A Generalization of the Fractional Stockwell Transform

It is well known that the Fourier transform is an efficient tool in signal processing. To address the limitations of the Fourier transform, researchers have explored various integral transforms that extend the classical Fourier transform. In [1], Namias introduced the fractional Fourier transform, which depends on a real parameter $\alpha$. When $\alpha ={\displaystyle \frac{\pi }{2}}$, this fractional Fourier transform reduces to the Fourier transform. The fractional Fourier transform has applications in optics [2], quantum mechanics [1,3], and signal processing [4]. The linear canonical transform [5], a generalization of both the Fourier and fractional Fourier transforms, is used in quantum mechanics [6] and optical systems [7].

The special affine Fourier transform (SAFT), introduced by Abe and Sheridan [8,9,10], stands out as a more general and versatile approach among the generalizations of the Fourier transform. The SAFT presents a time–frequency representation of a signal in quantum mechanics and optics, characterized by six real parameters $a,b,c,d,p,\mathrm{and}q$, with $ad-bc=1$ and $b\ne 0$. Notably, the SAFT includes the generalizations of the Fourier transform, such as fractional Fourier transform [1,2] and linear canonical transform [5,11], as special cases determined by specific parameter choices. It also includes pivotal transforms such as Laplace, Fresnel [12], Bargmann, Gauss–Weierstrass, offset Fourier, and offset fractional Fourier [13] within its framework. For more details, one can refer to [14]. Also, the special affine Fourier transform has been discussed on quaternion-valued functions [15] and on octonian-valued functions [16]. Using the SAFT, there are many hybrid transforms in the literature—for instance, see windowed SAFT [17], short-time SAFT [18], special affine wavelet transform [19], and special affine Stockwell transform [20].

In this paper, we introduce a hybrid integral transform involving SAFT and Stockwell transform. We now briefly recall the theory of Stockwell transform from the literature. Stockwell et al. [21] introduced the Stockwell transform, commonly referred as the S transform, which employs a Gaussian window function that dilates and translates to achieve localization of the signal $f\in L^2\left(\mathbb{R}\right)$. For a one-dimensional signal f, it is defined as follows. In 2008, Du et al. [22] introduced a Stockwell transform using a general window function $\varphi \in L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$ instead of a Gaussian function and obtained an inversion formula. The Stockwell transform with a general window function is given as followers: The Stockwell transform is widely applied in geophysics [23] to provide more detailed information about spectral (frequency) components.

Hybrid versions of the S transform with the kernels of the generalized Fourier transforms have been extensively studied. Xu and Guo [24] extended the classical Stockwell transform by introducing the fractional S transform, which incorporates a single real parameter. Du et al. [25] also explored the applications of fractional S transform in seismic data analysis. The paper in [26] introduced a fractional Stockwell transform with a general window function instead of a particular window function as in [24,25]. More recently, the fractional S transform has been investigated in higher-dimensional Euclidean spaces and quaternion function spaces [27,28]. In this direction, the linear canonical Stockwell transform (LCST) [29,30,31,32] and offset linear canonical Stockwell transform [33] have also been recently investigated.

This paper aims to develop a novel integral transform called generalized fractional Stockwell transform (GFST), by utilizing the kernel of the special affine Fourier transform (SAFT). Hence, the transform may also be referred as the special affine Stockwell transform (SAST). To achieve this, we apply the special affine convolution introduced in [18]. Using the SAFT kernel in the GFST definition, the transform gains greater flexibility and broadens the applicability of the Stockwell transform across various fields. It is worth noting that [20] addresses a form of the special affine Stockwell transform. As one of the SAFT parameters, q, is not included and the parameter p is involved only in the constant factor $e^{{\displaystyle \frac{i}{2B}}D{p}^2}$, the transform introduced in [20] is essentially a linear canonical S transform with an additional constant factor involving p.

The structure of this paper is organized as follows. Section 2 introduces the fundamental notations, definitions, and results. In Section 3, the definition of the generalized fractional Stockwell transform (GFST) and its key theorems are presented. This section emphasizes proving Parseval’s identity and, based on that, deriving the inversion formula for the GFST. Additionally, a theorem that characterizes the range of the GFST is included. Section 4 then addresses another key topic in the paper: the uncertainty principle. In this section, we demonstrate the uncertainty principle related to the GFST, using the uncertainty principle of the special affine Fourier transform and several supporting lemmas.

Throughout this article, the symbol $\mathbf{A}$ denotes the six real parameters $a,b,c,d,p,$ and q such that $ad-bc=1$ with $b\ne 0$ and the symbol $\mathit{\lambda }$ denotes the number $bq-dp$. We denote any $\mathbf{A}$ that satisfies the above-stated condition as the parameter matrix with a representation $\mathbf{A}=\left(\begin{array}{ccc}a\hfill & b\hfill & p\hfill \\ c\hfill & d\hfill & q\hfill \end{array}\right).$ Furthermore, we make use of the following notations from the paper [18] for easier computation. As usual, the set of reals, non-zero reals, and complex numbers are denoted by $\mathbb{R}$,${\mathbb{R}}^{\ast }$, and $\mathbb{C}$, respectively. The space of integrable and square-integrable complex valued functions on $\mathbb{R}$ are given by $L^1(\mathbb{R},\mathbb{C})$ and $L^2(\mathbb{R},\mathbb{C})$, respectively, with the norm and inner product We denote the Hilbert space of square-integrable complex valued functions on $\mathbb{R}\times {\mathbb{R}}^{\ast }$ by $L^2(\mathbb{R}\times {\mathbb{R}}^{\ast },\mathbb{C})$ with the inner product given by For $\kappa >0$ and $t\in \mathbb{R}$, the dilation and translation of a function f are given by respectively.

Next, we recall necessary definitions and required results. If the Fourier transform of f is given by then we have the following well-known results [34]:

According to Ref. [14], for $f\in L^1(\mathbb{R},\mathbb{C})$, the special affine Fourier transform (SAFT) of f, denoted by ${\mathcal{S}}_{\mathbf{A}}f$, is given by where $K_{\mathbf{A}}(x,y)={\displaystyle \frac{1}{\sqrt{2\pi \left|b\right|}}}{\Phi }_{a,b}\left(x\right){\Omega }_{\mathbf{A}}\left(y\right){\mu }_{(p-y),b}\left(x\right)$.

Notably, if ${\mathbf{A}}^{★}=\left(\begin{array}{ccc}d& -b& bq-dp\\ -c& a& cp-aq\end{array}\right)$, then ${\mathbf{A}}^{★}$ is also a parameter matrix whenever $\mathbf{A}$ is a parameter matrix and the inverse of ${\mathcal{S}}_{\mathbf{A}}$ is given by ${\mathcal{S}}_{{\mathbf{A}}^{★}}$. Precisely, the inversion formula for SAFT is given as follows.

For a given parameter matrix $\mathbf{A}$, $(y,\xi )\in \mathbb{R}\times {\mathbb{R}}^{\ast }$, and $0\ne g\in L^1(\mathbb{R},\mathbb{C})\cap L^2(\mathbb{R},\mathbb{C})$, we let $g_{y,\xi }^{\mathbf{A}}\left(x\right)=\left|\xi \right|g\left(\xi (x-y)\right){\Phi }_{a,b}\left(y\right)\overline{{\Phi }_{a,b}\left(x\right)}{\mu }_{p-\mathit{\lambda },b}(y-x){\mu }_{\xi ,b}\left(x\right)$, $\forall x\in \mathbb{R}$, and let $\mathit{\lambda }=bq-dp$.