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: it requires higher complexity computation (because FFT cannot 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.[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_x(t,f) = \int_{-\infty}^{\infty} x(\tau)|f|e^{- \pi (t- \tau)^2 f^2} e^{-j2 \pi f \tau} \, d \tau
x(\tau) = \int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}S_x(t,f)\, dt\right]\,e^{j2\pi f\tau}\, df

Modified 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 .

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.Computee^{-\pi \frac{p^2}{m^2}}for f=m\Delta_{F}

     Step3.Move X[p\Delta_{F}]toX[(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

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 have 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.

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.

Discussion

We can compare the S transform and short-time Fourier transform (STFT).[2] 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

See also

References

  1. ↑ Stockwell, RG, L Mansinha, and RP Lowe (1996). Localization of the complex spectrum: the S transform, IEEE Transactions on Signal Processing 44 (4), p 998-1001.
  2. ↑ 2.0 2.1 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.
  3. ↑ R. A. Brown and R. Frayne, "A fast discrete S-transform for biomedical signal processing", University of Calgary Seaman Family MR Research Centre Foothills Medical Centre, Canada. http://www.ncbi.nlm.nih.gov/pubmed/19163232
  4. ↑ Kelly Sansom, "Fast S Transform", University of Calgary, http://www.ucalgary.ca/news/utoday/may31-2011/computing
  5. ↑ http://sourceforge.net/projects/fst-uofc/
  6. ↑ 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
  7. ↑ Hongmei Zhu, and J. Ross Mitchell, "The S Transform in Medical Imaging," University of Calgary Seaman Family MR Research Centre Foothills Medical Centre, Canada.