TECHNICAL REPORT. Accuracy Enhancements for TDOA Estimation on Highly Resource Constrained Mobile Platforms

Transcription

1 Institut for Softwar Intgrat Systs Vanrbilt Univrsity Nashvill Tnnss 3735 TECHNICAL REPORT TR #: Titl: Authors: ISIS-3-4 Accuracy Enhancnts for TDOA Estiation on Highly Rsourc Constrain Mobil Platfors Kuar Chhokra, T Bapty, Jason Scott, Mitch Wilks Dat: /7/3 Copyright Vanrbilt Univrsity, 3

2 Accuracy Enhancnts for TDOA Estiation on Highly Rsourc Constrain Mobil Platfors Kuar Gaurav Chhokra, Thoor Bapty, Jason Scott, Mitch Wilks Institut for Softwar Intgrat Systs, Vanrbilt Univrsity {kuar, bapty, Tl: ( , Fax: ( Elctrical Enginring an Coputr Scinc, Vanrbilt Univrsity Abstract. Ovr th past fw yars, thr has bn an ins thrust in th fil of golocation, survillanc, tracking an location-awar systs an srvics. Th avnt of copact, low powr, high-procssing powr DSPs hav a possibl svral tasks which wr infasibl just a fw yars ago. TDOA stiats on such nrgy an rsourc constrain platfors suffrs fro th lack of a cohrnt sapling clock. W prsnt two rlat tchniqus of Dopplr-frquncy an ti shift corrction. Th tchniqus ar forally vlop, analyz. Thy ar thn copar fro an iplntation an prforanc prspctiv. I. Introuction Ovr th past fw yars, thr has bn an ins thrust in th fil of golocation, survillanc, tracking an location-awar systs an srvics. Th avnt of copact, low powr, high-procssing powr igital signal procssors hav a possibl svral tasks which wr consir infasibl just a fw yars ago. Th applications aboun in utilitarian (E9, prouctcustoiz srvics, in-builing tracking, tc. an ilitary sctors (survillanc, un-ann rconnaissanc, targt location tc. [-4]. Traitional TDOA / TOA systs us for golocation pn havily on fix array bas procssing an typically ploy high prcision, costly coponnts. Such systs also hav long stan off rangs, aking it ifficult to closly track an follow th ovnts of targts with low powr signals an in congst aras [5]. Proxiity to targt an obility of th tracking vics gathrs gratr rlvanc in urban nvironnts. Unann arial vhicls (UAVs an organic arial vhicls (OAVs quipp with golocation lctronics can countr th probls with long-stanoff systs. Thy can affor safty, scurity an stalth for th rconnaissanc ission. Howvr, thy pos intrsting tchnical constraints. Robustnss an rliability unr ilitary us conitions an that th vics b copact, rugg an chanically stabl. Stalth in rconnaissanc iplis a sall, silnt sign (i.. liit counication banwith whil, nuranc an usability an an xtnsiv oprating lif (i.. low powr. Ths UAVs ar iagin as a flt of ispnsabl obil vics, iplying a rlativly strong constraint on th anufacturing an oprating cost. Ths ral-worl constraints anifst thslvs in th algorithic an signal procssing oain as wll. Physically istinct, passiv RF an obil units iply th lack of a coon clock, thus obviating array bas procssing. Stalth prvnts th UAVs fro synchronizing on a rgular basis to aintain a coon clock, coplicating accurat TOA stiats. Also, a ruc counication banwith btwn th vics ngats th usual tchniqus of corrlating rciv signals for TDOA stiats. Ths constraints suggst a tchniqu whr uch of th tction, intification an TOA ar on locally an th actual targt locus coputation is rlgat to a rot bas station with banwith-constrain inputs fro th iniviual UAV snsors. Spcifically, ach UAV ust coput absolut TOA using a-priori inforation such as a signal tplat, (known signal faturs, synchronization squncs, tc. TOA procssing ust thn copnsat for rlativly low prcision GPS clocks an uniqu (possibly varying sapl rats at th snsors. This papr scribs two tchniqus for corrcting th rrors in TOA/TDOA stiats u to incongrunt sapling frquncis of th rciv an tplat signals. Sction II introucs an ols th ffcts of isparat sapling frquncis as a rlativ ti copaning (RTC probl athatically. Sction III introucs th Dopplr shift corrction tchniqu for narrow ban signals. Sction IV xplains th otivation for a ti oain corrction. Sction V vlops th ti-shift corrction tchniqu. Sction VI copars an contrasts th two tchniqus with rspct to

3 prforanc charactristics an coputational fficincy with fil ata. II a. Mathatical prliinaris F ( Sapling b A signal anating fro a rot sourc an onitor in th prsnc of nois at two spatially sparat snsors ay b athatically ol as: b Sapling f s x x ( t = f ( t + n ( t (a x t = f ( t + D + n ( (b ( t whr f(t, n (t an n (t ar ral, jointly stationary rano procsss. Th signal f(t is assu to b uncorrlat with nois n (t an n (t. D is th lay btwn th two rciv signals. Th traitional tchniqu of tcting a signal of intrst (tplat buri in a ata stra corrupt by aitiv Gaussian rano nois is to us a atch filtr or a gnraliz cross corrlator (GCC [6]. Sinc th GCC approach ay b viw as pr-filtring th two signals with whitning filtrs bfor a usual crosscorrlation, w focus on th corrlation tchniqu for siplicity. Th corrlation function ay b writtn as [7]: ( Rx x = x ( t x ( t + t ( Th valu of that axiizs ( provis an stiat of th lay D. II b. Scaling u to isparat sapling frquncis Sinc th signal procssing coponnts on th UAVs ar subjct to a wi gaut of oprating conitions, th oprating clock an sapling frquncis ar slo th sa as thos in th tst nvironnts. In particular, th frquncy at which th tplat was construct is gnrally iffrnt fro th frquncy at which th incoing signal is sapl. [8] an [9] provi a goo iscussion of th ffcts of RTC on th cross-corrlation opration. [] xtns th argunt to quaratic lays. Fig. illustrats th ffcts of iffrnt sapling frquncis. Th signal with a noinal Fourir transfor is sapl at two sapling frquncis, f s an f s, with f s < fs. Th sapl signal has iffrnt noraliz banwiths givn by (3 whr si = πfs i. b ˆ b = an b ˆ b = (3 s s f s b Figur iffrnt sapling frquncis la to frquncy copaning Clarly th highr sapling frquncy proucs a sallr noraliz banwith, i.., b < b. Whn viw in th ti oain, th iffring rlativ banwiths anifst thslvs as ti scaling artifacts. III a. Ti scaling an Dopplr shift Th isparat sapling frquncis of th tplat an th UAVs prouc corrlation artifacts, laing to rronous ti lay an TOA stiats. For rlativly narrow-ban signals an fairly siilar sapling frquncis, it is now shown that th RTC btwn th tplat an rciv signal ay b approxiat by Dopplr shifts. Wiss [] givs a goo xplanation of th narrow ban critrion. Th signals unr consiration fall wll within th oain of narrow ban rprsntation. Lt th tplat b rprsnt by f(t an th rciv signal by g(t. Both signals ar consir to b ral an continuous as this ass analysis. Lt th rciv signal suffr both a scaling, givn by th scal paratr s, an a ti lay, : g ( t = f ( s( t (4 Taking th Fourir transfor on both sis, w hav, G( = s / s Lt s = a, thn G( js (5 j( a = /( a /( a (6 if a <<, w hav + a a (7 j( a G( = ( + a ( + a (8 For narrow ban signals,

4 b + δ =, δ > (9 whr is th cntr raian frquncy an b is th banwith of th signal. Hnc th abov xprssion for G( ay b approxiat by: G j ja ( ( + a + a ( Noting that both a an ar sall, so that th scon xponntial ayb approxiat by unity, w gt, G j ( ( + a + a ( Taking th invrs Fourir transfor, w obtain, j ( t g ( t f ( t ( whr = a o = (-s o (3 Th abov xprssion inicats that for sall scal factors, a Dopplr shift can approxiat th ffct of scaling. III b. Copnsating for RTC with Dopplr shifts Obsrv that th for of th copan narrow ban signal closly rsbls th krnl of th crossabiguity function. Hnc, th cross-abiguity function is wll suit to corrct for th shift introuc u to rlativ copaning. [8, ] prsnt svral tchniqus of using cross-abiguity functions (CAFs for trining th rlvant lay an scal paratrs. For our cas, th ratio of th sapling frquncis of th two signals, s, is always known: th sapling frquncy of th tplat is known a priori, an can in fact b accuratly controll through corrct construction. Th instantanous sapling frquncy for th ata obtain at a snsor ay b stiat using svral introspction algoriths. Also, th cntr frquncy of th transitt signal ay b known a priori by construction. For arbitrary signals, it ay also b stiat as th an frquncy as fin in [, 3] by = F ( F ( (4 To stiat th TOA, a gnraliz cross corrlator (GCC ay b us on th RTC copnsat signals. As bfor, if f(t rprsnts th tplat an g(t th rciv signal, thn w crat a nw signal, f (t which is a frquncy shift vrsion of th tplat to copnsat for th currnt oprating frquncy of th snsor. f j ( t = f ( t t (5 Signals f (t an g(t ar thn f through atch filtr to obtain an stiat for th ti lay,. * = arg ax f ( u g ( u + t u (6 * jt = arg ax f ( u g ( u + t u (7 Th abov xprssion ay b viw as a crossabiguity function valuat at a givn shift. This circuvnts th n to coput svral CAFs for iffring valus of scal an lay. IV. Iplntation issus in Dopplr RTC corrction Th Dopplr shift tchniqu copnsats for RTC by oulating th tplat signal with a coplx xponntial as shown in (5. Usually, th coplx xponntial is coput using Eulr s xpansion: xp( j t = cos( t + j sin( t (8 For a iscrt ti syst, th abov valuation ust b prfor for ach sapling instant. Givn a tplat of lngth N, this proucs an N x vctor of sapls of th oulating coplx xponntial = cos( t + j sin( t (9 whr, t[ k] = k / is th vctor of sapling instants, s with k N, k Ζ an sapling frquncy s. Coputing th lnts of this coplx xponntial in trs of trigonotric functions is both coputationally an tporally intnsiv. Furthror, onc th vctor has bn coput, its application to th ral tplat vctor rquirs N coplx ultiplications. Exaining th structur an natur of this calculation an applying a fw nginring assuptions, w can significantly ruc this burn. For valus of s fairly clos to, or quivalntly, whn a is fairly sall an whn th prio ovr which th Dopplr corrction ust b ploy is liit (If it is larg, thn th ti-banwith prouct conition for narrow ban signals is violat, aking th Dopplr shift assuption invali, th prouct t is fairly sall. Thus, w can ak th following approxiations cos( θ an sin( θ θ ( xp( j t = + j( t (

5 Exprssing this as a vctor of iscrt sapls, w hav [ k] = + j [ k] ( t This coputation rprsnts norous savings in crating th corrction vctor,. Th structur of ach [k] also nabls us to apply th corrction fficintly. Th Dopplr shifting of th tplat ay b accoplish in a singl pass, laing to an O(N coputation routin. V. Altrnativ ti shift bas corrction Th Dopplr corrction tho copnsats for rlativ copaning by approxiating th frquncy scaling by a frquncy shift: th oulation of th tplat squnc by a coplx xponntial shifts th spctru of th tplat is to approxiatly lin up with that of th scal signal. Th Dopplr corrction has th avantag of offring a vry intuitiv an sipl tchniqu of copnsating for th iffrnt sapling frquncy btwn th nos an th tplats. Whil it also obviats th n to valuat th passiv abiguity function for a larg nubr of scals an shifts, it still suffrs fro a coputational prspctiv. As xplain arlir, th snsor nos ar constrain in ory an coputational powr. In particular, th TI C67x DSPs us to provi th signal procssing capabilitis hav a liit intrnal cach. Whn iplnting coputationally an tporally intnsiv oprations such as a GCC, it is iprativ that th ory accss rquirnts b controll to t har ral-ti constraints. Ensuring that th ata an rsult vctors fit in th fast intrnal ory is a provn an routinly ploy tchniqu [4]. Both th Dopplr corrction an approxiation tchniqus suffr in this rgar bcaus thy convrt norally ral ata vctors into coplx vctors, oubling th initial ory rquirnt an causing prforanc to gra. For.g., an 8K point ral-fft taks a significantly lssr ti than a coplx-fft of th sa lngth. Th prforanc loss accru ovr svral itrations can an th failur of a ral-ti constraint. Such a situation can occur in th coputation of th cross spctral nsity (CSD [5] of th rciv signal with th tplat. Lt us xain th rlativ copaning probl again. As bfor, lt g ( t = f ( s( t (3 whr, g(t, f(t ar, rspctivly, th rciv an tplat signals, s is th scaling factor an is th ti lay introuc. Th quations blow rcall th atch filtring opration. R gf ( u = g( t f ( t + u t (4 = arg ax g ( t f ( t + u t (5 whn s =, a valu of t = axiizs R gf. For th prsnt cas (s~, an narrow ban signals, th valu of t ovr a prio (T, T that axiizs R is givn by, T * ( T, T, t = arg ax f ( st s f ( t + u t (6a T ( T, T, t = u, s( t = ( u + t ( T, T, t = ( s t s (6b (6c Notic that this lay (T, T,t is not fix as in th prvious cas, but is sar in ti. W wish to approxiat this ti varying lay (T,T,t by a constant (T, T chosn accoring to a last squar rror critrion. Lt th su of th squar rrors, (, b givn by T T ( ( t t ( = ( t t = (7 T T whr (T, T is th intrval ovr which th two squncs ar copar. Using th Libniz intgral rul [6], w hav T ( = ( ( t t + T (8 ( ( T ( T ( ( T ( T For o, th sir optial valu, th LHS of (8 vanishs. T T T o t t = s s T T ( ( = (9 T Ovr a givn intrval [T, T ], th ffcts of rlativ copaning btwn f(t an g(t ay b iniiz in th last squars sns by constructing a ti-shift ~ t vrsion of f(t, not f (, as ~ o T T f ( t = f ( t + f ( t ( s s = + (3

6 In th iscrt oain, th abov quation ay b xprss as ~ ~ πn πn T f [ n] = f ( = f ( + ( s s, (3 s s whr s is th angular sapling frquncy of f. nos consist of an FRS rcivr connct to th rlvant lctronics. A global clock was riv using GPS signals on puls pr scon (PPS signals. Th GPS signals wr filtr an procss to ruc th ffcts of any tiing jittrs. Th analog signal f(t is usually not availabl to prit arbitrary ti-shifting of th tplat signal. woul thn n to b valuat by intrpolating at arbitrary ti instants. Whil this is achivabl, th intrpolation in th ti oain usually cos with a coputationally xpnsiv pric tag. ~ f CH GPS rcivr CH f(t-- tplat GPS rcivr CH An asir altrnativ ay b foun by igrating to th frquncy oain. Taking th Fourir transfor of th prcing quation, w obtain ~ j F ( = (3 whr = f ( t jt t is th Fourir transfor of f(t. Also not that w hav ropp th suprscript of in favor of brvity. As with th Dopplr corrction, it can b argu that th natur of an allow th corrction factor approxiation j to b coput as an j = + j (33 Th ti-shift bas approach has th following notabl avantags. Sallr ory foot print for coputation an ruc ory latncy: For ral signals, th Fourir transfor xhibits Hritian sytry. This ay b xploit to ruc th nubr of coputations rquir to coput th ntir FFT. It also ans that th input vctor to th FFT routin is a ral-vctor, rquiring half as uch ory as its coplx countrpart, rucing ory accss tis.. Ability to prcoput F ( : sinc th tplat is known by sign, F ( ay b coput offlin. This obviats its coputation at run-ti ntirly. This introucs hug savings in coputation ti. 3. Facility to fficintly iplnt corrction in th frquncy oain: As with th Dopplr shift corrction, w can utiliz th natur of th corrction to vis an fficint tho of applying th corrction in th frquncy oain. This has th a avantag of allowing an in situ coputation, bringing furthr ruction in ory accss latncis an coputation ti. VI. Exprintal Rsults Th figur shows th schatic stup on which ths algoriths wr iplnt an tst. Th snsor DSP Dtction, intification, RTC corrction + TOA snsor no DSP Dtction, intification, RTC corrction + TOA snsor no Bas station TOA TOA TDOA + locus algoriths Figur. Schatic tst stup. FRS raio transits a psuo rano squnc containing th tplat. Th two nos fin th tplat an sn TOA stiats to th rot bas station A -sc, 5 khz ban-liit psuo-rano squnc srv as th tplat. A signal sourc plac at a controll istanc fro both rcivrs transitt a squnc containing th tplat. Th snsor nos ha on-boar a pr-anufactur vrsion of th tplat. Th tction an TOA oprations wr prfor locally on th nos an th TOA stiats wr transfrr to a PC (bas-station containing th TDOA location trining algoriths. Th noinal sapling frquncy at all th nos was 48 khz. W prsnt th rsults of th various xprints in Figurs 3-5 an Tabl. VII. Conclusion Th availability of powrful low cost b DSPs with an iptus in wirlss counications has l to a vry larg intrst in vloping slf-contain istribut golocation systs. On such systs, location stiats using TDOA tchniqus suffr u to inconsistncis in anufacturing an oprating conitions. W hav prsnt for low banwith, highly rsourc constrain ral-ti b systs, th rlat tchniqus of Dopplr frquncy an ti shifting to copnsat for th avrs ffcts of iffrnt sapling frquncis on TOA/TDOA stiats. Whil th Dopplr shift corrction is foun

7 to b or accurat, th ti-shift tchniqu is or lucrativ fro a prforanc prspctiv. Histogra of lay rrors Ti shift corrction nubr of stiats Histogra of lay stiats no corrction ti lay rror in ns Mor Frquncy Figur 3 histogra of rror in TDOA stiats without any corrction appli Nubr of stiats Histogra of lay stiats Dopplr Shift corrction Mor Ti lay rror in ns Frquncy Figur 4 Histogra of rrors in TDOA stiats with Dopplr shift corrction, Rfrncs. Warrior, J t al; Thy know whr you ar ; IEE. Spctru, Vol.4, no.7, Jul 3; pp Kuar, S., Shphr, D.; SnsIT: Snsor inforation tchnology for th war fightr ; Proc. Intr. Conf. on Inforation Fusion, Vol., 3. Bahl, P t. al.; RADAR: an in-builing RF-bas usr location an tracking syst, INFOCOM, 9 th Ann. Jt. Conf. IEEE Cop. an Co. Soc. Proc. IEEE, Vol.,, pp Dran, C. t. al. Positioning GSM tlphons IEE. Co. Mag. Vol.36, Iss.4 Apr 998 pp , Stotts, L.B.: Unattn groun snsor rlat tchnologis; an ary prspctiv, Proc. SPIE. Vol. 44. ( 6. Knapp, C., Cartr, G.; Th gnraliz corrlation tho for stiation of ti lay ; IEE. Trans. Acous., Spch. Sig. Proc. Vol.4, Aug 976 pp nubr of stiats lay rror in nano sc Mor Frquncy Figur 5 Histogra of rrors in TDOA stiat whn ti-shift corrction is us Tabl Man an stanar viation of rrors in TDOA stiats Corrction typ Man rror (n St v. (n sc sc Non Dopplr shift Ti Shift Proakis J; Manolakis, D; Digital Signal Procssing, Principls Algoriths, an Applications, Prntic Hall, Thir E., Btz, J.; Effcts of uncopnsat rlativ ti copaning on a broa-ban cross corrlator IEEE Trans. Acous., Spch, Sig. Proc, Vol.33, Jun Rly, W.; Corrlation of signals having linar lay, Acoust. SOC. Ar., vol. 35, pp , Jan W. B. Aas t al.; Corrlator copnsation rquirnts for passiv ti lay stiation with oving sourc or rcivrs ; IEE. Trans. Acoust., Spch, Signal Procssing, vol. 8, pp.58-68, Apr Wiss, L.G.; Wavlts an wiban corrlation procssing, IEE. Sig Proc Mag Vol., Jan 994 pp3-3. Stin, S; Algoriths for Abiguity function procssing ; IEE. Trans Acou., Spch. & Sig. Proc. Vol.9, Iss.3, Jun 98 pp Ulan, R.; Graniotis, E; Wiban TDOA FDOA procssing using suation of short-ti CAF's, IEE. Trans. Sig. Proc. Vol.47, no, Dc Dharia A. t al; Signal Procssing Exapls Using th TMS3C67x Dig. Sig. Proc. Library (DSPLIB, Txas Instrunts, App. Rpt SPRA Wlch, P.; A tho bas on ti avraging ovr short, oifi prioogras; IEE. Transactions on Auio an Elctro., Vol.5, Iss., Jun 67 pp Libniz Intgral Rul, athworl.wolfra.co