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.
Figure 8.1:
Imaginary part of the frequency response
of the ideal digital differentiator plotted over
the unit circle in the plane (the real part being zero).

The discontinuity at
alone is enough to ensure that no
finiteorder 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
over the interval
, where
is the upper limit of human
hearing. For small guard
bands
, the
filter order required for a
given error tolerance is approximately inversely proportional to
.
Next Section: Digital Filter Design OverviewPrevious Section: Relation to Finite Difference Approximation