STATISTICAL MODELING OF BINAURAL SIGNAL AND ITS APPLICATION TO BINAURALSOURCE SEPARATION
1

Yuki Murota, 2 Daichi Kitamura, 3 Shoichi Koyama, 3 Hiroshi Saruwatari, 1 Satoshi Nakamura
1

2

Nara Institute of Science and Technology, Nara, JapanThe Graduate University for Advanced Studies (SOKENDAI), Kanagawa, Japan
3
The University of Tokyo, Tokyo, Japan
ABSTRACTThis paper addresses a new statistical model of binaural signals
and its application to eﬃcient binaural source separation. Binauralsource separation is always required to retain a spatial cue of theseparated sound, such as a head-related transfer function (HRTF).
However, the direct use of an HRTF is not realistic because thisinformation is normally not known in advance. To cope with thisproblem, first, we focus on the diﬀerence between signal probability
density functions at both ears, which can be blindly estimated byusing our previous work on higher-order statistics. Next, we derivea sound-localization-preserved generalized minimum mean-squareerror short-time spectral amplitude estimator. Objective and subjective experiments show the eﬃcacy of the proposed method in termsof spatial quality.
Index Terms— Binaural source separation, MMSE-STSA estimator, NMF, higher-order statistics, sound localization
1. INTRODUCTION
Audio signal separation has received much attention in signalprocessing research, and many studies have been published in thelast decade. Techniques for signal separation have been developed
for many audio applications, including target speech enhancement
for hearing-aid systems [1] and for controlling each source in amusic tune in interactive 3D audio systems [2]–[5]. In this paper,
we also address such an audio signal separation problem, especially
focusing on a signal provided in a binaural format [6].
Compared with simple multichannel signal processing, binaural signal separation includes a relatively diﬃcult task, namely, extraction of a specific sound while maintaining its spatial properties.
This is because deterioration of the spatial quality of the separated
sound has an adverse eﬀect on human’s 3D audio perception. Several
methods have been proposed for binaural signal separation, mainly
for blind speech separation and enhancement. To preserve a soundlocalization cue such as the interaural level diﬀerence, these methods [7]–[10] apply an equi-binaural spectral gain to both the left andright ears of the listener, which can be calculated via, e.g., Wiener
filtering (WF) [11] and the minimum mean-square error short-time
spectral amplitude (MMSE-STSA) estimator [12, 13]. These methods have a drawback that they do not take account of richer binaural information, e.g., waveform modification owing to diﬀraction
and reflection. Ideally, the best way to enhance it is to explicitly
use an important spatial cue, such as a head-related transfer function (HRTF) [6]. We have proposed an algorithm [14] to introduce
a user’s HRTF into a multichannel MMSE-STSA estimator but thismethod was not a blind system; the accurate measurement of the

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HRTF was required in advance, which is sometimes impossible inpractice.
2. CONTRIBUTION AND RELATION TO PRIOR WORKIn another context of signal separation such as informed source
separation, the authors have proposed a combination [15] of supervised nonnegative matrix factorization (SNMF) [16]–[19] anda prior-model-adapted generalized MMSE-STSA estimator [20] todeal with music signal separation. This method requires a certain
supervision such as a music scale of the target instrument, but it efficiently extracts a target sound composed with an arbitrary melody
from the observed monaural mixture. In this method, thanks to
the higher-order statistics analysis, hidden parameters of the target
statistical model can be estimated in each frequency subband.
Motivated by the above-mentioned prior work, in this paper,
we propose a new spatial-cue-aware binaural signal separation algorithm without knowing the user’s HRTF. The key idea and advantages of the proposed method are summarized as follows: (I) Insteadof using an HRTF, we introduce a target statistical model to express
the diﬀerence between user’s left- and right-ear signals. For instance,
if the target sound is located on the left-hand side, the extracted
signal in the left ear will obey a spiky probability density function(p.d.f.) but the right-ear signal will have a smooth p.d.f. becauseof many diﬀracted waves and the weak direct wave. (II) The statistical models for each ear can be accurately determined using only
observable data [15]. Thus, the proposed strategy is HRTF-blind anda new attempt at establishing a statistical HRTF approach. (III) Using the same idea as in [10], an equi-binaural spectral gain is derived
on the basis of statistical-HRTF-adapted generalized MMSE-STSA
estimators. This avoids the marked deterioration of spatial quality.
3. CONVENTIONAL METHOD
3.1. Single-Channel Music Signal Enhancement
In our previous work, we proposed single-channel music signal enhancement based on the generalized Bayesian estimator with automatic target prior adaptation [15]. Here, we use the SNMF-based
dynamic interference spectrogram estimator and closed-form parameter estimation for the statistical model of the target signal based onhigher-order statistics. The details are described below.
3.1.1. Music signal separation by generalized MMSE-STSA estimator with automatic target prior adaptation
We apply short-time Fourier analysis to the observed signal, which
is a mixture of target and interference signals, to obtain the time-

ICASSP 2015

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Amplitude

Amplitude(b) Right ear(a) Left ear

0.3
0.2

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0.05

0.1

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1

2
Time [s]

3

4

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1

2
Time [s]34Fig. 1. Diﬀerence in signals at left and right ears.
frequency-domain complex-valued signalx( f, τ) = s( f, τ)+n( f, τ),(1)where x( f, τ) is the observed signal, s( f, τ) is the target signal, n( f, τ)
is the interference signal, f is the frequency bin number, and τ is thetime-frame index.
For the generalized MMSE-STSA estimator, the amplitude
spectrum of the target signal is estimated on the basis of the MMSE
criterion under a certain target prior. The processed signal s̃( f, τ) via
the generalized MMSE-STSA estimator is given bys̃( f, τ) = G( f, τ)x( f, τ),√)1/β(ν( f, τ) Γ(ρ+0.5) Φ(0.5−ρ, 1, −ν( f, τ))G( f, τ) =,··γ( f, τ)Γ(ρ)Φ(1−ρ, 1, −ν( f, τ))

(2)
(3)

where Γ(·) is the gamma function, Φ(a, b; k) = F 1 (a, b; k) is the confluent hypergeometric function, β is the amplitude compression parameter, and()−1
ν( f, τ) = γ̃( f, τ)ξ̃( f, τ) 1 + ξ̃( f, τ) .
(4)
Here, ξ̃( f, τ) and γ̃( f, τ) are the estimated a priori and a posteriori
SNRs, respectively, which are defined asξ̃( f, τ) = αγ̃( f, τ − 1)G2 ( f, τ) + (1 − α)max[γ( f, τ) − 1, 0],(5)γ̃( f, τ) =|x( f, τ)|2 /Pñ ( f ),(6)

where Pñ ( f ) is the estimated interference power spectral density andα is the forgetting factor.
In the generalized MMSE-STSA estimator, the a priori statistical model of the target signal amplitude spectrum is set to the chi
distribution
p(x) = 2ϕρ Γ(ρ)−1 x2ρ−1 exp(−ϕx2 ),(7)

where p(x) is the p.d.f. of signal x in the amplitude domain, ϕ =ρ/E{|x|2 }, and ρ is the shape parameter. Here, ρ = 1 gives a Rayleighdistribution that corresponds to a Gaussian distribution in the time
domain, and a smaller value of ρ corresponds to a super-Gaussian
distribution signal.
In the generalized MMSE-STSA estimator, to calculate γ̃( f, τ),dynamic estimation is required if the interference signal is nonstationary, and estimation of the shape parameter ρ, which depends on
the type of target signal, is also required.
3.1.2. Interference estimation by SNMF
The following equation represents the decomposition model of
SNMF using the trained supervision components F( f, k):∑∑A( f, τ) = |x( f, τ)| ≈F( f, k)V(k, τ) +H( f, n)U(n, τ), (8)
k

n

495

Fig. 2. Conventional and proposed binaural models.
where F( f, k) is a nonnegative element of the supervised basis matrix
trained in advance, which comprises spectral patterns of the target
signal as column vectors, V( f, k) is a nonnegative element of an activation matrix that corresponds to F( f, k), H( f, n) represents a nonnegative element of the other basis matrix, which∑ comprises residual
spectral patterns that cannot be expressed by k F( f, k)V(k, τ), andU(n, τ) is a nonnegative element of the activation matrix that corresponds to H( f, n). Moreover, k is the basis index of F( f, k) andn is the basis index of H( f, n). The supervised basis matrix canbe trained using sample ∑sounds of the target signal in the training
process. Hence, ideally,k F( f, k)V(k, τ) represents the target sig∑
nal components and n H( f, n)U(n, τ) represents the other components∑ diﬀerent from the target signals after the decomposition. Thus,( n H( f, n)U(n, τ))2 is regarded as a good estimate of PÑ ( f ) in (6)in the time-frequency grids.
4. PROPOSED METHOD
4.1. Motivation and Strategy
In binaural source separation, it is desirable to use binaural cues suchas an HRTF to improve the separation performance. However, thisis diﬃcult because we cannot obtain the HRTF of an unknown user
blindly. Therefore, considering the HRTF from a statistical viewpoint, we eﬃciently express binaural cues using a statistical model
based on the chi distribution. First, in the binaural signal, the p.d.fs.
of the signals of each ear are assumed to be diﬀerent by the influence of the diﬀerence in the signals arriving at each ear, and we canobtain a binaural cue based on the diﬀerence in these p.d.fs. For example, Fig. 1 shows amplitude of left- and right-ear signals of celloin 6 kHz subband, where the sound comes from the left-hand side.
As shown in this figure, the left-ear signal contains several dominantcomponents, which yields spiky p.d.f., but nothing in the right ear.
Next, we introduce the chi distribution to represent the p.d.f. of each
ear. By applying the target prior adaption algorithm [15] to each ear,
it is possible to obtain suitable parameters of the p.d.fs. automatically. This means that we can convert the conventional deterministic
HRTF estimation problem into a parameter estimation problem of
the corresponding statistical model (see Fig. 2). This can also enable
adaptation to unknown users.
However, there is a problem when we apply this strategy
to a binaural source separation system. Generally speaking, astatistical-model-based source separation method (e.g., the generalized MMSE-STSA estimator) only provides the statisticallyfluctuating spectral gains for each of ears independently. However,the fluctuation of the gain function in interaural level diﬀerences atthe left and right ears causes the deterioration in sound localization.
To resolve this problem, we derive a new optimal spectral gain that

minimizes the residual interference power in terms of the MMSE
under the condition that the spectral gains are equivalent in bothears. Hereafter, we call this gain the equi-binaural optimal spectral
gain.
4.2. Signal Mixture Model
We consider a mixing model with two inputs, i.e., two ears, andassume that the observed signal contains the target signal and aninterference signal. Hereafter, the observed signal vector in the timefrequency domain, x( f, τ) = [xL ( f, τ), xR ( f, τ)]T , is given byx( f, τ) = h( f )s( f, τ) + n( f, τ),(9)where h( f ) = [hL ( f ), hR ( f )] is the column vector of the transfer
functions between the target source and each ear, s( f, τ) is the target
signal component, and n( f, τ) = [nL ( f, τ), nR ( f, τ)]T is the column
vector of the interference signal. Throughout this paper, the subscripts ∗ (∗ = {L, R}) represent the signals obtained at the left andright ears, respectively.
T

4.3. Derivation of Equi-Binaural Optimal Spectral Gain
The derivation of the equi-binaural optimal spectral gain is described
below. This is the extended version of [10] for a generalized costfunction, and can be formulated as the minimization problem of thefollowing error e:[{}2
e =E |hL ( f )s( f, τ)|β − (G( f, τ)|xL ( f, τ)|)β{}2 ]+ |hR ( f )s( f, τ)|β − (G( f, τ)|xR ( f, τ)|)β ,(10)where G( f, τ) is the equi-binaural spectral gain, which is considered
as a variable. The optimization problem based on (10) is given by
Gopt ( f, τ)[{}2= argmin E |hL ( f )s( f, τ)|β −{(G( f, τ)−GL ( f, τ))+GL ( f, τ)}β |xL ( f, τ)|βG( f,τ){}2 ]+ (|hR ( f )s( f, τ))β |−{(G( f, τ)−GR ( f, τ))+GR ( f, τ)}β |xR ( f, τ)|β[= argmin E {|hL ( f )s( f, τ)|β − (GL ( f, τ)|xL ( f, τ)|)β }2G( f,τ)+ {|hR ( f )s( f, τ)|β − (GR ( f, τ)|xR ( f, τ)|)β }2+ {(Gβ ( f, τ) − GβL ( f, τ))|xL ( f, τ)|β }2]+{(Gβ ( f, τ) − GβR ( f, τ))|xR ( f, τ)|β }2 + 2C ,(11)

where Gopt ( f, τ) is the equi-binaural optimal spectral gain to be estimated, and GL ( f, τ) and GR ( f, τ) are individual spectral gains for
the L and R ears, respectively, which are auxiliary parameters forcalculating an approximate solution of Gopt ( f, τ) because the direct
Bayesian estimation of Gopt ( f, τ) is diﬃcult. In addition, C is related
to the correlation between the estimation error and the observed signal in each channel when we estimate the target speech signals in
the L and R ears using the parameters GL ( f, τ) and GR ( f, τ), and isdefined by
C ={Gβ ( f, τ) − GβL ( f, τ)}· {(GL ( f, τ)|xL ( f, τ)|)β − |hL ( f, τ)s( f, τ)|β }|xL ( f, τ)|β
+ {Gβ ( f, τ) − GβR ( f, τ)}· {(GR ( f, τ)|xR ( f, τ)|)β − |hR ( f, τ)s( f, τ)|β }|xR ( f, τ)|β .

approximately considering C to be negligible. Hence, the residual
3rd and 4th terms, i.e., {(Gβ ( f, τ)−GβL ( f, τ))|xL ( f, τ)|β }2 +{(Gβ ( f, τ)−GβR ( f, τ))|xR ( f, τ)|β }2 , should be minimized. This problem can beformulated as
Gopt ( f, τ)[≒ argmin E {(Gβ ( f, τ) − GβLopt ( f, τ))|xL ( f, τ)|β }2G( f,τ)]+{(Gβ ( f, τ) − GβRopt ( f, τ))|xR ( f, τ)|β }2 ,(13)
subject to[]
GLopt ( f, τ) = argmin E {|hL ( f )s( f, τ)|β −(GL ( f, τ)|xL ( f, τ)|)β }2 , (14)GL ( f,τ)[]GRopt ( f, τ) = argmin E {|hR ( f )s( f, τ)|β −(GR ( f, τ)|xR ( f, τ)|)β }2 , (15)GR ( f,τ)

where GLopt ( f, τ) and GRopt ( f, τ) are the L- and R-ear optimal spectral gains, respectively.
To solve (13), we first obtain GLopt ( f, τ) and GRopt ( f, τ) from thegeneralized MMSE-STSA estimator in (14) and (15), then by substituting them into (13), we solve the following equation in G( f, τ):∂e=Gβ ( f, τ)|xL ( f, τ)|2β − GβLopt ( f, τ)|xL ( f, τ)|2β∂G( f, τ)+ Gβ ( f, τ)|xR ( f, τ)|2β − GβRopt ( f, τ)|xR ( f, τ)|2β
=0.

(16)

The solution of (16) is given by1/β
 β GLopt ( f, τ)|xL ( f, τ)|2β + GβRopt ( f, τ)|xR ( f, τ)|2β  .
Gopt ( f, τ) = |xL ( f, τ)|2β + |xR ( f, τ)|2β(17)4.4. Shape Parameter and Kurtosis
In (17), we need to calculate GLopt ( f, τ) and GRopt ( f, τ), which include a shape parameter ρ that should represent the a priori distribution of the target signal. In Sects. 4.4 and 4.5, we describe how toblindly estimate ρ.
Regarding the chi distribution p(x) in (7), the mth-order moment
can be written as∫ ∞Γ(ρ + m2 ) − m
(18)
ϕ 2.
µm (x) =xm p(x)dx =Γ(ρ)
0Then, the kurtosis κ of the chi distribution is calculated asκ = µ4 (x)/µ22 (x) = (ρ + 1)/ρ.

(19)

Therefore, the shape parameter ρ is given byρ = (κ − 1)−1 .

(20)

From this relation, the shape parameter of the target signal can be
estimated by obtaining its amplitude-domain kurtosis value. In general, however, it is diﬃcult to directly estimate the kurtosis of a target
signal because of its contamination by additive interference signals.
4.5. Estimation of Hidden Target Kurtosis and Gain Function(12)

We discuss the minimization of (11). First, the 1st and 2nd termson the right-hand side correspond to the problem of target signal
estimation in each ear. These terms can be minimized if we obtain
the optimal values of GL ( f, τ) and GR ( f, τ) using the generalized
MMSE-STSA estimator described in Sect. 3. Next, C in the 5th termon the right-hand side can be disregarded if the parameters GL ( f, τ)and GR ( f, τ) provide an accurate estimate of the target signals by

496

In our previous work, we proposed an algorithm for target kurtosis
estimation in additive signals, which can be derived from the closedform relation in higher-order statistics. In this algorithm, the resultant kurtosis of the target amplitude spectrum is estimated as∑∑∑µ4 (A∗ )−µ4 ( n (HU)∗ )+4µ22 ( n (HU)∗ )−4µ2 (A∗ )µ2 ( n (HU)∗ )
κ∗ =
,
∑
∑
µ22 (A∗ )+µ22 ( n (HU)∗ )−2µ2 (A∗ )µ2 ( n (HU)∗ )(21)

Φ(0.5 − (κL − 1)−1 , 1, −ν̃L ( f, τ))·Φ(1 − (κL − 1)−1 , 1, −ν̃L ( f, τ))()|xR ( f, τ)|2β (ν̃R ( f, τ))β/2 Γ (κR − 1)−1 + 0.5+(){|xL ( f, τ)|2β + |xR ( f, τ)|2β }γ̂Rβ ( f, τ)Γ (κR − 1)−1
}1/β
Φ(0.5 − (κR − 1)−1 , 1, −ν̃R ( f, τ)).
(22)·
Φ(1 − (κR − 1)−1 , 1, −ν̃R ( f, τ))Equi-gain WF

SNMF
Equi-gain MMSE-STSA
Generalized MMSE-STSA

Gain-min MMSE-STSA
Proposed method

87SDR [dB]

where we ignore the indexes f and τ for saving the space. For thedetailed derivation of (21), see Ref. [15].
The shape parameter of the target signal p.d.f. at each ear can be
estimated using the kurtosis and (20). Therefore, the equi-binaural
optimal spectral gain estimated by the proposed method is obtained
as follows by substituting (3) into (17):{ |x ( f, τ)|2β (ν̃ ( f, τ))β/2 Γ ((κ − 1)−1 + 0.5)
L
L
L
G̃opt ( f, τ) =
(
)
β
β
2β
2
{|xL ( f, τ)| + |xR ( f, τ)| }γ̂L ( f, τ)Γ (κL − 1)−1

6

5

4
-90

-75

-60

-45

-30

-15

0

15

30

45

60

75

90

Angle [deg]Fig. 3. Average SDRs for each method and each direction.

The final output is given by s̃∗ ( f, τ) = G̃opt ( f, τ)x∗ ( f, τ).
Proposedmethod

5. EVALUATION EXPERIMENTS

Generalized
MMSE-STSA

5.1. Experimental Conditions
In this experiment, we used four binaural instrumental signals,
namely, an oboe, clarinet, cello, and piano, where the target instrument s( f, τ) is the oboe (each melody part is depicted in [15]). Thesesignals were artificially generated by a MIDI synthesizer and the directions of arrival of these signals were set from −90◦ to 90◦ with
15◦ intervals by using the corresponding HRTF h( f ). The HRTFs
were selected from the open database of the “Samurai” dummy head[21]. The observed signals were produced by mixing two sources
selected from the target signal and the other three signals with thesame power. In this observed signal, the target and interference
signals were located in the same direction. In the estimation of
the interference signal using SNMF, we used artificial clean MIDI
sounds of the target instrument without an HRTF as supervision for
the training process. The training sounds contained two octave notesthat covered all the notes of the target signal in the observed signal.
The sampling frequency of all signals was 44.1 kHz. Spectrograms
were computed using a 92-ms-long rectangular window with an
11-ms-long overlap shift. Moreover, the number of trained bases
was 100 and the number of other bases was 50. The forgetting factor
α was 0.97, and the amplitude compression parameter β was 1.0.
5.2. Objective Experiment
In the objective experiment, we compared six methods, i.e., SNMF
(SNMF) [19], the equi-binaural spectral gain via WF (Equi-gainWF) [9], the equi-binaural spectral gain via the MMSE-STSA estimator (Equi-gain MMSE-STSA) [10], the minimum-gain-based
MMSE-STSA estimator (Gain-min MMSE-STSA) [8], the generalized MMSE-STSA estimators independently applied to each ear(Generalized MMSE-STSA), and the equi-binaural spectral gain
via the generalized MMSE-STSA estimator (Proposed method).
Every method uses the same SNMF as the interference estimator.
We used the signal-to-distortion ratio (SDR) defined in [22] as theevaluation score. The SDR indicates the overall quality of the separated target sound, and is high in the case of high separation, lowartificial distortion, and low spatial distortion.
Figure 3 shows the average SDRs for each method and each direction. From these results, we can confirm that the separation performance of the proposed method is better than those of the other

497

0

10

20

30 40 50 60 70
Preference score [%]8090Fig. 4. Result of XAB test for evaluation of spatial quality.
methods. This result indicates the eﬃcacy of introducing the flexible
a priori statistical model of the target signal and equi-binaural spectral gain. The simple MMSE-STSA estimators (Equi-gain MMSESTSA and Gain-min MMSE-STSA) also assume the fixed a priori
model of the Gaussian distribution but the assumption is not appropriate for representing a music signal and a spatial diﬀerence between both ears. In contrast, the proposed method automaticallychooses a spikier p.d.f. (ρ ≪ 1) for the ear closer to the source location and a smoother p.d.f. (ρ ≃ 1) for the opposite ear. These p.d.fs.
match the binaural target.
5.3. Subjective Experiment
We next conducted a subjective test to evaluate the performance ofthe proposed method, focusing on the human impression of the separated signal from the viewpoint of spatial quality. In the subjective
experiment, we employed the XAB method and compared two methods, i.e., Generalized MMSE-STSA and Proposed method. The participants in the experiment comprised four males and two females.
Figure 4 shows the result of the subjective experiment, whichindicates that the proposed method using the equi-binaural spectral
gain markedly outperforms Generalized MMSE-STSA. Therefore,
we confirmed the eﬀectiveness of using the equi-binaural spectral
gain to improve the spatial quality.
6. CONCLUSIONSIn this paper, to address the eﬀect of statistical models for bothears on binaural signal source separation, we applied the generalized MMSE-STSA estimator with automatic prior adaptation to abinaural signal. The proposed method of binaural signal separation using equi-binaural spectral gain can also improve the soundlocalization properties. From the results of and subjective experiments, it was found that the proposed method outperforms conventional methods from the viewpoint of separation performance andsound-localization preservation.

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