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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Delay-Line and Signal Interpolation
       Lagrange Interpolation
          Variable Filter Parametrizations
             Farrow Structure

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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].

Previous: Polynomials in the Delay
Next: Farrow Structure Coefficients

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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