A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

Abstract

The quaternion linear canonical transform is an important tool in applied mathematics and it is closely related to the quaternion Fourier transform. In this work, using a symmetric form of the two-sided quaternion Fourier transform (QFT), we first derive a variation on the Heisenberg-type uncertainty principle related to this transformation. We then consider the general two-sided quaternion linear canonical transform. It may be considered as an extension of the two-sided quaternion linear canonical transform. Based on an orthogonal plane split, we develop the convolution theorem that associated with the general two-sided quaternion linear canonical transform and then derive its correlation theorem. We finally discuss how to apply general two-sided quaternion linear canonical transform to study the generalized swept-frequency filters.

1. Introduction

The general two-sided quaternion Fourier transform was firstly proposed by Hitzer in [1,2]. As we all know, it is a generalized form of the two-sided quaternion Fourier transform. Hitzer further has defined the two-sided quaternion Fourier transform based on the orthogonal planes split and demonstrated some fundamental properties of this transform. In [3], Hitzer also extended general two-sided quaternion Fourier transform within the context of Clifford algebra, named general two-sided Clifford Fourier transform, and derived its convolution and correlation theorems.

In recent years, many researchers have shown interest in a kind of new signal representation tool, known as the quaternion linear canonical transformations (QLCT). It is well known that the QLCT is a nontrivial generalization of the classical linear canonical transform [4,5]. It also can be regarded as the generalization of the quaternion Fourier transform (QFT) because the QFT is a special case of the QLCT. Several useful properties of the QLCT have been extensively studied (see, e.g., [6,7,8,9,10,11] and references therein). As in the quaternion Fourier transform case [12,13,14,15,16,17,18], there are three different kinds of two-dimensional quaternion linear canonical transforms (QLCTs). They are called left-sided QLCT, right-sided QLCT, and two-sided QLCT, respectively.

In this paper, based on the symmetric decomposition of the two-sided quaternion Fourier transform (QFT), we first provide the derivation of a variation on the Heisenberg-type uncertainty principle related to this transformation. The uncertainty principle describes the interaction between a quaternion function and its QFT. To achieve this, we recall the component-wise uncertainty principle for the QFT and show that it is a special case of the proposed uncertainty principle. We then propose general two-sided quaternion linear canonical transform. We further provide a definition of its convolution operator. This definition is constructed by combining convolution definitions for the classical linear canonical transform and the quaternion Fourier transform. We establish its convolution theorem, which give a significant result of general two-sided quaternion linear canonical transform. This theorem is inspired by the work of Hitzer [19] who established the convolution theorem associated with general two-sided quaternion Fourier transform. Paper [20,21] proposed similar work for convolution theorem, but different type of the quaternion linear canonical transform which established the convolution theorem for the two-sided quaternion linear canonical transform. We finally derive correlation theorem of continuous quaternion signals associated with the general two-sided quaternion linear canonical transform.

The main content of this article is as follows. In Section 2, we recall the basic knowledge of quaternion algebra and orthogonal planes split that will be needed during the paper. Section 3 is devoted to the derivation of a variation on the Heisenberg-type uncertainty principle related to the two-sided quaternion Fourier transform. In Section 4 we recall the general two-sided quaternion Fourier transform. In Section 5 we discuss convolution definition for general two-sided quaternion linear canonical transform and obtain its convolution theorem. In Section 6, a correlation theorem related to general two-sided quaternion linear canonical transform is presented. In Section 7 an application of general two-sided quaternion linear canonical transform is studied. Lastly, the summary of this article is included in Section 8.

2. Quaternions

For the basic notations and definitions on quaternion algebra, see [22,23,24]. Quaternions are hypercomplex numbers, which requires an associative noncommutative four-dimensional algebra. They can be expressed as

$$\mathbb{H}=\{h=h_o+\mathit{i}{h}_a+\mathit{j}{h}_b+\mathit{k}{h}_c;h_a,h_a,h_b,h_c\in \mathbb{R}\},$$

where the elements $\mathit{i},\mathit{j}$, and $\mathit{k}$ have properties:

$$\begin{array}{c}\hfill {\mathit{i}}^2={\mathit{j}}^2={\mathit{k}}^2=\mathit{ijk}=-1.\end{array}$$

(1)

For every quaternion $h\in \mathbb{H}$, the scalar and vector parts of h are denoted by $Sc\left(h\right)=h_o$ and $vec\left(h\right)=\mathbf{h}=\mathit{i}{h}_a+\mathit{j}{h}_b+\mathit{k}{h}_c$, respectively.

The conjugate element $\overline{h}$ is given by

$$\overline{h}=h_o-\mathit{i}{h}_a-\mathit{j}{h}_b-\mathit{k}{h}_c,$$

(2)

which fulfills

$$\overline{hp}=\overline{p}\overline{h},\forall h,p\in \mathbb{H}.$$

From (2) we obtain the norm of a quaternion h in the form

$$|h|=\sqrt{h\overline{h}}=\sqrt{h_o^2+h_a^2+h_b^2+h_c^2}.$$

(3)

The modulus of a product of two quaternion obeys the property

$$|hp|=|h||p|.$$

It is not difficult to see that

$$Sc\left(h\right)\le |h|\mathrm{and}|\mathbf{h}|\le |h|.$$

(4)

Applying the conjugate (2) and the norm of h gives the inverse of nonzero quaternion h as

$$h^{-1}=\frac{\overline{h}}{{|h|}^2}.$$

The inner product for two quaternion-valued functions $h,g:{\mathbb{R}}^2\to \mathbb{H}$ is defined as

$$(h,g)={\int }_{{\mathbb{R}}^2}h\left(\mathit{z}\right)\overline{g\left(\mathit{z}\right)}d\mathit{z},d\mathit{z}=d{z}_{1}d{z}_2,$$

(5)

for all $\mathit{z}=(z_1,z_2)\in {\mathbb{R}}^2$. In particular, for $h=g$, we obtain

$${\parallel h\parallel }_{{\mathbb{L}}^2({\mathbb{R}}^2;\mathbb{H})}=\sqrt{(h,h)}={\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\right)}^{1/2}.$$

(6)

For $1\le q<\infty ,$ (6) becomes

$${\parallel h\parallel }_q={\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{q}d\mathit{z}\right)}^{1/q}.$$

(7)

Following Hitzer’s work [1], we introduce the definition of orthogonal 2D planes split (OPS) with respect to any two pure unit quaternions $I_1,I_2$ as follows

Definition 1.

Let $I_1,I_2\in \mathbb{H}$ be an arbitrary pair of pure quaternions $I_1,I_2,I_1^2=I_2^2=-1$, including the cases $I_1=\pm I_2$. For any $h\in \mathbb{H}$ we introduce the $h_{\pm }$ OPS split parts with respect to the two pure unit quaternion $I_1,I_2$ as

$$\begin{array}{c}\hfill h_{\pm }=\frac{1}{2}(h\pm I_{1}h{I}_2).\end{array}$$

(8)

Moreover one has for $\alpha ,\beta \in \mathbb{R}$

$$\begin{array}{c}\hfill e^{\alpha I_1}{h}_{\pm }{e}^{\beta I_2}=e^{(\alpha \mp \beta )I_1}{h}_{\pm }=h_{\pm }{e}^{(\beta \mp \alpha )I_2}.\end{array}$$

(9)

and

$$\begin{array}{c}\hfill e^{\beta I_2}\overline{h_{\pm }}{e}^{\alpha I_1}=e^{(\beta \mp \alpha )I_2}\overline{h_{\pm }}=\overline{h_{\pm }}{e}^{(\beta \mp \alpha )I_1}.\end{array}$$

(10)

Observe that for $\alpha =\pi /2,\beta =0$ and $\alpha =0,\beta =\pi /2$ Equation (9) will lead to

$$\begin{array}{c}\hfill I_1{h}_{\pm }=\mp h_{\pm }{I}_2,h_{\pm }{I}_2=\mp I_1{h}_{\pm }.\end{array}$$

(11)

3. A Variation on Heisenberg’s Inequality for Quaternion Fourier Transform

We start this part by introducing the definition of the two-sided quaternion Fourier transform (QFT) and present its useful properties. We use these results to obtain one of the main results in this study.

Definition 2.

Given h in $L^1({\mathbb{R}}^2;\mathbb{H})$. We define

$$\begin{array}{c}\hfill {\mathcal{F}}_H\{h\}\left(\mathit{u}\right)={\int }_{{\mathbb{R}}^2}{e}^{-\mathit{i}{u}_1{z}_1}h\left(\mathit{z}\right)e^{-\mathit{j}{u}_2{z}_2}d\mathit{z}.\end{array}$$

(12)

We call ${\mathcal{F}}_H\{h\}$ the two-sided quaternion Fourier transform of h.

Definition 3.

For any $h\in L^1({\mathbb{R}}^2;\mathbb{H}$ for which ${\mathcal{F}}_H\{h\}\in L^1({\mathbb{R}}^2;\mathbb{H})$, its inverse is defined by

$$\begin{array}{c}\hfill {\mathcal{F}}_H^{-1}\left\{{\mathcal{F}}_H\{h\}\right\}\left(\mathit{z}\right)=h\left(\mathit{z}\right)=\frac{1}{{\left(2\pi \right)}^2}{\int }_{{\mathbb{R}}^2}{e}^{\mathit{i}{u}_1{z}_1}{\mathcal{F}}_H\{h\}\left(\mathit{u}\right)e^{\mathit{j}{u}_2{z}_2}d\mathit{u}.\end{array}$$

(13)

Now we put $u_1=2\pi {\omega }_1$ and $u_2=2\pi {\omega }_2$ on the right-hand sides of (12) and (13), one can easily obtain

$${\mathcal{F}}_H\{h\}\left(\omega \right)={\int }_{{\mathbb{R}}^2}{e}^{-\mathit{i}2\pi {\omega }_1{z}_1}h\left(\mathit{z}\right)e^{-\mathit{j}2\pi {\omega }_2{z}_2}d\mathit{z},$$

(14)

and

$$h\left(\mathit{z}\right)={\int }_{{\mathbb{R}}^2}{e}^{\mathit{i}2\pi {\omega }_1{z}_1}{\mathcal{F}}_H\{h\}\left(\omega \right)e^{\mathit{j}2\pi {\omega }_2{z}_2}d\omega .$$

(15)

By (14) the decomposition of the quaternion function f will lead to

$$\begin{array}{c}\hfill {\mathcal{F}}_H\{h\}\left(\omega \right)={\int }_{{\mathbb{R}}^2}{e}^{\mathit{i}2\pi {\omega }_1{z}_1}(h_o\left(\mathit{z}\right)+\mathit{i}{h}_a\left(\mathit{z}\right)+\mathit{j}{h}_b\left(\mathit{z}\right)+\mathbf{k}{h}_c\left(\mathit{z}\right))e^{\mathit{j}2\pi {\omega }_2{z}_2}d\mathit{z}.\end{array}$$

(16)

In the symmetric form, the above identity may be rewritten as

$$\begin{array}{cc}\hfill {\mathcal{F}}_H\{h\}\left(\omega \right)& ={\int }_{{\mathbb{R}}^2}{e}^{\mathit{i}2\pi {\omega }_1{z}_1}(h_o\left(\mathit{z}\right)+\mathit{i}{h}_a\left(\mathit{z}\right)+\mathit{j}{h}_b\left(\mathit{z}\right)+\mathit{i}{h}_c\left(\mathit{z}\right)\mathit{j})e^{\mathit{j}2\pi {\omega }_2{z}_2}d\mathit{z}\hfill \\ \hfill & ={\mathcal{F}}_H\{h_o\}\left(\omega \right)+\mathit{i}{\mathcal{F}}_H\{h_a\}\left(\omega \right)+{\mathcal{F}}_H\{h_b\}\left(\omega \right)\mathit{j}+\mathit{i}{\mathcal{F}}_H\{h_c\}\left(\omega \right)\mathit{j}.\hfill \end{array}$$

(17)

Now we define the module of ${\mathcal{F}}_H\{f\}\left(\omega \right)$ as

$$\begin{array}{c}\hfill |{\mathcal{F}}_H{\{f\}\left(\omega \right)|}_H={\left(|{\mathcal{F}}_H\{h_o\}\left(\omega \right){|}^2+|{\mathcal{F}}_H\{h_a\}\left(\omega \right){|}^2+|{\mathcal{F}}_H\{h_b\}\left(\omega \right){|}^2+{|{\mathcal{F}}_H\{h_c\}\left(\omega \right)|}^2\right)}^{1/2}.\end{array}$$

(18)

Furthermore, we obtain the $L^p({\mathbb{R}}^2;\mathbb{H})$-norm

$$\parallel {\mathcal{F}}_H{\{h\}\parallel }_{H,p}={\left({\int }_{{\mathbb{R}}^2}{|{\mathcal{F}}_H\{h\}\left(\omega \right)|}_H^{p}d\omega \right)}^{1/p}.$$

(19)

Lemma 1.

Suppose that $h\in L^1({\mathbb{R}}^2;\mathbb{H})\cap L^2({\mathbb{R}}^2;\mathbb{H})$. If $\frac{\partial }{{\omega }_k}h$ with $k=1,2$ exists and belongs to $L^2({\mathbb{R}}^2;\mathbb{H})$, then for every $n\in \mathbb{N}$ we have

$${\mathcal{F}}_H^{-1}\left\{\frac{{\partial }^n}{\partial {\omega }_1^n}h\right\}\left(\mathit{z}\right)={(-\mathit{i}2\pi z_1)}^n{\mathcal{F}}_H\{h\}\left(\mathit{z}\right),$$

(20)

and

$${\mathcal{F}}_H^{-1}\left\{\frac{{\partial }^n}{\partial {\omega }_2^n}h\right\}\left(\mathit{z}\right)={\mathcal{F}}_H\{h\}\left(\mathit{z}\right){(-\mathit{j}2\pi z_2)}^n.$$

(21)

Proof. 

For $n=1$ applying (15) results in

$$\begin{array}{cc}\hfill {\mathcal{F}}_H^{-1}\left\{\frac{\partial }{\partial {\omega }_1}h\right\}\left(\mathit{z}\right)& ={\int }_{{\mathbb{R}}^2}{e}^{\mathit{i}2\pi {\omega }_1{z}_1}\frac{\partial }{\partial {\omega }_1}h\left(\omega \right)e^{\mathit{j}2\pi {\omega }_2{z}_2}d\omega \hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\frac{\partial }{\partial {\omega }_1}{e}^{\mathit{i}2\pi {\omega }_1{z}_1}h\left(\omega \right)e^{\mathit{j}2\pi {\omega }_2{z}_2}d\omega \hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\mathit{i}2\pi z_1{e}^{\mathit{i}2\pi {\omega }_1{z}_1}h\left(\omega \right)e^{\mathit{j}2\pi {\omega }_2{z}_2}d\omega \hfill \\ \hfill & =\mathit{i}2\pi z_1{\mathcal{F}}_H\{h\}(-z_1,-z_1)\hfill \\ \hfill & =(-\mathit{i}2\pi (-z_1)){\mathcal{F}}_H\{h\}(-z_1,-z_1)\hfill \\ \hfill & =(-\mathit{i}2\pi y_1){\mathcal{F}}_H\{h\}\left(\mathit{y}\right),y_i=-z_i,i=1,2,\hfill \end{array}$$

(22)

and

$$\begin{array}{cc}\hfill {\mathcal{F}}_H^{-1}\left\{\frac{\partial }{\partial {\omega }_2}h\right\}\left(\mathit{z}\right)& ={\int }_{{\mathbb{R}}^2}{e}^{\mathit{i}2\pi {\omega }_1{z}_1}h\left(\omega \right)\frac{\partial }{\partial {\omega }_2}{e}^{\mathit{j}2\pi {\omega }_2{z}_2}d\omega \hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{e}^{\mathit{i}2\pi {\omega }_1{z}_1}h\left(\omega \right)e^{\mathit{j}2\pi {\omega }_2{z}_2}\mathit{j}2\pi z_{2}d\omega \hfill \\ \hfill & ={\mathcal{F}}_H\{h\}(-z_1,-z_1)(-\mathit{j}2\pi (-z_2))\hfill \\ \hfill & ={\mathcal{F}}_H\{h\}\left(\mathit{y}\right)(-\mathit{j}2\pi y_2),y_i=-z_i,i=1,2.\hfill \end{array}$$

(23)

By repeating this process $n-1$ additional times, we obtain (20). Using a similar argument as Equation (20), we can obtain the proof for Equation (21). □

By Riesz’s interpolation theorem, we obtain the Hausdorff–Young inequality (see [25]), that is

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_H{\{h\}\parallel }_{H,p^{\prime }}\le {\parallel h\parallel }_p,\end{array}$$

(24)

where $1\le p\le 2$ with $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. Taking the inversion formula of the QFT on both sides of (24), one has

$$\begin{array}{c}\hfill {\parallel h\parallel }_{p^{\prime }}\le {\parallel {\mathcal{F}}_H^{-1}\{h\}\parallel }_{H,p}.\end{array}$$

(25)

Below we state and prove the uncertainty principle associated with the two-sided quaternion Fourier transform. Firstly, we recall the component-wise uncertainty principle for the QFT that will be established in next theorem.

Theorem 1.

Suppose that $h\in L^1({\mathbb{R}}^2;\mathbb{H})\cap L^2({\mathbb{R}}^2;\mathbb{H})$ and that $\frac{\partial }{z_k}h$ exists. Then

$${\int }_{{\mathbb{R}}^2}{z}_k^2{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}{\int }_{{\mathbb{R}}^2}{\omega }_k^2{|{\mathcal{F}}_H\{h\}\left(\omega \right)|}^{2}d\omega \ge \frac{1}{16{\pi }^2}{\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\right)}^2,k=1,2.$$

(26)

Remark 1.

It should be noticed that Theorem 1 is valid for all types of the QFT.

It can be observed that for $1\le p\le 2$ we may change $L^2$- norm to $L^p$- norm on left-hand side of (26) and obtain the next result.

Theorem 2.

Under the conditions as above, we have

$${\left({\int }_{{\mathbb{R}}^2}{z}_k^p{|h\left(\mathit{z}\right)|}^{p}d\mathit{z}\right)}^{1/p}{\left({\int }_{{\mathbb{R}}^2}{\omega }_k^p{|{\mathcal{F}}_H\{h\}\left(\omega \right)|}_H^{p}d\omega \right)}^{1/p}\ge \frac{1}{4\pi }{\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z},k=1,2.$$

(27)

Proof. 

Due to (4) and the Holder’s inequality, we have

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}& =-2Sc\left({\int }_{\mathbb{R}}{z}_{k}h\left(\mathit{z}\right)\overline{\frac{\partial }{\partial z_k}h\left(\mathit{z}\right)}dx\right)\hfill \\ \hfill & \le 2\left|{\int }_{\mathbb{R}}{z}_{k}h\left(\mathit{z}\right)\overline{\frac{\partial }{\partial z_k}h\left(\mathit{z}\right)}dx\right|\hfill \\ \hfill & \le 2{\left({\int }_{\mathbb{R}}{|z_{k}h\left(\mathit{z}\right)|}^{p}d\mathit{z}\right)}^{1/p}{\left({\int }_{\mathbb{R}}{|\frac{\partial }{\partial z_k}h\left(\mathit{z}\right)|}^{p^{\prime }}d\mathit{z}\right)}^{1/p^{\prime }}\hfill \\ \hfill & =2\parallel z_k{h\parallel }_p{\parallel \frac{\partial }{\partial z_k}h\parallel }_{p^{\prime }}.\hfill \end{array}$$

By virtue of the Hausdorff–Young inequality (25) we obtain

$$\begin{array}{cc}\hfill & \parallel \frac{\partial }{\partial z_k}{h\parallel }_{p^{\prime }}\hfill \\ \hfill & \le \parallel {\mathcal{F}}_H^{-1}\{\frac{\partial }{\partial {\omega }_k}h\}{\parallel }_{H,p}\hfill \\ \hfill & ={\left({\int }_{{\mathbb{R}}^2}\left(|{\mathcal{F}}_H^{-1}\{\frac{\partial }{\partial {\omega }_k}{h}_o\}{|}^p+|{\mathcal{F}}_H^{-1}\{\frac{\partial }{\partial {\omega }_k}{h}_a\}{|}^p+|{\mathcal{F}}_H^{-1}\{\frac{\partial }{\partial {\omega }_k}{h}_b\}{|}^p{|{\mathcal{F}}_H^{-1}\{\frac{\partial }{\partial {\omega }_k}{h}_c\}|}^p\right)d\mathit{z}\right)}^{1/p}.\hfill \end{array}$$

For $k=1$, we see from (20) that

$$\begin{array}{cc}\hfill & \parallel {\mathcal{F}}_H^{-1}\{\frac{\partial }{\partial {\omega }_1}h\}{\parallel }_{H,p}\hfill \\ \hfill & =({\int }_{{\mathbb{R}}^2}(|(-\mathit{i}2\pi z_1){\mathcal{F}}_H\{h_o\}{|}^p+{|(-\mathit{i}2\pi z_1){\mathcal{F}}_H\{h_a\}|}^p\hfill \\ \hfill & +|(-\mathit{i}2\pi z_1){\mathcal{F}}_H\{h_b\}{|}^p+{|(-\mathit{i}2\pi z_1){\mathcal{F}}_H\{h_c\}|}^p)d\mathit{z}{)}^{1/p}\hfill \\ \hfill & =2\pi {\left({\int }_{{\mathbb{R}}^2}\left(z_1^p|{\mathcal{F}}_H\{h_o\}{|}^p+z_1^p|{\mathcal{F}}_H\{h_a\}{|}^p+z_1^p|{\mathcal{F}}_H\{h_b\}{|}^p+z_1^p{|{\mathcal{F}}_H\{h_c\}|}^p\right)d\mathit{z}\right)}^{1/p}\hfill \\ \hfill & =2\pi {\left({\int }_{{\mathbb{R}}^2}{z}_1^p{|{\mathcal{F}}_H\{f\}|}_H^{p}d\mathit{z}\right)}^{1/p}.\hfill \end{array}$$

For $k=2$, we can apply similar arguments as above using (21), and obtain

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_H^{-1}\{\frac{\partial }{\partial {\omega }_2}h\}{\parallel }_{H,p}=2\pi {\left({\int }_{{\mathbb{R}}^2}{z}_2^p{|{\mathcal{F}}_H\{h\}|}_H^{p}d\mathit{z}\right)}^{1/p}.\end{array}$$

Hence,

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\le 4\pi {\parallel z_{k}h\parallel }_p{\left({\int }_{{\mathbb{R}}^2}{\omega }_k^p{|{\mathcal{F}}_H\{h\}|}_H^{p}d\omega \right)}^{1/p}.\end{array}$$

This ends the proof of the theorem. □

Remark 2.

The non-commutativity of the QFT kernel implies that Theorem 2 is slightly different to the right-sided QFT (compare to Theorem 12 of [26]). Below, Theorem 3 is not valid for the right-sided QFT and left-sided QFT.

Theorem 3.

If $h\in L^1({\mathbb{R}}^2;\mathbb{H})\cap L^2({\mathbb{R}}^2;\mathbb{H})$ and ${\mathcal{F}}_H\{h\}$ exists and is also in $L^2({\mathbb{R}}^2;\mathbb{H})$, then

$${\int }_{{\mathbb{R}}^2}(z_1^p+z_2^p)|h\left(\mathit{z}\right){|}^{p}d\mathit{z}{\int }_{{\mathbb{R}}^2}({\omega }_1^p+{\omega }_2^p){|{\mathcal{F}}_H\{h\}\left(\omega \right)|}^{p}d\omega \ge \frac{4}{{\left(4\pi \right)}^p}{\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\right)}^p,$$

(28)

for $1\le p\le 2$.

Proof. 

Simple computation shows that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}(z_1^p+z_2^p)|h\left(\mathit{z}\right){|}^{p}d\mathit{z}{\int }_{{\mathbb{R}}^2}({\omega }_1^p+{\omega }_2^p){|{\mathcal{F}}_H\{h\}\left(\omega \right)|}^{p}d\omega \hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{z}_1^p{|h\left(\mathit{z}\right)|}^{p}d\mathit{z}{\int }_{{\mathbb{R}}^2}{\omega }_1^p|{\mathcal{F}}_H{\{h\}\left(\omega \right)|}^{p}d\omega +{\int }_{{\mathbb{R}}^2}{z}_1^p{|h\left(\mathit{z}\right)|}^{p}d\mathit{z}{\int }_{{\mathbb{R}}^2}{\omega }_2^p{|{\mathcal{F}}_H\{h\}\left(\omega \right)|}^{p}d\omega \hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{z}_2^p{|h\left(\mathit{z}\right)|}^{p}d\mathit{z}{\int }_{{\mathbb{R}}^2}{\omega }_1^p|{\mathcal{F}}_H{\{h\}\left(\omega \right)|}^{2}d\omega +{\int }_{{\mathbb{R}}^2}{z}_2^p{|h\left(\mathit{z}\right)|}^{p}d\mathit{z}{\int }_{{\mathbb{R}}^2}{\omega }_2^p{|{\mathcal{F}}_H\{h\}\left(\omega \right)|}^{p}d\omega .\hfill \end{array}$$

By (26) we obtain

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^2}(z_1^p+z_2^p){|h\left(\mathit{z}\right)|}^{p}d\mathit{z}& {\int }_{{\mathbb{R}}^2}({\omega }_1^p+{\omega }_2^p){|{\mathcal{F}}_H\{h\}\left(\omega \right)|}^{p}d\omega \hfill \\ \hfill & \ge \frac{1}{{\left(4\pi \right)}^p}{\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\right)}^p+\frac{1}{{\left(4\pi \right)}^p}{\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\right)}^p\hfill \\ \hfill & +\frac{1}{{\left(4\pi \right)}^p}{\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\right)}^p+\frac{1}{{\left(4\pi \right)}^p}{\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\right)}^p\hfill \\ \hfill & =\frac{4}{{\left(4\pi \right)}^p}{\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\right)}^p.\hfill \end{array}$$

The proof is complete. □

It can be observed that for $p=2$ Theorem 3 changes to

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}{|\mathit{z}|}^2{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}{\int }_{{\mathbb{R}}^2}{|\omega |}^2{|{\mathcal{F}}_H\{h\}\left(\omega \right)|}^{2}d\omega \ge \frac{1}{4{\pi }^2}{\left({\int }_{{\mathbb{R}}^2}{|h\left(\mathit{z}\right)|}^{2}d\mathit{z}\right)}^2,\end{array}$$

(29)

which is directional uncertainty principle for the two-sided quaternion Fourier transform and right-sided quaternion Fourier transform [26].

4. General Two-Sided Quaternion Fourier Transform

In the following we introduce general two-sided quaternion Fourier transform, which is taken from some papers by Hitzer [2,3].

Definition 4.

For any $h\in L^1({\mathbb{R}}^2;\mathbb{H})$, the general two-sided quaternion Fourier transform of a quaternion function h with respect to two pure quaternions $I_1,I_2$ such that $I_1^2=I_2^2=-1$ is defined as

$$\begin{array}{c}\hfill {\mathcal{F}}^{I_1,I_2}\{h\}\left(\mathit{u}\right)={\int }_{{\mathbb{R}}^2}{e}^{-I_1{z}_1{u}_1}h\left(\mathit{z}\right)e^{-I_2{z}_2{u}_2}d\mathit{z},\end{array}$$

(30)

provided that the integral exists.

Especially, when $I_1=\mathit{i}$ and $I_2=\mathit{j}$ then (30) becomes

$$\begin{array}{c}\hfill {\mathcal{F}}_H\{h\}\left(\mathit{u}\right)={\int }_{{\mathbb{R}}^2}{e}^{-\mathit{i}{z}_1{u}_1}h\left(\mathit{z}\right)e^{-\mathit{j}{z}_2{u}_2}d\mathit{z},\end{array}$$

(31)

which is two-sided quaternion Fourier transform defined in (12).

For any ${\mathcal{F}}^{I_1,I_2}\{h\}\in L^1({\mathbb{R}}^2;\mathbb{H})$ the general quaternion Fourier transform mentioned above can be inverted by

$$\begin{array}{c}\hfill h\left(\mathit{z}\right)=\frac{1}{{\left(2\pi \right)}^2}{\int }_{{\mathbb{R}}^2}{e}^{I_1{z}_1{u}_1}{\mathcal{F}}^{I_1,I_2}\{h\}\left(\mathit{u}\right)e^{I_2{z}_2{u}_2}d\mathit{u}.\end{array}$$

(32)

From linearity of the integral (30) we obtain the general two-sided quaternion Fourier transform for the OPS split $h=h_{-}+h_{+}$ as

$$\begin{array}{cc}\hfill {\mathcal{F}}^{I_1,I_2}\{h\}\left(\mathit{u}\right)& ={\mathcal{F}}^{I_1,I_2}\{h_{-}+h_{+}\}\left(\mathit{u}\right)\hfill \\ \hfill & ={\mathcal{F}}^{I_1,I_2}\{h_{-}\}\left(u\right)+{\mathcal{F}}^{I_1,I_2}\{h_{+}\}\left(\mathit{u}\right).\hfill \end{array}$$

(33)

Definition 5.

Suppose that $h\in L^2({\mathbb{R}}^2;\mathbb{H})$ and $h_{\pm }=\frac{1}{2}(h\pm I_{1}h{I}_2)$. The general quaternion Fourier transform for the $h_{\pm }$ with respect to two linearly independent unit quaternions $I_1$ and $I_2$ is defined by

$$\begin{array}{c}\hfill {\mathcal{F}}_{\pm }^{I_1,I_2}\{h\}={\mathcal{F}}^{I_1,I_2}\{h_{\pm }\}={\int }_{{\mathbb{R}}^2}{e}^{-I_1{z}_1{u}_1}{h}_{\pm }{e}^{-I_2{z}_2{u}_2}d\mathit{z}.\end{array}$$

(34)

By using relation (9), the above identity can be rewritten in the form

$$\begin{array}{cc}\hfill {\mathcal{F}}^{I_1,I_2}\{h_{\pm }\}& ={\int }_{{\mathbb{R}}^2}{h}_{\pm }{e}^{-I_2(z_2{u}_2\mp z_1{u}_1)}d\mathit{z}\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & ={\int }_{{\mathbb{R}}^2}{e}^{-I_1(z_1{u}_1\mp z_2{u}_2)}{h}_{\pm }d\mathit{z}.\hfill \end{array}$$

(36)

5. General Two-Sided Quaternion Linear Canonical Transform

Because general two-sided QLCT is a generalization of two-sided QLCT, many useful properties of two-sided QLCT can be extended for general two-sided QLCT such as linearity, time shift, frequency shift, energy conservation, and uncertainty principles with some modifications. Another very important property of general two-sided QLCT is the convolution theorem. We first provide a definition of general two-sided QLCT and its relation to two-sided QLCT. We also present a theorem which describes the relationship between the orthogonal 2D plane split and general two-sided QLCT.

5.1. Definition of General Two-Sided QLCT

Definition 6

(General Two-sided QLCT). Suppose that $B_1=(a_1,b_1,c_1,d_1)=\left[\begin{array}{cc}{a}_1& b_1\\ c_1& d_1\end{array}\right]\in SL(2,\mathbb{R})$ and $B_2=(a_2,b_2,c_2,d_2)=\left[\begin{array}{cc}{a}_2& b_2\\ c_2& d_2\end{array}\right]\in SL(2,\mathbb{R})$. For any $h\in L^1({\mathbb{R}}^2;\mathbb{H})$, the general two-sided quaternion linear canonical transform of a quaternion function h with respect to two pure quaternions $I_1,I_2$ such that $I_1^2=I_2^2=-1$ is defined as

$$\begin{array}{c}\hfill {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{h\}\left(\mathit{u}\right)=\left\{\begin{array}{c}{\int }_{{\mathbb{R}}^2}{K}_{B_1}(z_1,u_1)h\left(\mathit{z}\right)K_{B_2}(z_2,u_2)d\mathit{z},\mathrm{for}{b}_1{b}_2\ne 0\hfill \\ \sqrt{d_1}{e}^{I_1\left(\frac{c_1{d}_1}{2}\right)u_1^2}h(d_1{u}_1,d_2{u}_2)\sqrt{d_2}{e}^{I_2\left(\frac{c_2{d}_2}{2}\right)u_2^2},\mathrm{for}{b}_1{b}_2=0,\hfill \end{array}\right.\end{array}$$

(37)

where the kernel functions of the general two-sided QLCT above are given by

$$\begin{array}{c}\hfill K_{B_1}(z_1,u_1)=\frac{1}{\sqrt{2\pi b_1}}{e}^{\frac{I_1}{2}\left(\frac{a_1}{b_1}{z}_1^2-\frac{2}{b_1}{z}_1{u}_1+\frac{d_1}{b_1}{u}_1^2-\frac{\pi }{2}\right)},\end{array}$$

(38)

and

$$\begin{array}{c}\hfill K_{B_2}(z_2,u_2)=\frac{1}{\sqrt{2\pi b_2}}{e}^{\frac{I_2}{2}\left(\frac{a_2}{b_2}{z}_2^2-\frac{2}{b_2}{z}_2{u}_2+\frac{d_2}{b_2}{u}_2^2-\frac{\pi }{2}\right)}.\end{array}$$

(39)

On the condition that the general two-sided QLCT parameters satisfy $b_1{b}_2=0$ or $b_1=b_2=0$, the general two-sided QLCT of a signal is essentially a quaternion chirp multiplication and is of no particular interest for our objective interests. Therefore, without loss of generality we focus mainly on the general two-sided QLCT in the case of $b_1{b}_2\ne 0$. For specific parameter matrices $B_1=B_2=(a_i,b_i,c_i,d_i)=(0,1,-1,0)$ with $i=1,2$, the general two-sided QLCT definition (37) is reduced to general two-sided QFT definition, that is,

$${\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{h\}\left(\mathit{u}\right)={\int }_{{\mathbb{R}}^2}\frac{e^{-I_1\frac{\pi }{4}}}{\sqrt{2\pi }}{e}^{-I_1{u}_1{z}_1}h\left(\mathit{z}\right)e^{-I_2{u}_2{z}_2}\frac{e^{-I_2\frac{\pi }{4}}}{\sqrt{2\pi }}d\mathit{z}=\frac{e^{-I_1\frac{\pi }{4}}}{\sqrt{2\pi }}{\mathcal{F}}^{I_1,I_2}\{h\}\left(\mathit{u}\right)\frac{e^{-I_2\frac{\pi }{4}}}{\sqrt{2\pi }}.$$

(40)

The inversion formula of the general two-sided QLCT is given by

$$\begin{array}{cc}\hfill {\mathcal{L}}_{B_1,B_2}^{-1,I_1,I_2}\left[{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{h\}\right]\left(\mathit{z}\right)& =h\left(\mathit{z}\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\overline{K_{B_1}(z_1,u_1)}{\mathcal{L}}_{B_1,B_2}\{h\}\left(\mathit{u}\right)\overline{K_{B_2}(z_2,u_2)}d\mathit{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{K}_{{B_1}^{-1}}(u_1,z_1){\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{h\}\left(\mathit{u}\right)K_{{B_2}^{-1}}(u_2,z_2)d\mathit{u}\hfill \\ \hfill & ={\int }_{\mathbb{R}}\frac{1}{\sqrt{2\pi b_1}}{e}^{\frac{I_1}{2}(\frac{a_1}{b_1}{z}_1^2-\frac{2}{b_1}{z}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2})}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{h\}\left(\mathit{u}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{\frac{I_2}{2}(\frac{a_2}{b_2}{z}_2^2-\frac{2}{b_2}{z}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2})}d\mathit{u},\hfill \end{array}$$

(41)

provided that the integral exists.

Based on Definition 5 we can easily obtain the following important theorem, which will be required to derive the main result in the sequel.

Theorem 4

(compare to [27]). Let $h\in L^2({\mathbb{R}}^2;\mathbb{H})$ be a quaternion function such that $h_{\pm }=\frac{1}{2}(h\pm I_{1}h{I}_2)$. The general two-sided quaternion linear canonical transform for the $h_{\pm }$ with respect to two linearly independent pure quaternions $I_1$ and $I_2$ can be expressed of the form

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{h_{\pm }\}\hfill \\ \hfill & =\frac{1}{\sqrt{2\pi b_2}}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{h}_{\pm }{e}^{\frac{I_2}{2}\left((\frac{a_2}{b_2}{z}_2^2-\frac{2}{b_2}{z}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2})\mp (\frac{a_1}{b_1}{z}_1^2-\frac{2}{b_1}{z}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2})\right)}d\mathit{z},\hfill \end{array}$$

(42)

and

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{h_{\pm }\}\hfill \\ \hfill & =\frac{1}{\sqrt{2\pi b_2}}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{\frac{I_1}{2}\left((\frac{a_1}{b_1}{z}_1^2-\frac{2}{b_1}{z}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2})\mp (\frac{a_2}{b_2}{z}_2^2-\frac{2}{b_2}{z}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2})\right)}{h}_{\pm }d\mathit{z},\hfill \end{array}$$

(43)

respectively.

5.2. Convolution Theorem for General Two-Sided QLCT

In the classical domain [28], a convolution operator is widely known as a signal processing algorithm in the theory of linear time-invariant systems. The output of any continuous time system is obtained by the convolution of the input signal with the system impulse response. The convolution is also important in designing the filters and in the reconstruction of the signals. According to these facts, the usefulness of convolution can be extended in a new domain. Furthermore, it is very interesting to consider the convolution definition and convolution theorem in the general two-sided QLCT domain. It shall be found that the convolution definition for the general two-sided QLCT is constructed by combining the quaternion convolution and the LCT convolution definitions [29,30].

Definition 7.

For any two quaternion functions $h,k\in L^1({\mathbb{R}}^2;\mathbb{H})$, we define the convolution and correlation operators related to the general two-sided QLCT as

$$\begin{array}{c}\hfill (h🟉k)\left(\mathit{z}\right)={\int }_{{\mathbb{R}}^2}h\left(\mathit{t}\right)e^{I_1\frac{a_1}{b_1}{z}_1(t_1-\frac{z_1}{2})}k(\mathit{z}-\mathit{t})e^{I_2\frac{a_2}{b_2}{z}_2(t_2-\frac{z_2}{2})}d\mathit{t},\end{array}$$

(44)

and

$$\begin{array}{c}\hfill (h\circ k)\left(\mathit{z}\right)={\int }_{{\mathbb{R}}^2}h\left(\mathit{t}\right)\overline{e^{I_1\frac{a_1}{b_1}{z}_1(t_1-\frac{z_1}{2})}k(\mathit{t}-\mathit{z})e^{I_2\frac{a_2}{b_2}{z}_2(t_2-\frac{z_2}{2})}}d\mathit{t},\end{array}$$

(45)

respectively.

The above definition enables us to build the following important theorem which describes how the convolution of two quaternion-valued functions interacts with its general two-sided QLCT.

Theorem 5.

Let $h,k\in L^1({\mathbb{R}}^2;\mathbb{H})$ be two quaternion-valued functions. If the decompositions of h and k are defined by

$$\begin{array}{c}\hfill h_{\pm }=\frac{1}{2}(h\pm I_{2}h{I}_1),k_{\pm }=\frac{1}{2}(k\pm I_{1}k{I}_2),\end{array}$$

(46)

then convolution of h and k related to the general two-sided QLCT can be expressed as

$$\begin{array}{cc}\hfill (h🟉k)\left(\mathit{z}\right)& ={\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{I_2,I_1}\{h_{-}\}\left(\mathit{u}\right)e^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{I_2,-I_1}\{h_{-}\}(u_1,-u_2)e^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{I_2,-I_1}\{h_{+}\}(-u_1,u_2)e^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{-I_2,-I_1}\{h_{+}\}(-\mathit{u})e^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}.\hfill \end{array}$$

(47)

Proof. 

Using an inverse transform of the general two-sided QLCT (41), we can rewrite the left-hand side of (47) as

$$\begin{array}{cc}\hfill & (h🟉k)\left(\mathit{z}\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}h\left(\mathit{t}\right){\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{I_1\frac{1}{2}\left(\frac{a_1}{b_1}{(z_1-t_1)}^2-\frac{2}{b_1}(z_1-t_1)u_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}{e}^{I_1\frac{a_1}{b_1}{z}_1(t_1-\frac{z_1}{2})}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k\}\left(\mathit{u}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{I_2\frac{1}{2}\left(\frac{a_2}{b_2}{(z_2-t_2)}^2-\frac{2}{b_2}(z_2-t_2)u_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}{e}^{I_2\frac{a_2}{b_2}{z}_2(t_2-\frac{z_2}{2})}d\mathit{t}d\mathit{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}h\left(\mathit{t}\right){\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k\}\left(\mathit{u}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u}.\hfill \end{array}$$

(48)

An application of two different splits to h and k defined by (46) gives us

$$\begin{array}{cc}\hfill & (h🟉k)\left(\mathit{z}\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\left(h_{-}\left(\mathit{t}\right)+h_{+}\left(\mathit{t}\right)\right){\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}\hfill \\ \hfill & \times \left({\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)+{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{h}_{-}\left(\mathit{t}\right)\frac{1}{\sqrt{2\pi b_1}}{e}^{I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}\hfill \\ \hfill & \times {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{h}_{-}\left(\mathit{t}\right)\frac{1}{\sqrt{2\pi b_1}}{e}^{I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}\hfill \\ \hfill & \times {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{h}_{+}\left(\mathit{t}\right)\frac{1}{\sqrt{2\pi b_1}}{e}^{I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}\hfill \\ \hfill & \times {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{h}_{+}\left(\mathit{t}\right)\frac{1}{\sqrt{2\pi b_1}}{e}^{I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}\hfill \\ \hfill & \times {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u},\hfill \end{array}$$

and thus

$$\begin{array}{cc}\hfill & (h🟉k)\left(\mathit{z}\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{I_2\frac{a_1{t}_1^2}{2b_1}}{e}^{I_2\frac{t_1{u}_1}{b_1}}{e}^{I_2\frac{d_1{u}_1^2}{2b_1}}{e}^{-I_2\frac{\pi }{4}}{h}_{-}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{I_1\frac{a_2{t}_2^2}{2b_2}}{e}^{I_1\frac{t_2{u}_2}{b_2}}{e}^{I_1\frac{d_2{u}_2^2}{2b_2}}{e}^{-I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{I_2\frac{a_1{t}_1^2}{2b_1}}{e}^{I_2\frac{t_1{u}_1}{b_1}}{e}^{I_2\frac{d_1{u}_1^2}{2b_1}}{e}^{-I_2\frac{\pi }{4}}{h}_{-}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{-I_1\frac{a_2{t}_2^2}{2b_2}}{e}^{-I_1\frac{t_2{u}_2}{b_2}}{e}^{-I_1\frac{d_2{u}_2^2}{2b_2}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_2\frac{a_1{t}_1^2}{2b_1}}{e}^{-I_2\frac{t_1{u}_1}{b_1}}{e}^{-I_2\frac{d_1{u}_1^2}{2b_1}}{e}^{I_2\frac{\pi }{4}}{h}_{+}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{I_1\frac{a_2{t}_2^2}{2b_2}}{e}^{I_1\frac{t_2{u}_2}{b_2}}{e}^{I_1\frac{d_2{u}_2^2}{2b_2}}{e}^{-I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_2\frac{a_1{t}_1^2}{2b_1}}{e}^{-I_2\frac{t_1{u}_1}{b_1}}{e}^{-I_2\frac{d_1{u}_1^2}{2b_1}}{e}^{I_2\frac{\pi }{4}}{h}_{+}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{-I_1\frac{a_2{t}_2^2}{2b_2}}{e}^{-I_1\frac{t_2{u}_2}{b_2}}{e}^{-I_1\frac{d_2{u}_2^2}{2b_2}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{t}d\mathit{u}.\hfill \end{array}$$

It means that we have

$$\begin{array}{cc}\hfill & (h🟉k)\left(\mathit{z}\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}[{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{I_2\frac{1}{2}\left(\frac{a_1}{b_1}{t}_1^2-\frac{2}{b_1}{t}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}{h}_{-}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{I_1\frac{1}{2}\left(\frac{a_2}{b_2}{t}_2^2-\frac{2}{b_2}{t}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}]d\mathit{t}{e}^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}[{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{I_2\frac{1}{2}\left(\frac{a_1}{b_1}{t}_1^2-\frac{2}{b_1}{t}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}{h}_{-}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{-I_1\frac{1}{2}\left(\frac{a_2}{b_2}{t}_2^2+\frac{2}{b_2}{t}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}]d\mathit{t}{e}^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}[{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_2\frac{1}{2}\left(\frac{a_1}{b_1}{t}_1^2+\frac{2}{b_1}{t}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}{h}_{+}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{I_1\frac{1}{2}\left(\frac{a_2}{b_2}{t}_2^2-\frac{2}{b_2}{t}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}]d\mathit{t}{e}^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}[{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_2\frac{1}{2}\left(\frac{a_1}{b_1}{t}_1^2+\frac{2}{b_1}{t}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}{h}_{+}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{-I_1\frac{1}{2}\left(\frac{a_2}{b_2}{t}_2^2+\frac{2}{b_2}{t}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}]d\mathit{t}{e}^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{I_2,I_1}\{h_{-}\}\left(\mathit{u}\right)e^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{g,-f}\{h_{-}\}(u_1,-u_2)e^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{-g,f}\{h_{+}\}(-u_1,u_2)e^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{-I_2,-f}\{h_{+}\}(-\mathit{u})e^{-I_1\frac{z_1{u}_1}{b_1}}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2\frac{z_2{u}_2}{b_2}}d\mathit{u}.\hfill \end{array}$$

The proof is complete. □

From Theorem 5 we know that when $B_1=B_2=(0,1,-1,0)$, the convolution theorem for the general two-sided QLCT above reduces to convolution theorem for general two-sided QFT [19], that is,

$$\begin{array}{cc}\hfill (h🟉k)\left(\mathit{z}\right)& ={\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{I_2,I_1}\{h_{-}\}\left(\mathit{u}\right)e^{-I_1{z}_1{u}_1}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2{z}_2{u}_2}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{I_2,-I_1}\{h_{-}\}(u_1,-u_2)e^{-I_1{z}_1{u}_1}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2{z}_2{u}_2}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{-I_2,I_1}\{h_{+}\}(-u_1,u_2)e^{-I_1{z}_1{u}_1}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)e^{-I_2{z}_2{u}_2}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{-I_2,-I_1}\{h_{+}\}(-\mathit{u})e^{-I_1{z}_1{u}_1}{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)e^{-I_2{z}_2{u}_2}d\mathit{u}.\hfill \end{array}$$

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6. Correlation Theorem for General Two-Sided QLCT

In the following we derive correlation theorem associated with the general two-sided QLCT. In this case we only apply two same splits to two quaternion function h and k as shown in the following theorem.

Theorem 6.

Let $h,k\in L^1({\mathbb{R}}^2;\mathbb{H})$ be two quaternion-valued functions. If the decompositions of h and k are defined by

$$\begin{array}{c}\hfill h_{\pm }=\frac{1}{2}(h\pm I_{1}h{I}_2),k_{\pm }=\frac{1}{2}(k\pm I_{1}k{I}_2),\end{array}$$

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then the correlation of h and k related to general two-sided QLCT can be expressed as

$$\begin{array}{cc}\hfill (h🟉k)\left(\mathit{z}\right)& ={\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{I_1,g}\{h_{-}\}\left(\mathit{u}\right)\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}{e}^{-I_1(\frac{z_1{u}_1}{b_1}+\frac{z_2{u}_2}{b_2})}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{f,-g}\{h_{-}\}(u_1,u_2)\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}{e}^{-I_1(\frac{z_1{u}_1}{b_1}-\frac{z_2{u}_2}{b_2})}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{-f,g}\{h_{+}\}(u_1,u_2)\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}{e}^{-I_1(\frac{z_1{u}_1}{b_1}+\frac{z_2{u}_2}{b_2})}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{-I_1,-g}\{h_{+}\}\left(\mathit{u}\right)\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}{e}^{-I_1(\frac{z_1{u}_1}{b_1}-\frac{z_2{u}_2}{b_2})}d\mathit{u}.\hfill \end{array}$$

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Proof. 

Taking into account inverse transform of the general two-sided QLCT (41), we have

$$\begin{array}{cc}\hfill & (h🟉k)\left(\mathit{z}\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}h\left(\mathit{t}\right){\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{1}{2}\left(\frac{a_2}{b_2}{(t_2-z_2)}^2-\frac{2}{b_2}(t_2-z_2)u_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}{e}^{-I_2\frac{a_2}{b_2}{z}_2(t_2-\frac{z_2}{2})}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k\}\left(\mathit{u}\right)}\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{-I_1\frac{1}{2}\left(\frac{a_1}{b_1}{(t_1-z_1)}^2-\frac{2}{b_1}(t_1-z_1)u_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}{e}^{-I_1\frac{a_1}{b_1}{z}_1(t_1-\frac{z_1}{2})}d\mathit{t}d\mathit{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}h\left(\mathit{t}\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{-I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{a_2{z}_2^2}{2b_2}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k\}\left(\mathit{u}\right)}\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{-I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{-I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{a_1{z}_1^2}{2b_1}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}.\hfill \end{array}$$

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Subsequent calculation yields

$$\begin{array}{cc}\hfill & (h🟉k)\left(\mathit{z}\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\left(h_{-}\left(\mathit{t}\right)+h_{+}\left(\mathit{t}\right)\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{-I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{a_2{z}_2^2}{2b_2}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\hfill \\ \hfill & \times \left(\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}+\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}\right)\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{-I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{a_1{z}_1^2}{2b_1}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{h}_{-}\left(\mathit{t}\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{-I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{a_2{z}_2^2}{2b_2}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\hfill \\ \hfill & \times \overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{-I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{a_1{z}_1^2}{2b_1}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{h}_{-}\left(\mathit{t}\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{-I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{a_2{z}_2^2}{2b_2}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\hfill \\ \hfill & \times \overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{-I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{a_1{z}_1^2}{2b_1}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{h}_{+}\left(\mathit{t}\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{-I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{a_2{z}_2^2}{2b_2}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\hfill \\ \hfill & \times \overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{-I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{a_1{z}_1^2}{2b_1}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{h}_{+}\left(\mathit{t}\right)\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{a_2{t}_2^2}{2b_2}}{e}^{I_2\frac{t_2{u}_2}{b_2}}{e}^{-I_2\frac{d_2{u}_2^2}{2b_2}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{a_2{z}_2^2}{2b_2}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\hfill \\ \hfill & \times \overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_1\frac{a_1{t}_1^2}{2b_1}}{e}^{I_1\frac{t_1{u}_1}{b_1}}{e}^{-I_1\frac{d_1{u}_1^2}{2b_1}}{e}^{I_1\frac{\pi }{4}}{e}^{-I_1\frac{a_1{z}_1^2}{2b_1}}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}.\hfill \end{array}$$

This leads to

$$\begin{array}{cc}\hfill & (h🟉k)\left(\mathit{z}\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_1\frac{a_2{t}_2^2}{2b_2}}{e}^{I_1\frac{t_2{u}_2}{b_2}}{e}^{-I_1\frac{d_2{u}_2^2}{2b_2}}{e}^{I_1\frac{\pi }{4}}{h}_{-}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{-I_2\frac{a_1{t}_1^2}{2b_1}}{e}^{I_2\frac{t_1{u}_1}{b_1}}{e}^{-I_2\frac{d_1{u}_1^2}{2b_1}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_1\frac{a_2{t}_2^2}{2b_2}}{e}^{I_1\frac{t_2{u}_2}{b_2}}{e}^{-I_1\frac{d_2{u}_2^2}{2b_2}}{e}^{I_1\frac{\pi }{4}}{h}_{-}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{I_2\frac{a_1{t}_1^2}{2b_1}}{e}^{-I_2\frac{t_1{u}_1}{b_1}}{e}^{I_2\frac{d_1{u}_1^2}{2b_1}}{e}^{-I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_2}}{e}^{I_1\frac{a_2{t}_2^2}{2b_2}}{e}^{-I_1\frac{t_2{u}_2}{b_2}}{e}^{I_1\frac{d_2{u}_2^2}{2b_2}}{e}^{-I_1\frac{\pi }{4}}{h}_{+}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{-I_2\frac{a_1{t}_1^2}{2b_1}}{e}^{I_2\frac{t_1{u}_1}{b_1}}{e}^{-I_2\frac{d_1{u}_1^2}{2b_1}}{e}^{I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_2}}{e}^{I_1\frac{a_2{t}_2^2}{2b_2}}{e}^{-I_1\frac{t_2{u}_2}{b_2}}{e}^{I_1\frac{d_2{u}_2^2}{2b_2}}{e}^{-I_1\frac{\pi }{4}}{h}_{+}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{I_2\frac{a_1{t}_1^2}{2b_1}}{e}^{-I_2\frac{t_1{u}_1}{b_1}}{e}^{I_2\frac{d_1{u}_1^2}{2b_1}}{e}^{-I_2\frac{\pi }{4}}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{t}d\mathit{u}.\hfill \end{array}$$

Consequently, we obtain

$$\begin{array}{cc}\hfill & (h🟉k)\left(\mathit{z}\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}[{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_1\frac{1}{2}\left(\frac{a_2}{b_2}{t}_2^2-\frac{2}{b_2}{t}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}{h}_{-}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{-I_2\frac{1}{2}\left(\frac{a_1}{b_1}{t}_1^2-\frac{2}{b_1}{t}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}]d\mathit{t}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}[{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{-I_1\frac{1}{2}\left(\frac{a_1}{b_2}{t}_2^2-\frac{2}{b_2}{t}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}{h}_{-}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{I_2\frac{1}{2}\left(\frac{a_1}{b_1}{t}_1^2-\frac{2}{b_1}{t}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}]d\mathit{t}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}[{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_2}}{e}^{I_1\frac{1}{2}\left(\frac{a_2}{b_2}{t}_2^2-\frac{2}{b_2}{t}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}{h}_{+}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{-I_2\frac{1}{2}\left(\frac{a_1}{b_1}{t}_1^2-\frac{2}{b_1}{t}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}]d\mathit{t}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}[{\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_2}}{e}^{I_1\frac{1}{2}\left(\frac{a_2}{b_2}{t}_2^2-\frac{2}{b_2}{t}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}{h}_{+}\left(\mathit{t}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_1}}{e}^{I_2\frac{1}{2}\left(\frac{a_1}{b_1}{t}_1^2-\frac{2}{b_1}{t}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}]d\mathit{t}{e}^{-I_2\frac{z_2{u}_2}{b_2}}\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}{e}^{-I_1\frac{z_1{u}_1}{b_1}}d\mathit{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{I_1,g}\{h_{-}\}\left(\mathit{u}\right)\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}{e}^{-I_1(\frac{z_1{u}_1}{b_1}+\frac{z_2{u}_2}{b_2})}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{f,-g}\{h_{-}\}(u_1,u_2)\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}{e}^{-I_1(\frac{z_1{u}_1}{b_1}-\frac{z_2{u}_2}{b_2})}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{-f,g}\{h_{+}\}(u_1,u_2)\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{-}\}\left(\mathit{u}\right)}{e}^{-I_1(\frac{z_1{u}_1}{b_1}+\frac{z_2{u}_2}{b_2})}d\mathit{u}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}{\mathcal{L}}_{B_1,B_2}^{-I_1,-g}\{h_{+}\}\left(\mathit{u}\right)\overline{{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{k_{+}\}\left(\mathit{u}\right)}{e}^{-I_1(\frac{z_1{u}_1}{b_1}-\frac{z_2{u}_2}{b_2})}d\mathit{u}.\hfill \end{array}$$

The proof is complete. □

7. An Application

In this part, we aim to consider a simple application of the general two-sided QLCT for studying the generalized swept-frequency filters. The output of swept-frequency filters (compare to [31,32]) is then defined as

$$\begin{array}{cc}\hfill y\left(\mathit{z}\right)& =e^{I_1\frac{c_1}{2}{z}_1^2}\left[\left(e^{-I_1\frac{c_1}{2}{z}_1^2}h\left(\mathit{z}\right)e^{-I_2\frac{c_2}{2}{z}_2^2}\right){*}_{q}r\left(\mathit{z}\right)\right]e^{I_2\frac{c_2}{2}{z}_2^2}\hfill \\ \hfill & =e^{I_1\frac{c_1}{2}{z}_1^2}{\int }_{{\mathbb{R}}^2}\left[\left(e^{-I_1\frac{c_1}{2}{\tau }_1^2}h\left(\tau \right)e^{-I_2\frac{c_2}{2}{\tau }_2^2}\right)r(\mathit{z}-\tau )\right]d\tau e^{I_2\frac{c_2}{2}{z}_2^2},\hfill \end{array}$$

(53)

where ${*}_q$ stands for quaternion convolution operator in the QFT domain. Assume that $r\left(\mathit{z}\right)$ is real impulse response of the shift-invariant filter. By taking the general two-sided QLCT on both sides of (53), we see that

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{y\}\left(\mathit{u}\right)\hfill \\ \hfill & =\frac{1}{\sqrt{2\pi b_1}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{e}^{-\frac{I_1}{2}\left(\frac{a_1}{b_1}{z}_1^2-\frac{2}{b_1}{z}_1{u}_1+\frac{d_1}{b_1}{u}_1^2+\frac{\pi }{2}\right)}\hfill \\ \hfill & \times \left[\left(e^{I_1\frac{c_1}{2}(z_1^2-{\tau }_1^2)}h\left(\tau \right)e^{I_2\frac{c_2}{2}(z_2^2-{\tau }_2^2)}\right)r(\mathit{z}-\tau )\frac{1}{\sqrt{2\pi b_2}}{e}^{-\frac{I_2}{2}\left(\frac{a_2}{b_2}{z}_2^2-\frac{2}{b_2}{z}_2{u}_2+\frac{d_2}{b_2}{u}_2^2+\frac{\pi }{2}\right)}d\mathit{z}\right]d\tau .\hfill \end{array}$$

If the matrix parameters are chosen as $B_1=(a_1,b_1,c_1,d_1)=(c_1,1,-1,0)$ and $B_2=(a_2,b_2,c_2,d_2)=(c_2,1,-1,0)$, then the above identity can be expressed as

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{y\}\left(\mathit{u}\right)\hfill \\ \hfill & =\frac{1}{\sqrt{2\pi b_1}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{e}^{-I_1\frac{c_1}{2}{z}_1^2}{e}^{-I_1{z}_1{u}_1}{e}^{-I_1\frac{\pi }{4}}\hfill \\ \hfill & \times \left[\left(e^{I_1\frac{c_1}{2}(z_1^2-{\tau }_1^2)}h\left(\tau \right)e^{I_2\frac{c_2}{2}(z_2^2-{\tau }_2^2)}\right)r(\mathit{z}-\tau )\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{c_2}{2}{z}_2^2}{e}^{-I_2{z}_2{u}_2}{e}^{-I_2\frac{\pi }{4}}d\mathit{z}\right]d\tau \hfill \\ \hfill & =\frac{1}{\sqrt{2\pi b_1}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{e}^{-I_1\frac{c_1}{2}{\tau }_1^2}{e}^{-I_1{z}_1{u}_1}{e}^{-I_1\frac{\pi }{4}}h\left(\tau \right)r(\mathit{z}-\tau )\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{c_2}{2}{z}_2^2}{e}^{-I_2{z}_2{u}_2}{e}^{-I_2\frac{\pi }{4}}d\mathit{z}d\tau .\hfill \end{array}$$

By changing the variables with $\mathit{z}-\tau =\mathit{y}$ in the above expression we have

$$\begin{array}{cc}\hfill {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{y\}\left(\mathit{u}\right)& =\frac{1}{\sqrt{2\pi b_1}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{e}^{-I_1\frac{c_1}{2}{\tau }_1^2}{e}^{-I_1{\tau }_1{u}_1}{e}^{-I_1\frac{\pi }{4}}{e}^{-I_1{y}_1{u}_1}h\left(\tau \right)r\left(\mathit{y}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2{\tau }_2{u}_2}{e}^{-I_2{y}_2{u}_2}{e}^{-I_2\frac{c_2}{2}{\tau }_2^2}{e}^{-I_2\frac{\pi }{4}}d\mathit{y}d\tau .\hfill \end{array}$$

(54)

Rewriting (54) as

$$\begin{array}{cc}\hfill {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{y\}\left(\mathit{u}\right)& =\frac{1}{\sqrt{2\pi b_1}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{e}^{-I_1\frac{c_1}{2}{\tau }_1^2}{e}^{-I_1{\tau }_1{u}_1}{e}^{-I_1\frac{\pi }{4}}{e}^{-I_1{y}_1{u}_1}{h}_{-}\left(\tau \right)r\left(\mathit{y}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2{\tau }_2{u}_2}{e}^{-I_2{y}_2{u}_2}{e}^{-I_2\frac{c_2}{2}{\tau }_2^2}{e}^{-I_2\frac{\pi }{4}}d\mathit{y}d\tau \hfill \\ \hfill & +\frac{1}{\sqrt{2\pi b_1}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{e}^{-I_1\frac{c_1}{2}{\tau }_1^2}{e}^{-I_1{\tau }_1{u}_1}{e}^{-I_1\frac{\pi }{4}}{e}^{-I_1{y}_1{u}_1}{h}_{+}\left(\tau \right)r\left(\mathit{y}\right)\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2{\tau }_2{u}_2}{e}^{-I_2{y}_2{u}_2}{e}^{-I_2\frac{c_2}{2}{\tau }_2^2}{e}^{-I_2\frac{\pi }{4}}d\mathit{y}d\tau \hfill \\ \hfill & =\frac{1}{\sqrt{2\pi b_1}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{e}^{-I_1\frac{c_1}{2}{\tau }_1^2}{e}^{-I_1{\tau }_1{u}_1}{e}^{-I_1\frac{\pi }{4}}{h}_{-}\left(\tau \right)\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{c_2}{2}{\tau }_2^2}{e}^{-I_2{\tau }_2{u}_2}{e}^{-I_2\frac{\pi }{4}}\hfill \\ \hfill & \times e^{-I_2{y}_1{u}_1}r\left(\mathit{y}\right)e^{-I_2{y}_2{u}_2}d\tau d\mathit{y}\hfill \\ \hfill & +\frac{1}{\sqrt{2\pi b_1}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{e}^{-I_1\frac{c_1}{2}{\tau }_1^2}{e}^{-I_1{\tau }_1{u}_1}{e}^{-I_1\frac{\pi }{4}}{h}_{+}\left(\tau \right)\frac{1}{\sqrt{2\pi b_2}}{e}^{-I_2\frac{c_2}{2}{\tau }_2^2}{e}^{-I_2{\tau }_2{u}_2}{e}^{-I_2\frac{\pi }{4}}\hfill \\ \hfill & \times e^{I_2{y}_1{u}_1}r\left(\mathit{y}\right)e^{-I_2{y}_2{u}_2}d\tau d\mathit{y}.\hfill \end{array}$$

(55)

Consequently, we finally arrive at

$$\begin{array}{cc}\hfill {\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{y\}\left(\mathit{u}\right)& ={\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{h_{-}\}\left(\mathit{u}\right){\int }_{{\mathbb{R}}^2}{e}^{-I_2{y}_1{u}_1}r\left(\mathit{y}\right)e^{-I_2{y}_2{u}_2}d\mathit{y}\hfill \\ \hfill & +{\mathcal{L}}_{B_1,B_2}^{I_1,I_2}\{h_{+}\}\left(\mathit{u}\right){\int }_{{\mathbb{R}}^2}{e}^{I_2{y}_1{u}_1}r\left(\mathit{y}\right)e^{-I_2{y}_2{u}_2}d\mathit{y}.\hfill \end{array}$$

(56)

8. Conclusions

In this work, based on the symmetric form of the two-sided quaternion Fourier transform, we have proved a variation on the Heisenberg-type uncertainty principle related to the two-sided quaternion Fourier transform. We also have introduced the general two-sided QLCT and provided its convolution definitions. In view of orthogonal plane split, we have derived convolution and correlation theorems related to general two-sided QLCT. We also have discussed its application to study the generalized swept-frequency filters.