Farrow Structure for Variable Delay FIR Filters

In §I.1.2, we noted that Lagrange interpolation is maximally flat in the frequency domain, while the Farrow filters Eq. (I.3) yielded a maximally flat error in the time domain. This section derives Farrow filters for which implement Lagrange interpolation [477]. Interestingly, the Farrow filters in this case are classic finite difference filters (low-order approximations to true differentiators).

As described in [477], solve the $N_\Delta$ equations

$$z^{-\Delta_i} = \sum_{k=0}^N C_k(z) \Delta_i^k, \quad i=1,2,\ldots,N_\Delta$$

for the $N+1$ FIR transfer functions $C_k(z)$, each order $N$ in general (must invert a constant Vandermonde matrix)

• Each coefficient of any Nth-order FIR interpolating filter can be expressed as an Nth-order polynomial in the fractional delay parameter $\Delta$ (see Farrow reference in [248])
• Each polynomial coefficient depends only on filter input samples and not on $\Delta\Rightarrow$ very convenient to modulate the delay parameter $\Delta$
• When the polynomial above is evaluated using Horner's rule, the resulting filter structure is as shown in the above figure