When avoiding delay-line interpolation and accepting the angles given by integer interspeaker delays, we should choose the sampling rate and speaker line-array spacing so that the angles available from each line array include the angles to the polygon vertices.

For an -sided polygon, the two needed angles are . The set of all vertex angles is , , where is sound speed and is the sampling interval. Thus, we need some integer to give , or

for some .

In the four-quadrant case (four line arrays defining a square) with speaker spacing in (example from §2.15 below), the sampling rate wants to be a multiple of , and kHz happens to be , so the 10th angle is very close to 45 degrees. For polygon sides and four-inch speaker spacing, we need to be a multiple of , and it so happens that 44.1 kHz is close to five times that ( ).

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`http://arxiv.org/abs/1911.07575`

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University