Ideal Differentiator (Spring Admittance)
Figure 8.1 shows a graph of the frequency response of the
ideal differentiator (spring admittance). In principle, a
digital differentiator is a filter whose frequency response
optimally approximates
for
between
and
. Similarly, a digital integrator must
match
along the unit circle in the
plane. The reason
an exact match is not possible is that the ideal frequency responses
and
, when wrapped along the unit circle in the
plane, are not ``smooth'' functions any more (see
Fig.8.1). As a result, there is no filter with a
rational transfer function (i.e., finite order) that can match the
desired frequency response exactly.
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The discontinuity at alone is enough to ensure that no
finite-order digital transfer function exists with the desired
frequency response. As with bandlimited interpolation (§4.4),
it is good practice to reserve a ``guard band'' between the highest
needed frequency
(such as the limit of human hearing) and half
the sampling rate
. In the guard band
, digital
filters are free to smoothly vary in whatever way gives the best
performance across frequencies in the audible band
at the
lowest cost. Figure 8.2 shows an example.
Note that, as with filters used for bandlimited
interpolation, a small increment in oversampling factor yields a much
larger decrease in filter cost (when the sampling rate is near
).
In the general case of Eq.(8.14) with
, digital filters
can be designed to implement arbitrarily accurate admittance transfer
functions by finding an optimal rational approximation to the complex
function of a single real variable





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Digital Filter Design Overview
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Relation to Finite Difference Approximation