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Convolution for the offset linear canonical transform with gaussian weight and its application

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This paper presents the convolution for the offset linear canonical transform (OLCT) with the Gaussian weight and its applications. The product theorem is also studied. In applications, some ways to design the filters in the OLCT domain as well as the multiplicative filter and the Gaussian filter are introduced. | VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 47-54 Original article Convolution for The Offset Linear Canonical Transform with Gaussian Weight and Its Application Quan Thai Ha1,*, Lai Tien Minh2, Nguyen Minh Tuan3 1 Faculty of Mathematics, Mechanics and Informatics, VNU Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam 2 Department of Mathematics, Hanoi Architectural University, Hanoi, Vietnam 3 Department of Mathematics, VNU University of Education, 144 Xuan Thuy, Hanoi, Vietnam Received 26 November 2018 Revised 25 February 2019; Accepted 15 March 2019 Abstract: This paper presents the convolution for the offset linear canonical transform (OLCT) with the Gaussian weight and its applications. The product theorem is also studied. In applications, some ways to design the filters in the OLCT domain as well as the multiplicative filter and the Gaussian filter are introduced. Keywords and phrase: Reconstruction, Shannon theorem, convolution, filter, signal, offset linear canonical transform, fractional Fourier transform, Fourier transform. 1. Introduction Throughout this paper we shall consider parameters a, b, c, d , u0 , 0 and i will be denoted the unit imaginary number. The Offset Linear Canonical Transform (OLCT) (see [1]) of a signal f t with real parameters A a, b, c, d , u0 , 0 , (ad bc 1) is defined as FA u : f t A u, t dt , b 0 , cd A f t u : i ( u u )2 i 0u de 2 0 f d u u , b 0 0 _ Corresponding author. E-mail address: haqt80@gmail.com https//doi.org/ 10.25073/2588-1124/vnumap.4300 47 (1.1) Q.T. Ha et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 47-54 48 where A (u, t ) : K Ae ( b 0 du0 ) u0 a d 2 1 u tu t 2 u t b 2b b b 2b i idu02 e 2b , and K A . 2 bi The inverse OLCT expression is given by f (t ) A 1 FA (u ) t C FA u u , t du , A 1 (1.2) where