Performance of Full-Rate Full-Diversity Space-Time Codes Under Quantization and Channel Estimation Error. Zhengwei Jiang

Transcription

1 Performance of Full-Rate Full-Diversity Space-Time Codes Under Quantization and Channel Estimation Error by Zhengwei Jiang A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 2010 by Zhengwei Jiang

2 Abstract Performance of Full-Rate Full-Diversity Space-Time Codes Under Quantization and Channel Estimation Error Zhengwei Jiang Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2010 In this work, we investigate the performance of full-rate full-diversity space-time codes (FRFD-STCs) under practical conditions. They were proposed recently, achieving the optimal diversity and multiplexing gain tradeoff. In this thesis, we first discuss the performance of FRFD-STCs in terms of bit error rate (BER) and frame error rate (FER) in the moderate SNR region. Different receivers are compared for these codes. The results show that the complexities of receivers are increased significantly in order to achieve the optimal performance. We also compare FRFD-STCs with spatial multiplexing, using the same transmission rate, for both un-coded and coded systems. The results show that FRFD-STCs have better performance than spatial multiplexing when the code rate is high and optimal or near-optimal detectors are implemented. Secondly, we investigate the issue of quantization in FRFD-STCs. We discuss the effect of the quantization error in the space-time encoding matrix. Our analysis and results show that the performance loss is negligible. Finally, we propose two receiver structures in the presence of imperfect channel state information (CSI). Variational expectation maximization (VEM) algorithm is used in our receivers. The two receivers use the VEM module as the channel estimator and MIMO detector respectively. The former structure is suitable for all kinds of STCs, while the latter one is specifically designed for FRFD-STCs. Both receivers are of low complexity, and have better performance than the methods proposed previously. ii

3 Acknowledgements I would like to first express my sincere gratitude to my thesis supervisor, Professor T. J. Lim, for his invaluable guidance and strong support throughout my M.A.Sc. program in University of Toronto. I have tremendously benefited from his unique vision, technical insights and practical sensibility. I deeply appreciate his strict training and precious advice in all aspects of my academic development. I thank my colleague Taiwen Tang, for valuable discussions on research and precious friendship that support me throughout the past two years. I wish him all the best in pursuing his dream in the future. I would like to thank my friends Pangzi, Junqi, Xiaofeng, Weiwei, Huahua and etc for their company and support, and also thank my friends Xiaoge, Jiawei, Duoge and Yunfeng, who are not here any more. I appreciate all the comfort from you when I feel horrible. I appreciate all the joy they have brought to me. I feel so blessed to have them on my side to unconditionally support me in my efforts towards my goals. Finally, I wish to send my gratitude to my parents. Their unreserved love and support have been the most important power that kept my perseverance throughout the past years. Without their care and encouragement from a distance, I would not have been so focused in pursuing my study. The spirit of this thesis is dedicated to them. iii

4 Contents 1 Introduction Background and Summary Literature Review Contributions of This Work Overview of the Thesis Background Diversity and Multiplexing Tradeoff Threaded Algebraic Space-Time Codes Perfect Space-Time Block Codes Variational Inference and Expectation Maximization Sub-Optimal Detection and Quantization of FRFD-STCs System Model Sub-Optimal Detection of FRFD-STCs Quantization Error in FRFD-STCs Summary Variational Expectation Maximization Receiver Problem Description VEM Algorithm for FRFD-STCs iv

5 4.2.1 VEM Algorithm using Discrete Model VEM Algorithm using Gaussian Model Comparison Between Two Models VEM as the Channel Estimator VEM as the MIMO Detector Simulation Convergence of VEM Receiver Performance Comparison in un-coded Systems Performance Comparison in Coded Systems Summary Conclusion Final Remarks Future Work Bibliography 64 v

6 List of Tables 3.1 Detector Comparison VEM iterative detection used as the channel estimator VEM iteration used as the MIMO detector vi

7 List of Figures 1.1 The conventional EM-based receiver The receiver applying VEM algorithm as the channel estimator The receiver applying VEM algorithm as the MIMO detector Diversity and Multiplexing Gain Tradeoff for general m, n The threading structure of TAST codes when N T = T = 4, where the vertical and horizontal axes denote the spatial and temporal dimensions and the numbers denote the indexes of the threads The bit error rate performance of perfect codes using different detection schemes in 2 2 MIMO channels without channel coding The bit error rate performance of TAST codes using different detection schemes in 2 2 MIMO channels without channel coding The bit error rate performance of PSTB, TAST, VBLAST codes using K-best and MMSE detections respectively in 2 2 MIMO channels The frame error rate performance of PSTB, TAST, VBLAST codes using K-best and MMSE detections respectively in 2 2 MIMO channels with the coding rate 1/ The frame error rate performance of PSTB, TAST, VBLAST codes using K-best and MMSE detections respectively in 2 2 MIMO channels with the coding rate 2/ vii

8 3.6 The achievable rates of PSTB and TAST in 2 2 MIMO channels, with and without quantization noise in the ST encoding matrices, in terms of bits per space-time codeword The BER performance of PSTB and TAST in 2 2 MIMO channels, with and without quantization noise in the ST encoding matrices The conventional EM-based receiver The receiver applying VEM algorithm as the channel estimator The receiver applying VEM algorithm as the MIMO detector The BER performances of PSTB codes, TAST codes and VBLAST codes over each VEM iteration in 2 2 MIMO channels The MSE of Channel Estimates over each VEM iteration in 2 2 MIMO channels, for PSTB codes, TAST codes and VBLAST codes The BER performance comparison between K-best detector and VEMassisted K-best detector, using PSTB and TAST codes respectively in 2 2 MIMO channels The BER performance comparison between VEM and MMSE detector, using PSTB and TAST codes respectively in 4 4 MIMO channels The FER performances of PSTB codes, TAST codes, VBLAST codes in 2 2 MIMO channels, compared between using the first receiver proposed (using VEM as channel estimator), using K-best detection and decoding in the presence of perfect CSI and using Iterative K-best Detection and Decoding in the presence of imperfect CSI (Iter=2) The MSE comparison of the channel estimation H c Ĥ c 2 F in 2 2 MIMO channels, between our proposed algorithm and EM based IDD receiver The FER performance comparison between VEM and IDD (iterative Turbo SIC receiver) in 4 4 MIMO channels, using PSTB and TAST codes respectively viii

9 4.11 The MSE comparison of the channel estimation H c Ĥ c 2 F in 4 4 MIMO channels, between our proposed algorithm and EM based IDD receiver.. 57 ix

10 Chapter 1 Introduction This work is concerned with the practical deployment of full rate full diversity spacetime codes (FRFD-STCs), in particular the design of receivers suitable for these codes. We propose two novel receiver structures with low computational complexity and strong robustness to channel estimation error. In this chapter, we will present the background of our work, and the main contributions of this work. Also we present the outline of the thesis. 1.1 Background and Summary Multiple-antenna transmission/reception has emerged as an important approach to achieve high spectral and power efficiency in wireless communications, due to its promise of increased channel capacity and communication reliability. For data transmission using multiple antennas over Rayleigh or Rician wireless channels, the family of codes called STCs (space time codes) was unveiled in [25], and a huge amount of work has been dedicated to this area since then. Assuming no channel information is available at the transmitter, we can use the multiinput multi-output (MIMO) structure as a means to increase the transmission reliability, e.g. the well known orthogonal STCs [24], but they cannot achieve the MIMO capacity at 1

11 Chapter 1. Introduction 2 the same time. On the other hand, spatial multiplexing is required to achieve the capacity of a MIMO channel [10]. Therefore, it is of practical importance to find a transmission scheme achieving both multiplexing and diversity simultaneously. The optimal tradeoff between the channel capacity (multiplexing gain) and reliability (diversity gain) has been revealed in [29], in what is now known as diversity and multiplexing gain tradeoff (DMT) theory. Following the introduction of DMT theory, several novel families of STCs have been proposed. Successive symbols are correlated through a linear transformation, and the symbols are split into streams, each of which is transmitted through all the antennas. These codes are named FRFD-STCs, in the sense that they are able to achieve the DMT limit. In [11, 6], linear threaded algebraic space time (TAST) codes were proposed and proven to be DMT optimal. In [21, 8], perfect space time block (PSTB) codes were also proven to achieve the DMT limit using sphere decoding (SD) [27]. In this thesis, we want to explore the performance of FRFD-STCs not just in terms of DMT, but in terms of error probability at moderate SNRs, in both un-coded and coded 1 systems, and we want to compare the performance with that of conventional STCs, like VBLAST (vertical Bell laboratories layered space time). It is also of practical importance to investigate the computational complexity we need to achieve the performance gain of FRFD-STCs, and the tradeoff between performance and complexity. Thus, we simulate different receivers for FRFD-STCs and discuss their computational complexities. The implementation of FRFD-STCs under realistic conditions will also be considered, where there is quantization, and channel uncertainty at the receiver. The design of receiver structure is the main focus of this thesis, where we assume channel uncertainty still exists after channel estimation with pilots. The ML (maximum likelihood) detection is too complicated to be implemented. Therefore, we appeal to EM (expectation maximization) algorithm. In order to further decrease the complexity at 1 In the sense of error control coding

12 Chapter 1. Introduction 3 the receiver, we use the variational EM (VEM) method. We design two receivers based on VEM. One of them provide excellent channel estimates that can be used by a softin soft-out MIMO detector to generate bit LLRs (log-likelihood ratios) for the channel decoder. The other is to use the VEM to generate bit LLRs, but we found that this was not as desirable because the LLR values generated were not accurate. 1.2 Literature Review In [29], DMT is proposed assuming that bit error probability is dominated by the outage probability in high SNR for the un-coded systems, and the authors point out that the outage probability can be decreased when symbols transmitted over successive time slots and different antennas are correlated together. Subsequently, TAST and PSTB codes were proposed respectively in [11, 6] and [21, 8], and they have been shown to meet the DMT limit. Besides, PSTB codes are also optimized with respect to lattice shaping, non-vanishing determinant and uniform average transmitted energy. Also shown in these work, their performance is better than previous works in terms of bit error rate (BER). The effect of channel uncertainty on receiver performance and design has been widely studied in the past. Here we are concerned with the design of an FRFD-STC receiver in the presence of residual channel estimation error. We assume that pilot symbols are transmitted to estimate the channel parameters prior to data transmission. However, even with ML channel estimation, the channel estimation error is still non-negligible. Previously, the detection of VBLAST have been studied in the presence of imperfect CSI. Given a Gaussian distributed channel estimation error, ML detection can be performed [23]. However this method is computationally prohibitive for practical implementations. A widely used alternative method is the expectation-maximization (EM) approach, where near-ml performance can be obtained after iterations between an E-step and a M-step. In [4], an exact EM procedures is implemented. Hard estimates of the symbols are produced

13 Chapter 1. Introduction 4 through iterations of sphere decoding (SD) and channel re-estimation. However, such an algorithm cannot produce the soft inputs essential in the decoding of powerful codes like LDPC (low-density parity check) and Turbo codes. Besides, SD detection is impractical for FRFD-STCs when the antenna number is moderately large (larger than 2), because the computational complexity of SD is at least the square of the signal vector length, which is the square of the antenna number in the FRFD-STCs case. In [17, 3], EMbased receivers are also proposed. The general frame work is shown in Fig The complexity of this scheme is very high, since channel decoding is performed multiple times, and performance in low SNR (signal to noise ratio) environment is still poor because the initial channel estimates are weak. Channel Output MIMO Detector LLR Turbo Decoder Channel Estimates Channel Estimator Figure 1.1: The conventional EM-based receiver 1.3 Contributions of This Work This thesis aims to gain deeper insights into the practical use of FRFD-STCs. Specifically, we study the performance in the presence of two realistic problems: quantization error and the absence of perfect channel state information (CSI) at the receiver. The contributions of this thesis are: 1. Due to complexity and energy constraints, the receivers cannot always afford a complicated and advanced MIMO detector, such as ML and SD detectors. Thus, we study the performance of the following detectors in this thesis: K-best detector

14 Chapter 1. Introduction 5 and MMSE (minimum mean square error) detector in the presence of perfect CSI. 2. According to the theoretical constructions of FRFD-STCs, irrational numbers are needed for the encoding and decoding. Due to quantization and limited-bit calculation, the numerical approximation of these numbers may distort the constellation shape, and as a result the STCs will suffer a performance loss. In this thesis, we present an analysis of the performance loss. 3. Extended from the VEM algorithm in [18], we propose two novel receivers. The first one uses the VEM algorithm as the channel estimator. The initial channel estimates are refined in the VEM module before being transferred to the MIMO detector as shown in Fig The second one uses the VEM method to produce the LLRs of the bits, with the channel estimates being updated iteratively inside the detector. This structure is shown in Fig The receivers are not only applicable to the FRFD-STCs, but also suitable to the conventional STCs, like vertical Bell laboratories layered space time (VBLAST). Channel Output MIMO Detector LLR Turbo Decoder Channel Estimates VEM Channel Estimator Figure 1.2: The receiver applying VEM algorithm as the channel estimator 1.4 Overview of the Thesis This thesis is organized as follows. In Chapter 2, we introduce the concept of the diversity and multiplexing gain tradeoff (DMT) theory, and the STCs which meet the

15 Chapter 1. Introduction 6 Channel Output E-STEP Update LLR LLR Turbo Decoder Channel Estimates M-STEP Update CSI Figure 1.3: The receiver applying VEM algorithm as the MIMO detector DMT limit. We also briefly describe the variational inference (VI) and expectation maximization (EM) algorithms, which we will use to design the receivers. In Chapter 3, we present the performances of FRFD-STCs in both un-coded and coded systems. The comparison between FRFD-STCs and VBLAST shows that a significant gain is achieved by FRFD-STC. We also present the analysis of the effect of quantization error on the encoding part of these codes, and the simulations show that the performance loss can be neglected, when 4-bit uniform quantization is taken for each real dimension, in a dynamic range. In Chapter 4, we describe the detection problem in the presence of channel uncertainty. Then we propose the variational EM detecting schemes, in which both the log likelihood ratio (LLR) and channel estimates are produced iteratively. Simulations show our proposed schemes are very suitable for FRFD-STCs. The complexity is much lower than previous EM-based joint iterative detection and decoding receiver, and the performance is improved at the same time. Finally, Chapter 5 concludes the thesis and discusses future possible work.

16 Chapter 2 Background This chapter introduces the concept of DMT (diversity and multiplexing gain tradeoff), which reveals the relation between the transmission rate and reliability in MIMO (multiple-input multiple output) systems. To achieve the DMT limit, several space time codes (STCs) have been proposed, and their coding procedures are presented in this chapter. The performances of these STCs are also studied. 2.1 Diversity and Multiplexing Tradeoff Conventionally, coding for MIMO systems was primarily concerned with either achieving the diversity gain (transmission reliability) or the spatial multiplexing gain (transmission rate). The former includes the well-known Alamouti scheme [1] and Orthogonal Space time block codes (OSTBC) [24], since they use multiple-antennas to increase the degrees of freedom and therefore combat fading. The latter includes the widely-used Bell Laboratories Layered Space Time (BLAST) scheme [10], which is also known as spatial multiplexing. In this case, the spatial degrees of freedom is limited, but the capacity of a channel with N T transmit, N R receive antennas in Rayleigh fading channels can be 7

17 Chapter 2. Background 8 written as [10] [ C(ρ) = E log det ( I + ρ HH )], N T where ρ is the signal to noise ratio (SNR) at the receiver antenna. At high SNR regime, the equation above can be approximated by C(ρ) = min{n T, N R } log ρ + O(1) Besides the two schemes above, there are also schemes that switch between them, depending on the communication environments. For the space-time structures above, the maximization of channel capacity always comes at the price of sacrificing the degrees of freedom and hence diversity, and vice-versa. It was not until [29] that the tradeoff between the transmission reliability and transmission rate was revealed, as the DMT theory. The authors assume that we have a scheme as a family of codes {C(ρ)}, one at each SNR level. Therefore, the transmission rate R(ρ), in bits Per Channel Use (PCU), can increase with the SNR. The spatial multiplexing gain r is then defined as and the diversity gain d is defined by R(ρ) lim ρ log ρ = r (2.1) P e (ρ) lim ρ log ρ where P e (ρ) is the average error probability. = d (2.2) The DMT theory allows us to understand the overall resources provided by multiantenna channels. Given a block length l, satisfying l N T +N R 1, the optimal tradeoff between multiplexing and diversity gain can be illustrated by the following piecewiselinear function: d(k) = (N T k)(n R k) (2.3) where k = 0, 1,... min{n T, N R }. In particular, d max = d(0) = N T N R, d min = d(min{n T, N R }) = 0. The DMT optimal curve for a general MIMO channel is as shown in Fig The

18 Diversity Gain: d(r)=-pe(snr)/logsnr Chapter 2. Background 9 (0,N T N R ) (1,(N T -1)(N R -1)) (min{n T,N R },0) Spatial Multiplexing Gain: r=r(snr)/logsnr Figure 2.1: Diversity and Multiplexing Gain Tradeoff for general m, n. optimal curve bridges the gap between the two design criteria before by connecting the two extreme points (0, N T N R ) and (min{n T, N R }, 0) together. Most of the previously well-known STCs cannot achieve the DMT theory limit, in the sense that they cannot achieve all the points in this figure. The DMT theory thus sets a framework for evaluating and comparing existing schemes, and for designing new schemes. 2.2 Threaded Algebraic Space-Time Codes In [11, 6], a family of novel STCs, known as TAST codes, was proposed, which achieved full-rate and full-diversity gain. The concept of space-time layering is integrated with algebraic component codes, where each component is assigned to a thread in the spacetime coding matrix. The main ideas behind this STC are described in this section. The main idea behind TAST codes is to introduce dependency among consecutive information symbols. We consider signaling over an N R N T MIMO channel. Before transmission, the information signals are mapped from a sequence of independent message symbols, say u = (u 1, u 2,..., u K ) T U K into an N T T 1 vector r(u) from the output

19 Chapter 2. Background 10 alphabet S N T T, where U is the QAM (quadrature amplitude modulation), PAM (pulse amplitude modulation) or HEX (hexagonal modulation) constellation and S is the singledimensional transmitted constellation. The output r(u) is then mapped into an N T T matrix T, i.e. the codeword used for transmission, and the mapping scheme is described below. The transmission rate of T is K/T symbols per channel use. In order to better illustrate the mapping procedures above, we introduce the concept of space-time threading [12]. Formally, a layer in an N T T space-time codeword is identified by an indexing set L I NT I T where I NT = 1, 2,..., N T, I T = 1, 2,..., T. The t th symbol interval on antenna n belongs to the layer L if and only if (n, t) L. A layer with full spatial and temporal spans will be referred to as a thread [12]. The sets of threads in TAST codes are designed so that each thread transmits a symbol using a different antenna during each symbol transmission interval. Assume the space time codeword consists of N threads, then the threaded layering set L = {L 1, L 2,...L N }, where L j = {( t + j 1 NT + 1, t ) : t = 0, 1,..., T 1 } for j = 1,..., N. (2.4) An example of this threading set is shown in Fig. 2.2 By space-time threading, we partition the space-time codes into multiple independent codes. The information vector u U K is first partitioned into N separate component vectors u j of length K j, j = 1, 2,...N, and then each of them is encoded independently using a component encoder r j mapping from U K j into S T j. Here we have K = K 1 +K K N and NT = T 1 + T T N. We mainly investigate the full-rate STCs, where N = N T and this assumption applies to the rest of the section. The output from each component encoder is then assigned to a thread. The remaining challenge is to construct the N T component encoders r 1, r 2,..., r NT, which map u 1,..., u NT into r 1 (u 1 ),..., r NT (u NT ). The composite space-time block code T constructed from r 1 (u 1 ),..., r NT (u NT ) achieves full spatial diversity. We may compare full-diversity DAST (diagonal algebraic space time) block codes [7] with the component

20 Chapter 2. Background 11 t=0 t=1 t=2 t=3 Ant. 1 Ant. 2 Ant. 3 Ant Figure 2.2: The threading structure of TAST codes when N T = T = 4, where the vertical and horizontal axes denote the spatial and temporal dimensions and the numbers denote the indexes of the threads. codes in T. A DAST code is obtained by rotating a k j dimensional information symbol vector using a K j K j real or complex matrix M j as M j u j. Each M j is constructed from an algebraic number field Q(θ) = {a + θb : a, b Q}, where Q is the rational number field and θ is an algebraic number of degree n. Compared with DAST codes, each component encoder r j in T also rotates the symbol vector using M j, and then scales the output M j u j using a properly chosen scalar, called the Diophantine number φ j C. φ j is chosen to achieve full diversity gain and maximize the coding gain for the composite code. The choice of the Diophantine numbers depends on the modulation scheme of the information signals. Hereto, we obtain the threads used for the construction of codeword T. By inducting the so-called Diophantine number, each thread can achieve full diversity whether other threads are present or not. Generally, each

21 Chapter 2. Background 12 thread can be denoted as r j (u j ) = φ j s j = φ j M j u j, (2.5) where j = 1, 2,...N T, r j (u j ) is the j th thread, u j is the j th symbol stream drawn from the constellation U, s j is the j th algebraic codeword (coded stream), φ j = φ (j 1)/N T is the Diophantine number assigned to the j th thread, for example φ = e iπ/6 for QPSK modulation. In this thesis, we mainly discuss the symmetric scenario, where we have K 1 = K 2 =... = K NT = T and all composite encoders share the same rotation matrix, i.e. M 1 = M 2 =... = M NT = M. To illustrate, we present two examples of full-rate and fulldiversity TAST codes, which will be investigated in this thesis. Example 1 : For N T = 2 transmit and N R = 2 receive antennas, two successive symbols are encoded together per thread, and there are two threads transmitted simultaneously. The TAST codeword is given by T = φ 1s 1 (1) φ 2 s 2 (2) φ 2 s 2 (1) φ 1 s 1 (2), (2.6) where s i (j) denotes the j th symbol transmitted from the i th coded stream, i.e. s 1 = (s 1 (1), s 1 (2)) T = M T AST (u 1 (1), u 1 (2)) T s 2 = (s 2 (1), s 2 (2)) T = M T AST (u 2 (1), u 2 (2)) T where u i (j) denotes the j th symbol from the i th signal stream drawn from U, and M T AST = 1 1 eiπ/4 2 1 e iπ/4. (2.7) Example 2 : For N T = 4 transmit and N R = 4 receive antennas, four successive symbols u i (1), u i (2), u i (3), u i (4) are encoded together per thread, and there are four

22 Chapter 2. Background 13 threads transmitted simultaneously. The TAST codeword is given by φ 1 s 1 (1) φ 4 s 4 (2) φ 3 s 3 (3) φ 2 s 2 (4) φ 2 s 2 (1) φ 1 s 1 (2) φ 4 s 4 (3) φ 3 s 3 (4) T =, (2.8) φ 3 s 3 (1) φ 2 s 2 (2) φ 1 s 1 (3) φ 4 s 4 (4) φ 4 s 4 (1) φ 3 s 3 (2) φ 2 s 2 (3) φ 1 s 1 (4) where s i (j) denotes the j th symbol transmitted from the i th coded stream, i.e. s 1 = (s 1 (1), s 1 (2), s 1 (3), s 1 (4)) T = M T AST (u 1 (1), u 1 (2), u 1 (3), u 1 (4)) T s 2 = (s 2 (1), s 2 (2), s 2 (3), s 2 (4)) T = M T AST (u 2 (1), u 2 (2), u 2 (3), u 2 (4)) T s 3 = (s 3 (1), s 3 (2), s 3 (3), s 3 (4)) T = M T AST (u 3 (1), u 3 (2), u 3 (3), u 3 (4)) T s 4 = (s 4 (1), s 4 (2), s 4 (3), s 4 (4)) T = M T AST (u 4 (1), u 4 (2), u 4 (3), u 4 (4)) T Here u i (j) denotes the j th symbol from the i th signal stream drawn from U, and 1 θ θ 2 θ 3 M T AST = 1 1 θ θ 2 θ 3, 2 1 iθ θ 2 iθ 3 1 iθ θ 2 iθ 3 (2.9) where θ = e iπ/8. We will present the performance of these codes in Chapter Perfect Space-Time Block Codes An n n STC is called a perfect code, if and only if it satisfies the following conditions. 1. Full rate: the code is a full-rate linear dispersion code [14], which use n 2 information symbols drawn from U.

23 Chapter 2. Background Non-vanishing determinant: a code has a non-vanishing determinant if there is a lower bound on the minimum determinant that does not depend on the constellation size, and the codes meet the rank criterion proposed by Tarokh [25]. Specifically, for every two distinct codeword matrices, say X 1 and X 2, the determinant of X = X 1 X 2, i.e. det( X X T ), prior to SNR normalization (explained in [29]), is lower-bounded by a positive constant. 3. Good constellation shaping: for each thread, the energy required to send the encoded symbols is analogous to that of sending the un-coded symbols directly, thus we do not introduce extra power in encoding the information symbols. 4. Uniform average transmitted energy: after space-time encoding, the average transmit energy per antenna in all time slots is unchanged. Perfect codes, presented in [8], [21], [9], are a subclass of linear dispersion space-time block codes (LD-STBCs) with the non-vanishing determinant (NVD) property. They are constructed from the so-called cyclic division algebra (CDA), described in Appendix A. With n transmit antennas, an n n STC codeword matrix will be obtained, sending n 2 symbols per codeword. For better understanding of the following content in this section, please refer to the Appendix for fundamental knowledge about cyclic division algebra. We construct a noncommutative algebra, denoted by A = (F/K, σ, γ), as follows: A = F ef... e n 1 F where F is a field extended from the central field K, e satisfies e n = γ and λe = eσ(λ) for λ F. Now we present the correspondence between an element x of the algebra A and a matrix X M n (F ), where M n (F ) denotes the set of n n matrixes with entries from F. The element x can be denoted as x = x 0 + ex e n 1 x n 1,

24 Chapter 2. Background 15 where x i F. We can find an isomorphism which maps x into the matrix given by X = x 0 γσ(x n 1 ) γσ 2 (x n 2 )... γσ n 1 (x 1 ) x 1 σ(x 0 ) γσ 2 (x n 1 )... γσ n 1 (x 2 ) x n 1 σ(x n 2 ) σ(x n 3 )... σ n 1 (x 0 ) (2.10) Now we present how a CDA-based space-time codeword can be constructed. Given the constellation U Q(i), where Q(i) = {a + bi : a, b Q, i = 1}, we choose the center field K to be Q(i), and F to be an n-degree cyclic Galois Extension of Q(i), say Q(i, θ), where n is the number of transmit antennas and θ is an irrational number. Q(i, θ) is defined as Q(i, θ) = {a + bθ : a, b Q(i)}. Let O K be the rings of algebraic integers in K. In order to guarantee the non-vanishing determinant property, we choose γ O K. In this case, we have O K = Z(i), where Z(i) = {a + bi : a, b Z, i = 1} and we choose γ = 1. Thus, we create the CDA, A = (Q(i, θ)/q(i), σ, 1), for constructing perfect codes. After this, we can construct the codeword according to (2.10). Finally, we shape the codeword by operating linear transform on each thread in order to make it power efficient. Example 3 : The Golden code [2] is a 2 2 perfect code. The PSTB code is defined from the cyclic algebra A = (F = (Q(i, 5)/Q(i), σ, i) with the generator embedding σ : 5 5. The algebraic integer ring of Q(i, 5) can be denoted as O F = {a + bθ a, b Q(i)},

25 Chapter 2. Background 16 where θ = (1 + 5)/2. The codeword before shaping 1 should be u 11 + u 12 θ u 21 + u 22 θ u 21 + u 22 σ(θ) u 11 + u 12 σ(θ), (2.11) where u ij denotes the j th symbol from the i th symbol stream drawn from the constellation U. After shaping, the 2 2 PSTB code is as follows where α = 1 + i iθ. Example 4 : X = 1 5 α(u 11 + u 12 θ) α(u 21 + u 22 θ) iσ(α)(u 21 + u 22 σ(θ)) σ(α)(u 21 + u 22 σ(θ)), (2.12) Now we consider the transmission of symbols over 4 antennas. We still choose the center field K to be Q(i). Let θ = 2 cos ( 2π ), and F to be Q(i, θ). We also choose 15 γ = i and the generator σ : θ 15 + θ 1 15 θ θ The corresponding CDA is given by A = (Q(i, 2 cos ( 2π )/Q(i), σ, i), i.e. 15 A = F ef e 2 F e 3 F The 4 4 PSTB codeword encodes 16 symbols x 0,..., x 15, drawn from U, so that the codeword is given by X = 3 diag ( M(x 4k, x 4k+1, x 4k+2, x 4k+3 ) ) E k, (2.13) k=0 where M is the generator matrix given by i i i i i i i i M = i i i i i i i i 1 The shaping procedure in STC is used to minimize the transmitting power

26 Chapter 2. Background 17 and E = γ Variational Inference and Expectation Maximization This section gives an introduction to variational inference (VI) and expectation maximization (EM), and after that we introduce the concept of variational expectation maximization (VEM). Variational inference is a technique for approximating complicated probability distributions, and in this thesis we will use it to simplify our joint detection and estimation problem. Suppose we have observed the data y, and we need the posterior distribution of x, p(x y). In some cases, this probability distribution is hard to manipulate, and we may want to use another distribution Q(x) to approximate it, i.e. Q(x) p(x y). The quality of the approximation is measured by the Kullback-Leibler divergence (KL divergence) between Q(x) and p(x y). The KL divergence or relative entropy [5] between two probability mass functions p(x) and q(x) is defined as follows and in our case we have D KL (q p) = x S x q(x) log q(x) p(x), (2.14) D KL (Q p) = x S x Q(x) log Q(x) p(x y). (2.15)

27 Chapter 2. Background 18 A straight-forward development of the equation above is D KL (Q p) = log p(y) + x S x Q(x) log Q(x) p(x, y). (2.16) If we denote L(Q) = Q(x) x Q(x) log, since p(y) is fixed, we have that we can minimize P (x,y) the KL divergence between Q(x) and p(x y) by minimizing the term L(Q). By choosing the distribution Q(x) carefully, we can make this problem tractable. Usually, we choose some distributions, such as Gaussian, which are fully described by a small number of parameters. Besides, we may use some other technique to further simplify the distribution we need to deal with, such as the mean-field approximation 2. In our case, say x is a vector of variables, containing x 1, x 2,...x n, then we approximate the distribution Q(x) assuming all the variables contained are independent, i.e. Q(x) n Q i (x i ). i=1 After these approximations, we will replace the original distribution p(x y) with n i=1 Q i(x i ) which is more suitable for analysis. In this thesis, we will use the variational method to simplify the EM algorithm. The EM algorithm [19] is an efficient iterative procedure to compute the maximum likelihood (ML) estimate when some data is hidden or missing. Generally, the EM algorithm consists of two steps: The E-step, and the M-step. In the E-step, the missing or hidden variables are estimated given the observed data and current estimate of the model parameters, using the conditional expectation. In the M-step, the likelihood function is maximized under the assumption that the hidden variable obtained from the E-step is accurate. This algorithm is guaranteed to increase the likelihood at each iteration, and hence convergence to a local maximum of the likelihood is assured. When the objective function is convex, the global maximum will be found. 2 This terminology arises in the field of statistical mechanics, and it can also be applied to inference theory.

28 Chapter 2. Background 19 Given a probability distribution p(x, y, z), where x is the parameter vector we are interested in, y is the observed data vector and z is the unobserved latent data vector or hidden variables. Based on this distribution, we would like to find the maximum likelihood estimate for x, and the problem is given by ˆx = max x p(x y) = max x z p(x z, y)p(z y)dz. (2.17) However, the distribution of the hidden variables p(z y) is also unknown, which should be calculated as p(z y) = p(x z, y)p(x y)dx. x Therefore, the problem goes to a dead end. Now we approach this problem, by approximating the distribution of z with the conditional distribution of z given y under the current estimate of the parameter x (t), i.e. p(z y, x (t) ). Therefore, we can solve the original problem (2.17) in an iterative method. The EM algorithm works as follows. Given the observed data, y, we wish to find the parameter x that maximizes the log likelihood, L(x) = log p(x y). We start the EM algorithm using some initial estimate of the parameter x (0), and then proceeds by iterating between the two following steps. E Step Compute the conditional distribution of z based on the estimate of the parameter x from the previous time, i.e. p (t) (z y, x (t 1) ). M Step Calculate the expected value of the log likelihood function based on the updated distribution of z, z p (t) (z y, x (t 1) ) log p(x z, y)dz, and then find the parameter x which maximizes this expectation, i.e. ˆx (t) = max p (t) (z y, x (t 1) ) log p(x z, y)dz. x z

29 Chapter 2. Background 20 Now we can introduce the concept of VEM [20] (variational expectation maximization). VEM can be regarded as the simplification of EM using variational technique. In the E-Step of EM algorithm, we need to update the distribution of z using the observed data y and the estimate of x. The problem is that, in some cases, this distribution is hard to manipulate such that the expectation is hard to calculate, or the optimal updating procedure is too complex to be implemented. Therefore, we want to approximate the original distribution p (t) (z y, x (t 1) ) with another distribution, Q (t) (z), which can be fully described by small number of parameters. Because p(z y, x (t 1) ) is approximated, the joint distribution of z and x is approximated as p(x, z y) = p(x z, y)p(z y) Q (t) (z)p(x z, y) Therefore, the VEM algorithm works as follows. We start the VEM algorithm using some initial estimate of the parameter x (0), and then proceeds by iterating between the two steps as follows. E Step Compute the parameters of the distribution Q (t) (z) such that the KL divergence between Q (t) (z) and p (t) (z y, x (t 1) ) is minimized. M Step Calculate the expected value of the log likelihood function based on the updated distribution of z, z p(x z, y)q (t) (z)dz, and then find the parameter x which maximizes this expectation, i.e. ˆx (t) = max Q (t) (z) log p(x z, y)dz. (2.18) x z

30 Chapter 3 Sub-Optimal Detection and Quantization of FRFD-STCs In this chapter, we will present the performance of the FRFD-STCs, using sub-optimal detectors. Quantization has always been an negative effect in the practical systems. Here we only consider the effect of the quantization noise at the transmitter for simplicity. Under finite precision, we will lose a certain amount of coding gain due to a decrease in the STC matrix determinant, but the diversity gain will not be affected since the STC matrix is still of full rank. The simulations show that the performance loss is actually negligible with 4 bits of quantization in each dimension. 3.1 System Model In Chapter 2, we have shown the construction of the FRFD-STCs. According to that, N T T consecutive symbols, are encoded by FRFD-STCs, and therefore N T T consecutive symbols are detected jointly at the receiver. Therefore, we denote the l th basebandequivalent space-time codeword transmitted and received over T channel uses as column vectors x(l) = [x 1,1 (l), x 1.2 (l),..., x 1,NT (l),..., x T,NT (l)] T 21

31 Chapter 3. Sub-Optimal Detection and Quantization of FRFD-STCs 22 y(l) = [y 1,1 (l), y 1.2 (l),..., y 1,NR (l),..., y T,NR (l)] T, where x(l) C T N T, y(l) C T N R, x t,n (l) denotes the symbol transmitted in the t th channel use over the n th transmit antenna and y t,n (l) denotes the signal received in the t th channel use over the n th receive antenna. Assuming that an error-control codeword spans LT channel uses and the channel remains constant over this period, the transmitted and received signal can be written in a T N T L and a T N R L matrix respectively X = [x(1),..., x(l)], Y = [y(1),..., y(l)]. Then the equivalent system is given by Y = (I T H c ) X + W = (I T H c ) GU + W, (3.1) where is the Kronecker product, I T is the T T identity matrix, H c is the N R N T channel matrix, G is the space-time encoding matrix, U U T N T L are the modulated symbols drawn from the symbol constellation U, and W is a matrix of i.i.d. Gaussian random variables. The decomposition of X into GU unifies various space-time coding schemes, e.g. for VBLAST G is the N T N T identity matrix and T = 1. We also identify G as an NT 2 N T 2 matrix and T = N T for PSTB and TAST codes, which is detailed in Chapter 2. For convenience, we rewrite G of TAST and PSTB codes for 2 2 MIMO channel here G T AST = M T AST 0 0 φ 1/2 M T AST, (3.2)

32 Chapter 3. Sub-Optimal Detection and Quantization of FRFD-STCs 23 and where i G P ST B = M P ST B 0, (3.3) φ 1/2 M P ST B M T AST = 1 1 eiπ/4 2 1 e iπ/4 M P ST B = 1 5 α αθ σ(α) σ(α)σ(θ), (3.4), (3.5) and θ = (1 + 5)/2, α = 1 + i iθ. 3.2 Sub-Optimal Detection of FRFD-STCs In [6, 2, 11, 21, 8], we have already seen that the FRFD-STCs have higher diversity gain and better performance in terms of the BER in un-coded systems. In this section, we would like to make a deeper investigation into the following aspects. Firstly, we present the performance of FRFD-STCs using non-ml detectors in un-coded systems, including K-best, MMSE-GDFE (minimum mean square error general decision feedback equalization) and MMSE-OSUC (minimum mean square error ordered successive cancelation). This is of practical significance because the allowable complexity at the receiver is usually limited. Secondly, we compare FRFD-STCs with VBLAST, which also has the full rate property. We present their performances in terms of BER in un-coded systems, and in terms of FER in coded systems. The following assumptions are made in this section. All the receivers have perfect knowledge of the channels, and no quantization error is considered in the systems. We first simulate the MIMO transmission scheme using PSTB and TAST codes respectively. Transmitted symbols are modulated using QPSK. Linear detectors and more

33 Chapter 3. Sub-Optimal Detection and Quantization of FRFD-STCs 24 advanced (SD, ML and K-best) detectors, are implemented in the simulations. The 2 2 MIMO channel with i.i.d. Rayleigh fading with zero mean and unit variance is studied here Bit Error Rate ML SD (Hassibi) MMSE GDFE K BEST SD (ZF) MMSE OSUC SNR Figure 3.1: The bit error rate performance of perfect codes using different detection schemes in 2 2 MIMO channels without channel coding In Fig. 3.1, the numerically optimized 2 2 PSTB codes is studied. The construction is presented in (2.12). The bit error rate (BER) in an un-coded system is presented for these codes using 6 different detection schemes. We present two methods of Sphere decoding (SD). One of them has achieved ML performance by using the zero-forcing (ZF) solution to set the radius, i.e. R = y Hˆx ZF, where y is the received signal vector, H is the MIMO channel and ˆx ZF denotes the detected signals using ZF detector. Suppose the ML solution is ˆx ML, since the distance

34 Chapter 3. Sub-Optimal Detection and Quantization of FRFD-STCs 25 between y and Hˆx ML is surely closer than that between y and Hˆx ZF, the ML solution is guaranteed to be located inside the sphere we choose. Thus, this method will always find the ML solution. The other SD detection scheme we use was introduced by Hassibi [15], which chooses a fixed radius. In this case, there is a chance that no point is located in this sphere, and thus this detector only gives a near ML solution. The complexity of SD is exponential in N R T in the worst case, but generally only polynomial in the size of the signal vector Bit Error Rate ML SD (Hassibi) MMSE GDFE K BEST SD (ZF) MMSE OSUC SNR Figure 3.2: The bit error rate performance of TAST codes using different detection schemes in 2 2 MIMO channels without channel coding The K-best detector [13] also gives near optimal solutions. We choose K = 10 here, i.e. 10 the best surviving paths are kept at each level of tree search. The value of K is proportional to the square of the length of the received signal vector, and thus detector complexity is much lower than that of ML detection, which is exponential in the size of signal vector, i.e.n T T. The comparatively low and fixed detection complexity makes this

35 Chapter 3. Sub-Optimal Detection and Quantization of FRFD-STCs 26 detector very attractive for the FRFD-STCs. Minimum mean square error (MMSE) decision feedback equalizer (DFE) and MMSE ordered successive cancelation (OSUC) were also simulated. We compare these linear detectors for the tradeoff between performance and complexity. The key idea of MMSE- DFE is layering striping where the symbol streams are successively detected and stripped away layer by layer. The detailed procedure is described in [22]. Because there is error propagation, the error rate performance is dominated by the first stream decoded by the receiver, whose transmit diversity gain is only 1. The complexity of MMSE-DFE is linear in N T T. Compared with MMSE-DFE, MMSE-OSUC [22] is a better but more complex receiver. The principle is that at the beginning of each stage, the stream with the highest SINR is selected for detecting and then canceled away from the remaining streams. Since the highest-sinr stream is detected first, the diversity gain of MMSE-OSUC is higher than that of MMSE-DFE, but the SINR comparison will require more computation than MMSE-DFE. Shown from Fig. 3.1, though these detectors are of comparatively low complexity, the performance lost is significant and the diversity gain is smaller than those using ML, SD and K-best detectors. The performance of 2 2 TAST codes are presented in Fig Different detection schemes are also compared for the TAST codes. Apparently, SD and K-best detectors still achieve ML or near-ml performance, and linear detectors lose the diversity gain compared with that using ML detector. Table 3.1: Detector Comparison ML SD(ZF) SD(Hassibi) K-best MMSE-DFE MMSE-OSUC Complexity O(e NtT ) O((N t T ) 2 ) O((N t T ) 2 ) O((N t T ) 2 ) O(N t T ) O(N t T ) Diversity Order N t N r N t N r N t N r N t N r N r N r In conclusion, we compare the complexities and diversity gains of different detectors for the FRFD-STCs in Table 3.1, and the detailed analysis can be found in [22]. We would also like to compare these FRFD-STCs against the conventional VBLAST

36 Chapter 3. Sub-Optimal Detection and Quantization of FRFD-STCs Bit Error Rate PSTB K best TAST K best VBLAST K best PSTB MMSE TAST MMSE VBLAST MMSE SNR Figure 3.3: The bit error rate performance of PSTB, TAST, VBLAST codes using K-best and MMSE detections respectively in 2 2 MIMO channels. codes, since all of them are full rate. In Fig. 3.3, MMSE and K-best BER curves are presented for these STCs in un-coded systems. PSTB and TAST codes generally have lower BER than VBLAST, especially when SNR is high, and their BER curves have steeper slopes. From that, we may conclude that PSTB and TAST codes have higher diversity gain than VBLAST, because the diversity gain is proportional to the slope of the BER curve in the high SNR region. Recall the definition of diversity in the DMT theory P e (SNR) lim SNR log SNR = d. This result matches the theoretical expectation exactly, i.e. PSTB and TAST codes are full-diversity, but VBLAST is not. We should also notice two things here: (i) Performance difference of these codes using MMSE detection is much less than that using K-best detection, which illustrates again that the MMSE detector will lose the diversity gain

37 Chapter 3. Sub-Optimal Detection and Quantization of FRFD-STCs Frame Error Rate PSTB K best TAST K best VBLAST K best PSTB MMSE TAST MMSE VBLAST MMSE SNR Figure 3.4: The frame error rate performance of PSTB, TAST, VBLAST codes using K-best and MMSE detections respectively in 2 2 MIMO channels with the coding rate 1/2. significantly. (ii) The performance improvements of the FRFD-STCs over VBLAST using K-best detector is at the cost of extra computational complexity, because the complexity of K-best is proportional to the square of N T T as shown in Table 3.1. In Fig. 3.4 and Fig. 3.5, we present the performance comparison between coded FRFD- STCs and VBLAST in terms of FER. Rate 1/2 and 2/3 Turbo codes are implemented respectively in the two figures, and we use parallel concatenated Turbo codes with (7,5) recursive system convolutional (RSC) code as the component code. Compared with the un-coded systems, the FER performance of these STCs are not significantly different from each other. From this figure, we may state that we should choose different STCs according to the detection computational complexity we can afford. For the K-best detection, when the code rate is low, these STCs have almost the same FER performances

38 Chapter 3. Sub-Optimal Detection and Quantization of FRFD-STCs Frame Error Rate PSTB K best TAST K best VBLAST K best PSTB MMSE TAST MMSE VBLAST MMSE SNR Figure 3.5: The frame error rate performance of PSTB, TAST, VBLAST codes using K-best and MMSE detections respectively in 2 2 MIMO channels with the coding rate 2/3. as shown in Fig When the code rate is high, FRFD-STCs have 1 db gain over the VBLAST at the FER of 10 2, as shown in Fig For MMSE detection, VBLAST have better performance over FRFD-STCs when the code rate is low and similar performance when the code rate is high. Notice that the MMSE detector loses the diversity gain of FRFD-STCs. Besides, the interference among the symbols for FRFD-STC is higher than that of VBLAST because more symbols are combined in one received signal vector. Therefore, MMSE detector produces a low signal to interference and noise ratio (SINR) for FRFD-STCs, which explains that VBLAST has similar or even better performance than FRFD-STCs using MMSE detection.