Single Platform Emitter Location

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1 Sigle Plarm Emier Lcai AOADF FOA Ierermeery TOA SBI LBI Emier Lcai is Tw Esimai Prblems i Oe: Esimae Sigal Parameers a Deed Emier s Lcai: a Time--Arrival TOA Pulses b Pase Ierermeery: Pase is measured bewee w diere sigals received a earby aeas SBI Sr Baselie Ierermeery aeas are clse eug geer a ase is measured wiu ambiguiy LBI Lg Baselie Ierermeery aeas are ar eug aar a ase is measured wi ambiguiy; ambiguiy reslved eier usig rcessig r s-called sel-reslved c Frequecy--Arrival FOA r Dler d Agle--Arrival AOA Use Sigal Parameers Measured a Several Isas Esimae Lcai

2 Frequecy-Based Lcai i.e. Dler Lcai Te Prblem Emier assumed -mvig ad a sii Trasmiig a radar sigal a ukw carrier requecy is Sigal is ierceed by a receiver a sigle aircra A/C dyamics are csidered be erecly kw as a uci ime Nav Daa: Psii ad elciy y z elaive mi bewee e T ad causes Dler si eceived carrier requecy diers rm rasmied carrier requecy Tus e carrier requecy e received sigal will cage wi ime Fr a give se av daa w e requecy cages deeds e rasmier s carrier requecy ad e emier s sii Parameer ecr: [ ] T is a uisace arameer eceived requecy is a uci ime as well as arameer vecr c y z

3 Make isy requecy measuremes a N : Prblem: Give isy requecy measuremes ad e av daa esimae Wa PDF mdel d we use r ur daa???? ~ v i i i I e TDOA/FDOA case we ad a ML esimar r TDOA/FDOA s we culd claim a e measuremes were asymically Gaussia. Because we e ad a well-seciied PDF r e TDOA/FDOA we culd e use ML r e lcai rcessig. Hwever ere we ave ML esimar r e isaaeus requecy s claimig a e is. req. esimaes are Gaussia is a bi a srec. S we culd:. Ourig ASSUME Gaussia ad e use ML arac. esr LS wic des eve require a PDF viewi!

4 B as ge us e eac same lace: Fid e esimae a miimizes N J [ ~ e i I we Assume Gaussia we culd cse: i New-as MLE arac: leads duble derivaives e measureme mdel i e. I we esr LS we culd cse eier: New-as arac wic i is case is ideical N- uder e Gaussia assumi Gauss-New arac wic eeds ly irs derivaives e measureme mdel i e. e i e ] We ll resr LS ad use Gauss-New

5 LS Arac: Fid e esimae suc a e crresdig cmued requecy measuremes i are clse e acual measuremes: Miimize J N i [ ~ ] i i Measured Frequecy Frequecy Cmued Usig Measured Nav ad Pr Assumed Lc. Frequecy Cmued Usig Measured Nav ad Gd Assumed Lc. Time

6 Te Slui Measureme mdel i is liear i! clsed rm slui New-as: Liearize e derivaive e cs uci Gauss-New: Liearize e measureme mdel ~ Tus: H! H v A Liear Mdel were H [ ] v Ge LS slui r udae ad e udae curre esimae: [ ] 4 T T H H H Uder e cdii a e requecy measureme errrs are Gaussia e e CLB r e rblem ca be sw be var{ } T H H Ca use is ivesigae errmace uder gemeries ieres eve we e measureme errrs are ruly Gaussia

7 Te Algrim Iiializai: Use e average e measured requecies as a iiial rasmier requecy esimae. T ge a iiial esimae e emier s cmes ere are several ssibiliies: Perrm a grid searc Use sme irmai rm aer sesr e.g. i er -bard sesrs ca give a rug agle use a geer wi a yical rage Pick several yical iiial lcais e.g. e i eac quadra wi sme yical rage Le e iiial esimae be [ ]

8 Ierai: Fr 0. Cmue e vecr rediced requecies a imes { N } usig e curre esimae ad e av i: z y c [ ] T N!. Cmue e residual vecr by subracig e rediced requecy vecr rm e measured requecy vecr: ~

9 . Cmue Jacbia mari H usig e av i ad e curre esimae: [ ] 4 H Deie: [ ] z y c [ ] z y y c [ ] z y z c 4

10 4. Cmue e esimae udae: T T H C H H C C is e cvariace e requecy measuremes; usually assumed be diagal wi measureme variaces e diagal I racice yu wuld imleme is iverse usig Sigular alue Decmsii SD due umerical issues H beig ear sigular MATLAB will give yu a warig we is is a rblem See e bk Numerical ecies 5. Udae e esimae usig

11 6. Ceck r cvergece slui: lk see i udae is small i sme seciied sese. I N Cverged g Se. I Cverged r Maimum umber ierais qui l & Se 7. Cmue Leas-Squares Cs Cverged slui C N ~ Ne: Tere is guaraee a is algrim will cverge i mig cverge a all i mig: i simly wader arud aimlessly ii scillae back ad r alg sme a r iii wader i cmlee divergece. I racical algrims i is a gd idea u ess i e cde ceck r suc ccurreces σ Tis las se is e de allw assessme w muc cidece yu ave i e slui. Tere are er ways assess cidece see discussi i C. 5 Numerical ecies

12 Simulai esuls wi 95% CLB Errr Ellises y meers Plarm Traecry meers 0 4