S transform

This article is about the time–frequency transform. For the mathematical use of this term, see Laplace transform.

S transform as a time–frequency distribution was developed in 1994 for analyzing geophysics data.^[1]^[2] 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.

A fast S Transform algorithm was invented in 2010.^[3] It reduces the computational time and resources by at least 4 orders of magnitude^[4] and is available to the research community under an open source license.^[5]

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

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

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

Modified Form

Spectrum Form

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

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

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}}}}

Implementation of Discrete Time S-transform

Below is the Pseudo code of the implementation.

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

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

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

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

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

Comparison with other Time-Frequency Analysis tools

Comparison with Gabor Transform

The only difference between Gabor Transform(GT) and S Transform is the window size. For GT, the windows size is a Gaussian function $(e^{{-\pi (t-\tau )^{2}}})$ , 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.

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 of Wigner Transform is the cross term which caused by the auto-correlation function in the Wigner Transform function. Cross term may cause noise and distortion to signal analysis. No cross term may appear by using the S-transform analysis which makes S Transform a suitable tool to design filter or signal modulation.

Comparison with the short-time Fourier transform

We can compare the S transform and short-time Fourier transform (STFT).^[2]^[6] 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^[7]
Magnetic resonance imaging (MRI)^[8]
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.^[9]
- 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

References

↑ Stockwell, RG; Mansinha, L; Lowe, RP (1996). "Localization of the complex spectrum: the S transform". IEEE Transactions on Signal Processing. 44 (4): 998–1001. doi:10.1109/78.492555.
1 2 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.
↑ Brown, RA; Frayne, R (2008). "A fast discrete S-transform for biomedical signal processing". Conf Proc IEEE Eng Med Biol Soc. 2008: 2586–9. doi:10.1109/IEMBS.2008.4649729. PMID 19163232.
↑ Kelly Sansom, "Fast S Transform", University of Calgary, http://www.ucalgary.ca/news/utoday/may31-2011/computing
↑ http://sourceforge.net/projects/fst-uofc/
↑ 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.
↑ Ditommaso, R, Mucciarelli M, Ponzo FC (2012). ANALYSIS OF NONSTATIONARY STRUCTURAL SYSTEMS BY USING A BAND-VARIABLE FILTER. Bulletin of Earthquake Engineering. doi:10.1007/s10518-012-9338-y. See also MATLAB file
↑ Hongmei Zhu, and J. Ross Mitchell, "The S Transform in Medical Imaging," University of Calgary Seaman Family MR Research Centre Foothills Medical Centre, Canada.
↑ Prakash K. Ray, et al. "Coherency determination in grid-connected distributed generation based hybrid system under islanding scenarios." Power and Energy (PECon), 2010 IEEE International Conference on. IEEE, 2010,doi:10.1109/PECON.2010.5697562

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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)
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