Single-Channel Blind Separation Using Adaptive Mode Separation-
Based Wavelet Transform and ICA Single-Channel Separation of the
MINA KEMIHA1, ABDELLAH KACHA2
1Automatic department. 2Electronic departement
Mohammed Seddik BenYahia University
BP 98 Ouled Aissa, 18000
JIJEL, ALGERIA
Abstract: - In this paper, a new method to solve the signal-channel blind source separation (SCBSS) problem
has been proposed. The method is based on combining the Adaptive Mode Separation-Based Wavelet
Transform (AMSWT) and the ICA-based single channel separation. First, the amplitude spectrum of the
instantaneous mixture signal is obtained via the Fourier transform. Then, the AMSWT is introduced to
adaptively extract spectral intrinsic components (SIC) by applying the variational scaling and wavelet
functions. The AMSWT is applied to every mode to obtain the time-frequency distribution. Then the time-
frequency distribution of the mixed signal is exploited. The ICA-based single-channel separation has been
applied on spectral rows corresponding to different time intervals. Finally, these components are grouped using
the -distance of Gaussian distribution . Objective measure of separation quality has been performed using
the scale-invariant (SI) parameters and compared with the existing method to solve SCBSS problem.
Experimental results show that the proposed method has better separation performance than the existed
methods, and the proposed method present a powerful method to solve de SCBSS problem.
Key-Words: -Signal-channel blind source separation. Adaptive Mode Separation-Based Wavelet Transform.
Spectral decomposition-based method. β-distance of Gaussian distribution
Received: April 25, 2021. Revised: March 16, 2022. Accepted: April 17, 2022. Published: May 18, 2022.
1 Introduction
Blind signal separation (BSS) consists to separate
source signal from mixed signals without any
information. BSS have wide range of applications
such as medical imaging and engineering [1-4],
image processing and speech recognition [5, 6], and
speech signal processing [7, 8], communication
systems [9], astrophysics [10], automatic
transcription or speech and musical instrument
identification [11], mechanical fault detection [12,
13].
In the literature, many approaches have
been proposed to solve the BSS problem. The most
popular is the independent component analysis
method (ICA). In [14] an algorithm based on phase
space reconstruction was proposed. In [15], an
algorithm composed of pseudo-multiple input
multiple output observation structure and
independent component analysis (ICA) was
proposed. In [16], an improved empirical mode
decomposition method for blind separation of
single-channel vibration signal mixtures was
proposed. The ICA is characterized by simplicity
and results quality. ICA technique is based on
linear transformation to find components from
multidimensional mixed data. The ICA is
performed on the hypothesis that the source signals
are statistically independent. The founded
components are statistically independent too.
A single channel source separation methods
overview is presented in [17]. Methods based on
spectral representation of the observed signal are
usually known as spectral decomposition-based
methods. Spectral decomposition-based methods
have been introduced by many authors. In [18]
nonnegative matrix factorization (NMF) method has
been applied on the Short Time Fourier Transform
(STFT) representation of a single-channel observed
signal, but the method requires the use of an
additional training data. In [19], wavelet transforms
and a combination of empirical mode decomposition
(EMD) and ICA has been proposed, but the wavelet
transforms require some predefined basis functions
to represent a signal. The EMD and its improved
algorithms are empirical, and there is no complete
mathematical theory basis [20]. In [21] the bark
scale aligned wavelet packet decomposition has
been introduced, after the Fourier transform, the
Gaussian mixture model (GMM) has been used in
separation step. In [22] a combination of various
single channel separation methods, a spectral
decomposition based techniques and model based
methods has been dissected.
In [23] a new Adaptive Mode Separation-
Based Wavelet Transform (AMSWT) has been
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proposed to seismic time–frequency analysis. The
novel time-frequency analysis approach is inspired
by the adaptive wavelet bank configuration to
empirical wavelet transform (EWT) [24-26] and the
spectral mode separation thought from variational
mode decomposition (VMD) [27]). The AMSWT
method consists to adaptively extract spectral
intrinsic components by solving a recursive
optimization problem. To obtain the spectral
boundaries for wavelets bank configuration, the
limited support of every spectral mode is
introduced. Then, the obtained spectral boundaries
for wavelets bank configuration built to highlight
the spectral information. The AMSWT method is a
fully adaptive approach without requiring prior
information.
In [28] a new method to solve the SCBSS
problem is proposed. The method is applied on the
time-frequency representation of a single-channel
observed signal. The ICA-based single-channel
separation has been applied on spectral rows
corresponding to different time intervals. The -
distance of Gaussian distribution is used to
measure the distance between time-frequency
domain components of the mixed signal obtained by
ICA, and finally, these components are grouped.
The grouping algorithm of the components return to
solve the optimization problem by minimizing the
negentropy of reconstructed constituent signals.
In this paper a new method has been
proposed to solve the SCBSS problem. The method
is based on combining the AMSWT [23] and the
ICA-based single channel separation method [28].
The time-frequency representation of a signal is
considered as a multichannel observed signal and
can be separated by ICA. After separation, the
statistically independent time-frequency
components are then grouped. The grouping using
the -distance of Gaussian distribution
The performance of the proposed method is
tested on real speech sounds chosen from available
databases and compared to the results obtained via
EMD based single-channel separation, the wavelets
based-single channel separation introduced in [19]
and the single-channel separation audio signals
based on variational mode decomposition (VMD).
The quality of the obtained separation results was
evaluated using the scale-invariant (SI) parameters
such as SI-SDR, SI-SAR, SI-SIR, which are
particularly recommended for single-channel
separation evaluation [29, 30].
The remaining content is composed of the
following parts: the second section gives the SCBSS
problem formulation; the third section introduces
adaptive mode separation-based wavelet transform;
the fourth section shows the ICA-based single
channel separation method; The fifth section present
the main steps of the proposed algorithm with the
application of this algorithm in the simulation
experiments and the comparison results with other
algorithms; finally, conclusions and discussions are
given in the fifth section.
2 SCBSS Problem Formulation
A general BSS problem can be mathematically
defined as follows: Let ()=[(),..,()] be
a vector of N independent sources at the discrete
time instant t. The vector ()=
[(),..,()]of the M observed instantaneous
mixtures is modeled as follow:
(
)
=
(
)
(1)
where is the (×)mixing matrix.
In the literature, the main BSS
classifications are defined such as: linear and
nonlinear BSS; instantaneous and convolutive BSS;
over complete and underdetermined BSS. For the
last classification, when the number of observed
signals is more than the number of independent
sources , this refers to over complete BSS. On the
other hand, when the number of observed signals
is smaller than the number of independent sources
, this becomes to underdetermined BSS.
In general case and for many practical
applications only one-channel recording is available.
This special case of instantaneous underdetermined
source separation problem termed as single channel
source separation is discussed in many papers. For
this special case, the conventional source separation
methods are not suitable.
The SCSS research area where the problem can be
simply treated as one observation instantaneous
mixed with several unknown sources:
(
)
=
(
)
(2)
where =1,.., denotes number of sources and
the goal is to estimate the sources () when only
the observation signal () is available. In
frequency domain, by applying the short time
Fourier transform (STFT). The mixture defined in
equation (2) becomes:
(
)
=
(
)
(3)
where denote the frequency. () design the
Fourier transform of the mixture signal () and
() is a (1) vector whose elements () are
the Fourier transforms of the source signals ().
Since the separation of the signal is performed
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frame by frame, the mixing model of each frame can
be written as :
(
,
)
=
(
,
)
(4)
where m denotes the frame index.
In [31] the original EMD description, a
mode is defined as a signal whose number of local
extrema and zero-crossings differ at most by one. In
most lately related works, the definition is changed
into so-called Intrinsic Mode Functions (IMF),
based on modulation criteria [31, 20].
3 Adaptive Mode Separation-Based
Wavelet Transform
The wavelet () is a function localized jointly in
time and frequency and with a zero mean. A mother
wavelet ,() defined as follow
,
(
)
=
1
√
(
−
)
(5)
Where and denote the translation and dilatation
parameters respectively.
The wavelet transform consists to perform the inner
product between the family of wavelets ,() and
the signal ().
=
〈
(
)
,
,
(
)
〉
=
(
)
1
√
(
−
)
(6)
The AMSWT perform the time-frequency
analysis by the variational scaling and wavelet
functions to every mode. So, the method is based on
the ADMM [31] solver and then defines a bank of
variational scaling functions and wavelets based on
the established spectral boundaries.
Therefore, the approximate coefficients and
detailed coefficients are obtained by the inner
product of the analyzed signal s with the variational
scaling function, and by the inner product of the
analyzed signal s with variational wavelets
respectively and expressed by the following
equations
(
0
,
)
=
〈
,
∅
〉
=
(
)
∅
(
−
)
(7)
and
(
,
)
=
〈
,
〉
=
(
)
(
−
)
(8)
In [23] the intrinsic modes () have
distinguishable features in the frequency domain
under the amplitude-modulated frequency-
modulated (AM-FM) assumption, using the
alternate direction method of multiplier (ADMM)
solver, the spectral modes can be adaptively
obtained, following how intrinsic mode functions
(IMF) are obtained, to estimate compact modes:
min
,
(
)
+
∗
(
)
.
.
=
(
)
(9)
Where () is the signal to be decomposed under
the constraint that over all modes should be the
input signal. (.) is a Dirac impulse. ()+
∗
() denotes the original data and its Hilbert
transform. , and denote the modes and their
central frequencies and the mode number
respectively. The spectral segmentation boundary
number can be empirically determinate using on the
following equation:
=
{
∈
ℤ
|
≥
2
ln
}
(10)
where presents the signal length and is the
scaling exponent determined by the detrended
fluctuation analysis (DFA) [ 32].
According to [23] the equation is solved using a
quadratic penalty term and the parameter that
denotes the Lagrangian multiplier for rendering the
problem unconstrained
(
,
,
)
=
∑
[
(
)
+
∗
(
)
]
+
〈
,
−
∑
〉
+
‖
−
∑
‖
.
(11)
therefore is determined recursively as
(
)
=
(
)
−
∑
(
)
+
1
+
2
(
−
)
(12)
where (),
() and () denote the Fourier
transform of the input signal (), the mode
function () and () respectively. denotes the
balancing parameter of the data-fidelity constraint.
The center frequencies
are updated as the
center of gravity of the corresponding mode’s power
spectrum using the following equation
=
∫
(
)
∫
(
)
(13)
Therefore, Instead of using a predefined wavelet
bank, we build adaptive wavelets banks using the
spectral modes and associated center frequencies
represent the intrinsic components.
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In [23] authors defined the boundaries between
each mode using the mode bandwidth and central
frequencies, Whereas, in the literature, some authors
are just used the average between the two central
frequencies as the spectral boundary, which does not
consider the spectral distribution.
We consider the ℎmode with the mean frequency
and a spectral bandwidth , then the boundary
between ℎ the and the +1 mode is given by
the following equation
=
+
+
−
2
(14)
we take =0 and =.
For the variational scaling functions and wavelets
based on the spectral boundaries: the authors use the
idea used in the construction of both Littlewood–
Paley and Meyer’s wavelets [33]. ∅
and
are
respectively defined by the following equation, with
is the parameter that ensures no overlap between
the two consecutive transitions.
∅
=
⎩
⎨
⎧
1
,
≤
(
1
−
cos
2
(
,
)
,
(
1
−
)
≤
≤
(
1
+
0
otherwise
(15
)
and
=
⎩
⎪
⎪
⎨
⎪
⎪
⎧
1
,
(
1
+
)
≤
≤
(
1
−
cos
2(,),(1−)≤≤
(
sin
2
(
,
)
,
(
1
−
)
≤
≤
(
0
otherwise
(1
6)
Where (,)={
[||−(1−)]}]
and () is an arbitrary function defined as follow:
(
)
=
0
,
≤
0
1
,
>
1
(
)
+
(
1
−
)
=
1
,
0
<
<
1
(17)
4 ICA-Based Single Channel
Separation Method
In [28] the authors propose a new method to solve
the SCBSS problem. The method is applied on the
time-frequency representation of a single-channel
observed signal. The time-frequency representation
is a non-linear transformation, the use of non-linear
ICA would be appropriate, but However, as
mentioned in [28, 34] under certain conditions
nonlinear BSS problem can be solved using linear
ICA.
Let () denote the signal in time domain,
using the Short Time Fourier Transform (STFT), the
signal is transformed in the frequency domain. The
transformation is performed frame by frame and
is the STFT time frame number. The STFT is the
x complex matrix of time frequency
representation, this matrix contain -rows
instantaneous signal spectra,
Let where =1,.., spectral
components obtained via the time-frequency
representation of a single channel signal. The
obtained are statistically independent. In this
step, the rows of the TFDmix matrix are treated as
individual channels in a multichannel signal. Then
the ICA is applied on this multichannel signal.
The ith row of denoted can be written as
= an ith time frequency component of a
mixed one-channel signal. The relation between Z
where =[] =1,.., and TFDmix is given as
the following equation
=
.
=
=
(18)
where Ais the (×)mixing matrix whose
elements ,where, is an ith column of .
The present the spectral bases. The columns of
describing time variation of are called time bases
and denoted by . The matrix denote the
product of the time basis and the spectral basis
is called ith time-frequency component.
The grouping of bases is performed
into subgroups by the grouping of time bases and
frequency bases as the following equation:
=
=
++⋯
+
(19)
Where ,.., are index sets obtained by
grouping bases.
In [28], to reduce computational
complexity, authors used only the bases
which have a specified variance of the mixed signal.
The grouping of bases consists to collecting
elements into clusters. The clustering is based on the
maximization of negentropy of separated
components. The ICA-based single-channel
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separation methods primarily use component
grouping based on similarity in time or frequency
domain. In [28] authors suggest the use of a time-
frequency structure to measure the similarity
features in both time and spectral domain.
5 Grouping Process
The grouping process is performed by clustering the
ith time-frequency distribution bases, or the
distance between bases, using the -distance
of Gaussian distribution [28]. The generalized
Gaussian distribution is expressed as following:
(
|
,
,
)
=
(
)
−
(
)
−
/
(
)
(20)
where denote the expected value. describes
the type of a random variable y, i.e., its deviation
from normal distribution where −1≤≤0.
present the standard deviation of a random variable
. The parameters () and () are given by the
following expression: ()=
()/
()
()/
and ()=
()
()/() where Γ is the
Gamma-Euler function.
=
,
−
(
,
)
(21)
the parameter is estimated by a posteriori
determination of the maximum of the , where , the
a posteriori distribution of the parameter is given
as [32, 35]: (|)∝(|)(), where (|)
denotes a data likelihood [32] and is given as the
following equation
(
|
)
=
(
)
−
(
)
−
/
(
)
(24)
where () present the a priori distribution of
the parameter [18, article de khedamy bih].
The statistically independent constituent signals
have the maximum negentropy [10,50]. So, the
grouping or the bases consists in maximizing
negentropy (negative entropy) of reconstructed
constituent signals , by finding of
reconstructed constituent signals , =
∑
with the maximum negentropy, the
bases can be grouped. Let is the normalized
Gaussian random variable (=0,=1) and
(.)is a nonlinear function of the random variable
usually having the form ()=
logcosℎ() ,
∈(1,2) or ()=−(
). The negentropy
function () is given by the following
equationexpression [35]: ()~()−
(). The negentropy function ()
approximation has numerous advantages such as
conceptual simplicity and rapid calculation rate
[35]. As a result, it is very often used as a cost
function in algorithms for solving ICA problems
[28].
6 Results and Discussion
To evaluate the performance of the proposed
approach, simulations are performed. The proposed
method has been applied on speech datasets selected
from TIMIT [36] and NOIZEUS [37] databases.
The instantaneous mixture is simulated by the
recordings of three sentences 1(t),2(t)and
3(t). The signals are pronounced by male and
female speakers and were recorded at the sampling
frequency . The instantaneous mixture is defined
by the following equation:
(
)
=
(
t
)
+
(
t
)
+
(
t
)
(22)
where , and are constants parameters. The
proposed method operates in the time-frequency
domain, and is summarized by the following steps
for each frame:
1. Compute the Short Time Fourier Transform
(STFT) of the observed signal ()
2. Apply the variational scaling and wavelet
functions to every mode to obtain the time-
frequency distribution using equation (7) and (8).
3. The input data for ICA is a spectrogram. The ICA
is applied on this multichannel signal (applied on
spectral rows corresponding to different time
intervals)
4. The -distance of Gaussian distribution is
used to measure the distance between time-
frequency domain components of the mixed signal
obtained by ICA.
5. Solve the optimization problem by minimizing
the negentropy of reconstructed constituent signals.
6. Reconstruct the appearance of the particular
source in the original signal.
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The time-frequency is considered as a random
variable, its distribution is given in parametric
terms. Therefore, it is poss
Figure 1. Flowchart of the proposed method
Thereafter, as an illustration example, the proposed
method is applied to separate an instantaneous
mixture defined by equation (22). The Fig.2 (a)
shows the three speech signals representation in
time domain. First, the observed single-channel
presented in Fig.2 (b) was transformed to the
frequency domain using the STFT. The Fig.2 (c)
presents the STFT of a frame of the observed
mixture. Then, for each frame, the AMSWT method
is introduced to obtain optimal spectral mode
separation; we apply the variational scaling and
wavelet functions to every mode to obtain the time-
frequency distribution using equation (7) and (8) as
illustrated by Fig. 2 (d).
Once the T-F distribution is obtained, the
spectrogram which is considered as a multichannel
observed signal is used as the input data for ICA-
based single channel separation. Then, as mentioned
in step 4 of the algorithm, the -distance of
Gaussian distribution is used to measure the
distance between time-frequency domain
components. Solving the optimization problem as
mentioned in step 5. For our example and for a
mode, the estimated spectral components are shown
in Fig. 2(e). For this frame, collecting elements into
clusters, the estimates frame of the signal ()is
illustrated in Fig.2(f). The estimated signals are
illustrated in Fig.2(g). As can be seen, the estimated
signals were similar to the original signal showed in
figure Fig.2 (a)
(
)
(
)
(
)
(a) Original sources time-Domain représentation.
Figure 2. illustration example
(b) Observed signal time-domain representation.
(c) FFT of the frame of the observed signal.
Figure 2. continued
0 5 10 15 20 25
-1
-0.5
0
0.5
1
Time (s)
0 5 10 15 20 25
-1
-0.5
0
0.5
1
Time (s)
0 5 10 15 20 25
-1
-0.5
0
0.5
1
Time (s)
0 5 10 15 20 25
-1
-0.5
0
0.5
1
Time (s)
0 1000 2000 3000 4000 5000 6000 7000 8000
-1
-0.5
0
0.5
1
Frequency (Hz)
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(d) Ttime-frequency distribution of one arbiter
chosen mode
Figure 2. continued
Objective measure of separation quality has been
performed. The performances of the proposed
method are compared with existing methods in the
literature such as the EMD signal-channel
separation [19], the wavelets signal-channel
separation presented in [19], and the single-channel
separation audio signals based on variational mode
decomposition (VMD).
(a)
(b)
©
(e) The time-frequency distribution of one arbiter
chosen mode.
(f) The time-frequency distribution of one arbiter
chosen mode
Estimated source
(
)
Estimated source
(
)
Estimated source
(
)
(g) The time-frequency distribution of one arbiter
chosen mode
Figure 2. continued
In [29, 30] a new method has been proposed, the
method is a simpler scale-invariant alternative for
single-channel separation evaluation by the
introduction a new parameters. These parameters
are called scale-invariant (SI) such as SI-SDR, SI-
SAR, SI-SIR, and they are particularly
recommended single-channel separation evaluation.
These parameters are defined by the usage of a
single coefficient to account for scaling
discrepancies. Let () is the original sources, and
() is the estimated source expressed as=
+ where can be decomposed as
=+ , where are the
source signals, and denotes the
Time
Freq uency
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Tim e
Freque ncy
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Tim e
Frequency
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Tim e
Freque ncy
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Tim e
Freque ncy
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Tim e
Frequency
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Time
Frequency
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Time
Frequency
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Time
Frequency
0 0.2 0.4 0.6 0 .8 1
0
0.2
0.4
0.6
0.8
1
Time
Frequency
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
1
1.5
2
2.5
Time (s)
0 5 10 15 20 25
1
1.5
2
2.5
Time (s)
0 5 10 15 20 25
1
2
Time (s)
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interferences from other sources, and
includes all other artifacts introduced by the
separation algorithm. The − is given as the
following equation : −= ||
|| where
=argmin|−|. The optimal scaling factor
for the target is obtained as =
‖‖ , and the
scaled reference is defined as =. The
performance criteria are given by the following
equation:
−
=
10
‖
‖
−
=
10
‖‖
‖
−
‖
(23)
The scale-invariant signal to interference ratio (SI-
SIR) is given by the following equation:
−
=
10
(24)
and the scale-invariant signal to artifacts ratio (SI-
SAR) is defined as follows:
−
=
10
(25)
Another performance measure has been evaluated;
the measure is expressed in terms of the relative root
mean squared error () given by the
following equation
=
(
)
−
(
)
(
)
100
[
%
]
(26)
where () denote the signal we want to extract and
() is the estimate of the signal. (In our case
=1,..,3). The speech dataset is corrupted at a
signal-to-noise ratio SNR=5 dB, then the SI-SIR,
SI- SAR and SI-SDR are evaluated. The obtained
results are showed in the Fig.3. A set of 4 noisy
mixtures are simulated by corrupting the clean
mixture at a signal-to-noise ratio (SNR) ranging
from 5 dB to 20 dB with a step of 5 dB, then the
RRMSE is evaluated. The Fig.4 shows the obtained
results. To discuss the relation between the frame
length and the -distance of Gaussian distribution
, the mean of the mean -distance of Gaussian
distribution has been evaluated for different
frame length (512, 1024, 2048, 4096 frame length),
the obtained results are showed in Fig.5. As shown,
the proposed method presents a better separation
quality then the exiting methods expressed by the
scale-invariant SI-SDR, SI-SAR, SI-SIR parameters
values. On the other hand, the relative roots mean
squared error () of the proposed method is
better than the of the existing methods for
different SNR values. The mean -distance of
Gaussian distribution for different frame length
shows that the combination of the AMSWT method
and the ICA-based single channel separation method
allows having better results and better separation
compared to existing methods. So, the proposed
method allows having better separation results then
the exiting methods in the literature. The use of the
AMSWT allow to generate a superior time–
frequency resolution because the wavelet bank is
adaptively built on the intrinsic spectral modes; and
the use of a time-frequency structure allows
measuring the similarity features in both time and
spectral domain also the -distance of Gaussian
distribution is a distance measure based on the
knowledge of the statistical nature of spectra of
original constituent signals of the mixed signal.
Figure 3. Comparison between the proposed
method and the EMD based single-channel
separation and the wavelets based-single channel
separation and the single-channel separation audio
signals based on variational mode decomposition
(VMD) in term of SI-SIR, SI-SAR, SI-SDR for
SNR=5 dB.
Proposed method
EMD based single-channel separation
Wavelets based-single channel separation
VMD based single-channel separation
SI-SIR SI-SDR SI-SAR
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Figure 4. Comparison of algorithms
performances for different SNR values
Figure 5. Comparison of the mean -distance of
Gaussian distribution for different frame length.
7 Conclusion
A new method to solve the signal-channel blind
source separation problem has been proposed. The
method is based on combining two powerful
methods such as the Adaptive Mode Separation-
Based Wavelet Transform (AMSWT) and the ICA-
based single channel separation. A new objective
measure of separation quality has been introduced to
evaluate the performance of the proposed method.
The evaluation parameters are called scale-invariant
(SI) such as SI-SDR, SI-SAR, SI-SIR. Simulation
results showed the good performance of the
proposed method compared to the exiting method.
0 5 10 15
0.04
SNR (dB)
RRMSE
Proposed method
EMD based single-channel separation
Wavelets based-single channel separation
VMD based single-channel separation
1
6
Frame length
Mean distance
EMD based single-channel separation
Wavelets based single-channel separation
Proposed method
VMD bas ed single-channel separation
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