Direction Finding and Positioning Algorithm with COLD- ULA Based on Quaternion Theory

Transcription

1 Journal of Communications Vol. 9, No., Octobr 4 Dirction Finding and Positioning Algoritm wit COLD- ULA Basd on Quatrnion Tory Lanmi Wang, Ziai Cn, and Guibao Wang Scool of Pysics and Optolctronic Enginring, Xidian Univrsity, xi an 77, Cina National Laboratory of Radar Signal Procssing, Xidian Univrsity, xi an 77, Cina lmwang@mail.xidian.du.cn; cz98968@63.com; gbwangxd@6.com Abstract Elctromagntic vctor snsor arrays av bn widly applid in t communication, radio, navigation, and so on. T application of lctromagntic vctor snsor antnna array will gratly improv t ovrall prformanc of t communication systm. In tis papr, a novl quatrnion- ESPRIT (stimation of signal paramtrs via rotational invarianc tcniqus) algoritm for dirction finding and positioning basd on COLD (cocntrd ortogonal loop and dipol) uniform linar array (COLD-ULA) is proposd. First, quatrnion data modl of COLD-ULA is dducd and constructd. Scond, t array string vctor and t angl of arrival (DOA) ar stimatd using a quatrnion ignvalu dcomposition of t data covarianc matrix. Finally, t stimation of polarization paramtrs ar rquird using t rlationsips btwn t dipols and t loops. T proposd tcniqu not only dcoupls t DOA stimation information from t polarization stimation information, but also improvs t ability of signal dtction. Morovr, t proposd tcniqu as t advantag of small amount of calculation and paramtr automatic matcing. Simulation rsults sow tat t prformanc of quatrnion mtod is obviously bttr tan tat of t long-vctor mtod. Indx Trms Array signal procssing, dirction of arrival, quatrnion-esprit, polarization, antnna array, subspac I. INTRODUCTION Elctromagntic vctor snsor (EMVS) array is a nw typ of array to obtain t lctromagntic signal spatial and polarization domain information, wic as t broad application prospct in t communication tcniqu, signal dtction tcniqu (SDT), mobil communication and so on []-[3]. Dirction of arrival (DOA) stimation basd on t EMVS array as bcom a ot topic in t rsarc fild of signal procssing, and it as mad many valuabl rsarc rsults [], [4]-[6], [8]-[], [5]-[]. T autors in litraturs []-[] av studid t dirction finding paramtr stimation of complt EMVS array, t paramtr stimation of incomplt EMVS as bn discussd in litraturs [3]-[8], t dirction finding and positioning paramtr stimation of Manuscript rcivd May 3, 4; rvisd Octobr 7, 4. Tis work was supportd by t National Scinc Fund undr Grant No. 695, No. 637 and t Fundamntal Rsarc Funds for t Cntral Univrsitis undr Grant No. K Corrsponding autor mail: lmwang@mail.xidian.du.cn. doi:.7/jcm spatially collocatd vctor-snsors composd of ortogonally orintd dipol(s) and /or loop(s) av also bn invstigatd by Wong & Yuan in [9], []. Various algoritms av bn dvlopd to stimat t DOA and polarization paramtrs of multipl lctromagntic signals. T first dirction-finding algoritms, xplicitly xploiting all six lctromagntic componnts, av bn dvlopd by Norai and Paldi [] and Li [3], rspctivly. T cross-product-basd DOA stimation algoritm was first adaptd to ESPRIT (stimation of signal paramtrs via rotational invarianc tcniqus) by Wong and Zoltowski [9]-[]. A uni-vctor-snsor ESPRIT algoritm was put forward in [6]. T maximum likliood approac was prsntd in []; t trilinar dcomposition algoritm was proposd in [4] and [5]; two distinct vrsions of ESPRIT stimators wr rportd in [3] and [8] rspctivly; t multipl signal classification (MUSIC) tcniqu was invstigatd in [9] and []. Howvr, t abov-mntiond stimation mtods ar basd on t long vctor data modl of EMVS. Undr tis condition, complx-valud vctors ar usd to rprsnt t output of ac EMVS in t array, and t collction of an EMVS array is arrangd via concatnation of ts vctors into a long vctor. Consquntly, t corrsponding algoritms somow dstroy t vctor natur of incidnt signals carrying multidimnsional information in spac, tim, and polarization. In rcnt yars a fw rsarc av bn mad on stimating t DOA of EMVS witin t algbraic systm tory for quatrnion and its xtnsion []-[8]. Quatrnion MUSIC tcniqu was proposd basd on t quatrnion formalism of t two componnt vctor snsor array in [5]. T tr componnt vctor snsor was xprssd as a biquatrnion numbr, tn biquatrnion-basd MUSIC was proposd in [6]. T six componnt lctromagntic (EM) vctor snsor array was rprsntd by a quad-quatrnion modl in [7]. T advantags of using quatrnion for EMVS is tat t local vctor natur of a EMVS array is prsrvd in multipl imaginary parts, and it could rsult in a mor compact rprsntation. T us of quatrnion allows us to skip t paramtrization stp usd in long-vctor tcniqus as it intrinsically includs t vctor dimnsion in t procss. Quatrnionic matrix oprations can provid a bttr subspac approximation tan t long-vctor approac. 4 Enginring and Tcnology Publising 778

2 Journal of Communications Vol. 9, No., Octobr 4 Compard wit t COLD (cocntrd ortogonal loop and dipol) pairs orintd along x-axis, y-axis and cocntrd dipol and dipol pairs, cocntrd loop and loop pairs orintd along x- and y-axis rspctivly, w can s tat COLD pairs along t z axis is mor asir to raliz t dcoupling of polarization and angl of arrival paramtrs bcaus of its simpl structur [3]-[8]. A novl quatrnion-esprit algoritm for stimating signal DOA and polarization using t COLD uniform linar array (COLD-ULA) is proposd in tis papr. In t algoritm, t DOA information and polarization information of incidnt signals ar dcoupld, rrors of DOA and polarization rin do not cumulat. Hnc, tis proposd algoritm: ) is computationally lss intnsiv tan many opn-form sarc mtods; ) producs closd-form solution wit no xtra computation ndd for signal paramtr stimats; 3) as t advantag of paramtr automatic matcing and witout spctral pak sarcing. T simulation rsults sow tat t prformanc of quatrnion mtod is bttr tan tat of long-vctor mtod. T papr is organizd as follows. In Sction II, a matmatical tory of quatrnion and its rlvant proprtis ar prsntd. Sction III introducs t quatrnion signal modl for six-componnt and twocomponnt vctor snsor. In Sction IV, t proposd quatrnion-esprit algoritm is introducd. Simulation rsults ar prsntd in Sction V. In Sction VI, conclusions ar givn. II. QUATERNION THEORY Quatrnion, wic xtnds imaginary numbrs into a four-dimnsional spac, bgan to b dvlopd by William Hamilton in 843. A complx numbr as two componnts: t ral and t imaginary part. T quatrnion as four componnts, i.., on ral part and tr imaginary parts, and can b rprsntd in Cartsian form as: q a ib jc k d a, b, c, d R () wr i, j and k ar complx oprators wic oby t following ruls. ij ji k jk kj i ki ik j i j k Quatrnion can also b xprssd as t following form: () q a ib c id j j (3) wic is known as t "Cayly-Dickson rprsntation". T quatrnion conjugat and t quatrnion modulus ar rspctivly givn by q a ib jc kd (4) q a b c d (5) Lt q i b q a ib jc kd, wr a, b, c, d i i i i i i i i i R, i =,, tn t addition of two quatrnion xprssd in trms of tir ral and imaginary parts is givn by i j k q q a a b b c c d d (6) From t ruls in quation (), it is clar tat multiplication is not commutativ, namly q q q q. T innr product of quatrnion can b rprsntd as: p, q H p q (7) If t innr product of quatrnion mts wit pq,, it is calld tat t quatrnion p and q H is ortogonal. For som rason, it is somtims usful to considr t quatrnion as composd of a vctor part and a scalar part, tus q can also b xprssd as wr t scalar part, S q q S q V q (8) S q is t ral part i.., a and t vctor part is a composit of tr imaginary componnts, i.., V q ai bj ck. T product of two quatrnions xprssd in trms of tir scalar and vctor parts is givn by S p V q V q V p qp S q S p V q V p S q V p wr and dnots t vctor dot cross products, rspctivly. It follows from tis tat t dot and cross products of two pur quatrnions m and n ar givn by (9) m n mn nm, m n mn nm () From t point of viw of algbra, quatrnion is an xtnsion of t complx fild. Quatrnion as mor compact data xprssion. A quatrnion contains mor tan two componnts tan tat of complx numbr, so tat it can contain mor information in on opration. III. SIGNAL AND DATA MODELS A six-componnt lctromagntic vctor snsor consists of tr spatially co-locatd idntical but ortogonally orintd, lctrically sort dipols and magntically small loops sparatly masuring all tr lctric-fild componnts and tr magntic-fild componnts of incidnt signals. Suc an lctromagntic vctor snsor can xploit any polarization divrsity among t impinging sourcs, as sown in Fig.. An idal lctromagntic vctor snsor was calld auxiliary lmnt. T auxiliary lmnt is idalizd, by ovrlooking all mutual coupling among its six collocatd constitunt antnnas. Suc an idalizd 4 Enginring and Tcnology Publising 779

3 Journal of Communications Vol. 9, No., Octobr 4 indpndnt of t sourc s dirction of arrival and it rquirs no prior information of azimut and lvation angls. Witout loss of gnrality, w assum tat t intr-lmnt spacing of t ULA is d, wr d <.5 and c f, wit f b t frquncy of t signals. vctor snsor s array manifold (also calld unit-powr lctromagntic fild vctor) would b a concatnation of t 3 lctric-fild vctor wit t 3 magntic-fild vctor, to b [3, 8]: x cos cos sin cos y cos sin sin sin j a,,, z () cos cos cos x sin cos cos sin y z sin d X Fig.. Uniform COLD linar array gomtry. T pas diffrncs btwn t M array lmnts and t origin constitut t spatial string vctor, i..: wr, π is t signal s lvation angl masurd from t positiv z -axis,, π dnots t azimut angl masurd form t positiv x-axis,, π / rprsnts t auxiliary polarization angl, and π,π symbolizs t polarization pas diffrnc. q k πd j sin( k ) π M d j sin( k ) (3) T joint xprssion of kz and kz ar indicatd as wit a quatrnion ck. ck kz ikz sin k sin k j k isin k cos k (4) T output of array rspons for t kt incidnt signal can b xprssd as follows: Fig.. A six-componnt lctromagntic vctor snsor. xk t ck q k sk t In tis papr, w mak t following assumptions. K far-fild narrowband compltly polarizd lctromagntic plan wav sourc signals in t x-z plan imping upon a uniform linar array (ULA), wic is composd of M M K idntical COLD pairs, as sown in Fig.. a k, k, k wr sk t is t kt incidnt signal and a k, k, k is dfind as ck q k. T rcivd data collctd by t COLD-ULA at tim t can b rprsntd as For t COLD pairs, t dipols paralll to t z-axis ar rfrrd to as t z-axis dipols and t loops paralll to t x-y plan as t x-y plan loops, rspctivly masuring t z-axis lctric fild componnts and t zaxis magntic fild componnts. T COLD pairs string vctor of t kt k K unit-powr X t AS t N t wr X t, S t, N t and A a, () wr k, π is t signal s lvation angl masurd from t positiv z-axis, k, π rprsnts t auxiliary polarization angl, and k -π,π symbolizs t polarization pas diffrnc. T z-axis lctric fild fild kz and t z-axis magntic kz bot involv t sam factor sin k, so, ak ar t xi, t xi, t Xi t, xi, M t s t s t S t sk t n i, t n i, t Ni t, ni, M t ait,, T a,, Ai T i, at,, i K K K wit ak a k, k, k, k =, polarization stimation basd on COLD pairs is 4 Enginring and Tcnology Publising (6) rcivd data, t uncorrlatd incidnt signals, t zroman additiv complx Gaussian nois and t string vctor matrix of incidnt signals, rspctivly, i.., lctromagntic sourc signal is t following vctor []: kz sin sin j k k k a k, k, k kz sin k cos k (5) 78,K.

4 Journal of Communications Vol. 9, No., Octobr 4 According to formulas (4), (5) and (6), t matrix A can b rwrittn as: A=A A i Lt ψ =E# Es EHsEs T is composd of t K ignvctors of corrsponding to t K largst ignvalus of matrix ψ. T tan K j K and A can b obtaind: stimations of A, A =E T, A E T, A =E T A s A AΦ arg kk πd () obtaind from A. Tir rlationsip is follow: R x E XXH AR s AH σ I () H is t k tan kk N N-K matrix composd of t rmaining N-K ignvctors of R x. Basd on t subspac tory, tr xists K K nonsingular matrix T, and t signal subspac can b xprssd xplicitly as k arg kk polarization (3),,,,3 impinging upon a COLD-ULA () wit M=5 snsors. T intr-lmnt spacing of ULA is.5. 4 snapsots ar usd in simulation xprimnts. In t first xprimnt, w considr a scnario wit 5 indpndnt Mont Carlo trials running on t corrsponding COLD ULA. W coos t first signal for tis xprimnt and t SNR is st to 5 db. T sts of valus of t DOA and polarization variabls av bn rprsntd in scattr diagrams (Fig. 3 to Fig. 6). From Fig. 4, it is sown tat almost all stimatd valus ar locatd in t vicinity of actual valu (3) From quation (3), it can b obtaind tat: (4) EH. 4 Enginring and Tcnology Publising (3) In tis sction, som simulations ar conductd to valuat t prformancs on DOA and polarization stimation by t proposd mtod. Tr uncorrlatd qual-powrd signals wit paramtrs,,,,6,,, 3,33, and constitut two nw matrics E and E rspctivly. According to t subspac tory, t signal subspac can b xprssd xplicitly as: HA A V. SIMULATION RESULTS T first M- rows and last M- rows of matrix E s (9) According to quation (3), t paramtrs stimation ar prsntd as sourc covarianc matrix. Lt E s b t N K matrix composd of t K ignvctors corrsponding to t K largst ignvalus of R x and lt En dnot t tan j Ω j K tan K complx conjugat transpos, σ indicating t wit nois powr and R s E S t SH t rprsnting t wr E# EH E (8) wr E# E T T Φ A Ω A A HA Ω T corrlation matrix of rcivd data X t is E AT AΦT (7) From t formulas (7) and (8) and using t tory and A can b of quatrnion, stimations of matrix A πd j sin K IV. QUATERNION-ESPRIT ALGORITHM E AT k sin (9) wr Es AT (6) From t quation (), t stimation of dirction of arrival is givn as: rlationsip btwn A and A can xprssd as wit E symbolizing t statistical man (5) Equation (5) implis tat t stimat of Φ is a matrix wos diagonal lmnts ar composd of t K largst ignvalus of matrix ψ and t full-rank matrix T first and last M rows of matrix A constitut two nw matrics A and A rspctivly, t j π d sin Φ EHsEs, tn formula (4) ψt T Φ wr tan Ω - can b rwrittn as (7) T rlationsip btwn A and A can xprssd as : (8) A A Ω j 78

5 Journal of Communications Vol. 9, No., Octobr 4 by using quatrnion mtod, t stimatd valu of On t contrary, from Fig. 5, t stimatd valus ar distributd in t rang of 43.,.75 and rang from t numrical rang (.75,.5 ). On t contrary, from Fig. 3, t stimatd valus using long 58.8,6.5. T stimatd rror using long vctor mtod is muc largr tan tat of quatrnion mtod. vctor mtod ar distributd in t rang of.5,.95. T stimatd rror using long vctor mtod is muc largr tan tat of quatrnion mtod ( ) ( ) ().5 Fig. 6. Polarization scattr diagram using quatrnion mtod. 3 Tims 4 5 In t scond xprimnt, w first compar t prformanc of quatrnion mtod wit long vctor mtod wit rspct to SNR. In t simulations, t signal-to-nois ratio (SNR) is from to db, 4 snapsots ar usd in ac of t 5 indpndnt Mont Carlo simulation xprimnts. T prformanc of standard dviation and probability of succss is illustratd, and t corrsponding rsults ar sown in Fig. 7- Fig. 9. Fig. 3. DOA scattr diagram using long vctor mtod ( ) Standard dviation of ().5 3 Tims 4 5 Fig. 4. DOA scattr diagram using quatrnion mtod Fig. 7. Standard dviation of DOA vrsus SNR Standard dviation of () ( ) () - - Fig. 5. Polarization scattr diagram using long vctor mtod. -3 From Fig. 6, it is sown tat almost valus ar locatd in t vicinity of,,6 by using quatrnion stimatd valu of and rang from rang 43.6,.4 and 59.5,6.5 4 Enginring and Tcnology Publising, all stimatd actual valu mtod, t Fig. 8. Standard dviation of APA vrsus SNR. t numrical T curvs wit star and circular data points in Fig. 7 to Fig. 9 rspctivly plot t standard dviation of, rspctivly. 78

6 Journal of Communications Vol. 9, No., Octobr 4 and, rspctivly stimatd by long vctor and t proposd quatrnion mtod, at various signal-to-nois ratio (SNR) lvls. T proposd quatrnion procdur is bttr tan long vctor. T stimation prcision at db basd on t quatrnion modl as improvd largr tan 4.89 for,. for,.5 for, compard wit tat of t long vctor mtod. T nancd prformanc is rootd in t spcial data modl of quatrnion. Standard dviation of ( ) Fig. 9. Standard dviation of APA vrsus SNR. Tn, w considr t probability of succss of DOA and polarization stimations. T rsult is sown in Fig. and Fig.. Probability of succss of Fig.. Probability of succss of DOA vrsus SNR. Probability of succss of polarization Fig.. Probability of succss of polarization vrsus SNR. T curvs wit star and circular data points in Fig. and Fig. rspctivly plot t probability of succss of DOA and polarization, rspctivly stimatd by long vctor and t proposd quatrnion mtod, at various signal-to-nois ratio (SNR) lvls. T proposd quatrnion procdur is bttr and mor robust tan long vctor procdur. VI. CONCLUSIONS A closd form stimation of DOA and polarization, basd on quatrnion-esprit algoritm is studid. T proposd algoritm can dcoupl DOA stimation from t polarization stimation. Bcaus in tis algoritm quatrnion is usd to rtain t vctor natur of ac vctor snsor, tr xists bttr stimatd accuracy of paramtrs tan tat of t traditional long vctor data modl mtod. ACKNOWLEDGMENTS Tis work was supportd by t National Scinc Fund (No.695) and t Fundamntal Rsarc Funds for t Cntral Univrsitis (K55377). T autors would lik to tank t anonymous rviwrs and t associatd ditor for tir valuabl commnts and suggstions tat improvd t clarity of tis manuscript. CONFLICT OF INTERESTS T autors dclar tat tr is no conflict of intrsts rgarding t publication of tis articl. REFERENCES [] T. Li and A. Norai, Maximum likliood dirction-of-arrival stimation of undrwatr acoustic signals containing sinusoidal and random componnts," IEEE Trans. Signal Procss., vol. 59, no., pp ,. [] A. Norai and E. Paldi, Vctor snsor array procssing for lctromagntic sourc localization, IEEE Trans. Signal Procss., vol. 4, no., pp , Fbruary 994. [3] J. Li, Dirction and polarization stimation using arrays wit small loops and sort dipols, IEEE Trans. Antnnas Propag., vol. 4, no. 3, pp , Marc 993. [4] X. F. Gong, Z. W. Liu, and Y. G. Xu, Rgularisd paralll factor analysis for t stimation of dirction-of-arrival and polarization wit a singl lctromagntic vctor-snsor, IET Signal Procss., vol. 5, no. 4, pp ,. [5] X. F. Zang, R. Z. Cao, and M. Zou, Noncircular-PARAFAC for D-DOA stimation of noncircular signals in arbitrarily spacd acoustic vctor- snsor array subjctd to unknown locations, EURASIP J. Adv. Signal Procss., vol. 3, no. 7, pp. -, 3. [6] J. W. Cao, L. Liu, and Z. Y. Lin, Joint DOA, rang, and polarization stimation in t frsnl rgion, IEEE Trans. Arosp. Elctron. Syst., vol. 47, no. 4, pp , Octobr. [7] K. T. Wong and M. D. Zoltowski, Uni-vctor-snsor ESPRIT for multisourc azimut, lvation, and polarization stimation, IEEE Trans. Antnnas Propag., vol., no., pp , Octobr 997. [8] M. D. Zoltowski and K. T. Wong, Closd-form ignstructurbasd dirction Finding using arbitrary but idntical subarrays on a spars uniform Cartsian array grid, IEEE Trans. Signal Procss., vol. 48, no. 8, pp. 5-, August. [9] K. T. Wong, L. Li, and M. D. Zoltowski, Root-MUSIC-basd dirction-finding and polarization stimation using divrsly 4 Enginring and Tcnology Publising 783

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