Differentiator Filter Bank

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

$$\begin{eqnarray*} x(n-\Delta) &=& x(n) -\Delta\, x^\prime(n) + \frac{\Delta^2}{2!} x^{\prime\prime}(n) + \cdots + \frac{(-\Delta)^k}{k!}x^{(k)}(n) + \cdots \nonumber \\ \;\longleftrightarrow\;&& X(z)\left[1 - \Delta\, D(z) + \frac{\Delta^2}{2!} D^2(z) + \cdots + \frac{(-\Delta)^k}{k!}D^k(z) + \cdots \right] \end{eqnarray*}$$

where $D(z)$ denotes the transfer function of the ideal differentiator, we see that the $m$ th filter in Eq.(4.10) should approach

$$C_m(z) \eqsp \frac{(-1)^m}{m!}D^m(z), \protect$$ (5.12)

in the limit, as the number of terms ${N_c}$ goes to infinity. In other terms, the coefficient $C_m(z)$ of $\Delta^m$ in the polynomial expansion Eq.(4.10) 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.(4.12) for finite $N$ gives a truncated Taylor series approximation of the ideal delay operator in the time domain [185, 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 $x(n)$ .^5.7 The approximation error in the time domain can be said to be maximally flat.

Farrow structures such as Fig.4.19 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 (replace $\Delta$ with $t$ ).