As Gerzon describes it, the dominance transformation is a Lorentz boost
in a certain direction, or a four-dimensional hyperbolic rotation,
between the first four ambisonic channels WXYZ. As far as I know, that
sort of thing only works neatly in plain 4D Minkowsky space, with the
possible extensions becoming approximate at higher finite orders of
surface spherical harmonical decomposition. Since I'm no relativistic
quantum physicist, I'm not too sure about how to even begin to calculate
with something like a relativistic atomic orbital above the first two S
and P stateS -- corresponding to a rigid boost of a higher order
decomposition, which we'd probably be after if we wanted to formalize
dominance for higher orders, and which is also beyond a strictly local
analysis in both the acoustic and EM fields.
What I *do* know though, is that in the 1D setting of pulse radar time
series, especially in connection with chirped pulses, hyperbolic phase
rotations used to be approximated piecewise using physical allpass
filters in various constant coefficients. Also, it's possible to piece
back together a perfect LTI response from a heterodyne system which
implements each band as a simpler non-LTI, SSB-modemed system.
Thus, long story short, is there a theory out there of heterodyning and
bandshifting akin to the SSB-radio one for spherical harmonics? It
certainly couldn't work as nicely as it does with normal frequencies,
since there e.g. just isn't enough room for the information if you
downshift from second order to first; conversely there *is* going down
linearly in normal frequency from (x,2x) to (0,x). But since the
spherical harmonic progression too is neatly *quadratic* instead of
linear, *maybe* there is somewhere a theory which lets us
*quadratically* transpose spatial harmonics, and so modulate willy
nilly?
Anybody privy to anything like that? Because if there is such a theory,
formally transposing the dominance operator one minimum level up could
clear the way to actually characterising and even implementing higher
order dominance for good.
--
Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front
+358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
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