A family of convolution-based generalized Stockwell transforms

Abstract

The main purpose of this paper is to introduce a family of convolution-based generalized Stockwell transforms in the context of time-fractional-frequency analysis. The spirit of this article is completely different from two existing studies (see D. P. Xu and K. Guo [Appl. Geophys. 9 (2012) 73–79] and S. K. Singh [J. Pseudo-Differ. Oper. Appl. 4 (2013) 251–265]) in the sense that our approach completely relies on the convolution structure associated with the fractional Fourier transform. We first study all of the fundamental properties of the generalized Stockwell transform, including a relationship between the fractional Wigner distribution and the proposed transform. In the sequel, we introduce both the semi-discrete and discrete counterparts of the proposed transform. We culminate our investigation by establishing some Heisenberg-type inequalities for the generalized Stockwell transform in the fractional Fourier domain.