Arbitrary-order non-causal lagrange-type variable fractional-delay digital filters

Liu and Wei have shown that even-order polynomial interpolation can be simplified as a variable fractional-delay (VFD) filtering problem so that the coefficients of an even-order VFD filter can be computed in a closed-form. However, the explicit formula is obtained through observing the coefficients when the polynomial degree is N = 2 and N = 4 without rigorous mathematical proof. In this paper, we derive an explicit formula for computing the coefficients of Lagrange-type VFD filters for both even-order and odd-order cases, and show that the Liu-Wei´s closed-form formula is a special case of the resulting general formula.