On 2013-06-03, Fons Adriaensen wrote:
Note the sqrt(k) factor in eq.(4). This the 3dB/oct factor I mentioned before. It arises because in the derivation of the driving function vertical line sources are replaced by point sources, and NOT because the resulting line array of secondary sources behaves as a line source.
True. But isn't that precise thing also happening within pantophony, if not in WFS's cylindrical symmetry, but in ambisonic's toroidal one, too?
For horizontal sources, there shouldn't be any problem, because it doesn't much matter where you place the transition from notional recording space to notional reproduction space. When you do separate the space in two using a curtain of secondary sources, you can do that pretty much willy-nilly even for monopole mics (and the corresponding re-emitters) if you just keep away from double arrivals and close proximity to anything that would approach them. That works for linear arrays as well as circular ones.
As such, yeah, impulsing the array coherently leads to "the wake" they are talking about in the paper in the WFS case, and it cannot lead to it in the case of a circular array, because in the latter ambisonic case all of the array points are equidistant from the sweet spot. But that's an edge case which only obtains when both of the arrays are reproducing a point source infinitely far away in the plane defined by the listener, the source, and the array, or infinitely away in the dual space, that is, directly above or below. That symmetry is broken as soon as you move away from those directions, and it's broken differently for the two systems. It'd again be inconsequential if we had a sphere shroud for ambisonic or an infinite plane for WFS, but since we don't, the end results become frequency dependent on the angle of arrival from the source to the two different kinds of emitter array, and their different axes of symmetry.
Or so I think. What am I missing?
So even if a very high order 2D AMB arrary looks just the same, it does not need those filters, and in WFS they are not there in order to obtain more 'plane' waves, but to have a flat frequency response of the system as a whole.
Of course they aren't. The driving function becomes funky simply to correct for the distance dependent delay and attenuation from source to the secondary emitter array to sink. Then it surfaces that you can get full temporal coherence in the horizontal plane for arbitrary point sources, but you can't get the distance-wise attenuation right except for a single distance.
But why shouldn't that same thing happen for pantophony, with its circular arrays? And why shouldn't it experience similar kinds of problems WFS does for *off-plane* sources? Those of course being much more of a problem with ambisonic, with its fully 3D, often recorded-in-natural space instead of synthesized nature?
And if you would build a full 3D WFS system (i.e. covering an entire enclosing surface with monopole secondary sources), these filters _do not go away_. Rather, they would be 6dB/oct in that case.
Yes, but in that case, everything you lose on the encoding filter side, you gain on the reconstruction filter side, so that the integral over an enclosing surface becomes a unity operator. The same is not true for a secondary emitter of most other kinds, and even if it is easily approximably true for in-plane primary sources, it need not be so in the general case. And not so even if you neglect the transverse contribution, running through the plane. That ain't going to be half as easy for a ring secondary emitter as it is for a line, because a line is a line from whatever viewpoint, whereas a circle can become an ellipse. Hitting something like that with an impulsive planewave from different angles is going to yield an entire second degree of freedom in how to reproduce, leading me to believe frequency dependency has to be present.
Another way to see this is that the monopole secondary sources need to be driven by the projection of the gradient of the field on the normal of the enclosing surface, and pressure and gradient magnitude have this 6dB/oct relation.
And in the end, you do necessarily need even the normal gradient as well, pointwise, as an independent degree of freedom. But I still don't see how this impacts the difference between WFS and pantophony. Sorry if I'm once again being a bit dense, but I just don't. :)
There is no similarity at this level between AMB and WFS, because even if the speaker arrays can be the same they are being driven in an entirely different way. The two systems do _not_ converge in the limit.
In the 3D limit they do. So isn't that yet another reason to believe what I'm saying is true? I mean, they'd pretty much *have* to converge even here if it was only about amplitude by distance, when the distance was set to the pantophonic array radius in the ambisonic case, and as the same radius for the circular variant of WFS. No?
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