Farrow Structure Coefficients

Beginning with a restatement of Eq.(4.9),

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

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

$$h_\Delta(n) \eqsp \underbrace{% \left[\begin{array}{ccccc} 1 & \Delta & \Delta^2 & \cdots & \Delta^{N_c}\end{array}\right]}_{\underline{V}_\Delta^T}\, \left[\begin{array}{c} C_n(0) \\ [2pt] C_n(1) \\ [2pt] \vdots \\ [2pt] C_n({N_c})\end{array}\right]$$

Making a row-vector out of the FIR coefficients gives

$$\underbrace{\left[\begin{array}{cccc}h_\Delta(0)\!&\!h_\Delta(1)\!&\!\cdots\!&\!h_\Delta(N)\end{array}\right]}_{\underline{h}_\Delta} \eqsp \underline{V}_\Delta^T \underbrace{ \left[\begin{array}{cccc} C_0(0) & C_1(0) & \cdots & C_N(0) \\ C_0(1) & C_1(1) & \cdots & C_N(1) \\ \vdots & \vdots & & \vdots \\ C_0({N_c}) & C_1({N_c}) & \cdots & C_N({N_c}) \end{array}\right]}_{\mathbf{C}}$$

or

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

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

$$\mathbf{H}_{\underline{\Delta}}\eqsp \mathbf{V}_{\underline{\Delta}}\mathbf{C}, \protect$$ (5.11)

where

$$\mathbf{H}_{\underline{\Delta}}\isdefs \left[\begin{array}{c} \underline{h}_{\Delta_0}^T \\ [2pt] \vdots \\ [2pt] \underline{h}_{\Delta_L}^T\end{array}\right] \qquad\mbox{and}\qquad \mathbf{V}_{\underline{\Delta}}\isdefs \left[\begin{array}{c} \underline{V}_{\Delta_0}^T \\ [2pt] \vdots \\ [2pt] \underline{V}_{\Delta_L}^T\end{array}\right].$$

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 $L\ge {N_c}$ , we have

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

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 $\mathbf{H}_{\underline{\Delta}}$ .

In MATLAB, the function fdesign.fracdelay can be used to design Farrow structures for fractional delay.