First-Order Allpass Interpolation
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
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.
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.
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 .
Linear Interpolation as Resampling