1 Introduction

Transcription

1 Published in IET Radar, Sonar and Navigation Reeived on 7th Otober 008 Revised on 3rd Otober 009 doi: /iet-rsn High-speed multi-target detetion with narrowband radar J. Su M. Xing G. Wang Z. Bao Key Laboratory o Radar Signal Proessing, Xidian University, Xi an, Shaanxi , People s Republi o China xmd@xidian.edu.n ISSN Abstrat: High-speed multi-target detetion is a hallenging problem in radar appliations. Typially, targets with high speed go through several range ells in the observation period, whih maes it more diiult to obtain eah target s power oherently aumulated or target detetion. In this study, novel multi-target detetion with a narrowband radar system is proposed. In order to remove range migration and obtain oherent integration o the target energy, the Keystone transorm is applied to the moving targets. However, beause o the high target veloity and the low radar pulse repetition requeny phase ambiguity will our, so the range migration will not be orreted properly. Then the phase ambiguity untion o the th target is ompensated, and the envelope o the th target onentrates in a ertain range ell. Following by requeny modulation rate searh a quadrati phase term is ompensated, and the signal energy is inally oherently aumulated by FT analysis. The target is deteted i the ratio o pea value to noise is higher than a predetermined threshold. For target detetion in low signal-to-noise ratio (SNR), the Clean tehnique is applied. The proposed algorithm is veriied by simulation and raw radar data results. 1 Introdution Target detetion is one o the most important appliations o radar systems. It is also a ey proedure or modern radar imaging. The perormane o target detetion diretly inluenes imaging quality and target identiiation. In order to detet high-speed small moving targets, a long oherent observation time is required. On the other hand, within a long observation period, the high-speed target will move aross several dierent range ells and with an unnown Doppler requeny. The shit o target in dierent range ells and the unnown Doppler requeny mae it is diiult to obtain the target s releting power oherently aumulated. Some ompensation methods are presented or small target detetion in [1 3], suh as Keystone transorm, range strething and joint time requeny proessing, and mathed Fourier transorm (MFT). In [4, 5] the oherent integration method o envelope migration ompensation based on Keystone transorm is studied, but the radial aeleration o the target is not taen into onsideration. For high-speed targets, it is highly possible that radial aeleration exists. At the same time, in the implementation o the Keystone transorm under ambiguous ondition the sin interpolation is used, so the omputational load is large. In addition, the methods o [4, 5] are only suitable or single target detetion. Some previous researh or multi-target detetion is provided in [6 10]. In[6, 7] a partile ilter and high-order spetrum orrelation are used or target detetion and traing. In [8] the perormane o split spetrum proessing or multi-target detetion is presented. In [9] the radial aeleration o a target is still not onsidered when implementing the oherent integration o signal energy. At the same time, Zhang et al. [9] have an assumption that the degree o ambiguity or eah target is nown, but in pratie it is not. The multi-target approah in [10] an simultaneously remove linear range migration regardless o target veloities, but it is limited or target detetion. Under the ondition that the multiple targets have dierent old ators it will not be valid or target detetion. In this paper, we propose a dierent approah or multi-target detetion. For multiple targets, i the signals an be separated rom eah other in range time domain or ross-range Doppler domain, the problem o multi-target detetion is similar to IET Radar Sonar Navig., 010, Vol. 4, Iss. 4, pp doi: /iet-rsn & The Institution o Engineering and Tehnology 010

2 that o single target detetion. In this paper, we will disuss the ase that the signals o multiple targets annot be separated rom eah other in range time domain or rossrange Doppler domain. The rest o the paper is organised as ollows. The eho harateristi is analysed in Setion. Setion 3 presents a Keystone transorm or undersampled data and a method or requeny modulation rate searh. The implementation o the old ator searh is disussed in Setion 4. Setion 5 is the target detetion algorithm. The simulation and raw data are presented in Setion 6. Finally, onlusions are given in Setion 7. Eho signal analysis Suppose the radar transmits a narrowband linear requeny modulation (LFM) signal as ollows where s(^t, t m ) ¼ ret! ^t exp(jpg^t ) exp(jp t) (1) T p 8 >< 1, juj 1 ret(u) ¼ >: 0, juj. 1 T p is the pulse width, g is the requeny modulation rate, is the arrier requeny, ^t ¼ t mt r is the ast time, m is the transmitted pulse number index, T r is the pulse repetition interval and t m ¼ mt r is the slow time. For the high-speed targets, suh as satellites and missiles, the eet o target veloity has to be onsidered. In the ollowing disussion, v denotes the target radial veloity and denotes the speed o light. Under the assumption that jvj=,, 1, the reeived baseband signal an be written as [11 13] be written as the ollowing s rm (^t, t m ) ¼ A 1 exp j 4p( þ d )R(t m ) sin D r ^t R(t m) (3) d g where sin(a) ¼ sin(pa)=pa is the sin untion and the last term o (3) represents the envelope o the target. In this ormula, D r denotes the bandwidth o the transmitted LFM signal, d ¼ v=l denotes the true Doppler requeny o the target and l denotes the radar wavelength. Equation (3) indiates that the target envelope has been shited rom its true position by d =g beause o its motion, and this oset is proportional to its true Doppler requeny d. Sine the oset is a onstant in eah pulse, it does not aet the proessing in parameter estimation and target detetion. Meanwhile, it is noted that the target envelope hanges with the slow time ater pulse ompression. In order to remove the eet o target motion and realise oherent aumulation o target energy, a Fourier transorm (FT) to the variable ^t is employed to transorm the ompressed signal into the range requeny domain, whih an be written as s rm ( r, t m ) ¼ A ret r exp j p d r B g exp j 4p ( r þ d þ )R(t m ) (4) where B is the signal bandwidth. The target model is illustrated in Fig. 1. The Y oordinate is the radar line o sight (RLOS). The angle between target diretion and RLOS is u. In this paper, we assume that u is a onstant in the observation time. The target veloity and aeleration are V e and A, respetively. The initial distane between the radar and target is R 0. Thus, the target slant range along radar LOS with respet to slow time t m satisies " # ^t (R(t s r (^t, t m ) ¼ A 0 ret m )=) exp j 4p R(t m ) T p " exp jpg ^t R(t # m) exp j 4p v^t () R(t m ) ¼ R 0 þ V e os(u)t m þ 1 A os(u)t m (5a) where A 0 denotes the target reletivity and R(t m ) denotes the distane between target and radar at slow time t m. Ater pulse ompression using the mathed ilter H (^t) ¼ ret(^t=t p ) exp( jpg^t ), the ompressed signal an Figure 1 Target model 596 IET Radar Sonar Navig., 010, Vol. 4, Iss. 4, pp & The Institution o Engineering and Tehnology 010 doi: /iet-rsn

3 Then we deine v ¼ V e os(u), a ¼ A os(u), where v and a are the target radial veloity and radial aeleration, respetively. Thus, (5a) an be rewritten as R(t m ) ¼ R 0 þ vt m þ 1 at m (5b) I we substitute (5b) into (4), then s rm ( r, t m ) an be rewritten as s rm ( r, t m ) ¼ A ret r exp(j B 0 ) exp(j 1 ) exp(j ) exp j p (6) d r g where 8 >< >: 0 ¼ 4p (1 þ b)r 0 1 ¼ 4p (1 þ b)v t m ¼ p (1 þ b)a tm, b ¼ r þ d There are our exponential terms in (6). The irst indiates target range eet to the signal. The seond is the Doppler term beause o the target s radial veloity. The third is the requeny modulation term indued by the target s radial aeleration and the last term results rom the high veloity o the target. The deinition o b in (6) shows that b is a untion o Doppler requeny, radar arrier requeny and range requeny. The ontribution o d to b an be negleted i the radar arrier requeny is very high, that is, the ratio o the true Doppler requeny to the radar arrier requeny is ar less than 1. Under this assumption, b an be simpliied as Then (6) an be rewritten as s rm ( r, t m ) ¼ A 3 exp jp d 0 r where b ¼ r (7) exp[ jp(1 þ b) d t m ] exp[ jp(1 þ b)g a tm] (8) A 3 ¼ A ret r exp j 4p R 0, g B a ¼ a l In (8), it is shown that the quadrati term and the irst-order term o slow time t m are all oupled with the range requeny r. The quadrati term o t m will result in signal energy deousing, and the irst-order term o t m will result in target range wal. Both o these two terms will bring diiulties to oherent aumulation o target energy. The ollowing subsetion will disuss the problem in more detail. For multiple targets, the eho in the range requeny rossrange time domain is s rm ( r, t m ) XK ¼1 A exp jp d r exp[ jp(1 þ b) d t m ] exp[ jp(1 þ b)g a tm] (9) where K is the number o targets, d and g a are the Doppler requeny and Doppler requeny modulation rate or the th target, respetively. 3 Keystone transorm or undersampled data and requeny modulation rate searh 3.1 Keystone transorm or undersampled data Owing to the high target speed and low radar pulse repetition requeny, however, it is highly possible that undersampling will our [14, 15]. In this situation, the th target true Doppler requeny an be expressed as d ¼ d0 þ n PRF (10) where d0 is the ambiguous Doppler requeny, n is the old ator and PRF is the radar pulse repetition requeny. Substituting (10) into (9), we an obtain s rm ( r, t m ) XK ¼1 A exp jp d exp[ jp(1 þ b) d0 t m ] exp( jpn PRF t m ) exp( jpbn PRF t m ) exp[ jp(1 þ b)g a tm] (11) It is worth pointing out that pn PRF t m in the third exponential term is a multiple o p. This term in (11) beomes exp( jpn PRF t m ) ¼ 1. So s rm ( r, t m ) an be written as s rm ( r, t m ) XK ¼1 A exp jp d exp[ jp(1 þ b) d0 t m ] exp( jpbn PRF t m ) exp[ jp(1 þ b)g a tm] (1) Owing to the high radial veloity o the target, range migration o the target may happen, whih impairs the r r IET Radar Sonar Navig., 010, Vol. 4, Iss. 4, pp doi: /iet-rsn & The Institution o Engineering and Tehnology 010

4 target s signal energy aumulation. To orret the range migration, a Keystone transorm is applied to the reeived signal in (1) with the ollowing variable transorm t m ¼ þ r t m (13) Then (1) an be rewritten as ollows with the Keystone transorm s rm ( r, t m ) XK A exp jp d r ¼1 exp( jp d0 t m ) exp jp g a 1 þ b t m H a ( r, t m ; n ) where H a ( r, t m ; n ) ¼ exp jpn PRF r t þ m r (14) (15) H a ( r, t m ; n ) is a phase ambiguity untion beause o undersampling, and the target detetion perormane will be degraded i this term is not ompensated properly. For the low-speed target, the onventional Keystone transorm will orret the target range migration properly, but or the high-speed target, it is the H a ( r, t m ; n ) term whih leads to the Keystone transorm invalidation. So H a ( r, t m ; n ) needs to be ompensated beore target detetion. In the narrowband environment r,,, so the term in (15) an be simply regarded as H a ( r, t m ; n ) ¼ exp jpn PRF r t m (16) For the same reason, the b in (14) also does not need to be onsidered. Assume that the old ator or the th target is nown, ater multiplying (14) with onjugate o (16), that is, the phase ambiguity untion o the th target is ompensated, and we have s rm ( r, t m ) A exp jp d r exp( jp d0 t m ) exp( jpg a t (17) m) þ s ross ( r, t m ) s ross ( r, t m ) XK l¼1,l= A l exp jp dl l r exp( jp dl0 t m ) exp( jpg al t m) (n exp jp l n )PRF r t m (18) Following that, an inverse Fourier transorm (IFT) is taen to the signal with variable r, whih transorms the signal into the range time domain, and we have s rm (^t, t m ) A 0 sin D r ^t d exp( jp d0 t m ) exp( jpg a t m) þ XK Al 0 sin D r ^t dl l l¼1,l= þ (n l n )PRF t m exp( jp dl0 t m ) exp( jpg al t m) (19) It an be seen rom (19) that only the envelope o the th target is lat, that is, the th target energy onentrated in a ertain range ell, whereas the envelopes o the other targets still vary with slow time t m. The reason is that the ompensation phase term only mathes to the phase ambiguity untion o the th target. The th target envelope is in the same range ell; however, it is still a quadrati phase term. So the Fourier analysis annot be used here or target energy aumulation. The quadrati phase term should be obtained and then ompensated. Following that the detetion an be ahieved by Fourier analysis. In the ollowing subsetion, the searh method or requeny modulation rate will be disussed. 3. Frequeny modulation rate searh The seleted ell an be expressed as s(t m ) ¼ A (t m ) exp( jp d0 t m ) exp( jpg a t m) þ e(t m ) (0) where e(t m ) is the signal o the other targets whose loations are in the same range ell as the th target. Generally, the dehirping method an be used to estimate the requeny modulation rate g a. Multiplying s(t m ) deined in (0) with an LFM signal with orm o exp(jpg 0 at m), the signal beomes a sinusoid signal when g 0 a equals to g a. Thereore g a an be estimated as ollows ^g a ¼ arg max jft[s(t m ) exp(jpg 0 at m)]j (1) g 0 a To redue the omplexity o requeny modulation rate searh, it an be implemented in several searhing steps. First, searh the requeny modulation rate with a larger searhing step and ind a rough estimation o it. Then searh this parameter around the estimated value with a smaller searhing step. Repeating the above proedure with several iterations, an aurate estimation o g a an be obtained. 598 IET Radar Sonar Navig., 010, Vol. 4, Iss. 4, pp & The Institution o Engineering and Tehnology 010 doi: /iet-rsn

5 4 Implementation In this setion, we will present some pratial details o the ompensation or the phase ambiguity untion inluding the hoie o the old ator under high- and low-signalto-noise ratio (SNR) onditions. 4.1 Choie o the old ator under high SNR In the above subsetion, we assumed that the old ator or the th target was nown, but in pratie it should be estimated rom the data. Without pre-nowledge o the target veloity, we an assume a maximum target veloity o 8000 m/s. The radar arrier requeny and the radar pulse repetition requeny are determined by the radar system, the maximum old ator an be nown. But whih old ator is the one or the th target? This is a question that should be studied. Reall that in the above subsetion, it is a meaningul phenomenon when the old ator equals that o the th target, the signal o the th target envelope loates in the same range ell. So it an be used here or old ator estimation. The Radon transorm is the integral along a straight line deined by distane rom the origin and angle o inlination ormed by the perpendiular to the line. Here we only need to integrate or angle o inlination 908. Then the signal an be expressed as sr n. I the old ator is proper, then the target energy an be aumulated and it will have a pea at sr n. But or the other targets, its energy an spread in several ells. For estimation o the old ator, the energy riterion an be used in the high-snr ondition and the ost untion an be expressed as ollows H (n) ¼ max(sr n ) () Fig. is the graph o the variation o the ost untion with the old ator when there are three targets. The dashed line is the ondition that the amplitudes o the three targets are all equal to 1. It an be seen there are three pea value regions, so it an be simply regarded that there are three targets. And the pea value o eah region an be regarded as the estimation o the target old ator. 4. Choie o the old ator under low SNR But or the low-snr ondition, the above estimation method based on the energy riterion will not validate. In Fig., the solid line is the ondition that the amplitudes o the three targets are 1, 0.6 and 0., respetively. There are only two pea value regions in the solid urve and this will lead to errors or target detetion. In the low-snr ondition, new riterion should be used or old ator estimation. But or the high SNR target, the old ator is exatly orret. Thereore we an heuristially apply the Clean [16] tehnique here. Assume that the old ator o a high SNR target is obtained, then the target s phase ambiguity untion an be ompensated. Following by requeny modulation rate searh a quadrati phase term an be obtained and ompensated, then the target energy an be oherently aumulated. Ater that the Clean tehnique an be used. The high SNR target an be eliminated by a band-stop ilter in the range time domain and ross-range Doppler domain, and then by a onjugate operation to renew the signals o the other targets, the operation inluding the phase ambiguity untion and the quadrati phase term. The above proedures are repeated to detet the next potential target. Fig. 3 is the graph o the ost untion against the old ator. From Fig. 3, it an be seen that the three targets are loated at their true old ators, the Clean tehnique is eetive. Figure Result or old ator searh or three targets with idential and non-idential amplitudes by the onventional method Figure 3 Result or old ator searh or three targets with non-idential amplitudes by the Clean method IET Radar Sonar Navig., 010, Vol. 4, Iss. 4, pp doi: /iet-rsn & The Institution o Engineering and Tehnology 010

6 5 Detetion and parameter omputation With the estimated requeny modulation rate g a, the quadrati phase term o the target an be ompensated so as to ahieve the signal energy oherent aumulation. Deine a quadrati phase ompensation untion as ollows H q (t m ) ¼ exp (jp ^g a t m) (3) Ater quadrati phase term ompensation, (19) an be expressed as s rm (^t, t m ) A 0 sin D r ^t d exp( jp d0 t m ) þ XK A l sin D r ^t dl l l¼1,l= þ (n l n )PRF t m exp( jp dl0 t m ) exp[ jp(g al g a )t m] (4) An IFT is then applied or target energy oherent aumulation and we have s rm (^t, a ) A 00 sin D r ^t d sin[t ( a þ d0 )] þ s ross (^t, a ) where T is the oherent aumulation time. (5) From this ormula, it is observed that the sin-lie point spread untion (PSF) is reonstruted. This is an advantage or target detetion. Based on (5), the targets are deteted i the ratios o their pea values to noise are larger than a given threshold, and the pea loation in ross-range orresponds to its ambiguous Doppler requeny ^ d0. With the estimated requeny modulation rate ^g a, old ator ^n and the ambiguous Doppler requeny ^ d0, the radial aeleration and radial veloity or the th target are estimated as ollows ^a ¼ l ^g aprf, ^v ¼ l( ^ d0 þ ^n PRF) (6) Notie that the estimation o ^ d0 may use an interpolation operation to obtain a more preise result. Based on the above disussion o the proposed algorithm or target detetion, a lowhart is shown in Fig. 4. There are some points to be explained. Figure 4 Flowhart o the proposed method or detetion 1. Sine the iteration operation is needed in the low-snr ondition, the omputational load will inrease a lot. But the Keystone transorm needs only one time; moreover, the phase ambiguity untion is easy to ompensate. Thereore the proposed algorithm is highly eiient or target detetion. In addition, or the detetion in the high-snr ondition, the iteration operation will not needed, so the omputational load will not be very large.. For the requeny modulation rate searh, the step size and the time should be balaned. As it has been pointed out in Setion 3., the step size should be iterated so as to obtain a high auray requeny modulation rate. 3. For the seond time ompensation o the phase ambiguity untion or the th target, sine the old ator has been Table 1 Simulation parameters Parameters Target A B C arrier requeny, GHz signal bandwidth, MHz sample requeny, MHz pulse repetition requeny, Hz initial range, m pulse number radial veloity, m/s radial aeleration, m/s satterer oeiient IET Radar Sonar Navig., 010, Vol. 4, Iss. 4, pp & The Institution o Engineering and Tehnology 010 doi: /iet-rsn

7 Figure 5 Simulation results a Mathed iltering b Keystone transorm Compensation o the phase ambiguity untion o the th target d Cleaned signal e FT-based detetion Radon transorm g Detetion with the proposed algorithm IET Radar Sonar Navig., 010, Vol. 4, Iss. 4, pp doi: /iet-rsn & The Institution o Engineering and Tehnology 010

8 obtained beore the irst ompensation so as to obtain a more preise estimate, only the old ator around the estimated value needs to be searhed. 6 Simulation and real data proessing The simulation parameters are listed in Table 1. The raw target eho is added to the omplex Gaussian white noise with SNR o 15 db. The simulation results are shown in Fig. 5. Fig. 5a is the mathed iltering result o the origin signal. It an be seen that the tra o the target C is wea, so a long integration time is needed. But in a long time, the radial veloity o the target will result in range migration. For the high-speed target, the range migration would be more severe. At the same time, radial aeleration will lead the quadrati signal term or the original signal and it is a disadvantage or target detetion. Fig. 5b is the result o the Keystone transorm. But beause o the ambiguous Doppler requeny o the target the linear range migration annot be orreted properly or the high-speed target. Fig. 5 is the result o an IFT in range requeny domain ater ompensation with the phase ambiguity untion o the th target. It is learly shown that the envelope o the th target is onentrated in a ertain range ell, so it is an advantage or oherent aumulation detetion target. Fig. 5d is the signal ater elimination o the strongest signal ontribution or the eho. It an be seen that the Clean tehnique is eetive. Fig. 5e is the FT-based detetion. It is shown that the energy o targets is spread around the whole range time ross-range Doppler domain. The targets are diiult to detet beause o the low SNR. Fig. 5 is the result o a Radon transorm o the mathed iltering signal. From the igure, it an be seen that the wea target will not be deteted as the Radon transorm is a non-oherent integration or target detetion. Fig. 5g shows the detetion result with the proposed algorithm. It is shown that the signal energy is eetively aumulated and the SNRs o the deteted targets are greatly improved. Note that this detetion result is the summation o three iterations. Next we will apply the proposed algorithm to the proessing o raw radar data. The radar operates at the L wave band, the signal bandwidth is 10 MHz, the pulse repetition requeny is 400 Hz and there are 000 pulses in the oherent proessing interval. Fig. 6 shows the proessing results. Fig. 6a is the original eho signal. It an be seen that the SNR or eah target is dierent and some target signals are muh weaer than the others. Fig. 6b shows the FT-based detetion. It an be seen that the strong target aets the detetion o the wea targets. Figure 6 Proessing results or raw data a Original eho b FT-based detetion Eho elimination o the strongest signal and then ompensation o the phase ambiguity untion o the sub-strongest target d Detetion with the proposed algorithm 60 IET Radar Sonar Navig., 010, Vol. 4, Iss. 4, pp & The Institution o Engineering and Tehnology 010 doi: /iet-rsn

9 Fig. 6 shows the result ater elimination o the strongest signal and then ompensation o the phase ambiguity untion o the seond target. It is shown that the envelope o the target has been onentrated in a ertain range ell, whih is a beneit or target energy oherent aumulation. Fig. 6d shows the detetion result with the proposed algorithm. It is shown that the signal energy is eetively aumulated and the SNRs or the targets detetion are improved. The three targets an be easily deteted. Note that the detetion result is the summation o three iterations. 7 Conlusions A high-speed target will ause target range ell migration beause o its high radial veloity, whih impairs target detetion. In order to remove the range migration, the Keystone transorm is applied. However, the high veloity o the target maes the radar pulse repetition requeny too low to mae the onventional Keystone transorm wor orretly beause o Doppler requeny ambiguity. In this paper, we proposed an undersampling Keystone transorm, and we ound that the phase ambiguity untion o the target will lead the target envelope migration. Based on the ompensation o the phase ambiguity untion and the quadrati phase term, a detetion algorithm or multiple targets with high speed was developed. The proposed algorithm was veriied by simulated and raw radar data. 8 Anowledgments The authors would lie to than the anonymous reviewers, whose omments will help to improve the quality o our manusript. This wor was supported by 973 program under grant 010CB731903, the program or New Century Exellent Talents in University (NCET ) and the National Deense Pre-researh Foundation o China under grant 9140C Reerenes [1] ZHANG S.S., ZENG T., LONG T., YUAN H.P.: Dim target detetion based on Keystone transorm. IEEE Int. Radar Con., 005, pp [] WANG J., ZHANG S.H., BAO Z.: On motion ompensation or wea radar releted signal detetion. The Sixth Int. Con. Signal Proessing, 00, vol. 1, pp [3] CHEN J.J., CHEN J., WANG S.L.: Detetion o ultra-high speed moving target based on mathed Fourier transorm. CIE Int. Con. Radar, 006, pp. 1 4 [4] YUAN S.J., WU T., MAO M., MEI G.J., WEI X.: Appliation researh o Keystone transorm in wea high-speed target detetion in low-prf narrowband hirp radar. Ninth Int. Con. Signal Proessing, 008, pp [5] LI Y., ZENG T., LONG T., WANG Z.: Range migration ompensation and Doppler ambiguity resolution by Keystone transorm. CIE 06 Int. Con. Radar, 006, pp. 1 4 [6] MORELANDE M.R., KREUCHER C.M., KASTELLA K.: A Bayesian approah to multiple target detetion and traing, IEEE Trans. Signal Proess., 007, 55, (5), pp [7] LIOU R.-J., AZIMI-SADJADI M.R.: Multiple target detetion using modiied high order orrelations, IEEE Trans. Aerosp. Eletron. Syst., 1998, 34, (), pp [8] TIAN Q., BILGUTAY N.M.: Statistial analysis o split spetrum proessing or multiple target detetion, IEEE Trans. Ultrasonis Ferroeletr. Freq. Control, 1998,45, (1), pp [9] ZHANG S.S., ZHANG W., WANG Y.: Multiple targets detetion in terms o Keystone transorm at the low SNR lever. IEEE Int. Con. Inormation and Automation, 008, pp. 1 4 [10] PERRY R.P., DIPIETRO R.C., FANTE R.L.: Coherent integration with range migration using Keystone ormatting. IEEE Radar Con., 007, pp [11] SHARIF M.R., ABEYSEKERA S.S.: Eiient wideband signal parameter estimation using a Radon-ambiguity transorm slie, IEEE Trans. Aerosp. Eletron. Syst., 007, 43, (), pp [1] SHARIF M.R., ABEYSEKERA S.S.: Eiient wideband sonar parameter estimation using a single slie o Radonambiguity transorm. IEEE Int. Con. Aoustis, Speeh, and Signal Proessing, 005, vol. 5, pp [13] KELLY E.J., WISHNER R.P.: Mathed-ilter theory or highveloity, aelerating targets, IEEE Trans. Military Eletron., 1965, 9, (1), pp [14] PERRY R.P., DIPIETRO R.C., FANTE R.L.: SAR imaging o moving targets, IEEE Trans. Aerosp. Eletron. Syst., 1999, 35, (1), pp [15] BAO Z., XING M.D., WANG T.: Radar imaging approahes (Eletronis Industry Press, Beijing, 005) [16] WANG Q., XING M.D., LU G.Y., BAO Z.: SRMF-CLEAN imaging algorithm or spae debris, IEEE Trans. Antennas Propag., 007, 55, (1), pp IET Radar Sonar Navig., 010, Vol. 4, Iss. 4, pp doi: /iet-rsn & The Institution o Engineering and Tehnology 010