2014 IEEE INTERNATIONAL WORKSHOP ON MACHINE LEARNING FOR SIGNAL PROCESSING, SEPT. 21–24, 2014, REIMS, FRANCEJOINT AUDIO SOURCE SEPARATION AND DEREVERBERATION BASED ON
MULTICHANNEL FACTORIAL HIDDEN MARKOV MODEL
Takuya Higuchi1) and Hirokazu Kameoka1),2)
1)

Graduate School of Information Science and Technology, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
2)
NTT Communication Science Laboratories, Nippon Telegraph and Telephone Corporation
3-1 Morinosatowakamiya, Atsugi, Kanagawa 243-0198, Japan{higuchi,kameoka}@hil.t.u-tokyo.ac.jp
ABSTRACTThis paper proposes a unified approach for jointly solving underdetermined source separation, audio event detection anddereverberation of convolutive mixtures. For monaural source
separation, one successful approach involves applying nonnegative matrix factorization (NMF) to the magnitude spectrogram of a mixture signal, interpreted as a non-negative matrix. Several attempts have recently been made to extend thisapproach to a multichannel case in order to utilize the spatial
correlation of the multichannel inputs as an additional clue forsource separation. The multichannel NMF assumes that an
observed signal is a mixture of a limited number of source signals each of which has a static power spectral density scaled
by a time-varying amplitude. We have previously proposed
an extension of this approach, in which the variations overtime of the spectral density and the total power of each source
is modeled by a hidden Markov model (HMM). This has allowed us to solve source activity detection and source separation simultaneously through model parameter inference.
While this method was based on an anechoic mixing model, the aim of this paper is to further extend the above approach to deal with reverberation by incorporating an echoic
mixing model into the generative model of observed signals.
Through an experiment of underdetermined source separation
under reverberant conditions, we confirmed that the proposed
method provided a 9.61 dB improvement compared with theconventional method in terms of the signal-to-interference ratio.
Index Terms: source separation, dereverberation, audio
event detection, non-negative matrix factorization, multichannel factorial hidden Markov model,
1. INTRODUCTION
Blind source separation (BSS) refers to a technique for separating out 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. To solve BSS problems, it is generally necessary to
make some assumptions about the sources, and formulate an
appropriate optimization problem based on criteria designedc978-1-4799-3694-6/14/$31.00 ⃝2014
IEEE

according to those assumptions. For example, if the observed
signals outnumber the sources, we can employ independent component analysis (ICA) [1] by assuming that the sources
are statistically independent of each other. However, in an underdetermined case, the independence assumption is too weak
to allow us to determine a unique solution and so directly applying ICA will not work well.
For monaural source separation, one successful approach
involves applying non-negative matrix factorization (NMF) tothe magnitude spectrogram of a mixture signal, interpreted asa non-negative matrix [2][3]. With this approach, the spectrogram of a mixture signal is factorized into the product of abasis matrix consisting of basis spectra and an activation matrix consisting of time-varying amplitudes associated with thebasis spectra. Several attempts have recently been made toextend this approach to a multichannel case in order to allow
for the use of the spatial correlation of multichannel inputs as an additional clue for separation, which have opened adoor to a new promising approach for underdetermined BSS [4][5]. This approach is based on the assumption that an
observed signal is a mixture of a limited number of source
signals each of which has a static power spectral density (i.e.,the basis spectrum) scaled by a time-varying amplitude. However, many source signals in real world are non-stationary innature and the variations of the spectral densities are muchricher in time. Another important fact is that many sources including speech tend to stay inactive for some while until theyswitch to an active mode. This implies that the total powerof a source may depend on its underlying state. To reasonably characterize such a non-stationary nature of source signals, we previously extended the multichannel NMF modelby modeling the transition of the set consisting of the spectral
densities and the total power of each source using a hidden
Markov model (HMM). We call this model the “multichannel
factorial hidden Markov model (MFHMM)” [6]. With thismodel, we are able to perform source separation and source
activity detection simultaneously. However, in the models
mentioned above, the length of the impulse response from asource to microphones is assumed to be sufficiently shorter
than the frame length of the STFT so that the observed signal
can be approximately modeled by an instantaneous mixture in
the time-frequency domain. To deal with reverberation, thispaper aims to further extend the above approach by incorporating an echoic mixing model into MFHMM. Parameter
inference of the present model allows us to simultaneously
solve source separation, source activity detection and dereverberation based on a unified maximum likelihood criterion.
The remainder of this paper is organized as follows: Sec.
2 formulates a generative model of a multichannel observed
signal under a reverberant condition, Sec. 3 presents the generative model of a source signal based on an HMM, Sec. 4
dervies a parameter estimation algorithm for the present model and Sec. 5 presents some results of a source separation
experiment.
2. ECHOIC MIXING MODELFirst we consider a situation where I source signals arerecorded by M microphones. Here, let ym (t) ∈ R be theobserved signal at the m-th microphone, and si (t) ∈ R be thesignal of the i-th source. The observed signal can be written
in the time domain:I ∫ ∞∑y(t) =ai (τ )si (t − τ )dτ,
(1)
i=1

−∞

where y(t) = (y1 (t), . . . , yM (t))T ∈ RM and ai (t) =(ai,1 (t), . . . , ai,M (t))T ∈ RM . ai,m (t) denotes the impulse
response between source i and microphone m. If we assume
that the length of the impulse response from a source to microphones is sufficiently shorter than the frame length of the
STFT, the observed signal can be approximated fairly well by
an instantaneous mixture in the time-frequency domain:y(ωk , tl ) =I∑ai (ωk )si (ωk , tl ),

(2)

i=1

where y(ωk , tl ) = (y1 (ωk , tl ), . . . , yM (ωk , tl ))T ∈ CM
and ai (ωk ) = (ai,1 (ωk ), . . . , ai,M (ωk ))T ∈ CM . 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-th source. 1 ≤ k ≤ K and
1 ≤ l ≤ L are the frequency and time indices, respectively.
ai (ωk ) denotes the frequency array response for source i at
frequency ωk . However, in a reverberant condition, the lengthof the impulse responses are relatively long and so an instantaneous mixture approximation is not always true. Therefore
we approximately express the observed signals as a form ofa convolution of the frequency array response and the source
signal in the time-frequency domain:y(ωk , tl ) ≈

I ∑
N
∑

ai (ωk , tn )si (ωk , tl − tn ).

(3)

i=1 n=00 ≤ n ≤ N is the time index of the frequency array responsein the time-frequency domain. Note that ai (ωk , t1 : tN ) denote the frequency array responses which correspond to theimpulse responses out of the frames of the STFT. This approximation is useful for dereverberation [7] and the validityof the approximation is experimentally shown. For convenience of notation, we hereafter use subscripts k, l and n to
indicate ωk , tl and tn respectively.

3. MULTICHANNEL FACTORIAL HMM
3.1. Generative process of observed signals
Here we describe the generative process of an observed signal
based on Eq. (3). If we assume that each source signal follows
a piecewise stationary Gaussian process, then si,k,l follows
a complex normal distribution with mean 0 and covariance2
σi,k,l,2
si,k,l |σi,k,l ∼ NC (si,k,l ; 0, σi,k,l
),

(4)

2
where σi,k,ldenotes the power spectral density of i-th sourceat frequency k and time l and NC (x; µ, Σ) ∝ exp(−(x −µ)H Σ−1 (x − µ)). From Eq. (3) and Eq. (4), y k,l is also
normally distributed such that

y k,l |a1:I,k,0:N , σi,k,l−N :l∑2∼ NC (y k,l ; 0,C i,k,n σi,k,l−n
),

(5)

i,n

conditioned on a1:I,k,0:N and σi,k,l−N :l where C i,k,n =ai,k,n aH
i,k,n . C i,k,n represents n-th spatial correlation matrixof the i-th source at frequency k.
3.2. Generative model for multichannel NMF [4, 5]
In the regular NMF model (see [8]), the power spectra of asource signal is assumed to be static up to a scale factor. We
can incorporate this assumption into the above model by setting
2
σi,k,l= wi,k hi,l ,(6)2
where σi,k,l
is assumed to be factorized into the product of thestatic power spectrum wi,k and the time-varying amplitude
hi,l . The generative model of si,k,l is thus rewritten as

si,k,l |wi,k , hi,l ∼ NC (si,k,l ; 0, wi,k hi,l ),(7)

conditioned on wi,k and hi,l .
3.3. Generative modeling of source signals using HMMs
As described above, the multichannel NMF model roughly
assumes that the power spectra of each sound source is static
up to a scale factor. However, many sound sources exhibit different spectra according to underlying “states” of the sources.
For example, the spectra of the sound of a piano note would
be different in “attack,” “decay,” “sustain” and “release” segments. Another important fact is that the total power of asource also depends on its underlying state. To reasonably
characterize such a non-stationary nature of source signals,
here we model the sequence of the power spectra and the total powers of each source using an HMM.
Now we introduce a latent variable zi,l ∈ {1, . . . , D} todenote a state of the i-th source at time l. The state sequencezi,1 , . . . , zi,L is assumed to follow a Markov chain:zi,l |zi,l−1 ∼ Categorical(zi,l ; ρzi,l−1 ),(8)where Categorical(x; y) = yx , ρd = (ρd,1 , . . . , ρd,D )denotes the transition probability of state d to each state
1, . . . , D, and ρ = (ρd,d′ )D×D denotes the transition matrix.
Here, we assume that hi,l follows a gamma distribution withhyperparameters determined according to zi,l ,
hi,l |zi,l ∼ Gamma(hi,l ; αzi,l βzi,l ),(9)where α1:D and β1:D are the shape and scale parameters of
α−1 −x/βea gamma distribution, and Gamma(x; α, β) = x Γ(α)β.
α
As we want hi,l to take a small value when zi,l is the “inactive” (i.e., silent) state, we set the hyperparameters of the
gamma distribution of that state so that it becomes a sparsityinducing distribution. As regards the gamma distributions of
the remaining states, we consider setting the hyperparameters so that they become uniform distributions. We expect that
this setting allows us to solve source separation and source
activity detection in a cooperative manner. Let us use wi,k,d
to denote the power spectrum of the i-th source at state d. The
power spectrum of the i-th source at time l is also assumed to
be determined according to zi,l . Thus, the generative modelof si,k,l is eventually written as
si,k,l |wi,k,1:D , hi,l , zi,l∼ NC (si,k,l ; 0, wi,k,zi,l hi,l ).

conditional distributions defined in Sec. 3, we can write thejoint distribution p(Y , Θ) asp(Y , Θ) ∝p(Y |Θ)p(H|Z)p(Z).

(13)

The objective function is defined as L(Θ) = log p(Θ|Y ).
Our goal is to obtain Θ̂ such thatΘ̂ = argmax log p(Θ|Y ).

(14)

Θ

By using Eqs. (12), (13) and (14), the current optimization
problem can be rewritten as(Θ̂ = argmax log p(Y |Θ)Θ)+ log p(H|Z) + log p(Z) .
(15)
According to the generative model defined in Sec. 3, log p(Y |Θ)is written aslog p(Y |Θ)−11∑=−(M log 2π + log |X̂ k,l | + y k,l H X̂ k,l y k,l ), (16)
2
k,l

(10)

Since the generative model of y k,l contains multiple HMMs
associated with the underlying sources, the overall model canbe viewed as a Factorial HMM. Our overall generative model
is given by Eqs. (8), Eqs. (9) andy k,l |a1:I,k,0:N , w1:I,k,1:D , h1:I,l−N :l , z1:I,l−N :l∑∼ NC (y k,l ; 0,C i,k,n wi,k,zi,l−n hi,l−n ), (11)i,nconditioned on a1:I,k,0:N , w1:I,k,1:D , h1:I,l−N :l and z1:I,l−N :l .
3.4. Related work
The present model is equivalent to the model proposed in [6]when the number N is set at 0, which leads to an anechoic
mixing model.

where X̂ k,l =∑i,n C i,k,n wi,k,zi,l−n hi,l−n .

4.2. Optimization algorithm based on an auxiliary function method
The optimization problem of maximizing L(Θ) with respectto Θ is difficult to solve analytically. However, we can invoke an auxiliary function approach to derive an iterative algorithm that searches for the estimate of Θ, as with [5]. To
apply an auxiliary function approach to the current optimization problem, the first step is to construct an auxiliary function
L+ (Θ, Λ) satisfying L(Θ) = maxΛ L+ (Θ, Λ). We refer toΛ as an auxiliary variable. It can then be shown that L(Θ) is
non-decreasing under the updates Θ ← argmaxΘ L+ (Θ, Λ)and Λ ← argmaxΛ L+ (Θ, Λ). The proof of this shall beomitted owing to space limitations. Thus, L+ (Θ, Λ) should
be designed as a function that can be maximized analyticallywith respect to Θ and Λ. Such a function can be constructed
as follows.

4. ALGORITHM FOR PARAMETER ESTIMATION
L(Θ)
4.1. Objective function
In this section, we describe a parameter estimation algorithm for our generative model based on an auxiliary function
method. The random variables of interest in our model are
W = w1:I,1:K,1:D , H = h1:I,1:L , C = C 1:I,1:K,0:N and
Z = z1:I,1:L . We denote the entire set of the above parameters as Θ. In the following, ρ is constants that is determined
experimentally. Our goal is to compute the posteriorp(Θ|Y ) =p(Y , Θ)
,
p(Y )

(12)

where Y = y 1:K,1:L is a set consisting of the time-frequency
components of observed multichannel signals. By using the≥L+ (Θ, Λ){ (−1
H1 ∑ ∑ tr(y k,l y k,l Ri,k,l,n C i,k,n Ri,k,l,n )=−2
wi,k,zi,l−n hi,l−n
i,n
k,l
)
}
−1
+ tr(U k,l C i,k,n )wi,k,zi,l−n hi,l−n + log |U k,l | − M∑{}+(αzi,l − 1) log hi,l − hi,l /βzi,l − αzi,l log βzi,li,l+ log p(Z),(17)

where Ri,k,l,n and U k,l are auxiliary∑ variables that satisfy
Hermitian positive definiteness and i,n Ri,k,l,n = I. We

denote the set of the auxiliary variables as Λ. tr(·) is the traceof a matrix. The equality L(Θ) = L+ (Θ, Λ) is satisfied when−1

Ri,k,l,n = C i,k,n wi,k,zi,l−n hi,l−n X̂ k,l ,U k,l = X̂ k,l .

(18)
(19)

Therefore, we can monotonically increase L by repeating
the following two steps.
1. Maximizing L+ with respect to R and U .
2. Maximizing L+ with respect to W , H, C and Z.

Fig. 1. Spectrogram of a mixture signal.

Step 1 consists in updating R and U using Eqs. (18) and(19). In step 2, we can obtain update rules of W , H, C by
setting the partial derivative of L+ with respect to each of theparameters at zero. The partial derivatives of L+ with respect
to W and H are given by∂L+∂wi,k,zi,l(−1
H1 ∑ tr(y k,l+n y k,l+n Ri,k,l+n,n C i,k,n Ri,k,l+n,n )=
22
wi,k,z
hi,l
i,l
l,n
)
− tr(U −1(20)k,l+n C i,k,n )hi,l ,
∂L+
∂hi,l

(
−1
H
1 ∑ tr(y k,l+n y k,l+n Ri,k,l+n,n C i,k,n Ri,k,l+n,n )=2
wi,k,zi,l h2i,lk,n)− tr(U −1C)w
i,k,zi,lk,l+n i,k,n+ (αzi,l − 1)/hi,l − 1/βzi,l ,

(21)

=

∑

tr(U −1k,l+n C i,k,n )wi,k,zi,l + 2/βzi,l .

(25)

k,n

The partial derivatives of L+ with respect to C is given by−1
H∑( C −1∂L+i,k,n Ri,k,l,n y k,l y k,l Ri,k,l,n C i,k,n=∂C i,k,nwi,k,zi,l−n hi,l−nl)− U −1
w
h
.
(26)
i,k,z
i,l−n
i,l−n
k,l

By setting this at zero, we obtain an algebraic Riccati equation:
C i,k,n Ai,k,n C i,k,n = B i,k,n ,

(27)

where
Ai,k,n =∑−1

wi,k,zi,l−n hi,l−n X̂ k,l ,l

B i,k,n = C i,k,n (∑wi,k,zi,l−n hi,l−n

l
−1

−1

respectively. By setting them at zero, we obtain the following
update rules:X̂ k,l y k,l y k,l H X̂ k,l )C i,k,n .

wi,k,zi,lvu∑tr(y k,l+n y k,l+n H Ri,k,l+n,n C −1ui,k,n Ri,k,l+n,n )
u l,n
hi,l
t
←
, (22)∑−1l,n tr(U k,l+n C i,k,n )hi,lWe can solve this equation by using a method in [5]. We
perform an eigenvalue decomposition of a 2M × 2M matrix[]
0−Ai,k,n,(29)−B i,k,n0hi,land let e1,i,k,n . . . eM,i,k,n be eigenvectors with negative
eigenvalues. It is guaranteed that there are exactly M negative
eigenvalues. Then, let us decompose the 2M -dimensional
eigenvectors as[]fem,i,k,n = m,i,k,n ,(30)
g m,i,k,n√(αzi,l − 1) + (αzi,l − 1)2 + µi,l νi,l
,
←
νi,l

(23)

where
µi,l
∑ tr(y k,l+n y k,l+n H Ri,k,l+n,n C −1
i,k,n Ri,k,l+n,n ),=
wi,k,zi,l
k,n

(24)
νi,l

(28)

for m = 1 . . . M where f m,i,k,n and g m,i,k,n are M dimensional vectors. We obtain the update rule for C i,k,nas
C i,k,n ← Gi,k,n F −1
i,k,n ,

(31)

Table 1. The SIRs of the three sources obtained by the conventional and proposed methods.
SIR [dB]
Proposed
Conventional

Fig. 2. (a)A spectrogram of the source signal of the stapler
recorded in an anechoic condition, (b)that of the separated
signal of the stapler obtained by the conventional method,(c)that of the separated and dereverbed signal of the stapler
obtained by the proposed method, and (d)the acoustic event
detection result obtained by the proposed method. Black indicates the state is assigned at the time.
where F i,k,n = [f 1,i,k,n , . . . , f M,i,k,n ] and Gi,k,n =[g 1,i,k,n , . . . , g M,i,k,n ].
L+ is equal up to constant terms to the sum of the log
posteriors of HMMs, when viewed as a function of Z. Thus,
we can invoke the Viterbi algorithm to search for the optimal
path zi,1 , . . . , zi,L for each i individually. Note that updating W , H, C and Z corresponds to solving the problems ofsource separation, source activity detection and dereverberation based on a unified objective function.
5. EXPERIMENTAL EVALUATION
We evaluated the performance of the present method in termsof the abilities of source separation, source activity detection

bell
43.35
32.37

phone
32.05
30.15

stapler
7.19-8.77

and dereverberation in a supervised case. We used a mixed
stereo signal as the experimental data, each of which we obtained by mixing the non-speech signals (sounds of a cell phone, a bell and a stapler) from the RWCP database[9]and was convolved with the measured room impulse response
from the RWCP database [9] (in which the distance between
the microphones was 2.83 cm and the reverberation time was
600 ms). Thus, the three signals were artificially located 60◦ ,90◦ and 130◦ from the microphones respectively. Fig. 1
shows a spectrogram of the observed signal in the reverberant
condition. The sampling rate was 32 kHz. To compute the
STFT components of the observed signal, the STFT frame
length was set at 16 ms and a Hamming window was usedwith an overlap length of 8 ms. We set the number of statesof HMMs D as 2. We expected that d = 1 was an inactive
state and d = 2 was an active state, by setting α1 and β1
as 1 and 10−2 respectively, and α2 and β2 as 1 and 1020 respectively.
√ The diagonal elements of C i,k,0 were initially all
set to 1/ M , and the off-diagonal elements were initially set
to zero. For n = 1,√. . . , N , the diagonal elements of C i,k,nwere set to 10−3 / M , and the off-diagonal elements werealso set to zero initially. We first learned W , H and ρ fromthe three clean source signal recorded in the anechoic condition with the proposed method starting from random initialmatrices W and H (therefore this experiment was a supervised source separation). The parameter estimation algorithm
was run for 150 iterations. In order to avoid an undesirable
local optima, we iterated the proposed algorithm 100 times
with setting N as 0, then gradually increased N up to 20 according to the iteration. We chose the method proposed in[6] as a comparison. This method equals to be our proposed
method when we set N = 0. The separated signal ŷ i,k,l was
obtained by Wiener filtering−1ŷ i,k,l = wi,k,zi,l hi,l C i,k,0 X̂ k,l y k,l .

(32)

As evaluation measures, we used the signal-to-interference
ratio (SIR) [10]. The SIR is expressed in decibels (dB), anda higher SIR indicates superior quality. The input SIR of thesounds of the bell, the cell phone and the stapler were -16.61,
16.58 and -39.17 [dB], respectively.
Table 1 shows the SIRs obtained by the conventional andproposed methods. The average of the SIRs obtained withthe proposed method was 9.61 [dB] more than that obtainedwith the conventional approach. These results show the effectiveness of the proposed method for source separation ina reverberant condition. Fig. 2 shows (a)a spectrogram ofthe source signal of the stapler recorded in an anechoic condition, (b)that of the separated signal of the stapler obtainedby the conventional method, (c)that of the separated and dereverbed signal of the stapler obtained by the proposed method,and (d)the acoustic event detection result obtained by the proposed method respectively. Black indicates the state is assigned at the time. We can see that the reverberant components were relatively removed by the proposed method andthe acoustic event was roughly detected.
6. CONCLUSION
This paper proposed a unified approach for jointly solving underdetermined source separation, audio event detection anddereverberation of convolutive mixtures based on MFHMM
with an echoic mixing model. Through an experiment of supervised source separation under reverberant conditions, we
confirmed that the proposed method provided a 9.61 dB improvement compared with the conventional method in termsof the signal-to-interference ratio.
7. ACKNOWLEDGEMENT
This work was supported by JSPS KAKENHI Grant Numbers
26730100.
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