4th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA2003), April 2003, Nara, JapanA ROBUST AND PRECISE METHOD FOR SOLVING THE PERMUTATION PROBLEM
OF FREQUENCY-DOMAIN BLIND SOURCE SEPARATION
Hiroshi Sawada

Ryo Mukai

Shoko Araki

Shoji Makino

NTT Communication Science Laboratories, NTT Corporation
2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan{sawada,ryo,shoko,maki}@cslab.kecl.ntt.co.jp
ABSTRACT

trices at adjacent frequencies [5]. Another approach is basedon direction of arrival (DOA) estimation in array signal processing. By analyzing the directivity patterns formed by aseparating matrix, source directions can be estimated andtherefore permutations can be aligned [6, 7]. If source signals are speech, we can employ the inter-frequency correlations of signal envelopes to align permutations [8, 9]. Eachof these approaches has different characteristics. They may
perform well under certain specific conditions but not others. Therefore, we believe that integrating some of theseapproaches is one way of obtaining better performance.
In this paper, we propose a new method for solving the
permutation problem, which incorporates two of the previous approaches. The first is the DOA approach, which is
described in Sec. 3. The second is based on inter-frequency
correlations, and is discussed in Sec. 4. Our new method
is proposed in Sec. 5. The experimental results reported inSec. 6 are very promising.
As another contribution, we propose a new way of estimating the direction of sources in Sec. 3.2. Unlike conventional methods [6, 7], it does not require the calculationof directivity patterns. Instead, it calculates the directionsof target signals directly from an estimated mixing matrix,
which is the inverse of a separating matrix obtained by ICA.

This paper presents a robust and precise method for solving
the permutation problem of frequency-domain blind source
separation. It is based on two previous approaches: the direction of arrival estimation and the inter-frequency correlation. We discuss the advantages and disadvantages of thetwo approaches, and integrate them to exploit their respective advantages. We also present a closed form formula toestimate the directions of source signals from a separating
matrix obtained by ICA. Experimental results show that our
method solved permutation problems almost perfectly fora situation that two sources were mixed in a room whose
reverberation time was 300 ms.
1. INTRODUCTION
Blind source separation (BSS) is a technique for estimating original source signals using only sensor observations,which consist of mixtures of the original signals. If the
mixture is instantaneous, we can directly employ independent component analysis (ICA) [1, 2] to separate the mixed
signals. In a real room environment, however, signals are
mixed in a convolutive manner with reverberations. This
makes the BSS problem difficult since we need a set of filters, not just scalars, to separate the signals. One of the major methods to obtain such separating filters is frequencydomain BSS [3–10], where a convolutive mixture in thetime domain is converted into multiple instantaneous mixtures in the frequency domain. Thus, we can apply ICA to
instantaneous mixtures in every frequency bin.
The problem with frequency-domain BSS is the indeterminacy of permutation that is inherent to ICA. We need tomap a separated signal at each frequency to a target source
signal so that we properly reconstruct the separated signal
in the time domain. Various approaches have been proposedto the permutation problem. Making separating matrices
smooth in the frequency domain is one solution. This has
been realized by averaging separating matrices with adjacent frequencies [3], limiting the filter length in the time
domain [4], or considering the coherency of separating ma-

2. FREQUENCY-DOMAIN BSS
Suppose that P source signals s p (t) are mixed and observedP 
at Q sensors xq (t) =k hqp (k)sp (t − k), wherep=1
hqp (k) represents the impulse response from source p tosensor q. The goal of BSS is to obtain separated signals
y1 (t), . . . , yP (t) that are estimates of the source signals s 1 (t),. . . , sP (t). The separating system typically consists of aset of FIR filters wrq (k) that produces separated signals
Q 
yr (t) = q=1 k wrq (k)xq (t − k).
This paper employs a frequency-domain approach where
frequency responses W rq (f ) of the separating filter w rq (k)are first calculated. By L-point short time DFT, time-domain
signals xq (t) are converted into frequency-domain time-series
signals Xq (f, m), where f = 0, fs /L, . . . , fs (L − 1)/L
(fs : sampling frequency), and m is the frame index. As-

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sume that X(f, m) is a Q-dimensional vector X(f, m) =[X1 (f, m), . . . , XQ (f, m)]T. To obtain frequency responses
Wrq (f ), we solve the ICA problem

Gain

Y(f, m) = W(f )X(f, m),where Y(f, m) = [Y1 (f, m), . . . , YP (f, m)]T and W(f )is a P × Q matrix whose elements are W rq (f ). Yr (f, m) isa frequency-domain representation of y r (t).
The ICA algorithm we use is the information maximization approach [1] combined with the natural gradient [2]. A
separating matrix W is improved by the learning rule

0.5

45

90
Direction (deg)

135

Gain

180

Y 176 Hz1
Y 176 Hz1where µ is a step-size parameter, · denotes the averaging
operator, and Φ(·) is a nonlinear function for a complex signal Yr = |Yr | ej·angle(Yr ) . We use∂
log p(|Yr |) ej·angle(Yr )Φ(Yr ) = −∂|Yr |
as a nonlinear function assuming that the density p(Yr ) is
independent of the angle [10].
An ICA solution has permutation ambiguity: if we permute the rows of W(f ), it is still a solution. Thus, we haveto align the rows of W(f ) so that Y r (f, m) at all frequencies correspond to the same source s p (t). After solving thispermutation problem, we obtain separating filters w rq (k) byapplying inverse DFT to W rq (f ).

2

0.5
0
0

45

90
Direction (deg)

135

180

Fig. 1. Directivity patterns
j2πf c−1 dq cos θp. If we regard θ p
Hqp (f ) = Q
q=1 Wrq (f ) · e
as a variable θ, the formula is expressed as
j2πf c−1 dq cos θ.
(3.2)
Br (f, θ) = Q
q=1 Wrq (f ) · e
It changes according to the direction θ, and thus is called a
directivity pattern.
Figure 1 shows directivity patterns for two sources. The
upper part (3156 Hz) shows that output Y 1 extracts a source
signal originating from around 45 ◦ and suppresses the othersignal coming from around 125 ◦ , which is called a null direction. With a similar consideration regarding Y 2 , we estimate the directions Θ(3156) = [45, 125]T of the target
signals. A simple way to solve the permutation problem isto permute W(f ) at each frequency so that the elements of
Θ(f ) are sorted. However, not every frequency bin gives us
such an ideal directivity pattern. The lower part of Fig. 1 isthe pattern at a low frequency (176 Hz). We see that the null
is not well formed for Y 1 and the null of Y 2 is in an obscuredirection. In fact, we cannot estimate Θ(176) or decide apermutation for this frequency with confidence.
Now we state three problems with this method: 1) directions of arrival cannot be well estimated at some frequencies, especially at low frequencies where the phase difference caused by a sensor spacing is very small, 2) the calculation of null directions by plotting directivity patterns istime consuming, and 3) estimating DOAs from null directions is difficult when there are more than two sources. The
first problem will be solved in Sec. 5, and the other two
problems will be solved in the next two subsections.

3. THE DIRECTION OF ARRIVAL APPROACH
In this section, we discuss how to estimate the directionsof source signals, and align permutations based on them. If
half the minimum wavelength of the source signals is longer
than the sensor spacing (e.g., conditions in Table 1), there
is no spatial aliasing. In most of such cases, each row ofW(f ) forms spatial nulls in the directions of jammer signals and extracts a target signal in another direction. Once
we estimate the directions Θ(f ) = [θ 1 (f ), . . . , θP (f )]T oftarget signals extracted by every row of W(f ), we can alignthe permutations according to Θ(f ).
3.1. Directivity patterns and null directions
We first review the method [6, 7] that solves the permutation problem by plotting the directivity pattern of each output Yr (f, m). Let dq be the position of sensor q (we assume
linearly arranged array sensors), and θ p be the direction ofsource sp (the direction orthogonal to the array is 90 ◦ ). In
beamforming theory [11], the frequency response of an impulse response hqp (t) is approximated as
−1

2

0
0

∆W = µ [I − Φ(Y)YH ] W,

Hqp (f ) = ej2πf c dq cos θp ,Y 3156 Hz
1
Y 3156 Hz

1

3.2. Novel way of estimating DOAs from W −1

(3.1)

where c is the propagation velocity. In this approximation,
we assume a plane wavefront and no reverberation. The
frequency response B rp (f ) from a source s p to a separated
Q
signal yr can be expressed as B rp (f ) =q=1 Wrq (f ) ·Instead of plotting directivity patterns and searching for theminimum as a null direction, we propose a new way of estimating the directions Θ(f ) of source signals directly fromthe inverse W −1 of separating matrix W.

506

After ICA is solved, we can estimate the frequency response of the mixing system by H(f ) = W −1 (f ). An
element Hqp (f ) of the matrix H(f ) may have an arbitrary
amplitude. Since the approximation (3.1) of the mixing system does not suit this situation, we remodel the mixing system with attenuation A qp (real-valued) and phase modulation ejϕp at the origin:−1
Hqp (f ) = Aqp ejϕp ej2πf c dq cos θp .
Here, we should take account of the scaling and permutation
ambiguities of ICA: the columns of H(f ) can have arbitraryscales, and be permuted arbitrarily. The scaling ambiguity
can be eliminated by taking the ratio between two elements
Hqp (f ) and Hq p (f ) corresponding to the same source p:

1
0

50

100

150

0

200

250

v1 1566 Hz
v2 1566 Hz

1

50

100

150

200

250

v 3516 Hz
1v2 3516 Hz

2
1

−1

Hqp /Hq p = Aqp /Aq p ej2πf c (dq −dq ) cos θp .
Then, taking the angle yields a formula for estimating θ p :angle(Hqp /Hq p ).
(3.3)
θp = cos−1
2πf c−1(dq − dq )Because of the permutation ambiguity, θ p may not correspond to sp but to another source signal. This formula does
not incur the computational cost to search a directivity pattern for the minimum. Moreover, it can be applied evenwhen there are more than two sources.

0

50

100
150
Frame index

200

250

Fig. 2. Envelopes at different frequencies
4. THE CORRELATION APPROACH
This section discusses an approach to permutation alignment based on inter-frequency correlations of speech signals [8, 9]. We use the envelope

3.3. Equivalence between θp and a null direction

vrf (m) = |Yr (f, m)|of a separated signal Y r (f, m) to measure correlations. Let
us define the correlation of two signals x(m) and y(m) as

For a two-source case, we prove that θ p calculated by (3.3)
is the same as a null direction that is the minimum of a directivity pattern. When |B r (f, θ)| is minimized, θ corresponds
to a null direction. Let α q = 2πf c−1 dq and f be omitted in(3.2). The value to be minimized isJ(θ) = Br (θ) · Br (θ)∗= (Wr1 ejα1 cos θ + Wr2 ejα2 cos θ ) ·∗ −jα1 cos θ∗ −jα2 cos θ
(Wr1
e
+ Wr2
e
).
Let α = α2 −α1 . The first and second derivatives are∗ −jα cos θdJ),dθ = −α sin θ·2 im(Wr1 Wr2 e
d2 J
dθ 2

v 1562 Hz
1v2 1562 Hzcor(x, y) = [x · y − x · y]/(σx · σy ),∗ −jα cos θ= −α cos θ·2 im(Wr1 Wr2e)∗ −jα cos θe)−α2 sin2 θ·2 re(Wr1 Wr2
where re and im extract the real and imaginary part of a
∗ −jα cos θcomplex, respectively. If angle(W r1 Wr2e) = π,
dJ
d2 J
is
zero
and
is
positive,
and
J(θ)
is
minimized.
Thus,
dθ
dθ 2
the null direction formed by the r-th row of W is given by∗angle(−Wr1 Wr2) = α cos θrnull ⇔angle(−Wr1 /Wr2 ).
θrnull = cos−1
2πf c−1 (d2 − d1 )Considering H21 = −W21 /det(W) and H11 = W22 /det(W),we see that θ1 and θ2null are the same:angle(H21/H11 )angle(−W21/W22 )= cos−1= θ2null .
θ1 = cos−1
2πf c−1 (d2 −d1 )2πf c−1(d2 −d1 )The derivation of θ rnull is based on derivatives. We haveanother derivation of θ rnull based on the graphical interpretation of a directivity pattern [12].

507

where · is an averaging operator and σ is a standard deviation. Based on this definition, cor(x, x) = 1, and cor(x, y) =0 if x and y are uncorrelated. Envelopes have high correlations at neighboring frequencies if separated signals correspond to the same speech signal. Figure 2 shows an example. Two envelopes v 11562 and v11566 , as well as v21562 andv21566 , are highly correlated. Thus, calculating such correlations helps us to align permutations.
Henceforth let π denote a permutation: {1, . . . , P } →{1, . . . , P }. A simple criterion for deciding the permutation
πf of frequency f is to maximize the sum of the correlations
between neighboring frequencies within distance D:f, vπgg (p) ), (4.1)
πf = argmaxπ |g−f |≤D p cor(vπ(p)where πg is the permutation at frequency g. This criterion isbased on local information and has a drawback in that mistakes in a narrow range of frequencies may lead to the complete misalignment of the frequencies beyond that point. To
avoid this problem, the method in [9] does not limit the frequency range in which correlations are calculated. It decides permutations one by one based on the criterion:f, g∈F vπgg (p) ), (4.2)
πf = argmaxπ p cor( vπ(p)where F is a set of frequencies in which the permutation
is decided. This method assumes high correlations of envelopes even between frequencies that are not close neigh-



✏

✏

/* Fix permutations by the DOA approach */for ( ∀ f ) {Θ(f ) = DOA(f, W(f ))Θs (f ) = sort(Θ(f ))}mΘ = Θs (f )f /* averaged directions */F = ∅ /* the set of fixed frequencies */for ( ∀ f ) {if ( confident(Θ(f ), mΘ , W(f )) ) {πf = getPermutation(Θ(f ))F = F ∪ {f }}}/* Fix permutations by neighboring correlations */while ( ∃ f ∈ F ) {for ( ∀ f ∈ F ) {f, vπgg (p) )cf = maxπ |g−f |≤D,g∈F p cor(vπ(p)f
πf = argmaxπ |g−f |≤D,g∈F p cor(vπ(p) , vπgg (p) )}i = argmaxf cf
F = F ∪ {i}ci = 0}✒/* Fix permutations by harmonic structure */for ( ∀ f ∈ F ) {Ha = setOfHarmonics(f );fcf = maxπ g∈Ha∩F p cor(vπ(p), vπgg (p) )if ( cf ≥ threshold ) {f, vπgg (p) )πf = argmaxπ g∈Ha∩F p cor(vπ(p)F = F ∪ {f }}}✒5. NEW ROBUST AND PRECISE METHOD

5.2. Exploiting the harmonic structure of signals✑

Fig. 4. Pseudo-code for the harmonic method

Our method benefits from both advantages: the robustnessof the DOA approach and the preciseness of the correlation
approach. Figure 3 shows the pseudo-code.
We first fix permutations at some frequencies where theconfidence of the DOA approach is sufficiently high. The
procedure confident decides whether the confidence is highenough. Our criteria for the decision are: 1) the number of
estimated directions is the same as the number of sources,
2) the directions Θ(f ) do not differ greatly from the averaged directions m Θ , 3) the SNR calculated by the frequency✑ responses Br (f, θp ) for each direction is sufficiently large.
Then, we decide the permutations for the remaining freFig. 3. Pseudo-code for the integrated method
quencies based on neighboring correlations without changing the permutations fixed by the DOA approach. The perbors. This assumption is not satisfied for all pairs of fremutations are decided in order of the sum of correlations
quencies, although a high correlation can be assumed forwith fixed frequencies g ∈ F within distance |g − f | ≤ D.
a fundamental frequency and its harmonics. As shown inThis method does not result in a large misalignment as
Fig. 2, vr1566 and vr3516 do not have a high correlation.
long as the permutations fixed by the DOA approach are
Therefore, this method still has a drawback in that permutacorrect. Moreover, the correlation part compensates for thetions may be misaligned at many frequencies.
lack of preciseness of the DOA approach.

In this section, we consider integrating the two approaches
discussed above to solve the permutation problem robustly
and precisely.

The method proposed above works very well in many cases.
However, there is a case where the DOA approach does notprovide any fixed permutation with confidence in a certain
range of frequencies. This occurs particularly at low frequencies where it is hard to estimate DOAs as discussed inSec. 3. In such a case, the proposed method has to align permutations for the range solely through the use of neighboring correlations, and may yield consecutive misalignments.
To cope with this problem, we exploit the harmonic structure of a speech signal. As alluded to in Sec. 4, there arestrong correlations among the envelopes of a fundamental
frequency f and its harmonics 2f, 3f and so forth. Suppose that the permutation is not fixed at frequency f but
fixed at frequencies 2f and 3f . If the correlationf, vπgg (p) )cf = maxπ g=2f,3f p cor(vπ(p)5.1. Integration of the two approaches
Let us begin by reviewing the characteristics of the two approaches.
robustness The direction of arrival (DOA) approach is robust since a misalignment at a frequency does not affect other frequencies. The correlation approach is
not robust as discussed in Sec. 4.
preciseness The DOA approach is not precise since the evaluation is based on the approximation of a mixing system as explained in Sec. 3. The correlation approachis precise as long as signals are well separated by ICA
since the measurement is based on separated signals.

is sufficiently large, we can fix the permutation at frequency

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Table 1. Experimental conditions
18

6 sec
120◦ and 50◦ (2 sources)d = 4 cm
TR = 300 ms
fs = 8 kHz
L = 2048∆f = fs /L = 3.90625 Hz
D = 3 · ∆fΦ(Yr ) = ej·angle(Yr )1614
SNR (dB)

Length of source signal
Direction of sources
Distance between 2 sensors
Reverberation time
Sampling rate
Frame size of DFT
Frequency resolution
Distance to take correlation
Nonlinear function

12
10
8
6

Average for 12 cases
9th case
Other 11 cases

4

f with confidence. Figure 4 shows the pseudo-code for the
harmonic method. The procedure setOfHarmonics provides
a set of harmonic frequencies of f , and threshold controls
the confidence.
To incorporate the above idea, the final version of our
method fixes all permutations with four steps:
1. By the DOA approach (the upper part of Fig. 3)

2

D

C2

C1

D+C1

D+C1+Ha Optimal

Fig. 5. Overall results
upper bounds of performance.
The performance with “D” is stable, but not sufficient.
The results with “C1” and “C2” are not stable and sometimes very poor, although most of the time they are very
good. Both integrated methods “D+C1” and “D+C1+Ha”
offer stable and good results. Especially, the method exploiting the harmonic structure “D+C1+Ha” offers almost
the same results as “Optimal”.
Now we examine the effectiveness of the integrated methods by looking at the 9th combination of speech signals in
detail. Figure 6 shows the SNRs at each frequency for “C1”,“D+C1” and “D+C1+Ha”. We see a large region (from 450
Hz to 1400 Hz) of permutation misalignments for the “C1”case, where permutations were decided only with neighboring correlations. We see in Fig. 7 that the correlations
around 1400 Hz do not have a large absolute value, and thecriterion based on (4.1) does not provide a clear-cut decision. Therefore, there was a decent possibility that the permutations were misaligned around 1400 Hz in this case.
With the “D+C1” method, the misalignments of the region (from 450 Hz to 1400 Hz) were corrected. This is because the DOA approach in the first step provided correct
permutations for some frequencies in the region. Figure 8
shows the DOA estimations for each frequency with confidence. We see many estimations from 450 Hz to 1400 Hz.
However, there was no DOA estimation with confidence at
frequencies lower than 250 Hz. This is why consecutive
misalignments occurred even for “D+C1”. As shown at thebottom of Fig. 6, the misalignments were corrected with the“D+C1+Ha” method. This shows the effectiveness of exploiting the harmonic structure for low frequencies.

2. By neighboring correlations (the lower part of Fig. 3)
with the exception that the while loop terminates if
the maximum of c f is too small.
3. By the harmonic method (Fig. 4)
4. By neighboring correlations (the lower part of Fig. 3)
again without the exception.
There are two important points to note as regards the final
version. The first is that the method becomes more robust
because of the exception in step 2. We do not fix the permutations for consecutive frequencies without high confidence.
The second point is that step 3 works well only if most of
the other permutations are fixed. This means that the harmonic method alone does not work well and we need steps
1 and 2 to fix most of the permutations.
6. EXPERIMENTAL RESULTS
We performed experiments to separate speech signals in areverberant environment whose conditions are summarizedin Table 1. We generated mixed signals by convolving aspeech signal sp (t) and an impulse response h qp (t) so that
we could calculate signal-to-noise ratios (SNRs) by
10 log[ r=p t yrp (t)2 ] − 10 log[ r=p t yrp (t)2 ],Q 
where yrp (t) = q=1 k wrq (k)xqp (t − k) and xqp (t) =k hqp (k)sp (t − k). Figure 5 shows the overall results ofthe experiments. We separated 12 combinations of speech
signals with 6 different methods for the permutation problem: the DOA approach “D”, the correlation approach basedon (4.1) “C1”, the correlation approach based on (4.2) “C2”,the first version of the integrated method (Fig. 3) “D+C1”,
the final version of the integrated method (Figs. 3 and 4)“D+C1+Ha”, and the optimal permutation “Optimal” basedon maximizing the SNR at each frequency. Although “Optimal” is not a realistic solution, we used it to estimate the7. CONCLUSION
We proposed a robust and precise method for solving the
permutation problem. Our method integrates two previous approaches: the DOA approach and the correlation approach. The criterion of the DOA approach is directions

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−20

SNR (dB)

0

C1
500

1000

1500

2000

2500

3000

3500

4000

20

0.6
0.4
0.2
with f+1∆fwith f+2∆fwith f+3∆f

0

0

−0.2

D+C1

−20
0

SNR (dB)

0.8

0Correlation

SNR (dB)

1
20

500

1000

1500

2000

2500

3000

3500

1350

4000

20

1400
Frequency

1450

1500

Fig. 7. Correlations with neighboring frequencies

0
D+C1+Ha
−20
0

500

1000

1500 2000 2500
Frequency

3000

3500

DOA of the first signal
DOA of the second signal

160

4000

140

Fig. 6. SNRs at each frequency
Degree120

which are absolute. This makes the approach robust. By
contrast, the criterion of the correlation approach is calculated from the separated signals themselves. This makes
the approach precise. Our proposed method benefits fromboth advantages. In our experiments, the proposed method
solved permutation problems almost perfectly under conditions whereby two sources were mixed in a room where T R= 300 ms. Future work will focus on experiments for caseswith more than two sources.

100
80
60
40
0

500

1000

1500 2000 2500
Frequency

3000

3500

4000

Fig. 8. DOA estimations with confidence[6] S. Kurita, H. Saruwatari, S. Kajita, K. Takeda, and F. Itakura,“Evaluation of blind signal separation method using directivity pattern under reverberant conditions,” in Proc. ICASSP
2000, June 2000, pp. 3140–3143.

8. ACKNOWLEDGEMENTS
We thank Prof. Hiroshi Saruwatari for valuable discussions
and for providing us with the impulse responses used in theexperiments, Dr. Tomohiro Nakatani for valuable discussions on the harmonic structure of speech, and Dr. Shigeru
Katagiri for continuous encouragement.

[7] M. Z. Ikram and D. R. Morgan, “A beamforming approach topermutation alignment for multichannel frequency-domain
blind speech separation,” in Proc. ICASSP 2002, May 2002,
pp. 881–884.
[8] J. Anemüller and B. Kollmeier, “Amplitude modulation
decorrelation for convolutive blind source separation,” inProc. ICA 2000, June 2000, pp. 215–220.

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[9] N. Murata, S. Ikeda, and A. Ziehe, “An approach to blindsource separation based on temporal structure of speech signals,” Neurocomputing, vol. 41, no. 1-4, pp. 1–24, Oct. 2001.

[2] S. Amari, “Natural gradient works efficiently in learning,”
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[10] H. Sawada, R. Mukai, S. Araki, and S. Makino, “Polar coordinate based nonlinear function for frequency-domain blindsource separation,” in Proc. ICASSP 2002, May 2002, pp.
1001–1004.

[4] L. Parra and C. Spence, “Convolutive blind separation of
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[11] B. D. Van Veen and K. M. Buckley, “Beamforming: a versatile approach to spatial filtering,” IEEE ASSP Magazine, pp.
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[5] F. Asano, S. Ikeda, M. Ogawa, H. Asoh, and N. Kitawaki,“A combined approach of array processing and independent
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[12] H. Sawada, R. Mukai, S. Araki, and S. Makino, “A robust
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