Channel estimation and synchronization for OFDM and OFDMA systems

CHANNEL ESTIMATION AND SYNCHRONIZATION FOR OFDM AND OFDMA SYSTEMS WANG ZHONGJUN NATIONAL UNIVERSITY OF SINGAPORE 2008 CHANNEL ESTIMATION AND SYNCHRONIZATION FOR OFDM AND OFDMA SYSTEMS WANG ZHONGJUN (M. Eng., National University of Singapore) (M. Sc., Shanghai Jiao Tong University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2008 Copyright c 2008, Wang Zhongjun To Grace Wang Ruiqi, my dearest daughter Acknowledgment I would like to thank my supervisors Professor Yan Xin and Professor George Mathew for their constant guidance and encouragement throughout the period of this research work. Without their help and advice completion of the thesis would not have been possible. I wish to thank Professor Xiaodong Wang, Columbia University, with whom I have had the good fortune to collaborate. I have benefited a lot from his inspirational guidance. I also wish to thank my mentor Mr. Masayuki Tomisawa and my fellow colleagues in Wipro Techno Centre (Singapore), for encouraging me to carry out my research work. Their understanding and support were essential to the completion of my study. Special thanks goes to my fellow graduate students Jinhua Jiang, Lan Zhang, Yan Wu and Feifei Gao who have always been willing to discuss and exchange ideas and help me a lot in my study. I owe them a great deal for their friendship. Last but not least, I thank my parents and my wife for their love and support which played an instrumental role in the completion of this project. i Contents Acknowledgment i Contents ii Summary vi List of Tables viii List of Figures ix List of Abbreviations xiii List of Symbols and Operators xvi Chapter 1. Introduction 1.1 Introduction to OFDM Based Systems . . . . . . . . . . . . . . . . . . . 1.2 Motivation for the Present Work . . . . . . . . . . . . . . . . . . . . . . 1.3 Contributions of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Channel Estimation in OFDM Systems . . . . . . . . . . . . . . 1.3.2 Phase Error Suppression for Multi-Band OFDM-UWB Systems . 1.3.3 CFO Estimation for SISO-OFDMA and MIMO-OFDMA Uplink 1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2. ML Channel Estimation in OFDM Systems 2.1 OFDM System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii 10 12 12 Contents 2.2 2.3 ML Channel Estimator and Performance . . . . . . . . . . . . . . . . . . 14 2.2.1 MSE of the ML Estimator . . . . . . . . . . . . . . . . . . . . . 15 2.2.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 16 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter 3. 3.1 Modified ML Channel Estimators 19 Modified ML Channel Estimator I - OMLE . . . . . . . . . . . . . . . . 20 3.1.1 Smoothing Matrix for OMLE . . . . . . . . . . . . . . . . . . . 20 3.1.2 Derivation of Optimum αi (k) . . . . . . . . . . . . . . . . . . . 21 Modified ML Channel Estimator II - IMLE . . . . . . . . . . . . . . . . 23 3.2.1 Smoothing Matrix for IMLE . . . . . . . . . . . . . . . . . . . . 23 3.2.2 Parameter Selection in IMLE . . . . . . . . . . . . . . . . . . . . 26 3.3 Advantages of Modified Estimators . . . . . . . . . . . . . . . . . . . . 27 3.4 System Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Chapter 4. Multi-Band OFDM-UWB System Model 33 4.1 Transmitter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 UWB Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Modeling of PHN, CFO and SFO at Receiver . . . . . . . . . . . . . . . 37 4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Chapter 5. Channel Estimation for Multi-Band OFDM-UWB Systems 42 5.1 Assumptions and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Stage – Primary CFR Estimation . . . . . . . . . . . . . . . . . . . . . 43 5.3 Stage – Enhanced CFR Estimation . . . . . . . . . . . . . . . . . . . . 45 5.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 iii Contents Chapter 6. Phase Error Suppression for Multi-Band OFDM-UWB Systems 6.1 61 SFO Estimation and Compensation . . . . . . . . . . . . . . . . . . . . . 61 6.1.1 Basic Algorithm for SFO Estimation . . . . . . . . . . . . . . . . 62 6.1.2 Weighted SFO Estimation . . . . . . . . . . . . . . . . . . . . . 64 6.1.3 Combined SFO Estimation . . . . . . . . . . . . . . . . . . . . . 65 6.1.4 Two-Dimensional SFO Compensation . . . . . . . . . . . . . . . 68 CPE Estimation and Correction . . . . . . . . . . . . . . . . . . . . . . . 69 6.2.1 Weighted CPE Estimation . . . . . . . . . . . . . . . . . . . . . 69 6.2.2 Smoothed CPE Estimation . . . . . . . . . . . . . . . . . . . . . 70 6.2.3 Analysis of MSE Reduction Performance . . . . . . . . . . . . . 72 6.2.4 CPE Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.2 Chapter 7. ML Estimation in OFDMA Systems 84 7.1 Signal Model for Generalized OFDMA Uplink . . . . . . . . . . . . . . 85 7.2 Existing ML Based CFO Estimators . . . . . . . . . . . . . . . . . . . . 87 7.3 Cram´er–Rao Bound (CRB) . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.4 Convergence Property of ML Estimation . . . . . . . . . . . . . . . . . . 92 7.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Chapter 8. New Approach for OFDMA Uplink CFO Estimation 8.1 8.2 94 Divide-and-Update Frequency Estimator (DUFE) . . . . . . . . . . . . . 94 8.1.1 Step – Primitive CFO Estimation . . . . . . . . . . . . . . . . . 95 8.1.2 Step – Divide-and-Update CFO Adjustment . . . . . . . . . . . 96 8.1.3 ˆ (i+1) ) . . . . . . . . . . . . . . . . . . . . . Computation of Φ(ω 98 Further Discussion on DUFE . . . . . . . . . . . . . . . . . . . . . . . . 100 8.2.1 Choices of G(·) . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.2.2 An Example Illustrating the Convergence Behavior of DUFE . . . 101 8.2.3 Remarks on Joint CFO and Channel Estimation . . . . . . . . . . 103 iv Contents 8.3 8.4 Performance and Complexity Comparison . . . . . . . . . . . . . . . . . 104 8.3.1 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . 104 8.3.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . . 110 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Chapter 9. CFO Estimation for MIMO-OFDMA Uplink Transmission 116 9.1 MIMO-OFDMA Signal Model . . . . . . . . . . . . . . . . . . . . . . . 116 9.2 Iterative CFO Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 118 9.3 Performance and Complexity Comparison . . . . . . . . . . . . . . . . . 119 9.4 9.3.1 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . 119 9.3.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . . 122 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Chapter 10. Conclusions 127 10.1 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 10.2 Directions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 130 Bibliography 132 List of Publications 144 Appendix A. Derivation of Optimum αh 146 Appendix B. Derivation of Peub and C24 147 B.1 Derivation of Peub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B.2 Derivation of C24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 ˆ(i) Appendix C. Derivation of E f (ej θm ) and MSEcpe (i) j θˆm C.1 Derivation of E f (e C.2 ) 151 . . . . . . . . . . . . . . . . . . . . . . . . . 151 Derivation of MSEcpe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 v Summary The development of robust and high-performance channel estimation and synchronization algorithms plays an important role in the area of multicarrier/multiuser wireless communications. In this dissertation, we investigate some critical issues associated with the development of these algorithms for orthogonal frequency division multiplexing (OFDM) and OFDM multiple-access (OFDMA) systems. This thesis consists of three parts. In the first part, the maximum likelihood (ML) solution for channel estimation in OFDM systems is investigated. The mean-squared error (MSE) performance of the conventional ML estimator (MLE) is analyzed and is shown to be linearly related to the effective length of channel impulse response (ELCIR). Tracking the variation in ELCIR is thus very important for conventional MLE for achieving optimum estimation. But, incorporating a run-time update of ELCIR into the ML estimator turns out to be computationally expensive. Therefore, a modified ML channel estimator, which systematically combines the ML estimation with a frequency-domain smoothing technique, is proposed. The proposed modification is presented in two forms, namely, optimum-smooth MLE (OMLE) and iterative-smooth MLE (IMLE). The proposed method introduces no extra complexity, and its performance has been proved using theoretical analysis and simulations to be robust to variation in ELCIR. Numerical results are provided to show the effectiveness of the proposed estimator under time-invariant and time-variant channel conditions. 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Mathew, “Iterative carrier-frequency offset estimation for generalized OFDMA uplink transmission,” IEEE Transactions on Wireless Communications, vol. 8, no. 3, pp. 1373–1383, Mar. 2009. 2. Z. Wang, Y. Xin, G. Mathew, and X. Wang, “A low complexity and efficient channel estimator for multi-band OFDM-UWB systems,” submitted to IEEE Transactions on Vehicular Technology (submitted in 2008 and revised/resubmitted in May 2009). 3. Z. Wang, Y. Xin, G. Mathew, and X. Wang, “Efficient phase error suppression for multi-band OFDM-based UWB systems,” submitted to IEEE Transactions on Vehicular Technology (submitted in 2008 and revised in May 2009). Conference Papers (published) 1. Z. Wang, Y. Xin, and G. Mathew “Carrier-frequency offset estimation for OFDMA uplink with generalized subcarrier-assignment,” in Proc. IEEE Int’l Conf. Commun. (ICC), Beijing, China, May 19–23, 2008, pp. 3490–3494. 2. Z. Wang, Y. Xin, and M. Tomisawa, “Design and analysis of channel estimation for multi-band OFDM-UWB systems,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), New Orleans, LA, Nov. 30–Dec. 4, 2008. 3. Z. Wang, Y. Xin, and M. Tomisawa, “Phase error suppression for multi-band OFDM-based UWB systems,” in Proc. IEEE Vehicular Technology Conf. (VTC), Singapore, May 11–14, 2008, 1072–1076. 144 List of Publications 4. Z. Wang and Y. Xin, “Carrier-frequency offset estimation for MIMO-OFDMA uplink transmission,” in Proc. IEEE Vehicular Technology Conf. (VTC), Singapore, May 11–14, 2008, 1712–1716. 5. Z. Wang, G. Mathew, Y. Xin, and M. Tomisawa, “A robust maximum likelihood channel estimator for OFDM systems,” in Proc. IEEE Wireless Commun. and Networking Conf. (WCNC), Hongkong, Mar. 11–15, 2007, pp. 169–174. 6. Z. Wang, G. Mathew, Y. Xin, and M. Tomisawa, “An iterative channel estimator for indoor wireless OFDM systems,” in Proc. IEEE Int’l Conf. on Commun. Sys. (ICCS), Singapore, Nov. 2006. 7. Z. Wang, W. Li, L. G. Yeo, Y. Yan, Y. Ting, and M. Tomisawa, “A technique for demapping dual carrier modulated UWB OFDM signals with improved performance,” in Proc. IEEE Vehicular Technology Conf. (VTC), Dallas, TX, Sept. 25–28, 2005, vol. 1, pp. 38–42. 8. Z. Wang, L. G. Yeo, W. Li, Y. Yan, Y. Ting, and M. Tomisawa, “A Novel FFT Processor for OFDM UWB Systems,” in Proc. IEEE Asia Pacific Conf. on Circuits and Sys. (APCCAS), Singapore, Dec. 4–7, 2006, pp. 374–377. 145 Appendix A Derivation of Optimum αh Assuming that the CFR, transmitted signal and additive white Gaussian noise are independent of each other, from (5.1) and (5.2), we have σ = E0 + E0 /(M1 · SNRr ) M1 r ˆ (1) (k)h∗ (k) = E hr (k) h ˆ (1) (k) ∗ = E0 E h r r r ˆ (1) (k)|2 = E0 + E |h r ˆ (1) (k ± ∆k ) h ˆ (1) (k) E h r r ∗ (A.1) (A.2) = E hr (k ± ∆k )h∗r (k) = corr(±∆k )E0 , ∆k ∈ Z21 (A.3) and ˆ (1) (k ∓ 1) ˆ (1) (k ± 1)h∗ (k) = E hr (k) h E h r r r ∗ = corr(±1)E0 (A.4) k ∈ Z0N −1 . Using (A.1) to (A.4) and the property that corr(∆k ) = corr∗ (−∆k ) [136], (2) we can derive MSEr from (5.2) and (5.3) as ˆ r (k) − hr (k)|2 } E{|h E0 (2) MSE(2) = r = − 2ℜ[4corr(1) − corr(2)] αh2 + (2) (6αh2 − 4αh + 1) . M1 · SNRr (A.5) (2) Next, we minimize MSEr with respect to αh . Setting to zero the gradient of MSEr with respect to αh , we get (2) ∂MSEr = 2(6 − 2ℜ[4corr(1) − corr(2)])αh + (12αh − 4)/(M1 · SNRr ) = 0. (A.6) ∂αh From (A.6), it is straightforward to obtain (5.5). 146 Appendix B Derivation of Peub and C24 B.1 Derivation of Peub In this appendix, we derive the approximate upper bound of average bit error probability of the proposed CFR-weighted detector, Peub . After the first stage of channel estimation, ˆ (2) h r (k), given by (5.3) can be expressed as ˆ (2) (k) = hr (k) + e¯r (k) h r (B.1) where the estimation error term, e¯r (k), is a complex normal random variable with mean 1,2 zero and variance σ ¯r2 = E0 /(M1 Rmse SNRr ). We note that e¯r (k) is uncorrelated with (i) ˆ (2) hr (k) and sm (k), but is correlated with h r (k). Referring to the discussion in [137, Section II-B], we can decompose e¯r (k) into two uncorrelated terms as e¯r (k) = eˆr (k) + ˆ (2) e˜r (k) = h σr2 /E0 + e˜r (k). Here, e˜r (k) is a complex normal random variable with r (k)˜ ˆ (2) mean zero and variance σ ˜r2 = E0 σ ¯r2 /(E0 + σ ¯r2 ), which is uncorrelated with h r (k). Thus, (B.1) can be rewritten as ˆ (2) (k) − e˜r (k). hr (k) = (1 − σ ˜r2 /E0 )h r (B.2) From (5.9), (5.1), (4.3) and (B.2), we have (i) (i) ˆ (2) (k) s(i) (k) − h ˆ (2) (k)s(i) (k)˜ zm (k)+zm (N −k) = (1 − σ˜r2 /E0 ) h er (k) r m r m ˆ (2) (N − k) [s(i) (k)]∗ ˆ (2) (k)υ (i) (k) + (1 − σ +h ˜r2 /E0 ) h r m r m ˆ (2) (N −k)[s(i) (k)]∗ e˜r (N −k)+ h ˆ (2) (N −k)υ (i) (N −k) −h r m r m 147 B.1 Derivation of Peub R/2 k ∈ Z1 −1 and k ∈ / {p(l)}Pl=0 , i ∈ Z50 , r = |i|3 + and m ∈ Z21 . , Let r0 = |2q|3 + and r1 = |2q + 1|3 + for q ∈ Z20 in the following. Obviously, (2q) (2q) (2q) (2q+1) (2q+1) zm (k) + zm (N − k) + zm sm (k), sm (2q+1) (k) + zm (N − k) is Gaussian distributed for given ˆ (2) ˆ (2) ˆ (2) ˆ (2) (k), h r0 (k), hr1 (k), hr0 (N−k) and hr1 (N−k). Following the fact that (i) (i) (i) |sm (k)|2 = 1, E{υm (k)} = 0, E{[υm (k)]2 } = 0, E{˜ er (k)} = 0, E{[˜ er (k)]2 } = 0, (2q+1) sm (2q) (x + x∗ ), it is straightforward to show that the (2q) (2q) (2q+1) (2q+1) ℜ[zm (k) + zm (N −k) + zm (k) + zm (N −k)] (k) = sm (k) and ℜ(x) = conditional mean and variance of are given by (2q) (2q) (2q+1) (2q+1) E ℜ[zm (k) + zm (N −k) + zm (k) + zm (N −k)] ˆ (2) ˆ (2) ˆ (2) ˆ (2) s(2q) m (k), hr0 (k), hr1 (k), hr0 (N−k), hr1 (N−k) ˆ (2) (N −k) ˆ (2) (N −k) +|h ˆ (2) (k) + h ˆ (2) (k) + h = (1− σ ˜r2 /E0 ) h r1 r0 r1 r0 ℜ[s(2q) m (k)] and (2q) (2q) (2q+1) (2q+1) Var ℜ[zm (k) + zm (N −k) + zm (k) + zm (N −k)] ˆ (2) ˆ (2) ˆ (2) ˆ (2) s(2q) m (k), hr0 (k), hr1 (k), hr0 (N−k), hr1 (N−k) ˆ (2) (N −k) ˆ (2) (N −k) +|h ˆ (2) (k) + h ˆ (2) (k) + h = (σr2 + σ ˜r2 ) h r1 r1 r0 r0 , respectively. Therefore, we get [139] (2q) (2q) (2q+1) (2q+1) Pr ℜ[zm (k) + zm (N −k) + zm (k) + zm (N −k)] <   = Q   ≈ Q ˆ (2) ˆ (2) ˆ (2) ˆ (2) ℜ[s(2q) m (k)] = c, hr0 (k), hr1 (k), hr0 (N−k), hr1 (N−k) (1 − σ ˜r2 /E0 )2 ˆ (2) h r0 (k) + 2 ˆ (2) ˆ (2) h r0 (N −k) +|hr1 (k) + σr2 + σ ˜r2 hr0 (k) + hr0 (N − k) + |hr1 (k) + hr1 (N −k) σr2 + σ ˜r2 The approximation in (B.3) follows from (B.2) since ˆ (2) (k) ≈ hr (k)/(1 − σ h ˜r2 /E0 ). r 148 ˆ (2) h r1 (N −k)  .     (B.3) (B.4) B.2 Derivation of C24 From Section 4.2, one can find that |hr (k)| corresponds to the shadowing factor X and thus is log-normal distributed, i.e., 20 log10 |hr (k)| ∼ N (0, σx2 ). Since |hr (k)| and (2q) (2q) (2q+1) |hr (N − k)| are not independent, Pr ℜ zm (k) + zm (N−k) + zm (2q+1) (k) + zm (N− (2q) ℜ sm (k) = c is upper-bounded under the assumption that |hr0 (k)| = k) < |hr0 (N −k)| = |hr1 (k)| = |hr1 (N −k)| (without both frequency-domain and time-domain diversities), i.e, (i) Pr u(i) m (k) = −1 ℜ[sm (k)] = c (2q) (2q) (2q+1) (2q+1) = Pr ℜ[zm (k)+zm (N −k) + zm (k)+zm (N −k)] < ℜ[s(2q) m (k)] = c ∞ x − x ≤√ (B.5) Q αp · 10 20 e 2σx2 dx 2πσx −∞ =2 σr2 +˜ σr2 where αp = (M1 R1,2 mse SNRr +1)SNRr . [(M1 R1,2 mse +1)SNRr +1]E0 (i) Assuming that ℜ[sm (k)] is equiprobably ±c, we have (i) (i) (i) Pr u(i) m (k)ℜ[sm (k)] < = Pr um (k) = −1 ℜ[sm (k)] = c (i) = Pr u(i) m (k) = ℜ[sm (k)] = −c . (i) (B.6) (i) A similar procedure can be applied for the derivation of Pr vm (k)ℑ[sm (k)] < , and we have (i) (i) (i) Pe = Pr u(i) m (k)ℜ[sm (k)] < = Pr vm (k)ℑ[sm (k)] < (B.7) which is approximately upper-bounded by Peub as shown in (5.15). B.2 Derivation of C24 Following the definition of C24 in Section 5.4, we have C24 ˆ (4) − hr )(h ˆ (2) − hr )H } E{(h r r = E{ hr } ∗ ˆ (4) ˆ (2) E{[h r (k)−hr (k)][hr (k)−hr (k)] } R/2 N −1 , k ∈ Z1 ∪ ZN = −R/2 . E0 (B.8) From (5.2) and (5.3), we have ˆ (2) (k) −hr (k) = αh [hr (k −1) +hr (k + 1)]−2αh hr (k) + ζ1, k ∈ ZR/2 ∪ZN −1 (B.9) h r N −R/2 149 B.2 Derivation of C24 where ζ1 is a zero-mean complex Gaussian random variable comprising the channel noise from the first stage of channel estimation. From (5.11) it follows that ˆ (4) (k) − hr (k) = βh [h ˆ (3) (k − 1) + h ˆ (3) (k + 1)] h r r r ˆ (3) (k) − hr (k), k ∈ ZR/2 ∪ ZN −1 . + (1 − 2βh )h r N −R/2 (B.10) Next, substituting (5.1) into (5.10) and neglecting the specific arrangement on pilot tones, we obtain ˆ (3) (k) = c h r M2 (q) (q) hr (k)s(q) m (k) um (k)−jvm (k) m=1 q=r−1 q=r+2 R/2 +ζ2 , k ∈ Z1 N −1 ∪ ZN −R/2 (B.11) where the noise term ζ2 = (q) (q) υm (k) u(q) m (k)−jvm (k) c M2 with E{ζ2 hr (k)} = m=1 q=r−1 q=r+2 and E{ζ1 ζ2 } = 0. Thus, from (B.11), it is straightforward to obtain   (1 − 2Pe )E0 , ∆k = (3) ∗ ˆ (k ± ∆k )h (k)} = E{h r r  corr(±∆k )(1 − 2Pe )E0 , ∆k ∈ {1, 2}. (B.12) Applying (B.9), (B.10) and (B.12) to (B.8), we have C24 = 2αh (1 − 2Pe ) (3βh − + ℜ[(1 − 4βh )corr(1) + βh · corr(2)]) −ℜ[corr(1)] + . (B.13) Further, following (5.14) and using MSEd ≈ MSEb , we have Pe = (MSEd − 1/SNRr )/4 ≈ (MSEb − 1/SNRr )/4 = (1/SNRb − 1/SNRr )/4. (B.14) Substituting (B.14) into (B.13), we obtain (5.20). 150 Appendix C (i) ˆm j θ Derivation of E f (e ) and MSEcpe ˆ(i) C.1 Derivation of E f (ej θm ) (i) (i1) m1 For notational brevity, we simply replace the subscripts/superscripts m , (i2) and m2 with the subscripts n , n−1 and n−2 , respectively, in all CPE related variables. Applying (6.25) and θn = φn + ϑn to (6.32), we obtain ˆ ˆ ˆ f (ej θn ) = (1 − αc )(ejφn ejϑn + wn ) + α1 f (ej θn−1 ) + α2 f (ej θn−2 ). (C.1) Applying ejϑn−1 = ν ∗ ejϑn to (C.1), we further obtain j θˆn E{f (e n jϑn )} = (1 − αc )e l=0 al (ν ∗ )n−l E{ejφl }, n≥2 (C.2) where an = 1, an−1 = α1 , al−2 = α1 al−1 +α2 al , l ∈ Zn3 , and a0 = (αc a1 +α2 a2 )/(1−αc ). ˆ ˜ ˆ In particular, for n ∈ Z10 , equation (6.32) reduces to f (ej θ0 ) = ej θ0 and f (ej θ1 ) = (1 − ˜ ˆ αc )ej θ1 + αc f (ej θ0 ). Thus, we have ˆ E f (ej θ0 ) = ejϑ0 E{ejφ0 }; ˆ E f (ej θ1 ) = (1 − αc )ejϑ1 E{ejφ1 } + αc ejϑ0 E{ejφ0 }. (C.3) Referring to the discussion in [141, Section III-A], from (4.5), we have E{ejφn (k) } = e−[n(N +Ng )+k]πβTs , 151 k ∈ Z0N −1 . (C.4) C.2 Derivation of MSEcpe Using (4.13), we obtain N E{ejφn } = N −1 E ejφn (k) = k=0 N N −1 Combining (C.2) and (C.3) by using (C.5) and E{ejθn } = ejϑn E{ejφn } yields     E{ejθn }, n=0    ˆ ∗ E f (ej θn ) = (1 − αc ) + αc λν E{ejθn }, n =      (1 − αc )ΩE{ejθn }, n≥2 where Ω = n l=0 (C.5) k=0 1−e−πβTs N . N (1−e−πβTs ) where λ0 = e−(N +Ng )πβTs and λ1 = e−[n(N +Ng )+k]πβTs = λ1 λn0 ∗ ∗ (C.6) ∗ al ( νλ0 )n−l . By computing Ω − α1 Ω νλ0 − α2 Ω( λν )2 , it is straightforward to obtain ∗ Ω= ∗ ∗ + ( νλ0 )n a0 − α1 a1 − α2 a2 − (α1 a0 + α2 a1 ) νλ0 − α2 a0 ( νλ0 )2 ∗ ∗ − α1 νλ0 − α2 ( λν )2 , n ≥ 2. (C.7) Furthermore, when n ≫ 1, we have Ω≈ 1− ∗ α1 λν . ∗ − α2 ( νλ0 )2 (C.8) Applying (6.28) and (C.8) to (C.6), we obtain (6.33). C.2 Derivation of MSEcpe The MSEcpe is given by ˆ MSEcpe = E |f (ej θn ) − ejθn |2 ˆ ˆ ˆ = + E |f (ej θn )|2 − E f (ej θn )e−jθn − E ejθn f ∗ (ej θn ) . ˆ ˆ Let A1 = E |f (ej θn )|2 and note that E |f (ej θn )|2 (C.9) ˆ ≈ E |f (ej θn−1 )|2 for n ≫ 1. From (C.1), we can obtain A1 = (1−αc )2 (1+σw2 ) + (α12 +α22)A1 ˆ ˆ +α1 (1−αc ) E ejθn f ∗ (ej θn−1 ) +E f (ej θn−1 )e−jθn ˆ ˆ +α2 (1−αc ) E ejθn f ∗ (ej θn−2 ) + E f (ej θn−2 )e−jθn 152 + α1 α2 A2 (C.10) C.2 Derivation of MSEcpe ˆ ˆ ˆ ˆ where A2 = E f (ej θn−1 )f ∗ (ej θn−2 ) +E f (ej θn−2 )f ∗ (ej θn−1 ) . Using the approximation ˆ ˆ ˆ ˆ E f (ej θn−1 )f ∗ (ej θn−2 ) ≈ E f (ej θn−2 )f ∗ (ej θn−3 ) , n ≫ 1, we obtain ˆ ˆ A2 = (1 − αc ) E ejθn f ∗ (ej θn−1 ) + E f (ej θn−1 )e−jθn + 2α1 A1 + α2 A2 . (C.11) Combining (C.10) and (C.11) yields A1 = 1−αc ˆ ˆ α1 E ejθn f ∗ (ej θn−1 ) + E f (ej θn−1 )e−jθn α0 ˆ ˆ + α2 (1−α2 ) E ejθn f ∗ (ej θn−2 ) +E f (ej θn−2 )e−jθn + (1 − α2 )(1−αc)(1+σw2 ) (C.12) where α0 = − α2 − α12 − α22 − α2 α12 + α23 . Moreover, from (C.1), we have ˆ ˆ ˆ E f (ej θn )e−jθn = (1 − αc ) + α1 E f (ej θn−1 )e−jθn + α2 E f (ej θn−2 )e−jθn (C.13) and ˆ ˆ ˆ E ejθn f ∗ (ej θn ) = (1 − αc ) + α1 E ejθn f ∗ (ej θn−1 ) + α2 E ejθn f ∗ (ej θn−2 ) . (C.14) By applying (C.12), (C.13) and (C.14) to (C.9), we can obtain MSEcpe = ˆ ˆ A0 + g1 E f (ej θn−1 )e−jθn + g2 E f (ej θn−2 )e−jθn α0 ˆ ˆ + g1 E ejθn f ∗ (ej θn−1 ) + g2 E ejθn f ∗ (ej θn−2 ) (C.15) where A0 = α0 (2αc − 1) + (1 − α2 )(1 − αc )2 (1 + σw2 ), g1 = α1 (1 − αc − α0 ) and g2 = α2 [(1 − α2 )(1 − αc ) − α0 ]. We note that E ej(θn−l−θn ) = E ej(φn−l −φn ) ej(ϑn−l−ϑn ) = (ν ∗ )l E ej(φn−l−φn ) , n ≥ 2, l ∈ Zn1 . (C.16) Using (C.16) and (C.1) in (C.15), we can obtain MSEcpe = A0 +(1−αc ) α0 n bn−l (ν ∗ )l E ej(φn−l −φn ) +ν l E ej(φn −φn−l ) l=1 153 , (C.17) C.2 Derivation of MSEcpe for n ≥ 2, where bn = g2 /α2 , bn−1 = g1 , bl−2 = α1 bl−1 + α2 bl , l ∈ Zn3 , and b0 = (αc b1 + α2 b2 )/(1 − αc ). Referring to [141, Section III-A], we have j(φn−l −φn ) E e E = N2 = N2 = N2 N −1 jφn−l (k) e k=0 N −1 N −1 N −1 e−jφn (k) k=0 E ejφn−l (k1 ) e−jφn (k2 ) k1 =0 k2 =0 N −1 N −1 e−πβTs |l(N +Ng )+k2 −k1 | , k1 =0 k2 =0 n ≥ 2, l ∈ Zn1 . (C.18) Further algebra on (C.18) reduces it to E ej(φn−l −φn ) = λ1 λ2 λl0 , n ≥ 2, l ∈ Zn1 (C.19) n ≥ 2, l ∈ Zn1 . (C.20) 2(1 − αc )g2 λ1 λ2 A0 + (1 − αc )λ1 λ2 (Ω1 + Ω2 ) − , α0 α2 n ≥ 2. (C.21) where λ2 = 1−eπβTs N . N (1−eπβTs ) Similarly, we can obtain E ej(φn −φn−l ) = λ1 λ2 λl0 , Combining (C.17), (C.19) and (C.20) yields MSEcpe = where Ω1 = n ∗ l l=0 bn−l (λ0 ν ) and Ω2 = n l l=0 bn−l (λ0 ν) . Following a similar procedure for obtaining Ω in (C.7), for n ≫ 1, we can derive Ω1 and Ω2 which are given by Ω1 ≈ bn + (bn−1 − α1 bn )λ0 ν ∗ , − α1 λ0 ν ∗ − α2 (λ0 ν ∗ )2 Ω2 ≈ bn + (bn−1 − α1 bn )λ0 ν . − α1 λ0 ν − α2 (λ0 ν)2 We note that Ω1 = Ω∗2 . Substitution of (C.22) into (C.21) yields (6.35). 154 (C.22) [...]... investigate channel estimation in OFDM systems including multi-band OFDM- based UWB systems Secondly, we investigate phase error mitigation 1 In an OFDM system, equalization is usually performed using a one-tap frequency-domain equalizer with low complexity 3 1.3 Contributions of This Thesis in multi-band OFDM- UWB systems Thirdly, we study CFO estimation in single-input single-output (SISO) OFDMA and MIMO -OFDMA. .. estimator and introduce an effective inter -OFDM- symbol smoothing scheme for enhancing the CPE tracking performance 1.3.3 CFO Estimation for SISO -OFDMA and MIMO -OFDMA Uplink Emerging as a promising technology for next generation wireless communication systems, the OFDMA has received considerable amount of research interest [10, 34–40] An appealing feature of OFDMA is its capability to mitigate the effects... our effort to ML approaches for joint estimation of CFO, timing error, and channel response of each active user in both single-input single-output (SISO) and multiple-input multiple-output (MIMO) OFDMA systems In particular, we focus our investigation on ML CFO estimation for the OFDMA uplink with generalized carrier-assignment scheme (GCAS), which is believed to be the most challenging task in OFDMA. .. Figures 2.1 NMSE performance for different values assumed for ELCIR 17 3.1 NMSE performance comparison of CMLE and OMLE 22 3.2 Illustration of the proposed IMLE 25 3.3 NMSE performance comparison of CMLE and IMLE 26 3.4 NMSE performance comparison for various channel estimation methods 29 3.5 FER performance comparison for various channel estimation methods... complete channel These pilots can also be used to 2 3 OFDM systems with FEC coding are usually called coded OFDM (COFDM) systems in the literature Channel estimation for OFDM is seldom performed in time-domain due to the multi-carrier nature of OFDM systems 4 1.3 Contributions of This Thesis track channel variations The blind schemes avoid the use of pilots, for achieving high spectral efficiency This is achieved... step further, we propose and analyze an innovative phase error suppression technique which is simple yet highly effective for SFO estimation and compensation as well as CPE tracking and correction The third part of the thesis consisting of Chapters 7 to 9, deals with CFO estimation for SISO -OFDMA and MIMO OFDMA uplink transmission Chapter 7 presents the signal model of GCAS based OFDMA uplink transmission... timing and frequency synchronization are in order In particular, the CFO’s of an OFDMA system, if not properly estimated and compensated, result in ICI and MAI at the receiver [16, 100, 105] In the downlink of an OFDMA system, the signals for different users are multiplexed by the same transmitter, and 8 1.3 Contributions of This Thesis each user performs frequency synchronization via estimating and correcting... thesis successfully addresses these deficiencies 1.3.1 Channel Estimation in OFDM Systems OFDM systems transform high-rate data signals, which would otherwise suffer from severe frequency selective channel fading, into a number of orthogonal components before transmission, with the bandwidth of each component being less than the coherence bandwidth of the channel By modulating them onto different subcarriers,... and βh = 0.05 ix 54 List of Figures 5.4 NMSE performance comparison for various channel estimation methods under (a) CM1, (b) CM2, (c) CM3, and (d) CM4 (Sim, Ana, and PD are abbreviations for simulation, analytical and proposed, respectively.) 5.5 58 FER performance comparison for various channel estimation methods under (a) CM1 & CM2, (b) CM3 & CM4 59... transmitted sequence and channel is the same as circular convolution As a result, the effects of ISI can be easily and completely eliminated Moreover, the approach enables the receiver to use fast signal processing transforms such as fast Fourier transform (FFT) for OFDM implementation Because of these properties, OFDM systems are more advantageous over single-carrier systems and become desirable for many applications . CHANNEL ESTIMATION AND SYNCHRONIZATION FOR OFDM AND OFDMA SYSTEMS WANG ZHONGJUN NATIONAL UNIVERSITY OF SINGAPORE 2008 CHANNEL ESTIMATION AND SYNCHRONIZATION FOR OFDM AND OFDMA SYSTEMS WANG. 4 1.3.1 Channel Estimation in OFDM Systems . . . . . . . . . . . . . . 4 1.3.2 Phase Error Suppression for Multi-Band OFDM- UWB Systems . 7 1.3.3 CFO Estimation for SISO -OFDMA and MIMO -OFDMA Uplink. NMSE performance comparison for various channel estimation methods. 29 3.5 FER performance comparison for various channel estimation methods. . . 30 3.6 FER performance comparison when the channel