{ "id": "epq20ng9fp", "slug": "stockwell-transform", "title": "Stockwell transform", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis" }, { "id": "physical.math.analysis.harmonic_analysis_and_transforms", "label": "Harmonic analysis, transforms, and integral equations", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis.harmonic_analysis_and_transforms" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms", "label": "Non-geographic · 1946 to 2000: Harmonic analysis, transforms, and integral equations", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis" }, { "id": "geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms", "label": "Harmonic analysis, transforms, and integral equations", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms" } ] } ], "excerpt": "The Stockwell transform, or S-transform, is a time-frequency analysis method introduced in 1996 that applies a frequency-dependent Gaussian window, used widely in seismology and power systems.", "snippet": "The Stockwell transform, or S-transform, is a time-frequency analysis method introduced in 1996 that applies a frequency-dependent Gaussian window, used widely in seismology and power systems.", "node": "physical.math.analysis.harmonic_analysis_and_transforms", "markdown": "# Stockwell transform\n\nThe Stockwell transform ([S-transform](https://www.edgepedia.org/s-transform)) is a linear time-frequency analysis method that applies a frequency-dependent Gaussian window to the [Fourier transform](https://www.edgepedia.org/fourier-transform) of a signal, producing a complex-valued local spectrum for non-stationary signals. It behaves as a short-time Fourier transform (STFT) whose window narrows with increasing frequency, or equivalently as a phase-corrected continuous wavelet transform, and it retains the absolutely referenced phase of each signal component.[1](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2014/HTML/papers/1569925853.pdf) • [2](https://arxiv.org/pdf/2603.06912) It was introduced in 1996 and is used widely in seismology,[3](https://imgw.univie.ac.at/fileadmin/user_upload/i_img/Geophyik/Publikationen_bis_2018/Tary_2014_Spectral.pdf) power systems, and biomedical signal processing.[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf)\n\n| Key fact | Value |\n|---|---|\n| Introduced | Stockwell, Mansinha, and Lowe, 1996, IEEE Trans. Signal Process. 44(4), 998–1001[5](https://iris.polito.it/retrieve/e384c42e-644b-d4b2-e053-9f05fe0a1d67/ACHA.pdf) • [6](https://github.com/claudiodsf/stockwell/blob/1d8158a3528a3d828337448bbf1ddffb1e236ec5/README.md) |\n| Gaussian window width | \\( \\sigma(f) = 1/\\|f\\| \\), proportional to the inverse of frequency[1](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2014/HTML/papers/1569925853.pdf) • [4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) |\n| Phase property | Absolutely referenced: kernel phase at \\( t = 0 \\) is zero[7](https://ar5iv.labs.arxiv.org/html/2101.06707) |\n| Invertibility | Time-averaging the local spectrum recovers the Fourier spectrum[1](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2014/HTML/papers/1569925853.pdf) |\n| Full discrete cost | \\( N^{2} \\) coefficients, \\( O(N^{3}) \\) computation for a length-N signal[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) |\n| Fast form | Discrete orthonormal Stockwell transform (DOST), \\( O(N \\log N) \\), nonredundant[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) • [8](https://doi.org/10.1109/tsp.2009.2028972) |\n| Main limitation | Fixed window shape gives poor energy concentration and poor time resolution at low frequencies[1](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2014/HTML/papers/1569925853.pdf) • [9](https://api.geophysical-press.com/uploads/file/asp/Vol32-1_art3.pdf) |\n\n## How it works\n\nFor a signal x(t), the transform is defined as\n\n\\[ ST(t,f) = \\int_{-\\infty}^{\\infty} x(t_1)\\, \\frac{|f|}{\\sqrt{2\\pi}}\\, e^{-f^{2}(t_1 - t)^{2}/2}\\, e^{-j \\cdot 2\\pi \\cdot f \\cdot t_1}\\, dt_1 \\]\n\nwhich is a STFT with a Gaussian (Gabor) window whose length is frequency dependent, giving variable time-frequency resolution similar to the wavelet transform.[7](https://ar5iv.labs.arxiv.org/html/2101.06707) In the equivalent wavelet-style form, \\( \\mathrm{ST}_x(t,f) = \\|f\\| \\int x(\\tau) \\bar{\\psi}(\\|f\\|(\\tau-t)) e^{-j 2 \\pi \\tau f} \\, d\\tau \\) for \\( f \\neq 0 \\), and at zero frequency the transform equals the average of the signal.[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) The original window is the Gaussian \\( \\psi(t) = (1/\\sqrt{2\\pi})\\, e^{-t^{2}/2} \\), whose width is proportional to the inverse of the frequency variable; the Gaussian is chosen because joint time-frequency resolution reaches the lower bound of the uncertainty principle.[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf)\n\nThree properties distinguish the output from a spectrogram or a wavelet transform. First, averaging the local spectrum over time recovers the ordinary Fourier spectrum, ∫ S_x(t,f) dt = X(f), so the transform keeps Fourier-transform properties such as uniform frequency sampling and frequency-independent spectral amplitudes, owing to normalization of the Gaussian taper to unit area.[1](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2014/HTML/papers/1569925853.pdf) • [3](https://imgw.univie.ac.at/fileadmin/user_upload/i_img/Geophyik/Publikationen_bis_2018/Tary_2014_Spectral.pdf) Second, in contrast to the continuous wavelet transform (CWT), the phase is absolutely referenced: the phase of the kernel functions at \\( t = 0 \\) is zero, so each local spectrum carries the phase of the signal component itself rather than a per-window phase.[7](https://ar5iv.labs.arxiv.org/html/2101.06707) • [4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) This matters in applications where phase information is sensitive, such as seismology and electroencephalography.[2](https://arxiv.org/pdf/2603.06912) Third, the basis function at a fixed frequency resembles a complex Morlet wavelet, but its envelope width changes with frequency, whereas the STFT window width is fixed.[10](https://pmc.ncbi.nlm.nih.gov/articles/PMC5796829/)\n\n## How it is done\n\nThe transform is computed voice by voice through the fast Fourier transform (FFT): for each frequency, the signal's Fourier spectrum is multiplied by a Gaussian window scaled to that frequency and transformed back, which is why the method became popular in seismic time-frequency analysis for its convenient computation via the Fourier transform.[9](https://api.geophysical-press.com/uploads/file/asp/Vol32-1_art3.pdf) For a signal of length N, the full discrete transform requires computing N² ST coefficients, and the total computational complexity is O(N³), which limits its use for large or higher-dimensional signals.[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) Fast algorithms reduce this: a fast \\( O(N \\log N) \\) algorithm computes a nonredundant set of Stockwell coefficients for admissible windows, extending an approach of Y. Wang and J. Orchard (2009),[5](https://iris.polito.it/retrieve/e384c42e-644b-d4b2-e053-9f05fe0a1d67/ACHA.pdf) and the fast DOST algorithms reduce the complexity from \\( O(N^{2} \\cdot \\log N) \\) to \\( O(N \\cdot \\log N) \\).[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) A Python package, `stockwell`, implements time-frequency analysis through the transform.[6](https://github.com/claudiodsf/stockwell/blob/1d8158a3528a3d828337448bbf1ddffb1e236ec5/README.md)\n\n## Origin\n\nThe S-transform has been applied to medical imaging, geophysics, and general signal processing.[5](https://iris.polito.it/retrieve/e384c42e-644b-d4b2-e053-9f05fe0a1d67/ACHA.pdf) The introducing paper is \"Localization of the complex spectrum: the S transform,\" IEEE Transactions on Signal Processing, 44(4), 998–1001.[6](https://github.com/claudiodsf/stockwell/blob/1d8158a3528a3d828337448bbf1ddffb1e236ec5/README.md) The transform was first derived as the \"phase correction\" of the continuous wavelet transform, and Stockwell and colleagues described it as a \"continuous wavelet transform with a phase shift.\"[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) • [3](https://imgw.univie.ac.at/fileadmin/user_upload/i_img/Geophyik/Publikationen_bis_2018/Tary_2014_Spectral.pdf) It builds on the cycle-octave transform work of Goupillaud, Grossmann, and Morlet (1984) in Geoexploration[11](https://doi.org/10.1016/0016-7142%2884%2990025-5) and on Daubechies' 1990 analysis of wavelet time-frequency localization in the IEEE Transactions on Information Theory.[12](https://doi.org/10.1109/18.57199)\n\n## Variants\n\nBecause the classic transform's fixed window shape can cause poor energy concentration in the time-frequency plane, generalized S-transforms introduce parameters that determine the shape and properties of the window.[1](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2014/HTML/papers/1569925853.pdf) Pinnegar and Mansinha published the S-transform with windows of arbitrary and varying shape in [Geophysics](https://www.edgepedia.org/geophysics) in 2003,[13](https://doi.org/10.1190/1.1543223) and Pinnegar introduced ST variations with arbitrary window shape for P-wave arrival detection.[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) Sejdić, Djurović, and Jiang proposed a window-width-optimized S-transform in the EURASIP Journal on Advances in Signal Processing in 2008,[14](https://doi.org/10.1155/2008/672941) with a generalized window width of \\( 1/\\|f\\|^{q} \\); Pei and Wang proposed linear scaling \\( p/\\|f\\| \\), and Liu developed adaptive S-transforms optimizing p and q.[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) One modified Gaussian window uses \\( \\sigma(f) = m \\cdot f^{p} + k \\cdot f^{r} \\) with parameters selected by a [genetic algorithm](https://www.edgepedia.org/genetic-algorithm) maximizing an energy concentration measure, while keeping the normalization that preserves invertibility.[1](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2014/HTML/papers/1569925853.pdf)\n\nOn the redundancy side, Stockwell's 2006 Digital Signal Processing paper formulated a basis for efficient representation of the S-transform,[15](https://doi.org/10.1016/j.dsp.2006.04.006) and Brown, Lauzon, and Frayne's 2009 IEEE Transactions on Signal Processing paper formulated a fast, invertible transform that samples the continuous S-transform spectrum nonredundantly.[8](https://doi.org/10.1109/tsp.2009.2028972) Synchrosqueezing versions sharpen the representation: Huang and Zhang published the synchrosqueezing S-transform (SS-ST) in Scientia Sinica Informationis in 2016,[16](https://doi.org/10.1360/n112015-00011) and Wang and colleagues published the synchrosqueezing generalized S-transform (SS-GST) in IEEE Geoscience and Remote Sensing Letters in 2018.[17](https://doi.org/10.1109/lgrs.2017.2789190) Ventosa and colleagues reformulated the transform from a wavelet point of view in the IEEE Transactions on Signal Processing in 2008.[18](https://doi.org/10.1109/tsp.2008.917029) A 2024 synchrosqueezing generalized phase-shifting S-transform (SS-GPST) phase-shifts the ST to obtain a frequency-invariant phase spectrum, addressing incompatibility between the ST and synchrosqueezing, and applies the result to ground-penetrating-radar detection.[19](https://www.mdpi.com/1424-8220/24/10/2981) Modified ST parameters must be optimized by expert users and tuned per application; no single modified ST was optimal for all signals tested in one 2024 comparison.[20](https://pmc.ncbi.nlm.nih.gov/articles/PMC11540225/)\n\n## Applications\n\nThe transform is valued in seismology, electroencephalography, medical imaging, mechanical vibration analysis, and image processing, where phase information is highly sensitive.[2](https://arxiv.org/pdf/2603.06912) It has also been applied in geophysical signal analysis, power system analysis, image compression, biomedical signal processing, and ocean wave analysis.[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf) Reported high-performance tasks include heart-sound analysis, power quality signals, and EEG signals.[1](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2014/HTML/papers/1569925853.pdf) In seismology it is used for time-frequency decomposition of seismic data,[9](https://api.geophysical-press.com/uploads/file/asp/Vol32-1_art3.pdf) and a 2024 sparse S-transform network (SSTNet) maps synthetic seismic traces to sparse S-transform spectra, with a knowledge-distillation variant (KD-SSTNet) computing sparse time-frequency spectra of field data without field training labels for seismic attenuation estimation in reservoir characterization.[21](https://www.sciopen.com/article/10.1016/j.petsci.2024.03.002) In biomedicine, a 2025 hybrid CNN-transformer model for arrhythmia detection without R-peak identification uses the Stockwell transform as its front end.[22](https://doi.org/10.1038/s41598-025-92582-9)\n\n## Limitations and alternatives\n\nThe classic transform's fixed window shape limits the resolution trade-off. The window width \\( \\sigma(f) = 1/\\|f\\| \\) gives higher frequency resolution at lower frequencies and higher time resolution at higher frequencies, but the fixed window shape leads to poor time resolution at low frequencies.[7](https://ar5iv.labs.arxiv.org/html/2101.06707) • [9](https://api.geophysical-press.com/uploads/file/asp/Vol32-1_art3.pdf) • [20](https://pmc.ncbi.nlm.nih.gov/articles/PMC11540225/) The original ST, which sets \\( \\sigma(f) = \\|f\\|^{-1} \\), cannot represent high-frequency components with satisfactory frequency-axis resolution, which motivated modified STs with \\( \\sigma(f) = \\|f\\|^{-r} \\) and related forms.[20](https://pmc.ncbi.nlm.nih.gov/articles/PMC11540225/) The factor \\|f\\| in the formula also makes the transform tend to emphasize higher-frequency content.[7](https://ar5iv.labs.arxiv.org/html/2101.06707) The full transform is highly redundant; the nonredundant DOST removes that redundancy but provides a rather coarse time-frequency representation, with frequency resolution proportionally scaled to the logarithm of frequency, that may not always be easy to interpret.[4](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf)\n\nAgainst alternatives, a published comparison rates the Stockwell transform's resolution as poor and frequency dependent with no artifacts, the STFT/Gabor and CWT as also poor with no artifacts, the smoothed pseudo Wigner-Ville distribution as good with occasional artifacts, and the Wigner-Ville distribution as excellent with strong artifacts.[7](https://ar5iv.labs.arxiv.org/html/2101.06707) In one benchmark, the CWT and synchrosqueezing failed to detect a low-amplitude spectral line near 2050 Hz that the S transform with a tuned width parameter, the STFT, basis pursuit, and short-time autoregressive methods detected.[3](https://imgw.univie.ac.at/fileadmin/user_upload/i_img/Geophyik/Publikationen_bis_2018/Tary_2014_Spectral.pdf) Separately, a 2015 review in Digital Signal Processing compared synchrosqueezed windowed-Fourier and wavelet transforms with their original forms, stating that \"the SWFT and SWT do not appear to provide any significant advantages over the original WFT and WT apart from a more visually appealing pictures.\"[23](https://dl.acm.org/doi/10.1016/j.dsp.2015.03.004)\n\n## References\n\n1. [Stockwell Transform Optimization Applied on the Detection of Split in Heart Sounds (Moukadem et al., EUSIPCO 2014)](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2014/HTML/papers/1569925853.pdf)\n2. [Stockwell transform on Gelfand pairs (arXiv mathematical paper)](https://arxiv.org/pdf/2603.06912)\n3. [Spectral estimation: What is new? What is next? (Tary et al., 2014, Geophysics review)](https://imgw.univie.ac.at/fileadmin/user_upload/i_img/Geophyik/Publikationen_bis_2018/Tary_2014_Spectral.pdf)\n4. [A Brief Overview of the S-transforms (RIMS Kôkyûroku, Kyoto University)](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2102-01.pdf)\n5. [Politecnico di Torino repository paper on the DOST basis and fast Stockwell algorithms (Applied and Computational Harmonic Analysis)](https://iris.polito.it/retrieve/e384c42e-644b-d4b2-e053-9f05fe0a1d67/ACHA.pdf)\n6. [stockwell Python package README](https://github.com/claudiodsf/stockwell/blob/1d8158a3528a3d828337448bbf1ddffb1e236ec5/README.md)\n7. [Fourier, Gabor, Morlet or Wigner: Comparison of Time-Frequency Transforms (Scholl, arXiv:2101.06707)](https://ar5iv.labs.arxiv.org/html/2101.06707)\n8. [Robert A. Brown, M. Louis Lauzon, Richard Frayne (2009). A General Description of Linear Time-Frequency Transforms and Formulation of a Fast, Invertible Transform That Samples the Continuous S-Transform Spectrum Nonredundantly. IEEE Transactions on Signal Processing.](https://doi.org/10.1109/tsp.2009.2028972)\n9. [Application of multi-synchrosqueezed generalized S-transform in seismic time-frequency analysis](https://api.geophysical-press.com/uploads/file/asp/Vol32-1_art3.pdf)\n10. [Comparison of time-frequency methods for analyzing stimulus frequency otoacoustic emissions](https://pmc.ncbi.nlm.nih.gov/articles/PMC5796829/)\n11. [Cycle-octave and related transforms in seismic signal analysis (Geoexploration, 1984)](https://doi.org/10.1016/0016-7142%2884%2990025-5)\n12. [I. Daubechies (1990). The wavelet transform, time-frequency localization and signal analysis. IEEE Transactions on Information Theory.](https://doi.org/10.1109/18.57199)\n13. [C. Robert Pinnegar, Lalu Mansinha (2003). The S -transform with windows of arbitrary and varying shape. Geophysics.](https://doi.org/10.1190/1.1543223)\n14. [Ervin Sejdić, Igor Djurović, Jin Jiang (2007). A Window Width Optimized S-Transform. EURASIP Journal on Advances in Signal Processing.](https://doi.org/10.1155/2008/672941)\n15. [R.G. Stockwell (2006). A basis for efficient representation of the S-transform. Digital Signal Processing.](https://doi.org/10.1016/j.dsp.2006.04.006)\n16. [Zhonglai HUANG, Jianzhong ZHANG (2016). Synchrosqueezing S-transform. Scientia Sinica Informationis.](https://doi.org/10.1360/n112015-00011)\n17. [Qian Wang and colleagues (2018). High-Resolution Seismic Time–Frequency Analysis Using the Synchrosqueezing Generalized S-Transform. IEEE Geoscience and Remote Sensing Letters.](https://doi.org/10.1109/lgrs.2017.2789190)\n18. [Sergi Ventosa and colleagues (2008). The $S$-Transform From a Wavelet Point of View. IEEE Transactions on Signal Processing.](https://doi.org/10.1109/tsp.2008.917029)\n19. [An Improved Synchrosqueezing S-Transform and Its Application in a GPR Detection Task (Sensors, 2024)](https://www.mdpi.com/1424-8220/24/10/2981)\n20. [Investigation and evaluation of cross-term reduction in masked Wigner-Ville distributions using S-transforms (PLoS ONE, 2024)](https://pmc.ncbi.nlm.nih.gov/articles/PMC11540225/)\n21. [Application of sparse S transform network with knowledge distillation in seismic attenuation delineation (Petroleum Science, 2024)](https://www.sciopen.com/article/10.1016/j.petsci.2024.03.002)\n22. [Donghyeon Kim and colleagues (2025). A novel hybrid CNN-transformer model for arrhythmia detection without R-peak identification using stockwell transform. Scientific Reports.](https://doi.org/10.1038/s41598-025-92582-9)\n23. [Linear and synchrosqueezed time-frequency representations revisited (Digital Signal Processing)](https://dl.acm.org/doi/10.1016/j.dsp.2015.03.004)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Harmonic analysis, transforms, and integral equations*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Oct 15, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/stockwell-transform", "markdown_url": "https://www.edgepedia.org/stockwell-transform.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Stockwell transform\", Edgepedia (EdgeChat), https://www.edgepedia.org/stockwell-transform. Edgepedia Community License 1.0.", "credit_md": "\"[Stockwell transform](https://www.edgepedia.org/stockwell-transform)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/stockwell-transform](https://www.edgepedia.org/stockwell-transform). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Stockwell transform\", Edgepedia (EdgeChat), https://www.edgepedia.org/stockwell-transform. Edgepedia Community License 1.0.", "speakable": "The Stockwell transform, or S-transform, is a time-frequency analysis method introduced in 1996 that applies a frequency-dependent Gaussian window, used widely in seismology and power systems." }