2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)MUSIC SIGNAL SEPARATION BASED ON BAYESIAN SPECTRAL AMPLITUDE
ESTIMATOR WITH AUTOMATIC TARGET PRIOR ADAPTATION†

Yuki Murota, † Daichi Kitamura, † Shunsuke Nakai, † Hiroshi Saruwatari,†
Satoshi Nakamura, ‡ Yu Takahashi, ‡ Kazunobu Kondo†

Nara Institute of Science and Technology, Nara, Japan‡
Yamaha Corporation, Shizuoka, Japan

ABSTRACT

2. CONTRIBUTION AND RELATION TO PRIOR WORKIn this paper, we propose a new approach for addressing music signal
separation based on the generalized Bayesian estimator with automatic prior adaptation. This method consists of three parts, namely,
the generalized MMSE-STSA estimator with a flexible target signal
prior, the NMF-based dynamic interference spectrogram estimator,and closed-form parameter estimation for the statistical model of thetarget signal based on higher-order statistics. The statistical model
parameter of the hidden target signal can be detected automaticallyfor optimal Bayesian estimation with online target-signal prior adaptation. Our experimental evaluation can show the eﬃcacy of the proposed method.

Motivated by the complementarity between the properties of the
MMSE-STSA estimator and SNMF, in this paper, we propose a
new approach based on the generalized Bayesian estimator with
automatic prior estimation suitable for music signal separation. Thismethod consists of three parts, namely, the generalized MMSESTSA estimator [12, 13] with a flexible target signal prior, the
SNMF-based interference spectrogram estimator, and a new closedform parameter estimation for the statistical model of the target
signal time sequence based on higher-order statistics.
Compared with the conventional methods, the proposed method
has the following advantages: (I) The target signal extraction is carried out via the generalized MMSE-STSA estimator so that the mixing of the time-frequency-domain complex-valued signals can beproperly considered without any approximation. (II) Thanks to the
SNMF-based spectrogram estimator, we can dynamically estimate
the nonstationary power spectra of the interference signal. (III) The
statistical model of the hidden target signal can be detected automatically only using the observable data, and can be used for optimal
Bayesian estimation with online target-signal prior adaptation.

Index Terms— Music signal separation, MMSE-STSA estimator, NMF, higher-order statistics
1. INTRODUCTION
In recent years, music signal separation has been a very active areaof signal-processing research. This technique is suitable for manypotential applications, e.g., controlling each source in a music signal in interactive 3D audio systems and realizing automatic musictranscription for each instrument player.
Common methods used for audio signal separation, which weremainly developed for speech enhancement, are nonlinear filtering algorithms such as Wiener filtering (WF) [1] and the minimum meansquare error short-time spectral amplitude (MMSE-STSA) estimator[2]. In particular, the MMSE-STSA estimator and its extended algorithms [3] are optimal Bayesian estimators based on the a priori
speech statistical model, resulting in a great improvement of soundquality. However, these methods cannot be eﬀectively applied tomusic signals because it is diﬃcult to deal with nonstationary interference signals. Also, the performance has strong dependence on
the selection of the a priori statistical model, which cannot be determined automatically.
As another means of signal separation, nonnegative matrix factorization (NMF) has been actively studied [4, 5, 6, 7], in which an
input spectrogram is decomposed into the product of a spectral basis matrix and its activation matrix. In particular, supervised NMF(SNMF) [8, 9, 10, 11], which includes a priori training with somesample sounds of a target instrument, can extract the target signal to some extent. Although SNMF can deal with nonstationary
source signals, it has an inherent drawback that the decomposition
includes approximation only valid for linear combinations of spectrograms, and is not strictly valid for time- (or time-frequency-) domain complex-valued mixtures. In addition, it is diﬃcult to implement the statistical model for time sequences.

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3. CONVENTIONAL METHOD
3.1. Generalized MMSE-STSA estimator [12, 13]
We apply short-time Fourier analysis to the observed signal, which
is a mixture of target and interference signals, to obtain the timefrequency-domain complex-valued signal
YR ( f, τ)+iYI ( f, τ) = (S R ( f, τ)+iS I ( f, τ))+(NR ( f, τ)+iNI ( f, τ)),(1)

where Y∗ ( f, τ) is the observed signal, S ∗ ({f, τ)} is the target signal,N∗ ( f, τ) is the interference signal, and ∗= R, I denote the real andimaginary parts of the signal, respectively. Also, f is the frequency
bin and τ is the frame index. The MMSE-STSA estimator provides
the amplitude spectrum of the target signal based on the MMSE criterion under the assumption that the amplitude spectrum of the target
signal obeys a Rayleigh distribution. The generalized MMSE-STSA
estimator [13] provides the amplitude spectrum of the target signal
under the assumption of a chi distribution.
3.2. SNMF [9]The mixture model of NMF approximately assumes the additivity of
amplitude (or power) spectrograms as√√√
YR2 ( f, τ)+YI2 ( f, τ) ≈ S R2 ( f, τ)+S I2 ( f, τ)+ NR2 ( f, τ)+NI2 ( f, τ) .
(2)

STFT

Y∗ ( f ,τ )~S ∗ ( f ,τ )Y∗ ( f ,τ )G ( f ,τ )Shape parameter
estimation
kurt target

G ( f ,τ )Spectral gain
calculation

Target kurtosis
estimation

ξ ( f ,τ )A posteriori
SNR estimation

PN~ ( f )The following equation represents the decomposition model of
SNMF using the trained supervision components F( f, k):√A( f, τ) = YR2 ( f, τ) + YI2 ( f, τ)∑∑≈H( f, n)U(n, τ),(9)F( f, k)V(k, τ) +~γ~ ( f ,τ )PN~ ( f )Interference signal
estimation by SNMF

4.2. Interference signal estimation by SNMF

ISTFT

A priori
SNR estimation

γ~ ( f ,τ )Fig. 1. Block diagram of proposed method.
SNMF [8, 9, 10] consists of two processes, namely, training of thetarget sound and separation of the observed signal. In the training
process, the supervised bases are trained as a dictionary of the target sound. Then, the observed spectrogram is decomposed into thetarget and other spectrograms using the supervised bases in the separation process.
4. PROPOSED METHOD
4.1. Generalized MMSE-STSA estimator
Figure 1 depicts a block diagram of the proposed method. In the nextpart, we describe these processes in detail.
In 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 byS˜∗ ( f, τ) = G( f, τ)Y∗ ( f, τ),√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))

n

k

(3)
(4)

where F( f, k) is a nonnegative element of the supervised basis matrix trained in advance, which involves 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∑ involves 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 can betrained using sample soundsof the target signal in the training pro∑cess. Hence, ideally,k F( f, k)V(k, τ) represents the target signal∑
components and n H( f, n)U(n, τ) represents the other components
diﬀerent from the∑target signals after the decomposition.
∑
We can use n H( f, n)U(n, τ) (or A( f, τ)− k F( f, k)V(k, τ)) as
the estimated amplitude spectrogram of the interference
signal for∑the generalized MMSE-STSA estimator. Thus, ( n H( f, n)U(n, τ))2
is regarded as a good estimate of PÑ ( f ) in (7) in the time-frequency
grids even if the interference sounds are nonstationary, which is common in actual music signals.
4.3. Target signal prior estimation
4.3.1. Shape parameter and kurtosis

where Γ(·) is the gamma function, Φ(a, b; k) = F 1 (a, b; k) is the confluent hypergeometric function, and()−1
ν( f, τ) = γ̃( f, τ)ξ̃( f, τ) 1 + ξ̃( f, τ) .
(5)
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],(6)γ̃( f, τ) =(YR2 + YI2 )/PÑ ( f ),(7)

where PÑ ( f ) is the estimated interference power spectral density andα is the forgetting factor.
In the generalized MMSE-STSA estimator, the a priori statisticalmodel of the target signal amplitude spectrum is set to [14]
(
)ρ
(
)
ρ
ρ
2
2ρ−1
2
p(x) =
x
exp
x
,
(8)
Γ(ρ) E[x2 ]E [x2 ]
where p(x) is the p.d.f. of signal x in the amplitude domain and ρ
is the shape parameter. Here, ρ = 1 gives a Rayleigh distributionthat corresponds to a Gaussian distribution in the time domain, anda smaller value of ρ corresponds to a super-Gaussian distribution
signal [15].
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. To solve these problems,
we propose the use of SNMF as the interference signal estimatorand estimate the shape parameter ρ using higher-order statistics ofthe target signal.

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Generally, we cannot obtain any a priori statistical model (8) fromthe training data (e.g., several octave notes of the target instrument)in SNMF because the statistical time structure is quite diﬀerent fromthat of the target signal S ∗ ( f, τ). Also, the target signal component∑k F( f, k)V(k, τ) in SNMF cannot be used because its accuracy is
not sometimes enough. Therefore, we inversely calculate the parameter of the target amplitude spectrogram in a data-driven manner,
utilizing two observable statistics of the input signal and interference
spectrogram estimated by SNMF.
Regarding the chi distribution p(x) in (8), the shape parameter ρ
can be written as follows:ρ = (µ4 /µ22 − 1)−1 ,(10)

where µ4 /µ22 is called the kurtosis and µm is the mth-order momentof the amplitude spectrum. µm is defined as∫ ∞xm p(x)dx.
(11)
µm =
0

From this relation, the shape parameter of the subjective target signal
can be estimated by obtaining its kurtosis value. In general, however,
it is diﬃcult to directly estimate the kurtosis of such a hidden target
signal because of the contamination by additive interference signals.
In the following subsections, a new algorithm of target kurtosis estimation is proposed for estimating of the shape parameter of thetarget p.d.f.
To cope with the mathematical problem that the mixing of thetarget and interference signals is additive but generally their higherorder moments are not additive, we introduce the cumulant, whichholds the additivity for additive variables. Meanwhile, in transformation from a waveform to its amplitude spectrum, the exponentiation operation is conducted but the cumulant does not have a straightforward relationship. In this case, we use the moment instead of the
cumulant. Thus, we propose to use moment-cumulant transformation.

1

 



4.3.2. Moment-cumulant transformation
In this section, we derive some formulas regarding moment-cumulant
transformation. They explicitly represent the relations between the
moment and cumulant in each order, which are useful for estimating
the kurtosis of the target amplitude spectrum.
First, the characteristic function ϕ x (it) of the random variable x
is defined as∫ ∞ϕ x (it) =eitx P(x)dx.
(12)¶ ED? E E E ED E E EE E EE E DD¶
´EEEEE E D
Fig. 2. Scores of each part.

observable, and (S R2 + S I2 )1/2 is a hidden value to be estimated. First,
we assume to obtain the following mth-order moments from the data:µm (YR ) = E[YRm ],µm (YI ) = E[YIm ],µm (NR ) = E[NRm ],µm (NI ) = E[NIm ].

−∞Then, we can define the mth-order moment µm (x) and the mth-order
cumulant κm (x) of x as follows:µm (x) =∂(m) ϕ x (it),∂it(m) t=0(13)κm (x) =∂(m) log ϕ x (it).
∂it(m)
t=0(14)

Next, polynomial forms of interrelations between the moment
and cumulant are derived below. From (13), the mth-order momentµm (x) can be rewritten as∂(m) exp(log ϕ x (it))∂it(m)
t=0∑∏=
exp(|π(m)|) (log ϕ x (it))[log ϕ x (it))](|B|)µm (x) =π(m)=B∈π(m)

∑ ∏

t=0

(15)

where we use a combinational form of Faà di Bruno’s formula,∂∏f (g(x))=f (|π(m)|) (g(x))[g(x))](|B|) ,
∂x(m)
π(m)
B∈π(m)(16)

where π(m) runs through the list of all partitions of a set of size m,
B ∈ π(m) means that B is one of the blocks into which the set ispartitioned, and |B| is the size of the set B.
In the same manner, from (14), the mth-order cumulant κm (x) is
given byκm (x) =

∑
π(m)

=

log(|π(m)|) (ϕ x (it))

∏

π(m)

B∈π(m)∑∏−(−1)|π(m)|−1 (|π(m)| − 1)!
µ|B| (NR ).
π(m)

(22)

B∈π(m)(23)

In the same manner, we can estimate the statistics of the squared
variable of S I . Given µm (S R2 ) and µm (S I2 ), we can calculate the cumulant of the power spectrum S R2 + S I2 asκm (S R2 + S I2 ) = κm (S R2 ) + κm (S I2 )∑∏= (−1)|π(m)|−1 (|π(m)| − 1)!
µ|B| (S R2 )π(m)B∈π(m)∑∏+(−1)|π(m)|−1 (|π(m)| − 1)!
µ|B| (S I2 ),
π(m)

(24)

B∈π(m)and the mth-order moment of the power spectrum is given by∑ ∏µm (S R2 + S I2 ) =κ|B| (S R2 + S I2 ).

(25)

π(m) B∈π(m)[ϕ x (it))](|B|)

B∈π(m)∑∏(−1)|π(m)|−1 (|π(m)| − 1)!
µ|B| (x).
π(m)κm (S R ) = κm (YR ) − κm (NR )∑∏= (−1)|π(m)|−1 (|π(m)| − 1)!
µ|B| (YR )π(2m) B∈π(2m)κ|B| (x),∑Generally, the cumulant has the property of additivity for additive
independent variables, i.e., κm (a + b) = κm (a) + κm (b). Using thisrelation and (17), we can estimate the cumulant of the real part of
the target signal S R =YR − NR as

The statistics of the squared variable of S R is given by∑ ∏µm (S R2 ) = µ2m (S R ) =κ|B| (S R ).

π(m) B∈π(m)

(m)

(18)
(19)
(20)
(21)

Furthermore, the mth-order moment of the amplitude spectrum (S R2 +S I2 )1/2 is

t=0

(17)

B∈π(m)

1µm ((S R2 + S I2 ) 2 ) = µ m2 (S R2 + S I2 ).

(26)

Using (18)–(26), we can estimate the resultant kurtosis of the target
amplitude spectrum as

4.3.3. Estimation of target signal kurtosis from observations
In this section, we estimate the amplitude-domain kurtosis of thetarget signal. First, we express the target kurtosis by using complexvalued variables as intrinsic parameters. After that, we show that the
kurtosis can be represented in the amplitude-spectrogram domain.
Hereafter, we ignore the indexes f, τ, k, and n of each signal. Only
the statistics of (YR + iYI ) and (NR2 + NI2 )1/2 obtained by SNMF are

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1

kurttarget =µ4 ((S R2 + S I2 ) 2 )
1µ22 ((S R2 + S I2 ) 2 )N (µm (YR ), µm (YI ), µm (NR ), µm (NI ))=,D (µm (YR ), µm (YI ), µm (NR ), µm (NI ))

(27)

12

(a)

11

11

10

10

SDR [dB]

SDR [dB]

12

WF + SNMF

98
7
6

MMSE-STSA + SNMF

Proposed method
16

(b)

(c)

14

SDR [dB]

SNMF

9
8

12
10

7

8

6

6

Fig. 3. Average SDRs for extraction of (a) clarinet signal, (b) oboe signal, and (c) cello signal for each method.
where

5. EXPERIMENTS
5.1. Experimental conditions

N (µm (YR ), µm (YI ), µm (NR ), µm (NI ))= µ4 (YR ) + µ4 (YI ) − µ4 (NR ) − µ4 (NI )+ 6µ22 (NR ) + 6µ22 (NI ) + 2µ2 (YR )µ2 (YI ) + 2µ2 (NR )µ2 (NI )− 6µ2 (YR )µ2 (NR ) − 6µ2 (YI )µ2 (NI )− 2µ2 (YR )µ2 (NI ) − 2µ2 (YI )µ2 (NR ),D (µm (YR ), µm (YI ), µm (NR ), µm (NI ))= µ22 (YR ) + µ22 (YI ) + µ22 (NR ) + µ22 (NI ) + 2µ2 (YR )µ2 (YI )− 2µ2 (YR )µ2 (NR ) − 2µ2 (YR )µ2 (NI ) − 2µ2 (YI )µ2 (NR )− 2µ2 (YI )µ2 (NI ) + 2µ2 (NR )µ2 (NI ).

(28)

(29)

Next, the sums of the 4th-order moments µ4 (YR ) + µ4 (YI ) andµ4 (NR ) + µ4 (NI ) are represented by the amplitude-domain kurtosis of
the observed signal spectrum and the interference signal spectrum as() µ4 (A)µ4 (YR ) + µ4 (YI ) = µ22 (YR ) + µ22 (YI ) + 2µ2 (YR )µ2 (YI ) 2µ2 (A)− 2µ2 (YR )µ2 (YI ),(30)∑() µ4 ( n HU)µ4 (NR ) + µ4 (NI ) = µ22 (NR ) + µ22 (NI ) + 2µ2 (NR )µ2 (NI ) 2 ∑µ2 ( n HU)− 2µ2 (NR )µ2 (NI ).

(31)

If we assume that the real and imaginary parts are i.i.d., µ2 (YR )equals µ2 (YI ) and µ2 (NR ) equals µ2 (NI ). Under this assumption and(15) and (17), we obtain the following relation for the SNMF output:1µ2 (A),21 ∑µ2 (NR ) = µ2 (NI ) = µ2 ( n HU).
2µ2 (YR ) = µ2 (YI ) =

(32)
(33)

Finally, using (30)–(33), (27) is rewritten as follows:
kurttarget =∑∑∑µ4 (A)−µ4 ( n HU)+4µ22 ( n HU)−4µ2 (A)µ2 ( n HU).
∑∑µ22 (A)+µ22 ( n HU)−2µ2 (A)µ2 ( n HU)(34)

The shape parameter of the target p.d.f. can be estimated using (10)and (34). Note that all the estimates can be obtained from the resultof SNMF without using any waveforms; this implies a good applicability to the combination with SNMF.

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In this experiment, we compared four methods, i.e., simple SNMF [9],WF with SNMF-based interference estimation (WF+SNMF) [16,17, 18, 19], the MMSE-STSA estimator with SNMF-based interference estimation (MMSE-STSA+SNMF), and the proposed
method. WF and the MMSE-STSA estimator utilized the interference spectrogram estimated by SNMF. We used three instrumental
signals, namely, an oboe, clarinet, and cello, as the target sounds(each melody part is depicted in Fig. 2). These signals were artificially generated by a MIDI synthesizer, and the observed signals
were produced by mixing two sources selected from three signals
with the same power. In estimation of the interference signal using
SNMF, we used the same MIDI sounds of the target instruments assupervision for the training process. The training sounds contained
two octave notes that cover all the notes of the target signal in theobserved signal. The spectrograms were computed using a 92-mslong rectangular window with a 46-ms overlap shift. Moreover, thenumber of trained bases was 100 and the number of other bases was
50. In the proposed method, the forgetting factor α and amplitude
compression parameter β were set to 0.1 and 1.0, respectively.
5.2. Experimental results and discussion
We used the signal-to-distortion ratio (SDR) defined in [20] as theevaluation score. SDR indicates the overall quality of the separated
target sound, showing high separation and less artificial distortion.
Figure 3 shows the average SDRs for each method and each target instrument. From the SDR results, we can confirm that the separation performance of the proposed method is better than those of
the other methods. This result indicates the eﬃcacy of introducing
the flexible a priori statistical model of the target signal. The simple MMSE-STSA estimator also assumes the fixed a priori model ofthe Gaussian distribution but the assumption is not appropriate for amusic target signal. In contrast, the proposed method almost always
uses more spiky p.d.f. (ρ < 1), which enhances the true sparsenessof the target music signal.
6. CONCLUSIONSIn this paper, we propose a new approach for addressing music signal separation based on the generalized Bayesian estimator with automatic prior adaptation. From the experimental evaluation, it isfound that the proposed method outperforms competitive methods,
namely, simple NMF, WF, and the MMSE-STSA estimator with afixed Gaussian prior.

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