Additive discrete linear canonical transform and other additive discrete operations

In this paper, we derive the discrete linear canonical transform (DLCT) that has the additivity property. It is the discrete counterpart of the continuous linear canonical transform (LCT). The LCT is a generalization of the Fourier transform (FT) and the fractional Fourier transform (FRFT) and is suitable for signal analysis. The discrete counterparts of the FT and the FRFT have already been derived. However, since the DLCT has four parameters {a, b, c, d}, it is hard to derive the DLCT that has the additivity property. In this paper, we use bilinear mapping together with the discrete time Fourier transform to derive the additive DLCT successfully. We can also use the similar method to derive the discrete 2-D non-separable LCT, the discrete fractional delay, the discrete fractional scaling, the discrete fractional differentiation, and the discrete geometric twisting operations that have the additivity property successfully.