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.

Figure 8.27: Structure of a phaser based on four second-order allpass filters.

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 [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

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

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.