### 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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Length Three FIR Loop Filter

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Phasing with First-Order Allpass Filters