Adaptive Design of OFDM Radar Signal With Improved Wideband Ambiguity Function

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1 928 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 58, NO. 2, FEBRUARY 200 Adaptiv Dsign of OFDM Radar Signal With Improvd Widband Ambiguity Function Satyabrata Sn and Ary Nhorai, Fllow, IEEE Abstract W propos an adaptiv tchniqu to dsign th spctrum of an orthogonal frquncy division multiplxing (OFDM) wavform to improv th radar s widband ambiguity function (WAF). h adaptiv OFDM signal yilds a bttr auto-corrlation function (ACF) that rsults into an improvd dlay (rang) rsolution for th radar systm. First, w dvlop a mutlicarrir OFDM signal modl and th corrsponding WAF at th output of th matchd filtr, mphasizing that th rcivd signal dpnds on th scattring paramtrs of th targt. hn, w dvis an optimization procdur to slct th OFDM wavform such that th volum of th corrsponding WAF bst approximats th volum of a dsird ambiguity function. W dmonstrat th improvmnt in th rsulting ambiguity function, along with th associatd ACF, through numrical xampls. W find that th optimization algorithm puts mor signal nrgy at subcarrirs in which th targt rspons is wakr. Indx rms Adaptiv wavform dsign, multifrquncy scattring, OFDM radar, widband ambiguity function. I. INRODUCION In this corrspondnc, w considr a mutlifrquncy radar that mploys an orthogonal frquncy division multiplxing (OFDM) signal [], and w comput its widband ambiguity function (WAF) [2], [3] including th ffcts of th targt rspons on th rcivd signal. h motivation for mploying multipl frquncis is that th diffrnt scattring cntrs of a targt rsonat diffrntly at ach frquncy, and this also allows us to dmonstrat th ffcts of targt rspons on th WAF. Morovr, th us of a mutlicarrir OFDM signal improvs th dlay-rsolution by a factor qual to th numbr of subcarrirs [4], [5, Ch. ]. In addition, w propos an algorithm to dsign th spctrum of th transmitting OFDM signal adaptivly in ordr to improv its ambiguity profil. A. Background h advantag of multicarrir radar signaling has bn wll stablishd in various applications, such as rmot snsing of clouds and prcipitation [6], dtction of landmins [7], intrprtation of an urban scn [8], tc. On of th ways to accomplish simultanous us of svral subcarrirs is th OFDM signaling schm, which mploys multipl orthogonal signals in th tim domain [9]. Although OFDM has bn laboratly studid and commrcializd in th digital communication fild [0], it has not so widly bn studid by th radar community apart from a fw rcnt fforts [] [4]. h ambiguity function for radar was originally introducd by Vill [5]; howvr, it is gnrally rfrrd to as Woodward s ambiguity Manuscript rcivd May 05, 2009; accptd August 23, First publishd Sptmbr 8, 2009; currnt vrsion publishd January 3, 200. h associat ditor coordinating th rviw of this manuscript and approving it for publication was Dr. Srgiy A. Vorobyov. his work was supportd by th Dpartmnt of Dfns undr th Air Forc Offic of Scintific Rsarch MURI Grant FA , and ONR Grant N h authors ar with th Dpartmnt of Elctrical and Systms Enginring, Washington Univrsity in St. Louis, St. Louis, MO 6330 USA (-mail: ssn3@s.wustl.du; nhorai@s.wustl.du). Color vrsions of on or mor of th figurs in this corrspondnc ar availabl onlin at Digital Objct Idntifir 0.09/SP function bcaus of his popular work [6], [7]. According to Woodward, an ambiguity function is dfind as a two-dimnsional corrlation btwn th transmittd narrowband signal and its tim-dlayd (rlatd to targt rang) and frquncy-shiftd (rlatd to targt vlocity) rcivd vrsion. Svral litratur intrprt th ambiguity function as a matchd filtr rspons [8, Ch. 4], [9, Ch. ], [20, Ch. 5], whras a fw othrs as a two-dimnsional point-sprad function [2], [22]. But in gnral, ths formulations of th ambiguity function ithr do not includ a scattring cofficint of th targt in th rcivd signal modl, or thy assum idntical valus for th scattring cofficints corrsponding to diffrnt dirctions and/or frquncis. In our work, w follow a similar formulation of th ambiguity function but only aftr including th ffct of diffrnt targt rsponss at diffrnt frquncis. Additionally, Woodward s vrsion of th ambiguity function dos not hold for larg-bandwidth signals, such as OFDM signals. argt movmnts rsult in ithr xpansion or comprssion in tim for th widband transmittd signal, and this ffct can no longr b approximatd by a simpl shift in frquncy. hrfor, in this work w follow th widband ambiguity function (WAF) introducd by Klly- Wishnr [2] and Spisr [3]. Diffrnt proprtis of th WAF, similar to thos of its narrowband countrpart, can b found in [23] [26]. B. Outlin In Sction II w dscrib th paramtric modls of th transmittd and rcivd signals; thn w comput thir WAF. W mphasiz that th rcivd signal and hnc th corrsponding WAF at th output of th matchd filtr dpnd on th scattring paramtrs of th targt. In Sction III w propos an optimization algorithm to comput an adaptiv OFDM wavform such that th volum of th corrsponding WAF bst approximats that of th dsird ambiguity function. Our numrical xampls, prsntd in Sction IV, dmonstrat th advantag of such an adaptiv wavform dsign. Sction V contains th conclusions and highlights of possibl futur work. II. SIGNAL MODEL AND WIDEBAND AMBIGUIY FUNCION In this sction, w first introduc th transmittd and rcivd signal modls of an OFDM signaling systm. Along with th dlay and Dopplr ffcts, th rcivd signal modl also incorporats th scattring cofficints of th targt at multipl frquncis. hn, w comput th xprssions of WAF for a singl puls and a cohrnt puls train. A. Signal Modl W considr a monostatic radar mploying an OFDM signaling systm [9] with L activ subcarrirs, a bandwidth of B Hz, and puls duration of sconds. Lt a [a 0 ;a ;...;a ] contain th complx wights transmittd ovr diffrnt subcarrirs, satisfying ja 2 l j. hn th complx nvlop of a singl puls can b rprsntd as s(t) a l l (t) whr l (t) j2lft () and f BL dnots th subcarrir spacing. Lt f c b th carrir frquncy of opration, th transmittd signal is givn by s(t) 2R a l j2f t (2) whr f l f c + lf rprsnts th lth subcarrir frquncy X/$ IEEE Authorizd licnsd us limitd to: WASHINGON UNIVERSIY LIBRARIES. Downloadd on January 2, 200 at :34 from IEEE Xplor. Rstrictions apply.

2 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 58, NO. 2, FEBRUARY Considring a far-fild point targt at distanc r and rlativ vlocity ~v with rspct to th radar, th rcivd signal in a nois-fr scnario can b writtn as y(t) p s((t 0 )) (3) whr + accounts for th strtching or comprssing in tim of th rflctd signal, (2c)hu; ~vi rprsnts th Dopplr sprading factor, u is th unit dirction-of-arrival (DOA) vctor, 2rc is th roundtrip dlay btwn th radar and th targt, and c is th spd of propagation. Hr h; i dnots th innr-product oprator ovr th ral vctor spac. Substituting (2) into (3), w gt y(t) 2 p R 2 p R 2 p R j2f (t0) a l a l j2lf(t0) 0j2f j2f t a l l ((t 0 )) 0j2f j2t j2f t whr f c rprsnts th Dopplr frquncy. Hnc, aftr dmodulation th complx nvlop of th rcivd signal is y(t) p a l l ((t 0 )) 0j2f j2t : (4) Howvr, not that nithr (3) nor (4) includs any paramtr rlatd to th targt scattring cofficint. hrfor, dnoting x [x0; x;...;x ] as a complx vctor containing th scattring cofficints of th targt at diffrnt subcarrirs, w modify th xprssions of th rcivd signal as y2(t) 2 p R and th complx nvlop as y2(t) p B. WAF of a Singl Puls x l a l j2f (t0) (5) x l a l l ((t 0 )) 0j2f j2t : (6) According to Klly Wishnr [2], th widband ambiguity function is dfind as (; ) p s anl (t)s 3 anl ((t 0 )) dt (7) 0 whr sanl(t) rprsnts th analytic signal corrsponding to s(t) and 3 is th complx conjugat oprator. On th assumption that th complx nvlop s(t) is prfctly bandlimitd in comparison with th carrir frquncy f c, which is tru for most radar signals [27], w can considr s anl (t) s(t) j2f t. hrfor, (7) can b simplifid to (; ) j2f p s(t)s 3 ((t 0 )) 0j2t dt : (8) 0 Somtims this xprssion is rfrrd to as th output of an optimum dtctor filtr matchd to zro dlay and zro Dopplr [22], with only th trm within th squar brackt in (8) labld as th WAF [28]. Altrnativly, in som othr litraturs th WAF is dfind as th magnitud or magnitud-squard of th trm within th squar brackt in (8) [29]. Now, incorporating th ffcts of th targt rspons in th rcivd signal, w rdfin th WAF at th output of th matchd filtr as follows: MF(; ; a; x) j2f p min a l l (t) x 3 l a 3 l 3 l ((t 0 )) 0j2t dt ; a l a 3 l x 3 l (; ) (9) whr min max(0;), max min(; + ), and j2f (; ) p l (t) 3 l ((t 0 )) 0j2t dt dnots th cross-ambiguity function btwn l (t) and l (t), having a similar form as (8). Not that in (9) w xplicitly paramtriz MF() in trms of x to mphasiz that it dpnds on th scattring cofficints of th targt. In addition, MF() also dpnds on a. his paramtrization is ralistic and also nabls us to adaptivly dsign th spctrum of th transmittd OFDM signal to improv th corrsponding ambiguity profil, which will b discussd in Sction III. W can furthr simplify (9) to th following (s Appndix A): whr MF(; ; a; x) (ml) MF (; ; a; x) +(sl) MF (; ; a; x) (0) (ml) MF (; ; a; x) p di x 3 l ja l j 2 j2[f 0f sinc [f l di ] ] (sl) MF (; ; a; x) p di l 6l x 3 l a l a 3 l sinc [ff l +(l2 0 l)f gdi ] j2[f 0ff +(l 0l )fg ] and di max 0 min, avg (max + min) 2. If w plot sparatly w can s that (ml) MF () producs th mainlob of th WAF whil (sl) MF () producs th sidlobs. Spcial Cas: For a convntional radar mploying a singl carrir frquncy f c, (0) can b simplifid to MF(; ; a; x) p di x 3 jaj 2 j2f (0 sinc [f c di ] ) and thrfor, th xprssion of th normalizd WAF, MF(; ; a; x) MF(0; 0;a; x) di (0) sinc (di ) 0j2 ; () di dos not dpnd on th scattring cofficints of th targt. Hr (0) di is di valuatd at 0. Authorizd licnsd us limitd to: WASHINGON UNIVERSIY LIBRARIES. Downloadd on January 2, 200 at :34 from IEEE Xplor. Rstrictions apply.

3 930 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 58, NO. 2, FEBRUARY 200 C. WAF of a Puls rain h Dopplr-rsolution of th ambiguity function is improvd whn w transmit a cohrnt puls train. h complx nvlop of a train of N idntical pulss can b dscribd as N0 s N (t) s (t 0 n PRI) (2) n0 whr PRI is th puls rptition intrval (PRI). hn, th transmittd signal is givn by s N (t) 2R 2R s N (t) j2f t N0 n0 s (t 0 n PRI) j2f t : (3) Using th xprssion of th analytic signal corrsponding to s N (t) in (7), w can formulat th WAF of a cohrnt puls train as follows: N (; ) p N0 0 n 0 N0 N0 N0 s (t 0 n PRI ) j2f t n 0 s 3 j2f (t0) ((t 0 ) 0 n 2 PRI ) dt + n2 0 n PRI ; 0j2(n 0n ) (4) whr () is th WAF of a singl puls as dfind in (8). S Appndix B for th drivation of (4). Following a similar typ of drivation, including th ffcts of th targt rspons in th rcivd signal, w can modify th xprssion of WAF for a cohrnt puls train as follows: whr MF (; ; a; x) N0 N0 MF + n 2 0 n PRI;;a; x 0j2f (n 0n ) ; N0 N0 a l a 3 l x 3 l + n2 0 n PRI ; 0j2f (n 0n ) ; a l a 3 l x 3 l (; ) (5) (; ) N0 N0 + n 2 0 n PRI; 0j2f (n 0n ) (6) dnots th cross-ambiguity function btwn two cohrnt puls trains of l (t) and l (t). In th rst of th corrspondnc w will considr th magnitud squard of (5), j MF ()j 2, as th xprssion of WAF. III. ADAPIVE WAVEFORM DESIGN In this sction, w dscrib an optimization approach to adaptivly dsign th spctrum of an OFDM signal such that th volum of th corrsponding WAF bst approximats th volum of a dsird ambiguity profil. h problm of synthsizing a wavform to satisfy a dsird ambiguity function has bn addrssd xtnsivly ovr th yars [4], [30] [37]. Wilcox [30] and Sussman [3] approachd this problm to approximat th dsird ambiguity profil in th last-squard (LS) sns. hir optimization procdur strtchs ovr th ntir (; ) plan, and hnc th rsultant wavform can produc an all-purpos ambiguity function that would b mor or lss suitabl for any radar applications [3]. Howvr, in many situations, it is not ncssary to hav a crtain ambiguity shap for all valus of and. Rcntly, Gladkova t al. [36], [37] xtndd Wilcox s LS approach, rstricting th optimization procdur ovr som limitd subrgions in th (; ) plan, particularly surrounding th mainlob. In this work, w sk to find an OFDM wavform satisfying ja lj 2, such that th rror btwn th volums of th rsulting WAF and dsird ambiguity function is th minimum. Instad of covring th ntir (; ) plan, w also limit th volum computations ovr a subrgion, R, containing th origin. Dnoting th dsird ambiguity function as opt(; ), w can stat th optimization problm as follows: a opt (x) argmin a subjct to j opt (; )j 2 0j MF (; ; a; x)j 2 d d; R ja l j 2 and ja l j 2 > 8 l (7) whr is a small positiv quantity (clos to zro) nsuring transmission ovr all L frquncy channls. W us numrical mthods to solv for th optimizd variabls, a opt, that dpnd on th scattring cofficints of th targt and hnc comply with th philosophy of adaptiv dsign. In som particular cass, whn j opt()j 2 < j MF ()j 2 holds tru ovr th ntir subrgion R (.g., opt (; )(; )), w can furthr simplify (7) to th following: a opt(x) argmin a subjct to a opt (x) argmin a j MF R (; ; a; x)j 2 d d; ja l j 2 and ja l j 2 > 8 l; a l a 3 l a 3 l a l x 3 l x l (; ) 3 (; )d d: (8) R Hnc, this lads to a minimization procdur having fourth ordr form, similar to that prsntd in [37], with th incorporation of th targt scattring cofficints. Howvr, from mathmatical prspctiv our mthod can b catgorizd as an L minimization, whil that of [37] as an L 2 minimization. h convrgnc tim of (7) strongly dpnds on th ara of subrgion R and th shap of th dsird ambiguity function opt (; ). If th knowldg of th targt rspons is known a priori, w can us an offlin computation of (7) xtnding ovr a largr ara of R and considring an impuls-shapd opt (). In ral-tim procssing, whn w nd to comput a opt basd on th stimatd valu of x from th Authorizd licnsd us limitd to: WASHINGON UNIVERSIY LIBRARIES. Downloadd on January 2, 200 at :34 from IEEE Xplor. Rstrictions apply.

4 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 58, NO. 2, FEBRUARY Fig.. Plots of widband ambiguity functions for (a) fixd and (b) adaptiv wavforms ovr a rgion R fjj ;jj(2 )g. prvious radar dwll, it would b practical to rstrict R to a smallr rgion (.g., jj ;jj (2 PRI )) and not to choos an idalistic shap of opt (). IV. NUMERICAL RESULS W prsnt simulation rsults to dmonstrat th advantag of adaptiv wavform dsign, in th form of an improvd ambiguity profil. W considr an OFDM radar oprating with th following spcifications: carrir frquncy f c GHz; availabl bandwidth B 25 MHz; numbr of subcarrirs L 4; subcarrir spacing f B(L + ) 25MHz; pulswidth f 40ns; puls rptition intrval PRI 20s; numbr of cohrnt pulss N 0. o valuat a opt w usd R : fjj ;jj (2 PRI)g and an impuls-lik opt (;) that has valu at th origin but zro vrywhr ls. W ralizd th ntris of x from a N (0; ) distribution. h rsults prsntd in Figs. and 2 wr obtaind using th numrical optimization solvr of MALAB and aftr avraging ovr 50 such indpndnt ralizations of x. W compard this adaptivly dsignd WAF with that obtaind from a fixd wavform that mploys a l p L 8l. Fig. 2. (a) Zro-Dopplr cuts (auto-corrlation functions) and (b) zro-dlay cuts of th widband ambiguity functions corrsponding to th adaptiv and fixd wavforms. Fig. (b) dpicts th WAF obtaind from th optimizd wavform following (7). his ambiguity function shows a considrabl improvmnt in comparison with th on shown in Fig. (a), which was obtaind from a fixd wavform. Numrically, th normalizd volum undr th ambiguity profil rducs from unity to h zro-dlay and zro-dopplr cut plots of ths ambiguity profils ar shown in Fig. 2. From Fig. 2(a) it is quit vidnt that th adaptiv wavform rsults in a much bttr auto-corrlation function. h first sidlob lvl of th ambiguity function corrsponding to th adaptiv wavform is 5.5 db down with rspct to thos of its countrpart for fixd wavform. h zro-dlay cut plots in Fig. 2(b) suggst that thr is no chang in th Dopplr rsolution, as w xpct, du to adaptiv wavform dsign. For furthr insight into th optimization procdur, w lookd into th nrgy distributions of th adaptiv wavform and targt rspons ovr diffrnt subcarrirs. For xampl, in a particular sampl run w had jx l j f6:42; :03; 4:23; 5:66g, and th optimization algorithm (7) rsultd in ja l j f0:24; 0:80; 0:42; 0:29g whn initializd with a l p L 0:50 8 l. From this w can mak two obsrvations: i) th sampl varianc of ja l x l j rducs from.06 to 0.3 and ii) th Authorizd licnsd us limitd to: WASHINGON UNIVERSIY LIBRARIES. Downloadd on January 2, 200 at :34 from IEEE Xplor. Rstrictions apply.

5 932 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 58, NO. 2, FEBRUARY 200 rdistribution nrgy occurs with mor signal nrgy to that particular subcarrir in which th targt rspons is wakr and lss signal nrgy to th subcarrir ovr which th targt rspons is alrady strongr. V. CONCLUSION W proposd an optimization algorithm to comput an adaptiv radar wavform such that th volum of th corrsponding widband ambiguity function bst approximats th volum of a dsird ambiguity function ovr a rgion in th dlay-dopplr plan. For this purpos, w considrd an OFDM radar and dvlopd th rcivd signal modl whil incorporating th scattring cofficints of th targt at multipl frquncis. W mphasiz that th xprssion of th widband ambiguity function at th output of a matchd filtr must includ th targt rspons along with dlay and Dopplr. W numrically dmonstratd th advantag of adaptiv wavform dsign. W concludd that th optimization algorithm puts mor signal nrgy at that particular subcarrir in which th targt rspons is wakr, thus incrasing th possibility of a bttr targt rturn. In futur work, w will includ svral othr charactristics of th widband ambiguity function, such as mainlob width and sidlob lvl, in th wavform dsigning critrion. APPENDIX A h trm within th squar brackt in (9) can b writtn as p a l j2l ft x 3 l a 3 0j2l f(t0) l 0 0j2t dt p x 3 l ja l j 2 j2lf 0j2(lf+f )t dt + p x 3 l a l a 3 j2l f l l 6l 0j2[(l f+f )+(l 0l )f]t dt: (A) Using th dfinitions of f l, min, max, di, avg, and th following intgration rsult: 0j2ft dt di sinc (f di ) 0j2f ; (A2) w gt th xprssions of (ml) MF (;;a; x) and (sl) (;;a; x). MF APPENDIX B h drivation of (4) involvs th following stps: N (;) p 0 N0 n 0 N0 n 0 s (t 0 n PRI) j2f t s 3 ((t 0 ) 0 n 2 PRI) j2f (t0) p N0 N0 j2f s (t 0 n PRI ) 0 s 3 ((t 0 ) 0 n 2 PRI) j2t dt: dt Using a chang of variabl with p t 0 n PRI w gt N (;) p N0 N0 j2f j2n s (p) 0 s 3 ((p + n PRI 0 ) 0 n 2 PRI) j2p dp N0 N0 j2n j2f (n 0n ) p j2f (+(n 0n ) ) s (p) 0 s 3 n 2 p 0 + PRI j2p dp : + n (B) h trm in th squar brackt has a similar form as (8) whn is rplacd with ( +(n 2 0 n )PRI), and n PRI + fc(n 2 0 n ) PRI f c (n 2 0 n ) PRI. Hnc, (B) rducs to (4). REFERENCES [] A. Pandharipand, Principls of OFDM, IEEE Potntials, vol. 2, no. 2, pp. 6 9, Apr [2] E. J. Klly and R. P. Wishnr, Matchd-filtr thory for high-vlocity acclrating targts, IEEE rans. Mil. Elctron., vol. 9, pp , Jan [3] J. Spisr, Wid-band ambiguity functions (Corrsp.), IEEE rans. Inf. hory, vol. 3, no., pp , Jan [4] N. Lvanon, Multifrquncy complmntary phas-codd radar signal, Proc. Inst. Elctr. Eng. Radar, Sonar, Navig., vol. 47, no. 6, pp , Dc [5] N. Lvanon and E. Mozson, Radar Signals. Hobokn, NJ: Wily- IEEE Prss, [6] N. Majurc, S. M. Sklsky, S. J. Frasir, and S. A. Rutldg, h advancd multi-frquncy radar (AMFR) for rmot snsing of clouds and prcipitation, prsntd at th 32nd Conf. Radar Mtorology, Albuqurqu, NM, 2004, Papr PR.6. [7] P. V. Gndrn, P. Hakkaart, J. V. Hijnoort, and G. P. Hrmans, A multi frquncy radar for dtcting landmins: Dsign aspcts and lctrical prformanc, in Proc. 3st Eur. Microwav Conf., 200, vol. 2, pp [8] M. L. Bryan, Intrprtation of an urban scn using multi-channl radar imagry, Rmot Sns. Environ., vol. 4, no., pp , 975. [9] S. B. Winstin and P. M. Ebrt, Data transmission by frquncy-division multiplxing using th discrt Fourir transform, IEEE rans. Commun. chnol., vol. 9, pp , Oct. 97. [0] B. L. Floch, R. Halbrt-Lassall, and D. Castlain, Digital sound broadcasting to mobil rcivrs, IEEE rans. Consum. Elctron., vol. 35, no. 3, pp , Aug [] G. E. A. Frankn, H. Nikookar, and P. V. Gndrn, Dopplr tolranc of OFDM-codd radar signals, in Proc. 3rd Eur. Radar Conf., Manchstr, U.K., Sp. 3 5, 2006, pp. 08. [2] D. S. Garmatyuk, Simulatd imaging prformanc of UWB SAR basd on OFDM, in Proc. IEEE 2006 Int. Conf. Ultra-Widband, Sp. 2006, pp [3] J. P. Stralka, Applications of orthogonal frquncy-division multiplxing (OFDM) to radar, Ph.D. dissrtation, h Johns Hopkins Univrsity, Baltimor, MD, [4] M. Sbt, A. Shikhi, and M. Naybi, Orthogonal frquncy-division multiplxing radar signal dsign with optimisd ambiguity function and low pak-to-avrag powr ratio, IE Radar, Sonar, Navig., vol. 3, no. 2, pp , Apr [5] J. Vill, héori t application d la notion d signal analytiqu, Cabls t ransmission, vol. 2, no., pp. 6 74, 948. [6] P. M. Woodward, Probability and Information hory, With Applications to Radar. Nw York: McGraw-Hill, 953. [7] P. M. Woodward, Radar ambiguity analysis, Royal Radar Establishmnt, Malvrn, U.K., ch. Rp. RRE ch. Not 73, 967. Authorizd licnsd us limitd to: WASHINGON UNIVERSIY LIBRARIES. Downloadd on January 2, 200 at :34 from IEEE Xplor. Rstrictions apply.

6 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 58, NO. 2, FEBRUARY [8] A. W. Rihaczk, Principls of High-Rsolution Radar. Nw York: McGraw-Hill, 969. [9] M. I. Skolnik, Introduction to Radar Systms, 2nd d. Nw York: Mc- Graw-Hill, 980. [20] N. Lvanon, Radar Principls. Nw York: Wily-Intrscinc, 988. [2] R. E. Blahut, W. Millr, Jr, and C. H. Wilcox, Radar and Sonar: Part I. Nw York: Springr-Vrlag, 99. [22] J.-C. Guy and M. R. Bll, Divrsity wavform sts for dlay-dopplr imaging, IEEE rans. Inf. hory, vol. 44, no. 4, pp , Jul [23] R. A. Alts, Som invarianc proprtis of th widband ambiguity function, J. Acoustic. Soc. Amr., vol. 53, no. 4, pp , Apr [24] L. H. Sibul and E. L. itlbaum, Volum proprtis for th widband ambiguity function, IEEE rans. Arosp. Elctron. Syst., vol. AES-7, no., pp , Jan. 98. [25] Z. b. Lin, Widband ambiguity function of broadband signals, J. Acoustic. Soc. Amr., vol. 83, no. 6, pp , Jun [26] D. C. Lush and D. A. Hudson, Ambiguity function analysis of widband radars, in Proc. 99 IEEE Nat. Radar Conf., Los Angls, CA, Mar. 2 3, 99, pp [27] P. Z. Pbls, Jr, Radar Principls. Nw York: Wily-Intrscinc, 998. [28] G. S. Antonio, D. R. Fuhrmann, and F. C. Roby, MIMO radar ambiguity functions, IEEE J. Sl. opics Signal Procss., vol., no., pp , Jun [29] A. I. Sinsky and C. P. Wang, Standardization of th dfinition of th radar ambiguity function, IEEE rans. Arosp. Elctron. Syst., vol. AES-0, no. 4, pp , Jul [30] C. H. Wilcox, h synthsis problm for radar ambiguity functions, Univ. of Wisconsin, Madison, WI, MRC ch. Summary Rp. 56, 960. [3] S. Sussman, Last-squar synthsis of radar ambiguity functions, IRE rans. Inf. hory, vol. 8, no. 3, pp , Apr [32] D. DLong and E. Hofstttr, On th dsign of optimum radar wavforms for cluttr rjction, IEEE rans. Inf. hory, vol. 3, no. 3, pp , Jul [33] C. Stutt and L. Spafford, A bst mismatchd filtr rspons for radar cluttr discrimination, IEEE rans. Inf. hory, vol. 4, no. 2, pp , Mar [34] R. d Buda, Signals that can b calculatd from thir ambiguity function, IEEE rans. Inf. hory, vol. 6, no. 2, pp , Mar [35] O. Arikan and D. Munson, im-frquncy wavform synthsis using a last-squars approach, in Proc. IEEE Int. Symp. Circuits Syst.,Nw Orlans, LA, May 3, 990, vol., pp [36] I. Gladkova and D. Chbanov, On a nw xtnsion of Wilcox s mthod, in Proc. 5th WSEAS Int. Conf. Appl. Math., Miami, FL, 2004, pp. 6. [37] I. Gladkova and D. Chbanov, On th synthsis problm for a wavform having a narly idal ambiguity surfac, prsntd at th 2004 Int. Radar Conf., oulous, Franc, Oct. 8 22, rack-bfor-dtct Stratgis for SAP Radars Danilo Orlando, Luca Vnturino, Marco Lops, and Giuspp Ricci Abstract In this corrspondnc w propos track-bfor-dtct (BD) stratgis for spac-tim adaptiv procssing (SAP) radars. As a prliminary stp w introduc th targt and nois modls in discrt-tim form. hn, rsorting to gnralizd liklihood ratio tst (GLR)-basd and ad hoc procdurs w driv dtctors for two diffrnt scnarios (a point bttr clarifid in th body of th corrspondnc). h prliminary prformanc assssmnt, conductd rsorting to Mont Carlo simulation, shows that th proposd procdurs might b viabl mans to implmnt arly dtction and track initiation of wak moving targts. Indx rms Constant fals alarm rat (CFAR), gnralizd liklihood ratio tst (GLR), spac-tim adaptiv radar dtction, track-bfor-dtct (BD), Vitrbi algorithm. I. INRODUCION raditional tracking algorithms ar dsignd assuming that th snsor provids a st of point masurmnts at ach scan. In a radar systm such masurmnts ar obtaind by thrsholding th output of a matchd filtr fd by a basband vrsion of collctd data []. An altrnativ approach, rfrrd to as track-bfor-dtct (BD), consists of fding th procssor with unthrsholdd data. BD-basd procdurs jointly procss svral conscutiv scans (or frams) and, rlying on a targt kinmatics or, simply, xploiting th physically admissibl targt transitions, jointly dclar th prsnc of a targt and, vntually, its track. A BD algorithm can improv track accuracy and follow low signal-to-nois ratio (SNR) targts at th pric of an incras of th computational complxity. Morovr, a BD schm nsuring th constant fals alarm rat (CFAR) proprty with rspct to th unknown statistics of th disturbanc controls th ovrall fals track accptanc probability (CFAR proprty at th track lvl). Most of BD algorithms hav bn proposd to dtct and track small moving objcts in optical imags corruptd by high cluttrd nois. hir us in connction with radar systms has rcivd lss attntion: for a dscription of th xisting rsults, s [2] [5]. A family of low-complxity powr-fficint BD procdurs has bn prsntd in [4]. hrin, th continuous-tim continuous-amplitud signal collctd by a puls Dopplr radar is discrtizd to rflct th sctorization of th covrag ara and th rang gating opration, and th gnralizd liklihood ratio tst (GLR) is solvd rsorting to a Vitrbi-lik tracking algorithm. h proposd algorithm has a complxity linar in th numbr of intgratd scans and in th tim on targt. h mphasis is on dtction prformanc mor than tracking: in fact, th GLR dos not rly on th targt kinmatics; it simply taks into account a maximum targt vlocity in ordr to dfin th admissibl targt transitions in rang and azimuth (th Dopplr is dalt with as a nuisanc quantity du to th considrd systm and targt paramtrs). Howvr, a rough stimat of th targt paramtrs is obtaind as a by-product of th construction of th targt statistic. Manuscript rcivd Novmbr 0, 2008; accptd August 2, First publishd Sptmbr 25, 2009; currnt vrsion publishd January 3, 200. h associat ditor coordinating th rviw of this manuscript and approving it for publication was Prof. Jan-Yvs ournrt. D. Orlando, L. Vnturino, and M. Lops ar with th DAEIMI, Univrsità dgli Studi di Cassino, I Cassino (Fr), Italy (-mail: danilo. orlando@unisalnto.it; l.vnturino@unicas.it; lops@unicas.it). G. Ricci is with th Dipartimnto di Inggnria dll Innovazion, Univrsità dl Salnto, 7300 Lcc (L), Italy (-mail: giuspp.ricci@unisalnto.it). Digital Objct Idntifir 0.09/SP X/$ IEEE Authorizd licnsd us limitd to: WASHINGON UNIVERSIY LIBRARIES. Downloadd on January 2, 200 at :34 from IEEE Xplor. Rstrictions apply.

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