Chinese Journal of Aeronautics, (2015), 28(6): 1718-1724

JOURNAL OF

AERONAUTICS

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cn www.sciencedirect.com

DOA and polarization estimation via signal reconstruction with linear polarization-sensitive arrays

Liu Zhangmeng *

School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China

Received 14 January 2015; revised 3 April 2015; accepted 3 August 2015 Available online 20 October 2015

KEYWORDS

Array signal processing; DOA estimation; Polarization estimation; Polarization sensitive array; Sparse Bayesian reconstruction

Abstract This paper addresses the problem of direction-of-arrival (DOA) and polarization estimation with polarization sensitive arrays (PSA), which has been a hot topic in the area of array signal processing during the past two or three decades. The sparse Bayesian learning (SBL) technique is introduced to exploit the sparsity of the incident signals in space to solve this problem and a new method is proposed by reconstructing the signals from the array outputs first and then exploiting the reconstructed signals to realize parameter estimation. Only 1-D searching and numerical calculations are contained in the proposed method, which makes the proposed method computationally much efficient. Based on a linear array consisting of identically structured sensors, the proposed method can be used with slight modifications in PSA with different polarization structures. It also performs well in the presence of coherent signals or signals with different degrees of polarization. Simulation results are given to demonstrate the parameter estimation precision of the proposed method.

© 2015 The Author. Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The polarization sensitive arrays (PSA) are able to collect the signal energy in different polarization directions and they have been widely used to improve the performance of direction-of-arrival (DOA) estimation (see Refs.1-6 and the references

therein). Many PSA with different polarization structures are now available in practical systems, such as the co-centered crossed-dipole pair (CCD) array, the co-centered orthogonal loop and dipole (COLD) array and the electromagnetic vector array (EVA) of super-resolution compact array radiolocation technology (SuperCART). However, the research on DOA estimation with PSA has lagged behind as many shortcomings still remain in the existing methods.

When 2-D DOA estimation and joint polarization estimation are required, the existing methods generally turn to the multi-dimensional searching techniques as the parameters can hardly be separated in the objective functions.7-9 Such a computationally prohibitive searching procedure greatly blocks the theoretical study and practical application of those

* Tel.: +86 731 84573490. E-mail address: liuzhangmeng@nudt.edu.cn Peer review under responsibility of Editorial Committee of CJA.

http://dx.doi.org/10.1016/j.cja.2015.09.005

1000-9361 © 2015 The Author. Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

methods. Therefore, some of the researches focus on the simpler 1-D estimation problems.10'11 In addition, as more than one parameter is usually concerned for each signal with the PSA, a possibly confusion-inducing procedure of variable-pairing is required for successful parameter estimation of multiple signals.7'10 Spatial and polarization smoothing techniques have also been combined with the ordinary subspace-based DOA estimation methods, so as to separate coherent sig-nals.11'12 However, these techniques can be realized only with particularly designed PSA, thus they have been blocked from widespread applications. Similar constraints lie in the method proposed for DOA estimation when completely and incompletely polarized signals coexist,13 as it can be used only on PSA with particular triangular geometries.

The sparsity of the incident signals in space is a comprehensive property in various array applications. Previous research based on the exploitation of such a property in scalar sensor arrays has witnessed significant performance improvements, especially in scenarios with low signal-to-noise ratio (SNR) and much limited snapshots.14-19 Among the existing sparsity-based DOA estimation methods, the ones based on the sparse Bayesian learning (SBL) technique20,21 exceed their counterparts in DOA estimation precision in adequate scenarios.17-19 In this paper, the SBL technique is introduced to solve the direction and polarization estimation problem with PSA. By exploiting the sparsity of the incident signals, the signal components contained in the differently polarized array measurements are reconstructed first, and then combined with respect to the sources to estimate the parameters of interest. In order for nota-tional facilitation, the proposed method is named reconstruction and combination of polarized signal components and ReCoP for short. It avoids the computationally prohibitive multi-dimensional searching procedure and the confusion-inducing variable-pairing procedure. With very slight modifications, it can be used in various linear PSA consisting of identically structured sensors, such as CCD arrays, COLD arrays and SuperCART, and it is also able to process coherent signals and signals with different degrees of polarization.

The rest of the paper consists of six parts. Section 2 reviews the observation model of the PSA, the SBL technique is introduced in Section 3 to reconstruct the polarized signal components, and those reconstructed signals are combined in Section 4 to estimate the direction and polarization parameters. Based on the differences during method implementation between the proposed method and its counterparts, Section 5 highlights some special properties of the proposed method. Section 6 demonstrates the performance of the proposed method via simulations and Section 7 concludes the whole paper.

2. Model formulation

Suppose that K transverse electromagnetic waves impinge onto an M-element PSA, the azimuths of the signals are #i,#2,... ,#K 2 [0, p], which are defined as the projections of the incident signal directions on the x-y plane to the x axis, and their elevations, defined as the signal directions to the z axis, are y1, u2,..., UK 2[0, p/2]. The sketch of a linear COLD array is shown in Fig. 1, where s(t) indicates the incident signal. When only 1-D DOA estimation is concerned, one should set u1 = U2 — ' " — UK — p/2.

Fig. 1 Sketch of a linear COLD array. The output of the PSA at time t is

~(t) = A«(t) + v(t)

where A — [a^, «i,v, «2,h, «2,v, •••, «K,h, «k,y], «¿,h = ak 0 Wk,h, ~k,v — ak 0 Wk,v, 0 represents the Kronecker product, the

subscripts

are used to indicate the horizontal

and vertical components, respectively, ak — [ej2pd1cos #k sin Ut/k , ej2pd2 cos #t sin ut/k ,'"' ,ej2pdM cos #t sin ut/k]T stands for the phase-shift vector depending on the array geometry, k is the signal wavelength, dm denotes the distance between the mth sensor and the reference point on the array axis, Wt h — NWt,h, Wt,h — [- sin #t , cos #t , 0 , cos Uk cos #t , cos Uk sin #t , - sin Ut]T, Wt,v — NWt,v, Wk,v — [cos Uk cos #k , cos Uk sin #k , - sin Uk , sin #k , — cos #k , 0]T, N indicates the polarization dimensions that the

array selects from the EVA, e.g., S —

for COLD arrays; «(t) — [«T(t), «T(t), ••• ,«J(t)]T, ak(t) — [sk,h(t),skv(t)]T, sk,h(t) and sk,v(t) represent the polarized components of the kth signal in two orthogonal polarization directions; v(t) is the additive white Gaussian noise independent of the signals with variance r2. For completely polarized signals, sk,h(t) and sk,v(t) are linearly dependent as «k(t) — [cos /k , sin /ke>gk]Tsk(t), with /k and gk representing the polarization angle and phase of the kth signal and sk(t) being the signal waveform.

By separating the outputs of the array elements with identical polarization directions from ~(t), P measurement vectors of x1(t), ,xP(t)eCMx1 can be obtained as

follows:

xp(t) — Yi ak(gTp«k(t)) + vp(t) — Awp(t) + vp(t) k—1

p — 1,2,... ,P

where P represents the number of polarization directions of each array sensor, xp(t) — GpX(t), Gp — IM 0 e^, IM denotes the identity matrix with dimension M x M, ep e CPx1 stands for a vector with its pth element being the only non-zero

one of 1, gkp — [Wk,h > Wk,v]TeP, wk,p(t)—gT,p«k(t), wp(t) —

[W1 ,p(t), W2 ,p ( t), ••• ,WK,p(t)]T, A — [a1, a2 , ••• ,ax], Vp(t) — Gpv(t). Eq. (2) indicates that the P measurement vectors consist of signals impinging from the same K directions and

independent noise. As K is generally very small, those vectors are much sparse from the perspective of spatial power distribution.

In practical applications, N snapshots are collected by the PSA at time instants of t — tx, t2 , ■■■, tN, the array output matrices in different polarization directions can thus be denoted by

Xp — AWp + Vp p — 1,2 , ... ;P (3)

where Xp — [xp(t:) , Xp(t2) ,■■■ ,Xp(tN)], Wp — [\Vp(ti) , Wp(t2) , ■■■, Wp(tN)] and Vp — [vp(ti), Vp(t2), ■■■, Vp(tN)]. Based on the sparsity of the signal components in the P groups of polarized measurements, the signals will be reconstructed first in Section 3, and then combined for DOA and polarization estimation in Section 4.

3. Reconstruction of polarized signal components

By extending the signal components in Eq. (3) to the potential spatial space of the sources via zero-padding, one can yield the spatially overcomplete formulation of the measurement matrices as follows:

Xp = AWp + Vp p — 1,2,..., P

where A — [«(£1), «(£2), ■■■, «(£l)], £ — cos # sin u, «(£) — [ej2pd1 i/k, ej2pd2i/i, ■■■, ej2pdMi/k]T, ¿^ £2, ■■■, are L discrete samples of £ within [—1,1] used to indicate the possible incident signals with moderately small quantization errors, which satisfy £1 < £2 < ■■■ < Wp — [Wp(ti), Wp(t2), ■ ■ ■,Wp(tN)], wp(t) is the zero-padded extension of wp(t), Wp has nonzero rows only corresponding to the «(£)'s associated with £ — cos #k sin uk.14-19 As X!, X2, ■■■, XP consist of the signals impinging from the same K directions, Wi, W2, ■■■, WP share identical sparsity profiles. The measurement matrices will be decomposed on the discrete grid of [£1, £2, ■■■, £L] in this section via sparse Bayesian reconstruction17-21 so as to recover the signal components.

As the signal powers are distributed with unknown ratios in different polarization directions, independent hyperparameters should be introduced during the Bayesian reconstruction process to represent the prior power spectrum of W1, W2, ■■■, WP along their rows, i.e.,

„(t)~N(0,Cp) t — tu t2,

where yp 2 RLx1, with its ith element of ypl being inverse-gamma distributed with parameters (al; bp ¡), i.e.,

yp - IG(a , bp, i) / (yp , i)_(ai+1) exp(—bp, i/yp,i) (6)

By combining the above model assumptions with the Gaussian distribution of the additive noise, one can conclude in the probability densities of X1; X2, ■■■ ,XP with respect to C1, C2 , ■■■, CP, r2, a and B via straightforward mathematical derivations, where a =[a1; a2 , ■■■ , aL]T, B — [bn b2, ■■■ , bP], bp — [bp, H bp, 2 , ■■■ ,bp ,L]T. The qth iteration of the SBL technique when used to reconstruct X1; X2 , ■■■ ,XP consists of the following updates, which are derived following the guidelines in,17-21

v(q+1) — »

bT — bp)+£ E[(wpq+1)(t„))2]

= bp;1 /(a)

i.q+1) //Jq+1)

where wpl(t) represents the lth element of wp(t), wpq+1)(t) is Gaussian distributed with mean ipq+l)(t) and variance Rp?+1), E[(wpq+1)(t))2] — (ipq+1)(t))2 + (Zip+l))lj with E[.] being the expectation operator and (•)l indicating the matrix element on the lth row and lth column, and

.,(?+1) (A =

(t) = ((r2)" ) l'pl+r)Axp(tn)

^ — ((CV + ((r2)(q))_1A ha f1

where Cpq — diag(yp?)) and diag(») is the diagonalization operator. The update strategy of r2 can be obtained similarly as that in Ref.19, i.e.,

(r2)(q+1) = (r2)(q) +

MNP xtE [||Xp - AW/+1)n2] (12)

The updating process approaches a stationary state gradually, and the variable values are exported as their final estimates when a predefined iteration termination condition is achieved. The locations of the significant peaks in yp indicate the signal directions.

As the spatial grids of ^ £2 , ■■■, are obtained via discrete sampling, quantization errors will be brought in if the £/s corresponding to the locations of the most significant peaks in y1; y2 , ■■■ ,yP are taken as the estimates of cos #k sin yk directly. In order to eliminate such errors, the reconstructed peak clusters around the signal directions generally consisting of several adjacent spectral lines should be replaced with a single line to approximate the signal components more compactly, which results in the following estimator

of cos #k sin uk 9:

arg min

in£[ln IQpl+tr^Rp)],

k — 1,2,... ,K (13)

where Rp — — XHXp, tr(») is the trace operator, Qp — Jp ,_k + çp(x)a(x)aH(x), a(x) is defined similarly as a(£) in Eq. (4), Jp-_k — A-kCp,-kAHk + r2lM, Cp-k — diag(Cp,_k), Cp and r2 are the estimates of cp and r2 when the SBL iteration is terminated, A _k and yp,_k equal A and yp, respectively, after removing the vectors and elements associated with the kth peak cluster (the peak clusters are selected in a similar way as that in Refs.17-19; 1p(x) stands for the coefficient of a(x)aH(x) in Qp obtained by minimizing the objective function in Eq. (13) for particular x, which can be derived following 17-19:

Çp(x) —

aH(s)J__k(Rp _ Jp,_k)J__-_ka(x) (aH(x)J___ka(x)f

— a)" + N

By scanning the scope of each peak cluster with required precision, the values of the s's that minimize Eq. (13) for each signal are taken as the estimates of fcos #k sin uk}k—1 2 ■ ■ K, which are denoted by s 1, s2 , ■■■, sK.

By substituting s1 ,s2,'" ,sK into Eq. (14), one can also obtain the power estimates of the K signals, denoted by {1p k j4—1 '2' ' ' K, embedded in the P groups of measurements.

fi J p-1, 2 ,""", P

The expression of 1p ka(sk)aH (sk) can then be used to represent the kth signal component in Qp, and then substituted into Eq. (10) to recover the corresponding signal waveform within Xp, which is denoted by sf) — [^(¿0 ,s^)(?2),''' ^(¿n)].

4. Estimation of direction and polarization parameters

4.1. DOA estimation

The model formulation given in Section 2 indicates that the kth signal component within Xp at time t is

sk)(t) — eJ[Wk, h , Wk, v ]«k(t)

Denoting

izk(t) — [sk1)(t) , sf(t) ,•••, skP)(t)]T I Zk — [Zk(t0,Zk(t2),- • ,Zk(tN)] and Uk — [«k(t1), «k(t2)r- ,«k(tN)], then

Zk — [Wk,h > Wk,v]Uk

As ak(t) — [cos /k , sin /kejgt] sk(t) for completely polarized signals, Eq. (17) can be rewritten in a more concrete form in such cases as follows,

Zk — [Wk,h > Wk,v][cos 0k > sin 0kejgk ] sk

where st — [st(t0,st(t2),''' ,st(ttf)]. Eqs. (17) and (18) indicate that whatever the degrees of polarization (with the definition given in Ref.13) are, the columns of Zk are embedded in the subspace spanned by Wk,h and Wk,v, and the rank of Zk is 1 in the case of completely polarized signals. Therefore, by denoting the eigenvector of Z^Z^1 corresponding to the largest eigenvalue by ck and Ck — [ck , Wk,h , Wk,v], one can easily conclude that Ck is rank deficient and the smallest eigenvalue of CHCk is 0. Such a conclusion holds for completely polarized, incompletely polarized and unpolarized signals, but is usable only in PSA with a polarization direction number no smaller than 3. The linear PSA with polarization direction number being 1 or 2 can be used for 1-D DOA estimation only, where the DOA estimates are obtained directly:

#k — cos-1 (st) (19)

In practical 2-D DOA estimation problems, only the estimates of Zk and ck, denoted by Zk and ck, can be obtained

/p k—1, 2,'", K

from the reconstructed signal components of {sk jp—1 2 ' P. The estimation error within ck deviates Ck — [ck , Wk h , Wk v] from a precise rank-deficient matrix, and the linear dependence among ck, Wk,h and Wk,v should be tested by checking the distance between the smallest eigenvalue of ¿HCt and 0, which leads to the following 2-D DOA estimator when the polarization direction number of the PSA is not smaller than 3,

[#t, Ut] — arg min Km^CC), subject to cos # sin u — st

where Kmin(CHC'k) stands for the smallest eigenvalue of ¿HQ;. Although both azimuth and elevation parameters are included in the objective function in Eq. (20), they can be estimated via 1-D searching by exploiting the constraint of cos # sin u — sk. For example, when scanning # within [0 p], the value of U can be determined according to u — sin-1(sk/ cos #), the pair of # and U are then used to formulate Wk h and Wk v and calculate Kmin(¿HCk), in this way the K signal directions are obtained from Eq. (20) with K 1-D searching.

4.2. Po/anzation estimation

The polarization parameters of completely polarized signals can also be estimated from the reconstructed signal components. In the case of 2-D DOA estimation, one can conclude from Eq. (18) that ck — X-1 x [Wk,h,Wk,v][cos/k ,sin/kejgt]T, with X being the normalizing factor, thus

(16) [Ck , Wk,h , Wk,v]

—X 1 cos /k

-X-1 sin /k ejgk

— Ck rk — 0

which indicates that rk — [1, -X-1 cos /k , -X-1 sin /kejgt]T is the eigenvector of Ck associated with the eigenvalue of 0. Such a conclusion can be exploited to estimate the polarization parameters based on the DOA estimates.

Denote the eigenvector of ¿HCt corresponding to the eigenvalue of K^^Ck) when [# , u] — [#t , Ut] by ?t, it can be deemed as an estimate of rk/||rk||2, thus the polarization parameters of the kth signal can be estimated based on Eq. (21):

/k — tan-1 | ?k(3)/?k(2)| gk — Arg( ?k(3)/?k(2))

where Arg(») is the argument operator.

When PSA with two polarization directions are used for 1-D DOA estimation of completely polarized signals, it also holds that Ck — X-1 x [Wk,h , Wk ,v][cos 0k , sin /kejgk ] . By substituting #k — cos-1( sk) into Wk,h and Wk,v to calculate Wk,h and Wk v, then

X 1[cos /k , sin /kejgk] — [Wk,h , Wk,v] Ck — rk

which finally concludes in the polarization parameter estimates of

Uk — tan-1 | rk(2)/rk(1)| \ gk — Arg(rk(2)/rk(1))

5. Further discussions on ReCoP

5.1. Implementation scheme

During the proposition of the new method, many implementation details are buried in the discussions. The implementation scheme is presented in Table 1 to make the method being more comprehensive.

Table 1 Implementation scheme of ReCoP.

Implementation scheme of ReCoP Input: X(t1) , ~(t2) ,... ,~(tw)

(1) Obtain X2 ,... ,XP according to Eqs. (2) and (3)

(2) Repeat Eqs. (7)-(12) until convergence

(3) Estimate £^£2,...,£K via Eq. (13) by combining Eq. (14) DOA estimation:

(4) In 1-D scenarios, estimate source DOA via Eq. (19)

(5) In 2-D scenarios, for t — 1 ,2 , ■■■, K

(5-1) Compute s'p — [^(¿1) ,skp)(i2) ,... ,s'kp\iN)] for p — 1 ,2 ,..., P in terms of their expectations via Eq. (10) (5-2) Form Z k according to Eq. (16) (5-3) Eigen-decompose ZkZ H to obtain its largest eigenvector c k

(5-4) Form Wk,h and Wt,v for each direction candidate during 1-D directional searching based on the constraint cos # sin u — £t, and output #t and Ut when the smallest eigenvalue of ¿HCt is minimized Polarization estimation:

(6) In 1-D scenarios, calculate Wt,h and Wt,v based on the DOA estimates, combine Wt,h and Wt,v with the largest eigenvector of ZtZH, i.e., ct to compute r't according to Eq. (23), finally estimate the polarization parameters via Eq. (24)

(7) In 2-D scenarios, denote the eigenvector corresponding to the minimized smallest eigenvalue of ¿HCt by rt (which is obtained in the step of (5-4)), then estimate the polarization parameters via Eq. (22)

5.2. Properties of ReCoP

As the proposed method of ReCoP follows a much different guideline from the previous counterparts, it also has some individual properties that should be highlighted.

(1) Adaptation to different array structures. Given a linear PSA consisting of identically structured sensors, no further constraint is required on the array geometry or polarization for the implementation of ReCoP. The only difference of ReCoP when used in differently structured

" 1 0 0 0 0 0

PSA lies in the variation of N; it is

for CCD arrays,

10 0 0

for COLD

1 0 0 0 0 0 0 0 1 0 arrays and /6 for EMV arrays.

(2) Free of multi-dimensional searching. After reconstructing the polarized signal components from the array outputs, only an 1-D searching procedure is included in ReCoP to realize 2-D DOA estimation based on Eq. (20) and the other procedures are realized via numerical calculations.

(3) Free of variable-pairing. The locations of the spectrum peaks in ,''', CP associate the variables of interest with respect to each signal automatically, and the direction and polarization parameters are then estimated by taking the signals into consideration sequentially.

(4) Separation of coherent signals. In ReCoP, the polarized signal components are recovered from the array outputs via sparse reconstruction, which has been shown by previous researches to own much enhanced adaptation to inter-signal correlation.14-19 Thus ReCoP is expected to perform well in the coherent scenarios.

(5) Adaptation to signals with different degrees of polarization. No assumption on the degree of polarization is introduced for the signals when using ReCoP for DOA estimation; it adapts to completely polarized, incompletely polarized and unpolarized signals inherently without any modifications.

6. Simulation results

Suppose that two equal-power signals impinge onto an 8-sensor linear PSA, and the array geometry is assumed to be uniform and inter-spaced by half-wavelength to facilitate the comparison between different methods. The 1-D DOA and polarization estimation performance of ReCoP is demonstrated first with COLD array by setting the signal parameters as (#,/ ,g) — {(60° ,30°,45°) ,(80° , 80° , 60°)j. Ten snapshots are collected by the array. During the spatial reconstruction procedure, ReCoP selects the discrete grid as {Pn P2 ,''' ,PLj — sin({-90° ,-89° ,'" , 90° j). The DOA estimation precision is evaluated by the average root-mean-square-error (RMSE) of the two signals, which is defined as

where I stands for the number of simulations in each particular scenario and #f) the DOA estimate of the fcth signal in the ith simulation. The estimation precision of the polarization parameters is evaluated by the RMSE of the angular distance between the location of (/k , gk) and its estimate on the sphere.10 Previous methods of ESPRIT10 and Quaternion-MUSIC (QMUSIC)8 are also implemented for performance comparison.

As the SNR of the two signals vary from 0 dB to 20 dB, the direction and polarization parameter estimation RMSE of ReCoP, ESPRIT and QMUSIC obtained from 100 simulations are shown in Fig. 2(a) and (b). ReCoP exceeds the other two methods in parameter estimation precision in the given scenarios and the predominance is very significant when the SNR is lower than 10 dB. The average implementation times of the three methods have also been compared during the simulations. The detailed time list has been excluded here due to space limitation, but the statistical results show that ReCoP spends about 10 s to obtain the DOA estimates in each simulation, which is approximately half of the implementation time of QMUSIC, but is much larger than that of ESPRIT (less than 1 s). The QMUSIC method is computationally most expensive because it requires multi-dimensional direction searching. ReCoP and ESPRIT are implemented via 1-D searching, but ReCoP is slower due to the iterative realization of Eqs. (7)-(12).

Then fix the SNR of the two signals at 10 dB and vary their correlation coefficient from 0 to 1, and then the DOA estimation RMSE of the three methods obtained from 100 simulations are shown in Fig. 3. ESPRIT and QMUSIC deteriorate significantly in DOA estimation precision for more heavily correlated signals, and they even fail completely in coherent scenarios. However, ReCoP shows much enhanced adaptation to correlated and coherent signals and it achieves high-precision DOA estimation even when the correlation coefficient approaches 1.

0 5 10 15 20 0 5 10 15 20

SNR (dB) SNR (dB)

(a) Direction (b) Polarization

Fig. 2 Parameter estimation RMSE with COLD array.

10-i-,-,-,-,-

0 0.2 0.4 0.6 0.8 1.0

Correlation coefficient

Fig. 3 DOA estimation RMSE of correlated signals.

signal azimuth and polarization parameters are kept unchanged from the previous simulations, the elevations are set to be 80° and 60°, respectively, and their SNRs are fixed at 10 dB. The degrees of polarization of the two signals10 are set to be 1 and 0.5, which indicate that they are completely and incompletely polarized signals, respectively. In such a scenario with differently polarized signals, the previous methods are not able to estimate the signal directions with linear PSA.13 The 2-D DOA estimation results of ReCoP in 30 randomly chosen simulations are shown in Fig. 4, where the red markers indicate the true signal directions. The azimuth estimation errors of ReCoP are all smaller than 1 ° and the elevation estimation errors are mostly smaller than 3°.

7. Conclusions

Fig. 4 2-D DOA estimation of coexisted completely and incompletely polarized signals.

(1) The signal components embedded in differently polarized array measurements are reconstructed and combined in this paper, so as to estimate the 1-D and 2-D directions of signals with different degrees of polarization and the polarization parameters of completely polarized signals.

(2) The proposed method is free of multi-dimensional searching and variable-pairing. It adapts to polarization sensitive linear arrays with different polarization directions if only the array sensors have identical structures.

(3) The proposed method also performs well in separating coherent signals and signals with different degrees of polarization.

(4) As is demonstrated by the simulation results, the direction and polarization estimation precisions of the proposed method exceed its counterparts in scenarios with typical settings, especially when the SNR is low.

The polarization parameter estimation precisions of the three methods vary in a similar way as the DOA precisions, thus the results are not listed here for conciseness.

In the following group of simulations, the COLD array is replaced by an EMV array, so as to demonstrate the performance of ReCoP in 2-D DOA estimation and separation of coexisted completely and incompletely polarized signals. The

Acknowledgments

This study was co-supported by the National Natural Science Foundation of China (No. 61302141) and the Special Fund for Doctoral Subjects in Higher Education Institutions of China (No. 20134307120023).

Z. Liu

References

1. Nehorai A, Paldi E. Vector-sensor array processing for electromagnetic source localization. IEEE Trans Signal Process 1994;42 (2):376-98.

2. Li J, Stoica P, Zheng D. Efficient direction and polarization estimation with a COLD array. IEEE Trans Antennas Propag 1996;44(4):539-47.

3. Wong KT, Zoltowski MD. Closed-form direction finding and polarization estimation with arbitrarily spaced electromagnetic vector-sensors at unknown locations. IEEE Trans Antennas Propag 2000;48(5):671-81.

4. Nordebo S, Gustafsson M, Lundback J. Fundamental limitations for DOA and polarization estimation with applications in array signal processing. IEEE Trans Signal Process 2006;54(10):4055-61.

5. Wong KT, Zoltowski MD. Self-initiating MUSIC-based direction finding and polarization estimation in spatio-polarizational beam-space. IEEE Trans Antennas Propag 2000;48(8):1235-45.

6. Yuan X. Estimating the DOA and the polarization of a polynomial-phase signal using a single polarized vector-sensor. IEEE Trans Signal Process 2012;60(3):1270-82.

7. Hua Y. A pencil-MUSIC algorithm for finding two-dimensional angles and polarizations using crossed dipoles. IEEE Trans Antennas Propag 1993;41(3):370-6.

8. Miron S, Bihan NL, Mars JI. Quaternion-MUSIC for vector-sensor array processing. IEEE Trans Signal Process 2006;54 (4):1218-29.

9. Costa M, Richter A, Koivunen V. DOA and polarization estimation for arbitrary array configurations. IEEE Trans Signal Process 2012;60(5):2330-43.

10. Li J, Compton RT. Angle and polarization estimation using ESPRIT with a polarization sensitive array. IEEE Trans Antennas Propag 1991;39(9):1376-83.

11. Li J, Compton RT. Angle and polarization estimation in a coherent signal environment. IEEE Trans Aerosp Electron Syst 1993;29(3):706-16.

12. He J, Jiang S, Wang J, Liu Z. Polarization difference smoothing for direction finding of coherent signals. IEEE Trans Aerosp Electron Syst 2010;46(1):469-80.

13. Ho KC, Tan KC, Nehorai A. Estimating directions of arrival of completely and incompletely polarized signals with electromagnetic vector sensors. IEEE Trans Signal Process 1999;47(10):2845-52.

14. Gorodnitsky IF, Rao BD. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Signal Process 1997;45(3):600-16.

15. Malioutov D, Cetin M, Willsky AS. A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Trans Signal Process 2005;53(8):3010-22.

16. Liu ZM, Huang ZT, Zhou YY. Direction-of-arrival estimation of wideband signals via covariance matrix sparse representation. IEEE Trans Signal Process 2011;59(9):4256-70.

17. Liu ZM, Huang ZT, Zhou YY. An efficient maximum likelihood method for direction-of-arrival estimation via sparse Bayesian learning. IEEE Trans Wireless Commun 2012;11(10):3607-17.

18. Liu ZM, Huang ZT, Zhou YY. Sparsity-inducing direction finding for narrowband and wideband signals based on array covariance vectors. IEEE Trans Wireless Commun 2013;12(8):3896-907.

19. Liu ZM, Zhou YY. A unified framework and sparse Bayesian perspective for direction-of-arrival estimation in the presence of array imperfections. IEEE Trans Signal Process 2013;61 (15):3786-98.

20. Tipping ME. Sparse Bayesian learning and the relevance vector machine. J Mach Learn Res 2001;1(1):211-44.

21. Tzikas DG, Likas AC, Galatsanos NP. The variational approximation for Bayesian inference. IEEE Signal Process Mag 2008;25 (1):131-46.

Liu Zhangmeng received the B.S. and Ph.D. degrees in information and communication engineering from National University of Defense Technology in 2006 and 2012 respectively, and then became a teacher there. His main research interests are array signal processing and passive localization.