On 2012-11-07, Tommaso Perego wrote:
I was wondering how, knowing the diameter of a speaker octagon, using
1st or 3rd Order ambisonics, to calculate precisely the dimensions of
the sweet spot area. Any ideas?
There is no unambiguous sweet spot. Even the arbitrarily cut-off,
approximative one depends on the precise rig layout and the precise
decoder being used.
In general, within the traditional Ambisonic theory, what delimits the
sweet spot is the area within which the energy vector magnitude
achievable by the rig and the decoder in unison stays above a certain
bound. That's because the velocity decode used at lower frequencies is
much easier to get right an sich. I'd say this holds at least for the
1st to 3rd orders you're talking about, at least if your rig and
decoding solution is close to any of the regular decodes we already know
about, and/or has been numerically optimized.
Within those bounds, the sweet spot is a circle (pantophony), or more
generally an ellipse (the more general 2D case) or even a general
ellipsoid (the 3D, or periphonic case). Its theoretical size does *not*
substantially vary above the minimum number of speakers needed for
spatially unaliased reproduction at any given order. That metric can be
derived directly from the radial Bessel functions associated with each
given order, minus any imperfections caused by a discrete rig (minimal
with a regular rig, much more if you go towards something like the ITU
setup, and pretty bad in the 3D case where you usually have to omit
beyond-horizon speakers altogether).
At the same time, the psychoacoustic optimization utilized even by plain
old ambisonic works over a much larger area. There the criterion becomes
evermore psychoacoustic the further away you go from the center of the
rig. What eventually breaks the image, there, is the differential
distance to your speakers. There it's the absolute distance to the rig
that matters, unlike with the sweet spot proper. What you'd want to mind
there is the temporal fusion threshold of incoherent sound sources,
which is somewhere in the vicinity of 10-40ms. Above that you'll get
separate arrivals/echo, and even if you approach that to within 20%,
you'll definitely hear combing.
That sort of stuff then partly (and surprisingly rapidly) goes away at
2nd to 3rd orders because the rig starts to speak only from the
direction of the incident sound instead of relying on antiphase signals
from the other side. But even then these off-centre effects have to be
calculated out in full if you want to know the true extent of the sweet
spot: near to rig edge you can still end up with two widely spaced
speakers crossing the fusion threshold or combing, especially since at
those higher orders, pretty much nobody can afford a dense right even if
they can afford high orders, plus pretty much everybody goes to higher
orders only when they already decided to do wide areas as well.
Finally, those combing artifacts and the like have additional funky
qualities. That's shown in that when you work with first order, the
minimum four speakers for pantophony and the minimum six for periphony
appear to work the best, unless you can go to tens or even hundreds of
speakers. The reason is that while the basic signal set is thoroughly
antialiased, a sparse rig still leads to spatial aliasing like combing
and even acutely perceived multiple arrivals away from the true
diffraction limited sweet spot. It surfaces that when than happens, the
fewer arrivals there are, the better it is for our ears,
psychoacoustically speaking. So, there is a kind of an unintuitive
Laffer curve working here, as far as speaker numbers go: you need the
minimum, or sometimes in periphonic work a bit more, but when you go
beyond that, you'd better be willing to go at last one or two whole
*magnitudes* beyond it before it sounds even as-good.
--
Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front
+358-50-5756111, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
___
Sursound mailing list
Sursound@music.vt.edu
https://mail.music.vt.edu/mailman/listinfo/sursound
