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Least Squares Equalizer Design under Consideration of Tail Effects
Stefan Goetze1, Markus Kallinger2, Alfred Mertins3, and Karl-Dirk Kammeyer1
1University of Bremen, Dept. of Communications Engineering, D-28334 Bremen, Email: goetze@uni-bremen.de
2Fraunhofer Institute for Integrated Circuits, D-91058 Erlangen, Email: markus.kallinger@iis.fraunhofer.de
3University of L¨
ubeck, Institute for Signal Processing, D-23538 L¨
ubeck, Email: alfred.mertins@isip.uni-luebeck.de
Abstract
Modern high-quality hands-free telecommunication sy-
stems have to cope with several real-world 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: Least-squares 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 coefficient-vectors
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 frequency-discrete room transfer function (RTF) and
the LRC-filter, 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 ·log10|E[m]| − ¯
EdB2.
Here the mean value of the logarithmic spectrum is given
by ¯
EdB = 1/(mmax −mmin)Pmmax
m=mmin 20 ·log10|E[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
x-axis 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 least-squares
equalization with an ideally known impulse response and
the unequalized case (EQ switched off) respectively.
noequalization
LSequalizerwitha-prioriinformation
-14
-12
-10-8
-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 video-conferencing 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.
