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Chapters

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

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Farrow Structure Coefficients

Beginning with a restatement of Eq. $\,$ (4.9),

$\displaystyle h_\Delta(n) \isdefs \sum_{m=0}^M c_n(m)\Delta^m, \quad n=0,1,2,\ldots,N,$

we can express each FIR coefficient $h_\Delta(n)$ as a vector expression:

$\displaystyle h_\Delta(n) \eqsp \underbrace{% \left[\begin{array}{ccccc} 1 & \... ...y}{c} C_n(0) \\ [2pt] C_n(1) \\ [2pt] \vdots \\ [2pt] C_n(M)\end{array}\right]$

Making a row-vector out of the FIR coefficients gives

$\displaystyle \underbrace{\left[\begin{array}{cccc}h_\Delta(0)\!&\!h_\Delta(1)\... ...\vdots \\ C_0(M) & C_1(M) & \cdots & C_N(M) \end{array}\right]}_{\mathbf{C}}$

$\displaystyle \underline{h}_\Delta \eqsp \underline{V}_\Delta^T \mathbf{C}. \protect$

We may now choose a set of parameter values ${\underline{\Delta}}^T=[\Delta_0,\Delta_1,\ldots,\Delta_L]$ over which an optimum approximation is desired, yielding the matrix equation

$\displaystyle \mathbf{H}_{\underline{\Delta}}\eqsp \mathbf{V}_{\underline{\Delta}}\mathbf{C}, \protect$

(5.11)

where

$\displaystyle \mathbf{H}_{\underline{\Delta}}\isdefs \left[\begin{array}{c} \un... ...elta_0}^T \\ [2pt] \vdots \\ [2pt] \underline{h}_{\Delta_L}^T\end{array}\right]$ and $\displaystyle \qquad \mathbf{V}_{\underline{\Delta}}\isdefs \left[\begin{array}... ...ta_0}^T \\ [2pt] \vdots \\ [2pt] \underline{V}_{\Delta_L}^T\end{array}\right].$

Equation (4.11) may be solved for the polynomial-coefficient matrix $\mathbf{C}$ by usual least-squares methods. For example, in the unweighted case, with $L\ge M$ , we have

$\displaystyle \zbox {\mathbf{C}\eqsp \left(\mathbf{V}_{\underline{\Delta}}^T\ma... ...ight)^{-1} \mathbf{V}_{\underline{\Delta}}^T \mathbf{H}_{\underline{\Delta}}.}$

Note that this formulation is valid for finding the Farrow coefficients of any

th-order variable FIR filter parametrized by a single variable $\Delta$ . Lagrange interpolation is a special case corresponding to a particular choice of $\mathbf{H}_{\underline{\Delta}}$ .

Previous: Farrow Structure
Next: Differentiator Filter Bank

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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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