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IEEE SIGNAL PROCESSING MAGAZINE [83] MARCH 2015

loudspeaker weights are chosen to ensure the implementation is

robust to driver positioning errors and changes in the acoustic

environment. The ACC problem can then be posed as

max Hg

(2a)

Hg Dsubject to

# (2b)

.g E

# (2c)

The objective and the constraints are summarized into a single

objective function represented using the Lagrangian [10],

() ( ) ( ),

max gHg Hg gLDE

=- -- -

(3)

where

m and

m are Lagrange multipliers to adjust the relative

importance of each condition (2b) and (2c). The solution that

maximizes the Lagrangian is obtained by taking the derivative of

with respect to g and equating it to zero, and is written as

[][],HH Ig HH g

+= (4)

which is recognized as a generalized eigenvector problem. The

optimum source strength vector g

is set as the eigenvector cor-

responding to the maximum eigenvalue of the matrix

.[(/)][]HIHHH

mm+

The ratio of Lagrange multipliers

mmm= determines the tradeoff between the performance and

array effort and must be chosen iteratively for the constraint on

the array effort to be satisfied. The formulation in (4) yields essen-

tially the same answer as that in [8], or the so-called indirect for-

mulation in [10], which diagonally loads the matrix

before

inverting it to improve the matrix condition number.

The formulation adopted here leads to a straightforward way

for demonstrating the connection between the ACC method and

the PM method, which will be explained next.

PRESSURE MATCHING

The PM method aims to reproduce a desired sound field in the

bright zone at full strength, while producing silence in other zones.

The idea comes from the traditional crosstalk-cancelation problem,

where small regions of personal audio are created by controlling

the pressure at discrete spatial points (microphone or listener posi-

tions). Multizone sound control is an extension of the traditional

approach with a sufficiently dense distribution of matching points

within all the zones. Given a target sound field

des

to be repro-

duced in the bright zone, the robust PM formulation can be written

using an

, PM objective along with the constraints on the sound

energy in the dark zones and the array effort constraint,

min Hg p

bdes

- (5a)

Hg Dsubject to

# (5b)

.g E

# (5c)

The problem can then be written as a Lagrangian cost function,

() ( ) ( ),

min gHgp Hg gLDE

b des d

=-+ -+ -

(6)

where again

m and

m are Lagrange multipliers. The solution

that minimizes L

is obtained by setting the derivative of L

with

respect to g to zero and is written as

[].HH HH Ig Hp

HH H

des

mm++= (7)

Equation (7) may be solved using an interior point algorithm to

choose appropriate values of

m and

m to satisfy the constraints

[11]. A simpler formulation is to set the parameter ,1

m = which

implies applying equal effort to matching the pressure in the

bright zone and minimizing the energy in the dark zone. This

gives the original formulation of multizone sound control as in

[12] but has an added robustness constraint on the array effort,

that is

[].gHHHH IHp

HH H

des

m=++

This solution is

also identical to that of the ACC method provided that 1) the

choice of target pressures in the bright zone is an ACC solution,

pHg

des b c

= and 2) identical constraints in E

and D

are met.

This demonstrates that the formulation in the PM approach to

sound field reproduction subsumes the ACC problem. Chang and

Jacobsen [13] investigated a combined solution of these two tech-

niques,

[( ) ] ( ) ,gHHHHHp11

HH H

descb

ll l=- + -

which

is actually same as the one presented in (7) with the regularization

term omitted. The tuning parameter

l is equivalent to the tuning

parameter .

m The design has been shown effective for reproduc-

ing plane wave sound fields at frequencies even above the Nyquist

frequency with good contrast control, thus providing the potential

to reduced the number of loudspeakers required and increase the

zone sizes and upper operating frequencies using the PM method.

The PM approach gives an explicit solution to obtain the loud-

speaker driving signals and does not require solving an eigenvec-

tor problem, as is required in the case of acoustic contrast

optimization. PM is especially suitable for the situation that differ-

ent constraints are imposed on each sound zone when the listen-

ers require different quality of listening experiences. However a

series of Lagrange multipliers need to be determined, and a gener-

alized eigenvalue solution is no longer possible. Instead convex-

optimization methods like the interior-point method should be

used [11]. The PM approach also imposes an objective on the

phase of reproduced sound fields within the bright zone, and thus

provides a better holographic image compared to the contrast

control method. Figure 2(b) demonstrates that the ACC method

always maintains a high level of contrast between the bright and

dark zone using a small array effort, but a high reproduction error

also indicates that the reproduced sound field may swirl around

the listener in different directions. On the other hand, the PM

approach achieves small reproduction error while higher contrast

may be obtained by choosing an appropriate desired sound field.

Preliminary perceptual tests were found to generally agree with

the simulation results however the source signal itself signifi-

cantly affects the system performance [14].

While the least squares solutions in the frequency domain

seems to provide a great deal of simplicity and flexibility, the

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