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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}} $

or

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

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 $ N$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}}$.
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