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1 Sensors,, 8-97; do:.339/s8 OPEN ACCESS sensors ISSN Artcle Extended Target Recognton n Cogntve Radar Networks Ymn We, uadong Meng *, Ymn Lu and Xqn Wang Department of Electronc Engneerng, Tsnghua Unversty, Beng 84, Chna; E-Mals: weym@mals.tsnghua.edu.cn (Y.M.W.); ymnlu@tsnghua.edu.cn (Y.M.L.); wangxq_ee@tsnghua.edu.cn (X.Q.W.) * Author to whom correspondence should be addressed; E-Mal: menghd@tsnghua.edu.cn; Tel.: ; Fax: Receved: 5 September ; n revsed form: 3 October / Accepted: November / Publshed: November Abstract: We address the problem of adaptve waveform desgn for extended target recognton n cogntve radar networks. A closed-loop actve target recognton radar system s extended to the case of a centralzed cogntve radar network, n whch a generalzed lkelhood rato (GLR) based sequental hypothess testng (ST) framework s employed. Usng Doppler veloctes measured by multple radars, the target aspect angle for each radar s calculated. The ont probablty of each target hypothess s then updated usng observatons from dfferent radar lne of sghts (LOS). Based on these probabltes, a mnmum correlaton algorthm s proposed to adaptvely desgn the transmt waveform for each radar n an ampltude fluctuaton stuaton. Smulaton results demonstrate performance mprovements due to the cogntve radar network and adaptve waveform desgn. Our mnmum correlaton algorthm outperforms the egen-waveform soluton and other non-cogntve waveform desgn approaches. Keywords: cogntve radar network; radar waveform desgn; target recognton. Introducton The mportance of radar target dentfcaton s wdely recognzed and t has become one of the maor concerns n radar survellance and homeland securty applcatons []. Normally, radar hgh range resoluton profle (RRP) s used as an mportant feature n radar automatc target recognton (ATR) [-5], snce t contans target structure sgnatures, such as target sze, scatterer dstrbuton, etc. [6].

2 Sensors, 8 In [7], a target mpulse response was ntroduced to model target scatterng behavor, and an optmal transmt waveform and recever flter par was proposed for extended target detecton n addtve Gaussan nose. By maxmzng the output sgnal-to-nose rato (SNR), Bell [7] derved an egen-waveform soluton under a total energy constrant. The egen-waveform soluton has been heurstcally extended to tackle the mult-extended-target dentfcaton problems n [8], where the transmt waveform s desgned to maxmze the average (weghted average, generally) Eucldean dstance or Mahalanobs dstance (n addtve colored nose) between dfferent hypotheses. Recently, aykn proposed the novel dea of cogntve radar [9], one of whose most mportant characterstcs s closed-loop operaton. Wth the feedback structure from the recever to the transmtter, waveforms can be adaptvely optmzed based on pror knowledge about targets and envronments to mprove system performance and effcency. Many pror attempts have focused on target recognton usng waveform adaptaton n cogntve radar. aykn [9] suggested that such a cogntve radar system can be represented usng a Bayesan formulaton whereby many dfferent hypotheses are gven a probablstc ratng. Based on ths dea, Goodman [] proposed the ntegraton of waveform desgn technques [8] wth a sequental-hypothess testng (ST) framework [] that controls when hard decsons may be made wth adequate confdence []. e also compared two dfferent waveform desgn technques for use wth actve sensors operatng n a target recognton applcaton. One consdered by Bell [7] s based on a maxmzaton of the mutual nformaton between a random target ensemble and the echo sgnal, and the other s based on egenvectors of the weghted autocorrelaton matrx proposed by Guerc [8] and Plla [3]. To make full use of the transmt energy under a maxmum modulus constrant, an adaptve sngle-tone waveform desgn algorthm was proposed n the same stuaton [4]. The target hypotheses were further extended to statstcal characterzaton by power spectral denstes n [5] where waveforms are matched to the target class rather than to ndvdual target realzatons. owever, several ssues must be consdered when applyng mpulse response to radar ATR. The most mportant of these s the well-known target-aspect senstvty [6]. Snce the mpulse response represents the proecton of the target scatterng behavor onto the radar lne of sght (LOS) [6], varaton n the target aspect wll lead to dfferent mpulse responses. Wthout a pror knowledge of the target aspect angle, 36-degree template matchng s nevtable, whch wll cause sgnfcant degradaton of recognton accuracy, especally n stuatons wth a large number of hypotheses. In most survellance applcatons, the target, such as an arcraft or shp, s movng and ts maor axs (headng drecton) s approxmately parallel to ts velocty vector [6]. Therefore, the aspect angle can be acqured by estmatng the target velocty va trackng. Ths approach works effectvely when dealng wth non-maneuverng (constant velocty and acceleraton) targets. owever, the ablty to handle maneuverng targets s stll lackng [7]. Another maor ssue s the ampltude senstvty of returned echoes. Ths comes from the fact that the ampltude of a returned echo s affected by multple factors such as target dstance, antenna gan, recever gan, and weather condtons [6]. Snce some of these factors are unpredctable and unstable, the ampltude of a returned echo s usually unknown and varable. Therefore, the sgnal models [7,8,] that assume the ampltude of returned echoes s determnstc and accurately known are not sutable for practcal applcatons.

3 Sensors, 83 In ths paper, we address the problem of extended target recognton n cogntve radar networks whose consttuton was descrbed by aykn [8]. A cogntve radar network system should ncorporate several radars workng together n a cooperatve manner wth the goal of realzng a remote-sensng capablty far n excess of what the radar components are capable of achevng ndvdually [8]. In our extended target recognton applcaton, the radar network can provde more robust detecton performance [9,], more accurate poston estmaton [], and the most mportantly, more relable target aspect angle for each radar. Snce the velocty of the target can be drectly estmated usng the Doppler frequences measured by the ndvdual radars, the ssue of targetaspect senstvty can be solved even for maneuverng targets. Also, because the radar statons are located across a large area, the sensor network s able to obtan returned echoes from multple aspects at the same tme, whch leads to a sgnfcant mprovement n the effcency and robustness of the recognton. The man contrbuton of ths paper s to extend the above mentoned closed-loop actve target recognton radar system [] to the case of a centralzed cogntve radar network. Once all radars have performed ther observatons, the target aspect angle for each radar s calculated. The ont probablty of each target hypothess s then updated usng all the observatons from dfferent radars based on ther aspect angles. The next transmt waveform for each radar s desgned accordng to the ont probabltes of the target hypotheses. Such nterrogaton repeats untl hard decsons can be made wth adequate confdence. We also contrbute by consderng the ampltude uncertanty of the returned echoes. The deal echo sgnal, whch s the convoluton of the transmt waveform wth the target mpulse response, s multpled by a random complex coeffcent n our sgnal model. A generalzed lkelhood rato (GLR) based ST framework n whch the unknown parameters are replaced wth ther maxmum lkelhood estmates (MLE) s employed to update the ont probabltes of target hypotheses nstead of the lkelhood rato based approach n []. Although the GLR test s not optmal, t appears to work qute well n practce. Fnally, the adaptve waveform desgn algorthm descrbed n [8,] s appled to the cogntve radar network. Because the egen-waveform soluton [7,8,] s no longer sutable for the ampltude fluctuaton stuaton, a mnmum correlaton algorthm s proposed and compared wth the algorthm based on average Eucldean dstance. In the next secton, we defne the problem and system model. The GLR based ST framework and the centralzed Bayesan update equatons are presented n Secton 3. In Secton 4, the mproved adaptve waveform desgn algorthm s detaled. Smulaton results are shown n Secton 5, and fnally, Secton 6 concludes the paper.. Problem Descrpton and Modelng We consder the target recognton problem n whch one of M possble targets s known to be present. The poston and velocty of the target are assumed to be known by the radars. Our obectve s to dentfy the target accurately and quckly. In ths secton, we frst descrbe the centralzed cogntve radar network framework for solvng the ssue of target-aspect senstvty. Then, a parametrc measurement model s developed by consderng the ampltude uncertanty of the echo.

4 Sensors, 84.. System Model In most radar survellance applcatons, targets are far from the radar staton and move horzontally, make t reasonable to assume that the targets and the radar staton are located n the same horzontal plane [-6]. A two-dmensonal target model s used n our analyss. As shown n Fgure, each target hypothess s characterzed by a set of mpulse responses g, t measured offlne from every aspect angle. The aspect angle s defned as the angle between the maor axs of the target (headng drecton) and the radar LOS. For most of the targets, the maor axs has approxmately the same drecton as ts velocty vector [6]. The aspect angle can be acqured by estmatng the target velocty vector and the poston relatve to the radar. Fgure. Defnton of target aspect angle. y radar LOS θ eadng drecton x A cogntve radar network wth N radars s ntroduced to solve the ssue of target-aspect senstvty. As shown n Fgure, the radar statons are located n the same two-dmensonal space n whch the target moves. The poston of the th radar staton s denoted as R ˆ ˆ Rx x Ry y, where ˆx and ŷ are the axs unt vectors and R x and R y are the axs weghts. The poston of the target s denoted as R. The relatve poston of the target wth respect to the th radar s then gven by r R R. Accordng to our defnton, for a target wth velocty vector v, the aspect angle of the th radar s expressed as arg r arg v, where the functon arg() returns the angle of a vector n the x-y coordnate system. Wth the observatons from multple radars located n dfferent places, the radar network can provde a more robust detecton performance [9,] and more accurate poston estmaton []. More mportantly, the velocty of the target can be drectly estmated based on the Doppler frequences measured from dfferent angles []. Therefore, the estmated aspect angle can be calculated by: ˆ arg ˆ r arg vˆ () where ˆr and ˆv are the estmated relatve poston and velocty.

5 Sensors, 85 Fgure. Top vew of the cogntve radar network n an x-y coordnate system. y Radar No. r Target V Radar No. r R R r N R R N Radar No.N x.. Measurement Model Only backscatterng s consdered n our network, and nterference among the radars s gnored. Ths makes the network easy to mplement snce each radar staton can operate usng a dfferent frequency band. For the th radar n the network, when s present, the echo sgnal s t s determned by the aspect angle and the transmtted waveform u t. It s gven by: f t s t ae g, t u t n t where g, t s the target mpulse response for the th hypothess at aspect angle, * denotes the convoluton operator, f s the target Doppler frequency at aspect angle, a s a random complex coeffcent representng the ampltude and ntal phase uncertantes of the echo sgnal, and n t s addtve whte complex Gaussan nose at the recever. The Doppler frequency f s gven by f f cos c v c, where f c s the carrer frequency and c s the speed of lght. Because most of the radar systems are dgtzed, and for the convenence of smulaton, we use a dscrete-tme formulaton to replace the model n Equaton (). The estmated Doppler speed ˆv and the estmated aspect angle ˆ are used to elmnate the Doppler phase shft nsde the echo sgnal by multplyng the nverse phase sequence wth frequency ˆ f ˆ ˆ f cos c v c. Although the estmated Doppler frequency f ˆ s dfferent from the real value f, the slght dfference can be gnored. After phase compensaton, the echo sgnal s then gven by: s g u n (3), where all the contnuous-tme sgnals are sampled usng the same samplng nterval T s, s the dgtzed random coeffcent gven by Ta s, u s an Lu complex vector representng the transmt waveform, g, s an L normalzed complex vector representng the mpulse response of g ()

6 Sensors, 86 at aspect angle, n s an Ls ( Ls Lu Lg ) complex vector representng the crcularly symmetrc zero-mean complex Gaussan nose wth known varance, and s s an L complex vector representng the receved sgnal of the th radar. The convoluton operaton n Equaton (3) can be replaced wth matrx multplcaton by defnng the convoluton matrx: G, g g g,,,,, g g L g g L g, g,,, g where G, s an Ls Lu complex matrx. Equaton (3) can therefore be wrtten as:, g L s G u n (5) The transmt waveform u s restrcted by the total energy constrant, whch s gven by: where E s the normalzed transmt energy of the th radar. 3. GLR Based Sequental ypothess Testng g s (4) u u E (6) One of the three maor characterstcs of cogntve radar s the preservaton of the nformaton content of radar returns [9]. aykn [9] suggests that ths can be realzed usng a Bayesan approach. Based on ths dea, Goodman [] proposed the ntegraton of waveform desgn technques [8] wth a ST framework []. The test s based on sequental observatons and updates runnng n a closed-loop. It updates the probablstc understandng of all the hypotheses after each llumnaton and then makes a decson on the next transmt sgnal. In ths secton, we extend the ST framework to the case of centralzed cogntve radar networks. In addton, snce we lack knowledge of the parameter, a GLR based ST framework s used nstead of the one based on lkelhood rato. 3.. Ampltude Factor and GLR In our sgnal model Equaton (5), the varance of addtve nose can be measured offlne for each radar, but we stll lack the knowledge of the parameter. The lkelhood rato test cannot be appled to ths problem. Instead, we use the GLR test n whch the unknown parameter s replaced wth ts MLE. Although the GLR test s not optmal, t appears to work qute well n practce [].

7 Sensors, 87 When the aspect angle s known for Equaton (5), the MLE of parameter under ˆ and the generalzed lkelhood functon under,,, s [3]: G u G u G u s (7) s gven by: p s ; ˆ, exp s ˆ G ˆ u s G u (8) Ls,, The lkelhood functon s used to update the probablty of each target hypothess untl a decson has been made. In practce, we use the estmated ˆ from Equaton () to replace the unknown. Snce the scatterer dstrbuton changes slowly wth respect to the aspect angle (the poston and ntensty of scatterers reman approxmately unchanged wthn ten degrees [4]), our approxmaton s reasonable. Ths also gves us an opportunty to reduce the number of target templates from dfferent aspect angles stored n the knowledge base. A database wth the template from every three degrees s suffcent for practcal applcatons [5]. By replacng the unknown wth the estmated ˆ, the generalzed lkelhood functon s then gven by: ˆ ˆ ˆ p s ; ˆ, exp s ˆ G u s ˆ G u (9), Ls, 3.. Centralzed Bayesan Updates and Sequental Test Snce no nterference exsts among the radars n the network, the ont lkelhood functon under s the product of all the lkelhood functons n (9) as gven by: N s,, s ; ˆ ˆ N s, p p () where N s the total number of radars n the network or the number of radars coverng the target poston f the detecton area s not completely covered by the network. After every radar has performed an observaton, the lkelhood functons are gathered to update the probablty of each target hypothess. If we let P (k-) ( ) represent the probablty for before the k th observaton, the posteror probablty after executng the k th observaton s gven by: P k M N k k k ˆ ˆ k s P p ;, N k k k ; ˆ ˆ k P p s, The radar network contnuously nterrogates the target channel and updates the probablty of each hypothess untl the tme when hard decsons can be made wth adequate confdence. Let mn, for m n be the desred probablty of ncorrectly selectng n gven that m s true []. The GLR between m and n can be calculated as: ()

8 Sensors, 88 The experment termnates and k P k k m mn, () P n s selected to be true when the condton: k, n n, for all n (3) n, s met for some []. If the condton s not met for any of the hypotheses, another llumnaton cycle commences. 4. Adaptve Waveform Desgn Another maor characterstc of cogntve radar s the feedback structure from the recever to the transmtter [9]. Based on the pror probablty of each hypothess obtaned from prevous tests, the transmt waveform can be optmzed to enhance system performance and effcency. In ths secton, we provde two adaptve waveform desgn technques for extended target recognton. One s the egen-waveform soluton proposed by Guerc [8] and Goodman []. The waveform s desgned to maxmze the weghted average Eucldean dstance between all the target hypotheses where the probabltes of the hypotheses are used as the weghtng coeffcents. Ths works qute well n stuatons where the ampltude of the deal echo s known a pror. owever, the method s not sutable for stuatons wth ampltude fluctuaton where the parameter s unknown. To solve ths problem, we propose a mnmum correlaton algorthm for waveform desgn. 4.. Egen-waveform Soluton In stuatons wth no ampltude fluctuaton ( ), a provably optmal transmt waveform for the M case s derved by Guerc [8]. The transmt sgnal should maxmze the Eucldean dstance between the mean values of the lkelhood functons gven by:,,,, d u G G G G u (4) The unknown parameter s set to n Equaton (4), whch ndcates that the ampltude of the deal echo s known a pror. The predcted target aspect angle wth respect to the th radar at the tme of upcomng transmsson s denoted by. Predcton s requred when the target s movng at hgh speed. Otherwse, t can be replaced wth the latest estmated ˆ. The optmal waveform under energy constrant Equaton (6) whch maxmzes Equaton (4) s the egenvector correspondng to the maxmum egenvalue of the target autocorrelaton matrx defned as:,,,, Ω G G G G (5) When M, the autocorrelaton matrx s suggested to be n the form: M M m, n, m, n, (6) m, n m nm Ω G G G G

9 Sensors, 89 where mn, s a weghtng factor [8]. The transmt waveform s desgned to maxmze the weghted average Eucldean dstance of each bnary par, whch s the egenvector correspondng to the maxmum egenvalue of matrx Ω. Usng the target probabltes, Goodman [] compared two weghtng factor mn, optons, m, n Pm Pn and m, n PP m n, and found that the second weghtng coeffcent provdes better performance. owever, snce we have no dea about the value of unknown parameter, t should be set to f the egen-waveform soluton s appled drectly. 4.. Mnmum Correlaton Algorthm To solve the problem of ampltude senstvty, we suggest usng the mnmum correlaton crteron nstead of the maxmum Eucldean dstance. Fgure 3 shows the multdmensonal space for the receved sgnal under two target hypotheses, where Fgure 3(a) represents the stuaton wthout ampltude fluctuaton and Fgure 3(b) represents the one wth ampltude fluctuaton. Fgure 3. Multdmensonal space for receved sgnal under two target hypotheses. The stuaton wthout ampltude fluctuaton s shown n (a) and the stuaton wth ampltude fluctuaton s shown n (b). Gu s Gu s Gu Gu a b As shown n Fgure 3(a), snce the deal echo sgnals Gu and Gu are exactly known, the lkelhood rato between the two hypotheses s determned by the Eucldean dstance between s and Gu and the Eucldean dstance between s and Gu. Ths leads to the dea of desgnng the transmt waveform to maxmze the weghted average Eucldean dstance. owever, n stuatons wth ampltude fluctuaton, shown n Fgure 3(b), the GLR between the two hypotheses s determned by the perpendcular dstance from s to the axs of Gu and the axs of Gu. The transmt waveform should be desgned to make the axs of Gu and Gu perpendcular to each other, or, n other words, to mnmze the correlaton between Gu and Gu. When M, accordng to the egen-waveform soluton, the transmt waveform u s desgned to mnmze the weghted average of correlaton between each bnary par. Thus, n our optmzaton approach we am to acheve: M M arg mn PP m n u, m, n m nm u u G G u (7) subect to the constrant n (6), where P m and P n are the posteror probabltes of target hypothess and n. m

10 Sensors, 9 5. Results In ths secton, we demonstrate the benefts of a cogntve radar network for extended target recognton by comparng t to one wthout a feedback structure. We also compare the performance of the dfferent adaptve waveform desgn approaches descrbed n Secton 4. To evaluate the performance of the closed-loop system, 5 dfferent sets of targets are generated. Each set ncludes M 4 target hypotheses. For each hypothess, a two-dmensonal target wth multple reflecton centers s randomly generated accordng to: L,,,,,, g (8) x y l x x l y y l l where the number of reflecton centers L 5, the reflecton coeffcents, l are the samples of a zero-mean complex Gaussan dstrbuton wth unt varance, () denotes the Drac delta functon, and the locatons of the reflecton centers are the samples of unform dstrbuton n a crcular regon. The dameter of the crcle equals the length of the target mpulse response L g. From every aspect angle, the mpulse response represents the proecton of the reflecton centers onto the radar LOS [6], whch s gven by: L cos sn g t t (9),, l x,, l y,, l l Snce the Drac delta functon n Equaton (9) s not practcal and cannot be sampled n dscrete tme, the contnuous mpulse response s fltered usng an deal low-pass flter wth bandwdth B and sampled wth an nterval of Ts. The elements of the mpulse response vector g, are gven by: L n n L g snc cos sn (),, l g x,, l y,, l l where n, Lg and snc() denotes the normalzed snc functon. The mpulse response vector s then normalzed to unt energy. The specfed error rate n ST s mn,. for all hypotheses and the pror probablty P s set to / M for every. The length of all mpulse responses and waveform vectors s Lu Lg 3. In the observaton process, the addtve nose n s randomly generated wth varance and the ampltude factor s the sample of a zero-mean complex Gaussan dstrbuton wth unt varance. 5.. Adaptve Waveform Desgn For Fgure 4, three radars (located at R ˆ ˆ x y, R ˆ ˆ x y, and R ˆ ˆ 3 x y km) form the network. The target s located at R xˆ yˆ km and s movng wth velocty v xˆ yˆ m/s. The poston and velocty of the target are assumed to be exactly known by the radars. Fgure 4(a) shows the average number of teratons requred for each waveform desgn approach to reach a decson as a functon of transmt energy whle Fgure 4(b) shows the correct recognton rates of the decsons. Snce GLR s used n the Bayesan formulaton nstead of the lkelhood functon, the correct rates for dfferent methods are no longer the same when the same desred ncorrect probabltes

11 Avg. number of llumnatons Correct recognton rates Sensors, 9 are set. For each target set, two waveforms, a smple pulse and an egen waveform, are used for comparson wth the waveforms obtaned by the proposed mnmum correlaton algorthm. A smple pulse s defned as u EL,,, T u whch s a constant n the transmt duraton. It does not change accordng to knowledge acqured from the envronment, whch represents a non-cogntve radar system. The egen-waveform soluton adaptvely changes the transmt waveform to maxmze the weghted average Eucldean dstance between each hypothess par, where the weghtng factor s the product of pror probabltes. In our algorthm, the waveform s desgned to mnmze the weghed correlaton between the hypotheses. Both the egen-waveform soluton and the mnmum correlaton algorthm update the system s understandng of the target after each observaton and then optmze the waveform to match that understandng. To show the mportance of usng the GLR test n stuatons wth ampltude fluctuaton, the three waveform desgn technques are also nvolved n a statstcal model msmatch. Wth the same observatons, the generalzed lkelhood functon Equaton (9) s replaced wth the lkelhood functon that assumes the ampltude factor s known to be, whch s not true. Fgure 4. (a) Average number of llumnatons to reach a decson and (b) correct recognton rates at the tme when a decson has been made vs. energy per llumnaton for dfferent waveform desgn approaches Smple pulse Egen soluton Mn correlaton Smple pulse(msmatch) Egen soluton(msmatch) Mn correlaton(msmatch) Smple pulse Egen soluton Mn correlaton Smple pulse(msmatch) Egen soluton(msmatch) Mn correlaton(msmatch) Energy per llumnaton (a) Energy per llumnaton (b) As shown n Fgure 4(b), for all three methods wthout consderng actual ampltude fluctuatons (msmatch), the correct recognton rates are approxmately equal to.5 ( M ), whch s the same as the probablty of blnd random selecton. From Fgure 4, t s clearly demonstrated that both of the closed-loop waveform desgn methods perform better than the approach transmttng a smple pulse. Wth approxmately the same number of llumnatons, the egen-waveform soluton acheves much hgher correct rates. Meanwhle, the mnmum correlaton method not only reduces the average number of llumnatons but also enhances recognton accuracy. Nevertheless, t s dffcult to make a udgment between the two. Compared wth the egen-waveform soluton, our method requres fewer llumnatons, but performs hgher probabltes of falure.

12 Normalzed Power Normalzed Power Correct recognton rates Correct recognton rates Sensors, 9 To make a far udgment, an addtonal experment s performed, n whch all three methods execute the same number of llumnatons. No hard decson s made by the ST framework. The probablty of each hypothess s updated repeatedly after each llumnaton usng the Bayesan formulaton. Once the number of llumnatons reaches the maxmum, the hypothess wth the greatest probablty s selected to be true. Fgure 5 presents the correct recognton rates of 5 dfferent sets after sx llumnatons and eght llumnatons. Our method shows the hghest recognton accuracy among the three whle the approach transmttng a smple pulse shows the lowest. In addton, for each waveform desgn approach, the greater the number of llumnatons s, the hgher the correct recognton rates become. Fgure 5. Correct recognton rates after sx (a) or eght (b) llumnatons vs. energy per llumnaton. The hypothess wth maxmum probablty s selected to be true after sx (a) or eght (b) llumnatons Smple pulse 6 llumnatons.5 Egen soluton 6 llumnatons Mn correlaton 6 llumnatons Energy per llumnaton (a) Smple pulse 8 llumnatons.5 Egen soluton 8 llumnatons Mn correlaton 8 llumnatons Energy per llumnaton (b) Fgure 6. Waveform spectra compared to target spectral varance. Egen-waveform soluton s shown n (a) and the mnmum correlaton algorthm s shown n (b)..9 Target spectral varance Egen soluton.9 Target spectral varance Mn correlaton Normalzed Frequency (a) Normalzed Frequency (b) In Fgure 6, the spectrum of both the waveforms desgned by the egen-waveform soluton and the mnmum correlaton algorthm are compared wth the weghted spectral dfferences between the four mpulse responses. To maxmze the weghted Eucldean dstance, egen waveform focuses most of ts energy on the maxmum response frequency, snce the Fourer transform preserves the Eucldean

13 Average dstance Average correlaton Sensors, 93 dstance between sgnal and ts spectra. owever, our mnmum correlaton algorthm seems to have no sgnfcant relaton wth the target spectral varance, snce t ams to acheve mnmum correlaton between the echo sgnals. The average Eucldean dstance and the average correlaton between the deal echoes generated by dfferent methods are shown n Fgure 7, where 5 target mpulse response sets are randomly generated accordng to prevous descrptons as test samples. Fgure 7. Comparson of (a) the average Eucldean dstance and (b) the average correlaton between the deal echoes for dfferent waveform desgn technques..4. Smple pulse Egen soluton Mn correlaton.9.8 Smple pulse Egen soluton Mn correlaton Impulse response set (a) Impulse response set (b) As seen n Fgure 7, the egen waveforms clearly produce echoes wth largest average dstance among all the transmt waveforms. owever, the egen soluton also causes hghest correlaton n the echo set. Wth the presence of unknown parameter descrbed n our sgnal model (5), echoes wth hgh correlaton can hardly be dstngushed from each other, snce the ampltude and the ntal phase of the sgnal no longer contan any nformaton. The echoes of smple pulse show a volatle average dstance snce no adaptaton s performed. A more stable average dstance s acqured by our mnmum correlaton algorthm because the transmt energy s wdely dstrbuted n the passband. Despte the fact that the echoes of our algorthm show lower average dstance than that of the egen soluton, our algorthm outperforms any other method n the comparson of average correlaton, whch leads to better recognton performance n the stuatons wth ampltude fluctuaton. The deal echoes n a scenaro wth only two target hypotheses are presented n Fgure 8 to show the characterstcs of dfferent waveform desgn technques n a more ntutve way. As we can see, the echoes correspondng to the egen soluton have more energy than the echoes correspondng to the mnmum correlaton algorthm, and the dstance between the egen soluton echoes s also much larger than the dstance between the mnmum correlaton echoes, whch s consstent wth the results shown n Fgure 7(a). owever, sgnfcant correlaton s found between the egen waveform echo and the egen waveform echo. It s very dffcult to dstngush the two sgnals, f the egen soluton echo s multpled by a factor. The echo sgnals have a tendency to be opposte to each other about the

14 Avg. number of llumnatons Correct recognton rates Ampltude Ampltude Sensors, 94 orgn n the multdmensonal space, snce the egen soluton ams to maxmze the Eucldean dstance between the two sgnals. Fgure 8. Ideal echoes generated by (a) the egen-waveform soluton and (b) the mnmum correlaton algorthm n a scenaro wth only two target hypotheses..4.3 Egen soluton echo Egen soluton echo.4.3 Mn correlaton echo Mn correlaton echo Range cell (a) Range cell (b) 5.. Estmaton Varance of Target Aspect Angle In prevous experments, the poston and velocty of the target are assumed to be known by the radars, whch means that the target aspect angle for the th radar n Equaton () s known. To show the nfluence of the estmaton varance of aspect angle on system performance, the drecton of the estmated target velocty ˆv s assumed to be the sample of a unform dstrbuton on arg v, arg v, where s the nterval of the unform dstrbuton. Fgure 9. (a) Average number of llumnatons to reach a decson and (b) correct recognton rates at the tme when a decson has been made vs. energy per llumnaton for dfferent estmaton varances of the target aspect angle =3 = = =3 = Energy per llumnaton (a).7 5 =3 = =.55 =3 = Energy per llumnaton (b)

15 Avg. number of llumnatons Correct recognton rates Sensors, 95 In Fgure 9, the proposed waveform desgn approach s tested wth 5 target sets n stuatons wth dfferent, whle the other condtons are the same as those from the prevous experments. As shown n ths fgure, hgher correct recognton rates are acheved wth fewer average llumnatons n the stuaton of hgher target aspect angle accuracy. If we assume that the varance of the measured Doppler velocty s the same for each radar, hgher accuracy n target velocty can be acqured by ncreasng the number of radars n the network, whch wll lead to better system performance. It s also clear that the average number of llumnatons and the correct recognton rates for both and 3 are very close to each other. The feasblty of buldng a knowledge base wth target templates every three degrees s proved once agan Number of Radars Fgure shows the performance of the proposed waveform desgn algorthm appled to centralzed cogntve radar networks wth dfferent numbers of radars. The target s stll located at R xˆ yˆ km and s movng wth velocty v xˆ yˆ m/s whle the radars are located at R ˆ ˆ x y, R ˆ ˆ x y, R ˆ ˆ 3 x y, R ˆ ˆ 4 x y, R ˆ ˆ 5 x y, R ˆ ˆ 6 x y, and R ˆ ˆ 7 x y km. The frst N radars n the queue are selected to work whle the others reman dle. We also assume that the poston and the velocty of the target are exactly known. As shown n Fgure (b), the recognton accuraces are approxmately the same for dfferent number of radars n the network. In Fgure (a), the average number of llumnatons apparently decreases monotoncally wth ncreasng numbers of radars n the network. owever, the total energy transmtted from all the radars to reach a decson stll ncreases. Fgure. (a) Average number of llumnatons to reach a decson and (b) correct recognton rates at the tme when a decson has been made vs. energy per llumnaton for dfferent numbers of radars n the network N= N=3 N=4 N=5 N= N= N=3 N=4 N=5 N= Energy per llumnaton (a) Energy per llumnaton (b)

16 Sensors, Conclusons We have extended the dea of ntegratng waveform desgn technques wth a ST framework for target recognton [] to the case of a centralzed cogntve radar network. Several ssues, ncludng the target-aspect senstvty and the echo ampltude fluctuaton, have been consdered and solved. The GLR was employed n the ST framework to update the ont probabltes of target hypotheses because of the unknown ampltude factor. The performance of three waveform desgn approaches, a non-adaptve method transmttng a smple pulse, the egen-waveform soluton, and our mnmum correlaton algorthm, are compared usng smulaton. The advantage of both adaptve waveform desgn technologes based on the latest knowledge about the target was substantal, and our mnmum correlaton algorthm outperformed the egen-waveform soluton. Moreover, the nfluence of system parameters on recognton performance s shown by smulatons usng dfferent estmaton varances of target aspect angle and dfferent number of radars. Acknowledgements Ths work was supported n part by the Natonal Natural Scence Foundaton of Chna (No. 6957). References and Notes. Mtchell, R.A.; Dewall, R. Overvew of hgh range resoluton radar target dentfcaton. In Proceedngs of Automatc Target Recognton Workng Group, Target Recognton Workng Group: Monterey, CA, USA, November L,.J.; Yang, S.. Usng range profles as feature vectors to dentfy aerospace obects. IEEE Trans. Antennas Propag. 993, 3, Zyweck, A.; Bogner, R.E. Radar target classfcaton of commercal arcraft. IEEE Trans. Aerosp. Electron. Syst. 996,, Mtchell, R.A. Robust gh Range Resoluton Radar Target Identfcaton Usng a Statstc Feature Based Classfer wth Feature Level Fuson. Ph.D. Dssertaton, Unversty of Dayton, Dayton, O, USA, December Jacobs, S.P.; O Sullvan, J.A. Automatc target recognton usng sequences of hgh resoluton radar range profles. IEEE Trans. Aerosp. Electron. Syst.,, Du, L.; Lu,.; Bao, Z.; Xng, M. Radar RRP target recognton based on hgher order spectra. IEEE Trans. Sgnal Process. 5, 7, Bell, M.R. Informaton theory and radar waveform desgn. IEEE Trans. Inform. Theory 993, 5, Guerc, J.R.; Plla, S.U. Theory and applcaton of optmum transmt-receve radar. In Proceedngs of the IEEE Internatonal Radar Conference, Washngton, DC, USA, May ; pp aykn, S. Cogntve radar: A way of the future. IEEE Sg. Process. Mag. 6,, 3-4.

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