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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$.

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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