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Elevation, azimuth, and polarization estimation with nested electromagnetic vectorsensor arrays via tensor modeling
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 153 (2020)
Abstract
In this paper, we address the joint estimation problem of elevation, azimuth, and polarization with nested array consists of complete sixcomponent electromagnetic vectorsensors (EMVS). Taking advantage of the tensor permutation, we convert the sample covariance matrix of the receive data into a tensorial form which provides enhanced degreeoffreedom. Moreover, the parameter estimation issue with the proposed model boils down to a Vandermonde constraint Canonical Polyadic Decomposition problem. The structured least squares estimation of signal parameters via rotational invariance techniques is tailored for joint autopairing elevation, azimuth, and polarization estimation, ending up with a computational efficient method that avoids exhaustive searching over spatial and polarization region. Furthermore, the sufficient uniqueness analysis of our proposed approach is addressed, and the stochastic CramérRao bound for underdetermined parameter estimation is derived. Simulation results are given to verify the effectiveness of the proposed method.
1 Introduction
Electromagnetic vectorsensor (EMVS) has been widely used in a variety of applications such as localization, tracking, and beamforming [1–3]. A complete sixcomponent EMVS contains spatially colocated three identical orthogonally electric dipoles and magnetic loops that could measure all sixcomponents of the electromagnetic field [4, 5]. The model of EMVS was investigated in [6]. Different from scalarsensor arrays, EMVS arrays that are composed of multiple EMVSs with particular configurations could detect both directionofarrival (DOA), i.e., elevation and azimuth, and polarization of incident sources. This polarization diversity brings us lots of advantages, such as resolving sources from the same DOA as long as they have different polarization states, providing better resolution ability, offering extra degreesoffreedom (DOFs), and improving estimation performance [7–10].
In order to utilize the aforementioned benefits, the DOA and polarization estimation with EMVS arrays could be cast as a multipleparameter estimation problem which, however, turns out to be much more complicated than the scalarsensor array case. Usually, they demand a timeconsuming multidimensional searching procedure [11]. Also, the autopairing issue of different parameters cannot be neglected. Moreover, the steering matrix of an EMVS always has irregular structure, which aggravates the difficulty in the parameter estimation techniques. Several matrixbased DOA and polarization estimation methods have been proposed with the EMVS array. Most of them are the extensions of the existing DOA estimators with scalarsensor arrays by taking into the physical structure of EMVSs into consideration. In [6], a vector crossproduct approach to estimate the Poynting vector of the sources was presented. Eigenvectorbased parameter estimators such as multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRIT) were developed in [12, 13], through utilizing either spatial or temporal invariance properties. These approaches show superior resolution abilities and estimation accuracy with tolerable computational burdens.
The multidimensional model, i.e., tensor model, was first linked with array processing under the framework with Canonical Polyadic Decomposition (CPD) in [14]. Interestingly, the tensor modeling techniques reverse the curse of multidimensional problems into blessing, which could achieve several benefits such as autoparing of parameters, relaxed uniqueness condition, to name a few. Subsequently, tensorbased approaches were introduced to the EMVS array whose received data embodies multidimensional structure. In [15], the identifiability was analyzed with an EMVS array. More general CPDbased approach for EMVS array were developed in [16–19]. In parallel, higherorder singular value decomposition (HOSVD) which represents tensorbased lowrank approximation method, generalizes the concept of matrixbased SVD. HOSVDbased parameter estimators are able to provide higher estimation accuracy [20, 21]. The socalled tensorMUSIC approaches were developed with the EMVS array in [22, 23], which require exhaustive multidimensional searching procedures.
Compared with uniform linear array (ULA), difference coarrays such as nested and coprime arrays are able to provide more DOFs [24–26]. The nested array could construct virtual arrays that obtain as large as \(\mathcal {O}(M^{2})\) DOFs with M physical sensors, which greatly improves the identifiability as well as resolvability and reduces mutual coupling effects between sensors. The stochastic CramérRao bound (CRB) when the number of sources is more than the number of sensors was derived separately by [27–29]. It is worth noting that the aforementioned methods are only suitable for uncorrelated sources.
A nested array which is equipped with EMVSs was first developed by Han et al. in [30]. The nested EMVS array provides more DOFs and better resolvability than the EMVS ULA. However, the DOA and polarization vectors were not decoupled, meaning that the nested EMVS array requires a prior knowledge of the polarization state before performing MUSIC method to avoid exhaustive searching. A tensor model for nested EMVS array that could efficiently decouple the DOA and polarization was constructed in [31]. This method divides the nested EMVS array into several subarrays with respect to different polarization states. Then, it builds a tensor composed of multiple local covariances, followed by the CPD to obtain DOA and polarization estimations.
In this paper, we propose a novel methodology for tensor modeling and parameter estimation by means of utilizing the relationship between tensor permutation and the array structure. The proposed approach has several novelties and advantages: (1) A tensor model of nested EMVS array model is established through the property of tensor permutation, which is different from the complex produce proposed in [31]. (2) The proposed estimator is capable of achieving the autopairing of DOA and polarization state. (3) It is proved that the proposed method guarantees more DOFs through uniqueness condition analysis. (4) An underdetermined stochastic CRB for the nested EMVS array is derived as a benchmark for performance evaluation.
The remainder of this paper is organized as follows. Section 2 introduces the basic mathematical notations and constructs the tensor model of a nested EMVS array. In Section 3, we briefly review the Han’s tensor modeling method and then propose our tensor modeling approach through multilinear algebra. In Section 4, we devise CPD and the structured least squares (SLS) ESPRIT method for joint DOA and polarization estimation. In Section 5, we analyze the sufficient uniqueness condition of the proposed method and derive the underdetermined stochastic CRB of the nested EMVS array. In Section 6, we give simulation results to demonstrate the effectiveness of our propose method. Section 7 concludes this work.
2 Problem formulation
2.1 Notations
Definition 1
Tensor vectorization A tensor vectorization of an Ndimensional tensor \(\boldsymbol {\mathcal {A}}\in \mathbb {C}^{I_{1}\times I_{2}\times \cdots \times I_{N}}\) is denoted as \(\text {vec}(\boldsymbol {\mathcal {A}})\in \mathbb {C}^{\prod _{n=1}^{N}I_{n}\times 1}\) where vec(·) represents the vectorization operator.
Property 1
If an Ndimensional tensor \(\boldsymbol {\mathcal {A}}\in \mathcal {C}^{I_{1}\times \ldots \times I_{N}}\) could be expressed as the outerproduct of a sequence of vectors \(\mathbf {a}_{n}\in \mathbb {C}^{I_{n}\times 1}, n=1,2,\ldots,N\), namely,
where ∘ denotes the outer product, then, its tensor vectorization has the following structure as
where ⊙ represents the KhatriRao product.
Definition 2
Tensor matrization A matrix unfolding of an Ndimensional tensor \(\boldsymbol {\mathcal {A}}\in \mathbb {C}^{I_{1}\times I_{2}\times \cdots \times I_{N}}\) along nmode is denoted as \(\boldsymbol {\mathcal {A}}_{(n)}\in \mathbb {C}^{I_{n}\times I_{n+1}\ldots I_{N}I_{1}\ldots I_{n1}}\).
Definition 3
The nmode tensormatrix product The nmode product of a tensor \(\boldsymbol {\mathcal {A}}\in \mathbb {C}^{I_{1}\times I_{2}\times \cdots \times I_{N}}\) and a matrix \(\mathbf {D}\in \mathbb {C}^{J_{n} \times I_{n}}\) along the nth mode is given by
where \(\boldsymbol {\mathcal {C}}\in \mathbb {C}^{I_{1}\times \ldots \times I_{n1}\times J_{n}\times I_{n+1}\times \ldots \times I_{N}}\) and \(\boldsymbol {\mathcal {C}}_{(n)}=\mathbf {D}\boldsymbol {\mathcal {A}}_{(n)}\).
2.2 Signal model
For a sixcomponent EMVS, it consists six spatially colocated antennas, i.e., three orthogonally electric dipoles and three orthogonally magnetic loops. We adopt e_{k}=△[e_{xk},e_{yk},e_{zk}]^{T} and h_{k}=△[h_{xk},h_{yk},h_{zk}]^{T} to denote the kth source’s electromagnetic field characterized by electric and magnetic triads along x, y, and zaxis, respectively. The diagram of an EMVS under Cartesian coordinates is shown in Fig. 1.
We assume that there are no mutual coupling effects within each EMVS. Thus, the physical polarization vector of kth source observed by an EMVS is a collection of e_{k} and h_{k}
where θ_{k}∈[0,π),ϕ_{k}∈(0,2π],γ_{k}∈[0,π/2),η_{k}∈[−π,π) denotes the kth source’s elevation measured from positive vertical zaxis, azimuth, auxiliary polarization angle, and polarization phase difference, respectively. The (·)^{T} stands for the transpose. Throughout this work, we assume that different sources have different polarization states.
Then, the normalized Poynting vector g_{k} is given by
where (·)^{∗}, ×, and · represent the complex conjugation, Cartesian product, and ℓ_{2}norm, respectively. We use μ_{k},ν_{k},ω_{k} for the directioncosine functions along the x, y, and zaxis, respectively.
In this paper, we consider a typical twolevel nested array which is composed of two concatenated ULAs with different inner EMVS spacings. The EMVS linear array is placed along the yaxis with a total of M EMVSs as illustrated in Fig. 2. The small ULA has M_{1} sensors with a halfwavelength spacing, whereas the large one has a total of M_{2} sensors with an intersensor spacing of \((M_{1}+1)\frac {\lambda }{2}\). Thus, the positions of EMVSs in the nested array are given as
According to [24], the DOFs for a nested array with identical scalarsensors could be determined as
Assume that there are K narrowband farfield completed polarized signals impinging on this array. As a result, the spatial steering vector of the nested EMVS array for the kth source is given by
where λ denotes the wavelength of the sources and z_{k} denotes the kth element of z. Note that (8) only stands for the spatial relationship among all M EMVSs in the array without taking the physical property of EMVS into account. The tth sample vector of whole array gives as
where \(\mathbf {A}_{\mathrm {p}}\in \mathbb {C}^{6M\times K}\) represents the steering matrix of the whole nested EMVS array where s(t) and n(t) represent K signals’ waveforms at time t and the additive temporally and spatially white Gaussian noise, respectively.
In this work, we assume that the sources are uncorrelated stationary white Gaussian process. Meanwhile, the additive noise obeys independent and identically distributed (IID) Gaussian distribution, i.e., \(n(t) \sim CN(0, \sigma _{n}^{2}\mathbf I)\) with \(\sigma _{n}^{2}\) being the noise variance. Furthermore, the signal and noise are uncorrelated. The covariance matrix is calculated as
where D denotes the source covariance matrix, \(\mathbb {E}[\cdot ]\) represents the mathematical expectation. Since the sources are statistically uncorrelated, we have
where \(\mathbf d=\left [\sigma ^{2}_{1},\sigma ^{2}_{2},\ldots,\sigma ^{2}_{K}\right ]^{T}\) with \(\sigma _{k}^{2}\) being the power of the kth source and diag(·) denotes the diagonalization operator that forms an K×1 vector into a K×K square matrix with elements on its main diagonal and zeroes elsewhere. Note that the covariance matrix cannot be exactly obtained since the number of samples is finite. Instead, we use the sample covariance matrix (SCM) as
For simplicity, we use the covariance matrix for derivation of the proposed method. However, it should be kept in mind that the SCM is different from the covariance matrix. In consequence, the signal covariance matrix R_{s} will not have a perfect diagonal structure. This phenomenon may degrade the performance of the proposed method as we will see in the simulations later.
By vectorizing the covariance matrix, an apertureenlarged array is obtained, and its observation is given as
where 1=vec(I_{6M}). Note that the steering matrix \(\mathbf {a}_{p}^{*}\odot \mathbf {a}_{p}\in \mathbb {C}^{36M^{2}\times K}\) has a larger size, which extends the array aperture. However, the sources d become coherent since s(t) are stationary process. To circumvent this problem, several rank restoration methods, e.g., spatialsmoothing technique and compressive sensingbased approaches, have been suggested. These strategies might not be appropriate for the situation of the nested EMVS array. This is because the apertureextended array does not have the ULA structure, which is prohibiting the application of spatialsmoothing technique. Note that the spatial matrix a and polarization matrix P are alternating arranged as depicted in (18), which are not decoupled. As a matter of fact, the EMVS array can be described as a multidimensional model. This thereby motivates us to establish the tensor model for the observations of the EMVS array.
3 Tensor modeling
3.1 Han’s tensor modeling approach
The measurement matrix of the whole EMVS array at tth sample is obtained through matrizating (9) as
where mat(·) denotes the matrization operator, \(\mathbf {Y}(t)\in \mathbb {C}^{M\times 6}\), and \(\boldsymbol {\mathcal {A}}\in \mathbb {C}^{M\times 6\times K}\) represents the steering tensor as shown in Fig. 3.
The 3mode matrix unfolding of the steering tensor is equivalent to the steering matrix of EMVS as
Based on [30], the fourthdimensional covariance tensor \(\boldsymbol {\mathcal {R}}\) is constructed as
We could regard (23) as a tensor extension of the covariance matrix. To further investigate the structure of \(\boldsymbol {\mathcal {R}}\), each element in \(\boldsymbol {\mathcal {R}}\) could be expressed as
To be specific, we have
Then perform 2mode matrization of \(\boldsymbol {\mathcal {R}}\), yielding
where \(\boldsymbol {\mathcal {A}}_{\text {nest}}\in \mathbb {C}^{6M^{2}\times 6\times K}\) stands for the steering tensor and
Comparing with (17), (26) has a similar form but with a nested steering tensor of size 6M^{2}×6×K. Taking one horizontal slice of (30) since the DOA and polarization state are coupled in the steering tensor as
where \(\boldsymbol {\mathcal {A}}_{\text {nest}}^{l}\in \mathbb {C}^{6M^{2}\times K}, l=1,\ldots,6\) represents the subnested steering tensor associated with the lth polarization state in p_{k}. The polarization state and the spatial structure are coupled in the first dimension of the lth subnested steering tensor, and it has the following form
and a_{l} denotes one slice of the nested steering tensor associated with lth polarization state
where \(\mathbf {a}_{l}\in \mathbb {C}^{M\times K}, l=1,2,\ldots,6\). In a word, the tensor modeling of nested EMVS array could be regarded as folding one of the polarization dimension of the matrixbased steering matrix into a higher dimension. Note that the spatial and polarization dimensions of the nested steering tensor are still coupled.
3.2 Proposed tensor modeling method
Different from the existing tensor modeling methods in [30] and [31], we introduce the property of tensor permutation to establish a tensor model of the nested EMVS array.
Definition 4
Tensor permutation Consider an Ndimensional tensor, a permutation operator of this tensor is denoted as π=[π_{1},π_{2},…,π_{N}], where π_{n}∈{1,2,…,N},n=1,2,…,N. An Ndimensional tensor \(\boldsymbol {\mathcal {C}}\in \mathbb {C}^{I_{1}\times I_{2}\ldots \times I_{N}}\) with indexes i_{1},i_{2},…,i_{N} after permutation leads to \(\boldsymbol {\mathcal {C}}_{\pi }\in \mathbb {C}^{I_{\pi _{1}}\times I_{\pi _{2}}\ldots \times I_{\pi _{N}}}\)
For example, we define a permutation as π=[3,2,1]. Then, a threedimensional tensor \(\boldsymbol {\mathcal {C}}\in \mathbb {C}^{I_{1}\times I_{2}\times I_{3}}\) after permutation is \(\boldsymbol {\mathcal {C}}_{\pi }\in \mathbb {C}^{I_{3}\times I_{2}\times I_{1}}\) as shown in Fig. 4.
Property 2
If an Ndimensional tensor \(\boldsymbol {\mathcal {C}}^{I_{1}\times \cdots \times I_{N}}\) could be expressed as a outerproduct of N vectors, we have the following expression after permutation π=[π_{1},π_{2},…,π_{N}] as
This property is proved through using Property 1 and the definition of tensor permutation. It represents a special case of rankone tensors, which is based the fact that tensor permutation will maintain the interrelationship of each vectors.
Recalling (17), we obtain
According to Property 1, we formulate (33) into a fourdimensional tensor as
where \(\boldsymbol {\mathcal {G}}_{\text {nest}}\in \mathbb {C}^{M\times 6\times M\times 6}\). Then, by setting π:(1,3,2,4), \(\boldsymbol {\mathcal {G}}_{\text {nest}}\) after permutation π gives
Performing 1mode unfolding of \(\boldsymbol {\mathcal {G}}_{\pi }\in \mathbb {C}^{M\times M\times 6\times 6}\) into a threedimensional tensor yields
where \(\boldsymbol {\mathcal {G}}_{\pi }^{(1)}\in \mathbb {C}^{M^{2}\times 6\times 6}\). The spatial and polarization vectors are arranged in order instead of placing alternatively as in (33). Until this step, a threedimensional tensor with decoupled spatial and polarization factors is constructed. However, a^{∗}(θ_{k})⊙a(θ_{k}) does not obey the Vandermonde structure and incurs several repeated spatial phase factors as pointed out in [30]. Here, we directly remove the repeated ones to reduce the dimension of a^{∗}(θ_{k})⊙a(θ_{k}) while keeping the DOFs unchanged. The selecting and permutation matrices are defined as J_{s} and J_{p}, respectively. Then, we operate them on \(\boldsymbol {\mathcal {G}}_{\pi }\) to remove the repeated and arrange the spatial phase factors in order as
where \(\boldsymbol {\mathcal {G}}\in \mathbb {C}^{\bar {M}\times 6\times 6}\) and
Note that the deleting and selecting matrices are determined by the spatial structure of the nested array and thus can be calculated offline once.
To be specific, each element in \(\boldsymbol {\mathcal {G}}\) is expressed as
where \(i_{1}=1,\ldots,\bar {M}, i_{2}=1,\ldots,6, i_{3}=1,\ldots,6\) and δ stands for the Dirac function such that
Since we have already known the positions of noise items as given in (41), the noise could be easily eliminated from \(\boldsymbol {\mathcal {G}}\) to further improve the signaltonoise ratio (SNR) in real world applications. Performing SVD to (16), the singular value matrix is given as \(\mathbf \Sigma =\text {diag}(\hat {\sigma }_{1}^{2},\ldots,\hat {\sigma }_{6M}^{2})\). The singular values are decreasingly ordered as \(\hat {\sigma }_{1}^{2}\geq \cdots \geq \hat {\sigma }_{K}^{2}>\hat {\sigma }_{K+1}^{2}=\ldots =\hat {\sigma }_{6M}^{2}\). Thus, an estimate of the noise power could be given as
It should be emphasized that although the proposed tensor model has a similar form as that in [31], the physical properties behind the model are quite different. In our proposed model, the matrix factors represent spatial and polarization states while dropping the temporal information. Note that the powers of K sources are trivial parameters which will not influence the estimate of parameters.
4 DOA and polarization estimation
4.1 Elevation estimation
In this subsection, elevation estimation is firstly achieved. Then, we use the results to obtain azimuth and polarization estimates in the next subsection. The proposed tensor modeling approach constructs a threedimensional tensor which is able to exploit the spatially correlation structure inherent in the EMVS array data. To this end, we use B, P, and P^{∗}, respectively, to stand for matrix factors. Thus, we could use CPD to estimate these factors directly. The CPD of \(\boldsymbol {\mathcal {G}}\) could achieve the estimates of B and P up to permutation and scaling [14]. The Alternating Least Squares (ALS) algorithm is usually applied to conduct the general CPD without a prior knowledge of the structure of matrix factors. It is worth noting that since B obeys the Vandermonde structure, this problem at hand could be solved through Vandermonde constrained CPD (VCPD) [17], which has more a relaxed uniqueness condition.
The noiseless matrix form of \(\boldsymbol {\mathcal {G}}\) could be obtained as 1mode unfolding, which is given by
where
Consider (43), we divide G into L overlapped submatrices, each of them has size M_{s}×36 to obtain the spatialsmoothing of G along the spatial dimension. Firstly, we define the lth selection matrix as
The augmented covariance G_{s} is then constructed as
where \(M_{s}=\bar {M}L+1\) and
Through the constuction of augmented covariance matrix, we restore the rank of G to achieve a better estimate of the signal subspace. Performing SVD to G_{s}, a lowrank approximation is obtained as
It follows from (50) and (51) that
where T represents a fullrank matrix. To proceed, we need to define the selection matrices J_{s1} and J_{s2}, which are defined as
Recall that the steering matrix B_{s} obeys Vandermonde structure, which indicates the steering matrix has spatial invariance property. The elevation is estimated by applying (52) and (53)
However, there exists subspace perturbation in both handsides. Here, we use the total least squares (TLS) solution to eliminate the influence of subspace perturbation as follows
where ΔU_{s1} and ΔU_{s2} represent the perturbation of signal subspace of both handsides. Note that the left and right handsides of (55) are highly structured and share many elements, which in turn means that they share the same noises. In order to suppress the noise shared in the subarrays, the structured least squares ESPRIT (SLSESPRIT) is tailored for enhancing the performance of elevation finding. In particular, defining define \(\mathbf U_{s}^{\text {SLS}}=\mathbf U_{s}+\Delta \mathbf U_{s}\) and the residual matrix as \(\mathbf F\left (\mathbf U_{s}^{\text {SLS}},\Phi \right)=\mathbf {J}_{s1}\mathbf U_{s}^{\text {SLS}}\Phi \mathbf {J}_{s2}\mathbf U_{s}^{\text {SLS}}\) as the residual matrix, the SLSESPRIT method attempts to minimize
where γ is the weighting factor which is used to avoid trivial solutions. Note that we use the iteration method to solve (56). The rth iteration step is formulated as
Neglecting the secondorder term of (58), we have
where
Thus, the updates of Φ_{r} and \(\mathbf U_{s,r}^{\text {SLS}}\) could be expressed in a closedform, expressed as
where (·)^{†} represents the pesudoinverse operator. After several iterations, the solution of (56) is achieved. Although the proposed SLSVCPD method is not optimal, it realizes a tradeoff between computational burden and estimation accuracy.
4.2 Azimuth and polarization joint estimation
In the above subsection, the estimates of elevations have been determined. This allows us to construct \(\mathbf B(\hat {\boldsymbol {\theta }})\), leading to the LS estimate of H as
where \(\hat {\mathbf H}\) consists of K vectors, that is
Recalling (45), the kth column of H is indeed an outer product of polarization vectors. Dividing \(\hat {\mathbf h}_{k}\) into six equally sized concatenated vectors and stack them along column to form a square matrix, we have
Performing SVD to mat(h_{k}), an approximate estimate of the polarization information vector is obtained as
where \(\hat {\mathbf u}_{k}\) stands for the singular vector associated with the largest singular value.
We now have achieved an estimate of the polarization vector. According to (4), the estimated Poynting vector of kth source is
It follows that the estimated azimuth is calculated as
Substituting \(\hat {\boldsymbol {\theta }}_{k}\) and \(\hat {\boldsymbol {\phi }}_{k}\) into (4), we achieve the estimate of \(\hat {\boldsymbol {\Xi }}_{k}\). The polarization parameters associated with kth signal are hence calculated as
where
Note that DOA and polarization associated with the same source is autopaired since they share the same eigenvector. Thus, no extra pairing step is needed. Table 1 concludes the DOA and polarization estimation procedure.
5 Identifiability and CRB
5.1 Identifiability
In this subsection, we first review the sufficient uniqueness condition with EMVS array from the CPD perspective. Note that an Ndimensional tensor is defined as a rankone tensor if it could be written into a outer product of N vectors. Thus, \(\boldsymbol {\mathcal {G}}\) is rankK since it is the minimal number of K rankone tensor as given in (38). Consider a threedimensional \(\boldsymbol {\mathcal {X}}\in \mathbb {C}^{I_{1}\times I_{2}\times I_{3}}\) with rankK which means that it is a sum of K outer product of three vectors, which could be expressed as
The uniqueness condition of a tensor with CPD relies on KrusKal’s condition. The Kruskalrank, termed as kr, is defined as the maximum number of J such that every J columns of a matrix are linearly independent. And it is denoted as kr(a)=J. Thus, the KrusKal’s condition provides a sufficient uniqueness condition for a threedimensional tensor. That is, three matrix factors satisfy
a, B, and C are unique up to scaling and permutation [14].
For a complete sixcomponents EMVS, its observation data always has kr(P)≥3. Thus, for an Melements EMVS ULA, the upper bound for the uniqueness condition when sources are uncorrelated gives
Usually, it holds that rank(P)≥4 for arbitrary DOA and polarization parameters. So the maximum number of identifiable sources is
Comparing with the scalarsensor ULA, the EMVS array could resolve more sources. Now, we consider the nested EMVS array, the sufficient uniqueness condition with the proposed model (38) yields
Since B is a Vandermonde matrix, it is straightforward to obtain \(kr(\mathbf B)=\bar {M}\). Also, the manifold of a sixcomponent EMVS is free of rank2 ambiguity, which indicates that kr(P)≥3 [32]. Substituting kr(P)=3 into (76) to obtain the upper bound of identifiability with CPD yields
Comparing with (75), the nested EMVS array obviously provides more DOFs. However, the uniqueness condition could be further relaxed if we take the Vandermonde structure of B and ESPRIT method together into consideration. According to the results in [17], we could directly obtain the upper bound of identifiability after applying spatial smoothing as
It should be noted that the proposed method perhaps cannot resolve the maximum number of sources provided by the upper bound since there exists errors between the SCM and covariance matrix when the number of samples is finite, and SNR is not sufficient high. For more discussions about this phenomenon, readers can refer to [27].
5.2 Underdetermined CRB
The stochastic CRB with EMVSs array model has been well discussed in literature. However, the existing CRB could not be directly applied to the underdetermined case in which the number of sources excesses the number of EMVSs. Under the assumption that R_{s} is a diagonal matrix, we extend the stochastic CRB for underdetermined DOA estimation with scalarsensor array to the EMVSs array cases. Indeed, the derived CRB generalizes the existing results so that the scalarsensor array could be regarded as a special case of the EMVS array. For simplicity, the variance of noise is assumed to be known. The parameter vector composed by DOAs, polarization, and source powers is defined as
According to [33], the (m,n)th element of the Fisher information matrix (FIM) is given by
where tr(·) represents the trace operator. Since tr(aB)=vec(a^{T})^{T}vec(B) and vec(aXB)=(B^{T}⊗a)vec(X), we have
where r=△vec(R), ⊗ denotes the Kronecker product. The firstorder derivatives of r with respect to each element of ζ is defined as
According to (33), it is straightforward to get
where \(\mathbf {a}_{nest}^{(1)}=\mathbf {a}^{(1)^{*}}_{p}\odot \mathbf {a}_{p}+\mathbf {a}^{*}_{p}\odot \mathbf {a}^{(1)}_{p}\) and
To proceed, we need to define the following matrices
Then, the FIM could be expressed into a compact form as
The underdetermined stochastic CRB of θ,ϕ,γ,η with nested EMVS array is calculated as
where \(\Pi _{\mathbf Q_{2}}^{\perp }=\mathbf I\mathbf Q_{2}(\mathbf Q_{2}^{H}\mathbf Q_{2})^{1}\mathbf Q_{2}^{H}\), and binv(·) stands for blockwise inverse operator which takes inverse of the block matrices with size K×K on the main block diagonal of (87).
6 Simulation results and discussions
This section gives various numerical examples to show the performance of the proposed SLSVCPD method. First, we evaluate root mean square error (RMSE) performance of our proposed method for elevation, azimuth, and polarization estimations. Second, we compare the proposed approach with the stateoftheart tensorbased schemes which use the SLSVCPD technique and EMVS array. Throughout all simulations, a nested EMVS array with M=6 EMVSs is adopted, which is depicted in Fig. 2. The position of these EMVSs are placed on \(\mathbf z=\frac {\lambda }{2}[1,2,3,4,8,12]^{T}\).
The RMSE of elevation estimation is defined as
Note that RMSEs of azimuth and polarization are computed by the same formula, but the variables are different. In addition, the SNR is defined as \(10\text {log}_{10}\left (\\mathbf {A}_{p}\mathbf {S} \_{F}^{2}/ \ \mathbf {N} \_{F}^{2}\right)\) where ·_{F} represents the Frobenius norm.
6.1 Performance analysis
In Figs. 5 and 6, we present the RMSE performances of elevations, azimuth, and polarization, respectively. The parameters of two sources are set as θ=[30^{∘},40^{∘}], ϕ=[30^{∘},60^{∘}], γ=[20^{∘},45^{∘}], and η=[15^{∘},30^{∘}], respectively. It is seen that the RMSE performances improve when SNR or the number of samples increases.
In Fig. 7, we study the influence of the number of iterations on the proposed SLSVCPD method. In this simulation, the parameters of four sources are randomly chosen as θ=[49.0^{∘},3.8^{∘},44.8^{∘},74.3^{∘}], ϕ=[103.1^{∘},101.4^{∘},33.2^{∘},67.7^{∘}], γ=[14.7^{∘},46.9^{∘},65.2^{∘},33.7^{∘}], and η=[−67.0^{∘},−101.7^{∘},−114.2^{∘},30.5^{∘}]. The number of receive samples is set as T=100. The curves of SLSVCPD with iteration number, i.e., n=1 and n=3, are also plotted for comparison. We observe that the proposed SLSVCPD method outperforms the LSVCPD scheme as expected. Also, the performance of SLSVCPD remains unchanged as the number of iterations increases, which indicates that the proposed SLSVCPD method has convergenced within one iteration.
6.2 Overdetermined case
In this subsection, we focus on the elevation estimation performance in the overdetermined case. To examine the performance of the nested EMVS array, two array configurations are considered, i.e., ULA and nested array. The number of elements in the EMVS ULA is set as M=6 with K=2 impinging sources. We compare CPD with EMVS ULA, namely, ULACPD [18], tensorMUSIC [30], SSCPD [31], and proposed SLSVCPD methods using the nested EMVS array. Note that the tensorMUSIC, SSCPD, and SLSVCPD methods are constructed using nested array, and thereby, we conceal the item “nested array” for simplification. Note that the SLSVCPD method only uses one iteration step. The number of spatial smoothing is set as L=2 for both SSCPD and SLSVCPD methods. For comparison, the polarization states of sources are prior known to the tensorMUSIC method to reduce the computational burden. Also, the stochastic CRB with nested EMVS array is plotted as a benchmark. The MonteCarlo trials for each simulation are set as 1000.
In Fig. 8, we compare the elevation RMSEs of all methods while varying SNR from –4 dB to 20 dB for a fixed T=50 samples. The parameters of two sources are the same as those in Fig. 5. It is seen that the SLSVCPD approach results in the lowest RMSE among the aforementioned schemes. The nested EMVS array offers more DOFs comparing with the EMVS ULA. Thus, we can observe better RMSE performance. However, the RMSEs of parameter estimation methods with nested EMVS array converges to nonzero value since there exists errors according to (17) when the source covariance matrix is not strictly diagonal [27]. In addition, we also give average CPU run time for each method for comparison. The ULACPD, tensorMUSIC, SSCPD, and SLSVCPD methods demand 0.23 s, 3.31 s, 0.47 s, and 0.51 s, respectively. It could be seen that the proposed SLSVCPD method has a modest computation complexity among four method. The tensorMUSIC method requires multidimensional searching over the whole parameter space, which needs the most computations.
In Fig. 9, we examine the elevation RMSEs of all methods while varying the number of samples from T=20 to T=200 with fixed SNR=10 dB. The parameters of sources remain the same as those in Fig. 5. It can be observed that the SLSVCPD algorithm is clearly advantageous when compared with the other methods.
In Fig. 10, we compare the elevation resolvabilities of all the above approaches, T=50 samples are used and 500 MonteCarlo trials have been carried out. Two spatially closely sources are assumed to impinge on the array with θ=[20^{∘},22^{∘}], ϕ=[30^{∘},30^{∘}], γ=[0^{∘},15^{∘}], and η=[15^{∘},30^{∘}]. The resolvable condition defined for the elevation estimation is given as [34]
where \(\hat {\theta }_{k}, k=1,2\), represents the estimated elevation of two sources. We can see that the SLSVCPD method provides the best resolvability among all the schemes. SLSVCPD, ULACPD, and SSCPD methods attain one in terms of probability of successful resolution when SNR ≥12 dB. However, the TensorMUSIC method fails to provide the reliable detection even though SNR becomes sufficiently large.
6.3 Underdetermined case
In this subsection, the underdetermined case where the number of sources is more than the number of elements in the original EMVS array. Recall that an Melements EMVS ULA could resolve up to M+2 sources. Here, we consider an M=6 elements nested EMVS array with K=10 impinging sources, which is called the underdetermined case. In this scenario, the ULACPD method [15, 18] cannot resolve these sources. The parameters of ten sources are randomly chosen. We examine the performance of the ALSCPD with our proposed model (38), SSCPD and SLSVCPD methods based on the nested EMVS array. The tensorMUSIC method is no longer included since the polarization state is very difficult to be a prior known when the number of sources is so large. Also, the computational burden of tensorMUSIC method becomes intolerable.
Figure 11 shows the elevation RMSEs of three methods versus SNR for T=2000 samples. It is obvious that the SLSVCPD method offers the lowest RMSE especially when SNR <8 dB. When SNR >16 dB, the ALSCPD, SSCPD, and SLSCPD schemes have almost the same RMSE performance. The RMSE curves saturate as SNR is sufficiently high. This phenomenon is caused by the approximation of (17) since the SCM cannot be strictly diagonal when the number of samples is finite as we discussed in the overdetermined case.
In Fig. 12, we study the elevation RMSEs versus the number of samples with SNR=10 dB. The proposed approach outperforms the other two algorithms when T<1600. Moreover, it is observed that the SLSVCPD and SSCPD methods result in the almost same performance when the number of samples is greater than 1600.
7 Methods
A tensor modeling method for elevation, azimuth, and polarization estimation with nested electromagnetic vectorsensor arrays is proposed. Since the signals are uncorrelated, we build the SCM into a CP model through tensor permutation. The spatial and polarization information are separated. Then, the SLSVCPD method is implemented on this model to calculate the elevation and azimuth. Next, the polarization is estimated based on the structure of the vectorsensor.
8 Conclusions
The issue of joint elevation, azimuth, and polarization estimation in underdetermined case with the nested EMVS array has been addressed in this work. This array could be modeled into a highdimensional tensor. The property of tensor permutation is introduced to separate the spatial and polarization vectors which are originally coupled in the steering matrix. This allows us to develop a tensor model of the nested EMVS array with extended spatial aperture. Furthermore, since the spatial and polarization are decoupled, which enables an efficient computational method for autopairing parameter estimation, avoiding exhaustive multiparameter searching. We also investigate the uniqueness condition offered by the proposed SLSVCPD approach. Besides, the underdetermined stochastic CRB with nested EMVS array is derived. Numerical examples confirm the superiority of our proposed method.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Abbreviations
 ALS:

Alternating least squares
 CRB:

Cramé
 rRao bound; CPD:

Canonical polyadic decomposition
 DOA:

Directionofarrival
 DOF:

Degreeoffreedom
 EMVS:

Electromagnetic vectorsensor
 ESPRIT:

Estimation of signal parameters via rotational invariance techniques
 HOSVD:

Higherorder singular value decomposition
 IID:

Independent and identically distributed
 MUSIC:

Multiple signal classification
 RMSE:

Root mean square error
 SCM:

Sample covariance matrix
 SLS:

Structured least square
 SNR:

Signaltonoise ratio
 ULA:

Uniform linear array
 VCPD:

Vandermonde constrained CPD
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Funding
This work was supported by the Key Program of National Natural Science Foundation of China (61831009, U1713217).
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MYC proposed the method and wrote the manuscript. XPM advised on the method and checked the simulations. LH revised the manuscript. The authors read and approved the final manuscript.
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Cao, MY., Mao, X. & Huang, L. Elevation, azimuth, and polarization estimation with nested electromagnetic vectorsensor arrays via tensor modeling. J Wireless Com Network 2020, 153 (2020). https://doi.org/10.1186/s13638020017648
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DOI: https://doi.org/10.1186/s13638020017648