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Physical Audio Signal Processing
    Transfer Function Models
       Frequency-Response Matching Using Digital Filter Design Methods
          Ideal Differentiator (Spring Admittance)

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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 interpolation (§4.4), 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_... ...ega)^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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Next: Digital Filter Design Overview

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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