Product Theorem for Quaternion Fourier Transform

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1 Int. Journal of Math. Analysis, Vol. 8, 204, no. 2, 8-87 HIKAI Ltd, Product Theorem for Quaternion ourier Transform Mawardi Bahri Department of Mathematics, Hasanuddin University Jl. Perintis Kemerdekaan KM 0, Makassar 90245, Indonesia Copyright c 204 Mawardi Bahri. 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 This paper presents in some detail the quaternion ourier transform (QT of the product of two quaternion functions. It is shown that the proposed product theorem for the QT is closely related to the convolution in the quaternion ourier domain. Mathematics Subject Classification: 52, 42C40 Keywords: quaternion ourier transform, convolutionm, correlation Introduction The quaternion ourier transform (QT which is considered as a generalization of the classical ourier transform (T has recently been extensively used and discussed as a very efficient mathematical tool in signal processing for signals [, 5]. Many properties of generalized transform are already known, such as translation, modulation, differentiation, convolution, correlation, the Parseval and Plancherel formula, and uncertainty principle (see, for example, [2, 3, 4]. The properties are extensions of the corresponding version of the T with the some modifications. The most important property of the QT for signal processing applications is convolution theorem. It describes the relationship between convolution of two quaternion functions and the QTs. This property is in fact closely related to the product theorem in the quaternion ourier domain.

2 82 Mawardi Bahri It is well known that in the ourier domain the product theorem states that the ourier transform of the product of two real and complex functions is the convolution of their ourier transforms. In this paper we propose an extension of the product theorem for the QT. This property describes how the QT relates to product of two quaternion functions. 2 Quaternion The quaternion, which is a type of hypercomplex number, was formally introduced by Hamilton in 843. It is a generalization of complex number to a 4D algebra and is denoted by H. Every element of H can be written in a hypercomplex form as follows H {q q 0 iq jq 2 kq 3 : q 0,q,q 2,q 3 }. ( Here the three different imaginary parts obey the following multiplication rules: ij ji k, jk kj i, ki ik j, i 2 j 2 k 2 ijk. (2 or a quaternion q q 0 iq jq 2 kq 3 H, q 0 is called the scalar part of q denoted by Sc(q and a pure quaternion q denoted by Vec(q iq jq 2 kq 3. Any quaternion q can be written as q q e μθ, q q0 2 q 2 q2 2 q3, 2 (3 where θ arctan Sc(q /Vec(q, 0 θ π is the eigen angle or phase of q and μ is any pure unit quaternion such that μ 2 When q,q is a unit quaternion. Proposition 2.. If p and q are two pure quaternions, then p and q are parallel (p q if and only if pq qp, p and q are perpendicular (p q if and only if pq qp. 3 Main esults In this section, we begin by introducing a definition of the quaternion ourier transform (QT. Definition 3. (QT. Let f be in L 2 ( ; H. Then quaternion ourier transform of the function f is given by q {f}(ω f(xe μ! x dx, dx dx dx 2, (4 where ω, x.

3 Product theorem for quaternion ourier transform 83 Theorem 3.2 (Inverse QT. Suppose that f be in L 2 ( ; H and q {f} L ( ; H. Then inverse transform of the QT is given by f(x (2π 2 q {f}(ωe μ! x dω. (5 Definition 3.3 (Quaternion Convolution. Let f,g L 2 ( ; H. The convolution of two quaternion functions f and g is denoted by f g and is defined by (f g(x f(tg(x t dt. (6 It is not difficult to see that (f g( x f(tg(t x dt. (7 Definition 3.4 (Quaternion Correlation. Let f,g L 2 ( ; H be two quaternion functions. The correlation of f and g is defined by (f g(x f(yg(x y dy. (8 The main result of this paper is the following theorem, which describes the relationship between the product of two quaternion functions and its QT. Theorem 3.5. Let f,g L 2 ( ; H. quaternion functions f and g is given by Then the QT of product of two q {fg}(ω (2π (( q{g} 2 q {f 0 }(ωi( q {g} q {f }(ω ( q {g} q {f 2 }(ω( q {g} q {f 3 }(ω. (9 Proof. Applying the QT definition (4 yields q {fg}(ω f(xg(xe μ! x dx 2 ( f(x (2π 2 2 q {g}(ue μu x du e μ! x dx (f 0 (xif (xf 2 (xf 3 (x ( 2 (2π 2 q {g}(ue μu x du e μ! x dx

4 84 Mawardi Bahri i i (2π q{g}(u 4 q {f 0 }(ve μv x e μu x e μ! x dv du dx (2π q{g}(u 2 4 q {f }(ve μv x e μu x e μ! x dv du dx (2π q{g}(u 2 4 q {f 2 }(ve μv x e μu x e μ! x dv du dx (2π q{g}(u 2 4 q {f 3 }(ve μv x e μu x e μ! x dv du dx ( (2π q{g}(u 4 q {f 0 }(v e μ(vu! x dx dv du ( 2 (2π q{g}(u 2 4 q {f }(v e μ(vu! x dx dv du ( 2 (2π q{g}(u 2 4 q {f 2 }(v e μ(vu! x dx dv du ( 2 (2π q{g}(u 2 4 q {f 3 }(v e μ(vu! x dx dv du i (2π q{g}(u 2 q {f 0 }(vδ(v u ω dv du (2π q{g}(u 2 2 q {f }(vδ(v u ω dv du (2π q{g}(u 2 2 q {f 2 }(vδ(v u ω dv du (2π q{g}(u 2 2 q {f 3 }(vδ(v u ω dv du (2π q{g}(u 2 q {f 0 }(ω u du i (2π q{g}(u 2 2 q {f }(ω u du (2π q{g}(u 2 2 q {f 2 }(ω u du (2π q{g}(u 2 2 q {f 3 }(ω u du, which completes the proof of the theorem. As an immediate consequence of the above theorem, we get the following corollary. Corollary 3.6. Let f,g L 2 ( ; H. Assume that the QT of g is a real-

5 Product theorem for quaternion ourier transform 85 valued function, then Theorem 3.5 will reduce to q {fg}(ω (2π 2 ( q{f} q {g}(ω. (0 Proof. An application of the QT definition (4 we easily obtain q {fg}(ω f(xg(xe μ! x dx 2 ( (2π 2 2 q {f}(ve μv x dv q {g}(ue μu x du e μ! x dx. The assumption allows us to interchange the position of kernel e μv x and the function q {g}. Therefore, we get q {fg}(ω As desired. (2π 4 q{f}(v q {g}(ue μu x e μv x e μ! x du dv dx (2π 2 q{f}(v q {g}(uδ(v u ω du dv (2π 2 q{f}(v q {g}(ω v dv. The following theorem provides an alternative form of Theorem 3.5. Theorem 3.7. Let f,g L 2 ( ; H. If g g is a pure quaternion function, then we have q {fg}(ω (2π 2 ( (q {f} q {g,μ }(ω( q {f} q {g,μ }( ω. Proof. Because g is a pure quaternion function, then using Preposition 2. we may decompose g with respect to the axis μ into g,μ g,μ. It means that we have q {fg}(ω f(xg(xe μ! x dx 2 ( (2π 2 2 q {f}(ve μv x du g(xe μ! x dx

6 86 Mawardi Bahri ( (2π 2 2 q {f}(ve μv x du (g,μ (xg,μ (xe μ! x dx (2π q{g}(ug 2 2,μ (x e μv x e μ! x dv dx (2π q{g}(ug 2 2,μ (x e μv x e μ! x dv dx (2π q{g}(u 2 4 q {g,μ }(v e μu x e μv x e μ! x du dv dx (2π q{g}(u 2 4 q {g,μ }(v e μu x e μv x e μ! x du dv dx (2π q{g}(u 2 2 q {g,μ }(vδ(v u ω dv du (2π q{g}(u 2 2 q {g,μ }(vδ(u v ω dv du (2π q{g}(u 2 2 q {g,μ }(ω u dv (2π q{g}(u 2 2 q {g,μ }(u ω dv. This proves the theorem. Acknowledgments This work is partially supported by Hibah Penelitian Kompetisi Internal 203 (No. 0/UN4-.42/LK.26/SP-UH/203 from the Hasanuddin University, Indonesia. eferences [] T. A. Ell and S. J. Sangwine, Hypercomplex ourier transform of color images, IEEE Trans. Signal. Process., 6( (2007, [2] M. Bahri, E. Hitzer, A. Hayashi, and. Ashino, An uncertainty principle for quaternion ourier transform, Comput. Math. Appl., 56(9 (2008, [3] M. Bahri,. Ashino and. Vaillancourt, Convolution theorems for quaternion ourier transform: properties and applications, Abstract and Applied Analysis, vol. 203, Article ID 62769, 203, 0 pages. [4] M. Bahri and Surahman, Discrete quaternion ourier transform and properties, Int. Journal of Math. Analysis, 7(25 (203,

7 Product theorem for quaternion ourier transform 87 [5] C. Zhu, Y. Shen, and Q. Wang, New fast algorithm for hypercomplex decomposition and cross-correlation, Journal of Systems Engineering and Electronics, 2(3 (200, eceived: December 5, 203