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arxiv: 2605.17964 · v1 · pith:P564AM6Tnew · submitted 2026-05-18 · 📡 eess.AS

Fractional-Order Subband p-Norm Adaptive Filter via Transformation Nearest Kronecker Product Decomposition for Active Noise Control

Jianhong Ye , Haiquan Zhao , Shaohui Lv , Yang Zhou This is my paper

Pith reviewed 2026-05-20 00:50 UTC · model grok-4.3

classification 📡 eess.AS
keywords adaptive filteractive noise controlfractional ordersubbandp-normKronecker productalpha-stable noisetransformation decomposition
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The pith

A transformation nearest Kronecker product decomposition enables a fractional-order subband p-norm adaptive filter that lowers both misadjustment and computational cost in active noise control.

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

The paper seeks to overcome limitations of the standard normalized subband p-norm algorithm when dealing with non-Gaussian inputs, alpha-stable noise where the stability index is at most 1, and sparse systems. It introduces the NKP-FoNSPN by applying nearest Kronecker product decomposition together with fractional-order stochastic gradient descent, and derives bounds on the fractional order beta. A transformation-based version called TNKP-FoNSPN further cuts complexity, and filtered-x variants are developed for active noise control applications. If correct, these methods would allow more reliable noise cancellation in real-world settings with impulsive or sparse disturbances using less processing power.

Core claim

By integrating fractional-order moments into the subband p-norm framework and using a nearest Kronecker product or its transformation decomposition, the NKP-FoNSPN and TNKP-FoNSPN algorithms achieve lower steady-state misadjustment and multiplication cost than prior methods, while providing theoretical bounds for the parameter beta and effective performance in alpha-stable noise and sparse identification tasks.

What carries the argument

The transformation nearest Kronecker product (TNKP) decomposition, which restructures the adaptive filter coefficients to reduce the number of multiplications required in the update for specific filter lengths while preserving the benefits of the fractional-order p-norm error measure.

If this is right

  • When the fractional-order parameter beta equals 1, the algorithm simplifies to the NKP-NSPN or the non-decomposed FoNSPN.
  • Filtered-x versions NKP-FxFoNSPN and TNKP-FxFoNSPN extend the approach to active noise control by accounting for the secondary path.
  • Simulations with pink, helicopter, gunshot, pile driver, and traction substation noise confirm lower error and cost.
  • Real single-channel duct and simulated multi-channel ANC systems validate the noise reduction capability.

Where Pith is reading between the lines

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

  • The TNKP technique may apply to other Kronecker-based adaptive algorithms to achieve similar complexity reductions in different filtering tasks.
  • Improved handling of alpha less than or equal to 1 noise could benefit applications like speech enhancement in noisy environments.
  • Lower multiplication cost might facilitate implementation in embedded systems for portable noise cancellation devices.

Load-bearing premise

The derived theoretical bounds for the fractional-order parameter beta continue to hold for the input signals and noise distributions encountered in practice without additional stabilization.

What would settle it

Measure the steady-state misadjustment of TNKP-FoNSPN in a new experiment with alpha-stable noise of alpha=0.8 and a sparse system, and check if it remains below that of the NKP version as predicted.

Figures

Figures reproduced from arXiv: 2605.17964 by Haiquan Zhao, Jianhong Ye, Shaohui Lv, Yang Zhou.

Figure 1
Figure 1. Figure 1: Structure of NKP-FoNSPN. 3. Proposed NKP-FoNSPN algorithms To address the limitations of the conventional NSPN algorithm outlined in Remark 1, this section introduces the NKP-FoNSPN and TNKP-FoNSPN algorithms. Furthermore, we establish the theoretical valid range for the fractional￾order parameter 𝛽 and conduct a comprehensive computational complexity analysis comparing the proposed methods with state-of-t… view at source ↗
Figure 2
Figure 2. Figure 2: NMSD curves of the NKP-NLMS and NLMS algorithms. Similarly, the weight update formula of sub-filter 𝒎̂ 𝑘,2 can be described as 𝒎̂ 𝑘+𝑟,2 = 𝒎̂ 𝑘,2 − 𝜇 ∑ 𝑁 𝑗=1 𝑔(𝑒𝑘,𝑗,2 )Re{ (−𝒙𝑘,𝑗,1 ) 𝛽 } ||𝒙𝑘,𝑗,1 ||𝑝 𝑝 , (44) where 𝑔(𝑒𝑘,𝑗,2 ) = sgn(𝑒𝑘,𝑗,2 )|𝑒𝑘,𝑗,2 | 𝑝−𝛽 represents the nonlinear error term. As a summary, (21), (24), (25), (27), (30), (43), and (44) constitute the proposed NKP-FoNSPN algorithm. Notably, the n… view at source ↗
Figure 3
Figure 3. Figure 3: Computational complexity of the NSPN [37], FoNLMP [41], FoNSPN, FoMVC [46], NKP-GHSAF [54], NKP-RLS [53], NKP-NSPN, and NKP-FoNSPN algorithms. (a) The computational complexity of multiplications versus the parameter 𝑄; (b) the computational complexity of powers versus the parameter 𝑄; (c) the computational complexity of multiplications versus the parameter 𝐷; (d) the computational complexity of powers vers… view at source ↗
Figure 4
Figure 4. Figure 4: Structure of NKP-FxFoNSPN. Based on (54), the filtered subband input signal can be expressed as 𝑿 ′ 𝑘,𝑠 △ = [ 𝒙 ′ 𝑘,1 , 𝒙 ′ 𝑘,2 , ..., 𝒙 ′ 𝑘,𝑁 ] = 𝑿 ′ 𝑘 𝑭, (55) where 𝑿 ′ 𝑘,𝑠 denotes the filtered subband input matrix. To enhance the conciseness of the paper, we directly extend the learning rules of the NKP-FoNSPN and TNKP￾FoNSPN algorithms to develop the NKP-FxFoNSPN and TNKP-FxFoNSPN algorithms for ANC ap… view at source ↗
Figure 5
Figure 5. Figure 5: IR of the unknown system 𝒎0 with 𝐷 = 500. (a) network echo path is chosen from G.168 Recommendation [72] and (b) acoustic echo path [73]. 0 5 10 15 Input samples, k (a) 104 -30 -25 -20 -15 -10 -5 0 5 10 NMSD (dB) N=2, r=2 N=4, r=4 N=8, r=8 N=4, r=2 N=4, r=8 N=4, r=16 0 0.5 1 1.5 2 2.5 3 3.5 4 Input samples, k (b) 104 -30 -25 -20 -15 -10 -5 0 5 NMSD (dB) =0.00007 (Method-I) =0.0007 (Method-I) =0.07 (Method-… view at source ↗
Figure 6
Figure 6. Figure 6: NMSD curves of the NKP-FoNSPN algorithm for different 𝑁, 𝑟, and 𝜄. 𝛼 = 1.5 and 𝛾 = 1∕60 for the additive noise; 𝜇 = 0.01, 𝑝 = 1.4, 𝛽 = 1.1, and 𝑄 = 2 for the NKP-FoNSPN algorithm. 5.1. System Identification [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: NMSD curves of the NKP-FoNSPN algorithm for different 𝛽. Refering to [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Testing the effectiveness of the transformation NKP decomposition technique. (a) The TNKP-NLMS algorithm and (b) the TNKP-FoNSPN algorithm. 𝛼 = 2, 𝛾 = 1∕60, and 𝑄 = 2 for [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of NMSD learning curves for 𝛼-stable noise with 𝛼 = 1.5 and 𝛾 = 1∕60. 𝑄 = 2 for NKP decomposition; 𝑝 = 1.4 for the NSPN algorithm; 𝑝 = 1.4 and 𝛽 = 1.1 for the FoNSPN, FoNLMP, NKP-FoNSPN, and TNKP-FoNSPN algorithms; ̂𝜌 = 0.6, 𝛼 ′ = 1.4, and 𝜏 = 0.2 for the FoMVC algorithm; 𝜆 = 0.99 and 𝛼 ′ = 2 for the NKP-GHSAF algorithm. 0 5 10 15 Input samples, k 104 -25 -20 -15 -10 -5 0 5 10 15 NMSD (dB) NSPN(… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of NMSD learning curves for 𝛼-stable noise with 𝛼 = 0.75 and 𝛾 = 1∕60. 𝛼 ′ = 1.1 for the FoMVC algorithm; 𝑝 = 0.7 and 𝛽 = 0.65 for the FoNSPN, FoNLMP, NKP-FoNSPN, and TNKP-FoNSPN algorithms. Refering to [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of NMSD learning curves for Cauchy noise input with 𝛼 = 1 and 𝛾 = 1∕10 and 𝛼-stable noise with 𝛼 = 0.75 and 𝛾 = 1∕60. 𝛼 ′ = 1.1 for the FoMVC algorithm; 𝑝 = 0.7 and 𝛽 = 0.65 for the FoNSPN, FoNLMP, NKP-FoNSPN, and TNKP-FoNSPN algorithms. Refering to [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of NMSD learning curves for 𝛼-stable noise with 𝛼 = 0.75 and 𝛾 = 1∕60. (a) 𝛼 ′ = 1.1 for the FoMVC algorithm; 𝜆 = 0.9 and 𝛼 ′ = 0.5 for the NKP-GHSAF algorithm; 𝑝 = 0.7 and 𝛽 = 0.65 for the FoNSPN, FoNLMP, NKP-FoNSPN, and TNKP-FoNSPN algorithms. Refering to [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: (a) IR of the non-low-rank system; (b) Comparison of NMSD learning curves for 𝛼-stable noise with 𝛼 = 0.75 and 𝛾 = 1∕60. 𝐷1 = 8, 𝐷2 = 5, 𝑄 = 4 for NKP decomposition; 𝑝 = 1.4 for the NSPN algorithm; 𝑝 = 0.7 and 𝛽 = 0.65 for the FoNSPN, FoNLMP, and NKP-FoNSPN algorithms; ̂𝜌 = 0.6, 𝛼 ′ = 1.4, and 𝜏 = 0.2 for the FoMVC algorithm; 𝜆 = 0.99 and 𝛼 ′ = 2 for the NKP-GHSAF algorithm. 5.2. Acoustic Echo Cancellatio… view at source ↗
Figure 14
Figure 14. Figure 14: Structure of the adaptive echo canceler. 0 1 2 3 4 5 6 Input samples, k (a) 104 -20 -15 -10 -5 0 5 10 NMSD(dB) FoNLMP ( =0.008) FoMVC ( =0.007) FoNSPN ( =0.08) NKP-NSPN ( =0.0055) NKP-GHSAF ( =0.007) NKP-FoNSPN ( =0.0055) TNKP-FoNSPN ( =0.0055, b=0.08) 0 1 2 3 4 5 6 Input samples, k (b) 104 -25 -20 -15 -10 -5 0 5 10 NMSD (dB) FoNLMP ( =0.05) FoMVC ( =0.0007) FoNSPN ( =0.05) NKP-NSPN ( =0.0055) NKP-GHSAF (… view at source ↗
Figure 15
Figure 15. Figure 15: NMSD learning curves of different algorithm in AEC scenarios. (a) 𝛼-stable noise with 𝛼 = 0.75 and 𝛾 = 1∕60, 𝑝 = 1 for the NKP-NSPN algorithm, 𝜌 = 0.75 for the FoMVC algorithm, and refering to [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Time-domain IRs of ANC system [1]. (a) Primary path and (b) secondary path. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 104 -0.1 0 0.1 ANC OFF FxLMS 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 104 -0.1 0 0.1 ANC OFF FxFoNLMP 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 104 -0.1 0 0.1 ANC OFF FxGMCC 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 104 -0.1 0 0.1 Amplitude ANC OFF FxAPLEHS 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 104 -0.1 0 0.1 ANC OFF FxFoNSPN 0 0.5 1… view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of ANC algorithms under pink noise. (a) The noise reduction results and (b) the ANR curves of several algorithms. 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 ANC OFF FxLMS 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 ANC OFF FxFoNLMP 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 ANC OFF FxGMCC 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 Amplitude ANC OFF FxAPLEHS 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 ANC OFF FxFoNSPN 0 1 2 3 4 5 6 7 8 9 10 … view at source ↗
Figure 18
Figure 18. Figure 18: Comparison of ANC algorithms under real-world gunshots. (a) The noise reduction results and (b) the ANR curves of several algorithms [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: further validates the effectiveness of the proposed algorithms using real-world helicopter and pile driver noise. Consistent with the resultings in Figs. 17 and 18, the proposed TNKP-FxFoNSPN algorithm demonstrates superior noise reduction performance compared to all competing methods. 𝐷. Real Duct ANC Acoustic Models J. Ye et al.: Preprint submitted to Elsevier Page 22 of 28 [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 20
Figure 20. Figure 20: The single-channel duct ANC system. 0 200 400 600 800 1000 Number of samples -5 0 5 Amplitude 10-3 Primary path 0 100 200 300 400 500 Number of samples -0.02 -0.01 0 0.01 Amplitude Secondary path [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Time-domain IRs of the simulated primary and secondary paths [64]. 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 ANC OFF FxLMS 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 ANC OFF FxFoNLMP 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 ANC OFF FxGMCC 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 ANC OFF FxAPLEH 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 Amplitude ANC OFF FxFoNSPN 0 1 2 3 4 5 6 7 8 9 10 104 -1 0 1 ANC OFF NKP-FxFoNSPN 0 1 2 3 4 5 6 7 8 9 10 Inpu… view at source ↗
Figure 22
Figure 22. Figure 22: Comparison of ANC algorithms under real traction substation noise. (a) The noise reduction results and (b) the ANR curves of several algorithms. 𝑝 = 0.9 and 𝛽 = 0.8 for the FxFoMLNP algorithm; 𝑝 = 1.5 and 𝜂 = 0.2 for the FxGMCC algorithm; 𝜒 = 0.01 and 𝑃 = 8 for the FxAPLEHS algorithm; 𝑝 = 0.9 and 𝛽 = 0.8 for the FxFoNSPN algorithm; 𝑄 = 5, 𝑝 = 0.9, and 𝛽 = 0.8 for the NKP-FxFoNSPN and TNKP-FxFoNSPN algorit… view at source ↗
Figure 23
Figure 23. Figure 23: Diagram of a general multi-channel active noise control system [PITH_FULL_IMAGE:figures/full_fig_p024_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The ANR𝑠 curves of the algorithms in the multi-channel active noise control system. 𝐷 = 20, 𝐷1 = 5, 𝐷2 = 4, and 𝑄 = 4 for the NKP technology. 𝜇 = 0.0009 for the CFxLMS algorithm; 𝜇 = 0.0004, 𝑝 = 1.1, and 𝜂 = 0.2 for the MFxGMCC algorithm; 𝑝 = 1.2, 𝛽 = 1.1 and 𝜇 = 0.001 for the MFxFoNLMP algorithm; 𝑝 = 1.2, 𝛽 = 1.1, and 𝜇 = 0.004 for the MFxFoNSPN algorithm; 𝑝 = 1.2, 𝛽 = 1.1, and 𝜇 = 0.009 for the NKP-MFxF… view at source ↗
read the original abstract

The conventional normalized subband p-norm (NSPN) algorithm achieves robustness in $\alpha$-stable noise ($1<\alpha \leq 2$) by utilizing low-order error moments. However, its performance degrades significantly under three scenarios: (1) non-Gaussian inputs, (2) $\alpha$-stable noise with $0<\alpha \leq 1$, and (3) sparse system identification. To address these limitations, this paper proposes a fractional-order NSPN algorithm based on the nearest Kronecker product (NKP) decomposition and fractional-order stochastic gradient descent, termed NKP-FoNSPN. Theoretical bounds for the fractional-order parameter $\beta$ are also derived. Notably, when $\beta=1$, the NKP-FoNSPN reduces to a new NKP-NSPN algorithm, while its non-NKP decomposition variant becomes the fractional-order NSPN (FoNSPN) algorithm. Furthermore, a novel transformation-based NKP (TNKP) decomposition technique is designed, which exhibits lower computational complexity than conventional NKP for specific filter structures. The resulting TNKP-based FoNSPN (TNKP-FoNSPN) achieves lower steady-state misadjustment and multiplication cost compared with the NKP-FoNSPN algorithm. Additionally, complete computational complexity analyses are provided. For active noise control (ANC) scenarios, we develop filtered-x variants: NKP-FxFoNSPN and TNKP-FxFoNSPN. From the former, two additional variants are derived: NKP-FxNSPN and FxFoNSPN. Simulations using diverse noise sources (pink, helicopter, gunshot, pile driver, and traction substation noise) demonstrate the superiority of the proposed algorithms. Finally, we validate their noise reduction performance in a real constructed single-channel duct ANC and a simulated multi-channel ANC systems.

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

2 major / 2 minor

Summary. The paper introduces fractional-order normalized subband p-norm (FoNSPN) adaptive filters using nearest Kronecker product (NKP) and transformation-based NKP (TNKP) decompositions for active noise control (ANC). It derives theoretical bounds for the fractional-order parameter β, notes that β=1 recovers the NKP-NSPN algorithm, provides complexity analyses, develops filtered-x variants (NKP-FxFoNSPN, TNKP-FxFoNSPN), and demonstrates through simulations with pink, helicopter, gunshot, pile driver, and traction substation noise, as well as real single-channel duct and simulated multi-channel ANC, that the proposed methods achieve lower steady-state misadjustment and computational cost.

Significance. Should the performance improvements and theoretical bounds hold under the tested conditions, this work advances robust adaptive filtering techniques for ANC in environments with α-stable noise (including 0<α≤1), non-Gaussian inputs, and sparse systems. The inclusion of physical validation in a constructed duct ANC system and complete computational complexity analyses are notable strengths that enhance the practical applicability of the proposed TNKP-FoNSPN and its variants.

major comments (2)
  1. [Theoretical bounds for β] The derivation of bounds for the fractional-order parameter β via stochastic gradient analysis assumes moment conditions that are fragile precisely when 0<α≤1 and for sparse impulse responses. The simulations with gunshot, pile-driver, and traction-substation noise do not indicate whether the selected β values fall within these bounds or if implicit stabilization was employed, which is critical for validating the claimed convergence and lower misadjustment over NKP-FoNSPN.
  2. [TNKP decomposition and complexity] The assertion that TNKP-FoNSPN achieves lower multiplication cost than NKP-FoNSPN for specific filter structures requires more detailed breakdown in the complexity analysis, particularly comparing the transformation steps to standard NKP for the filter lengths used in the ANC experiments.
minor comments (2)
  1. [Abstract] The abstract states superiority but would benefit from including at least one quantitative performance metric or reference to error bars from the simulations to better support the claims.
  2. [Simulation results] Parameter values such as the specific choices for β, p-norm order, and subband numbers should be explicitly tabulated or listed for each experiment to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help us improve the clarity and rigor of the manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Theoretical bounds for β] The derivation of bounds for the fractional-order parameter β via stochastic gradient analysis assumes moment conditions that are fragile precisely when 0<α≤1 and for sparse impulse responses. The simulations with gunshot, pile-driver, and traction-substation noise do not indicate whether the selected β values fall within these bounds or if implicit stabilization was employed, which is critical for validating the claimed convergence and lower misadjustment over NKP-FoNSPN.

    Authors: We appreciate the referee's observation on the assumptions underlying the derivation of the bounds for β. The stochastic gradient analysis in Section III relies on the existence of finite moments of the error signal, which can indeed be delicate for α-stable processes when 0<α≤1. In the reported simulations, β was selected empirically within the range permitted by the derived inequalities for each noise type, and the algorithm's built-in normalization provides the necessary stabilization without additional mechanisms. To address the concern, we will revise the manuscript to explicitly list the β values employed in each experiment (including the gunshot, pile-driver, and traction-substation cases) and add a short discussion clarifying the relationship between the theoretical bounds and practical choices under low-α conditions. revision: yes

  2. Referee: [TNKP decomposition and complexity] The assertion that TNKP-FoNSPN achieves lower multiplication cost than NKP-FoNSPN for specific filter structures requires more detailed breakdown in the complexity analysis, particularly comparing the transformation steps to standard NKP for the filter lengths used in the ANC experiments.

    Authors: We agree that a finer-grained breakdown would strengthen the complexity section. The current manuscript states the overall multiplication counts and notes the advantage of TNKP for particular filter lengths arising from the transformation. In the revision we will insert an expanded table that itemizes the multiplications required by the transformation matrix operations in TNKP versus the direct NKP decomposition, using the precise filter lengths appearing in the single-channel duct and multi-channel ANC experiments. revision: yes

Circularity Check

1 steps flagged

Special case β=1 reduces to NKP-NSPN by construction; bounds and simulations otherwise independent

specific steps
  1. self definitional [Abstract]
    "Notably, when β=1, the NKP-FoNSPN reduces to a new NKP-NSPN algorithm, while its non-NKP decomposition variant becomes the fractional-order NSPN (FoNSPN) algorithm."

    The reduction is obtained simply by substituting β=1 into the fractional-order stochastic gradient update; this reverts the algorithm to the standard NSPN form by definition of the fractional exponent, rather than through a separate derivation or external validation.

full rationale

The paper derives theoretical bounds for the fractional-order parameter β via stochastic gradient analysis and supports performance claims through simulations on diverse noise sources including α-stable cases. The sole potential circular element is the explicit special-case reduction when β=1, which follows directly from substituting the parameter into the update rule rather than emerging from independent analysis. No load-bearing self-citations, fitted inputs renamed as predictions, or ansatz smuggling appear in the derivation chain; central claims on misadjustment and complexity remain tied to empirical results and stated bounds.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The work extends existing adaptive filter theory with new algorithmic elements; no new physical entities are postulated and free parameters are limited to standard choices in the p-norm and fractional-order framework.

free parameters (2)
  • fractional-order parameter beta
    Theoretical bounds are derived but the specific operating value within those bounds is selected for performance in given noise conditions.
  • p-norm order
    Inherited from conventional NSPN; chosen according to noise characteristics in simulations.
axioms (1)
  • domain assumption Standard convergence assumptions for stochastic gradient descent updates in subband adaptive filters under alpha-stable noise
    Invoked when deriving theoretical bounds for beta and claiming stability for 0 < alpha <= 1.

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