DEEP NEURAL NETWORKS FOR SINGLE CHANNEL SOURCE SEPARATION
Emad M. Grais, Mehmet Umut Sen, Hakan Erdogan

arXiv:1311.2746v1 [] 12 Nov 2013

Faculty of Engineering and Natural Sciences,
Sabanci University, Orhanli Tuzla, 34956, Istanbul.
{grais, umutsen, haerdogan}@sabanciuniv.edu
ABSTRACT
In this paper, a novel approach for single channel source separation
(SCSS) using a deep neural network (DNN) architecture is introduced. Unlike previous studies in which DNN and other classifiers
were used for classifying time-frequency bins to obtain hard masks
for each source, we use the DNN to classify estimated source spectra
to check for their validity during separation. In the training stage, the
training data for the source signals are used to train a DNN. In the
separation stage, the trained DNN is utilized to aid in estimation of
each source in the mixed signal. Single channel source separation
problem is formulated as an energy minimization problem where
each source spectra estimate is encouraged to fit the trained DNN
model and the mixed signal spectrum is encouraged to be written
as a weighted sum of the estimated source spectra. The proposed
approach works regardless of the energy scale differences between
the source signals in the training and separation stages. Nonnegative
matrix factorization (NMF) is used to initialize the DNN estimate
for each source. The experimental results show that using DNN initialized by NMF for source separation improves the quality of the
separated signal compared with using NMF for source separation.
Index Terms— Single channel source separation, deep neural
network, nonnegative matrix factorization.
1. INTRODUCTION
Single channel audio source separation is an important and challenging problem and has received considerable interest in the research
community in recent years. Since there is limited information in the
mixed signal, usually one needs to use training data for each source
to model each source and to improve the quality of separation. In
this work, we introduce a new method for improved source separation using nonlinear models of sources trained using a deep neural
network.
1.1. Related work
Many approaches have been introduced so far to solve the single channel source separation problem. Most of those approaches
strongly depend on training data for the source signals. The training
data can be modeled using probabilistic models such as Gaussian
mixture model (GMM) [1, 2, 3], hidden Markov model (HMM) or
factorial HMM [4, 5, 6]. These models are learned from the training
data and usually used in source separation under the assumption that
the sources appear in the mixed signal with the same energy level
as they appear in the training data. Fixing this limitation requires
complicated computations as in [7, 8, 9, 10, 11, 12]. Another approach to model the training data is to train nonnegative dictionaries
for the source signals [13, 14, 15]. This approach is more flexible

with no limitation related to the energy differences between the
source signals in training and separation stages. The main problem
in this approach is that any nonnegative linear combination of the
trained dictionary vectors is a valid estimate for a source signal
which may decrease the quality of separation. Modeling the training
data with both nonnegative dictionary and cluster models like GMM
and HMM was introduced in [16, 17, 18, 19] to fix the limitation
related to the energy scaling between the source signals and training
more powerful models that can fit the data properly. Another type of
approach which is called classification-based speech separation aims
to find hard masks where each time-frequency bin is classified as
belonging to either of the sources. For example in [20], various classifiers based on GMM, support vector machines, conditional random
fields, and deep neural networks were used for classification.

1.2. Contributions
In this paper, we model the training data for the source signals using a single joint deep neural network (DNN). The DNN is used as
a spectral domain classifier which can classify its input spectra into
each possible source type. Unlike classification-based speech separation where the classifiers are used to segment time-frequency bins
into sources, we can obtain soft masks using our approach. Single
channel source separation problem is formulated as an energy minimization problem where each source spectral estimate is encouraged
to fit the trained DNN model and the mixed signal spectrum is encouraged to be written as a weighted sum of the estimated source
spectra. Basically, we can think of the DNN as checking whether
the estimated source signals are lying in their corresponding nonlinear manifolds which are represented by the trained joint DNN.
Using a DNN for modeling the sources and handling the energy differences in training and testing is considered to be the main novelty
of this paper. Deep neural network (DNN) is a well known model
for representing the detailed structures in complex real-world data
[21, 22]. Another novelty of this paper is using nonnegative matrix
factorization [23] to find initial estimates for the sources rather than
using random initialization.

1.3. Organization of the paper
This paper is organized as follows: In Section 2 a mathematical formulation for the SCSS problem is given. Section 3 briefly describes
the NMF method for source separation. In Section 4, we introduce
our new method. We present our experimental results in Section 5.
We conclude the paper in Section 6.

2. PROBLEM FORMULATION
In single channel source separation problems, the aim is to find estimates of source signals that are mixed on a single channel y(t). For
simplicity, in this paper we assume the number of sources is two.
This problem is usually solved in the short time Fourier transform
(STFT) domain. Let Y (t, f ) be the STFT of y(t), where t represents
the frame index and f is the frequency-index. Due to the linearity of
the STFT, we have:
Y (t, f ) = S1 (t, f ) + S2 (t, f ),

(2)

We can write the magnitude spectrogram in matrix form as follows:
Y ≈ S1 + S2.

(3)

where S 1 , S 2 are the unknown magnitude spectrograms of the
source signals and need to be estimated using the observed mixed
signal and the training data.
3. NMF FOR SUPERVISED SOURCE SEPARATION
In this section, we briefly describe the use of nonnegative matrix
factorization (NMF) for supervised single channel source separation.
We will relate our model to the NMF idea and we will use the source
estimates obtained from using NMF as initilization for our method,
so it is appropriate to introduce the use of NMF for source separation
first.
To find a suitable initialization for the sources signals, we use
nonnegative matrix factorization (NMF) as in [23]. NMF [25] factorizes any nonnegative matrix V into a basis matrix (dictionary) B
and a gain matrix G as
V ≈ BG.
(4)
The matrix B contains the basis vectors that are optimized to allow the data in V to be approximated as a linear combination of its
constituent columns. The solution for B and G can be found by
minimizing the following Itakura-Saito (IS) divergence cost function [26]:
min DIS (V || BG) ,
B ,G

(5)

where
DIS (V || BG) =

X
a,b

!
V a,b
V a,b
− log
−1 .
(BG)a,b
(BG)a,b

This divergence cost function is a good measurement for the perceptual difference between different audio signals [26]. The IS-NMF
solution for equation (5) can be computed by alternating multiplicative updates of G and B as follows:

V
2
(BG)

 ,
G←G⊗
1
BT
BG


V
GT
2
(BG)

B←B⊗ 
,
1
GT
BG
BT

S train
≈ B i Gtrain
, ∀i ∈ {1, 2} ,
i
i

(1)

where S1 (t, f ) and S2 (t, f ) are the unknown STFT of the first and
second sources in the mixed signal. In this framework [13, 24], the
phase angles of the STFT were usually ignored. Hence, we can approximate the magnitude spectrum of the measured signal as the sum
of source signals’ magnitude spectra as follows:
|Y (t, f )| ≈ |S1 (t, f )| + |S2 (t, f )| .

where 1 is a matrix of ones with the same size of V , the operation ⊗ is an element-wise multiplication, all divisions and (.)2 are
element-wise operations. The matrices B and G are usually initialized by positive random numbers and then updated iteratively using
equations (6) and (7).
In the initialization stage, NMF is used to decompose the frames
for each source i into a multiplication of a nonnegative dictionary
B i and a gain matrix Gi as follows:
(8)

where S train
is the nonnegative matrix that contains the spectroi
gram frames of the training data of source i. After observing the
mixed signal, we calculate its spectrogram Y psd . NMF is used to
decompose the mixed signal’s spectrogram matrix Y psd with the
trained dictionaries as follows:


G1
Y psd ≈ [B 1 , B 2 ] G or Y psd ≈ [B 1 B 2 ]
.
(9)
G2
The only unknown here is the gains matrix G since the dictionaries
are fixed. The update rule in equation (6) is used to find G. After
finding the value of G, the initial estimate for each source magnitude
spectrogram is computed as follows:
Ŝ init1 =

B 1 G1
B 2 G2
⊗ Y , Ŝ init2 =
⊗Y,
B 1 G1 + B 2 G2
B 1 G1 + B 2 G2
(10)

where ⊗ is an element-wise multiplication and the divisions are done
element-wise.
The magnitude spectrograms of the initial estimates of the
source signals are used to initialize the sources in the separation
stage of the DNN approach.
4. THE METHOD
In NMF, the basic idea is to model each source with a dictionary, so
that source signals appear in the nonnegative span of this dictionary.
In the separation stage, the mixed signal is expressed as a nonnegative linear combination of the source dictionaries and separation is
performed by taking the parts corresponding to each source in the
decomposition.
The basic problem in NMF is that each source is modeled to lie
in a nonnegative cone defined by all the nonnegative linear combinations of its dictionary entries. This assumption may be a limiting
assumption usually since the variability within each source indicates
that nonlinear models may be more appropriate. This limitation led
us to consider nonlinear models for each source. It is not trivial to
use nonlinear models or classifiers in source separation. Since deep
neural networks were recently used with increased success in speech
recognition and other object recognition tasks, they can be considered as superior models of highly variable real-world signals.
We first train a DNN to model each source in the training
stage. We then use an energy minimization objective to estimate the
sources and their gains during the separation stage. Each stage is
explained below.



4.1. Training the DNN
(6)

(7)

We train a DNN that can classify sources present in the mixed signal. The input to the network is a frame of normalized magnitude
spectrum, x ∈ Rd . The neural network architecture is illustrated in
Figure 1. There are two outputs in the DNN, each corresponding to
a source. The label of each training instance is a binary indicator

function, namely if the instance is from source one, the first output
label f1 (x) = 1 and the second output label f2 (x) = 0. Let nk
be the number of hidden nodes in layer k for k = 0, . . . , K where
K is the number of layers. Note that n0 = d and nK = 2. Let
W k ∈ Rnk ×nk−1 be the weights between layers k − 1 and k, then
the values of a hidden layer hk ∈ Rnk are obtained as follows:
hk = g(W k hk−1 ),

(11)

1
where g(x) = 1+exp(−x)
is the elementwise sigmoid function. We
skip the bias terms to avoid clutter in our notation. The input to the
network is h0 = x ∈ Rd and the output is f (x) = hK ∈ R2 .

𝑓1 (𝒙)

𝑓2 (𝒙)

𝒉𝟒
𝑾𝟒
𝒉𝟑

corresponding source model in the DNN.
E1 (x) = (1 − f1 (x))2 + (f2 (x))2 ,

(12)

E2 (x) = (f1 (x))2 + (1 − f2 (x))2 .

(13)

Basically, we expect to have E1 (x) ≈ 0 when x comes from source
one and vice versa. We also define the following energy function
which quantifies the energy of error caused by the least squares difference between the mixed spectrum y and its estimate found by
linear combination of the two source estimates x1 and x2 :
Eerr (x1 , x2 , y, u, v) = ||ux1 + vx2 − y||2 .

(14)

Finally, we define an energy function that measures the negative energy of a variable, R(θ) = (min(θ, 0))2 .
In order to estimate the unknowns in the model, we solve the
following energy minimization problem.

𝑾𝟑

(x̂1 , x̂2 , û, v̂) =

argmin E(x1 , x2 , y, u, v),

(15)

{x1 ,x2 ,u,v}

𝒉𝟐
𝑾𝟐
𝒉𝟏

𝑾𝟏
𝒙

where
E(x1 , x2 , y, u, v) = E1 (x1 ) + E2 (x2 ) + λEerr (x1 , x2 , y, u, v)
X
+β
R(θi )
i

(16)

Fig. 1. Illustration of the DNN architecture.

Training a deep network necessitates a good initialization of the
parameters. It is shown that layer-by-layer pretraining using unsupervised methods for initialization of the parameters results in superior performance as compared to using random initial values. We
used Restricted Boltzmann Machines (RBM) for initialization. After initialization, supervised backpropagation algorithm is applied
to fine-tune the parameters. The learning criteria we use is leastsquares minimization. We are able to get the partial derivatives with
respect to the inputs, and this derivative is also used in the source
separation part. Let f (.) : Rd → R2 be the DNN, then f1 (x)
and f2 (x) are the scores that are proportional to the probabilities of
source one and source two respectively for a given frame of normalized magnitude spectrum x. We use these functions to measure how
much the separated spectra carry the characteristics of each source
as we elaborate more in the next section.
4.2. Source separation using DNN and energy minimization
In the separation stage, our algorithm works independently in each
frame of the mixed audio signal. For each frame of the mixed signal
spectrum, we calculate the normalized magnitude spectrum y. We
would like to express y = ux1 + vx2 where u and v are the gains
and x1 and x2 are normalized magnitude spectra of source one and
two respectively.
We formulate the problem of finding the unknown parameters
θ = (x1 , x2 , u, v) as an energy minimization problem. We have a
few different criteria that the source estimates need to satisfy. First,
they must fit well to the DNN trained in the training stage. Second,
their linear combination must sum to the mixed spectrum y and third
the source estimates must be nonnegative since they correspond to
the magnitude spectra of each source.
The energy functions E1 and E2 below are least squares cost
functions that quantify the fitness of a source estimate x to each

is the joint energy function which we seek to minimize. λ and β are
regularization parameters which are chosen experimentally. Here
θ = (x1 , x2 , u, v) = [θ1 , θ2 , . . . , θn ] is a vector containing all the
unknowns which must all be nonnegative. Note that, the nonnegativity can be given as an optimization constraint as well, however
we obtained faster solution of the optimization problem if we used
the negative energy function penalty instead. If some of the parameters are found to be negative after the solution of the optimization
problem (which rarely happens), we set them to zero. We used the
LBFGS algorithm for solving the unconstrained optimization problem.
We need to calculate the gradient of the DNN outputs with respect to the input x to be able to solve the optimization problem. The
(x)
gradient of the input x with respect to fi (x) is given as ∂f∂ix
=
q 1,i for i = 1, 2, where,
q k,i = W Tk (q k+1,i ⊗ hk ⊗ (1 − hk )),
fi (x)(1 − fi (x))wTK,i ,
th

nK−1

(17)

and q K,i =
where wK,i ∈ R
contains
the weights between i node of the output layer and the nodes at the
previous layer, in other words the ith row of W K .
The flowchart of the energy minimization setup is shown in Figure 2. For illustration, we show the single DNN in two separate
blocks in the flowchart. The fitness energies are measured using the
DNN and the error energy is found from the summation requirement.
Note that, since there are many parameters to be estimated and
the problem is clearly non-convex, the initialization of the parameters is very important. We initialize the estimates x̂1 and x̂2 from
the NMF result after normalizing by their `2 -norms. û is initialized
by the `2 -norm of the initial NMF source estimate ŝ1 divided by the
`2 -norm of the mixed signal y. v̂ is initialized in a similar manner.
After we obtain (x̂1 , x̂2 , û, v̂) as the result of the energy minimization problem, we use them as spectral estimates in a Wiener
filter to reconstruct improved estimates of each source spectra, e.g.
for source one we obtain the final estimate as follows:

Fig. 2. Flowchart of the energy minimization setup. For illustration,
we show the single DNN in two separate blocks in the flowchart.

ŝ1 =

(ûx̂1 )2
⊗ y.
(ûx̂1 )2 + (v̂ x̂2 )2

(18)

5. EXPERIMENTS AND DISCUSSION
We applied the proposed algorithm to separate speech and music signals from their mixture. We simulated our algorithm on a collection
of speech and piano data at 16kHz sampling rate. For speech data,
we used the training and testing male speech data from the TIMIT
database. For music data, we downloaded piano music data from piano society web site [27]. We used 39 pieces with approximate 185
minutes total duration from different composers but from a single
artist for training and left out one piece for testing. The magnitude
spectrograms for the speech and music data were calculated using
the STFT: A Hamming window with 480 points length and 60%
overlap was used and the FFT was taken at 512 points, the first 257
FFT points only were used since the conjugate of the remaining 255
points are involved in the first points.
The mixed data was formed by adding random portions of the
test music file to 20 speech files from the test data of the TIMIT
database at different speech to music ratio. The audio power levels
of each file were found using the “speech voltmeter” program from
the G.191 ITU-T STL software suite [28].
For the initialization of the source signals using nonnegative matrix factorization, we used a dictionary size of 128 for each source.
For training the NMF dictionaries, we used 50 minutes of data for
music and 30 minutes of the training data for speech. For training the
DNN, we used a total 50 minute subset of music and speech training
data for computational reasons.
For the DNN, the number of nodes in each hidden layer were
100-50-200 with three hidden layers. Sigmoid nonlinearity was used
at each node including the output nodes. DNN was initialized with
RBM training using contrastive divergence. We used 150 epochs for
training each layer’s RBM. We used 500 epochs for backpropagation
training. The first five epochs were used to optimize the output layer
keeping the lower layer weights untouched.
In the energy minimization problem, the values for the regularization parameters were λ = 5 and β = 3. We used Mark Schmidt’s
minFunc matlab LBFGS solver for solving the optimization problem
[29].

Performance measurements of the separation algorithm were
done using the signal to distortion ratio (SDR) and the signal to
interference ratio (SIR) [30]. The average SDR and SIR over the
20 test utterances are reported. The source to distortion ratio (SDR)
is defined as the ratio of the target energy to all errors in the reconstructed signal. The target signal is defined as the projection
of the predicted signal onto the original speech signal. Signal to
interference ratio (SIR) is defined as the ratio of the target energy to
the interference error due to the music signal only. The higher SDR
and SIR we measure the better performance we achieve. We also
use the output SNR as additional performance criteria.
The results are presented in Tables 1 and 2. We experimented
with multi-frame DNN where the inputs to the DNN were taken
from L neighbor spectral frames for both training and testing instead of using a single spectral frame similar to [15]. We can see that
using the DNN and the energy minimization idea, we can improve
the source separation performance for all input speech-to-music ratio
(SMR) values from -5 to 5 dB. In all cases, DNN is better than regular NMF and the improvement in SDR and SNR is usually around
1-1.5 dB. However, the improvement in SIR can be as high as 3
dB which indicates the fact that the introduced method can decrease
remaining music portions in the reconstructed speech signal. We
performed experiments with L = 3 neighboring frames which improved the results as compared to using a single frame input to the
DNN. For L = 3, we used 500 nodes in the third layer of the DNN
instead of 200. We conjecture that better results can be obtained if
higher number of neighboring frames are used.
Table 1. SDR, SIR and SNR in dB for the estimated speech signal.
SMR
dB
-5
0
5

NMF
SDR
1.79
4.51
7.99

SIR
5.01
8.41
12.36

DNN
SNR
3.15
5.52
8.62

SDR
2.81
5.46
8.74

L=1
SIR
7.03
9.92
13.39

SNR
3.96
6.24
9.24

SDR
3.09
5.73
8.96

L=3
SIR
7.40
10.16
13.33

SNR
4.28
6.52
9.45

Table 2. SDR, SIR and SNR in dB for the estimated music signal.
SMR
dB
-5
0
5

NMF
SDR
5.52
3.51
0.93

SIR
15.75
12.65
9.03

DNN
SNR
6.30
4.88
3.35

SDR
6.31
4.23
1.79

L=1
SIR
18.48
16.03
12.94

SNR
7.11
5.60
3.96

SDR
6.67
4.45
1.97

L=3
SIR
18.30
15.90
13.09

SNR
7.43
5.88
4.17

6. CONCLUSION
In this work, we introduced a novel approach for single channel
source separation (SCSS) using deep neural networks (DNN). The
DNN was used in this paper as a helper to model each source signal.
The training data for the source signals were used to train a DNN.
The trained DNN was used in an energy minimization framework to
separate the mixed signals while also estimating the scale for each
source in the mixed signal. Many adjustments for the model parameters can be done to improve the proposed SCSS using the introduced
approach. Different types of DNN such as deep autoencoders and
deep recurrent neural networks which can handle the temporal structure of the source signals can be tested on the SCSS problem. We
believe this idea is a novel idea and many improvements will be possible in the near future to improve its performance.

7. REFERENCES
[1] T. Kristjansson, J. Hershey, and H. Attias, “Single microphone
source separation using high resolution signal reconstruction,”
in IEEE International Conference Acoustics, Speech and Signal Processing (ICASSP), 2004.

[15] E. M. Grais and H. Erdogan, “Single channel speech music
separation using nonnegative matrix factorization with sliding
window and spectral masks,” in Annual Conference of the International Speech Communication Association (InterSpeech),
2011.

[2] A. M. Reddy and B. Raj, “Soft Mask Methods for singlechannel speaker separation,” IEEE Transactions on Audio,
Speech, and Language Processing, vol. 15, Aug. 2007.

[16] E. M. Grais and H. Erdogan, “Regularized nonnegative matrix factorization using gaussian mixture priors for supervised
single channel source separation,” Computer Speech and Language, vol. 27, no. 3, pp. 746–762, May 2013.

[3] A. M. Reddy and B. Raj, “A Minimum Mean squared error estimator for single channel speaker separation,” in International Conference on Spoken Language Processing (InterSpeech), 2004.

[17] E. M. Grais and H. Erdogan, “Gaussian mixture gain priors for
regularized nonnegative matrix factorization in single-channel
source separation,” in Annual Conference of the International
Speech Communication Association (InterSpeech), 2012.

[4] T. Virtanen, “Speech recognition using factorial hidden
markov models for separation in the feature space,” in International Conference on Spoken Language Processing (InterSpeech), 2006.

[18] E. M. Grais and H. Erdogan, “Hidden Markov Models as priors for regularized nonnegative matrix factorization in singlechannel source separation,” in Annual Conference of the International Speech Communication Association (InterSpeech),
2012.

[5] S. T. Roweis, “One Microphone source separation,” Neural
Information Processing Systems, 13, pp. 793–799, 2000.
[6] A. N. Deoras and A. H. Johnson, “A factorial hmm approach to
simultaneous recognition of isolated digits spoken by multiple
talkers on one audio channel,” in IEEE International Conference Acoustics, Speech and Signal Processing (ICASSP), 2004.
[7] M. H. Radfar, W. Wong, W. Y. Chan, and R. M. Dansereau,
“Gain estimation in model-based single channel speech separation,” in In proc. IEEE International Workshop on Machine
Learning for Signal Processing (MLSP, Grenoble, France),
Sept. 2009.
[8] M. H. Radfar, R. M. Dansereau, and W. Y. Chan, “Monaural speech separation based on gain adapted minimum mean
square error estimation,” Journal of Signal Processing Systems,Springer, vol. 61, no. 1, pp. 21–37, 2008.
[9] M. H. Radfar and R. M. Dansereau, “Long-term gain estimation in model-based single channel speech separation,” in in
Proc. of IEEE Workshop on Applications of Signal Processing
to Audio and Acoustics (WASPAA, New Paltz, NY), 2007.

[19] E. M. Grais and H. Erdogan,
“Spectro-temporal postenhancement using MMSE estimation in NMF based singlechannel source separation,” in Annual Conference of the International Speech Communication Association (InterSpeech),
2013.
[20] Y. Wang and DL. Wang, “Cocktail party processing via structured prediction,” in Advances in Neural Information Processing Systems (NIPS), 2012.
[21] Geoffrey E. Hinton, Simon Osindero, and Yee-Whye Teh, “A
fast learning algorithm for deep belief nets,” Neural Comput.,
vol. 18, no. 7, pp. 1527–1554, July 2006.
[22] Geoffrey Hinton, Li Deng, Dong Yu, George Dahl, Abdel rahman Mohamed, Navdeep Jaitly, Andrew Senior, Vincent Vanhoucke, Patrick Nguyen, Tara Sainath, and Brian Kingsbury,
“Deep neural networks for acoustic modeling in speech recognition,” Signal Processing Magazine, 2012.
[23] E. M. Grais and H. Erdogan, “Single channel speech music
separation using nonnegative matrix factorization and spectral
masks,” in International Conference on Digital Signal Processing, 2011.

[10] A. Ozerov, C. Fvotte, and M. Charbit, “Factorial scaled hidden
markov model for polyphonic audio representation and source
separation,” in IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), Mohonk, NY, 2009.

[24] T. Virtanen, “Monaural sound source separation by nonnegative matrix factorization with temporal continuity and
sparseness criteria,” IEEE Transactions on Audio, Speech, and
Language Processing, vol. 15, pp. 1066–1074, Mar. 2007.

[11] M. H. Radfar, W. Wong, R. M. Dansereau, and W. Y. Chan,
“Scaled factorial Hidden Markov Models: a new technique for
compensating gain differences in model-based single channel
speech separation,” in IEEE International Conference Acoustics, Speech and Signal Processing (ICASSP), 2010.

[25] D. D. Lee and H. S. Seung, “Algorithms for non-negative matrix factorization,” Advances in Neural Information Processing
Systems, vol. 13, pp. 556–562, 2001.

[12] J. R. Hershey, S. J. Rennie, P. A. Olsen, and T. T. Kristjansson, “Super-human multi-talker speech recognition: A graphical modeling approach,” Computer Speech and Language, vol.
24, no. 1, pp. 45–66, Jan. 2010.
[13] M. N. Schmidt and R. K. Olsson, “Single-channel speech separation using sparse non-negative matrix factorization,” in International Conference on Spoken Language Processing (InterSpeech), 2006.
[14] E. M. Grais and H. Erdogan,
“Spectro-temporal postsmoothing in NMF based single-channel source separation,”
in European Signal Processing Conference (EUSIPCO), 2012.

[26] C. Fevotte, N. Bertin, and J. L. Durrieu, “Nonnegative matrix
factorization with the itakura-saito divergence. With application to music analysis,” Neural Computation, vol. 21, no. 3,
pp. 793–830, 2009.
[27] URL, “http://pianosociety.com,” 2009.
[28] URL, “http://www.itu.int/rec/T-REC-G.191/en,” 2009.
[29] URL, “http://www.di.ens.fr/ mschmidt/software/minfunc.html,”
2012.
[30] E. Vincent, R. Gribonval, and C. Fevotte, “Performance measurement in blind audio source separation,” IEEE Transactions
on Audio, Speech, and Language Processing, vol. 14, no. 4, pp.
1462–69, July 2006.

