On Optimal Derivative DSP Operators for Sampled Data

Motivated by partial differential equation (PDE) models in signal processing, an l₂ approach to derivative calculation is introduced based on sampled data. This approach utilizes regularity constraints on the continuous-domain signal that are already embedded in the PDE model. In particular, the continuous-domain input signal is assumed to belong to a reproducing kernel Hilbert space and the sampling process (ideal or non-ideal) is shown to correspond to an appropriate orthogonal projection. The values of the derivative function are shown to correspond to a set of inner product calculations, giving rise to a minimax solution for an h approximation problem. Several matrix operators are then demonstrated for 1D and 2D cases, found to be superior to the backward-forward difference approach.