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.

本文的主要目的是在时分频率分析的背景下介绍基于卷积的广义Stockwell变换。本文的精神与两项现有研究完全不同（请参见DP Xu和K. Guo [ Appl。Geophys。9（2012）73-79]和SK Singh [ J. Pseudo-Differ。Oper。Appl。4（2013）251–265]），在某种意义上，我们的方法完全依赖于与分数傅立叶变换相关的卷积结构。我们首先研究广义Stockwell变换的所有基本属性，包括分数Wigner分布与拟议变换之间的关系。在续篇中，我们介绍了拟议变换的半离散和离散对应。通过建立分数傅里叶域中的广义Stockwell变换的一些Heisenberg型不等式，我们结束了我们的研究。