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Physical Audio Signal Processing
    Artificial Reverberation
       FDN Reverberation
          Delay-Filter Design

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Delay-Filter Design

Let $t_{60}(\omega)$ denote the desired reverberation time at radian frequency $\omega$ , and let denote the transfer function of the lowpass filter to be placed in series with delay line . 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 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

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

denote the

th pole. (Recall that all poles of the lossless prototype are on the unit circle.) If the per-sample loss filter

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

is close to 1 along the unit circle, then we have the approximation that the

th pole moves from $z=e^{j\omega_iT}$ to

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

where

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

In other words, when $z^{-1}$ is replaced by $G(z)z^{-1}$ , where

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 $t_{60}(\omega_i)$ :