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Speakers as Spatial Samples

The polar pattern for a microphone or loudspeaker is its gain along a circle of constant radius away from the diaphragm/driver. For a spherical-wave ``point-source'', the polar pattern is simply a constant at each radius, e.g., $ p(\theta) =p_1/r$ , where $ p_1$ denotes the pressure-scaling at $ r=1$ and $ r$ denotes the distance from the center of the source.

Since our speaker arrays are typically flat, we need to calculate a slice through the polar pattern along a listening line (or plane) which we will take to be parallel to the array and to the $ x$ axis, as shown in Fig.15. The polar-pattern slice is then be considered as one sample (interpolation kernel) used to reconstruct the soundfield at a distance $ z_l$ from the array.

Figure 15: Geometry of $ x$ -slice through polar pattern to evaluate the effective sampling kernel along a line parallel to the speaker array.
\resizebox{0.5\textwidth }{!}{\includegraphics{eps/polarpat.eps}}

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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