Transmit Waveform Selection for Polarimetric MIMO Radar Based on Mutual Information Criterion

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Sensors & Transducers ol. 5 Specal Issue Deceber 3 pp. 33-38 Sensors & Transducers 3 by IFSA hp://www.sensorsporal.

1 Sensors & Transducers ol. 5 Specal Issue Deceber 3 pp Sensors & Transducers 3 by IFSA hp:// Trans Wavefor Selecon for Polarerc MIMO Radar Based on Muual Inforaon Creron ajng CUI * ong JIANG Xaohu TANG College of Councaon Engneerng Jln Unversy 537 Nanhu Road Changchun 3 Chna Tel.: E-al: hanbngyan45@63.co jangh@jlu.edu.cn xaohu_ @6.co Receved: 6 Sepeber 3 /Acceped: 5 Ocober 3 /Publshed: 3 Deceber 3 Absrac: Trans wavefor selecon s essenal o prove he perforance of a polarerc ulple-npu ulple-oupu MIMO) radar syse. In hs paper an nforaon heorec creron based ehod s proposed o selec he polarzaon wavefors of he rans array. The desgn creron s o nze he uual nforaon MI) beween he radar reurn sgnal a wo adjacen coheren processng nervals. Based on hs creron he wavefor o be ransed a nex nerval s seleced accordng o he reurn sgnal a curren nerval. The creron guaranees ha he MIMO radar can cach ore new nforaon abou he arge a nex e. A sple analyc for s aaned when all hese processes are proper Gaussan processes. An erave opzaon algorh based on alernang projecon s proposed o realze he procedure of wavefor selecon. The algorh grealy reduces he aoun of copuaon. Nuercal exaples are conduced o llusrae he effecveness of he proposed ehod. Copyrgh 3 IFSA. Keywords: MIMO radar Wavefor desgn Polarzaon wavefor Muual nforaon creron Alernang projecon.. Inroducon Mulple-npu ulple-oupu MIMO) radar [] has he advanage of flexble ransng bea desgn n whch dverse fors of sgnal can be ransed by ul-anennas. The syse overcoes he perforance declnaon of radonal phased array radar brough by he arge RCS scnllaon. By nroducng polarzaon sensve arrays o he MIMO fraework he polarerc MIMO radar has recenly drawn consderable aenon n whch he polarzaon nforaon of sgnals can be fully ulzed. The polarerc MIMO radar syse offers boh spaal and polarzaon dverses ha furher proves he perforance of radar arge deecon [] and paraeer esaon [3]. Coonly a radar reles on wavefor o realze s full poenal. Thus s a crucal ask o desgn wavefor o unleash s advanages. The desgn ehod based on uual nforaon creron MIC) has been suded by researchers n recen years. Shannon uual nforaon MI) s a easure of he nforaon beween a rando varable or vecor) and anoher rando varable or vecor). Early n 993 M. R. Bell frs proposed he ehod of wavefor desgn aed a provng he esaon and denfcaon capably usng uual nforaon beween radar echoes and scaerng arges where he waer-fllng algorh was eployed [4]. For wavefor desgn of MIMO radar Yang Yang e al deonsraed he equaly of he wo ehods of wavefor desgn usng axu Arcle nuber P_SI_469 33

2 Sensors & Transducers ol. 5 Specal Issue Deceber 3 pp condonal MI creron and he nu ean square error MMSE) creron n he condons of deernsc he power specru densy and ha he power specral densy s led [5][6] and gave a ore praccal and splfed ehod n whch only fne saple values of power specral densy PSD) of pulse response are needed whch has grealy splfed he algorh. Based on hese a knd of wavefor ha ees he wo crera was desgned va alernang projecon algorh [7]. Then under he condon ha he power specral densy s unknown he nax robus wavefor desgn was explored based on MI and MMSE esaon and he conclusons were gven deonsrang ha he opzaon soluons are dfferen n he wo suaons [8]. I s shown ha he MIC s an effecve creron for polarzaon ype selecon n synhec aperure radar SAR) and ha he MI aans a sple analyc for for Gaussan-dsrbued scaerer and nose [9]. MIMO radar feaures wavefor dversy. The wavefor desgn of MIMO radar has been under nensve sudy n recen years whle he wavefor desgn of polarerc MIMO radar s rare o be nvesgaed. The research n [9] provdes a plausble soluon whch s used n sngle-npu sngle-oupu SISO) polarerc radar for polarzaon saes selecon. In hs arcle we exend he odel o he polarerc MIMO radar. An nforaon heorec creron based ehod s proposed o selec he polarzaon wavefors of he rans array. The desgn creron s o nze he uual nforaon beween he radar reurn sgnal a wo adjacen coheren processng nervals. Based on hs creron he wavefor o be ransed a nex nerval s seleced accordng o he reurn sgnal a curren nerval. The creron guaranees ha he MIMO radar can cach ore new nforaon abou he arge a nex e. A sple analyc for s aaned when all hese processes are proper Gaussan processes. An erave opzaon algorh based on alernang projecon s proposed o realze he procedure of wavefor selecon. The algorh grealy reduces he aoun of copuaon. Noaon: Throughou hs paper we use bold upper case leers o denoe arces and bold lower case leers o sgnfy colun vecors. Superscrps T { } and{ } denoe coplex conjugae ranspose and ranspose of a arx respecvely and { } denoes he nverse-square-roo arx operaor.. Polarerc MIMO Radar Model Consder a polarerc MIMO radar syse wh M ranser anennas and N recever anennas each of whch consss of horzonal and vercal coponens. A pon arge s locaed n far feld. The ransng ecevng sgnals are boh n he for of full polarzaon. Then he receved baseband sgnal can be descrbed as where r = + w ) denoes a 4 MN vecor coposed of MN coplex-valued scaerng coeffcens n he ground regon of neres and each ground scaerer s characerzed by he four eleens of scaerng arx n he convenonal horzonal ) and vercal ) axes a he scaerer locaon. The arx MN 4MN denoes ransng sgnal odel arx wh M ransng anennas. The whe nose w s a MN zero-ean Gaussandsrbued coplex-valued rando vecor ha s uncorrelaed over space and e and wh covarance arx R ww = E[ ww ]. The receved sgnal r s a colun vecor whch descrbes he echoes of N recever anennas. I s defned as T N T M T MN T T = ) ) ) ) r r r r r ) n where r denoes an echo vecor wh respec o he sgnal fro he h ranser anenna o he nh recever anenna. The arge scaerng coeffcen vecor has he for of T N T M T MN T T ) ) ) ) = 3) Each e of n s also a colun vecor.e. n n n n [ ] T = 4) whch s he scaerng daa beween he h ranser anenna and nh recever anenna. Wh respec o he eleens of polarzaon scaerng arx n Eq. 4 he frs polarzaon-specfc subscrp represens he rans coponen and he second polarzaon-specfc subscrp represens he receve coponen. descrbes he feaures of arge whch depends on scaerer scenaro. A he recever each polarerc anenna can capure wo pars of recevng sgnals fro wo orhogonal drecons naely and n n n [ r r ] T n r = 5) r = h + h + w 6a) n n n n n n r = h + h + w 6b) n where varables h and h represen he coplexvalue scalar odulaon sgnal coponens for and ransed by he h anennas. 34

3 Sensors & Transducers ol. 5 Specal Issue Deceber 3 pp Snce polarzaon s he locus of he elecrc feld vecor as a funcon of e descrbes he drecon of wave oscllaon n he plane perpendcular o he drecon of propagaon. Addonally he locus s coonly an ellpse whch can be descrbed by s polarzaon l angle π π [ ] and s polarzaon ellpcy angle π π [ ]. Assue ha all sgnals 4 4 have noralzed apludes hus he elecrc feld vecor can be defned as Eq. 7 usng he geoerc descrpors h E = = Q ) h ) h 7) where = cos sn Q ) s a roaon arx sn cos and cos h ) = s an ellpcy vecor. E j sn can be reforulaed no nforaon s a easure of he sascal correlaon beween wo rando varables. Denoe MI r r ) as he uual nforaon beween wo rando vecors r. The relaonshp beween enropy and uual nforaon can be wren as MI r r ) = r ) r r ) = r ) r ) ) r MI r r ) = r ) + r ) r ) ) r where r ) denoes he enropy of he rando vecor r r r ) denoes condonal enropy of j r condoned on r j and r rj ) denoes he jon enropy of he rando vecors r j. For a Gaussan connuous dsrbuon hese easures are defned respecvely as [] ) r = ln[ p r )] p r ) dr 3) h h = h h So Eq. 5 can be rewren as 8) j = ln[ p r rj )] p r rj dr drj r r ) ) 4) r ) = ln[ p r r )] p r r ) drd r r 5) r + n n n = w 9) Therefore he ransng sgnal arx n Eq. conssng of he varable of polarzaon wavefor we wan o desgn has a paroned srucure of he for = M M ) where each represens a 4 arx of zeros and subarx s repeaed N es along he an block dagonal. I s noed ha he polarzaon wavefor paraeers and s ncorporaed n so we a a selecng he opal value for every and o ake he polarerc MIMO radar syse perfor bes. 3. Shannon MI and MI Creron 3.. Shannon Muual Inforaon As we know enropy can be used o descrbe he average uncerany of a rando varable and uual In hese defnons naural logarhs are ulzed whch s convenen when Gaussan dsrbuon s nvolved. 3.. Muual Inforaon Creron A coplex rando vecor s s called srcly) T proper f Css = E[ ss ] []. Assue s also a zero-ean Gaussan-dsrbued coplex-valued vecor wh covarance arx R = E[ ]. The rando vecors and w are ndependen of each oher and each s assued o be an analyc sgnal. Thus each s a proper rando vecor and ads a coplex Gaussan probably densy funcon PDF) denoed as p [ ] wh he for of p ) [ R ] M = π R exp 6) Based on above assupons can be nferred fro Eq. ha he vecor r s also a proper process and ads a coplex Gaussan PDF. Accordngly he MI beween r could aan a sple for whch s proved n [9]. Le r denoe he receved sgnal vecor for he wavefor { ) } a hs coheren processng nernal denoe he receved sgnal vecor for ) o be ransed a he he wavefor { } 35

4 Sensors & Transducers ol. 5 Specal Issue Deceber 3 pp nex nernal. The space-e whenng ransforaons of vecors r are defned as alernang projecon we can refer o leraure []. Our algorh can be saed as Table. X = 7) R r Table. Alernang projecon algorh. Z = R r 8) So he cross-covarance arx of hese wo rando vecors s wh E [ ZX ] 9) = R ZX = R R R R = E[ rr ] = R + R ) R = E[ rr ] = R ) R = E[ rr ] = R + R ) ww ww where R R and R are he auo-covarance and cross-covarance arces of echoes respecvely. Le d l [ ] denoe he l s sngular value of he cross-covarance arx R hen uual ZX nforaon beween proper rando varables r and r has he for of MN ) = ln[ d l ] l = MI r r 3) We hope o selec desrable wavefor o be ransed a nex e fro he alernave wavefor canddaes n order o ake he radar capure ore new nforaon abou he arge. I s o say ha he wavefor o be seleced guaranees he reurn sgnals dfferen fro he oen reurns as possble naely o nze he uual nforaon beween r. ence he uual nforaon creron can be expressed as MIC = n { MI r r )} ) 4) where and are he paraeer vecors of wavefor o be ransed fro M anennas a nex e. 4. Wavefor Selecon Process va Alernang Projecon To selec wavefor effecvely we presen an erave opzaon algorh n hs secon ha feaures an alernang projecon approach whn M subspaces. For he convergence properes of Inal and le = M. = ; = = whle end condon s no sasfed do for =:M l l l l Le = = l = and k k k k = = k = + M. Fnd π π [ ] π π [ ] o nze MI; 4 4 end for = + ; end whle The an dea of hs erave algorh s o fnd he opal paraeers ) for he h anenna n he fne canddae se whle paraeers of oher M anennas anan her laes values unchanged. Once he paraeers of one anenna have changed he opal values of paraeers of he ohers ay change as well. Thus we opze paraeers of each denson of anenna alernaely unl he end condon s sasfed. 5. Sulaon and Analyss In hese secon we use he exaple of he odel wh M = ranser anennas and N = recever anennas o llusrae he wavefor desgn soluon derved. Assue ha s space-e uncorrelaed naely he arge scaerng coeffcens beween dfferen anennas n dfferen es s ndependen and dencally-dsrbued. Thus he covarance arx R s block dagonal and n our exaple can be descrbed as R R R = 33 R 44 R 5) where R = [ ) ] s he covarance arx beween he h ransng anenna and he h recevng anenna. In realsc scene R can be easured however we can no oban he value of snce s a rando sascal varable. ere we drecly gve he value of o oban R for noaonal splcy hey are gven as 36

5 Sensors & Transducers ol. 5 Specal Issue Deceber 3 pp = [.+ j..3 + j j j. ] T [ j j j j ] [ j j j j ] [ j j j j ] = T = T = T Assue ha he ransng sgnals are copleely polarzed so hey are he funcons of polarzaon l angle and polarzaon ellpcy angle. Fro Eq. 4 we know ha here are 4 varables o be seleced n our sulaon naely. They are he paraeers of ransng sgnals fro he frs and he second ranser anennas respecvely. ere we assue ha and are respecvely chosen fro π π π π π + ) + 8 ) and 8 8 π π π π π + ) + 8 ) I s clear ha here are oally 9 4 knds of wavefor n our exaple. If he nuber of ranser anennas s ore han wll be uch greaer. ow o selec he wavefor effecvely becoes a proble. ere an erave algorh va alernang projecon s eployed o solve. Assue he polarzaon paraeers of he ransed wavefor a curren nerval are π = = and π = =. We use alernang 4 projecon approach n he wo subspaces ) and ) o selec he opal paraeers sasfyng MIC. So he proble of wavefor desgn can be break no wo sub-probles. Frs choose and n he subspace ) o nze MI r r ) wh he laes and nvolved. Nex usng he new values of and renew and n subspace ) o nze MI r r ). Alernaely do he wo seps above unl he values of are all unchanged n wo adjacen eraons. In hs case we oban he opal wavefor. Addonally he nal values can be arbrary snce he resuls of hs erave algorh are rrelevan o hese nal daa. Based on he daa gven above he sulaon resuls ndcae ha he nal MI value s wh = =. 397 = =.397. owever he value of MI ay flucuae a lle each e for he rando nose even under he sae wavefor paraeers. Our fndngs are llusraed n Fg. and Fg.. Noe ha Fg. presens he fac when = =. 397 he nal MI value s locaed a = =.397. Furherore fro Fg. s also verfed ha he bes resuls of he four paraeers sll have he sae values wh = =. 397 unchanged.usng alernang projecon approach he aoun of calculaon s reduced o n 9 where n s he nuber of alernang eraons. Obvously 4 s uch less han 9. uual nforaon value uual nforaon value X:.7854 Y: Z: ellpcy angle of anennas l angle of anennas Fg.. The selecon of and when s fnally opzed and 7.8 X:.7854 Y: Z: ellpcy angle of anennas l angle of anennas Fg.. The selecon of and when and s fnally opzed. 6. Conclusons In hs paper a soluon o polarzaon wavefor selecon for polarerc MIMO radar s presened. The creron s based on he uual nforaon beween he radar reurn sgnals a wo adjacen nsans of e. I can aan a sple for for he proper Gaussan case. The opal rans polarzaon wavefor s effecvely desgned accordng o he second-order sascs of he arge. Furher an effecve wavefor opzaon ehod based on alernang projecon s proposed. I s an erave algorh whch has grealy reduced he 37

Sensors & Transducers ol. 5 Specal Issue Deceber 3 pp. 33-38 calculaon aoun. The nuercal exaples gven above have llusraed he effecveness of he proposed algorh.

6 Sensors & Transducers ol. 5 Specal Issue Deceber 3 pp calculaon aoun. The nuercal exaples gven above have llusraed he effecveness of he proposed algorh. The opzaon ehod can be also used n wreless sensor neworks [3 4]. Acknowledgeens The research s suppored by he Naonal Naural Scence Foundaons of Chna 674 and 63758). References []. J. L P. Soca and Y. Xe On probng sgnal desgn for MIMO radar n Proceedngs of he 4 h Asloar Conference on Sgnals Syses and Copuers nved) Pacfc Grove CA Ocober 6. []. S. Gognen and A. Nehora Polarerc MIMO Radar Wh Dsrbued Anennas for Targe Deecon IEEE Transacons on Sgnal Processng ol. 58 Issue 3 March pp [3]. W. N. Guo M. L. Yang B. X. Chen and G. M. Zheng Jon DOA and polarzaon esaon usng MUSIC ehod n polarerc MIMO radar n Proceedngs of he IET Inernaonal Conference on Radar Syses Radar ) Oc. pp. -4. [4]. M. R. Bell Inforaon heory adar wavefor desgn IEEE Transacon on Inforaon Theory ol. 39 Issue 5 Sep. 993 pp [5]. Y. Yang and R. S. Blu MIMO radar wavefor desgn n Proceedngs of he 4 h Workshop on Sascal Sgnal Processng 7 pp [6]. Y. Yang and R. S. Blu MIMO Radar Wavefor Desgn Based on Muual Inforaon and Mnu Mean-Square Error Esaon IEEE Transacon on Aerospace and Elecronc Syses ol. 43 Issue 7 pp [7]. Y. Yang R. S. Blu Z. S. e and D. R. Fuhrann MIMO Radar Wavefor Desgn va Alernang Projecon IEEE Transacons on Sgnal Processng ol. 58 Issue 3 March pp [8]. Y. Yang and R. S. Blu Mnax robus wavefor desgn for MIMO radar n he presence of PSD unceranes n Proceedngs of he 4 s Annual Conference on Inforaon Scences and Syses Balore MD Uned Saes 7 pp [9]. J. R. Roan J. W. Garnha and P. Anonk Polarzaon Dversy Usng Muual Inforaon n Proceedngs of he IEEE Radar Conference Aprl 7 pp []. P. Beckann Probably n Councaon Engneerng arcour Brace & World Inc. New York NY USA 967. []. P. J. Schreer and L. L. Scharf Second-order analyss of proper coplex rando vecors and processes IEEE Transacons on Sgnal Processng ol. 5 Issue 3 March 3 pp []. J. A. Tropp I. S. Dhllon R. W. eah and T. Sroher Desgnng srucured gh fraes va an alernang projecon ehod IEEE Transacons on Inforaon Theory ol. 5 Issue Jan. 5 pp [3]. I. Ullah Q. Ullah F. Ullah and S. Shn Sensor- Based Auonoous Robo Navgaon wh Dsance Conrol Journal of Copuaonal Inellgence and Elecronc Syses ol. Issue pp [4]. D. Lohan and S. ara Dsrbued Copung Paradgs for CSIP n Wreless Sensor Neworks: A Coparave Revew Journal of Copuaonal Inellgence and Elecronc Syses ol. Issue pp Copyrgh Inernaonal Frequency Sensor Assocaon IFSA). All rghs reserved. hp:// 38

Normal Random Variable and its discriminant functions

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