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Basic idea: Each FIR filter coefficient becomes a polynomial in the delay parameter $\Delta$ :

$\displaystyle h_\Delta(n)$	$\displaystyle \isdef$	$\displaystyle \sum_{m=0}^P c_n(m)\Delta^m, \quad n=0,1,2,\ldots,N$
$\displaystyle \Leftrightarrow \; H_\Delta(z)$	$\displaystyle \isdef$	$\displaystyle \sum_{n=0}^N h_\Delta(n)z^{-n}$
	$\displaystyle =$	$\displaystyle \sum_{n=0}^N \left[\sum_{m=0}^P c_n(m)\Delta^m\right]$
	$\displaystyle =$	$\displaystyle \sum_{m=0}^P \left[\sum_{n=0}^N c_n(m) z^{-n}\right]\Delta^m$
	$\displaystyle \isdef$	$\displaystyle \sum_{m=0}^P C_m(z) \Delta^m \protect$	(K.1)

Such a parametrization of a variable filter as a polynomial in fixed filters

is called a Farrow structure [133,515], When the polynomial is evaluated using Horner's rule, the efficient structure of Fig.K.10 is obtained.

**Figure K.10:** Farrow structure for implementing parametrized filters as a fixed-filter polynomial in the varying parameter.
$\includegraphics[width=\twidth]{eps/farrow}$

Since, in the time domain, a Taylor series expansion of $x(n-\Delta)$ about time gives

$\displaystyle x(n-\Delta)$	$\displaystyle =$	$\displaystyle x(n) -\Delta x^\prime(n) + \frac{\Delta^2}{2!} x^{\prime\prime}(n) + \cdots + \frac{(-\Delta)^k}{k!}x^{(k)}(n) + \cdots$
	$\displaystyle \leftrightarrow$	$\displaystyle X(z)\left[1 - \Delta D(z) + \frac{\Delta^2}{2!} D^2(z) + \cdots + \frac{(-\Delta)^k}{k!}D^k(z) + \cdots \right]$	(K.2)