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Circular PBAP Truncates to VBAP

In the limit as the number of polygon sides $ N$ becomes large, we obtain a circular array, having only one speaker representing each line array in the truncated PPBAP case. Furthermore, the available angles (when avoiding interpolation) are simply the speaker angles. This coincides with zeroth-order VBAP (Vector Based Amplitude Panning) (Pulkki, 2001,1997).

In practical VBAP, sources are typically enlarged to more than a single speaker, both to make a sonically larger source, and so that they pan more smoothly from one location to the next around the ring (or dome, etc.). This of course gets us into interpolation, and can be viewed as such. The multispeaker interpolation strategies of VBAP can be applied to PBAP, both to the interior polygon/circle as in VBAP, and more generally to the line arrays creating an $ N$ -sided polygon. In the latter case, PBAP can be viewed as a sweet-spot enlarging strategy for VBAP. The longer the tangential line arrays, the straighter the plane waves emitted, and the less sensitivity to listener position downstream from the plane waves.

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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