We saw the ideal digital differentiator frequency response in Fig.8.1, where it was noted that the discontinuity in the response at made an ideal design unrealizable (infinite order). Fortunately, such a design is not even needed in practice, since there is invariably a guard band between the highest supported frequency and half the sampling rate .
Figure 8.2 illustrates a more practical design specification for the digital differentiator as well as the performance of a tenth-order FIR fit using invfreqz (which minimizes equation error) in Octave.9.12 The weight function passed to invfreqz was 1 from 0 to 20 kHz, and zero from 20 kHz to half the sampling rate (24 kHz). Notice how, as a result, the amplitude response follows that of the ideal differentiator until 20 kHz, after which it rolls down to a gain of 0 at 24 kHz, as it must (see Fig.8.1). Higher order fits yield better results. Using poles can further improve the results, but the filter should be checked for stability since invfreqz designs filters in the frequency domain and does not enforce stability.9.13
Fitting Filters to Measured Amplitude Responses
Digital Filter Design Overview