MODELING OF A ROOM TRANSFER FUNCTIONUSING COMMON ACOUSTICAL POLES
Yoichi HANEDA, Shoji MAKINO, and Yutaka KANEDANIT Human Interface Laboratories,
3-9- 11, Midoricho, Musashino-shi, Tokyo, 180 Japan

Abstract
A new method is proposed for modeling a roomtransfer function (RTF) by using estimated common
acoustical poles that correspond to resonance propertiesof a room. These poles are estimated as common valuesof the multiple RTFs corresponding to different source
and receiver positions. This ‘common-acoustical-pole andzero’ (CAPZ) model requires far fewer variable parameters
to represent RTFs than conventional all-zero or pole/zeromodels. This model was applied to an acoustic echo
canceller and to head-related transfer functions. At low
frequencies, the acoustic echo canceller based on thismodel converges 1.5 times faster than the one based on
the all-zero model. Head-related transfer functions thathave resonance characteristics of the external ear are also
successfully modeled by the proposed model.
1. Introduction
A room transfer function (RTF) expresses thetransmission characteristics of a sound between a source
and a receiver. Modeling an RTF is a key technique formany applications, such as acoustic echo cancellers(ARCS) and sound field controllers.
An all-zero model (MA model) or pole/zero model(ARMA model) is usually used for modeling an RTF,
but they require a large number of parameters whichchange in a complex manner when the position of thesource or receiver changes. These shortcomings slow theconvergence of an ARC based on the conventionalmodels.
A possible solution to this problem would be toestimate parameters that remain constant despite RTF
variations, and use them when modeling an RTF [l-3].
Figure 1 shows the idea of this solution. The
conventional all-zero or pole/zero models require different
sets of parameters Di to represent different RTFs Hi(i= 1,2,3 ,..., M). In the proposed model, common
parameters l/A are first estimated from multiple RTFs.
Then the RTFs are represented by l/A and by different
sets of parameters Bi (i=l,2,3,...Jf),
which have far

fewer parameters than the sets Di of the conventionalmodels.
This paper proposes a new modeling method that uses
estimated acoustical poles of a room as common constantparameters. The proposed model is applied to two
applications, an ARC and the modeling of a head-relatedtransfer function. Computer simulations demonstrate theadvantage of the proposed model.
2. Common-Acoustical-Poleand Zero Model
Acoustical poles are common to all RTFs, observedwith different source and receiver positions in a room [4].
These poles correspond to the resonance frequencies(eigenfrequencies) and their Q-factors. They depend on the
acoustical conditions of a room, such as its shape andabsorption coefficients, but they do not depend on thepositions of the source and receiver .
The proposed common-acoustical-poleand zero(CAF’Z) model is thus represented in pole/zero form andconventional model

RI%

proposed modelFig. 1 Modeling of RTFs corresponding to different
positions of the source and receiver.
The
conventional all-zero or pole/zero models requiredifferent sets of parameters Di to represent different
RTFs Hi (i=1,2,3 ,...,M). The proposed model
reduces the number of parameters in the sets Bi by
extracting parameters l/A that are common to HIHM.

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ARMA form by z-transform:~out@~P)n(l-61

C bi(R)r-’
=i=On

9 (1)

bi(RL)&k-i) . (3)ai g(RL, k-i)-$i-0%ut(RL9 k) is, however,i=li=lk)

Finding ais and his that minimize output error
known to be difficult [5].
Therefore, the “equation error” seq(RL.1k1 is unroduced:

2+x aiz-iPiZ-‘)h(RL,k) - ~(RL,= ~(RL, k) +iQQz

C Z- an ( l-qi(R) r-‘)p i=’
G(R,z) ==R: positions of a source and a receiver,G(R,z): CAP2 model of an RTF corresponding to eachR,
pi: common acoustical pole,qi(R): zero for each R,
P: order of poles,~lq(RL,k)= h(Rt, k) +i

ai h(Rt, k-i)-;bi(RL) &k-i).
i=O

i=l

(4)

Note that tire difference between Eq. (3) and Eq. (4) is in
the second term of the right-hand side, where g(RL, k) is@aced by h(RL,k).
Now, the common AR coefficients are estimated as
those that minimize the following error index I.

Q,Ql,Qz:mlerofzeros, <Q= QI + Qz>,

C: constant,
Ui: common AR coefficient,hi(R): MA coefficient for each R.
With the proposed model, once the common acousticalpoles are estimated, only zeros are estimated for different
RTFs. With the conventional pole/zero model, on theother hand, both poles and zeros are always estimated foreach RTF. Estimated poles in the conventional pole/zeromodel are usually different for different RTFs for thefollowing reason.
3. Estimation of common acoustical poles
All acoustical poles cannot necessarily be observed in asingle RTF, even though all the acoustical poles arecommon to all the RTFs in a room. This is because
zeros that depend on the source and receiver positions
influence or cancel some poles [3], thus causing
erroneous estimation of poles. Common acoustical poles
should therefore be estimated from multiple RTFs
corresponding to different Rs.
Common acoustical poles are estimated as common
AR coefficients, which are equivalent to the poles asshown in Eq. (1). According to Eq. (l), the impulse
response of the CAP2 model for RL, ~(RL, k), isexpressed as~(RL, k) = - iai g(RL, k-i) + 5i-li-0M : number of impulse responses.
The orders of P and Q in Eq. (4) are chosen to minimize
the sum of P and Q for the predetermined values ofsufficiently small I.
Figures 2(a) and 2(b) show one example of the poles
estimated from a single RTF and from 30 RTFs. Because
acoustical poles could be obtained theoretically for arectangular room, RTFs were simulated by assuming a
rectangular room (6.7 x 4.3 x 3.0 m, wall reflection
coefficient 95 %) and using the image method. Thesymbols ( A ) indicate estimated poles and the symbols(0) indicate the theoretical acoustical poles. The rp
1.00 )

IFrequency (Hz)(a) Poles estimated from a single RTF.

bi(RL)&k-i) , (2)1.00 )

125

1estimated
6 6Qp p’,$Oz sg *‘p A*
0 theoretical

L: index for the variation of R,

k: discrete time,
6: unit pulse function.

0.95 1
0The error (output error) between the actual impulse
response h(RL, k) and the model impulse response~(RL, k) is represented asII-214

Fiquency(Hz)I125(b) Poles estimated from 30 RTFs withdifferent positions of the source and receiver.
Fig. 2 Comparison of estimated (A) and theoretical (0)acoustical poles.

(vertical axis in Fig.2) represents the absolute values ofcomplex poles. The poles estimated from a single RTF
do not fit the theoretical poles (Fig.2(a)). On the otherhand, many acoustical poles are well estimated frommultiple RTFs (Fig.Z(b)). Summarizing this result: 1)
common acoustical poles are better estimated frommultiple RTFs than from a single RTF, 2) the proposed
method is effective for estimating common acousticalpoles.
When P and Q are large, however, a lot ofcomputational power is needed to calculate the AR
coefficients that minimize error index Z of Eq. (5). In
such a situation, a set of AR coefficients that minimize
Ceq(RL, k) of Eq. (4) is first obtained for each RTF.
Common AR coefficients are then obtained by averaging
each set of AR coefficients:
a= i$ a,@~.)
Lo(i = 1,2,3 ,.., P) .

(6)

The theoretical background of these averaged AR
coefficients is left for future study, but the estimatedpoles fit the theoretical poles as well as they do whenestimated by minimizing the error index I.
4. Performance of the proposed model
The performance of the proposed model was evaluatedby using it in two applications, the modeling of an RTF
for an acoustic echo canceller and the modeling of a headrelated transfer function. Both transfer functions can bemodeled by the CAPZ model.
4.1. Acoustic echo canceller
We used the proposed model in a series-parallel-type
acoustic echo canceller (AEC) that had a fixed filter withestimated common AR coefficients and an adaptive filter
with variable MA coefficients (Fig.3). The block diagramin Fig.3 was simulated in a computer by using measured
impulse responses.
Because the proposed model for RTFs is especiallyeffective at low frequencies, its performance wasevaluated in the frequency band from 60 to 800 Hz. The
sampling frequency was 2 kHz. Impulse responses ofdifferent source and receiver positions were measured in aroom. The room volume was 80 m3 and its reverberation
time was 0.6 s.
The number of coefficients was 250 for the fixed filter,and 450 for an adaptive filter. These numbers werechosen to provide 35 dB of stationary echo return loss
enhancement (ERLE), where ERLE is defined as the ratioof echo power to residual echo power. Because the ordersof the filters are so large, the common AR coefficientswere obtained using Eq.(6) with ten measured impulse
responses. The common AR coefficients were copied to
the fixed filter and the normalized LMS algorithm wasused for the adaptive filter.
An AEC based on the conventional all-zero model with
only an adaptive filter was also evaluated and found torequire 800 coefficients to achieve 35 dB of stationary
ERLE. The proposed AEC needs only about half as
many adaptive filter coefficients as the conventional AEC
needs.
The performances of both AECs were evaluated bysimulating the convergence speed of the ERLE. Because
the convergence becomes faster as the number of adaptive
filter coefficients becomes smaller, the proposed AEC
converges about 1.5 times faster than the conventional
AEC (Fig. 4).

ERLE
ow
40 1

1

Iterations
iI

Fig. 4 ERLE (echo return loss enhancement)of two AECs. Solid line: proposed AEC with afixed filter (AR part: 250 coefficients) and an
adaptive filter (MA part: 450 coefficients).
Dashed line: conventional all-zero AEC with anadaptive filter (MA part: 800 coefficients).

::

Fig. 3 Acoustic echo canceller that has a fixed filter withestimated common AR coefficients.
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4.2. Head-related transfer function
A head-related transfer function (HRTF) represents thetransmission characteristics from a sound-source to thecar of a listener. HRTF changes when the source
direction changes. Humans localize a sound-source basedon this difference in HRTFs for different directions. The
modeling of an HRTF is important for such applications
as the reproduction of a virtual sound image. The modelwith fewer parameters have been investigated [6].
HRTFs have a resonance system composed of thepinna and ear canal [7]. This resonance system can beconsidered as the common acoustical poles in all theHRTFs corresponding to different sound directions. The
proposed model can thus be applied, by considering thatthe parameter R in Eqs. (l)-(6) corresponds to the source
direction 8.
Modeling of HRTFs was evaluated by calculating thefollowing modeling error E(8):m Conventional

e&Listenerindependentfor sound directionof the source direction

8.

5.

Conclusions
We have modeled a rmrn transfer function by using
common acoustical poles, which are constant when theposition of source or receiver changes. These poles are
estimated as common AR coefficients
either byminimizing an equation error for multiple impulse
responses or by averaging the AR coefficients derived
from each impulse response.
The proposed common-acoustical-pole and zero model
requires fewer parameters that depend on the source andreceiver positions than the conventional models require.
An acoustic echo canceller based on the proposed model
requires half as many adaptive filter coefficients andconverges about 1.5 times faster than an acoustic echo
canceller based on the conventional all-zero model. For
modeling head-related transfer functions, which areimportant for binaural cues of sound localization, theproposed model reduces by l/3 the number of parametersthat depend on the source directions.

k) : output error,h(6. k): measured impulse response
N: impulse response length.

source direction 8 (deg.)

The modeling errors of the conventional
allFig. 5zero model (60 MA coefficients)(blackbars) and the
proposed common acoustical pole and zero model (20commonARcoefficients;
40
variableMAcoefficients)(shadedbars) for head-relatedtransferfunctions.
The
commonAR coefficientsare(7).5,&e,

q proposed
1

Source0,

The output error .cOUt(tl,k)here is defined by Eq. (3),where 8 is replaced by RL.
Eighteen HRTFs with different source directions in thehorizontal plane were measured in an anechoic room. Thefrequency band was from 100 Hz to 18 kHz. The
sampling frequency was 45 kHz. The number of common
AR coefficients, 20, and the number of different MA
coefficients, 40, were predetermined. The common AR
coefficients were then estimated with twelve of theeighteen measured HRTFs by minimizing error index Z ofEs. (5).
Another six HRTFs were modeled using the CAPZ
model with the 20 common AR coefficients estimatedabove, and 40 variable MA coefficients estimated for each
HRTF. They were compared to the conventional all-zeromodel with 60 variable MA coefficients. To represent
HRTFs with different source directions, the proposedmodel of an HRTF was given Z/3 as many variable
parameters that depend on source directions as the all-zeromodel had.
The results of modeling errors of both models areshown in Fig. 5. Although using a small number ofvariable parameters, the modeling errors of the proposedmodel are as good as those of the conventional all-zeromodel. These results confvm that the proposed modelwith common acoustical poles successfully represents theReferences[l] Y. Haneda, S. Makino, Y. Kaneda, “Estimation of Polesin Room Acoustic
Transfer
Functions,”
Proc. Spring
meeting of Acoust. Sot. Jpn., p. 393, March 1991.
[2] J. Mourjopoulos,
A. E. Tsopanoglou.
N. D. Fakotakis,“A Vector Quantization
Approachfor Room Transfer
Function Classification,”
ICASSP91, p.3593, May 1991.
[3] N. Koizumi, R. H. Lyon, “On the model order for theidentificationof acoustic systems,” ASA 118th meeting,
J.A.S.A, 86, Suppl. 1, S3. 1989.
[4] For example, H. Kuttruf. ” Room Acoustics,” Elsevier
Applied Science, London, 1991.
[5] L. Ljung and T. Soderstrom,‘Theory and Practice of
Recursive Identification,”
MIT Press, Cambridge, 1983.
[6] F. Asano, Y. Suzuki, S. Kono, T. Sone, “Simulation andModification of Head-related Transfer Function Using ARMAmodel,” Proc. Autumn meeting of Acoust. Sot. Jpn.. p.367,
October 1987.
[7] J. Blauert. “Spatial Hearing,” MIT Press, Cambridge,
1983.

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