WIDE AREA, FINE RESOLUTION SAR FROM MULTI-APERTURE RADAR ARRAYS

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1 WIDE AREA, FINE RESOLUTION SAR FROM MULTI-APERTURE RADAR ARRAYS James M. Stles (1) ad Natha A. Goodma (2) (1) Radar Systems ad Remote Sesg Laboratory The Uversty of Kasas 2335 Irvg Hll Road, Lawrece, Kasas USA Phoe: , Fax: Emal: (2) Departmet of Electrcal & Computer Egeerg The Uversty of Arzoa 1230 E. Speedway Blvd., Tucso, Arzoa USA Phoe: , Fax: Emal: ABSTRACT Wdebad Mult-statc radar provdes scatterg observatos over frequecy, tme, ad space, thus allowg for measuremets delay, Doppler, ad agle-of arrval (.e., spatal frequecy). Statoary targets ambguous delay ad/or Doppler caot be physcally collocated, so that ther spatal resposes wll ot geerally be ambguous or eve sgfcatly correlated. As a result, a mult-statc radar ca be desged such that the sesor space-tme-frequecy ambguty fucto produces o large (e.g., gratg) lobes over a large llumated area. The sze of ths uambguous magg area s depedet o the umber of spatal apertures mplemeted the wde bad desg, creasg sze as the umber of spatal elemets s creased. The costrat o ths desg s the value of the measuremet clutter rak, as compared to the sesor measuremet dmeso. Essetally, the umber of llumated resoluto cells caot exceed the umber of depedet measured samples. Addtoal spatal elemets (apertures) ca crease the measuremet dmeso wthout affectg clutter rak. If a suffcet umber of spatal elemets are avalable, a SAR mage ca be formed for a dstrbuted target of ay clutter rak (.e., resoluto ad mage area). It ca be show that a ecessary codto s that the total aperture area must exceed the mmum aperture area codto mposed o tradtoal (sgle aperture) SAR. However, a multaperture desg, ths total area does ot determe, or lmt, the extet of the llumated area. The sze of each dvdual elemet sets the llumato area, ad thus addg spatal elemets to a mult-statc SAR desg wll geerally mprove sesor performace, wthout alterg the llumato area or resoluto sze. For cotguous spatal arrays, we fd that SAR mages ca be formed by smply extedg tradtoal correlato processg over space ad tme. For sparse, radom arrays, we fd that SAR mage formato requres a MMSE (.e., Weer) estmator. The computatoal cost of ths lear estmator ca be reduced by ether mplemetg a reduced rak verso, or by applyg a teratve Weer processor (.e. a Kalma flter). For all these SAR processors, we fd that mage qualty mproves as the umber of spatal elemets s creased. 1. INTRODUCTION Recetly, there has bee sgfcat terest the possblty of costructg a dstrbuted space-bore sesor from a collecto of dvdual small satelltes [1-3]. It has bee show that a cluster of satelltes wll rema grouped together orbt, provdg they are placed specfc geometres. The resultg formato flyg of these satelltes provdes terestg opportutes remote sesg, as ths cluster of satelltes would essetally form oe large sesor wth tremedous spatal extet (e.g., hudreds of meters to klometers). For example, a collecto of dvdual radar satelltes could be clustered to effectvely produce a sgle radar wth a very large, albet sparse, array [4-7]. Several research questos arse from ths oto. How ca the dvdual radars be combed to produce a optmal overall sesor? What would the performace beefts of ths sesor be, terms of both SAR ad Movg Target Idcato (MTI)? How should the sgal processg be performed?

2 The beeft of a multple satellte radar sesor s two-fold. Frst, the large spatal extet of the vrtual satellte provdes excellet agle-of-arrval formato about a llumated area. For example, the sesor would provde multple baseles for terferometrc processg. Secodly, multple radar satelltes, each wth a coheret recever, greatly crease the amout of depedet data collected over a gve tme. As more satelltes (.e., recevers) are added to the array, more depedet formato about the llumated area ca be collected. Ths paper wll show that, f properly processed, ths formato ca be used to produce fe-resoluto SAR mages over a arbtrarly large area or swathwdth. However, tradtoal matched flter approaches to SAR mage formato wll typcally ot provde adequate SAR magg results. Istead, t wll be show that a Weer flter approach s ecessary to process the spacetme measuremets of a mult-aperture SAR system. 2. THE RADAR MEASUREMENT MODEL I ths secto, a mathematcal model descrbg the space-tme measuremets of a mult-aperture SAR sesor s preseted. Ths model wll be used to develop the SAR processors preseted the remag sectos. The arrowbad, complex measuremet of a SAR aperture at spatal locato r ad at tme t ca be expressed as: rrt (,) = γ ()(;,; xhxrtr, t )( sr, t ) dtdrdx + rt (,) A V 0 T 0 = γ ( x) ρ( x; r, t) dx + ( r, t) A (1) I ths represetato, sr (,) t descrbes the trasmtted sgal, whch for a mult-aperture case ca be a o-separable fucto space ad tme. The fucto hx ( ; r, tr ;, t ) effectvely defes the measuremet space-tme mpulse respose for a pot target located at x o the llumated surface A. Ths fucto cludes the trasmtter ad recever resposes, as well as the propagato from the trasmtter (at r, t ), to target, ad back to the recever (at r,t). Note that because of SAR moto, ths mpulse respose s ot tme-varat. The fucto γ 0 ( x ) s a ormalzed scatterg parameter that defes the complex scatterg of the dstrbuted target across the llumated surface A, ad r (,) t s recever ose. After evaluatg the tegrato over all trasmtter tme T ad spatal volume V, the respose rrt (,) ca lkewse be descrbed as the secod equato (1). I ths expresso, the receved sgal ca be vewed as a superposto of the resposes from each pot of the llumated surface ( ρ (;,) xrt ), weghted by the complex scatterg ampltude of the surface ( γ 0 ( x )). Sce the complex trasmt sgal sr (,) t wll be costraed (approxmately) both tme ad frequecy, we ca accurately express the trasmt sgal terms of a fte umber of space-tme bass fuctos ϕ (,) rt: N srt (,) s ϕ (,) rt (2) Thus, the space-tme trasmt code s completely specfed by a N dmesoal data vector s, whose elemets cosst of the N complex values s. Addtoally, the scatterg of the dstrbuted target ca be descrbed as the scatterg from a collecto of I dscrete scatterers, each wth a complex scatterg coeffcet of γ = γ0 ( x ) A, where A defes a secto of surface area that s less tha or equal to the resoluto of the radar. To accurately represet the radar measuremet, the area A of each surface cell must be small eough such that the umber of llumated cells I exceeds the clutter rak of the radar measuremet. Clutter rak s approxmately equal to the umber of resoluto cells llumated by the radar, ad thus the area A, must be less tha or equal to the SAR resoluto o the surface [8]. As a result of these approxmatos, the tegrato of (1) ca be wrtte a lear algebrac form:

3 r = γ Hs+ = γ ρ + (3) The vectors r ad are M-dmesoal vectors whose elemets are the space ad tme sampled values of rr (,) t ad r (,) t respectvely. Essetally, the vector r cotas the complex sampled data collected by a mult-aperture radar over tme T. The elemets of matrx H are: m m m V T H = h( x ; r, t ; r, t ) ϕ ( r, t ) dt dr (4) It s mportat to ote that sce the mpulse respose hx ( ; r, tr ;, t ) s tme-varat, the matrx H s ot crculat. The vector ρ = Hs s the ormalzed space-tme respose of the clutter target at surface locato x, ad equato (3) aga shows that radar measuremet s modeled as weghted superposto of resposes from dssmlar targets. Ths s more cocsely wrtte by formg a I dmesoal vector γ of elemets γ, ad a M by I matrx Ρ whose colums cosst of vectors ρ : ι r = Ργ + (5) T γ γ1 γ2 γ I ad ρ1 ρ2 I where = P = ρ. I addto to beg more ameable to umerc computato, the beeft of ths represetato s that lear algebrac techques ca be drectly appled to the radar estmato problem. 3. SINGLE-APERTURE SAR IMAGE FORMATION Gve that the radar parameters (e.g., posto, velocty, wavelegth, trasmt sgal, atea patter) are accurately kow, the ormalzed target respose vectors (ad therefore matrx P) are kow a pror. It s surface scatterg, ρ ι descrbed by complex vector γ that s ukow, ad thus the SAR problem ca be smply stated as estmatg the vector γ, gve some measuremet vector r. Ths paper s cocered wth determg effectve lear estmators, ad ay lear estmato process ca be wrtte as a multplcato of r wth matrx W, so that W s regarded as the lear estmator: ˆ γ = Wr (6) where γˆ s the estmate of vector γ. Note the row vectors of matrx W (defed as w) provde a lear estmator for each surface clutter cell (.e., SAR pxel). A estmate of the scatterg coeffcet from the -th surface target ca thus be foud by takg the er product of w ad the measured vector r: ˆ γ = w r (7) It s evdet from (6) that a ecessary codto for provdg a depedet estmate of γ s that the dmeso of r must meet or exceed that of γ (M > I ). I other words, the umber of depedet measuremets must exceed the clutter rak (umber of llumated resoluto cells). It s ths fact that ultmately lmts the sze of a stadard SAR

4 mage. As the llumato area s creased, the clutter rak creases utl t surpasses the measuremet dmeso M the result s the mmum aperture requremet for SAR sesors. The smplest ad perhaps most commo lear estmator s a correlato processor, also kow as a matched flter. The weght vectors w of ths lear estmator are gve as: w = ρ 2 ρ (8) Recall that the elemets of these vectors are complex measuremets across space ad tme, ad thus (6) descrbes a space-tme matched flter. Geerally speakg, ths weght vector ca be separated to two depedet operatos, oe spatal (.e., beamformg) ad oe temporal (.e., stadard tme-frequecy SAR processg). As a test case, a stadard spacebore SAR scearo usg oe atea elemet was derved ad used to calculate the values of matrx P (.e., every ρ ). A flat Earth approxmato was used, ad the dstrbuto of scatterers across a llumated secto of the surface was assumed to correspod to the mage Fg. 1. Each pxel of the mage correspods to a equvaletly located -th resoluto cell o the surface, ad the magtude of the pxel was used to specfy the magtude of the complex scatterg coeffcet γ. The phase of each γ was radomly selected over 2π radas. Fg. 1. Image represetg the magtude of the scatterg across a llumated surface, for purposes of the smulatos preseted ths paper The resultg measuremet vector r was calculated, alog wth a radom vector, to specfy the system ose. Ths measuremet vector was processed wth (6) to geerate the SAR mage of Fg. 2a. The mage s obvously of poor qualty, a result that was predctable sce the llumated area ths scearo was too large (.e., the SAR aperture was too small), so that the clutter rak greatly exceeded the measuremet dmeso s the mmum aperture sze requred by SAR. If the llumato area s decreased (the aperture s creased) to a suffcetly large value, a uambguous mage ca be created, as show Fg. 2b. The problem wth ths mage, of course, s that t does ot cover the spatal extet of the orgal problem, as the larger SAR aperture llumates a smaller rego of the Earth s surface. From ths perspectve, t would appear that the soluto to creatg a uambguous mage over the etre spatal extet represeted by Fg. 1 s to crease the dmeso of r;.e., to take more depedet measuremets. The dmeso of r (assumg each measuremet s depedet) s equal to the tme-badwdth product (BT) of the receved sgal, where badwdth B s approxmately equal to the badwdth of the trasmtted sgal, ad T s aperture tme, or Coheret Processg Iterval (CPI). Icreasg ether of these two parameters would deed result more depedet measuremets, ad therefore a larger uambguous SAR mage could be costructed. The problem s that ths mage

5 would be larger terms of umber of resoluto cells, but ot spatal extet. Icreasg sgal badwdth ad processg tme also mproves rage ad Doppler resoluto respectvely. Therefore, a mage of equal sze but wth fer resoluto ca be costructed, but ot oe of greater spatal extet. Aga, the result s the well-kow lmt o SAR mage area (e.g., swathwdth). (a) (b) Fg. 2. The ambguous SAR mage resultg whe exceedg the maxmum llumated area lmt (a), ad whe costrag the llumato to create a uambguous, but spatally small mage (b). Both a large ad uambguous mage s desred. 4. MULTI-APERTURE SAR IMAGE FORMATION The soluto, therefore, to costructg a fe-resoluto SAR mage over a large spatal extet s to crease the umber of depedet samples collected by the sesor, wthout modfyg the resoluto of the radar. I other words, the measuremet must be creased wthout chagg the measuremet clutter rak. Sce sesor resoluto s determed by temporal (.e., delay ad Doppler) formato, t s evdet that addtoal spatal formato ca be used to crease the dmeso of measured vector r, whle preservg the fudametal resoluto of the radar. For example, f the aperture used to produce the uambguous mage of Fg. 2b s subdvded to create a array of e elemets, ad each elemet s mplemeted wth a coheret recever, the the umber of depedet measuremets collected creases e tmes. The elemets of the radar receve vector r are ow measuremets space ad tme, rather tha tme aloe. As a result, the dmeso of r creases as a factor of N to approxmately NBT, where N s the umber of elemets the array (for ths case N=9). Wth ths addtoal spatal formato, a uambguous mage ca be formed over the etre spatal extet of the orgal scearo, wth the result provded Fg. 3. Correlato processg was lkewse used for ths example, but ow ths correlato was performed over measuremets space-tme, rather tha o tme data aloe. Ths processg effectvely performs both stadard delay-doppler processg of the data across tme ad spatal beamformg processg of the array formato. Fg 3. Image resultg from the smulated measuremets of a mult-aperture SAR. Space-tme correlato processg was used, essetally provdg both delay-doppler ad spatal beamformg processg.

6 Ths pot s emphaszed by Fg. 4. Say the mage of Fg. 1 s replaced wth a surface cosstg of a sgle scatterg target, placed precsely at the ceter of the llumated area. Fg. 4a dsplays the resultg mage f a sgle receve aperture were mplemeted, showg clearly the locatos of the eght targets ambguous delay ad doppler. Effectvely, ths mage dsplays the ambguty (or pot spread) fucto of the sesor, projected oto the llumated surface. Alteratvely, Fg. 4b dsplays the mage f oly the spatal data of the e-elemet array were used. Ths mage essetally dsplays the array beamformg patter, projected oto the mage area. Clearly, the resoluto assocated wth ths patter s poor compared to delay-doppler. However, ote that the ma lobe s suffcetly arrow to reject the delay-doppler ambguous targets show Fg. 4a. As a result, the mage created from processg the etre space-tme measuremet vector r s dsplayed Fg. 4c. Effectvely, ths s the ambguty fucto of the overall magg sesor, a respose that approaches the deal thumbtack fucto. Note ths space-tme ambguty fucto s effectvely the multplcato of the delay-doppler fucto wth the spatal beamformg fucto. The result shows that the resposes from targets the llumated area ca be correlated delay-doppler, provded that they are ucorrelated spatally (ad vce versa). (a) (b) (c) Fg. 4. The sesor delay-doppler ambguty fucto (a), the sesor spatal ambguty fucto (.e., beamformg patter) (b), ad the combed space-tme ambguty fucto (c). It should be oted that a Mult-Aperture SAR (M-SAR) stll retas the mmum aperture area requremet that s assocated wth stadard SAR. If the total aperture area s too small, the beamwdth of the beamformg patter s suffcetly arrow to reject targets ambguous delay-doppler. The dfferece betwee M-SAR ad stadard SAR s that ths lmt o aperture area does ot lkewse lmt, or otherwse determe, the sze of the llumated area or the extet of the resultg mage. For ay sesor resoluto, a mage or swathwdth of arbtrarly large extet may be obtaed by subdvdg the orgal aperture (wth suffcet area) to creasgly smaller (ad thus more umerous) aperture elemets. 5. SPARSE MULTI-APERTURE SAR IMAGE FORMATION For the dstrbuted radar cocept, the spatal array (.e., satellte costellato) would be radomly ad sparsely populated, as opposed to the cotguous array case cosdered the prevous secto. A sparse array scearo was thus developed, wth the relatve elemet posto radomly selected over a costraed area. Itally, space-tme correlato processg was appled to the smulated measuremets, wth the result show Fg. 5. Fg. 5. A poor qualty SAR mage resultg from a sparse, mult-aperture SAR, usg correlato processg.

7 Clearly, correlato processg s suffcet for formg qualty SAR mages wth sparse spatal arrays. The reaso for ths s see whe examg the beamformg patter of the sparse array, as show Fg. 6a. Although the large spatal extet of the array provdes a arrow ma beam, the sparse ature of the array leads to a patter wth sgfcat sdelobes. The resultg space-tme ambguty fucto s dsplayed Fg. 6b, ad demostrates that the ceter target s partally correlated wth other targets wth the llumated area. As a result, the SAR mage s of poor qualty. Aother way of demostratg ths problem s to drectly evaluate (6) for the matched flter case (7). The resultg estmate of γ s: ρρ ι j ρι = + j j ρ ρ γˆ γ γ (9) Note ths estmate cossts of three terms. The frst s the correct scatterg value γ ; therefore the remag two terms represet estmate error. The last term represets error due to ose. The correlato processor mmzes ths term, by maxmzg the receved eergy wth respect to ose. The secod term represets error due to the correlato of the weght vector wth all other llumated targets. The correlato flter does othg to mmze ths term, ad therefore, f the resposes from the llumated targets are sgfcatly correlated, a correspodgly large error wll result. Geerally, for the sparse array case, the resposes of some resoluto cells wll be sgfcatly correlated, ad thus a dfferet lear estmator must be mplemeted. (a) (b) (c) Fg 6. The beamformg patter created by the sparse spatal array (a), ad the poor space-tme ambguty fucto that results from ths (b). 4.1 The Maxmum Lkelhood Estmator The Maxmum Lkelhood (ML) estmator provdes the estmate of γ that maxmzes the lkelhood fucto p(r γ), where p(r γ) s the codtoal probablty desty fucto (pdf) of r gve γ. For equato (3), t ca be show that ths estmate s acheved by mplemetg the lear estmator: Wml = Ρ 1 (10) where P ~1 s the pseudo verse of matrx P. The ML estmator provdes a weght vector w that s orthogoal to all other llumated targets (.e., w ρ 0 ). Essetally, the ML estmator correlates r wth the compoet of the target j = vector ρ that resdes a subspace orthogoal to the resposes of all other resoluto cells. Thus, ths estmator mmzes the error due to clutter, where clutter s defed as the resposes from all other resoluto cells. There are, however, two requremets for satsfactory results whe mplemetg the ML estmator. The frst s that the matrx P be well codtoed, so that that the verse P ~1 ca be effectvely determed. Ths requremet ca geerally be satsfed f the dmeso of r far exceeds the measuremet clutter rak. I other words, the requremet ca be satsfed f the umber of apertures the sparse array sgfcatly exceeds the mmum requred for proper mage formato. The secod requremet relates to system ose. The ML estmator mmzes the estmate error due to clutter, but does othg to mmze the error due to ose. As a result, the estmate error (.e., the SAR mage) quckly degrades as measuremet SNR decles.

8 4.2 The Mmum Mea-Squared Error Estmator Gve that we have some a pror kowledge of the radar SNR, a mmum mea-squared error estmator ca be mplemeted, whch ca be show to be: W = σ Ρ [ ΡΡ σ + σ I] mmse γ γ 1 (11) 2 where σ s the expected value of γ γ 2 2, ad σ s the expected value of k 2, where k s a elemet of ose vector. It ca be show that ths soluto s lkewse the Maxmum A Posteror (MAP) estmator, whch s the soluto that maxmzes the fucto p(γ r), where p(γ r) s the codtoal pdf of γ gve r.. The resultg magg s dsplayed Fg. 7, ad demostrates a marked mprovemet over the correlato processor of Fg. 5. The smulato data used both Fgs. 5 ad 7 are detcal, the oly dfferece betwee the results were the lear estmators used to process that data. Fg. 7. A qualty SAR mage resultg from a sparse, mult-aperture SAR, usg MMSE processg. Ths estmator s the dscrete mplemetato of a Weer flter ad mmzes the estmato error due to both ose ad clutter. I other words, f the correlato processor maxmzes sgal to ose, ad the ML estmator maxmzes sgal to clutter, the MMSE estmator ca be sad to maxmze sgal to terferece, where terferece s defed as the summato of both clutter ad ose eergy. Accordgly, ths estmator provdes SAR mages superor to both correlato ad ML processg for all SNR. It should be lkewse oted that the ML ad MMSE estmators are true space-tme solutos, that the resultg weght vectors are ot separable space ad tme. I cotrast to the correlato processor, these solutos caot be vewed or terpreted as a tme-frequecy Weer flter followed by a spatal Weer flter. The cost of mplemetg ths processg s the addtoal complexty determg the lear estmator W mmse. However, t should be oted that oce W mmse s calculated, the basc matrx multplcato W mmse r s o more complex tha ether the correlato or the ML estmator. Addtoally, for large problems, the matrx verse operato ca be problematc. Reduced rak methods ca be mplemeted [9], a techque used to produce all the results ths paper. The tme data at each recever were processed wth a matched flter, ad the the ML ad MMSE processg was completed o the remag spatal data. I ths maer, the rak of the matrces to be verted was o greater tha the umber of array elemets, typcally less tha 20. A alteratve approach s to apply a teratve soluto of the Weer flter [10], commoly kow as the Kalma flter. For ths mplemetato, the measuremet data vector r s frst sub-dvded to multple segmets. The ovato vector ad Kalma ga s determed for each measuremet segmet, ad the SAR mage estmate s updated. Whe all the data segmet of vector r are used, the resultg mage s detcal to the Weer (MMSE) soluto appled to the etre measuremet vector r. The beeft mplemetg the teratve (.e., Kalma) soluto s strctly computatoal effcecy, as the Kalma ga calculato requres a verso of a matrx whose dmeso s equal to the dmeso

9 of each data segmet. The result s that the MMSE SAR mage s formed by determg the verse of may small matrces, as opposed to oe large matrx verso. The result s sgfcat computatoal savgs, partcularly whe the dmeso of the measuremet data vector r s very large. 6. ERROR ANALYSIS A umber of smulatos were coducted for varous sesor scearos to evaluate the performace of the varous processg techques as a fucto of measuremet SNR ad umber of recever apertures. The evaluato crtero was the mea-squared error of the scatterg values γ, as compared to the actual values dsplayed by Fg. 1. Fg. 8 dsplays ths error as a fucto of the umber of receve apertures used by the sesor. For each case, a sparse aperture desg was mplemeted, where the postos of the apertures were radomly selected wth a costraed rego. A mmum of e apertures was requred to collect a suffcet umber of depedet samples for ths test case, so the horzotal axs of Fg. 8 begs at e. 0 5 o Matched * Orthogoal + MMSE Normalzed MSE (db) MMSE *ML o correlato Number of Apertures Fg. 8. SAR mage error as a fucto of umber of recever elemets. The best results occur whe the umber of elemets sgfcatly exceeds the mmum umber requred for mage formato. Fg. 8 shows that mage qualty for all three processors mproves as the umber of recevers creases, wth partcularly sharp mprovemet the ML ad MMSE results. As stated earler, extedg the umber of receve apertures beyod the requred mmum mproves the codto of the matrx that must verted, ad thus mproves the resultg mage from MMSE ad, partcularly, ML estmato. Fg. 9 dsplays the performace of the estmators as a fucto of SNR. The SNR value preseted s the SNR of a sgle radar measuremet; other words, the average SNR of a sgle elemet vector r. The fgure demostrates the superorty of the MMSE processor at all SNR. It lkewse shows the sestvty of the ML to SNR, approachg MMSE performace as SNR creases, but degradg rapdly as SNR decreases. Fally, the plot demostrates that, ulke the ML or MMSE estmator, the correlato processor s ot a effcet estmator. I other words, the estmato error assocated wth the correlato processor does ot coverge to zero as SNR approaches fty o Matched * Orthogoal + MMSE 0 Normalzed MSE (db) MMSE *ML o correlato SNR (db) Fg. 9. SAR mage error as a fucto of measuremet SNR. Note the MMSE processor provdes the best results for all SNR.

10 7. CONCLUSIONS The dea of a dstrbuted, mult-aperture spacebore sesor represets a ew paradgm for SAR sesor desg. The stregth of ths dea s the addto of a wde volume of spatal samples to the tradtoal delay-doppler SAR formato. Although the resultg spatal array s sparse ad radom, the resultg spatal data s suffcet to decorrelate surface target resposes that are otherwse perfectly correlated (ambguous) delay ad Doppler. As a result, a suffcetly populated mult-aperture sesor ca be used to form uambguous, fe-resoluto SAR mages over a wde surface area or swathwdth. However, mage formato ca be accomplshed oly by mplemetato of a space-tme Weer flter, as opposed to the tradtoal correlato processg used stadard SAR sesors. ACKNOWLEDGEMENT Ths worked was fuded by the US Ar Force Offce of Scetfc Research (AFOSR), grat # AFOSR F REFERENCES 1. Zetocha P., et al, Commadg ad cotrollg satellte clusters, IEEE Itellget Systems, vol. 15, o. 6, pp. 8-13, Nov., Ktts C., et al, Emerald: a low-cost spacecraft msso for valdatg formato flyg techologes, Proc IEEE Aerospace Cof., Aspe, CO, pp Yedavall, R.K., Sparks, A.G.; Satellte formato flyg cotrol desg based o hybrd cotrol system stablty aalyss, Proc. 2000Amerca Cotrol Coferece, Vol. 3, , Massoet D., et al, A wheel of passve radar mcrosats for upgradg exstg SAR projects, Proc. IEEE 2000 It. Geosc. Rem. Ses. Symp., Hoolulu, HI, Massoet D., Capabltes ad lmtatos of the terferometrc cartwheel, IEEE Tras. Geosc. Rem. Ses., vol. 39, o. 3, , Goodma N., L S.C., Rajakrsha D., ad Stles J.M., Processg of multple recever, spacebore arrays for wde-area SAR, IEEE Trasactos o Geoscece ad Remote Sesg, vol. 40, , Das A., Cobb R., ad Stallard M., TechSat 21: a revolutoary cocept dstrbuted space based sesg, Proc. of the AIAA Defese ad Cvl Space Programs Coferece ad Exhbt, Hutsvlle, AL, AIAA , Goodma N. ad Stles J.M., Sythetc Aperture Characterzato of Radar Satellte Costellatos, Proceedgs of the IEEE Iteratoal Geoscece ad Remote Sesg Symposum, vol. 1, , Hog M.L. ad Xao W., Performace of reduced-rak lear terferece suppresso, vol. 47, o. 5, , Kay S.M., Fudametals of Statstcal Sgal Processg: Estmato Theory, Pretce Hall, Saddle Rver, NJ, 1993.