On 2013-05-31, Daniel Courville wrote:
I always insist on recording 'Z', and then almost never end up using
it...
Not even to "look" down or up in a stereo decode? I use the Z quite
often (if not always) when recording large ensemble and the SF mic is
more than 10 feet off the floor.
BTW, did you ever look through what Z really does to you *encoding*
equations? Formally, in order to arrive at proper pantophony you always
have to either reject Z fully or purposely subtract it from the whole
B-format signal set. Otherwise, even assuming perfect coincidence, your
W will have directionally aliased components from the above and the
below. For the most part that isn't noticeable in e.g. concert work
where you have a wide and loud array of early arrivals right in the
horizontal layer. But the theoretical error easily bite even with a
simple walkaround where not everything is in the horizontal plane,
including close vertical room modes.
That particular problem has then also been used to attac the ambisonic
system as a whole. I believe I told about Christof Faller's analysis of
"why ambisonic can't work", a few years back, didn't I? Which I followed
live at what is now Aalto University, and then Teknillinen korkeakoulu
(lit. "technical high school", formally "Helsinki Polytechnic").
How Christof saw it was much from the WTF point of view. There, if you
reproduce a point source in the horizontal plane only using a horizontal
array of speakers, you will get the angle of arrival right, but the
normal attenuation suddenly acquires an extra 3dB/per normalized
distance factor. In WFS they purposely compensate for that with their
linear and rectangular arrays. But very few analyses really go into
where that factor comes from, or how it could be avoided, or what it's
really about. The pantophonic analysis of ambisonic doesn't go there
either, even if it really, *really* should.
The basic problem is that you just can't in 3D space radiate a 3D
soundfield which fails to collimate in at least x dimensions, where
y=3-x is the effective dimension of your array. If you use a monopole,
it'll always attenuate as 1/r^2. If you use a line array or any variant
of it like a circular array (any pantophonic array), it will still
remain uncollimated in the third dimension. As such, power will be
radiating away from a 2D array not in 2D, but in the enveloping 3D, and
it will be felt within the array as a dropping off of power by distance.
In WFS theory, the dropoff is steeper, because all of the speakers have
to be on the horizontal plane. I believe it's ten 6dB per natural unit
of length. But even in arbitrarily dense, ideal, ambisonic arrays, where
some collimation from the above and the below will be taking place even
with purely XY (zonal?) harmonics, because they're always a bit spread
out even vertically, not all of that problem can be remedied.
The only way to really get that distance calculation right is to employ
periphony, so that in the limit you can reproduce true plane waves from
your array. Or at least do that within your array. If you try to do
anything of the sort even at infinite order, in 2D, energy will "bleed
of into the third dimension" and suddenly the system reduces to a
circular variant of WFS, with a single critical distance from the array
which does the two-dimensional attenuation right.
That much I think I know for sure. So what am I not too sure about?
Well, theoretically you can also expand your soundfield in cylindrical
harmonics. Instead of going with the two spherical Fourier functions and
a Bessel radial part, you can go with a second kind of Bessel part and a
circular Fourier part, which is then just a pure Fourier series in two
neatly orthogonal coordinates, instead of spherical surface harmonic
nasty one. It's mathematically given that is what happens when you
expand the natural solution to the Helmholtz equation in this second
coordinate system. It's going to be much easier, it's going to fit very
well with the spherical surface harmonic decomposition, but the two
different radial, Bessel terms won't match.
If you had the kind of natural vertical line emitters available which a
direct realization of the cylindrical transfrom described, ÿou'd never
suffer from decollimation in the z-direction, and so your soundfields
wouldn't have any decay by distance. That'd work for both WFS and
pantophonic HOA at the same time.
But it wouldn't work for periphonic HOA at all, because it's really
difficult to build *anything* spherically symmetrical 3D stuff out of 2D
symmetrical cylinders. At the same time it's pretty easy to build
everything wavelike out of either planewaves (arbitrarily, via basic,
rectangular Fourier theory), or point sources (under the additional far
field Sommerfeld radiation condition, which takes the place of the
straight forward L^2 norm when doing this kind of an integral; from the
viewpoint of Kirchof-Helmholtz integrals and the pointwise Huygens
principle).
So, in 2D, embedded in 3D, you're really want to use basic emitters
which aren't points, but lines. In 3D, within a 3D enveloping space,
you¨'d want to use just points.
My question to gurus Robert Green, Filippo Fazi, and perhaps both Daniel
Courville and Eric Benjamin is, could it perhaps be shown that this sort
of a calculus also leads to the differing constants in decoding UHJ and
pantophonics? If not directly, then via some circuitous route? Because
if it works out, maybe we could finally rest aside the annoying little
thingie that BHJ as UHJ's horizontal variant ain't roundtrippable with
B-format? That way also advance a bit towards an understanding of higher
order compatibility formats, and in the process, of how to optimally and
scalably encode ambisonics?
--
Sampo Syreeni, aka decoy - [email protected], http://decoy.iki.fi/front
+358-50-5756111, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
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