1 Introduction. Keywords: Passive radar, likelihood function, Fisher information, tracking, estimation.

Transcription

1 1h Inernaional Conference on Inforaion Fusion Seale, WA, USA, July 6-9, 9 Inforaion Analysis in Passive Radar Neworks for Targe Tracking 1 Gokhan Soysal, A. Onder Bozdogan and Mura Efe Elecronics Engineering Deparen Faculy of Engineering Ankara Universiy Ankara,Turkey {soysal, bozdogan, efe}@ eng.ankara.edu.r Absrac - In his paper, he aoun of inforaion ha can be eraced fro a passive radar nework is analyzed based on Fisher inforaion. The sudy ais o presen a easure of perforance for arge racking wih respec o varying sensor nubers and geoery. Adequae nuber and geoery of receivers for differen locaions of illuinaors of opporuniy have been proposed o achieve an esiaed arge sae wih predeerined level of error. Addiionally, racking zones where racking error can be susained below a predeerined level have been esablished. Obained resuls could prove useful while designing a passive radar nework based on racking accuracy requireens. Keywords: Passive radar, likelihood funcion, Fisher inforaion, racking, esiaion. 1 Inroducion Passive radar syses are a varian of bisaic radars ha accoplish arge deecion and arge racking funcions wihou eiing any energy bu by eploiing illuinaors of opporuniy as a source of radar ransission. The growh in he nuber of RF eissions for TV and radio broadcass in addiion o erresrial and space based counicaions have creaed diversiy in he signal ypes ha would be eploied by passive radars. The use of eission a VHF and UHF frequencies akes passive radars able o deec sealhy arges easier han he odern acive radars which do no use his par of he specru. Due o hese advanages and advances in cheap copuaional power, here has been significan increen in he nuber of sudies on passive deecion in he las decade, which focus on wo ain subjecs naely, arge deecion and arge racking. Targe deecion proble in passive neworks is analyzed hrough wavefor properies of he signal ypes available for hese neworks by eans of aiu deecable range, available range and range rae resoluions [1 3]. On he oher hand, principal probles of arge racking wih bisaic/ulisaic passive easureens are rack iniiaion, daa associaion, arge classificaion/recogniion and rack ainenance [4-8]. Perforance of any racker is affeced by arge deecion process by eans of durabiliy of deecion in ie and accuracy of he obained easureens. While, deecing a arge in every scan guaranies o have inforaion abou he arge sae, accuracy of he observed easureens will deerine he accuracy of he esiaed arge sae ha can be achieved by he racking algorih. Accuracy of he easureens can be hough as he aoun of inforaion abou arge sae and can be odeled hrough he use of Fisher inforaion [9]. In his view, in a passive radar nework he aoun of inforaion conained in arge originaed easureen abou arge sae would depend on he nuber of ransier receiver pairs (which we will call a `sensor`), locaions of he pairs and he arge sae. In his sudy he relaion beween deecion and racking in passive neworks has been used o deerine he effec of nuber of sensors and sensor geoery on he inforaion gahered fro he arge originaed easureen hrough he analysis of Fisher Inforaion Mari (FIM). Variaion of available inforaion depending on nuber/geoery of sensors and arge saes can be oniored by observing he FIM. Diagonal eleens of he inverse of FIM, which is known as he Craer-Rao Lower Bound (CRLB) are he variances of he paraeers o be esiaed, square roo of which can be defined as he uncerainy peraining o hese paraeers. The obained uncerainy, hen, can be uilized o define a bound for achievable accuracy of esiaed arge sae in passive radar neworks. In his paper, an analyical sudy, based on he FIM, regarding he perforance of passive radar neworks is presened. The sudy ais o presen he perforance liis wih respec o varying sensor nubers and geoery. For his purpose, a single ransier uli receiver passive radar nework wih he capabiliy of easuring bisaic range and range rae has been odeled and Fisher inforaion has been copued independen of arge deecion and racking where he arge was assued o be oving on a grid defined in 3D Caresian space. FIMs have 1 This work is suppored by Ankara Universiy Scienific Research progra hrough conrac No: 8B ISIF 1115

2 been copued based on saic paraeer esiaion a every possible locaion of he arge for differen configuraions (differen nuber of sensors and differen sensor geoery) in a passive radar nework. The inforaion conained in he easureens observed fro he nework has been analyzed in ers of oal posiion and velociy uncerainies as well as he noralized coverage area. Furherore, 3D racking zones are copued ha have an uncerainy level less han a pre-defined value for boh posiion and velociy. Also he disribuion of he uncerainies in hese zones has been presened. The paper is organized as follows: In secion II derivaion of likelihood funcion is given and Fisher inforaion ari based on saic paraeer esiaion is derived for 3D arge sae. Passive radar nework configuraions and siulaion resuls are presened in secion III. Finally in secion IV a brief suary and discussion of he resuls are given. Likelihood Funcion and he Fisher Inforaion Mari Assue ha θ is he vecor valued paraeer ha conains posiions, y, z and velociies v, vy, vz in 3D Caresian space o be esiaed. Lr and L are locaions of he receiver and ransier respecively. Le θ = [ v y vy z vz] T (1) L = [ y z ] T () r r r r L = [ y z ] T (3) and R denoe he disances beween he receiver and he arge of ineres and beween he ransier and arge of ineres respecively. Assue ha range rae of he arge wih respec o he receiver and ransier are R ɺ r and respecively. Then, one can wrie bisaic range and range rae easureens generaed by a passive ransier receiver pair as r = R εr (4) rɺ = Rɺ r Rɺ εr ɺ (5) where r ( r ) ( r ) ( r ) R = y y z z (6) ( ) ( ) ( ) R = y y z z (7) ( r ) v ( y yr ) vy ( z zr ) vz Rɺ r = (8) R r ( ) v ( y y ) vy ( z z ) vz Rɺ = (9) R Here ε r and ε r ɺ are easureen errors which are assued o be uually independen Gaussian processes wih zero ean and variances and R ɺ ɺ respecively. Assuing ha ha and ɺ are known, hen he likelihood funcion of he paraeer θ can be wrien as Λ ( r, rɺ θ,, ) = 1 1 ep( ) ep( ) π ( R r) ( Rɺ r Rɺ rɺ ) π (1) In he presence of ore han one ransier-receiver pairs he likelihood funcion can be defined as he uliplicaion of he individual likelihood funcions of each pair under he assupion ha easureen errors are uually independen, as given by Eq. (11) k Λ ( Z θ, Z ) = Λ( r, rɺ θ,, ) (11) i= 1 i i i i In Eq. (11), Z is se of easureens acquired fro he sensors a ie and Z is se of sandard deviaions peraining o he easureens. For he described syse he Fisher inforaion ari is defined as, FIM = E θ θ ln( Λ( θ )) θ = θ (1) where θ is he gradien operaor. Logarih of he likelihood funcion for a single ransier-receiver pair is ϕ = ln( Λ ( θ )) = ( R R r) ( Rɺ Rɺ rɺ ) (13) ln( π ɺ ) r r hen he Fisher inforaion ari is wrien as ϕ ϕ vz FIM = E ϕ ϕ v z vz (14) Derivaion of he eleens of FIM is given in he Appendi for a single sensor. However, i can be easily epanded o uliple sensor case by using Eq. (11) as su of FIMs. 3 Passive Radar Configuraions and Siulaion Resuls In his sudy, single ransier uli receiver passive radar nework wih capabiliy of easuring bisaic range and range rae has been odeled. The analysis carried ou here is independen of arge deecion and racking processes. The ulisaic passive nework is assued o oupu bisaic range and range rae easureens ha are generaed by each ransier-receiver pair and odeled as corruped by addiive whie Gaussian noise wih known ean and 1116

3 variance. In he analysis, plane earh odel is used and coverage area of he passive radar nework is kep liied in boh X and Y aes in he inerval of (-k, k). I was assued ha arge is oving in a 4 long discree grid in boh X and Y aes and arge oveen is assued independen beween wo locaions. Targe was assued o be oving in 3D Caresian space according o sae vecor [ ɺ y yɺ z zɺ ] T a a fied aliude. Measureens obained fro each ransier-receiver pair for 3D arge sae have been given as. r r r R = ( ) ( y y ) ( z z ) r ( ) ( y y ) ( z z ) ε (15) ( r ) ( y yr ) y ( z zr ) z Rɺ ɺ ɺ ɺ = ( r ) ( y yr ) ( z zr ) ( ) ɺ ( y y ) yɺ ( z z ) zɺ (16) εrɺ ( ) ( y y ) ( z z ) In Eqs. (15) and (16), subscrips r and indicae he locaion of receiver and ransier in 3D Caresian space respecively. Addiive error ers ε r and ε r ɺ are odeled as whie Gaussian processes wih zero ean and variances of 1 and 1( / sec) respecively. A passive radar nework consising of single ransier uli receivers is odeled in wo principal configuraions of ransier-receiver locaions. Receivers are locaed o for a circle. Two differen ransier locaions are assued in he circular receiver configuraion of which one is a he cener of receiver circle and he oher is ou of i. Thus, boh having a ransier in a conrolled erriory and a ransier which is locaed ou of he hoeland borders are odeled. In he second configuraion he ransier is locaed a 75 k fro he cener of receiver circle and a 45 aziuh angle easured counerclockwise fro he X ais easured. Afer fied ransier locaions are deerined, several differen ransier-receiver geoeries are generaed by changing he nuber of receivers and he radius of he receiver circle. The nuber of receivers has been changed fro 3 o 7 which corresponds o 6 separaion beween adjacen receivers in he 3 receiver case and 5 separaion beween adjacen receivers in he 7 receiver case. For he firs configuraion, he salles radius of he receiver circle is se o 1 k and i is increased o 1 k by 5 k increens. Radius of he receiver circle is also varied beween 1-5 k wih 5 k increens for he configuraion in which ransier was locaed ouside he circle. For all configuraions, aliude of receivers and ransiers was assued o be sae and se o he ground level. Variaion of he Fisher inforaion, i.e. changes in uncerainy, w.r.. he arge sae and ransier-receiver configuraion have been analyzed in ers of Toal Posiion Uncerainy (TPU), Toal Velociy Uncerainy (TVU) and Noralized Coverage Area (NCA) hrough siulaions. TPU and TVU are copued for he k h posiion of he arge on he grid as follows, 1 k k k k k k U k = diag( FIM k ) = Σ Σ Σ yy Σ yy Σ zz Σ ɺɺ ɺɺ zz ɺɺ (17) k k k P k = Σ Σ yy Σ zz (18) k k k k ɺɺ yy ɺɺ zz ɺɺ (19) V = Σ Σ Σ 1 N Pk N k = 1 TPU = () 1 N Vk N k = 1 TVU = (1) where FIM k and diag indicae Fisher inforaion ari a he k h posiion and he diagonal eleens respecively. However, no all arge posiions (cells of grid) conribue o suaions defined by Eqs. () and (1). Only he cells for which he copued uncerainy value is less han 3 in posiion and 4 /sec in velociy for all aes have been aken ino accoun. Thus, zones where posiion and velociy of he arge can be esiaed wih significan error level are deerined. NCA was deerined as he raio of area of cells which have significan uncerainy levels as defined above o area of receiver circle as given by Eq. () c c N π r Nr NCA = = () π r r where, r, rc, N are radius of he receiver circle, radius of a cell and he nuber of significan cells respecively. 3.1 Siulaion Resuls Siulaion resuls for 3D arge sae are presened in his subsecion. I is assued ha arge is oving a consan aliude of 5 k and is saionary a he grid poin during he easureen eracion process. For he firs configuraion where he ransier is locaed a he cener, variaion of TPU and TVU wih respec o he change in he nuber of receivers and radius of receiver circle are shown in Figs. 1 and. Toal uncerainies in posiion and velociy reduce eponenially wih he increasing nuber of receivers for all radiuses. This is epeced due o fac ha increase on aoun of inforaion by deploying ore ransierreceiver pairs and gahering ore easureens leads o ore accurae sae esiaion. However, i is seen eplicily in he Figures ha while changing he nuber of receivers fro 3 o 1 provides significan iproveen in uncerainy, conribuion fro he addiion of a new receiver reduces rapidly afer he 1 h receiver. This oucoe is suppored by he NCA analysis which is presened in Fig. 3. As i is seen in Fig. 3 a rapid growh in NCA has been obained while increasing he nuber of sensors fro 3 o 1 whereas he growh rae decreases for each sensor added afer he 1 h. 1117

4 Anoher resul worh enioning is he effec of radius of he receiver circle. Saller radius achieves bigger NCA in coparison o he larger receiver circle. For insance he NCA has been.5 ies he area of he receiver circle wih he 3 k receiver circle when he aiu nubers of receivers is deployed. 1% coverage of he region bounded by receiver circle could be achieved by eploiing 6, 8 and 15 receivers for 3k, 5k and 75k receiver radius respecively. Analysis of TPU, TVU and NCA has shown ha growh in he nuber of ransier-receiver pairs deployed in a passive radar nework has he advanage of collecing ore inforaion fro he arge of ineres, i.e. esiaing arge sae ore accuraely and covering a greaer region. However in real world ipleenaions, uilizing a big nuber of ransier-receiver pairs increases he counicaion cos and he easureen o rack daa associaion proble becoes alos unsolvable in real ie. Therefore, a feasible passive radar syse should be designed o have he appropriae nuber of sensors o cover he region of ineres wih accepable accuracy while allowing copuaions o be carried ou in real ie. Such a syse can be designed by uilizing he resuls presened in Figs.1-3. As enioned previously 1% coverage of he region of ineres wih uncerainies in posiion and velociy less han 3 and 4 /sec respecively can be achieved by eploiing 6 receivers locaed on a circle wih radius 3 k. For furher analysis, deailed siulaion resuls have been obained for he 6 receiver configuraion. Firs of all, bounds of he NCA have been invesigaed by epanding he radius of he receiver circle in 1 k, 1 k inerval wih 5k increens and he resuls are given in Fig. 4. Toal Posiion Uncerainiy Radius of Circle 3k Radius of Circle 5k Radius of Circle 75k Nuber of Receivers Figure 1. Variaion of posiion uncerainy wih respec o he nuber of receivers. Toal Velociy Uncerainiy Radius of Circle 3k Radius of Circle 5k Radius of Circle 75k Nuber of Receivers Figure. Variaion of velociy uncerainy wih respec o he nuber of receivers. Noralized Coverage Area Radius of Circle 3k Radius of Circle 5k Radius of Circle 75k Nuber of Receivers Figure 3. Variaion in he noralized coverage area wih respec o he nuber of receivers. As shown in Fig. 4, he NCA is greaer han 1, i.e. bigger han he area of he receiver circle, while he radius of he circle varies beween 1 k and 37 k. The coverage, hen, decreases o % beyond 37 k radius for he fied nuber of receivers. The NCA is approiaely 1.5 when he radius is 3k which is a fine coverage and iage of his region has been given in Fig. 5. The region depiced in Figure 5 ay be hough as he racking zone where he upper bound of racking error is 3 in posiion and 4 /sec in velociy around he edges of he region. Disribuion of posiion uncerainies wihin he racking zone has been given for X, Y and Z aes in Figs. 6, 7 and 8 respecively. While uncerainies on X and Y aes vary in he inerval of 3-8, he uncerainy on Z ais varies in a wider range (3-3) and hese resuls show ha bounds of he racking zone are deerined by he uncerainy on Z ais. Siilar disribuion paerns and higher uncerainy level on Z ais have been obained for velociy uncerainies and resuls have been given in Figs

5 Noralized Coverage Area Radius of Circle (k) Figure 4. Variaion of he NCA wih respec o he increasing radius of he receiver circle Figure 7. Disribuion of posiion uncerainy on Y Figure 5. Copued racking zone where posiion uncerainy is less han 3 and velociy uncerainy is less han 4 /sec in X, Y, Z aes (aserisks indicae receiver locaions and penagra indicaes ransier locaion) -5 5 Figure 8. Disribuion of posiion uncerainy on Z Figure 6. Disribuion of posiion uncerainy on X. 35 Figure 9. Disribuion of velociy uncerainy on X. Siulaion resuls for he configuraion in which ransier is locaed ouside he receiver circle have showed ha, he change in he locaion of ransier has a grea effec on he size of he coverage region as well as uncerainy levels. 1119

6 Coverage region of a passive radar nework coprising 6 receivers locaed on a circle wih radius 3 k is shown in Fig beyond which a circular gap occurs due o higher uncerainies in posiion and velociy as depiced in Fig. 14. Moreover, variaions of posiion and velociy uncerainies wih respec o he nuber of receivers for he ransier ouside he receiver circle case are given in Figs. 15 and 16 respecively. These resuls were obained for case where radius of receiver circle is 3 k Figure 1. Disribuion of velociy uncerainy on Y Figure 1. Copued racking zone for he 6 receiver configuraion Figure 11. Disribuion of velociy uncerainy on Z. As i is seen fro he figure, he region lying beween he ransier and cener of he circle could no be covered by he passive radar nework. In order o cover his region, one can locae anoher ransier a 5 easured CCW fro he X ais ha resuls in 1 ransier-receiver pairs. However, his is no feasible under he assupion ha eission sources are illuinaors of opporuniy and hey are no under conrol. Anoher soluion o his proble would be o increase he nuber of receivers unil region bounded by he receiver circle is fully covered. Adequae nuber of receivers has been found o be 1 where he coverage area for his configuraion is depiced in Fig. 13. Furher analysis has been carried ou in order o deerine he achievable aiu radius of he receiver circle for he 1 receiver configuraion. Circle radius has been epanded fro 3 k o 5 k by 1 k increen and variaion of he covered region observed. Siulaions have revealed ha he achievable aiu radius for his configuraion is 4 k Figure 13. Copued racking zone for he 1 receiver configuraion If one ade a coparison beween he posiion uncerainies obained by configuraions one and wo, i would be seen ha here is approiaely 1 difference in posiion uncerainies for he wo configuraions whils second configuraion deploys ore receivers han firs one. Siilar resuls are obained for he velociy uncerainy as well. Alhough he siulaion resuls presened here depic a circular sensor geoery, siilar resuls would be obained for any syerical sensor geoery where he iniu nuber of sensors should be 6, i.e., heagon, ocagon ec. 11

7 -5 4 Conclusions In his sudy, posiion and velociy uncerainies in a passive radar nework have been analyzed in ers of Fisher inforaion. For his purpose, Fisher inforaion ari based on saic paraeer esiaion has been derived and differen nubers of sensors and sensor geoery have been considered. The obained uncerainy can be uilized o define a bound for achievable accuracy of he esiaed arge sae in passive radar neworks Figure 14. Copued racking zone beyond he achievable aiu receiver circle radius Toal Posiion Uncerainy Nuber of Receivers Figure 15. Variaion of he posiion uncerainy (ransier ouside he receiver circle case) Toal Velociy Uncerainy Nuber of Receivers Figure 16. Variaion of he velociy uncerainy (ransier ouside he receiver circle case) References [1] C. J. Baker, H. D. Griffihs and I. Papousis, Passive coheren locaion radar syses. Par 1: Perforance predicion, IEE Proceedings, Radar, Sonar and Navigaion, volue 15, No.3, pp , June 5. [] C. J. Baker, H. D. Griffihs and I. Papousis, Passive coheren locaion radar syses. Par : Wavefor properies, IEE Proceedings, Radar, Sonar and Navigaion, volue 15, No.3, pp , June 5. [3] P. E. Howland, D. Maksiiuk and G. Reisa, FM radio based bisaic radar, Proceedings, Radar, Sonar and Navigaion, volue 15, No.3, pp , June 5. [4] M. Tobias, A. D. Laneran, A probabilisic densiy bases uli arge racker using uliple bisaic range and velociy easureens, Proceedings of 36. Souhern Syposiu of Syse Theory, pp. 5-9, 4. [5] M. Daun, C. R. Berger, Track iniiaion in a ulisaic DAB/DVB-T Nework, 11 h Inernaional Conference of Inforaion Fusion, pp. 1-8, June 8. [6] M. Daun, W. Koch, Mulisaic arge racking for non cooperaive illuinaion by DAB/DVB T, Radar Conference, IEEE, pp. 1-6, 8. [7] X. Yin, T. Pedersen, e al, A single sage racking algorih for ulisaic DVB T passive radar syses, Digial Signal Processing Workshop and 5 h IEEE Signal Processing Educaion Workshop, pp , January, 9. [8] A.O. Bozdogan, G. Soysal and M. Efe, Mulisaic Targe Tracking Using Bisaic Range Range Rae Measureens o appear in Fusion 9 proceedings. [9] Y. Bar Shalo, X. R. Li, T. Kirubarajan, Esiaion wih Applicaion o Tracking and Navigaion Wiley Inerscience Publicaion, Newyork,

8 APPENDIX Derivaion of Eleens of FIM ϕ 1 r E = 1 v v ( ( r r R R ϕ 1 y yr y y y E = 1 vy vy ( y y r r R R ϕ 1 z zr z z z E = 1 vz vz ( z z ( y y ( z z r r R R ϕ 1 r v ɺ v v ( r r ( R r R y y vy ɺ v v ( r r ( R r R ϕ 1 z zr z z vz ɺ v v ( r r ( R r R ϕ 1 z zr z z y vz ɺ vy vy ( y yr r ( y y R r R ϕ 1 r y v ɺ vy vy ( y yr r ( y y R r R y y y vy ɺ vy vy ( y yr r ( y y R r R y y z vy R r ɺ r ( z z R r R ϕ 1 r z v ɺ r ( z z R r R ϕ 1 z zr z z z vz ɺ r ( z z R r R ϕ 1 r y yr y y y R r R r 1 v v ( r r ( R ɺ vy vy ( y yr r ( y y R r R ϕ 1 r z zr z z z R r R r 1 v v ( r r ( R ɺ r ( z z R r R y y z zr z z y z R r R r 1 vy vy ( y yr r ( y y R ɺ r ( z z R r R ϕ 1 r E = v ɺ y y E = vy ɺ ϕ 1 z zr z z E = vz ɺ ϕ 1 r y yr y y r r E = v vy ɺ R R R R ϕ 1 r z zr z z r r E = v vz ɺ R R R R y y z zr z z r r E = vy vz ɺ R R R R 11