Open Access Improved Chirp Scaling Algorithm Based on Fractional Fourier Transform and Motion Compensation

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1 Send Orders for Reprints to The Open Automtion nd Control Systems Journl, 15, 7, Open Aess Improved Chirp Sling Algorithm Bsed on Frtionl Fourier Trnsform nd Motion Compenstion Tn Gewei *, Pn Gungwu nd Lin Wei Shool of Informtion Siene nd Engineering, Huqio University, Ximen, Fujin, P.R. Chin Abstrt: Frtionl Fourier trnsform (FrFT is new tool of time-frequeny nlysis, nd good one to ompress hirp signl nd to proess non-sttionry motion error. Aiming t the problem of the onventionl SAR imging lgorithm bsed on FFT nd two-step motion ompenstion, tht the effet of non-sttionry error elimintion is not obvious nd the improvement of resolution is restrited, n improved Chirp Sling imging lgorithm bsed on FrFT nd motion ompenstion is put forwrd in this pper, whih is expeted to eliminte the influene of rnge motion error, so s to obtin high qulity SAR imges. Simultion results nd imging results of rel SAR dt show tht the proposed lgorithm n eliminte the influene of rnge motion error effetively (the rel SAR dt provided by Institute of Eletronis, Chinese Ademy of Sienes. Keywords: Frtionl Fourier trnsform, Non-sttionry motion error, Motion ompenstion, CS lgorithm, High resolution. 1. INTRODUCTION Motion error is ruil ftor to limit irborne SAR resolution improvement. It is neessry to tke mesures to ompenste the motion error used by ir turbulene to obtin high qulity SAR imges [1-3]. The two-dimensionl sptil-vrint property of SAR motion error is hllenge to motion ompenstion tehnique. Two-step motion ompenstion lgorithm n eliminte the influene of rnge motion errors, but the effet of ompenstion is not obvious when motion errors re non-sttionry. Frtionl Fourier trnsform (FrFT is proposed by V. Nmis in 198 [4], whih is new time-frequeny nlysis tool. Compred to Fourier trnsform, FrFT hs inomprble superiority in proessing non-sttionry signls. Chirp signl n be well foused fter frtionl Fourier trnsform with the speifi rottion ngle whih provides possibility to hieve high resolution nd high ury in SAR imging. Espeilly when signl nd interferene soure re oupled to eh other, good seprtion effet in frtionl domin n be obtined [5-7]. There hve been litertures putting forwrd SAR imging lgorithm ombining with FrFT. Literture [8, 9] proposed n improving CS imging lgorithm bsed on FrFT. Literture 1 nd 11 lso proposed ombintion of FrFT nd RD imging lgorithm. These reserh results show tht FrFT n ontribute to the improvement of SAR resolution. *Address orrespondene to this uthor t the Shool of Informtion Engineering, Huqio University, Ximen, Fujin, 3611, P.R. Chin; Tel: ; E-mil: tngewei7@163.om In order to eliminte the influene of motion error more effetively, in the pper, FrFT is ombined with SAR imging lgorithm nd motion ompenstion tehnology, through utilizing the superiority of FrFT in proessing hirp signl nd non-sttionry signl, so s to obtin high qulity SAR imges.. FRACTIONAL FOURIER TRANSFORM Frtionl Fourier trnsform is the generl form of Fourier trnsform, nd its trnsform kernel is exp{ j# ( t + u ot"! ut s" } " $n# ( 1! jot" * # * K p (t,u = (1 * % (t! u * " &n'# 1 + *% (t + u " + # &n'# where p π π α = p is the order of FrFT, when α = the FrFT beomes the Fourier trnsform. Frtionl Fourier trnsform nd its inverse trnsform for signl x( t is defined s: = = (, X u Fp x t u x t Kp t u dt = = (, xt F p X u t X uk p tudu Frtionl Fourier trnsform for hirp signl j kt xt = e π is s follows: = p (, = X u x t K t u dt jπu ot α jπkt jπt ot α jπutsα Ae e e dt ( ( Benthm Open

2 43 The Open Automtion nd Control Systems Journl, 15, Volume 7 Gewei et l. where 1 j otα A =. π When otα = k, p= rot ( k = p,whih is the opt π optiml order of FrFT. Thus, jπku jπutsα X u = Ae e dt A = = = sα ( s jπ ku Ae δ π αu δ u Cδ u So the optiml-order frtionl Fourier trnsform of hirp signl is n impulse signl. FrFT hs the following time-shift nd frequeny-shift properties, F [( s t+ τ ]( u = e F [(]( s t u+ τ os α (4 p jπτ sin α os α + j πτ u sin α p F s t e u = F s t u (5 jπρt jπρ sinα osα + jπρuosα p[( ]( e p[(]( ρsin α F p [e j!"t s(t + # ](u = F p [s(t](u + # os$ % " sin$ - ' &exp j! # sin$ os$ + # usin$ + "uos* 1., ( %" sin$ os$ % "# sin $ + 3 Aording to frequeny-shift property, there is, jπ ( kt + vt F [e ]( u = Cδ u vsinα popt if the hirp signl is disrete sequene, then the optiml k order is popt = rot. π fs N Just using the optiml-order frtionl Fourier trnsform, hirp signls will be well foused, where there is no need of dditionl mthing ompression, thereby improving proessing effiieny. 3. MOTION ERROR MODEL AND SAR ECHO SIG- NAL Motion error in rnge diretion is divided into rngeindependent nd rnge-dependent motion errors [1]. The Δ r t, r, nd the prt prt unrelted to rnge slnt is lled tht is dependent on the hnge of rnge slnt is lled Δ r t, r. Rnge motion error refers to the differene of the v nerest distne from one trget to rel trk nd the nerest distne from the trget to the nominl trk [13]. (, (, (, Δ rtr =Δ r tr +Δ r tr (7 m m v where r m is the nerest slnt rnge of swth bnd enter, wr is the width of swth bnd, r is the nerest slnt rnge of the trget P under nominl trk. Suppose tht SAR trnsmit hirps to n observed sene, their ehoes fter the demodultion re m m (6 τ Rt Rt sd ( t, τ; r = σ( x, r ret exp jπk τ T p t t 4 Rt π ret exp j Ts λ where τ is fst time in the slnt rnge diretion, t is slow time long the rdr flight pth, nd λ re speed of light nd rdr wvelength respetively, nd k is the hirp rte. is the retngle funtion, in whih Tp nd Ts re the ret pulse durtion nd syntheti perture time respetively. Rt is the instnt rnge from the rdr to point trget in the observed sene, whih inludes motion error. (, (, (, Rt = R t +Δ rtr = R t +Δ r tr +Δ r tr (9 m m v where (8 R t = r + v t t is instnt rnge from rdr to the point trget under nominl trk. Ignoring zimuth motion errors, rnge-fourier trnsform is first performed by the sttionry phse priniple, the result is: f r t t f r sd( t, fr; r = C1 σ( x, r ret ret exp jπ Br Ts k π 4 πδrm t, rm 4 Δrv t, r exp j fr exp j fr π 4π R 4, 4, πδrm t rm Δrv t r exp j exp j exp j λ λ λ where f r is rnge frequeny. (1 Rnge-independent motion errors n use the eho envelope dely errors (the seond exponentil item nd zimuth phse errors (the fifth exponentil item, whih should be ompensted long with rnge ompress. Rngedependent motion error (the third nd sixth exponentil item is relted to slnt rnge. In rel SAR dt ll point trgets re spred out in rnge diretion nd zimuth diretion, so ompenstion to suh error must be implemented when rnge ompression nd the rnge ell migrtion orretion (RCMC re ompleted nd zimuth ompression is unfinished. 4. IMPROVED CS ALGORITHM BASED ON FRAC- TIONAL FOURIER TRANSFORM AND MOTION COMPENSATION Proessing flow of the improved CS lgorithm bsed on FrFT nd motion ompenstion is shown in Fig. (1. Firstly, perform rnge-fourier trnsform for rw dt nd the result is formul 1, whih when multiplying with H n omplete first-order motion ompenstion, nd moo1 the phse ompenstion funtion is: 4π H t f r j f f r t r 1(, ; = exp ( + Δ (, moo r m r m m (11

3 Improved Chirp Sling Algorithm The Open Automtion nd Control Systems Journl, 15, Volume First-order motion ompenstion Chirp sling operte H t r H ( f CS, f r; r ref moo1, m RCMC H rm (, τ ρ Rw dt rnge FT rnge IFT zimuth FT Foused imge zimuth FrFT zimuth FT zimuth IFT rnge FrFT Hmoo(; t r Seond-order motion ompenstion H f r, v Residul phse ompenstion Fig. (1. Proessing flow of the improved CS lgorithm bsed on FrFT nd motion ompenstion. So rnge-independent motion errors re removed from the rw dt. After first-order motion ompenstion, the eho signl is, ( R r t t 4π +Δ v sd ( t, τ; r = σ( x, r ret exp j Ts λ is ( R r ( R r τ +Δ v +Δ v ret exp jπk τ T p Then performing zimuth-fourier trnsform, the result Sd ( f,!;r = C "# x,r $ f "ret ' & % B d ( exp - * j 4+ R f ;r 1., 3 $! * R( f "ret ;r ' & % T p ( exp * j+ k! * R f ;r - $ ' 1 &. m % ( 3 (1 where f is Doppler frequeny, C is omplex onstnt, r +Δr λ f R( f ; r = = ( r +Δ r [ 1 + C ( f ], γ ( f = 1, v k m v v s γ f ( f 3 1 γ =, KSRC =. 1 k + 1 K 1 γ f λr SRC Next CS opertion is performed, the orresponding proessing funtion is: { } (, τ; exp π ( ; ( τ τ H f r = j k f r C f f (13 CS ref m ref s ref where ref ref r is referene rnge, [ 1 C ( f ] r τ ref = +. s Then multiplying formul 13 with formul 1, the result fter CS opertion is: (, τ; = (, τ; (, τ; Sd f r Sd f r H f r CS CS ref ( r rv γ f 4 π +Δ = C σ( x, r ret exp j f Bd λ 1 exp Θ ; + Tp { j ( f r } ret τ r rref Cs ( f exp jπkm 1+ Cs ( f τ ( r + rref Cs ( f (14 where Θ is residul phse used by CS opertion, 4π Θ ( f ; r = k, 1 m f rref Cs f + Cs f r rref (15 From formul 14, it is known tht the migrtion trk of point trget of different rnge is sme s tht of referene point. So bulk RCMC n be performed. Using the fousing properties of FrFT in proessing hirp signl, nd its time-shift nd frequeny-shift properties, the rnge ompression nd RCMC re shown s follows. Formul 14 is rewritten s hirp signl with time-shift, ( r rv 4π +Δ SdCS ( f, τ; r = sd ( f, τ + τ; r exp j γ f jθ (16 λ = C "# "ret { ' ((! +! } s d f,! +! ;r exp * j+ k m $ % 1+ C s f where r r C ( f $ 1 ' & (! +! T %& p ( (17 τ = ( + ref s, nd the RCM term is r C. ref rm s Aording to formul 6, phse ompenstion funtion j H τ, ρ = e πρτ is needed to offset the influene of RCM term. Then rnge-frtionl Fourier trnsform of the optiml order popt = rot ( km ( 1+ Cs is performed, the result is: π H rm ( f,! = F p " # Sd CS f,! +! ;r $ % u F p " # s d ( f,! +! ;r e j&'! $ %( u. (exp j 4& ( r + *r v, f + 1 = C 3 sin( u +! os5 ' sin5 ( j-. 6 exp j&! sin5 os5 + u! sin5 + u' os5 9 8 ; ' sin5 os5 '! sin5 : 4. (exp j 4& r + *r v, f j- 3 4 (18

4 434 The Open Automtion nd Control Systems Journl, 15, Volume 7 Gewei et l. Tble 1. Simultion prmeters. Prmeter Vlue Prmeter Vlue Crrier Wvelength.4m Forwrd Veloity 18ms Bndwidth for Trnsmit Signl 3MHz Durtion of Syntheti Aperture 5.1s Pulse Durtion of Trnsmit Signl 1µs Pulse Repetition Frequeny 1Hz Smpling Frequeny in Rnge Diretion 4MHz Rnge-independent Motion Errors t In order to eliminte the influene of RCM term, it requires rref Cs osα ρsinα = 4 ρ = r C otα = r k C 1+ C ( τ, ρ ref s ref m s s H = e rm 4π j rref kmcs ( 1+ Cs τ The next step is to ompenste to the residul phse, so the phse ompenstion funtion is: = exp! j" # (. ' H v f,r - & sin$ os$ + u# sin$ + u% os$ 1 +!% sin$ os$! %# sin$ * (19 + j, f,r 3 In the following, seond-order motion ompenstion in two-dimensionl time-domin is performed, nd the ompenstion funtion is: 4π Hmoo(, t r = exp j Δrv (; t r λ ( After this step, rnge-dependent motion errors n be removed. The lst step is to rry out zimuth-frft in rnge- Doppler domin, so + sd(, t u; r = Sd ( f, u; r K ( f, t df p r + f 4π r = C4 sin ( τ ret exp j γ( f Bd λ { π ( β β } exp j f + t ot f ts df r 4π r where Sd( f, u; r = C3sin u exp j γ ( f, λ whih is the signl in rnge-doppler domin fter seond- λf 1 λf order motion ompenstion. γ ( f = 1 1 v v. λπr Choosing ot β =, λπr p ot opt = rt v π v, then r 4πr λπr sd( t, u; r = C4sin ( τ exp j exp j t λ v + f ret exp{ jπ ft sβ} df (1 Bd r = C5 sin τ sin ( s βt 5. SIMULATIONS AND SAR IMAGING RESULTS BASED ON REAL DATA FOR THE PROPOSED AL- GORITHM 5.1. Simultions of Point Trget for the Proposed Algorithm Simultions of point trget with motion error for the lgorithm put forwrd in this pper re s follows; the simultion prmeters re shown in Tble 1. The rnge motion error is non-sttionry rndom motion error whose men vlue is exponentil funtion, so rnge- Δ r t, r =Δr t, r Δ r t, r. dependent motion error is v m m Fig. ( is simultion omprison of the onventionl CS lgorithm bsed on FFT nd the improved CS lgorithm bsed on FrFT nd motion ompenstion bout point trget with motion error. From the simultion results, it is known tht both of the lgorithms n eliminte the influene of motion error effetively. The zimuth impulse responses fter ompenstion re lose to tht when there is no motion error. Fig. ( nd d indites tht the min-lobes of impulse response both in the rnge nd zimuth diretion fter proessing using the proposed lgorithm in this pper, re nrrowed thn tht tking the onventionl FFT-bsed CS lgorithm, whih proves tht the new lgorithm n improve the resolution nd obtin good fousing effet. At the sme time, piture ( lso shows tht the first side-lobe of rnge impulse response for the proposed lgorithm flls below -35dB, whih for the onventionl lgorithm only rehes bout -5dB nd there is burr in -11dB, ll of whih proves tht the effet of suppression side-lobe nd nti-interferene for the proposed lgorithm is obvious. The performne omprison of the two lgorithms is shown in Tble nd Tble 3.

5 Improved Chirp Sling Algorithm The Open Automtion nd Control Systems Journl, 15, Volume zimuth impulse response -1 no error FFT-twostep-MoCo - -3 db Azimuth ( Comprison of zimuth impulse response when no error nd when error exists but proessed tking the onventionl motion ompenstion lgorithm zimuth impulse response -1 no error FrFT-twostep MoCo - -3 db Azimuth (b Comprison of zimuth impulse response when no error nd when error exists but proessed using the proposed ompenstion lgorithm Fig. (. Contd

6 436 The Open Automtion nd Control Systems Journl, 15, Volume 7 Gewei et l. rnge ompression -1 FFT-twostepMoCo-CS FrFT-twostepMoCo-CS - db Rnge ( Comprison of rnge impulse response fter the two ompenstion lgorithms proessed zimuth ompression -1 FFT-twostepMoCo-CS FrFT-twostepMoCo-CS - -3 db Azimuth (d Comprison of zimuth impulse response fter the two ompenstion lgorithms proessed Fig. (. Comprison of point trget simultions for the onventionl nd the proposed lgorithm.

7 Improved Chirp Sling Algorithm The Open Automtion nd Control Systems Journl, 15, Volume Tble. Performne omprison of zimuth impulse response. Min Lobe ISLR PSLR No Error 1.5m db -3.1dB FFT-two-step Motion Compenstion CSA 1.96m dB -1.38dB FrFT- two-step Motion Compenstion CSA 1.85m -3.19dB dB Tble 3. Performne omprison of rnge impulse response. Min Lobe ISLR PSLR No Error 1.195m db -9.11dB FFT- two-step Motion Compenstion CSA m dB -.88dB FrFT- two-step Motion Compenstion CSA 1.9m -4.45dB -8.86dB ( SAR imge with motion error Fig. (3. Contd

8 438 The Open Automtion nd Control Systems Journl, 15, Volume 7 Gewei et l. (b SAR imge fter FFT-two step motion ompenstion CSA ( SAR imge fter FrFT-two step motion ompenstion CSA Fig. (3. Comprison of imging results before nd fter motion ompenstion (resolution: 3m 3m.

9 Improved Chirp Sling Algorithm The Open Automtion nd Control Systems Journl, 15, Volume ( SAR imge fter FFT-two-step motion ompenstion CS lgorithm proessing (b SAR imge fter FrFT- two-step motion ompenstion CS lgorithm proessing Fig. (4. Imging results of X-SAR dt ombined motion ompenstion (resolution:.5m.5m. 5.. Comprison of Imging Results of Rel L-SAR Dt Fig. (3 is imging result of the rel SAR dt with motion errors. The blurring imge shows there re obvious qudrti phse errors. Proessing result for suh SAR dt using the onventionl CS lgorithm bsed on FFT nd two-step motion ompenstion is shown s Fig. (3b. Due to the elimintion of most motion errors, imge resolution is improved signifintly, but some res re with more detils (suh s prts of the irle line, the imge is not ler nd imge resolution deteriortes beuse of the residul phse error. The proessing result with the improved CS imging lgorithm bsed on FrFT nd two-step motion ompenstion tehnology is shown in Fig. (3. As the detiled informtion inreses, the imge resolution is further improved Comprison of Imging Results of Rel X-SAR Dt Fig. (4 shows imging results of X-SAR dt. Sine the motion error is stble nd the zimuth bem is nrrow, so the proessing results using the onventionl lgorithm nd using the proposed lgorithm re pproximtely the sme. But in some res with more detils, Fig. (4b is lerer thn Fig.

10 44 The Open Automtion nd Control Systems Journl, 15, Volume 7 Gewei et l. (4, whih further proves tht the proposed CS lgorithm bsed on FrFT nd motion ompenstion n effetively enhne the resolution of SAR imges. CONCLUSION Motion error is ruil ftor to limit irborne SAR resolution improvement. Due to the ontinuous pursuit of modern SAR for high resolution, how to overome the motion error used by ir turbulene beomes n urgent problem for rdr workers. The improved CS imging lgorithm bsed on frtionl Fourier trnsform nd two-step motion ompenstion proposed in this pper n redue vrious motion errors effetively, espeilly it n eliminte the imge blurring used by non-sttionry motion errors. The reserh results of the pper provide n effetive solution sheme to motion error proessing. CONFLICT OF INTEREST The uthors onfirm tht this rtile ontent hs no onflit of interest. ACKNOWLEDGEMENTS This pper is supported by the nturl siene foundtion of Fujin provine of 13 projet (the number is 13J14. REFERENCES [1] Y.Li, Xing-dong Ling, C. Ding, L. Zhou, L. Chen, nd W. Hong, A motion ompenstion pproh integrted in the omeg-k lgorithm for irborne SAR, IEEE Interntionl Conferene on Imging Systems nd Tehniques, Mnhester, United Kingdom, pp , 1. [] Y. Mo, M. Xing, L. Wei, Y. Li, nd W. Hong, Error nlysis of SAR motion ompenstion, IEEE Interntionl Conferene on Imging Systems nd Tehniques, Mnhester, United Kingdom, 1, pp [3] H. Guo, nd Y. Li, Studying tmospheri turbulene effets on irrft motion for irborne SAR motion ompenstion requirements, IEEE Interntionl Conferene on Imging Systems nd Tehniques, Mnhester, United Kingdom, 1, pp [4] V. Nmis, The frtionl Fourier trnsform nd its pplition in quntum mehnis, Order of IMA Journl of Applied Mthemtis, vol. 5, pp , 198. [5] Q. Lin, T. Rn, Z. Siyong, nd W. Yue, Adptive time-vrying filter for liner FM signl in frtionl Fourier domin, 6 th IEEE Interntionl Conferene on Signl Proessing, Beijing, Chin, pp ,. [6] L. Durk, nd S. Aldirmz, Adptive frtionl Fourier domin filtering, Signl Proessing, vol. 9. no. 4, pp , 1. [7] S. E. D. Igor, nd S. L. Jubi, Frtionl Fourier trnsform s signl proessing tool: n overview of reent developments, Signl Proessing, vol. 91, no. 6, pp , 11. [8] A. S. Amein, A new hirp sling lgorithm bsed on the frtionl Fourier trnsform, IEEE Signl Proessing Letters, vol. 1, no. 1, pp , 5. [9] S. Ahmed, J.J. Amein, nd J.J. Sorghn, Azimuth frtionl trnsformtion of the frtionl hirp sling lgorithm, IEEE Trnstions on Geosienes nd Remote Sensing, vol. 44, no. 1, pp , 6. [1] C. Clemente, nd J. J. Sorghn, Rnge Doppler nd hirp sling proessing of syntheti perture rdr dt using the frtionl Fourier trnsform, IET Signl Proessing, vol. 6, no. 5, pp , 1. [11] M.G. El-Mshed., nd M.I. Dessouky, Trget imge enhnement in rdr imging using frtionl Fourier trnsform, Sense Imging, vol. 13, pp , 1. [1] R. Lnri, nd G. Fornro, A short disussion on the ext ompenstion of the SAR rnge-dependent rnge ell migrtion effet, IEEE Trnstions on Geosiene nd Remote Sensing, vol. 35, pp , [13] T. Ge-wei, nd D. Yun-ki, The two-dimensionl sptil-vrint properties of irborne SAR motion error nd its ompenstion, Journl of Eletronis nd Informtion Tehnology, vol. 31, no., pp , 9. Reeived: September 16, 14 Revised: Deember 3, 14 Aepted: Deember 31, 14 Gewei et l.; Liensee Benthm Open. This is n open ess rtile liensed under the terms of the Cretive Commons Attribution Non-Commeril Liense ( whih permits unrestrited, non-ommeril use, distribution nd reprodution in ny medium, provided the work is properly ited.