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A joint DOA and polarization estimation method based on the conformal polarization sensitive array from the sparse reconstruction perspective
EURASIP Journal on Advances in Signal Processing volume 2022, Article number: 93 (2022)
Abstract
The conformal polarization sensitive array (CPSA) is formed by placing some vector sensors on the conformal array and it has a wide range of practical application in direction of arrival (DOA) and polarization parameters estimation. However, due to the diversity of each sensor’s direction, the performance of the conventional parameters estimation methods based on the CPSA would decrease greatly, especially under the low signal to noise ratio (SNR) and limited snapshots. In order to solve this problem, a unified framework and sparse reconstruction perspective for joint DOA and polarization estimation based on CPSA is proposed in this paper. Specifically, the array received signal model of the CPSA is formulated first and the twodimensional spatial sparsity of the incident signals is then exploited. Subsequently, after employing the singular value decomposition method to reduce the dimension of array output matrix, the variational sparse Bayesian learning and orthogonal matching pursuit methods are utilized to solve the source DOA estimation, respectively. Finally, the polarization parameters are obtained by the minimum eigenvector method. Simulation results demonstrate that the novel approaches can provide improved estimation accuracy and resolution with low SNR and limited snapshots.
1 Introduction
The joint directionofarrival (DOA) and polarization estimation of the polarization sensitive array (PSA) is one of the classical topics in array signal processing and has been widely used in radar, telecommunications, seismology [1]. In most scenarios, these estimation methods are proposed based on the linear array or planar array. Recently, the conformal array has aroused great concern. A conformal array [2, 3] is generally an array amounted with radiating sensors on the curvature surface. The conformal array has many benefits including reduction of aerodynamic drag, wide angle coverage, space saving, reduction of radar crosssection and so on. Complicated surface structure of CPSA will lead to the nonuniformity of element pattern and polarization characteristics, which brings lots of new challenges to parameters estimation. Therefore, the research on parameters estimation of CPSA is relatively significant.
At present, the DOA and polarization estimation methods based on the conformal array or the PSA are mainly according to their similar characteristics with the conventional array signal model and extended from the traditional parameter estimation techniques to conformal and PSA array. As a result, various approaches for cylindrical conformal arrays have been suggested to carry out twodimensional (2D) DOA estimation, such as multiple signal classification (MUSIC) [4] and signal parameters via rotational invariance technique (ESPRIT) methods [5]. However, due to the direction diversity of the antenna, these conventional methods demand large signal to noise ratio (SNR) and snapshots. In practical electromagnetic environment, the increasingly dense signals and jamming signals, and more mobility targets will cause the received signal to face the problems of low SNR and small snapshots, which would cause the algorithm performance to deteriorate or even fail. In order to solve this problem, by considering the multidimensional structure of the array received data, the tensor technique is utilized to the MUSIC algorithm for cylindrical conformal array to improve the estimation performance [6, 7] proposed a 2D DOA estimation method by using the nested array on the cylindrical conformal arrays, which shows better performance. Nonetheless, the method in [6] is computationally much expensive as it requires spectrum peak searching, while the method in [7] needs to place the antenna in an elaborate design.
Recently, the emerging sparse reconstruction methods for DOA estimation, including \({\ell _p}\)norm \(\left( {0 \le p \le 1} \right)\) methods [8,9,10,11,12], orthogonal matching pursuit (OMP) methods [13, 14] and the sparse Bayesian learning (SBL) methods [15,16,17,18], have aroused a lot of attention in DOA estimation. The essential idea of these algorithms is that the directions of incident source are substantially sparse in the spatial domain, which is intrinsically different from the subspacebased algorithms. Related methods have been shown to gain much enhanced performance over the subspacebased methods in the condition of low SNR and limited snapshots. In [19], a joint DOA, power and polarization estimation method using the cocentered orthogonal loop and dipole array is proposed by utilizing the signal reconstruction method. By exploiting the sparsity of the incident signals in the spatial domain, [20] proposed a novel method to estimated the DOA and polarization parameters by using SBL method. In [21], a novel offgrid hierarchical blocksparse Bayesian method for DOA and polarization parameters estimation was presented to improve the estimation accuracy [22]. Addressed a DOA and polarization estimation method based on the spatially separated polarization sensitive array under the SBL framework. However, these methods mentioned above focus on the estimation problem based on linear array or planar array, and do not consider the CPSA. As a result, the DOA and polarization estimation based on sparse reconstruction framework for CPSA needs further research.
This paper follows sparse reconstruction to address the joint DOA and polarization estimation based on the CPSA. By using the 2D joint sparsity signals of incident signals, a comprehensive array model is established first and a singular value decomposition (SVD) method is then used to reduce the dimension of array output matrix. And then, a variational sparse Bayesian learning (VSBL)based method named CPSAVSBL and OMPbased method named CPSAOMP are proposed to realize DOA estimation. Finally, the polarization parameters are obtained by the minimum eigenvector method. The scenario when cylinder CPSA is taken as example, so as to show how the CPSAVSBL and CPSAOMP realize DOA and polarization estimation jointly. Numerical examples also be provided to show how the performance of the proposed methods in DOA and polarization estimation and resolution under low SNR and limited snapshots.
Notations: Bolditalic letters are used to represent vectors and matrices. Lower case letters denotes scalars. \({\mathcal {C}}\) denotes complex numbers. \({\left( \cdot \right) ^{  1}}\), \({\left( \cdot \right) ^{\mathrm{T}}}\) and \({\left( \cdot \right) ^{\mathrm{H}}}\) denote the inverse operation, transpose operation and conjugate transpose operation, respectively. \(\left\langle \cdot \right\rangle\) denotes the statistical expectation. \(\bullet\), \(\otimes\) and \(\odot\) denote Hadamard Product, Kronecker product and KhatriRao Product, respectively. For a vector \({\varvec{x}}\), \(\mathrm{{diag}}\left( {{\varvec{x}}} \right)\) is diagonal matrix with \({\varvec{x}}\) being its diagonal. \({\left\ \cdot \right\ _2}\) represents 2norm.
2 2D sparse signal model
As shown in Fig. 1, suppose that there are K narrowband farfield true sources \(\left( {{\theta _k},{\varphi _k},{\gamma _k},{\eta _k}} \right) , k = 1,2, \cdots ,K\), impinging on an arbitrary CPSA with M electromagnetic vector sensor (EMVS), where \({\theta _k}\) is azimuth and \({\varphi _k}\) is elevation, \({\gamma _k} \in \left[ {0,{\pi / 2}} \right)\) and \({\eta _k} \in \left[ {  \pi ,\pi } \right)\) are the polarization auxiliary angle and polarization phase difference angle of the kth signal, respectively. \({{\varvec{{{r}}}}_i} = {\left[ {{x_i},{y_i},{z_i}} \right] ^{\mathrm{T}}}, \mathrm{{ }}i\mathrm{{ = 1,2,}} \cdots \mathrm{{,}}M\) is position vector of ith sensor. The propagation direction vector of the signal is \({\varvec{{{u}}}} = \left[ {\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta } \right] ^{\mathrm{T}}\). Each EMVS can output three electricfield vectors \({\varvec{E}} = \left[ {{{{\varvec{e}}}_x},{{{\varvec{e}}}_y},{{{\varvec{e}}}_z}} \right]\) and three magneticfield vectors \({\varvec{H}} = \left[ {{{{\varvec{h}}}_x},{{{\varvec{h}}}_y},{{{\varvec{h}}}_z}} \right]\) [23]. For the kth fully polarization wave, the polarization steering vector of each sensor can be written as
where \({\varvec{b}}\left( {{\theta _k},{\varphi _k},{\gamma _k},{\eta _k}} \right) \in {{\mathcal {C}}^{6 \times 1}}\), \({{\varvec{{v}}}}\left( {{\theta _k},{\varphi _k}} \right)\) and \({\varvec{\rho } }\left( {\gamma _k ,\eta _k } \right) \mathrm{{ }}\) represent the source’s spatial information and polarization information, respectively.
\(O\left( {X,Y,Z} \right)\) represents the global coordinate system in Fig. 1, and accordingly, \({{O'}_i}\left( {X',Y',Z'} \right)\) represents the local coordinate system of ith element [24]. Hence, \({\theta }\) is azimuth and \({\varphi }\) is elevation in the global coordinate system, and accordingly, \({\theta '}\) and \({\varphi '}\) are azimuth and elevation in the local coordinate system, respectively. Given the effect of the CPSA curvature, the element pattern and polarization characteristics of each sensor is not uniform any more. There exists a rotation matrix \({\varvec{R}}\) in [25], so that the pattern of ith sensor can be transformed from local coordinate system \({{g}}_i\left( {\theta ',\varphi '} \right)\) to global coordinate system \({{g}}_i\left( {\theta ,\varphi } \right)\). Therefore, the array steering vector of the kth signal can be written as
where \({\varvec{a}}\left( {{\theta _k},{\varphi _k}} \right) \in {{\mathcal {C}}^{M \times 1}}\) denotes the spatial steering vector, \({{\varvec{g}}\left( {{\theta _k},{\varphi _k}} \right) \in {{\mathcal {C}}^{M \times 1}}}\) denotes the element pattern in the global coordinate system. \({{\varvec{d}}}\left( {{\theta _k},{\varphi _k}} \right) = {{\varvec{g}}}\left( {{\theta _k},{\varphi _k}} \right) \bullet {{\varvec{a}}}\left( {{\theta _k},{\varphi _k}} \right)\) and its expression is formulated as follows
where \(\lambda\) is the carrier wavelength. As shown in Fig. 2 [26], \({f_i}\) is the direction diagram of the ith sensor, \({u_\theta }\) and \({u_\varphi }\) are unit vectors, \({k_\theta }\) and \({k_\varphi }\) are the polarization parameters of the true sources. \({g_{i{\theta }}}\) and \({g_{i{\varphi }}}\) are the response of ith sensor to the \({\theta '}\) and \({\varphi '}\) in the local coordinate system [27]. Thus, the received data model is expressed as
where \(\bar{{\varvec{x}}}\left( t \right) = \left[ {{\bar{x}_1}\left( t \right) , \cdots ,{{\bar{x}}_{6M}}\left( t \right) } \right] ^{\mathrm{{T}}} \in {{\mathcal {C}}^{6M \times 1}}\) and \(\bar{{\varvec{n}}}\left( t \right) = \left[ {{{\bar{n}}_1}\left( t \right) , \cdots ,{{\bar{n}}_{6M}}\left( t \right) } \right] ^{\mathrm{{T}}} \in {{\mathcal {C}}^{6M \times 1}}\) are the received data vector and complex Gaussian noise vector, respectively. Consider that all the K true sources are incident on the CPSA [28]. As a result, the array steering matrix is
where \({{\varvec{D}}}\left( {\theta ,\varphi } \right) = {{\varvec{G}}}\left( {\theta ,\varphi } \right) \bullet {{\varvec{A}}}\left( {\theta ,\varphi } \right)\) and \({{\varvec{B}}}\left( {\theta ,\varphi ,\gamma ,\eta } \right) = {{\varvec{V}}}\left( {\theta ,\varphi } \right) {{\varvec{Q}}}\left( {\gamma ,\eta } \right)\). \({\varvec{I}}\) is the \(K \times K\) identity matrix. \({\bar{{\varvec{A}}}}\) is a \(6M \times K\) matrix denoting the response of the array. It is seen that \({\varvec{G}}\) denotes \(M \times K\) element pattern matrix of K signals, \({\varvec{A}}\) denotes the \(M \times K\) fullrank steering matrix, \({\varvec{B}}\) denotes the \(6 \times K\) polarization steering matrix, \({{\varvec{V}}}\left( {\theta ,\varphi } \right)\) denotes \(6 \times 2K\) spatial information matrix of \({\varvec{B}}\), \({{\varvec{Q}}}\left( {\gamma ,\eta } \right)\) denotes \(2K \times K\) polarization information matrix of \({\varvec{B}}\).
Hence, the received array data model is given by
For simplicity, (15) can be rewritten as
where \(\bar{{\varvec{S}}} = {\left[ {{{\varvec{s}}_1},{{\varvec{s}}_2}, \cdots ,{{\varvec{s}}_K}} \right] ^{\mathrm{{T}}}}\), \(\bar{{\varvec{N}}} = {\left[ {{{\varvec{n}}_1},{{\varvec{n}}_2}, \cdots ,{{\varvec{n}}_{6M}}} \right] ^{\mathrm{{T}}}}\). \(\bar{{\varvec{S}}}\) and \(\bar{{\varvec{N}}}\) denote \(K \times T\) source and \(6M \times T\) additive white Gaussian noise, respectively.
In order to use sparse reconstruction method to DOA estimation, generally [29], we can uniformly sample the range of azimuth and elevation to formulate a DOA set \(\left\{ {\left. {\left( {\tilde{\varvec{\theta }} ,\tilde{\varvec{\varphi }} } \right) } \right\} } \right. = \left\{ {\left. {\left( {{{\tilde{\theta }}_1},{{\tilde{\varphi }}_1}} \right) , \cdots ,\left\{ {\left. {\left( {{{\tilde{\theta }}_{{J_{\tilde{\theta }}}}},{{\tilde{\varphi }}_{{J_{\tilde{\varphi }}}}}} \right) } \right\} } \right. } \right\} } \right.\) with \(J = {J_{\tilde{\theta }}}{J_{\tilde{\varphi }}} \gg K\). Assume that all the true sources lie on a fixed DOA set. For simplicity, the sparsity of the polarization information is not considered. The proposed methods are derived in the multiple measurement vectors. Therefore, the observation matrix \({{\varvec{X}}}\) with the snapshots is T can be presented as
It is noted that \(\tilde{{\varvec{A}}}\) is called the overcomplete dictionary and the number of \(\tilde{{\varvec{A}}}\) columns is much larger than that of rows. Similar to (8), it can be found that \(\tilde{{\varvec{A}}}\) not only depends spatial parameters but also depends polarization parameters. \({\varvec{{S}}}\) is called sparse direction weights, each sparse weight has nonzero value only at the true source directions. Compared with nonsparse method, sparse reconstruction method can obtain the the source targets DOA with smaller reconstruction error. It expolits the observation matrix to reconstruct the signals and then the reconstructed signals are used to realize the DOA and polarization parameters estimation.
3 Methods
The sparse reconstruction methods [30, 31] make use of the sparsity of signal in the spatial domain to reconstruct true signals. In this section, we propose a sparsityprompting CPSAVSBL method to realize DOA estimation based on the MMV array output formulation given in (17). Meanwhile, the CPSAOMP method is also proposed.
3.1 Dimensionality reduction of received data
From (17), it is easy to note that the large dimension of the received data will degrade the computational efficiency. So, SVD is conducted on \({{\varvec{X}}}\), we have
where \({\varvec{{U}}}\), \(\varvec{\varSigma }\) and \({\varvec{{V}}}\) is denoted as the left singular matrix, eigenvalue matrix and right singular matrix, respectively. Let \({\varvec{{V}}} = \left[ {{{\varvec{{V}}}_1},{{\varvec{{V}}}_2}} \right]\), \({\varvec{{V}}}\) is divided into \({{\varvec{{V}}}_1}\) and \({{\varvec{{V}}}_2}\) matrix according to the first K and the rest \(TK\) columns. In the presence of noise, we have \({\varvec{{X}}}{\varvec{{V}}}=\left[ {{{\varvec{{X}}}_{\text {sv}}},{\varvec{{X}}}{{\varvec{{V}}}_2}} \right]\) by the SVD, where \({{{\varvec{{X}}}_{\text {sv}}}}\) preserves most true sources information while the \({\varvec{{X}}}{{\varvec{{V}}}_2}\) is abandoned. Denote \({\varvec{{Y}}}={{{\varvec{{X}}}_{\text {sv}}}}\), \({{\varvec{{S}}}_{\text {sv}}} = {\varvec{{S}}}{{\varvec{{V}}}_1}\) and \({{\varvec{{N}}}_{\text {sv}}} = {\varvec{{N}}}{{\varvec{{V}}}_1}\), then we have
where \({{\varvec{S}}_{\text {sv}}}\) and \({{\varvec{N}}_{\text {sv}}}\) are the new matrices of sparse direction weights and measurement noise, respectively. \({\varvec{{Y}}} \in {{\mathcal {C}}^{6M{K^2} \times K}}\) represents the observation data matrix after reducing the dimension of \({{\varvec{X}}}\), and \(K \le T\). Hence, by using \({\varvec{{Y}}}\) in the following signal reconstruction process, the computational complexity of the algorithm is reduced greatly.
3.2 The proposed CPSAVSBL method
The posterior distributions of all unknown parameters can be calculated by using the Bayesian criterion [32], we have
Assume that all the variables are mutually independent, the joint probability density function (PDF) \(p\left( {{\varvec{{Y}}},{\varvec{{S}}}_{\text {sv}},{\varvec{\alpha }} ,\beta } \right)\) is expressed as
where \({\varvec{\alpha }}\) and \(\beta\) are hyperparameters, \(\tau \ge 0\) is shape parameter and \(\nu \ge 0\) denotes rate parameter. The directed graphical model [33] of the factorization of the joint PDF is depicted in Fig. 3. \({\varvec{{N}}}_{\text {sv}}\) obeys complex zeromean stationary Gaussian noise with known variance \({\sigma ^2}\). Thus, we have
As a result, a Gaussian likelihood function model of \({\varvec{{Y}}}\) can be obtained by
The nonzero rows of \({\varvec{{S}}}_{\text {sv}}\) contain all the angular information of the incident signals. The CPSAVSBL method is employed to update the hidden variables and hyperparameters to approximate the posterior probability [34]. We apply a threelayer hierarchical prior [35] to \({\varvec{{S}}}_{\text {sv}}\). A zeromean complex Gaussian (Gauss) distribution imposed on \({\varvec{{S}}}_{\text {sv}}\) as the first layer of prior
where \({\varvec{\varLambda } ^{\mathrm{{ }}  1}} = \mathrm{{diag}}\left( {{1 \big / {{\alpha _1}}},{1 \big / {{\alpha _2}}}, \cdots {{,1} \big / {{\alpha _j}, \cdots }},{1 \big /{{\alpha _J}}}} \right)\), \({\alpha _j} = {\sigma _j}^{2}\) and \({\alpha _j}\) is the jth noise precision. The exponential (Exp) distribution is applied to the hyperparameter \(\alpha\) as the secondlayer prior
The hyperparameter \(\beta\) obeys the chisquare (Chi2) distribution as the thirdlayer prior
However, \(p\left( {{\varvec{{S}}}_{\text {sv}},{\varvec{\alpha }} ,\beta \left {\varvec{{Y}}} \right. } \right)\) is not easy to solve. CPSAVSBL minimizes KullbackLeibler (KL) divergence [36] between approximate PDF \(q\left( \varTheta \right)\) and its posterior \(p\left( {{\varvec{{S}}}_{\text {sv}},{\varvec{\alpha }} ,\beta \left {\varvec{{Y}}} \right. } \right)\), where \(\varTheta = \left\{ {{\varvec{{S}}}_{\text {sv}},{\varvec{\alpha }} ,\beta } \right\}\). The independent variables in \(q\left( \varTheta \right)\) can be rewritten into the product form of the distribution function which is described as \(q\left( \varTheta \right) = q\left( {{\varvec{{S}}}_{\text {sv}},{\varvec{\alpha }} ,\beta } \right) = q\left( {\varvec{{S}}}_{\text {sv}} \right) q\left( {\varvec{\alpha }} \right) q\left( \beta \right)\). We can get the mean, variance of \({\varvec{{S}}}_{\text {sv}}\), and hyperparameters through the following steps.
The posterior distribution \(q\left( {\varvec{{S}}}_{\text {sv}} \right)\) is updated by \(q\left( {\varvec{{S}}}_{\text {sv}} \right) \propto {\left( {p\left( {{\varvec{{Y}}},{\varvec{{S}}}_{\text {sv}},{\varvec{\alpha }} ,\beta } \right) } \right) _{q\left( {\varvec{\alpha }} \right) }}\), where \({\left( \cdot \right) _{q\left( {\varvec{\alpha }} \right) }}\) denotes the subset of \(\varTheta\) that removes \({\varvec{\alpha }}\). \(p\left( {{\varvec{{Y}}},{\varvec{{S}}}_{\text {sv}},{\varvec{\alpha }} ,\beta } \right)\) can be obtained by the joint distribution of \(p\left( {{\varvec{{Y}}}\left {\varvec{{S}}}_{\text {sv}} \right. } \right)\) and \(p\left( {{\varvec{{S}}}_{\text {sv}}\left {\varvec{\alpha }} \right. } \right)\). We can acquire the approximate posterior for \(q\left( {\varvec{{S}}}_{\text {sv}} \right)\) as
The parameters of the approximate posterior in (27) can be renewed according to
The \(q\left( {\varvec{\alpha }} \right)\) is updated according to \(q\left( {\varvec{\alpha }} \right) \propto {\left( {p\left( {{\varvec{{S}}}_{\text {sv}}\left {\varvec{\alpha }} \right. } \right) p\left( {{\varvec{\alpha }} \left \tau \right. ,\beta } \right) } \right) _{q\left( {\varvec{{S}}}_{\text {sv}} \right) q\left( \beta \right) }}\) and can be approximated as the product of the PDF of the generalized inverse Gaussian distribution, and thus, we have
where \(\left\langle {h_j^2} \right\rangle = \mu _j^2 + {\varGamma _j}\), \({\kappa _p}\left( \cdot \right)\) is referred to the third kind Bessel function with order p. When n = 1, the estimated \(\varGamma\) and \(\left\langle \varLambda \right\rangle\) in (29) are obtained.
The posterior distribution \(q\left( \beta \right)\) is updated by \(q\left( \beta \right) \propto {\left( {p\left( {\alpha \left \tau \right. ,\beta } \right) p\left( {\beta \left \nu \right. } \right) } \right) _{q\left( \alpha \right) }}\) and complies with Gamma distribution as
Thus, the mean value of \(\beta\) can be expressed as
CPSAVSBL firstly carries out the initialization of constant variables in the selected distribution function, and then iteratively update the hidden variables and hyperparameters in turn until the set convergence conditions are satisfied [37]. The CPSAVSBL algorithm in this paper is summarized in Table 1.
3.3 The proposed CPSAOMP method
In the CPSAOMP method, \({{\varvec{S}}}\) can be reconstructed from the known matrix \({\varvec{Y}}\) and \(\tilde{{\varvec{A}}}\) based on the idea of dictionary atomic matching. In each iteration of CPSAOMP, a column of \(\tilde{{\varvec{A}}}\) is selected by orthogonal projection method, which is most relevant to the \({\varvec{{Y}}}\) . Then we subtract off its contribution to \({\varvec{{Y}}}\) and iterate over the remaining residual \({\varvec{{w}}}_{re{s^i}}^i\). The algorithm will identify the index set of the correct column when the residual is zero. The minimization of \({\varvec{{w}}}_{re{s^i}}^i\) is selected by as follows
where \({\varvec{{w}}}_{re{s^i}}^i\) is the residual matrix in the ith iteration, \(\tilde{{\varvec{{A}}}}_{re{s^i}}^i\) denotes the remaining part of \(\tilde{{\varvec{A}}}\) after the ith iteration. The execution steps of CPSAOMP algorithm are shown in Table 2.
Some remarks of specific implementation of CPSAVSBL and CPSAOMP are shown as follows.
Remark 1
The number of target sources K is used as known condition. Estimation K as a significant study topic is outside the scope of this paper. Furthermore, the scenarios of coherent sources and colored noise are not considered.
Remark 2
Compared with the conventional twolayer prior, the CPSAVSBL method benefits from the selected threelayer prior (GaussExpChi2) with the property of a sharp peak at the origin and heavy tails. It has the effect of prompting the sparsity of its solutions.
Remark 3
In this paper, CPSAVSBL and CPSAOMP methods only discussed the CPSAbased structure. It is noted that the proposed methods can be generalized to arbitrary array geometries.
Remark 4
According to empirical, the hyperparameters \(\alpha\) and \(\beta\) are initialized to the \(\frac{1}{T}\sum \nolimits _{t = 1}^K {\tilde{{\varvec{A}}}{\varvec{{Y}}}\left( t \right) }\) and 0.1, respectively. \(\tau = 1.5\) and \(\nu = 1\) are set in the third level prior distribution. \(\varepsilon = {10^{  3}}\) and \({i_{\max }} = 2000\) represent the termination threshold and maximum iterative number, respectively.
3.4 Refined DOA estimation
The mean and variance of \(\bar{{\varvec{S}}}\) can be output in \(\left\{ {\left[ {{\mu _1},{\varGamma _1}} \right] ,} \right. \left[ {{\mu _2},{\varGamma _2}} \right] , \cdots ,\left. {\left[ {{\mu _J},{\varGamma _J}} \right] } \right\}\) after the iteratively update termination criterion \(\left\ {{{\left\langle {{\alpha _j}} \right\rangle }^{\left( i \right) }}  {{\left\langle {{\alpha _j}} \right\rangle }^{\left( {i  1} \right) }}} \right\ /\left\ {{{\left\langle {{\alpha _j}} \right\rangle }^{\left( {i  1} \right) }}} \right\ < \varepsilon\) or \(i > {i_{\max }}\) is reached. The powers from different directions are obtained by substituting the estimated \(\left[ {{ \varvec{\mu }},{ \varvec{\varGamma } }} \right]\) into the power function, where the power of the jth impinging signal in the \(\varvec{\varOmega }\) is expressed as \({{\hat{P}}_j}\)
We use all the estimated signal powers to form a power spectrum. Similar to other spectralsearchingbased methods, the DOA is estimated by finding the positions of the highest peaks of the spectrum. Suppose that the grid indices of the highest K power values are the position of the true sources. The estimated K DOA will be \(\left\{ {\left[ {{{\hat{\theta }}_1},{{\hat{\varphi }}_1}} \right] ,\left[ {{{\hat{\theta }}_2},{{\hat{\varphi }}_2}} \right] , \cdots ,\left[ {{{\hat{\theta }}_K},{{\hat{\varphi }}_K}} \right] } \right\}\)
3.5 The polarization parameters estimation
Consider that the true sources \(\bar{{\varvec{{S}}}} = \left[ {{{\varvec{s}}_1},{{\varvec{s}}_2}, \cdots ,{{\varvec{s}}_K}} \right] ^{\mathrm{{T}}}\) impinge on the surface of the CPSA as shown in Fig. 1. Then the steering vector of the kth source is
where \({\varvec{{g}}}\left( {{\theta _k},{\varphi _k}} \right) =\left[ {{g_1}\left( {{\theta _k},{\varphi _k}} \right) ,{g_2}\left( {{\theta _k},{\varphi _k}} \right) , \cdots ,{g_M}\left( {{\theta _k},{\varphi _k}} \right) } \right]\), \({g_i}\left( {{\theta _k},{\varphi _k}} \right)\) is the response of unit signal by the ith array element to the kth signal, and \({\mathop {{{\varvec{a}}}}\limits ^{\frown }} \left( {{\theta _k},{\varphi _k}} \right) = {\left[ {{{\hbox {e}}^{  j\frac{{2\pi }}{\lambda }{{\varvec{u}}_k} \bullet {{{\varvec{r}}}_1}}},{{\hbox {e}}^{  j\frac{{2\pi }}{\lambda }{{\varvec{u}}_k} \bullet {{{\varvec{r}}}_2}}}, \ldots ,{{\hbox {e}}^{  j\frac{{2\pi }}{\lambda }{{\varvec{u}}_k} \bullet {{{\varvec{r}}}_M}}}} \right] ^{\mathrm{T}}}\) is phase delay vector.
The array manifold matrix of the CPSA is shown as
The received data model of polarization parameters estimation could be written as
In order to catch the noise subspace \({{\varvec{{U}}}_\text {N}}\) of \(\hat{{\varvec{{X}}}}\), by taking the eigenvalue decomposition of the covariance matrix \({\varvec{{R}}} = \frac{1}{L}{\hat{{\varvec{{X}}}}}{{\hat{{\varvec{{X}}}}}^\mathrm{{H}}}\) and let the K eigenvectors corresponding to the K significant eigenvalues to form the signal subspace \({{\varvec{{U}}}_\text {S}}\). Hence, \({{\varvec{{U}}}_\text {N}}\) can be expressed as
where \({{\varvec{{I}}}_{MT}}\) symbolizes \(MT \times MT\) identity matrix. The K estimated DOA in section 3.4 are substituted into the constructed conventional spatial spectrum function to obtain the K functions as follows
The polarization parameters can be calculated by the eigenvector corresponding to the minimum eigenvalue of \({\varvec{{W}}}\) [38].
where \({\varvec{\rho } _k}\left( i \right)\) denotes the ith element in the eigenvector.
4 The computational cost analysis
It is obviously that the proposed methods based on the sparse model will be more time consuming than the subspacetype methods. The reason is the dimension of the output matrix \({{\varvec{X}}}\) in sparse signal model is too large to iteratively update the mean, variance and hyperparameters in (28), (29), (30) and (32) quickly. As a result, SVD is utilized to reduce the dimension of \({{\varvec{X}}}\) to improve computational efficiency before the sparse component learning and inner product.
As far as the computational complexity, CPSAVSBL in this paper mainly derives from solving for the hidden variables and hyperparameters. Among the mean and variance of \({\varvec{{S}}}_{\text {sv}}\) is approximately \(2{J^3}  2{J^2} + 6{J^2}M + 4J{M^2} + {M^3}  3JM\) and \({J^3} + 2{J^2}M + 2JMK  JM\), respectively. Then the hyperparameters \(\left\langle {{\alpha _j}} \right\rangle\) and \(\left\langle {{\alpha _j}^{  1}} \right\rangle \mathrm{{ }} \left( j = 1,2, \cdots ,J \right)\) are about 4J and 10J. The most time consuming of the CPSAOMP is the selection of atoms in the dictionary \(\tilde{{\varvec{A}}}\), and the computational complexity of each inner product is MJ. An appended computational load for the SVD of \({{\varvec{X}}}\) is \(6M{K}{T^2}\). The computational complexity of covariance matrix and eigenvalue decomposition in TensorMUSIC and MUSIC is \(\mathrm{{36}}{M^2}T\) and \(21\mathrm{{6}}{M^3}\), and the spectral searching of subspace methods is \({n^2}\left[ {6M\left( {6M  K} \right) } \right]\). In addition to, TensorMUSIC also needs higher order singular value decomposition. The computational complexity of the above methods is given in Table 3 when an arbitrary conformal array with M=25, K=2, J=16471, T=50 and n=5000, where n is the number of angle searches. The computational complexity is represented as \(O_t\).
5 Results and discussion
In this section, the simulation results of CPSAVSBL and CPSAOMP based on CPSA are analyzed, and compared with MUSIC and TensorMUSIC. As shown in Fig. 4, the array structure in the simulation is a cylindrical carrier and there are \({M_c} \times {M_z}\) sensors are uniformly distributed over the surface of the cylinder. The distance between the adjacent sensors along the Zaxis is \(\frac{\lambda }{2}\). For simplicity, we assume \({M_z} = 5\) and \({M_c} = 5\). The range of azimuth and elevation is sampled with \(1^{\circ }\) interval to form the direction set \(\left\{ {\left. {\left( {\tilde{\varvec{\theta }} ,\tilde{\varvec{\varphi }} } \right) } \right\} } \right.\). The RMSE of DOA estimation by independent Monte Carlo simulations is defined as
where Q denotes the number of Monte Carlo simulations, \({u_k}\) is one of the parameters \(\left( {{\theta _k},{\varphi _k},{\gamma _k},{\eta _k}} \right)\), and \({{\hat{u}}_{kq}}\) is the estimation of \({u_k}\) in the qth simulation.
5.1 The estimation accuracy versus SNR and the number of snapshots
In this simulations, the DOA estimation accuracy is evaluated by RMSE. The RMSE of the DOA estimation methods versus the SNR and number of snapshots are shown in Figs. 5 and 6, respectively. Consider two uncorrelated incident signals with DOA and polarization parameters \(\left( {\theta ,\varphi ,\gamma ,\eta } \right)\) are \(\left( {  {{14}^ \circ },{{35}^ \circ },{{25}^ \circ },{{10}^ \circ }} \right)\) and \(\left( { {{10}^ \circ },\mathrm{{5}}{\mathrm{{0}}^ \circ },{{50}^ \circ },{{45}^ \circ }} \right)\), respectively. The number of snapshots is fixed at 50. The RMSE versus SNR from \(1\) to 25 dB for the four methods is presented in Fig. 5, and the RMSE versus snapshots number from to 10 to 150 while SNR=5 dB is presented in Fig. 6. In each experiment, we run 300 Monte Carlo simulations. From Figs. 5 and 6, it can be seen that, the RMSE performance of four methods are improved with the SNR and snapshots increasing. In Fig. 5, the proposed CPSAVSBL has a more accurate DOA estimation than other methods at every SNR. The performance of the proposed CPSAOMP is apparently superior to MUSIC and TensorMUSIC while the SNR is relatively high. In Fig. 6, the proposed CPSAVSBL offers the best performance than that of CPSAOMP, MUSIC and TensorMUSIC. CPSAOMP provides an estimation with zero error when the number of snapshots is relatively large. We further examine the RMSE of polarization parameters estimation. For all the methods, it can be seen that the RMSE continuously decrease when SNR and the number of snapshots increase. It is observed that the RMSE \(\gamma\) and \(\eta\) of the proposed methods consistently outperform the TensorMUSIC and MUSIC methods in Fig. 7. From Fig. 8, the estimation performance of the proposed CPSAVSBL is better than subspace methods at arbitrary number of snapshots. The CPSAOMP performs better than the MUSIC and TensorMUSIC methods in the large number of snapshots.
5.2 The resolution performance versus SNR and number of snapshots
In this subsection, the resolution performance is evaluated by the probability of successful detection. For the MUSIC and TensorMUSIC methods, the “successful trial” is defined in this simulation as when both the \(\left {\hat{\theta } \theta } \right < {2^ \circ }\) and \(\left {\hat{\varphi } \varphi } \right < {2^ \circ }\). Set the two uncorrelated incident signals with DOA and polarization parameters \(\left( {\theta ,\varphi ,\gamma ,\eta } \right)\) are \(\left( {  {8^ \circ },{{45}^ \circ },{{10}^ \circ },{{30}^ \circ }} \right)\) and \(\left( {{2^ \circ },{{45}^ \circ },{{30}^ \circ },{{20}^ \circ }} \right)\), respectively. Set the number of snapshots is 50, the probability of successful detection of four methods varying SNR from \(5\) to 20 dB in Fig. 9. Figure 10 shows the probability of successful detection for SNR=5 dB by varying the snapshots from 20 to 200 under the two uncorrelated incident signals with DOA and polarization parameters \(\left( {\theta ,\varphi ,\gamma ,\eta } \right)\) are \(\left( {  {4^ \circ },{{45}^ \circ },{{10}^ \circ },{{30}^ \circ }} \right)\) and \(\left( {{6^ \circ },{{45}^ \circ },{{30}^ \circ },{{20}^ \circ }} \right)\). It is seen from Figs. 9 and 10, both the probability of successful detection improve with the increasing of the SNR and the number of snapshots, whereas the proposed methods have higher resolution performance than the conventional MUSIC and TensorMUSIC methods, which gives us a strong evidence of the effectiveness of the proposed methods.
6 Conclusions
In this paper, we have illustrated sparse reconstruction methodbased DOA and polarization joint estimation for CPSA. By exploiting 2D spatial sparsity of incident signals, an array output model that is applicable to arbitrary CPSA is obtained. In order to improve the computational efficiency of the proposed methods, SVD is used to reduce the dimension of the array output matrix. The CPSAVSBL and CPSAOMP methods are proposed for DOA estimation. The minimum eigenvector method is used to obtain the polarization information of signals. As a result, both the estimation accuracy and the resolution of the proposed methods have been validated by a lot of simulations. The simulation results show that the proposed methods have low RMSE and optimal resolution than the stateoftheart methods especially with low SNR and limited snapshots. In future work, we will focus on the multidimensional structural information in the array received data.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 CPSA:

Conformal polarization sensitive array
 DOA:

Direction of arrival
 2D:

Twodimensional
 SNR:

Signal to noise ratio
 VSBL:

Variarional sparse Bayesian learning
 OMP:

Orthogonal matching pursuit
 PSA:

Polarization sensitive array
 SBL:

Sparse Bayesian learning
 EMVS:

Electromagnetic vector sensor
 CPSAVSBL:

CPSA variational sparse Bayesian learning
 CPSAOMP:

CPSA orthogonal matching pursuit
 MUSIC:

Multiple signal classification
 ESPRIT:

Estimation of signal parameters via rotational invariance techniques
 RMSE:

Root meansquare error
 PDF:

Probability density function
 KL:

KullbackLeibler
 SVD:

Singular value decomposition
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Acknowledgements
This work was supported in part by the National Science Foundation for Young Scientists of China under Grant 61801308 and the Liaoning Revitalization Talents Program under Grant XLYC1907195 and the Aeronautical Science Foundation under Grant 2020Z017054001 and the foundation of Shandong Province under Grant ZR2019BF046.
Funding
This work was supported in part by the National Science Foundation for Young Scientists of China under Grant 61801308 and the Liaoning Revitalization Talents Program under Grant XLYC1907195 and the Aeronautical Science Foundation under Grant 2020Z017054001 and the foundation of Shandong Province under Grant ZR2019BF046.
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XL proposed the original idea of the full text. LJ designed and implemented the simulation experiments. PX drafted the manuscript and was a major contributor in writing the manuscript. All authors read and approved the final manuscript.
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Lan, X., Jiang, L. & Xu, P. A joint DOA and polarization estimation method based on the conformal polarization sensitive array from the sparse reconstruction perspective. EURASIP J. Adv. Signal Process. 2022, 93 (2022). https://doi.org/10.1186/s13634022009277
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DOI: https://doi.org/10.1186/s13634022009277