S transform

In mathematics, the S transform usually refers to the Laplace transform. However, S transform as a time frequency distribution was developed in 1994 for analyzing geophysics data. In this way, the S transform is a generalization of the Short-time Fourier transform, 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: it requires higher complexity computation (because FFT can't be used), and the clarity is worse than Wigner distribution function and Cohen's class distribution function.

A fast S Transform algorithm was invented in 2010. It reduces the computational time and resources by at least 4 orders of magnitude and is available to the research community under an open source license.

Mathematical 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.

$S_x(t,f) = \int_{-\infty}^{\infty} x(\tau)|f|e^{- \pi (t- \tau)^2 f^2} e^{-j2 \pi f \tau} \, d \tau$

Discussion

Based on work by R. G. Stockwell et al., the S transform and STFT are compared. 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.

(Experiment results haven't been uploaded due to author's permission not yet been granted)

Applications

• Dynamic Identification of Soil and Structures (see http://roccoditommaso.xoom.it)
• Band-Variable Filter to extract a single mode of vibration from a non-linear system (http://roccoditommaso.xoom.it/index_file/MATLAB%20FILE/matlab%20file.htm)
• Magnetic Resonance imaging (MRI)
• 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.
  • S transform can 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
  • Exploration seismology
  • Global seismology

See also

• Laplace transform
• wavelet transform
• Short-time Fourier transform

References

• Ditommaso R., Mucciarelli M., Ponzo F. C. (2012). ANALYSIS OF NONSTATIONARY STRUCTURAL SYSTEMS BY USING A BAND-VARIABLE FILTER. Bulletin of Earthquake Engineering. DOI: 10.1007/s10518-012-9338-y.
• 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 03, 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.
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• Hongmei Zhu, PhD and J. Ross Mitchell, PhD, "The S Transform in Medical Imaging," University of Calgary Seaman Family MR Research Centre Foothills Medical Centre, Canada.
• 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.
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