Comparison of superimposition andsparse models in blind source separation
by multichannel Wiener filter
Ryutaro Sakanashi∗ , Shigeki Miyabe† , Takeshi Yamada‡ and Shoji Makino§∗ Graduate School of Systems and Information Engineering, University of Tsukuba, Japan†§ Life Science Center of Tsukuba Advanced Research Alliance, University of Tsukuba, Japan†‡§ Faculty of Engineering, Information and Systems, University of Tsukuba, Japan

E-mail: ∗ sakanashi@mmlab.cs.tsukuba.ac.jp, {† miyabe,§ maki}@tara.tsukuba.ac.jp, ‡ takeshi@cs.tsukuba.ac.jp

Abstract—Multichannel Wiener filter proposed by Duonget al. can conduct underdetermined blind source separation(BSS) with low distortion. This method assumes that the observed
signal is the superimposition of the multichannel source images
generated from multivariate normal distributions. The covariancematrix in each time-frequency slot is estimated by an EM
algorithm which treats the source images as the hidden variables.
Using the estimated parameters, the source images are separated
as the maximum a posteriori estimate. It is worth nothing that
this method does not assume the sparseness of sources, which
is usually assumed in underdetermined BSS. In this paper weinvestigate the effectiveness of the three attributes of Duong’s
method, i.e., the source image model with multivariate normal
distribution, the observation model without sparseness assumption, and the source separation by multichannel Wiener filter. We
newly formulate three BSS methods with the similar source image
model and the different observation model assuming sparseness,
and we compare them with Duong’s method and the conventional
binary masking. Experimental results confirmed the effectivenessof all the three attributes of Duong’s method.

I. I NTRODUCTION
Source separation is an essential technology for handsfree speech recognition and understanding of the sound environment by computer in real environments. In past decades
there has been a rapid progress in the research field of blindsource separation (BSS), which does not require prior information, such as source position and voice activity detection[1]. Independent component analysis [2] is representative of
BSS, but this method cannot be applied to underdetermined
conditions where the number of sources is larger than that of
microphones. Since stereo recording devices are used widely,
underdetermined BSS technique is demanded so that we canapply BSS to two-channel recordings of arbitrary number ofsources.
A typical approach to underdetermined BSS is to assume
sparseness among sources [3]. Sparseness is the nature ofsignals whose energy is concentrated in some frequencies andalmost zero in the other frequencies at each short time period.
Thus the BSS methods of this type assume that each timefrequency slot of speech mixture is dominated by one of thesources. The most representative of those methods is timefrequency masking [3], which identifies the dominant source ineach time-frequency slot, and separate each source by masking
the time-frequency slots where the other sources are dominant.
Such identification is done by clustering the multichannel
features such as inter-channel level or phase differences withk-means clustering [4], EM algorithm [5], etc.
There have been various works related with the timefrequency masking. Araki et al. proposed combination withminimum variance distortionless response beamformer [6]to reduce musical noise caused by masking [7]. Also, Isoet al. [8] used the time-frequency binary masking [9] for
initialization of an initialization-sensitive BSS method using
multichannel Wiener filter proposed by Duong et al. [10],whose details are discussed in this paper.
The problem of the time-frequency binary masking is degradation of speech quality caused by modeling mismatch of thesparseness assumption, which is satisfied approximately but
not perfectly. While most of the underdetermined BSS techniques assume sparseness of speech, Duong et al. proposed
low-distortion underdetermined BSS using the multichannel
Wiener filter without the sparseness assumption [10]. Thismethod models source image, which is the multiplication ofeach source and its related transfer functions. The source
image is modeled by multivariate normal distribution whoseparameterization assumes stationarity of the transfer systemand quasi-stationarity of the sources. The multichannel Wiener
filter is designed using the parameters estimated by an EM
algorithm. The details of the underlying probabilistic model is
not described in the original paper [10], but as pointed in [11],the source images are dealt as hidden variables, and the sumof the hidden source images are constrained to equal to theobserved signal. With this constraint, superimposition of themultiple sources in the mixed signals is modeled effectively.
The obtained multichannel Wiener filtering corresponds tomaximum a priori (MAP) estimate of the source images. The
detailed probabilistic modeling is described in this paper.
As described above, Duong’s method has various interesting
attributes. However, it has not been well-investigated which
factor is effective for the high-quality source separation. In thispaper, we analyze in detail some of these attributes which areconsidered to be the reason why Duong’s method can achieve

high quality source separation. In particular, we focus attention
on the following three, i.e., 1) the source image model bymultivariate normal distribution, 2) source separation by themultichannel Wiener filter, and 3) use of EM algorithm developed without assumption of sparseness. In order to confirm
the effectiveness of these factors, we formulate alternative
underdetermined BSS methods with the same source image
model as Duong’s method by multivariate normal distribution
and the different observation model of sparse source image occurrences but superimposition of multiple source images. We
design the following three BSS schemes using the estimatedparameters: A) separation using the posterior probability thatsource image is active, B) separation by maximum likelihood
binary mask which corresponds to maximum a posteriori
estimation of sparse source images, and C) separation by
an alternative multichannel Wiener filter designed using the
expectation of the sources in the sparse model. We comparedthese three methods with Duong’s method and MENUET
which is a general binary masking method, and discuss theperformances from the following three viewpoints; i) validityof the source image model with multivariate normal distribution by comparing the typical binary mask MENUET andsparse binary mask formulated, ii) effectiveness of separationby the multichannel Wiener filter by comparing the above
three methods A), B), and C), and iii) effectiveness of source
image superposition model not assuming the sparseness bycomparing the newly formulated sparse Wiener filter andDuong’s method. As a result, effectiveness of all the above
three attributes of Duong’s method has been confirmed.
In Section 2, we describe the observation model dealt in thispaper. We review Duong’s method in Section 3. The sparsemodel which is a comparative approach is formulated in Section 4. In Section 5, we perform a comparison experiment ofsource separation to confirm the validity of Duong’s method.
Finally, Section 6 concludes this paper.

and the source image cj (n, f ) is assumed to be generated from the zero-mean multivariate normal distribution
Nc (cj (n, f ); 0, Rcj (n, f )):II. O BSERVATION M ODELp(cj (n, f )|θ(f )) = Nc (cj (n, f ); 0, Rcj (n, f )).

Here we describe the observation model of Duong’s method
and sparse models we newly formulate in this paper. First, theobserved signal is expressed asHere the probability density function Nc (c; µ, Σ) of the Idimensional random variable c arose from the multivariate
normal distribution with the mean µ and the covariance matrix
Σ is given byT

x(n, f ) = [x1 (n, f ), ..., xI (n, f )]≈J∑

hj (f )sj (n, f ),

(1)

j=1
T

hj (f ) = [h1j (f ), ..., hIj (f )] .

(2)

where sj (n, f ) is the source signal, hj (f ) is the transfer
function vector of the j-th source whose entry hij represents
the transfer function from the j-th source to the i-th microphone, J is the number of the sources, I is the number of the
microphones, and (·)T is the transpose. Also, we define thecomponent of the j-th source reaching at the microphones as
the source image cj (n, f ), which contains spatial informationsuch as reverberation, expressed as
cj (n, f ) = hj (f )sj (n, f ).

(3)

In this paper, we discuss the problem to estimate the source
image cj (n, f ) of each source.
III. M ULTICHANNEL W IENER F ILTER B Y D UONG et al.
Here we explain about the multichannel Wiener filter by
Duong et al., from the viewpoint of maximum likelihood
estimation problem, which was not explicitly formulated in[10].
A. Probabilistic Modeling of Source Image
Originally, the source image can be expressed as (3), but
assuming the time variation of the transfer function effectedby the longer reverberation than the length of the analysis
frames, Eq. (3) is rewritten as
cj (n, f ) = hj (n, f )sj (n, f ),(4)

where hj (n, f ) is defined as the time-varying transfer function. For the probabilistic modeling of the source image
cj (n, f ), two assumptions are introduced. First, under theassumption that the sources does not change the position, we
regard the spatial correlation to be stationary and each of the jth source is given the time-invariant spatial correlation matrix
Rj (f ). Second, assuming speech is non-stationary but quasistationary, we denote the time-varying variance of the j-thsource with νj (n, f ). Under these assumptions, the covariancematrix Rcj (n, f ) of cj (n, f ) is regarded as the product of
Rj (f ) and νj (n, f ) as
Rcj (n, f ) = νj (n, f )Rj (f ),

(5)

(6)

p(c; µ, Σ) = Nc (c; µ, Σ)
(
)
1
≜ I
exp −(c − µ)H Σ−1 (c − µ) , (7)π det(Σ)
where det(·) is the determinant of the square matrix, and(·)H is the complex conjugate transpose. Note that the spatial
correlation matrix Rj (f ) should be expressed as Rj (f ) =hj (f )hH
j (f ) with its rank one when time-invariant transfer
function hj (n, f ) is assumed. However, considering the effectsof such long reverberation which does not fit in the analysis
frames, Rj (f ) is assumed to be full-rank considering the time
variation of the transfer function vector hj (n, f ). Maximum
likelihood estimation of the generative model of cj (n, f ) andx(n, f ) is described in the following section.

Mj (n, f )B. Derivation of EM Algorithm
Assuming the observed signal x(n, f ) appears as the superimposition of the source images cj (n, f ), j = 1, ..., J asx(n, f ) =J∑

cj (n, f )⇔ cj (n, f ) = x(n, f ) −H−1+ Rcj (n,f )R−1
x (n,f )x(n,f )x (n,f )Rx (n,f )Rcj (n,f ).
(13)

The EM algorithm to estimate the parameters is obtained by
setting the partial differential of the Q-function to be zero.

j=1
J−1
∑

= Rcj (n, f ) − Rcj (n, f )R−1
x (n, f )Rcj (n, f )cj (n, f ),

(8)

j=1

the joint probability of x(n, f ) and cj (n, f ) is derived from
Eqs. (7) and (8) as

C. Parameter Estimation and Source Separation
First, set appropriate initial values of νj (n, f ) and Rj (f ),and initialize Rcj (n, f ) and Rx (n, f ) as follows.
Rcj (n, f ) = νj (n, f )Rj (f ),p(C(n, f ), x(n, f )|θ(f ))
=

J−1
∏
j=1

(
)
Nc cj (n, f ); 0, Rcj (n, f )· Nc x(n, f ) −J−1∑

Rx (n, f ) =Rcj (n, f ).

(14)
(15)

j=1


cj (n, f ); 0, RcJ (n, f ) ,

J
∑

(9)

After the above initialization, the E-step estimates the aposteriori covariance matrix Rcj (n, f ) as

j=1

where C(n, f ) = {c1 (n, f ), ..., cj (n, f ), ..., cJ (n, f )}. Moreover the likelihood of the observation is obtained as∫p(x(n, f )|θ(f )) = p(C(n, f ), x(n, f )|θ(f ))dCJ (n, f )= Nc (x(n, f ); 0, Rx (n, f )),(10)

Wj (n, f ) = Rcj (n, f )R−1
x (n, f ),ĉj (n, f ) = Wj (n, f )x(n, f ),R̂cj (n, f ) = ĉj (n, f )ĉHj (n, f ) + (I − Wj (n, f ))Rcj (n, f ).
(18)
The update of the M-step is given by

by marginalization of all the source images, where CJ (n, f ) ={c1 (n, f ), ..., cJ−1 (n, f )}, and Rx (n, f ) is covariance matrixof observation matrix x(n, f ), given as
Rx (n, f ) =J∑νj (n, f )Rj (f ).

1tr(R−1j (f )R̂cj (n, f )),
I
N
1
1 ∑
R̂c (n, f ),Rj (f ) =N n=1 νj (n, f ) jνj (n, f ) =(11)

Rcj (n, f ) = νj (n, f )Rj (f ),j=1With these probability density functions, the Q-function isformulated as()· log p C(n, f ), x(n, f )|θ̄(f ) dCJ (n, f )N ∫∑p(C(n, f ), x(n, f )|θ(f ))p (x(n, f )|θ(f ))n=1()· log p C(n, f ), x(n, f )|θ̄(f ) dCJ (n, f )

N
J
∑
∑
−IJ log π − I=
log ν̄j (n, f )νj (n, f )Rj (f ),

(20)
(21)
(22)

j=1

p(cj (n, f )|x(n, f ), θ(f ))(= Nc cj (n, f ); Rcj (n, f )R−1
x (n, f )x(n, f ),=n=1

Rx (n, f ) =

J
∑

(19)

where I is the I × I identity matrix. After the convergence of the iteration of the E- step and M-step,
the source image is estimated by using the parameters.
Marginalization of p(C(n, f )|x(n, f ), θ(f )) about c1 (n, f ),..., cj−1 (n, f ), cj+1 (n, f ), ..., cJ (n, f ) gives

Q(θ(f ), θ̄(f ))N ∫∑=p (C(n, f )|x(n, f ), θ(f ))
n=1

(16)
(17)

(

)
(
)−1 )−1
,
R−1
(n,
f
)
+
R
(n,
f
)
−
R
(n,
f
)
x
cj
cj

(23)

j=1

−

J
∑

log det(R̄j (f ))

j=1
J
∑

and each source image can be estimated by the multichannel
Wiener filter represented as

1
−
ν̄
(n,
f)
j=1 j


)
−1
Tr Mj (n, f )R̄j (f )  ,(cj (n, f ) = Rcj (n, f )R−1
x (n, f )x(n, f ),
(12)

(24)

which corresponds both to the expectation and the MAP
estimate.

IV. F ORMULATION O F T HE S EPARATION M ETHOD A ND
O BSERVATION M ODEL B Y A SSUMPTION O F S PARSENESS
In this section, for the comparison to verify the effectivenessof the model of Duong’s method, we formulate new source
separation methods which have the similar source image modelto Duong’s method but with an alternative assumption that theobservation is generated from the sparse distribution of thesource images.
A. Problem Establishment
We assume that in each time-frequency slot only one sourceis active, and we denote the index of the active source inthe time-frequency slot (n, f ) as z(n, f ). We define the priorprobability of the j-th source to be active asp(z(n, f ) = j) = µj (f ),J∑µj (f ) = 1.

(25)

j=1

Under this assumption, the observed signal x(n, f ) is modeledasx(n, f ) = cz(n,f ) (n, f ),(26)
say, the sparse observation model. Similarly to Duong’s
method, the generation of the source image cj (n, f ) of thismodel is represented asp (cj (n, f )|z(n, f ) = j, θ(f ))= Nc (cj (n, f ) ; 0, νj (n, f ) Rj (f )) ,(27)

by assuming that the covariance matrix Rcj (n, f ) is givenby the product of the time-varying variances νj (n, f ) andthe time-invariant spatial correlation matrix Rj (f ), andthe source image cj (n, f ) follows the normal distribution
Nc (x(n, f ); 0, νj (n, f )Rj (f )). We represent the model parameters by θ(f ) = {Rj (f ), νj (n, f ), µj (f ), for j = 1, ..., J}.
B. Model Parameter Estimation∏
Likelihoodn p(x(n, f )|θ(f )) of the observed signalx(n, f ) to maximize the model parameters θ(f ) are definedas∏p (x(n, f )|θ(f ))
n

=

J
∏∑

µj (f )Nc (x(n, f ); 0, νj (n, f )Rj (f )) .

and satisfiesJ∑

mj (n, f ) = 1.

Qf (θ(f ), θ̄(f ))∑=mj (n, f ) log µj (f )Nc (x(n, f ); 0, νj (n, f )Rj (f ))n,j(mj (n, f ) log µj (f ) − I log π − I log νj (n, f )n,j)x(n, f )H R̄−1
j (f )x(n, f )− log det R̄j (f ) −,νj (n, f )
(

)

(30)

(31)

j=1

The EM algorithm is derived by setting the partial derivativeof this Q-function by each parameter to be zero.
In the M-step, the variances νj (n, f ), the spatial covariancematrix Rj (f ) and the prior probability µj (f ) are updated as
xH (n, f )R−1
j (f )x(n, f ),I∑ mj (n,f )H
n νj (n,f ) x(n, f )x (n, f )∑Rj (f ) ←,mj (n, f )n∑
mj (n, f )µj (f ) ← ∑ n.
n,j ′ mj ′ (n, f )νj (n, f ) ←

(32)
(33)
(34)

In E-step, the posterior probabilities are updated.
mj (n, f ) ← ∑µj (f )Nc (x(n, f ); 0, νj (n, f )Rj (f )).
′′′j ′ µj (f )Nc (x(n, f ); 0, νj (n, f )Rj (f ))(35)

C. Source Separation
Here we formulate three source separation schemes using
the estimated parameters.
1) Sparse soft mask using the posterior probability: The
first scheme is to use the posterior probability mj (n, f )directly as the mask.
cj (n, f ) = mj (n, f )x(n, f ).

(36)2) Sparse binary mask: The second scheme is using thebinary mask Mj (n, f ) to regard the observation as the exclusive occurrence of one source with the maximum posterior
probability mj (n, f ), given by1 if j ∗ = argmax mj (n, f )jMj (n, f ) =,(37)
0 otherwise
cj (n, f ) = Mj (n, f )x(n, f ),This maximum likelihood estimation can be obtained by EM
algorithm to maximize the Q-function with the hidden variablez(n, f ) is given below.

=

mj (n, f ) = p (z(n, f ) = j|x(n, f ), Rj (f )νj (n, f )) ,

(28)

n j=1

∑

where mj (n, f ) behaving as the time-frequency soft mask is
the posterior probability of z(n, f ) given as(38)

which corresponds to the MAP estimate of the source image
cj (n, f ) to maximize p(c1 (n, f ), ..., cJ (n, f )|x(n, f ), R1 (f ),..., RJ (f ), ν1 (n, f ), ..., νJ (n, f )).
3) Sparse Wiener filter: In the third scheme, we design a multichannel Wiener filter similar to the one inDuong’s method, using the expectation of each variancemj (n, f )νj (n, f );∑Rx (n, f ) =mj (n, f )νj (n, f )Rj (f ),
(39)
j

(29)

Rcj (n, f ) = mj (n, f )νj (n, f )Rj (f ),ĉj (n, f ) = Rcj (n, f )R−1
x (n, f )x(n, f ).

(40)
(41)

In the next section, we conduct experiments to compare
Duong’s method without the assumption of sparseness togeneral binary mask MENUET and three methods with theassumption of sparseness formulated in this section.

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V. E XPERIMENTS
A. Experimental Condition
We compare the performances of MENUET the generalbinary masking method, Duong’s method and the three source
separation methods derived from the sparseness assumptions.
We evaluated the source separation performance of six mixtures of three speeches chosen from four male and four femaleutterances as listed in Table I.
We used two-channel microphone array with the interelement spacing of 2.15 cm. The distances between the sources
and the microphone array are 1.0 m. The horizontal angles ofthe sources are −40◦ , 0◦ and 30◦ . The sampling frequency is
16 kHz, and the frame size is 1024 samples. The EM algorithms are terminated after 50 iterations. Objective evaluation
scores of separation performance are the following 4 distortion
measures proposed in [12].
• SDR (Signal to Distortion Ratio): comprehensive distortion• ISR (Source Image to Spatial distortion Ratio): linear
distortion• SIR (Source to Interference Ratio): distortion by the rest
reduction of the other source• SAR (Sources to Artifacts Ratio): non-linear distortion
The unit is dB, and the higher values show the better performance. We used the medhod in [8] to generate the initialvalues and to align the permutation for Duong’s method andthe formulated methods. The other experimental conditions arelisted in Table II.
TABLE IC OMBINATION OF SIGNAL MIXTURE

1
2
3
4
5
6

−40◦
Male1
Female1
Female2
Female1
Male4
Male1

0◦
Female2
Male2
Female4
Male1
Female2
Male3

30◦
Female3
Female4
Male3
Male4
Male2
Female3

TABLE IIE XPERIMENTAL CONDITION
sourcessource direction
number of microphones
distance between the two microphones
FFT size
sampling frequency
iteration of EM algorithm
performance measures

SiSEC 2011−40◦ ，0◦ ，30◦
22.15 cm
1024 point
16000 kHz
50 times
SDR, ISR, SIR, SAR

^Z

/^Z

^/Z

^Z

Fig. 1. Experimental result

B. Discussion
The experimental results are shown in Fig. 1. Duong’s
method marked good score in SIR, which indicates degree ofseparation, and the highest score in SDR, which measures theoverall quality of the separated signal. Thus Duong’s method
is the best in this comparison. We discuss in detail in thefollowing.
In order to verify the validity of the multivariate modeling ofsource image, we discuss the comparison between MENUET
and the sparse binary mask. Since all the evaluation scores are
similar, the multivariate model seems to have no remarkableside effect and its validity is confirmed.
In order to verify which source separation scheme is effective, we compare the three methods formulated by the sparsemodel. The sparse soft mask has the lowest scores. And while
the sparse binary mask marks higher scores in ISR and SIR
than the sparse Wiener filter, the sparse Wiener filter marked
the higher scores in SDR and SAR. The assumption of thebinary mask, which regards that in each time-frequency slot
the estimated dominant source solo generates the observed
signal, is effective for yielding high SIR, the source separation
score. However, this assumption is not strictly satisfied, and theresultant error causes high distortion, as can be seen in the low
SAR of the sparse binary mask. Because of the high distortion,
SDR of the binary mask, which measures the overall qualityof the separated signal, is not very high in spite of the high
SIR. In contrast, the sparse Wiener filter has low distortion
as can be seen in its high SAR. Thus SDR of the sparse
Wiener filter is higher than that of the sparse binary maskeven with the lower SIR, and the sparse Wiener filter has thebetter source separation performance than the sparse binaryand soft maskings. Therefore we can conclude that Wiener
filter performs the high-quality source separation with lower
distortion than masking.
Moreover, in order to verify the effectiveness of source
image superposition model not assuming the sparseness, we
compare the performances of the formulated sparse Wiener
filter and Duong’s method. These two methods has similar
SAR and causes similarly low amounts of distortion. However,Duong’s method marks higher scores in SIR and ISR, andas a result Duong’s method has higher SDR than the sparse
Wiener filter. Thus we can say that Duong’s method has
better source separation performance than the sparse Wiener
filter, and the best one among those compared methods. As
we discussed in Sect. III-C, Duong’s method estimates theparameters of the multichannel Wiener as the optima of theassumed probabilistic model in two senses of expectation and
MAP. However, the parameters of the sparse Wiener filter isestimated indirectly in its probabilistic model combining theestimations of soft mask and the covariance matrices, as shownin Sect. IV-C. Therefore, we confirmed the effectiveness of theparameter estimation by Duong’s method without assuming
sparseness both theoretically and experimentally.
As a result of the above comparisons, we can conclude thatmultichannel Wiener filter causes low distortion, and by effective estimation of the parameters employing the multivariate
model without assuming sparseness, Duong’s method performs
high-quality and high-performance source separation.
VI. C ONCLUSION
To investigate the factor of the low-distortion source separation by Duong’s method, we analyzed the three characteristic
attributes of Duong’s method, i.e., source image model withmultivariate normal distribution, separation by multichannel
Wiener filter and source observation model of superimpositionof source images without assuming sparseness among sources.
For this analysis, we formulated three alternative BSS methods
by sparse soft mask, sparse binary mask and sparse Wiener
filter, and the performances are compared with the generalbinary masking method MENUET and Duong’s method. Allof these methods are derived from the same source image
model of Duong’s method and different observation model
where the sources are not superimposed but a single source
image appears in each time-frequency slot assuming source
sparseness. As a result, effectiveness of all the above three
attributes of Duong’s method has been confirmed.
ACKNOWLEDGMENT
This work was supported by MEXT/JSPS KAKENHI Grant
Number 23240023. We would like to thank Dr. Masato Togami
of Central Research Laboratory, Hitachi, Ltd., who gave us
fruitful suggestions about the probabilistic model of Duong’s
method.
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