Taking the z transform of Eq.(4.9) yields
Since is an th-order FIR filter, at least one of the must be th order, so that we need . A typical choice is .
Such a parametrization of a variable filter as a polynomial in fixed filters is called a Farrow structure [134,502]. When the polynomial Eq.(4.10) is evaluated using Horner's rule,5.5 the efficient structure of Fig.4.19 is obtained. Derivations of Farrow-structure coefficients for Lagrange fractional-delay filtering are introduced in [502, §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 .
Farrow Structure Coefficients
Polynomials in the Delay