Hypercomplex Signal Energy Concentration in the Spatial and Quaternionic Linear Canonical Frequency Domains
HHypercomplex Signal Energy Concentration in theSpatial and Quaternionic Linear Canonical FrequencyDomains
Cuiming Zou a , Kit Ian Kou b, ∗ a Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa,Macao, China. Email: [email protected] b Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa,Macao, China.
Abstract
Quaternionic Linear Canonical Transforms (QLCTs) are a family of integral trans-forms, which generalized the quaternionic Fourier transform and quaternionicfractional Fourier transform. In this paper, we extend the energy concentrationproblem for 2D hypercomplex signals (especially quaternionic signals). The mostenergy concentrated signals both in 2D spatial and quaternionic linear canonicalfrequency domains simultaneously are recently recognized to be the quaternionicprolate spheroidal wave functions (QPSWFs). The improved definitions of QP-SWFs are studied which gave reasonable properties. The purpose of this paperis to understand the measurements of energy concentration in the 2D spatial andquaternionic linear canonical frequency domains. Examples of energy concen-trated ratios between the truncated Gaussian function and QPSWFs intuitivelyillustrate that QPSWFs are more energy concentrated signals.
Keywords:
Quaternionic linear canonical transforms, energy concentration,quaternionic Fourier transform, quaternionic prolate spheroidal wave functions. ∗ Corresponding author
Email address: [email protected] (Kit Ian Kou)
Preprint submitted to Elsevier October 5, 2018 a r X i v : . [ m a t h . C A ] S e p . Introduction The energy concentration problem in the time-frequency domain plays a cru-cial role in signal processing. The foundation of this problem comes from 1960sthe research group of bell labs [1]. The problem states that for any given signal f with its Fourier transform (FT) F ( f )( ω ) : = √ π (cid:90) ∞−∞ f ( t ) e − i ω t dt , (1.1)the energy ratios of the duration and bandwidth limiting of the signal f , i.e., α f : = (cid:82) τ − τ | f ( t ) | dt (cid:82) ∞−∞ | f ( t ) | dt and β f : = (cid:82) σ − σ |F ( f )( ω ) | d ω (cid:82) ∞−∞ |F ( f )( ω ) | d ω of f ( t ) both in fixed time [ − τ, τ ] and frequency[ − σ, σ ] domains, satisfy the following inequalityarccos α f + arccos β f ≥ arccos (cid:112) λ . (1.2)Let E f : = (cid:82) ∞−∞ | f ( t ) | dt be the total energy of f . By the Parseval theorem [2], theenergy in time and frequency domains are equal, i.e., E f = E F ( f ) . Without loss ofgenerality, we consider the unit energy signals throughout this paper, i.e., E f = λ in Eq. (1.2) is the eigenvalue of the zero order pro-late spheroidal wave functions (PSWFs). The PSWFs are originally used to solvethe Helmhotz equation in prolate spheroidal coordinates by means of separation ofvariables [3, 4]. In 1960s, Slepian et al. [5, 6, 7] found that PSWFs are solutionsfor the energy concentration problem of bandlimited signals [2]. Their real-valuedPSWFs are solutions of the integral equation (cid:90) τ − τ f ( x ) sin σ ( x − y ) π ( x − y ) dx = α f ( y ) , (1.3)where α are eigenvalues of PSWFs. Here [ − τ, τ ] and [ − σ, σ ] are the fixed timeand frequency domains, respectively. Important properties of PSWFs are givenin [5, 8, 9, 10, 11]. The following properties follow form the general theory ofintegral equations and are stated without proof.1. Eq.(1.3) has solutions only for real, positive values eigenvalues α n . Thesevalues is a monotonically decreasing sequence, 1 > α > α > ... > α n >... >
0, such that lim n →∞ α n = α n there corresponds only one eigenfunction ψ n ( x ) with a constantfactor. The functions { ψ n ( x ) } ∞ n = form a real orthonormal set in L ([ − τ, τ ]; R ).2. An arbitrary real σ -bandlimited function f ( x ) can be written as a sum f ( x ) = Σ ∞ n = a n ψ n ( x ) , for all x ∈ R , where a n : = (cid:82) R f ( x ) ψ n ( x ) dx .These properties are useful in solving the energy concentration problem and otherapplications [12, 13, 14, 15]. Slepian et al. [8] naturally extended them to higherdimension and discussed their approximation in some special case in the followingyears. After that, the works on this functions are slowly developed until 1980sa large number of engineering applied this functions to signal processing, suchas bandlimited signals extrapolation, filter designing, reconstruction and so on[16, 17, 18].The PSWFs have received intensive attention in recent years. There are manye ff orts to extend this kind of functions to various types of integral transformations.Pei et al. [12, 19] generalized PSWFs associated with the finite fractional Fouriertransform (FrFT) and applied to the sampling theory. Zayed et al. [20, 21] gen-eralized PSWFs not only associated with the finite FrFT but also associated withthe linear canonical transforms (LCTs) and applied to sampling theory. Zhao etal. [13, 22] discussed the PSWFs associated with LCTs in detail and presented themaximally concentrated sequence in both time and LCTs-frequency domains. Thewavelets based PSWFs constructed by Walter et al. [14, 15, 23] have some desir-able properties lacking in other wavelet systems. Kou et al. [24] developed thePSWFs with noncommutative structures in Cli ff ord algebra. They not only gen-eralized the PSWFs in Cli ff ord space (CPSWFs), but also extended the transformto Cli ff ord LCT. But they just gave some basic properties of this functions andhave not discussed details of the energy relationship for square integrable signals.In this paper, we consider the energy concentration problem for hypercomplexsignals, especially for quaternionic signals [25, 26] associated with quaternionicLCTs (QLCTs) in detail. The improvement definition of QPSWFs are consideredfor odd and even quaternionic signals. The study is a great improvement on theone appeared in [24].The QLCT is a generalization of the quaternionic FT (QFT) and quaternionicFrFT (QFrFT). The QFT and QFrFT are widely used for color image processingand signal analysis in these years [27, 28, 29, 30]. Therefore, it has more degreesof freedom than QFT and QFrFT, the performance will be more advanced in colorimage processing.In the present paper, we generalize the 1D PSWFs under the QLCTs to thequaternion space, which are referred to as quaternionic PSWFs (QPSWFs). The3mproved definition of QPSWFs associated with the QLCTs is studied and theirsome important properties are analyzed. In order to find the relationship of ( α f , β f )for any square integrable quaternionic signal, we show that the Parseval theoremand studied the energy concentration problem associated with the QLCTs. In par-ticularly, we utilize the quaternion-valued functions multiply two special chirpsignals on both sides as a bridge between the QLCTs and the QFTs. The maingoal of the present study is to develop the energy concentration problem asso-ciated with QLCTs. We find that the proposed QPSWFs are the most energyconcentrated quaternionic signals.The body of the present paper is organized as follows. In Section 2 and 3, somebasic facts of quaternionic algebra and QLCTs are given. Moreover, the Parsevalidentity for quaternionic signals associated with the (two-sided) QLCTs are pre-sented. In Section 4, the improved definition and some properties of QPSWFsassociated with QLCTs are discussed. The Section 5 presents the main results,it includes two parts. In subsection 5.1, we introduce the existence theorem forthe maximum energy concentrated bandlimited function on a fixed spatial domainassociated with the QLCTs. In subsection 5.2, we discuss the energy extremalproperties in fixed spatial and QLCTs-frequency domains for any quaternionicsignal. In particular, we give an inequality to present the relationship of energyratios for any quaternionic signal, which is analogue to the high dimensional realsignals. Moreover, examples of energy concentrated ratios between the truncatedGaussian function and QPSWFs are presented, which can intuitively illustrate thatQPSWFs are the more energy concentrated signals. Finally, some conclusion aredrawn in Section 6.
2. Quaternionic Algebra
The present section collects some basic facts about quaternions [31, 32], whichwill be needed throughout the paper.For all what follows, let H be the Hamiltonian skew field of quaternions : H : = { q = q + i q + j q + k q | q , q , q , q ∈ R } , (2.4)which is an associative non-commutative four-dimensional algebra. The basiselements { i , j , k } obey the Hamilton’s multiplication rules: i = j = k = − ij = − ji = k , jk = − kj = i , ki = − ik = j , and the usual component-wise defined addition. In this way the quaternionic al-gebra arises as a natural extension of the complex field C .4he quaternion conjugate of a quaternion q is defined by q : = q − i q − j q − k q , q , q , q , q ∈ R . We write Sc ( q ) : = ( q + q ) = q and Vec ( q ) : = ( q − q ) = i q + j q + k q , whichare the scalar and vector parts of q , respectively. This leads to a norm of q ∈ H defined by | q | : = (cid:112) qq = (cid:112) qq = ( q + q + q + q ) . Then we have pq = q · p , | q | = | q | , | pq | = | p || q | , for any p , q ∈ H . By (2.4), aquaternion-valued function or, briefly, an H -valued function f : R → H can beexpressed in the following form: f ( x , y ) = f ( x , y ) + i f ( x , y ) + j f ( x , y ) + k f ( x , y ) , where f i : R → R ( i = , , , f in the following symmetric form [33]: f ( x , y ) = f ( x , y ) + i f ( x , y ) + f ( x , y ) j + i f ( x , y ) j . (2.5)Properties (like integrability, continuity or di ff erentiability) that are ascribed to f have to be fulfilled by all components f i ( i = , , , L p ( R ; H ) (1 ≤ p < ∞ ) consist of all H -valued functions in R under leftmultiplication by quaternions, whose p -th power is Lebesgue integrable in R : L p ( R ; H ) : = f (cid:12)(cid:12)(cid:12)(cid:12) f : R → H , (cid:107) f (cid:107) L p ( R ; H ) : = (cid:32)(cid:90) R | f ( x , y ) | p dxdy (cid:33) / p < ∞ . In this work, the left quaternionic inner product of f , g ∈ L ( R ; H ) is defined by < f , g > L ( R ; H ) : = (cid:90) R f ( x , y ) g ( x , y ) dxdy . (2.6)The reader should note that the norm induced by the inner product (2.6), (cid:107) f (cid:107) = (cid:107) f (cid:107) L ( R ; H ) : = < f , f > L ( R ; H ) = (cid:90) R | f ( x , y ) | dxdy . coincides with the L -norm for f , considered as a vector-valued function.The angle between two non-zero functions f , g ∈ L ( R ; H ) is defined byarg( f , g ) : = arccos (cid:32) Sc ( < f , g > ) (cid:107) f (cid:107) (cid:107) g (cid:107) (cid:33) . (2.7)The superimposed argument is well-defined since, obviously, it holds | Sc ( < f , g > ) | ≤ | < f , g > L ( Ω ; H ) | ≤ (cid:107) f (cid:107) (cid:107) g (cid:107) . . The Quaternionic Linear Canonical Transforms (QLCTs) The LCT was first proposed by Moshinsky and Collins [34, 35] in the 1970s.It is a linear integral transform, which includes many special cases, such as theFourier transform (FT), the FrFT, the Fresnel transform, the Lorentz transformand scaling operations. In a way, the LCT has more degrees of freedom and ismore flexible than the FT and the FrFT, but with similar computational costs asthe conventional FT. Due to the mentioned advantages, it is of natural interestto extend the LCT to a quaternionic algebra framework. These extensions leadto the
Quaternionic Linear Canonical Transforms (QLCTs). Due to the non-commutative property of multiplication of quaternions, there are di ff erent typesof QLCTs. As explained in more detail below, we restrict our attention to the two-sided QLCTs [36, 37] of 2D quaternionic signals in this paper.
Definition 3.1 (Two-sided QLCTs)
Let A i = (cid:32) a i b i c i d i (cid:33) ∈ R × be a matrix pa-rameter such that det( A i ) = , for i = , . The two-sided QLCTs of signals f ∈ L (cid:84) L ( R ; H ) are given by L ( f )( u , v ) : = (cid:90) R K i A ( x , u ) f ( x , y ) K j A ( y , v ) dxdy , (3.8) where the kernel functions are formulated byK i A ( x , u ) : = √ i π b e i (cid:18) a b x − b xu + d b u (cid:19) , for b (cid:44) , √ d e i ( c d ) u , for b = , (3.9) and K j A ( y , v ) : = √ j π b e j (cid:18) a b y − b yv + d b v (cid:19) , for b (cid:44) , √ d e j ( c d ) v , for b = . (3.10)It is significant to note that when A = A = (cid:32) − (cid:33) , the QLCT of f reduces to √ i π F ( f )( u , v ) √ j π , where F ( f )( u , v ) : = (cid:90) R e − i xu f ( x , y ) e − j yv dxdy (3.11)6s the two-sided QFT of f . Note that when b i = i = , b i > i = ,
2) throughoutthe paper.
Remark 3.1
Let b , b (cid:44) . Using the Euler formula for the quaternionic linearcanonical kernel we can rewrite Eq. (3.8) in the following form: L ( f )( u , v ) = − i √ i π √ b b ( P + i P + P j + i P j ) ( − j (cid:112) j ) , whereP : = (cid:90) R f ( x , y ) cos (cid:32) a b x − b xu + d b u (cid:33) cos (cid:32) a b y − b yv + d b v (cid:33) dxdy , P : = (cid:90) R f ( x , y ) sin (cid:32) a b x − b xu + d b u (cid:33) cos (cid:32) a b y − b yv + d b v (cid:33) dxdy , P : = (cid:90) R f ( x , y ) cos (cid:32) a b x − b xu + d b u (cid:33) sin (cid:32) a b y − b yv + d b v (cid:33) dxdy , P : = (cid:90) R f ( x , y ) sin (cid:32) a b x − b xu + d b u (cid:33) sin (cid:32) a b y − b yv + d b v (cid:33) dxdy . The above equation clearly shows how the QLCTs separate real signals f ( x , y ) into four quaternionic components, i.e., the even-even, odd-even, even-odd andodd-odd components of f ( x , y ) . From Eq. (3.8) if f ∈ L (cid:84) L ( R ; H ), then the two-sided QLCTs L ( f )( u , v )has a symmetric representation L ( f )( u , v ) = L ( f )( u , v ) + L ( f )( u , v ) i + L ( f )( u , v ) j + i L ( f )( u , v ) j , where L ( f i ) ( i = , , ,
3) are the QLCTs of f i and they are H -valued functions.Under suitable conditions, the inversion of two-sided quaternionic linear canon-ical transforms of f ( u , v ) can be defined as follows. Definition 3.2 (Inversion QLCTs)
Suppose that f ∈ L (cid:84) L ( R , H ) . Then theinversion of two-sided QLCTs of f ( u , v ) are defined by L − ( f )( x , y ) : = (cid:90) R K i A − ( x , u ) f ( u , v ) K j A − ( y , v ) dudv , (3.12) where A − i = (cid:32) d i − b i − c i a i (cid:33) and det( A − i ) = for i = , . Note that the QLCTs of f multiple the chirp signals 2 π √ b i e − i d b u on the leftand e − j d b v (cid:112) b j on the right can be regarded as the QFT on the scale domain.Since 2 π (cid:112) b i e − i d b u L ( f )( u , v ) e − j d b v (cid:112) b j (3.13) = (cid:90) R e − i b xu (cid:18) e i a b x f ( x , y ) e j a b y (cid:19) e − j b yv dxdy = F ( ˜ f ) (cid:32) ub , vb (cid:33) , where ˜ f ( x , y ) : = e i a b x f ( x , y ) e j a b y is related to the parameter matrix A i , i = , Lemma 3.1 (Relation Between QLCT and QFT)
Let A i = (cid:32) a i b i c i d i (cid:33) ∈ R × be a real matrix parameter such that det( A i ) = for i = , . The relationshipbetween two-sided QLCTs and QFTs of f ∈ L (cid:84) L ( R ; H ) are given by F ( ˜ f ) (cid:32) ub , vb (cid:33) = π (cid:112) b i e − i d b u L ( f )( u , v ) e − j d b v (cid:112) b j , (3.14) where ˜ f ( x , y ) = e i a b x f ( x , y ) e j a b y .3.3. Energy Theorem Associated with QLCTs This subsection describes energy theorem of two-sided QLCTs [38], whichwill be applied to derive the extremal properties of QLCTs in Section 5.
Theorem 3.1 (Energy Theorem of the QLCTs)
Any 2D H -valued function f ∈ L ( R , H ) and its QLCT L ( f ) are related by the Parseval identity (cid:107) f (cid:107) = (cid:107)L ( f ) (cid:107) . (3.15) Proof.
For f ∈ L ( R , H ), direct computation shows that ||L ( f ) || = (cid:90) R L ( f )( u , v ) L ( f )( u , v ) dudv = Sc (cid:34)(cid:90) R L ( f )( u , v ) L ( f )( u , v ) dudv (cid:35) . ||L ( f ) || = Sc (cid:34)(cid:90) R (cid:32)(cid:90) R K i A ( x , u ) f ( x , y ) K j A ( y , v ) dxdy (cid:33) L ( f )( u , v ) dudv (cid:35) = Sc (cid:34)(cid:90) R K i A ( x , u ) f ( x , y ) K j A ( y , v ) L ( f )( u , v ) dxdydudv (cid:35) = (cid:90) R Sc (cid:104) K i A ( x , u ) f ( x , y ) K j A ( y , v ) L ( f )( u , v ) (cid:105) dxdydudv . With Sc ( qp ) = Sc ( pq ) for any p , q ∈ H and K i A = K i A − , K j A = K j A − , we have ||L ( f ) || = (cid:90) R Sc (cid:104) f ( x , y ) K j A ( y , v ) L ( f )( u , v ) K i A ( x , u ) (cid:105) dxdydudv = (cid:90) R Sc (cid:20) f ( x , y ) K i A − ( x , u ) L ( f )( u , v ) K j A − ( y , v ) (cid:21) dxdydudv = Sc (cid:90) R f ( x , y ) (cid:90) R K i A − ( x , u ) L ( f )( u , v ) K j A − ( y , v ) dudvdxdy = Sc (cid:34)(cid:90) R f ( x , y ) f ( x , y ) dxdy (cid:35) = (cid:90) R f ( x , y ) f ( x , y ) dxdy = (cid:107) f (cid:107) . Hence this completes the proof. (cid:3)
Theorem 3.1 shows that the energy for an H -valued signal in the spatial domainequals to the energy in the QLCTs-frequency domain. The Parseval theorem al-lows the energy of an H -valued signal to be considered on either the spatial do-main or the QLCTs-frequency domain, and exchange the domains for conveniencecomputation. Corollary 3.1
The energy theorem of f and ˜ f ( x , y ) = e i a b x f ( x , y ) e j a b y associ-ated with their QFT is given by (cid:107) f (cid:107) = (cid:107) ˜ f (cid:107) = (cid:107)F ( f ) (cid:107) . (3.16)
4. The Quaternionic Prolate Spheroidal Wave Functions
In the following, we first explicitly present the definition of PSWFs associatedwith QLCTs. 9 .1. Definitions of QPSWFs
Consider the 1D PSWFs [2, 5, 8, 24], let us extend the PSWFs to the quater-nionic space associated with QLCTs.
Definition 4.1 (QPSWFs)
The solutions of the following integral equation in L ( R ; H ) λ n i n − ψ n ( u , v ) j n − : = (cid:90) τ K i A (cid:48) ( x , u ) ψ n ( x , y ) K j A (cid:48) ( y , v ) dxdy , (4.17) are called the quaternionic prolate spheroidal wave functions (QPSWFs) { ψ n ( x , y ) } ∞ n = associated with QLCTs. Here, the complex valued λ n are the eigenvalues corre-sponding to the eigenfunctions ψ n ( x , y ) . The real parameter matrix A (cid:48) i : = (cid:32) ca i b i cc i cd i (cid:33) with a i d i − b i c i = , b i (cid:44) , for i = , . The real constant c is a ratio aboutthe frequency domain σ : = [ − σ, σ ] × [ − σ, σ ] and the spatial domain τ : = [ − τ, τ ] × [ − τ, τ ] , where c : = στ , < c < ∞ . Eq. (4.17) is named the finiteQLCTs form of QPSWFs. Note that for simplicity of presentation, we write (cid:82) τ − τ (cid:82) τ − τ = (cid:82) τ and (cid:82) σ − σ (cid:82) σ − σ = (cid:82) σ . Remark 4.1
The solutions of this integral equation in Eq. (4.17) are well estab-lished in some special cases. (i)
In the square region τ = [ − τ, τ ] × [ − τ, τ ] , if QLCTs are degenerated to 2DFourier transform (FT), then QPSWFs becomes the 2D real PSWFs, whichis given by λ n ψ n ( u , v ) = (cid:90) τ e i cxu ψ n ( x , y ) e i cyv dxdy . Here, if ψ n ( x , y ) is separable, i.e., ψ n ( x , y ) = ψ n ( x ) ψ n ( y ) , then the 2D PSWFscan be regarded as the product of two 1D PSWFs. To aid the reader, see [5]for more complete accounts of this subject. (ii) In a unit disk, the QLCTs are degenerated to 2D FT, then the QPSWFs be-tween the circular PSWFs [25] λ n ψ n ( u , v ) = (cid:90) x + y ≤ e i ( cxu + cyv ) ψ n ( x , y ) dxdy . Remark 4.2
We call the right-hand side of Eq. (4.17) is the finite QLCTs. How-ever, det( A (cid:48) i ) = , i = , only for the c = . There is a scale factor c added to theparameter matrix, which is di ff erent from the definition of QLCTs. .2. Properties of QPSWFs Some important properties of QPSWFs will be considered in this part, whichare crucial in solving the energy concentration problem.
Proposition 4.1 (Low-pass Filtering Form in τ ) Let ψ n ( x , y ) ∈ L ( R ; H ) bethe QPSWFs associated with their QLCTs and ˜ ψ ( x , y ) : = e i ca b x ψ n ( x , y ) e j ca b y .Then { ˜ ψ n ( x , y ) } ∞ n = are solutions of the following integral equation µ n ˜ ψ n ( u , v ) = (cid:90) τ ˜ ψ n ( x , y ) sin σ ( x − u ) π ( x − u ) sin σ ( y − v ) π ( y − v ) dxdy , (4.18) where µ n : = c b b λ n for n = , , · · · are the eigenvalues corresponding to ˜ ψ n ( x , y ) and a i d i − b i c i = , b i (cid:44) , for i = , , and c : = στ , < c < ∞ . Eq. (4.18)is named the low-pass filtering form of QPSWFs associated with QLCTs. Proof.
We shall show that Eq. (4.18) is derived by the Eq. (4.17). Straightforwardcomputations of the right-hand side of Eq. (4.18) show that (cid:90) τ ˜ ψ n ( x , y ) sin σ ( x − u ) π ( x − u ) sin σ ( y − v ) π ( y − v ) dxdy = (cid:90) τ sin σ ( u − x ) π ( u − x ) e i ca b x ψ n ( x , y ) e j ca b y sin σ ( v − y ) π ( v − y ) dxdy . Applying the following two important equations [39] to the last integral,12 π (cid:90) σ − σ e i xu dx = sin( σ u ) π u and 12 π (cid:90) σ − σ e j yv dy = sin( σ v ) π v , (4.19)then we have (cid:90) τ sin σ ( u − x ) π ( u − x ) e i ca b x ψ n ( x , y ) e j ca b y sin σ ( v − y ) π ( v − y ) dxdy = π ) (cid:90) τ (cid:90) σ e i v ( u − x ) e i ca b x ψ n ( x , y ) e j ca b y e j v ( v − y ) dv dv dxdy = π ) (cid:90) σ e i v u (cid:34)(cid:90) τ e − i cxb ( b v c ) e i ca b x ψ n ( x , y ) e j ca b y e − j cyb ( b v c ) dxdy (cid:35) e j v v dv dv . Combining Eq.(4.17) with the parameter matrices A (cid:48) i = (cid:32) ca i b i cc i cd i (cid:33) , i = ,
2, and11 (cid:48) i = − cd i b i b i cc i − ca i b i , i = ,
2, we have1(2 π ) (cid:90) σ e i v u (cid:34)(cid:90) τ e − i cxb ( b v c ) e i ca b x ψ n ( x , y ) e j ca b y e − j cyb ( b v c ) dxdy (cid:35) e j v v dv dv = π (cid:90) σ e i v u (cid:34) λ n e − i cd b ( b v c ) i n (cid:112) cb ψ n (cid:16) b v cb , b v cb (cid:17) (cid:112) cb j n e − j cd b ( b v c ) (cid:35) e j v v dv dv = π λ n (cid:90) σ e i v u (cid:20) e − i cd b ( v c ) i n (cid:112) cb ψ n (cid:16) v c , v c (cid:17) (cid:112) cb j n e − j cd b ( v c ) (cid:21) e j v v dv dv = π λ n c (cid:112) b b i n (cid:34)(cid:90) τ e i w cu e − i cd b w ψ n ( w , w ) e − j cd b w e j w cv dw dw (cid:35) j n = λ n c (cid:112) b b i n λ n e − i c ( − a b
12 )2 b ( ub ) ( − i ) n √ i (cid:112) cb i ψ n (cid:16) ub b , vb b (cid:17) (cid:112) cb j ( − j ) n (cid:112) j e − j c ( − a b
22 )2 b ( vb ) j n = λ n c b b e i ca b u ψ n ( u , v ) e j ca b v = c b b λ n ˜ ψ n ( u , v ) = : µ n ˜ ψ n ( u , v ) . The proof is complete. (cid:3)
Remark 4.3
For the specific parameters a i = , b i = , c i = − , d i = , i = , ,Eq. (4.18) becomes the low-pass form of QPSWFs associated with QFT (cid:90) τ ψ n ( x , y ) sin σ ( x − u ) π ( x − u ) sin σ ( y − v ) π ( y − v ) dxdy = c λ n ψ n ( u , v ) . To obtain the following property, we shall show a special convolution theoremof any H -valued signal and real-valued signal. Lemma 4.1
Let f ∈ L ( R ; H ) and g ∈ L ( R ; R ) associated with their QFT F ( f ) and F ( g ) with f = f + i f + f j + i f j , where f i ∈ L ( R ; R ) , i = , , , and F ( g ) ∈ L ( R ; R ) . The convolution of f and g is defined as ( f ∗ g )( s , t ) : = (cid:90) R f ( x , y ) g ( s − x , t − y ) dxdy . (4.20) Then the QFT for f ∗ g holds F ( f ∗ g ) ( u , v ) (4.21) = F ( f + i f )( u , v ) F ( g )( u , v ) + F ( f j + i f j )( u , v ) F ( g )( − u , v ) . roof. Let s − x = m and t − y = n , straightforward computation the QFT inEq.(3.11) of the convolution between f and g shows that F (cid:32)(cid:90) R f ( x , y ) g ( s − x , t − y ) dxdy (cid:33) ( u , v ) = (cid:90) R e − i su (cid:32)(cid:90) R f ( x , y ) g ( s − x , t − y ) dxdy (cid:33) e − j tv dsdt = (cid:90) R e − i ( x + m ) u (cid:32)(cid:90) R f ( x , y ) g ( m , n ) dxdy (cid:33) e − j ( y + n ) v dmdn = (cid:90) R e − i mu e − i xu (cid:2) ( f ( x , y ) + i f ( x , y )) + ( f ( x , y ) j + i f ( x , y ) j ) (cid:3) g ( m , n ) e − j yv e − j nv dxdydmdn . With e − i j = j e i , the last integral becomes (cid:90) R e − i xu ( f ( x , y ) + i f ( x , y )) e − i mu g ( m , n ) e − j nv e − j yv dxdydmdn + (cid:90) R e − i xu ( f ( x , y ) j + i f ( x , y ) j ) e i mu g ( m , n ) e − j nv e − j yv dxdydmdn = (cid:90) R e − i xu ( f ( x , y ) + i f ( x , y )) F ( g )( u , v ) e − j yv dxdy + (cid:90) R e − i xu ( f ( x , y ) j + i f ( x , y ) j ) F ( g )( − u , v ) e − j yv dxdy . Since we have known F ( g ) is real-valued, then we have (cid:90) R e − i xu ( f ( x , y ) + i f ( x , y )) e − j yv F ( g )( u , v ) dxdy + (cid:90) R e − i xu ( f ( x , y ) j + i f ( x , y ) j ) e − j yv F ( g )( − u , v ) dxdy = F ( f )( u , v ) F ( g )( u , v ) + i F ( f )( u , v ) F ( g )( u , v ) + F ( f )( u , v ) F ( g )( − u , v ) j + i F ( f j )( u , v ) F ( g )( − u , v ) . This completes the proof. (cid:3)
Note that if the real signal g ( x , y ) = g ( − x , y ) with F ( g ) is real valued, then F ( g )( u , v ) = F ( g )( − u , v ). It means that F ( f ∗ g ) ( u , v ) = F ( f + i f )( u , v ) F ( g )( u , v ) + F ( f j + i f j )( u , v ) F ( g )( − u , v ) = F ( f + i f + f j + i f j )( u , v ) F ( g )( u , v ) = F ( f )( u , v ) F ( g )( u , v ) . emark 4.4 The convolution theorems for quaternion Fourier transform was givenin [40]. Lemma 4.1 is following the idea of Theorem 13 and Lemma 14 in [40].For completeness, we proof the convolution formula in Eq. (4.21).
Proposition 4.2
Let ψ n ( x , y ) ∈ L ( R ; H ) be the QPSWFs associated with QLCTsand ˜ ψ n ( x , y ) = e i ca b x ψ n ( x , y ) e j ca b y , { ˜ ψ n ( x , y ) } ∞ n = satisfies ˜ ψ n ( u , v ) = (cid:90) R ˜ ψ n ( x , y ) sin σ ( x − u ) π ( x − u ) sin σ ( y − v ) π ( y − v ) dxdy , (4.22) which extends the integral of ˜ ψ n ( x , y ) from τ to R . Proof.
Let p τ ( x , y ) : = (cid:40) , ( x , y ) ∈ τ , , otherwise , Eq. (4.18) is actually a convolution of p τ ( x , y ) ˜ ψ n ( x , y ) with two-dimensional sinc kernel sin( σ x ) π x sin( σ y ) π y as follows µ n ˜ ψ n ( u , v ) = (cid:90) R p τ ( x , y ) ˜ ψ n ( x , y ) sin σ ( x − u ) π ( x − u ) sin σ ( y − v ) π ( y − v ) dxdy . (4.23)Denote φ ( x , y ) : = p τ ( x , y ) ˜ ψ n ( x , y ). From Lemma 4.1, let g ( x , y ) = sin σ x π x sin σ y π y , the g ( x , y ) = g ( − x , y ) and its QFT p σ ( u , v ) is real valued function. Then taking QFTto the both sides of Eq. (4.23), we have µ n F ( ˜ ψ n )( u (cid:48) , v (cid:48) ) = F ( φ )( u (cid:48) , v (cid:48) ) p σ ( u (cid:48) , v (cid:48) ) . (4.24)Immediately, we obtain that F ( ˜ ψ n )( u (cid:48) , v (cid:48) ) = | u (cid:48) | > σ and | v (cid:48) | > σ , i.e., F ( ˜ ψ n )( u (cid:48) , v (cid:48) ) = F ( ˜ ψ n )( u (cid:48) , v (cid:48) ) p σ ( u (cid:48) , v (cid:48) ) . (4.25)Here p σ ( x , y ) : = (cid:40) , ( x , y ) ∈ σ , , otherwise . From Lemma 4.1, taking the inverse QFT onboth sides of the above equation, it follows that ˜ ψ n satisfies Eq. (4.22), whichextends the integral domain of ˜ ψ n ( x , y ) from τ to R . (cid:3) The Propositions 4.3 and 4.4 follow from the general theory of integral equa-tions of Hermitian kernel and are stated without proof [10, 11].
Proposition 4.3 (Eigenvalues)
Eq. (4.18) has solutions for real or complex µ n .These values are a monotonically decreasing sequence, > | µ | > | µ | > ... > | µ n | > ... , and satisfy lim n →∞ | µ n | = . roposition 4.4 (Orthogonal in τ ) For di ff erent eigenvalues { µ n } ∞ n = , the corre-sponding eigenfunctions { ˜ ψ n ( x , y ) } ∞ n = are an orthonormal set in τ , i.e., (cid:90) τ ˜ ψ n ( x , y ) ˜ ψ m ( x , y ) dxdy = (cid:40) µ n , m = n , , otherwise . (4.26) Proposition 4.5 (Orthogonal in R ) The eigenfunctions { ˜ ψ n ( x , y ) } ∞ n = form an or-thonormal system in R , i.e., (cid:90) R ˜ ψ n ( x , y ) ˜ ψ m ( x , y ) dxdy = (cid:40) , m = n , , otherwise . (4.27) Proof.
Combining Eq. (4.18), the orthogonality in R can be immediately de-duced as follows (cid:90) R ˜ ψ n ( x , y ) ˜ ψ m ( x , y ) dxdy = (cid:90) R (cid:32) µ n (cid:90) τ ˜ ψ n ( s , t ) sin σ ( x − s ) π ( x − s ) sin σ ( y − t ) π ( y − t ) dsdt (cid:33)(cid:32) µ m (cid:90) τ ˜ ψ m ( u , v ) sin σ ( x − u ) π ( x − u ) sin σ ( y − v ) π ( y − v ) dudv (cid:33) dxdy = µ n µ m (cid:90) τ (cid:90) τ ˜ ψ n ( s , t ) ˜ ψ m ( u , v ) dsdtdudv (cid:32)(cid:90) R sin σ ( x − s ) π ( x − s ) sin σ ( y − t ) π ( y − t ) sin σ ( x − u ) π ( x − u ) sin σ ( y − v ) π ( y − v ) dxdy (cid:33) = µ n µ m (cid:90) τ (cid:32)(cid:90) τ ˜ ψ n ( s , t ) sin σ ( s − u ) π ( s − u ) sin σ ( t − v ) π ( t − v ) dsdt (cid:33) ˜ ψ m ( u , v ) dudv = µ m (cid:90) τ ˜ ψ n ( u , v ) ˜ ψ m ( u , v ) dudv = (cid:40) , m = n , , otherwise . (cid:3)
5. Main Results
In the present section, we will consider the energy concentration problem ofbandlimited H -valued signals in fixed spatial and QLCTs-frequency domains. Thedefinitions and notations of bandlimited H -valued signals associated with QLCTsand QFT are introduced in the following.15 efinition 5.1 ( σ -bandlimited H -valued signal associated with QLCTs) an H -valued signal f ( x , y ) with finite energy is σ -bandlimited associated with QLCTs,if its QLCTs vanishes for all ( u , v ) outside the region σ , i.e., L ( f )( u , v ) = , for ( u , v ) ∈ R \ σ . (5.28) Denote B σ the set of σ -bandlimited H -valued signals associated with QLCTs,i.e., B σ : = (cid:26) f ∈ L ( R ; H ) (cid:12)(cid:12)(cid:12)(cid:12) supp ( L ( f ( u , v ))) ∈ σ (cid:27) . (5.29) Definition 5.2 ( σ -bandlimited H -valued signal associated with QFT) an H -valuedsignal f ( x , y ) with finite energy is σ -bandlimited associated with QFT, if its QFTvanishes for all ( u , v ) outside the region σ , i.e., F ( f )( u , v ) = , for ( u , v ) ∈ R \ σ . (5.30) Denote ˜ B σ the set of the σ -bandlimited H -valued signals associated with QFT,i.e., ˜ B σ : = (cid:26) f ∈ L ( R ; H ) (cid:12)(cid:12)(cid:12)(cid:12) supp ( F ( f ( u , v ))) ∈ σ (cid:27) . (5.31)Note that the relationship between QLCT and QFT for an H -valued signal f F ( ˜ f ) (cid:32) ub , vb (cid:33) = π (cid:112) b i e − i d b u L ( f )( u , v ) e − j d b v (cid:112) b j . That is to say if f ∈ B σ , then for the F ( ˜ f ) (cid:16) ub , vb (cid:17) , ( u , v ) is also in σ , that means( ub , vb ) ∈ [ − σ b , σ b ] × [ − σ b , σ b ] = : ˜ σ . (5.32)Then ˜ f ∈ ˜ B ˜ σ , because F ( ˜ f )( u , v ) = , for ( u , v ) ∈ R \ ˜ σ . (5.33)Now we pay attention to the energy concentration problem associated withQLCTs. To be specific, the energy concentration problem associated with QLCTsaims to obtain the relationship of the following two energy ratios for any H -valued16ignal f with finite energy in a fixed spatial and QLCTs-frequency domains, i.e., τ and σ , α f : = (cid:107) p τ f (cid:107) (cid:107) f (cid:107) and β f : = (cid:107) p σ L ( f ) (cid:107) (cid:107) L ( f ) (cid:107) . (5.34)By the Parseval identity in Eq. (3.16), the two ratios can also be obtained by α f = (cid:107) p τ ˜ f (cid:107) (cid:107) ˜ f (cid:107) and β f = (cid:107) p ˜ σ F ( ˜ f ) (cid:107) (cid:107) F ( ˜ f ) (cid:107) . (5.35)Note that the value of α f and β f are real values in [0 , σ -Bandlimited Signals In this part, we only consider the energy problem for f ∈ B σ , i.e., β f = f ∈ ˜ B ˜ σ . Concretely speaking, given an unit energy ˜ f ∈ ˜ B ˜ σ , the energyconcentration problem is finding the maximum of α f , i.e.,max ˜ f ( x , y ) ∈ ˜ B ˜ σ α f = (cid:107) p τ ˜ f (cid:107) . (5.36)Denote the maximum α f as follows α max : = max ˜ f ( x , y ) ∈ ˜ B ˜ σ α f . (5.37)Let ˜ f τ ( x , y ) : = p τ ( x , y ) ˜ f ( x , y ), we can also reformulate α f as follows α f = (cid:90) R ˜ f τ ( x , y ) ˜ f ( x , y ) dxdy . (5.38)We conclude that the maximum α max can be taken if ˜ f τ ( x , y ) = µ ˜ f ( x , y ). To derivethis fact, the generally cross-correlation function ρ f g of f and g ∈ L ( R ; C ) wasintroduced at first [2], ρ f g ( s , t ) : = (cid:90) R f ( x , y ) g ( s + x , t + y ) dxdy . (5.39)Consider the ( s , t ) = (0 , ρ f g (0 , = (cid:82) R f ( x , y ) g ( x , y ) dxdy . From thecomplex-valued Schwarz’s inequality, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R f ( x , y ) g ( x , y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) R | f ( x , y ) | dxdy (cid:90) R | g ( x , y ) | dxdy . (5.40)17he | ρ f g (0 , | takes the maximum value if f ( x , y ) = kg ( x , y ), where k is a constant.Similarly, we can define the cross-correlation function ρ fg of H -valued signals f and g ∈ L ( R ; H ) as follows ρ fg ( s , t ) : = (cid:90) R f ( x , y ) g ( s + x , t + y ) dxdy . (5.41)Since the quaternionic Schwarz’s inequality also holds. Then to get the maximumvalue of ρ fg (0 , f and g satisfies f ( x , y ) = γ g ( x , y ),where γ is a constant. Here, we find that α f = ρ ˜ f τ ˜ f (0 , . (5.42)To achieve the maximum α f , the two functions should be the same except a con-stant factor. For this reason, there exists a constant µ such that ˜ f τ ( x , y ) = µ ˜ f ( x , y ).Let F ( ˜ f τ ) and F ( ˜ f ) are the QFT for ˜ f τ and ˜ f , respectively. Taking QFT toboth sides of the equation ˜ f τ ( x , y ) = µ ˜ f ( x , y ), we have F ( ˜ f τ )( u , v ) = µ F ( ˜ f )( u , v ) . (5.43)Since ˜ f ∈ ˜ B ˜ σ , then ˜ f τ ( x , y ) is also in ˜ B ˜ σ , i.e., F ( ˜ f τ )( u , v ) p ˜ σ ( u , v ) = µ F ( ˜ f )( u , v ) . (5.44)From Lemma 4.1, taking the inverse QFT to the above equation, we have (cid:90) R ˜ f τ ( s , t ) sin σ ( x − s ) π ( x − s ) sin σ ( y − t ) π ( y − t ) dsdt = µ ˜ f ( x , y ) . (5.45)Substituting ˜ f τ ( s , t ) = p τ ( s , t ) ˜ f ( s , t ) to the above equation, we have (cid:90) τ ˜ f ( s , t ) sin σ ( x − s ) π ( x − s ) sin σ ( y − t ) π ( y − t ) dsdt = µ ˜ f ( x , y ) , (5.46)which is the low-pass filter form of QPSWFs.Now we show that σ -bandlimited H -valued signals satisfying the low-passfilter form Eq. (5.46) can reach the maximum α max . Theorem 5.1
If the eigenvalues of the integral equation (cid:90) τ ˜ f ( s , t ) sin σ ( x − s ) π ( x − s ) sin σ ( y − t ) π ( y − t ) dsdt = µ ˜ f ( x , y ) , (5.47) have a maximum µ , then α max = µ max . The eigenfunction corresponding to µ max isthe function such that α max are reached. roof. For any σ -bandlimited signal ˜ f , construct a function ˜ s ( x , y ) as follows ˜ s ( x , y ) : = (cid:90) τ ˜ f ( s , t ) sin σ ( x − s ) π ( x − s ) sin σ ( y − t ) π ( y − t ) dsdt . (5.48)Let the QFT of ˜ s ( x , y ) as F ( ˜ s )( u , v ), it follows that F ( ˜ s )( u , v ) = F ( ˜ f τ )( u , v ) p ˜ σ ( u , v ) . It means that ˜ s ∈ ˜ B ˜ σ .Denote the energy ratio α s for ˜ s ( x , y ) in the fixed spatial domain τ as follows α s = E s (cid:90) R p τ ( x , y ) ˜ s ( x , y ) ˜ s ( x , y ) dxdy . (5.49)We conclude that for any ˜ f ∈ ˜ B ˜ σ , α f cannot exceed the α s . Direct computationsshow that the energy of the signal ˜ s ( x , y ) is given as follows E s = (cid:90) R ˜ s ( x , y ) ˜ s ( x , y ) dxdy = (cid:90) R F ( ˜ s )( u , v ) F ( ˜ s )( u , v ) dudv = Sc (cid:34)(cid:90) ˜ σ F ( ˜ f τ )( u , v ) F ( ˜ s )( u , v ) dudv (cid:35) = Sc (cid:34)(cid:90) R ˜ f τ ( x , y ) ˜ s ( x , y ) dxdy (cid:35) = Sc (cid:34)(cid:90) R p τ ( x , y ) ˜ f ( x , y ) ˜ s ( x , y ) dxdy (cid:35) . On the other hand, we consider that α f E f = (cid:90) R ˜ f τ ( x , y ) ˜ f ( x , y ) dxdy = Sc (cid:34)(cid:90) R F ( ˜ f τ )( u , v ) F ( ˜ f )( u , v ) dudv (cid:35) = Sc (cid:34)(cid:90) ˜ σ F ( ˜ s )( u , v ) F ( ˜ f )( u , v ) dudv (cid:35) . Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ˜ σ F ( ˜ s )( u , v ) F ( ˜ f )( u , v ) dudv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:107) p ˜ σ F ( ˜ s ) (cid:107) (cid:107) p ˜ σ F ( ˜ f ) (cid:107) , (5.50)19nd (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Sc (cid:34)(cid:90) ˜ σ F ( ˜ s )( u , v ) F ( ˜ f )( u , v ) dudv (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ˜ σ F ( ˜ s )( u , v ) F ( ˜ f )( u , v ) dudv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , simplifying the above three inequalities, we obtain that( α f E f ) ≤ E s E f , from which it follows that α f E f ≤ E s . We also have the following result for E s E s = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Sc (cid:34)(cid:90) R p τ ( x , y ) ˜ f ( x , y ) ˜ s ( x , y ) dxdy (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.51) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R p τ ( x , y ) ˜ f ( x , y ) ˜ s ( x , y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) R p τ ( x , y ) ˜ f ( x , y ) ˜ f ( x , y ) dxdy (cid:90) R p τ ( x , y ) ˜ s ( x , y ) ˜ s ( x , y ) dxdy . Here, we take p τ ( x , y ) into two parts, i.e., (cid:16) (cid:112) p τ ( x , y ) (cid:17) , and use the Schwarzinequality for the above inequality. Clearly, ( E s ) ≤ ( α f E f )( α s E s ), then E s ≤ α f α s E f . Summarizing, we have α f E f ≤ E s ≤ α f α s E f . That means, for any ˜ f ∈ ˜ B ˜ σ , α f ≤ α s .If α f = α s , then Eq. (5.50) and Eq. (5.51) must be equalities. This is at-tained only by setting ˜ s ( x , y ) = α f ˜ f ( x , y ) with E s = α f E f . It means that ˜ f is aneigenfunction of Eq. (5.47) and α f is the corresponding eigenvalue, i.e., µ = α f .At last, we will show that α max = µ max , and the eigenfunction correspondingto µ max is the function such that α max is reached. By definition of 0 ≤ α f ≤
1, there exists a maximum α f and we denote the maximum α f as α max and thecorresponding signal as ˜ f ( x , y ). As we have shown, the α max corresponding the20igenfunction satisfies ˜ s ( x , y ) = α f ˜ f ( x , y ). Here, ˜ s ( x , y ) = α f ˜ f ( x , y ) = α f ˜ f ( x , y )corresponds to the maximum eigenvalue of µ max . Hence, µ max ≤ α max .In order to prove that µ max = α max , it su ffi ces to show that ˜ f ( x , y ) is an eigen-function of the integral equation Eq. (5.47), or equivalently, that with ˜ S ( x , y )defined as ˜ s ( x , y ) in Eq. (5.48) with α S = α max . Obviously, α S ≥ α max and α S ≤ α max , because α max is maximum by assumption. The proof is complete. (cid:3) Theorem 5.1 shows that for arbitrary unit energy σ -bandlimited H -valued sig-nal associated with QLCTs the maximum value of α f can be achieved by theQPSWFs. In fact, from the symmetry theorem of Fourier theory [2], there is alsoa similar integral equation for time-limited signals, which have the maximum β ˜ f .The prove of this conclusion is similar to Theorem 5.1. Corollary 5.1
If the eigenvalues of the integral equation (cid:90) ˜ σ F ( ˜ f )( u , v ) sin τ ( x − u ) π ( x − u ) sin τ ( y − v ) π ( y − v ) dudv = µ F ( ˜ f )( x , y ) . (5.52) have a maximum µ , then β ˜ f have a maximum number β max and β max = µ max . Theeigenfunction corresponding to µ max is the function such that β max are reached. The Eq. (5.52) is equivalent to Eq. (5.47) with u = σ s τ and v = σ t τ . In this section, we will discuss the relationship of ( α f , β f ) in Eq. (5.34) fromthree cases:(1) f ( x , y ) is a σ -bandlimited signal associated with QLCTs.(2) f ( x , y ) is a τ -time-limited signal.(3) f ( x , y ) is an arbitrary signal.The first case follows form the general theory of the f ∈ B σ in Section 5.1.As we have known ˜ f is in ˜ B ˜ σ when f ∈ B σ , i.e., β f =
1. From Theorem 5.1,we know that the maximum α f equals the maximum eigenvalue µ in Eq. (5.47).Using the expansion for the ˜ f ∈ ˜ B ˜ σ , ˜ f ( x , y ) = Σ ∞ n = a n ˜ ψ n ( x , y ) , where a n : = (cid:82) R ˜ f ( x , y ) ψ n ( x , y ) dxdy . It is clear that α f = (cid:82) τ ˜ f ( x , y ) ˜ f ( x , y ) dxdy = Σ ∞ n = µ n a n ≤ µ Σ ∞ n = a n = µ . Hence, α f ≤ µ . If α f = µ , then ˜ f ( x , y ) = ˜ ψ ( x , y ). If α f < µ ,21hen we can find a signal f ∈ B σ whose energy ratio in spatial domain equals α f ,and in this case, ˜ f ( x , y ) is not unique.The second case means α f =
1. From the property of symmetry of the QLCTwe conclude that all the properties for signals f ∈ B σ have corresponding time-limited counterparts. Reversing ( x , y ) and ( u , v ), we conclude that β f ≤ µ . Spe-cially, if β f = µ , then ˜ f ( x , y ) = p τ ( x , y ) ˜ ψ ( x , y ) √ µ .For the third case, considering arbitrary signals with α f <
1, we aim to findthe maximum β f and the corresponding signal f ( x , y ). If α f ≤ µ , as we noted inthe case of f ∈ B σ , we can find ˜ f ∈ ˜ B ˜ σ with energy ration α f , hence, β max = α f > µ . Theorem 5.2
The maximum β max of β f must satisfy the following equation arccos (cid:112) β f + arccos √ α f = arccos √ µ , (5.53) where µ is the largest eigenvalues of Eq. (4.18) and the corresponding ˜ f for themaximum β max is given by ˜ f ( x , y ) = (cid:115) − α f − µ p τ ( x , y ) ˜ ψ ( x , y ) + (cid:32) (cid:114) α f µ − α f (cid:33) ˜ ψ ( x , y ) . (5.54) Proof.
Before giving the proof to Eq. (5.53), we first need to present the followingfact. Given a function ˜ f with spatial projection p τ ˜ f and frequency projection p ˜ σ F ( ˜ f ), we construct a new function as follows ˜ f ( x , y ) : = ap τ ˜ f ( x , y ) + b F − (cid:16) p ˜ σ F ( ˜ f ) (cid:17) ( x , y ) , (5.55)where a and b are two constants such that the energy of g ( x , y ) is minimum, where g ( x , y ) : = ˜ f ( x , y ) − ˜ f ( x , y ) . (5.56)Denote α f , β f and α f , β f the energy ratios for ˜ f ( x , y ) and ˜ f ( x , y ) as Eq. (5.34),respectively. We conclude that α f ≥ α f , β f ≥ β f .Suppose the energy of ˜ f ( x , y ) equals to 1 and we rewrite α f , β f as follows α f = (cid:68) p τ ˜ f , p τ ˜ f (cid:69) ,β f = (cid:68) p ˜ σ F ( ˜ f ) , p ˜ σ F ( ˜ f ) (cid:69) . (5.57)From the orthogonality principle [2], it follows that (cid:68) p τ ˜ f , g (cid:69) = , and (cid:68) F − (cid:16) p ˜ σ F ( ˜ f ) (cid:17) , g (cid:69) = , (5.58)22hich means (cid:68) ˜ f , g (cid:69) =
0. Meanwhile, we have E f of ˜ f by E f : = (cid:68) ˜ f , ˜ f (cid:69) = − E g . (5.59)Now we denote two energy for the projection of g ( x , y ) as follows E p τ g : = (cid:68) p τ g , p τ g (cid:69) , and E p ˜ σ F ( g ) : = (cid:68) p ˜ σ F ( g ) , p ˜ σ F ( g ) (cid:69) . (5.60)The E p τ f and E p ˜ σ F ( f ) will be simply written as E τ and E ˜ σ in the following, re-spectively. Since ˜ f ( x , y ) = ˜ f ( x , y ) − g ( x , y ), we have (cid:68) p τ ˜ f , p τ ˜ f (cid:69) = α f E f = α f + E τ , (cid:68) p ˜ σ F ( ˜ f ) , p ˜ σ F ( ˜ f ) (cid:69) = β f E f = β f + E ˜ σ . (5.61)Therefore, α f ≥ α f and β f ≥ β f . That means, in order to get the maximum β f ,we can formula a function as follows ˜ f = ap τ ˜ f + b F − (cid:16) p ˜ σ F ( ˜ f ) (cid:17) . (5.62)Taking QFT to both sides for Eq. (5.62) and then taking frequency projection, wehave F (cid:16) ˜ f (cid:17) p ˜ σ = ap ˜ σ F (cid:16) ˜ f (cid:17) ∗ (cid:32) sin( τ u ) π u sin( τ v ) π v (cid:33) + bp ˜ σ F ( ˜ f ) . (5.63)Rearranging this formula, we obtain that(1 − b ) F (cid:16) ˜ f (cid:17) p ˜ σ = ap ˜ σ (cid:34) F (cid:16) ˜ f (cid:17) ∗ (cid:16) sin( τ u ) π u sin( τ v ) π v (cid:17)(cid:35) . (5.64)Taking inverse QFT to the above equation, we have1 − ba F − (cid:16) F (cid:16) ˜ f (cid:17) p ˜ σ (cid:17) (5.65) = F − (cid:32) F (cid:16) ˜ f (cid:17) ∗ (cid:32) sin( τ u ) π u sin( τ v ) π v (cid:33)(cid:33) ∗ (cid:32) sin( σ x ) π x sin( σ y ) π y (cid:33) . On the other hand, taking the spatial projection to Eq. (5.62), we get p τ ( x , y ) ˜ f ( x , y ) = ap τ ( x , y ) ˜ f ( x , y ) + b F − (cid:16) p ˜ σ F ( ˜ f ) (cid:17) ( x , y ) p τ ( x , y ) . (5.66)23earranging this equation, it becomes(1 − a ) p τ ( x , y ) ˜ f ( x , y ) = bp τ ( x , y ) F − (cid:16) p ˜ σ F ( ˜ f ) (cid:17) ( x , y ) . (5.67)Taking QFT on both sides to the above equation, it follows that(1 − a ) F (cid:16) ˜ f (cid:17) ∗ (cid:32) sin( τ u ) π u sin( τ v ) π v (cid:33) = b (cid:16) p ˜ σ F ( ˜ f ) (cid:17) ∗ (cid:32) sin( τ u ) π u sin( τ v ) π v (cid:33) . (5.68)Applying Eq. (5.65) and Eq. (5.68), we have1 − ba F − (cid:16) F (cid:16) ˜ f (cid:17) p ˜ σ (cid:17) = F − (cid:32) F (cid:16) ˜ f (cid:17) ∗ (cid:32) sin( τ u ) π u sin( τ v ) π v (cid:33)(cid:33) ∗ (cid:32) sin( σ x ) π x sin( σ y ) π y (cid:33) = b − a F − (cid:32)(cid:16) p ˜ σ F ( ˜ f ) (cid:17) ∗ (cid:32) sin( τ u ) π u sin( τ v ) π v (cid:33)(cid:33) ∗ (cid:32) sin( σ x ) π x sin( σ y ) π y (cid:33) = b − a F − (cid:16) p ˜ σ F ( ˜ f ) (cid:17) p τ ( x , y ) ∗ (cid:32) sin( σ x ) π x sin( σ y ) π y (cid:33) . Simplifying the above equality, we obtain that(1 − a )(1 − b ) ab F − (cid:16) F (cid:16) ˜ f (cid:17) p ˜ σ (cid:17) = p τ ( x , y ) F − (cid:16) F (cid:16) ˜ f (cid:17) p ˜ σ (cid:17) ∗ (cid:32) sin σ x π x sin σ y π y (cid:33) . From above equality, we find that F − (cid:16) F (cid:16) ˜ f (cid:17) p ˜ σ (cid:17) is one of QPSWFs for Eq. (4.18)and the corresponding eigenvalue is (1 − a )(1 − b ) ab . By the relationship between ˜ f and F − (cid:16) F (cid:16) ˜ f (cid:17) p ˜ σ (cid:17) in Eq. (5.67), we conclude that ˜ f in Eq. (5.62) can be rewritten as ˜ f ( x , y ) = A ˜ ψ ( x , y ) + Bp τ ( x , y ) ˜ ψ ( x , y ) . (5.69)Now, we compute the inner product of the above equation with ˜ f and p τ ˜ f respec-tively. Since E f = ˜ f , we have1 = A + µ B + AB µ, (5.70) α f = ( A + B ) µ. Then we have A = (cid:113) − α f − µ and B = (cid:113) α f µ − (cid:113) − α f − µ . It follows that β f = (cid:104) p ˜ σ F ( ˜ ψ ) , p ˜ σ F ( ˜ ψ ) (cid:105) = ( A + B µ ) . (5.71)24ith √ α f = cos θ and √ µ = cos θ , the parameters become A = sin θ sin θ and B = cos θ cos θ − sin θ sin θ . That means (cid:112) β f = sin θ sin θ + (cid:16) cos θ cos θ − sin θ sin θ (cid:17) cos θ = cos( θ − θ ) , (5.72)from which it follows thatarccos (cid:112) β f + arccos √ α f = arccos √ µ. (5.73)In order to get the maximal β f , we must take the largest µ = µ . The correspond-ing function is ˜ f ( x , y ) = (cid:115) − α f − µ ˜ ψ ( x , y ) + ( (cid:114) α f µ − (cid:115) − α f − µ ) p τ ˜ ψ ( x , y ) . (5.74)The proof is complete. (cid:3) Until now we have discussed all the relationships of ( α f , β f ), as well as thesignals to reach the maximum value of β f for di ff erent conditions of α f . Example 5.1
Now we give some comparison examples to intuitively illustrate theconcentration levels of QPSWFs associated with QLCTs. The widely used Gaus-sian function will be compared with QPSWFs. In Theorem 5.1, we have shownthat QPSWFs are the most energy concentred σ -bandlimited signals.Now, a σ -bandlimited Gaussian function is constructed at first. Consider thetruncated Gaussian function g ( x , y ) in QLCTs-frequency domain as followsG ( u , v ) = p σ ( u , v ) e − ( u + v ) (cid:107) p σ e − ( u + v ) (cid:107) , (5.75) where G ( u , v ) is the QLCT of g ( x , y ) . Obviously, G ( u , v ) has unit energy. This σ -bandlimited Gaussian function g ( x , y ) in spatial domain becomes g ( x , y ) = (cid:107) p σ e − ( u + v ) (cid:107) L − (cid:16) p σ ( u , v ) e − ( u + v ) (cid:17) . (5.76) As for the QPSWFs, by means of the classical one-dimensional PSWFs of zeroorder we now construct a special QPSWF as follows ψ ( x , y ) = ϕ ( x ) ϕ ( y ) (cid:107) ϕ ( x ) ϕ ( y ) (cid:107) , (5.77)25 igure 1: σ -bandlimited g ( x , y ) associated with QFT and the modulus of ψ ( x , y ) in timeand QFT-frequency domains. where ϕ is the first one-dimensional zero order PSWF. Here, we construct theQPSWF under the condition of c = . The QLCTs for the QPSWF becomes L ( ψ ) ( u , v ) = (cid:107) ϕ ( x ) ϕ ( y ) (cid:107) L ( ϕ ( x ) ϕ ( y )) . (5.78) For both of the σ -bandlimited signals above, the energy ratios β equal to inQLCT-frequency domain. The energy ratio pair in spatial and frequency in thecomparison is noted as ( α, β ) : = ( α f , β f ) .In Fig. 1 and Fig. 2, we will show two pairs of the energy ratios α for g ( x , y ) and ψ ( x , y ) in spatial domain associated with QLCT with two kinds of di ff erentparameter matrices. In Fig. 1 we set the parameter matrices of QLCT A i = igure 2: σ -bandlimited g ( x , y ) associated with QLCTs and the modulus of ψ ( x , y ) intime and QLCT-frequency domains with a = a = , b = b = , c = c = − , d = d = . (cid:32) − (cid:33) , i = , , which is already a QFT. In this case, the energy ratios α for g ( x , y ) and ψ ( x , y ) are very close. However, in Fig. 2 we set the parametermatrices of QLCT A i = (cid:32) . − (cid:33) , i = , . In this case, the energy ratio α for g ( x , y ) is . and α for ψ ( x , y ) is . . In fact, we just change theparameters a i , i = , from to . . That means, for QPSWFs the energy is moreconcentred then truncated Gaussian function.As for the τ time-limited function, there are the similar results like σ -bandlimitedcases. We also list two pairs of the energy ratios β for g ( x , y ) and ψ ( x , y ) inQLCT-frequency domains in Fig. 3 and Fig. 4. In Fig. 3 we also set the parame- igure 3: τ -time-limited g ( x , y ) associated with QFT and the modulus of ψ ( x , y ) in timeand QFT-frequency domains. ter matrices of QLCT to be the QFT. The parameter matrices of QLCT in Fig. 4is the same as that in Fig. 2. In this two pair cases, you may see the energy ratios β for g ( x , y ) and ψ ( x , y ) are very close. But one more thing di ff erent from Fig.1 and Fig. 2 is that the energy ratios β for g ( x , y ) and ψ ( x , y ) associated withQFT is smaller than the energy ratios β for g ( x , y ) and ψ ( x , y ) associated withthe second parameter matrices. That means, the parameter matrices of QLCT isvary important. In some sense, for specific conditions the results for QLCT willbe better than QFT. igure 4: σ -time-limited g ( x , y ) associated with QLCT and the modulus of ψ ( x , y ) intime and QLCT-frequency domains with a = a = . , b = b = , c = c = − , d = d =
6. Conclusion
This paper presented a new generalization of PSWFs, namely QPSWFs, whichare the optimal H -valued signals for the energy concentration problem associatedwith the QLCTs. We developed the definition of the QPSWFs associated withQLCTs and established various properties of them. In order to find the energydistribution of ( α f , β f ) for any H -valued signals, we not only derive the Parsevalidentity associated with (two-sided) QLCTs, but also show that the maximum α f for σ -bandlimited signals associated with QLCTs in a fixed spatial domain mustbe QPSWFs. 29 cknowledgments The authors acknowledges financial support from the National Natural Sci-ence Foundation of China under Grant (No. 11401606,11501015), University ofMacau (No. MYRG2015-00058-FST and No. MYRG099(Y1-L2)-FST13-KKI)and the Macao Science and Technology Development Fund (No. FDCT / / / Aand No. FDCT / / / A3).
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