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.

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“…So far in the literature, we have not seen any paper on the MRA that is associated with the ST [3], FrST [15], or LCST [18]. This paper deals with the novel way of defining an MRA and constructing some novel ONB for $$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$$ from the newly defined MRA.…”

Two-sided quaternion wave-packet transform and the quantitative uncertainty principles

“…Although the Gabor representations are quite handy, however, such representations are not adequate for signals having high frequency components for shorter durations and low frequency components for longer durations, leading to the birth of time-scale integral transform, often known as the wavelet transform [11,26,33,36]. As of now, several generalizations of the classical wavelet transform have been reported in recent years including the fractional wavelet transform [32,34,35], linear canonical wavelet transform [28,29], quadratic-phase and special affine wavelet transform [30]. Owing to the lucid nature and close resemblance with the conventional Fourier transform, the wavelet transforms have fascinated the mathematical, physical, chemical, biological and engineering communities with their versatile applicability [37,38].…”