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1.State Key Laboratory of Complex Electromagnetic Environment Effects（CEMEE）on Electronics and Information System，Luoyang 471003，P.R.China；2.College of Electronic and Information Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 211106，P.R.China

（Received 20 December 2019；revised 18 June 2020；accepted 18 September 2020）

Abstract:The problem of joint direction of arrival（DOA）and polarization estimation for polarization sensitive coprime planar arrays（PS-CPAs）is investigated，and a fast-convergence quadrilinear decomposition approach is proposed.Specifically，we first decompose the PS-CPA into two sparse polarization sensitive uniform planar subarrays and employ propagator method（PM）to construct the initial steering matrices separately.Then we arrange the received signals into two quadrilinear models so that the potential DOA and polarization estimates can be attained via quadrilinear alternating least square（QALS）.Subsequently，we distinguish the true DOA estimates from the approximate intersecting estimations of the two subarrays in view of the coprime feature.Finally，the polarization estimates paired with DOA can be obtained.In contrast to the conventional QALS algorithm，the proposed approach can remarkably reduce the computational complexity without degrading the estimation performance.Simulations demonstrate the superiority of the proposed fast-convergence approach for PS-CPAs.

Key words：polarization sensitive arrays；coprime planar arrays；direction of arrival（DOA）estimation；polarization estimation；quadrilinear decomposition；fast convergence

0 Introduction

Polarization characteristic of electromagnetic wave has played an important role in target detection and recognition［1］.Polarization sensitive arrays（PSAs）have been widely utilized in vital applications such as radar，navigation and wireless communications［2］.Compared with the traditional arrays with scalar sensors，PSAs with electromagnetic vector sensors（EMVSs）offer desirable improvements in array performance［3］.Various angle-polarization estimation methods for PSAs have been proposed［4-5］，including multiple signal classification（MUSIC）algorithm［6］，estimating signal parameters via rotational invariance techniques（ESPRIT）［7］，etc.However，the compact structures of most PSAs with inter-element spacing no more than half-wavelength restrict the resolution.

Recently，the coprime arrays［8-11］，a newly emerged typical sparse array structure，have attracted more and more concerns for their inherent advantages over the uniform arrays，e.g.enlarged array aperture，increased degrees of freedom and improved estimation performance.Varieties of methods have been developed for conducting direction of arrival（DOA）estimation for coprime arrays.In Ref.［9］，a representative method of phase ambiguity elimination is proposed.The traditional MUSIC algorithm for uniform arrays is extended to coprime planar arrays in Ref.［10］，whereas the two-dimensional（2D）total spectral search（TSS）brings a tremendous amount of computing.In order to reduce the computational burden，Ref.［11］converts the 2D TSS into one-dimensional partial spectral search.

However，the existing studies mainly take the scalar coprime arrays into account，where the significant polarization characteristic of the electromagnetic wave is neglected.Moreover，the subspace-based methods usually omit the structural characteristic of received signals.Tensor algebra-based tools are effective in improving the estimation performance due to its excellent anti-noise capacity［12］.Parallel factor（PARAFAC）technique［13］，a typical tensor-based decomposition，has been turned out to be computationally efficient in multi-parameter estimation by factorizing the tensor data and employing the least squares（LS）estimation.Quadrilinear decomposition algorithm［14］，has been successfully applied in DOA and polarization estimation.Unfortunately，the conventional quadrilinear decomposition-based algorithm suffers from heavy computational burden.

In this paper，we investigate the problem of joint multi-parameter estimation for polarization sensitive coprime planar arrays（PS-CPAs）and derive a fast-convergence quadrilinear decomposition approach.The main contributions are as follows：（1）We take the polarization sensitive coprime planar arrays into consideration，which can take full advantages of coprime arrays and polarization sensitive arrays to enhance the array performance and achieve better engineering applicability.（2）We develop a connection between the DOA and polarization estimation problem for PS-CPAs and quadrilinear decomposition problem，which utilizes the structural characteristic of received signal data and thereby construct it as two quadrilinear models.（3）We propose a fast-convergence quadrilinear decomposition approach for PS-CPAs，where an initial estimation with PM is exploited to construct the initial matrices and significantly reduces the complexity.Furthermore，the proposed approach outperforms ESPRIT and PM in parameter estimation.

1 Data Model

Suppose that a PS-CPA configuration consists of two uniform planar subarrays（UPAs）withMi×Mi(i=1，2)crossed short dipoles.The distances of adjacent sensors of the subarray withM1×M1sensors isd1=M2λ/2，while the other withM2×M2sensors isd2=M1λ/2，whereM1andM2are coprime integers andλis the wavelength.The two subarrays share the same element at the origin of coordinates.A PS-CPA configuration is displayed in Fig.1 as an example.

Fig.1 PS-CPA configuration with M1=3 and M2=4

AssumeK(K＜min{M21，M22})far-field uncorrelated signals impinge on the PS-CPA from{(θk，φk)|k=1，2，…，K}.Define thatθk∈[0，π/2]is the elevation angle，φk∈[0，π]the azimuth angle.γk∈[0，π/2]andηk∈[-π，π]are polarization parameters of thek-th signal.Define a transformation asuk=sinθkcosφk，vk=sinθksinφkfor simplification.

Considering that the PS-CPA can be decomposed into two polarization sensitive uniform planar arrays（PS-UPAs），we process the signal data with the two PS-UPAs separately and elaborate on the proposed approach with the subarray ofMi×Mi(i=1，2)crossed short dipoles.

The output of thei-th subarray can be presented by［5］

whereB=[b1，b2，…，bK]∈CL×Krepresents the signal matrix，bk=[bk(1)，bk(2)，…，bk(L)]T；Lthe number of snapshots；Ni∈C2M2i×Lthe received noise which is zero-mean white Gaussian independent with signals.the steering matrix，whereis the steering vector；anddenote the Kronecker product and the Khatri-Rao product，respectively；(·)Tis the operation of transpose；Ai，x=[ai，x1，ai，x2，…，ai，xK]andAi，y=[ai，y1，ai，y2，…，ai，yK]represent the steering matrices corresponding to thex-andy-axis direction，respectively.Andai，x(uk)andai，y(vk)are the steering vectors and can be expressed as

andS=[s1，s2，…，sK]T∈C2×Kis the polarization matrix，where the polarization vectorskis

Eq.（1）can be written as

whereNi=Miin the subarrays of PS-CPA considered；Dm(A)produces a diagonal matrix formed by them-th row ofA.To describe the quadrilinear model more exhaustively，we use the subarray withMi×NiEMVSs to illustrate it.Xi，m，nin Eq.（5）can be denoted as the quadrilinear model［14］

whereai，m，k，ai，n，krepresent the（m，k）-th，（n，k）-th items inAi，x，Ai，y，respectively；sp，kstands for the（p，k）-th element ofSis；bl，kthe（l，k）-th element ofB；andni，m，n，p，lthe（m，n，p，l）-th element ofNiwhich is regarded as a four-array matrix.The other rearranged matrices can be derived from the structural characteristics of the quadrilinear model as

2 The Proposed Approach

To accelerate convergence and reduce complexity effectively，instead of initializing the loading matrices randomly like the conventional QALS method，we first make an initial estimation with PM to construct the initialAi，xandAi，y，and then iteratively update the four loading matrices in turn according to QALS until the convergence.The coprime relationship between the two subarrays is exploited to remove the ambiguity.Finally，the polarization parameters can be obtained with the previous estimates.

2.1 Initialization with PM

whereXi，1means the firstKrows ofXi；andXi，2the remaining rows.Define

whereΦi，p=diag{pi，1，pi，2，…，pi，K}andpi，k=.diag(·)represents a diagonal matrix consisting of the included elements as diagonal elements.Then we haveby partitioning，wherePi，aandPi，bare the first 2Mi(Mi-1)rows and the last 2Mi(Mi-1)rows of the matrix，respectively.The initial estimatesofcan be sequentially achieved，which refers to thek-th eigenvalue of，where||·||?stands for pseudo-inverse.Meanwhile，we can obtain the eigenvectors.In the noise-free case，，whereΠis a permutation matrix，andΠ-1=Π.Accordingly，the estimate ofGiis

whereΦi，q=diag{qi，1，qi，2，…，qi，K}andqi，k=.Similarly，we achieve the initial estimatesofby partitioningand getΠΦi，qΠ-1.

2.2 Quadrilinear decomposition

Herein，we initialize the steering matricesAi，xandAi，ywithandto speed the convergence.And the initial polarization matrixSand signal matrixBare constructed randomly.

According to Eq.（1），the costing function ofBin the quadrilinear model is

According to Eq.（8），the LS fitting forAi，xis

In a similar way，the LS fitting forSis

where is the noisy signal.The LS update forSis where，andare the previous estimats.

The sum of squared residuals（SSR）is defined as SSRk=，wherer，lrepresent the（r，l）-th elements of matrixThe convergence rate is denoted as SSRrate=(SSRk-SSRk-1)SSRk-1.According to Eqs.（16），（18），（20）and（22），we repeatedly perform the updating process ofandwith LS until SSRrate＜ε，whereεis a certain small value［16］.

Thereafter，we achieve the estimates as

whereΠirepresents a permutation matrix，which may lead to permutation ambiguity.AndΔi，1，Δi，2，Δi，3，Δi，4are the diagonal scaling matrices satisfyingΔi，1Δi，2Δi，3Δi，4=IK，which may lead to scale ambiguity.Vi，1，Vi，2，Vi，3andVi，4represent the estimation errors.Since the permutation matrixΠiin Eqs.（23）—（26）is the same，the permutation ambiguity makes no difference to parameters pairing.And the scale ambiguity can be resolved via normalization.

2.3 Least squares estimation

where angle(·)represents the operation of getting the phase angle.Then we use LS criterion to estimateukandvk.The LS fitting isQici，1=hi，ukandQici，2=hi，vk，where，ci，2=and

According to LS criterion，we can obtain

Then the estimates of(uk，vk)can be achieved by

2.4 Ambiguity elimination

We elucidate the generation and elimination method of phase ambiguity in this section.

Assume a single source impinges on the subarray withMi×MiEMVSs from the direction(θt，φt).Denote(θa，φa)as one of the ambiguous DOAs.The period of exponential function 2πimplies that［11］

whereut=sinθtcosφt，vt=sinθtsinφt，ua=sinθacosφa，va=sinθasinφa，di=Mjλ/2(i，j∈{1，2}，i≠j)，ki，u∈Z，ki，v∈Z.Constraintsut，ua∈[-1，1]，vt，va∈[0，1]，0＜+＜1，0＜+＜1 have to be satisfied.

From Eqs.（34），（35），we have

AsM1andM2are coprime integers，there uniquely existk1，u=k2，u=0 andk1，v=k2，v=0 making Eq.（36）valid，which reveals that the true DOA estimates can be uniquely distinguished from the intersecting estimations of the two subarrays.Since it is impractical for two subarrays containing completely coincident estimates，the closest ones are exactly required.Similar conclusions can be obtained in the case of multi-source.

2.5 Parameter estimation

With the unambiguous estimates defined as，，we obtain the true estimatesvia

Since the ambiguity elimination process makes the pairing of，andinvalid，we must determine the pairing of，andbefore calculating the polarization parameters.

By this means，the pairing ofandis achieved，so are，andDefineas the polarization matrix paired with，Since the two subarrays correspond to the same polarization matrix，we adopt an average operation as

Subsequently，the polarization parameters can be determined from

where

3 Performance Analysis

3.1 Convergence analysis

We compare the convergence performance of the proposed approach and the conventional QALS algorithm and illustrate the iteration times of the two subarrays in Fig.2，where we set SNR=10 dB，L=100.Define DSSR=SSRn-SSRc，where SSRndenotes the value of SSR corresponding to then-th iteration and SSRcthe value of SSR at convergence.

It is explicitly indicated in Fig.2 that the proposed approach requires fewer iterations to converge than the conventional QALS algorithm.A faster convergence speed can lead to a lower computational complexity.

Fig.2 Convergence comparison of different algorithms

3.2 Complexity analysis

As the two subarrays of the PS-CPA possess the similar complexity form，the complexity calculation of the subarray withMi×Misensors of our approach is shown as follows.The initialization with PM costs about1，2).Each iteration of QALS needsO

We list the complexity of the proposed approach and the conventional QALS algorithm in Table 1，wheren1andn2are the number of iterations of the former and the latter，respectively.

Fig.3 displays the complexity comparison between PM and each iteration of QALS in the PSCPA.We can conclude that although PM process is involved in initialization，the complexity of PM is lower than that of one iteration of QALS.Fig.4 displays the complexity comparison versus snapshots，whereK=2，n1=5，n2=100.It is observed from Figs.3，4 that the proposed approach can reduce the complexity remarkably.

Table 1 Computational complexities of different algorithms

Fig.3 Complexity comparison between PM and each iteration of QALS in the PS-CPA

Fig.4 Complexity comparison of different snapshots between the proposed algorithm and the conventional QALS algorithm

3.3 Advantages

（1）The proposed approach achieves a fast convergence by employing PM as the initialization，which remarkably reduces the computational complexity.

（2）The proposed approach obtains the same parameter estimation performance as the conventional QALS algorithm but owns much lower computational burden，and outperforms PM and ESPRIT.

（3）The proposed approach in PS-CPA has a higher estimation accuracy than that in PS-UPA，owing to the larger array aperture.

4 Simulation Results

In this section，we perform 500 Monte-Carlo simulations to evaluate the parameter estimation performance.The root mean square error（RMSE）is defined by

Suppose thatK=2 signals impinge on the PSCPA from(θ1，φ1)=(10°，30°)，(θ2，φ2)=(20°，40°)，and their corresponding polarization parameters are(γ1，η1)=(7°，1 5°)，(γ2，η2)=(17°，25°).

4.1 Parameter estimation results

We examine the validity of the proposed approach and the scatter plots of parameter estimation is shown in Fig.5，whereM1=4，M2=5，L=200 and SNR=20 dB.As illustrated in Fig.5，the approach is effective in estimating multi-parameters.

Fig.5 Estimation results of the proposed approach

4.2 RMSE comparison of different algorithms

We present the parameter estimation performance comparison of the proposed approach，the conventional QALS，ESPRIT，PM algorithms and the Cramer-Rao Bound［17］in Fig.6，whereM1=7，M2=9 andL=300.It is clear from Fig.6 that the proposed approach has the same estimation performance as the conventional QALS algorithm but with a faster convergence，which shows that the proposed approach can guarantee estimation accuracy while reducing complexity effectively.By contrast，the proposed approach outperforms ESPRIT and PM，as it utilizes the structural characteristic of received signal and applies the quadrilinear alternating least square method.

Fig.6 Parameter estimation performance of different algorithms

4.3 RMSE comparison with different arrays

We give the RMSE results of the proposed approach in PS-CPA and PS-UPA in Fig.7 to compare array performance，whereM1=4，M2=5，L=300.Consider a PS-UPA with 5×8 sensors so that the two arrays have the same number of elements for fair.As depicted in Fig.7，the approach in PS-CPA has superior estimation performance to that in PS-UPA，as the PS-CPA enables a larger array aperture with the same number of elements.

Fig.7 Parameter estimation performance of different arrays

5 Conclusions

This paper focuses on PS-CPAs and proposes a fast-convergence quadrilinear decomposition approach for DOA and polarization estimation with the PS-CPAs.To accelerate convergence of the conventional quadrilinear decomposition algorithm，the proposed approach first employs PM as the initialization，and then arranges the receive data into two quadrilinear models to perform QALS.Thereafter，the phase ambiguity can be eliminated and the polarization estimates paired with DOA can be achieved by utilizing the previous estimations.Simulations demonstrate the superiority of the proposed approach in terms of computational complexity and estimation performance.

AcknowledgementThis work was supported by the Open Research Fund of the State Key Laboratory of Complex Electromagnetic Environment Effects on Electronics and Information System（No.CEMEE2019Z0104B）.

AuthorsDr.XU Xiong is a research assistant of the State Key Laboratory of Complex Electromagnetic Environment Effects（CEMEE）on Electronic and Information System.He got his Ph.D.degree from University of Electronic Science and Technology of China（UESTC）in 2012.His research interests include Signal Processing and Machine Learning.

Prof.ZHANG Xiaofei received his M.S.degree from Wuhan University in 2001 and Ph.D.degree in communication and information systems from Nanjing University of Aeronautics and Astronautics in 2005.He is now a full professor at College of Electronic and Information Engineering of Nanjing University of Aeronautics and Astronautics.His research is focused on array signal processing and communication signal processing.

Author contributionsDr.XU Xiong designed the study and analyzed the performance.Ms.SHEN Jinqing provided the simulation results and wrote the manuscript.Mr.ZHU Beizuo contributed to the discussion and background.Prof.ZHANG Xiaofei provided guidance of the study.

Competing interestsThe authors declare no competing interests.