Direction of Arrival Estimation Using the Extended Kalman Filter

Transcription

1 SEI 7 4 th Iteatioal Cofeece: Scieces of Electoic, echologies of Ifomatio elecommuicatios Mach 5-9, 7 UISIA Diectio of Aival Estimatio Usig the Exteded alma Filte Feid Haabi*, Hatem Chaguel*, Ali Ghasallah* *Equipe d Electoique, Dépatemet de Physique, Faculté des Scieces de uis 9, El-Maa uis. feidsamia@yahoo.f Chaguelh@yahoo.f ali.ghasallah@fst.u.t Abstact: he techique fo estimatig the paametes of multiple waves povides a coveiet tool fo the aalysis of multiple-waves fields evetually fo actual applicatios to mobile commuicatios. Based o the exteded alma filte (EF), we aalye a ecusive pocedue fo -dimesioal diectios of aival (DOA) estimatio we will employ the two-l-shape aays. A ew space-vaiable model which we call a spatial state equatio is peseted usig aay elemet locatios icidet agles. I this pape we biefly ecapitulate the most impotat featues of the exteded alma filte (EF). he pefomace of the poposed appoach is examied by a simulatio study with thee sigals model. he simulatio esults show a good estimate pefomace. ey wods: elevatio agles, aimuth agles, spatial state equatio, exteded alma filte. IRODUCIO he authos have poposed to use the alma filte techique fo adaptive ateas Geg 4], showed the advatage of fast covegece though simulatio the piciple of iitial paamete selectio. he poblem of estimatig the two-dimesioal diectios of aival (DOAs), amely, the aimuth elevatio agles, of multiple souces was the topic of seveal eseaches eel 89- Wu 3- Li 9- Liu 5]. We will employ the two-l-shape aays that showed ayem 5] bette pefomaces tha the oe L-shape Hua 9] the paallel shape aays Macos 93]. I this pape the detectio estimatio of the paametes of multiple waves ae discussed. Beig impotat i ada commuicatios systems iuma 9], the techique fo estimatig the paametes of multiple waves is cuetly the subject of extesive ivestigatio. We will peset a method fo estimatig the elevatio aimuth aivig waves based o the exteded alma filte Jiua 95]. he geeal chaacteistics of a alma filte such as most-lielihood covegece have bee discussed i seveal papes Shu 3]. O the cotay, we deive a suitable space-vaiable model i the peset pape which we call a spatial state equatio usig aay elemet locatios icidet agles. Simulatio esults show the pefomace of ou method eve at low SRs. he est of the pape is ogaied as follows: he data model is peseted i sectio, the ew spatial state equatio fo icidet waves is developed i sectio, i sectio 3 the exteded alma filte is ecapitulated, sectio 4 pesets the -D diectio of aival estimatio algoithm, sectio 5 shows simulatio esults sectio 6 maes coclusios.. Data Model Coside the two-l-shape uifom liea aay (ULAs) i the x- the y- plaes show i fig. with ite-elemet equals d, usig thee aay elemets placed o the x, y axes. Each liea aay cosists of elemets. he elemet placed at the oigi is commo fo efeecig puposes. Suppose that thee ae aow b souces, S(, with the same wavelegth λ impigig o the aay, such that th souce has a elevatio agle θ a aimuth agle φ, =,.,. We put the complex base-b epesetatio of - -

2 SEI7 the sigal eceived by the th elemet of oe subaay as x ( ( =,... ), the sigal souces ae fa apat fom the subaay. he subaay output vecto at the sapshot t is the give by: θ x t ) = ( x ( t ), x ( t ),... x ( t )) ( = a ( θ ) s ( t ) (), t=,..,p = Φ y is the taspose, a( θ ) is the steeig vecto defied by : x a( θ ) = (, e ϕ j,..., j ( ) e ϕ ) ] () Figue : he -L-shape aay cofiguatio used fo the joit aimuth elevatio ( φ, θ ) DOA estimatio with ϕ depeds o the positio of the subaay. he eceived sigal is give by: y ( t ) Hx ( t ) + η ( t ) = (3) Whee η is a -dimesioal complex white oise vecto with mea eo covaiaceσ I, I is the idetity matix of sie. H= ] (4) with -compoets. At the (+) th elemet positio, the icidet wave vecto is deived fom that at the th elemet positio by : x ( ) = A ( θ ) x ( ) + (5). A ew spatial state equatio fo icidet waves Equatio (5) could be called a spatial state equatio. Coespodigly, (3) is called a measuemet equatio. he elemet of X, A y ae all complex i geeal. We efomulate the poblem ivolvig complex quatities i tems of eal quatities, i ode to cay out the paamete estimatio of icidet waves. Let the eal the imagiay pats of x, be deoted by espectively, the eal imagiay pats of exp( jφ ) be deoted by α α, espectively. Cotiuig i this mae we have x,, + j, = (7) exp( (8) jφ ) = α + jα Whee θ = θ, θ... θ ] = φ... φ ], exp( jϕ ) A( ) =,... exp( jϕ ) θ (6) It follows that the L-compoet complex vecto X() ca be completely epeseted by the followig eal vecto X () with L-compoets X ( ) =,,...,, By this way, we ca ewite (5) i tems of eal vecto as: ] X ( + ) = A ( α ) X ( ) (9) - -

3 SEI7 Whee A α α α α ( α ) =... () α α α α... h(.) ae oliea matix fuctios. We will assume f(.) et h(.) ae diffeetiable. W( η( ae vectos with white Gaussia om elemets of eo mea, thei covaiace matices ae as follows: Cov Cov ( W ( t ) W ( t ) ( η ( t ) η ( t ) ) ) = Q = R (7) he exteded alma filte is descibed by the followig equatios: α = α α... α ], =,, he A ( α ) is L L squae matix, coespodigly, (3) ca be ewitte as: ξˆ ( t + / t ) = f ( t, ξˆ ( t )) ξˆ ( t / t ) = ξˆ ( t / t ) + ( t ) V (8) ( t ) h ( t, ξˆ ( t ))] (9) Y ( ) = H ( ) X ( ) ( ) () + Whee Y ))] ( ) = Re( y( )) Im( y( () ( = P( / ) () ˆ( ξ t / t )) P( t / t ) ˆ( ξ t / t )) P( t / ) = P( t / t ) ( ˆ( ξ t / t )) + R] ˆ( ξ t / t ) P( t / t )) () H... =... (3) P( t / t + ) = F ˆ( ξ t / ) P( t / F ˆ( ξ t / ) + Q () ( ) η ))] = Re( η ( )) Im( ( (4) H is L, Re(a) Im(a) i he sie of (4) epeset the eal imagiay pats of a complex vaiable a. 3. he Exteded alma Filte he alma filteig appoach fo a o liea system is based o the fist ode lieaiatio of a oliea state equatio usig a pevious estimate as the cete of the updated liea aylo appoximatio. he oliea system is give by Whee ˆ( t / ξ ˆ( t / t ) ˆ ξ ( t ξ deote the coditioal mea estimates of ), based o the measuemets V( V(t-), espectively. F ξ t / ) ˆ( ξ t / t ) deote the followig Jacobea matices evaluated at the values of ˆ ξ ( t / ˆ ξ ( t / t ), espectively. (, ˆ Fξ ( / )) = f (, ξ ( )) ξ ξ = ξ ˆ( (3) ξ ( t + ) = f ξ ( ) + W ( (5) V ( t ) = h ( t, ξ ( t )) + η ( t ) (6) (, ˆ ξ( / )) = h(, ξ( )) (4) ξ ξ= ξ ξ (t ) : a state vecto, V(: measuemet vecto, W(: pocess oise, η(: measuemet oise, f(.) ( deotes the alma gai matix. P(t/ - 3 -

4 SEI7 P(t/t-) deote the estimatio eo covaiace matices. he above exteded alma filte ca also be used to estimate uow paametes i a liea system. I the followig, we apply it to the poblem whee a vecto α is give by: 4. he -D diectio of aival estimatio algoithm 4.. Estimatio of the elevatio agle Let v( be the x sigal eceived at the liea subaay i the axes at the sapshot t. X ( t + ) = Λ( α ) X ( W ( (5) + Y ( t + ) = C( α ) X ( + η( (6) v t ) = ( v ( t ), v ( t ),... v ( t )) ( = a ( θ ) s ( t ) + η = ( t ), t=,..,p (3) he matices Λ ( α) et C(α ) ae of fiite dimesios. It is assumed that the matix elemets ae diffeetiable with espect toα. W( η( ae idepedet of α. he exteded alma filte to detemie the uow paamete vecto α ow ca be obtaied by extedig the state vecto by X to the vecto ξ( ) icludig the paamete vecto α =α () which depeds o the paamete, h X ( ξ ( = (7) α( Equatios (5) (6) ae ewitte as follows: Whee a( θ ) = (, ϕ, e jϕ j ( ),..., e ϕ ) ] (33) π ( ) d cos θ = (34) λ θ is the elevatio agle of the th souce sigal. η ( is the additive White Gaussia oise of the th souce sigal at the sapshot t. 4.. Estimatio of the aimuth agle Let x( be the x sigal eceived at the liea subaay i the x axes at the sapshot t. W ( t ) ξ ( t + ) = f ξ ( t )) + (8) Y ( t ) = h ( t, ξ ( t )) + η ( t ) (9) x t ) = ( x ( t ), x ( t ),... x ( t )) ( = a ( θ x ) s ( t ) + η x = t=,..,p ( t ) (35) Whee Whee f ( t, ξ ( t )) = Λ ( α ) X α ( t ) (3) a( θ ) = (, e j ϕ,..., j ( ) e ϕ ) ] (36) h ( t, ξ ( t )) = C ( α ) X ( t ) (3) ϕ, π ( ) d siθ cosφ = (37) λ Because (8) (9) have the same foms as (5) (6), the exteded alma filte i (8)-() ca be applied to (8) (9). It shows that the paamete vecto α ca be estimated by the exteded alma filte. φ is the aimuth agle of the th souce sigal. ηx( is the additive White Gaussia oise of the th souce sigal at the sapshot t i the x subaay with mea eo covaiace σ I. I the same way we estimate the aimuth agle - 4 -

5 SEI7 φˆ usig the y axis subaay. he x sigal eceived y y( at the liea subaay i the y axes at the sapshot t ca be ewitte as: y = ( y (, y (,... y ( ) ( = = Whee a ( θ ) s ( t ) + η t=,..,p y y ( t ), (38) by H Y ( m ) i (), espectively. ) Extedig the state vecto by icludig the uow icidet agles i the state vecto. It is obtaied fom (), (7) (7) as follows: ξ ( ) =,,.., α α.. α ] (4) 3) Adaptig the exteded state vecto (4) to (8) (9), to obtai the oliea state equatio. 4) Usig the oliea state equatio, the exteded alma filte (8)... () is caied out to obtai the estimatio of the elevatio o the aimuth aival waves. a( φy θ, ) = (, e ϕ j,..., j ( ) e ϕ ) ] (39) π ( ) d siθ siφ ϕ = (4) λ Fially, the aimuth agle estimatio φˆ ca be witte ayem 5] as: ( ˆ φ + ˆ x φy ) if both ˆ φ ˆ x φy ae eal ˆ φ if ˆ x φy is complex ˆ (4) φ = ˆ φy if ˆ φx is complex Failue if both ae complex he poblem to estimate the elevatio agle the aimuth agle φ i () o α θ α i (8), ca be solved with the same pocedue as paametes estimatio i liea systems. We should apply a exteded alma filte appoach by meas of spatial state equatios of the icidet waves as metioed above. he pocedue is summaied as follows: ) Defiig a space-vaiable vecto expessig the icidet waves, to obtai a spatial state equatio of the icidet waves. he matix Λ ( α) X h () i (5) is eplaced by A ( α ) X ( m ) i (9), espectively. Ad the matix C(α ) Yh ( ) i (6) 5. Simulatio Results Compute simulatios have bee coducted to evaluate the -D DOA estimatio pefomace of the poposed method. he paametes used i the simulatio ae as follows: he sesos displacemet d is tae to be half the wave legth of the sigal waves. he icidet waves ae thee waves =3, with diectios of aival DOA ( θ ), =... he additive oise is white Gaussia pocesses. he L= is the umbe of sapshots pe tial. All the waves ae assumed to be diect waves the powe levels ae a costat, 3. 5, espectively. he measuemet oise η( i (6) is assumed to be.. he pocess oise W( is assumed to be eo as the state (5) epesets the elemets state at the same time. he covegece of estimated values of the elevatio aimuth agles ae plotted i fig. fig.3, espectively. he abscissa is the umbe of iteatios, meas the umbes of ecessay elemets. I figues. (a) (b) the waves ae located at (6,º), (7,3º) (8,4º), espectively sigal to oise atio (SR)= db. his is a example of the most commo case because the icidet agles diffeeces ae medium, accodigly, the covegece is ot bad. It is show that the estimated values steadily covege to the actual values. About iteatios ae equied to achieve a steady state fo the elevatio agles. I othe wods, out of elemets fo the axes aay ae used fo covegece. Fo the aimuth agles, oly 85 iteatios ae equied to obtai the covegece to the eal aimuth agles. I figues.3 (a) (b), thee waves ae located at (7,º), (74,4º) (78,8º), espectively. Despite the poximity of the elevatio aimuth agles, the covegece chaacteistics of agles estimatios show o degadatio fom fig.. Although the vaiatios of the elevatio agles ae lage the covegece speeds ae slowe tha i fig., the estimates eos ae quite small afte achievig the covegece state afte 8 iteatios

6 SEI7 Figues.4 (a) (b) show the histogam plots fo the joit elevatio aimuth agles, espectively, fo a sigle souce with DOA located at (6,º) sigal to oise atio (SR)= db, by usig the exteded alma filte of the -L shape aays. We obseve that the method gives a vey close joit DOA estimatio the clea peas appea aoud (6, ). o failue ca be obseved i the estimatio of the -D diectios of aival, but it is clea that the poposed algoithm impoves the pefomace sigificatly it also educes the estimatio eo of both the aimuth elevatio agles. DOA(elevatio)(deg.) estimted elevatio eal elevatio 9 8 estimted elevatio eal elevatio 6 3 umbe of iteatio (a) DOA(elevatio)(deg.) 7 6 DOA(aimuth)(deg.) estimted imuth eal aimuth 4 umbe of iteatio 4 (a) estimted imuth eal aimuth 5 3 umbe of iteatio (b) Figue.3: Covegece esults fo the souces located at (7,º), (74,4º) (78,8º), espectively. SR =db. (a) Elevatio agles. (b) Aimuth agles. DOA(aimuth)(deg.) 3 Histogamme agle elevatio - umbe of iteatio (b) iteatio Figue.: Covegece esults fo the souces located at (6,º), (7,3º) (8,4º), espectively. SR =db. (a) Elevatio agles. (b) Aimuth agles DOA(elevatio)(deg.) (a) - 6 -

7 SEI7 iteatio Histogamme agle aimuth 3 DOA(aimuth)(deg.) (b) Figue.4: Histogam of elevatio DOA estimatios fo a sigle souce of DOA at (6, ), SR = db, (a) Elevatio agles. (b) Aimuth agles. 6. COCLUSIOS: A atea aay cofiguatio was poposed usig the exteded alma filte method, fo the -D aimuth elevatio agle estimatio poblem. he esults obtaied ae summaied as follows. () A space vaiable model, called spatial state equatio is deived, usig the elemets aay of the - L shape atea. () We show that the exteded alma filte algoithm, which ca be applied to all aay cofiguatios, is simple applicable to pactical cases. (3) he poposed scheme educes the estimatio eo of both the aimuth elevatio agles it shows a good pefomace at low SRs at the poximity of the waves. It should be oted that the idea of the poposed appoach is diffeet fom the well-ow MUSIC o the PM algoithms. he agles of the aival waves ae diectly estimated fom the sigal eceived at each aay elemet, though MUSIC PM algoithms obtai the estimatio based o the spectal aalysis techiques. he pefomace of the poposed appoach is ot affected by the chage of the icidet waves duig the computig time of the algoithm, because the sigals at each aay elemet ae collected at the same samplig time. REFERECES: Geg Welch Gay Bishop, A Itoductio to the alma Filte, Depatmet of Compute Sciece Uivesity of oth Caolia at Chapel Hill, Apil 4. Hua Y.,.. Saa, D.D.Weie a L-shape aay fo estimatig -D diectios of wave aival, IEEE as. Ateas Popagatio 39 () (99) Jiua W.,.aao H.ojio, estimatio of aival waves usig a exteded alma filte IEICE as. Vol. E78-B-II, o., ov.995. iuma., M.Aai, M.Ogawa,.Yamada,.Iagai, Estimatio of diectio of aival l delay pofile of multi-path waves by MUSIC algoithm fo idoo adio commuicatios IEICE as., vol.j73-b-ii,o.,pp ,ov.99. eel P., E. Depette, A two dimesioal vesio of matix pecil method to solve the DOA poblem, Euopea cofeece o cicuit theoy desig, pp , 989. Li P., B. Yu, J. Su a ew method fo two dimesioal aay, sigal pocessig 47(3), p , Decembe 995. Liu., J. Medel aimuth elevatio diectio fidig usig abitay aay geometies, IEEE as. Atea popagatios, Vol. 53, o. 4, APRIL 5 Macos S., calibatio of a distoted towed aay usig a popagato opeato, J. Acoust. Soc.Ame. 93(4), p , mach 993. Shu C., Geeal wo Stage Exteded alma Filtes, IEEE as. Automatic cotol, Vol.48, o., p.89-93, Febuay 3. ayem., H. wo, L-shape--D aival agle estimatio with popagato method, IEEE as. vol.53.o5, May 5. Wu Y., G. Liao, H.C.So a fast algoithm fo -D diectio of aival estimatio sigal pocessig. 83 (8), p , august