A Compact 120
Independent Element Spherical Loudspeaker Array with
Programmable Radiation Patterns
Rimas Avizienis1, Adrian
Freed2, and Peter Kassakian3 and David
Wessel4
Center for New Music and Audio Technologies(CNMAT) UC Berkeley
1750 Arch Street, Berkeley, CA 94720, USA
1 rimas [at] cnmat [dot] berkeley [dot] edu
2 adrian [at] cnmat [dot] berkeley [dot] edu
3 kassak [at] cnmat [dot] berkeley [dot] edu
4 wessel [at] cnmat [dot] berkeley [dot] edu
Abstract:
We describe the geometric and
engineering design challenges that were overcome to create a
new compact, 10-inch diameter spherical loudspeaker array with
integrated class-D amplifiers and a 120 independent channel
digital audio interface using Gigabit Ethernet.
A special hybrid geometry is used that
combines the maximal symmetry of a triangular-faceted
icosahedron with the compact planar packing of 6 circles on an
equilateral triangle ("billiard ball packing"). Six custom
1.25inch drivers developed by Meyer Sound Labs are mounted on
each of 20 aluminum triangular circuit boards. Class D
amplifiers for the six speakers are mounted on the other side
of each board. Two pentagonal circuit boards in the icosahedron
employ Xilinx Spartan 3E FPGA's to demultiplex digital audio
signals from incoming Gigabit Ethernet packets and process them
before feeding the class-D modulators. Processing includes
scaling, delaying, filtering and limiting.
1.
Introduction
The array we describe has the largest
driver count of a series we have constructed based on the
platonic solids [1, 6]. These arrays can be used to synthesize
radiation patterns of musical instruments, to create a monopole
source for room measurements and to create directional beams
and nulls. The mathematical basis of the signal processing
applications for these arrays has been published already [4,
5]. In this paper we focus on the geometric considerations and
engineering design constraints that led to our latest
array. We also briefly discuss the systems-level control
software for the array. Details of the directivity control
software itself and measurements of the array will be part of a
future publication when we complete careful acoustic
characterization and optimizations.
2.
Geometric Constraints
The key parameter to achieving good
directivity control at high frequencies is to maximize the
number of closely spaced drivers. The quality of control
depends also on how evenly spaced the drivers are. The 2D
version of this problem is easy: centering circular
loudspeakers at the vertices of equilateral triangles
tiling the plane yields the most efficient arrangement in
a space filling sense. For a spherical array we have to embrace
the challenges of the regular polygonal tilings in 3D and
exploit the many useful properties of the platonic solids:
equal spacing of neighboring face centers, faces tangential to
the same insphere of the solid, and tangent points at the
center of each face. The symmetries of these shapes often allow
us to rotate a desired response without recomputing a filter
design by simply changing amplitude scaling factors.
Figure 1. The Platonic Solids
2.1. Volumetrically
efficient Platonic Solids
As well as having fewer faces the
tetrahedron, cube and octahedron are much less efficient in
terms of volume than the largest Platonic solids: the
dodecahedron and icosahedron. Since most practical drivers for
this application are circular the type of polygonal face is
also relevant as the output power of the overall system will
depends on how large a driver can be installed in each face.
The platonic solid that yields the best volume efficiency and
area efficiency is the dodecahedron. For this reason we built a
"wide-band" dodecahedral array using long-throw 4-inch drivers
with independent baffle chambers to improve low-frequency
response [7].
Figure 2. Dodecahedral Speaker Array
Although we realized the basic wideband
goal for monopole and dipole patterns, the driver spacing
proved insufficient for high quality directivity control at
high frequencies and for higher spherical harmonics [6]. The
only solution to this is to increase the number of independent
drivers. Since the icosahedron offers only a minor improvement
over the dodecahedron we had to abandon the equally-spaced
driver constraint and settle for a geometric arrangement with
approximately even spacing.
2.2. Circular
Packings on the Sphere
The first design we explored is the
optimal packing of circles on the sphere using the tabulations
of optimal spherical codes that are readily available [2].
Although this approach is very efficient in terms of area and
volume, numerous practical problems would have to be solved. It
is hard to build accurate spheres and accurately place drivers
on them and the packing is so area efficient it is difficult to
find enough room to attach a mounting or suspension structure.
Finally, the lack of symmetries complicates the control
software.
2.3. Geodesic
Spheres
Another interesting geometry is the
geodesic triangulation of a polyhedron such as the 80-faced,
triangulated-in-4 icosahedron.
Figure 3. Geodesic sphere approximation
This does yield a solution with some
symmetries and a hemispherical "seem" facilitating construction
and maintenance. Unfortunately some of the face triangles are
not equilateral and have varying dihedral angles negatively
impacting area efficiency and directivity control quality.
2.4. Archimedian
Solids
We can avoid the dihedral angle issue of
the geodesics by using some of the larger members of the
13-strong Archimedian solid family, i.e., the 36-sided
truncated icosahedron (soccer ball) and the 62-sided
rhombicosidodecahedron.
Figure 4 . A Truncated Icosahedron
Figure 5. A Rhombicosidodecahedron
These solids have faces with more than
one regular polygonal shape. This requires different driver
sizes or some compromise in area and volume efficiency to use
the largest single driver diameter that fits all the face
shapes.
3.
Engineering Constraints
3.1.
Amplification
The final geometry we selected arose by
considering the implications of a further practical constraints
especially the lack of commercially available compact
amplifiers to drive large speaker arrays. The amplifiers for
our dodecahedral speaker are larger than the speaker itself and
the wiring between them was an expensive, multi-core cable. By
integrating the amplifiers within the array we avoid the long
cable, increase reliability and lower cost of ownership of the
system.
Flat circuit boards don't stack very
efficiently within spherical structures and flexible circuit
boards would have been challenging as we have to cool the
electronics efficiently. We decided to attach flat circuit
boards to aluminum plates that serve as a structural element to
mount the speakers on, as heat sinks and EMI shield. Since
efficient class-D controller chips are readily available for 6
or 8 channels we sought an appropriately shaped single "module"
from which to assemble a polyhedron [3]. The best unit we found
is simply an equilateral triangle. This allows us to use the
optimal symmetric packing of circles in equilateral triangles,
i.e., the "billiard ball" packings.
3.2. Class-D Chip
Set Channel Efficiency
The "billiard ball" packings follow the
sequence 1, 3, 6, 10, (n+n*n)/2. With 1.25inch diameter drivers
the space-efficient match for the class-D chip set we used was
6; 20 of these equilateral triangle modules form the
tetrahedron from which we built our 120 speaker array.
Figure 6. Controller side of 6-channel Class-D Amp.
Each amplifier board contains a Class-D
controller (bottom of Figure 6), balanced output reconstruction
filters (LCR) and 6 MOSFET driver chips (Figure 7). The
controller does clock regeneration, digital to pulse conversion
and thermal monitoring of the driver chips.
Figure 7. Driver and Heat-sink side of 6-channel Amp.
Two parallel pentagonal circuit boards
(Figure 8) near the center of the icosahedron employ Xilinx
Spartan 3E FPGA's to demultiplex digital audio signals from
incoming ethernet packets and process them before feeding the
class-D modulators.
Using two cards allows us to split the
in two parts that resemble broken dinosaur egg shells and to
rationalize the extensive wiring required between the control
and amplifier boards for power, clock and data distribution.
Each half of the array can be operated independently and they
sit conveniently and stably on a bench for testing.
Figure 8. One of two Controller/interface Boards
A special daughter card for the
controller board was developed to allow digital connections to
an Audio Precision audio analyzer. This was essential for
optimizing the circuit board layout of the amplifier boards.
This layout and the output filter component choice turned out
to be critical and very challenging parts of the project. PCB
trace lengths and routing play an important role in achieving
the potential of class-D amplifiers and the triangular board
constraint made it impossible to adapt layouts already debugged
by the chipset supplier and other builders using rectangular
circuit boards. The close proximity of all the parts and high
power high frequency traces made it essential to carefully
shield the jitter attenuating clock recovery circuits in the
controller.
3.3. Energy Storage
Component Density
We considered a 60 driver array using a
"diamond" module of two equilateral triangles with three
drivers on each. It would have the interesting property that
all drivers have the same spatial relationship to their
neighbors throughout the array and according to simulations
would perform quite well. It would however have to be
significantly smaller than our 120 speaker array allowing too
little room for the large passive components and digital
interfacing circuitry required as illustrated in figure 9.
Figure 9. Volume-filling Toroidal Inductors and Decoupling
Capacitors
3.4. Cooling, Cable
Access and Mountability
The 120 driver array has less symmetry
than a 60 driver array in the sense that drivers in our array
fall into two equivalence classes, with vertex and edge
relationships.
The optimal planar packing results in
small gaps in coverage at the 10 vertices of the array. We
could attempt to fill these gaps with additional drivers but
chose instead to exploit them to provide air flow through the
array and to provide access for cable and mounting/suspension
options (see Figure 10).
Figure 10. Pentagonal access holes at the vertices
Meyer Sound Labs. contributed the custom
built silk-dome drivers we use in the array. They are 1.25
inches in diameter and have a very narrow circular surround.
This kind of driver would usually be used with a crossover
filter as a tweeter in a conventional speaker. We operate these
drivers at lower frequencies where they are not acoustically
efficient so we have to protect them from thermal overloads and
keep them within physical excursion limits. This is very
important in arrays of this type because large power levels are
required to create the cancellations necessary to achieve the
desired radiation patterns. With the number of drivers
involved, individual closed-loop temperature and excursion
control would have been prohibitively expensive and complex so
we have put the protection into the FPGA algorithmically using
multiband limiters carefully designed to protect the array and
also allow for short-term power peaks typical of musical
material [4].
Although class-D amplifiers reach
efficiencies of90% or more this is usually at high output power
levels, (in our case 20-40W/channel). Our continuous power
output level goal is 5W/driver where efficiencies are
approximately 50%. The number of drivers in a relatively
small space create significant thermal design constraints that
we satisfy by a combination of the vertex holes to provide
convective airflow and the use of aluminum and thermally
conductive heat sink interfacing components.
3.5. Non-Ideal
Polar Response of the Drivers
Mounting the drivers on triangular faces
results in the vertex drivers being more distant from the
center of the insphere of the array than the edge drivers. Also
the line of rotational symmetry of the drivers does not pass
through the center of the insphere.
We could have compensated for this using
an offsetting "rack" for each driver, but for this and the
other compromises mentioned above we have chosen to exploit the
power of our convex optimization software and the affordable
real-time signal processing integrated into the control
electronics. We have to implement these features anyway to
compensate for the actual polar radiation responses of the
drivers. Our confidence in this approach is supported by Meyer
Sound Labs reputation for manufacturing very consistent,
high-quality drivers and the simulation results in Figures
11-14 that illustrate the potential directivity control quality
using the error analysis methodology developed for our
dodecahedron [6]. These plots show worse case errors for
steered spherical harmonics of varying degrees at varying
frequencies. The darker lines at lower heights on the plots
indicate desirable values (the least error from ideal spherical
harmonic synthesis).
Figure 11. Twelve Driver Dodecahedron
Figure 12. Twenty Driver Icosahedron
Figure 13. 60 Driver Icosahedron Simulation
Figure 14. 120 Driver Icosahedron
Simulation
4.
Control Firmware and Software
4.1. Firmware
The FPGA firmware is developed in VHDL
and also by direct synthesis from matlab. Processing includes
configuration, monitoring, gain, delay, filtering and limiting.
The firmware also implements input and output assembly,
disassembly and buffering of ethernet packets that flow over a
Gigabit Ethernet to host computers
4.2. Host
Software
We chose Gigabit Ethernet because it is
a relatively simple industry standard and is efficiently
implemented in modern computers: most modern laptops and
motherboards provide a contention free path to the main
processor for ethernet traffic. We have found it necessary to
make small modifications to the ethernet drivers in OS/X and
linux to avoid priority inversion problems and move data
reliably and with low latency from audio applications to the
array. We have also demonstrated the capability of combining
packets from multiple host computers for applications where the
load of the steering algorithms is spread over a cluster of
independent computers.
5.
Conclusion
The spherical array described here is an
essential tool for exploration of the effectiveness of
programmable directivity algorithms. It will guide and inspire
new and unusual applications in acoustic measurement and
artistic endeavors. By carefully characterizing its performance
we will be able to guide designs of future high-driver-count
arrays.
6.
acknowledgements
This work was supported by Meyer Sound
Labs. and the UC Discovery Grant in Digital Media from the
University of California's Industry-University Cooperative
Research Program (IUCRP).
7.
REFERENCES
1.
Baird, J., Meyer, J., Meyer, P., Kassakian, P. and Wessel, D.,
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2.
Conway, J.H., Sloane, N.J.A. and Bannai, E. Sphere packings,
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3.
Fuse, T. Unit origami : multidimensional transformations. Japan
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4.
Kassakian, P. Convex Approximation with Applications in
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5.
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Programming with Applications to Beamforming and
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6.
Kassakian, P. and Wessel, D., Characterization of Spherical
Loudspeaker Arrays. in 117th Audio Engineering Society
Convention, (San Francisco, 2004), AES.
7.
Kassakian, P. and Wessel, D., Design of Low-Order Filters for
Radiation Synthesis. in 115th Audio Engineering Society
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