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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

(5.12) 
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
thorder differentiator as the polynomial order increases.
For any finite
, we expect
to be close to some scaling of
the
thorder differentiator. Choosing
as in Eq.(4.12)
for finite
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
derivatives of the interpolation error
are zero at
.^{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
oneparameter filter variation in terms of several constant
filters. The same basic idea of polynomial expansion has been applied
also to timevarying filters (replace
with
).
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