Reconstruction of multidimensional bandlimited signals from multichannel samples in linear canonical transform domain

Abstract

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. In this study, the authors address the problem of signal reconstruction from the multidimensional multichannel samples in the LCT domain. Firstly, they pose and solve the problem of expressing the kernel of the multidimensional LCT in the elementary functions. Then, they propose the multidimensional multichannel sampling (MMS) for the bandlimited signal in the LCT domain based on a basis expansion of an exponential function. The MMS expansion which is constructed by the ordinary convolution structure can reduce the effect of the spectral leakage and is easy to implement. Thirdly, based on the MMS expansion, they obtain the reconstruction method for the multidimensional derivative sampling and the periodic non-uniform sampling by designing the system filter transfer functions. Finally, the simulation results and the potential applications of the MMS are presented. Especially, the application of the multidimensional derivative sampling in the context of the image scaling about the image super-resolution is discussed.

A generalized convolution theorem for the special affine Fourier transform and its application to filtering

“…Simultaneously, as the generalization of FT, the relevant theory of SAFT has been developed including the convolution theorem, uncertainty principle, sampling theory and so on [16][17][18], which are generalizations of the corresponding properties of the FT, FRFT and LCT [3,9,[19][20][21][22][23][24][25][26][27][28]. Conventional convolution operations for FT are fundamental in the theory of linear timeinvariant (LTI) system [9].…”

A Bayesian grouplet transform

“…Grouplets provide stable geometrical image representation using an orthogonal weighted Haar lifting, where a multi-scale association field is used. Both bases and tight frames have been proposed and tested on several image processing examples such as denoising and super-resolution [20,21].…”

Uncertainty principles associated with the offset linear canonical transform

“…The offset linear canonical transform (OLCT) is known as a six parameter ( a , b , c , d , τ , η ) class of linear integral transform, which is a time‐shifted and frequency‐modulated version of the linear canonical transform (LCT) with four parameters ( a , b , c , d ) . The two extra parameters, ie, time shifting τ and frequency modulation η , make the OLCT more general and flexible, and thereby the OLCT can apply to most electrical and optical signal systems.…”

“…The offset linear canonical transform (OLCT) [1][2][3][4] is known as a six parameter (a, b, c, d, , ) class of linear integral transform, which is a time-shifted and frequency-modulated version of the linear canonical transform (LCT) with four parameters (a, b, c, d). [5][6][7][8][9][10][11] The two extra parameters, ie, time shifting and frequency modulation , make the OLCT more general and flexible, and thereby the OLCT can apply to most electrical and optical signal systems. It basically says that the Fourier transform (FT), the fractional Fourier transform (FrFT), the Fresnel transform (FnT), the LCT, and many other widely used linear integral transforms in signal processing and optics are all special cases of the OLCT.…”