On Uncertainty Principle for Quaternionic Linear Canonical Transform

and Applied Analysis 3 on each of the components of the distribution. We will use the following results: 1̂ (ω 1 , ω 2 ) = (2π) 2 δ (ω 1 , ω 2 ) , (14) ̂ (i−|α|Dαδ) (ω 1 , ω 2 ) = ω α1 1 ω α2 2 , (15) where α = (α 1 , α 2 ), |α| = α 1 + α 2 , D = (∂/∂x 1 ) α1 (∂/∂x 2 ) α2 , and δ is the usual Dirac delta function. In the following we introduce the LCT for 2D quaternionic signals. 3. LCTs of 2D Quaternionic Signals The LCT was first introduced in the 70s and is a fourparameter class of linear integral transform, which includes among its many special cases the FT, the fractional Fourier transform (FRFT), the Fresnel transform, the Lorentz transform, and scaling operations. In a way, the LCT has more degrees of freedom and is more flexible than the FT and the FRFT, but with similar computation cost as the conventional FT [41]. Due to the mentioned advantages, it is natural to generalize the classical LCT to the quaternionic algebra. 3.1. Definition. Using the definition of the LCT [33, 42], we extend the LCT to the 2D quaternionic signals. Let us define the left-sided and right-sided LCTs of 2D quaternionic signals. Definition 1 (left-sided and right-sided LCTs). Let A i = [ ai bi ci di ] ∈ R × 2 be a matrix parameter such that det(A i ) = 1, for i = 1, 2. The left-sided and right-sided LCTs of 2D quaternionic signals f ∈ L(R;H) are defined by

The QFT plays a vital role in the representation of (hypercomplex) signals.It transforms a real (or quaternionic) 2D signal into a quaternion-valued frequency domain signal.The four components of the QFT separate four cases of symmetry into real signals instead of only two as in the complex FT.In [25] the authors used the QFT to proceed color image analysis.The paper [26] implemented the QFT to design a color image digital watermarking scheme.The authors in [27] applied the QFT to image preprocessing and neural computing techniques for speech recognition.Recently, certain asymptotic properties of the QFT were analyzed and straightforward generalizations of classical Bochner-Minlos theorems to the framework of quaternionic analysis were derived [28,29].In this paper, we study the uncertainty principle for the QLCT and the generalization of the QFT to the Hamiltonian quaternionic algebra.
The classical LCT being a generalization of the FT, was first proposed in the 1970s by Collins [30] and Moshinsky and Quesne [31].It is an effective processing tool for chirp signal analysis, such as the parameter estimation, sampling progress for nonbandlimited signals with nonlinear Fourier atoms [32], and the LCT filtering [33][34][35].The windowed LCT [36], with a local window function, can reveal the local LCT-frequency contents, and it enjoys high concentrations and eliminates cross terms.The analogue of the Poisson summation formula, sampling formulas, series expansions, Paley-Wiener theorem, and uncertainly relations is studied in [36,37].In view of numerous applications, one is particularly interested in higher-dimensional analogues to Euclidean space.The LCT was first extended to the Clifford analysis setting in [38].It was used to study the generalized prolate spheroidal wave functions and their connection to energy concentration problems [38].In the present work, we study the QLCT which transforms a quaternionic 2D signal into a quaternion-valued frequency domain signal.Some important properties of the QLCT are analyzed.An uncertainty principle for the QLCT is established.This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains.To the best of our knowledge, the study of a Heisenberg-type uncertainty principle for the QLCT has not been carried out yet.The results in this paper are new in the literature.The main motivation of the present study is to develop further general numerical methods for differential equations and to investigate localization theorems for summation of Fourier series in the quaternionic analysis setting.Further investigations and extensions of this topic will be reported in a forthcoming paper.
The paper is organized as follows.Section 2 gives a brief introduction to some general definitions and basic properties of quaternionic analysis.The LCT of 2D quaternionic signal is introduced and studied in Section 3. Some important properties such as Parseval's and inversion theorems are obtained.In Section 4, we introduce and discuss the concept of QLCT and demonstrate some important properties that are necessary to prove the uncertainty principle for the QLCT.The classical Heisenberg uncertainty principle is generalized for the QLCT in Section 5.This principle prescribes a lower bound on the product of the effective widths of quaternionvalued signals in the spatial and frequency domains.Some conclusions are drawn in Section 6.

Preliminaries
The quaternionic algebra was invented by Hamilton in 1843 and is denoted by H in his honor.It is an extension of the complex numbers to a 4D algebra.Every element of H is a linear combination of a real scalar and three orthogonal imaginary units (denoted, resp., by i, j, and k) with real coefficients where the elements i, j, and k obey Hamilton's multiplication rules For every quaternionic number  =  0 +,  = i 1 +j 2 +k 3 , the scalar and nonscalar parts of  are defined as Sc() :=  0 and NSc() := , respectively.Every quaternion  =  0 +  has a quaternionic conjugate  =  0 − .This leads to a norm of  ∈ H defined as Let || and  (∈ R) be polar coordinates of the point ( 0 , ) ∈ H that corresponds to a nonzero quaternion  =  0 + . can be written in polar form as where  0 = || cos , || = || sin ,  = arctan(||/ 0 ), and  = /||.If  ≡ 0, the coordinate  is undefined; so it is always understood that  ̸ = 0 whenever  = arg  is discussed.The symbol   , or exp(), is defined by means of an infinite series (or Euler's formula) as where  is to be measured in radians.
Quaternions can be used for three-or four-entry vector analyses.Recently, quaternions have also been used for color image analysis.For  =  0 + i 1 + j 2 + k 3 ∈ H, we can use  1 ,  2 , and  3 to represent, respectively, the , , and  values of a color image pixel and set  0 = 0.
For  = 1 and 2, the quaternion modules   (R 2 ; H) are defined as For two quaternionic signals ,  ∈  2 (R 2 ; H) the quaternionic space can be equipped with a Hermitian inner product, whose associated norm is As a consequence of the inner product (9), we obtain the quaternionic Cauchy-Schwarz inequality      Sc (⟨, ⟩  2 (R for any ,  ∈  2 (R 2 ; H).
In [39,40] a Clifford-valued generalized function theory is developed.In the following, we adopt the definition that  is called a tempered distribution, if  is a continuous linear functional from S := S(R 2 ) to H, where S(R 2 ) is the Schwarz class of rapidly decreasing functions.The set of all tempered distributions is denoted by S  .If  ∈ S  , we denote this value for a test function  by writing using square brackets.(In the literature one often sees the notation ⟨, ⟩, but we shall avoid this, since it does not completely share the properties of the inner product.)This is equivalent to the one defined in [39] using modules and enables us to define Fourier transforms on tempered distributions, by the formula which is just to perform Fourier transform on each of the components of the distribution.We will use the following results: where  = ( 1 ,  2 ), || =  1 +  2 ,   = (/ 1 )  1 (/ 2 )  2 , and  is the usual Dirac delta function.
In the following we introduce the LCT for 2D quaternionic signals.

LCTs of 2D Quaternionic Signals
The LCT was first introduced in the 70s and is a fourparameter class of linear integral transform, which includes among its many special cases the FT, the fractional Fourier transform (FRFT), the Fresnel transform, the Lorentz transform, and scaling operations.In a way, the LCT has more degrees of freedom and is more flexible than the FT and the FRFT, but with similar computation cost as the conventional FT [41].Due to the mentioned advantages, it is natural to generalize the classical LCT to the quaternionic algebra.

Definition.
Using the definition of the LCT [33,42], we extend the LCT to the 2D quaternionic signals.Let us define the left-sided and right-sided LCTs of 2D quaternionic signals.
Definition 1 (left-sided and right-sided LCTs).Let   = [         ] ∈ R 2 × 2 be a matrix parameter such that det(  ) = 1, for  = 1, 2. The left-sided and right-sided LCTs of 2D quaternionic signals  ∈  1 (R 2 ; H) are defined by respectively.Note that, for   = 0 ( = 1, 2), the LCT of a signal is essentially a chirp multiplication and it is of no particular interest for our objective in this work.Hence, without loss of generality, we set   ̸ = 0 in the following sections unless stated otherwise.Therefore where the kernel functions respectively.

Properties.
The following proposition summarizes some important properties of the kernel functions  i  1 (and  j  2 ) of the left-sided (and right-sided) LCTs which will be useful to study the properties of LCTs, such as the Plancherel theorem.Proposition 2. Let the kernel function   be defined by (19) or (20).
The proofs of properties (i) to (iii) follow from definitions (19) and (20).The proof of property (iv) can be found in [33,35].
Note that some properties of the LCT for 2D quaternionic signals follow from the one-dimensional case [35,42].
(i) Additivity: (ii) Reversibility: (iii) Plancherel Theorem (right-sided LCT): If ,  ∈ S, then In particular, with  = , we get the Parseval theorem; that is, Proof.By Fubini's theorem, property (iv) of Proposition 2 establishes the additivity property (i) of left-sided LCTs, The proof of the right-sided LCT    :=  j  is similar.Reversibility property (ii) is an immediate consequence of additivity property (i) once we observe that  1 =   and  2 =  −1  .To verify property (iii), applying Fubini's theorem, it suffices to see that where we have used (14).
Notice that the left-sided and right-sided LCTs of quaternionic signals are unitary operators on  2 (R 2 ; H).In signal analysis, it is interpreted in the sense that (right-sided) LCT of quaternionic signal preserves the energy of a signal.Remark 4. Note that the Plancherel theorem is not valid for the two-sided or left-sided LCT of 2D quaternionic signal.For this reason, we study the right-sided LCT of 2D quaternionic signals in the following.
It is worth noting that when  1 =  2 = [ 0 1 −1 0 ], the leftsided and right-sided LCTs of  reduce to the left-sided and right-sided FTs of .That is, Here are the left-sided FT and right-sided FT of , respectively.We now formulate the linear canonical integral representation of a 2D quaternionic signal .
Theorem 5 (linear canonical inversion theorem).Suppose that  ∈  1 (R 2 ; H), that  is continuous except for a finite number of finite jumps in any finite interval, and that (, ) = (1/2)((, +) + (, −)) for all  and .Then for every  0 and  where  has (generalized) left and right partial derivatives.In particular, if  is piecewise smooth (i.e., continuous and with a piecewise continuous derivative), then the formula holds for all  0 and uniformly in .

𝐼 (𝑠, 𝑡
and rewrite this expression by inserting the definition of Switching the order of integration is permitted, because the improper double integral is absolutely convergent over the strip (, ) ∈ R × [−, ], and in the last step we have put  0 −  = .Using the formula we can write Now let  > 0 be given.Since we have assumed that  ∈  1 (R 2 ; H), there exists a number  such that Changing the variable, we find that The last integral in (33) can be split into three terms: The term  3 tends to zero as  > 0 and  → ∞, because of (35).The term  2 can be estimated: In the term  1 we have the function (, ) = ((,  0 − ) − (,  0 −))/.This is continuous except for jumps in the interval R × (0, ), and it has the finite limit (, 0+) = (/)  (,  0 ) as  ↘ 0; this means that  is bounded uniformly in  and thus integrable on the interval.By the Riemann-Lebesgue lemma, we conclude that  1 → 0 as  → ∞.All this together gives, since  can be taken as small as we wish, A parallel argument implies that the corresponding integral over (−∞, 0) tends to (,  0 +) uniformly in .Taking the mean value of these two results, we have completed the proof of the theorem.Remark 6.If  i  () ∈  1 (R 2 ; H), then (29) can be written as the absolutely convergent integral The following lemma gives the relationship between the left-(right-) sided LCTs and Left-(right-) sided FTs of .
be a matrix parameter such that det(  ) = 1, for  = 1, 2. Let  ∈  1 (R 2 ; H); then one has Proof.By the definition of  i  () in ( 17), a direct computation shows that Similarly, by the definition of  j  () in ( 18), we obtain (41).
The LCT can be further generalized into the offset linear canonical transform (offset LCT) [33,43,44].It has two extra parameters which represent the space and frequency offsets.The basic theories of the LCT have been developed including uncertainty principles [20,45], convolution theorem [42,46], the Hilbert transform [11,47], sampling theory [32,42], and discretization [41,48,49], which enrich the theoretical system of the LCT.On the other hand, since the LCT has three free parameters, it is more flexible and has found many applications in radar system analysis, filter design, phase retrieval, pattern recognition, and many other areas [35,42].

QLCTs of 2D Quaternionic Signals
4.1.Definition.This section leads to the quaternionic linear canonical transforms (QLCTs).Due to the noncommutative property of multiplication of quaternions, there are many different types of QLCTs: two-sided QLCTs, left-sided QLCTs, and right-sided QLCTs.
where the kernels  i  1 and  j  2 are given by ( 19) and ( 20), respectively.
Due to the validity of the Plancherel theorem, we study the right-sided QLCTs of 2D quaternionic signals in this paper.
It is significant to note that when  1 =  2 = [ 0 1 −1 0 ], the QLCT of  reduces to the QFT of .We denote it by Remark 11.In fact, the right-sided QLCTs defined above can be generalized as follows: where e 1 = e 1,i i + e 1,j j + e Equation ( 45) is the special case of (47) in which e 1 = i and e 2 = j.

Let us give an example to illustrate expression (45).
Example 13.Consider the quaternionic distribution signal, that is, the QLCT kernel of ( 45) It is easy to see that the QLCT of  is a Dirac quaternionic function; that is, 4.2.Properties.This subsection describes important properties of the QLCTs that will be used to establish the uncertainty principles for the QLCTs.We now establish a relation between the right-sided LCTs and the right-sided QLCTs of 2D quaternion-valued signals.
Theorem 15 (the Plancherel theorems of QLCTs).For  = 1, 2, let   ∈ S; the inner product (8) of two quaternionic module functions and their QLCTs is related by In particular, with  1 =  2 = , we get the Parseval identity; that is, Proof.By the inner product (8) and definition of right-sided QLCTs (45), a straightforward computation and Fubini's theorem show that where we have used the Plancherel theorem of right-sided LCTs (24) and formula (14).
Remark 16.Note that the Plancherel theorem is not valid for the two-sided or left-sided QLCT of quaternionic signals.For this reason, we choose to apply the right-sided QLCT of 2D quaternionic signals in the present paper.
Theorem 15 shows that the total signal energy computed in the spatial domain equals the total signal energy in the quaternionic domain.The Parseval theorem allows the energy of a quaternion-valued signal to be considered on either the spatial domain or the quaternionic domain and the change of domains for convenience of computation.
To proceed with, we prove the following derivative properties.

Uncertainty Principles for QLCTs
In signal processing much effort has been placed in the study of the classical Heisenberg uncertainty principle during the last years.Shinde and Gadre [9] established an uncertainty principle for fractional Fourier transforms that provides a lower bound on the uncertainty product of real signal representations in both time and frequency domains.Korn [50] proposed Heisenberg-type uncertainty principles for Cohen transforms which describe lower limits for the time frequency concentration.In the meantime, Hitzer et al. [51][52][53][54] investigated a directional uncertainty principle for the Clifford-Fourier transform, which describes how the variances (in arbitrary but fixed directions) of a multivector-valued function and its Clifford-Fourier transform are related.On our knowledge, a systematic work on the investigation of uncertainty relations using the QLCT of a multivector-valued function has not been carried out.
In the following we explicitly prove and generalize the classical uncertainty principle to quaternionic module functions using the QLCTs.We also give an explicit proof for the Gaussian quaternionic functions (the Gabor filters) to be indeed the only functions that minimize the uncertainty.We further emphasize that our generalization is nontrivial because the multiplication of quaternions and the quaternionic linear canonical kernel are both noncommutative.For this purpose we introduce the following definition.Definition 18.For  = 1, 2, let ,    ∈  2 (R 2 ; H) and L i,j  (),   L i,j  () ∈  2 (R 2 ; H).Then the effective spatial width or spatial uncertainty Δ  of  is evaluated by where Var  () is the variance of the energy distribution of  along the   -axis defined by Similarly, in the quaternionic domain we define the effective spectral width as where  =  0 + i 1 + j 2 + k 3 ∈ H, for  = 1, 2, are quaternionic constants and  1 ,  2 ∈ R are positive real constants.
Then the QLCT of  is given by This shows that the QLCT of the Gaussian quaternionic function is another Gaussian quaternionic function.
(67) The combination of the two spatial uncertainty principles above leads to the uncertainty principle for the 2D quaternionic signal ( 1 ,  2 ) of the form Equality holds in (68) if and only if  is a 2D Gaussian function; that is.
where  1 ,  2 are positive real constants and Proof.Applying (58) in Lemma 17 and using the Schwarz inequality (10), we have Using the exponential form of a 2D quaternionic signal (6), let where  = ( and therefore we have the uncertainty principle as given by (67) and (68).We finally show that the equality in (67) and ( 68) is satisfied if and only if  is a Gaussian quaternionic function.
Since the minimum value for the uncertainty product is   /2, we can ask what signals have that minimum value.The Schwarz inequality (10) becomes an equality when the two functions are proportional to each other.Hence, we take  = −, where  is a quaternionic constant and the −1 has been inserted for convenience.We therefore have (80)

𝜕
The only way this can be zero is if NSc(  ) = 0 and hence   must be a real number.We then have where  1 ,  2 are positive real constants since  ∈ S and we have included the appropriate normalization  = ‖‖  2 (R 2 ;H) ( 1  2 / 2 ) 1/4 .
Since the 2D Gaussian function ( 1 ,  2 ) of (81) achieves the minimum width-bandwidth product, it is theoretically a very good prototype waveform.One can therefore construct a basic waveform using spatially or frequency-scaled versions of ( 1 ,  2 ) to provide multiscale spectral resolution.Such a wavelet basis construction derived from a Gaussian quaternionic function prototype waveform has been realized, for example, in the quaternionic wavelet transforms in [55].The optimal space-frequency localization is also another reason why 2D Clifford-Gabor bandpass filters were suggested in [56].

Conclusion
In this paper we developed the definition of QLCT.The various properties of QLCT such as partial derivative, the Plancherel, and Parseval theorems are discussed.Using the well-known properties of the classical LCT, we established an uncertainty principle for the QLCT.This uncertainty principle states that the product of the variances of quaternionvalued signals in the spatial and frequency domains has a lower bound.It is shown that only a 2D Gaussian signal minimizes the uncertainty.With the help of this principle, we hope to contribute to the theory and applications of signal processing through this investigation and to develop further general numerical methods for differential equations.The results in this paper are new in the literature.Further investigations on this topic are now under investigation and will be reported in a forthcoming paper.

Figure 4
Figure 3 It enables us to write the polar form (4) in exponential form more compactly as 1,k k and e 2 = e 2,i i + e 2,j j + e 2,k k so that e 1,i e 2,i + e 1,j e 2,j + e 1,k e 2,k = 0.
because by (73), we see that is the only way we can actually get the value of   /2.