School of Computer Engineering, Nanyang Technological University, Singapore

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1 NARROWBAND DETECTION IN OCEAN WIT IMPULSIVE NOISE USING AN ACOUSTIC VECTOR SENSOR ARRAY V. N. ari 1, Aad G. V. 2, P. V. Nagesha 2 ad A. B. Premkumar 1 1 School of Computer Egieerig, Nayag Techological Uiversity, Sigapore Departmet of Electrical Commuicatio Egieerig, Idia Istitute of Sciece, Bagalore , Idia ABSTRACT This paper presets the formulatio ad aalysis of some methods for arrowbad detectio of uderwater acoustic sources i impulsive oise usig a array of acoustic vector sesors. Sice the array sigal vector is ukow due to the ukow locatio of the source, detectio is based o the geeralized likelihood ratio test which ivolves estimatio of the sigal vector. Differet detectors use differet sigal models which yield differet sigal estimators. It is show that the trucated subspace detector (T), which uses a trucated ormal mode model, yields the best performace. Idex Terms- subspace detectio, geeralized likelihood ratio test, shallow ocea, acoustic vector sesor 1. INTRODUCTION This paper addresses the problem of arrowbad detectio of a acoustic source i a shallow ocea usig a array of acoustic vector sesors (AVS). Detectio of sources usig a sesor array may be achieved usig a optimal Neyma- Pearso (NP) detectio strategy if the sigal field at the array is kow [1]. owever, the sigal is ukow i a geeral detectio sceario because the source locatio is ukow, ad the evirometal parameters may ot be kow fully. ece detectio of the sigal is doe by a geeralized likelihood ratio test (GLRT), which ivolves estimatio of the sigal vector ad oise parameters at the array. Oe form of this test leads to the formulatio of a subspace detectio algorithm [2]. Two detectors based o the GLRT approach, have bee preseted recetly for AVS arrays [, 4], viz. the subspace detector () ad the approximate sigal form detector (ASFD). These were formulated uder the assumptio that the ambiet oise is Gaussia. owever, ambiet oise i the ocea ofte has a heavy-tailed distributio [5]. Therefore it is of iterest to develop effective methods of detectio i such a eviromet. GLRT detectio has earlier bee employed i the case of spherically ivariat radom vector oise [6] ad geeralized Gaussia oise with iterferece [7]. It is well-kow that AVS arrays have a better directiofidig capability tha the covetioal APS arrays [8, 9]. This performace advatage arises because a AVS measures ot oly the acoustic pressure but also all compoets of particle velocity at a poit i space. Recet work has show that this advatage of a AVS array ca also be exploited to obtai better detectio performace tha that achieved by a APS array [, 4]. I this paper we exted the ad ASFD techiques to uderwater sigal detectio i heavy-tailed oise usig a AVS array. The oise distributio is modeled as geeralized Gaussia (GG). We also preset a ew techique called trucated subspace detector (T). Each of these detectors is based o a differet model of the array sigal vector. Differet models lead to differet estimates of the sigal vector. It is show that the ormalized mea square sigal estimatio error (MSE) of T is sigificatly lower ad hece the T provides a sigificatly better performace tha both ad ASFD. The paper is orgaized as follows. The data model is preseted i Sectio 2. The alterative sigal models ad correspodig detectors are formulated i Sectio ad expressios for the associated MSEs are derived. A comparative performace aalysis of the detectors is preseted i Sectio 4. Coclusios are preseted i Sectio DATA MODEL Cosider the detectio of a arrowbad source with ceter frequecy f located i a shallow ocea, usig a N elemet uiform horizotal liear AVS array. We shall cosider three outputs produced by each AVS, viz. the acoustic pressure ad two orthogoal horizotal compoets of particle velocity. The vertical compoet of particle velocity is ot cosidered sice it is foud to icrease complexity without yieldig ay sigificat additioal improvemet i performace. The sesor array is assumed to be at depth z a i the far-field regio with respect to the source, which is located at rage r, depth z s, ad azimuth agle ϕ measured with respect to the axis of the array. Let x, s ad w deote, respectively, the N x 1 data vector, sigal vector ad oise vector at the AVS array; ad let x, s ad w deote, respectively, the x1 data vector, sigal vector ad oise vector at the th AVS. Assumig the ocea to be rage-idepedet with water desity ρ ad soud speed profile c(z), we ca express s as s = [p 2 ρc(z a )v x 2 ρc(z a )v y ] T, (1) where p ad (v x, v y ) deote, respectively, the complex

2 amplitudes of acoustic pressure ad horizotal (x, y) compoets of particle velocity of the sigal at the th AVS. The vectors w ad x are defied similarly. The particle velocity compoets are scaled by the factor 2 ρc(z a ) to esure that all the elemets have the same dimesio. The sigal vector s ca be writte i terms of the ormal modes of the oceaic waveguide as [9]. s = A(ϕ)b, (2) A(ϕ) = [a 1 (ϕ) a M (ϕ)], () a m (ϕ) = c m (ϕ) d m (ϕ), (4) c m (ϕ) = [1 exp(ik m dcos(ϕ)) exp(i(n-1)k m dcos(ϕ))] T (5) d m (ϕ) = [1 2 ξ m cos(ϕ) 2 ξ m si(ϕ)] T, (6) ξ m = k m /k(z a ) = k m c(z a )/2πf; m = 1, M, (7) b = [b 1 b M ] T, (8) b B z exp r jk r k r ; m 1,... M, (9) m m s m m m where k m, δ m ad ψ m (z) are respectively the waveumber, atteuatio coefficiet ad eigefuctio of the m th mode, M is the umber of modes, d is the iter-sesor spacig, B is a costat, ad deotes the Kroecker product. The colums of A(ϕ) are the modal steerig vectors, ad the elemets of b are the amplitudes of the modes. The ambiet oise i the ocea is assumed to be spatially white ad heavy-tailed, modeled by the followig circular complex geeralized Gaussia (GG) PDF with variace σ 2 f GG N N J( ) K( ) w exp, 0< 2, 2 w (10) 1 /2 (4 / ) (4 / ) J( ), K( ). 2 2 [ (2 / )] 2 (2 / ). GLRT DETECTION IN SALLOW OCEAN.1 Formulatio of the detectio problem The detectio problem ca be cast i the form of the followig hypothesis testig problem: 0 : x = w 1 : x = s+ w. (11) The joit likelihood fuctios of the array data vectors uder hypotheses 0 ad 1 are give by f (x, 0 ) = f GG (x), (12) f (x, 1 ) = f GG (x - s), (1) where f GG (.) is defied i (10). The likelihood ratio L(x) = f(x; 1 )/f(x; 0 ) (14) yields the followig test statistic ad decisio rule N 1 x ( ) ( ) ( ), (15) TUD x x s Decide 1 if T UD (x) > γ(p FA ) (16) where x() ad s() deote the th elemet of x ad s respectively, ad γ(p FA ) is the threshold correspodig to probability of false alarm P FA [1]. But (16) is a urealizable detector (UD) sice it requires the kowledge of the sigal vector s that depeds o the ukow source locatio. owever, the performace of UD may be cosidered as a upper boud o the performace of ay realizable detector. Whe some parameters of the sigal s or the oise w are ukow, they ca be estimated by maximizig the likelihood fuctios with respect to these ukow parameters. Let Θ j deote the ukow parameter vector uder hypothesis j, j=0,1. The ratio of the maximized likelihoods called the geeralized likelihood ratio, is give by G ; ˆ 1; 1 ; ˆ 0; 0 L x f x Θ f x Θ, (17) where Θˆ j is the maximum likelihood estimate (MLE) of Θ j. Simplificatio of the geeralized likelihood ratio yields the GLRT test statistic. I sectios.2 ad., two ovel detectors based o the GLRT approach are preseted..2. Trucated Subspace detector If N > M, where N is the umber of sesors i the array ad M is the umber of ormal modes, the colums of the modal steerig matrix A(ϕ) are liearly idepedet ad the N-dimesioal array sigal vector s belogs to the M- dimesioal modal subspace V M (ϕ) spaed by the colums of A(ϕ). A subspace detector () based o this property of s was proposed i [] for arrowbad detectio i Gaussia oise. A simpler formulatio of the is preseted here for the geeral case of o-gaussia oise. We ca represet s i the alterative form s = A(ϕ)b = Q(ϕ) β =[q 1 (ϕ).. q M (ϕ)] β, (18) where Q(ϕ) is a uitary matrix which may be obtaied by QR decompositio of A(ϕ) as A(ϕ) = Q(ϕ) R. (19) I (18), β = Rb is a trasformed versio of the ukow mode amplitude vector b. The sigal vector s belogs to the modal subspace V M (ϕ) defied as V M (ϕ) = spa{q 1 (ϕ) q M (ϕ)}, (20) so that {q 1 (ϕ) q M (ϕ)} costitutes a orthoormal basis of V M (ϕ). For a give ϕ, the subspace V M (ϕ) is kow if the modal waveumbers {k m ; m = 1,, M} are kow. Thus the problem of estimatig the N-dimesioal sigal vector s is reduced to the simpler problem of estimatig the M- dimesioal mode amplitude vector β. Maximizatio of the likelihood fuctio f (x; 1 ) = f(x;β,ϕ, 1 ) with respect to the ukow M-dimesioal vector β yields the equatio N 1 ˆ 2 ˆ x (21) x r ( ) ( ) [ ( ) r ( ) ( )] 0, where r (ϕ) is a row vector deotig the th row of Q(ϕ), ad β ˆ deotes the coditioal MLE of β for a give ϕ. The

3 followig closed form solutio of (21) is readily obtaied if α = 2: β ˆ = Q (ϕ)x. (22) As o closed form solutio of (21) is available whe α 2, we use (22) as a approximatio to the coditioal MLE of β. The approximate MLEs of ϕ ad s ca ow be writte as ˆ argmax{ x Q( ) Q ( ) x}, (2) ˆ sˆ Q( ) Q ( ˆ ) x, (24) ad the test statistic of the is thus give by N 1 T x x( ) x( ) r ( ) ( ) x. ˆ ˆ Q (25) The ormalized mea square error (MSE) of the sigal estimate s ca be foud from (18) ad (24) as ˆ E[( s sˆ) ( s sˆ)] M, N (26) [ ] where is the sigal-to-oise ratio (SNR) ad E is the expectatio operator. The ca be employed oly if the colums of A(ϕ) are liearly idepedet, i.e. if M N. Sice the umber of modes M icreases as the frequecy f is icreased [9], the applicability of the is limited by a upper cut-off frequecy f c which depeds o the legth of the array; a shorter array limits the applicability of the to a lower cut-off frequecy. Moreover, the suffers degradatio i performace as f is icreased eve if f < f c, because a icrease i M leads to a icrease i the MSE ε, as show by (26). I order to exted the applicability of the to shorter arrays/higher frequecies, as well as arrest the degradatio associated with a icrease i the umber of modes, we propose a detector which uses a trucated model of the sigal vector obtaied by projectig s o a modal subspace V M of smaller dimesio M ', where M ' < M: V M (ϕ) = spa{q 1 (ϕ) q M (ϕ)}. (27) The set of spaig vectors of the trucated subspace V M (ϕ) is a subset of the set of spaig vectors of the full modal subspace V M (ϕ). The trucated sigal vector is give by s Q' ( ) ' [ q ( )... q ( )] '. (28) 1 M ' O replacig s by s' i (1) ad the adoptig the same procedure as for the subspace detector, we get the followig expressios for the coditioal MLE of β '(ϕ), the MLEs of ϕ ad s', ad the test statistic of the trucated subspace detector ˆ '( ) ' β Q ( ) x, (29) ˆ T argmax{ x Q'( ) Q ' ( ) x }, (0) sˆ' Q'( ˆ ) Q ' ( ˆ ) x, (1) T T ' ˆ ˆ N T T T 1 T ( x) [ x( ) x( ) r ( ) Q ' ( ) x ]. (2) where r ( ˆ ) is the th row of Q '( ˆ ). It ca be show ' T T that the ormalized MSE of the estimate s ˆ ' is E[( s sˆ') ( s sˆ')] T T,1 T,2 E[ s' s'] M' 1,. N T,1 T,2, () (4) The ormalized MSE ε T has two compoets. The first compoet ε T,1 is due to trucatio of the ormal mode expasio of the sigal vector. The secod compoet ε T,2 is due to oise, ad it is aalogous to ε defied i (26). Plots of ε T versus are show i Figs. 1 (a), (b) ad (c) for SNR = 0, 10 ad 20 db respectively, whe a sigal of frequecy 50 z is received by a 10-elemet AVS LA. The evirometal parameters are as specified at the begiig of Sectio 4. For this chose set of parameters, M = 15. We ca draw the followig coclusios from (), (4) ad Fig. 1. The oise-iduced error compoet ε T,2 decreases liearly i proportio to as is reduced. The trucatio-iduced error compoet icreases very slowly as the trucatio is icreased (i.e. as is reduced) ad becomes sigificat oly whe is close to 1. This behavior ca be attributed to the fact that (i) the modal vectors {a 1 (ϕ) a M (ϕ)} i the expasio of s are highly correlated, ad (ii) amplitudes { } of the discarded higher order modes are quite small due to faster atteuatio of the higher order modes. The total MSE ε T is miimum at a optimal value of which is very close to 1. The optimal value of icreases very slowly with icreasig SNR. It follows that the performace of T may be expected to be sigificatly better tha that of ad that the choice =1 is optimal or ear-optimal. This predictio is cofirmed by the result preseted i Sectio 4. The use of a trucated sigal model i T also has the additioal advatages of (i) reducig the eed for chael iformatio to modal waveumbers of the first few modes oly, ad (ii) reducig the computatioal complexity... Approximate sigal form detector A simple detector called the approximate sigal form detector (ASFD) may be formulated [4] by exploitig the fact that the modal waveumbers {k m ; m = 1,, M} are very close to oe aother. Usig the approximatio ξ m = k m /k(z a ) 1 i (7), we get the followig approximate expressio for the array sigal vector s"= G(ϕ)p, G(ϕ) = I N [1 2 cos(ϕ) 2 si(ϕ)] T, (5) where p = [p 1 p N ] T, ad I N deotes the NxN idetity matrix. O replacig s by s" i (1) ad maximizig the resultat likelihood fuctio f(x; 1 ) = f(x; p, ϕ, 1 ) with respect to p leads to the equatio

4 Fig. 1: ε T versus M' for ϕ = o f = 50 z, M = 15 ad (a) SNR = 0 db, (b) SNR = 10 db, ad (c) SNR = 20 db. 2 N ˆ x ˆ 1 g ( ) p [ g ( ) g ( ) p g ( ) x( )] 0, (6) where g (ϕ) is a row vector deotig the th row of G(ϕ), ad ˆp is the ML estimate of p. If α = 2, the solutio of (6) is give by 1 pˆ ( G ( ) G( )) G ( ) x G ( ) x / (7) No closed form solutio of (6) is available whe α 2, but (7) may still be used as a approximatio to the coditioal MLE. The correspodig MLEs of bearig ϕ ad sigal vector s '', ad the test statistic of the ASFD are give by ˆ ASFD { x G( ) G ( ) x}, (8) ˆ sˆ G( ) G ( ˆ ) x /. (9) ASFD N 1 ASFD TAS D ( x) [ x( ) x( ) s ˆ( ) ]. (40) F The ormalized MSE of the sigal estimate ˆ s is give by E[( s sˆ'') ( s sˆ'')] ASFD ASFD,1 ASFD,2 E[ s'' s''] 1 1,. ASFD,1 ASFD,2, (41) (42) We ote that ε ASFD also has two compoets. The first compoet ε ASFD,1, which is due to icorrect modelig of the sigal vector, is very small sice the error i the approximatio ξ m 1 is very small. The secod compoet ε ASFD,2, which is due to oise, is aalogous to ε ad ε T,2 defied i (26) ad (4) respectively. The oise-iduced error compoet ε ASFD,2 is lower tha ε whe M > N. ece, for a give N, the ASFD is expected to perform better tha the at higher frequecies. The T is expected to perform better tha ASFD at all frequecies sice ε T< ε ASFD. These predictios are cofirmed by the simulatio results show i Sectio 4. owever, the ASFD has the advatage of ot requirig ay prior iformatio o the modal waveumbers of the chael. 4. PERFORMANCE ANALYSIS We have evaluated the performace of the detectors through Fig. 2: P D vs. P FA at SNR = -10 db. GG (α = 0.5) oise. (a) f = 50 z, (b) f = 50 z. Fig. : (a) Normalized sigal estimatio MSE vs. frequecy, ad (b) P D (at P FA = 0.01) vs. frequecy for GG (α = 0.5) oise with -5 db SNR. simulatios usig a 10-sesor horizotal AVS array with half-wavelegth spacig i a Pekeris chael [9] with the followig parameters: ocea depth h = 70 m, soud speed i water c = 1500 m/s, bottom soud speed c b = 1700 m/s, bottom atteuatio δ = 0.5 db/wavelegth, desity ratio ρ b /ρ = 1.5. The array depth is z a = 40 m, the source is at a rage r = 5 km, depth z s = 40 m, azimuth ϕ = o. The SNR = -10 db i Fig. 2, ad -5 db i Figs. ad 4. I Figs. 2 ad, the evirometal oise is GG (α = 0.5) distributed. The probability of false alarm is fixed at P FA = 0.01 i Figs. ad 4. T results are show for M ' = 1, which is the value that maximizes the probability of detectio P D of the T for the curret simulatio parameters. The plots i Fig. 2 show the receiver operatig characteristics (variatio of P D with P FA ) at a array SNR = - 10 db, for the UD (solid lie), (dotted lie), T (dashed lie) ad ASFD (dot-dashed lie) whe the sigal frequecy is (a) 50 z (correspodig to M = 2 modes) ad (b) 50 z (M = 15). It ca be see from Fig. 2 that amog the realizable detectors cosidered, the T provides the best performace. The ASFD provides a better performace tha the at the higher frequecy, eve though the former uses o iformatio about the chael. I Fig. we study, i greater detail, the variatio of performace of the detectors with frequecy ad the relatio betwee detector performace ad sigal estimatio error.

5 Fig. 4: P D vs. parameter α of GG oise at P FA = 0.01 for f = 50 z at SNR = -5 db. Fig. (a) shows the variatio of the sigal estimatio MSE of the ASFD, ad T methods with frequecy, ad Fig (b) shows the correspodig variatio of the P D of these detectors. It is see that there is a very good egative correlatio betwee MSE ad P D across all detectors ad at all frequecies. The errors ε T ad ε ASFD are almost idepedet of frequecy, ad ε T < ε ASFD. Accordigly, the detectio performace of T is uiformly better tha that of ASFD. The error ε keeps icreasig with frequecy due to the icrease i the umber of modes M, ad cosequetly the performace of keeps degradig as frequecy is icreased. At very low frequecies the performace of is oly margially iferior to that of T ad better tha that of ASFD. At higher frequecies, ASFD performs better tha. I Fig. 4 we study the variatio of P D of the detectors with the parameter α of the evirometal oise. We cosider two cases whe the parameters σ 2 ad α are kow (solid lies) ad ukow (dashed lies). Whe the oise parameters are ukow, detectio is doe by obtaiig ML estimates of the ukow oise parameters. Figure 4 idicates that the performace of all detectors keeps improvig as α is reduced (i.e. as impulsiveess of evirometal oise icreases). This is so because, for a give SNR, most of the eergy of a impulsive oise resides i the outlier values of oise. The detectors are able to effectively reject these impulsive compoets ad thus achieve better performace i more impulsive oise. The differece betwee the dashed ad solid lies represets the degradatio i detector performace due to lack of kowledge of the oise parameters. This degradatio is lower i the case of the T as compared to the ASFD ad the, showig the greater robustess of T. If the oise PDF is estimated i advace usig secodary data [4], this degradatio ca be reduced. Overall, the T is show to be the most effective detectio scheme over the whole rage of heavy-tailed GG oise PDFs. 5. CONCLUSION This paper proposes two ew AVS array-based detectio schemes for sources i shallow ocea eviromets cotamiated by impulsive oise, viz. the trucated subspace detector (T) ad the approximate sigal form detector (ASFD). The detectors ivolve estimatio of the array sigal vector which is ukow due to the ukow locatio of the source. The detectio performace is show to be related to the ormalized mea square sigal estimatio error (MSE) of the associated GLRT. Thus the MSE provides a simple idicator to gauge the effectiveess of a detector without usig Mote Carlo simulatios. The subspace detector () degrades i performace with icreasig frequecy, due to a associated icrease i the MSE. The T ad ASFD methods are formulated to obtai improved detectio performace by reducig the sigal estimatio MSE. The T is show to be the most effective ad robust detectio scheme for a wide rage of oise PDFs, if the modal waveumbers of the ocea chael are kow. If this iformatio is ukow, however, detectio may be doe usig the ASFD. As the oise becomes more impulsive, the performace of the detectors improves because they are able to discard more effectively the outlier values arisig from oise. 6. ACKNOWLEDGMENT: This work was partly supported by a grat from Natioal Istitute of Ocea Techology, Cheai, Idia, uder the Ocea Acoustics Programme. REFERENCES [1] S. M. Kay, Fudametals of Statistical Sigal Processig, Vol.II: Detectio Theory, Pretice-all, Upper Saddle River, New Jersey, [2] S. Kraut, L. L. Scharf, L. T. McWhorter, Adaptive subspace detectors, IEEE Trasactios o Sigal Processig, vol. 49, o. 1, pp. 1-16, [] K. M. Krisha, G. V. Aad, "Narrowbad detectio of acoustic source i shallow ocea usig vector sesor array," Proc. OCEANS 2009, MTS/IEEE Biloxi, pp.1-8, Oct [4] V. N. ari, G. V. Aad, A. B. Premkumar, A. S. Madhukumar, "Uderwater Sigal Detectio i Partially kow Ocea usig Short Acoustic Vector Sesor Array," Proc. OCEANS 2011, Satader, pp. 1-9, Jue [5] F.W.Machell, C.S.Perod, G.E.Ellis, "Statistical characteristics of ocea acoustic oise process", i: E. J. Wegma, S. C. Schwartz, J. B. Thomas (Eds.), Topics i No-Gaussia Sigal Processig, Spriger, Berli pp , [6] N. B. Pulsoe, R. S. Raghava, Aalysis of a Adaptive CFAR Detector i No-Gaussia Iterferece, IEEE Trasactios o Aerospace ad Electroic Systems, vol. 5, o., pp , [7] M. N. Desai ad R. S. Magoubi, Robust Gaussia ad o- Gaussia matched subspace detectio, IEEE Trasactios o Sigal Processig, vol. 51, o. 12, pp , 200 [8] M. awkes ad A. Nehorai, Acoustic vector-sesor beamformig ad Capo directio estimatio, IEEE Trasactios o Sigal Processig, vol. 46, o. 9, pp , [9] K. G. Nagaada, G. V. Aad, Subspace itersectio method of high resolutio bearig estimatio i shallow ocea usig acoustic vector sesors, Sigal Processig, vol. 90, o. 1, pp , 2010.