Frequency-Response Matching Using
Digital Filter Design Methods

The lumped modeling methods discussed in §K.3 and §K.4 are order preserving. As a result, they suffer from severe approximation error such as frequency warping and perhaps artificial damping as well. By allowing the order to increase in the digital model, it is possible to obtain arbitrarily accurate models of individual masses and springs insofar as their frequency response is concerned. A possible drawback is that the precise physical interpretation is lost for the internal state of the filter.

Given force inputs and velocity outputs, the frequency response of an ideal mass was given in Eq.$\,$(K.1.2) as

$$\Gamma_m(j\omega) = \frac{1}{m j\omega}$$

and the frequency response for a spring was given by Eq.$\,$(K.1.3) as

$$\Gamma_k(j\omega) = \frac{j\omega}{k}$$

Thus, an ideal mass is an integrator and an ideal spring is a differentiator. The modeling problem for masses and springs can thus be posed as a problem in digital filter design given the above desired frequency responses.

Consider the case of a spring model (differentiator). In audio applications, it is usually desirable to evaluate numerical approximations in the frequency domain. This is because the ear acts to a first approximation like a spectrum analyzer [546]. In the $s$ plane, an ideal differentiator can be characterized as a linear filter with transfer function $H(s)=s$. A transfer function $H(s)=s$ implies an amplitude response

$$G(\omega) = \vert\omega\vert$$

and a phase response

$$\Theta(\omega) = \sigma(\omega) \pi/2,$$

where $\sigma(\omega)$ is $1$ for positive $\omega$ and $-1$ for negative $\omega$. Thus, the amplitude response of the ideal differentiator gives a $6$ dB per octave boost, and the phase response provides a quarter cycle phase advance at each positive frequency.

In discrete time, the Laplace transform is replaced by the $z$ transform [337]. For continuous-time spectrum analysis, the frequency axis appears along the vertical coordinate in the $s$-plane (the ``$j\omega$ axis''). For discrete-time spectrum analysis, the frequency axis appears along the unit circle in the $z$-plane. Ideally, in converting from continuous time to discrete time, the $j\omega$ axis in the $s$ plane should be mapped to the unit circle in the $z$ plane without aliasing. This can be accomplished using the bilinear transform (discussed in §K.4).

The bilinear transform suggests that a single time derivative should be replaced by the recursion

$${\dot y}(t,x) \approx\frac{2}{T}\left[y(t,x) - y(t-T,x)\right] - {\dot y}(t-T,x)$$ (Q.3)

in place of the FDA recursion $[y(t,x)-y(t-T,x)]/T$. Note that this new recursion is that of a first-order allpass filter while the FDA recursion is a first-order finite-impulse-response (FIR) filter [366]. The second derivative with respect to time would be computed according to the bilinear transform by two of these allpass filters in series, or they can be combined into one second-order allpass recursion as

$$\begin{eqnarray*} {\ddot y}(t,x) &\approx& \left(\frac{2}{T}\right)^2 \left[y(t... ...\\ & & \qquad\qquad {} - 2 {\dot y}(t-T,x) - {\dot y}(t-2T,x). \end{eqnarray*}$$

As discussed in §K.4.1, the bilinear transform of an ideal differentiator has, unlike the FDA version, a pole at $z=-1$ which is right on the unit circle and undamped.

See Fig.G.14 for a graph of the frequency response of the ideal differentiator and associated discussion.