... Arrays¹

Published 2019-11-18: http://arxiv.org/abs/1911.07575

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... point.²

A nice overview of ambisonics references appears on the Web:
http://www.york.ac.uk/inst/mustech/3d_audio/gerzonrf.htm

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...BerkhoutEtAl93,³

https://en.wikipedia.org/wiki/Wave_field_synthesis

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... avoided.⁴

Ambisonics can also approach physical completeness of the soundfield, as the order (number of spherical harmonics) increases, but most practical systems are fairly low order. (The highest order used at CCRMA is presently seven.) First-order ambisonics is essentially stereo (representable as a monopole plus one left-right dipole) augmented to include front-back and top-down dipoles.

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... (1690).⁵

https://www.britannica.com/science/Huygens-principle

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... approximation;⁶

Since every source-free soundfield can be constructed as a superposition of plane waves, it follows that solving the plane-wave synthesis problem is quite general.

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... sphere''.⁷

This analogy for PBAP sources as point-sources (stars) on the celestial sphere motivates the alternate name ``Star Field Synthesis (SFS).'' Viewing a starfield through a window is a vivid analogy for PBAP using a rectangular speaker array. However, diffraction effects are much more significant in SFS (especially at low frequencies) than when viewing light through a rectangular aperture, so the analogy can be misleading.

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...band-limited⁸

https://ccrma.stanford.edu/~jos/resample/

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...PASP.⁹

https://ccrma.stanford.edu/~jos/pasp/Delay_Line_Signal_Interpolation.html

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... center-front.¹⁰

http://acousticslab.org/psychoacoustics/PMFiles/Module07a.htm

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...PASP.¹¹

https://ccrma.stanford.edu/~jos/pasp/Software_Delay_Line.html

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... direction.¹²

The analogous condition for Wave Field Synthesis is to use only secondary sources that are ``illuminated'' by the virtual source being rendered (Ahrens, 2012).

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... perception.¹³

http://en.wikipedia.org/wiki/Head-related_transfer_function

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...¹⁴

http://en.wikipedia.org/wiki/Sound_localization#ITD_and_ILD.
In (Gerzon, 1974), citing Rayleigh from 1907, the low-frequency crossover is given as 700 Hz. It is also noted in (Gerzon, 1974) that pinnae filtering is thought to be important above 5 kHz.

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...SASP.¹⁵

https://ccrma.stanford.edu/~jos/sasp/Spectrum_Analysis_Windows.html

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... error.¹⁶

https://ccrma.stanford.edu/~jos/sasp/Window_Design_Linear_Programming.html

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...Wierstorf2012a.¹⁷

https://github.com/sfstoolbox/sfs

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... excursion.¹⁸

To see this, consider that the speaker is much smaller than the wavelength produced at its low end, so it can be regarded as an approximate ``point source'' in that frequency range. From the theory of a point source (Morse and Ingard, 1968, p. 310), the peak pressure-amplitude from a sinusoidally oscillating point source is proportional to the peak volume acceleration from the source, which is in turn proportional to the radial acceleration (second time-derivative of spherical radius) for any small sphere used to model the simple source. (Any sphere much smaller than a wavelength in diameter will do.) Let the spherical radius acceleration be denoted $a(t)=A\cos(\omega t)$ , which is proportional to the far-field pressure a fixed distance away. Then the peak radial excursion of the spherical surface is given by $A/\omega^2$ , and keeping the excursion fixed while decreasing $\omega$ by one octave reduces the far-field pressure by a factor of four, or $-12$ dB. When operating as a cell of a planar array, on the other hand, far-field pressure is proportional to the driver surface velocity instead of acceleration. In that case, only 6 dB per octave is lost integrating velocity to get displacement.

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... purpose.¹⁹

https://ccrma.stanford.edu/~jos/pasp/Vector_Wavenumber.html

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... reconstruction.²⁰

The 2D Fourier transform of $1/r$ can be shown to be $1/k_r=\lambda_r/(2\pi)$ :
http://sepwww.stanford.edu/public/docs/sep103/jon3/paper_html/node3.html

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... above.²¹

This spectral partition issue is related to the classic ``panning problem'' in which low frequencies see a 3dB boost relative to high frequencies, due to coherent versus noncoherent summation from stereo speakers for an off-axis listener: https://ccrma.stanford.edu/~jos/sasp/Panning_Problem.html

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....²²

https://adriftjustoffthecoast.wordpress.com/2013/06/06/2d-fourier-transform-of-the-unit-disk/, http://www.robots.ox.ac.uk/~az/lectures/ia/lect2.pdf

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... four-way:²³

https://www.acs.psu.edu/drussell/Demos/BaffledPiston/BaffledPiston.html

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... cm.²⁴

These are ``octave spaced'' from midrange to super-tweeter, with an extra large woofer. To adhere to octave spacing, the woofer diameter could be changed to 24 cm and the lower frequencies could be taken over by a subwoofer where the woofer leaves off.

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... radius.²⁵

This implies the lower speaker diameter is 1.75 wavelengths at crossover while the upper speaker diameter is 0.7 (midrange) or 0.88 (tweeter and super-tweeter) wavelengths at crossover. The geometric means of these two diameters in wavelengths are 1.1 (woofer-midrange) and 1.24 (other two crossovers) wavelengths. The general tradeoff is that driving with diameter less than a wavelength is inefficient (below cutoff), but yields nicely omnidirectional radiation, while driving with diameter much larger than a wavelength becomes highly directional.

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... divisions.²⁶

We avoid the term ``Huygens Octave Array'' (HOA), which would be nice to use for planar arrays as in Fig.9, due to the common use of HOA as ``Higher Order Ambisonics''. It is convenient to organize multiresolution line arrays into panels, so not much is lost.

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... etc.²⁷

2019-09-30: https://en.wikipedia.org/wiki/Subwoofer

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... field.²⁸

http://acousticslab.org/psychoacoustics/PMFiles/Module07a.htm

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... away.²⁹

A great demonstration of this effect in stereo, is to turn on a gated sinusoid in, say, the right speaker, cross-fade the signal over to the left speaker, then pull out the right speaker cable and hand it to the befuddled listener who still hears the tone coming out of the right speaker. Thanks to Bill Putnam for showing me this one.

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... system.³⁰

https://www.svsound.com/blogs/svs/strengths-and-pitfalls-of-big-subwoofer-drivers

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... listeners,³¹

See International Standard ISO-7029:2017 (3rd edition): https://www.iso.org/standard/42916.html

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... zone.''³²

A single-frequency soundfield is always highly correlated at distances less than a quarter wavelength or so, even when it is randomly constructed as a sum of plane waves from all directions with random phases (a ``diffuse field'' (Smith, 2010; Pierce, 1989; Beranek, 1986)). Even in a richly reverberant environment, one can imagine ``correlation bubbles'' on the order of the wavelength at each frequency.

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... documentation.³³

https://sfs.readthedocs.io/en/3.2/problem/

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... formulas.³⁴

For a definition of ``frequency response'' and related terms, see, e.g., (Smith, 2007a).

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... circle.³⁵

Matlab for these figures is available at
https://ccrma.stanford.edu/~jos/huygens/matlab.tgz

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