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IEEE SIGNAL PROCESSING MAGAZINE [87] MARCH 2015

infinite norm. Equation (12) can be solved analytically for the

pp2

case where it reduces to a generalized Rayleigh quo-

tient. In general, (12) is solved using the steepest descent meth-

ods [25]. A relaxed multichannel approach using least squares

[26] and minimax metrics [24] may include regularizations to

reduce the array effort below that of the ratio-based approach in

[25]. These approaches are compared in Figure 5 for simulation

with

L 3= and Q 2= in a room with a reverberation time of

250 ms using only 150 ms-long reshaping filters. The ratio-based

approach shown is for

,p 10

= ,p 01

= and 1,000 steepest

descent iterations.

Impulse response reshaping, in principle, can be applied to

the PM- and modal-space approaches of creating personal sound

zones. More robust and efficient filters can be obtained than

equalization by canceling the undesirable late reverberation

while leaving in some beneficial early reflections. Unfortunately

this problem must be formulated in the time domain, which

results in a computationally intractable massive multichannel

problem. The future development of lower-complexity convex

optimization algorithms may permit practical solutions to these

large problems.

DIRECTIONAL SOURCES

The use of directional sources can provide advantages over con-

ventional loudspeakers, whose directivity is omnidirectional at low

frequencies and is not typically controllable. Directional sources

that provide multiple modes of sound radiation can be used with

active compensation to produce sound arriving from angles where

there are no sources by reflecting sound from room surfaces and

can also cancel unwanted reverberation (Figure 6).

In a multilistener situation, a single directional loudspeaker

can reduce unwanted radiation of sound to other listeners by max-

imizing the direct sound to the intended recipient relative to the

reverberant field. A loudspeaker with directivity

D and radiating

acoustic power W in an ideal Sabinian space produces a direct

sound intensity /( )I WDr4

dir

r= and a reverberant sound inten-

sity of /,I W R4

rev

where /( )RS1

ee=-

is the room con-

stant, S the room surface area, and

the mean absorption

coefficient of the room surfaces. The direct to reverberant inten-

sity ratio is thus

DRR

(13)

Increasing the directivity then allows the direct sound at the lis-

tener to be increased relative to the reverberant sound. Equiva-

lently, the reverberant field is reduced by /.DRR1

100 200 300

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Time (ms)

Magnitude (dB)

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Time (ms)

Magnitude (dB)

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Time (ms)

(a) (b)

(c)

Magnitude (dB)

Delivered

Crosstalk

[FIG5] The shortening of impulse responses to 50 ms in a room of reverberation time 250 ms using (a) relaxed multichannel least

squares, (b) the relaxed minimax approach in [24], and (c) the ratio optimization approach of [25].

Room

Walls

Zone

Reflections

Higher-Order

Loudspeaker

(a)

(b)

[FIG6]

A demonstration of the higher-order loudspeaker in (a) a

cylindrical baffle and (b) the schematic plot of its behavior.

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