Enhancement of acoustic tomography using spatial and frequency diversities
 Ali Mansour^{1}Email author
https://doi.org/10.1186/168761802012225
© Mansour; licensee Springer. 2012
Received: 28 July 2011
Accepted: 13 June 2012
Published: 24 October 2012
Abstract
This article introduces several contributions to enhance an important application such as acoustic tomography (AT), using mainly the spatial and spectral diversities of underwater acoustic signals. Due to their inherited properties, (i.e. spareness, nonstationarity or cyclostationarity, wideband frequency range, wide range of power, etc.), the process of underwater acoustic signals becomes a real challenge for many scientists and engineers who are involved in studies related to the ocean. For various applications, these studies require huge and daily information. AT techniques remain fast and cheap ways to obtain such data. Nowadays, active acoustic tomography (AAT), is communally used to generate powerful and repetitive acoustic sources. Recently, researchers have been attracted by an alternative way, called passive acoustic tomography (PAT), which uses acoustic opportune signals of their environment. PAT techniques are mainly used for ecological, economical and other reasons such as military applications. With PAT, no signal is emitted; therefore, problems become more challenging. The number and positions of existent sources are unknown, and sensors measure mixtures of available sources. Algorithms based on time or frequency domains are widely deployed to classify, identify, and study received signals in AAT applications. For PAT, researchers employ multiple sensors in order to add an extra dimension, (such as space). This article focuses on approaches used in space along with time or frequency to extract information, improve performances, and simplify the overall architecture. This article explains the use of signal processing and statistical approaches to solve problems raised using PAT and discusses the experimental results. The review of the literature offers a big variety of algorithms to deal with classic AAT problems. Therefore, only problems related to PAT have been considered herein.
Keywords
Introduction
Oceans cover more than 70% of the earth surface, roughly containing 97% of all our water supply and playing a major role in global climate regulation and economical systems.
Acoustic tomography (AT) is used in many civil or military applications such as: mapping underwater surfaces, oceanographical, meteorological applications, (to measure the temperature, the salinity, the motion and the depth of the water), to improve sonar technology, as well as other applications. Many algorithms[1] have been developed to deal with active acoustic tomography (AAT).
 1.
Emitting sources similar to natural sounds or noises: a set of artificial signals imitating natural sounds, (whales, dolphins, etc.), or noises, (waves, ships, etc.), are generated. The main advantage of such approach remains in the control of the sources and their positions, (similar to active methods). In order to achieve this, researchers could imitate the timefrequency signature of nature signals. However, this method is not totaly discreet as the generated signals may have different high order statistics (HOS), instantaneous power, or frequency than original signals. Besides, artificial signals generally can be characterized by specific patterns, (periodicity, time or statistical coherence, fixed positions, and deterministic motions, etc.). These specific patterns can be used to unmask hidden emitted signals.
 2.
Using natural signals: by completely relying on existing natural signals, a PAT system with a high discreet level can be achieved. However, main drawbacks of such system are the lack of information, (number, positions or natures of sources, etc).
 3.
Applying hybrid systems: by mixing the previous two strategies, better performances and good discretion levels could be achieved. However, that will results in more complex emitterreceiver systems.
On the one hand, it seems that the 2nd strategy is the more attractive one, (completely discreet systems and no emitters). On the other hand, the problems raised in this case are more challenging because of the total lack of information about the sources. In order to reduce the complexity of this problem, we investigate several advanced signal processing techniques and statistical approaches. In fact, let us assume that we are able to estimate the number of the sources, separate the sources form their mixing observed signals, and evaluate their statistical properties. In this case, the identification of the channel could be, also, investigated. Therefore, remaining PAT problems become very similar to AAT.
The article’s primary purpose is to discuss the preprocess observed mixed signals to extract maximum information about the sources, then, we can apply classic algorithms to deal with residual problems. This article is organized as follows: Section “Acoustic oceanic tomography”, describes AAT and PAT, briefly; Section “Assumption and background, contains the assumptions and mathematical models; Section “Preprocessing systems”, presents the preliminary studies; Section “Adaptive HOS estimators, proposes new HOS estimators, in order to enhance the spatial diversity of original sources; Section “Spatial diversity and independence discrimination criteria, discusses several criteria, so as to exploit the spatial or the spectral diversity of our signals; Section “Blind separation of observed acoustic signals”, presents independent component analysis, (ICA), algorithms to separate mixed observed signals; Section “Experimental results”, shows experimental results; and Section “Conclusion”, presents the conclusions.
Acoustic oceanic tomography
Acoustic tomography’s goal is to get a fast and cheap monitoring of water mass and subbottom characteristics. This monitoring requires an inversion 2step procedure[2]. First, estimate the acoustic properties, (such as the water column sound speed profile, 3D structure of internal tides in water masses, geoacoustic parameters of the seafloor), from the measurement of a known propagated acoustic waveform between fixed sources and receivers. Second, infer some ocean physical parameters from these estimated acoustic characteristics.
Active acoustic tomography
To perform oceanic tomography, an active acoustic emission is propagated between an emitter and a set of receivers on an horizontal track of about 10 km long. Frequencies involved in tomography range is from 30 Hz to a few kHz, whereas, power range is from 180 to 220 dB.
First works in tomography have been only considered deep water channel, (depth deeper than 1 km). In this case and in order to estimate underwater acoustic transmission channel parameters, acoustic refraction is the main physical phenomenon which should be considered.
In the mid 1990s, scientists have extend their interests to shallow water, (i.e. depth less than 300 m),[3].
In shallow water, an acoustic propagation encounters numerous interactions with the sea surface and the sea floor. Therefore, new techniques had to be developed such as ‘matched field processing’ in[4] and the ‘matched impulse response processing’ in[5]. In their applications, a single input multiple output, (SIMO), configuration is used to extract channel information.
To get efficient results in a SIMO configuration, a large number of sensors should be used which means increasing the experimental setting. To tackle the last problem and using frequency diversity, researchers proposed “matched impulse response processing” methods. In the last case, a wide band signal should be emitted, but a single distant hydrophone could be enough as a receiver. The main idea of such technique consists of estimating the channel impulse response by applying maximum likelihood or matched filter estimations on the known emitted and received signals[6]. Once the channel response filter has been estimated, other features such as time delay or magnitude of arrivals could be extracted. The last features could be used in order to estimate water column and subbottom properties.
Passive acoustic tomography
Active acoustic tomography strongly relies on the possibility to emit powerful acoustic signals in the ocean. Major problems can arise. powerful emissions need a heavy power supply which can drastically limit the efficiency of autonomous monitoring systems, thereby causing drastic harm to marine mammals and disturbing their behavior. Finally in a warfare context, some constraints about covertness may exit in the acoustic process. To overcome these problems, the concept of PAT has recently emerged in the community.
Passive acoustic tomography consists in estimating acoustic properties by using natural opportunity sources present in the channel at the time of interest without using active emission. Surface noise created by breaking waves, ship noise, and marine mammal calls are three kinds of opportunistic sources which are under the scope of passive tomography[7].
The main drawbacks of PAT are the lack of information about the number, positions, and nature of emitted signals. With more than two sources many actual tomography algorithms can’t give satisfactory results. Many others don’t work well or at all when the emitted signals are wide band signals[8]. Some algorithms take into consideration the position of acoustic sound emitters[9]. Typically, in real world PAT applications, underwater acoustic signals are generated by various moving sources whose number and positions are hardly, (or impossible), to be identified, (as in the case of shoal of fish or wave noises). It is obvious that PAT is a quite difficult technique requiring substantial effort in signal processing to tackle the unknowns of source position and emitted waveform as well as to separate the sources present simultaneously in the channel before switching them toward a dedicated blind inversion processor.
Assumption and background
In PAT applications, the sources are obviously signals of opportunities which have various properties such as spatial diversity, different probability density functions (pdf), different temporal or spectral structures, different timefrequency signatures, etc. These properties can be used at different level of the separation stage. However, in PAT applications, simple and cheap systems are often used which means that linear multisensor antenna are not recommended. Mainly, for this reason, ICA algorithms will be of great importance to reach our goal. ICA algorithms can successfully handle multiinput multioutput (MIMO) channel.
In a previous work[10], an extensive experimental study has been conducted in order to classify and characterize many recorded anthropogenic signals, (made by human activities as boats, ships, or submarine noises, etc.), and natural signals, (mainly animals sounds or natural noises, such as waves etc.). According to that study, one can add to the above mentioned features, the following ones:

Recorded signals are affected by a background ocean noise which can be considered as an additive white Gaussian noise (AWGN).

Some signals have a very weak kurtosis[11].

Almost all of the signals are nonstationary signals with more or less cyclic behavior as boat noises.

Natural signals are very sparse ones and artificial ones are very noisy.
The above mentioned properties have been considered to select appropriate ICA algorithms.
Underwater acoustic channel
Underwater sounds are produced by natural or artificial phenomena through forced mass injection leading to inhomogeneous wave equations which can be converted to frequency domain[12]. The frequencydomain wave equation is called the Helmholtz’s equation which gives us an underwater sound propagation model. A general solution of the Helmholtz’s equation is very difficult to obtain. Therefore, researchers use simplified propagation models, (such as the ray theory, the mode theory, the parabolic model, the hybrid model, etc.), according to their applications[13]. The choice of a propagation model depends on many parameters such as wave frequency, the depth of the sea, etc. In our case, (shallow water, i.e. the channel depth is about few hundred meters), our frequency range is from 300 to 10 KHz, the ray theory was the more appropriate propagation model.
The above equation is an empirical relationship satisfied when 0 ≤ T ≤ 30, 30 ≤ S ≤ 40, and D ≤ 8000. In shallow deep underwater channels[15], (depth less than 300 m), where emitters and receivers are not so close to the water surface nor to the bottom and the distances among emitters and receivers are less than 3 Km, the sound speed could be approximated by a constant.
The reflected acoustic waves on the bottom of the propagation channel depend on many parameters such as the composition and the geometrical properties of the bottom[16].
The reflected acoustic waves on the top of the propagation channel, i.e. the water surface, depend, also, on many parameters such as the wind, the wave frequency as well as the swell properties[16]. For this reason, the water surface can’t be considered as a flat surface. Therefore, the direction of a reflected acoustic wave is dispersed in the space. However in average term, reflected acoustic waves can be considered as obtained by a flat surface with absorption coefficients[15]. In our model, a flat surface is considered and random coefficients are added to characterize other unknown parameters.
where$\left(\right)close="">{f}_{T}=21.9\phantom{\rule{0.3em}{0ex}}\ast \phantom{\rule{0.3em}{0ex}}1{0}^{\left(6\frac{1520}{T+273}\right)}$, (in kHz), T is the water temperature, (°C), S = 3.5% is the water salinity, (in the ocean S ≈ 35g/l), P_{ w } is the water pressure, (in kg/m^{2}), A = 2.34 ∗ 10^{−6}and B = 3.38 ∗ 10^{−6}.
From physical point of view, an acoustic ray represents a propagation trajectory of an emitted signal between the source, (emitter), and the receiver. In many cases, the channel depth is limited in size which means that the propagation is multirays. Each ray may be bent by refraction if the sound speed is a function of depth and range. Ray trajectories and sound speed profile allow us to compute propagation times. In addition ray trajectories, water attenuation, boundaries roughness and subbottom properties allow us to compute the signal magnitude.
From a computational view point, ray trajectory is computed by solving the ‘Eikonal equation’ but signal magnitude is obtained as a result of ‘Transport equation’[12]. As general and analytical solutions of Eikonal and transport equations do not exist, researchers use approximate and simulated results[18].
Mathematical models
Here H(i) denotes the q × p real constant matrix corresponding to the impulse response of the channel at time i and S(n−i) is the source vector at time (n−i).
Preprocessing systems
As it was mentioned before that the processing of acoustic signals is a very challenging problem. To enhance our processing algorithms, pre and post processing systems have been proposed.
Pre & postprocessing
Our sources are bounded in frequency domain. Therefore, a lowpass filter was extremely helpful for us to reduce the impact of the AWGN and, then, achieve better performances. It is worth mentioning that only three tested algorithms have given satisfactory results. These three algorithms, (for further details see the following references[19–23]), were dedicated to separate nonstationary sources (audio or music signals). The last two algorithms[22, 23], which be called in the following SOS[22] and Parra and Alvino[23], are implemented in frequency domain using discrete frequency adapted filters.
Estimation of source number
It is obvious that the number of sources is an input parameter. ICA algorithms can cope with an overestimate number of sources, (extra separated signals should be residual noises). However, an underestimation of that number can affect seriously overall performances[24]. For this reason, a rough estimation of that number should be considered. To roughly estimate the source number, few approaches have been considered and briefly discussed. Hereinafter, the channel is assumed overdetermined, (i.e. q > p).
Here Σ_{ S } stands for the unknown and invertible diagonal covariance matrix of the statistically independent sources. For noise free channel, the rank of Σ_{ X } becomes equal to the rank of Σ_{ S }otherwise the number of sources[25].
With an AWGN channel, Σ_{ X }becomes a full rank matrix. Without loss of generality, let us assume that noise components have the same variance, then, the q singular values λ_{ i }of Σ_{ X }will have different values except the last q − p ones. Normally, the first p singular values are linked to signal space and the last q − p ones are related to the noise space. In order to apply this method, one should deal with two problems: How can we estimate the covariance matrix of nonstationary signals an what is the optimal threshold between the two sets of singular values? The estimation of covariance matrices has been conducted over slippery estimation windows, see Section “Adaptive HOS estimators’. Concerning the threshold, it can be easily set when the signal to noise ratio (SNR) is relatively high. Unfortunately, the SNR is our case is not high enough, (i.e. SNR > 2 dB). Therefore, different thresholds have been considered:

If q > p + 5, one can easily set a threshold as the limit between two sets of singular values. This approach requires a very good SNR and q >> p.

To improve the first approach, normalized singular values have been considered, (i.e. λ_{ i }have been divided by the maximum λ_{ i }). Experimental results showed that a threshold can be easily set using normalized singular values when SNR is higher than 10 dB and the signatures of sources are relatively the same, (the signature of the i th source on the j th sensor is the power received by that sensor from that source. Therefore, the signature of a source depends on the source power and the channel parameters.). The last two assumptions can’t be, always, satisfied in our application.

Another method was considered: first, the singular values λ_{ i }should be sorted in descending order; second, sorted λ_{ i }should be divided by λ_{2}. Finally, the number of sources is considered as the number of normalized λ_{ i }> ε, where ε depends on SNR. Experimentally, we obtained satisfactory results for SNR higher than 4 dB and ε > 0.1.

By considering that the signals are close to Gaussian ones, one can use Akaike’s information criterion, (AIC), to set the threshold. Even though the gaussianity assumption is a strong one, (underwater acoustic signals are very strong nonstationary signals which can not be considered as gaussian signals), Karhunen et al.[26] shows that obtained results are still satisfactory.
where$\left(\right)close="">{\mathcal{S}}_{N+M}\left(n\right)$ stands for the extended signal vector and T_{ N }(H) is the Sylvester matrix which is full rank under some mild assumptions[27].
where H_{s+}is the hypothesis that number of sources is higher than s and the threshold T_{ s } should be set so that the allowable probability of false alarm can be achieved.
A main advantage of the last algorithm comparing to previous approaches is that this algorithm can be applied even though the noise are spatially correlated and that it can give a confidence level for the estimated number. The main drawback is the computational effort. In fact, with 2N + 1 receivers, one can only estimate a source number up to N. In the following, we consider that the number of sources is already estimated.
Adaptive HOS estimators
In order to exploit spatial diversity, many blind or semiblind separation; or identification algorithms uses HOS, in time or frequency domain. For this reason, the estimation of cross cumulants and moments up to the fourth order have been investigated in this section, further details are given in Appendix 1.
k_{4}(X) is a consistent biased estimator of Cum_{4}(X). In previous studies[30], we proposed and compared estimators for autocumulants of second and fourth orders. Here, we propose new adaptive HOS estimators for fourth order crosscumulants which can be applied on underwater acoustic signals which are nonstationary signals.
where 0 < λ < 1 is a forgotten factor. To evaluate the performances of last estimators, some simulations have been conducted using a nonstationary zeromean signals. For example, let S(n) be a nonstationary signal that consists of four parts:

S_{1} is an uniform signal in [1, 1] with 8,000 samples.

S_{2} is Gaussian with unit variance and 5,000 samples.

S_{3} is an uniform signal in [2, 2] with 3,000 samples.

S_{4} is Gaussian with a standard deviation$\left(\right)close="">\sigma =\sqrt{2}$ and 4,000 samples.
Using S, two other signals have been generated X(n) = S(n) and Y(n) = S^{3}(n), (it is clear that x_{ i }and y_{ i }are i.i.d and that x_{ i }depends on y_{ i }). Using the definition of cumulants and the properties S, we can prove that:

For uniform parts,$\left(\right)close="">{\text{Cum}}_{31}(X,Y)=\frac{2}{35}{a}^{6}$, here a is the maximum amplitude.

For the Gaussian parts, Cum_{31}(X,Y) = 6σ^{6}.
Finally, x_{ N }and y_{ N } in Equation (15) have been replaced by their average over a small estimation window, (10 to 50 samples). The above proposed estimators can be improved by considering non iid samples. However, in the last case, a stochastic model with transition probability should be considered. The last statement is beyond the scoop of this manuscript and it will be considered in a future study. Hereinafter, HOS are estimated at different stages using the estimators described in this section.
Spatial diversity and independence discrimination criteria
where s_{2}(n) represents a mixture of all sources except the first one s_{1}(n). The filter$\left(\right)close="">{h}_{i}(z)={h}_{i}(0)+{h}_{i}(1){z}^{1}+\cdots +{h}_{i}({m}_{i}){z}^{{m}_{i}}$ is a residual separation filter. The separation is considered achieved when the norm of the residual error h_{2}(z)∗s_{2}(n) becomes much less than the one of the separated signal h_{1}(z)∗s_{1}(n). In addition, the identification or classification of underwater acoustic signals is extraordinarily difficult step because these signals are nonstationary and nonintelligible sparse signals with low variable kurtosis. In this context, the classification of ICA algorithms according to the separation quality becomes a difficult and important task.
The following discrimination criteria can be optimized to maximize the spatial diversity or the independence among estimated signals. At the same time, they can be very useful to quantify the separation achievement. In the last case, these criteria are called performance indices.
Modified crosstalk
To apply the crosstalk, one should have original sources. Therefore, this performance index cannot be applied in real situation where sources are unknown. However it is very useful in simulations.
It is well known that sources can be separated from a convolutive mixture up to a permutation and up to a scalar filter. Therefore, the last definition D_{ r } is useless for the BSS convolutive mixture, see Equation (16), since it doesn’t take into consideration the power ratio between the filtered version of the signal ξ_{1} = h_{1}(z)∗s_{1}(n) and the residual error h_{2}(z)∗s_{2}(n).
Our experimental results show that for a low order channel filter, (<20), this performance index can be used efficiently. When the order of channel is larger than 20, computing time becomes a big issue.
Mutual information
where U = (u_{1},…,u_{ n })^{ T } is a random vector and P_{ U }(V) (resp.$\left(\right)close="">{p}_{{u}_{i}}({v}_{i}))$ are the joint, (resp. marginal), PDF. In the context of BSS problem, the joint and the marginal PDF are unknown but they can be estimated.
Here$\left(\right)close="">{\widehat{\Pi}}_{U}(i)$ is the joint PDF estimator and$\left(\right)close="">{\widehat{\Pi}}_{{u}_{k}}(j)$ is the marginal PDF estimator. Good results have been obtained with stationary signals, but we couldn’t get similar results for underwater acoustic signals.
Quadratic dependence
Here h is an integrable function from$\left(\right)close="">{\mathbb{R}}^{n}$ to$\left(\right)close="">\mathbb{R}$ which satisfies the following two conditions[34]:

h is a non zero almost everywhere and a positive function.

For analytical FCF Φ(Ω), h should be positive around zero and vanish elsewhere.
Here$\left(\right)close="">\mathbb{K}$ is a square integrable kernel function that its Fourier transform should be non zero almost everywhere and$\left(\right)close="">{\sigma}_{{X}_{i}}$ is a scale factor, (i.e. a positive function only depends on the PDF of X_{ i }).
 (1)
Gaussian Kernel $\left(\right)close="">{\mathbb{K}}_{1}(x)=exp({x}^{2})$
 (2)
Square Gaussian Kernel $\left(\right)close="">{\mathbb{K}}_{2}(x)=\frac{1}{{(1+{x}^{2})}^{2}}$
 (3)
Inverse of Square Gaussian Kernel second derivative function $\left(\right)close="">{\mathbb{K}}_{3}(x)=\frac{420{x}^{2}}{{(1+{x}^{2})}^{2}}$
In our experimental studies, the best results were obtained using the Gaussian Kernel. In fact, the Gaussian Kernel gives the largest possible difference between the quadratic independence measure applied on a vector A with i.i.d uniformly independent components and the quadratic independence measure applied on a vector B = MA, here M is a full rank mixing matrix. The main drawback of such performance index is the important computing time.
Nonlinear Kernel decorrelation
We call Cov(X,Y) and Var(X) respectively the covariance and the variance of X and Y . It is worth mentioning that$\left(\right)close="">\mathbb{F}$ is a vectorial space of all functions applied from$\left(\right)close="">\mathbb{R}$ to$\left(\right)close="">\mathbb{R}$ which contents all Fourier transform basis, (i.e. the exponential functions exp(jwx), with$\left(\right)close="">w\in \mathbb{R}$).$\left(\right)close="">{\rho}_{\mathbb{F}}$ means the independence between X and Y .
Using different kernels, Gaussian, Polynomial and Hermite functions, the NLdecorrelation is applied on source and mixed signals
Signals  Mixture model  NLdecorrelation of sources  NLdecorrelation of mixed signals 

i.i.d uniform PDF  Instantaneous  Kernel ‘Gaussian’ − 23.4  Kernel ‘Gaussien’ −5.8319 
Kernel ‘poly’ − 25.5  Kernel ‘poly’ 8.1  
Kernel ‘hermite’ − 22.4  Kernel ‘hermite’ −20.4  
Four acoustic signals  Instantaneous convolutive  Kernel ‘poly’ − 33.4  Kernel ‘poly’ 3.2 
2000 samples  Kernel ‘poly’ −14.9817  
Four acoustic signals  Instantaneous convolutive  Kernel ‘poly’ − 31.3  Kernel ‘poly’ 8.8 
4∗10^{5} samples  Kernel ‘poly’ −13.2 
Simplified nonlinear decorrelation
 (1)
‘Gauss’: Gaussian kernel.
 (2)
‘poly’: 6th order polynomial Kernel which its coefficients are the components of an unitary vector.
 (3)
‘atan’: Saturation kernel using arctangent function.
 (4)
‘tanh’: Saturation kernel using hyperbolic tangent function.
Simplified NLdecorrelation applied on source and mixed signals using different kernels
Signals  Mixture model  NLdecorrelation of sources  NLdecorrelation of mixed signals 

i.i.d  Kernel ‘Gaussian’ − 66.3211  Kernel ‘Gaussian’ −40.6513  
Uniform PDF  Instantaneous  Kernel ‘poly’ −49.2054  Kernel ‘poly’ −6.6205 
uniform  Kernel ‘atan’ −63.2202  Kernel ‘atan’ −0.0802  
Kernel ‘tanh’ −52.5625  Kernel ‘tanh’ 0.1597  
Four acoustic signals  Instantaneous  Kernel ‘atan’ −40.7142  Kernel ‘atan’ 1.5864 
2000 samples  Convolutive  Kernel ‘atan’ −31.8532  
Four acoustic signals  Instantaneous  Kernel ‘tanh’ −86.6931  Kernel ‘tanh’ 1.0391 
4∗10^{5} samples  Convolutive  Kernel ‘tanh’ −57.4885 
Independence measure based on the FCF
where g is an adequately chosen function[37],$\left(\right)close="">{X}^{\prime}={\mathrm{\Phi}}^{1}\left(\frac{8X3}{8n+2}\right)$ is the approximation of the score function of X, and Φ(X) is the PDF of zero mean and unite variance Gaussian signal. Our experimental studies show that the computing time is the main drawback of this performance index. We should mention that for stationary signals, this performance index is consistent. Unfortunately, the last intersting property is useless in our application since the acoustic signals are nonstationary signals.
Here k(t,s) is a bounded estimation window. Our experimental studies show that:

The obtained values depend on original sources. This inconvenient is common to previous performance indices.

For beta random variables, good results have been obtained. On the other hand, we noticed bad results for uniform random signals.

For acoustic signals, we noticed good results for instantaneous mixture and bad ones for convolutive mixtures.

Computing time is crucial.
Crosscumulants
where Γ = (Perfc(X_{ i },X_{ j })) and$\left(\right)close="">\text{Off}(\mathrm{\Gamma})=\sum _{i\ne j}{\gamma}_{\mathit{\text{ij}}}^{2}$. Good results have been obtained using this performance index on instantaneous or convolutive mixture of acoustic signals. However, the computing time is relatively important.
Blind separation of observed acoustic signals
In previous study[39], we implemented and tested some instantaneous ICA algorithms. According to that study, good results, at least in instantaneous mixture of acoustic underwater signals, can be obtained using ICA algorithms based on HOS or dedicated to nonstationary signals. The algorithms discussed in this section have been selected according to our previous study.
In real applications of PAT, hydrophones could record mixed signals. In order to apply classic AAT algorithms, one should, first, separate the recorded mixed signals. It was mentioned that in PAT applications, MIMO configuration is quite possible. In this case, the sources could be generated and recorded at different locations. This spatial diversity could be translated into statistical independence. Since the early of 1990s, ICA, has been considered as a set of important signal processing tools[40–42]. By assuming that the unknown p emitted signals, (i.e. sources), are statistically independent from each other, ICA consists on retrieving a set of independent signals, (output signals), from the observation of unknown mixtures of the p sources. It was proved that the output signals can be the sources up to a factor, (or filter), scale and up to a permutation[43].
Due to long and sparse impulse response of acoustic underwater channels and acoustic underwater signals’ features, (i.e. nonstationary, close to Gaussian, sparseness, etc.), see Section “Assumption and background’, many ICA algorithms couldn’t achieve the separation of sources in our application. Every selected and implemented algorithm has been evaluated using the following steps: we, first, used the same, (or similar), signals to the ones originally proposed by the authors of that algorithm. Second, an algorithm should be run over some simulated scenarios using a set of nonstationary signals, (normally speech signals), in memoryless or simple convolutive channels. Algorithms that give good, (or at least satisfactory), results in the first two stages have been selected in our project.
Best experimental results were obtained using two frequency domain ICA Algorithms[22, 23] based on the minimization of second order statistics criteria in frequencydomain. These two algorithms exploit the spatial and the spectral diversity of the original signals. In the following, the major tested algorithms are briefly described.
Blind estimation of time delay
It is worth to be mentioned that the authors proposed in[44] another version of their algorithm. However, we didn’t implemented the latest version of the algorithm, for the simple reason that the first version of algorithm didn’t give satisfactory results in our application. In fact, underwater acoustic channel is more complex than the model considered by the authors.
Nguyen’s algorithms
In the early 1990s, Nguyen and Jutten[45–47] were the first to propose an ICA algorithm to separate a convolutive mixture of speech signals. The first version of the algorithm consists on the minimization of a cost function as the mathematical expectation of an odd nonlinear function evaluated over the estimated signals. Later on, they proposed another cost function as the sum of fourth order crosscumulants. To prevent a matrix invertible problem, they proposed a recursive structure which can only deal with a mixture of two sources. The latest constraint can be easily avoided by using our recursive system proposed in[48]. In addition, the algorithms proposed by Nugyen et al. can be, easily, implemented and they have been used to separate speech signals. For these reasons, we decided to implement these algorithms.
In addition to different versions originally proposed by the authors, we implemented hybrid structures, (i.e. a minimization of cost function based on a weighted sum of their different cost functions). Unfortunately, our experimental studies show that the algorithm, in all implemented versions, is not helpful to reach our goal. In fact the performance of the separation were not satisfactory due to the particularity of our application. It is worth mentionning that the convergence of the algorithm was a critical point in many cases.
Natural gradient applied to entropy maximization
In order to characterize and localize the developing of material defects, acoustic emission analysis (AEA) is used. To improve the performance of their AEA, Kosel et al.[49] have processed observed signals by using an ICA algorithm proposed earlier by Amari and Cardoso[50] based on the natural gradient minimization algorithm proposed in[51], and introduced independently by Cardoso and Laheld[52] under the name of relative gradient.
In the context of our project, many simulations have been conducted. According to our experimental studies, these algorithms can render good results for stationary signals and for relatively short channel filters, (i.e. low order filters). Unfortunately, divergence problems or non satisfactory results were often observed when the signals were sparse non stationary ones and the channel filter was very long as in our application.
Blind separation of non stationary signals
 (1)
H(z) is a full rank stable filter matrix and it has no zero on the unit circle.
 (2)
The sources are zeromean nonstationary signals.
 (3)
The sources have different autocovariance r _{ i }(n,m) = E(s _{ i }(n)s _{ i }(n−m)) which should be a time function.
where$\left(\right)close="">L\in \mathbb{Z}$ stands for time delay and R_{ Y }(n) = E(Y(n)Y^{ T }(n)).
The convergence needed a huge number of samples. Besides, obtained results were not always satisfactory. The performances of the algorithm depended on the source signals as well as the transmission channel. The algorithm was a time and memory consuming.
A frequency domain method for BSS of convolutive audio mixture (SOS)
where X_{ im }(w) is the Fourier transform of the observed signals, and J is the number of estimated windows such that L_{ J }< L_{ m } and J L_{ J }> L_{ m }.
where F_{ R }(w,m) and F_{ I }(w,m) are the real and the imaginary parts of Equation (36). Finally, the minimization was done using a conjugate gradient algorithm.
Convolutive blind separation of nonstationary sources
Experimental results
Using the structure proposed in Figure2, many simulations have been conducted. Generally, over 500,000–1,000,000 samples were needed to achieve the separation. The original sources were sampled at 44 KHz. In almost all the simulations, the separation of artificial or natural signals have been successfully achieved. In these simulations, we have set the channel depth between 100 to 500 m, the distances among the sources or the sensors were among 30 to 100 m, the distances among the different sources and the divers sensors are from 1,500 to 2,500 m, the number of sensors is strictly higher than the number of sources.
Conclusion
In this article, several signal processing contributions applied on real world application such as the PAT, have been presented. Many simulations have been conducted and experimental studies showed the necessity of considering preprocessing and post processing of the observed signals in order to achieve properly the separation of the sources.
Many algorithms have been implemented and tested. However, few algorithms which are dedicated to the separation of nonstationary signals, give us satisfactory results. In a real scenario of warfare applications, the use of any ICA algorithm becomes very challenging. In fact, many ICA algorithms can not achieve satisfactory results when:

Most of the signals are close to Gaussian ones.

Sources have very inhomogeneous power, (the power ratio can be up to a dozen of dB).

SNR can be very limited depending on operational situations.

Even though ICA algorithms can handle convolutive mixtures. However, in our applications, the channel filter orders can be up to few thousand. At the same time, such a filter is a very sparse one. In fact, just few filter parameters do not vanish.
Our future work consists on developing an ICA algorithm which can use other features of acoustic signals such as sparseness along with nonstationarity, etc.
Appendix
Appendix 1: HOS estimators
Arithmetic estimators
where$\left(\right)close="">\hat{{\mu}_{r}}\left\{k\right\}$ is the estimator of the r th order moment at the k th iteration.
Exponential estimators
Appendix 2: adaptive unbiased estimator of 4th order cumulant
Declarations
Acknowledgements
A part of this work was supported by the French Military Center for Hydrographic & Oceanographic Studies, (SHOM i.e. Service Hydrographique et Océanographique de la Marine, Centre Militaire d’Océanographie).
Authors’ Affiliations
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