A wavelet-based study on phase and magnitude relationships of the Stockwell transform

This paper establishes a relationship between the phase and log-magnitude of the Stockwell transform (S-transform). The proposed relationship is derived by defining the S-Transform in terms of wavelet functions. The proposed work is an extension of the study [Holighaus N., Koliander G., Průša Z., Abreu L. D., Characterization of analytic wavelet transforms and a new phaseless reconstruction algorithm, IEEE Trans. Signal Process. 67(15):3894–3908, 2019] carried out to establish a relationship between the phase and magnitude of the continuous wavelet transform. Our methodology exploits the relationship between partial derivatives of the real and imaginary parts of the wavelet and S-transform for a couple of window functions (Gaussian and bi-Gaussian). Apart from the continuous case, these relationships are explicitly shown for the discrete version of the S-transform.