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Combining Line Arrays to make Polygons

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

For an $ N$ -sided polygon, the two needed angles are $ \pm\pi/N$ . The set of all vertex angles is $ \arcsin(ncT/X)$ , $ n=0,\pm1,\pm2,\dots$ , where $ c$ is sound speed and $ T=1/f_s$ is the sampling interval. Thus, we need some integer $ n$ to give $ \sin(\pi/N)\approx ncT/X$ , or

$\displaystyle f_s(n) = n \frac{\csc\left(\frac{\pi}{N}\right)}{X/c}

for some $ n=1,2,\ldots\,$ .

In the four-quadrant case (four line arrays defining a square) with speaker spacing $ X=4$ in (example from §2.15 below), the sampling rate wants to be a multiple of $ 4802.2$ , and $ 48$ kHz happens to be $ 9.9954\times 4802.2$ , so the 10th angle is very close to 45 degrees. For $ N=8$ polygon sides and four-inch speaker spacing, we need $ f_s$ to be a multiple of $ 8873.3$ , and it so happens that 44.1 kHz is close to five times that ($ 4.97$ ).

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``A Spatial Sampling Approach to Wave Field Synthesis: PBAP and Huygens Arrays'', by Julius O. Smith III, Published 2019-11-18:
Copyright © 2020-05-15 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University