On Sun, Jan 23, 2011 at 12:39:19AM +0100, Jörn Nettingsmeier wrote: > http://stackingdwarves.net/public_stuff/linux_audio/tmt10/TMT2010_J%c3%b6rn_Nettingsmeier-Higher_order_Ambisonics-Slides.pdf
Nice ! But I don't really agree with some of the reasoning :-) There's a logical 'jump' in there which is not explained (and it would be hard to explain it): A. In your first example (the Kirchhoff-Helmholtz Integral) the zillion mics sample the sound field on a surface surrounding the listener. This is pure WFS. B. Then you try to do the same thing with less mics, exploiting their polar patterns. But here the mics have to be coincident. not on a surface surrounding the listener. Now I don't think you can say that (B) is some form of (A) scaled down to a practical size. It is something completely different. There is no continuous way from (B) to (A) by increasing the number of channels - the mics would remain coincident and not on the surface, even in the 'infinite' limit case. The basis of (B) is not Kirchoff-Helmholtz, but Fourier-Bessel, which is the expansion of a sound field in spherical coorinates. Doing this it turns out that the angular dependency is given by the spherical harmonic functions, and the radial one by the spherical Bessel functions. The angular part, the spherical harmonics, can be seen as a sort of spectral transform acting on the surface of a sphere. Just as a 1-D FT maps a cyclic function (i.e. a function defined on a circle) to a discrete spectrum, the spherical harmonics represent the discrete spectrum of a function defined on the sphere (i.e. a function of direction in 3-D space). The reason why this is not a 2-D FT is because a 2-D FT has a torus [*] as its domain, not a sphere: on a torus azim and elev are independent, on a sphere they are not. So each component of an AMB signal set is really an element of the discrete spatial spectrum, not a sample of space or the sphere. There is also a fundamental difference in the way a tetra mic and spherical high order mics work, and it is similar to the difference between (A) and (B). A conventional tetra mic would still work if the capsules were really coincident, it uses the polar patterns of the mics and the output signals correspond to spatial spectrum components, this is (B). An HOA mic consisting of omni capsules in free space samples the space, this is (A). Clearly the capsules must not be coincident. Its signals do not represent spatial spectrum components and have to be processed by a 'spherical harmonics transform' to provide B-format. An HOA mic with omni capsules on a solid body (e.g. Eigenmike) is a mix of the two, it's (A) at LF and (B) at HF were diffraction by the solid body makes each individual capsule directional. An AMB decoder is in fact doing the inverse transform: from a spectral representation to speakers signals which are 'samples in space'. Filippo Fazi does decoding for signals from an HOA mic in a different way: he takes the original spatial samples, and expands them using the Fourier-Bessel integral directly to speaker signals, without ever going into the spatial spectral domain. It's a form of extra- polation in fact and clearly illustrates the potential instability (which when going to B-format instead must be handled by limiting the frequency ranges of the higher order signals). [*] Suppose you have function defined on an infinite 2-D plance, i.e. a function of x,y. You could take the 2-D FT of this, providing a spectrum. Now assume the function is cyclic in both x and y, with period 2 pi. So it consists of an infinite number of identical square tiles. Its FT is then discrete, consisting of isolated points. We take a single tile (which still contains all information) and bend it so the top and bottom ends meet, giving a cylinder. This makes the y-axis cyclic. Now we bend the cylinder so its two ends meet, making the x-axis cyclic. The result is a torus. Each point on it can be located by its original x,y values, which now can be interpreted as azimuth and elevation, both of them having a range of 2 pi and fully independent. You can't warp a square into a sphere this way, it's a fundamentally different type of surface on which azim and elev are not independent. So it's spectrum is not a 2-D FT. It turns out to be spherical harmonics. Ciao, -- FA There are three of them, and Alleline. _______________________________________________ Sursound mailing list Sursound@music.vt.edu https://mail.music.vt.edu/mailman/listinfo/sursound
