EURASIP Journal on Applied Signal Processing 2003:11, 1157–1166
c 2003 Hindawi Publishing Corporation

Equivalence between Frequency-Domain Blind Source
Separation and Frequency-Domain Adaptive
Beamforming for Convolutive Mixtures
Shoko Araki
NTT Communication Science Laboratories, NTT Corporation, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan
Email: shoko@cslab.kecl.ntt.co.jp

Shoji Makino
NTT Communication Science Laboratories, NTT Corporation, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan
Email: maki@cslab.kecl.ntt.co.jp

Yoichi Hinamoto
Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5 Takayama-cho,Ikoma, Nara 630-0192, Japan
Email: yoichi-h@is.aist-nara.ac.jp

Ryo Mukai
NTT Communication Science Laboratories, NTT Corporation, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan
Email: ryo@cslab.kecl.ntt.co.jp

Tsuyoki Nishikawa
Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5 Takayama-cho,Ikoma, Nara 630-0192, Japan
Email: tsuyo-ni@is.aist-nara.ac.jp

Hiroshi Saruwatari
Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5 Takayama-cho,Ikoma, Nara, 630-0192, Japan
Email: sawatari@is.aist-nara.ac.jp
Received 2 December 2002 and in revised form 16 March 2003
Frequency-domain blind source separation (BSS) is shown to be equivalent to two sets of frequency-domain adaptive beamformers(ABFs) under certain conditions. The zero search of the oﬀ-diagonal components in the BSS update equation can be viewed as
the minimization of the mean square error in the ABFs. The unmixing matrix of the BSS and the filter coeﬃcients of the ABFs
converge to the same solution if the two source signals are ideally independent. If they are dependent, this results in a bias for thecorrect unmixing filter coeﬃcients. Therefore, the performance of the BSS is limited to that of the ABF if the ABF can use exact
geometric information. This understanding gives an interpretation of BSS from a physical point of view.
Keywords and phrases: blind source separation, convolutive mixtures, adaptive beamformers.

1.

INTRODUCTION

Blind source separation (BSS) is an approach for estimating
source signals si (t) using only the information on mixed signals x j (t) observed at each input channel. BSS can be applied
to achieve noise-robust speech recognition and high-quality

hands-free telecommunication. It might also become one of
the cues for auditory scene analysis.
Several methods have been proposed for BSS of convolutive mixtures [1, 2]. Some approaches consider the impulse responses of a room h ji as FIR filters, and estimate
those filters in the time domain [3, 4, 5]; other approaches

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EURASIP Journal on Applied Signal Processing

transform the problem into the frequency domain to solve
an instantaneous BSS problem for every frequency simultaneously [6, 7]. Here, we consider the BSS of convolutive mixtures of speech in the frequency domain.
In this paper, we provide an interpretation of BSS froma physical point of view showing the equivalence between
frequency-domain BSS and two sets of frequency-domain
adaptive beamformers (ABFs).
Signal separation by using a noise cancellation framework with signal leakage into the noise reference was discussed in [8, 9]. These studies showed that the least squares
criterion is equivalent to the decorrelation criterion of anoise-free signal estimate and a signal-free noise estimate.
The error minimization was shown to be completely equivalent to a zero search in the cross correlation.
Inspired by the discussions in [8, 9], but apart from thenoise cancellation framework, we attempt to compare thefrequency-domain BSS problem with the frequency-domain
ABF framework. In earlier work, Dinc and Bar-Ness [10] andCardoso and Souloumiac [11] indicated the connection between blind identification and beamforming in a narrowband context. Kurita et al. [12] and Parra and Alvino [13] utilized the relationship between BSS and ABFs to achieve better
BSS performance; however, they did not discuss this relationship theoretically. We discuss this relationship more closely
and more quantitatively, focusing on BSS with second-order
statistics (SOS), and we show that BSS and ABFs have equivalent functions despite their completely diﬀerent adaptation
procedures. Moreover, we provide a physical understandingof frequency-domain BSS [14]. From the equivalence between BSS and ABFs, we can make it clear that the physical behavior of BSS is to reduce jammer signal by forming aspatial null in the jammer direction. Knaak and Filbert [15]have also provided a somewhat quantitative discussion of therelationship between frequency-domain ABF and frequencydomain BSS. Beyond their discussions, in this paper, we arealso able to explain the eﬀect of collapse of the independence
assumption in BSS.
In Section 2, we summarize the framework of frequencydomain BSS for convolutive mixtures. In Section 3, thefrequency-domain ABF is summarized. In Section 4, weshow the equivalence between BSS and ABFs theoretically.
In Section 5, we confirm this equivalence and the limitation with experiments using measured impulse responses ina real room and six combinations of male and female speech.
Section 6 concludes this paper.
2.

FREQUENCY-DOMAIN BSS OF CONVOLUTIVEMIXTURES OF SPEECH

In real environments, the signals are aﬀected by reverberation and observed by the microphones. Therefore, N signals
recorded by M microphones are modeled asN P
i=1 p=1

h ji (p)si (n − p + 1)

( j = 1, . . . , M),

H11

W11
mic. 1

H12
H21
S2

W21

mic. 2

H22

W22

X2

Mixing system

(1)

Y1

W12

Y2

Unmixing system

Figure 1: BSS system configuration.

where si is the source signal from a source i, x j is the signal
received by a microphone j, and h ji is the P-taps impulse
response from source i to microphone j.
2.2.

Unmixed signal model

In order to obtain unmixed signals, we estimate unmixing
filters wi j (k) of Q-taps, and the unmixed signals are obtainedasyi (n) =QM 

wi j (q)x j (n − q + 1) (i = 1, . . . , N).

(2)

j =1 q=1

The unmixing filters are estimated such that the unmixed
signals become mutually independent.
In this paper, we consider a two-input, two-output convolutive BSS problem, that is, N = M = 2 (Figure 1).
2.3.

Frequency-domain approach

The frequency-domain approach to convolutive mixtures isto transform the problem into an instantaneous BSS problem
in the frequency domain [6, 7]. Using a T-point short-time
Fourier transformation for (1), we obtainX(ω, m) = H(ω)S(ω, m),(3)

where ω denotes the frequency, m represents the timedependence of the short-time Fourier transformation,S(ω, m) = [S1 (ω, m), S2 (ω, m)]T is the source signal vector,
and X(ω, m) = [X1 (ω, m), X2 (ω, m)]T is the observed signal
vector. We assume that the (2 × 2) mixing matrix H(ω) is invertible and that H ji (ω) = 0. Also, H(ω) does not depend ontime m.
The unmixing process can be formulated in a frequency
bin ω:Y(ω, m) = W(ω)X(ω, m),

2.1. Mixed signal modelx j (n) =

X1

S1

(4)

where Y(ω, m) = [Y1 (ω, m), Y2 (ω, m)]T is the estimatedsource signal vector and W(ω) represents a (2 × 2) unmixing matrix at frequency bin ω. The unmixing matrix W(ω)
is determined so that Y1 (ω, m) and Y2 (ω, m) become mutually independent. The above calculation is carried out at eachfrequency independently. In this paper, we consider the DFT
frame size T to be equal to the length Q of the unmixing filter.

Equivalence between BSS and ABF

1159

2.4. Frequency-domain BSS of convolutive
mixtures using SOS
In [9], it is pointed out that nonstationary signals provide
enough additional information to enable us to estimate all
Wi j (ω). Some authors have utilized SOS for mixed speech
signals [16, 17].
The source signals S1 (ω, m) and S2 (ω, m) are assumed to
be zero mean, nonstationary, and mutually uncorrelated.
In order to determine W(ω) so that Y1 (ω, m) andY2 (ω, m) become mutually uncorrelated, we seek a W(ω)
that diagonalizes the covariance matrices RY (ω, k) simultaneously for all time blocks k:
RY (ω, k) = W(ω)RX (ω, k)W∗ (ω)= W(ω)H(ω)Λs (ω, k)H∗ (ω)W∗ (ω)= Λc (ω, k),

S1

H22

S2

X2
(a) ABF for a target S1 and a jammer S2 .
S1

H11
X1

W22

(b) ABF for a target S2 and a jammer S1 .

Figure 2: Two sets of ABF-system configurations.

(6)Λs (ω, k) is the diagonal covariance matrix of the source signals that is diﬀerent for each k, and Λc (ω, k) is an arbitrary
diagonal matrix.
The diagonalization of RY (ω, k) can be written as an
overdetermined least squares problem:
W(ω)



oﬀ-diagW(ω)RX (ω, k)W∗ (ω)2 ,(7)

First, we consider the case of a target S1 and a jammer S2[see Figure 2a]. When target S1 = 0, the output Y1 (ω, m) is
expressed as
Y1 (ω, m) = W(ω)X(ω, m),W(ω) = W11 (ω), W12 (ω)

3.1. ABF for target S1 and jammer S2
In order to estimate the coeﬃcients Wi j of an ABF, we minimize the output signal power when a jammer is active but atarget is not.





T

(9)

X(ω, m) = X1 (ω, m), X2 (ω, m) .

To minimize jammer S2 (ω, m) in the output Y1 (ω, m)when target S1 = 0, the mean square error J(ω) is introducedas

J(ω) = E Y12 (ω, m)= W(ω)E X(ω, m)X∗ (ω, m) W∗ (ω)

FREQUENCY-DOMAIN ABF

Here, we consider the frequency-domain ABF which can remove a jammer signal. Since our aim is to separate two signals S1 and S2 with two microphones, we use two sets of ABFs
(see Figure 2). That is, an ABF that forms a null directivity
pattern towards source S2 by using filter coeﬃcients W11 andW12 , and an ABF that forms a null directivity pattern towards
source S1 by using filter coeﬃcients W21 and W22 . Note that
the ABF can be adapted when only a jammer exists but a target does not exist, and that the direction of the target or theimpulse responses from the target to the microphones should
be known. In this section, we attach more importance to an
intuitive explanation of the ABF mechanism than to a strict
mathematical explanation.

(8)

where

k

where  · 2 is the squared Frobenius norm. In order to avoid
a trivial solution, W(ω) = 0, we use a constraint, for example, k diagW(ω)RX (ω, k)W∗ (ω)2 = c or W(ω)2 = c,where c is a positive constant. While these constraints for determining a nontrivial W(ω) give rise to a diﬀerent solution,
they still have the same function.
3.

0
Y2

X2

M −1

arg min

W21

S2

where ∗ denotes the conjugate transpose and, RX is the covariance matrix of X(ω), represented as follows:
1 
RX (ω, k) =X(ω, Mk + m)X∗ (ω, Mk + m),
M m=0

Y1
W12

H21

(5)

0

W11
X1

H12

(10)

∗

= W(ω)R(ω)W (ω),

where E[·] is the expectation operator and

X (ω, m)X1∗ (ω, m) X1 (ω, m)X2∗ (ω, m)
R(ω) = E 1. (11)
X2 (ω, m)X1∗ (ω, m) X2 (ω, m)X2∗ (ω, m)
By diﬀerentiating the cost function J(ω) with respect to
W and setting the gradient to zero, we obtain (hereafter(ω, m) and (ω) are omitted for convenience)∂J(ω)= 2RW∗ = 0.
∂W

(12)

Using X1 = H12 S2 , X2 = H22 S2 , we get
W11 H12 + W12 H22 = 0.

(13)With (13) only, we have a trivial solution W11 = W12 =0. Therefore, an additional constraint should be added to1160

EURASIP Journal on Applied Signal Processing

ensure that target signal S1 is in the output Y1 , that is,

Y1 = W11 H11 + W12 H21 S1 = c1 S1 ,(14)

These paths are shown in Figure 3. Here, a and d represent
the paths for targets, and b and c are the paths for jammers.
4.1.

which leads to
W11 H11 + W12 H21 = c1 ,(15)

where c1 is an arbitrary complex constant. In the ABF framework, this constraint is usually approximately given by thesteering vector under the condition that the direction of atarget signal is known. This constraint can also be given by
the measured impulse responses from a target source to microphones. In this paper, we assume that the target direction
or impulse responses between a target and microphones are
known correctly.
The ABF solution is derived from the simultaneous equations (13) and (15).
In practice, R is a positive definite matrix due to the effect of ambient noise and a finite length DFT. Here, however, we consider the ideal case. That is, we assume that R
is not invertible. Moreover, for a practical ABF, W is calculated by solving the constrained minimization problem; theconstraint is included in advance. Therefore, (13) usually includes an estimation error and does not become 0 in a strict
sense. Although we should evaluate and compare this error
for ABF and BSS quantitatively, in this paper, we stress thequalitative equivalence between ABFs and BSS.
3.2. ABF for target S2 and jammer S1
Similarly, for a target S2 , a jammer S1 , and an output Y2 (seeFigure 2b), we obtain
W21 H11 + W22 H21 = 0,

(16)

W21 H12 + W22 H22 = c2 .

(17)

When S1 = 0 and S2 = 0

We now analyze what is occurring in the BSS framework. After convergence, the expectation of the oﬀ-diagonal component E[Y1 Y2∗ ] is expressed asE Y1 Y2∗

2



 ∗
∗
= ad ∗ E S1 S∗
2 + bc E S2 S1 ∗  2
 2 2∗+ ac E S1 + bd E S2

= 0.

(22)

Since S1 and S2 are assumed to be uncorrelated, the firstand second terms become zero. Then, the BSS adaptation
should drive the third term of (22) to zero for all time blocks
k. That is, (22) is an identical equation with regard to E[S21 ]and E[S22 ] for all time blocks k. This leads toac∗ = bd∗ = 0.

(23)Case 1. When a = c1 , c = 0, b = 0, and d = c2 ,
W11 W12
W21 W22









H11 H12c1 0.
=
H21 H220 c2(24)This equation is identical to (18) in ABFs.
Case 2. When a = 0, c = c1 , b = c2 , and d = 0,


W11 W12
W21 W22









H11 H120 c2=.
H21 H22c1 0(25)This equation leads to a permutation solution Y1 = c2 S2 ,Y2 = c1 S1 ; the estimated source signal components are recovered with a diﬀerent order.

3.3. Two sets of ABFs
By combining (13), (15), (16), and (17), we can summarize
the simultaneous equations for two sets of ABFs as follows:

W11 W12
W21 W22

4.







(18)

As we showed in (7), the SOS-BSS algorithm works to minimize oﬀ-diagonal components in(see (5)) for all time blocks k. Using H and W, the outputs
Y1 and Y2 are expressed in each frequency bin as
Y2 = cS1 + dS2 ,





(20)

where




(19)

Y1 = aS1 + bS2 ,a b
W11 W12=c d
W21 W22

H11 H120 0=.
H21 H22c1 c 2(26)(21)

W11 W12
W21 W22









H11 H12c1 c2.
=
H21 H220 0(27)This equation leads to an undesirable solution Y1 = c1 S1 +c2 S2 , Y2 = 0.
Note that Cases 3 and 4 do not appear in general because
we assume that H(ω) is invertible and H ji (ω) = 0. That is, if
a = 0, then b = 0 (Case 2), and if c = 0, then d = 0 (Case 1).
4.2.



H11 H12.
H21 H22Case 4. When a = c1 , c = 0, b = c2 , and d = 0,

Y Y∗ Y Y∗
E 1 1∗ 1 2∗ ,Y2 Y1 Y2 Y2W11 W12
W21 W22

This equation leads to an undesirable solution Y1 = 0, Y2 =c1 S1 + c2 S2 .

EQUIVALENCE BETWEEN BSS AND ABFs







H11 H12c1 0.
=
H21 H220 c2Case 3. When a = 0, c = c1 , b = 0, and d = c2 ,When S1 = 0 and S2 = 0

BSS can adapt even if there is only one active source. In thiscase, only one set of ABF is achieved.

Equivalence between BSS and ABF

1161

X1

S1

H11

W11

H12
H21
S2

X2

H22

X1

S1

Y1

H11

W11

W12

H12

W12

W21

H21
Y2

W22

S2

W22

Y2

H11

W11

Y1

W12

H22
(b)
X1

X1

S1

H11

W11

H12

W12

H12

W21

H21

H21

Y1

X2

S2

W21

X2

(a)

S1

Y1

H22

Y2

W22

S2

W21

X2

H22

Y2

W22

(c)

(d)

Figure 3: Paths in (21).

When S2 = 0, we haveCase 7. When b = 0 and d = c2 ,Y1 = aS1 ,Y2 = cS1 ,







(28)

then




 

E Y1 Y2∗ = E aS1 c∗ S∗1 = ac∗ E S21 = 0,(29)

W11 W12
W21 W22(30)



W11 W12
W21 W22










H11 H12c1 −=,H21 H22
0 −

(31)



 



Y1
c1 −
=
Y2
0 −



(32)

Case 6. When c = c1 and a = 0,


W11 W12
W21 W22









H11 H120 −=.
H21 H22c1 −(33)This equation leads to the permutation solution which is
Y1 = 0, Y2 = c1 S1 :






 

Y1
0 −
=
Y2
c1 −







Y1
− 0
=
Y2
− c2

W11 W12
W21 W22




S1
c1 S1
.
=
0
0



(35)

 





0
0
.
=
S2
c2 S2

(36)









H11 H12− c2=.
H21 H22− 0(37)This equation leads to the permutation solution

where − shows a don’t care. Since S2 = 0, the output can bederived correctly, Y1 = c1 S1 , Y2 = 0, as follows:Case 8. When b = c2 and d = 0,Case 5. When c = 0 and a = c1 ,




− 0
H11 H12.
=
H21 H22− c2

We can obtain the result

and therefore, the BSS adaptation should driveac∗ = 0.







Y1
− c2
=
Y2
− 0

 





0
c2 S2
=
.
0
S2

(38)

The values c1 and c2 in Sections 3 and 4 are not the same
due to the scaling problem in BSS: the estimated source signal
components are recovered with a diﬀerent gain in diﬀerent
frequency bins. Although the outputs obtained by BSS arefiltered versions of the source signals, the behavior whereby
they make a null towards the jammer signal is still the sameas the two sets of ABFs. Moreover, we can scale the output
signals in the same way as the constraint in an ABF (15) and(17) by using the directivity pattern obtained by the unmixing matrix (e.g., with the method described in Section 5.3).



S1
0
=
.
c1 S1
0

4.3. When S1 = 0 and S2 = 0
Similarly, only one set of ABF is achieved in this case.

(34)

5.
5.1.

EXPERIMENTS AND DISCUSSIONS
Limitation of frequency-domain BSS

Frequency-domain BSS and frequency-domain ABFs areequivalent (see (18) and (24)) in an ideal case if the inde-

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EURASIP Journal on Applied Signal Processing
45
40

5.73 m3.12 m2.15 m
4 cm

1.15 m30◦
40◦

SIR [dB]

Loudspeakers(height 1.35 m)

1.56 m

ABF

35
30
25
20

BSS

15

1.15 m10
5

Microphones(height 1.35 m)

32

64

128

Room height 2.70 m

256
512
Frame size

1024

2048

(a) TR = 0 ms.

Figure 4: Layout of the room used in experiments.
9ABF

5.1.1. Simulation conditions and evaluation
measurement
We compared the separation performance of BSS with thatof an ABF. These experiments were conducted using speech
data convolved with impulse responses recorded in two environments specified by diﬀerent reverberation times: TR =0 millisecond and 300 milliseconds. Since the sampling rate
was 8 kHz, 300 milliseconds correspond to 2400 taps. The
size of the room used to measure the impulse responses was
5.73 m × 3.12 m × 2.70 m and the distance between the loudspeakers and microphones was 1.15 m (Figure 4). We used a
two-element array with an interelement spacing of 4 cm. The
speech signals arrived from two directions, −30◦ and 40◦ . As
the original speech, we used two sentences spoken by two
male and two female speakers. The investigations were carried out for six combinations of speakers. The length of the
speech data was about eight seconds. We used the first threeseconds of the data for learning, and the entire eight seconds
for separation. We changed the DFT frame size T from 32
to 2048 and investigated the performance for each condition.
The frame shift was half the frame size T, and the analysis
window was a Hamming window. To evaluate the performance, we used the signal to interference ratio (SIR), defined

SIR [dB]

pendence assumption ideally holds (see (22)). If not, the firstand second terms of (22) behave as a bias when calculating
the correct coeﬃcients a, b, c, and d in (22). We have shownin [18] that a long frame size works poorly in frequencydomain BSS for speech data of a few seconds. This is because
when we use a long frame, the number of samples in eachfrequency bin becomes small. This makes the estimation of
statistics, such as the zero mean and independent assumptions, diﬃcult [19]. Therefore, the first and second terms of(22) are not equal to zero. Therefore, the upper bound of the
BSS performance is given by that of the ABF. However, notethat BSS does not need the absence of a target signal: BSS canadapt in the presence of target and jammer and also in thepresence of only one active source, whereas an ABF can beadapted only when there is a jammer but no target. Note alsothat an ABF needs to know the array manifold and the target
direction but BSS does not need these for the adaptation.

8
7
6

BSS

5
4

32

64

128

256
512
Frame size

1024

2048

(b) TR = 300 ms.

Figure 5: Results of SIR for diﬀerent frame sizes. The solid lines are
for ABF and the broken lines are for BSS. (a) Nonreverberant test
(TR = 0 ms), (b) reverberant test (TR = 300 ms).

as follows:
SIRi = SIROi − SIRIi ,

SIROi = 10 log  ωωSIRIi = 10 log  ωω2

Aii (ω)Si (ω)
2,
Ai j (ω)S j (ω)

(39)

2

Hii (ω)Si (ω)
2,
Hi j (ω)S j (ω)

where A(ω) = W(ω)H(ω) and i = j. SIR means the ratio of atarget-originated signal to a jammer-originated signal. These
values were averaged over all six combinations with respect
to the speakers, and SIR1 and SIR2 were averaged.
The ABF we used was that proposed by Frost [20].
5.1.2. Simulation results
Figure 5 shows the separation performance of BSS and the
ABF. With BSS, when the frame size was too long, the separation performance deteriorated. This is because the number of samples in each frequency bin is too small to estimate
the statistics correctly when the frame size is long [19]. In
this case, the first and second terms of (22) are not equal
zero and behave as a bias noise as mentioned in Section 5.1.
Therefore, the performance is degraded when we use a long
frame in BSS.

Equivalence between BSS and ABF

1163

10
0
Gain [dB]Gain [dB]

10
0
−20
−40

4

−60

2

−

20 40 06 80 0 0
Angle (
9
deg.)

y (k
enc
u
q
Fre

1

BSS TR = 300 ms
0−9−80 60
− −40 200 20−
40 60 0 0
8 90
Angle (
deg.)

)
Hz

(a)

3
2
1

Fre

ncy
que

z)
( kH

(b)

10

10
0

0
Gain [dB]Gain [dB]

4

−40

3

BSS TR = 0 ms
0−9−80 60
− −40 20
0

−20

−20
−40

4

−60

ABF TR = 0 ms
0−9−80 60
− −40 200 20−
40 60 08 90 0
Angle (
deg.)

2

y (k
enc
u
q
Fre

4

−40

ABF TR = 300 ms
0−9−80 60
− −40 200 20−
40 60 08 90 0
Angle (
deg.)

3
1

−20

)
Hz

(c)

3
2

y (k
enc
u
q
Fre

1

)
Hz

(d)

Figure 6: Directivity patterns (a) obtained by BSS (TR = 0 ms), (b) obtained by BSS (TR = 300 ms), (c) obtained by ABF (TR = 0 ms), and(d) obtained by ABF (TR = 300 ms).

By contrast, an ABF does not employ the assumption of
independence of the source signals. With the ABF, therefore,
the separation performance increased as the frame size became longer. Figure 5 confirms that the performance of the
BSS is limited by that of the ABF.
5.2. Physical interpretation of BSS
Now, we can understand the behavior of BSS as two sets of
ABFs. Figure 6 shows the directivity patterns obtained by BSS
and ABF. Figures 6a and 6b are the directivity patterns obtained by BSS after solving the permutation and scaling problem with the method described in Section 5.3, and Figures 6c
and 6d show the directivity patterns by W obtained by ABF.
When TR = 0, a sharp spatial null is obtained with both BSS
and ABF (see Figures 6a and 6c). When TR = 300 milliseconds, the directivity pattern becomes duller (see Figures 6band 6d).
BSS removes the sound from the jammer direction andreduces the reverberation of the jammer signal to some extent [21] in the same way as an ABF does. This understanding clearly explains the poor performance of the BSS in a real
acoustic environment with a long reverberation.
The BSS was shown to outperform a null beamformer
that forms a steep null directivity pattern towards a jammer[21, 22]. It is well known that an adaptive beamformer outperforms a null beamformer in long reverberation. Our understanding also clearly explains the result.
Although the ABF and BSS procedures are diﬀerent, theiressential behavior is the same: they make a null towards the
jammer direction. The relationship between ABF and BSS issummarized in Table 1.
5.3.

Improvement in separation performancewith equivalence of BSS and ABFs

So far, we have described the equivalence of BSS and ABFs:
an unmixing system obtained by BSS removes the soundfrom the jammer direction in the same way as ABFs do.
In order to improve the separation performance of BSS, weshould exploit this relationship between BSS and ABFs. In
this section, we outline our successful examples of achieving
this.
Permutation and scaling solution with directivity patterns
A scaling and permutation problem occurs in frequencydomain BSS, that is, the estimated source signal components
are recovered with a diﬀerent order and gain in diﬀerent frequency bins. When we know the array manifold, we can solve

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Table 1: The relationship between ABF and BSS.
ABF

Prior knowledge
Adaptation
Sensitivity to independence
Behavior

Array manifold and look direction or
acoustic transfer function are needed
When only jammer exist
Insensitive (however sensitive
to double-talk errors)

WheneverHighly sensitive

Make a null towards the jammer direction and reduce the jammer signal

the permutation and scaling problem in frequency-domain
BSS with directivity patterns obtained by the unmixing system W(ω) [12]. First, from the directivity pattern obtainedby W(ω), we estimate the source directions and reorder therow of W(ω) so that the directivity pattern forms a null towards the same direction in all frequency bins, then we normalize the row of W(ω) so that the target direction gains become 0 dB.
Source direction estimation with directivity pattern
After solving the permutation and scaling problem, we canroughly estimate the source directions by analyzing the null
directions, for example, clustering and averaging the null directions for all frequency bins.
Initial value of unmixing system with null beamformers
Because the solution of BSS makes a spatial null towards a
jammer, we can use this characteristics for designing the initial value of an unmixing system. As an initial value, we canuse constraint null beamformers, which can make a sharp
null towards a given jammer and maintain the gain and phaseof a given target direction.
We can apply this method to frequency-domain BSS [23],time-domain BSS [24], and subband-domain BSS [23].
Design of appropriate microphone spacing
for each frequency [25]If the spacing is longer than half the wavelength, spatial aliasing occurs: nulls are formed in several directions. By contrast,
when the sensors are very closely spaced, the phase diﬀerence
at a low frequency becomes too small and it becomes diﬃcult
to obtain good separation. Generally speaking, a long spacing is suitable for low frequencies and a short spacing for highfrequencies. If we arrange sensors according to frequency, we
can obtain better BSS performance.
6.

BSS
Not needed in itself, but to solve the permutation/scaling
problem, some is needed (e.g., array manifold)

CONCLUSION

We provided an interpretation of BSS from a physical pointof view showing the equivalence between frequency-domain
BSS and two sets of frequency-domain ABFs. The unmixing
matrix of the BSS and the filter coeﬃcients of the ABFs converge to the same solution in the ideal case if the two source
signals are ideally independent. If they are not independent,
the dependency results in bias noise in estimating the cor-

rect unmixing filter coeﬃcients. Therefore, the performanceof the BSS is limited by that of the ABF. Moreover, BSS mainly
removes sound from the jammer direction. Since we can understand the behavior of BSS as two sets of ABFs, BSS reduces
the reverberation of the jammer signal to some extent in thesame way as an ABF. This understanding clearly explains thepoor performance of the BSS in a real acoustic environment
with long reverberation.
ACKNOWLEDGMENT
We would like to thank Drs. Shigeru Katagiri and Kiyohiro
Shikano for their continuous encouragement.
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Shoko Araki received the B.E. and M.E. degrees in mathematical engineering and information physics from the University ofTokyo, Tokyo, Japan, in 1998 and 2000, respectively. Her research interests include array signal processing, blind source separation applied to speech signals, and auditory
scene analysis. She is a member of the IEEEand the Acoustical Society of Japan (ASJ).
Shoji Makino received the B.E., M.E., andPh.D. degrees from Tohoku University,
Sendai, Japan, in 1979, 1981, and 1993,
respectively. He joined NTT in 1981. He
is now an Executive Manager of the NTT
Communication Science Laboratories. His
research interests include blind source separation of convolutive mixtures of speech,
acoustic signal processing, and adaptive filtering and its applications. He received thePaper Award of the IEICE in 2002, the Paper Award of the ASJ in
2002, the Achievement Award of the IEICE in 1997, and the Outstanding Technological Development Award of the ASJ in 1995. He
is the author or coauthor of more than 170 articles in journals andconference proceedings and has been responsible for more than 140
patents. He is a member of the Conference Board of the IEEE SP Society and an Associate Editor of the IEEE Transactions on Speech
and Audio Processing. He is a member of the Technical Committee
on Audio and Electroacoustics as well as on Speech of the IEEE SP
Society. Dr. Makino is a senior member of the IEEE, a member ofthe ASJ, and the IEICE.
Yoichi Hinamoto was born in Kobe, Japan
in 1979. He received the B.E. degree in electrical and electronic engineering from the
University of Tokushima in 2001 and M.E.
degree in information science from Nara Institute of Science and Technology (NAIST)in 2003. Presently, he is a candidate for
the Ph.D. degree in the Graduate School ofInformatics, Kyoto University. His researchinterests include digital signal processing
and adaptive filter algorithm. He is a member of the IEICE andthe ASJ.
Ryo Mukai received the B.S. and M.S. degrees in information science from the University of Tokyo, Tokyo, Japan, in 1990 and
1992, respectively. His research interests include digital signal processing and blindsource separation. He is a member of theIEEE, the ACM, the IEICE, the IPSJ, and the
ASJ.

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Tsuyoki Nishikawa was born in Mie, Japan
in 1978. He received the B.E. degree in electronic system and information engineeringfrom Kinki University in 2000 and the M.E.
degree in information and science fromNara Institute of Science and Technology(NAIST) in 2002. He is now a Ph.D. student
at Graduate School of Information Science,
NAIST. His research interests include arraysignal processing and blind source separation. He is a member of the IEEE, the IEICE, and the AcousticalSociety of Japan.
Hiroshi Saruwatari was born in Nagoya,Japan in 1967. He received the B.E., M.E.,
and Ph.D. degrees in electrical engineering from Nagoya University, Nagoya, Japan,
in 1991, 1993, and 2000, respectively.
He joined Intelligent Systems Laboratory,
SECOM Co.,Ltd., Mitaka, Tokyo, Japan, in
1993, where he engaged in the research anddevelopment on the ultrasonic array system
for the acoustic imaging. He is currently an
Associate Professor of Graduate School of Information Science,
Nara Institute of Science and Technology (NAIST). His research interests include array signal processing, blind source separation, andsound field reproduction. He received the Paper Award from IEICE
in 2001. He is a member of the IEEE, the IEICE, and the AcousticalSociety of Japan (ASJ).

EURASIP Journal on Applied Signal Processing