Pith. sign in

REVIEW 3 major objections 4 minor 49 references

This paper claims that a phase-based algebraic estimator built from the ratio of two consecutive Cauchy-Paul wavelet transforms recovers the instantaneous frequency of a linear chirp exactly at any scale, and uses this to decode non-sinusoi

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 21:48 UTC pith:H63BECZA

load-bearing objection The order-ratio estimator is a clean, useful specialization of phase-derivative IF methods, but the 'rigorously exact' chirp claim is asymptotic at best, the impulse prefactor has a slip, and the EEG demo is too thin. the 3 major comments →

arxiv 2607.15953 v1 pith:H63BECZA submitted 2026-07-17 physics.bio-ph

Cauchy-Paul wavelet transforms revisited: A framework for intermittent non-sinusoidal oscillations

P. Argoul , J. Taillard , F. Argoul This is my paper
classification physics.bio-ph MSC 42C4094A12
keywords Cauchy-Paul waveletcontinuous wavelet transforminstantaneous frequency estimationnon-sinusoidal oscillationssleep spindlesEEGchirpphase-based estimator
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper rebuilds the Cauchy-Paul wavelet with a physical time scale so that the transform carries consistent dimensions, then proves that three natural frequency references—peak, centroid, and energy-weighted—are strictly ordered because the wavelet spectrum is positively skewed. Its central move is a phase-based instantaneous-frequency estimator that replaces phase unwrapping with an algebraic ratio of two consecutive-order wavelet transforms. For linear chirps the estimator is claimed to be exact at every scale, because the chirp-induced error term is purely imaginary and is killed by taking the real part. The authors validate the estimator on synthetic transients, harmonics, and chirps, then use it to isolate sleep spindles in real EEG, arguing the wavelet's asymmetric spectrum matches the non-sinusoidal shape of spindles.

Core claim

The discovery is the estimator f^(3)_I(b,a) = (m+1)/a Re(W_{m+1}/W_m) (for τ=1, L1 normalization): a single algebraic ratio of the CWT at orders m and m+1 that yields the instantaneous frequency without ridge optimization or phase unwrapping. The paper derives it from the identity ∂_b W_m = (2iπ/(aτ))(A_m/A_{m+1}) W_{m+1}, which turns the phase derivative into a ratio of two transforms of the same signal. For a linear chirp, a second-order stationary-phase calculation gives a structural error term that is purely imaginary, so the real-part operator removes it and f_I(b,a)=f_i(b) on every scale; for nonlinear chirps exactness degrades to an asymptotic result along the ridge, and cross-compone

What carries the argument

The recurrence identity of the Cauchy-Paul family: differentiating the mother wavelet with respect to the translation parameter is equivalent to computing the CWT at the next integer order m+1, up to a known constant (Eq. 26). This maps the phase derivative to the ratio W_{m+1}/W_m, and the real part of that ratio gives the instantaneous frequency directly. The paper's second piece of machinery is the stationary-phase expansion of the CWT for a chirp, which shows that the finite-chirp correction to the ratio is imaginary to leading order, enabling the exactness claim.

Load-bearing premise

The proof of chirp exactness rests on a second-order stationary-phase approximation whose neglected higher-order terms are asserted, without proof, to have zero real part; if they contribute a real component, the estimator develops a scale-dependent bias.

What would settle it

Run the f^(3)_I estimator on a linear chirp with chirp rate α large enough that α·a·τ/f_i is of order 1 at two different scales a and a' (same time b), using high-accuracy numerical quadrature rather than the paper's asymptotic formula. If the two estimates differ by more than numerical precision, or if the real part of the error term ε in Eq. (54) is nonzero, the claim of exactness at any scale is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For any signal modeled as a linear chirp over short windows, the instantaneous frequency can be read off a single CWT scale slice, without argmax ridge search, making the estimator computationally cheap and scale-independent.
  • Raising the wavelet order m sharpens frequency selectivity and suppresses inter-component interference exponentially fast (O(e^{-m χ})), so overlapping rhythms such as spindle and delta bands can be separated by increasing m.
  • The strict ordering Q^(0) < Q^(1) < Q^(2) means the choice of pseudo-frequency mapping (peak vs centroid vs energy) changes the effective time-frequency localization; this choice must be reported and standardized for reproducible spindle analyses.
  • Because the chirp-bias term is imaginary, the same two CWT slices encode both instantaneous frequency (real part) and chirp rate (imaginary part), opening a potential joint estimator for both quantities.
  • On real N2-stage EEG, the estimator produces flat phase-velocity plateaus near 12.5 Hz and near 2.8 Hz that identify genuine oscillatory components regardless of the analysis scale, as opposed to noise which follows the identity line.

Where Pith is reading between the lines

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

  • The exactness at any scale likely does not survive beyond linear chirps: the paper's own Section 5.1.6 shows nonlinear chirps reintroduce a real error part. A sharp next step would be to compute the explicit next-order term in the stationary-phase expansion for a cubic phase and verify that the scale-dependent bias appears at O(α' / ω^2), giving a quantitative bound on when 'exact' fails.
  • The spectral-asymmetry match between the Cauchy-Paul wavelet and spindle waveforms is argued qualitatively; a controlled test would compare detection sensitivity and phase-error statistics against symmetric Gaussian-windowed wavelet approaches on the same labeled spindle dataset, since the paper reports only one subject and one electrode.
  • The dimensional consistency argument suggests that τ can be fixed by the physical sampling context; this could allow cross-study comparisons of spindle chirp slopes if researchers adopt the same τ and frequency-reference convention.
  • The imaginary part of W_{m+1}/W_m is proportional to the chirp rate (Eq. 56); although the paper does not develop it, this suggests a direct chirp-rate estimator from the same two slices, which could be validated on synthetic chirps before application to spindle deceleration.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper revisits the Cauchy-Paul (Klauder) wavelet transform, introducing a time scale τ into the frequency-domain definition to achieve dimensional consistency, and derives three reference frequencies (peak, centroid, energy-weighted) with associated quality factors. The main methodological contribution is an algebraic instantaneous-frequency estimator f_I^(3)(b,a) = (m+1)/a Re(W_{m+1}/W_m) (Eq. 40) that avoids phase unwrapping. The authors claim that this estimator tracks the instantaneous frequency of a linear chirp 'rigorously exactly' at any scale (Section 5.1.3, Eq. 57), validate it on synthetic impulses, sinusoids, chirps, and multicomponent signals, and apply it to sleep-spindle EEG recordings.

Significance. If the exact-chirp claim were rigorously established, the estimator would be a computationally attractive tool for time-frequency analysis of non-stationary, non-sinusoidal signals, particularly in EEG. The paper provides useful analytical expressions for wavelet characteristics (Section 2, Tables 1-2), a parameter-free derivation of the order-ratio estimator for pure sinusoids, and a plausible exponential-decay bound for cross-component interference (Section 4.2, Eq. 35). The pure-sinusoid case is exact by construction, and the multi-component analysis is sensible. However, the central claim of 'rigorously exact' chirp tracking rests on an unverified stationary-phase cancellation, and one of the validation equations contains a factor-of-2π error. These issues undermine the confidence in the headline result and require substantial revision.

major comments (3)
  1. [§5.1.3, Eqs. (53)–(57), Table 5] The claim that f_I^(3)(b,a)=f_i(b) 'rigorously exactly' for a linear chirp at any arbitrary scale is not established. The derivation uses a second-order stationary-phase approximation around f_s = f_i(b)+i aτ α/(2π), and the structural error ε is asserted to be purely imaginary without bounding the neglected higher-order terms or the boundary contributions from the bounded chirp. The stationary-phase expansion is asymptotic, so Eq. (57) is at best an asymptotic result, not an identity. The authors' own §5.1.6 acknowledges that residual real parts appear for non-linear chirps; the same mechanism leaves uncontrolled corrections for linear chirps. A rigorous error bound or an exact evaluation (e.g., via the Fresnel/parabolic-cylinder form of the linear-chirp CWT) is needed to support the exactness claim, or the claim must be weakened.
  2. [§5.1.1, Eq. (48)] The impulse-signal estimator contains a spurious 1/(2π) factor. From the general estimator (40) and the ratio W_{m+1}/W_m = a/(a-2iπb) (Eq. 45), the result should be f_I^(3)(b,a) = (m+1)a/(a^2+4π^2 b^2), with f_I^(3)(0,a)=(m+1)/a. Equation (48) instead gives (m+1)a/[2π(a^2+4π^2 b^2)], differing by 1/(2π). This inconsistency indicates a re-derivation of the prefactors is needed, and it casts doubt on the algebraic accuracy of the surrounding derivations.
  3. [§5.1.3, Fig. 3 (right column)] The numerical validation of the chirp exactness is partly circular: the same stationary-phase approximation used to derive the estimator is also used to evaluate the CWT integrals that the numerical implementation presumably follows. The agreement displayed in Fig. 3 therefore does not independently confirm exactness. An independent test—e.g., high-accuracy numerical quadrature of the CWT, a non-asymptotic expression, or a comparison over a wide range of chirp rates α and scales a—is required to substantiate the 'rigorously exact' claim.
minor comments (4)
  1. [§5.1.1, text after Eq. (48)] Typo: 'yields the maximum maximum value' should be 'yields the maximum value'.
  2. [Fig. 2 caption] The caption lists '(d) Temporal bandwidth' and then '(d) Temporal and spectral bandwidths product'; the second should likely be '(e)'.
  3. [§6, Q-factor definition] In Section 6 the quality factor is defined as Q=f[0]/σ_f, whereas Section 2.5 and Table 2 define Q[0]=f[0]/Δf[0] with Δf[0]=2σ_f. This factor-of-2 discrepancy should be reconciled or explicitly stated as a convention change.
  4. [§2.3, Table 2] The centroid frequency is called the 'first moment of the spectral energy density' but the defined quantity uses |ψ̂|^2 as the weight; this is the mean of the energy distribution, not the first moment of the amplitude. Clarify the terminology.

Circularity Check

0 steps flagged

No circularity: the algebraic estimator is derived from exact CWT identities; the chirp-exactness issue is an unproven asymptotic-bound concern, not a self-referential fit.

full rationale

The derivation chain is self-contained. Eq. (40) follows from the exact differentiation identity ∂_b W_m = (2iπ/(aτ))(A_m/A_{m+1}) W_{m+1} and the phase-derivative estimator (18)-(23); nothing is fitted. For a pure sinusoid, Eqs. (49) and (52) give W_{m+1}/W_m = aν0/(m+1), so f^(3)_I = ν0 is an algebraic consequence of the CWT definition, not an input. The linear-chirp claim (Eqs. 53-57) is an asymptotic stationary-phase calculation: ε is read off the same expansion and claimed purely imaginary, but the paper does not bound the neglected higher-order terms; this is a rigor/correctness gap, which the paper itself acknowledges for non-linear chirps in Sec. 5.1.6, not a circularity, because ε is not a fitted parameter and Eq. (57) is not used to define ε. Self-citations (refs. 4, 27, 32) provide historical/contextual support, but the core algebra is re-derived in the manuscript, so they are not load-bearing. The EEG plateau (Sec. 6, Fig. 7) is an external-data illustration, not a parameter fitted to make the method succeed. The internal inconsistency in Eq. (48) (extra 1/(2π) relative to Eq. (40)) is an algebraic typo to correct, but does not amount to a prediction reducing to its input by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities. The free parameters are the wavelet order m and the time scale τ, both chosen by hand rather than fitted. The main hidden assumptions are the multicomponent analytic-signal model and the validity of the stationary-phase approximation for the chirp-exactness claim.

free parameters (3)
  • wavelet order m = 4, 32, 64, 128 (chosen by hand in applications)
    The order m tunes the quality factor and time-frequency trade-off; the paper selects different m values for different analysis goals (m=4 for time, m=32 for frequency) without a data-driven selection criterion.
  • time scale τ = 1 (fixed after being introduced)
    τ is introduced to make arguments dimensionless and then set to 1 in all applications, so it merely rescales the frequency axis.
  • target reference frequency in EEG (ν0) = 12 Hz
    The EEG analysis extracts frequency lines at 12 Hz; this is chosen from the known spindle band (11-16 Hz) as a detection target, not fitted to the data.
axioms (4)
  • domain assumption Multicomponent signal model: the analyzed signal is a linear sum of slowly-modulated analytic components with well-separated instantaneous frequencies (Eq. 30).
    The estimator's behavior for multi-component signals and the interference-decay analysis assume this model; real EEG may deviate strongly from separated analytic modes.
  • domain assumption Second-order stationary-phase approximation is accurate enough to neglect higher-order real terms for linear chirps (Eqs. 53-57).
    The 'exact' chirp tracking claim rests on this approximation; the paper does not bound the next-order terms, so the approximation is assumed valid.
  • standard math Standard Gamma-integral identities, the König-Huygens theorem, and the Hilbert transform analytic-signal framework hold.
    These are classical results invoked without proof in Sections 2, 3, and the supplementary material.
  • domain assumption The wavelet order m is an integer greater than 1/2 so that factorial expressions Γ(2m+1)=(2m)! apply.
    The paper uses integer factorial identities in the normalization and moment formulas; the analysis does not generalize to non-integer m without modification.

pith-pipeline@v1.3.0-alltime-deepseek · 33177 in / 14456 out tokens · 151276 ms · 2026-08-01T21:48:53.339642+00:00 · methodology

0 comments
read the original abstract

This paper revisits the continuous wavelet transform framework by establishing a rigorous physical and dimensional formulation of the Cauchy-Paul mother wavelet, tailored specifically for intermittent, non-sinusoidal electrophysiological oscillations. Departing from conventional, purely mathematical definitions, we introduce a characteristic time scale $\tau$ into the frequency-domain formulation of the mother wavelet. This parameter ensures strict dimensional consistency by maintaining dimensionless functional arguments, thereby confining the physical dimension solely to the multiplicative normalization constant under both $L^1(\mathbb{R})$ and $L^2(\mathbb{R})$ norms. A sharp dimensional and structural analysis of the resulting wavelet is conducted. We demonstrate that the spectral asymmetry inherent to the Cauchy-Paul wavelet dictates a strict mathematical hierarchy between three alternative reference frequencies: the peak ($L^\infty$), the centroid ($L^1$), and the energy-weighted ($L^2$) frequencies. Each frequency definition yields distinct quality factors ($Q$) and time-bandwidth characteristics that govern the time-frequency localization trade-off. To track multi-component EEG sleep pattern signals, a phase-based algebraic estimator is deployed alongside an advanced ridge-extraction method. The robust tracking performance and morphological adaptability of the proposed Cauchy-Paul framework are first numerically validated on synthetic transients, harmonics, and chirps, and subsequently applied to real, non-stationary EEG recordings to successfully isolate and decipher the non-sinusoidal signatures of sleep spindles.

Figures

Figures reproduced from arXiv: 2607.15953 by F. Argoul, J. Taillard, P. Argoul.

Figure 1
Figure 1. Figure 1: Representations of the Cauchy-Paul mother wavelet in time 𝜓𝑚 (𝑡) and in frequency ̂𝜓𝑚 (𝑓) for increasing values of 𝑚 and 𝜏 = 1. Gray lines: | |𝜓𝑚 (𝑡) | | , black lines: ℜ(𝜓𝑚 (𝑡)), dashed lines: ℑ(𝜓𝑚 (𝑡)). The temporal Δ𝑡𝜓𝑚 and spectral Δ𝑓 [0] 𝜓𝑚 bandwidths are reported with red lines. The red (blue) dots correspond to 𝑓 [0] 𝜓𝑚 and 𝑓 [1] 𝜓𝑚 respectively. The 𝑄 values are computed with the relation 𝑄[0] = 𝑓 … view at source ↗
Figure 2
Figure 2. Figure 2: Representation of the variations of the different Cauchy-Paul wavelet spectral and temporal characteristics with 𝑚 (and 𝜏 = 1). (a) Central frequency 𝑓 [𝑐] 𝜓𝑚 . (b) Spectral bandwidth Δ𝑓 [𝑐] 𝜓𝑚 . (c) Quality factor 𝑄. (d) Temporal bandwidth Δ𝑡𝜓𝑚 . (d) Temporal and spectral bandwidths product Δ𝑡𝜓𝑚 Δ𝑓 [𝑐] 𝜓 . (e) Estimated number of temporal oscillations 𝑁𝑜𝑠𝑐 = Δ𝑡𝜓𝑚 𝑓 [𝑐] 𝜓𝑚 . The three centering methods are… view at source ↗
Figure 3
Figure 3. Figure 3: Time-frequency analysis and direct frequency tracking of a single-component bounded harmonic signal (left) and single-component bounded linear chirp signal (right) using the Cauchy-Paul CWT with 𝑚 = 32 and 𝜏 = 1. (a) Time-domain representation of a bounded 12 Hz harmonic signal 𝑥(𝑡). (b) Scalogram displaying log10(|𝜓𝑚 [𝑥]|). (c) Wavelet phase distribution Φ(𝜓𝑚 [𝑥]). (d) Local instantaneous frequency dist… view at source ↗
Figure 4
Figure 4. Figure 4: Time-frequency analysis and direct frequency tracking of a two-component bounded signal (sum of two sines) using the Cauchy-Paul CWT with 𝜏 = 1, 𝑚 = 32 (left) and 𝑚 = 128 (right). (a) Time history of the analyzed bi-harmonic signal 𝑥(𝑡). (b) Scalogram displaying the log-magnitude distribution log10 |𝜓𝑚 [𝑥]| in the time-frequency plane (log2 (𝑓) vs. Time), with the black horizontal lines indicating the 12 … view at source ↗
Figure 5
Figure 5. Figure 5: Frequency profiles of wavelet transform modulus and instantaneous frequency for a sine function (for 𝑏 = 8 s) (left column) and for a sum of two sine functions (right column). (a) log10 |𝜓𝑚 [𝑥]|(𝑓). (b) 𝑓𝐼 (𝑓). 5.1.6. Some remarks on the behavior of the estimator under non-linear chirps For a complex non-linear chirp, the higher-order time-derivatives of the instantaneous frequency do not vanish. Conseque… view at source ↗
Figure 6
Figure 6. Figure 6: Time-frequency analysis and instantaneous frequency tracking of a non-stationary sleep EEG recording (Stage N2) using the Cauchy-Paul continuous wavelet transform (CWT) with order 𝑚 = 4 (left) and 𝑚 = 32 (right). The continuous analysis window spans from 𝑡min = 14095 s to 𝑡max = 14120 s. (a) Raw EEG signal 𝑥(𝑡) displaying characteristic micro-events and transient sleep spindles. (b) Scalogram representatio… view at source ↗
Figure 7
Figure 7. Figure 7: Frequency profiles of wavelet transform modulus and instantaneous frequency extracted from the EEG signal of [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

49 extracted references · 24 canonical work pages

  1. [1]

    TheIllustratedWaveletTransformHandbook:IntroductoryTheoryandApplicationsinScience,Engineering,Medicine and Finance

    Addison,P.S.,2002. TheIllustratedWaveletTransformHandbook:IntroductoryTheoryandApplicationsinScience,Engineering,Medicine and Finance. Institute of Physics Publishing, Bristol and Philadelphia. doi:10.1201/9781420033397

    work page doi:10.1201/9781420033397 2002

  2. [2]

    Sleep Spindles in Humans: Insights from Intracranial EEGandUnitRecordings

    Andrillon, T., Nir, Y., Staba, R.J., Ferrarelli, F., Cirelli, C., Tononi, G., Fried, I., 2011. Sleep Spindles in Humans: Insights from Intracranial EEGandUnitRecordings. TheJournalofNeuroscience31,17821–17834. URL:https://www.jneurosci.org/lookup/doi/10.1523/ JNEUROSCI.2604-11.2011, doi:10.1523/JNEUROSCI.2604-11.2011

    work page doi:10.1523/jneurosci.2604-11.2011 2011

  3. [3]

    Use of the continuous wavelet transform for processing seismic structural responses, Rhodes Island, Greece

    Argoul, P., Fragiadakis, M., Argoul, F., 2025. Use of the continuous wavelet transform for processing seismic structural responses, Rhodes Island, Greece. pp. 4827–4847. URL:https://www.eccomasproceedia.org/conferences/thematic-conferences/ compdyn-2025/12777, doi:10.7712/120125.12777.27734

    arXiv 2025

  4. [4]

    Instantaneous indicators of structural behavior based on Cauchy wavelet analysis

    Argoul, P., Le, T.P., 2003. Instantaneous indicators of structural behavior based on Cauchy wavelet analysis. Mechanical Systems and Signal Processing 17, 243–250. URL:https://linkinghub.elsevier.com/retrieve/pii/S0888327002915578, doi:10.1006/ mssp.2002.1557

    arXiv 2003

  5. [5]

    Oscillatory-quality of sleep spindles links brain state with sleep regulation and function

    Blanco-Duque,C.,Bond,S.A.,Krone,L.B.,Dufour,J.P.,Gillen,E.C.P.,Purple,R.J.,Kahn,M.C.,Bannerman,D.M.,Mann,E.O.,Achermann, P., Olbrich, E., Vyazovskiy, V.V., 2024. Oscillatory-quality of sleep spindles links brain state with sleep regulation and function. Science Advances 10, eadn6247. URL:https://www.science.org/doi/10.1126/sciadv.adn6247, doi:10.1126/scia...

    work page doi:10.1126/sciadv.adn6247 2024

  6. [6]

    Estimating and interpreting the instantaneous frequency of a signal

    Boashash, B., 1992. Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals. IEEE Proceedings 80, 520–538. doi:10.1109/5.135376

    doi:10.1109/5.135376 1992

  7. [7]

    Fourier, hilbert and wavelet-based signal analysis: are they really different approaches? Journal of Neuroscience Methods 137, 321–332

    Bruns, A., 2004. Fourier, hilbert and wavelet-based signal analysis: are they really different approaches? Journal of Neuroscience Methods 137, 321–332. doi:10.1016/j.jneumeth.2004.03.002

    work page doi:10.1016/j.jneumeth.2004.03.002 2004

  8. [8]

    Bicoherence of intracranial EEG in sleep, wakefulness and seizures

    Bullock, T., Achimowicz, J., Duckrow, R., Spencer, S., Iragui-Madoz, V., 1997. Bicoherence of intracranial EEG in sleep, wakefulness and seizures. ElectroencephalographyandClinicalNeurophysiology103,661–678. URL:https://linkinghub.elsevier.com/retrieve/ pii/S0013469497000874, doi:10.1016/S0013-4694(97)00087-4

    work page doi:10.1016/s0013-4694(97)00087-4 1997

  9. [9]

    PracticalTime-FrequencyAnalysis:GaborandWaveletTransforms,withanImplementation in S

    Carmona,R.,Hwang,W.L.,Torrésani,B.,1998. PracticalTime-FrequencyAnalysis:GaborandWaveletTransforms,withanImplementation in S. Academic Press

    1998

  10. [10]

    Onthestationaryphaseapproximationofchirpspectra,in:ProceedingsoftheIEEE-SPInternational Symposium on Time-Frequency and Time-Scale Analysis, IEEE, Pittsburgh, PA, USA

    Chassande-Mottin,É.,Flandrin,P.,1998. Onthestationaryphaseapproximationofchirpspectra,in:ProceedingsoftheIEEE-SPInternational Symposium on Time-Frequency and Time-Scale Analysis, IEEE, Pittsburgh, PA, USA. pp. 117–120. doi:10.1109/TFSA.1998.721375

    arXiv 1998

  11. [11]

    Individualized temporal patterns drive human sleep spindle timing

    Chen, S., He, M., Brown, R.E., Eden, U.T., Prerau, M.J., 2025. Individualized temporal patterns drive human sleep spindle timing. Proceedings of the National Academy of Sciences 122, e2405276121. URL:https://pnas.org/doi/10.1073/pnas.2405276121, doi:10.1073/pnas.2405276121

    work page doi:10.1073/pnas.2405276121 2025

  12. [12]

    font-variant:small-caps

    Coelho, J., Degros, H., Micoulaud-Franchi, J., Sagaspe, P., d’Incau, E., Galvez, P., Berthomier, C., Philip, P., Taillard, J., 2025. Threshold Values of Sleep Spindles Features in Healthy Adults Using Scalp- <span style="font-variant:small-caps;">EEG</span> and Associations With Sleep Parameters. Annals of Clinical and Translational Neurology 12, 1276–129...

    work page doi:10.1002/acn3.70055 2025

  13. [13]

    A better way to define and describe Morlet wavelets for time-frequency analysis

    Cohen, M.X., 2019. A better way to define and describe Morlet wavelets for time-frequency analysis. NeuroImage 199, 81–86. URL: https://linkinghub.elsevier.com/retrieve/pii/S1053811919304409, doi:10.1016/j.neuroimage.2019.05.048

    work page doi:10.1016/j.neuroimage.2019.05.048 2019

  14. [14]

    Brain Oscillations and the Importance of Waveform Shape

    Cole, S.R., Voytek, B., 2017. Brain Oscillations and the Importance of Waveform Shape. Trends in Cognitive Sciences 21, 137–149. URL: https://linkinghub.elsevier.com/retrieve/pii/S1364661316302182, doi:10.1016/j.tics.2016.12.008

    work page doi:10.1016/j.tics.2016.12.008 2017

  15. [15]

    Cycle-by-cycle analysis of neural oscillations

    Cole, S.R., Voytek, B., 2019. Cycle-by-cycle analysis of neural oscillations. Journal of Neurophysiology 122, 846–861

    2019

  16. [16]

    Sleep spindles: an overview

    De Gennaro, L., Ferrara, M., 2003. Sleep spindles: an overview. Sleep Medicine Reviews 7, 423–440. URL:https://linkinghub. elsevier.com/retrieve/pii/S1087079202902522, doi:10.1053/smrv.2002.0252. P. Argoul et al.:Preprint submitted to ElsevierPage 23 of 26 Cauchy-Paul wavelet revisited

    arXiv 2003

  17. [17]

    Sleep spindles comprise a subset of a broader class of electroencephalogram events

    Dimitrov, T., He, M., Stickgold, R., Prerau, M.J., 2021. Sleep spindles comprise a subset of a broader class of electroencephalogram events. SLEEP 44, zsab099. URL:https://academic.oup.com/sleep/article/doi/10.1093/sleep/zsab099/6226812, doi:10.1093/ sleep/zsab099

    work page doi:10.1093/sleep/zsab099/6226812 2021

  18. [18]

    AdaptiveSingle-ChannelEEGArtifactRemovalWithApplicationstoClinicalMonitoring

    Dora,M.,Holcman,D.,2022. AdaptiveSingle-ChannelEEGArtifactRemovalWithApplicationstoClinicalMonitoring. IEEETransactions on Neural Systems and Rehabilitation Engineering 30, 286–295. URL:https://ieeexplore.ieee.org/document/9694664/, doi:10. 1109/TNSRE.2022.3147072

    arXiv 2022

  19. [19]

    Sleep Spindles: Mechanisms and Functions

    Fernandez, L.M.J., Lüthi, A., 2020. Sleep Spindles: Mechanisms and Functions. Physiological Reviews 100, 805–868. URL:https: //journals.physiology.org/doi/10.1152/physrev.00042.2018, doi:10.1152/physrev.00042.2018

    arXiv 2020

  20. [20]

    Time-Frequency/Time-Scale Analysis

    Flandrin, P., 1999. Time-Frequency/Time-Scale Analysis. volume 10. Academic Press, San Diego

    1999

  21. [21]

    Fourier and wavelet spectrograms seen as smoothed Wigner-Ville distributions, in: Wavelets and Applications, Masson, Marseille France

    Flandrin, P., Vidalie, B., Rioul, O., 1992. Fourier and wavelet spectrograms seen as smoothed Wigner-Ville distributions, in: Wavelets and Applications, Masson, Marseille France. pp. 93–103

    1992

  22. [22]

    Decomposition of Hardy functions into square integrable wavelets of constant shape

    Grossmann, A., Morlet, J., 1984. Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM Journal on Mathematical Analysis 15, 723–736

    1984

  23. [23]

    Extraction of instantaneous frequencies from ridges in time–frequency representations of signals

    Iatsenko, D., McClintock, P., Stefanovska, A., 2016. Extraction of instantaneous frequencies from ridges in time–frequency representations of signals. Signal Processing 125, 290–303. URL:https://linkinghub.elsevier.com/retrieve/pii/S0165168416000451, doi:10.1016/j.sigpro.2016.01.024

    work page doi:10.1016/j.sigpro.2016.01.024 2016

  24. [24]

    The New Field of Network Physiology: Building the Human Physiolome

    Ivanov, P.C., 2021. The New Field of Network Physiology: Building the Human Physiolome. Frontiers in Network Physiology 1, 711778. URL:https://www.frontiersin.org/articles/10.3389/fnetp.2021.711778/full, doi:10.3389/fnetp.2021.711778

    arXiv 2021

  25. [25]

    Instantaneous frequencyestimationbasedon synchrosqueezingwavelettransform

    Jiang,Q.,Suter, B.W.,2017. Instantaneous frequencyestimationbasedon synchrosqueezingwavelettransform. SignalProcessing138, 167–

    2017

  26. [26]

    Wavelets from coherent states: II, in: Wavelet Applications VI, SPIE

    Klauder, J.R., 1999. Wavelets from coherent states: II, in: Wavelet Applications VI, SPIE. pp. 2–10. doi:10.1117/12.342924

    work page doi:10.1117/12.342924 1999

  27. [27]

    Continuous wavelet transform for modal identification using free decay response

    Le, T.P., Argoul, P., 2004. Continuous wavelet transform for modal identification using free decay response. Journal of Sound and Vibration 277,73–100.URL:https://linkinghub.elsevier.com/retrieve/pii/S0022460X03012124,doi:10.1016/j.jsv.2003.08.049

    work page doi:10.1016/j.jsv.2003.08.049 2004

  28. [28]

    Spindle oscillations emerge at the critical state of electrically coupled networks in the thalamic reticular nucleus

    Li, S., Wang, C., Wu, S., 2024. Spindle oscillations emerge at the critical state of electrically coupled networks in the thalamic reticular nucleus. Cell Reports 43, 114790. URL:https://linkinghub.elsevier.com/retrieve/pii/S2211124724011410, doi:10.1016/ j.celrep.2024.114790

    arXiv 2024

  29. [29]

    Practical Approach to Electroencephalography

    Libenson, M.H., 2025. Practical Approach to Electroencephalography. Ii ed., Elsevier - Health Sciences Division, Philadelphia, PA

    2025

  30. [30]

    Elementanalysis:awavelet-basedmethodforanalysingtime-localizedeventsinnoisytimeseries

    Lilly,J.M.,2017. Elementanalysis:awavelet-basedmethodforanalysingtime-localizedeventsinnoisytimeseries. ProceedingsoftheRoyal Society A: Mathematical, Physical and Engineering Sciences 473, 20160776. URL:https://royalsocietypublishing.org/doi/10. 1098/rspa.2016.0776, doi:10.1098/rspa.2016.0776

    arXiv 2017

  31. [31]

    A wavelet tour of signal processing: the sparse way

    Mallat, S., 2009. A wavelet tour of signal processing: the sparse way. 3rd ed ed., Elsevier/Academic Press, Amsterdam Boston

    2009

  32. [32]

    Continuous Cauchy Wavelet Transform of XAFS Spectra

    Munoz, M., Farges, F., Argoul, P., 2005. Continuous Cauchy Wavelet Transform of XAFS Spectra. Physica Scripta T115, 221. URL: https://iopscience.iop.org/article/10.1238/Physica.Topical.115a00221, doi:10.1238/Physica.Topical.115a00221

    work page doi:10.1238/physica.topical.115a00221 2005

  33. [33]

    Cross-spectral methods for processing speech

    Nelson, D.J., 2001. Cross-spectral methods for processing speech. The Journal of the Acoustical Society of America 110, 2575–2592. URL: https://asa.scitation.org/doi/10.1121/1.1402616, doi:10.1121/1.1402616

    work page doi:10.1121/1.1402616 2001

  34. [34]

    Sleep spindle detection using time-frequency sparsity, in: 2014 IEEE Signal ProcessinginMedicineandBiologySymposium(SPMB),IEEE,Philadelphia,PA,USA.pp.1–6

    Parekh, A., Selesnick, I.W., Rapoport, D.M., Ayappa, I., 2014. Sleep spindle detection using time-frequency sparsity, in: 2014 IEEE Signal ProcessinginMedicineandBiologySymposium(SPMB),IEEE,Philadelphia,PA,USA.pp.1–6. URL:http://ieeexplore.ieee.org/ document/7002965/, doi:10.1109/SPMB.2014.7002965

    arXiv 2014

  35. [35]

    DetectionofK-complexesandsleepspindles(DETOKS)usingsparseoptimiza- tion

    Parekh,A.,Selesnick,I.W.,Rapoport,D.M.,Ayappa,I.,2015. DetectionofK-complexesandsleepspindles(DETOKS)usingsparseoptimiza- tion. JournalofNeuroscienceMethods251,37–46. URL:https://linkinghub.elsevier.com/retrieve/pii/S0165027015001533, doi:10.1016/j.jneumeth.2015.04.006

    work page doi:10.1016/j.jneumeth.2015.04.006 2015

  36. [36]

    Sleep neurophysiological dynamics through the Lens of multitaper spectral analysis

    Prerau, M.J., Brown, R.E., Bianchi, M.T., Ellenbogen, J.M., Purdon, P.L., 2017. Sleep neurophysiological dynamics through the Lens of multitaper spectral analysis. Physiology 32, 60–92. URL:https://www.physiology.org/doi/10.1152/physiol.00062.2015, doi:10.1152/physiol.00062.2015

    arXiv 2017

  37. [37]

    (Eds.), 2018

    Schomer, D.L., Lopes da Silva, F.H., Niedermeyer, E. (Eds.), 2018. Niedermeyer’s electroencephalography: basic principles, clinical applications, and related fields. Seventh edition ed., Oxford University Press, New York, NY

    2018

  38. [38]

    Quantifying chirp in sleep spindles

    Schönwald, S.V., Carvalho, D.Z., Dellagustin, G., De Santa-Helena, E.L., Gerhardt, G.J., 2011. Quantifying chirp in sleep spindles. Journal of Neuroscience Methods 197, 158–164. URL:https://linkinghub.elsevier.com/retrieve/pii/S0165027011000525, doi:10.1016/j.jneumeth.2011.01.025

    work page doi:10.1016/j.jneumeth.2011.01.025 2011

  39. [39]

    Time-frequency characteristics and dynamics of sleep spindles in WAG/Rij rats with absence epilepsy

    Sitnikova, E., Hramov, A.E., Grubov, V., Koronovsky, A.A., 2014. Time-frequency characteristics and dynamics of sleep spindles in WAG/Rij rats with absence epilepsy. Brain Research 1543, 290–299. URL:https://linkinghub.elsevier.com/retrieve/pii/ S0006899313014960, doi:10.1016/j.brainres.2013.11.001

    work page doi:10.1016/j.brainres.2013.11.001 2014

  40. [40]

    Sleep spindles and spike–wave discharges in EEG: Their generic features, similarities and distinctions disclosed with Fourier transform and continuous wavelet analysis

    Sitnikova, E., Hramov, A.E., Koronovsky, A.A., Van Luijtelaar, G., 2009. Sleep spindles and spike–wave discharges in EEG: Their generic features, similarities and distinctions disclosed with Fourier transform and continuous wavelet analysis. Journal of Neuroscience Methods 180,304–316. URL:https://linkinghub.elsevier.com/retrieve/pii/S0165027009001976,doi...

    work page doi:10.1016/j.jneumeth.2009 2009

  41. [41]

    Transient oscillation dynamics during sleep provide a robust basis for electroencephalographic phenotyping and biomarker identification

    Stokes, P.A., Rath, P., Possidente, T., He, M., Purcell, S., Manoach, D.S., Stickgold, R., Prerau, M.J., 2023. Transient oscillation dynamics during sleep provide a robust basis for electroencephalographic phenotyping and biomarker identification. Sleep 46, zsac223. URL: https://academic.oup.com/sleep/article/doi/10.1093/sleep/zsac223/6701543, doi:10.1093...

    work page doi:10.1093/sleep/zsac223/6701543 2023

  42. [42]

    Sleep in Normal Aging, Homeostatic and Circadian Regulation and Vulnerability to Sleep Deprivation

    Taillard, J., Gronfier, C., Bioulac, S., Philip, P., Sagaspe, P., 2021. Sleep in Normal Aging, Homeostatic and Circadian Regulation and Vulnerability to Sleep Deprivation. Brain Sciences 11, 1003. URL:https://www.mdpi.com/2076-3425/11/8/1003, doi:10.3390/ brainsci11081003

    2021

  43. [43]

    Analyse Continue par Ondelettes

    Torresani, B., 2012. Analyse Continue par Ondelettes. EDP Sciences. P. Argoul et al.:Preprint submitted to ElsevierPage 24 of 26 Cauchy-Paul wavelet revisited

    2012

  44. [44]

    Stage-independent, single lead EEG sleep spindle detection using the continuous wavelet transform and localweightedsmoothing

    Tsanas, A., Clifford, G.D., 2015. Stage-independent, single lead EEG sleep spindle detection using the continuous wavelet transform and localweightedsmoothing. FrontiersinHumanNeuroscience9. URL:http://journal.frontiersin.org/article/10.3389/fnhum. 2015.00181/abstract, doi:10.3389/fnhum.2015.00181

    arXiv 2015

  45. [45]

    Théorie et applications de la notion de signal analytique

    Ville,J.A.,1958. TheoryandApplicationsoftheNotionofComplexSignal. TranslationT-92.RandCorporation. TranslatedfromtheFrench original: “Théorie et applications de la notion de signal analytique”,Câbles et Transmission, 2(1):61–74, 1948

    1958

  46. [46]

    Sleep-spindle detection: crowdsourcing and evaluating performance of experts, non-experts and automated methods

    Warby,S.C.,Wendt,S.L.,Welinder,P.,Munk,E.G.S.,Carrillo,O.,Sorensen,H.B.D.,Jennum,P.,Peppard,P.E.,Perona,P.,Mignot,E.,2014. Sleep-spindle detection: crowdsourcing and evaluating performance of experts, non-experts and automated methods. Nature Methods 11, 385–392. URL:https://www.nature.com/articles/nmeth.2855, doi:10.1038/nmeth.2855

    work page doi:10.1038/nmeth.2855 2014

  47. [47]

    Regulation of peripheral glucose levels during human sleep

    Yang, X., Fedumenti, F.T., Niethard, N., Hallschmid, M., Born, J., Rauss, K., 2025. Regulation of peripheral glucose levels during human sleep. SLEEP 48, zsaf042. URL:https://academic.oup.com/sleep/article/doi/10.1093/sleep/zsaf042/8030559, doi:10.1093/sleep/zsaf042

    work page doi:10.1093/sleep/zsaf042/8030559 2025

  48. [48]

    peak frequency

    Zygierewicz, J., 1999. High resolution study of sleep spindles. Clinical Neurophysiology 110, 2136–2147. URL:https://linkinghub. elsevier.com/retrieve/pii/S1388245799001753, doi:10.1016/S1388-2457(99)00175-3. P. Argoul et al.:Preprint submitted to ElsevierPage 25 of 26 Supplementary Material1 Supplementary Material Appendices and Technical Annexes for the...

    work page doi:10.1016/s1388-2457(99)00175-3 1999

  49. [181]

    URL:https://linkinghub.elsevier.com/retrieve/pii/S0165168417300907, doi:10.1016/j.sigpro.2017.03.007

    work page doi:10.1016/j.sigpro.2017.03.007 2017