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Least Squares Equalizer Design under Consideration of Tail Eﬀects
Stefan Goetze1, Markus Kallinger2, Alfred Mertins3, and KarlDirk Kammeyer1
1University of Bremen, Dept. of Communications Engineering, D28334 Bremen, Email: goetze@unibremen.de
2Fraunhofer Institute for Integrated Circuits, D91058 Erlangen, Email: markus.kallinger@iis.fraunhofer.de
3University of L¨
ubeck, Institute for Signal Processing, D23538 L¨
ubeck, Email: alfred.mertins@isip.uniluebeck.de
Abstract
Modern highquality handsfree telecommunication sy
stems have to cope with several realworld problems,
such as corruption of the desired signal by additive noise,
acoustic echoes and reverberation. This paper addresses
the mutual impacts of the subsystems for Acoustic Echo
Cancellation and Listening Room Compensation (LRC).
In acoustic systems for LRC the equalizer is placed in
front of the loudspeaker. An estimate of the room im
pulse response (RIR) is necessary for the equalizer to
compensate for the inﬂuence of the RIR at the positi
on of the reference microphone where the human user
is located. Since the RIR is identiﬁed by the acoustic
echo canceller (AEC) anyway, its estimate can be used
to design the equalizer. The quality of equalization in
dependence of the degree of system identiﬁcation will be
investigated in this contribution. Furthermore the inﬂu
ence of the equalizer on an echo canceller is analyzed.
Listening Room Compensation
Figure 1shows the basic setup for an LRC ﬁlter cEQ pre
ceding the RIR hwhose inﬂuence has to be compensated.

+
eEQ[k]
cEQ
d
h
s[k]x[k]y[k]
ˆy[k]
near end room
Figure 1: Leastsquares equalizer for Listening Room Com
pensation.
By minimizing the mean square error of
eEQ[k] = sT[k]HcEQ −sT[k]d(1)
with the deﬁnitions
s[k] = [ s[k], s[k−1], ... , s[k−Lh−Lc,EQ + 2] ]T(2)
cEQ =cEQ,0, cEQ,1, ... , cEQ,Lc,EQ−1T(3)
d= [ 0, ..., 0
{z }
k0
, d[0], d[1], ..., d[Ld−1],0, ..., 0
 {z }
Lh+LcEQ
−1−Ld−k0
]T(4)
and the convolution matrix Hof dimension (Lh+Lc,EQ −
1×Lh) we get the well known least squares equalizer
cEQ =H+d(5)
for a white noise input s[k]. In (5)H+denotes the Moore
Penrose pseudoinverse of the channel matrix and dis
the desired system which should be approximated by the
concatenated system cEQ H. Here dis chosen as a 10th
order butterworth bandpass with band limits at 200Hz
and 3700Hz for a sampling frequency of fs= 8kHz. The
lengths of the RIR, the LRC ﬁlter and the desired system
dare denoted by Lh,Lc,EQ and Ld, respectively.
System Identiﬁcation by an Acoustic Echo
Canceller
For LRC an estimate of the RIR in eq. (5) is needed
which can be delivered by the AEC since the estimate of
the echo ˆ
ψ[k] is obtained by system identiﬁcation anyway.
+
cEQ[k]
cAEC[k]
h[k]
s[k]x[k]
ˆ
ψ[k]
ψ[k]eAEC[k]
near end room
4000
300020001000
h[k]
k
1
0.5
0
0
LcLt
Figure 2: System for Listening Room Compensation with an
Acoustic Echo Canceller for system identiﬁcation.
Since the length Lhof the RIR which has to be identiﬁed
is greater than the length Lcof the identiﬁcation ﬁlter,
the system identiﬁcation will be biased for a nonwhite
input x[k]. This is known from echo cancellation as the
tail eﬀect [1]. We split up the RIR into a part hc[k] which
can be modeled by the AEC and a tail ht[k] according
to Figure 2. By minimizing the power of the AEC error
Ee2
AEC[k]with the error signal
eAEC[k] = hT
c[k]xc[k]−cT
AEC[k]xc[k] + hT
t[k]xt[k] (6)
and the signal and coeﬃcientvectors
xc[k] = [x[k], x[k−1] , ... , x[k−Lc+ 1]]T(7)
xt[k] = [x[k−Lc], ... , x[k−Lc−Lt+ 1]]T(8)
hc[k] = [h0[k], h1[k], ... , hLc−1[k]]T(9)
ht[k] = [hLc[k], hLc+1[k], ... , hLh−1[k]]T(10)
cAEC[k] = [cAEC,0[k], cAEC,1[k], ..., cAEC,Lc−1[k]]T(11)
we obtain
cAEC[k] = hc[k] + Exc[k]xT
c[k]−1Exc[k]xT
t[k]ht[k].
(12)
From equation (12) we see that the exact identiﬁcation
of the RIR is only possible for a white input signal since
Exc[k]xT
t[k]is zero only for a white input. The more
the early part of the input signal xc[k] is correlated to
the late part of the input signal xt[k] the stronger the
inﬂuence of the tail ht[k] is. As we can see from Figure 2
further correlation is caused by the equalizer in the input
path of the AEC.
Simulation Results
The RIR was simulated with a reverberation time of
τ60 = 300 ms. The ﬁlter orders of the AEC and the
equalizer (EQ) were 1024 and 2048, respectively. As in
put signals white Gaussian noise and a recorded speech
signal (male speaker) were used.
AEC convergence
The convergence of the AEC is inﬂuenced by the addi
tional coloration introduced by the EQ.
DdB [k]
k
30
25
20
15
10
5
2.5
2
1.5
1
0.5
0
0
·104
noise input, EQ off
noise input, EQ on
speech input, EQ off
speech input, EQ on
Figure 3: Relative System Misalignment DdB[k]
Figure 3shows the relative system misalignment
DdB[k] = 10 ·log10
h[k]−cAEC[k]2
h[k]2(13)
with the quadratic vector norm h[k]2=hT[k]h[k] for
the two input signals s[k] (white noise or speech) and for
the cases of active and inactive EQ. If the EQ is switched
oﬀ and the system input is white the AEC reaches the
best system identiﬁcation and the fastest convergence. If
we switch on the EQ, both convergence and maximum
system identiﬁcation decrease. This is due to the correla
tion introduced by the EQ ﬁlter as we can see from (12).
The same tendency can be observed for speech input.
Inﬂuence of the AEC on the EQ
For evaluation of the LRC subsystem we use the er
ror criterion after [2]. The spectrum of the concaten
ated system of cEQ [k] and h[k] can be calculated by
E[m] = H[m]·CEQ[m] with H[m] and CEQ [m] being
the frequencydiscrete room transfer function (RTF) and
the LRCﬁlter, respectively. The variance of E[m] gives a
measure for the spectral ﬂatness of the equalized system
and thus for the quality of equalization:
σ2
E=1
mmax −mmin
mmax
X
m=mmin
20 ·log10E[m] − ¯
EdB2.
Here the mean value of the logarithmic spectrum is given
by ¯
EdB = 1/(mmax −mmin)Pmmax
m=mmin 20 ·log10E[m].
The limits mmin and mmax are chosen to match the
Discrete Fourier Transform (DFT) bins at 200Hz and
3700Hz respectively because this is our desired equali
zation area speciﬁed by the reference system d.
Figure 4shows the variance σ2
Ein dependance of the sy
stem misalignment of the AEC for the white noise and
the speech input signal. It should be mentioned that the
xaxis is ﬂipped so that high values for DdB , which indi
cate bad convergence, are left and smaller values indica
ting good convergence are right. The two horizontal lines
at σ2
E= 1.05 and σ2
E= 14.34 indicate the leastsquares
equalization with an ideally known impulse response and
the unequalized case (EQ switched oﬀ) respectively.
noequalization
LSequalizerwithaprioriinformation
14
12
108
6
4
20
10
0
white noise
speech
σ2
E(DdB )
DdB
Figure 4: Variance σ2
Eof the equalized system depending on
the degree of system identiﬁcation
The system shows the same behavior for both white noise
and speech input. If the AEC shows poor convergence,
which means that the system misalignment is high, the
EQ introduces further distortion to the loudspeaker si
gnal and should better be switched oﬀ in such periods.
The better the system identiﬁcation is the more the va
riance decreases which indicates a good equalization.
Conclusion
In this contribution we analyzed the mutual inﬂuences of
the videoconferencing subsystems Listening Room Com
pensation and Acoustic Echo Cancellation. The quality
of system identiﬁcation and thus of echo reduction was
shown in dependance of the coloration introduced by the
equalizer. Furthermore the quality of equalization was
analyzed in dependance of the degree of system identiﬁ
cation. Using these results it is possible to inﬂuence the
adaptation of one of the subsystems by analyzing the
other to archive a better overall performance.
Literatur
[1] J. Benesty, D. R. Morgan, and M. M. Sondhi. A
Better Understanding and an Improved Solution to
the Speciﬁc Problems of Stereophonic Acoustic Echo
Cancellation. IEEE Trans. on Speech and Audio Pro
cessing, 6(2):156–165, Mar 1998.
[2] J. N. Mourjopoulos. Digital Equalization of Room
Acoustics. Journal of the Audio Engineering Society,
42(11):884–900, November 1994.