Low Probability Identification Performance in Radar Network System by using Fuzzy Chance-Constrained Programming

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1 Lo obably denfcaon efomance n Rada eok Sysem by usng Fuzzy Chance-Consaned ogammng 1 M..Saln M.Sc., M.hl., Ms.S..Subha M.E., 3 M. D.M.huchean, 1 Reseach Schola, Depamen of Mahemacs, L..Gov.College onne, Madas Unvesy, chenna-5. Asssan ofesso, Depamen of ECE, AVCE, Kanyakuma. 3 Assocae ofesso, Depamen of Mahemacs, L..Gov.College onne. Absac n hs pape, he poblem of lo pobably of denfcaon (LD mpovemen fo ada ok sysems s nvesgaed. Fsly, he secuy nfomaon s deved o evaluae he LD pefomance fo ada -ok. hen, hou any po knoledge of hosle ne-cep eceve, a novel fuzzy chance-consaned pogammng (FCC based secuy nfomaon opmzaon scheme s pesened o acheve enhanced LD pefomance n ada ok sysems, hch focuses on mnmzng he achevable muual nfomaon (M a necepo, hle he aanable M ouage pobably a ada -ok s enfoced o be geae han a specfed confdence level. Regadng o he complexy and unceany of elecomagc envonmen n he moden balefeld, he apezodal fuzzy numbe s used o descbe he heshold of achevable M a ada ok based on he cedbly heoy. Fnally, he FCC model s ansfomed o a csp equvalen fom h he popey of apezodal fuzzy numbe. umecal smulaon esuls demonsang he pefomance of he poposed saegy ae povded. Keyods Secuy nfomaon, lo pobably of denfcaon (LD, poe allocaon, fuzzy chance-consaned pogammng (FCC, and ada ok sysems. 1. noducon Rada ok achecue, hch ofen efes o dsbued mulple-npu mulple -oupu (MMO ada s poneeed by Fshe n [] and has dan consdeable aenons due o s advanage of sgnal and spaal dveses. Moeove, ada ok oupefoms adonal monosac ada n age deecon, localzaon accuacy and nfomaon exacon. Recen yeas have nessed an nceasng nees on he ada ok confguaon hch has been exensvely suded fom vaous pespecves. he auhos n [4] consde he opmal avefom desgn fo MMO ada n coloed nose based on mzaon of muual nfomaon (M and elave enopy. Yang and Blum pesen o ada avefom desgn scheme s h consans on ave-fom poe [5]: he one s mzaon of he M beeen he age mpulse esponse and he efleced avefom, he ohe s mnmzaon of he mnmum mean-squae eo (MMSE n esmang he age mpulse esponse. A novel o-sage avefom opmzaon algohm fo dsbued MMO ada s poposed n [6], hee s demonsaed ha hs mehod can povde gea pefomance mpovemen n age nfomaon exacon. n [7], he auhos pesen hee poe allocaon cea negang popagaon losses no dsbued MMO ada sgnal model: mzng he M, mnmzng he MMSE and mzng he echo enegy. Sh e al. n [8], [9] nvesgae he lo pobably of necep (L opmzaon saeges n ada ok confguaons fo he fs me, hch ae shon o be effecve o enhance he L pefomance fo ada oks. n ecen yeas, pusng hgh physcal-laye (HY secuy s becomng a cenal ssue n eless communcaons, n hch sececy capacy s ulzed as a mec fo sececy communcaon pefomance [10]. n [11], he auhos sudy he use of afcal nefeence n mzng sececy capacy, hee a poon of he ansmng poe s allocaed o boadcas he nfomaon sgnal h enough poe o guaanee a cean sgnal-onefeence-plus-nose ao (SR fo he nended eceve, hle he es of he poe s ulzed o boadcas afcal nefeence o am he passve eavesdoppes. Zhou e al. n [1] nvesgae he poblem of secue communcaon n fadng channels. 577

2 Whle [13] poposes an opmzaon saegy fo achevng secuy ove mulple-npu sngle-oupu (MSO channels by beam fomng and afcal nefeence combned h he poeced zone. Mukheee and Sndlehus model he neacons beeen he legmae ansme and acve eavesdoppe as a o-peson zeo-sum game [14]. he auhos n [15] pesen a muluse schedulng algohm o mpove he cognve ansmsson secuy. Fuhe, Wang e al. n [16] popose secuy nfomaon faco o evaluae ada ado fequency (RF sealh, hee s llusaed abone ada RF sealh effecs based on secuy nfomaon faco concep unde some condons. Sh e al. exend he ok n [16] and povde secuy nfomaon based opmal poe allocaon scheme fo LD pefomance n ada oks [16], [18]. Hoeve, mos eseaches on sececy capacy ae manly oads mzng he sececy ae fo communcaon h guaaneeng sysem equemens. he use of secuy nfomaon fo LD pefomance n ada -ok sysems has aely been suded pevously, hch movaes us o consde hs poblem. n addon, he moden elecomagc envonmen s becomng moe and moe complcaed, lage dffcules fo ada msson ae caused by amouns of uncean facos n eleconc afae, hch canno be compleely solved by sochasc heoy. he heoy of fuzzy se has dan consdeable aenons snce hs concep as naed by Zadeh [18] n n 00, Lu [17] poposed he concep of cedbly measue, and esablshed he heoy of fuzzy chance-con-saned pogammng (FCC, hch s a banch of mahemacs fo sudyng fuzzy phenomena. hs pape ll nvesgae he FCC based secuy nfomaon opmzaon fo LD enhancemen n ada oks. he man conbuons of hs pape ae summazed as follos. Fsly, e deve an analycal closed-fom expesson of secuy nfomaon. Secondly, hen he po knoledge of necep eceve s unavalable, a novel FCC based secuy nfomaon opmzaon algohm s fomulaed o mnmze he achevable M a necep eceve, hle h e achevable M ouage pobably a ada ok s enfoced o be geae han a specfed confdence level. Regadng o he complexy and unceany of elecomagc envonmen n he moden eleconc afae, he apezodal fuzzy numbe s ulzed o descbe he heshold of achevable M a ada ok. Fnally, he FCC model s ansfomed o a csp equvalen fom h he popey of cedbly heoy. umecal smulaons ae povded o demonsae ha ou poposed algohm can mpove he LD pefomance fo ada oks o defend agans passve necep eceves. o he bes of auhos knoledge, no leaue dscussng FCC based secuy nfomaon opmzaon fo mpoved LD pefomance n ada ok sysems as conduced po o hs ok. he emande of hs pape s oganzed as follos. Secon noduces he basc conceps of cedbly heoy and he sysem model fo ada ok. We fs deve he analycal closed - fom expesson of secuy nfomaon h coopeave ammng (CJ fo ada -ok n Sec. 3 and fomulae he FCC based secuy nfomaon opmzaon algohm fo ada ok sys-em. elmnaes and Sysem Model.1 Cedbly heoy he heoy of fuzzy se has eceved close aenon by he scenfc communy ove he las seveal decades, hch as poneeed by Zadeh va membeshp funcon n n 1978, Zadeh pesened he concep of possbly measue, hch s ulzed o measue a fuzzy se. Alhough possbly measue has been dely used n boh heoy and pacce, has no self- dualy popey. n 00, Lu poposed he concep of cedbly measue o defne a self-dual measue. Afe ha, Lu esablshed he cedbly heoy n 004, hch s a banch of mahemacs fo sudyng fuzzy phenomena. Some basc conceps of cedbly heoy ae povded n he follong. Defnon.1: (Lu & Lu [17] le Θ be a nonempy se, and he poe se of Θ. he se funcon C s called a cedbly measue f sasfes he follong fou axoms: Axom 1: C{Θ}=1. Axom : C{A} C {B}, heneve A B. c Axom 3: CA CA 1, fo any even A fo any evens A h sup, CA 0.5. hen, he ple,,c s called a cedbly space. Axom 4: C A sup CA, A Defnon.: (Lu [1] A fuzzy vaable s a measuable funcon fom a cedbly space,,c o he se of eal numbes. Defnon.3: (Lu [1] Le be a fuzzy vaable defned on he cedbly space,,c. hen s membeshp funcon s deved fom he cedbly measue by: ( x C x 1, x (1 578

3 heoem.1 (Cedbly nveson heoem: (Lu [1] Le be a fuzzy vaable h membeshp funcon (x. hen fo any se B of eal numbes, e have: 1 C B sup ( x 1 sup ( x ( xb c xb Defnon.4: (Lu [1] he cedbly dsbuon : 0,1 of a fuzzy vaable s defned by: ( x C ( x. (3 ha s, (x s he cedbly ha he fuzzy vaable akes a value less han o equal o x.. Rada eok SR Equaon Le us consde a ada ok sysem h ans-mes and eceves, hch can be boken don no ansme-eceve pas each h a bsac componen conbung o he eney of he ada ok sgnal-o-nose ao (SR, as depced n Fg. 1. he ada ok sysem has a common pecse knoledge of space and me. n addon, s oh ponng ou ha ohogonal polyphase codes ae ulzed n ada ok sysem, hch have a lage man lobe-o-sde lobe ao. hese codes have a moe complcaed sgnal sucue makng hade o necep and denfy by a no coopeave necep eceve. he ada ok SR can be calculaed by summng up he SR of each ansm-eceve pa as n [1]: SR m1 n1 4 G G m m n mn m 3 komnbmfnrmrn ansm anenna gan, Gn s he n h eceve anenna gan, σmn s he ada coss secon (RCS of he age fo he mh ansme and nh eceve, λm s he m h ansmed avelengh, k s Bolzmann s consan, omn s he ecevng sysem nose empeaue a he n h eceve, Bm s he banddh of he mached fle fo he mh ansmed avefom, Fn s he nose faco fo he nh eceve, Lmn s he sysem loss beeen he mh ansme and nh eceve, Rm s he dsance fom he mh ansme o he age and Rn s he dsance fom he age o he nh eceve. L mn (4.3 Rada eok Sgnal Model Le K denoe he dscee me ndex, hen e can expess he ada ok sgnal model as: Y XH W (5 hee R X x x, x,... x C, 3 K 1 K, s he se of ansmsson sequences, H h 1, h, h3, h 4,..., h C efes o he pah gan fo ada ok sysem, K W 1,, 3,..., C epesens he sysem nose, and he eceved sgnal can be Y y,..., K 1, y, y3 y C Fo en as convenence, s assumed ha he nose W does no depend on he ansmed avefom X, and H and W ae muually ndependen. 3. oblem Fomulaon 3.1 Secuy nfomaon fo Rada eok Sysems Wh he defnon of M n [16], e can oban he M beeen he ansmng sgnal of ada ok X and he backscae sgnal Y as follos: ( X, Y H( Y m1 n1 H( Y X H( Y m g GmG ln1 n RmR n ( m1 1 m g p ln1 mn n H( W (6 hee (X, Y s he M beeen Y and X, H(Y s he enopy of backscae sgnal, and H(W s he enopy of Gaussan he nose. Smlaly, e can expess he M beeen he ansmng sgnal of ada ok X and he eceved sgnal of necep eceve Y as: 579

4 ( X, Y H ( Y H ( Y X mgmgn ln 1 m1 Rm (7 hee Gn s he anenna gan of necep eceve, σ denoes he nose covaance of necep eceve. he achevable M a necepo can be degaded by he CJ sgnals hle he ada ok sysem s unaffeced. Wh he consdeaon of CJ, e can modfy as follos: ( X, Y m1 mgmgn ln 1 G G n R m (8 R (, n hee s he oal ansmng poe fo CJ sgnal, G s he anenna gan of coopeave amme, R s he dsance fom he age o coopeave amme.ognang fom he sececy capacy n eless communcaons, e defne secuy nfomaon o measue he LD pefomance fo ada ok sysem. sec (, ( n (, ggg ln 1 4 R GGn ln1 n G G R m R (9 hee [x]+ = (0, x. has been poned ou n [16], [18] ha sec > 0 means ha ada ok s n compleely secue sae hle ackng age, and ha he lage he achevable secuy nfomaon sec obaned, he bee LD pefomance o fnsh he sysem msson. 3. FCC Based Secuy nfomaon Opmzaon n paccal applcaons, ould be mpossble o suppose ha any po nfomaon abou he hosle necep eceve s avalable, such as he sensvy of necepo, he pocessng gan, e al. hee he age and he necep eceve ae sepaaed a dffeen places. Wh he devaon of secuy nfomaon as (9, e can obseve ha secuy nfomaon s based on sasfyng M consans boh a ada ok and a necep eceve. Wh a pope choce of he poe allocaon, a pefecly secung channel can be devsed such ha: h ( (10 n (, 0 hee δh s he pedefned heshold of M a ada ok. Equaon (10 means asympocally pefec secuy n ada ok sysem. he secuy nfomaon opmzaon saegy can be summazed as follos: 1 Specfy he desed M heshold δh fo ada ok sysem, hch s ulzed as a mec fo age deecon pefomance. Allocae some ansmsson poe o acheve he desed M heshold δh fo age deecon. 3 Mnmze he achevable M a necep eceve n(, by dsbung he emanng ansmsson poe o yeld as much nefeence as possble, hle guaaneeng ha he CJ sgnal s desgned o be compleely ohogonal o ada modulang sgnal and geneaed o am he necep eceve hou affecng he ada ok. Hence, he secuy nfomaon opmzaon fo enhanced LD pefomance can be fomulaed as: subec mn, ( 0, o n ( (, o ada ok, s he mum ansmng poe fo ada modulang sgnal. Whle egadng o he complexy and unceany of elecomagc envonmen n he moden eleconc afae, he pedefned heshold of M a ada ok δh ould be uncean. Heen, a fuzzy vaable δh fuzzy s ulzed o evaluae he pedefned heshold of M a ada ok. Based on he conceps of cedbly heoy, he achevable M ouage pobably a ada ok s enfoced o be geae han a specfed confdence level α, ha s: 580 h (11

5 h C ( fuzzy (1 heefoe, e have he FCC based secuy nfomaon opmzaon fo LD enhancemen n ada ok as: mn n (,, C h subec o ( (0, o (13 Wh he FCC model (13, e can obseve ha nceasng he confdence level leads o enlagng he feasble se of he ue poblem, hch n un may esul n deceasng of he opmal value of he ue poblem [1]. s also oh ponng ou ha hee exss a escve elaonshp beeen he confdence level and he achevable M a necep eceve. 3.3 he Csp Equvalen Fom of FCC Model he FCC model (17 s a fuzzy lnea pogammng, hch can be ansfomed no he csp equvalen fom. n hs pape, e se δh fuzzy = (a, b, c, d (a < b c < d o be a apezodal fuzzy vaable. Defnon 3.1: (Lu, Zhao & Wang [19] By a apezodal fuzzy vaable, e mean ha he fuzzy vaable fully deemned by he quaduple ( 1,, 3, 4 of csp numbes h ( by: x 1, f 1 x 1 1, f x 3 ( x (14 x 4, f 3 x , else Defnon 3.: (Lu, Zhao & Wang [19] he cedbly dsbuon of a apezodal fuzzy vaable,,, s: ( , f x 1 x 1, f 1 x ( 1 1 ( x, f x (15 3 x 3 4, f 3 x 4 ( 4 3 1, f 4 x heoem 3.1: (Lu [0] f s a apezodal fuzzy umbe,,, ( fo he ( gven confdence level 0.5,1, he follong equvalen ansfomaon can be deved as: C x x ( 1 (, C C x x ( ( 1. h ( ( ( c ( 1 d. fuzzy (16 he poposed FCC based secuy nfomaon algohm (13 could be ansfomed no he follong csp equvalen fom: mn, n subec o (0, (, ( ( c o hs pape has poposed a novel FCC based secuy nfomaon opmzaon algohm o acheve mpoved LD pefomance n ada ok sysems hou any po knoledge of non-coopeave necep eceve, hose pupose s o mnmze he achevable M a necepo, hle he achevable M ouage pobably a ada ok s enfoced o be geae han a ( 1 d (17 oblem (13 akes ada ok msson h fuzzy no consdeaon because ada ok sysem mus accomplsh s msson n moden balefeld. So fa, e have compleed he achevable secuy nfomaon devaon and he FCC based secuy nfomaon opmzaon fo LD enhancemen n ada ok sysems. n ha follos, some numecal smulaons ae povded o sho he feasbly and effecveness of ou pesened algohm. 4. Conclusons

6 specfed confdence level. s oh ponng ou ha ou poposed algohm s pesened by smple analycal closed-fom expesson. Smulaon esuls demonsae ha ou poposed algohm s effecve o enhance LD pefomance fo ada ok o defend agans passve necepo aacks. Fo fuue eseach, ohe opmzaon cea need o be addessed o mpove LD pefomance fo ada ok sysems. [9] SH, C. G., WAG, F., SELLAHURA, M., ZHOU, J. J. L opmzaon fameok fo age ackng n ada ok achecues usng nfomaon-heoec cea. [10] WYER, A. X. he eap channel. he Bell Sysem echncal Jounal, 1975, vol. 54, no. 8, p Refeences [1] ACE,. E. Deecng and Classfyng Lo obably of necep Rada. Boson: Aech House, 009, p [] FSHER, E., HAMOVCH, A., BLUM, R. S., CM, L. J., CHZHK, D., VALEZUELA, R. A. Spaal dvesy n adas: models and deecon pefomance. EEE ansacons on Sgnal ocessng, 006, vol. 54, no. 3, p DO: /S RADOEGEERG, VOL. 4, O. 1, ARL [3] HAMOVCH, A. M., BLUM, R. S., CM, L. J. JR. MMO ada h dely sepaaed anennas. EEE Sgnal ocessng Magazne, 008, vol. 5, no. 1, p [4] AG, B., AG, J., EG, Y.. MMO ada avefom desgn n coloed nose based on nfomaon heoy. EEE ansacons on Sgnal ocessng, 010, [5] YAG, Y., BLUM, R. S. MMO ada avefom desgn based on muual nfomaon and mnmum mean-squae eo esmaon. EEE ansacons on Aeospace and Eleconc Sysem, 007. [6] CHE, Y. F., JSURE, Y., YUE, C., CHEW, Y. H., DG, Z. G. Adapve dsbued MMO ada avefom opmzaon based on muual nfomaon. EEE ansacons on Aeospace and Eleconc Sysem, 013, vol. 49, no., p DO: /AES [7] SOG, X. F., WLLE,., ZHOU, S. L. Opmal poe allocaon fo MMO adas h heeogeneous popagaon losses. [8] SH, C. G., ZHOU, J. J., WAG, F. Lo pobably of necep opmzaon fo ada ok based on muual nfomaon. [11] SWDLEHURS, A. L. Fxed SR soluons fo he MMO eap channel. n EEE nenaonal Confeence on Acouscs, Speech and Sgnal ocessng CASS 009. ape, 009, p DO: /CASS [1] ZHOU, X. Y., MCKAY, M. R. Secue ansmsson h afcal nose ove fadng channels: achevable ae and opmal poe allocaon. EEE ansacons on Vehcula echnology, 010, vol. 59, no. 8, p DO: /V [13] ROMERO-ZURA,., MCLERO, D., GHOGHO, M.,SWAM, A. HY laye secuy based on poeced zone and afcal nose. EEE Sgnal ocessng Lees, 013, vol. 0, no. 5, p DO: /LS [14] MUKHERJEE, A., SWDLEHURS, A. L. Jammng games n he MMO eap channel h an acve eavesdoppe. EEE ansacons on Sgnal ocessng, 013, vol. 61, no. 1, p [15] ZOU, Y. L., WAG, X. B., SHE, W. M. hyscal-laye secuy h muluse schedulng n cognve ado oks. EEE ansacons on Communcaons, 013, vol. 61, no. 1, p o [16] SH, C. G., ZHOU, J. J., WAG, F., CHE, J. LD opmzaon h secuy nfomaon n ada ok. ndusal Eleconcsand Engneeng, 014, vol. 93, p [17] LU, B. D., ZHAO, R. Q., WAG, G. Unceany pogammngh Applcaon. Beng: snghua Unvesy ess, 003, p (n Chnese [4] LU, B. D. heoy and acce of Unceany ogammng.hedelbeg: hyscal-velag, 00, p [18] ZADEH, L. A. Fuzzy ses. nfomaon and Conol, 1965, 58

7 [19] LU, B. D., ZHAO, R. Q., WAG, G. Unceany pogammngh Applcaon. Beng: snghua Unvesy ess, 003, p (n Chnese [0] LU, B. D. heoy and acce of Unceany ogammng.hedelbeg: hyscal-velag, 00, p [1] LU, B. D. Unceany heoy: An noducon o s AxomacFoundaons. Beln: Spnge-Velag, 004, p