#### 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 .

**Next Section:**

Convolution Interpretation

**Previous Section:**

Fractional Delay Filtering by Linear Interpolation