On Tue, Mar 26, 2019 at 09:59:45PM +0000, Martin Dupras wrote:

> However, no one has answered my question. I will ask again.
> 
> Can someone point me to any published resources that explain the principles
> and theoretical equations required to in theory convert from 2nd order
> A-format to B-format? I can find resources for 1st order but not for 2nd
> order.

First, there is no such thing as a standard 2nd order A-format. All
higher order AMB mics I know of use different capsule geometries.
Also some are on a solid sphere, some are open, and that changes
things as well.

Secondly, for second and higher order things become much more
complex. One of the reasons for this is that normal capsules
don't have much 2nd or higher order components in their polar
pattern. Whatever there is are imperfections or the result of
diffraction on the mic body. That means you can only obtain
the higher order components by exploiting the finite distance
between the capsules and any diffraction, and things become
very frequency dependent as a result of this.

Basically this a system inversion problem, basic matrix maths.

The theoretical polar patterns of the mics, or IR measurements
in a number of directions, define a matrix. Inverting this matrix
and then multiplying the result with another one defining the
B-format polar patterns, you obtain the A/B conversion matrix. 

This sound simple, but in practice it isn't. The inversion will
be frequency dependent and ill-conditioned at lower frequencies.
The theoretically 'perfect' result won't be usable, it may require
extreme gains and amplify noise and small errors in the original
data out of proportion.

You need to deal with this in a practical way. This is were
the 'secret sauce' is, it is a careful balancing act between
conflicting requirements. There are some mathematical tricks
to help with this but that would lead us very far. The simple
ones (presented in some papers) don't work well in practice.

The basic theory you'll find in several papers on higher order
Ambisonic theory. Warning: this becomes very mathematical, and
there is no other way.

Ciao,

-- 
FA
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