arxiv: 2605.01684 · v1 · submitted 2026-05-03 · 📊 stat.AP

Recognition: unknown

Data-driven time-frequency tessellation for signals with oscillatory amplitude envelopes and instantaneous frequency, with application to photoplethysmograhy

Hau-Tieng Wu, Jennifer Laine

Pith reviewed 2026-05-09 16:48 UTC · model grok-4.3

classification 📊 stat.AP

keywords time-frequency analysisphotoplethysmogramrespiratory signal extractionamplitude modulationinstantaneous frequencytessellationnon-harmonic modelbiomedical signal processing

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The pith

TETRIS recovers respiratory signals from PPG by tessellating the time-frequency plane along the cardiac instantaneous frequency.

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

The paper develops TETRIS, a time-frequency analysis method for signals whose main oscillation has amplitude and frequency that are themselves modulated by slower oscillations. With PPG as the example, where breathing imprints on the cardiac component through both amplitude and frequency changes, the approach divides the time-frequency plane into regions guided by the estimated cardiac instantaneous frequency and processes each region to isolate the breathing effects more cleanly. This produces several different surrogate respiratory signals extracted straight from the PPG waveform. The authors supply theoretical support for the adaptive partition handling and confirm the gains on semi-synthetic test signals built to match the model. Readers may care because PPG sensors are already common in clinics and wearables, so better extraction could expand non-invasive breathing monitoring without added hardware.

Core claim

The central claim is that, for signals obeying a generalized adaptive non-harmonic model, tessellating the time-frequency plane along the estimated instantaneous frequency of the primary component and then applying integrated shifting and adaptive processing to each resulting partition produces a refined representation that more effectively recovers the secondary oscillatory modulations governing amplitude and frequency, thereby enabling improved direct reconstruction of multiple surrogate respiratory signals from PPG data.

What carries the argument

TETRIS (Tessellation-based Ensembled Time-Frequency Representation via Integrated Shifting), which partitions the time-frequency plane along the estimated instantaneous frequency of the cardiac component and adaptively processes each partition to enhance the visibility of respiratory modulations.

If this is right

Multiple surrogate respiratory signals can be reconstructed directly from PPG data.
The tessellation produces a refined time-frequency representation that better isolates respiratory modulation of the cardiac amplitude.
Theoretical justification supports the adaptive processing applied within each time-frequency partition.
Performance gains are confirmed through validation on semi-synthetic signals that follow the model.
The same framework can be applied to other signals obeying the generalized adaptive non-harmonic model.

Where Pith is reading between the lines

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

The tessellation strategy could be tested on other physiological recordings that contain nested oscillatory dynamics, such as certain blood-pressure or EEG traces.
Making the partitioning step fully data-driven without an initial frequency estimate might broaden the method to signals where the primary instantaneous frequency is harder to locate first.
Real-time versions running on wearable PPG hardware could support continuous extraction of breathing information in ambulatory settings.

Load-bearing premise

The observed signals obey a generalized adaptive non-harmonic model in which both the amplitude and instantaneous frequency of the primary component are themselves governed by additional oscillatory dynamics.

What would settle it

A side-by-side test on real PPG recordings in which the respiratory signals recovered by TETRIS do not match independent reference breathing measurements more accurately than those obtained from standard non-tessellated time-frequency methods would falsify the claimed improvement.

Figures

Figures reproduced from arXiv: 2605.01684 by Hau-Tieng Wu, Jennifer Laine.

**Figure 1.** Figure 1: Example of a signal satisfying the gANHM. (a) The signal, with its amplitude modulation (AM) overlaid as a blue dashed curve. (b) The TFR obtained via STFT. (c) The TFR obtained via SST. (d) Same as (c), with ϕ ′ 1 (t) superimposed in red and ϕ ′ 1 (t) ± ϕ ′ 2 (t) in blue. examining higher-order cardiac harmonics due to fast IF changes, as indicated by the magenta arrow. 3.2. Tessellation-based Ensembled T… view at source ↗

**Figure 2.** Figure 2: A real PPG signal is shown in the top panel (scaled by 1/1000). Bottom panels, from left to right: STFT with a 10 s window, SST with a 10 s window, STFT with a 28 s window, SST with a 28 s window, and SST with a 28 s window with the IHR and its harmonics overlaid in red, and the IRR together with l-th cardiac harmonics shifted by ± IRR overlaid in magenta. The IRR is confirmed with the simultaneously recor… view at source ↗

**Figure 3.** Figure 3: Example of a semi-real PPG with RIAV. Here, (a) is the signal, with the RIAV superimposed as the blue dashed curve; (b) is the TFR determined by STFT; (c) is the TFR determined by SST, where the superimposed red arrows indicates the curve kϕ′ 1 (t) − ϕ ′ 0 (t), for k = 1, 2, 3; (d) replicates (c), with ϕ ′ 1 (t) and ϕ ′ 1 (t)±ϕ ′ 0 (t) superimposed as solid red and blue curves, 2ϕ ′ 1 (t) and 2ϕ ′ 1 (t)±ϕ … view at source ↗

**Figure 4.** Figure 4: Example of a semi-real PPG with RIAV and RIFV. Here, (a) is the signal; (b) is the TFR determined by STFT with red arrows indicating ϕ ′ 1 (t) ± ϕ ′ 0 (t); (c) is the TFR determined by SST with red arrows indicating ϕ ′ 1 (t) ± ϕ ′ 0 (t); (d) replicates (c), with ϕ ′ 1 (t) and ϕ ′ 1 (t) ± ϕ ′ 0 (t) superimposed as solid red and blue curves, 2ϕ ′ 1 (t) and 2ϕ ′ 1 (t) ± ϕ ′ 0 (t) as dashed curves, and 3ϕ ′ 1… view at source ↗

**Figure 5.** Figure 5: Top, from left to right, S10,f↓0 , S10,f↓1 , S10,f↓2 , S10,f↓3 , and T10 of the PPG signal shown in view at source ↗

**Figure 6.** Figure 6: Various reconstructed respiratory waveform from PPT. In (b), the blue curve is the traditional RIIV (tRIIV) and the magenta curve is the traditional RIAV (tRIAV). ABD is superimposed as the gray curve in (d)-(g). The red circles are detected expiration starting times from THO. Clifton. Breathing rate estimation from the electrocardiogram and photoplethysmogram: A review. IEEE reviews in biomedical enginee… view at source ↗

read the original abstract

Biomedical signals often comprise multiple non-sinusoidal oscillatory components whose amplitude modulation (AM) and instantaneous frequency (IF) may themselves be governed by additional (second-order) oscillatory dynamics with time-varying amplitude and frequency. We introduce a novel time-frequency (TF) analysis framework, {\em Tessellation-based Ensembled Time-Frequency Representation via Integrated Shifting} (TETRIS), designed based on the proposed generalized adaptive non-harmonic model to leverage second-order oscillatory information in this class of signals. We present the model and algorithm using the photoplethysmogram (PPG) as a canonical example, whose cardiac component is known to encode respiratory information in both AM and IF, and demonstrate how respiratory signals can be recovered from PPG. The central idea of TETRIS is to partition the TF plane along the estimated IF of the cardiac component and to process each partition adaptively to enhance representation quality. This tessellation enables a refined time-frequency representation (TFR), allowing more effective recovery of the respiratory modulation governing the AM of the cardiac component. We provide theoretical justification for the proposed method and validate its performance on semi-synthetic signals. Finally, we demonstrate that TETRIS enables improved reconstruction of multiple surrogate respiratory signals directly from PPG data. While the model and algorithm are developed with a focus on PPG, the framework is flexible and has potential to be applied to other signals.

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.

Desk Editor's Note private letter to a colleague

TETRIS tessellates the TF plane along estimated cardiac IF to pull respiratory modulations from PPG under a second-order oscillatory model, but gains rest on that model actually fitting real signals.

read the letter

The core of this paper is TETRIS, a new time-frequency method that partitions the plane using the estimated instantaneous frequency of the main component and processes each piece to sharpen the representation of amplitude and frequency modulations that are themselves oscillatory. They apply it to PPG to recover multiple surrogate respiratory signals. The model they introduce is a generalized adaptive non-harmonic one that explicitly builds in second-order dynamics for both AM and IF, and the algorithm follows from that by doing integrated shifting within the tessellated regions. They supply some theoretical backing, run semi-synthetic tests, and show results on real PPG data. That last part is the practical hook for biomedical signal work. The tessellation idea is distinct enough from plain synchrosqueezing or EMD variants that it is not just a routine tweak. The semi-synthetic validation and the PPG demonstration give it some grounding. The main limitation is that everything flows from the assumption that the observed signals really do have those clean second-order oscillatory modulations on top of the primary component. Semi-synthetic data generated from the same family will naturally favor the method, so it is not clear how much the gains survive when breathing is irregular, noise is structured differently, or the IF estimate drifts. The abstract does not spell out checks that the estimated AM/IF on real PPG actually follow the assumed oscillatory form, and tuning details for the IF estimator are light. This is aimed at people working on adaptive TFRs for physiological monitoring. A reader who already uses synchrosqueezing or similar tools on non-stationary biomedical signals would see a concrete new option worth testing. It is solid enough on novelty and application to deserve a serious referee, mainly to pressure-test the model fit on real data and the reproducibility of the partitioning step.

Referee Report

2 major / 2 minor

Summary. The paper introduces TETRIS, a time-frequency tessellation framework based on a generalized adaptive non-harmonic model in which both amplitude and instantaneous frequency of the primary oscillatory component are themselves governed by additional (second-order) oscillatory dynamics. Using PPG as the canonical example, it claims theoretical justification for partitioning the TF plane along the estimated cardiac IF, adaptive processing of partitions to refine the TFR, and improved recovery of multiple surrogate respiratory signals directly from PPG, supported by semi-synthetic validation and a demonstration on real data.

Significance. If the model assumptions hold and the reported gains are robust, the work could meaningfully advance non-invasive respiratory monitoring from single-channel PPG by leveraging second-order structure in AM/IF; the explicit theoretical justification and semi-synthetic validation are strengths that distinguish it from purely empirical approaches.

major comments (2)

[Abstract and model section] Abstract and model section: the central performance claims (improved respiratory reconstruction) rest on the assumption that observed PPG signals obey the proposed generalized adaptive non-harmonic model with second-order oscillatory AM/IF; the manuscript provides no explicit diagnostics or checks (e.g., spectral analysis of estimated IF/AM on real data) confirming this structure holds beyond the semi-synthetic cases, which is load-bearing for the generalizability claim.
[Validation section] Validation section: semi-synthetic tests are described as generated from the model family, but without reported details on data-generation parameters, noise models, or exclusion criteria, it remains unclear whether reconstruction gains would persist for real PPG exhibiting non-oscillatory modulations or irregular breathing that violates the second-order assumption.

minor comments (2)

[Algorithm description] Clarify the precise definition of the tessellation boundaries and the integrated-shifting operator (including any tuning parameters of the underlying IF estimator) to support reproducibility.
[Results] Add a brief comparison table of reconstruction metrics against standard TFR methods (e.g., synchrosqueezing) on the same semi-synthetic and real datasets.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which help clarify the presentation and strengthen the manuscript. We address each major comment point by point below.

read point-by-point responses

Referee: [Abstract and model section] Abstract and model section: the central performance claims (improved respiratory reconstruction) rest on the assumption that observed PPG signals obey the proposed generalized adaptive non-harmonic model with second-order oscillatory AM/IF; the manuscript provides no explicit diagnostics or checks (e.g., spectral analysis of estimated IF/AM on real data) confirming this structure holds beyond the semi-synthetic cases, which is load-bearing for the generalizability claim.

Authors: We agree that the manuscript would benefit from explicit verification that the second-order oscillatory structure is present in real PPG recordings. The semi-synthetic experiments are constructed from the model family using parameters informed by observed PPG statistics, and the real-data demonstration shows improved surrogate respiratory signal recovery, but we did not include spectral diagnostics (such as periodograms or coherence analysis) of the extracted AM and IF on the real recordings. In the revised version we will add a dedicated subsection providing such spectral analysis on the real PPG dataset to directly support the model assumptions and generalizability. revision: yes
Referee: [Validation section] Validation section: semi-synthetic tests are described as generated from the model family, but without reported details on data-generation parameters, noise models, or exclusion criteria, it remains unclear whether reconstruction gains would persist for real PPG exhibiting non-oscillatory modulations or irregular breathing that violates the second-order assumption.

Authors: We acknowledge that the current description of the semi-synthetic data generation is insufficiently detailed for full reproducibility and assessment of robustness. The manuscript states that signals are generated from the model family but omits specific parameter ranges, noise models, and selection/exclusion rules. We will expand the validation section to report these details explicitly (e.g., ranges for second-order frequencies and amplitudes, SNR levels, and any filtering or exclusion criteria applied). On the question of performance when the second-order assumption is violated, the real-data results already reflect typical PPG variability; we will add a brief limitations paragraph discussing expected behavior under strong violations such as highly irregular breathing. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the TETRIS derivation chain

full rationale

The paper proposes a new generalized adaptive non-harmonic model for signals with second-order oscillatory AM and IF, then derives the TETRIS tessellation framework directly from that model to partition the TF plane along the cardiac IF and enhance respiratory modulation recovery. Theoretical justification follows from the model's assumptions, and validation uses semi-synthetic signals plus real PPG demonstrations. No load-bearing step reduces by construction to a fitted input, self-citation, or renamed known result; the central reconstruction claim rests on the independent model and algorithm rather than tautological redefinition of inputs. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on a newly proposed generalized adaptive non-harmonic model whose assumptions are not independently verified outside the paper.

axioms (1)

domain assumption Biomedical signals of interest can be represented as a sum of components whose amplitude and instantaneous frequency are themselves modulated by second-order oscillatory processes.
This modeling choice is the foundation for the tessellation and adaptive processing steps.

pith-pipeline@v0.9.0 · 5556 in / 1229 out tokens · 39110 ms · 2026-05-09T16:48:31.666219+00:00 · methodology

discussion (0)

Reference graph

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