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arxiv: 2605.23499 · v1 · pith:5F52VH2Bnew · submitted 2026-05-22 · 📡 eess.SP

Outlier-Robust unscented Kalman filter based on generalized correntropy induced

Jinhui Hu , Haiquan Zhao , Yi Peng This is my paper

Pith reviewed 2026-05-25 03:35 UTC · model grok-4.3

classification 📡 eess.SP
keywords unscented Kalman filtergeneralized correntropyrobust estimationnon-Gaussian noisenonlinear state estimationsquare root decompositioniterative filtering
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The pith

A square-root unscented Kalman filter based on generalized correntropy bounds estimation error under non-Gaussian noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes an iterative square root unscented Kalman filter using the generalized correntropy induced criterion to solve nonlinear state estimation problems with non-Gaussian noise. The GCI framework allows robust adaptation across different noise environments due to its insensitivity to kernel bandwidth. The method includes a nonlinear error generalization model for correcting measurement errors and square-root decomposition for maintaining numerical stability. Theoretical analysis proves bounded estimation variance, and experiments demonstrate superior robustness compared to other methods in nonlinear systems, vehicle navigation, and power systems.

Core claim

The SR-GCI-IUKF constructs a nonlinear error generalization model that dynamically corrects measurement-induced errors during the state update phase, while the generalized correntropy induced criterion provides intrinsic kernel bandwidth insensitivity. Rigorous error dynamics analysis establishes theoretical stability guarantees with bounded estimation variance under non-Gaussian disturbances, and the square-root implementation preserves covariance positive definiteness.

What carries the argument

The generalized correntropy induced (GCI) criterion, which characterizes higher-order noise statistics with kernel bandwidth insensitivity for robust adaptation in diverse noise environments.

If this is right

  • Estimation variance remains bounded under non-Gaussian disturbances.
  • Covariance matrix positive definiteness is preserved throughout recursive operations.
  • Estimation accuracy increases in strongly nonlinear regimes through the error correction model.
  • Stronger robustness is observed in land vehicle navigation and power system state estimation tasks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The bandwidth insensitivity may allow deployment without environment-specific retuning of parameters.
  • Similar GCI-based structures could be adapted to other nonlinear estimators such as extended Kalman filters.
  • Systems with varying noise statistics could require less manual intervention in filter tuning.

Load-bearing premise

The generalized correntropy framework inherently provides kernel bandwidth insensitivity that enables robust adaptation without retuning across diverse noise environments.

What would settle it

An experiment where the filter requires kernel bandwidth retuning to maintain performance across significantly different noise environments would falsify the insensitivity claim.

Figures

Figures reproduced from arXiv: 2605.23499 by Haiquan Zhao, Jinhui Hu, Yi Peng.

Figure 1
Figure 1. Figure 1: RMSE for position and velocity in case of noise e (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: RMSE for position and velocity in case of noise [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: RMSE for position and velocity in case of noise [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

Conventional Kalman filtering (KF) approaches exhibit significant limitations in addressing nonlinear state estimation problems contaminated by non-Gaussian noise disturbances. To overcome these challenges, this work proposes a robust iterative square root unscented Kalman Filter based on the generalized correntropy induced (SR-GCI-IUKF). While sharing the maximum correntropy criterion's (MCC) ability to characterize higher-order noise statistics, the proposed GCI framework exhibits intrinsic kernel bandwidth insensitivit a critical advantage enabling robust adaptation to diverse complex noise environments through its generalized kernel structure. For nonlinear state estimation challenges, the algorithm constructs a nonlinear error generalization model that dynamically corrects measurement-induced errors during the state update phase, thereby significantly enhancing estimation accuracy in strongly nonlinear regimes. Furthermore, the square-root decomposition implementation ensures numerical robustness by preserving covariance matrix positive definiteness throughout recursive operations. Theoretical stability guarantees are established through rigorous error dynamics analysis, demonstrating bounded estimation variance under non-Gaussian disturbances. Finally, experiments are carried out in nonlinear systems, land vehicle navigation systems as well as power system FASE to compare other robust algorithms, and it is determined that the proposed algorithm has stronger robustness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes the SR-GCI-IUKF, an iterative square-root unscented Kalman filter based on the generalized correntropy induced (GCI) criterion, for nonlinear state estimation under non-Gaussian noise. It claims that the GCI framework supplies intrinsic kernel bandwidth insensitivity enabling adaptation without retuning, constructs a nonlinear error generalization model for dynamic correction during the update, employs square-root decomposition to preserve positive definiteness, establishes theoretical stability via error dynamics analysis showing bounded estimation variance, and reports stronger robustness than other algorithms in experiments on nonlinear systems, land vehicle navigation, and power system FASE.

Significance. If the error-dynamics analysis rigorously establishes bounded variance and demonstrates that the GCI loss renders the bound independent of kernel bandwidth, the work would advance robust nonlinear filtering by reducing the need for retuning across noise environments, with potential value for navigation and power-system applications. The square-root implementation is a standard numerical safeguard.

major comments (1)
  1. [Abstract] Abstract, paragraph on GCI advantages: the assertion that the GCI framework exhibits 'intrinsic kernel bandwidth insensitivity' as a 'critical advantage' enabling robust adaptation 'without retuning across diverse noise environments' is presented without a derivation showing that this property follows from the error-dynamics analysis or that the stability bound is independent of the kernel bandwidth parameter. The abstract states that the analysis demonstrates bounded estimation variance under non-Gaussian disturbances, but supplies no explicit connection establishing bandwidth independence; this link is load-bearing for the central robustness-without-retuning claim.
minor comments (1)
  1. [Abstract] Typo: 'insensitivit' should read 'insensitivity'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment on the abstract. We address the point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph on GCI advantages: the assertion that the GCI framework exhibits 'intrinsic kernel bandwidth insensitivity' as a 'critical advantage' enabling robust adaptation 'without retuning across diverse noise environments' is presented without a derivation showing that this property follows from the error-dynamics analysis or that the stability bound is independent of the kernel bandwidth parameter. The abstract states that the analysis demonstrates bounded estimation variance under non-Gaussian disturbances, but supplies no explicit connection establishing bandwidth independence; this link is load-bearing for the central robustness-without-retuning claim.

    Authors: The error-dynamics analysis in Section IV derives a bound on the estimation error variance that holds under the GCI criterion for non-Gaussian disturbances. The generalized kernel structure in the GCI loss (Eq. 3) produces the bandwidth-insensitivity property because the criterion remains effective over a range of kernel parameters without explicit retuning, unlike fixed-bandwidth MCC. We acknowledge that the abstract does not explicitly connect the derived bound to this independence. We will revise the abstract to add a concise statement referencing the analysis result and its implication for bandwidth independence. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The abstract asserts that the GCI framework has 'intrinsic kernel bandwidth insensitivity' as a 'critical advantage' and that 'theoretical stability guarantees are established through rigorous error dynamics analysis,' but supplies no equations, self-citations, or derivation steps that reduce any prediction or result to its own inputs by construction. No fitted-input-called-prediction, self-definitional, or load-bearing self-citation patterns are present in the provided text. The central claims remain independent assertions whose grounding cannot be inspected for circular reduction; this is the normal case of an unevaluated paper receiving score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; the method appears to rest on standard UKF assumptions plus the unverified claim that generalized correntropy supplies bandwidth insensitivity. No explicit free parameters, axioms, or invented entities are detailed.

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Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    EKF uses first order Taylor expansion to linearize the nonlinear system to achieve the problem of state estimation in nonlinear systems

    and the Unscented Kalman Filter (UKF) [3] have been developed. EKF uses first order Taylor expansion to linearize the nonlinear system to achieve the problem of state estimation in nonlinear systems. On the basis, IEKF [5] was developed to further improve the performance of EKF in facing nonlinear systems. The UKF, on the other hand, employs the unscented...

    work page 2025

  2. [2]

    is more robust than the traditional CI, which is largely due to its greater flexibility in parameter selection and the ability to adapt to different noise types. Although C I and GCI both use kernel-based loss functions to suppress outliers, GCI introduces additional tuning p arameters, usually including kernel bandwidth and shape parameters, to better de...

    work page

  3. [3]

    Its generalized kernel structure makes th e algorithm insensitive to kernel bandwidth, significantly improving its adaptability to complex noise environments

    The SR -GCI-IUKF algorithm was developed based on GCI to overcome the limitations of traditional KF in non - Gaussian noise and strongly non-linear systems. Its generalized kernel structure makes th e algorithm insensitive to kernel bandwidth, significantly improving its adaptability to complex noise environments. Furthermore, by constructing a nonlinear ...

    work page

  4. [4]

    To address the paramet er tuning challenge for the shape and scale parameters in GCI, an adaptive selection strategy based on information potential matching is proposed

    An automated parameter adaptation strategy is designed, and a systematic computational complexity analysis of the algorithm is provided. To address the paramet er tuning challenge for the shape and scale parameters in GCI, an adaptive selection strategy based on information potential matching is proposed. Concurrently, a detailed analysis of the computati...

    work page

  5. [5]

    This theoretically ensures the algorithm's reliable convergence and numerical stability i n complex noise and highly nonlinear scenarios

    Through rigorous error dynamics analysis, it is proven that the estimation error of SR -GCI-IUKF remains bounded in the mean square sense under non -Gaussian disturbances. This theoretically ensures the algorithm's reliable convergence and numerical stability i n complex noise and highly nonlinear scenarios. To demonstrate the comprehensive advantages of ...

    work page

  6. [6]

    ()  is the th − element of the matrix and 2()    = + − is the scale correction factor

    Predict ( ) ( ) 1| 1 1| 11| 1 1| 1 1| 1 1| 1 0 ( ) 1, , ( ) 1, , 2 n NN N N N              −− −−− − − − −− −− −  = = + + =   + + = + x X x S xS (6) where 1| 1−−S is obtained by Cholesky for |1 −P , 1| 1−−P represents the state covariance matrix. ()  is the th − element of the matrix and 2()    = + − is the scale c...

    work page

  7. [7]

    filter stability

    Update Similarly, 21N + sigma points was obtained in neighborhood of a |1 −x using the UT transform. ( ) ( ) |1 |1| 1 | 1 |1 |1 0 ( ) 1, , ( ) 1, , 2 N NN N N N              − −−− − − −  = = + + =   + + = + x X x S xS (10) TABLE I COMPARISON OF ROBUST KALMAN FILTERING ALGORITHMS Algorithm Noise Resistance Nonlinearity Handli...

    work page 2042

  8. [8]

    A New Approach to Linear Filtering and Prediction Problems,

    Tamer Basar, "A New Approach to Linear Filtering and Prediction Problems," Control Theory: Twenty-Five Seminal Papers (1932-1981) , IEEE, 2001, pp.167-179

    work page 1932

  9. [9]

    Stochastic stability condition for the extended Kalman filter with intermittent observations,

    X. Liu, L. Li, L. Zhen, T. Fernando, and H. H. C. Iu, "Stochastic stability condition for the extended Kalman filter with intermittent observations, " IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 64, no. 3, pp. 334–338, Mar. 2017

    work page 2017

  10. [10]

    Robust stable iterated unscented Kalman filter based on maximum Correntropy criterion,

    H. Zhao, B. Tian, B. Chen, "Robust stable iterated unscented Kalman filter based on maximum Correntropy criterion, " Automatica, vol. 142, 2022

    work page 2022

  11. [11]

    The iter ated Kalman filter update as a gauss-newton method,

    B. M. Bell and F. W. Cathey, "The iter ated Kalman filter update as a gauss-newton method, " IEEE Transactions on Automatic Control, vol. 38, no. 2, pp. 294–297, Feb. 1993

    work page 1993

  12. [12]

    A Robust Iterated Extended Kalman Filter for Power System Dynamic State Estimation,

    J. Zhao, M. Netto and L. Mili, "A Robust Iterated Extended Kalman Filter for Power System Dynamic State Estimation," IEEE Transactions on Power Systems, vol. 32, no. 4, pp. 3205-3216, Jul.2017

    work page 2017

  13. [13]

    Performance analysis of improved iterated cubature Kalman filter and its application to GNSS/INS,

    B. Cui, X. Chen, Y. Xu, H. Huang, X. Liu, "Performance analysis of improved iterated cubature Kalman filter and its application to GNSS/INS," ISA Transactions, vol. 66, pp. 460-468, 2017

    work page 2017

  14. [14]

    Maximum correntropy Kalman filter,

    B. Chen, X. Liu, H. Zhao, Jose C. Principe, "Maximum correntropy Kalman filter," Automatica, vol. 76, pp. 70-77, 2017

    work page 2017

  15. [15]

    Maximum Correntropy Extended Kalman Filtering for Power System Dynamic State Estimation,

    S. Mohiuddin and J. Qi, "Maximum Correntropy Extended Kalman Filtering for Power System Dynamic State Estimation," 2019 IEEE Power & Energy Society General Meeting (PESGM), Atlanta, GA, USA, 2019, pp. 1-5

    work page 2019

  16. [16]

    Robust Power System State Estimation with Minimum Error Entropy Unscented Kalman Filter,

    L. Dang, B. Chen, S. Wang, W. Ma and P. Ren, "Robust Power System State Estimation with Minimum Error Entropy Unscented Kalman Filter," IEEE Transactions on Instrumentation and Measur ement, vol. 69, no. 11, pp. 8797-8808, Nov. 2020

    work page 2020

  17. [17]

    An Improved Iterative Maximum Correntropy Cubature Kalman Filter,

    Q. Dou, T. Du, L. Guo and J. Yang, "An Improved Iterative Maximum Correntropy Cubature Kalman Filter," 2021 China Automation Congress (CAC), Beijing, China, 2021, pp. 2582-2586

    work page 2021

  18. [18]

    Robust state of charge estimation for Li -ion batteries based on cubature Kalman filter with generalized maximum Correntropy criterion

    W. Ma, P. Guo, X. Wang, Z. Zhang, S. Peng, B. Chen, "Robust state of charge estimation for Li -ion batteries based on cubature Kalman filter with generalized maximum Correntropy criterion", Energy, vol. 260, 2022

    work page 2022

  19. [19]

    Generalized minimum error entropy Kalman filter for non -Gaussian noise

    J. He, Gang Wang, Huijun Yu, JunMing Liu, Bei Peng, "Generalized minimum error entropy Kalman filter for non -Gaussian noise", ISA Transactions, vol. 136, 2023

    work page 2023

  20. [20]

    Robust Power System Forecasting -Aided State Estimation With Generalized Maximum Mixture Correntropy Unscented Kalman Filter,

    H. Zhao and B. Tian, "Robust Power System Forecasting -Aided State Estimation With Generalized Maximum Mixture Correntropy Unscented Kalman Filter," IEEE Transactions o n Instrumentation and Measurement, vol. 71, pp. 1-10, 2022

    work page 2022

  21. [21]

    Fixed-Point Minimum Error Entropy With Fiducial Points,

    Y. Xie, Y. Li, Y. Gu, J. Cao and B. Chen, "Fixed-Point Minimum Error Entropy With Fiducial Points," IEEE Transactions on Signal Processing, vol. 68, pp. 3824-3833, 2020

    work page 2020

  22. [22]

    Cubature Kalman Filter Under Minimum Error Entropy With Fiducial Points for INS/GPS Integration,

    L. Dang, B. Chen, Y. Huang, Y. Zhang and H. Zhao, "Cubature Kalman Filter Under Minimum Error Entropy With Fiducial Points for INS/GPS Integration," IEEE/CAA Journal of Automatica Sinica, vol. 9, no. 3, pp. 450-465, March 2022

    work page 2022

  23. [23]

    Chandrasekhar-Based Maximum Correntropy Kalman Filtering With the Adaptive Kernel Size Selection,

    M. V. Kulikova, "Chandrasekhar-Based Maximum Correntropy Kalman Filtering With the Adaptive Kernel Size Selection," in IEEE Transactions on Automatic Control, vol. 65, no. 2, pp. 741 -748, Feb. 2020

    work page 2020

  24. [24]

    Adaptive Multikernel Size-Based Maximum Correntropy Cubature K alman Filter for the Robust State Estimation,

    J. Shao, W. Chen, Y. Zhang, F. Yu and J. Chang, "Adaptive Multikernel Size-Based Maximum Correntropy Cubature K alman Filter for the Robust State Estimation," in IEEE Sensors Journal, vol. 22, no. 20, pp. 19835-19844, 15 Oct.15, 2022

    work page 2022

  25. [25]

    Outlier -Robust Iterative Extended Kalman Filtering,

    Y. Tao and S. S. -T. Yau, "Outlier -Robust Iterative Extended Kalman Filtering," IEEE Signal Processing Letters, vol. 30, pp. 743-747, 2023

    work page 2023

  26. [26]

    Generalized Correntropy induced metric based total least squares for sparse system identification

    J. Zhao, J. A. Zhang, H. Zhang, Q. Li, "Generalized Correntropy induced metric based total least squares for sparse system identification", Neurocomputing, vol. 467, 2022

    work page 2022

  27. [27]

    The square -root unscented Kalman filter for state and parameter -estimation,

    R. Van der Merwe and E. A. Wan, "The square -root unscented Kalman filter for state and parameter -estimation," 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), Salt Lake City, UT, USA, 2001, pp. 3461 -3464 vol.6

    work page 2001

  28. [28]

    Square root unscented Kalm an filter with modified measurement for dynamic state estimation of power systems,

    L. Dang, W. Wang, and B. Chen, "Square root unscented Kalm an filter with modified measurement for dynamic state estimation of power systems, " IEEE Transactions on Instrumentation and Measurement, vol. 71, pp. 1–13, 2022

    work page 2022

  29. [29]

    Square -Root Approach for Chandrasekhar -Based Maximum Correntropy Kalman Fil tering,

    M. V. Kulikova, "Square -Root Approach for Chandrasekhar -Based Maximum Correntropy Kalman Fil tering," IEEE Signal Processing Letters, vol. 26, no. 12, pp. 1803-1807, Dec. 2019

    work page 2019

  30. [30]

    Maximum Correntropy Square - Root Cubature Kalman Filter for Non -Gaussian Measurement Noise,

    J. He, C. Sun, B. Zhang and P. Wang, "Maximum Correntropy Square - Root Cubature Kalman Filter for Non -Gaussian Measurement Noise," IEEE Access, vol. 8, pp. 70162-70170, 2020

    work page 2020

  31. [31]

    A New Look at Boundedness of Error Covariance of Kalman Filtering,

    W. Li, G. Wei, D. Ding, Y. Liu and F. E. Alsaadi, "A New Look at Boundedness of Error Covariance of Kalman Filtering," IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 48, no. 2, pp. 309-314, Feb. 2018

    work page 2018

  32. [32]

    Uniform Detectability-Aided Boundedness Analysis of Error Covariances of Kalman Filter for Time -Varying Systems,

    W. Li, G. Wei and F. Yang, "Uniform Detectability-Aided Boundedness Analysis of Error Covariances of Kalman Filter for Time -Varying Systems," IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 52, no. 8, pp. 4798-4806, Aug. 2022

    work page 2022

  33. [33]

    On the Stabili ty Bounds of Kalman Filters for Linear Deterministic Discrete -Time Systems,

    M. Haring and T. A. Johansen, "On the Stabili ty Bounds of Kalman Filters for Linear Deterministic Discrete -Time Systems," IEEE Transactions on Automatic Control, vol. 65, no. 10, pp. 4434-4439, Oct. 2020

    work page 2020

  34. [34]

    Accessed: 2021

    Power Systems Test Case Archive. Accessed: 2021. [Online]. Available: http://www.ee.washington.edu/research/pstca/

    work page 2021

  35. [35]

    Updating the inverse of a matrix,

    W. W. Hager, "Updating the inverse of a matrix," SIAM Rev., vol. 31, no. 2, pp. 221–239, 1989

    work page 1989

  36. [36]

    Abur and A

    A. Abur and A. G. Exposito, Power System State Estimation: Theory and Implementation. New York, NY, USA: Marcel Dekker, 2004

    work page 2004

  37. [37]

    Identification of nonlinear dynamic system using a novel recurrent wavelet neural network based on the pipelined architecture ,

    H Zhao, S Gao, Z He, X Zeng, W Jin, T Li , "Identification of nonlinear dynamic system using a novel recurrent wavelet neural network based on the pipelined architecture ," IEEE Transactions on Industrial Electronics, vol. 61, no. 8, pp. 4171-4182

    work page

  38. [38]

    Low-complexity nonlinear adaptive filter based on a pipelined bilinear recurrent neural network,

    H Zhao, X Zeng, Z He, "Low-complexity nonlinear adaptive filter based on a pipelined bilinear recurrent neural network," IEEE Transactions on Neural Networks, vol. 2, no. 9, pp. 1494-1507

    work page

  39. [39]

    Memory proportionate APA with individual activation factors for acoustic echo cancellation,

    H Zhao, Y Yu, S Gao, X Zeng, Z He, "Memory proportionate APA with individual activation factors for acoustic echo cancellation," IEEE/ACM transactions on audio, speech, and language processing , vol. 22, no. 6, 1047

    work page

  40. [40]

    Robust power system forecasting -aided state estimation with generalized maximum mixture correntropy unscented Kalman filter,

    H Zhao, B Tian , "Robust power system forecasting -aided state estimation with generalized maximum mixture correntropy unscented Kalman filter," IEEE Transactions on Instrumentation and Measurement, vol. 71, pp.1-10

    work page

  41. [41]

    Robust stable iterated unscented Kalman filter based on maximum correntropy criterion,

    H Zhao, B Tian, B Chen, "Robust stable iterated unscented Kalman filter based on maximum correntropy criterion," Automatica, vol. 142, 110410

    work page

  42. [42]

    A Family of Robust Generalized Adaptive Filters and Application for Time-Series Prediction,

    Y. Peng, H. Zhao and J. Hu, "A Family of Robust Generalized Adaptive Filters and Application for Time-Series Prediction," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 72, no. 12, pp7902-7914, Dec. 202551

    work page

  43. [43]

    Functional link neural network cascaded with Chebyshev orthogonal polynomial for nonlinear channel equaliza tion,

    H. Zhao and J. Zhang, "Functional link neural network cascaded with Chebyshev orthogonal polynomial for nonlinear channel equaliza tion," Signal Processing, vol. 88, no. 8, pp. 1946-1957, 2008

    work page 1946

  44. [44]

    Adaptive reduced feedback FLNN filter for active control of nonlinear noise processes,

    H. Zhao, X. Zeng, and J. Zhang, "Adaptive reduced feedback FLNN filter for active control of nonlinear noise processes," Signal Processing, vol. 90, no. 3, pp. 834-847, 2010

    work page 2010

  45. [45]

    Adaptively combined FIR and functional link artificial neural network equalizer for nonlinear communication 14 channel,

    H. Zhao and J. Zhang, "Adaptively combined FIR and functional link artificial neural network equalizer for nonlinear communication 14 channel," IEEE Transactions on Neural Networks, v ol. 20, no. 4, pp. 665674,Apr. 2009

    work page 2009

  46. [46]

    A Fast Robust Adaptive Filter using Improved Data -Reuse Method,

    Y. Peng, H. Zhao and J. Hu, "A Fast Robust Adaptive Filter using Improved Data-Reuse Method," IEEE Transactions on Signal Processing, doi: 10.1109/TSP.2026.3685279

    work page doi:10.1109/tsp.2026.3685279 2026