Polynomial Parametrization of Interpolating Filter $h_\Delta$

Basic idea: Each FIR filter coefficient $h_n$ becomes a polynomial in the delay parameter $\Delta$:

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

$$\Leftrightarrow \; H_\Delta(z) \isdef \sum_{n=0}^N h_\Delta(n)z^{-n}$$

$$= \sum_{n=0}^N \left[\sum_{m=0}^P c_n(m)\Delta^m\right]$$

$$= \sum_{m=0}^P \left[\sum_{n=0}^N c_n(m) z^{-n}\right]\Delta^m$$

$$\isdef \sum_{m=0}^P C_m(z) \Delta^m \protect$$ (I.1)

Such a parametrization of a variable filter as a polynomial in fixed filters $C_m(z)$ is called a Farrow structure [124,477], When the polynomial is evaluated using Horner's rule, the efficient structure of Fig. I.4 is obtained.

Figure I.4: Farrow structure for implementing parametrized filters as a fixed-filter polynomial in the varying parameter.

Since, in the time domain, a Taylor series expansion of $x(n-\Delta)$ about time $n$ gives

$$x(n-\Delta) = x(n) -\Delta x^\prime(n) + \frac{\Delta^2}{2!} x^{\prime\prime}(n) + \cdots + \frac{(-\Delta)^k}{k!}x^{(k)}(n) + \cdots$$

$$\leftrightarrow X(z)\left[1 - \Delta D(z) + \frac{\Delta^2}{2!} D^2(z) + \cdots + \frac{(-\Delta)^k}{k!}D^k(z) + \cdots \right]$$ (I.2)

where $D(z)$ denotes the transfer function of the ideal differentiator, we see that the $m$th filter in Eq. (I.1) should approach

$$C_m(z) = \frac{(-1)^m}{m!}D^m(z), \protect$$ (I.3)

in the limit, as the number of terms $P$ goes to infinity. In other terms, the coefficient $C_m(z)$ of $\Delta^m$ in the polynomial expansion Eq. (I.1) must become proportional to the $m$th-order differentiator as the polynomial order increases. For any finite $N$, we expect $C_m(z)$ to be close to some scaling of the $m$th-order differentiator. Choosing $C_m(z)$ as in Eq. (I.3) for finite $N$ gives a truncated Taylor series approximation of the ideal delay operator in the time domain [172, p. 1748]. Such an approximation is ``maximally smooth'' in the time domain, in the sense that the first $N$ derivatives of the interpolation error are zero at $x(n)$.^I.2 The approximation error in the time domain can be said to be maximally flat.

Farrow structures such as Fig. I.4 may be used to implement any one-parameter filter variation in terms of several constant filters. The same basic idea of polynomial expansion has been applied also to time-varying filters ( $\Delta\leftarrow t$).