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Extension from a 2D Listening Plane to 3D

Since PBAP converges to VBAP when using many line arrays truncated to the enclosed polygon, a simple VBAP-style extension to 3D is to place a new speaker directly overhead and a second new speaker directly below. Then, elevation cues can be imparted by mixing in the above or below speaker according to a psychoacoustically measured panning law between the array and an out-of-plane speaker above or below.

The proper extension of PBAP to 3D is of course obtained using sampled plane waves arriving at the correct 3D angles, instead of cylindrical waves sampled in the listening plane for 2D PBAP from a line array. That means our line arrays must be replaced by planar speaker arrays, and the polygonal listening space becomes a polyhedron, or sphere in the limit. This of course also reduces to non-interpolating spherical VBAP when the planar arrays are truncated down to one point in each plane, and normal VBAP interpolation can be used as described above here as well. As in the 2D case (§2.7 on page [*]), 3D PBAP can be used as a sweet-spot enlarger for 3D VBAP.

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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