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arxiv: 2211.09284 · v5 · pith:INSQFQONnew · submitted 2022-11-17 · 📡 eess.SP · cs.NA· math.NA· stat.ME

Iterative execution of discrete and inverse discrete Fourier transforms with applications for signal denoising via sparsification

H. Robert Frost This is my paper

Pith reviewed 2026-05-24 11:18 UTC · model grok-4.3

classification 📡 eess.SP cs.NAmath.NAstat.ME
keywords iterative DFTsparsificationsignal denoisingperiodic spikesGaussian noiseFourier transformsuncertainty principle
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The pith

Iterative application of sparsification to DFT and IDFT outputs recovers periodic spike signals from Gaussian noise by converging on stable real-domain sparsity patterns.

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

The paper presents a family of algorithms that repeat discrete Fourier transforms and their inverses. A key member applies a sparsification step in both the time and frequency domains, drawing on the DFT uncertainty principle, until the sparsity pattern in the real domain stabilizes. Simulations demonstrate that this process can extract periodic spikes buried in Gaussian noise and matches or exceeds the performance of other denoising techniques. A reader would care if the approach provides a practical, transform-based route to denoising without requiring explicit signal models beyond sparsity.

Core claim

The sparsification variant of the iterative DFT/IDFT family recovers a periodic spike signal in the presence of Gaussian noise, with convergence when real-domain sparsity reaches a stable pattern, and shows competitive denoising performance in simulation studies.

What carries the argument

The sparsification operation applied after each DFT and IDFT step in both domains, iterated until real-domain sparsity stabilizes.

If this is right

  • The algorithm converges when real-domain sparsity hits a stable pattern.
  • It recovers periodic spike signals embedded in Gaussian noise.
  • Denoising performance is competitive with existing methods in simulations.
  • General convergence properties hold for the iterative DFT/IDFT family.

Where Pith is reading between the lines

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

  • The same iteration structure might apply to other signals that are sparse in time but structured in frequency.
  • Automatic threshold selection rules could be derived from the observed convergence behavior.
  • The supplied R package makes direct testing on new sparse signals straightforward.

Load-bearing premise

The input signal must be sparse enough in the real domain and the sparsification thresholds must be set so that the iteration reaches a stable sparsity pattern.

What would settle it

Run the iterative sparsification procedure on a periodic spike train plus Gaussian noise and observe whether the output fails to recover the original spikes or never reaches a stable sparsity count.

Figures

Figures reproduced from arXiv: 2211.09284 by H. Robert Frost.

Figure 1
Figure 1. Figure 1: Mean iterations until convergence for random length [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mean iterations until convergence for random length 500 vectors of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Output of the IterativeFT method on x vector simulated according to the model detailed in Section 4.1. The top panels show the periodic spike signal and Gaussian noise. The remaining panels show the input data and output from the h() function after each iteration with the error relative to the spike signal captured as a dashed red line and quantified as mean squared error (MSE). 7 [PITH_FULL_IMAGE:figures… view at source ↗
Figure 4
Figure 4. Figure 4: Average MSE ratio relative to the number of cycles, b, captured in the input [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average MSE ratio after each iteration of the algorithm. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Average mean squared error (MSE) ratio relatieve to the variance of the Gaussian noise, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Output of the IterativeFT method on a matrix simulated according to Section 4.2. The top [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Output of the IterativeFT method and comparison denoising methods on a single vector [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Signal denoising performance for the uniform spike model relative to spike signal frequency. [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Signal denoising performance for the uniform spike model relative to signal-to-noise ratio. [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Output of the IterativeFT method and comparison denoising methods on a single vector [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Signal denoising performance for the alternating sign spike model relative to spike signal [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Signal denoising performance for the alternating sign spike model relative to signal-to-noise [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Output of the IterativeFT method and comparison denoising methods on a single vector [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Signal denoising performance for the alternating size spike model relative to spike signal [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Signal denoising performance for the alternating size spike model relative to signal-to-noise [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
read the original abstract

We describe a family of iterative algorithms that involve the repeated execution of discrete and inverse discrete Fourier transforms. One interesting member of this family is motivated by the discrete Fourier transform uncertainty principle and involves the application of a sparsification operation to both the real domain and frequency domain data with convergence obtained when real domain sparsity hits a stable pattern. This sparsification variant has practical utility for signal denoising, in particular the recovery of a periodic spike signal in the presence of Gaussian noise. General convergence properties and denoising performance relative to existing methods are demonstrated using simulation studies. An R package implementing this technique and related resources can be found at https://hrfrost.host.dartmouth.edu/IterativeFT.

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 manuscript presents a family of iterative algorithms based on repeated execution of the discrete Fourier transform (DFT) and inverse DFT (IDFT). One variant applies sparsification (thresholding) operations in both the real (time) and frequency domains, motivated by the DFT uncertainty principle, with the iteration asserted to converge once the sparsity pattern in the real domain reaches a stable state. This sparsification variant is proposed for denoising periodic spike signals in additive Gaussian noise. Convergence properties and denoising performance are evaluated through simulation studies, and an R package is provided for implementation.

Significance. If the convergence behavior and denoising results hold, the work offers a conceptually simple iterative procedure for recovering signals that are sparse in both domains, with the R package providing a concrete, reproducible implementation that facilitates further testing and application. The simulation-based evidence for competitive performance against existing methods is a positive aspect, though the absence of theoretical convergence guarantees or bounds on recovery error limits the broader impact relative to methods with provable recovery conditions.

major comments (2)
  1. [§2] §2 (Sparsification Variant Description): No explicit, a priori procedure is given for selecting the two sparsification thresholds or for declaring convergence independently of post-hoc observation of the final real-domain sparsity pattern. This is load-bearing for the central recovery claim, as the asserted denoising of the periodic spike train is conditional on threshold choices whose failure modes are not characterized.
  2. [§4] §4 (Simulation Studies): The reported denoising performance lacks details on threshold sensitivity, error bars across noise realizations, or explicit baseline implementations, which undermines assessment of whether the competitive results are robust or dependent on favorable post-hoc parameter choices.
minor comments (2)
  1. [Abstract] The abstract and introduction could more clearly delineate the full family of iterative DFT/IDFT algorithms versus the specific sparsification member.
  2. [§2] Notation for the sparsification operator and iteration index should be introduced consistently before first use in the algorithm description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We agree that the two major comments identify areas where the manuscript can be strengthened for clarity and reproducibility. We outline our responses below and will incorporate revisions in the next version of the manuscript.

read point-by-point responses

  1. Referee: [§2] §2 (Sparsification Variant Description): No explicit, a priori procedure is given for selecting the two sparsification thresholds or for declaring convergence independently of post-hoc observation of the final real-domain sparsity pattern. This is load-bearing for the central recovery claim, as the asserted denoising of the periodic spike train is conditional on threshold choices whose failure modes are not characterized.

    Authors: We acknowledge that the current manuscript does not supply an explicit a priori rule for choosing the two sparsification thresholds or an independent convergence criterion. In the revision we will add a dedicated subsection that specifies a practical selection heuristic based on the expected number of non-zero time-domain coefficients (derived from the known periodicity of the target spike train) together with an estimate of the noise variance; thresholds are then set to retain that number of coefficients after sorting. Convergence will be declared when the support of the real-domain vector remains unchanged over a fixed number of consecutive iterations (e.g., five). We will also include a brief discussion of failure modes, such as over-sparsification when the assumed spike count is too low. These additions will make the procedure fully specified before any post-hoc inspection of results. revision: yes

  2. Referee: [§4] §4 (Simulation Studies): The reported denoising performance lacks details on threshold sensitivity, error bars across noise realizations, or explicit baseline implementations, which undermines assessment of whether the competitive results are robust or dependent on favorable post-hoc parameter choices.

    Authors: We agree that the simulation section would benefit from greater detail on robustness. The revised manuscript will report mean and standard deviation of denoising error across at least 100 independent noise realizations for each SNR level, include a sensitivity plot showing performance as each threshold is varied by ±20 % around the nominal value, and provide explicit pseudocode or R-package function calls for every baseline method so that the comparisons can be reproduced exactly. These changes will allow readers to judge whether the reported gains are stable under reasonable parameter variation. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic iteration with external simulation validation

full rationale

The paper presents an iterative family of DFT/IDFT algorithms with a sparsification variant whose convergence criterion is explicitly stated as reaching a stable real-domain sparsity pattern. This is an algorithmic definition, not a self-referential fit or prediction. Denoising performance is assessed via simulation studies against existing methods, with no equations, fitted parameters, or self-citations shown that reduce the central claim to its own inputs by construction. The method is self-contained as a procedure whose outputs are evaluated externally.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not introduce new physical entities or free parameters beyond standard DFT operations and an implicit sparsification threshold; no explicit axioms beyond the DFT uncertainty principle are stated.

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