Phasing with 2nd-Order Allpass Filters
The allpass structure proposed in [429] provides a convenient means for generating nonuniformly spaced notches that are independently controllable to a high degree. An advantage of the allpass approach even in the case of uniformly spaced notches (which we call flanging, as introduced in §5.3) is that no interpolating delay line is needed.
The architecture of the phaser based on second-order allpasses is
shown in Fig.8.27. It is identical to that in
Fig.8.23 with each first-order allpass being replaced by
a second-order allpass. I.e., replace
in
Fig.8.23 by
, for
, to get
Fig.8.27. The phaser will have a notch wherever the phase
of the allpass chain is at
(180 degrees). It can be shown that
these frequencies occur very close to the resonant frequencies of the
allpass chain [429].
It is therefore convenient to use a single conjugate pole pair in each
allpass section, i.e., use second-order allpass sections of the form


and is the radius of each pole in the complex-conjugate pole pair,
and pole angles are
. The pole angle can be interpreted as
where
is the resonant frequency and
is the sampling interval.
Phaser Notch Parameters
To move just one notch, the tuning of the pole-pair in the
corresponding section is all that needs to be changed. Note that
tuning affects only one coefficient in the second-order allpass
structure. (Although the coefficient appears twice in the
transfer function, it only needs to be used once per sample in a
slightly modified direct-form implementation [449].)
The depth of the notches can be varied together by changing the gain of the feedforward path.
The bandwidth of individual notches is mostly controlled by the distance of the associated pole-pair from the unit circle. So to widen the notch associated with a particular allpass section, one may increase the ``damping'' of that section.
Finally, since the gain of the allpass string is unity (by definition of allpass filters), the gain of the entire structure is strictly bounded between 0 and 2. This property allows arbitrary notch controls to be applied without fear of the overall gain becoming ill-behaved.
Phaser Notch Distribution
As mentioned above, it is desirable to avoid exact harmonic spacing of the notches, but what is the ideal non-uniform spacing? One possibility is to space the notches according to the critical bands of hearing, since essentially this gives a uniform notch density with respect to ``place'' along the basilar membrane in the ear. There is no need to follow closely the critical-band structure, so that simple exponential spacing may be considered sufficiently perceptually uniform (corresponding to uniform spacing on a log frequency scale). Due to the immediacy of the relation between notch characteristics and the filter coefficients, the notches can easily be placed under musically meaningful control.
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Phasing with First-Order Allpass Filters