Taking the z transform of Eq.(4.9) yields
When the polynomial Eq.(4.10) is evaluated using Horner's rule,5.6the efficient Farrow structure [135,506] depicted in Fig.4.19 is obtained. Derivations of Farrow-structure coefficients for Lagrange fractional-delay filtering are introduced in [506, §3.3.7].
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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.