Delay-Filter Design

Let $t_{60}(\omega)$ denote the desired reverberation time at radian frequency $\omega$, and let $H_i(z)$ denote the transfer function of the lowpass filter to be placed in series with delay line $i$. The problem we consider now is how to design these filters to yield the desired reverberation time. We will specify an ideal amplitude response for $H_i(z)$ based on the desired reverberation time at each frequency, and then use conventional filter-design methods to obtain a low-order approximation to this ideal specification.

Since losses will be introduced by the substitution $z^{-1}\leftarrow G(z)z^{-1}$, we need to find its effect on the pole radii of the lossless prototype. Let

$$p_i \isdef e^{j\omega_iT}$$

denote the $i$th pole. (Recall that all poles of the lossless prototype are on the unit circle.) If the per-sample loss filter $G(z)$ were zero phase, then the substitution $z^{-1}\leftarrow G(z)z^{-1}$ would only affect the radius of the poles and not their angles. If the magnitude response of $G(z)$ is close to 1 along the unit circle, then we have the approximation that the $i$th pole moves from $z=e^{j\omega_iT}$ to

$$p_i = R_i e^{j\omega_iT}$$

where

$$R_i = G\left(R_i e^{j\omega_iT}\right) \approx G\left(e^{j\omega_iT}\right).$$

In other words, when $z^{-1}$ is replaced by $G(z)z^{-1}$, where $G(z)$ is zero phase and $\vert G(e^{j\omega})\vert$ is close to (but less than) 1, a pole originally on the unit circle at frequency $\omega_i$ moves approximately along a radial line in the complex plane to the point at radius $R_i \approx G\left(e^{j\omega_iT}\right)$.^3.15

The radius we desire for a pole at frequency $\omega_i$ is that which gives us the desired $t_{60}(\omega_i)$:

$$R_i ^{t_{60}(\omega_i)/T} = 0.001$$

Thus, the ideal per-sample filter $G(z)$ satisfies

$$\left\vert G(\omega)\right\vert^{t_{60}(\omega)/T} = 0.001$$

The lowpass filter in series with a length $M_i$ delay line should therefore approximate

$$H_i(z) = G^{M_i}(z)$$

which implies

$$\left\vert H_i(e^{j\omega T})\right\vert^{\frac{t_{60}(\omega)}{M_iT}} = 0.001.$$

Taking $20\log_{10}$ of both sides gives

$$20 \log_{10}\left\vert H_i(e^{j\omega T})\right\vert = -60 \frac{M_i T}{t_{60}(\omega)}. \protect$$ (3.5)

This is the same formula derived by Jot [220] using a somewhat different approach.

Now that we have specified the ideal delay-line filter $H_i(e^{j\omega T})$, any number of filter-design methods can be used to find a low-order $H_i(z)$ which provides a good approximation to satisfying Eq.$\,$(2.5). Examples include the functions invfreqz and stmcb in Matlab. Since the variation in reverberation time is typically very smooth with respect to $\omega$, the filters $H_i(z)$ can be very low order.