A frequency domain proof for the equivalence of the maximally flat FIR fractional delay filter and the Lagrange interpolator

One of the most important properties of the maximally flat (MF) FIR fractional delay (FD) filter is its equivalence with the Lagrange interpolator for uniformly sampled signals. In this article, to provide the required background for the reader, we first propose a straightforward algebraic proof for this equivalence. This proof is given by simply demonstrating that the system of linear equations governing the maximally flatness property of this filter is the same as the one from which the coefficients of the Lagrange interpolator are computed. We then present the main contribution of the paper, which is a frequency domain proof for the same equivalence. In contrast with its classic counterparts which typically deploy the system of equations characterizing the maximally flatness property of the MFFD filter, the proof presented here is just based on the definition of this property. The aim of the article is to shed more light on the important equivalence it discusses by revisiting it from a new perspective.