適用於多輸入多輸出與多輸入多輸出中繼系統中最大似然接收機之前置編碼器設計

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(1)國立交通大學電信工程研究所博士論文適用於多輸入多輸出與多輸入多輸出中繼系統中最大似然接收機之前置編碼器設計 Precoder Designs for Maximum-Likelihood Detectors in MIMO and MIMO Relay Systems. 研究生：林鈞陶指導教授：吳文榕中. 華. 民. 國. 101. 博士年. 8. 月.

(2) 適用於多輸入多輸出與多輸入多輸出中繼系統中最大似然接收機之前置編碼器設計 Precoder Designs for Maximum-Likelihood Detectors in MIMO and MIMO Relay Systems 研究生：林鈞陶. Student: Chun-Tao Lin. 指導教授：吳文榕博士. Advisor: Dr. Wen-Rong Wu. 國立交通大學電信工程研究所博士論文 A Dissertation Submitted to Institute of Communication Engineering College of Electrical and Computer Engineering National Chiao Tung University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Communication Engineering Hsinchu, Taiwan 2012 年 8 月.

(3) 適用於多輸入多輸出與多輸入多輸出中繼系統中最大似然接收機之前置編碼器設計. 研究生: 林鈞陶. 指導教授: 吳文榕博士. 國立交通大學電信工程研究所博士班摘要. 前置編碼已被認為是一種可以有效改善多輸入多輸出 (Multiple-input multiple-output；MIMO)系統中傳輸品質的技術。一般而言，前置編碼的設計與接收機的類型有關。對於最大似然 (Maximum-likelihood；ML)接收機而言，最佳的前置編碼設計準則是最大化系統的自由距離，然而最佳解的導出相當困難，因此目前多數文獻僅討論其次佳解，本論文旨在探討對應 ML 接收機的前置編碼器設計問題。在論文的第一部分，我們首先考慮簡化之前置編碼，也就是傳送端天線選擇，最佳的天線選擇準則必需透過高運算量的窮舉法才能得到，為了避免此問題，研究者利用奇異值分解(Singular value decomposition；SVD)以及 QR 分解 (QR decomposition；QRD)推導出自由距離的下界。我們提出以 QRD 為基礎之天線選擇方法，我們證明了 QRD 選擇法會優於傳統的 SVD 法，此外我們進一步提出基底轉換的方法使得 QRD 選擇法能夠更接近最佳解。除了傳送端天線選擇之外，我們提出的方法也可適用於其他應用，例如接收端天線選擇、傳送端與接收端聯合天線選擇、以及在 MIMO 中繼系統中天線選擇。模擬結果顯示， iii.

(4) 我們提出的方式在上述應用中均能提供接近最佳的表現。. 除了天線選擇之外，最近有研究者提出一種基於 X-架構之次佳前置編碼方式，此方法的概念在於先利用 SVD 得到平行子通道，再利用兩兩配對的方式得到多個 2×2 的子系統，這樣的作法允許我們僅需設計 2×2 的子前置編碼器，因而得以在降低編碼的複雜度，此外 X-架構前置編碼的方式同時可提供一個低複雜度的 ML 接收機。然而，目前現有文獻中所提出的 X-架構前置編碼器的設計均仰賴數值法以及查表的方式，使得現有的方法在實際應用上困難度與複雜度均相對增加。在論文的第二部分我們提出一個簡單但有效的方法來解決此問題，我們的方法所得到的前置編碼矩陣具有解析解，另外，我們也探討如何將 X-架構前置編碼延伸至 MIMO 中繼系統中聯合前置編碼器設計的問題。模擬結果顯示我們提出的方法比現有的前置編碼法可以更有效的改善系統效能。. 前置編碼器的計算需要完整的通道資訊，因此一般都在接收端完成，在實際系統中，前置編碼的實現是透過從碼書中選取一個最佳的碼字，再經由迴授通道將該碼字的索引回傳至傳送端。在論文的最後一個部分，我們將討論如何建構X-架構前置編碼所需要的碼書，有別於傳統的前置編碼，X-架構前置編碼器需要兩種碼書，一種是用於一么正矩陣，另一種是用於子編碼器的矩陣，使用於么正矩陣的碼書所面臨的問題在於量化的矩陣不能讓系統保有X的架構，低複雜度ML接收機也因此不復存在，針對此問題，我們先證明我們所提出的X-架構前置編碼器仍然有效，接著我們提出一個低複雜的接收機架構來解決偵測的問題。模擬結果顯示，我們所提出的方法除了可以使用低複雜度的接收機外同時也可有效改善系統效能。. iv.

(5) Precoder Designs for Maximum-Likelihood Detectors in MIMO and MIMO Relay Systems Student:. Chun-Tao Lin. Advisor: Dr. Wen-Rong Wu. National Chiao Tung University Institute of Communication Engineering Abstract Precoding has been considered a promising technique in multiple-input multiple-output (MIMO) transmission. In general, the design criterion depends on the detector used at the receiver. For the maximum-likelihood (ML) detector, the criterion is known to maximize the free distance. Unfortunately, the derivation of the optimum solution is difficult, and suboptimum solutions have then been developed. In this dissertation, we study the precoder design for the ML detector in MIMO and MIMO relay systems. In the first part of this dissertation, we consider a simplified precoding scheme, namely, transmit antenna selection. To maximize the free distance, it is necessary to conduct exhaustive search for the selection pattern. To avoid the problem, lower bounds of the free distance derived with the singular value decomposition (SVD) or QR decomposition (QRD) were developed. We propose a QRD-based selection method maximizing the corresponding lower bound. With some matrix properties, we theoretically prove that the lower bound yielded by the QRD is tighter than that by the SVD. We then further propose a v.

(6) basis-transformation method so that the lower bound yielded by the QRD can be further tightened. The proposed method is also extended to antenna selection in amplify-and-forward (AF) MIMO relay systems, and other types of selections such as receive antenna selection, and joint transmit and receive antenna selection. Simulations show that the lower bound that the proposed methods evaluate can approach the true free distance closely. As mentioned, the optimum precoder for the ML detector is difficult to derive. Recently, a simple design method, referred to as X-structured precoding, was proposed to solve the problem. This method first adopts the SVD to transform the MIMO channel into parallel subchannels. Then, the subchannels are paired to obtain a set of. . subsystems and. . sub-. precoders can be designed. Due to this special structure, the ML detection in the receiver can be conducted on. . subsystems, reducing computational complexity significantly. Several. methods have been developed to solve the X-structured precoder. However, most of them use numerical searches to find their solutions and require table look-ups during the run time. In the second part of this dissertation, we propose a simple but effective method to solve the problems. The proposed precoder has a simple closed-form expression and no numerical searches and table look-ups are required. We also extend the proposed method in joint source/relay precoders design in two-hop AF MIMO relay systems. With the proposed source subprecoder, the joint design problem can be significantly simplified. Simulations show that the proposed X-structured precoding for MIMO relay systems significantly outperforms other types of precoding methods. Calculation of the precoder requires full channel state information and is conducted in the receiver in general. In real-world applications, a codebook is designed for the precoder, and only the index of the codeword is fed back. In the final part of this dissertation, we investigate the codebook design problem in X-structured precoding. Unlike the conventional precoding, X-structured precoding requires two codebooks, one for a unitary matrix and the other for. . subprecoders. The challenges are that the quantized unitary matrix cannot yield the X-structure and the receiver cannot conduct ML detection on vi. . subsystems. We show that the proposed.

(7) X-structure precoding scheme can still be used, and propose low-complexity detection schemes to solve the detection problem. Simulation results show that the proposed method can effectively reduce the computational complexity of the receiver and at the same time improve the system performance.. vii.

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(9) Acknowledgements. During my Ph.D. program, I would like to show my gratitude to many people. First, I would like to deeply thank my advisor, Prof. Wen-Rong Wu, for his kindly guidance. He spent a lot of time in discussing the problems I encountered in my research, providing valuable suggestions, and teaching me how to write technical papers. Under his enthusiastic instruction, I have learned not only the method to do research, but also the optimistic attitude. At this moment, I have to say Prof. Wu is the key person whom I am most grateful to. Second, I am grateful to all the members in our Lab for their valuable discussions and help in academic research; they are Chun-Fang Lee, Din-Hwa Huang, Chi-Han Lee, Sheng-Lung Cheng, and all other master students. Especially, I deeply appreciate Dr. Chao-Yuan Hsu, Dr. Fan-Shuo Tseng, and Dr. Hung-Tao Hsieh for their encouragement and suggestions. Besides, thank you all who help me prepare my oral defense; you are Jacky, Yangyang, Wranky, and other master students. Also, I would like to thank all my friends who ever encouraged or helped me. Thank you Hsueh-Shu Huang for your kindly help in preparing travel information about conferences. Last, but not least, I would like to show my deep gratitude to my family, especially my dear mother. Thank you all for your economic and spirit support in the period. Without your supports and encouragements, I cannot finish my Ph.D. program.. x.

(10) Contents Chinese Abstract. iii. English Abstract. v. Acknowledgements. x. Contents. xi. List of Tables. xvi. List of Figures. xvii. 1. Introduction. 1. 2. Antenna Selection for ML Detectors in Spatial-Multiplexing MIMO and MIMO Relay Systems 2.1. 2.2. 9. Lower bounds for free distance . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 2.1.1. System and Signal Models . . . . . . . . . . . . . . . . . . . . . . . .. 10. 2.1.2. Lower Bound with SVD-Based Method . . . . . . . . . . . . . . . . .. 11. 2.1.3. Lower Bound with QRD-Based Method . . . . . . . . . . . . . . . . .. 12. Proposed Basis-Transformation Method . . . . . . . . . . . . . . . . . . . . .. 15. 2.2.1. Permutation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 2.2.2. Transformation Matrix in Lattice Reduction . . . . . . . . . . . . . . .. 18. xi.

(11) 2.2.3 2.3. 2.4. 2.5. Cascade of Permutation and LR Matrices . . . . . . . . . . . . . . . .. 20. Implementation Issues and Complexity Comparisons . . . . . . . . . . . . . .. 21. 2.3.1. Givens Rotations Method . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 2.3.2. Efficient Permutations . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 2.3.3. Complexity Comparisons . . . . . . . . . . . . . . . . . . . . . . . . .. 23. Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 2.4.1. Receive and Joint Transmit/Receive Antenna Selection . . . . . . . . .. 23. 2.4.2. Antenna Selection in MIMO Relay Systems . . . . . . . . . . . . . . .. 25. 2.4.3. Sphere Decoding Algorithm . . . . . . . . . . . . . . . . . . . . . . .. 27. Simulations and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 3 X-Structured Precoding for ML Detectors in Spatial-Multiplexing MIMO and MIMO Relay Systems. 43. 3.1. System Models and Problem Formulation . . . . . . . . . . . . . . . . . . . .. 44. 3.2. Existing Subprecoders for MIMO Systems . . . . . . . . . . . . . . . . . . . .. 46. 3.2.1. Complex-Valued Subprecoder Design . . . . . . . . . . . . . . . . . .. 46. 3.2.2. Real-Valued Subprecoder Design . . . . . . . . . . . . . . . . . . . .. 48. 3.2.3. Orthogonal Subprecoder Design . . . . . . . . . . . . . . . . . . . . .. 49. Proposed Subprecoders for MIMO Systems . . . . . . . . . . . . . . . . . . .. 49. 3.3.1. GMD-Based Subprecoder with Rank-Deficiency . . . . . . . . . . . . .. 50. 3.3.2. Extension to MIMO Systems for. . . . . . . . . . . . . . . . . .. 52. 3.3.3. Complexity Comparisons . . . . . . . . . . . . . . . . . . . . . . . . .. 53. Joint Precoders Design for MIMO Relay Systems . . . . . . . . . . . . . . . .. 54. 3.4.1. Problem Formulation and Source Precoder Design . . . . . . . . . . .. 54. 3.4.2. Relay Precoder Design . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. Simulations and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66. 3.5.1. Performance Comparisons for MIMO Systems . . . . . . . . . . . . .. 66. 3.5.2. Performance Comparisons for MIMO Relay Systems . . . . . . . . . .. 67. 3.3. 3.4. 3.5. xii.

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(13) 4 Limited-Feedback for X-Structured Precoding in Spatial-Multiplexing MIMO Systems. 77. 4.1. System Models and Problem Formulation . . . . . . . . . . . . . . . . . . . .. 78. 4.2. Quantization for X-Structured Precoding. . . . . . . . . . . . . . . . . . . . .. 79. 4.3. Low-Complexity MIMO Detection . . . . . . . . . . . . . . . . . . . . . . . .. 84. 4.4. Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87. 5 Conclusions. 95. Appendix A. 99. A.1 SVD-Based Lower Bound with Transformed Symbol Vectors . . . . . . . . . .. 99. A.2 Evaluation of (2.27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 A.3 Proof of (2.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Appendix B. 105. B.1 Proof of (3.57) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 B.2 Proof of (3.58) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 B.3 Derivation of (3.68) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B.4 Derivation of (3.80) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Bibliography. 111. xiv.

(14) List of Tables 2.1. Algorithm of proposed QRD-based BT-P method . . . . . . . . . . . . . . . .. 31. 2.2. Operations of CLLL algorithm . . . . . . . . . . . . . . . . . . . . . . . . . .. 33. 2.3. Algorithm of proposed QRD-based BT-C method . . . . . . . . . . . . . . . .. 34. 2.4. Algorithm of proposed QRD-based BT-E method . . . . . . . . . . . . . . . .. 35. 2.5. Complexity comparisons for different antenna selection methods . . . . . . . .. 36. 3.1. Complexity comparisons for X-structured precoding . . . . . . . . . . . . . .. 69. 3.2. Detection complexity comparisons for X-structured precoding . . . . . . . . .. 70. 3.3. Joint source/relay precoders design with source power allocation . . . . . . . .. 76. 4.1. Complexity comparisons for detection methods . . . . . . . . . . . . . . . . .. 89. xvi.

(15) List of Figures 2.1. 2.2. System model for transmit antenna selection in a spatial-multiplexing MIMO system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31. LR for. 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.3. . 2.4. System model of a two-hop AF MIMO relay system. . . . . . . . . . . . . . .. 2.5. BER performance comparison for transmit antenna selection. computations for. . . . . . . . . . . . . . . . . . . . . . . . . . . "!#%$'&(*)+, ,. .- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BER performance comparison for receive antenna selection "!/0'&(*) 0$ , and .- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BER performance comparison for joint transmit/receive antenna selection 1! 2 &(*)/43 , and 5.- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and. 2.6. 2.7. 2.8. 35. 37. 38. 39. BER performance comparison for antenna selection in a two-hop MIMO relay system. 2.9. 34. *!6437&(*)98:37&(*)/3 , and 5.- .. . . . . . . . . . . . . . . . . .. BER performance comparison for transmit antenna selection with SDA. "!:. 40. $'&(*)/ , and 5.- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. 3.1. System model for a precoded spatial-multiplexing MIMO system. . . . . . . .. 69. 3.2. System model for a precoded spatial-multiplexing MIMO relay system. . . . .. 70. 3.3. BLER performance comparisons for MIMO systems with. . "!. ,. *)/. , and. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii. 71.

(16) 3.4 3.5 3.6 3.7. BLER performance comparisons for MIMO systems with. 43 .. 4.2 4.3 4.4 4.5. 3 , *);43 , =<+3 , and 43 ). 3 , *);43 , =<+3 , and 43 ).. "!#. . . . . . . . . . . . . . . . . . . . . . . . . .. BLER performance comparisons for MIMO relay systems with 16-QAM (. "!. . . . . . . . . . . . . . . . . . . . . . . . . .. 72 73 74. Algorithm 3.1: Joint source/relay precoders design without source power allo-. >. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75. System model for a limited-feedback precoding MIMO system. . . . . . . . . .. 89. Performance comparisons for different sizes of. 90. ?A@ . . . . . . . . . . . . . . . . Performance comparisons for different values of BDC . . . . . . . . . . . . . . . Performance comparisons for unprecoded and precoded systems with 1!EF3 , *);3 , and 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91 92. Detection complexity comparisons for unprecoded systems with different values of. A1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. BLER performance comparisons for MIMO relay systems with 4-QAM (. cation 4.1. "!643 , *);3 , and. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Symbol constellations in which. S. GH :JI @ -;KGL NMOIDP"Q (R LR. ,. . 93. : original. constellation, : transformed constellation). . . . . . . . . . . . . . . . . . . . 101. xviii.

(17) Chapter 1 Introduction. I. N recent years, the multiple-input multiple-output (MIMO) technique has been widely adopted. in wireless communication systems. In MIMO systems, it is well-known that transmis-. sion with spatial multiplexing can provide higher spectral efficiency without bandwidth expan-. sion [1]. However, its performance heavily depends on the condition number of the channel matrix [2]. For ill-conditioned channels, the performance of a MIMO system can be degraded seriously. Precoding is an effective method to overcome this problem. The precoder design problem has been extensively studied in the literature. With the mutual information criterion, a power allocation method referred to as mercury/water-filling [3] was proposed for parallel Gaussian channels, and the obtained precoder, being diagonal, is shown to be the optimum power allocation scheme. For general MIMO channels, the optimum precoder has also been studied in [4]. However, the computational complexity of the optimum precoder can be very high in solving a so-called fixed-point equation. Except for mutual information, there are also other criteria used for the precoder design. Precoders that maximize the signal-to-noise ratio (SNR) or achieve minimum mean square error (MMSE) were developed for linear receivers in [5–8]. Although the computational complexity of linear receivers is low, the performance is often not satisfactory. Two non-liear receivers are well known: successive interference cancellation (SIC) and maximum-likelihood (ML) detection. The optimum precoder 1.

(18) for the QR-SIC receiver, minimizing the block error rate (BLER), has been solved with geometric mean decomposition (GMD) [9, 10]. In addition, the optimum precoder for the MMSE-SIC receiver was also solved with uniform channel decomposition (UCD) [11]. To further improve the performance, bit loading was jointly considered with precoding in MIMO transceiver design in [12, 13]. It is known that the performance of an ML receiver is dominated by the minimum distance of received signal constellations, referred to as free distance. This suggest that the design criterion for the ML detector is equivalent to maximizing the free distance. Unfortunately, this optimization problem is known to be difficult; the optimum ML precoder remains unsolved. In this dissertation, we study the precoder design for the ML detector in MIMO and MIMO relay systems. In the first part of this dissertation, we first consider a simplified precoding scheme, namely, transmit antenna selection. Consider a MIMO system with antennas, and. A!. transmit antennas,. transmitted bit-streams. In transmit antenna selection, only. selected for signal transmission though there are. A!. =). receive. antennas are. antennas. The idea of this scheme comes. from the fact that the cost of antennas is low while that of radio-frequency (RF) chains is relatively high. With a feedback channel, the transmitter can conduct the optimum selection such that the performance of the. =). system can approach that of the un-selected. A! *). system.. Since the number of the RF chains is reduced, the implementation cost of the MIMO system can be reduced. Many methods for antenna selection have been proposed. In [14], antenna selection maximizing the capacity was considered and performance was analyzed. Several selection criteria for linear receivers were proposed in [15], including post signal-to-noise ratio (SNR) maximization and mean-square-error (MSE) minimization. It is known that nonlinear receivers can outperform linear receivers. However, antenna selection for a nonlinear receiver is also more involved. In [16], a selection method for ordered-successive-interference-cancellation (OSIC) was proposed, and in [17], a selection method for the ML receiver was proposed. The method in [17] minimizes the union bound of the error rate and its computational complexity can be very high. As mentioned, the performance of the ML receiver is determined by the free dis2.

(19) tance, and we can then maximize the free distance in the selection problem. With this method, the authors in [18] conjectured that the diversity for antenna selection in the ML receiver is. *):T*!U. . Unfortunately, finding the free distance requires an exhaustive search, and the. computational complexity can be prohibitively high. To reduce the computational complexity, a singular value decomposition (SVD) based method was proposed in [18]. The SVD-based method selects the antenna subset that maximizes the smallest singular value of the channel matrix. It was shown that this singular value can serve as a lower bound of the free distance. An alternative lower bound using QR decomposition (QRD) was also derived in the literature [10]. It is obtained with the smallest diagonal entry in the R-factor of the channel matrix, where the R-factor is the upper-triangular matrix obtained with the QRD of the channel matrix. Although the computational complexity of the SVD-based or QRD-based method is low, the tightness of the lower bounds have not been analyzed before. To improve the performance of the selection, we propose a QRD-based selection method maximizing the smallest diagonal entry in the R-factor of the channel matrix. We theoretically prove that the lower bound achieved with the QRD-based method is tighter than that with the SVD-based one. The tightness of the lower bound is related to the spread of the diagonal entries in the R-factor. If the spread is smaller, the lower bound is tighter. This property motivates us to propose a basis-transformation method further tightening the lower bound of the QRDbased method. The idea is to find a basis transformation for the transmitted symbol vector so that the spread can be reduced. We propose two basis-transformation matrices to do the work. The first one is the permutation matrix, and the second one is the transformation matrix used in lattice reduction (LR) [19, 20]. The LR technique has been used in antenna selection [21] for performance improvement. The selection method in [21] is designed for a linear detector operating in the basis-transformed domain. In our method, LR is used only for the derivation of the transformation matrix. An ML detector operating in the original basis is used at the receiver. Therefore, the role of LR is much different from that in [21]. We theoretically prove that the proposed basis transformations can further tighten the lower bound obtained with the original 3.

(20) QRD-based method. The basis-transformation method needs extra QRDs, and the computational complexity will be increased. We propose an efficient permutation method and the use of Givens rotations to reduce the computational complexity of our selection methods. Except for transmit antenna selection, we also consider the applications of the proposed algorithms in other scenarios which include receive antenna selection [22–24], joint transmit/receive antenna selection [25], and antenna selection in amplify-and-forward (AF) MIMO relay systems. For receive antenna selection, no feedback is required and this will be a great advantage in high mobility environments. In some scenarios, the number of receive antenna elements may be limited due to the size constraint. Joint transmit/receive antenna selection provides a solution to this problem. MIMO relay systems have been extensively studied recently since they can provide range extension or diversity enhancement for MIMO systems [26,27]. In general, it is desirable to minimize the hardware complexity in a relay. Antenna selection is then a good candidate for the performance enhancement in the system. As mentioned, the optimum precoder for the ML detector is difficult to derive. Recently, a simple suboptimum method was proposed to solve the problem. It first uses SVD to transform the MIMO channel into parallel subchannels. Then, the subchannels are paired to obtain a set of. V. MIMO subsystems, and. V. subprecoders are designed to maximize the free distance. in the MIMO subsystems. We refer this approach as precoding with X-structure which has been considered in [28–30]. Precoding with X-structure not only facilitates the precoder design, but also yields a low-complexity ML receiver since only. 3. 1W. MIMO subsystems have to be dealt. with. For -QAM, the optimum complex-valued subprecoder was first found in [31]. The result was extended to higher QAM constellations in [32], but the optimality is no longer held. An orthogonal subprecoder derived from a rotation matrix was proposed in [29]. The advantage of this approach is that the precoder has a simple closed-form expression. However, the closed-. 3. form expression of the optimum rotation angle is available only for -QAM. A numerical search is required for the angle in higher QAM constellations. Recently, an optimum real-valued subprecoder was developed [30]. It also requires the numerical search for the optimum precoder 4.

(21) though simpler. Note that both orthogonal and real-valued precoders require table look-ups during run time. In the second part of this dissertation, we propose a design method for the Xstructured precoder. The main idea is using the GMD method in the subprecoder design. With the proposed method, no numerical searches are required in the design phase and no table lookups are required during run time. In addition, the existing subprecoders are only valid for subsystems. In other words, the subprecoder designed for more than. ;". A. has to be numerically. resolved [33]. On the contrary, the GMD solution can be easily extended to the subprecoder with higher dimension. Simulation results show that the performance of the proposed method is almost as good as the existing methods. In recent years, cooperative communications have been considered a promising method to improve the performance of point-to-point MIMO systems [26, 27]. By employing relays between the source and destination, the signal can be transmitted via the source-to-relay and then relay-to-destination link. Multiple antennas can be equipped at each node to form a MIMO relay system. In the MIMO relay system, both the source and relay nodes can conduct precoding, referred to as joint source/relay precoding. Several joint precoders designs for AF MIMO relay systems have been proposed. In [34, 35], the precoders maximizing the channel capacity were developed. In [36–42], the precoders using the MMSE criterion were designed for the linear receivers. To the best of our knowledge, the joint precoders design in the MIMO relay system with an ML receiver has not been considered before. We then extend the proposed methods for MIMO systems to MIMO relay systems. However, the problem becomes much more involved in this case and a closed-form solution is difficult to obtain. We then propose iterative methods to derive the source and relay precoders, individually and repeatedly. First assume that the relay precoder is given; the source precoder design can then be easily solved by the proposed MIMO precoder. Then with the solved source precoder, the relay precoder can be solved and updated. The derivation of the relay precoder for a given source precoder, however, is much more complicated due to the fact that the MIMO relay channel is a nonlinear and complicated function of the relay precoder. We then propose two methods overcoming the problem such that 5.

(22) the relay precoder can be efficiently solved with Karash-Kuhn-Tucker (KKT) conditions [43]. Simulation results show that the proposed method significantly outperform existing methods. To derive a precoder, CSI is generally required. The precoder can be calculated in the transmitter or receiver. If it is calculated in the transmitter, CSI must be fed back, and if it is calculated in the receiver, the coefficients of the precoder must be fed back. In real-world applications, feedback of the perfect CSI or precoder is difficult. To have lower distortion, the feedback of the precoder is generally preferred. In general, the coefficients of the precoder are not fed back individually. Instead, a codebook is designed for the precoder, and only the index of the codeword is fed back. This approach is referred to as limited-feedback precoding in the literature [44–48]. In the final part of this dissertation, we investigate the codebook design problem in X-structured precoding. Due to the quantization error, the channel matrix cannot be fully diagonalized and the X-structure cannot be maintained. Therefore, the subprecoders developed in [28–30] may not be applicable. We show that the proposed subprecoders can still be used in the limited-feedback system. Unlike the conventional precoding, X-structured precoding requires two codebooks, one for a unitary matrix diagonalizing the channel matrix and the other for the subprecoders. We propose using vector quantization (VQ) [49] to construct the codebook for unitary matrices. As for the subprecoder, the proposed methods only require a rotation angle and quantized angles can serve as the codebook. The other problem in the limitedfeedback system is that the receiver cannot conduct ML detection on the. X. subsystems.. We then propose interference-cancelation-based low-complexity detection schemes to solve the problem. This method combines. DY. ML detection combined with SIC. Simulation results. show that the proposed method can effectively reduce the computational complexity of the receiver and at the same time improve the system performance. This dissertation is organized as follows. Chapter 2 considers the antenna selection scheme and proposes the basis-transformation method to achieve near-optimum performance. Chapter 3 details the proposed X-structured precoding for MIMO and MIMO relay systems. Chapter 4 extends the use of the proposed subprecoder to limited-feedback systems. Finally, Chapter 5 6.

(23) draws conclusions and outlines possible future works.. 7.

(24) 8.

(25) Chapter 2 Antenna Selection for ML Detectors in Spatial-Multiplexing MIMO and MIMO Relay Systems In this chapter, we first consider the transmit antenna selection in spatial-multiplexing MIMO systems. For ML detection, it is well-known that optimum selection criterion is to choose the antenna subset giving the maximum free distance. However, the optimum solution is difficult to obtain since evaluating free distance requires an exhaustive search. To reduce the computational complexity, we resort to maximizing an lower bound of free distance, instead of free distance itself. In Section 2.1, we propose using a QRD-based lower bound as the selection criterion, and show that the lower bound yielded by QRD is tighter than that by SVD. However, the QRDbased lower bound may not be tight enough when the size of the MIMO system becomes large. In Section 2.2, we then propose a basis-transformation method to further tighten the QRD-based lower bound. In Section 2.3, we consider the issue of computational complexity and propose some low complexity methods for implementation. In Section 2.4, we show that the proposed methods can be easily extended to other applications. Finally, we evaluate the performance of the proposed methods in Section 2.5. 9.

(26) § 2.1 Lower bounds for free distance § 2.1.1 System and Signal Models. A ! ceive antennas, as described in Figure 2.1. Let A!E

(27) , , *) ZK. =) re * ! , and [ denote the =) channel matrix. In transmit antenna selection, the receiver first selects transmit antennas according to a selection criterion, where is the number of transmitted bit streams. Let each transmit antenna subset be represented by an index \ . Then, via a feedback channel, the receiver Consider a spatial-multiplexing wireless MIMO system with. transmit antennas and. sends the index of the optimum antenna subset back to the transmitter. Finally, the transmitter uses the selected antennas for signal transmission. Note that there are. A ) at antenna c and def a@g&habij&jkjkjkl&ha nm P each of which corresponds to an. MIMO channel. Let where. Oko- m. ab. M^]`P _ Q. antenna subsets,. be the symbol transmitted. represents the transpose operation. The. corresponding received signal vector can then be expressed as. where. [q. p [qldDTsr. (2.1). is the channel matrix corresponding to the selected antenna subset and. Gaussian noise vector. Assume that each entry of. r. r. is the. 1) ut. is identically and independently distributed. *) * ) ]`x identity matrix. The ML detector searches all possible symbol vectors to obtain an estimate d y. (i.i.d.) with the covariance matrix of such that. where. IDP. v (i w `] x , where v i. is the noise variance and. w. is an. dzy |{(H}~ p U[qjd is a set consisting of all possible transmitted symbol vectors. We also define. d. (2.2). IP. as. the symbol-vector constellation of . It is well known that the performance of ML detection in high SNR depends on the free distance defined as. G. free. ({H;}~ j [q/dUd- 10. (2.3).

(28) where. Oko-. represents the Hermitian operation and. dXUd -. is the difference vector. The free. distance represents the minimum distance of the received signal constellation. Therefore, the optimum antenna selection criterion [18] for the ML receiver is equivalent to choosing the antenna subset whose. [q. gives the maximum free distance. We can compute the free distance of. each candidate channel matrix using (2.3), and then choose the antenna subset with the largest. G. free .. This optimum solution can be found by an exhaustive search over all possible. M9]`P _ Q. candi-. date channel matrices and all difference vectors. However, this exhaustive search requires very high computational complexity when considering a large number of transmitted bit-streams with a large-constellation modulation scheme. A suboptimum approach is considered to minimize a lower bound of the free distance, instead of the free distance itself.. § 2.1.2 Lower Bound with SVD-Based Method. [ q be an *) full column-rank matrix with its SVD given as [qV * , where is unitary matrix, and is an A) diagonal matrix. an *) *) unitary matrix, is an The non-zero entries of are the singular values of [q . Define the symbol constellation of a as I; , and the minimum distance of I+ as GL :JI;-: O( ({j}(~ Og ab+Ua` k (2.4) Let. Also define the minimum distance of the symbol-vector constellation,. IP. , as. GL NM9I P QV ({H;}~ j dUd k (2.5) In a spatial-multiplexing MIMO system, a ’s are usually uncorrelated. Thus, we can have GL NMOI P Q 4{}~GL :JI@-&(GL :JI;i-&jkjkjkl&(GL JI P -(/k (2.6) Note that if ab ’s are correlated, (2.6) is not valid in general. With (2.6), G NMOIDP"Q can be easily computed for QAM constellations. Let a QAM symbol be represented by ¡7¢VT£H¡H¤ t t where ¡.¢¥,§¦ &¦*'&j¨¨¨g and ¡H¤¥,§¦ &¦*'&j¨¨¨g . We then have G. E I@-*©GL :JI;ig-* ¨¨¨H5GL :JI P - and GL M IDP Q 4{}~V & &jkjkjkl& V . 11.

(29) Using the Rayleigh-Ritz theorem, the SVD-based lower bound of the free distance was derived in [18] as. Z5ª P LG M I P Q (2.7) where ª P is the minimum singular value of the matrix [q . Note that (2.7) is different from that in [18] by a factor of . The reason for this is that the free distance in our application is a relative not absolute value. Since all [q ’s are of the same dimension, scaling the free distance will not change the selection result. Thus, the factor is omitted for simplicity. The lower bound in (2.7) indicates that the free distance can be evaluated with ª P and GL NMOIDPAQ . It is simple to see that the value of G. M IDP Q is the same for each [q . Thus, with the SVDbased method, only the minimum singular value of each [Dq is required to compute, and the G. free. computational complexity can be reduced dramatically. However, the main problem for the SVD-based method is that the lower bound (2.7) may not be tight enough. An alternative lower bound of the free distance derived from the QRD was developed in [10]. In this dissertation, we propose the use of this lower bound for solving the antenna selection problem.. § 2.1.3 Lower Bound with QRD-Based Method The matrix. [q. can be factorized in the form of. [qD¬«u. , where. «. is an. =) . column-. upper-triangular matrix with positive real-valued wise orthonormal matrix and is an diagonal entries as ®® ° ° ´¶µµ ° ®® @9± @ 9@ ± i²kjkjk @9± P µµ ®® ³ ° ih± i²kjkjk ° ih± µµ P k © .. .. .. .. ¯ ³. ³. . ° . · jk kjk P ± P As mentioned, the matrix is also referred to as the R-factor [10] of [Dq . Let denote the ¸ th diagonal entry of . Via this decomposition, we can have another lower bound of the free 12.

(30) distance as. G where. . free. Zº¹Y»{¼ }¼ ~ l½¾GL M I P Q ¿ lGL NMOI P Q. represents the minimum diagonal entry in the matrix. (2.8). . . Thus, we propose the. use of (2.8) as a selection criterion, referred to as QRD-based method. In what follows, we show that the lower bound obtained with the QRD-based method is tighter than that with the SVD-based method.. . ÀÁÂ[¾q [ q , we can have its eigenvalue diagonal matrix whose decomposition expressed as ÀÃÄzÅ=¾ , where Å is an nonzero entries are the eigenvalues of À . It is known that a positive-definite matrix can also be decomposed by the Cholesky factorization in the form of À©5Æ1Ç Æ , where Ç is an unit lower-triangular matrix expressed as diagonal matrix and Æ is an ®® t ³ kjkjk ³ ´¶µµµ ®® ®® È ih± @ t kjkjk ³ µµµ Æ¿ .. .. . . .. k . . . . · ¯ È È t P ± @ P ± i²kjkjk The diagonal entries of Ç are also referred to as Cholesky values [50]. Consider two sequences v ÉÊv@g&vì&jkjkjkl&v P - and G É G'@g&(G.il&jkjkjkl&(G P - consisting of the eigenvalues and Cholesky values of À respectively. Note that the entries of both sequences v and G are arranged in the descending order so that v@DZ¬vìZËkjkjk;ZÌv P and G'@ZÌG.iXZÁkjkjk;ZÌG P . With the above definitions, we can have the following proposition. full column-rank matrix [q with its QRD and SVD exProposition 2.1: For an =) pressed as [que«u and [quÂ *z respectively, the inequality ª PºÍ holds true For an arbitrary. positive-definite matrix. for all channel realizations.. Proof: It is known that the Cholesky factorization of a positive-definite matrix is unique [51] and can be expressed as. ÀË0Æ1Ç Æ 13. . With the QRD. [que«u. À©[Wq [q. , we can also.

(31) have. ÀÎ4 © Ç u. is an Let Ï ÐÊÏ@&Ïjij&jkjkjkl&Ï P where. u. (2.9). u. unit upper-triangular matrix and. Ç . is an. . diagonal matrix.. G Ð G @ &(G i &jkjkjkl&(G P - denote the diagonal entries of and Ç ¸ t & &jkjkjkg& . Furthermore, using the i respectively. From (2.9), we know that Ï G for all and. uniqueness property of the Cholesky factorization, it can be seen that the entries of the sequence. G exactly represent the Cholesky values of À . Definition 2.1: Let Ñ ¿ Ñj@g&(Ñ(ij&jkjkjkl&(Ñ - and Ò ¿ Òj@&(Òij&jkjkjkl&(Ò - be two positive, real-valued P P sequences satisfying Ñj@+Z4Ñ(iZ%kjkjk7Z4Ñ and Òj@;Z4ÒiZ%kjkjk7ZÒ . We say that Ñ majorizes Ò in P P Ó Ó the product sense [52] if Ô Ô Ñ(*Z Ò (2.10) (Õ@ (Õ@ t & &jkjkjkg& & and with equality when Ö, . for all Ö Lemma 2.1: For a positive-definite matrix ÀÐ©[¾q [q , the sequence v majorizes the seÓ Ó quence G in the product sense, i.e., Ô Ô v` Z G. (2.11) (Õ@ (Õ@ t & &jkjkjkg& & and with equality when Ö6% . This lemma and its proof can be found for all Ö6 in [50, 52].. G are also with the descending order. Then, by Lemma Ó Ó Ó Ó 2.1, we can obtain Ô Ô Ô Ô v`*Z G. G Ï i (2.12) (Õ@ (Õ@ (Õ@ (Õ@ t & &jkjkjkg& & and with equality when Ö=Á k For a positive-definite matrix À× for all Ö= [q [q , it is known that vbDÌª i , where ª` is the ¸ th largest singular value of [q . We thus Ï. Assume that the entries of both and. 14.

(32) Ó. have. Ö t & Note that Ï P for all. Ó. Ó. Ô i Ô Ô ª v`*Z Ï i (2.13) ( Õ@ ( Õ@ (Õ@ &jkjkjkg& & and with equality when Ö, k From (2.13), we arrive at that ª iP Í Ï Pi . ¿ . We can have ª P0Í since both ª P and are positive values,. which completes the proof.. § 2.2 Proposed Basis-Transformation Method In the previous section, we see that the lower bound obtained with the QRD-based method is tighter than that with the SVD-based method. In this section, we propose a method for further tightening the lower bound of the free distance. To do that, we first observe how the channel matrix affects the tightness of the bound in (2.8). From [10], it can be seen that when the diagonal entries of. . are all equal, the equality in (2.8) will hold. We then conjecture that. the tightness of the lower bound is related to the spread of the diagonal entries in the R-factor,. -value spread. The -value spread is the value of 6ØJÙ divided by that of where 6ØJÙ is the maximum diagonal entry in the R-factor. If the -value spread is. defined as the. smaller, the bound in (2.8) is tighter. Now, considering the signal part in (2.1), we have. p 5[qld 4a@Úbqj± @TabiÚbqj± iÛT4¨¨¨ÜTa P Úbqj± P (2.14) ¸ where Úbqj± is the th column of [q . Thus, p can be seen as a vector expanded by a basis formed by the columns of [q , i.e., Úbqj± @&(Úbqj± ij&j¨¨¨&(Úbqj± , and the corresponding coordinate is P a@g&habi&j¨¨¨&ha P m . With an invertible matrix Ý , we can rewrite (2.14) as p [qÝ/Ý/Þ @ d Äß[ q d ß (2.15) s. s. s. 15.

(33) d ß ¬Ý Þ @ d . Thus, the basis is transformed to the columns of ß[ q , and the corresponding coordinate is dY ß ¬ aß @g& abß i&j¨¨¨& a ß P nm . If a proper Ý can be chosen such that the -value spread of the R-factor in ß[ q is reduced, then the bound in (2.8) can be tightened. where. ß[ qD©[qÝ. and. This is the basic concept of our basis-transformation method. Using the idea described above, we rewrite the free distance as. G. ({H;}~ j [q/dXUd - ({H;}~ j 6à [qÝ/Ý/Þ @ dXUd - à à à á á {Há ;}~ á á à ß[ q/ dXß U d ß - à (2.16) à à @ where ßI P is the symbol-vector constellation reshaped by Ý Þ . In Appendix A.1, we show that the lower bound yielded by the SVD in (2.7) is still valid. However, G L/;ßI - becomes diffißI P"Q% â {}~=ãHGL NM+ßI @hQ:&jkjkjkGL NM+ßI P Q'ä where GL LE;ßI -+ cult to find since in general G. NM+D {}~ abß +U a ß . On the other hand, the lower bound yielded by the QRD in (2.8) is no longer valid. From [10], we see that the lower bound in (2.8) is derived with the assumption of (2.6). @ Thus, if let GL .EßI -¾Ä{}~GL LEßI @-&(GL LE;ßI i-&j¨¨¨GL LEßI P -( , we can have a new lower free. bound from (2.8) as. G. free. Zågß jGL LEßI @ -k. (2.17). G. LEßI @ - â GL .JIDP"- in general. We cannot conclude that (2.17) is tighter than ß 1ZÂ . The other problem with (2.17) is that the value of GH L ßI @ (2.8) even when . As mentioned,. is no longer easy to obtain. This is because after the transformation, the signal constellation in each dimension is significantly expanded. The challenge in the basis-transformation method is to find. Ý. such that. G. .EßI @ -. can be easily computed and at the same time. GH .EßI @ -æ. GL LJIDP1- . The constraints are stringent and cannot be satisfied by most of the transformations.. Fortunately, we have found two matrices that can do the job. 16.

(34) § 2.2.1 Permutation Matrix. 1) candidate channel matrix [q , we can have åç different column permutation patterns. Let èz t c åç . denote the permutation matrix corresponding to the c th permutation pattern, where Í Í The first transformation matrix we propose is the permutation matrix. For a given. We can rewrite the free distance as. G è m 5 è¾Þ @. free. ({H;}~ j 6à [qè è¾Þ @ é dUd - à à à ({H;}~ j 6à [qè è m dX U d - à à à. (2.18). è is an orthogonal matrix. Note that the combinations of è m dUd are the same as those of dXUd - , which allows us to rewrite (2.18) as G ({H;}~ j [qè 'dXUd - k (2.19) Let denote the R-factor of [qÜè . The free distance can be bounded as G Zå jGL NMOI P Q:k (2.20) For a given [q , we can obtain åç different R-factors with åç permutations. Denote as where. since. free. free. per. the R-factor having the largest minimum diagonal entry. Then, the minimum diagonal entry of. . per. can be used in the lower bound in (2.8). Then, we have. Z jGL NMOI P Q F (2.21) where 5{êÜë*h X@9 §&Ü ih §&j¨¨¨&Ü P=ì - . In other words, we find the optimum permutation pattern such that the smallest -value spread can be obtained. We summarize G. free. per. per. the proposed basis-transformation method with the permutation matrix (denoted as QRD-based BT-P) in Table 2.1. Assume that following inequality as. èW@+ w P. . Then,. X@9 =© . Clearly, we can have the. Zå §k per. (2.22). The inequality in (2.22) indicates that this permutation method can improve the tightness of (2.8). 17.

(35) § 2.2.2 Transformation Matrix in Lattice Reduction The second transformation matrix we found is the basis-transformation matrix used in LR. The LR technique has been successfully applied to MIMO systems for enhancing the detection performance [19, 20]. The basic idea is to find a new basis for the transmitted symbol vector, and then signal detection is conducted in the basis-transformed domain. In this subsection, we propose using the transformation matrix used in LR to obtain tighter lower bound of the. [ qD. free distance. Note that the ML detection is still conducted in the original basis. With. Úbqj± @&(Úbqj± il&jkjkjkl&(Úbqj± P , we can describe an . í. as. î í;Úbqj± @&(Úbqj± ij&jkjkjkl&(Úbqj± P P j Úbqj± (Õ@ï is a complex integer. The vector set Úqj± @&(Úbqj± il&jkjkjkl&(Úbqj± P . where. ð. -dimensional lattice. (2.23). í. is a basis spanning . Let. ï be an invertible matrix whose entries are all complex integers, and we have [ ± qV[q ð LR. Note that all entries of. [ ±q LR. ð Þ@. F Ú ± qj± @(&(Ú ± qj± il&jkjkjkl&(Ú ± qj± P .k LR. LR. (2.24). LR. are also complex integers. Thus, the column vectors of the matrix. also form a basis for the same lattice. The LR method finds a basis whose elements are as. orthogonal as possible, and at the same time the magnitudes of the basis elements are as short as possible. An example of LR with. ñ. is illustrated in Figure 2.2. As we can see, the. reduced basis vectors will have shorter length, and the orthogonality of. [. LR. ±q. is improved also.. Several algorithms to implement LR have been proposed in the literature. Among them, the complex Lenstra-Lenstra-Lovász (CLLL) algorithm is most widely used since its computational complexity is lower. We then use CLLL as our LR algorithm. The operation of the CLLL algorithm is summarized in Table 2.2. As we can see in the table, a parameter. ò4¥²M i@ & t Q. is defined in CLLL. This parameter determines the orthogonality of the transformed channel matrix. [ ± q . A larger ò LR. will make. [. LR. ±q. closer to an orthogonal matrix. However, the CLLL. algorithm will require more iterations to converge and its computational complexity is higher. 18.

(36) Now, we can conduct LR on. [q. to obtain. ð. . Rewrite the free distance in (2.3) as. ({H;}~ j 6à [q ð=ð Þ @ dXUdn- à à à ({ }~ [ ± /q d Ud - (2.25) Ud ð Þ @ dUd - and IDP represents the symbol-vector constellation after the where d ± q obtaining [ ± qe« . Using transformation. Thus, we can conduct the QRD on [ G. free. LR. LR. LR. LR. LR. LR. LR. LR. LR. LR. LR. LR. LR. LR. LR. (2.17), we can have a lower bound of the free distance as. where. GL :JI @ LR. G. free. Zå lGL NMOI @ Q LR. (2.26). LR. is expressed as. GL M I @ Q 4{}~GL LJI ± @h-&(GL LJI ± i-&j¨¨¨&(GL LJI ± P -(Hk (2.27) @ - can be seen as equal to G. NM9I P Q . As a result, only In Appendix A.2, we show that GH :JI needs to be evaluated in the comparison of (2.26) and (2.8). full column-rank matrix [q , if LR is conducted with the Proposition 2.2: For an =) LR. LR. LR. LR. LR. LR. CLLL algorithm, the following inequality is held;. Zå §k. (2.28). LR. Proof: See Appendix A.3. Thus, for a candidate channel matrix, the lower bound obtained with (2.26) will be always tighter than or equal to the original QRD-based lower bound. We can then select the antenna subset whose corresponding channel matrix has the maximum. . Note that the singular LR. values of a matrix are invariant under the column permutation operation. Therefore, the tightness of the SVD-based lower bound cannot be improved by the transformation with permutation. ð. matrices. As for the transformation with the LR matrix , the minimum singular value may also be enlarged. However, as proved in Proposition 2.1, the resultant minimum singular value will be still smaller than. . LR. 19.

(37) § 2.2.3 Cascade of Permutation and LR Matrices To improve performance further, we can consider the cascade of a permutation and a LR matrix as another transformation matrix. Let. ó ð è . The free distance can then be expressed as. ({H;}~ j 6à [qó ó#Þ @ dXUd - à (2.29) à ð à ({H;}~ j 6à [q è éè m ð Þ @ édUd - à k (2.30) à à ð With the effective channel matrix [q è , the QRD-based lower bound of the free distance can G. free. be tightened even further. In this chapter, we denote QRD-based BT-C as the proposed QRDbased method with the basis transformation using the cascade of permutation and LR matrices.. ± ð as the R-factor of [q è and as the R-factor having the largest minimum diagonal en± ’s. In other words, {êÜë*h ± @9 §&Ü ± ih §&j¨¨¨&Ü ± P=ì - . try among all Table 2.3 summarizes the operations of the proposed QRD-based BT-C method. Denote. LR. cas. LR. cas. LR. LR. LR. Thus, we can have the following inequality:. Zå §k cas. From (2.31) and (2.22), we see that the lower bound of since. ¾Zô . LR. (2.31). LR. cas. This, however, does not imply that. only say that the probability of. Z¬ cas. per. is larger than that of. ¾Zô . cas. per. We can. is larger. Furthermore, we can exchange. the cascading order. In other words, we can let the transformation matrix be this case, however, we have to conduct LR for each. per. . ó#¬è ð . In. [Dqè , and the resultant computational. complexity is then higher. Finally, we can even use a transformation matrix by cascading a series of permutation and LR matrices. For example, we can have a transformation matrix in the form of. ð @è ð i(è , where t c åç and t c åç . Simulations show that this may Í Í Í Í. not be required. With only one-level cascading, the performance of the selection is very close to the optimum solution. 20.

(38) § 2.3 Implementation Issues and Complexity Comparisons The computational complexity of the basis-transformation method will be increased due to extra transformations and QRD operations. To reduce the computational complexity, we proposed several efficient methods for real-world implementations.. § 2.3.1 Givens Rotations Method To reduce the computational complexity of the QRDs, we propose using Givens rotations [53]. [qj± i be another matrix obtained by exchanging two neighbor columns of [ qj± @ . We seek to find Di of [qj± i without using another complete QRD. Denote õè as a permutation matrix conducting a to compute each. . Assume that [qj± @,«@X@. is available via a complete QRD. Let. column-exchange operation on two neighbor columns, i.e.,. [qj± i/%«@OX@ Ì õè %«@ Xõ @. Xõ @ into a upper-triangular matrix again. Since õè only exchanges two neighbor columns of z@ , we can upper-triangulize X õ @ by a simple Givens rotation matrix ö¾@ . Then, ö@ Xõ @NF÷ , where ÷ is a where. Xõ @. (2.32). is a near upper-triangular matrix. Now, all we have to do is to transform. upper-triangular matrix. Thus we can rewrite (2.32) as. [qj± i/,«@(ö @ ö@ Xõ @:,«ui÷. (2.33). «ui×«@ö @ is a unitary matrix. From (2.33), we know that «i(÷ is the QRD of [qj± i , and ÷ is equal to Di . In other words, we obtain Di by simply left-multiplying a Givens rotation matrix on X õ @ rather than by performing a complete QRD on [qj± i . Therefore, we can. where. dramatically reduce the computational complexity of the proposed basis-transformed method when conducting the permutation operations. Note that applying Givens rotations will not affect the performance of the proposed selection methods. Figure 2.3 illustrates an example (for. ) how each . can be derived with Givens rotations. 21. .

(39) § 2.3.2 Efficient Permutations. [q .. Consider a Rayleigh flat-fading MIMO channel matrix. «u. With the QRD, we have. . It has been shown that the square value of each diagonal entry in. . [q. is independently. iø,ù J úT t U ¸ - , where ù J úT t U ¸ t U ¸ - . Thus, the expectation i denotes the Gamma distribution with mean û J ÄJ T i of , which is equal to one, is the smallest. As a result, has the highest probability P P to be the minimum diagonal entry of . Also note that Givens rotations have to be conducted distributed with Gamma distribution [54]. That is,. sequentially. Using these two properties, we propose an efficient permutation method to reduce the required computational complexity. The idea is to conduct permutations in a local rather than global manner. Define an integer the first. ß. columns of. [q. ß. and. ý ß ü¬. . With Givens rotations, permutating. will not change the resultant values of. h PVþ ÿ @ &Ü `PVþ ÿ i &jkjkjkl&Ü P - .. P is the minimum value, these permutations are totally useless. Thus, we choose the last ÃU ß columns of [q for local permutations. If ß is chosen to be much smaller than , the computational complexity can be reduced significantly. Note that local. This is to say if. permutations will result in some performance loss. In [54, 55], it has been theoretically shown that. û J iP . tends to be larger when the columns of. [q. are exchanged according the norm-. ascending order. This is referred to as pre-ordering. Thus, we can combine this pre-ordering with our efficient permutation method to compensate the performance loss. The idea of efficient permutation can also be applied to the QRD-based BT-C scheme. As shown in Appendix A.3, the. -value spread of . LR. will be effectively reduced with the. CLLL algorithm. Thus, the values of the diagonal entries in. . LR. can become very close. Pre-. ordering is then not required. From simulations, we found that due to some special properties of CLLL, using the first. ß. neighbor columns for local permutations can provide slightly better. performance. For this reason, when conducting efficient permutations for the QRD-based BT-C scheme, we choose the first. ß. columns of. [q ð. . The operation of the QRD-based BT-C method. implemented with efficient permutations (denoted as QRD-based BT-E) is shown in Table 2.4. 22.

(40) § 2.3.3 Complexity Comparisons In this dissertation, we use the number of floating operations (FLOPS) required in an algorithm as the measure for computational complexity. Many algorithms for conducting the QRD and SVD have been developed [53]. In general, the QRD requires less FLOPS than the SVD does [53]. Thus, the QRD-based selection scheme not only has better performance, but also requires lower computational complexity. In Table 2.5, we summarize the order of computational complexity in each proposed method.. c &j¨¨¨& åç in either the QRD-based BT-P or QRD-based BT-C method requires J l åço- FLOPS, which can be reduced to J , åçoAs we can see, computing. . min. for. with Givens rotations. It has been shown that the computational complexity of the CLLL algo-. J l- [56]. Therefore, the total computational complexity of the QRD-based BT-C T , åço- . The term , åç may grow rapidly when scheme with Given rotations is : rithm is. becomes very large, which dominates the total computational complexity. This problem can be solved with the efficient permutation method proposed in Section 2.3.2. In the QRD-based BTE scheme,. ß. can be chosen much smaller than. , and the resultant computational complexity. ß ç Q ø J j- . Although the required complexity will can be further reduced to M9 :T F be increased when LR is considered, the overall complexity is still much lower than that of the exhaustive search.. § 2.4 Other Applications § 2.4.1 Receive and Joint Transmit/Receive Antenna Selection Antenna selection can also be conducted at the receiver. As we did in transmit antenna selection, we can select a subset of receive antennas according to a performance criterion. Here, we select. out of. *). receive antenna elements. Note that the receiver does not have to feed the index of. the selected antenna subset back to the transmitter, which is a significant advantage. However, 23.

(41) the performance of receive antenna selection may be inferior to transmit antenna selection for the same number of candidate channel matrices. We now give a simple example to illustrate. A! &(*)K , and Ã , where the channel matrix can be expressed as ®® ´ µµ ® @9± @ @9± i µ [ ¯ ih± @ ih± i · k ±@ ±i W t From the diversity point of view, we can treat a MIMO channel [ as two separate sub-channels, each of which is obtained from a column of [ . The reason is that either aÛ@ or abi z MIMO system since the ML detection is adopted at has the same diversity as the original this property. Consider a MIMO system with. the receiver. This facilitates a simple performance analysis for the antenna selection problem. For receive antenna selection, denoted as. a@. have three candidate channel columns for selection, which are. ´ ´ @ 9 ± @ Ú @: ¯ · , 6Ú i; ¯ 9@ ± @ · ih± @ ±@. ´ hi ± @ · k , and Ú ¯ ±@. It can be seen that there exists correlation between any two columns since those candidate channel columns have the common entries. This correlation will degrade the performance of receive antenna selection.. . Now, consider transmit antenna selection in a MIMO system with. "!*ú'&(*)D. , and. , where the channel matrix can be expressed as. ´ [Ì ¯ 9@ ± @ 9@ ± i 9@ ± · k ih± @ ih± i ih± . a @ also has three candidate channel columns given by ´ ´ ´ @9± @ · , Ú6i; ¯ @9± i · , and Ú ¯ @9± · k ih± i ih± ih± @. For transmit antenna selection,. . Ú @: ¯ . Note here that three independent columns are available for selection, which is different from receive antenna selection. This clearly indicates that transmit antenna selection can outperform 24.

(42) receive antenna selection for the same number of candidate channel matrices. However, as mentioned, conducting antenna selection at the transmitter side requires feedback overhead. Thus, there is a trade-off between the feedback requirement and diversity performance. Also note that increasing the number of receive antennas may not be always possible due to the receiver size constraint. Therefore, we can then consider joint transmit/receive antenna selection, conducting antenna selection at both the transmitter and the receiver side simultaneously, to achieve the optimum tradeoff. Consider an. 9M ]`x Q P. =) *!. MIMO channel. [. , where. =)z

(43) ú. *!

(44) º. and. possible candidate channel matrices. It is worth noting that we only need. bits for feedback, where. . . M ]`P _ Q

(45) i M9]`P _ Q. . We have. denotes the smallest integer larger than . Besides, joint selection. scheme may provide more candidate channel matrices for a fixed number of total antennas. For. =!3 , *);43 , and 5 , we have sixteen candidate channel matrices, and only 2, two bits are required for feedback. If we conduct pure transmit antenna selection with 1!# *)+, , and 5 , the number of the candidates is reduced to ten. Furthermore, the required. example, if. bits for feedback will be increased to four. Note that the total numbers of antenna elements are the same for these two cases, i.e.,. A!`Ts*); .. § 2.4.2 Antenna Selection in MIMO Relay Systems Recently, cooperative communications have drawn a great deal of attention in wireless transmission. With the additional relay nodes, spatial diversity can be effectively enhanced. Multiple antennas can be placed at the source, the relays, and the destination. Such a cooperative system is referred to as a MIMO relay system. We now extend the proposed methods to antenna selection in MIMO relay systems. In this dissertation, we only consider a two-hop amplify-andforward (AF) system, as illustrated in Figure 2.4. As mentioned, we can consider antenna selection at each node for performance improve-. A)98 antennas. Let [ be the *)98 *! be the * ) *)98 relay-to-destination channel matrix.. ment. Assume that the relay node is equipped with source-to-relay channel matrix and. [. RD. 25. SR.

(46) In the AF relay scheme, signal transmission is divided into two phases. Denote. \#@. and. \ì. as the. indices of the candidate channel matrices in Phase I and Phase II, respectively. With antenna selection at both source and relay nodes, the source transmits the signal. an . channel. [ ±q » SR. d. to the relay through. during Phase I. Note that in a two-hop system, the destination cannot. receive the transmitted signal from the source in Phase I. Thus, the received signal at the relay can be expressed as. p @:5[ ± q » dTYr SR. II, the relay amplifies and retransmits. SR ,. p@. where. r. through an. is a white Gaussian noise vector. In Phase. . SR. channel. [. RD. ± q . The corresponding. p i , can then be expressed as ± q6Ê[ ± q » dDTsr -Tsr. received signal at the destination, denoted by. p i+[ where. r. RD. RD. SR. SR. (2.34). RD. is also a white Gaussian noise vector. Equivalently, (2.34) can be written as. p i;5[ ± qldTsr (2.35) where [ ± q"F[ ± qh[ ± q » , r F[ ± q hr T r , and \ depends on \@ and \ì . Note that the equivalent noise vector r is colored with the covariance matrix of qVv i ¹ [ ± q h[ ± q T v ii w ½ (2.36) v P i and v i are the variances of r and r , respectively. To conduct ML detection, the where v eq. eq. RD. SR. eq. RD. eq. SR. RD. eq. RD. SR. SR. RD. RD. RD. SR. SR. RD. equivalent noise vector must be whitened and this can be achieved by left-multiplying a matrix. q on p i , where. qVË¹b[ ± qh[ ± q T RD. RD. » v ii w ½ Þ k v P RD. (2.37). SR. After the whitening process, the resultant received signal can be expressed as. p i Üq [ ± qldT qÜr (2.38) q p i . Since the covariance of qr becomes a scaled identity matrix v i w , the where p i P eq. eq. eq. SR. selection schemes described in Section 2.1 and 2.2 can be directly used to enhance the system performance. 26.

(47) § 2.4.3 Sphere Decoding Algorithm The sphere decoding algorithm (SDA) is an efficient method to realize the ML detection in MIMO systems. In this subsection, we demonstrate the use of our proposed selection methods for SDA. Considering the signal model in (2.1), the idea of SDA is to search a subset of. IzP. such that. p U [qjd Í where. . [qV%«u. is the radius of the searching sphere. First, we conduct the QRD on . Since. «. (2.39). [Dq. yielding. is a unitary matrix, (2.39) can be rewritten as. p U [qld p U «ud Y p U d (2.40) Í where p e« p . Let denote the c th entry of p . With the upper-triangular structure of , we can further rewrite (2.40) as. p UY[qld i H U ° ± a i T H @ U ° 9@ ± a U ° @9± @9a @! i T4¨¨¨ P P P P Þ P Þ P P P Þ P Þ P Þ iP (2.41) Í The expression of (2.41) allows a tree search operation, starting with a , for finding the solution P candidates. Then, the candidate with the minimum distance is chosen as the output.. As we can see, the QRD operation is required in the SDA. Thus, the QRD processing unit can be shared with proposed antenna selection methods. However, if we adopt other selection methods such as the SVD-based or capacity-based method [18], extra circuits are required to conduct the SVD or calculate the channel capacity. Thus, with the proposed methods, the implementation complexity of the receiver can be effectively reduced. The capacity-based method, maximizing the capacity of the channel matrix, is described as follows. For a given candidate 27.

(48) [q , the channel capacity is expressed as "

(49) $i #&%')( w + P T * [ q [q-,. channel matrix. *. (2.42). where is the average SNR. The method then evaluates (2.42) for each candidate channel matrix, and selects the antenna subset having the maximum channel capacity. The computational complexity of the capacity-based method is. J g- , mainly arising from the matrix multipli-. cation and the determinant computation in (2.42). One additional overhead for the method is that the estimation of the noise variance is required.. § 2.5 Simulations and Discussions In this section, we report simulations evaluating the performance of our proposed selection methods. The simulation setup is described as follows. A flat-fading MIMO channel. [. is. used; its entries are assumed to be i.i.d. complex Gaussian random variables with zero mean and unit variance. The modulation scheme is QPSK, and the detection method is ML. Besides,. ò. the parameter. in the CLLL algorithm is set as 0.99. In our simulations, several selection. methods are compared, including 1) the SVD-based method, 2) the capacity-based method, 3) the QRD-based method, 4) the QRD-based BT-P method, 5) the QRD-based BT-C method, 6) the QRD-based BT-E method, and 7) the optimum method realized with an exhaustive search. Figure 2.5 shows the bit error rate (BER) performance of transmit antenna selection in the MIMO system. Here,. =!+e$ , *)Ae , and e . As we can see, the QRD-based method. indeed outperforms the SVD-based method. The performance of the capacity-based method is comparable to that of the QRD-based scheme. However, the capacity-based method requires additional information of the noise variance. The QRD-based BT-C method can outperform the. t³ Þ . We also observe that the QRD-based BT-C method provides the near-optimum performance. These results indicate that the -value spread can be SVD-based by. t k. dB at the BER of. reduced effectively with the QRD-based BT-C method. The performance of CLLL depends on. ò. ò. the parameter . As mentioned, if is smaller (close to 0.5), the computational complexity will 28.