2D DOA and Polarization Estimation Using Parallel Synthetic Coprime Array of Non-Collocated EMVSs

Abstract

For target detection and recognition in a complicated electromagnetic environment, the two-dimensional direction-of-arrival and polarization estimation using a polarization-sensitive array has been receiving increased attention. To efficiently improve the performance of such multi-parameter estimation in practice, this paper proposes a parallel synthetic coprime array with reduced mutual coupling and hardware cost saving and then presents a dimension-reduction compressive sensing-based estimation method. For the proposed array, the polarization types, numbers, and positions of antennas in each subarray are jointly considered to effectively mitigate mutual coupling in the physical array domain and to both enhance degrees of freedom and extend the aperture in the difference coarray domain with the limited physical antennas. By exploring the array configuration, the parameter estimation can be formulated as a block-sparse signal reconstruction problem, and then the one-dimensional sparse reconstruction algorithm is only used once to achieve multi-parameter estimation with automatic pair-matching. The theoretical analysis and simulation results are provided to demonstrate the superior performance of the proposed array and method over the existing techniques.

1. Introduction

The polarization sensitive array (PSA), as a typical kind of vector-sensor array, has received much attention in remote sensing, communication, radar, and sonar [1,2,3], since it can provide spatial and polarized information about electromagnetic (EM) signals for target detection and recognition. The estimation of two-dimensional (2D) direction-of-arrival (DOA) and polarization parameters (polarization angle and polarization phase difference) is an important issue for PSAs [4]. Many methods have been explored for such multi-parameter estimation with high-accuracy in theory, e.g., subspace-based algorithms [5,6,7] and compressive sensing algorithms [8,9,10]. Nevertheless, PSAs usually require a lot of antennas to obtain multi-parameter information, which can cause the hardware cost to skyrocket. As the result, the performance of those methods will be degraded in practice with limited antennas. Additionally, PSAs often suffer from two kinds of mutual coupling, i.e., inter-element coupling (IEC) and inter-polarization coupling (IPC), which further degrade the estimation performance [11].

Recently, two kinds of nonuniform scalar–vector arrays (i.e., nested arrays [12,13,14] and coprime arrays [15,16,17]) have been explored with closed-form expressions for their antenna positions and the number of degrees of freedom (DOFs). By exploiting the difference coarray, a nonuniform array with N physical antennas can provide $O\left(N^2\right)$ DOFs, to improve the performance of DOA estimation. It is noted that, compared with coprime arrays, nested arrays usually suffer from comparatively strong mutual coupling due to their dense-level subarray, resulting in performance loss.

Based on this, coprime PSAs are investigated to alleviate such problems for performance degradation of DOA and polarization estimation. Nevertheless, similar to coprime arrays, some antenna pairs with small separations used to obtain a larger aperture with increased DOFs in the difference coarray domain still exist and introduce IEC for coprime PSAs [18]. Moreover, the existing coprime PSAs usually consist of spatially collocated electromagnetic vector sensors (EMVSs), e.g., three-parallel coprime PSA (TPC-PSA) [19] and parallel coprime PSA (PCP-PSA) [20]. However, such spatially collocated EMVSs in arrays have strong IPC and can further enhance the effect of IEC [11,21]. Furthermore, compared to coprime arrays, the existing coprime PSAs still require lots of antennas due to the polarization domain, such as coprime EMVS arrays [22,23,24]. One can see that, owing to six-component EMVSs, the number of antennas in such coprime EMVS arrays is six times that of their parent coprime array with a similar aperture.

In order to break such limitations for 2D DOA and polarization estimations, this paper proposes a parallel synthetic coprime PSA (PSC-PSA) composed of spatially separated two-component EMVSs with three subarrays and then provides a compressive sensing-based method with only one-dimensional (1D) sparse reconstruction. To our knowledge, the existing coprime PSA designs first set the coprimality between spacings of two/three multi-polarized subarrays to enhance DOFs. By contrast, the proposed array design first optimizes the allocation of the antenna numbers of the subarrays for aperture extension due to the configuration of three single-polarization subarrays, and then rearranges antenna spacings to satisfy coprimality for DOF maximization. Based on this, for various two-component EMVSs (i.e., dipole–dipole pair, dipole–loop pair, and loop–loop pair), a systematic procedure is provided to determine the polarization types, numbers, and positions of antennas for each single-polarization subarray, in order to obtain reduced mutual coupling in the physical array domain and enhanced DOFs and apertures in the difference coarray domain. The multi-parameter estimation is formulated as a block-sparse signal reconstruction problem with parameter separation. The Bayesian compressive sensing (BCS)-based method is given to solve the problem. The contributions of this work include the following three aspects:

(a)

Due to a shared subarray and the non-collocated two-component EMVSs, the PSC-PSA can save at least half the number of antennas compared with the existing arrays in the case of a similar aperture and DOF in the coarray domain. The Cramér–Rao bound (CRB) derivation then shows that the former can provide better estimation performance with the same number of antennas.

(b)

Mutual coupling in the physical array domain can be efficiently reduced by spatial separation of antennas and extended displacements between subarrays. The separated antennas along the subarray direction are different array elements for IPC reduction. The displacements are extended in 2D (e.g., x- and y-axes) for mitigating IEC and further suppressing IPC.

(c)

Due to the array configuration, the polarization phase difference only exists in phase for one item in the 2D polarization domain, and then angle information can be inserted in phase for the other one. In this case, a block-sparse model is derived to achieve multi-parameter separation. The 1D sparse reconstruction-based method is then only used once to obtain multi-parameter estimation with automatic pair-matching, which avoids huge complexity.

The remainder of this paper is as follows. Section 2 reviews the polarized signal model, the concept of the difference coarray, and the mutual coupling model. The proposed array with the analysis of mutual coupling is presented in detail in Section 3. Then, the proposed method for 2D DOA and the polarization estimation is provided in Section 4. Section 5 gives the derivation of CRB. In Section 6, the simulation results of the proposed array and methods are shown in comparison with the existing techniques. Conclusions are drawn in Section 7.

In this paper, vectors are denoted by boldface lowercase letters, e.g., $\mathbf{a}$; matrices are denoted by boldface capital letters, e.g., $\mathbf{A}$; operators ${(·)}^{*}$, ${(·)}^T$, and ${(·)}^H$, respectively, denote complex conjugate, transpose, and conjugate transpose; $\lceil a\rceil$ rounds a number to the nearest integer, with $\lceil a\rceil \ge a$; $\mathrm{E}(·)$, $\mathrm{vec}(·)$, and ${(·)}^{-1}$ are the mathematical expectation, vectorization operator, and matrix inverse, respectively; $\mathrm{angle}(·)$ stands for the phase-finding operator; $\mathrm{diag}\left(\mathbf{a}\right)$ is diagonal matrix whose elements are given by $\mathbf{a}$; $|·|$ is the absolute value operation; ⊙ is the Khatri–Rao product; ⊗ is the Kronecker product; ${\left[\mathbf{a}\right]}_g$ and ${\left[\mathbf{a}\right]}_{\mathrm{end}}$ denote the gth and last elements of $\mathbf{a}$; ${\left[\mathbf{A}\right]}_{i,j}$ is the $(i,j)$th element of $\mathbf{A}$; ${\mathbf{I}}_N$ is an $N\times N$ identity matrix; and $\max (a,b)$ stands for choosing the maximum value between a and b.

2. Preliminaries

2.1. Polarized Signal Model

Assume that K narrowband far-field EM waves impinge on a parallel coprime PSA composed of spatially collocated six-component EMVSs, i.e., three orthogonally oriented dipoles and three orthogonally oriented magnetic loops are collocated in a point-like geometry. The array consists of two uniform linear arrays (ULAs) in the $x-y$ plane, where subarray 1 contains M six-component EMVSs with an inter-antenna spacing of $Nd$, and subarray 2 has N six-component EMVSs with the spacing of $Md$. Here, M and N are coprime, and d denotes the half-wavelength. The three-dimensional view and top view of the configuration of the typical existing PSA are shown in Figure 1a and Figure 1b, respectively. The antenna positions in subarray 1 are ${\mathbb{D}}_1=\{(d_{i,x},d_{i,y})=(0,d_{1}d)|d_1\in {\mathbb{P}}_1,1\le i\le 6M\}$, with $d_{i,x}$ and $d_{i,y}$ respectively denoting the position of the ith antenna along the x- and y-axes, whereas the positions in subarray 2 are ${\mathbb{D}}_2=\{(d_{i,x},d_{i,y})=(d,d_{2}d)|d_2\in {\mathbb{P}}_2,1\le i\le 6N\}$, where ${\mathbb{P}}_1=[0,N,\dots ,N(M-1)]$ and ${\mathbb{P}}_2=[0,M,\dots ,M(N-1)]$ stand for the antenna position sets along the y-axis. Hence, the antenna positions of the parallel coprime PSA are $\mathbb{D}\left(j\right)=\left\{(d_{j,x},d_{j,y})\right|(d_{j,x},d_{j,y})\in ({\mathbb{D}}_1\cup {\mathbb{D}}_2),1\le j\le 6(M+N)\}$, and the signal model of this array is expressed as follows:

$$\begin{array}{c}\hfill \mathbf{x}\left(t\right)={\mathbf{A}}_{sp}\mathbf{s}\left(t\right)+\mathbf{n}\left(t\right)=\left[\begin{array}{c}{\mathbf{A}}_{sp,1}\\ {\mathbf{A}}_{sp,2}\end{array}\right]\mathbf{s}\left(t\right)+\left[\begin{array}{c}{\mathbf{n}}_1\left(t\right)\\ {\mathbf{n}}_2\left(t\right)\end{array}\right],\end{array}$$

(1)

where $\mathbf{s}\left(t\right)$ stands for the signals, and ${\mathbf{n}}_1\left(t\right)$ and ${\mathbf{n}}_2\left(t\right)$ denote the white Gaussian noise. The EM signals $\mathbf{s}\left(t\right)$ are assumed to be temporally white and uncorrelated with each other. Also, consider that the white Gaussian noise is uncorrelated with the signals. Then, ${\mathbf{A}}_{sp,1}={\mathbf{A}}_p\odot {\mathbf{A}}_{s,1}$ and ${\mathbf{A}}_{sp,2}={\mathbf{A}}_p\odot {\mathbf{A}}_{s,2}$ respectively denote the array manifolds based on subarrays 1 and 2. Here, ${\mathbf{A}}_p$ and ${\mathbf{A}}_{s,1}$ (or ${\mathbf{A}}_{s,2}$) respectively denote the polarized and spatial manifolds as follows:

$$\begin{array}{cc}\hfill {\mathbf{A}}_p& =[{\mathbf{a}}_p\left({\theta }_1,{\varphi }_1,{\gamma }_1,{\eta }_1\right),\cdots ,{\mathbf{a}}_p\left({\theta }_K,{\varphi }_K,{\gamma }_K,{\eta }_K\right)],\hfill \\ \hfill {\mathbf{A}}_{s,1}& =[{\mathbf{a}}_{s,1}\left({\theta }_1,{\varphi }_1\right),\cdots ,{\mathbf{a}}_{s,1}\left({\theta }_K,{\varphi }_K\right)],\hfill \\ \hfill {\mathbf{A}}_{s,2}& =[{\mathbf{a}}_{s,2}\left({\theta }_1,{\varphi }_1\right),\cdots ,{\mathbf{a}}_{s,2}\left({\theta }_K,{\varphi }_K\right)]\Phi (\theta ,\varphi ),\hfill \end{array}$$

(2)

with

$${\mathbf{a}}_{s,1}\left({\theta }_k,{\varphi }_k\right)={[1,e^{-j\pi {\left[{\mathbb{P}}_1\right]}_{2}sin{\theta }_{k}sin{\varphi }_k},\dots ,e^{-j\pi {\left[{\mathbb{P}}_1\right]}_{M}sin{\theta }_{k}sin{\varphi }_k}]}^T,$$

(3)

$${\mathbf{a}}_{s,2}\left({\theta }_k,{\varphi }_k\right)={[1,e^{-j\pi {\left[{\mathbb{P}}_2\right]}_{2}sin{\theta }_{k}sin{\varphi }_k},\dots ,e^{-j\pi {\left[{\mathbb{P}}_2\right]}_{N}sin{\theta }_{k}sin{\varphi }_k}]}^T,$$

(4)

$$\Phi (\theta ,\varphi )=\mathrm{diag}(e^{-j\pi sin{\theta }_{1}cos{\varphi }_1},\dots ,e^{-j\pi sin{\theta }_{K}cos{\varphi }_K}),$$

(5)

$$\begin{array}{cc}\hfill {\mathbf{a}}_p\left({\theta }_k,{\varphi }_k,{\gamma }_k,{\eta }_k\right)& =\left[\begin{array}{c}{e}_{x,k}\\ e_{y,k}\\ e_{z,k}\\ h_{x,k}\\ h_{y,k}\\ h_{z,k}\end{array}\right]=\left[\begin{array}{cc}-sin{\varphi }_k& cos{\theta }_{k}cos{\varphi }_k\\ cos{\varphi }_k& cos{\theta }_{k}sin{\varphi }_k\\ 0& -sin{\theta }_k\\ cos{\theta }_{k}cos{\varphi }_k& sin{\varphi }_k\\ cos{\theta }_{k}sin{\varphi }_k& -cos{\varphi }_k\\ -sin{\theta }_k& 0\end{array}\right]\left[\begin{array}{c}cos{\gamma }_k\\ sin{\gamma }_k{e}^{j{\eta }_k}\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{c}-sin{\varphi }_{k}cos{\gamma }_k+cos{\theta }_{k}cos{\varphi }_{k}sin{\gamma }_k{e}^{j{\eta }_k}\\ cos{\varphi }_{k}cos{\gamma }_k+cos{\theta }_{k}sin{\varphi }_{k}sin{\gamma }_k{e}^{j{\eta }_k}\\ -sin{\theta }_{k}sin{\gamma }_k{e}^{j{\eta }_k}\\ cos{\theta }_{k}cos{\varphi }_{k}cos{\gamma }_k+sin{\varphi }_{k}sin{\gamma }_k{e}^{j{\eta }_k}\\ cos{\theta }_{k}sin{\varphi }_{k}cos{\gamma }_k-cos{\varphi }_{k}sin{\gamma }_k{e}^{j{\eta }_k}\\ -sin{\theta }_{k}cos{\gamma }_k\end{array}\right],\hfill \end{array}$$

(6)

where ${\theta }_k\in (0,\pi /2)$ and ${\varphi }_k\in (-\pi /2,\pi /2)$ are the pitch and azimuth angles of the kth incident signal, and ${\gamma }_k\in (0,\pi /2)$ and ${\eta }_k\in (-\pi ,\pi )$ respectively denote the polarization angle and the polarization phase difference. According to the signal model in Equation (1), an example for the existing array with 42 antennas ($M=3$ and $N=4$) is given to demonstrate the positions of the antennas, as shown in

Table 1. The antenna positions of the existing coprime array with 42 antennas ($M=3$ and $N=4$).

	Subarray 1			Subarray 2
	j = 1,4,7 10,13,16	j = 2,5,8 11,14,17	j = 3,6,9 12,15,18	j = 19,23,27 31,35,39	j = 20,24,28 32,36,40	j = 21,25,29 33,37,41	j = 22,26,30 34,38,42
$\mathbb{D}\left(j\right)$	(0,0)	(0,4d)	(0,8d)	(d,0)	(d,$3d$)	(d,$6d$)	(d,$9d$)

. It is noted that the configuration of the prototype coprime array in Figure 1 can be easily replaced by its generalized versions, such as a coprime array with compressed inter-element spacing and a coprime array with displaced subarrays [25].

2.2. Difference Coarray

For a nonuniform linear array composed of several subarrays (e.g., ${\mathbb{P}}_1$ and ${\mathbb{P}}_2$), the antenna positions of its difference coarray can be defined as follows [26,27]:

$$\begin{array}{c}\hfill \mathbb{L}=\{l_i-l_j|l_i,l_j\in \mathbb{P}={\mathbb{P}}_1\cup {\mathbb{P}}_2\},\end{array}$$

(7)

where $l_i$ denotes the position of the ith antenna of the array. Specifically, we obtain:

$$\begin{array}{c}\hfill \mathbb{L}=\{\underset{{\mathbb{L}}_{12}}{\underbrace{\pm ({\left[{\mathbb{P}}_1\right]}_i-{\left[{\mathbb{P}}_2\right]}_j)}}\}\cup \{\underset{{\mathbb{L}}_{11}}{\underbrace{{\left[{\mathbb{P}}_1\right]}_i-{\left[{\mathbb{P}}_1\right]}_h}}\}\cup \{\underset{{\mathbb{L}}_{22}}{\underbrace{{\left[{\mathbb{P}}_2\right]}_j-{\left[{\mathbb{P}}_2\right]}_k}}\},\end{array}$$

(8)

where ${\left[{\mathbb{P}}_1\right]}_i$ or ${\left[{\mathbb{P}}_2\right]}_i$ denotes the position of the ith antenna in subarray 1 or 2, with $1\le i,h\le M,0\le j,k\le N$.

It is obvious that the difference coarray consists of a cross-difference coarray (whose antenna positions are given by ${\mathbb{L}}_{12}$) and self-difference coarrays (whose antenna positions are given by ${\mathbb{L}}_{11}\cup {\mathbb{L}}_{22}$). For example, in the case of Table 1, the antenna positions of the self-difference coarrays of subarrays 1 and 2 are ${\mathbb{L}}_{11}=[-8-4048]d$ and ${\mathbb{L}}_{22}=[-9-6-30369]d$, respectively. The antenna positions of the cross-difference coarray between subarrays 1 and 2 are ${\mathbb{L}}_{12}=[-9-8-6-5-4-3-2-1012345689]d$.

2.3. Mutual Coupling Model

In practical applications, the effect of mutual coupling is unavoidable, especially when the spacing between adjacent antennas is minimal. By introducing mutual coupling, Equation (1) can be reformulated as follows:

$$\mathbf{x}\left(t\right)={\mathbf{C}}_{sp}{\mathbf{A}}_{sp}\mathbf{s}\left(t\right)+\mathbf{n}\left(t\right).$$

(9)

Due to the spatially collocated EMVSs, the mutual coupling matrix for the existing array is given as follows:

$${\mathbf{C}}_{sp}={\mathbf{C}}_p\otimes {\mathbf{C}}_s,$$

(10)

where

$${\left[{\mathbf{C}}_p\right]}_{i,j}=\left\{\begin{array}{cc}1,\hfill & i=j,\hfill \\ c_p,\hfill & i\ne j.\hfill \end{array}\right.$$

(11)

Here, $c_p$ is the polarized mutual coupling coefficient, which satisfies $\left|c_p\right|\le 1$. On the other hand, inspired by the B-band mode for the mutual coupling model in ULA, the spatial mutual coupling matrix ${\mathbf{C}}_s$ is written as follows:

$${\left[{\mathbf{C}}_{\mathbf{s}}\right]}_{i,j}=\left\{\begin{array}{cc}1,\hfill & i=j,\hfill \\ c_{s,{\overline{d}}_{i,j}},\hfill & i\ne j\mathrm{and}{\overline{d}}_{i,j}\le B,\hfill \\ 0,\hfill & i\ne j\mathrm{and}{\overline{d}}_{i,j}>B,\hfill \end{array}\right.$$

(12)

where ${\overline{d}}_{i,j}=\sqrt{{(d_{i,x}-d_{j,x})}^2+{(d_{i,y}-d_{j,y})}^2}$, and the spatial mutual coupling coefficients $c_{s,1},c_{s,2},\cdots ,c_{s,B}$ satisfy $1>\left|c_{s,1}\right|>\cdots >\left|c_{s,B}\right|>0$. Additionally, $c_{s,|{\overline{d}}_{i,j}-{\overline{d}}_{i,j}^{\prime }|}$ is inversely proportional to the inter-antenna spacing, i.e., $c_{s,{\overline{d}}_{i,j}}/c_{s,{\overline{d}}_{i,j}^{\prime }}={\overline{d}}_{i,j}^{\prime }/{\overline{d}}_{i,j}$.

Therefore, to access the effect of both IEC and IPC, the coupling leakage Q is provided to evaluate the total mutual coupling, i.e., [18,28]:

$$\begin{array}{c}\hfill Q=\parallel {\mathbf{C}}_{sp}-{\mathbf{I}}_{6(M+N)}{\parallel }_F/\parallel {\mathbf{C}}_{sp}{\parallel }_F,\end{array}$$

(13)

where ${\parallel ·\parallel }_F$ stands for the Frobenius norm. One can see that $\parallel {\mathbf{C}}_{sp}-{\mathbf{I}}_{6(M+N)}{\parallel }_F$ is the energy of all the off-diagonal coupling coefficients, which characterizes the amount of mutual coupling. Therefore, Q can be regarded as the energy ratio between the two components and is called the mutual coupling ratio in this paper. In theory, mutual coupling is reduced as Q decreases.

3. Proposed Array

3.1. PSC-PSA

It is noted that, since polarization is bivariate, an EMVS of at least two differently polarized antennas can achieve polarimetry reception of an EM signal. In this case, there exist $6!/(2!4!)=15$ two-component EMVSs configurations, i.e., two components are chosen out of a total of six EM components. These configurations are divided into three types with various orientations, i.e., a dipole–dipole pair, a dipole–loop pair, and a loop–loop pair. Among these possible pairs, polarization estimation based on an orthogonal loop and dipole (OLD) pair is independent of the DOA of the signal [9], which is convenient for demonstrating the proposed estimation method. Therefore, the PSC-PSA composed of three subarrays with OLD pairs is selected as a way to demonstrate the configuration of the proposed array, as shown in Figure 2. Subarray 1 comprises $W_1$ dipole antennas with spacing $Nd$, subarray 2 consists of $W_2$ loop antennas with spacing $Md$, and subarray 3 contains $W_2$ dipole antennas with spacing $Md$. Using the origin of the coordinate axis as a reference, subarray 2 has $D_{1}d$ displacement along the y-axis, and subarray 3 has a half-wavelength and $D_{2}d$ displacement along the x- and y-axes, respectively. Further, M and N are coprime, d is the half-wavelength, and $D_{1}d$ and $D_{2}d$ are constants that can be used for reducing mutual coupling and increasing the array flexibility.

The antenna positions of the proposed array are denoted as $\overline{\mathbb{D}}\left(j\right)=\left\{(d_{j,x},d_{j,y})\right|(d_{j,x},d_{j,y})$$\in ({\overline{\mathbb{D}}}_1\cup {\overline{\mathbb{D}}}_2\cup {\overline{\mathbb{D}}}_3),1\le j\le W_1+2W_2\}$, where ${\overline{\mathbb{D}}}_1$, ${\overline{\mathbb{D}}}_2$, and ${\overline{\mathbb{D}}}_3$ respectively represent the antenna positions of subarrays 1, 2, and 3; that is:

$$\begin{array}{cc}\hfill {\overline{\mathbb{D}}}_1& =\left\{(0,d_{1}d)\right|d_1\in {\overline{\mathbb{P}}}_1=[0,N,\cdots (W_1-1)N]\},\hfill \\ \hfill {\overline{\mathbb{D}}}_2& =\left\{(0,d_{2}d)\right|d_2\in {\overline{\mathbb{P}}}_2=[0,M,\cdots (W_2-1)M]+D_1\},\hfill \\ \hfill {\overline{\mathbb{D}}}_3& =\left\{(d,d_{3}d)\right|d_3\in {\overline{\mathbb{P}}}_3=[0,M,\cdots (W_2-1)M]+D_2\}.\hfill \end{array}$$

(14)

Since the self-difference coarray is contained by the cross-difference coarray for a coprime array [27], we mainly focus on the cross-difference coarray of the PSC-PSA. In this case, the positions of the first and last antennas in the cross-difference coarray between subarrays 1 and 2 are located at $-\left((W_1-1)N\right)$ and $(W_2-1)M+D_1$, respectively, and the aperture of this cross-difference coarray (normalized by d) is $F_1=(W_1-1)N+(W_2-1)M+D_1$. Similarly, the positions of the first and last antennas in the cross-difference coarray between subarrays 1 and 3 are located at $-\left((W_1-1)N\right)$ and $(W_2-1)M+D_2$, respectively, and thus, the corresponding normalized aperture is $F_2=(W_1-1)N+(W_2-1)M+D_2$. Hence, the normalized aperture of the difference coarray of the PSC-PSA is $F=\max (F_1,F_2)=(W_1-1)N+(W_2-1)M+\max \{D_1,D_2\}$.

For the fixed total number W, the allocation of the number of antennas for each subarray used for efficiently extending the aperture can be transformed into an optimization problem:

$$\begin{array}{c}\hfill (W_1,W_2)=\arg \max \left\{F\right\}\mathrm{s}.\mathrm{t}.W=W_1+2W_2.\end{array}$$

(15)

First, a lemma is given to help determine $W_1$ and $W_2$.

Lemma 1.

When $W_2=\lceil (2W+1)/8\rceil$ or $\lceil (2W+1)/8\rceil -1$, the coarray aperture F can be maximized.

Proof. 

For a coprime array, one subarray usually consists of M antennas with spacing N, while the other one is composed of N antennas with spacing M. In this case, F is rewritten as follows:

$$\begin{array}{c}\hfill F=(W_1-1)W_2+(W_2-1)W_2+\max \{D_1,D_2\}.\end{array}$$

(16)

Substituting $W_1=W-2W_2$ in (16), the derived function of F related to $W_2$ is given as follows:

$$\begin{array}{c}\hfill \partial F/\partial W_2=-8W_2+2W+1.\end{array}$$

(17)

Letting $\partial F/\partial W_2=0$, the extreme point of $W_2$ can be achieved, i.e., $W_2=(2W+1)/8$. Since $W_2$ is an integer, the maximum coarray aperture can then be achieved with $W_2=\lceil (2W+1)/8\rceil$ or $\lceil (2W+1)/8\rceil -1$.

□

Note that the coprimality of $W_1$ and $W_2$ is not taken into consideration in this proof, which may lead to DOF loss in the coarray domain. Define $\mathrm{L}(W_1,W_2)$ as the least common multiple of $W_1$ and $W_2$. As a consequence, the coprimality of the spacings M and N can be achieved by

$$\begin{array}{c}\hfill M=k_1{W}_1/\mathrm{L}(W_1,W_2),N=k_2{W}_2/\mathrm{L}(W_1,W_2),\end{array}$$

(18)

where $k_1$ and $k_2$, as control coefficients to increase DOFs and aperture in the coarray domain, meet the following two conditions:

$$\begin{array}{cc}\hfill & \mathrm{L}(k_1,W_2)=1,\mathrm{L}(k_2,W_1)=1,\hfill \end{array}$$

(19a)

$$\begin{array}{cc}\hfill & M>N,M\ge W_2,\mathrm{or}N>M,N\ge W_1.\hfill \end{array}$$

(19b)

In this case, the coprimality can be used to illustrate that the coarray has $W_1{W}_2$ distinct elements, i.e., $W_1{W}_2$ DOFs, when using $W_1+W_2$ antennas. Correspondingly, the extended coarray aperture is given as follows:

$$F=[(W_1-1)k_2{W}_2+(W_2-1)k_1{W}_1]/\mathrm{L}(W_1,W_2)+\max \{D_1,D_2\}.$$

(20)

Furthermore, compared with the existing parallel coprime PSAs [19,20], the PSC-PSA can effectively save the cost of antennas due to the shared subarray (i.e., subarray 1) in the space domain. Additionally, in the polarization domain, the collocated EMVSs in the existing arrays are replaced by one antenna with a certain polarization type in PSC-PSA, which further saves the cost. Hence, with the similar aperture and DOFs in the coarray domain, the PSC-PSA can save at least half the number of antennas compared to the existing parallel coprime PSAs.

Remark 1.

Note that other pairs (e.g., the dipole–dipole pair and the loop–loop pair) can also be applied to the proposed array configuration, and thus there are 15 different types of the proposed array due to the various polarization types of the subarrays. The parameter estimation and CRB based on these proposed arrays will be shown in detail in the following.

3.2. Mutual Coupling Analysis

Due to the non-collocated EMVSs, the mutual coupling matrix ${\overline{\mathbf{C}}}_{sp}$ for the PSC-PSA is given as follows:

$${\left[{\overline{\mathbf{C}}}_{sp}\right]}_{m,n}=\left\{\begin{array}{cc}{c}_p{c}_{s,{\overline{d}}_2},\hfill & {\overline{d}}_2\le B,\hfill \\ 0,\hfill & {\overline{d}}_2>B,\hfill \end{array}\right.$$

(21)

where

$${\overline{d}}_2=\sqrt{{(d_{m,x}-d_{n,x})}^2+{(d_{m,y}-d_{n,y})}^2}.$$

(22)

Comparing Equations (11), (12) and (21), some observations are given as follows:

(a)

Generally, the first three coefficients $c_s$ dominate the effect of mutual coupling for nonuniform scalar–vector arrays [18], i.e., $c_{s,1}$, $c_{s,2}$, and $c_{s,3}$. In this case, for the same number of antennas W, the number of mutual coupling coefficients for the existing arrays is at least $W+32$, while that for the proposed array is reduced to at most 12 due to the shared subarray and displacements between subarrays in both the x- and y-axes. This shows that the PSC-PSA can eliminate mutual coupling with the decreased number of coupling coefficients.

(b)

On the other hand, considering both the displacements and spatial separation between differently polarized antennas along the x- and y-axes, the values of coefficients $c_p{c}_s$ for the PSC-PSA are much smaller than those (e.g., $c_p$ or $c_s$) for the existing arrays. This reflects that the former can mitigate mutual coupling by decreasing the values of the coupling coefficients.

4. 2D DOA and Polarization Estimation

4.1. Block Sparse Model

According to the configuration of the PSC-PSA with dipole–loop pairs, as shown in Figure 2, the signal models of the three subarrays can be given as follows:

$$\begin{array}{cc}\hfill {\mathbf{x}}_{\mathbf{1}}\left(t\right)& ={\mathbf{A}}_1\mathbf{D}\mathbf{s}\left(t\right)+{\mathbf{n}}_1\left(t\right),\hfill \\ \hfill {\mathbf{x}}_{\mathbf{2}}\left(t\right)& ={\mathbf{A}}_2\mathbf{L}\mathbf{s}\left(t\right)+{\mathbf{n}}_2\left(t\right),\hfill \\ \hfill {\mathbf{x}}_3\left(t\right)& ={\mathbf{A}}_3{\mathbf{\Phi }}_1\mathbf{D}\mathbf{s}\left(t\right)+{\mathbf{n}}_3\left(t\right),\hfill \end{array}$$

(23)

where $\mathbf{s}\left(t\right)$ are supposed to be temporally white and uncorrelated with each other, and

$$\mathbf{D}=\mathrm{diag}(e_{z,1},\cdots ,e_{z,K})=-\mathrm{diag}(sin{\theta }_{1}sin{\gamma }_1{e}^{j{\eta }_1},\cdots ,sin{\theta }_{K}sin{\gamma }_K{e}^{j{\eta }_K}),$$

(24)

$$\mathbf{L}=\mathrm{diag}(h_{z,1},\cdots ,h_{z,K})=-\mathrm{diag}(sin{\theta }_{1}cos{\gamma }_1,\cdots ,sin{\theta }_{K}cos{\gamma }_K),$$

(25)

$${\mathbf{\Phi }}_{\mathbf{1}}=\mathrm{diag}(e^{-j\pi cos{\beta }_1},\cdots ,e^{-j\pi cos{\beta }_K}),$$

(26)

$${\mathbf{A}}_i=[{\mathbf{a}}_i\left({\alpha }_1\right),{\mathbf{a}}_i\left({\alpha }_2\right),\cdots ,{\mathbf{a}}_i\left({\alpha }_K\right)]$$

(27)

$$\begin{array}{c}\hfill {\mathbf{a}}_i\left({\alpha }_k\right)={[e^{-j\pi cos\left({\alpha }_k\right){\left[{\overline{\mathbb{P}}}_i\right]}_1},\cdots ,e^{-j\pi cos\left({\alpha }_k\right){\left[{\overline{\mathbb{P}}}_i\right]}_{\mathrm{end}}}]}^T,\end{array}$$

(28)

with $i=1,2,3$. Here, for the kth signal, we have

$$\begin{array}{c}\hfill cos{\alpha }_k=sin{\theta }_{k}sin{\varphi }_k,cos{\beta }_k=sin{\theta }_{k}cos{\varphi }_k,\end{array}$$

(29)

where ${\alpha }_k$ is the angle between the kth signal and the y-axis, and ${\beta }_k$ is the angle between the kth signal and the x-axis. In this case, the cross-correlation matrix between subarray 1 and subarray 2 is given as follows:

$${\mathbf{R}}_{12}=\mathrm{E}\left[{\mathbf{x}}_1\left(t\right){\mathbf{x}}_2^H\left(t\right)\right]={\mathbf{A}}_1\mathbf{D}{\mathbf{R}}_s{\mathbf{L}}^H{\mathbf{A}}_2^H,$$

(30)

where the signal correlation matrix ${\mathbf{R}}_s=\mathrm{E}\left[\mathbf{s}\left(t\right){\mathbf{s}}^H\left(t\right)\right]=\mathrm{diag}[{\sigma }_1^2,\dots ,{\sigma }_K^2]$ is a diagonal matrix, with ${\sigma }_k^2$ denoting the power of the kth signal.

Vectorizing ${\mathbf{R}}_{\mathbf{12}}$, we obtain

$${\mathbf{r}}_{12}=\mathrm{vec}\left({\mathbf{R}}_{12}\right)={\mathbf{A}}_2^{*}\odot {\mathbf{A}}_1{\mathbf{p}}_{12}={\mathbf{A}}_{12}\left(\alpha \right){\mathbf{p}}_{12},$$

(31)

where

$${\mathbf{A}}_{12}\left(\alpha \right)={\mathbf{A}}_2^{*}\odot {\mathbf{A}}_1=[{\mathbf{a}}_{12}\left({\alpha }_1\right),\cdots ,{\mathbf{a}}_{12}\left({\alpha }_K\right)],$$

(32)

$${\mathbf{a}}_{12}\left({\alpha }_k\right)={\mathbf{a}}_2^{*}\left({\alpha }_k\right)\otimes {\mathbf{a}}_1\left({\alpha }_k\right),$$

(33)

$${\mathbf{p}}_{12}={[{sin}^2{\theta }_{1}cos{\gamma }_{1}sin{\gamma }_1{e}^{j{\eta }_1}{\sigma }_1^2,\cdots ,{sin}^2{\theta }_{K}cos{\gamma }_{K}sin{\gamma }_K{e}^{j{\eta }_K}{\sigma }_K^2]}^T.$$

(34)

Also, the correlation matrix between subarrays 1 and 3 is

$${\mathbf{R}}_{13}=\mathrm{E}\left[{\mathbf{x}}_1\left(t\right){\mathbf{x}}_3^H\left(t\right)\right]={\mathbf{A}}_1\mathbf{D}{\mathbf{R}}_s{\mathbf{D}}^H{\mathbf{\Phi }}_1^H{\mathbf{A}}_3^H.$$

(35)

Vectorizing ${\mathbf{R}}_{13}$, we obtain

$${\mathbf{r}}_{13}=\mathrm{vec}\left({\mathbf{R}}_{13}\right)={\mathbf{A}}_3^{*}\odot {\mathbf{A}}_1{\mathbf{p}}_{13}={\mathbf{A}}_{13}\left(\alpha \right){\mathbf{p}}_{13},$$

(36)

where

$${\mathbf{A}}_{13}\left(\alpha \right)={\mathbf{A}}_3^{*}\odot {\mathbf{A}}_1=[{\mathbf{a}}_{13}\left({\alpha }_1\right),\cdots ,{\mathbf{a}}_{13}\left({\alpha }_K\right)],$$

(37)

$${\mathbf{a}}_{13}\left({\alpha }_k\right)={\mathbf{a}}_3^{*}\left({\alpha }_k\right)\otimes {\mathbf{a}}_1\left({\alpha }_k\right),$$

(38)

$${\mathbf{p}}_{13}={[{sin}^2{\theta }_1{sin}^2{\gamma }_1{e}^{j\pi cos{\beta }_1}{\sigma }_1^2,\cdots ,{sin}^2{\theta }_K{sin}^2{\gamma }_K{e}^{j\pi cos{\beta }_K}{\sigma }_K^2]}^T.$$

(39)

Comparing ${\mathbf{A}}_{12}$ and ${\mathbf{A}}_{13}$, we have

$${\mathbf{A}}_{13}\left(\alpha \right)={\mathbf{A}}_{12}\left(\alpha \right){\mathbf{\Phi }}_2,$$

(40)

where

$${\mathbf{\Phi }}_2=\mathrm{diag}(e^{-j\pi (D_2-D_1)cos{\alpha }_1},\cdots ,e^{-j\pi (D_2-D_1)cos{\alpha }_K}).$$

(41)

It is shown from (40) that ${\mathbf{r}}_{\mathbf{12}}$ and ${\mathbf{r}}_{\mathbf{13}}$ have the same spatial support. Then, a block signal model for the received data can be constructed as follows:

$$\begin{array}{cc}\hfill \mathbf{z}& =\mathrm{vec}\left({\left[{\mathbf{r}}_{12}^T{\mathbf{r}}_{13}^T\right]}^T\right)\hfill \\ \hfill & =\mathrm{vec}\left({\left[{\left({\mathbf{A}}_{12}\left(\alpha \right){\mathbf{p}}_{12}\right)}^T{\left({\mathbf{A}}_{12}\left(\alpha \right){\mathbf{\Phi }}_2{\mathbf{p}}_{13}\right)}^T\right]}^T\right)\hfill \\ \hfill & ={[{\left({\mathbf{B}}_1{\mathbf{e}}_1\right)}^T,\dots ,{\left({\mathbf{B}}_K{\mathbf{e}}_K\right)}^T]}^T,\hfill \end{array}$$

(42)

where

$${\mathbf{B}}_k={\mathbf{a}}_{12}\left({\alpha }_k\right)\otimes {\mathbf{I}}_2,$$

(43)

and

$$\begin{array}{cc}\hfill & {\mathbf{e}}_k={\left[\begin{array}{cc}{\left[{\mathbf{p}}_{12}\right]}_k& {\left[{\mathbf{\Phi }}_2\right]}_k{\left[{\mathbf{p}}_{13}\right]}_k\end{array}\right]}^T\hfill \\ \hfill & ={\left[\begin{array}{c}{sin}^2{\theta }_{k}cos{\gamma }_{k}sin{\gamma }_k{e}^{j{\eta }_k}{\sigma }_k^2\\ {sin}^2{\theta }_k{sin}^2{\gamma }_k{e}^{j\pi [cos{\beta }_k-(D_2-D_1)cos{\alpha }_k]}{\sigma }_k^2\end{array}\right]}^T.\hfill \end{array}$$

(44)

One can see that, owing to the configuration of the proposed array, the polarization phase difference and spatial information are respectively provided in the phases of distinct items of one block ${\mathbf{e}}_k$, indicating that the latter can be obtained in the polarization domain and is not coupled by the former due to parameter separation. It is shown that the remaining one DOF in phase of the polarization domain can be used for obtaining the spatial angle in order to achieve 2D DOA estimation.

Then, by discretizing the angle range of interest as a uniform grid $\Theta =({\overline{\alpha }}_1,\dots ,{\overline{\alpha }}_{\overline{K}})$ with spacing $\Delta \alpha$ and $\overline{K}\gg K$, the model in (42) is transformed into a block-sparse signal model:

$$\overline{\mathbf{z}}=\mathbf{G}\left(\alpha \right)\mathbf{H},$$

(45)

where

$$\mathbf{G}\left(\alpha \right)=[{\tilde{\mathbf{G}}}_1,\dots ,{\tilde{\mathbf{G}}}_{\overline{K}}],$$

(46)

$${\tilde{\mathbf{G}}}_{\overline{k}}={\mathbf{a}}_{12}\left({\alpha }_{\overline{k}}\right)\otimes {\mathbf{I}}_2,$$

(47)

$$\mathbf{H}=[{\mathbf{h}}_1,\dots ,{\mathbf{h}}_{\overline{K}}],$$

(48)

$${\mathbf{h}}_{\overline{k}}={\left[\begin{array}{cc}{sin}^2{\theta }_{\overline{k}}cos{\gamma }_{\overline{k}}sin{\gamma }_{\overline{k}}{e}^{j{\eta }_{\overline{k}}}{\sigma }_{\overline{k}}^2& {sin}^2{\theta }_{\overline{k}}{sin}^2{\gamma }_{\overline{k}}{e}^{-j\pi (cos{\beta }_{\overline{k}}+(D_2-D_1)cos{\alpha }_{\overline{k}})}{\sigma }_{\overline{k}}^2\end{array}\right]}^T.$$

(49)

4.2. BCS-Based Estimation Method with 1D Sparse Reconstruction

Based on (45), a BCS-based method is provided for multi-parameter estimation. By using BCS [29], a block-sparsity vector ${\mathbf{h}}_{\overline{k}}$ and its support (index set of non-zero block elements) can be recovered from the measurement $\mathbf{z}$ under some recovery conditions. Therefore, ${\alpha }_k$ of the kth signal is estimated from the index set of non-zero block elements, denoted as ${\widehat{\alpha }}_k$, and the other parameters of this signal can be estimated from the corresponding block elements ${\widehat{\mathbf{h}}}_{\overline{k}}$, given as follows:

$$\begin{array}{cc}\hfill {\widehat{\beta }}_k& =arccos(-\mathrm{angle}\left({\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_2\right)/\pi -(D_2-D_1)cos{\widehat{\alpha }}_k),\hfill \\ \hfill {\widehat{\gamma }}_k& =arctan(|{\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_2|/|{\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_1|),\hfill \\ \hfill {\widehat{\eta }}_k& =\mathrm{angle}\left({\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_1\right).\hfill \end{array}$$

(50)

Hence, 2D DOA of the kth signal can be estimated by (29), i.e.,

$$\begin{array}{cc}\hfill {\widehat{\theta }}_k& =arcsin\left(\sqrt{{(cos{\widehat{\alpha }}_k)}^2+{(cos{\widehat{\beta }}_k)}^2}\right),\hfill \\ \hfill {\widehat{\varphi }}_k& =arctan\left({\displaystyle \frac{cos{\widehat{\alpha }}_k}{cos{\widehat{\beta }}_k}}\right).\hfill \end{array}$$

(51)

Based on the above discussion, one can see that 1D sparse reconstruction is only used once, and then the estimation of four parameters is achieved with automatic pairing, which effectively reduces the complexity for the multi-parameter estimation case.

Remark 2.

The estimation results of the proposed array composed of dipole–loop pairs are given as (50).

On the other hand, taking the pair of $e_{y,k}$ and $e_{z,k}$ as an example, the estimation results of the proposed array consisting of dipole–dipole pairs are provided as follows:

$$\begin{array}{cc}\hfill {\widehat{\beta }}_k& =arccos(\mathrm{angle}\left({\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_2\right)/\pi +(D_2-D_1)\cos {\widehat{\alpha }}_{\overline{k}}),\hfill \\ \hfill {\widehat{\gamma }}_k& =arctan\left|{\displaystyle \frac{{\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_{2}cos{\widehat{\varphi }}_{\overline{k}}}{-V_1}}\right|,\hfill \\ \hfill {\widehat{\eta }}_k& =\mathrm{angle}\left(V_1\right),\hfill \end{array}$$

(52)

where

$$\begin{array}{cc}\hfill V_1& ={\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_{1}sin{\widehat{\theta }}_{\overline{k}}+{\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_2{e}^{-j\pi (cos{\widehat{\beta }}_{\overline{k}}-(D_2-D_1)\cos {\widehat{\alpha }}_{\overline{k}})}cos{\widehat{\theta }}_{\overline{k}}sin{\widehat{\varphi }}_{\overline{k}}\hfill \\ \hfill & =-{sin}^2{\widehat{\theta }}_{\overline{k}}cos{\widehat{\varphi }}_{\overline{k}}sin{\gamma }_{\overline{k}}cos{\gamma }_{\overline{k}}{e}^{j{\eta }_{\overline{k}}}{\sigma }_{\overline{k}}^2,\hfill \end{array}$$

(53)

$${\left[{\mathbf{h}}_{\overline{k}}\right]}_1=-sin{\theta }_{\overline{k}}cos{\theta }_{\overline{k}}sin{\varphi }_{\overline{k}}{sin}^2{\gamma }_{\overline{k}}-sin{\theta }_{\overline{k}}cos{\varphi }_{\overline{k}}sin{\gamma }_{\overline{k}}cos{\gamma }_{\overline{k}}{e}^{j{\eta }_{\overline{k}}}{\sigma }_{\overline{k}}^2,$$

(54)

$${\left[{\mathbf{h}}_{\overline{k}}\right]}_2={\sin }^2{\theta }_{\overline{k}}{\sin }^2{\gamma }_{\overline{k}}{e}^{j\pi (cos{\beta }_{\overline{k}}-(D_2-D_1)\cos {\alpha }_{\overline{k}})}{\sigma }_{\overline{k}}^2.$$

(55)

Taking the pair of $h_{x,k}$ and $h_{z,k}$ as an example, the estimation results of the proposed array composed of loop–loop pairs are expressed as follows:

$$\begin{array}{cc}\hfill {\widehat{\beta }}_k& =arccos(\mathrm{angle}\left({\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_2\right)/\pi +(D_2-D_1)cos{\widehat{\alpha }}_{\overline{k}}),\hfill \\ \hfill {\widehat{\gamma }}_k& =arctan\left|{\displaystyle \frac{-V_2}{{\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_{2}sin{\widehat{\varphi }}_{\overline{k}}}}\right|,\hfill \\ \hfill {\widehat{\eta }}_k& =-\mathrm{angle}\left(V_2\right),\hfill \end{array}$$

(56)

where

$$\begin{array}{cc}\hfill V_2& ={\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_{1}sin{\widehat{\theta }}_{\overline{k}}+{\left[{\widehat{\mathbf{h}}}_{\overline{k}}\right]}_2{e}^{-j\pi (cos{\widehat{\beta }}_{\overline{k}}-(D_2-D_1)\cos {\widehat{\alpha }}_{\overline{k}})}cos{\widehat{\theta }}_{\overline{k}}cos{\widehat{\varphi }}_{\overline{k}}\hfill \\ \hfill & =-{sin}^2{\widehat{\theta }}_{\overline{k}}sin{\widehat{\varphi }}_{\overline{k}}sin{\gamma }_{\overline{k}}cos{\gamma }_{\overline{k}}{e}^{-j{\eta }_{\overline{k}}}{\sigma }_{\overline{k}}^2,\hfill \end{array}$$

(57)

$${\left[{\mathbf{h}}_{\overline{k}}\right]}_1=-sin{\theta }_{\overline{k}}cos{\theta }_{\overline{k}}cos{\varphi }_{\overline{k}}{cos}^2{\gamma }_{\overline{k}}-sin{\theta }_{\overline{k}}sin{\varphi }_{\overline{k}}sin{\gamma }_{\overline{k}}cos{\gamma }_{\overline{k}}{e}^{-j{\eta }_{\overline{k}}}{\sigma }_{\overline{k}}^2,$$

(58)

$${\left[{\mathbf{h}}_{\overline{k}}\right]}_2={\sin }^2{\theta }_{\overline{k}}{\cos }^2{\gamma }_{\overline{k}}{e}^{j\pi (cos{\beta }_{\overline{k}}-(D_2-D_1)\cos {\alpha }_{\overline{k}})}{\sigma }_{\overline{k}}^2.$$

(59)

4.3. Computational Complexity

For the first step of the block-sparse model, the complexity load of two cross-correlation matrices is about $O\left(2W_1{W}_{2}L\right)$, with L being the number of snapshots. The complexity of the estimation based on the BCS is about $O\left(\kappa W_1{W}_2{\overline{K}}^2\right)$, with $\kappa$ denoting the number of iterations. Hence, the total computational complexity of the one-dimensional sparse reconstruction method is

$$O(2W_1{W}_{2}L+\kappa W_1{W}_2{\overline{K}}^2).$$

(60)

5. CRB

The CRB can be developed from a linear sparse scalar-sensor array [30] to a parallel sparse PSA. Recalling (23), the covariance matrix of the PSC-PSA can be written as follows:

$$\mathbf{R}=\mathrm{E}\left[\mathbf{x}\left(t\right){\mathbf{x}}^H\left(t\right)\right]=\mathbf{A}{\mathbf{R}}_s{\mathbf{A}}^H+{\sigma }_n^2{\mathbf{I}}_{W_1+2W_2},$$

(61)

where ${\sigma }_n^2$ is the power of noise, $\mathbf{x}\left(t\right)={[{\mathbf{x}}_1^T\left(t\right),{\mathbf{x}}_2^T\left(t\right),{\mathbf{x}}_3^T\left(t\right)]}^T$, ${\mathbf{R}}_s=\mathrm{E}\left[\mathbf{s}\left(t\right){\mathbf{s}}^H\left(t\right)\right]$, and $\mathbf{A}=[{\mathbf{a}}_1,\dots ,{\mathbf{a}}_K]={[{\left({\mathbf{A}}_1\mathbf{D}\right)}^T,{\left({\mathbf{A}}_2\mathbf{L}\right)}^T,{\left({\mathbf{A}}_3{\mathbf{\Phi }}_1\mathbf{D}\right)}^T]}^T$. Here, ${\mathbf{a}}_k$ can be written as follows:

$${\mathbf{a}}_k=\left[\begin{array}{c}{\mathbf{a}}_1({\theta }_k,{\varphi }_k)e_{z,k}\\ {\mathbf{a}}_2({\theta }_k,{\varphi }_k)h_{z,k}\\ {\overline{\mathbf{a}}}_3({\theta }_k,{\varphi }_k)e_{z,k}\end{array}\right],$$

(62)

where ${\mathbf{a}}_1({\theta }_k,{\varphi }_k)$ and ${\mathbf{a}}_2({\theta }_k,{\varphi }_k)$ are given in (28), and

$$\begin{array}{c}\hfill {\overline{\mathbf{a}}}_3\left({\alpha }_k\right)={[e^{-j\pi (sin{\theta }_{k}sin{\varphi }_k{\left[{\overline{\mathbb{P}}}_i\right]}_1+sin{\theta }_{k}cos{\varphi }_k)},\cdots ,e^{-j\pi (sin{\theta }_{k}sin{\varphi }_k{\left[{\overline{\mathbb{P}}}_i\right]}_{\mathrm{end}}+sin{\theta }_{k}cos{\varphi }_k)}]}^T.\end{array}$$

(63)

In this case, the CRB for the PSC-PSA is given as follows:

$$\mathrm{CRB}\left(\theta ,\varphi ,\gamma ,\eta \right)={\left({\mathbf{B}}^H{\mathbf{U}}^{1/2}{\mathbf{P}}_{{\mathbf{U}}^{1/2}\tilde{\mathbf{A}}}^{\perp }{\mathbf{U}}^{1/2}\mathbf{B}\right)}^{-1}/L,$$

(64)

where $\tilde{\mathbf{A}}=\mathbf{A}\odot \mathbf{A}$, $\mathbf{U}={\mathbf{R}}^{-T}\otimes {\mathbf{R}}^{-1}$, and ${\mathbf{P}}_{\mathbf{Y}}^{\perp }=\mathbf{I}-\mathbf{Y}{\left({\mathbf{Y}}^H\mathbf{Y}\right)}^{-1}{\mathbf{Y}}^H$ stands for the orthogonal complement of $\mathbf{Y}$. Here, $\mathbf{B}$ is a matrix composed of derivation and can be expressed as follows:

$$\begin{array}{c}\hfill \mathbf{B}=\left[\begin{array}{ccccccccc}{\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_1}{\partial {\theta }_1}}{\sigma }_1^2& {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_1}{\partial {\varphi }_1}}{\sigma }_1^2& {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_1}{\partial {\gamma }_1}}{\sigma }_1^2& {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_1}{\partial {\eta }_1}}{\sigma }_1^2& \dots & {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_K}{\partial {\theta }_K}}{\sigma }_K^2& {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_K}{\partial {\varphi }_K}}{\sigma }_K^2& {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_K}{\partial {\gamma }_K}}{\sigma }_K^2& {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_K}{\partial {\eta }_K}}{\sigma }_K^2\end{array}\right],\end{array}$$

(65)

where ${\tilde{\mathbf{a}}}_k={\mathbf{a}}_k^{*}\otimes {\mathbf{a}}_k$ and

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_k}{\partial {\theta }_k}}& ={\left({\displaystyle \frac{\partial {\mathbf{a}}_k}{\partial {\theta }_k}}\right)}^{*}\otimes {\mathbf{a}}_k+{\mathbf{a}}_k^{*}\otimes {\displaystyle \frac{\partial {\mathbf{a}}_k}{\partial {\theta }_k}},\hfill \\ \hfill {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_k}{\partial {\varphi }_k}}& ={\left({\displaystyle \frac{\partial {\mathbf{a}}_k}{\partial {\varphi }_k}}\right)}^{*}\otimes {\mathbf{a}}_k+{\mathbf{a}}_k^{*}\otimes {\displaystyle \frac{\partial {\mathbf{a}}_k}{\partial {\varphi }_k}},\hfill \\ \hfill {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_k}{\partial {\gamma }_k}}& ={\left({\displaystyle \frac{\partial {\mathbf{a}}_k}{\partial {\gamma }_k}}\right)}^{*}\otimes {\mathbf{a}}_k+{\mathbf{a}}_k^{*}\otimes {\displaystyle \frac{\partial {\mathbf{a}}_k}{\partial {\gamma }_k}},\hfill \\ \hfill {\displaystyle \frac{\partial {\tilde{\mathbf{a}}}_k}{\partial {\eta }_k}}& ={\left({\displaystyle \frac{\partial {\mathbf{a}}_k}{\partial {\eta }_k}}\right)}^{*}\otimes {\mathbf{a}}_k+{\mathbf{a}}_k^{*}\otimes {\displaystyle \frac{\partial {\mathbf{a}}_k}{\partial {\eta }_k}}.\hfill \end{array}$$

(66)

A further discussion of (66), based on the proposed array composed of dipole–loop pairs, dipole–dipole pairs, or loop–loop pairs, is provided in Appendix A.

6. Simulation Results

In this section, the superiority of the proposed technique is verified with the comparison of the existing arrays, e.g., TPC-PSA [19] and PCP-PSA [20], in the presence of mutual coupling.

6.1. Mutual Coupling Ratio

In the first experiment, we evaluate the effect of mutual coupling on the proposed and contrastive arrays by using the mutual coupling ratio defined in (13). The mutual coupling ratios for various arrays under different numbers of antennas are given in Figure 3. The mutual coupling coefficients are set as $c_1=0.1e^{-j\frac{\pi }{3}}$, with $\left|c_l\right|=\left|c_1\right|/l$, $1<l\le 3$, and $c_p=0.1e^{-j\frac{\pi }{2}}$. One can see from Figure 3 that all mutual coupling ratios decrease with an increasing number of antennas due to the fact that the inter-antenna spacings are enlarged with the growth in antenna number for coprime array configurations. Obviously, the mutual coupling ratio of the proposed array is much lower than that of the existing arrays, showing that the former can provide lower mutual coupling compared with the latter.

6.2. Scatter Diagrams

In the second experiment, consider that seven uncorrelated signals impinging on the PSC-PSA are located at $(\theta ,\varphi )=[({29}^{\circ },-{51}^{\circ }),({35}^{\circ },-{81}^{\circ }),({36}^{\circ },{45}^{\circ }),({39}^{\circ },{66}^{\circ }),({39}^{\circ },-{24}^{\circ }),$$({46}^{\circ },{13}^{\circ }),({55}^{\circ },-{59}^{\circ })]$ with $(\gamma ,\eta )=[({25}^{\circ },{105}^{\circ }),({35}^{\circ },{45}^{\circ }),({45}^{\circ },{95}^{\circ }),({85}^{\circ },-{50}^{\circ }),({75}^{\circ },$${125}^{\circ }),({60}^{\circ },{75}^{\circ }),({55}^{\circ },-{115}^{\circ })]$, all with equal power. The SNR for this case is 20 dB, the number of snapshots is 2000, and the mutual coupling coefficients are the same as that in the previous experiment. The results of 2D DOA and polarization estimations are given in Figure 4a and Figure 4b, respectively. It is obvious that the estimated values are very close to the true values for all signals, demonstrating that the proposed technique can provide high-precision multi-parameter estimation due to its extended aperture and reduced mutual coupling.

Furthermore, Figure 5 plots the comparison of scatter diagrams in the scenario with closely spaced signals to show the superiority of the proposed array in terms of resolution. The two black lines represent true values of DOAs of two signals that are located at $({\theta }_1,{\varphi }_1)=({60}^{\circ },{35}^{\circ })$ and $({\theta }_2,{\varphi }_2)=({62}^{\circ },{37}^{\circ })$, with $({\gamma }_1,{\eta }_1)=({30}^{\circ },{35}^{\circ })$ and $({\gamma }_2,{\eta }_2)=({50}^{\circ },{45}^{\circ })$. In this condition, the SNR is set as 20 dB, and the number of snapshots is 4000. It is obvious that the proposed array can perform DOA estimation for closely spaced signals, demonstrating that it has better resolution performance.

6.3. RMSE

Consider two uncorrelated signals with $({\alpha }_1,{\beta }_1,{\gamma }_1,{\eta }_1)$$=({45}^{\circ },{68}^{\circ },{50}^{\circ },{35}^{\circ })$ and $({\alpha }_2,{\beta }_2,$${\gamma }_2,{\eta }_2)=({55}^{\circ },{36}^{\circ },{40}^{\circ },{45}^{\circ })$, in the presence of mutual coupling. To measure the estimation performance, the root mean square error (RMSE) is given as follows:

$$\begin{array}{c}\hfill \mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{KV}}\sum _{k=1}^K\sum _{v=1}^V{({\widehat{\psi }}_{k,v}-{\psi }_k)}^2},\end{array}$$

(67)

where ${\psi }_k$ and ${\widehat{\psi }}_{k,v}$ stand for the true and estimated parameters of the kth signal in the spatial and polarized domains, respectively, and V denotes the number of Monte Carlo trials.

In the third experiment, the RMSEs of DOA and polarization estimations for various arrays are plotted in Figure 6a and Figure 6b, respectively. The number of snapshots for the experiment is 600, the number of Monte Carlo trials is 200, and the mutual coupling coefficients are the same as that in the first experiment. It can be seen that the performance of multi-parameter estimation depending on the PSC-PSA is much better than that depending on the existing arrays, showing that the former can significantly enhance the estimation performance. The estimation performance under the mutual coupling scenario is close to that without mutual coupling and CRBs, which further verifies the superiority of the PSC-PSA for mutual coupling mitigation and aperture extension.

In the fourth experiment, the RMSEs of DOA and polarization estimation versus the number of snapshots are shown in Figure 7a and Figure 7b, respectively. In this case, the SNR is set as 5 dB, the number of Monte Carlo trials is 200, and the mutual coupling coefficients are the same as that in the first experiment. It is observed that the performance of DOA and the polarization estimation based on the proposed array is superior to that based on the existing arrays, which also shows that the former can provide high-accuracy estimation performance. Similarly, with successful mutual coupling reduction and aperture expansion, the estimation performance of PSC-PSA is close to the CRB.

In the fifth experiment, the RMSE performance comparison of the existing and proposed arrays with and without mutual coupling is shown in

Table 2. RMSE performance comparison between the existing and proposed arrays with or without mutual coupling.

	With Mutual Coupling		Without Mutual Coupling
	2D DOA Estimation	Polarization Estimation	2D DOA Estimation	Polarization Estimation
TPC-PSA	${2.45}^{\circ }$	${3.38}^{\circ }$	${1.69}^{\circ }$	${0.52}^{\circ }$
PCP-PSA	${1.36}^{\circ }$	${3.88}^{\circ }$	${0.51}^{\circ }$	${2.91}^{\circ }$
Proposed	${0.23}^{\circ }$	${0.66}^{\circ }$	${0.22}^{\circ }$	${0.56}^{\circ }$
Proposed with strong mutual coupling	${0.27}^{\circ }$	${0.98}^{\circ }$	-	-

. The SNR for this case is 5 dB, the number of snapshots is 600, the number of Monte Carlo trials is 200, and the mutual coupling coefficients are the same as that in the first experiment. The observations show that the estimation performance based on TPC-PSA and PCP-PSA without mutual coupling is much better than that with mutual coupling, indicating that the effect of mutual coupling on such arrays is huge. Additionally, the estimation performance based on the proposed array without mutual coupling is similar to that with mutual coupling, and both performances are superior to those based on TPC-PSA and PCP-PSA without mutual coupling. This further illustrates that the proposed array can significantly reduce mutual coupling for better estimation performance.

Furthermore, we consider a strong mutual coupling case, i.e., $c_1=0.3e^{-j\frac{\pi }{3}}$, with $\left|c_l\right|=\left|c_1\right|/l$, $1<l\le 3$, and $c_p=0.1e^{-j\frac{\pi }{2}}$. In this case, the estimation results based on the proposed array are given in the last row of Table 2. The other simulation conditions are similar to those of the previous experiment. Compared with the RMSE based on the proposed array shown in the penultimate row of Table 2, one can see that the estimation performance based on the proposed array degrades with enhanced mutual coupling. On the other hand, the performance based on the proposed array in the presence of strong mutual coupling is still better than that based on the existing arrays, which illustrates that the former has superior performance in terms of mutual coupling mitigation.

7. Conclusions

This paper presents a novel PSC-PSA and a BCS-based method for enhancing the performance of multi-parameter estimation in reality. A systematic procedure is given for the proposed array design to achieve mitigated mutual coupling, extended aperture, and increased DOFs. The parameter estimation is formulated as a block-sparse signal reconstruction problem, and then a 1D sparse reconstruction-based estimation method is only used once to handle the problem. Simulation results show the superiority of the proposed array and method in the case of the practical mutual coupling model. In the future, it will be of interest to combine array motion and the proposed array design to further extend array aperture and improve DOFs. Another future direction of interest would be the extension of the proposed arrays to a three-dimensional array for further improving estimation performance. These extensions are currently under investigation.