# Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform

Authors: Mawardi Bahri; Muh. Saleh Arif Fatimah
Journal: Journal of Applied Mathematics
 (2017)

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## Abstract

The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT.


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## Body

## 1. Introduction

The quaternion Fourier transform (QFT) is a nontrivial generalization of the real and complex classical Fourier transforms (FT) using quaternion algebra. Many useful properties of the QFT were obtained such as shift, modulation, convolution, correlation, differentiation, energy conservation, and uncertainty principle. It was first introduced in [1] for the analysis of 2D linear time-invariant partial differential systems and then applied in color image processing [2, 3]. It is a natural question to extend the QFT to the linear canonical transform (LCT) domains and then it is the so-called quaternionic linear transform (QLCT). This extension is constructed by substituting the kernel of the QFT with the kernel of the LCT. A number of useful properties of the QLCT have been investigated including shift, orthogonality relation, reconstruction formula, and Heisenberg uncertainty principle (see, for example, [4–6] and the references given therein).In [7], the author studied that the fractional Fourier transform (FrFT) can be reduced to the classical Fourier transform. Based on this fact, some properties of the FrFT can be derived very easily from those of the classical Fourier transform by simple change variable. Recently, in [8], the authors developed this idea to derive an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by using the fundamental relationship between the QLCT and the QFT [8]. In [9], the authors proposed the quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) and obtain its fundamental properties. In the present paper, we first establish the basic relationship between the QWVD-LCT and the QFT. We then show that some fundamental properties of the QWVD-LCT such as inversion formula and Moyal formula can be obtained by combining this relation and the properties of the QFT. We investigate that the QWVD-LCT can be reduced to the quaternion windowed Fourier transform and the continuous quaternion Fourier transform. We finally establish the relationship between the QAF-LCT and the QFT which enables us to derive some useful properties of the QAF-LCT.


## 2. Preliminaries

In the preliminaries we remind the reader of some facts of quaternions, the quaternion Fourier transform, and the quaternion linear canonical transform.


### 2.1. Basic Facts about Quaternions

The quaternion algebra overR, denoted by H, is an associative noncommutative four-dimensional algebra:(1)H=q=q0+iq1+jq2+kq3;q0,q1,q2,q3∈R,which 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, q0 is called thescalar part of q denoted by Sc(q) and iq1+jq2+kq3 is called thevector (orpure) part of q. The vector part of q is conventionally denoted by q. Let p,q∈H and p,q be their vector parts, respectively. Equation (2) yields the quaternionic multiplication qp as(3)qp=q0p0-q·p+q0p+p0q+q×p, where q·p=(q1p1+q2p2+q3p3) and q×p=i(q2p3-q3p2)+j(q3p1-q1p3)+k(q1p2-q2p1).The quaternion conjugate ofq, given by(4)q¯=q0-iq1-jq2-kq3,q0,q1,q2,q3∈R, is an anti-involution; that is,(5)qp¯=p¯q¯.From (4), we obtain the norm or modulus of q∈H defined as(6)q=qq¯=q02+q12+q22+q32.It is not difficult to see that(7)qp=qp,∀p,q∈H.Furthermore, it is easily seen that(8)pqr=rqp,∀p,q,q∈H.Using conjugate (4) and the modulus of q, we can define the inverse of q∈H∖{0} as(9)q-1=q¯q2,which shows that H is a normed division algebra.Now we observe that(10)Scpq-=12pq-+qp-=q0p0+q·p.This leads to the cyclic multiplication; that is,(11)Scpqr=Scrqp=Scqpr,∀p,q,r∈H.We define an inner product for quaternion-valued functionsf,g:R2→H as follows:(12)f,g=∫R2fxgx¯dx,dx=dx1dx2with symmetric real scalar part(13)f,g=12f,g+g,f=Sc∫R2fxg¯xdx.In particular, for f=g, we obtain the L2(R2;H)-norm(14)f=f,f=∫R2fx2dx1/2.This gives the Lp(R2;H)-norm(15)fp=∫R2fxpdx1/p,1≤p<∞.


### 2.2. Quaternion Linear Canonical Transform and Its Basic Properties

In this section, we briefly discuss the definition of the two-sided QFT and the two-sided quaternion linear canonical transform (QLCT) (for simplicity of notation, we write the QFT and QLCT instead of the two-sided QFT and the two-sided QLCT, resp., in the next section). We further collect some basic properties of the QLCT, which will be very useful later on.Definition 1.
The QFT off∈L1(R2;H) is the transform Fq{f}:R2→H given by the integral(16)Fqfω=∫R2e-iω1x1fxe-jω2x2dx,where x=x1e1+x2e2, ω=ω1e1+ω2e2 and the quaternion exponential product e-iω1x1e-jω2x2 is the quaternion Fourier kernel. Here Fq is called the quaternion Fourier transform operator.Definition 2.
Iff∈L1(R2;H) and Fq{f}∈L1(R2;H), then the inverse transform of the QFT is given by(17)fx=Fq-1Fqfx=12π2∫R2eiω1x1Fqfωejω2x2dω,where Fq-1 is called the inverse QFT operator.A useful property of the QFT is stated in the following lemma, which is needed to derive Moyal formula of the quaternion Wigner-Ville distribution associated with the linear canonical transform (QWVD-LCT).Lemma 3 (QFT Parseval).
Letf,g∈L1(R2;H)∩L2(R2;H). The relation between f,g and their QFT is given by(18)f,g=Fqf,Fqg.In particular, with f=g, we get the QFT version of the Plancherel formula; that is,(19)f2=Fqf2.Definition 4 (QLCT definition).
Suppose thatA1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) are real matrix parameters satisfying det(A1)=det(A2)=1. The QLCT of a quaternion signal f∈L1(R2;H) is defined by(20)LA1,A2Hfω=∫R2KA1x1,ω1fxKA2x2,ω2dx,where the kernel functions of the QLCT above are given by(21)KA1x1,ω1=12πb1ei/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2,forb1≠0d1eic1d1/2ω12,forb1=0,KA2x2,ω2=12πb2ej/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2,forb2≠0d2ejc2d2/2ω22,forb2=0.From the definition of the QLCT, we can see easily that, whenb1b2=0 and b1=b2=0, the QLCT of a signal is essentially a quaternion chirp multiplication. Therefore, in this work, we always assume that b1b2≠0.Lemma 5.
The QLCT of a signalf can be seen as the QFT of the signal f in the following form:(22)LA1,A2Hfω=e-iπ/42πb1eid1/2b1ω12Fqeia1/2b1x12fxeja2/2b2x22ω1b1,ω2b2e-jπ/42πb2ejd2/2b2ω22.It is worth noting that ifA1=A2=(ai,bi,ci,di)=(0,1,-1,0) for i=1,2, (22) will reduce to the QFT definition; that is,(23)LA1,A2Hfω=∫R2e-iπ/42πe-iω1x1fxe-jω2x2e-jπ/42πdx=e-iπ/42πFqfωe-jπ/42π.Theorem 6 (QLCT Parseval).
Two quaternion functionsf,h∈L1(R2;H)∩L2(R2;H) are related to their QLCT via the Parseval formula, given as(24)f,h=LA1,A2Hf,LA1,A2Hh.When f=h, we get(25)f2=LA1,A2Hf2.Proof.
For a detailed proof of the above theorem, we refer the reader to [8].


## 2.1. Basic Facts about Quaternions

The quaternion algebra overR, denoted by H, is an associative noncommutative four-dimensional algebra:(1)H=q=q0+iq1+jq2+kq3;q0,q1,q2,q3∈R,which 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, q0 is called thescalar part of q denoted by Sc(q) and iq1+jq2+kq3 is called thevector (orpure) part of q. The vector part of q is conventionally denoted by q. Let p,q∈H and p,q be their vector parts, respectively. Equation (2) yields the quaternionic multiplication qp as(3)qp=q0p0-q·p+q0p+p0q+q×p, where q·p=(q1p1+q2p2+q3p3) and q×p=i(q2p3-q3p2)+j(q3p1-q1p3)+k(q1p2-q2p1).The quaternion conjugate ofq, given by(4)q¯=q0-iq1-jq2-kq3,q0,q1,q2,q3∈R, is an anti-involution; that is,(5)qp¯=p¯q¯.From (4), we obtain the norm or modulus of q∈H defined as(6)q=qq¯=q02+q12+q22+q32.It is not difficult to see that(7)qp=qp,∀p,q∈H.Furthermore, it is easily seen that(8)pqr=rqp,∀p,q,q∈H.Using conjugate (4) and the modulus of q, we can define the inverse of q∈H∖{0} as(9)q-1=q¯q2,which shows that H is a normed division algebra.Now we observe that(10)Scpq-=12pq-+qp-=q0p0+q·p.This leads to the cyclic multiplication; that is,(11)Scpqr=Scrqp=Scqpr,∀p,q,r∈H.We define an inner product for quaternion-valued functionsf,g:R2→H as follows:(12)f,g=∫R2fxgx¯dx,dx=dx1dx2with symmetric real scalar part(13)f,g=12f,g+g,f=Sc∫R2fxg¯xdx.In particular, for f=g, we obtain the L2(R2;H)-norm(14)f=f,f=∫R2fx2dx1/2.This gives the Lp(R2;H)-norm(15)fp=∫R2fxpdx1/p,1≤p<∞.


## 2.2. Quaternion Linear Canonical Transform and Its Basic Properties

In this section, we briefly discuss the definition of the two-sided QFT and the two-sided quaternion linear canonical transform (QLCT) (for simplicity of notation, we write the QFT and QLCT instead of the two-sided QFT and the two-sided QLCT, resp., in the next section). We further collect some basic properties of the QLCT, which will be very useful later on.Definition 1.
The QFT off∈L1(R2;H) is the transform Fq{f}:R2→H given by the integral(16)Fqfω=∫R2e-iω1x1fxe-jω2x2dx,where x=x1e1+x2e2, ω=ω1e1+ω2e2 and the quaternion exponential product e-iω1x1e-jω2x2 is the quaternion Fourier kernel. Here Fq is called the quaternion Fourier transform operator.Definition 2.
Iff∈L1(R2;H) and Fq{f}∈L1(R2;H), then the inverse transform of the QFT is given by(17)fx=Fq-1Fqfx=12π2∫R2eiω1x1Fqfωejω2x2dω,where Fq-1 is called the inverse QFT operator.A useful property of the QFT is stated in the following lemma, which is needed to derive Moyal formula of the quaternion Wigner-Ville distribution associated with the linear canonical transform (QWVD-LCT).Lemma 3 (QFT Parseval).
Letf,g∈L1(R2;H)∩L2(R2;H). The relation between f,g and their QFT is given by(18)f,g=Fqf,Fqg.In particular, with f=g, we get the QFT version of the Plancherel formula; that is,(19)f2=Fqf2.Definition 4 (QLCT definition).
Suppose thatA1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) are real matrix parameters satisfying det(A1)=det(A2)=1. The QLCT of a quaternion signal f∈L1(R2;H) is defined by(20)LA1,A2Hfω=∫R2KA1x1,ω1fxKA2x2,ω2dx,where the kernel functions of the QLCT above are given by(21)KA1x1,ω1=12πb1ei/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2,forb1≠0d1eic1d1/2ω12,forb1=0,KA2x2,ω2=12πb2ej/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2,forb2≠0d2ejc2d2/2ω22,forb2=0.From the definition of the QLCT, we can see easily that, whenb1b2=0 and b1=b2=0, the QLCT of a signal is essentially a quaternion chirp multiplication. Therefore, in this work, we always assume that b1b2≠0.Lemma 5.
The QLCT of a signalf can be seen as the QFT of the signal f in the following form:(22)LA1,A2Hfω=e-iπ/42πb1eid1/2b1ω12Fqeia1/2b1x12fxeja2/2b2x22ω1b1,ω2b2e-jπ/42πb2ejd2/2b2ω22.It is worth noting that ifA1=A2=(ai,bi,ci,di)=(0,1,-1,0) for i=1,2, (22) will reduce to the QFT definition; that is,(23)LA1,A2Hfω=∫R2e-iπ/42πe-iω1x1fxe-jω2x2e-jπ/42πdx=e-iπ/42πFqfωe-jπ/42π.Theorem 6 (QLCT Parseval).
Two quaternion functionsf,h∈L1(R2;H)∩L2(R2;H) are related to their QLCT via the Parseval formula, given as(24)f,h=LA1,A2Hf,LA1,A2Hh.When f=h, we get(25)f2=LA1,A2Hf2.Proof.
For a detailed proof of the above theorem, we refer the reader to [8].


## 3. Quaternion Wigner-Ville Distribution and Quaternion Ambiguity Function in Linear Canonical Transform Domains

Let us introduce the 2D quaternion Wigner-Ville distribution (QWVD) and quaternion ambiguity function QAF [10]. According to the QWVD and QAF definitions and the QLCT definition, we obtain a definition of the quaternion Wigner-Ville distribution associated with the linear canonical transform (QWVD-LCT) and the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) (see [9]). We establish the fundamental relationship between the QWVD-LCT and QFT. Applying this relation and the properties of the QFT, we in detail derive the inverse transform formula and Moyal’s formula for the QWVD-LCT and the QAF-LCT, where the proof of the properties is quite different from one proposed in [9]. We also study the relationship among the QWVD-LCT, the quaternion windowed Fourier transform, and the continuous quaternion Fourier transform.


### 3.1. Main Properties of QWVD-LCT and Relation among QAF-LCT, QWFT, and CQWT

Definition 7.
The cross quaternion Wigner-Ville distribution of two-dimensional functions (or signals)f,g∈L2(R2;H) is given by(26)Wf,gx,ω=∫R2e-iω1τ1fx+τ2g¯x-τ2e-jω2τ2dτ,provided that the integral exists.It should be remembered that the kernel of the cross QWVD in (26) does not commute with quaternion functions f and g so that several properties of the classical Wigner-Ville distribution (WVD) are not valid in the cross QWVD [10].Definition 8.
The cross quaternion ambiguity function of two-dimensional functions (or signals)f,g∈L2(R2;H) is given by(27)Af,gx,ω=∫R2e-iω1τ1fτ+x2g¯τ-x2e-jω2τ2dτ,provided that the integral exists.Definition 9.
Suppose thatA1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) are real matrix parameters satisfying det⁡(A1)=det⁡(A2)=1. The cross QWVD-LCT and cross QAF-LCT of a quaternion signal f,g∈L1(R2;H) are defined by(28)Wf,gA1,A2x,ω=∫R2KA1τ1,ω1fx+τ2g¯x-τ2KA2τ2,ω2dτ(29)Af,gA1,A2x,ω=∫R2KA1τ1,ω1fτ+x2g¯τ-x2KA2τ2,ω2dτ.Here the kernel functions of the above transforms are given by(30)KA1τ1,ω1=12πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2,forb1≠0d1eic1d1/2ω12,forb1=0,KA2τ2,ω2=12πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2,forb2≠0d2ejc2d2/2ω22,forb2=0.It can be directly seen that if we writehf,g(x,τ)=f(x+τ/2)g¯(x-τ/2), we immediately obtain(31)Wf,gA1,A2x,ω=∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2fx+τ2g¯x-τ212πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dτ=∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2hf,gx,τ12πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dτ=LA1,A2Hhf,gω.This tells us that the cross QWVD-LCT is in fact the QLCT of the function hf,g(x,τ) with respect to τ. This fact is very important in proving Moyal’s formula for the cross QWVD-LCT. Similarly, we also get(32)Af,gA1,A2x,ω=LA1,A2HRf,gω,where Rf,g is given by (33)Rf,gx,τ=fτ+x2g¯τ-x2.The following result presents an inequality related to the cross QWVD-LCT.Lemma 10.
Suppose thatf∈Lp(R2;H), g∈Lq(R2;H) with 1/p+1/q=1. Then we have(34)Wf,gA1,A2x,ω≤12πb1b2fpgq.Proof.
We straightforwardly obtain from (28) that (35)Wf,gA1,A2x,ω=∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2fx+τ2g¯x-τ2×12πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dτ≤∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2fx+τ2pdτ1/p×∫R2g¯x-τ212πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2qdτ1/q.Letting x+τ/2=y and x-τ/2=z, we immediately obtain(36)Wf,gA1,A2x,ω=12πb1b2∫R2fx+τ2pdτ1/p∫R2g¯x-τ2qdτ1/q=12πb1b2∫R2fypdy1/p∫R2g¯zqdz1/q.Hence, the result follows.Observe first that, forp=q=2, (34) will reduce to(37)Wf,gA1,A2x,ω≤12πb1b2fg,which shows that Wf,gA1,A2(x,ω) is bounded on L2(R2;H).Lemma 11.
The cross QWVD-LCT of a signalf with matrix parameters A1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) can be reduced as the QFT of the signal f in the following form:(38)Wf,gA1,A2x,ω=e-iπ/42πb1eid1/2b1ω12Fqeia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22ω1b1,ω2b2e-jπ/42πb2ejd2/2b2ω22.Proof.
By calculation directly, we get(39)Wf,gA1,A2x,ω=12πb1∫R2ei1/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2fx+τ2g¯x-τ212πb2ej1/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dτ=e-iπ/42πb1eid1/2b1ω12∫R2e-iτ1ω1/b1eia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22e-jτ2ω2/b2dτe-jπ/42πb2ejd2/2b2ω22=e-iπ/42πb1eid1/2b1ω12Fqeia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22ω1b1,ω2b2e-jπ/42πb2ejd2/2b2ω22,where the last line follows directly from (28).For abbreviation, we use the notation(40)hf,g~x,τ=e-iπ/42πb1eia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22e-jπ/42πb2.Therefore, we can write (38) in the form(41)Fqhf,g~x,τω1b1,ω2b2=e-id1/2b1ω12Wf,gA1,A2x,ωe-jd2/2b2ω22.In the below theorem by combining the properties of the QFT and the fundamental relation between the QFT and the QWVD-LCT, we provide a new proof of reconstruction formula for the QWVD-LCT.Theorem 12 (reconstruction formula for QWVD-LCT).
The inverse transform of the cross the QWVD-LCT of the signalf∈L2(R2;H) is given by(42)fx=1g¯0∫R2KA1-1x1,ω1Wf,gA1,A2x2,ωKA2-1x2,ω2dωprovided that g¯(0)≠0.Proof.
From the inverse transform of the QFT (17), it follows that(43)hf,g~x,τ=12π2∫R2eiω1τ1/b1Fqhf,g~x,τω1b1,ω2b2ejω2τ2/b2dωb.In view of (40) and (41), we easily get(44)e-iπ/42πb1eia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22e-jπ/42πb2=12π2∫R2eiω1τ1/b1Fqhf,g~x,τω1b1,ω2b2ejω2τ2/b2dωb=12π2∫R2eiω1τ1/b1e-id1/2b1ω12Wf,gA1,A2x,ωe-jd2/2b2ω22ejω2τ2/b2dωb.Subsequently,(45)fx+τ2g¯x-τ2=∫R2eiπ/42πb1e-ia1/2b1τ12eiω1τ1/b1e-id1/2b1ω12Wf,gA1,A2x,ωejπ/42πb2e-jd2/2b2ω22ejω2τ2/b2e-ja2/2b2τ22dω=∫R212πb1e-i/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2Wf,gA1,A2x,ω12πb2e-j/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dω=∫R2KA1-1τ1,ω1Wf,gA1,A2x,ωKA2-1τ2,ω2dω.Letting τ/2=x, the above expression will lead to(46)f2xg¯0=∫R2KA1-12x1,ω1Wf,gA1,A2x,ωKA2-12x2,ω2dω,and the final result can be obtained by letting 2x=s; that is,(47)fs=1g¯0∫R2KA1-1s1,ω1Wf,gA1,A2s2,ωKA2-1s2,ω2dω,which completes the proof.We now observe that from (45), we straightforwardly obtain the time marginal property of the QWVD-LCT; that is,(48)fx+τ2g¯x-τ2=∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2Wf,gA1,A2x,ω12πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dω.If we set τ=0, the above identity will reduce to(49)fxg¯x=∫R2eiπ/42πb1e-id1/2b1ω12Wf,gA1,A2x,ωejπ/42πb2e-jd2/2b2ω22dω,and if we set τ=0 and f=g, (48) becomes(50)fx2=∫R2eiπ/42πb1e-id1/2b1ω12WfA1,A2x,ωejπ/42πb2e-jd2/2b2ω22dω.Integrating (49) and (50) with respect to the x-variable gives(51)∫R2fxg¯xdx=∫R2∫R2eiπ/42πb1e-id1/2b1ω12Wf,gA1,A2x,ωejπ/42πb2e-jd2/2b2ω22dωdx∫R2fx2dx=∫R2∫R2eiπ/42πb1e-id1/2b1ω12WfA1,A2x,ωejπ/42πb2e-jd2/2b2ω22dωdx.Applying Moyal’s formula for the QFT and the relation between the QFT and the QWVD-LCT, we establish general Moyal’s formula of the QWVD-LCT (compared to [9]).Theorem 13 (Moyal’s formula for the QWVD-LCT).
Letf1,f2,g1,g2∈L2(R2;H) be quaternion-valued signals. Then the following equation holds:(52)Sc∫R2∫R2Wf1,g1A1,A2x,ωWf2,g2A1,A2x,ω¯dωdτ=f1g1¯,g2¯,f2.Proof.
Applying Parseval’s formula of the QFT (18) to ω-integral into the left-hand side of (52) yields(53)∫R2hf1,g1~,hf2,g2~dτ=∫R2Fqhf1,g1~,Fqhf2,g2~dτ=1b1b2∫R2Sc∫R2Fqhf1,g1~ωbFqhf2,g2~ωb¯dωdτ.Using the cyclic multiplication symmetry (11) yields(54)∫R2hf1,g1~,hf2,g2~dτ=1b1b2Sc∫R2∫R2e-id1/2b1ω12Wf1,g1A1,A2x,ωe-jd2/2b2ω22e-id1/2b1ω12Wf2,g2A1,A2x,ωe-jd2/2b2ω22¯dωdτ=1b1b2Sc∫R2∫R2e-id1/2b1ω12Wf1,g1A1,A2x,ωe-jd2/2b2ω22ejd2/2b2ω12Wf2,g2A1,A2x,ω¯eid1/2b1ω12dωdτ=1b1b2Sc∫R2∫R2Wf1,g1A1,A2x,ωWf2,g2A1,A2x,ω¯dωdτ.The right-hand side of the above identity can be rewritten in the form(55)∫R2hf1,g1~,hf2,g2~dτ=Sc∫R2∫R2hf1,g1~x,τhf2,g2~x,τ¯dxdτ=Sc∫R2∫R2e-iπ/42πb1eia1/2b1τ12f1x+τ2g1¯x-τ2eja2/2b2τ22e-jπ/42πb2×e-iπ/42πb1eia1/2b1τ12f2x+τ2g2¯x-τ2e-jπ/42πb2eja2/2b2τ22¯dxdτ=1b1b2Sc∫R2∫R2f1x+τ2g1¯x-τ2f2x+τ2g2¯x-τ2¯dxdτ.Making the change of variables y=x+τ/2 and z=x-τ/2 and applying Fubini’s theorem we obtain(56)∫R2hf1,g1~,hf2,g2~dτ=1b1b2Sc∫R2∫R2f1yg1¯zg2zf2¯ydydz=1b1b2Sc∫R2f1y∫R2g1¯zg2zdzf2¯ydy=1b1b2f1g1¯,g2¯,f2.By comparing the last line of (54) with the last line of (56) finishes the proof of the theorem.Based on the above theorem, we obtain the following consequences:(i)
Ifg1=g2, then(57)Sc∫R2∫R2Wf1,g1A1,A2x,ωWf2,g1A1,A2x,ω¯dωdτ=g12f1,f2.(ii)
Iff1=f2, then(58)Sc∫R2∫R2Wf1,g1A1,A2x,ωWf1,g2A1,A2x,ω¯dωdτ=f1g1¯,g2¯,f1.(iii)
Iff1=f2 and g1=g2, then(59)∫R2∫R2Wf1,g1A1,A2x,ω2dωdτ=f12g12.


### 3.2. Relationship between QAF-LCT and QWFT

In this subsection, we introduce the quaternionic windowed Fourier transform (QWFT) and obtain the relationship between the QAF-LCT and QWFT. Following [11, 12], we define the QWFT as follows.Definition 14 (QWFT).
Letϕ∈L2(R2;H) be a fixed nonzero quaternion window function. The QWFT of f∈L2(R2;H) with respect to ϕ is defined to be the quaternion function on phase space R2×R2 given by(60)Sfx,ω=∫R2e-iω1τ1fxg¯x-τe-jω2τ2dτ.
Before presenting the main result of this subsection, let us introduce the followingcarrier definition and its properties (see [13, 14]).Definition 15 (carrier).
Given two quaternionsp and q, we define the right Cr and left Cl carrier operators as(61)Crpq=qp,qClp=pq.Lemma 16.
Carriers (61) above satisfy the following properties with p∈H:(62)Crp¯=Clp¯,Clp¯=Crp¯.We now describe the relationship between the QAF-LCT and the QWFT in the following theorem.Theorem 17.
The QAF-LCT of a quaternion signalf∈L2(R2;H) can be expressed by the QWFT in the form(63)Af,gA1,A2x2,ω2=4e-iπ/42πb1eid1/8b1ω12+1/2b1ω1x1-3a1/2b1x12Sfx,ω1+2a1x1b1,ω2+2a2x2b2×4e-jπ/42πb2ejd2/8b2ω22+1/2b2ω2x2-3a2/2b2x22.Proof.
According to the definition of QWVD-LCT (28), we obtain the following by making the change of variable:(64)Af,gA1,A2x2,ω2=42πb1∫R2eia1/2b1τ12-1/2b1τ1ω1+d1/8b1ω12-π/2fτ2+x2g¯τ2-x212πb2eja2/2b2τ22-1/2b2τ2ω2+d2/8b2ω22-π/2dτ=4e-iπ/42πb1eid1/8b1ω12∫R2e-iτ1-x1/2ω1/b1ei2a1/b1τ1-x1/22fτg¯τ-xej2a2/b2τ2-x2/22e-jτ2-x2/2ω2/b2dτ×4e-jπ/42πb2ejd2/8b2ω22=4e-iπ/42πb1eid1/8b1ω12+1/2b1ω1x1-3a1/2b1x12∫R2e-iω1+2a1x1/b1τ1ei2a1/b1x1-τ12fxg¯x-τ×ej2a2/b2x2-τ22e-jω2+2a2x2/b2τ2dτ4e-jπ/42πb2ejd2/8b2ω22+1/2b2ω2x2-3a2/2b2x22.With the help of (61) and (62), we rewrite the above identity as(65)Af,gA1,A2x2,ω2=4e-iπ/42πb1eid1/8b1ω12+1/2b1ω1x1-3a1/2b1x12∫R2e-iω1+2a1x1/b1τ1fx×e-j2a2/b2x2-τ22gx-τCre-i2a1/b1x1-τ12¯×e-jω2+2a2x2/b2τ2dτ4e-jπ/42πb2ejd2/8b2ω22+1/2b2ω2x2-3a2/2b2x22.According to the definition of QWFT (60), we choose the quaternion window function(66)gx=e-j2a2/b2x22gxCre-i2a1/b1x12and obtain(67)Af,gA1,A2x2,ω2=4e-iπ/42πb1eid1/8b1ω12+1/2b1ω1x1-3a1/2b1x12Sfx,ω1+2a1x1b1,ω2+2a2x2b2×4e-jπ/42πb2ejd2/8b2ω22+1/2b2ω2x2-3a2/2b2x22.The proof is complete.


### 3.3. Relationship between QAF-LCT and CQWT

Before proving the relationship between the QAF-LCT and the continuous quaternion wavelet transform (CQWT), we first introduce the definition of the CQWT (see [15–17]).Definition 18 (CQWT).
The CQWT of a quaternion functionf∈L2(R2;H) with respect to the quaternion mother wavelet ψ is defined by(68)Tψfa,b=∫R2fx1aψx-ba¯dx.Here the family of the quaternion wavelets ψa,b is defined by(69)ψa,bx=1aψx-ba,a∈R+.This definition will lead to the following result.Theorem 19.
The QAF-LCT off∈L2(R2;H) can be reduced to the CQWT(70)Af,gA1,A2x,ω=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1Tψf1,xe-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2.Proof.
Applying the definition of the QAF-LCT (29) and following [18], we easily get (71)Af,gA1,A2x,ω=e-iπ/42πb1∫R2eia1/2b1τ12-1/b1τ1ω1+d1/2b1ω12-π/2fτ+x2g¯τ-x2e-jπ/42πb2eja2/2b2τ22-1/b2τ2ω2+d2/2b2ω22-π/2dτ=e-iπ/42πb1∫R2eia1/2b1τ12-1/b1τ1ω1+d1/2b1ω12-π/2fτ+x2g¯τ+x2-xe-jπ/42πb2eja2/2b2τ22-1/b2τ2ω2+d2/2b2ω22-π/2dτ=e-iπ/42πb1∫R2eid1/2b1ω12ei1/b1τ1′-x1/2ω1eia1/2b1τ1′-x1/22ω1fτ′g¯τ′-x×e-jπ/42πb2ejd2/2b2ω22ej1/b2τ2′-x2/2ω2eja2/2b2τ2′-x2/22ω1dτ′=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1∫R2eia1/2b1τ1′-x12-iτ1′-x12ω1-a1x1/2b1fτ′g¯τ′-x×eja2/2b2τ2′-x22-jτ2′-x22ω2-a2x2/2b2dτ′e-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2.Again applying (61) and (62), we obtain (72)Af,gA1,A2x,ω=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1∫R2fτ′×e-ja2/2b2τ2′-x22+jτ2′-x22ω2-a2x2/2b2gτ′-xCre-ia1/2b1τ1′-x12+iτ1′-x12ω1-a1x1/2b1¯×dτ′e-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1∫R2fτ′ψ¯τ′-xdτ′e-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1Tψf1,xe-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2,where the family of the quaternion wavelets ψ1,b is given by(73)ψ1,bx=e-ja2/2b2τ2-x22+jτ2-x22ω2-a2x2/2b2gτ-xCre-ia1/2b1τ1-x12+iτ1-x12ω1-a1x1/2b1¯.This is the desired result.


## 3.1. Main Properties of QWVD-LCT and Relation among QAF-LCT, QWFT, and CQWT

Definition 7.
The cross quaternion Wigner-Ville distribution of two-dimensional functions (or signals)f,g∈L2(R2;H) is given by(26)Wf,gx,ω=∫R2e-iω1τ1fx+τ2g¯x-τ2e-jω2τ2dτ,provided that the integral exists.It should be remembered that the kernel of the cross QWVD in (26) does not commute with quaternion functions f and g so that several properties of the classical Wigner-Ville distribution (WVD) are not valid in the cross QWVD [10].Definition 8.
The cross quaternion ambiguity function of two-dimensional functions (or signals)f,g∈L2(R2;H) is given by(27)Af,gx,ω=∫R2e-iω1τ1fτ+x2g¯τ-x2e-jω2τ2dτ,provided that the integral exists.Definition 9.
Suppose thatA1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) are real matrix parameters satisfying det⁡(A1)=det⁡(A2)=1. The cross QWVD-LCT and cross QAF-LCT of a quaternion signal f,g∈L1(R2;H) are defined by(28)Wf,gA1,A2x,ω=∫R2KA1τ1,ω1fx+τ2g¯x-τ2KA2τ2,ω2dτ(29)Af,gA1,A2x,ω=∫R2KA1τ1,ω1fτ+x2g¯τ-x2KA2τ2,ω2dτ.Here the kernel functions of the above transforms are given by(30)KA1τ1,ω1=12πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2,forb1≠0d1eic1d1/2ω12,forb1=0,KA2τ2,ω2=12πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2,forb2≠0d2ejc2d2/2ω22,forb2=0.It can be directly seen that if we writehf,g(x,τ)=f(x+τ/2)g¯(x-τ/2), we immediately obtain(31)Wf,gA1,A2x,ω=∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2fx+τ2g¯x-τ212πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dτ=∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2hf,gx,τ12πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dτ=LA1,A2Hhf,gω.This tells us that the cross QWVD-LCT is in fact the QLCT of the function hf,g(x,τ) with respect to τ. This fact is very important in proving Moyal’s formula for the cross QWVD-LCT. Similarly, we also get(32)Af,gA1,A2x,ω=LA1,A2HRf,gω,where Rf,g is given by (33)Rf,gx,τ=fτ+x2g¯τ-x2.The following result presents an inequality related to the cross QWVD-LCT.Lemma 10.
Suppose thatf∈Lp(R2;H), g∈Lq(R2;H) with 1/p+1/q=1. Then we have(34)Wf,gA1,A2x,ω≤12πb1b2fpgq.Proof.
We straightforwardly obtain from (28) that (35)Wf,gA1,A2x,ω=∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2fx+τ2g¯x-τ2×12πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dτ≤∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2fx+τ2pdτ1/p×∫R2g¯x-τ212πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2qdτ1/q.Letting x+τ/2=y and x-τ/2=z, we immediately obtain(36)Wf,gA1,A2x,ω=12πb1b2∫R2fx+τ2pdτ1/p∫R2g¯x-τ2qdτ1/q=12πb1b2∫R2fypdy1/p∫R2g¯zqdz1/q.Hence, the result follows.Observe first that, forp=q=2, (34) will reduce to(37)Wf,gA1,A2x,ω≤12πb1b2fg,which shows that Wf,gA1,A2(x,ω) is bounded on L2(R2;H).Lemma 11.
The cross QWVD-LCT of a signalf with matrix parameters A1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) can be reduced as the QFT of the signal f in the following form:(38)Wf,gA1,A2x,ω=e-iπ/42πb1eid1/2b1ω12Fqeia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22ω1b1,ω2b2e-jπ/42πb2ejd2/2b2ω22.Proof.
By calculation directly, we get(39)Wf,gA1,A2x,ω=12πb1∫R2ei1/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2fx+τ2g¯x-τ212πb2ej1/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dτ=e-iπ/42πb1eid1/2b1ω12∫R2e-iτ1ω1/b1eia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22e-jτ2ω2/b2dτe-jπ/42πb2ejd2/2b2ω22=e-iπ/42πb1eid1/2b1ω12Fqeia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22ω1b1,ω2b2e-jπ/42πb2ejd2/2b2ω22,where the last line follows directly from (28).For abbreviation, we use the notation(40)hf,g~x,τ=e-iπ/42πb1eia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22e-jπ/42πb2.Therefore, we can write (38) in the form(41)Fqhf,g~x,τω1b1,ω2b2=e-id1/2b1ω12Wf,gA1,A2x,ωe-jd2/2b2ω22.In the below theorem by combining the properties of the QFT and the fundamental relation between the QFT and the QWVD-LCT, we provide a new proof of reconstruction formula for the QWVD-LCT.Theorem 12 (reconstruction formula for QWVD-LCT).
The inverse transform of the cross the QWVD-LCT of the signalf∈L2(R2;H) is given by(42)fx=1g¯0∫R2KA1-1x1,ω1Wf,gA1,A2x2,ωKA2-1x2,ω2dωprovided that g¯(0)≠0.Proof.
From the inverse transform of the QFT (17), it follows that(43)hf,g~x,τ=12π2∫R2eiω1τ1/b1Fqhf,g~x,τω1b1,ω2b2ejω2τ2/b2dωb.In view of (40) and (41), we easily get(44)e-iπ/42πb1eia1/2b1τ12fx+τ2g¯x-τ2eja2/2b2τ22e-jπ/42πb2=12π2∫R2eiω1τ1/b1Fqhf,g~x,τω1b1,ω2b2ejω2τ2/b2dωb=12π2∫R2eiω1τ1/b1e-id1/2b1ω12Wf,gA1,A2x,ωe-jd2/2b2ω22ejω2τ2/b2dωb.Subsequently,(45)fx+τ2g¯x-τ2=∫R2eiπ/42πb1e-ia1/2b1τ12eiω1τ1/b1e-id1/2b1ω12Wf,gA1,A2x,ωejπ/42πb2e-jd2/2b2ω22ejω2τ2/b2e-ja2/2b2τ22dω=∫R212πb1e-i/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2Wf,gA1,A2x,ω12πb2e-j/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dω=∫R2KA1-1τ1,ω1Wf,gA1,A2x,ωKA2-1τ2,ω2dω.Letting τ/2=x, the above expression will lead to(46)f2xg¯0=∫R2KA1-12x1,ω1Wf,gA1,A2x,ωKA2-12x2,ω2dω,and the final result can be obtained by letting 2x=s; that is,(47)fs=1g¯0∫R2KA1-1s1,ω1Wf,gA1,A2s2,ωKA2-1s2,ω2dω,which completes the proof.We now observe that from (45), we straightforwardly obtain the time marginal property of the QWVD-LCT; that is,(48)fx+τ2g¯x-τ2=∫R212πb1ei/2a1/b1τ12-2/b1τ1ω1+d1/b1ω12-π/2Wf,gA1,A2x,ω12πb2ej/2a2/b2τ22-2/b2τ2ω2+d2/b2ω22-π/2dω.If we set τ=0, the above identity will reduce to(49)fxg¯x=∫R2eiπ/42πb1e-id1/2b1ω12Wf,gA1,A2x,ωejπ/42πb2e-jd2/2b2ω22dω,and if we set τ=0 and f=g, (48) becomes(50)fx2=∫R2eiπ/42πb1e-id1/2b1ω12WfA1,A2x,ωejπ/42πb2e-jd2/2b2ω22dω.Integrating (49) and (50) with respect to the x-variable gives(51)∫R2fxg¯xdx=∫R2∫R2eiπ/42πb1e-id1/2b1ω12Wf,gA1,A2x,ωejπ/42πb2e-jd2/2b2ω22dωdx∫R2fx2dx=∫R2∫R2eiπ/42πb1e-id1/2b1ω12WfA1,A2x,ωejπ/42πb2e-jd2/2b2ω22dωdx.Applying Moyal’s formula for the QFT and the relation between the QFT and the QWVD-LCT, we establish general Moyal’s formula of the QWVD-LCT (compared to [9]).Theorem 13 (Moyal’s formula for the QWVD-LCT).
Letf1,f2,g1,g2∈L2(R2;H) be quaternion-valued signals. Then the following equation holds:(52)Sc∫R2∫R2Wf1,g1A1,A2x,ωWf2,g2A1,A2x,ω¯dωdτ=f1g1¯,g2¯,f2.Proof.
Applying Parseval’s formula of the QFT (18) to ω-integral into the left-hand side of (52) yields(53)∫R2hf1,g1~,hf2,g2~dτ=∫R2Fqhf1,g1~,Fqhf2,g2~dτ=1b1b2∫R2Sc∫R2Fqhf1,g1~ωbFqhf2,g2~ωb¯dωdτ.Using the cyclic multiplication symmetry (11) yields(54)∫R2hf1,g1~,hf2,g2~dτ=1b1b2Sc∫R2∫R2e-id1/2b1ω12Wf1,g1A1,A2x,ωe-jd2/2b2ω22e-id1/2b1ω12Wf2,g2A1,A2x,ωe-jd2/2b2ω22¯dωdτ=1b1b2Sc∫R2∫R2e-id1/2b1ω12Wf1,g1A1,A2x,ωe-jd2/2b2ω22ejd2/2b2ω12Wf2,g2A1,A2x,ω¯eid1/2b1ω12dωdτ=1b1b2Sc∫R2∫R2Wf1,g1A1,A2x,ωWf2,g2A1,A2x,ω¯dωdτ.The right-hand side of the above identity can be rewritten in the form(55)∫R2hf1,g1~,hf2,g2~dτ=Sc∫R2∫R2hf1,g1~x,τhf2,g2~x,τ¯dxdτ=Sc∫R2∫R2e-iπ/42πb1eia1/2b1τ12f1x+τ2g1¯x-τ2eja2/2b2τ22e-jπ/42πb2×e-iπ/42πb1eia1/2b1τ12f2x+τ2g2¯x-τ2e-jπ/42πb2eja2/2b2τ22¯dxdτ=1b1b2Sc∫R2∫R2f1x+τ2g1¯x-τ2f2x+τ2g2¯x-τ2¯dxdτ.Making the change of variables y=x+τ/2 and z=x-τ/2 and applying Fubini’s theorem we obtain(56)∫R2hf1,g1~,hf2,g2~dτ=1b1b2Sc∫R2∫R2f1yg1¯zg2zf2¯ydydz=1b1b2Sc∫R2f1y∫R2g1¯zg2zdzf2¯ydy=1b1b2f1g1¯,g2¯,f2.By comparing the last line of (54) with the last line of (56) finishes the proof of the theorem.Based on the above theorem, we obtain the following consequences:(i)
Ifg1=g2, then(57)Sc∫R2∫R2Wf1,g1A1,A2x,ωWf2,g1A1,A2x,ω¯dωdτ=g12f1,f2.(ii)
Iff1=f2, then(58)Sc∫R2∫R2Wf1,g1A1,A2x,ωWf1,g2A1,A2x,ω¯dωdτ=f1g1¯,g2¯,f1.(iii)
Iff1=f2 and g1=g2, then(59)∫R2∫R2Wf1,g1A1,A2x,ω2dωdτ=f12g12.


## 3.2. Relationship between QAF-LCT and QWFT

In this subsection, we introduce the quaternionic windowed Fourier transform (QWFT) and obtain the relationship between the QAF-LCT and QWFT. Following [11, 12], we define the QWFT as follows.Definition 14 (QWFT).
Letϕ∈L2(R2;H) be a fixed nonzero quaternion window function. The QWFT of f∈L2(R2;H) with respect to ϕ is defined to be the quaternion function on phase space R2×R2 given by(60)Sfx,ω=∫R2e-iω1τ1fxg¯x-τe-jω2τ2dτ.
Before presenting the main result of this subsection, let us introduce the followingcarrier definition and its properties (see [13, 14]).Definition 15 (carrier).
Given two quaternionsp and q, we define the right Cr and left Cl carrier operators as(61)Crpq=qp,qClp=pq.Lemma 16.
Carriers (61) above satisfy the following properties with p∈H:(62)Crp¯=Clp¯,Clp¯=Crp¯.We now describe the relationship between the QAF-LCT and the QWFT in the following theorem.Theorem 17.
The QAF-LCT of a quaternion signalf∈L2(R2;H) can be expressed by the QWFT in the form(63)Af,gA1,A2x2,ω2=4e-iπ/42πb1eid1/8b1ω12+1/2b1ω1x1-3a1/2b1x12Sfx,ω1+2a1x1b1,ω2+2a2x2b2×4e-jπ/42πb2ejd2/8b2ω22+1/2b2ω2x2-3a2/2b2x22.Proof.
According to the definition of QWVD-LCT (28), we obtain the following by making the change of variable:(64)Af,gA1,A2x2,ω2=42πb1∫R2eia1/2b1τ12-1/2b1τ1ω1+d1/8b1ω12-π/2fτ2+x2g¯τ2-x212πb2eja2/2b2τ22-1/2b2τ2ω2+d2/8b2ω22-π/2dτ=4e-iπ/42πb1eid1/8b1ω12∫R2e-iτ1-x1/2ω1/b1ei2a1/b1τ1-x1/22fτg¯τ-xej2a2/b2τ2-x2/22e-jτ2-x2/2ω2/b2dτ×4e-jπ/42πb2ejd2/8b2ω22=4e-iπ/42πb1eid1/8b1ω12+1/2b1ω1x1-3a1/2b1x12∫R2e-iω1+2a1x1/b1τ1ei2a1/b1x1-τ12fxg¯x-τ×ej2a2/b2x2-τ22e-jω2+2a2x2/b2τ2dτ4e-jπ/42πb2ejd2/8b2ω22+1/2b2ω2x2-3a2/2b2x22.With the help of (61) and (62), we rewrite the above identity as(65)Af,gA1,A2x2,ω2=4e-iπ/42πb1eid1/8b1ω12+1/2b1ω1x1-3a1/2b1x12∫R2e-iω1+2a1x1/b1τ1fx×e-j2a2/b2x2-τ22gx-τCre-i2a1/b1x1-τ12¯×e-jω2+2a2x2/b2τ2dτ4e-jπ/42πb2ejd2/8b2ω22+1/2b2ω2x2-3a2/2b2x22.According to the definition of QWFT (60), we choose the quaternion window function(66)gx=e-j2a2/b2x22gxCre-i2a1/b1x12and obtain(67)Af,gA1,A2x2,ω2=4e-iπ/42πb1eid1/8b1ω12+1/2b1ω1x1-3a1/2b1x12Sfx,ω1+2a1x1b1,ω2+2a2x2b2×4e-jπ/42πb2ejd2/8b2ω22+1/2b2ω2x2-3a2/2b2x22.The proof is complete.


## 3.3. Relationship between QAF-LCT and CQWT

Before proving the relationship between the QAF-LCT and the continuous quaternion wavelet transform (CQWT), we first introduce the definition of the CQWT (see [15–17]).Definition 18 (CQWT).
The CQWT of a quaternion functionf∈L2(R2;H) with respect to the quaternion mother wavelet ψ is defined by(68)Tψfa,b=∫R2fx1aψx-ba¯dx.Here the family of the quaternion wavelets ψa,b is defined by(69)ψa,bx=1aψx-ba,a∈R+.This definition will lead to the following result.Theorem 19.
The QAF-LCT off∈L2(R2;H) can be reduced to the CQWT(70)Af,gA1,A2x,ω=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1Tψf1,xe-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2.Proof.
Applying the definition of the QAF-LCT (29) and following [18], we easily get (71)Af,gA1,A2x,ω=e-iπ/42πb1∫R2eia1/2b1τ12-1/b1τ1ω1+d1/2b1ω12-π/2fτ+x2g¯τ-x2e-jπ/42πb2eja2/2b2τ22-1/b2τ2ω2+d2/2b2ω22-π/2dτ=e-iπ/42πb1∫R2eia1/2b1τ12-1/b1τ1ω1+d1/2b1ω12-π/2fτ+x2g¯τ+x2-xe-jπ/42πb2eja2/2b2τ22-1/b2τ2ω2+d2/2b2ω22-π/2dτ=e-iπ/42πb1∫R2eid1/2b1ω12ei1/b1τ1′-x1/2ω1eia1/2b1τ1′-x1/22ω1fτ′g¯τ′-x×e-jπ/42πb2ejd2/2b2ω22ej1/b2τ2′-x2/2ω2eja2/2b2τ2′-x2/22ω1dτ′=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1∫R2eia1/2b1τ1′-x12-iτ1′-x12ω1-a1x1/2b1fτ′g¯τ′-x×eja2/2b2τ2′-x22-jτ2′-x22ω2-a2x2/2b2dτ′e-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2.Again applying (61) and (62), we obtain (72)Af,gA1,A2x,ω=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1∫R2fτ′×e-ja2/2b2τ2′-x22+jτ2′-x22ω2-a2x2/2b2gτ′-xCre-ia1/2b1τ1′-x12+iτ1′-x12ω1-a1x1/2b1¯×dτ′e-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1∫R2fτ′ψ¯τ′-xdτ′e-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2=e-iπ/42πb1eid1/2b1ω12-ω1x1/2b1+a1x12/8b1Tψf1,xe-jπ/42πb2ejd2/2b2ω22-ω2x2/2b2+a2x22/8b2,where the family of the quaternion wavelets ψ1,b is given by(73)ψ1,bx=e-ja2/2b2τ2-x22+jτ2-x22ω2-a2x2/2b2gτ-xCre-ia1/2b1τ1-x12+iτ1-x12ω1-a1x1/2b1¯.This is the desired result.


## 4. Relationship between QAF-LCT and QFT

By a similar argument as in the proof of the Lemma11, one may establish the following lemma, which describes the basic relationship between the QAF-LCT and the QFT.Lemma 20.
The cross QAF-LCT of a signalf with matrix parameters A1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) can be reduced as the QFT of the signal f in the following form:(74)Af,gA1,A2x,ω=e-iπ/42πb1eid1/2b1ω12Fqeia1/2b1τ12fτ+x2g¯τ-x2eja2/2b2τ22ω1b1,ω2b2e-jπ/42πb2ejd2/2b2ω22.Applying (74) and inversion formula for the QFT (17), we easily obtain the following theorem.Theorem 21 (reconstruction formula for QAF-LCT).
The inverse transform of the cross the QAF-LCT of the signalf∈L2(R2;H) is given by(75)fx=1g¯0∫R2KA1-1x1,ω1Af,gA1,A2x2,ωKA2-1x2,ω2dω,provided that g¯(0)≠0.In the same kind of reasoning as Theorem13, one can easily derive Moyal’s formula of the QAF-LCT as follows.Theorem 22 (Moyal’s formula for QAF-LCT).
Letf1,f2,g1,g2∈L2(R2;H) be quaternion-valued signals. Then the following equation holds:(76)Sc∫R2∫R2Af1,g1A1,A2x,ωAf2,g2A1,A2x,ω¯dωdτ=f1g1¯,g2¯,f2.


## 5. Conclusion

In this paper, we have presented the quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) and the quaternion ambiguity function associated with linear canonical transform (QAF-LCT). We provide alternative proof of well-known properties of the QWVD-LCT and QAF-LCT such as inversion formula and Moyal formula. The proof of the properties can be obtained using the properties of the QFT and relationship among the QWVD-LCT, QAF-LCT, and the QFT. We also studied that the QAF-LCT can be reduced to the QWFT and the CQWT.



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