A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

and Applied Analysis 3 We learn from the above equation that q±eμθ = e∓μθq±, (17) where cos θ = q0 󵄨󵄨󵄨󵄨q󵄨󵄨󵄨󵄨 , sin θ = √q2 1 + q2 2 + q2 3 󵄨󵄨󵄨󵄨q󵄨󵄨󵄨󵄨 . (18) In particular, taking μ = i and ] = j, (14) becomes q = q+ + q−, q± = 1 2 (q ± iqj) . (19)


Introduction
The quaternion Fourier transform (QFT) is an extension of the classical two-dimensional Fourier transform (FT) [1][2][3][4] in the framework of quaternion algebra.It plays an important role in the representation of the two-dimensional quaternion signals.A number of useful properties of the extended transform are generalizations of the corresponding properties of the FT with some modifications (see, e.g., [5][6][7][8][9][10][11][12][13][14][15]).The QFT has found many applications in color image processing and signal analysis; we refer the reader to [16][17][18][19] and the references mentioned therein.An extension of the QFT in the framework of the classical linear canonical transform (LCT) (see [20][21][22]), known as the quaternion linear canonical transform, has received much attention in recent years.Due to the several definitions of the QFT, there are basically three ways of obtaining the quaternion linear canonical transform (QLCT): the right-sided quaternion linear canonical transform, the left-sided quaternion linear canonical transform, and the two-sided quaternion linear canonical transform.The right-sided quaternion linear canonical transform is obtained by substituting the Fourier kernel with the right-sided QFT kernel in the LCT definition, and so on.Recent works related to some important properties of the QLCT such as Parseval's theorem, reconstruction formula, and component-wise uncertainty principles were also published [18,[23][24][25].It was found that the properties of the QLCT are extensions of the corresponding version of the QFT with some modifications.
On the other hand, the uncertainty principle plays one important role in signal processing.It describes a function and its FT, which cannot both be simultaneously sharply localized.One example of this is the Heisenberg uncertainty principle concerning position and momentum wave functions in quantum physics.In signal processing, an uncertainty principle states that the product of the variances of the signal in the time and frequency domains has a lower bound.Up till now, several attempts have been made to extend the uncertainty principles associated with the QFT and QLCT domains.The component-wise and directional uncertainty principles associated with the QFT were proposed in [11].In [26,27], the authors established a component-wise uncertainty principle for the QLCT and proved that the equality is achieved for optimal quaternion Gaussian function.Recently, the authors [23] proposed the logarithmic uncertainty principle associated with the QLCT which is the generalization of the logarithmic uncertainty principle for the QFT.
Therefore, the main objective of the present paper is to establish full uncertainty principle for the two-sided QLCT, which is a new general form of component-wise uncertainty principle for the two-sided QLCT.This uncertainty principle is derived using the connection between the QFT and QLCT.

Preliminaries
In this section, let us briefly recall some basic definitions and properties of the quaternions (for more details, see [28]).
2.1.Quaternions.The quaternions, a generalization of complex numbers, are members of a noncommutative division algebra.The set of quaternions is denoted by H. Every element of H can be written in the following form: with the units i, j, k, which obey the following: (2) For a quaternion  =  0 +i 1 +j 2 +k 3 ∈ H,  0 is simply called the scalar part of  denoted by Sc() and q = i 1 + j 2 + k 3 is called the vector part of  denoted by Vec().
Let ,  ∈ H and p, q be their vector parts, respectively.Equation (2) yields the quaternionic multiplication  as where Analogously to the complex case, a quaternionic conjugation  is given by which leads to the anti-involution; that is, With the help of ( Using conjugate (5) and the modulus of q, we can define the inverse of  ∈ H\{0} as which shows that H is a normed division algebra.
In quaternionic notation, we may define an inner product for quaternion-valued functions ,  : R 2 → H as follows: with symmetric real scalar part In particular, for  = , we obtain the  2 (R 2 ; H)-norm A quaternion module  2 (R 2 ; H) is then defined as 2.2.Split Quaternion and Properties.In this section, we study some of the basic formulas of split quaternion (see [9]), which will be used to prove the fundamental results in the sequel.
Definition 1.For two quaternion square roots , ] such that  2 = ] 2 = −1, one may express a quaternion  as For the special case of  = ], any quaternion  may be split up into commuting and anticommuting parts with respect to ; that is, It easily seems that the commuting and anticommuting parts satisfy the interesting properties: We learn from the above equation that where In particular, taking  = i and ] = j, (14) becomes The above gives This leads to the following modulus identity: Furthermore, one can obtain

Relationship between Quaternion Fourier Transform (QFT) and Quaternion Linear Canonical Transform (QLCT)
In this section, we introduce the QFT and its relationship to the QLCT.We begin by introducing different types of the QFT.
Lemma 4. The QLCT of a signal  with matrix parameters  1 = ( 1 ,  1 ,  1 ,  1 ) and  2 = ( 2 ,  2 ,  2 ,  2 ) can be seen as the QFT of the signal  in the following form: We introduce This implies that (33) can be rewritten in the form Further, we have the following lemma which describes a relation between the right-sided QLCT and the two-sided QLCT of 2D quaternion-valued signals.
Lemma 5.For  ∈  1 (R 2 ; H), one has where the matrix parameters The following lemma allows us to represent the rightsided QLCT to the single right-sided QFT of 2D quaternionvalued signals.

A Version of Uncertainty Principle Associated with QLCT
The classical uncertainty principle of harmonic analysis describes 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).Let us now establish a version of the uncertainty principle associated with the QLCT.However, before proceeding the statement of this main result, we need to introduce a modified uncertainty principle for the QFT as follows (see [29] for more details).
Theorem 7 (the two-sided QFT uncertainty principle).Let S(R 2 ; H) be the quaternion Schwartz space.If the quaternionvalued function  ∈ S(R 2 ; H), then the following inequality holds: (40) Theorem 7 has been recently generalized in the context of the QLCT by the authors of [30].Our generalization is given by the following theorem.

Theorem 9 (the two-sided QLCT uncertainty principle). Under the assumptions of Theorem 7, one has
where f(x) is defined by (43) Proof.By replacing  by ℎ defined by (34) on both sides of (39), we immediately get Let  = /b; then, (44) becomes In view of (34), we obtain This leads to Inserting (35) into (47), we easily obtain where f(x) is defined in (42).Therefore, the proof is complete. 6

Abstract and Applied Analysis
The above theorem is also valid for the right-sided QLCT as described in the following lemma. where Furthermore, we obtain the   (R Thus, the theorem is completely proved.

Conclusion
In this paper, we derived a version of the uncertainty principle for the QLCT using a relation between the QFT and the QLCT.We presented Hausdorff-Young inequality associated with the QLCT.This inequality is very useful for establishing a variation on Heisenberg's uncertainty principle related to the QLCT.