the optimum beamforming matrix ¿lter Aoptz (f) of relay z is obtained as Aoptz (f) = √ p(f)c(f) σ2np(f)‖hz(f)‖2 + σ2vσ2s‖gz(f)‖2 + σ2nσ2v h ∗ z(f)gHz (f) , z = 1, . . . , NR. (4.52) Eq. (4.52) reveals that the optimal IIR FF¸BF matrix ¿lters for all considered receiver structures can be interpreted as the concatenation of a ¿lter matched to the source¸ relay and the relay¸destination link with frequency response h∗z(f)gHz (f) and a second ¿lter whose frequency response √ p(f)c(f)/(σ2np(f)‖hz(f)‖2 +σ2vσ2s‖gz(f)‖2 +σ2nσ2v) 100Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays depends on the power allocation, and thus on the particular equalizer used at the destination. Note that Aoptz (f) of relay z depends on the CIRs of all source¸relay, all relay¸destination, and the source¸destination channels via power allocation factor p(f). 2) Optimum Power Allocation: Before we formulate the power allocation problem for the three considered receiver structures in a uni¿ed way, we ¿rst introduce the following de¿nitions: SDFE(f) , ln(M(f)), SLE(f) , −1/M(f), and SMF(f) , M(f), (4.53) with M(f) , qT (f)(σ2nΓ(f) + σ2vD(f)/p(f))−1q∗(f) + ξ, (4.54) where for DFE and LE ξ is de¿ned after (4.35) and ξ = 0 for the MF receiver. Based on these de¿nitions, the equalizer output SINRs (4.34)¸(4.36), the original optimization problem (4.43), and the optimal frequency response direction in (4.45), we can formulate the power allocation problem as max p(f) 1/2∫ −1/2 SX(f) df (4.55a) s.t. 1/2∫ −1/2 p(f) df ≤ P (4.55b) p(f) ≥ 0, (4.55c) where X = DFE, X = LE, and X = MF for DFE, LE, and an MF at the receiver, respectively. Since ∂M(f)/∂p(f) < 0 and ∂SX(f)/∂M(f) > 0 for M(f) > 0 and X ∈ 101Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays {DFE, LE, MF}, the power allocation problem is convex for all considered equalizer structures. The Lagrangian of problem (4.55) is given by L(p(f), µ) = 1/2∫ −1/2 SX(f) df − µ 1/2∫ −1/2 p(f) df , (4.56) where µ ≥ 0 is the Lagrangian multiplier. The corresponding Lagrange dual function is D(µ) = max p(f) L(p(f), µ) = maxp(f) 1/2∫ −1/2 (SX(f)− µp(f)) df = 1/2∫ −1/2 max p(f) (SX(f)− µp(f)) df . (4.57) The last step in (4.57) can be established because the total power constraint (4.55b) is implicitly captured by the dual variable µ and the maximization over p(f) can be moved inside the integration. Therefore, for a given µ, p(f) can be obtained from max p(f) SX(p(f)) = SX(f)− µp(f) (4.58) or equivalently S ′X(f) , ∂SX(f)∂p(f) = µ. (4.59) S ′X(f) can be easily computed for all considered equalization schemes. In particular, we obtain S ′DFE(f) , M ′(f)/M(f), S ′LE(f) , M ′(f)/M2(f), and S ′MF(f) , M ′(f), (4.60) 102Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays where M ′(f) , ∂M(f)∂p(f) = q T (f)D(f)(σ2np(f)Γ(f) + σ2vD(f))−2 q∗(f). (4.61) Note that constraint (4.55c), which has been ignored in (4.57), can be taken into ac- count by evaluating S ′X(f) , ∂SX(f)/∂p(f) for p(f)→ 0+. In particular, since S ′X(f) is a monotonic decreasing function of p(f) for all considered equalization schemes, for a given µ, S ′X(f) = µ does not have a positive solution if limp(f)→0+ S ′X(f) < µ, and we set p(f) = 0 in this case. Otherwise, we ¿nd p(f) from (4.59) by using e.g. the bisection search method [1] 3 . On the other hand, the optimal value µ = µopt that ensures the power constraint is satis¿ed can be found iteratively by another bisec- tion search. More speci¿cally, if the corresponding total power PR = ∫ 1/2 −1/2 p(f)df is less than the maximum power P for a given µ, the Lagrange multiplier µ has to be decreased, whereas it is increased if PR > P . We note that since the frequency axis is real valued, in practice, f has to be discretized in −1/2 ≤ f ≤ 1/2 to make the problem computationally tractable. A summary of the numerical algorithm for ¿nding the optimal power allocation, popt(f), for discrete frequency points for the three considered equalization schemes is given in Table 4.1. Applying popt(f) found with the algorithm in Table 4.1 in (4.52), yields the optimal FF¸BF ¿lter frequency response Aoptz (f) for relay z, 1 ≤ z ≤ NR. Although we concentrate in this section on the case where the direct source¸destination link is not exploited for detection, with a minor modi¿cation our equalization results are valid if the source¸destination link is also used. In particular, for the latter case, our journal paper [82] provides the details. 3 Note that algorithms with faster convergence, e.g. Newton's method, can be used as long as the condition p(f) ≥ 0 is satis¿ed. 103Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays Table 4.1: Numerical algorithm for ¿nding the optimum power allocation p(f) for IIR FF¸BF ¿lters at the relays. X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver, respectively. Termination constant  and frequency spacing ∆f have small values (e.g.  = 10−5, ∆f = 10−5). i denotes the iteration index. 1 Let i = 0, N = d1/∆fe, and fn = −1/2 + (n− 1) ∆f , 1 ≤ n ≤ N . Initialize l = 0 and u = maxf limp(f)→0+ S ′X(f). 2 Update µ by µ = (l + u)/2. 3 For n = 1 to N If limp(fn)→0+(S ′X(fn)− µ) < 0, set p(fn) = 0, otherwise compute p(fn) by solving S ′X(fn) = µ with the bisectional search method [1]. 4 If ∑N n=1 p(fn)∆f > P , l = µ, else u = µ. 5 If u− l > , goto Step 2; else p(fn), 1 ≤ n ≤ N , are the optimal power allocation parameters, and µ is the optimum Lagrange multiplier µopt. 4.4.2 Optimal FIR FF¸BF with Equalization In practice, it is not possible to implement the IIR FF¸BF ¿lters discussed in the previous section since they would require the feedback of an in¿nite number of ¿lter coeÁcients from the destination to the relays. However, the performance achievable with these IIR FF¸BF ¿lters provides a useful upper bound for the FIR FF¸BF ¿lters considered in this section. In particular, the performance of the IIR solution can be used for optimizing the FIR BF¸FF length to achieve a desired trade¸o¾ between the amount of feedback and performance. We note that although FIR FF¸BF ¿lters are considered in this section, in order to be able to exploit the simple SINR expressions in (4.34)¸(4.36), we still assume that the equalizers at the destination employ IIR ¿lters. With FIR FF¸BF ¿lters of length La at the relays, the length of the equivalent CIR heq[k] (4.4) is given by Leq = La +Lg +Lh− 2. In this case, the Fourier transform of heq[k] can be expressed as Heq(f) = dH(f)HGDa (4.62) 104Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays with d(f) , [1 ej2pif . . . ej2pif(Leq−1)]T . FIR FF¸BF coeÁcient vector a, H, and GD are de¿ned in Section 4.3 after (4.6), respectively. The noise whitening ¿lter in the FIR case is given by W (f) = (σ2naH ¯Γ(f)a + σ2v)−1/2 (4.63) with ∑NR z=1 M2z La × ∑NR z=1 M2z La block diagonal matrix ¯Γ(f) , diag { ¯Γ1(f), . . . , ¯ΓNR(f) } of rank ∑NR z=1 Mz, where ¯Γz(f) , ˘HHz ( IMz ⊗ ¯d(f) )( IMz ⊗ ¯d(f) )H ˘Hz is an M2z La×M2z La matrix of rank Mz. ˘Hz is de¿ned after (4.10), and ¯d(f) , [1 ej2pif . . . ej2pif(Lh+La−2)]T . Therefore, after noise whitening, the frequency response of the overall channel is H ′eq(f) = dH(f)HGDa ( σ2na H ¯Γ(f)a + σ2v ) −1/2 . (4.64) We note that for a practical implementation, the noise whitening ¿lter does not have to be implemented. Instead, the noise correlation can be directly taken into account for equalizer ¿lter design [83]. However, in order to be able to exploit the simple existing expressions for the SINR of the equalizer output given in [83, 84], it is advantageous to assume the presence of a whitening ¿lter for FIR BF¸FF ¿lter design. Similar to the IIR case in (4.33), also for the FIR case it is convenient to introduce the de¿nition Z(a) , |H ′eq(f)|2 = a HGHDHHd(f)dH(f)HGDa σ2na H ¯Γ(f)a + σ2v . (4.65) Note, however, that this is a slight abuse of notation since while the argument of Z(a(f)) in (4.33) is a vector containing all frequency responses of the IIR FF¸BF ¿lters, the argument of Z(a) in (4.65) is a vector containing all FIR FF¸BF coeÁcients. Replacing Z(a(f)) now in the SINR expressions in (4.34)¸(4.36) by Z(a) from (4.65), we obtain the SINRs SINRX(a), where X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver, 105Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays respectively. This allows us to formulate the FIR FF¸BF ¿lter optimization problem in a uni¿ed manner: max a SINRX(a) (4.66a) s.t. aHDa ≤ P , (4.66b) where the power constraint (4.66b) is the same as in (4.14b). Although problem (4.66) formally looks very similar to problem (4.39), it is substantially more diÁcult to solve. The main reason for this lies in the fact that the optimization variable a(f) in (4.39) can be chosen freely for each frequency f , whereas the coeÁcient vector a in (4.66) is ¿xed for all frequencies. To simplify the power constraint, we introduce v , D1/2a. Furthermore, it is not diÁcult to see that at optimality, the power constraint in (4.66b) is ful¿lled with equality, i.e., aHDa = vHv = P . With this identity, we obtain M(v, f) , Z(a) + ξ , v H ¯J(f)v vH ¯X(f)v (4.67) where ¯J(f) , D−H/2 ¯Φ(f)D−1/2 + ξσ 2 v P INRLa , (4.68) ¯X(f) , σ2nD−H/2¯Γ(f)D−1/2 + σ 2 v P INRLa , (4.69) ¯Φ(f) , GHDHHd(f)dH(f)HGD + ξσ2n¯Γ(f). (4.70) 106Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays Now, we can rewrite optimization problem (4.66) in equivalent form as max v 1/2∫ −1/2 SX(v, f) df (4.71a) s.t. vHv = P , (4.71b) where SDFE(v, f) , ln(M(v, f)), SLE(v, f) , −1/M(v, f)), and SMF(v, f) , M(v, f). (4.72) The FIR FF¸BF optimization problem in (4.71) is a diÁcult non¸convex optimization problem. To substantiate this claim, we consider the special case of DFE and discretize the integral in (4.71a). This leads to the new equivalent problem max vHv=P N∏ i=1 vH ¯J(fi)v vH ¯X(fi)v , (4.73) where fi , −1/2+(i−1)/N and N denotes the number of sampling points. The objective function in (4.73) is a product of generalized Rayleigh quotients. Unfortunately, it is well known that the maximization of a product of generalized Rayleigh quotients is a diÁcult mathematical problem which is not well understood and a solution is not known except for the trivial case N = 1, cf. e.g. [67, 68]. Therefore, we also do not expect to ¿nd a simple solution for optimization problem (4.71). Similar statements apply for the optimization problems resulting for LE and an MF receiver. In order to obtain a practical and simple method for ¿nding a locally optimal solution for the FIR BF¸FF coeÁcient vectors, we propose a gradient algorithm (GA). In iteration 107Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays Table 4.2: Gradient algorithm (GA) for calculation of near¸optimal FIR FF¸BF ¿lter vector a. Termination constant  has a small value (e.g.  = 10−5). i denotes the iteration index and δi is the adaptation step size chosen through a backtracking line search [1]. 1 Let i = 0 and initialize vector v with some v0 ful¿lling vH0 v0 = P . 2 Update the vector v: DFE: vi+1 = vi + δi [∫ 1/2 −1/2 ( ¯J(f) vHi ¯J(f)vi − ¯X(f) vHi ¯X(f)vi ) df ] vi; LE: vi+1 = vi − δi [∫ 1/2 −1/2 vHi ¯J(f)vi ¯X(f)− vHi ¯X(f)vi ¯J(f)( vHi ¯J(f)vi)2 df] vi; MF: vi+1 = vi + δi [∫ 1/2 −1/2 vHi ¯X(f)vi ¯J(f)− vHi ¯J(f)vi ¯X(f)( vHi ¯X(f)vi)2 df] vi. (Note that normalization of vector vi+1 is not necessary since v H i+1vi = P .) 3 Compute SINRX(vi+1) based on (4.34)¸(4.36). 4 If |SINRX(vi+1)− SINRX(vi)| < , goto Step 5, otherwise increment i→ i + 1 and goto Step 2. 5 vi+1 is the desired vector, and the corresponding optimum FF¸BF ¿lter is a = D−1/2vi+1. i+1, the GA improves vector vi from iteration i in the direction of the steepest ascent [1] 1/2∫ −1/2 ∂SX(v, f) ∂v df (4.74) of the objective function in (4.71a). The GA for the three considered equalization schemes is summarized in Table 4.2. Although, in principle, the GA may not be able to ¿nd the globally optimal solution, extensive simulations have shown that for the problem at hand the performance achievable with GA is practically independent of the initialization v0. More importantly, for suÁciently large FIR ¿lter lengths La, the solution found with the GA closely approaches the performance of the optimal IIR FF¸BF ¿lter. This suggests that the solution found by the GA is at least near optimal. Exemplary simulation results con¿rming these claims are provided and discussed in the next section. 108Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays We note that we can again accommodate the case where the source¸destination channel is exploited for detection, please refer to our journal paper [82] for details. 4.5 Simulation Results In this section, we present simulation results for the SINR and the BER of a cooperative network with FF¸BF. Throughout this section we assume σ2n = σ 2 v = 1 and P = 1. This allows us to decompose the CIRs as hi,z[k] = √γh ¯hi,z[k] and gi,z[k] = √γg g¯i,z[k], where γh and γg denote the transmitter SNRs of the relay¸destination and the source¸relay links, respectively. The normalized CIRs ¯hi[k] and g¯i[k] include the e¾ects of multipath fading and path¸loss. All IIR and FIR FF¸BF ¿lters were obtained using the methods introduced in Sections 4.3 and 4.4. The locations of the source, the destination, and the relays are shown in Fig. 4.2, where the numbers on top and beside the arrows indicate the normalized distance between the nodes. We consider the following three cooperative relay network setups: 1) NR = 1 relays with M1 = 5 at location (c); 2) NR = 2 relays with M1 = 2 and M2 = 3 at locations (a) and (e), respectively; and 3) NR = 5 relays with Mz = 1, 1 ≤ z ≤ NR, at locations (a)¸(e), respectively. The normalized distance between the source and the destination is equal to 2 and the normalized horizontal distance between the source and the relays is d. A path¸loss exponent of 3 with reference distance dref = 1 is assumed. The CIR coeÁcients of all links are modeled as independent quasi¸static Rayleigh fading with Lg = Lh = 5 and following an exponential power delay pro¿le p[k] = 1 σt Lx−1∑ l=0 e−k/σtδ[k − l] , (4.75) where Lx ∈ {Lg, Lh} and σt characterizes the delay spread [85]. All results shown were 109Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays (c) (b) (d) (e) 2 − dd source destination 1/4 1/4 1/4 1/4 (a) Figure 4.2: Locations of source, destination, and relays in simulation. averaged over 100, 000 independent realizations of the fading channels. 4.5.1 FF¸BF without Equalization Optimal Decision Delay: First, we consider the optimal decision delay for FF¸BF without equalization. In theory, the decision delay parameter k0 can be optimized for each channel realization. However, it is not practical to search for the optimal delay k0 for every channel realization. In practice, it is preferable to ¿nd a value for k0 which works well for given channel statistics. Fig. 4.3 shows the average SINR vs. decision delay k0 for FIR FF¸BF without equalization for σt = 2 and σt = 7. The FIR FF¸BF ¿lters were optimized for the SINR Maximization Under Relay Power Constraint criterion in Section 4.3.1. We assume network setup 3), d = 1, and γg = γh = 10 dB. As can be observed, for σt = 2, the optimal k0 is equal to 2, 3, and 6 for ¿lter length La = 1, 3, and 7, respectively. In comparison, for σt = 7, the optimal k0 is equal to 5, 5, and 7 for La = 1, 3, and 7, respectively. In other words, the larger the channel delay spread σt, i.e., the more frequency selective the channel, the larger the optimal delay k0. Fig. 4.3 also shows that increasing the FF¸BF ¿lter length is highly bene¿cial for the achievable maximum average SINR. For the remaining results presented in this section, we will adopt the optimal values for k0. 110Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 2 4 6 8 10 12 14 −4 −2 0 2 4 6 8   FIR FF−BF w/o EQ (L a =1) FIR FF−BF w/o EQ (L a =3) FIR FF−BF w/o EQ (L a =7) σt = 2 σt = 7 k0 A v erag e SIN R [dB ] Figure 4.3: Average SINR vs. decision delay k0 for FIR FF¸BF without equalization (EQ) at the destination. The FF¸BF ¿lters were optimized for SINR maximization under joint relay power constraint. Exponentially decaying channel power delay pro¿le with Lg = Lh = 5, d = 1, NR = 5, Mz = 1, z ∈ {1, 2, . . . , 5} and γg = γh = 10 dB. SINR Optimization: Figs. 4.4 and 4.5 show the average SINR vs. distance d for FF¸ BF for joint relay and joint source¸relay power constraints, respectively. Relay network setups 1) ¸ 3) were adopted. The FF¸BF matrix ¿lters were generated using the results in Section 4.3.1 and 4.3.3, respectively. For both considered constraints FF¸BF relaying enables considerable performance gains compared to direct transmission except for the case with La = 1, NR = 5, and Mz = 1, z ∈ {1, 2, . . . , 5}. Direct transmission is preferable only if the relay is located either closed to the source or the destination (small d or large d). The joint source¸relay power constraint can yield signi¿cant performance gains if the relays are 111Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 5 10 15   FIR FF−BF w/o EQ (L a  = 1) FIR FF−BF w/o EQ (L a  = 3) FIR FF−BF w/o EQ (L a  = 5) FIR FF−BF w/o EQ (L a  = 7) NR = 1, M1 = 5 NR = 2, M1 = 2, M2 = 3 NR = 5, M1=M2=M3=M4=M5=1 no relay (MMSE−DFE) d A v erag e SIN R [dB ] Figure 4.4: Average SINR vs. distance d for FIR FF¸BF without equalization (EQ) at the destination. The FF¸BF matrix ¿lters were optimized for a joint relay power constraint. Exponentially decaying channel power delay pro¿le with σt = 2 and Lg = Lh = 5, and γg = γh = 10 dB. Results for direct transmission with transmit power P = 2 at the source are also included. close tothe source or close to the destination, respectively, by Àexibly allocating more or less power to the source. Furthermore, Figs. 4.4 and 4.5 also show that it is preferable to have fewer relays with more antennas than more relays with fewer antennas. 112Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 5 10 15   FIR FF−BF w/o EQ (L a  = 1) FIR FF−BF w/o EQ (L a  = 3) FIR FF−BF w/o EQ (L a  = 5) FIR FF−BF w/o EQ (L a  = 7) NR = 1, M1 = 5 NR = 2, M1 = 2, M2 = 3 NR = 5, M1=M2=M3=M4=M5=1 no relay (MMSE−DFE) d A v erag e SIN R [dB ] Figure 4.5: Average SINR vs. distance d for FIR FF¸BF without equalization (EQ) at the destination. The FF¸BF matrix ¿lters were optimized for a joint source¸relay power constraint. Exponentially decaying channel power delay pro¿le with σt = 2 and Lg = Lh = 5, and γg = γh = 10 dB. Results for direct transmission with transmit power P = 2 at the source are also included. Power Minimization: Fig. 4.6 shows the total source and relay transmit power, PR + σ2s , vs. the minimum required SINR γ at the destination for di¾erent relay network setups. The FF¸BF matrix ¿lters are generated based on the results in Sections 4.3.2 and 4.3.4, respectively. Similar to [50], we have only included simulation points which guarantee feasibility of the optimization problem for more than 50 % of the channels. The total source and relay transmit power is computed by averaging over the feasible runs. The probability that this problem is feasible is shown in Fig. 4.7. From Figs. 4.6 and 113Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays −10 −5 0 5 10 15 20 25 0 5 10 15 20 25   NR=5, Mz=1 (Joint Relay Power Min.) NR=5, Mz=1 (Joint Source−Relay Power Min.) NR=2, M1=2, M2=3 (Joint Relay Power Min.) NR=2, M1=2, M2=3 (Joint Source−Relay Power Min.) FIR FF−BF w/o EQ (L a =1) FIR FF−BF w/o EQ (L a =5) γ [dB] P R + σ 2 s [dB ] Figure 4.6: Total average source and relay transmit power vs. required SINR γ for FIR FF¸BF without equalization (EQ) at the destination for relay power minimization and joint source¸relay power minimization. Exponentially decaying power delay pro¿le with σt = 2 and Lg = Lh = 5, d = 1, and γg = γh = 10 dB. 4.7, we observe that joint source¸relay transmit power minimization and multiple¸antenna relays can lead to signi¿cant power savings. Fig. 4.6 also reveals that increasing La can substantially reduce the total source and relay transmit power. 114Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays −10 −5 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   NR=5, Mz=1 (Joint Relay Power Min.) NR=5, Mz=1 (Joint Source−Relay Power Min.) NR=2, M1=2, M2=3 (Joint Relay Power Min.) FIR FF−BF w/o EQ (L a =1) FIR FF−BF w/o EQ (L a =3) FIR FF−BF w/o EQ (L a =5) γ [dB] F easibili t y Probabili t y Figure 4.7: Feasibility probability vs. required SINR γ for FIR FF¸BF without equaliza- tion (EQ) at the destination for relay power minimization and joint source¸relay power minimization. Exponentially decaying power delay pro¿le with σt = 2 and Lg = Lh = 5, d = 1, and γg = γh = 10 dB. 4.5.2 FF¸BF with Equalization Convergence of the GA: We ¿rst investigate the convergence of the proposed GA for optimization of the FIR FF¸BF ¿lters. We assume MMSE¸DFE at the destination and relay network setup 3) (i.e., NR = 5 relays with Mz = 1, 1 ≤ z ≤ NR, at locations (a)¸ (e), respectively). The CIRs of all involved channels are given by g¯1,z[k] = ¯h1,z[k] = 1/√5, 0 ≤ k < 5, 1 ≤ z ≤ 5, with Lg = Lh = 5 and γg = γh = 10 dB. Fig. 4.8 shows the achievable 115Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 10 20 30 40 50 60 70 6.25 6.3 6.35 6.4 6.45 6.5 6.55 6.6 6.65   IIR FF−BF (MMSE−DFE) GA (all−one vector) GA (random vector) FIR FF−BF (MMSE−DFE, L a =1) FIR FF−BF (MMSE−DFE, L a =3) FIR FF−BF (MMSE−DFE, L a =7) iteration number SIN R [dB ] Figure 4.8: SINR vs. iteration number i of GA given in Table 4.2 for FIR FF¸BF with MMSE¸DFE at the destination. γg = γh = 10 dB, Lg = Lh = 5, and g¯1,z[k] = ¯h1,z[k] = 1/√5, 0 ≤ k < 5, 1 ≤ z ≤ 5. NR = 5 relays with Mz = 1, 1 ≤ z ≤ NR, at locations (a)¸(e), respectively. For comparison the SINR for IIR FF¸BF is also shown. SINR vs. iteration number i for initialization of the GA with a normalized random vector and a normalized all¸one vector for di¾erent FIR ¿lter lengths La, respectively. Note that the adaptation step size, δi, is obtained from a backtracking line search, cf. Table 4.2. After a suÁciently large number of iterations, the SINR converges to the same constant value for both initializations. The steady¸state SINR increases with increasing La and for suÁciently large FIR ¿lter lengths La, the steady¸state SINR approaches the SINR of IIR FF¸BF. Similar observations were made for other random and deterministic initializations of the proposed GA. Thus, for all results shown in the remaining ¿gures, the GA in Table 116Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 4.2 was initialized with a normalized all¸one vector. Filter Design for a Fixed Test Channel: In order to get some insight into the e¾ect that di¾erent equalization schemes have on the IIR and FIR FF¸BF ¿lter design, we consider next a cooperative network with NR = 1 single antenna relay and assume a simpli¿ed test channel with Lg = Lh = 2 and g¯1,1[k] = ¯h1,1[k] = 1/√2, k ∈ {0, 1}, i.e., all involved channels are identical and their frequency response has a zero at frequency f = 1/2, cf. Fig. 4.9. We also choose identical transmitter SINRs γg = γh = 10 dB for all channels. In Fig. 4.9, we show the magnitude of the optimal IIR FF¸BF ¿lter frequency response |Aopt1 (f)| vs. frequency f . We consider the cases where the destination is equipped with ZF¸DFE, MMSE¸DFE, ZF¸LE, MMSE¸LE, and an MF receiver. Interestingly, although the frequency responses for all equalization schemes have the same structure, cf. (4.52), due to di¾erences in the optimal relay power allocation, p(f), the FF¸BF ¿lter frequency response for the ZF¸LE case exhibits a completely di¾erent behavior than the frequency responses for the other equalization schemes. In particular, since a zero in the frequency response of the overall channel, consisting of the source¸relay channel, the FF¸BF ¿lter, and the relay¸destination channel, would lead to in¿nite noise enhancement in a linear zero¸ forcing equalizer at the destination, the FF¸BF ¿lter design tries to avoid this problem by enhancing frequencies around f = 1/2. Note that the resulting scheme would still have a very poor performance since most of the relay power is allocated to frequencies where the overall channel is poor. In contrast, the other considered equalization strategies inherently avoid in¿nite noise enhancement at the destination even if the overall channel has zeros. Thus, in these cases, the optimal FF¸BF ¿lters avoid allocating signi¿cant amounts of power to frequencies around f = 1/2. This is particularly true for the MMSE equalizers and the MF receiver. The former allocate the power such that there is an optimal tradeo¾ 117Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2   IIR FF−BF (MF) IIR FF−BF (MMSE−DFE) IIR FF−BF (ZF−DFE) IIR FF−BF (MMSE−LE) IIR FF−BF (ZF−LE) Magnitud e f Test Channel ∣∣∣Aopt1 (f) ∣∣∣ Figure 4.9: Frequency responses of IIR FF¸BF ¿lters for γg = γh = 10 dB, NR = 1 single antenna relay, Lg = Lh = 2, and g¯1,1[k] = ¯h1,1[k] = 1/√2, k ∈ {0, 1}. For comparison the frequency response of the test channel is also shown. between residual ISI and noise enhancement in the equalizer output signal, whereas the latter, idealized receiver is not a¾ected by residual ISI. Fig. 4.10 compares the frequency responses of the IIR FF¸BF ¿lter and FIR FF¸BF ¿lters of various lengths assuming MMSE¸DFE at the receiver. As expected, as the FIR FF¸BF ¿lter length La increases, the degree to which the FIR frequency response approxi- mates the IIR frequency response increases. Although Fig. 4.10 suggests that relatively long FIR FF¸BF ¿lters are required to closely approximate the IIR ¿lters, subsequent results 118Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5   IIR FF−BF (MMSE−DFE) FIR FF−BF (MMSE−DFE, L a =3) FIR FF−BF (MMSE−DFE, L a =5) FIR FF−BF (MMSE−DFE, L a =7) FIR FF−BF (MMSE−DFE, L a =30) Magnitud e f Figure 4.10: Frequency responses of IIR FF¸BF ¿lter and FIR FF¸BF ¿lters of various lengths for MMSE¸DFE at the receiver. All channel parameters are identical to those in Fig. 4.9. will show that short FIR FF¸BF ¿lters suÁce to closely approach the SINR performance of IIR FF¸BF ¿lters. SINR Performance for Fading Channels: In Fig. 4.11, we show the average SINR vs. distance d for various FF¸BF ¿lter and equalization designs for relay nework setup 2) (i.e., NR = 2, M1 = 2, and M2 = 3). We compare the performance of the proposed FF¸BF matrix ¿lter design with MMSE¸DFE and without equalizer at the destination. Interestingly, while for short FIR FF¸BF ¿lters equalization at the transceivers results 119Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1 2 3 4 5 6 7 8 9 10 11 12   IIR FF−BF (MF) IIR FF−BF (MMSE−DFE) IIR FF−BF (MMSE−LE) FIR FF−BF (MMSE−LE) FIR FF−BF w/o EQ FIR FF−BF (L a =1) FIR FF−BF (L a =3) FIR FF−BF (L a =5) no relay (MMSE−DFE) no relay (MMSE−LE) d A v erag e SN R [dB ] No Relay Figure 4.11: Average SINR vs. distance d for FF¸BF with MMSE¸LE, MMSE¸DFE, and an MF receiver at the destination. NR = 2 relays with M1 = 2 and M2 = 3, exponentially decaying power delay pro¿le with σt = 2 and Lg = Lh = 5, and γg = γh = 10 dB. For comparison the SINRs of FF¸BF without (w/o) equalization (EQ) at the destination and without relaying are also shown, respectively. in large performance gains, FIR FF¸BF without equalization with large La approaches the same performance as FIR FF¸BF with equalization. We note that for a given ¿lter length La the feedback requirements and the relay complexity for the proposed FIR FF¸ BF schemes with or without equalization are identical. Fig. 4.11 also shows that as La increases, the performance of FIR FF¸BF approaches the performance of IIR FF¸BF with MMSE¸DFE at the destination. For IIR FF¸BF ¿lters, Fig. 4.11 shows that the loss of MMSE¸DFE compared to an idealized MF receiver, which is the ultimate performance bound for any equalizer architecture, exceeds 1 dB only for d < 0.4. 120Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 0 1 2 3 4 5 6 7 8 9 10 11 0 2 4 6 8 10 12   IIR FF−BF (MF) IIR FF−BF (MMSE−DFE) IIR FF−BF (MMSE−LE) FIR FF−BF (L a =1) FIR FF−BF (L a =3) FIR FF−BF (L a =5) no relay (MMSE−DFE) no relay (MMSE−LE) A v erag e SN R [dB ] σt FIR FF¸BF (MMSE¸DFE) FF¸BF w/o EQ No Relay Figure 4.12: Average SINR vs. decay parameter σt for FF¸BF with MMSE¸LE, MMSE¸ DFE, and an MF receiver at the destination. NR = 2 relays with M1 = 2 and M2 = 3, distance d = 1, exponentially decaying power delay pro¿le with Lg = Lh = 5, and γg = γh = 10 dB. For comparison the SINRs of FF¸BF without (w/o) equalization (EQ) at the destination and without relaying are also shown, respectively. Impact of Decay Parameter σt: In Fig. 4.12, we investigate the impact of decay parameter σt on the performance of FF¸BF for d = 1 and γg = γh = 10 dB. We note that the CIR coeÁcients of the test channel decay the faster (i.e., the channel is less frequency selective), the smaller σt is. As a special case, the channel becomes frequency Àat when σt = 0. Fig. 4.12 shows that when the channel becomes frequency Àat, i.e., σt = 0, all relaying schemes provide the same average SINR performance. We also observe that the performance of suÁciently long FF¸BF ¿lters is practically not a¾ected by the 121Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays 0 5 10 15 20 10−4 10−3 10−2 10−1   NR=2, M1=2, M2=3, IIR FF−BF (MF, analytical) NR=2, M1=2, M2=3, IIR FF−BF (MMSE−LE, analytical) NR=2, M1=2, M2=3, IIR FF−BF (MMSE−DFE, analytical) NR=2, M1=2, M2=3, FIR FF−BF (MMSE−DFE) NR=2, M1=2, M2=3, FIR FF−BF w/o EQ FIR FF−BF (L a  = 1) FIR FF−BF (L a  = 5) γg = γh [dB] FF¸BF w/o EQ FF¸BF with EQ BE R Figure 4.13: Average BER of BPSK vs. transmit SNR γ for FF¸BF with MMSE¸LE, MMSE¸DFE, and an MF receiver at the destination. Exponentially decaying power delay pro¿le with σt = 2 and Lg = Lh = 5. For comparison the BER of FF¸BF without (w/o) equalization (EQ) at the destination is also shown. frequency selectivity of the channel if MMSE¸LE or MMSE¸DFE are employed at the destination. The idealized MF receiver with IIR FF¸BF bene¿ts even slightly from more frequency selectivity (larger σt) because of the additional diversity o¾ered by the channel. In contrast, FF¸BF without equalization at the receiver is adversely a¾ected by increased frequency selectivity and is even outperformed by direct transmission without relay (but with equalization at the destination) for σt > 11. BER Performance for Fading Channels: Fig. 4.13 shows BERs of BPSK modu- lation vs. transmit SNR, γ = γg = γh, for FIR and IIR FF¸BF matrix ¿lters. We adopt 122Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays cooperative relay network setup 2), and assume σt = 2 and d = 1. The BERs for FIR FF¸BF matrix ¿lters were simulated by implementing MMSE¸DFE with FIR equalization ¿lters of lengths 4 × Leq, which caused negligible performance degradation compared to IIR equalization ¿lters. The BERs for IIR FF¸BF were obtained by approximating the BER of BPSK transmission by BERX = Q(√2SINRX) [84], where X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver at the destination, respectively. Fig. 4.13 shows that equalization at the destination is very bene¿cial in terms of the achievable BER and large performance gains are realized compared to FF¸BF without equalization. Also, for IIR FF¸BF matrix ¿lters MMSE¸LE and MMSE¸DFE receivers achieve practically identical BERs and the gap to the idealized MF receiver is less than 0.6 dB. This gap could potentially be closed by trellis¸based equalizers, such as decision¸feedback sequence estimation, at the expense of an increase in complexity. 4.6 Conclusions In this chapter, we considered FF¸BF for frequency¸selective cooperative relay networks with one source, multiple multi¸antenna relays, and one destination. In contrast to prior work, we assumed that the destination is equipped with either a slicer or a simple equalizer such as a linear or a decision feedback equalizer. For both cases, the FF¸BF ¿lters at the relays were optimized for maximization of the SINR at the equalizer output under a joint relay power constraint. Additionally, for the simple slicer case we also considered the optimization of the FF¸BF ¿lters for minimization of the total transmit power subject to a QoS constraint to guarantee a certain level of performance. For the slicer case, we obtained closed¸form solutions and eÁcient numerical methods for computation of the optimal FIR FF¸BF matrix ¿lters. For IIR FF¸BF ¿lters, we found a uni¿ed expression for the frequency response of the optimal ¿lters valid for LE, DFE, and 123Chapter 4. Cooperative FF¸BF with Multiple Multi¸Antenna Relays an idealized MF receiver. We proposed a simple algorithm with guaranteed convergence for optimization of the power allocation factor included in the optimal frequency response. For FIR FF¸BF ¿lters, we showed that a diÁcult non¸convex optimization problem results and proposed a simple and eÁcient gradient algorithm to ¿nd near¸optimal ¿lter coeÁcients. Our simulation results con¿rmed that (1) the performance gap between FF¸BF ¿lters with LE/DFE and FF¸BF ¿lters with an idealized MF receiver is relatively small implying that little can be gained by employing more complex trellis¸based equalization schemes at the destination, (2) relatively short FIR FF¸BF ¿lters closely approach the performance of IIR FF¸BF ¿lters for all considered receiver structures con¿rming the near¸optimal performance of the proposed gradient algorithm for FIR ¿lter optimization, (3) for a given total number of antennas it is preferable to have the antennas concentrated in few relays rather than having many relays with few antennas, (4) if short FIR FF¸BF ¿lters are used and/or few relays are employed, equalization at the destination is bene¿cial; 5) if long FIR FF¸BF ¿lters are employed, the simple slicer destination with optimized decision delay closely approaches the same performance as destinations with equalizers. 124Chapter 5 Two¸Way Filter¸and¸Forward Beamforming for Frequency¸Selective Channels with Multiple Single Antenna Relays 5.1 Introduction Drawing from the ¿ndings on one¸way relaying in the previous chapter, we investigate FF¸BF for two¸way cooperative relay networks in this chapters. Particularly, we consider FF¸BF for two¸way cooperative networks with two transceivers communicating with each other over frequency¸selective channels via multiple single¸antenna relays using the so¸ called MABC protocol. Thereby, we consider two cases for the receive processing at the tranceivers: (1) a simple slicer is used without equalization and (2) LE or DFE is performed. The resulting FF¸BF ¿lter design problems are substantially more challenging than those for one¸way relaying in the previous chapter and [50, 82], since one ¿lter at the relay has to be optimized to achieve a certain level of performance at two receivers. In particular, we consider the following design problems. For both case (1) and case (2), we optimize the FF¸BF ¿lters at the relays for a SINR balancing objective under a relay transmit power 125Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays constraint, i.e., maximization of the worst transceiver SINR. Additionally, for case (1) we also consider the optimization of the FF¸BF ¿lters for minimization of the total transmit power subject to two QoS constraints to guarantee a certain level of performance. For case (1), we convert the resulting optimization problems into convex SOCP problems for which eÁcient o¾¸the¸self interior point algorithms are available for ¿nding global optimal solutions. For case (2), it does not seem possible to ¿nd an exact solution to the problem. However, we provide an upper bound and an achievable lower bound for the optimization problem, and our results show that the gap between both bounds is small. In addition, for case (2), we also consider the problem of minimizing the sum of the MSEs of the outputs of the equalizers, which allows for an exact solution. Our simulation results show that while transceivers with equalizers always achieve a superior performance, the gap to transceivers employing simple slicers decreases with in- creasing FF¸BF ¿lter length and increasing number of relays. Furthermore, for suÁciently long FF¸BF ¿lters and a suÁciently large number of relays, transceivers with and without equalizers lead to an SINR loss of less than one decibel compared to an idealized matched ¿lter receiver, which constitutes a performance upper bound for all receiver structures. The remainder of this chapter is organized as follows. In Section 5.2, the adopted system model is presented. The optimization of FIR FF¸BF ¿lters for transceivers without and with equalization is presented in Sections 5.3 and 5.4, respectively. Simulation results are provided in Section 5.5, and some conclusions are drawn in Section 5.6. 5.2 System Model We consider a relay network with two transceiver nodes and NR relay nodes. All network nodes have a single antenna. A block diagram of the discrete¸time overall transmission system in equivalent complex baseband representation is depicted in Fig. 5.1. The adopted 126Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays 2nd Transmission Interval v1[k] s1[k] s2[k] 1st Transmission Interval v2[k] ni[k] sˆ1[k]sˆ2[k] EQ/Slicer EQ/Slicer Relay 1 Relay NR Relay 1 Relay NR g1[k] h1[k] gNR[k] g1[k] h1[k] hNR[k]gNR[k] hNR[k] gi[k] hi[k] ai[k] ai[k] gi[k] hi[k] Figure 5.1: Cooperative two¸way network with two transceiver nodes and NR relay nodes. EQ is the equalizer at the transceivers. sˆ1[k] and sˆ2[k] are estimated received symbols at TC2 and TC1, respectively. two¸way MABC relay protocol involves only two transmission intervals. In the ¿rst inter- val, the two transceivers transmit their packets simultaneously to the relays, and in the second interval, the relays process the packets and broadcast them to the two transceiver nodes. The discrete¸time CIRs between transceiver 1 (TC1) and relay i, gi[k], 0 ≤ k ≤ Lg − 1, and between transceiver 2 (TC2) and relay i, hi[k], 0 ≤ k ≤ Lh − 1, contain the combined e¾ects of transmit pulse shaping, the continuous¸time channel, receive ¿ltering, and sampling. Here, Lg and Lh denote the lengths of the TC1¸relay and the TC2¸relay channels, respectively. 127Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays Similar to [50, 57, 58], we assume in this chapter that both transceivers have perfect knowledge of all channels in the network. This can be accomplished by having separate training phases for all involved nodes, where they transmit training symbols. In this way, both transceivers can estimate their respective CIRs to the relays and the over all channels hi[k] ∗ gi[k] to the other transceiver. TC1 can then obtain the channel between TC2 and relay i from hi[k] ∗ gi[k] and gi[k], i = 1, . . . , NR, via deconvolution or via a low rate feedback channel from TC2. TC2 can obtain hi[k], i = 1, . . . , NR, in a similar manner. Subsequently, one of the two transceivers computes the optimal FF¸BF ¿lters adopting the algorithms proposed in Sections 5.3 and 5.4 and broadcasts the ¿lter coeÁcients ai[k] to relay i and the other transceiver via an error¸free and zero¸delay feedback channel. 5.2.1 FF¸BF at Relays In the ¿rst phase of transmission, TCj transmits the i.i.d. symbols sj[k], j ∈ {1, 2}, which are taken from a scalar symbol alphabet A such as phase¸shift keying (PSK) or quadrature amplitude modulation (QAM), and have variance σ2sj , E{|sj[k]|2}, j ∈ {1, 2}. The signal received at the ith relay, i = 1, . . . , NR, is given by yi[k] = gi[k] ∗ s1[k] + hi[k] ∗ s2[k] + ni[k] , (5.1) where ni[k] denotes AWGN with variance σ2n , E{|ni[k]|2}. The FF¸BF ¿lter impulse response coeÁcients of relay i for the second transmission interval are denoted by ai[k], −ql ≤ k ≤ qu. For IIR FF¸BF ¿lters ql → ∞ and qu → ∞ and for FIR FF¸BF ¿lters ql = 0 and qu = La − 1, where La is the FIR BF ¿lter length. 128Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays The signal transmitted from the ith relay during the second interval can be expressed as ti[k] = ai[k] ∗ yi[k] = ai[k] ∗ gi[k] ∗ s1[k] + ai[k] ∗ hi[k] ∗ s2[k] + ai[k] ∗ ni[k] , i = 1, . . . , NR. (5.2) 5.2.2 Transceiver Processing The signal received at TC2 during the second time interval is given by 4 r˜2[k] = NR∑ i=1 hi[k] ∗ ti[k] + v2[k] = NR∑ i=1 hi[k] ∗ ai[k] ∗ gi[k] ∗ s1[k] + NR∑ i=1 hi[k] ∗ ai[k] ∗ hi[k] ∗ s2[k] + NR∑ i=1 hi[k] ∗ ai[k] ∗ ni[k] + v2[k] , (5.3) where vj[k] denotes AWGN with variance σ2vj , j ∈ {1, 2}. It is noteworthy that since s2[k], hi[k], and ai[k], i = 1, . . . , NR, are known at TC2, the second term on the right hand side of (5.3) can be subtracted from r˜2[k] before the residual signal r2[k] is further processed to extract the information symbols s1[k]. Similar considerations hold for TC1. Thus, the residual received signal at TCj can be expressed as rj[k] = heq[k] ∗ si[k] + v′j[k] , j ∈ {1, 2} , (5.4) where i = 1 if j = 2 and i = 2 if j = 1 and we introduced the equivalent CIR between TC1 and TC2 heq[k] , NR∑ i=1 hi[k] ∗ ai[k] ∗ gi[k] , (5.5) 4 Note that during the ¿rst time interval the two transceivers do not receive any signal, since we assumed that there is no direct link between them. 129Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays and the e¾ective noise v′1[k] , NR∑ i=1 gi[k] ∗ ai[k] ∗ ni[k] + v1[k] , (5.6) v′2[k] , NR∑ i=1 hi[k] ∗ ai[k] ∗ ni[k] + v2[k] . (5.7) We note that v′j[k], j ∈ {1, 2}, is colored noise because of the ¿ltering with the TC¸relay CIRs and the FF¸BF ¿lters. 5.3 FIR FF¸BF without Equalization In practice, it is conceivable that the transceiver nodes cannot a¾ord an equalizer due to size and/or power limitation. This may be valid for applications such as sensor networks with battery powered sensors. This case is considered in this section and the transceivers are assumed to apply only simple slicers for detection. We note that FF¸BF ¿lter optimiza- tion for transmit power minimization in two¸way relaying networks has been considered independently in [86]. In particular, [86] deals with FF¸BF for two¸way relaying without equalization at the transceiver and is closely related to this Section 5.3.2, where relay power minimization under SINR constraints are considered. However, [86] only considers the case of power minimization under SINR constraints but not the case of max¸min SINR maxi- mization under a power constraint, which will be discussed in Section 5.3.1. Furthermore, a decision delay was not considered in [86]. As has been shown in Chapter 4 for one¸way relaying, such decision delay parameter leads to signi¿cant performance improvements. The vector containing the coeÁcients of the equivalent CIR between TC1 and TC2, 130Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays heq , [heq,2[0] heq,2[1] . . . heq,2[La + Lg + Lh − 3]]T , can be rewritten as heq = NR∑ i=1 H i ¯Giai , HGDa (5.8) whereH , [H1 . . . HNR ], GD , diag { ¯G1, . . . , ¯GNR } , and a , [aT1 . . . aTNR ]T . (La+Lg+ Lh− 2)× (La +Lg− 1) matrix H i and (La +Lg− 1)×La matrix ¯Gi are column¸circulant matrices with [hi[0] . . . hi[Lh − 1] 0TLa+Lg−2]T and [gi[0] . . . gi[Lg − 1] 0TLa−1]T in the ¿rst columns, respectively, and ai , [ai[0] ai[1] . . . ai[La − 1]]T . Matrix H can be separated into a vector hk0 and a sub¸matrix Hk0 , i.e., vector hTk0 of length (La + Lg − 1)NR is the k0th row of matrix H, and Hk0 , [H]ij, i ∈ {1, . . . , k0 − 1, k0 + 1, . . . (La + Lg + Lh − 2)}, j ∈ {1, . . . , (La + Lg − 1)NR}. Therefore, for j = 2 and i = 1, the ¿rst term in (5.4) can be decomposed into a signal part and an ISI part heq[k] ∗ s1[k] = heq[k0]s1[k − k0] + La+Lg+Lh−3∑ l=0, l 6=k0 heq[l]s1[k − l] = hTk0GDas1[k − k0] ︸ ︷︷ ︸ desired signal + sT1 [k]Hk0GDa ︸ ︷︷ ︸ ISI (5.9) where s1[k] = [s1[k] . . . s1[k− k0 + 1] s1[k− k0 − 1] . . . s1[k− (La + Lg + Lh − 3)]]T , and k0 is the slicer decision delay at the transceiver. We note that for one¸way relaying such a decision delay was not introduced in [50]. However, as will be shown in Section 5.5, for two¸way relaying a decision delay is highly bene¿cial. The power of the desired signal and the ISI can be obtained as E {∣∣hTk0GDas1[k − k0]∣∣2} = σ2s1aHGHD h∗k0hTk0GDa (5.10) 131Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays and E {∣∣sT1 [k]Hk0GDa∣∣2} = σ2s1aHGHDHHk0Hk0GDa , (5.11) respectively. Similarly, v′2[k] in (5.7) can be rewritten as v′2[k] = NR∑ i=1 nTi [k]H iai + v2[k] , nT [k]HDa + v2[k] (5.12) with row vector n[k] , [nT1 [k] . . . nTNR [k]]T of length (La+Lh−1)NR and (La+Lh−1)NR× NRLa matrix HD , diag { ¯H1, . . . , ¯HNR } , where ni[k] , [ni[k] . . . ni[k−(La+Lh−2)]]T and ¯H i is a (La + Lh − 1) × La column¸circulant matrix with vector [hi[0] . . . hi[Lh − 1] 0TLa−1]T in the ¿rst column. The noise power is obtained as E{|v′2[k]|2} = σ2naHHHDHDa + σ2v2 . (5.13) The SINR at TC2 can be obtained by combining (5.9)¸(5.11), and (5.13) and is given by SINRslicer, 2 (a) , E {∣∣hTk0GDas1[k − k0]∣∣2} E { |sT1 [k]Hk0GDa|2 } + E{|v′2[k]|2} = σ2s1a HGHD h∗k0hTk0GDa σ2s1a HGHDHHk0Hk0GDa + σ2naHHHDHDa + σ2v2 . (5.14) Similarly, the SINR at TC1 is given by SINRslicer, 1 (a) = σ 2 s2 aHHHDg∗k0gTk0HDa σ2s2a HHHDGHk0Gk0HDa + σ2naHGHDGDa + σ2v1 , (5.15) where gTk0 is the k0th row of matrix G and matrix Gk0 is matrix G without the k0th row. Here, G , [G1 . . . GNR ] with (La + Lg + Lh− 2)× (La + Lh− 1) column¸circulant matrix 132Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays Gi which has vector [gi[0] . . . gi[Lg − 1] 0TLa+Lh−2]T in the ¿rst column. From (5.2), the total relay transmit power, PR(a), in the second transmission interval is given by PR(a) = NR∑ i=1 E {|ti[k]|2}= aHDa (5.16) with D , σ2s1GHDGD + σ2s2HHDHD + σ2nILaNR . In the following two subsections, we will optimize the FIR FF¸BF ¿lters for (a) max- imization of the minimum transceiver SINR at the slicer output under a relay transmit power constraint and (b) minimization of the transmit power under individual transceiver SINR constraints, respectively. The decision delay k0 is assumed to be ¿xed for ¿lter opti- mization. We will show in Section 5.5 that the choice of k0 can have a substantial impact on performance. 5.3.1 Max¸min Criterion Under Relay Power Constraint First, we consider the optimization of the FF¸BF ¿lters for maximization of the worst transceiver SINR subject to a maximum relay power of P . This problem is of interest when the power available at the relays is limited and the aim is to maximize the QoS given this strict system restriction [50, 58, 87]. The corresponding optimization problem can be formulated as max a min {SINRslicer, 1 (a) , SINRslicer, 2 (a)} (5.17a) s.t. aHDa ≤ P . (5.17b) 133Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays Equivalently, problem (5.17) can be rewritten as max a t (5.18a) s.t. SINRslicer, 1 (a) ≥ t (5.18b) SINRslicer, 2 (a) ≥ t (5.18c) aHDa ≤ P . (5.18d) Realizing that HHDg∗k0 = GHD h∗k0 , we let q , HHDg∗k0 , and reformulate problem (5.18) as max c t (5.19a) s.t. cH ¯V (t)c ≤ cH q¯q¯Hc (5.19b) cH ¯W (t)c ≤ cH q¯q¯Hc (5.19c) cHc ≤ P + 1 (5.19d) [c]NRLa+1 = 1 , (5.19e) where c , [ (D1/2a)T 1 ]T , q¯ , [ (D−H/2q)T 0 ]T , ¯V (t) , tσ2v1 σ2s2 diag { D−H/2V D−1/2, 1 } with V , σ 2 s2 σ2v1 HHDGHk0Gk0HD + σ2n σ2v1 GHDGD, and ¯W (t) , tσ 2 v2 σ2s1 diag { D−H/2WD−1/2, 1 } with W , σ 2 s1 σ2v2 GHDHHk0Hk0GD + σ2n σ2v2 HHDHD. Note that multiplying the optimal c by ejθ, where θ is an arbitrary phase, does not a¾ect the objective function or the constraints for problem (5.19). Therefore, we can assume that q¯Hc is a real number without loss of generality. Thus, for a given t, problem (5.19) can be 134Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays transformed into the convex SOCP feasibility problem min c − t (5.20a) s.t. ∥∥∥ ¯V 1/2(t)c ∥∥∥ ≤ q¯Hc (5.20b) ∥∥∥ ¯W 1/2(t)c ∥∥∥ ≤ q¯Hc (5.20c) ‖c‖ ≤ √P + 1 (5.20d) [c]NRLa+1 = 1 . (5.20e) Consequently, for a given t, problem (5.20) can be eÁciently solved using interior point methods [88] and a bisectional search can be used to ¿nd the optimal t [1]. Since the optimal FF¸BF ¿lter vector aopt can be directly obtained from the solution of (5.20), we have provided an eÁcient procedure for computation of the optimal FF¸BF ¿lter vector. 5.3.2 Relay Power Minimization Under SINR Constraints Another relevant problem is the minimization of the relay transmit power under SINR constraints. This problem is of interest when we want to satisfy a required QoS with minimum relay transmitted power [50, 58, 89]. The corresponding optimization problem can be formulated as min a aHDa (5.21a) s.t. SINRslicer, 1 (a) ≥ γ1 (5.21b) SINRslicer, 2 (a) ≥ γ2 . (5.21c) 135Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays Equivalently, problem (5.21) can be reformulated as min c cHc− 1 (5.22a) s.t. cH q¯q¯Hc ≥ cH ¯V (γ1)c (5.22b) cH q¯q¯Hc ≥ cH ¯W (γ2)c , (5.22c) where c, ¯V (·), ¯W (·), and q¯ are de¿ned after (5.19). By exploiting again the fact that multiplying the optimal c by ejθ does not a¾ect the objective function or the constraints of problem (5.22), we can assume q¯Hc is a real number without loss of generality and transform problem (5.22) into an SOCP problem min c τ (5.23a) s.t. ∥∥∥ ˜V 1/2 c ∥∥∥ ≤ q¯Hc (5.23b) ∥∥∥ ˜W 1/2 c ∥∥∥ ≤ q¯Hc (5.23c) ‖c‖ ≤ τ (5.23d) [c]NRLa+1 = 1 . (5.23e) The SOCP problem (5.23) can again be eÁciently solved using interior point methods [88]. 5.4 FF¸BF with Equalization If only a simple slicer is employed at the transceivers, the FF¸BF ¿lters at the relays are burdened with equalizing both TC¸relay channels. By implementing equalizers a the transceivers some of the processing burden is shifted from the relays to the transceivers, which leads to better performance at the expense of an increase in complexity. However, 136Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays for some applications, such as the GSM and EDGE communication network, the increased complexity at the transceivers is acceptable, since these systems also use equalizers if relaying is not applied. From (5.4) we observe that a cooperative two¸way relay network with FF¸BF can be modeled as an equivalent SISO system. Therefore, as long as the transceivers know the CIRs of all involved channels and the coeÁcients of the FF¸BF ¿lter, the same equalization techniques as for point¸to¸point single¸antenna transmission can be used [83]. Here, we consider LE and DFE optimized according to the conventional ZF and MMSE criteria [84, 90]. Throughout this section we assume that the transceivers employ LE or DFE with IIR equalization ¿lters. In a practical implementation, FIR equalization ¿lters are used, of course. However, suÁciently long FIR ¿lters will approach the performance of IIR ¿lters arbitrarily close. Assuming IIR equalization ¿lters has the advantage that relatively simple and elegant expressions for the SINR at the equalizer output exist [83, 84]. For FF¸BF, we consider both IIR ¿lters, which provide performance bounds, and FIR ¿lters, which are required for practical implementation. 5.4.1 Optimal IIR FF¸BF with Equalization In order to be able to exploit the SINR expressions in [83, 84], we ¿rst whiten the noise impairing the signal received at the transceivers. The power spectral densities of the noises v′1[k] and v′2[k] at the two transceivers are given by Φv′j(f) = σ2naH(f)Γj(f)a(f) + σ2vj , j ∈ {1, 2} , (5.24) where Γ1(f) , diag{|G1(f)|2, . . . , |GNR(f)|2}, Γ2(f) , diag{|H1(f)|2, . . . , |HNR(f)|2}, and a(f) , [A1(f), . . . , ANR(f)]T . Gi(f) , F{gi[k]}, Hi(f) , F{hi[k]}, and Ai(f) , F{ai[k]} 137Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays denote the frequency responses of the TC1¸ith relay channel, TC2¸ith relay channel, and the FF¸BF ¿lter at the ith relay, respectively. Therefore, the whitening ¿lter for v′j[k] is given by Wj(f) = ( σ2na H(f)Γj(f)a(f) + σ2vj ) −1/2 . (5.25) After whitening, the frequency response of the equivalent overall channel at transceiver j can be obtained as H ′eq,j(f) , Wj(f)F{heq,j[k]} = qT (f)a(f) ( σ2na H(f)Γj(f)a(f) + σ2vj ) −1/2 , j ∈ {1, 2} , (5.26) where heq,j[k] , hj[k]∗aj[k]∗ gj[k], q(f) , [Q1(f) . . . QNR(f)]T and Qi(f) , Hi(f)Gi(f). Note that, after whitening, the power spectral density of the noise at the output of the whitening ¿lter at TCj, n′j[k], is Φn′j(f) = 1. For TCj, we can express the SINRs at the outputs of a decision feedback and a linear equalizer as [83, 84] SINRDFE, j(a(f)) = Psj exp    1/2∫ −1/2 ln (|H ′eq,j(f)|2 + ξj)df  − χ , (5.27) and SINRLE, j(a(f)) = Psj   1/2∫ −1/2 (|H ′eq,j(f)|2 + ξj)−1 df−1 − χ , (5.28) 138Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays respectively, where |H ′eq,j(f)|2 = a H(f)q∗(f)qT (f)a(f) σ2na H(f)Γj(f)a(f) + σ2vj . (5.29) In (5.27) and (5.28), we have χ = 0, ξ1 = ξ2 = 0 and χ = 1, ξ1 = 1/σ2s2 , ξ2 = 1/σ2s1 if the equalization ¿lters are optimized based on a ZF and an MMSE criterion, respectively. Also, we de¿ne Ps1 , σ2s2 and Ps2 , σ2s1 . Similarly, if only a single isolated symbol is transmitted, the SINR at the output of an MF is given by [5] SINRMF, j(a(f)) = Psj 1/2∫ −1/2 |H ′eq,j(f)|2 df . (5.30) The MF SINR, SINRMF, j(a(f)), constitutes an upper bound for the SINR achievable with any realizable receiver structure [5] and can be used to quantify the suboptimality of simple equalizers such as LE and DFE. From (5.2), the total relay transmit power, PR(a(f)), is given by PR(a(f)) = NR∑ i=1 1/2∫ −1/2 Φti(f)df = 1/2∫ −1/2 aH(f)D(f)a(f)df , (5.31) where Φti(f) , |Ai(f)|2 ( σ2s1 |Gi(f)|2 + σ2s2 |Hi(f)|2 + σ2n ) is the power spectral density of the transmit signal ti[k] at the ith relay, and D(f) , σ2s1diag{|G1(f)|2, . . . , |GNR(f)|2} + σ2s2diag{|H1(f)|2, . . . , |HNR(f)|2}+ σ2nINR . Max¸min Criterion Under Relay Power Constraint In analogy to Section 5.3.1, we consider ¿rst the optimization of the FF¸BF ¿lters a(f) at the relays for maximization of the minimum transceiver SINR at the output of DFE/LE/MF receivers under a relay transmit power constraint. Formally, the resulting optimization 139Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays problem can be formulated as max a(f) min {SINRX,1(a(f)), SINRX,2(a(f))} (5.32a) s.t. 1/2∫ −1/2 aH(f)D(f)a(f)df ≤ P , (5.32b) where X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver, respectively. Unfortunately, problem (5.32) is very diÁcult to solve because of the structure of the SINR expressions in (5.27), (5.29), and (5.30) and the fact that Γ1(f) 6= Γ2(f) in (5.29). Here, we provide a tight upper bound and tight achievable lower bound for the solution of (5.32). The basic idea of the proposed bounds is to compute two beamformers where each one maximizes the SINR at one transceiver under the power constraint. In other words, we consider the problem max aj(f) SINRX,j(aj(f)) (5.33a) s.t. 1/2∫ −1/2 aHj (f)D(f)aj(f)df ≤ P , (5.33b) where j ∈ {1, 2}. Let aoptj (f) denote the optimum solution for problem (5.33). Since (5.33) is equivalent to the optimization of the FF¸BF ¿lters for two one¸way relaying systems, we can draw from the results in Chapter 4 and [82]. In particular, a opt j (f), j ∈ {1, 2, }, can be eÁciently obtained with the algorithm summarized in Table 4.1. Based on these FF¸BF ¿lters we are able provide upper and lower bounds for the optimal solution of problem (5.32). In particular, the performance upper bound is given by SINRup , min { SINRX,1(aopt1 (f)), SINRX,2(aopt2 (f)) } (5.34) 140Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays and the achievable lower bound is SINRlow , max { min { SINRX,1(aopt1 (f)), SINRX,2(aopt1 (f)) } , min { SINRX,1(aopt2 (f)), SINRX,2(aopt2 (f)) }} , (5.35) where the (in general) suboptimal solution to problem (5.32) is given by the argument of the SINR on the right hand side of (5.35) after the max and min operations. If SINRX,1(aopt1 (f)) ≤ SINRX,2(aopt1 (f)) or SINRX,2(aopt2 (f)) ≤ SINRX,1(aopt2 (f)), which typ- ically occurs if the relays are closer to one transceiver than the other, cf. Section 5.5, SINRup = SINRlow and the optimal solution for problem (5.32) is obtained. Otherwise, SINRup 6= SINRlow and the obtained solution is suboptimal. However, even in this case the gap between the upper and the lower bounds is typically only a fraction of a decibel. Thus, we have provided a close¸to¸optimal solution to problem (5.32). The small gap between both bounds can be explained by the fact that the only di¾erence between the equivalent TC1¸TC2 and TC2¸TC1 channels is the noise correlation in (5.24), which has a minor impact on the design of the FF¸BF ¿lters. Minimization of the Sum of MSEs As an alternative FF¸BF ¿lter optimization criterion we consider the minimization of the sum of the MSEs (error variances) at the output of the equalizers at the two transceivers. This criterion allows for an exact solution for ZF¸LE but not for the other considered equalization schemes. Thus, we concentrate on the ZF¸LE case in the following. For ZF¸LE, the MSE at the output of the equalizer at TCj is given by σ2LE,j(a(f)) = 1/2∫ −1/2 |H ′eq,j(f)|−2df . (5.36) 141Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays i.e., SINRLE,j(a(f)) = Psj/σ2LE,j(a(f)). The considered optimization problem can be ex- pressed as min a(f) JLE(a(f)) = 2∑ j=1 σ2LE,j(a(f)) (5.37a) s.t. 1/2∫ −1/2 aH(f)D(f)a(f)df ≤ P. (5.37b) Exploiting (5.29) the objective function (5.37a) can be expressed as JLE(a(f)) = 1/2∫ −1/2 σ2na H(f)(Γ1(f) + Γ2(f))a(f) + σ2v1 + σ2v2 aH(f)q∗(f)qT (f)a(f) df . (5.38) Next, we introduce matrix Γ(f) , σ2n σ2v1+σ 2 v2 (Γ1(f) + Γ2(f)) and restate problem (5.37) in equivalent form as max a(f)   1/2∫ −1/2 ( aH(f)q∗(f)qT (f)a(f) aH(f)Γ(f)a(f) + 1 ) −1 df   −1 (5.39a) s.t. 1/2∫ −1/2 aH(f)D(f)a(f)df ≤ P. (5.39b) Since Γ(f) is a diagonal matrix, problem (5.37) is of the same form as the ZF¸LE SINR maximization problem for one¸way relaying considered in Chapter 4 and [82]. Thus, the exact solution to (5.37) can be computed with the algorithm provided in Table 4.1. 142Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays 5.4.2 FIR FF¸BF Filter Optimization Since the IIR FF¸BF ¿lters would require an in¿nite amount of feedback, they are mostly useful to establish performance bounds for practical FIR FF¸BF ¿lters. We emphasize that although FIR FF¸BF ¿lters are considered in this section, the equalizers at the transceivers are still assumed to employ IIR ¿lters. Assuming FIR FF¸BF ¿lters of length La at the relays the length of the equivalent CIR heq[k] is given by Leq = La + Lg + Lh − 2 and its Fourier transform can be expressed as Heq(f) = dH(f)Qa (5.40) with d(f) , [1 ej2pif . . . ej2pif(Leq−1)]T , FIR FF¸BF coeÁcient vector a , [a1[0] a1[1] . . . a1[La−1] a2[0] . . . aNR [La− 1]]T , and Leq×NRLa matrix Q , [Q1 . . . QNR ], where Qi is an Leq×La column¸circulant matrix with vector [( ˜H ig˜i)T 0TLa−1]T in the ¿rst column. Here, ˜H i is an (Lh+Lg−1)×Lg column¸circulant matrix with vector [hi[0] . . . hi[Lh−1] 0TLg−1]T in the ¿rst column and g˜i , [gi[0] . . . gi[Lg − 1]]T . We apply again noise whitening which transforms Heq(f) into the equivalent frequency responses of TC1 and TC2: H ′eq,j(f) = dH(f)Qa ( σ2na H ˜Γj(f)a + σ2vj ) −1/2 , j ∈ {1, 2}, (5.41) with LaNR × LaNR block diagonal matrices ˜Γj(f) , diag { ˜Γj,1(f), . . . , ˜Γj,NR(f) } , j ∈ {1, 2}, where ˜Γ1,i(f) , ˜GHi ˜d(f)˜dH(f) ˜Gi and ˜Γ2,i(f) , ˜HHi ˜d(f)˜dH(f) ˜H i are La × La matrices of rank 1. Here, ˜Gi and ˜H i are (Lg +La−1)×La and (Lh +La−1)×La column¸ circulant matrices with vectors [gi[0] . . . gi[Lg−1] 0TLa−1]T and [hi[0] . . . hi[Lh−1] 0TLa−1]T in the ¿rst columns, respectively, and ˜d(f) , [1 ej2pif . . . ej2pif(Lh+La−2)]T . The noise power spectral density at the output of the noise whitening ¿lter is again equal to one. 143Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays In the following, we will discuss the optimization of the FF¸BF coeÁcient vector a for the two criteria considered in Section 5.4.1. Max¸min criterion under relay power constraint: Similar to the IIR case, an exact solution of the max¸min FIR FF¸BF ¿lter optimization problem does not seem possible. Instead, we use the same approach as in Section 5.4.1 and compute the FIR ¿lters for two one¸way relaying setups having equivalent channel frequency responses H ′eq,1(f) and H ′eq,2(f), respectively. Comparing the equivalent frequency response in (5.41) with the corresponding fre- quency response for the one¸way relaying case in (4.64) or [82, Eq. (38)], we conclude that optimal FIR FF¸BF coeÁcient vectors a opt 1 and a opt 2 required for evaluation of the upper and lower bounds in (5.34) and (5.35) can be computed with the algorithm given in Table 4.2. Thus, a close¸to¸optimal solution for max¸min optimization of the FIR FF¸BF ¿lters for the two¸way relaying is available. Minimization of the sum of MSEs: For FIR BF¸FF with ZF¸LE receivers, the sum MSE can be written as JLE(a) = (σ2v1 + σ2v2) 1/2∫ −1/2 ( aHQHd(f)dH(f)Qa aH ˜Γ(f)a + 1 ) −1 df , (5.42) where ˜Γ , σ2n σ2v1+σ 2 v2 (˜Γ1(f) + ˜Γ2(f)). Now, the FIR FF¸BF ¿lter optimization problem can be written as max a 1/JLE(a) (5.43a) s.t. aHDa ≤ P , (5.43b) where D is de¿ned after (5.16). Problem (5.43) is of the same form as the FIR FF¸BF 144Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays ¿lter optimization problem for one¸way relaying with ZF¸LE at the receiver. Thus, we can again use the algorithm given in Table 4.2 to ¿nd the optimal vector aopt. 5.5 Simulations In this section, we present simulation results for the SINR and the BER of a cooperative two¸way relay network with FF¸BF. Throughout this section we assume σ2n = σ 2 v1 = σ2v2 = 1 and P = 1. This allows us to decompose the CIRs as gi[k] = √γg g¯i[k] and hi[k] = √γh ¯hi[k], where γg and γh denote the transmitter SNRs of the TC1¸relay and TC2¸relay links, respectively. The normalized CIRs ¯hi[k] and g¯i[k] include the e¾ects of multipath fading and path¸loss. All IIR and FIR FF¸BF ¿lters were obtained using the methods developed in Sections 5.3 and 5.4, respectively. In this section, unless stated otherwise, we consider the cooperative relay network shown in Fig. 5.2 with NR = 5 relays at locations (a)¸(e). The normalized distance between the two transceivers is equal to 2 and the normalized horizontal distance between TC1 and the relays is d. A path¸loss exponent of 3 with reference distance dref = 1 is assumed. The CIR coeÁcients of all links are modeled as independent quasi¸static Rayleigh fading with Lg = Lh = 5 and following an exponential power delay pro¿le p[k] = 1 σt ∑Lx−1 l=0 e −k/σtδ[k− l], where Lx ∈ {Lg, Lh} and σt characterizes the delay spread [85]. 5.5.1 Relay Power Minimization for FF¸BF without Equalization Fig. 5.3 shows the total relay transmit power, PR(a), vs. the minimum required SINR γ1 and γ2 at the transceivers for γ1 = γ2. We adopted σt = 2, d = 1, NR = 5, and γg = γh = 10 dB. Similar to [50], we have only included simulation points which guarantee feasibility of the optimization problem for more than 50 % of the channels. The total relay transmit power is computed by averaging over the feasible runs. The probability that the problem 145Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays (b) (d) (e) 2− dd Transceiver 1 1/4 1/4 1/4 1/4 (a) Transceiver 2 (c) Figure 5.2: Locations of TC1, TC2, and the relays in the simulations. −6 −4 −2 0 2 4 6 8 10 12 −15 −10 −5 0 5 10 15   FIR FF−BF w/o EQ (L a =1) FIR FF−BF w/o EQ (L a =3) FIR FF−BF w/o EQ (L a =5) FIR FF−BF w/o EQ (L a =7) FIR FF−BF w/o EQ (L a =9) FIR FF−BF w/o EQ (L a =11) P R (a )[dB ] γ1 = γ2 [dB] Figure 5.3: Total average relay transmit power vs. required SINRs γ1 and γ2 for FIR FF¸BF without equalization at the transceivers. The FF¸BF ¿lters were optimized for minimization of the relay transmit power. Exponentially decaying power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. 146Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays −2 0 2 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1   FIR FF−BF w/o EQ (L a =1) FIR FF−BF w/o EQ (L a =3) FIR FF−BF w/o EQ (L a =5) FIR FF−BF w/o EQ (L a =7) FIR FF−BF w/o EQ (L a =9) FIR FF−BF w/o EQ (L a =11) F easibili t y Probabili t y γ1 = γ2 [dB] Figure 5.4: Feasibility probability vs. required SINRs γ1 and γ2 for FIR FF¸BF without equalization at the transceivers. The FF¸BF ¿lters were optimized for minimization of the relay transmit power. Exponentially decaying power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. is feasible is shown in Fig. 5.4. From Figs. 5.3 and 5.4, we observe that increasing La substantially reduces the total required relay transmit power and increases the probability that the problem is feasible especially for higher SINR requirements. 147Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 1 2 3 4 5 6 7 8 9 10   IIR FF−BF (MF Lower Bound) IIR FF−BF (MF Upper Bound) IIR FF−BF (MMSE−DFE Lower Bound) IIR FF−BF (MMSE−DFE Upper Bound) FIR FF−BF, L a =1 FIR FF−BF, L a =5 FIR FF−BF, L a =11 FIR FF−BF (MMSE−DFE Lower Bound) FIR FF−BF (MMSE−DFE Upper Bound) FIR FF−−BF (w/o EQ) d A v erag e mi n{ SIN R 1 , SIN R 2 }[dB ] Figure 5.5: Average worst¸case SINR at the transceivers vs. distance d for FF¸BF with/without equalization at the transceivers. The FF¸BF ¿lters were optimized for maximization of the minimum transceiver SINR. Exponentially decaying channel power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. 5.5.2 Max¸min SINR Optimization for FF¸BF with and without Equalization In Figs. 5.5 and 5.6, we show the average SINR at the transceivers vs. distance d for various FF¸BF ¿lter designs at the relays and various transceiver structures. We adopted σt = 2, NR = 5, and γg = γh = 10 dB. The FF¸BF ¿lters were optimized for maximization of the minimum transceiver SINR. 148Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 3 4 5 6 7 8 9 10   SINR @ Transceiver 2 SINR @ Transceiver 1 FF−BF Lower Bound FF−BF Upper Bound IIR FF−BF (MF) IIR FF−BF (ZF−LE) FIR FF−BF (ZF−LE, L a =5) FIR FF−BF (w/o EQ, L a =5)A v erag e SIN R [dB ] d Figure 5.6: Average SINR at transceivers vs. distance d for FF¸BF with/without equal- ization (EQ) at the transceivers. The FF¸BF ¿lters were optimized for maximization of the minimum transceiver SINR. Exponentially decaying channel power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. In Fig. 5.5, we show the minimum transceiver SINR and observe that the performance gap between the upper and lower bounds for FF¸BF with equalization is very small for both IIR and FIR FF¸BF ¿lters. The performance gap is largest for d = 1 and IIR ¿lters. However, even in this case the gap is less than 0.3 dB suggesting that the ¿lters obtained from the achievable lower bound are close¸to¸optimal. Furthermore, Fig. 5.5 shows that transceivers employing MMSE¸DFE closely approach the performance of idealized MF receivers if IIR FF¸BF ¿lters are adopted implying that little can be gained by employing 149Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays more complex trellis¸based receivers compared to MMSE¸DFE. Also, as the length of the FIR FF¸BF ¿lters increases, FIR FF¸BF approaches the performance of IIR FF¸BF. For La = 5 and MMSE¸DFE receivers, the gap between both schemes is less than 0.6 dB over considered range of distances d. Interestingly, while for short FIR FF¸BF ¿lters equalization at the transceivers results in large performance gains, FIR FF¸BF without equalization with La = 11 achieves practically the same performance as FIR FF¸BF with MMSE¸DFE with La = 5. We note that FF¸BF with MMSE¸DFE with La = 11 slightly outperforms FIR FF¸BF without equalization with La = 11 but the corresponding curve is not shown in Fig. 5.5 for clarity. Fig. 5.6 shows the average SINRs at both transceivers with and without equalization and also the performance upper and lower bounds for the case of equalization. Note that since we show average SINRs, for a given d, the minimum transceiver SINR in Fig. 5.6 does not necessarily coincide with the (average) performance lower bound. For example, at d = 1, the probability that TC1 or TC2 contributes to the minimum SINR is half and half. As expected, TC2 enjoys a higher SINR than TC1 when the relays are close to TC1, and vice versa. We also note that even simple ZF¸LE at the transceivers can approach the performance of an idealized MF receiver up to less than one decibel. 5.5.3 Max¸min SINR vs. Minimum Sum MSE Optimization for FF¸BF with ZF¸LE In Fig. 5.7, we compare the average SINRs at both transceivers for ZF¸LE at the transceivers with max¸min and minimum sum MSE FF¸BF optimization. We adopted σt = 2, NR = 5, and γg = γh = 10 dB. For the max¸min criterion, Fig. 5.7 shows the SINRs obtained from the achievable lower bound. As can be observed, both criteria achieve very similar SINRs at both transceivers for both IIR and FIR equalizers. Since the minimum sum MSE opti- 150Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −1 0 1 2 3 4 5 6 7 8 9 10   SINR @ Transceiver 2 SINR @ Transceiver 1 IIR FF−BF (Min Sum MSE) FIR FF−BF (Min Sum MSE, L a =7) FIR FF−BF (Min Sum MSE, L a =1) IIR FF−BF (Max−Min) FIR FF−BF (Max−Min, L a =7) FIR FF−BF (Max−Min, L a =1)A v erag e SIN R [dB ] d Figure 5.7: Average SINR at transceivers vs. distance d for FF¸BF with ZF¸LE at the transceivers. Exponentially decaying channel power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. mization requires only the computation of one FF¸BF ¿lter, its complexity is roughly half of that of the max¸min optimization. Thus, in practice, the minimum sum MSE criterion may be preferable if ZF¸LE is employed at the transceivers. 5.5.4 Impact of Number of Relays NR In Fig. 5.8, we investigate the impact of the number of relays NR on the performance of various FF¸BF and equalizer designs for σt = 2, d = 1, and γg = γh = 10 dB. We assume 151Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16   IIR FF−BF (MF Lower Bound, Max−Min) IIR FF−BF (MF Upper Bound, Max−Min) IIR FF−BF (MMSE−DFE Lower Bound, Max−Min) IIR FF−BF (MMSE−DFE Upper Bound, Max−Min) FIR FF−BF (L a =1) FIR FF−BF (L a =5) FIR FF−BF (L a =11) FIR FF−BF (MMSE−DFE, Max−Min) FIR FF−BF (ZF−LE, Min Sum MSE) FIR FF−BF (w/o EQ) NR A v erag e mi n{ SIN R 1 , SIN R 2 }[dB ] Figure 5.8: Average SINR vs. number of relays NR for FF¸BF with MMSE¸DFE, ZF¸LE, MF, and slicer (no equalizer) receivers at the transceivers. Exponentially decaying power delay pro¿le with σt = 2, Lg = Lh = 5, d = 1, and γg = γh = 10 dB. all the relays are located at location (c) of Fig. 5.2. We show results for MMSE¸DFE, MF, and slicer (no equalizer) receivers with the FF¸BF ¿lters optimized for maximization of the minimum transceiver SINR. For MMSE¸DFE and MF receivers with IIR FF¸BF ¿lter the performance upper and lower bounds introduced in Section 5.4.1 are shown. For the FIR case only the achievable lower bound is shown for clarity. In addition, we show the average SINR for ZF¸LE with FF¸BF ¿lters optimized under the sum MSE criterion. We observe from Fig. 5.8 that for all values of NR the gap between the upper and lower bound 152Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays for max¸min FF¸BF ¿lter optimization with equalization is very small. As NR increases the gap between the simple slicer receiver and the MMSE¸DFE receiver diminishes. In fact, the slicer receiver with FIR FF¸BF ¿lters of lenght La = 11 closely approaches the performance of MMSE¸DFE with IIR FF¸BF ¿lter and FIR FF¸BF ¿lters of length La = 11 (which is not shown for clarity), but outperforms MMSE¸DFE with FIR FF¸BF ¿lters of length La = 5. 5.5.5 BER Performance for Fading Channels Figs. 5.9 and 5.10 show BERs of BPSK modulation vs. transmit SNR, γ = γg = γh, for FIR and IIR FF¸BF ¿lters. The BERs for FIR FF¸BF ¿lters were simulated by implementing ZF¸LE and MMSE¸DFE receivers with FIR equalization ¿lters of lengths 4 × Leq, which caused negligible performance degradation compared to IIR equalization ¿lters. The BERs for IIR FF¸BF were obtained by approximating the BER of BPSK transmission by BERX = Q (√ 2SINRX ) , where X = DFE, X = LE, and X = MF for DFE, LE, and MF receivers at the transceivers, respectively. The BERs for FIR FF¸BF with equalization and max¸min criterion were generated by using the FF¸BF ¿lters from the achievable lower bound. Here, the BER is averaged over 100,000 channel realizations. We consider a network with σt = 2, NR = 5, and d = 1. Fig. 5.9 shows that MMSE¸DFE with IIR FF¸BF at the relays closely approaches the performance of a MF receiver with IIR¸BF. Furthermore, FIR FF¸BF ¿lters of moderate length (La = 5) approach the performance of IIR FF¸BF ¿lters up to less than one decibel if MMSE¸DFE is employed at the transceivers. The same performance can also be achieved without equalization at the transceivers but with longer FF¸FB ¿lters (La = 11). At BER = 10−5, slicer (no equalizer) receivers with La = 11 achieve only 0.4 dB performance lost comparing with the performance of the MMSE¸DFE receivers with FIR FF¸BF and 153Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays 0 2 4 6 8 10 12 14 16 10−4 10−3 10−2 10−1   Analytical IIR FF−BF (MF Lower Bound) Analytical IIR FF−BF (MF Upper Bound) Analytical IIR FF−BF (MMSE−DFE, Max−Min) FIR FF−BF (L a =1) FIR FF−BF (L a =5) FIR FF−BF (L a =11) FIR FF−BF (MMSE−DFE) FIR FF−BF (w/o EQ) γg = γh [dB] BE R Figure 5.9: Average BER of BPSK vs. transmit SINR γ for FF¸BF with MMSE¸DFE, MF, and slicer receiver at the transceivers. BERs for FIR FF¸BF with EQ and IIR FF¸ BF with MMSE¸DFE were generated using the FF¸BF ¿lters from the achievable lower bound of the max¸min criterion. Exponentially decaying power delay pro¿le with σt = 2, Lg = Lh = 5, NR = 5, and d = 1. La = 11. Fig. 5.10 reveals that for ZF¸LE at the transceivers, FF¸BF ¿lters according to the max¸min and sum MSE criteria achieve a similar BER performance. Furthermore, the performance gap between the MF receiver and the simple ZF¸LE receiver with IIR FF¸BF ¿lters is less than one decibel which suggests again that simple LE and DFE equalizers are suÁcient to achieve a close¸to¸optimal performance. 154Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays 0 2 4 6 8 10 12 14 16 10−4 10−3 10−2 10−1   Analytical IIR FF−BF (MF, Max−Min) Analytical IIR FF−BF (ZF−LE, Max−Min) FIR FF−BF (L a =1) FIR FF−BF (L a =5) FIR FF−BF (ZF−LE, Max−Min) FIR FF−BF (ZF−LE, Sum Min MSE) γg = γh [dB] BE R Figure 5.10: Average BER of BPSK vs. transmit SINR γ for FF¸BF with ZF¸LE and MF receiver at the transceivers. For the min¸max criterion, BERs were generated using the FF¸BF ¿lters from the achievable lower bound. Exponentially decaying power delay pro¿le with σt = 2, Lg = Lh = 5, NR = 5, and d = 1. 5.6 Conclusions In this chapter, we have investigated FF¸BF for two¸way relaying networks employing single¸carrier transmission over frequency¸selective channels. For the processing at the transceivers, we considered two di¾erent cases: (1) a simple slicer without equalization and (2) LE or DFE. For the ¿rst case, we optimized FIR FF¸BF ¿lters at the relays for maximization of the minimum transceiver SINR subject to a relay power constraint and for minimization of the total relay transmit power subject to two QoS constraints. 155Chapter 5. Two¸Way FF¸BF with Multiple Single Antenna Relays Both problems can be transformed into convex SOCP problems, which can be eÁciently solved with standard numerical methods. For the second case, we optimized FIR and IIR FF¸BF ¿lters for maximization of the minimum transceiver SINR and, in case of ZF¸ LE, also for minimization of the sum MSE at the equalizer outputs of both transceivers. For the max¸min criterion, we established an upper and an achievable lower bound for the original problem. Both optimization problems were solved by transforming them into one¸way relay problems and leveraging corresponding results from Chapter 4. From our simulation results, we can draw the following conclusions: for max¸min optimization with equalization, the gap between the upper bound and the achievable lower bound is very small rendering the obtained solution close¸to¸optimal; and for ZF¸LE the max¸min and the minimum sum MSE criteria lead to as similar performance. Thus, the two proposed architectures allow us to trade relay complexity and transceiver complexity. For networks with powerful relays and low¸complexity transceivers, long FF¸ BF ¿lters and a simple slicer may be implemented at the relays and the transceivers, respectively. In contrast, for networks with powerful transceivers and simple relays, it is preferable to implement short FF¸BF ¿lters and equalizers at the relay and the transceivers, respectively. 156Chapter 6 Summary of Thesis and Future Research Topics In this ¿nal chapter, we summarize our results and highlight the contributions of this thesis in Section 6.1. In Section 6.2, we also propose ideas for future related research. 6.1 Summary of Results This thesis as a whole has focused on beamforming design for next generation wireless communication systems, namely: (1) a novel TD transmit beamforming scheme for MIMO¸ OFDM systems; (2) cooperative AF¸BF schemes with multiple multi¸antenna relays and multi¸antenna source; (3) one¸way cooperative FF¸BF schemes for frequency¸selective channels with multiple multi¸antenna relays; (4) two¸way cooperative FF¸BF schemes for frequency¸selective channels with multiple single¸antenna relays. In the following, we brieÀy review the main results of each chapter. In Chapter 2, we have proposed a novel TD approach to BF in MIMO¸OFDM systems. The C¸BFFs have been optimized for maximization of the AMI and minimization of the BER, respectively, and eÁcient algorithms for recursive calculation of the optimum C¸ BFFs have been provided for both criteria. For the case of a ¿nite¸rate feedback channel a GVQ algorithm has been introduced for codebook design. Simulation results for the IEEE 802.11n Channel Model B have con¿rmed the excellent performance of TD¸BF and have 157Chapter 6. Summary of Thesis and Future Research Topics shown that TD¸BF achieves a more favorable performance/feedback rate trade¸o¾ than traditional FD¸BF. In Chapter 3, we have considered AF¸BF for cooperative networks with one multi¸ antenna source, multiple multi¸antenna relays, and one single¸antenna destination for three di¾erent power constraints. In particular, we have considered the cases of individual relay power constraints, a joint power constraint for all relays, and a joint source¸relay power constraint. For a given BF vector at the source, we have fully characterized the optimal AF¸BF matrices for all three constraints. Furthermore, optimal and sub¸optimal methods for optimization of the source BF vectors have been provided. Simulation results show that increasing the number of antennas at the source is particularly bene¿cial if the relays are located far away from the source. In contrast, increasing the number of antennas at the relays or the number of relays is always bene¿cial regardless of the location of the relays. In Chapter 4, we investigated FF¸BF for one¸way relay networks with multiple multi¸ antenna relays and single¸carrier transmission over frequency¸selective channels. The FF¸ BF matrix ¿lters at the relays were optimized for the cases where (1) a simple slicer without equalization and (2) LE/DFE were employed at the destination. For the ¿rst case, we obtained closed¸form solutions and eÁcient numerical methods for computation of the optimal FIR FF¸BF matrix ¿lters. For the second case, we obtained an elegant method for calculation of the optimal IIR FF¸BF matrix ¿lters and an eÁcient numerical algorithm for calculation of near¸optimal FIR FF¸BF matrix ¿lters. In Chapter 5, we have investigated FF¸BF for two¸way relay networks employing single¸carrier transmission over frequency¸selective channels. Multiple single antenna re- lays are assumed in the network. For the processing at the transceivers, we again considered two di¾erent cases: (1) a simple slicer without equalization and (2) LE or DFE. For the 158Chapter 6. Summary of Thesis and Future Research Topics ¿rst case, we optimized FIR FF¸BF ¿lters at the relays for maximization of the minimum transceiver SINR subject to a relay power constraint and for minimization of the total relay transmit power subject to two QoS constraints. Both problems can be transformed into convex SOCP problems, which can be eÁciently solved with standard numerical methods. For the second case, we optimized FIR and IIR FF¸BF ¿lters for maximization of the mini- mum transceiver SINR and, in case of ZF¸LE, also for minimization of the sum MSE at the equalizer outputs of both transceivers. For the max¸min criterion, we established an upper and an achievable lower bound for the original problem. Both optimization problems were solved by transforming them into one¸way relay problems and leveraging corresponding results from Chapter 4. 6.2 Future Work Future wireless communication networks will have to strive for higher data rates and more reliable communication, and at the same time, cope with a tremendous growth in the num- ber of users. This brings about several technical problems such as a higher interference level as well as a major decrease in available bandwidth per user. The above issues have raised serious concerns on whether existing network topologies are able to cope with the challenges introduced by future applications. In Chapters 3¸5, we have considered beam- forming for cooperative networks, and proposed several innovative beamforming schemes for such networks. However, cooperative communication system design is a vast research area and many problems are still unsolved. Cooperative communications may also be combined with the cognitive radio concept [91]. Since the wireless spectrum is a scarce and costly resource, the diÁculty in obtaining spectrum allocations is becoming a hindrance to innovation. This problem has prompted regulatory bodies to allow unlicensed terminals, known as cognitive radios, to use previ- 159Chapter 6. Summary of Thesis and Future Research Topics ously allocated spectrum if they can avoid causing interference to the incumbent licensees. The combination of cognitive radios and cooperative communications has the potential to revolutionize the wireless industry. In the following, we propose some ideas for further research that are similar to or can be based on the results of this thesis. 6.2.1 Two¸way Relaying with Multiple Multi¸antenna Relays One immediate extension of the current work is on beamforming schemes for two¸way MABC relaying with multiple multi¸antenna relays. As a matter of fact, we have already made preliminary but encouraging progress on such topic. In [92], we assume single¸carrier transmission and frequency¸selective channels. The relays are equipped with FF¸BF ma- trix ¿lters in contrast with FF¸BF ¿lters in Chapter 5. As shown in Chapter 5, the performance of a simple slicer with optimized decision delay can closely approach the per- formance of transceivers with equalizers. Therefore, we assume that a simple slicer is employed at each of the transceivers in [92]. We optimize the FF¸BF matrix ¿lters at the relays for (1) a SINR balancing objective under a relay transmit power constraint, i.e. maximization of the worst transceiver SINR, and (2) minimization of the total relay trans- mit power subject to two QoS constraints to guarantee a certain level of performance. We show that the optimization problems are diÁcult to solve in general. However, by relaxing the rank constraints, we convert the optimization problems to semide¿nite programming (SDP) problems, which provide certi¿ed numerical upper bounds for the original problems. Subsequently, we show that the original problems can be approximated as convex SOCP problems by strengthening the constraints. It is noteworthy that the SOCP approximation method does not impose any rank relaxations. Simulations reveal that the close¸to¸optimal SOCP approximation method provides practically the same performance as the SDP rank relaxation method. In future work, we can leverage the ¿nding for slicer transceiver in [92] 160Chapter 6. Summary of Thesis and Future Research Topics and conduct research on transceivers equipped with equalizers. 6.2.2 Cooperative Communications for Multi¸user Systems Next generation mobile communication systems have to be able to provide reliable com- munications for a large number of users within a cell and on cell edges. Multi¸user MIMO schemes can provide a substantial gain in network downlink throughput by allowing mul- tiple users to communicate in the same frequency (or OFDM subcarrier) and time slots [93]. The combination of multi¸user MIMO¸OFDM beamforing and relaying is a promis- ing technique for performance enhancement for next generation wireless communications. Although some preliminary research has been already conducted on MIMO¸OFDM re- laying system [94, 95] and multi¸user MIMO relaying systems [96], there are still many interesting open problems such as resource allocation and protocol design. Since di¾erent users interfere with each other in a multi¸user MIMO relaying systems, maximizing the performance of a particular user may degrade the performance of the other users. To deal with this problem in a systematic way, a constraint optimization framework for the design of multi¸user cooperative beamforming communications should be developed. This will optimally allocate system resources (time, frequency, and beamforming direction) to all the users, permit the maximization of the performance of certain (preferred) users while guaranteeing a certain minimum performance for other (secondary) users. For example, preferred users may be those who have an ongoing call, whereas secondary users are those who are just in the process of establishing a connection. 6.2.3 Synchronization for Cooperative Communications Perfect timing is assumed in most of the literature, e.g. [28]¸[31], for analyzing the perfor- mance of cooperative communications. However, in practices, perfect timing is an unrealis- 161Chapter 6. Summary of Thesis and Future Research Topics tic assumption due to the distributed nature of the cooperative networks. Therefore, time synchronization is a critical issue for any cooperative network. Since cooperative network usually consists of two transmission phases, it is diÁcult to provide a precise clock reference for all the signal coming from distributed users with di¾erent prospectives. Literature on synchronization for cooperative networks is very sparse. Recent publication [97] consid- ered frequency o¾set estimation and correction for AF and DF cooperative networks, and [98] proposed timing resynchronization algorithms for AF cooperative networks. However, both paper considered single antenna equipped relays in Àat fading channels, and many open questions are still unanswered, e.g. the impact of synchronization error in frequency¸ selective channels. Thus, time synchronization problem should be investigated and special attention should be given to signaling schemes which are robust against synchronization errors. 6.2.4 Cooperative Communications for Cognitive Radio Beamforming for cognitive radio has attracted considerable attention recently, cf. e.g. [99, 100] and references therein. The combination of cooperative communications with cogni- tive radio would allow for relaying retransmissions to occur in temporarily idle licensed frequency bands, hence considerably reducing the inherent overhead per channel use. 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