# S transform

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S transform

In mathematics, the S transform is usually realized as a Laplace transform. However, S transform in the time frequency distribution was developed in 1994 for analyzing geophysics data. [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.] 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.

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.

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

Discussion

Based on R. G. Stockwell et Al.'s work, 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 of the S transform with the STFT. The S transform characteristic of frequency dependent resolution allows the detection of the high frequency burst. On the other hand, as STFT consist of a constant window width, it leads to the result having poorer definition. As 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

* Magnetic Resonance imaging (MRI)
* Power System Disturbance Recognition (This is based on Jaya Bharata Reddy et Al.'s work)
** 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 multi resolution analysis.

ee also

* Laplace transform
* wavelet transform
* Short-time Fourier transform

References

* J. J. Ding, "time-frequency analysis and wavelet transform course note," the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
* R. G. Stockwell, L. Mansinha, and R. P. Lowe, "Localization of the complex spectrum: the S transform," IEEE Trans. Signal Processing, vol.44, no.4, pp.998-1001, Apr.1996.
* 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.
* 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 , McGrawHill Book Company, New York, 1978
* E. O. Brigham, The Fast Fourier Transform , Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1974
* L. Cohen, “Time-frequency distributions - A review”, Proc. IEEE, vol. 77, no. 7, July 1989
* I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis”, IEEE Trans. on Information Theory, vol. 36, no. 5, Sept. 1990
* M. Farge, “Wavelet transforms and their application to turbulence”, Annual Review of Fluid Mechanics, vol. 24, pp. 395-457, 1992
* D. Gabor, “Theory of communication”, J. Inst. Elect. Eng. , vol. 93, no. 3, pp. 429-457, 1946
* P. Goupillaud, A. Grossmann, and J. Morlet, “Cycle-octave and related transforms in seismic analysis”, Geoexploration, vol. 23 pp. 85-102, 1984
* F. Hlawatsch and G. F. Boudreuax-Bartels, “Linear and quadratic timefrequency signal representations”, IEEE SP Magazine, pp. 21-67, April 1992
* O. Rioul and M. Vetterli, “Wavelets and signal processing”, IEEE SP Magazine, vol. 8 pp. 14-38, 1991
* R. K. Young, Wavelet Theory and its Applications, Kluwer Academic Publishers, Dordrecht,1993

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