Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. Many theories for this transform are already known, but the uniform sampling theorem, as well as the sampling rate conversion theory about arbitrary lattices sampling in the LCT domain are still to be determined. Focusing on these issues, this paper carefully investigates arbitrary lattices sampling, the sampling with separable matrices and nonseparable matrices, to obtain uniform sampling theorem and the sampling rate conversion theory in the LCT domain. Firstly, the spectral expression of the discrete-time signal sampled via arbitrary lattice is deduced in the LCT domain. Based on it we propose the alias-free sampling relationship between two matrices and present the perfect reconstruction expressions for bandlimited signals in the LCT domain. Secondly, for further research on discrete signals to obtain sampling rate conversion theory, we define the multidimensional discrete time linear canonical transform (MDTLCT), as well as the convolution for the MDTLCT. Thirdly, the formulas of multidimensional interpolation and decimation via integer matrices in the LCT domain are derived. Then, based on the results of interpolation and decimation, we make analyses of the sampling rate conversion via rational matrices in the LCT domain, including spectral analyses and the formulas in time domain. Finally, simulation results and the potential applications of the theories are also presented.