00 Conference on Informaton Scences and Systems, The Johns opkns Unversty, March 3, 00 Pulse Extracton for Radar Emtter Locaton Mark L. Fowler, Zhen Zhou, and Anupama Shvaprasad Department of Electrcal Engneerng State Unversty of New York at Bnghamton Bnghamton, NY 390 Emal: mfowler@bnghamton.edu Abstract Two related data compresson methods for radar sgnals are descrbed and analyzed. The methods use the sngular value decomposton (SVD) of a data matrx contanng one pulse n each row to explot pulseto-pulse redundancy. By usng a rank-one approxmaton to the data matrx t s possble to acheve compresson ratos typcally of the order of several 0 s and sometmes over 50: for typcal radar types whle mantanng accurate locaton; the amount of compresson depends on the radar s parameters. Alternatvely, by retanng a sngle sngular vector as an extracted prototype pulse, t s possble to use t for matched flter processng, thus provdng an mproved method for dong noncoherent tme-of-arrval (TOA) processng for a slght cost of a small amount of addtonal data to be sent. I. INTRODUCTION A common way to locate electromagnetc emtters s to measure the tme-dfference-of-arrval (TDOA) and the frequency-dfference-of-arrval (FDOA) between pars of sgnals receved at geographcally separated stes []. For radar emtters there are (at least) two alternatves for measurng the TDOA/FDOA values. The coherent method measures the TDOA/FDOA by coherently cross-correlatng the sgnal pars [], and the noncoherent method measures the tme-of-arrval (TOA) of the pulses (and possbly frequency-of-arrval (FOA) also) at each platform and then combnes the TOA (and possbly the FOA) measurements made at two platfo nto TDOA (and possbly FDOA) estmates [8]. From a theoretcal vewpont, the coherent method has a clear advantage because t more completely utlzes the nformaton embedded n the receved sgnals. Furthermore, the noncoherent approach requres that the SNR at each platform be hgh enough that the pulses can be detected usng smple thresholdng of the leadng edge because no matched flter s avalable. After pulse detecton the TOA/FOA can be measured (TOA s usually measured usng leadng edge methods or pulse centrod methods, whle varous frequency estmaton technques are used to measure FOA). On the other hand, coherent methods explot the tme-bandwdth processng gan [] to allow, n prncple, operaton at much lower SNRs at all platfo; although n practce at least one platform generally needs to be at an SNR hgh enough to detect the pulses to allow sgnal acquston processng such as dentfyng the presence of a sgnal of nterest. owever, coherent processng does have a serous drawback: sgnal samples receved at one platform must be transmtted over a data lnk to another platform n order to perform cross correlaton. Because these lnks are rarely completely allocated to the sole task of transferrng data for locaton processng, the allocated lnk rate usually s nsuffcent to accomplsh ths n a tmely manner especally for the radar locaton case wth ts wde bandwdths. To mtgate ths, varous data compresson approaches have been proposed [3] [7], although they have been desgned for the generc sgnal case and can t fully explot the characterstcs of radar sgnals. For the radar case, the fact that (at least) one platform wll be operatng at an SNR hgh enough to detect pulses can be exploted as a frst step towards reducng the transferal by not sendng the samples between detected pulses. owever, even wth ths reducton the transferal tme s stll excessve gven allocated rates for current and projected data lnks. In ths paper, rather than approachng the problem drectly from the perspectve of data compresson, we wll propose a way to extract a prototype pulse from the hgh- SNR platform and then send that pulse to the other platfo (possbly together wth some sde nformaton), where t can be used for measurng the TDOA/FDOA values. As a result we get a sgnfcant amount of data compresson, but vewng the scheme more as a prototype pulse extracton leads to some alternatve vewponts. We dscuss two dfferent ways to use the extracted prototype: one based on noncoherent TOA methods (whch we wll call sem-coherent ) and one based on coherent TDOA/FDOA methods. The semcoherent method uses the extracted pulse as a pulse-matched flter allowng mproved pulse detecton and TOA measurement; ths allows all but one platform to be at low SNR, although not as low as for coherent processng. On the other hand, the coherent method uses the prototype pulse together wth some sde nformaton (a sequence of pulse phases, magntudes, and tmes) to reconstruct a complete pulse tran at the other platform that s sutable for cross correlatng wth that platform s locally receved pulse tran. We wll show how to use the sngular value decomposton (SVD) together wth some pulse algnment processng to effectvely extract the prototype pulse. We then wll dscuss how to explot the prototype pulse n the sem-coherent and coherent approaches. II. EXTRACTIING TE PULSE The proposed extracton approach s based on the fact that a radar emts a tran of smlar pulses. Because modern
radars can change modes we assume that prelmnary subtran-extracton processng has grouped the sgnal of nterest nto one or more subtrans, each havng pulses from the same mode of operaton such processng s a standard part of any electronc warfare system (ths processng also removes pulses from other emtters) [9]. We extract a prototype pulse for each subtran dentfed at the hgh-snr platform, although for smplcty of dscusson we wll assume here that there s a sngle subtran. The pulses n a subtran look very much alke except that they may have random phase shfts f the radar s not pulse-to-pulse phase coherent. Subtran extracton provdes a seres of smlar pulses separated by ther orgnal pulse repetton nterval (PRI); the separaton s ndcated n the fgure n te of number of samples N, N, etc., whch wll be nearly-equal ntegers. The subtran pulses are then gated and the nterpulse samples are removed and the numbers of samples removed between the pulses (N, N, etc.) are extracted as sde nformaton; the resultng seres of pulses can be thought of as a pulse tran havng an artfcally short PRI. (Ths gatng can be vewed as prelmnary compresson but note that all compresson ratos stated n ths paper do not nclude ths gatng-based compresson.) Fnally, the gated pulse tran s processed to extract the prototype pulse and any other requred sde nformaton, whch s then sent over the data lnk along wth the gatng-extracted pulse separatons. The mathematcal bass of the prototype extracton processng s the sngular value decomposton (SVD) [0]. If we put the gated pulses of the baseband equvalent sgnal for the receved pulse tran nto a matrx wth one pulse per row we would have a rank one matrx f () there were no nose or propagaton effects, () the pulses were perfectly tme algned.e. perfect leadng-edge detecton & gatng, and () the radar s PRI were an nteger multple of the samplng nterval T. All but the frst of these causes of ncreased rank can be mtgated by performng tme algnment on the pulses n the pulse matrx. The needed tme algnment values can be determned by computng pulse-to-pulse cross correlatons. The pulses are then tme algned usng a fractonal delay FIR flter [] although we are currently nvestgatng the use of DFT based algnment. By usng ths algnment step the resultng pulse matrx s closer to beng a rank-one matrx than t was before the algnment. Let p denote the number of pulses n the pulse subtran and let n denote the number of samples per pulse kept after gatng; then the total number of samples s pn. If we denote the p n (algned) pulse matrx by P, ts SVD s r P σ u v, () where r s the rank of P, u s the th left sngular vector, v s ermtan transpose of the th rght sngular vector, and σ s the th sngular value, ordered such that σ σ+. Each term n the sum n () s a rank-one matrx. If we truncate ths sum to only k < r te we get the rank-k matrx k P σ u v () k that best approxmates P n the sense that the sum of the squares of the elements of P - P k s smaller than for any other rank-k matrx. Note that n our case the matrx contans the pulses and therefore ths approxmaton gves the smallest mean square error (MSE) between the orgnal pulse tran and the approxmate pulse tran formed by concatenatng the de-algned rows of P k. As we wll see later, ths mnmum MSE property s the bass for usng the SVD for both of the two vewponts consdered here. In the perfect-sgnal scenaro, where there s no nose and no pulse msalgnment, P s rank one and σ 0 for. owever, when sgnal perturbatons are present, P has hgher rank, but stll has a few domnant sngular values. Therefore, to get maxmum compresson we strve to approxmate P by a matrx havng a low rank whle havng a small MSE n fact we wll approxmate t wth a rank-one matrx. The effect of the tme algnment s to concentrate the energy of the pulse matrx nto the frst sngular value, producng a matrx that s closer to a rank one matrx. The effect of the nose on the sngular values s unformly spread across all the sngular values ths s n fact a known result that s exploted by prevous applcatons of the SVD to sgnal processng problems. Thus, when we truncate the SVD to k te as n () we are throwng away all the nose that exsts n the thrown away sngular values, and f we ve done our job rght we have thrown away very lttle of the sgnal because t s mostly concentrated n the sngular values that we keep. The effect of ths s to ncrease the SNR of the reconstructed sgnal found by concatenatng the rows of P k nto a pulse tran; thus, not only do we compress the sgnal but we get an mprovement n SNR rather than a degradaton due to compresson! Ths smultaneous compresson and nose reducton wll be demonstrated n the next secton after further dscussng the compresson/decompresson processng. To extract a prototype pulse we consder the case where we truncate the SVD to a sngle term (k ) to get P ; we ll demonstrate later that the accuracy acheved wth k s excellent. To specfy P we need the p vector u (.e., the st left sngular vector), the n vector v (.e., the st rght sngular vector), and the scalar σ (.e., the st sngular value). Note, then, that v s the same sze as a pulse (n samples). Thus, we can nterpret vector v as a sngle prototype pulse that has been extracted from the orgnal pulse tran, whch s a nce vewpont gven that the radar s recever would process ts receved pulse tran usng a pulse template as a matched flter; ths leads to what we call a sem-coherent approach. Alternatvely, we can vew the concatenaton of the rows of P as formng an approxmaton to gated and algned pulse tran, whch together wth the algnment and gatng sde nfo can be used to create an approxmaton of the orgnal pulse tran. These two vewponts wll be explored n the next secton. III. EXPLOITING TE PULSE A. Sem-Coherent Method For smplcty here we consder the TDOA-only case, where we assume that there s no relatve moton between emtter and platfo. In the sem-coherent method we use
the prototype pulse p pp (k) as a matched flter to detect the pulses at the low SNR platform and to measure the TOA values for each pulse n the receved pulse stream. The prototype pulse s also used to measure the TOA values at the platform where t was extracted. Then the correspondng TOA values from the two platfo are subtracted to form a sequence of TDOA values that are then averaged to gve the desred TDOA estmate. To do ths processng we need to extract a prototype pulse from the subtran receved at the hgh SNR platform. Smply choosng one of the detected pulses as the prototype pulse s possble, although some rules would have to be used to select the best canddate that would lkely be dffcult to do n the presence of fadng, emtter scannng, and multpath. owever, we ll see that the SVD method gves a mathematcally-based rule to generate a prototype pulse by explotng all the data n the pulse subtran. We seek a prototype pulse that s hghly correlated (postve or negatve snce we check the magntude of the matched flter output because we are not assured that the emtter s pulse-to-pulse coherent) wth all the pulses n the receved pulse subtran. One crtera that could be used to fnd such a pulse s to seek a prototype pulse that maxmzes the sum of the magntude squares of the correlatons. In other words, we seek p pp (k) such that f p (k) are the pulses n the receved pulse subtran, then we maxmze C < p, p pp >, (3) where we have used vectors to represent the correspondng pulses. owever, we now show that ths s equvalent to choosng the prototype pulse vector such that t has unt norm and mnmzes E p α p (4) for approprately chosen α. Mnmzng ths over the choce of p pp and α can be done n two steps: fnd the mnmzng α for a gven p pp and then fnd the mnmzng * unt-norm p pp. The mnmzng α are α < p, p pp >, for whch we get E pp < p,p pp > p, (5) whch s mnmzed when C n (3) s maxmzed. owever, recallng that the SVD approxmate matrx mnmzes the MSE between the elements of the matrx and the approxmaton, from (4) t s clear that E s mnmzed when we choose p pp v and. σ u where α s the th element of.. Thus, we see that extractng a prototype pulse usng the SVD-based method descrbed above satsfes our crtera of maxmzng (3). In the sem-coherent method, the sde nformaton needed n addton to the prototype pulse s the TOA measurements made at the hgh SNR platform usng the pulse-matched flter. Of course, ths sde nformaton s not truly sde nformaton n ths case because they have to be sent for the noncoherent method, too. After the prototype pulse s receved at the other platform t s used as a matched flter to detect subtran pulses from the receved sgnal and to measure ther TOA values, whch are then combned wth the TOA values sent from the extractng platform. The major advantage of dong ths s that unlke noncoherent TOA-based processng, ths method uses matched flter processng rather than leadng-edge processng to measure the TOA. Ths enables operaton at a lower SNR at the nonextractng platform than s possble wth conventonal noncoherent methods. In addton t gves a compresson rato on the order of np CR sem p (6) n f we gnore any possblty of further codng the samples of the prototype pulse. Ths compares the amount of data needed for the coherent method wth no compresson to the amount needed for the proposed sem-coherent method. Thus, we see that the amount of compresson s the number of pulses to be used n the processng. Because the number of pulses needed to acheve suffcent TDOA accuracy can range between several tens of pulses to several thousands of pulses (dependng on the radar type as well as other system consderatons), ths method can gve extremely large compresson ratos. B. Coherent Method In the coherent method we seek a reconstructed pulse tran that would mnmze the MSE between t and the orgnal pulse tran before t was compressed. For a gven compresson rato (e.g., for a gven number of te retaned n ()) ths clearly s the pulse tran formed from the dealgned rows of the approxmatng pulse matrx P k, due to the SVD mnmzng MSE. For k, ths s equvalent to mnmzng E n (4). Thus, we see that the sem-coherent method and the coherent method have the same mnmzaton requrement and t s met through fndng the SVD-based rank-one approxmate matrx P. In the coherent method, n addton to the prototype pulse we need sde nformaton consstng of the values of the left-sngular vector u, the fractonal tme algnments, and the number of samples between adjacent pulses that were removed by gatng. The left-sngular vector u s used to reassemble the truncated SVD form of the pulse matrx (up to the scalng factor of σ ), after whch the tme algnment nformaton s used to undo the tme algnment and, fnally, zeros are nserted n place of the gatng-removed sgnal samples. Thus, for the cost of a modest amount of sde nformaton t s possble to reconstruct a MSEmnmzng pulse tran sutable for coherent crosscorrelaton. In partcular, the approxmatng matrx P s formed from v P σ u, (7) from whch t s clear that each row n P s a complex-valued T scalar multple of v, where the complex scalar for the th
row s the th element n u tmes σ ; t s also clear that σ does nothng more than ampltude scale the entre reconstructed pulse tran and can therefore be omtted. Thus, we can change (7) to ~ P u, (8) v from whch we see that u holds the reconstructon ~ magntudes and phases. Fnally, the rows of P have to be tme shfted to undo the algnment processng tme shfts, after whch the results are assembled nto a pulse tran (wth zeros nserted between pulses to undo the effect of pulse gatng) that s cross-correlated wth the pulse tran receved at the other platform. Thus, the nformaton that s needed to reconstruct the sgnal s:. The n rght sngular vector (RSV) v (.e., the prototype pulse). The p left sngular vector (LSV) u (.e., the reconstructon magntudes and phases) 3. The p- tme shfts 4. The p- numbers of nserted zeros (N, N, N p-) Usng ths data at the other platform the reconstructed pulse tran can be formed and then cross-correlated wth the sgnal data receved locally at that platform to estmate the TDOA/FDOA. ow much compresson can we get from ths scheme? We frst consder the case where a sngle pulse s put nto each row of P, but we wll see that t s often better to put multple pulses per row. Thus we have that P s p n, where p s the number of pulses (.e. rows) n the pulse matrx and n s the number of samples per pulse. Thus, f no compresson s used there are np samples to be sent. To send P k n () we only need to send: () k sngular values, whch requres k values, () k left sngular vectors each havng p elements, whch requres kp values, and () k rght sngular vectors each havng n elements, whch requres kn values. The total number of values needed to specfy the reduced-rank SVD approxmate P k s k(+p+n) values; f k we don t need the sngular value so ths becomes p+n. Assumng that we use the same number of bts for each element as we dd for the sgnal samples, the compresson rato s CR np coho k( p +, (9) + n) where k s the # of sngular vectors retaned, p s the number of pulses to be processed, and n s the number of samples per pulse. Results on how to code the SVD extracted nformaton s gven n [] as well as results that show for the coherent approach that the CR s maxmzed when the pulse matrx s made square. Thus, t may be desrable n many cases to put more than one pulse per row when usng the coherent approach. When that s done the compresson rato for the coherent method becomes An addtonal advantage of the coherent approach s that the SVD s nose reducng property provdes a reconstructed sgnal that s an mproved verson of the orgnal rather than a compresson-degraded verson. Thus, a large compresson rato s acheved wthout the usual degradaton assocated wth lossy compresson methods. Furthermore, the coherent method allows the low-snr platfo to operate at lower SNR values than for the sem-coherent method because the coherent processng provdes the maxmum tme-bandwdth gan. C. Performance Results Monte Carlo smulatons were performed to demonstrate the capablty of the sem-coherent processng method. Two varatons of the sem-coherent method were smulated to provde nsght nto the performance: () usng a nose-free prototype and () usng the SVD-extracted prototype. Obvously the frst varaton s not possble n practce but s ncluded here solely for comparson purposes. The smulaton results are shown n Fgure for the case of 0 pulses of 40 samples each. The sgnal-to-nose rato for the sgnal beng compressed (denoted as DNR) was taken as 0 db, the sgnal-to-nose rato for the sgnal not beng compressed (denoted SNR) vared from db to 30 db; 00 Monte Carlo smulaton runs were performed. The fgure shows results for the sem-coherent method usng () the SVD-prototype, () the nose-free (NF) prototype taken drectly from the nose-free pulse tran, and () the coherent cross-correlaton method wthout compresson (for reference). From the fgure t s seen that as long as SNR 0 db (for ths case) there s no dfference between the three methods. To see why, consder the NF-prototype case where n s the number of samples n a pulse. When the NF-prototype s cross-correlated wth pulse tran # to estmate the th TOA, the varance s gven by [] σ TOA,, () (πβ ) nsnr where β s the (or Gabor) bandwdth of the sgnal. At the other platform we can wrte DNR αsnr for some α, so the correspondng TOA estmate on the other pulse tran (#) has varance σ TOA, (πβ. () ) nαsnr The resultng TDOA estmate (assumng that the two TOA s are uncorrelated, whch s only an approxmaton) has varance gven by TDOA, σtoa, + σtoa, ( + ) σtoa, σ α (3) After averagng these we get that np np CR coho, (0) k( + np ) k
σ TDOA, sem p σ (πβ (πβ TDOA, ) npsnr/( + ) np( SNR + DNR ) α ) (4) On the other hand, f we compute the TDOA usng the coherent method (wthout compresson) the varance s gven by [] σ TDOA, coho (5) (πβ ) np( + + ) SNR DNR SNR DNR Recall that DNR must be hgh enough to detect the pulses pror to gatng, say DNR 0 db, for whch (5) can be shown to be approxmately equal to the varance n (4). owever, there s an addtonal constrant that must be met for (4) and (5) to be vald: the cross-correlaton output SNR must exceed about 5 db. For sem-coherent processng, (4) s vald f the output SNR n () must exceed 5 db: ths requres that nsnr 5 db. owever, for coherent processng, (5) s vald f np ( + ) 5 db (6) + SNR DNR SNR DNR For the SNR/DNR scenaros here ths reduces to npsnr 5 db whch shows that the coherent method can work at an SNR that s p tmes lower than for sem-coherent. In other words, the coherent method has a processng gan of np whereas the sem-coherent method has a processng gan of only n. For the example gven n Fgure (n 40, p 0) the sem-coherent method should break down when SNR < db whle the coherent method doesn t break down untl SNR < db. Ths explans why the sem-coherent smulaton result devates from the coherent result at SNR db as shown n Fgure. To gve a rough dea of how the accuracy of noncoherent processng compares we use [8] σ TDOA, nonco, (7) SNR where t R s the rse tme of the pulse. For the smulaton n Fgure the rse tme was about µs, whch at an SNR of 5 db gves a TDOA error of about 350 ns, whch s extremely large compared to the results obtaned usng the sem-coherent and coherent methods. Monte Carlo smulatons were also performed to demonstrate the capablty of the coherent processng method. Due to space lmtatons we show only the TDOA results; the FDOA results are smlar n nature. The TDOA accuracy results are shown n Fgure, where t s seen that usng the SVD method actually mproves the accuracy despte the fact that t requres much less data transferal; at moderate SNR values the mprovement s on the order of a 3 db mprovement n effectve SNR. The case consdered here t R had p 50 pulses and n 43 samples/pulse gvng a compresson rato of 3:. Table gves some dea of the large compresson ratos achevable for typcal scenaros. Four emtter scenaros are lsted wth typcal pulse wdths (PW), pulse repetton ntervals (PRI), bandwdths (BW), number of complex samples per pulse, and the number of pulses that would typcally be processed n order to get desrable accuraces. In addton, the resultng compresson ratos (after gatng) are computed usng the equatons gven above. From these results t s clear that these methods can gve very large compresson ratos. Furthermore, f SNR s hgh enough then the sem-coherent and coherent methods have the same TDOA accuracy; however, the coherent method can operate at lower SNR values due to ts hgher tme-bandwdth gan. REFERENCES [] P. C. Chestnut, Emtter locaton accuracy usng TDOA and dfferental doppler, IEEE Trans. Aero. and Electr. Systems, vol. AES-8, pp. 4-8, March 98. [] S. Sten, Algorthms for ambguty functon processng, IEEE Trans. Acoust., Speech, and Sgnal Processng, vol. ASSP-9, pp. 588-599, June 98. [3] M. L. Fowler, Data compresson for TDOA/DD-based locaton system, US Patent #5,99,454 ssued Nov. 3, 999, Lockheed Martn Federal Systems. [4] G. Desjardns, TDOA/FDOA technque for locatng a transmtter, US Patent #5,570,099 ssued Oct. 9, 996, Lockheed Martn Federal Systems. [5] M. L. Fowler, Coarse quantzaton for data compresson n coherent locaton systems, IEEE Trans. Aero. and Electr. Systems, vol. 36, no. 4, pp. 69 78, Oct. 000. [6] M. L. Fowler, Data compresson for emtter locaton systems, Conference on Informaton Scences and Systems, Prnceton Unversty, March 5-7, 000, pp. WA7b-4 WA7b-9. [7] M. L. Fowler, Decmaton vs. quantzaton for data compresson n TDOA systems, n Mathematcs and Applcatons of Data/Image Codng, Compresson, and Encrypton III, Mark S. Schmalz, Edtor, Proceedngs of SPIE Vol. 4, pp. 56 67, San Dego, CA, July 30 August 4, 000. [8] R. G. Wley, Electronc Intellgence: The Intercepton of Radar Sgnals. Artech ouse, 985. [9] R. G. Wley, Electronc Intellgence: The Analyss of Radar Sgnals, nd Edton. Artech ouse, 993. [0] T. K. Moon and W. C. Strlng, Mathematcal Methods and Algorthms for Sgnal Processng. Prentce all, 000. [] T. I. Laakso, V. Valmak, M. Karjalanen, and U. K. Lane, Splttng the unt delay, IEEE Sg. Proc. Mag., vol. 3, no., pp. 30-60, January 996. [] M. L. Fowler and Z. Zhou, Data compresson va pulse-to-pulse redundancy for radar emtter locaton, submtted to Mathematcs and Applcatons of Data/Image Codng, Compresson, and Encrypton IV, SPIE Annual Symposum, August,00.
RMS TDOA Error (ns) 5 4 3 Cross Correlaton SVD Method NF Prototype DNR 0 db 00 MC Runs Fgure 0-5 0 5 0 5 0 5 30 SNR (db) 0-8 RM S TDO A Error (sec) 0-9 W thout SVD W th SVD 0-0 -5 0 5 0 5 0 5 30 35 40 Input SNR (db) Fgure Table PW (µs) PRI (µs) BW (M z) # Pulses p Samples/ pulse n CR sem Eq. (6) CR coho Eq. (0) w/ k 0.5 600 4.0 80 300 6 80 300 0.7.0.5 0.0,500 4,000 8,500 4,000 54.5 67. 6.5 70.5 50,500 4 50,500 38.5 94.6 9.0 40.8 60 400 38 60 400 3.6 6.4