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 (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.
Contents
- Definition
- Modified Form
- Implementation of Discrete Time S transform
- Comparison with Gabor Transform
- Comparison with Wigner Transform
- Comparison with the short time Fourier transform
- Applications
- References
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.
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.
Modified Form
The above definition implies that the s-transform function can be express as the convolution of
Applying the Fourier Transform to both
From the Spectrum Form of S-tansform, we can derive the discrete time S-transform.
Let
The Discrete time S-transform can then be expressed as:
Implementation of Discrete Time S-transform
Below is the Pseudo code of the implementation.
loop{ Step2.Compute
Step3.Move
Step4.Multiply Step2 and Step3
Step5.IDFT(
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
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 with the Wigner Transform is the cross term, which stems from the auto-correlation function in the Wigner Transform function. This cross term may cause noise and distortions in signal analyses. S-transform analyses avoid this issue.
Comparison with the short-time Fourier transform
We can compare the S transform and short-time Fourier transform (STFT). 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.