### First-Order Allpass Interpolation

A delay line interpolated by a first-order allpass filter is drawn in Fig.4.3.

Intuitively, ramping the coefficients of the allpass gradually ``grows'' or ``hides'' one sample of delay. This tells us how to handle resets when crossing sample boundaries.

The difference equation is

Thus, like linear interpolation, first-order allpass interpolation requires only one multiply and two adds per sample of output.

The transfer function is

At low frequencies (), the delay becomes

Figure 4.4 shows the *phase delay* of the first-order
digital allpass filter for a variety of desired delays at dc. Since
the amplitude response of any allpass is 1 at all frequencies, there
is no need to plot it.

The first-order allpass interpolator is generally controlled by setting its dc delay to the desired delay. Thus, for a given desired delay , the allpass coefficient is (from Eq.(4.3))

*pole-zero cancellation*! Due to inevitable round-off errors, pole-zero cancellations are to be avoided in practice. For this reason and others (discussed below), allpass interpolation is best used to provide a delay range lying wholly above zero,

*e.g.*,

Note that, unlike linear interpolation, allpass interpolation is not suitable for ``random access'' interpolation in which interpolated values may be requested at any arbitrary time in isolation. This is because the allpass is

*recursive*so that it must run for enough samples to reach steady state. However, when the impulse response is reasonably short, as it is for delays near one sample, it can in fact be used in ``random access mode'' by giving it enough samples with which to work.

The `STK` class implementing allpass-interpolated delay is
`DelayA`.

#### Minimizing First-Order Allpass Transient Response

In addition to approaching a pole-zero cancellation at , another
undesirable artifact appears as
: The *transient
response* also becomes long when the pole at gets close to
the unit circle.

A plot of the impulse response for
is shown in
Fig.4.6. We see a lot of ``ringing'' near half the sampling rate.
We actually should expect this from the *nonlinear-phase
distortion* which is clearly evident near half the sampling rate in
Fig.4.4. We can interpret this phenomenon as the signal
components near half the sampling rate being delayed by different
amounts than other frequencies, therefore ``sliding out of alignment''
with them.

For audio applications, we would like to keep the impulse-response duration short enough to sound ``instantaneous.'' That is, we do not wish to have audible ``ringing'' in the time domain near . For high quality sampling rates, such as larger than kHz, there is no issue of direct audibility, since the ringing is above the range of human hearing. However, it is often convenient, especially for research prototyping, to work at lower sampling rates where is audible. Also, many commercial products use such sampling rates to save costs.

Since the time constant of decay, in samples, of the impulse response of a pole of radius is approximately

For example, suppose 100 ms is chosen as the maximum allowed at a sampling rate of . Then applying the above constraints yields , corresponding to the allowed delay range .

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