Constraints on additivity of the 1D discrete linear canonical transform

The continuous linear canonical transform (LCT) can describe a wide variety of paraxial (quadratic phase) first-order optical systems. Digital algorithms to numerically calculate the LCT are therefore important in modeling the field propagations and are also of interest for many digital signal-processing applications. The continuous LCT is additive (and unitary), but discretization can destroy additivity. In this paper, the general constraint sufficient to ensure the discrete LCTs are additive is derived. Often, we wish to decompose the transform into a series of more computationally efficient steps. Having previously discussed the unitarity of such algorithms, in this paper we consider how our additivity constraint applies to the direct method (DM) and spectral method (SM) algorithms. Examples are presented showing how to correct nonadditive calculations and to appropriately choose parameters.