Differentiator Filter Bank
Since, in the time domain, a Taylor series expansion of
about time
gives
![\begin{eqnarray*}
x(n-\Delta)
&=& x(n) -\Delta\, x^\prime(n)
+ \frac{\Delta^2...
...D^2(z) + \cdots
+ \frac{(-\Delta)^k}{k!}D^k(z) + \cdots \right]
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img1101.png)
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













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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Farrow Structure Coefficients