Discrete complex linear canonical transform based on super-differential operators

Abstract As a generalized form of linear canonical transform (LCT), complex linear canonical transform (CLCT) is also a powerful tool in optics, signal processing and so on. However, the existing discrete algorithm of CLCT is relatively deficient. In view of this, we propose a new method to solve discrete CLCT. To be precise, we define a new method of discrete CLCT by means of the theory of super-differential operators. Firstly, we decompose the parameter matrix of CLCT appropriately. Secondly, the concepts of coordinate multiplication operators and differential operators are used to represent each part of the decomposition. The application of operator theory can not only make the new discrete algorithm more approximate to the continuous transform, but also the obtained discrete CLCT matrix has Fourier duality relation. In the final simulation, we obtain the corresponding CLCT results by changing the conditions.

[1]  Deyun Wei,et al.  Random discrete linear canonical transform. , 2016, Journal of the Optical Society of America. A, Optics, image science, and vision.

[2]  Deyun Wei,et al.  The dual extensions of sampling and series expansion theorems for the linear canonical transform , 2015 .

[3]  Aykut Koç,et al.  Efficient computation of quadratic-phase integrals in optics. , 2006, Optics letters.

[4]  Yuan-Min Li,et al.  Convolution and Multichannel Sampling for the Offset Linear Canonical Transform and Their Applications , 2019, IEEE Transactions on Signal Processing.

[5]  Z. Zalevsky,et al.  The Fractional Fourier Transform: with Applications in Optics and Signal Processing , 2001 .

[6]  J. Sheridan,et al.  Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach , 1994 .

[7]  Ran Tao,et al.  New sampling formulae related to linear canonical transform , 2007, Signal Process..

[8]  Billur Barshan,et al.  Optimal filtering with linear canonical transformations , 1997 .

[9]  José A Rodrigo,et al.  Optical system design for orthosymplectic transformations in phase space. , 2006, Journal of the Optical Society of America. A, Optics, image science, and vision.

[10]  Bing-Zhao Li,et al.  Sampling in the Linear Canonical Transform Domain , 2012 .

[11]  Kecheng Zhang,et al.  Aliased polyphase sampling theorem for the offset linear canonical transform , 2020 .

[12]  Cagatay Candan,et al.  Digital Computation of Linear Canonical Transforms , 2008, IEEE Transactions on Signal Processing.

[13]  Zhi-Chao Zhang,et al.  New convolution structure for the linear canonical transform and its application in filter design , 2016 .

[14]  Bin Du,et al.  Uncertainty relations for signal concentrations associated with the linear canonical transform , 2018, Digit. Signal Process..

[15]  Kurt Bernardo Wolf.  Canonical transforms. II. Complex radial transforms , 1974 .

[16]  Yong Li,et al.  New convolution theorem for the linear canonical transform and its translation invariance property , 2012 .

[17]  Zhi-Chao Zhang.  Variance analysis of linear canonical Wigner distribution , 2020, Optik.

[18]  Peter Kramer,et al.  Complex Extensions of Canonical Transformations and Quantum Mechanics , 1975 .

[19]  C. Shih,et al.  Optical interpretation of a complex-order Fourier transform. , 1995, Optics letters.

[20]  Liyun Xu,et al.  Multichannel Consistent Sampling and Reconstruction Associated With Linear Canonical Transform , 2017, IEEE Signal Processing Letters.

[21]  Zhi-Chao Zhang,et al.  Uncertainty principle for linear canonical transform using matrix decomposition of absolute spread matrix , 2019, Digit. Signal Process..

[22]  K. Yosida.  Operational Calculus: A Theory of Hyperfunctions , 1984 .

[23]  Soo-Chang Pei,et al.  Fast Discrete Linear Canonical Transform Based on CM-CC-CM Decomposition and FFT , 2016, IEEE Transactions on Signal Processing.

[24]  Deyun Wei,et al.  Multichannel sampling theorem for bandpass signals in the linear canonical transform domain , 2014 .

[25]  Feng Zhang,et al.  Discrete linear canonical transform computation by adaptive method. , 2013, Optics express.

[26]  Haldun M. Ozaktas,et al.  Discrete Linear Canonical Transform Based on Hyperdifferential Operators , 2018, IEEE Transactions on Signal Processing.

[27]  Joseph Shamir,et al.  Fourier optics described by operator algebra , 1980 .

[28]  Yi Chai,et al.  Reconstruction theorem for bandlimited signals from noisy samples in the offset linear canonical transform domain , 2018, Optik.

[29]  Ran Tao,et al.  Convolution theorems for the linear canonical transform and their applications , 2006, Science in China Series F: Information Sciences.

[30]  Deyun Wei,et al.  Generalized Sampling Expansions with Multiple Sampling Rates for Lowpass and Bandpass Signals in the Fractional Fourier Transform Domain , 2016, IEEE Transactions on Signal Processing.

[31]  Yi Chai,et al.  The extrapolation of bandlimited signals in the offset linear canonical transform domain , 2019, Optik.

[32]  Zhi-Chao Zhang.  Linear canonical transform's differentiation properties and their application in solving generalized differential equations , 2019, Optik.

[33]  S. Gabarda,et al.  Chirp multiplexing and filtering in the offset linear canonical transform domain , 2020 .

[34]  J. B. Lima,et al.  Eigenstructure and fractionalization of the quaternion discrete Fourier transform , 2020 .

[35]  Kurt Bernardo Wolf,et al.  Canonical transforms. I. Complex linear transforms , 1974 .

[36]  Myung K. Kim,et al.  Digital computation of the complex linear canonical transform. , 2011, Journal of the Optical Society of America. A, Optics, image science, and vision.

[37]  Wenwen Yang,et al.  Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain , 2019, J. Frankl. Inst..

[38]  Deyun Wei,et al.  Novel convolution and correlation theorems for the fractional Fourier transform , 2016 .

[39]  Amalia Torre.  Linear and radial canonical transforms of fractional order , 2003 .

[40]  You He,et al.  Radon-Linear Canonical Ambiguity Function-Based Detection and Estimation Method for Marine Target With Micromotion , 2015, IEEE Transactions on Geoscience and Remote Sensing.

[41]  K. Wolf,et al.  Structure of the set of paraxial optical systems. , 2000, Journal of the Optical Society of America. A, Optics, image science, and vision.

[42]  John J. Healy,et al.  Linear canonical transforms : theory and applications , 2016 .

[43]  Deyun Wei,et al.  New product and correlation theorems for the offset linear canonical transform and its applications , 2018, Optik.

[44]  Naitong Zhang,et al.  Multichannel Sampling and Reconstruction of Bandlimited Signals in Fractional Fourier Domain , 2010, IEEE Signal Processing Letters.

[45]  Ran Tao,et al.  On Sampling of Band-Limited Signals Associated With the Linear Canonical Transform , 2008, IEEE Transactions on Signal Processing.

[46]  Lambertus Hesselink,et al.  Fast and accurate algorithm for the computation of complex linear canonical transforms. , 2010, Journal of the Optical Society of America. A, Optics, image science, and vision.

[47]  A. Anderson.  Canonical Transformations in Quantum Mechanics , 1992, hep-th/9205080.

[48]  Kurt Bernardo Wolf,et al.  Integral transforms in science and engineering , 1979 .

[49]  Mj Martin Bastiaans.  Wigner distribution function and its application to first-order optics , 1979 .