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.

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

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 .

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University