SKT-DLVT Processing. Jing Tian, Wei Cui, and Si-liang Wu

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1 Prmeter Estimtion of Ground Moving Trgets Bsed on SKT-DLVT Proessing Jing Tin, Wei Cui, nd Si-ling Wu Abstrt It is well known tht the motion of ground moving trget my indue the rnge ell migrtion, spetrum spred nd veloity mbiguity during the imging time, whih mkes the imge smered. To eliminte the influene of these ftors on imge fousing, novel method for prmeter estimtion of ground moving trgets, known s SKT-DLVT, is proposed in this pper. In this method, the mentl keystone trnsform (SKT) is used to orret the rnge wlk of trgets simultneously, nd new trnsform, nmely, Doppler Lv s trnsform (LVT) is pplied on the zimuth signl to estimte the prmeters. Theoretil nlysis onfirms tht no interpoltion is needed for the proposed method nd the trgets n be well foused within limited serhing rnge of the mbiguity number. The proposed method is pble of obtining the urte prmeter estimtes effiiently in the low signl-to-noise rtio (SNR) senrio with low omputtionl burden nd memory ost, mking it suitble to be pplied in memory-limited nd rel-time proessing systems. The effetiveness of the proposed method is demonstrted by both simulted nd rel dt. Index Terms ground moving trget, mentl keystone trnsform (SKT), Doppler Lv s trnsform (LVT), prmeter estimtion. I INTRODUCTION Syntheti perture rdr (SAR) hs been widely used in mny ivilin nd militry pplitions inluding moving trget imging nd identifition for its bility of informtion quisition. The position shift of moving trget in onventionl SAR imge use the imge defousing; therefore, detetion, prmeter estimtion, 1

2 imging nd relotion of moving trgets hve reeived onsiderble ttention in the rdr imging ommunity [1-3]. For SAR system with ground moving-trget indition (GMTI) [4, 5], rdr motion nd long dwell time produe the lrge zimuth bndwidth whih n be used to improve the signl-to-noise rtio (SNR). However, lrge rnge ell migrtion (RCM), spetrum spred nd veloity mbiguity my our [6-10], resulting in imge defousing when the onventionl SAR imging lgorithms re pplied to the observed sene with moving trgets. Severl methods hve been proposed to refous the moving trget. These methods n be lssified into two types. In the first type, trgets should be deteted before prmeter estimtion [11-14]. The imge n be well foused with the estimtes of motion prmeters, whih re hieved first by exploiting the rnge migrtion indued by the motion. However, these methods perform poorly in the se of the lrge RCM, spetrum spred or veloity mbiguity. In the seond type, the moving trgets n be well imged without priori knowledge of motion prmeters. The motion prmeters of trgets n be estimted by optimizing the qulity of the trget imge signture in [15]. During the fousing proess, the RCM is ompletely orreted. However, it hs hevy omputtionl burden when the prmeter serhing rnge is lrge. Keystone trnsform (KT) bsed methods hve been proposed in [6][16][17]. However, these methods nnot orret the RCM ompletely in the se of veloity mbiguity, thereby impting the energy integrtion nd prmeter estimtion. A -D mthed filtering method hs been proposed in [4], whih n orret the RCM without priori knowledge of the

3 urte motion prmeters. However, the zimuth defousing my our without knowing the long-trk veloity informtion. An instntneous-rnge-doppler method bsed on dermp-keystone proessing hs been proposed in [9], whih n fous moving trget t n rbitrrily hosen zimuth time without speifi knowledge of its urte motion prmeters. This method n eliminte the RCM of multiple trgets simultneously nd solve the problems of Doppler spetrum spred nd veloity mbiguity. However, the position of the trgets nnot be obtined diretly nd the Doppler spetrum spred nnot be ompensted ompletely due to the mismth of dermp funtion for the trget with lrge zimuth veloity, whih would further ffet the preision of prmeter estimtes nd zimuth fousing. The sling proessing nd frtionl Fourier trnsform (SPFRFT) method proposed in [18] n be pplied for the ompenstion of RCM nd Doppler spetrum spred, however, it hs hevy omputtionl burden sine it needs 3-dimensionl serhing. A new trnsform, i.e., streth keystone-wigner trnsform (SKWT) hs been proposed to estimte the motion prmeters, whih n resolve the RCM, veloity mbiguity nd spetrum spred [19]. However, it suffers from hevy omputtionl burden, biliner trnsform nd non-oherent integrtion. In [0], new prmeter estimtion method bsed on KT nd Lv s trnsform (LVT) [1] hs been proposed to redue the omputtionl burden. It hs similr estimtion preision to the frtionl Fourier trnsform (FrFT), yet n be implemented without using ny serhing opertion. For the liner frequeny modulted (LFM) signls over long-time durtion, however, the omputtionl omplexity nd memory ost of LVT 3

4 re huge, resulting in inessible requirement for DSP hips nd unsuitble for rel-time proessing. To del with the problems of lrge RCM, Doppler spetrum spred nd veloity mbiguity for the moving trget, tke the bility of proessor, the dt rte nd the improvement of SNR into ount, with s little priori knowledge s possible, this pper proposes mentl keystone trnsform (SKT) nd Doppler LVT bsed method (SKT-DLVT), whih borrows the ide of ment to redue the omputtionl burden nd storge memory ost. In this method, the SKT is used to orret the rnge wlk nd the Doppler LVT is pplied on the zimuth signl to estimte the prmeters. The mjor steps of the Doppler LVT inlude: 1) the fst Fourier trnsform (FFT) is pplied on the zimuth signl within eh ment; ) the sme frequeny resolution bins of eh ment re seleted to onstrut new series; 3) Doppler KT is employed to orret the frequeny wlk ross the ments; 4) inter-ment LVT is implemented to obtin the prmeter estimtes. Unlike the onventionl methods in [15-17, ], the RCM orretion of multiple moving trgets in this pper is rried out simultneously. The proposed estimtor is urte for the trgets with Doppler spetrum spred nd veloity mbiguity, whih does not suffer from hevy omputtionl burden by pplying the prllel proessing on the SKT nd the Doppler LVT. In ddition, the serhing of the mbiguity number is only within limited serhing rnge. It is fesible, simple nd suitble to be pplied in memory-limited nd rel-time proessing systems. The reminder of this pper is orgnized s follows. Setion II estblishes the 4

5 mthemtil model of eho signl. Setion III desribes the proposed prmeter estimtion method for both slow nd fst moving trgets. In Setion IV, some pplition onsidertions, suh s the implementtion of SKT, the riterion to hoose the number of ments, the mrginl veloity, the prmeter estimtion strtegy for multiple moving trgets, the output SNR, the omputtionl omplexity nd memory ost, re nlyzed in detil. Setion V proesses the simulted nd rel dt to vlidte the proposed method. Setion VI onludes the pper. II SIGNAL MODELING This setion derives the signl model for trget moving with uniform retiliner motion while ignores the higher order motion. The geometry reltionship between the flying pltform nd the moving trget is shown in Fig. 1, in whih V, v nd v denote the veloity of the pltform, the long- nd ross-trk veloities of trget, respetively. R B is the nerest rnge between the pltform nd the trget, t is the slow time. Aording to the geometry, the instntneous slnt rnge Rt () between the pltform nd the trget n be expressed s [4, 6] R t Vtv t R v t R v t V v t R (1) () ( ) ( B ) B ( ) ( B) Vt R() t v R B v Fig.1. Geometry of moving trget. 5

6 Assume the rdr dopts LFM wveforms, i.e., s t T f t () T(, ) ret( p)exp(jπ )exp[jπ ( )] where is the fst time, i.e., the rnge time; t nt( n0,1, N1) is the slow time; T is the pulse repetition time; N is the number of oherent integrted pulses; ret(x) is the window funtion nd equl to one for x 1 or zero if otherwise; T p is the pulse width; f is the rrier frequeny; nd is the modultion rte. The reeived bsebnd signl fter rnge ompression n be expressed s [4] R() t R() t st (, ) Gwt ()sinb exp j4 (3) where is the bksttering oeffiient of the trget, G is the rnge ompression gin, wt () is the zimuth window funtion [4], B is the bndwidth of the signl, is the light speed, nd f is the wvelength. Substituting (1) into (3) yields R v t V v t R st (, ) Gwt ()sinb RB vt( V v) t ( RB) exp j4 B ( ) ( B) Trnsforming st (, ) into the rnge-frequeny nd zimuth-time domin yields (4) Gw() t f f f St (, f) ret expj4 RB vt ( Vv) t ( RB) B B (5) It n be seen from (4) tht the problems of the lrge RCM, Doppler spetrum spred nd veloity mbiguity nnot onentrte the energy of the trget ompletely, thereby mking the imge smered. In the next setion, we desribe new prmeter estimtion method. For the slow moving trget without veloity mbiguity, this pproh n obtin the estimtes of trgets without priori knowledge of the motion 6

7 informtion. For the fst moving trget, i.e., in the presene of veloity mbiguity, this pproh n estimte the prmeters of trgets with one-dimensionl serhing of the mbiguity number. III METHOD FOR PARAMETER ESTIMATION A. Prmeter Estimtion Method for the Slow Moving Trget We first present the lgorithm for the trget without veloity mbiguity [9], whih stisfies v PRF 4, PRF 4 with PRF=1 T. Sine the lrge RCM of the trget would ffet the preision of prmeter estimtion, the orretion of the lrge RCM should be implemented first using the proposed SKT. The zimuth signl in (5) is firstly divided into ments with equl length, i.e., the zimuth time of NT is divided into P ments (the riterion to hoose the number of ments is disussed in the Setion IV-B lter). Then substituting the sling formul of the KT, i.e., t f f f t [3, 4], into the -th ( 1,,..., P ) ment yields Gw( t ) f St (, f) ret B B f f f ( V v ) f exp j4 R v t t f f R f f Gw( t ) f ret B B B B f f f f ( V v ) f exp j4 RB j4 vt j 1 t RB f (6) where t [( 1) NT P: ( 1) NT P( N P 1) T]. Performing the inverse Fourier trnsform on St (, f) with respet to f yields 7

8 R V v t R st (, ) Gwt ( )sin B B ( ) ( ) B R v t V v t R exp j4 B ( ) ( ) B (7) From (7), we find tht the liner RCM hs been removed ompletely. However, the qudrti RCM remins, whih is relted with ( V v) t ( R ) B. This term hs minor influene on the RCM for C-bnd stellite SAR systems. However, for L-bnd stellites, the qudrti prt is reltively lrge. In this sitution, the qudrti RCM n be removed effiiently in the zimuth frequeny domin [5]. After the qudrti RCM orretion, the resulting signl is written s R ( ) ( ) B RB vt V v t R B st (, ) Gwt ( )sin exp 4 B j (8) It n be seen from (8) tht ll the trgets sty in the right rnge ells fter rnge migrtion orretion, whih n improve the preision of estimtion nd further obtin the well-foused imge. And it is obvious tht the reeived signl from ll stters in one rnge ell n be modeled s multi-omponent LFM signl fter rnge ompression nd motion ompenstion. To obtin the urte estimtes of the veloities nd elertions of trgets, we need to estimte the prmeters of the LFM signl preisely. For simpliity, (8) n be further expressed s xt j t t ( ) exp 0 1 (9) R where Gw( t )sin B V v RB ( ). B, 0 R B, 1 v nd 8

9 For multi-omponent LFM signls, the onventionl time-frequeny trnsform [6-30] suffers from performne degrdtion (even ineffetive) beuse of the ross terms nd the low-resolution problems in the low SNR senrio. These problems n be solved by pplying the LVT for prmeter estimtion over the rnge ells. The LVT is ble to obtin urte prmeter estimtes without using ny serhing opertion. This method breks through the trdeoff between resolution nd ross terms. For the LFM signls over long-time durtion, however, the omputtionl omplexity nd memory ost of LVT re huge, whih would restrit its pplitions. Therefore, new Doppler LVT method is proposed for prmeter estimtion. The ore steps ontin the mentl FFT proessing of LFM signls nd the inter-ment LVT pplied on the new series onstruted by the sme frequeny resolution bins of eh ment. Define tq qt s the intr-ment time where q 0,1,..., N P 1 nd NP is the number of smples within eh ment, nd define t ( p1) NT P s the inter-ment time with p 1,,..., P. Then the zimuth time t is rewritten s p t t t. Ignoring the hnge of frequeny within eh ment intervl, x( t ) q p n be pproximted s xt t j t t t t ( q, p) exp 0 1( q p) ( q p) exp j 0 1( tq tp) ( tqtp tp) exp j 1 t ptq jp (10) with t t. p 0 1 p p The FFT of x( tq, tp) with t q is omputed to be 9

10 N sin t f x( f, t ) exp( j )exp j 1 t f 1 p q P N q p p 1 p q sin P 1tp f q It n be seen from (11) tht the pek position of spetrum envelope vries with t p of eh ment. And the frequeny wlk, whih is lrger thn one frequeny resolution bin, would ffet the preision of prmeter estimtion. Tht is to sy, we need to orret the frequeny wlk when NT P NT (11) holds. Then new Doppler KT is proposed to orret the frequeny wlk. Sine the onventionl rnge KT is implemented in the rnge-frequeny nd zimuth-time domin, the Doppler KT should be relized in the intr-ment time nd inter-ment time domin ordingly. Eqution (10) n be rewritten s xt ( q, tp) exp j 0 1( t q NTtp) ( t q NTt ) p t p (1) where t q tq NT. Substituting the sling expression ( t NT) t NTt into (1) yields q p p NT x( t q, tp) exp j0exp j1t q NTexp j1 t p tq NT NT exp jnttpexp j tp t q NT (13) Sine t q NT, (13) n be further expressed s j NTtp jtp x( t, t) exp j exp j t NT exp jt q p 0 1 q 1 p exp exp (14) The FFT of x( t q, t p) with t q is omputed to be x( f, t) sin N ( f) P exp j N 1 f exp j 10 1 q q p 1 q 0 sin ( 1 f q ) P j 1NT j 1 NTt p jtp exp exp exp (15)

11 From (15), it n be seen tht the frequeny wlk is orreted ompletely nd the energy of trget hs been onentrted into the frequeny ell whose frequeny stisfies f q 1. The zimuth signl remins n LFM signl with the frequeny NT nd the hirp rte. Then pplying LVT on x( f, t ) with respet to 1 q p t p yields fˆ, ˆ, fˆ rg mxlvt q x( f q, tp) t p (16) f,, fq Hene the prmeters n be estimted by ˆ ˆ ˆ ˆ ˆ 1 f NT (17) where f ˆq is the orse estimte of 1, whih stisfies ˆ IP P IP P N N fq,, I :1: 1 NT NT NT NT P P, nd ˆf is the refined estimte of ˆ ˆ 1 NT. It should be noted tht the estimted frequeny ˆf nd hirp rte ˆ stisfy fˆ [ P (4 NT), P (4 NT)] nd ˆ [ P ( NT), P ( NT)], respetively. Generlly, the vilble hirp rte is bout [ P ( NT), P ( NT)], however, the vilble frequeny 1 NT my be within the rnge of [ P (4 NT),1 ( T )] or [ 1 ( T), P (4 NT)]. Then modified method is proposed to estimte the prmeters of trgets preisely. Aording to the estimted f ˆq nd â, we n lulte the orse estimte of ˆ ˆ NT, whih stisfies 1 ˆ AP P AP P N N f ˆ q NT,, A :1: 1 NT NT NT NT P P nd then onstrut the serhing frequeny funtion 11

12 f fˆ k P ( NT), k A1:1: A 1 within the rnge of serh mb _ in mb _ in fˆ q ˆ NT. The orresponding serhing veloity is omputed to be v ( f NT). However, in the rel sitution, k mb _ in is seleted to be serh serh ˆ kmb _ in ( A1) 1:1: ( A1) 1 to ensure the orretness of prmeter estimtes. Then onstruting the phse-ompensted funtion Hom(, t f) expj4 ( f f) vserhtˆ t 4 with v serh nd multiplying it by (5) yields S (, t f) S(, t f) H (, t f) om om Gw() t f f f ret exp 4 j RB v vserht B B (18) Hene v n be determined by solving v rg mx sum IFFT S ( t, f) (19) vserh t f om TABLE I SYSTEM PARAMETERS FOR SIMULATION System Prmeters Vlues Wvelength (GHz) 10 Rnge bndwidth (MHz) 8 Pulse repetition frequeny (Hz) 000 Rnge smpling frequeny (MHz) 0 Pulse width (us) 0 In the following, the effetiveness of the proposed method is exmined under the idel irumstne. The simultion prmeters re listed in Tble I. The reltive rdil veloity nd elertion between the slow moving trget nd the rdr pltform re v 10m s nd 0.9 m s, respetively. Fig. () shows the trjetory of the trget fter rnge ompression. It is obvious tht the signl energy spreds over severl rnge ells. We perform the SKT to orret the RCM nd obtin the result in Fig. (b). It is observed tht the RCM is eliminted ompletely. Then mentl FFT 1

13 is pplied to the zimuth signl with the number of ments of 56 nd the frequeny wlk ours shown in Fig. (). The detiled riterion to hoose the number of ments is disussed in Setion IV-B. The Doppler KT is used to orret the frequeny wlk nd the result is shown in Fig. (d), from whih it n be noted tht the frequeny wlk is removed ompletely. After LVT, s shown in Fig. (e), the trget is well foused. The frequeny nd hirp rte with the vlue of 19.3Hz nd 61.55Hz s, respetively, re lso estimted. Fig. (f) shows the serhing result of the inner mbiguity number within the rnge from PRF to PRF, in whih the inner mbiguity number n be esily determined with the vlue of k _ 14. Aording to the forementioned nlysis, the finl estimtes of the reltive rdil veloity nd elertion between the slow moving trget nd the rdr pltform re mb in m s nd 0.93 m s, respetively. () (b) 13

14 () (d) (e) (f) Fig.. Simultion results of the slow moving trget. () Trjetory fter rnge ompression. (b) Trjetory fter SKT. () Trjetory fter FFT pplied on the zimuth signl within eh ment. (d) Trjetory fter Doppler KT. (e) Result of LVT. (f) Estimtion of inner mbiguity number k mb _ in within the rnge from PRF to PRF. B. Prmeter Estimtion Method for the Fst Moving Trget For fst moving trget, its Doppler frequeny will exeed the mission PRF. In this se, the trget spetrum will be overlpped by the mission PRF. The fst moving trget stisfies v k mb _ out PRF PRF 4,PRF 4, where k _ 0 denotes the mbiguity number reltive to PRF. In this sitution, the forementioned SKT nnot del with the RCM ompletely. The veloity of trget n be written s mb out 14

15 v k v v (0) mb _ out mb 0 where v PRF is the blind veloity nd v0 [ v, v ]. Applying mb SKT to orret the liner RCM nd removing the qudrti RCM in the zimuth frequeny domin, we n get mb mb RB k _ outvmbt st (, ) Gwt ( )sin B R v t V v t R exp j4 B ( ) ( ) B (1) It n be noted tht the trjetory in the rnge-time nd zimuth-time domin exhibits liner feture nd its slope is proportionl to the mbiguity number. Therefore, the RCMC/integrtion method n be well dopted to estimte the slope [5]. This estimtor is formulted s NT N ( 1) ( 1) T P P P 4 f f( k, ) IFFT FFT s( t, ) exp j k v t mb _ out mb _ out mb f 1 NT t ( 1) P () where IFFT denotes the inverse fst Fourier trnsform. Then the entropy of n imge is employed to determine the estimted vlue nd evlute the estimtion performne. The mbiguity number n be estimted by Ek ( ) log 1 k_ out rg mx k mb _ out Ek ( mb _ out ) f( kmb _ out, ) f( kmb _ out, ) mb _ out f( kmb _ out, ) f( kmb _ out, ) (3) Wht should be pointed out is tht the mbiguity number n be estimted urtely nd the omputtionl lod is reltively low beuse the number vlue is n integer. 15

16 By performing the entropy of n imge, we n get relible result of mbiguity number in medium- to high- SNR senrios, however, we nnot obtin the right estimte in low SNR senrio. Aordingly, n improved method is proposed to estimte the mbiguity number. The phse-ompensted funtion is first onstruted s 4 f H ( k, t ) exp j k v t om mb _ out mb _ out mb (4) Multiplying (4) by the signl fter RCM orretion in the rnge-frequeny nd zimuth-time domin yields Gw( t ) f 4 f S ( t, f) ret exp j kmb out k out vmbt B B f f f f ( V v ) exp j4 R j4 v t j t R Applying the IFFT on S( t, f) B B with respet to f yields R k k v t st (, ) Gwt ( )sin B RB vt ( V v ) ( ) t R B exp j4 B _ out mb _ out mb Then the Doppler LVT is pplied on (6) with respet to t nd the mbiguity (5) (6) number k _ out is estimted s k, fˆ, ˆ, fˆ rgmx LVT x( f, t, k ) (7) _ out q q p mb _ out kmb _ out, f,, f t q p where x( f, t, k ) is the derived zimuth signl of st (, ) orresponding to q p mb _ out different mbiguity number (,, ) is the sme k mb _ out. The derivtion of x fq tp kmb _ out s tht in Setion III-A. It n be seen from (7) tht the urte prmeter estimtes n be obtined if the prmeter k mb _ out is mthed with the mbiguity number k _ out 16. Otherwise, the

17 smering result will be obtined. The prmeters n be further estimted by ˆ ˆ ˆ ˆ ˆ 1 f NT (8) where f ˆq is the orse estimte of 1 within the rnge from 1 T to 1 T, whih stisfies ˆ IP P IP P N N fq,, I :1: 1 NT NT NT NT P P, nd ˆf is the refined estimte of ˆ 1 ˆ NT. It should be noted tht the equivlent intervl where ˆ ˆ 1 NT is loted in turns into [ k T P (4 NT), k T P (4 NT)], however, the vilble frequeny mb _ out mb _ out NT my be within the rnge of [ kmb _ out T P (4 NT), kmb _ out T 1 ( T)] or 1 [ k T1 ( T), k T P (4 NT)]. mb _ out mb _ out Aording to the estimted f ˆq nd â, we n lulte the orse estimte of ˆ ˆ NT, whih stisfies 1 kmb _ out ˆ kmb _ out AP P kmb _ out AP P N N f ˆ q NT,, A :1: 1 T T NT NT T NT NT P P nd then onstrut the serhing frequeny funtion f fˆ k P ( NT ), k Nk P A 1:1:Nk P A 1 serh mb _ in mb _ in mb _ out mb _ out within the rnge of fˆ ˆ q NT. The orresponding serhing veloity is omputed to be v ( ˆ serh fserh NT). However, in the rel sitution, k mb _ in is seleted to k Nk P( A1) 1:1:Nk P( A1) 1 to ensure the be mb _ in mb _ out mb _ out orretness of prmeter estimtes. The subsequent steps re the sme s tht in Setion III-A. Fig. 3 shows the result of the proposed method for the fst moving trget with the reltive rdil veloity of v 40 m s nd the reltive rdil elertion of 17

18 0.9 m s. Fig. 3() shows the trjetory of the trget fter rnge ompression. It n be seen tht the signl energy spreds over lrge number of rnge ells during the exposure time. Fig. 3(b) shows the result fter RCM orretion is performed. It is observed tht the RCM nnot be well mitigted beuse of the mbiguous veloity. Fig. 3() shows the reiprol of the entropy of the RCMC/integrtion. The mbiguity number n be esily determined with the right vlue of k _ 1. Fig. 3(d) shows the result of RCM orretion fter the phse ompenstion with the estimted mbiguity number, from whih it n be seen tht the lrge RCM is eliminted ompletely. Then mentl FFT is pplied on the zimuth signl with the number of ments of 56 nd the frequeny wlk ours shown in Fig. 3(e). The Doppler KT is used to orret the frequeny wlk nd the result is shown in Fig. 3(f), from whih it n be noted tht the frequeny wlk is removed ompletely. After LVT, s shown in Fig. 3(g), the trget is well foused. The frequeny nd hirp rte with the vlue of 19.3Hz nd 61.55Hz s, respetively, re lso estimted. Fig. 3(h) shows the serhing result of the inner mbiguity number within the frequeny rnge entered t PRF from PRF to 3PRF, in whih the inner mbiguity number n be esily determined with the vlue of k _ 46. Aording to the forementioned nlysis, mb in the finl estimtes of the reltive rdil veloity nd elertion between the fst out moving trget nd the rdr pltform re m s nd 0.93 m s, respetively. 18

19 () (b) () (d) (e) (f) 19

20 (g) (h) Fig. 3. Simultion results of the fst moving trget. () Trjetory fter rnge ompression. (b) Trjetory fter RCM orretion. () Estimtion of the mbiguity number k _ out. (d) Trjetory fter RCM orretion nd mbiguity number ompenstion. (e) Trjetory fter FFT pplied on the zimuth signl within eh ment. (f) Trjetory fter Doppler KT. (g) Result of LVT. (h) Estimtion of the inner mbiguity number k within the rnge from PRF to 3PRF. mb _ in IV APPLICATIONS AND DISCUSSIONS A. Implementtion of the SKT It is worth to mention tht the onventionl KT ligns the pek position of the eho envelope in eh pulse repetition time (PRT) of eh ment to tht in the first PRT of tht ment [16, 31]. We present the KT proessing in eh ment during the exposure time in the following. After rng ompression, the reeived signl of the -th ment in the rnge-frequeny nd zimuth-time domin n be expressed s Gw( t ) f St (, f) ret B B f f exp j4 RB vt ( V v) t ( R ) B (9) where t [( 1) NT P: ( 1) NT P( N P 1) T]. 0

21 Let t be t t ( 1) NT P with t (0 : N P 1) T nd in in 1,,..., P. Then (9) is further expressed s Gw tin ( 1) NT P f St ( in,, f) ret B B f f ( ) in ( 1) NT V v t NT P exp j4 RB v tin ( 1) P RB Gwtin ( 1) NT P f ret B B f f ( V NT v ) ( 1) NT P exp j4 RB v( 1) P RB f f tin tin( 1) NT P exp j4 RB vtin ( V v) RB (30) Substituting the sling ftor t in f f f t in into the signl of the -th ment yields St (,, f) in Gw tin ( 1) NT P f ret B B f f exp j4 RB v( 1) NT P ( V v) (( 1) NT P) ( RB) 4 ( V v) NT ( V v) f exp j v ( 1) t inexp j 1 t in RP B RB f Tke the Sin interpoltion to relize KT nd we hve (31) N P f S( m,, f ) S( m,, f )sin mm m1 f f (3) Performing IFFT on St (,, f) in with f yields 1

22 st (,, ) R V v t NT P R Gw( t )sin B R v t V v t R exp j4 ( ) ( 1) ( B ) B ( ) ( ) B where t t ( 1) NT P nd in R R v NT P V v NT P R. B ( 1) ( ) ( 1) ( B ) It n be seen from (33) tht fter SKT opertion, the pek position of the envelope of different ment is ligned to different rng ells, whih degrdes the performne of the proposed method. An intuitive method to del with this problem is mking the position of lignment of KT proessing ontrollble, thereby migrting the pek position of the envelope of different ment to the sme rnge ell. Hene, modified reliztion of SKT is proposed to ensure (6) holds. The SKT implementtion using Sin interpoltion is modified s (33) N P f N N Sm (,, f) Sm (,, f)sin m( 1) m( 1) (34) m1 f f P P By using (34), the pek position of the envelope in eh PRT of eh ment is ligned to tht in the first PRT of the first ment. Sine KT is essentilly uniform resmpling or liner sling, it n be rried out effiiently by hirp trnsform [4], Chirp-Z trnsform [3] or sled fst FT [33]. In these trnsforms, the sling ftor is updted with the rnge frequeny. These implementtions re interpoltion free nd use only omplex multiplitions. Fig. 4 shows the flowhrt of the SKT through Chirp-Z trnsform.

23 Sm (,, f) Sm (,, f) e f f j P m N f e f f f j P m j ( 1) m N f f e e f f j P m N f Fig. 4. The flowhrt of the SKT through Chirp-Z trnsform. B. the Criterion to Choose the Number of Segments As disussed bove, we need to do ment proessing of the zimuth signl to estimte the prmeters. And the hoie of the number of ments would ffet the integrtion gin of eh ment nd the ury of the prmeter estimtion. The riterion of deiding the number of ments is given s follows. Aording to the ssumption of the derivtion, i.e., negleting the hnge of frequeny during the intervl of eh ment, nd to hieve the integrtion gin with the lowest integrtion loss in eh ment, the integrtion intervl NT P of eh ment should be less thn 1 f ( NT P) with f ( NT P) denoting the hnge of frequeny within the intervl of NT P. Therefore, we obtin the seletion riterion NT P 1 (35) Generlly speking, the hoie of P is trdeoff mong the eptble performne degrdtion, the tolerble omputtionl omplexity nd memory ost, under the ondition of stisfying (35). C. Proessing for Moving Trgets with Mrginl Veloity The spetrum of the moving trgets with mrginl veloity [9] is split into two prts 3

24 by the mission PRF nd spns the neighboring PRF bnds. After rnge migrtion orretion, the signl in the rnge-time nd zimuth-time domin n be represented s st (, ) s1( t, ) (, ) s t RB ( k _ out 1) vmbt s1( t, ) Gw 1 ( t )sin B RB vt ( V v ) ( ) t R B exp j4 RB k _ outvmbt s( t, ) Gw( t )sin B RB vt ( V v ) ( ) t R B exp j4 (36) where G 1 nd G re the gin of the rnge ompression for the two prts [loted t the k _ out 1th nd k _ outth PRF], respetively. From (36), it is evident tht, two stright lines exist with different slopes expressed s ( k 1) v 1 k _ outvmb _ out mb (37) If k in the onstruted ompenstion funtion stisfies k _ k _ 1, the mb _ out mb out out signl energy of s (, ) 1 t n be umulted effetively nd the orret prmeter estimtes n be obtined, while the signl energy of s ( t, ) nnot be umulted ompletely, resulting in the defoused trget. In the sme wy, if kmb _ out in the onstruted ompenstion funtion stisfies kmb _ out k _ out, the signl energy of s ( t, ) n be umulted effetively nd the orret prmeter estimtes n be obtined, while s (, ) 1 t will be defoused. 4

25 Tht is, lthough the signl energy of different prt n be umulted individully, the energy of eh prt is less thn the totl energy. This phenomenon is disdvntgeous to the prmeter estimtion. To void these defiienies, preproessing should be performed on the signl of trget before estimting prmeters in this se. For the trget with the Doppler bndwidth smller thn 1( T ), Doppler shifting by 1 ( T ) is implemented to ensure the spetrum of the signl is not split into two prts. The urte implementtion onsists of the following mjor steps. First, the ompenstion funtion is onstruted s (, ) exp f f Hom3 t f j t Tf (38) And then multiplying (38) by (5) yields Gw() t f f f St (, f) ret expj4 RB v t( Vv) t ( RB) B B 4T (39) It is observed tht the trget spetrum beomes n entire prt. After tht, the proposed method in Setion III is pplied on (39) to hieve the estimtes of the prmeters. The simulted dt is employed to exmine the orretness in this generl se. Fig. 5 shows the results of moving trget with mrginl veloity. The result fter SKT opertion is shown in Fig. 5(), from whih we n find two trjetories with different slopes. Fig. 5(b) shows the signl fter zimuth spetrum ompression, from whih it n be seen tht the spetrum is split into two prts. Fig. 5() shows the trjetory of the trget fter Doppler shifting. From this figure, it n be seen tht the trjetory turns into stright line. Fig. 5(d) shows the result of zimuth spetrum ompression 5

26 pplied on the signl fter Doppler shifting. It n be seen tht the Doppler spetrum is not split into two prts fter shifting. () (b) () (d) Fig.5. Simultion results of moving trget with mrgin veloity. () Trjetory fter SKT. (b) Compressed zimuth spetrum. () Trjetory fter SKT nd Doppler shifting. (d) Compressed zimuth spetrum fter Doppler shifting. D. Proessing for Multiple Moving Trgets From the forementioned nlysis, it is known tht the proposed method n diretly fous slow moving trget without knowing its motion prmeters; while for fst moving trget, we just need to know its mbiguity number. For multiple moving trgets with the sme mbiguity number, phse ompenstion funtion is onstruted 6

27 with (4) nd the preise prmeter estimtes n be hieved simultneously. While for multiple moving trgets with different mbiguity number, the phse ompenstion ftors should be onstruted respetively. In this wy, moving trget is expeted to be well foused fter ompensting the phse relted with the mbiguity number nd to be defoused by mismthed ftor. The mismthing of (4) will result in residul liner RCM nd thus introdue defousing. In this se, the different onstruted phse ompenstion funtion H ( k _, t ) nd H (, t f ) re om mb out employed to hieve the prmeter estimtes of eh trget. If the sttering intensities of multiple trgets differ signifintly, the len tehnique [34] is employed to improve the preision of the estimtes. To investigte the effetiveness of the proposed method, three trgets re set to be loted in the sme rnge ell. The reltive rdil veloity nd elertion between om the trgets nd the pltform re 10m s, 10m s, 9m s, 0.9 m s, 0.93m s nd 0.93m s, respetively. It is seen tht the reltive rdil veloities of trget 1 nd trget nd the reltive rdil elertions of trget nd trget 3 re, respetively, identil, whih re seleted to better explin how the new pproh works. Figs. 6() nd 6(b) re the results of the pulse ompression nd the proposed method for the three trgets, respetively. It is obvious tht the three trgets n be well foused nd the estimtes of the three trgets re m s, m s, m s, m s, 0.93 m s nd 0.93 m s, respetively. 7

28 () (b) Fig. 6. Results for multiple trgets. Result fter () pulse ompression nd (b) Doppler LVT. E. Computtionl Complexity nd Memory Cost To redue the omputtionl omplexity nd memory ost, ment proessing is introdued for prmeter estimtion. The SKT nd Doppler LVT opertion n be implemented through prllel proessing to redue the storge memory ost. In ddition, the opertion of sliding window ould be used to selet the dt of Doppler LVT proessing, whih n further redue the storge memory requirement for prmeter estimtion over long observtion intervl. In mny prtil rdr systems, the seletion riteri of sliding window n be found in [35]. In wht follows, the omputtionl omplexity of the proposed Doppler LVT nd the diret LVT (DT-LVT) in [0] will be nlyzed. As to the intr-ment FFT, P N N log P P multiplitions re needed. For the Doppler KT nd the 7 inter-ment LVT opertion, log N P log N P nd P P multiplitions re needed, respetively. Therefore, the overll omplexity of the Doppler LVT is 7 1 N N(Plog P log P log ). And the omplexity of the DT-LVT is P 8

29 N log N. Defining s the omplexity rtio of the Doppler LVT to DT-LVT, the omplexity rtio is omputed to be Plog P Nlog N ording to the forementioned nlysis. Tking N 4096 nd P 56 for exmple, the redued omplexity n be 4.17%, whih suggests tht the omplexity of the new pproh is redued signifintly, mking this pproh more suitble for rel-time proessing. F. Output SNR The detetion performne n be exmined in terms of output SNR; therefore, the output SNR of the proposed method is derived nd nlyzed. Aording to [0], the output SNR fter DT-LVT opertion is limited by SNR N NN NN 4 N NSNRin out1 NSNRin 4 1 (40) where SNR is the input SNR of the zimuth signl, nd N is the in N number of pulses during the exposure time. Next we derive the output SNR of the proposed method. It is indited in (15) tht the energy of the trget in eh ment hs been onentrted into the frequeny resolution bin stisfying f q 1 fter intr-ment FFT nd its output SNR is SNR N N FFT N P P, where N N P P nd N denote the power of signl nd noise fter intr-ment FFT, respetively. After the frequeny wlk orretion nd the inter-ment LVT opertion, the output SNR is limited by SNR N P 4 P P P P out N N N P P N P N 4 (41) 9

30 (41) n be further simplified s SNR N NN NN 4 N NSNRin out NSNRin 4 1 (4) From (40) nd (4), it n be seen tht the lower limit of the new pproh is equl to tht of DT-LVT. It should be noted tht the dditionl SNR loss of ment proessing is not onsidered during the theoretil derivtion of SNR out. In prtil pplitions, the slloping loss exists in intr-ment FFT opertion. However, it n be deresed through windowed FFT opertion or FFT with zero-pdding. V EXPERIMENTAL RESULTS In this setion, some results with simulted nd rel dt re presented to vlidte the performne of the proposed lgorithm nd omprisons re performed between the proposed SKT-DLVT nd the method in [0] for the slow nd fst moving trgets. A. Simulted Dt The prmeters used in the simultion re listed in Tble I. The signl is embedded in omplex white Gussin noise nd the input SNR of the trget is SNR [ 44: : 30]dB. For eh input SNR vlue, 500 trils re performed to lulte the root-men-squre errors (RMSE) of the estimtes of the trget for the SKT-DLVT nd the method in [0]. Figs. 7() nd 7(b) show the RMSE of the veloity nd elertion estimtes for the slow moving trget with v 10m s nd 0.9 m s, respetively. Figs. 8() nd 8(b) show the RMSE of the veloity nd elertion estimtes for the fst moving trget with v 40 m s nd 0.9 m s, respetively. 30

31 () (b) Fig.7. RMSE of () veloity nd (b) elertion ginst input SNRs vi the SKT-DLVT nd the method in [0] for the slow moving trget. () (b) Fig.8. RMSE of () veloity nd (b) elertion ginst input SNRs vi the SKT-DLVT nd the method in [0] for the fst moving trget. It n be seen from Figs. 7(), 7(b), 8() nd 8(b) tht the SKT-DLVT hs similr performne of prmeter estimtion to the method in [0]. Aording to the nlysis in Setion IV-E, ompred with the method in [0], the SKT-FLVT hs signifintly redued omputtionl omplexity, whih mkes it fesible for the rel-time proessing systems. B. Rel Dt 31

32 Prt of the RADARSAT-1 Vnouver sene dt [3] were seleted to verify our proposed method nd nlysis. The system prmeters of these dt re given in Tble II nd the proposed proedure is performed on the seleted trget (lbeled in the Fig. 9()). Fig. 9(b) shows the result fter SKT, from whih it n be seen tht the lrge RCM nnot be eliminted ompletely beuse of the veloity mbiguity. Fig. 9() shows the reiprol of the entropy of the RCMC/integrtion. The mbiguity number n be esily determined with the vlue of k _ 6. Fig. 9(d) shows the SKT result fter the phse ompenstion with the estimted mbiguity number, from whih it n be seen tht the lrge RCM is eliminted ompletely. After DLVT, s shown in Fig. 9(e), the trget is well foused. And the veloity nd elertion with the vlue mb out of m s nd m s, respetively, re lso estimted. The orresponding frequeny nd hirp rte re equl to Hz nd Hz s, respetively, whih is onsistent with the results of the onventionl prmeter estimtion method, thereby verifying the effetiveness of the new pproh. TABLE II SYSTEM PARAMETERS FOR RADARSAT DATA System prmeters Vlues Crrier frequeny (GHz) 5.3 Rnge bndwidth (MHz) Pulse repetition frequeny (Hz) Rnge smpling frequeny (MHz) Pulse width (us) Doppler entriod frequeny (Hz) Azimuth hirp rte (Hz/s)

33 () (b) () (d) (e) Fig. 9. Result of the Rel dt vi the proposed method. () Trjetory fter rnge ompression. (b) Trjetory fter SKT. () Ambiguity number estimtion of the trget. (d) Trjetory fter SKT nd mbiguity number ompenstion. (e) Result of DLVT. VI CONCLUSIONS 33

34 This pper hs introdued prmetri estimtion method for the ground moving trgets. For the slow moving trgets with unmbiguous veloity, it n estimte the prmeters of trgets simultneously without speifi knowledge on the trgets motion. While for the fst moving trgets, i.e., in the presene of veloity mbiguity, only its mbiguity number, whih n be well estimted by lulting the imge entropy (in medium- to high- SNR senrios) or serhing diretly within the limited rnge (in low SNR senrio), is needed, to hieve the prmeter estimtes preisely. The new pproh does not suffer from the onsiderble troublesome ross-term interferene, mking it work well for multiple trgets. The SKT nd DLVT re inherently suitble for prllel implementtion nd the omputtions n be prllelized to run on multiple proessors with the sme (or very similr) progrm nd t the sme durtion. It n hieve the preise prmeter estimtion in low SNR senrio beuse of its effetive oherent integrtion. The performne of the proposed lgorithm hs been vlidted by experimentl results of simulted dt nd rel dt, whih shows tht the proposed lgorithm serves s good ndidte for GMTI. In the ner future, lgorithms will be designed for prmeter estimtion of trgets with high-order omplex motion (i.e., the existene of the long-trk elertion nd time-vrying ross-trk elertion). REFERENCES [1] M. Soumekh, Syntheti Aperture Rdr Signl Proessing with MATLAB Algorithms, New York: Wiley, [] J. C. Curlnder nd R. N. MDonough, Syntheti Aperture Rdr: System nd Signl Proessing, 34

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36 estimtion lgorithm of fst moving trgets vi single-ntenn irborne SAR system," IEEE Geosi. Remote Sens. Lett., vol. 9, no. 5, pp , Sep. 01. [11] P. A. C. Mrques nd J. M. B. Dis, "Veloity estimtion of fst moving trgets using single SAR sensor," IEEE Trns. Aerosp. Eletron. Syst., vol. 41, no. 1, pp , Jn [1] G. Li, X. G. Xi, J. Xu, nd Y. N. Peng, "A veloity estimtion lgorithm of moving trgets using single ntenn SAR," IEEE Trns. Aerosp. Eletron. Syst., vol. 45, no. 3, pp , Jul [13] J. Xu, G. Li, Y. N. Peng, X. G. Xi, nd W. Yong-Ling, "Prmetri veloity syntheti perture rdr: signl modeling nd optiml methods," IEEE Trns. Geosi. Remote Sens., vol. 46, no. 9, pp , Sep [14] G. Y. Wng, X. G. Xi, V. C. Chen, nd R. L. Fielder, "Detetion, lotion, nd imging of fst moving trgets using multifrequeny ntenn rry SAR," IEEE Trns. Aerosp. Eletron. Syst., vol. 40, no.1, pp , Jn [15] J. K. Jo, "Theory of syntheti perture rdr imging of moving trget," IEEE Trns. Geosi. Remote Sens., vol. 39, no. 9, pp , Sep [16] R. P. Perry, R. C. DiPietro nd R. L. Fnte, "SAR imging of moving trgets," IEEE Trns. Aerosp. Eletron. Syst., vol. 35, no. 1, pp , Jn [17] J. G. Yng, X. T. Hung, J. Thompson, T. Jin, nd Z. M. Zhou, "Low-frequeny ultr-widebnd syntheti perture rdr ground moving trget imging," IET Rdr Sonr Nvig., vol. 5, no. 9, pp , De [18] R. To, N. Zhng nd Y. Wng, "Anlysing nd ompensting the effets of rnge nd Doppler frequeny migrtions in liner frequeny modultion pulse ompression rdr," IET Rdr Sonr Nvig., vol. 5, no. 1, pp. 1-, Jn

37 [19] X. L. Lv, M. D. Xing, C. R. Wn, nd S. H. Zhng, "ISAR imging of mneuvering trgets bsed on the rnge entroid Doppler tehnique," IEEE Trns. Imge Proess., vol. 19, no. 1, pp , Jn [0] J. Tin, W. Cui, X. Lv, S. Wu, J. Hou, nd S. Wu, "Joint estimtion lgorithm for multi-trgets' motion prmeters," IET Rdr Sonr Nvig., 014. [1] X. L. Lv, G. A. Bi, C. R. Wn, nd M. D. Xing, "Lv's distribution: priniple, implementtion, properties, nd performne," IEEE Trns. Signl Proess., vol. 59, no. 8, pp , Aug [] J. Yng, X. Hung, T. Jin, J. Thompson, nd Z. Zhou, "New pproh for SAR imging of ground moving trgets bsed on keystone trnsform," IEEE Geosi. Remote Sens. Lett., vol. 8, no. 4, pp , Jul [3] Y. F. Wu, G. C. Sun, X. G. Xi, M. D. Xing, nd Z. Bo, "An improved SAC lgorithm bsed on the rnge-keystone trnsform for Doppler rte estimtion," IEEE Geosi. Remote Sens. Lett., vol. 10, no. 4, pp , Jul [4] D. Y. Zhu, Y. Li nd Z. D. Zhu, "A keystone trnsform without interpoltion for SAR ground moving-trget imging," IEEE Geosi. Remote Sens. Lett., vol. 4, no. 1, pp. 18-, Jn [5] I. G. Cumming nd S. Li, "Improved slope estimtion for SAR Doppler mbiguity resolution," IEEE Trns. Geosi. Remote Sens., vol. 44, no. 3, pp , Mr [6] A. W. Lohmnn nd B. H. Soffer, "Reltionships between the Rdon-Wigner nd frtionl Fourier trnsforms," J. Opt. So. Amer. A, vol. 11, pp , Jun [7] J. C. Wood nd D. T. Brry, "Rdon trnsformtion of time-frequeny distributions for nlysis of multiomponent signls," IEEE Trns. Signl Proess., vol. 4, no. 11, pp , Nov [8] S. Brbross, "Anlysis of multiomponent LFM signls by ombined Wigner-Hough 37

38 trnsform," IEEE Trns. Signl Proess., vol. 43, no. 6, pp , Jun [9] V. C. Chen nd W. J. Mieli, "Time-vrying spetrl nlysis for rdr imging of mneuvering trgets," Pro. Inst. Elet.Eng. Rdr, Sonr Nvig., vol. 145, no. 5, pp. 6-68, Ot [30] M. Wng, A. K. Chn nd C. K. Chui, "Liner frequeny-modulted signl detetion using Rdon-mbiguity trnsform," IEEE Trns. Signl Proess., vol. 46, no. 3, pp , Mr [31] Y. Li, T. Zeng, T. Long, nd Z. Wng, "Rnge migrtion ompenstion nd Doppler mbiguity resolution by keystone trnsform," in Pro. CIE Int. Rdr Conf., Shnghi, Chin, Ot , 006, pp [3] R. Lnri, "A new method for the ompenstion of the SAR rnge ell migrtion bsed on the hirp z-trnsform," IEEE Trns. Geosi. Remote Sens., vol. 33, no. 5, pp , Sep [33] K. Ntroshvili, O. Loffeld, H. Nies, A. M. Ortiz, nd S. Knedlik, "Fousing of generl bistti SAR onfigurtion dt With -D inverse sled FFT," IEEE Trns. Geosi. Remote Sens., vol. 44, no. 10, pp , Ot [34] T. Jenho nd B. D. Steinberg, "Redution of sidelobe nd spekle rtifts in mirowve imging: the CLEAN tehnique," IEEE Trns. Antenns Propgt., vol. 36, no. 4, pp , Apr [35] Y. S. Wei, N. Xu nd Y. H. Hou, "Study on long-time integrtion lgorithm for wek spe-borne rdr trget," Systems Engineering nd Eletronis, vol. 9, no. 10, pp , Ot Jing Tin ws born in Shndong, Chin, in November She reeived the B.Eng. nd M.S. degrees both in eletroni engineering, from Xidin University, in 006 nd 009, respetively. She is urrently working towrds the Ph.D. degree in the Shool of Informtion nd Eletronis, Beijing Institute of Tehnology, Beijing. Her reserh interests inlude moving-trget detetion, prmeter estimtion nd imging. 38

39 Wei Cui ws born in Inner Mongoli Muniiplity, Chin in He reeived the Ph.D. degrees in eletronis engineering from Beijing Institute of Tehnology in 003. From 003 to 005, he worked s post-dotor in Rdr Reserh Institute in Beijing Institute of Tehnology, where his reserh minly onentrted on rdr system nd VLSI implementtion of rdr signl proessing. Now, he worked s n professor nd supervisor for dotorte students in Beijing Institute of Tehnology. His reserh interests inlude spe trget detetion nd loliztion, rry signl proessing, nd VLSI design. Siling Wu ws born in Anhui Provine, Chin, in He reeived his Ph.D. degree from Hrbin Institute of Tehnology in 1995 nd then worked s post-dotor in Rdr Reserh Institute in Beijing Institute of Tehnology from 1996 to He is now professor nd supervisor for dotorte students in Beijing Institute of Tehnology nd is senior member of Chinese Institute of Eletronis. His reserh interests inlude rdr system nd theory, stellite nvigtion nd pplition of modern signl proessing. 39