High-Frequency-Damping Lowpass

High-Frequency Damping Lowpass:

$$H_h(z) \;=\;\frac{1-p_h}{1-p_hz^{-1}}$$

For $t_{60}$ at ``HF Damping'' frequency $f_h$ to be half of middle-band $t_{60}$ (gain $g_m$ ), we require

$$\left\vert H_h\left(e^{j2\pi f_hT}\right)\right\vert \;=\; \left\vert\frac{1-p_h}{1-p_he^{-j2\pi f_hT}}\right\vert \;=\;g_m$$

Squaring and normalizing yields a quadratic equation:

$$p_h^2 + b\,p_h +1=0$$

Solving for $p_h$ using the quadratic formula yields

$$p_h \;=\;-\frac{b}{2} - \sqrt{\left(\frac{b}{2}\right)^2 - 1},$$

where

$$-\frac{b}{2} \;=\;\frac{1-g_m^2\cos(2\pi f_h T)}{1-g_m^2} > 1,$$

Discard unstable solution $-b/2 + \sqrt{(b/2)^2 - 1} > 1$

To ensure $\vert g_m\vert<1$ , GUI keeps middle-band $t_{60}$ finite