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Four-Quadrant PBAP

If four line arrays are arranged in a ``tic tac toe board'' configuration (or perhaps just a square) enclosing the listener in its central square, as shown in Fig.5, then each line array need only cover a $ \pm 45$ degree range, which is more uniformly sampled in angle.

Figure: Ray coverage for the same example considered in Fig.4 using four line arrays enclosing the listener at the center of a ``tic tac toe board'' configuration. The rays at $ \pm 45$ are drawn thicker.
\resizebox{0.8\textwidth }{!}{\includegraphics{eps/showangles2.eps}}

Four-Quadrant PBAP uses only a fourth of the speakers for each plane wave. Infinitely long line arrays emit cylindrical waves, which are equivalent (ignoring the wake) to plane waves in one listening plane passing through the cylindrical axis. However, more practical truncated line arrays can benefit from using more than a fourth of the speakers. It is intuitively obvious that at least half of the speakers could help construct a desired plane wave at the listener--all speakers having a radial component in the desired direction.12 At a single listening point, as in ambisonics, all of the speakers can be put to work toward approximating the desired soundfield pressure and velocity versus frequency at that point.

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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