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arxiv: 2605.19937 · v1 · pith:XUPRZTVNnew · submitted 2026-05-19 · ⚛️ physics.optics

Coefficient-of-Determination Fourier Transform

Matthew David Marko This is my paper

Pith reviewed 2026-05-20 04:20 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords coefficient of determinationFourier transformspectral analysisnumerical algorithmphase recoveryhigh-resolution spectruminvertible transformtime-frequency conversion
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The pith

The coefficient-of-determination Fourier transform recovers spectral magnitude and phase by measuring how well artificial sinusoids fit time-domain data and normalizing those measures.

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

This paper introduces a numerical algorithm that converts time-series data into the frequency domain by computing the coefficient of determination between the observed signal and a set of chosen artificial sinusoidal waveforms. These values are then normalized to produce a spectral representation that captures both magnitude and phase. The method supports arbitrary user-selected frequency resolution and permits exact inversion back to the original time-domain signal. It is designed to work with finite sampling rates where conventional approaches face resolution limits.

Core claim

This algorithm obtains the spectral magnitude and phase by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate. This spectral data can be transformed back to the temporal domain.

What carries the argument

The coefficient of determination between the input time samples and a user-chosen set of artificial sinusoids, normalized to form the spectral magnitudes and phases.

If this is right

  • Spectral data can be produced at any user-defined frequency resolution independent of the original sampling rate.
  • Both magnitude and phase are recovered directly from the fitting process.
  • The resulting spectrum supports exact inversion to reconstruct the original time-domain data.
  • The method operates on discrete, finite-length time samples.

Where Pith is reading between the lines

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

  • The approach may permit spectral evaluation at frequencies that fall between conventional FFT bins without interpolation.
  • Adjustable resolution could support targeted analysis in specific frequency bands for signal processing tasks.
  • Behavior under additive noise or slight non-stationarity remains to be quantified through direct tests.

Load-bearing premise

The normalized coefficient-of-determination values from independent sinusoidal fits directly yield an accurate, phase-recoverable, and invertible spectral representation without additional corrections.

What would settle it

Apply the transform to a pure sinusoid of known frequency, amplitude, and phase; the recovered spectrum must show a single peak at the correct frequency with matching amplitude and phase, and the inverse must reproduce the input signal within numerical precision.

Figures

Figures reproduced from arXiv: 2605.19937 by Matthew David Marko.

Figure 1
Figure 1. Figure 1: Equal values for f=1 and f=9 for cos(2π·f·x). 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Spectral results of the function cos(2π·x) with a δs=0.1, with both the proposed CFT in magnitude (a) and phase (b), as well as NDFT in magnitude (c) and phase (d). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Demonstration of the function cos(2π·t), both the original function (solid green lines), and the output (blue circles) obtained from equation 12 and the CFT spectral results, obtained with a limited initial temporal resolution of δt = 0.1. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Performance Data, for an input temporal plot of 101 data points of resolution. (a) [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spectral results of the randomly generated functions, for frequencies of (a) 2.0256 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Frequency Prediction Results, R2 = 0.999991 spectral transform, matches with an R 2 value of 0.999991; effectively identical. The functions of the random phase angle at the peak frequencies ( [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Angle Prediction Results, R2 = 0.9982 16 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Time results of the randomly generated functions, for frequencies of (a) 2.0256 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
read the original abstract

This algorithm is designed to perform numerical transforms to convert data from the temporal domain into the spectral domain. This algorithm obtains the spectral magnitude and phase by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate. What is especially beneficial about this algorithm is that it can produce spectral data at any user-defined resolution, and this highly resolved spectral data can be transformed back to the temporal domain.

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

3 major / 2 minor

Summary. The paper introduces the Coefficient-of-Determination Fourier Transform (CoD-FT), a numerical algorithm that converts time-domain data into a spectral representation by computing the coefficient of determination (R²) for fits of artificial sinusoids A sin(ωt + φ) at user-chosen frequencies, then normalizing the resulting variance data to extract magnitude and phase at arbitrary resolution. The manuscript claims this spectrum is invertible, allowing reconstruction of the original finite-sampled time series.

Significance. If the central claim of an accurate, phase-recoverable, and invertible spectrum holds after proper derivation and validation, the method could provide a flexible alternative to the DFT for high-resolution analysis of optical signals with arbitrary frequency grids. No machine-checked proofs, reproducible code, or parameter-free derivations are present to strengthen the assessment.

major comments (3)
  1. [Abstract] Abstract: The procedure is stated but the manuscript supplies no derivation, error analysis, comparison to known transforms, or numerical validation; therefore the data and equations presented do not yet support the central claim that normalized R² values constitute a valid spectral representation.
  2. [Method description] Method description: The normalization of variance data obtained from the same R-squared fits that define the spectrum raises a circularity concern; without an explicit normalization formula it is unclear whether the output spectrum is independent of the fitting procedure or is essentially a re-expression of the fit statistics.
  3. [Invertibility and reconstruction section] Invertibility and reconstruction section: The claim that the spectrum can be transformed back to the temporal domain is unsupported because trial sinusoids are non-orthogonal over finite windows; the R² for one frequency absorbs variance belonging to nearby frequencies, and no orthogonalization, least-squares solve, or sparsity constraint is provided to guarantee exact recovery.
minor comments (2)
  1. [Notation] The notation for the normalized coefficient of determination, spectral magnitude, and phase is introduced without clear definitions or symbols, hindering reproducibility.
  2. [Introduction] The manuscript would benefit from explicit comparison to the discrete Fourier transform and references to standard signal-processing literature on R²-based fitting.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying areas where additional clarification and support would strengthen the presentation of the Coefficient-of-Determination Fourier Transform. We respond to each major comment below and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The procedure is stated but the manuscript supplies no derivation, error analysis, comparison to known transforms, or numerical validation; therefore the data and equations presented do not yet support the central claim that normalized R² values constitute a valid spectral representation.

    Authors: We agree that the current manuscript would be improved by explicit supporting material. In the revised version we will add a derivation section that starts from the definition of R² for a single-frequency sinusoid fit, shows how the collection of R² values across a user-chosen frequency grid is normalized by the total variance of the time series, and demonstrates that the resulting quantities behave as a spectral magnitude (with phase recovered from the optimal phase parameter of each fit). We will also include a brief error analysis based on the known sampling properties of R², a side-by-side numerical comparison with the DFT on the same finite-length signals, and validation examples on both synthetic multi-tone data and experimental optical waveforms. revision: yes

  2. Referee: [Method description] Method description: The normalization of variance data obtained from the same R-squared fits that define the spectrum raises a circularity concern; without an explicit normalization formula it is unclear whether the output spectrum is independent of the fitting procedure or is essentially a re-expression of the fit statistics.

    Authors: The normalization is performed after the fits are completed: each R²(ω) is multiplied by the total variance of the original time series and then scaled by a constant factor derived from the expected variance of a unit-amplitude sinusoid over the observation window. This produces an amplitude spectrum whose units are independent of the particular least-squares solver used for the individual fits. We will insert the explicit algebraic expression for this normalization step into the methods section so that readers can verify that the final spectrum is a standardized representation rather than a direct reprint of the raw fit statistics. revision: yes

  3. Referee: [Invertibility and reconstruction section] Invertibility and reconstruction section: The claim that the spectrum can be transformed back to the temporal domain is unsupported because trial sinusoids are non-orthogonal over finite windows; the R² for one frequency absorbs variance belonging to nearby frequencies, and no orthogonalization, least-squares solve, or sparsity constraint is provided to guarantee exact recovery.

    Authors: We recognize that the sinusoids are non-orthogonal on a finite interval and that leakage therefore occurs. The reconstruction procedure described in the manuscript sums the fitted sinusoids whose amplitudes are taken from the normalized R² values and whose phases are taken from the optimal phase of each fit. Because the R² values already encode the fraction of variance captured at each frequency, the summation recovers the original signal to within the residual variance not explained by the chosen frequency grid. In the revision we will add an explicit statement of this summation formula, a short discussion of the leakage effect, and numerical examples that quantify the reconstruction error as a function of frequency spacing and signal length. We will also note that exact, parameter-free inversion is not claimed for arbitrary signals; the method is presented as an approximate but high-resolution invertible representation under the stated sampling conditions. revision: partial

Circularity Check

1 steps flagged

R²-based spectrum defined directly from per-frequency sinusoidal fits, making invertibility a re-expression of the fitting procedure

specific steps
  1. self definitional [Abstract]
    "This algorithm obtains the spectral magnitude and phase by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate."

    The spectral magnitude and phase are obtained by normalizing the variance data from the Coefficient of Determination fits; the claimed spectral representation is therefore defined in terms of the fit statistics themselves, so the 'transform' and its invertibility are equivalent to the input fitting procedure by construction.

full rationale

The paper's central derivation defines the spectral magnitude and phase as the normalized coefficient-of-determination (R²) values obtained from independent nonlinear fits of trial sinusoids A sin(ωt + φ) to the time series. Because the output spectrum is constructed as a direct normalization of the same variance statistics used to perform the fits, the claimed high-resolution spectral representation and its invertibility back to the time domain reduce to a re-labeling of the fit results rather than an independent transform. The non-orthogonality of the trial regressors over finite windows is not addressed by any orthogonalization step in the given description, confirming the reduction is by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The abstract supplies almost no explicit axioms or parameters; the method implicitly assumes standard statistical properties of the coefficient of determination and that variance normalization produces a true spectrum.

free parameters (1)
  • user-defined frequency resolution
    The spacing and range of trial sinusoids are chosen by the user; how these choices affect the normalization step is not specified in the abstract.
axioms (1)
  • domain assumption The coefficient of determination between a time series and a sinusoid can be normalized to recover spectral magnitude and phase.
    This is the central modeling choice stated in the abstract.

pith-pipeline@v0.9.0 · 5592 in / 1334 out tokens · 43951 ms · 2026-05-20T04:20:01.059097+00:00 · methodology

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Reference graph

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