A clean signal reconstruction approach for coherently combining multiple radars

Transcription

1 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 hps://doi.og/ /s EURASIP Jounal on Advancs in Signal Pocssing RESEARCH Opn Accss A clan signal consucion appoach fo cohnly combining mulipl adas Xinghua Liu 1*, Zhnhai Xu 1,XiangLiu 2, Siwi Chn 1 and Shunping Xiao 1 Absac Disibud adas hav h ponial o combin cohnly fo achiving a high signal-o-nois aio SNR) whil mainaining a moda annna siz. Th ky o cohnly combining mulipl adas is obaining accua cohn paams CPs), which a usd o adjus h ansmiing/civing im and phas of ach ada. On appoach fo CP simaion is o ansmi ohogonal wavfoms. Howv, idally, ohogonal wavfoms occupying h sam fquncy band may no b found in pacic. Coss-colaion ngy lakag xiss bwn non-ohogonal wavfoms, which siously impais h accua acquisiion of CPs. To solv his poblm, w popos a clan signal consucion appoach fo CP simaion. This appoach consucs clan chos by gadually sipping ou h coss-colaion ngy lakag wih a consucion-liminaion-consucion famwok. And CPs a obaind fom hs consucd clan chos. Sinc h majoiy of coss-cocion ngy lakags a liminad, nhancd CP simaion pfomanc can b achivd. Vifid simulaions a dsignd fo a dual ada scnaio. Rsuls show ha h poposd appoach significanly impovs h pfomanc of CP simaion whil ducing h SNR quimn fo cohnly combining mulipl adas. Kywods: Muli-ada cohn combinaion, Cohn paams, Clan signal consucion, All-pol modl 1 Inoducion A singl ada has a hoical maximum signal-o-nois aio SNR) fo a givn siz ag a a givn ang, which dicly affcs ada s abiliy o dc, ack, and idnify h ag [1]. Th a mainly wo soluions o ais SNR. On is o dvlop highly snsiiv ada wih a lag annna. Howv, such lag annna sysms a cosly and no asily anspoabl. Alnaivly, by joinly pocssing h ansmid and civd signals fom sval co-locad small indpndn adas, i is possibl o combin hm ino a cohnly funcioning sysm fo a significanly nhancd SNR [1 4]. Whn N adas a cohnly combind, an N 3 SNR gain ov a singl ada can b acquid, i.., full cohnc is achivd on boh ansmiing and civing [5]. Cohnly combining mulipl adas is challnging, spcially widband adas. Signal pocssing in ms of iming and phas adjusmns should b caid ou o *Cospondnc: xinghua217@163.com Xinghua Liu and Zhnhai Xu conibud qually o his wok. 1 Sa Ky laboaoy of CEMEE, Naional Univsiy of Dfnc Tchnology, Yanwachi 137, Changsha, Popl s Rpublic of China Full lis of auho infomaion is availabl a h nd of h aicl caliba h incohnc inducd by dispaa popagaion pahs and synchonizaion os. If laiv posiions of h adas a known o a facion of a wavlngh, accua im dlay o phas cocions can b implmnd in accodanc wih his pio knowldg [2, 6]. Howv, masuing accua laiv posiions is difficul and subjc o incasing os as h spaaion of ada annnas incass, if h adas a no laivly fixd [1]. Fuhmo, only knowing h laiv spaaion bwn h adas is no sufficin fo mulipl adas cohn combinaion, sinc i dos no accoun fo h innal lcical diffnc causd by synchonizaion os. Hnc, an addiional calibaion pocss is ndd. Anoh appoach lis upon piodically spaaing h monosaic and bisaic chos a ach ada [3]. Th monosaic cho cosponds o is ansmid ada signal, whil h bisaic cho cosponds o a ansmid ada signal fom h oh ada. By compaing h laiv im and phas laionship bwn hs uns, cohnc paams CPs), i.., fin im and phas cocing valus, can b simad. Ths paams a hn usd o adjus h ansmiing/civing im and Th Auhos) Opn Accss This aicl is disibud und h ms of h Caiv Commons Aibuion 4.0 Innaional Licns hp://caivcommons.og/licnss/by/4.0/), which pmis unsicd us, disibuion, and poducion in any mdium, povidd you giv appopia cdi o h oiginal auhos) and h souc, povid a link o h Caiv Commons licns, and indica if changs w mad.

2 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 Pag 2 of 11 phas of ach ada fo obaining h full cohnc SNR gain. A ky fau of his appoach is ha laiv posiions of ada a no ciical and cohnly combining mulipl adas is achivd via ag-basd calibaion. Inspid by such ida, lvan woks a spinging up, including CPs Cam-Rao bound CRB) and h hoical pfomanc bounds analyz [7 9], schm fo dual ada cohn combinaion basd on sppd-fquncy signal [10 12], and dual ada cohn combinaion laboaoy and fild xpimns [13, 14]. To spaa h monosaic and bisaic uns, wo diffn ways hav bn pod: kping hs chos spaad in h im domain, i.., using im-division muliplxing TDM) chniqu [1] o using ohogonal wavfoms [2, 3]. To no, h scaing spons of h ag is commonly im-vaying and using TDM chniqus will undmin h cohnc among spaad chos. Hnc, h la appoach is a b choic. Whn ohogonal wavfoms a ansmid o sima CPs, h monosaic and bisaic chos can b spaad by machd filing. Concpually, hs ohogonal wavfoms should hav h sam opaing bandwidh and cn fquncy. Ohwis, CP simas may b subjc o addiional os du o h fquncy-slciv ag scaing spons. Asid fom h common spcal quimn, h wavfoms mus b as naly ohogonal as possibl. Howv, i is had o find idally ohogonal wavfoms occupying h sam fquncy band in pacic. Coss-colaion ngy lakag bwn nonidally ons canno b ignod, which will affc h accua acquisiion of CPs. In his pap, o h bs of ou knowldg, h influnc of non-ohogonal wavfoms on CP simaion is analysd in dail fo h fis im. Thoical analyss dmonsa ha h coss-colaion ngy lakag smmd fom wavfom non-ohogonaliy will inoduc addiional simaion os and poblms. Sinc h coss-colaion ngy lakag is h dominan ason fo impaiing CP simas, a CP simaion appoach basd on clan signal consucion is poposd. Th basic ida of his appoach is o consuc h clan signal ha is no o aly infd by coss-colaion ngy lakag. Onc h clan singl is consucd, impovd CP simas can b obaind via hs consucd clan signals. And h ffcivnss of ou poposd appoach is vifid by a dual ada cohn combinaion xampl ansmiing up- and down-chip wavfoms o sima CPs. Th maind of his pap is oganizd as follows. Scion 2 inoducs h pincipl of cohnly combining mulipl adas. Scion 3 psns h CP simaion famwok and discusss h influnc of nonohogonal wavfoms on CP acquisiion. Th clan signal consucion appoach fo simaing CPs and validad simulaions a daild in Scions 4 and 5. Conclusions a givn in Scion 6. 2 Pincipl of cohnly combining mulipl adas As adas a disibud, hoically, ach adaansmid signal canno supimpos cohnly a h ag du o dispaa popagaion pahs fom adas o h ag and synchonizaion os Th diffnc of ag scaing chaacisics among diffn adas canbappoximalynglcd,sincadasaassumd o b naly co-locad compad o h ag dcion ang). Fo h sam ason, incohnc also xiss among h chos civd by adas. And such incohnc among signals can b chaacizd as im and phas misalignmns. Hnc, o cohnly combin mulipl adas, i.., achiv boh cohn-on-ansmi and cohn-on-civ, signal pocssing concning im and phas adjusmns should b caid ou o caliba h incohnc inducd by h abov asons. Pu i mo concly, if h signal ansmid by ada 1 fnc ada) aivs a h ag af h signal ansmid by ada2,popimandphasadjusmnsfoada2a ndd o slow down and ad is ansmiing im and phas appopialy. L s κ k ) j2πf c κ k )+jϕ k dno h ansmid signal of ada k, s) is h common basband signal of all adas, κ k is h im synchonizaion o of ada k compad o h fnc clock, and ϕ k is h iniial phas of ada k. Thn, hs signals popagad o h ag can b xpssd as s κ k τ k ) j2πf c κ k τ k )+jϕ k, k = 1,, N 1) wh, f c is h cai fquncy, τ k = R k /c, R k is h ang fom ada k o h ag fo h muli-sca ag modl cas, R k dnos h ang fom ada k o h ag scaing cn), and c is h spd of ligh. Appanly, fom 1), haivalimandphasofhs signals a misalignd. To nhanc h adio fquncy RF) lcic fild impinging upon h ag, i.., achiv cohn-on-ansmi, fin im and phas adjusmns a ncssay. Wihou loss of gnaliy, if ada 1 is h fnc ada, i is ssnial o adjus h ansmiing im and phas of oh adas. And h adjusd signals can b xpssd as s κ k + τk) j2πf c κ k )+jϕ k j φ k, k = 2,, N,wh τ k = τ k τ 1 ) + κ k κ 1 ) φ k = 2πf c[ τ k τ 1 ) + κ k κ 1 )] +ϕ k ϕ 1 ) a h im and phas cocing valus ansmi CPs) of ada k.tono,asmodnadasaassumdobusd, 2)

3 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 Pag 3 of 11 in which digial wavfom gnaos allow indpndn conolofimandphas. L u) j2πf c dno h flcd signal a h ag, hn his flcd signal civd by ada l af downconvsion can b xpssd as assuming ach ada can aliz cohn cpion) u τ l + κ l ) j2πf cτ l +j2πf c κ l jϕ l, l = 1,, N 3) Fo h sam ason, o achiv cohn-onciv, appopia im and phas shifs a also ndd fo h cho civd by oh adas xcp ada 1. And h adjusing cho can b xpssd as u τ l + κ l + τl ) j2πf c τ l +j2πf c κ l jϕ l j φ l, l = 2,, N,wh τ l = τ l τ 1 ) κ l κ 1 ) φ l = 2πf c [ τ l τ 1 ) κ l κ 1 )] ϕ l ϕ 1 ) dno h im and phas cocing valus civ CPs) of ada l. No ha h supscip is h abbviaion fo ansmi, whil h supscip is h abbviaion fo civ. Appanly, τ1 = τ1 = { φ1 = φ1, h maining CPs o b simad a τ k, τl, φ k, } { φ l, k, l = 2,, N. Fo convninc, τ k, τl } { and φ k, φl } a collcivly calld im and phas CPs. 3 Poblm fomulaion 3.1 Signal modl and CP simaion famwok Obviously, as h xisnc of synchonizaion os, ansmi and civ CPs of sam ada a no idnical. To sima CPs, signals civd by ach ada hav o b spaad ino monosaic and bisaic chos fisly. I is h ason why ohogonal wavfoms a ansmid. Assuming h is a ag wih Q-fixd non-scinillaing scas in h spac, in h sa of ach ada ansmiing ohogonal wavfoms, h signal civd by ada l af down-convsion can b xpssd as appoximaion is don basd on h abov assumpion ha adas a naly co-locad) l ) Q k=1 = ξ N 4) N ξ q s k τ lk τ q ) j2πfcτ lk+ τ q )+jϕ k jϕ l + w l ) s k k=1 τ11 τk τ l ) j φk +j φ l +w l ), l =1,, N wh ξ q dnos h scaing cofficin of sca q; τ lk = τ k + τ l + κ k κ l is h popagaion im fom ada k, o ag scaing cn, o ada l; τ q is h cosponding wo-way popagaion dlay fom sca q o ag scaing cn in h ada lin of sigh LOS); s k ) dnos h ohogonal wavfom ansmid by ada k; w l ) dnos h nois inoducd duing ada l cpion, which is assumd o b a zo-man, complx 5) whi Gaussian andom pocss; s ) = Q ξ q s k τ q ) j2πf c τ q ;andξ = j2πf cτ 11. By machd filing MF), h N civd signals { l ), l = 1,, N} can b spaad ino N 2 chos. And ach spaad cho chaacizs a disinc popagaion pah fom ada k o ada l, which can b fomulad as y lk ) = l ) s k ) = ξa k τ11 τk τ l ) j φk +j φ l N + ξ C k k τ11 τk τ l ) j φ k +j φl k =1 k =k + w MF lk ), l, k =1,, N wh A k ) = Q ξ q A k τ q ) j2πf c τ q ; C k k ) = Q ξ q C k k τ q ) j2πf c τ q ; A k ) is h auocolaion funcion of s k ); C kk ) is h coss-colaion funcion bwn s k ) and s k ), i.., h coss-colaion ngy lakag m; and w MF lk ) = w l) s k ), is h convoluion opaion, and ) is h complx conjuga opaion. Af spaaion, cho pai nds o b cafully slcd o sima CPs. Fo h dual ada cas, h spaad cho s is {y 11 ), y 12 ), y 21 ), y 22 )}. Compaingy 11 ) wih y 12 ), hi civing pahs a idnical, whil ansmiing pahs a no. Hnc, by compaing h im and phas { diffncs bwn his cho pai, ansmi CPs τ 2, φ2 } can b simad. Manwhil, compaing h cho pai {y 21 ), y 22 )}, sam ansmi CPs can b also simad. Th ulima ansmi CP simas will b h fusion of hs wo simas. And h cosponding CP simaion famwok is shown in Fig. 1, which can b asily xndd o civ CP simaion and N-ada cas. On his basis, a CP simao calld coss-colaion CC) pocssing is poposd [3, 10]. In his simao, h slcd cho pai is coss-colad. Th pak of h coss-colaion oupu dfins h cosponding im CPs, and h phas a h pak of h coss-colaion oupu dfind h cosponding phas CPs. This pocssing can b bifly fomulad as follows k = N N 1 k,l, ˆφ k = N N 1 l=1 l=1 N l = N 1 l,k, N ˆφ l = N 1 k=1 k=1 { )} ag x k,l k,l 6) { )} 7) ag x l,k l,k wh k,l and l,k dno h paks of cho pai cosscolaion oupus x k,l τ) = y lk )y l1 τ)d and x l,k τ) = y lk )y 1k τ)d, ishimspanofh ada civ ang ga s Fig. 1), and ag{ } dnos h phas of a complx agumn.

4 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 Pag 4 of 11 Fig. 1 Famwok of ansmi CP simaion fo dual ada cas 3.2 Influnc of ansmiing non-ohogonal wavfoms on CP simaion To no, CC is poposd on h assumpion of ansmiing idally ohogonal wavfoms. In pacic, idally ohogonal wavfoms occupying sam fquncy band a no xisn. In his cas, CP simaion pfomanc of CC may b impaid. And daild analyss a illusad in h following Rciv CP simaion Assuming SNR nds o infiniy, fom 6), x l,k τ) can b divd as ξ 2 is omid) x l,k τ) = j φ l f k τ l )[ f k τ) ] d 8) wh f k ) = A k τ11 τk) j φk + N k =1 k =k C k k τ11 τk ) j φ k. Obviously, h pak of x l,k τ) is τ l, i.., l,k = ) τ l. By subsiuing l,k ino 8), w hav x lk k,l = Mk j φ l,whmk R dnos h ffciv simaion ngy. Thn, accoding o 7), w hav l = τ l, ˆφ l = φ l 9) Tansmi CP simaion Und h sam assumpion, x k,l τ) can b divd as ξ 2 is omid) x k,l τ) = j φ k A k τ11 τk)[ A 1 τ 11 τ) ] + g k, τ)d 10) wh g k, τ) is h m lad o coss-colaion ngy lakag fo mo dails, s in Appndix A). Divid 10) ino wo pas: h m wih a pak a τk and h disubanc x k,l τ) j φ k A k T RP + j φ T RP τ11 τk )[ A k τ 11 τ) ] d k A k τ11 τk )[ A 1 τ 11 τ) ] d+ g k,τ)d } {{ } disubanc 11) wh T RP dnos h im span of h ag ang pofil RP) wihou blank magin s Fig. 1). And in his im span, A k ) A 1 ). Gnally, h ampliud of A k ) in h mainlob gion is fa ga han ha of A k ) in h sidlob gion and C k k ). Evn if h disubanc xiss, h pak of x k,l τ) will li in h nighbohood of poin τ k,hais k,l = τ k + ε k, ε k [ 0, εk max ] 12) To no, εk max 0, whos valu is dmind by h slcion of ohogonal wavfoms A k ) and C k k ))andh gomic aangmn of adas τk, k = 2,, N). By subsiuing 12) ino11), w hav x k,l ) k,l = j φ k A k τ11 τ )[ k A k τ11 τk ε )] k d T RP +Jk ε k) = j φ k Ek + F k ε k) + Jk ε k) wh E k < T RP A k 13) τ11 τk )[ A k τ11 τk )] d R dnos h ffciv simaion ngy, Fk ε k) C dnos h disubanc causd by pak posiion dviaion, and Jk ε k) = τ11 τk) T RP A k [ A 1 τ11 τk ε ) k)] d + g k, τ k + ε k d C dnos h disubanc inoducd by coss-colaion ngy lakag and h mismach bwn A 1 ) and A k ). Combining 7), 12), and 13), w hav

5 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 Pag 5 of 11 k = τ k + ε k, ˆφ k = φ k + ag[ F k ε k) ] + ag [ J k ε k) ] 14) Thoughou h abov analyss, ansmiing nonohogonal wavfoms siously affc h simaion pfomanc of CC. And such influncs can b oulind as follows: 1) Whn SNR, using CC, h simad civ CPs a unbiasd s 9)), whil h simad ansmicps a no s 14)). And hi simad biass a dmind by h slcion of ohogonal wavfoms and h gomic aangmn of adas; 2) Sinc > T RP, i is asy o vify ha M k > E k, which implis ha simaing ansmi CPs is mo nois-snsiiv han simaing civ CPs. 4 Mhods As analyzd in Scion 3, coss-colaion ngy lakags bwn non-ohogonal wavfoms siously impai h accua acquisiion of CPs. If such ngy lakags can b liminad bfo CP simaion, i.., consuc clan signals and us hm o sima CPs, h impacs inoducd by non-ohogonal wavfoms will disappa. Inspid by his, a nw appoach fo simaing CPs is poposd in his scion. In h following, w fis inoduc h clan signal consucion famwok. Thn, an addiional pocss calld p-mach pocssing is laboad as an nhancmn of ou poposd appoach. Finally, w show h ovall CP simaion pocss. 4.1 Clan signal consucion Fom 6), sinc coss-colaion ngy lakags in h spaad cho oigina fom h psnc of chos of oh wavfoms in h mixd cpion signal, by liminaing hs chos fom h iniial cpion signal, a clan spaad cho can b consucd. Spcifically, ak h clan consucion of y ln ) as an xampl, coss-colaion ngy lakags inoducd fom h chos of {s 1 ),, s N 1 )} in y ln ) can b suppssd by consucing and liminaing hs chos fom h iniially mixd cpion signal l ). Und his considaion, a clan signal consucion appoach is dsignd wih a consucion-liminaion-consucion R-E-R) famwok, which can b illusad as follows: 1. Rconsucion h fom on).mfis implmnd on l ) in h fquncy domain o spaa h cho of s 1 ). Th spaad cho, i.., h spcum of y l1 ), will b Y l1 f ) = R l f ) S1 f ) Q τ = ξ ξ q j2πf τq j2πfc τq S 1 f ) 2 j2πf 11 + τ ) 1 + τ l j φ 1 +j φ l [ ] N +S1 f ) V lk k f ) + W l f ) =2 15) wh f [ B/2, B/2], B is h bandwidh of h Q ohogonal wavfoms, V lk f ) = ξ ξ q j2πf τ q ) j2πf c τ q S k f ) j2πf τ 11 + τ k + τl j φ k +j φl is h coss-colaion ) ngy lakag m, R l f ), S k f,andwl f ) a h spcum of l ), s k ),and w l ), spcivly. A high fquncis, accoding o h gomical diffacion hoy GDT), h ada backsca fom a ag can b accualy psnd by an all-pol modl [15 17]. L X l1 f ) = Y l1 f )/ S 1 f ) 2 mov h ffc of wavfom spcal nvlop on solving all-pol modl). Af discizaion, X l1 f ) can b fomulad as X l1 f m ) = Q d q l1 q ) m p l1 }{{} all-pol modl + S 1 m) S 1 m) 2 [ N ] V lk m) + W l m) k =2 wh d q l1 = ξ ξ q jπb τ 11 + τ1 + τ l + τ q) j2πf c τ q j φ 1 +j φ l and p q l1 = j2π f τ 11 + τ1 + τ l + τ q) a h pols and hi ampliuds, f m = B/2 + m f, m = 1,, M 1, f = B/M. By solving h paamic modl in 16), Y l1 f ) can b consucd as Ŷ c l1 f m) = ˆQ l1 ˆd l1ˆp q q ) m ) S1 l1 fm 2 16) 17) wh ˆd q l1, ˆpq l1,and ˆQ q l1 a h solvd paams mo dails in Appndix B); 2. Eliminaion. Alhough h solvd paams in 17) a conaminad by V k m), k = 2,, N, Ŷ c l1 f m) sill svs h majoiy ngy of Y l1 f m ). Gnally spaking, Ŷ c l1 f m) can b considd as an appoximaion of Y l1 f m ) in h las-squas sns. To no, h supscip c in Ŷ c l1 f m) psns h wod conaminad. And h cho of s 1 ) in l ) can b consucd as Ŷ c l1 f m)s 1 f m )/ S 1 f m ) 2. Thn, by liminaing i fom h iniially civd mixd signal, is inoducd coss-colaion ngy lakag will b suppssd. This pocss is fomulad as [ ] Y l1 f m ) Ŷl1 c f S1 f m ) m) S 1 f m ) 2 18) wh h poduc m S 1 f m )/ S 1 f m ) 2 dnos h invs MF pocssing conay o muliplying S 1 f m).

6 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 Pag 6 of 11 By muliplying 18) wih S 2 f m), h cho of s 2 ) can b spaad ) [ ] Ỹ l2 fm = Y l1 f m ) Ŷl1 c f S1 f m ) ) m) S 1 f m ) 2 S 2 fm 19) To disinc wih h spaad cho Y l2 f m ) via dic MF, his on is ildd. Wih h sam pocss, h cho of s 2 ) in l ) can b also consucd and liminad. Doing his iaiv consucion and liminaion pocdus unil only h cho of s N ) mains in l ), h liminaion wok is don; 3. Rconsucion h la on). By muliplying h final liminad signal wih S N f m), w can obain Ỹ ln f m ). Applying h sam consucion wok o Ỹ ln f m ),whav Q ) LN X ln fm = d q q ) m S ln p ln + N m) S N m) 2 δ ln m) 20) wh X ln f m ) = Ỹ ln f m )/ S N f m ) 2 and δ ln m) conains h nois and liminaing siduals. Sinc h cho of {s 1 ),, s N 1 )} has bn liminad fom h mixd cpion signal l ), coss-colaion ngy lakag ms in 20) disappa. And h clan signal Ŷ 1N f m ) can b consucd by solving all-pol in 20). Essnially, ach im fo solving all-pol modl can b sn as a filing pocss, i svs h signal pa whil liminaing h disubanc coss-colaion ngy lakag, nois and liminaion siduals). Thfo, wih his addiional consucion sp, ah han dic MF, his disubanc can b fuh suppssd. Thn, using h invs fas Foui ansfom IFFT), is cosponding im domain signal ŷ 1N m ) can b also covd. Algoihm 1: R-E-R famwok fo clan signal consucion Inpu : civd mixd cho l m ), l = 1,, N, cicl shif vco α fom 1 N Oupu: consucd clan signal ŷ lk m ), l, k = 1,, N 1 l 1; 2 α [..., N 1, N,1,2,..., k 1, k] T ; 3 pa 4 R l f m ) FFT l m ); 5 Ỹ lα1) f m ) = R l f m )Sα1) f m); 6 Ŷlα1) c f m) ˆQ lα1) ˆd q lα1) ˆp lα1)) q m Sα1) f m ) 2 ; 7 fo i 2 o N do 8 Ỹ lαi) f m ) [ Ỹ lαi 1) f m ) Ŷlαi 1) c f m)] S αi) f m) S αi 1) f m ) S αi 1) f m ) 2 ; 9 Ŷlαi) c m) ˆQ lαi) ˆd q lαi) pq lαi) )m S αi) f m ) 2 ; 10 nd 11 Ŷ lk f m ) ŶlαN) c m); 12 ŷ lk m ) IFFT Ŷ lk f m ); 13 l l + 1; 14 unil l N; 15 un ŷ lk m ) As mniond abov, w only psn h clan consucion pocdus lad o ŷ 1N m ). Gnal consucion pocdus a oulind as Algoihm 1. Noic ha h clan signal canno b consucd in on sp and iaiv consucion and liminaion pocdus a ncssay. Such sp-by-sp consucion and liminaion is h majo fau of ou poposd clan signal consucion appoach. Paiculaly, fo h dual ada cas, h visualizd R-E-R famwok is shown in Fig. 2. Af consucing clan chos {ŷlk m ), l, k = 1,, N }, CPs will b simad accoding o 7), i.., CC mhod. Fo disincion, w f o Fig. 2 R-E-R famwok fo dual ada cas

7 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 Pag 7 of 11 a c b d is basd on h assumpion mniond ali; ha is, vy ada displays sam ag scaing chaacisics. In oh wods, h laiv phas bwn pols of ach consucd clan cho is appoximaly fixd, in spi of dispaa absolu pol phas, which can b acd in 16). Assuming pol A and pol B a h aw pol ss smmd fom wo diffn consucd clan chos, h machd pol pais can b slcd by: Fig. 3 Schmaic diagam of p-mach pocssing. a Phas disibuion of aw pol ss bfo p-mach pocssing. Pols makd wih a coss dno h mismachd pols. b Quanifid phas squnc of boh aw pol ss. c Bs mach posiion. d Slcd pol pais wih h minimum pol phas alignd ou poposd CP simaion appoach as consucd coss-colaion RE-CC) pocssing. 4.2 P-mach pocssing In pacic, h numb of pols Q is usually unknown. Th Akaik Infomaion Ciion ALC) [18] and minimum dscipion lngh MDL) [19] aavailablo sima Q. Bu Q simad in 20) cosponding o diffn consucd clan chos is no always idnical s Fig. 3a). This is spcially u fo al masumns chaacizd by a colod nois and infnc. As pols conain h whol infomaion of signal, mismachd pols may caus xa dviaion fom h u CPs. Thfo, w popos a mhod o slc h machd pol pais fom aw pol ss as h nhancmn of RE- CC, calld p-mach pocssing. Th slcion pincipl 1. Quanify. Dividing h pol phas span of pols A and B ino sval bins, h lngh of bin γ should saisfy ε<γ < min o nsu ha a mos on pol xising in a bin. ε is h maximum simad dviaion of h pol phas and min is h minimum pol phas span of h aw pol ss s Fig. 3a). And h bin ampliud quals h cosponding pol s nomalizd ampliud. If no pol falls ino a bin, h ampliud is zo s Fig. 3b). 2. Slc. Find h maximum coss-colaion oupu posiion of h wo quanifid squncs as h bs mach posiion s Fig. 3c). Thn, w ain h posiion coincidn pols in boh squncs as slcd pol pais s Fig. 3d). To conclud, combind h R-E-R clan signal consucion famwok wih h p-maching pocssing, h ovall flow of h RE-CC is shown in Fig Simulaion suls and discussion To valida h ffcivnss of ou poposd appoach, w dsign sval simulaions in dual ada cohnly combining scnaio. Th dual ada, ada 1 and ada 2, a locad a d/2,0) and d/2,0) spcivly, wh d = 7 m. W choos up- and down-chip wavfoms as h ansmid ohogonal wavfom s o spaa h Fig. 4 Pocss flow diagam of RE-CC

8 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 Pag 8 of 11 Tabl 1 Paams usd in simulaions Rada Rada 1 Rada 2 Signal modulaion Up-chip Down-chip Chip duaion Signal bandwidh Cai fquncy Sampling fquncy Tim synchonizaion o Iniial phas Rang ga 1 μs 300 MHz 10 GHz X-band) 1 GHz 0 s 0 ad [ R 0 100, R ] m monosaic and bisaic chos and sima CPs. Rada 1 acs as h fnc ada and ansmis an up-chip wavfom,whilada2ansmisadown-chipwavfom. Assum ha h is a ag wih fiv scas in h ang dicion, whos scaing cn is locad a R 0 cos θ, R 0 sin θ), θ = 60, R 0 = 200 km, and R 0 d. Oh paams usd in h simulaion a lisd in Tabl CP simaion pfomanc Accoding o h discussion in Scion 2,inhisdualada cohnly combining scnaio, h CPs o b simad a { τ2, τ 2, φ 2, 2} φ. In simulaions, h ffc of nois on CP simaion is fuh considd. And h nois is addd as complx whi Gaussian andom pocss, whos innsiy is s basd on h SNR of a singl ada SNR in MF ). Manwhil, w assum boh adas hav h sam SNR in MF. Bsids, fo valuaing h CP simaion pfomanc, h oo man squa o RMSE) is inoducd, which is dfind as h RMSE of τ2 ) RMSE τ = 1 M c [ τ 2 M 2 i) 2 i)] 2 21) c i=1 wh M c is h numb of Mon Calo MC) uns, τ2 i) and 2 i) a h u and simad CPs of un i. In ou simulaion, 200 MC uns a xcud p ach SNR in MF sampld valu fo ach un o capu h avag simaing pfomanc. Fo compaison, w compa h poposd RE-CC wih h CC and h TDM-CC. TDM-CC is h cas spaaing monosaic and bisaic chos using TDM chniqu and simaing CPs via CC. To no, whn TDM chniqu is applid, h monosaic and bisaic chos a saggd in im. And no coss-colaion ngy lakags xis in spaad chos, which is quivaln o using idally ohogonal wavfoms. In his cas, ansmi CP simaion and civ CP simaion a idnical. Figu 5 psnshrmsesfoallhhmhods. Noicd ha h civ CPs simad via CC a asympoically unbiasd and hi RMSEs appoach h CRB abiaily clos in high SNR in MF, whil h ansmi CPs simad via CC a no, hi RMSEs dvia away fom h CRB. Also, a h sam SNR in MF,hRMSEs of simad ansmi CPs a always ga han hos of simad civ CPs whn CC is usd. Th ason lis in h xisnc of coss-colaion ngy lakags inoducd by non-idally ohogonal up- and down-chip wavfoms. I is hs lakags ha caus h diffnc in ansmi and civ CP simaion. Simulaion suls coincid wih h analyss in Scion 3.2. Bsids, such simaion poblms a basically solvd by RE-CC, whos pfomanc is compaabl o ha of TDM-CC, which is h obainabl opimal pfomanc using CC und h cun condiions. Simulaion suls vify h ffcivnss of RE-CC. To fuh illusa h ason bhind RE-CC fo h nhancmn of CP simaion pfomanc, Fig. 6 compas h spaad chos fom ada 1 cpion bfo and af clan consucion, whn SNR in MF = db. As up- and down-chip wavfoms a no idally a b Fig. 5 RMSEs of simad CPs vsus SNR in MF. a Tim CPs. b Phas CPs

9 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 Pag 9 of 11 a b Fig. 6 Spaad chos compaison bfo and af clan consucion. a Th cho of up-chip wavfom. b Th cho of down-chip wavfom ohogonal, coss-colaion ngy lakags xis bfo consucion, which a psnd as flucuan sidlobs in Fig. 6 s h black lin). Onc a clan signal is consucd, hos sidlobs disappa s h d lin). And RP appoximaly mains h sam, which implis ha no infomaion loss occud. This is h ason why impovd CP simas can b obaind using RE-CC. 5.2 Dual ada cohn combinaion pfomanc compaison Nx, o g a cla undsanding of cohnly combining bnfis, w valua h pfomanc of h dual ada sysm af cohn combinaion, whos incohnc is calibad via h CP simas obaind by RE-CC andcc,spcivly.onccpsasimad,oaliz consuciv infnc impinging upon h ag, boh adas should ansmi idnical wavfom. In ou simulaion, h up-chip wavfom is ansmid by boh adas a his sag. Oh paams main unchangd. Figu 7 compas h fnc and combind signal, whn SNR in MF = db. Th fnc signal is h cho ansmid and civd by only ada 1, whil h combind signal is h synhic cho af dual ada cohn combining. In Fig. 7a, h dual ada sysm is combind by RE-CC. Compaing h combind and fnc signal, h is a makd dop in nois lvl fo h combind signal. Tha is, h dual ada sysm obains h SNR gain of cohnon-ansmi and cohn-on-civ.tovaluahissnr gain, 200 MC ials a caid ou. And h calculad SNR gain is 8.37 db, appoaching h idal lvl 9 db 10log 2 3 ). By conas, in Fig. 7b, h dual ada sysm is combind via CC. Compad o h fnc signal, a sligh nois lvl dop sill xiss in h combind signal. Bu RPs of h wo signals no long ovlap. In his cas, h SNR gain is also calculad, which is 4.52 db. Appanly, und his siuaion, using RE-CC can obain high SNR gain han CC. Fuhmo, fom hs simulaion suls, i can b infd ha using h appoach ansmiing ohogonal wavfoms and simaing CPs o cohnly combin mulipl adas is condiional. Onc h SNR of singl ada is oo low, h muual incohnc among adas canno b calibad via h inaccua simad CPs. In his cas, cohnly combining adas is in vain. 6 Conclusions In his pap, w poposd a novl appoach RE-CC) fo cohnly combining mulipl adas, basd on a clan signal consucion famwok R-E-R). This mhod mainly solvs h impaid simaion pfomanc of CPs suling fom ansmiing non-ohogonal wavfoms. Thoical analyss and simulaion suls indica ha h influnc of ansmiing non-ohogonal wavfoms a b Fig. 7 Compaison bwn h fnc and combind signal. a Dual ada cohn combinaion alizd by RE-CC. b Dual ada cohn combinaion alizd by CC

10 Liu al. EURASIP Jounal on Advancs in Signal Pocssing 2018) 2018:47 Pag 10 of 11 on CP acquisiion can b appoximaly ignod af mploying RE-CC. In addiion, fuh compaisons a caid ou on dual ada cohn combinaion pfomanc. Rsuls show ha, und h sam condiions, using RE-CC can obain high cohn combinaion SNR gain han using CC. Fuhmo, i is woh o no ha h applicaion of h R-E-R famwok is no limid o consuc h clan signal fom h signal mixd wih up- and downchip wavfoms and, fo a signal mixd wih gnal ohogonal wavfoms,.g., h ohogonal phas-coding wavfoms, i is also ffciv. Manwhil, in conay o h pfc ohogonal wavfom dsign, R-E-R famwok offs an alnaiv amp fo coss-colaion ngy lakag suppssion, which is xpcd o b xndd ino much wid ada applicaions. Bsids, whn h ag is moving, CPs simad via h pvious puls will no long adap h puls nx. Thfo, how o pdic h CPs o adap h succssiv puls will b an insing subjc fo fuu sach. Appndix A: Expssion of g k, τ)in 10) In his appndix, w giv h daild xpssion of g k, τ). Accoding o 6)and τ1 = φ 1 = 0, w hav y l1 τ) = ξa 1 τ11 τl τ ) j φ l +ξ N τ11 τk τl τ ) j φ k +j φl C k k 1 =2 22) Sinc y lk )y ) l1 τ)d = y lk + τ l y l1 T G + τ l τ ) d holds, w can xpss x k,l τ) in h following simplifid fom ξ 2 is omid) x k,l τ) = y lk )y l1 τ)d j φ = k A k τ 11 τ k )[ A 1 τ 11 τ) ] + gk, τ)d wh [ g k, τ) = A k τ11 τk ) N j φk C k k 1 τ11 τk τ ) ] j φ k =2 + N C k k τ11 τk ) j φ [ k A 1 τ 11 τ) ] k =1 k =k + N C k k k =1 k =k τ11 τ k ) j φ k [ N k =2 23) C k 1 τ11 τk τ ) ] j φ k 24) Appndix B: Pocdus fo solving all-pol modl Th all-pol modl in 16) can b solvd wih h following fou-sp pocss: 1 Us h sampling daa X l1 m) o consuc h Hankl maix X l1 0) X l1 1) X l1 M L) X l1 1) X l1 2) X l1 M L + 1) H l1 = X l1 L 1) X l1 L) X l1 M 1) wh L is h lngh of h colaion window lngh. Gnally, L = M/3 ; 2 Applying h singula-valu dcomposiion SVD) o xpss H l1 = U l1 S l1 V H l1, and h modl od Q l1 can b dmind by h dcomposd singula-valu maic S l1 ; 3 Accoding o h simad ˆQ l1, U l1,andv l1, signal o nois subspac can b buil. On h basis of hs subspacs, h simaion of signal paams via oaional invaianc chniqus ESPRIT) [20]and h oo mulipl signal classificaion oo-music) [21] a availabl o sima pols p q l1, q = 1,, ˆQ l1 ; 4 Onc h pols a known, h ampliud ms d q l1, q = 1,, ˆQ l1 a simad by fiing h all-pol modl o h daa X l1 m) using a lina las-squaslls) o a non-lina las-squas NLLS) algoihm. Abbviaions ALC: Akaik Infomaion Ciion; CC: Coss-colaion; CPs: Cohn paams; CRB: Cam-Rao bound; ESPRIT: Esimaion of signal paams via oaional invaianc chniqus; GDT: Gomical diffacion hoy; IFFT: Invs fas Foui ansfom; LLS: Lina las-squas; LOS: Lin of sigh; MDL: Minimum dscipion lngh; MF: Machd filing; NLLS: Non-lina las-squas; RE-CC: Rconsucd coss-colaion; R-E-R: Rconsucion-liminaion-consucion; RF: Radio fquncy; RP: Rang pofil; RMSE: Roo man squa o; oo-music: Roo mulipl signal classificaion; SNR: Signal-o-nois aio; SVD: Singula-valu dcomposiion; TDM: Tim-division muliplxing Acknowldgmns Th auhos would lik o hank h anonymous viws fo hi valuabl commns and suggsions. Funding This wok was suppod in pa by h Naional Naual Scinc Foundaion of China und Gan Availabiliy of daa and maials Plas conac auho fo daa quss. Auhos conibuions All h auhos hav paicipad in wiing h manuscip and hav visd h final vsion. All auhos ad and appovd h final manuscip. Comping inss Th auhos dcla ha hy hav no comping inss. Auho dails 1 Sa Ky laboaoy of CEMEE, Naional Univsiy of Dfnc Tchnology, Yanwachi 137, Changsha, Popl s Rpublic of China. 2 Elconic Engining Insiu, Huangshan Road 460, Hfi, Popl s Rpublic of China.

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