DATA COMPRESSION FOR EMITTER LOCATION

Transcription

1 Coneence on Inomaton Scences and Systems, Pnceton Unvesty, Mach 5-7, DATA COMPRESSION FOR EMITTER LOCATION Mak L Fowle Depatment o Electcal Engneeng State Unvesty o New Yok at nghamton nghamton, NY 9 mowle@bnghamtonedu ASTRACT Locatng emttes by coss-coelatng eceved sgnals to compute the tme-deence-o-aval (TDOA) and the euency-deence-o-aval (FDOA) eues that sgnal data eceved at one platom be tanseed to the othe platom Oten the data lnk used has nsucent bandwdth to accomplsh the tanse wthn the tme euement, and theeoe use o data compesson s needed Ths pape outlnes a useul pogesson n compesson technues om those that consde only mean suae eo to those that consde the tue mpact on the estmated TDOA/FDOA accuacy Wthn ths context, specc esults ae pesented o two compesson appoaches o TDOA/FDOA systems The st s the applcaton o block-adaptve uantzes (AQ) to the eal and magnay pats o the complex baseband sgnal to be tanseed The second uses a wavelet tansom togethe wth an adaptvely allocated set o uantzes; also, cetan wavelet coecents can be elmnated wth lowe mpact on the TDOA/FDOA accuacy than expected om a mean-suae uantzaton pont o vew INTRODUCTION An eectve way to locate electomagnetc emttes s to measue the tme-deence-o-aval (TDOA) and the euency-deence-o-aval (FDOA) between pas o sgnals eceved at geogaphcally sepaated stes []-[] The measuement o TDOA/FDOA between these sgnals s done by coheently coss-coelatng the sgnal pas [], [], and eues that the sgnal samples o the two sgnals ae avalable at a common ste, whch s geneally accomplshed by tanseng the sgnal samples ove a data lnk om one ste to the othe ste An mpotant aspect o ths that s not wdely addessed n the lteatue s that oten the avalable data lnk ate s nsucent to accomplsh the tanse wthn the tme euement unless some om o lossy data compesson s employed Fo the case o Gaussan sgnals and noses, the esults n [4] establsh bounds on the ate-dstoton peomance o the TDOA/FDOA poblem and compae them to the peomance achevable usng scala uantzes, whee dstoton s measued n tems o lost due to compesson Howeve, these esults ae not applcable when locatng ada and communcaton emttes because the sgnals encounteed ae not Gaussan The two nosy sgnals eceved ae complex-valued baseband sgnals gven by sˆ( s( k ) + n( k ) [ s ( + js ( ] + [ n ( + jn ( ] dˆ( d( + v( [ d ( + jd ( ] + [ v ( + jv ( ] whee s( and d( ae the complex baseband sgnals o nteest and n( and v( ae complex whte Gaussan noses, each wth eal and magnay pats notated as ndcated The sgnal d( s a delayed and dopple shted veson o s( The sgnal-tonose atos () o these two sgnals ae denoted and DNR, espectvely To coss coelate these two sgnals one o them (assumed to be s ˆ( hee) s compessed, tanseed to the othe ste, and then decompessed beoe coss-coelaton Ate lossy compesson, sgnal s ˆ( has o, and the ate coss-coelaton s gven by cc WT + + DNR DNR whee WT s the tme-bandwdth poduct (o coheent pocessng gan), wth W beng the nose bandwdth o the eceve and T beng the duaton o the eceved sgnal [] Note that the post-coelaton s domnated by the smalle o and DNR The accuaces o the TDOA/FDOA estmates ae whee ms Dms στ π σ ν π D ms ms s the sgnal s ms (o Gabo) bandwdth n Hz, cc cc, () s the sgnal s ms (o Gabo) duaton n seconds []; we (non-talc) epesents an aconym o sgnal-to-nose ato; (talc) epesents the o s ˆ( WA7b- 4

2 wll ee collectvely to these two sgnal paametes as the sgnal s ms wdths In ths pape we pesent esults o two deent compesson appoaches o TDOA/FDOA estmaton The st, called R/I uantzaton, apples block-adaptve uantzes (AQ) to the eal and magnay pats o the sgnal [7], and the second uses a wavelet tansom togethe wth an adaptvely allocated set o uantzes [8] A AQ technue has been appled to the compesson o synthetc apetue aay (SAR) sgnals [5], whee blockadaptve uantzes desgned o Gaussan sgnals ae appled to the eal and magnay components Howeve, o the TDOA/FDOA case consdeed hee, the sgnal s pd s not known no s t expected to be Gaussan Thus, st, we deve bounds on the peomance o the AQ technue when appled to ada and communcaton sgnals and nvestgate the mpact on TDOA/FDOA estmaton Smulaton esults ae compaed to these bounds to show that they ae ndeed vable Second, we develop a wavelet-based algothm and demonstate though smulatons that t outpeoms the R/I uantzaton appoach o sgnals o nteest Fnally, the wavelet-based method s bely dscussed n a settng that consdes a tadeo between uantzaton and decmaton o TDOA/FDOA emtte locaton Ths dea s genealzed to the ollowng: cetan wavelet coecents can be elmnated wth less mpact on the TDOA/FDOA accuacy than expected om a mean-suae uantzaton eo pont o vew R/I QUANTIZATION METHOD The R/I uantzaton methods decease the numbe o bts used to epesent sgnal samples by convetng each sample to a om that s moe coasely uantzed; ths s done on a blockby-block bass by scalng the eal and magnay samples n a block by scale actos γ and γ, espectvely, and then oundng to the desed numbe o bts Fo analyss puposes we consde that the compessed sgnal s unscaled beoe cosscoelaton We consde two ways o scalng: () Multply- Scalng: multplyng the samples n a block by an appopate postve acto chosen to set to s all the non-sgn bts o the lagest-magntude sample n the block, and () Sht-Scalng: let-shtng the bts o the samples n a block by an amount chosen to sht the most-sgncant o the lagest-magntude sample n the block nto the MS poston Thus, sht-scalng s multply-scalng wth a scalng acto that s a powe o two The uantzed sgnal can be consdeed to have a b + bt epesentaton o the om S Lb, whee S denotes the sgn bt, denotes the bnay pont, and the s denote the b bts used to epesent the magntude b Then σ / s the vaance o the uantzaton nose [6] Ate scalng and uantzng the eal pat o s ˆ(, we get s ~ ( s ( + n ( k ) + e( / γ whee e (k ) s the uantzaton nose; a smla expesson o the magnay pat can be wtten Usng ths and denng the sgnal peak-to-ms paametes { s ( } max{ s ( k )} max P s P s t s possble to show [7] that o the sgnals o nteest and that the o the complex sgnal ate compesson and decompesson s bounded by < + Fom ths we see that the degadaton due to uantzaton s moe ponounced when s lage o when s lage (o both); o couse, when s lage a lage loss mght be moe easly toleated by the system Note that lage values o educe the bound on the post-uantzaton ; lage coesponds to sgnals that ae peaky such as speech, whle small coesponds to sgnals that ae not such as FM sgnals Fo multply-scalng ths uppe bound can be vey nealy acheved Howeve, when usng sht-scalng thee s no guaantee that ths bound wll be even close to beng acheved Shtscalng can only assue that the scaled veson o the lagest sgnal sample les n [ /,), whch leads to the lowe bound gven n + 4 < + These bounds can be used to establsh bounds on the coss-coelato output wth uantzaton that, togethe wth smulatons, show that t s possble to educe the uantzaton to 4 bts pe eal pat sample and 4 bts pe magnay pat sample and sue neglgble educton n output (see Fgue ) and n TDOA/FDOA accuacy (see Fgue dscussed late) Fewe than 4 bts has been ound to be unsatsactoy due to the excessve nonlneaty o the uantzaton Fgue shows the theoetcal and smulated esults o both the uantzed and the nonuantzed cases o a smulated sngle-sdeband sgnal wth 4 Fo the case shown t s clea that thee s lttle mpact on the output ; ths s tue n () WA7b- 5

3 geneal when P s / σ SQR s kept hghe than ethe o DNR Ths s easly seen by ecognzng that so that the output s + SQR T cc SQR DNR DNR SQR DNR Thus, cc s domnated by the lowest o, DNR, and SQR Thus, when ethe (o both) o the sgnals has low, the addtonal mpact o modeate uantzaton s neglgble I both sgnals have hgh then even modeate uantzaton has a sgncant mpact, but that s a case whee pehaps the degadaton can be toleated WAVELET TRANSFORM METHOD The wavelet tansom compesson algothm [8] conssts p o beakng the sgnal nto blocks o N samples, applyng an L-level wavelet tansom to each block o L<p (e, stoppng the cascade o wavelet tansom lte bank stages at the level whee the lte outputs have goupng the esultng N wavelet coecents nto K L subblocks o N L N N / elements), samples each, and adaptvely uantzng each o these subblocks Fo the complex baseband sgnals used hee, ths pocedue s appled ndependently to the eal and the magnay components The subblocks o the wavelet coecents ae omed wthn wavelet scale levels as ollows: the N/ wavelet tansom coecents om the st lte bank stage ae gouped nto L subblocks o coecents each, the N/4 wavelet tansom coecents om the second lte bank stage ae gouped nto L subblocks o coecents each,, and nally the wavelet tansom coecents om the last lte bank stage om a sngle subblock, and the scalng coecents om the last stage also om a sngle subblock Each one o these subblocks s uantzed wth a uantze desgned to acheve the desed level o uantzaton nose The choce o these uantzes s made easy by the act that the wavelet tansom peseves enegy; ths popety can be used to show that the pope choce o the uantze cell wdth s gven by P SQR x, whee SQR s the desed sgnal-to-uantzaton nose ato and s the powe o the nput sgnal x(n) (n ths case, ethe Px s sˆ ( k ) o ˆ ( k ) ) Thus, to obtan a desed SQR, the uantzes { Q,Q, QK } should each have a uantzaton step sze gven by Then the numbe o bts k used by the k th uantze s chosen to assue that the esultng uantze coves the ange o the k th subblock Ths leads to the ule k ( [ max{ k W } ]- log + ), log x whee the maxmum s taken ove the wavelet coecents n the k th block and the symbol a means Athe smallest ntege not less than that s lage than a ;@ ths means that when the expesson n paentheses n the euaton o k s negatve we set k In addton to sendng the uantzed wavelet coecents, ths scheme eues sendng sde nomaton to the eceve about the numbe o bts used o each uantze as well as the step sze used I the maxmum numbe o bts used by any o the subblocks s max, then the allowable uantzes ae those that use between and max bts, o a total o max + deent uantzes; the numbe o bts eued to specy whch o these s used o a specc subblock s log ( + ) bts Snce ths must be done o each o the K max K max subblocks, we eue log ( + ) bts o sde nomaton; sde nomaton on the uantze step sze also must be sent, whch wll be no moe than the numbe o bts to whch the ognal sgnal s uantzed (we have assumed 8 bts hee) So the total amount o sde nomaton s R sde K x log ( max +) +8 (bts) Smulatons have shown that t s possble to lmt max to 7 bts In ths appoach, the wavelet tansom s used togethe wth bt allocaton to povde a means o educng the numbe o bts pe (eal o magnay) sample wth neglgble degadaton o the TDOA/FDOA accuacy Ths scheme accepts a specc desed sgnal-to-uantzaton ato (SQR) and attempts to mnmze the numbe o bts needed to acheve that SQR value In pactce, the desed SQR can be set ethe () to be oughly eual to the estmated o the sgnal to ensue that the mpact o the compesson on the TDOA/FDOA accuacy s neglgble, o () to some xed a po value An algothm paamete that can be adjusted s called mn ; t s possble to set all values o k, as detemned above, that ae below some speced value mn to zeo Ths helps to elmnate wavelet coecents that contan only nose, and thus WA7b- 6

4 helps to educe the amount o nomaton that must be tansmtted Inceasng mn causes a lage numbe o coecents to be set to zeo and can theeoe ncease the compesson ato wth only a small mpact on accuacy The R/I uantzaton and wavelet compesson methods descbed above have ocused on mnmzng the mean-suae eo (MSE) due to compesson Howeve, because the goal s to estmate TDOA/FDOA, the mnmum MSE cteon s not the most appopate one because t als to ully explot how the sgnal s stuctue mpacts the paamete estmates ecause TDOA/FDOA accuacy depends not only on but also on the sgnal s ms bandwdth and ms duaton (see ()), compesson appoaches that can educe the amount o data whle neglgbly mpactng the sgnal s ms wdths ae desed Accodngly, one ntent o nceasng mn n the wavelet method s to emove small wavelet coecents that may contbute nsgncantly to the sgnal s ms wdths The wavelet tansom appoach s a natual tool to enable emovng tme-euency components o the sgnal that contbute vey lttle to the sgnal s ms wdths These deas ae natual genealzatons o esults we have obtaned to detemne the coect balance between uantzaton and decmaton o data compesson o TDOA systems, whch we pesent hee to llustate the potental o a moe geneal wavelet-based appoach We now nvestgate ths tade-o between decmaton and uantzaton Assume that we uantze the complex sgnal samples usng bts ( o the eal pat, o the magnay pat) and that we sample t at Fs complex samples/second I the sgnals ae collected o T seconds, then the total numbe o bts collected s TFs System euements oten specy a xed length o tme, T l, o the data tansmsson ove the lnk at a ate R l bts/second Then the total numbe o bts collected must be able to be sent ove the lnk at ate R l n no moe than lnk tme T l ; thus, the constant to satsy s TFs RlTl Euvalently, we dene R Rl Tl / T as an eectve ate and assume eualty n the constant we get R F s () Now consde that the eceved sgnals ae lteed and decmated to a bandwdth o W (now F s W o ctcal samplng o the complex sgnals ate decmaton) and assume that the sgnals specta ae lat so that the two s don t depend on W Ate lteng and decmaton, the sgnal to be tans- mtted s uantzed usng bts pe complex sample ( bts o the eal pat and bts o the magnay pat ) The esult s that the decmated and uantzed sgnal has, om (), appoxmate gven by ( ) + and the output then depends on the lteed bandwdth and the uantzaton level accodng to ( W ) o, W whee DNR s the o T (), (4) W T + + () DNR () DNR e (5) dˆ (, the sgnal that s not uantzed Usng Euaton (5) n Euaton () gves a bound on TDOA accuacy that depends on the bandwdth ate decmaton ( W Hz) and the uantzaton level ( bts), that s Usng W R/ om the Aate-bandwdth-bts@ constant n () wth σ τ 8 W / T e () F W gves s / [ / b e ( ) ], σ τ () (6) 8 / R T whee t s eally the backeted tem that s o nteest hee, snce t shows the tadeo between decmaton and uantzaton It s mpotant to emembe that (6) ncludes the atebandwdth-bts constant, so o a xed R, nceasng necessaly deceases W, and vce vesa The nonbacketed tem n (6) just scales the esult up o down dependng on the values o the system paametes eectve ate R and collecton tme T Howeve, one mpotant nsght does come om the st tem: the bound on στ vaes as the -/ powe o the ate R ; thus, you double the allowable ate you get almost thee tmes bette accuacy, and you uaduple the allowable ate you get eght tmes bette accuacy The eason that nceasng the data ate mpoves the accuacy s because we have constaned the tme avalable to tansmt the data, so nceasng the data ate allows an ncease n the amount o nomaton about the sgnal that can be tansmtted Ths s an mpotant nsght nto the system desgn ssues To compute the backeted tem n (6) we st compute () usng (4) Then t s used n (6) to compute the backeted tem o a patcula set o values o the paametes WA7b- 7

5 á,,, and DNR Plots o the backeted tem n (6) vesus, paametezed by á,, and DNR eveal the pope way to choose the optmal value o ; that s, how to tadeo decmaton and uantzaton Snce the value o R does not aect these cuves, the optmal level o uantzaton s not set by the allowable data ate Instead, the optmal degee o uantzaton s set by the nteplay between the s o the two sgnals and the peak acto o the sgnal to be uantzed Once ths level o uantzaton s detemned, the appopate amount o decmaton s dened by detemnng the allowable amount o bandwdth by solvng () o the samplng ate F s gven the allowable data ate and the numbe o bts detemned above Fgue shows a plot o the backeted tem n (6) o the case o eectve ate R kbps, sgnal W MHz, ognal 8 bts, 5,, DNR The desed opeatng pont s whee the cuve s at a mnmum; howeve, note that o ths case the mnmum eues use o a vey small numbe o bts, whee the theoy o uantzaton used hee beaks down Thus, n ths case we would choose 4 bts as the best opeatng pont and the sgnal would be decmated to 5 Hz Fo that choce the esultng σ τ s 7 tmes lowe than that o decmaton only; note that uantzaton alone can not meet the lnk euements o ths case I nstead the sgnals had (eg, and FM sgnal),, and DNR the esultng σ τ would be nealy tmes lowe o combned uantzaton and decmaton compaed to decmaton alone Ths nvestgaton shows that compesson algothms that balance uantzaton nose and the educton o ms wdths can be moe eectve than just uantzaton In othe wods, to undestand the peomance o the wavelet technue we must eally consde how the uantzaton s zeong o the coecents mpacts the sgnals ms wdths, and to mpove ts peomance we should seek to zeo-out wavelet coecents that contbute nsgncantly to the sgnal s ms wdths 4 SIMULATION RESULTS Smulatons ae used to demonstate the peomance o the wavelet tansom usng adaptve uantzaton method These smulatons also made use o the compesson-coecton method poposed n [9], n whch po to sendng the compessed sgnal t s coss-coelated wth ts ognal veson and the locaton o the peak o ths coelaton suace s then sent to the othe platom whee t s subtacted om the peak locatons o the suace computed thee Such an appoach s vey eectve at emovng bas mpated by the compesson method The esults pesented hee ae o the case o a ada pulse tan whose samples between pulses have been emoved by a pe-compesson detecton pocedue; tmng pontes ae also sent to allow eassemblng the pulses nto the ognal tmng elatonshps The pulse tans ae complex baseband lnea FM sgnals havng a pulse wdth o 4 µs and a euency devaton o ±7 MHz, and conssted o 496 samples geneated at 4 MSPS usng 8 bts/sample o the eal samples and 8 bts/sample o the magnay samples The sgnal that was not compessed had an o DNR 4 d; the sgnal that was compessed had s po to compesson n the ange,4 d [ ] The R/I uantzaton scheme used multply scalng on blocks o 8 samples and uantzed the samples to 4 bts/sample o the eal samples and 4 bts/sample o the magnay samples The wavelet tansom method used a tansom havng sze o N 48 and L 8 levels Thus, the numbe o subblocks pe tansom was 56, each havng 8 samples pe subblock The values SQR d and mn wee used Fgue shows thee plots Each plot shows thee cuves: no compesson, wavelet tansom (WT) compesson, and R/I uantzaton The st two plots show the acheved TDOA and FDOA accuaces, espectvely, as a uncton o the compessed sgnal s o the R/I and wavelet tansom (WT) methods The thd s a plot o the acheved compesson atos vs the compessed sgnal s The mpact o R/I uantzaton on the TDOA/FDOA accuaces can be neglgble when a : compesson ato (4 bt uantzaton) s used The wavelet method, howeve, can acheve a much lage compesson ato but at the expense o about 5% lage TDOA eo on but vtually no degadaton n the FDOA eo 5 REFERENCES [] P C Chestnut, Emtte locaton accuacy usng TDOA and deental dopple, IEEE Tans Aeo and Electonc Systems, vol AES-8, pp 4-8, Mach 98 [] S Sten, Deental delay/dopple ML estmaton wth unknown sgnals, IEEE Tans Sg Poc, vol 4, pp 77-79, August 99 [] S Sten, Algothms o ambguty uncton pocessng, IEEE Tans Acoust, Speech, and Sgnal Pocessng, vol ASSP-9, pp , June 98 [4] D J Matthesen and G D Mlle, Data tanse mnmzaton o coheent passve locaton systems, Repot No ESD-TR-8-9, A Foce Poject No 4, June 98 [5] U enz, K Stodl, and A Moea, A compason o seveal algothms o SAR aw data compesson, IEEE Tans Geosc Remote Sensng, vol, pp 66-76, Sept 995 [6] Y Neuvo, Dgtal lte mplementaton consdeatons, n S Mta and J Kase (Eds), Handbook o Dgtal Sgnal Pocessng, New Yok: John Wley & Sons, 99 [7] M L Fowle, Coase uantzaton o data compesson n coheent locaton systems, unde evson o IEEE Tansactons on Aeospace and Electonc Systems [8] M L Fowle, Data compesson o TDOA/DD-based locaton system, US Patent #5,99,454 ssued Nov, 999, Lockheed Matn Fedeal Systems WA7b- 8

6 [9] G Desjadns, TDOA/FDOA technue o locatng a tansmtte, US Patent #5,57,99 ssued Oct 9, 996, Lockheed Matn Fedeal Systems Peomance Facto Sold: Theoetcal ounds - Quantzed Dash-Dot: Smulaton - Quantzed Dashed: Theoy - Not Quantzed Ccles: Smulaton - Not Quantzed Fgue : ounds and Smulaton Results o Output usng R/I Appoach b_ Nomalzed Delay Eo Nomalzed Dopple Eo Compesson Rato 5 5 R/I WT No Compesson SQR d DNR 4 d (d) WT SQR d DNR 4 d (d) R/I No Compesson (d) SQR d DNR 4 d b_d (bts) Fgue :Smualtons o TDOA/FDOA Accuacy usng R/I Quantzaton and Wavelet Method Fgue : "Peomance Facto" o Decmaton vs Quantzaton Tadeo WA7b- 9