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IEEE SIGNAL PROCESSING MAGAZINE [86] MARCH 2015

matrices are large and may have issues with computational

requirements (for filtered x-RLS) and convergence rates (for fil-

tered x-LMS). Poor convergence can be solved using eigenspace

adaptive filtering [22] by performing a generalized singular value

decomposition (SVD) to diagonalize the system. Unfortunately the

SVD still incurs a high computational cost.

Fortunately, the problem can be solved computationally and

efficiently by using a wave-domain approach. If the microphones

are arranged over a uniform circular array of radius

r and the

sources are arranged over a concentric uniform circular array,

then the anechoic-room ATF matrix may be parameterized

,HUJKV

(11)

where

C is a matrix of ATF coefficients corresponding to the

anechoic room, K is a diagonal matrix of Hankel functions, and

V is a tall Vandermonde matrix (2-D) or a spherical harmonic

matrix (3-D). Matrix V possesses the property ,VV I

= provided

that at least one loudspeaker is present for each mode to be con-

trolled, i.e.,

L N

$ or .L N

The wave-domain adaptive filtering (WDAF) approach trans-

forms the signals at the microphones and the loudspeaker signals

into the wave domain through the transform

and ,T

then

adaptively calculates the mode-domain compensation signals

(),C w

and transforms the compensated loudspeaker signals back

using the inverse transform T

as depicted in Figure 3. If the

compensation filter matrix ()C w

is forced to be diagonal, then

each of its diagonal entries can be determined from decoupled

adaptive filters. This would explicitly solve the problems of compu-

tational complexity that appeared in multipoint compensation

techniques. While it is straightforward to choose

and T

to do

so, in reality T

cannot always be chosen without a priori knowl-

edge of the ATF matrix. However, [22] and [23] show that the sys-

tem can be partially diagonalized by choosing

,VT

= ,VT

and .UT

SYSTEM IDENTIFICATION OF THE ATF MATRIX

The ATFs change in a room as people move about and as tempera-

ture changes. Since active room compensation in particular is

sensitive to this phenomenon, it is better if the ATFs are deter-

mined adaptively. Similar to active listening room compensation,

this task can be performed efficiently in the wave domain where

transforms are used to part-diagonalize the reverberant-room ATF

matrix [23].

The advantages of WDAF and the mode-domain approaches are

that 1) sound pressure is controlled over the entire control region

and not just at specific points and 2) they represent the problem

with a reduced number

1 (or )MN

1 of parameters,

which reduces the complexity and reduces the correlation in the

elements of the ATF matrix since the filters are part decoupled.

This helps speed the convergence of adaptive filtering.

Since many more microphones and loudspeakers are required

for a 3-D control zone, active room compensation is more practi-

cally deployed in 2-D scenarios. However, 2-D compensation can-

not satisfactorily correct for roof and floor reflections, so sound

absorbers must be employed to eliminate these effects.

IMPULSE RESPONSE RESHAPING

Multiple listening zones may also be achieved by using crosstalk

cancelation. Here, each of

Q signal is delivered to a listening posi-

tion while canceling the crosstalk paths to the remaining Q 1-

positions using L loudspeakers and, for monaural signals, M 1=

microphone in each zone. As shown in Figure 4, this problem is

solved by implementing crosstalk-cancelation filters. The basic

idea of the impulse response reshaping approach is that fully

equalizing the delivered paths is unnecessary. It is more robust

and efficient to reshape these impulse responses.

Using impulse response reshaping, the early reflections of the

delivered paths are reinforced while late reverberation and crosstalk

are minimized [25]. Here, by defining windows on these desired

and undesired ATF components

()

and w

()

respectively in each

zone ,q the ratio of undesired-to-desired components is minimized

,min log

{

(12)

where r

{

represents the global impulse response given a concate-

nated vector of crosstalk cancelation filters [, , ]gg g

{{ {

and

a block-Toeplitz matrix H

{

representing the room impulse

responses, i.e., ,rHg=

{

,WwwDiag

() ()

f_ and W

.(,, )wwDiag

() ()

f Different p

- and p

-norms may be chosen

for the desired and undesired components, but it has been shown

to be perceptually favorable to choose norms that approximate the

g(ω)

C(ω)

H(ω)

(ω)

H(ω)

e(ω)

I(ω)

g′(ω)

2N +1 2N + 1

2N + 1

–

˘g

[FIG3] The listening room compensation using WDAF. The

free-field transformed loudspeaker excitation signals g

are

used in a reverberant room with the filter matrix C

compensate for the ATFs in matrix .H

[FIG4] Crosstalk cancelation for delivering a time-domain signal

s to the top microphone while canceling the signals at the

remaining Q 1- microphones.

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