4th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA2003), April 2003, Nara, JapanSUBBAND BASED BLIND SOURCE SEPARATIONWITH APPROPRIATE PROCESSING FOR EACH FREQUENCY BAND
Shoko Araki † Shoji Makino † Robert Aichner †‡ Tsuyoki Nishikawa ? Hiroshi Saruwatari ?
†NTT Communication Science Laboratories, NTT Corporation,
2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japane-mail: shoko@cslab.kecl.ntt.co.jp‡
University of Erlangen-Nuremberg, Cauerstrasse 7, 91058 Erlangen, Germany?
Graduate School of Information Science, Nara Institute of Science and Technology,
8916-5 Takayama-cho, Ikoma, Nara 630-0192, Japan
ABSTRACT
We propose subband-based blind source separation(BSS) for convolutive mixtures of speech. This is motivated by the drawback of frequency-domain BSS, i.e.,
when a long frame with a fixed frame-shift is used fora few seconds of speech, the number of samples ineach frequency bin decreases and the separation performance is degraded. In our proposed subband BSS,
(1) by using a moderate number of subbands, a sufficient number of samples can be held in each subband, and (2) by using FIR filters in each subband, we
can handle long reverberation. Subband BSS achieves
better performance than frequency-domain BSS. Moreover, subband BSS allows us to select the separation
method suited to each subband. Using this advantage,
we propose efficient separation procedures that takethe frequency characteristics of room reverberation andspeech signals into consideration, (3) by using longer
unmixing filters in low frequency bands, and (4) byadopting overlap-blockshift in BSS’s batch adaptation
in low frequency bands. Consequently, subband processing appropriate for each frequency bin is successfully realized with the proposed subband BSS.
1. INTRODUCTION
Blind source separation (BSS) is an approach that
estimates original source signals using only information
on mixed signals observed in each input channel. This
technique can be used for noise robust speech recognition and high-quality hearing aid systems.
We consider the BSS of speech signals in a realenvironment, i.e., the BSS of convolutive mixturesof speech. Several methods have been proposed forachieving the BSS of convolutive mixtures [1, 2]. In areal environment, signals are mixed with their reverberation. In order to separate such complicated mixtures, we need to estimate unmixing filters of several
thousand taps. Moreover, in a real environment, an
impulse response does not remain unchanged even for499

several seconds. Therefore, we have to estimate unmixing filters with short mixed speech signals.
We have shown, however, that the performance becomes poor with frequency-domain BSS when we apply
a long frame to several seconds of speech to estimate
a long unmixing filter that can cover realistic reverberation [3]. This is because when we use a longer
frame for a few seconds of speech mixtures, the number of samples in each frequency bin becomes smalland, therefore, we cannot correctly estimate the statistics in each frequency bin.
Motivated by this fact, we propose a method of BSS
that employs subband processing. Hereafter, we callthis method subband BSS. With this method, we canchoose a moderate number of subbands. Therefore, we
can maintain a sufficient number of samples in eachsubband. The subband BSS also allows us to estimate
FIR filters as unmixing filters in each subband due tothe down-sampling procedure. Therefore, we can obtain an unmixing filter long enough to cover reverberation.
Furthermore, we propose an efficient separation procedure taking into consideration the frequency characteristics of room reverberation and speech signals, i.e.,
using longer unmixing filters and the overlap-blockshifttechnique only in low frequency bands.
Previous studies have used subband processingfor BSS. One approach used subband BSS to reduce
computational complexity [4]. However, their subband framework suffered from large aliasing distortion,
therefore, they failed to obtain a good result. We utilize a polyphase filterbank with oversampling, which is
widely used in the echo-canceller area, and our aim is
to maintain the number of samples in each subband.
Some other authors [5, 6] utilized a scalar coefficientfor the unmixing system in each subband. However,
we use FIR filters as the unmixing system in each subband so as to estimate sufficiently long unmixing filters
to cover the reverberation.
The organization of this paper is as follows. Section 2

S1

S2

X1

H11

W11

H12

mic. 1

W12

H21

mic. 2

W21

H22
Mixing system

X2

W22

Y1

Y2

Unmixing system

Figure 1: BSS system configuration.
describes the framework of the BSS of convolutive mixtures of speech. In Sec. 3, we show the configurationof subband BSS and mention implementation issues.
Section 4 describes experimental conditions, and Sec.
5 reports experiments conducted to confirm the validityof this method. In Sec. 5.2, we propose the use of efficient techniques only for low frequency subbands where
the SIR is worse than at high frequencies. This considers the frequency characteristics of room reverberationand speech signals. The final section concludes thispaper.
2. BSS OF CONVOLUTIVE MIXTURES
In real environments, the observed signals are affected by reverberation. Therefore, N s signals recordedby Nm microphones are modeled as
xj (n) =Ns P
XXhji (k)si (n − k + 1) (j = 1, · · · , Nm ), (1)i=1 k=1

where si is the source signal from a source i, x j is thesignal observed by a microphone j, and h ji is the P taps impulse response from source i to microphone jwhich is modeled by an FIR filter.
To obtain unmixed signals, we estimate unmixing
filters wij (k) of Q-taps, and the unmixed signals are
obtained as below:
yi (n) =Nm Q
XX

wij (k)xj (n − k + 1) (i = 1, · · · , Ns ). (2)statistics difficult, and therefore, we cannot obtain thesufficient separation performance.
Motivated by this fact, we propose subband BSS.
With this method, we can choose the number of subbands and, therefore, we can maintain a sufficient number of samples in each subband by selecting a moderate
number of subbands. Subband BSS also allows us to estimate FIR filters as unmixing filters in each subband,
because the subband analysis stage contains the downsampling procedure. Therefore, we should be able toobtain an unmixing filter long enough to cover reverberation. Furthermore, we can execute different separation operations which are suitable for each subband.
3.1. Configuration of Subband BSS
The subband BSS system is composed of three parts:
a subband analysis stage, a BSS stage, and a subband
synthesis stage (Fig. 2).
First, in the subband analysis stage, input signals
xj (n) are divided into N subband signals X j (k, m)(k = 0 · · · , N − 1), where k is the subband index, mis the time index, and N is the number of subbands.
We used a polyphase filterbank [8] here. Because signals are band-limited in each subband, we can apply
decimation at the down-sampling rate R. In the analysis/synthesis stage, we also utilized single sideband(SSB) modulation/demodulation [9]. We obtain the
SSB modulated signals XjSSB (k, m) in each subband.
Then, time-domain BSS is executed on XjSSB (k, m)
in each subband. Because SSB modulation is performed in the analysis stage, we can implement the
time-domain BSS algorithm without expanding it to
a complex value version. Since we employ downsampling, short FIR filters are sufficient to separate
the subband signals in each subband. Thus SSB modulated unmixed signals YiSSB (k, m) are obtained in eachsubband.
Finally, unmixed signals yi (n) are obtained by synthesizing each unmixed signal YiSSB (k, m).

j=1 k=1

The unmixing filters are estimated so that the unmixed
signals become mutually independent.
In this paper, we consider a two-input, two-output
convolutive BSS problem, i.e., Ns = Nm = 2 (Fig. 1).
3. SUBBAND BASED BSS
In the frequency domain, when we use a longer framewith a fixed frame shift rate to prepare an unmixing
filter long enough to cover reverberation, it becomesdifficult to maintain a sufficient number of data samples to estimate the statistics of the source signals ineach frequency bin [7]. This makes the estimation of

500

3.2. Time-domain BSS Implementation
We can use any time-domain BSS algorithm for subband BSS. Here, we describe the algorithm we used inour experiment. To simplify the notation, X jSSB (k, m)
and YiSSB (k, m) are written as xj (n) and yi (n), respectively.
3.2.1. Time-domain BSS algorithm
In this paper, we used an algorithm based on thetime-delayed decorrelation for non-stationary signals[10]. The adaptation rule of the p-th iteration used toSUBBAND ANALYSIS-0WN

SEPARATION

jω mR

f(n)

R

-n

WN

∆
X1(0,m) e

Re[.]

jω mR

f(n)

R

∆
X1(1,m) e

SSB

(0,m)
TDBSS

SSB
1

X

Re[.]

0

-jω∆mRSSB

X1

SUBBAND SYNTHESIS

(1,m)

Y1

(0,m)

e

Y1(0,m)

WN
R

g(n)

R

g(n)

-jω∆mR

Y
TDBSS

n

e

SSB
1

Y1(1,m)

(1,m)

WN

x1(n)
-nk

WN

jω∆mR

f(n)

R

-n(N-1)NW

X1(k,m)

e

Re[.]
jω∆mR

f(n)

R

-0

WN

X1(N-1,m) e

SSB

X1

TDBSS
SSB
N-1
1

X

Re[.]

(

f(n)

R

-n

WN

Re[.]

jω mR

f(n)

R

∆
X2(1,m) e

Y1 (k,m)
SSB
N-1
1

,m)

Y

(

,m)

Y1(k,m)e

Y1(N-1,m)

R

SSB

(0,m)

Y2 (0,m)e

Y2(0,m)

R

g(n)

0

WN
R

g(n)

X

Re[.]

R

g(n)

(1,m)

Y

n

e

SSB
2

(1,m)

Y2(1,m)

WN

x2(n)
-nk

WN

jω∆mR

f(n)

R

-n(N-1)NW

X2(k,m)

e

jω∆mR

f(n)

R

X2(N-1,m) e

analysis

X

Re[.]

SSB

(k,m)

Y2 (k,m)(e

Y2(k,m)

WN
R

g(n)

R

g(n)

-jω∆mR

SSB
N-1
2

X

Re[.]

nk

-jω∆mR

SSB
2

SSB
N-1
2

,m)

Y

SSB modulation

(

,m)

e

Y2(N-1,m)

SSB demodulation

Y2(n)n(N-1)

WN-jω∆mR

SSB
2

Re[.]

g(n)

-jω∆mR

SSB

X2

Y1(n)

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jω mR

∆
X2(0,m) e

SSB

(k,m)

Re[.]
nk

-jω∆mRen(N-1)

WN

synthesis
j2π/N

N: # of subband, R: down-sampling rate, WN=e, ω∆=2π/N
f(n): LPF for analysis, g(n): LPF for synthesis, Re[.]: real part

Figure 2: System configuration of subband BSS. TDBSS: time-domain BSS.
obtain the optimal unmixing filters w(k) = {wij (k)} is
B

∆

p

(k) =

αX
{(diagRby (0))−1 (diagRby (−k))
Bb=1−(diagRby (0))−1 Rby (−k)} ∗

= −α

"

R21 (k)p∗ w21(k)
R11 (0)p
R12 (k)∗w(k)11
R22 (0)p(k)

R21 (k)p∗ w22(k)
R11 (0)R12 (k)
p
∗
w
(k)
12
R22 (0)

(3)

#

,(4)

where Rby (k) = {Rij (k)} represents the covariance matrix of outputs y(n) = [y1 (n), y2 (n)]T in the b-th analysis block with time delay k, and α is a step-size parameter. The unmixing filter length in each subband isdetermined by the filter length of the fullband version
and the down-sampling rate in the subband analysis
stage. Note that the algorithm we used here is a batchalgorithm, i.e., the algorithm runs by using all the dataon each iteration.
3.2.2. Initial value of unmixing filters
We have shown that the solution of BSS behaves asa set of adaptive beamformers, which make a spatial
null towards a jammer [11]. Based on this fact, we can

501

use constraint null beamformers as the initial value ofan unmixing system w, which can make a sharp null
towards a given jammer and maintain the gain andphase of a given target direction. Without such an
initial value, the time-domain BSS algorithm does notconverge at all for the long unmixing filters, which arenecessary for the separation in a real environment [12].
Moreover, we can mitigate the permutation problem
with this initial value.
To design the initial value, first, we assume thatthe mixing system H = {h ji } represents only the timedifference of sound arrival τ ji with respect to the midpoint between microphones. In the frequency domain,
H(ω) is modeled as H(ω) = {h ji (ω)} = {exp (jωτji )},dwhere τji = cj sin θi , dj is the position of the j-th microphone, θi is the direction of the i-th source, and cis the speed of sound. Then we calculate the inverseof H at each frequency, W (ω) = H −1 (ω). Next, weconvert this W (ω) = {Wij (ω)} into the time domain,wij (k) = IFFT(Wij (ω)), and then obtain the initialvalue in each subband by applying subband analysis tothese wij (k). Here, we adopted θi =±60◦ as the initialvalues.

3.2.3. Solving the permutation and scaling problem
Due to our initialization method described in Sec.
3.2.2, we did not observe any permutation ambiguity.
However, the scaling problem occurred, i.e., the estimated source signal components had a different gain in
the different subbands.
To solve this problem, we use the directivity patternobtained by w [13]. First, we estimate the source directions from the directivity patterns in each frequency
bin. In order to scale the signals of each frequency bin,
we normalize the rows of W (ω) so that the gains of thetarget directions become 0 dB in each frequency bin.
After converting these rescaled unmixing filters to thetime domain, we execute a subband analysis. We thenrescale the unmixing filters wij so that they have thesame power as the subband analyzed rescaled unmixing
filters in each subband.

Separation experiments were conducted using speech
data convolved with impulse responses recorded in areal room. The room size was 5.73 m × 3.12 m × 2.70
m and the distance between the loudspeakers and microphones was 1.15 m. The reverberation time was
TR = 300 ms. We used a two-element array withan inter-element spacing of 4 cm. The speech signals
arrived from two directions, −30 ◦ and 40◦ . As theoriginal speech, we used two sentences spoken by two
male and two female speakers. We investigated threecombinations of speakers: male-male, male-female, andfemale-female. The sampling rate was 8 kHz. The
data length was three seconds for adaptation and about
eight seconds for separation. In order to evaluate theperformance, we used the signal to interference ratio(SIR), defined as

SIROi
SIRIi

4.2. Conventional frequency-domain BSS
In the next section, we compared the separation performance of subband BSS with frequency-domain BSS
to show the drawback of frequency-domain BSS. The
frequency-domain BSS iteration algorithm was



H
H
∆

4. EXPERIMENTAL CONDITIONS

SIRi

where N is the number of subbands (0-2π). The prototype low-pass filter used in the analysis was f (n) =nπ) with a length of 6N and in the synthesis wassinc( N/2nπg(n) = sinc( R/2) of length 6R (see Fig. 2). Here, thenumber of subbands N = 64 and the down-sampling
rate R = 16.
For the time-domain BSS, we estimated the unmixing filters wij of 64 and 128 taps in each subband. The
step-size for adaptation α was 1.0×10 −4 and the number of blocks B was fixed at 20 for three seconds ofspeech.

= SIROi − SIRIi
P|Aii (ω)Si (ω)|2,= 10 log P ω|Aij (ω)Sj (ω)|2Pω
|Hii (ω)Si (ω)|2= 10 log P ω,2
ω |Hij (ω)Sj (ω)|

where A(ω) = W (ω)H(ω) and i 6= j. SIR means
the ratio of a target-originated signal to a jammeroriginated signal.
4.1. Subband System
In the analysis stage, in order to avoid the aliasing influence, the SSB-modulated subband signals were notcritically sampled, but two-times oversampled. Thatis, the down-sampling rate R was given by R = N4 ,

502

i (ω) = ηdiag hΦ( )i −hΦ( )ii (ω),

where Y =Y (ω, m), superscript H denotes the conjugate transpose and h·i denotes the time average. As
the nonlinear function Φ(·), we used Φ(Y ) = tanh(g ·abs(Y ))ej ark(Y ) , where g is a parameter to control thenonlinearity. We fixed the frame shift at half the DFT
frame size T , so that the number of samples in thetime-frequency domain were equal. As the initial valueof the unmixing matrix, we utilized W (ω) = H −1 (ω)
mentioned in Sec. 3.2.2.
5. EXPERIMENTS AND DISCUSSIONS

5.1. Separation performance of subband BSS
Figure 3 shows the separation result and the value ofthe average correlation coefficient CC between source
signals averaged over all frequency bins and subbands.
We used unmixing filters w of 64 and 128 taps in eachsubband and since the down-sampling rate R was 16,
this corresponds to 1024 and 2048 taps in the fullband,
respectively. N = 64 subbands with decimation R = 16
corresponds to T = 32 in frequency-domain BSS withregard to the down-sampling rate.
In frequency-domain BSS, CC becomes large and the
independent assumption seems to collapse as frame size
T becomes large. This is because the number of samples in each frequency bin becomes small. Therefore,
the performance degraded when we used unmixing filters of 2048 taps (i.e., frame size T = 2048).
This limitation does not hold for subband BSS where
we could even improve the separation performance
when using unmixing filters with a length of 2048. It
is possible to use long filters because the correlation
coefficient remains sufficiently small in each subband.
Another possible reason for the superior performance of40

TR=300ms, N=64, 64taps/band

35
30

SIR [dB]

25
20
15
10
Speaker combination

5

Female-female
Male-male
Male-female

0
-5

0

5

10

15

20

25

30

Subband number

Figure 3: Separation performance of frequency-domain
BSS (white bars) and subband BSS (gray bars). CC: average correlation coefficient. TR = 300 ms.

subband BSS is that the permutation problem does notarise in the subbands. Even if it occurs between subbands, it can be solved more easily than in frequencydomain BSS, simply because there are fewer problems
in subband BSS than in frequency-domain BSS.
5.2. Further improvement
dependent processing

withfrequency-

In subband BSS, we can vary the method of estimating the unmixing filter in each subband. In thissubsection, we propose a technique to improve separation performance by concentrating on low frequency
bands.
Generally speaking, the SIR is worse in low frequency bands as shown in Fig. 4, in which we plot
the SIR values for each subband for three combinationsof speakers. One of the reasons for poor performanceat low frequencies is that an impulse response is usually longer and therefore it is more difficult to separate
signals in low frequency bands than in high frequency
bands. Since speech signals have high power in low
frequency bands, it is important to improve the separation performance in low frequency bands to obtainbetter overall separation performance.
5.2.1. Longer unmixing filters in low frequency bands
One possible way to improve the SIR in low frequency bands is to estimate longer unmixing filters
in low frequency bands. From this, we propose using
longer unmixing filters for low frequency bands (bands0-5). The row labeled “no-overlap” in Table 1 shows
the separation performance for each unmixing filter
length. Here, we used unmixing filters of 32, 64 and128 taps in each subband.
Let us look at (A)-(C) in Table 1. It is conceivablethat the 32 taps long unmixing filter cannot cover re-503

Figure 4: SIR in each subband.
verberation in low frequency bands. In this case, evenwhen we used long unmixing filters only in low frequency bands, the separation performance was greatly
improved. However, when we used 128 taps in low frequency bands, the separation performance degraded.
This may be because the number of samples in eachsubband is too small to allow us to estimate a 128 taps
unmixing filter precisely. The proposal in Sec. 5.2.2
will overcome this problem.
5.2.2. Overlap-blockshift in low frequency bands
Another possible way to improve the SIR in low frequency bands is to utilize the overlap-blockshift in thetime-domain BSS stage for low frequency bands. Using the overlap-blockshift, we can increase outwardly
the number of samples in each subband, and estimate
unmixing filters more precisely. This is because we canestimate statistics correctly using sufficiently long data.
Since our time-domain BSS algorithm (4) divides signals into B blocks to utilize the non-stationarity of signals, we can divide signals into blocks with an overlap as long as the non-stationarity is expressed among
blocks. Note that this overlap-blockshift is executed in
the BSS stage, i.e., after the decimation for subband
analysis.
In Table 1, the columns show the SIR obtainedby the overlap-blockshift only for low frequency bands(bands 0-5). Overlap(×2) and overlap(×4) means thatthe block-shift rate was 1/2 and 1/4 of the block size,
respectively. When we used the overlap-blockshift only
for low frequency bands, we obtained better separation performance. With quadruple overlap-blockshift,
we can estimate the unmixing filters of 128 taps in low
frequency bands, and we obtained the best separation
performance [see (E) in Table 1]. Even if we used 128
taps for all frequency bands [(F) in Table 1], the performance does not increase compared with the case (E)when we used 64 taps in bands 6-32. The use of 128
taps in all subbands does not lead to improved sep-

Table 1: Separation performance of subband BSS. (A)(F) the overlap-blockshift was executed only for bands 0-5,
and (G) and (H) the overlap-blockshift was executed for all
subbands
overlap (x2)overlap (x4)

(A)
(B)
(C)

band 0-5 band 6-32

32
64
128

32
32
32

no-overlap

5.82
8.86
8.71

9.61
9.75

10.09

(D)
(E)

64
128

64
64

10.31
10.28

10.80
11.21

12.01

(F)

128

128

10.28

11.22

12.00

(G)
(H)

64
128

64
128

10.31
10.28

10.88
11.22

10.99
12.14

( tap )

10.79

( SIR [dB] )aration performance, and for implementing the algorithm in a computationally efficient way we can restrict
ourselves to 64 taps in the higher frequency subbands.
Furthermore, when the overlap-blockshift was used inall subbands [see (G) and (H) in Table 1], the SIR increase was very small compared with the SIR for (D)and (F) in Table 1. This means that, by limiting theoverlap method only to low frequency bands, we cansave the computation cost without degrading the separation performance.
It should be noted that the SIR values in this section
are for 500 iterations, and the unmixing filters w ij areroughly rescaled so that they have the same power as
the power of the initial value in each subband.
By using long unmixing filters and the overlapblockshift technique only in low frequency bands, we
can efficiently separate the convolutive mixtures ofspeech. Such frequency-appropriate processing is impossible in time- and frequency-domain BSS.
6. CONCLUSION
Subband processing was applied to BSS for convolutive
mixtures of speech. This proposal was motivated by thefact that the separation performance is degraded whena long frame size is used for several seconds of speechin frequency-domain BSS. Our proposed subband BSS
can (1) maintain a sufficient number of samples to estimate the statistics in each subband and (2) estimate
an unmixing filter long enough to cover the reverberation. We confirmed in experiments that subband BSS
is effective. Furthermore, we showed that (3) we can
improve the separation performance with long unmixing filters and (4) the overlap-blockshift technique only
in low frequency bands.
ACKNOWLEDGMENTS
We would like to thank Dr. Y. Haneda, Mr. A. Nakagawa,and Mr. S. Sakauchi for their help with the SSB subband.

504

We also thank Dr. W. Kellermann for his collaboration and
Dr. S. Katagiri for his continuous encouragement.

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