Convolution, Correlation and Uncertainty Principle in the One-Dimensional Quaternion Quadratic-Phase Fourier Transform Domain

Abstract

In this paper, we present a novel integral transform known as the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT). We first define the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT) of integrable (and square integrable) functions on $\mathbf{R}$. Later on, we show that 1D-QQPFT satisfies all the respective properties such as inversion formula, linearity, Moyal’s formula, convolution theorem, correlation theorem and uncertainty principle. Moreover, we use the proposed transform to obtain an inversion formula for two-dimensional quaternion quadratic-phase Fourier transform. Finally, we highlight our paper with some possible applications.

1. Introduction

Recently, Castro et al. [1,2] introduced a superlative generalized version of the Fourier transform(FT) coined as quadratic-phase Fourier transform (QPFT) which has overthrown all the applicable signal processing tools as it provides a unified analysis of both transient and non-transient signals in an easy and insightful fashion. We refer to [3,4] for information about integral transforms.

For parameter $\mu =(A,B,C,D,E),$ the QPFT of any signal $f\in L^1\left(\mathbb{R}\right)$ is defined as

$${\mathcal{Q}}_{\mu }\left[f\right]\left(w\right)=\frac{1}{\sqrt{2\pi }}\int f\left(t\right){\Lambda }_{\mu }(t,w)dt,$$

(1)

where ${\Lambda }_{\mu }(t,w)$ is called quadratic-phase Fourier kernel and is given by

$${\Lambda }_{\mu }(t,w)=exp\left\{i(A{t}^2+Btw+C{w}^2+Dt+Ew)\right\}$$

(2)

with $A,B,C,D,E\in \mathbf{R},B\ne 0.$

Later on, much research has been carried out on quadratic-phase Fourier transform (see [5,6,7,8]). In the prospect of signal processing, it is evident that any signal processing tool converts the time-domain signals into frequency-domain. Further, in signal processing, convolution of two functions [9,10,11,12,13] is a most useful tool in constructing a filter for denoising the given noisy signals (see [14]).

In past decades, hypercomplex algebra has become a leading area of research with its applications in color image processing, image filtering, watermarking, edge detection and pattern recognition (see [15,16,17,18,19,20,21]). The Cayley–Dickson algebra of order four is labeled as quaternions which has wide applications in optical and signal processing. The extension of the Fourier transform in quaternion algebra is known as quaternion Fourier transform (QFT) [22] which is believed to be the substitute for the commonly used two-dimensional complex Fourier transform (CFT). The QFT has a wide range of applications: see [23,24]. The quaternion linear canonical transform (QLCT) and the quaternion offset linear canonical transform (QOLCT) are the generalized versions of the QFT and QLCT, respectively, which are both effective signal processing tools. The modern era of information processing is in dire need of quaternionic valued signals and, therefore, is a very hot area of research (see [25,26,27,28]). Recently, Bhat and Dar [29] introduced quaternion quadratic-phase Fourier transform Q-QPFT as a generilization of QFT. Since then it has gained much popularity in signal processing community [30,31,32,33]. Many fundamental properties of Q-QPFT are known, but the theory about quaternion one-dimensional QPFT as far as we know is still in its infancy. Therefore, it is worthwhile to study the theory of quaternion one-dimensional QPFT which can be productive for signal processing theory and applications.

In this paper, our main objective is to introduce the novel integral transform called the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT) and study its properties, such as inversion formula, linearity, Moyal’s formula, convolution theorem, correlation theorem, Heisenberg–Pauli–Weyl uncertainty inequality and inversion formula for two-dimensional quaternion QPFT.

This paper is organized as follows: In Section 2, some general definitions and basic properties of quaternions are summarized. The definition and the properties of the 1D-QQPFT are studied in Section 3. In Section 4 we obtain inversion formula for the two-dimensional quadratic-phase Fourier transform. In Section 5, some potential applications of the proposed transform are highlighted. In Section 6, conclusions are drawn.

2. Preliminaries

Here, we introduce the already known results which are required in subsequent sections.

2.1. Quadratic-Phase Fourier Transform

In this subsection we introduce the quadratic-phase Fourier transform, which is a neoteric addition to the classical integral transforms and we also give its inversion formula and some other classical results which are already present in literature.

Definition 1.

Given a parameter $\mu =(A,B,C,D,E),$ the QPFT of any signal $f\in L^1\left(\mathbb{R}\right)$ is defined by

$${\mathcal{Q}}_{\mu }\left[f\right]\left(w\right)=\frac{1}{\sqrt{2\pi }}\int f\left(t\right){\Lambda }_{\mu }(t,w)dt,$$

(3)

where the quadratic-phase Fourier kernel ${\Lambda }_{\mu }(t,w)$ is given by

$${\Lambda }_{\mu }(t,w)=exp\left\{i(A{t}^2+Btw+C{w}^2+Dt+Ew)\right\}$$

(4)

with $A,B,C,D,E\in \mathbf{R},B\ne 0.$

Theorem 1.

The inversion formula of the quadratic-phase Fourier transform is given by

$$f\left(t\right)=\frac{\left|B\right|}{\sqrt{2\pi }}\int {\mathbf{Q}}_{\mu }\left[f\right]\left(w\right)\overline{{\Lambda }_{\mu }(t,w)}dw.$$

(5)

Using the inversion theorem, we can obtain the Parseval’s relation given by

$$\langle f,g\rangle =\left|B\right|\langle {\mathbf{Q}}_{\mu }\left[f\right],{\mathbf{Q}}_{\mu }\left[g\right]\rangle$$

(6)

Theorem 2

([5,6]). Let $f,g\in L^2\left(\mathbf{R}\right)$ and $\alpha ,\beta ,\tau \in \mathbf{R}$ then

• ${\mathbf{Q}}_{\mu }[\alpha f+\beta g]\left(w\right)=\alpha {\mathbf{Q}}_{\mu }\left[f\right]\left(w\right)+\beta {\mathbf{Q}}_{\mu }\left[g\right]\left(w\right).$
• ${\mathbf{Q}}_{\mu }\left[f\right(t-\tau \left)\right]\left(w\right)=exp\{-i(A{\tau }^2+B\tau w+D\tau \left)\right\}{\mathbf{Q}}_{\mu }[exp\{-2iA\tau t\left\}f\right(t\left)\right]\left(w\right).$
• ${\mathbf{Q}}_{\mu }\left[f(-t)\right]\left(w\right)={\mathbf{Q}}_{{\mu }^{\prime }}\left[f\left(t\right)\right](-w)$, where ${\mu }^{\prime }=(A,B,C,-D,-E).$
• ${\mathbf{Q}}_{\mu }[exp\{i\alpha t\left\}f\right(t\left)\right]\left(w\right)=exp\left\{i\right({\alpha }^2+2\alpha Bw+\alpha EB\left)\frac{1}{B}\right\}{\mathbf{Q}}_{\mu }\left[f\right]\left(w+\frac{A}{B}\right).$
• ${\mathbf{Q}}_{\mu }\left[\overline{f\left(t\right)}\right]\left(w\right)=\overline{{\mathbf{Q}}_{-\mu }\left[f\left(t\right)\right]\left(w\right)},$ where $-\mu =(-A,-B,-C,-D,-E).$

Theorem 3

(QPFT Convolution [5]). If $f,g\in L^2\left(\mathbf{R}\right)$ then

$$\begin{array}{c}\hfill {\mathbf{Q}}_{\mu }\left[f{\ast }_{\mu }g\right(t\left)\right]\left(w\right)=exp\{-i(C{w}^2+Ew)\}{\mathbf{Q}}_{\mu }\left[f\right]\left(w\right){\mathbf{Q}}_{\mu }\left[g\right]\left(w\right)\end{array}$$

where

$$\left(f{\ast }_{\mu }g\right)\left(t\right)=\frac{1}{\sqrt{2\pi }}{\int }_{\mathbf{R}}f(z)g(t-z)exp\left\{2iAz\right(t-z\left)\right\}dz.$$

(7)

Theorem 4

(Heisenberg–Pauli–Weyl inequality [6]). Let ${\mathbf{Q}}_{\mu }\left[f\right]$ be the quadratic-phase Fourier transform of any signal f, then the following inequality holds:

$$\begin{array}{c}\hfill {\int }_{\mathbf{R}}{w}^2|{\mathbf{Q}}_{\mu }\left[f\right]{|}^{2}dw{\int }_{\mathbf{R}}{t}^2{\left|f\left(t\right)\right|}^{2}dt\ge {\left\{\frac{1}{2\left|B\right|}{\int }_{\mathbf{R}}{\left|f\left(t\right)\right|}^{2}dt\right\}}^2.\end{array}$$

2.2. Quaternions

Let $\mathbf{R}$ and $\mathbf{C}$ be the usual set of real numbers and set of complex numbers, respectively. The division ring of quaternions in the honor of Hamilton, is denoted by $\mathbf{H}$ and is defined as

$$\begin{array}{ccc}\hfill \mathbf{H}& =& \{a+e_{1}b+e_{2}c+e_{3}d:a,b,c,d\in \mathbf{R}\}\hfill \\ & =& \{(a+e_{1}b)+e_2(c-e_{1}d):a,b,c,d\in \mathbf{R}\}\hfill \\ & =& \{z_1+e_2{z}_2:z_1,z_2\in \mathbf{C}\}\left(CayleyDicksonform\right)\hfill \end{array}$$

where $e_1,e_2,e_3$ satisfy Hamilton’s multiplication rule

$$e_1{e}_2=-e_2{e}_1=e_3,e_2{e}_3=-e_3{e}_2=e_1,$$

$$e_3{e}_1=-e_1{e}_3=e_2\mathrm{and}{e}_1^2=e_2^2=e_3^2=-1.$$

Every member of $\mathbf{H}$ is known as quaternion. In quaternion algebra addition, multiplication, conjugate and absolute value of quaternions are defined by

$$(a_1+e_2{a}_2)+(b_1+e_2{b}_2)=(a_1+b_1)+e_2(a_2+b_2),$$

$$(a_1+e_2{a}_2)(b_1+e_2{b}_2)=(a_1{b}_1-{\overline{a}}_2{b}_2)+e_2(a_2{b}_1+{\overline{a}}_1{b}_2),$$

$${(a_1+e_2{a}_2)}^c={\overline{a}}_1-e_2{a}_2,$$

$$|a_1+e_2{a}_2|=\sqrt{{\left|a_1\right|}^2+{\left|a_2\right|}^2},$$

here, ${\overline{a}}_k$ is the complex conjugate of $a_k$ and $|a_k|$ is the modulus of the complex number $a_k,k=1,2.$ For all $a=a_1+e_2{a}_2,b=b_1+e_2{b}_2\in \mathbf{H},$ the following properties of conjugate and modulus and multiplicative inverse are well known.

$${\left(a^c\right)}^c=a,{(a+b)}^c=a^c+b^c,{\left(ab\right)}^c=b^c{a}^c,$$

$${\left|a\right|}^2=a{a}^c{=\left|a_1\right|}^2{+\left|a_2\right|}^2,\left|ab\right|=\left|a\right|\left|b\right|,$$

$$a^{-1}=\frac{\overline{a}}{{\left|a\right|}^2}.$$

We denote $L^p(\mathbf{R},\mathbf{H}),$ the Banach space of all quaternion-valued functions f satisfying

$${\parallel f\parallel }_p={\left(\int |f_1{\left(t\right)|}^p+{\left|f_2\left(t\right)\right|}^{p}dt\right)}^{1/p}<\infty ,p=1,2.$$

Moreover, on $L^2(\mathbf{R},\mathbf{H})$ the inner product

$$\langle f,g\rangle =\int f\left(t\right){\left[g\left(t\right)\right]}^{c}dt,$$

where integral of a quaternion valued function is defined by

$$\int (f_1+e_2{f}_2)\left(t\right)dt=\int f_1\left(x\right)dt+e_2\int f_2\left(x\right)dt,$$

whenever the integral exists.

3. Quaternion One-Dimensional Quadratic-Phase Fourier Transform

In this section, we will introduce the definition of quaternion one-dimensional quadratic-phase Fourier transform (1D-QQPFT) by using [34,35,36,37]. Prior to that we note $e_1,e_2$ and $e_3$ (or equivalently $i,j,k$) denote the three imaginary, units in the quaternion algebra [38].

Definition 2.

The 1D-QQPFT of any signal $f\in L^1(\mathbf{R},\mathbf{H})$ with respect a parameter $\mu =(A,B,C,D,E)$ is defined by

$${\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\left(t\right)\right]\left(w\right)=\frac{1}{\sqrt{2\pi }}\int f\left(t\right){\Lambda }_{\mu }^{e_2}(t,w)dt$$

(8)

where

$${\Lambda }_{\mu }^{e_2}(t,w)=exp\left\{e_2(A{t}^2+Btw+C{w}^2+Dt+Ew)\right\}.$$

(9)

With $A,B,C,D,E\in \mathbf{R},B\ne 0.$ Now we can find that if $f\left(t\right)$ is a real-valued signal in (8), then we can interchange the kernel (9) and (2).

By appropriately choosing parameters in

$\mu =(A,B,C,D,E)$ the 1D-QQPFT(8) gives birth to the following existing time-frequency transforms:

• For $\mu =(0,1,0,0,0)$, the 1D-QQPFT (2) boils down to the quaternion one-dimensional Fourier Transform [34]
• For $\mu =(A/2B,-1/B,C/2B,0,0)$ and multiplying the right side of (9) by $1/\sqrt{e_{2}B}$ the 1D-QQPFT(2) reduces to the quaternion one-dimensional linear canonical transform [37].
• For $\mu =(cot\theta /2,-csc\theta ,cot\theta /2,0,0),\theta \ne 0$ and multiplying the right side of (9) by $\sqrt{1-e_{2}cot\theta },$ the 1D-QQPFT (2) reduces to the quaternion one-dimensional fractional Fourier transform [35].

Example 1.

The 1D-QQPFT of the signal $f\left(t\right)=exp\{-k{t}^2\}$, $k\ge 0$ is given by

$$\begin{array}{ccc}\hfill & & {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\left(t\right)\right]\left(w\right)\hfill \\ \hfill & & =\frac{1}{\sqrt{2\pi }}\int f\left(t\right){\Lambda }_{\mu }^{e_2}(t,w)dt\hfill \\ \hfill & & =\frac{1}{\sqrt{2\pi }}\int exp\{-k{t}^2\}exp\left\{e_2(A{t}^2+Btw+C{w}^2+Dt+Ew)\right\}dt\hfill \\ \hfill & & =\frac{1}{\sqrt{2\pi }}exp\left\{e_2(C{w}^2+Ew)\right\}\int exp\{-k{t}^2\}exp\left\{e_2(A{t}^2+Btw+Dt)\right\}dt\hfill \\ \hfill & & =\frac{1}{\sqrt{2\pi }}exp\left\{e_2(C{w}^2+Ew)\right\}\hfill \\ \hfill & & \times \int exp\left\{-(k+e_{2}A)\left[t+e_2\frac{{(Bw+D)}^2}{2(k+e_{2}A)}\right]\right\}dtexp\left\{-\frac{{(Bw+D)}^2}{4(k+e_{2}A)}\right\}.\hfill \end{array}$$

(10)

Now, with the help of Gaussian integral, (10) yields

$$\begin{array}{ccc}\hfill & & {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\left(t\right)\right]\left(w\right)\hfill \\ & & =\frac{1}{\sqrt{2\pi }}exp\left\{e_2(C{w}^2+Ew)\right\}\sqrt{\frac{\pi }{k+e_{2}A}}exp\left\{-\frac{{(Bw+D)}^2}{4(k+e_{2}A)}\right\}\hfill \\ & & =exp\left\{e_2(C{w}^2+Ew)\right\}\sqrt{\frac{1}{2(k+e_{2}A)}}exp\left\{-\frac{{(Bw+D)}^2}{4(k+e_{2}A)}\right\}.\hfill \end{array}$$

Definition 3

(Inversion). The inverse can be expressed in the form

$$\begin{array}{ccc}\hfill f\left(t\right)& =& {\left\{{\mathbf{Q}}_{\mu }^{\mathbf{H}}\right\}}^{-1}\left[{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\right]\left(t\right)\hfill \\ \hfill & =& \frac{\left|B\right|}{\sqrt{2\pi }}\int {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right)\overline{{\Lambda }_{\mu }^{e_2}(t,w)}dw.\hfill \end{array}$$

(11)

Definition 4.

Let $f=f_1+e_2{f}_2$ be a quaternion valued signal in $L^1(\mathbf{R},\mathbf{H}),$ then the quaternion quadratic-phase Fourier transform is defined as

$${\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\left(t\right)\right]\left(w\right)={\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_1\left(t\right)\right]\left(w\right)+e_2{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_2\left(t\right)\right]\left(w\right).$$

(12)

By above definition, it is consistent with the quadratic-phase Fourier transform on $L^1(\mathbf{R},\mathbf{C}).$ Now it is clear from the definition of quaternion quadratic-phase Fourier transform and the properties of quadratic-phase Fourier transform on $L^1(\mathbf{R},\mathbf{H}),$ that ${\mathbf{Q}}_{\mu }^{\mathbf{H}}\left({\mathbf{Q}}_{\nu }^{\mathbf{H}}\left[f\right]\right)={\mathbf{Q}}_{\mu +\nu }^{\mathbf{H}}\left[f\right]$ and ${\left\{{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\right\}}^{-1}={\mathbf{Q}}_{-\mu }^{\mathbf{H}}\left[f\right]$ for every signal $f\in L^1(\mathbf{R},\mathbf{H}).$

Theorem 5.

The quaternion quadratic-phase Fourier transform ${\mathbf{Q}}_{\mu }^{\mathbf{H}}$ is $\mathbf{H}-$ linear on $L^1(\mathbf{R},\mathbf{H}).$

Proof. 

Let us consider two quaternion signals $f=f_1+e_2{f}_2$ and $g=g_1+e_2{g}_2$ in $L^1(\mathbf{R},\mathbf{H});$ now, by the linearity of ${\mathbf{Q}}_{\mu }^{\mathbf{H}}$ on $L^1(\mathbf{R},\mathbf{C}),$ we obtain

$$\begin{array}{ccc}& & {\mathbf{Q}}_{\mu }^{\mathbf{H}}[f+g]\left(w\right)\hfill \\ & & ={\mathbf{Q}}_{\mu }^{\mathbf{H}}[(f_1+e_2{f}_2)+(g_1+e_2{g}_2)]\left(w\right)\hfill \\ & & ={\mathbf{Q}}_{\mu }^{\mathbf{H}}[(f_1+g_1)+e_2(f_2+g_2)]\left(w\right)\hfill \\ & & ={\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_1\right]\left(w\right)+{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\left(w\right)\hfill \\ & & +e_2\left({\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_2\right]\left(w\right)+{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\left(w\right)\right)\hfill \\ & & =\left({\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_1\right]\left(w\right)+e_2{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_2\right]\left(w\right)\right)+\left({\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\left(w\right)\right.\hfill \\ & & +\left.e_2{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\left(w\right)\right)\hfill \\ & & ={\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right)+{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g\right]\left(w\right).\hfill \end{array}$$

Now, to prove $\mathbf{H}-$ linearity, we let $q=q_1+e_2{q}_2\in \mathbf{H}$ and $f=f_1+e_2{f}_2\in L^1(\mathbf{R},\mathbf{H})$ be arbitrary, then we have

$$\begin{array}{ccc}\hfill {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[e_{2}f\right]\left(w\right)& =& {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[e_2(f_1+e_2{f}_2)\right]\left(w\right)\hfill \\ & =& {\mathbf{Q}}_{\mu }^{\mathbf{H}}[e_2{f}_1-f_2]\left(w\right)\hfill \\ & =& e_2{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_1\right]\left(w\right)-{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_2\right]\left(w\right)\hfill \\ & =& e_2\left({\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_1\right]\left(w\right)+e_2{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_2\right]\left(w\right)\right)\hfill \\ & =& e_2{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[qf\right]\left(w\right)& =& {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[q_{1}f\right]\left(w\right)+{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[e_2{q}_{2}f\right]\left(w\right)\hfill \\ & =& q_1{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right)+e_2{q}_2{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right)\hfill \\ & =& (q_1+e_2{q}_2){\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right)\hfill \\ & =& q{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right).\hfill \end{array}$$

Which completes the proof. □

Theorem 6

(Moyal’s formula). Let $f,g\in L^1(\mathbf{R},\mathbf{H})\cap L^2(\mathbf{R},\mathbf{H})$ be two signals functions with ${\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\in L^1(\mathbf{R},\mathbf{H})$, $\langle f,g\rangle =\left|B\right|\langle {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right],{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g\right]\rangle$.

Proof. 

For $f,g\in L^1(\mathbf{R},\mathbf{H})\cap L^1(\mathbf{R},\mathbf{H})$ with ${\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\in L^1(\mathbf{R},\mathbf{H})$,

$$\begin{array}{ccc}& & \langle f,g\rangle \hfill \\ \hfill & & =\int f\left(t\right){\left[g\left(t\right)\right]}^{c}dt\hfill \\ \hfill & & =\frac{\left|B\right|}{\sqrt{2\pi }}\int \int {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right)\overline{{\Lambda }_{\mu }^{e_2}(t,w)}dw{\left[g\left(t\right)\right]}^{c}dt\left(by\left(\right)\right)\hfill \\ \hfill & & =\frac{\left|B\right|}{\sqrt{2\pi }}\int \int {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right)\overline{{\Lambda }_{\mu }^{e_2}(t,w)}{\left[g\left(t\right)\right]}^{c}dtdw\hfill \\ \hfill & & =\frac{\left|B\right|}{\sqrt{2\pi }}\int {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right)\int {\left\{g\left(t\right){\Lambda }_{\mu }^{e_2}(t,w)\right\}}^{c}dtdw\hfill \\ \hfill & & =\frac{\left|B\right|}{\sqrt{2\pi }}\int {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right){\left\{\int g\left(t\right){\Lambda }_{\mu }^{e_2}(t,w)dt\right\}}^{c}dw\hfill \\ \hfill & & =\left|B\right|\int {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right){\left\{\frac{1}{\sqrt{2\pi }}\int g\left(t\right){\Lambda }_{\mu }^{e_2}(t,w)dt\right\}}^{c}dw\hfill \\ \hfill & & =\left|B\right|\int {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right){\left\{{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g\right]\left(w\right)\right\}}^{c}dw\hfill \\ \hfill & & =\left|B\right|\langle {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right],{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g\right]\rangle ,\hfill \end{array}$$

which completes the proof. □

Lemma 1.

For $f\in L^p(\mathbf{R},\mathbf{C}),p=1,2$, we have ${\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[\overline{f}\right]\left(w\right)=\overline{{\mathbf{Q}}_{-\mu }^{\mathbf{H}}\left[f\right]\left(w\right)}.$

Proof. 

It follows from Definition 2 that

$$\begin{array}{cc}\hfill {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[\overline{f}\right]\left(w\right)& =\frac{1}{\sqrt{2\pi }}\int \overline{f\left(t\right)}{\Lambda }_{\mu }^{e_2}(t,w)dt\\ \hfill & =\frac{1}{\sqrt{2\pi }}\int \overline{f\left(t\right)\overline{{\Lambda }_{\mu }^{e_2}(t,w)}}dt\\ \hfill & =\frac{1}{\sqrt{2\pi }}\int \overline{f\left(t\right){\Lambda }_{-\mu }^{e_2}(t,w)}dt\\ \hfill & =\overline{\frac{1}{\sqrt{2\pi }}\int f\left(t\right){\Lambda }_{-\mu }^{e_2}(t,w)dt}\\ \hfill & =\overline{{\mathbf{Q}}_{-\mu }^{\mathbf{H}}\left[f\right]\left(w\right)},\end{array}$$

which completes the proof. □

Remark 1.

The Lemma 1 can also be written as ${\mathbf{Q}}_{-\mu }^{\mathbf{H}}\left[\overline{f}\right]\left(w\right)=\overline{{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right)}.$

Definition 5.

For $f\in L^2(\mathbf{R},\mathbf{H})$ and $g\in L^1(\mathbf{R},\mathbf{H}),$ define

$$\left(f{\otimes }_{\mu }g\right)=(f_1{\ast }_{\mu }{g}_1-{\mathbf{Q}}_{-2\mu }^{\mathbf{H}}\left[{\overline{f}}_2{\ast }_{\mu }{g}_2\right])+e_2(f_2{\ast }_{\mu }{g}_1+{\mathbf{Q}}_{-2\mu }^{\mathbf{H}}\left[{\overline{f}}_1{\ast }_{\mu }{g}_1\right])$$

(13)

where ${\otimes }_{\mu }$ is the proposed definition of convolution.

Remark 2.

${\mathbf{Q}}_{-2\mu }^{\mathbf{H}}={\mathbf{Q}}_{-\mu }^{\mathbf{H}}\circ {\mathbf{Q}}_{-\mu }^{\mathbf{H}}$ is the composition of ${\mathbf{Q}}_{-\mu }^{\mathbf{H}}$ with itself.

Theorem 7

(Convolution theorem). Let $f,g$ be two given signal functions such that $f\in L^2(\mathbf{R},\mathbf{H})$ and $g\in L^1(\mathbf{R},\mathbf{H}),$ then for all $w\in \mathbf{R}$ we have

$${\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f{\otimes }_{\mu }g\right]\left(w\right)={\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right){\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g\right]\left(w\right)exp\{-e_2(C{w}^2+Ew)\}.$$

(14)

Proof. 

By applying Definition 5 and Theorem 3, we have

$$\begin{array}{ccc}& & {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f{\otimes }_{\mu }g\right]\left(w\right)\hfill \\ \hfill & & ={\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[(f_1{\ast }_{\mu }{g}_1-{\mathbf{Q}}_{-2\mu }^{\mathbf{H}}\left[{\overline{f}}_2{\ast }_{\mu }{g}_2\right])\right]\left(w\right)\hfill \\ \hfill & & +e_2{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[(f_2{\ast }_{\mu }{g}_1+{\mathbf{Q}}_{-2\mu }^{\mathbf{H}}\left[{\overline{f}}_1{\ast }_{\mu }{g}_1\right])\right]\left(w\right)\hfill \\ \hfill & & ={\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_1\right]\left(w\right){\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\left(w\right)exp\{-e_2(C{w}^2+Ew)\}\hfill \\ \hfill & & -{\mathbf{Q}}_{\mu }^{\mathbf{H}}{\mathbf{Q}}_{-2\mu }^{\mathbf{H}}\left[{\overline{f}}_2\right]\left(w\right){\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\left(w\right)exp\{-e_2(C{w}^2+Ew)\}\hfill \\ \hfill & & +e_2\left\{{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_2\right]\left(w\right){\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\left(w\right)exp\left\{e_2(C{w}^2+Ew)\right\}\right.\hfill \\ \hfill & & \left.+{\mathbf{Q}}_{\mu }^{\mathbf{H}}{\mathbf{Q}}_{-2\mu }^{\mathbf{H}}\left[{\overline{f}}_1\right]\left(w\right){\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\left(w\right)exp\{-e_2(C{w}^2+Ew)\}\right\}\hfill \\ \hfill & & =\left\{\left[{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_1\right]{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]-{\mathbf{Q}}_{-\mu }^{\mathbf{H}}\left[{\overline{f}}_2\right]{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\right]\left(w\right)\right.\hfill \\ \hfill & & \times \left.e_2\left[{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_2\right]{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]-{\mathbf{Q}}_{-\mu }^{\mathbf{H}}\left[{\overline{f}}_1\right]{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\right]\left(w\right)\right\}\hfill \\ \hfill & & \times exp\{-e_2(C{w}^2+Ew)\}\hfill \\ \hfill & & =\left\{\left[{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_1\right]{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]-\overline{{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_2\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\right]\left(w\right)\right.\hfill \\ \hfill & & \times \left.e_2\left[{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_2\right]{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]+\overline{{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f_1\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\right]\left(w\right)\right\}\hfill \\ \hfill & & \times exp\{-e_2(C{w}^2+Ew)\}\hfill \\ \hfill & & ={\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\left(w\right){\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g\right]\left(w\right)exp\{-e_2(C{w}^2+Ew)\},\hfill \end{array}$$

which completes the proof. □

Remark 3.

By appropriate choice of parameters in $\mu =(A,B,C,D,E),$ Theorem 7 yields the corresponding convolution theorem for all the integral transforms ranging from the 1D quaternion Fourier transform to the much recent 1D quaternion special affine Fourier transforms.

Prior to establishing the correlation theorem for the proposed 1D-QQPFT, we have the following lemma:

Lemma 2

(QPFT Correlation). If $f,g\in L^2\left(\mathbf{R}\right)$ then

$$\begin{array}{ccc}& & {\mathbf{Q}}_{\mu }\left[f{\circ }_{\mu }g\left(t\right)\right]\left(w\right)\hfill \\ \hfill & & =exp\left\{e_2(C{w}^2+Ew)\right\}\overline{{\mathbf{Q}}_{{\mu }^{\prime }}\left[f\right]\left(w\right)}{\mathbf{Q}}_{\mu }\left[g\right]\left(w\right)\hfill \end{array}$$

where ${\mu }^{\prime }=(-A,B,C,D,E)$ and

$$\left(f{\circ }_{\mu }g\right)\left(t\right)=\frac{1}{\sqrt{2\pi }}{\int }_{\mathbf{R}}\overline{f\left(z\right)}g(t+z)exp\left\{2e_{2}Az(t+z)\right\}dz.$$

(15)

Proof. 

From the Definition 1, we have

$$\begin{array}{ccc}& & {\mathbf{Q}}_{\mu }\left[f{\circ }_{\mu }g\left(t\right)\right]\left(w\right)\hfill \\ & & =\frac{1}{\sqrt{2\pi }}{\int }_{\mathbf{R}}\left(f{\circ }_{\mu }g\right)\left(t\right)exp\left\{e_2(A{t}^2+Btw+C{w}^2+Dt+Ew)\right\}dt\hfill \\ & & =\frac{1}{{\left(\sqrt{2\pi }\right)}^2}{\int }_{\mathbf{R}}{\int }_{\mathbf{R}}\overline{f\left(z\right)}g(t+z)exp\left\{2e_{2}Az(t+z)\right\}\hfill \\ & & \times exp\left\{e_2(A{t}^2+Btw+C{w}^2+Dt+Ew)\right\}dzdt\hfill \\ & & =\frac{1}{{\left(\sqrt{2\pi }\right)}^2}{\int }_{\mathbf{R}}{\int }_{\mathbf{R}}\overline{f\left(z\right)}g\left(u\right)exp\left\{2e_{2}Azu\right\}\hfill \\ & & \times exp\left\{e_2[A{(u-z)}^2+B(u-z)w+C{w}^2+D(u-z)+Ew]\right\}dzdu\hfill \\ & & =\frac{1}{\sqrt{2\pi }}{\int }_{\mathbf{R}}g\left(u\right)exp\left\{e_2[A{u}^2+Buw+C{w}^2+Du+Ew]\right\}du\hfill \\ & & \times \frac{1}{\sqrt{2\pi }}{\int }_{\mathbf{R}}\overline{f\left(z\right)}exp\left\{e_2[A{z}^2-Bzw-C{w}^2-Dz-Ew]\right\}dz{e}^{e_2(C{w}^2+Ew)}\hfill \\ & & =exp\left\{i(C{w}^2+Ew)\right\}\frac{1}{\sqrt{2\pi }}\hfill \\ & & \times {\int }_{\mathbf{R}}\overline{f\left(z\right)}exp\{-e_2[(-A)z^2+Bzw+C{w}^2+Dz+Ew]\}dz\hfill \\ & & \times \frac{1}{\sqrt{2\pi }}{\int }_{\mathbf{R}}g\left(u\right)exp\left\{e_2[A{u}^2+Buw+C{w}^2+Du+Ew]\right\}du\hfill \\ & & =exp\left\{e_2(C{w}^2+Ew)\right\}\frac{1}{\sqrt{2\pi }}\hfill \\ & & \times {\int }_{\mathbf{R}}\overline{f\left(z\right)exp\left\{e_2[(-A)z^2+Bzw+C{w}^2+Dz+Ew]\right\}}dz\hfill \\ & & \times \frac{1}{\sqrt{2\pi }}{\int }_{\mathbf{R}}g\left(u\right)exp\left\{e_2[A{u}^2+Buw+C{w}^2+Du+Ew]\right\}du\hfill \\ & & =exp\left\{e_2(C{w}^2+Ew)\right\}\overline{{\mathbf{Q}}_{{\mu }^{\prime }}\left[f\right]\left(w\right)}{\mathbf{Q}}_{\mu }\left[g\right]\left(w\right),\hfill \end{array}$$

which completes the proof. □

Definition 6.

For $f\in L^2(\mathbf{R},\mathbf{H})$ and $g\in L^1(\mathbf{R},\mathbf{H}),$ define

$$\begin{array}{ccc}& & \left(f{\odot }_{\mu }g\right)\hfill \\ & & =(f_1{\circ }_{\mu }{g}_1-{\mathbf{Q}}_{-2{\mu }^{\prime }}^{\mathbf{H}}\left[{\overline{f}}_2{\circ }_{\mu }{g}_2\right])+e_2(f_2{\circ }_{\mu }{g}_1+{\mathbf{Q}}_{-2{\mu }^{\prime }}^{\mathbf{H}}\left[{\overline{f}}_1{\circ }_{\mu }{g}_2\right])\hfill \end{array}$$

where ${\mu }^{\prime }=(-A,B,C,D,E)$ and ${\odot }_{\mu }$ is the proposed definition of correlation.

Theorem 8

(Correlation theorem). Let $f,g$ be two given signal functions such that $f\in L^2(\mathbf{R},\mathbf{H})$ and $g\in L^1(\mathbf{R},\mathbf{H}),$ then for all $w\in \mathbf{R}$ we have

$${\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f{\odot }_{\mu }g\right]\left(w\right)=\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f\right]\left(w\right)}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g\right]\left(w\right)exp\left\{e_2(C{w}^2+Ew)\right\}$$

(16)

Proof. 

By applying Definition 6 and Lemma 2, we have

$$\begin{array}{ccc}& & {\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f{\odot }_{\mu }g\right]\left(w\right)\hfill \\ \hfill & & ={\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[(f_1{\circ }_{\mu }{g}_1-{\mathbf{Q}}_{-2{\mu }^{\prime }}^{\mathbf{H}}\left[{\overline{f}}_2{\circ }_{\mu }{g}_2\right])\right]\left(w\right)\hfill \\ \hfill & & +e_2{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[(f_2{\circ }_{\mu }{g}_1+{\mathbf{Q}}_{-2{\mu }^{\prime }}^{\mathbf{H}}\left[{\overline{f}}_1{\circ }_{\mu }{g}_2\right])\right]\left(w\right)\hfill \\ \hfill & & =\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f_1\right]\left(w\right)}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\left(w\right)exp\left\{e_2(C{w}^2+Ew)\right\}\hfill \\ \hfill & & -\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}{\mathbf{Q}}_{-2{\mu }^{\prime }}^{\mathbf{H}}\left[{\overline{f}}_2\right]\left(w\right)}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\left(w\right)exp\left\{e_2(C{w}^2+Ew)\right\}\hfill \\ \hfill & & +e_2\left\{\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f_2\right]\left(w\right)}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\left(w\right)exp\left\{e_2(C{w}^2+Ew)\right\}\right.\hfill \\ \hfill & & \left.+\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}{\mathbf{Q}}_{-2{\mu }^{\prime }}^{\mathbf{H}}\left[{\overline{f}}_1\right]\left(w\right)}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\left(w\right)exp\left\{e_2(C{w}^2+Ew)\right\}\right\}\hfill \\ \hfill & & =\left\{\left[\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f_1\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]-\overline{{\mathbf{Q}}_{-{\mu }^{\prime }}^{\mathbf{H}}\left[{\overline{f}}_2\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\right]\left(w\right)\right.\hfill \\ \hfill & & \times \left.e_2\left[\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f_2\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]-\overline{{\mathbf{Q}}_{-{\mu }^{\prime }}^{\mathbf{H}}\left[{\overline{f}}_1\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]\right]\left(w\right)\right\}\hfill \\ \hfill & & \times exp\left\{e_2(C{w}^2+Ew)\right\}\hfill \\ \hfill & & =\left\{\left[\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f_1\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]-\overline{\left[\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f_2\right]}\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\right]\left(w\right)\right.\hfill \\ \hfill & & \times \left.e_2\left[\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f_2\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_1\right]+\overline{\left[\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f_1\right]}\right]}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g_2\right]\right]\left(w\right)\right\}\hfill \\ \hfill & & \times exp\left\{e_2(C{w}^2+Ew)\right\}\hfill \\ \hfill & & =\overline{{\mathbf{Q}}_{{\mu }^{\prime }}^{\mathbf{H}}\left[f\right]\left(w\right)}{\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[g\right]\left(w\right)exp\left\{e_2(C{w}^2+Ew)\right\},\hfill \end{array}$$

which completes the proof. □

Theorem 9

(1D-QQPFT Uncertainty inequality). Let ${\mathbf{Q}}_{\mu }^{\mathbf{H}}\left[f\right]\in L^2(\mathbf{R},\mathbf{H})$ and $f,tf\left(t\right)\in L^2(\mathbf{R},\mathbf{H}).$ Then, we have

$$\begin{array}{c}\hfill {\int }_{\mathbf{R}}{w}^2|{\mathbf{Q}}_{\mu }{\left[f\right]|}^{2}dw{\int }_{\mathbf{R}}{t}^2{\left|f\left(t\right)\right|}^{2}dt\ge \frac{1}{{8\left|B\right|}^2}{\left({\int }_{\mathbf{R}}{\left|f\left(t\right)\right|}^{2}dt\right)}^2.\end{array}$$

Proof. 

For $f=f_1+e_2{f}_2$ in $L^2(\mathbf{R},\mathbf{H}),$ we have ${\parallel f\parallel }_2^2=\parallel f_1{\parallel }_2^2+{\parallel f_2\parallel }_2^2.$ Thus

$$\begin{array}{ccc}& & {\displaystyle {\int }_{\mathbf{R}}{w}^2|{\mathbf{Q}}_{\mu }{\left[f\right]|}^{2}dw{\int }_{\mathbf{R}}{t}^2{\left|f\left(t\right)\right|}^{2}dt}\hfill \\ \hfill & & {\displaystyle ={\int }_{\mathbf{R}}\left(w^2|{\mathbf{Q}}_{\mu }\left[f_1\right]{|}^2+w^2{|{\mathbf{Q}}_{\mu }\left[f_2\right]|}^2\right)dw}\hfill \\ \hfill & & {\displaystyle \times {\int }_{\mathbf{R}}\left(t^2|f_1{|}^2+t^2{\left|f_2\right|}^2\right)dt}\hfill \\ \hfill & & {\displaystyle \ge {\int }_{\mathbf{R}}{w}^2|{\mathbf{Q}}_{\mu }\left[f_1\right]{|}^{2}dw+{\int }_{\mathbf{R}}{t}^2{\left|f_1\right|}^2}\hfill \\ \hfill & & {\displaystyle +{\int }_{\mathbf{R}}{w}^2|{\mathbf{Q}}_{\mu }\left[f_2\right]{|}^{2}dw+{\int }_{\mathbf{R}}{t}^2{\left|f_2\right|}^2.}\hfill \end{array}$$

Now, using Theorem 4, we obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_{\mathbf{R}}{w}^2|{\mathbf{Q}}_{\mu }{\left[f\right]|}^{2}dw{\int }_{\mathbf{R}}{t}^2{\left|f\left(t\right)\right|}^{2}dt}\hfill \\ \hfill & & \ge {\left\{{\displaystyle \frac{1}{2\left|B\right|}{\int }_{\mathbf{R}}{\left|f_1\left(t\right)\right|}^{2}dt}\right\}}^2+{\left\{{\displaystyle \frac{1}{2\left|B\right|}{\int }_{\mathbf{R}}{\left|f_2\left(t\right)\right|}^{2}dt}\right\}}^2\hfill \\ \hfill & & =\frac{1}{{4\left|B\right|}^2}\left\{{\left({\displaystyle {\int }_{\mathbf{R}}{\left|f_1\left(t\right)\right|}^{2}dt}\right)}^2+{\left({\displaystyle {\int }_{\mathbf{R}}{\left|f_2\left(t\right)\right|}^{2}dt}\right)}^2\right\}\hfill \\ \hfill & & \ge \frac{1}{{8\left|B\right|}^2}{\left({\displaystyle {\int }_{\mathbf{R}}|f_1{\left(t\right)|}^{2}dt+{\int }_{\mathbf{R}}{\left|f_2\left(t\right)\right|}^{2}dt}\right)}^2\hfill \\ \hfill & & =\frac{1}{{8\left|B\right|}^2}{\left({\displaystyle {\int }_{\mathbf{R}}{\left|f\left(t\right)\right|}^{2}dt}\right)}^2\hfill \end{array}$$

which completes the proof. □

4. Application to the 2-Dimensional Quaternion Quadratic-Phase Fourier Transform

In Section 3, we used 1D-QQPFT to define various topics but the main motive to define the proposed transform is to obtain an inversion formula for the two-dimensional quaternion quadratic-phase Fourier transform (2D-QQPFT) which will lay the foundation for applications of two-dimensional quaternion quadratic-phase Fourier transform in signal processing. Prior to that we shall define two-dimensional quaternion quadratic-phase Fourier transform(2D-QQPFT). Let us begin:

Definition 7.

The 2D-QQPFT of a quaternion-valued signal function $f\in L^1({\mathbf{R}}^2,\mathbf{H}),$ with respect to parameter ${\mu }_s=(A_s,B_s,C_s,D_s,E_s)$ for $s=1,2$ is defined as

$$\begin{array}{c}\hfill {\mathbf{Q}}_{{\mu }_1,{\mu }_2}^{\mathbf{H}}\left[f\right]\left(\mathbf{w}\right)=\frac{1}{2\pi }{\int }_{{\mathbf{R}}^2}f\left(\mathbf{t}\right){\Lambda }_{{\mu }_1}^{{\mathbf{e}}_1}(t_1,w_1){\Lambda }_{{\mu }_2}^{{\mathbf{e}}_2}(t_2,w_2)d\mathbf{t}\end{array}$$

where $\mathbf{w}=(w_1,w_2)\in {\mathbf{R}}^2$, $\mathbf{t}=(t_1,t_2)\in {\mathbf{R}}^2$ and ${\Lambda }_{{\mu }_1}^{{\mathbf{e}}_1}(t_1,w_1)$ and ${\Lambda }_{{\mu }_2}^{{\mathbf{e}}_2}(t_2,w_2)$ are kernel signals given by

$$\begin{array}{ccc}\hfill & & {\Lambda }_{{\mu }_1}^{{\mathbf{e}}_1}(t_1,w_1)\hfill \\ & & =exp\left\{e_1(A_1{t}_1^2+B_1{t}_1{w}_1+C_1{w}_1^2+D_1{t}_1+E_1{w}_1)\right\}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & {\Lambda }_{{\mu }_2}^{{\mathbf{e}}_2}(t_2,w_2)\hfill \\ & & =exp\left\{e_2(A_2{t}_2^2+B_2{t}_2{w}_2+C_2{w}_2^2+D_2{t}_2+E_2{w}_2)\right\}\hfill \end{array}$$

where $A_s,B_s,C_s,D_s,E_s\in \mathbf{R},B_s\ne 0$ and $s=1,2.$

Theorem 10.

Let $f\in L^2({\mathbf{R}}^2,\mathbf{H})$ and ${\mathbf{Q}}_{{\mu }_s}^{\mathbf{H}}\left[f\right]\in L^1({\mathbf{R}}^2,\mathbf{H}).$ Then, 2-D QQPFT is invertible with inverse

$$\begin{array}{ccc}\hfill & & f\left(\mathbf{t}\right)\hfill \\ & & ={\left\{{\mathbf{Q}}_{{\mu }_1,{\mu }_2}^{\mathbf{H}}\right\}}^{-1}\left[{\mathbf{Q}}_{{\mu }_1,{\mu }_2}^{\mathbf{H}}\left\{f\right\}\right]\left(\mathbf{t}\right)\hfill \\ \hfill & & =\frac{|B_1{B}_2|}{2\pi }{\int }_{{\mathbf{R}}^2}{\mathbf{Q}}_{{\mu }_1,{\mu }_2}^{\mathbf{H}}\left[f\right]\}\left(\mathbf{w}\right){\Lambda }_{-{\mu }_2}^{{\mathbf{e}}_2}(t_1,w_1){\Lambda }_{-{\mu }_1}^{{\mathbf{e}}_1}(t_2,w_2)d\mathbf{w}.\hfill \end{array}$$

Proof. 

From the Definition 1D-QQPFT and the 2D-QQPFT, we have

$$\begin{array}{ccc}\hfill {\mathbf{Q}}_{{\mu }_1,{\mu }_2}^{\mathbf{H}}\left\{f\right\}\left(\mathbf{w}\right)& =& \int {\mathbf{Q}}_{{\mu }_1}^{\mathbf{H}}\left\{f\right\}\left(w_1\right){\Lambda }_{{\mu }_2}^{e_2}(t_2,w_2)d{t}_2\hfill \\ \hfill & =& {\mathbf{Q}}_{{\mu }_2}^{\mathbf{H}}\left[{\mathbf{Q}}_{{\mu }_1}^{\mathbf{H}}\left\{f\right\}\right]\left(\mathbf{w}\right).\hfill \end{array}$$

Now it is given that $f\in L^2({\mathbf{R}}^2,\mathbf{H})$ and ${\mathbf{Q}}_{{\mu }_s}^{\mathbf{H}}\left[f\right]\in L^1({\mathbf{R}}^2,\mathbf{H});$ therefore, we have

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbf{R}}^2}{\mathbf{Q}}_{{\mu }_1,{\mu }_2}^{\mathbf{H}}\left\{f\right\}\left(\mathbf{w}\right){\Lambda }_{-{\mu }_2}^{e_2}(t_2,w_2){\Lambda }_{-{\mu }_1}^{e_1}(t_1,w_1)d\mathbf{w}}\hfill \\ \hfill & & {\displaystyle =\int \int {\mathbf{Q}}_{{\mu }_2}^{\mathbf{H}}\left[{\mathbf{Q}}_{{\mu }_1}^{\mathbf{H}}\left\{f\right\}\right]\left(\mathbf{w}\right){\Lambda }_{-{\mu }_2}^{e_2}(t_2,w_2)d{w}_2}\hfill \\ \hfill & & \times {\Lambda }_{-{\mu }_1}^{e_1}(t_1,w_1)d{w}_1.\hfill \end{array}$$

Using (11), we obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_{{\mathbf{R}}^2}{\mathbf{Q}}_{{\mu }_1,{\mu }_2}^{\mathbf{H}}\left\{f\right\}\left(\mathbf{w}\right){\Lambda }_{-{\mu }_2}^{e_2}(t_2,w_2){\Lambda }_{-{\mu }_1}^{e_1}(t_1,w_1)d\mathbf{w}}\hfill \\ \hfill & & {\displaystyle =\frac{\sqrt{2\pi }}{|B_2|}\int {\mathbf{Q}}_{{\mu }_1}^{\mathbf{H}}\left\{f\right\}\left(w_1\right){\Lambda }_{-{\mu }_1}^{e_1}(t_1,w_1)d{w}_1}\hfill \\ \hfill & & =\frac{2\pi }{|B_1{B}_2|}f\left(\mathbf{t}\right),\hfill \end{array}$$

which completes the proof. □

5. Potential Applications

The QQPFT which is a generalized version of the QFTs has gained its ground intermittently and profoundly influenced several disciplines of science and engineering, including harmonic analysis, quantum mechanics, differential equations, optics, pattern recognition, and so on [29,30]. The 1D-QQPFT can be widely applied in computer graphics, computer vision, robotics and even in astrophysics [39,40]. The 1D-QQPFT can be used in the recovery of bandlimited quaternion-valued signals in 1D-QQPFT domain from noisy samples; it is based on the oversampling theorem and without adding too much complexity, a reconstruction algorithm for bandlimited signals in 1D-QQPFT domain from noisy observations is obtained [41]. The potential application can be found in the detection of linear frequency-modulated(LFM) and non-transient quaternion-valued signals [31]. Moreover, the 1D-QQPFT can have a potential application in image preprocessing, color analysis and neural computing techniques for speech recognition [42]. On the other hand, the convolution and correlation type operators are very important mathematical objects which are used in the modelling of a great diversity of applied problems in signals, images and optics, especially in the design and implementation of multiplicative filters in the 1D-QQPFT [43]. The potential application can be found in in the theory of linear time-invariant (LTI) system [44].

The uncertainty inequalities have potential applications in signal processing as they can estimate the lower bound of integral [19,29,31]. The estimation of band-widths is another possible application of the proposed transform [45]. Theorem 9 can be used in the estimation of effective band-width in the 1D-QQPFT domain which states that the band-width of a system that performs 1D-QQPFT cannot be narrower than $\frac{1}{T_{\mu }^{2}8B^2}$, where $T_{\mu }$ is the spread of the signal in the time domain.

6. Conclusions

In this paper, we have proposed the definition of the novel integral transform known as the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT) which is the embodiment of several well-known signal processing tools. We then obtained Moyal’s formula and convolution theorem for the proposed transform. We then used theory 1D-QQPFT to obtain an inversion formula for two-dimensional quaternion quadratic-phase Fourier transform. Our future work on the two-sided quaternion quadratic-phase Fourier transform and two-sided Gabor quaternion quadratic-phase Fourier transform is in progress.