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 <a href="/facts/Short-time_Fourier_transform/XHo1t2m6">short-time Fourier transform</a> (STFT), extending the <a href="/facts/Continuous_wavelet_transform/nGUoUAKX">continuous wavelet transform</a> 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 <a href="/facts/Gabor_transform/dbbHrA1l">Gabor transform</a>. However, the S transform has its own disadvantages: the clarity is worse than <a href="/facts/Wigner_distribution_function/l5lUlUDV">Wigner distribution function</a> and <a href="/facts/Cohen%27s_class_distribution_function/zS9J0gHy">Cohen's class distribution function</a>.
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 <a href="/facts/Open_source_license/FlFTwHmd">open source license</a>.
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.

S transform open-in-new

S transform