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Farrow Structure

Taking the z transform of Eq.(4.9) yields

$\displaystyle h_\Delta(n)$ $\displaystyle \isdef$ $\displaystyle \sum_{m=0}^{N_c}c_n(m)\Delta^m, \quad n=0,1,2,\ldots,N$  
$\displaystyle \Longleftrightarrow \quad
H_\Delta(z)$ $\displaystyle \isdef$ $\displaystyle \sum_{n=0}^N h_\Delta(n)z^{-n}$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^N \left[\sum_{m=0}^{N_c}c_n(m)\Delta^m\right]z^{-n}$  
  $\displaystyle =$ $\displaystyle \sum_{m=0}^{N_c}\left[\sum_{n=0}^N c_n(m) z^{-n}\right]\Delta^m$  
  $\displaystyle \isdef$ $\displaystyle \sum_{m=0}^{N_c}C_m(z) \Delta^m.
\protect$ (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 $ {N_c}\ge N$ . A typical choice is $ {N_c}=N$ .

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].

Figure 4.19: Farrow structure for implementing parametrized filters as a fixed-filter polynomial in the varying parameter.

As we will see in the next section, Lagrange interpolation can be implemented exactly by the Farrow structure when $ {N_c}=N$ . For $ {N_c}<N$ , 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 ($ \Delta$ above), where the polynomial coefficients are fixed (time-invariant) filters ($ C_m(z)$ 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.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University