Zita-Rev1 Delay-Line Filters

In zita-rev1, the damping filter for each delay line consists of a low-shelf filter $H_l(z)$ [452],^4.20in series with a unique first-order lowpass filter $H_h(z)$ that sets the high-frequency $t_{60}$ to be half that of the middle-band at a particular frequency $f_h$ (specified as ``HF Damping'' in the GUI). Since the filter $H_h$ is constrained to be a lowpass, $t_{60}(f)<t_{60}(f_h)$ for $f>f_h$ , i.e., the decay time gets shorter at higher frequencies.

Viewing the resulting damping filter $H_d(z)=H_l(z)H_h(z)$ as a three-band filter bank (§3.7.5), let $g_0$ and $g_m$ denote the desired band gains at dc and ``middle frequencies'', respectively.^4.21 Then the low shelf may be set for a desired dc-gain of $g_0/g_m$ , and its input (or output) signal multiplied by $g_m$ to obtain the resulting filter

$$H_l(z) \eqsp g_m + (g_0-g_m)\frac{1-p_l}{2}\frac{1+z^{-1}}{1-p_lz^{-1}},$$

where $p_l$ denotes the (real) first-order lowpass pole, given by [452]

$$p_l \isdefs \frac{1-\pi f_1T}{1+\pi f_1T},$$

where $f_1$ specifies (in Hz) the crossover point between ``low'' and ``middle'' frequencies, and $T$ denotes the sampling interval as usual.

The lowpass filter $H_h(z)$ is also first order, and to provide half the middle-band $t_{60}$ at the beginning of the ``high'' band, the lowpass should ``break'' to a gain of $g_m$ at the ``HF Damping'' frequency $f_h$ specified in the GUI. A unity-dc-gain one-pole lowpass has the form [452]

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

where the pole $p_h$ must be found to give a gain of $g_m$ at frequency $f_h$ :

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

Squaring and normalizing yields a quadratic equation of the form $p_h^2 + b\,p_h +1=0$ . Solving for $p_h$ using the quadratic formula yields

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

where

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

and the unstable solution $-b/2 + \sqrt{(b/2)^2 - 1} > 1$ is discarded. To ensure $\vert g_m\vert<1$ , the GUI must limit the middle-band $t_{60}$ to finite values. (The upper limit is presently $8$ seconds for both low and middle frequencies.)