logo

SCIENCE CHINA Information Sciences, Volume 61, Issue 11: 112210(2018) https://doi.org/10.1007/s11432-018-9461-3

Distributed bias-compensated normalized least-mean squares algorithms with noisy input

More info
  • ReceivedJan 8, 2018
  • AcceptedMay 3, 2018
  • PublishedOct 18, 2018

Abstract

In this paper, we study the problem of distributed normalized least-mean squares (NLMS) estimation over multi-agent networks, where all nodes collaborate to estimate a common parameter of interest. We consider the situations that all nodes in the network are corrupted by both input and output noise. This yields into biased estimates by the distributed NLMS algorithms. In our analysis, we take all the noise into consideration and prove that the bias is dependent on the input noise variance. Therefore, we propose a bias compensation method to remove the noise-induced bias from the estimated results. In our development, we first assume that the variances of the input noise are known a priori and develop a series of distributed-based bias-compensated NLMS (BCNLMS) methods. Under various practical scenarios, the input noise variance is usually unknown a priori, therefore it is necessary to first estimate for its value before bias removal. Thus, we develop a real-time estimation method for the input noise variance, which overcomes the unknown property of this noise. Moreover, we perform some main analysis results of the proposed distributed BCNLMS algorithms. Furthermore, we illustrate the performance of the proposed distributed bias compensation method via graphical simulation results.


References

[1] Golub G H, van Loan C F. An analysis of the total least squares problem. SIAM J Numer Anal, 1980, 17: 883-893 CrossRef Google Scholar

[2] Feng D Z, Zhang X D, Chang D X. A fast recursive total least squares algorithm for adaptive FIR filtering. IEEE Trans Signal Process, 2004, 52: 2729-2737 CrossRef ADS Google Scholar

[3] So H C. Modified LMS algorithm for unbiased impulse response estimation in nonstationary noise. Electron Lett, 1999, 35: 791-792 CrossRef Google Scholar

[4] Feng D Z, Bao Z, Zhang X D. Modified RLS algorithm for unbiased estimation of FIR system with input and output noise. Electron Lett, 2000, 36: 273-274 CrossRef Google Scholar

[5] Sardellitti S, Barbarossa S. Distributed RLS estimation for cooperative sensing in small cell networks. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICCASP), Vancouver, 2013. 5283--5287. Google Scholar

[6] Lorenzo P D, Isufi E, Banelli P, et al. Distributed recursive least squares strategies for adaptive reconstruction of graph signals. In: Proceedings of the 25th European Signal Processing Conference (EUSIPCO), Kos, 2017. 2289--2293. Google Scholar

[7] He X K, Xue W C, Fang H T. Consistent distributed state estimation with global observability over sensor network. 2017,. arXiv Google Scholar

[8] He X, Hu C, Xue W. On event-based distributed Kalman filter with information matrix triggers. IFAC-PapersOnLine, 2017, 50: 14308-14313 CrossRef Google Scholar

[9] Lopes C G, Sayed A H. Incremental adaptive strategies over distributed networks. IEEE Trans Signal Process, 2007, 55: 4064-4077 CrossRef ADS Google Scholar

[10] Mostafapour E, Hoseini A, Nourinia J, et al. Channel estimation with adaptive incremental strategy over distributed sensor networks. In: Proceedings of the 2nd International Conference on Knowledge-Based Engineering and Innovation (KBEI), Tehran, 2015. 803--807. Google Scholar

[11] Lopes C G, Sayed A H. Distributed adaptive incremental strategies: formulation and performance analysis. In: Proceedings of IEEE International Conference on Acoustics Speech and Signal Processing Proceedings (ICASSP), Toulouse, 2006. 584--587. Google Scholar

[12] Shi L, Zhao H. Variable step-size distributed incremental normalised LMS algorithm. Electron Lett, 2016, 52: 519-521 CrossRef Google Scholar

[13] Schizas I D, Mateos G, Giannakis G B. Distributed LMS for consensus-based in-network adaptive processing. IEEE Trans Signal Process, 2009, 57: 2365-2382 CrossRef ADS Google Scholar

[14] Braca P, Marano S, Matta V. Running consensus in wireless sensor networks. In: Proceedings of the 11th International Conference on Information Fusion (ICIF), Cologne, 2008. Google Scholar

[15] Kar S, Moura J M F. Distributed consensus algorithms in sensor networks with imperfect communication: link failures and channel noise. IEEE Trans Signal Process, 2009, 57: 355-369 CrossRef ADS Google Scholar

[16] Bogdanović N, Plata-Chaves J, Berberidis K. Distributed diffusion-based LMS for node-specific parameter estimation over adaptive networks. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, 2014. 7223--7227. Google Scholar

[17] Nassif R, Richard C, Chen J, et al. Diffusion LMS over multitask networks with noisy links. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Shanghai, 2016. 4583--4587. Google Scholar

[18] Lee H S, Kim S E, Lee J W. A variable step-size diffusion LMS algorithm for distributed estimation. IEEE Trans Signal Process, 2015, 63: 1808-1820 CrossRef ADS Google Scholar

[19] Cattivelli F S, Lopes C G, Sayed A H. Diffusion recursive least-squares for distributed estimation over adaptive networks. IEEE Trans Signal Process, 2008, 56: 1865-1877 CrossRef ADS Google Scholar

[20] Lopes C G, Sayed A H. Diffusion least-mean squares over adaptive networks: formulation and performance analysis. IEEE Trans Signal Process, 2008, 56: 3122-3136 CrossRef ADS Google Scholar

[21] Mateos G, Schizas I D, Giannakis G B. Distributed recursive least-squares for consensus-based in-network adaptive estimation. IEEE Trans Signal Process, 2009, 57: 4583-4588 CrossRef ADS Google Scholar

[22] Cattivelli F S, Sayed A H. Diffusion LMS strategies for distributed estimation. IEEE Trans Signal Process, 2010, 58: 1035-1048 CrossRef ADS Google Scholar

[23] Bertrand A, Moonen M, Sayed A H. Diffusion bias-compensated RLS estimation over adaptive networks. IEEE Trans Signal Process, 2011, 59: 5212-5224 CrossRef ADS Google Scholar

[24] Bertrand A, Moonen M. Consensus-based distributed total least squares estimation in Ad hoc wireless sensor networks. IEEE Trans Signal Process, 2011, 59: 2320-2330 CrossRef ADS Google Scholar

[25] Bertrand A, Moonen M, Sayed A H. Diffusion-based bias-compensated RLS for distributed estimation over adaptive sensor networks. In: Proceedings of the 19th European Signal Processing Conference (EUSIPCO), Barcelona, 2011. 1025--1029. Google Scholar

[26] Abdolee R, Champagne B, Sayed A H. A diffusion LMS strategy for parameter estimation in noisy regressor applications. In: Proceedings of the 20th European Signal Processing Conference (EUSIPCO), Bucharest, 2012. 749--753. Google Scholar

[27] Sayed A H. Adaptive Filters. Hoboken: Wiley, 2008. Google Scholar

[28] Haykin S. Adaptive Filter Theory. Englewood Cliffs: Prentice-Hall, 2002. Google Scholar

[29] Nagumo J, Noda A. A learning method for system identification. IEEE Trans Autom Control, 1967, 12: 282-287 CrossRef Google Scholar

[30] Lou J, Jia L J, Tao R. Distributed incremental bias-compensated RLS estimation over multi-agent networks. Sci China Inf Sci, 2017, 60: 032204 CrossRef Google Scholar

  • Figure 1

    (Color online) Network topology for incremental strategy.

  • Figure 2

    (Color online) Network topology for consensus strategy.

  • Figure 3

    (Color online) Network topology for ATC diffusion strategy.

  • Table 1   Different BCNLMS algorithms with different choices of matrices $\boldsymbol{A}_0$, $\boldsymbol{A}_1$, $\boldsymbol{A}_2$
    Algorithm $\boldsymbol{A}_0$ $\boldsymbol{A}_1$ $\boldsymbol{A}_2$
    $\text{Non-cooperative~BCNLMS}$ $\boldsymbol{I}$ $\boldsymbol{I}$ $\boldsymbol{I}$
    $\text{Consensus~BCNLMS}$ $\boldsymbol{A}$ $\boldsymbol{I}$ $\boldsymbol{I}$
    $\text{ATC~BCNLMS}$ $\boldsymbol{I}$ $\boldsymbol{I}$ $\boldsymbol{A}$

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1