# 2D-DOA Estimation for EMVS Array with Nonuniform Noise

Authors: Chang-Xin Cai; Guan-Jun Huang; Fang-Qing Wen; Xin-Hai Wang; Lin Wang
Journal: International Journal of Antennas and Propagation
 (2021)

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## Abstract

Electromagnetic vector sensor (EMVS) array is one of the most potential arrays for future wireless communications and radars because it is capable of providing two-dimensional (2D) direction-of-arrival (DOA) estimation as well as polarization angles of the source signal. It is well known that existing subspace algorithm cannot directly be applied to a nonuniform noise scenario. In this paper, we consider the 2D-DOA estimation issue for EMVS array in the presence of nonuniform noise and propose an improved subspace-based algorithm. Firstly, it recasts the nonuniform noise issue as a matrix completion problem. The noiseless array covariance matrix is then recovered via solving a convex optimization problem. Thereafter, the shift invariant principle of the EMVS array is adopted to construct a normalized polarization steering vector, after which 2D-DOA is easily estimated as well as polarization angles by incorporating the vector cross-product technique and the pseudoinverse method. The proposed algorithm is effective to EMVS array with arbitrary sensor geometry. Furthermore, the proposed algorithm is free from the nonuniform noise. Several simulations verify the improvement of the proposed method.


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## Body

## 1. Introduction

Sensor array is one of the most important infrastructures in wireless communication and radar detection [1–4]. Among the various branches in array signal processing, direction-of-arrival (DOA) estimation is the most canonical one and has aroused much attention. The principle of DOA estimation is to estimate the direction of the incoming source via the phase characteristics between sensors, and it is a highly nonlinear problem. Many efforts have been devoted to tackling this issue, for instance, an estimation approach to signal parameters with the rotational invariance technique (ESPRIT) [5, 6], Capon, multiple signal classification (MUSIC) [7–9], propagator method (PM) [10], maximum-likelihood (ML) [11], and tensor method [12–14]. Usually, the spectrum search counterparts, such as MUSIC, are always inefficient. Besides, they hardly avoid the off-grid problem. ESPRIT, however, is much more efficient than the spectrum search frameworks because it can acquire closed-form solutions to the parameter estimation issue.A majority of the current studies focus on how to estimate the one-dimensional (1D) DOA from the scalar sensor arrays, e.g., uniform linear array (ULA). In practice, two-dimensional (2D) DOA may be more attractive. To pursue 2D-DOA estimation with the traditional scalar sensors, nonlinear sensor geometries are necessary [15–17], e.g., L-shape array, circular geometry, and rectangular manifold. Unfortunately, scalar sensor arrays often suffer from the sensor position error. Thus, complex array calibration is indispensable. Unlike scalar sensors, a single electromagnetic vector sensor (EMVS) is capable of providing 2D-DOA estimation [18]. Moreover, it is able to offer additional polarization angles of the source, and such characteristic may be very helpful in detecting stealth source [19]. Besides, an EMVS array with N sensors occupies more degree of freedom (DOF) than a scalar array, and thus, it provides more accurate estimation result than the latter. Furthermore, it has been proven that parameter estimation using EMVS array is insensitive to sensor positions [20], giving rise to the fact that the EMVS is more flexible than the traditional scalar sensor arrays.It should be emphasized that the angle estimation issue using EMVS array is often more complex than that using the scalar sensors, since it involves 2D-DOA (azimuth angle and elevation angle) and 2D polarization angle (polarization phase difference and auxiliary polarization angle). In [19], the vector cross product was proposed, and the angles therein were obtained from the Poynting vector of the polarization steering vector. In [20], the ESPRIT-like algorithm was introduced. Therein, the concept of normalized Poynting vector was proposed to estimate the 2D-DOA, which was insensitive to the sensor positions and free from the sensor position error. In [21], the ULA-configured EMVS architecture was presented, and another ESPRIT estimator was derived. Unlike [20], the elevation angle was achieved by using ESPRIT, and the azimuth angle was estimated by using vector cross product. Likewise, in [22–26], the methods of combining the subspace approach and vector cross product were investigated. To avoid the eigendecompositon in the subspace approaches, a PM-like algorithm was presented in [27]. To further exploit the multidimensional structure in EMVS array, tensor algorithms were also been investigated in [28]. Besides, some efforts have been devoted to the active radar system with EMVS arrays [29–31], which brings new insights to target detection.Nevertheless, it should be noticed that the subspace-based approaches achieve good performance with Gaussian white noise. In practice, the array noise may be nonuniform due to hardware nonideality. The nonuniform noise issue has been extensively stressed in scalar sensor array [32–35], but little attention has been paid to EMVS array. Therefore, we revisit the 2D-DOA estimation in EMVS array with nonuniform noise in this paper. An improved ESPRIT algorithm is presented. It eliminates the nonuniform noise via constructing a reduced covariance matrix, after which both noise covariance and parts of the signal covariance are removed. The recovery of the noiseless covariance matrix is recast as a matrix completion issue and is accomplished via solving a convex optimization problem. Thereafter, the ESPRIT idea is adopted to construct the normalized polarization steering vector. 2D-DOA, as well as polarization parameters, is then achieved by combining the vector cross product and the least squares (LS) technique. Our algorithm is effective in the scenario with arbitrary array geometry. Numerical simulations are designed to verify its effectiveness.


## 2. Preliminaries and the Data Model

### 2.1. EMVS Preliminaries

For a complete EMVS, it consists of six colocated antennas: three electric dipoles, and three magnetic loops. The dipoles and loops, respectively, sense the information of the electric field and magnetic field. Considering that a far-field source signal impinges on a single EMVS, the polarization responses of the six components can be expressed as(1)b=b1b2b3︸e∈ℂ3×1b4b5b6︸m∈ℂ3×1T,=Dp,where ⋅T denotes transpose. D and p are, respectively, given by(2)D=cosϕcosθ−sinϕsinϕcosθcosϕ−sinθ0−sinϕ−cosϕcosθcosϕ−sinϕcosθ0sinθand(3)p=sinγejηcosγ,where e and m denote the electric steering vector and the magnetic steering vector, respectively; θ, ϕ, γ, and η denote, respectively, the elevation angle, the azimuth angle, the auxiliary polarization angle, and the polarization phase difference. D∈ℂ6×2 denotes the direction-only matrix, and p∈ℂ2×1 denotes the polarization-only vector, respectively. Moreover, the Poynting vector between e and p satisfies [20](4)ee∗p∗p=sinθcosϕsinθsinϕcosθ,where ⋅∗ denotes the conjugate, ⋅ returns the absolute value, and ∗ denotes the vector cross product.


### 2.2. Data Model

Let us consider anN-element EMVS array. Without loss of generality, let the coordinate of the n-th EMVS be rn=xn,yn,znT. Suppose that K far-field signals appear in the array. Let θk, ϕk, γk, and ηk stand for the k-th (k=1,2,…,K) angle parameters. The array signal can be written as [19](5)yt=∑k=1Kak⊗bkskt+nt,where t is the time index; ⊗ denotes the Kronecker product, ak=e−jπτ1,k,e−jπτ2,k,…,e−jπτN,kT, where τn,k=rnTgk/λ∈ℂN×1 denotes the k-th spatial response (steering) vector with gk=cosϕt,ksinθt,k,sinϕt,ksinθt,k,cosθt,kT and λ is the carrier wavelength; and bk=Dkvk denotes the polarization response vector associated with the k-th target. skt accounts for the k-th signal; nt denotes the array noise. Let A=a1,a2,…,aK∈ℂN×K and B=b1,b2,…,bK∈ℂ6×K. Equation (6) can also be formulated as(6)yt=A⊙Bst+nt,=Cst+nt,where the symbol ⊙ stands for the Khatri–Rao product, st=s1t,s2t,…,sKtT, and C=A⊙B. Suppose that the noise nt is uncorrelated with the signal st; then, the covariance matrix of yt is given by(7)R=EytyHt,=CRsCH+Rn,=R˜+Rn,where E⋅ is to acquire the mathematical expectation and ⋅H denotes Hermitian transpose.Rs=EstsHt, Rn=EntnHt, and R˜=CRsCH. In the presence of uncorrelated source signals, Rs=diagλ1,λ2,…,λK, where diag⋅ accounts for the diagonalization operation and λk is the power of the k-th source. Moreover, since the noise is nonuniform, its covariance matrix is then given by(8)Rn=diagσ12,σ22,…,σ6N2,where σ6n−1+q2 denotes the noise power corresponding to the q-th component of the n-th EMVS. In practical applications, we can estimate R via L samples as(9)R^=1L∑t=1LytyHt.Our objective here is to estimate the angles fromR^.


## 2.1. EMVS Preliminaries

For a complete EMVS, it consists of six colocated antennas: three electric dipoles, and three magnetic loops. The dipoles and loops, respectively, sense the information of the electric field and magnetic field. Considering that a far-field source signal impinges on a single EMVS, the polarization responses of the six components can be expressed as(1)b=b1b2b3︸e∈ℂ3×1b4b5b6︸m∈ℂ3×1T,=Dp,where ⋅T denotes transpose. D and p are, respectively, given by(2)D=cosϕcosθ−sinϕsinϕcosθcosϕ−sinθ0−sinϕ−cosϕcosθcosϕ−sinϕcosθ0sinθand(3)p=sinγejηcosγ,where e and m denote the electric steering vector and the magnetic steering vector, respectively; θ, ϕ, γ, and η denote, respectively, the elevation angle, the azimuth angle, the auxiliary polarization angle, and the polarization phase difference. D∈ℂ6×2 denotes the direction-only matrix, and p∈ℂ2×1 denotes the polarization-only vector, respectively. Moreover, the Poynting vector between e and p satisfies [20](4)ee∗p∗p=sinθcosϕsinθsinϕcosθ,where ⋅∗ denotes the conjugate, ⋅ returns the absolute value, and ∗ denotes the vector cross product.


## 2.2. Data Model

Let us consider anN-element EMVS array. Without loss of generality, let the coordinate of the n-th EMVS be rn=xn,yn,znT. Suppose that K far-field signals appear in the array. Let θk, ϕk, γk, and ηk stand for the k-th (k=1,2,…,K) angle parameters. The array signal can be written as [19](5)yt=∑k=1Kak⊗bkskt+nt,where t is the time index; ⊗ denotes the Kronecker product, ak=e−jπτ1,k,e−jπτ2,k,…,e−jπτN,kT, where τn,k=rnTgk/λ∈ℂN×1 denotes the k-th spatial response (steering) vector with gk=cosϕt,ksinθt,k,sinϕt,ksinθt,k,cosθt,kT and λ is the carrier wavelength; and bk=Dkvk denotes the polarization response vector associated with the k-th target. skt accounts for the k-th signal; nt denotes the array noise. Let A=a1,a2,…,aK∈ℂN×K and B=b1,b2,…,bK∈ℂ6×K. Equation (6) can also be formulated as(6)yt=A⊙Bst+nt,=Cst+nt,where the symbol ⊙ stands for the Khatri–Rao product, st=s1t,s2t,…,sKtT, and C=A⊙B. Suppose that the noise nt is uncorrelated with the signal st; then, the covariance matrix of yt is given by(7)R=EytyHt,=CRsCH+Rn,=R˜+Rn,where E⋅ is to acquire the mathematical expectation and ⋅H denotes Hermitian transpose.Rs=EstsHt, Rn=EntnHt, and R˜=CRsCH. In the presence of uncorrelated source signals, Rs=diagλ1,λ2,…,λK, where diag⋅ accounts for the diagonalization operation and λk is the power of the k-th source. Moreover, since the noise is nonuniform, its covariance matrix is then given by(8)Rn=diagσ12,σ22,…,σ6N2,where σ6n−1+q2 denotes the noise power corresponding to the q-th component of the n-th EMVS. In practical applications, we can estimate R via L samples as(9)R^=1L∑t=1LytyHt.Our objective here is to estimate the angles fromR^.


## 3. The Proposed Approach

### 3.1. Principle of Traditional Eigendecompositon

It is well known that when Gaussian white noise (uniform noise) exists, the noise powers fulfill(10)σ12=σ22=⋯=σ6N2=σ2,where σ2 is a constant, and then, the noise covariance becomes(11)Rn=σ2I6N,where IM stands for the M×M identity matrix. If we ignore the noise item in equation (7), the eigendecomposition of R is given by(12)R˜=∑n=1KαkukukH,=UsΣsUsH,where αk is the k-th eigenvalue, uk is the associated eigenvector, and uk1Huk2=0 is for any k1≠k2; Σs=diagα1,α2,…,αK denotes the eigenvalue matrix, and Us=u1,u2,…,uK∈ℂ6N×K is called the signal subspace, which spans the same subspace as A⊙B. Namely, there is a full-rank matrix T∈ℂK×K such that(13)Us=CT.Also, the noiseless covariance matrix can be expressed as(14)R˜=∑n=1KαkukukH+∑n=16N−K0⋅vnvnH,=UsΣs1UsH+Un06N−K6N−KUnH,where vn is the eigenvector from the null subspace of Us, i.e., vnHUs=01×K, and 0M×N represents the M×N full zero matrix. Σs1=diagα1+σ2,α2+σ2,…,αK+σ2, and Un=v1,v2,…,v6N−K∈ℂ6N×6N−K is called the noise subspace. Since the identity matrix can be formulated as the product of arbitrary unitary matrix and its Hermitian transpose, the noisy R can be written as(15)R=∑n=1Kαk+σ2ukukH+∑n=16N−Kσ2ununH,=UsΣs1UsH+UnΣnUnH,where Σn=diagσ2,σ2,…,σ2∈ℂ6N−K×6N−K. The results in equation (15) reveal that the uniform noise would not destroy the eigendistribution of the signal. However, in the presence of nonuniform noise, the noise power is not unique, so the conclusion in equation (15) will be untenable. It is necessary for us to denoise before further processing.


### 3.2. Denoising

LetΩ be a set that records the nonzero entities of Rn, i.e.,(16)Ω=m,m|m=1,2,…,6N.We define a sampling operatorSΩ⋅ that picks up the elements of the matrix in the blanket with indexes in Ω, for example, SΩR=R¯∈C6N×6N such that(17)R¯m,n=Rm,n,m,n∈Ω,0,m,n∉Ω,where Rm,n denotes the (m, n)-th entity of R and is similar to others. Since Rn is a diagonal matrix, we have Rn=SΩRn. The effect of the noise can be easily removed via the following reduced covariance matrix:(18)R⌣=R−SΩR,=R˜−SΩR˜.The abovementioned denoising procedure is illustrated in Figure1. However, the abovementioned denoising procedure can also destroy the structure of R˜, so we need to recover R˜ from R⌣. Next, let us focus on the diagonal element of R˜. It can be deduced that the m-th (m = 6 (n − 1) + q) diagonal entity of R˜ is(19)R˜m,m=∑k=1KCm,kλkCHm,k,=∑k=1KCm,k2λk,=∑k=1Ke−jπτn,kBq,k2λk,=∑k=1KBq,k2λk.Figure 1
Illustration of the proposed denoising principle.We defineχq=∑k=1KBq,k2λk and let χ=diagχ1,χq,…,χq∈ℂ6×6, and then, we have(20)R⌣=Rs−I6⊗χ.SinceRs is a low-rank matrix, the noiseless recovery problem can be formulated as [32](21)minrankRs.t.SΩR=R⌣,where rankR returns the rank of R. To recover, the noiseless covariance coincides with the concept of matrix completion. Noting that the rank optimization is a nonconvex issue, it is usually replaced by the relaxed nuclear norm constraint ⋅∗, i.e.,(22)minR∗s.t.SΩR=R⌣.In practice,R⌣ is replaced by its estimation, denoted by R^1. Since there exists error between R⌣ and R^1, a boundεis usually set, and the issue in equation (22) is transformed into(23)minR∗s.t.SΩR−R^1≤ε.In practice,ε is usually chosen according to the noise tolerance; in this paper, it is set to 10−4. The abovementioned optimization can be easily accomplished via the convex toolboxes, e.g., cvx. After that, the eigendecompositon can be performed, and then, the estimation of the signal subspace Us is accomplished.


### 3.3. Parameter Estimation

Actually, the following rotational invariance relations exist:(24)AD1B=AD2BΦ1,2,AD1B=AD3BΦ1,3,⋮AD1B=AD6BΦ1,6,where DnB returns a diagonal matrix, the diagonals of which are the n-th row of B, and Φ1,q=diagβ11,q,β21,q,…,βK1,q, q=2,3,…,6, βk1,q=bkq/bk1. Next, we define(25)Jq=IM⊗i6,q,where i6,q accounts for the q-th column of I6. The relations in equation (24) become(26)J1A⊙B=J2A⊙BΦ1,2,J1A⊙B=J3A⊙BΦ1,3,⋮J1A⊙B=J6A⊙BΦ1,6.Inserting equation (26) into equation (13) yields(27)J1Us=J2UsT−1Φ1,2T,J1Us=J3UsT−1Φ1,3T,⋮J1Us=J6UsT−1Φ1,6T,where ⋅−1 denotes the inverse. In other words, we have(28)J2Us†J1Us=T−1Φ1,2T,TJ3Us†J1UsT−1=Φ1,3,⋮TJ6Us†J1UsT−1=Φ1,6,where the superscript ⋅† denotes the pseudoinverse. Performing eigendecomposition on J2Us†J1Us, we can get the eigenvalues of the matrix and the associated eigenvectors, which reveal the estimation of Φ1,2 and the estimation of T (denoted as T^). Calculating the left parts of equation (28) (except the first row), one can get the estimation of Φ1,3, Φ1,4, Φ1,5, and Φ1,6, respectively.It has been pointed out in [20] that bk can be written as(29)bk=bk11βk1,2⋮βk1,6.It is easy to find that(30)bk11βk1,2βk1,3∗bk1βk1,4βk1,5βk1,6∗=bk121βk1,2βk1,3∗βk1,4βk1,5βk1,6∗.Letek=bk1,bk2,bk3T and hk=bk4,bk5,bk6T. Then, we have(31)ekek∗hk∗hk=sinθkcosϕksinθksinϕkcosθk.According to equation (30), we can get(32)ukvkwk≜bk1bk∗1bk1bk∗1⋅1βk1,2βk1,31βk1,2βk1,3∗βk1,4βk1,5βk1,6∗βk1,4βk1,5βk1,6,≜1βk1,2βk1,31βk1,2βk1,3∗βk1,4βk1,5βk1,6∗βk1,4βk1,5βk1,6,=sinθkcosϕksinθksinϕkcosθk.Obviously,bk12 is a constant, and it has been removed by normalizing calculation. Let the estimations of uk, vk, and wk be u^k, v^k, and w^k, respectively. 2D-DOA can be estimated by(33)θ^k=arccosw^k,ϕ^k=arctanv^k/u^k.Once the 2D-DOA estimation has been accomplished, the polarization parameters can be estimated via the least squares approach in [30]. The details are omitted for simplicity.


## 3.1. Principle of Traditional Eigendecompositon

It is well known that when Gaussian white noise (uniform noise) exists, the noise powers fulfill(10)σ12=σ22=⋯=σ6N2=σ2,where σ2 is a constant, and then, the noise covariance becomes(11)Rn=σ2I6N,where IM stands for the M×M identity matrix. If we ignore the noise item in equation (7), the eigendecomposition of R is given by(12)R˜=∑n=1KαkukukH,=UsΣsUsH,where αk is the k-th eigenvalue, uk is the associated eigenvector, and uk1Huk2=0 is for any k1≠k2; Σs=diagα1,α2,…,αK denotes the eigenvalue matrix, and Us=u1,u2,…,uK∈ℂ6N×K is called the signal subspace, which spans the same subspace as A⊙B. Namely, there is a full-rank matrix T∈ℂK×K such that(13)Us=CT.Also, the noiseless covariance matrix can be expressed as(14)R˜=∑n=1KαkukukH+∑n=16N−K0⋅vnvnH,=UsΣs1UsH+Un06N−K6N−KUnH,where vn is the eigenvector from the null subspace of Us, i.e., vnHUs=01×K, and 0M×N represents the M×N full zero matrix. Σs1=diagα1+σ2,α2+σ2,…,αK+σ2, and Un=v1,v2,…,v6N−K∈ℂ6N×6N−K is called the noise subspace. Since the identity matrix can be formulated as the product of arbitrary unitary matrix and its Hermitian transpose, the noisy R can be written as(15)R=∑n=1Kαk+σ2ukukH+∑n=16N−Kσ2ununH,=UsΣs1UsH+UnΣnUnH,where Σn=diagσ2,σ2,…,σ2∈ℂ6N−K×6N−K. The results in equation (15) reveal that the uniform noise would not destroy the eigendistribution of the signal. However, in the presence of nonuniform noise, the noise power is not unique, so the conclusion in equation (15) will be untenable. It is necessary for us to denoise before further processing.


## 3.2. Denoising

LetΩ be a set that records the nonzero entities of Rn, i.e.,(16)Ω=m,m|m=1,2,…,6N.We define a sampling operatorSΩ⋅ that picks up the elements of the matrix in the blanket with indexes in Ω, for example, SΩR=R¯∈C6N×6N such that(17)R¯m,n=Rm,n,m,n∈Ω,0,m,n∉Ω,where Rm,n denotes the (m, n)-th entity of R and is similar to others. Since Rn is a diagonal matrix, we have Rn=SΩRn. The effect of the noise can be easily removed via the following reduced covariance matrix:(18)R⌣=R−SΩR,=R˜−SΩR˜.The abovementioned denoising procedure is illustrated in Figure1. However, the abovementioned denoising procedure can also destroy the structure of R˜, so we need to recover R˜ from R⌣. Next, let us focus on the diagonal element of R˜. It can be deduced that the m-th (m = 6 (n − 1) + q) diagonal entity of R˜ is(19)R˜m,m=∑k=1KCm,kλkCHm,k,=∑k=1KCm,k2λk,=∑k=1Ke−jπτn,kBq,k2λk,=∑k=1KBq,k2λk.Figure 1
Illustration of the proposed denoising principle.We defineχq=∑k=1KBq,k2λk and let χ=diagχ1,χq,…,χq∈ℂ6×6, and then, we have(20)R⌣=Rs−I6⊗χ.SinceRs is a low-rank matrix, the noiseless recovery problem can be formulated as [32](21)minrankRs.t.SΩR=R⌣,where rankR returns the rank of R. To recover, the noiseless covariance coincides with the concept of matrix completion. Noting that the rank optimization is a nonconvex issue, it is usually replaced by the relaxed nuclear norm constraint ⋅∗, i.e.,(22)minR∗s.t.SΩR=R⌣.In practice,R⌣ is replaced by its estimation, denoted by R^1. Since there exists error between R⌣ and R^1, a boundεis usually set, and the issue in equation (22) is transformed into(23)minR∗s.t.SΩR−R^1≤ε.In practice,ε is usually chosen according to the noise tolerance; in this paper, it is set to 10−4. The abovementioned optimization can be easily accomplished via the convex toolboxes, e.g., cvx. After that, the eigendecompositon can be performed, and then, the estimation of the signal subspace Us is accomplished.


## 3.3. Parameter Estimation

Actually, the following rotational invariance relations exist:(24)AD1B=AD2BΦ1,2,AD1B=AD3BΦ1,3,⋮AD1B=AD6BΦ1,6,where DnB returns a diagonal matrix, the diagonals of which are the n-th row of B, and Φ1,q=diagβ11,q,β21,q,…,βK1,q, q=2,3,…,6, βk1,q=bkq/bk1. Next, we define(25)Jq=IM⊗i6,q,where i6,q accounts for the q-th column of I6. The relations in equation (24) become(26)J1A⊙B=J2A⊙BΦ1,2,J1A⊙B=J3A⊙BΦ1,3,⋮J1A⊙B=J6A⊙BΦ1,6.Inserting equation (26) into equation (13) yields(27)J1Us=J2UsT−1Φ1,2T,J1Us=J3UsT−1Φ1,3T,⋮J1Us=J6UsT−1Φ1,6T,where ⋅−1 denotes the inverse. In other words, we have(28)J2Us†J1Us=T−1Φ1,2T,TJ3Us†J1UsT−1=Φ1,3,⋮TJ6Us†J1UsT−1=Φ1,6,where the superscript ⋅† denotes the pseudoinverse. Performing eigendecomposition on J2Us†J1Us, we can get the eigenvalues of the matrix and the associated eigenvectors, which reveal the estimation of Φ1,2 and the estimation of T (denoted as T^). Calculating the left parts of equation (28) (except the first row), one can get the estimation of Φ1,3, Φ1,4, Φ1,5, and Φ1,6, respectively.It has been pointed out in [20] that bk can be written as(29)bk=bk11βk1,2⋮βk1,6.It is easy to find that(30)bk11βk1,2βk1,3∗bk1βk1,4βk1,5βk1,6∗=bk121βk1,2βk1,3∗βk1,4βk1,5βk1,6∗.Letek=bk1,bk2,bk3T and hk=bk4,bk5,bk6T. Then, we have(31)ekek∗hk∗hk=sinθkcosϕksinθksinϕkcosθk.According to equation (30), we can get(32)ukvkwk≜bk1bk∗1bk1bk∗1⋅1βk1,2βk1,31βk1,2βk1,3∗βk1,4βk1,5βk1,6∗βk1,4βk1,5βk1,6,≜1βk1,2βk1,31βk1,2βk1,3∗βk1,4βk1,5βk1,6∗βk1,4βk1,5βk1,6,=sinθkcosϕksinθksinϕkcosθk.Obviously,bk12 is a constant, and it has been removed by normalizing calculation. Let the estimations of uk, vk, and wk be u^k, v^k, and w^k, respectively. 2D-DOA can be estimated by(33)θ^k=arccosw^k,ϕ^k=arctanv^k/u^k.Once the 2D-DOA estimation has been accomplished, the polarization parameters can be estimated via the least squares approach in [30]. The details are omitted for simplicity.


## 4. Algorithmic Analyses

### 4.1. Important Remarks

Remark 1: as described in the context, the proposed method is insensitive to thern, which means it is suitable for arbitrary sensor geometry. Besides, it is insensitive to the sensor position error as well.Remark 2: it is well known that the uniform noise is a special case of the nonuniform noise. Therefore, the proposed algorithm is effective in the white noise scenario.Remark 3: as explained in [30], all the estimated parameters θk,ϕk,ηk,γk are one-to-one paired.Remark 4: since the matrix completion would not hurt the rank and the dimension of the covariance matrix, the proposed algorithm can identify the same number of sources in [20].


### 4.2. Stochastic CRB

LetRn=Qq, where q=q1,q2⋯qPTis a real vector that parameterizes Rn. From the derivations in [35], we can get the stochastic CRB on 2D-DOA and polarization angle, which are given by(34)CRB=1LH−MT−1MT−1,with(35)H=2ReD˜H∏F˜⊥D˜⊕RcF˜HR˜−1F˜Rc⊗14×4,M=2ReJTD˜dH∏F˜⊥⊗R˜−1F˜RcTQ˜∗JTD˜pH∏F˜⊥⊗R˜−1F˜RcTQ˜∗,T=2ReQ˜HR˜−T⊗∏F˜⊥Q˜−Q˜H∏F˜⊥T⊗∏F˜⊥Q˜,where F˜=Q−1/2C, ∏F˜⊥=I6N−∏F˜ with ∏F˜=F˜F˜†, And D˜=D˜d,D˜p, where D˜d=Q−1/2Dd andD˜p=Q−1/2Dp with Dd=∂f1/∂θ1,…,∂fK/∂θK,∂f1/∂ϕ1,…,∂fK/∂ϕK and Dp=∂f1/∂η1,…,∂fK/∂ηK,∂f1/∂γ1,…,∂fK/∂γK, respectively, where fk denotes that with the k-th column of C. R˜=Q−1/2RQ−1/2. J=vece1e1T,vece2e2T,…,veceKeKT, where ek denotes the k-th column of IK and vec⋅ denotes the vectorization. Q˜=vecQ˜′1,vecQ˜′2,…,vecQ˜′P with Q˜′p=Q−1/2Qp′Q−1/2, Q˜′p=∂Q/∂qp.


## 4.1. Important Remarks

Remark 1: as described in the context, the proposed method is insensitive to thern, which means it is suitable for arbitrary sensor geometry. Besides, it is insensitive to the sensor position error as well.Remark 2: it is well known that the uniform noise is a special case of the nonuniform noise. Therefore, the proposed algorithm is effective in the white noise scenario.Remark 3: as explained in [30], all the estimated parameters θk,ϕk,ηk,γk are one-to-one paired.Remark 4: since the matrix completion would not hurt the rank and the dimension of the covariance matrix, the proposed algorithm can identify the same number of sources in [20].


## 4.2. Stochastic CRB

LetRn=Qq, where q=q1,q2⋯qPTis a real vector that parameterizes Rn. From the derivations in [35], we can get the stochastic CRB on 2D-DOA and polarization angle, which are given by(34)CRB=1LH−MT−1MT−1,with(35)H=2ReD˜H∏F˜⊥D˜⊕RcF˜HR˜−1F˜Rc⊗14×4,M=2ReJTD˜dH∏F˜⊥⊗R˜−1F˜RcTQ˜∗JTD˜pH∏F˜⊥⊗R˜−1F˜RcTQ˜∗,T=2ReQ˜HR˜−T⊗∏F˜⊥Q˜−Q˜H∏F˜⊥T⊗∏F˜⊥Q˜,where F˜=Q−1/2C, ∏F˜⊥=I6N−∏F˜ with ∏F˜=F˜F˜†, And D˜=D˜d,D˜p, where D˜d=Q−1/2Dd andD˜p=Q−1/2Dp with Dd=∂f1/∂θ1,…,∂fK/∂θK,∂f1/∂ϕ1,…,∂fK/∂ϕK and Dp=∂f1/∂η1,…,∂fK/∂ηK,∂f1/∂γ1,…,∂fK/∂γK, respectively, where fk denotes that with the k-th column of C. R˜=Q−1/2RQ−1/2. J=vece1e1T,vece2e2T,…,veceKeKT, where ek denotes the k-th column of IK and vec⋅ denotes the vectorization. Q˜=vecQ˜′1,vecQ˜′2,…,vecQ˜′P with Q˜′p=Q−1/2Qp′Q−1/2, Q˜′p=∂Q/∂qp.


## 5. Simulation Results

In this subsection, the Monte Carlo simulation is utilized to assess the estimation accuracy. We consider an arbitraryN-element EMVS receives array configuration, and we assume that K=3 far-field sources impinge on the array, whose parameters are θ=10°,20°,30°, ϕ=20°,30°,40°, γ=30°,40°,50°, and η=25°,35°,45°. Moreover, we suppose that L snapshots have been collected. Each result relies on 200 experiments. In the simulation, the signal-to-noise ratio (SNR) is defined as the ration of the power of the two components in equation (6). Two measures are adopted: one is the root mean square error (RMSE), and the other is the probability of successful detection (PSD). In all the simulations, the noise is randomly generated with powers which are uniformly chosen from [1 100000].Example 1.
We give the scattering figures of the proposed algorithm withN = 4 and L = 500, where the SNR is set to 20 dB. Herein, the EMVS is randomly placed in a three-dimensional space with (xn,yn,zn) fulfilling a uniform distribution with interval [−0.5λ, 0.5λ]. Figure 2 shows the results of direction angle estimation and polarization parameters estimation. Clearly, all the angles can be correctly estimated and automatically paired. It is evident that our estimator is effective in a nonuniform noise scenario.Figure 2
Scattering figures of our algorithm withN = 4, L = 500, and SNR = 20 dB: (a) 2D-DOA estimation; (b) polarization parameter estimation.
(a)(b)Example 2.
We present the RMSE and PSD curves of the proposed estimator. Herein, a successful trial is recognized if the absolute error of the estimated angle is smaller than1∘. In Figure 3, we plot the average estimation performance on the direction angle estimate (labelled with the suffix “−d”) and polarization angle estimate (labelled with the suffix “−p”), where N = 4 and L = 500. In contrast, the RMSE results of ESPRIT in [20] as well as the CRB are added. From the result, we can observe that the RMSE performance of both the proposed estimator and ESPRIT is improved with the increasing SNR, while the PSD of both estimators reaches 100%, once SNR is larger than a threshold (e.g., 25 dB). Besides, it depicts that the proposed estimator provides more accurate parameter estimation performance for direction angle estimation when SNR <10 dB. However, the improvement is not obvious as for polarization parameter estimation.Figure 3
RMSE performance and PSD performance versus SNR (a) RMSE vs. SNR (b) PSD vs. SNR.
(a)(b)Example 3.
We plot the average RMSE performance and the average PSD curves with different snapshot numberL in Figure 4, while N and SNR are set to 4 and 10 dB, respectively. Notably, both the proposed estimator and ESPRIT provide better estimation performance with larger L. Similar to the previous observation, the proposed algorithm outperforms the ESPRIT algorithm. The improvement benefits from the fact that the proposed algorithm can eliminate the noise, while traditional ESPRIT algorithm cannot avoid the effect of nonuniform noise.Figure 4
RMSE performance and PSD performance versusL (a) RMSE vs. L (b) PSD vs. L.
(a)(b)


## 6. Conclusions

In this paper, we investigate the issue of angle estimation using EMVS array with arbitrary sensor geometry and nonuniform noise. A matrix completion-based algorithm has been proposed, which first eliminates the effect of nonuniform via solving a convex problem. After the noiseless covariance matrix has been recovered, the traditional subspace method is utilized to estimate the signal subspace, and the ESPRIT idea is adopted for 2D-DOA estimation. Our algorithm is robust to nonuniform noise and sensor position error. It should be pointed out that the tensor structure has not been exploited. More attention should be paid to this topic to further increase the estimation accuracy.



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*Source: 9053864-2021-08-19.xml*