A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transform

and Applied Analysis 3 is essentially a quaternion chirp multiplication. Therefore, in this work we always assume that b 1 b 2 ̸ = 0. As a special case, when A 1 = A 2 = (a i , b i , c i , d i ) = (0, 1, −1, 0) for i = 1, 2, LCT definition (17) reduces to the QFT definition. That is,


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][2][3][4][5][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][8][9][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.

Abstract and Applied Analysis
The quaternion conjugate of , given by is an anti-involution; that is, From ( 4) we obtain the norm or modulus of  ∈ H defined as Using conjugate (4) and the modulus of , we can define the inverse of  ∈ H \ {0} as 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) functions ,  : R 2 → H as with symmetric real scalar part In particular, for  = , we obtain the  2 (R 2 ; H)-norm:

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 twosided QFT in the next section).We discus some properties, which will be used to prove the uncertainty principle.
Definition 1.The QFT of  ∈  1 (R 2 ; H) is the transform F  {} : R 2 → H given by the integral where x =  1 e 1 +  2 e 2 ,  =  1 e 1 +  2 e 2 , and the quaternion exponential product  −i 1  1  −j 2  2 is the quaternion Fourier kernel.Here F  is called the quaternion Fourier transform operator.
Definition 2. If  ∈  1 (R 2 ; H) and F  {} ∈  1 (R 2 ; H), then the inverse transform of the QFT is given by where F −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][13][14][15][16].
Lemma 3 (QFT Parseval).The quaternion product of ,  ∈  1 (R 2 ; H) ∩  2 (R 2 ; H) and its QFT are related by In particular, with  = , we get the quaternion version of the Plancherel formula; that is, Based on the definition of the QFT mentioned above, we consider the two-sided QLCT which is defined as follows.
We need the following important result (compare to [17,18]), which will be useful in proving Theorem 15.
Proof.Indeed, we have It means that Or, equivalently, 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 functions , ℎ ∈  1 (R 2 ; H) ∩  2 (R 2 ; H) are related to their QLCT via the Parseval formula, given as For  = ℎ, one has Proof.From Parseval formula (15), it follows that Applying the cyclic multiplication symmetry, we get On the other hand, The proof is complete.
It is interesting to describe the relationship between the QLCT and QFT as shown in the following example.
From the definition of QLCT (17), we easily obtain Abstract and Applied Analysis 5 Using the QFT of the Gaussian function, We immediately obtain (36)

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 that  ∈  1 (R2 ; H).Then Proof.It is not difficult to see that Now applying (38) gives Therefore, by making the change of variable  1 +  1 / 1 =  1 in the above identity, we immediately obtain Abstract and Applied Analysis This means that lim Similarly we can prove lim Theorem 10 (continuity).
Proof.Simple computations show that By the Lebesgue dominated convergence theorem, we may conclude that

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 which is and a right linearity property with specific constants Theorem 11 (shift property).Given a quaternion function  ∈  2 (R 2 ; H), let  k (x) denote the shifted (translated) function defined by  k (x) = (x − k), where k ∈ R 2 .Then one gets Abstract and Applied Analysis 7 Proof.Taking into account the definition of QLCT ( 17), we get By making the change of a variable x−k = m, we easily obtain Therefore, we further get Applying the definition of the QLCT (17), the above expression can be rewritten in the form We notice that Because     −     = 1, then     /  − 1/  =   for  = 1, 2. It means that we get By the above equalities, we finally arrive at This completes the proof of theorem.
Next, we are concerned with the behavior of the QLCT under modulation.
Theorem 12 (modulation property).Let M  0  be modulation operator defined by M  0 (x) =  i 1  0 (x) j 2 V 0 with  0 =  0 e 1 + V 0 e 2 .Then Proof.From Definition 4, it follows that Subsequent calculations reveal that Hence, This is desired result.
Theorem 13 (time-frequency shift).If quaternion function  ∈  2 (R 2 ; H), then one gets Proof.The proof directly follows from two previous theorems.
The above properties of the QLCT are summarized in Table 1.

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).

Example 8 .
Let us now compute the QLCT of the twodimensional Gaussian function