On 2022-02-25, Eero Aro wrote:
On Angelo's site, although he says there that this procedure is
outdated:
http://pcfarina.eng.unipr.it/Aurora/conversion_between_uhj_and_b.htm
While I appreciate Angelo's work beyond pretty much all others, right
here I'd argue there *is* no conversion from UHJ to B-format, unless the
precise variant of UHJ being talked about is PHJ, i.e. fully linearly
invertible.
We pretty much never have that as source material. I've never seen or
heard a PHJ recording myself. Just BHJ.
Much of that material wasn't captured on a SoundField either, which
means it was captured under a cylindrically symmetrical, spatially
widely sampled setup. Not via a compact array like the SoundField, or
even if it was done by Blumlein technique in the horizontal
(pantophonic) plane, it does *not* project down to horizontal ambisonics
the way you'd think it does.
Christoph Faller I think once showed this at Aalto University, here,
while not really getting that the problem was precisely between the
spherical and cylindrical decompositions being mutually indecomposable.
The only guys I've seen doing this right are the NFC-HOA people (Daniel,
Nicol & Moreau), but then they never did the cylindrical analysis
either, but just the neater spherical one. Whereupon they don't much
help come up with pantophonic spatial sampling theorems, either.
You first do a UHJ to B-format conversion by using the convolvers,
then use a B-format to speakers decoder VST plugin, of which there are
several to different kinds of layouts and number of speakers.
Angelo's analysis talks about WXYZ, but then suddenly drops to WXY. This
is okay in the free field, because of separability. But it isn't okay
for anything else, including binaural work. Or in fact UHJ itself.
Because UHJ is decidedly *not* rotationally invariant with respect to
the horizontal plane. You can see this from the encoding equations: Left
and Right take three different channels from the soundfield, W, X and Y,
via S (sum) and D (difference). S is easy, but D takes in
j(a*W+b*X)+c*Y. The encoding locus is decidedly off symmetrical.
Then we do even the same thing for T. In Q, only Z, which means that PHJ
is symmetrical over WXY modulo Z. But it isn't symmetrical over WXY
only. In fact this is because of Gerzon's analysis of how you can trade
off localisation at the front for them humming, reactive fields at the
back Angelo so much talked about at the time, in his automotive work.
Where the reactivity/phase mismatch isn't as easily heard.
But the thing is, now the solution, even at the level of the transfer
format, UHJ/BHJ, cannot be symmetrical even in the horizontal plane,
anymore. And so, if you want to invert it into something like B-format,
the problem is going to be difficult. Evenmore, there's going to be an
*extra* problem on the way.
Because we already know at least that separate orders of the ambisonic
decomposition lead to different optimum decoding equations. Different
cutoffs of the shelf filters. You can't mix and match them. And based on
the above, you can't even mix and match pantophonic and periphonic
pickups, in the limit, because even in the case of impinging wavefronts,
they don't behave the same in sensing and reconstruction. (The WFS guys
showed even more, using spherical Bessel and Hankel functions, how the
entire solution ought to behave at the surface of the rig, and how there
is a natural length scale to the rig diameter, if we want to go to
holophony.)
So, how do you "invert" UHJ?
You do not. It is a highly underdetermined problem, unless you have a
PHJ signal to work with. It's not to be tackled via simple convolution.
Not even if you have three horizontal channels to work with, in
quadrature, because typically they do not work well with regard to the
fourth channel everybody thinks is of no consequence...yet is. (Faller
showed this live to me; the fourth dimension leads to bleedoff of energy
from a wave, at 1/r rate. This has also been shown in WFS work,
separately.)
Also, when you pseudo-invert such a problem, it will lead to noise
amplification. Both in time as the typical kind of noise, but in
inverting the UHJ encoding matrix, to amplification of directional
errors as well. And phasing, in the frontal direction, where it is most
annoying.
Is there then a solution which takes UHJ into an equivariant inverse
over the sphere or the circle, as the two variants of B-format?
No there is not. Because once you projected down one dimension, you
already broke the so called topological "ball theorem": "no ball in
Euclidean space can be continuously embeded onto a ball of smaller
dimension". This holds for complex linear spaces as well, of dimension
two times, and a fortiori for complex entire, harmonic functions. As in
bandlimited in space, time and angle functions, such as those we'd like
to handle in ambisonic.
So, if you want to invert such thingies even in an ambiguous, arbitrary
way, you cannot really do so using the typical shift-invariant linear
machinery. That is prohibited by topological, symmetry considerations.
What you need to do is go into nonlinear variational calculus. Using
nonlinear perceptual measures, rate-distortion theory, and the like.
--
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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