Since high-quality fractional-delay filtering is expensive, it is worth considering restriction to angles-of-arrival corresponding to integer delays (in samples). If the speaker-to-speaker spacing along a line array is , then the speaker-to-speaker delay for a plane wave at angle-of-incidence is , where denotes sound speed. Thus, an angle-of-arrival corresponds to an integer speaker-to-speaker delay (in samples) when
Note that increasing the speaker spacing for a given temporal sampling rate gives more integer-delay angles . However, doing this also decreases the stage-width (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 inter-speaker delay makes all the delays integer. Finally, this can all be implemented as a single delay line with a tap (non-interpolating) for each speaker signal. For moving sources, to avoid clicks, moving taps should be cross-faded from one integer delay to the next in the usual way (Smith, 2010).9
Solving Eq.(5), the collection of angles corresponding to integer inter-speaker delays (in samples) is
Figure 4 depicts the available geometric rays of plane-wave propagation for this example. Thicker rays are drawn for 0 degrees (directly in front) and degrees (full left and right).
If the 21 angles-of-arrival across a stage listed in Eq.(7) are deemed sufficient, then PBAP is essentially free: just provide the appropriate integer adjacent-speaker 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 circular-buffer pointer-increment each sampling instant (Smith, 2010).11