1

Underdetermined BSS With Multichannel Complex
NMF Assuming W-Disjoint Orthogonality of Source
Kazuma Takeda†‡ , Hirokazu Kameoka∀‡ , Hiroshi Sawada‡ ,Shoko Araki‡ , Shigeki Miyabe† , Takeshi Yamada† , Shoji Makino†† Graduate School of Systems and Information Engineering, University of Tsukuba‡ NTT Communication Science Laboratories, NTT Corporation∀ Graduate School of Information Science and Technology, The University of Tokyo

Abstract—This paper presents a new method for underdetermined Blind Source Separation (BSS), based on a concept called
multichannel complex non-negative matrix factorization (NMF).
The method assumes (1) that the time-frequency representationsof sources have disjoint support (W-disjoint orthogonality ofsources), and (2) that each source is modeled as a superposition ofcomponents whose amplitudes vary over time coherently acrossall frequencies (amplitude coherence of frequency components)
in order to jointly solve the indeterminacy involved in thefrequency domain underdetermined BSS problem. We confirmed
experimentally that the present method performed reasonably
well in terms of the signal-to-interference ratio when the mixing
process was known.
Index Terms—Underdetermined blind source separation(BSS),
non-negative matrix factorization(NMF), multichannel audio, Wdisjoint orthogonality(W-DO), Expectation-maximization (EM)algorithm

I. I NTRODUCTION
LIND Source Separation (BSS) is a technique for reconstructing source signals from microphone inputs on theassumption that the source signals and transfer characteristics
between sources and microphones are unknown.
In the following, we consider a situation where K source
signals are captured by M microphones. Here, let ym (ω, t) be
the short-time Fourier transform (STFT) component observedat the m-th microphone, and sk (ω, t) be the STFT componentof the k-th source. 1 ≤ ω ≤ Ω and 1 ≤ t ≤ T are thefrequency and time indices, respectively. We assume that thelength of the impulse response from a source to a microphone
is sufficiently shorter than the frame length of the STFT, inwhich case the observed signals can be approximated fairly
well by an instantaneous mixture in the frequency domain:y(ω, t) = A(ω)s(ω, t) + n(ω, t),(1)
TM
where y(ω, t) = (y1 (ω, t), . . . , yM (ω, t))∈ C
ands(ω, t) = (s1 (ω, t), . . . , sK (ω, t))T ∈ CK . A(ω) =(am,k (ω))M ×K = (a1 (ω), . . . , aK (ω)) ∈ CM ×K denotes
a mixing matrix that consists of the frequency responseam,k (ω) of the transfer characteristics, which is assumed tobe time-invariant. n(ω, t) is assumed to comprise all kindsof components that cannot be expressed by the instantaneous
mixture representation (e.g., background noise and reverberant
components). BSS based on this observation model is calledfrequency domain BSS, which allows for a fast implementation compared with BSS that use a time domain convolutive
mixture model.
To estimate the unknown mixing matrix and source signals
solely from observed signals, we must make some assumptions
about sources, 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],B

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which estimates the separation matrix (the inverse of themixing matrix) such that the independence of the source
estimates is maximized. For an underdetermined case where
there are fewer observations than sources, there are an infinite
number of solutions for source signals even if a mixed process
is known. Hence, the independence assumption is too weak
to allow us to determine a unique solution. For this reason,in underdetermined situations, we need a stronger assumption
than independence such as sparseness [2]–[6]. Moreover, even
if separation is achieved in each frequency bin ω so that we obtain K components s1 (ω, t), . . . , sK (ω, t), which component
belongs to which source remains ambiguous. Therefore, some
additional clue is required to make it possible to group together
the components considered to originate from the same source.
This kind of problem is usually called a permutation problem.
Various approaches to tackling the permutation problem havealready been proposed. They include approaches that use thedirection-of-arrival information of sources [7], the correlationof amplitudes in sources between different frequency bins [8],and the combination of the above two pieces of information
and a harmonicity constraint [9].
As mentioned above, the frequency domain underdetermined BSS problem simultaneously involves indeterminacy
in terms of the inverse mixing process and permutation ambiguity. To effectively avoid the indeterminacy, we find it
natural to introduce a unified framework incorporating (1) an
observation model (mixed process) that assumes sparseness ofsources in the time-frequency domain, and (2) a source model
consisting of components each of whose amplitudes varies
over time coherently across frequencies. We have introduced
a framework called “multichannel complex NMF” based on
this concept (which is an extension of the complex NMF
framework proposed in [10] to a multichannel case), and havealready proposed several different methods [11], [12]. Ozerov
introduces a similar concept in [13], where an attempt is madeto extend non-negative matrix factorization (NMF) to a multichannel case. It is important to note here that the multichannel
complex NMF framework differs from Ozerov’s framework inthat it uses a complex factor representation introduced in [10]to model the means of Gaussian source components whereas
Ozerov’s uses a non-negative factor representation to model
the variances.
As a contribution to this series of approaches, this paper
proposes a multichannel complex NMF method assuming theW-disjoint orthogonality of the source components, which canbe thought of as one example of the notion of sparseness.
II. M ULTICHANNEL C OMPLEX NMF
A. Assumptions on sources
We henceforth assume the use of the observation model
given by (1) and also assume that n(ω, t) is a complex

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Gaussian random variable that is uncorrelated in time, spaceand frequency:n(ω, t) ∼ NC (0, σ 2 (ω)I).
(2)
Moreover, we assume∥ak (ω)∥22 = 1, a1,k (ω) ≥ 0, 1 ≤ k ≤ K,(3)
to remove the scaling indeterminacy and an ambiguity in thesign between A(ω) and s(ω, t). We assume the following in
relation to the source signals:
Assumption 1 (Amplitude coherence of sources). Each sourceis expressed as the sum of I elements each of whose amplitudes
varies over time coherently across frequencies.
Let us now consider expressing the STFT component of thei-th element of the k-th source using a time-invariant spectral
shape Hki (ω), a time-varying gain Uki (t) and a time-varying
phase spectrum ϕik (ω, t), such thatixik (ω, t) = Hki (ω)Uki (t)ejϕk (ω,t) .
We assume for convenience∑
Hki (ω) = 1,

whereµk (ω, t) :=i

Hki (ω)Uki (t)ejϕk (ω,t) ,

(11)
(12)

(
)
2
Λ := diag ν12 , . . . , νK.
(13)
According to (2), (8) and (10), the conditional probability
density function of y(ω, t) given values ofθ := {ak (ω), πk , σ 2 (ω), νk2 }1≤k≤K,1≤ω≤Ω ,
(14)
i
i
i
πk := {Hk (ω), Uk (t), ϕk (ω, t)}1≤i≤Ik ,1≤ω≤Ω,1≤t≤T ,is described as:(p(y(ω, t)|k̂(ω, t), θ) = NC A(ω)Z k̂(ω,t) µ(ω, t),)
A(ω)Z k̂(ω,t) ΛZ k̂(ω,t) A(ω)H + σ 2 (ω)I . (15)(4)

According to (8), the conditional probability density functionof y(ω, t) given the values of θ, k̂(ω, t) and s(ω, t) is:(5)p(y(ω, t)|s(ω, t), k̂(ω, t), θ)()= NC A(ω)Z k̂(ω,t) s(ω, t), σ 2 (ω)I . (16)to eliminate the scaling indeterminacy between Hki (ω) andUki (t). We note that since Hki (ω) is time-invariant and Uki (t)is frequency-invariant, using the above representation amounts
to assuming that the amplitude |xik (ω, t)| is constrained to varyover time coherently across frequencies. By using (4), we can
now construct a source model incorporating Assumption 1:
Ik∑(6)
sk (ω, t) =xik (ω, t) + ϵk (ω, t).
i=1ϵk (ω, t) is used to denote a modeling error for each source.
Since a signal model expressed as the sum of (4) is basedon the complex factor model employed in the complex NMF
framework [10], we call a BSS framework based on the source
model given by (6) a “multichannel complex NMF”.
Next, we make an assumption about the mixing process.
Assumption 2 (Sparseness of sources). Only one source isactive in each time-frequency bin.
This assumption is called the W-disjoint orthogonality (WDO) [2], which is known to help solve underdetermined BSS
problems effectively, especially for sparse sources. Now, let
us use k̂(ω, t) to denote the (unknown) active source index attime-frequency bin (ω, t). Then, with this assumption, (1) canbe rewritten as:y(ω, t) = ak̂(ω,t) (ω)sk̂(ω,t) (ω, t) + n(ω, t)(7)(8)

Here, we have used Z k̂(ω,t) to denote a K × K matrix whose
component at row n and column m is given by δn,k̂(ω,t) ·δm,k̂(ω,t) (where δn,m is the Kronecker delta).

B. Probability density function of observed signals
Based on the above assumptions, we now derive the probability density function of the observed signals. Let ϵk (ω, t)
be a Gaussian noise with variance νk2 :ϵk (ω, t) ∼ NC (0, νk2 ).
(9)Hence, s(ω, t) is normally distributed such that()
s(ω, t) ∼ NC µ(ω, t), Λ ,(10)978-1-4577-0254-9/11/$26.00 ©2011 IEEE

Ik
∑
i=1

ω

= A(ω)Z k̂(ω,t) s(ω, t) + n(ω, t).

()Tµ(ω, t) := µ1 (ω, t), . . . , µK (ω, t) ,III. M AXIMUM L IKELIHOOD E STIMATION OF θ U SING
THE EM A LGORITHM
Here we derive an iterative algorithm that searches for the
maximum likelihood (ML) estimate of θ given the observed
data y := {y(ω, t)}1≤ω≤Ω,1≤t≤T . By treating y as incomplete
data and a set consisting of y, s = {s(ω, t)}1≤ω≤Ω,1≤t≤T
and k̂ = {k̂(ω, t)}1≤ω≤Ω,1≤t≤T as complete data, the ML
problem under consideration can be viewed as an incomplete
data problem, which can be dealt with using the ExpectationMaximization (EM) algorithm. The first step when devising
an EM algorithm is to define the Q function. The Q function
is defined as the conditional expectation of the complete datalog-likelihood log p(c|θ) where c := {y, s, k̂} with respect tothe latent variables s(ω, t) and k̂(ω, t), given the values ofthe observed data y(ω, t) and the model parameters θ = θ′ .
Hence, the Q function Q(θ, θ′ ) is⟨⟩Q(θ, θ′ ) = log p(c|θ) p(s,k̂|y,θ=θ′ )
(17)
⟨⟨
⟩
⟩
= log p(c|θ) p(s|k̂,y,θ=θ′ ) p(k̂|y,θ=θ′ )
(18)
⟨⟨
⟩
⟩
θ
= log P (y|s, k̂, θ) p(s|k̂,y,θ=θ′ ) p(k̂|y,θ=θ′ )⟨⟨⟩⟩+ log P (s|θ) p(s|k̂,y,θ=θ′ ) p(k̂|y,θ=θ′ ) .
(19)
The notation ⟨f (ξ)⟩p(ξ) is used to indicate the expectation off (ξ) with respect to ξ that is assumed have the distributionξp(ξ). = denotes equality up to terms not depending on ξ. We
give the concrete form of each term of (19) below.
Firstly, ⟨⟨log p(y|s, k̂, θ)⟩p(s|k̂,y,θ=θ′ ) ⟩p(k̂|y,θ=θ′ ) can be
written as:⟨⟨⟩⟩log p(y|s, k̂, θ) p(s|k̂,y,θ=θ′ ) p(k̂|y,θ=θ′ )∑θ
log σ 2 (ω)= −T M

807

ω

−

∑

K
∑

ω,t k̂(ω,t)=1

mk̂(ω,t) ( y)tr F k̂(ω,t) (ω, t) ,2
σ (ω)

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where F yk̂(ω,t) (ω, t) is given by
F yk̂(ω,t) (ω, t) :=y(ω, t)y(ω, t)H(21)−A(ω)Z k̂(ω,t) s̄k̂(ω,t) (ω, t)y(ω, t)H

the Q function is maximized with respect to A(ω) whenA(ω) =Γ(ω)Σ(ω)−1 ,
(29)
∑
∑
H
Γ(ω) :=y(ω, t)mk̂ s̄k̂ (ω, t) Z k̂ ,(30)t

−Z k̂(ω,t) A(ω)H y(ω, t)s̄k̂(ω,t) (ω, t)HΣ(ω) :=+Z k̂(ω,t) A(ω)H A(ω)Z k̂(ω,t) Rk̂(ω,t) (ω, t),

t

with
′

s̄k̂(ω,t) (ω, t) :=µ (ω, t) + Gk̂(ω,t) (ω)y(ω, t),

(22)
′

H

Rk̂(ω,t) (ω, t) :=s̄k̂(ω,t) (ω, t)s̄k̂(ω,t) (ω, t) + Λ− Gk̂(ω,t) (ω)A′ (ω)Z k̂(ω,t) Λ′ ,
′

′

(23)

H

Gk̂(ω,t) (ω) :=Λ Z k̂(ω,t) A (ω)
(24)
( ′
)
−1
A (ω)Z k̂(ω,t) Λ′ Z k̂(ω,t) A′ (ω)H + σ ′2 (ω)I.
In the above, we have used s̄k̂(ω,t) (ω, t) and Rk̂(ω,t) (ω, t) todenote ⟨s(ω, t)⟩p(s|y,k̂,θ=θ′ ) and ⟨s(ω, t)s(ω, t)H ⟩p(s|y,k̂,θ=θ′ ) ,respectively. We note that Gk̂(ω,t) (ω) has the same form as
the Wiener filter. mk̂(ω,t) denotes the probability that the k-thsource is active at time-frequency bin (ω, t):mk̂(ω,t) := p(k̂(ω, t)|y(ω, t), θ = θ′ ).
′

(25)

′

ξ corresponds to the value of ξ when θ = θ . On the other
hand, ⟨⟨log P (s|θ)⟩p(s|k̂,y,θ=θ′ ) ⟩p(k̂|y,θ=θ′ ) is written as:⟨⟨⟩⟩log P (s|θ) p(s|k̂,y,θ=θ′ ) p(k̂|y,θ=θ′ )∑θ= −T
log |Λ|−K∑)(
mk̂(ω,t) tr Λ−1 F sk̂(ω,t) (ω, t) ,ŝ(ω, t) :=−µ(ω, t)s̄k̂(ω,t) (ω, t)H−s̄k̂(ω,t) (ω, t)µ(ω, t)H+µ(ω, t)µ(ω, t)H .
(27)
Altogether, the term involving θ in the Q function is expressedas the sum of (20) and (26).
A. E stepIn the E step, we substitute the newest value of θ, updated
at the previous M step, into θ′ . Specifically, we update thevalues of s̄k̂(ω,t) (ω, t), Rk̂(ω,t) (ω, t) and mk̂(ω,t) using (22),(23) and (25). If we assume p(k̂(ω, t)|θ) = 1/K, (25) can be
written in a more concise formmk̂(ω,t) :=p(k̂(ω, t)|y(ω, t), θ = θ′ )′′k′ =1 p(y(ω, t)|k̂(ω, t) = k , θ = θ )K∑

mk̂(ω,t) s̄k̂(ω,t) (ω, t),∑ j
Hki (ω) ← Hki (ω) ∑ t,
Uki (t)Hk (ω)Ukj (t)
t

∑

,

(28)

which can be computed using the distribution given by (15).
B. M Step
Now, we consider updating θ to values such that Q(θ, θ′ ) is
guaranteed to be non-decreasing. First, we consider fixing all
the parameters in θ except for A(ω) at constant values. Then,

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(33)

(34)

j

Hki (ω)|[ŝ(ω, t)]k |∑ jUki (t) ← Uki (t) ∑ ω,
Hki (ω)Hk (ω)Ukj (t)ωiϕk (ω, t) ← arg([ŝ(ω, t)]k ).

F sk̂(ω,t) (ω, t) :=Rk̂(ω,t) (ω, t)(31)k̂(ω,t)=1(26)

where F sk̂(ω,t) (ω, t) is given by= ∑K

mk̂ Z k̂ Rk̂ (ω, t)Z k̂ .

k̂where [·]k indicates the k-th element of a vector. We note that(32) has exactly the same form as the objective function ofComplex NMF [10]. Thus, we can locally maximize Q(θ, θ′ )with respect to πk through an iterative algorithm described in[10], which consists of performing the following updates:∑
Uki (t)|[ŝ(ω, t)]k |ω,t k̂(ω,t)=1p(y(ω, t)|k̂(ω, t), θ = θ′ )k̂Care must be taken that when A(ω) is updated by (29), thecondition (3) is not necessarily satisfied and so it is necessary
to perform an appropriate normalization. Next, we consider
updating πk to values such that Q(θ, θ′ ) is guaranteed to be
non-decreasing. The term related to πk in the Q function is∑2
π 1
Q(θ, θ′ ) =k 2[ŝ(ω, t)]k − µk (ω, t) ,(32)
νk ω,t

ω

∑

∑∑

(35)

j

(36)
Readers are referred to [10] for the derivation process of(34), (35) and (36). Note that the update rules for σ 2 (ω)
and νk2 (ω) can also be obtained analytically by setting thepartial derivatives of Q(θ, θ′ ) at zero, but for convenience, we
fixed σ 2 (ω) and νk2 (ω) at constant values in the following
experiments.
C. Obtaining separated signals
One natural choice for an estimate of separated signals
would be ⟨s(ω, t)⟩p(s(ω,t)|y (ω,t),θ) . We can notice from⟨s(ω, t)⟩p(s(ω,t)|y (ω,t),θ)= ⟨⟨s(ω, t)⟩p(s(ω,t)|y (ω,t),k̂(ω,t),θ) ⟩p(k̂(ω,t)|y (ω,t),θ)= ⟨s̄k̂(ω,t) (ω, t)⟩p(k̂(ω,t)|y (ω,t),θ)=K∑

mk̂(ω,t) s̄k̂(ω,t) (ω, t) = ŝ(ω, t),(37)k̂(ω,t)=1

that ⟨s(ω, t)⟩p(s(ω,t)|y (ω,t),θ) amounts exactly to (33).
IV. EXPERIMENTS
We evaluated the performance of the proposed method
in terms of source separation ability. Although the present
method is intended to solve all the indeterminacy involvedin the frequency-domain underdetermined BSS problem, we
decided to preliminarily confirm the effect of the present
method only in terms of the indeterminacy with respect to theinverse mixing process. Thus, we assumed an oracle situationsuch that the true mixing matrix A(ω) is given.

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Frequency (kHz)2.0

intended to solve the indeterminacy involved in the underdetermined BSS problem. An experimental evaluation showedthat reasonably good separation was obtained with the present
method if the mixing matrix was known.

1.5

1.0

0.5

R EFERENCES

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Basis spectrum H1i (ω) of Source 1 (corresponding to the
piano sound). H11 (ω), · · · , H110 (ω) are each displayed in a column.

Analysis, John Wiley & Sons, 2001.

Figure. 1.

0

2

4

6

8

10

Time (s)Figure. 2.

Elementwise time-varying gain U1i (t) of Source 1. U11 (t),· · · , U110 (t) are each displayed in a row.

Table. ISIR S OF EACH SOURCE BEFORE AND AFTER SEPARATION [ D B]
Source1: Piano
Source2: Vocal
Source3: Drum

Before
-6.8
-6.1
2.9

After
17.7
15.4
14.0

We used a 10-second stereo musical signal with a sampling
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length of 64ms. The EM algorithm was run for 50 iterations.
The mixing matrix A(ω) was set at the true value, and allother parameters were initialized randomly. The number ofbasis spectra was set at I1 = · · · = IK = 10. The separated
signals were defined as (36). For an objective performance
criterion, we used the Source-to-Interference Ratio (SIR) [14]for a quantitative evaluation of the distortion caused by othersignals. The SIR is expressed in decibels (dB), and a higher
SIR indicates superior quality. We compared the SIRs of thesignals prior to separation with that of the separated signals
obtained with the present method. The result can be seen in
Tab I. The basis spectrum of the piano sound, Hki (ω), areshown in Fig 1, and the time-varying gain for each basis
spectrum, Uki (t), is shown in Fig 2.
Tab I confirms that there were significant improvements in
the SIRs. From Fig 1, 2, we observe a harmonic structure ineach basis spectrum and a fast rise followed by a fast decay ineach gain function. This should imply that the element signals
are successfully estimated such that each one is associated witha single piano note. We also confirmed through a number of
evaluations with different I1 , . . . , IK settings that I1 = · · · =IK = 10 gave the best performance. Overall, we confirmedthat reasonably good separation was obtained with the present
method if the mixing matrix A(ω) was known.

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V. CONCLUSION
This paper proposed a multichannel complex NMF method
based on the W-disjoint orthogonality of sources, which is978-1-4577-0254-9/11/$26.00 ©2011 IEEE

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