A Clipping-Based Selective-Tap Adaptive Filtering Approach to Stereophonic Acoustic Echo Cancellation

Tarbiat Modares Univ., Tehran, Iran
IEEE Transactions on Audio Speech and Language Processing (Impact Factor: 2.48). 09/2011; 19(6):1826 - 1836. DOI: 10.1109/TASL.2010.2102752
Source: IEEE Xplore
ABSTRACT
Stereophonic acoustic echo cancellation remains one of the challenging areas for tele/video-conferencing applications. However, the existence of high interchannel coherence between the two input signals for such systems leads to considerable degradation in misalignment convergence of the adaptive filters. We propose a new algorithm for improving the convergence performance and steady-state misalignment by considering robustness to the source position in the transmission room. We achieve this by exploiting the inherent decorrelating properties of selective-tap adaptive filtering as well as employing a variable clipping threshold for the unselected taps. Simulation results using colored noise and speech signals show an improvement over existing algorithms both in terms of convergence rate as well as steady-state normalized misalignment.

Full-text

Available from: Mojtaba Lotfizad
1826 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 19, NO. 6, AUGUST 2011
A Clipping-Based Selective-Tap Adaptive
Filtering Approach to Stereophonic
Acoustic Echo Cancellation
Mehdi Bekrani, Andy W. H. Khong, Member, IEEE, and Mojtaba Lotfizad
Abstract—Stereophonic acoustic echo cancellation remains one
of the challenging areas for tele/video-conferencing applications.
However, the existence of high interchannel coherence between the
two input signals for such systems leads to considerable degrada-
tion in misalignment convergence of the adaptive filters. We pro-
pose a new algorithm for improving the convergence performance
and steady-state misalignment by considering robustness to the
source position in the transmission room. We achieve this by ex-
ploiting the inherent decorrelating properties of selective-tap adap-
tive filtering as well as employing a variable clipping threshold
for the unselected taps. Simulation results using colored noise and
speech signals show an improvement over existing algorithms both
in terms of convergence rate as well as steady-state normalized mis-
alignment.
Index Terms—Center clipping, convergence rate, interchannel
coherence, partial updating, selective-tap, stereophonic acoustic
echo cancellation (SAEC).
I. INTRODUCTION
T
HERE has been increasing interest in employing stereo-
phonic audio communication systems for video/tele-con-
ferencing, home entertainment, and E-learning applications
in order to achieve better perception of sound [1], [2]. Such
systems have become increasingly popular since a stereophonic
audio system provides spatial information, leading to better
perception of the transmitted speech as well as improving the
ambience of the transmission room. These systems mitigate,
to a certain extent, the cocktail party problem that exists in
a multiparty conferencing scenario [2]. One of the problems
that should be addressed in such systems is the cancellation
of stereophonic acoustic echo using a pair of adaptive filters.
Stereophonic acoustic echo cancellation (SAEC) has issues
that are considerably more challenging to overcome than the
Manuscript received October 09, 2009; revised June 18, 2010; accepted
November 29, 2010. Date of publication February 10, 2011; date of current
version June 03, 2011. This work was supported in part by the Singapore
National Research Foundation Interactive Digital Media R&D Program under
research grant NRF2008IDM-IDM004-010 and in part by the Research Insti-
tute for ICT. The associate editor coordinating the review of this manuscript
and approving it for publication was Prof. Sharon Gannot.
M. Bekrani was with Tarbiat Modares University, Tehran, Iran. He is now with
the School of Electrical and Electronic Engineering, Nanyang Technological
University, Singapore 639798 (e-mail: mbekrani@ntu.edu.sg).
A. W. H. Khong is with Nanyang Technological University, Nanyang Tech-
nological University, Singapore 639798 (e-mail: andykhong@ntu.edu.sg).
M. Lotfizad is with the Department of Electrical and Computer Engineering,
Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran (e-mail: lot-
fizad@modares.ac.ir).
Digital Object Identifier 10.1109/TASL.2010.2102752
monophonic case [3]. The fundamental problem is the poor
mismatch between adaptive filter coefficients and the receiving
room acoustic impulse responses. It has been shown [4] that, in
a practical scenario, the adaptive filter misalignment converges
poorly, leading to a performance degradation. The misalign-
ment problem is caused by the high interchannel coherence that
exists between the two transmitted stereo signals [4].
A variety of methods have been proposed to address the mis-
alignment problem. These methods revolve around reducing
the interchannel coherence between the two transmitted signals.
One of the first methods involved adding controlled quantities
of independent noise to each input channel [5], or modulating
them [6]. These two approaches are however not feasible since
distortion can be heard even when the added noise level is very
low [7]. Another approach involves the use of comb filtering
[8], where frequency components of the left and right channels
are separated in order to reduce the interchannel coherence.
Although this method improves the performance of the adap-
tive filters, it degrades the quality of the stereophonic sound,
especially at lower frequencies.
To address the disadvantages of the methods described above,
a preprocessor has been proposed to add a nonlinear (NL) func-
tion of the transmitted signal in each channel to the signal it-
self [4], [9]. This method employs a half-wave rectifier and is
attractive in terms of improving the misalignment behavior of
the adaptive filters as well as its simplicity in implementation.
Although less distortion was introduced compared to the other
techniques discussed above, this distortion is still found to be
objectionable in some cases for music applications [4], [10].
An adaptive nonlinearity control was subsequently proposed to
maintain the desired level of misalignment and to minimize the
audio distortion [11].
It is apparent by now that algorithms proposed for SAEC
need to decorrelate the transmitted signals without degrading
the quality of the speech signals or destroying the stereophonic
image of the transmission room. In view of this, the use of time-
varying all-pass filtering of the stereophonic signals has been
proposed [12] with the aim of signal decorrelation while main-
taining the stereophonic perception. The use of psychoacoustic
properties to reduce perceived distortion while achieving signal
decorrelation has also been proposed, including the use of spec-
trally shaped random noise [7], [13], gain controlled phase dis-
tortion [14] and the combination of comb filtering and all-pass
filtering with respect to the masking effect [15]. These methods
exploit a perceptual property of the human auditory system,
called “noise masking.”
1558-7916/$26.00 © 2010 IEEE
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BEKRANI et al.: CLIPPING-BASED SELECTIVE-TAP ADAPTIVE FILTERING APPROACH TO SAEC 1827
More recent advances in SAEC research involve a decorre-
lation procedure for the adaptive weight update [10], [16]–[19].
These approaches decorrelates the tap-input vectors of the adap-
tive filters as opposed to decorrelating the transmitted signals.
Among these methods, the exclusive-maximum (XM) tap-selec-
tion algorithm proposed in [17] appeared to be an attractive solu-
tion. This algorithm achieves update signal decorrelation by en-
suring that only an exclusive set of filter coefficients from each
channel is selected for adaptation. In order to reduce any degra-
dation in convergence performance due to this subselection pro-
cedure, the XM tap-selection strategy further ensures that the
energies of these exclusive tap inputs are maximized. The XM
tap selection has been incorporated with the NL half-wave rec-
tifier and the resulting XMNL normalized least-mean-square
(XMNL-NLMS) algorithm has been shown to achieve a higher
rate of misalignment convergence compared to that of nonlinear
NLMS (NL-NLMS) [17].
We propose to further improve the convergence performance
of the XM tap-selection algorithm. This motivation is derived
from the degradation in convergence performance of the XM
tap-selection algorithm when the interchannel coherence be-
tween the two tap-input vectors is relatively low. This can occur,
for example, when the source in the transmission room is lo-
cated away from the centroid of the stereophonic microphone
pair. We note that the robustness issue of XMNL-NLMS to
the source position has not been investigated and that, in this
work, we present insight into this problem. Utilizing this new
knowledge, we propose to improve the misalignment conver-
gence of XMNL-NLMS by employing a center-clipping algo-
rithm so that the low interchannel coherence and maximization
of tap-input energy criteria can be jointly optimized. The pro-
posed algorithm ensures that the misalignment convergence and
steady-state misalignment of the adaptive filters will be robust
to the source position in the transmission room.
In [7] and references therein, the authors evaluated the adap-
tive signal decorrelation filter as a preprocessor and reported
that
complete decorrelation in the frequency domain cannot be
achieved unless one or both of the stereophonic signals are zero
at every frequency. This process is undesirable since it destroys
the stereophonic image of the transmitted signals, which is im-
portant to the listeners in the receiving room. Therefore, they
concluded that complete decorrelation is not applicable in prac-
tice. Our proposed method, as opposed to the technique dis-
cussed in [7], does not apply decorrelation filtering to the trans-
mitted signals. We instead operate on the tap-input vector of
the adaptive filters. Therefore, similar to XMNL-NLMS, the
stereophonic image is preserved. We also note that the decor-
related tap-input vectors of both algorithms may prevent some
of the adaptive filter coefficients from adaptation in some iter-
ations. However, this effect will not significantly degrade the
convergence of weights since any significant reduction in the in-
terchannel coherence due to our center-clipping approach will
bring about an improvement in convergence rate. It is also im-
portant to note that, similar to the approach in [17], the use of the
NL preprocessor is required to provide a solution to the ill-posed
SAEC problem, while our proposed center-clipping approach
improves the convergence rate and robustness of XMNL-NLMS
to the source position.
Fig. 1. Stereophonic acoustic echo cancellation for teleconferencing applica-
tion.
II. R
EVIEW OF
STEREOPHONIC
ACOUSTIC ECHO
CANCELLATION
Fig. 1 shows the stereophonic acoustic echo canceller. For
simplicity, we consider only one microphone in the receiving
room, since similar analysis can be applied to the other channel
[4]. Microphones in the transmission room receive signals pro-
duced by the sound source
via acoustic impulse responses
and giving transmitted signals and , re-
spectively, where
for Channel and is the length of the transmission room im-
pulse responses, while
is defined as the transpose operator.
The transmitted signal to the receiving room for Channel
can
then be expressed as
(1)
where
. These
signals produce an echo
in the receiving room given by
(2)
where
is
the
th channel receiving room impulse response and
is the
th channel tap-input vector while is the length of .
Similar to single-channel AEC, adaptive filters are employed
to estimate
and . In this paper, we assume, similar
to that of [4], [17], that the adaptive filters are each of length
,
which is of the same length as that of
and . For real-
istic applications where the adaptive filters are shorter than that
of
, residual echo will be transmitted back to the transmis-
sion room due to the unmodeled “tails” of
.
The error between the echo and its estimate can then be ex-
pressed as
(3)
where
, 1, 2, is
the vector of adaptive filter coefficients for the
th channel.
A. Nonlinear Normalized Least-Mean-Square Algorithm
In order to efficiently reduce
, adaptive algorithms are
employed for SAEC. Defining
as the expectation operator,
the NLMS algorithm [20] is the result of minimizing
,
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1828 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 19, NO. 6, AUGUST 2011
and is popular because of its simplicity in computational effi-
ciency and ease of implementation. The two-channel NLMS al-
gorithm is expressed as
(4)
where
and
are the concatenated two-channel
tap-input vector and filter coefficient vector, respectively, while
is the step-size, which controls the rate of convergence, and
is a regularization parameter to prevent division by zero.
Unlike single-channel AEC, estimation of
and
for the stereophonic case is challenging. As shown in [4], when
, the adaptive filters coefficients is of the form
(5)
where
, 1, 2 are vectors given
that
is appended with zeros and is a scalar
quantity. Equation (5) indicates that multiple solutions exist and
that
and are linearly related to and .
The dependency of
and on these multiple solu-
tions due to
and is known as the non-uniqueness
problem. In practical cases where
, the non-uniqueness
problem is mitigated [4]. However, even in such cases, direct ap-
plication of the standard adaptive filtering is normally not suc-
cessful because
and are highly correlated, giving
rise to an ill-conditioned system identification problem. As a
result, the misalignment convergence of the adaptive filters is
impaired significantly. This degradation is known as the mis-
alignment problem [4].
In order to address the misalignment problem, a nonlinear
(NL) preprocessor is proposed [4], [21]. This preprocessor op-
erates on
and such that the modified transmitted
signals
and are given by
(6)
(7)
where
controls the amount of nonlinearity to be added. It has
been shown in [4] that a value of
is a good compromise
between speech quality and misalignment convergence of the
NLMS algorithm.
Due to its simplicity in implementation and the low distortion
introduced, the NL preprocessor has become an intrinsic part of
SAEC and has been incorporated into several recently proposed
algorithms for SAEC [10], [17]. For the remainder of this paper,
the NLMS algorithm employing this NL preprocessing will be
referred to as NL-NLMS.
B. Exclusive-Maximum (XM) Tap-Selection Algorithm
The XMNL-NLMS update [17] can be expressed as
(8)
where
1, 2 is the channel index,
is the NL-pre-
Fig. 2. Locations of the source and the microphone pair in the transmission
room.
processed tap-input vector of the
th channel defined
by (6) and (7),
, while
is a tap-selection matrix and
with elements
given by
otherwise
(9)
otherwise
(10)
where
, denote the elemental indices of
and , respectively, and ,
.
As can be seen, the XM algorithm [17] incorporates a tap-se-
lection scheme that reduces the interchannel coherence by se-
lecting exclusive filter coefficients for updating in each channel.
It is important to note that the selected tap inputs are only used
for updating the coefficients and hence no distortion is intro-
duced. However, with any tap-selection updating strategy, con-
vergence performance of the adaptive filters will be reduced.
This degradation is then minimized by jointly maximizing the
norm of the selected tap inputs across both channels. The use
of the NL preprocessor is required to provide a solution to the
ill-posed SAEC problem, while the XM approach improves the
convergence rate of NL-NLMS. As a result of this combination,
which we refer to as XMNL, better misalignment convergence
of the adaptive filter can be achieved. Alternatively, the XM tap
selection can be seen as an effective approach to achieve good
misalignment convergence with lower distortion brought about
by a smaller nonlinearity factor
.
III. E
FFECT OF SOURCE POSITION ON MISALIGNMENT
CONVERGENCE
One of the problems that has yet been considered for
XMNL-NLMS is its robustness to the position of the source
in the transmission room. We now illustrate the misalignment
convergence of XMNL-NLMS by considering different source
positions. Fig. 2 shows an experimental setup where two mi-
crophones are placed at
positions (2.7,2,1.5) m and
(3,2,1.5) m in a room with dimensions 7 m
7m 4m.We
vary the
position of the source starting from the front of the
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BEKRANI et al.: CLIPPING-BASED SELECTIVE-TAP ADAPTIVE FILTERING APPROACH TO SAEC 1829
Fig. 3. Misalignment convergence of the XMNL-NLMS algorithm (solid) for
different positions of the source, as compared to NL-NLMS (dashed).
array centroid at m to the front of the right micro-
phone at
m. The positions in the receiving room are (3, 2,
1.5) m for the microphone and (2.85, 1.8, 1.6) m and (2.4, 1.1,
1.7) m for the two loudspeakers. For the purpose of illustration,
all room impulse responses are generated synthetically using
the method of images [22] such that
and
samples and these synthetic impulse responses have lengths
that correspond to their reverberation times. At a sampling
rate of
Hz, this corresponds to 23 ms and 36 ms,
respectively. A stationary colored noise source signal
is
obtained by filtering white Gaussian noise through a low-pass
finite impulse response (FIR) filter with coefficients given
by
[23] which was chosen to generate a
speech-like spectrum. The convergence of the algorithms is
quantified by the normalized misalignment
(11)
where
. Fig. 3 shows the misalign-
ment convergence of XMNL-NLMS averaged over ten indepen-
dent trials, for the different source positions. The convergence
performance of NL-NLMS for
m, when it is directly
in front of the microphone pair centroid, has been included for
comparison. Additional tests conducted have shown that the
misalignment convergence of NL-NLMS for various source po-
sitions is comparable to that shown in Fig. 3. A step-size of
was used for XMNL-NLMS, while the step-size of
NL-NLMS was adjusted to
so that its steady-state mis-
alignment reaches that of XMNL-NLMS. As shown in Fig. 3,
XMNL-NLMS outperforms the full-update NL-NLMS when
the source is directly in front of the microphone pair centroid at
m. On the contrary, the convergence rate of XMNL-
NLMS is reduced significantly when the source is located away
from the microphone pair centroid such as when
m.
To gain further insight into the degradation in convergence
rate of XMNL-NLMS with respect to the source position, we
consider both the interchannel coherence as well as the ratio of
selected tap-input energy to the total tap-input energy.
Fig. 4. Average interchannel coherence of the NL-NLMS (dashed) and
XMNL-NLMS (solid) algorithms for various positions of the source.
A. Effect of XM Tap Selection on Interchannel Coherence
We first investigate the effect of XM tap-selection on the in-
terchannel coherence for various source positions. We denote
the XM subselected tap-input vector in (8) as
(12)
The interchannel coherence between
and is then
defined by
(13)
where
is the cross power spectrum between and
while is the normalized frequency. For the same condi-
tion as in Figs. 3 and 4 shows the mean interchannel coherence
between
and , across different frequencies for var-
ious source positions, obtained by averaging over ten indepen-
dent trials.
As can be seen, when the source is in front of the microphone
pair centroid
m , the XM tap-selection criterion is
utilized efficiently to decorrelate input vectors
and ,
giving a low interchannel coherence of 0.43. Due to this effi-
cient decorrelation, a good misalignment convergence shown in
Fig. 3 is achieved. For this source location, the modest amount of
degradation due to tap selection does not significantly outweigh
the benefits brought about by the reduction in interchannel co-
herence due to the exclusivity criterion. Fig. 5(a) shows an ex-
ample of the XM selected taps
and in this case. For
clarity, we show only the first 80 samples of
and ,
each of length
samples. As can be seen, most of the se-
lected taps in the first channel correspond to elements in
being greater than zero, whereas for the second channel, most
of the active taps correspond to elements in
being less
than zero. This is due to the intrinsic effect of NL preprocessing,
which increases the magnitude of the positive elements in the
first channel and the negative elements in the second channel.
As a result of XM tap selection on
and , low inter-
channel coherence of 0.43 is achieved as shown in Fig. 4.
As shown in Fig. 4, the interchannel coherence between
and employing XM tap selection increases from
m to approximately m. To understand this increase
in interchannel coherence even after XM tap selection is ap-
plied, Fig. 5(b) shows a plot of
and for m.
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1830 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 19, NO. 6, AUGUST 2011
Fig. 5. Selected tap inputs
and for the XMNL-NLMS algorithm
for various source position
. (a)
m. (b)
m. (c)
m.
As can be seen, the effect of NL preprocessing on XM tap se-
lection reduces and the similarity between
and in-
creases. The increase in interchannel coherence between
and in turn reduces the convergence rate for XMNL-
NLMS, as can be seen from Fig. 3.
Fig. 5(c) illustrates
and when the source is in
front of the right microphone at
m. Due to the source
being further away from the left microphone, elements in
have magnitudes much lower than those of . In addition,
it is expected that
differs from . As a result of this
difference, the interchannel coherence is lower for
m
compared to
m, as shown in Fig. 4. It is therefore expected
that the misalignment convergence of XMNL-NLMS should in-
crease when the source moves from
mto m.
On the contrary, however, the XMNL-NLMS convergence rate
continues to reduce with increasing
position, as can be seen
from Fig. 3. In Section IV, we gain better insight into this con-
tradictory behavior by studying the effect of XM tap selection
on the energies of the active tap inputs for different source po-
sitions.
B. Effect of XM Tap Selection on Tap-Input Energies
In order to further illustrate how XM tap selection affects the
energies of the tap inputs, we employ the
-ratio criterion [17]
defined as
(14)
where
for the XM tap se-
lection, with elements defined by (9) and (10). Our intention is
not to show the exact relationship between
and the misalign-
ment convergence rate, but to illustrate that the loss of tap-input
energy has an undesirable effect on the convergence rate of the
adaptive filters. Fig. 6 illustrates how
varies with source po-
sition for XMNL-NLMS and NL-NLMS, obtained by averaging
over all frames in the signal where each frame is calculated using
Fig. 6. Variation of against source position for NL-NLMS (dashed) and
XMNL-NLMS (solid).
(14). As can be seen, NL-NLMS has
for all source po-
sitions since all tap-inputs are used for weight update. On the
other hand, for XMNL-NLMS,
and increases with
position of the source.
We can now see from Figs. 4 and 6 why XMNL-NLMS
achieves poorer convergence performance when the source
is far from the centroid of the microphone pair: although the
interchannel coherence is relatively low, the
-ratio is not
sufficiently high to reduce the degradation in convergence rate
due to tap selection. As a consequence of this conflict between
the need to reduce interchannel coherence and maximization
of tap-input energies, the overall result is a reduction in con-
vergence rate of XMNL-NLMS, as can be seen from Fig. 3.
On the other hand, when
m, the degradation in con-
vergence rate due to a reduction in
is offset by a significant
reduction in interchannel coherence, as shown in Figs. 6 and
4, respectively. As a result of this joint effect, good overall
convergence performance can be obtained for XMNL-NLMS,
as can be seen in Fig. 3.
As an additional note, the position of the source affects not
only the misalignment convergence of XMNL-NLMS, but also
its steady-state value. As can be seen from Fig. 3, the steady-
state normalized misalignment is higher than that of NL-NLMS
for increasing
position since the weight update is performed
using only a fraction of the tap inputs. This causes an additional
error in the weight update, resulting in an increase in the steady-
state normalized misalignment.
IV. C
ENTER-CLIPPING APPROACH TO SAEC
We now propose a center-clipping algorithm that has the
ability to reduce the interchannel coherence according to
the source location in the transmission room. We exploit the
decorrelation properties of XM tap selection, similar to that of
XMNL-NLMS. In addition we propose an error-based com-
pensation technique that addresses the additional steady-state
normalized misalignment resulting from XM tap selection.
A. Center-Clipping Exclusive Maximum Tap Selection
We propose to apply center-clipping to the tap-input vectors
in order to increase the energies of the “inactive” tap inputs
when the interchannel coherence between
and is
relatively low, such as when the source is in front of one of the
microphones. This is to reduce the degradation in misalignment
convergence brought about by the XM tap-selection process. As
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BEKRANI et al.: CLIPPING-BASED SELECTIVE-TAP ADAPTIVE FILTERING APPROACH TO SAEC 1831
Fig. 7. Schematic diagram of the proposed structure.
will be shown in Section IV-D, the clipping threshold is based
on indirect estimation of the similarity between the energies of
and . This similarity reflects how close the source
is to the microphone centroid, which in turn affects the con-
vergence behavior. The proposed approach ensures a soft-op-
timization constraint, which makes it robust to source position.
A schematic diagram of the proposed method is as shown in
Fig. 7.
The proposed clipping-based XMNL-NLMS algorithm
(cXMNL-NLMS) updates the filter coefficients using
(15)
(16)
where
is defined in (6) and (7), is the XM tap-
selection matrix with diagonal elements defined in (9) and (10),
and the
matrix is defined by
(17)
The matrix
is used to identify the tap-input elements
not selected by XM. The purpose of the proposed center-clip-
ping strategy is to increase the energies of tap inputs corre-
sponding to the “inactive” (unselected) taps. We achieve this by
first defining
as the clipped vector whose elements are computed by (18),
as shown at the bottom of the page, where
controls the
amount of clipping for
. In Section V, we discuss how this
clipping threshold can be determined for our SAEC application.
B. Effect of
on Tap-Input Energy
The range of
for each tap-input vector can be
bounded between zero and the maximum magnitude of any el-
ement within that vector, i.e.,
, where
(19)
Fig. 8. Variation of and
against
for the XM tap-
selection algorithm and center-clipping algorithm, respectively.
It can be seen from (18) that when ,wehave
, which results in the second term of (16)
having values equivalent to the unselected tap inputs, so that
and the proposed cXMNL-NLMS algorithm be-
comes the full-update NL-NLMS algorithm. On the other hand,
when
,wehave , causing the
second term of (16) to vanish, hence reducing cXMNL-NLMS
to XMNL-NLMS.
In order to illustrate how the clipping threshold
affects
the energies of the tap-input vectors
and , we em-
ploy a
-ratio criterion similar to the one defined by (14)
(20)
where the subscript
in denotes center-clipped signals.
Fig. 8 illustrates how
and vary with ,
using signals generated by convolving the previously defined
colored speech-like noise sequence with
and when
the source position is at (2.85,1.85,1.6) m. If
for both
channels,
and therefore the convergence behavior of
the center-clipped cXMNL-NLMS algorithm will be equivalent
to NL-NLMS. On the other hand, when
,
we have
and hence the performance of the proposed
cXMNL-NLMS algorithm will be equivalent to XMNL-NLMS.
C. Effect of Clipping Threshold on Interchannel Coherence
As illustrated in Section III, the misalignment conver-
gence of XMNL-NLMS depends on both the tap-input
energy as well as the interchannel coherence between
and . As such, we investigate the effect of on
the interchannel coherence between
and .We
convolve the same colored Gaussian noise sequence with
and , where , with the source po-
sitioned at coordinates (2.85,1.85,1.6) m. A total of ten
(18)
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1832 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 19, NO. 6, AUGUST 2011
Fig. 9. Interchannel coherence versus clipping threshold for colored noise
signal.
Fig. 10. Variation of against horizontal position of the source in the
transmission room for four cases of vertical positions
.
independent trials are averaged and Fig. 9 illustrates how the
mean interchannel coherence across frequency varies with
, where we constrain and such
that
. As can be observed,
the interchannel coherence reduces with increasing values of
clipping threshold. This is as expected since with increasing
, less energy will be allocated to the unselected taps
brought about by the XM tap-selection criterion, thus reducing
the similarity between
and , which as a conse-
quence causes a reduction in interchannel coherence.
D. Soft-Decision Rule for Clipping Threshold
As shown in Figs. 8 and 9, a high value of will re-
duce both
and interchannel coherence. As described in
Section III, a reduction in interchannel coherence is crucial
when the source position is near the centroid of the microphone
array pair, while the need to increase
becomes important
when the source is nearer to one microphone. Hence,
enables a tradeoff between interchannel decorrelation and
degradation of misalignment convergence due to tap selection.
Therefore, high
is desirable when the source is near the
centroid of the microphone pair, while low
is desirable
when the source is near one of the microphones. As was pointed
out, the similarity between the energies of
and
contributes to the high convergence rate of XMNL-NLMS
when the source is near the microphone pair centroid. On the
other hand, considerable difference in the energies as well as
low interchannel coherence between
and cause the
convergence rate of the XMNL-NLMS to reduce significantly.
Fig. 11. Clipping threshold
versus
.
We therefore propose to use the difference between absolute
values of the two channels as a measure of energy dissimilarity.
It is foreseeable that when the source is near the microphone
array centroid, the relative absolute values of
and
are approximately equal. On the contrary, when the source is
nearer to one microphone, the relative absolute values of these
tap-input vectors differ from each other. Thus, the dissimilarity
measure is defined as
(21)
where
, 1, 2, is a moving average of the ab-
solute values of the input elements given by
(22)
Here, we use
to smooth so as to avoid the effects
of instantaneous changes of
on the value of the clipping
threshold. We see that
and that is close
to 0 when the received energy of the two microphones are ap-
proximately equal and approaches 1 when the received energy
from one microphone is much larger than that of the other mi-
crophone.
Fig. 10 shows four illustrative examples of how
varies
with different
and positions of the source in the transmission
room. In these examples, the room dimensions and the coordi-
nates of microphones are given as shown in Fig. 2. As expected,
the value of
is small when the source is close to the cen-
troid at
m. On the contrary, is large when the
source is near Microphone 2 at
m.
We now incorporate
into cXMNL-NLMS by varying the
value of
as a function of . Since we desire a low
when the source is nearer to one microphone and vice versa,
should reduce with increasing . We therefore propose
a piecewise linear mapping
(23)
The relation between
and is plotted in Fig. 11. For
speech signals, we use values
and which
were determined empirically.
We note from (23) that
is independently determined for
each channel, and indirectly depends on the relative position of
the source and the microphones. In addition, when
,
such as when the source is near the microphone pair centroid,
increases with reducing . This reduces the effect of
the second term in (16) and as a result, the proposed algorithm
converges in the same manner as XMNL-NLMS. On the other
Page 7
BEKRANI et al.: CLIPPING-BASED SELECTIVE-TAP ADAPTIVE FILTERING APPROACH TO SAEC 1833
Fig. 12. Comparison of and for NL-NLMS, XMNL-NLMS, and
cXMNL-NLMS.
hand, when increases, such as when the source is near one
of the microphones,
reduces to zero. As shown in Fig. 8,
this has the effect of increasing the energy of the unselected taps,
which in turn reduces the degradation in misalignment conver-
gence due to XM tap selection in situations where reducing the
interchannel coherence cannot further improve the convergence
rate of the adaptive algorithm.
Fig. 12 further illustrates how degradation in
, and
consequently the convergence performance due to tap se-
lection, can be reduced by incorporating
into the
proposed cXMNL-NLMS algorithm. As described earlier in
Section III-B, the degradation of convergence performance
for XMNL-NLMS when the source is far away from the mi-
crophone pair centroid
is due to a reduction of
compared to NL-NLMS. In this scenario, the proposed
cXMNL-NLMS algorithm ensures that
is closer to the
value 1 achieved by NL-NLMS. On the other hand, when the
source is near the centroid
m , cXMNL-NLMS at-
tains the beneficial properties of the XM tap-selection strategy
to maximally decorrelate the tap-input vectors. The overall
joint result is a fast converging cXMNL-NLMS that is robust
to the source position in the transmission room.
V. E
NHANCEMENT OF THE STEADY-STATE PERFORMANCE
As noted from Fig. 3 and Section III, the steady-state nor-
malized misalignment of XMNL-NLMS is higher than that of
NL-NLMS. This is due to the unselected filter coefficients in-
troducing an error during adaptation since there is now a mis-
match between
, which drives the unknown system, and
the selective tap-input vector
. This error occurs
regardless of the source position. It is therefore expected that
cXMNL-NLMS also suffers from increased steady-state mis-
alignment since the proposed clipping method generates a signal
that is different from .
To illustrate this, we consider input vectors
defined by
(17) each of length
. A colored noise source signal
generated as described in Section III, is positioned at coordi-
nates (2.89,1.85,1.6) m. As before, the room is of dimension
as shown in Fig. 2. This steady-state normalized misalignment
is achieved by allowing the algorithm to reach its steady-state
and averaging over the last 5000 samples. Fig. 13 shows how
the steady-state normalized misalignment varies with normal-
ized clipping thresholds
.
Fig. 13. Relation between steady-state misalignment and normalized clipping
threshold in NL-NLMS, XMNL-NLMS, and cXMNL-NLMS, when the source
is at (2.89,1.85,1.6) m.
As can be seen, the steady-state normalized misalignment for
NL-NLMS is
29 dB, and 24 dB for XMNL-NLMS. If we
employ
defined by (23) for the above source position,
we obtain
, so the proposed cXMNL-
NLMS algorithm gives an additional 1 dB of steady-state nor-
malized misalignment improvement over XMNL-NLMS.
As a final improvement, we propose to enhance the
steady-state normalized misalignment performance of
cXMNL-NLMS. We note that the steady-state performance of
cXMNL-NLMS depends on the source position and therefore
when
, we need to reduce the additional steady-state
normalized misalignment. Hence, we propose to reduce
to zero after convergence of the mean-square error (MSE).
To estimate the convergence of the algorithm, we employ the
following recursive relation for estimating the MSE [24]
(24)
where
is related to the time-constant of the averaging
process. We consider
for our experiments. Hence,
when
reaches below a lower limit, will be set to zero.
In this case, the normalized misalignment reduces towards the
steady-state misalignment of NL-NLMS. We therefore propose
to incorporate
into (23) giving (25),
otherwise
(25)
where
is an empirically-derived lower limit that aims to
achieve a low level of MSE after convergence. Fig. 14 shows
an example of MSE and misalignment convergence for a single
trial when the source is at coordinates (2.87,1.85,1.6) m. As can
be seen, setting
based on (23) brings about a higher initial
convergence rate than NL-NLMS, while reducing
to zero
using (25) after MSE convergence will bring about additional
reduction in steady-state normalized misalignment. Additional
tests revealed that although the convergence of MSE occurs
before convergence of misalignment, exact knowledge of MSE
is not required. The proposed cXMNL-NLMS algorithm is
summarized in Table I.
Page 8
1834 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 19, NO. 6, AUGUST 2011
Fig. 14. Estimated MSE and misalignment for the cXMNL-NLMS algorithm.
TABLE I
T
HE cXMNL-NLMS ALGORITHM
VI. FURTHER SIMULATION AND EXPERIMENTAL RESULTS
We evaluate, by way of further simulation, the performance
of cXMNL-NLMS under different source positions. In order
to simulate the SAEC system, impulse responses
,
, , and were generated using the method
of images [22]. To evaluate the robustness of the algorithms,
TABLE II
S
PECIFICATIONS OF THE
SIMULATED ENVIRONMENT IN
SAEC
we fixed the location of the microphones while the source
position in the transmission room was varied across three
cases shown in Table II. A sampling rate of
Hz
was used throughout the experiment. The source signal was
generated by filtering a white Gaussian noise signal through a
low-pass finite impulse response (FIR) filter with coefficients
, as was used in Section III.
We compare the convergence performance of the pro-
posed cXMNL-NLMS algorithm with NL-NLMS and
XMNL-NLMS. Since the steady-state normalized misalign-
ment for XMNL-NLMS varies with the source position, we
chose its step-size so that its steady-state normalized mis-
alignment reaches that of NL-NLMS and cXMNL-NLMS
when the source position is in front of the microphone array
centroid at (2.85,1.85,1.6) m. This corresponds to
for both NL-NLMS and cXMNL-NLMS and for
XMNL-NLMS. White Gaussian noise (WGN) is added to
to achieve an dB. For all simulations, we have used
dB for cXMNL-NLMS. The normalized misalign-
ment curves, obtained by averaging over ten independent trials,
are plotted for Cases 1, 2, and 3 (Table II) in Fig. 15(a)–(c)
respectively.
Fig. 15(a) shows the convergence performance of the algo-
rithms where the source is directly in front of the right micro-
phone. In this case,
and are significantly different
and hence a high value of
defined in (21) is expected.
As shown in Fig. 11, this translates to a low
, and con-
sequently, as shown in (18),
. As a result,
the convergence performance of cXMNL-NLMS is equivalent
to that of NL-NLMS. The proposed cXMNL-NLMS algorithm
thus achieves an initial convergence of nearly 8 dB better than
XMNL-NLMS and reaches a steady-state normalized misalign-
ment of 4 dB lower as expected.
Fig. 15(b) shows convergence results when the source posi-
tion is mid-way between the microphone pair centroid and the
right microphone at (2.88,1.85,1.6) m. Now, the interchannel
coherence increases relative to the previous case and as can be
seen from this result, cXMNL-NLMS achieves the highest rate
Page 9
BEKRANI et al.: CLIPPING-BASED SELECTIVE-TAP ADAPTIVE FILTERING APPROACH TO SAEC 1835
Fig. 15. Normalized misalignment of the NL-NLMS, XMNL-NLMS, and
cXMNL-NLMS algorithms for a colored noise source signal. (a) Source directly
in front of right microphone at (3,1.9,1.55) m. (b) Source at (2.88,1.85,1.6) m.
(c) Source in the center of microphone pair at (2.85,1.85,1.6) m.
of initial convergence, improving that of NL-NLMS by nearly 4
dB during initial convergence. We note that when compared to
XMNL-NLMS, cXMNL-NLMS achieves approximately 3 dB
improvement during initial convergence and about 2 dB lower
steady-state normalized misalignment.
Finally, when the source position is in front of the micro-
phone pair centroid at coordinates (2.85,1.85,1.6) m,
and
are similar and the interchannel coherence between
and is high. As can be seen from Fig. 15(c), the conver-
gence of cXMNL-NLMS achieves the highest rate of conver-
gence with an improvement of approximately 4 dB over that
of XMNL-NLMS and nearly 10 dB over that of NL-NLMS. In
terms of steady-state normalized misalignment, the NL-NLMS
algorithm requires nearly 10 s more than that of cXMNL-NLMS
to reach
30 dB.
To further illustrate the convergence performance of the
proposed cXMNL-NLMS algorithm, we simulated the SAEC
system using a speech signal as shown in Fig. 16. In this ex-
ample, the speech signal is sampled at 11 025 Hz and a WGN is
added to
to achieve an SNR dB. The position of the
source in the transmission room is (2.88,1.85,1.6) m. As can be
seen from this result, cXMNL-NLMS achieves approximately
6 dB lower misalignment than NL-NLMS and 4 dB lower than
XMNL-NLMS during initial convergence.
We consider using recorded impulse responses, where the di-
mensions of the transmission room is 6.5 m
8.75 m 2.65 m,
the source was positioned at (3.25,4.37,1.15) m while the
two microphones were placed at (3.11,2.37,1.2) m and
(3.39,2.37,1.2) m for Case 1 and at (3.25,2.37,1.2) m and
(2.83,2.37,1.2) m for Case 2. The estimated reverberation time
was 280 ms. These impulse responses were of length 3087
samples and subsequently truncated to 512 samples. Fig. 17
Fig. 16. Normalized misalignment for the NL-NLMS, XMNL-NLMS, and
cXMNL-NLMS algorithms when the source is at (2.88,1.85,1.6) m for a speech
signal.
Fig. 17. Illustration of measured transmission room impulse response .
Fig. 18. Normalized misalignment of the NL-NLMS, XMNL-NLMS, and
cXMNL-NLMS algorithms for real room impulse responses. (a) Case 1: source
in the center of microphone pair. (b) Case 2: source approximately in front of
the right microphone.
shows one of the measured impulse responses in the trans-
mission room. For this experiment, the sampling frequency,
step-sizes as well as the SNRs were the same as that of the
previous simulations. The results are shown in Fig. 18. As
can be seen from Fig. 18(a), cXMNL-NLMS achieves nearly
3 dB improvement in convergence performance compared to
XMNL-NLMS when the source is in front of the microphone
centroid. In Fig. 18(b), when the source is in front of the right
microphone, the proposed algorithm achieves nearly 6 dB
improvement in convergence compared to XMNL-NLMS.
VII. C
ONCLUSION
We presented a new approach to improve the misalignment
convergence as well as the steady-state performance and ro-
bustness of adaptive filters for SAEC. This approach retains
the decorrelation properties of the XM selective-tap algorithm
Page 10
1836 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 19, NO. 6, AUGUST 2011
when the source is located near the microphone centroid, but
employs a variable center-clipping threshold whose value is de-
rived based on the absolute values of the received microphone
signals in order to work better, when the source is located closer
to one of the microphones. The proposed approach achieves
better convergence performance for different source positions
in comparison to both NL-NLMS and XMNL-NLMS.
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Mehdi Bekrani was born in Gorgan, Iran, in 1979.
He received the B.Sc. degree from Ferdowsi Uni-
versity of Mashhad, Mashad, Iran, in 2002, and the
M.Sc. and Ph.D. degrees from Tarbiat Modares Uni-
versity, Tehran, Iran, in 2004 and 2010, respectively,
all in electrical engineering.
He is currently a Research Fellow at Nanyang
Technological University, Singapore. His current
research interests include acoustic signal processing
and their applications.
Andy W. H. Khong (M’06) received the B.Eng. de-
gree from Nanyang Technological University, Singa-
pore, in 2002 and the Ph.D. degree from the Depart-
ment of Electrical and Electronic Engineering, Im-
perial College London, London, U.K., in 2005. His
Ph.D. research was mainly on partial-update and se-
lective-tap adaptive algorithms with applications to
mono- and multi-channel acoustic echo cancellation
for hands-free telephony.
He is currently an Assistant Professor in the
School of Electrical and Electronic Engineering,
Nanyang Technological University, Singapore. Prior to that, he served as
a Research Associate in the Department of Electrical and Electronic En-
gineering, Imperial College London, from 2005 to 2008. His postdoctoral
research involved the development of signal processing algorithms for vehicle
destination inference as well as the design and implementation of acoustic
array and seismic fusion algorithms for perimeter security systems. He has also
published works on acoustic blind channel identification and equalization for
speech dereverberation. His other research interests include human-computer
interfaces, source localization, speech enhancement, and blind deconvolution.
Mojtaba Lotfizad was born in Tehran, Iran, in 1955.
He received the B.S. degree in electrical engineering
from AmirKabir University of Technology, Tehran,
in 1980, and the M.S. and Ph.D. degrees from the
University of Wales, Cardiff, U.K., in 1985 and 1988,
respectively.
He joined the Department of Electrical and
Computer Engineering, Tarbiat Modares University,
Tehran, Iran. He has also been a Consultant to sev-
eral industrial and governmental organizations. His
current research interests are in signal processing,
adaptive filtering, speech processing, and specialized processors.
Page 11
  • Source
    • "Here, the channel acoustic response parameters are updated adaptively to produce an estimate of echo. Among different adaptive filter algorithms, the gradient based least mean squares (LMS) algorithm and its modifications, such as normalized LMS (NLMS) and variable step size LMS (VLMS) algorithms, are widely used for their satisfactory performances, less computational burden, and ease of implementation [2, 9, 10]. A faster algorithm is the recursive least mean squares (RLS) algorithm which is, however, computationally expensive [9]. "
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  • [Show abstract] [Hide abstract] ABSTRACT: One of the efficient solutions for the identification of long finite-impulse response systems is the three-level clipped input LMS/RLS (CLMS/CRLS) adaptive filter. In this paper, we first derive the convergence behavior of the CLMS and CRLS algorithms for both time-invariant and time-varying system identification. In addition, we employ results arising from this analysis to derive the optimal step-size and forgetting factor for CLMS and CRLS. We show that these optimal step-size and forgetting factor allow the algorithms to achieve a low steady-state misalignment.
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