Genealized Came-Rao Bound fo Join Esimaion of Tage Posiion Velociy fo Acive Passive Rada Newoks Qian He, Membe, IEEE, Jianbin Hu, Rick S. Blum, Fellow, IEEE, Yonggang Wu axiv:1510.02551v1 [mah.st] 9 Oc 2015 Absac In his pape, we deive he Came-Rao bound (CRB) fo join age posiion velociy esimaion using an acive o passive disibued ada newok unde moe geneal, pacically occuing, condiions han assumed in pevious wok. In paicula, he pesened esuls allow nonohogonal signals, spaially dependen Gaussian eflecion coefficiens, spaially dependen Gaussian clue-plus-noise. These bounds allow designes o compae he pefomance of hei developed appoaches, which ae deemed o be of accepable complexiy, o he bes achievable pefomance. If hei developed appoaches lead o pefomance close o he bounds, hese developed appoaches can be deemed good enough. A paicula ecen sudy whee algoihms have been developed fo a pacical ada applicaion which mus involve nonohognal signals, fo which he bes pefomance is unknown, is a gea example. The pesened esuls in ou pape do no make any assumpions abou he appoximae locaion of he age being known fom pevious age deecion signal pocessing. In addiion, fo siuaions in which we do no know some paamees accuaely, we also deive he mismached CRB. Numeical invesigaions of he mean squaed eo of he maximum likelihood esimaion ae employed o suppo he validiy of he CRBs. In ode o demonsae he uiliy of he povided esuls o a opic of gea cuen inees, he numeical esuls focus on a passive ada sysem using he Global Sysem fo Mobile communicaion (GSM) cella sysem. Index Tems Disibued newoked ada, genealized Came-Rao bound (CRB), Global Sysem fo Mobile communicaion (GSM), MIMO ada, paamee esimaion, passive ada. I. Inoducion The focus of his pape is on new Came-Rao bounds (CRB) fo esimaion of age posiion velociy fom disibued ada newoks, someimes called MIMO ada sysems o mulisaic ada sysems [1], [2], [3], [4], [5], [6], [7], [8], opeaing unde moe geneal, pacically occuing, condiions han assumed in pevious wok. In paicula, he pesened esuls allow nonohogonal signals, spaially The wok of Q. He, J. Hu, Yonggang Wu was suppoed by he Naional Naue Science Foundaion of China unde Gans No. 61102142 61371184, he Inenaional Science Technology Coopeaion Exchange Reseach Pogam of Sichuan Povince unde Gan No. 2013HH0006, he Fundamenal Reseach Funds fo he Cenal Univesiies unde Gan No. ZYGX2013J015. The wok of R. S. Blum was suppoed by he Naional Science Foundaion unde Gan No. ECCS-1405579. Q. He, J. Hu, Yonggang Wu ae wih Univesiy of Eleconic Science Technology of China, Chengdu, Sichuan 611731 China (emails: qianhe@uesc.edu.cn, 1205858265@qq.com, wyguesc@163.com). R. S. Blum is wih Lehigh Univesiy, Behlehem, PA 18015 USA (email: blum@eecs.lehigh.edu). dependen Gaussian eflecion coefficiens, spaially dependen Gaussian clue-plus-noise, which ae cases of gea pacical inees. In fac, one could ague ha all of hese condiions ae ue in any eal sysem. The iniial CRBs we pesen ae applicable o boh acive passive ada sysems, povided he signals of oppouniy in he passive sysems ae assumed o be pefecly esimaed fom, fo example, he diec pah ecepion. These esuls fuhe assume all he paamees of he obsevaions model ae known, including he covaiance maices of he zeo-mean Gaussian noiseplus-clue eflecion coefficiens. The mismached CRB esuls given in his pape even allow cases whee he model assumed by he esimaion algoihm is incoec, including cases whee he model fo he diec pah signal may involve unmodeled noise, inefeence, o some ohe impefecion. Similaly, he eflecion coefficiens, noise, o inefeence may be incoecly modeled he mismached CRB will sill povide a lowe bound on pefomance. While boh passive acive ada sysems ae of gea inees, passive ada sysems may have aaced even geae aenion ove he pas few yeas due o he emendous advanages hey povide fom using exising communicaion signals o implemen a ada, essenially boowing he aleady exising ansmie infasucue poviding no eleconic evidence ha a ada is opeaing in a given aea. Passive ada, as he name implies, is a ada sysem which eceives only. Insead of acively ansmiing signals, i woks passively by gaheing signals fom non-coopeaive illuminaos of oppouniy eflecions fom objecs in he monioed aea o make decisions o povide infomaion abou ages. Since ansmies ae no equied in a passive ada, i has he advanages of low implemenaion coss, sealh, he abiliy o opeae in a wide fequency b wihou concens of causing inefeence o exising wieless sysems. Fo hese easons, passive ada sysems have aaced he aenion of he inenaional ada communiy. Passive ada sysems based on FM [9] digial illuminaos (DAB/DVB-T) [10], o saellie-bone illuminaos [11], WIFI [12] Global Sysem fo Mobile communicaion (GSM) [13] signals have been peviously invesigaed mainly fom pooypes o measuemens o vey simple analyical models. The facos ha affec he deecion pefomance of passive coheen locaion ada sysems ae discussed in [14]. The ambiguiy funcions of a se of off-ai measuemens of signals ha may be used fo passive coheen locaion (PCL) ada sysems ae pesened
analyzed in [15]. The poblem of age deecion in passive MIMO ada (PMR) newoks compised of non-coopeaive ansmies mulichannels is addessed in [16]. As descibed lae, he CRB is a lowe bound, in a ceain sense, on he covaiance maix of all unbiased esimaos. I is a useful ool fo evaluaing he bes possible esimaion pefomance of a ada sysem. A deivaion of he sochasic CRB is povided in [17]. The CRB expessions fo he esimaion of ange (ime delay), velociy (Dopple shif) diecion of a poin age using an acive ada o sona aay ae given in [18]. The CRB of DOA esimaion of a nonsaionay age fo a MIMO ada wih colocaed anennas fo a geneal ime division muliplexing (TDM) scheme is compued in [19]. The CRB fo bisaic ada channels is deived in [20], which also explois he elaionship beween he ambiguiy funcion he CRB. Came-Rao-like bounds fo he esimaion of a deeminisic paamee in he pesence of om nuisance paamees ae deived in [21], [22]. Fo he case of muliple ansmi eceive anennas employed in a disibued acive ada seing, [24] descibes he CRB unde he assumpion of ohogonal signals, spaially independen eflecion coefficiens, spaially independen clue-plus-noise. Fo esimaion of he posiion velociy of a single age using a passive ada, he CRB ambiguiy funcions ae consideed in [25] fo a muliple ansmie eceive ada, bu only fo he case whee a single ansmie eceive pai is seleced fom among a much lage se of possible pais. This wok does no conside he effec of signal nonohogonaliy o spaially dependen eflecion coefficiens o noise. Unde he same assumpions employed in [24], he CRB has been deived fo passive ada seings wih well esimaed signals of oppouniy in [26], [27]. Thus, none of he published wok has given he CRB fo he impoan pacical case of nonohogonal signals, spaially dependen eflecion coefficiens, spaially dependen noise fo join age posiion velociy esimaion pefomance using a disibued passive o acive ada newok employing all signals available fom he muliple ansmi eceive anenna pahs in an opimum manne. This esul is exemely useful since i descibes he bes achievable pefomance fo some impoan cases fo he fis ime. Knowing his bes achievable pefomance allows designes o compae he pefomance of hei developed appoaches, o hese bounds. If hei developed appoaches lead o pefomance close o he bounds, hese developed appoaches can be deemed good enough while hese developed appoaches ae ypically consained o have accepable complexiy. The vey ecen wok in [23] povides an excellen example whee hese esuls can be exemely useful. In [23], a vey pacical scenaio is consideed whee a numbe of ansmies of oppouniy send digial TV signals ha can no be accuaely modeled by assuming he ansmies send a se of nonohogonal signals. The wok in [23] pesens an ineesing subopimum algoihm fo implemening a ada employing hese nonohogonal signals. Howeve, i is no known how fa he pefomance of he suggesed appoach in [23] is fom he opimum achievable pefomance. Such infomaion would be exemely useful in judging if he appoach suggesed in [23] povides a good adeoff in ems of pefomance complexiy. Simila quesions aise in many elaed pacical applicaions, some of which involve acive adas. In his pape, we conside hese moe geneal cases deive a genealized CRB mismached CRB fo join locaion velociy esimaion in passive acive disibued ada newoks. The pesened esuls do no assume he appoximae locaion of he age is known fom pevious age deecion signal pocessing, unlike he pevious esuls employing opimum pocessing using all available anennas [24], [26], [27]. A closed-fom Fishe infomaion maix (FIM) is pesened. In a few epesenaive cases, he genealized o mismached CRB is numeically compaed wih he mean-squaed eo (MSE) fom maximum likelihood (ML) esimaion o show consisency a highe signal-o-noise aio (SNR). We use GSM signals as illuminaos fo ou numeical passive ada invesigaions. The es of his pape is oganized as follows. The signal model fo acive passive disibued ada newoks is pesened in Secion II. The ML esimae is analyzed in Secion II-A. In Secion III, he genealized CRB is deived. In Secion IV, we deive he mismahed CRB. Pefomance analysis numeical examples ae pesened in Secion V. Finally, conclusions ae dawn in Secion VI. Thoughou his pape, he noaion fo anspose is (.), while ha fo complex conjugae is (.) H. The symbol Diag{ } denoes a block diagonal maix wih he maices in he baces being he diagonal blocks, CN(µ, R) denoes a complex Gaussian disibuion wih mean veco µ covaiance maix R,E θ,α { } implies aking expecaion wih espec o he pobabiliy densiy funcion (pdf) p ( θ, α), T ( ) denoes he ace of a maix, epesens he Konecke poduc, R( ) means aking he eal pa, epesens he Hadamad poduc, vec( ) denoes he column vecoizing opeao which sacks he columns of a maix in a column veco. II. Signal Model Conside a disibued ada sysem wih M widely spaced single anenna ansmi saions N widely spaced single anenna eceive saions, locaed a (x m, y m), m = 1,..., M (x n, y n ), n = 1,..., N in a wo-dimensional Caesian coodinae sysem, especively. The lowpass equivalen imesampled vesion of he signal ansmied fom he mh ansmi saion a ime insan kt s is E m s m (k,α m ), whee T s is he sampling peiod, k (k=1,..., K) is an index unning ove he diffeen ime samples, α m denoes a veco of paamees needed o descibe he wavefom, he wavefom is nomalized using K k=1 s m (k,α m ) 2 T s = 1. Le E m denoe he enegy ansmied by he mh ansmi anenna. Then he eceived wavefom a he nh eceive a ime kt s is M E m P 0 n (k)= ζ d 2 nm s m (kt s τ nm,α m ) e j2π f nmkt s + w n (k), m=1 m d2 n (1)
wheeτ nm, f nm, ζ nm epesen he ime delay, Dopple shif, eflecion coefficien coesponding o he nmh pah, especively. The vaiable d m denoes he disance beween he age he mh ansmie, while d n denoes he disance beween he age he nh eceive. The em w n (k) denoes clue-plus-noise a he nh eceive a ime kt s. The eceived signal sengh a d m =d n =1 is E m P 0, so P 0 denoes he aio of eceived enegy a d m =d n =1 o ansmied enegy. The eflecion coefficienζ nm is assumed o be consan ove he obsevaion ineval o have a known complex Gaussian saisical model [28]. Assume he posiion (x, y) velociy (v x, v y ) of he age ae deeminisic unknowns. The disances d m d n ae expessed in ems of (x, y) as (x d m = m x )2 + ( y m y )2, (2) d n = (x n x )2 + ( y n y )2. (3) The ime delayτ nm is also a funcion of he unknown age posiion (x, y) (x m x )2 + ( y m y )2 (x + n x )2 + ( y n y ) 2 τ nm = c = d m+ d n, (4) c whee c denoes he speed of ligh, The Dopple shif f nm is a funcion of he unknown age posiion (x, y) velociy (v x, v y ) given by f nm = v ( x x m x ) ( + v y y m y ) + v ( x x n x ) ( + v y y n y ), λd n λd m whee λ denoes he wavelengh. Define an unknown paamee veco θ ha collecs he paamees o be esimaed (5) θ= [ x, y, v x, v y ]. (6) The obsevaions fom he nh eceive can be expessed as n = [ n (1), n (2),, n (K)] (7) = U n ζ n +w n, (8) whee U n is a K M maix ha collecs he ime delayed Dopple shifed signals a he nh eceive as whee u nm (k)= U n = [u n (1),u n (2),...,u n (K)], (9) u n (k)=[u n1 (k),u n2 (k),, u nm (k)], (10) E m P 0 s dm 2 m (kt s τ nm,α m )e j2π f nmkt s. (11) d2 n The M 1 eflecion coefficien veco ζ n can be expessed as ζ n = [ ζ n1,,ζ nm ]. Denoe he veco of noise samples a he nh eceive as w n = [w n (1),, w n (K)]. The obsevaions fom he se of all eceives can be wien as ] = [ 1, 2,, N =Sζ+w, (12) whee S collecs he ime delayed Dopple shifed signals fom all pahs S=Diag{U 1,U 2,...,U N }. (13) The ζ in (12) collecs eflecion coefficiens fo all pahs ζ= [ ζ 1,,ζ N], (14) i is assumed ha ζ is a complex Gaussian disibued veco wih zeo mean covaiance maix R=E{ζζ H }, i.e. ζ CN (0,R). The w in (12) denoes he clue-plusnoise veco w= [ w 1,,w N], (15) which is assumed o be complex Gaussian disibued wih zeo mean covaiance maix Q=E{ww H }, i.e., w CN (0, Q). Assume ha he noise veco w is independen fom he eflecion coefficien veco ζ. A. Maximum Likelihood Esimaion In his he nex secion (Secions II III), we assume S ( hus α), Q, R ae known o he esimaion algoihm. We addess ohe cases lae. Using he signal model in (12) he fac ha he linea combinaion of wo Gaussian vecos is also Gaussian, he likelihood funcion condiioned on he wavefom paamee veco can be obained as p ( θ,α)= whee C denoes he covaiance maix α=[α 1,...,α M ], (16) 1 π KN de(c) exp( H C 1 ), (17) C=E { (Sζ+w) (Sζ+w) H} =E { Sζζ H S H +ww H} = SRS H +Q. (18) The log-likelihood funcion can be wien as L ( θ,α)=ln p ( θ,α) = H C 1 ln (de (C)) KN ln (π). (19) Neglecing he las consan em of he second line in (19) assuming known o pefecly esimaed α, he (ML) esimae of he unknown paamee veco θ can be calculaed as ˆθ ML = ag max L( θ,α) θ { = ag max H C 1 ln (de (C)) }. (20) θ
III. Genealized Came-Rao Bound In his secion, we povide he CRB fo joinly esimaing he age locaion (x, y) velociy ( v x, v y ) fo he case whee S ( hus α), Q, R ae known o he esimaion algoihm. The fis sep in obaining he CRB is o compue he FIM, which is a 4 4 maix elaed o he second ode deivaives of he log-likelihood funcion J (θ α)=e θ,α { θ L ( θ,α) [ θ L ( θ,α)] }. (21) Consideing he likelihood is a funcion ofτ nm, f nm, d m, d n (n = 1,, N, m = 1,, M), which depend on θ = [ x, y, vx, v y ], we define an inemediae paamee veco ϑ=[τ,f,d,d ] (22) =[τ 11,τ 12,,τ NM, f 11, f 12,,f NM, d 1, d 2,, d M, d 1, d 2, d N ] whee τ = [τ 11,τ 12,,τ NM ], f = [ ] f 11, f 12,,f NM, d = [d 1, d 2,, d M ] d = [d 1, d 2,, d N ] collec he unknown ime delays, Dopple shifs, disance paamees, especively. Accoding o he chain ule, he FIM can be deived by J (θ α)= ( θ ϑ ) J (ϑ α) ( θ ϑ ), (23) whee J (ϑ α)=e ϑ,α { ϑ L ( ϑ,α) [ ϑ L ( ϑ,α)] }. A. Calculaion of θ ϑ Recalling (6) (22), we have [ F G θ ϑ D D = 0 H 0 0 whee F= G= H= D = τ 11 τ 11 f 11 f 11 τ 12 τ 12 f 12 f 12 f 11 f 12 v x v x f 11 f 12 v y v y d 1 d 1 d 2 d 2 τ NM τ NM f NM f NM f NM v x f NM v y d M d M, ], (24), (25), (26), (27), (28) d 1 d 2 d D = N d 1 d 2 d N. (29) Using calculaions dawing on (2)-(5), he elemens of he maices in (25)-(29) will be descibed as a nm = τ nm = 1 ( x x m + x ) x n, (30) c d m d n b nm = τ nm = 1 ( y y ) m + y y n, (31) c d m d n e nm = f ( nm = v x 1 + 1 ) λ d m d ( n x + m x ) [ ( vx x λ(d m ) 3 m x ) ( + v y y m y )] ( x + n x ) [ ( vx x λ(d n ) 3 n x ) ( + v y y n y ) ], (32) g nm = f ( nm = v y 1 + 1 ) λ d m d ( n y + m y ) [ ( vx x λ(d m ) 3 m x ) ( + v y y m y )] ( y + n y ) [ ( vx x λ(d n ) 3 n x ) ( + v y y n y ) ], (33) β nm = f nm v x κ nm = f nm v y υ m = d m l m = d m = x m x λd m = y m y λd m + x n x λd n, (34) + y n y λd n, (35) = x x m d m, (36) = y y m d m, (37) η n = d n = x x n d n, (38) ψ n = d n = y y n d n. (39) Noe ha a nm, b nm, e nm, g nm,β nm,κ nm,υ m, l m,η n ψ n ae deemined by he age posiion velociy, as well as he posiion of he eceives ansmies. B. Calculaion of J(ϑ α) Accoding o he likelihood funcion in (19), he i jh elemen of he FIM fo he paamee veco ϑ is given by [29] ( [J(ϑ α)] i j = T C 1 C C 1 C ). (40) ϑ i ϑ j Using he following ideniies, [30] T (ABXY )= ( vec ( Y )) ( X A ) vec (B) (41) T (AB)=T (BA), (42) we can ewie (40) as ( C [J (ϑ α)] i j =T C 1 C ) C 1 ϑ i ϑ j ( ) H Cvec ( = C C 1)( ) C vec, (43) ϑ i ϑ j
whee C vec = vec (C). Calculaion of he deivaives fuhe simplificaion of (43) ae povided in Appendix A. Then we can ge he final equaion A 11 A 12 A 13 A 14 N M N M A J (θ α)= 21 A 22 A 23 A 24, (44) p=1 q=1 n=1 m=1 A 31 A 32 A 33 A 34 A 41 A 42 A 43 A 44 whee A 11 = a pq (a nm (J ττ ) c,d + e nm (J fτ ) c,d + υ m (J d τ ) m,d /N+η n (J d τ ) n,d /M) + e pq (a nm (J τf ) c,d + e nm (J ff ) c,d + υ m (J d f) m,d /N+η n (J d f ) n,d /M) (45) +υ q /N(a nm (J τd ) c,q + e nm (J fd ) c,q + υ m (J d d ) m,q /N+η n (J d d ) n,q /M) +η p /M(a nm (J τd ) c,p + e nm (J fd ) c,p + υ m (J d d ) m,p /N+η n (J d d ) n,p /M), A 12 = A 21 = b pq (a nm (J ττ ) c,d + e nm (J fτ ) c,d + υ m (J d τ ) m,d /N+η n (J d τ ) n,d /M) + g pq (a nm (J τf ) c,d + e nm (J ff ) c,d + υ m (J d f) m,d /N+η n (J d f) n,d /M) + l q /N(a nm (J τd ) c,q + e nm (J fd ) c,q + υ m (J d d ) m,q /N+η n (J d d ) n,q /M) +ψ p /M(a nm (J τd ) c,p + e nm (J fd ) c,p + υ m (J d d ) m,p /N+η n (J d d ) n,p /M), A 13 = A 31 =β pq [a nm (J τf ) c,d + e nm (J ff ) c,d + υ m (J d f) m,d /N+η n (J d f) n,d /M], A 14 = A 41 = k pq [a nm (J τf ) c,d + e nm (J ff ) c,d + υ m (J d f) m,d /N+η n (J d f) n,d /M], A 22 = b pq (b nm (J ττ ) c,d + g nm (J fτ ) c,d + l m (J d τ ) m,d /N+ψ n (J d τ ) n,d /M) + g pq (b nm (J τf ) c,d + g nm (J ff ) c,d + l m (J d f) m,d /N+ψ n (J d f ) n,d /M) + l q /N(b nm (J τd ) c,q + g nm (J fd ) c,q + l m (J d d ) m,q /N+ψ n (J d d ) n,q /M) +ψ p /M(b nm (J τd ) c,p + g nm (J fd ) c,p + l m (J d d ) m,p /N+ψ n (J d d ) n,p /M) A 23 = A 32 =β pq [b nm (J τf ) c,d + g nm (J ff ) c,d + l m (J d f ) m,d /N+ψ n (J d f ) n,d /M], A 24 = A 42 = k pq [b nm (J τf ) c,d + g nm (J ff ) c,d + l m (J d f ) m,d /N+ψ n (J d f ) n,d /M], (46) (47) (48) (49) (50) (51) A 33 =β pq β nm (J ff ) c,d, (52) A 34 = A 43 = k pq β nm (J ff ) c,d, (53) A 44 = k pq k nm (J ff ) c,d, (54) whee c= M(n 1)+m d=m(p 1)+q. J ττ, J τf, J fτ, J τd, J d τ, J τd, J d τ, J ff, J fd, J d f, J fd, J d f, J d d, J d d, J d d, J d d ae defined in (72). I should be noed ha, he esuls obained hee, say (44)-(54), ae a highly nonival exension of he pevious esuls in [24]. Unfounaely, hey ae, as one migh expec, consideably moe complicaed bu hey descibe he bes possible esimaion pefomance in non-ideal scenaios ha ae of gea pacical inees in he following sense. Given any unbiased esimao ˆθ of an unknown paameeθ based on an obsevaion veco, when α is assumed known fixed, we have [29] MSE=E θ,α { (ˆθ θ)(ˆθ θ) } CRB(θ α)=j 1 (θ α). (55) which is he sad CRB fo veco paamees whee A B meansa B is posiive semidefinie, MSE is he mean squaed eo maix of he unbiased esimao. IV. Came-Rao Bound Fo Mismached Case In ode o find an ML esimae o use he CRB esul in (55), now called he genealized CRB (GCRB), we mus know he acual values of he signal maix S ( hus α) fom (13), he eflecion coefficiens covaiance maix R descibed nea (14), he noise covaiance maix Q descibed nea (15). Hee, we assume he esimaion algoihm employs incoec values fo hese maices denoed by S 0, R 0, Q 0 especively. The incoec values S 0, R 0, Q 0 migh be obained fom some inaccuae esimaion. Given he esimaion algoihm uses hese incoec values S 0, R 0, Q 0, we find a lowe bound on he esimaion pefomance using some ecenly published wok [31]. In he case descibed, he assumed likelihood funcion is 1 p 0 ( θ,α)= π KN exp( H C 1 0 ), (56) dec 0 whee C 0 = S 0 R 0 S H 0 + Q 0. To avoid confusion wih he GCRB, we denoe he acual values of he signal maix fom (13), he eflecion coefficiens covaiance maix descibed nea (14), noise covaiance maix descibed nea (15) by S 1, R 1, Q 1. Thus, he acual likelihood funcion is 1 p 1 ( θ,α)= π KN exp( H C 1 1 ) (57) dec 1 whee C 1 = S 1 R 1 S 1 H +Q 1. Accoding o [31], we know ha MSE mis CRB mis (θ α)=j mis 1 (θ α). (58) whee MSE mis, CRB mis (θ α) J mis (θ α) denoe he MSE, CRB FIM maices unde mismached siuaion, {( p0 ( θ,α) ) 2 J mis (θ α)=e p1 ( θ,α) p 1 ( θ,α) } θ log p 0 ( θ,α)[ θ log p 0 ( θ,α)] (59) Nex noe ha θ log p( θ,α)=( θ ϑ ) ϑ log p( ϑ,α) p 0 ( θ,α)= p 0 ( ϑ,α) so ha J mis (θ α)=( θ ϑ )J mis (ϑ α)( θ ϑ ), (60)
whee {( p0 ( ϑ,α) ) 2 J mis (ϑ α)=e p1 ( ϑ,α) p 1 ( ϑ,α) } ϑ log p 0 ( ϑ,α)[ ϑ log p 0 ( ϑ,α)] (61) Calculaion of he deivaives fuhe simplificaion of (61) is omied due o similaiy o he case wihou mismach. V. Numeical Examples In his secion, examples ae pesened which demonsae he use of he GCRB he mismached CRB pesened in he pevious secion o bound he pefomance of disibued ada newoks which employ muliple widely spaced ansmies eceives o joinly esimae age posiion velociy. Fo beviy, we focus on examples which employ signals ha ae moe applicable fo passive ada. Iniially, we descibe pefomance when he ansmied signals ae eihe known o whee he ansmied signals of oppouniy ae esimaed pefecly fom a diec pah ecepion. Lae we conside cases whee his is no ue. We also assume ha he posiions of he ansmies eceives ae exacly known. Fo passive ada cases, hese assumpions allow us o descibe he bes possible pefomance ha can be obained unde he bes cicumsances. I is easy o employ ou bounds fo cases whee all paamees o be included in he veco α ae known hus he bound in (55) is applicable. Howeve, he veco α migh include a om bi sequence which conains infomaion being ansmied. In ode o avoid pesening a CRB fo evey possible bi sequence (α), we quoe he expeced CRB aveaged ove all bi sequences (ECRBOB), assuming each bi sequence o be pefecly esimaed. Fom (55), ECRBOB(θ)=E α {CRB(θ α)} (62) clealy bounds he coesponding covaiance maix aveaged ove all bi sequences. Fo he bes case, when he bi sequence in α is pefecly esimaed, he ECRBOB is a good indicao of pefomance. Fo example, i descibes how he sysem paamees, such as he numbe of anennas, he geomey, he wavefoms impac pefomance, assuming accuae esimaion of α. One can use he ECRBOB o opimize any paamee of inees. Conside a age moving wih velociy (50, 30) m/s is pesen a (15.15, 10.1275) km. To define a geneal es se up ha is easy o descibe fo geneal M N, each ansmi eceive (single anenna) saion is locaed 7 km fom he efeence poin (15,10) km. The M ansmi saions ae unifomly disibued in angle ove he ange [0, 2π), i.e, he angle of he m-h ansmie is ϕ m = 2π(m 1)/M, m = 1,, M. The N eceive saions ae also unifomly disibued in angle ove he ange [0, 2π), i.e. he angle of he n-h eceive saion is ϕ n = 2π(n 1)/N, n = 1,, N, whee he angles ae measued wih espec o he hoizonal axis oiginaed a he efeence poin as illusaed in Figue 1. Suppose E 1 = E 2 =... = E M = E. Fix S CNR = 10 log(( N Mm=1 n=1 σ 2 nm EP 0/dm 2 d2 n )/(Nσ2 w ), called he signalo-clue-plus-noise aio (SCNR), wheeσ 2 nm =E{ζ nmζ H nm } 1.6 1.4 1.2 1 0.8 0.6 0.4 x 10 4 180 o 90 o efeence poin (15000,10000) 270 o Receive saion Tansmi saion Tage 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Fig. 1: Paamee se up fo a disibued ada newok wih M= 2 N= 3. σ 2 w=e{w n (1)w n (1) H }==E{w n (K)w n (K) H }. We se σ 2 nm = 1 fo all n m, P 0= 1. To be elevan o a passive ada sysem, he signals consideed ae hose employed by he popula Global Sysem fo Mobile (GSM) Communicaions sysem. The baseb ansmied wavefoms ae Gaussian minimum shif keying (GMSK) signals [13] Nc k s m (k,α m )=A m exp j c mi z(kt s it p )T s e j2πm f kt s, i=1 whee α m = [ c m1,...,c mnc ], z()= π 2T p j=1 0 o x 10 4 (63) { [ ] [ ]} 2πB 2πB ϑ ( T P ) ϑ, (64) ln 2 ln 2 ϑ[]= ( 1/ 2π ) e τ2 /2 dτ, T p is he bi duaion, B denoes he 3 db bwidh of he Gaussian pefile used in he GMSK modulaos, c mi { 1, 1} is he ih (i = 1,..., N c ) binay daa bi of he mh ansmied wavefom, N c denoes he numbe of bis conained in he obsevaion ineval, A m is he nomalizaion faco, f=f k+1 f k is he fequency offse beween diffeen signals of oppouniy wih neighboing fequencies. In he simulaions, we geneae c mi = 1 o 1 omly wih he same pobabiliy of 0.5. To model a GSM sysem, assume T p = 577µs, BT p = 0.3, N c = 16, he caie fequency f c = 900 MHz f= 3 KHz (ohogonal signals) o f = 300 Hz (nonohogonal signals). I should be noiced ha he bwidh is only 520Hz N c = 16 in he simulaion because of he huge calculaion complexiy. Figue 2 shows he cases wih M= 2, N= 3 M=5, N= 4 fo spaially independen efecion coefficiens, spaially independen noise, nonohogonal signals. The solid dashed cuves show he oo ECRBOB (RECRBOB) he oo mean squaed eo (RMSE) of he ML esimaion, especively, in he cases invesigaed. I is seen ha all cuves show ha he RMSE deceases as he signal-o-clue-plusnoise aio (SCNR) is inceased. In suppo of he coecness of ou deived CRBs, all RMSE cuves show he exisence of a
RMSE 10 8 10 6 10 4 10 2 RMSE fo M=5,N=4 RMSE fo M=2,N=3 RMSE fo x RMSE fo y RMSE fo vx RMSE fo vy RECRBOB fo x RECRBOB fo y RECRBOB fo vx RECRBOB fo vy RECRBOB 10 4 10 3 10 2 10 1 10 0 N c =16 RECRBOB fo x RECRBOB fo y RECRBOB fo vx RECRBOB fo vy 10 1 10 0 RECRBOB fo M=5,N=4 RECRBOB fo M=2,N=3 10 2 5 0 5 10 15 20 25 30 35 40 45 SCNR (db) Fig. 2: RMSE, RECRBOB vesus SCNR fo a passive disibued ada newok wih M = 2, N = 3 M = 5, N = 4, spaially independen efecion coefficiens, spaially independen noise, nonohogonal signal. 10 2 N c =64 10 3 5 0 5 10 15 20 25 30 35 40 45 SCNR (db) Fig. 3: RECRBOB vesus SCNR fo a passive disibued ada newok wih M=2, N= 3, spaially independen efecion coefficiens, spaially independen noise, ohogonal signals. heshold, above which he RMSE sas o become close o he RECRBOB in value slope. We can see ha he heshold in he case wih M= 5, N= 4 is 15 db, while he heshold in he case wih M= 2, N= 3 is 20 db. The MSE cuves fo M= 5, N= 4 have a significanly lowe heshold han hose fo M= 2, N= 3, appaenly due o he addiional ansmi eceive saions, while he educion in RECRBOB he RMSE above heshold due o employing M= 5, N= 4 insead of M= 2, N= 3 is significanly smalle. Also, in he high SCNR egion, RMSE is close o RECRBOB fo he case wih M= 5, N= 4 han fo he case wih M= 2, N= 3. Inceasing he ime duaion of he signals, N c in (63) also povides benefis as one would expec. While Figue 2 consides he case of N c = 16, Figue 3 shows a compaison beween RECRBOB fo N c = 16 RECRBOB fo N c = 64 fo he same case of M = 2, N = 3, spaially independen efecion coefficiens, spaially independen noise, ohogonal signals. Hee we can see ha if we incease N c, he RECRBOB will be significan educed. In he following cases, o educe complexiy, we employ N c = 16. 1) Ohogonal Signals Nonohogonal Signals: In his secion, we focus on he effec of he nonohogonaliy of he diffeen ansmied signals. We conside he siuaion of spaially independen eflecion coefficiens, spaially independen noise, nonohogonal signals. The ohe facos ae he same as in Figue 2. The sysem consideed in Figue 4 has M=2ansmies N = 3 eceives. The ed blue cuves in his figue coespond o he cases wih ohogonal nonohogonal signals, especively. We see ha he heshold obained using he ohogonal signals is 15 db while he heshold obained using he nonohogonal signals is 20 db. Thus he heshold fo he nonohogonal signals esed is highe han he heshold fo he ohogonal signals esed. I is also seen ha he RECRBOB of he ohogonal signals esed is smalle han ha fo he nonohogonal signals esed ove he whole egion of SCNR shown. So boh he RMSE RECRBOB indicae ha he ada can achieve bee RMSE 10 8 10 6 10 4 10 2 10 0 RECRBOB fo ohogonal RMSE fo ohogonal RECRBOB fo nonohogonal RMSE fo nonohogonal RMSE fo x RMSE fo y RMSE fo vx RMSE fo vy RECRBOB fo x RECRBOB fo y RECRBOB fo vx RECRBOB fo vy 10 2 5 0 5 10 15 20 25 30 35 40 45 SCNR (db) Fig. 4: RMSE, RECRBOB vesus SCNR fo a passive MIMO ada wih M=2 N= 3, spaially independen efecion coefficiens, spaially independen noise. pefomance if he wavefoms ae close o being ohogonal in he case consideed. 2) Spaially Dependen Reflecion Coefficiens: In his secion, we conside he siuaion of spaially dependen eflecion coefficiens, spaially independen noise, ohogonal signals. The elemens of he covaiance maix R descibing he coelaion beween he diffeen eflecion coefficiens ae geneaed wih [24] whee R = R=R R (65) ρ 11. ρ N1 ρ M1 ρ 1N.... ρ NN, (66) ρ nn = exp( φ ) nn, (67) ρ 11 ρ 1M R =....., ρ MM (68)
10 8 10 6 ϖ= ϖ=0.1 ϖ=0.0110 4 ϖ=0.1 ϖ= ϖ=0.01 RMSE fo x RMSE fo y RMSE fo vx RMSE fo vy RECRBOB fo x RECRBOB fo y RECRBOB fo vx RECRBOB fo vy 10 6 10 5 10 4 Case 1 Case 2 10 3 Case 3 Case 3 Case 2 Case 1 RMSE fo x RMSE fo y RMSE fo vx RMSE fo vy RECRBOB fo x RECRBOB fo y RECRBOB fo vx RECRBOB fo vy RMSE 10 2 ϖ= ϖ=0.1 ϖ=0.01 10 0 ϖ= ϖ=0.1 ϖ=0.01 RMSE 10 2 Case 1 10 1 Case 2 Case 3 10 0 Case 3 Case 2 Case 1 10 1 10 2 5 0 5 10 15 20 25 30 35 40 45 SCNR (db) Fig. 5: RMSE, RECRBOB vesus SCNR fo a passive MIMO ada wih M= 2 N= 3, spaially dependen eflecion coefficiens, spaially independen noise, ohogonal signals. ρ mm = exp( φ mm ). (69) The symbol φ nn denoes he sepaaion angle beween he nh n h ansmie-o-age pahs, φ mm denoes he sepaaion angle beween he mh m h age-o-eceive pahs, ses he exponenial decay in coelaion wih angle. Fom he model, i is easy o see ha lage implies less dependency fo fixed φ nn φ mm. We conside =0.01, 0.1, in he figues. Hee = implies ha he eflecion coefficiens ae independen. Figue 5 shows he compaison of RECRBOB RMSE fo diffeen when all of he ohe paamees ae he same as in Figue 2. We can see ha he hesholds fo he cases wih =, 0.1, 0.01 ae 15 db, 20 db 25 db, especively. Thus, less dependency leads o a moe favoable heshold such ha he RECRBOB is achievable a lowe SCNR. Above heshold, all he cuves ae elaively close. The esuls imply ha he dependency of he eflecion coefficiens does no have emendous impac on he ada esimaion pefomance, povided we opeae above heshold. Howeve, wih less dependency he ada can opeae a lowe SCNR while sill achieving an accepable pefomance level. 3) Gaussian Spaially Dependen Noise: In his secion, we conside he siuaion of spaially independen eflecion coefficiens, spaially dependen noise, ohogonal signals. The elemens of he noise covaiance maix Q ae geneaed wih he following model Q=σ 2 w Q I K, (70) whee he nn h elemen of Q is assumed o be Q nn = exp{ d nn γ} (71) (x whee d nn = n ) 2+ ( ) x n y 2,γses n y n he exponenial decay in coelaion wih disance, I K denoes a K K ideniy maix. Fom he model we can see lageγesuls in less dependency fo fixed d nn. We conside he siuaions 10 2 5 0 5 10 15 20 25 30 35 40 45 SCNR (db) Fig. 6: RMSE, RECRBOB vesus SCNR fo a passive MIMO ada wih M= 2 N= 3, spaially independen eflecion coefficiens, dependen noise, ohogonal signals. ofγ=0.000005, 0.00001, assume all he ohe paamees ae he same as in Figue 2. Heeγ= implies ha he noise componens ae independen. Figue 6 shows he compaison of he RECRBOB RMSE fo diffeen values ofγ. Case 1, Case 2 Case 3 especively epesens γ =, 0.00001, 0.000005. I is obseved ha he hesholds fo cases wihγ=, 0.00001, 0.000005 ae 15 db, 10 db, 5 db, especively. Thus, moe dependency leads o a moe favoable heshold such ha he RECRBOB is achievable a lowe SCNR. Above he heshold, we see ha γ = 0.000005 has he smalles RECRBOB whileγ= has he lages RECRBOB, which means lage dependency can lead o lowe RECRBOB. In he cases consideed in Figue 6, coelaed noise leads o bee pefomance. 4) Inaccuae Signal Esimaion: Now conside he case whee he ansmied signals ae no esimaed pefecly, possibly fom he diec pah ecepions. Le n nm (k), n = 1,, N, m=1,, M, k=1,, K denoe an independen idenically disibued sequence of complex Gaussian noise samples, each wih zeo mean vaiance 0.1 which models he esimaion eo in he signal using E m P 0 u nm (k)= [s dm 2 m (kt s τ nm,α m )+n nm (k)]e j2π f nmkt s (72) d2 n Then (72) is used o fom S wih he equaions (9), (10), (13) aleady given in he pape, we call his mismached S S 0. The undisoed S obained his way, bu wihou addiive noise, is called S 1. This is exacly a case whee he model we employ in ou esimaion algoihm is mismached so he RECRBOB mis esuls fom Secion IV become applicable. The esuling aveage RECRBOB mis RMSE mis, afe aveaging ove he noise using a Mone Calo simulaion, ae ploed in Figue 7. Fom he figue, we can see ha he RECRBOB mis povides an infomaive lowe bound 1 on RMSE mis in his case. 1 We have veified ha he unaveaged values of RECRBOB mis also povide a lowe bound o he unaveaged values of RMSE mis.
RMSEmis 10 6 10 5 10 4 10 3 10 2 10 1 10 0 RMSEmis fo x RMSEmis fo y RMSEmis fo vx RMSEmis fo vy RECRBOBmis fo x RECRBOBmis fo y RECRBOBmis fo vx RECRBOBmis fo vy 10 1 5 0 5 10 15 20 SCNR (db) Fig. 7: RMSEmis, RECRBOBmis vesus SCNR fo a passive MIMO ada wih M=2 N= 3, spaially independen eflecion coefficiens, independen noise, ohogonal signals. Noe ha all he ohe deails of he sysem analyzed in Figue 7, excep fo his signal mismach, ae he same as in Figue 2. VI. Conclusions In his pape, we sudied he pefomance of join age posiion velociy esimaion using a disibued ada newok unde moe geneal condiions han assumed in pevious wok. A eceived signal model has been developed fo acive passive ada wih M ansmi N eceive saions. The ML esimae he exac CRB expession ae deived fo possibly nonohogonal signals, spaially dependen Gaussian eflecion coefficiens, spaially dependen Gaussian clue-plus-noise. Fo cases in which some paamees (fo example he ansmied signal fom diec pah ecepion) ae no esimaed coecly, we also deive he mismached CRB. Numeical esuls ae given o illusae he use of he CRB mismached CRB. The numeical esuls show vaious cases wih signals of oppouniy aken fom a GSM wieless communicaion sysem. I was shown ha in he paicula cases invesigaed, he nonohogonaliy of signal degaded he esimaion pefomance boh in ems of RECRBOB in ems of he heshold above which he RMSE sas o become close o he RECRBOB in value slope. Deceasing he dependency beween he diffeen eflecion coefficiens led o a moe favoable heshold such ha he ada can opeae a lowe SCNR while sill achieving an accepable pefomance level. Above heshold he dependency of he eflecion coefficiens had lile impac on he esimaion pefomance, povided a well pefoming esimaion appoach (nealy CRB achieving) is employed. In some specific examples, i was also shown ha an incease in he dependency beween he noise samples a diffeen anennas led o bee esimaion pefomance in ems of he heshold RECRBOB. The wok hee can be genealized in seveal diecions. The CRB, a igh bound only fo he high SCNR egion being limied o unbiased esimaos, is incapable of chaaceizing he heshold value o accuaely descibing he low-scnr esimaion pefomance of esimaos. In his egad, we need bee analyical ools which can pedic esimaion pefomance fo low SCNR. The Ziv-Zakai bound is one pomising appoach, which will be sudied in ou fuue wok. The Ziv-Zakai bound also allows pio infomaion o be incopoaed ino he esimaion. Appendix A Calculaion of J(ϑ α) Accoding o (23) (43), we can obain he FIM of he veco ϑ as J ττ J τf J τd J τd J J (ϑ α)= fτ J ff J fd J fd, (73) J d τ J d f J d d J d d J d τ J d f J d d J d d whee J ττ = Jτ HJ τ, J τf = J H fτ = J τ HJ f, J τd = J H d τ = Jτ H J d, J τd = Jd H τ = J τ H J d, J ff = Jf HJ f, J fd = J H d f = J H f J d, J fd = J H d f = J H f J, J d d d = J H d J d, J d d = J H d d = J H d J d, J d d = J H d J d, J τ = ( C /2 C 1/2) C vec τ, (74) J f = ( C /2 C 1/2) C vec f, (75) J d = ( C /2 C 1/2) C vec d, (76) J d = ( C /2 C 1/2) C vec d. (77) Then we efomulae he J (ϑ α) in a somewha moe explici maix fom. Fis we deive J ττ, le s i z i denoe he ih column of S R, especively, such ha S= [s 1,,s MN ] R=[z 1,,z MN ]. Noe ha R is a Hemiian maix, i.e., R=[z 1,,z MN ] H. Then, we have C = τ nm ( SRS H +Q ) τ nm = S RS H +SR SH τ nm τ nm =s τ i zh i S H +Sz i (s τ i )H, (78) whee i= M(n 1)+m fo n=1,, N m=1,, M, s τ i= s [ i unm (1) = e n,, u ] nm (K), (79) τ nm τ nm τ nm whee e n is an N 1 column veco wih zeo eveywhee excep fo a 1 in he nh eny u nm (k) E m P 0 s m (kt s τ nm,α m ) = e j2π f nmkt s. (80) τ nm dm 2 d2 n τ nm
Accoding o he following ideniy [30] ( X A ) vec (B)=vec (ABX), (81) we can obain J τnm = ( C /2 C 1/2) vec (C) τ ( nm =vec C 1/2 C ) C 1/2 τ nm = vec { C 1/2( s τ i zh i S H +Sz i (s τ i )H) C 1/2} =vec ( ) V i +Vi H, (82) wheev i = C 1/2 s τ i zh i S H C 1/2. Using (82) he following ideniy [30] vec ( A ) vec (B)=T (AB)=T (BA), (83) we can deive he i jh elemen of J ττ as follows [J ττ ] i j = vec { } Hvec { } V i +Vi H V j +V j H = T {( )( )} V i +V H i V j +V H j = 2R{T (V i V j + V H i V j )} = 2R { z H i S H C 1 s τ j zh j SH C 1 s τ i+ (s τ i )H C 1 s τ j zh j SH C 1 Sz i }, (84) whee j= M(n 1)+m fo n = 1,, N m = 1,, M. Then, accoding o (84), we can efomulae J ττ in he fom of a maix J ττ = 2R { Y S τ (Y S τ ) + (S τ ) H C 1 S τ (Y SR) }, (85) whee S τ = [ s τ 1,,sτ MN] Y = RS H C 1. Similaly, we can obain J τf = 2R { Y S f (Y S τ ) + (S τ ) H C 1 S f (Y SR) }, (86) J ff = 2R {Y S f ( Y S f) + ( S f ) H C 1 S f (Y SR) }, (87) whee S f = [ s f 1 MN],,sf, s f i = s [ i unm (1) = e n,, u ] nm (K), (88) f nm f nm f nm u nm (k) = j2πkt s u nm (k). (89) f nm Nex we deive J d, C = (SRSH +Q) d m d m = S RS H +SR SH d m d m = (s m zh m +s m+m zh m+m ++s m+(n 1)M zh m+(n 1)M )SH +S(z m (s m )H +z m+m (s m+m )H ++z m+(n 1)M (s m+(n 1)M )H ), (90) whee s m+(n 1)M = s m+(n 1)M d m = e n [ u nm(1), u nm(2) d m d m, u nm(k) ], n=1,, N d m (91) u nm (k) Em P 0 = d m dm 2 d s(kt s τ nm,α m )e j2π f nmkt s. (92) n I can be deived ha J dm =(C /2 C 1/2 ) C vec d m =vec(c 1/2 C d m C 1/2 ) =vec{c 1/2 ((s m zh m +s m+m zh m+m + +s m+(n 1)M zh m+(n 1)M )SH +S(z m (s m )H +z m+m (s m+m )H + +z m+(n 1)M (s m+(n 1)M )H ))C 1/2 } =vec(l m +l H m), (93) whee l m = C 1/2 (s m zh m + s m+m zh m+m + + s m+(n 1)M zh m+(n 1)M )SH C 1/2. Then, we obain [J d d ] mm = vec(l m +l H m )H vec(l m +l H m ) = T{(l m +lm H )(l m +lh m )} = 2R{T(l m l m +lm H l m )} N N = 2R{ ((z m+(n 1)M ) H S H C 1 s m +(n 1)M (z m +(n 1)M) H S H n=1 n =1 C 1 s m+(n 1)M + (s m+(n 1)M )H C 1 s m +(n 1)M (z m +(n 1)M) H S H C 1 Sz m+(n 1)M )}. (94) Refomulae J d d in he fom of a maix N N J d d = 2R{ (ℵ n S H C 1 I n (ℵ n S H C 1 I n ) n=1 n =1 + (I n ) H C 1 I n (ℵ n S H C 1 S(ℵ n ) H ) )}, (95) whee ℵ n = (z 1+(n 1)M,,z M+(n 1)M ) H, I n = (s 1+(n 1)M,,s M+(n 1)M ). Similaly, we can deive N J d τ= 2R{ (ℵ n S H C 1 S τ (RS H C 1 I n ) n=1 + (I n ) H C 1 S τ (RS H C 1 S(ℵ n ) H ) )}, (96) N J d f= 2R{ (ℵ n S H C 1 S f (RS H C 1 I n ) n=1 + (I n ) H C 1 S f (RS H C 1 S(ℵ n ) H ) )}. (97)
To deive d n, we employ C = (SRSH +Q) d n d n = S RS H +SR SH d n d n = (s 1+(n 1)M zh 1+(n 1)M +s 2+(n 1)M zh 1+(n 1)M + +s M+(n 1)M zh M+(n 1)M )SH +S(z 1+(n 1)M (s 1+(n 1)M )H +z 2+(n 1)M (s 2+(n 1)M )H ++z M+(n 1)M (s M+(n 1)M )H ), (98) whee λ m = (z m,,z m+(n 1)M ) H, m = (s m,, s m+(n 1)M ). Similaly, we can obain M J d τ= 2R{ ( λ m S H C 1 S τ (RS H C 1 m ) m=1 + ( m ) H C 1 S τ (RS H C 1 S( λ m ) H ) )} (104) M J d f= 2R{ ( λ m S H C 1 S f (RS H C 1 m ) m=1 + ( m ) H C 1 S f (RS H C 1 S( λ m ) H ) )}, (105) whee s m+(n 1)M = s m+(n 1)M d n = e n [ u nm(1), u nm(2), u nm(k) ], m=1,, M, d n d n d n (99) u nm (k) Em P 0 = s(kt d n d m dn 2 s τ nm,α m )e j2π f nmkt s, (100) We can hen deive J dn = (C /2 C 1/2 ) C vec d n = vec(c 1/2 C C 1/2 ) d n = vec{c 1/2 ((s 1+(n 1)M zh 1+(n 1)M +s 2+(n 1)M zh 2+(n 1)M + +s M+(n 1)M zh M+(n 1)M )SH +S((s 1+(n 1)M )H z 1+(n 1)M + (s 2+(n 1)M )H z 2+(n 1)M + + (s M+(n 1)M )H z M+(n 1)M ))C 1/2 } = vec(w n + w H n ), (101) whee w n = C 1/2 (s 1+(n 1)M zh 1+(n 1)M + s 2+(n 1)M zh 2+(n 1)M + +s M+(n 1)M zh M+(n 1)M )SH C 1/2. Then, we can obain [J d d ] nn = vec(w n + w H n )H vec(w n +l H n ) = T{(w n + wn H )(w n + wh n )} = 2R{T(w n w n + wn H w n )} M M = 2R{ ((z m+(n 1)M ) H S H C 1 s m +(n 1)M (z m +(n 1)M) H S H m=1 m =1 C 1 s m+(n 1)M + (s m+(n 1)M )H C 1 s m +(n 1)M (z m +(n 1)M) H S H C 1 Sz m+(n 1)M )}. (102) The esul of (102) can be efomulaed as M M J d d = 2R{ ( λ m S H C 1 m ( λ m S H C 1 m ) m=1 m =1 + ( m ) H C 1 m ( λ m S H C 1 S( λ m ) H ) )}, (103) M N J d d = 2R{ ( λ m S H C 1 I n (ℵ n S H C 1 m ) m=1 n=1 + ( m ) H C 1 I n (ℵ n S H C 1 S( λ m ) H ) )} (106) Refeences [1] T. Aiomaki V. Koivunen, MIMO ada filebank design fo inefeence miigaion, Poceedings of 2014 IEEE Inenaional Confeence on Acousics, Speech Signal Pocessing (ICASSP), pp.5297-5301, May 2014. [2] P. Wang, H. Li, B. Himed, A paameic moving age deeco fo disibued MIMO ada in non-homogeneous envionmen, IEEE Tansacions on Signal Pocessing, vol. 61, no. 9, pp. 2282 2294, May 2013. [3] Y. Yu, A. Peopulu, H.V. Poo, CSSF MIMO Rada: Compessivesensing sep-fequency based MIMO ada, IEEE Tansacions on Aeospace Eleconic Sysems, vol. 48, no. 2, pp. 1490 1504, Ap 2012. [4] M. Rossi, A. Haimovich, Y. Elda, Spaial compessive sensing in MIMO ada wih om aays, in Poceedings of 46h Annual Confeence on Infomaion Sciences Sysems (CISS), pp. 1-6, Ma 2012. [5] X. Song, P. Wille, S. Zhou, P. Luh, The MIMO Rada Jamme Games, IEEE Tansacions on Signal Pocessing, vol. 60, no. 2, pp. 687 699, Feb 2012. [6] Y. Zhang, M. Amin, B. Himed, Join DOD/DOA esimaion in MIMO ada exploiing ime-fequency epesenaions, EURASIP Jounal on Advances in Signal Pocessing, Special issue on Advanced in Time-Fequency Aay Pocessing of Nonsaionay Signals, vol. 2012, no. 1, 2012. [7] H. Godich, A. Peopulu, H.V. Poo, Powe allocaion saegies fo age localizaion in disibued muliple-ada achiecues, IEEE Tansacions on Signal Pocessing, vol. 59, no. 7, pp. 3226 3240, Jul 2011. [8] J. Li P. Soica, MIMO ada wih collocaed anennas, IEEE Signal Pocessing Magazine, vol. 24, no. 5, pp. 106 114, Sep 2007. [9] P. Howl, D. Maksimiuk, G. Reisma, FM adio based bisaic ada, in Poc. Ins. Elec. Eng., Rada Sona Navig., vol. 152, no. 3, pp. 107C115, Jun. 2005. [10] M. Glende, J. Heckenbach, H. Kuschel, S. Mulle, J. Schell, C. Schumache, Expeimenal passive ada sysems using digial illuminaos (DAB/DVB-T), in Poc. In. Rada Symp. (IRS), Cologne, Gemany, 2007, pp. 5C7. [11] H. Giffihs, C. Bake, J. Baube, N. Kichen, M. Teagus, Bisaic ada using saellie-bone illuminaos, in Poc. RADAR, pp. 1C5, Oc. 2002. [12] H. Guo, K. Woodbidge, C. Bake, Evaluaion of WIFI beacon ansmissions fo wieless based passive ada, in Poc. IEEE Rada Conf. RADAR, pp. 1C6, 2008. [13] D. Tan, H. Sun, Y. Lu, M. Lesugie, H. Chan, Passive ada using global sysem fo mobile communicaion signal:heoy, implemenaion measuemens, Rada, Sona Navigaion,IEE Poceedings -, vol. 152, no. 3, pp. 116-123, June 2005. [14] H. Giffihs C. Bake, Passive coheen locaion ada sysems. Pa 1: pefomance pedicion, Rada, Sona Navigaion,IEE Poceedings -, vol. 152, no. 3, pp. 153-159, June 2005.
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