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 (
).
http://arxiv.org/abs/1911.07575
.