Specific Sample Shapes (Speaker Radiation Patterns)

Perhaps the most obviously ideal sampling kernel for a speaker centered at within a line-array with cell-width is the rectangular pulse:

That is, each speaker in the array pushes a unit volume of air in one second (unit area). Such a speaker array can be regarded as a series of contiguous pistons, each of width .

The unit-area pulse function is not bandlimited. A more graceful choice is the ``spatial sinc function,'' which is the ideal sampling kernel used in bandlimited audio sampling:

where

sinc

It is well known that the sinc function is the Fourier transform of a rectangular pulse, and vice versa. Therefore, rectangular-pulse samples at the line array propagate and diffract to become overlapping sinc functions in the far field, while sinc-shaped radiation patterns diffract to become adjacent or overlapping rectangular spatial bands in the far field. Of course, a true sinc function can never be implemented precisely because it is infinite in spatial extent. Still, it is interesting to imagine disjoint spatial samples in the far field, and consider whether approaching that might be desirable in some situations, such as delivering different program material to different listening positions (a current problem in automotive sound).

The most typical example in practice is a *circular* driver--an
ordinary circular speaker cone. In this case, the far field shape is
the so-called ``Airy function'' involving Bessel function
.^{22}

In addition to the rectangular piston and sinc function, or circular driver and Airy pattern, any spectrum-analysis window , or its Fourier transform , having the constant-overlap-add (COLA) property

constant

can be used as a spatial sampling kernel, with the added stipulation that should have an effective width on the order of (one spatial sample) in order that spatial resolution be maximized. In other words, it suffices for the speaker radiation gains at a particular distance away from the array to overlap-and-add to a constant gain-versus-position at the speaker spacing used. In the limit of infinite sampling density,

Spatial oversampling gives more flexibility in the choice of radiation pattern. For example, doubling the spatial sampling rate allows the radiation patterns to overlap by an additional factor of two with no loss of spatial resolution along the line array. On the other hand, that factor of two could be given to the aliasing cutoff frequency, which is probably a better use of it.

As in audio interpolation, *windowed sinc* interpolation can be
used in practice (Smith and Gossett, 1984), if a speaker driver can be
devised to generate that radiation pattern at some distance from the
speaker.

The spatial sampling kernel should ideally be frequency
*independent*, but this is never the case for typical speaker
systems. Instead, typical speakers look like point sources at low
frequencies, radiate efficiently and widely at wavelengths comparable
to the speaker diameter, and begin to ``spotlight'' increasingly at
higher frequencies (where the diameter is multiple wavelengths).

Due to the naturally narrowing spatial beam-width with increasing
frequency for typical speaker drivers, a sufficient sampling density
for high frequencies corresponds to heavy oversampling (spatially) at
low frequencies. PBAP and its extensions therefore should either
be implemented in separate frequency bands (multiband PBAP is
discussed below), or using a new kind of speaker having a
frequency-independent radiation pattern that sums to a constant when
the speaker outputs are all added together at any point of the
listening region. One solution is to approximate a point source
(see Appendix B), for which the sampling kernel is a substantially
identical sphere for all speaker drivers much smaller than a
wavelength. Such speakers radiate inefficiently, but they are already
widely used at the low-frequency end in practice (subwoofers are
smaller than most of the wavelengths they must produce). The main
problem with a single set of point-source-approximation drivers is
that they must be packed very densely for the high frequencies and
also have long-throw excursion for low frequencies--expensive.
Furthermore, there is always intermodulation distortion in any
wideband driver (*e.g.*, Doppler shift of high-frequency components by
low-frequency excursion, which nobody apparently compensates).

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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