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, 2011 Abstract Multipleinput multipleoutput (MIMO) and relaying are two promising techniques which will be employed in next generation wireless communication systems. Transmit beamforming (BF) and receive combining are simple yet popular methods for performance enhancement 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 timedomain BF (TDBF) scheme which uses cyclic BF lters (CBFFs). Both perfect and partial channel state information at the transmitter are considered. The CBFFs are optimized for maximum average mutual information per subcarrier and minimum average uncoded bit error rate. We show that TDBF has a more favorable performance/feedback rate tradeo than previously proposed frequencydomain BF schemes. Secondly, BF for oneway cooperative networks with multiple multiantenna amplify andforward relays in frequencynonselective channels is considered. The source BF vector and the amplifyandforward BF matrices at the relays are optimized for maximization of the signaltointerferenceplusnoise ratio (SINR) at the destination under three dierent power constraints. We show the benets of having multiple antennas at the source and/or multiple multiantenna relays. Subsequently, we investigate lterandforward BF (FFBF) for oneway relay networks in frequencyselective channels. For the processing at the destination, we investi- ii Abstract gate two dierent cases: a simple slicer, and a linear equalizer (LE) or a decisionfeedback equalizer (DFE). For both cases, we optimize the FFBF 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 FFBF matrix lters for minimization of the total transmit power under a quality of service constraint. Leveraging results from oneway relaying, we also investigate FFBF for twoway 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 maxmin problem. iii Preface Chapters 25 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, TimeDomain Transmit Beamforming for MIMOOFDM Systems with Finite Rate Feedback. munications, • 57(9): 28282838, Sept. 2009. Y. Liang, R. Schober, and W. Gerstacker, TimeDomain Transmit Beamforming for MIMOOFDM Systems. In Conference (GLOBECOM), • IEEE Transactions on Com- Proceedings of the IEEE Global Telecommunications Washington DC, USA, Nov. 2007. Y. Liang, R. Schober, and W. Gerstacker, Minimum BER Transmit Beamforming for MIMOOFDM Systems with Finite Rate Feedback. In International Conference on Communications (ICC), Proceedings of the IEEE Beijing, China, May 2008. iv Preface Four papers related to Chapter 3 have been published: • Y. Liang and R. Schober, Cooperative AmplifyandForward Beamforming with Multiple MultiAntenna Relays. IEEE Transactions on Communications, 59(9): 26052615, Sept. 2011. • Y. Liang and R. Schober, Cooperative AmplifyandForward Beamforming for OFDM Systems with Multiple Relays. In on Communications (ICC), • Proceedings of the IEEE International Conference Dresden, Germany, Jun. 2009. Y. Liang and R. Schober, Cooperative AmplifyandForward Beamforming with invited paper). MultiAntenna Source and Relays ( In Proceedings of the Third International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), • Aruba, Dec. 2009. Y. Liang and R. Schober, AmplifyandForward MultiAntenna Beamforming with Joint SourceRelay Power Constraint. In nology Conference (VTC), Proceedings of the IEEE Vehicular Tech- Ottawa, Canada, Sept. 2010. Three papers related to Chapter 4 have been published: • Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, Cooperative FilterandForward Beamforming with Equalization for FrequencySelective Channels. Wireless Commun., • IEEE Trans. on 10(1): 228239, Jan. 2011. Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, FilterandForward Beamforming with Multiple MultiAntenna Relays for FrequencySelective Channels ( paper). In invited Proceedings of the International ICST Conference on Communications and Networking in China (Chinacom), Aug. 2010. v Preface • Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, Cooperative FilterandForward Beamforming with Equalization for FrequencySelective Channels. In of the IEEE Global Telecommunications Conference (GLOBECOM), Proceedings 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, TwoWay FilterandForward Beamforming for FrequencySelective Channels. tions on Wireless Communications, • Accepted by the IEEE Transac- Oct. 2011. Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, Cooperative TwoWay Filter andForward Beamforming for FrequencySelective Channels. In Proceedings of the IEEE International Conference on Communications (ICC), Kyoto, Japan, Jan. • 2011. Y. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, TwoWay FilterandForward Beamforming with Multiple MultiAntenna Relays for FrequencySelective Channels. In preparation, Oct. 2011. vi Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv List of Abbreviations Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii xxvii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Beamforming for MIMO Systems 1.2 Cooperative Relay Network 1.3 TwoWay Cooperative Relay Network 1.4 1.5 1 . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . 7 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 10 vii Table of Contents 2 TimeDomain Transmit Beamforming for MIMOOFDM Systems 2.1 Introduction 2.2 System Model 2.3 2.4 2.5 2.6 2.7 . . 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Transmitter Processing for TDBF . . . . . . . . . . . . . . . . . . 16 2.2.2 MIMO Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.3 Receiver Processing 2.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Feedback Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Maximum AMI Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Formulation of the Optimization Problem 2.3.2 Solution of the Optimization Problem for Lg = Nc . . . . . . . . . 21 2.3.3 Solution of the Optimization Problem for L g < Nc . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Minimum BER Criterion . . . . . . . . . . . . . . 2.4.1 Formulation of the Optimization Problems 2.4.2 Solution of the Optimization Problems for Lg = Nc . . . . . . . . . 28 2.4.3 Solution of the Optimization Problems for Lg < Nc . . . . . . . . . 29 . . . . . . . . . . . . . . . . . . . . 32 . . . . . . . . . . . . . . . . . . . . . . 32 . . . . . . . . . . . . . . . . . . . . . . . 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 FiniteRate Feedback and Comparison 2.5.1 FiniteRate Feedback Case 2.5.2 Comparison with FDBF Simulation Results Conclusions . . . . . . . . . . . . . 20 27 . . . . 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 viii Table of Contents 3 Cooperative AFBF with Multiple MultiAntenna Relays . . . . . . . 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1 Introduction 3.2 System Model and Optimization Problem 3.3 3.4 3.5 3.6 . . . . . . . . . . . . . . . . . . 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.1 System Model 3.2.2 Formulation of the Optimization Problem Optimal AFBF Matrices . . . . . . . . . . . . . . 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.1 AFBF with Individual Power Constraints for Relays 3.3.2 AFBF with Joint Power Constraint for Relays 3.3.3 AFBF with Joint Power Constraint for Source and Relays 3.3.4 Comparison of the Solutions for the Dierent Constraints Optimal BF Vector at the Source . . . . . . . 53 . . . . . . . . . . . 57 . . . . 58 . . . . . 59 . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.1 AFBF with Individual Power Constraints for Relays 3.4.2 AFBF with Joint Power Constraint for Relays 3.4.3 AFBF with Joint Power Constraint for Source and Relays 3.4.4 Comparison of the Solutions and CSI Feedback Requirements Simulation Results . . . . . . . 60 . . . . . . . . . . . 62 . . . . 65 . . . 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.1 Comparison of Source BF Vector Optimization Methods 3.5.2 Impact of Network Parameters on Performance 3.5.3 Impact of Power Constraints on Performance Conclusions . . . . . 69 . . . . . . . . . . . 74 . . . . . . . . . . . . 76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 Cooperative FFBF with Multiple MultiAntenna Relays . . . . . . . 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.1 FFBF at Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.2 Processing at Destination 86 4.1 Introduction 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . ix Table of Contents 4.2.3 4.3 4.4 4.5 4.6 Feedback Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . FIR FFBF without Equalization . . . . . . . . . . . . . . . . . . . . . . SINR Maximization Under Relay Power Constraint . . . . . . . . . 90 4.3.2 Relay Power Minimization Under SINR Constraint . . . . . . . . . 91 4.3.3 SINR Maximization Under SourceRelay Power Constraint . . . . 92 4.3.4 SourceRelay Power Minimization Under SINR Constraint . . . . 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 FFBF with Equalization 4.4.1 Optimal IIR FFBF with Equalization . . . . . . . . . . . . . . . 94 4.4.2 Optimal FIR FFBF with Equalization . . . . . . . . . . . . . . . 104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Simulation Results 4.5.1 FFBF without Equalization 4.5.2 FFBF with Equalization Conclusions . . . . . . . . . . . . . . . . . . . . . 110 . . . . . . . . . . . . . . . . . . . . . . . 115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 . . . . . . . . . 125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.2.1 FFBF at Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.2.2 Transceiver Processing . . . . . . . . . . . . . . . . . . . . . . . . . 129 FIR FFBF without Equalization . . . . . . . . . . . . . . . . . . . . . . . 130 5.3.1 Maxmin Criterion Under Relay Power Constraint . . . . . . . . . 133 5.3.2 Relay Power Minimization Under SINR Constraints . . . . . . . . 135 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1 Introduction 5.2 System Model 5.4 5.5 87 4.3.1 5 TwoWay FFBF with Multiple Single Antenna Relays 5.3 87 FFBF with Equalization 5.4.1 Optimal IIR FFBF with Equalization 5.4.2 FIR FFBF Filter Optimization Simulations . . . . . . . . . . . . . . . . 137 . . . . . . . . . . . . . . . . . . . 143 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 x Table of Contents 5.5.1 Relay Power Minimization for FFBF without Equalization 5.5.2 Maxmin SINR Optimization for FFBF with and without Equalization 5.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 148 Maxmin SINR vs. Minimum Sum MSE Optimization for FFBF with ZFLE 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NR 150 5.5.4 Impact of Number of Relays . . . . . . . . . . . . . . . . . . . 151 5.5.5 BER Performance for Fading Channels . . . . . . . . . . . . . . . . 153 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary of Thesis and Future Research Topics 155 . . . . . . . . . . . . . . 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.1 Summary of Results 6.2 Future Work 6.2.1 Twoway Relaying with Multiple Multiantenna Relays . . . . . . 160 6.2.2 Cooperative Communications for Multiuser Systems . . . . . . . . 161 6.2.3 Synchronization for Cooperative Communications . . . . . . . . . . 161 6.2.4 Cooperative Communications for Cognitive Radio . . . . . . . . . 162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Bibliography xi List of Tables 2.1 Calculation of the optimum CBFFs g for the maximum AMI and the mini- mum average BER criterion using a GA, respectively. Termination constant ϵ has a small value (e.g. ϵ = 10−4 ). i denotes the iteration and adaptation step size necessary for the GA. 2.2 δi is the . . . . . . . . . . . . . . . . . . Feedback Requirements for TDBF, ideal FDBF, and FDBF with modied spherical (MS), Grassmannian (GS), and geodesic (GD) interpolation. 3.1 Gradient algorithm for calculation of source BF vector joint relay power constraints. The denitions of gˆ gˆ a small value (e.g. ϵ = 10−5 ). k gradk and the gradient denotes the iteration index and ak ϵ has is the adaptation step size chosen through a backtracking line search [1]. . . . . . Gradient algorithm for calculation of source BF vector g ϵ = 10−5 ). k 62 and power allo- cation for joint sourcerelay power constraint. Termination constant a small value (e.g. 34 for individual and depend on the power constraint, cf. Section 3.4. Termination constant 3.2 26 denotes the iteration index and ak ϵ has is the adaptation step size chosen through a backtracking line search [1]. . . . . . 67 xii List of Tables 4.1 Numerical algorithm for nding the optimum power allocation FFBF lters at the relays. ∆f have small values (e.g. iteration index. 4.2 for IIR X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver, respectively. Termination constant spacing p(f ) ϵ and frequency ϵ = 10−5 , ∆f = 10−5 ). i denotes the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Gradient algorithm (GA) for calculation of nearoptimal FIR FFBF lter vector a. Termination constant ϵ has a small value (e.g. ϵ = 10−5 ). i denotes the iteration index and δi a backtracking line search [1]. . . . . . . . . . . . . . . . . . . . . . . . . . is the adaptation step size chosen through 108 xiii List of Figures 1.1 Dierent relaying protocols. (a) decodeandforward (DF) relaying pro- tocol, and (b) amplifyandforward (AF) protocol. from the source, after decoding, 1.2 xr : xs : signal transmitted from the relay, T Sx : time slot x, and a: Dierent twoway relaying protocols. xˆs : scaling factor. signal broadcasted regenerated signal . . . . . . . . . . . 5 (a) Bidirectional oneway relaying protocol, (b) Time Division Broadcast (TDBC) protocol, and (c) Multiple Access Broadcast (MABC) protocol. 1 (TC1), s2 : s1 : signal transmitted from transceiver signal transmitted from transceiver 2 (TC2), and processed version of the received signals. 2.1 . . . . . . . . . . . . . . . . . . . 8 MIMOOFDM system with TDBF. P/S: Paralleltoserial conversion. S/P: Serialtoparallel conversion. CE: Channel estimation. 2.2 f (s1 , s2 ): . . . . . . . . . . . 17 AMI of TDBF (AMI criterion), MSFDBF [2], and GDFDBF [3] with perfect CSI. NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. For comparison the AMIs for ideal FDBF and singleinput singleoutput (SISO) transmission (NT 2.3 = 1, NR = 1) are also shown. . . . . . . . . . . . AMI of TDBF (AMI criterion) vs. number of feedback bits update. NT = 2, NR = 1, Nc = 512, Es /N0 = 10 B 37 per channel dB, and IEEE 802.11n Channel Model B. For comparison the AMIs for GDFDBF with codebooks from [4] are also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 xiv List of Figures 2.4 BER of coded MIMOOFDM system with TDBF (AMI criterion), MS FDBF [2], and GDFDBF [3]. Perfect CSI and niterate feedback, 2, NR = 1, Nc = 512, Rc = 1/2, and IEEE 802.11n Channel Model B. For comparison the BERs for ideal FDBF and SISO transmission (NT NR = 1) 2.5 are also shown. NT = = 1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Average BER of uncoded MIMOOFDM system with TDBF. Minimum average BER criterion (solid lines) and maxmin criterion (dashed lines), perfect CSI, NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. For comparison the BERs for ideal FDBF and SISO transmission (N T 2.6 = 1, N R = 1) are also shown. . . . . . . . . . . . . . . . . . . . . . . . Average BER of uncoded MIMOOFDM system with TDBF (average BER criterion) vs. number of feedback bits for CBFF optimization. B per channel update. GA was used NT = 2, NR = 1, Nc = 512, Eb /N0 = 10 IEEE 802.11n Channel Model B. 2.7 41 dB, and . . . . . . . . . . . . . . . . . . . . . . . 42 Average BER of uncoded and coded MIMOOFDM system with TDBF (average BER criterion). GA was used for CBFF optimization and Lg = 2 is valid for all curves shown. Perfect CSI (bold lines) and niterate feedback channel, 2.8 NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. 43 Average BER of uncoded and coded MIMOOFDM 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 xv List of Figures 3.1 Cooperative network with one multiantenna source, multiple multiantenna relays, and one singleantenna destination. element of source BF vector vector n1,i denotes the ith g . ni,µ , 1 ≤ µ ≤ Mi , is the µth element of noise i, 1 ≤ i ≤ NR . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Locations of source, destination, and relays in simulation. . . . . . . . . . . 70 3.3 CDF of the instantaneous SINR for AFBF with (PC) at relay gi , 1 ≤ i ≤ NT , joint relay power constraint and one relay located at (a) and (e), respectively. Results for dierent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A pathloss exponent of 3, d=1 3.4 are assumed. and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CDF of the instantaneous SINR for AFBF with constraint (PC) NT = 2, and NR 71 joint sourcerelay power relays. Results for dierent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A pathloss exponent of 3, 3.5 NT = 2, and d=1 are assumed. The relays are located at (a) and (e) for NR = 2, (a)(e) for (a)(e) with 2 relays at each location for NR = 10. . . . . . . . . . . . . . . CDF of the instantaneous SINR for AFBF with straints (PCs) and NR = 5 NR = 5, and 72 individual relay power con- singleantenna relays at locations (a)(e). Re- sults for dierent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A pathloss exponent of 3 and d=1 are assumed. . . . . . . . . . . . . . . . . . . . . . . . . . . 73 xvi List of Figures 3.6 Average SINR vs. distance (PC) d for AFBF with and dierent numbers of antennas NT joint relay power constraint at the source. A pathloss 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Average SINR vs. distance straints (PCs) d for AFBF with 75 individual relay power con- and dierent 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 also shown. 3.8 and the SINR for relay selection are . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average SINR vs. distance straint (PC) P =2 d for AFBF with 76 joint sourcerelay power con- and dierent numbers of relays and dierent numbers of relay antennas. A pathloss 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. . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Average mutual information (AMI) in (bits/s/Hz) vs. distance d 77 for two dierent network setups and dierent 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 pathloss exponent of 4 is assumed. g. A For comparison the average mutual information without relaying for a source transmit power of P =2 average mutual information for relay selection are also shown. and the . . . . . . . 78 xvii List of Figures 3.10 Average BER vs. σd2 /σn2 for two dierent network setups and dierent power constraints. The relays are in locations (a) and (e) for for NR = 5. NR = 2 and (a)(e) The proposed gradient methods are used for computation of the source BF vector g. A pathloss exponent of 3 and d=1 are assumed. AFBF: 16QAM. Direct transmission: QPSK, source transmit power 4.1 P = 2. 79 Cooperative network with one singleantenna source, multiple multiantenna relay nodes, and one singleantenna 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 FFBF without equalization (EQ) at the destination. The FFBF lters were optimized for SINR maximization under joint relay power constraint. Lg = Lh = 5, d = 1, NR = 5, Mz = 1, nel power delay prole with z ∈ {1, 2, . . . , 5} 4.4 and γg = γh = 10 Average SINR vs. distance at the destination. σt = 2 and d . . . . . . . . . . . . . . . . . . . . Average SINR vs. distance and d γg = γh = 10 P =2 dB. Results for direct at the source are also included. . . σt = 2 and 112 for FIR FFBF without equalization (EQ) at the destination. The FFBF matrix lters were optimized for a with joint Exponentially decaying channel power delay prole Lg = Lh = 5, relay power constraint. 111 for FIR FFBF without equalization (EQ) transmission with transmit power 4.5 dB. The FFBF matrix lters were optimized for a relay power constraint. with Exponentially decaying chan- joint source Exponentially decaying channel power delay prole Lg = Lh = 5, and transmission with transmit power γg = γh = 10 P =2 dB. Results for direct at the source are also included. . . 113 xviii List of Figures 4.6 Total average source and relay transmit power vs. required SINR FFBF without equalization (EQ) at the destination for imization and joint sourcerelay power minimization. caying power delay prole with γg = γh = 10 4.7 dB. Feasibility probability vs. required SINR relay power minimization. σt = 2 and and for FIR relay power min- Exponentially de- Lg = Lh = 5, d = 1, and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tion (EQ) at the destination for 4.8 σt = 2 γ for FIR FFBF without equaliza- relay power minimization and joint source Exponentially decaying power delay prole with Lg = Lh = 5, d = 1, SINR vs. iteration number γ 114 i and γg = γh = 10 dB. . . . . . . . . . . . . 115 of GA given in Table 4.2 for FIR FFBF with γg = γh = 10 dB, Lg = Lh = 5, and √ ¯ 1,z [k] = 1/ 5, 0 ≤ k < 5, 1 ≤ z ≤ 5. NR = 5 relays with Mz = 1, g¯1,z [k] = h MMSEDFE at the destination. 1 ≤ z ≤ NR , at locations (a)(e), respectively. For comparison the SINR for IIR FFBF is also shown. 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . 116 γg = γh = 10 dB, NR = 1 √ ¯ 1,1 [k] = 1/ 2, k ∈ {0, 1}. g¯1,1 [k] = h Frequency responses of IIR FFBF lters for single antenna relay, L g = L h = 2, and For comparison the frequency response of the test channel is also shown. . 118 4.10 Frequency responses of IIR FFBF lter and FIR FFBF lters of various lengths for MMSEDFE at the receiver. All channel parameters are identical to those in Fig. 4.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 xix List of Figures 4.11 Average SINR vs. distance d for FFBF with MMSELE, MMSEDFE, and an MF receiver at the destination. NR = 2 relays with M1 = 2 exponentially decaying power delay prole with and γg = γh = 10 σt = 2 and and M2 = 3, Lg = Lh = 5, dB. For comparison the SINRs of FFBF without (w/o) equalization (EQ) at the destination and without relaying are also shown, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Average SINR vs. decay parameter σt for FFBF with MMSELE, MMSE DFE, and an MF receiver at the destination. and 120 NR = 2 relays with M1 = 2 M2 = 3, distance d = 1, exponentially decaying power delay prole with Lg = Lh = 5, and γg = γh = 10 dB. For comparison the SINRs of FFBF without (w/o) equalization (EQ) at the destination and without relaying are also shown, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Average BER of BPSK vs. transmit SNR γ 121 for FFBF with MMSELE, MMSEDFE, and an MF receiver at the destination. Exponentially decaying power delay prole with σt = 2 and Lg = Lh = 5. For comparison the BER of FFBF without (w/o) equalization (EQ) at the destination is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Cooperative twoway network with two transceiver nodes and nodes. EQ is the equalizer at the transceivers. relay sˆ1 [k] and sˆ2 [k] are estimated received symbols at TC2 and TC1, respectively. 5.2 NR 122 . . . . . . . . . . . . . . . Locations of TC1, TC2, and the relays in the simulations. . . . . . . . . . 127 146 xx List of Figures 5.3 Total average relay transmit power vs. required SINRs FFBF without equalization at the transceivers. γ1 and γ2 for FIR The FFBF lters were optimized for minimization of the relay transmit power. Exponentially decaying power delay prole with γg = γh = 10 5.4 dB. σt = 2, Lg = Lh = 5, d = 1, NR = 5, and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feasibility probability vs. required SINRs γ1 and γ2 146 for FIR FFBF without equalization at the transceivers. The FFBF lters were optimized for minimization of the relay transmit power. Exponentially decaying power delay prole with 5.5 σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 Average worstcase SINR at the transceivers vs. distance d dB. . . 147 for FFBF with/without equalization at the transceivers. The FFBF lters were optimized for maximization of the minimum transceiver SINR. Exponentially decaying channel power delay prole with NR = 5, 5.6 and γg = γh = 10 dB. σt = 2, Lg = Lh = 5, d = 1, . . . . . . . . . . . . . . . . . . . . . . . . Average SINR at transceivers vs. distance ization (EQ) at the transceivers. 148 d for FFBF with/without equal- The FFBF lters were optimized for maximization of the minimum transceiver SINR. Exponentially decaying channel power delay prole with γg = γh = 10 5.7 dB. σt = 2, Lg = Lh = 5, d = 1, NR = 5, and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average SINR at transceivers vs. distance d 149 for FFBF with ZFLE at the transceivers. Exponentially decaying channel power delay prole with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. . . . . . . . . . 151 xxi List of Figures 5.8 Average SINR vs. number of relays NR for FFBF with MMSEDFE, ZF LE, MF, and slicer (no equalizer) receivers at the transceivers. Exponentially decaying power delay prole with γh = 10 5.9 dB. σt = 2, Lg = Lh = 5, d = 1, and γg = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average BER of BPSK vs. transmit SINR γ 152 for FFBF with MMSEDFE, MF, and slicer receiver at the transceivers. BERs for FIR FFBF with EQ and IIR FFBF with MMSEDFE were generated using the FFBF lters from the achievable lower bound of the maxmin criterion. Exponentially decaying power delay prole with σt = 2, Lg = Lh = 5, NR = 5, 5.10 Average BER of BPSK vs. transmit SINR MF receiver at the transceivers. γ and d = 1. 154 for FFBF with ZFLE and For the minmax criterion, BERs were generated using the FFBF lters from the achievable lower bound. Exponentially decaying power delay prole with and d = 1. σt = 2, Lg = Lh = 5, NR = 5, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 xxii List of Abbreviations 3GPP 3rd Generation Partnership Project 4G The Fourth Generation AF AmplifyandForward 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 CBFF Cyclic Beamforming Filter CC Convolutional Code CDF Cumulative Distribution Function CIR Channel Impulse Response CP Cyclic Prex CSI Channel State Information CSIT CSI at the Transmitter DF DecodeandForward DFE DecisionFeedback Equalizer DFT Discrete Fourier Transform DSTC Distributed SpaceTime Coding xxiii List of Abbreviations EDGE Enhanced Data Rates for GSM Evolution EQ Equalization FCC Federal Communications Commission FEC Forward Error Correction FFBF FilterandForward 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 Innite Impulse Response ISI InterSymbol Interference LE Linear Equalizer LTE (3GPP) Long Term Evolution MABC Multiple Access Broadcast MF Matched Filter MIMO MultipleInput MultipleOutput MLSE Maximum Likelihood Sequence Estimation xxiv List of Abbreviations MMSE Minimum Mean Square Error MS Modied Spherical MSE Mean Square Error NPhard NonDeterministic PolynomialTime 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 Semidenite Programming SINR SignaltoInterferenceplusNoise Ratio SISO SingleInput SingleOutput SIMO SingleInput MultipleOutput SOCP SecondOrder Cone Programming SM Spatial Multiplexing SNR SignaltoNoise Ratio STBC SpaceTime Block Code STTC SpaceTime Trellis Code TC Transceiver TD Time Domain TDBC Time Division Broadcast WiMAX Worldwide Interoperability for Microwave Access WLAN Wireless Local Area Network xxv List of Abbreviations ZF Zero Forcing xxvi Notation (·)T Transpose (·)H Hermitian transpose (·)∗ Complex conjugate 0X Allzero column vector of length IX X ×X [X]ij Element of matrix det(·) Matrix determinant diag{x1 , . . . xN } Diagonal matrix with diag{X 1 , . . . , X N } Block diagonal matrix with ℜ{·} Real part of a complex number ℑ{·} 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 vec(·) Vectorization of a matrix (stacking the columns of trace(·) Trace of a matrix x+ max(0, x) ⊗ Kronecker product X identity matrix X in row i and column x1 , . . . , x N j on the main diagonal X 1 , . . ., X N on the main diagonal eigenvector of a matrix · on top of each other) xxvii Notation ⊕ Kronecker sum ∗ Discretetime convolution E{·} Statistical expectation F{x[k]} Fourier transform of discretetime signal x[k], i.e. ∞ ∑ x[k]e−j2πf k k=−∞ Q(·) Gaussian Qfunction [5] A≽B A−B is positive semidenite xxviii Acknowledgments 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 profound and distinguished professor, Dr. Schober sets an example of being a great researcher. The knowledge and training I got from him benet me then, now and forever. I am much indebted for his patience and encouragement over the years. guidance, this thesis would not be possible. Without his support and 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 eort 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 parentsinlaw, Wendong Tian and Zhichun xxix Acknowledgments 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. xxx Dedication To My Family xxxi Chapter 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. Multipleinputmultipleoutput (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 briey review beamforming (BF) for MIMO wireless systems. In Section 1.2, we discuss oneway relaying protocols, and in Section 1.3, we introduce twoway relaying protocols. We briey 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 signicant performance gains over conventional single antenna based solutions [6][8]. These MIMO systems enable a spatial diversity gain, a spatial multiplexing 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 1 Chapter 1. Introduction paths in space (dierent transmit antennas) and by utilizing appropriate combining techniques at the receiver. Several schemes have been proposed to exploit the spatial diversity gain. For example, spacetime block codes (STBC) [9, 10] and spacetime trellis codes (STTC) [11] are wellknown transmit diversity techniques, which lead to improved link reliability. 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 spacetime (VBLAST) [12] and the diagonal Bell Labs layered spacetime (DBLAST) [13] schemes. The aforementioned techniques to achieve a spatial diversity gain or a spatial multiplexing gain are based on the socalled openloop conguration, where only the receiver has knowledge of the communication channel. Recent research shows that system performance can be further enhanced by socalled closedloop 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 oered by MIMO systems to mitigate the eects 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 niterate 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 frequencyselective channels into a number of parallel narrowband frequencyat channels [20]. In such a MIMOOFDM system, spatial multiplexing, 2 Chapter 1. Introduction spacetime coding, and other signal processing algorithms are usually employed in order to approach the MIMO channel capacity. MIMOOFDM 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]. Transmit BF techniques proposed for narrowband channels can be easily extended to broadband MIMOOFDM systems by applying independent BF in each subcarrier [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 moderatetolarge number of subcarriers Nc (e.g. Nc ≥ 64). Since the fading gains as well as the corresponding BF vectors are correlated across OFDM subcarriers, in [2] it was proposed to reduce the amount of feedback by only feeding back the BF vectors for a small number of subcarriers. The remaining BF vectors are obtained by modied spherical interpolation. This approach signicantly reduces the required amount of feedback at the expense of some loss in performance. The required number of feedback bits of this frequencydomain BF (FDBF) scheme can be further reduced by postprocessing 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 FDBF schemes the required amount of feedback to achieve a certain performance is tied to the number of OFDM subcarriers. This may be problematic in OFDM systems with a large number of subcarriers and stringent limits on the aordable amount of feedback. Therefore, this motivates us to propose a novel timedomain (TD) approach to BF in MIMOOFDM systems in Chapter 2. 3 Chapter 1. Introduction 1.2 Cooperative Relay Network Multipleantenna 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 onehalf 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 amplifyandforward (AF) relaying and decodeandforward (DF) relaying [30, 31]. An illustration of the halfduplex 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 (AFSM) relaying for singlerelay networks with multiple an- 4 Chapter 1. Introduction Relay Relay xs T S1 xs xs Source xs Destination Source Relay Destination Relay xr = axs xr = x ˆs T S2 Source Destination Source (a) Destination (b) Figure 1.1: Dierent relaying protocols. (a) decodeandforward (DF) relaying protocol, and (b) amplifyandforward (AF) protocol. signal transmitted from the relay, x, and a: xˆs : xs : signal broadcasted from the source, regenerated signal after decoding, T Sx : xr : time slot 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 ecient and popular approach to exploit the spatial diversity oered by the channel. AFBF for wireless relay networks was considered in [35][44] and ref- erences therein. In particular, AFBF for networks with one singleantenna source and multiple singleantenna 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 signaltointerferencenoise ratio (SINR) maximization problem at the destination involved only the optimization of one scalar BF gain for each relay. In contrast, in [37, 38], AFBF for a network with a single relay and multiple antennas at the relay and the source was investigated and closedform solutions for the BF vector at the source and 5 Chapter 1. Introduction the AFBF matrix at the relay were provided. Furthermore, in [43, 44], the performance of AFBF with multiple antennas at the source and one singleantenna relay was investigated. 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 OFDMbased cooperative networks is investigated in [45][48]. In [45], a relay network with one transmitterreceiver 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. ltering for relay networks with one relay node was introduced in [46]. Linear In [49], a time domain AFBF scheme for cooperative OFDM networks with multiple relays is proposed. As mentioned above, BF for cooperative networks with singlecarrier transmission over frequencynonselective channels and multicarrier transmission over frequencyselective channels has been studied extensively in the literature. In contrast, the literature on BF (and other relay processing techniques as well) for singlecarrier transmission over frequencyselective channels is very sparse. Nevertheless, wireless channels are typically frequency selective and multicarrier 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 linear power ampliers required for OFDM may be prohibitive. To compensate for the effect of frequencyselective channels, lterandforward (FF) beamforming (FFBF) for frequencyselective channels is proposed in [50]. However, in [50], only a simple slicer was 6 Chapter 1. Introduction employed at the destination requiring the FFBF lters at the relays to equalize both the sourcerelay and the relaydestination channels. This motivates us to consider the case if a simple linear equalization (LE) or decisionfeedback 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 LTEAdvanced [52]. The goal of utilizing cooperative communications in both standards is to increase the data rates available to celledge users and to increase coverage at a given data rate. 1.3 TwoWay Cooperative Relay Network Most of the published results on distributed beamforming consider a oneway relaying protocol 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 twoway relaying, the relays cooperate with each other to establish reliable bidirectional communication between two transceivers [53, 54]. The capacity and achievable rate region for twoway relaying protocols have been studied in [55, 56] and references therein. The choice between oneway and twoway relaying mainly depends on the application. Oneway relaying is of interest for unidirectional communication, whereas twoway relaying is preferable for bidirectional communication. Various protocols for twoway relaying exist in the literature. The most common two way relaying protocols are the bidirectional oneway 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 oneway relaying operations, as shown in Fig. 1.2 (a). However, 7 Chapter 1. Introduction TC1 T S1 Relay TC1 Relay TC2 TC1 s1 s2 XOR(s1 , s2 ) s2 T S3 Relay s1 s1 T S2 T S4 TC2 s1 TC2 s2 f (s1 , s2 ) f (s1 , s2 ) XOR(s1 , s2 ) s2 (a) (b) (c) Figure 1.2: Dierent twoway relaying protocols. (a) Bidirectional oneway relaying protocol, (b) Time Division Broadcast (TDBC) protocol, and (c) Multiple Access Broadcast (MABC) protocol. s1 : signal transmitted from transceiver 1 (TC1), from transceiver 2 (TC2), and f (s1 , s2 ): s2 : signal transmitted 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 eciency point of view. A singlerelay 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 eciency. A detail comparison of the bidirectional oneway relaying protocol and the singlerelay 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 ecient 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 twoway relay networks. In [57, 59, 8 Chapter 1. Introduction 60], a twoway relay network with single multiantenna relays is considered, whereas [58] considered a network with multiple single antenna relays. However, all the aforementioned papers consider frequencynonselective channels. This motivates us to investigate twoway relaying schemes for frequencyselective 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 timedomain approach to BF in MIMOOFDM systems. The proposed timedomain BF scheme employs cyclic BF lters. Simulation results conrm the excellent performance of the proposed scheme and show that timedomain BF has a more favorable performance/feedback rate tradeo than previously proposed frequencydomain BF schemes. 2. We propose BF schemes for cooperative networks with one multiantenna source, multiple multiantenna AF relays, and one singleantenna destination. For a given BF vector at the source, we nd the optimal AFBF matrices at the relays for each of the three considered power constraints, namely individual relay power constraint, joint relay power constraint and joint sourcerelay power constraint. Several numerical methods for nding the optimal source BF vectors are also proposed. 3. We investigate FFBF for oneway relay networks employing singlecarrier transmission over frequencyselective channels. We consider two cases for the receive processing at the destinations: (1) a slicer and (2) LE/DFE. For both cases, we optimize the FFBF lters for maximization of the SINR under a transmit power constraint. 9 Chapter 1. Introduction In addition, for case (1), we also optimize the FFBF lters for minimization of the transmit power under a QoS constraint, respectively. We nd closedform/near optimal solutions for the IIR and FIR FFBF matrix lters at the relays. 4. Drawing from the ndings on oneway relaying, we investigate FFBF for twoway relay networks with multiple singleantenna 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 FFBF 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 FFBF 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 singledata stream, timedomain BF scheme for MIMOOFDM systems which uses cyclic BF lters (CBFFs). Assuming perfect CSI at the transmitter, the CBFFs are optimized for two dierent criteria, namely, maximum average mutual information (AMI) per subcarrier and minimum average uncoded bit error rate (BER). If the CBFF length Lg is equal to the number of subcarriers Nc , closedform solutions to both optimization problems exist. For the practically relevant case Lg < Nc we present numerical methods for calculation of the optimum CBFFs for both criteria. Using a global vector quantization (GVQ) approach, the CBFFs are quantized 10 Chapter 1. Introduction for practical niterate feedback channels. Simulation results for typical IEEE 802.11n channels conrm the excellent performance of the proposed scheme and show that TD BF has a more favorable performance/feedback rate tradeo than previously proposed FDBF schemes. In Chapter 3, we consider BF for cooperative networks with one multiantenna source, multiple multiantenna AF relays, and one singleantenna destination. The source BF vector and the AFBF matrices at the relays are optimized for maximization of the SINR at the destination under three dierent power constraints. In particular, we consider individual 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 AFBF matrices for a given BF vector at the source. For the cases of individual and joint relay power constraints, closedform solutions for the AFBF matrices are provided, respectively. Furthermore, for the case of a joint source relay power constraint, the direction of the AFBF matrices is derived in closed form and an ecient 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 sourcerelay power constraints, the resulting problem can be transformed into a nonconvex 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 ecient suboptimal optimization methods for the source BF vector. Simulation results show the benets of having multiple antennas at the source and/or multiple multiantenna relays and illustrate the performance dierences introduced by the three dierent power constraints. In Chapter 4, we investigate FFBF for relay networks employing singlecarrier transmission over frequencyselective channels. In contrast to prior work, which concentrated 11 Chapter 1. Introduction on multiple singleantenna relay nodes, we consider networks employing multiple multi antenna relay nodes. For the processing at the destination, we investigate two dierent cases: (1) a simple slicer without equalization and (2) a LE or a DFE. For both cases, we optimize the FFBF 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 FFBF matrix lters for minimization of the total transmit power under a quality of service (QoS) constraint. For the rst case, we obtain closedform solutions for the optimal FIR FFBF matrix lters, whereas for the second case, we provide the optimal solution for IIR FFBF matrix lters, and an ecient gradient algorithm for recursive calculation of nearoptimal FIR FFBF matrix lters. Our simulation results reveal that for a given total number of antennas in the network, a small number of multipleantenna relays can achieve signicant performance gains over a large number singleantenna relays. In Chapter 5, we consider FFBF for twoway relay networks employing singlecarrier transmission over frequencyselective channels. We adopt the MABC protocol for twoway relaying with singleantenna relays is assumed. Similar to the oneway 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 dierent cases: (1) a simple slicer without equalization and (2) LE/DFE. For the rst case, we optimize FIR FFBF lters, respectively, for maximization 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 secondorder cone programming (SOCP) problem, which can be eciently solved using standard tools. For the second case, we optimize IIR and FIR FFBF lters for maxmin optimization of the SINR, and for transceivers with zeroforcing LE, also for minimization of the sum MSE at the equalizer outputs of both transceivers. Leveraging 12 Chapter 1. Introduction results from FFBF for oneway relaying, we establish an upper and an achievable lower bound for the maxmin problem and an exact solution for the sum MSE problem. Since the gap between the upper and the lower bound for the maxmin problem is small, a closeto optimal solution is obtained. Our simulation results reveal that the performance of FFBF without equalization at the transceivers crucially depends on the slicer decision delay and transceivers with slicers can closely approach the performance of transceivers with equalizers provided that the FFBF lters are suciently long and a sucient number of relays is deployed. Finally, Chapter 6 summarizes the contributions of this thesis and outlines areas of future research. 13 Chapter 2 TimeDomain Transmit Beamforming for MIMOOFDM Systems 2.1 Introduction As pointed out in Chapter 1, transmit BF and receiver combining are simple yet ecient techniques for exploiting the benets of MIMOOFDM systems [14]. Several FDBF 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 subcarriers, [2] proposed to reduce the amount of feedback by only feeding back the BF vectors for a small number of socalled pilot sub carriers. The remaining BF vectors are obtained by modied spherical interpolation. This approach signicantly reduces the required amount of feedback at the expense of some loss in performance. The required number of feedback bits of this FDBF scheme can be further reduced by postprocessing 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 FDBF schemes the required amount of feedback to achieve a certain performance is tied to the number of OFDM subcarriers. In this chapter, we propose a novel TD approach to BF in MIMOOFDM systems . In this chapter, we only consider singlestream BF which is sometimes also referred to as maximal ratio transmission. We note, however, that the concept of TDBF can also be extended to multistream BF which is also referred to as spatial multiplexing. 14 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM 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 subcarriers is typically much larger than the number of nonzero channel impulse response (CIR) coecients. Therefore, tackling the problem directly in the TD is a natural choice. The proposed TDBF scheme employs CBFFs of length Lg ≤ Nc . The CBFFs are optimized for maximization of the AMI and for minimization of the average uncoded BER, respectively. While other CBFF optimization criteria are certainly possible (e.g., maximum cuto 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 capacityachieving) 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 (dierent) nonlinear eigenvalue problems for the CBFF coecient vectors, and we show that closedform solutions to both problems exist for Lg = Nc . However, for the practically more interesting case of L g < Nc , a closedform solution does not exist for either problem, and we provide ecient numerical methods for calculation of the CBFFs. Furthermore, for the case of a niterate 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 TDBF and FDBF [2, 3, 27]. We note that TD preprocessing for MIMOOFDM has been considered in dierent contexts before. For example, TDBF schemes with one scalar BF weight per antenna (as opposed to CBFFs) have been proposed to reduce the number of inverse discrete Fourier transforms (IDFTs) required at the transmitter of MIMOOFDM systems, cf. e.g. [63] and references therein. Similarly, cyclic delay diversity, which is a simple form of spacetime coding, cf. e.g. [64, 65], may be viewed as a TD MIMOOFDM preprocessing technique. 15 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems Furthermore, BF with linear BFFs has been considered for singlecarrier transmission over frequencyselective channels and DFE at the receiver in [62]. However, the concept of employing CBFFs for (limited feedback) BF in MIMOOFDM 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 CBFFs 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 niterate feedback TDBF and a detailed comparison between TDBF and FDBF 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 MIMOOFDM system with and Nc NT transmit antennas, NR receive antennas, OFDM subcarriers. The block diagram of the discretetime 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 TDBF The modulated symbols have variance 1]]T D[n], 0 ≤ n < Nc , are taken from a scalar symbol alphabet A and 2 σD = E{|D[n]|2 } = 1. The transmit symbol vector x [x[0] x[1] . . . x[Nc − after the IDFT operation can be represented as x W D, (2.1) 16 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems D[0] IDFT C-BFF D[Nc − 1] Add CP & P/S Remove CP & S/P Y [0] DFT & CE C1∗ [0] Y [Nc − 1] n1 [k] C1∗ [Nc − 1] C-BFF Add CP & P/S Remove CP & S/P DFT & CE CN∗ R [0] nNR [k] CN∗ R [Nc − 1] Feedback Channel Figure 2.1: MIMOOFDM system with TDBF. P/S: Paralleltoserial conversion. S/P: Serialtoparallel conversion. CE: Channel estimation. D [D[0] D[1] . . . D[Nc − 1]]T ∑Nc −1 j2πnk/Nc . n=0 D[n]e where √1 Nc At transmit antenna nt sequence gnt [k], 0 ≤ k < Lg , 1 ≤ nt ≤ NT , and x[k] W is the unitary IDFT matrix [66], i.e., x[k] = is ltered with a CBFF with impulse response of length Lg ≤ Nc . The resulting OFDM symbol after cyclic ltering is given by ¯ nt x, snt = G where ¯ nt G is an Nc × Nc (2.2) columncirculant matrix with rst column [gnt [0] gnt [1] . . . gnt [Lg − 1]]T . [g Tnt 0TNc −Lg ]T , g nt We note that in practice the cyclic ltering in (2.2) can be implemented using the following three simple steps: 1. Add a cyclic prex (CP) of length ¯ Lg −1 to x to generate x [x[Nc −Lg +1] . . . x[Nc − 1] xT ]T . gnt [k], 0 ≤ k < Lg , to ∑Lg −1 gnt [κ]¯ x[k −κ] [¯ snt [0] s¯nt [1] . . . s¯nt [Nc +Lg −2]]T , where s¯nt [k] = κ=0 2. Pass the elements of generate and ¯nt s ¯ x through a x¯[k], 0 ≤ k < Nc + Lg − 1, linear lter with coecients are the elements of ¯ x and x¯[k] = 0 for k < 0. 17 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems 3. Remove the CP from ¯nt s to obtain After cyclic ltering a CP is added to than of L − 1, ¯ nt , G where L snt = [¯ snt [Lg − 1] . . . s¯nt [Nc + Lg − 2]]T . snt . We assume that the CP length is not smaller is the length of the CIR. We note that due to the cyclic structure TDBF does not aect the length requirements of the CP, i.e., the required CP length for TDBF is identical to that for singleantenna transmission. 2.2.2 MIMO Channel We model the wireless channel as a frequencyselective and spatially correlated MIMO channel. The spatial correlations may be introduced by insucient antenna spacing. The channel between transmit antenna response hnt nr [l], 0 ≤ l < L. nt and receive antenna nr is characterized by its impulse Note that the impulse response coecients 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 TDBF does not aect the processing at the receiver, i.e., standard OFDM receiver processing is applied. After CP removal the discretetime received signal at receive antenna nr , 1 ≤ nr ≤ NR , can be modeled as r nr = NT ∑ ¯ nt x + nnr , ¯ nt nr G H (2.3) nt =1 where ¯ nt nr H 1] 0TNc −L ]T is an and Nc ×Nc columncirculant matrix with rst column [hnt nr [0] . . . hnt nr [L− nnr is an additive white Gaussian noise (AWGN) vector whose entries 18 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems nnr [k], 0 ≤ k < Nc , and variance are independent and identically distributed (i.i.d.) with zero mean σn2 . After DFT we obtain at antenna nr NT ∑ H Rnr = W r nr = H nt nr Gnt D + N nr , (2.4) nt =1 where ¯ nt nr W = diag{Hnt nr [0] . . . Hnt nr [Nc − 1]}, Gnt W HH H nt nr diag{Gnt [0] . . . Gnt [Nc − 1]}, and N nr 0 ≤ n < Nc , W H nnr = [Nnr [0] . . . Nnr [Nc − 1]]T . are i.i.d. AWGN samples with variance and the CBFF gains Gnt [n] ¯ nt W = W HG σn2 . The The FD channel gains Nnr [n], Hnt nr [n] are given by Hnt nr [n] L−1 ∑ hnt nr [l]e−j2πnl/Nc , (2.5) l=0 Lg −1 Gnt [n] ∑ gnt [l]e−j2πnl/Nc . (2.6) l=0 Considering now the vector C[n] nth subcarrier and assuming an NR dimensional receive combining [C1 [n] . . . CNR [n]]T , with (2.4) the combined received signal can be expressed as Y [n] = C H [n]H[n]G[n]D[n] + C H [n]N [n], where NR × NT GNT [n]]T , and matrix N [n] H[n] contains Hnt nr [n] in row nr [N1 [n] . . . NNR [n]]T . has perfect knowledge of maximizes the SNR of H[n], 0 ≤ n < Nc . Y [n] is given by 0 ≤ n < Nc , and column nt , G[n] (2.7) [G1 [n] . . . In this chapter, we assume that the receiver In this case, the combining vector C[n] = H[n]G[n] C[n] that (maximalratio combining). 19 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM 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 innite capacity, the receiver sends the unquantized CBFF vector mitter (perfect CSI case). [g T1 . . . g TNT ]T , g, g 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 predesigned CBFF vector codebook of size N = 2B , where gˆ n is an NT Lg dimensional vector. [h11 [0] h11 [1] . . . h11 [L − 1] h21 [0] . . . hNT NR [L − 1]]T h dress n to the trans- of the codeword (CBFF vector) G {ˆ g 1 , gˆ 2 , . . . , gˆ N } For a given channel vector the receiver determines the ad- gˆ n ∈ G , 1 ≤ n ≤ N , which maximizes the prescribed optimality criterion (maximum AMI or minimum BER). Subsequently, index is sent to the transmitter which then utilizes g = gˆ n n for BF. Similar to [2, 3, 27] we assume that the feedback channel is errorfree and has zero delay. 2.3 Maximum AMI Criterion In this section, we optimize the CBFFs for maximization of the AMI per subcarrier. After rigorously formulating the optimization problem, we present a closedform solution for Lg = Nc and numerical methods for computation of the optimum CBFFs for L g < Nc . 2.3.1 Formulation of the Optimization Problem Assuming i.i.d. Gaussian input symbols nth D[·], the mutual information (in bit/s/Hz) of the subcarrier is given by [66] C[n] = log2 (1 + SNR[n]) . (2.8) 20 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems For maximalratio combining the SNR of the nth subcarrier can be obtained from (2.7) as SNR[n] We note that G[n] = 1 H G [n]H H [n]H[n]G[n]. σn2 (2.9) can be expressed as G[n] = F [n]g, where the nt th 1 ≤ nt ≤ NT , row of with carrier depends on NT × NT Lg f [n] g and matrix F [n] (2.10) is given by [0T(nt −1)Lg f T [n] 0T(NT −nt )Lg ], [1 e−j2πn/Nc . . . e−j2π(Lg −1)n/Nc ]T . Therefore, the AMI per sub ∑Nc −1 1 is given by C = n=0 C[n]. The optimization problem can Nc now be formulated as max g s.t. N c −1 ∑ C[n] (2.11) g g = 1, (2.12) n=0 H where (2.12) is a transmit power constraint. 2.3.2 Solution of the Optimization Problem for Lg = Nc Although in practice ful to rst consider Lg ≪ Nc is desirable to minimize the amount of feedback, it is insight- Lg = Nc since in this case a closedform solution to the optimization problem in (2.11), (2.12) exists. In addition, the solution for formance upper bound for the practically relevant case F [F T [0] . . . F T [Nc − 1]]T is invertible, and for a given Lg = Nc Lg < Nc . G For serves as a per- Lg = Nc matrix [GT [0] . . . GT [Nc − 1]]T the 21 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems CBFF vector g can be obtained from g = F −1 G, (2.13) cf. (2.10). This means (2.11) and (2.12) are equivalent to N c −1 ∑ max G ( log2 n=0 H ) 1 H H 1 + 2 G [n]H [n]H[n]G[n] σn G G = Nc . s.t. (2.14) (2.15) The solution to this equivalent problem can be obtained as G[n] = α[n]E max [n], where λmax [n], and α[n] is obtained from √ α[n] = x+ max(0, x) and λ N c −1 ( ∑ n=0 G[n], 0 ≤ n < Nc , ( Nc σn2 1 1 − λ Nc λmax [n] )+ , (2.17) is the solution to the waterlling equation σn2 Once (2.16) E max [n] is that eigenvector of matrix H H [n]H[n] which corresponds to the maximum eigenvalue where 0 ≤ n < Nc , 1 1 − λ Nc λmax [n] )+ = 1. (2.18) g can be obtained from (2.13). has been calculated, the optimum Therefore, in this case, TDBF is equivalent to ideal FDBF with waterlling which is not surprising since for Lg = Nc there are as many degrees of freedom in the TD as there are in the FD. 22 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM 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 . For convenience we rewrite (2.11), (2.12) as max g s.t. with NT Lg × NT Lg matrix M [n] N c −1 ∑ n=0 H ( log2 1 1 + 2 g H M [n]g σn ) g g=1 (2.19) (2.20) F H [n]H H [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 coecients ˜ L(g) N∏ c −1 n=0 ( ) g H σn2 I NT Lg + M [n] g , gH g (2.21) which is a wellknown dicult 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 CBFF vector. 1) Relaxation of the Optimization Problem: A popular approach for solving nonconvex optimization problems is to transform the original nonconvex problem into a convex one by relaxing the constraints [1]. This leads in general to a suboptimum (but often close tooptimum) solution for the original problem. For the problem at hand we may dene a 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. 23 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems matrix gg H S and rewrite (2.19), (2.20) as max S s.t. N c −1 ∑ ( log2 det I NR n=0 trace{S} ) 1 H H + 2 H[n]F [n]SF [n]H [n] σn ≤ 1, (2.23) S ≽ 0, rank{S} where S ≽ 0 means that S (2.22) (2.24) = 1, (2.25) is a positivesemidenite matrix. equality is satised in (2.23) when S It is easy to show that is optimal. The equivalent optimization problem in (2.22)(2.25) is still nonconvex 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 semidenite 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, nonconvex 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 nonconvex problem. Unfortunately, the amount of time to solve the relaxed optimization problem strongly depends on Nc , and for medium numbers of subcarriers (e.g. Nc ≥ 64) standard optimiza- tion software (e.g. yalmip and SeDuMi) 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 subcarriers is small (e.g. 2) Gradient Algorithm: The Lagrangian of (2.19), (2.20) can be formulated as L(g) = N c −1 ∑ n=0 where µ Nc < 64). ( log2 ) 1 H 1 + 2 g M [n]g − µg H g, σn denotes the Lagrange multiplier. (2.26) The optimum CBFF vector has to fulll 24 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems ∂L(g)/∂g ∗ = 0NT Lg , which leads to the nonlinear eigenvalue problem [N −1 c ∑ ] σn2 n=0 M [n] g = µg. + g H M [n]g (2.27) σn2 → ∞) the optimum CBFF vector can be obtained from (2.27) ∑Nc −1 eigenvector of n=0 M [n] which corresponds to the maximum eigen- For very low SNRs (i.e., as the unitnorm value of that matrix, i.e., a closedform 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 optimization 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 dierent 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 procedure outlined in [69, p. 41], which optimizes the step size starting from an initial value contraction factor backtracking line search δi in each iteration. Thereby, δi = δ¯ > 0 the step size is gradually reduced as δi ← ρ δi with ρ ∈ (0, 1) until the socalled Armijo condition with constant c is fullled [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 25 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems Table 2.1: Calculation of the optimum CBFFs g for the maximum AMI and the minimum average BER criterion using a GA, respectively. Termination constant ϵ has a small value −4 (e.g. ϵ = 10 ). i denotes the iteration and δi is the adaptation step size necessary for the GA. i=0 1 Let 2 Update the CBFF vector: [N −1 c ∑ AMI: BER: 3 and initialize the CBFF vector with some g0 fullling gH 0 g 0 = 1. ] M [n] g˜ i+1 = g i + δi gi σn2 + g H i M [n]g i n=0 ] [N −1 ) ( c ∑ c2 H g˜ i+1 = g i + δi exp − 2 g i M [n]g i M [n] g i σn n=0 Normalize the CBFF: g˜ g i+1 = √ i+1 ˜ i+1 g˜ H i+1 g H 1 − |g i+1 g i | < ϵ, goto Step 5, otherwise 4 If 5 g i+1 increment i→i+1 and goto Step 2. is the desired CBFF vector. if the termination constant (dened in Table 2.1) is set to ϵ = 10−4 . However, in practice, the speed of convergence of the GA is not critical, since in the realistic niterate feedback case, the GA is only used to nd the CBFF codebook, which is done oline. 2.4 Minimum BER Criterion The main criterion considered for CBFF optimization in this section is the BER averaged over all subcarriers. However, we will also consider the minimization of the maximum sub carrier BER for optimization of the CBFFs. Besides the additional insight that this second BER criterion oers, it also provides a useful starting point for numerical computation of the minimum average BER CBFF lters, cf. Section 2.4.3. 26 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems 2.4.1 Formulation of the Optimization Problems While closedform expressions for the BER or/and symbol error rate exist for most regular signal constellations such as M ary M ary quadrature amplitude modulation (M QAM) and phaseshift keying (M PSK), these expressions are quite involved which is not desirable for CBFF optimization. Therefore, we adopt here the simple yet accurate BER approximations from [70], which allow us to express the approximate BER of the nth subcarrier as BER[n] where the ≈ c1 exp (−c2 SNR[n]) , nth subcarrier SNR is dened in (2.9) and c1 constants. For example, for square M QAM we have and (2.28) c2 c1 = 0.2 are modulation dependent and c2 3/[2(M − 1)] [70]. Throughout this chapter we assume that all subcarriers use the same modulation scheme. 1 Nc 1) Average BER Criterion: The (approximate) average BER is given by BER = ∑Nc −1 n=0 BER[n]. Consequently, the minimum average BER optimization problem can be formulated as N c −1 ∑ min g n=0 H g g = 1. s.t. 2) MaxMin Criterion: BER[n] (2.29) (2.30) Since the exponential function is monotonic, we observe from (2.28) that minimizing the maximum subcarrier BER is equivalent to maximizing the minimum subcarrier SNR. The resulting maxmin problem becomes max min g ∀n s.t. SNR[n] (2.31) g H g = 1. (2.32) 27 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems Since for high SNR, the maximum subcarrier 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 matrix F is invertible, i.e., for a given 1) Average BER Criterion: min s.t. can be obtained from (2.13). Eq. (2.13) implies that (2.29) and (2.30) are equivalent to N c −1 ∑ G G the CBFF vector g Lg = Nc n=0 H ( ) c2 H H exp − 2 G [n]H [n]H[n]G[n] σn (2.33) G G = Nc . (2.34) Formulating (2.33) and (2.34) as a Lagrangian, it can be shown that the optimum is again proportional to E max [n], given by √ α[n] = where λ i.e., (2.16) is still valid. However, now [ ( )]+ λmax [n] ln , c2 λmax [n] λ σn2 α[n] G[n] in (2.16) is (2.35) is the solution to the waterlling problem ]+ Nc −1 [ σn2 ∑ ln (λmax [n]/λ) = 1. c2 Nc n=0 λmax [n] For high SNR, i.e., can be calculated as σn2 ≪ 1, λmax [n] > λ, 0 ≤ n < Nc , (2.36) holds and the subcarrier BER ∑ c −1 BER[n] = c1 λ/λmax [n], where λ = exp([ N n=0 (ln(λmax [n])/λmax [n]) − ∑ c −1 c2 Nc /σn2 ]/[ N n=0 1/λmax [n]]), cf. (2.9), (2.28), (2.35), and (2.36). This means for high SNR the subcarrier BER is inversely proportional to the maximum subcarrier eigenvalue λmax [n]. 28 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems 2) MaxMin Criterion: Exploiting (2.13) also for the maxmin criterion, it can be shown that the optimum solution has again the general form given by (2.16) with ( α[n] = Nc −1 λmax [n] ∑ 1 Nc n=0 λmax [n] This means that for the maxmin criterion and SNR[n] = Nc /(σn2 ∑Nc −1 n=0 1/λmax [n]). Lg = Nc )− 12 . (2.37) all subcarrier SNRs are equal to Therefore, in contrast to the minimum average BER solution, for the maxmin solution all subcarriers 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: min g s.t. where M [n] For convenience we rewrite (2.29), (2.30) as N c −1 ∑ n=0 H ( c2 exp − 2 g H M [n]g σn g g = 1, ) (2.38) (2.39) was dened 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 29 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems to the problem. In particular, letting again min N c −1 ∑ S s.t. n=0 S = gg H we can rewrite (2.38), (2.39) as ( ( ) c2 exp − 2 trace H[n]F [n]SF H [n]H H [n] σn trace{S} ≤ 1, ) (2.40) (2.41) S ≽ 0, (2.42) rank{S} = 1. (2.43) The equivalent optimization problem (2.40)(2.43) is still nonconvex 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 to its maximum eigenvalue. strongly depends on (e.g. Nc , S which corresponds Furthermore, the complexity of the relaxed problem again and becomes prohibitive for a moderate number of subcarriers Nc ≥ 64). 2) MaxMin Criterion: For the maxmin criterion, we may rewrite (2.31), (2.32) as max g min g H M [n]g (2.44) g H g = 1, (2.45) ∀n s.t. 30 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems which constitutes a quadratic objective quadratic constraint (QOQC) NPhard problem [71]. This problem can be restated in equivalent form as [71] max s.t. t (2.46) trace{S} ≤ 1, trace{M [n]S} (2.47) ≥ t, ∀n, (2.48) S ≽ 0, rank{S} (2.49) = 1. (2.50) By dropping the rank condition (2.50) the optimization problem (2.46)(2.50) can be relaxed 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 dominated by Lg and not by Nc . Since we are mainly interested in the case where relaxed problem for the maxmin criterion can be solved even for large Nc Lg ≪ Nc , the (e.g. Nc ≥ 256) using standard software (e.g. SeDuMi). 3) Gradient Algorithm: Unfortunately, for both relaxed optimization problems pre- sented in this section the resulting eigenvector of S S has a high rank most of the time, and the dominant is a suboptimum solution which may entail a signicant performance degradation. However, a GA may be used to recursively improve the initial CBFF vector found through relaxation. In Table 2.1, we provide the GA for the average BER criterion since this is our primary BERrelated criterion. However, if the average BER SDP problem (2.40)(2.42) cannot be solved since the number of subcarriers Nc is too large, we use the solution found for the maxmin 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- 31 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM 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 FiniteRate Feedback and Comparison In this section, we briey discuss codebook design for niterate feedback channels based on the GVQ algorithm in [62]. Furthermore, we also compare TDBF with interpolation based FDBF [2, 3, 27]. 2.5.1 FiniteRate Feedback Case G Vector quantization can be used to design a codebook of size N for the niterate feedback channel case, cf. Section 2.2.4. Here, we adopt the GVQ algorithm introduced in [62]. For this purpose a set Thereby, the CIRs of the H {h1 , h2 , . . . , hT } NT NR Ldimensional nth vector hn MIMO channel realization. corresponding CBFF vector g = g¯ n of T channel vectors hn is generated. contains the CIR coecients of all NT NR For each of these channel realizations the 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 {¯ g 1 , g¯ 2 , . . . , g¯ T }. The vector quantizer can then be represented as a function Ideally, this function is optimized for minimization of the MQE GT Q: GT → G . mean quantization error T 1∑ d(Q(¯ g i ), g¯ i ), T i=1 (2.51) 32 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems where d(ˆ g m , g¯ i ) denotes the distortion caused by quantizing g¯ i ∈ GT to gˆ m ∈ G . The distortion measure depends on the optimization criterion and is given by d(ˆ g m , g¯ i ) − N c −1 ∑ ( ) log2 1 + SNR(ˆgm ,hi ) [n] (2.52) n=0 and d(ˆ g m , g¯ i ) N c −1 ∑ ( ) exp −c2 SNR(ˆgm ,hi ) [n] (2.53) n=0 for the maximum AMI and the minimum BER criterion, respectively. Here, SNR(ˆ g m ,hi ) [n] is dened in (2.9) and the subscripts indicate that gˆ m and hi , respectively. G[n] and H[n] have to be calculated for With this denition 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 oline optimization of the codebook is completed, transmitter and the receiver. For a given channel realization CBFF gˆ m ∈ G h G is conveyed to the the receiver selects that 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 FDBF We compare TDBF with FDBF in terms of feedback requirements and computational complexity. 1) Feedback Requirements: The required number of complex feedback symbols S for TDBF, interpolationbased FDBF with modied spherical (MSFDBF) [2], Grassmannian (GSFDBF) [27], and geodesic (GDFDBF) [3] interpolation, and ideal FDBF are summarized in Table 2.2, where [2], i.e., Nc /K K denotes the cluster size in interpolationbased FDBF is the number of subcarriers for which CSI is assumed to be available at the 33 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems Table 2.2: Feedback Requirements for TDBF, ideal FDBF, and FDBF with modied spherical (MS), Grassmannian (GS), and geodesic (GD) interpolation. BF Scheme Number of Complex Feedback Symbols per Frame S = Nc NT S = NKc (NT + 1) S = NKc NT S = NT Lg Ideal FDBF MSFDBF [2] GSFDBF [27] and GDFDBF [3] Proposed TDBF transmitter. We will use S to compare the feedback requirements of TDBF and FDBF in Section 2.6. 2) Computational Complexity: The calculation of the CBFFs and the GVQbased codebook design for the proposed TDBF scheme are more involved than the calculation of the BF weights and the codebook design method adopted in [2, 3, 27] for FDBF, 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 eort for CBFF calculation and codebook design can be neglected. The interpolation of BF weights in FDBF has to be done in every frame. The interpolation complexity is generally proportional to Nc but strongly depends on the interpolator used. For example, modied spherical interpolation requires a grid search whereas Grassmannian and geodesic interpolation do not. Assuming a codebook of size index at the receiver requires evaluation of N and N Nc /K N selecting the beamformer distortion measures for TDBF and interpolationbased FDBF, respectively. However, a fair quantitative comparison of the associated complexities is dicult since the required N to achieve a similar performance may be very dierent in both cases. Similar to [63] we assume that the inverse IDFTs and the BF itself dominate the complexities of TDBF and FDBF. As is customary in the literature, we adopt the required 34 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM 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 since FDBF requires NT Nc log2 (Nc )/2 IFFT operations and complex multiplications. NT Nc Therefore, complex multiplications for BF, a total of MFD = NT Nc log2 (Nc ) + NT Nc 2 (2.54) complex multiplications are obtained. In contrast, assuming a straightforward TD implementation of convolution, MTD = Nc log2 (Nc ) + Lg NT Nc 2 (2.55) complex multiplications are required for TDBF. A comparison of MFD and MTD shows that the complexity of TDBF is lower than that of FDBF if Lg < For example, assuming Nc = 512 NT − 1 log2 (Nc ) + 1. 2NT subcarriers and BF requires a lower complexity than FDBF for (2.56) NT = 2, 3 ≤ NT < 9, Lg ≤ 3, Lg ≤ 4, and and Lg ≤ 5, NT ≥ 9 TD 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 MIMOOFDM with TDBF. 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- 35 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems bination of BICM and OFDM has been adopted in various recent standards, cf. e.g. [20]. However, rst we briey discuss the parameters used in our simulations. 2.6.1 Simulation Parameters Throughout this section we consider a MIMOOFDM system with transmit antennas, NR = 1 receive antenna, and Nc = 512 symbols D[·] Rc = 1/2, or NT = 3 OFDM subcarriers. If BICM is employed, the data bits are encoded with the quasistandard code of rate NT = 2 (171, 133)8 convolutional possibly punctured, interleaved, and Gray mapped to the data [20, 25]. At the receiver standard Viterbi soft decoding is applied. For all BER results 16QAM was used. For practical relevance we adopted for our simulations the IEEE 802.11n Channel Model B with a transmit antenna spacing of L=9 λ0 /2, where λ0 assuming a carrier frequency of 2.5 GHz and is the wavelength [72]. All simulation results were averaged over 100,000 independent channel realizations. For Lg < Nc the CBFF vectors were calculated with the algorithms given in Table 2.1. The allones vector and the solution of the relaxed maxmin problem were used for initialization of the GAs for the maximum AMI and the minimum BER criterion, respectively. For Lg = Nc (equivalent to ideal FDBF) the closedform solutions for the CBFF provided in Sections 2.3.2 and 2.4.2 were used. For the niterate feedback case the CBFF 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 TDBF with AMIoptimized CBFFs and compare its performance with that of MSFDBF [2] and GDFDBF [3], respectively. We note that in [2] an AMI criterion is used for interpolator optimization, whereas the interpolator optimization in [3] 36 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems 5 ideal FD−BF TD−BF (Lg = 1, S = 2) 4.5 TD−BF (Lg = 2, S = 4) TD−BF (Lg = 4, S = 8) MS−FD−BF (K MS−FD−BF (K GD−FD−BF (K GD−FD−BF (K NT=1, NR=1 4 AMI (bit/s/Hz) 3.5 = = = = 256, S = 6) 64, S = 24) 256, S = 4) 64, S = 16) 3 2.5 2 1.5 0 2 4 6 Es /N0 [dB] 8 10 12 14 Figure 2.2: AMI of TDBF (AMI criterion), MSFDBF [2], and GDFDBF [3] with perfect CSI. NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. For com- parison the AMIs for ideal FDBF and singleinput singleoutput (SISO) transmission (NT = 1, NR = 1) are also shown. is not directly tied to the AMI or BER. Throughout this subsection Fig. 2.2 shows the AMI per subcarrier vs. N0 : Es /N0 (Es : NT = 2 is valid. energy per received symbol, power spectral density of underlying continuoustime passband noise process) for the proposed TDBF, MSFDBF, and GDFDBF for the case of perfect CSI at the transmitter. To facilitate a fair comparison between TDBF with CBFFs of length and FDBF with cluster size K, Lg 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 37 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems 3.8 3.7 3.6 TD−BF (Finite−Rate Feedback, Lg = 1) TD−BF (Perfect CSI, Lg = 1) AMI (bit/s/Hz) 3.5 TD−BF (Finite−Rate Feedback, Lg = 2) TD−BF (Perfect CSI, Lg = 2) TD−BF (Finite−Rate Feedback, Lg = 3) 3.4 TD−BF (Perfect CSI, L = 3) g GD−FD−BF (Finite−Rate Feedback, K = 512) GD−FD−BF (Finite−Rate Feedback, K = 256) 3.3 3.2 3.1 0 1 2 3 4 5 6 7 B Figure 2.3: AMI of TDBF (AMI criterion) vs. number of feedback bits update. NT = 2, NR = 1, Nc = 512, Es /N0 = 10 B per channel dB, and IEEE 802.11n Channel Model B. For comparison the AMIs for GDFDBF with codebooks from [4] are also shown. BF provides a better performance/feedback tradeo than interpolationbased FDBF. For example, TDBF with S = 6 (K = 256) and S = 2 (Lg = 1) outperforms MSFDBF and GDFDBF with S = 4 (K = 256), is necessary to outperform TDBF with respectively. MSFDBF with S = 8 (Lg = 4) S = 24 (K = 64) which performs only less than 0.5 dB worse than ideal FDBF. In Fig. 2.3, we consider the AMI of TDBF with niterate feedback channel as a 38 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM 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 TDBF with perfect CSI (dashed lines) and the AMIs for GDFDBF with the best known codebooks from [4] and K = 256. For B=0 K = 512 and the codebook has just one entry and no feedback is required. As can be observed from Fig. 2.3, niterate feedback TDBF 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 GDFDBF scheme is signicantly worse than that of the TDBF scheme for the same number of feedback bits. From further simulations we have observed that GDFD BF requires more than with 7 B = 80 feedback bits and feedback bits to achieve the same performance as TDBF Lg = 3. Fig. 2.4 shows the BERs of a coded MIMOOFDM system (Rc BF, MSFDBF, and GDFDBF vs. Eb /N0 , where Eb = 1/2) employing TD denotes the average energy per information bit. Both perfect CSI and niterate feedback are considered. With perfect CSI at the transmitter, at a BER of 0.8 dB and S = 64, 0.77 10−4 the performance of TDBF with dB worse than that of MSFDBF with S = 48 S=6 is about and GDFDBF with respectively. However, in case of niterate feedback the performance of TDBF with B = 7 is slightly better than that of GDFDBF with B = 64 with B = 80, where we adopted the codebooks from [4] for GDFDBF and MSFDBF, and MSFDBF respectively. 2.6.3 Minimum BER Criterion Now, we shift our attention to TDBF with BERoptimized CBFFs. NT = 2 is still valid. Assuming perfect CSI we show in Fig. 2.5 the average BERs for the average BER criterion and the maxmin criterion, respectively. As expected, for Lg = Nc (ideal FD 39 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems −1 10 −2 10 −3 BER 10 Ideal FD−BF TD−BF (L = 3, S = 6) g −4 MS−FD−BF (K = 32, S = 48) GD−FD−BF (K = 16, S = 64) TD−BF (L =3, B=7, Finite−Rate Feedback) 10 g MS−FD−BF (K=32, B=80, Finite−Rate Feedback) GD−FD−BF (K=16, B=64, Finite−Rate Feedback) N =1, N =1 T R −5 10 0 2 4 6 Eb/N0 [dB] 8 10 12 Figure 2.4: BER of coded MIMOOFDM system with TDBF (AMI criterion), MSFD BF [2], and GDFDBF [3]. Nc = 512, Rc = 1/2, Perfect CSI and niterate feedback, NT = 2, NR = 1, and IEEE 802.11n Channel Model B. For comparison the BERs for ideal FDBF and SISO transmission (NT = 1, NR = 1) are also shown. BF) the average BER criterion leads to a lower average BER than the maxmin criterion. However, the dierence between both criteria is less than 1 dB at and Lg = 5 BER = 10−3 . For Lg = 1 we show the average BER obtained for the relaxed maxmin 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 oered by the second antenna 40 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems −1 10 −2 10 −3 BER 10 Ideal FD−BF, (Average BER Criterion) Ideal FD−BF (Max−Min Criterion) TD−BF, (Perfect CSI, GA, L =1) g TD−BF, (Perfect CSI, GA, Lg=2) −4 10 TD−BF, (Perfect CSI, GA, L =3) g TD−BF, (Perfect CSI, GA, Lg=4) TD−BF, (Perfect CSI, GA, L =5) g TD−BF, (Max−Min Criterion, L =1) g TD−BF, (Max−Min Criterion, Lg=5) NT = 1, NR = 1 0 5 10 15 Eb/N0 Figure 2.5: [dB] 20 25 30 Average BER of uncoded MIMOOFDM system with TDBF. Minimum average BER criterion (solid lines) and maxmin criterion (dashed lines), perfect CSI, NT = 2, NR = 1, Nc = 512, and IEEE 802.11n Channel Model B. For comparison the BERs for ideal FDBF 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 maxmin solution. In this case, the BER approaches the BER of the limiting loss compared to Lg = Nc case as Lg Lg = Nc = 512 increases. For example, for is less than 1.5 dB at Lg = 5 the performance BER = 10−3 . In Fig. 2.6, we investigate the eect of a niterate feedback channel on the average BER. In particular, we show the average BER as a function of the number of feedback bits B 41 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems 0.026 Finite−Rate Feedback, Lg = 1 Perfect CSI, Lg = 1 0.024 Finite−Rate Feedback, Lg = 2 Perfect CSI, L = 2 g Finite−Rate Feedback, Lg = 3 0.022 Perfect CSI, Lg = 3 0.02 BER 0.018 0.016 0.014 0.012 0.01 0 1 2 3 4 5 6 7 B Figure 2.6: Average BER of uncoded MIMOOFDM system with TDBF (average BER B per channel update. GA was used for CBFF NT = 2, NR = 1, Nc = 512, Eb /N0 = 10 dB, and IEEE 802.11n Channel criterion) vs. number of feedback bits optimization. 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, niterate 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 Therefore, smaller Lg Lg . are preferable if only few feedback bits can be aorded. In Fig. 2.7 we show the average BER for uncoded and coded (Rc = 1/2) transmission 42 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems TD−BF, (Perfect CSI, L = 2) g TD−BF, (L = 2, B = 0) g TD−BF, (Lg = 2, B = 1) −1 10 TD−BF, (L = 2, B = 3) g TD−BF, (L = 2, B = 7) g uncoded −2 10 coded (Rc = 1/2) −3 BER 10 −4 10 0 5 10 15 Eb/N0 Figure 2.7: [dB] 20 25 30 Average BER of uncoded and coded MIMOOFDM system with TDBF Lg = 2 is valid for all Perfect CSI (bold lines) and niterate feedback channel, NT = 2, NR = 1, (average BER criterion). GA was used for CBFF optimization and curves shown. Nc = 512, and IEEE 802.11n Channel Model B. with niterate feedback TDBF and TDBF with perfect CSI, respectively. CBFFs of length Lg = 2 Eb /N0 = 10 were used in all cases and the CBFF vector codebook was optimized for dB. Interestingly, for coded transmission signicantly fewer feedback bits are required to approach the performance of the perfect CSI case than for uncoded transmission. 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. 43 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems AMI Criterion, L g = 2 −1 10 Average BER Criterion, Lg = 2 AMI Criterion, L g = 4 Average BER Criterion, Lg = 4 AMI Criterion, L g = Nc Average BER Criterion, Lg = Nc −2 10 uncoded −3 BER 10 coded (Rc = 3/4) coded (Rc = 1/2) −4 10 −5 10 0 2 4 6 8 Eb/N0 10 [dB] 12 14 16 18 20 Figure 2.8: Average BER of uncoded and coded MIMOOFDM system employing TDBF 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 MIMOOFDM systems employing minimum average BER (dashed lines) and maximum AMI (solid lines) TDBF, respectively. We assume perfect CSI, NT = 3, Lg = 2, 4, and Nc (ideal FDBF). As one would expect, for uncoded transmission the minimum average BER criterion yields a 44 Chapter 2. TimeDomain Transmit Beamforming for MIMOOFDM Systems signicantly 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 MIMOOFDM systems. The CBFFs have been optimized for maximization of the AMI and minimization of the BER, respectively, and ecient algorithms for recursive calculation of the optimum CBFFs have been provided for both criteria. In contrast to existing FDBF schemes, for TDBF the number of complex feedback symbols to be conveyed to the transmitter is independent from the number of OFDM subcarriers. For the case of a niterate feedback channel a GVQ algorithm has been introduced for codebook design. Simulation results for the IEEE 802.11n Channel Model B have conrmed the excellent performance of TDBF and have shown that TDBF achieves a more favorable performance/feedback rate tradeo than FDBF. 45 Chapter 3 Cooperative AmplifyandForward Beamforming with Multiple MultiAntenna Relays 3.1 Introduction In the previous chapter, we have introduced a novel TDBF scheme for direct pointto 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, AFBF for wireless relay networks was considered in [35][44] and [73]. AF BF for networks with one singleantenna source and multiple singleantenna 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], AFBF for a network with a single relay and multiple antennas at the relay and the source was investigated and closedform solutions for the BF vector at the source and the AFBF 46 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays matrix at the relay were provided. Furthermore, in [43, 44], the performance of AFBF with multiple antennas at the source and one singleantenna 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 nonconvex optimization problems for the AFBF matrices at the relays and the BF vector at the source. We note that multiple multiantenna 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 AFBF for networks with one multiantenna source (e.g. a base station), multiple multiantenna relays, and one singleantenna 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 AFBF matrices at the relays are optimized under three dierent power constraints. In particular, we consider the cases of individual relay power constraints, a joint power constraint for all relays, and a joint sourcerelay power constraint. This chapter makes the following contributions: • For a given BF vector at the source, we nd the optimal AFBF matrices at the relays for each of the three considered power constraints. In particular, we provide closedform solutions for the AFBF matrices for the individual and joint relay power constraints, respectively. For the joint sourcerelay power constraint, we derive the direction of the AFBF 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 sourcerelay 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 nonconvex, it can be eciently 47 Chapter 3. Cooperative AFBF with Multiple MultiAntenna 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 ecient suboptimal methods for computation of the optimal source BF vector. • To implement the proposed AFBF scheme, the source node has to acquire the channel state information of all sourcerelay channels and the Euclidean norm of each relaydestination 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 sourcerelay and relaydestination 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 sourcerelay power constraint). • Our simulation results conrm that the proposed suboptimal optimization methods for the source BF vector achieve a closetooptimal performance. Furthermore, our results show that increasing the number of antennas at the source is highly benecial if the sourcerelay channels have a lower SNR than the relaydestination channels. In contrast, increasing the number of relays or the number of relay antennas is always benecial. 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 AFBF 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. 48 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays n1,1 B1 H1 d f1 g1 n1,M1 f NR nNR ,1 gNT r n2 H NR B NR nNR ,MNR Figure 3.1: Cooperative network with one multiantenna source, multiple multiantenna relays, and one singleantenna destination. source BF vector 1 ≤ i ≤ NR . g . ni,µ , 1 ≤ µ ≤ Mi , gi , 1 ≤ i ≤ NT , denotes the ith element of µth element of noise vector n1,i at relay i, is the 3.2 System Model and Optimization Problem We consider the downlink of a relay network with one source node, destination node. NR relays, and one A block diagram of the discretetime 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 1 ≤ i ≤ NR , i, 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. 49 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays 3.2.1 System Model In the rst transmission interval, the source transmits the elements of NT dimensional vector, x = g d, over its NT antennas, where g denotes the NT dimensional lated symbol taken from a scalar symbol alphabet signal received at the Mi (3.1) A with variance i, 1 ≤ i ≤ NR , antennas of relay BF vector, and σd2 d is the modu- E{|d|2 } = 1. can be modeled as q i = H i x + n1,i , where [H i ]µν , 1 ≤ µ ≤ Mi , 1 ≤ ν ≤ NT , source and antenna variance µ of relay i, The (3.2) is the channel gain between antenna and the elements of vector n1,i ν of the represent AWGN with σ12 . In the second transmission interval, relay i transmits the si = B i q i over antenna µ, 1 ≤ µ ≤ Mi , where Bi is an µth element of vector (3.3) Mi × Mi AFBF matrix. The received signal at the destination node is given by r= NR ∑ f Ti si + n2 , (3.4) i=1 where the µth element of µ, 1 ≤ µ ≤ Mi , of relay i Mi dimensional vector fi is the channel gain between antenna and the destination node, and n2 is AWGN with variance σ22 . 50 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays Combining (3.1)(3.4), the received signal at the destination node can be expressed as r= NR ∑ i=1 f Ti B i H i g d + NR ∑ f Ti B i n1,i + n2 = f T B D Hg d + f T B D n1 + n2 , i=1 with relaydestination channel vector BD diag{B 1 , . . . , B NR }, ( H TNR ]T , (3.5) ∑NR and relay noise vector i=1 f [f T1 . . . f NR ]T , Mi ) × NT n1 AFBF block diagonal matrix sourcerelay channel matrix H [H T1 . . . [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 |f T B D Hg|2 SINR = . ∥f T B D ∥22 σ12 + σ22 (3.6) The design problem considered in this chapter is the optimization of the BF vector the source and the AFBF matrices B i , 1 ≤ i ≤ NR , g at 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, B i , 1≤i≤NR s.t. SINR (3.7a) Power Constraints (3.7b) For the power constraints, we consider three dierent scenarios: Constraint I (Individual Power Constraints for Relays): If the transmit power of the 51 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays source and each relay is limited, the power constraints are given by ∥g∥22 ≤ P1 , (3.8a) ∥B i H i g∥22 + σ12 ∥B i ∥2F ≤ P2,i , where P1 and P2,i 1 ≤ i ≤ NR , (3.8b) 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 , NR ∑ ( (3.9a) ) ∥B i H i g∥22 + σ12 ∥B i ∥2F ≤ P2 , (3.9b) i=1 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 ∑ ( ) ∥B i H i g∥22 + σ12 ∥B i ∥2F ≤ P, (3.10) i=1 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 dierent constraints 52 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays in (3.8)(3.10). 3.3 Optimal AFBF Matrices It is convenient to solve problem (3.7) in two steps. In Subsections 3.3.13.3.3, we determine the optimal AFBF matrices B i , 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 dene vector H i g , 1 ≤ i ≤ NR . ui 3.3.1 AFBF with Individual Power Constraints for Relays Combining (3.7) and (3.8) we obtain the optimization problem ∑NR max B i ,1≤i≤NR s.t. σ12 ∑NR i=1 T i=1 f i B i ui 2 (3.11a) ∗ 2 f Ti B i B H i f i + σ2 H 2 2 uH i B i B i ui + σ1 ||B i ||F ≤ P2,i , 1 ≤ i ≤ NR , where we have ignored the source power constraint (3.8a) since Next, we introduce the denitions Qi uTi ⊗ I Mi . wi u∗i ⊗ f ∗i , bi g (3.11b) is assumed to be xed. vec{B i }, T i I Mi ⊗ f Ti , and With these denitions, we can rewrite problem (3.11) in equivalent form as ∑NR max bi ,1≤i≤NR s.t. ∑NR i=1 wH i bi 2 H 2 bH i T i T i bi + σ2 ) ( H 2 2 I bi ≤ P2,i , bH Q Q + σ i 1 Mi i i σ12 (3.12a) i=1 1 ≤ i ≤ NR . (3.12b) 53 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays For the next step, we introduce matrix JH i Ji as 2 J i , which is obtained from matrix QH i Qi +σ1 I Mi2 via Cholesky decomposition, and vector √ P˜2,i xi , yi where P˜2,i = ||y i ||22 and xi yi J i bi . yi Vector can be represented is a unit norm vector. Now, we can restate problem (3.12) as ∑NR √ ˜ −1 P2,i wH i J i xi i=1 ( H −1 H x σ12 P˜2,i J −H i T i T iJ i + i=1 i max ∑NR P˜2,i ,xi ,1≤i≤NR ||xi ||22 = 1, P˜2,i ≤ P2,i , s.t. Assuming that the powers 1 ≤ i ≤ NR , to xi P˜2,i , 1 ≤ i ≤ NR , 2 σ22 NR ) I Mi2 xi (3.13a) 1 ≤ i ≤ NR . (3.13b) are xed, we can nd direction vectors xi , that maximize (3.13a) by dierentiating the objective function with respect and by accounting for the constraint ||x||22 = 1 by using Lagrange multipliers. After some algebraic manipulations, this leads to ( xi = αi where αi and βi H −1 J −H i T i T iJ i + βi I Mi2 )−1 J −H i wi , (3.14) 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 denitions of ( xi = αi J i ( J i, T i, TH i Ti + and wi in (3.14), we obtain βi (QH i Qi + σ12 I Mi2 ) )−1 wi = αi J i (I Mi ⊗ f Ti )H (I Mi ⊗ f Ti ) + βi (uTi ⊗ I Mi )H (uTi ⊗ I Mi ) + σ12 βi I Mi2 ))−1 ∗ ( ( (ui ⊗ f ∗i ) , = αi J i (I Mi ⊗ f ∗i f Ti ) + βi (u∗i uTi + σ12 I Mi ) ⊗ I Mi where we have used the identity (A ⊗ B)(C ⊗ D) = AC ⊗ BD simplied by introducing the Kronecker sum [77]. xi )−1 wi (3.15) can be further (A ⊗ I M ) + (I M ⊗ B) = A ⊕ B in (3.15) and 54 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays exploiting the relation [78] (M ⊕ N )−1 = M ∑ M ∑ ¯i⊗n ¯ j )H (mi ⊗ nj )(m λi (M ) + λj (N ) i=1 j=1 where NH, ¯ i, mi , ni , m and ¯i n denote the eigenvectors of M ×M , (3.16) matrices M, N, MH, and respectively. This leads to ( )−1 ∗ xi = αi J i βi (u∗i uTi + σ12 I Mi ) ⊕ f ∗i f Ti (ui ⊗ f ∗i ) ( ∗ )( ∗ )H ui ui f ∗i f ∗i ⊗ ⊗ ||ui ||2 ||f i ||2 ||ui ||2 ||f i ||2 = αi J i (u∗i ⊗ f ∗i ) 2 2 ||f i ||2 + βi (||ui ||2 + σ12 ) αi = J i (u∗i ⊗ f ∗i ) . 2 2 2 ||f i ||2 + βi (||ui ||2 + σ1 ) Exploiting (3.17) along with bi = √ P˜2,i J −1 xi , i (3.17) we obtain for the AFBF matrix Bi the expression B i = ci f ∗i uH i , where complex scalar ci 1 ≤ i ≤ NR , (3.18) has to be optimized taking into account the perrelay power constraint. Eq. (3.18) reveals that under a perrelay power constraint eigenbeamforming with respect to the sourcerelay and the relaysource channel is optimal. For the special case where the source and all relays have only a single antenna, i.e., fi and ui are scalars, this result has already been derived in [39]. Substituting (3.18) into problem (3.11), it is obvious that all phase θ to achieve the maximum SINR, i.e., ci = |ci |ejθ . ci have to have the same The resulting optimization 55 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays problem is given by (∑ )2 |ci |∥f i ∥2 ∥ui ∥2 ∑ R 2 2 2 σ12 N i=1 |ci | ∥f i ∥2 + σ2 √ P2,i |ci | ≤ , 1 ≤ i ≤ NR . ∥ui ∥22 + σ12 NR i=1 max |ci | s.t. (3.19a) (3.19b) Problem (3.19) is equivalent to the power allocation problem for relaying with multiple singleantenna relays, which was solved in [39]. For completeness, we provide the solution here using the notation of this chapter. Dene √ ∥ui ∥2 ∥ui ∥22 + σ12 √ ϕi = P2,i ∥f i ∥2 and sort of ϕi in descending order (1, . . . , NR ). ϕτ1 ≥ ϕτ2 ≥ · · · ≥ ϕτNR , where (3.20) (τ1 , . . . , τNR ) is an ordering The optimal solution to problem (3.19) is given by [39] √ ci = P2,i ejθ , ∥ui ∥22 +σ12 i = τ1 , . . . , τj , ∥f i ∥2 i = τj+1 , . . . , τNR , κj ∥ui ∥2 ejθ , where κj (3.21) ∑ P2,τm ∥f τm ∥22 σ22 + σ12 jm=1 ∥u 2 2 τm ∥2 +σ1 √ ∑ P2,τm ∥f τm ∥2 ∥uτm ∥2 √ σ22 jm=1 2 2 (3.22) ∥uτm ∥2 +σ1 and j is the smallest index such that For a given source BF vector g, κj < ϕ−1 τj+1 . (3.18) and (3.21) fully specify the optimal AFBF matrices for multiple multiantenna relays with individual relay power constraints. 56 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays 3.3.2 AFBF with Joint Power Constraint for Relays Considering (3.13) and taking into account the dierences between constraints (3.8) and (3.9), the optimization problem for the joint relay power constraint can be rewritten as ( P˜2 ,y ||y||22 ≤ P2 , s.t. where ||y||22 = P˜2 ≤ P2 , y diag{T 1 , . . . , T NR }, achieved for and P˜2 = P2 , y H J −H wwH J −1 y ) y H σ12 J −H T H T J −1 + σ22 /P˜2 I M y max (3.23b) diag{J 1 , . . . , J NR }, T Jb, b = [bT1 . . . bTNR ]T , J ∑NR 2 i=1 Mi . We observe from M (3.23a) (3.23a) that the maximum is 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 where c σ12 J −H T H T J −1 is a complex scaling factor. b = J −1 y σ2 + 2 IM P2 )−1 J −H w , (3.24) Using similar operations as in (3.15)(3.17) and we obtain for the optimal BF matrix for AF relays with a joint power constraint B i = c si f ∗i uH i , 1 ≤ i ≤ NR , (3.25) where si c P2 , 1 ≤ i ≤ NR , P2 ∥f i ∥22 σ12 + ∥ui ∥22 σ22 + σ12 σ22 (N )−1/2 R ∑ P2 ∥f i ∥22 ∥ui ∥22 (∥ui ∥22 + σ12 ) ejθ 2 2 2 2 2 2 2 i=1 (P2 ∥f i ∥2 σ1 + ∥ui ∥2 σ2 + σ1 σ2 ) (3.26) (3.27) 57 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays with arbitrary phase θ. We note that the proposed solution for the AFBF matrix includes the result in [35] as a special case if the source and all relays have only a single antenna. 3.3.3 AFBF with Joint Power Constraint for Source and Relays For the joint sourcerelay power constraint, problem (3.7a), (3.10) can be rewritten as max P1 ,g,B i ,1≤i≤NR s.t. |f T B D u|2 ∥f T B D ∥22 σ12 + σ22 (3.28a) ∥g∥22 ≤ P1 (3.28b) NR ∑ ∥B i g i ∥22 + σ12 ∥B i ∥2F ≤ P − P1 . (3.28c) i=1 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 AFBF 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 s.t. NR ∑ i=1 P1 (P − P1 ) Γ1,i Γ2,i P1 Γ1,i + (P − P1 ) Γ2,i + 1 (3.29a) 0 ≤ P1 ≤ P , (3.29b) where we have introduced the equivalent sourcerelay SNR the equivalent relaydestination SNR Γ2,i ||f i ||2 /σ22 . Γ1,i ||ui ||2 /(σ12 ||g||2 ) and It is easy to show that the second derivative of the objective function (SINR) in (3.29a) with respect to P1 is always negative: R ∑ ∂ 2 SINR 2Γ1,i Γ2,i (P Γ1,i + 1) (P Γ2,i + 1) = − < 0 , when 0 ≤ P1 ≤ P. 3 ∂P12 [P Γ + (P − P )Γ + 1] 1 1,i 1 2,i i=1 N (3.30) 58 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays Therefore, the objective function is concave and the optimum power allocation can be obtained with a simple bisectional search method based on [1] R ∂SINR ∑ Γ1,i Γ2,i [−P12 (Γ1,i − Γ2,i ) − 2 (P Γ2,i + 1) P1 + P (P Γ2,i + 1)] = =0. 2 ∂P1 [P Γ + (P − P )Γ + 1] 1 1,i 1 2,i i=1 N (3.31) For the special case when there is only one relay in the cooperative network, a closedform solution for the optimal P1 is obtained as √ (P Γ1,1 +1)(P Γ2,1 +1)+(P Γ2,1 +1) Γ2,1 −Γ1,1 P1 = P/2 √ (P Γ1,1 +1)(P Γ2,1 +1)−(P Γ2,1 +1) Γ1,1 −Γ2,1 , if Γ1,1 < Γ2,1 , , if 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 sourcerelay and the relaydestination channels by allocating more power to the weaker channel. This result is intuitively pleasing since the performance of twohop links is limited by the SNR of the weaker link. 3.3.4 Comparison of the Solutions for the Dierent Constraints A comparison of (3.18) and (3.25) shows that the optimal AFBF matrices for all power constraints can be expressed as and ci = c, 1 ≤ i ≤ NR , B i = ci si f ∗i uH i , 1 ≤ i ≤ NR , where si = 1, 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 sourcerelay and the relaydestination 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 sourcerelay power constraints the relays and the 59 Chapter 3. Cooperative AFBF with Multiple MultiAntenna 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 AFBF matrices obtained in Section 3.3 constitute the solution to problem (3.7). In Subsections 3.4.13.4.3, we propose optimal and suboptimal solutions for the BF vector g for the case NT > 1 assuming that the optimal AFBF matrices obtained in Subsections 3.3.13.3.3 are adopted at the relays, respectively. In Subsection 3.4.4, we discuss the feedback requirements of the proposed AFBF scheme. 3.4.1 AFBF 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 dicult and least tractable problem. Considering (3.19) and using the optimal g is the solution to the following optimization problem (∑ max SINR(g) = s.t. ||g||22 = P1 , g where we have made the dependence of on g ui = H i g , ci )2 |ci (g)|∥f i ∥2 ∥H i g∥2 ∑ R 2 2 2 σ12 N i=1 |ci (g)| ∥f i ∥2 + σ2 NR i=1 (3.33a) (3.33b) on g explicit, cf. (3.21). Since SINR(g) depends 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 60 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays of g. 1) Ad hoc Method: One suboptimal solution is to perform eigenbeamforming at the source with respect to the average sourcerelay channel. This means we choose dominant eigenvector of matrix 2) Gradient Method: ∑NR i=1 HH i Hi and normalize it to g as the ||g||22 = P1 . 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. SINR(g) of in (3.33a) with respect to [ℜ{g}T ℑ{g}T ]T g¯ g Since the derivative of is cumbersome, we express the SINR as a function and use a gradient estimate given by [79] 1[ ∇g¯ SINR(¯ g) = (SINR(¯ g + δe1 ) − SINR(¯ g − δe1 )) 2δ . . . (SINR(¯ g + δe2NT ) − SINR(¯ g − δe2NT )) where δ is a small positive constant and ei has a one in position i, ]T 1 ≤ i ≤ 2NT , (3.34) and zeros in all other positions (ith unit norm vector). This leads to the gradient algorithm given in Table 3.1, where gˆ k g¯ k and gradk ∇g¯ SNR(ˆ g k ). 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 4NT vectors g¯ k ± δei , 1 ≤ i ≤ 2NT , ci , 1 ≤ i ≤ NR , have to be computed for all using (3.19). 61 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays gˆ for individual and joint gradk depend on the power −5 small value (e.g. ϵ = 10 ). k Table 3.1: Gradient algorithm for calculation of source BF vector relay power constraints. The denitions of gˆ and the gradient ϵ has a ak is the adaptation step size chosen through a backtracking constraint, cf. Section 3.4. Termination constant denotes the iteration index and line search [1]. k=0 1 Let and initialize vector 2 Update the BF vector: gˆ 0 with solution of ad hoc method g˜ k+1 = gˆ k + ak gradk 3 Rescale the BF vector: 4 If 5 gˆ k+1 √ gˆ k+1 = P1 g˜ k+1 /||˜ g k+1 ||2 H 1 − |ˆ g k+1 gˆ k |/P1 < ϵ, goto Step 5, otherwise increment counter k and goto Step 2 is the desired BF vector 3.4.2 AFBF 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 s.t. P2 ∥f i ∥22 ∥H i g∥22 P2 ∥f i ∥22 σ12 + ∥H i g∥22 σ22 + σ12 σ22 (3.35a) ||g||22 ≤ P1 . (3.35b) For the following it is convenient to rewrite the objective function in (3.35a) as SINR = where ei NR NR ∑ ei P2 ∑ 2 ∥f ∥ − , i 2 2 HA g σ2 i=1 g i i=1 σ12 Pσ22 ∥f i ∥22 (P2 ∥f i ∥22 + σ22 ) independent of 2 g. and Ai σ22 H H i Hi + (3.36) σ12 (P2 ∥f i ∥22 P1 + σ22 )I NT are The rst term in (3.36) is the SINR achieved with beamforming in pointtopoint 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 AFBF with distributed relays and not BF for the relaydestination channel with colocated antennas. Consequently, maximization problem 62 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays (3.35) is equivalent to the following minimization problem min g s.t. For the special case of the optimal g NR = 1 NR ∑ i=1 ei g H Ai g (3.37a) ||g||22 ≤ P1 . relay having M1 (3.37b) antennas it is obvious from (3.37) that is simply the dominant eigenvector of matrix HH 1 H 1. This result is not new and has already been mentioned in [38]. However, here we are interested in the more dicult case of multiple relays, for which a solution has not been provided before. We note that for NR > 1 (3.37) is a dicult nonconvex optimization problem. In the following, we provide the optimal and three suboptimal solutions to problem (3.37) which dier in their complexity and performance. 1) Transformation Method: Problem (3.37) can be transformed into the following poly- nomial programming problem NR ∑ min ¯ , ti , 1≤i≤NR g ti (3.38a) ℜ{Ai } −ℑ{Ai } ti g¯ T g¯ ≥ ei , ℑ{Ai } ℜ{Ai } i=1 s.t. 1 ≤ i ≤ NR g¯ T g¯ ≤ P1 , where g¯ = [ℜ{g}T ℑ{g}T ]T . still nonconvex, for small (3.38c) Although the polynomial programming problem in (3.38) is NT and small NR (e.g., NT = 2 and by using the GloptiPoly or SOSTOOLS software [75, 76]. NR = 2), it can be solved In this case, we can indeed obtain the globally optimal solution to the AFBF problem. However, for large NR (3.38b) NT and nding the globally optimal solution with the aforementioned software tools does not 63 Chapter 3. Cooperative AFBF with Multiple MultiAntenna 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 relaydestination channels have a much SNR than the sourcerelay channels, i.e., P2 ||f i ||22 σ22 to see from (3.35a) that the optimal BF vector ∑NR i=1 HH i Hi normalized to ||g||22 = P1 . g ≫ 1 ||H i g||22 , σ12 1 ≤ i ≤ NR , higher it is easy is the dominant eigenvector of matrix 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 fullled. We note that for the case where the relaydestination channels have a much lower SNR than the sourcerelay channels, the objective function in (3.35a) becomes independent of g 3) Gradient Method: and optimization of the BF vector at the source is not necessary. 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 [∑ NR ˆ k )2 gH k Ai g i=1 ei Ai /(ˆ ] gˆ k . gˆ k gk and gradk 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 good" suboptimal strategy to achieving a high SINR is to maximize the minimum value of g H Ai g , 1 ≤ i ≤ NR . This results in a new (relaxed) optimization problem: max g min i, 1≤i≤NR s.t. 1 H g Ai g ei (3.39a) ||g||22 ≤ P1 . (3.39b) 64 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays The maxmin problem in (3.39) can be easily relaxed to a semidenite programming (SDP) problem and eciently solved using SeDuMi in Matlab [71]. 3.4.3 AFBF 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 s.t. (P − P1 )∥f i ∥22 ∥H i g∥22 (P − P1 )∥f i ∥22 σ12 + ∥H i g∥22 σ22 + σ12 σ22 (3.40a) ||g||22 ≤ P1 (3.40b) 0 ≤ P1 ≤ P . (3.40c) Clearly, this nonconvex problem is in general more dicult 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 optimal direction of matrix g/||g||2 HH 1 H 1. 2 λmax (H H 1 H 1 )/σ1 NR = 1 relay, it can be observed from (3.40a) that the of the source BF vector is given by the dominant eigenvector The corresponding optimal power and Γ2,1 = ||f 1 ||22 /σ22 . P1 is given by (3.32), where For the general case of NR > 1, Γ1,1 = a closedform 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- 65 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays nomial programming problem NR ∑ max ¯ , P1 , ti ,1≤i≤NR g s.t. where again ti (3.41a) ( ) ℜ{H i } −ℑ{H i } (P − P1 )∥f i ∥22 − ti σ22 g¯ T g¯ ℑ{H i } ℜ{H i } ( ) ≥ ti σ12 (P − P1 )∥f i ∥22 − σ22 , 1 ≤ i ≤ NR (3.41b) g¯ T g¯ ≤ P1 (3.41c) 0 ≤ P1 ≤ P , (3.41d) i=1 g¯ = [ℜ{g}T ℑ{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 nonconvexity, 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: ∑NR i=1 HH i Hi As an ad hoc solution, we may adopt the dominant eigenvector of 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 smallscale and largescale 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. 66 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays Table 3.2: Gradient algorithm for calculation of source BF vector g and power allocation for joint sourcerelay power constraint. Termination constant ϵ has a small value (e.g. ϵ = 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 g 0 , 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 = g k + ak 3 NR H 2 i=1 ei Ai /(g k Ai g k ) ] gk Rescale the BF vector: g k+1 = 4 [∑ √ P1,k g˜ k+1 /||˜ g k+1 ||2 Find the optimal power allocation P1,k+1 and P2,k+1 = P − P1,k+1 g k+1 for based on (3.31) using the bisectional search method and compute the SNRk+1 |SNRk+1 − SNRk | < ϵ, corresponding 5 If goto Step 6, otherwise increment counter k and goto Step 2 6 g k+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 3.4.13.4.3 show that for the special case of relay, the optimal source BF vector (note that for NR = 1 g. can be found in closed form for all three constraints the individual power constraint is identical to the joint power constraint for the relays). to obtain g NR = 1 In contrast for NR > 1 numerical methods have to be used 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: the source BF vector g. We rst consider the feedback necessary for computation of 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 67 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays relay channels channel f i. Subsequently, relay knowledge of g H i , 1 ≤ i ≤ NR , Hi and and each relay can estimate its own relaydestination i feeds back real number ||f i ||22 , 1 ≤ i ≤ NR , ||f i ||22 to the source. With the the source can compute the optimal BF vector 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 AFBF matrix can be expressed as B i = ci si f ∗i uH i , where sourcerelay and all relaydestination links and destination channels of relay constraint. i si ci depends on the channel gains of all depends on the sourcerelay and relay only. The specic values of ci and si depend on the power 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 (eective) sourcerelay channel si , ui = H i g . while the source can compute particular form of ci ci . Thus, relay 1 ≤ i ≤ NR , and ui and can compute The additional feedback requirements depend on the and are slightly dierent for the three considered power constraints. For the individual relay power constraints, one complex number i knows f i ci ci depends on i and the source has to feedback to each relay, cf. (3.18). For the joint relay power constraint, and the source has to broadcast only one complex number c ci = c, to all relays, cf. (3.27). For the joint sourcerelay power constraint, the source has to broadcast complex constant c and the power P2 (which aects si in this case) to all relays. Overall the feedback requirements for the proposed AFBF 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 crosslayer resource allocation. 68 Chapter 3. Cooperative AFBF with Multiple MultiAntenna 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 AFBF. For all mutual information and SINR results presented in this section we assume a cooperative network with σ12 = σ22 = 0.1. For the individual relay power constraints, the joint relay power constraint, and the joint sourcerelay power constraint, we use (P1 P2 = 1), and P = 2, = 1, P2,i = 1/NR , 1 ≤ i ≤ NR ), (P1 = 1, 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 pathloss exponent of over 100, 000 3 is assumed and all results were averaged independent realizations of the fading channels unless specied otherwise. The optimal BF vectors at the source and the optimal AFBF 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 multirelay BF, we compare the performance 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 optimal AFBF 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 optimization methods. For this purpose, we show in Figs. 3.33.5 cumulative distribution functions 69 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays (a) 1/4 (b) (c) d source 1/4 2−d 1/4 destination (d) 1/4 (e) 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 xaxis. Since the optimal source beamforming vectors can be computed with the proposed transformation methods only for joint relay and joint sourcerelay 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. Subsequently, 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 dierent source BF vector optimization methods proposed for the joint relay power constraint. There are source and one relay at locations (a) and (e), respectively. the cases M1 = M2 = 1 and M1 = 2, M2 = 3, NT = 2 antennas at the For the relays we consider 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 suered by the relaxation method and the ad hoc method is larger for M2 = 1 than for M1 = 2, M2 = 3. M1 = Relay selection suers from a signicant loss in 70 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays 1 0.9 0.8 0.7 M1 = M2 = 1 CDF 0.6 0.5 M 1 = 2, M 2 = 3 0.4 0.3 0.2 Global Optimum Gradient Method Ad hoc Method Relaxation Method Relay Selection (Per−Relay PC) 0.1 0 2 4 6 8 10 12 14 SINR [dB] Figure 3.3: CDF of the instantaneous SINR for AFBF with (PC) joint relay power constraint and one relay located at (a) and (e), respectively. Results for dierent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A pathloss 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 sourcerelay power constraint for and NR singleantenna relays for d = 1. For NR = 2 NT = 2 antennas at the source 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 71 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays 1 Global Optimum Gradient Method w/ Random Initialization Gradient Method Ad hoc Method Relay Selection (Joint Source−Relay PC) 0.9 0.8 0.7 NR = 10 CDF 0.6 0.5 NR = 5 0.4 NR = 2 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 SINR [dB] Figure 3.4: CDF of the instantaneous SINR for AFBF with constraint (PC) and NR joint sourcerelay power relays. Results for dierent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A pathloss exponent of 3, NR = 2, NT = 2, and d = 1 are assumed. The relays are located at (a) and (e) NR = 5, and (a)(e) with 2 relays at each location for NR = 10. for (a)(e) for that additional random initializations cannot signicantly improve the performance of the gradient method, which suggests that the gradient method initialized with the solution of the ad hoc method is closetooptimal 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 suered by relay selection increases 72 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays 1 0.9 0.8 NT = 2 0.7 CDF 0.6 NT = 10 0.5 0.4 0.3 NT = 5 0.2 Gradient Method Gradient Method w/ Random Initialization Ad hoc Method Relay Selection (Per−Relay PC) 0.1 0 5 6 7 8 9 10 11 12 13 14 SINR [dB] Figure 3.5: CDF of the instantaneous SINR for AFBF with individual relay power constraints (PCs) and NR = 5 singleantenna relays at locations (a)(e). Results for dierent optimization methods for the source BF vector for multiple relays are shown and compared with relay selection. A pathloss 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 dierent source BF vector optimization methods for antenna relays located at positions (a)(e) in Fig. 3.2 for d = 1. NR = 5 single For the gradient method, the performance gain achievable with additional random initializations is negligible even for NT = 10 source antennas. However, the performance loss suered by the ad hoc method 73 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays compared to the gradient method increases with increasing number of source antennas. For example, for a CDF value of 0.5, the performance dierence 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 AFBF with dierent 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 multirelay AFBF 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 suered by relay selection is between 1 and 2 dB. Increasing the number of source antennas is benecial for both constraints unless the relays are located close to the source. In the latter case, the relaydestination channel is the performance bottleneck and more source antennas cannot improve performance. If only available, BF is not used at the source (i.e., g/||g||2 = 1). For NT = 1 NT = 2 source antenna is and NT = 5 source antennas the gradient methods achieve the highest SINRs in both gures. Fig. 3.6 shows that while the maxmin relaxation method outperforms the ad hoc method for small the ad hoc method is preferable for large d (e.g. d ≥ 1.4 for NT = 5). d, In the latter case, the SINR of the sourcerelay channels is much lower than that of the relaydestination 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 antennas. In Fig. 3.8, we show the average SINR vs. distance d for AFBF with NT = 5 source antennas for the joint sourcerelay power constraint. For the case with two relays (in positions (a) and (e)) increasing the number of relay antennas from M1 = M2 = 1 to 74 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays 12 10 8 Average SINR [dB] 6 4 2 NT = 1 BF N = 2 (no relay) T BF NT = 5 (no relay) 0 AF−BF NT = 1, M1 = 2, M2 = 3 AF−BF NT = 2, M1 = 2, M2 = 3 AF−BF NT = 5, M1 = 2, M2 = 3 −2 −4 Gradient Method Ad hoc Method Relaxation Relay Selection (Per−Relay PC) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d d for AFBF with joint relay power constraint NT at the source. A pathloss exponent of 3 is For comparison the SNR without relaying for a source transmit power of P = 2 Figure 3.6: Average SINR vs. distance (PC) and dierent numbers of antennas assumed. and the SINR for relay selection are also shown. M1 = 2, M2 = 3 signicantly 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. case the AFBF matrices have latter case they have only This can be explained by the fact that in the former 9 + 4 = 13 elements that can be optimized whereas in the 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 singleantenna 75 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays 12 10 8 Average SINR [dB] 6 4 2 N =1 T BF NT = 2 (no relay) BF NT = 5 (no relay) 0 AF−BF NT = 1, M1 = 2, M2 = 3 AF−BF NT = 2, M1 = 2, M2 = 3 AF−BF NT = 5, M1 = 2, M2 = 3 −2 Gradient Method Ad hoc Method Relay Selection (Per−Relay PC) −4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d d for AFBF with individual relay power constraints NT at the source. A pathloss exponent of 3 is For comparison the SNR without relaying for a source transmit power of P = 2 Figure 3.7: Average SINR vs. distance (PCs) and dierent numbers of antennas assumed. and the SINR for relay selection are also shown. relays than for multiantenna relays. 3.5.3 Impact of Power Constraints on Performance In Fig. 3.9, we compare the average mutual information of AFBF for the three considered power constraints and dierent 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 76 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays 12 11 10 9 Average SINR [dB] 8 7 6 5 BF N = 5 (no relay) T AF−BF N = 5, M = 1, M = 1 T 4 1 2 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) 3 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d Figure 3.8: Average SINR vs. distance straint (PC) d for AFBF with joint sourcerelay power con- and dierent numbers of relays and dierent numbers of relay antennas. A pathloss 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 dierence between the joint relay power constraint and the individual relay power constraints is comparatively small. In contrast, the joint sourcerelay power constraint can yield signicant performance gains if the relays are close to the source or close to the destination, respectively, by exibly allocating more or less 77 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays Joint Source−Relay PC Joint Relay PC Individual Relay PC Relay Selection (Per−Relay PC) Relay Selection (Joint Source−Relay PC) N = 5, M = 2, M = 3 2 T 1 2 NT = 2, M1 = M2 = M3 = M4 = M5 = 1 BF with NT = 5 (no relay) BF with N = 2 (no relay) T AMI (bit/s/Hz) 1.5 1 0.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d Figure 3.9: Average mutual information (AMI) in (bits/s/Hz) vs. distance d for two dierent network setups and dierent power constraints (PCs). The relays are in locations (a) and (e) for NR = 2 and (a)(e) for NR = 5. g. A for computation of the source BF vector The proposed gradient methods are used pathloss 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 16ary quadrature amplitude modulation (16QAM) for the three considered power constraints. For comparison we also show the BER for direct transmission with quaternary phaseshift 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 78 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays Joint Source−Relay PC Joint Relay PC Individual Relay PC Relay Selection (Per−Relay PC) Relay Selection (Joint Source−Relay PC) N = 5, M = 2, M = 3 −1 10 T 1 2 NT = 2, M1 = M2 = M3 = M4 = M5 = 1 QPSK BF with NT = 5 (no relay) QPSK BF with N = 2 (no relay) −2 10 T −3 BER 10 −4 10 0 2 4 6 8 10 12 14 16 σd2 /σ12 Figure 3.10: Average BER vs. σd2 /σn2 for two dierent network setups and dierent power constraints. The relays are in locations (a) and (e) for NR = 2 The proposed gradient methods are used for computation of the source BF d = 1 are power P = 2. pathloss exponent of 3 and QPSK, source transmit NR = 5. vector g . A and (a)(e) for assumed. AFBF: 16QAM. Direct transmission: of source transmit antennas, AFBF yields signicant performance gains in termsof the achievable BER compared to direct transmission and relay selection. Thereby, the achievable BER is the lower, the less restrictive the power constraints are, i.e., for a given SINR, the joint sourcerelay power constraint yields a lower BER than the joint relay power constraint and the joint relay power constraint yields a lower BER than the individual relay 79 Chapter 3. Cooperative AFBF with Multiple MultiAntenna Relays power constraints. 3.6 Conclusions In this chapter, we have considered AFBF for cooperative networks with one multi antenna source, multiple multiantenna relays, and one singleantenna destination for three dierent 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 sourcerelay and relaydestination channels for implementation of the optimal AFBF 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 characterized the optimal AFBF matrices for all three constraints. Furthermore, for small scale networks with joint relay or joint sourcerelay power constraints the optimal source BF vector can be found using polynomial programming. For large scale networks and networks with individual relay power constraints ecient suboptimal ad hoc and gradient methods for optimization of the source BF vectors have been provided. Simulation results conrm the closetooptimal performance of the proposed gradient methods and show that the relative performance of the three considered power constraints signicantly depends on the network topology. Furthermore, our results show that increasing the number of antennas at the source is particularly benecial 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 benecial regardless of the location of the relays. 80 Chapter 4 Cooperative FilterandForward Beamforming for FrequencySelective Channels with Multiple MultiAntenna 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 cooperative networks in frequencyselective channels. Particularly, in this chapter, we consider oneway cooperative networks with one singleantenna source, one singleantenna destination, and multiple multiantenna relay nodes. We assume singlecarrier transmission and frequencyselective channels. Relaying schemes for singlecarrier transmission over frequencyselective channels have received little attention in the literature so far with [50, 81] being two notable exceptions. Specically, a cooperative lterandforward (FF) BF technique was proposed and optimized 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 81 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays is available [50]. We note that FF relaying for frequencyat channels was also considered in [46]. For the frequencyselective case, distributed spacetime block coding at the relays and equalization at the destination has been proposed in [81]. Distributed spacetime coding does not require full CSI but has a worse performance than FFBF. In this chapter, we investigate cooperative FFBF for frequencyselective channels for the case where the destination node has either (1) a simple slicer without equalization or (2) enough processing power to perform lowcomplexity 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 multiantenna relays and equalization at the destination. This chapter makes the following contributions: • For the simple slicer case, we optimize the FFBF 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 closedform solution for the optimal FIR FFBF matrix lters at the relays. • For the LE/DFE case, we assume FIR and IIR lters at the relays. We optimize FFBF for maximization of the SINR at the output of LE and DFE as well as an idealized matched lter (MF) receiver ignoring any intersymbol 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 achievable with more complex equalization schemes such as maximum likelihood sequence estimation (MLSE). • For IIR FFBF with equalization, we show that the frequency response vector of the optimal FFBF lters can be decomposed into a unitnorm direction vector and a scalar power allocation factor across frequencies. We provide a unied closedform 82 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays solution for the direction vector valid for all three considered equalization receiver structures and an ecient numerical method with guaranteed convergence for the power allocation. • For FIR FFBF with equalization, we show that the FFBF lter optimization problem is related to a dicult mathematical problem for which an exact solution in closed form does not seem to exist. Therefore, we provide an ecient numerical method for recursive calculation of the optimum FIR FFBF lters. • Our simulation results show that (1) the performance of FFBF without equalization at the destination crucially depends on the slicer decision delay, (2) with the same FFBF lter length, the addition of simple LE and DFE equalizers at the destination node yields large performance gains compared to FFBF with a slicer, (3) if long FIR FFBF lters are employed, the simple slicer receiver with optimized decision delay closely approaches the same performance as equalizers, (4) relatively short FIR FFBF lters with equalization suce to closely approach the performance of IIR FFBF lters, (5) the gap between FFBF 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 FFBF 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. 83 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 4.2 System Model We consider a relay network with one singleantenna source node, and one singleantenna destination node. NR multiantenna relays, A block diagram of the discretetime 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 (FFBF 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. ith In Fig. 4.1, the discretetime CIRs between the source and the relay, gi,z [k], 0 ≤ k ≤ Lg − 1, hi,z [k], 0 ≤ k ≤ Lh − 1, and between the ith antenna of relay z antenna of the and the destination, contain the combined eects of transmit pulse shaping, the continuoustime channel, receive ltering, and sampling. Here, Lg Lh and lengths of the sourcerelay and the relaydestination CIRs, respectively. we assume that relay g z [k] z z th has [g1,z [k] . . . gMz ,z [k]]T . Mz antennas and dene hz [k] denote the Furthermore, [h1,z [k] . . . hMz ,z [k]]T and In the following, we describe the processing performed at the relays and the destination in detail. 4.2.1 FFBF at Relays The signal received at the ith antenna, i = 1, . . . , Mz , of the z th 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) 84 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays Relay 1 g 1[k] n1,z [k] h1[k] Relay z g z [k] s[k] hz [k] EQ/Slicer Az [k] sˆ[k] n0[k] g NR [k] nMz ,z [k] hNR [k] Relay NR Figure 4.1: Cooperative network with one singleantenna source, multiple multiantenna relay nodes, and one singleantenna 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. with variance the z th σs2 symbols taken from a scalar symbol alphabet E{|s[k]|2 }, relay with variance and σn2 ni,z [k] Az [k], −ql ≤ k ≤ qu , matrix lters where La antenna, ql → ∞ and E{|ni,z [k]|2 }. with elements qu → ∞ aji,z [k] on row of the z th j relay, z are denoted by Mz ×Mz and column i. For IIR FFBF and for FIR FFBF lters is the FIR FFBF matrix lter length. j = 1, . . . , Mz , such as PSK or QAM denotes the AWGN at the ith receive antenna of The FFBF matrix lter impulse response coecients of relay matrix A ql = 0 and qu = La − 1, The signal transmitted by the z = 1, . . . , NR , j th during the second time interval can be expressed as tj,z [k] = Mz ∑ i=1 aji,z [k] ∗ yi,z [k] = Mz ∑ i=1 Mz ∑ aji,z [k] ∗ gi,z [k] ∗ s[k]+ aji,z [k] ∗ ni,z [k] . (4.2) i=1 85 Chapter 4. Cooperative FFBF with Multiple MultiAntenna 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 ∑ Mz ∑ hj,z [k] ∗ tj,z [k] + n0 [k] = heq [k] ∗ s[k] + n[k] , (4.3) z=1 j=1 where n0 [k] is AWGN with variance E{|n0 [k]|2 }. σv2 source and destination and the eective noise heq [k] NR ∑ Mz ∑ hj,z [k] ∗ z=1 j=1 and n[k] NR ∑ Mz ∑ hj,z [k] ∗ z=1 j=1 respectively. Note that Az [k]. n[k] is colored n[k] Mz ∑ The equivalent CIR heq [k] between are given by aji,z [k] ∗ gi,z [k] , (4.4) i=1 Mz ∑ aji,z [k] ∗ ni,z [k] + n0 [k] , (4.5) i=1 noise because of the ltering of ni,z [k] by hz [k] and Eq. (4.3) shows that a cooperative relay network with FFBF 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 pointtopoint singleantenna transmission can be used. Here, we consider two cases: (1) The destination makes a decision based on equalization. (2) The destination rst equalizes r[k] r[k] without using LE or DFE optimized under zeroforcing (ZF) and minimum meansquared error (MMSE) criteria before making a decision [5]. The optimization of the corresponding FFBF matrix lters will be discussed in Sections 4.3 and 4.4, respectively. 86 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 4.2.3 Feedback Channel We assume that the destination estimates the relaydestination CIRs hi,z [k], 0 ≤ k ≤ Lh − 1, 1 ≤ i ≤ NR Similarly, relay and 1 ≤ z ≤ NR , estimate its own sourcerelay CIR during a training phase. gi,z [k], 0 ≤ k ≤ Lg − 1, i and forwards the estimate to the destination node. Alternatively, the destination may directly estimate the combined CIR of the sourcerelay and relaydestination channels, hi,z [k] ∗ gi,z [k] if relay i retransmits the training signal received from the source. The destination can then extract hi,z [k] ∗ gi,z [k] and hi,z [k] gi,z [k] from via deconvolution. Subsequently, the destination node computes the FFBF lters using the CSI of all links and feeds back the lter coecients 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 frequencyat and frequencyselective channels, cf. e.g. [35, 39, 42, 44, 50]. 4.3 FIR FFBF without Equalization In this section, we consider the case where the destination node cannot aord 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 FFBF 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 multiantenna relays in Sections 4.3.1 and 4.3.2 are extensions of the results for singleantenna relays given in [50]. Joint sourcerelay power constraints as considered in Sections 4.3.3 and 4.3.4 were not discussed in [50]. Also, for relaying with 87 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays single antenna relays a decision delay k0 was not considered in [50], i.e., k0 = 0. k0 as will be shown in Section 4.5, the proper choice of decision delay However, 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 ∑ HGD a, ¯ za ¯z HzG (4.6) z=1 where H { } ¯ 1, . . . , G ¯N , diag G R [H 1 . . . H NR ], GD (La + Lg + Lh − 2) × (La + Lg − 1)Mz Lg − 1)Mz × Mz2 La matrix (La + Lg − 1) × La matrix are column circular matrices with ¯ i,z G with and H can be separated into one vector ∑NR z=1 Mz vector hTk0 is the row with [H 1,z H 2,z . . . H Mz ,z ], (La + and Mz2 La × 1 vector az [aij,z [0] aij,z [1] . . . aij,z [La − 1]]T . aij,z [gi,z [0] . . . gi,z [Lg − 1] 0TLa −1 ]T in the rst columns, respectively. (La + Lg − 1) [aT1 . . . aTNR ]T a (La + Lg + Lh − 2) × (La + Lg − 1) matrix H i,z 1] 0TLa +Lg −2 ]T Matrix Hz ¯ 1,z . . . G ¯ Mz ,z ], I Mz ⊗ [G ¯z G [aT11,z aT12,z . . . aT1Mz ,z aT21,z . . . aTMz Mz ,z ]T Moreover, matrix and k0 hk0 of and [hi,z [0] . . . hi,z [Lh − and one submatrix Hk0 , Hk0 ∑NR z=1 i.e., length [H]ij , i ∈ {1, . . . , k0 − and 1, k0 + 1, . . . (La + Lg + Lh − 2)}, j ∈ {1, . . . , (La + Lg − 1) Hk0 , Mz }. Therefore, the rst term in (4.3) can be decomposed into a signal part and an ISI part La +Lg +Lh −3 heq [k] ∗ s[k] = heq [k0 ]s[k − k0 ] + ∑ heq [l]s[k − l] l=1, l̸=k0 = hTk0 GD as[k − k0 ] + sT [k]Hk0 GD a desired signal with (4.7) ISI 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 88 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays the ISI can be obtained as E { and E respectively. Similarly, { hTk0 GD as[k − k0 ] sT [k]Hk0 GD a n[k] n[k] = 2 2 } } H ∗ = σs2 aH GD hk0 hTk0 GD a (4.8) H H = σs2 aH GD Hk0 Hk0 GD a , (4.9) in (4.5) can be rewritten as NR ∑ ˘ za ¯ z + n0 [k] nz [k]H ˘ + n0 [k] N [k]Ha (4.10) z=1 ∑NR T T T (L + L − 1)M row vector n[k] [n [k] . . . n [k]] and a h z 1 N z=1 z=1 (La + R { } ∑NR 2 ˘ 1, . . . , H ˘ N . Moreover, nz [k] [nT [k] ˘ diag H Lh − 1)Mz × z=1 Mz La matrix H 1,z R with length ∑NR . . . nTMz ,z [k]]T with ¯ 1,z . . . I Mz ⊗ H ¯ Mz ,z ], H with [ni,z [k] . . . ni,z [k − (La + Lh − 2)]]T , ni,z [k] where (La + Lh − 1) × La [hi,z [0] . . . hi,z [Lh − 1] 0TLa −1 ]T matrix ¯ i,z H and ˘z H [I Mz ⊗ is column circular matrix in the rst column. The noise power can be obtained as ˘ H Ha ˘ + σ2 . E{|n[k]|2 } = σn2 aH H v (4.11) From (4.8), (4.9), and (4.11), the SINR at the destination can be obtained as SINR (a) = hTk0 GD as[k 2 } − k0 ] } E |sT [k]Hk0 GD a|2 + E{|n[k]|2 } { E { aH W 1 a = aH W 2 a + aH W 3 a + 1 with W1 H ∗ hk0 hTk0 GD /σv2 , W 2 σs2 GD H H Hk0 Hk0 GD /σv2 , σs2 GD (4.12) and W3 ˘ H H/σ ˘ 2. σn2 H v 89 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays From (4.2), the total transmit power, PR (a) = PR (a), of the relays can be obtained as NR ∑ Mz ∑ { } E |tj,z [k]|2 = aH Da , (4.13) z=1 j=1 with D H σs2 GD GD + σn2 I La ∑NR z=1 Mz2 . In the following, we will formulate various FFBF 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 FFBF 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 SINR (a) (4.14a) a s.t. aH Da ≤ P . By letting w D 1/2 a, where D 1/2 is the Cholesky decomposition of (4.14b) D, the optimization problem in (4.14) can be reformulated as a generalized eigenvalue problem. The optimum w can be obtained as { } −H/2 −1/2 P u Q−1 D W D 1 1 √ −H/2 H ∗ P Q−1 GD hk0 1 D = √ , −H/2 H ∗ hTk0 GD D −1/2 Q−2 D G h 1 D k0 wopt = √ (4.15) 90 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays where Q1 D −H/2 (W 2 + W 3 ) D −1/2 + vector of matrix X. 1 I ∑NR M 2 and P La z=1 z u{X} is the principle eigen- Therefore, the maximum SINR can be obtained as ( )−1 1 σs2 T H ∗ hk0 , = 2 hk0 GD W 2 + W 3 + D GD σv P SINRmax (4.16) and the corresponding optimum FFBF matrix lter in vector form is given as aopt √ ( )−1 H ∗ P W 2 + W 3 + P1 D GD hk0 = √ . −H/2 H ∗ hTk0 GD D −1/2 Q−2 D G h 1 D k0 (4.17) 4.3.2 Relay Power Minimization Under SINR Constraint Here, we optimize the FFBF 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 multipleantenna relays. The optimization problem can be formulated as min PR (a) = aH Da (4.18a) s.t. aH W 1 a ≥γ, aH W 2 a + aH W 3 a + 1 (4.18b) a where γ is the QoS requirement (minimal required SINR) at the destination. We let D 1/2 a again and note that the above problem is infeasible when Q2 ) −γW 3 D −1/2 , is negative semidenite. If the problem is feasible, D ( −H/2 w= W 1 − γW 2 the optimum FFBF matrix lter can be obtained as ( aopt = γ λmax {Q2 } )1/2 D −1/2 u{Q2 } (4.19) 91 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays and the corresponding minimum relay power is Pmin = γ . λmax {Q2 } (4.20) 4.3.3 SINR Maximization Under SourceRelay 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 sourcerelay transmit power constraint. We note that the joint sourcerelay 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 max2 a, σs s.t. aH W 1 a aH W 2 a + aH W 3 a + 1 (4.21a) aH Da + σs2 ≤ P (4.21b) The optimal solution can be found with a divideandconquer method. In particular, if we assume that σs2 is xed, problem (4.21) is identical to problem (4.14). The optimum FFBF matrix lter is obtained as √ ( D(σs2 ) P −σs2 )−1 H ∗ GD hk0 P− + W3 + aopt = √ , 2 )D −H/2 (σ 2 )G H h∗ hTk0 GD D −1/2 (σs2 )Q−2 (σ 1 s s D k0 σs2 W 2 (σs2 ) (4.22) and the corresponding maximum SINR is given by SINRmax (σs2 ) )−1 ( σs2 T D(σs2 ) H ∗ 2 hk0 = 2 hk0 GD W 2 (σs ) + W 3 + GD σv P − σs2 (4.23) 92 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays where Q1 (σs2 ) Note that ( ) D −H/2 (σs2 ) W 2 (σs2 ) + W 3 D −1/2 (σs2 ) + D, W 1 , and W2 dened earlier depend on problem is to nd the optimal σs2 such that max σs2 , 0≤σs2 ≤P 1 I ∑NR . P − σs2 La z=1 Mz2 σs2 SINRmax (σs2 ) in this case. (4.24) The remaining is maximized, i.e. SINRmax (σs2 ) . (4.25) Problem (4.25) can be easily solved by a grid search or other numerical methods given in [1]. 4.3.4 SourceRelay Power Minimization Under SINR Constraint In this case, the goal is to minimize the joint sourcerelay transmit power subject to a destination SINR constraint. The optimization problem can be formulated as min2 aH Da + σs2 (4.26a) s.t. aH W 1 a ≥γ aH W 2 a + aH W 3 a + 1 (4.26b) a, σs Again, we assume that σs2 is xed, and the resulting problem is identical to problem (4.18). If the problem is feasible, the optimum FFBF matrix lter is given by ( aopt = γ λmax {Q2 (σs2 )} )1/2 D −1/2 (σs2 )u{Q2 (σs2 )} , (4.27) 93 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays where λmax {·} denotes maximum eigenvalue of a matrix, and the corresponding minimum joint sourcerelay transmit power is Pmin = with Q2 (σs2 ) γ + σs2 λmax {Q2 (σs2 )} (4.28) D −H/2 (σs2 ) (W 1 (σs2 ) − γW 2 (σs2 ) − γW 3 ) ×D −1/2 (σs2 ). The remaining op- timization problem is γ + σs2 λmax {Q2 (σs2 )} (4.29a) s.t. λmax {Q2 (σs2 )} > 0 . (4.29b) min 2 σs Note that λmax {Q2 (σs2 )} = 0 when σs2 = 0. Therefore, (4.29) due to the fact that (4.29b) is satised only if σs2 = 0 σs2 > 0. has been ignored in problem Problem (4.29) can be easily solved by numerical methods given in [1]. 4.4 FFBF 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, suciently 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 FFBF 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 94 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays (4.5) can be obtained as Φn (f ) = σn2 NR ∑ Mz ∑ Mz ∑ z=1 i=1 2 Hj,z (f )Aji,z (f ) + σv2 = σn2 aH (f )Γ(f )a(f ) + σv2 (4.30) j=1 ∑ R 2 Mz2 × N diag {Γ1 (f ), . . . , ΓNR (f )}, where Γz (f ) z=1 Mz square matrix Γ(f ) ( ∗ ) hz (f )hTz (f ) ⊗ I Mz and hz (f ) [H1,z (f ), . . . , HMz ,z (f )]T . The frequency response of with ∑NR z=1 the relaydestination channel corresponding to the F{hj,z [k]}. Hj,z (f ) in vector where a(f ) Aji,z (f ) at relay z The frequency responses of the FFBF matrix lters are collected [aT1 (f ) . . . aTNR (f )]T F{aji,z [k]} W (f ) for n[k] W (f ) = σn2 = [A11,z (f ) A12,z (f ) . . . AMz Mz ,z (f )]T , az (f ) ith receive antenna and the j th transmit antenna. The can be easily obtained as ( with denotes the frequency response of the FFBF matrix lter corresponding to the whitening lter j th antenna of the z th relay is given by NR ∑ Mz ∑ Mz ∑ z=1 i=1 2 −1/2 Hj,z (f )Aji,z (f ) + σv2 j=1 σn2 aH (f )Γ(f )a(f ) + σv2 and we denote the output of the whitening lter by )−1/2 r′ [k]. , (4.31) Taking into account the whitening, the frequency response of the equivalent overall channel can be obtained as ′ Heq (f ) with q(f ) ( )−1/2 T W (f )F{heq [k]} = σn2 aH (f )Γ(f )a(f ) + σv2 q (f )a(f ) hz (f ) ⊗ g z (f ), g z (f ) [q T1 (f ) . . . q TNR (f )]T , q z (f ) GMz ,z (f )]T , Gi,z (f ) F{gi,z [k]}, and hz (f ) spectral density of the noise component, [G1,z (f ) G2,z (f ) . . . [H1,z (f ) H2,z (f ) . . . HMz ,z (f )]T . n′ [k], of r′ [k] is (4.32) The power Φn′ (f ) = 1. In the remainder of this section, we formulate and solve the IIR FFBF lter opti- 95 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays mization problems for LE, DFE, and an idealized matched lter (MF) receiver in a unied manner. After introducing Z(a(f )) ′ |Heq (f )|2 = aH (f )q ∗ (f )q T (f )a(f ) , σn2 aH (f )Γ(f )a(f ) + σv2 (4.33) we can express the SINRs at the outputs of a decision feedback and a linear equalizer as [83, 84] SINRDFE (a(f )) = σs2 exp ∫1/2 ln (Z(a(f )) + ξ) df −1/2 −χ (4.34) and SINRLE (a(f )) = σs2 ∫1/2 −1 (Z(a(f )) + ξ)−1 df − χ, (4.35) −1/2 respectively. In (4.34) and (4.35), we have χ = 0, ξ = 0 and χ = 1, ξ = 1/σs2 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] ∫1/2 SINRMF (a(f )) = σs2 Z(a(f )) df. (4.36) −1/2 It is not dicult to show that regardless of how the FFBF 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 96 Chapter 4. Cooperative FFBF with Multiple MultiAntenna 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 FFBF matrix lters for maximization of the SINRs at the output of the considered equalizers. To make the problem well dened, we constrain the relay transmit power, PR = 1/2 NR ∑ Mz ∫ ∑ z=1 j=1 where Φtj,z (f ) σs2 | PR , which is given by ∫1/2 aH (f )D(f )a(f )df Φtj,z (f )df = −1/2 (4.38) −1/2 ∑Mz 2 2 i=1 Aji,z (f )Gi,z (f )| + σn ∑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 j th antenna of z th the relay, D(f ) σs2 GH (f )G(f ) + σn2 I ∑NR z=1 and Gz (f ) Mz2 , G(f ) diag {G1 (f ), . . . , GNR (f )}, I Mz ⊗ g Tz (f ). Formally, the IIR FFBF lter optimization problem can now be stated as max a(f ) SINRX (a(f )) (4.39a) ∫1/2 aH (f )D(f )a(f ) df ≤ P, s.t. (4.39b) −1/2 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 D 1/2 (f )a(f ), where which can be expressed as v(f ) = √ p(f )u(f ) v(f ) without loss of generality, p(f ) denotes the power of v(f ) and u(f ) has unit norm, ∥u(f )∥2 = 1. Furthermore, 97 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays we introduce ¯ ¯ Z(v(f )) = Z( √ p(f )u(f )) Z(a(f )), which is given by aH (f )q ∗ (f )q T (f )a(f ) uH (f )J (f )u(f ) ¯ = Z(v(f )) = 2 H σn a (f )Γ(f )a(f ) + σv2 uH (f )X(f )u(f ) (4.40) with rank one, positive semidenite matrix J (f ) = p(f )D −1/2 (f )q ∗ (f )q T (f )D −1/2 (f ) (4.41) and full rank, positive denite matrix X(f ) = σn2 p(f )D −1/2 (f )Γ(f )D −1/2 (f ) + σv2 I NR . Introducing (√ ) SINRX (v(f )) = SINRX p(f )u(f ) SINRX (a(f )), (4.42) we can restate prob- lem (4.39) in equivalent form as max p(f ),u(f ) SINRX (√ ) p(f )u(f ) (4.43a) ∫1/2 p(f ) df ≤ P s.t. (4.43b) −1/2 p(f ) ≥ 0. (4.43c) In the following, we provide a unied solution to problem (4.43) valid for all three considered equalization schemes. 1) Optimum IIR FFBF 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 98 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays optimizing the resulting SINR expression with respect to p(f ). Furthermore, for all three considered receiver structures, the SINR monotonically increasing in p(f ), Z¯ (√ we can maximize the SINR u(f ) respect to uopt (f ), ) p(f )u(f ) f. is . Thus, for any given power allocation SINRX (v(f )) by maximizing Z¯ for all frequencies SINRX (v(f )) (√ ) p(f )u(f ) with Hence, the optimal FFBF direction vector, can be found from the following optimization problem max u(f ) Z¯ (√ ) uH (f )J (f )u(f ) p(f )u = H . u (f )X(f )u(f ) (4.44) Problem (4.44) is a generalized eigenvalue problem for which a closedform 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 ) , where c(f ) is a realvalued scaling factor which is given by c(f ) = √ The maximum Z¯ (4.45) Z¯ 1 q T (f )D −1/2 (f )X −2 (f )D −1/2 (f )q ∗ (f ) (√ ) p(f )u(f ) achievable with uopt (f ) . (4.46) is (√ ) p(f )uopt (f ) = p(f )q T (f )D −1/2 (f )X −1 (f )D −1/2 (f )q ∗ (f ) )−1 ∗ ( q (f ) . = p(f )q T (f ) σn2 p(f )Γ(f ) + σv2 D(f ) (4.47) Now, we can express the optimal FFBF lter frequency response vector (for a given 99 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays power allocation), aopt (f ), aopt (f ) = as √ ( )−1 ∗ p(f )c(f ) σn2 p(f )Γ(f ) + σv2 D(f ) q (f ) . (4.48) z , aopt z (f ), can be simpli- From (4.48), the optimum individual FFBF lter of relay ed as aopt z (f ) )−1 ( √ 2 2 2 H 2 2 q ∗z (f ) = p(f )c(f ) σn p(f )Γz (f ) + σv σs Gz (f )Gz (f ) + σn σv I Mz2 ( √ ])−1 [ ] [ = p(f )c(f ) σn2 p(f ) h∗z (f )hTz (f ) ⊕ σv2 σs2 g ∗z (f )g Tz (f ) + σn2 σv2 I Mz × (h∗z (f ) ⊗ g ∗z (f )) √ p(f )c(f ) (h∗z (f ) ⊗ g ∗z (f )) = 2 . σn p(f )∥hz (f )∥2 + σv2 σs2 ∥g z (f )∥2 + σn2 σv2 (4.49) (4.50) The transformation from (4.49) to (4.50) is accomplished by exploiting the relation [78] −1 (M ⊕ N ) = N ∑ N ∑ ¯i⊗n ¯ j )H (mi ⊗ nj ) (m i=1 j=1 where and ¯ i, m i , ni , m and N H , respectively. relay z ¯i n λi (M ) + λj (N ) denote the eigenvectors of N ×N , (4.51) matrices M, N, MH, Therefore, the optimum beamforming matrix lter Aopt z (f ) of is obtained as √ Aopt z (f ) = p(f )c(f ) h∗z (f )g H z (f ) , 2 2 2 2 2 2 2 σn p(f )∥hz (f )∥ + σv σs ∥g z (f )∥ + σn σv z = 1, . . . , NR . (4.52) Eq. (4.52) reveals that the optimal IIR FFBF matrix lters for all considered receiver structures can be interpreted as the concatenation of a lter matched to the source relay and the relaydestination link with frequency response lter whose frequency response h∗z (f )g H z (f ) and a second √ p(f )c(f )/(σn2 p(f )∥hz (f )∥2 + σv2 σs2 ∥g z (f )∥2 + σn2 σv2 ) 100 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays depends on the power allocation, and thus on the particular equalizer used at the destination. Note that Aopt z (f ) of relay z depends on the CIRs of all sourcerelay, all relaydestination, and the sourcedestination 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 unied way, we rst introduce the following denitions: SDFE (f ) ln(M (f )), SLE (f ) −1/M (f ), and SMF (f ) M (f ), (4.53) with M (f ) where for DFE and LE q T (f )(σn2 Γ(f ) + σv2 D(f )/p(f ))−1 q ∗ (f ) + ξ, ξ is dened after (4.35) and ξ = 0 (4.54) for the MF receiver. Based on these denitions, 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 ∫1/2 max p(f ) SX (f ) df (4.55a) p(f ) df ≤ P (4.55b) −1/2 ∫1/2 s.t. −1/2 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 ∈ 101 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays {DFE, LE, MF}, the power allocation problem is convex for all considered equalizer structures. The Lagrangian of problem (4.55) is given by ∫1/2 L(p(f ), µ) = ∫1/2 SX (f ) df − µ −1/2 where p(f ) df , (4.56) −1/2 µ ≥ 0 is the Lagrangian multiplier. The corresponding Lagrange dual function is ∫1/2 D(µ) = max L(p(f ), µ) = max p(f ) p(f ) (SX (f ) − µp(f )) df −1/2 ∫1/2 max (SX (f ) − µp(f )) df . = (4.57) p(f ) −1/2 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 moved inside the integration. Therefore, for a given max p(f ) µ, p(f ) p(f ) can be can be obtained from SX (p(f )) = SX (f ) − µp(f ) (4.58) or equivalently SX′ (f ) SX′ (f ) ∂SX (f ) = µ. ∂p(f ) (4.59) can be easily computed for all considered equalization schemes. In particular, we obtain ′ SDFE (f ) M ′ (f )/M (f ), ′ SLE (f ) M ′ (f )/M 2 (f ), ′ and SMF (f ) M ′ (f ), (4.60) 102 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays where M ′ (f ) ( )−2 ∗ ∂M (f ) = q T (f )D(f ) σn2 p(f )Γ(f ) + σv2 D(f ) q (f ). ∂p(f ) (4.61) Note that constraint (4.55c), which has been ignored in (4.57), can be taken into account by evaluating SX′ (f ) ∂SX (f )/∂p(f ) for p(f ) → 0+ . is a monotonic decreasing function of a given we set µ, SX′ (f ) = µ p(f ) = 0 In particular, since SX′ (f ) p(f ) for all considered equalization schemes, for does not have a positive solution if in this case. Otherwise, we nd p(f ) limp(f )→0+ SX′ (f ) < µ, and from (4.59) by using e.g. the ! bisection search method [1] . On the other hand, the optimal value µ = µopt that ensures the power constraint is satised can be found iteratively by another bisection search. More specically, if the corresponding total power less than the maximum power P for a given decreased, whereas it is increased if µ, −1/2 ≤ f ≤ 1/2 ∫ 1/2 −1/2 the Lagrange multiplier µ p(f )df is has to be PR > P . We note that since the frequency axis is real valued, in practice, in PR = f has to be discretized 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 FFBF lter frequency response Aopt z (f ) for relay z , 1 ≤ z ≤ NR . Although we concentrate in this section on the case where the direct sourcedestination link is not exploited for detection, with a minor modication our equalization results are valid if the sourcedestination link is also used. In particular, for the latter case, our journal paper [82] provides the details. ! Note that algorithms with faster convergence, e.g. Newton's method, can be used as long as the condition p(f ) ≥ 0 is satised. 103 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays Table 4.1: Numerical algorithm for nding the optimum power allocation p(f ) for IIR FFBF lters at the relays. X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver, respectively. Termination constant ϵ and frequency spacing −5 −5 (e.g. ϵ = 10 , ∆f = 10 ). i denotes the iteration index. 1 2 3 ∆f have small values i = 0, N = ⌈1/∆f ⌉, and fn = −1/2 + (n − 1) ∆f , 1 ≤ n ≤ N . ′ Initialize l = 0 and u = maxf limp(f )→0+ SX (f ). Update µ by µ = (l + u)/2. For n = 1 to N ′ If limp(fn )→0+ (SX (fn ) − µ) < 0, set p(fn ) = 0, ′ otherwise compute p(fn ) by solving SX (fn ) = µ Let with the bisectional search method [1]. 4 5 ∑N p(fn )∆f > P , l = µ, else u = µ. If u − l > ϵ, goto Step 2; else p(fn ), 1 ≤ n ≤ N , are the optimal power allocation parameters, and µ is the optimum Lagrange multiplier µopt . If n=1 4.4.2 Optimal FIR FFBF with Equalization In practice, it is not possible to implement the IIR FFBF lters discussed in the previous section since they would require the feedback of an innite number of lter coecients from the destination to the relays. However, the performance achievable with these IIR FFBF lters provides a useful upper bound for the FIR FFBF lters considered in this section. In particular, the performance of the IIR solution can be used for optimizing the FIR BFFF length to achieve a desired tradeo between the amount of feedback and performance. We note that although FIR FFBF 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 FFBF 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 )HGD a (4.62) 104 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays with d(f ) [1 ej2πf . . . ej2πf (Leq −1) ]T . FIR FFBF coecient vector a, H, and GD are dened in Section 4.3 after (4.6), respectively. The noise whitening lter in the FIR case is given by ) ( ¯ )a + σ 2 −1/2 W (f ) = σn2 aH Γ(f v (4.63) { } ∑NR 2 2 ¯ ) diag Γ ¯ 1 (f ), . . . , Γ ¯ N (f ) L block diagonal matrix Γ(f L × M M a a z z R z=1 z=1 )( )H ∑NR H( ¯ ¯ ) H ¯ ˘ ˘ z is an M 2 La ×M 2 La of rank H I Mz ⊗ d(f z I Mz ⊗ d(f ) z z z=1 Mz , where Γz (f ) with ∑NR matrix of rank ˘z Mz . H is dened after (4.10), and ¯ ) d(f [1 ej2πf . . . ej2πf (Lh +La −2) ]T . Therefore, after noise whitening, the frequency response of the overall channel is ( ) ′ ¯ )a + σ 2 −1/2 . Heq (f ) = dH (f )HGD a σn2 aH Γ(f v (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 BFFF lter design. Similar to the IIR case in (4.33), also for the FIR case it is convenient to introduce the denition Z(a) ′ |Heq (f )|2 = H H aH GD H d(f )dH (f )HGD a . ¯ )a + σ 2 σn2 aH Γ(f v (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 FFBF lters, the argument of Z(a) in (4.65) is a vector containing all FIR FFBF coecients. now in the SINR expressions in (4.34)(4.36) by SINRX (a), Z(a) Replacing Z(a(f )) from (4.65), we obtain the SINRs where X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver, 105 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays respectively. This allows us to formulate the FIR FFBF lter optimization problem in a unied manner: max SINRX (a) (4.66a) a s.t. aH Da ≤ 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 dicult to solve. The main reason for this lies in the fact that the optimization variable be chosen freely for each frequency f, whereas the coecient vector a a(f ) in (4.39) can in (4.66) is xed for all frequencies. To simplify the power constraint, we introduce v D 1/2 a. Furthermore, it is not dicult to see that at optimality, the power constraint in (4.66b) is fullled with equality, i.e., aH Da = v H v = P . With this identity, we obtain M (v, f ) Z(a) + ξ ¯ (f )v vH J ¯ )v v H X(f (4.67) where ¯ (f ) J ¯ ) X(f ¯ ) Φ(f ξσv2 I NR La , P 2 ¯ )D −1/2 + σv I N La , σn2 D −H/2 Γ(f P R ¯ ). G H HH d(f )dH (f )HGD + ξσ 2 Γ(f ¯ )D −1/2 + D −H/2 Φ(f D n (4.68) (4.69) (4.70) 106 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays Now, we can rewrite optimization problem (4.66) in equivalent form as ∫1/2 max SX (v, f ) df v (4.71a) −1/2 s.t. v H v = 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 FFBF optimization problem in (4.71) is a dicult nonconvex 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 v H v=P where fi −1/2 + (i − 1)/N and N N ∏ ¯ (fi )v vH J H X(f ¯ i )v , v i=1 (4.73) 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 dicult 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 BFFF coecient vectors, we propose a gradient algorithm (GA). In iteration 107 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays Table 4.2: vector a. index and Gradient algorithm (GA) for calculation of nearoptimal FIR FFBF lter ϵ has a small value (e.g. ϵ = 10−5 ). i denotes the iteration Termination constant δi is the adaptation step size chosen through a backtracking line search [1]. i=0 1 Let 2 Update the vector 3 4 and initialize vector v with some v0 fullling vH 0 v0 = P . v: [ ] ∫ 1/2 ( ¯ ¯ ) ) J (f ) X(f DFE: v i+1 = v i + δi ¯ (f )v i − v H X(f ¯ )v i df v i ; vH J −1/2 i i [∫ ] 1/2 ¯ (f )v i X(f ¯ ) − v H X(f ¯ )v i J ¯ (f ) vH J i i LE: v i+1 = v i − δi df v i ; ( H )2 ¯ −1/2 v J (f )v i i [∫ ] 1/2 ¯ )v i J ¯ (f ) − v H J ¯ (f )v i X(f ¯ ) vH X(f i i MF: v i+1 = v i + δi df v i . ( H )2 ¯ −1/2 v i X(f )v i H (Note that normalization of vector v i+1 is not necessary since v i+1 v i = P .) Compute SINRX (v i+1 ) based on (4.34)(4.36). If |SINRX (v i+1 ) − SINRX (v i )| < ϵ, goto Step 5, otherwise increment i → i + 1 and goto Step 2. 5 v i+1 is the desired a = D −1/2 v i+1 . vector, and the corresponding optimum FFBF lter is i + 1, the GA improves vector v i from iteration ∫1/2 i in the direction of the steepest ascent [1] ∂SX (v, f ) df ∂v (4.74) −1/2 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 More importantly, for suciently large FIR lter lengths La , v0. the solution found with the GA closely approaches the performance of the optimal IIR FFBF lter. This suggests that the solution found by the GA is at least near optimal. Exemplary simulation results conrming these claims are provided and discussed in the next section. 108 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays We note that we can again accommodate the case where the sourcedestination 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 σn2 = σv2 = 1 and P = 1. This √ ¯ √ hi,z [k] = γh h γg g¯i,z [k], where γh i,z [k] and gi,z [k] = network with FFBF. Throughout this section we assume allows us to decompose the CIRs as and γg denote the transmitter SNRs of the relaydestination and the sourcerelay links, respectively. The normalized CIRs ¯ i [k] h and g¯i [k] include the eects of multipath fading and pathloss. All IIR and FIR FFBF 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 NR = 1 relays nodes. We consider the following three cooperative relay network setups: 1) with M1 = 5 at location (c); 2) and (e), respectively; and 3) NR = 2 relays with M1 = 2 and M2 = 3 at locations (a) 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 exponent of 3 with reference distance dref = 1 d. A pathloss is assumed. The CIR coecients of all links are modeled as independent quasistatic Rayleigh fading with Lg = Lh = 5 and following an exponential power delay prole p[k] = where Lx ∈ {Lg , Lh } and σt Lx −1 1 ∑ e−k/σt δ[k − l] , σt l=0 (4.75) characterizes the delay spread [85]. All results shown were 109 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays (a) 1/4 (b) (c) d source 1/4 2−d destination 1/4 (d) 1/4 (e) Figure 4.2: Locations of source, destination, and relays in simulation. averaged over 100, 000 independent realizations of the fading channels. 4.5.1 FFBF without Equalization Optimal Decision Delay: First, we consider the optimal decision delay for FFBF without equalization. In theory, the decision delay parameter channel realization. k0 can be optimized for each every channel realization. In practice, it is preferable to nd a value for well for given channel statistics. σt = 2 and σt = 7. 4.3.1. We assume network setup 3), In comparison, for is equal to σt = 7, which works d = 1, and γg = γh = 10 k0 The FIR FFBF lters were SINR Maximization Under Relay Power Constraint σt = 2, the optimal k0 k0 for Fig. 4.3 shows the average SINR vs. decision delay for FIR FFBF without equalization for optimized for the k0 However, it is not practical to search for the optimal delay criterion in Section dB. As can be observed, for 2, 3, and 6 for lter length La = 1, 3, and 7, respectively. the optimal k0 is equal to 5, 5, and respectively. In other words, the larger the channel delay spread selective the channel, the larger the optimal delay k0 . 7 for La = 1, 3, and 7, σt , i.e., the more frequency Fig. 4.3 also shows that increasing the FFBF lter length is highly benecial for the achievable maximum average SINR. For the remaining results presented in this section, we will adopt the optimal values for k0 . 110 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays FIR FF−BF w/o EQ (La=1) 8 FIR FF−BF w/o EQ (L =3) a FIR FF−BF w/o EQ (La=7) σ =2 t σt = 7 6 Average SINR [dB] 4 2 0 −2 −4 2 4 6 8 10 12 14 k0 Figure 4.3: Average SINR vs. decision delay k0 for FIR FFBF without equalization (EQ) at the destination. The FFBF lters were optimized for SINR maximization under joint relay power constraint. Exponentially decaying Lg = Lh = 5, d = 1, NR = 5, Mz = 1, z ∈ {1, 2, . . . , 5} SINR Optimization: channel power delay prole with and γg = γh = 10 dB. Figs. 4.4 and 4.5 show the average SINR vs. distance d for FF BF for joint relay and joint sourcerelay power constraints, respectively. Relay network setups 1) 3) were adopted. The FFBF matrix lters were generated using the results in Section 4.3.1 and 4.3.3, respectively. For both considered constraints FFBF 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 sourcerelay power constraint can yield signicant performance gains if the relays are 111 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 15 FIR FF−BF w/o EQ (La = 1) FIR FF−BF w/o EQ (La = 3) FIR FF−BF w/o EQ (La = 5) FIR FF−BF w/o EQ (La = 7) N = 1, M = 5 R 1 NR = 2, M1 = 2, M2 = 3 NR = 5, M1=M2=M3=M4=M5=1 no relay (MMSE−DFE) Average SINR [dB] 10 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d d for FIR FFBF without equalization (EQ) at the destination. The FFBF matrix lters were optimized for a joint relay power constraint. Exponentially decaying channel power delay prole with σt = 2 and Lg = Lh = 5, and γg = γh = 10 dB. Results for direct transmission with transmit power P = 2 at the source Figure 4.4: Average SINR vs. distance are also included. close tothe source or close to the destination, respectively, by exibly 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. 112 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 15 FIR FF−BF w/o EQ (L = 1) a FIR FF−BF w/o EQ (L = 3) a FIR FF−BF w/o EQ (L = 5) a FIR FF−BF w/o EQ (L = 7) a N = 1, M = 5 R 1 N = 2, M = 2, M = 3 R 1 2 N = 5, M =M =M =M =M =1 R 1 2 3 4 5 no relay (MMSE−DFE) Average SINR [dB] 10 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d Figure 4.5: Average SINR vs. distance d for FIR FFBF without equalization (EQ) at joint sourcerelay power constraint. Exponentially decaying channel power delay prole with σt = 2 and Lg = Lh = 5, and γg = γh = 10 dB. Results for direct transmission with transmit power P = 2 the destination. The FFBF matrix lters were optimized for a at the source are also included. Power Minimization: PR + σs2 , setups. Fig. 4.6 shows the total source and relay transmit power, vs. the minimum required SINR γ at the destination for dierent relay network The FFBF 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 113 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 25 20 PR + σs2 [dB] 15 10 5 N =5, M =1 (Joint Relay Power Min.) R z N =5, M =1 (Joint Source−Relay Power Min.) R z NR=2, M1=2, M2=3 (Joint Relay Power Min.) N =2, M =2, M =3 (Joint Source−Relay Power Min.) 0 R 1 2 FIR FF−BF w/o EQ (L =1) a FIR FF−BF w/o EQ (L =5) a −10 −5 0 γ 5 10 15 20 25 [dB] γ for relay power minimization Figure 4.6: Total average source and relay transmit power vs. required SINR FIR FFBF without equalization (EQ) at the destination for and joint sourcerelay power minimization. Exponentially σt = 2 and Lg = Lh = 5, d = 1, and γg = γh = 10 dB. decaying power delay prole with 4.7, we observe that joint sourcerelay transmit power minimization and multipleantenna relays can lead to signicant power savings. Fig. 4.6 also reveals that increasing La can substantially reduce the total source and relay transmit power. 114 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 1 NR=5, Mz=1 (Joint Relay Power Min.) N =5, M =1 (Joint Source−Relay Power Min.) R 0.9 z N =2, M =2, M =3 (Joint Relay Power Min.) R 1 2 FIR FF−BF w/o EQ (La=1) 0.8 FIR FF−BF w/o EQ (L =3) a FIR FF−BF w/o EQ (La=5) Feasibility Probability 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −10 −5 0 5 γ 10 15 20 25 30 [dB] γ for FIR FFBF without equalizafor relay power minimization and joint sourcerelay power decaying power delay prole with σt = 2 and Lg = Lh = 5, Figure 4.7: Feasibility probability vs. required SINR tion (EQ) at the destination minimization. Exponentially d = 1, and γg = γh = 10 dB. 4.5.2 FFBF with Equalization Convergence of the GA: We rst investigate the convergence of the proposed GA for optimization of the FIR FFBF lters. We assume MMSEDFE at the destination and relay network setup 3) (i.e., NR = 5 relays with Mz = 1, 1 ≤ z ≤ NR , (e), respectively). The CIRs of all involved channels are given by at locations (a) √ ¯ 1,z [k] = 1/ 5, g¯1,z [k] = h 0 ≤ k < 5, 1 ≤ z ≤ 5, with Lg = Lh = 5 and γg = γh = 10 dB. Fig. 4.8 shows the achievable 115 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 6.65 6.6 6.55 6.5 SINR [dB] 6.45 6.4 6.35 IIR FF−BF (MMSE−DFE) GA (all−one vector) GA (random vector) FIR FF−BF (MMSE−DFE, L =1) 6.3 a FIR FF−BF (MMSE−DFE, La=3) FIR FF−BF (MMSE−DFE, La=7) 6.25 10 20 30 40 50 60 70 iteration number i of GA given in Table 4.2 for FIR FFBF with ¯ 1,z [k] = MMSEDFE at the destination. γg = γh = 10 dB, Lg = Lh = 5, and g ¯1,z [k] = h √ 1/ 5, 0 ≤ k < 5, 1 ≤ z ≤ 5. NR = 5 relays with Mz = 1, 1 ≤ z ≤ NR , at locations Figure 4.8: SINR vs. iteration number (a)(e), respectively. For comparison the SINR for IIR FFBF is also shown. SINR vs. iteration number i for initialization of the GA with a normalized random vector and a normalized allone vector for dierent FIR lter lengths the adaptation step size, δi , La , respectively. Note that is obtained from a backtracking line search, cf. Table 4.2. After a suciently large number of iterations, the SINR converges to the same constant value for both initializations. The steadystate SINR increases with increasing suciently large FIR lter lengths La , La and for the steadystate SINR approaches the SINR of IIR FFBF. 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 116 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 4.2 was initialized with a normalized allone vector. Filter Design for a Fixed Test Channel: In order to get some insight into the eect that dierent equalization schemes have on the IIR and FIR FFBF lter design, we consider next a cooperative network with simplied test channel with Lg = Lh = 2 NR = 1 and single antenna relay and assume a √ ¯ 1,1 [k] = 1/ 2, k ∈ {0, 1}, g¯1,1 [k] = h 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 FFBF lter frequency response |Aopt 1 (f )| vs. frequency f. We consider the cases where the destination is equipped with ZFDFE, MMSEDFE, ZFLE, MMSELE, and an MF receiver. Interestingly, although the frequency responses for all equalization schemes have the same structure, cf. (4.52), due to dierences in the optimal relay power allocation, p(f ), the FFBF lter frequency response for the ZFLE case exhibits a completely dierent 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 sourcerelay channel, the FFBF lter, and the relaydestination channel, would lead to innite noise enhancement in a linear zero forcing equalizer at the destination, the FFBF 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 innite noise enhancement at the destination even if the overall channel has zeros. Thus, in these cases, the optimal FFBF lters avoid allocating signicant 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 tradeo 117 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 2 Test Channel 1.8 1.6 1.4 Magnitude 1.2 1 IIR FF−BF (MF) IIR FF−BF (MMSE−DFE) IIR FF−BF (ZF−DFE) IIR FF−BF (MMSE−LE) IIR FF−BF (ZF−LE) 0.8 Aopt 1 (f ) 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f √ γg = γh = 10 dB, NR = 1 single ¯ g¯1,1 [k] = h1,1 [k] = 1/ 2, k ∈ {0, 1}. For comparison the Figure 4.9: Frequency responses of IIR FFBF lters for antenna relay, Lg = Lh = 2, and 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 aected by residual ISI. Fig. 4.10 compares the frequency responses of the IIR FFBF lter and FIR FFBF lters of various lengths assuming MMSEDFE at the receiver. As expected, as the FIR FFBF 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 FFBF lters are required to closely approximate the IIR lters, subsequent results 118 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 0.5 Magnitude 0.4 0.3 0.2 IIR FF−BF (MMSE−DFE) FIR FF−BF (MMSE−DFE, La=3) 0.1 FIR FF−BF (MMSE−DFE, La=5) FIR FF−BF (MMSE−DFE, L =7) a FIR FF−BF (MMSE−DFE, La=30) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f Figure 4.10: Frequency responses of IIR FFBF lter and FIR FFBF lters of various lengths for MMSEDFE at the receiver. All channel parameters are identical to those in Fig. 4.9. will show that short FIR FFBF lters suce to closely approach the SINR performance of IIR FFBF lters. SINR Performance for Fading Channels: vs. distance 2) (i.e., d In Fig. 4.11, we show the average SINR for various FFBF lter and equalization designs for relay nework setup NR = 2, M1 = 2, and M2 = 3). We compare the performance of the proposed FFBF matrix lter design with MMSEDFE and without equalizer at the destination. Interestingly, while for short FIR FFBF lters equalization at the transceivers results 119 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 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 (La=1) 11 10 FIR FF−BF (L =3) a FIR FF−BF (L =5) a 9 no relay (MMSE−DFE) no relay (MMSE−LE) 8 Average SNR [dB] 7 6 5 4 No Relay 3 2 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d d for FFBF with MMSELE, MMSEDFE, and NR = 2 relays with M1 = 2 and M2 = 3, exponentially σt = 2 and Lg = Lh = 5, and γg = γh = 10 dB. For Figure 4.11: Average SINR vs. distance an MF receiver at the destination. decaying power delay prole with comparison the SINRs of FFBF without (w/o) equalization (EQ) at the destination and without relaying are also shown, respectively. in large performance gains, FIR FFBF without equalization with large La approaches the same performance as FIR FFBF 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 FFBF approaches the performance of IIR FFBF with MMSEDFE at the destination. For IIR FFBF lters, Fig. 4.11 shows that the loss of MMSEDFE compared to an idealized MF receiver, which is the ultimate performance bound for any equalizer architecture, exceeds 1 dB only for d < 0.4. 120 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 12 10 FIR FFBF (MMSEDFE) Average SNR [dB] 8 IIR FF−BF (MF) IIR FF−BF (MMSE−DFE) IIR FF−BF (MMSE−LE) FIR FF−BF (L =1) FFBF w/o EQ 6 a FIR FF−BF (L =3) a FIR FF−BF (L =5) a no relay (MMSE−DFE) no relay (MMSE−LE) 4 2 No Relay 0 0 1 2 3 4 5 6 7 8 9 10 11 σt σt for FFBF with MMSELE, 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 prole with Lg = Lh = 5, and γg = γh = 10 dB. For comparison the SINRs of FFBF without (w/o) equalization (EQ) Figure 4.12: Average SINR vs. decay parameter at the destination and without relaying are also shown, respectively. Impact of Decay Parameter σt : parameter σt In Fig. 4.12, we investigate the impact of decay on the performance of FFBF for d = 1 and γg = γh = 10 dB. We note that the CIR coecients of the test channel decay the faster (i.e., the channel is less frequency selective), the smaller at when σt = 0, σt = 0 . σt is. As a special case, the channel becomes frequency Fig. 4.12 shows that when the channel becomes frequency at, i.e., all relaying schemes provide the same average SINR performance. We also observe that the performance of suciently long FFBF lters is practically not aected by the 121 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays 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) −1 10 N =2, M =2, M =3, FIR FF−BF (MMSE−DFE) R 1 2 N =2, M =2, M =3, FIR FF−BF w/o EQ R 1 2 FIR FF−BF (La = 1) FIR FF−BF (La = 5) −2 10 FFBF w/o EQ −3 BER 10 −4 10 FFBF with EQ 0 5 10 γg = γh 15 20 [dB] Figure 4.13: Average BER of BPSK vs. transmit SNR γ for FFBF with MMSELE, MMSEDFE, and an MF receiver at the destination. Exponentially decaying power delay prole with σt = 2 and Lg = Lh = 5. For comparison the BER of FFBF without (w/o) equalization (EQ) at the destination is also shown. frequency selectivity of the channel if MMSELE or MMSEDFE are employed at the destination. The idealized MF receiver with IIR FFBF benets even slightly from more frequency selectivity (larger σt ) because of the additional diversity oered by the channel. In contrast, FFBF without equalization at the receiver is adversely aected 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: lation vs. transmit SNR, γ = γg = γh , Fig. 4.13 shows BERs of BPSK modu- for FIR and IIR FFBF matrix lters. We adopt 122 Chapter 4. Cooperative FFBF with Multiple MultiAntenna Relays cooperative relay network setup 2), and assume σt = 2 and d = 1. The BERs for FIR FFBF matrix lters were simulated by implementing MMSEDFE with FIR equalization lters of lengths 4 × Leq , which caused negligible performance degradation compared to IIR equalization lters. The BERs for IIR FFBF 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 benecial in terms of the achievable BER and large performance gains are realized compared to FFBF without equalization. Also, for IIR FFBF matrix lters MMSELE and MMSEDFE 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 trellisbased equalizers, such as decisionfeedback sequence estimation, at the expense of an increase in complexity. 4.6 Conclusions In this chapter, we considered FFBF for frequencyselective cooperative relay networks with one source, multiple multiantenna 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 FFBF 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 FFBF 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 closedform solutions and ecient numerical methods for computation of the optimal FIR FFBF matrix lters. For IIR FFBF lters, we found a unied expression for the frequency response of the optimal lters valid for LE, DFE, and 123 Chapter 4. Cooperative FFBF with Multiple MultiAntenna 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 FFBF lters, we showed that a dicult nonconvex optimization problem results and proposed a simple and ecient gradient algorithm to nd nearoptimal lter coecients. Our simulation results conrmed that (1) the performance gap between FFBF lters with LE/DFE and FFBF lters with an idealized MF receiver is relatively small implying that little can be gained by employing more complex trellisbased equalization schemes at the destination, (2) relatively short FIR FFBF lters closely approach the performance of IIR FFBF lters for all considered receiver structures conrming the nearoptimal 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 FFBF lters are used and/or few relays are employed, equalization at the destination is benecial; 5) if long FIR FFBF lters are employed, the simple slicer destination with optimized decision delay closely approaches the same performance as destinations with equalizers. 124 Chapter 5 TwoWay FilterandForward Beamforming for FrequencySelective Channels with Multiple Single Antenna Relays 5.1 Introduction Drawing from the ndings on oneway relaying in the previous chapter, we investigate FFBF for twoway cooperative relay networks in this chapters. Particularly, we consider FFBF for twoway cooperative networks with two transceivers communicating with each other over frequencyselective channels via multiple singleantenna 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 FFBF lter design problems are substantially more challenging than those for oneway 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 FFBF lters at the relays for a SINR balancing objective under a relay transmit power 125 Chapter 5. TwoWay FFBF 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 FFBF 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 ecient otheself 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 increasing FFBF lter length and increasing number of relays. Furthermore, for suciently long FFBF lters and a suciently 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 FFBF 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 discretetime overall transmission system in equivalent complex baseband representation is depicted in Fig. 5.1. The adopted 126 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays Relay 1 g1[k] h1[k] gi[k] s1[k] hi[k] s2[k] ni[k] ai[k] gNR [k] hNR [k] Relay NR 1st Transmission Interval Relay 1 g1[k] sˆ2[k] h1[k] gi[k] EQ/Slicer hi[k] ai[k] v1[k] EQ/Slicer sˆ1[k] v2[k] gNR [k] hNR [k] Relay NR 2nd Transmission Interval Figure 5.1: Cooperative twoway network with two transceiver nodes and EQ is the equalizer at the transceivers. sˆ1 [k] and sˆ2 [k] NR relay nodes. are estimated received symbols at TC2 and TC1, respectively. twoway MABC relay protocol involves only two transmission intervals. In the rst interval, 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 discretetime CIRs between transceiver 1 (TC1) and relay 1, and between transceiver 2 (TC2) and relay i, gi [k], 0 ≤ k ≤ Lg − i, hi [k], 0 ≤ k ≤ Lh − 1, contain the combined eects of transmit pulse shaping, the continuoustime channel, receive ltering, and sampling. Here, Lg and Lh denote the lengths of the TC1relay and the TC2relay channels, respectively. 127 Chapter 5. TwoWay FFBF 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] relay i to the other transceiver. TC1 can then obtain the channel between TC2 and from hi [k] ∗ gi [k] and feedback channel from TC2. gi [k], i = 1, . . . , NR , TC2 can obtain via deconvolution or via a low rate hi [k], i = 1, . . . , NR , in a similar manner. Subsequently, one of the two transceivers computes the optimal FFBF lters adopting the algorithms proposed in Sections 5.3 and 5.4 and broadcasts the lter coecients to relay i ai [k] and the other transceiver via an errorfree and zerodelay feedback channel. 5.2.1 FFBF at Relays In the rst phase of transmission, TCj transmits the i.i.d. symbols are taken from a scalar symbol alphabet A such as phaseshift keying (PSK) or quadrature amplitude modulation (QAM), and have variance received at the ith relay, i = 1, . . . , NR , sj [k], j ∈ {1, 2}, which σs2j E{|sj [k]|2 }, j ∈ {1, 2}. The signal is given by yi [k] = gi [k] ∗ s1 [k] + hi [k] ∗ s2 [k] + ni [k] , where ni [k] denotes AWGN with variance σn2 E{|ni [k]|2 }. The FFBF lter impulse response coecients of relay interval are denoted by ai [k], −ql ≤ k ≤ qu . and for FIR FFBF lters ql = 0 and (5.1) i for the second transmission For IIR FFBF lters qu = La − 1, where La ql → ∞ and qu → ∞ is the FIR BF lter length. 128 Chapter 5. TwoWay FFBF 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 r˜2 [k] = = NR ∑ i=1 NR ∑ hi [k] ∗ ti [k] + v2 [k] hi [k] ∗ ai [k] ∗ gi [k] ∗ s1 [k] + i=1 NR ∑ + NR ∑ hi [k] ∗ ai [k] ∗ hi [k] ∗ s2 [k] i=1 hi [k] ∗ ai [k] ∗ ni [k] + v2 [k] , (5.3) i=1 where hi [k], vj [k] denotes AWGN with variance σv2j , j ∈ {1, 2}. and ai [k], i = 1, . . . , NR , It is noteworthy that since s2 [k], are known at TC2, the second term on the right hand side of (5.3) can be subtracted from r˜2 [k] to extract the information symbols before the residual signal s1 [k]. r2 [k] is further processed Similar considerations hold for TC1. Thus, the residual received signal at TCj can be expressed as rj [k] = heq [k] ∗ si [k] + vj′ [k] , where i=1 if j =2 and i=2 if TC1 and TC2 heq [k] j =1 NR ∑ j ∈ {1, 2} , (5.4) and we introduced the equivalent CIR between hi [k] ∗ ai [k] ∗ gi [k] , (5.5) i=1 " 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. 129 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays and the eective noise v1′ [k] v2′ [k] NR ∑ i=1 NR ∑ gi [k] ∗ ai [k] ∗ ni [k] + v1 [k] , (5.6) hi [k] ∗ ai [k] ∗ ni [k] + v2 [k] . (5.7) i=1 We note that vj′ [k], j ∈ {1, 2}, is colored noise because of the ltering with the TCrelay CIRs and the FFBF lters. 5.3 FIR FFBF without Equalization In practice, it is conceivable that the transceiver nodes cannot aord 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 FFBF lter optimization for transmit power minimization in twoway relaying networks has been considered independently in [86]. In particular, [86] deals with FFBF for twoway 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 maxmin SINR maximization 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 oneway relaying, such decision delay parameter leads to signicant performance improvements. The vector containing the coecients of the equivalent CIR between TC1 and TC2, 130 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays [heq,2 [0] heq,2 [1] . . . heq,2 [La + Lg + Lh − 3]]T , heq heq = NR ∑ can be rewritten as HGD a ¯ i ai H iG (5.8) i=1 where H [H 1 . . . H NR ], GD { } ¯ 1, . . . , G ¯ N , and a diag G R Lh − 2) × (La + Lg − 1) matrix H i matrices with length H (La + Lg − 1) × La [hi [0] . . . hi [Lh − 1] 0TLa +Lg −2 ]T columns, respectively, and Matrix and ai is the matrix are columncirculant [gi [0] . . . gi [Lg − 1] 0TLa −1 ]T k0 th hk0 and a submatrix row of matrix H, and Hk0 1, k0 + 1, . . . (La + Lg + Lh − 2)}, j ∈ {1, . . . , (La + Lg − 1)NR }. i = 1, ¯i G in the rst [ai [0] ai [1] . . . ai [La − 1]]T . can be separated into a vector (La + Lg − 1)NR and [aT1 . . . aTNR ]T . (La +Lg + Hk0 , i.e., vector hTk0 of [H]ij , i ∈ {1, . . . , k0 − Therefore, for j=2 and the rst term in (5.4) can be decomposed into a signal part and an ISI part La +Lg +Lh −3 heq [k] ∗ s1 [k] = heq [k0 ]s1 [k − k0 ] + ∑ heq [l]s1 [k − l] l=0, l̸=k0 = hTk0 GD as1 [k − k0 ] + sT1 [k]Hk0 GD a desired signal where k0 (5.9) ISI s1 [k] = [s1 [k] . . . s1 [k − k0 + 1] s1 [k − k0 − 1] . . . s1 [k − (La + Lg + Lh − 3)]]T , and is the slicer decision delay at the transceiver. We note that for oneway relaying such a decision delay was not introduced in [50]. However, as will be shown in Section 5.5, for twoway relaying a decision delay is highly benecial. The power of the desired signal and the ISI can be obtained as E { hTk0 GD as1 [k − k0 ] 2 } H ∗ hk0 hTk0 GD a = σs21 aH GD (5.10) 131 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays and E respectively. Similarly, { sT1 [k]Hk0 GD a v2′ [k] v2′ [k] = 2 } H H = σs21 aH GD Hk0 Hk0 GD a , (5.11) in (5.7) can be rewritten as NR ∑ nTi [k]H i ai + v2 [k] nT [k]HD a + v2 [k] (5.12) i=1 with row vector n[k] NR La matrix HD and ¯i H is a 1] 0TLa −1 ]T [nT1 [k] . . . nTNR [k]]T of length (La +Lh −1)NR and (La +Lh −1)NR × { } ¯ 1, . . . , H ¯ N , where ni [k] [ni [k] . . . ni [k −(La +Lh −2)]]T diag H R (La + Lh − 1) × La columncirculant matrix with vector [hi [0] . . . hi [Lh − in the rst column. The noise power is obtained as H E{|v2′ [k]|2 } = σn2 aH HD HD a + σv22 . (5.13) The SINR at TC2 can be obtained by combining (5.9)(5.11), and (5.13) and is given by { 2 } E − k0 ] { } 2 E |sT1 [k]Hk0 GD a| + E{|v2′ [k]|2 } SINRslicer, 2 (a) hTk0 GD as1 [k H ∗ σs21 aH GD hk0 hTk0 GD a = . H H H σs21 aH GD Hk0 Hk0 GD a + σn2 aH HD HD a + σv22 (5.14) Similarly, the SINR at TC1 is given by H ∗ T σs22 aH HD g k0 g k0 HD a , SINRslicer, 1 (a) = 2 H H H H GD a + σv21 σs2 a HD Gk0 Gk0 HD a + σn2 aH GD where Here, g Tk0 G is the k0 th row of matrix [G1 . . . GNR ] with G and matrix Gk0 is matrix G without the (5.15) k0 th row. (La + Lg + Lh − 2) × (La + Lh − 1) columncirculant matrix 132 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays Gi which has vector [gi [0] . . . gi [Lg − 1] 0TLa +Lh −2 ]T From (5.2), the total relay transmit power, is given by in the rst column. PR (a), in the second transmission interval NR ∑ { } PR (a) = E |ti [k]|2 = aH Da (5.16) i=1 with D H H σs21 GD GD + σs22 HD HD + σn2 I La NR . In the following two subsections, we will optimize the FIR FFBF lters for (a) maximization 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 Maxmin Criterion Under Relay Power Constraint First, we consider the optimization of the FFBF 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)} s.t. aH Da ≤ P . (5.17a) (5.17b) 133 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays Equivalently, problem (5.17) can be rewritten as max t (5.18a) a Realizing that H ∗ H ∗ HD g k0 = GD hk0 , s.t. SINRslicer, 1 (a) ≥ t (5.18b) SINRslicer, 2 (a) ≥ t (5.18c) aH Da ≤ P . (5.18d) we let q H ∗ HD g k0 , and reformulate problem (5.18) as max t (5.19a) c s.t. cH V¯ (t)c ≤ cH q¯ q¯ H c where with c V [ (D 1/2 a)T σs22 H H HD Gk0 Gk0 HD σv21 σs21 H H G H H G σv22 D k0 k0 D W ]T 1 , q¯ + + (5.19b) ¯ (t)c ≤ cH q¯ q¯ H c cH W (5.19c) cH c ≤ P + 1 (5.19d) [c]NR La +1 = 1 , (5.19e) [ (D −H/2 q)T 2 σn G H G , and σv21 D D ]T 0 , V¯ (t) ¯ (t) W tσv22 σs21 { } diag D −H/2 V D −1/2 , 1 { } diag D −H/2 W D −1/2 , 1 with tσv21 σs22 2 σn H HD HD . σv22 Note that multiplying the optimal c by ejθ , where θ is an arbitrary phase, does not aect the objective function or the constraints for problem (5.19). Therefore, we can assume that q¯ H c is a real number without loss of generality. Thus, for a given t, problem (5.19) can be 134 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays transformed into the convex SOCP feasibility problem min c s.t. −t 1/2 V¯ (t)c ≤ q¯ H c (5.20b) ¯ 1/2 (t)c ≤ q¯ H c W (5.20c) ∥c∥ ≤ Consequently, for a given t, (5.20a) √ P +1 (5.20d) [c]NR La +1 = 1 . (5.20e) problem (5.20) can be eciently solved using interior point methods [88] and a bisectional search can be used to nd the optimal optimal FFBF lter vector aopt t [1]. Since the can be directly obtained from the solution of (5.20), we have provided an ecient procedure for computation of the optimal FFBF 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 aH Da a s.t. SINRslicer, 1 (a) ≥ γ1 SINRslicer, 2 (a) ≥ γ2 . (5.21a) (5.21b) (5.21c) 135 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays Equivalently, problem (5.21) can be reformulated as min cH c − 1 (5.22a) c s.t. cH q¯ q¯ H c ≥ cH V¯ (γ1 )c ¯ (γ2 )c , cH q¯ q¯ H c ≥ cH W where ¯ (·), c, V¯ (·), W and multiplying the optimal c q¯ by are dened after (5.19). ejθ (5.22b) (5.22c) By exploiting again the fact that does not aect the objective function or the constraints q¯ H c of problem (5.22), we can assume is a real number without loss of generality and transform problem (5.22) into an SOCP problem min τ (5.23a) c s.t. 1/2 V˜ c ≤ q¯ H c (5.23b) ˜ 1/2 c ≤ q¯ H c W (5.23c) ∥c∥ ≤ τ (5.23d) [c]NR La +1 = 1 . (5.23e) The SOCP problem (5.23) can again be eciently solved using interior point methods [88]. 5.4 FFBF with Equalization If only a simple slicer is employed at the transceivers, the FFBF lters at the relays are burdened with equalizing both TCrelay 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, 136 Chapter 5. TwoWay FFBF 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 twoway relay network with FFBF can be modeled as an equivalent SISO system. Therefore, as long as the transceivers know the CIRs of all involved channels and the coecients of the FFBF lter, the same equalization techniques as for pointtopoint singleantenna 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, suciently 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 FFBF, we consider both IIR lters, which provide performance bounds, and FIR lters, which are required for practical implementation. 5.4.1 Optimal IIR FFBF 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 v1′ [k] and v2′ [k] at the two transceivers are given by Φvj′ (f ) = σn2 aH (f )Γj (f )a(f ) + σv2j , where a(f ) Γ1 (f ) diag{|G1 (f )|2 , . . . , |GNR (f )|2 }, Γ2 (f ) [A1 (f ), . . . , ANR (f )]T . Gi (f ) F{gi [k]}, Hi (f ) j ∈ {1, 2} , (5.24) diag{|H1 (f )|2 , . . . , |HNR (f )|2 }, and F{hi [k]}, and Ai (f ) F{ai [k]} 137 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays denote the frequency responses of the TC1ith relay channel, TC2ith relay channel, and the FFBF lter at the ith relay, respectively. Therefore, the whitening lter for vj′ [k] is given by ( )−1/2 Wj (f ) = σn2 aH (f )Γj (f )a(f ) + σv2j . (5.25) After whitening, the frequency response of the equivalent overall channel at transceiver j can be obtained as ′ Heq,j (f ) where heq,j [k] Wj (f )F{heq,j [k]} )−1/2 ( , = q T (f )a(f ) σn2 aH (f )Γj (f )a(f ) + σv2j hj [k] ∗ aj [k] ∗ gj [k], q(f ) [Q1 (f ) . . . QNR (f )]T j ∈ {1, 2} , and Qi (f ) (5.26) 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] ∫1/2 SINRDFE, j (a(f )) = Psj exp ( ′ ) ln |Heq,j (f )|2 + ξj df −1/2 − χ, (5.27) and SINRLE, j (a(f )) = Psj ∫1/2 −1 ( ′ |Heq,j (f )|2 + ξj )−1 df − χ, (5.28) −1/2 138 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays respectively, where ′ |Heq,j (f )|2 = In (5.27) and (5.28), we have aH (f )q ∗ (f )q T (f )a(f ) . σn2 aH (f )Γj (f )a(f ) + σv2j χ = 0, ξ1 = ξ2 = 0 and (5.29) χ = 1, ξ1 = 1/σs22 , ξ2 = 1/σs21 if the equalization lters are optimized based on a ZF and an MMSE criterion, respectively. Also, we dene Ps1 σs22 Ps2 and σs21 . Similarly, if only a single isolated symbol is transmitted, the SINR at the output of an MF is given by [5] ∫1/2 SINRMF, j (a(f )) = Psj ′ |Heq,j (f )|2 df . (5.30) −1/2 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 )) = 1/2 NR ∫ ∑ i=1 where Φti (f ) PR (a(f )), ∫1/2 aH (f )D(f )a(f )df , Φti (f )df = −1/2 ti [k] at the ith (5.31) −1/2 ( ) |Ai (f )|2 σs21 |Gi (f )|2 + σs22 |Hi (f )|2 + σn2 the transmit signal is given by relay, and D(f ) is the power spectral density of σs21 diag{|G1 (f )|2 , . . . , |GNR (f )|2 } + σs22 diag{|H1 (f )|2 , . . . , |HNR (f )|2 } + σn2 I NR . Maxmin Criterion Under Relay Power Constraint In analogy to Section 5.3.1, we consider rst the optimization of the FFBF 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 139 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays problem can be formulated as min {SINRX,1 (a(f )), SINRX,2 (a(f ))} max a(f ) (5.32a) ∫1/2 aH (f )D(f )a(f )df ≤ P , s.t. (5.32b) −1/2 where X = DFE, X = LE, and X = MF for DFE, LE, and an MF receiver, respectively. Unfortunately, problem (5.32) is very dicult to solve because of the structure of the SINR expressions in (5.27), (5.29), and (5.30) and the fact that Γ1 (f ) ̸= 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) ∫1/2 aH j (f )D(f )aj (f )df ≤ P , s.t. (5.33b) −1/2 where j ∈ {1, 2}. Let aopt j (f ) denote the optimum solution for problem (5.33). Since (5.33) is equivalent to the optimization of the FFBF lters for two oneway relaying systems, we can draw from the results in Chapter 4 and [82]. In particular, aopt j (f ), j ∈ {1, 2, }, can be eciently obtained with the algorithm summarized in Table 4.1. Based on these FFBF 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 { } opt min SINRX,1 (aopt 1 (f )), SINRX,2 (a2 (f )) (5.34) 140 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays and the achievable lower bound is SINRlow { { } opt max min SINRX,1 (aopt 1 (f )), SINRX,2 (a1 (f )) , { }} opt min SINRX,1 (aopt , 2 (f )), SINRX,2 (a2 (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. opt SINRX,1 (aopt 1 (f )) ≤ SINRX,2 (a1 (f )) or opt SINRX,2 (aopt 2 (f )) ≤ SINRX,1 (a2 (f )), If which typ- ically occurs if the relays are closer to one transceiver than the other, cf. Section 5.5, SINRup = SINRlow SINRup ̸= SINRlow and the optimal solution for problem (5.32) is obtained. Otherwise, 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 closetooptimal solution to problem (5.32). The small gap between both bounds can be explained by the fact that the only dierence between the equivalent TC1TC2 and TC2TC1 channels is the noise correlation in (5.24), which has a minor impact on the design of the FFBF lters. Minimization of the Sum of MSEs As an alternative FFBF 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 ZFLE but not for the other considered equalization schemes. Thus, we concentrate on the ZFLE case in the following. For ZFLE, the MSE at the output of the equalizer at TCj is given by ∫1/2 2 (a(f )) = σLE,j ′ (f )|−2 df . |Heq,j (5.36) −1/2 141 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays i.e., 2 SINRLE,j (a(f )) = Psj /σLE,j (a(f )). The considered optimization problem can be ex- pressed as min JLE (a(f )) = a(f ) 2 ∑ 2 σLE,j (a(f )) (5.37a) j=1 ∫1/2 aH (f )D(f )a(f )df ≤ P. s.t. (5.37b) −1/2 Exploiting (5.29) the objective function (5.37a) can be expressed as ∫1/2 JLE (a(f )) = ( ) σn2 aH (f ) Γ1 (f ) + Γ2 (f ) a(f ) + σv21 + σv22 df . aH (f )q ∗ (f )q T (f )a(f ) (5.38) −1/2 Next, we introduce matrix Γ(f ) 2 σn (Γ1 (f ) σv21 +σv22 + Γ2 (f )) and restate problem (5.37) in equivalent form as max a(f ) ∫1/2 ( aH (f )q ∗ (f )q T (f )a(f ) aH (f )Γ(f )a(f ) + 1 )−1 −1 df (5.39a) −1/2 ∫1/2 aH (f )D(f )a(f )df ≤ P. s.t. (5.39b) −1/2 Since Γ(f ) is a diagonal matrix, problem (5.37) is of the same form as the ZFLE SINR maximization problem for oneway 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. 142 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays 5.4.2 FIR FFBF Filter Optimization Since the IIR FFBF lters would require an innite amount of feedback, they are mostly useful to establish performance bounds for practical FIR FFBF lters. We emphasize that although FIR FFBF lters are considered in this section, the equalizers at the transceivers are still assumed to employ IIR lters. Assuming FIR FFBF lters of length heq [k] is given by La Leq = La + Lg + Lh − 2 at the relays the length of the equivalent CIR and its Fourier transform can be expressed as Heq (f ) = dH (f )Qa with d(f ) [1 ej2πf . . . ej2πf (Leq −1) ]T , FIR FFBF coecient vector a1 [La − 1] a2 [0] . . . aNR [La − 1]]T , and Leq × NR La an (5.40) matrix in the rst column and is in the rst column. Here, [hi [0] . . . hi [Lh −1] 0TLg −1 ]T columncirculant matrix with vector g˜ i [a1 [0] a1 [1] . . . [Q1 . . . QNR ], where Qi Q ˜ i g˜ i )T 0T ]T Leq ×La columncirculant matrix with vector [(H La −1 ˜ i is an (Lh +Lg −1)×Lg H a [gi [0] . . . gi [Lg − 1]]T . We apply again noise whitening which transforms Heq (f ) into the equivalent frequency responses of TC1 and TC2: ( )−1/2 ′ ˜ j (f )a + σ 2 Heq,j (f ) = dH (f )Qa σn2 aH Γ , vj with La NR × La NR {1, 2}, where block diagonal matrices ˜ 1,i (f ) Γ matrices of rank 1. Here, ˜ )d ˜ H (f )G ˜i ˜ H d(f G i ˜i G and circulant matrices with vectors ˜i H are ˜ j (f ) Γ and ˜ 2,i (f ) Γ (Lg + La − 1) × La ˜ ) d(f (5.41) { } ˜ j,1 (f ), . . . , Γ ˜ j,N (f ) , j ∈ diag Γ R [gi [0] . . . gi [Lg −1] 0TLa −1 ]T in the rst columns, respectively, and j ∈ {1, 2}, ˜ )d ˜ H (f )H ˜i ˜ H d(f H i and and are La × La (Lh + La − 1) × La column [hi [0] . . . hi [Lh −1] 0TLa −1 ]T [1 ej2πf . . . ej2πf (Lh +La −2) ]T . The noise power spectral density at the output of the noise whitening lter is again equal to one. 143 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays In the following, we will discuss the optimization of the FFBF coecient vector a for the two criteria considered in Section 5.4.1. Maxmin criterion under relay power constraint: Similar to the IIR case, an exact solution of the maxmin FIR FFBF 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 oneway relaying setups having equivalent channel frequency responses and ′ Heq,2 (f ), ′ Heq,1 (f ) respectively. Comparing the equivalent frequency response in (5.41) with the corresponding frequency response for the oneway relaying case in (4.64) or [82, Eq. (38)], we conclude that optimal FIR FFBF coecient vectors aopt 1 and aopt 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 closetooptimal solution for maxmin optimization of the FIR FFBF lters for the twoway relaying is available. Minimization of the sum of MSEs: For FIR BFFF with ZFLE receivers, the sum MSE can be written as ∫1/2 ( JLE (a) = (σv21 + σv22 ) −1/2 where ˜ Γ 2 σn ˜ 1 (f ) (Γ σv21 +σv22 ˜ 2 (f )). +Γ aH QH d(f )dH (f )Qa ˜ )a + 1 aH Γ(f )−1 df , (5.42) Now, the FIR FFBF lter optimization problem can be written as max 1/JLE (a) a s.t. aH Da ≤ P , where D (5.43a) (5.43b) is dened after (5.16). Problem (5.43) is of the same form as the FIR FFBF 144 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays lter optimization problem for oneway relaying with ZFLE at the receiver. can again use the algorithm given in Table 4.2 to nd the optimal vector Thus, we aopt . 5.5 Simulations In this section, we present simulation results for the SINR and the BER of a cooperative σn2 = σv21 = √ gi [k] = γg g¯i [k] and twoway relay network with FFBF. Throughout this section we assume σv22 = 1 hi [k] = and √ P = 1. ¯ i [k], γh h This allows us to decompose the CIRs as where γg and γh denote the transmitter SNRs of the TC1relay and TC2relay links, respectively. The normalized CIRs ¯ i [k] h and g¯i [k] include the eects of multipath fading and pathloss. All IIR and FIR FFBF 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 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 exponent of 3 dref = 1 with reference distance an exponential power delay prole p[k] = 1 σt d. A pathloss is assumed. The CIR coecients of all links are modeled as independent quasistatic Rayleigh fading with σt NR = 5 relays ∑Lx −1 l=0 Lg = Lh = 5 and following e−k/σt δ[k − l], where Lx ∈ {Lg , Lh } and characterizes the delay spread [85]. 5.5.1 Relay Power Minimization for FFBF without Equalization Fig. 5.3 shows the total relay transmit power, and γ2 at the transceivers for γ1 = γ2 . PR (a), We adopted vs. the minimum required SINR γ1 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 145 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays (a) 1/4 (b) (c) d Transceiver 1 1/4 2−d Transceiver 2 1/4 (d) (e) 1/4 Figure 5.2: Locations of TC1, TC2, and the relays in the simulations. 15 FIR FF−BF w/o EQ (La=1) FIR FF−BF w/o EQ (L =3) a FIR FF−BF w/o EQ (L =5) a FIR FF−BF w/o EQ (L =7) 10 a FIR FF−BF w/o EQ (L =9) a FIR FF−BF w/o EQ (La=11) PR (a) [dB] 5 0 −5 −10 −15 −6 −4 −2 0 2 4 γ1 = γ2 6 8 10 12 [dB] Figure 5.3: Total average relay transmit power vs. required SINRs γ1 and γ2 for FIR FFBF without equalization at the transceivers. The FFBF lters were optimized for minimization of the relay transmit power. with σt = 2, Lg = Lh = 5, d = 1, NR = 5, Exponentially decaying power delay prole and γg = γh = 10 dB. 146 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays 1 0.9 0.8 Feasibility Probability 0.7 0.6 0.5 0.4 0.3 FIR FF−BF w/o EQ (La=1) 0.2 FIR FF−BF w/o EQ (La=3) FIR FF−BF w/o EQ (L =5) a 0.1 FIR FF−BF w/o EQ (La=7) FIR FF−BF w/o EQ (La=9) FIR FF−BF w/o EQ (L =11) a 0 −2 0 2 4 6 8 γ1 = γ2 of the relay transmit power. Lg = Lh = 5, d = 1, NR = 5, 12 14 γ1 and γ2 for FIR FFBF without The FFBF lters were optimized for minimization Exponentially decaying power delay prole with and is feasible is shown in Fig. 5.4. 16 [dB] Figure 5.4: Feasibility probability vs. required SINRs equalization at the transceivers. 10 γg = γh = 10 σt = 2, dB. 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. 147 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays Average min{SINR1 , SINR2 } [dB] 10 9 8 7 6 5 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, La=1 4 3 FIR FF−BF, La=5 FIR FF−BF, L =11 2 a FIR FF−BF (MMSE−DFE Lower Bound) FIR FF−BF (MMSE−DFE Upper Bound) FIR FF−−BF (w/o EQ) 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d Figure 5.5: Average worstcase SINR at the transceivers vs. distance with/without equalization at the transceivers. d for FFBF The FFBF lters were optimized for maximization of the minimum transceiver SINR. Exponentially decaying channel power delay prole with σt = 2, Lg = Lh = 5, d = 1, NR = 5, and γg = γh = 10 dB. 5.5.2 Maxmin SINR Optimization for FFBF with and without Equalization In Figs. 5.5 and 5.6, we show the average SINR at the transceivers vs. distance d for various FFBF lter designs at the relays and various transceiver structures. We adopted NR = 5, and γg = γh = 10 σt = 2 , dB. The FFBF lters were optimized for maximization of the minimum transceiver SINR. 148 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays 10 9 8 Average SINR [dB] 7 6 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 =5) 5 4 a FIR FF−BF (w/o EQ, L =5) a 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d Figure 5.6: Average SINR at transceivers vs. distance d for FFBF with/without equal- ization (EQ) at the transceivers. The FFBF lters were optimized for maximization of the minimum transceiver SINR. Exponentially decaying channel power delay prole 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 FFBF with equalization is very small for both IIR and FIR FFBF 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 closetooptimal. Furthermore, Fig. 5.5 shows that transceivers employing MMSEDFE closely approach the performance of idealized MF receivers if IIR FFBF lters are adopted implying that little can be gained by employing 149 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays more complex trellisbased receivers compared to MMSEDFE. Also, as the length of the FIR FFBF lters increases, FIR FFBF approaches the performance of IIR FFBF. La = 5 For and MMSEDFE receivers, the gap between both schemes is less than 0.6 dB over considered range of distances d. Interestingly, while for short FIR FFBF lters equalization at the transceivers results in large performance gains, FIR FFBF without equalization with La = 11 MMSEDFE with La = 5. achieves practically the same performance as FIR FFBF with We note that FFBF with MMSEDFE with outperforms FIR FFBF without equalization with La = 11 La = 11 slightly 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 ZFLE at the transceivers can approach the performance of an idealized MF receiver up to less than one decibel. 5.5.3 Maxmin SINR vs. Minimum Sum MSE Optimization for FFBF with ZFLE In Fig. 5.7, we compare the average SINRs at both transceivers for ZFLE at the transceivers with maxmin and minimum sum MSE FFBF optimization. We adopted and γg = γh = 10 σt = 2, NR = 5, dB. For the maxmin 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- 150 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays 10 9 8 7 6 Average SINR [dB] 5 4 3 SINR @ Transceiver 2 SINR @ Transceiver 1 IIR FF−BF (Min Sum MSE) FIR FF−BF (Min Sum MSE, La=7) 2 1 FIR FF−BF (Min Sum MSE, La=1) IIR FF−BF (Max−Min) FIR FF−BF (Max−Min, L =7) 0 a FIR FF−BF (Max−Min, La=1) −1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d Figure 5.7: Average SINR at transceivers vs. distance d for FFBF with ZFLE at the transceivers. Exponentially decaying channel power delay prole with 5, d = 1, NR = 5, and γg = γh = 10 σt = 2, Lg = Lh = dB. mization requires only the computation of one FFBF lter, its complexity is roughly half of that of the maxmin optimization. Thus, in practice, the minimum sum MSE criterion may be preferable if ZFLE 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 various FFBF and equalizer designs for σt = 2, d = 1, and NR on the performance of γg = γh = 10 dB. We assume 151 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays Average min{SINR1 , SINR2 } [dB] 16 14 12 10 8 6 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 =1) 4 a FIR FF−BF (La=5) FIR FF−BF (L =11) 2 a FIR FF−BF (MMSE−DFE, Max−Min) FIR FF−BF (ZF−LE, Min Sum MSE) FIR FF−BF (w/o EQ) 0 2 4 6 8 10 12 14 16 NR Figure 5.8: Average SINR vs. number of relays NR for FFBF with MMSEDFE, ZFLE, MF, and slicer (no equalizer) receivers at the transceivers. Exponentially decaying power delay prole 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 MMSEDFE, MF, and slicer (no equalizer) receivers with the FFBF lters optimized for maximization of the minimum transceiver SINR. For MMSEDFE and MF receivers with IIR FFBF 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 ZFLE with FFBF 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 152 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays for maxmin FFBF lter optimization with equalization is very small. As NR increases the gap between the simple slicer receiver and the MMSEDFE receiver diminishes. fact, the slicer receiver with FIR FFBF lters of lenght La = 11 In closely approaches the performance of MMSEDFE with IIR FFBF lter and FIR FFBF lters of length La = 11 (which is not shown for clarity), but outperforms MMSEDFE with FIR FFBF 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, for FIR and IIR FFBF lters. γ = γg = γh , The BERs for FIR FFBF lters were simulated by implementing ZFLE and MMSEDFE receivers with FIR equalization lters of lengths 4 × Leq , lters. which caused negligible performance degradation compared to IIR equalization The BERs for IIR FFBF 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 FFBF with equalization and maxmin criterion were generated by using the FFBF 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 MMSEDFE with IIR FFBF at the relays closely approaches the performance of a MF receiver with IIRBF. Furthermore, FIR FFBF lters of moderate length (La = 5) approach the performance of IIR FFBF lters up to less than one decibel if MMSEDFE is employed at the transceivers. The same performance can also be achieved without equalization at the transceivers but with longer FFFB lters (La BER = 10−5 , slicer (no equalizer) receivers with La = 11 achieve only 0.4 = 11). At dB performance lost comparing with the performance of the MMSEDFE receivers with FIR FFBF and 153 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays −1 10 −2 10 −3 BER 10 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 (La=1) −4 10 FIR FF−BF (La=5) FIR FF−BF (L =11) a FIR FF−BF (MMSE−DFE) FIR FF−BF (w/o EQ) 0 2 4 6 8 γg = γh 10 12 14 16 [dB] Figure 5.9: Average BER of BPSK vs. transmit SINR γ for FFBF with MMSEDFE, MF, and slicer receiver at the transceivers. BERs for FIR FFBF with EQ and IIR FF BF with MMSEDFE were generated using the FFBF lters from the achievable lower bound of the maxmin criterion. Exponentially decaying power delay prole with Lg = Lh = 5, NR = 5, and σt = 2, d = 1. La = 11. Fig. 5.10 reveals that for ZFLE at the transceivers, FFBF lters according to the maxmin and sum MSE criteria achieve a similar BER performance. Furthermore, the performance gap between the MF receiver and the simple ZFLE receiver with IIR FFBF lters is less than one decibel which suggests again that simple LE and DFE equalizers are sucient to achieve a closetooptimal performance. 154 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays −1 10 −2 10 −3 BER 10 Analytical IIR FF−BF (MF, Max−Min) Analytical IIR FF−BF (ZF−LE, Max−Min) FIR FF−BF (La=1) −4 10 FIR FF−BF (La=5) FIR FF−BF (ZF−LE, Max−Min) FIR FF−BF (ZF−LE, Sum Min MSE) 0 2 4 6 8 γg = γh 10 12 14 16 [dB] Figure 5.10: Average BER of BPSK vs. transmit SINR γ for FFBF with ZFLE and MF receiver at the transceivers. For the minmax criterion, BERs were generated using the FFBF lters from the achievable lower bound. Exponentially decaying power delay prole with σt = 2, Lg = Lh = 5, NR = 5, and d = 1. 5.6 Conclusions In this chapter, we have investigated FFBF for twoway relaying networks employing singlecarrier transmission over frequencyselective channels. For the processing at the transceivers, we considered two dierent cases: (1) a simple slicer without equalization and (2) LE or DFE. For the rst case, we optimized FIR FFBF 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. 155 Chapter 5. TwoWay FFBF with Multiple Single Antenna Relays Both problems can be transformed into convex SOCP problems, which can be eciently solved with standard numerical methods. For the second case, we optimized FIR and IIR FFBF 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 maxmin criterion, we established an upper and an achievable lower bound for the original problem. Both optimization problems were solved by transforming them into oneway relay problems and leveraging corresponding results from Chapter 4. From our simulation results, we can draw the following conclusions: for maxmin optimization with equalization, the gap between the upper bound and the achievable lower bound is very small rendering the obtained solution closetooptimal; and for ZFLE the maxmin 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 lowcomplexity 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 FFBF lters and equalizers at the relay and the transceivers, respectively. 156 Chapter 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 AFBF schemes with multiple multiantenna relays and multiantenna source; (3) oneway cooperative FFBF schemes for frequencyselective channels with multiple multiantenna relays; (4) twoway cooperative FFBF schemes for frequencyselective channels with multiple singleantenna relays. In the following, we briey review the main results of each chapter. In Chapter 2, we have proposed a novel TD approach to BF in MIMOOFDM systems. The CBFFs have been optimized for maximization of the AMI and minimization of the BER, respectively, and ecient algorithms for recursive calculation of the optimum C BFFs have been provided for both criteria. For the case of a niterate feedback channel a GVQ algorithm has been introduced for codebook design. Simulation results for the IEEE 802.11n Channel Model B have conrmed the excellent performance of TDBF and have 157 Chapter 6. Summary of Thesis and Future Research Topics shown that TDBF achieves a more favorable performance/feedback rate tradeo than traditional FDBF. In Chapter 3, we have considered AFBF for cooperative networks with one multi antenna source, multiple multiantenna relays, and one singleantenna destination for three dierent power constraints. In particular, we have considered the cases of individual relay power constraints, a joint power constraint for all relays, and a joint sourcerelay power constraint. For a given BF vector at the source, we have fully characterized the optimal AFBF matrices for all three constraints. Furthermore, optimal and suboptimal 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 benecial 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 benecial regardless of the location of the relays. In Chapter 4, we investigated FFBF for oneway relay networks with multiple multi antenna relays and singlecarrier transmission over frequencyselective 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 closedform solutions and ecient numerical methods for computation of the optimal FIR FFBF matrix lters. For the second case, we obtained an elegant method for calculation of the optimal IIR FFBF matrix lters and an ecient numerical algorithm for calculation of nearoptimal FIR FFBF matrix lters. In Chapter 5, we have investigated FFBF for twoway relay networks employing singlecarrier transmission over frequencyselective channels. Multiple single antenna relays are assumed in the network. For the processing at the transceivers, we again considered two dierent cases: (1) a simple slicer without equalization and (2) LE or DFE. For the 158 Chapter 6. Summary of Thesis and Future Research Topics rst case, we optimized FIR FFBF 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 eciently solved with standard numerical methods. For the second case, we optimized FIR and IIR FFBF lters for maximization of the minimum transceiver SINR and, in case of ZFLE, also for minimization of the sum MSE at the equalizer outputs of both transceivers. For the maxmin criterion, we established an upper and an achievable lower bound for the original problem. Both optimization problems were solved by transforming them into oneway 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 number 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 35, we have considered beamforming 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 diculty 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- 159 Chapter 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 Twoway Relaying with Multiple Multiantenna Relays One immediate extension of the current work is on beamforming schemes for twoway MABC relaying with multiple multiantenna relays. As a matter of fact, we have already made preliminary but encouraging progress on such topic. In [92], we assume singlecarrier transmission and frequencyselective channels. The relays are equipped with FFBF trix lters in contrast with FFBF lters in Chapter 5. ma- As shown in Chapter 5, the performance of a simple slicer with optimized decision delay can closely approach the performance of transceivers with equalizers. Therefore, we assume that a simple slicer is employed at each of the transceivers in [92]. We optimize the FFBF 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 transmit power subject to two QoS constraints to guarantee a certain level of performance. We show that the optimization problems are dicult to solve in general. However, by relaxing the rank constraints, we convert the optimization problems to semidenite programming (SDP) problems, which provide certied 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 closetooptimal 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] 160 Chapter 6. Summary of Thesis and Future Research Topics and conduct research on transceivers equipped with equalizers. 6.2.2 Cooperative Communications for Multiuser Systems Next generation mobile communication systems have to be able to provide reliable communications for a large number of users within a cell and on cell edges. Multiuser MIMO schemes can provide a substantial gain in network downlink throughput by allowing multiple users to communicate in the same frequency (or OFDM subcarrier) and time slots [93]. The combination of multiuser MIMOOFDM beamforing and relaying is a promising technique for performance enhancement for next generation wireless communications. Although some preliminary research has been already conducted on MIMOOFDM relaying system [94, 95] and multiuser MIMO relaying systems [96], there are still many interesting open problems such as resource allocation and protocol design. Since dierent users interfere with each other in a multiuser 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 multiuser 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 performance of cooperative communications. However, in practices, perfect timing is an unrealis- 161 Chapter 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 dicult to provide a precise clock reference for all the signal coming from distributed users with dierent prospectives. Literature on synchronization for cooperative networks is very sparse. Recent publication [97] considered frequency oset 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 at fading channels, and many open questions are still unanswered, e.g. the impact of synchronization error in frequency selective channels. 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Beamforming schemes for next generation wireless communication systems Liang, Yangwen 2011
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Title | Beamforming schemes for next generation wireless communication systems |
Creator |
Liang, Yangwen |
Publisher | University of British Columbia |
Date | 2011 |
Date Issued | 2011-10-13 |
Description | Multiple-input multiple-output (MIMO) and relaying are two promising techniques which will be employed in next generation wireless communication systems. Transmit beamforming (BF) and receive combining are simple yet popular methods for performance enhancement 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 filters (C-BFFs). Both perfect and partial channel state information at the transmitter are considered. The C-BFFs are optimized for maximum average mutual information per sub-carrier and minimum average uncoded bit error rate. We show that TD-BF has a more favorable performance/feedback rate trade-off than previously proposed 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 different power constraints. We show the benefits of having multiple antennas at the source and/or multiple multi-antenna relays. Subsequently, we investigate filter-and-forward BF (FF-BF) for one-way relay networks in frequency-selective channels. For the processing at the destination, we investigate two different cases: a simple slicer, and a linear equalizer (LE) or a decision-feedback equalizer (DFE). For both cases, we optimize the FF-BF matrix filters at the relays for maximization of the SINR under a transmit power constraint, and for the first case we consider additionally optimization of the FF-BF matrix filters 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. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2011-10-13 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial 3.0 Unported |
DOI | 10.14288/1.0072285 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2011-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc/3.0/ |
URI | http://hdl.handle.net/2429/37957 |
Aggregated Source Repository | DSpace |
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