Example

• Start with a pole at dc (digital integrator):
  $$H(z) = \frac{1}{1-z^{-1}} \;\leftrightarrow\;[1,1,1,\ldots]$$

• Move it from radius 1 to radius 0.9 using $z^{-1}\leftarrow 0.9\,z^{-1}$ :
  $$H(z) = \frac{1}{1-0.9\,z^{-1}} \;\leftrightarrow\;[1,\,0.9,\,0.81,\,\ldots]$$

• Now progress from radius 0.9 to $0.8$ using
  $$z^{-1}\leftarrow 0.9\frac{1+\alpha z^{-1}}{1+\alpha}z^{-1}$$
  with $0.8=(1-\alpha)/(1+\alpha)$
  $\Rightarrow\alpha=(1-0.8)/(1+0.8)=1/0.9$ :
  $$\begin{eqnarray*} H(z) &=& \frac{1}{1-0.9\,\frac{1+\alpha z^{-1}}{1+\alpha}z^{-1}} \;=\;\frac{1}{1-0.9\,\frac{0.9+z^{-1}}{0.9+1}z^{-1}}\\ [10pt] &=& \frac{1}{1-\frac{0.81}{1.9}z^{-1}+ \frac{1}{1.9}z^{-2}} \end{eqnarray*}$$
  

Desired Pole Radius

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

$$R_k ^{t_{60}(\omega_k)/T} = 0.001$$

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

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

$\Rightarrow$

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

or

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

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