{ "id": "epqwpme0wy", "slug": "s-transform", "title": "S-transform", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.algorithms" }, { "id": "technology.computing.ai.numerical_string_geometry", "label": "Numerical, string, and geometric algorithms", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.numerical_string_geometry" }, { "id": "technology.computing.ai.numerical_string_geometry.fourier_signal_transforms", "label": "Fourier and signal transforms", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.numerical_string_geometry.fourier_signal_transforms" } ], "geo": [ { "id": "geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.fourier_signal_transforms", "label": "Non-geographic · 1946 to 2000: Fourier and signal transforms", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.fourier_signal_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.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology" }, { "id": "geo.nongeo.t1946.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing" }, { "id": "geo.nongeo.t1946.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai" }, { "id": "geo.nongeo.t1946.technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.algorithms" }, { "id": "geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry", "label": "Numerical, string, and geometric algorithms", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry" }, { "id": "geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.fourier_signal_transforms", "label": "Fourier and signal transforms", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.fourier_signal_transforms" } ] } ], "excerpt": "The S-transform, or Stockwell transform, is a linear time-frequency transform with a Gaussian window that narrows at high frequencies, introduced in 1996 and used in power quality and seismic analysis.", "snippet": "The S-transform, or Stockwell transform, is a linear time-frequency transform with a Gaussian window that narrows at high frequencies, introduced in 1996 and used in power quality and seismic analysis.", "node": "technology.computing.ai.numerical_string_geometry.fourier_signal_transforms", "markdown": "# S-transform\n\nThe S-transform ([Stockwell transform](https://www.edgepedia.org/stockwell-transform)) is a linear time-frequency transform that analyzes a signal with a Gaussian window whose width scales inversely with frequency, producing an invertible, frequency-dependent spectral representation of nonstationary signals.[1](https://digital.csic.es/bitstream/10261/19699/1/2007-Simon-DIST.pdf) It sits between the short-time [Fourier transform](https://www.edgepedia.org/fourier-transform) ([STFT](https://www.edgepedia.org/short-time-fourier-transform)) and the continuous wavelet transform (CWT), with narrower windows at higher frequencies and wider windows at lower frequencies, while keeping the time-origin-referenced phase of the STFT.[2](https://imgw.univie.ac.at/fileadmin/user_upload/i_img/Geophyik/Publikationen_bis_2018/Tary_2014_Spectral.pdf) • [1](https://digital.csic.es/bitstream/10261/19699/1/2007-Simon-DIST.pdf) Its output, the S-matrix, is a complex-valued time-frequency array whose integral over time at each frequency recovers the Fourier spectrum of the signal.[3](https://www.crewes.org/Documents/ResearchReports/2009/CRR200970.pdf)\n\n| Key fact | Detail |\n|---|---|\n| Introducing paper | R.G. Stockwell, L. Mansinha, R.P. Lowe, \"Localization of the complex spectrum: the S transform,\" IEEE Transactions on Signal Processing 44(4):998–1001, 1996[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC3910128/) |\n| Window | Gaussian with frequency-dependent standard deviation \\( \\sigma(f) = 1/|f| \\)[5](http://imedlab.org/pdfs/papers/frequency-wwost.pdf) |\n| Defining property | Time average of the S-transform equals the Fourier transform of the signal (frequency-marginal property)[5](http://imedlab.org/pdfs/papers/frequency-wwost.pdf) |\n| Relation to CWT | A CWT with a phase correction that references phase to the time origin[6](https://www.sciencedirect.com/science/article/abs/pii/S0165168414002618) |\n| Discrete cost | \\( O(N^{2} \\log_{2} N) \\) for an N-point signal, reducible to \\( O((N/2+1)N \\log_{2} N) \\) using FFT conjugate symmetry[7](https://www.mdpi.com/2227-9717/13/3/743) |\n| Invertibility | Perfectly invertible in the continuous domain; in the discrete domain both inverses carry side effects that depend on signal length and scaling[1](https://digital.csic.es/bitstream/10261/19699/1/2007-Simon-DIST.pdf) |\n| Main uses | Power quality disturbance classification,[8](https://doi.org/10.1049/iet-spr.2019.0042) seismic spectral decomposition,[9](https://library.seg.org/doi/10.1190/geo2015-0425.1) EEG seizure detection,[10](https://www.maths.lu.se/fileadmin/maths/personal_staff/mariasandsten/TFkompver4.pdf) and machinery fault diagnosis[11](https://iopscience.iop.org/article/10.1088/1361-6501/ad0e59) |\n\n## How it works\n\nFor a signal \\( x(t) \\), the standard S-transform convolves the signal with a Gaussian window modulated by a Fourier kernel:[10](https://www.maths.lu.se/fileadmin/maths/personal_staff/mariasandsten/TFkompver4.pdf)\n\n\\[ S_{x}(\\tau, f) = \\int_{-\\infty}^{+\\infty} x(t)\\, \\frac{\\|f\\|}{\\sqrt{2\\pi}}\\, e^{- (t-\\tau)^{2} f^{2} / 2}\\, e^{-i 2\\pi f t}\\, dt \\]\n\nThe Gaussian window has standard deviation \\( \\sigma(f) = 1/\\|f\\| \\), so the window is wide in time at low frequencies (giving high frequency resolution) and narrow at high frequencies (giving high time resolution).[7](https://www.mdpi.com/2227-9717/13/3/743) • [5](http://imedlab.org/pdfs/papers/frequency-wwost.pdf) The transform can be derived from the [Gabor transform](https://www.edgepedia.org/gabor-transform) (the Gaussian-window STFT) simply by making the window's standard deviation a function of frequency.[3](https://www.crewes.org/Documents/ResearchReports/2009/CRR200970.pdf)\n\nTwo properties distinguish it from neighboring methods. First, integrating the S-transform over time returns the Fourier transform \\( X(f) \\) of the signal, a frequency-marginal property the wavelet transform lacks; the S-transform does not satisfy the corresponding time-marginal property.[5](http://imedlab.org/pdfs/papers/frequency-wwost.pdf) Second, it is a CWT with a phase correction: the Fourier kernel references the phase of every time-frequency coefficient to the time origin, whereas the CWT references phase locally within each window.[6](https://www.sciencedirect.com/science/article/abs/pii/S0165168414002618) Stockwell, Mansinha, and Lowe described the transform as a \"continuous wavelet transform with a phase shift.\"[2](https://imgw.univie.ac.at/fileadmin/user_upload/i_img/Geophyik/Publikationen_bis_2018/Tary_2014_Spectral.pdf) A consequence of the \\( \\|f\\| \\) amplitude factor is that the transform emphasizes higher-frequency content.[12](https://ar5iv.labs.arxiv.org/html/2101.06707)\n\n## How it is done\n\nIn the discrete implementation, the signal's DFT is computed once, and a Gaussian window is applied to the DFT spectrum at each frequency bin:[6](https://www.sciencedirect.com/science/article/abs/pii/S0165168414002618)\n\n\\[ S[m, k] = \\sum_{l=0}^{N-1} X[l+k]\\, W(l, k)\\, e^{j 2\\pi l m / N}, \\qquad W(l, k) = e^{- 2\\pi^{2} l^{2} / k^{2}} \\]\n\nwhere \\( k \\) is the frequency index and \\( m \\) the time index; the zero-frequency entry \\( S[m, 0] \\) equals the mean of the time-domain signal.[6](https://www.sciencedirect.com/science/article/abs/pii/S0165168414002618) The computational complexity for an N-point signal is \\( O(N^{2} \\log_{2} N) \\), which FFT conjugate symmetry reduces to \\( O((N/2+1)N \\log_{2} N) \\).[7](https://www.mdpi.com/2227-9717/13/3/743) Sparsity-based optimization of the window parameters can regularize sudden changes in frequency content at the same computational complexity as the nonoptimized algorithm.[9](https://library.seg.org/doi/10.1190/geo2015-0425.1)\n\nTwo inverses exist. The frequency inverse sums the S-transform over all times to recover the Fourier transform of the signal, and is exact in the continuous domain. The time inverse sums over frequency and returns the time-domain signal directly, but contains an approximation that can be made arbitrarily small.[1](https://digital.csic.es/bitstream/10261/19699/1/2007-Simon-DIST.pdf) In the finite discrete domain neither inverse is exact; the side effects depend on the number of points and a tunable scaling factor, and the frequency inverse requires as many frequency samples as time samples, which is inefficient for long geophysical records where only a few frequencies matter.[1](https://digital.csic.es/bitstream/10261/19699/1/2007-Simon-DIST.pdf)\n\n## Origin\n\nIt builds on two traditions. Elementary signals occupy the smallest possible area in the two-dimensional time-frequency \"information diagram,\"[10](https://www.maths.lu.se/fileadmin/maths/personal_staff/mariasandsten/TFkompver4.pdf) and the Gabor expansion and transform are the basis from which the S-transform is derived by frequency-dependent window scaling.[3](https://www.crewes.org/Documents/ResearchReports/2009/CRR200970.pdf) On the wavelet side, Goupillaud, Grossmann, and Morlet published \"Cycle-octave and related transforms in seismic signal analysis\" in 1984,[13](https://doi.org/10.1016/0016-7142%2884%2990025-5) and Daubechies published \"The wavelet transform, time-frequency localization and signal analysis\" in 1990.[14](https://doi.org/10.1109/18.57199) Mansinha, Stockwell, and Lowe extended the method to two-dimensional spectral localization in 1997.[15](https://doi.org/10.1016/s0378-4371%2896%2900487-6)\n\n## Variants\n\nMost variants modify the Gaussian window while keeping the transform's structure.\n\n- Constant-factor window. Mansinha, Stockwell, and Lowe (1997) introduced a constant factor \\( k \\) in the window standard deviation; larger \\( k \\) buys frequency resolution at the cost of time resolution, with \\( k = 3 \\) suggested.[3](https://www.crewes.org/Documents/ResearchReports/2009/CRR200970.pdf) A variable-factor version with \\( k \\) rising linearly from 1 at zero frequency to 6 at Nyquist shows better simultaneous time-frequency resolution than both the Gabor transform and the standard S-transform.[3](https://www.crewes.org/Documents/ResearchReports/2009/CRR200970.pdf)\n- Arbitrary and asymmetric windows. Pinnegar and Mansinha (2003) generalized the transform to windows of arbitrary and varying shape,[16](https://doi.org/10.1190/1.1543223) including the Bi-Gaussian S-transform.\n- Window-width-optimized ST. Sejdić, Djurović, and Jiang (2008) introduced a parameter \\( p \\) controlling window width, \\( S^{p}_{x}(t, f) = \\int x(\\tau)\\, \\|f\\|^{p}/\\sqrt{2\\pi}\\, e^{- (t-\\tau)^{2} f^{2p} / 2}\\, e^{-j 2\\pi f \\tau}\\, d\\tau \\), with \\( p \\) optimized per frequency to maximize a concentration measure; the result concentrates energy better than the standard S-transform, the STFT, and the pseudo Wigner–Ville distribution.[5](http://imedlab.org/pdfs/papers/frequency-wwost.pdf) A related exponent form uses \\( \\sigma_{f} = \\|f\\|^{-r} \\), where for \\( f > 1 \\) Hz, \\( 0 < r < 1 \\) widens the time window and \\( r > 1 \\) narrows it.[17](https://journals.plos.org/plosone/article/file?id=10.1371%2Fjournal.pone.0310721&type=printable)\n- Kaiser window. Replacing the Gaussian with a frequency-dependent Kaiser window (parameter \\( \\alpha(f) = \\pi f \\)) keeps nearly the same autoterm widths while narrowing the mainlobe and reducing sidelobes.[18](http://imedlab.org/pdfs/papers/kaiser-st.pdf)\n- Generalized S-transform. A three-parameter Gaussian window (width factor \\( k \\), a tradeoff \\( m \\) between STFT and ST behavior, and a rate \\( s \\) of width change with frequency) can be tuned adaptively, for example by minimizing Renyi entropy.[19](https://api.geophysical-press.com/uploads/file/asp/Vol32-1_art3.pdf)\n- Reformulations and fast forms. Ventosa and colleagues (2008) recast the transform from a wavelet point of view;[20](https://doi.org/10.1109/tsp.2008.917029) Stockwell (2006) gave a basis for efficient representation of the S-transform;[21](https://doi.org/10.1016/j.dsp.2006.04.006) Wang and Orchard (2009) formulated the fast discrete orthonormal Stockwell transform;[22](https://doi.org/10.1137/080737113) and Brown, Lauzon, and Frayne (2009) formulated a fast, invertible transform that samples the continuous S-transform spectrum nonredundantly.[23](https://doi.org/10.1109/tsp.2009.2028972) Assous and Boashash (2012) evaluated a modified S-transform for time-frequency synchrony analysis.[24](https://doi.org/10.1186/1687-6180-2012-49)\n- Sparse and concentrated forms. Sattari, Gholami, and Siahkoohi (2013) built adaptive sparse time-frequency decomposition on the transform,[25](https://doi.org/10.1190/geo2012-0550.1) Radad, Gholami, and Siahkoohi (2015) introduced an S-transform with maximum energy concentration for nonstationary seismic deconvolution,[26](https://doi.org/10.1016/j.jappgeo.2015.04.010) and Huang, Zhang, Zhao, and Sun (2015) introduced the synchrosqueezing S-transform for seismic spectral decomposition.[27](https://doi.org/10.1109/tgrs.2015.2466660)\n\n## Applications\n\nA 2019 IET Signal Processing review presents the S-transform as gathering the positive qualities of the STFT and the wavelet transform in a single function for analyzing time-varying signals in power quality applications, while noting that a final solution for extracting low- and high-frequency information from time-varying signals is not yet available. In geophysics, the transform serves seismic spectral decomposition and thin-bed and fault identification; the adaptive sparse S-transform-based method produces instantaneous complex attributes superior to those from adaptive sparse STFT, robust adaptive windowed [Hilbert transform](https://www.edgepedia.org/hilbert-transform), and the conventional Hilbert transform on data with thin beds, trapped gas reservoirs, and faults.[9](https://library.seg.org/doi/10.1190/geo2015-0425.1) In biomedical signal processing, published applications include detection of epileptic seizures from EEG and double-talk detection for acoustic echo cancellation.[10](https://www.maths.lu.se/fileadmin/maths/personal_staff/mariasandsten/TFkompver4.pdf) In machinery monitoring, synchroextracting generalizations target rotating-machinery fault diagnosis under variable-speed conditions.[11](https://iopscience.iop.org/article/10.1088/1361-6501/ad0e59) Recent work mostly combines the transform with other tools: a fast S-transform feeding an improved CNN-LSTM hybrid model maintained power quality disturbance identification accuracy above 97% for single disturbance types in strong noise and above 95% for mixed multi-type disturbances,[7](https://www.mdpi.com/2227-9717/13/3/743) and a synchrosqueezing generalized phase-shifting S-transform has been applied to ground-penetrating-radar detection.[28](https://www.mdpi.com/1424-8220/24/10/2981)\n\n## Limitations and alternatives\n\nThe S-matrix is redundant: an N-point signal yields an N-by-N complex array, and the frequency inverse requires as many frequency slots as time slots, a major inconvenience for large data sets.[1](https://digital.csic.es/bitstream/10261/19699/1/2007-Simon-DIST.pdf) [Discretization](https://www.edgepedia.org/discretization) introduces side effects in both inverses, and filtering choices interact with them: the frequency ST–inverse combination for time-dependent filtering causes time-localization problems, while the time ST–inverse combination for frequency-dependent filtering causes smoothing; artifacts of one inverse add to those of the other, so mixing the two should be avoided.[1](https://digital.csic.es/bitstream/10261/19699/1/2007-Simon-DIST.pdf)\n\nThe fixed assignment \\( \\sigma(f) = 1/\\|f\\| \\) applies the same standard deviation to all signal components at a given frequency,[5](http://imedlab.org/pdfs/papers/frequency-wwost.pdf) and the original transform cannot represent high-frequency components with satisfactory frequency-axis resolution.[17](https://journals.plos.org/plosone/article/file?id=10.1371%2Fjournal.pone.0310721&type=printable) The Gaussian window and DFT leakage give poor frequency resolution and energy concentration in both domains, motivating analytic-DCT-based reformulations.[6](https://www.sciencedirect.com/science/article/abs/pii/S0165168414002618) As a linear transform, it is unaffected by the number of signal components, and the time-frequency distributions of different components stack linearly; but under the [Heisenberg uncertainty principle](https://www.edgepedia.org/heisenberg-uncertainty-principle) it cannot achieve high precision in both time and frequency simultaneously.[29](https://www.sciencedirect.com/science/article/abs/pii/S0926985123002252) The S-transform has frequency-dependent resolution: its time window narrows as frequency increases, and at a fixed frequency the resolution is constant across time, whereas the CWT's resolution varies across the scalogram.[30](https://link.springer.com/article/10.1007/s11760-026-05421-3)\n\nAgainst alternatives: a comparative study rates the Stockwell transform's time-frequency resolution as poor and frequency-dependent but free of artifacts, and recommends it when variable resolution with fixed phase alignment is required; the Wigner–Ville distribution has excellent resolution but strong cross-term artifacts, and the smoothed pseudo Wigner–Ville has good resolution with occasional artifacts.[12](https://ar5iv.labs.arxiv.org/html/2101.06707) The window-width-optimized variant achieves higher energy concentration than the STFT and the pseudo Wigner–Ville distribution.[5](http://imedlab.org/pdfs/papers/frequency-wwost.pdf)\n\n## References\n\n1. [The S-Transform and Its Inverses: Side Effects of Discretizing and Filtering](https://digital.csic.es/bitstream/10261/19699/1/2007-Simon-DIST.pdf)\n2. [Spectral estimation, What is new? What is next? (Tary et al., 2014)](https://imgw.univie.ac.at/fileadmin/user_upload/i_img/Geophyik/Publikationen_bis_2018/Tary_2014_Spectral.pdf)\n3. [Variable-factor S-transform seismic data analysis](https://www.crewes.org/Documents/ResearchReports/2009/CRR200970.pdf)\n4. [The S-Transform of Distributions](https://pmc.ncbi.nlm.nih.gov/articles/PMC3910128/)\n5. [A Window Width Optimized S-Transform (Sejdić, Djurović, Jiang, EURASIP JASP 2008)](http://imedlab.org/pdfs/papers/frequency-wwost.pdf)\n6. [S-transform based on analytic discrete cosine transform for time–frequency analysis](https://www.sciencedirect.com/science/article/abs/pii/S0165168414002618)\n7. [Power Quality Disturbance Classification Strategy Based on Fast S-Transform and an Improved CNN-LSTM Hybrid Model](https://www.mdpi.com/2227-9717/13/3/743)\n8. [S-transform: from main concepts to some power quality applications (IET Signal Processing, 2019), paper record](https://doi.org/10.1049/iet-spr.2019.0042)\n9. [High-resolution seismic complex trace analysis by adaptive fast sparse S-transform](https://library.seg.org/doi/10.1190/geo2015-0425.1)\n10. [Time-Frequency Analysis (lecture notes, Maria Sandsten, Lund University)](https://www.maths.lu.se/fileadmin/maths/personal_staff/mariasandsten/TFkompver4.pdf)\n11. [A novel time-frequency analysis method for fault diagnosis based on generalized S-transform and synchroextracting transform](https://iopscience.iop.org/article/10.1088/1361-6501/ad0e59)\n12. [Fourier, Gabor, Morlet or Wigner: Comparison of Time-Frequency Transforms](https://ar5iv.labs.arxiv.org/html/2101.06707)\n13. [Cycle-octave and related transforms in seismic signal analysis (Geoexploration, 1984)](https://doi.org/10.1016/0016-7142%2884%2990025-5)\n14. [I. Daubechies (1990). The wavelet transform, time-frequency localization and signal analysis. IEEE Transactions on Information Theory.](https://doi.org/10.1109/18.57199)\n15. [Pattern analysis with two-dimensional spectral localisation: Applications of two-dimensional S transforms (Physica A Statistical Mechanics and its Applications, 1997)](https://doi.org/10.1016/s0378-4371%2896%2900487-6)\n16. [C. Robert Pinnegar, Lalu Mansinha (2003). The S -transform with windows of arbitrary and varying shape. Geophysics.](https://doi.org/10.1190/1.1543223)\n17. [Investigation and evaluation of cross-term reduction in masked Wigner-Ville distributions using S-transforms](https://journals.plos.org/plosone/article/file?id=10.1371%2Fjournal.pone.0310721&type=printable)\n18. [S-Transform with Frequency Dependent Kaiser Window](http://imedlab.org/pdfs/papers/kaiser-st.pdf)\n19. [Application of multi-synchrosqueezed generalized S-transform in seismic time frequency analysis](https://api.geophysical-press.com/uploads/file/asp/Vol32-1_art3.pdf)\n20. [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)\n21. [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)\n22. [Yanwei Wang, Jeff Orchard (2009). Fast Discrete Orthonormal Stockwell Transform. SIAM Journal on Scientific Computing.](https://doi.org/10.1137/080737113)\n23. [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)\n24. [Said Assous, Boualem Boashash (2012). Evaluation of the modified S-transform for time-frequency synchrony analysis and source localisation. EURASIP Journal on Advances in Signal Processing.](https://doi.org/10.1186/1687-6180-2012-49)\n25. [Hamid Sattari, Ali Gholami, Hamid R. Siahkoohi (2013). Seismic data analysis by adaptive sparse time-frequency decomposition. Geophysics.](https://doi.org/10.1190/geo2012-0550.1)\n26. [Mohammad Radad, Ali Gholami, Hamid Reza Siahkoohi (2015). S-transform with maximum energy concentration: Application to non-stationary seismic deconvolution. Journal of Applied Geophysics.](https://doi.org/10.1016/j.jappgeo.2015.04.010)\n27. [Zhong-lai Huang and colleagues (2015). Synchrosqueezing S-Transform and Its Application in Seismic Spectral Decomposition. IEEE Transactions on Geoscience and Remote Sensing.](https://doi.org/10.1109/tgrs.2015.2466660)\n28. [An Improved Synchrosqueezing S-Transform and Its Application in a GPR Detection Task](https://www.mdpi.com/1424-8220/24/10/2981)\n29. [Time-synchroextracting of generalized S-transform and its application in fault identification](https://www.sciencedirect.com/science/article/abs/pii/S0926985123002252)\n30. [Enhancing heart sound signal denoising: unveiling the impact of time-frequency transformation in U-Net performance](https://link.springer.com/article/10.1007/s11760-026-05421-3)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods › Numerical, string, and geometric algorithms › Fourier and signal transforms*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 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/s-transform", "markdown_url": "https://www.edgepedia.org/s-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": "\"S-transform\", Edgepedia (EdgeChat), https://www.edgepedia.org/s-transform. Edgepedia Community License 1.0.", "credit_md": "\"[S-transform](https://www.edgepedia.org/s-transform)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/s-transform](https://www.edgepedia.org/s-transform). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"S-transform\", Edgepedia (EdgeChat), https://www.edgepedia.org/s-transform. Edgepedia Community License 1.0.", "speakable": "The S-transform, or Stockwell transform, is a linear time-frequency transform with a Gaussian window that narrows at high frequencies, introduced in 1996 and used in power quality and seismic analysis." }