New Uncertainty Principles in the Linear Canonical Transform Domains Based on Hypercomplex Functions

Abstract

In this paper, we obtain uncertainty principles associated with the linear canonical transform (LCT) of hypercomplex functions. First, we derive the uncertainty principle for hypercomplex functions in the time and LCT domains. Moreover, we exploit the uncertainty principle in two LCT domains. The lower bounds are related to the LCT parameters and the covariance, and the uncertainty principle presented herein is sharper than what has been presented in the existing literature.These tighter bounds can be obtained using hypercomplex chirp functions for a Gaussian envelope. Finally, we verify the validity of the uncertainty principles through some examples and discuss several potential applications of the new results in signal processing.

1. Introduction

Uncertainty principles have a wide range of useful applications in different fields; for example, Dar and Bhat derived the Donoho–Stark uncertainty principle, Hardy’s uncertainty principle, Beurling’s uncertainty principle, and the logarithmic uncertainty principle for the short-time quaternion offset linear canonical transform [1]. Kais obtained Hardy’s uncertainty principle for the Gabor transform and solve the sharpness problem [2]. Samanta and Sarkar studied an uncertainty principle of finite linear combinations of Fourier–Wigner transforms and proved a finite rank theorem for the Weyl transform [3]. In [4], the logarithmic uncertainty principle, the entropic uncertainty principle, the Lieb uncertainty principle, and the Donoho–Stark uncertainty principle associated with the quaternion windowed linear canonical transform were obtained, and the authors discussed a numerical example and a potential application to signal recovery.Nicola and Trapasso studied a new type of uncertainty principle for the short-time Fourier transform that forbids the arrangement of an arbitrary “bump with fat tail” profile [5]. Zhang derived the N-dimensional Heisenberg’s uncertainty principle of the fractional Fourier transform, and its effectiveness was illustrated by applications in the effective estimation of spreads in time–frequency analysis and optical systems analysis [6]. In [7], some equivalent formulations of the sign uncertainty principle were established. The uncertainty principle for vector-valued functions is studied in [8]. In [9,10], uncertainty principles of the graph Fourier transform were proposed for representing and processing signals on directed graphs.

The uncertainty principle states that the spread product of a function $x\left(t\right)$ in the time domain ${\Delta }_t^2$ and the frequency domain ${\Delta }_w^2$ has a lower bound:

$${\Delta }_t^2{\Delta }_w^2\ge {\displaystyle \frac{1}{4}},$$

where

$${\Delta }_t^2={\int }_{\mathbb{R}}{(t-\overline{t_x})}^2\mid \mathit{x}\left(t\right){\mid }^2\mathrm{d}\mathit{t},$$

is the spread of the function in the time domains and

$${\Delta }_w^2={\int }_{\mathbb{R}}{(w-\overline{w_x})}^2\mid \widehat{\mathit{x}}\left(w\right){\mid }^2\mathrm{d}\mathit{w},$$

is the spread of the function in the frequency domains. Here, the Fourier transform (FT) [5,6] of the function x is defined as

$$\begin{array}{c}\hfill \widehat{\mathit{x}}\left(w\right)={\int }_{\mathbb{R}}x\left(t\right){\mathrm{e}}^{-2\pi itw}\mathrm{d}\mathit{t},\end{array}$$

the mean time is denoted by

$$\begin{array}{c}\hfill \overline{t_x}={\int }_{\mathbb{R}}t\mid x\left(t\right){\mid }^2\mathrm{d}\mathit{t},\end{array}$$

and the mean frequency is given by

$$\begin{array}{c}\hfill \overline{w_x}={\int }_{\mathbb{R}}w\mid \widehat{x}\left(w\right){\mid }^2\mathrm{d}\mathit{w}.\end{array}$$

On the basis of Heisenberg’s uncertainty principle of the FT, several scholars have obtained the uncertainty principles of the FT for different types of functions. From the covariance matrix, two forms of the lower bound of the covariance uncertainty product of a real signal in two free element transform domains are obtained [11]. Afzal et al. [12] derived a stronger uncertainty principle of the FT for the complex signals, and the lower bound of the inequality was not a constant but was related to the covariance of the signal. Naoki [13] investigated the uncertainty principles of the short-time FT for a complex signal and studied signal recovery problems according to the uncertainty principles. Moreover, based on complex signals, Mahesh Krishna [14] obtained an uncertainty principle of the FT showing that the lower bound is related to absolute covariance. Zhang [15] obtained the uncertainty principle under the N-dimensional FT of a complex signal, and the result was a sharper uncertainty inequality with a tighter lower bound than the classical N-dimensional lower bound. In addition, based on the obtained results, the applications in signal processing were discussed. Fei et al. [16] established a new uncertainty principle of the Dunkl transform associated with a complex signal. Ren et al. [17] studied several sharper uncertainty principles of a quaternion signal. A hypercomplex function is a more generalized function type than a complex function [18]. Based on hypercomplex functions, Yang et al. [19] obtained the uncertainty principles in the context of the FT and derived the uncertainty principle related to the covariance and absolute covariance in [20]. In addition, the uncertainty principles associated with the geometric Fourier transform of Clifford ambiguity functions [21,22] were exploited.

The linear canonical transform (LCT) [23,24,25,26,27] is a general transform of the FT. In recent years, the uncertainty principles associated with the LCT have been extensively studied and applied. Feng et al. [28] studied time frequency resolution analysis and signal energy concentrations by the weighted uncertainty principles of the LCT. Adrian [29] obtained several uncertainty relations of the LCT and discussed their implications in some common optical systems. The uncertainty principle of the LCT can be applied in signal processing associated with propagation in the LCT domain [30,31]. In two existing papers [32,33], signal recovery problems have been discussed according to the uncertainty principle of the LCT. Based on the uncertainty principle, Xu et al. [34] analyzed the frequency resolution of a signal in the LCT domain. We studied biquaternion signal recovery by the Donoho-Stark uncertainty principle associated with the biquaternion offset LCT [35]. Sharma [36] obtained the uncertainty principle associated with the LCT of a real signal. In [37], three different types of uncertainty principles in the LCT domain were derived. The results of the study showed that the lower bound was related to the parameters of the matrix, and their physical explanations were given. The complex signal moments and uncertainty principles in the LCT domain were studied [38]. In [39], the uncertainty principle bounds of the LCT were related to the phase and the amplitude of the complex signal. Shi [40] proved that the lower bounds of uncertainty principles for complex signals hold for arbitrary LCTs. Dang et al. [41] derived tighter lower bounds of the uncertainty principles for a complex signal in the LCT context. In addition, the uncertainty principles for the short-time LCT of a complex signal were studied in different papers [42]. To date, most studies on the uncertainty principle of the LCT have focused on real functions and complex functions.

Due to the extensive research on the uncertainty principle of the LCT, Kou and Yang [43] investigated the uncertainty principle for hypercomplex functions associated with the LCT. Tighter lower bounds were given for effective width products of hypercomplex functions in the time domain and the LCT domain, and these lower bounds can be obtained by Gaussian functions.

In this paper, we further study the uncertainty principles of the LCT under hypercomplex functions. First, according to the properties of hypercomplex functions, we obtain the uncertainty principle in the time and LCT domains. The lower bound obtained is related to the absolute covariance and matrix parameters. The result is sharper than what has been presented in the existing literature. Then, we derive the uncertainty principle of the LCT for hypercomplex functions in two LCT domains, and get the conditions that make the equality sign true in the inequality. Finally, we illustrate the validity of the obtained results by examples and introduce potential applications of the uncertainty principles of the LCT in signal processing.

The structure of this paper is as follows: In Section 2, we present several properties of Clifford algebra and the FT. The definition and properties of the LCT for hypercomplex functions are given in Section 3. In Section 4, we obtained two types of uncertainty principles associated with hypercomplex functions in the LCT domains. Two examples are provided in Section 5. In Section 6, potential applications are introduced. Some conclusions are drawn in Section 7.

2. Preliminaries

Hypercomplex signals are generated by Clifford algebras [44], and so it is necessary to understand the properties of Clifford algebras to study hypercomplex functions. In this section, we mainly review Clifford algebras and the FT.

2.1. Clifford Algebras

Let the Euclidean space [44] be given by

$${\mathbb{E}}^n=\{\mathbf{t}\mid \mathbf{t}=t_1{\mathbf{e}}_1+\cdots +t_n{\mathbf{e}}_n,t_i\in \mathbb{R},i=1,2,\cdots ,n\},$$

where these basic elements ${\mathbf{e}}_1,\cdots ,{\mathbf{e}}_n$ satisfy

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{e}}_i{\mathbf{e}}_j=\left\{\begin{array}{cc}-{\mathbf{e}}_j{\mathbf{e}}_i,\hfill & i\ne j\hfill \\ 1,\hfill & i=j\hfill \end{array}\right.,\end{array}\end{array}$$

and every element in ${\mathbb{E}}^n$ is a real vector.

The paravector space [44] is defined by

$${\mathbb{P}}^n=\{t_0+\mathbf{t}\mid t_0\in \mathbb{R},\mathbf{t}\in {\mathbb{E}}^n\},$$

where every element in ${\mathbb{P}}^n$ is a real paravector.

For $\tilde{t}=t_0+\mathbf{t}={\sum }_{i=0}^n{t}_i{\mathbf{e}}_i\in {\mathbb{P}}^n$ and $\tilde{r}=r_0+\mathbf{r}={\sum }_{i=0}^n{r}_i{\mathbf{e}}_i\in {\mathbb{P}}^n$, we have

$$\begin{array}{c}\hfill \tilde{t}\tilde{r}=(t_0+\mathbf{t})(r_0+\mathbf{r})=(t_0{r}_0+\mathbf{t}·\mathbf{r})+(t_0\mathbf{r}+r_0\mathbf{t})+(\mathbf{t}\wedge \mathbf{r}),\end{array}$$

where ${\mathbf{e}}_0=1$ is the unit element of the algebra. $\mathbf{t}·\mathbf{r}=-{\sum }_{i=1}^n{t}_i{r}_i={\displaystyle \frac{1}{2}}(\mathbf{t}\mathbf{r}+\mathbf{r}\mathbf{t}),$ and $\mathbf{t}\wedge \mathbf{r}={\sum }_{i<j}{\mathbf{e}}_i{\mathbf{e}}_j(t_i{r}_j-t_j{r}_i)={\displaystyle \frac{1}{2}}(\mathbf{t}\mathbf{r}-\mathbf{r}\mathbf{t}).$

The Clifford algebras $C{l}_{0,n}=C{l}_n$ is expressed by the orthonormal basis ${\mathbf{e}}_1,\cdots ,{\mathbf{e}}_n$ of the real n-dimensional Euclidean space, generated by the multiplication rule:

$$\begin{array}{c}\hfill {\mathbf{e}}_i{\mathbf{e}}_j+{\mathbf{e}}_j{\mathbf{e}}_i=-2{\delta }_{ij},i,j=1,2,\cdots ,n,\end{array}$$

where ${\delta }_{ij}$ denotes the Kronecker delta function [44].

They are the associative and non-commutative algebras on the field $\mathbb{R}\left(\mathbb{C}\right)$ [44]. For $k=\{1\le i_1<i_2<\cdots <i_l\le n\},1\le l\le n$, let $\hslash ={\sum }_k{\hslash }_k{\mathbf{e}}_k\in C{l}_{0,n},$ where ${\mathbf{e}}_k={\mathbf{e}}_{i_1}{\mathbf{e}}_{i_2}\cdots {\mathbf{e}}_{i_l}$.

The conjugation of ${\mathbf{e}}_k$ is defined by [44]

$$\begin{array}{c}\hfill \overline{{\mathbf{e}}_k}=\overline{{\mathbf{e}}_{i_l}}\cdots \overline{{\mathbf{e}}_{i_1}},\end{array}$$

where $\overline{{\mathbf{e}}_i{\mathbf{e}}_j}=-{\mathbf{e}}_i{\mathbf{e}}_j$ and $\overline{{\mathbf{e}}_i}=-{\mathbf{e}}_i$. Then, we can obtain $\overline{\mathbf{t}}=-\mathbf{t}$ and $\overline{\mathbf{t}\wedge \mathbf{r}}=-\mathbf{t}\wedge \mathbf{r}$.

Let $\hslash ={\sum }_k{\hslash }_k{\mathbf{e}}_k$ and $\hslash ={\sum }_k{\hslash }_k{\mathbf{e}}_k$. The inner product $\langle \hslash ,\hslash \rangle$ is given by [44]

$$\langle \hslash ,\hslash \rangle =\sum _k{\hslash }_k{\hslash }_k;$$

if $\hslash =\hslash$, then $|\hslash |={\langle \hslash ,\hslash \rangle }^{{\displaystyle \frac{1}{2}}}={\left({\sum }_k\mid {\hslash }_k{\mid }^2\right)}^{{\displaystyle \frac{1}{2}}}.$

The Clifford-valued modules $L^p({\mathbb{E}}^n,C{l}_{0,n})$ are given by [44]

$$L^p({\mathbb{E}}^n,C{l}_{0,n}):=\left\{x:{\mathbb{E}}^n\to C{l}_{0,n}\mid {\parallel x\parallel }_{L^p({\mathbb{E}}^n,C{l}_{0,n})}^p={\int }_{{\mathbb{E}}^n}\mid x\left(\mathbf{t}\right){\mid }^p\mathrm{d}\mathbf{t}<\infty \right\},$$

where $p=1,2$.

For any function $x\left(\mathbf{t}\right)\in ({\mathbb{E}}^n,C{l}_{0,n})$ [44,45], we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x\left(\mathbf{t}\right)& =x_0\left(\mathbf{t}\right)+x_1\left(\mathbf{t}\right){\mathbf{e}}_1+x_2\left(\mathbf{t}\right){\mathbf{e}}_2+\cdots +x_n\left(\mathbf{t}\right){\mathbf{e}}_n\hfill \\ \hfill & =\mid x\left(\mathbf{t}\right)\mid {\mathbf{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\hfill \\ \hfill & =\Im \left(\mathbf{t}\right){\mathbf{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)},\hfill \end{array}\end{array}$$

(1)

where

$$\Im \left(\mathbf{t}\right)=\mid x\left(\mathbf{t}\right)\mid ={(x_0^2\left(\mathbf{t}\right)+x_1^2\left(\mathbf{t}\right)+\cdots +x_n^2\left(\mathbf{t}\right))}^{{\displaystyle \frac{1}{2}}},$$

$$\wp \left(\mathbf{t}\right)={\displaystyle \frac{x_1\left(\mathbf{t}\right){\mathbf{e}}_1+x_2\left(\mathbf{t}\right){\mathbf{e}}_2+\cdots +x_n\left(\mathbf{t}\right){\mathbf{e}}_n}{{(x_1^2\left(\mathbf{t}\right)+x_2^2\left(\mathbf{t}\right)+\cdots +x_n^2\left(\mathbf{t}\right))}^{\frac{1}{2}}}},$$

while

$$\begin{array}{c}\hfill \aleph \left(\mathbf{t}\right)=arctan{\displaystyle \frac{{(x_1^2\left(\mathbf{t}\right)+x_2^2\left(\mathbf{t}\right)+\cdots +x_n^2\left(\mathbf{t}\right))}^{\frac{1}{2}}}{x_0\left(\mathbf{t}\right)}}\in [0,\pi ]\end{array}$$

is the phase angle and ${\mathbf{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}$ is the phase vector.

Let Clifford-valued functions $x,y\in L^2({\mathbb{E}}^n,C{l}_{0,n})$ [44], we have

$$\begin{array}{c}\hfill {\langle x,y\rangle }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}=Sc\left[{\int }_{{\mathbb{E}}^n}x\left(\mathbf{t}\right)\overline{y\left(\mathbf{t}\right)}\mathrm{d}\mathbf{t}\right],\end{array}$$

where $Sc\left[{\int }_{{\mathbb{E}}^n}x\left(\mathbf{t}\right)\overline{y\left(\mathbf{t}\right)}\mathrm{d}\mathbf{t}\right]$ is the real part of ${\int }_{{\mathbb{E}}^n}x\left(\mathbf{t}\right)\overline{y\left(\mathbf{t}\right)}\mathrm{d}\mathbf{t}$.

When $x=y$,

$$\begin{array}{c}\hfill {\parallel x\parallel }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}={\left({\int }_{{\mathbb{E}}^n}\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}\right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

2.2. Fourier Transform of Hypercomplex Functions

In this subsection, we will present some definitions and properties related to FT [20,44].

Definition 1

([20,44]). Let a hypercomplex function $x\in L^1({\mathbb{E}}^n,C{l}_{0,n})$. Then the FT of x is defined as

$$\begin{array}{c}\hfill \mathcal{F}\left\{x\right\}\left(\mathbf{w}\right)={\displaystyle \frac{1}{\sqrt{{\left(2\pi \right)}^n}}}{\int }_{{\mathbb{E}}^n}{\mathrm{e}}^{-\mathbf{i}\langle \mathbf{t},\mathbf{w}\rangle }x\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}$$

(2)

where $\langle \mathbf{t},\mathbf{w}\rangle =t_1{w}_1+\cdots +t_n{w}_n$. $t_i,w_i\in \mathbb{R}$ and $i=1,\cdots ,n$.

The inverse transform of the FT is given by [20,44]

$$\begin{array}{c}\hfill x\left(\mathbf{t}\right)={\displaystyle \frac{1}{\sqrt{{\left(2\pi \right)}^n}}}{\int }_{{\mathbb{E}}^n}{\mathrm{e}}^{\mathbf{i}\langle \mathbf{t},\mathbf{w}\rangle }\mathcal{F}\left\{x\right\}\left(\mathbf{w}\right)\mathrm{d}\mathbf{w}.\end{array}$$

(3)

Let $x,y\in L^2({\mathbb{E}}^n,C{l}_{0,n})$. Then, the Plancherel theorem of the FT is presented by [20,44]

$$\begin{array}{c}\hfill Sc\left[{\int }_{{\mathbb{E}}^n}x\left(\mathbf{t}\right)\overline{y\left(\mathbf{t}\right)}\mathrm{d}\mathbf{t}\right]=\mathrm{Sc}\left[{\int }_{{\mathbb{E}}^{\mathrm{n}}}\mathcal{F}\left\{\mathrm{x}\right\}\left(\mathbf{w}\right)\overline{\mathcal{F}\left\{\mathrm{y}\right\}\left(\mathbf{w}\right)}\mathrm{d}\mathbf{w}\right].\end{array}$$

(4)

When $x=y$,

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}={\int }_{{\mathbb{E}}^{\mathrm{n}}}\mid \mathcal{F}\left\{\mathrm{x}\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}.\end{array}$$

(5)

Lemma 1

([20,44]). Let the hypercomplex function $x\left(\mathbf{t}\right)=\Im \left(\mathbf{t}\right){\mathbf{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\in L^1\cap L^2\left({\mathbb{E}}^n\right)$ and ${\displaystyle \frac{\partial }{\partial t_m}}x\in L^2({\mathbb{E}}^n,{\mathbb{P}}^n)$, for $t_m,w_m\in \mathbb{R}$, $m=1,2,\cdots ,n$. Then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\int }_{{\mathbb{E}}^n}{w}_m^2\mid \mathcal{F}\left\{x\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}\hfill \\ & ={\int }_{{\mathbb{E}}^n}{\left[{\displaystyle \frac{\partial }{\partial t_m}}\Im \left(\mathbf{t}\right)\right]}^2\mathrm{d}\mathbf{t}+{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\Im }^2\left(\mathbf{t}\right)\mid \left({\displaystyle \frac{\partial }{\partial {\mathrm{t}}_{\mathrm{m}}}}{\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right)\left({\mathrm{e}}^{-\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right){\mid }^2\mathrm{d}\mathbf{t}.\hfill \end{array}\end{array}$$

(6)

3. Linear Canonical Transform of Hypercomplex Functions

Definition 2

([43]). Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\in {\mathbb{R}}^{2\times 2}$ be a matrix parameter satisfying $\det \left(A\right)=1$. The LCT of a hypercomplex function $x\in L^2({\mathbb{E}}^n,C{l}_{0,n})$ is defined as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right)=\left\{\begin{array}{cc}{\int }_{{\mathbb{E}}^n}x\left(\mathbf{t}\right)K_A(\mathbf{t},\mathbf{w})\mathrm{d}\mathrm{t},\hfill & b\ne 0\hfill \\ \sqrt{d}{\mathrm{e}}^{\mathbf{i}{\displaystyle \frac{cd}{2}}{\mathbf{w}}^2}x\left(d\mathbf{w}\right),\hfill & b=0\hfill \end{array}\right.\end{array},\end{array}$$

(7)

where $K_A(\mathbf{t},\mathbf{w})$ is a transform kernel function of the LCT and is denoted by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill K_A(\mathbf{t},\mathbf{w})=\sqrt{{\displaystyle \frac{1}{\mathbf{i}{\left(2\pi \right)}^{n}b}}}{\mathrm{e}}^{\mathbf{i}{\displaystyle \frac{a}{2b}}{\left|\mathbf{t}\right|}^{\mathbf{2}}-\mathbf{i}{\displaystyle \frac{1}{b}}\langle \mathbf{t},\mathbf{w}\rangle +\mathbf{i}{\displaystyle \frac{d}{2b}}{\left|\mathbf{w}\right|}^{\mathbf{2}}}.\end{array}\end{array}$$

(8)

If $b=0$, the LCT of $x\left(\mathbf{t}\right)$ is a chirp multiplication, and it is not the subject of our study. Without loss of generality, we suppose that $b\ne 0$ in this paper.

Notice that when $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, the LCT of $x\left(\mathbf{t}\right)$ reduces to the FT of $x\left(\mathbf{t}\right)$. That is to say,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \sqrt{\mathbf{i}}{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right)=\mathcal{F}\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right).\end{array}\end{array}$$

(9)

The LCT of $x\left(\mathbf{t}\right)$ reduces to the fractional Fourier transform (FRFT) [44] of $x\left(\mathbf{t}\right)$, when $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right]$.

According to the definition of the LCT, the additive property of the LCT [46] is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{L}}_B\left\{{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\right\}={\mathcal{L}}_{BA}\left\{x\left(\mathbf{t}\right)\right\},\end{array}\end{array}$$

(10)

where $B=\left[\begin{array}{cc}{a}_1& b_1\\ c_1& d_1\end{array}\right]\in {\mathbb{R}}^{2\times 2}$ is a matrix parameter.

Lemma 2

(Inverse transform for the LCT). Let the hypercomplex function $x\left(\mathbf{t}\right)=\Im \left(\mathbf{t}\right){\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\in L^2({\mathbb{E}}^n,C{l}_{0,n}),{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\in L^2({\mathbb{E}}^n,C{l}_{0,n})$. Then the inverse transform of the LCT is given by

$$\begin{array}{c}\hfill x\left(\mathbf{t}\right)={\mathcal{L}}_{A^{-1}}\left\{{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\right\}\left(\mathbf{t}\right)={\int }_{{\mathbb{E}}^n}{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right)K_{A^{-1}}(\mathbf{w},\mathbf{t})\mathrm{d}\mathbf{w},\end{array}$$

(11)

where $A^{-1}=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\in {\mathbb{R}}^{2\times 2}$ is a matrix parameter.

Proof. 

From the Formula (7), we obtain the following result:

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{E}}^n}{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right)K_{A^{-1}}(\mathbf{w},\mathbf{t})\mathrm{d}\mathbf{w}\hfill \\ & ={\int }_{{\mathbb{E}}^n}\left({\int }_{{\mathbb{E}}^n}x\left(\mathbf{s}\right)K_A(\mathbf{s},\mathbf{w})\mathrm{d}\mathbf{s}\right){\mathrm{K}}_{{\mathrm{A}}^{-1}}(\mathbf{w},\mathbf{t})\mathrm{d}\mathbf{w}.\hfill \end{array}$$

According to Formula (8), the above formula becomes

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{E}}^n}{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right)K_{A^{-1}}(\mathbf{w},\mathbf{t})\mathrm{d}\mathbf{w}\hfill \\ & ={\int }_{{\mathbb{E}}^n}x\left(\mathbf{s}\right){\mathrm{e}}^{\mathbf{i}{\displaystyle \frac{a}{2b}}({\mathbf{s}}^2-{\mathbf{t}}^2)}\delta (\mathbf{s}-\mathbf{t})\mathrm{d}\mathbf{s}\hfill \\ \hfill & =x\left(\mathbf{t}\right).\hfill \end{array}$$

This completes the proof. □

Lemma 3

((Plancherel theorem) [43]). If two hypercomplex functions $x,y\in L^2({\mathbb{E}}^n,C{l}_{0,n})$, then the Plancherel theorem can be presented by

$$\begin{array}{c}\hfill {\langle x,y\rangle }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}={\langle {\mathcal{L}}_A\left\{x\right\},{\mathcal{L}}_A\left\{y\right\}\rangle }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}.\end{array}$$

(12)

In particular, if $x=y$, we obtain Parseval’s theorem:

$$\begin{array}{c}\hfill {\parallel x\parallel }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}^2={\parallel {\mathcal{L}}_A\left\{x\right\}\parallel }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}^2.\end{array}$$

Based on the definitions of the FT and the LCT, the relationship between the LCT and the FT is obtained [43]

$$\begin{array}{c}\hfill \sqrt{\mathbf{i}b}{\mathrm{e}}^{-\mathbf{i}{\displaystyle \frac{d}{2b}}{\mathbf{w}}^{\mathbf{2}}}{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right)=\mathcal{F}\left\{x\left(\mathbf{t}\right){\mathrm{e}}^{\mathbf{i}{\displaystyle \frac{a}{2b}}{\mathbf{t}}^{\mathbf{2}}}\right\}\left({\displaystyle \frac{\mathbf{w}}{b}}\right).\end{array}$$

(13)

4. Uncertainty Principles for Linear Canonical Transform of Hypercomplex Functions

First, we will present the uncertainty principle of the FT in the directional case [20].

Lemma 4.

Suppose a hypercomplex function $x\left(\mathbf{t}\right)=\Im \left(\mathbf{t}\right){\mathbf{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\in L^2\left({\mathbb{E}}^n\right)$ and ${\parallel x\parallel }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}=1$, if $t_{j}x$ and ${\displaystyle \frac{\partial x\left(\mathbf{t}\right)}{\partial t_j}}$ exist for $j=1,\cdots ,n$. Then, the uncertainty principle of the FT is given by

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathbf{w}}^2\mid \mathcal{F}\left\{\mathit{x}\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}\ge {\displaystyle \frac{{\mathrm{n}}^2}{4}}+{\mathrm{COV}}_{\mathbf{t}}^2,\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{COV}}_{t_j}={\int }_{{\mathbb{E}}^n}\mid t_j\left({\displaystyle \frac{\partial }{\partial t_j}}{\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right)\left({\mathrm{e}}^{-\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right)\mid {\Im }^2\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}\end{array}$$

and the absolute covariance is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{COV}}_{\mathbf{t}}=\sum _{j=1}^n{\mathrm{COV}}_{t_j}.\end{array}\end{array}$$

The equality holds if and only if

$$\begin{array}{c}\hfill x\left(\mathbf{t}\right)={\mathrm{e}}^{-{\displaystyle \frac{\gamma }{2}}{\mathbf{t}}^2}{\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)},\end{array}$$

and

$$\begin{array}{c}\hfill \lambda \mid t_j\mid =\mid \left({\displaystyle \frac{\partial }{\partial t_j}}{\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right)\left({\mathrm{e}}^{-\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right)\mid ,\end{array}$$

where $\gamma >0$ and $\lambda >0$.

Next, according to the above lemma, we obtain the uncertainty principle of hypercomplex functions in the time domain and the LCT domain.

Theorem 1.

Let $x\left(\mathbf{t}\right)=\Im \left(\mathbf{t}\right){\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\in L^1\left({\mathbb{E}}^n\right)\bigcap L^2\left({\mathbb{E}}^n\right)$ be a hypercomplex function and ${\parallel x\parallel }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}=1$, if $t_{j}f$ and ${\displaystyle \frac{\partial x\left(\mathbf{t}\right)}{\partial t_j}}$ exist for $j=1,\cdots ,n$. Then, we have

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathbf{w}}^2\mid {\mathcal{L}}_{\mathrm{A}}\left\{\mathrm{x}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}\ge {\mathrm{b}}^2\left({\displaystyle \frac{{\mathrm{n}}^2}{4}}+{\mathrm{COV}}_{\mathbf{t}}^2\right),\end{array}$$

(14)

where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{COV}}_{t_j}={\int }_{{\mathbb{E}}^n}\mid t_j\left({\displaystyle \frac{\partial }{\partial t_j}}{\mathrm{e}}^{\phi \left(\mathbf{t}\right)\vartheta \left(\mathbf{t}\right)}\right)\left({\mathrm{e}}^{-\phi \left(\mathbf{t}\right)\vartheta \left(\mathbf{t}\right)}\right)\mid {\Im }^2\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}\end{array}$$

and the absolute covariance is ${\mathrm{COV}}_{\mathbf{t}}={\sum }_{j=1}^n{\mathrm{COV}}_{t_j}.$

The equality holds if and only if

$$\begin{array}{c}\hfill x\left(\mathbf{t}\right)={\mathrm{e}}^{-\mathbf{i}{\displaystyle \frac{a}{2b}}{\mathbf{t}}^{\mathbf{2}}}{\mathrm{e}}^{-{\displaystyle \frac{\gamma }{2}}{\mathbf{t}}^2}{\mathrm{e}}^{\phi \left(\mathbf{t}\right)\vartheta \left(\mathbf{t}\right)},\end{array}$$

and

$$\begin{array}{c}\hfill \lambda \mid t_j\mid =\mid \left({\displaystyle \frac{\partial }{\partial t_j}}{\mathrm{e}}^{\phi \left(\mathbf{t}\right)\vartheta \left(\mathbf{t}\right)}\right)\left({\mathrm{e}}^{-\phi \left(\mathbf{t}\right)\vartheta \left(\mathbf{t}\right)}\right)\mid ,\end{array}$$

where $\gamma >0$ and any real number $\lambda >0$.

Proof. 

Let $h\left(\mathbf{t}\right)=x\left(\mathbf{t}\right){\mathrm{e}}^{\mathbf{i}{\displaystyle \frac{a}{2b}}{\mathbf{t}}^{\mathbf{2}}}$. By Formula (1), $h\left(\mathbf{t}\right)=\Im \left(\mathbf{t}\right){\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\in L^1\left({\mathbb{E}}^n\right)\bigcap L^2\left({\mathbb{E}}^n\right)$.

According to Lemma 4, we have

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid h\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathbf{w}}^{\prime }2\mid \mathcal{F}\left\{\mathit{h}\right\}\left({\mathbf{w}}^{\prime }\right){\mid }^2\mathrm{d}{\mathbf{w}}^{\prime }\ge {\displaystyle \frac{{\mathrm{n}}^2}{4}}+{\mathrm{COV}}_{\mathbf{t}}^2,\end{array}$$

that is to say,

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid h\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\left({\displaystyle \frac{\mathbf{w}}{\mathrm{b}}}\right)}^2\mid \mathcal{F}\left\{\mathit{h}\right\}\left({\displaystyle \frac{\mathbf{w}}{\mathrm{b}}}\right){\mid }^2\mathrm{d}\left({\displaystyle \frac{\mathbf{w}}{\mathrm{b}}}\right)\ge {\displaystyle \frac{{\mathrm{n}}^2}{4}}+{\mathrm{COV}}_{\mathbf{t}}^2,\end{array}$$

where ${\mathbf{w}}^{\prime }={\displaystyle \frac{\mathbf{w}}{b}}={\displaystyle \frac{1}{b}}(w_1,w_2,\cdots ,w_n)\in {\mathbb{R}}^n$.

Based on Formula (13), the above formula becomes

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\left({\displaystyle \frac{\mathbf{w}}{\mathrm{b}}}\right)}^2\mid \sqrt{\mathbf{i}\mathrm{b}}{\mathrm{e}}^{-\mathbf{i}{\displaystyle \frac{\mathrm{d}}{2\mathrm{b}}}{\mathbf{w}}^{\mathbf{2}}}{\mathcal{L}}_{\mathrm{A}}\left\{\mathrm{x}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\left({\displaystyle \frac{\mathbf{w}}{\mathrm{b}}}\right)\ge {\displaystyle \frac{{\mathrm{n}}^2}{4}}+{\mathrm{COV}}_{\mathbf{t}}^2.\end{array}$$

The equality holds if and only if $x\left(\mathbf{t}\right){\mathrm{e}}^{\mathbf{i}{\displaystyle \frac{a}{2b}}{\mathbf{t}}^{\mathbf{2}}}={\mathrm{e}}^{-{\displaystyle \frac{\gamma }{2}}{\mathbf{t}}^2}{\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)},$ that is to say,

$x\left(\mathbf{t}\right)={\mathrm{e}}^{-\mathbf{i}{\displaystyle \frac{a}{2b}}{\mathbf{t}}^{\mathbf{2}}}{\mathrm{e}}^{-{\displaystyle \frac{\gamma }{2}}{\mathbf{t}}^2}{\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}.$ Hence, we obtain the result. □

Corollary 1.

When $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, Theorem 1 reduces to Lemma 4.

Corollary 2.

When $A=\left[\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right]$, Theorem 1 reduces to the uncertainty principle of the FRFT in the spatial case [44].

Theorem 2.

Let $x\left(\mathbf{t}\right)=\Im \left(\mathbf{t}\right){\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\in L^1\left({\mathbb{E}}^n\right)\bigcap L^2\left({\mathbb{E}}^n\right)$ be a hypercomplex function and ${\parallel x\parallel }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}=1$, if $t_{j}f$ and ${\displaystyle \frac{\partial x\left(\mathbf{t}\right)}{\partial t_j}}$ exist for $j=1,\cdots ,n$. Then, we obtain

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{\mathbf{v}}^2\mid {\mathcal{L}}_M\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{v}\right){\mid }^2\mathrm{d}\mathbf{v}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathbf{w}}^2\mid {\mathcal{L}}_{\mathrm{A}}\left\{\mathrm{x}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}\ge {\mathrm{b}}^2\left({\displaystyle \frac{{\mathrm{n}}^2}{4}}+{\mathrm{COV}}_{\mathbf{t}}^2\right),\end{array}$$

(15)

where $A^{\prime }=\left[\begin{array}{cc}{a}^{\prime }& b\\ c^{\prime }& d\end{array}\right]\in {\mathbb{R}}^{2\times 2}$, $a\ne a^{\prime }$, $a^{\prime }d-b{c}^{\prime }=1$, $M=A^{\prime -1}A=\left[\begin{array}{cc}1& 0\\ c{a}^{\prime }-a{c}^{\prime }& 1\end{array}\right]$, $\mathbf{v}=(v_1,v_2,\cdots ,v_n)\in {\mathbb{R}}^n$, and the absolute covariance of every variable is given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{COV}}_{t_j}={\int }_{{\mathbb{E}}^n}\mid t_j\left({\displaystyle \frac{\partial }{\partial t_j}}{\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right)\left({\mathrm{e}}^{-\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right)\mid {\Im }^2\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{COV}}_{\mathbf{t}}=\sum _{j=1}^n{\mathrm{COV}}_{t_j}.\end{array}\end{array}$$

The equality holds if and only if

$$\begin{array}{c}\hfill x\left(\mathbf{t}\right)={\mathrm{e}}^{-\mathbf{i}{\displaystyle \frac{a}{2b}}{\mathbf{t}}^{\mathbf{2}}}{\mathrm{e}}^{-{\displaystyle \frac{\gamma }{2}}{\mathbf{t}}^2}{\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)},\end{array}$$

and

$$\begin{array}{c}\hfill \lambda \mid t_j\mid =\mid \left({\displaystyle \frac{\partial }{\partial t_j}}{\mathrm{e}}^{\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right)\left({\mathrm{e}}^{-\wp \left(\mathbf{t}\right)\aleph \left(\mathbf{t}\right)}\right)\mid ,\end{array}$$

where $\gamma >0$ and $\lambda >0$.

Proof. 

Let $h\left(\mathbf{t}\right)=x\left(\mathbf{t}\right){\mathrm{e}}^{\mathbf{i}{\displaystyle \frac{a}{2b}}{\mathbf{t}}^{\mathbf{2}}}$. According to Formula (3), we have

$$\begin{array}{c}\hfill \mid h\left(\mathbf{t}\right){\mid }^2=\mid {\displaystyle \frac{1}{\sqrt{{\left(2\pi \right)}^n}}}{\int }_{{\mathbb{E}}^n}\mathcal{F}\left\{h\left(\mathbf{t}\right)\right\}\left({\mathbf{w}}^{\prime }\right){\mathrm{e}}^{\mathbf{i}\langle \mathbf{t},{\mathbf{w}}^{\prime }\rangle }\mathrm{d}{\mathbf{w}}^{\prime }{\mid }^2.\end{array}$$

Let ${\mathbf{w}}^{\prime }={\displaystyle \frac{\mathbf{w}}{b}}$; then,

$$\begin{array}{c}\hfill \mid h\left(\mathbf{t}\right){\mid }^2=\mid {\displaystyle \frac{1}{\sqrt{{\left(2\pi \right)}^n}}}{\int }_{{\mathbb{E}}^n}\mathcal{F}\left\{h\left(\mathbf{t}\right)\right\}({\displaystyle \frac{\mathbf{w}}{b}}){\mathrm{e}}^{\mathbf{i}\langle \mathbf{t},{\displaystyle \frac{\mathbf{w}}{b}}\rangle }\mathrm{d}({\displaystyle \frac{\mathbf{w}}{\mathrm{b}}}){\mid }^2.\end{array}$$

By Formula (13), the above formula becomes

$$\begin{array}{cc}\hfill & \mid h\left(\mathbf{t}\right){\mid }^2\hfill \\ & =\mid {\displaystyle \frac{1}{\sqrt{{\left(2\pi \right)}^n}}}{\int }_{{\mathbb{E}}^n}\sqrt{\mathbf{i}b}{\mathbf{e}}^{-\mathbf{i}{\displaystyle \frac{d}{2b}}{\mathbf{w}}^{\mathbf{2}}}{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mathrm{e}}^{{\displaystyle \frac{\mathbf{i}\langle \mathbf{t},\mathbf{w}\rangle }{b}}}\mathrm{d}({\displaystyle \frac{\mathbf{w}}{\mathrm{b}}}){\mid }^2\hfill \\ & =\mid {\displaystyle \frac{1}{\sqrt{-{\left(2\pi \right)}^{n}b\mathbf{i}}}}{\int }_{{\mathbb{E}}^n}{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mathrm{e}}^{{\displaystyle \frac{\mathbf{i}\langle \mathbf{t},\mathbf{w}\rangle }{b}}-\mathbf{i}{\displaystyle \frac{d}{2b}}{\mathbf{w}}^{\mathbf{2}}}\mathrm{d}\mathbf{w}{\mid }^2\hfill \\ & =\mid {\displaystyle \frac{{\mathrm{e}}^{-\mathbf{i}\frac{a^{\prime }}{2b}{\mathbf{t}}^{\mathbf{2}}}}{\sqrt{-{\left(2\pi \right)}^{n}b\mathbf{i}}}}{\int }_{{\mathbb{E}}^n}{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mathrm{e}}^{-\mathbf{i}{\displaystyle \frac{a^{\prime }}{2b}}{\mathbf{t}}^{\mathbf{2}}+{\displaystyle \frac{\mathbf{i}\langle \mathbf{t},\mathbf{w}\rangle }{b}}-\mathbf{i}{\displaystyle \frac{d}{2b}}{\mathbf{w}}^{\mathbf{2}}}\mathrm{d}\mathbf{w}{\mid }^2,\hfill \end{array}$$

where $A^{\prime }=\left[\begin{array}{cc}{a}^{\prime }& b\\ c^{\prime }& d\end{array}\right]\in {\mathbb{R}}^{2\times 2}$, $a\ne a^{\prime }$, and the condition $a^{\prime }d-b{c}^{\prime }=1$ is satisfied.

According to Formula (11), we obtain

$$\begin{array}{cc}\hfill \mid h\left(\mathbf{t}\right){\mid }^2& =\mid {\mathcal{L}}_{A^{\prime -1}}\left\{{\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\right\}{\mid }^2;\hfill \end{array}$$

by Formula (10),

$$\begin{array}{cc}\hfill \mid h\left(\mathbf{t}\right){\mid }^2& =\mid {\mathcal{L}}_{A^{\prime -1}A}\left\{x\left(\mathbf{t}\right)\right\}{\mid }^2\hfill \\ & =\mid {\mathcal{L}}_M\left\{x\left(\mathbf{t}\right)\right\}{\mid }^2,\hfill \end{array}$$

where $M=A^{\prime -1}A$, and

$$\begin{array}{cc}\hfill M& =\left[\begin{array}{cc}d& -b\\ -c^{\prime }& a^{\prime }\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\hfill \\ & =\left[\begin{array}{cc}1& 0\\ c{a}^{\prime }-a{c}^{\prime }& 1\end{array}\right].\hfill \end{array}$$

According to the Euler formula,

$$\begin{array}{c}\hfill \mid x\left(\mathbf{t}\right){\mid }^2=\mid h\left(\mathbf{t}\right){\mid }^2=\mid {\mathcal{L}}_M\left\{x\left(\mathbf{t}\right)\right\}{\mid }^2.\end{array}$$

Based on Theorem 1, we can obtain the result. □

Based on Theorem 2, the uncertainty principle in two domains of the LCT can generate the uncertainty principle of the FRFT in two fractional Fourier transform domains [44], when $A=\left[\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right]$ and $M=\left[\begin{array}{cc}cos\beta & sin\beta \\ -sin\beta & cos\beta \end{array}\right]$.

5. Example

Next, we prove the validity of Theorem 1 in two cases through the following examples.

Case 1: Let a hypercomplex function be given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill x\left(\mathbf{t}\right)={({\displaystyle \frac{z}{\pi }})}^{{\displaystyle \frac{n}{4}}}{\mathrm{e}}^{-{\displaystyle \frac{z}{2}}\langle \mathbf{t},\mathbf{t}\rangle }{\mathrm{e}}^{\mathbf{i}{\displaystyle \frac{a}{2b}}\langle \mathbf{t},\mathbf{t}\rangle }{\mathrm{e}}^{\phi {\displaystyle \frac{\langle \mathbf{t},\mathbf{t}\rangle }{2}}},\end{array}\end{array}$$

(16)

where $\phi$ is a vector-valued constant, $\mathit{z}$ is a positive real number, and ${\parallel x\parallel }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}=1$ is unit energy.

First, we calculate the following formula:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{t}_j^2\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}={\displaystyle \frac{1}{2\mathit{z}}}.\end{array}\end{array}$$

(17)

Moreover, according to the Formula (13), we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{w}_j^2\mid {\mathcal{L}}_A\left\{x\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}={\mathrm{b}}^2({\displaystyle \frac{\mathit{z}}{2}}+{\displaystyle \frac{1}{2\mathit{z}}}),\end{array}\end{array}$$

(18)

where the equation is derived from Formula (6).

Then, we obtain

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{t}_j^2\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathit{w}}_{\mathrm{j}}^2\mid {\mathcal{L}}_{\mathrm{A}}\left\{\mathit{x}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}={\displaystyle \frac{{\mathrm{b}}^2}{4}}(1+{\displaystyle \frac{1}{{\mathit{z}}^2}}).\end{array}$$

On the other hand,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{COV}}_{t_j}={\displaystyle \frac{1}{2\mathit{z}}}.\end{array}\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{t}_j^2\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathit{w}}_{\mathrm{j}}^2\mid {\mathcal{L}}_{\mathrm{A}}\left\{\mathit{x}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}={\mathrm{b}}^2({\displaystyle \frac{1}{4}}+{\mathrm{COV}}_{{\mathrm{t}}_{\mathrm{j}}}^2).\end{array}$$

Hence,

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathbf{w}}^2\mid {\mathcal{L}}_{\mathrm{A}}\left\{\mathit{x}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}={\mathrm{b}}^2({\displaystyle \frac{{\mathrm{n}}^2}{4}}+{\mathrm{COV}}_{\mathbf{t}}^2).\end{array}$$

Thus we have verified that the equation of Theorem 1 is true.

Case 2: Suppose a hypercomplex function

$$\begin{array}{c}\hfill \begin{array}{c}\hfill y\left(\mathbf{t}\right)={({\displaystyle \frac{\mathit{z}}{\pi }})}^{{\displaystyle \frac{n}{4}}}{\mathrm{e}}^{-{\displaystyle \frac{\mathit{z}}{2}}\langle \mathbf{t},\mathbf{t}\rangle }{\mathrm{e}}^{\mathbf{i}{\displaystyle \frac{a}{2b}}\langle \mathbf{t},\mathbf{t}\rangle }{\mathrm{e}}^{{\mathit{z}}_1{t}_1{\mathbf{e}}_{\mathbf{1}}},\end{array}\end{array}$$

(19)

where $\mathit{z}$ is a positive real number, ${\mathit{z}}_1$ is a real number, and ${\parallel y\parallel }_{L^2({\mathbb{E}}^n,C{l}_{0,n})}=1$ is unit energy.

According to Formula (17), we have

$$\begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{t}_j^2\mid y\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}={\displaystyle \frac{1}{2\mathit{z}}}.\end{array}$$

When $j=1$, it follows from Formula (18) that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{w}_1^2\mid {\mathcal{L}}_A\left\{y\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}={\mathrm{b}}^2({\displaystyle \frac{\mathit{z}}{2}}+{\mathit{z}}_1^2),\end{array}\end{array}$$

(20)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{COV}}_{t_1}={\displaystyle \frac{|{\mathit{z}}_1|}{\sqrt{\pi \mathit{z}}}};\end{array}\end{array}$$

thus,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{t}_1^2\mid y\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathit{w}}_1^2\mid {\mathcal{L}}_{\mathrm{A}}\left\{\mathit{y}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}>{\mathrm{b}}^2({\displaystyle \frac{1}{4}}+{\mathrm{COV}}_{{\mathrm{t}}_1}^2).\end{array}\end{array}$$

(21)

In addition, for $j=2,\cdots ,n$, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{w}_j^2\mid {\mathcal{L}}_A\left\{y\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}={\mathrm{b}}^2{\displaystyle \frac{\mathit{z}}{2}},\end{array}\end{array}$$

(22)

and ${\mathrm{COV}}_{t_j}=0.$

Then,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{t}_j^2\mid y\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathit{w}}_{\mathrm{j}}^2\mid {\mathcal{L}}_{\mathrm{A}}\left\{\mathit{y}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}={\mathrm{b}}^2({\displaystyle \frac{1}{4}}+{\mathrm{COV}}_{{\mathrm{t}}_1}^2).\end{array}\end{array}$$

(23)

For $j=1,\cdots ,n$, by combining Formulas (21) and (23), we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{t}_j^2\mid y\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathit{w}}_{\mathrm{j}}^2\mid {\mathcal{L}}_{\mathrm{A}}\left\{\mathit{y}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}>{\mathrm{b}}^2({\displaystyle \frac{1}{4}}+{\mathrm{COV}}_{{\mathrm{t}}_{\mathrm{j}}}^2).\end{array}\end{array}$$

(24)

Hence

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid y\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{E}}^{\mathrm{n}}}{\mathbf{w}}^2\mid {\mathcal{L}}_{\mathrm{A}}\left\{\mathit{y}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}>{\mathrm{b}}^2({\displaystyle \frac{{\mathrm{n}}^2}{4}}+{\mathrm{COV}}_{\mathbf{t}}^2).\end{array}\end{array}$$

(25)

The above formula show that the inequality in Theorem 1 true.

6. Potential Applications

In this section, several potential applications of the uncertainty principles associated with the LCT are discussed to show the significance and useful of the theorems in signal processing [33,47].

The uncertainty principles of the LCT show that when the signal $x\left(\mathbf{t}\right)$ is a Gaussian signal, the minimum value $b^2\left({\displaystyle \frac{n^2}{4}}+CO{V}_{\mathbf{t}}^2\right)$ can be achieved. However, for a non-Gaussian signal, the lower bound $b^2\left({\displaystyle \frac{n^2}{4}}+CO{V}_{\mathbf{t}}^2\right)$ of the uncertainty principles cannot be obtained, and the spread product is greater than the lower bound. Hence, uncertainty principles can be applied to estimate the bandwidths by the lower bound [33]. For instance, when ${\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}$ is given, based on Theorem 1, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{{\mathbb{E}}^n}{\mathbf{w}}^2\mid {\mathcal{L}}_A\left\{\mathit{y}\left(\mathbf{t}\right)\right\}\left(\mathbf{w}\right){\mid }^2\mathrm{d}\mathbf{w}& \ge {\displaystyle \frac{b^2({\displaystyle \frac{n^2}{4}}+{\mathrm{COV}}_{\mathbf{t}}^2)}{{\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid y\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}}}\hfill \\ & \ge {\displaystyle \frac{b^2{n}^2}{4{\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid y\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}}}.\hfill \end{array}\end{array}$$

(26)

It shows that only Gaussian signals can obtain the second term in the above inequality. That is to say, the middle term of the above inequality is usually not reachable.

The LCT is not only an integral transform in mathematics but also an affine transform associated with the time-frequency plane [33]. The uncertainty principle of LCT is also applied to the spectral analysis of affine modulation systems [47]. For instance, in affine modulation schemes, spectral efficiency plays a very important role in the effective implementation of the LCT [33,47]. If the pulse is compressed in the time–frequency plane, it is crucial to study the effective bandwidth of the LCT domain [47]. The uncertainty principles in this paper are combined with the multi-channel interpolation technique to discuss the spread spectrum problem in the LCT domain, which has theoretical significance and application value [33].

The uncertainty principle is closely related to the study of optical signal processing [30,31]. The uncertainty principle of the LCT can be used to deal with optical systems associated with propagation in the LCT domain [30,31]. Here, we present an example to prove the application associated with the uncertainty principle of the LCT in the field of optics. Specifically, we use an example based on wave propagation through an aperture setting [48,49]. After passing through the aperture, the field has a certain effective width ${\int }_{{\mathbb{E}}^n}{\mathbf{t}}^2\mid x\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}$, which is determined by the aperture width. According to Theorem 1, by propagating some distance, the transverse distribution of the field can be described by simulation of the LCT. Therefore, the uncertainty principle of the LCT has important application prospects for signal recovery in the hypercomplex domain.

7. Conclusions

In this paper, we studied two kinds of uncertainty principles associated with the LCT for hypercomplex functions. According to the relationship between the FT and the LCT, the inverse transform of the LCT was given. Based on the uncertainty principle of the FT, we obtained a new uncertainty principle of the LCT for hypercomplex functions in the time and LCT domains. The conditions under which the equality holds were presented. The obtained results can be regarded as the general form of the FT and FRFT. Then, we exploited an uncertainty principle in two LCT domains by the inverse transform of the LCT. This paper shows that these uncertainty principles were related to the absolute covariance, which can be obtained from Gaussian signals. In addition, the advantages of our results were analyzed, and the differences between the obtained results and the existing literature were also compared. Finally, the validity of the obtained result in two cases were proved using examples. Several potential applications in applied mathematics and signal processing were discussed.