On 2013-04-27, Sampo Syreeni wrote:
(And actually doesn't even get solved properly unless you impose such a cutoff. In most papers that cutoff is imposed by accident by restricting the analysis to the order of the spherical harmonical decomposition that the system aims at; bit mistake: in the process you end up assuming the pressure field is smooth to that order, in precisely the sense that the system requires, which it in reality mostly isn't.)
Oh, finally, something that Filippo raised "ages ago" and took some time for me to understand properly... When you work with the linear wave equation, usually imposing a frequency limit buys you every mathematical nicety you could ever want. In fact, it buys you a bit more than you even bargained for: bandlimited signals are analytic, so that knowing them fully in any full neighbourhood theoretically allows you to do analytic continuation. That stuff can lead to some truly unwieldy consequences like acausal knowledge, unless you're careful and do something like restrict yourself to minimum phase systems only.
Yet, as it happens, bandlimitation still doesn't buy us all we want/need for ambisonic work. The reason is that incomparability of the Fourier-Bessel and ordinary planar Fourier bases I already mentioned. If you impose a bandlimit over time, that'll get translated into a spatial bandlimit in terms of the usual planar Fourier series, because of the constant speed of wave propagation inherent in the linear wave equation. Thanks to the multidimensional Fourier transform being a rotational homomorphism in group theoretical terms, the end result is that once you've imposed a bandlimit in time, in terms of the usual Fourier transform you can be sure that no spatial bandlimit is violated either, provided you just sample our pressure field densely enough. And it's enough to just impose that bandlimit pointwise; spatial sampling commutes with the temporally defined anti-alias filter in this case.
The same does not hold for Fourier-Bessel analysis at all. Even the lowest frequency planar sinusoid will always have a full, nonzero to infinity F-B series. That means any realistic soundfield will be of much higher intrinsic degree than the system you're using to capture it, so that you a) have to consider directional-spatial aliasing rejection and b) you can't implement it cheaply except at first order where intrinsically, physically first order capsules (which average all of the higher order components away by their nature) are available and so few are needed that they fit into a compact array which is small enough to cover the majority of the useful auditory bandwidth without hitting spacing issues.
-- Sampo Syreeni, aka decoy - [email protected], http://decoy.iki.fi/front +358-50-5756111, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2 _______________________________________________ Sursound mailing list [email protected] https://mail.music.vt.edu/mailman/listinfo/sursound
