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.
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 over , denoted by , is an associative noncommutative four-dimensional algebra:which obeys the following multiplication rules:For a quaternion , is called the scalar part of denoted by and is called the vector (or pure) part of . The vector part of is conventionally denoted by . Let and , be their vector parts, respectively. Equation (2) yields the quaternionic multiplication aswhere and .
The quaternion conjugate of , given byis an anti-involution; that is,From (4) we obtain the norm or modulus of defined asIt is not difficult to see thatFurthermore, it is easily seen thatUsing conjugate (4) and the modulus of , we can define the inverse of aswhich shows that 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 aswith symmetric real scalar partIn particular, for , we obtain the -norm:
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 of is the transform given by the integralwhere , , and the quaternion exponential product is the quaternion Fourier kernel. Here is called the quaternion Fourier transform operator.
Definition 2. If and , then the inverse transform of the QFT is given bywhere 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 of and its QFT are related byIn 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.
Definition 4 (QLCT). Let and be two matrix parameters satisfying , . The QLCT of a quaternion signal is defined bywhere the kernel functions of the QLCT are given by, respectively,
From the definition of the QLCT, we can see easily that when and , the QLCT of a signal is essentially a quaternion chirp multiplication. Therefore, in this work we always assume that . As a special case, when for , LCT definition (17) reduces to the QFT definition. That is,where is the QFT of 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 function with matrix parameters and can be reduced to the QFTwherewith
Proof. Simple computations using Definition 4 show thatThen, multiplying both sides of (24) by results inThis is the desired result.
Theorem 6. If and , then the inverse transform of the QLCT can be derived from that of the QFT.
Proof. Indeed, we haveIt means thatOr, 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 are related to their QLCT via the Parseval formula, given asFor , one has
Proof. From Parseval formula (15), it follows thatApplying the cyclic multiplication symmetry, we getOn 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.
Example 8. Let us now compute the QLCT of the two-dimensional Gaussian function with .
From the definition of QLCT (17), we easily obtainUsing the QFT of the Gaussian function, We immediately obtain
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 that . Then
Proof. It is not difficult to see that Now applying (38) givesTherefore, by making the change of variable in the above identity, we immediately obtainThis means thatSimilarly we can prove
Theorem 10 (continuity). If , then is continuous on .
Proof. Simple computations show thatBy the Lebesgue dominated convergence theorem, we may conclude thatwhen . This proves that is continuous on . Again since (43) is independent of , is, in fact, uniformly continuous on .
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 constantswhich isand a right linearity property with specific constants
Theorem 11 (shift property). Given a quaternion function , let denote the shifted (translated) function defined by , where . Then one gets
Proof. Taking into account the definition of QLCT (17), we getBy making the change of a variable , we easily obtainTherefore, we further getApplying the definition of the QLCT (17), the above expression can be rewritten in the formWe notice thatBecause , then for . It means that we getBy the above equalities, we finally arrive atThis completes the proof of theorem.
Next, we are concerned with the behavior of the QLCT under modulation.
Theorem 12 (modulation property). Let be modulation operator defined by with . Then
Proof. From Definition 4, it follows thatSubsequent calculations reveal thatHence,This is desired result.
Theorem 13 (time-frequency shift). If quaternion function , then one gets
Proof. The proof directly follows from two previous theorems.
The above properties of the QLCT are summarized in Table 1.
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 that . If and , then one has
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 that , and that , , . Then, the following inequality holds:
Proof. Substituting the quaternion function by defined by (21) on both sides of (62), we easily obtainNow setting , we further haveand thusHence,By inserting (23) into (66), we immediately obtainSimplifying it givesThis 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 that . If and , then one has
Proceeding as in the proof of Theorem 15, we obtain the QLCT directional uncertainty principle as follows.
Theorem 17. Suppose that and and . Then the following inequality is satisfied:
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The first author is partially supported by Hibah Penelitian Kompetisi Internal Tahun 2014 (no. 1454/UN4.20/PL.9/2014) from the Hasanuddin University, Indonesia. The second author is partially supported by JSPS. KAKENHI (C)25400202 of Japan.