FUNDAMENTAL LIMITATION OF
FREQUENCY DOMAIN BLIND SOURCE SEPARATIONFOR CONVOLVED MIXTURE OF SPEECH
Shoko Araki †Shoji Makino †Ryo Mukai †Tsuyoki Nishikawa ‡Hiroshi Saruwatari ‡†NTT Communication Science Laboratories
2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan
Email: shoko@cslab.kecl.ntt.co.jp‡
Nara Institute of Science and Technology
8916-5 Takayama-cho, Ikoma, Nara 630-0101, Japan
ABSTRACT
Despite several recent proposals to achieve Blind
Source Separation (BSS) for realistic acoustic signals,
the separation performance is still not enough. In particular, when the length of an impulse response is long,the performance is highly limited. In this paper, weconsider the reason for the poor performance of BSS ina long reverberation environment. First, we show that
it is useless to be constrained by the condition P  T ,where T is the frame size of FFT and P is the lengthof a room impulse response. We also discuss the limitation of frequency domain BSS, by showing that thefrequency domain BSS framework is equivalent to two
sets of frequency domain adaptive beamformers.
1. INTRODUCTION
Blind Source Separation (BSS) is an approach toestimate original source signals si (t) using only the information of the mixed signals x j (t) observed in each
input channel. This technique is applicable to the realization of noise robust speech recognition and highquality hands-free telecommunication systems. It may
also become a cue for auditory scene analysis.
To achieve BSS of convolutive mixtures, several
methods have been proposed [1, 2]. In this paper, weconsider the BSS of convolutive mixtures of speech in
the frequency domain [3, 4], for the sake of mathematical simplicity and the reduction of computational complexity.
There have been a lot of proposals to achieve BSS
in a realistic room environment, however, the separation performance is still not enough. In this paper, weconsider the reason for the poor performance of BSS ina long reverberation environment.
First, we discuss the frame size of FFT used in frequency domain BSS. It is commonly believed that theframe size T must be P  T to estimate an unmixing
matrix for a P -point room impulse response [5, 6]. We
point out that this is not the case for BSS, and showthat a smaller frame size is much better, even for longroom reverberation.
Next, we discuss the limitation of frequency domain
BSS, by showing the equivalence between frequency do-

X1
H11

S1

H12

W11
mic. 1

H21
S2

X2

H22

Y1

W12
W21
W22

Y2

mic. 2

mixing system

unmixing system

Figure 1: BSS system conﬁguration.
main BSS framework and two sets of beamformers. To
date, signal separation by using a noise cancellation
framework with signal leakage into the noise reference
has been discussed in [7, 8]. In these papers it is shownthat the least squares criterion is equivalent to the decorrelation criterion of a noise free signal estimate anda signal free noise estimate. Inspired by their discussions, but apart from the noise cancellation framework,
we attempt to see the frequency domain BSS problem
with a frequency domain adaptive microphone array,i.e., Adaptive Beamformer (ABF) framework.
2. FREQUENCY DOMAIN BSS OF
CONVOLUTIVE MIXTURES OF SPEECH
The signals recorded by M microphones are givenbyxj (n) =N Phji (p)si (n−p+1) (j = 1, · · ·, M ), (1)i=1 p=1

where si is the source signal from a source i, xj is thereceived signal by a microphone j, and hji is a P -point
impulse response from source i to microphone j. In thispaper, we consider a two-input, two-output convolutive
BSS problem, i.e., N = M = 2 (Fig. 1).
The frequency domain approach to convolutive mixtures is to transform the problem into an instantaneous
BSS problem in the frequency domain [3, 4]. Using
T -point short time Fourier transformation for (1), we
obtain,X(ω, m) = H(ω)S(ω, m).
(2)

132

5.73 m

0.6
0.4
0.2
0
-0.2

loudspeakers(height : 1.35 m)

1.56 m3.12 m1.15 m-0.4

30

2.15 m40

4 cm

1.15 m[dB]

room height : 2.70 m

Figure 2: Layout of a room used in experiments.

Y (ω, m) = W (ω)X (ω, m),

3.1. Conditions for experiments
3.1.1. Learning algorithm
For the calculation of the unmixing matrix W (ω)
in (3), we use an algorithm based on the minimizationof the Kullback-Leibler divergence [3, 10]. The optimal W (ω) is obtained by using the following iterativeequation:i

Y Y  −Φ(Y )Y W (ω),(4)+η diag Φ( )

Y Y
Y

1
1 + exp(−

Y

(R)

0.3

used in experiments (TR = 300 ms).

H

H

3.1.2. Conditions for experiments
Separation experiments were conducted using speech
data convolved with impulse responses recorded in threeenvironments speciﬁed by diﬀerent reverberation times:
TR = 0 ms, 150 ms (P = 1200), and 300 ms (P =2400).
The layout of the room we used to measure theimpulse responses is shown in Fig. 2. We used a twoelement array with inter-element spacing of 4 cm. The
speech signals arrived from two directions, −30◦ and40◦. An example of a measured room impulse response
used in our experiments is shown in Fig. 3.
In these experiments, we changed the frame size
T from 32 to 2048 and investigated the performancefor each condition. The sampling rate was 8 kHz, theframe shift was half of frame size T , and the analysis
window was a Hamming window. To solve the permutation problem, we used the blind beamforming algorithm proposed by Kurita et al [10].
3.1.3. Evaluation measure
In order to evaluate the performance for diﬀerent
frame sizes T with diﬀerent reverberation times TR , weused the noise reduction rate (NRR), deﬁned as theoutput signal-to-noise ratio (SNR) in dB minus the input SNR in dB.

i

NRRi

where = (ω, m), · denotes the averaging operator,
i is used to express the value of the i-th step in theiterations, and η is the step size parameter. In addition,
we deﬁne the nonlinear function Φ(·) as
Φ( ) =

0.2

Figure 3: An example of measured impulse response h11(3)

3. EXPERIMENTS
It is commonly believed that the frame size T mustbe P  T to estimate an unmixing matrix for a P point room impulse response [5, 6]. In this section, weinvestigate this point for BSS, and show that there isan optimal frame size for BSS [9].

W (ω)

0.1

0.3

time (s)where X(ω, m) = [X1 (ω, m), X2 (ω, m)]T is the observed
signal at frequency bin ω, Y (ω, m) = [Y1 (ω, m), Y2 (ω, m)]T
is the estimated source signal, and W (ω) represents
a (2×2) unmixing matrix. W (ω) is determined sothat Y1 (ω, m) and Y2 (ω, m) become mutually independent. The above calculations are carried out in eachfrequency independently.



0

0.2

)

+j

1
1 + exp(−

Y ),
(I)

-20
-30
-40
-50
-60

where S(ω, m) = [S1 (ω, m), S2 (ω, m)]T . We assume
that a (2×2) mixing matrix H(ω) is invertible, andHji (ω) = 0.
The unmixing process can be formulated in a frequency bin ω:

i+1 (ω) =

0.1

0
-10

microphones(height : 1.35 m)

W

0

(5)

where Y (R) and Y (I) are the real part and the imaginary part of Y , respectively.

SNROi

SNROi − SNRIi
|Aii (ω)Si (ω)|2= 10 log  ω|Aij (ω)Sj (ω)|2
ω

=

(6)



SNRIi

=

|Hii (ω)Si (ω)|2|Hij (ω)Sj (ω)|2ω

10 log  ω(7)where A(ω) = W (ω)H(ω) and i = j. These values were averaged for the whole six combinations withrespect to the speakers, and NRR1 and NRR2 wereaveraged for the sake of convenience.

133

1

8 s learning

3 s learning
128 256 512
frame size T

10(b)
TR = 150 ms
981024 2048

8 s learning

0.1

Y1 and Y2

3s
8s

S1 and S2
0.01

7
6
5

32

3 s learning64

128 256
512
frame size T

1024 2048

Figure 5: Relationship between T and the correlation co32

64

128 256 512
frame size T

1024 2048

8.5
8 (c) TR = 300 ms
NRR [dB]

X1 and X2
correlation coefficient J(T)NRR [dB]

NRR [dB]24
22 (a)
20
18
16
14
12
10
8 TR = 0 ms
632
64

8 s learning

7
6

3 s learning

5
4
32

64

128 256 512
frame size T

1024 2048

Figure 4: Results of NRR for diﬀerent frame sizes. The
solid lines are for 8 seconds learning, and the dotted lines
are for 3 seconds learning. Separation was executed for the
8 seconds long data.
3.2. Experimental results
The experimental results are shown in Fig. 4. The
lengths of the mixed speech signals were about eight
seconds each. We used the beginning three seconds(dotted lines) or entire eight seconds (solid lines) of
the mixed data for learning according to (4), and theentire eight seconds data for separation.
In the case of the three seconds learning, the maximum NRR was obtained when T = 128 [Fig. 4(a)] in
non-reverberant tests. In reverberant tests, the maximum NRR was obtained using T =512 when TR =150 ms[Fig. 4(b)], and T R =300 ms [Fig. 4(c)]. A short frame
was found to function far better than a long frame,
even for long room reverberation.
For the longer learning data, i.e., the eight seconds
data, the results were slightly diﬀerent. In this case,
we obtained a better separation performance than thethree seconds learning case. Furthermore, in comparison with the three seconds learning case, the peak performance appeared when we used a longer frame size
T . With an overly long frame size, however, the performance become poor even when we used longer learning
data.
Even for long room reverberation, the condition
P  T is useless, and a shorter frame size T is best.

eﬃcient. The solid lines are for data of 8 seconds, and thedotted lines are for data of 3 seconds. TR =0 ms.

4. DISCUSSION
4.1. Optimum frame size for frequency domain
BSS
In the previous section, we showed that a longer
frame size T fails. In this section, we discuss the reason
why both a short frame and a long frame fail.
In the frequency domain BSS framework, the signal
we can use is not x(n) but X(ω, m). If the frame size
T is long, the number of data in each frequency bin
becomes few. This causes assumptions to collapse, likethe zero mean assumption.
As an example of such a collapse, we observed thatthe independency of two original signals went downwhen the frame size became longer. Figure 5 shows
the relationship between the frame size T and the correlation coeﬃcient:1 |rω |,T
T

J (T ) =

(8)

ω

where

rω = 

m(U1 (ω, m) − U1 (ω))(U2 (ω, m) − U2 (ω))[U1 (ω, m) − U1 (ω)]2m



[U2 (ω, m) − U2 (ω)]2m(9)

where U represents a mean value, and U is original
signal S, observed signal, X or separated signal Y .
Although a correlation coeﬃcient does not show independency directly, we use this measure as an index ofindependency. As Fig. 5 shows, the independency decreases when the frame size T becomes longer. In thesecases, because the data length gets shorter, the assumption of independency does not hold for the two original
sources. This is a reason why the long frame failed.
On the other hand, if we use a short frame, theframe cannot cover the reverberation, therefore, theseparation performance is limited.
In frequency domain BSS, the optimum frame size
is decided by the trade-oﬀ between maintaining the as-134.

sumption of independency and covering the whole reverberation.

W11

S1

4.2. Length of learning data and separation performance
In 3.2, we obtained a better performance usingeight seconds data than using three seconds data. The
reason for this result is also explained by Fig. 5. With
eight seconds data, independency is better maintained
than with three seconds data. Therefore, we can obtaina better performance using the former. Furthermore,
the optimum frame size changes when we use learning
data of diﬀerent length, because the optimum frame
size is decided by the trade-oﬀ we mentioned in 4.1.

W12
H22

S2

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

S1

H11

(b) ABF for a target S2 and a jammer S1 .
Figure 6: Two sets of ABF system conﬁgurations.
The diagonalization of R Y (ω, k) can be written as
an overdetermined least-squares problem,W (ω)R (ω, k)W (ω)||

diag||W (ω)R (ω, k)W (ω)|| = 0,
s.t.,

arg min5.1. Frequency domain BSS of convolutive mixtures using Second Order Statistics (SOS)In this section, we use a second order statistics(SOS) BSS algorithm for convenience. It is well known
that a decorrelation criterion is insuﬃcient to solve
problems. In [8], however, it is pointed out that nonstationary signals can provide enough additional information to estimate all W ij . Some authors have utilized
SOS for convolutively mixed speech signals [5, 6].
Source signals S1 (ω, m) and S2 (ω, m) are assumed
to be zero mean and mutually uncorrelated:

∗

s

=

Λc (ω, k),(11)where RX is the covariance matrix of X(ω) as follows,
1 

R (ω, k) = M
X

M −1

m=0

Y1 (ω, m)=

W (ω)X (ω, m),

(14)

where

W (ω) = [W (ω), W (ω)], X (ω, m) = [X (ω, m), X (ω, m)] .
12

1

T

2

To minimize jammer S 2 (ω, m) in output Y1 (ω, m)when target S1 = 0, mean square error J(ω) is introduced as
J (ω)

X (ω, M k + m)X (ω, M k + m), (12)∗and Λc (ω, k) is an arbitrary diagonal matrix.

2

5.2.1. ABF null towards S2First, we consider the case of target S1 and jammer
S2 [Fig. 6(a)]. When target S 1 = 0, output Y1 (ω, m)
is expressed as

11

∗

∗

Here, we consider frequency domain adaptive beamformers (ABF), which can remove jammer signals. Since
our aim is to separate two signals S1 and S2 with two
microphones, two sets of ABF are used (Fig. 6). Notethat an ABF can be adapted only when a jammer exists
but a target does not exist.

(10)

X

(13)

k

5.2. Frequency domain adaptive beamformer

where ∗ denotes the conjugate transpose, and Λs (ω, k)
is a diﬀerent diagonal matrix for each block k.
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 k,Y2

where ||x||2 is the squared Frobenius norm.

m=0

∗

∗

X

k

∗

R (ω, k) = W (ω)R (ω, k)W (ω)= W (ω)H (ω)Λ (ω, k)H (ω)W (ω)||oﬀ-diagX

R (ω, k) = M1  S(ω, M k + m)S (ω, M k + m)
Λs (ω, k),

0
Y2

X2

W (ω)

=

W21
W22

S2

In this section, in order to discuss the fundamental limitation of frequency domain BSS, we show thatfrequency domain BSS is equivalent to two sets of frequency domain adaptive beamformers (ABF). Fromthe equivalence between BSS and ABF, we can conclude that the performance of BSS is upper boundedby that of ABF [11].

M −1

X1

H12

5. EQUIVALENCE BETWEENFREQUENCY DOMAIN BSS AND
FREQUENCY DOMAIN ABF

S

0
Y1

X1

H12

=
=
=

E[Y12 (ω, m)](ω)E[ (ω, m) ∗ (ω, m)](ω)R(ω) ∗ (ω),

W
W

X

W

where E is the expectation and

135

X

W (ω)
∗

(15)

R(ω) = E

X1 (ω, m)X1∗(ω, m)
X2 (ω, m)X1∗(ω, m)

X1 (ω, m)X2∗(ω, m)
X2 (ω, m)X2∗(ω, m)[see (11)]. Using H and W , outputs Y1 and Y2 areexpressed in each frequency bin as follows,. (16)

By diﬀerentiating cost function J(ω) with respect to
W and setting the gradient equal to zero
∂J (ω)= 2R∂

W

W = 0,∗E

X1 X2
X2 X2∗

W11
∗
W12

a b
c d

(17)

=

0
0

.

=
=

W11 X1 + W12 X2
W11 H11 S1 + W12 H21 S1 = c1 S1 ,

=

W11
W21

W12
W22

H11
H21

H12
H22

.

(28)

We now analyze what is going on in the BSS framework. After convergence, the expectation of the oﬀ-diagonal component E[Y1 Y2∗ ] is expressed as(19)

E[Y1 Y2∗ ](20)= ad∗E[S1 S2∗ ] + bc∗ E[S2 S1∗ ] + (ac∗ E[S12 ] + bd∗E[S22 ])
= 0.
(29)
Since S1 and S2 are assumed to be uncorrelated, theﬁrst term and the second term become zero. Then,the BSS adaptation should drive the third term of (29)
to be zero. By squaring the third term and setting it
equal to zero(ac∗ E[S12 ] + bd∗ E[S22 ])2which leads to
W11 H11 + W12 H21 = c1 ,= a2 c∗2 (E[S12 ])2 + 2abc∗ d∗E[S12 ]E[S22 ] + b2 d∗2 (E[S22 ])2
=0
(30)

(21)

where c1 is an arbitrary complex constant. Since H12and H22 are unknown, the minimization of (15) withadaptive ﬁlters W11 and W12 is used to derive (19) withconstraint (21). This means that the ABF solution isderived from simultaneous equations (19) and (21).

(30) is equivalent toac∗ = bd∗ = 0,

W21 H11 + W22 H21
W21 H12 + W22 H22

=
=

0
c2 .

(22)
(23)

5.2.3. Two sets of ABF
By combining (19), (21), (22), and (23), the simultaneous equations for the two sets of ABF are summarized as
H11
H21

W12
W22

H12
H22

c1
0

=

0
c2

.

(24)

5.3. Equivalence between BSS and ABF
As we showed in (13), the SOS BSS algorithm
works to minimize oﬀ-diagonal components inE

Y1 Y1∗Y2 Y1∗Y1 Y2∗Y2 Y2∗,(25)abc∗ d∗ = 0.

(31)

CASE 1: a = c1 , c = 0, b = 0, d = c 2

5.2.2. ABF null towards S1
Similarly for target S 2 , jammer S1 , and output Y2[Fig. 6(b)], we obtain

W11
W21

(26)
(27)

(18)

With (19) only, we have trivial solution W 11 =W12 =0.
Therefore, an additional constraint should be added toensure target signal S1 in output Y1 . With this constraint, output Y1 is expressed as
Y1

aS1 + bS2
cS1 + dS2 ,5.3.1. When S1 = 0 and S2 = 0

Using X1 = H12 S2 , X2 = H22S2 , we get
W11 H12 + W12 H22 = 0.

=
=

where

we obtain the equation to solve as follows [(ω, m), etc.,
are omitted for convenience],

 ∗

∗
∗
X1 X1
X2 X1∗

Y1
Y2

W11
W21

W12
W22

H11
H21

H12
H22

=

c1
0

0
c2

(32)

This equation is exactly the same as that of the ABF(24).
CASE 2: a = 0, c = c1 , b = c2 , d = 0
W11
W21

W12
W22

H11
H21

H12
H22

=

0
c1

c2
0

(33)

This equation leads to the permutation solution which
is Y1 = c2 S2 , Y2 = c1 S1 .
Note that the undesirable solutions (i.e., a = 0, c =c1 , b = 0, d = c2 and a = c1 , c = 0, b = c2 , d = 0) do notappear, since we assume that H(ω) is invertible andHji (ω) = 0.
If the uncorrelated assumption between S1 (ω) andS2 (ω) collapses, the ﬁrst and second terms of (29) become the bias noise to get the correct coeﬃcients a, b, c, d.
5.3.2. When S1 = 0 and S2 = 0
The BSS can adapt, even if there is only one active
source. In this case, only one set of ABF is achieved.

136

0

Gain

(b)
10

0

-20

(d)
10

0

80
90

1
40
60

.)

(c)

0

c

en

qu

fre

Gain

0

4

DOA(deg

.)

1
80
90

0

0
-2

0

0

0

-6

-4

y

nc

ue

q
fre

3

2
20

0

4

-40

)

z
(kH

40
60

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1
80
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DOA(deg

40
60

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20

-2

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-6

-4

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-40

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-20

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-4
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-2
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c

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and P is the length of a room impulse response. Thisis because the lack of data causes the collapse of theassumption of independency between the two original
signals in each frequency bin when the data length isshort, or when a longer frame size T is used.
Next, we show that frequency domain BSS is equivalent to two sets of frequency domain ABF. Because
ABF (and BSS) mainly considers the direct sound by
making a null towards jammer direction, the separation performance is fundamentally limited. This understanding clearly explains the poor performance of
BSS in a real room with long reverberation.
ACKNOWLEDGEMENTS

Figure 7: Directivity patterns (a) obtained by an NBF,(b) obtained by BSS (TR =0 ms), (c) obtained by BSS(TR =150 ms), and (d) obtained by BSS (TR =300 ms). DOA
means direction of arrival. T =256, three second learning.

We would like to thank Dr. Shigeru Katagiri for
his continuous encouragement.

5.4. Fundamental limitation of frequency domain BSS[1] A. J. Bell and T. J. Sejnowski, “An informationmaximization approach to blind separation and blind
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pp. 1129–1159, 1995.
[2] S. Haykin, Unsupervised adaptive filtering. John Wiley & Sons, 2000.
[3] S. Ikeda and N. Murata, “A method of ICA in timefrequency domain,” Proc. ICA99, pp. 365-370, Jan.
1999.
[4] P. Smaragdis, “Blind separation of convolved mixtures in the frequency domain,” Neurocomputing, vol.
22, pp. 21-34, 1998.
[5] L. Parra and C. Spence, “Convolutive blind separation of non-stationary sources,” IEEE Trans. Speech
Audio Processing, vol. 8, no. 3, pp. 320-327, May
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[6] M. Z. Ikram and D. R. Morgan, “Exploring
permutation inconsistency in blind separation of
speech signals in a reverberant environment,” Proc.
ICASSP2000, pp. 1041-1044, June 2000.
[7] S. Garven and D. Compernolle, “Signal separation
by symmetric adaptive decorrelation: stability, convergence, and uniqueness,” IEEE Trans. Signal Processing, vol. 43, no. 7, pp. 1602-1612, July 1995.
[8] E. Weinstein, M. Feder, and A. V. Oppenheim,“Multi-channel signal separation by decorrelation,”
IEEE Trans. Speech Audio Processing, vol. 1, no. 4,
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[9] S. Araki, S. Makino, T. Nishikawa, and H.
Saruwatari, “Fundamental limitation of frequency
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[10] S. Kurita, H. Saruwatari, S. Kajita, K. Takeda, and
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[11] S. Araki, S. Makino, R. Mukai, and H. Saruwatari,“Equivalence between frequency domain blind source
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REFERENCES

Frequency domain BSS and frequency domain ABF
are shown to be equivalent [see (24) and (32)] if the independent assumption ideally holds [see (29)]. We canform only one null towards the jammer in the case of
two microphones. Figure 7 shows directivity patterns
obtained by a null beamformer (NBF) and BSS; Fig.
7(a) shows a directivity pattern obtained by an NBF
that forms a steep null directivity pattern towards a
jammer under the assumption of the jammer’s direction being known. In Fig. 7, (b),(c), and (d) are drawn
by W by BSS: (b) is drawn by W when TR = 0, (c)is drawn by W when TR = 150 ms, and (d) is drawn
by W when TR = 300 ms. When TR = 0, a sharp
null is obtained like with an NBF. When T R is long,the directivity pattern is comparatively duller; however, we can draw a directivity pattern. Although BSS
and ABF can reduce reverberant sounds to some extent[12], they mainly remove the sounds from the jammer
direction. This understanding clearly explains the poor
performance of BSS in a real room with long reverberation.
Moreover, as we have shown in section 3, a long
frame size works poorly in frequency domain BSS for
speech data of a few seconds. This is because when weuse a long frame, the assumption of independency between S1 (ω) and S2 (ω) does not hold in each frequency;
this is caused by the lack of the number of data in eachfrequency bin. Therefore, the performance of BSS is
upper bounded by that of ABF.
Our discussion here is essentially also true for BSS
with Higher Order Statistics (HOS), and will be extended to it shortly.
6. CONCLUSIONSIn this paper, we discuss why the separation performance is poor when there is long reverberation.
First, we show that it is useless to be constrained by
the condition P  T , where T is the frame size of FFT137