Adaptive subspace detection based on two-step dimension reduction in the underwater waveguide

De-zhi Kong(孔德智),Chao Sun(孫超),,Ming-yang Li(李明楊),Lei Xie(謝磊)

a School of Marine Science and Technology,Northwestern Polytechnical University,Xi’an,710072,PR China

b Key Laboratory of Ocean Acoustics and Sensing,Ministry of Industry and Information Technology,Xi’an,710072,PR China

c College of Information Science and Electronic Engineering,Zhejiang University,Hangzhou,310027,China

d College of Oceanic and Atmospheric Sciences,Ocean University of China,Qingdao,266100,China

Keywords: Underwater waveguide Adaptive subspace detection Dimension reduction Processing gain

ABSTRACT In the underwater waveguide,the conventional adaptive subspace detector(ASD),derived by using the generalized likelihood ratio test(GLRT)theory,suffers from a signi ficant degradation in detection performance when the samplings of training data are de ficient.This paper proposes a dimension-reduced approach to alleviate this problem.The dimension reduction includes two steps:firstly,the full array is divided into several subarrays;secondly,the test data and the training data at each subarray are transformed into the modal domain from the hydrophone domain.Then the modal-domain test data and training data at each subarray are processed to formulate the subarray statistic by using the GLRT theory.The final test statistic of the dimension-reduced ASD(DR-ASD)is obtained by summing all the subarray statistics.After the dimension reduction,the unknown parameters can be estimated more accurately so the DR-ASD achieves a better detection performance than the ASD.In order to achieve the optimal detection performance,the processing gain of the DR-ASD is deduced to choose a proper number of subarrays.Simulation experiments verify the improved detection performance of the DR-ASD compared with the ASD.

1.Introduction

An important task for sonar is to detect the presence or absence of an underwater acoustic source against noise background utilizing an array of hydrophones.According to adiabatic normal mode theory[1],the signal radiated from the source impinging the array after a propagation through an underwater waveguide lies in the modal subspace spanned by the sampled modal information of the array.Thus,the problem of detecting an underwater source of unknown location and level can be conveniently modeled as detecting a subspace signal against noise background[2-6].In most realistic situations,the noise covariance matrix is unknown,and estimated from training data to produce adaptive subspace detection.In this paper,we investigate the issue of adaptive subspace detection utilizing a vertical linear array(VLA)in the underwater waveguide.

The adaptive procedure was firstly applied to radar detection and proposed in the dimension-1[7-9],and later generalized to multidimensional subspace[10-13],and these corresponding detectors were well summarized in Ref.[12].It is not hard to apply the adaptive subspace detection to an underwater[14]where the signal subspace is the modal subspace.More generally,the array may only cover a small portion of the waveguide and the modal information is undersampled[15,16].Then the signal substantially lies in an effective modal subspace with a dimension less than or equal to the number of hydrophones(i.e.the dimension of test data)even if the number of modes is much larger than that of hydrophones.Based on the effective modal subspace,the adaptive subspace detector(ASD)can be derived by using the generalized likelihood ratio test(GLRT)theory.

In the adaptive procedure,the training data(also known as secondary data),that is,a sequence of signal-free noise snapshots,are sampled and used to estimate the noise covariance matrix,the estimation accuracy of which mainly depends on the number of training snapshots.To achieve a detection performance within 3 dB of the optimal adaptive processing,the number of homogeneous training snapshots is required to be more than double the dimension of the test[7].However,this requirement can be dif ficult to meet in practice,especially for an array with a large number of hydrophones or an inhomogeneous environment.The de ficiency of training snapshots can lead to a signi ficant degradation of the ASD in detection performance compared with the known noise covariance matrix case.

To solve the problem of de ficient training snapshots,many methods have been developed,such as data editing[17,18]and knowledge-aided(KA)[19,20].The data editing is used to remove the heterogeneity from the available snapshots before training.In KA approaches,some a priori knowledge about the radar’s operating environment is incorporated with the estimation of the noise covariance matrix and enhances the estimation accuracy.The KA processing used for underwater waveguide remains to be exploited.Some other improved versions of the ASDare presented as well by employing the method of orthogonal partition[21]or the Krylov subspace technique[22].Nevertheless,these above methods are mainly used for radar system but not underwater environment.

A promising solution to underwater environment is dimensionreduced processing[23,24]in which test data and training data are processed in a lower dimension.Then the noise covariance matrix is estimated in a lower dimension,which reduces the amount of unknown parameters in the noise covariance matrix and hence help improve the estimation accuracy of the noise covariance matrix.Subarray processing[25-27]is a straightforward way to reduce the dimension of the problem and has been studied by several literatures.In Ref.[25],the full array is divided into several nonoverlapping and contiguous subarrays of equal length.For the subarray data,Ref.[26]presented the coherent subarray detector and the noncoherent subarray detector,respectively.Another dimension-reduced approach is to map the test data into a lowerdimensional subspace through a transformation matrix prior to detection,and the transformation matrix is designed by avoiding the signal loss as little as possible[28].In Ref.[16],the authors propose a modal-domain adaptive subspace detector(MD-ASD)by mapping the test data into the effective modal subspace without loss of signal energy.The MD-ASD achieves a better detection performance compared with the conventional ASD.

In this paper,a two-step dimension reduction is developed to tackle the problem of de ficient training snapshots in adaptive subspace detection.In the proposed method,the first step is carried out by subarray division which is treated in the same fashion as Ref.[25];at the second step,the test data and the training data from each subarray are mapped into the effective modal space by an orthogonal matrix.For each subarray,the dimension-reduced test data and training data are treated by using the GLRT theory to formulate a subarray statistic.Finally,the dimension-reduced adaptive subspace detector(DR-ASD)test statistic is attained by summing all the subarray statistics.Through the step-two dimension reduction,the unknown parameters can be estimated more accurately in the GLRT procedure,and the computational burden is also remarkably reduced.

As a matter of fact,the work of this paper is the extension of our previous study in Ref.[16].Subarray processing is exploited to further improve the detection performance of the MD-ASD.Nevertheless,the subarray processing also brings about some loss of array gain compared with the full-array processing.This indicates that the number of subarrays has a crucial in fluence on the detection performance of the DR-ASD.Unlike other papers not studying the in fluence of the number of subarray,we provide a manner for the DR-ASD to choose the optimal number of subarrays based on several approximate assumptions.Employing the chosen number of subarrays,the DR-ASD is designed and able to achieve a signi ficant improvement of detection performance compared with the ASD or even the MD-ASD.

The remainders of this paper are arranged as follows.In section 2,the problem formulation involved with the signal and the noise model is introduced,based on which the conventional ASD is given.The dimension-reduced process is presented and the DR-ASD test statistic is derived in section 3.In section 4,we analyze the statistical distribution of the proposed DR-ASD and the processing gain of DR-ASD is derived to choose the optimal number of subarrays.Simulation experiments are performed to validate the improved detection performance of DR-ASD compared with ASD in section 5.Some conclusions are summarized in the final section.

2.Problem formulation

Assume that an acoustic source is located in an underwater waveguide(shallow water or deep ocean),and the source frequency is known but the source location is not.The signals radiated from the source are received by a VLA withNhydrophones.Then the received data can be expressed in frequency domain as

wherer,s,andnareN×1 column vectors denoting the received data,signal,and noise respectively,nobeys a complex gaussian distribution with mean zero and covariance matrixR(denoted byn～CN（0，R)).The signal-to-noise ratio(SNR)of the received data is defined as[3]

where the superscript’H’means the conjugate transpose operation.

2.1.Signal model

According to the adiabatic normal mode theory,the acoustic signal propagates through the underwater waveguide in the form of a series of normal modes and can be expressed as the product of the sampled mode matrix and the mode amplitude

where the complex numberμdenotes the unknown amplitude and phase of the source,ΦtheN×Mmode matrix consisting of sampled modal information,atheM×1 mode amplitude vector which contains the location information of the source and hence is unknown.Inputting environmental parameters,the mode functions can be calculated by sound field programs(such as KRAKEN).

The sampled modal information contained in the mode matrix often suffers undersampling when the VLA only covers a small portion of the water column.Under this circumstance,the mode matrix is ill-conditioned and has very small singular values.By singular value decomposition(SVD),i.e.,Φ=UΛVHand omitting those very small singular values,we reconstruct the mode matrix as[5,16].

whereunandvnare column vectors of the unitary matricesUandV,respectively,λnis the corresponding singular value;the subscript e means effective,diag denotes the diagonalization operation,pdenotes the number of the effective singular values.When the number of modes is much larger than that of hydrophones or the array aperture is large enough[5],it may happen thatp=Nwhich means there are no very small singular values in the mode matrix.In this paper,the situation of 1

whereae=ΛeVHeais thep×1 column vector denoting the effective modal amplitude.From(5),it shows the signal lies in thepdimensional subspace〈Ue〉.

2.2.Noise model

The background noise is modeled as the sum of uncorrelated component and correlated component,which come from self noise of the sonar platform and ambient noise of the ocean environment,respectively.Assuming the self noise and the ambient noise are mutually independent,the noise covariance matrix consists of two components and can be written as

According to Ref.[30],the ambient noise,including windgenerated noise and ship noise,has its covariance matrix as

whereAdenotes the modal amplitude covariance matrix ofM×M.Its elementA（i，j）are calculated by integrating over all noise sources

whereS（r）denotes the noise source spectral amplitude at a fixed frequency,the superscript*the conjugation operation,andRthe radius of the noise source area.From(7),the ambient noise also lies in the modal space〈Φ〉for experiencing a propagation through the same waveguide as the source signal.

2.3.Adaptive subspace detector

The detection problem can be described with a binary hypothesis test

whereθ=μaeis called the location parameter because of containing the location information of the source.The null hypothesisH0represents the signal is absent while the alternative hypothesisH1represents the signal is present.In practice,the noise covariance matrix is estimated from the training data which are independent and identically distributed and obey the same statistical distribution as the noise.Assume that a sequence of training snapshots are sampled and expressed as

wherexl～CN（0，R）denotes thelth training snapshot,Lthe number of training snapshots,Sthe sample covariance matrix(SCM).

Applying the GLRT theory,the test statistic of the ASD is obtained as

The detection performance of the ASD relies on the snapshot numberL.When the training snapshots are de ficient,the ASD will suffer a signi ficant degradation of detection performance.In addition,there comes with huge real-time computational burden on the matrix inversion(S-1)when the dimension of the test data is high.

3.Dimension-reduced ASD

In this section,a dimension-reduced ASD is proposed to alleviate the snapshot-de ficient problem.In the proposed detector,the dimension of the test data is reduced before proceeding to the hypothesis test.The dimension reduction contains two steps which are operated separately in the hydrophone domain and the modal domain.Firstly,the array is divided into several continuous and nonoverlapping subarrays of equal length.Secondly,the test data of each subarray are transformed into the modal domain.Then applying the GLRT theory to each subarray test data,and the DRASD is finally obtained by summing all the subarray statistics.

Assume that the VLA is divided intoKsubarrays each of which hasNshydrophones,i.e.,N=K·Ns.For thekth subarray,the test data and training data are denoted by

whererkis theNs×1 test data vector,UktheNs×pk(pkis the subspace dimension)signal subspace matrix,θkthepk×1 location parameter vector,nk～CN（0，Rk）theNs×1 noise vector,xk，ltheNs×1 training snapshot vector.The SNR of thekth subarray is then

Utilizing the orthogonality among columns of the signal subspace matrix,we transform the test data and the training data into the modal domain denoted by

wherezk，mk～CN（0，Mk），yk，landQkare the modal-domain counterparts,andQkthe modal-domain covariance matrix and SCM,respectively.

After two-step dimension reduction,the hypothesis test can be written as

Then the joint PDFs of test data and training data（zk，Yk）can be expressed as

whereR0，kandR1，kare covariance matrix combining the test data with the training data given as,respectively,

It is found from(15),the detection procedure is processed in a much lower dimension.The dimension of the test data is firstly reduced by subarray division fromNtoNs,then secondly reduced by modal-domain transformation fromNstopk.

The MLEs ofMkunderH0andH1are solved as,respectively,

Then the MLE ofθkunderH1is equivalent to the optimization problem

The MLE ofθkis easily solved as

Combining（18）～（20）with（16）,it yields the GLRT statistic of thekth subarray

As shown in(24),it only needs one operation of the matrix inversion in the subarray statistic whereas it needs twice in the ASD statistic(seen in(11)).Also considering the dimension of the matrix inversion is decreased fromNtopk,thus the computational burden is signi ficantly reduced.Then by incoherently summing the subarray statistics,the test statistic of the DR-ASD is attained as

whereUDRandQare the block diagonal matrices as follows

where blkdiag（·）denotes the block diagonalization operation.The flow block chart of the DR-ASD statistic is given in Fig.1 to clarify the proposed two-step detection procedure.

From the above deduction,the DR-ASD procedure is virtually an incoherent subarray processing in modal domain.The incoherent subarray processing aims to diminish the unknowns in the noise covariance matrix to be estimated;for instance,the number of unknowns reduces fromN2to,which bene fits the estimation of the noise covariance matrix with de ficient training snapshots[26].The detection procedure in modal domain makes the estimate of the location parameter independent of the sample covariance,which improves the estimation accuracy of the location parameter[16].

On the other hand,what the subarray processing brings about is the change of the total dimensions of the signal subspaces,i.e.,fromptopDR=.Provided with an underwater waveguide environment,pandpDRcan be calculated beforehand.Through some calculation results(which is later shown in the simulation experiment),it turns out thatpDRis larger thanpand grows with the increase of the subarray numberK.The increase inpDRis to disadvantage of the DR-ASD procedure due to bringing about more unknowns to be estimated in the location parameters(i.e.,θ1，θ2，…).Therefore,subarray division may be not always necessary.

4.Analysis of detection performance

Fig.1.The flow block chart of the detection procedure based on two-step dimension reduction.

It has been veri fied that modal-domain transformation always bene fits the detection performance in Ref.[16]whereas subarray division does not.Thus it is critical to decide whether subarray division needs to be carried out or not.In this section,we investigate the statistical distribution of the DR-ASD.Then based on some approximate assumptions,the processing gain of the DR-ASD is derived according to the statistical property of the DR-ASD.Finally,the number of subarrays can be chosen by maximizing the processing gain.

4.1.Statistical distribution of the DR-ASD

Ref.[12]presents an approach to analyze how adaptive detectors are distributed by statistically identical decomposition.Through statistical decomposition,thekth subarray statistic can be represented by two independent random variables[16]hk，1andhk，2

hk，1andhk，2obey central and noncentral chi-square distributions as follows,respectively

where 2pkand 2（L-pk+1）are degree of freedoms(DoFs),βkis noncentral parameter

Scaled by a scalar,?kin(24)obeys anFdistribution underH0and a noncentralFdistribution underH1

From(27),the DR-ASD statistic is actually the sum ofK Fdistributed random variables.

4.2.Processing gain of the DR-ASD

The detection performance of a detector can be quanti fied by its output SNR.The output SNR can be calculated by[31]

where T denotes a test statistic,E［·］a expectation operation,D［·］denotes a variance operation.

To derive the output SNR of the DR-ASD,several approximate assumptions are made as follows:

[1.]

1.when the training data are sufficient,theseF-distributed variables?1，…，?Kare mutually independent;assume that?1，…，?Kare approximately independent in the finite training data case;

2.when the number of subarrays is not large,it is approximately assumed that the sum of subarray SNRs is approximately equivalent to the full-array SNR,that is,;

3.the depth of the array in the underwater waveguide has small in fluence on the dimension of the signal subspacepwhich mostly relies on the array aperture[4];that is,pkmostly relies on the number of subarrays which determines the subarray aperture.Considering the subarrays with an equal length,it can be approximately assumed to havep1=p2=…=pK=psub.

Based on these assumptions,we derive the output SNR of DRASD(seen in Appendix)which is written as

whereGdenotes the processing gain of DR-ASD.WhenK=1,it comes up to the case of the full array,where the dimension of test data is only reduced by the modal-domain transformation.From(29),the processing gain of the DR-ASD is determined by these variablespsub,LandK.For a given array con figuration,psubis determined byK.Therefore,the processing gain is determined by the number of subarrays for given training data.Then the number of subarrays can be chosen by maximizingGin(29)for a given number of training snapshots.

5.Simulation experiment

Some numerical results from simulation experiment are shown to demonstrate the improved detection performance of the DR-ASD compared with the ASD.The underwater waveguide for simulation experiment is set in a deep-ocean environment which is shown in Fig.2.The sound velocity pro file in water layer is generated with the Munk curve.The acoustic source is located at depth of 200 m and range of 50 km from the array with a center frequency of 400 Hz.The array is con figured in the deep ocean with the first hydrophone depth of 200 m and hydrophone spacing of 1 m.The sound field and mode functions are calculated by KRAKEN.PFAis set as 0.01,andPDof the detector is calculated via the Monte-Carlo method.

Fig.2.Deep-ocean environment and corresponding parameters.

5.1.Signal subspace

Some features of the signal subspace are firstly shown here.The number of hydrophones is set asN=60.The modal information is sampled by the array and contained in the mode matrix fromwhich the signal subspace is constructed.Through SVD of the mode matrix,the normalized singular values of the mode matrix are obtained and shown in Fig.3.It shows in Fig.3 that a fraction of singular values are very small and approaching zero.The critical value is set as 0.01,and then the signal subspace is obtained to have the dimension ofp=24.When the full array is divided into subarrays,the signal subspace of each subarray can be obtained likewise.

Fig.3.The normalized singular values of the mode matrix when using the full array.

In the processing gain,the sum of the signal subspace dimensions of subarrays,i.e.,pDR=∑k=Kk=1pk,is an important factor.It is related with the number of subarrays and shown in Fig.4 varying with various numbers of subarrays.From Fig.4,the sum of the signal subspace dimensions increases when the number of subarrays grows,which is to the disadvantage of the processing gain.Therefore,the number of subarrays need to be chosen properly.

5.2.Approximate assumptions

Three approximate assumptions are made to derive the processing gain of the DR-ASD.Here we demonstrate that these assumptions are reasonable by some numerical results.

5.2.1.The independence of the subarray statistics

To verify the approximate independence among the subarray statistics,that is,theF-distributed variables?1，?2，…，?K,we calculate the correlation coef ficient of(?i，?j).The correlation coef ficient

where Cov［·，·］denotes the covariance operation.Each correlation coef ficient is computed by monte carlo experiment of 10,000 trails.The full array withNhydrophones is divided intoKsubarrays,in whichK=3，4，5，6.The correlation coef ficient of?1and?kis shown in Fig.5.It indicates from Fig.5 that the correlation coef ficients are all smaller than 0.1 or even less than 0.01.Considering the weak correlation,therefore these subarray statistics can be approximated to be mutually independent.

Fig.4.The sum of signal subspace dimensions of subarrays varying with various numbers of subarrays.

Fig.5.The correlation coef ficient of?1 and?j,the number of training snapshots L=40,the number of hydrophones N=60.

5.2.2.Subarray SNRs

Fig.6.The sum of subarray SNRs when K=1，2，3，4，5，6,the number of hydrophones N=60.

The sum of subarray SNRs,i.e.,is calculated and depicted in Fig.6 when the full array is divided into various numbers of subarrays,i.e.,K=1，2，3，4，5，6.Specially,K=1 means the SNR of the full array.It is found that the sum of subarray SNRs is approximately equivalent to the SNR of the full array,i.e.,=snr.

5.2.3.Subarray signal subspace

Fig.7.The signal subspace dimension when K=2，3，4，5，6,the number of hydrophones N=60.

In Fig.7,the signal subspace dimension of a subarray,i.e.,pk,is calculated and depicted when the full array is divided into various numbers of subarrays,i.e.,K=2，3，4，5，6.It is veri fied thatp1=p2=…=pK.

5.3.Detection performance of the DR-ASD

We exhibit the detection performance improvement of the DRASD compared with the ASD through curves of detection probability versus SNR.The detection probabilities are calculated using the Monte Carlo experiment.For a givensnr,10,000 trials are conducted to obtain a.The SNR is presented in the form of its logarithm,namely 10lgsnr.

The number of subarrays determines the DR-ASD statistic and its detection performance,and the optimal number of subarrays is dependent on the number of training snapshots.For various numbers of training snapshots,the curves of the processing gain of the DR-ASD versus the number of subarrays are exhibited in Fig.8,in which’DR-ASD,K=1’means the full array case and is equivalent to the MD-ASD in Refs.[16].From Fig.8,we can see that the processing gain degrades when the number of training snapshots decreases and varies with the number of subarrays.Also it shows that the optimal number of subarrays changes for different numbers of training snapshots.When the number of training snapshots is small,likeL=40 or 60,the subarray processing(K)is the optimal choice.When the number of training snapshots increases,likeL=100,the full array processing becomes the optimal choice.

To verify the results of the processing gain above,the curves of detection probability versus SNR are depicted in Fig.9～11.In Figs.9 and 10,the ASD curve is also plotted as a contrast.It shows that the DR-ASD always outperforms the ASD signi ficantly in detection performance.When the number of training snapshotsL=100,the DR-ASD reaches its best detection performance atK=1;when the number of training snapshotsL=60,the DR-ASD reaches its best detection performance atK=3 and it is also noted that the ASD does not work any longer.In Fig.11,the DR-ASDs with various numbers of subarrays are compared.The choice ofK=4 is optimal and much better than that ofK=1.Furthermore,these results in Fig.9～11 are all consistent with that in Fig.8.Therefore,it demonstrates that our derived processing gain of the DR-ASD is effective for the choice of the optimal number of subarrays.

Fig.8.The processing gain of the DR-ASD versus various numbers of subarrays when the number of training snapshots L=100，60，40,respectively,the number of hydrophones N=60.

Fig.9.The detection probabilities of the DR-ASD and the ASD versus SNR when the number of subarrays K=1，5,respectively,the number of training snapshots L=100,the number of hydrophones N=60.

Fig.10.The detection probabilities of the DR-ASD and the ASD versus SNR when the number of subarrays K=1，3,respectively,the number of training snapshots L=60,the number of hydrophones N=60.

Fig.11.The detection probability of the DR-ASD versus SNR when the number of subarrays K=4，1，6,respectively,the number of training snapshots L= 40,the number of hydrophones N=60.

6.Conclusions

In this paper,an improved adaptive detection procedure based on a two-step dimension reduction is proposed to alleviate the de ficiency of training data.The proposed DR-ASD has less computational burden and achieves a much better detection performance than the ASD or even the MD-ASD with de ficient training snapshots.The detection performance of the DR-ASD is determined by the number of subarrays and is quanti fied by the processing gain of the DR-ASD.The derived processing gain are simply related with the number of subarrays and the number of training snapshots,and simulation results shows it coincides with the detection probability of the DR-ASD.The optimal detection performance of the DR-ASD can be achieved by maximizing the processing gain.When the number of training snapshots is not quite small,the DR-ASD does not need subarray division;otherwise,subarray division is better choice for the DR-ASD.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.

Acknowledgment

The authors would like to acknowledge the National Natural Science Foundation of China(Grant No.11534009,11974285)to provide fund for conducting this research,and also acknowledge the anonymous reviewers to review this paper.

Appendix

For aF-distributed random variableT0,that is,

wherem,ndenote the degree of freedom andλdenotes the noncentral parameter,its expectation and variance are then

Combining(27)and(32)and,it yields

Then for the DR-ASD statistic,utilizing the approximation relationp1≈p2…=psub,we have

Substituting（34）into(28)and utilizing the relationβk=2snrk,it yields the output SNR of DR-ASD

whereGis the processing gain of DR-ASD.