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.

Figure 8.27: Structure of a phaser based on four second-order allpass filters.
\includegraphics[width=4.2in]{eps/allpassphaser}

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 $ \hbox{AP}_{1}^{\,g_i}$ in Fig.8.23 by $ \hbox{AP}_{2}^{\,g_i}$, for $ i=1,2,3,4$, to get Fig.8.27. The phaser will have a notch wherever the phase of the allpass chain is at $ \pi$ (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

$\displaystyle H(z) \eqsp \frac{a_2 + a_1 z^{-1} + z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}
$

where

\begin{eqnarray*}
a_1 &=& -2R\cos(\theta)\\
a_2 &=& R^2
\end{eqnarray*}

and $ R$ is the radius of each pole in the complex-conjugate pole pair, and pole angles are $ \pm\theta$. The pole angle can be interpreted as $ \theta=\omega_c T$ where $ \omega_c$ is the resonant frequency and $ T$ 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 $ a_1$ 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