# A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transform

Authors: Mawardi Bahri; Ryuichi Ashino
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

We provide a short and simple proof of an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by considering the fundamental relationship between the QLCT and the quaternion Fourier transform (QFT). We show how this relation allows us to derive the inverse transform and Parseval and Plancherel formulas associated with the QLCT. Some other properties of the QLCT are also studied.


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## Body

## 1. Introduction

It is well-known that the traditional linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. It can be regarded as a generalization of many mathematical transforms such as the Fourier transform, Laplace transform, the fractional Fourier transform, and the Fresnel transform. Many fundamental properties of this extended transform are already known, including shift, modulation, convolution, and correlation and uncertainty principle, for example, in [1–6].Recently, there are so many studies in the literature that are concerned with the generalization of the LCT within the context of quaternion algebra, which is the so-called quaternion linear canonical transform (QLCT) (see, e.g., [7–10]). They also established some important properties of the QLCT such as inversion formula and the uncertainty principle. An application of the QLCT to study of generalized swept-frequency filters was presented in [11]. In this paper, we will focus on the two-dimensional case and provide a new proof of uncertainty principle associated with the QLCT, the ones proposed in [8], the proof of which is much simpler using the component-wise and directional uncertainty principles for the QFT [12, 13]. Therefore, before proving this main result, we first derive the fundamental relationship between the QLCT and QFT. Using the relation, we obtain useful properties of the QLCT such as inverse transform and Parseval formula associated with the QLCT.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)q¯p¯=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,r∈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.It is convenient to introduce an inner product for quaternion-valued (in the rest of the paper, we will always consider quaternion function) functionsf,g:R2→H as(10)f,g=∫R2fxgx¯dx,dx=dx1dx2,with symmetric real scalar part(11)f,g=12f,g+g,f=Sc∫R2fxg¯xdx.In particular, for f=g, we obtain the L2(R2;H)-norm:(12)f=f,f=∫R2fx2d2x1/2.


## 2. Quaternion Linear Canonical Transform

In this section we begin by defining the two-sided QFT (for simplicity of notation we write the QFT instead of the two-sided QFT in the next section). We discus some properties, which will be used to prove the uncertainty principle.Definition 1.
The QFT off∈L1(R2;H) is the transform Fq{f}:R2→H given by the integral(13)Fqfω=12π2∫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(14)fx=Fq-1Fqfx=12π2∫R2eiω1x1Fqfωejω2x2dω,where Fq-1 is called the inverse QFT operator.An important property of the QFT is stated in the following lemma, which is needed to prove Parseval formula of the QLCT. For more details of the QFT, see [12–16].Lemma 3 (QFT Parseval).
The quaternion product off,g∈L1(R2;H)∩L2(R2;H) and its QFT are related by(15)f,gL2R2;H=Fqf,FqgL2R2;H.In particular, with f=g, we get the quaternion version of the Plancherel formula; that is,(16)fL2R2;H2=FqfL2R2;H2.Based on the definition of the QFT mentioned above, we consider the two-sided QLCT which is defined as follows.Definition 4 (QLCT).
LetA1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) be two matrix parameters satisfying det⁡(As)=asds-bscs=1,  s=1,2. The QLCT of a quaternion signal f∈L1(R2;H) is defined by(17)LA1,A2Hfω=∫R2KA1x1,ω1fxKA2x2,ω2dx,b1b2≠0d1eic1d1/2ω12fd1ω1,x2KA2x2,ω2,b1=0,b2≠0d2ejc2d2/2ω22fx1,d2ω2KA1x1,ω1,b1≠0,b2=0d1d2eic1d1/2ω12fd1ω1,d2ω2ejc2d2/2ω22,b1=b2=0,where the kernel functions of the QLCT are given by, respectively,(18)KA1x1,ω1=12πb1ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2,b1≠0,(19)KA2x2,ω2=12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2,b2≠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. As a special case, when A1=A2=(ai,bi,ci,di)=(0,1,-1,0) for i=1,2, LCT definition (17) reduces to the QFT definition. That is,(20)LA1,A2Hfω=∫R2e-iπ/42πfxe-iω1x1e-jπ/42πe-jω2x2dx=e-iπ/4Fqfωe-jπ/4,where Fq{f} is the QFT of f given by (13).We need the following important result (compare to [17, 18]), which will be useful in proving Theorem 15.Theorem 5.
The QLCT of a quaternion functionf∈L1(R2;H) with matrix parameters A1=(a1,b1,c1,d1) and A2=(a2,b2,c2,d2) can be reduced to the QFT(21)Fqgfω=12π2∫R2e-ix1ω1gfxe-jx2ω2dx,where(22)Fqgfω=F~bω,gfx=e-iπ/4b1f~xe-jπ/4b2,f~x=eia1/2b1x12fxeja2/2b2x22,with(23)F~ω=12π2∫R2e-ix1ω1/b1gfxe-jx2ω2/b2dx,F~ω=e-id1/2b1ω12LA1,A2Hfωe-jd2/2b2ω22.Proof.
Simple computations using Definition4 show that(24)LA1,A2Hfω=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx=e-iπ/42πb1eid1/2b1ω12∫R2e-ix1ω1/b1eia1/2b1x12fxeja2/2b2x22e-jπ/42πb2je-jx2ω2/b2ejd2/2b2ω22dx=e-iπ/42πb1eid1/2b1ω12∫R2e-ix1ω1/b1f~xe-jx2ω2/b2dxe-iπ/42πb2ee-jπ/4d2/2b2ω22.Then, multiplying both sides of (24) by e-id1/2b1ω12e-jd2/2b2ω22 results in(25)e-id1/2b1ω12LA1,A2Hfωe-jd2/2b2ω22=e-iπ/42πb1∫R2e-ix1ω1/b1f~xe-jx2ω2/b2dxe-jπ/42πb2=12π∫R2e-ix1ω1/b1e-iπ/4b1f~xe-iπ/4b2e-jx2ω2/b2dx12π=12π2∫R2e-ix1ω1/b1gfxe-jx2ω2/b2dx=Fqgfωb.This is the desired result.Theorem 6.
Iff∈L1(R2;H) and LA1,A2H{f}∈L1(R2;H), then the inverse transform of the QLCT can be derived from that of the QFT.Proof.
Indeed, we have(26)gfx=12π2∫R2eix1ω1Fqgfωejx2ω2dω=12π2∫R2eix1ω1F~bωejx2ω2dω=1b1b22π2∫R2eix1ω1/b1F~ωejx2ω2/b2dω=1b1b22π2∫R2eix1ω1/b1e-id1/2b1ω12LA1,A2Hfωe-jd2/2b2ω22ejx2ω2/b2dω.It means that(27)e-iπ/4b1eia1/2b1x12fxeja2/2b2x22e-jπ/4b2=1b1b22π2∫R2eix1ω1/b1e-id1/2b1ω12LA1,A2Hfωe-jd2/2b2ω22ejx2ω2/b2dω.Or, equivalently,(28)fx=∫R212πb1e-ia1/2b1x12eix1ω1/b1e-id1/2b1ω12eiπ/4LA1,A2Hfω12πb2e-ja2/2b2x22ejx2ω2/b2e-jd2/2b2ω22e-iπ/4dω=∫R2KA1x1,ω1¯LA1,A2HfωKA2x2,ω2¯dω,which is inverse transform of the QLCT. This proves the theorem.In following we give an alternative proof of Parseval formula for the QLCT (cf. [8]).Theorem 7 (QLCT Parseval).
Two quaternion functionsf,h∈L1(R2;H)∩L2(R2;H) are related to their QLCT via the Parseval formula, given as(29)f,hL2R2;H=LA1,A2Hf,LA1,A2HhL2R2;H.For f=h, one has(30)fL2R2;H2=LA1,A2HfL2R2;H2.Proof.
From Parseval formula (15), it follows that(31)gf,gh=Fqgf,Fqgh=Sc∫R2FqgfωFqghω¯dω=1b1b2Sc∫R2FqgfωbFqghωb¯dω.Applying the cyclic multiplication symmetry, we get(32)gf,gh=1b1b2Sc∫R2e-id1/2b1ω12LA1,A2Hfωe-jd2/2b2ω22e-id1/2b1ω12LA1,A2Hhωe-jd2/2b2ω22¯dω=1b1b2Sc∫R2e-id1/2b1ω12LA1,A2Hfωe-jd2/2b2ω22e-jd2/2b2ω12LA1,A2Hhω¯e-id1/2b1ω12dω=1b1b2Sc∫R2LA1,A2HfωLA1,A2Hhω¯dω.On the other hand,(33)gf,gh=Sc∫R2gfxghx¯dx=Sc∫R2e-iπ/4b1f~xe-jπ/4b2e-iπ/4b1h~xe-jπ/4b2¯dx=1b1b2Sc∫R2f~xh~x¯dx=1b1b2Sc∫R2eia1/2b1x12fxeja2/2b2x22eia1/2b1x12hxeja2/2b2x22¯dx=1b1b2Sc∫R2fxhx¯dx.The proof is complete.It is interesting to describe the relationship between the QLCT and QFT as shown in the following example.Example 8.
Let us now compute the QLCT of the two-dimensional Gaussian functionf(x)=e-(k1x12+k2x22) with k1,k2>0.From the definition of QLCT (17), we easily obtain(34)LA1,A2Hfω=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω+d1/b1ω2-π/2fxej1/2a2/b2x22-2/b2x2ω+d2/b2ω2-π/212πb2dx=12πb1∫Re-k1x12ei1/2a1/b1x12-2/b1x1ω+d1/b1ω2-π/2dx112πb2∫Re-k2x22ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω2-π/2dx2=12πb1ei1/2d1/b1ω2∫Re-1/2b12k1b1-ia1x12e-iω/b1x1e-iπ/2dx112πb2ej1/2d2/b2ω2∫Re-1/2b22k2b2-ja2x22e-jω/b2x2e-jπ/2dx2.Using the QFT of the Gaussian function, (35)Fqfω=∫R2e-iω1x1e-k1x12+k2x22e-jω2x2dx=πk1k2e-ω12/4k1+ω22/4k2.We immediately obtain(36)LA1,A2Hfω=12πb1iei1/2d1/b1ω122πb12k1b1-ia1e-ω12/2b12k1b1-ia112πb1ej1/2d2/b2ω222πb22k2b2-ja2e-ω22/2b22k2b2-ja2=1a1+2k1b1eω12/2c1+2k1id1/2k1b1-ia11a2+2k2b2eω22/2c2+2k2jd2/2k2b2-ja2.


## 3. Properties of the QLCT

In this section we present useful properties of the QLCT in detail. We see that the results are generalizations of the properties of the LCT [5, 19]. In [9], the authors derived the asymptotic behavior of the QLCT. In the following, we shall provide a different proof of the results using the QLCT kernel properties.


### 3.1. Asymptotic Behavior of the QLCT

Like the classical Fourier transform, the Riemann-Lebesgue lemma is also valid for the QLCT, expressed as follows.Theorem 9 (Riemann-Lebesgue lemma).
Suppose thatf∈L1(R2;H). Then(37)limω1→∞⁡LA1,A2Hfω=0,limω2→∞⁡LA1,A2Hfω=0.Proof.
It is not difficult to see that(38)e-iω1x1/b1=-e-iω1/b1x1+b1π/ω1,e-jω2x2/b2=-e-jω2/b2x2+b2π/ω2.Now applying (38) gives(39)LA1,A2Hfω=∫R212πb1ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx=-∫R212πb1ei1/2a1/b1x12-2/b1x1+b1π/ω1ω1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx.Therefore, by making the change of variable x1+b1π/ω1=t1 in the above identity, we immediately obtain(40)LA1,A2Hfω=-∫R212πb1ei1/2a1/b1t1-b1π/ω12-2/b1t1ω1+d1/b1ω12-π/2ft1-b1πω1,t212πb2ej1/2a2/b2t22-2/b1t2ω1+d1/b2ω22-π/2dt=12∫R212πb1ei1/2a1/b1t12-2/b1t1ω1+d1/b1ω12-π/2ft12πb2ej1/2a2/b2t22-2/b2t2ω2+d2/b2ω22-π/2dx-∫R212πb1ei1/2a1/b1t1-b1π/ω12-2/b1t1ω1+d1/b1ω12-π/2ft1-b1πω1,t212πb2ej1/2a2/b2t22-2/b2t2ω2+d2/b2ω22-π/2dt=12∫R212πb1ei1/2a1/b1t12-2/b1t1ω1+d1/b1ω12-π/2ft-eia1/2b1-2t1b1π/ω1+b1π/ω12ft1-b1πω1,t212πb2ej1/2a2/b2t22-2/b2t2ω2+d2s/b2ω22-π/2dt.This means that(41)limω1→∞⁡LA1,A2Hfω≤14πb1b2limω1→∞⁡∫R2ft-eia1/2b1-2t1b1π/ω1+b1π/ω12ft1-b1πω1,t2dt=0.Similarly we can prove(42)limω2→∞⁡LA1,A2Hfω=0.Theorem 10 (continuity).
Iff∈L1(R2;H), then LA1,A2H{f}(ω) is continuous on R2.Proof.
Simple computations show that(43)LA1,A2Hfω+h-LA1,A2Hfω=∫R212πb1ei1/2a1/b1x1+h12-2/b1x1+h1ω1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x2+h22-2/b2x2+h2ω2+d2/b2ω22-π/2dx-12πb1ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx≤∫R212πb1ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2ei1/2a1/b12x1h1+h12-2/b1ω1h1fxej1/2a2/b22x2h2+h22-2/b2ω2h2-fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx=12πb1b2∫R2ei1/2a1/b12x1h1+h12-2/b1ω1h1fxej1/2a2/b22x2h2+h22-2/b2ω2h2-fxdx≤1πb1b2∫R2fxdx.By the Lebesgue dominated convergence theorem, we may conclude that(44)LA1,A2Hfω+h-LA1,A2Hfω⟶0when h→0. This proves that LA1,A2H{f}(ω) is continuous on R2. Again since (43) is independent of ω, LA1,A2H{f}(ω) is, in fact, uniformly continuous on R2.


### 3.2. Useful Properties of the QLCT

Due to the noncommutativity of the kernel of the QLCT, we only have a left linearity property with specific constants(45)α,β∈q∣q=q0+iq1,q0,q1∈R,which is(46)LA1,A2Hαf+βgω=αLA1,A2Hfω+βLA1,A2Hgω,and a right linearity property with specific constants(47)α′,β′∈q∣q=q0+jq2,q0,q2∈R.Theorem 11 (shift property).
Given a quaternion functionf∈L2(R2;H), let τkf(x) denote the shifted (translated) function defined by τkf(x)=f(x-k), where k∈R2. Then one gets(48)LA1,A2Hτkfω=e-ia1c1k12/2+ic1k1ω1LA1,A2Hfω1-a1k1,ω2-a2k2e-ja2c2k22/2+jc2k2ω2.Proof.
Taking into account the definition of QLCT (17), we get(49)LA1,A2Hτkfω=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2fx-k12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx.By making the change of a variable x-k=m, we easily obtain(50)LA1,A2Hτkfω=∫R212πb1ei1/2a1/b1m1+k12-2/b1m1+k1ω1+d1/b1ω12-π/2fm12πb2ej1/2a2/b2m2+k22-2/b2m2+k2ω2+d2/b2ω22-π/2dm=12πb1∫R2ei1/2a1/b1m12-2/b1m1ω1-k1a1+d1/b1ω12-π/2ei1/2a1/b1k12ei-1/22k1ω1/b1fm12πb2ej1/2a2/b2m22-2/b2m2ω2-k2a2+d2/b2ω22-π/2ej1/2a2/b2k22ej-1/22k2ω2/b2dm.Therefore, we further get(51)LA1,A2Hτkfω=12πb1∫R2ei1/2a1/b1m12-2/b1m1ω1-k1a1+d1/b1ω1-k1a1+k1a12-π/2ei1/2a1/b1k12ei-1/22k1ω1/b1fm2πb2ej1/2a2/b2m22-2/b2m2ω2-k2a2+d2/b2ω2-k2a2+k2a22-π/2ej1/2a1/b1k12ej-1/22k2ω2/b2dm=ei1/2d1/b12ω1-k1a1k1a1+k1a12ei1/2a1/b1k12ei-1/22k1ω1/b1∫R212πb1ei1/2a1/b1m12-2/b1m1ω1-k1a1+d1/b1ω1-k1a12fm12πb2ej1/2a2/b2m22-2/b2m2ω2-k2a2+d2/b2ω2-k2a22dmej1/2d2/b22ω2-k2a2k2a2+k2a22ej1/2a2/b2k22ej-1/22k2ω2/b2.Applying the definition of the QLCT (17), the above expression can be rewritten in the form(52)LA1,A2Hτkfω=ei1/2d1/b12ω1-k1a1k1a1+k1a12ei1/2a1/b1k12ei-1/22k1ω1/b1LA1,A2Hfω1-a1k1,ω2-a2k2ej1/2d2/b22ω1-k2a2k2a2+k2a22ej1/2a2/b2k22ej-1/22k2ω2/b2.We notice that(53)ei1/2d1/b12ω1-k1a1k1a1+k1a12ei1/2a1/b1k12ei-1/22k1ω1/b1=eik1ω1d1a1/b1-1/b1e-i1/2k12a1d1a1/b1-1/b1,ej1/2d2/b22ω2-k2a2k2a2+k2a22ej1/2a2/b2k22ej-1/22k2ω2/b2=ejk2ω2d2a2/b2-1/b2e-j1/2k22a2d2a2/b2-1/b2.Because aidi-bici=1, then diai/bi-1/bi=ci for i=1,2. It means that we get(54)ei1/2d1/b12ω1-k1a1k1a1+k1a12ei1/2a1/b1k12ei-1/22k1ω1/b1=eik1ω1c1e-ia1k12/2c1.By the above equalities, we finally arrive at(55)LA1,A2Hτkfω=eik1ω1c1e-ia1k12/2c1LA1,A2Hfω1-a1k1,ω2-a2k2ejk2ω2c2e-ja2k22/2c2.This completes the proof of theorem.Next, we are concerned with the behavior of the QLCT under modulation.Theorem 12 (modulation property).
LetMω0f be modulation operator defined by Mω0f(x)=eix1u0f(x)ejx2v0 with ω0=u0e1+v0e2. Then(56)LA1,A2HMω0fω=LA1,A2Heix1u0fxejx2v0ω=e-ib1d1u02/2+id1u0ω1LA1,A2Hfω1-u0b1,ω2-v0b2e-jb2d2v02/2+jd2v0ω2.Proof.
From Definition4, it follows that(57)LA1,A2HMω0fω=∫R212πb1ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2eix1u0fxejx2v012πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12+2u0x1-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22+2v0x2-π/2dx=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1-u0b1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2-v0b2+d2/b2ω22-π/2dx.Subsequent calculations reveal that(58)LA1,A2HMω0fω=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1-u0b1+d1/b1ω1-u0b1+u0b12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2-v0b2+d2/b2ω2-v0b2+v0b22-π/2dx=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1-u0b1+d1/b1ω1-u0b12+2ω1-u0b1u0b1+u02b12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2-v0b2+d2/b2ω2-v0b22+2ω2-v0b2v0b2+v02b22-π/2dx.Hence,(59)LA1,A2HMω0fω=eiω1-u0b1u0d1+id1u02b1/2LA1,A2Hω1-u0b1,ω2-v0b2ejω2-v0b2v0d2+jd2v02b2/2=eiω1u0d1e-i2b1d1u02-b1d1u02/2LA1,A2Hω1-u0b1,ω2-v0b2ejω2v0d2e-j2b2d2v02-b2d2v02/2=eiω1u0d1e-ib1d1u02/2LA1,A2Hω1-u0b1,ω2-v0b2ejω2v0d2e-jb2d2v02/2.This is desired result.Theorem 13 (time-frequency shift).
If quaternion functionf∈L2(R2;H), then one gets(60)LA1,A2HMω0τkfω=LA1,A2Heix1u0fx-kejx2v0ω=e-ia1c1k12+b1d1u02/2+ic1k1+d1u0ω1-ib1c1k1u0LA1,A2Hfω1-a1k1-u0b1,ω2-a2k2-v0b2e-ja2c2k22+b2d2v02/2+jc2k2+d2v0ω2-jb2c2k2v0.Proof.
The proof directly follows from two previous theorems.The above properties of the QLCT are summarized in Table1.Table 1
Properties of the QLCT off,g∈L2(R2;H), where α,β,α′,β′∈H are constants and ω0=u0e1+v0e2∈R2.
Property
Quaternion func.
QLCT
Left linearity
α
f
+
β
g
α
L
A
1
,
A
2
H
f
ω
+
β
L
A
1
,
A
2
H
g
ω
Right linearity
f
α
′
+
g
β
′
L
A
1
,
A
2
H
f
ω
α
′
+
L
A
1
,
A
2
H
g
ω
β
′
Shift
f
x
-
k
e
-
i
a
1
c
1
k
1
2
/
2
+
i
c
1
k
1
ω
1
L
A
1
,
A
2
H
f
ω
1
-
a
1
k
1
,
ω
2
-
a
2
k
2
e
-
j
a
2
c
2
k
2
2
/
2
+
j
c
2
k
2
ω
2
Modulation
e
i
x
1
u
0
f
x
e
j
x
2
v
0
e
-
i
b
1
d
1
u
0
2
/
2
+
i
d
1
u
0
ω
1
L
A
1
,
A
2
H
f
ω
1
-
u
0
b
1
,
ω
2
-
v
0
b
2
e
-
j
b
2
d
2
v
0
2
/
2
+
j
d
2
v
0
ω
2
Time-frequency
e
i
x
1
u
0
f
x
-
k
e
j
x
2
v
0
e
-
i
a
1
c
1
k
1
2
+
b
1
d
1
u
0
2
/
2
+
i
(
c
1
k
1
+
d
1
u
0
)
ω
1
-
i
b
1
c
1
k
1
u
0
L
A
1
,
A
2
H
f
ω
1
-
a
1
k
1
-
u
0
b
1
,
ω
2
-
a
2
k
2
-
v
0
b
2 ⋅e-ja2c2k22+b2d2v02/2+jc2k2+d2v0ω2-jb2c2k2v0
Gaussian function
e
-
k
1
x
1
2
+
k
2
x
2
2
1
a
1
+
2
k
1
b
1
i
e
ω
1
2
/
2
c
1
+
2
k
1
i
d
1
/
2
k
1
b
1
-
i
a
1
1
a
2
+
2
k
2
b
2
j
e
ω
2
2
/
2
c
2
+
2
k
2
j
d
2
/
(
2
k
2
b
2
-
j
a
2
)
1
a
2
+
2
k
2
b
2
j
X
∑
X
X


## 3.1. Asymptotic Behavior of the QLCT

Like the classical Fourier transform, the Riemann-Lebesgue lemma is also valid for the QLCT, expressed as follows.Theorem 9 (Riemann-Lebesgue lemma).
Suppose thatf∈L1(R2;H). Then(37)limω1→∞⁡LA1,A2Hfω=0,limω2→∞⁡LA1,A2Hfω=0.Proof.
It is not difficult to see that(38)e-iω1x1/b1=-e-iω1/b1x1+b1π/ω1,e-jω2x2/b2=-e-jω2/b2x2+b2π/ω2.Now applying (38) gives(39)LA1,A2Hfω=∫R212πb1ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx=-∫R212πb1ei1/2a1/b1x12-2/b1x1+b1π/ω1ω1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx.Therefore, by making the change of variable x1+b1π/ω1=t1 in the above identity, we immediately obtain(40)LA1,A2Hfω=-∫R212πb1ei1/2a1/b1t1-b1π/ω12-2/b1t1ω1+d1/b1ω12-π/2ft1-b1πω1,t212πb2ej1/2a2/b2t22-2/b1t2ω1+d1/b2ω22-π/2dt=12∫R212πb1ei1/2a1/b1t12-2/b1t1ω1+d1/b1ω12-π/2ft12πb2ej1/2a2/b2t22-2/b2t2ω2+d2/b2ω22-π/2dx-∫R212πb1ei1/2a1/b1t1-b1π/ω12-2/b1t1ω1+d1/b1ω12-π/2ft1-b1πω1,t212πb2ej1/2a2/b2t22-2/b2t2ω2+d2/b2ω22-π/2dt=12∫R212πb1ei1/2a1/b1t12-2/b1t1ω1+d1/b1ω12-π/2ft-eia1/2b1-2t1b1π/ω1+b1π/ω12ft1-b1πω1,t212πb2ej1/2a2/b2t22-2/b2t2ω2+d2s/b2ω22-π/2dt.This means that(41)limω1→∞⁡LA1,A2Hfω≤14πb1b2limω1→∞⁡∫R2ft-eia1/2b1-2t1b1π/ω1+b1π/ω12ft1-b1πω1,t2dt=0.Similarly we can prove(42)limω2→∞⁡LA1,A2Hfω=0.Theorem 10 (continuity).
Iff∈L1(R2;H), then LA1,A2H{f}(ω) is continuous on R2.Proof.
Simple computations show that(43)LA1,A2Hfω+h-LA1,A2Hfω=∫R212πb1ei1/2a1/b1x1+h12-2/b1x1+h1ω1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x2+h22-2/b2x2+h2ω2+d2/b2ω22-π/2dx-12πb1ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx≤∫R212πb1ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2ei1/2a1/b12x1h1+h12-2/b1ω1h1fxej1/2a2/b22x2h2+h22-2/b2ω2h2-fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx=12πb1b2∫R2ei1/2a1/b12x1h1+h12-2/b1ω1h1fxej1/2a2/b22x2h2+h22-2/b2ω2h2-fxdx≤1πb1b2∫R2fxdx.By the Lebesgue dominated convergence theorem, we may conclude that(44)LA1,A2Hfω+h-LA1,A2Hfω⟶0when h→0. This proves that LA1,A2H{f}(ω) is continuous on R2. Again since (43) is independent of ω, LA1,A2H{f}(ω) is, in fact, uniformly continuous on R2.


## 3.2. Useful Properties of the QLCT

Due to the noncommutativity of the kernel of the QLCT, we only have a left linearity property with specific constants(45)α,β∈q∣q=q0+iq1,q0,q1∈R,which is(46)LA1,A2Hαf+βgω=αLA1,A2Hfω+βLA1,A2Hgω,and a right linearity property with specific constants(47)α′,β′∈q∣q=q0+jq2,q0,q2∈R.Theorem 11 (shift property).
Given a quaternion functionf∈L2(R2;H), let τkf(x) denote the shifted (translated) function defined by τkf(x)=f(x-k), where k∈R2. Then one gets(48)LA1,A2Hτkfω=e-ia1c1k12/2+ic1k1ω1LA1,A2Hfω1-a1k1,ω2-a2k2e-ja2c2k22/2+jc2k2ω2.Proof.
Taking into account the definition of QLCT (17), we get(49)LA1,A2Hτkfω=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2fx-k12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx.By making the change of a variable x-k=m, we easily obtain(50)LA1,A2Hτkfω=∫R212πb1ei1/2a1/b1m1+k12-2/b1m1+k1ω1+d1/b1ω12-π/2fm12πb2ej1/2a2/b2m2+k22-2/b2m2+k2ω2+d2/b2ω22-π/2dm=12πb1∫R2ei1/2a1/b1m12-2/b1m1ω1-k1a1+d1/b1ω12-π/2ei1/2a1/b1k12ei-1/22k1ω1/b1fm12πb2ej1/2a2/b2m22-2/b2m2ω2-k2a2+d2/b2ω22-π/2ej1/2a2/b2k22ej-1/22k2ω2/b2dm.Therefore, we further get(51)LA1,A2Hτkfω=12πb1∫R2ei1/2a1/b1m12-2/b1m1ω1-k1a1+d1/b1ω1-k1a1+k1a12-π/2ei1/2a1/b1k12ei-1/22k1ω1/b1fm2πb2ej1/2a2/b2m22-2/b2m2ω2-k2a2+d2/b2ω2-k2a2+k2a22-π/2ej1/2a1/b1k12ej-1/22k2ω2/b2dm=ei1/2d1/b12ω1-k1a1k1a1+k1a12ei1/2a1/b1k12ei-1/22k1ω1/b1∫R212πb1ei1/2a1/b1m12-2/b1m1ω1-k1a1+d1/b1ω1-k1a12fm12πb2ej1/2a2/b2m22-2/b2m2ω2-k2a2+d2/b2ω2-k2a22dmej1/2d2/b22ω2-k2a2k2a2+k2a22ej1/2a2/b2k22ej-1/22k2ω2/b2.Applying the definition of the QLCT (17), the above expression can be rewritten in the form(52)LA1,A2Hτkfω=ei1/2d1/b12ω1-k1a1k1a1+k1a12ei1/2a1/b1k12ei-1/22k1ω1/b1LA1,A2Hfω1-a1k1,ω2-a2k2ej1/2d2/b22ω1-k2a2k2a2+k2a22ej1/2a2/b2k22ej-1/22k2ω2/b2.We notice that(53)ei1/2d1/b12ω1-k1a1k1a1+k1a12ei1/2a1/b1k12ei-1/22k1ω1/b1=eik1ω1d1a1/b1-1/b1e-i1/2k12a1d1a1/b1-1/b1,ej1/2d2/b22ω2-k2a2k2a2+k2a22ej1/2a2/b2k22ej-1/22k2ω2/b2=ejk2ω2d2a2/b2-1/b2e-j1/2k22a2d2a2/b2-1/b2.Because aidi-bici=1, then diai/bi-1/bi=ci for i=1,2. It means that we get(54)ei1/2d1/b12ω1-k1a1k1a1+k1a12ei1/2a1/b1k12ei-1/22k1ω1/b1=eik1ω1c1e-ia1k12/2c1.By the above equalities, we finally arrive at(55)LA1,A2Hτkfω=eik1ω1c1e-ia1k12/2c1LA1,A2Hfω1-a1k1,ω2-a2k2ejk2ω2c2e-ja2k22/2c2.This completes the proof of theorem.Next, we are concerned with the behavior of the QLCT under modulation.Theorem 12 (modulation property).
LetMω0f be modulation operator defined by Mω0f(x)=eix1u0f(x)ejx2v0 with ω0=u0e1+v0e2. Then(56)LA1,A2HMω0fω=LA1,A2Heix1u0fxejx2v0ω=e-ib1d1u02/2+id1u0ω1LA1,A2Hfω1-u0b1,ω2-v0b2e-jb2d2v02/2+jd2v0ω2.Proof.
From Definition4, it follows that(57)LA1,A2HMω0fω=∫R212πb1ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12-π/2eix1u0fxejx2v012πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22-π/2dx=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1+d1/b1ω12+2u0x1-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2+d2/b2ω22+2v0x2-π/2dx=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1-u0b1+d1/b1ω12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2-v0b2+d2/b2ω22-π/2dx.Subsequent calculations reveal that(58)LA1,A2HMω0fω=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1-u0b1+d1/b1ω1-u0b1+u0b12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2-v0b2+d2/b2ω2-v0b2+v0b22-π/2dx=12πb1∫R2ei1/2a1/b1x12-2/b1x1ω1-u0b1+d1/b1ω1-u0b12+2ω1-u0b1u0b1+u02b12-π/2fx12πb2ej1/2a2/b2x22-2/b2x2ω2-v0b2+d2/b2ω2-v0b22+2ω2-v0b2v0b2+v02b22-π/2dx.Hence,(59)LA1,A2HMω0fω=eiω1-u0b1u0d1+id1u02b1/2LA1,A2Hω1-u0b1,ω2-v0b2ejω2-v0b2v0d2+jd2v02b2/2=eiω1u0d1e-i2b1d1u02-b1d1u02/2LA1,A2Hω1-u0b1,ω2-v0b2ejω2v0d2e-j2b2d2v02-b2d2v02/2=eiω1u0d1e-ib1d1u02/2LA1,A2Hω1-u0b1,ω2-v0b2ejω2v0d2e-jb2d2v02/2.This is desired result.Theorem 13 (time-frequency shift).
If quaternion functionf∈L2(R2;H), then one gets(60)LA1,A2HMω0τkfω=LA1,A2Heix1u0fx-kejx2v0ω=e-ia1c1k12+b1d1u02/2+ic1k1+d1u0ω1-ib1c1k1u0LA1,A2Hfω1-a1k1-u0b1,ω2-a2k2-v0b2e-ja2c2k22+b2d2v02/2+jc2k2+d2v0ω2-jb2c2k2v0.Proof.
The proof directly follows from two previous theorems.The above properties of the QLCT are summarized in Table1.Table 1
Properties of the QLCT off,g∈L2(R2;H), where α,β,α′,β′∈H are constants and ω0=u0e1+v0e2∈R2.
Property
Quaternion func.
QLCT
Left linearity
α
f
+
β
g
α
L
A
1
,
A
2
H
f
ω
+
β
L
A
1
,
A
2
H
g
ω
Right linearity
f
α
′
+
g
β
′
L
A
1
,
A
2
H
f
ω
α
′
+
L
A
1
,
A
2
H
g
ω
β
′
Shift
f
x
-
k
e
-
i
a
1
c
1
k
1
2
/
2
+
i
c
1
k
1
ω
1
L
A
1
,
A
2
H
f
ω
1
-
a
1
k
1
,
ω
2
-
a
2
k
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## 4. Heisenberg Uncertainty Principle for QLCT

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot be sharply localized simultaneously. In quantum mechanics, the uncertainty principle asserts that one cannot at the same time be certain of the position and of the velocity of an electron (or any particle) [20]. Let us now give an alternative proof of the Heisenberg type uncertainty principle for the QLCT, which is recently studied in [8] (the uncertainty principle of the QCT was proved using the exponential form of a 2D quaternion function and proposed proof of this paper uses the relationship between the QFT and QLCT). However, before proceeding with the statement of this main result, we need to introduce the component-wise uncertainty principle for the QFT as follows (see [12] for more details).Theorem 14 (the QFT component-wise uncertainty principle).
Suppose thatf∈L1(R2;H)∩L2(R2;H). If ∂f/∂xk and ωk∂f/∂xk∈L2(R2;H), then one has(61)∫R2xk2fx2dx∫R2ωk2Fqfω2dω≥14∫R2fx2dx2,k=1,2.The generalization of the above uncertainty principle to the the QLCT domain is given by the following theorem (for more detailed information, see [8]).Theorem 15 (the QLFT component-wise uncertainty principle).
Assume thatf∈L1(R2;H)∩L2(R2;H), ∂f/∂xk∈L2(R2;H) and that LA1,A2H{f}, ωkLA1,A2H{f}∈L2(R2;H),  k=1,2. Then, the following inequality holds:(62)∫R2xk2fx2dx∫R2ωl2LA1,A2Hfω2dω≥bk24∫R2fx2dx2,k=1,2.Proof.
Substituting the quaternion functionf by gf defined by (21) on both sides of (62), we easily obtain(63)∫R2xk2gfx2dx∫R2ωk2Fqgfω2dω≥14∫R2gfx2dx2.Now setting ω=ω/b, we further have(64)∫R2xk2e-iπ/4b1f~xe-jπ/4b22dx∫R2ωk2bk2Fqgfωb2dωb≥14∫R2e-iπ/4b1f~xe-iπ/4b22dx2,and thus(65)∫R2xk2b1b22f~x2dx∫R2ωk2bk2Fqgfωb2dω≥14b1b22∫R2f~x2dx2.Hence,(66)∫R2xk2b1b22eia1/2b1x12fxeja2/2b2x222dx∫R2ωk2bk2Fqgfωb2dω≥14b1b22∫R2eia1/2b1x12fxeja2/2b2x222dx2.By inserting (23) into (66), we immediately obtain(67)∫R2xk2b1b22eia1/2b1x12fxeja2/2b2x222dx∫R2ωk2bk2e-id1/2b1ω12LA1,A2Hfωe-jd2/2b2ω222dω≥14b1b22∫R2eia1/2b1x12fxeja2/2b2x222dx2.Simplifying it gives(68)∫R2xk2fx2dx∫R2ωk2LA1,A2Hfω2dω≥bk24∫R2fx2dx2.This finishes the proof of theorem.It is not difficult to check that directional uncertainty principle for the QFT takes the following form (cf. [21, 22]).Theorem 16.
Suppose thatf∈L1(R2;H)∩L2(R2;H). If ∂f/∂xk and ωk∂f/∂xk∈L2(R2;H), then one has(69)∫R2x2fx2dx∫R2ω2Fqfω2dω≥∫R2fx2dx2.Proceeding as in the proof of Theorem15, we obtain the QLCT directional uncertainty principle as follows.Theorem 17.
Suppose thatf∈L1(R2;H)∩L2(R2;H) and LA1,A2H{f} and ω2LA1,A2H{f}∈L2(R2;H). Then the following inequality is satisfied:(70)∫R2x2fx2dx∫R2ω2LA1,A2Hfω2dω≥b2∫R2fx2dx2.



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*Source: 5874930-2016-01-06.xml*