There is an extensive literature on microphone- and speaker-arrays for
audio measurement and reconstruction (Ahrens, 2012; Pulkki, 2017). Let
denote the
number of microphones, and
the number of speakers. When
,
we have monaural recording and playback, while
describes
stereo, etc.
There are many approaches to making larger microphone and/or
speaker-arrays (
and/or
greater than 2). Since only a small
number of speakers is affordable in typical practice, we are normally
very concerned with human *perception* of spatial sound
(Blauert, 1997), informing stereophonic, quadraphonic, and more generally
*ambisonic* sound systems (Cooper, 1972; Gerzon, 1985,1974).
Ambisonics extends stereo and quad with an expansion of the soundfield
in terms of *spherical harmonic* basis functions centered on
one listening point.^{2} Such systems must deal with the psychoacoustics of direction and timbre
perception in various frequency ranges and for various geometries.

Given a very large number of microphones and speakers, it is possible
to approximate complete reconstruction of the soundfield in a given
space. The best known approach to this problem is *Wave Field
Synthesis*
(WFS) (Berkhout et al., 1993),^{3}also called ``acoustic holography,'' or
``holophony'' (Berkhout, 1988). WFS reproduces (or synthesizes) a
recorded soundfield *physically*, so that psychoacoustic
questions can in principle be avoided.^{4} However, for best results at
minimum expense, psychoacoustic considerations remain important.

The basic idea of an ``acoustic curtain'' for reconstructing
soundfields in principle was described by Harvey
Fletcher (1934), and at that time, two or three speakers
was considered an adequate psychoacoustic
approximation (Steinberg, 1934). Generating wave propagation
from spherical waves (``secondary sources'') emitted along the
wavefront is the essence of Huygens' Principle
(1690).^{5} Both
Huygens and Fletcher called for a *continuum* of wavefront
samples. The theory of *bandlimited sampling* was introduced
by Nyquist (1928), which, together with basic wave theory,
can be considered the basis of this paper.

Deriving WFS begins with the Kirchhoff-Helmholtz integral (or in
simplified form from the Rayleigh integral), which expresses any
source-free acoustic field as a sum of
contributions--called secondary sources--from the boundary of
any enclosing surface (Firtha, 2018; Pierce, 1989; Berkhout et al., 1993; Ahrens, 2012). The same basic approach is used by the well known *Boundary
Element Method* (BEM) for numerically computing a wavefield from its
values along a boundary surface (Kirkup, 2007). The secondary
sources in WFS aim to reconstruct (in the listening zone) the same
soundfield produced by the original (primary) sources ``on stage.''
In practice, the secondary sources are simplified from an enclosing
sphere down to (typically) a ring of loudspeakers in a line array
around the listening space (which should not be reverberant and
ideally anechoic--a major goal of Berkhout's WFS formulation was to
include the reverberant as well as the direct soundfield). There are
many variations on the details of deriving a practical WFS system, and
some of them get close to the sampling-based point-of-view taken here.
However, there does not appear to be a WFS paper which formulates the
problem as basic soundfield sampling and reconstruction from samples
(spatial analog-to-digital and digital-to-analog conversion).
As a result, differences in final implementation do emerge, as will be
brought out below.

Far-Field WFS (FFWFS) is the limiting form of WFS in which the sources
are many wavelengths away from the recording mics and listening
audience. By adopting this simplifying assumption, which is not
restrictive in many applications, we can derive FFWFS very easily and
clearly from *sampling theory*, and this is where we begin below.

Download huygens.pdf

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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