Iterative receiver design for broadband wireless communication systems via expectation maximization (EM) based algorithms

ITERATIVE RECEIVER DESIGN FOR BROADBAND WIRELESS COMMUNICATION SYSTEMS VIA EXPECTATION MAXIMIZATION (EM) BASED ALGORITHMS THE-HANH PHAM NATIONAL UNIVERSITY OF SINGAPORE 2007 ITERATIVE RECEIVER DESIGN FOR BROADBAND WIRELESS COMMUNICATION SYSTEMS VIA EXPECTATION MAXIMIZATION (EM) BASED ALGORITHMS THE-HANH PHAM (B Eng., Hanoi University of Technology) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 To my beloved mother and wife Acknowledgement I would like to express my sincere gratitude and appreciation to my supervisors Dr Nallanathan Arumugam, Dr Ying-Chang Liang and Dr Balakrishnan Kannan for their valuable guidance and constant encouragement throughout my Ph.D course My thanks also go to my colleagues in the ECE-I2 R Wireless Communications Laboratory at the Department of Electrical and Computer Engineering for their friendship and help Special thanks go to Cao Wei and “superman” Gao Feifei Finally, I would like to thank my family for their understanding and support I acknowledge my mother who has sacrificed herself for my happiness I also would like to thank my wife, a precious gift from the heaven, who has shared in my happiness and sadness ii Contents Acknowledgement ii Contents ii Summary vii List of Figures x List of Tables xi List of Symbols xii List of Abbreviations xiv Introduction 1.1 Motivations 1.2 Objectives and Contributions 1.3 Organization of the thesis Overview of Expectation Maximization (EM), Expectation Conditional Maximization (ECM) and Space-Alternating Generalized EM (SAGE) Algorithms 2.1 Expectation Maximization Algorithm iii CONTENTS 2.1.1 The Algorithm 2.1.2 Some examples 10 2.1.3 Basic Theory of the EM Algorithm 15 2.2 Expectation Conditional Maximization (ECM) Algorithm 17 2.3 Space-Alternating Generalized Expectation Maximization (SAGE) Algorithm 2.4 19 Summary 22 Joint Channel and Frequency Offset Estimation for Distributed MIMO Flat-Fading Channels 23 3.1 Introduction 23 3.2 System Model and ML Estimation 25 3.3 Proposed Iterative Joint Channel and Frequency Offsets Estimators 27 3.3.1 Algorithm 1: ECM Based Approach 28 3.3.2 Algorithm 2: SAGE-ECM Based Approach 32 Simulation Results 35 3.4.1 Example 1: × system with fixed channel and fixed offset 35 3.4.2 Example 2: × system, fading channel and fixed offset 40 3.4.3 Example 3: fading channel and random offset 42 Summary 44 3.4 3.5 Joint Channel Estimation and Data Detection for SIMO Systems 45 4.1 Introduction 45 4.2 System Model 47 4.3 Proposed Iterative Receiver 49 4.3.1 E-step 49 4.3.2 CM-step 51 Computational Complexity 53 4.4 iv CONTENTS 4.4.1 Step 54 4.4.3 Step 54 4.4.4 Step 54 Simulation Results 55 4.5.1 Initialization 56 4.5.2 4.6 53 4.4.2 4.5 Step Main Results 57 Summary 63 Doubly Iterative Receiver for Block-based Transmissions with EM-based Channel Estimation 64 5.1 Introduction 64 5.2 Overview of BI-GDFE receiver 68 5.3 Iterative Receiver for SCCP, MC-CDMA and CP-CDMA 70 5.3.1 System Models 70 5.3.2 Proposed Iterative Receiver 74 5.3.3 Cram´ r-Rao Lower Bound e 77 5.3.4 Simulation Results 79 Iterative Receiver for MIMO-IFDMA 88 5.4.1 System Model 88 5.4.2 Proposed Iterative Receiver 93 5.4.3 Cram´ r-Rao Lower Bound e 97 5.4.4 Simulation Results 98 5.4 5.5 Summary 105 Conclusions and Future works 106 6.1 Conclusions 106 6.2 Future works 107 v CONTENTS Appendix A 119 Appendix B 121 Appendix C 123 vi Summary Wireless communication systems are good choices to satisfy the growing demands on high-rate, high-quality communications for today’s users Due to the severe propagation environment, the quality of communication relies heavily on the channel information at the receiving side In this thesis, the Expectation Maximization (EM) algorithm, an iterative algorithm to find the Maximum-Likelihood (ML) estimates, is used to design iterative receivers in wireless communications More explicitly, in this thesis, the EM algorithm is used to estimate the channel coefficient as well as the frequency offset in Multi-Input Multi-Output (MIMO) systems with a general assumption of having multiple frequency offsets It is also used for joint channel estimation and data detection in Single-Input Multi-Output (SIMO) systems under the correlated noise environment The channel estimation and detection in the popular block-based transmission such as Single carrier cyclic-prefix (SCCP), Multicarrier code division multiple access (MC-CDMA), Cyclic-prefix code division multiple access (CP-CDMA) and Interleaved frequency division multiple access (IFDMA) are also investigated using the EM algorithm vii List of Figures 2.1 Illustration of many-to-one mapping from X to Y The point y is the image of x 2.2 An overview of the EM algorithm After initialization, the E-step and M-step are alternated until the parameter has converged (no more change in the estimate) 11 3.1 MIMO with frequency offsets system model 25 3.2 Comparison of MSE performances of w1,2 of [1], [2], ECM and SAGEECM algorithms 3.3 36 Comparison of MSE performances of h1,2 of [2], ECM and SAGEECM algorithms 37 3.4 Average number of iterations of ECM and SAGE-ECM algorithms 38 3.5 Comparison of MSE performances of w1,2 of [1], [2] and SAGE-ECM algorithm for different values of P 3.6 Comparison of MSE performances of h1,2 of [2] and SAGE-ECM algorithm for different values of P 3.7 39 Comparison of average number of iterations of SAGE-ECM algorithm for different values of P 3.8 39 40 Comparison of MSE performances of w1,2 of [1], [2], and SAGE-ECM algorithms for transmit antennas system viii 41 BIBLIOGRAPHY [24] E Telatar, “Capacity of multi-antenna gaussian channels,” AT&T Bell Labs Intern Tech Memo., Jul 1995 [25] G J Foschini, “Layered space-time architechture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech J., vol 1, no 2, pp 41–59, 1996 [26] Z Liu, G Giannakis, and B Hughes, 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to mobile radio transmission,” in Proc of IEEE ICUPC’98, Oct 1998, pp 1267–1272 115 BIBLIOGRAPHY [67] R Dinis, D Falconer, C T Lam, and M Sabbaghian, “A multiple access scheme for the uplink of broadband wireless systems,” in Proc of IEEE GLOBECOM’04, vol 6, Nov 2004, pp 3808–3812 [68] B Anderson and J Moore, Optimal Filtering Cliffs, N.J., 1979 116 Prentice-Hall, Inc, Englewood List of Publications The-Hanh Pham, A Nallanathan, and Ying-Chang Liang,“Joint Channel and Frequency Offset Estimation for distributed MIMO Flat-Fading Channels,” to appear in IEEE Trans on Wireless Communications, 2008 The-Hanh Pham, A Nallanathan, and Ying-Chang Liang,“A Computationally Efficient Joint Channel Estimation and Data Detection for SIMO Systems,” submitted to IEEE Trans on Wireless Communications, revised 05/2007 The-Hanh Pham, Ying-Chang Liang, and A Nallanathan, “Joint Channel Estimation and Data Detection for MIMO IFDMA Systems,” submitted to IEEE Trans on Signal Processing The-Hanh Pham, Ying-Chang Liang and A Nallanathan, “Doubly Iterative Receiver for Block Transmissions with EM-Based Channel Estimation,” submitted to IEEE Trans on Communications The-Hanh Pham, Ying-Chang Liang and A Nallanathan, “Doubly Iterative Receiver for Single carrier Cyclic-Prefix Transmissions with EM-Based Channel Estimation,” to appear in Proc of IEEE GLOBECOM’07, Washington D.C., U.S.A, November 2007 The-Hanh Pham, A Nallanathan and Ying-Chang Liang, “An EM-Based Joint Channel Estimation and Data Detection for SIMO Systems,” in Proc of IEEE APCCAS’06, Singapore, April, 2006 117 The-Hanh Pham, A Nallanathan and Ying-Chang Liang, “Joint Channel Estimation and Data Detection for SIMO Systems: An EM-based Approach,” in Proc of IEEE ICC’06, Istanbul, Turkey, June, 2006 118 Appendix A The input-output relationship of the considered system is defined in (4.3) over a block of T symbols and is written (in Section 4.3) as y = Sh + n (A-1) The conditional mean (and covariance) of h given y, S and Σ are ˆ ˆ ˆ ˆ h = hT (1) hT (2) · · · hT (T ) = K hS H SK hS H + K n T −1 E h|y, S, Σ y (A-2) and ˆ Kh ˆ ˆ E (h − h)(h − h)H |y, S, Σ = K h − K hS H SK hS H + K n −1 SK h (A-3) Equations (A-2) and (A-3) are well-known conditional expectation formulas in the Gaussian case (e.g., [68]) Because of the linearity of (A-1), (y − Sh) given y, S and Σ is a conditionally ˆ Gaussian random vector with mean (y−S h) and covariance matrix of SK hS H Thus, ˆ ˆ ˆ E (y − Sh)(y − Sh)H |y, S, Σ = S K hS H + (y − S h)(y − S h)H (A-4) 119 Furthermore, E −(y − Sh)H K −1 (y − Sh) y, S, Σ n = E −tr K −1 (y − Sh)(y − Sh)H y, S, Σ n = − tr K −1 E (y − Sh)(y − Sh)H y, S, Σ n ˆ ˆ ˆ = − tr K −1 S K hS H + (y − S h)(y − S h)H n ˆ ˆ ˆ = − tr K −1 S K hS H − tr (y − S h)H K −1 (y − S h) n n 120 (A-5) Appendix B In this appendix we present the derivations for the algorithm which takes signal as the missing data in the SIMO system under fast-fading channels in Chapter The receive signal at them t for the fast fading channels is written as y(t) = h(t)s(t) + n(t), t = 1, 2, · · · , T (B-1) According to [22], the missing data would be {s(t)}T The parameter we want t=1 to estimate is θ = {h(t)}T , Σ) t=1 We define the complete data is X = {y(t)}T , {s(t)}T The EM-based t=1 t=1 estimation algorithm consists of two steps as follows E-step In this step, we determine ˆ ˆ ˆ Q θ|θ[m] = E log f X|{h(t)}T , Σ {y(t)}T , {h[m] (t)}T , Σ[m] t=1 t=1 t=1 = C1 + E log f {y(t)}T {s(t)}T , {h(t)}T , Σ t=1 t=1 t=1 ˆ ˆ {y(t)}T , {h[m] (t)}T , Σ[m] (B-2) t=1 t=1 where C1 is a constant which does not relate to θ We have T f {y(t)}T t=1 T = t=1 {s(t)}T , {h(t)}T , Σ t=1 t=1 = f (y(t)|s(t), h(t), Σ) t=1 exp − y(t) − h(t)s(t) |πΣ| 121 H Σ−1 y(t) − h(t)s(t) (B-3) Therefore (B-2) can be written as ˆ Q(θ|θ [m] ) = −T log|Σ| T − t=1 E tr Σ−1 y(t)y H (t) − y(t)s∗ (t)hH (t) − h(t)(y(t)s∗ (t))H ˆ ˆ {y(t)}T , {h[m] }T , Σ[m] t=1 t=1 + |s(t)|2 h(t)hH (t) T = − T log|Σ| − tr Σ−1 y(t)y H (t) t=1 ˆ ˆ − E y(t)s∗ (t)|{y(t)}T , {h[m] (t)}T , Σ[m] h(t)H t=1 t=1 ˆ ˆ − h(t) E y(t)s∗ (t)|{y(t)}T , {h[m] (t)}T , Σ[m] t=1 t=1 −h(t)h(t)H H (B-4) Here we use the |s(t)|2 = assumption M-step In this step, the updated value of θ is determined by ˆ ˆ θ [m+1] = arg max Q θ|θ [m] (B-5) θ From [22], the updated value of {h(t)}T is determined by t=1 |C| ˆ h[m+1] (t) = y(t) c∗ ρ[m] (t), u u (B-6) u=1 where ρ[m] (t) u ˆ ˆ exp 2ℜ y H (t)(Σ[m] )−1 h[m] (t)cu = ˆ ˆ 2ℜ y H (t)(Σ[m] )−1 h[m] (t)cl |C| l=1 exp (B-7) The updated value of Σ is given by ˆ Σ[m+1] = T T y(t)y (t) − T T ˆ ˆ h[m+1] (t)(h[m+1] (t))H H t=1 122 t=1 (B-8) Appendix C In this appendix, we derive the CRLB for the channel vector h in the model of (5.45) over the T blocks The model of (5.45) is repeated here for convenience y(t) = S(t)h + n(t), t = 1, · · · , T, (C-1) Instead of separating the complex vector h into real and imaginary parts, we simplify the derivation by applying derivatives with respect to the complex vector h itself The CRLB states the lower bound for the vector h as follows CRLB(h) = tr I −1 (h) , (C-2) where I(h) is the Fisher information matrix [64] I(h) = E ∂ log f {y(t)}T h t=1 ∂h ∂ log f {y(t)}T h t=1 ∂h H (C-3) From (C-1), we have the conditional probability density function of {y(t)}T t=1 given h: f {y(t)}T t=1 1 h = exp − 2 ) N NR T (πσn σn T t=1 y(t) − S(t)h , (C-4) where we assume that the signal is known (in the form of the matrix S(t), t = 1, 2, · · · , T ) Therefore, the probability density function is not conditioned on S(t), t = 123 1, 2, · · · , T After differentiating the logarithm of (C-4) with respect to h, we obtain ∂ ∂ log f {y(t)}T h = t=1 ∂h ∂h − σn T t=1 y(t) − S(t)h T = σn = σn H S (t)y(t) t=1 T ∗ − σn T S H (t)S(t)h t=1 y H (t) − hH S H (t) S(t) t=1 ∗ T (C-5) Then the Fisher information matrix can be obtained as I(h) = E ∂ log f {y(t)}T |h t=1 ∂h =E σn T y H (t) − hH S H (t) S(t) t=1 σn = = E (σn )2 σn ∂ log f {y(t)}T |h t=1 ∂h T × T t=1 H y H (t) − hH S H (t) S(t) ∗ T ∗ t=1 S T (t) y H (t) − hH S H (t) y H (t) − hH S H (t) S ∗ (t) T S T (t)S ∗ (t) (C-6) t=1 Replacing (5.46) into (C-6), we have I(h) = (I NR ⊗ C) , σn (C-7) where  T F  C=   FT T t=1 DT T t=1 (t)D ∗ (t) ∗ F ···F T D T T (t)D ∗ (t) F ∗ · · ·F T N T t=1 DT (t)D ∗ T (t) N ∗  F    (C-8)   T D T T (t)D ∗ T (t) F ∗ N N t=1 The above derivation of CRLB can also be applied to the single-input singleoutput SCCP, MC-CDMA and CP-CDMA systems considered in Section 5.3 The general system model of those systems is given as in (5.13) y(t) = D(t)F L h + n(t), t = 1, 2, · · · , T 124 (C-9) From this equation and (C-6), we have the Fisher information matrix for the channel vector h as follows I(h) = σn T F T DT L t=1 (t)D ∗ (t)F ∗ L = 2FT σn L 125 T D T (t)D ∗ (t) F ∗ (C-10) L t=1 .. .ITERATIVE RECEIVER DESIGN FOR BROADBAND WIRELESS COMMUNICATION SYSTEMS VIA EXPECTATION MAXIMIZATION (EM) BASED ALGORITHMS THE-HANH PHAM (B Eng., Hanoi... recommendations for future works Chapter Overview of Expectation Maximization (EM), Expectation Conditional Maximization (ECM) and Space-Alternating Generalized EM (SAGE) Algorithms 2.1 Expectation Maximization. .. Overview of Expectation Maximization (EM), Expectation Conditional Maximization (ECM) and Space-Alternating Generalized EM (SAGE) Algorithms 2.1 Expectation Maximization Algorithm