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Interpolation Accuracy

In a uniform speaker array (line or plane) that can be considered a ``spatial digital to analog converter (D/A)'', the speaker's radiation pattern $ g(\theta)$ plays the role of ``sampling kernel,'' or ``reconstruction lowpass-filter impulse-response'' used in D/A conversion.

A speaker's radiation pattern is naturally given as a function of angle relative to the speaker's central axis. It may therefore appear to be a problem that our acoustic ``samples'' are changing (diffracting) as they propagate away from the speaker. However, this is ok as long as there is any distance from the speaker array at which the soundfield has been reconstructed (both pressure and velocity versus position). Beyond that, the wavefront ``takes care of itself'' (consider the more complete Huygens-Fresnel principle (Miller, 1991)).

In the far-field case (PBAP), we observe only traveling plane waves, where pressure and velocity are proportional to each other. In the case of a plane wave impinging on the array at angle $ \theta=0$ , the velocity is obviously in the correct direction by symmetry. For an angled plane wave, it is easy to show that the velocity angle is correct far downstream from the array, since the contribution of each point-source along the array becomes a plane wave, and the lines of constant phase are along the desired angle due to the relative timing of the point sources. Having the pressure of a plane wave propagating in the desired angle means that its velocity points in that direction as well.

It is more difficult to show velocity reconstruction in the near-field case (general HA), where diverging wavefronts are sampled and re-emitted by the array. In this case, we need to show that the pressure samples add up to give the both the correct pressure and velocity for continuing the spherical wavefront expansion from the source. We have found simulation results to be helpful in the absence of analytical results (see Appendix A). There is no error in the timing of the secondary wavefronts, making the simulation results look great (a reconstructed plane wave looks very planar), but there is in principle amplitude error along the synthesized wavefront caused by the individualized $ 1/R$ factors. This error declines in relative terms downstream from the array.

While the instantaneous pressure $ p({\ensuremath \underline{x}})$ along a line or plane does not determine the corresponding velocity, unless we can assume a traveling plane wave, etc., the wavefront pressure $ p({\ensuremath \underline{x}},t)$ over a nonzero time interval does determine the particle velocity $ u({\ensuremath \underline{x}},t)$ . In fact, the wave equation itself can be integrated to compute it. The time interval creates an interval of pressure history which allows the pressure gradient to be calculated (in a progressive wave), and the pressure gradient drives the velocity in the absence of a coincident source. Specifically, the sound velocity is given by the time-integral of the pressure-gradient divided by the air's mass-density $ \rho$ (Newton's second law of motion $ f=ma$ , which, in the wave equation, appears as $ \partial p(x,t)/\partial x =
-\rho \partial u(x,t)/\partial t$ , neglecting high-order terms (Morse and Ingard, 1968, p. 243)).

It thus suffices to reconstruct sound pressure as a function of time and position along any parallel line (or plane) in front of the array. Since we assume no sources along that line or plane, the velocity is determined by the pressure in any spatial neighborhood.

In the case of the line array, we are only concerned about the velocity vector lying in the plane determined by the line array and any vector from the line array to the sources or listeners, all of which are assumed to lie in one plane.

For a planar microphone/speaker array, the velocity vector can point anywhere in the half-space from the array toward the listeners.

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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