Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform

Abstract

The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT.

1. Introduction

The quaternion Fourier transform (QFT) is a nontrivial generalization of the real and complex classical Fourier transforms (FT) using quaternion algebra. Many useful properties of the QFT were obtained such as shift, modulation, convolution, correlation, differentiation, energy conservation, and uncertainty principle. It was first introduced in [1] for the analysis of 2D linear time-invariant partial differential systems and then applied in color image processing [2, 3]. It is a natural question to extend the QFT to the linear canonical transform (LCT) domains and then it is the so-called quaternionic linear transform (QLCT). This extension is constructed by substituting the kernel of the QFT with the kernel of the LCT. A number of useful properties of the QLCT have been investigated including shift, orthogonality relation, reconstruction formula, and Heisenberg uncertainty principle (see, for example, [4–6] and the references given therein).

In [7], the author studied that the fractional Fourier transform (FrFT) can be reduced to the classical Fourier transform. Based on this fact, some properties of the FrFT can be derived very easily from those of the classical Fourier transform by simple change variable. Recently, in [8], the authors developed this idea to derive an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by using the fundamental relationship between the QLCT and the QFT [8]. In [9], the authors proposed the quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) and obtain its fundamental properties. In the present paper, we first establish the basic relationship between the QWVD-LCT and the QFT. We then show that some fundamental properties of the QWVD-LCT such as inversion formula and Moyal formula can be obtained by combining this relation and the properties of the QFT. We investigate that the QWVD-LCT can be reduced to the quaternion windowed Fourier transform and the continuous quaternion Fourier transform. We finally establish the relationship between the QAF-LCT and the QFT which enables us to derive some useful properties of the QAF-LCT.

2. Preliminaries

In the preliminaries we remind the reader of some facts of quaternions, the quaternion Fourier transform, and the quaternion linear canonical transform.

2.1. Basic Facts about Quaternions

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 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.

Now we observe thatThis leads to the cyclic multiplication; that is,

We define an inner product for quaternion-valued functions as follows:with symmetric real scalar partIn particular, for , we obtain the -normThis gives the -norm

2.2. Quaternion Linear Canonical Transform and Its Basic Properties

In this section, we briefly discuss the definition of the two-sided QFT and the two-sided quaternion linear canonical transform (QLCT) (for simplicity of notation, we write the QFT and QLCT instead of the two-sided QFT and the two-sided QLCT, resp., in the next section). We further collect some basic properties of the QLCT, which will be very useful later on.

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.

A useful property of the QFT is stated in the following lemma, which is needed to derive Moyal formula of the quaternion Wigner-Ville distribution associated with the linear canonical transform (QWVD-LCT).

Lemma 3 (QFT Parseval). Let . The relation between and their QFT is given byIn particular, with , we get the QFT version of the Plancherel formula; that is,

Definition 4 (QLCT definition). Suppose that and are real matrix parameters satisfying . The QLCT of a quaternion signal is defined bywhere the kernel functions of the QLCT above are given by

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 .

Lemma 5. The QLCT of a signal can be seen as the QFT of the signal in the following form:

It is worth noting that if for , (22) will reduce to the QFT definition; that is,

Theorem 6 (QLCT Parseval). Two quaternion functions are related to their QLCT via the Parseval formula, given asWhen , we get

Proof. For a detailed proof of the above theorem, we refer the reader to [8].

3. Quaternion Wigner-Ville Distribution and Quaternion Ambiguity Function in Linear Canonical Transform Domains

Let us introduce the 2D quaternion Wigner-Ville distribution (QWVD) and quaternion ambiguity function QAF [10]. According to the QWVD and QAF definitions and the QLCT definition, we obtain a definition of the quaternion Wigner-Ville distribution associated with the linear canonical transform (QWVD-LCT) and the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) (see [9]). We establish the fundamental relationship between the QWVD-LCT and QFT. Applying this relation and the properties of the QFT, we in detail derive the inverse transform formula and Moyal’s formula for the QWVD-LCT and the QAF-LCT, where the proof of the properties is quite different from one proposed in [9]. We also study the relationship among the QWVD-LCT, the quaternion windowed Fourier transform, and the continuous quaternion Fourier transform.

3.1. Main Properties of QWVD-LCT and Relation among QAF-LCT, QWFT, and CQWT

Definition 7. The cross quaternion Wigner-Ville distribution of two-dimensional functions (or signals) is given byprovided that the integral exists.

It should be remembered that the kernel of the cross QWVD in (26) does not commute with quaternion functions and so that several properties of the classical Wigner-Ville distribution (WVD) are not valid in the cross QWVD [10].

Definition 8. The cross quaternion ambiguity function of two-dimensional functions (or signals) is given byprovided that the integral exists.

Definition 9. Suppose that and are real matrix parameters satisfying . The cross QWVD-LCT and cross QAF-LCT of a quaternion signal are defined byHere the kernel functions of the above transforms are given by

It can be directly seen that if we write , we immediately obtainThis tells us that the cross QWVD-LCT is in fact the QLCT of the function with respect to . This fact is very important in proving Moyal’s formula for the cross QWVD-LCT. Similarly, we also getwhere is given by

The following result presents an inequality related to the cross QWVD-LCT.

Lemma 10. Suppose that , with . Then we have

Proof. We straightforwardly obtain from (28) that Letting and , we immediately obtainHence, the result follows.

Observe first that, for , (34) will reduce towhich shows that is bounded on .

Lemma 11. The cross QWVD-LCT of a signal with matrix parameters and can be reduced as the QFT of the signal in the following form:

Proof. By calculation directly, we getwhere the last line follows directly from (28).

For abbreviation, we use the notationTherefore, we can write (38) in the form

In the below theorem by combining the properties of the QFT and the fundamental relation between the QFT and the QWVD-LCT, we provide a new proof of reconstruction formula for the QWVD-LCT.

Theorem 12 (reconstruction formula for QWVD-LCT). The inverse transform of the cross the QWVD-LCT of the signal is given byprovided that .

Proof. From the inverse transform of the QFT (17), it follows thatIn view of (40) and (41), we easily getSubsequently,Letting , the above expression will lead toand the final result can be obtained by letting ; that is,which completes the proof.

We now observe that from (45), we straightforwardly obtain the time marginal property of the QWVD-LCT; that is,If we set , the above identity will reduce toand if we set and , (48) becomesIntegrating (49) and (50) with respect to the -variable givesApplying Moyal’s formula for the QFT and the relation between the QFT and the QWVD-LCT, we establish general Moyal’s formula of the QWVD-LCT (compared to [9]).

Theorem 13 (Moyal’s formula for the QWVD-LCT). Let be quaternion-valued signals. Then the following equation holds:

Proof. Applying Parseval’s formula of the QFT (18) to -integral into the left-hand side of (52) yieldsUsing the cyclic multiplication symmetry (11) yieldsThe right-hand side of the above identity can be rewritten in the formMaking the change of variables and and applying Fubini’s theorem we obtainBy comparing the last line of (54) with the last line of (56) finishes the proof of the theorem.

Based on the above theorem, we obtain the following consequences:(i)If , then(ii)If , then(iii)If and , then

3.2. Relationship between QAF-LCT and QWFT

In this subsection, we introduce the quaternionic windowed Fourier transform (QWFT) and obtain the relationship between the QAF-LCT and QWFT. Following [11, 12], we define the QWFT as follows.

Definition 14 (QWFT). Let be a fixed nonzero quaternion window function. The QWFT of with respect to is defined to be the quaternion function on phase space given byBefore presenting the main result of this subsection, let us introduce the following carrier definition and its properties (see [13, 14]).

Definition 15 (carrier). Given two quaternions and , we define the right and left carrier operators as

Lemma 16. Carriers (61) above satisfy the following properties with :

We now describe the relationship between the QAF-LCT and the QWFT in the following theorem.

Theorem 17. The QAF-LCT of a quaternion signal can be expressed by the QWFT in the form

Proof. According to the definition of QWVD-LCT (28), we obtain the following by making the change of variable:With the help of (61) and (62), we rewrite the above identity asAccording to the definition of QWFT (60), we choose the quaternion window functionand obtainThe proof is complete.

3.3. Relationship between QAF-LCT and CQWT

Before proving the relationship between the QAF-LCT and the continuous quaternion wavelet transform (CQWT), we first introduce the definition of the CQWT (see [15–17]).

Definition 18 (CQWT). The CQWT of a quaternion function with respect to the quaternion mother wavelet is defined byHere the family of the quaternion wavelets is defined by

This definition will lead to the following result.

Theorem 19. The QAF-LCT of can be reduced to the CQWT

Proof. Applying the definition of the QAF-LCT (29) and following [18], we easily get Again applying (61) and (62), we obtain where the family of the quaternion wavelets is given byThis is the desired result.

4. Relationship between QAF-LCT and QFT

By a similar argument as in the proof of the Lemma 11, one may establish the following lemma, which describes the basic relationship between the QAF-LCT and the QFT.

Lemma 20. The cross QAF-LCT of a signal with matrix parameters and can be reduced as the QFT of the signal in the following form:

Applying (74) and inversion formula for the QFT (17), we easily obtain the following theorem.

Theorem 21 (reconstruction formula for QAF-LCT). The inverse transform of the cross the QAF-LCT of the signal is given byprovided that .

In the same kind of reasoning as Theorem 13, one can easily derive Moyal’s formula of the QAF-LCT as follows.

Theorem 22 (Moyal’s formula for QAF-LCT). Let be quaternion-valued signals. Then the following equation holds:

5. Conclusion

In this paper, we have presented the quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) and the quaternion ambiguity function associated with linear canonical transform (QAF-LCT). We provide alternative proof of well-known properties of the QWVD-LCT and QAF-LCT such as inversion formula and Moyal formula. The proof of the properties can be obtained using the properties of the QFT and relationship among the QWVD-LCT, QAF-LCT, and the QFT. We also studied that the QAF-LCT can be reduced to the QWFT and the CQWT.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is partially supported by WCU Percepatan Publikasi Internasional Tahun 2016 (no. 3834/UN4.21/PL.09/2016) from the Hasanuddin University, Indonesia.