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A Convolution and Correlation Theorem for the Linear Canonical Transform and Its Application

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Abstract

As a generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, but the correlation theorem, similar to the version of the Fourier transform (FT), is still to be determined. In this paper, firstly, we introduce a new convolution structure for the LCT, which is expressed by a one dimensional integral and easy to implement in filter design. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, based on the new convolution structure, the correlation theorem is derived, which is also a one dimensional integral expression. Last, as an application, utilizing the new convolution theorem, we investigate the sampling theorem for the band limited signal in the LCT domain. In particular, the formulas of uniform sampling and low pass reconstruction are obtained.

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Authors and Affiliations

  1. National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology, Harbin, 150001, P.R. China

    Deyun Wei & Qiwen Ran

  2. Natural Science Research Center, Harbin Institute of Technology, Harbin, 150001, P.R. China

    Qiwen Ran

  3. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China

    Yuanmin Li

Authors
  1. Deyun Wei
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  2. Qiwen Ran
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  3. Yuanmin Li
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Correspondence to Deyun Wei.

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Wei, D., Ran, Q. & Li, Y. A Convolution and Correlation Theorem for the Linear Canonical Transform and Its Application. Circuits Syst Signal Process 31, 301–312 (2012). https://doi.org/10.1007/s00034-011-9319-4

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  • DOI: https://doi.org/10.1007/s00034-011-9319-4

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