Editing S transform

{{About|the time–frequency transform|the mathematical use of this term|Laplace transform}}
'''''S''''' '''transform''' as a time–frequency distribution was developed in 1994 for analyzing geophysics data.<ref>{{cite journal | last1 = Stockwell | first1 = RG | last2 = Mansinha | first2 = L | last3 = Lowe | first3 = RP | year = 1996 | title = Localization of the complex spectrum: the S transform | journal = IEEE Transactions on Signal Processing | volume = 44 | issue = 4| pages = 998–1001 | doi=10.1109/78.492555| bibcode = 1996ITSP...44..998S | citeseerx = 10.1.1.462.1500 | s2cid = 30202517 }}</ref><ref name="Stockwell PhD">Stockwell, RG (1999). ''S''-transform analysis of gravity wave activity from a small scale network of airglow imagers. PhD thesis, University of Western Ontario, London, Ontario, Canada.</ref> In this way, the ''S'' transform is a generalization of the [[short-time Fourier transform]] (STFT), extending the [[continuous wavelet transform]] and overcoming some of its disadvantages. For one, modulation sinusoids are fixed with respect to the time axis; this localizes the scalable Gaussian window dilations and translations in ''S'' transform. Moreover, the ''S'' transform doesn't have a cross-term problem and yields a better signal clarity than [[Gabor transform]]. However, the ''S'' transform has its own disadvantages: the clarity is worse than [[Wigner distribution function]] and [[Cohen's class distribution function]].{{citation needed|date=January 2016}}

A fast ''S'' transform algorithm was invented in 2010.<ref>{{cite book |pmid=19163232 | doi=10.1109/IEMBS.2008.4649729 | volume=2008 | year=2008 | pages=2586–9 | last1 = Brown | first1 = RA | last2 = Frayne | first2 = R| title=2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society | chapter=A fast discrete S-transform for biomedical signal processing | isbn=978-1-4244-1814-5 | s2cid=29974786 }}</ref><ref name=":0">{{Cite journal|last1=Brown|first1=Robert A.|last2=Lauzon|first2=M. Louis|last3=Frayne|first3=Richard|date=January 2010|title=A General Description of Linear Time-Frequency Transforms and Formulation of a Fast, Invertible Transform That Samples the Continuous S-Transform Spectrum Nonredundantly|journal=IEEE Transactions on Signal Processing|volume=58|issue=1|pages=281–290|doi=10.1109/tsp.2009.2028972|bibcode=2010ITSP...58..281B|s2cid=16074001|issn=1053-587X}}</ref> It reduces the computational complexity from O[N<sup>2</sup>·log(N)] to O[N·log(N)] and makes the transform one-to-one, where the transform has the same number of points as the source signal or image, compared to storage complexity of N<sup>2</sup> for the original formulation.<ref name=":0" /><ref>Kelly Sansom, "Fast S Transform", University of Calgary, https://www.ucalgary.ca/news/utoday/may31-2011/computing</ref> An implementation is available to the research community under an [[open source license]].<ref>{{cite web| url = http://sourceforge.net/projects/fst-uofc/ |title = Fast S-Transform download {{!}} SourceForge.net| date=13 August 2018 }}</ref><ref name=":1">{{Citation |last=Brown |first=Robert |title=robb-brown/GFT |date=2024-12-01 |url=https://github.com/robb-brown/GFT?tab=readme-ov-file |access-date=2024-12-10}}</ref>

A general formulation of the S transform<ref name=":0" /> makes clear the relationship to other time frequency transforms such as the Fourier, short time Fourier, and wavelet transforms.<ref name=":0" />

==Definition==
There are several ways to represent the idea of the ''S'' transform. In here, ''S'' transform is derived as the phase correction of the continuous wavelet transform with window being the [[Gaussian function]].
*S-Transform
:<math> S_x(t,f) = \int_{-\infty}^\infty x(\tau)|f|e^{- \pi (t- \tau)^2 f^2} e^{-j2 \pi f \tau} \, d \tau </math>
*Inverse S-Transform
:<math>x(\tau) = \int_{-\infty}^\infty \left[\int_{-\infty}^{\infty}S_x(t,f)\, dt\right]\,e^{j2\pi f\tau}\, df</math>

==Modified form==
*Spectrum Form
The above definition implies that the s-transform function can be expressed as the convolution of <math>( x(\tau) e^{-j2 \pi f \tau} )</math> and <math>( |f|e^{- \pi t^2 f^2} )</math>.<br>
Applying the [[Fourier transform]] to both <math>( x(\tau) e^{-j2 \pi f \tau} )</math> and <math>( |f|e^{- \pi t^2 f^2} )</math> gives
:<math> S_x(t,f) = \int_{-\infty}^\infty X(f+\alpha)\,e^{-\pi\alpha^2 /f^2}\,e^{j2\pi\alpha t}\, d\alpha  </math>.
*Discrete-time S-transform
From the spectrum form of S-transform, we can derive the discrete-time S-transform. <br>
Let <math>t = n\Delta_T\,\, f = m\Delta_F\,\, \alpha = p\Delta_F</math>, where <math>\Delta_T</math> is the sampling interval and <math>\Delta_F</math> is the sampling frequency.<br>
The Discrete time S-transform can then be expressed as: <br>
:<math>S_x(n\Delta_T\, ,m\Delta_F) = \sum_{p=0}^{N-1} X[(p+m)\,\Delta_F]\,e^{-\pi\frac{p^2}{m^2}}\,e^{\frac{j2pn}{N}}</math>

==Implementation of discrete-time S-transform==
Below is the Pseudo code of the implementation.<br>
   Step1.Compute <math>X[p\Delta_{F}]\,</math> <br>
   loop over m (voices)
      Step2.Compute <math>e^{-\pi \frac{p^2}{m^2}}</math>for <math>f=m\Delta_{F}</math><br>
      Step3.Move <math>X[p\Delta_{F}]</math> to <math>X[(p+m)\Delta_{F}]</math><br>
      Step4.Multiply Step2 and Step3 <math>B[m,p] = X[(p+m)\Delta_{F}]\cdot e^{-\pi \frac{p^2}{m^2}}</math><br>
      Step5.IDFT(<math>B[m,p]</math>).
   Repeat.}

==Comparison with other time–frequency analysis tools==

===Comparison with Gabor transform===
The only difference between the Gabor transform (GT) and the S transform is the window size. For GT, the windows size is a Gaussian function <math>( e^{-\pi (t-\tau)^2} )</math>, meanwhile, the window function for S-Transform is a function of f. With a window function proportional to frequency, S Transform performs well in frequency domain analysis when the input frequency is low. When the input frequency is high, S-Transform has a better clarity in the time domain. As table below. 
{| class="wikitable"
|-
| Low-frequency || Bad clarity in time domain || Good clarity in frequency domain
|-
| High-frequency || Bad clarity in frequency domain|| Good clarity in time domain
|}  
This kind of property makes S-Transform a powerful tool to analyze sound because human is sensitive to low frequency part in a sound signal.

===Comparison with Wigner transform===
The main problem with the Wigner Transform is the cross term, which stems from the auto-correlation function in the Wigner Transform function. This cross term may cause noise and distortions in signal analyses. S-transform analyses avoid this issue.

===Comparison with the short-time Fourier transform ===
We can compare the ''S'' transform and short-time Fourier transform (STFT).<ref name="Stockwell PhD" /><ref>E. Sejdić, I. Djurović, J. Jiang, "Time-frequency feature representation using energy concentration: An overview of recent advances," ''Digital Signal Processing'', vol. 19, no. 1, pp. 153-183, January 2009.</ref> First, a high frequency signal, a low frequency signal, and a high frequency burst signal are used in the experiment to compare the performance. The S transform characteristic of frequency dependent resolution allows the detection of the high frequency burst. On the other hand, as the STFT consists of a constant window width, it leads to the result having poorer definition. In the second experiment, two more high frequency bursts are added to crossed chirps. In the result, all four frequencies were detected by the S transform. On the other hand, the two high frequencies bursts are not detected by STFT. The high frequencies bursts cross term caused STFT to have a single frequency at lower frequency.

==Applications==
* Signal filterings<ref>{{cite journal |url=http://roccoditommaso.xoom.it/index_file/Band%20Variable%20Filter.pdf |doi=10.1007/s10518-012-9338-y |title=Analysis of non-stationary structural systems by using a band-variable filter |date=2012 |last1=Ditommaso |first1=Rocco |last2=Mucciarelli |first2=Marco |last3=Ponzo |first3=Felice Carlo |journal=Bulletin of Earthquake Engineering |volume=10 |issue=3 |pages=895–911 |bibcode=2012BuEE...10..895D }}. See also [http://roccoditommaso.xoom.it/index_file/MATLAB%20FILE/matlab%20file.htm MATLAB file]</ref>
* [[Magnetic resonance imaging]] (MRI)<ref>Hongmei Zhu, and J. Ross Mitchell, "The S Transform in Medical Imaging," University of Calgary Seaman Family MR Research Centre Foothills Medical Centre, Canada.</ref>
* Power system disturbance recognition
** ''S'' transform has been proven to be able to identify a few types of disturbances, like voltage sag, voltage swell, momentary interruption, and oscillatory transients.<ref>{{cite book | doi=10.1109/PECON.2010.5697562 | chapter=Coherency determination in grid-connected distributed generation based hybrid system under islanding scenarios | title=2010 IEEE International Conference on Power and Energy | date=2010 | last1=Ray | first1=Prakash K. | last2=Mohanty | first2=Soumya R. | last3=Kishor | first3=Nand | last4=Dubey | first4=Harish C. | pages=85–88 | isbn=978-1-4244-8947-3 }}</ref>
** ''S'' transform also be applied for other types of disturbances such as notches, harmonics with sag and swells etc.
** ''S'' transform generates contours which are suitable for simple visual inspection. However, wavelet transform requires specific tools like standard [[multiresolution analysis]].
* Geophysical signal analysis
** [[Reflection seismology]]
** [[Seismology|Global seismology]]

==See also==
* [[Laplace transform]]
* [[Wavelet transform]]
* [[Short-time Fourier transform]]

==References==
{{reflist|30em}}

==Further reading==
* Rocco Ditommaso, Felice Carlo Ponzo, Gianluca Auletta (2015). Damage detection on framed structures: modal curvature evaluation using Stockwell Transform under seismic excitation. Earthquake Engineering and Engineering Vibration. June 2015, Volume 14, Issue 2, pp 265–274.
* Rocco Ditommaso, Marco Mucciarelli, Felice C. Ponzo (2010). S-Transform based filter applied to the analysis of non-linear dynamic behaviour of soil and buildings. 14th European Conference on Earthquake Engineering. Proceedings Volume. Ohrid, Republic of Macedonia. August 30 – September 3, 2010. (downloadable from http://roccoditommaso.xoom.it)
* M. Mucciarelli, M. Bianca, R. Ditommaso, M.R. Gallipoli, A. Masi, C Milkereit, S. Parolai, M. Picozzi, M. Vona (2011). FAR FIELD DAMAGE ON RC BUILDINGS: THE CASE STUDY OF NAVELLI DURING THE L’AQUILA (ITALY) SEISMIC SEQUENCE, 2009. [[Bulletin of Earthquake Engineering]]. {{doi|10.1007/s10518-010-9201-y}}.
* J. J. Ding, "Time-frequency analysis and wavelet transform course note," the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
* Jaya Bharata Reddy, Dusmanta Kumar Mohanta, and B. M. Karan, "Power system disturbance recognition using wavelet and s-transform techniques," Birla institute of Technology, Mesra, Ranchi-835215, 2004.
* B. Boashash, "Notes on the use of the wigner distribution for time frequency signal analysis", IEEE Trans. on Acoust. Speech. and Signal Processing, vol. 26, no. 9, 1987
* R. N. Bracewell, The Fourier Transform and Its Applications, McGraw Hill Book Company, New York, 1978
* E. O. Brigham, ''The Fast Fourier Transform'', Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1974
* {{cite journal | last1 = Cohen | first1 = L. | year = 1989 | title = Time-frequency distributions—A review | journal = Proc. IEEE | volume = 77 | issue = 7 | pages = 941–981 | doi = 10.1109/5.30749 | citeseerx = 10.1.1.1026.2853 }}
* I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis", ''IEEE Trans. on Information Theory'', vol. 36, no. 5, Sept. 1990
* {{cite journal | last1 = Farge | first1 = M. | year = 1992 | title = Wavelet transforms and their application to turbulence | url = https://zenodo.org/record/1302376| journal = Annual Review of Fluid Mechanics | volume = 24 | pages = 395–457 | doi=10.1146/annurev.fluid.24.1.395}}
* D. Gabor, "Theory of communication", J. Inst. Elect. Eng., vol. 93, no. 3, pp.&nbsp;429–457, 1946
* {{cite journal | last1 = Goupillaud | first1 = P. | last2 = Grossmann | first2 = A. | last3 = Morlet | first3 = J. | year = 1984 | title = Cycle-octave and related transforms in seismic analysis | journal = Geoexploration | volume = 23 | pages = 85–102 | doi=10.1016/0016-7142(84)90025-5}}
* F. Hlawatsch and G. F. Boudreuax-Bartels, 1992 "Linear and quadratic timefrequency signal representations", IEEE Signal Processing Magazine, pp.&nbsp;21–67
* {{cite journal | last1 = Rioul | first1 = O. | last2 = Vetterli | first2 = M. | year = 1991 | title = Wavelets and signal processing | url =https://infoscience.epfl.ch/record/33910/files/RioulV91.pdf | journal =  IEEE Signal Processing Magazine| volume = 8 | issue = 4| pages = 14–38 | doi=10.1109/79.91217| bibcode = 1991ISPM....8...14R | s2cid = 13266737 }}
* R. K. Young, Wavelet Theory and its Applications, Kluwer Academic Publishers, Dordrecht,1993

[[Category:Integral transforms]]
[[Category:Fourier analysis]]
[[Category:Time–frequency analysis]]