Phasing with 2nd-Order Allpass Filters

The allpass structure proposed in [433] 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 [433].
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

where

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.

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