Stress-Sensitive Frequency-Domain Transformation

• Stress-sensitive frequency-domain transformation is a suite of methods that remap data into spectral, time-frequency, or modal coordinates to highlight stress-induced changes.
• These approaches are applied across geophysical, physiological, and mechanical contexts, enabling analysis of seismic precursors, ECG stress levels, and vibration anomalies.
• They employ techniques such as adjacent-band spectral contrast, reassignment, and dynamic mode decomposition to enhance the visibility of stress proxies while suppressing irrelevant variations.

Stress-sensitive frequency-domain transformation denotes a class of transformations that seek to make stress-conditioned structure more observable by remapping data into spectral, time-frequency, or spatiotemporal modal coordinates. In the most explicit sense, the term is instantiated by the Shakibay Senobari Frequency-domain Transform, or SSF, which compares adjacent spectral bands to isolate stress-related changes in seismic noise (Senobari, 29 Aug 2025). In a broader sense, recent work uses analogous frequency-domain or time-frequency-domain mappings to expose physiological stress in ECG, condition-sensitive dynamics in vibration, transient perturbations in nonstationary signals, and the modal organization of evolving thermal stress fields (Ahmad et al., 2020). This suggests a unifying view: stress sensitivity is often enhanced when absolute amplitude is de-emphasized in favor of spectral contrast, localized oscillatory structure, reassigned concentration, or modal decomposition.

1. Scope and conceptual boundaries

The phrase does not denote a single canonical transform. Instead, the literature spans several distinct but related constructions. One group of methods is directly tied to evolving stress in a medium or a stress field itself. SSF is designed to “amplify stress-sensitive changes in seismic wavefields while suppressing variability associated with source properties,” and dynamic mode decomposition in thermal-crystal plasticity is applied directly to the evolving von Mises stress field under cyclic thermal loading (Senobari, 29 Aug 2025). Another group is indirect: ECG-based multi-level stress assessment treats psychological stress as the target state and uses frequency-domain and time-frequency-domain images derived from R-R-interval-based signal images; downsampled synchrosqueezing for vibration analysis is better interpreted as a condition-sensitive transform whose output can reflect load, spindle state, chatter onset, or other mechanically relevant changes rather than stress as a material field (Ahmad et al., 2020).

A recurring distinction is therefore between direct stress observables and stress proxies. In seismic noise analysis, the claimed mechanism is that evolving fault stress modifies elastic properties, attenuation, scattering, and microcrack state, which in turn tilt or reshape the local spectrum. In ECG, the objective is not physical stress in matter but multi-level affective stress assessment from cardiac dynamics. In vibration and transient analysis, the relevant signatures are time-varying instantaneous frequencies, weak modulation, nonstationary harmonics, transients, group delay structure, or mode-specific oscillation patterns that may change under load or damage. This suggests that the field is unified more by representation strategy than by a single definition of stress.

A second boundary concerns the meaning of “frequency-domain.” Some methods operate on raw or lightly processed time series; others transform intermediate representations. The ECG study applies a 2D DFT and a 2D Gabor transform not to the original 1D ECG waveform but to a 2D signal image formed by stacking interpolated R-R segments. The thermal-stress study does not use a Fourier spectrum at all; it maps stress-field histories into dynamic modes with associated modal frequencies and growth or decay rates. The common element is therefore not a specific basis but a change of representation intended to reveal latent stress-conditioned structure.

2. Mathematical archetypes

One archetype is the adjacent-band spectral contrast transform. SSF is defined as

$$SSF(t_0,f)=\log \frac{P(t_0,f,\delta f)}{P(t_0,f+\Delta f,\delta f)},$$

where $P(t_0,f,\delta f)$ is the average power spectral density in the band centered at $f$, $\delta f$ is the bandwidth, and $\Delta f$ is the offset to the neighboring comparison band (Senobari, 29 Aug 2025). The method is explicitly interpreted as capturing the local logarithmic slope of spectral energy across adjacent bands. Its design principle is local differencing: broadband source fluctuations, station gain, or amplitude scaling are attenuated because numerator and denominator are nearby in frequency, while small stress-related spectral tilts are enhanced.

A second archetype is the spectral transformation of constructed images. In multi-level ECG stress assessment, the spatial-domain signal image $f(m,n)$ is transformed by a 2D DFT,

$$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$

and by a 2D Gabor wavelet transform,

$$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$

The DFT yields a frequency-domain image, whereas the Gabor transform yields a time-frequency-like image intended to preserve both local structure and oscillatory content (Ahmad et al., 2020). In this formulation, stress sensitivity is not attributed to a single spectrum alone, but to the complementarity of spatial, spectral, and localized spectral views.

A third archetype is reassignment and concentration. The downsampled STFT-based synchrosqueezing transform estimates an instantaneous frequency for each time-frequency coefficient and reassigns energy toward the corresponding ridge:

$$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$

The wavelet-based time-reassigned synchrosqueezing transform instead uses a group-delay-based time reassignment operator,

$$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$

and moves coefficients along time rather than frequency (He et al., 2020). The first construction is optimized for nonstationary vibration; the second is tailored to transient signals with multivalued instantaneous frequency but single-valued group delay as a function of frequency. In both cases, stress sensitivity arises through concentration: weak state-sensitive or perturbation-sensitive structures become less diffuse after reassignment.

A fourth archetype is spatiotemporal modal decomposition. In the thermal-stress study, the evolving stress field is represented as

$P(t_0,f,\delta f)$0

with modal growth and decay rates and frequencies

$P(t_0,f,\delta f)$1

This is not a classical spectrum of a scalar signal; it is a low-dimensional spectral description of the stress field itself, with each mode carrying both a spatial stress pattern and a temporal frequency (Ohashi et al., 19 May 2026).

3. ECG-derived frequency-domain representations for multi-level stress assessment

The ECG study addresses a specific gap: most prior ECG-based stress work performs binary recognition, whereas the reported system targets five stress levels on an in-house Ryerson Multimedia Research Laboratory dataset collected from 15 participants during a 6-minute virtual-reality roller coaster simulation (Ahmad et al., 2020). ECG was recorded from a chest-worn sensor at 256 Hz. Stress labels were assigned in 30-second segments with five levels: $P(t_0,f,\delta f)$2 Very Low, $P(t_0,f,\delta f)$3 Low, $P(t_0,f,\delta f)$4 Above Average, $P(t_0,f,\delta f)$5 High, and $P(t_0,f,\delta f)$6 Very High. The paper uses only ECG, even though ECG, GSR, and respiration were collected, to emphasize simplicity and applicability.

The core representation is not the raw waveform but an R-R-interval-derived signal image. The ECG is split into segments between successive R peaks; because these segments have unequal sample length, interpolation is applied so that each has the same number of samples while preserving R-R interval and HRV characteristics. The equal-length segments are then stacked row by row to form a 2D signal image, interpreted as capturing temporal correlation among successive R-R segments. All images are resized to $P(t_0,f,\delta f)$7. The paper does not report any explicit denoising or artifact-removal stage, any named R-peak detector, or a precise normalization formula. It also does not specify whether the final DFT input is complex-valued, magnitude-only, log-magnitude, phase, or power.

Three modality-specific image representations are then processed by CNNs: the spatial-domain signal image, the frequency-domain DFT image, and the time-frequency-domain Gabor image. The fusion is explicitly decision-level fusion rather than input-level or feature-level fusion. Reported training settings are momentum $P(t_0,f,\delta f)$8, initial learning rate $P(t_0,f,\delta f)$9, learning-rate drop factor $f$0, drop period $f$1, $f$2 regularization $f$3, and mini-batch size $f$4. The train/test split is $f$5, repeated 10 times, with augmentation expanding the training set to 1350 images and 55 test images.

The reported accuracies are central to the topic. Raw ECG with a 1D CNN reaches $f$6; signal images only reach $f$7; FFT images only reach $f$8; Gabor images only reach $f$9; and decision-level fusion reaches $\delta f$0 (Ahmad et al., 2020). Two conclusions are explicit. First, the major performance gain comes from R-R-based signal-to-image conversion rather than from transform-domain conversion alone. Second, neither the DFT nor the Gabor representation individually outperforms the spatial-domain signal image. The authors explain that transform domains can lose information that varies with time, and abrupt changes near R-R intervals “cannot be modeled accurately in transform domains.” The strongest claim is therefore not that the frequency domain is superior in isolation, but that it contributes complementary information when fused with the original signal image.

This result corrects a common misconception. A stress-sensitive frequency-domain transformation need not be individually dominant to be useful. In this case, the frequency-domain image and the Gabor image are auxiliary modalities that improve multi-level stress classification only in a multidomain late-fusion framework. The paper’s under-documented elements—DFT encoding choice, Gabor parameters, R-peak detection details, and subject-independent evaluation—remain material limitations for reproducibility and generalizability.

4. Condition-sensitive vibration analysis and transient reassignment

In large-scale vibration analysis, the most relevant transform is the fast implementation of STFT-based synchrosqueezing transform based on time downsampling, frequency downsampling, selective reassignment, frequency subdivision, and an approximate direct inverse (He et al., 2020). The paper does not claim that the method directly estimates mechanical stress or stress fields. Rather, it presents a high-resolution, reconstructive time-frequency transform that can reveal time-varying instantaneous frequencies, weak speed fluctuations, nonstationary harmonics, modulation, chatter components, and other condition-sensitive dynamics. The time hop is controlled by $\delta f$1, the effective frequency resolution by $\delta f$2 and $\delta f$3, and selective reassignment can be restricted to a band $\delta f$4 with optional subdivision factor $\delta f$5 to recover reassignment precision inside that band.

The quantitative efficiency gains are substantial. For the simulation, full SST with $\delta f$6 requires $\delta f$7 s, whereas $\delta f$8 requires $\delta f$9 s, $\Delta f$0 requires $\Delta f$1 s, $\Delta f$2 requires $\Delta f$3 s, $\Delta f$4 requires $\Delta f$5 s, and $\Delta f$6 requires $\Delta f$7 s. Selective reassignment further reduces runtime from $\Delta f$8 s to $\Delta f$9 s for $f(m,n)$0, and from $f(m,n)$1 s to $f(m,n)$2 s for $f(m,n)$3. In the aero-engine case, downsampled SST completes in about $f(m,n)$4 s and resolves high-frequency speed fluctuations of the L4 shaft; in spindle chatter analysis, a $f(m,n)$5–$f(m,n)$6 Hz restricted-band SST completes in about $f(m,n)$7 s and identifies chatter onset at about $f(m,n)$8 s (He et al., 2020). The paper’s practical conclusion is that frequency decimation cannot be too aggressive, or SST degenerates toward STFT.

The wavelet-based time-reassigned synchrosqueezing transform and its iterative refinement, WTMSST, address a different signal class: transient, multicomponent signals whose behavior is better characterized by group delay than by a single-valued instantaneous frequency (Dong et al., 2022). The method begins from a frequency-domain signal model,

$f(m,n)$9

and reassigns wavelet coefficients in time using the group-delay candidate $$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$0. The continuous WTSST is

$$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$1

while WTMSST iteratively refines the reassignment candidate. For second-order phase behavior, the refined candidate converges to the true group delay as the number of refinement steps increases. The transform remains reconstructive, and the paper provides concentration and reconstruction analyses for weakly varying group delay, then shows why WTSST alone becomes biased when group delay curvature is strong.

The experimental evidence is oriented toward transient feature extraction rather than explicit stress estimation. WTMSST yields more concentrated representations than WT, SST, or SET on simulations, bearing fault signals, ECG arrhythmia, and GW190521, and the time-frequency envelope spectrum identifies a bearing pulse interval matching the theoretical $$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$2 ms fault characteristic spacing (Dong et al., 2022). A plausible implication is that the method is a strong front-end for stress-sensitive monitoring when the relevant perturbation manifests as dispersive arrival shifts, transient mode splitting, or localized burst structure rather than smooth frequency ridges.

5. SSF and the seismic interpretation of stress-sensitive spectral change

The most explicit formulation of stress-sensitive frequency-domain transformation appears in the seismic-noise study that introduces the Shakibay Senobari Frequency-domain Transform, abbreviated SSF (Senobari, 29 Aug 2025). Its central idea is not to track absolute spectral power but to compare the power in adjacent narrow frequency bands using a logarithmic ratio. The stated goal is to amplify stress-sensitive changes in seismic wavefields while suppressing variability associated with source properties. The physical rationale is that fault loading changes elastic properties, wave velocity, attenuation, scattering, and microcrack evolution, and these path effects slightly tilt or reshape the local spectrum. SSF therefore acts as a local logarithmic spectral-slope detector.

The workflow is explicit. In laboratory acoustic-emission experiments, AE is recorded continuously at 4 MHz, downsampled by a factor of 4, segmented into non-overlapping windows of about $$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$3 s, and processed with Welch PSD using a Hamming window of length 2048 with $$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$4 overlap over 200 Hz to 500 kHz, divided into 100 linearly spaced bins, giving $$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$5 kHz. For seismic data, continuous waveforms are segmented into two-minute, non-overlapping windows; only amplitude sensitivity is corrected; Welch PSD is computed over either $$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$6–$$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$7 Hz with 98 linearly spaced bins for 20 Hz data or $$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$8–$$F(k,l) = \sum_{m=0}^{a-1}\sum_{n=0}^{b-1}{f(m,n)}e^{-j2\pi(\frac{km}{a}+\frac{ln}{b})},$$9 Hz with 150 linearly spaced bins for data sampled above 40 Hz, with $$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$0 Hz in both cases. The seismic transform averages two adjacent-band offsets, $$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$1 Hz and $$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$2 Hz. Post-processing is causal and backward-looking: for one setting, a median over 30 samples, or 1 hour, followed by a mean over 60 samples, or 2 hours; for another, a median over 120 samples, or 4 hours, followed by a mean over 360 samples, or 12 hours.

The reported phenomenology is distinctive. In laboratory shear experiments P4581 and P5198, the transform exhibits cyclic patterns aligned with measured shear stress and normal stress. The paper states that global stacks reveal shear-stress signatures, local low-frequency stacks track shear stress, and higher frequencies reflect normal-stress variations. In seismic case studies spanning Pawnee, Turkey-Syria, Gorkha, Tohoku, Kilauea, Denali, and Maule, the recurring patterns are arc-like trajectories, accelerations toward extrema, and occasionally V-shaped spikes. The abstract summarizes the lead time as hours to days before rupture, while the body expands this to days to weeks for several events and months for Maule and Tohoku. Maule shows a “wake-up” perturbation nearly four months before rupture, and Tohoku shows months-long departure from baseline with a pronounced run-up in the final week. The figures are often based on the component that displays the most visually prominent precursory features.

The robustness argument is largely qualitative. The paper reports negligible differences when full instrument response removal is tested, emphasizes multistation coherence, aligns the event time to the start of the relevant two-minute window to avoid post-event contamination, and discusses possible confounders such as anthropogenic noise, narrow-band interference, local site effects, nonlinear soil response, and seasonal temperature shifts. Yet it does not provide ROC curves, precision-recall, false alarm rates, p-values, confidence intervals, surrogate-data tests, or formal significance analysis (Senobari, 29 Aug 2025). The result is therefore strongest as a retrospective and physically motivated visual phenomenology rather than a statistically validated operational forecasting system.

6. Dynamic mode decomposition of thermal stress fields

A different realization of stress-sensitive transformation arises in thermal-crystal plasticity under cyclic thermal loading, where the transformed quantity is the stress field itself rather than a measured waveform (Ohashi et al., 19 May 2026). The study considers a polycrystalline 316L stainless steel microstructure subjected to sinusoidal temperature boundary conditions,

$$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$3

with $$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$4, $$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$5, and $$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$6. The coupled model solves mechanical equilibrium, heat conduction, isotropic thermal expansion, thermally activated slip, and dislocation-density-based hardening. The stress observable analyzed by DMD is the von Mises stress field.

The frequency interpretation is organized by the Fourier number,

$$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$7

with reported values approximately $$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$8 for 200 Hz, $$GWT =Ae^{-\pi\sigma^2(x^2+y^2)}e^{(j2\pi F(xcos\omega+ysin\omega)+\theta)}.$$9 for 20 Hz, and $$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$0 for 2 Hz. At 2 Hz, the temperature field is nearly uniform and the spatially averaged von Mises stress is almost periodic with negligible plastic strain accumulation. At 20 Hz, stress oscillations grow larger, two local maxima appear per cycle, and moderate plasticity accumulates. At 200 Hz, the response is strongly non-stationary, the initial cycle differs from later cycles, the two local maxima per cycle separate more widely, and the largest plastic strain accumulation occurs, about twice that at 20 Hz. Spatially, stress at 200 Hz is broadly distributed through the interior except near boundaries, whereas 20 Hz shows stronger concentration and more localized plasticity near the center in some regions.

Dynamic mode decomposition then maps the stress-field history into spatiotemporal modes. Snapshot matrices $$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$1 and $$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$2 satisfy $$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$3, with exact DMD modes

$$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$4

and modal representation

$$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$5

Continuous modal frequencies and growth or decay rates are extracted from $$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$6. Hankel DMD extends this by time-delay embedding with maximum embedding dimension $$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$7. The stress field is interpolated onto a $$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$8 grid, snapshots are taken every 10 time steps, and 492 snapshots are used for DMD after discarding the trivial zero-stress initial state. Using the top 20 greedy-ordered modes, the reconstruction error with $$\hat{\omega}_x[m,k] = \xi[k] - \Im\!\left\{ \frac{S_x^{g'}[m,k]}{S_x^g[m,k]} \right\}, \qquad T_x[m,l] = \sum_{k:\,|\hat{\omega}_x[m,k]-\xi[l]|\le \Delta\xi/2} S_x^g[m,k].$$9 is reduced to about $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$0 or less in all frequency cases.

The spectral interpretation is revealing. In all cases, the first mode has $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$1 Hz, but its growth rate differs sharply: at 200 Hz, $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$2, indicating non-stationary stress-level increase; at 20 Hz, $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$3, close to steady; at 2 Hz, $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$4, consistent with periodic steady-state behavior. The second mode lies near $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$5: $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$6 Hz for 200 Hz, $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$7 Hz for 20 Hz, and $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$8 Hz for 2 Hz. Harmonic deviation, quantified by $$\hat t(a,b)=b+\Re\left[\frac{aW_x^{g'}(a,b)}{W_x(a,b)}\right],$$9, remains below $P(t_0,f,\delta f)$00 at 2 Hz, reaches about $P(t_0,f,\delta f)$01 at 20 Hz, and increases up to $P(t_0,f,\delta f)$02 at 200 Hz. The transformation therefore exposes a transition from quasi-steady harmonic response to increasingly inharmonic, history-dependent, multi-component stress dynamics.

This construction is stress-sensitive in a direct and literal sense. The transformed object is the evolving stress field, and each DMD mode combines a spatial stress pattern with a temporal frequency and growth or decay rate. The result is not merely a compressed simulation output; it is a modal organization of how thermal-cycle frequency changes stress localization, synchronization, and plastic accommodation.

7. Misconceptions, adjacent uses, and unresolved issues

A common misconception is that any frequency-domain transformation associated with stress is a direct stress estimator. Several papers explicitly reject that interpretation. The downsampled SST paper is “not a direct stress estimator,” but rather a fast, high-resolution, reconstructive transform for condition-sensitive vibration analysis. The ECG multidomain paper estimates psychological stress level, not material or structural stress. The seismic SSF paper is physically motivated by evolving shear and normal stress, but it does not provide a calibrated inverse mapping from transform values to stress tensor components or an operational forecasting rule (He et al., 2020). The literature therefore supports a narrower claim: many transforms are stress-sensitive because stress-related changes alter spectral, time-frequency, or modal structure, not because the transform itself returns stress as a state variable.

A second misconception is that frequency-domain representations are necessarily superior to spatial or time-domain representations. The ECG results directly contradict that simplification: signal images alone outperform FFT images alone and Gabor images alone, while fusion of all three is best (Ahmad et al., 2020). In the seismic paper, visually prominent precursor trajectories do not by themselves establish false alarm performance, because no ROC curves, p-values, or surrogate tests are reported. In the vibration paper, aggressive frequency downsampling can destroy the advantage over STFT. In the transient-wavelet paper, WTSST becomes biased for strong group-delay curvature and WTMSST is needed to recover concentration (Senobari, 29 Aug 2025).

A third boundary concerns transferred uses of the phrase in adjacent fields. Frequency-domain regularization in CNNs learns layer-specific spectral masks to improve robustness to adversarial perturbations, corruption, and domain shift, but the “stress” there is perturbation or application-scenario transformation rather than physical stress (Guo et al., 2020). Dense pruning of pointwise convolutions in the frequency domain learns per-channel thresholds over DCT coefficients based on each channel’s sensitivity to frequency pruning, again using “sensitivity” in a computational rather than physical sense (Buckler et al., 2021). The DS-UWB FDR receiver converts structured sign patterns across frames into energy localization in transform bins, a transferable idea for transform-domain sensitivity amplification, but it is not a stress-sensing method (Abri et al., 2012). These papers are best read as analogues: they show how frequency-domain mappings can concentrate informative structure and suppress nuisance variability, which is the same representational logic exploited by physically stress-sensitive transformations.

Across the core literature, unresolved issues are consistent. Small datasets and non-subject-independent splits limit generalizability in ECG stress assessment; under-documented transform parameters complicate exact reproduction in ECG, SST, and seismic workflows; approximate inverses and parameter sensitivity remain central in reassignment methods; and seismic precursor results remain primarily qualitative rather than statistically benchmarked (Ahmad et al., 2020). A plausible implication is that future work will be judged less by whether a transformation is spectral in name and more by whether it preserves interpretability, reports sufficient implementation detail, and demonstrates robustness under realistic null conditions and cross-domain generalization.