# A Convolution Theorem Related to Quaternion Linear Canonical Transform

Authors: Mawardi Bahri; Ryuichi Ashino
Journal: Abstract and Applied Analysis
 (2019)

---

## Abstract

We introduce the two-dimensional quaternion linear canonical transform (QLCT), which is a generalization of the classical linear canonical transform (LCT) in quaternion algebra setting. Based on the definition of quaternion convolution in the QLCT domain we derive the convolution theorem associated with the QLCT and obtain a few consequences.


---

## Body

## 1. Introduction

The linear canonical transform (LCT) plays an important role in various fields of optics and signal processing. In some papers, the LCT is also known as the affine Fourier, the ABCD, and Moshinsky-queue transforms. The LCT can be considered as a generalization of many mathematical transforms, such as Fourier, Laplace, fractional Fourier, and Fresnel transforms. Many fundamental properties of the LCT have been investigated like translation, modulation, convolution, correlation, and uncertainty principles (see, e.g., [1–8]).The quaternion linear canonical transform (QLCT) is a generalization of the linear canonical transform (LCT) using quaternion algebra. According to definitions of the quaternion Fourier transform (QFT), there are basically two ways of obtaining the QLCT: the (right-sided) quaternion linear canonical transform and the (two-sided) quaternion linear canonical transform. The (right-sided) quaternion linear canonical transform is obtained by substituting the Fourier kernel with the right-sided QFT kernel in the LCT definition. Some important properties of the quaternion linear canonical transform such as the Parseval’s theorem, reconstruction formula, and uncertainty principles are also discussed (see [9–14] and the references mentioned therein). However, there is no literature for establishing the convolution theorem associated with the QLCT as far as we know.Therefore, it is worthwhile to study the convolution theorems associated with the QLCT, which can be useful in signal processing theory and application. Our main objective of the present paper is to establish convolution theorems for the QLCT which are generalizations of the related classical ones. We will accomplish this task by using the properties of quaternions and combining the LCT convolution and the QFT convolution definition [15]. In the beginning, we make a definition of the QLCT and obtain the relationship between the QLCT and QFT. Based on the convolution definitions of the LCT and QFT, we propose a new definition convolution for the QLCT and obtain its convolution theorem. We emphasize that the proposed convolution definition is different from the one studied in [16]. The definition uses the kernel of the classical fractional Fourier transform which is commutative with quaternion signals.


## 2. Basic Facts about Quaternion Algebra and Quaternion Fourier Transform

Quaternions are hypercomplex numbers, which can be written in the following form(1)H=q=q0+iq1+jq2+kq3;q0,q1,q2,q3∈R,where {1,i,j,k} is a basis of H and obeys the following multiplication rules: (2)ij=-ji=k,jk=-kj=i,ki=-ik=j,i2=j2=k2=ijk=-1.For a quaternion q=q0+iq1+jq2+kq3∈H, the conjugate q¯ of the quaternion q is given by(3)q¯=q0-iq1-jq2-kq3.and satisfies (4)qp¯=p¯q¯.From (3) we obtain the norm or modulus of q∈H defined as(5)q=qq¯=q02+q12+q22+q32.It is not difficult to see that (6)qp=qp,∀p,q∈H.Like in complex case, the inverse of q∈H∖{0} is given by (7)q-1=q¯q2.Every quaternion-valued functionf:R2→H can be written as(8)fx=f0x+if1x+jf2x+kf3x,x∈R2.where f0,f1,f2, and f3 are real-valued functions. A quaternion module Lp(R2;H) is then defined as(9)LpR2;H=f∣f:R2→H,∫R2fpdx<∞,1≤p<∞.Definition 1.
The QFT off∈L1(R2;H) is the transform Fq{f}∈L1(R2;H) given by the integral(10)Fqfω=∫R2e-iω1x1fxe-jω2x2dx,dx=dx1dx2.Here Fq is called the quaternion Fourier transform operator or the quaternion Fourier transform.Definition 2.
The inverse QFT ofg∈L1(R2;H) is the transform Fq-1{g}∈L1(R2;H) given by the integral (11)Fq-1gx=12π2∫R2eiω1x1gωejω2x2dω.where ω∈R2 and dω=dω1dω2. Here Fq-1 stands for the inverse QFT operator.


## 3. Quaternion Linear Canonical Transform and Its Convolution Theorem

In this section we first introduce the two-dimensional quaternion linear canonical transform (QLCT). We then make a convolution definition in the QLCT domain and derive a convolution theorem related to the QLCT.


### 3.1. Definition of QLCT

Based on the definition of the two-sided quaternion Fourier transform (QFT) and its properties [17–22], we obtain a definition of the QLCT. We also can derive useful properties of the QLCT using fundamental relationship between the QFT and QLCT. Denote by SL(2,R) the special linear group of degree 2 over R, that is, the group of all real 2×2 matrices with determinant one. Let(12)As=as,bs,cs,ds=asbscsds∈SL2,R,s=1,2.When b1b2≠0, we define the kernel KAs of the QLCT by (13)KA1x1,ω1=12πb1iei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12KA2x2,ω2=12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22.Observe that we can write the imaginary units above of the form(14)i=eiπ/4,j=ejπ/4.These facts yield(15)12πb1i=e-iπ/42πb1,12πb2j=e-jπ/42πb2.Definition 3 (QLCT definition).
The QLCT of a quaternion signalf∈L1(R2;H) is defined by(16)FA1,A2Hfω=∫R2KA1x1,ω1fxKA2x2,ω2dx,b1b2≠0d1d2eic1d1/2ω12fd1ω1,d2ω2ejc2d2/2ω22,b1b2=0.Becauseeic1d1/2ω12 and ejc2d2/2ω22 arechirp signals in signal processing, then we always work for the case b1b2≠0. The inverse transform of the QLCT above is then described by(17)fx=∫R2KA1-1ω1,x1FA1,A2HfωKA2-1ω2,x2dω.This form is equivalent to(18)fx=1-2πb1i∫R2e-i1/2a1/b1x12-2/b1x1ω1+d1/b1ω12FA1,A2Hfω1-2πb2j×e-j1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dω,where A1-1=(d1,-b1,-c1,a1) and A2-1=(d2,-b2,-c2,a2).It directly follows from (8) and (15) that(19)FA1,A2Hfω=12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12f0x+if1+jf2x+kf3x×12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx=12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12f0x12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx+12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12if1x12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx=12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12f2xj12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx=12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12if3xj12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx=FA1,A2Hf0ω+iFA1,A2Hf1ω+FA1,A2Hf2ωj+iFA1,A2Hf3ωj.In the rest of the paper, we always assume that FA1,A2H{fi} for i=0,1,2,3 are real-valued function or FA1,A2H{fi}∈L1(R2;R).Theorem 4.
Iff(x)∈L1(R2;H), then FA1,A2H{f}(ω) is continuous on R2.Proof.
See [9].Similarly, one can obtain the following result.Theorem 5.
IfFA1,A2H{f}(ω)∈L1(R2;H), then f(x) is continuous on R2.


### 3.2. Convolution Theorem for QLCT

In the following we first define the convolution for the QLCT. It is an extension of the convolution definition from the LCT (see [5, 6]) to the QLCT domain. We then investigate how the QLCT behaves under convolutions.Definition 6.
For any two quaternion functionsf,g∈L1(R2;H), we define the convolution operator of the QLCT as(20)f⋆gx=∫R2eia1/b1t1t1-x1ftgx-teja2/b2t2t2-x2dt.As a direct consequence, we get the convolution theorem associated with the QLCT, which is expressed asTheorem 7.
Letf,g∈L1(R2;H) be two quaternion-valued functions. Then we have the QLCT of the convolution of f and g in the form (21)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2Hf0ω+iFA1,A2Hf1ωFA1,A2Hg0ω+jFA1∗,A2Hg2ω+FA1,A2∗Hf0ω+iFA1,A2∗Hf1ωiFA1,A2Hg1ω+kFA1∗,A2Hg3ω+jFA1∗,A2Hf2ω+kFA1∗,A2Hf3ωFA1∗,A2Hg0ω+jFA1,A2Hg2ω+jFA1∗,A2∗Hf2ω+kFA1∗,A2∗Hf3ωiFA1∗,A2Hg1ω+kFA1,A2Hg3ωe-jd2ω22/2b22πb2j,where the matrix parameters(22)A1∗=a1-b1c1d1andA2∗=a2-b2c2d2.Proof.
It directly follows from (16) and (20) that (23)FA1,A2Hf⋆gω=∫R2∫R212πb1iei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12eia1/b1t1t1-x1ftgx-t×12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22eja2/b2t2t2-x2dtdx=∫R2∫R212πb1ieia1/b1t1t1-x1eia1/2b1x12e-i2/2b1x1ω1eid1/2b1ω12ftgx-t×12πb2jeja2/b2t2t2-x2eja2/2b2x22e-j2/2b2x2ω2ejd2/2b2ω22dtdx.Letting x-t=z, (23) can be rewritten as (24)FA1,A2Hf⋆gω=∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×f0t+if1t+jf2t+kf3tg0z+jg2z+ig1z+kg3z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt=∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×f0t+if1tg0z+jg2z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt+∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×(f0t+if1tig1z+kg3z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt+∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×jf2t+kf3tg0z+jg2z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt+∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×jf2t+kf3tig1z+kg3z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt.Simplifying this result we have(25)FA1,A2Hf⋆gω=∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1z12/2b1e-iz1ω1/b1f0t+if1tg0z+jg2z×∫R212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2eja2z22/2b2e-jz2ω2/b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1z12/2b1e-iz1ω1/b1(f0t+if1tig1z+kg3z×∫R212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2eja2z22/2b2e-jz2ω2/b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1z12/2b1e-iz1ω1/b1jf2t+kf3tg0z+jg2z×∫R212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2eja2z22/2b2e-jz2ω2/b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1z12/2b1e-iz1ω1/b1jf2t+kf3tig1z+kg3z×∫R212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2eja2z22/2b2e-jz2ω2/b2dzdt. Applying properties of kernel function of the QLCT we obtain (26)FA1,A2Hf⋆gω=∫R2eia1t12/2b1e-it1ω1/b1f0t+if1t12πb1ieia1z12/2b1e-iz1ω1/b1eid1ω12/2b1g0z+jg2z×∫R212πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b2eja2t22/2b2e-jt2ω1/b2dzdt+∫R2eia1t12/2b1e-it1ω1/b1f0t+if1t12πb1ieia1z12/2b1e-iz1ω1/b1eid1ω12/2b1ig1z+kg3z×∫R212πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b2eja2t22/2b2e-jt2ω1/b2dzdt+∫R2eia1t12/2b1e-it1ω1/b1jf2t+kf3t12πb1ie-ia1z12/2b1eiz1ω1/b1e-id1ω12/2b1g0z+jg2z×∫R212πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b2eja2t22/2b2e-jt2ω1/b2dzdt+∫R2eia1t12/2b1e-it1ω1/b1jf2t+kf3t12πb1ie-ia1z12/2b1eiz1ω1/b1e-id1ω12/2b1ig1z+kg3z×∫R212πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b2eja2t22/2b2e-jt2ω1/b2dzdt.Now multiplying both sides of the above equation by 1/2πb1ieid1ω12/2b1 and 1/2πb2jejd2ω22/2b2 gives (27)12πb1ieid1ω12/2b1FA1,A2Hf⋆gωejd2ω22/2b212πb2j=∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1t12πb1ieia1z12/2b1e-iz1ω1/b1eid1ω12/2b1×∫R2g0z+jg2z12πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1t12πb1ieia1z12/2b1e-iz1ω1/b1eid1ω12/2b1×∫R212πb2jig1z+kg3zeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3t12πb1ie-ia1z12/2b1eiz1ω1/b1e-id1ω12/2b1×∫R2g0z+jg2z12πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3t12πb1ie-ia1z12/2b1eiz1ω1/b1e-id1ω12/2b1×∫R2ig1z+kg3z12πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt=∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1t×FA1,A2Hg0ω+jFA1∗,A2Hg2ω12πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1t×iFA1,A2Hg1ω+kFA1∗,A2Hg3ω12πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3t×∫R2FA1∗,A2Hg0ω+jFA1,A2Hg2ω12πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3t×iFA1∗,A2Hg1ω+kFA1,A2Hg3ω12πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt.Based on Definition 3, the required result follows.Remark 8.
If we use the matrix parametersA1=A2=a1-10 with a≠0. Then (21) will lead to(28)FA1,A2Hf⋆gω=2πiFA1,A2Hf0ω+iFA1,A2Hf1ωFA1,A2Hg0ω+jFA1∗,A2Hg2ω+FA1,A2∗Hf0ω+iFA1,A2∗Hf1ωiFA1,A2Hg1ω+kFA1∗,A2Hg3ω+jFA1∗,A2Hf2ω+kFA1∗,A2Hf3ωFA1∗,A2Hg0ω+jFA1,A2Hg2ω+jFA1∗,A2∗Hf2ω+kFA1∗,A2∗Hf3ωiFA1∗,A2Hg1ω+kFA1,A2Hg3ω2πj.Observe that the above form is quite similar to the convolution theorem associated with the QFT [15]. Now we investigate some consequences of Theorem 7, which are given in the following results.Lemma 9.
Letf,g∈L1(R2;H) be two quaternion-valued functions. If we assume that FA1,A2H{g}∈L1(R2;R), then we have(29)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2Hf0ω+iFA1,A2Hf1ωFA1,A2Hgω+jFA1∗,A2Hf2ω+kFA1∗,A2Hf3ωFA1∗,A2Hgω2πb2je-jd2ω22/2b2.Moreover, if FA1,A2H{f}∈L1(R2;R), then we have(30)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2HfωFA1,A2Hg0ω+jFA1∗,A2Hg2ω+FA1∗,A2HfωiFA1,A2Hg1ω+kFA1∗,A2Hg3ω2πb2je-jd2ω22/2b2.Proof.
We only verify the identity (29) and the others are quite similar. Direct computations yield (31)FA1,A2Hf⋆gω=∫R2∫R212πb1iei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12eia1/b1t1t1-x1ftgx-t×12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22eja2/b2t2t2-x2dtdx=∫R2∫R212πb1ieia1/b1t1t1-x1eia1/2b1x12e-i2/2b1x1ω1eid1/2b1ω12ftgx-t×12πb2jeja2/b2t2t2-x2eja2/2b2x22e-j2/2b2x2ω2ejd2/2b2ω22dtdx.By making the change of variable x-t=v of the above, we get (32)FA1,A2Hf⋆gω=∫R2∫R212πb1ie-ia1/b1t1v1ei1/2a1/b1v1+t12e-iv1+t1ω1/b1eid1ω12/2b1ftgv×12πb2je-ja2/b2t2v2ej1/2a2/b2v2+t22e-jv2+t2ω2/b2ejd2ω22/2b2dtdv=∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1×f0t+if1t+jf2t+kf3tgv×12πbjjeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b2dtdv=∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1f0t+if1tgv×12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b2dtdv+∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1jf2t+kf3tgv×12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b2dtdv.Multiplying this result by 1/2πb1ieid1ω12/2b1 and 1/2πb2jejd2ω22/2b2, we easily obtain (33)12πb1ieid1ω12/2b1FA1,A2Hf⋆gω12πb2jejd2ω22/2b2=∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1eid1ω12/2b1f0t+if1tgv×eja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b212πb2jejd2ω22/2b2dtdv+∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1eid1ω12/2b1×jf2t+kf3tgv12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b2ejd2ω22/2b2dtdv=∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1tFA1,A2Hgω×12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2dt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3tFA1∗,A2Hgω×12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2dt.The definition of the QLCT (16) finally gives(34)12πb1ieid1ω12/2b1FA1,A2Hf⋆gω12πb1jejd2ω22/2b2=FA1,A2Hf0ω+iFA1,A2Hf1ωFA1,A2Hgω+jFA1∗,A2Hf2ω+kFA1∗,A2Hf3ωFA1∗,A2Hgω.This finishes the proof of the lemma.Lemma 10.
Suppose thatf,g∈L1(R2;H) has the form(35)fx=f0x+if1x+jf2x+kf3x,gx=g0x+jg2x.If FA1∗,A2H{f}∈L1(R2;R) is satisfied, then we have(36)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2HfωFA1,A2Hgω2πb2je-jd2ω22/2b2=2πb1ie-id1ω12/2b1FA1,A2HgωFA1,A2Hfω2πb2je-jd2ω22/2b2.Remark 11.
It should be noticed iff,g∈L1(R2;H) takes the form(37)fx=jf2x+kf3x,gx=g0x+ig1x+jg2x+kg3x,then it holds (38)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2HfωFA1∗,A2Hgω2πb2je-jd2ω22/2b2=2πb1ie-id1ω12/2b1FA1∗,A2HgωFA1,A2Hfω2πb2je-jd2ω22/2b2,where FA1,A2H{f},FA1,A2H{g}∈L1(R2;R).


## 3.1. Definition of QLCT

Based on the definition of the two-sided quaternion Fourier transform (QFT) and its properties [17–22], we obtain a definition of the QLCT. We also can derive useful properties of the QLCT using fundamental relationship between the QFT and QLCT. Denote by SL(2,R) the special linear group of degree 2 over R, that is, the group of all real 2×2 matrices with determinant one. Let(12)As=as,bs,cs,ds=asbscsds∈SL2,R,s=1,2.When b1b2≠0, we define the kernel KAs of the QLCT by (13)KA1x1,ω1=12πb1iei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12KA2x2,ω2=12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22.Observe that we can write the imaginary units above of the form(14)i=eiπ/4,j=ejπ/4.These facts yield(15)12πb1i=e-iπ/42πb1,12πb2j=e-jπ/42πb2.Definition 3 (QLCT definition).
The QLCT of a quaternion signalf∈L1(R2;H) is defined by(16)FA1,A2Hfω=∫R2KA1x1,ω1fxKA2x2,ω2dx,b1b2≠0d1d2eic1d1/2ω12fd1ω1,d2ω2ejc2d2/2ω22,b1b2=0.Becauseeic1d1/2ω12 and ejc2d2/2ω22 arechirp signals in signal processing, then we always work for the case b1b2≠0. The inverse transform of the QLCT above is then described by(17)fx=∫R2KA1-1ω1,x1FA1,A2HfωKA2-1ω2,x2dω.This form is equivalent to(18)fx=1-2πb1i∫R2e-i1/2a1/b1x12-2/b1x1ω1+d1/b1ω12FA1,A2Hfω1-2πb2j×e-j1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dω,where A1-1=(d1,-b1,-c1,a1) and A2-1=(d2,-b2,-c2,a2).It directly follows from (8) and (15) that(19)FA1,A2Hfω=12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12f0x+if1+jf2x+kf3x×12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx=12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12f0x12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx+12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12if1x12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx=12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12f2xj12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx=12πb1i∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12if3xj12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22dx=FA1,A2Hf0ω+iFA1,A2Hf1ω+FA1,A2Hf2ωj+iFA1,A2Hf3ωj.In the rest of the paper, we always assume that FA1,A2H{fi} for i=0,1,2,3 are real-valued function or FA1,A2H{fi}∈L1(R2;R).Theorem 4.
Iff(x)∈L1(R2;H), then FA1,A2H{f}(ω) is continuous on R2.Proof.
See [9].Similarly, one can obtain the following result.Theorem 5.
IfFA1,A2H{f}(ω)∈L1(R2;H), then f(x) is continuous on R2.


## 3.2. Convolution Theorem for QLCT

In the following we first define the convolution for the QLCT. It is an extension of the convolution definition from the LCT (see [5, 6]) to the QLCT domain. We then investigate how the QLCT behaves under convolutions.Definition 6.
For any two quaternion functionsf,g∈L1(R2;H), we define the convolution operator of the QLCT as(20)f⋆gx=∫R2eia1/b1t1t1-x1ftgx-teja2/b2t2t2-x2dt.As a direct consequence, we get the convolution theorem associated with the QLCT, which is expressed asTheorem 7.
Letf,g∈L1(R2;H) be two quaternion-valued functions. Then we have the QLCT of the convolution of f and g in the form (21)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2Hf0ω+iFA1,A2Hf1ωFA1,A2Hg0ω+jFA1∗,A2Hg2ω+FA1,A2∗Hf0ω+iFA1,A2∗Hf1ωiFA1,A2Hg1ω+kFA1∗,A2Hg3ω+jFA1∗,A2Hf2ω+kFA1∗,A2Hf3ωFA1∗,A2Hg0ω+jFA1,A2Hg2ω+jFA1∗,A2∗Hf2ω+kFA1∗,A2∗Hf3ωiFA1∗,A2Hg1ω+kFA1,A2Hg3ωe-jd2ω22/2b22πb2j,where the matrix parameters(22)A1∗=a1-b1c1d1andA2∗=a2-b2c2d2.Proof.
It directly follows from (16) and (20) that (23)FA1,A2Hf⋆gω=∫R2∫R212πb1iei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12eia1/b1t1t1-x1ftgx-t×12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22eja2/b2t2t2-x2dtdx=∫R2∫R212πb1ieia1/b1t1t1-x1eia1/2b1x12e-i2/2b1x1ω1eid1/2b1ω12ftgx-t×12πb2jeja2/b2t2t2-x2eja2/2b2x22e-j2/2b2x2ω2ejd2/2b2ω22dtdx.Letting x-t=z, (23) can be rewritten as (24)FA1,A2Hf⋆gω=∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×f0t+if1t+jf2t+kf3tg0z+jg2z+ig1z+kg3z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt=∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×f0t+if1tg0z+jg2z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt+∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×(f0t+if1tig1z+kg3z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt+∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×jf2t+kf3tg0z+jg2z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt+∫R212πb1ie-ia1/b1t1z1ei1/2a1/b1z1+t12-2/b1z1+t1ω1+d1/b1ω12×jf2t+kf3tig1z+kg3z×∫R212πb2je-ja2/b2t2z2ej1/2a2/b2z2+t22-2/b2z2+t2ω2+d2/b2ω22dzdt.Simplifying this result we have(25)FA1,A2Hf⋆gω=∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1z12/2b1e-iz1ω1/b1f0t+if1tg0z+jg2z×∫R212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2eja2z22/2b2e-jz2ω2/b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1z12/2b1e-iz1ω1/b1(f0t+if1tig1z+kg3z×∫R212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2eja2z22/2b2e-jz2ω2/b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1z12/2b1e-iz1ω1/b1jf2t+kf3tg0z+jg2z×∫R212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2eja2z22/2b2e-jz2ω2/b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1z12/2b1e-iz1ω1/b1jf2t+kf3tig1z+kg3z×∫R212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2eja2z22/2b2e-jz2ω2/b2dzdt. Applying properties of kernel function of the QLCT we obtain (26)FA1,A2Hf⋆gω=∫R2eia1t12/2b1e-it1ω1/b1f0t+if1t12πb1ieia1z12/2b1e-iz1ω1/b1eid1ω12/2b1g0z+jg2z×∫R212πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b2eja2t22/2b2e-jt2ω1/b2dzdt+∫R2eia1t12/2b1e-it1ω1/b1f0t+if1t12πb1ieia1z12/2b1e-iz1ω1/b1eid1ω12/2b1ig1z+kg3z×∫R212πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b2eja2t22/2b2e-jt2ω1/b2dzdt+∫R2eia1t12/2b1e-it1ω1/b1jf2t+kf3t12πb1ie-ia1z12/2b1eiz1ω1/b1e-id1ω12/2b1g0z+jg2z×∫R212πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b2eja2t22/2b2e-jt2ω1/b2dzdt+∫R2eia1t12/2b1e-it1ω1/b1jf2t+kf3t12πb1ie-ia1z12/2b1eiz1ω1/b1e-id1ω12/2b1ig1z+kg3z×∫R212πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b2eja2t22/2b2e-jt2ω1/b2dzdt.Now multiplying both sides of the above equation by 1/2πb1ieid1ω12/2b1 and 1/2πb2jejd2ω22/2b2 gives (27)12πb1ieid1ω12/2b1FA1,A2Hf⋆gωejd2ω22/2b212πb2j=∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1t12πb1ieia1z12/2b1e-iz1ω1/b1eid1ω12/2b1×∫R2g0z+jg2z12πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1t12πb1ieia1z12/2b1e-iz1ω1/b1eid1ω12/2b1×∫R212πb2jig1z+kg3zeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3t12πb1ie-ia1z12/2b1eiz1ω1/b1e-id1ω12/2b1×∫R2g0z+jg2z12πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3t12πb1ie-ia1z12/2b1eiz1ω1/b1e-id1ω12/2b1×∫R2ig1z+kg3z12πb2jeja2z22/2b2e-jz2ω2/b2ejd2ω22/2b212πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt=∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1t×FA1,A2Hg0ω+jFA1∗,A2Hg2ω12πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1t×iFA1,A2Hg1ω+kFA1∗,A2Hg3ω12πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3t×∫R2FA1∗,A2Hg0ω+jFA1,A2Hg2ω12πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3t×iFA1∗,A2Hg1ω+kFA1,A2Hg3ω12πb2jeja2t22/2b2e-jt2ω1/b2ejd2ω22/2b2dzdt.Based on Definition 3, the required result follows.Remark 8.
If we use the matrix parametersA1=A2=a1-10 with a≠0. Then (21) will lead to(28)FA1,A2Hf⋆gω=2πiFA1,A2Hf0ω+iFA1,A2Hf1ωFA1,A2Hg0ω+jFA1∗,A2Hg2ω+FA1,A2∗Hf0ω+iFA1,A2∗Hf1ωiFA1,A2Hg1ω+kFA1∗,A2Hg3ω+jFA1∗,A2Hf2ω+kFA1∗,A2Hf3ωFA1∗,A2Hg0ω+jFA1,A2Hg2ω+jFA1∗,A2∗Hf2ω+kFA1∗,A2∗Hf3ωiFA1∗,A2Hg1ω+kFA1,A2Hg3ω2πj.Observe that the above form is quite similar to the convolution theorem associated with the QFT [15]. Now we investigate some consequences of Theorem 7, which are given in the following results.Lemma 9.
Letf,g∈L1(R2;H) be two quaternion-valued functions. If we assume that FA1,A2H{g}∈L1(R2;R), then we have(29)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2Hf0ω+iFA1,A2Hf1ωFA1,A2Hgω+jFA1∗,A2Hf2ω+kFA1∗,A2Hf3ωFA1∗,A2Hgω2πb2je-jd2ω22/2b2.Moreover, if FA1,A2H{f}∈L1(R2;R), then we have(30)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2HfωFA1,A2Hg0ω+jFA1∗,A2Hg2ω+FA1∗,A2HfωiFA1,A2Hg1ω+kFA1∗,A2Hg3ω2πb2je-jd2ω22/2b2.Proof.
We only verify the identity (29) and the others are quite similar. Direct computations yield (31)FA1,A2Hf⋆gω=∫R2∫R212πb1iei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12eia1/b1t1t1-x1ftgx-t×12πb2jej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22eja2/b2t2t2-x2dtdx=∫R2∫R212πb1ieia1/b1t1t1-x1eia1/2b1x12e-i2/2b1x1ω1eid1/2b1ω12ftgx-t×12πb2jeja2/b2t2t2-x2eja2/2b2x22e-j2/2b2x2ω2ejd2/2b2ω22dtdx.By making the change of variable x-t=v of the above, we get (32)FA1,A2Hf⋆gω=∫R2∫R212πb1ie-ia1/b1t1v1ei1/2a1/b1v1+t12e-iv1+t1ω1/b1eid1ω12/2b1ftgv×12πb2je-ja2/b2t2v2ej1/2a2/b2v2+t22e-jv2+t2ω2/b2ejd2ω22/2b2dtdv=∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1×f0t+if1t+jf2t+kf3tgv×12πbjjeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b2dtdv=∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1f0t+if1tgv×12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b2dtdv+∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1jf2t+kf3tgv×12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b2dtdv.Multiplying this result by 1/2πb1ieid1ω12/2b1 and 1/2πb2jejd2ω22/2b2, we easily obtain (33)12πb1ieid1ω12/2b1FA1,A2Hf⋆gω12πb2jejd2ω22/2b2=∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1eid1ω12/2b1f0t+if1tgv×eja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b212πb2jejd2ω22/2b2dtdv+∫R2∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1eia1v12/2b1e-iv1ω1/b1eid1ω12/2b1×jf2t+kf3tgv12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2eja2v22/2b2e-jv2ω2/b2ejd2ω22/2b2dtdv=∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1f0t+if1tFA1,A2Hgω×12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2dt+∫R212πb1ieia1t12/2b1e-it1ω1/b1eid1ω12/2b1jf2t+kf3tFA1∗,A2Hgω×12πb2jeja2t22/2b2e-jt2ω2/b2ejd2ω22/2b2dt.The definition of the QLCT (16) finally gives(34)12πb1ieid1ω12/2b1FA1,A2Hf⋆gω12πb1jejd2ω22/2b2=FA1,A2Hf0ω+iFA1,A2Hf1ωFA1,A2Hgω+jFA1∗,A2Hf2ω+kFA1∗,A2Hf3ωFA1∗,A2Hgω.This finishes the proof of the lemma.Lemma 10.
Suppose thatf,g∈L1(R2;H) has the form(35)fx=f0x+if1x+jf2x+kf3x,gx=g0x+jg2x.If FA1∗,A2H{f}∈L1(R2;R) is satisfied, then we have(36)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2HfωFA1,A2Hgω2πb2je-jd2ω22/2b2=2πb1ie-id1ω12/2b1FA1,A2HgωFA1,A2Hfω2πb2je-jd2ω22/2b2.Remark 11.
It should be noticed iff,g∈L1(R2;H) takes the form(37)fx=jf2x+kf3x,gx=g0x+ig1x+jg2x+kg3x,then it holds (38)FA1,A2Hf⋆gω=2πb1ie-id1ω12/2b1FA1,A2HfωFA1∗,A2Hgω2πb2je-jd2ω22/2b2=2πb1ie-id1ω12/2b1FA1∗,A2HgωFA1,A2Hfω2πb2je-jd2ω22/2b2,where FA1,A2H{f},FA1,A2H{g}∈L1(R2;R).



---

*Source: 3749387-2019-05-28.xml*