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IEEE SIGNAL PROCESSING MAGAZINE [88] MARCH 2015
Standard loudspeakers typically have insufficient directivity to
provide a significant enhancement of direct sound in a reverberant
space. High directivity can be achieved using traditional array
techniques such as delay and sum beamforming, but the array size
must be large at low frequencies to achieve significant directivity.
For practical use, superdirectional arrays are required, which
achieve higher directivities than an array with uniform amplitude
weightings [27]. Superdirectivity can be achieved using linear dif-
ferential arrays, where the transducer weights have alternating
signs, or by using circular and spherical arrays, where the weights
are obtained from trigonometric or spherical harmonic functions,
respectively. Such loudspeakers are termed higher-order sources
(HOSs) and can produce multiple radiation patterns that are
described by cylindrical or spherical harmonics.
Because superdirectional arrays are compact relative to their
directivity, they may be built into a single unit, and we therefore
assume here that a directional source is a single unit, typically of
similar dimension to a standard loudspeaker. This section consid-
ers the design of directional loudspeakers and their application to
maximum directivity and then focuses on the advantages of using
arrays of sources, which allow greater control of sound fields over
wide areas and are particular suitable for establishing personal
sound zones.
SPHERICAL ARRAYS
The sound field produced by an arbitrary source of maximum
radius
a positioned at the origin and radiating a complex fre-
quency ()exp it~ is represented in the wave domain as in
(S2b) [17]
(,,,) () () (,), ,pr w wh krY r a
()
N
0
2
$iz a iz=
o
n
no
o
o
o
o
n
=-=
//
(14)
where ()hkr
()2
o
is the spherical Hankel function of the second
kind, i.e., the radial function to represent the mode amplitude at r
and ()w
a
o
n
are sound field coefficients. Similar to the dimension-
ality analysis in the wave domain, we will assume that the directiv-
ity of the source can be described by a maximum order
N so that
.[, ]N0!o
The most direct method for constructing a loudspeaker that
can produce a controllable directivity is to mount a number of
drivers in a spherical baffle of radius
a [28]. The general behavior
of such a source is most simply explained by deriving the sound
field due to a sphere with arbitrary surface velocity
,,, ()(,),vtwe wY
ss
it
ss
N
0
iz g iz=
~
o
n
o
n
no
o
o
=-=
^h
//
(15)
where (, )
ss
iz is the driver position on the sphere. The exterior
field has the general form of (14). The expansion coefficients are
found by calculating the radial velocity for the general case, and
requiring that they equal (15), i.e.,
o
()
()
wic
hka
w
()2
at
g
=-
o
n
o
n
l
^h
and the sound field, including the effect of mass-controlled driv-
ers, is
o
(, , , , ) ( )
()
()
,, .
pr tw
k
ice
w
hka
hkr
Yra
()
()
it
N
2
2
0
# $
iz
t
g
iz
=-
~
o
n
no
o
o
o
o
n
=-=
l
^h
//
Hence, each coefficient of the surface velocity produces a cor-
responding mode of radiation whose polar response is governed
by a spherical harmonic.
The normalized magnitude of the mode responses for orders
0–5 are shown in Figure 7(a). For all modes greater than order
,0
o = the response reduces at low frequencies. All modes of
order
o become active at a frequency approximately given by
ka
o= or
.f
a
c
2
r
o
= (16)
This means that it is not possible to create high-order directivities
at low frequencies. The spherical loudspeaker is omnidirectional at
low frequencies and can produce increasing directivities as more
modes become active above frequencies given by (16).
In practice, the surface velocity in (15) must be approximated
using a discrete array of
L
0
drivers positioned on the sphere. Ide-
ally the drivers are positioned so that they are spaced equally
from each other which produces the most robust approximation
to the integration over the sphere required to approximate each
spherical harmonic. This is possible if the drivers are placed in
the center of the faces of platonic solids, allowing up to 20 drivers
(for the icosahedron). Higher numbers of drivers can be used
using numerically optimized integration nodes for the sphere.
A simple way to model the discrete approximation is to
assume each driver is a point source. The sound field due to a
point source on a sphere then models a single driver, and the
sound fields due to
L
0
point sources allows the calculation of the
total field. However, this approach ignores the directivity of each
driver, which becomes significant at high frequencies. A more
accurate model of the drivers that is mathematically tractable is
to model each one as a spherical cap vibrating radially [28].
The sampling of the sphere means that the spherical loud-
speaker is unable to generate spherical harmonic terms above
the spatial Nyquist frequency of the array. This may be derived
by noting that there are a total of
()N 1N
2
=+ spherical har-
monics up to order .N Controlling this number of modes
using L
0
loudspeakers is possible for .L N
0
$ At a given fre-
quency, the maximum-mode order that can be radiated is
.Nka= Hence, the spatial Nyquist frequency is
()
.f
a
c L
2
1
, D3
0
Nyq
r
=
-
(17)
The number of drivers required for a sphere of radius a to produce
Nth-order directional responses up to a frequency f is given by
.L
c
af2
1
D3
2
r
=+
cm
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