The extrapolation of bandlimited signals in the offset linear canonical transform domain

Abstract The extrapolation theorem is a fundamental and important theory in signal analysis. As the offset linear canonical transform (OLCT) has proven to be a novel and effective method in signal processing and optics, a variety of properties and theories of the OLCT have been well studied. However, there are still no papers considered the extrapolation theorem for the OLCT bandlimited signals. Therefore in this paper, the extrapolation of the bandlimited signals associated with the OLCT based on the Gerchberg–Papoulis (GP) method has been presented for the first time. First, a useful theorem for the generalized prolate spheroidal wave function (GPSWFs) has been introduced. Then, the extrapolation of bandlimited signals in the OLCT domain has been derived based on this theorem by using the GP algorithm. Moreover, a fast computation of the extrapolation theorem in the OLCT domain also has been attained. Finally, the numerical results have been carried out to show the effective and useful of the proposed results.

[1]  A. Papoulis.  A new algorithm in spectral analysis and band-limited extrapolation. , 1975 .

[2]  Kaiyu Qin,et al.  Convolution, correlation, and sampling theorems for the offset linear canonical transform , 2014, Signal Image Video Process..

[3]  Ran Tao,et al.  Convolution and correlation theorems for Wigner-Ville distribution associated with the offset linear canonical transform , 2018 .

[4]  John T. Sheridan,et al.  Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation. , 1994, Optics letters.

[5]  Kaiyu Qin,et al.  Multichannel Sampling of Signals Band-Limited in Offset Linear Canonical Transform Domains , 2013, Circuits Syst. Signal Process..

[6]  Yigang He,et al.  Analysis of A-stationary random signals in the linear canonical transform domain , 2018, Signal Process..

[7]  John J Healy,et al.  Constraints on additivity of the 1D discrete linear canonical transform. , 2015, Applied optics.

[8]  J. Sheridan,et al.  Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization. , 2014, Journal of the Optical Society of America. A, Optics, image science, and vision.

[9]  A. Jain,et al.  Extrapolation algorithms for discrete signals with application in spectral estimation , 1981 .

[10]  Xuejun Sha,et al.  Linear Canonical Matched Filter: Theory, Design, and Applications , 2018, IEEE Transactions on Signal Processing.

[11]  Yan-Na Zhang,et al.  Φ-linear Canonical Analytic Signals , 2018, Signal Process..

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

[13]  Luca Gemignani.  Fast Inversion of Hankel and Toeplitz Matrices , 1992, Inf. Process. Lett..

[14]  Adrian Stern.  Sampling of compact signals in offset linear canonical transform domains , 2007, Signal Image Video Process..

[15]  Xuejun Sha,et al.  Filter Design for Constrained Signal Reconstruction in Linear Canonical Transform Domain , 2018, IEEE Transactions on Signal Processing.

[16]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[17]  R. Gerchberg.  Super-resolution through Error Energy Reduction , 1974 .

[18]  Christiane Quesne,et al.  Linear Canonical Transformations and Their Unitary Representations , 1971 .

[19]  Chai Yi,et al.  Nonuniform Sampling Theorems for Bandlimited Signals in the Offset Linear Canonical Transform , 2018 .

[20]  Liang Zhao,et al.  Unitary discrete linear canonical transform: analysis and application. , 2013, Applied optics.

[21]  Girish S. Agarwal,et al.  The generalized Fresnel transform and its application to optics , 1996 .

[22]  Giancarlo Pedrini,et al.  From speckle pattern photography to digital holographic interferometry [Invited]. , 2013, Applied optics.

[23]  Feng Li,et al.  Diagnosis of Gear Faults by Using the Linear Canonical Transform in Acceleration and Deceleration Processes , 2018, IEEE Access.

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

[25]  Kit Ian Kou,et al.  Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis , 2013 .

[26]  Yi Chai,et al.  Spectral Analysis of Sampled Band-Limited Signals in the Offset Linear Canonical Transform Domain , 2015, Circuits Syst. Signal Process..

[27]  Wei Zhang,et al.  A generalized convolution theorem for the special affine Fourier transform and its application to filtering , 2016 .

[28]  Xiaoping Liu,et al.  Error Analysis of Reconstruction From Linear Canonical Transform Based Sampling , 2018, IEEE Transactions on Signal Processing.

[29]  Xuejun Sha,et al.  Extrapolation of Bandlimited Signals in Linear Canonical Transform Domain , 2012, IEEE Transactions on Signal Processing.

[30]  Yi Chai,et al.  Reconstruction of digital spectrum from periodic nonuniformly sampled signals in offset linear canonical transform domain , 2015 .

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

[32]  Soo-Chang Pei,et al.  Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms. , 2003, Journal of the Optical Society of America. A, Optics, image science, and vision.

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

[34]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

[35]  Lei Huang,et al.  Nonuniform Sampling Theorems for Bandlimited Signals in the Offset Linear Canonical Transform , 2018, Circuits Syst. Signal Process..

[36]  Tianqi Zhang,et al.  Extrapolation of discrete bandlimited signals in linear canonical transform domain , 2014, Signal Process..

[37]  Tianqi Zhang,et al.  Unified approach to extrapolation of bandlimited signals in linear canonical transform domain , 2014, Signal Process..

[38]  John J. Healy,et al.  Unitary Algorithm for Nonseparable Linear Canonical Transforms Applied to Iterative Phase Retrieval , 2017, IEEE Signal Processing Letters.

[39]  Lei Huang,et al.  The properties of generalized offset linear canonical Hilbert transform and its applications , 2017, Int. J. Wavelets Multiresolution Inf. Process..

[40]  Soo-Chang Pei,et al.  Generalized prolate spheroidal wave functions for optical finite fractional Fourier and linear canonical transforms. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[41]  Yi Chai,et al.  Sampling theorems for bandlimited random signals in the offset linear canonical transform domain , 2017 .