Scholarly article on topic 'Computationally Efficient DOA and Polarization Estimation of Coherent Sources with Linear Electromagnetic Vector-Sensor Array'

Computationally Efficient DOA and Polarization Estimation of Coherent Sources with Linear Electromagnetic Vector-Sensor Array Academic research paper on "Electrical engineering, electronic engineering, information engineering"

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Academic research paper on topic "Computationally Efficient DOA and Polarization Estimation of Coherent Sources with Linear Electromagnetic Vector-Sensor Array"

Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2011, Article ID 490289,10 pages doi:10.1155/2011/490289

Research Article

Computationally Efficient DOA and Polarization Estimation of Coherent Sources with Linear Electromagnetic Vector-Sensor Array

Zhaoting Liu,1 Jing He,2 and Zhong Liu1

1 Department of Electronic Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China

2 Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada H3G 2W1

Correspondence should be addressed to Zhaoting Liu, liuzhaoting@163.com Received 3 September 2010; Revised 10 December 2010; Accepted 16 January 2011 Academic Editor: Ana Perez-Neira

Copyright © 2011 Zhaoting Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper studies the problem of direction finding and polarization estimation of coherent sources using a uniform linear electromagnetic vector-sensor (EmVS) array. A novel preprocessing algorithm based on EmVS subarray averaging (EVSA) is firstly proposed to decorrelate sources' coherency. Then, the proposed EVSA algorithm is combined with the propagator method (PM) to estimate the EmVS steering vector, and thus estimate the direction-of-arrival (DOA) and the polarization parameters by a vector cross-product operation. Compared with the existing estimate methods, the proposed EVSA-PM enables decorrelation of more coherent signals, joint estimation of the DOA and polarization of coherent sources with a lower computational complexity, and requires no limitation of the intervector sensor spacing within a half-wavelength to guarantee unique and unambiguous angle estimates. Also, the EVSA-PM can estimate these parameters by parameter-space searching techniques. Monte-Carlo simulations are presented to verify the efficacy of the proposed algorithm.

1. Introduction

A typical electromagnetic vector-sensor (EmVS) consists of six component sensors configured by two orthogonal triads of dipole and loop antennas with the same phase center. Therefore, an EmVS can simultaneously measure the three components of the electric field and the three components of the magnetic field. Since its introduction into signal processing community [1, 2], a significant number of research has been done on EmVS array processing [3-19]. For application considerations, different types of EmVS containing part of the six sensors are devised and manufactured [3, 20, 21].

In the study of direction finding applications, conventional eigenstructure-based source localization techniques have been extended to the case of the EmVS array. ESPRIT/ MUSIC algorithms using EmVS arrays obtain thorough

investigations [10-12, 16-19]. The signal subspace and noise subspace are usually constructed by decomposing the column space of the data correlation matrix with the eigen-decomposition (or singular value decomposition) techniques [22, 23]. Because the decomposing process is computationally intensive and time consuming, the eigen-structure-based techniques may be unsuitable for many practical situations, especially when the number of vector sensors is large and/or the directions of impinging sources should be tracked in an online manner.

Furthermore, the eigenstructure-based direction finding techniques using the EmVS arrays usually assume incoherent signals, that is, that the signal covariance matrix has full rank. This assumption is often violated in scenarios where multi-path exists. Coherent signals could reduce the rank of signal covariance matrix below the number of incident signals, and hence, degrade critically the algorithmic performance.

To deal with the coherent signals using the EmVS array, a polarization smoothing algorithm (PSA) has been proposed to restore the rank of signal subspace [19]. The PSA does not reduce the effective array aperture length and has no limit to array geometries. However, the PSA-based method has non-negligible drawbacks. (1) It assumes the intervector sensor spacing within a half-wavelength to guarantee unique and unambiguous angle estimates; (2) it is not able to estimate the polarization of impinging electromagnetic waves; (3) the EmVS type limits the maximum number of resolvable coherent signals.

In this paper, we employ a uniform linear EmVS array to perform parameter estimation of coherent sources. Firstly, to decorrelate the coherent sources, an EmVS sub-array averaging-based pre-processing (EVSA) algorithm is developed. Then the EVSA algorithm is coupled with the propagator method (PM) [24, 25] to estimate parameters of the coherent sources without eigen-decomposition or singular value decomposition unlike the ESPRIT/MUSIC-based methods. By using the vector cross-product of the electric field vector estimate and the magnetic field vector estimate, the proposed EVSA-PM can estimate both the DOA and polarization parameters, hence, can overcome the drawbacks of the PSA-based algorithms to some extent. The vector cross-product estimator is valid to a six-component EmVS array. For the array comprising any types of EmVSs, the EVSA-PM with parameter-space searching techniques is developed to estimate the parameters. The EVSA-PM can be regarded as an extension of the subspace-based method without eigendecomposition (SUMWE) [26] to the case of the EmVS arrays. The SUMWE is also a PM-based method, which estimates the DOA of coherent sources using unpolarized scalar sensors by an iterative angle searching. However, the proposed methods make use of more available electromagnetic information, and hence, should outperform the SUMWE algorithm in accuracy and resolution of DOA estimation.

The rest of this paper is organized as follows. Section 2 formulates the mathematical data model of EmVS array. Section 3 develops the proposed EmVS-PM. Section 4 presents the simulation results to verify the efficacy of the EmVS-PM. Section 5 concludes the paper.

2. Mathematical Data Model

Assume that K narrowband completely polarized coherent signals impinge upon a uniform linear EmVS array with M vector sensors (M > 2K), and the array is neither mutual coupling nor cross-polarization effects. The K is known in advance and the kth incident source is parameterized {9k, fk, Yk, nk}, where 0 < 9k < n/2 denotes the kth source's elevation angle measured from the vertical z-axis, 0 < fk < 2n represents the kth source's azimuth angle, 0 < yk < n/2 refers to the kth source's auxiliary polarization angle, and -n < nk < n symbolizes the kth source's polarization phase difference. For a six-component EmVS, the steering vector of

the kth unit-power electromagnetic source signal produces the following 6 X 1 vector:

c(Ok, fk, Yk, nk)

"ci,k" ex,k

C2,k ey,k

C3,k def ez,k

C4,k hx,k

C5,k hy,k

_C6,k_ _hz,k_

cos fk cos 9k

sin fk

sin fk cos 9k cos fk

- sin 9k

- sin fk cos fk

cos fk cos 9k - sin fk cos 9k sin 9k

sin Ykejnk cos yk

'g(ïk flk)

where ek d= [ex,k,ey,k,ez,k]T and hk d= [hx,k,hy,k,hz,k]T denote the electric field vector and the magnetic field vector, respectively.

The intersensor spatial phase factor for the kth incident signal and the mth vector sensor is qm(6k, fk) == ej2n(xmuk +ymVk)/x, where uk == sin 6k cos fk and vk == sin 6k sin fk signify the direction cosines along the x-axis and y-axis, respectively. (xm,ym) is the location of the mth vector sensor, X equals the signals' wavelength. Denoting the spacing between adjacent vector sensors as (Ax, Ay), we have xm = xi + (m - l)Ax, ym = yi + (m - l)Ay. The 6 X 1 measurement vector corresponding to the mth vector sensor can be expressed as

Xm(f)d= [xm,l(t),..., xm,6(t)] T

= X %m(6k, fO C( 6k, fk, Yk, Vk)sk (t) + Wm(t),

where wm(t) = [wm>i(f),...,wm,6(t)] is the additive zero-mean complex noise and independent to all signals. xmn„(t) and wm,n(t) refer to the measurement and the noise corresponding to the mth vector sensor's nth component, respectively; sk(t) represents the kth source's complex envelope. Without loss of generality, we consider the signals {sk(t)} are all coherent so that they are all some complex multiples of a common signal s1(t). Then, under the flat-fading multipath propagation, they can be expressed as Sk (t) = fiks1(t) [26,27], where fa is the multipath coefficient that represents the complex attenuation of the kth signal with respect to the first one = 1 and ¡5k = 0).

For the entire vector-sensor array, the array manifold, a( Ok, fk, Yk, nk) e C6Mx1,is given by

a(8k, fk, Yk, nk) == q(Ok, fk) ® c(Ok, fk,Yk, , (3)

where ® symbolizes the Kronecker product operator,

q(8k, fk) =f [^1(8k, fk),..., qM (Ok, fk )]r. With a total of K signals, the entire 6M X 1 output vector measured by the EmVS array at time t has the complex envelope represented as

z(t) = [xT (t),..., xM (t)]T

= £ a( Ok, fk, Yk, nk)sk (t) + n(t) (4)

= As(t) + n(t),

where A e C6MxK, s(t) e CKX1, n(t) e C6Mx1, and A = [a(81,f1,Y1,n1),...,a(Ok,fK,Yk,nK)]; s(t) = [s1(t),..., sk(t)]T, n(t) = [wf(t),...,wf (t)]T.

3. Algorithm Development

This section is devoted to the algorithm development. Section 3.1 develops the EVSA algorithm, Section 3.2 describes EVSA-PM algorithm for estimating both DOA and polarization parameters from the available EmVS steering vector estimates and Section 3.3 is for parameters estimation by parameter-space searching techniques.

3.1. EVSA Algorithm. Let us consider the subarray averaging scheme with a linear EmVS array, which is divided into L overlapping subarrays with K vector sensors and the Ith subarray comprises the lth to (i + K - 1)th vector sensor, where L = M - K +1. We use the first vector sensor as a reference (x1 = 0, y1 = 0), and then the corresponding 6K X 1 signal vector is given as

zi(t)d=f [xf(t),...,xf+K_1(t)]T = A0Dl-1 s(t)+ ni(t), (5)

where D e CK xK, and D d= diag^^^y^,..., ej2n(Ax"K+VK )/A); A0 e C6K xK contains the first 6K rows of A;

def f T T

ni(t) = [wf (t),...,wf+K-1(t)] . We can calculate the cross-correlation vector fln e C6Kx1 between zi(t) and xM,n(t)

fUn = E{Zi(t)xM,n(t)}

= A0D1-1 e{ s(t)sH (t)j aM,n + e{ ni (t)<,nj

pM,nrsA0Dl 1 p, l = 1, ..., L - 1; n = 1, ... ,6,

where £{■} denotes the expectation, rs = £{s1(t)s*(t)},

pi,n == pHa*n, ai,n == [qi(O1,f1)cn,1,...,qi(Ok,fK)cn,K]T,

p = [ft,..., fa]f. Similarly, the cross-correlation vector fln e C6Kx1 between Zi(t) and x1,n(t) is as follows

<pUn = E{zi (t)x*n(t)j

= p1,nrsAoDi 1 p, i = 2,..., L; n = 1,... ,6. Let us rewrite the vector fln as a 6 x K matrix

(i,n d= [J1 Vm..., Jk fin]

= pM,nrs[ A 1Di-1p,..., A k Di-1 p] = PMnnrsA i[ p,..., DK-1 p]

pM,n rsAiBQf,

where Jk = [06,6(k-1),I6,06,6(K-k)]; B = diag(ft,...,fa); Ai is the 6 x K matrix with the column Ckqi(Ok,fk), k = 1,...,6; Q is the K x K matrix with the column [q1(Ok, fk),..., qK (Ok, fk)]1. Similarly, the vector f Un can be rewritten as

ti,n = J1 fi,n,...,JkfJ = P1,nrsAiBQT. (9)

Therefore, concatenating (i,n for i = 1,..., L - 1 and (i,n for i = 2,..., L, respectively, we can get two correlation matrices

Rn d=f [of,n, ®ln,..., oTl-1),^T = PM,nrsABQf, Rn = [Of,n, (f,n,..., ®f,nJ = P1,nrsABDQT,

where Rn e C6(L-1)xK, Rn e C6(L-1)xK, and A d= [Af,...,

Af^r includes the first 6(L - 1) rows of A. With (10), the EmVS subarray averaging (EVSA) matrix can be formulated

R1,..., R6, R1,..., R^J = Afi,

def = |R1,

where fi = rsB[pM,1QT,..., Pm,6Qt, pUDQr,..., p^DQr ]. Note that B and D are diagonal matrices with nonzero diagonal elements, and Q is full rank when all sources impinge with the distinct incident directions. Then the Rn and Rn are of rank K, and hence, R is of rank K and can be used to estimate the DOA and the polarization parameters of the coherent sources.

In realistic cases where only a finite number of snapshots are available, the cross-correlation vector fln and fln can

be estimated as fl n = X(=1 Zi(t)xM n(t)/S and fl n = 1 Zi (t)x*n(t)/S, where S denotes the number of snapshots. With f! n and f! n, the matrix R is accordingly obtained using (8)—(11) .

Note that the proposed EVSA algorithm can also be used to the case of partly coherent or incoherent signals. To see this, we assume that the first K1(1 < K1 < K) incident

signals are coherent and the others are uncorrelated with these signals and with each other. Then after some algebraic manipulations, we can obtain

where e; is the 6(L - 1) dimensional unit vector whose z'th element is 1 and other elements are zero. In addition, we define

rv/V m ÎVÎV TT rr<

Rn = pu,nrSl AB QT + ARAM,nQT,

_ , rv T-. ÎV iV IT T-.

Rn = P1,nrs1 AB DQT + ADRAj|n QT,

Re = ETR = Aefi,

d=f Er A:

[AT1,..., A^] ,

where p,n

f Kn, P

= ...,i3Kl,0,...,0]T, B

diag(ft,..., fe,0,...,0), rSk = E{sk(t)s*k(t)}, R = diag(0,

..., rSKi+iy..., rSK ), Al,n = diag( qi (0^ fl) cn,1,... , qi (Ok , fK)cn,K). It is easy to find that the rank of Rn and Rn still equals K when all sources impinge with the distinct incident directions.

Remarks. (1) The proposed EVSA algorithm is still effective in the case of partly coherent or incoherent sources in which there exist two incoherent sources with the same incident directions but with the distinct polarizations. As shown in the appendix, the matrix R defined in (11) has full rank. However, neither the PSA [19] nor the SUMWE [26] algorithm can be so.

(2) The EVSA algorithm needs low computations. As seen from (6) and (7), the EVSA only needs compute the cross-correlations, which require 72(L - 1) cross-correlation operations. However, most of EmVS direction finding algorithms require to compute the correlations of all array data with (6M) correlation operations.

(3) The EVSA-based method may estimate both DOA and polarization parameters, while the PSA-based one can only estimate the DOA parameters because of the polarization smoothing.

(4) From (11), the EVSA algorithm can decorrelate more coherent sources than the PSA can do. The EVSA algorithm can decorrelate up-to L - 2 coherent sources regardless of EmVS's types, while the PSA can only decorrelate 6 coherent sources for six-component EmVS array, 4 for quadrature polarized array [19] and 2 for dual polarized array [19]. By coupling the forward/backward (FB) averaging technique [27], the maximum number of the coherent signals decorrelated by the PSA is doubled, however, it is only valid for the case of the symmetric array, for instance, uniform linear array, to which the proposed method is limited.

where Ae e C6(i-1)xK, Ae,n e C(i-1)xK(n = 1, ...,6) is a submatrix whose kth column is given as qe(0k, fk)cn,k

with qe(0k, fk) d= [<î1(0k, fk),..., qi-1(0k, fk)]T. These submatrices are related with each other by

: Ae,1 An,

where An e CKXK and An == diag(dn,1,..., dn,K) with dn,k == cn,k/c1,k denoting the kth source's invariant factor between the first and the nth EmVS component. We can divide Ae,n into

Ae,n Ae,n

, n = 1,... ,6,

where A$ e CKxK and a(2« e C(i-1-K)xK. Therefore, Ae,n can be rewritten as

u tt def ua(2)\ t / a(1)i t / a(2k T , A (1K T , a (2)\ T-¡T

where U = [(Ae,1) ,(A^,2) ,(Ae,2^) ,...,(Ae,6) ,(Ae,6) ] . Obviously, Aej^n! is a matrix with full rank. The K X (6L - 6 -K) propagator matrix P can be defined as a unique linear operator which relates the matrices A(e1,1) and U through the equation

We partition Pw into Pw = [Pf, PT,..., PT1 ]T, where P1 to P11 have the dimensions identical to Ae«, A^, Ae13, A^, Ae« ,Ae24, Ae«, Ae25, Ae16), and Ae2, respectively. Thus, we have

PiA(1? - A(2?

P1 Ae,1 = Ae,1 ,

3.2. EVSA-PM Algorithm for Estimating Parameters from the EmVS Steering Vector. The EVSA-PM algorithm performs the estimation of the coherent sources' DOA and polarization parameters by using the vector cross-product operation of the estimated electric field vector and magnetic field vector. For this purpose, we define an exchange matrix

E = [e1, e7,..., e6(L-2)+1, e2, e8,..., e6(L-2)+2,..., e6, e12,..., e6(i-1^,

P2n-1Ae« = Ae21 An, n = 2,... ,6.

Equations (20) and (21) together yield

P|P2n-1 = a(11aJ Adi

n = 2,..., 6,

where f denotes the Pseudo inverse.

Equation (22) suggests that the matrices PfP2n-1 (n = 2,..., 6) have the same set of eigenvectors and the corresponding eigenvalues lead to the invariant factors of the same sources. Hence, we can obtain the eigenvalue pairs by

< 10-1 o

SNR (dB)

0.5A -x- 2A

-+- 4A -©- 8A

R1 < 10-1

SNR (dB)

--- 0.5A -*- 2A

-+- 4A -e- 8A

(a) (b)

Figure 1: DOA estimates RMSE of the proposed EVSA-PM against SNRs. (a) Source 1, (b) source 2.

matching the eigenvectors of the different matrices P[ P2n-1 (n = 2,...,6) [11]. With the estimated c(0k,fk,Yk,nk) = [1,d2,k,...,d6,k]T, the Poynting vector estimates can be obtained by the vector cross-product operation and then the DOA and polarization parameters are estimated from the normalized Poynting vectors [11]. For a dipole triad array or loop triad array, the estimates of the electric field vector ek or the magnetic field vector hk can be done in the same way. In this case, the DOA and polarization parameter estimates can be obtained using the amplitude-normalized estimates of the electric or magnetic field steering vector [3].

In order to calculate the propagator matrix P, we divide the matrix Re into Re = [Rf1, Rf2] , where Re1 and Re2 consist of the first K rows and the last 6L - 6 - K rows of Re. In the noise-free case, we have PHRe1 = Re2. In the noise case, a least squares solution can be used to estimate P

P = (Re1 Re1 rH .

3.3. EVSA-PM Algorithm for Estimating Parameters by Angle Searching. The EVSA-PM is also applied to the uniform linear array comprising any types of identical EmVSs. In the case, the estimates of DOA and polarization parameters cannot be extracted from the estimates of the steering vectors. However, they are obtainable by the use of parameter-space searching techniques. We here use two-dimensional angle searching to estimate the DOA.

Consider N-component EmVS array (2 < N < 6), then the matrix Ae in (15) can be rewritten as Ae = [Af,1, ..., AfN]T e CN(L-1)xK, and Ae,n can also be rewritten as

Ae,n = Qefl n, n = 1,..., N,

where Qe = [qe(01, f1),..., qe(0K, fK)] e C(L-1)xK, n n = diag(cn,1,...,cn,K) e CKxK.

Defining gn =f [0l-1,(l-1)(«-1),Il-1,0l-1,(l-1)(n-n)] e R(L-1)XN(L-I)y we have Rg Xn=l%nRe = QeTTil, where

n = Xn=i un. Partitioning Rg into Rg = [RjiRj2]T> where Rg1 and Rg2 consist of the first K rows and the last L -1 - K rows of Rg, we have the propagator matrix P =

(Rg1 RH1) 1 Rg1 RH2. Then the source's DOA parameters can be estimated as

{dky<pk} = arg minqf (dy<p)WHqe(dy<p)y (25)

where W d=f [PT, -Il-1-k]t.

4. Simulations

We conduct computer simulations to evaluate the performances of the proposed EVSA-PM. Comparison with the PSA based [19] PM (PSA-PM) and the SUMWE algorithm [26] is also made. For proposed EVSA-PM algorithm, the parameter estimates shown in Figures 1-5 are extracted from the EmVS steering vector, and those shown in Figure 6 are obtained by angle searching. The performance metrics used is the root mean square errors (RMSEs) of the sources' 2-D DOA and the polarization parameters estimates, where the RMSE of kth source's 2-D DOA estimate is defined as

RMSE* = --

--- 0.5A -*- 2A

SNR (dB)

- + - 4A -e- 8A

0.5A -*- 2A

SNR (dB)

- + - 4A -e- 8A

(a) (b)

Figure 2: Polarization state estimates RMSE of the proposed EVSA-PM against SNRs. (a) Source 1, (b) source 2.

3 10-1

10-3 -10

10 20 SNR (dB)

a- EVSA-PM (A = 4A) SUMWE

EVSA-PM (A = A/2) - CRB

PSA-PM

< 10-1 o

10 20 SNR (dB)

-e- EVSA-PM (A = 4A) SUMWE

-©- EVSA-PM (A = A/2) - CRB

-a- PSA-PM

(a) (b)

Figure 3: DOA estimate RMSEs of EVSA-PM, PSA-PM, and SUMWE against SNRs. (a) Source 1, (b) source 2.

and the RMSE of kth source's polarization state estimate is defined as

RMSE* =

z \ Xtfejc - m)

where 0e,k, , ye,k, and ne,k symbolize the eth Monte Carlo trial's estimates for the kth source's directions and polarization states and E is the total Monte Carlo trials. In the simulations, E = 500.

Figures 1 and 2 plot the RMSEs of the sources' DOA and polarization estimates against signal-to-noise ratio (SNR) levels using the EVSA-PM. The SNR is defined as SNR = (1/K) XK=1 \2/&2> where a2 is the noise power lever. Two equal-power narrowband coherent signals impinge with parameters 01 = 75o, (pi = 35o, yi = 45o, n = -90o, 02 = 80o, p2 = 30o, y2 = 45o, and n2 = 90o, and the multipath coefficient is set to fi2 = exp(j * 50o). The uniform linear array consists of 12 six-component EmVSs. The intervector

sensor spacing is set as A = yA^ + A2 = 0.5A,2A,4A, and 8A, respectively. The snapshot number is 300. It is seen from that both DOA and polarization estimation errors decreases as the SNR increases. Also, the increase of intervector sensor spacing, which results in the array aperture extension,

Snapshot number

EVSA-PM (A = 4A) -*- SUMWE

EVSA-PM (A = A/2) - CRB

PSA-PM

~ -i---_ ^

—------ ~

Snapshot number

-B- EVSA-PM (A = 4A) -*- SUMWE

-e- EVSA-PM (A = A/2) - CRB

-a- PSA-PM

(a) (b)

Figure 4: DOA estimate RMSEs of EVSA-PM, PSA-PM and SUMWE against the number of snapshots. (a) Source 1, (b) source 2.

70 60 50 40 30 20 10 0

65 66 67 68 69 70 71 72 73 74 75 Elevation angle

65 66 67 68 69 70 71 72 73 74 75 Elevation angle

65 66 67 68 69 70 71 72 73 74 75 Elevation angle

Figure 5: The histogram of the estimated elevation using the three methods. (a) EVSA-PM; (b) PSA-PM; (c) SUMWE.

contributes to the estimation accuracy enhancement. Since the estimation of DOA and polarization is extracted from the EmVS steering vector, which contains no time-delay phase factor, we can obtain more accurate but unambiguous estimates of coherent source using an aperture extension array without a corresponding increase in hardware and software costs [12].

Figures 3 and 4 make the comparison between the proposed algorithm with PSA-PM and SUMWE under different SNRs and number of snapshots. The impinging signal parameters are same as in Figures 1 and 2. We use 300 snapshots in Figure 3 and set SNR = 20 dB in Figure 4. For the proposed algorithm, a uniform linear array with 8 dipole-triads, separated by A = A/2 and 4A is considered. For the PSA-PM, we use an L-shape geometry, with 8 dipole-triads uniformly placed along x-axis for estimating Uk and 8 dipole-triads uniformly placed along y-axis for estimating Vk. For the SUMWE, we use an L-shape geometry, with 12 unpolarized scalar sensors uniformly placed along x-axis for estimating Uk and 12 unpolarized scalar sensors uniformly placed along y-axis for estimating Vk. Hence, the hardware costs of the SUMWE and the presented algorithm are comparable. The intersensor displacement for the PSA-PM and SUMWE is a half-wavelength, since these two algorithms would suffer angle ambiguities when two sensors are spaced over a half-wavelength. The curves in these two figures unanimously demonstrate that the proposed EVSA-PM with A = 4A can offer performance superior to those of the PSA-PM and SUMWE.

From the computational complexity analysis, the major computational costs involved in the three algorithms are the calculation of the corresponding propagator and correlation matrix, and the numbers of multiplications required by the EVSA-PM, the PSA-PM, and SUMWE are in the order of O(3M1KF + 18(M1 - 1)F) « 174F, O(2M1KF + 6M2F) ~ 416F, and O(2M2KF + 4(m2 - 1)F) ~ 92F, respectively, where Mi = 8, M2 = 12, and F denotes the number of snapshots. Therefore, the proposed EVSA-PM also is more computationally efficient than the PSA-PM.

The proposed EVSA-PM can fully exploit polarization diversity to resolve closely spaced sources with distinct polarizations. To verify this performance, we assume two incident coherent sources with parameters 91 = 70°, 02 = 70.5°, f1 = 90°, f2 = 90°, Y1 = 45°, y2 = 45°, m = -90°, and n2 = 90°. Others simulation conditions are the same as that in Figure 4, except that the SNR is set at 35 dB. Figure 5 shows the histogram of the estimated elevation using the three methods based on 500 independent trials. From the figure, we can observe that the proposed EVSA-PM can resolve the closely spaced sources. However, the other two methods fail.

Figure 6 plots the spatial spectrum to present comparison of the maximum numbers of coherent signals, which can be, respectively, resolved by the proposed algorithm, the SUMWE, the PSA-PM, and the PSA-FB-PM which combines the PSA with the FB averaging technique [27]. We consider a uniform linear array comprised of 20 unpolarized scalar sensors for the SUMWE and 20 quadrature polarized vector

-a- EVSA-PM -e- PSA-FB-PM

-b- PSA-PM -k- SUMWE

Figure 6: Spatial spectrum of EVSA-PM, PSA-PM, PSA-FB-PM, and SUMWE for nine coherent sources.

sensors [19] (i.e., N = 4, M = 20) for all the other three algorithms and estimate the sources' direction by angle searching. The intervector sensor spacing of array is a half-wavelength. Like [19], we assume zero elevation incident angle (8k = 90°) and randomly chosen polarizations for all sources, and set SNR = 15 dB.

Nine equal power, coherent sources with the azimuth incident angles 35°, 50°, 65°, 80°, 90°, 100°, 110°, 125°, and 140° are considered, and the corresponding multipath coefficients ft = exp(j * 10°(k - 1)), k = 1,...,9. This figure shows that the proposed EVSA-PM and the SUMWE successfully resolve the nine coherent signals, while the PSA-PM, and the PSA-FB-PM fail to do so. This is due to the factor that the PSA-PM and the PSA-FB-PM, respectively, only can resolve min(N, M - 1) = 4andmin(2N, M - 1) = 8 coherent sources at most, while the proposed EVSA-PM can resolve L - 2 coherent sources (L = M - K +1), and the maximum number of coherent signals resolved using the SUMWE is equal to that using the EVSA-PM.

5. Conclusions

This paper employs a linear electromagnetic vector-sensor array to propose a novel pre-processing algorithm for decorrelating the coherent signals by electromagnetic vector-sensor subarray averaging, and combine it with the propagator method to estimate the DOA and polarization of coherent sources without eigen-decomposition into signal/noise subspaces. Compared with the existing estimate algorithms, the proposed algorithm makes use of more available electromagnetic information, hence, has an improved estimation performance. It does not necessarily require the intervector sensor spacing of a half-wavelength, enable decorrelation of more coherent signals, and joint estimation of DOA and polarization of coherent sources.

Appendix

From (12), we can obtain

[R1,..., R6] = AFG,

where F = diag (^ ft,... ,^^1,^+1 qM(0K1+1, fK1+1),...,

rSKqM(0K, fK))

r-o T ~ 1 T

pM,1h1 pM,2h1

PM,1hK1

PM,2hK1

c1,K1+1hK1+1 c2,K1+1hK1+1

c1,KhTK

hk == [q1 (0k, fk), ...,qK (0k, fk)Y

c2,KhTK

pM,6h1

PM,6hK1

.. c6,K1+1hK1+1

c6,KhTK

The matrix A is of full column rank due to the distinct polarizations (although there are two sources from the same direction). The diagonal matrix F has full rank. If the two sources have the same incident directions but with the distinct polarizations, and are uncorrelated with each other (i.e., the two sources are not all included in the set consisting of the first K1 coherent sources), the K X6K matrix G is of full row rank. Therefore, in this scenario, the matrix [ R1,..., R6 ] is of rank K. Similarly, the matrix [R1,..., R6] also is of rank K. Thus, the matrix R defined in (11) still has full rank.

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