4th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA2003), April 2003, Nara, Japan

ON-LINE TIME-DOMAIN BLIND SOURCE SEPARATIONOF NONSTATIONARY CONVOLVED SIGNALS
Robert Aichner, Herbert Buchner

Shoko Araki, Shoji Makino

Multimedia Communications
and Signal Processing
University of Erlangen-Nuremberg
Cauerstr. 7, D-91058 Erlangen, Germany{aichner, buchner}@LNT.de

NTT Communication Science Laboratories
NTT Corporation
2-4 Hikaridai, Seika-cho, Soraku-gun,Kyoto 619-0237, Japan{shoko, maki}@cslab.kecl.ntt.co.jp
algorithm to the time-varying mixing system with a sufficiently
fast convergence of the algorithm. We deal with both problems
in this paper and propose a new geometric initialization method
to improve the separation under reverberant conditions. The second problem is addressed by introducing a generalized weighting
factor to allow us to track time-variant environments.
This paper is organized as follows. In Sec. 2 we present a modified cost function with a generalized weighting factor. We showthat the inherent normalization avoids stability problems. Thereafter we derive the natural gradient adaptation for the proposedcost function and present a recursive formulation for an on-line
time-domain gradient algorithm. We then introduce a new geometric initialization method. After that we address the whitening
problem for time-domain BSS algorithms. In Sec. 3 we compare
simulation results for the proposed on-line algorithm with thoseof its off-line counterpart and another on-line algorithm in the frequency domain.

ABSTRACTIn this paper we propose a time-domain gradient algorithm that exploits the nonstationarity of observed signals and recovers the original sources by simultaneously decorrelating time-varying secondorder statistics. By introducing a generalized weighting factor
in our cost function we can formulate an on-line algorithm thatcan be applied to time-varying multipath mixing systems. A further benefit is the possibility of implementing updates in a recursive manner and thus reduce computational complexity. We showthat this method inherently possesses an adaptive step size andhence avoids stability problems. Furthermore we present a newgeometric initialization for time-domain gradient algorithms that
improves separation performance in strongly reverberant environments. In our experiments we compared the separation performance of the proposed algorithm with those of its off-line counterpart and another multiple-decorrelation-based on-line algorithm inthe frequency domain for real-world speech mixtures.

2. TIME-DOMAIN GRADIENT ALGORITHM
1. INTRODUCTION
2.1. Convolutive BSS model
Blind source separation (BSS) refers to the problem of recovering signals from several observed linear mixtures. The adjective“blind” stresses the fact that the source signals are not observed
and that no information is available on the multi-path mixing process. This lack of a priori knowledge about the mixing system iscompensated for by a statistically strong but physically plausible
assumption of independence. The absence of prior information is
actually the strength of the BSS model. Thus BSS has received
considerable attention in the recent years, and many algorithms
have been proposed [1, 2, 3], mainly in relation to scalar mixing
systems.
We focus on BSS for acoustic signals, which is characterized
by a convolutive mixing process and therefore is an even morechallenging topic. Among the unresolved problems are two of particular interest. Firstly, in strong reverberant environments most of
the current algorithms fail to achieve sufficient separation performance. This is due to the fact that most algorithms try to invert
the multi-path mixing system by adapting multi-path finite impulse response (FIR) filters. Thus in highly echoic rooms a verylarge number of filter coefficients need to be identified. Secondly,
in real-world scenarios, the multi-path mixing system is usually
time-variant because of moving sources, moving sensors or changing environments. This requires a continuous adaptation of the

In real environments, signals arrive at the sensors with different
time delays due to reflections. This scenario is referred to as amulti-path environment and can be described as a finite impulse
response (FIR) convolutive mixture:
P −1

x (t) =

H (k) s (t − k)

(1)

k=0

where s(t) = [s1 (t), ..., sn (t)]T captures the n mutually independent source signals and x(t) = [x1 (t), ..., xm (t)]T are the mixed
signals obtained by the microphones. The superscript T denotes
transposition. The mixing system H is an m × n matrix consistingof channel impulse responses hij (k) (i = 1, ..., m, j = 1, ..., n)that are modeled by FIR filters of length P with the filter coefficients k = 0, ..., P − 1.
To obtain the estimated sources y(t) = [y1 (t), ..., yn (t)]T ,we seek a n × m matrix W of FIR filters of length L operatingon the sensor measurements x(t) such that the components of theoutput vector y(t) are statistically independent:
L−1y(t) =

W(k)x(t − k)
k=0

987

(2)

where the unmixing matrix W(k) is defined as

 w (k) · · ·W(k) =



11

..
.

..

.
···

wn1 (k)


w (k)
..
 .
.

Kawamoto et al. [5] and used to separate convolutive mixtures.
As a measure of uncorrelatedness, we use the cost function proposed in [5] and modify it for use with block processing methods.
(b)Therefore we introduce Ryy (k) to represent the correlation matrixof y(t) in the b-th analysis block with time delay k. The correla(b)tion matrix defined by Ryy (k) = E(b) y(t)yT (t + k) is calculated using a block processing procedure, where E(b) [x] denotes
the time average of x for the b-th block.
Additionally, with regard to deriving an on-line time-domain
gradient algorithm, we introduce a generalized weighting factor
β(m, b) where m and b denote block indices.

1m

(3)

wnm (k)`We introduce W(z) as the z-transform of the unmixing filter coefficient W(k) with k = 0, ..., L − 1:
L−1

W(z) =

W(k)z −k ,

(4)

a

k=0

b

where L denotes the length of the unmixing filter and z −1 isused as the unit-delay operator for convenience, i.e., z −k x(t) =x (t − k).
To recover the original source signals from the observed mixtures, BSS uses the mutual independence of the original sources.
However, the mutual independence cannot resolve the following
two ambiguities:

J1 m, W(z)

c

=

1
2

m

b=1

β(m, b) log det diag R(b)yy (0)

d

e

f

e

fhg

(6)

− log det R(b)yy (0)The operator diag X denotes the diagonal elements of the matrix
X.
The segmentation of the observed mixtures into blocks ensuresthat we are calculating the cross-correlations at multiple times.
The non-negative cost function becomes zero only when yi (t)and yj (t) are uncorrelated for all the local analysis blocks, i.e.,E(b) [yi (t)yj (t)] = 0 (i, j = 1, ..., n; i 6= j, b = 1, ..., m). This
corresponds to a simultaneous diagonalization of multiple correlation matrices at different times t.

1. The order of the recovered source signals y(t) can be arbitrarily permuted as the mutual independence of the sources
is unaffected by the permutation.
2. The recovered signals y(t) may be arbitrarily filtered witha diagonal matrix Λ where each element on the diagonal
represents an FIR filter of length P.
Thus we can express the recovered signals y(t) as:y(t) = PΛ ∗ s(t)2.3. Weighting factor(5)

In adaptive filter signal processing it is customary to introduce
a weighting factor β(m, b) into the definition of the cost function [6]. The weighting factor, which we normalize so thatmb=1 β(m, b) = 1, allows the more recent samples to havegreater influence on the output error, allowing the tracking of timevariant acoustic environments. The introduced weighting factor isbased on the block indices m,b. Special forms of the weighting fac-

where * denotes convolution.
In the remainder of the paper we consider a two-speaker, twomicrophone BSS scenario as shown in Fig. 1

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2.2. Cost function
The assumption of independence causes the correlation matricesof the sources Rss (t, k) = E s(t)sT (t + k) to become diagonal matrices. Acoustic sources are assumed to be nonstationary,i.e., the auto-correlations of sources change independently overtime t. Hence the correlation matrix of the outputs Ryy (t, k) =E y(t)yT (t + k) also varies over time t. Thus, if we force estimated outputs y(t) to be uncorrelated at every time t, we obtain amuch stronger condition than mere decorrelation for a single time
instant t, and this enables us to separate the sources.
Matsuoka et al. [4] used this principle for separating
scalar mixtures. This method was subsequently extended by

`

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

Fig. 1. BSS model.

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


a

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

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Fig. 2. Weighting factors for adaptive algorithms. (a) exponential
forgetting factor, (b) sliding window of length K.

tor that are commonly used are sliding windows or the exponential

988

the nonwhiteness of the source signals which lead to improved
separation.
With regard to an on-line algorithm, in our update equation (10) we use the exponential forgetting factor defined in (7).
This results inweighting factor (forgetting factor) (Fig. 2) defined byβ(m, b) = (1 − λ)λ

m−b

(7)

where λ is a positive constant close to, but less than 1.
This on-line formulation of algorithms including a generalized weighting factor is also similar to off-line or so-called batch
methods. For a batch method, the entire observed signal has to beprocessed using the sliding window. After the block processing of
the entire signal, the output signals are generated by applying thefinal unmixing filters to the sensor signals. Hence the algorithm
can be interpreted as one iteration of a batch method, as we haveno continuous separation of the observed signals.
In [7] we presented an off-line algorithm based on the costfunction (6) and an additional modification. This approach leads
to good results in highly reverberant environments.
In this paper, we consider the on-line counterpart with an exponential forgetting factor, as it can be formulated in a recursive
manner that is computationally simple. In the following paragraphwe will provide an on-line formulation of the update rule.

Wm (k) = α∆Wm (k) + Wm−1 (k)
where
m

c

Qb (k)αβ(m, b)e R (0) f
(b)
yy

T

−1

(b)
yy

−1

(b)
yy

(b)
yy

m−1 (z)(13)

and k = 0, ..., L − 1 denotes the filter tap index of the unmixing
filter W. Equation (12) can be formulated in a recursive manner
to reduce the computational complexity. Another benefit is the
reduced memory requirements, since only the preceding filter tap
matrix has to be saved for the update.
∆Wm (k) = λ∆Wm−1 (k) + (1 − λ)Qb (k)∂J1 m, W (z)T
W (z −1 )W(z),∂W (k)

m=e diag R (0)f=e diag R (k) − R (k)fhg W

(14)

(8)
Thus, by using the exponential forgetting factor, we obtain an online time-domain gradient algorithm that can be applied to timevariant multipath environments.

where the symbol ∝ means “proportional to” and k = 0, ..., L − 1
denotes the filter tap index. The adaptation rule for an off-line
algorithm using the natural gradient is derived in [9]. For a costfunction including a general weighting factor β(m, b) (6) this results in:Wm (k)(12)b=1

We use the natural gradient adaptation, as introduced by Amari[8], which provides an isotropic convergence independent of themixing matrix H, to minimize the cost function:b(1 − λ)λm−b Qb (k)∆Wm (k) =2.4. Natural gradient adaptation∆Wm (k) ∝ −(11)

2.5. Inherent normalization
Another approach for measuring the uncorrelatedness of the estimated output signals is the cost function defined by:R(b)yy (k)b=1e− diag R(b)yy (0)

f

−1

(9)

(b)

where Ryy (k) = E(b) [y(t)yT (t + k)], the index b denotes block
processing and k · k2 is the Frobenius norm. This cost function
exploits the nonstationarity and nonwhiteness of the sources andis widely used (see, e.g., [10]). To achieve fast convergence in thiscase it is usually necessary to consider second order gradient expressions. A proper Newton-Raphson update requires the inverseof the Hessian. This is both tedious to derive and computationallyvery demanding. Thus there are many heuristic approaches withwhich to solve this normalization problem.
The cost function J1 , however, shows an inherent normalization. It can be seen in the update equation (9) that the correlation(b)(b)
matrix Ryy (k) is scaled by the inverse of Ryy (0)T and the in(b)verse of diag Ryy (0), respectively.
Similarily we can observe that the off-diagonal componentsof the correlation matrix in the modified update rule (10), which(b)(b)
are expressed by diag Ryy (k) − Ryy (k), are normalized by the(b)inverse of diag Ryy (0). This means a normalization to the shorttime power of the output signals. We observed that this inherent
normalization ensures a fast convergence of the algorithm, thus
making an adaptive step size becomes unnecessary.

where α is a step size parameter, m denotes the current block andk is the filter tap index. Equation (9) converges if the off-diagonal(b)components of Ryy (0) are minimized for all blocks. We confirmed in experiments that if we consider only time-delay k = 0,
we cannot achieve separation in a strongly reverberant environment. Therefore, we modify (9) to the following equation to eval(b)uate the off-diagonal components of Ryy (k) for all time delays
k = 0, ..., L − 1:m

Wm (k)=αβ(m, b)e diag R (0)f
(b)
yy

−1

b=1

e diag R (k) − R (k)fhg
(b)
yy

(b)
yy

(15)

b,k



+Wm−1 (k)

2
(b)
kR(b)
yy (k) − diag Ryy (k)k

J2 (W) =R(b)yy (k) W m−1 (z)(10)

Wm−1 (z) + Wm−1 (k)This update equation simultaneously diagonalizes the correlation(b)matrices Ryy (k) for multiple blocks (depending on the shapeof β(m, b)) and additionally decorrelates the cross-correlations at
multiple time lags k. Hence we are using the nonstationarity and989

2.6. Utilization of geometric beamforming

2.7. Dewhitening of output signals

The convergence and the separation performance of gradientbased algorithms are greatly influenced by the initial value, especially if we need long FIR filters to model the room impulse
responses. To solve this problem we propose a new approach forcalculating the initial value by adding geometric information on
the positions of the microphones and the assumed positions of the
speakers. This method has so far only been applied to frequencydomain methods [11] where the geometric initialization showedsuperior separation especially for a large number of microphones.
The equivalence of adaptive beamformers and BSS shown in[12] was our motivation to use a beamformer technique for initializing the adaptation algorithm. We assume the sources to bespatially separated so that we can employ a null beamformer withbeams that place spatial zeros at the orientations of interfering
sources. For our experiments with a two-speaker, two-microphone
scenario, we assume two sources with angles of θi = ±60o , measured with respect to the normal vector of the microphone array.
We calculate the null beamformer for both target and jammer
signal configurations with respect to the angle of the interfering
source and the microphone positions. The delays resulting fromthe null beamformer are then used as cross path filters w12 andw21 . As these delays are fractions of the sampling time, we haveto use a sinc function for interpolation. The filter in the straightpath w11 , w22 is initialized as a unit impulse. The small components that exist in addition to the unit impulse (Fig. 3) are the resultof disregarding frequency components smaller than 62.5 Hz. They
were disregarded because we cannot calculate a sharp spatial null
for low frequencies due to the small microphone spacing of 4 cm.
Additionally we incorporate a delay of L2 to access both future
and past values of the observed signals. Thus a noncausal filter
is formed that allows us to identify non-minimum phase systems.
In [7] we showed for a batch algorithm that the new initialization

In Sec. 2.1 we pointed out that the recovered source signals possess
two ambiguities; the signal order may be permuted and the outputs
can be arbitrarily filtered. The latter ambiguity causes a change inthe output signal spectrum. Higher frequencies are usually emphasized whereas lower frequencies are attentuated and thus the spectrum becomes whitened. The flattened spectrum gives the speech
signals an unnatural timbre. Although this ambiguity can arise
in both time-domain and frequency-domain BSS, it is seldom addressed in time-domain BSS. To simplify the explanation, here we
consider the problem in the frequency domain. Frequency-domain
BSS decomposes the convolutive mixing system in multiple scalar
mixing systems in each frequency bin. The arbitrary filtering corresponds to an arbitrary scaling in each frequency bin. Before recomposing the signals from each frequency bin, the arbitrary scaling has to be resolved because otherwise no separation is achieved.
To overcome the filtering ambiguity, we apply post-filters toour system. These filters are based on the method for removing
the amplitude ambiguity in frequency-domain BSS proposed by
Ikeda [13]. To the authors’ knowledge, no post-processing method
using this principle has been proposed for time-domain BSS.
The general idea is to transfer the separated output signals
y (t) into the frequency domain and then solve the problem of
irregular amplitude for each frequency bin. The observed mixtures X (ω) are described by X (ω) = H (ω) S (ω) and X (ω) =W−1 (ω) Y (ω). We can assume that, when sources are set at almost the same distance from a microphone array, the amplitudesof all the elements of the mixing filter matrix H (ω) are equal,
because the attenuations of all observed sound signals are nearly
equal due to the small microphone spacing. The inverse of the unmixing matrix W (ω) is denoted as W−1 (ω) = Wij−1 (ω) . For
the two-speaker, two-microphone scenario we obtain:

1`

X1 (ω) =X1 (ω) =X2 (ω) =X2 (ω) =

0.2

w11

0.8

0

w12

-0.2

0.6

-0.4

0.4

−1
−1
W11
(ω)Y1 (ω) + W12(ω)Y2 (ω)
1 · S1 (ω) + 1 · S2 (ω)
−1
−1
W21
(ω)Y1 (ω) + W22(ω)Y2 (ω)
1 · S1 (ω) + 1 · S2 (ω)

a

(16)
(17)
(18)
(19)

-0.6
0.2

We can now rescale the output signal Y1 (ω) in each frequency−1
bin by multiplying Y1 (ω) with the factor W11(ω) so that theamplitude of X1 (ω) in (16) is equal to the amplitude of X1 (ω)
in (17). The same process is applied to the output signal Y2 (ω)
so that the amplitudes of X2 (ω) in (18) and (19) are the same.
Thus we obtain a dewhitening matrix V (ω), which is defined as
V (ω) = diag W−1 (ω). The dewhitened output signals Ỹ (ω)
are obtained by

-0.8
0
0

10

20

30

40

50

60

-1

0.2

1

0

0.8

w21

-0.2
-0.4

0

10

20

30

40

50

60

w22

0.6
0.4

-0.6
0.2
-0.8
0
-1

0

10

20

30

40

50

60

0

10

20

30

40

50

60

Ỹ(ω) = V(ω)Y(ω) = V(ω)W(ω)X(ω).
Fig. 3. Initial value W for an unmixing filter length of L = 64
taps.

(20)

The post-processing filter emphasizes low frequencies to restore
the original spectral content of the source signals (Fig. 4). The
algorithm without dewhitening returns higher SIR values due tothe emphasis of high frequencies. The smaller separation capability of our BSS system for low frequencies is not taken into account. Thus the dewhitening process is corrects the higher signalto-interference ratios. We can observe that the obtained spectrum
complies well with the original spectral content.

method improves the separation significantly and that we can estimate long unmixing FIR filters in the time domain that can cover
the entire reverberation. As shown in Sec. 3 we could also confirm this experimentally when using the on-line version with an
exponential forgetting factor.

990

35

5.73 mWhitened separated signal
Original source signal
30

loudspeakers(height : 1.35 m)

25

1.56 m1.15 m3.12 mPower in dB

20

15

10

30

2.15 m40

4 cm

1.15 m5

microphones(height : 1.35 m)

0

room height : 2.70 m

-5
0

500

1000

1500

2000
2500
Frequency in Hz

3000

3500

4000

(a) Whitened spectrum
40

Fig. 5. Layout of the reverberant room used in the experiments.

Dewhitened separated signal
Original source signal
3530with an overlap factor of 4. We applied the new geometric initialization method to both classes, but for the two-source, twomicrophone scenario we could only achieve an SIR improvement
for the time-domain algorithms.
Due to the filtering ambiguity the output signals of the online time-domain algorithm showed a flattened spectrum. The
frequency-domain method avoids the scaling ambiguity in eachfrequency bin by enforcing a unity gain constraint for the diagonal
filters. However, we observed that this does not fully prevent thewhitening of the spectrum of the separated output signals. Figure 6
shows the average SIR for six different combinations of source
signals for the investigated algorithms. We observed that the ge-

Power in dB

25

20

15

10

5

0

-5
0

500

1000

1500

2000
2500
Frequency in Hz

3000

3500

4000

(b) Dewhitened spectrum
Fig. 4. Spectrum of output signal before (a) and after (b) postprocessing.

87

3. EXPERIMENTS AND RESULTS

6

3.1. Experimental conditions

5

SIR in dB

We conducted the experiments using speech data convolved with
the impulse responses of a real room (Fig. 5), with a reverberation
time TR = 300 ms. Since the sampling frequency was 8 kHz,the reverberation time corresponded to a room impulse responseof 2400 taps. We used a two-element microphone array with aninter-element spacing of 4 cm. The speech signals arrived fromtwo different directions, −30o and 40o . We used two sentences
spoken by two male and two female speakers selected from the
ASJ continuous speech corpus as source signals [14]. We used six
combinations of speakers with a signal length of seven seconds.
To evaluate the performance, we used the signal-to-interference
ratio (SIR), defined as the ratio of the signal power of the target
signal to the signal power from the jammer signal. The SIR wascontinuously calculated for each block by using the post-processeddewhitened separated output signals.

4

3

2

1

0

−1

0

1

2

3

4

5

6

7

time t in seconds

Fig. 6. Average SIR depicted for the time-domain on-line algorithm without (solid line) and with (dashed line) the dewithening
postprocessing method. In comparison the frequency-domain online algorithm without postprocessing (dash-dotted line) [10]. Reverberation time T60 = 300 ms.

3.2. Experimental results
We compare the performance of the proposed on-line algorithmwith that of its off-line counterpart [7] and with the second on-line
decorrelation algorithm in the frequency domain presented in [10].
To compare the two on-line algorithms we set the unmixing
filter length at L = 512 taps. For the time-domain algorithm wechose a blocklength of 1024 samples with an overlap factor of 4
and used an exponential forgetting factor with λ = 0.9999. The
frequency-domain BSS algorithm also had an FFT length of 1024

ometric initialization of the time-domain algorithm leads to fast
convergence. Moreover it is seen that the postprocessing method,which restores the original spectral content of the sources, reduces
the SIR. This results from the fact that the dewhitening process emphasizes the low frequencies, which exhibit a smaller SIR causedby strong reverberation.

991

6. REFERENCES

Figure 6 also shows that the frequency-domain on-line algorithm converges to a lower SIR than with the time-domain method.
By contrast, the frequency-domain algorithm is also applicable tostrongly reverberant environments where no a priori informationon the array geometry is available.
The on-line BSS algorithm can exploit the observed signals
only once and yet it achieves good separation. The off-line algorithm on the other hand can repeatedly access their values for each
iteration. This further improves the separation performance seenin Fig. 7 which shows the SIR for the batch algorithm with andwithout the postprocessing method. For the off-line algorithm weset the unmixing filter length at L = 512 taps, and we chose ablocklength of 1024 samples. This corresponds to the parameters
we used for the on-line algorithm.

[1] J.-F. Cardoso, “Blind signal separation: statistical principles,” in Proc. IEEE, Oct. 1998, vol. 9, pp. 2009–2025.
[2] S. Haykin, Ed., Unsupervised Adaptive Filtering, Volume 1
Blind Source Separation. John Wiley & Sons, 2000.
[3] A. Hyvaerinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons, 2001.
[4] K. Matsuoka, M. Ohya, and M. Kawamoto, “Neural net forblind separation of nonstationary signals,” IEEE Trans. Neural Networks, vol. 8, no. 3, pp. 411–419, 1995.
[5] M. Kawamoto, K. Matsuoka, and N. Ohnishi, “A method ofblind separation for convolved non-stationary signals,” Neurocomputing, vol. 22, pp. 157–171, 1998.

14[6] S. Haykin, Adaptive Filter Theory, Prentice Hall Inc., Englewood Cliffs, NJ, 1996.

12[7] R. Aichner, S. Araki, S. Makino, T. Nishikawa, and
H. Saruwatari,“Time-domain blind source separationof non-stationary convolved signals by utilizing geometricbeamforming,” in Proc. Neural Networks for Signal Processing, June 2002, pp. 445–454.

SIR in dB108[8] S.-I. Amari, “Natural gradient works efficiently in learning,”
Neural Computation, vol. 10, pp. 251–276, 1998.

6[9] T. Nishikawa, H. Saruwatari, and K. Shikano, “Comparisonof time-domain ICA, frequency-domain ICA and multistage
ICA for blind source separation,” in Proc. European Signal
Processing Conference, Sep. 2002, vol. 2, pp. 15–18.

4

2

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55

[10] L. Parra and C. Spence, “On-line convolutive source separation of non-stationary signals,” Journal of VLSI Signal
Processing, vol. 26, no. 1/2, pp. 39–46, Aug. 2000.

60

Number of iterations[11] L. Parra and C. Alvino, “Geometric source separation: Merging convolutive source separation with geometric beamforming,” in Proc. Neural Networks for Signal Processing, 2001,
pp. 273–282.

Fig. 7. Average SIR of the off-line time-domain algorithm without (solid line) and with the dewhitening postprocessing method(dashed line). Reverberation time T60 = 300 ms.

[12] S. Araki, S. Makino, R. Mukai, and H. Saruwatari, “Equivalence between frequency-domain blind source separation andfrequency-domain adaptive null beamformers,” in Proc. Eurospeech, Sept. 2001, pp. 2595–2598.

4. CONCLUSION[13] S. Ikeda and N. Murata, “A method of ICA in timefrequency-domain,” in Proc. ICA, Jan. 1999, pp. 365–371.

We proposed a time-domain gradient algorithm with a generalized
weighting function. Using an exponential forgetting factor, we introduced a recursive on-line BSS algorithm, which can be applied
to time-varying multipath environments. We presented a new geometric initialization for time-domain algorithms, which showed
good results for strongly reverberant environments. Our simulation results showed that both on-line and off-line methods are capable of separating real-world speech signals in echoic surroundings.

[14] T. Kobayashi, S. Itabashi, S. Hayashi, and T. Takezawa, “ASJ
continuous speech corpus for research,” J. Acoust. Soc. Jpn.,
vol. 48, no. 12, pp. 888–893, 1992, (in Japanese).

5. ACKNOWLEDGMENTS
We thank Prof. Walter Kellermann for his continuous encouragement and constructive criticism. We are also grateful to Prof. Hiroshi Saruwatari from Nara Institute of Science and Technology
for providing the room impulse responses.

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