Abstract and Applied Analysis
Volume 2016 (2016), Article ID 5874930, 11 pages
http://dx.doi.org/10.1155/2016/5874930

Research Article

A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transform

Mawardi Bahri¹ and Ryuichi Ashino²

¹Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia
²Division of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, Japan

Received 20 August 2015; Accepted 9 December 2015

Academic Editor: Jozef Banas

Copyright © 2016 Mawardi Bahri and Ryuichi Ashino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 as

where and .

The quaternion conjugate of , given by

is an anti-involution; that is,

From (4) we obtain the norm or modulus of defined as

It is not difficult to see that

Furthermore, it is easily seen that

Using conjugate (4) and the modulus of , we can define the inverse of as

which 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 as

with symmetric real scalar part

In 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 integral

where , , 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 by

where 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 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.

Definition 4 (QLCT). Let and be two matrix parameters satisfying ,  . The QLCT of a quaternion signal is defined by

where 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 QFT

where

with

Proof. Simple computations using Definition 4 show that

Then, multiplying both sides of (24) by results in

This 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 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 are related to their QLCT via the Parseval formula, given as

For , 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.

Example 8. Let us now compute the QLCT of the two-dimensional Gaussian function with .

From the definition of QLCT (17), we easily obtain

Using 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) gives

Therefore, by making the change of variable in the above identity, we immediately obtain

This means that

Similarly we can prove

Theorem 10 (continuity). If , then is continuous on .

Proof. Simple computations show that

By the Lebesgue dominated convergence theorem, we may conclude that

when . 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 constants

which is

and 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 get

By making the change of a variable , 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 , then for . 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 be modulation operator defined by with . 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 , then one gets

Proof. The proof directly follows from two previous theorems.

The above properties of the QLCT are summarized in Table 1.

Table 1: Properties of the QLCT of , where are constants and .

Property
Quaternion func.
QLCT

Left linearity

Right linearity

Shift

Modulation

Time-frequency

Gaussian function

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 obtain

Now setting , we further have

and thus

Hence,

By inserting (23) into (66), we immediately obtain

Simplifying it gives

This 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.

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