Huygens Arrays (HA)

We have so far considered only PBAP and its variants, which can be
considered Far-Field Wave Field Synthesis (FFWFS), in which each source contributes a
plane wave to each listening point. This is pretty general, in
principle, because, as is well known, every source-free soundfield can
be expressed as a sum of plane waves at various amplitudes, phases,
and directions of arrival. In fact, Fourier transform methods can be
used for this
purpose.^{19}Therefore, a straightforward path from FFWFS to full-fledged WFS is
to decompose any desired soundfield into a sum of plane waves, and
then generate those plane waves using FFWFS/PBAP. There are many
known methods for so-called Plane-wave Decomposition (PWD)
(Pulkki, 2017).

A more direct extension of PBAP toward WFS is based again on simple
*sampling* of the acoustic source wave, but now allowing
*spherical waves* instead of only plane waves in the soundfield
reconstruction. We could call this *Sphere-Based Range and Angle
Panning* (SBRAP). However, reconstructing a wavefront as a
superposition of spherical waves is essentially the idea of
*Huygens' Principle*. We therefore choose the name Huygens Array
(HA) for the extension of PBAP to include spherical as well as
planar wavefronts.

For constructing a Huygens Array, each virtual source is at a known location in 3D space:

According to the basic sampling assumption in PBAP, each speaker location is also a microphone location, so we can denote the th speaker/mic location by , . Different mic distributions are obtainable via spatial resampling as before.

In PBAP, each virtual source was characterized by an angle of arrival , which determined the inter-speaker delay

in seconds along the line array, where denotes the center-to-center speaker spacing. To generalize from plane waves to spherical waves, we need both a delay and a gain describing the acoustic ray from source to speaker .

Let denote the amplitude of source at a distance one meter from its center. (Each source is assumed to be a point source for now; distributed sources can be modeled as weighted sums of point sources.) Then for the delays we have

where denotes the Euclidean norm of , denotes sound speed, and is the digital audio sampling interval, as before. For the gains we have

and, if desired, lowpass-filtering due to air absorption can be included:

where is a standard air absorption filter corresponding to propagating meters through air at some assumed standard conditions (humidity level being the most important) (Smith, 2010).

- Linear Huygens Arrays
- Sampling Spreading Loss
- Virtual Sources in Front of the Array
- Undersampled Huygens Arrays become VBAP
- More General Huygens Arrays
- Interpolation Accuracy
- Evolving Radiation Pattern (Sampling Kernel) due to Diffraction
- Specific Sample Shapes (Speaker Radiation Patterns)

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`http://arxiv.org/abs/1911.07575`

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University