On 2013-04-25, Robert Greene wrote:
I would appreciate an explanation of this. If I may say so, I do not believe it. There are not enough degrees of freedom to record the transient arrival times. One only has four degrees of freedom from the four microphone pickups.
One has only four degrees. However they are fully independent from each other. In a planewave you know they will always arrive at quadrature. At the same time first order reproduction always mixes in a standing wave component in addition to the propagating contribution. At that one precious spot in the centre you can't tell the difference. Everywhere else you can, even at the lowest frequencies.
It doesn't hurt at low frequencies, but at higher ones, it often does, and no matter what your frequency range is, that reactive part always fucks up the arriving wavefront. That whole thing has even its own theory behind it, in the form of the convergence of the Fourier-Bessel series over a sphere -- something which is fully about convergence in L^2 norm, and something which has absolutely no idea about direction of propagation. This for instance is why the analysis of synthetic sources inside the rig is so spotty, in the literature, even after the NFC-HOA papers.
How does anyone think that this is enough to record a soundfield in the neighborhodd of a point?
It necessarily is if you think purely about the pressure field. There the pointwise pressure plus three velocity components always dictate the close-by pressure gradient as well. But while they do so, they don't dictate the velocity field. That can be freely added on subject to the condition that it agrees at the coincident point where you chose to put your Soundfield in. If you do the math, that leaves you three full degrees of freedom in velocity undetermined outside of your measuring point, and the classical ambisonic decoding solution then takes full and unabashed advantage of that latitude. The velocity/systematic decode is half and half about propagation and on oscillating solution, by mathematical necessity.
Do calculate it out. Take a single frequency, two points symmetrically placed out from the absolute sweet spot, and then calculate what happens not only to the pressure field, but to the velocity one as well. There's going to be a reactive component even for a single, pure planewave. Simply because we're working with first order POA, which has been formulated in terms of the kind of transform it always was.
To my mind it makes not much sense to suppose that the first order reconstruction is correct in a neighborhood of the listener but higher order is correct in a larger neighborhood--not literally correct.
It literally is correct, because at the same time you go up in Fourier order, you also add to the Bessel one. HOA is about Fourier-Bessel decompositions after all, not one or the other in isolation. Just as when you go from the lower atomic orbitals to the higher ones and then have to shift the primary shell at some point, similarly adding orders to ambisonic shifts from directional information at one order to a whole new, additive set, from time to time. When it does so, just as in quantum physics you shift to a new primary shell, with new side quantum numbers, and the new primary shell is then spatially more extended by mathematical necessity, so it goes with ambisonic as well. And it's not just a funny coincidence: that funny quantum mechanics solution in a spherical potential is mathematically exactly the same thing as ambisonic's orders are; it's not just the directionally advancing capacity that proceeds here (the Fourier part of the series) but the radially advancing part as well (the Bessel part; that which in QM determines the existence of the primary quantum shells' dimension, and which in ambisonic determines the area within which the representation can be losslessly sampled in space).
This seems "metaphysically" impossible. Where in the set up is there any length scale? What would determine the size of the neighborhood?
Do the dimensional calculus. Whatever units you start with, eventually you'll end up dividing centimetres per second with centimetres per kilogram times kilogram, times Hertz, plus a couple of weird angular thingies on the way, so suddenly you just get 1/1=1. Usually the most annoying part being that in the end, Hz==1/s, with no unit at all in the numerator.
But surely no one really believes that the playback is perfectly correct around the listener and then at a little greater distance it is not? This defies all credibility in mathematical terms.
No, it doesn't. This would be impossible if we talked about a single, scalar, analytic-because-of-bandlimited pressure field. But it doesn't hold in a field described by f:R^3->R^4 signature, with a distinguished point, where you only fix the f:R^3->R part; the first coordinate on the image side.
No relatively simple physical process produces exactly a correct answer over a small interval and then suddenly does not over a large interval.
So the reason you fail is that you don't consider the dimensionality of the problem. It's self-evident that you can't do what you say, in one dimension. But within even POA we always talk about a linear function fromm dimension three real space to at least dimension four. It's that latter four which gets you; what you think is singular is just comparable to the square root of minus one or something like that, there, which seems singular but is actually just very well-behaved math given a couple of extra dimensions. (In the analysis of ambisonic reconstruction, a full infinity of degrees of freedom, of course.)
(Functions which are 0 over a region of space and then are not 0 elsewhere of course exist mathematically but they do not turn up in physical problems very much).
To reiterate, we don't need such functions here at all. If you want such things, I can give you spherically standing waves with nasty spherical null spheres. Both in pressure and in velocity, though not at the same time.
Incidentally, Blumlein did not think it worked either. That is what "shuffling: is for as I understand it. I realize that Blumlein stereo is not horizontal Ambisonics because the omni pickup part is missing.
Incidentally I don't understand Blumlein's contribution this way, either. As I understand it, Alain-dearest tried to do two separate things at the same time: 1) the psychoacoustic thing which led people later on to try things such as ORTF, and 2) correct via (analog, then) signal processing means the problems with his time's difficult mics, towards a coincidence setup.
My understanding of Blumlein's work is that he was an early trier-out, who tried out such about everything that the time's microphone's could produce. By ear. Based on that and theory, he settled on crossed dipoles, because in his time nothing happened at the back to fuck up the frontal image. Later on Blumlein stereo has been equated with crossed cardioids as well, once the backstage went live; that's a mistake, and something I tried (was it with Martin Leese) to reflect while doing the .ogg channel mapping spec.
Then, Blumlein early on also did experiments with slightly separated mics. Most likely starting with the fact that sensitive-enough mics at that time were rather sizable. Be as it may, he did the experiments, and put out the theory of the "Blumlein shuffler". It was a mid-side frequency selective analog circuit, conditioned on the distance between the two stereo capsules. Essentially it did in an elementary, analog form what the A-format to B-format circuitry does bigger and more accurately, from the start of the ambisonic age. Blumlein's circuit was less perfect and parametrized, whereas SoundFields are fixed, more developed and all round more useful.
To my mind the nastiest thing is that the Blumlein shuffler eventually got reduced to pure mid-side processing, with no temporal content. In that form it can be found in most consoles even today. And it's totally useless that way, for the stereo processing it was originally about; nowadays we talk about ORTF and whatnot coincident or near-coincident mic techniques, while we never notice that the age old Blumlein shuffler proper could actually help us process such signals better and easier than we now do.
Just sayin... ;) -- 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
