Unified approach for underdetermined BSS, VAD,
dereverberation and DOA estimation withmultichannel factorial HMM
Takuya Higuchi∗ and Hirokazu Kameoka∗†∗ Graduate School of Information Science and Technology,

The University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo, 113-8656, Japan† NTT Communication Science Laboratories, NTT Corporation,
Morinosato Wakamiya 3-1, Atsugi, Kanagawa, 243-0198, Japan
Abstract—This paper proposes a novel method for simultaneously solving the problems of underdetermined blind source
separation (BSS), source activity detection, dereverberation anddirection-of-arrival (DOA) estimation by introducing an extension of the “multichannel factorial hidden Markov model (MFHMM).” The MFHMM is an extension of the multichannel nonnegative matrix factorization (NMF) model, in which the basis
spectra are allowed to vary over time according to the transitionsof the hidden states. This model has allowed us to perform source
separation, source activity detection and dereverberation in a
unified manner. In our previous model, the spatial covarianceof each source has been treated as a model parameter. This
has led the entire generative model to have an unnecessarily
high degree of freedom, and thus the parameter inference has
been prone to getting trapped into undesired local optima. To
reasonably restrict the solution space of the spatial covariancematrix of each source, we propose to describe it as a weighted
sum of the fixed spatial covariance matrix corresponding to
the discrete set of DOAs. Through the parameter inference, theproposed model allows us to simultaneously solve the problems ofunderdetermined BSS, source activity detection, dereverberation
and DOA estimation. Experimental results revealed that the
proposed method was superior to a previous method in terms of
the signal-to-distortion ratios of separated signals.

Index Terms: source separation, dereverberation, hidden
Markov model, non-negative matrix factorization, DOA
I. I NTRODUCTION
Blind source separation (BSS) refers to a technique forseparating out individual source signals from microphone array
inputs when the transfer characteristics between the sources
and microphones are unknown. To solve BSS problems, it
is generally necessary to make some assumptions about thesources, and formulate an appropriate optimization problem
based on criteria designed 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 independentof each other. However, in an underdetermined case, theindependence 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
as a non-negative matrix [2][3]. Up to now, several attempts
have been made to extend this approach to a multichannelcase in order to allow for the use of spatial information as anadditional clue for separation, which have opened a door toa new promising approach for underdetermined BSS [4][5].
Moreover, we previously proposed the multichannel factorial
hidden Markov model [6], where non-stationary spectra of asource are modeled by using a HMM. Furthermore, we extended the MFHMM for dereverberation [7], by approximating
the multichannel observed signal recorded in a reverberant
condition as a form of a convolution of the frequency array
response and the source signal in the time-frequency domain.
Thus, we modeled the impulse response out of the frame of
STFT by introducing a time sequence of frequency response
arrays in the time-frequency domain. However, in the model
we mentioned above, we do not make any assumption fora spatial correlation matrix and the degree of freedom of
the spatial correlation matrix is very high. This sometimes

makes the spatial correlation matrix trapped into undesirable
local optima. Generally, it is known that the spatial correlation
matrix of the direct sound of a point source has a certain
structure described by the source s direction. One way tomodel the spatial correlation matrix would be to use a mixture
model of matrices parametrized by all possible direction ofarrivals (DOAs), as in [8][9]. Another way would be to use
a linear sum of those matrices, as in [10]. For reverberant
conditions, the latter model would be more relevant than theformer, since the spatial correlation matrix could be contaminated by the components of early reflections. In this paper,
we incorporate this model of a spatial correlation matrix into
our previous model, and approximate the impulse responseboth within and out of the frame of the STFT in the timefrequency domain. Through the parameter inference of our
new generative model, we can simultaneously perform source
separation, source activity detection, dereverberation and DOA
estimation based on a unified maximum likelihood criterion.
II. M ULTICHANNEL FACTORIAL HIDDENM ARKOV MODEL [7]
A. 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:y(t) =I ∑∞∑ai (τ )si (t − τ ),(1)

i=1 τ =0

where y(t) = (y1 (t), . . . , yM (t))T ∈ RM and ai (t) =(a1,1 (t), . . . , a1,M (t))T ∈ RM . ai,m (t) denotes the impulse
response between source i and microphone m. In a reverberant
condition, the length of the impulse responses are relatively
long and so an instantaneous mixture approximation is notalways true. Therefore we approximately express the observed
signals as a form of a convolution of the frequency array
response and the source signal in the time-frequency domain.
y(ωk , tl ) ≈

I ∑
T
∑

ai (ωk , τ )si (ωk , tl − τ ).

(2)

i=1 τ =0

Here, let ym (ωk , tl ) ∈ C be the short-time Fourier transform(STFT) component observed at the m-th microphone, andsi (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 , τ ) denotes the frequency array
response for source i at frequency ωk and time τ . 0 ≤ τ ≤ T
is the time index of the frequency array response in thetime-frequency domain. Note that ai (ωk , 1 : T ) denote thefrequency array responses which correspond to the impulse
responses out of the frames of the STFT. This approximation
is useful for dereverberation [11] and the validity of theapproximation is experimentally shown. For convenience ofnotation, we hereafter use subscripts k and l to indicate ωkand tl respectively.

B. Generative process of observed signals
Here we describe the generative process of an observed
signal based on Eq. (2). We assume that the source signal
si,k,l follows a complex normal distribution with mean 0 and2
covariance σi,k,l.

Unlike our previous model [13], the amplitudes of the basis
spectra also depend on the hidden states in the present model,
making it possible to perform audio event detection and source
separation simultaneously through parameter inference. Our
overall generative model is given by Eqs. (7), Eqs. (8) and2
si,k,l |σi,k,l ∼ NC (si,k,l ; 0, σi,k,l),

y k,l |a1:I,k,1:T , w1:I,k,1:Q , h1:I,l−T :l , z1:I,l−T :l∑∼ NC (y k,l ; 0,C i,k,τ wi,k,zi,l−τ hi,l−τ ),(3)

Then, from Eq. (2), y k,l is also normally distributed such that
y k,l |a1:I,k,0:T , s1:I,k,l−T :l∑2∼ NC (y k,l ; 0,C i,k,τ σi,k,l−τ
),

(10)

i,τ

(4)

conditioned on a1:I,k,0:T ,w1:I,k,1:Q , h1:I,l−T :l and z1:I,l−T :l .

conditioned on a1:I,k,0:T and s1:I,k,l−T :l where C i,k,τ =H −1ai,k,τ aH(x − µ)).
i,k,τ and NC (x; µ, Σ) ∝ exp(−(x − µ) ΣC i,k,τ is often called a spatial correlation matrix.

III. M ODELING A SPATIAL CORRELATION MATRIX BASEDON DOA

C. Generative modeling of source signals using HMMs
We now describe the generative process of the source signal
si,k,l using a HMM. First, we assume each signal has a specific2spectrum and utilize NMF for σi,k,l, which is an expected2
value of power of si,k,l . σi,k,l is factorized as

If we assume that a source is far from the microphones, thefrequency array response has a certain structure in the timefrequency domain depending on Direction of Arrival (DOA).
Therefore, we can express a spatial correlation matrix by using
the DOAs [8][9][10]. Specifically, with M = 2 microphones,
the spatial correlation matrix for a source at direction θ suchthat 0 ≤ θ ≤ π is defined as a function of ω depending on θi,τ2
σi,k,l= wi,k hi,l ,where wi,k and hi,l are non-negative variables. wi,1:K represents a specific structure of spectrum that the source signal
has. hi,l means the activation of the spectrum in time l. The
generative model of si,k,l is rewritten as
si,k,l |wi,k , hi,l ∼ NC (si,k,l ; 0, wi,k hi,l ),(6)

conditioned on wi,k and hi,l . Under the condition of Eq. (6),the generative model of the single-channel observed signal isequivalent to that proposed in [12].
In many cases, a sound signal consists of several spectra andthe source’s spectrum varies according to the source’s state.
Now we introduce latent variable zi,l ∈ {1, . . . , Q} to denote astate of i-th source in time l. The state sequence zi,1 , . . . , zi,Lfollows a Markov chain:zi,l |zi,l−1 ∼ Categorical(zi,l ; ρzi,l−1 ),(7)
where Categorical(x; y) = yx , ρq = (ρq,1 , . . . , ρq,Q ) denotes
the transition probability of state q to each state 1, . . . , Q,and ρ = (ρq,q′ )Q×Q denotes the transition matrix. Then, we
assume hi,l follows a gamma distribution which has different
parameters according to zi,l [6],hi,l |zi,l ∼ Gamma(hi,l ; αzi,l βzi,l ),(8)where α1:Q and β1:Q 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. The power spectrumof the i-th source at time l is also assumed to be determinedaccording to zi,l . Thus, the generative model of si,k,l iseventually written as
si,k,l |wi,k,1:Q , hi,l , zi,l∼ NC (si,k,l ; 0, wi,k,zi,l hi,l ).

[(5)
J (θ, ω) =

]

1

e

ȷωB cos θ/C

∗

(11)

where 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 of source i is known, the spatial correlation matrix
for the direct wave should be equal to J (θi , ωk ). However,C i,k,0 would not be equal to J (θi , ωk ) because of reverberant
components in the frame of the STFT. Therefore, we introduce
a discrete set of O possible directions ϑ1 , . . . , ϑO and weightvariables di,1 . . . di,O , then we express a spatial correlation
matrix using J (θi , ωk ) and di,o as;
C i,k,0 =∑di,o J (ϑo , ωk ).

(12)

o

∑
di,1 . . . di,O are non-negative and satisfy
o di,o = 1 [10].
We expect that an relatively large value in di,1 . . . di,O would
indicate the DOA of the direct wave of the i-th source.
IV. A LGORITHM FOR PARAMETER ESTIMATION
A. 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 , H = h1:I,1:L ,C = C 1:I,1:K,1:T D = d1:I,1:O
and Z = z1:I,1:L . We denote the entire set of the above
parameters except Z as Θ. In the following, ρ, α and β areconstants that is determined experimentally.
The∑ objective function L(Θ) is defined as L(Θ) =log Z p(Θ, Z|Y ), where Y = y 1:K,1:L is a set consistingof the time-frequency components of observed multichannel
signals. Our goal is to obtain Θ̂ such that

(9)With a similar motivation, we have previously proposed modeling single-channel mixture signals using a factorial HMM,
where the basis spectrum of each latent source signal isallowed to vary over time as a result of state transitions [13].

[1 eȷωB cos θ/C ]Θ̂ = argmax logΘ∑p(Θ, Z|Y ).

(13)

Z

By using the conditional distributions defined in Sec. II, weTABLE I
T HE EXPERIMENTAL CONDITION .

can rewrite Θ̂ as( ∑Θ̂ = argmax logp(Y |W , H, C, D, Z)
Θ

∑

+ log

Z

p(H|Z) + log

Z

∑

)
p(Z)

where X̂ k,l =

Z

(14)

Z

∑

i,τ C i,k,τ wi,k,zi,l−τ hi,l−τ,zi,l−τ .

L(Θ)≥L+ (Θ, Λ)((tr(y k,l y k,l H Ri,k,l,0,q J −11∑ ∑i,k,o Ri,k,l,0,q )
λq,i,l=−2di,o wi,k,qi,l hi,lk,l i,q,o)+ tr(U −1
k,l J i,o )di,o wi,k,qi,l hi,l
(

O
30

T
3

α
1

RT=380ms
Proposed
Method in [7]Input

B. Optimization algorithm based on an auxiliary function
methodThe 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 to Λas 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.

∑

Q
5

β
β1 = 10−1 , β2:10 = 1010

TABLE II
T HE AVERAGE INPUT SDR S /SIR S [dB] AND THESE OBTAINED BY OURPREVIOUS AND PRESENT METHODS .

∑(∑ 1= argmin(− log|X̂ k,l |
2
Θ
Z
k,l
∑
−1
1
− log
exp y k,l H X̂ k,l y k,l )
2
∑Z
∑
)
+ log
p(H|Z) + log
p(Z) ,
Z

RT=380 ms

tr(y k,l y k,l H Ri,k,l,τ,q C −1
i,k,τ Ri,k,l,τ,q )

SDR
-4.17
-6.49
-40.33

SIR
5.90
1.94
-4.22

Therefore, we can indirectly maximize L by repeating thefollowing two steps.
1) Maximizing L+ with respect to R, U and λ.
2) Maximizing L+ with respect to W , H, C and D.
Step 1 consists in updating R and U using Eqs. (16) and(17). With respect to λ, we can apply the Forward-Backward
algorithm. The update rule of λ is given by∑
λq,i,l = Fq,i,l Bq,i,l /Fq,i,l Bq,i,l ,(19)q

Fq,i,l = p(Θ|qi,l )Bq,i,l =∑∑Fq,i,l−1 ρqi,l−1 ,qi,l ,(20)

qi,l−1

Bq,i,l+1 p(Θ|qi,l+1 )ρqi,l ,qi,l+1 ,(21)

qi,l+1

wherep(Θ|qi,l )
( (
(
−1
H
1 ∑ ∑ tr(y k,l y k,l Ri,k,l,0,q J i,k,o Ri,k,l,0,q )∝ exp −
2di,o wi,k,qi,l hi,l
i,o
k,l
)
+ tr(U −1
k,l J i,k,o )di,o wi,k,qi,l hi,l+∑ ( tr(y k,l+τ y k,l+τ H Ri,k,l+τ,τ,q C −1
i,k,τ Ri,k,l+τ,τ,q )

wi,k,qi,l hi,l))−1
+ tr(U k,l+τ C i,k,τ )wi,k,qi,l hi,l))∑(+(αqi,l − 1) log hi,l − hi,l /βqi,l − αqi,l log βqi,l.
i,τ ̸=0i,l(22)
wi,k,qi,l−τ hi,l−τ
In step 2, we can obtain update rules of W , H, C and D by))
setting the partial derivative of L+ with respect to each of the+ tr(U −1
C
)w
h
+
log
|U
|
−
M
parametersat zero. Specifically, the partial derivatives of L+
i,k,qi,l−τ i,l−τk,lk,l i,k,τ
with respect to D is given by
(
)
∑
(
∑
tr(y k,l y k,l H Ri,k,l,0,q J −1
λq,i,l (αqi,l − 1) log hi,l − hi,l /βqi,l − αqi,l log βqi,l+∂L+i,k,o Ri,k,l,0,q )
=
λ
q,i,l
2
i,l,q
∂di,o
di,o wi,k,qi,l hi,l
∑
k,l,q
)
λq,i,l log p(Z),
(15)
+
−1
q
− tr(U k,l J i,k,o )wi,k,qi,l hi,l .
(23)+

λq,i,li,q,τ ̸=0

where Ri,k,l,τ,q , U k,l and λq,i,l are auxiliary variables.
Ri,k,l,τ and U k,l satisfy Hermitian∑positive definiteness and∑i,τ Ri,k,l,τ,q = I. λq,i,l satisfiesq λq,i,l = 1. We denotethe set of the auxiliary variables as Λ. tr(·) is the trace of amatrix. The equality L(Θ) = L+ (Θ, Λ) is satisfied when−1

Ri,k,l,τ,q = C i,k,τ wi,k,qi,l−τ hi,l−τ X̂ k,l ,U k,l = X̂ k,l ,
λq,i,l = p(qi,l |Θ).

(16)
(17)
(18)

According to the conditions for extremal values, update rules
are written asvu∑tr(y k,l y k,l H Ri,k,l,0 J −1ui,k,o Ri,k,l,0 )u k,l,q λq,i,l
wi,k,qi,l hi,ltdi,o ←. (24)∑−1
k,l,q λq,i,l tr(U k,l J i,k,o )wi,k,qi,l hi,lNote that to estimate W , H, C, D and λ = λ1:Q,1:I,1:L
means to solve the problems of source separation, source
activity detection, dereverberation and DOA estimation simultaneously.

8000
8000
7000
7000
6000

frequency[Hz]frequency[Hz]

6000
5000

4000

3000

5000

4000

3000

2000
2000
1000
1000
0

0
1

2

3

4

5

6

1

2

3

time[s]

4

5

6

4

5

6

time[s]Fig. 1. An example of a spectrogram of the observed signal in the reverberant
condition (the reverberation time was 380 ms).

8000

7000

frequency[Hz]

6000

5000

4000

3000

2000

1000

0
1

2

3
time[s]1

Fig. 2. Initial condition of d (top) and an example of the results of d withthe echoic signals (bottom).

1

q

2

V. R ELATED WORKS
Our proposed model is equal to the model proposed in [7]except Eqs. (12). The expression of Eqs. (12) was proposedin [10] for the spatial matrix of multichannel NMF with themean-square-error criterion. We incorporate this model of thespatial correlation matrix into our previous generative model, which means that we model reverberation in the frame of
the STFT by using the DOAs. The estimation of D allows us
to estimate the DOAs, and we derive update rules of D for
optimizing our objective function. Moreover, we marginalize
Z in Sec. IV, while the algorithm proposed in [7] allows us
only to obtain the optimal sequence of Z by applying theViterbi algorithm.
VI. E XPERIMENTAL EVALUATION
We evaluated the performance of the present method interms of the abilities of source separation. We used 10 mixed
stereo signals (therefore the number of the microphones Mis 2) as the experimental data, each of which we obtainedby mixing three speech signals (speech of a male and two
females) from the ATR database [14] and was convolved withthe measured room impulse response from the RWCP database[15] (in which the distance between the microphones was
11.48 cm and the reverberation time was 380 ms). Thus, thethree signals were artificially located 30◦ , 90◦ and 130◦ fromthe microphones respectively. Fig. 1 shows a spectrogram ofthe observed signal (the reverberation time was 380 ms). The
sampling rate was 16 kHz. To compute the STFT componentsof the observed signal, the STFT frame length was set at 64 ms
and a Hamming window was used with an overlap length of 48
ms. We set the parameters as Table I. We expected that q = 1 isan inactive state and q = 2 : 5 are active states. D was initially
set as Fig. 2 (top). For τ = √1, . . . , T , The diagonal elements of
C i,k,1:T were set to 10−1 / M , and the off-diagonal elements
were also set to zero initially. W was initially randomized.
H was set as 1 initially. ρq,q′ was set as 1/Q The parameter
estimation algorithm was run for 30 iterations. In order toavoid an undesirable local optima, we first set T as 0 andgradually increased T up to 3 with the iteration. We chose themethod proposed in [7] as a comparison. The separated signalŷ i,k,l was obtained by Wiener filtering∑−1λq,i,l wi,k,qi,l hi,l C i,k,0 X̂ k,l y k,l .
(25)ŷ i,k,l =q

As evaluation measures, we used the signal-to-distortion ratio(SDR) and the signal-to-interference ratio (SIR) [16]. The

3
0.5
4
5
0
1

2

3

4

5

6

time[s]Fig. 3. A spectrogram of an anechoic source signal (top), that of a separated
and dereverbed signal with the proposed method (middle) and the result of
λ which corresponds to the estimated activity of the source (bottom). q = 1
corresponds to the inactive state.

SDR and SIR are expressed in decibels (dB), and a higher
SDR(/SIR) indicates superior quality.
Table II shows the average SDRs and SIRs obtained byour previous and present methods. The average SDR andSIR obtained with the present method were superior to theseobtained with conventional method. Fig. 2 (bottom) shows anexample of d obtained by the proposed method. We can seethat the DOA of the sources were estimated roughly by the
proposed method. Fig. 3 shows a spectrogram of an anechoic
source signal (top), that of a separated and dereverbed signal
with the proposed method (middle) and the result of λ whichcorresponds to the estimated activity of the source (bottom).
We expected q = 1 is an inactive state by setting thehyperparameters of the gamma distribution properly (Table I),
and the result shows the voice activity was roughly detected.
VII. C ONCLUSION
In this paper we extend the MFHMM and propose a
unified approach for source separation, source activity detection, dereverberation and DOA estimation. Specifically, we
describe a spatial correlation matrix by using weight variableand kernels and incorporate into the MFHMM. Through theestimation of the parameters of the overall generative model,
we can simultaneously performed source separation, source
activity detection, dereverberation and DOA estimation. The
experiment showed that the proposed algorithm were superior
to our previous method in terms of the signal-to-distortion ratiowith echoic mixed signals.
VIII. ACKNOWLEDGEMENT
This work was supported by JSPS KAKENHI Grant Number 26730100, 26280060.

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