Reassigning and synchrosqueezing the Stockwell Transform: Complementary proofs
Résumé
The S-transform is an efficient tool for time-frequency analysis that gained interest in many signal processing
application fields. This is due to its ability to produce an analysis with a frequency dependent resolution, as for the wavelet
transform, while keeping a direct relation with the Short-Time Fourier Transform (STFT). In this paper, the S-transform is first
briefly recalled with its properties and its relationships with the STFT and the Continuous Wavelet Transform (CWT). Then, we
propose to enhance the S-transform using the reassignment and the synchrosqueezing methods, which have been already successfully
applied to the STFT and the CWT. Thanks to these methods, we provide a “sharpen” time-frequency representation from the
S-transform while allowing a signal reconstruction thanks to the synchrosqueezing method. The Levenberg-Marquardt algorithm
is also used to provide reassignment and synchrosqueezing operators for the S-transform that allow a user to adjust the
strength of the resulting time-frequency distribution. We also derive for the S-transform a second-order synchrosqueezing
operator that improves the time-frequency localization of the synchrosqueezed S-transform by considering signals with frequency-modulated
components. Finally, numerical results obtained with the reassigned and synchrosqueezed S-transforms are compared
with the Gabor STFT and the Morlet CWT.
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