On 2022-04-06, Fons Adriaensen wrote:
I ask because theory-wise pantophonic and periphonic soundfields
shouldn't be captured the same way,
There is no such thing as a 'pantophonic sound field'. We don't live
in Flatland. All microphones operate in 3D space.
Some more than others. Interferometric boom mics try to operate in 1D
land. You know, the ones which attenuate everything off-axis.
The ideal pantophonic mic array does the same in 2D. It cuts off
everything above horizon, sharply. Then the ideal pantophonic array
reproduces cylindrical waves, matched.
That's the only way the recording and reproduction conditions ever truly
match. Faller's math showed that on blackboard, and the NFC-HOA work
made it even clearer, if between the lines.
The way *I* interpret the mess is that pantophony is a mirage; there can
be only periphony within the ambisonic framework. We might argue about
its order, or about how truly regular the capture and reproduction rigs
need to really be, but in the end, the spherical harmonical framework
underlying ambisonic just doesn't pan out except in full 3D.
Unless you mean a sound field with all sources in the horizontal plane
of the mic. Then it is just a subset of a 3D sound field, and a
conventional AMB mic will capture it correctly.
Actually it will not. It's easy to fall prey to the idea of planar
symmetry, but the 3D soundfield doesn't behave like that. For infinitely
far-away sources in the horizontal plane the symmetry idea holds, but
only because such sources constitute planewaves when they hit the mic.
Any closer sources even in the horizontal plane are near-field, and
constitute a near-field component even in the third dimension.
Recreating them to first order calls for Z as well. And if you cut out
Z...we're back to Christoff's skeptical analysis of the system: the
apparent amplitude of a horizontal source takes on a 1/r term within the
reproduction rig, because some of the sound energy is leaking off the
plane.
The effect was analysed in WFS work, earlier. In there they used
synthesized point sources and -- planar array that they had -- actually
compensated actually for the term in software. Obviously nothing like
that can be done in whole ambisonic soundfields.
nor do they represent even the same encoding system.
How does a sound field 'represent an encoding system' ? Your statement
may be grammatically correct but it has no meaning.
Sorry, I'm being vague as usual. The exact statement would be that
pantophony is the theory of cylindrically symmetrical solutions to the
wave equation, whereas periphony concerns itself with spherically
symmetrical ones. The continuous symmetry group respected by the former
is that of the one-sphere by a line, and the one respected by the second
is that of the two-sphere. By well known topological reasoning, you
cannot continuously, much less differentiably or linearly, embed the
latter into the former, since it's of lower dimension. Neither can you
work around the problem using any linear representation, as would be the
case with circular/cylindrical harmonics and spherical harmonics.
Working down from that idea, there can *be* *no* consistent definition
of what a sound source off the horizontal plane means in pantophony.
There's just no way to define it without having it vanish somewhere, or
suddenly flip sign. And since we know you need planewaves of all
directions in order to decompose any monopole... You might think you can
forget about such behavior, but in the sense of the enveloping wave
equation you actually cannot: every nearby point source in the
horizontal plane requires a Z-wise component. (More on that later.)
It might not be easy to see, because we're used to dealing with
pointwise pressure fields only; the acoustical equation instead of the
full wave equation. But then only in the case of infinitely far off
sources can we assume that pressure and velocity are in phase, in the
plane of symmetry. That is the far field/plane wave assumption. With
near field sources, the outwards radiating field from a point source is
reactive at each point. Pressure and velocity are *not* in phase, and
the vector describing energy transfer (in EM I think the Poynting
vector) is *not* in the plane, but outwards from the source, all round.
So what happens is that while the pressure field is fully symmetric in
the horizontal plane, there still has to be a Z component in order to
recreate the field to full first order.
You can't capture that with a periphonic array, nor can you recreate it
using a periphonic rig. The only way periphony can work is by
approximating infinitely far off sources i.e. planewaves. There
pantophony can work, and that's how it was analyzed by Gerzon, no less.
But if you want to approximate near fields -- any fully general fields
-- you run into a topological singularity (pretty much the hairy ball
theorem): there's just no way to map down the 3D soundfield into a 2D
compatibility signal set.
Try it out if you don't believe me. Follow the math in the NFC-HOA
papers to first order, with a horizontal near source. What you'll get is
a W signal with its directional velocities and pressure slighly out of
phase (remember, this is not the acoustical approximation anymore, but
the full soundfield, with four fully independent components per point;
the stuff Angelo Farina's reactive field work incited way back). What it
does is counteract from above and below that 1/r term in amplitude
Faller naïvely thought would topple the whole pantophonic framework.
The same applies to decoder design. Unless your speakers are
infinitely long vertical line sources and radiate only in the
horizontal plane, your system is a 3D one.
...and that's precisely why pantophony is an idea born dead. We don't
have infinite vertical line sources, nor microphone arrays which mimic
their directional patterns. The only thing we really have is 3D mic
arrays and 3D rigs.
Let's not pretend there is some 2D thingy anywhere there. Because while
a noble aspiration, it's also a topological impossibility. Separate
dimensions just do not and cannot mesh like that. I believe it'd be
better to center ambisonic work around how to do with lesser vertical
resolution -- which is topologically speaking somewhat workable -- than
to think pantophonic formats would ever actually live upto Gerzon-like
rigor.
--
Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front
+358-40-3751464, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
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