Variable FIR Interpolating Filter

Basic idea: Each FIR filter coefficient $h_n$ becomes an order ${N_c}$ polynomial in the delay parameter $\Delta$ :

$$\begin{eqnarray*} h_\Delta(n) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \sum_{m=0}^{N_c}c_n(m)\Delta^m, \quad n=0,1,2,\ldots,{N_h}\\ \Leftrightarrow \; H_\Delta(z) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \sum_{n=0}^{N_h}h_\Delta(n)z^{-n} \\ &=& \sum_{n=0}^{N_h}\left[\sum_{m=0}^{N_c}c_n(m)\Delta^m\right]z^{-n}\\ &=& \sum_{m=0}^{N_c}\left[\sum_{n=0}^{N_h}c_n(m) z^{-n}\right]\Delta^m \\ &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \sum_{m=0}^{N_c}C_m(z) \Delta^m \end{eqnarray*}$$

• More generally: $H_\Delta(x) = \sum_m \alpha(\Delta)C_m(z)$
  where $\alpha(\Delta)$ is provided by a table lookup
• Basic idea applies to any one-parameter filter variation
• Also applies to time-varying filters ( $\Delta\leftarrow t$ )