Data Compression for Multiple Parameter Estimation with Application to TDOA/FDOA Emitter Location

Transcription

1 Data Compresson for Multple Parameter Estmaton wth Applcaton to TDOA/FDOA Emtter Locaton Mo Chen and Mark L. Fowler Department of Electrcal and Computer Engneerng State Unversty of New York at Bnghamton Bnghamton, NY 390 Abstract Because a prmary task of mult-sensor systems s to make estmates based on the data collected and shared throughout the sensor system, t s mportant to desgn data compresson methods that reduce the volume of data to be shared whle causng only mnmal degradaton of the qualty of these estmates. An mportant aspect not specfcally consdered prevously n compresson research for sensor systems s that sensor systems generally have to make multple estmates from the data. Furthermore, t s unrecognzed n the lterature that these multple estmates generally have conflctng compresson requrements and that fndng the rght way to balance these conflcts s crucal. The key tools we brng to bear on ths area are: () the use of a Fshernformaton-based dstorton measure that s desgned specfcally for multple estmates, and () the use of numercal optmzaton methods to acheve desred compresson trade-offs among the multple estmates. We frst develop results that support usng the trace of the Fsher nformaton matrx (FIM) as a dstorton measure for the smultaneous multple-parameter estmaton problem. We then apply these results to the problem of TDOA/FDOA-based locaton estmaton of an RF emtter. We show that wthn ths problem there s a fundamental trade-off between the mpact of compresson on TDOA estmaton and the mpact of compresson on FDOA estmaton; furthermore, we show that ths trade-off can be addressed by adaptng the compresson scheme to the sensor-emtter geometry usng a geometry-adaptve compresson scheme. Index Terms: Data Compresson, Estmaton, Fsher Informaton, TDOA/FDOA, Emtter Locaton, Electronc Warfare, Sensor Networks Correspondng Author: Mark Fowler Department of Electrcal and Computer Engneerng State Unversty of New York at Bnghamton P. O. Box 6000 Bnghamton, NY Phone: Fax: E-mal: mfowler@bnghamton.edu Ths work was supported n part by the Ar Force Offce of Scentfc Research through Grant number FA and through the Ar Force Summer Faculty Fellowshp Program.

2 Page of Mark Fowler From: Sent: Monday, September 9, 008 6:0 PM To: Cc: Subject: TAES RR Decson Letter "Data Compresson for Multple Parameter Estmaton wth Applcaton to TDOA/FDOA Emtter Locaton Systems" Dear Prof. Fowler: On behalf of the IEEE Transactons on Aerospace and Electronc Systems (TAES), I am pleased to accept your above referenced manuscrpt for publcaton. The IEEE AESS Edtoral Offce wll contact you wth the requrements for your Fnal Submsson Package, respondng n a tmely manner wll mnmze publcaton delays. Thank you for your contrbuton to the IEEE Transactons on Aerospace and Electronc Systems. Sncerely, 9/30/008

3 I. Introducton Because a prmary task of sensor systems s to make estmates based on the data collected throughout the sensor system, t s mportant to desgn compresson methods that can sgnfcantly reduce the nter-sensor communcaton cost whle causng mnmal degradaton of the qualty of these estmates. Although data compresson for dstrbuted sensor systems has been prevously consdered to some degree for sngle estmate tasks, an mportant aspect consdered here but not prevously consdered s that sensor systems often have to make multple smultaneous estmates from the collected data. A further new aspect addressed here s that multple estmates generally have conflctng compresson requrements and fndng the rght way to balance these conflcts can be crucal. To provde a clear focus on these ssues as well as to demonstrate relevance of the results, we apply these deas to the problem of locatng an RF emtter usng tme-dfference-of-arrval (TDOA) and frequency-dfference-of-arrval (FDOA) measurements [] [3]. A key to developng compresson for the multple-estmaton problem s to use a dstorton measure that accurately reflects the mpact of compresson on estmaton. Classcal compresson methods for nonaudo/vdeo sgnals are typcally based on mnmzng the sgnal mean-square error (MSE): N MSE= xn xn N ( [ ] [ ]) where x[n] s the orgnal sgnal and x[ n] n= 0 s the de-compressed verson of the sgnal. However, for estmaton tasks the sgnal MSE s only part of what determnes the accuracy t only captures the mpact of the compresson on the SNR. Alternatvely, usng Fsher nformaton (FI) as a dstorton measure (to be maxmzed) allows the compresson algorthm to better assess the mportance to the estmaton task of each pece of sgnal data some peces are mportant and should be allocated more bts, whle other peces are less mportant and can be allocated fewer bts (or maybe even none) whle stll capturng the SNR s role (because FI typcally depends on SNR). For example, n [] we showed that for the task of estmatng TDOA t s possble for a weak DFT coeffcent far from the sgnal s center band to be more mportant than a strong DFT coeffcent near the sgnal s center band. Yet, usng MSE dstorton would lkely allocate fewer

4 bts to a weak DFT coeffcent and more bts to a strong one regardless of each coeffcent s frequency. See our earler paper [] for further dscusson. Although Fsher nformaton has been appled before as a dstorton measure (e.g., [],[3],[4],[5],[6]), our nterest here s to explore the trade-off ssues that arse n the multple-estmate problem. Some past results are avalable on compresson for the mult-parameter case of TDOA/FDOA estmaton [4],[5],[6],[7],[8],[9],[0],[]; only our prelmnary work n [0] addresses the trade-offs between compresson for TDOA/FDOA and the work presented here s an extenson of that. Matthesen and Mller [4] consdered the case of whte Gaussan sgnals (unrealstc for most RF emtters) and establshed rate-dstorton bounds (where dstorton s sgnal MSE) for scalar quantzaton n the TDOA-only problem. A method usng block-adaptve scalar quantzaton was proposed [5] and analyzed [6], but stll assessed the mpact of the compresson only through ts effect on sgnal MSE dstorton. Breakng from the sgnal-mse-based dstorton approach, we used the Cramer-Rao lower bound (CRLB) to better capture the effect compresson would have on TDOA accuracy [7]. However, we found [8] that optmzng ths CRLB-based dstorton measure was dffcult even for the TDOA-only case. Recently we have shown [] that a Fsher-nformaton-based approach avods these dffcultes and we developed a method for the general sngle-parameter estmaton problem; applcatons to TDOA-only and FDOA-only were consdered as specal cases. Another recent effort [9] used an alternatve dstorton vewpont specfcally developed for the TDOA-only problem, whch re-cast the TDOA estmaton problem nto a decson problem and desgned a tme-doman scalar quantzer that maxmzes the mutual nformaton between the receved sgnals. Many researchers have addressed compresson for general parameter estmaton; some major examples are [],[3],[4],[5],[6],[7]. Most of these restrct the encoder to a scalar quantzer [],[3],[4],[7] rather than the more powerful transform codng framework. Only [5] and [6] consder general compresson structures and multple parameters, but they mostly focus on establshng nformaton theoretc results and bounds and don t explore the trade-offs nvolved among parameters. In short, the problem of developng

5 3 compresson algorthms to address the trade-offs nherent n smultaneous multple estmaton tasks has not prevously been addressed. Fsher nformaton s a well-known concept n estmaton theory [8] that captures the qualty of data relatve to a parameter estmaton problem. For many practcal estmaton problems the FI does not depend on θ; however, n general the FI can depend on the parameter vector values, whch makes t unclear how to use t for a compresson measure when t s the parameter vector that s to be estmated. However, for such a case a mnmax approach (as suggested n []) can be approprate, although we won t pursue that avenue here. Alternatvely, we propose to frst get a rough estmate of the parameters usng a small amount of data, whch are then used to evaluate the Fsher nformaton. In Secton III-A we consder the problem of estmatng TDOA/FDOA, whose Fsher nformaton does not depend on the parameters to be estmated. However, n Secton III-B when we consder compresson for the full geo-locaton problem we fnd that the geo-locaton Fsher nformaton does depend on the unknown parameter (e.g., the locaton); we use the small amount of data approach. For the sngle parameter case, the FI s a scalar and has been used as a dstorton measure for data compresson: [3] and [4] derve scalar quantzers that maxmze the scalar FI; n [] we derved transformcodng schemes that maxmze the scalar FI. However, when multple parameters are to be estmated the FI becomes the Fsher nformaton matrx (FIM) and t s not obvous how to use ths matrx as a dstorton functon. The determnant of the FIM has been used as an objectve functon to be maxmzed n the soluton of engneerng problems (see e.g., [],[],[3]). Ths s an ntutvely pleasng approach but has two drawbacks for our applcaton: () t doesn t allow trade-offs between the parameters and () t doesn t have the form of an addtve dstorton measure and therefore s hard to optmze. In fact, the latter drawback lmted the use of the determnant n [] and [] to a suboptmal approach. See [33] for further dscusson of the use of the determnant as a dstorton measure. Instead of usng the determnant of the FIM we advocate the use of the weghted trace of the FIM because t enables the compresson algorthm to make trade-offs between the compresson mpact on the qualty of the

6 4 multple parameters. Ths need for trade-offs was motvated by our nterest n compresson for TDOA/FDOA emtter locaton, where the mportance of TDOA accuracy vs. FDOA accuracy depends on the geometry between the sensors and emtter: n some geometres a large degradaton n FDOA accuracy (n TDOA accuracy for some other geometres) wll cause no loss n locaton accuracy. Thus, n these cases we need not spend compresson bts n an effort to mnmze degradaton n FDOA accuracy (n TDOA accuracy n some other cases); ths then should reduce the amount of data needed to acheve a desred level of locaton accuracy. In ths paper we frst develop a general methodology that uses a weghted trace of the FIM to allow compresson trade-offs between multple parameters and then show how ths methodology provdes a sgnfcant mprovement n compresson performance for TDOA/FDOA emtter locaton applcatons. For concseness we consder the problem of collectng data at Sensor # that s compressed and sent to Sensor # where t s decompressed and then used wth Sensor # data to perform the estmaton tasks; another scenaro that could arse s sendng data to some central processng center where estmaton processng s done ether wth the compressed data alone or n conjuncton wth other data at the processng center. Our nterest n TDOA/FDOA locaton drves us to consder the former case. The compresson scheme s transform codng that uses a lnear nvertble (preferably orthogonal) transform whose output coeffcents are quantzed by a set of quantzers {Q }; for the TDOA/FDOA locaton problem we wll focus on the case where ths transform s a tme-frequency representaton of the sgnal, whch ncludes varous types of flter banks as a sub-category. Each transform coeffcent s evaluated va the FIM to assess ts mportance for the estmaton tasks and then bts are allocated among quantzers on that bass. The man challenge we address s to fnd methods to properly determne the optmal allocaton of the bts to enact the desred trade-offs. These trade-offs are then exploted to allow the compresson to be adapted to the geometry between sensors and emtter. The paper s organzed as follows. Secton II addresses how a weghted trace of the FIM can be used as a dstorton measure n a compresson algorthm for the general smultaneous mult-parameter estmaton problem that enables trade-offs among the parameters. Secton III apples these deas to the problem of

7 5 compresson for smultaneous TDOA/FDOA estmaton for emtter locaton. Secton IV presents conclusons of the work. II. Compresson for Smultaneous Estmates A. Transform Codng Framework The transform codng framework we use here s the same one we used n []; we provde some detals to support our further development here. Suppose we have two sensor nodes that ntercept a common sgnal and wsh to share data to estmate a p parameter vector θ. At sensor node S k we model the receved N sgnal vectors x k as x = s ( θ) + w, k =,, () k k where s k (θ) s an unknown determnstc N sgnal vector dependent on the unknown determnstc parameter p vector θ, and w k s a zero-mean, Gaussan nose vector (N ) wth w and w ndependent. We assume the covarance matrx Σ of w s known or estmated. Here we use an orthonormal (ON) transform for H N the lnear nvertble transform. The untary matrx of ths transform wll be denoted as Φ wth rows { φ n } n= that form an ON bass. Expandng data vector x wth respect to ths ON bass gves the coeffcents k N { n} n= χ. The coeffcents χ are then quantzed usng a set of bt allocatons = [ b b b ] n N b. The postcompresson sgnal s x = χˆn φ ˆ n n. Groupng these quantzed coeffcents nto vector form gves χˆ = Φx = ξ ( θ) + ω = ξ ( θ) + ν + ε, () where vector ξ θ) = Φs ( ) holds the selected sgnal coeffcents ξ n (θ ), vector ω holds the correspondng ( θ nose coeffcents ω n and has covarance matrx Σ, and ε s the quantzaton nose vector, assumed ndependent of the recever nose ω, wth known covarance matrx Q = dag{ q, q, q }, N. We proceed as we

8 6 dd n our earler paper and use a Gaussan approxmaton for the PDF of ν for the case of mult-bt quantzaton and use an exact form for -bt quantzaton []. The allocaton of bts to the coeffcents s chosen to optmze a dstorton measure subject to a rate constrant b n R where R s the total number of bts avalable for codng. Classcally, the dstorton n measure s chosen to be sgnal MSE; here we use an estmaton-approprate measure based on the Fsher nformaton matrx. B. Fsher Informaton Matrx as a Dstorton Measure To smplfy the dscusson, but wthout loss of generalty, we wll focus on the two parameter case. As n [] we have assumed that the sensor noses are ndependent and therefore the total FIM s the sum of the FIMs at the two sensors; thus, we only need to consder the effect of compresson on the FIM of the data at sensor #. The FIM for θ based on the data from sensor # s the matrx J wth elements gven by Re{ H s J= G Σ G } where = s θ θ G s an N matrx of the sgnal s senstvtes to the parameters [8]. = The FIM specfes an nformaton ellpse va J θ θ T wth sem-axes along the FIM s egenvectors and whose lengths are proportonal to the square roots of the egenvalues; the larger ths ellpse the better the data supports the estmaton. Lossy compresson of the data vector x makes the data nferor for estmaton of θ and thus degrades the FIM, whch shrnks and (perhaps) rotates the nformaton ellpse. From the model n () and the fact that the FIM s nvarant under transformaton of the data by a untary matrx, the post-compresson FIM s gven by Jˆ J = ˆ J where the ^ symbol ndcates after compresson and Jˆ Jˆ = Re ~ H ~ { G [ Σ + Q] G} ˆ ~ G = ΦG., (3) In the sngle-parameter case [] the way to proceed was clear: compress the sgnal to a gven bt budget R whle maxmzng Ĵ (the only FIM element for the sngle-parameter case). But how should we proceed n the

9 7 two-parameter case? What does t mean to maxmze a matrx? To answer ths we explot the nformatonellpse nterpretaton of the FIM. Maxmzng the area of the FIM ellpse seems desrable; however, we wll see that ths does not allow trade-offs on the mpact of compresson between the two parameters. Alternatvely, maxmzng the permeter of the FIM ellpse seems attractve; as we wll show, ths allows trade-offs on the mpact of compresson between the two parameters. In Secton III we explot the ablty to perform such tradeoffs to develop a better compresson method for the TDOA/FDOA emtter locaton applcaton. The area of the FIM ellpse s A = C A λ λ, where C A s a constant and λ s the th egenvalue of the FIM; because the area nvolves a product of the egenvalues t s clear that maxmzng the area s equvalent to maxmzng det( J ˆ ) = Jˆ ˆ ˆ ˆ J JJ. Unfortunately, t does not allow the possblty of settng an mportance-weghtng on the accuracy of the two parameter estmates due to ts dependence on the product ˆ J ˆ J : swtchng the weghtng has no effect as seen by ( Jˆ )( βjˆ ) ( βjˆ )( αjˆ ) α =. The need for mportanceweghtng arses when the user wshes to (or needs to) favor the accuracy of one parameter at the expense of the others. Such trade-offs between parameters are mportant n TDOA/FDOA problems because for some emtterrecever geometres the geo-locaton may depend more on TDOA accuracy than FDOA accuracy (or vce versa); ths wll be explored further n Secton III. Ths tradablty can also be mportant n general parameter estmaton problems where a user may place a premum on the accuracy for a subset of the parameters. The permeter of the ellpse s an alternatve to the area as a measure of the sze of the FI ellpse, but t s qute complcated to compute exactly; however, we can use an approxmaton gven by CP λ+ λ, where C P s a constant. Because the sum of the egenvalues equals the sum of dagonal elements t s clear that maxmzng the permeter s approxmately equvalent to maxmzng the trace tr{ J ˆ} = J ˆ ˆ + J. Unlke the determnant, t does allow mportance-weghtng on the accuracy of the two parameters: we use as our dstorton measure a weghted trace ( ) parameter satsfyng 0 α. + α Jˆ wtr{ J ˆ} = αjˆ, where α s an mportance-controllng

10 8 Because the weghted trace measure does not contan the off-dagonal elements of the FIM, an ssue of concern s the effect that compresson can have on the tlt of the ellpse, whch s of most concern when the FIM ellpse s hghly eccentrc. The tlt of the ellpse mpacts the correlaton between the estmates and an undesrable change n ths correlaton could cause problems when the post-compresson estmates are used for further estmaton processng. We know that the effect of quantzaton wll be that ˆ J > J, but we also need to characterze the effect of compresson on the tlt of the ellpse (.e., the correlaton of the estmates). The followng theorem shows that the mpact on the tlt s not a concern because the post-compresson FIM ellpse wll always resde nsde the orgnal FIM ellpse; thus, for the crtcal case of a hghly eccentrc orgnal ellpse, compresson s not able to greatly change the correlaton between the estmates. Theorem: For the transform codng framework outlned above, the post-compresson FIM Ĵ gven n (3) has an nformaton ellpse θ T Jˆ θ = that les nsde the orgnal FIM ellpse θ T J θ =. Proof: See the Appendx. Our proposed approach, then, s to compress the data collected at sensor # so as to mantan the largest weghted trace of the FIM whle meetng the constrant on the rate; that s, α ˆ ( α) ˆ max J ( b) + J ( b ) subject to b R, (4) b where Jˆ ( b ) s the th element of the post-compresson FIM showng explctly ts dependence on the bt allocaton b and where the maxmzaton s done over all allocatons b that satsfy the rate constrant. Of course, n practce we do not have the sgnal model equaton needed to exactly compute the FIM so we need to use the data-computed verson of the FIM (computed wth the data replacng the sgnal model) as we dd n []; see the dscusson after (7) and (8) for a specfc example of data-computed FIM for the TDOA/FDOA case. The mnmzaton of the weghted trace of the data-computed FIM can be done usng the same numercal Lagrange approach [0] that we used n []. N n=

11 9 To summarze ths secton, we have proposed usng a weghted trace of the data-computed FIM as a dstorton measure for compressng data collected for use n the estmaton of multple parameters. The weghtng factors appled to the weghted trace can be set to assert mportance of the varous parameter accuraces. Wthout further knowledge of how the parameter estmates mght be jontly exploted by the user, ths weghted trace measure provdes a vable and useful dstorton measure. However, f the post-compresson parameter estmates are to be jontly used to make further estmates then t s preferable to apply the weghtedtrace-of-the-fim measure to the fnal estmaton problem; such a scenaro s llustrated n Secton III-B. III. Applcaton to TDOA/FDOA Estmaton To provde a clear focus on the ssues as well as to ensure maxmal relevance of the results, jont TDOA/FDOA estmaton for emtter locaton s used to llustrate our deas. A bref descrpton of TDOA/FDOA locaton s gven here to provde the context for the followng dscusson. The goal s to use multple RF sensors (here assumed to be movng) to locate a non-cooperatve RF emtter (here assumed statonary) by explotng the delay and Doppler effects on the receved sgnals. However, because the sensors do not know the tme of transmsson of the sgnal nor do they know the transmtted frequency of the sgnal, t s not possble for a sngle sensor to estmate the absolute delay and Doppler of ther receved sgnal. Instead, pars of sensors are used to estmate the relatve delay and Doppler (.e., the TDOA and FDOA) between ther respectve receved sgnals; the maxmum lkelhood estmator for TDOA/FDOA s to cross-correlate the two receved sgnals to compute the ambguty surface (also known as the cross-correlaton surface) and then extract the TDOA/FDOA estmates as the locaton of the peak of the ambguty surface [],[3]. Usng the TDOA/FDOA estmates from several pars of sensors, the system then estmates the locaton of the emtter []. Ths shows that TDOA/FDOA emtter locaton s a two-stage estmaton problem: () estmate TDOA/FDOA from pars of receved sgnals and () estmate locaton from the set of estmated TDOA/FDOA values. In the frst stage the two receved sgnals must be avalable at one ste n order to compute the ambguty surface; the data communcaton to accomplsh ths s what drves the need for compresson. In

12 0 subsecton III-A we demonstrate how the weghted trace of the FIM can be used to compress data whle enablng a trade-off between the mpact on TDOA and FDOA accuraces acheved n the frst stage; ths gnores the fact that a subsequent estmate s made jontly usng the TDOA/FDOA estmates ths s done here to () llustrate the man deas of Secton II wthout excessve cloudng from the subsequent second-stage processng and () demonstrate that there s a fundamental trade-off between compressng to acheve TDOA accuracy vs. to acheve FDOA accuracy. Ths latter nsght s mportant when understandng the mpact of compresson on the fnal task of estmatng the emtter locaton. Furthermore, the results n subsecton III-A wll be lnked to the more complete approach dscussed n subsecton III-B. In subsecton III-B we address optmzng the compresson wth respect to the end-to-end FIM of the geo-locaton estmate. These results wll demonstrate the mportance of trade-offs n enablng adaptaton of the compresson to the geometry among the sensors and the emtter. Before proceedng we frst llustrate the nteractons between geometry, TDOA/FDOA accuraces, and locaton accuracy. The locaton accuracy depends on the accuraces of the TDOA and FDOA estmates as well as the geometry of the sensors wth respect to the emtter. Some general qualtatve nsght on why the geometry mpacts TDOA and FDOA accuracy s gven n [4]; here we dscuss some qualtatve nsght relevant to the data compresson trade-offs between TDOA and FDOA. Fgure llustrates -D locaton error ellpses determned usng theoretcal covarance analyss [] for two pars of sensors n each of three geometry cases. For each geometry case the fgure shows () error ellpse for locatng usng only TDOA measurements, () error ellpse for locatng usng only FDOA measurements, and error ellpse for locatng usng both TDOA and FDOA. Note that n general the error ellpse for combned TDOA/FDOA wll be (approxmately) crcumscrbed wthn the ntesecton of the TDOA-only and FDOA-only ellpses. For the geometry used n Fgure (a), the locaton error ellpse usng only FDOA and the locaton error ellspe usng only TDOA are orented so that ther ntersecton s smaller than ether ellspe thus, the locaton error ellspe usng both TDOA and FDOA (nearly nscrbed n the ntersecton) s smaller; f data compresson changes the sze of ether the TDOA-only ellpse or the FDOA-only ellpse t wll change the sze of the TDOA/FDOA ellpse.

13 Thus, for ths geometry the locaton accuracy acheved usng both TDOA and FDOA depends on both the TDOA accuracy and the FDOA accuracy. In ths case, f compresson causes a degradaton n ether TDOA accruacy or n FDOA accuracy (or both) we would get a degradaton n locaton accuracy when usng TDOA/FDOA; ths represents a case where the compresson should strve to equally preserve TDOA and FDOA accuracy. In contrast, for the geometry used n Fgure (b), the locaton error ellpse for FDOA-only s much larger than and completely encloses the locaton error ellspe for TDOA-only thus when we use both TDOA and FDOA to locate the target the resultng locaton error ellpse s not much better than usng TDOAonly. In ths case we would get approxmately the same locaton accuracy even f the FDOA accuracy were sgnfcantly worse than shown; ths represents a case where the compresson should strve to mantan TDOA accuracy but could allow a large degradaton n FDOA accuracy. Fnally, for the geometry used n Fgure (c), the locaton error ellpse for TDOA-only s much larger than and completely encloses the locaton error ellspe for FDOA-only thus when we use both TDOA and FDOA to locate the target the resultng locaton error ellpse s not much better than usng FDOA-only. In ths case we would get approxmately the same locaton accuracy even f the TDOA accuracy were sgnfcantly worse than shown; ths represents a case where the compresson should strve to mantan FDOA accuracy but could allow a large degradaton n TDOA accuracy.

14 Results for Geometry # TDOA Only y error FDOA Only Both x error (a) Results for Geometry # Results for Geometry #3 y error TDOA Only y error TDOA Both Only x error FDOA Only Both x error FDOA Only (b) (c) Fgure : Effect of geometry on the relatve mportance of TDOA and FDOA accuracy. Each part of the fgure shows locaton error ellpses for three cases: () locatng usng only TDOA estmates dotted, () locatng usng only FDOA estmates dashed-dotted, and () locatng usng both TDOA and FDOA sold. Part (a) shows the case of a geometry where TDOA accuracy and FDOA accuracy are both mportant; part (b) shows the case of a geometry where only TDOA accuracy s mportant; part (c) shows the case of a geometry where only FDOA accuracy s mportant. A. Compresson for Trade-Offs Between TDOA and FDOA We consder the compresson of sgnal data at sensor S for subsequent transmsson to sensor S where the TDOA/FDOA estmates are computed. The contnuous-tme sgnal model for two passvely-receved complex baseband sgnals at sensors S and S havng an unknown TDOA of τ and FDOA of v s gven by

15 3 x () t = s( t ( t + τ /)) e + w () t j( ν0 + ν/) t 0 x () t = s( t ( t τ /)) e + w () t j( ν0 ν/) t 0 Tme-doman samples of these sgnals are used to estmate the TDOA/FDOA between the two sgnals. The TDOA Fsher nformaton measure s [] J π = m S [ m], (5) N / Nσ m= N/ where S [m] s the DFT of the sgnal at sensor S (wth the DFT frequences runnng over both negatve and postve frequences); t should be noted that because the DFT s an orthogonal, but not orthonormal, transform the DFT nose varance s N σ, whch accounts for the /N n front of the summaton. The FDOA Fsher nformaton measure s [] J N / π = n σ n= N / s [ n], (6) where s [n] s the sgnal data at sensor S. These equatons ndcate that compresson requrements for TDOA and FDOA are conflctng because the TDOA Fsher nformaton depends on the DFT coeffcents whle the FDOA Fsher nformaton depends on the sgnal samples. To perform the FIM-based compresson dscussed n Secton II we need a transform that allows smultaneous access to both the tme and frequency domans so that our bt allocaton to the transform coeffcents can affect both J and J. An orthonormal wavelet packet transform s a good choce because t provdes not only tme-frequency resoluton for our TDOA/FDOA trade-offs, but also provdes energy compactness that s mportant for the compresson effcency. Ideally, then, we desre to re-express the FIM expressons n (5) and (6) n terms of the wavelet packet coeffcents. In prncple, t s possble to wrte a mathematcal equaton for the wavelet packet coeffcents and then consder the dervatves wth respect to parameters, as needed for the FIM. Ths, however, quckly becomes ntractable (lkely even mpossble to get a usable closed-form expresson). Thus, we resort to

16 4 explotng ntutve relatonshps between the wavelet packet coeffcents on one hand and the DFT coeffcents and orgnal tme-doman samples on the other hand. Let { ψnm, } be an orthonormal wavelet packet bass set; each bass vector s concentrated on a dfferent tmefrequency regon of the sgnal n selects where n tme the bass vector s concentrated and m selects where n s frequency the bass vector s concentrated. Let {, nm} c be the correspondng coeffcents for the sgnal data s [n]. Although t s mpossble to relate each coeffcent to a sngle DFT coeffcent or sngle tme-doman samples, as shown n [5] [7], wavelet coeffcents n the same frequency band have the tendency to be assocated only wth a certan frequency range and wavelet coeffcents n the same tme band have a tendency to correspond to the same tme range. That s the set { nm } s the content of the sgnal at sngle frequency set by m o ; lkewse, the set { n, m} c for a fxed m o can be thought of as representng s, o c for a fxed n o can be thought of as representng the content of the sgnal at a sngle tme set by n o. From ths dscusson and notng that (5) ndcates that the TDOA FI J depends on a quadratc weghtng over frequency we can approxmately compute J from the wavelet packet coeffcents usng o π s m n, m σ m n J f c =, (7) s where the f m are approprately specfed measures of the frequency locaton of c, nm, and /N n (5) s not needed because the wavelet packet s assumed orthonormal. Note that the nner sum n ths expresson sums over all tmes at each frequency; thus, each nner sum captures the energy at the m th frequency and thus acts lke S [m] n (5) and s thus subsequently quadratcally weghted n frequency to mmc the quadratc weghtng seen n (5). Lkewse, notng that (6) ndcates that the FDOA FI J depends on a quadratc weghtng over tme we can approxmately compute J from the wavelet packet coeffcents usng π s n n, m σ n m J t c =, (8)

17 5 s where the t n are approprately specfed measures of the tme locaton of c, nm. Note that the nner sum n ths expresson sums over all frequences at each tme; thus, each nner sum captures the energy at the n th tme and thus acts lke s [n] n (6) and s thus subsequently quadratcally weghted n tme to mmc the quadratc s weghtng seen n (6). Recall that the coeffcents c nm, used above are specfed n terms of the nose-free sgnal s [n]. However, because we only know the nosy data x [n] we must use the data-computed versons of (7) x and (8); that s (7) and (8) should be computed usng wavelet packet coeffcents c nm, of the nosy data samples x [n] rather than the coeffcents of the unavalable nose-free samples s [n]. In addton to usng the data-computed versons, the effect of quantzaton must now be addressed. Each wavelet packet coeffcent s quantzed and the effect of that quantzaton on the FIM elements are captured n x c nm, ˆ ( ) = π fm m n σ + q ( bnm, ) J b, (9) and x c nm, = π tn n m σ + q bnm, Jˆ ( b ) ( ), (0) where q ( b nm, ) s the varance of the quantzaton nose arsng when b n,m bts are allocated to the n,m coeffcent (see [] for specfc detals) and b s the vector of all the ndvdual bt allocatons b n,m (put nto the vector n, say, lexcographc order). Some coeffcents may have zero bts allocated to them and thus are not sent; therefore they have zero contrbuton to the FIM ther zero contrbuton to (9) and (0) s enforced by settng ther quantzaton nose varance to nfnty so that the correspondng term n the summatons s zero. The evaluaton of the quantzaton nose varances are descrbed n [] and the mnmzaton of the weghted trace of the data-computed FIM can be done usng the same numercal Lagrange approach [0] that we used n []. The weghted-trace-based TDOA/FDOA dstorton measure we use s then D ( b, α) = αjˆ ( b) + ( α) Jˆ ( b ), () wtr

18 6 where b s the vector of all the bt allocatons and where we have suppressed the π that s common between the two terms n (). Bts are allocated to the coeffcents usng the method of [0] to maxmze () for a gven rate constrant. To compare the proposed scheme wth a tradtonal scheme, we also allocated bts to the wavelet packet coeffcents to mnmze MSE under the bt constrant (see [] for detals). Although the developed scheme s applcable to all varetes of sgnals, we use a lnear FM radar sgnal to llustrate the method. A 3-level wavelet packet transform s performed and 8 subbands are produced; each subband s parttoned nto 8 blocks wth a sngle quantzer used wthn each block. Moreover, to focus attenton on the lossy compresson performance, no entropy codng s appled after quantzaton. The smulaton results n Fgure show the nherent trade-off that s controlled by the choce of α, whose value controls whether the operatng pont of the algorthm ( OP Ponts ) favors TDOA accuracy ( TDOA Optmal ), favors FDOA accuracy ( FDOA Optmal ), or balances them to acheve the closest operaton to the no compresson case ( Balanced ); for comparson we nclude the operatng pont for optmzng our transform coder wth respect to MSE. The MSE result was acheved usng the same transform codng structure as for the other results but the bt allocaton was numercally optmzed usng MSE as the dstorton crtera. The ablty of the FIM-based method to outperform the MSE approach stems from the fact that the FIM captures the nherent mportance f the tme-frequency coeffcents to the task of estmatng TDOA/FDOA whle MSE captures only the effect of compresson on the post-compresson SNR. The ablty to use the FIM to accomplsh trades between TDOA accuracy and FDOA accuracy comes from the fact that compresson for TDOA accuracy should mantan bandwdth (note that the quadratc weghtng n (5) favors keepng components far from the band center) whle compresson for FDOA accuracy should mantan duraton (note that the quadratc weghtng n (6) favors keepng components far from the tme center). B. Compresson for Locaton Processng The prevous subsecton appled the weghted trace of the FIM method of Secton II to the TDOA/FDOA FIM. As mentoned prevously ths was done to llustrate the applcaton of the weghted trace and, most mportantly, to demonstrate the fundamental trade-offs between TDOA accuracy and FDOA accuracy when

19 7 compresson s used; n partcular, t was shown that adjustng α enables one to acheve a trade-off between TDOA and FDOA. In ths sub-secton we broaden our compresson vewpont and look at the mpact that compresson has on the locaton problem as a whole rather than smply lookng at ts mpact on the frst stage of a cascade of two estmaton problems. In ths broader context we can stll apply the results of Secton II, although now we apply t to the overall end-to-end geo-locaton FIM rather than to the TDOA/FDOA FIM. Balanced Fgure : Trade-off between TDOA and FDOA accuraces as α s vared for compresson rato 3: and SNR = 5 db & SNR = 5 db; symbol denotes the operatonal pont (α = 0.5) closest to that wthout compresson. For smplcty of notaton we consder only -D geo-locaton: all sensors and the emtter are n a -D x-y plane wth the emtter at the locaton p = [ x, y ] T. We assume that there are P sensors that are networked nto e e J = P/ dsjont pars. Wthn each par one sensor shares ts collected sgnal data wth the other sensor where the receved data s cross correlated wth ts own ntercepted sgnal to estmate the TDOA/FDOA for the par. Many connecton types are possble. Dsjont pars s a common one; another common one s to use a centralnode that s pared wth each of the other sensors. Each type has ts pros and cons as far as communcaton, locaton accuracy, and other ssues (e.g., fault-tolerance [8]). It s beyond the scope of the current paper to

20 8 The resultng set of TDOA/FDOA estmates from the pars are then used to compute the locaton estmate p ˆ = [ xˆ, yˆ ] T ; t s mmateral to ths dscusson where ths second-stage computaton s performed. e e It should be noted that ths two-stage approach has been the classcal approach to emtter locaton processng []: frst estmate from the sgnal(s) some sgnal parameter(s) that depend on the emtter locaton and then use a set of these estmated parameters to estmate the emtter locaton. Although t s possble to formulate a sngle-stage estmaton problem that takes n all the sgnal data and from t drectly estmates the locaton, such an approach leads to algorthms whose structure and complexty s qute dauntng. However, there has been some work toward ths drecton (e.g., [9],[30]) but so far the results are lmted to only narrowband sgnals. Nonetheless, the two-stage approach remans the most vable one for the foreseeable future; furthermore, t s lkely that even though a sngle-stage method does not explctly estmate the TDOA/FDOA, the way n whch data compresson affects TDOA/FDOA characterstcs should stll have an mpact on the sngle-stage performance. The th sensor par uses ts ntercepted sgnal data to compute ts TDOA/FDOA estmate ˆ ˆ τ τ wτ θ =,,,, J vˆ = + = v, wv where ˆ τ and v ˆ are the TDOA/FDOA estmates, τ and v are the TDOA/FDOA values, and w τ and wv are the random TDOA/FDOA measurement errors. Because the pars are dsjont and the sensor recever noses are assumed uncorrelated between sensors, the estmates θ ˆ and θ ˆ j are uncorrelated for j. Furthermore, because the TDOA/FDOA estmates are obtaned usng the maxmum lkelhood (ML) estmator of cross correlaton, the asymptotcal propertes of ML estmators [8] state that [ wτ, w v ] asymptotcally follows the Gaussan dstrbuton; that s wτ wv a ~ (0, J ) N, θ = τ, v ], [ explore the nteractons between connectvty and compresson; nstead we focus on the dsjont pars case because t s used n practce and s a good vehcle to llustrate the aspects of trade-offs va compresson.

21 9 where J s the Fsher nformaton matrx for the th TDOA/FDOA par. In lght of ths result we wll assume a Gaussan dstrbuton. Lettng H f x = f x TDOA, e FDOA, e f TDOA, y f e FDOA, y e t can be shown [] that the FIM of the geo-locaton estmaton s gven by H T T Jgeo = J dag{,,, J } H,,H J J J HJ J = HJH = T. () Equaton () shows that the locaton-estmaton FIM J geo depends not only on the TDOA/FDOA-estmaton FIMs J but also strongly depends on the geometry of the emtter and the sensors as captured n the Jacoban matrces H ; to stress ths pont we call the matrces H the geometry matrces. We now can apply the results from Secton II drectly to the locaton-estmaton FIM gven n (); because there s usually no reason to favor x-locaton accuracy over y-locaton accuracy, here we use the standard trace as the dstorton measure rather than weghted trace. Let for notatonal purposes H H = H,, H H,, J J = J,, J J,, then J { geo} = tr J = ( H, + H,) J, + ( H, + H,) J, + ( H,H, + H,H,) J,, (3) where the frst two terms nsde the sum depend on the dagonal elements of the TDOA/FDOA FIM but the last term depends on the off-dagonal elements. Our goal s to perform the compresson to maxmze the postcompresson value of tr{ J geo}. As before, to provde a usable dstorton measure, the trace n (3) must be computed from the data avalable wthn the set of sensors. Thus, the TDOA/FDOA FIMs J must be computed

22 0 from the receved data; except we now also need the off-dagonal elements n J, whch wll be dscussed later. Lkewse, we need a way to compute the H, but that requres knowledge of the true locaton p = [ x, y ] T, whch s unknown. However, we can frst send a small amount of data enough to roughly estmate the locaton and the roughly determne the geometry matrces H. There are optons on how to obtan ths small amount of data: () use the compresson method of Secton III-A wth α = 0.5 and constran the allocaton to use a small total number of bts or () smply send a small number of data samples (.e., a short tme record). Another vable opton n practce s that the locaton system may be cued by another sensor system that provdes a very rough locaton for the emtter (e.g., the emtter s nsde ths km by km box, fnd t precsely ). Another scenaro s that the TDOA/FDOA system has already located the emtter prevously and s re-nterceptng ts sgnal to locate agan to better accuracy [3]. Usng the same notaton as before to ndcate data-computed versons that nclude the effect of quantzaton, notate the roughly estmated geometry matrces and the data-computed elements of the TDOA/FDOA FIM as follows Hˆ Hˆ Jˆ ( b) Jˆ ( b) Hˆ ˆ = ( ) = b b,,,,, J b ˆ ˆ ˆ ˆ H, H, J, ( ) J,( ) e e where the on-dagonal elements Jˆ, ( b ) and Jˆ,( b ) are computed accordng to (9) and (0) and the computaton of the off-dagonal elements of J ˆ ( b) wll be dscussed later. Then the data-computed postquantzaton verson of the geo-locaton FIM trace becomes J { } tr Jˆ ( b) = ( Hˆ + Hˆ ) Jˆ ( b) + ( Hˆ + Hˆ ) Jˆ ( b) + ( Hˆ Hˆ + Hˆ Hˆ ) Jˆ ( b ). (4) geo,,,,,,,,,,, = If we assume that TDOA and FDOA are uncoupled,.e., J 0, (even though t may not be true) then (4) corresponds to () wth α chosen accordng to, = α = ( Hˆ + Hˆ )/( Hˆ + Hˆ + Hˆ + Hˆ ). (5),,,,,, Ths drectly leads to the followng smple scheme where the compresson s adapted to the geometry.

23 Step : The central node sets the desred compresson rato and ts assocated operatonal compresson ponts based on the requrement of energy consumpton and latency. Then t randomly pcks J pars of nodes to get J measurements of TDOA and FDOA. In the begnnng, one of the nodes (,), J sends a small amount of ts data (compressed as n Secton III-A usng α=0.5) to ts pared sensor (,), where rough measurements of TDOAs ~ τ } and FDOAs v ~ } are measured. { d, { d, Step : ~ τ } and v~ } from the selected par are sent to the central node, where the rough geometry matrces { d, { d, Hˆ are estmated. Step 3: In terms of the estmated geometry matrces H ˆ, the central node determnes the α usng (5) for each sensor par and sends them to the pars. Step 4: Each node (,) compresses ts data accordng the specfed compresson rato and ts evaluated α and sends the compressed verson of ts sensed data to the node (,) n ts par. Step 5: Each node (,) decompresses the data from node (,) and computes the TDOA/FDOA estmates whch are then sent to the central node, where the fnal locaton of the emtter s estmated. To llustrate the performance of ths approach, smulatons were performed as follows. An emtter can be located anywhere n a -D 5x5 km area, wthn whch the sensors are spread. Four sensors are randomly pcked to form two pars to estmate TDOA/FDOA. Only 56 samples are sent n Step to roughly estmate the geometry. Fnally, the remander of the 4096 samples are compressed and shared between each par. For comparson, the 4096 data samples are compressed usng the MSE measure. In the smulaton, we use the rato of the area of crcular error probable (CEP) [] wth compresson to that wthout compresson mnus (whch corresponds to the relatve ncrease n CEP) as the metrc. We estmate CEP usng 0.75 Trace{ P }, where P s The CEP s defned as the radus of the crcle that has ts center at the mean and contans half the realzatons of the random vector.

24 the covarance matrx of the geo-locaton error [], and we estmate P from the Monte Carlo smulaton results; note that P s the nverse of the geo-locaton FIM. There were 000 Monte Carlo runs performed for each compresson rato and the average values are shown n Fgure 3 along wth the performance acheved usng MSE-optmzed compresson (usng the same wavelet packet transform). Over the range of compresson rato values consdered, the results n Fgure 3 show a roughly 5x mprovement n CEP for our geometry-adaptve method relatve to one that uses the wavelet packet transform but optmzed accordng to MSE 3. Fgure 3: Effect of compresson rato on CEP when usng the geometry adaptve method whle assumng that the TDOA/FDOA FIM off-dagonal terms are zero. For the above method we assumed that J 0 ; however, ths condton s not met n general. And, = although the above algorthm provdes sgnfcant mprovement over MSE, there mght be further potental by maxmzng the trace of the geo-locaton FIM gven n (4). In dong so, we need to be able to accurately 3 The MSE results are obtaned usng standard the MSE dstorton measure for compresson; the same wavelet packet transform was used for the MSE approach; no geometry adaptaton s possble for the MSE approach.

25 3 evaluate the data-computed versons of the off-dagonal elements of the TDOA/FDOA FIM. Based on wellknown understandng of the structure of the FIM for radar delay-doppler estmaton and the structures n (9) and (0) we mght conjecture that the cross-terms can be numercally approxmated as x c nm, ( ) = ˆ ( ) π m n m n σ + q bnm, Jˆ b J b f t. (6) ( ) However, (6) s not able to compute the off-dagonal terms J and J correctly by usng a standard wavelet packet transform because each channel n ths wavelet packet transform has both postve and negatve frequency content. For (6), wth ts t n f m cross-term, t s crucal that we have a so-called one-sded flter bank that has ndvdual postve and negatve frequency channels. We used a smple method, outlned n [3], to develop such flters. The advantage of ths flter bank approach s that exstng routnes for desgnng and mplementng orthogonal PR flter banks can be used wth mnor modfcatons. However, we have demonstrated that these orthogonal flter banks have dffcultes properly evaluatng the TDOA/FDOA FIM elements [3],[33]. Thus, the short-tme Fourer transform (STFT) was nvestgated as a replacement for the complex- valued orthogonal PR flter bank [3]. Although the STFT enables qute accurate evaluaton of the FIM, the STFT s not well-suted for compresson due to the fact that t s not an orthogonal representaton. Thus, as shown n Fgure 4, we use t smply as an auxlary parallel mechansm n place of the wavelet packet coeffcents n (9), (0) and (6) to evaluate the TDOA/FDOA FIM whch s then used n (4) for bt allocaton to the wavelet packet transformed coeffcents. In the approach shown n Fgure 4, each x one-sded flter bank output sample (.e., transform coeffcent) c nm, has ts contrbuton to the FIM evaluated usng STFT x samples that occupy the same tme-frequency regon as the flter bank output sample. In partcular, c n (9), (0) and nm, x (6) gets replaced by the sum of magntude-squared STFT values that le n the same tme-frequency regon as c nm,. Further detals are gven n [3] and [33]. At ths pont ths approach s ad hoc and s more art than scence; although work s underway to precsely characterze the approach.

26 4 Ths approach of allocatng bts to maxmze the trace of the geo-locaton post-compresson FIM n (4) s explored here va smulatons. The geometry matrces are evaluated n the same way as n the prevous method and the TDOA/FDOA elements are computed usng ether the STFT or drectly n terms of the one-sded wavelet packet. Two typcal geometres of the two par of sensors and emtters are chosen to llustrate the mprovement obtaned by accurate assessment of FIM. These two geometres are shown n Fgure 5(a) and Fgure 6(a), respectvely. Fgure 5(a) corresponds to the case when TDOA and FDOA are both mportant for the locaton of emtter, whle Fgure 6(a) corresponds to the case when FDOA s more mportant for one par of sensor whle both TDOA and FDOA are mportant to the other par. Compute STFT Evaluate FIM Elements Q x T-F Flter Bank Q Q N Fgure 4: Our compresson framework usng parallel auxlary STFT processng to evaluate the FIM elements. Smulaton results are shown n Fgure 5(b) and Fgure 6(b), where the trace of the error covarance matrx of locaton s used as the geo-locaton accuracy measure to compare the dfferent algorthms under nvestgaton. The label STFT-WP FIM represents the case of usng the one-sded wavelet packet transform codng wth parallel auxlary STFT to evaluate the FIM whle WP FIM represents usng the one-sded wavelet packet transform codng to drectly compute the elements of the FIM from the one-sded wavelet packet coeffcents. Results labeled wtr FIM represent the performance of the wavelet packet transform codng that s based on the weghted trace dstorton measure n (). As mentoned above, the wtr FIM gnores

27 5 the effect of the off-dagonal element J, n (4) and chooses α as ( H + H ) /( H + H + H + H ). For comparson the performance wthout compresson and,,,,,, wth MSE-based compresson (allocatng bts to the same one-sded wavelet packet coeffcents to mnmze MSE). Clearly, the STFT-WP FIM approach s slghtly better than the WP FIM approach, whle both acheved sgnfcant mprovement over the wavelet packet codng to mnmze MSE. Accordng to the Schwarz nequalty, ( H, + H, ) J, + ( H, + H, ) J, s greater than or equal to ( H, H, + H,H, ) J,, whch means that the on-dagonal elements of J geo domnant the off-dagonal elements and t mght be good enough to use only wtr FIM because t has lower computatonal complexty than STFT-WP FIM. Ths s verfed by Fgure 5(b) and Fgure 6(b) where wtr FIM s very close to the performance represented by STFT- WP FIM, at least at low compresson ratos n Fgure 5(b) and more unformly n Fgure 6(b). The dfferences n the relatve performance between the methods that use the full form n (4) (.e., STFT-WP FIM and WP FIM ) versus the one that uses only the on-dagonal terms n (4) are due to dfferences n the geometres for these two cases. IV. Conclusons Other researchers have proposed and explored the use of Fsher nformaton as a dstorton measure for compresson. We have extended ths lne of research by proposng the use of the weghted trace of the FIM as a well-suted dstorton measure for the case where multple parameters are consdered; we note, though, that the trace of the FIM has been used n other areas (e.g., control problems). We have shown that for the estmaton of multple parameters there s generally a trade-off between the mpact of compresson among the varous estmaton accuraces and that usng a weghted trace allows the user to accomplsh trade-offs among the estmaton accuraces of the multple parameters.