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

Viewing the resulting damping filter
as a
three-band filter bank (§3.7.5), let
and
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
, and its input (or output) signal
multiplied by
to obtain the resulting filter

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

where specifies (in Hz) the crossover point between ``low'' and ``middle'' frequencies, and denotes the sampling interval as usual.

The lowpass filter is also first order, and to provide half the middle-band at the beginning of the ``high'' band, the lowpass should ``break'' to a gain of at the ``HF Damping'' frequency specified in the GUI. A unity-dc-gain one-pole lowpass has the form [452]

where the pole must be found to give a gain of at frequency :

Squaring and normalizing yields a quadratic equation of the form . Solving for using the quadratic formula yields

where

and the unstable solution is discarded. To ensure , the GUI must limit the middle-band to finite values. (The upper limit is presently seconds for both low and middle frequencies.)

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University