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 .