Since highquality fractionaldelay filtering is expensive, it is worth considering restriction to anglesofarrival corresponding to integer delays (in samples). If the speakertospeaker spacing along a line array is , then the speakertospeaker delay for a plane wave at angleofincidence is , where denotes sound speed. Thus, an angleofarrival corresponds to an integer speakertospeaker delay (in samples) when
Note that increasing the speaker spacing for a given temporal sampling rate gives more integerdelay angles . However, doing this also decreases the stagewidth (or supported bandwidth) by the same factor.
It is clearly inaudible to shift the location of each virtual source so that the time delay to the nearest speaker is an integer number of samples. Then having an integer number of samples for each interspeaker delay makes all the delays integer. Finally, this can all be implemented as a single delay line with a tap (noninterpolating) for each speaker signal. For moving sources, to avoid clicks, moving taps should be crossfaded from one integer delay to the next in the usual way (Smith, 2010).^{9}
Solving Eq.(5), the collection of angles corresponding to integer interspeaker delays (in samples) is
(6) 
Figure 4 depicts the available geometric rays of planewave propagation for this example. Thicker rays are drawn for 0 degrees (directly in front) and degrees (full left and right).

If the 21 anglesofarrival across a stage listed in Eq.(7) are deemed sufficient, then PBAP is essentially free: just provide the appropriate integer adjacentspeaker delays for each source in the sum for each speaker. As is well known, an integer delay is an computation, requiring only a single read, write, and circularbuffer pointerincrement each sampling instant (Smith, 2010).^{11}
http://arxiv.org/abs/1911.07575
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