MA Huihui and TAO Haihong

National Laboratory of Radar Signal Processing，Xidian University，Xi'an 710071，China

Abstract: In this paper，a sparse nonuniform rectangular array based on spatially spread electromagnetic vector sensor (SNRASSEMVS) is introduced，and a method for estimating 2D-direction of arrival (DOA) and polarization is devised. Firstly，according to the special structure of the sparse nonuniform rectangular array (SNRA)，a set of accurate but ambiguous direction-cosine estimates can be obtained. Then the steering vector of spatially spread electromagnetic vector sensor (SSEMVS) can be extracted from the array manifold to obtain the coarse but unambiguous direction-cosine estimates. Finally，the disambiguation approach can be used to get the final accurate estimates of 2DDOA and polarization. Compared with some existing methods，the SNRA configuration extends the spatial aperture and refines the parameters estimation accuracy without adding any redundant antennas，as well as reduces the mutual coupling effect.Moreover，the proposed algorithm resolves multiple sources without the priori knowledge of signal information，suffers no ambiguity in the estimation of the Poynting vector，and pairs the x-axis direction cosine with the y-axis direction cosine automatically. Simulation results are given to verify the effectiveness and superiority of the proposed algorithm.

Keywords: sparse nonuniform rectangular array (SNRA)，spatially spread electromagnetic vector sensor (SSEMVS)，directioncosine，polarization，mutual coupling.

1. Introduction

The electromagnetic vector sensor array is superior to the conventional scalar array [1，2]due to the polarization diversity，which can maximumly exploit the propagation information of the electromagnetic wave and improves the performance of the radar system. Direction of arrival(DOA) and polarization parameters estimation is a key issue of electromagnetic vector sensor array processing，which has drawn an increasing attention in the past decades. At the beginning，the DOA and polarization parameters estimation algorithm is only the extension of the DOA estimation method based on the conventional scalar array. For example，in [3-5]，the traditional multiple signal classification (MUSIC) algorithm and the estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm are extended to parameters estimation methods for the electromagnetic vector sensor array. Recently，some particular DOA and polarization parameters estimation techniques for electromagnetic vector sensor array have been developed. In [6-9]，a vector-cross-product algorithm，which has low computational complexity，was proposed. In [10，11]，the polarization smoothing algorithms were presented to cope with correlated sources. In [12-14]，various parameters estimation algorithms based on quaternion algebra were developed，which are more robust to array error.

However，the array geometries in [6-14]were all based on the collocated electromagnetic vector sensor，which will introduce serious mutual coupling effect and sharply decrease the parameters estimation performance.To alleviate this problem，a spatially non-collocated electromagnetic vector sensor (SNC-EMVS) was proposed in[15]，whose components are distributed along two parallel lines，which can reduce the mutual coupling effect.However，it is strict with the location of antenna components. To relax this condition，Li et al. [16]put forward a spatially spread electromagnetic vector sensor (SSEMVS)array，and a joint DOA and polarization parameters estimation algorithm was presented. However，it can only resolve five sources at most and has poor parameters estimation performance. To solve this problem，a sparse uniform array based on SNC-EMVS was proposed in[17]，and a joint 2D-DOA and polarization parameters estimation method was devised，but it requires parameters pair matching operation，and has ambiguity in the estima-tion of the Poynting vector，which causes unsatisfactory parameters estimation results. Considering the nonuniform sparse array，Wang et al. [18]presented a DOA estimation method for the nonuniform scalar L-shaped array，which requires pair-matching operation and has low parameters estimation accuracy. Moreover，it cannot measure the polarization information. On account of the shortcomings of the above methods，a sparse nonuniform rectangular array SSEMVS (SNRA-SSEMVS) is proposed in this paper，and an algorithm is devised accordingly. The proposed method avoids suffering ambiguity in the estimation of direction finding，pairs the parameters automatically，alleviates the mutual coupling effect and improves the estimation accuracy without adding any antennas. Moreover，the use of SSEMVS also relaxes the condition of array configuration in [17].

The rest of the paper is organized as follows. Section 2 introduces the signal model. In Section 3，the proposed method is described and a computational complexity comparison between the proposed method and the reduced dimensional (RD)-MUSIC is presented. In Section 4，simulations are conducted to validate the performance of our method. Section 5 draws the conclusion.

2. The proposed signal model

2.1 SSEMVS array geometry

The SSEMVS is composed of three orthogonal dipoles and three orthogonal loops that are spatially spread in space as shown in Fig. 1. The position vectors of the three orthogonal oriented dipole antennas and loop antennas arePex，Pey，Pez，Phx，PhyandPhz，respectively. The locations of all antennas need to satisfy the following relationship:

Fig. 1 Sketch of a single SSEMVS

whereis the electric field vector，is the magnetic field vector，d(u，v) is the spatial steering vector of SSEMVS，λis the wavelength，andPs=[u，v，w]Tis the unit Poynting vector of the incident source withu=sinθcosφ，v=sinθsinφandw=cosθ respresenting the impinging source's direction-cosines respectively along thexaxis，theyaxis，and thezaxis. Herein，θ ∈ [0，π]denotes the incident source's elevation-angle，φ ∈ [0，2π) symbolizes the azimuth-angle，refers to the auxiliary polarization angle，η ∈ [-π，π) represents the polarization phase difference，and ⊙ symbolizes the element-wise multiplication between two vectors. Note that Θ (θ，φ) depends only on the arrival-angles，whereasg(γ，η) depends only on the polarization parameters.

2.2 Structure of SNRA-SSEMVS

The SNRA composed of SSEMVS is depicted in Fig. 2，where the black dot represents the SSEMVS array. ConsiderKnarrowband and independent signal sources impinging on this array，the received data of the (m，n)th SSEMVS at timetcan be expressed as

whereis the steering vector of thekth signal of the(m，n)th SSEMVS;is the phase shift factor，and the spacing between the first two adjacent SSEMVSs along thexaxis and theyaxis areDXandDY，respectively;sk(t) is thekth signal;nm，n(t)is a complex Gaussian white noise vector. The received signal can be expressed as

wherebkrepresents thekth source’s whole steering vector of the SNRA-SSEMVS，ukandvkdenote thekth signal’s direction cosines ofxandyaxes，γkand ηkdenote thekth signal’s auxiliary polarization angle and polarization phase difference.CM×1is the steering vector on thexaxis，qy(vk)=is the steering vector on theyaxis，s(t)=[s1(t)，···，sK(t)]T∈ CK×1is the signal vector，andn(t)∈ C6MN×1is the additional Guassian white noise. The array manifold can be represented as

where the superscript T represents the transpose operation and ⊗ symbolizes the kronecker product.

Fig. 2 Sketch of SNRA-SSEMVS

3. Algorithm description

Based on the array depicted in Section 2，a joint 2D-DOA and polarization estimation algorithm is proposed，and the procedure is shown in Fig. 3.

Fig. 3 Algorithm flowchart

In the proposed algorithm，the covariance matrixcan be calculated firstly，and the signal sub-spacesE1andE2can be obtained by the eigenvalue decomposition technique，then the array manifoldcan be estimated. Regard the whole nonuniform rectangular array as a set of line arrays parallel to thexaxis，and all the line arrays on the plane can be converted to a line array located on thexaxis by making phase compensation. According to the relationship among the elements on thexaxis，the ambiguous direction-cosine estimatecan be obtained. In a similar way，the ambiguous direction-cosine estimatecan also be estimated. After extracting the steering vectorof the SSEMVS from，a pair of coarse direction-cosine estimatescan be derived by the modified vector cross-product algorithm. Next，the fine and unambiguous direction cosine estimates can be obtained by using the disambiguation approach. Finally，the high accurate estimation of 2D-DOA and polarization angles can be achieved.

3.1 Accurate but ambiguous direction cosine estimates based on ESPRIT

Uni-vector-sensor ESPRIT [16]forms a temporal invariance by two time-delayed sets of data collected from the SNRA-SSEMVS，thekth source impinging upon the array would contribute to two 6MNdatasets. One isbks(tn，fk)，n=1，···，L，and the other can be expressed as the following from:

where the steering vectorbkis defined as in (5)，ΔTis a constant time delay，ands(tn，fk) has the following form:

wherePk，fk，and φkare the power，frequency，and random phase of thekth source，respectively.

With a total ofKincident sources and additive zeromean white Gaussian noise，the received data of the nonuniform rectangular array can be written as

Then the entire 1 2MN×Ldataset can be expressed as

whereZ1andZ2are the 6MN×Ldatasets sampled at{t1，···，tL}and {t1+ΔT，···，tL+ΔT}，respectively. The two signal subspacesE1andE2can be estimated fromby selecting the eigenvectors associated with the largestKeigenvalues. A uniqueK×Knonsingular matrixTexists and relatesEi(i=1，2)with the array manifold matrix as

where φ =diagej2πf1ΔT，···，ej2πfKΔT. Based on (12)，the following equation holds:

where + denotes the pseudo-inverse，Tis the right eigenvector of ψ，and the diagonal elements of φ is composed of the eigenvalues of ψ. Thus，the array manifold can be estimated as

Moreover，the frequency estimation can be derived as

According to the estimation of array manifold，a matrixthat includes the spatial steering vector of thekth signal can be derived，with its (m，n)th element being

whereA[m:n，k]represents a column vector composed of elements from themth to thenth row of thekth column of the matrixA.A./Brepresents the division of corresponding elements of matrixAand matrixB.

In order to get the estimation of the spatial steering vectorqykalong theyaxis，the entire nonuniform rectangular array can be regarded as a combination of line arrays parallel to the theyaxis，then by making phase compensation for each line array and making an average，qykcan be obtained according to the following equations.

Similarly，the entire array can also be considered as a combination of line arrays parallel to thexaxis，thenqxkcan be estimated as

From (19) and (21)，the accurate but ambiguous direction cosine estimatescan be obtained as follows:

3.2 Coarse direction cosine estimates based on the modified vector-cross-product algorithm

It can be seen that the estimation ofcan be derived from the first six rows of，and the estimation performance can be further improved by taking average of all the steering vector estimates of SSEMVS. Then the steering vector of SSEMVScan be estimated as

whereA{i:j} is a matrix consisting of elements from themth to thenth row of，the superscript * denotes the complex conjugation，

It was claimed in [16]that different directions of the incident signals can result in the same cross product vector when their unit Poynting vectors have the same modulus but different signs. To overcome this problem，Li et al. [16]proposed a modified vector-cross-product algorithm to avoid suffering ambiguity in the estimation of the unit Poynting vector. In the following，the brief procedure is provided.

Firstly，the normalized cross productccan be calculated fromwith the vector-cross-product algorithm as follows:

Psis the unit Poynting vector of the incident source，gx，gy，andgzare calledGparameters of the SSEMVS.Owing to the ambiguous estimation of direction cosine in[15]and [17]，the entry-wise modulus ofccan be used to construct eight unit Poynting vectors，which can be expressed as

Then substitute all of these vectors into (32) and choose the one with the least fitting error as the final estimation of the Poynting vector. Therefore，the coarse direction cosine estimates can be obtained as

3.3 Yield of closed-form DOA and polarization estimates with high accuracy

By using the disambiguation approach [15]，the final estimation of direction cosine can be achieved as

With the estimation of direction-cosine，the elevation and azimuth angles can be calculated subsequently. The following eight cases are discussed with the difference of Poynting vectors.

As a result，the corresponding polarization parameters can be estimated as

where ∠ denotes the angle of the ensuring entity.

3.4 Computational complexity analysis

The main computational complexity of the proposed method includes: (i) calculation of array covariance mat-rix withO{(12MN)2L} flops，(ii) eigen decomposition operation to obtain signal subspaces withO{(12MN)3}flops，(iii) estimation of the array manifolds withO{[2·6MNK2+K3]+4·6MNK2+K3}flops，(iv) estimation of the array manifold for SSEMVS array withO{3MN·6K2}flops. Overall，the{ computational load of the proposed }algorithm isO(12MN)2L+(12MN)3+2K3+54MNK2.

Table 1 shows the computational complexity comparison of the reduced dimensional polarimetric MUSIC algorithm (RD-MUSIC) and the proposed algorithm，where Δθis the search range of elevation angle，and Δ φ is the search range of azimuth angle. It can be seen that the computational complexity of the proposed algorithm is relatively lower.

Table 1 Computational complexity of the proposed algorithm and RD-MUSIC

4. Simulation results

In this section，several simulations are conducted to validate the performance of the proposed algorithm. In the following simulations，the configuration of SSEMVS is shown in Fig. 4. Stars refer to dipole antennas，circles denote the loop antennas. The corresponding parametersG=[gxgygz]Tis of the following form:

Fig. 4 Geometry illustrations of SSEMVS

4.1 Four-dimensional parameters estimation and pairing

It is assumed that the size of the SNRA is 5 ×6，M=5 andN=6. The SNR is set as 20 dB，and the number of snapshots isL=500. We considerK=7 targets，

A total of 100 Monte Carlo simulations are conducted and the results are shown in Fig. 5. It can be seen that the DOA and polarization angles are well estimated and correctly paired.

Fig. 5 Four-dimensional parameters estimations of the proposed algorithm

4.2 Parameters estimation performance versus SNR

SupposeM=5，N=6，DX=4λ，DY=8λ，and suppose there are two sources with

L=500，and the simulation results are obtained by 1 000 Monte Carlo experiments. The root mean squared error (RMSE) can be used to measure the estimation performance，which is defined as

Fig. 6 RMSEs versus SNR of parameters estimation

It can be seen that the parameters estimation performance of the proposed algorithm is superior to the RD-MUSIC in [17]and can approach to the Cramér-Rao bounds (CRBs) more effectively. It is worth mentioning that the parameters estimation performance of the RD-MUSIC is unsatisfactory when the SNR is lower than 5dB. However，for our proposed method，it can provide a high parameters estimation accuracy even at low SNRs.

4.3 Parameters estimation performance versus the number of data samples

The SNR is fixed at 20 dB，and the other settings are the same as those in Section 4.2. When the number of snapshots varies from 200 to 1 200，the RMSEs of parameter estimations are shown in Fig. 7. Again，it is seen that the proposed algorithm still has obvious advantages over the RD-MUSIC for all available snapshots and approaches to the related CRBs more effectively.

Fig. 7 RMSEs versus snapshots of parameters estimation

4.4 Parameters estimation performance of different array structures

In this section，the CRBs of the proposed array is compared with three different array configurations presented in [16-18]，respectively. Fig. 8 plots the RMSEs versusDX=DY，the SNR is fixed as 20 dB，and other simulation conditions are the same as those in Section 4.2. Fig. 9 describes the RMSEs versus SNR，and other settings are fixed as those in Section 4.2. As it can be seen，the 2D-DOA estimation performance gets better when the inter-sensor spacing gets larger，while the RMSEs of the polarization parameters estimation remain the same with the increase of intersensor spacing. From Fig. 8 and Fig. 9，we can also see that the proposed algorithm has obvious advantages over the other methods. The reason is that the sparse nonuniform array configuration extends the array aperture and hence refines the parameter estimation accuracy.

Fig. 8 CRBs of the four-dimensional parameters estimation versus intersensor spacing of L-shaped array，SNC-EMVS，and the proposed array

Fig. 9 CRBs of four-dimensional parameters estimations versus SNR of SSEMVS，L-shape array，SNC-EMVS，and the proposed array

Moreover，compared to the method in [17]，the mutual coupling effect and hardware cost can also be reduced.The reason for the unsatistactory parameters estimation performance in [17]is that there exists ambiguity in the estimation of the unit Poynting vector，which leads to the incorrect estimation of 2D-DOA and polarization angles.

5. Conclusions

In this paper，a method has been developed for 2D-DOA and polarization estimation in the SNRA composed of SSEMVS. Compared with some existing methods，the proposed one avoids suffering ambiguity in the estimation of direction finding，improves the parameters estimation accuracy without adding any redundant antennas，and reduces the mutual coupling effect. Moreover，compared to RD-MUSIC，the proposed method has relatively lower computational complexity. In addition，the 2D-DOA and polarization angles can also be paired automatically.