The practical calculation of optical correlation filters in correlators that use spatial light modulators with discrete elements is based on the assumption that the image on the input modulator can be modeled as a modulated 2D comb function or 'bed of nails'. A 2D discrete Fourier transform (DFT) is used to calculate a filter that is also modeled as a modulated bed of nails. The sample values in the comb array are assigned to pixel values in the filter. This approach actually gives fairly good qualitative results in modeling correlation behavior. However, it cannot account in detail for the finite size of pixel elements. The DFT approach has problems when modeling modulators whose pixels' center positions cannot be aligned with corresponding sample values. A modified DFT algorithm and an interpolation scheme for modeling these situations is given. As a practical application of the method, we look at modeling an optical correlator whose pixels are not centered at positions that correspond the DFT sample values.
"Modeling discrete modulators for optical correlation", Proc. SPIE 2565, Optical Implementation of Information Processing, (28 August 1995); doi: 10.1117/12.217655; https://doi.org/10.1117/12.217655