Orthogonalized First-Order Delay-Filter Design

In [218], first-order delay-line filters of the form

$$H_i(z) \eqsp g'_i \frac{1-p_i}{1-p_iz^{-1}}$$

are proposed. Clearly $g_i=g'_i\cdot(1-p_i)$ . This form has the advantage that the dc gain is always $H_i(1)=g'_i$ for all (stable) values of $p_i$ . Therefore, we can set $g'_i$ to give a desired reverberation time at dc, and not have to change it when $p_i$ is varied to modulate the high-frequency decay rate. As in the previous section, from Eq.(3.9), we obtain

$$g'_i \eqsp 10^{-3 M_i T / t_{60}(0)}.$$

A calculation given in [218] arrives at

$$p_i \eqsp \frac{\mbox{ln}(10)}{4}\log_{10}(g_i)\left(1-\frac{1}{\alpha^2}\right)$$

where

$$\alpha \isdef \frac{t_{60}(\pi/T)}{t_{60}(0)} \protect$$ (4.10)

denotes the ratio of reverberation time at half the sampling rate divided by the reverberation time at dc.^4.17