Sun Yannan, Li Bingzhao, Tao Ran. Research progress on discretization of linear canonical transform[J]. Opto-Electronic Engineering, 2018, 45(6): 170738. doi: 10.12086/oee.2018.170738
Citation: Sun Yannan, Li Bingzhao, Tao Ran. Research progress on discretization of linear canonical transform[J]. Opto-Electronic Engineering, 2018, 45(6): 170738. doi: 10.12086/oee.2018.170738

Research progress on discretization of linear canonical transform

    Fund Project: Supported by National Natural Science Foundation of China (61671063) and Foundation for Innovative Research Groups of the National Natural Science Foundation of China (61421001)
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  • Linear canonical transformation (LCT) is a generalization of the Fourier transform and fractional Fourier transform. The recent studies have shown that LCT is widely used in optics, signal processing and applied mathematics, and the discretization of the LCT becomes vital for the applications of LCT. Since the discretization of LCT cannot be obtained by directly sampling in time domain and LCT domain, the discretization of the LCT becomes the focus of investigation recently. Based on the development history of LCT discretization, a review of important research progress and current situation of discretization of the LCT is presented in this paper. Meanwhile, the connection among different discretization algorithms and the future development direction are given. It is of great reference value for researchers to fully understand the LCT discretization method and can further promote its engineering applications.
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  • Overview:Linear canonical transformation (LCT) is a generalized form of linear integral transform, which is a three-parameter linear integral transform. The LCT unifies a variety of transforms from the well-known Fourier transform (FT), fractional Fourier transform (FRFT) and Fresnel transform (also known as chirp convolution (CC)) to simple operations such as scaling and chirp multiplication (CM). The LCT is an important tool in optics because a broad class of optical systems including thin lenses, sections of free space in the Fresnel approximation, sections of quadratic graded-index media, and arbitrary concatenations of any number of these, sometimes referred to as first-order optical systems the paraxial light propagation can be modeled by the LCT. Besides, as a generalization of the transforms mentioned above, the LCT could be more useful and attractive in many signal processing applications, such as filter design, radar system, time-frequency analysis, phase reconstructions, and so on. On the other hand, the LCT can also be used in the fields of application mathematics, such as solution of differential equations. Therefore, the LCT has attracted a considerable amount of attention in many areas. In order to promote the applications of LCT, the discretization becomes the key vital issue of the LCT. Since the discretization of LCT cannot be obtained by directly sampling in time domain and LCT domain, it has been investigated recently. After the continuous LCT has been introduced, the definition and implementation of the discrete LCT (DLCT) have been widely considered by many researchers. Based on the development history of LCT discretization, a review of important research progress and current situation of discretization methods in the last nearly two decades is presented in this paper. The discretization algorithms include, discrete-time LCT, linear canonical series, discrete linear canonical transform. In this paper, the existing discretization methods are divided into three categories, directly discrete LCT, which appeared to have been first undertaken by Pei and Ding; operator decomposition LCT, which decomposed into products of these special operator combinations through the benefit of the additive property of the LCT; base decomposition fast discrete LCT, which was also first utilized by Hennelly and Sheridan to fast-implement DLCT; and other discrete LCT. Meanwhile, the connection among different discretization algorithms and the future development direction are given. It provides important reference value for researcher in the related fields, and can further promote its engineering application. With the deepening of research, LCT will be more and more widely used in practical applications.

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