`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 _______________________________________________ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp