S transform

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

${\displaystyle S_{x}(t,f)=\int _{-\infty }^{\infty }x(\tau )|f|e^{-\pi (t-\tau )^{2}f^{2}}e^{-j2\pi f\tau }\,d\tau }$

• Inverse S-Transform

${\displaystyle x(\tau )=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }S_{x}(t,f)\,dt\right]\,e^{j2\pi f\tau }\,df}$

• Spectrum Form

The above definition implies that the s-transform function can be express as the convolution of ${\displaystyle (x(\tau )e^{-j2\pi f\tau })}$ and ${\displaystyle (|f|e^{-\pi t^{2}f^{2}})}$.
Applying the Fourier transform to both ${\displaystyle (x(\tau )e^{-j2\pi f\tau })}$ and ${\displaystyle (|f|e^{-\pi t^{2}f^{2}})}$ gives

${\displaystyle S_{x}(t,f)=\int _{-\infty }^{\infty }X(f+\alpha )\,e^{-\pi \alpha ^{2}/f^{2}}\,e^{j2\pi \alpha t}\,d\alpha }$.

• Discrete-time S-transform

From the spectrum form of S-transform, we can derive the discrete-time S-transform.
Let ${\displaystyle t=n\Delta _{T}\,\,f=m\Delta _{F}\,\,\alpha =p\Delta _{F}}$, where ${\displaystyle \Delta _{T}}$ is the sampling interval and ${\displaystyle \Delta _{F}}$ is the sampling frequency.
The Discrete time S-transform can then be expressed as:

${\displaystyle S_{x}(n\Delta _{T}\,,m\Delta _{F})=\sum _{p=0}^{N-1}X[(p+m)\,\Delta _{F}]\,e^{-\pi {\frac {p^{2}}{m^{2}}}}\,e^{\frac {j2pn}{N}}}$

Below is the Pseudo code of the implementation.

  Step1.Compute ${\displaystyle X[p\Delta _{F}]\,}$ 

  loop over m (voices)
     Step2.Compute ${\displaystyle e^{-\pi {\frac {p^{2}}{m^{2}}}}}$for ${\displaystyle f=m\Delta _{F}}$

     Step3.Move ${\displaystyle X[p\Delta _{F}]}$ to ${\displaystyle X[(p+m)\Delta _{F}]}$

     Step4.Multiply Step2 and Step3 ${\displaystyle B[m,p]=X[(p+m)\Delta _{F}]\cdot e^{-\pi {\frac {p^{2}}{m^{2}}}}}$

     Step5.IDFT(${\displaystyle B[m,p]}$).
  Repeat.}

• Signal filterings[8]
• Magnetic resonance imaging (MRI)[9]
• 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.[10]
  • 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
  • Global seismology

• Laplace transform
• Wavelet transform
• Short-time Fourier transform

• 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
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• I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis", IEEE Trans. on Information Theory, vol. 36, no. 5, Sept. 1990
• Farge, M. (1992). "Wavelet transforms and their application to turbulence". Annual Review of Fluid Mechanics. 24: 395–457. doi:10.1146/annurev.fluid.24.1.395.
• D. Gabor, "Theory of communication", J. Inst. Elect. Eng., vol. 93, no. 3, pp. 429–457, 1946
• Goupillaud, P.; Grossmann, A.; Morlet, J. (1984). "Cycle-octave and related transforms in seismic analysis". Geoexploration. 23: 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 SP Magazine, pp. 21–67
• Rioul, O.; Vetterli, M. (1991). "Wavelets and signal processing" (PDF). IEEE SP Magazine. 8 (4): 14–38. Bibcode:1991ISPM....8...14R. doi:10.1109/79.91217. S2CID 13266737.
• R. K. Young, Wavelet Theory and its Applications, Kluwer Academic Publishers, Dordrecht,1993