On 2013-04-20, Eric Carmichel wrote:
Does beam forming or delay techniques to create additional first-order patterns from the omnidirectional mics change up the design (and math) from arrays using intrinsically cardiod mic elements?
Not per se, as we can see from designs like the Eigenmike. However, it does lead to hugely wasteful use of microphones compared to letting physics do the basic job of discriminating between waves going in opposite directions.
Especially so since there's a topological problem with extended arrays which aren't internally sampled to the spatial aliasing limit. (I think it was Filippo who first raised this one.) If you think about a spherical array of omnis on the one hand and omnis mounted on a sphere (or cardioids) on the other, the latter has a topological hole in the center whereas the former does not. The sphere of directions has that hole too, so if you forget about W for a second, those holes topologically have to map to each other when going from the array signals to the spherical harmonic representation (finite dimensional linear maps are continuous, so they preserve holes). That means that your A-format to B-format transform will necessarily end up mimicing that singularity, and that's expensive in mic terms. (As a whole, the hole ain't there because once you add W, it kind of patches over it. But that doesn't much help you when you consider the derivation of each 1+ order signal in isolation.)
The nasty side effect of that is that knowing the field over a sphere doesn't uniquely determine the field inside the sphere, whereas knowing the field and that there's a rigid ball in the center (the singularity, the hole) in addition does get you there. The same goes for knowing both the pressure field and its normal velocity, or knowing the outwards pointing cardioid response, because in both cases you can easily synthesize the singularity. If you only work with monopoles in the free field, eventually you'll end up simulating the dipole (velocity) component, at the cost of extra mics, plus spatial aliasing concerns force you to oversample in space. Effectively you have to go look inside the sphere in order to fully determine the field there, whereas it would have been enough to look just at the border if there was some physical mechanism which gained direct access to the velocity component of the field, like an obstruction giving rise to a cardioid response, a genuine velocity measurement like a MicroFlown, or that rigid sphere in the center which mixes pressures and velocities by imposing a boundary condition of zero normal velocity.
Okay, that's almost abstract nonsense and it took ages for me to grasp what was going on, so let's look at it from another perspective. How could two fields with the same boundary pressure field happen? The secret is a special kind of standing wave with full spherical symmetry. If you have that, you can place one of the antinodes (a full sphere) right at your microphone rig boundary. If the mics are monopoles, they won't sense the field at all, so that means any field with this property can be freely added to any other field without the A-format changing. If you had access to normal velocity as well, all of the information would be there, but if you only have pressure sensitive monopoles, you'll necessarily have to place some of them inside the sphere to tell the difference. And if you think about it, that special field is not just a harmless corner case: it's composed of all the modes of a spherical cavity under the boundary constraint of zero pressure (plus a matching, uniquely determined field outside the sphere); if you want to tame them all, you pretty much have to sample the whole interior to some highish spatial frequency limit.
The same can't happen once you put a singularity in the middle, which is what outwards pointing cardioids too end up doing. Then the remaining modes are topologically speaking those of a sphere, not those of a ball, and monopoles suffice to capture them. I'm also pretty sure this is connected to some nasty algebraic weirdness (spin groups as double covers of their orthogonal counterparts) through the NFC-HOA papers, because in the latter the sphere of directions is covered twice with not only an incoming but also an outgoing field (cardioids pick one, a full Kirchhoff-Helmholtz integral can represent both) which too patches the hole. Dunno how the details might work out, though.
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