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IEEE SIGNAL PROCESSING MAGAZINE [89] MARCH 2015

For example, a third-order speaker with radius . ma 01= and a

Nyquist frequency of 4 kHz would require 70 drivers. This is a

large number of drivers, and motivates the investigation of sim-

pler approaches such as cylindrical and line arrays.

CYLINDRICAL ARRAYS

A simpler approach may be taken if the directional loudspeaker is

only required to produce directivity in a 2-D plane. This is com-

monly the case for sound reproduction in the home, where stereo

and 5.1-surround formats are ubiquitous. A circular array requires

fewer drivers than a spherical array for the same spatial Nyquist

frequency. To see this, consider a sphere where

drivers are

placed on the equator instead of equally spaced around the sphere.

This arrangement allows for the generation of sectorial spherical

harmonics, where

||,on= which produce radiation with lobes

only in the (, )xy plane. The driver spacing is now /a L2

r and

the spatial Nyquist frequency is

()

c L

, D2

Nyq

(18)

The number of drivers for a given 2-D spatial frequency is

af4

Comparing (18) with (17), the 2-D Nyquist frequency can be

much higher than the 3-D Nyquist frequency for the same

number of drivers. The limitation of the circular array is that

the transducer layout does not provide sufficient vertical direc-

tivity at high frequencies, and the source begins to produce

unwanted radiation lobes in elevation. To reduce these lobes,

the transducers must either have greater aperture in elevation

or a line array must be used to control the vertical directivity.

Since a line array is more effective when mounted on a cylinder

than on a sphere, a practical alternative to the spherical array

for the 2-D case is a cylindrical baffle in which multiple

circular arrays are mounted (Figure 6). Such a geometry can

still use fewer transducers than the spherical case, for the

same spatial Nyquist frequency.

The radiation of sound for the cylindrical case can be approxi-

mated by assuming that the cylinder is infinite and that each

driver is represented as a surface velocity distribution in height

and azimuth angle

z [29]. Its produced mode responses are

shown in Figure 7(b). The responses are similar to those for the

spherical source, and the activation frequencies are the same. The

limitation of this analysis is that, in practice, a truncated cylinder

must be used leading to variations of the mode response magni-

tude around the infinite cylinder values due to diffraction from the

ends of the cylinder.

LINE ARRAYS

The simplest array for providing high directivity is a line array,

which produces an axisymmetric polar response. While this does

not provide the full control of 3-D or 2-D radiation that the spheri-

cal and cylindrical arrays do, it is sufficient for maximizing the

direct to reverberant ratio. It has the same limitation as the circu-

lar and spherical arrays, that is difficult to create high-order

responses at low frequencies. However, the line array allows an

order

N response to be produced using L N 1

transducers

as opposed to ()N 1

+ using a spherical array or N21+ for a cir-

cular array (assuming no vertical directivity control). The maxi-

mum directivity produced in 3-D is [30]

().DN1

An order N loudspeaker with this directivity will produce the

maximum direct to reverberant ratio for an on-axis listener. The

simplest case,

,N 1= results in a polar response ( ) .p 025

i =+

.(),cos075

which has a directivity of four [7]. The first-order

response can be implemented using N 2= coupled or uncoupled

drivers, or more simply, using a single driver and controlling the

[FIG7] The normalized magnitude of the mode responses of (a) a spherical source and (b) a cylindrical source for orders 0–5.

–10

–20

–30

–40

–50

–60

Frequency (Hz)

(a)

–10

–20

–30

–40

–50

–60

Frequency (Hz)

(b)

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