We have approached the problem of soundfield synthesis from the point of view of spatial sampling theory. Starting with the simplest case of far-field sources, we derived Planewave-Based Angle Panning (PBAP) which can be considered both a special-case of wave field synthesis (WFS) for distant sources without reverberation, and a path for enlarging the ``sweet-spot'' in Vector-Base Amplitude Panning (VBAP). Linear, planar, and more generally distributed arrays were considered, with the most general case termed Huygens Arrays (HA). Considerations of basic sampling theory led to straightforward practical guidelines on how to use the arrays, such as determining speaker spacing and size, speaker radiation pattern (interpolation kernel), maximum source angle, and how closely a source can approach the array. Finally, various multiband systems were considered, allowing each driver to focus on a particular band, such as an octave band. Multiband linear arrays, for use with conventional stereo and/or subwoofer(s) at low frequencies, were found to be particularly attractive and relatively practical.
Interesting next steps include studying the overlap-add properties of the near- and far-field radiation patterns for circular and other commonly used drivers. The SFS software (linked in Appendix A) could be adapted for this purpose. The simple case of point-source drivers is given a start in Appendix B.
The shape of a spatial sample (speaker radiation pattern) can be optimized for various purposes. One is to make it easy to produce using conventional drivers, such as the simple matrix of square pistons considered in §3.8. Another is to optimize far-field considerations, such as minimizing cross-talk between angular directions (also considered in §3.8). Other optimization criteria can be imagined. For example, if the center of the array at a particular listening distance is given special status, then the array can be optimized for that region in various ways, including minimizing the spatial width of the radiation pattern and thence array truncation effects.