2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)UNDERDETERMINED BLIND SEPARATION AND TRACKING OF MOVING SOURCESBASED ON DOA-HMM
Takuya Higuchi† , Norihiro Takamune† , Tomohiko Nakamura† and Hirokazu Kameoka†‡†

Graduate School of Information Science and Technology, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan‡NTT Communication Science Laboratories, NTT Corporation,
3-1 Morinosatowakamiya, Atsugi, Kanagawa 243-0198, Japan
ABSTRACTThis paper deals with the problem of the underdetermined blindseparation and tracking of moving sources. In practical situations, sound sources such as human speakers can move freely and soblind separation algorithms must be designed to track the temporalchanges of the impulse responses. We propose solving this problem through the posterior inference of the parameters in a generativemodel of an observed multichannel signal, formulated under the assumption of the sparsity of time-frequency components of speech
and the continuity of speakers’ movements. Specifically, we describe a generative model of mixture signals by incorporating a generative model of a time-varying frequency array response for eachsource, described using a path-restricted hidden Markov model (HMM). Each hidden state of the present HMM represents the direction of arrival (DOA) of each source, and so we call it a “DOAHMM.” Through the posterior inference of the overall generativemodel, we can simultaneously track the DOAs of sources, separate
source signals and perform permutation alignment. The experimentshowed that the proposed algorithm provided a 6.20 dB improvement compared with the conventional method in terms of the signalto-interference ratio.
Index Terms— Underdetermined blind separation, moving
sources, direction of arrival, hidden Markov model, variational
inference
1. INTRODUCTION
Blind source separation (BSS) refers to a technique for separatingout individual source signals from microphone array inputs when
the transfer characteristics between the sources and microphones are
unknown. The best-known commercial application of BSS techniques is their use in teleconferencing systems. In practical situations, sound sources and microphones are likely to move duringsound recording, and the transfer characteristics between the sources
and the microphones can change accordingly. Many conventional BSS algorithms have been developed on the assumption that the array
response is time-invariant, and thus they do not work satisfactorily
when the sources or microphones move. This paper deals with a BSS problem in a situation where the array response has the potential
to change over time according to the movements of the sources.
To solve BSS problems, it is generally necessary to make someassumptions about the sources, and formulate an appropriate optimization problem based on criteria designed according to thoseassumptions. For example, if the observed signals outnumber thesources, we can employ independent component analysis (ICA) [1]by assuming that the sources are statistically independent of eachother. However, in an underdetermined case, the independence assumption is too weak to allow us to determine a unique solution andso directly applying ICA will not work well.

978-1-4799-2893-4/14/$31.00 ©2014 IEEE

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One successful approach for underdetermined BSS is to utilize
the fact that the time-frequency components of speech are near zero at most of the time-frequency points [2–9]. This implies that the
time-frequency components of speech rarely overlap each other evenwhen multiple speakers are speaking simultaneously. Hence, themain focus of this approach is how to design a time-frequency maskwith which we can extract only the components of target speech fromthe mixture. To exploit the sparse nature of speech, we must convert
observed signals to a time-frequency representation. In contrast to atime domain formulation of the BSS problem, a time-frequency domain formulation requires us to solve an additional problem called
the permutation alignment problem. That is, we must group together
the separated components of different frequency bins that are considered to originate from the same source in order to construct a
separated signal. To solve the BSS and permutation alignment problems simultaneously, we have previously proposed constructing ahierarchical generative model consisting of generative models of anobserved signal and the array response for each source. This model
has allowed us to perform source separation, permutation alignment and direction-of-arrival estimation of the sources simultaneously
through the posterior inference [8]. However, this approach has relied on the assumption of fixed source positions.
This paper proposes extending our previous approach to deal
with an underdetermined BSS problem allowing for moving sources.
Specifically, we formulate a generative model of mixture signals by
incorporating a generative model of the time-varying frequency array response for each source, described using a path-restricted hidden Markov model (HMM). Each hidden state of the present HMM
represents the direction of arrival (DOA) of each source, and thus wecall it a “DOA-HMM.” Through the posterior inference of the proposed generative model, we can simultaneously track the DOAs ofsources, separate source signals and perform permutation alignment. Sec. 2 reviews our previous generative model designed under astatic source position assumption [8] and Sec. 3 extends this model
to allow for moving sources by introducing a DOA-HMM.

2. GENERATIVE MODEL FOR STATIC SOURCES
2.1. Mixing modelFirst we consider a situation where I fixed source signals are recorded by M microphones. Here, let ym (ωk , tl ) ∈ C be the short-time
Fourier transform (STFT) component observed at the m-th microphone, and si (ωk , tl ) ∈ C be the STFT component of the i-thsource. 1 ≤ k ≤ K and 1 ≤ l ≤ L are the frequency and time
indices, respectively. If we assume that the length of the impulse response from a source to microphones is sufficiently shorter than theframe length of the STFT, the observed signal can be approximatedfairly well by an instantaneous mixture in the frequency domain:y(ωk , tl ) =I∑ai (ωk )si (ωk , tl ) + n(ωk , tl ),

(1)

i=1

where y(ωk , tl ) = (y1 (ωk , tl ), . . . , yM (ωk , tl ))T ∈ CM ands(ωk , tl ) = (s1 (ωk , tl ), . . . , sI (ωk , tl ))T ∈ CI . ai (ωk ) denotes
the frequency array response for source i at frequency ωk , whichis assumed to be time-invariant throughout this section. n(ωk , tl )is assumed to contain all kinds of components such as backgroundnoise and reverberant components, which cannot be represented by
the instantaneous mixture representation.
We now utilize the sparseness of speech and assume that only
one source is active in each time-frequency bin. By using zk,l ∈{1, . . . , I} to denote the (unknown) active source index at timefrequency bin (ωk , tl ), Eq. (1) can be rewritten asy(ωk , tl ) = azk,l (ωk )s(ωk , tl ) + n(ωk , tl ).

(2)

Notice that the subscript i is dropped from si (ωk , tl ) in Eq. (2) as
it is no longer necessary since we are assuming si (ωk , tl ) = 0 fori ̸= zk,l . For convenience of notation, we hereafter use subscripts k
and l to indicate ωk and tl respectively.

to h(θi , ωk ). Since the DOAs are unobservable, we regard the DOA
of each source as a latent variable and further consider describing its
generative process.
We now introduce a discrete set of D possible directions,ϑ1 , . . . , ϑD , which are all assumed to be constants. For instance,ϑd is defined as ϑd = (d − 1)π/D, (d = 1, . . . , D), which means
dividing π into D equal angels. We then assume that each source
signal propagates from one of these directions. First, we consider thegenerative process of the DOA θi of source i. For each source i, an
index ci of direction is drawn according to a categorical distribution
ρi = (ρi,1 , . . . , ρi,D )ci |ρi ∼ Categorical(ci ; ρi ),(5)

where Categorical(x; y) = yx . Given ci , θi is given as ϑci . Since ai,k may deviate from h(ϑci , ωk ) due to such factors as theplane wave assumption and the narrow band instantaneous mixture approximation, we assume that the frequency array response
ai,k is generated from a complex normal distribution with meanh(ϑci , ωk ), given ci ,(a)ai,k |ci ∼ NC (ai,k ; h(ϑci , ωk ), Σk ),

(6)

(a)

where Σk denotes the covariance of the complex normal distribution, which is assumed to be a constant.

2.2. Generative process of observed signals
Here we describe the generative process of an observed signal basedon Eq. (2). We assume that the noise component nk,l follows a com(n)plex normal distribution with mean 0 and covariance Σk . Then,from Eq. (2), y k,l is also normally distributed such that(n)y k,l |a1:I,k , sk,l , zk,l ∼ NC (y k,l ; azk,l ,k sk,l , Σk ),(3)

conditioned on a1:I,k = {a1,k , . . . , aI,k }, sk,l and zk,l , whereNC (x; µ, Σ) ∝ exp(−(x − µ)H Σ−1 (x − µ)). Moreover, we assume that zk,l derives from a discrete uniform distribution independently among all k and l.

3. GENERATIVE MODEL FOR MOVING SOURCES
In the model described in Sec. 2, the position of each source isassumed to be fixed and hence the frequency array response ai,kdoes not depend on time l. However, in practical situations, soundsources can move during sound recording and so the frequency array response ai,k may vary accordingly. To allow a time-varying
frequency array response, we represent the frequency array response
as a time sequence ai,k,1 , . . . , ai,k,L . Eqs. (2), (3) should thus be
rewritten as
y k,l = azk,l ,k,l sk,l + nk,l ,(n)y k,l |a1:I,k,l , sk,l , zk,l ∼ NC (y k,l ; azk,l ,k,l sk,l , Σk ).

2.3. Generative process of frequency array responses
We now describe the generative process of the frequency array response ai,k by introducing a latent variable indicating the DOA ofeach source.
Thus far we have treated ai,k as an independent parameter across k. If the index i indicates an identical source acrossω1 , . . . , ωK , ai,k will have a certain structure that can be describedusing the property of acoustic wave propagation. We therefore expect that the incorporation of an appropriate constraint into ai,kwould help solve both the permutation alignment problem and thefrequency-wise source separation problem simultaneously throughparameter inference. If each source is assumed to be located far fromthe microphones so that the signal can be treated approximately asa plane wave, the time difference between the microphones depends
only on the DOA of the source. Since the time delay between two
microphones corresponds to the phase difference of the frequency
response of the microphone array, the complex array response can be
expressed explicitly by using the DOAs. Specifically, with M = 2
microphones, the complex array response for a source at direction θsuch that 0 ≤ θ ≤ π is defined as a function of ω depending on θ[]1h(θ, ω) = ȷωB cos θ/C ,(4)ewhere ȷ is the imaginary unit, B [m] is the distance between the two
microphones, and C [m/s] is the speed of sound. If the DOA θi ofsource i is known, the frequency array response ai,k should be equal

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(7)
(8)

In the same way, we replace the DOA indicator variable ci with atime sequence ci,1 , . . . , ci,L and assume(a)ai,k,l |ci,l ∼ NC (ai,k,l ; h(ϑci,l , ωk ), Σk ).

(9)Now, if we simply treat ci,1 , . . . , ci,L as independent free variables
we must solve a BSS problem independently for each time l. Thisimplies the need to solve another permutation problem in the time
direction. Namely, we must first solve source separations in a frameby-frame manner and then group together the separated signals over
all l’s that are considered to originate from the same source. However, it would be better if we could join these two processes, as they
are intrinsically interdependent. To solve the permutation alignment
in the time direction, frame-by-frame source separation and permutation alignment in the frequency direction simultaneously, we needcertain assumptions about the time sequence of the DOAs.
Here we assume that each source moves continuously and thus
the DOA of each source varies continuously. That is, the DOA ofa source at time l is assumed to be close to that at time l − 1.
To incorporate this assumption into our generative model, we propose modeling the time sequence of the frequency array responses ai,k,1 , . . . , ai,k,L using an HMM in which the direction indicesd = 1, . . . , D are regarded as the hidden states (Fig. 1). Thus, Eq. (9) can be seen as a state emission density. The state sequenceci,1 , . . . , ci,L follows a Markov chain:ci,l |ci,l−1 ∼ Categorical(ci,l ; ρci,l−1 ),(10)intractable as directly computing the posterior. Although it may appear that no progress has been made, restricting q(Θ) to belong to a
family of distributions with a simpler form than p(Θ|Y ) allows us
to obtain principled approximate solutions.
For our model, we define the set of approximate distributions Q
as those that factor as follows:
Q = {q : q(A)q(S)q(Z)q(C)}.

(15)

This approximation is often called a naive mean-field approximation.
4.2. Coordinate AscentFig. 1. The time-varying frequency array response for each source ismodeled by an HMM. The DOA indices d = 1, . . . , D correspond tothe hidden states and the time sequence of frequency array responses
ai,k,1 , . . . , ai,k,L corresponds to the output sequence of the HMM,
respectively.
where ρd = (ρd,1 , . . . , ρd,D ) denotes the transition probability ofstate d to each state 1, . . . , D, and ρ = (ρd,d′ )D×D denotes thetransition matrix. Note that we can straightforwardly constrain the
DOAs to be continuously time-varying by setting the transition probabilities from state d to states d, d − 1 and d + 1, respectively, atreasonably large values. Overall, our new generative model is given
by Eqs. (8), (9) and (10).
4. APPROXIMATE POSTERIOR INFERENCE
4.1. Variational Bayesian Approach
In this section, we describe an approximate posterior inference algorithm for our generative model based on variational inference. The
random variables of interest in our model are A = a1:I,1:K,1:L ,S = s1:K,1:L , Z = Z1:K,1:L and C = c1:I,1:L . We denote the en(n)tire set of the above parameters as Θ. In the following, ρ, Σ1:K and(a)Σ1:K are constants that are determined experimentally. Our goal is
to compute the posteriorp(Y , Θ),p(Θ|Y ) =p(Y )We now present an algorithm for solving the optimization problem
described in (12) and (14). Unfortunately, the optimization problem
is non-convex, and it is difficult to find the global optimum. However, we can use a simple coordinate ascent algorithm to find a local
optimum. Note that (13) can be written as∫q(Θ)
KL(q(Θ)∥p(Θ|Y )) = q(Θ)logdΘ + log p(Y ).
p(Y , Θ)(16)
As the log evidence log p(Y ) is fixed with respect to q(Θ), minimizing the first term, which is known as the (negative) variational freeenergy, amounts to minimizing the KL divergence of p(Θ|Y ) fromq(Θ). In the mean-field approximation of the posterior, the algorithm can optimize one factor at a time while fixing all other factors.
It can be shown using the calculus of variations that the “optimal”
distribution for each of the factors can be expressed as:q̂(X) ∝ expE Θ\X [log p(Y , Θ)],where X indicates one of the factors and E Θ\X [logp(Y , Θ)] is the
expectation of the joint probability of the data and latent variables,
taken over all variables except X. The update equations for thevariational distributions are given in the following form:

(11)

(12)

but to obtain the exact posterior p(Θ|Y ), we must compute p(Y ),which involves many intractable integrals.
We can express this posterior variationally as the solution to an
optimization problem:
argmin KL(q(Θ)||p(Θ|Y )),∏q̂(A) =

(13)q∈Q

where KL(·∥·) denotes the Kullback-Leibler (KL) divergence between its two arguments, i.e.,
∫
q(Θ)
dΘ.
(14)
KL(q(Θ)∥p(Θ|Y )) = q(Θ)log
p(Θ|Y )
Indeed, if we let Q be the family of all distributions over Θ, thesolution to the optimization problem is the exact posterior p(Θ|Y ),since KL divergence is minimized when its two arguments are exactly equal. Of course, solving this optimization problem is just as3217

NC (ai,k,l ; mi,k,l , Γi,k,l ),(18)

2
NC (sk,l ; µk,l , σk,l
),

(19)

q̂(zk,l ), q̂(zk,l = i) = ϕi,k,l ,

(20)

i,k,l

∏

q̂(S) =

where Y = y 1:K,1:L is a set consisting of the time-frequency components of observed multichannel signals. By using the conditional
distributions defined in Sec. 2 and 3, we can write the joint distribution p(Y , Θ) asp(Y , A, S, Z, C) = p(Y |A, S, Z)p(Z)p(A|C)p(C),

(17)

k,l

∏

q̂(Z) =

k,l

where
2
2
Γ−1
i,k,l =(ϕi,k,l (|µk,l | + σk,l ))Σk

(n)−1

(a)−1

+ Σk

,

(21)

(n)−1
mi,k,l =Γi,k,l (Σkϕi,k,l µ∗k,l y k,l
(a)−1

+ Σk

∑

q̂(ci,l = d)h(ϑd , ωk )),

(22)

d

∑
1
(n)−1
],
=
ϕi,k,l tr[(mi,k,l mH
i,k,l + Γi,k,l )Σk
2
σk,l
i
∑
(n)−1
2
y k,l ,
µk,l =σk,l(ϕi,k,l mH
i,k,l )Σk

(23)
(24)

i
(n)−1

φi,k,l =exp(2Re[µk,l y H
k,l Σk− (|µk,l |2

mi,k,l ]

(n)−1
2
]),
+ σk,l)tr[(mi,k,l mH
i,k,l + Γi,k,l )Σk(25)

φi,k,lϕi,k,l = ∑.
i φi,k,l(26)

For updating q̂(C), we can use the forward-backward algorithm.
The update equation of q̂(C) is given as:q̂(C) =∏i,l∑α(ϑci,l )β(ϑci,l ),α(ϑci,l )β(ϑci,l )
ϑc

(27)

i,l

where α and β denote the forward and backward variables, that canbe computed using the emission probabilities q̂(ai,k,l |ϑd ):∑α(ϑci,l ) = q̂(ai,k,l |ϑci,l )α(ϑci,l−1 )ρci,l−1 ,ci,l ,(28)
β(ϑci,l ) =∑Table 1. The average output SIRs and standard deviations of thethree sources by the conventional and proposed methods.
movingmovingSIR(±SD) [dB]source A
source B
fixed source
Proposed
4.82(±3.94)
6.07(±3.24)
8.16(±1.50)
Conventional-1.10(±1.37) -2.00(±1.70) 3.55(±0.78)

ϑci,l−1β(ϑci,l+1 )q̂(ai,k,l+1 |ϑci,l+1 )ρci,l ,ci,l+1 . (29)

ϑci,l+1

In the sense of variational inference, we can obtain “optimal”q̂(ai,k,l |ϑd ) as:q̂(ai,k,l |ϑd ) ∝expE ai,k,l [log p(ai,k,l |ϑd )](a)−1=exp(−tr[(mHi,k,l mi,k,l + Γi,k,l )Σk(a)−1+ 2Re[h(ϑd , ωk ) Σkmi,k,l ](a)−1h(ϑd , ωk )).
− h(ϑd , ωk )H Σk]Fig. 2. Example of the ground truth DOA trajectories (dashed lines)
and the estimated DOA trajectories (solid lines).

H

(30)

Note that q̂(ai,k,l ) and p(ai,k,l |ϑd ) are both expressed as complex
normal distributions.
Finally, the STFT components of the i-th separated signal canbe obtained by multiplying µk,l by ϕi,k,l . Since q(Θ) is an approximation of the true posterior p(Θ|Y ), ϕi,k,l µk,l corresponds to an
approximation of the minimum meansquare error estimatorof the[]i-th source signal, i.e., ŝi,k,l = E 1[z,k,l = i]sk,l |Y ≃ E[z,k,l =i|Y ]E[sk,l |Y ] = ϕi,k,l µk,l , where 1[·] denotes the indicator function that takes the value 1 if its argument is true and 0 otherwise.
5. EXPERIMENTAL EVALUATION
We evaluated the performance of the present method in terms of theability to separate moving sources and track their DOAs. We used
ten mixed stereo signals as the experimental data, each of which
we obtained by mixing the speech signals of one static female speaker and two moving male speakers. The static source was obtained from the ATR Japanese speech database [10] and was convolved with the measured room impulse response from the RWCP
database [11] (in which the distance between the microphones was
2.83 cm and the reverberation time was 0 ms). We selected the twomoving sources from ten different moving sources obtained from the
RWCP database [11]. The sampling rate was 16 kHz. To compute
the STFT components of the observed signal, the STFT frame length
was set at 64 ms and a Hamming window was used with an overlap(n)(a)
length of 16 ms. Σk and Σk were set at I and 101.5 × I, respectively. D was set at 180. In the experiments, we fixed source signals(ωk , tl ) as y1 (ωk , tl ). This may be reasonable since the noise wasrelatively lower than the speech, and could help prevent s(ωk , tl )from being trapped in local optima. Moreover, our preliminary experiments revealed that q̂(C) was likely to be trapped in local optima
due to the spatial aliasing that occurs at high frequencies. To avoid
this, we adopted the following procedure: we first ran the variational inference algorithm using only the low-frequency region of theobserved signals, after which we gradually increased the frequency
range to the Nyquist frequency during the iterations. The variational
inference algorithm was run for 100 iterations. q̂(z1:K,1:L = i) was
initially set equally at 1/3 for i = 1, 2, 3. As for q̂(c1:3,1:L = d),q̂(c1,1:L = 46), q̂(c2,1:L = 91) and q̂(c3,1:L = 136) were set at relatively large values. The estimated DOAs were obtained from q̂(C)as the most possible direction at each time. We chose the method3218

proposed in [8] as a comparison. This method assumed that thesource directions were fixed. As an evaluation measure, we used
the signal-to-interference ratio (SIR) [12]. The SIR is expressed indecibels (dB), and a higher SIR indicates superior quality. The average input SIRs(±SD) dB of moving source A, moving source B andthe fixed source were -4.81(±1.15), -5.22(±1.03) and 0.29(±0.78)
dB, respectively. The SIRs were calculated until the time at which
the shortest signal of the three source signals was finished.
Table 1 shows the average SIRs and standard deviations for theten mixed signals obtained by the conventional and proposed methods. The average SIRs of the proposed method were superior tothose of the conventional method for each signal, especially for moving signals. The total average of the SIRs obtained with the proposed method was 6.20 dB more than that obtained with the conventional approach. These results show the effectiveness of the proposed method for BSS of moving sources. Examples of separated signals are available at http://www.hil.t.u-tokyo.ac.
jp/˜higuchi/demo/Examples.htm. In this experiment, theDOAs of different sources hardly overlapped each other. Another experiment revealed that if the DOAs of different sources overlappedeach other, the proposed method did not work well because it separates each signal based on its DOA. Fig. 2 shows a DOA estimation
result. The DOAs were almost all estimated correctly from 0.5 to
3.0 seconds, when all of the speakers were speaking.
6. CONCLUSION
This paper proposed a novel BSS approach that simultaneously estimates the directions of moving sources, separates the sources basedon sparseness of speech and performs permutation alignment. Focusing on the fact that the DOAs of a source tend to change gradually in practical situations, we modeled a time-varying frequency
array response for each source as a path-restricted HMM. Each hidden state of the HMM represented the DOA of each source and we
integrated the assumption of the smoothness of the DOA into thetransition probabilities of the HMM. The experiment showed that the
proposed algorithm provided a 6.20 dB improvement compared withthe conventional algorithm as regards the signal-to-interference ratio
and estimated the DOAs of sources when two of the three sourceswere moving.

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