  A simple approach to microphone- and speaker-arrays is described in which the
microphone array is regarded as a sampling grid for the acoustic field, and the
corresponding speaker-array is treated as a "spatial digital to analog
converter" that reconstructs the acoustic field from its spatial samples.
Advantages of this approach include ease of understanding and teaching, ease of
deployment, effective practical guidelines for deployment, and significant
computational savings in special cases. In particular, in the far-field case
(acoustic sources many wavelengths away from a linear array of speakers) it is
possible to quantize source angles slightly so that no processing per speaker
is required beyond pure integer delay. Smoothly moving sources are obtained
using well known delay-line interpolation techniques such as linear
(cross-fading) and Lagrange (polynomial) interpolation between/among speakers.
We call the far-field line-array case Planewave-Based Angle Panning (PBAP), in
reference to the well-known Vector-Based Amplitude Panning (VBAP) family of
techniques, some of which are derived here as special cases: When speakers
undersample the acoustic field, the result may be considered a form of VBAP,
and VBAP is also obtained as a limiting case of polygonal PBAP arrays truncated
to the polygon perimeter. Spatial samples need not be on a linear array,
leading to a simple spatial audio system we call Huygens Arrays (HA). HAs are
quite general for sources located behind the speaker array, which no longer
needs to be linear, and the sources are no longer restricted to the far field.
Multiband and hybrid arrays employing VBAP (or stereo) and subwoofer(s) are
discussed, using sampling theory to inform the choices of crossover
frequencies.
