### Digital Differentiator Design

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}

**Next Section:**

Fitting Filters to Measured Amplitude Responses

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Digital Filter Design Overview