#### Differentiator Filter Bank

Since, in the time domain, a Taylor series expansion of about time gives

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

in the limit, as the number of terms goes to infinity. In other terms, the coefficient of in the polynomial expansion Eq.(4.10) must become proportional to the

*th-order differentiator*as the polynomial order increases. For any finite , we expect to be close to some scaling of the th-order differentiator. Choosing as in Eq.(4.12) for finite gives a

*truncated Taylor series approximation*of the ideal delay operator in the time domain [184, p. 1748]. Such an approximation is ``maximally smooth'' in the time domain, in the sense that the first derivatives of the interpolation error are zero at .

^{5.6}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 (
).

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