Hi, Am 24.12.2011 um 07:38 schrieb Sampo Syreeni <de...@iki.fi>: > I wouldn't dare to claim I have a solution to this overall problem, of > course. But one part of it interests me above most, and seems to me to be a > stepping stone to more general solutions as well: the problem with > ill-conditioned decoder matrices. They after all come from too irregularly > spaced speakers, either in space, or with regard to the highest order > spherical harmonical function being decoded.
A big part of the problem is even obvious in a continuous representation: if we were able to synthesize spherical harmonics on a dense (nearly continuous) surrounding loudspeaker setup with gaps that were left uncovered by loudspeakers, we could try to do so by driving the setup with the spherical harmonic patterns. However, the analysis of this continuous pattern with holes would differ from the original. If we try to pre-condition the synthesized pattern so that the analysis matches the original wish, this pre-conditioning is also ill-conditioned when the gaps are big. Abstractly, this explains the biggest reason for ill-conditioning on 3D arrangements. One solution is to build Slepian functions from the spherical harmonics for the domain between the gaps, as suitable analysis-synthesis pair (see papers of last Ambisonics Symposia). However, everything stays in the L2 norm sense. > In there I somehow feel one of the numerical L^1 optimization methods such as > basis pursuit could perhaps be brough to bare, in a dual formulation. > Especially because of the connection with the usual L^2 norm, so essential to > the HF optimization problem, via regularization. > Has anybody ever worked with something like this? I mean, even if it's > numerical and not closed-form, this sort of stuff at least has solid > convergence proofs behind it and all. And at least my hind-brain tells me it > could lead to a solid, systematic means of controlling undue gain in even > highly irregular rigs. You will find papers from Nicolas Epain and his colleagues using the keyword compressive sensing or sampling. They did some work that can be seen as a starting point for L1 norm based rendering with sparsity in space. For decoding, I assume that sparsity in the SH domain is more reasonable... someone should try... There will be a paper in the first acta acustica issue in the coming year that improves our work on decoding in Graz. We fixed the L2 norm for Ambisonic decoding there by avoiding mode matching or sampling. This is relevant for outside sweet spot, reverberant field, and HF, and gets rid of the uncontrolled loudness problem of ill-conditioned decoders. It finally yields decoding with constant "energy" (as it was called usually) for all playback arrangements. This was the biggest drawback of Ambisonics compared to other panning strategies, which can now be solved. > Finally, at time there's been some talk of "forbidden harmonics" in some of > the controlled opposites kinda work. Could somebody perhaps tell me how that > theory came to be? Because I have a serious feeling it could be systematized > and placed into the general framework I seem to be seeing glimpses of, above. Are there?- I would rather say that harmonics become linearly dependent. But you have to employ SVD or eigendecomposition to see which combinations of harmonics are weakly represented in your loudspeaker setup; it often is a complicated complex of combinations. The talk was about forbidden frequencies somewhen, a purely numerical/mathematical problem that can be circumvented and lies mainly in the boundary integral formulations that cannot easily separate interior and exterior problems. It does not occur with other formulations and will never occur with listening environments of realistic electro and room acoustic accuracy. Best regards and a Happy New Year Franz Zotter Institut für Elektronische Musik und Akustik Kunstuniversität Graz _______________________________________________ Sursound mailing list Sursound@music.vt.edu https://mail.music.vt.edu/mailman/listinfo/sursound
