Taking the z transform of Eq.
(4.9) yields
Such a parametrization of a variable filter as a polynomial in fixed filters is called a Farrow structure [135,504]. When the polynomial Eq. (4.10) is evaluated using Horner's rule,^{5.6} the efficient structure of Fig.4.19 is obtained. Derivations of Farrow-structure coefficients for Lagrange fractional-delay filtering are introduced in [504, §3.3.7].
As we will see in the next section, Lagrange interpolation can be implemented exactly by the Farrow structure when . For , approximations that do not satisfy the exact interpolation property can be computed [149].