On the Security of Angle of Arrival Estimation

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1 On he Securiy of Angle of Arrival Esimaion Amr Abdelaziz, C. Emre Koksal and Hesham El Gamal Deparmen of Elecrical and Compuer Engineering The Ohio Sae Universiy Columbus, Ohio arxiv: v1 [cs.it] 2 Jul 2016 Absrac Angle of Arrival AoA esimaion has found is way o a wide range of applicaions. Much aenion have been paid o sudy differen echniques for AoA esimaion and is applicaions for jamming suppression, however, securiy vulnerabiliy issues of AoA esimaion iself under hosile aciviy have no been paid he same aenion. In his paper, he problem of AoA esimaion in Rician fla fading channel under jamming condiion is invesigaed. We consider he scenario in which a receiver wih muliple anenna is rying o esimae he AoA of he specular line of sigh LOS componen of signal received from a given single anenna ransmier using a predefined raining sequence. A jammer equipped wih muliple anennas is rying o inerrup he AoA esimaion phase by sending an arbirary signal. We derive he opimal jammer and receiver sraegies in various scenarios based on he knowledge of he opponen sraegies and he available informaion abou he communicaion channel. In all scenarios, we derive he opimal jammer signal design as well as is opimal power allocaion policy. The resuls show he opimaliy of he raining based Maximum Likelihood ML AoA esimaor in case of randomly generaed jamming signal. We also show ha, he opimal jammer sraegy is o emi a signal idenical o he predefined raining sequence urning he esimaion process ino a highes power compeiion scenario in which he deeced AoA is he one for he ransmiing eniy of higher power. The obained resuls are suppored by he provided compuer simulaion. Index Terms AoA Esimaion; Opimal Jamming; CRB; ML Esimaor. I. INTRODUCTION AoA esimaion is one of he mos imporan applicaions of array signal processing. I has found is way o a wide range of applicaions in miliary, navigaion, radar, law enforcemen and some commercial communicaion sysems. Ani-jamming is one of he mos ineresing applicaions of AoA esimaion. Recenly in [1], an ani-jamming mechanism for receivers operaing in a cogniive radio nework using AoA esimaion combined wih adapive beamforming is presened. The use of AoA esimaion in wireless physical layer auhenicaion is anoher promising applicaion. In [2], Xiong and Jamieson proposed a novel signal processing echnique leverage muli-anenna receiver o profile he direcions a which a cerain ransmission arrives, using his AoA informaion o consruc highly sensiive signaures ha wih very high probabiliy uniquely idenify each clien. Despie he fac ha much aenion paid o he enhancemen of AoA esimaion and he promising resuls repored abou is various applicaions, performance measure on AoA esimaion under jamming aacks did no receive he same aenion. To ha end, we develop a framework in which we evaluae he opimal aack sraegies o degrade he AoA esimaion performance. Subsequenly, we find he opimal esimaor under jamming aack and evaluae is performance. We clearly idenify he condiions under which he aacker successfully manages o derail he esimaor and he cases in which he aack can be overcome by he legiimae receiver. In he general problem of AoA esimaion, various signal models have been invesigaed in lieraure based on he number of esimaed signal sources, modeling of noise process, cooperaion beween emiing and receiving nodes and implicaions of mulipah environmen. In a non cooperaive environmen, where he emied signal is unknown, he subspace echniques such as muliple signal classifier MUSIC [3] and esimaion of signal parameers via roaional invarian echnique ESPRIT [4] were developed wih no knowledge required abou he emied signal. Performance analysis of condiional and uncondiional ML-AoA esimaor have been presened in [5]. There, condiional ML esimaor model he unknown emied signal as a random process where i is modeled as a deerminisic unknown parameer in he uncondiional ML esimaor. A crucial requiremen for all he aforemenioned AoA esimaion echniques in a non cooperaive regime ha he number of signals ha can be esimaed has o be less han he receiver array size. However, in a hosile jamming environmen or in a varying inerference condiions his condiion will fail if he rank of he received signal plus jamming/inerference exceeds he array size. Meanwhile, in a cooperaive environmen, where ransmiers and receivers share a predefined signal waveform, i was shown in [6] ha exploiaion of he known signal waveform enables a successfull esimaion of number of sources greaer han he array size using ML esimaor. Despie of his considerable advanage, a huge compuaional complexiy of he ML esimaor is considered a major disadvanage as i requires a K-dimensional search algorihm for esimaing AoAs of K inciden signals. In [6] his problem have been resolved by he so called decoupled ML DEML esimaor which urns he K-dimensional search problem ino K one-dimensional search problem. Anoher advanage for he work herein ha he resuls were obained for an arbirary inerference covariance marix. In his work, we explore he effec of Rician fading environmen on AoA esimaion in he presence of a hosile jamming aciviy. In Rician fading environmen, he received signal can be decomposed ino wo componens; one is he specular componen resuls from he LOS pah and he oher is he diffuse componen due o mulipah reflecions, or generally he non-line of sigh componen NLOS. The LOS componen

2 Fig. 1. Sysem Model can be considered fixed while he NLOS componen can be bes described as a Rayleigh fading channel. For AoA esimaion purposes, only he specular LOS componen is considered as a signal of ineres SOI while he diffuse NLOS componen is considered as an undesired inerference. This fac urns he AoA esimaion process ino a challenging ask due o he inheri correlaion beween specular and diffuse componen carrying he same signal waveform. Despie he resuls for AoA esimaion for an arbirary inerference covariance marix, under jamming condiions hese resuls doesn hold direcly. Tha is because, in conras o arbirary inerference, he jamming process is an inended hosile aciviy ha may have he abiliy o adop, maneuver or change sraegies o achieve he inended goal. Hence, based on he capabiliies, availabiliy of informaion abou he arge receiver design and CSI, we analyze boh opimal jammer and, on he oher side, receiver sraegies in a game heoreic approach. II. SYSTEM MODEL AND PROBLEM STATEMENT In he res of his paper we use boldface uppercase leers for random vecors/marices, uppercase leers for heir realizaions, bold face lowercase leers for deerminisic vecors and lowercase leers for is elemens. While,. denoes conjugae of complex number,. denoes conjugae ranspose, I N denoes ideniy marix of size N, r. denoes marix race operaor, var. denoes variance of random variable, de. denoes marix deerminan operaor and 1 m n denoes a m n marix of all 1 s. A. Sysem model As depiced in Fig.1, we consider he scenario where a mobile receiver is rying o esimae he AoA of he he signal emied from a cerain fixed ransmier equipped wih a single anenna in he presence of a muliple anenna jammer. The ransmier sends a predefined raining sequence, x C L, from which he AoA will be esimaed. The receiver is equipped wih a uniform linear array ULA anenna ha consiss of n r elemens placed along a linear array wih neighboring anennas spaced a a disance d r. On he oher side, he jammer is equipped wih a ULA anenna of size n j. We assume a narrowband sysem under fla fading wih a single significan channel ap. The discree baseband equivalen channel for he signal received by he legiimae receiver a he l h ime slo can be expressed as: Y[l] = H [l]x [l] + H j [l]x j [l] + N[l], 1 where x [l] C is he l h symbol of he predefined raining sequence. The raining sequence x = [x 1,.., x L ] T is consrained by boh an insananeous maximum power consrain x [l] 2 P max and a sum power consrain r x x P o. The jammer signal X j [l] is n j dimensional and i saisfies a oal power consrain rq j [l] P j., where Q j [l] = X j [l]x j [l]. Also, H [l] C nr 1 is he channel coefficiens vecor beween ransmier and receiver, H j [l] C nr nj is he channel coefficiens marix beween jammer and receiver a he l h ime index and N[l] C nr 1 is an independen zero mean circular symmeric complex random vecor, N CN 0, R n where R n = σni 2 nr. Thus, we can define he Signal o Jamming and Noise Raio SJNR as follows: H [l]x [l] 2 SJNR = H j [l]x j [l] 2. + σn 2 While he resuls we will drive are valid for all saionary and ergodic channel models, we will illusrae our examples on he Rician fading channel. Therefore, we nex give some basics of he Rician fading channels. In Rician fading model, he received signal can be decomposed ino wo componens; one is he specular componen resuls from he LOS pah and he oher is he diffuse componen due o mulipah reflecions. or generally he non-line of sigh componen NLOS. The LOS componen can be considered fixed while he NLOS componen can be bes described as a Rayleigh fading channel. Since boh ransmier and jammer channels are Rician, we give he channel descripion for boh ransmier and jammer channels wih he subscrips, j are dropped. We consider a general ransmier and receiver wih N and N r array sizes. H = H LOS + H NLOS, 2 where H LOS and H NLOS represens he LOS and NLOS componens respecively and k 1 H LOS = 2 + j ˆΨ 1 + k 2 ˆΨ = a r θa φ H NLOS 1 = kĥ, where k is he Ricean facor, a r θ and a φ are he anenna array seering vecors a receiver and ransmier respecively, θ and φ are he AoA and he angle of deparure AoD of

3 he ransmied signal respecively, as shown in Fig.1. In case of single anenna ransmier, we consider a φ = 1. There are no prior disribuion assumed for θ and φ. Therefore, he associaed esimaion problem is non-bayesian. For he ULA configuraion, he enries of he seering vecors are given by aθ = [ 1 z z 2... z N 1] T d sinθ j2π z = e λ, 4 where λ,d, and N are he wavelengh of he cener frequency of he ransmied signal, array elemens spacing and size respecively. While Ĥ CN 0, I N r N represens he channel coefficiens marix for he NLOS signal componen. We paramerize he conribuion of he NLOS and LOS componens o he signal wih σ = 1/21 + k, µ = k/1 + k, respecively and choose µ 2 + 2σ 2 = 1 for simpliciy. I worh menioning ha, AWGN and Rayleigh fading channels are in fac a limiing cases of he Rician fading channel. B. Aack Model In his paper we consider wo differen jamming scenarios, signal-unaware and signal-aware jamming aacks. By signal-unaware jamming we refer o he scenario in which he jammer has no knowledge of he esimaor, ˆθ, he raining sequence x and he realizaion of he channels, H [l] and H j [l]. Signal-unaware jammer only knows he channels disribuions. Meanwhile, he signal-aware jammer knows he esimaor, he raining sequence, and has perfec knowledge of he realizaion of boh channels. C. Problem Saemen Wih he above seup we consider he problem in which he receiver wih he knowledge of he predefined raining paern, x, has he objecive o choose an esimaor ˆθY ha has he minimum possible esimaor variance under all possible jamming sraegies for he choice of X j [l], l {1,..., L}, subjec o is power consrain. On he oher side, by an appropriae design of is ransmied signal, he jammer seeks o maximize he receiver esimaor variance over all possible receiver sraegies subjec o is power consrain. Hence, we can formulae our problem as follows: min ˆθ,x x [l] 2 P max rx X P o max X j[l] rq j[l] P j l 1,..,L var III. BASIC LIMITS UNDER NO ATTACK ˆθ Y. 5 In his secion, we evaluae he basic limis of AoA esimaion performance in he absence of he aacker. In paricular, we find a lower bound o he soluion of he following problem: min ˆθ,x x [l] 2 P max rx X P o var ˆθ Y. 6 The soluion of 6 is he CRB for AoA esimaion by definiion. To evaluae he CRB, we sar by inroducing Z[l] = H NLOS [l]x [l] + N[l], 7 where Z incorporaes all undesired inerfering componens of he received signal. Since he receiver objecive is o esimae he AoA of he LOS componen, he NLOS diffuse componen is also considered as an undesired signal. Noe ha Z[l] CN 0 nr 1, R z [l]. Accordingly, he poserior disribuion of he observaion Y is given as follows: f Y/H LOS,x,Z Y = 1 L i=1 deπr z[l] exp { 1 L Y[l] H LOS [l]x [l] R z [l] 1 i=1 Y[l] H LOS [l]x [l] }, 8 which yields he following log-likelihood funcion LY = ln deπr z [l] i=1 r R 1 z Y H LOS x Y H LOS x, 9 where R z = 1 L L i=1 R z[l]. Furher, I can be shown ha, he CRB of AoA esimaion is given by where CRB = 1 2 [ L Re µ x [l] ˆD Gθ ˆDx ] 1 [l]µ 1 + k =, 10 2Lk P max ˆD Gθ ˆD ˆD = R 1/2 z D D = a/ θ Gθ = [I aa a 1 a ] where he dependence of a on θ was dropped for ease of noaion. We noe ha, as k, only LOS componen is presen and R z σ 2 ni nr. Noe ha, his resul is in agreemen wih ha derived in [6]. Also, one can show ha his was also discussed in [6] efficien esimaor exiss only asympoically in he array size for any choice of x ha saisfies he power consrain. Moreover, he ML esimaor given by: ˆθY = max θ a R 1 z B 2 a R 1 z a = min B R 1/2 z GθR 1/2 z B 11 θ

4 achieves he CRB wih equaliy asympoically in he large array size limi, where B = R xyr 1 xx C nr 1 12 R xy = 1 x [l]y [l] C 1 nr L R xx = 1 L x [l]x [l] C. The resuls obained in his secion will be useful in he subsequen analysis in he res of his paper. IV. OPTIMAL JAMMER STRATEGIES In his secion we evaluae he opimal jammer sraegies in he wo differen aack scenarios described in Secion II. We sar wih he signal-unaware jamming scenario hen we consider he wors case of jamming in which he jammer is aware wih he receiver sraegies. A. Opimal Sraegy for The Signal-Unaware Jammer In his subsecion we consider he scenario where he only informaion available o he jammer is he saisical disribuion of boh channels. Here, he jammer has no knowledge on he esimaor, ˆθY, or he raining sequence, x. Hence, he aacker will arge he opimal esimaor performance, assuming ha he receiver is using he asympoically efficien esimaor. Noe ha his aacker sraegy also opimal for he aacker if he receiver is using he ML esimaor. Thus, i will consider firs he minimizaion problem in 6 whose soluion is he CRB is given in Eq. 10 for any arbirary inerference covariance marix R z which can be evaluaed as R z = P max 1 + k + σ 2 n I nr + E [ H j Q j H ] j 13 we derive his resuls in Appendix C. Hence, he jammer objecive is o find X j and Q j ha boh maximize he CRB expression and saisfy he power consrain. I can be formulaed as follows: X j, Q j = arg min D R 1/2 z GθR 1/2 z D, 14 rq j P j where he dependence on ime slo index l was dropped for ease of noaion. We give he opimal jamming sraegy as he soluion of Eq. 14 in he following heorem: Theorem 1. Given ha saisical channel disribuion is available a he jammer, he opimal jamming signal, X j [l], ha provides he soluion of Eq. 14 is a Gaussian vecor wih independen enries generaed according o X j [l] CN 0, Q j l {1, 2,.., L}, where Q j is a diagonal marix wih Q i,i j = { Pj min, n j [ Pj n j k j1 + k j n r1 + n j k j 15 and 0 k j < wih [x] + = max{0, x}. Proof: The proof is given in Appendix A. We noe ha, his resul is in agreemen wih he opimal power allocaion policy derived in [7] for a ransmier aiming o maximize i s muual informaion over a Rician MIMO channel. I also worh noing ha, as he Rician facor, k j 0, we noice ha Q j P j /n j I nj, which is he uniform power allocaion policy. Thus, for he relaively small values of he Rician facor, k j, he uniform power allocaion policy is near opimal even if he channel disribuion is unavailable a he jammer. The influence of using he uniform power allocaion policy raher han he opimal one will be discussed in more deails in Secion VI. B. Opimal Sraegy for The Signal-Aware Jammer In he previous secion, we showed ha wih no knowledge abou boh receiver sraegies and CSI, he opimal jammer sraegy is o generae a Gaussian signal wih power allocaion policy as defined in 15. In his secion, we consider he case where boh he raining sequence, x, and he esimaor, ˆθY, are known o he jammer. Also, perfec CSI abou boh channels in he form of channel realizaions, H [l] and H j [l] l {1, 2,.., L}, are assumed o be known o a signalaware jammer. In his case, he jammer objecive is o find X j and Q j ha boh maximize he error of he esimaor ˆθY and saisfy he power consrain. Since he ML esimaor is an asympoically efficien esimaor, hen, o maximize he ML esimaor variance, a signal-aware jammer can consider maximizing he CRB based on is available informaion. Thus, he opimal jammer sraegies are again he soluion of Eq. 14 aking ino consideraion he knowledge of he receiver sraegies. We give he opimal jammer sraegies in he following heorem: Theorem 2. Given ha boh he raining sequence, x [l], and perfec CSI available a he jammer, he opimal jamming signal ha provides he soluion of Eq. 14 can be found as: Q i,i j = { + µ λ 1 il if 1 < i n, 16 0 if n < i n j where λ 1l, λ 2l,..., λ nl are he eigenvalues of H j [l]h j [l] wih n = min{n r, n j }, µ is a consan chosen o saisfy he power consrain and X i j { [l] = + x[l] µ λ 1 il x [l] 2 if 1 < i n if n < i n j Proof: The proof is given in Appendix B. Comparing he opimal jamming sraegies in signal-unaware o ha in he signal-aware jamming scenario we observe ha, he knowledge of he raining sequence x is a considerable advanage o he jammer. I grans he jammer he abiliy } [ k j 1 + k j Pj n j + k ] j1 + k j + o design is signal aligned o he raining sequence. Under i = 1his scenario for equal jammer and ransmier power, he n r1 + n j k j n j n r1 + n j k j ] +, 1 < i n j ML specrum oupu from he ML-AoA esimaor shall be maximized in boh he ransmier and jammer direcions as can be seen in Fig. 4. Meanwhile, as he jammer power

5 exceeds ha of he ransmier, he ML specrum is maximized in he jammer direcion raher han he receiver one. Thus we conclude ha, a signal aware jammer urns he AoA esimaion process ino a higher power compeiion in which he ML esimaor oupus an esimae o he AoA of he ransmiing eniy of higher power. A more deails abou his scene are given in Secion VI. We have sudied he opimal jamming sraegies in boh aack scenarios, we now urn our aenion o he opimal receiver sraegies in wo scenarios, known and unknown jammer sraegies. V. OPTIMAL AOA ESTIMATOR In his secion, firs evaluae he opimal esimaor under he oy case in which he jammer sraegies and CSI are known o he receiver. Wih he insighs drawn, we solve he case in which he aacker sraegy, as well as CSI is unknown o he receiver. A. Opimal AoA Esimaor Wih Known Jammer Sraegies Assuming he jammer sraegies, X j [l] l, and perfec CSI availabiliy a he receiver, he noise, diffuse componen and jamming covariance marix is fully characerized a he receiver. Tha is because boh channels realizaion and ransmied signals are known o he receiver. Thus, he receiver can subsiue R z direcly ino 11 o form he ML esimaor. Simulaion resuls provided in Secion VI show he superior performance of ML esimaor wih known jammer sraegies. I shows ha he knowledge of jammer sraegies provides a considerable performance enhancemen even for he case of signal-aware jammer. More discussion abou hese resuls will be provided in Secion VI. The effec of unknown jamming sraegies will be discussed in he nex secion. B. Opimal AoA Esimaor Wih Unknown Jammer Sraegies In his secion, he receiver is assumed o know only he saisical disribuion of he channels. In Secion IV-B, we showed ha he wors case jamming sraegy is o mae is ransmied signal o he arge raining sequence. In such scenario he inerference covariance marix can be evaluaed as follows P max R z = + σn 2 I nr + P j ˆΥ, k where ˆΥ C nr nr is as defined in 23 he derivaion of his resul is given in Appendix C. Direc subsiuion from 18 ino 10 and 11 yields he CRB and he associaed ML esimaor soluion for he opimizaion problem given in 5 for any choice of x ha saisfies he power consrains. Simulaion resuls provided shows ha he ML esimaor is inefficien agains a signal-aware jammer despie he advanage of having he channel saisical disribuion. As can be seen in Figures 7, 8 and 9, he normalized ML specrum is maximized owards he direcion of he ransmiing eniy of higher power direcion even if saisical channel disribuion is available a he receiver. VI. SIMULATION RESULTS The simulaion resuls provided in his secion is based on he following simulaion seup: Array Size. Boh receiver and jammer are of array size n r = n j = 4 while he ransmier has a single anenna. Transmier Signal. The predefined raining sequence x, is generaed from a zero mean, uni variance complex Gaussian random variable, fixed once chosen and shared o he receiver. Jammer Signal. In he Signal-Unaware scenario, he Jammer signal X j [l], is generaed from a zero mean, uni variance complex Gaussian random variable and scaled o saisfy he power consrain. Meanwhile, in he Signal-Aware scenario i is chosen idenical o he raining sequence. Communicaion Channels. The communicaion channels, H and H j, are generaed according o Equaions 2 and 3. The enries of he channel marix of he Rayleigh par of he channel are generaed from a zero mean, uni variance complex Gaussian random variable and hen scaled each by he corresponding value of σ. A. Evaluaion of The CRB In Fig.2, he CRB is ploed as a funcion of he AoA, θ, in logarihmic scale for a raining sequence of lengh 64 a a signal o noise raio SNR = 15dB. While he jammer is emiing a random signal wih power budge equal o ha of he ransmier, which implies SJNR < 0dB. The figure provides comparison beween he CRB for he jamming free environmen o ha for a jammer wih uniform power allocaion policy and also for a jammer uses he opimal power allocaion policy derived in Eq. 15. Noe ha, he uniform power allocaion policy is opimal for k j = 0, hence, for relaively small k j values he uniform power allocaion policy provides a perform close o opimal. To highligh he difference in heir performance, we use k j = 10dB. As can be seen in Fig. 2, he use of he opimal power allocaion policy derived in Eq. 15 has a superior performance compared o he uniform power allocaion scheme in case of large values of he Rician facor k j. Similar comparison is given in Fig. 3 excep ha he jammer is emiing a signal idenical o he raining sequence wih four imes he power available a he ransmier. Noe ha, he use of he raining sequence as a jamming signal did no affec he CRB. Tha is because he ploed CRB is for any unbiased esimaor, however, using a jamming signal idenical o he raining sequence adds bias o he ML esimaor. B. ML Esimaor Performance under Jamming To evaluae he performance of he ML esimaor under jamming condiions, we provide simulaion resuls for differen scenarios of jamming sraegies as well as he amoun of informaion available a he receiver. Firs we sar by he case where no channel knowledge is available a he receiver. In Fig.

6 saisical channel disribuion informaion is available a he receiver as seen in Figures 7, 8 and 9. In case of perfec CSI availabiliy abou boh channels a he receiver, we see in Figures 10, 11 and 12 ha he ML esimaor have a considerable performance even for an increased jamming power. Fig. 2. CRB as a funcion of he AoA in log scale, SNR = 15dB, P j = P, k j = 10dB Fig. 4. ML Normalized Specrum wihou CSI, θ = 12, θ j = 50 and P j = P Fig. 3. CRB as a funcion of he AoA in log scale, SNR = 15dB, P j = 4P, k j = 10dB 4, he ransmier is locaed is a 12, he jammer is locaed a 50 and he jamming power equal o ha of he ransmier. We noice ha, For he signal-unaware jamming scenario where he opimal jamming signal is Gaussian generaed as saed in Theorem 1, he ML oupu is maximized in he direcion of he SOI as of he case of he jamming free model. As he jammer aligns is signal o he predefined raining paern, he ML esimaor sars o bias owards he signal wih higher power. This will be clear in Fig. 5, where he ransmier is locaed is a 43, he jammer is locaed a 63 and he jamming power is wice ha of he ransmier. While in Fig. 6 he ransmier is locaed is a 23 and he jammer is locaed a 40 and he jamming power is four imes he ransmier power. We see ha in case of random jamming, he effec of jamming power appear as an increased esimaor error. Meanwhile, as he jammer aligns is signal o he raining sequence, he esimaor is biased owards he jammer signal raher han ransmier one. The ML esimaor exhibis he same behavior when only Fig. 5. ML Normalized Specrum wihou CSI, θ = 43, θ j = 63 and P j = 2P VII. CONCLUSION The vulnerabiliy of AoA esimaion o hosile jamming aciviy is sudied. we considered he problem of AoA esimaion in Rician fla fading channel under jamming condiion. We showed ha in case of unknown receiver sraegy, he opimal jamming signal is Gaussian. Moreover, if he jammer have he knowledge of he receiver sraegy, is opimal signal design is o align is signal o he raining paern used for AoA esimaion. Also, opimal power allocaion policy based on he amoun of informaion available a he jammer

7 Fig. 6. ML Normalized Specrum wihou CSI, θ = 23, θ j = 40 and P j = 4P Fig. 9. ML Normalized Specrum wih Saisical Channel disribuion, θ = 23, θ j = 40 and P j = 4P Fig. 7. ML Normalized Specrum wih Saisical Channel disribuion, θ = 12, θ j = 50 and P j = P Fig. 10. ML Normalized Specrum wihou CSI, θ = 12, θ j = 50 and P j = P Fig. 8. ML Normalized Specrum wih Saisical Channel disribuion, θ = 43, θ j = 63 and P j = 2P Fig. 11. ML Normalized Specrum wih perfec CSI, θ = 43, θ j = 63 and P j = 2P

8 Fig. 12. ML Normalized Specrum wih perfec CSI, θ = 23, θ j = 40 and P j = 4P abou he communicaion channel is provided. I is shown ha, wih CSI availabiliy, he waer filling power allocaion policy is opimal while he uniform power allocaion is opimal where no CSI available. From he receiver poin of view, i was shown ha he raining based ML-AoA esimaor has a superior performance in subjec o he availabiliy of perfec CSI. Compuer simulaion resuls are provided o suppor and demonsrae he obained resuls. I showed he robusness of raining based ML-AoA esimaor under random jamming condiions. I also demonsraed he highes power compeiion in which he deeced AoA is for he ransmiing eniy of higher power for a jammer wih idenical signal. In conclusion, he ML esimaor works efficienly in all jamming scenarios in case of known jammer sraegies. Meanwhile, in case of unknown jamming sraegies, signal-aware jammer affecs he ML performance o a large exen. a X j, Q j = arg min r rq j P j APPENDIX A PROOF OF THEOREM 1 We sar from he Signal-unaware jammer objecive given in Eq. 14 which is equivalen o DR 1/2 z GθR 1/2 z D b = arg min r rq j P j c = arg min r rq j P j R 1/2 z D DR 1/2 z Gθ ΣR 1 z Gθ, 19 where b follows from he cyclic invarian propery of he marix race operaor and c follows from 2 nr 1 Σ = D 2πdr cosθ D = i 2, 20 λ i=0 which holds for he ULA configuraion. Subsiuing 13 and ignoring he consan erms as well as he erms ha are independen o X j and Q j we ge X j, Q j = arg max r rq j P j a = arg max r rq j P j E [ H j Q j H j ] E [ H j H jq j ], 21 where a follows from he cyclic invarian propery of he marix race operaor ogeher wih he fac ha race and expecaion operaors commue. Firs, we need o evaluae he disribuion of X j ha maximizes he above expression. We know ha he enries of he channel marix are Gaussian disribued as described in Secion II. Hence, expanding he race operaor in he above expression and differeniaing wih respec o he disribuion of X j yields he enries of X j should also be Gaussian disribued. Now, we need o show ha he power allocaion policy given in Theorem 1 is opimal. Based on he availabiliy of channel saisical informaion, we evaluae [ ] E H j H j = n r Υ, 22 where Υ is n j n j marix and is given by: 1 + k j k j... k j Υ = 1 k j 1 + k j... k j 1 + k j k j k j k j For any value of 0 k <, Υ is non-singular, hus all i s eigenvalue are non-zero. The n j eigenvalues of Υ are given by [8] : λ i = 1 + n j k j 1 + k j if i = k j if 1 < i n j 0 k < 24 We apply he eigenvalue decomposiion o ge Υ = UΛU where Λ C nj nj is a diagonal marix having he eigenvalues in 24 as is diagonal enries, and U C nj nj is a uniary marix composed of he eigenvecors of Υ. Define Q j = U Q j U, we ge he following alernaive expression for 21 X j, Q j = arg max r E [ n r Qj Λ ]. 25 rq j P j The above expression is maximized for he choice of Q j o be diagonal. The soluion of he diagonal enries Q j is found by he waer filling algorihm as follows: Q i,i j = { Pj k j 1 + k j min, n j n r1 + n j k j [ Pj k j1 + k j n j n r1 + n j k j } n j + ] + if 1 < i n j [ Pj k ] j1 + k j + if i = 1 n j n r1 + n j k j 26

9 where 0 k j < and [x] + = max{0, x}. We noe ha, he CRB evaluaed for Q j is he same as i evaluaed for Qj. Tha is because H j U has he same disribuion as H j. Thus, he rsul of Theorem 1 follows immediaely. APPENDIX B PROOF OF THEOREM 2 We ake ino consideraion ha, channel realizaions of boh ransmier and jammer channel are known o a signal-aware jammer, hus, we follow he proof of Theorem 1 excep ha we drop he expecaion operaor. We sar by evaluaing R z in such case R z = 1 L H NLOS [l]x [l] + H j [l]x j [l] + N[l]. H NLOS [l]x [l] + H j [l]x j [l] + N[l] = 1 L H NLOS + H NLOS [l]x [l]x j [l]h j [l] [l]x [l]x [l]h NLOS [l] + H j [l]x j [l]x [l]h NLOS [l] + H j [l]x j [l]x j [l]h j [l] + σ2 ni nr 27 Saring from Eq. 19 we subsiue 27, ignore he consan erms as well as he erms ha are independen o X j and Q j and drop he dependence on ime slo index for ease of noaion, we ge X j, Q j = arg max r rq j P j a = arg max r rq j P j H NLOS H NLOS x X j H j + H jq j H j x X j H j + r H j H jq j, 28 where a follows from he cyclic invarian propery ogeher wih he lineariy of he marix race operaor. Wihou he power consrain, i is sraighforward Using Cauchy-Schwarz inequaliy o see ha he firs race expression in 28 is maximized for X j [l] whose enries are all equal o x [l], i.e., he jammer aligns is signal o ha of he ransmier signal. I remains o show ha he power allocaion policy saed in Theorem 2 is opimal. Considering he maximizaion of he second race expression, we apply he eigenvalue decomposiion o ge H j H j = UΛU where Λ C nj nj is a diagonal marix having he eigenvalues λ 1, λ 2,..., λ n as is firs n diagonal enries and he oher n j n enries are zeros where n = min{n r, n j }, and U C nj nj is a uniary marix composed of he eigenvecors of H j H j. Define Q j = U QU, we ge he following alernaive expression for 28 X j, Q j = arg max r QΛ. 29 rq j P j The above expression is maximized for he choice of Q j o be diagonal. The soluion of he diagonal enries Q j is found by he waer filling algorihm and he heorem follows. APPENDIX C DERIVATION OF R z The inerference, jamming and noise covariance marix R z can be evaluaed as follows a R z = 1 E [ H NLOS [l]x [l] + H j [l]x j [l] + N[l] L. H NLOS [l]x [l] + H j [l]x j [l] + N[l] ] b = 1 L E [ H NLOS [l]x [l]x [l]h NLOS [l] ] + E [ H j [l]x j [l]x j [l]h j [l]] + E [ N[l]N [l] ] c P max = I nr + 1 E [ H j [l]q j [l]h j 1 + k L [l]] + σni 2 nr, where a is a direc subsiuion form he definiion of he covariance marix, in b we used he fac ha signals, channels and noise of boh ransmier and jammer are independen. Also, we assumed he channels disribuion o be ime invarian. While in c, we used he disribuion of he NLOS componen of he received signal provided in Secion II. Now using Eq. 22, he resul given in Eq. 18 is immediae. REFERENCES [1] A. S. and G. Umamaheswari, Performance analysis of ani-jamming echnique using angle of arrival esimaion in crn s, in Signal Processing, Communicaion and Neworking ICSCN, rd Inernaional Conference on, March 2015, pp [2] J. Xiong and K. Jamieson, Securearray: Improving wifi securiy wih finegrained physical-layer informaion, in Proceedings of he 19h Annual Inernaional Conference on Mobile Compuing & Neworking, ser. MobiCom 13. New York, NY, USA: ACM, 2013, pp [Online]. Available: hp://doi.acm.org/ / [3] R. Schmid, Muliple emier locaion and signal parameer esimaion, IEEE Transacions on Anennas and Propagaion, vol. 34, no. 3, pp , Mar [4] R. Roy and T. Kailah, Espri-esimaion of signal parameers via roaional invariance echniques, IEEE Transacions on Acousics, Speech, and Signal Processing, vol. 37, no. 7, pp , Jul [5] P. Soica and A. Nehorai, Performance sudy of condiional and uncondiional direcion-of-arrival esimaion, IEEE Transacions on Acousics, Speech, and Signal Processing, vol. 38, no. 10, pp , Oc [6] J. Li, B. Halder, P. Soica, and M. Viberg, Compuaionally efficien angle esimaion for signals wih known waveforms, IEEE Transacions on Signal Processing, vol. 43, no. 9, pp , Sep [7] E. Telaar, Capaciy of muli-anenna gaussian channels, European ransacions on elecommunicaions, vol. 10, no. 6, pp , [8] S. K. Jayaweera and H. V. Poor, On he capaciy of muliple-anenna sysems in rician fading, IEEE Transacions on Wireless Communicaions, vol. 4, no. 3, pp , 2005.

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