
S transform

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.
A fast "S" Transform algorithm was invented in 2010. It reduces the computational complexity from O[N2·log(N)] to O[N·log(N)] and makes the transform one-to-one, where the transform has the same number of points as the source signal or image, compared to storage complexity of N2 for the original formulation. An implementation is available to the research community under an open source license.
A general formulation of the S transform makes clear the relationship to other time frequency transforms such as the Fourier, short time Fourier, and wavelet transforms.
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 formula_3 and formula_4.<br>
Applying the Fourier Transform to both formula_3 and formula_4 gives 
From the Spectrum Form of S-transform, we can derive the discrete time S-transform. <br>
Let formula_8, where formula_9 is the sampling interval and formula_10 is the sampling frequency.<br>
The Discrete time S-transform can then be expressed as: <br>
Implementation of Discrete Time S-transform.
Below is the Pseudo code of the implementation.<br>
 Step1.Compute formula_12 <br>
 loop{
 Step2.Computeformula_13for formula_14<br>
 Step3.Move formula_15toformula_16<br>
 Step4.Multiply Step2 and Step3 formula_17<br>
 Step5.IDFT(formula_18).
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 formula_19, 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. 
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.
References.
[[Category:Integral transforms]]
[[Category:Fourier analysis]]
[[Category:Time–frequency analysis]]