Parameter Dynamic Estimation in Near-Field Bistatic MIMO Radar System

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1 Jouna of Communicaions Vo. No. Novembe 5 Paamee Dynamic Esimaion in Nea-Fied Bisaic MIMO Rada Sysem i i and ianshuang Qiu Infomaion Engineeing Coege Daian Univesiy Daian China Daian Univesiy of echnoogy Daian 3 China Emai: ffsimpe@3.com; qiush@du.edu.cn Absac his pape poposes a nove agoihm of paamee dynamic esimaion in nea-fied Bisaic Muipe-Inpu Muipe-Oupu (MIMO) Rada. When a age ies in he neafied egion he nea-fied wavefon is spheica and he pane waves assumpion is no vaid. he exising mehods and signa mode based on he fa-fied assumpion ae no appicabe o his siuaion. In his pape we popose a new nea-fied signa mode in nea-fied bisaic MIMO Rada. Fisy Doppe fequency is esimaed dynamicay by seaching he peak of he faciona ambiguiy funcion. he mehod of pojecion appoximaion subspace acking based on he faciona ambiguiy funcion is poposed o esimae he azimuh ange and eevaion ange. Fuhemoe he Camé-Rao bound fo age paamee esimaion is deived. Paamee esimaion pefomances ae evauaed and sudied heoeicay and via simuaions. Index ems Nea-Fied bisaic MIMO ada Doppe fequency faciona ambiguiy funcion dynamic esimaion I. INRODUCION Muipe-Inpu Muipe-Oupu (MIMO) ada as a new ype of ada has become a ho eseach aea owing o is poenia advanages. A MIMO ada sysem consiss of boh ansmi and eceive sensos having he abiiy o ansmi ohogona wavefoms. When muipe eceived signas ae ohogona hus sepaabe a he eceive MIMO pocessing may be pefomed o achieve spaia and signa wavefom divesiy. A pesen age ocaizaion and paamee esimaion has eceived much eseach aenion in bisaic MIMO ada sysem []-[]. In he as sevea decades age ocaizaion is egaded as a key opic in aay signa pocessing and has eceived consideabe aenion. Numeous agoihms [5] [] have been poposed o dea wih he pobem fo paamee esimaion of fa-fied unde he pane waves hypohesis. oweve when a age ies in he nea-fied egion he nea-fied wavefon is spheica and he pane waves assumpion is no vaid. he above menioned mehods based on fa-fied assumpion ae no appicabe o his siuaion. hus he pefomance of he conveniona mehod based on pane waves assumpion may degeneae. Receny numeous mehods [7]-[] Manuscip eceived Decembe 7 ; evised Novembe 5. his wok was suppoed by he Naiona Science Foundaion of China unde Gans and 7. Coesponding auho emai: ffsimpe@3.com doi:.7/jcm...- have been deveoped fo nea-fied age ocaizaion such as he maximum ikeihood mehod he wodimensiona (-D) MUSIC mehod he pah foowing mehod and he highe-ode ESPRI mehod [9]-[]. As menioned above hese mehods have obained good pefomance in ems of paamee esimaion when Doppe fequency is assumed as ime-invaian and Diecion of Aiva (DOA) ae esimaed non-dynamic. In fac due o he hee dimensiona moion chaaceisics of he ages he eceived signas may conain no ony a quadaic em bu aso a cubic em in hei phase funcions and azimuh ange and eevaion ange is dynamic vaiaion. heefoe i is no appopiae ha Doppe fequency is assumed as imeinvaian and he ange is assumed as non- dynamicay vaiaion [] [3]. In his case he exising signa modes and mehods can no effecivey sove his pobem caused by he 3D moion of he age and povide an opima souion. o ovecome he dawbacks of he adiiona modes and mehods his pape mainy sudies he paamee esimaion of ime-vaying Doppe fequency and dynamic vaiaion of he azimuh ange and eevaion ange in nea-fied bisaic MIMO ada sysem. his pape poposes a new signa mode since he exising ones ae no appopiae o appoximae he paamees in his case. A pesen we sedom find he sudy of join dynamic esimaion fo Doppe azimuh ange and eevaion ange in nea-fied bisaic MIMO ada which shoud be sudied deepy fo age acking and age ocaizaion. So in eseach backgound of he pobem of inefeence ocaizaion we sudy paamee esimaion dynamic in nea-fied bisaic MIMO ada. his pape is oganized as foows. In Secion II a new signa mode is poposed. he anaysis on he Faciona Ambiguiy Funcion (FAF) is pesened in Secion III. he paamee esimaion and aipane acking agoihm ae descibed in Secion IV. In Secion V he pefomance of he poposed mehod is sudied hough exensive numeica simuaions. Finay concusions ae dawn in Secion VI. II. E PROPOSED SIGNA MODE In his pape he age acking and ocaizaion is eaized by empoying he pincipe of paamee esimaion in nea-fied bisaic MIMO ada sysem. Fig. 5 Jouna of Communicaions

2 Jouna of Communicaions Vo. No. Novembe 5 whee is Konecke poduc and iusaes he nea-fied bisaic MIMO ada sysem. he consideed bisaic MIMO ada is composed of Q ansmi anennas and N eceive anennas wih an ineeemen spacing of. D is he base ine disance beween he ansmi efeence eemen and he eceive efeence eemen. and ae azimuh ange A A A A a... aq aq aq... aq and eevaion ange coesponding o he ansmi aay. and ae azimuh ange and eevaion ange A a... an coesponding o he eceive aay. he ansmiing anennas emi ohogona wavefoms xq. he echo of an an... an Q sn xq exp j 3 q R Rw Rw w w R w w O... Y Receive Aay is eaed as andom inefeence []. Fo a given deay R has he chaaceisics of x inea fequency moduaion signa which conains noise. Accoding o he definiion of Faciona Fouie ansfom FRF [] [5] faciona ambiguiy funcion (FAF) of he signa namey he FRF of (5) Fig.. Nea-fied Bisaic MIMO ada sysem. Since he ansmied waves ae ohogona wih each ohe and ohonoma hee exis wo condiions such as fo k... Q and FAFs m can be wien as q... Q. A each eceiving anenna hese ohogona wavefoms can be exaced by Q mached fies. he oupu of he mh mached fie a he nh eceive anenna can be expessed as yqn exp j 3 aq an wn. FAF m R K m d 5 Jouna of Communicaions () whee is he oaion ange and m is he fequency in FRF domain K ( m) is he kene funcion of he FRF. K ( m) can be expessed as () ( j co ) exp j co -m csc n K m exp j m co ( m) n (n ) ( m) he veco of a oupu of M mached fie can be expessed as Y AD wn (5) whee denoes he deay and (xp yp ) xq Rw (xp y p zp ) xq xk q k and b exp j 3a3 a a a3 3 mean Gaussian whie noise wih vaiance w. x' () signa is defined by D an exp j n sin cos is he seeing veco of he n h eemen of he eceive. he noise wn ( ) is assumed o be independen zeo- D w whee b is he signa ampiude and ai i 3 ae signa phase facos he ampiude and phase facos ae ea and unknown. Insananeous auocoeaion funcion R of he q h eemen of he ansmie seeing veco O'... Y' ansmi Aay ( ) b exp j a a a a3 3 numbe of age and denoe he Iniia Doppe Fequency (IDF) Doppe Fequency Rae (DFR) and Doppe Fequency Rae of Change (DFRC). aq exp j q sin cos is he z Assume ha he cubic phase signa is modeed as whee denoes he ada coss-secion is he R III. FRACIONA AMBIGUIY FUNCION () aq an wn z' D exp j 3 he n h eceiving anenna is (3) 3 (7)

3 Jouna of Communicaions Vo. No. Novembe 5 Accoding o (5) - (7) we can see ha m b j j m 3 exp j a a3 FAF co exp co a 3 co exp j d a mcsc W m whee W( m ) denoes he FAF of noise. By seaching he peak of FAF m foowing expessions m ag max FAF m () we can obain he m FAF m b ( j co ) 3 exp j m co a a3 co a3 m asin IV. PARAMEERS DYNAMIC ESIMAION BASED ON FAF In his secion he sudy of paamee esimaion is made by aking he signa y as an exampe. he signa (9) qn yqn denoes he exaced signas yqn coesponding o he h age. as yqn can be expessed 3 exp y j qn a a w q n n A. Doppe Fequency Paamees Esimaion () Accoding o () and () he insananeous y can auocoeaion funcion R of signa be wien as whee y R y y y qn qn qn exp j 3 j f Rw w exp q n w yq n w w R y w is eaed as a andom inefeence. Accoding o () he FAF of y () can be wien as y FAF m R K m d y qn qn exp j co mcsc d W( m ) () 3 ( j co ) exp j m co f () whee W( m ) denoes FAF of noise. () aises he peak vaue when and m mee he condiions shown in (3) as co sin m FAF y m ( jco ) (3) 3 exp j m co f Accoding o (3) we can obain as e Define co m sin () 3 qn exp exp j. R as he Fouie ansfom of y y j y heefoe he IDF can be esimaed by p f y (5) y. ag max Y () f B. Azimuh Ange and Eevaion Ange Dynamic Esimaion Bin Yang [] poposed he mehod of pojecion appoximaion subspace acking (PAS) based on he eas squaes esimaion. On he base of he sudy of he mehod of PAS his pape poposes a nove mehod of FAF-PAS based on he faciona ambiguiy funcion. Accoding o (3) and () we can define zqn 3 as z exp j w. (7) qn n he insananeous coss-coeaion funcion z is defined as R beween y and yz qn whee qn qn j a a R R y z yz qn qn qn 3 exp 3 q n w w qn w z w w R y w is eaed as andom inefeence. () he coss-ambiguiy funcions yz qn m yqn and z can be expessed as beween qn yz qn j a a m = R K m d yz qn ( co ) 3 exp co q n j m co exp j d mcsc W m (9) 5 Jouna of Communicaions

4 Jouna of Communicaions Vo. No. Novembe 5 whee W( m ) denoes he faciona ambiguiy funcion of he noise. he m yz qn aises he peak vaue when boh and m mee he condiions shown in () and he posiion of he peak is aso ocaed a m m is. he peak vaue of he yz a yz qn m ( j co ) 3 exp j m co () a q n Accoding o (3) and () we can obain m a a FAF m yz q n () y A ( m ) he faciona coss-ambiguiy funcion yz MCAF m beween () and (7) can be expessed as m m MCAF yz yz yz m () Because he ampiudes of of diffeen ages ae vey ow a ( m ) in FRF domain hese signas ae no consideed as andom inefeing. heefoe () can be ewien as m a a FAF m MCAF yz q n y Seecing peak poins MCAF m yz (3) fo... as obseved daa a he eceive he oupu of he nh eceive anenna in he FRF domain can be expessed as n MCAF MCAF yz yz m MCAF n n m () yz he oupu of a eceiving anenna based on can be expessed as yz... yz N ;...;... yz Q yz QN (5) Boh eceive subaays R and R consuced in his pape can be expessed by R N A D N () R Q A D N (7) Nex he poposed mehod of FAF-PAS is inoduced in deai by aking he subaay R as an exampe. he maix W denoes he signa subspace of obsevaion veco R. his signa subspace is esimaed by he minimizaion of a cos funcion J he FAF-PAS agoihm. W in FAF-PAS () n - () J W R n W y n () -PAS n whee denoes he fogeing faco. y n W R n veco. J FAF-PAS () is pojecion appoximaion W is minimized if W C C (9) Ry We appy R y C n R y yy n n n Ry yy C R y C n y y n n n C y y yy (3) (3) W o he spaia specum of wo dimensiona Muipe Signa Cassificaion (D-MUSIC) agoihm [] P A WW A (3) Seaching speca peak of P we can eaize and eevaion ange he azimuh ange dynamica ecusive esimaion. Simiay he agoihm of FAF-PAS is appied o obsevaion veco R. We can obain as P (33) A WW A Seaching speca peak of P we can eaize and eevaion ange he azimuh ange dynamica ecusive esimaion. C. Camé-Rao Bound fo he Poposed Signa Mode his secion pesens he Came-Rao bound of paamee esimaion. he veco of a oupu of M mached fie can be expessed as K N (3) whee is Konecke poduc K + [ k k ]... NQ k A D... 5 Jouna of Communicaions 5

5 Jouna of Communicaions Vo. No. Novembe 5 A A A eceive aay especivey ae obained by simuaion expeimens. he numbe of Mone Cao uns is 5 in a simuaions. Simuaion : ime seies In his simuaion he signa o noise aio is se as SNR db. Fig. pos he esimaion pefomance of Doppe. Fig. 3 (a)-(b) show cuves of azimuh ange and eevaion ange dynamica ecusive esimaos wih ime. Fom Fig. and Fig. 3 we can find ha he poposed mehod has good esimaion pefomance. In he foowing simuaion expeimens we sudy he esouion capabiiy and esimaion accuacy of he poposed mehod he PARAFAC mehod[] and he SDOE mehod [9] as a funcion of he Signa o Noise Raio (SNR). and A a... am A a... an N is he noise veco. he seven paamees o be esimaed ae he Doppe fequency paamees he azimuh ange and eevaion ange which fom he paamees veco ξ such ha ξ denoes Esimaed vaue ue vaue 7 (35) Doppe(z) whee he anspose of a veco. Suppose ha he numbe of snapshos is N s. he Fishe infomaion maix (FIM) [] [9] fo esimaing ξ can be obained as In his case eemen i j of he Fishe infomaion maix (FIM) fo he N s obsevaions can be shown o be equa o K ξ j Qn (3) ence he expession fo he CRB can be expessed as CRB ξ (37) V. SIMUAION RESUS he consideed bisaic MIMO ada is composed of M ansmi anennas and N eceive anennas wih an ineeemen spacing of.5m. he base ine disance beween he ansmi efeence eemen and he eceive efeence eemen is D 5Km. he azimuh ange and eevaion ange eaive o ansmi aay and 5 Jouna of Communicaions (s) 3 5 azimuh ange eevaion ange ue vaue (s) (a) azimuh ange and eevaion ange of ansmi aay( ) K ij Re ξ i Ns Fig.. Esimaion pefomance of Doppe azimuh ange and eevaion ange of eceive aay( ) azimuh ange eevaion ange ue vaue (s) (b) Fig. 3. Esimaion pefomance of ange and aipane ocaion

6 Jouna of Communicaions Vo. No. Novembe 5 Simuaion : Signa-noise-aio SNR Since he age has hee dimensiona moion saes he eceived signas efeced fom hese ages ofen conain cubic phase. his pape poposes a new signa mode since he exising ones ae no appopiae o appoximae he paamees in his case. Fo a given deay he insananeous auocoeaion funcion of he signa wih cubic phase is an expession of inea fequency moduaion signa. ence his pape poposes a new appoach o esimae hese paamees in FRF domain. We esimae Doppe fequency paamees by seaching peak of he faciona ambiguiy funcion. We aso deveop wo sub-aay modes o accuaey esimae he azimuh ange and eevaion ange by empoying he FAF-PAS agoihm. Fuhemoe we deive he Camé-Rao bound fo age paamee esimaion. In Simuaion esus secion we discuss he effecs of vaious paamees in ems of he RMSE incuding he numbe of snapshos and he SNR. We aso compae he pefomance of he poposed mehod wih ha of he ohes. he coecness and effeciveness of he poposed mehod ae veified wih he compue simuaion. RMSE of D-ange in ansmi aay (db) eevaion ange-poposed mehod azimuh ange-poposed mehod eevaion ange-sdoe mehod azimuh ange-sdoe mehod eevaion ange-crb azimuh ange-crb SNR (db) (a) ansmi aay RMSE of D-ange in eceive aay (db) eevaion ange-poposed mehod azimuh ange-poposed mehod eevaion ange-sdoe mehod azimuh ange-sdoe mehod eevaion ange-crb azimuh ange-crb SNR (db) ACKNOWEDGMENS his wok was pay suppoed by he Naiona Science Foundaion of China unde Gans 55 7 and 39 and he Naiona Science and echnoogy Suppo Pogam and he Gan BAJB. (b) Receive aay Fig.. RMSEs of ange esimaion REFERENCES - RMSE of he Doppe (db) [] - poposed mehod PARAFAC Doppe-CRB -3 [] [3] SNR (db) [] Fig. 5. RMSEs of Doppe esimaion [5] Fig. and Fig. 5 show he RMSE of azimuh ange eevaion ange and Doppe fequency vesus SNR. Fom Fig. he pefomance of he poposed mehod is significany bee han he SDOE mehod. Fom Fig. 5 we can find ha he poposed mehod has good pefomance. Fom Fig. 5 he pefomance of he poposed mehod is bee han he PARAFAC mehod. [] [7] VI. CONCUSION [] In his pape he faciona ambiguiy funcion in FRF domain is appied o he join esimaion of Doppe fequency paamees he azimuh ange and eevaion ange in nea-fied bisaic MIMO ada sysem. 5 Jouna of Communicaions [9] 7 E. Fishe A. aimovich and R. Bum MIMO ada: An idea whose ime has come in in Poc. IEEE Rada Confeence Newak NJ USA pp J. F. i X. F. Zhang and X. Gao A join scheme fo ange and aay gain-phase eo esimaion in bisaic MIMO ada IEEE Geoscience and Remoe Sensing ees vo. no. pp. 7 June 3. A. Faina and A. M. esugie Gues edioia: Specia issue on bisaic and MIMO adas and hei appicaions in suveiance and emoe sensing IE Rada Sona & Navigaion vo. no. pp Febuay.. S. Qiu N. Xia and J. C. i An inefeence ocaizaion agoihm based on Gaussian appoximaion paice fieing wih sabe disibuion noise Signa Pocessing vo. no. 9 pp. -53 Sepembe. R. Schmid Muipe emie ocaion and signa paamee esimaion IEEE ansacions on Anennas and Popagaion vo. 3 no. 3 pp. 7- Mach 9. R. Roy and. Kaiah ESPRI-Esimaion of signa paamees via oaiona invaiance echniques IEEE ansacions on Acousics Speech and Signa Pocessing vo. 37 no. 7 pp Juy 99. W. Zhi and M. Y. W. Chia Nea-Fied souce ocaizaion via symmeic subaays IEEE Signa Pocessing ees vo. no. pp. 9- June 7. D.. iao Scaeing and imaging of nonineay oaded anenna sucues in haf-space envionmens IEEE ansacions on Anennas and Popagaion vo. no. pp. 3 Augus.. G. Wang A new 5-D paamees Join esimaion mehod fo nea fied souces in Poc. nd Inenaiona Confeence on

7 Jouna of Communicaions Vo. No. Novembe 5 Compue Science and Newok echnoogy Changchun China pp [] E. Gosicki K. Abed-Meaim and Y. ua A weighed inea pedicion mehod fo nea-fied souce ocaizaion IEEE ansacions on Signa Pocessing vo. 53 no. pp Ocobe 5. [] P.. eong. D. Abhayapaa and. A. amahewa Muipe 3D Fa-Fied/Nea-Fied moving age ocaizaion using wideband echo chip signas IEEE ansacions on Signa Pocessing vo. no. pp. 3 5 Novembe. [] J. Y. Zhang Z. D. Zheng and X. B. i An agoihm fo DOD- DOA and Doppe fequency join esimaing of bisaic MIMO ada Jouna of Eeconics & Infomaion echnoogy vo. 3 no. pp. 3- Augus. [3] J. i P. Soica and W. Robes On paamee idenifiabiiy of MIMO ada IEEE Signa Pocessing ees vo. no. pp Decembe 7. [] S.. iu. Shan R. ao Y. D. Zhang G. Zhang F. Zhang and Y. Wang Spase discee faciona fouie ansfom and is appicaions IEEE ansacions on Signa Pocessing vo. no. pp Decembe. [5] R. ao N. Zhang and Y. Wang Anaysing and compensaing he effecs of ange and Doppe fequency migaions in inea fequency moduaion puse compession ada IE Rada Sona & Navigaion vo. 5 no. pp. - Januay. [] Y. Bin Pojecion appoximaion subspace acking IEEE ansacion on Signa Pocessing vo. 3 no. pp Januay 995. [7].. uang Y.. Wu. C. So Y. D. Zhang and. uang Muidimensiona sinusoida fequency esimaion using subspace and pojecion sepaaion appoaches IEEE ansacions on Signa Pocessing vo. no. pp Ocobe. [] A.. Swindehus and P. Soica Maximum ikeihood mehods in ada aay signa pocessing Poceedings of he IEEE vo. no. pp. - Febuay 99. [9] P.. Sun. Jun and S. Wan Came-Rao bound of join esimaion of age ocaion and veociy fo coheen MIMO ada Jouna of Sysems Engineeing and Eeconics vo. 5 no. pp Augus. i i was bon in eiongjiang China on Juy She eceived he B.S. and M.S. degees fom iaoning noma univesiy of China in and 5 especivey. She eceived he Ph.D. degees fom Daian Univesiy of echnoogy in 3. e eseach ineess incude aay signa pocessing and esimaing paamee esimaion in MIMO ada sysems. ianshuang Qiu was bon in Jiangsu China on Augus 95. e eceived he B.S. degee fom ianjin Univesiy in 93 he M.S. degee fom Daian Univesiy of echnoogy in 993 and he Ph.D. degee fom Souheasen Univesiy in 99. e is cueny a pofesso in he Depamen of Eeconic Engineeing Daian Univesiy of echnoogy. is eseach ineess incude Non-Gaussian signa pocessing adapive signa pocessing and biomedica signa pocessing. 5 Jouna of Communicaions

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission