Blind Separation of Underwater Acoustic Signals
N. Benchekroun, and A. Mansour, Member, IEEE
Abstract- Acoustic tomography is an issue of great importance
in many applications. Few algorithms have been dedicated to
the passive acoustic tomography of a single input single output
(SISO) channel. As matter of fact, most of those algorithms can
not be applied in a real situation i.e. for a Multi-Input Multi-
Output (MIMO) channel. In this paper, we discussed at first
the features of the acoustic channel and signals then we
proposed an architecture to separate acoustic signals issued
from an acoustic realistic channel. The proposed structures
used algorithms based on independent component analysis
(ICA). We should mention here that many algorithms have
implemented and tested but only two algorithms give good
results. The latter algorithms minimize two different second
order statistic criteria in the frequency domain. Finally, some
simulations have been presented and discussed.
I. INTRODUCTION
To estimate ocean physical parameters (such as temperature
distribution, currents, sediment structure), The Ocean
Acoustic Tomography (OAT) is widely used. Acoustic
tomography is used in many civil or military applications
such as: Mapping underwater surfaces, meteorological
applications, Warfare. OAT basic principle relies on
dependence of the acoustic propagation on the spatial
distribution of the ocean parameters, in particular
temperature. Two kind of OAT can be used: Classical active
OAT and Passive ones. In active OAT, a typical sound
source in known fixed position should be emitted. At the
receiver, one can estimate the channel parameters by using
the properties of the received signal. Many algorithms [11,
12] have been developed to deal with active acoustic
tomography.
Recently, the Passive Acoustic Tomography (PAT) [10]
which is an acoustic tomography variant where the usual
cooperative acoustic source is replaced by a non-cooperative
noise source as for example a ship of opportunity, has taken
an increased importance because it exhibits three main
advantages: suits the Submarine Acoustic Warfare
Applications, avoids the need of both economically and
operationally expensive signal sources and allows the
investigation of areas as wide as those swept by the moving
ships and finally doesn’t perturb the ecological system. In
typically real world PAT applications, underwater signals
are generated by various sources in motion whose number’s
and positions are hardly identified. With more than two
sources, many actual tomography algorithms can’t give
pleasant results. Many others don’t work well or at all when
the emitted signals are large band [14].
The Blind Separation of Sources (BSS) problem consists on
retrieving unknown mixed independent sources from their
observed mixture. To reach that goal, researchers use
Independent Component Analysis (ICA) methods. Actually,
the last methods are ones of the most up-to-date methods in
signal processing. These methods can be applied in many
fields including speech processing, data communication,
biomedical signal processing, radar, sonar as well as the
surveillance and control of airport and sea traffic. In this
paper, we apply BSS algorithms in PAT in order to improve
and simplify the PAT algorithms as well as the processing of
the received signals.
II. ACOUSTIC CHANNEL MODEL AND
ACOUSTIC SIGNALS
A. Acoustic Channel Model
Under some mild assumptions [12], acoustic submarine
channel can be considered as a multiple paths which, in
frequency domain, each of them can be defined by a
complex constant gain (i.e. a real lag
i
τ
and a real gain C
i
).
For a classic Single Input Single Output (SISO) Channel, the
relationship between the emitted signal
, which will be
called later on as source signal and the received (or observed
signal
)
is given by:
)(ts
(tx
(1)
∑
=
+−=
M
i
ii
tntsCtx
0
)()()(
τ
Where n(t) stands for an Additive White Gaussian Noise
(AWGN) and M is the channel order or the number of paths.
In general case, the channel can be considered as Multiple
Input Multiple Output (MIMO) channel whose mathematical
model is given by the following equation:
(2)
∑
=
+−=
M
i
nNinSiHnX
0
)()()()(
Where
is the )(nX
1
×
q
vector of observed signals, is
the
)(nS
1
×
p
vector of sources, N(n) is a AWGN
1
×
q
vector
and H is the mixing system where H(z) = (h
ij(z)) is a
qp
×
complex polynomial matrix. In the following, we assume
that the channel is a linear and causal one and that the
coefficients h
ij(z) are FIR filters.
It is obvious that for PAT applications, Blind Source
Separation (BSS) problem can be very helpful. In BSS
terminology, the above problem can be solved by
considering the blind separation of linear mixed signals.
Linear mixtures include the following two models:
- Convolutive Mixtures: When the channel is a multi-path
with memory channel (i.e. an echo real channel). This type
of mixture can represent the general case of acoustic
underwater channel, see equation (2).
- Instantaneous Mixtures: this model is a simplified
convolutive one, especially when the channel is a memory-
less channel (without echo). The last case can be observed in
deep water environment. Then, equation (2) can be rewritten
as:
(3)
In our study, it is the general case (2) which will be
considered. The instantaneous case has been studied in [1].
Convolutive mixture algorithms are generally time
consuming algorithms. Few of them have been developed in
the literature [13] and are dedicated to specific tasks and
signals. In our knowledge, none of them has been optimized
to deal with our problem.
)()()( tNtHStX +=
- After that, the same algorithm should be tested on simple
mixture of acoustic signals.
In practise, our simulated underwater acoustic channel used
the ray theory as a propagation model which is the more
appropriate model to our application. In our model, a sand
bottom had been considered and some random coefficients
have been added to characterize varieties on the top and the
bottom of the channel. Finally, an acoustic model proposed
in [15] was used to consider the acoustic propagation effect.
B.
Acoustic Signals
Generally, Independent Component Analysis (ICA)
algorithms use only the independence assumption of the
sources. In PAT applications, the sources are some signals
of opportunities. Extensive experimental studies have been
conducted by a research engineer in our laboratory to
classify and characterize many recorded artificial (made by
human activities as boats, ships or submarine noises,etc.)
and natural (mainly animals sounds or noises) signals. That
study is of extreme important to ours. In fact, according to
the characterization study, one can conclude the following
facts:
- All the recorded signals have a background ocean noise
which can be considered as an Additive White Gaussian
Noise (AWGN).
- Many signals are Gaussians or close to Gaussians ones.
- Even if all the signals are non-stationary signals, some of
them have more or less periodic components as boat noises.
- Natural signals are very sparse ones and artificial ones are
very noisy.
The above mentioned properties were very useful to select
appropriate ICA algorithms.
III.
SEPARATION STRUCTURES APPLIED TO
ACOUSTIC SIGNALS
A.
Background and Assumptions
The selected algorithms have been tested using the
following three steps:
- To valid our implementation, we use same (or similar)
signals used by the authors of the algorithm, and we try to
obtain same (or similar) results shown by the authors.
- At the end, we try the algorithm on real signals which
cross our simulated underwater acoustic channel.
In our experimental studies, we found that at the third step
none of the tested algorithms can unfortunately achieve a
satisfactory separation according to a set of performance
indexes [16]. For this reason, a complete separation structure
has been implemented using the following pre- and post-
processing modules of the signals see Fig. 1:
- The Low Pass Filter helps us to reduce the impact of
AWGN.
- The filter bank is to improve the frequency resolution of
the frequency algorithms.
- The recovering module of the signals is based on second
order statistics and it uses the correlation of the signals in
time or frequency domain.
Fig.1. The proposed structure
Using these pre- and post-processings, we found that among
the tested algorithms, only two [2, 7] have given satisfactory
results. They were dedicated to separate non-stationary
sources (audio or music signals) and will be called in the
following SOS [2] and Parra [7]. Both of them are
implemented in frequency domain and are using discrete
frequency adapted filter.
B. A Frequency Domain Method For Blind Source
Separation Of Convolutive Audio Mixture (SOS)
K. Rahbar et al. in [2, 6] propose an algorithm which
minimize a criterion based on the cross-spectral density
matrix of the observed signals. For non-stationary signals,
the latter matrix depends of frequency and time epoch m:
Let
),( mP
x
ω
represent the cross-spectral density matrix of
the observed signal at frequency
ω
and time epoch m. Based
on the above assumptions, the main idea of this approach
comes from the following equation:
(4)
IHmPHmP
s
x
2
)(),()(),(
σωωωω
+=
+
Where
),( mP
s
ω
is a diagonal matrix which represents the
cross-spectral density matrix of the sources at epoch m and
is the power of N(t). In practise,
2
σ
ω
has to be discretized
as
K
k
k
π
ω
2
=
where K is the total number of frequency
samples. For q>p,
can be estimated from the smallest
eigenvalue of the matrix
. Therefore, we consider
the following noise free case:
2
σ
),(
~
mP
x
ω
(5)
)(),()(),(
ωωωω
+
= HmPHmP
s
x
To do so, the authors of [2] developed a two stages
algorithm. The first stage employs joint diagonalization [3,
4, 5] of the set of cross power spectral density matrices
, m = 0,…, M-1 at each frequency
),( mP
x
ω
k
ω
, over M
epochs, to estimate the mixing system up to a permutation
and diagonal scaling ambiguity at each frequency bin. In [2],
the authors used the following least-squares based joint
diagonalization criterion for the case when a sample estimate
of each
is available:
),( mP
x
ω
F
K
k
M
m
kkkkx
mB
BmBmP
k
∑∑
−
=
−
=
+
Λ
Λ−
1
0
1
0
2
^
1)(),(
)(),()(),(min
ωωωω
ω
(6)
Where
)(
k
B
ω
is an estimate of the mixing system )(
k
H
ω
,
is a sample estimate of the observed signal
cross spectral density matrix at frequency bin
),(
^
mP
kx
ω
k
ω
and time
epoch m, and
),( m
k
ω
Λ is a diagonal matrix, representing
the unknown cross-spectral density matrix of the sources at
each epoch m. In the second stage of the algorithm, the
authors propose a novel solution for solving the permutation
problem which exploits the cross-frequency correlation
between diagonal values of
),( m
k
ω
Λ and ),(
1
m
k+
Λ
ω
.
Finally, having
)(
k
B
ω
= )(
k
H
ω
Π
D(
k
ω
) at each
frequency bin (
is a permutation matrix and
D(
pp
R
×
∈Π
k
ω
) represents a frequency-dependent diagonal matrix),
we can calculate the separating matrix
)(
k
W
ω
from the
following equation:
(7)
)()(
kk
BW
ωω
+
=
Where
) is the pseudo inverse of matrix
k
B
ω
(
+
)(
k
B
ω
.
C.
Convolutive Blind Separation of Non-Stationary
Sources (Parra)
The main idea of the algorithm [7] proposed by L. Parra and
C. Spence is similar to the previous ones [2] and [8]. This
algorithm will be called later “Parra”.
The main difference
between the two approaches remains on the considered
estimation model. The authors prefer instead of estimating a
forward model B of H and finding a stable inverse to
directly estimate a stable multi-path backward FIR model
W. They wish to find model sources with cross-power-
spectral-density satisfying:
(8)
)()],(),()[(),(
^^
ωωωωω
H
n
xs
WmmPWm Λ−=Λ
A multipath model W that satisfies these equations for M
epochs simultaneously can be found with a Least Square
estimate:
2
11
1)(
,0)(
,,
^^^
),(minarg,,
∑∑
==
=
<<>=
ΛΛ
=ΛΛ
K
k
M
m
W
KQW
W
ns
mErW
ii
ns
ω
ω
ττ
(9)
Where
),()()],(),()[(),(
^
mWmmPWmEr
s
H
n
x
ωωωωωω
Λ−Λ−=
The additional time domain constraint on the filter size Q of
W relative to the frame size K, i.e.
0)( =
τ
W
KQ
<
<>
∀
τ
will restrict the solutions to be continuous in the frequency
domain and solve the frequency permutation problem. The
same idea was used in [9].
IV.
EXPERIMENTAL RESULTS
Many simulations have been conducted. Generally, we need
over 600000-1200000 samples to achieve the separation.
The original sources are sampled at 44 KHz. In almost all
the simulations, the separation of artificial or natural
mixtures have been successfully achieved. In these
simulations, we have set the channel depth between 100 to
500m, the distances among the sources or the sensors are
from 30 to 300 m, the distances among the different sources
and the diverse sensors are from 1500 to 2500 m, the
number of sensors (3 in all the simulations) is strictly greater
than the number of sources (2).
Fig. 2 represents the experimental results obtained by
applying the structure (SOS) proposed in fig.1 to separate a
mixture of acoustic signals (Whale and ship). We should
mention here, that good results have been obtained by only
applying SOS algorithm except for some configurations
notably when the sources are close to the water surface. For
the latter cases, we found that the Parra algorithm before
SOS algorithm could improve the overall results.
Fig. 2. Experimental results: the first line shows respectively the original
and the estimated sources, the second line contains the observed signals (the
sources are: a whale sound and a boat noise)
Best results have been obtained when both algorithms Parra
and SOS are used and the number of sensors is strictly
greater than the number of sources, as shown in Fig3.
Fig.3. The general structure
Fig. 4 shows the experimental results obtained by applying
the general structure (Parra+SOS) proposed in fig.3 to
separate a mixture of acoustic signals (two ships) convolved
in an acoustic channel.
Fig. 4. Experimental results: The figure shows respectively the original, the
estimated sources at the first line and the observed signals at the second line
(the sources are: two boat noises).
Fig. 5
Fig. 5 shows the experimental results obtained by the
different algorithms (Parra, SOS or Parra + SOS) applied to
the same acoustic sources and the same configuration of the
acoustic channel at each simulation. In this figure, a
normalized performance index based on nonlinear
decorrelation is used [16]. This index is forced to be zero for
the mixture values and 1 for the sources.
V. CONCLUSION
In this paper, we presented a general structure using BSS
algorithms applied on a real word application which is the
Passive Acoustic Tomography (PAT). After many
simulations, we obtained experimental results that showed
the necessity of considering pre-processing and post
processing which have to be applied to the observed signals
in order to achieve properly the separation of the sources.
Many algorithms have been implemented and tested on our
application but only few BSS algorithms dedicated to the
separation of non-stationary signals gave satisfactory results.
Our future work consists on developing a BSS algorithm
which can use other features of acoustic signals.
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![Fig. 5 shows the experimental results obtained by the different algorithms (Parra, SOS or Parra + SOS) applied to the same acoustic sources and the same configuration of the acoustic channel at each simulation. In this figure, a normalized performance index based on nonlinear decorrelation is used [16]. This index is forced to be zero for the mixture values and 1 for the sources.](/figures/fig-5-shows-the-experimental-results-obtained-by-the-2wadxt0w.webp)