HIGHER ORDER DIRECTION FINDING WITH POLARIZATION DIVERSITY: THE PD-2q-MUSIC ALGORITHMS

15th European Signa Processing Conference (EUSIPCO 2007), Poznan, Poand, September 3-7, 2007, copyright by EURASIP HIGHER ORDER DIRECTION FINDING WITH POLARIZATION DIVERSITY: THE PD-2-MUSIC ALGORITHMS Pasca Chevaier (1), Anne Ferréo (1), Laurent Abera (2,3) and Gwenaë Birot (2,3) (1) Thaes-Communications, EDS/SPM, 160 Bd Vamy BP 82, Coombes, F-92704, France ; (2) INSERM, U642, Rennes, F-35000, France ; (3) Université de Rennes 1, LTSI, Rennes, F-35000, France ; pasca.chevaier@fr.thaesgroup.com, anne.ferreo@fr.thaesgrooup.com aurent.abera@univ-rennes1.fr, gwenae.birot@univ-rennes1.fr ABSTRACT Some 2-th ( 2) order extensions of the MUSIC method, expoiting the information contained in the 2-th ( 2) order statistics of the data and caed 2-MUSIC methods, have been proposed recenty for direction finding of non Gaussian signas. These methods are asymptoticay robust to a Gaussian background noise whose spatia coherence is unknown and offer increasing resoution and robustness to modeing errors jointy with an increasing processing capacity as increases. However, 2-MUSIC methods have been mainy deveoped for arrays with space diversity ony and cannot put up with arrays of sensors diversey poarized. The purpose of this paper is to introduce, for arbitrary vaues of ( 1), three extensions of the 2-MUSIC methods abe to put up with arrays having poarization diversity, which gives rise to the socaed PD-2-MUSIC (Poarization Diversity 2-MUSIC) agorithms. These agorithms are shown to increase resoution, robustness to modeing errors and processing capacity of 2-MUSIC methods in the presence of diversey poarized sources from arrays with poarization diversity. 1. INTRODUCTION Some 2-th ( 2) order extensions of the MUSIC [13] method, expoiting the information contained in the 2-th ( 2) order statistics of the data and caed 2-MUSIC methods, have been proposed recenty for direction finding (DF) of non Gaussian signas [4-5]. Note that the 4-MUSIC method proposed in [12] is a particuar case of 2-MUSIC methods for = 2. The 2-MUSIC methods with > 1 are asymptoticay robust to a Gaussian background noise whose spatia coherence is unknown and generate a virtua increase of both the number of sensors and the effective aperture of the considered array, introducing the higher order (HO) virtua array (VA) concept [2], which generaizes the fourth order (FO) one s introduced previousy in [7] and [3]. A conseuence of this property is that, despite of their higher variance [1], 2-MUSIC methods are shown in [5] to have resoution, robustness to modeing errors and processing capacity increasing with. However, 2-MUSIC ( 2) agorithms have been mainy deveoped for arrays of sensors with space diversity ony (i.e. arrays with identica sensors at different ocations), or for sources with the same poarization, and cannot put up with arrays of sensors having diverse poarizations. The expoitation of arrays with poarization diversity, possiby in addition to the space diversity, is very advantageous since for such arrays, mutipe signas may be resoved on the basis of poarization as we as direction of arriva (DOA). However, most of methods which are currenty avaiabe for DF from arrays with poarization diversity expoit ony the information contained in the second order (SO) statistics of the observations, among which we find [8] [11]. HO methods of this kind are very scarce [10]. In order to increase the performance of the 2-MUSIC agorithms in the presence of sources having different poarizations, the purpose of this paper is to introduce, for arbitrary vaues of ( 1), three extensions of the 2-MUSIC methods abe to put up with arrays having poarization diversity, which gives rise to the so-caed PD-2-MUSIC agorithms. For a given vaue of, these agorithms are shown in this paper to increase the resoution, the robustness to modeing errors and the processing capacity of the 2- MUSIC methods in the presence of diversey poarized sources and from an array with poarization diversity. 2. HYPOTHESES AND DATA STATISTICS 2.1. Hypotheses We consider an array of N narrow-band (NB) potentiay different sensors and we ca x(t) the vector of compex enveopes of the signas at the output of these sensors. Each sensor is assumed to receive the contribution of P zero-mean stationary NB sources, which may be statisticay independent or not, corrupted by a noise. We assume that the P sources can be divided into G groups, with Pg sources in the group g, such that the sources in each group are assumed to be statisticay dependent, but not perfecty coherent, whie sources beonging to different groups are assumed to be statisticay independent. Of course, P is the sum of a the Pg over a the groups. Under these assumptions, the observation vector can approximatey be written as foows 2007 EURASIP 257

15th European Signa Processing Conference (EUSIPCO 2007), Poznan, Poand, September 3-7, 2007, copyright by EURASIP x(t) P i = 1 m i (t) a(θ i, β i ) + η(t) = G g = 1 A g m g (t) + η(t) (1) where η(t) is the noise vector, assumed zero-mean, stationary and Gaussian, m i (t) is the compex enveope of the source i, θ i = Δ (θ i, ϕ i ) where θ i and ϕ i are the azimuth and the eevation anges of source i, β i is a (2 x 1) vector characterizing the state of poarization of source i and whose components wi be defined hereafter, a(θ i, β i ) is the steering vector of source i, A g is the (N x Pg) matrix of the steering vectors of the sources beonging to group g, m g (t) is the associated (Pg x 1) vector with corresponding m i (t). In particuar, in the absence of couping between sensors, assuming a pane wave propagation, component n of vector a(θ i, β i ), denoted a n (θ i, β i ), can be written, in the genera case of an array with space and poarization diversity, as [6] a n (θ i, β i ) = f n (θ i, β i ) exp{j2π[x n cos(θ i )cos(ϕ i )+y n sin(θ i )cos(ϕ i )+z n sin(ϕ i )]/λ} (2) In (2), λ is the waveength, (x n, y n, z n ) are the coordinates of sensor n of the array, f n (θ i, β i ) is a compex number corresponding to the response of sensor n to a unit eectric fied coming from the direction θ i and having the state of poarization β i [6]. Let β i1 and β i2 be two distinct poarizations for the source i (for exampe vertica and horizonta) and a 1 (θ i ) Δ = a(θi, β i1 ) and a 2 (θ i ) Δ = a(θ i, β i2 ) be the corresponding steering vectors for DOA θ i. We assume that the vectors a 1 (θ) and a 2 (θ) can be cacuated anayticay or measured by caibration whatever the vaue of θ. Considering an arbitrary poarization β i for the source i, the compex eectric fied of the atter can be broken down into the sum of two compex fieds, each arriving from the same direction, and having the poarizations β i1 and β i2 [6]. The steering vector a(θ i, β i ) is then the weighted sum of the steering vectors a 1 (θ i ) and a 2 (θ i ) given by a(θ i, β i ) = β i1 a 1 (θ i ) + β i2 a 2 (θ i ) = Δ A 12 (θ i ) β i (3) In (3), A 12 (θi) is the (N x 2) matrix of the steering vectors a1(θi) and a2(θi), βi1 and βi2 are compex numbers such that βi1 2 + βi2 2 = 1 and βi is the (2 x 1) vector with components βi1 and βi2, which can be written, to within a phase term, as βi = [cosγi, e jφi sinγi] T, where γi and φi are two anges characterizing the poarization of source i and such that (0 γi π/2, π φi < π). 2.2. Data statistics The 2-th ( 1) order DF methods considered in this paper expoit the information contained in the (N xn ) 2 th order covariance matrix, C 2,x, whose entries are the 2 th order cumuants of the data, Cum[x i 1 (t),, x i (t), x i +1 (t)*,, x i 2 (t)* ] (1 i j N) (1 j 2), where * corresponds to the compex conjugation. However, the previous entries can be arranged in C 2,x in different ways, indexed by an integer such that (0 ), as it is expained in [2] [5], and giving rise, under hypotheses of section 2.1, to the C2,x() matrix given by [5] G C 2,x () = C 2,x g () + η 2 V() δ( 1) (4) g = 1 In (4), η 2 is the mean power of the noise per sensor, V() is the (N x N) spatia coherence matrix of the noise for the arrangement indexed by, such that Tr[V()] = N, Tr[.] means Trace, δ(.) is the Kronecker symbo. The (N xn ) matrix C 2,xg () contains the 2-th order cumuants of x g (t) for the arrangement indexed by, which can be written as C 2,x g () [A g Ag * ( ) ]C2,m g ()[A g Ag * ( ) ] H (5) In (5), C 2,mg () is the (P g x Pg ) matrix of the 2-th order cumuants of m g (t) for the arrangement indexed by, H corresponds to the conjugate transposition, is the Kronecker product and A g is the (N x Pg ) matrix defined by Ag Δ = A g A g... A g with a number of Kronecker product eua to 1. Note that it is shown in [2] and verified in this paper that the parameter determines in particuar the maxima processing power of PD-2-MUSIC agorithms. In situations of practica interests, the 2-th order statistics of the data, Cum[x i 1 (t),, x i (t), x i+1 (t)*,, x i 2 (t)* ], are not known a priori and have to be estimated from L sampes of data, x(k) Δ = x(kt e ), 1 k L, where T e is the sampe period, in a way that is competey described in [5] and which is not recaed here. 3.1. Hypotheses 3. PD-2Q-MUSIC ALGORITHMS To deveop PD-2-MUSIC agorithms for the arrangement indexed by, we need extra assumptions: H1) 1 g G, P g < N H2) 1 g G, A g Ag * ( ) has fu rank Pg H3) P(G, ) Δ G = P < N g= 1 g H4) Ã Δ, = [A 1 A1 * ( ),, AG AG * ( ) ] has fu rank P(G, ) 3.2. KP-PD-2-MUSIC agorithm Noting r 2,mg () the rank of C 2,mg () (r 2,mg () P g ), we deduce from H1 and H2 that C 2,xg () for > 1 has aso rank r 2,mg (). Hence, using H4 and for > 1, matrix C 2,x () has a rank r 2,x () eua to the sum, over a the groups g, of r 2,mg () and such that r 2,x () < N from H3. As matrix C 2,x () is Hermitian, but not positive definite, we deduce from the previous resuts that C 2,x () has r 2,x () non zero eigenvaues and N r 2,x () zero eigenvaues for > 1. The eigendecomposition of C 2,x (), for > 1, gives C 2,x ()=U 2,s ()Λ 2,s ()U 2,s () H +U 2,n ()Λ 2,n ()U 2,n () H (8) where Λ 2,s () is the diagona matrix of the non zero eigenvaues of C 2,x (), U 2,s () is the unitary matrix of the associ- 2007 EURASIP 258

15th European Signa Processing Conference (EUSIPCO 2007), Poznan, Poand, September 3-7, 2007, copyright by EURASIP ated eigenvectors, Λ 2,n () is the diagona matrix of the zero eigenvaues of C 2,x () and U 2,n () is the unitary matrix of the associated eigenvectors. As C 2,x () is Hermitian, a the coumns of U 2,s () are orthogona to a the coumns of U 2,n (). Moreover, Span{U 2,s ()} = Span{Ã, } when the matrices C 2,m g (), 1 g G, are fu rank whereas Span{U 2,s ()} Span{Ã, } otherwise. Defining a, (θ, β) Δ = a(θ, β) a(θ, β) * ( ) and noting (θ ig, β ig ) the DOA and poarization parameters of the ith source in the gth group, it can be easiy verified that, in a cases, the vector a, (θ ig, β ig ), aways beongs to Span{U 2,s ()}. Conseuenty, a vectors {a, (θ ig, β ig ), 1 i Pg, 1 g G} are orthogona to the coumns of U 2,n () and are soutions of the foowing euation a, (θ, β) H U 2,n () U 2,n () H a, (θ, β) = 0 (9) Using (3) into (9), removing the redundancy of β Δ, = [β β * ( ) ] and normaizing the eft hand side of (9) to obtain no minima in the absence of sources, the probem of sources DOA estimation by the PD-2-MUSIC agorithm for the arrangement then consists to find the P sets of parameters (θ^i, β^i) = (θ^i, ϕ^i, γ^i, φ^i), (1 i P), which are soution of the foowing euation [β β * ] H Q,,1 (θ)[β β * ] [β β * ] H Q,,2 (θ)[β β = 0 (10) * ] In (10), the ((+1)( +1) x (+1)( +1)) Q,,1 (θ) and Q,,2 (θ) matrices are defined by Q,,1 (θ) = Δ (11) [B B ] H A 12,, (θ) H U 2,n ()U 2,n () H A 12,, (θ) [B B ] Q,,2 (θ) = Δ [B B ] H A 12,, (θ) H A 12,, (θ)[b B ] (12) where A 12,, (θ) is eua to A 12 (θ) A 12 (θ) * ( ), β is the ((+1)x1) vector with components β [j] = β 1 j+1 β2 j 1, β 1 and β 2 are components of β and B is the (2 x(+1)) matrix such that β = B β. For sources with unknown poarizations, the previous agorithm has to impement a searching procedure in both DOA and poarization, which is very compex. We thus imit the use of this agorithm to situations where the poarization of sources is known and we ca this agorithm KP-PD-2-MUSIC agorithm (Known Poarization PD-2-MUSIC). In practica situations, Q,,1 (θ) is not known and has to be estimated from the eigenvaue decomposition of an estimate of C 2,x (), by repacing U 2,n () by its estimate. In this case, the estimated eft-hand side of (10) has to be minimized over θ. 3.3. UP-PD-2-MUSIC agorithms For sources with unknown poarizations, a simpe way to remove the searching procedure with respect to the poarization parameter consists, for any fixed DOA, to minimize the eft-hand side of euation (10) with respect to vector β, = [β β * ], as it is proposed in [8] for = 1. This gives rise to the UP-PD-2-MUSIC (Unknown Poarization PD-2- MUSIC) agorithms whose first version for the arrangement indexed by, caed UP-PD-2-MUSIC()-1, consists to find DOA which cance the pseudo-spectrum given by P UP-PD-2-Music()-1 (θ) = Δ λ,,min (θ) (13) In (13), λ,,min (θ) is the minimum eigenvaue of matrix Q,,1 (θ) in the metrics Q,,2 (θ). Note that the associated optima vector β,, noted β,,min(θ), corresponds to the associated eigenvector. Note that one way in which the eigenvaue λ,,min (θ) can be computed is by determining the minimum root of the foowing euation det[q,,1 (θ) λ Q,,2 (θ) ] = 0 (14) where det[x] means determinant of X. Thus, for each vaue of θ, searching in poarization space has been avoided by finding the roots of an euation of order (+1)( +1), which corresponds to a substantia reduction in computation, at east for sma vaues of. We deduce from (14) and [9], that finding θ such that λ = λ,,min (θ) is zero is euivaent to find θ such that det[q,,2 (θ) 1 Q,,1 (θ)] = det[q,,1 (θ)] / det[q,,2 (θ)] = 0. A second version of the UP-PD-2- MUSIC agorithm for the arrangement indexed by, caed UP-PD-2-MUSIC()-2, consists to find DOA which cance the pseudo-spectrum given by P UP-PD-2-Music()-2 (θ) Δ det[q,,1 (θ)] = (15) det[q,,2 (θ)] which aows a compexity reduction with respect to (13). Again, in practica situations, Q,,1 (θ) is not known and has to be estimated from the eigenvaue decomposition of an estimate of C 2,x (). The probem then consists to find the P sets of parameters θ^i = (θ^i, ϕ^i), (1 i P), for which estimates of (13) or (15) are minimized. 4. IDENTIFIABILITY The maximum number of sources to be processed by a given DP-2-MUSIC agorithm is obtained for statisticay independent sources. Under this assumption, assuming no couping between the sensors, we deduce from the HO VA theory [2] that the KP-PD-2-MUSIC agorithm for the arrangement indexed by is abe to process up to P max =N 2 1 sources, provided that the associated VA has no ambiguities up to order N 2 1, where N 2 is the number of different virtua sensors of the associated VA. It has been shown in [2] that N 2 is a function of, and the true array of N sensors. Besides, tabe IV of [2] shows, for a genera array with space and poarization diversities having sensors arbitrary ocated with different responses, the expression of an upperbound, N max (2,), of N 2 as a function of N for 2 4 and severa vaues of. A very usefu case for practica situations, which was not studied in [2], corresponds to an array of N = 2M sensors composed of two subarrays of M sensors having orthogona poarizations. Two kinds of such arrays are considered in this section and correspond to arrays for which the sensors of the two subarrays are either coocated or not. Tabe 1 shows, for non coocated and coocated 2007 EURASIP 259

15th European Signa Processing Conference (EUSIPCO 2007), Poznan, Poand, September 3-7, 2007, copyright by EURASIP subarrays respectivey, the expression of N max (2,) as a function of N for = 2 and severa vaues of. Note that this upper-bound corresponds to N 2 in most cases of array geometry with no particuar symmetry, which is in particuar the case for uniform circuar array of M vectoria sensors with two components, when M is a prime number. In addition, tabe 1 shows the expression of N 2 as a function of N=2M for = 2 and severa vaues of, for an array composed of two coocated and orthogonay poarized Uniformy spaced Linear Array (ULA) of M identica sensors. Non-coocated subarrays coocated subarrays Coocated ULA subarrays 2 N max (2,) N max (2,) N 2 4 2 N(N +1)/2 3N(N+2)/8 3(N 1) 4 1 N 2 N+2 N 2 2N+4 4(N 1) Tabe 1 N max (2, ) as a function of N = 2M for = 2 and severa vaues of, and for arrays with two orthogonay poarized subarrays Under the same assumptions, foowing the same reasoning as that presented in [8], we deduce that the UP-PD-2- MUSIC agorithms for the arrangement indexed by are abe to process up to P max = N 2 (+1)( +1) sources, which corresponds to N 2 for = 1, resut obtained in [8]. 5. SIMULATIONS The resuts of the previous sections are iustrated in this section through computer simuations. To do so, two criterions [5] are computed, from 300 reaizations, in the foowing in order to uantify the uaity of the associated DOA estimation: a Probabiity of Non-Aberrant Resuts (PNAR) generated by a given method for a given source and the corresponding Root Mean Suare Error (RMSE). The sources are assumed to have a zero eevation ange and to be zeromean stationary sources corresponding to QPSK sources samped at the symbo rate and fitered by a raise cosine puse shaped fiter with a ro-off = 0.3. Moreover, the statistica arrangement index used hereafter is =1 and no modeing error is considered. Figure 1 PNAR resuts of source 2 as a function of sampes Figure 2 RMSE resuts of source 2 as a function of sampes 5.1. Overdetermined mixtures of sources We consider a Uniform Circuar Array (UCA) of N = 6 crossed-dipoes with a radius r such that r = 0.3 λ. One dipoe is parae to the x-axis whereas the other is parae to the z-axis. Three of these crossed-dipoes are combined to generate a right sense circuar poarization in the y-axis whie the three other dipoes are combined to generate a eft sense circuar poarization in the y-axis. The array is then composed of two orthogonay poarized overapped (noncoocated) circuar subarrays of M = 3 sensors so that adjacent sensors aways have different poarizations. In this context, two QPSK sources with the same input SNR eua to 5 db are received by the array. They are assumed to be weaky separated in both DOA and poarization, such that (θ 1, γ 1, φ 1 ) = (50, 45, 0 ) and (θ 2, γ 2, φ 2 ) = (60, 45, 10 ) respectivey. Under these assumptions, figures 1 and 2 show the variations, as a function of the number of sampes, of the PNAR criterion for the source 2, PNAR2, and the associated RMSE2 criterion (we obtain simiar resuts for the source 1), at the output of the 12 methods mentioned on figure 1. For 2- MUSIC, 4-MUSIC and 6-MUSIC agorithms, the 6 sensors of the UCA are assumed to be identica with responses of sensor 1. Figures 1 and 2 show, at east for UP agorithms, better performance of methods expoiting both HO statistics and poarization discrimination and increasing performance with for UP-PD-2-MUSIC methods. 5.2. Underdetermined mixtures of sources To iustrate the capabiity of PD-2-MUSIC (>1) agorithms to process underdetermined mixtures of sources, we imit the number of sensors of the previous circuar array to N = 3 sensors. Under these assumptions, we deduce from tabes II and IV of [2], and section 4 that KP-PD-4-MUSIC, UP-PD-4-MUSIC, KP-PD-6-MUSIC and UP-PD-6-MUSIC can process up to P max =7, P max =4, P max =14 and P max =9 sources, respectivey. We deduce from tabes VI and VII of [2] that 4-MUSIC and 6-MUSIC can process up to P max =6 and P max =11 sources, respectivey. We assume now that the array receives 4 statisticay independent QPSK sources with 2007 EURASIP 260

15th European Signa Processing Conference (EUSIPCO 2007), Poznan, Poand, September 3-7, 2007, copyright by EURASIP the same input SNR eua to 15 db and DOA and poarization parameters eua to (θ 1, γ 1, φ 1 ) = (15, 45, -75 ), (θ 2, γ 2, φ 2 ) = (45, 45, 0 ), (θ 3, γ 3, φ 3 ) = (95, 22.5, 75 ), (θ 4, γ 4, φ 4 ) = (122.5, 45, 150 ) respectivey. Under these assumptions, figures 3 and 4 show the variations, as a function of the number of sampes, of the ower PNAR and the highest RMSE resuts, among a the sources, at the output of the eight methods, namey the 2-MUSIC, the KP-PD-2- MUSIC and the UP-PD-2-MUSIC-m agorithms for {2,3} and for m {1,2}. Note the capabiity of PD-2- MUSIC methods to process underdetermined mixtures of sources provided P P max. Note the poor performance of 2-MUSIC methods for the considered scenario due to the ow input power of the weakest source at the output of the sensors. Better performance woud be obtained for higher vaues of sampes. Figure 3 Minima PNAR resuts of sources Figure 4 Maxima RMSE resuts of sources 6. CONCLUSION Three versions of the 2-MUSIC ( 1) agorithm abe to put up with arrays having poarization diversity and giving rise to PD-2-MUSIC agorithms have been presented in this paper for severa arrangements of the 2-th order data statistics. The first version is we-suited for sources with known poarization whereas the two others do not assume any knowedge about the sources poarization. For a given vaue of, these agorithms have been shown to increase the resoution, the robustness to modeing errors (at east for severa poory anguary separated sources) and the processing capacity (at east for VA without any HO ambiguities) of the 2-MUSIC method in the presence of diversey poarized sources and from an array with poarization diversity. Moreover, despite a higher variance in the statistics estimation, performance of UP-PD-2-MUSIC agorithms have been shown to generay increase with when some resoution is reuired. This occurs in particuar for sources which are poory separated in both DOA and poarization. This resut shows off for these scenarios the interest to jointy expoit poarization diversity and HO statistics for DF. Finay, identifiabiity issue of a of these methods has been addressed. 7. REFERENCES [1] J.F. Cardoso, E. Mouines, "Asymptotic performance anaysis of DF agorithms based on fourth-order cumuants", IEEE Trans. Sig. Proc., vo. 43, pp. 214-224, January 1995. [2] P. Chevaier, L. Abera, A. Ferréo, P. Comon, "On the Virtua Array concept for higher order array processing", IEEE Trans. Sig. Proc., vo. 53, pp. 1254-1271, Apri 2005 [3] P. Chevaier, A. Ferréo, "On the virtua array concept for the fourth-order direction finding probem", IEEE Trans. Sig. Proc., vo. 47, pp. 2592-2595, September 1999 [4] P. Chevaier, A. Ferréo, "Procédé et dispositif de goniométrie à haute resoution à un ordre pair arbitraire", Patent N 05.03180, Apri 2005. [5] P. Chevaier, A. Ferréo, L. Abera, "High resoution direction finding from higher order statistics: the 2-MUSIC agorithm", IEEE Trans. Sig. Proc., vo. 54, pp. 2986-2997, Aug. 2006. [6] R.T. Compton, JR., "Adaptive Antennas - Concepts and Performance", Prentice Ha, Engewood Ciffs, New Jersey, 1988. [7] M.C. Dogan, J.M. Mende, "Appications of cumuants to array processing - Part I : Aperture extension and array caibration", IEEE Trans. Sig. Proc., vo. 43, N 5, pp. 1200-1216, May 1995. [8] E.R. Ferrara, Jr, T.M. Parks, "Direction finding with an array of antennas having diverse Poarizations", IEEE Trans. Ant. Prop., vo. 31, pp. 231-236, March 1983 [9] A. Ferreo, E. Boyer, P. Larzaba, "Low-Cost agorithm for some bearing estimation methods in presence of separabe nuisance parameters", Eectronics Letters, vo. 40, pp. 966-967, Juy 2004. [10]E. Gönen, J.M. Mende, "Appications of cumuants to array processing - Part VI : Poarization and Direction of Arriva estimation with Minimay Constrained Arrays", IEEE Trans. Sig. Proc., vo. 47, pp. 2589-2592, September 1999. [11] S. Miron, N. Le Bihan, J.I. Mars, "Quaternion-MUSIC for vector-sensor array processing", IEEE Trans. Sig. Proc., vo. 54, N 4, pp. 1218-1229, Apri 2006. [12] B. Porat, B. Friedander, "Direction finding agorithms based on higher order statistics", IEEE Trans. Sig. Proc., vo. 39, pp. 2016-2024, September 1991. [13] R.O. Schmidt, "Mutipe emitter ocation and signa parameter estimation", IEEE Trans. Ant. Prop., vo. 34, pp. 276-280, March 1986. 2007 EURASIP 261