Delay-Filter Design

Let

• $t_{60}(\omega)$ = desired reverberation time at frequency $\omega$
• $H_i(z)$ = lowpass filter for delay-line $i$

How do we design $H_i(z)$ to achieve $t_{60}(\omega)$ ?

Let

$$p_i \mathrel{\stackrel{\Delta}{=}}e^{j\omega_iT}$$

denote the $i$ th pole of the lossless prototype. Neglecting phase in the loss filter $G(z)$ , the substitution

$$z^{-1}\leftarrow G(z)z^{-1}$$

only affects the pole radius, not angle.

Assuming $G(e^{j\omega T})\approx 1$ , pole $i$ moves from $z=e^{j\omega_iT}$ to

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

where

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

Desired Pole Radius

Pole radius $R_i$ and $t_{60}$ are related by

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

The ideal loss filter $G(z)$ therefore satisfies

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

The desired delay-line filters are therefore

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

$\Rightarrow$

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

or

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

Now use invfreqz or stmcb, etc., in Matlab to design low-order filters $H_i(z)$ for each delay line.