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

MULTISTAGE ICA FOR BLIND SOURCE SEPARATION OF REAL ACOUSTICCONVOLUTIVE MIXTURE
Tsuyoki NISHIKAWA † , Hiroshi SARUWATARI † ,†Kiyohiro SHIKANO † , Shoko ARAKI ‡ , and Shoji MAKINO ‡Graduate School of Information Science, Nara Institute of Science and Technology
8916-5 Takayama-cho, Ikoma-shi, Nara, 630-0192, JAPAN‡NTT Communication Science Laboratoriess, NTT Corporation
2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0237 JAPANE-Mail: {tsuyo-ni, sawatari, shikano}@is.aist-nara.ac.jp,{shoko, maki}@cslab.kecl.ntt.co.jp

In order to resolve the problems, we propose a new BSS
algorithm called multistage ICA (MSICA), in which FDICA
and TDICA are combined. By using the proposed method,
we can achieve a superior separation performance even under heavily reverberant conditions. The results of the signal
separation experiments reveal that the separation performance of the proposed algorithm is superior to those of theconventional ICA-based BSS methods.

ABSTRACT
We propose a new algorithm for blind source separation(BSS), in which frequency-domain independent component
analysis (FDICA) and time-domain ICA (TDICA) are combined to achieve a superior source-separation performance
under reverberant conditions. Generally speaking, conventional TDICA fails to separate source signals under heavily
reverberant conditions because of the low convergence in
the iterative learning of the separation system. On the other
hand, the separation performance of conventional FDICA
also degrades seriously because the independence assumption of narrow-bin signals collapses when the number of
frequency bins increases. In the proposed method, the separated signals of FDICA are regarded as the input signals
for TDICA, and we can remove the residual crosstalk components of FDICA by using TDICA. The experimental results obtained under the reverberant condition reveal thatthe separation performance of the proposed method is superior to those of TDICA- and FDICA-based BSS methods.

2. SOUND MIXING MODEL
In general, the observed signals in which multiple source
signals are convoluted with room impulse responses are obtained by the following equation:x(t) =P−1X

a(τ )s(t − τ ),

(1)

τ =0

where x(t) = [x1 (t), · · · , xK (t)]T is the observed signal vector and s(t) = [s1 (t), · · · , sL (t)]T is the source signal vector(see Fig. 1) . K is the number of array elements (microphones) and L is the number of multiple sound sources. In
this study, we deal with the case of K = L = 2. Also,a(τ ) = [aij (τ )]ij ([·]ij denotes the matrix in which ij-th element is [·]) is the mixing filter matrix. P is the length of
the impulse response which is assumed to be an FIR filterof thousands of taps because we introduce a model to deal
with the arrival lags among the elements of the microphonearray and the room reverberations.

1. INTRODUCTION
Blind source separation (BSS) is the approach taken to estimate the original source signals using only the informationof the mixed signals observed in each input channel. This
technique is applicable to the realization of high-quality
hands-free speech recognition system. The BSS methods
based on independent component analysis (ICA) [1, 2] canbe classified into two groups in terms of the processing domain, i.e., frequency-domain ICA (FDICA) in which thecomplex-valued separation matrix is calculated in the frequency domain [3, 4], and time-domain ICA (TDICA) in
which the separation FIR filter matrix is calculated in thetime domain [5, 6]. The recently developed BSS techniques
can achieve a good source-separation performance under artificial or short reverberant conditions. However, the performances of these methods under heavily reverberant conditions degrade seriously because of the following problems.
(1) In conventional FDICA, the separation performance issaturated before reaching a sufficient performance level because we transform the fullband signals into the narrowband signals and the independence assumption collapses ineach narrow-band [7]. (2) In conventional TDICA, the convergence degrades because the iterative learning rule becomes more complicated as the reverberation increases [6].

3. CONVENTIONAL ICA AND PROBLEMS
3.1. FDICA
The conventional BSS based on FDICA is conducted withthe following steps: (1) transform the observed fullband
signals into the narrow-band signals, (2) optimize the separation matrix in each frequency bin, and (3) reconstruct the
fullband separated signal from the narrow-band separated
signals. FDICA has the following advantages and disadvantages.
Advantages:(F1) It is easy to converge the separation filter in iterative
ICA learning because we can simplify the convolutive mixture down to simultaneous mixtures by thefrequency transform.

523

Mixing system

FDICA

TDICA

X1( f,t)Sourcesignalss 1 (t)Observed
signalsf(f)

Z ( f,t)=W ( f )X( f,t) Separated
y(t) = Σ w(τ)z(t-τ)τ
signals from FDICA

fst-DFTa 11(τ)f

Z1 (f,t)a 12(τ)X( f,t)(f)

W (f )a 22(τ)IDFTz(t)Z ( f,t)a 21(τ)s 2 (t)Separated
signals

Z2 (f,t)y1 (t)

(t)

w (τ)

y(t)
y2 (t)

IDFT

Microphonex(t) = Σ a(τ)s(t-τ)τ

st-DFTX2( f,t)
f

f

(t)

(f)

Optimize w (τ)
so that y1 (t) and y2 (t)are mutually independent

Optimize W ( f )so that Z1 ( f,t) and Z2 ( f,t)
are mutually independent

Figure 1: Blind source separation procedure performed in multistage ICA.
Disadvantages:(F2) The separation performance is saturated before reaching a sufficient performance level because the independence assumption collapses in each narrow-band[7] (see, e.g., Sect. 5.3).
(F3) Permutation among source signals and indeterminacyof each source gain in each frequency bin.
Regarding the disadvantage (F3), various solutions have already been proposed [3, 8, 9, 10]. However, the collapse of
the independence assumption, (F2), is a serious and inherent problem, and this prevents us from applying FDICA ina real acoustic environment with a long reverberation.

MSICA is conducted with the following steps. In thefirst stage, we perform FDICA to separate the source signals to some extent with the high-stability advantages (F1)of FDICA. In the second stage, we regard the separated signals of FDICA as the input signals for TDICA, and we remove the residual crosstalk components of FDICA by using
TDICA. Finally, we regard the output signals of TDICA as
the resultant separated signals. MSICA can achieve a highstability and a separation performance superior to that ofconventional FDICA and TDICA. In the following sections,
we describe details of the ICA-learning rules for each stage.

3.2. TDICA

In the first-stage ICA, we introduce the fast-convergence
FDICA proposed by one of the authors [4]. We perform thesignal separation procedure as described below (see FDICA
in Fig. 1).
In FDICA, first, the short-time analysis of observed signals x(t) is conducted by frame-by-frame discrete Fouriertransform (DFT). By plotting the spectral values in a frequency bin of each microphone input frame by frame, we
consider them as a time series. Hereafter, we designate thetime series as X (f, t) =[X1 (f, t), · · · , XK (f, t)]T . Next, we
perform signal separation using the complex-valued inverseof the mixing matrix, W (f) (f ), so that the L time-series(f)(f)
output Z (f, t) = [Z1 (f, t), · · · , ZL (f, t)]T becomes mutually independent; this procedure can be given as

4.1. First-Stage ICA: FDICA

In the conventional BSS based on TDICA, each element of
the separation filter matrix is represented as a FIR filter.
We can optimize this separation filter system, by using the
fullband observed signals themselves. TDICA has the following advantages and disadvantages.
Advantages:(T1) We can treat the fullband speech signals where theindependence assumption of sources usually holds.
Disadvantages:(T2) The convergence degrades under reverberant conditions because the iterative rule for FIR filter learning
is complicated.
It is known that TDICA works only in the case of mixtureswith a short-tap FIR filter, i.e., less than 100 taps. Also,
TDICA fails to separate source signals under real acoustic
environments because of the disadvantages (T2).

Z (f, t)=W (f) (f )X (f, t).

(2)

We perform this procedure with respect to all frequency
bins. Finally, by applying the inverse DFT and the overlapadd technique to the separated time series Z (f, t), we reconstruct the resultant source signals in the time domain,z(t).
In conventional FDICA , the optimal W (f) (f ) is obtained by the following iterative equation [3]:4. PROPOSED METHOD: MSICA
As described above, the conventional ICA methods havesome disadvantages. However, note that the advantages
and disadvantages of FDICA and TDICA are mutually complementary, i.e., (F2) can be resolved by (T1), and (T2) canbe resolved by (F1). Hence, in order to resolve the disadvantages, we propose a new algorithm, MSICA, in which
FDICA and TDICA are combined (see Fig. 1).

h





(f)
W (f)
Φ(Z (f, t))Z H (f, t) t
i+1 (f ) = W i (f ) + α diagi(f)− Φ(Z (f, t))Z H (f, t) t W i (f ),
(3)

524

where h·it denotes the time-averaging operator, i is used toexpress the value of the i-th step in the iterations, and α
is the step-size parameter. Also, we define the nonlinear
vector function Φ(·) asΦ(Z (f, t)) ≡ [Φ(Z1 (f, t)), · · · , Φ(ZL (f, t))]T ,Φ(Zl (f, t)) ≡where β is the step-size parameter. Since the Eq. (8) eval(b)uates only off-diagonal of hy (t)y (t)T it , we confirmed thatthe iterative equation of Eq. (8) could not achieve a superior separation performance under the reverberant condition (see Sect. 5.4). Namely, the source separation is notachieved by only using nonstationarity of signals. Therefore we use not only nonstationarity of signals but also
time-delayed decorrelation approach. We expand Eq. (8)to the following equation to evaluate the off-diagonal of(b)hy (t)y (t − τ + d)T it for all time delays τ − d :

(4)

(R)
(I)
tanh(Zl (f, t))+j · tanh(Zl (f, t)), (5)where Re[Zl (f, t)] and Im[Zl (f, t)] are the real and imaginary parts of Zl (f, t), respectively.

[TDICA2]

4.2. Second-Stage ICA: TDICAw(t2)i+1 (τ )The output signals from TDICA (i.e., the separated signalsof MSICA) can be given asy(t) =

Q−1

X

w(t) (τ )z(t − τ ),β X X n(b) −1
diag y (t)y (t)T t
B
b=1 d=0(b)(b) −1· diag y (t)y (t − τ + d)T t − diag y (t)y (t)T t
o
(b)
(t2)
· y (t)y (t − τ + d)T t wi (d),
(9)

= wi

(t2)

(6)

τ =0

B Q−1

(τ ) +

where y (t) = [y1 (t), · · · , yL (t)]T is the resultant separated(t)signal vector of MSICA, w(t) (τ ) = [wij (τ )]ij is the separation filter matrix, and Q is the length of the separation filter. The selection of TDICA is an important issue
because the quality of resultant separated signals is determined by TDICA. In this study, we introduce three TDICA
algorithms, i.e.,

Choi proposed the TDICA algorithm which optimizes
the separation filter by minimizing the KLD between the
joint probability density function and the marginal probability density function of the separated signals [12]. The
KLD is given by

TDICA1: TDICA based on simultaneous decorrelation of
nonstationary signals,


KLD w(t) (τ ) =Z

TDICA2: TDICA based on combination of TDICA1 andtime-delayed decorrelation approach, and

l=1 t=0

TDICA3 [12]: TDICA based on minimization of KullbuckLeibler divergence (KLD),
and compare these TDICA algorithms.
First, we drive the TDICA1 algorithm. This optimizes
the separation filter by minimizing the nonnegative costfunction which takes the minimum value only when thesecond-order cross-correlation becomes zero if the source
signals are nonstationary. The cost function is defined asfollows (this cost function has been already proposed by
Kawamoto et al. [11], however their derivation of learning
rule includes mathematical error):

(t3)
w(t3)
(τ )
i+1 (τ ) = widiag h(y (t))y (t − τ + d)T it
d=0
o
(t3)
−h(y (t))y (t − τ + d)T it wi (d).
(11)

+α

t

w

(t1)
i+1 (τ ) =

w

β
(t1)
(τ ) +iy(t)y· y (t)y
·

B Q−1
X
X n

B

y(t)y

(b)
(t)T tb=1 d=0(b)(t − τ + d)T t − diag (t)
o
(b)
(t1)
(t − τ + d)T t
(d),
i

w

X





(y(t)) ≡ tanh(y1 (t)), · · · , tanh(yL (t)) T .

(12)

In this study, we compare the MSICA1, MSICA2,
and MSICA3 in which FDICA followed by TDICA1,TDICA2, and TDICA2, respectively.
5. EXPERIMENTS AND RESULTS

−1y y(t)T (b)tQ−1nwhere we define the nonlinear vector function (·) as(b)

where B is the number of local analysis blocks and h·it
denotes the time-averaging operator for the b-th local analysis block. Equation (7) becomes zero only when yi (t) andyj (t) are uncorrelated for all of the local analysis blocks.
The correct iterative equation to minimize Q w(t) (τ ) is
given by [6][TDICA1]

where p(·) is the joint probability density function, q(·)
is the marginal probability density function, and T is thelength of the separated signals. With a nonholonomic constraint, the iterative equation of the separation filter to minimize the KLD(w(t) (τ )) is given as (hereafter we designate
the iterative equation as “TDICA3”):[TDICA3](b) 
B det diag y (t)y (t)T t
1 X
log
, (7)
Q w(t) (τ ) =(b)
2B
det y (t)y (t)Tb=1

p(y (t))p(y (t)) log L T −1dy (t), (10)YYq(yl (t))

5.1. Experimental Setup

−1

A two-element array with the interelement spacing of 4 cm
is assumed. We determined this interelement spacing byconsidering that the spacing should be smaller than half of

(8)

525

2.15 mNoise Reduction Rate [dB]

3.12 m1.56 m

5.73 mLoudspeakers
1.15 m(Height : 1.35 m)-30
40

Microphone
array(Height : 1.35 m)(Spacing: 0.04m)(Room height : 2.70 m)

10
8
6
4
2
0

32

64

128 256 512 1024 2048 4096
Number of Frequency Bins

Figure 3: Relation between separation performances andthe number of frequency bins in conventional FDICA.

Figure 2: Layout of reverberant room used in experiments.

Jthe minimum wavelength to avoid the spatial aliasing effect; it corresponds to 8.5/2 cm in 8 kHz sampling. The
speech signals are assumed to arrive from two directions,−30◦ and 40◦ . The distance between the microphone arrayand the loudspeakers is 1.15 m (see Fig. 2). Two kindsof sentences, those spoken by two male and two female
speakers selected from the ASJ continuous speech corpus
for research, are used as the original speech samples. The
sampling frequency is 8 kHz and the length of speech islimited to within 3 seconds. Using these sentences, we obtain 12 combinations with respect to speakers and source
directions. In these experiments, we use the following signals as the source signals: the original speech convolved
with the impulse responses specified by the reverberation
time of 300 ms. This corresponds to 2400-taps FIR filter
in 8 kHz. The impulse responses are recorded in a variable
reverberation time room as shown in Fig. 2.

0.03
0.02
0.01
0

32

64

128 256 512 1024 2048 4096
Number of Frequency Bins

Figure 4: Relation between the number of frequency bins
and the value of J defined by Eq. (16), which corresponds
to the independence of subband signals.

5.3. Relation between Separation Performance andNumber of Frequency Bins in FDICA

5.2. Objective Evaluation Score

In order to confirm the independence problem of narrowband signals in FDICA ((F2) described in Sect. 3.1), we
carried out the preliminary experiment under the following
analysis conditions. The number of frequency bins (frame
length in DFT) is set to be from 32 to 4096, the frame
shift is 16 taps, the window function is a Hamming window,
the number of iterations in ICA is 30, and the step-size
parameter α for iterations is set to be 1.0 × 10−5 .
Figure 3 shows the NRR results for different numbers of
frequency bins in FDICA. As shown in Fig. 3, the NRR of
FDICA obviously degrades when the number of frequency
bins becomes too large, and the separation performance issaturated before reaching a sufficient performance. This isbecause we transform the fullband signals into the narrowband signals and the independence assumption collapses ineach frequency bin, particularly when the number of frequency bins is large.
In order to confirm the fact, we newly define the following objective measure to quantify an independence, andinvestigate the relation between the number of frequency
bins and the independence among subband signals.
D


J =
diag Φ(Z (f, t))Z H (f, t) t
E,
(16)
− Φ(Z (f, t))Z H (f, t) t

Noise reduction rate (NRR), defined as the output signalto-noise ratio (SNR) in dB minus input SNR in dB, is usedas the objective evaluation score in this experiment. The
SNRs are calculated under the assumption that the speech
signal of the undesired speaker is regarded as noise. The
NRR is defined as
2

1 X
(O)
(I)
SNRl − SNRl ,
(13)
NRR ≡
2
l=1
P
2
f |Hll (f )Sl (f )|
(O)
SNRl
,
(14)
= 10 log10 P
2
f |Hln (f )Sn (f )|P
2
f |All (f )Sl (f )|(I)= 10 log10 P
,
(15)
SNRl
2
f |Aln (f )Sn (f )|
(O)

0.08
0.07
0.06
0.05
0.04

(I)

and SNRl are the output SNR and thewhere SNRl
input SNR, respectively, and l 6= n. Also, Sl (f ) is thefrequency-domain representation of the source signal, sl (t),Hij (f ) is the element in the i th row and the j th column ofthe matrix H (f ) = W (MSICA) (f )A(f ) where W (MSICA) (f )denotes the entire separation process in MSICA including
both FDICA and TDICA and A(f ) is the mixing matrix
which corresponds to the frequency-domain representationof the room impulse responses described in Sect. 2.

f526

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High-Independence 

Robust against Reverberation

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MSICA1, MSICA2, and MSICA3.

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Figure 5: Trade-off relation among the independence of subband signals and robustness against reverberation.
TDICA1 and TDICA2, we divided the signals equally into
B parts (B = 1–10). We chose the optimal B and number of iterations for each combination of speaker because
the convergence is different for every combination. As for
the FDICA part in MSICA, the analysis conditions are thesame as those given in Sect. 5.3, except for the number of
frequency bins (which is fixed at 1024 bins).
Figure 6 shows the NRR results in the MSICA1, MSICA2
and MSICA3. Figure 6 shows that TDICA1 can not achievea signal separation under the reverberant condition. Comparing MSICA1 with MSICA2 in Fig. 6, we confirm that
TDICA2 can achieve a superior separation performance to
TDICA1. These results show that it is necessary to evaluate correlations of different times to achieve a superior
performance. TDICA3 can also achieve a superior separation performance to TDICA1. The separation performancesof MSICA2 and MSICA3 are superior to that of FDICA
and the improvements of the separation performance from
FDICA are not so different. Thus, the use of FDICA with
TDICA2 or TDICA3 is effective in the proposed MSICA
structure.

where k · k is frobenius norm of matrix. This measure J
is a part of the iterative equation (3) and has no dimension. Therefore the absolute value of J is meaningless itself,
however, the relative value between the different numbersof frequency bins is important. If narrow-band signals become mutually independent, the measure J becomes zero.
Also we can consider that the independence of subband
signals is high when J is small. In order to evaluate theindependence of real narrow-band speech signals, we carried out the experiment in which the input signal, Y (f, t),
in Eq. (16) is regarded as the perfectly separated sources,
i.e., original speech samples. Figure 4 shows the relation
between the number of frequency bins and the value of J
which corresponds to the independence of subband signals.
Figure 4 shows that the independence decreases as the number of frequency bins increases, especially when the numberof frequency bins is large.
Above-mentioned experimental results clarify the disadvantage that the separation performance is saturated in
FDICA because we transform the fullband signals into thenarrow-band signals. We should lengthen the separation
filter (or FFT length for analysis) when we confront with along reverberation. In this case, however, the independenceof subband signals decreases. Thus, there is a trade-off
relation among the independence of subband signals androbustness against reverberation as shown in Figure 5. On
the basis of these results, we should cascade another signal
processing analysis, e.g., TDICA, with FDICA to obtainthe further separation performances.

5.5. Comparison between MSICA2 and MSICA3

In order to compare MSICA2 and MSICA3 in detail, we
evaluate the separation performances of MSICA3 and MSICA2
with the various number of local blocks B. In TDICA2 partin MSICA2, the utilization of large B is effective because
it can evaluate the nonstationarity of the signals. Figure 7
shows the NRR results in the MSICA3 and MSICA2 withthe various B. The separation performance is improved as
the B is increased, however, the larger B is not the bestfor improving the separation performance. In addition, the5.4. Comparison of Separation Performance between
separation performances of MSICA3 and MSICA2 with the
MSICAsoptimal B are not so different. Therefore, MSICA3 is more
We carried out the experiments using MSICA1, MSICA2,
feasible than MSICA2 because MSICA3 does not requireand MSICA3 to evaluate the contribution of TDICA1, TDICA2, the estimation of the optimal B.
and TDICA3 for improving the separation performances
under reverberant conditions. The analysis conditions of
5.6. Discussion on Combination Order in MSICA
these experiments are as follows: the filter length Q is
As described in the previous section, the combination of
set to be 2048taps, the maximum number of iterations is
FDICA and TDICA can contribute to the improvement
500, and the step-size parameter β for iterations is set toof separation. In this combination, the advantage (F1) of
be 5.0 × 10−4 for TDICA1, 1.0 × 10−3 for TDICA2, andFDICA is useful in the initial step of separation procedure
1.0 × 10−6 for TDICA3. As for the local analysis block for

527 ONP

and the advantage (T1) of TDICA is also useful in the laterstep. Therefore we use FDICA as the first-stage ICA and
TDICA as the second-stage ICA. In order to confirm theavailability of this combination order, we compare the proposed combination with the combination in which TDICA
is used in the first stage and FDICA is used in the second
stage ( hereafter we designate this swapped combination as”MSICA4”).

M

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The experiment of MSICA4 was carried out in the following manner. As for TDICA part in MSICA4, TDICA2
is used, the number of local analysis blocks, B, is fixed at
3, and the filter length is 8 taps improved best the separation performance in TDICA2. This TDICA2 can obtainthe NRR of 6.0 dB. As for FDICA part in MSICA4, theanalysis conditions are the same as those given in Sect. 5.3,
except for the number of frequency bins (which is fixed at1024 bins). As the result, the NRR of 7.5 dB is obtained in
MSICA4, and this performance is better than that of simple
TDICA but is poorer than those of MSICA2, MSICA3, andsimple FDICA. In MSICA4, the separation performance isstill improved by using FDICA in the second stage, however, the separation performance is saturated because of
the disadvantage (F2) of FDICA. MSICA4 can not achieve
the separation performance of 9.4 dB which corresponds to
NRR of simple FDICA. This reason is that FDICA in thispaper uses the beamforming technique and the directivity
pattern of the array which provide a good initial value of theseparation matrix to improve the convergence [4], however,such kind of information is no longer valid in the combination order of MSICA4 because we can not know the effective
positions of the array elements after the first-stage TDICA
and can not depict the directivity pattern. Thus the separation performance of MSICA4 is almost equal to that of a
raw FDICA without the beamforming technique (from [4]we can see the NRR of about 7.5 dB at the 30-iteration
point). This fact indicates that the swapped combination
order of MSICA4 has no contribution to the improvementof the separation performance, and the proposed combination order of MSICA2 and MSICA3 (FDICA in the firststage and TDICA in the second stage) is essential.

Q

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!#"%$&('*),+-'/.10!+/"!+-'2.43 57698:;;<Figure 7: Comparison of separation performance between
MSICA3 and MSICA2 with the various B.
7. ACKNOWLEDGEMENT
The authors are grateful to Dr. Mitsuru Kawamoto of Shimane University and Mr. Robert Aichner of NTT. CO.,LTD for their greateful discussions. This work was partly
supported by CREST (Core Research for Evolutional Science and Technology) of JST (Japan Science and Technology Corporation).
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6. CONCLUSION
In this paper, we propose a new algorithm for BSS, inwhich FDICA and TDICA are combined to achieve a superior source-separation performance under reverberant conditions. In TDICA part in MSICA, we compare (1) TDICA
based on simultaneous decorrelation of nonstationary signals, (2) TDICA based on combination of (1) and timedelayed decorrelation approach, and (3) TDICA based onminimization of KLD. The results of the signal separation
experiments reveal that the separation performances of (1)and conventional FDICA are not so different and the separation performances of (2) and (3) are superior to that ofconventional FDICA. Therefore, the combination of FDICA
and (2) or (3) is inherently effective for improving the separation performance. Specifically, the proposed method can
improve the SNR by about 3.0 dB over that of FDICA for
an average of 12 speaker-combinations.

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