On 2013-04-26, Fons Adriaensen wrote:
(Okay, this one is long and filled with intuition-beyond-verified-math.
Take it with a grain of salt, even if I think there's a point or two
there..)
Nobody claims there's a hard border between the 'correctly
reconstructed' area and the rest. If you're close enough the error
will be small, just as x is a reasonable approximation to sin(x) for
small x, adding an x^3 term will give a better one, etc. The length
scale is wavelenght.
The same obviously goes for all series expansions, not just
Fourier-Bessel, unless it just so happens what you're reconstructing is
already 100% some truncated, partial sum of the series you're using. For
F-B and most other useful series that is rarely the case.
Robert, if you're looking for the precise place where the scale enters
into the equations, it's via the radial Bessel term. As you increase the
order, the span (also effective support) of the Bessel functions
included thus far grows linearly in distance from the origin as well,
and sets the distance from the sweet spot upto which the square norm
error stays within some given bound.
In the classical POA framework you don't see the Bessel term directly
because everything is built up from directivity patterns and other
distinctly coincident concepts. But if you go through a basis change
from plane waves to Fourier-Bessel, the radial dependency falls out
naturally and works just as it does in the explicit form utilized by
e.g. NFC-HOA and spherical WFS work. Alternatively you could just solve
the wave equation in spherical coordinates using separation of
variables; Bessel functions constitute the most natural basis for
expressing the radial part under square norm.
What originally messed at least me up was two things: the high
directionality of higher order systems which is just unphysical at a
single point (precisely four degrees of freedom there), and the fact
that we have at least three different decoders going even with POA
(systematic, max-Re and in-phase). The first problem is resolved
precisely by including the radial Fourier component as well, because it
shows higher directivity always goes along with a larger volume around
the central point, for both reconstruction and pickup design. The second
one is about the mode of convergence we choose for the F-B series, much
like you can sacrifice fast convergence of ordinary Fourier series to
control Gibbs's phenomenon: systematic gives you optimal square norm
convergence but leads to oscillating components, in-phase kills
oscillation but convergences a lot slower especially at the start of the
series where the relevant directional components are badly localised
over the sphere, and max-Re is somewhere in between.
From this perspective the basic length scale enters analogously to how
the Nyquist frequency enters conventional sampling theory. When you set
the sampling frequency, you at the same time set the upper limit to what
can be represented faithfully, and rom there on anything above half F_s
folds cyclically. Despite the fact that in POA we conceptually work with
infinitely extended soundfields, we do so roughly using a Fourier-Bessel
series, not the continuous transform version of the construct. If we
used the latter, there would be no intrinsic length scale, but with the
series you actually have to fix one of the Bessel functions to a given
scale. Or to make the analogy even more exact, of the four different
forms of usual Fourier analysis, we're not using the transform
(continuous in time and frequency), the series (periodic in time,
discrete in frequency), or the DFT (periodic and discrete in both time
and frequency), but the discrete time/shift Fourier transform (discrete
in time, periodic in frequency).
In that case the spatial sampling frequency sets the scale above which
spatial-directional frequencies fold -- in mic design we get poor
rejection of higher order components present in the field, in sparse
rigs what happens is exactly the same thing that happens in spherical
WFS above some given frequency. The effect is exactly the same as it is
with the sampling theorem, only it's expressed in a spherically
symmetric form so that most setups both in mics and in playback rigs
violate the sampling constraint to some degree. (Neither of the usual
planar Fourier basis, truncated, nor the Fourier-Bessel basis, again
truncated, is a subset of the other to any finite degree of truncation.
They both span the whole of L^2(R^3) in the limit, but approach that
limit in fully incomparable ways, even in the usual square norm. I
believe this is where the confusion regarding spatial derivatives and
spherical harmonical components came from, years ago, e.g. in the form
of -- was it -- the ten component "chi-format"; the latter mixed and
matched two different series, with confusing results.)
(I believe this is also the reason Gerzon mentions Gaussian summation
formulae in his work on the classical tetrahedral Soundfield mic: the
symmetry inherent in tetrahedra, used together with pickups that have an
ideal first order directivity pattern because of physical reasons, buys
you extraordinarily high directional rejection of out-of-band components
which only breaks down above the asymmetry caused by capsule spacing.
That spacing is effectively what sets the length scale/spatial bandwidth
limit within POA. You have to do some nasty mathematical gymnastics in
order to show that for real, but at least my third lobe says the length
scale is definitely there, despite being masked by the act that in the
very special case of ideal first order POA, you could do without as far
as mic design goes. There the representation is essentially scale-free
and only references against whatever frequency you happening to be
talking about, leading us to talk about it all in relative terms, like
illposed transform matrices, noise amplification and sensitivity.)
The reason that limit is not part of the orthodox theory is once again
an artifact caused by the fully coincident analysis framework and the
funkiness that comes from the fact that we can exchange the first (and
only the first) order components for point velocities, without
considering the extended field at all. POA truly is a special case. When
you go to higher orders you actually ought to be doing it the NFC-HOA
way, and when you do, the spatial length scale necessarily enters
explicitly, in the form of a fixed array diameter. If you resize the
array, you have to do the analog of bandlimited interpolation in the
spherical domain -- implemented in NFC-HOA as the summation over, was it
now, the B(.,.) transfer functions. Those effectively work as
directional anti-imaging interpolation when you go to higher D, and as a
directional anti-alias when you do the opposite.
Then, just as it happens in more usual forms of resampling, it's
difficult to see the precise effect before you go into the transform
domain: there the effect is a brickwall filter, while in the dual base
domain the operation seems to be just a soft blurring which stretches
the half energy cutoff of the basis functions further out. That's then
why you don't easily see what the real effect is if you work in the base
domain, as we do in the classical POA derived theory; there we don't
consider the hard sampling criterion at all but only deal with the
sensitivity issues caused by the approaching band edge at each temporal
frequency, and sometimes get surprised by the nastiness caused by
unrejected spatial alias or reactive fields. (The latter are just fine
from a theoretical viewpoint, but can bring in unexpected free degrees
of freedom and so break e.g. mic designs which didn't expect them. That
problem is particular to higher order, because at first order, physics
means there is no essential difference between pressure gradient, the
first order Fourier-Bessel expansion of the pressure field, and
velocity.)
And of course all of that is doubly difficult to to see when you start
out with the central, coincident viewpoint: there there is no explicit
array to consider, so no array diameter either. The Bessel terms are
fully implicit and have to be dug out by intention if you want to see
them. The conventional theory gives you absolutely no hint that you
should look in that direction before you consider a finite spherical
mic/rig consisting of monopoles and/or first order (cardioid, fig-8)
mics (the best we can do with stock components) which can no longer be
considered coincident. Suddenly you have to consider the
spatial-directional sampling effects because in order to get the nice
harmony we get from Gaussian interpolation would necessiate physical
mics with at least second order, near-ideal directional selectivity.
That you can't buy in a store, so you have to synthesize the response
from simpler atoms, and suddenly your math goes haywire because it now
shows all of the nasty radial, dependent on your spatial-directional
sampling cutoff dependent terms as well. (And actually doesn't even get
solved properly unless you impose such a cutoff. In most papers that
cutoff is imposed by accident by restricting the analysis to the order
of the spherical harmonical decomposition that the system aims at; bit
mistake: in the process you end up assuming the pressure field is smooth
to that order, in precisely the sense that the system requires, which it
in reality mostly isn't.)
--
Sampo Syreeni, aka decoy - [email protected], http://decoy.iki.fi/front
+358-50-5756111, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
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