New sampling formulae related to linear canonical transform

Linear canonical transform (LCT) is an integral transform with four parameters a, b, c, d and has been shown to be a powerful tool for optics, radar system analysis, filter design, phase retrieval, pattern recognition, and many other applications. Many well-known transforms such as the Fourier transform, the fractional Fourier transform, and the Fresnel transform can be seen as special cases of the linear canonical transform. In this paper, new sampling formulae for reconstructing signals that are band-limited or time-limited in the linear canonical transform sense have been proposed. Firstly, the sampling theorem representation of band-limited signals associated with linear canonical transform from the samples taken at Nyquist rate is derived in a simple way. Then, based on the relationship between the Fourier transform and the linear canonical transform, the other two new sampling formulae using samples taken at half the Nyquist rate from the signal and its first derivative or its generalized Hilbert transform are obtained. The well-known sampling theorems in Fourier domain or fractional Fourier domain are shown to be special cases of the achieved results. The experimental results are also proposed to verify the accuracy of the obtained results. Finally, discussions about these new results and future works related to the linear canonical transform are proposed.