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Farrow Structure

Taking the z transform of Eq.$ \,$(4.9) yields

$\displaystyle h_\Delta(n)$ $\displaystyle \isdef$ $\displaystyle \sum_{m=0}^M c_n(m)\Delta^m, \quad n=0,1,2,\ldots,N$  
$\displaystyle \Longleftrightarrow \quad H_\Delta(z)$ $\displaystyle \isdef$ $\displaystyle \sum_{n=0}^N h_\Delta(n)z^{-n}$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^N \left[\sum_{m=0}^M c_n(m)\Delta^m\right]z^{-n}$  
  $\displaystyle =$ $\displaystyle \sum_{m=0}^M \left[\sum_{n=0}^N c_n(m) z^{-n}\right]\Delta^m$  
  $\displaystyle \isdef$ $\displaystyle \sum_{m=0}^M C_m(z) \Delta^m \protect$ (5.10)

Since $ H_\Delta(z)$ is an $ N$th-order FIR filter, at least one of the $ C_m(z)$ must be $ N$th order, so that we need $ M\ge N$. A typical choice is $ M=N$.

Such a parametrization of a variable filter as a polynomial in fixed filters $ C_m(z)$ 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].

Figure 4.19: Farrow structure for implementing parametrized filters as a fixed-filter polynomial in the varying parameter.
\includegraphics[width=\twidth]{eps/farrow}

As we will see in the next section, Lagrange interpolation can be implemented exactly by the Farrow structure when $ M=N$. For $ M<N$, approximations that do not satisfy the exact interpolation property can be computed [148].

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Farrow Structure Coefficients
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Polynomials in the Delay