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 -,
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