Taking the z transform of Eq.(4.9) yields
When the polynomial Eq.(4.10) is evaluated using Horner's rule,^{5.6}the efficient Farrow structure [135,504] depicted in 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 [].
In summary, the Farrow structure was obtained by writing the variable FIR-filter transfer-function as a polynomial in the control-variable ( above), where the polynomial coefficients are fixed (time-invariant) filters ( above). In this form, it is clear that the response of the resulting variable filter is always well defined, being a variable mix of static filters at all times.