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First-Order Allpass Interpolation

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

Figure 4.3: Allpass-interpolated delay line.
\includegraphics[width=\twidth]{eps/delayai}

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

\begin{eqnarray*} {\hat x}(n-\Delta) \isdef y(n) &=& \eta \cdot x(n) + x(n-1) - ... ...y(n-1) \\ &=& \eta \cdot \left[ x(n) - y(n-1)\right] + x(n-1). \end{eqnarray*}

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

The transfer function is

$\displaystyle H(z) = \frac{\eta + z^{-1}}{1 + \eta z^{-1}}. \protect$ (5.2)

At low frequencies ($ z\to 1$), the delay becomes

$\displaystyle \Delta \approx \frac{1-\eta}{1+\eta} \protect$ (5.3)

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.

Figure 4.4: Allpass-interpolation phase delay for a variety of desired delays (exact at dc).
\includegraphics[width=\twidth]{eps/allpass1}

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

$\displaystyle \eta \approx \frac{1-\Delta}{1+\Delta} $

From Eq.$ \,$(4.2), we see that the allpass filter's pole is at $ z=-\eta$, and its zero is at $ z=-1/\eta$. A pole-zero diagram for $ \Delta =0.1$ is given in Fig.4.5. Thus, zero delay is provided by means of a 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.,

$\displaystyle \Delta\in[0.1,1.1] \;\longleftrightarrow\; \eta\in[-0.05,0.82] $

Figure 4.5: Allpass-interpolator pole-zero diagram for $ \Delta =0.1$.
\includegraphics[width=4.8in]{eps/ap1pz}
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 $ z=-1$, another undesirable artifact appears as $ \Delta\to0$: The transient response also becomes long when the pole at $ z=-\eta$ gets close to the unit circle.

A plot of the impulse response for $ \Delta =0.1$ 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.

Figure 4.6: Impulse response of the first-order allpass interpolator for $ \Delta =0.1$.
\includegraphics[width=\twidth]{eps/ap1ir}

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 $ f_s/2$. For high quality sampling rates, such as larger than $ f_s=40$ 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 $ f_s/2$ 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 $ R$ is approximately

$\displaystyle \frac{\tau}{T} \approx \frac{1}{1-R}, $

and since a 60-dB decay occurs in about 7 time constants (``$ t_{60}$'') [451, p. 38], we can limit the pole of the allpass filter to achieve any prescribed specification on maximum impulse-response duration.

For example, suppose 100 ms is chosen as the maximum $ t_{60}$ allowed at a sampling rate of $ f_s=10,000$. Then applying the above constraints yields $ \eta\leq 0.993$, corresponding to the allowed delay range $ [0.00351,1.00351]$.

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