WAVEFORMS, WAVES AND MATHEMATICAL MODELING OF RADAR SIGNAL FORMATION PROCESS

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1 WAVEFOMS, WAVES AND MAHEMAICAL MODELING OF ADA SIGNAL FOMAION POCESS Andon Dimirov Lazarov Burgas Free Universiy ВЪЛНОВИ ФОРМИ, ВЪЛНИ И МАТЕМАТИЧЕСКО МОДЕЛИРАНЕ НА ПРОЦЕСА НА ФОРМИРАНЕ НА РАДАРНИ СИГНАЛИ Андон Димитров Лазаров Absrac: In he presen work heoreical descripion of waveforms, waves and mahemaical model of radar signal formaion is addressed. Based on he complex exponenial funcion coninuous and finie waveforms, waves and radar signals are analyically described and graphically illusraed. adar imagining geomery is presened and analyical described. Key words: radar waveforms, elecromagneic wave, radar signal modeling 1. Inroducion ecenly imaging radar echnology aracs excepional aenion. Imaging radar is very imporan for many miliary and civilian applicaions including auomaic arge recogniion (A) of non cooperaive arge [1-5], balefields awareness [6], developmen as well as mainenance of low observable aircrafs and arge characerizaion, plane imaging in radio asronomy [7, 1]. Compared wih convenional radars, imaging radars improve deecion and racking performance, and exclusive arge idenificaion. Imaging radars uilize range-doppler principle o obain and desired image of he objec. he range resoluion of he radar image is realized by he bandwidh of he ransmied radar signal, as he cross-range resoluion is achieved from he gradien of he Doppler frequency specrum generaed by he relaive displacemen of he arge wih respec o he radar sysem of observaion. I is a vas reason o deep he knowledge in principles of elecromagneic waveforms generaion, wave propagaion and radar signal formaion. he main objecives of he paper: 1. Mahemaical Models of Elecromagneic Coninuous and Finie Waveforms and Microwaves.. adar Signal Formaion 3. adar Imaging Geomery he paper is organized as follows. In Secion mahemaical models of elecromagneic coninuous and finie waveforms and microwaves. In Secion 3 radar signal formaion is considered. In Secion 4 radar imaging geomery is described. In Secion 4 conclusions are made.

2 . Waveforms and wave processes a. Coninuous waveform and waves Coninuous waveform he waveforms are elecrical oscillaions called signals and generaed by generaors. he waveforms generae elecromagneic waves while propagaing hrough maerial and immaerial environmen. Boh of hem can be monochromaic (unmodulaed) and wide bandwidh (modulaed) waveforms. he waveforms can be divided ino coninuous waveforms and finie waveforms. While propagaing coninuous waveforms cause coninuous waves as finie waveforms cause finie waves. Fig.1. Consider radar generaed waveforms and emied elecromagneic waves o a poin objec placed a disance, refleced elecromagneic waves o he radar (Fig.1). A mahemaical model of a coninuous waveform wih ideniy ampliude is he complex exponenial funcion (1) s( ) exp j cos j sin exp j cos ; Imexp j sin e ; c 8 is he angular frequency; c 3.10 m/s denoes he speed of ligh; denoes he wavelengh. he imaginary par of he waveform is presened in Fig.. Fig. Coninuous wave A mahemaical model of a coninuous wave wih ideniy ampliude is he complex exponenial funcion () s ( ) exp j( cos ( ) j sin ( ) eexp j ( ) cos( ) ; Imexp j( ) sin( ),

3 is he ime delay of he wave fron measured on he range direcion. c c Subsiue in (), hen (3) s( ) exp j( k) cos( k) j sin( k), k is he wave number. In Fig. 3 an imaginary par of he coninuous wave process as a funcion of he ime measured in a paricular range disance is presened. Fig. 3 he disribuion of he wave on he range direcion in paricular momens 1, and is presened in Fig. 4. he displacemen of he wave fron in momens 1, and can be observed. b. Finie waveforms and waves Fig Finie waveforms Finie monochromaic waveform Consider finie monochromaic waveform. he mahemaical model of a single finie monochromaic waveform wih uniy ampliude is he complex exponenial funcion (4) s ( ) rec exp( j)

4 1,0 1, (5) rec 0, 0 0, 1 is he ime duraion of he finie monochromaic waveform. Finie linier frequency modulaed (LFM) waveform Consider finie linier frequency modulaed (LFM) waveform. he mahemaical model of a single finie LFM waveform wih uniy ampliude is he complex exponenial funcion (6) s ( ) rec expj( b ) is he ime duraion of he LFM waveform, b is he LFM rae. (7) Denoe rec 1,0 1, 0, 0 is he recangular funcion. 0, 1 (8) es ( ) rec cosj( b ), and Ims( ) rec sinj( b ) c. Waveform rain. Monochromaic waveform rain he mahemaical model of a monochromaic waveform rain wih uniy ampliude is he complex exponenial funcion (9) s ( ) rec exp( j) mod p is he slow ime, p p is he fas ime, p is he pulse repeiion period, p is he number of he generaed waveform (pulse). LFM waveform rain he mahemaical model of a finie LFM waveform rain wih uniy ampliude is he complex exponenial funcion (10) s ( ) rec exp j( b )

5 mod p is he slow ime, p p is he fas ime, p is he pulse repeiion period, p is he number of he generaed waveform (pulse). Denoe (11) es ( ) rec cosj( b ), and Ims ( ) rec sinj( b ) In Fig. 5 he imaginary par of he finie LFM waveform rain wih uniy ampliude is presened. d. Finie waves Fig. 5 Monochromaic wave A mahemaical model of a single finie monochromaic wave wih uniy ampliude is he complex exponenial funcion (1) s (, ) rec exp j 1,0 1, (13) rec 0, 0 0, 1 Finie LFM wave he mahemaical model of a single finie LFM wave wih uniy ampliude is he complex exponenial funcion (14) rec s (, ) rec exp j b 1,0 1, 0, 0 is he recangular funcion. 0, 1

6 e. Wave rains Finie monochromaic wave rain he mahemaical model of a finie monochromaic wave rain wih uniy ampliude and ime delay is he finie complex exponenial funcion (15) s (, ) rec exp j mod p is he slow ime, p p is he fas ime, p is he pulse repeiion period, and p is he number of he emied wave. Finie LFM wave rain he mahemaical model of a finie LFM wave rain wih uniy ampliude and ime delay is he finie complex exponenial funcion (16) s (, ) rec exp j b mod p is he slow ime, p p is he fas ime, p is he pulse repeiion period, and p is he number of he emied wave. Fig. 6 he imaginary par of he finie LFM wave rain wih uniy ampliude as a funcion of he ime measured in a paricular range disance is presened in Fig Signal formaion models a. Signal formaion model wih finie monochromaic waves he radar emis elecromagneic wave rains o he objec. he objec is presened as an assembly of poin scaerers placed on he reflecing objec s surface. he radar receiver processes signals induced by waves refleced by objec s poin scaerers. he ime delay from a paricular poin scaerer is proporional o, is he radar-objec disance. If he objec is moving rough radar paern he disance is a funcion of slow ime pp. In general case he displacemen of he objec in he fas ime is negligible, hen is a funcion of p, i.e. p ) or. he monochromaic finie signal from a paricular ijkh poin p ( p scaerer wih inensiy a ijk can be wrien as

7 (17) ijk s ( ) a expj ijk p p. ijk rec ijk, p he recangular funcion is defined by he following expression ijk 0, 0 ijk ijk (18) rec 1,0 1, ijk 0, 1 ijk ijk denoes he ime delay of he propagaion of he wave from he radar o he c ijkh poin scaerer of he objec and back o he radar; ijk denoes he disance measured from he radar o he ijkh poin scaerer of he objec. he model of a monochromaic finie signal from he objec is a geomerical sum of signals from all poin scaerers and can be wrien as (19) ijk s ( ) a expj p ijk p. ijk rec ijk, p b. Signal formaion model wih finie LFM waves he model of a LFM c finie signal from a paricular ijkh poin scaerer can be wrien as ijk p ijk (0) ( ) sijk ( ) aijk rec exp j, p pp. b ijk he model of a LFM finie signal from he objec can be expressed as (1) ijk ijk s ( ) aijk rec exp j, p p ijk p. ijk ( p 4. adar Imaging Geomery b he main geomerical characerisic is he disance from he radar o a paricular poin scaerer from he objec space which can be defined be he following geomery. he radar scenario is depiced in 3-D coordinae sysem 0 xyz (Fig. 7). he objec, presened as an assembly of poin scaerers is depiced in coordinae sysem 0 ' XYZ. he posiion of he radar is defined by disance vecor r. he curren posiion of he ijkh poin scaerer is defined by he following vecor equaion )

8 () ijk r0 A ijk p) ( ) is he curren vecor disance from he radar o he origin of he r0 ( r 00 p ' XYZ coordinae sysem 0 of he objec, 00( p ) is he curren vecor disance of from he origin of he coordinae sysem 0 xyz o he origin of he coordinae sysem 0' XYZ of he objec space. Fig Conclusion In he presen work heoreical descripion of waveforms, waves and mahemaical model of radar signal formaion has been considered. Based on he complex exponenial funcion coninuous and finie waveforms, waves and radar signals have been analyically described and graphically illusraed. adar imagining geomery has been presened and analyical described. Acknowledgemen he work is suppored by Projec NAO CLG: ESP.EAP.CLG , and Projec ESA C1P eferences [1] Novak, L. M. A comparison of 1-D and -D algorihms for radar arge classificaion. In Proceedings of he 1991 IEEE Inernaional Conference on Sysems Engineering, Fairborn, OH, Aug. 1-3, 1991, 6-1. [] Menon, M. M., Boudreau, E.., Kolodzy, P. J. An auomaic ship classificaion.sysem for ISA imagery. he MI Lincoln Laboraory Journal, 6, (1993), [3] Boha, E. C. Classificaion of aerospace arges using super resoluion LSA images. In Proceedings of he I994 IEEE Souh Afncan Symposium on Communicaions and Signal Processing, Sellenbosch, Souh Africa, Oc. 4, 1994,

9 [4] Fechner,., Hansche,., and anger,. Classificaion of objecs in ISA imagery using arificial neural neworks. In Proceedings of SPIE: Applicai ions and Science of Arificial Neural Neworks II, Orlando, FL, Apr. 9-1, 1996, [5] Musman, S., Ken; D., and Bachmainn, C. Auomaic recogniion of ISA ship images. IEEE ransacions O Aerospace and Elecronic Sysems, 3,4 (Oc. 1996), [6] Fennel], M.., and Wishner,. P. Balefield awareness via synergisic SA and MI exploiaion. IEEE Aerospace and Elecronic Sysems Magazine, 13,, (Feb. 1998), [7] Bouladakis, G., K. Skrapas, Panaiois Frangos, ime frequency analysis of radar signals, O / SE 080 In. Sym. on arge Idenificaion and ecogniion Using F Sysems, Oslo, Norway (004). [8] Chen, V. C., S. Qian. Join ime-frequency ransform for radar range-doppler Imaging, IEEE ransacions on Aerospace and Elecronic Sysems, AES-34,, 1998, pp [9] Qian, S., V. C. Chen. ISA moion compensaion via adapive join ime-frequency echnique, IEEE ransacions on Aerospace and Elecronic Sysems, AES-34,, 1998, pp [10] Palmer, J., J. Homer, I.D. Longsaff, M. Marorella, B.Lileon. ISA imaging using an emulaed mulisaic radar sysem, IEEE, ransacion n AES, vol. 41, 4, 005, pp [11] S. W. Werness, Carrara, L. Joyce, and D. Franczak. Moving arge imaging algorihm for SA daa, IEEE ransacions on Aerospace and Elecronic Sysems, vol. AES-6, No 1, (1990) [1] Marorella, M., F. Berizzi. ime windowing for highly focused ISA image reconsrucion. IEEE ransacion on AES vol. 41, 3, 005, pp