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. 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 to O 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. 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. The above definition implies that the s-transform function can be express as the convolution of ( x ( τ ) e − j 2 π f τ ) {displaystyle (x( au )e^{-j2pi f au })} and ( | f | e − π t 2 f 2 ) {displaystyle (|f|e^{-pi t^{2}f^{2}})} .Applying the Fourier Transform to both ( x ( τ ) e − j 2 π f τ ) {displaystyle (x( au )e^{-j2pi f au })} and ( | f | e − π t 2 f 2 ) {displaystyle (|f|e^{-pi t^{2}f^{2}})} gives From the Spectrum Form of S-transform, we can derive the discrete time S-transform. Let t = n Δ T f = m Δ F α = p Δ F {displaystyle t=nDelta _{T},,f=mDelta _{F},,alpha =pDelta _{F}} , where Δ T {displaystyle Delta _{T}} is the sampling interval and Δ F {displaystyle Delta _{F}} is the sampling frequency.The Discrete time S-transform can then be expressed as: Below is the Pseudo code of the implementation. The only difference between Gabor Transform (GT) and S Transform is the window size. For GT, the windows size is a Gaussian function ( e − π ( t − τ ) 2 ) {displaystyle (e^{-pi (t- au )^{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 has a better clarity in the time domain. As table below.