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REVIEW 4 major objections 7 minor 26 references

Optimal step-size of least mean absolute fourth algorithm in low SNR

T0 review · 4 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A time-updated, MSD-minimizing step size makes the least mean fourth adaptive filter converge to zero steady-state misalignment, even at low SNR and under non-Gaussian noise.

desk verdict The OPLMF step-size rule is an oracle method that requires the very MSD it is supposed to minimize, so the claimed 20–60 dB gains are not established for real implementations; the derivation is a legitimate but flawed extension of the authors' prior LMAT work. read the letter →

arxiv 1908.08165 v1 pith:JCY5MV5D submitted 2019-08-22 eess.SP cs.ITmath.IT

Sihai Guan , Chun Meng , Bharat Biswal This is my paper · ORCID
classification eess.SPcs.ITmath.IT MSC 94A12
keywords adaptivefilteringleastmeanfourthoptimalstep-sizemean-squaredeviationlowsignal-to-noiserationon-Gaussiannoisesystemidentificationtime-varyingsystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes an optimized least mean absolute fourth (OPLMF) adaptive filter for system identification in low signal-to-noise-ratio conditions. At each iteration the step-size is chosen to minimize the next mean-square deviation, producing the update $\mu(n) = 3\,\mathrm{MSD}(n)(\sigma_\rho^2 + \sigma_x^2 \mathrm{MSD}(n))/\mu_{f,n}$. The paper derives the mean convergence and steady-state analysis from a recursive MSD model and shows that the steady-state MSD becomes zero, so the excess mean-square error reduces to the noise variance $\sigma_\rho^2$. If correct, this gives a low-complexity LMF variant that tracks time-varying systems under Gaussian, Uniform, Binary, Rayleigh, and Poisson noises better than the compared normalized and quotient-form variable step-size LMF algorithms.

What carries the argument

The load-bearing object is the time-varying MSD recursion of Eq. (18), a cubic polynomial in $\mathrm{MSD}(n)$ whose coefficients depend on filter length $L$, input variance $\sigma_x^2$, step-size $\mu$, and the noise moments $\mathbb{E}[\rho^4(n)]$ and $\mathbb{E}[\rho^6(n)]$. The optimal step-size is obtained by setting the partial derivative $\partial \mathrm{MSD}(n+1)/\partial \mu = 0$ under the assumption that $\mathrm{MSD}(n)$ is known, which produces Eq. (29). The recursion is also used to derive the stability bound $0 < \mu(n) < \sigma_\rho^2/(5(L+2)\sigma_x^2 \mathbb{E}[\rho^4(n)])$ and to propagate the predicted MSD in Table 1. The paper uses the known moment formulas for Gaussian, Uniform, Binary, Rayleigh, and Poisson noises to instantiate the fourth and sixth noise moments.

What would settle it

Run OPLMF on the paper's low-SNR identification tasks while estimating $\mathrm{MSD}(n)$ online from the filter's error signal instead of using the true coefficient error $\|\mathbf{W}_O(n)-\mathbf{W}(n)\|^2$; if the measured steady-state MSD then remains clearly above zero, or the filter diverges, the claimed zero steady-state MSD holds only when an oracle supplies the state.

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Extended reading notes

Core claim

The central claim is that the step-size, rather than being a fixed tuning constant, can be treated as a state-dependent control that makes the LMF algorithm's misalignment vanish. Starting from the coefficient-error recursion $\mathbf{V}(n+1)=\mathbf{V}(n)+\mu \mathbf{X}(n)e(n)^3$, the paper derives a scalar recursion for $\mathrm{MSD}(n)=\mathbb{E}[\mathbf{V}^T(n)\mathbf{V}(n)]$ by approximating the fourth-, sixth-, and higher-order moments of the Gaussian input. Minimizing $\mathrm{MSD}(n+1)$ with respect to $\mu$ at every time $n$ yields Eq. (29), and inserting this step-size back into the recursion leaves an equation whose only steady-state solution is $\mathrm{MSD}(\infty)=0$. The companion statement is that the steady-state EMSE equals the additive noise variance, $\mathrm{EMSE}(\infty)=\sigma_\rho^2$, meaning the adaptive filter contributes no excess error. The paper validates this by seven Monte Carlo system-identification experiments at SNR levels from 0 to 3 dB, reporting steady-state MSD values far below NLMF and VSSLMFQ baselines.

Load-bearing premise

The whole scheme presumes that at iteration $n$ the filter knows its current mean-square deviation $\mathrm{MSD}(n)$ and the noise moments $\mathbb{E}[\rho^4(n)]$ and $\mathbb{E}[\rho^6(n)]$, quantities that in real system identification depend on the very unknown system being identified.

Editorial extensions

If this is right

  • Steady-state misalignment vanishes: in the model, $\mathrm{MSD}(\infty)=0$, so the adaptive filter's excess error disappears and only the ambient noise variance remains.
  • The same step-size rule applies across noise types; only the fourth and sixth noise moments in Eq. (29) change, so no redesign is needed for Gaussian, Uniform, Binary, Rayleigh, or Poisson noise.
  • The algorithm needs no divisions in its update and about $2L+16$ multiplications per iteration, making it cheaper than the VSSLMFQ baseline while giving lower steady-state MSD.
  • The automatically large initial step-size followed by a shrinking step-size provides fast initial convergence and fine tracking in slowly time-varying systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A practical implementation still needs a running estimate of $\mathrm{MSD}(n)$; because the paper's simulations supply the true coefficient error, a real identifier would have to estimate MSD from data, and the zero steady-state result would likely degrade to a small nonzero floor.
  • The same 'minimize next MSD' construction transfers naturally to other error-power cost functions, such as least mean absolute third or mixed-norm filters, where analogous closed-form step-sizes may exist.
  • A testable extension is to replace the known noise moments by online estimates (sample fourth and sixth moments of the error); if those estimates are accurate, OPLMF should track time-varying systems at even lower SNR without oracle knowledge.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The manuscript proposes an optimized least mean absolute fourth (OPLMF) algorithm for adaptive system identification at low signal-to-noise ratio. The step-size is chosen at each iteration to minimize a mean-square deviation (MSD) recursion, Eq. (18), leading to the formula in Eq. (29). The authors claim that the resulting steady-state MSD is zero, assert this via Eq. (31), and support the claim with seven system-identification experiments comparing OPLMF with NLMF and VSSLMFQ under Gaussian, Uniform, Binary, Rayleigh, and Poisson noises. The paper also includes a computational-complexity comparison and a brief stability discussion.

Significance. A step-size rule that optimally trades convergence speed against steady-state misalignment for a high-order error-power filter could be a useful contribution, especially for low-SNR and non-Gaussian-noise regimes. The paper covers a relevant problem class and provides a broad simulation study, and the complexity comparison in Table 2 is a positive feature. However, the central construction depends on the true weight-error MSD, which is not available in practice, and the theoretical derivation relies on heuristic moment closures without error control. As it stands, the paper demonstrates an oracle-assisted design rather than an implementable algorithm, so the claimed practical superiority over NLMF and VSSLMFQ is not established.

major comments (4)
  1. [Section 2, Eq. (29) and Table 1] The step-size formula uses MSD(n) = E[||W_O(n) - W(n)||^2], the true weight-error norm of the unknown system. Table 1 updates MSD(n) with the model recursion, but this recursion must be initialized; with W(0) = 0, the required initial value is MSD(0) = ||W_O||^2, which depends on the unknown system. No data-driven estimator for MSD or for the noise moments E[rho^4] and E[rho^6] is supplied. In every Section 4 experiment, the known W_O is used both to schedule mu(n) and to evaluate the MSD curves, so the reported 20-60 dB improvements over NLMF and VSSLMFQ are conditional on information a real system identifier does not possess.
  2. [Section 2, Eqs. (12)-(18)] The core recursion Eq. (18) is obtained through ad-hoc Gaussian input moment closures, for example E[(X^T V)^4] is replaced by 6 sigma_x^4 (E[V^T V])^2 and higher-order terms are factorized as E[(V^T V)^k] approximately equal to (E[V^T V])^k. The manuscript gives no error bound or supporting argument for these closures, and the input is assumed Gaussian while the algorithm is then optimized for non-Gaussian noises. The claimed agreement between simulation and theory in Figs. 1(B)-7(B) is not independent validation, because the theory curve is generated from the same approximate recursion that defines the step-size.
  3. [Section 2, Eq. (31)] The conclusion MSD(infinity) = 0 is asserted after 'several computations' and 'Combine Eq. (23)', but the intermediate algebra is not shown. Eq. (31) has the structure MSD(infinity)^2 times a bracket equals zero, so it also admits a nonzero root; the manuscript does not prove that the bracket cannot vanish or that the zero root is the fixed point actually attained. Because the step-size depends on MSD(n), the fixed-point analysis must account for this coupling explicitly, which the paper does not do.
  4. [Section 2, Eqs. (26)-(29)] The minimization leading to the 'optimal' step-size sets partial MSD(n)/partial mu = 0, treating the current MSD as independent of the step-size. In reality MSD(n) depends on all previous step-sizes, so this is a heuristic approximation. The manuscript does not state it as such or assess its effect on the optimality claim, and therefore the label 'optimal' is not formally justified.
minor comments (7)
  1. [Throughout] There are multiple typographical errors and inconsistencies: 'read' should be 'red' in figure captions, 'discription' should be 'description', 'Passion' should be 'Poisson', and 'OPLMAT' is used instead of 'OPLMF' in several places.
  2. [Section 2, Eq. (22) and Eq. (19)] Equation numbering is confusing: Eq. (19) is first defined as g(L, mu, sigma_x) and later reused for the reduced first-order recursion, and Eq. (22) repeats a formula already derived. Please renumber consistently.
  3. [Section 2, Eq. (23)] The text says gamma is a small positive number to keep the denominator finite, but then states (1 - 1/(2L)) <= gamma < 1, which is not derived and seems inconsistent with 'small'; the range should be justified or corrected.
  4. [Section 4] The correlated input is defined by y(n) = 0.5 y(n) + x(n), which is self-referential and presumably should be y(n) = 0.5 y(n-1) + x(n); please correct and define the filtering operation precisely.
  5. [Table 3] In Experiment 2 the VSSLMFQ result is described as divergent, but Table 3 does not show an entry; please clarify whether the algorithm diverged and how the table was compiled in that case.
  6. [Figures 1(B)-7(B)] The 'MSD error' is described as the difference between simulation and Theory (Eq. (30)), but Eq. (30) is a recursion and not a closed-form curve; please specify exactly how the theoretical MSD trajectory is computed in the figures.
  7. [Eqs. (29)-(31)] The notation mu_f,n is used in Eq. (29), while Eqs. (30) and (31) use mu_f,infinity without a definition; avoid switching subscripts without explanation.

Circularity Check

1 steps flagged · score 6.0 of 10

OPLMF's step-size is driven by the same MSD it claims to predict; the zero steady-state result is an artifact of minimizing that same MSD recursion, and simulations feed the true unknown system into the algorithm.

  1. fitted input called prediction [Section 2, Eq. (29), Eq. (18), Eq. (31); Table 1; Section 4 simulation protocol]
    "The optimal step-size is then given by μ(n)=3MSD(n)(σρ2+σx2MSD(n))/μf,n ... MSD(n+1)=MSD(n){1+15(L+2)μ2σx4E[ρ4(n)]−6μσx2σρ2}+... ; Combine Eq. (23), we know MSD(∞)=0 ... MSD(n)=10log10(‖WO(n)−W(n)‖22) is used to measure the performance."

    Eq. (29) makes the step-size a function of MSD(n), the very quantity whose steady-state value the paper then 'derives' as zero from the same recursion Eq. (18) that was minimized. The fixed point Eq. (31) is obtained by substituting the minimizer back into Eq. (18), so MSD(∞)=0 is a consequence of the design objective, not an independent result. In the experiments, the known WO is used both to drive this MSD-based step-size and to compute the reported MSD; hence the OPLMF gains over NLMF/VSSLMFQ are conditional on oracle access to the target system. No estimator for MSD or the noise moments is given, so the 'prediction' is the input under another name.

full rationale

The algebraic minimization leading from Eq. (18) to Eq. (29) is not itself logically circular: it is a valid first-order condition. The circularity enters at the level of what the paper calls a prediction and an algorithm. μ(n) is defined in terms of the exact MSD, which is unknown in system identification; Table 1 requires MSD(0)=||WO||2 and the noise moments E[ρ4], E[ρ6], σρ2, and no estimator is supplied. The steady-state result MSD(∞)=0 is a consequence of plugging the minimizer back into the same approximate recursion, so it is a property of the model used to design the step-size, not an independent first-principles bound. In all seven experiments the known WO is used both to drive and to evaluate the MSD, making the empirical superiority over NLMF and VSSLMFQ in-sample/oracle rather than a test of an implementable algorithm. There is no load-bearing self-citation or imported uniqueness theorem; the issue is the self-referential use of MSD as both input and output. This is partial circularity, not a fully vacuous derivation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on Gaussian input moment closures, zero-odd-moment noise, known noise moments, and a model-based MSD that is both target and input. These are standard adaptive-filtering assumptions but are not all satisfied by the paper's own simulation settings, and the MSD dependency is not resolved by a practical estimator.

free parameters (2)
  • γ (input-variance smoothing factor) = 0.98 in simulations; constraint (1-1/(2L)) ≤ γ < 1
    Introduced in Eq. (23) to estimate σ_x²(n); no optimality criterion is derived, and the value is chosen by hand following NPVSS-NLMS [25].
  • MSD(0) (initial weight-error norm squared) = ||W_O||² = 1.18 for the fixed system W_O=[0.8,0.2,-0.7,0.2,0.1] used in all experiments
    Required by Table 1 to start the MSD recursion; it depends on the unknown W_O and is only available in simulations where the true system is known. Its value affects the entire step-size trajectory.
assumptions (4)
  • domain assumption X(n) is zero-mean Gaussian with covariance R, and R is approximated by σ_x² I (spatially white input).
    Used to evaluate the Gaussian moment expressions in Eqs. (12)-(16); the correlated-input simulations violate the white-input part, so the constants (15L+90), (3L+12), and (L+2) are approximate.
  • domain assumption ρ(n) is zero-mean, independent of X and V, with zero odd-order moments and known σρ², E[ρ⁴], E[ρ⁶].
    Required for the MSD expansion in Eq. (11); Experiments 6 and 7 use Rayleigh and Poisson noises, which have nonzero means, so this assumption is violated in the validation experiments.
  • ad hoc to paper Moment factorization: E[||V||^{2k}] ≈ (E[||V||²])^k, and MSD(n) is known or updated by the model recursion.
    Used at Eqs. (12)-(16) to close the recursion and in Table 1 to propagate MSD(n); no bound on the approximation error is given, and the algorithm needs MSD(0), which depends on the unknown W_O.
  • standard math The noise-moment formulas for Uniform, Rayleigh, Binary and Poisson from reference [26] are correct.
    Used in Section 2 for the steady-state results; the printed Poisson E[ρ⁶] expression appears truncated (missing λ⁵ and λ⁶ terms), so the computed step-sizes for Experiment 7 are questionable.

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Pith. "Pith review of Optimal step-size of least mean absolute fourth algorithm in low SNR." pith.science (2026). https://pith.science/paper/JCY5MV5D

@misc{pith2026190808165,
  author       = {Pith},
  title        = {Pith review of: Optimal step-size of least mean absolute fourth algorithm in low SNR},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JCY5MV5D}},
  note         = {Machine review of arXiv:1908.08165}
}
read the original abstract

There is a need to improve the capability of the adaptive filtering algorithm against Gaussian or multiple types of non-Gaussian noises, time-varying system, and systems with low SNR. In this paper, we propose an optimized least mean absolute fourth (OPLMF) algorithm, especially for a time-varying unknown system with low signal-noise-rate (SNR). The optimal step-size of OPLMF is obtained by minimizing the mean-square deviation (MSD) at any given moment in time. In addition, the mean convergence and steady-state error of OPLMF are derived. Also the theoretical computational complexity of OPLMF is analyzed. Furthermore, the simulation experiment results of system identification are used to illustrate the principle and efficiency of the OPLMF algorithm. The performance of the algorithm is analyzed mathematically and validated experimentally. Simulation results demonstrate that the proposed OPLMF is superior to the normalized LMF (NLMF) and variable step-size of LMF using quotient form (VSSLMFQ) algorithms.

Figures

Figures reproduced from arXiv: 1908.08165 by the authors.

Figure 2
Figure 2. Fig.2 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Fig.3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Fig.4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: Fig.5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Fig.6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Works this paper leans on

26 extracted references · 25 canonical work pages

  1. [1]

    A.H. Sayed. Adaptive Filters [M]. Wiley, Hoboken, 2008

    work page 2008

  2. [2]

    Normalised spline adaptive filtering algorithm for nonlinear system identification [J]

    Guan S, Zhi L. Normalised spline adaptive filtering algorithm for nonlinear system identification [J]. Neural Processing Letters, 2017, 46(5):595-607

    work page 2017

  3. [3]

    Diffusion normalized Huber adaptive filtering algorithm [J]

    Zhi L, Guan S. Diffusion normalized Huber adaptive filtering algorithm [J]. Journal of the Franklin Institute, 2018, 355(8):3812-3825

    2018

  4. [4]

    Functional organization of the human 4D Nucleome [J]

    Chen H, Chen J, Muir L A, et al. Functional organization of the human 4D Nucleome [J]. Proceedings of the National Academy of Sciences of the United States of America, 2015, 112(26):8002-7

    work page 2015

  5. [5]

    Adaptive waveform learning: a framework for modeling variability in neurophysiologica l signals [J]

    Hitziger S, Clerc, Maureen, Saillet, Sandrine, et al. Adaptive waveform learning: a framework for modeling variability in neurophysiologica l signals [J]. IEEE Transactions on Signal Processing, 2017, 65(16):4324-4338

    work page 2017

  6. [6]

    The least mean fouth (LMF) adaptive algorithm and its family [J]

    Walach E, Widrow B. The least mean fouth (LMF) adaptive algorithm and its family [J]. IEEE Transactions on Information Theory, 1984, 30(2):275-283

    work page 1984

  7. [7]

    Nonparametric variable step-size LMAT algorithm [J]

    Sihai GUAN, Zhi Li. Nonparametric variable step-size LMAT algorithm [J]. Circuits Systems & Signal Processing, 2017, 36(3):1322-1339

    work page 2017

  8. [8]

    An optimized NLMS algorithm for system identification [J]

    Ciochină S, Paleologu C, Benesty J. An optimized NLMS algorithm for system identification [J]. Signal Processing. 2016, 118:115-121

    work page 2016

Show all 26 references
  1. [9]

    Optimal step size of least mean absolute third algorithm [J]

    Guan S, Li Z. Optimal step size of least mean absolute third algorithm [J]. Signal, Image and Video Processing, 2017, 11(6):1105-1113

    2017

  2. [10]

    An improved statistical analysis of the least mean fourth (LMF) adaptive algorithm [J]

    Hubscher P I, Bermudez J C M. An improved statistical analysis of the least mean fourth (LMF) adaptive algorithm [J]. IEEE Transactions on Signal Processing, 2003, 51(3):664-671

    2003

  3. [11]

    Probability of divergence for the least -mean fourth algorithm [J]

    Nascimento V H, Bermudez J C M. Probability of divergence for the least -mean fourth algorithm [J]. IEEE Transactions on Signal Processing, 2006, 54(4):1376 - 1385

    2006

  4. [12]

    A mean-square stability analysis of the least mean fourth adaptive algorithm [J]

    Hubscher P I, Bermudez J C M, et al. A mean-square stability analysis of the least mean fourth adaptive algorithm [J]. IEEE Transactions on Signal Processing, 2007, 55(8):4018-4028

    2007

  5. [13]

    Stochastic anal ysis of a stable normalized least mean fourth algorithm for adaptive noise canceling with a white Gaussian reference [J]

    Eweda, Bershad. Stochastic anal ysis of a stable normalized least mean fourth algorithm for adaptive noise canceling with a white Gaussian reference [J]. IEEE Transactions on Signal Processing, 2012, 60(12):6235-6244

    2012

  6. [14]

    Dependence of the stability of the least mean fourth algorithm on target weights non-stationarity [J]

    Eweda E. Dependence of the stability of the least mean fourth algorithm on target weights non-stationarity [J]. IEEE Transactions on Signal Processing, 2014, 62(7):1634-1643

    2014

  7. [15]

    Combination of recursive least p -norm algorithms for robust adaptive filtering in alpha-stable noise [J]

    Navia-Vazquez A, Arenas -Garcia J. Combination of recursive least p -norm algorithms for robust adaptive filtering in alpha-stable noise [J]. IEEE Transactions on Signal Processing, 2012, 60(3):1478-1482

    2012

  8. [16]

    Robust adaptive filtering algorithms for α-stable random processes [J]

    Aydin, G., Arikan, O., Cetin, A.E. Robust adaptive filtering algorithms for α-stable random processes [J]. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 1999, 46(2):198-202

    1999

  9. [17]

    Convergence and steady -state analysis of the normalized least mean fourth algorithm [J]

    Zerguine A. Convergence and steady -state analysis of the normalized least mean fourth algorithm [J]. Digital Signal Processing, 2007, 17(1):17-31

    2007

  10. [18]

    Convergence and tracking analysis of a variable normalised LMF (XE-NLMF) algorithm [J]

    Zerguine A, Chan M K, et al. Convergence and tracking analysis of a variable normalised LMF (XE-NLMF) algorithm [J]. Signal Processing, 2009, 89(5):778 - 790

    2009

  11. [19]

    Global stabilization of the least mean fourth algorithm [J]

    Eweda E. Global stabilization of the least mean fourth algorithm [J]. IEEE Transactions on Signal Processing, 2012, 60(3):1473-1477

    2012

  12. [20]

    Variable step-size NLMS and affine projection algorithms [J]

    Shin H C, Sayed A H, et al. Variable step-size NLMS and affine projection algorithms [J]. IEEE Signal Processing Letters, 2004, 11(2):132-135

    2004

  13. [21]

    A noise constrained least mean fourth (NCLMF) adaptive algorithm [J]

    Zerguine A, Moinuddin M, et al. A noise constrained least mean fourth (NCLMF) adaptive algorithm [J]. Signal Processing, 2011, 91(1):136-149

    2011

  14. [22]

    Bermudez

    Bershad N J, José C.M. Bermudez. Mean-square stability of the normalized least- mean fourth algorithm for white Gaussian inputs [J]. Digital Signal Processing, 2011, 21(6):694-700

    2011

  15. [23]

    Variable step -size LMS algorithm with a quotient form [J]

    Zhao S, Man Z, Khoo S, et al. Variable step -size LMS algorithm with a quotient form [J]. Signal Processing, 2009, 89(1):67-76

    2009

  16. [24]

    A robust and stable variable step-size design for the LMF using quotient form [J]

    Syed Muhammad Asad, et al. A robust and stable variable step-size design for the LMF using quotient form [J]. Signal Processing, 2019, 162:196-210

    2019

  17. [25]

    A nonparametric VSSNLMS algorithm [J]

    Benesty J, Rey H, Vega L R, et al. A nonparametric VSSNLMS algorithm [J]. IEEE Signal Processing Letters, 2006, 13(10):581-584

    2006

  18. [26]

    Mathematical handbook of formulas and tables [M]

    Spiegel, M.R. Mathematical handbook of formulas and tables [M]. McGraw-Hill, New York, 2012

    2012

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