#### Classic Analog Phase Shifters

Setting in Eq.(8.19) gives the frequency response of the analog-phaser transfer function to be

*break frequency*of the allpass section.

^{9.23}

Figure 8.24a shows the phase responses of four first-order analog allpass filters with set to . Figure 8.24b shows the resulting normalized amplitude response for the phaser, for (unity feedfoward gain). The amplitude response has also been normalized by dividing by 2 so that the maximum gain is 1. Since there is an even number (four) of allpass sections, the gain at dc is . Put another way, the initial phase of each allpass section at dc is , so that the total allpass-chain phase at dc is . As frequency increases, the phase of the allpass chain decreases. When it comes down to , the net effect is a sign inversion by the allpass chain, and the phaser has a notch. There will be another notch when the phase falls down to . Thus, four first-order allpass sections give two notches. For each notch in the desired response we must add two new first-order allpass sections.

From Fig.8.24b, we observe that the first notch is near
Hz. This happens to be the frequency at which the first allpass pole
``breaks,'' *i.e.*,
. Since the phase of a first-order
allpass section at its break frequency is , the sum of the
other three sections must be approximately
.
Equivalently, since the first section has ``given up'' radians
of phase at
, the other three allpass sections
*combined* have given up radians as well (with the second
section having given up more than the other two).

In practical operation, the break frequencies must *change
dynamically*, usually periodically at some rate.

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Classic Virtual Analog Phase Shifters

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Matlab Code for Inverse Filtering