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Recently,tensor models (or multi-way arrays) play a vital role in many applications,such as wireless communication systems,blind source separation,machine learning,signal (audio,image,speech) processing,chemometrics,data mining, arithmetic complexity,environmental sciences,etc.Parallel factor (PARAFAC) analysis,also known as canonical polyadic decomposition,is a common name for low rank decomposition of tensors.A traditional way to fit the PARAFAC model is the alternating least squares (ALS) algorithm,which can transform a nonlinear optimization problem into some independent linear least squares problems.However,the ALS scheme for computing the decomposition of the tensor is known to converge slowly if one or some modes include nearly collinear columns.Particularly,if the collinearity is presented in all modes,the ALS will end in a convergence bottleneck.Hence,it is necessary to develop a robust and fast algorithm to compute the decomposition of the tensor.In this paper,a novel channel estimation algorithm using the Levenberg Marquardt (LM) method based on a third-order tensor model is presented in a multi-user uplink amplify-and-forward (AF) relay system.As the relay nodes all operate with half-duplex mode to aid the transmission,the overall transmission period is partitioned into two transmission subprocesses.In the first transmission sub-process,the users transmit channel training sequence to the relay nodes.This stage requires time block once.During the second transmission sub-process,a set of diagonal amplifying factor matrices are utilized by the relay nodes to amplify the received data.Then,the relay nodes transmit each of the amplified data to the base station.This stage requires time blocks K times.With the help of the channel training sequence and the relay amplifying factor matrices,the received data at the base station can be stacked up into a third-order PARAFAC model. And then based on this tensor model an LM channel estimation algorithm is proposed to provide the individual channel state information of both user-to-relay and relay-to-base station channel links.As the channel sequence is transmitted by the users only once,the proposed scheme has a higher spectral efficiency than the case that the channel sequence is transmitted K times by the users.Numerical experiments are shown to demonstrate the efficacy of the proposed LM channel estimation algorithm.The results are as follows.Firstly,the LM approach has the same channel estimation performance as the bilinear alternating least-squares method.Secondly,the proposed estimator yields much faster convergence speed when the relay amplifying factor matrix is a random matrix or a highly collinear one.Finally,the proposed scheme performs well in both independent identically distributed channels and correlated channels scenarios,which means that the proposed channel estimator can provide the robust and reliable feature for multi-user uplinks AF relay systems.
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Keywords:
- tensor model /
- channel estimation /
- Levenberg Marquardt algorithm /
- amplify-and-forward relay
[1] Sanguinetti L, D'Amico A A, Rong Y 2012 IEEE J. Sel. Areas Commun. 30 1331
[2] Hammerstrom I, Wittneben A 2007 IEEE Trans. Wirel. Commun. 6 2798
[3] Rong Y 2010 IEEE Commun. Lett. 14 390
[4] Munoz M O, Vidal J, Agustin A 2007 IEEE Trans. Signal Process 55 2593
[5] Zhou J, Jiang H, Hisakazu K, Shao G F 2014 Acta Phys. Sin. 63 140506(in Chinese)[周杰, 江浩, 菊池久和, 邵根富2014物理学报63 140506]
[6] Ma L, Liu S Z, Qiao G 2015 Acta Phys. Sin. 64 154304(in Chinese)[马璐, 刘凇佐, 乔钢2015物理学报64 154304]
[7] Sidiropoulos N D, Giannakis G B, Bro R 2000 IEEE Trans. Signal Process 48 810
[8] Kruskal J B 1977 Linear Algebra. Appl. 18 95
[9] Xiao H L, Ouyang S, Nie Z P 2009 Acta Phys. Sin. 58 3685(in Chinese)[肖海林, 欧阳缮, 聂在平2009物理学报58 3685]
[10] de Almeida A L F, Fernandes C A, Da Costa D 2013 IEEE Signal Process. Lett. 20 697
[11] Du J H, Yuan C W, Hu Z W, Lin H Y 2015 IEEE Commun. Lett. 19 1961
[12] Rong Y, Khandaker M R, Xiang Y 2012 IEEE Trans. Wirel. Commun. 11 2224
[13] Du J H, Yuan C W, Zhang J B 2015 IET Commun. 9 737
[14] De Almeida A L F, Favier G, Ximenes L R 2013 IEEE Trans. Signal Process 61 1895
[15] Marquardt D 1963 SIAM J. Appl. Math. 11 431
[16] Nion D, De Lathauwer L 2008 IEEE Trans. Signal Process 56 5567
[17] Tomasi G, Bro R 2006 Comp. Stat. Data Anal. 50 1700
[18] Madsen K, Nielsen H B, Tingleff O 2016 IET Commun. 10 995
[19] Ximenes L R, Favier G, De Almeida A L F, Silva Y C 2014 IEEE Trans. Signal Process 62 3604
[20] Shiu D, Foschini G, Gans M J, Kahn J 2000 IEEE Trans. Commun. 48 502
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[1] Sanguinetti L, D'Amico A A, Rong Y 2012 IEEE J. Sel. Areas Commun. 30 1331
[2] Hammerstrom I, Wittneben A 2007 IEEE Trans. Wirel. Commun. 6 2798
[3] Rong Y 2010 IEEE Commun. Lett. 14 390
[4] Munoz M O, Vidal J, Agustin A 2007 IEEE Trans. Signal Process 55 2593
[5] Zhou J, Jiang H, Hisakazu K, Shao G F 2014 Acta Phys. Sin. 63 140506(in Chinese)[周杰, 江浩, 菊池久和, 邵根富2014物理学报63 140506]
[6] Ma L, Liu S Z, Qiao G 2015 Acta Phys. Sin. 64 154304(in Chinese)[马璐, 刘凇佐, 乔钢2015物理学报64 154304]
[7] Sidiropoulos N D, Giannakis G B, Bro R 2000 IEEE Trans. Signal Process 48 810
[8] Kruskal J B 1977 Linear Algebra. Appl. 18 95
[9] Xiao H L, Ouyang S, Nie Z P 2009 Acta Phys. Sin. 58 3685(in Chinese)[肖海林, 欧阳缮, 聂在平2009物理学报58 3685]
[10] de Almeida A L F, Fernandes C A, Da Costa D 2013 IEEE Signal Process. Lett. 20 697
[11] Du J H, Yuan C W, Hu Z W, Lin H Y 2015 IEEE Commun. Lett. 19 1961
[12] Rong Y, Khandaker M R, Xiang Y 2012 IEEE Trans. Wirel. Commun. 11 2224
[13] Du J H, Yuan C W, Zhang J B 2015 IET Commun. 9 737
[14] De Almeida A L F, Favier G, Ximenes L R 2013 IEEE Trans. Signal Process 61 1895
[15] Marquardt D 1963 SIAM J. Appl. Math. 11 431
[16] Nion D, De Lathauwer L 2008 IEEE Trans. Signal Process 56 5567
[17] Tomasi G, Bro R 2006 Comp. Stat. Data Anal. 50 1700
[18] Madsen K, Nielsen H B, Tingleff O 2016 IET Commun. 10 995
[19] Ximenes L R, Favier G, De Almeida A L F, Silva Y C 2014 IEEE Trans. Signal Process 62 3604
[20] Shiu D, Foschini G, Gans M J, Kahn J 2000 IEEE Trans. Commun. 48 502
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