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All-pass filters

Sometimes a filter is applied to get a desired phase change, rather than to alter the amplitudes of the frequency components of a sound. To do this we would need a way to design a filter with a constant, unit frequency response but which changes the phase of an incoming sinusoid in a way that depends on its frequency. We have already seen in Chapter 7 that a delay of length $d$ introduces a phase change of $- d \omega$, at the angular frequency $\omega $. Another class of filters, called all-pass filters, can make phase changes which are more interesting functions of $\omega $.

To design an all-pass filter, we start with two facts: first, an elementary recirculating filter and an elementary non-recirculating one cancel each other out perfectly if they have the same gain coefficient. In other words, if a signal has been put through a one-zero filter, either real or complex, the effect can be reversed by sequentially applying a one-pole filter, and vice versa.

The second fact is that the elementary non-recirculating filter of the second form has the same frequency response as that of the first form; they differ only in phase response. So if we combine an elementary recirculating filter with an elementary non-recirculating one of the second form, the frequency responses cancel out (to a flat gain independent of frequency) but the phase response is not constant.

To find the transfer function, we choose the same complex number $P<1$ as coefficient for both elementary filters and multiply their transfer functions:

\begin{displaymath}
H(Z) = {{
{\overline{P} - {Z^{-1}}}
} \over {
{1 - P{Z^{-1}}}
}}
\end{displaymath}

The coefficient $P$ controls both the location of the one pole (at $P$ itself) and the zero (at $1/\overline{P}$). Figure 8.23 shows the phase response of the all-pass filter for four real-valued choices $p$ of the coefficient. At frequencies of $0$, $\pi $, and $2\pi $, the phase response is just that of a one-sample delay; but for frequencies in between, the phase response is bent upward or downward depending on the coefficient.

Figure 8.23: Phase response of all-pass filters with different pole locations $p$. When the pole is located at zero, the filter reduces to a one-sample delay.
\begin{figure}\psfig{file=figs/fig08.23.ps}\end{figure}

Complex coefficients give similar phase response curves, but the frequencies at which they cross the diagonal line in the figure are shifted according to the argument of the coefficient $P$.


next up previous contents index
Next: Applications Up: Designing filters Previous: Impulse responses of recirculating   Contents   Index
Miller Puckette 2006-12-30