A Convolution Theorem Related to Quaternion Linear Canonical Transform
We introduce the two-dimensional quaternion linear canonical transform (QLCT), which is a generalization of the classical linear canonical transform (LCT) in quaternion algebra setting. Based on the definition of quaternion convolution in the QLCT domain we derive the convolution theorem associated with the QLCT and obtain a few consequences.
The linear canonical transform (LCT) plays an important role in various fields of optics and signal processing. In some papers, the LCT is also known as the affine Fourier, the ABCD, and Moshinsky-queue transforms. The LCT can be considered as a generalization of many mathematical transforms, such as Fourier, Laplace, fractional Fourier, and Fresnel transforms. Many fundamental properties of the LCT have been investigated like translation, modulation, convolution, correlation, and uncertainty principles (see, e.g., [1–8]).
The quaternion linear canonical transform (QLCT) is a generalization of the linear canonical transform (LCT) using quaternion algebra. According to definitions of the quaternion Fourier transform (QFT), there are basically two ways of obtaining the QLCT: the (right-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. Some important properties of the quaternion linear canonical transform such as the Parseval’s theorem, reconstruction formula, and uncertainty principles are also discussed (see [9–14] and the references mentioned therein). However, there is no literature for establishing the convolution theorem associated with the QLCT as far as we know.
Therefore, it is worthwhile to study the convolution theorems associated with the QLCT, which can be useful in signal processing theory and application. Our main objective of the present paper is to establish convolution theorems for the QLCT which are generalizations of the related classical ones. We will accomplish this task by using the properties of quaternions and combining the LCT convolution and the QFT convolution definition . In the beginning, we make a definition of the QLCT and obtain the relationship between the QLCT and QFT. Based on the convolution definitions of the LCT and QFT, we propose a new definition convolution for the QLCT and obtain its convolution theorem. We emphasize that the proposed convolution definition is different from the one studied in . The definition uses the kernel of the classical fractional Fourier transform which is commutative with quaternion signals.
2. Basic Facts about Quaternion Algebra and Quaternion Fourier Transform
Quaternions are hypercomplex numbers, which can be written in the following form where is a basis of and obeys the following multiplication rules: For a quaternion , the conjugate of the quaternion is given byand satisfies From (3) we obtain the norm or modulus of defined asIt is not difficult to see that Like in complex case, the inverse of is given by
Every quaternion-valued function can be written aswhere , and are real-valued functions. A quaternion module is then defined as
Definition 1. The QFT of is the transform given by the integralHere is called the quaternion Fourier transform operator or the quaternion Fourier transform.
Definition 2. The inverse QFT of is the transform given by the integral where and . Here stands for the inverse QFT operator.
3. Quaternion Linear Canonical Transform and Its Convolution Theorem
In this section we first introduce the two-dimensional quaternion linear canonical transform (QLCT). We then make a convolution definition in the QLCT domain and derive a convolution theorem related to the QLCT.
3.1. Definition of QLCT
Based on the definition of the two-sided quaternion Fourier transform (QFT) and its properties [17–22], we obtain a definition of the QLCT. We also can derive useful properties of the QLCT using fundamental relationship between the QFT and QLCT. Denote by the special linear group of degree 2 over , that is, the group of all real matrices with determinant one. LetWhen , we define the kernel of the QLCT by Observe that we can write the imaginary units above of the formThese facts yield
Definition 3 (QLCT definition). The QLCT of a quaternion signal is defined by
Because and are chirp signals in signal processing, then we always work for the case . The inverse transform of the QLCT above is then described byThis form is equivalent towhere and .
Theorem 4. If , then is continuous on .
Proof. See .
Similarly, one can obtain the following result.
Theorem 5. If , then is continuous on .
3.2. Convolution Theorem for QLCT
In the following we first define the convolution for the QLCT. It is an extension of the convolution definition from the LCT (see [5, 6]) to the QLCT domain. We then investigate how the QLCT behaves under convolutions.
Definition 6. For any two quaternion functions , we define the convolution operator of the QLCT as
As a direct consequence, we get the convolution theorem associated with the QLCT, which is expressed as
Theorem 7. Let be two quaternion-valued functions. Then we have the QLCT of the convolution of and in the form where the matrix parameters
Proof. It directly follows from (16) and (20) that Letting , (23) can be rewritten as Simplifying this result we have Applying properties of kernel function of the QLCT we obtain Now multiplying both sides of the above equation by and gives Based on Definition 3, the required result follows.
Remark 8. If we use the matrix parameters with . Then (21) will lead to
Lemma 9. Let be two quaternion-valued functions. If we assume that , then we haveMoreover, if , then we have
Proof. We only verify the identity (29) and the others are quite similar. Direct computations yield By making the change of variable of the above, we get Multiplying this result by and , we easily obtain