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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 $ H(e^{j\omega T})$ optimally approximates $ j\omega $ for $ \omega T$ between $ -\pi$ and $ \pi$ . Similarly, a digital integrator must match $ 1/j\omega$ along the unit circle in the $ z$ plane. The reason an exact match is not possible is that the ideal frequency responses $ j\omega $ and $ 1/j\omega$ , when wrapped along the unit circle in the $ z$ 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 $ H(e^{j\omega T})=j\omega $ of the ideal digital differentiator plotted over the unit circle in the $ z$ plane (the real part being zero).
\includegraphics[scale=0.9]{eps/f_ideal_diff_fr_cropped}

The discontinuity at $ z=-1$ alone is enough to ensure that no finite-order digital transfer function exists with the desired frequency response. As with bandlimited interpolation4.4), it is good practice to reserve a ``guard band'' between the highest needed frequency $ f_{\mbox{\tiny max}}$ (such as the limit of human hearing) and half the sampling rate $ f_s/2$ . In the guard band $ [f_{\mbox{\tiny max}},f_s/2]$ , digital filters are free to smoothly vary in whatever way gives the best performance across frequencies in the audible band $ [0,f_{\mbox{\tiny max}}]$ 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 $ 2f_{\mbox{\tiny max}}$ ).

In the general case of Eq.$ \,$ (8.14) with $ s=j\omega$ , 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 $ \omega $

$\displaystyle H(e^{j\omega}) \eqsp \frac{B(j\omega)}{A(j\omega)} \eqsp \frac{b_M
(j\omega)^M + \cdots b_1 j\omega + b_0}{a_N (j\omega)^N + \cdots a_1
j\omega + a_0}
$

over the interval $ -\omega_{\mbox{\tiny max}}\leq \omega \leq \omega_{\mbox{\tiny max}}$ , where $ \omega_{\mbox{\tiny max}}T<\pi$ is the upper limit of human hearing. For small guard bands $ \delta\isdeftext \pi-\omega_{\mbox{\tiny max}}T$ , the filter order required for a given error tolerance is approximately inversely proportional to $ \delta$ .


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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