We have so far not used any assumptions regarding the microphone/speaker array to be used.

The sampling analysis of §2 on page made use of the far-field
assumption in obtaining a spatial aliasing limit that depended only on
the source angle
and spatial frequency
. Generalizing to
the *near field* (arbitrary source *distances*) means that
the sampling analysis is applicable only *locally* along the
array. That is, the wavelength seen by the line array depends on both
the source angle and the *distance* of the source to the array
(or equivalently, the relative distance of the source to the array and
to the listening point). For example, the line from the source normal
to a horizontal array (see
Fig.3 on page ) is at angle
, which is always oversampled
by the array. Points on the array far away from the normal line,
however, see an angle approaching
degrees to the right and
degrees to the left. If a source touches the array at
, then
all of array points other than the point at
see a right angle
(
degrees). This behavior means we cannot set a limited
stage-angle to avoid spatial aliasing like we did in the far-field
case (§2). We can now accept a 180-degree stage, or limit
the closeness and layout-width of the sources to obtain a worst-case
angle limit (maximally close to the array at the edge of the allowed
stage), and treat that as before.

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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