 Research
 Open Access
 Published:
A single triangular SSEMVS aided highaccuracy DOA estimation using a multiscale Lshaped sparse array
EURASIP Journal on Advances in Signal Processing volume 2019, Article number: 44 (2019)
Abstract
We propose a new array configuration composed of multiscale scalar arrays and a single triangular spatially spread electromagneticvectorsensor (SSEMVS) for highaccuracy twodimensional (2D) directionofarrival (DOA) estimation. Two scalar arrays are placed along xaxis and yaxis, respectively, each array consists of two uniform linear arrays (ULAs), and these two ULAs have different interelement spacings. In this manner, these two scalar arrays form a multiscale Lshaped array. The two arms of this Lshaped scalar array are connected by a sixcomponent SSEMVS, which is composed of a spatially spread dipoletriad plus a spatially spread looptriad. All the interelement spacings in our proposed array can be larger than a halfwavelength of the incident source, thus to form a sparse array to mitigate the mutual coupling across antennas. In the proposed DOA estimation algorithm, we perform the vectorcrossproduct algorithm to the SSEMVS to obtain a set of lowaccuracy but unambiguous direction cosine estimation as a reference; we then impose estimation of signal parameters via rotation invariant technique (ESPRIT) algorithm to the two scalar arrays to get two sets of highaccuracy but cyclically ambiguous direction cosine estimations. Finally, the coarse estimation is used to disambiguate the fine but ambiguous estimations progressively and therefore a multipleorder disambiguation algorithm is developed. The proposed array enjoys the superiority of low redundancy and low mutual coupling. Moreover, the thresholds of the intersensor spacings utilized in the proposed array are also analyzed. Simulation results validate the performance of the proposed array geometry.
Introduction
In the field of array signal processing, the directionofarrival (DOA) estimation accuracy of the incident sources is proportional to the aperture of the antenna array, and therefore an array with a larger aperture is desired [1]. However, to avoid the phase ambiguity in DOA estimation, it is generally believed that the spacing between adjacent antennas should not be greater than λ/2, where λ denotes the wavelength of the incident signal [1, 2]. In this way, a large aperture array usually requires more antennas and thus increases the cost as well as the mutual coupling between antennas. In order to mitigate this issue, various sparse array configurations and the corresponding DOA estimation algorithms have been developed. One type of sparse array is constructed by multiple widely separated subarrays [3–5], and the corresponding estimation of signal parameters via rotation invariant technique (ESPRIT)based algorithms which used the dualsize or multiplesize invariance within these arrays were developed therein. Another type is designed to obtain as many as degreesoffreedom (DOFs) to resolve more sources than sensors, such as the minimumredundancy array [6], the nested array [7], and the coprime array [8]. Their DOA estimation algorithms focused on using the high order statistic characteristics of the received data of the sparse array to increase the number of DOF and thus often required a large computational workload.
In the meantime, the electromagneticvectorsensor (EMVS) [9] has received extensive attention in array signal processing recently as well as other polarization antenna arrays [10–16]. EMVS can not only provide the DOA estimation of the signal, but can also give the polarization information. An EMVS usually consists of three orthogonally oriented dipoles and three orthogonally oriented loops to measure the electric field and the magnetic field of the incident source [17]. Unfortunately, due to the collocated geometry, the mutual coupling across the EMVS components affects the performance of the algorithm severely. In 2011, Wong and Yuan [18] proposed a SSEMVS which consists of six orthogonally oriented but spatially noncollocating dipoles and loops. This SSEMVS reduces the mutual coupling between antenna components, and the developed algorithm retains the effectiveness of the vectorcrossproduct algorithm [9]. Following this, various spatially spread polarization antenna arrays have been proposed [19–23]. Li et al. [24] presented many geometry configurations of the SSEMVS and a nonlinear programmingbased DOA estimation algorithm. Yuan [25] proposed the way how the four/five spatial noncollocated dipoles/loops were placed to estimate multisource azimuth/elevation direction finding and polarization. The array configuration of the SSEMVS was further investigated in [11, 26].
Most recently, there are some research on the combination of EMVS and sparse array and the corresponding parameter estimation algorithms. For example, Han et al. [27] developed a nested vectorsensor array, He et al. [28] proposed a nested crossdipole array, and Rao et al. [29] proposed a new class of sparse vectorsensor arrays. Various compositions of sparse acoustic vectorsensor arrays to estimate the elevationazimuth angles of coherent sources were presented in [30]. In [21], we proposed a multiscale sparse array with each sensor unit consisting of one SSEMVS, which is capable of estimating the 2D directions and polarization information of the source simultaneously. However, the estimation accuracy for one of the two direction cosines is limited (by the aperture of a single SSEMVS) since the sparse array is only extended along one axis. Furthermore, the unit of the aforementioned array is a sixcomponent SSEMVS, and therefore, the cost and redundancy of the whole array are still high.
In order to tackle the limitation of the sparse array developed in [21], in this paper, we propose a new array geometry composed with multiscale scalar arrays and a single triangular SSEMVS, and develop the corresponding 2D DOA estimation algorithm. The proposed array consists of an Lshaped scalar array and a triangular SSEMVS. The two arms of the Lshaped scalar array are connected by a triangular SSEMVS, which is placed in such a way that the vectorcrossproduct algorithm can be applied on it for DOA estimation. The scalar sensors in each arm of the Lshaped array can be divided into two uniform linear subarrays with different intersensor spacings. Owing to the spatially spread geometry of the SSEMVS and the different intersensor spacings of the two subarrays, we can obtain multiple estimates of target parameters. From the SSEMVS, we can obtain an unambiguous but lowaccuracy estimates and a relatively highaccuracy but ambiguous estimates of incident sources using the vectorcrossproduct algorithm [18]. In addition, we can obtain two highaccuracy but cyclically ambiguous estimates of desired direction cosines by applying the ESPRIT algorithm to the corresponding two subarrays in the Lshaped array, respectively. Following this, we develop a threeorder disambiguation method to obtain the final highaccuracy and unambiguous estimates of target DOA.
The proposed array integrates the advantages of sparse (scalar) array and SSEMVS in reducing mutual coupling and achieving highaccuracy DOA estimation. Moreover, we only use a single SSEMVS along with the Lshaped scalar array to achieve highaccuracy DOA estimation, and thus the cost, the redundancy of the proposed array, and the computational workload of the corresponding DOA estimation algorithm decrease significantly.
The rest of this paper is organized as follows. Section 2 describes the proposed array geometry. Section 3 develops the proposed algorithm for DOA estimation. In Section 4, numerical examples are provided to show the effectiveness and advantages of the proposed array and algorithm. Section 5 concludes the paper.
Array geometry
Triangular spatiallyspread electromagneticvectorsensor
Figure 1 depicts the array configuration for the triangular SSEMVS used in our paper, where one dipole e_{y} is placed at the origin (of the Cartesian coordinate system) and the other two dipoles are placed along xaxis and yaxis. The distance between e_{x} and e_{y} is Δ_{x,y}, and the distance between e_{y} and e_{z} is Δ_{y,z}. The loops of the SSEMVS are placed in such a way that the vectorcrossproduct algorithm can be adopted for DOA estimation, i.e., \(\overrightarrow {e_{y}e_{x}}=\overrightarrow {h_{y}h_{x}}\) and \(\overrightarrow {e_{y}e_{z}}=\overrightarrow {h_{y}h_{z}}\) [11], where \(\overrightarrow {xy}\) denotes a vector from point x to point y and h_{y} is located at (x_{h},y_{h},z_{h}). The positions of the three dipoles and the three loops form two rightangled triangles, and thus we name it as the triangular SSEMVS. It is worth noting that both Δ_{x,y} and Δ_{y,z} can be larger than a halfwavelength of the signal. Therefore, the SSEMVS itself is a sparse array.
Besides, the configuration of the SSEMVS used in [21] is based on two parallel lines. It can only expand in one direction; the estimation accuracy for another direction cosine is limited. By contrast, the triangular SSEMVS depicted in Fig. 1 has two direction extensions. Therefore, this configuration can provide relatively higher accuracy directioncosine estimates for the two direction consines along the x and yaxis, respectively, and thus higher accuracy estimates for θ (elevation angle) and ϕ (azimuth angle) through the vectorcrossproduct algorithm. Thereby, it is reasonable to use this configuration of SSEMVS to extend the aperture of the array by constructing a 2D Lshaped array.
Consider a farfield source, located at elevation angle θ∈[0,π] and azimuth angle ϕ∈[0,2π), with polarization parameters (γ,η), where γ refers to the auxiliary polarization angle and η represents the polarization phase difference. The array manifold of the triangular SSEMVS in Fig. 1, a, can be denoted by the electricfield vector e=[e_{x},e_{y},e_{z}]^{T} and the magneticfield vector h=[h_{x},h_{y},h_{z}]^{T} by taking account of the interdipole/loop spacings {Δ_{x,y},Δ_{y,z}},
where
and λ represents the wavelength of the signal, the superscript (.)^{T} is the transposition operator, ⊙ denotes Hadamard (elementwise) product, \(j=\sqrt {1}\), and
represent the direction cosines along the x, y, and zaxis, respectively.
Design of proposed array
Figure 2 demonstrates our proposed array configuration composed of an Lshaped sparse scalar array and a single triangular SSEMVS. The triangular SSEMVS is located at the origin and two scalar arrays are placed along the xaxis and yaxis, respectively. The antennas on the two arms of the Lshaped array are oriented differently, i.e., along with e_{x} and e_{z}, respectively. Each arm of the Lshaped array consists of two subarrays. Taking the arm along with the yaxis as an example, the first subarray, which consists of the first n_{1} dipoles (including the e_{x} in the triangular SSEMVS located at the origin), is placed with intersensor spacing D_{1}>Δ_{x,y}≫λ/2; the second subarray, which consists of the last n_{2} dipoles, is placed with an even larger intersensor spacing D_{2}=m_{1}D_{1}, where m_{1} is an integer. Futhermore, we can see that the first subarray and the triangular SSEMVS share a same e_{x}. Similarly, the arm of the Lshaped array along with the xaxis consists of two subarrays with intersensor spacings D_{3}>Δ_{y,z}≫λ/2 and D_{4}=m_{2}D_{3}, respectively, where m_{2} is an integer; the first subarray and the triangular SSEMVS share a same e_{z}. It should be noticed that the dipoles placed along the xaxis and the yaxis are of different orientations, and they are the same as the dipoles of the triangular SSEMVS along with the corresponding axis.
We take the scalar array placed along the yaxis as the example again to illustrate the design idea of the proposed array. The triangular SSEMVS can provide a coarse estimate of v by applying the vectorcrossproduct algorithm. And the estimation result can be used as a reference for solving the ambiguity problem of the v estimate from the first subarray in the scalar array. Therefore, the intersensor spacing of the first subarray can be larger than λ/2. The aperture of the first subarray is much larger than the interdipole/loop spacings of the triangular SSEMVS and thus we can obtain a finer estimate of the v with the first subarray. Similarly, the disambiguated estimation result of the first subarray can be adopted as the reference of the second subarray and finally a highaccuracy v estimation result is obtained. Similarly, we can use the same method for the scalar array placed along the xaxis to obtain a highaccuracy u estimation result. Finally, the highaccuracy angular estimation results can be calculated with the highaccuracy u and v estimations. Besides, the intersensor spacings of the two scalar arrays are much larger than λ/2, and thus the apertures and the angle estimation accuracy of the proposed array will be better than the Lshaped array with λ/2 intersensor spacing [31] and the Lshaped nested array [32] with λ/2 intersensor spacing in the first subarray. These probabilities will be verified in Section 4 through extensive simulation experiments. In addition, owing to that the scalar arrays are extended along xaxis and yaxis at the same time, 2D highaccuracy DOA estimations can be obtained simultaneously. This can not be reached by the multiscale EMVS array proposed in [21], where the multiscale aperture extension is only in one axis. Furthermore, only a single SSEMVS (along with scalar sensors) instead of many SSEMVSs are adopted in the proposed array, and thus the cost and redundancy of the array will be decreased dramatically.
Array manifold and signal model
The array manifold of the scalar array placed along the yaxis is
where ⊗ denotes the Kronecker product, a is defined in Eq. (1), a[1] is the first row of a, and thus \(\boldsymbol {a}_{y} \in {\mathbb C}^{N_{1}\times 1}\) with N_{1}=n_{1}+n_{2}.
Similarly, the array manifold of the scalar array placed along the xaxis is
where a[3] is the third row of a, and \(\boldsymbol {a}_{x} \in {\mathbb C}^{N_{2}\times 1}\) with N_{2}=n_{3}+n_{4}.
Following this, the array manifold of the proposed array is
where a_{y}[2:N_{1}] consists (N_{1}−1) rows in a_{y}, i.e., from the second row to last row of a_{y}, and a_{x}[2:N_{2}] consists (N_{2}−1) rows in a_{x}, i.e., from the second row to last row of a_{x}. Therefore, \(\boldsymbol {b} \in {\mathbb C}^{N\times 1}\) with N=N_{1}+N_{2}+4.
In a multiple sources scenario with K incident signals, the received data of the proposed sparse array at time t is
where \(\boldsymbol {b}_{k} \in {\mathbb C}^{N\times 1}\) represents the array manifold of the kth signal and \(\mathbf {B} = [\boldsymbol {b}_{1},\boldsymbol {b}_{2}, \dots, \boldsymbol {b}_{K}] \in {\mathbb C}^{N\times K}\). s(t)=[s_{1}(t),s_{2}(t),…,s_{K}(t)]^{T} denotes the incident signal vector, and n(t) signifies the additive white Gaussian noise.
Considering L time snapshots, we can form the received data matrix
The following task is to estimate the DOA of these K sources from \(\mathbf {X} \in {\mathbb C}^{N \times L}\), which will be described in detail below.
Procedure of multiscale DOA estimation algorithm
As described in Section 2.2, we can obtain multiple estimates of the direction cosines along yaxis and xaxis by the received data of the triangular SSEMVS and the two arms of the Lshaped array. However, some of the estimates are cyclically ambiguous and we will use the coarse estimates to disambiguate the ambiguous estimates step by step. The procedure of the entire algorithm is shown in Algorithm 1.
In the following, we give the detailed derivation and progress of the DOA estimation algorithm.
ESPRITbased method to estimate the two sets of highaccuracy but cyclically ambiguous v and two sets of highaccuracy but cyclically ambiguous u
The array covariance matrix can be calculated by the maximum likelihood estimation
where the superscript ^{H} is the Hermitian operator. Following [4], let \(\mathbf {E}_{s} \in {\mathbb C}^{N \times K}\) be the signal subspace matrix composed of the K eigenvectors corresponding to the K largest eigenvalues of \(\hat {\mathbf {R}}\). And E_{s} has the same signal subspace with the manifold matrix B and thus
where T denotes an unknown K×K nonsingular matrix. According to the composition of the proposed array, we divide the manifold matrix B into three parts, i.e., B_{1}, B_{y}, and B_{x}, where \(\mathbf {B}_{1} \in {\mathbb C}^{6 \times K}\) is composed of the top six rows of B (corresponding to the triangular SSEMVS), \(\mathbf {B}_{y} \in {\mathbb C}^{N_{1} \times K}\) is composed of the first row of B and (N_{1}−1) rows from the seventh row (corresponding to the senors on the yaxis), and \(\mathbf {B}_{x} \in {\mathbb C}^{N_{2} \times K}\) is composed of the third row of B and (N_{2}−1) rows from the (N_{1}+5)th row (corresponding to the senors on the xaxis). In this way, B_{1}, B_{y}, and B_{x} signify the manifold matrices of the SSEMVS and the two scalar arrays, respectively. Similarly, we can divide the signal subspace matrix E_{s} into three parts with the same method, i.e., \(\mathbf {E}_{s_{1}}\), \(\mathbf {E}_{s_{y}}\), and \(\mathbf {E}_{s_{x}}\). Thus, according to the relationship between array manifold matrix and signal subspace [33] described in Eq. (10), we have
After this, we deal with \(\mathbf {E}_{s_{y}}\) and \(\mathbf {E}_{s_{x}}\) separately to get two sets of the highaccuracy but cyclically ambiguous estimates of v and u. Let us take \(\mathbf {E}_{s_{y}}\) as an example to demonstrate the derivation. According to the different intersensor spacings of the scalar array whose unit is e_{x}, we divide the \(\mathbf {E}_{s_{y}}\) into two parts, i.e., \(\mathbf {E}_{s_{y,1}}\) and \(\mathbf {E}_{s_{y,2}}\), where \(\mathbf {E}_{s_{y,1}} \in {\mathbb C}^{n_{1} \times K}\) and \(\mathbf {E}_{s_{y,2}} \in {\mathbb C}^{n_{2} \times K}\) correspond to subarray 1 and subarray 2, respectively. Recalling Fig. 2, both subarray 1 and subarray 2 are uniform arrays, so the ESPRIT algorithm can be used to \(\mathbf {E}_{s_{y,1}}\) and \(\mathbf {E}_{s_{y,2}}\) to obtain two sets of the highaccuracy but cyclically ambiguous estimates of v, respectively. The process is consistent with that developed in [21]. Since the intersensor spacing D_{1} and D_{2} are both larger than λ/2, two sets of highaccuracy but cyclically ambiguous yaxis direction cosine estimations \(\hat {v}_{k}^{\text {fine,1}}\) and \(\hat {v}_{k}^{\text {fine,2}}\) can be derived.
In addition, it can be seen from [21, 34] that due to the same column permutation of T, these two sets of v estimations \(\{\hat {v}_{k}^{\text {fine, 1}}\}_{k=1}^{K}\) and \(\{\hat {v}_{k}^{\text {fine, 2}}\}_{k=1}^{K}\) are paired automatically.
Besides, we can obtain two sets of highaccuracy but cyclically ambiguous u estimations, \(\hat {u}_{k}^{\text {fine,1}}\) and \(\hat {u}_{k}^{\text {fine,2}}\) by applying similar process to \(\mathbf {E}_{s_{x}}\). And the u estimations, \(\hat {u}_{k}^{\text {fine,1}}\) and \(\hat {u}_{k}^{\text {fine,2}}\), are also paired automatically.
Vectorcrossproduct algorithm to estimate the unambiguous but lowaccuracy v and u and the relatively highaccuracy but ambiguous v and u
According to [18, 24], we need to get the estimate of the array manifold in order to apply the vectorcrossproduct algorithm to process the data of the triangle SSEMVS. And recalling Eq. (11), we can estimate the manifold matrix of the SSEMVS with
where \(\hat {\mathbf {B}}_{1} = \left [\hat {\boldsymbol {a}}_{1}, \dots, \hat {\boldsymbol {a}}_{K}\right ]\) and \(\hat {\boldsymbol {a}}_{k}\) is the estimation of array manifold of kth source at the triangular SSEMVS.
The following step is to apply the vectorcrossproduct algorithm to \(\hat {\boldsymbol {a}}_{k}\). For convenience, we set θ∈[0,π/2], ϕ∈[0,π/2), and we omit the source index k and recall Eq. (1), where we have \(\boldsymbol {a} = \left [ {\tilde {\boldsymbol {e}}}^{T}, {\tilde {\boldsymbol {h}}}^{T}\right ]^{T}\) with
According to the vectorcross product algorithm of the triangular SSEMVS [11], we have,
where × denotes the vectorcross product and p is calculated from \(\hat {\boldsymbol {a}}\).
From the Poynting vector of kth source p_{k} derived in Eq. (17), we can obtain the unambiguous but lowaccuracy estimations of {u_{k},v_{k},w_{k}} by
where [ ]_{i} extracts ith element of the vector inside [ ], and   denotes the absolute value of the entity inside  .
In the following, we estimate the relatively highaccuracy estimation of u and v from the displacement of the dipoles/loops within the triangular SSEMVS, i.e., Δ_{x,y} and Δ_{y,z}. From p, we can get
where ⊙ denotes the Hadamard (elementwise) product. Based on Eq. (19), we have one set of relatively highaccuracy but ambiguous estimations of u and v by
It is worth mentioning that the unambiguous but lowaccuracy estimations \(\{u_{k}^{\text {coarse}}\}_{k=1}^{K}\) and the relatively highaccuracy but ambiguous estimations \(\{\hat {u}_{k}^{\text {fine, 0}}\}_{k=1}^{K}\) are paired automatically, and due to the same T in Eq. (14), all u estimations have been paired, the same for v. Moreover, for θ and ϕ in other angular ranges, the changes are the plus or minus signs in Eqs. (18) and (19) [11].
Disambiguate the estimations of u and v and calculate the final estimates of θ and ϕ
As can be seen from the above Sections 3.1 and 3.2, for both u and v, there are three sets of highaccuracy but ambiguous estimations and one set of unambiguous but lowaccuracy estimation. The three sets of ambiguous estimations correspond to different levels of ambiguity, and a threeorder disambiguation method is utilized here.
We take v as the example to demonstrate the derivation and the process for u is similar. Recalling Fig. 2, we know the ambiguity of \(\hat {v}_{k}^{\text {fine,0}}\), \(\hat {v}_{k}^{\text {fine,1}}\), and \(\hat {v}_{k}^{\text {fine,2}}\) correspond to Δ_{x,y}, D_{1}, and D_{2}, respectively. And due to the fact that D_{2}>D_{1}>Δ_{x,y}, the order of solving ambiguity should be \(\hat {v}_{k}^{\text {fine,0}}\), \(\hat {v}_{k}^{\text {fine,1}}\), \(\hat {v}_{k}^{\text {fine,2}}\) step by step.
Disambiguate \(\hat {v}_{k}^{\text {fine,0}}\) with \(v_{k}^{\text {coarse}}\)
With \(v_{k}^{\text {coarse}}\) as the reference value, the ambiguity of \(\hat {v}_{k}^{\text {fine,0}}\) is solved, and the result can be obtained by
where \(\left \lceil \left (1\hat {v}_{k}^{\text {fine,0}}\right)\frac {\Delta _{x,y}}{\lambda } \right \rceil \le l_{1} \le \left \lfloor \left (1\hat {v}_{k}^{\text {fine,0}}\right) \frac {\Delta _{x,y}}{\lambda }\right \rfloor \) with ⌈ε⌉ denoting the smallest integer not less than ε and ⌊ε⌋ referring to the largest integer not more than ε [35].
Disambiguate \(\hat {v}_{k}^{\text {fine,1}}\) with \(v_{k}^{\text {fine, 0}}\)
With \(v_{k}^{\text {fine, 0}}\) as the reference value, the ambiguity of \(\hat {v}_{k}^{\text {fine,1}}\) is solved, and the result can be obtained by
where \(\left \lceil \left (1\hat {v}_{k}^{\text {fine,1}}\right)\frac {D_{1}}{\lambda } \right \rceil \le l_{2} \le \left \lfloor \left (1\hat {v}_{k}^{\text {fine,1}}\right) \frac {D_{1}}{\lambda }\right \rfloor \).
Disambiguate \(\hat {v}_{k}^{\text {fine,2}}\) with \(v_{k}^{\text {fine, 1}}\)
Finally, we can disambiguate \(\hat {v}_{k}^{\text {fine,2}}\) with \(v_{k}^{\text {fine, 1}}\) derived above to estimate the final highaccuracy and unambiguous estimation of \(v_{k}^{\text {final}}\):
where \(\left \lceil \left (1\hat {v}_{k}^{\text {fine,2}}\right)\frac {D_{2}}{\lambda } \right \rceil \le l_{3} \le \left \lfloor \left (1\hat {v}_{k}^{\text {fine,2}}\right) \frac {D_{2}}{\lambda }\right \rfloor \).
Similar to the above three steps, we can get the final highaccuracy and unambiguous estimation of \(u_{k}^{\text {final}}\) by replacing {Δ_{x,y},D_{1},D_{2}} with {Δ_{y,z},D_{3},D_{4}}, respectively.
After getting the unambiguous and highaccuracy estimation of {u,v}, we can get the highaccuracy DOA estimation of kth source by (3) and the results are
Analysis of the three intersensor spacings
Larger intersensor spacing brings in larger aperture and further leads to higher direction estimation accuracy. At the same time, it makes the disambiguation more difficult. There is a threshold in the process of disambiguation [36]. When the intersensor spacing value is larger than the threshold, the probability of successful disambiguation will break down. Therefore, we analyze the threshold of the intersensor spacing by analyzing the success probability of the disambiguation process.
Let us take v as an example to demonstrate the derivation again. According to the proposed array configuration shown in Fig. 2, there are three scales, i.e., {Δ_{x,y},D_{1},D_{2}} for v. Thus, we utilize the threeorder disambiguation process in Section 3.3 to obtain v^{final}. Take the Δ_{x,y} as an example, recalling Eq. (23), only by satisfying the following equation
can the disambiguation process be successful. The value of \(\left v_{k}^{\text {ref}}  v_{k}^{\text {coarse}}\right \) is the estimation error of the \(v_{k}^{\text {coarse}}\). We hereby assume that the angle estimation error follows a Gaussian distribution [37]. According to the distribution function of the normal process [38], the probability of the sample error falling into the scope of 3σ is about 99.85%, where σ is the standard deviation of the samples. Thus, when the root mean square error (RMSE) of \(v_{k}^{\text {coarse}}\) satisfies
we consider that the disambiguation process is successful. Therefore, we can calculate the threshold of Δ_{x,y} by
We can obtain the threshold of D_{1} and D_{2} using the similar method. Furthermore, considering the practical applications, we can only obtain the CramérRao bounds (CRB) of each parameter rather than RMSE. Thus, we can substitute the RMSE of \(v_{k}^{\text {fine,0}}\) and \(v_{k}^{\text {fine,1}}\) with their CRB to calculate the thresholds of D_{1} and D_{2}. However, due to the CRB is much less than the RMSE, the calculated values of thresholds of D_{1} and D_{2} will be far larger than the actual values. And this probility will be verified in Section 4.
Similar to v, we can obtain the corresponding thresholds of {Δ_{y,z},D_{3},D_{4}} for u.
Owing to the fact that RMSE is related to signaltonoise ratio (SNR), the snapshot number and the source direction, we will analyze the influence of these factorsin Section 4.
The derivation of the CRB for the new array is similar to that in [21], and we will use the corresponding equations therein to derive the CRB in the following simulations.
Simulation results and discussion
In this section, we conduct simulations to verify the effectiveness and performance of the proposed array geometry and algorithm. For simplicity, we set θ∈[0,π/2], ϕ∈[0,π/2). The coordinate of the h_{y} of the SSEMVS is (x_{h},y_{h},z_{h})=(7.5λ,7.5λ,5λ). The RMSE of parameter estimation is defined as
where \(\hat {\alpha }_{m}\) is the estimation of mth trial of parameter α, and M is the number of Monte Carlo trials. We assume that the number of sources is known a priori in the following simulations.
Parameter estimation results
In the first example, we consider that there are N_{1}=12e_{x}’s placed along the yaxis direction and N_{2}=12e_{z}’s placed along the xaxis direction. The first six e_{x}’s compose the subarray 1 with intersensor spacing D_{1}=35λ; the rest of the e_{x}’s constitute the subarray 2 with intersensor spacing D_{2}=7D_{1}=245λ. Besides, the first six e_{z}’s compose the subarray 3 with intersensor spacing D_{3}=35λ; the rest of the e_{z}’s constitute the subarray 4 with intersensor spacing D_{4}=7D_{3}=245λ. For the triangular SSEMVS, Δ_{x,y}=Δ_{y,z}=5λ. There are K=2 puretone incident sources with unit power, which have the numerical frequency f=(0.537,0.233), elevation θ=(42^{∘},35^{∘}), azimuth ϕ=(55^{∘},52^{∘}), the auxiliary polarization angle γ=(36^{∘},60^{∘}), and the polarization phase difference η=(80^{∘},70^{∘}) impinging on the array. The number of snapshots is L=200 and SNR = 10 dB. The noise is a complex Gaussian white noise vector with zero mean and covariance matrix σ^{2}I. Figure 3 shows the estimation results of the proposed algorithm with 200 Monte Carlo trials. We can see that the spatial parameters of all targets are correctly paired and estimated.
Parameter estimation performance
In order to further exploit the performance of the proposed array, we hereby conduct various simulations with different parameters of the array and sources.
Performance versus SNR
In the first example, we consider the performance of parameter estimation versus SNR. Figure 4 a shows the RMSE of all estimates of u, i.e., u^{fine,0}, u^{fine,1}, and u^{final} of the proposed array versus SNR compared with u^{coarse} and the CRB. Figure 4 b shows the RMSE of all estimates of v, i.e., v^{fine,0}, v^{fine,1}, and v^{final} of the proposed array versus SNR compared with v^{coarse} and the CRB. It can be observed that both u^{final} and v^{final} improve significantly from their coarse estimates, u^{coarse} and v^{coarse}, respectively; both of them are getting closer to their CRB. Moreover, the disambiguation described in Section 3.3 is similar to that of dualsize ESPRIT [4]. There exists a SNR threshold in the process of disambiguation [39]. The parameter estimation performance will be degraded significantly if the SNR is lower than the threshold. When SNR is larger than this threshold, the performance improves dramatically, and the performance is getting better with the increase of SNR. From Fig. 4, we can see that the SNR threshold of u and v are 7 dB and 6 dB, respectively.
In addition, we compare the proposed array with the array configuration in [32] which has the same number of scalar sensors, and the array configuration in [21] which has the same number of SSEMVSs in yaxis. Figure 5 a and b shows the RMSE of u and v estimates versus SNR for all three arrays, respectively. Comparing with the 2D nested scalar array in [32], the proposed array has a much larger aperture extension and lower mutual coupling; comparing with the linear multiscale SSEMVS array in [21], the proposed array has a much larger aperture extension in xaxis. We can observe from Fig. 5 a that the performance of the proposed array of u estimation is higher than those of the two other arrays when SNR is larger than the threshold. That is because the array aperture of the proposed array in xaxis is much larger than the two other arrays. From Fig. 5 b, we can observe that the performance of the proposed array of v estimation is a little worse than that of the array configuration in [21]. However, the SNR threshold of the proposed array is far smaller (7 dB).
Moreover, we consider another configuration of the proposed array in which the array is extended along yaxis and zaxis, respectively. And the triangular SSEMVS of this configuration is placed along yaxis and zaxis, respectively. The DOA estimation process of this array is similar to that of the proposed array, except that the corresponding direction cosines change from u and v to v and w. Using the same simulation conditions as in Section 4.1, we compare the parameter estimation performance of this array configuration with the proposed array. The results are given in Fig. 6. We can observe that RMSEs of u of these two configurations are similar, and the same behavior happens for v of the proposed array and w of another configuration. But we still can see that the accuracy of the proposed array are marginally better than the other configuration when SNR is large enough, i.e., > 8 dB.
As the arriving angle estimation is determined by u and v jointly, in Fig. 7, we show the RMSEs of the estimated θ and ϕ of all array configurations versus SNR and the CRB of the proposed array. It can be seen that the SNR threshold of θ and ϕ of the proposed array are both 7 dB, the lowest one of the threshold of u and v. Moreover, the performance of the proposed array is the best in all array configurations. Therefore, our proposed array is a good tradeoff of mutual coupling, estimation accuracy, and robustness (lower SNR threshold) to noise.
Performance versus snapshot number
In the next example, we consider the performance of DOA estimation versus snapshot number L. Figure 8 a and b show the RMSEs of θ and ϕ estimation of all array configurations versus L at SNR =10 dB, respectively. We can see that the parameter estimation performance of the proposed array improves with the increase of snapshots, and the performance of the proposed array is the best among all array configurations once again.
Performance versus intersensor spacing
In the third example, we consider the performance of parameter estimation versus intersensor spacings. We take one target as an example, and set Δ_{x,y}=Δ_{y,z} in the SSEMVS with SNR =10 dB. The elevation of the target is θ=35^{∘}, azimuth ϕ=52^{∘}, the auxiliary polarization angle γ=36^{∘}, and the polarization phase difference η=80^{∘}. As mentioned in Section 3.4, there is a threshold of intersensor spacing. Figure 9 shows the RMSE of u^{fine,0} and v^{fine,0} of the proposed array versus Δ_{x,y}. Recalling Eq. (31), we can obtain the threshold of Δ_{y,z} at SNR=10 dB is \(\Delta _{y,z}^{t}=7.15\lambda \). From Fig. 9, the threshold of Δ_{y,z} is approximately 6λ. Thus, according to the obtained threshold and practical applications, we set Δ_{y,z}=5λ. The same method can be performed for Δ_{x,y}, \(\Delta _{x,y}^{t}=8.04\lambda \), and the threshold of Δ_{x,y} is approximately 8λ from Fig. 9. Therefore, the method derived in Section 3.4 for calculating the thresholds of different intersensor spacings is effective. Similar to Δ_{y,z}, we set Δ_{x,y}=5λ.
In the second simulation, we set D_{1}=D_{3}, and the RMSE of u^{fine,1} and u^{fine,1} of the proposed array versus D_{1} is shown in Fig. 10. Similar to Δ_{y,z}, we can calculate the threshold of D_{1} and D_{3}. But there is little different, as mentioned in Section 3.4, we utilize the CRB of u^{fine,0} and v^{fine,0} instead of the RMSE to calculate the threshold of D_{1} and D_{3}. And the calculated value are \(D_{1}^{t}=161.5\lambda \) and \(D_{3}^{t}=197.5\lambda \). From Fig. 10, we can obtain these threshold values (\(D_{1}^{t}=76\lambda \), \(D_{3}^{t}=72\lambda \)). We know that CRB is much smaller than RMSE. Therefore, we should set D_{1} and D_{3} much smaller than the calculated threshold values. Thereby, we set D_{1}=D_{3}=35λ.
In the third simulation, we set D_{2}=D_{4} and plot the RMSE of u^{final} and u^{final} of the proposed array versus D_{2} in Fig. 11. Similar to D_{1}, we can calculate the threshold of D_{2} and D_{4}, \(D_{2}^{t}=8047.6\lambda \) and \(D_{4}^{t}=7857.9\lambda,\) by CRB. From Fig. 11, we can obtain these threshold values \(\left (D_{2}^{t}=2500\lambda, D_{4}^{t}=2300\lambda \right)\). Again, we should set D_{2} much smaller than the calculated threshold values, and considering the practical applications, we set D_{2}=D_{4}=245λ.
Threshold of intersensor spacing versus SNR
We investigate the threshold of intersensor spacing versus SNR. Take one target as example, and we set the elevation of the target as θ=35^{∘}, azimuth ϕ=52^{∘}, the auxiliary polarization angle γ=36^{∘}, and the polarization phase difference η=80^{∘}. Other simulation conditions remain the same with Section 4.1. Figure 12 shows the thresholds of Δ_{y,z} and Δ_{x,y} versus SNR. It is seen that the thresholds of Δ_{y,z} and Δ_{x,y} both increases as SNR increases. And the thresholds of {D_{1},D_{3}} and {D_{2},D_{4}} are shown in Figs. 13 and 14, respectively. The results are similar to Fig. 12.
Threshold of SNR versus arriving angle
Lastly, we consider the threshold of SNR in the disambiguation process versus the signal arriving angle. Take one target as the example, we set the auxiliary polarization angle of the target γ=36^{∘} and the polarization phase difference η=80^{∘}. We set another angle equals 45^{∘} when we analyze one angle. Other simulation conditions remain the same as in Section 4.1. Figure 15 shows the threshold of SNR of u and v versus θ and ϕ. We can see that the threshold of SNR is approximately symmetrical with 90^{∘} for θ and symmetrical with 0^{∘} for ϕ. As we set θ∈[0,π/2] and ϕ∈[0,π/2), the threshold of SNR is in a lower range when the target is located in θ∈[20^{∘},70^{∘}] and ϕ∈[20^{∘},70^{∘}].
Conclusions
In this paper, a new array configuration composed of multiple sparse scalar arrays and a single triangle electromagneticvectorsensor is proposed, which enjoys the superiorities of both the spatially spread electromagneticvectorsensor and the sparse array. The new array can provide four direction cosine estimates with gradually improved accuracies, which are along the xaxis and yaxis, respectively. Based on this, we developed the algorithm for directionofarrival estimation, which utilizes the approach of threeorder disambiguation. We have analyzed the thresholds of the intersensor spacings in the four uniform scalar subarrays and conducted extensive simulations to validate them. We compare the performance of the direction cosine estimations of our array with the 2D nested scalar array and the linear multiscale SSEMVS array. These results demonstrated that our proposed array geometry enjoys the optimal tradeoff on estimation accuracy, mutual coupling, and robustness to noise. Moreover, since only a single SSEMVS is used with other scalar sensors, the proposed array has achieved a good performance with small redundancy, less elements, and low cost.
Abbreviations
 2D:

Twodimensional
 DOA:

Directionofarrival
 DOF:

Degreeoffreedom
 CRB:

CramérRao bounds
 EMVS:

Electromagneticvectorsensor
 ESPRIT:

Estimation of signal parameters via rotation invariant technique
 RMSE:

Root mean square error
 SSEMVS:

Spatially spread electromagneticvectorsensor
 ULAs:

Uniform linear arrays
References
 1
H. L. Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory (John Wiley & Sons, New York, 2004).
 2
X. Yuan, Directionfinding wideband linear fm sources with triangular arrays. IEEE Trans. Aerosp. Electron. Syst.48(3), 2416–2425 (2012).
 3
K. T. Wong, M. D. Zoltowski, in Proceedings of the 39th Midwest Symposium on Circuits and Systems. Sparse Array Aperture Extension with DualSize Spatial Invariances for ESPRITbased Direction Finding (IEEEIA, 1996), pp. 691–6942.
 4
V. I. Vasylyshyn, in European Radar Conference, 2005. Unitary ESPRITbased DOA estimation using sparse linear dual size spatial invariance array (IEEEParis, 2005), pp. 157–160.
 5
A. L. Swindlehurst, B. Ottersten, R. Roy, T. Kailath, Multiple invariance ESPRIT. IEEE Trans. Signal Process.40(4), 867–881 (1992).
 6
A. Moffet, Minimumredundancy Linear Arrays. Antennas Propag. IEEE Trans.16(2), 172–175 (1968).
 7
P. Pal, P. P. Vaidyanathan, Nested arrays: A novel approach to array processing with enhanced degrees of freedom. Signal Process. IEEE Trans.58(8), 4167–4181 (2010).
 8
P. P. Vaidyanathan, P. Pal, Sparse sensing With coprime samplers and arrays. Signal Process. IEEE Trans.59(2), 573–586 (2011).
 9
A. Nehorai, P. Tichavsky, Crossproduct algorithms for source tracking using an EM vector sensor. IEEE Trans. Signal Process.47(10), 2863–2867 (1999).
 10
X. Yuan, Estimating the DOA and the polarization of a polynomialphase signal using a single polarized vectorsensor. IEEE Trans. Signal Process.60(3), 1270–1282 (2012).
 11
X. Yuan, Diversely polarized antennaarray signal processing. PhD thesis (The Hong Kong Polytechnic University, 2012).
 12
X. Yuan, in 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Polynomialphase signal source tracking using an electromagnetic vectorsensor (Kyoto, 2012), pp. 2577–2580.
 13
X. Yuan, K. T. Wong, Z. Xu, K. Agrawal, Various compositions to form a triad of collocated dipoles/loops, for direction finding and polarization estimation. IEEE Sensors J.12(6), 1763–1771 (2012).
 14
X. Yuan, Quad compositions of collocated dipoles and loops: For direction finding and polarization estimation. IEEE Antennas Wirel. Propag. Lett.11:, 1044–1047 (2012).
 15
X. Yuan, K. T. Wong, K. Agrawal, Polarization estimation with a dipoledipole pair, a dipoleloop pair, or a looploop pair of various orientations. IEEE Trans. Antennas Propag.60(5), 2442–2452 (2012).
 16
Z. Xu, X. Yuan, CramerRao bounds of angleofarrival and polarisation estimation for various triads. IET Microw. Antennas Propag.6(15), 1651–1664 (2012).
 17
K. T. Wong, M. D. Zoltowski, Univectorsensor ESPRIT for multisource azimuth, elevation, and polarization estimation. IEEE Trans. Antennas Propag.45(10), 1467–1474 (1997).
 18
K. T. Wong, X. Yuan, ‘Vector crossproduct directionfinding’ with an electromagnetic vectorsensor of six orthogonally oriented but spatially noncollocating dipoles/loops. IEEE Trans. Signal Process.59(1), 160–171 (2011).
 19
X. Yuan, in 2011 IEEE Statistical Signal Processing Workshop (SSP). CramerRao bound of the directionofarrival estimation using a spatially spread electromagnetic vectorsensor (Nice, 2011), pp. 1–4.
 20
X. Yuan, Spatially spread dipole/loop quads/quints: For direction finding and polarization estimation. IEEE Antennas Wirel. Propag. Lett.12:, 1081–1084 (2013).
 21
M. Yang, J. Ding, B. Chen, X. Yuan, A multiscale sparse array of spatially spread electromagneticvectorsensors for direction finding and polarization estimation. IEEE Access. 6:, 9807–9818 (2018).
 22
X. Yuan, Coherent sources direction finding and polarization estimation with various compositions of spatially spread polarized antenna arrays. Signal Process.102:, 265–281 (2014).
 23
F. Luo, X. Yuan, Enhanced ‘vectorcrossproduct’ directionfinding using a constrained sparse triangulararray. EURASIP J. Adv. Signal Process.2012(1) (2012). https://doi.org/10.1186/168761802012115.
 24
Y. Li, J. Q. Zhang, An enumerative nonlinear programming approach to direction finding with a general spatially spread electromagnetic vector sensor array. Signal Process.93(4), 856–865 (2013).
 25
X. Yuan, Spatially spread dipole/loop quads/quints: For direction finding and polarization estimation. IEEE Antennas Wirel. Propag. Lett.12:, 1081–1084 (2013).
 26
M. Ji, X. Gong, Q. Lin, in 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD). A multiset approach for direction finding based on spatially displaced electromagnetic vectorsensors (IEEEZhangjiajie, 2015), pp. 1824–1828.
 27
K. Han, A. Nehorai, Nested vectorsensor array processing via tensor modeling. IEEE Trans. Signal Process.62(10), 2542–2553 (2014).
 28
J. He, Z. Zhang, T. Shu, W. Yu, Direction finding of multiple partially polarized signals with a nested crossdiople array. IEEE Antennas Wirel. Propag. Lett.16:, 1679–1682 (2017).
 29
S. Rao, S. P. Chepuri, G. Leus, in 2015 IEEE 6th International Workshop on Computational Advances in MultiSensor Adaptive Processing (CAMSAP). DOA estimation using sparse vector sensor arrays (IEEECancun, 2015), pp. 333–336.
 30
X. Yuan, Coherent source directionfinding using a sparselydistributed acoustic vectorsensor array. IEEE Trans. Aerosp. Electron. Syst.48:, 2710–2715 (2012).
 31
J. Liang, D. Liu, Joint elevation and azimuth direction finding using Lshaped array. IEEE Trans. Antennas Propag.58(6), 2136–2141 (2010).
 32
C. Niu, Y. Zhang, J. Guo, Interlaced doubleprecision 2D angle estimation algorithm using Lshaped nested arrays. IEEE Signal Process. Lett.23(4), 522–526 (2016).
 33
R. Roy, T. Kailath, ESPRITEstimation of signal parameters via rotational invariance techniques. Acoust. Speech Signal Process. IEEE Trans.37(7), 984–995 (1989).
 34
J. Li, M. Shen, D. Jiang, in 2016 IEEE 5th AsiaPacific Conference on Antennas and Propagation (APCAP). DOA estimation based on combined ESPRIT for coprime array (IEEEKaohsiung, 2016), pp. 117–118.
 35
V. I. Vasylyshyn, in First European Radar Conference, 2004. EURAD. Closedform DOA estimation with multiscale unitary ESPRIT algorithm (IEEEAmsterdam, 2004), pp. 317–320.
 36
F. Athley, Threshold region performance of maximum likelihood direction of arrival estimators. IEEE Trans. Signal Process.53(4), 1359–1373 (2005).
 37
B. Ottersten, M. Viberg, T. Kailath, Performance analysis of the total least squares ESPRIT algorithm. IEEE Trans. Signal Process.39(5), 1122–1135 (1991).
 38
D. C. Montgomery, G. C. Runger, Applied Statistics and Probability for Engineers (John Wiley & Sons, New York, 2010).
 39
C. D. Richmond, MeanSquared Error and Threshold SNR Prediction of MaximumLikelihood Signal Parameter Estimation With Estimated Colored Noise Covariances. IEEE Trans. Inf. Theory. 52(5), 2146–2164 (2006).
Acknowledgements
This work is partly supported by the National Natural Science Foundation of China under Grant 61571344, in part by the Foundation of Shanghai Academy of Spaceflight Technology under Grant SAST2016093 and in part by the Fund for Foreign Scholars in University Research and Teaching Programs the 111 Project under Grant B18039.
Author information
Affiliations
Contributions
JD, MLY, BXC, and XY conceived and designed the study. JD and MLY performed the experiments. JD and MLY wrote the paper. MLY, BXC, and XY reviewed and edited the manuscript. All authors read and approved the manuscript.
Corresponding author
Correspondence to Minglei Yang.
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
Keywords
 Multiple scale arrays
 Electromagneticvectorsensor
 Directionofarrival (DOA) estimation
 Sparse array