A Spatial Sampling Approach to Wave Field Synthesis:
PBAP and Huygens Arrays¹

Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA)

Abstract:

A simple approach to microphone- and speaker-arrays is described in which the microphone array is regarded as a sampling grid for the acoustic field, and the corresponding speaker-array is treated as a ``spatial digital to analog converter'' that reconstructs the acoustic field from its spatial samples. Advantages of this approach include ease of understanding and teaching, ease of deployment, effective practical guidelines for deployment, and significant computational savings in special cases. In particular, in the far-field case (virtual acoustic sources many wavelengths away from a linear array of speakers) it is possible to quantize source angles slightly so that no processing per speaker is required beyond pure integer delay. Smoothly moving sources are obtained using well known delay-line interpolation techniques such as linear (cross-fading) and Lagrange (polynomial) interpolation between/among speakers. We call the far-field line-array case Planewave-Based Angle Panning (PBAP), in reference to the well-known Vector-Base Amplitude Panning (VBAP) family of techniques, some of which are derived here as special cases: When speakers undersample the acoustic field, the result may be considered a form of VBAP, and VBAP is also obtained as a limiting case of polygonal PBAP arrays truncated to the polygon perimeter. Spatial samples need not be on a linear array, leading to a simple spatial audio system we call Huygens Arrays (HA). HAs are quite general for sources located behind the speaker array, which no longer needs to be linear, and the sources are no longer restricted to the far field. Multiband and hybrid arrays employing VBAP (or stereo) and subwoofer(s) are discussed, using sampling theory to inform the choices of crossover frequencies. In summary, various sound spatialization techniques are discussed, spanning VBAP, PBAP, Huygens Arrays, and special cases of Wave Field Synthesis (WFS). All may be unified under the general topic of spatial acoustic field sampling, and all were suggested by attempts to derive WFS systems as properly bandlimited spatial interpolators.