Convolution theorems for the linear canonical transforms

The linear canonical transforms (LCTs) are a Lie group of transforms including the Fresnel and Fourier transforms that describe scalar wave propagation in quadratic phase systems. As such, they are useful in system analysis and design, and their discretisations are important for opto-numerical systems, e.g. numerical reconstruction algorithms in digital holography. An important topic in the literature is therefore the generalization of Fourier transform properties for the LCTs. A number of authors have proposed convolution theorems for the linear canonical transform, with different goals in mind. In this paper, we compare those methods, with particular attention being paid to the consequences of discretization. In a similar way to how discrete convolution associated with the DFT differs from that associated with the Fourier transform, we must take the chirp-periodic nature of discrete LCTs into account when determining the discrete convolution associated with LCTs. This work is of significance for the simulation of VanderLugt correlators, which have been used for optical implementations of neural networks, and for optical filtering operations and coherent optical signal processing in general.