The dual extensions of sampling and series expansion theorems for the linear canonical transform

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. In this paper, first, we present much briefer and more direct derivations of sampling relation for time-limited signal in LCT domain. Moreover, based on the sampling relation expansion in the LCT domain, the equivalent of the classical Fourier series (FS) for the LCT is derived directly. Then, the discrete-time linear canonical transform (DTLCT) is proposed through the dual extension of linear canonical series (LCS), which is the generalization of discrete-time Fourier transform (DTFT).