Tacking/Fusion and Deghosting with Dopple Fequency fom Two Passive Acoustic Sensos Rong Yang, Gee Wah Ng DSO National Laboatoies 2 Science Pak Dive Singapoe 11823 Emails: yong@dso.og.sg, ngeewah@dso.og.sg Yaakov Ba-Shalom Depatment of ECE Univesity of Connecticut Stos, CT 6269, USA Email: ybs@eng.uconn.edu Abstact It is known that using two passive sensos that povide beaings-only infomation cannot eliminate false intesections in a multi-taget envionment, and esults in ghost tacks. In this pape, we popose a tacking/fusion and deghosting method using two stationay acoustic sensos. The existing appoaches fo beaings-only infomation equie the numbe of sensos to be geate than o equal to thee. The method poposed in this pape makes taget tacking possible in a two-senso system though making use of fequency infomation. This method geneates tentative tacks fom all possible beaing tack combinations, and distinguishes taget tacks fom ghost tacks using a 2- D assignment algoithm. Tentative tacks ae initiated using an iteated Least Squaes (LS) algoithm, and updated by an Unscented Kalman Filte (UKF) based on Dopple-Beaing Tacking (DBT) appoach. The likelihood atios of tentative tacks, which ae computed based on tack innovation pobability density functions, ae assigned as costs in the 2-D assignment algoithm. Simulation tests have been conducted to validate the effectiveness of the poposed method. Keywods Deghosting, passive tacking, Dopplie-beaing tacking I. INTRODUCTION Passive tacking is a challenging poblem, as taget ange cannot be obtained fom senso measuements diectly. Taget position has to be computed fom some special methods. One appoach is the so called Taget Motion Analysis (TMA), which equies a senso deployed on a maneuveing platfom. The taget tajectoy is then estimated using a Beaings-Only Tacking (BOT) algoithm [8] [1]. The BOT was futhe developed to the Dopple-Beaing Tacking (DBT) algoithm in naowband sona applications [9] [7] [4]. The DBT algoithm tacks taget position and emitted fequency fom beaings and Dopple shifted fequencies, and a taget can be localized even when the platfom is not maneuveing. Anothe appoach to obtain taget positions is though tiangulation of synchonous beaings detected by multiple sensos [1] [5] [12] [11] [6]. The tiangulation technique, which geneates composite measuements (as pat of configuation III fusion) [3], needs moe than one passive senso woking collaboatively. The pesent pape focuses on the tiangulation technique. The main challenge of the tiangulation appoach is deghosting. In a multi-taget scenaio, tiangulation points fom beaing lines consist of taget points and ghost points, and the taget points need to be distinguished fom ghost points though a deghosting algoithm. One conventional deghosting method is though compaing hinge angles [11] [6]. A hinge angle is the angle between the efeence plane and the taget plane. The efeence plane is fomed by the locations of two sensos and a pedefined efeence point, and the taget plane is constucted by two sensos and an angula vecto (beaing and elevation) detected by a senso. Since hinge angle computation needs measued elevation, this method cannot wok on beaingonly measuements. A moe obust appoach is to conside the poblem as an S-D assignment poblem [1] [5]. The appoach matches the beaing measuements fom all the sensos, and computes the cost of each combination by the logaithm of likelihood atio. Howeve, this appoach equies the numbe of sensos, S, to be geate than o equal to thee. This is because ghosts and tagets cannot be distinguished fom two-senso tiangulation. The constaint makes S-D assignment poblem (S 3) to be an NP-had poblem. A Lagangian elaxation was suggested to solve the poblem by a seies of 2-D assignment poblems. To avoid the NP-had poblem, featueaided deghosting method has been consideed. A deghosting method using beaings and acoustic signals was poposed in [12]. This method can wok in a two-senso system. A simple had condition was applied to likelihoods of acoustic signals fo deghosting. In this pape, we develop a featued-aided deghosting appoach. Fequency infomation is utilized in tacking/fusion and deghosting in a two-stationay-senso system. The oveall tacking system achitectue is shown in Fig. 1. It consists of two beaing tackes, a tentative taget tacke, a deghosting component and a taget tack management component. A bief desciption fo each component is given in the following: The beaing tackes tack beaings and Dopple shifted fequencies. A Kalman filte is used to obtain Beaing Tacks (BT) consisting of estimated beaings, Dopple shifted fequencies, and thei measued values. To avoid common measuements in diffeent BTs, a 2-D assignment algoithm is applied in the tack to measuement association. The tentative taget tacke fuses the BT infomation to geneate and maintain tentative taget tacks based on all possible BT combinations. The tacking poblem is fomulated based on the DBT appoach. Tacks ae initiated by an iteated LS algoithm, and ae updated
x, y Taget postion in x and y coodinates; ẋ, ẏ Taget speed in x and y coodinates; x s i, ys i Position of senso i, and i {1, 2}; x i = x xs i Taget position elative to senso i in x coodinate; yi = y ys i Taget position elative to senso i in y coodinate; i Taget ange to senso i; ṙ i Taget adial velocity to senso i; f e Taget emitted fequency; T Time inteval; s Sound speed in the ai (343m/s). Fig. 1. Two-senso tacking system achitectue by an UKF. The deghosting component selects taget tacks fom all tentative tacks. The poblem is fomulated as a 2- D assignment poblem based on the assumption that a BT can only contibute to one taget. The cost of each tentative tack is the negative log Likelihood Ratio (LR) of the tack. The taget tack management function ceates and maintains taget tacks fom the selected/deghosted tentative tacks. Note that tacking/estimation is not pefomed in this component, since it has been done in the tentative taget tacke. The taget tack state is updated fom the tentative tack diectly. The management logic used hee includes: a taget tack is ceated if a new tentative tack has been selected in the cuent and pevious time cycles continuously in a sliding window; a taget tack is updated by its tentative tack if the tentative tack is selected at the cuent time; a taget tack is pedicted if its tentative tack is not selected o deleted; a taget tack is deleted if its tentative tack is not updated fo a cetain peiod of time. In this pape, we focus on the tentative taget tacke and the deghosting. Fo simplicity, we tack in 2-D and assume that the popagation delays ae the same fo all the sensos. The notations used in the pape ae listed in the following: The stuctue of the est of the pape is as follows. Section II descibes the tentative taget tacke in detail. It includes an iteated LS tack initiation and a DBT algoithm. Section III fomulates the deghosting poblem as a 2-D assignment poblem, and descibes the costs of tentative tacks. Simulation esults and conclusions ae in Section IV and Section V, espectively. II. TENTATIVE TARGET TRACKING Tentative taget tacking is pefomed afte beaing tacking in each time cycle. Fist of all, BT pais ae fomed fom all the combinations of BTs. A BT pai consists of two BTs, which ae fom two diffeent sensos. A validation pocess is caied out to filte out those pais which cannot tiangulate o whose tiangulation points ae out of detection ange. Tentative taget tacks ae then updated by the validated BT pais. The following subsections descibe the state estimation and the tack initiation of tentative tacks. A. Tentative taget tack state estimation The tentative taget tack estimation is fomulated based on the DBT appoach. The state vecto is defined as x(k) = [ x(k) ẋ(k) y(k) ẏ(k) f e (k) ] T (1) whee k is the time index. The measuement vecto is z(k) = [ b m i 1 (k) f m i 1 (k) b m i 2 (k) f m i 2 (k) ] T (2) whee b m i 1 (k) and fi m 1 (k) ae measued beaing and Dopple fequency in BT i 1 fom senso 1, b m i 2 (k) and fi m 2 (k) ae measued beaing and Dopple fequency in BT i 2 fom senso 2, and BTs i 1 and i 2 belong to a valid BT pai. Note that the measued beaings and fequencies (not thei estimated values) attached to BTs ae used to pefom state estimation. This is because the estimated beaings and fequencies have eos that ae coelated ove time, and using them as measuements conflicts with the assumption of Kalman-based filte on measuements. The state tansition is modified fom the White Noise Acceleation (WNA) model [2] to include the emitted fequency x(k) = F x(k 1) + Γv(k 1) (3) whee v is white Gaussian pocess noise with covaiance Q, and 1 T 1 F = 1 T (4) 1 1
Γ = The measuement model is 1 2 T 2 T 1 2 T 2 T 1 (5) z(k) = h[x(k)] + w(k) (6) whee w is white Gaussian measuement noise with covaiance R = diag( σ 2 b m σ 2 f m σ 2 b m σ 2 f m ) (7) σ bm and σ fm ae the standad deviation of measuement eos on beaing and fequency espectively, and h( ) is given by ( ) x h 1 = actan 1 (k) y1 (k) (8) ( h 2 = 1 ṙ1(k) ) f e (k) (9) s ( ) x h 3 = actan 2 (k) y2 (k) (1) ( h 4 = 1 ṙ2(k) ) f e (k) (11) s with 1 (k) = ẋ(k)x 1(k) + ẏ(k)y1(k) 1 (k) 2 (k) = ẋ(k)x 2(k) + ẏ(k)y2(k) 2 (k) (12) (13) Note that the measuement model mentioned above is fo when both sensos have measuements. If only one senso has detection, the measuement model should be adjusted to o with h( ) = [ h 1 h 2 ] T (14) h( ) = [ h 3 h 4 ] T (15) R = diag( σ 2 b m σ 2 f m ) (16) Since the measuement model is nonlinea, the unscented Kalman filte is selected to pefom state estimation. B. Tentative tack initiation The ecusive state estimation descibed in section II-A equies an initial state and its eo covaiance. The iteated LS algoithm is poposed to compute them adaptively based on a batch of the fist n measuements fom two sensos. Let Z n = [ z(1) T z(n) T ] T (17) be the fist n measuements fom time 1 to n, and x(n) = [ x(n) ẋ(n) y(n) ẏ(n) f e (n) ] T (18) be the state at time n, the iteated LS algoithm is based on Z n = h n [x(n)] + w n (19) whee w n is the Gaussian measuement noise on Z n with covaiance R n, and the function h n ( ) can be obtained by the following two steps: whee Step 1: x[k, x(n)] computes the state at time k fom x(n), whee k {1,, n}, is the time index. It is based on the assumptions that the taget is moving at a constant velocity (ẋ, ẏ) with fixed emitted fequency f e, and is given by x(k) = x(n) (n k)ẋ x s i (2) ẋ(k) = ẋ(n) = ẋ (21) y(k) = y(n) (n k)ẏ yi s (22) ẏ(k) = ẏ(n) = ẏ (23) f e (k) = f e (n) = f e (24) Step 2: z[k, x(k)] computes z at time k fom x(k). It can be obtained fom the measuement model descibed in (8) (11). Z n is then built fom z accoding to (17). The Jacobian matix of h n ( ) is defined as H n = h n( ) = = h (25) 3 y 1 ( 1 ) 2 (26) = (n k)t y 1 ( 1 ) 2 (27) = x 1 ( 1 ) 2 (28) = (n k)t x 1 ( 1 ) 2 (29) = (3) = f e y 1 s( 1 ) 3 (y 1ẋ x 1ẏ) (31) = f e x 1 s 1 + (n k)t f e y 1 s( 1 ) 3 (y 1ẋ x 1ẏ) (32)
= f e x 1 s( 1 ) 3 (y 1ẋ x 1ẏ) (33) = f e y 1 s 1 (n k)t f e x 1 s( 1 ) 3 (y1ẋ x 1ẏ) (34) = 1 x 1ẋ + y1ẏ s 1 (35) To simplify the expessions, the time index (k) is omitted fom x 1(k), y1(k) and 1 (k) in (26) (35). The expessions fo,,,,,,,, and ae simila to (26) (35) with eplacement of x 1, y 1 and 1 by x 2, y 2 and 2, espectively. The estimate ˆx(n) and its eo covaiance P(n) ae computed by the iteated LS algoithm descibed in [2], which is given by ˆx j+1 (n) = ˆx j (n) + P j (n)(hn) j T R 1 n [Z n h n (ˆx j (n))] (36) P j+1 (n) = [(H j+1 n ) T R 1 n H j+1 n ] 1 (37) whee j is the iteation index. III. DEGHOSTING This section descibes the deghosting algoithm. Fig. 2 shows a two-taget scenaio in a two-senso system. The tagets ae detected and two beaing tacks ae fomed in each senso. It esults in fou tentative taget tacks being geneated, but only two of them ae taget tacks. The aim of deghosting is to distinguish the taget tacks fom ghost tacks. Fig. 2. Deghosting fom two acoustic sensos The poblem can be fomulated as a 2-D assignment poblem, that is, to minimize a cost function with a set of constaints. The cost function [1] is defined as J = n 1 n 2 i 1 =1 i 2 =1 c i1 i 2 (38) and the constaints ae n 1 = 1 fo all i 2 = 1, 2,, n 2 (39) i 1 =1 n 2 = 1 fo all i 1 = 1, 2,, n 1 (4) i 2 =1 whee n 1 and n 2 ae the numbes of beaing tacks at senso 1 and senso 2 espectively, c i1i 2 is the cost of the tentative tack fomed by BT i 1 fom senso 1 and BT i 2 fom senso 2, and is binay vaiable defined as { 1 if the tentative tack is a taget tack = (41) if the tentative tack is a ghost tack If i 1 i 2 is an invalid pai, infinity is assigned to c i1 i 2. The 2-D assignment poblem can be solved in quasipolynomial time using some existing algoithms, such as Auction o Joncke-Volgenant-Castanon (JVC) algoithm. The main focus hee is to compute the tack cost c i1 i 2, which is the cumulative negative log likelihood atio given by c i1 i 2 (k) = c i1 i 2 (k 1) ln(λ) (42) whee Λ is the likelihood atio given by [3] ( ) 2 PD λ e N (ν;, S) two-detection update Λ = (1 P D )P D λ e N (ν 1 ;, S 1 ) one-detection update (1 P D ) 2 no detection (43) whee P D is the pobability of detection, λ e is the spatial density of the extaneous measuements, ν and S ae the innovation and its covaiance with two-detection update espectively (namely, both sensos have measuements), and ν 1 and S 1 ae the innovation and its covaiance with one-detection update espectively (namely, only one senso has detection). Once the costs of all tentative tacks ae computed, the taget tacks ae selected though solving the 2-D assignment poblem. The selected tacks ae then used to update taget tacks o to fom new taget tacks. IV. SIMULATIONS Two scenaios shown in Fig. 3 and Fig. 4 ae used to veify the poposed tacking/fusion and deghosting algoithms. The scenaios simulate helicoptes being detected by a twoacoustic-senso system. The helicoptes emit constant fequencies of 2Hz fom the main otos. The emitted fequency is used in simulation only, and it is unknown to the detection and tacking system. Thee ae thee tagets in each scenaio. Scenaio 1 consists of a stationay taget and two tagets moving with diffeent velocities. The thee tagets in scenaio 2 have the same velocity. Scenaio 2 is a difficult scenaio, as the taget and ghost tacks have simila motion and Dopple shift, and it is difficult to distinguish them. In the simulation tests, the sampling inteval of the two synchonized sensos is T = 1s. The fequency eo is Gaussian with standad deviation σ f m =.1Hz. The Gaussian beaing eos ae set to thee levels, namely σ b m {.1 o,.3 o,.5 o }. The pocess noise covaiance (3 3) is Q = diag( 1 1.1 ) (44)
TABLE II. DEGHOSTING AND TRACKING PERFORMANCE IN SCENARIO 2 gound taget tack false tack σ b m P D tuth num. du. du. tk. num. du. du. ( o ) (s) (s) (%) bk. (s) (%) 1. 188 3.8 174.9 93..8 2.5 23. 12.3.1.9 188 4.1 173.6 92.4 1.1 3. 28.9 15.4.8 188 4.2 172. 91.5 1.2 3. 31.1 16.5 1. 188 4.8 167.9 89.3 1.8 4.3 47.8 25.4.3.9 188 4.9 167.3 89. 1.9 4.1 5.4 26.8.8 188 5.2 164.9 87.7 2.2 4.3 55.9 29.7 1. 188 6.4 158.1 84.1 3.4 5.4 72.6 38.6.5.9 188 6.2 155.9 82.9 3.2 5.5 74. 39.3.8 188 6.3 155.4 82.6 3.3 5.4 74.2 39.5 aveage 188 5.1 165.6 88.1 2.1 4.2 5.9 27.1 Fig. 3. Scenaio 1 taget/false tack du.(%): aveage atio of the total duation of taget (o false) tacks to the total duation of gound tuth pe un; taget tack bk.: aveage numbe of beaks in taget tacks pe un. The esults of scenaio 1 and scenaio 2 ae shown in Table I and Table II, espectively. We use the fist case in Table I, which is tested unde σ b m =.1 o and P D = 1., to futhe illustate the pefomance metics. In this case, we obtained 31 tacks ove the whole duation of the scenaio fom 1 uns, and all of them ae taget tacks. The aveage numbe of taget tacks pe un is then 3.1, and the aveage numbe of false tacks pe un is.. Thee ae total 1 tack beaks in the 1 uns, so the aveage taget tack beaks pe un is then.1. The pecentage of taget/false tack duation ae also displayed in Fig. 5 and Fig. 6 fo scenaio 1 and scenaio 2, espectively. Fig. 4. Scenaio 2 The false alam ate fo each senso is set to 2 pe second ove 18 o. The tentative tack initial length n is set to 4. The memoy coefficient is set to.7. TABLE I. DEGHOSTING AND TRACKING PERFORMANCE IN SCENARIO 1 gound taget tack false tack σ b m P D tuth num. du. du. tk. num. du. du. ( o ) (s) (s) (%) bk. (s) (%) 1. 188 3.1 183.6 97.7.1....1.9 188 3.3 182.1 96.8.3....8 188 3.7 179.8 95.6.7... 1. 188 3.3 183.6 97.6.3....3.9 188 3.3 183.1 97.4.3..3.2.8 188 3.4 182.9 97.3.4..3.1 1. 188 3.5 183.5 97.6.5..1..5.9 188 3.5 183.3 97.5.5....8 188 3.8 179.8 95.7.8.1 1.1.6 aveage 188 3.4 182.4 97..4..2.1 The simulation tests show the pefomance obtained fom 1 Monte Calo uns. The pefomance metics listed below ae obtained fom aveage of 1 uns: taget/false tack num.: aveage of total numbe of taget (o false) tacks pe un; taget/false tack du.(s): aveage of total duation of taget (o false) tacks in seconds pe un; Fig. 5. The duations of the taget tacks and false tacks in scenaio 1 Fig. 6. The duations of the taget tacks and false tacks in scenaio 2 Fom the esults, we can obseve that the tagets ae well tacked in scenaio 1. The tagets ae almost fully coveed by tacks. The taget tack duation is fom 95.7% to 97.7%. The taget tack continuity is vey good as the numbe of tack beaks is less than 1. The numbe of false tacks is close to. We can say that the poposed algoithm has vey good pefomance in scenaio 1.
12 1 8 time index:28 Taget tk Selected tentative tk Gound tuth Beaing line Beaing tk 2 Fig. 7. The duations of the taget tacks and false tacks fo the last 36 time cycles in scenaio 2 6 4 2 1 6 9 8 5 7 Howeve, the pefomance is not that good in scenaio 2. The aveage taget tack duation deceases to 88.1%, and the false tack duation inceases to 27.1%. The aveage numbe of tack beaks also inceases to 2.1. The pefomance becomes wose when the beaing eo inceases and P D deceases. This is because the tentative tacks in this scenaio have simila velocities and Dopple shifts. It esults in simila innovation pdfs, and leads to simila assignment costs in deghosting. Howeve, the diffeences among the innovation pdfs become lage when tagets get close to the sensos, and the algoithm yields bette deghosting pefomance. To validate this, the duations of taget and false tacks fo the last 36 time cycles ae computed fom 1 Monte Calo uns. The esults ae shown in Fig. 7. It can be seen the pefomance is impoved significantly. The tagets ae well tacked with vey small amount of false tacks. To futhe illustate the esults intuitively, Figs. 8 11 show taget tacking sceen shots at time 8, 28, 33 and 62 in one un of scenaio 2. The esults ae obtained with the following paamete settings: P D = 1, false alam ate is 2 pe second, σ b m =.5 o, and σ f m =.1Hz. We can see that thee false tacks ae geneated at time 8. The tagets at ange aound 7km ae not tacked. Deghosting pefomance is poo at this ange. At time 28, the thee tacks at ange aound 5km can be tacked by tacks 6, 8 and 9. Fou false tacks exist due to wong deghosting befoe. At time 33, the false tacks ae all dopped, and the thee tagets tacks ae well maintained until end of the simulation un. 12 1 8 6 4 2 time index:8 Taget tk Selected tentative tk Gound tuth Beaing line Beaing tk Senso1 8 6 4 2 2 4 6 1 3 2 Senso2 Fig. 8. Tacking esult in scenaio 2 at time 8. Thee false tacks ae geneated in fa ange. The tagets ae not tacked. Fom the esults, we can conclude that the poposed Senso1 Senso2 8 6 4 2 2 4 6 Fig. 9. Tacking esult in scenaio 2 at time 28. The thee tagets ae tacked by tacks 6, 8 and 9. Fou false tacks exist due to wong deghosting befoe. 12 1 8 6 4 2 time index:33 Taget tk Selected tentative tk Gound tuth Beaing line Beaing tk 6 8 8 6 4 2 2 4 6 9 Senso1 Senso2 Fig. 1. Tacking esult in scenaio 2 at time 33. All false tacks ae dopped, and the thee taget tacks ae well maintained. 12 1 8 6 4 2 time index:62 Taget tk Selected tentative tk Gound tuth Beaing line Beaing tk 6 8 8 6 4 2 2 4 6 9 Senso1 Senso2 Fig. 11. Tacking esult in scenaio 2 at time 62. Thee taget tacks ae well maintained. deghosting and tacking algoithms can tack tagets vey well when tentative tacks have lage diffeences in innovation pdfs, which leads to lage diffeences in deghosting assignment costs. False tacks and tack loss ae seldom obseved in this
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