S transform as a time–frequency distribution was developed in 1994 for analyzing geophysics data.
For one, modulation sinusoids are fixed with respect to the time axis; this localizes the scalable Gaussian window dilations and translations in S transform.
However, the S transform has its own disadvantages: the clarity is worse than Wigner distribution function and Cohen's class distribution function.
[citation needed] A fast S transform algorithm was invented in 2010.
[3][4] 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.
[4][5] An implementation is available to the research community under an open source license.
The above definition implies that the s-transform function can be expressed as the convolution of
The Discrete time S-transform can then be expressed as: Below is the Pseudo code of the implementation.
For GT, the windows size is a Gaussian function
, 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.
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 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.
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.
On the other hand, the two high frequencies bursts are not detected by STFT.