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Directionofarrival of strictly noncircular sources based on weighted mixednorm minimization
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 225 (2018)
Abstract
In this paper, a superresolution directionofarrival (DoA) algorithm for strictly noncircular sources is introduced. The proposed algorithm is based on subspaceweighted mixednorm minimization. Firstly, we augment the array aperture for efficiently exploiting the noncircularity of signal source. Then, we transform the augmented array matrix to the real array matrix due to the centroHermitian of the augmented array matrix. To this end, a subspaceweighted mixednorm minimization problem is formulated for the DoA estimation. In the proposed algorithm, we utilize singular value decomposition (SVD) to reduce the dimension of matrix, which improves the computational efficiency. We design the weighted scheme by utilizing the orthogonality of the noise subspace and the array manifold dictionary, which increases the reliability of the sparse DoA estimation. As shown by simulations, the proposed algorithm outperforms the stateoftheart algorithms in difficult scenarios, such as low signaltonoise ratio, small snapshots, and correlated source. Moreover, the proposed algorithm exhibits a superior performance for the DoA estimation in the underdetermined case.
1 Introduction
Direction finding has gained great interest in array signal processing field over the past decades, which is widely used in radar, underwater acoustics, wireless communication, and seismic [1–3]. In this area, most of the studies have assumed that the signal follows complex circular Gaussian distribution, such as multiple signal classification (MUSIC) [4], estimation of signal parameters via rotational invariance technique (ESPRIT) [5]. However, this is not an accurate assumption; in practical application systems, many signals follow complex noncircular Gaussian distribution, such as binary phase shift keying (BPSK), Mary amplitude shift keying (MASK), and binary pulse amplitude modulated (PAM). Therefore, how to exploit the secondorder noncircularity of complex signals to improve the angle estimation performance becomes an urgent issue in signal processing [6–8].
So far, a large number of subspacebased parameter estimation algorithm, which exploited the noncircular property of the signal, had been proposed in the literatures, for example, noncircular multiple signal classification (NCMUSIC) [9], polynomial rooting NCMUSIC (NCRootMUSIC) [10], fourthorder NCRootMUSIC (NCRootFOMUSIC) [11], and unitary ESPRIT for noncircular sources (NCunitaryESPRIT) [12], which aim to increase degree of freedom (DoF) and improve angular estimation accuracy. For example, the work in [13] utilized the conjugation information of the partial received signal to extend the virtual aperture as well as the joint virtual array, by using a forward spatial smoothing technique in order to handle the coherent source. The main challenge for the superresolution directionofarrival (DoA) estimation is to resolve the closely spaced sources with smallsize sample, and even to resolve the coherent source. Compared with the conventional highresolution estimation algorithms, source localization algorithms which based on sparse signal recovery (SSR), showed the predominant performance in the estimation accuracy [14–16]. Specifically, the authors in [17] proposed an l_{1}SVD algorithm which uses singular value decomposition (SVD) of the measurements to reduce the data dimension. The authors in [18] presented a subspaceweighted l_{2,1}SVD algorithm which exploits the MUSIClike spectrum to design the weighted matrix. Furthermore, the authors in [19] proposed a unitary subspaceweighted l_{2,1}SVD algorithm by applying the spatial smoothing technique, which doubles the number of snapshots and benefits the DoA estimation accuracy. Moreover, by employing unitary transformation, the unitary l_{2,1}SVD algorithm puts the sparse constraint on realdata matrix and further reduces the computational complexity. For the past two decades, the noncircular SSRbased algorithms have attracted a great number of researchers’ attention, [20–22], and by exploiting the noncircular property of the signal sources, the performance of the DoA estimation algorithms can be effectively improved.
In this paper, we propose a weighted subspace mixednorm DoA estimation algorithm for noncircular signal. The algorithm makes the most of the noncircularity of the sources and formulates the DoA estimation as a weighted subspace mixednorm minimization problem. The offered improvement mainly displays in two aspects: For one thing, the algorithm takes advantage of the strictly noncircular sources, enhances the DoF to two times of the number of sensors, which increases the number of detectable sources and decreases the estimation error. For another, the subspace weighting scheme exploits the relationship between the overcomplete dictionary and the noise subspace, which improves the reliability of weighted coefficient and further enforces the sparsity of the signal. Results verify that our proposed algorithm provides superior performance than l_{1}SVD and SW l_{2,1}SVD algorithms under small snapshots and low SNR regime. Moreover, the proposed algorithm offers the ability to estimate DoAs in underdetermined conditions.
The rest of the paper is organized as follows. In Section 2, we introduce the system model and problem description. In Section 3, we present a weighted subspace mixednorm DoA estimation algorithm. In Section 4, we provide the detailed experimental simulations and discussion. Finally, we conclude the whole paper in Section 6.
Notation: A vector and a matrix are denoted by a and A, respectively. For a matrix D, D^{∗} denotes the conjugate, D^{T} and D^{H} account for transpose and conjugate transpose, respectively, [D]_{i,k} is the ith row and the kth column element in matrix D, D_{i,·} is the elements of the ith row in D. ∥D∥_{F} accounts for the Frobenius norm, \(\phantom {\dot {i}\!}[\textbf {u}]^{(l_{2})}\) stands for a vector u whose ith entries equals to the l_{2}norm of its ith row. I_{M} denotes an M×M identity matrix. Π_{M} accounts for the M×M exchange matrix with elements 1_{s} in its antidiagonal and zeros elsewhere, and diag(u) represents a diagonal matrix, whose diagonal elements consist of the vector u.
2 System model and problem description
2.1 System model
Consider an uniform linear array (ULA) with M isotropic elements, whose interelement space is halfwavelength d=λ/2 and λ accounts for the carrier wavelength. K independent farfield narrowband sources impinge on the ULA s_{k}(t) from the distinct directions θ_{1},θ_{2},⋯,θ_{K}, as depicted in Fig. 1. Therefore, the received signal array vector y(t) is given by
in which A(θ)=[a(θ_{1}),a(θ_{2}),⋯,a(θ_{K})] represents array manifold matrix with M×K dimension, and \(\textbf {a}(\theta _{k})=\left [1,e^{j 2{\pi } \frac {d}{\lambda }\sin (\theta _{k}) },\cdots,e^{j2{\pi }(M1) \frac {d}{\lambda }\sin (\theta _{k})}\right ]^{T}\) accounts for an array steering vector with M×1 dimension, s(t)=[s_{1}(t),⋯,s_{K}(t)]^{T} is the noncircular incident signal vector with M×1 dimension, and \(\textbf {n}(t)=[n_{1}(t),n_{2}(t),\cdots,n_{M}(t)]^{T} \in \mathbb {C}^{M \times 1} \) is the additive noise vector, whose entries follow the circularly symmetric complex Gaussian distribution (CSCG) with zero mean and variance \(\sigma _{n}^{2}\) [23–25].
For convenience, the sign t is omitted and N snapshots are collected, then the model of (1) can be expressed as a matrix form, which holds that
in which Y=[y(1),⋯,y(N)] is a received matrix of K×N dimension, and the source symbol matrix S and the noise matrix N are defined the same way as Y.
When the received signal is a strictly secondorder noncircular signal, the complex symbol amplitudes of each received signal locate on a rotated line in the complex plane. According to this property, the signal S is denoted as an array form as follows:
where S_{r} accounts for a realvalued symbol vector of K×1 dimension, and \(\pmb \Phi = \text {diag}\left \{ \left [e^{j\phi _{1}},e^{j\phi _{2}},\cdots,e^{j\phi _{K}}\right ]\right \}\), ϕ_{k}∈[0^{∘},180∘], and ϕ is the complex phase shift, called noncircular phase [26, 27]. Here, we just consider the strictly secondorder noncircular signal case with the noncircular phase is zero.
2.2 Problem description
The conventional SSRbased algorithm for source localization depends on the array model in (2). In order to describe the DoA estimation problem through a sparse representation framework, an overcomplete dictionary \( \textbf {A} (\tilde {\pmb \theta })\) with the M×P dimension needs to be constructed by discretizing the spatial angle range [−90∘,90∘] at the angular grid \( \tilde {\pmb \theta } = \left [{\tilde \theta }_{1}, {\tilde \theta }_{2},\cdots,{\tilde \theta }_{P} \right ]^{T}\), P is the number of atoms in dictionary. Assuming that the angle grid point is dense enough, in general P≫K, such that the true angle exactly lies in each grid. Then, the P×N dimensional rowsparse signal matrix \( \tilde {\textbf {S}} =[\tilde {\textbf {s}}_{1}, {\tilde {\textbf {s}}}_{2},\cdots, {\tilde {\textbf {s}}}_{P}]^{T}\) can be defined as:
where the kth row corresponds to the signal comes from a source at \( {\tilde \theta }_{k} \). Obviously, when the rows of \(\tilde {\textbf {S}}\) are nonzero elements, the real DoA information can be attained. The system model in (2) can be formulated as a sparse representation problem
Under the sparse representation framework, the matrix \(\tilde {\textbf {S}}\) in (5) can be solved by using the sparse constrained minimization problem, which satisfied the following expression
where
and δ(∥X_{i,·}∥_{2}>0) is an indication function,
η is an error fitting bound and \(\eta ^{2} \geq \ \textbf {N} \_{F}^{2}\). Here, we select the upper value of ∥N∥_{2} with a 99% confidence interval as the value of η.
Unfortunately, the problem in (6) is an NPhard problem and the optimal solution can be found only with an exponential complexity. However, we can arrive at a suboptimal solution through a simpler way, by using a convex relaxation technique, which replaces the l_{2,0} norm by its closest convex surrogate l_{2,1} norm. In this case, (6) can be transformed to
where
This is a l_{2,1} mixednorm which defined by the Eq. (10). The problem in (9) can be solved through standard convex optimization techniques.
Nevertheless, the SSRbased DoA algorithm in (10) does not take into account the noncircular property of the signal. It is necessary to make a deeper study on how to effectively exploit the statistic property of the sources to improve the estimation performance of the SSRbased DoA algorithm.
3 The proposed algorithm
In this section, we present a SSRbased DoA estimation algorithm which is based on weighted mixednorm minimization. The algorithm mainly include three steps: the first step is augmenting array matrix and spatial smoothing processing, the second step is realvalue matrix transformation and reducing the dimension of the matrix via SVD, and the last step is utilizing the weighted mixednorm minimization to estimate the DoAs.
3.1 Augmented array aperture and spatial smoothing processing
In order to take full use of noncircular property of signal sources, we stack the received measurement matrix of the array and its complex conjugate counterpart and construct an augmented matrix \(\textbf {Y}^{(nc)}\in \mathbb {C}^{2M\times N}\) as:
which can be simplified as
Under the above consideration, we consider the first element on the left of the array as the phase reference. According to the centersymmetric characteristic of the ULAs [28], it holds that
where Δ_{A} accounts for a unitary diagonal matrix, which is related to the phase reference of the array, whose nonzero elements consist of the last row of A. Combined with the formula (13), the array manifold matrix of augmented array can be further simplified as
Herein, we consider the case that the noncircular phase is zero, then (14) can be converted to
and (12) can be represented as:
We observe that the B^{(nc)}(θ) has the centersymmetric characteristic, so B^{(nc)}(θ) holds that
in which Δ_{B} is a unitary diagonal matrix, in which the definition is the same as the definition of Δ_{A}.
Furthermore, incorporating the spatial smoothing technique and with the help of identical formula in (17), the new measurement Y^{(nc)} can be converted to a centroHermitian matrix \({\textbf {Z}}\in \mathbb {C}^{2M\times 2N}\) as:
where
and
3.2 Unitary transformation and SVD
Aiming to reduce the computation complexity, we will utilize the centroHermitian of Z to convert the complex data to real data by unitary transformation. The received realdata matrix can be constructed as:
Here, we define the evenorder unitary matrix as
It is worth noting that the matrix \(\textbf {Z}^{r} \in \mathbb {R}^{{2M} \times K}\) is the realvalue matrix due to the centroHermitian property of the matrix Z, whose unitary transformation is real matrix.
Furthermore, in order to derive a reduced 2M×K dimensional signal space, we represent the dominant component by using the K largest singular vectors of the matrix Z^{r}, which is corresponding to the signal subspace. We perform SVD operation on the matrix Z^{r} results to
where Σ is singular value matrix with the dimension the 2M×N, U and V are called left and right singular vector matrix for Z^{r}, respectively, which are the orthogonal matrixes. The singular vectors of matrix U corresponding to nonzero singular value form the signal subspace U_{s}. Ideally, the number of nonzero singular value of matrix Z^{r} is K. The 2M−K singular vectors of matrix U corresponding to the smaller singular values form the noise subspace U_{n}.
Let
where D_{K}=[I_{K},0]^{T} with I_{K} being the identity matrix of K×K dimension and 0 being the zero matrix of K×(N−K) dimension. Also, we define \(\textbf {S}_{SV} =\hat {\textbf {S}}_{0}\textbf {Q}_{2T} \textbf {V}_{s}\) and \(\textbf {N}_{SV} =\textbf {Q}_{2M}^{H} {\textbf {N}^{(nc)}}\textbf {Q}_{2T} \hat {\textbf {V}}_{s}\); we can obtain
Now, we can apply the sparse recovery framework to this model that reduces the dimension of the equation.
3.3 DoA estimation based on weighted mixednorm minimization
The same as the discussion in Section 2.2, the array manifold dictionary \( D_{\bar {\theta }} =\left [\textbf {B}^{(nc)}\left (\bar {\theta }_{1}\right),\cdots, \textbf {B}^{(nc)}\left (\bar {\theta }_{L}\right)\right ]\) can be constructed by extending the B^{(nc)}(θ) on spatial angular grid \(\left \{\bar {\theta }_{1},\bar {\theta }_{l},\cdots, \bar {\theta }_{L}\right \}\: (L \gg K)\). Now, the DoA estimation can be reformulated as a sparse signal recovery problem:
where \(\textbf {D}^{(nc)} = \textbf {Q}_{2M}^{H} \textbf {D}\), β is a new error fitting bound, \(\beta ^{2} \geq \\hat {\textbf {N}}_{SV} \_{F}^{2}\) which can be determined distribution with χ^{2} degrees of freedom 2M×K and 99% probability [17].
As mentioned above, the l_{2,1} norm is used in (26) as an approximation to l_{2,0} norm; therefore, the solution is suboptimal. In order to increase the accuracy of this solution, we adopt the subspaceweighted strategy. Specifically, in D^{(nc)}, there are K steering vectors which corresponded to the actual targets. Based on the orthogonality between the steering vector D^{(nc)} and the noise subspace U_{n}, for the true target DoA, θ_{k} holds that \( \left [{\textbf {D}^{(nc)}}^{H} \textbf {U}_{n} \right ]^{l_{2}} \rightarrow 0\) when N→∞. Then, the weighted vector is given by:
Also, the vector w can be divided into the following two parts:
where w^{s} corresponds to the true DoAs, and w^{n} consists of the remaining elements of w. The reweighted vector can be written as w=[w_{1},w_{2},⋯,w_{2M}]^{T}, with w_{i} being the reweighted coefficient, and i=1,2,⋯,2M. The values in w^{s} and w^{n} satisfy w^{s}(k)→0 and 0<w^{n}(j)≤1, respectively, and w^{s}(k)<w^{n}(j).
Now, we can formulate a new weighted l_{2,1} mixednorm minimization for the DoA estimation as:
where
w_{l} accounts for the lth element of w. It is worth noting that (29) is treated as a secondorder cone program (SOCP) problem, which is efficiently checked by utilizing standard software packages, as the CVX [29], which is used in this paper.
Once we attain the estimation of \(\hat {\textbf {S}}_{SV}\), the spatial spectrum can be caculated by averaging the rows of \(\hat {\textbf {S}}_{SV}\) (i.e., the solution of (29))
The angular information \(\hat {\boldsymbol {\theta }}\) can be estimated by searching the nonzero rows of \(\hat {\textbf {S}}_{SV}\), which corresponding to the K peaks of \(\hat {\textbf {p}}\). The proposed algorithm is summarized in Algorithm 1.
3.4 Computational complexity analysis
For the subspacebased algorithm, the computational complexity of MUSIC is \(\mathcal {O}\left (M^{2} N +M^{2} L\right)\), whose main computational cost is the spectral search. The computational complexity of NCMUSIC is \(\mathcal {O}\left (4M^{2} N + 16M^{3} 4M^{2} K + 4\left (M^{2} + 2M\right)L^{2}\right)\). For the l_{1}SVD algorithm, from the conclusion in [17], we know that the main computational cost is the sparse recovery process, which solves (9) via the SOCP, the computational complexity of is \(\mathcal {O}(K L)^{3}\). For the SW l_{2,1}SVD algorithm, we consider the computation cost in the formulation of the weighted matrix w, which requires \( \mathcal {O}\left (M^{2} N + M^{3} + L M(MK)\right)\). We note that the l_{1}SVD algorithm is realized that complex multiplication costs four times as much as that of real multiplication [12]; in the proposed algorithm, we transform the complexvalued problem into a realvalued one by the realvalued transformation, and the computational complexity is reduced by a quarter, which means that the computational cost of the sparse recovery process in (29) can be decreased as \(\mathcal {O}\left (\frac {1}{4} K L\right)^{3}\). In the proposed algorithm, we also use the weighted strategy; due to aperture expansion, the computation cost of the formulation of the weighted matrix w requires \( \mathcal {O}((2M)^{2} N + (2M)^{3} + 2LM(2MK))\). In Table 1, we have given the computational complexity of the subspacebased algorithm and the sparsebased algorithm. From Table 1, we observe that the computational complexity of the proposed algorithm is lower than other sparsebased algorithms. Although the proposed algorithm is higher than the subspacebased algorithms, it can work in the smallsize snapshots and the underdetermined case.
4 Experimental simulations and discussion
This section provides the MonteCarlo simulations to validate the efficiency of the subspaceweighted mixednorm algorithm and further compares the performance with the MUSIC [4], NCMUSIC [9], l_{1}SVD [17], and SW l_{2,1}SVD [18]. Experimental simulations are performed by Intel(R) Core(TM) i77700 CPU @ 3.60 GHz, RAM 8G, MATLAB 2017a.
In simulations, an ULA with number M=10 of isotropic sensors is considered, and the interelement spacing is halfwavelength. The noncircular signal is BPSK modulation signal. For each snapshot and individual signal at the sensor, the input SNR is defined as \(\textrm {SNR}= 10\log _{10} \left ({ \\textbf {S}\_{F}^{2}} /{ \ \textbf {N}\_{F}^{2}}\ \right)\). In all algorithm, the angle grid is uniform which divided from − 90 to 90∘ with step interval 0.1∘. Unless otherwise specified, N=200 snapshots are collected.
To evaluate the DOA estimation performance, we use the rootmeansquare error (RMSE) as the performance indicator, which is defined as:
where \(\hat {\theta }_{i}\) and θ_{i} represent the estimated and true DoA of the ith signal in the qth trial, respectively. Q accounts for the number of MonteCarlo trials. All simulation results derived through Q=500 MonteCarlo trials.
We evaluate the DoA estimation performance of the proposed algorithm in the underdetermined case. In this simulation, 11 farfield narrowband BPSK signals uniformly distribute between − 55 to 55∘. The SNR is fixed to 0 dB. From Fig. 2, it can be seen that the proposed algorithm can work well in underdetermined DOA estimation, i.e., the proposed algorithm is capable of handling more sources than sensors. In general, we assume that the number of sources is smaller than the number of sensors in the subspacebased algorithms. When the number of sensors is M, the subspacebased algorithm at most identifies the M 1 source, that is incapable of resolving more sources than sensors. From Fig. 2, we can conclude that the proposed algorithm presents better performance when the number of sources is greater than the number of sensors. The reason is that the proposed algorithm does not only increase the degree of freedom by exploiting the noncircular property of the signal source, but also employ the weighted scheme which utilizes the relationship of the noise subspace and array manifold dictionary and further enhances the reliability of the sparse DoA estimation.
Figure 3 shows the RMSE of DoA estimation against the SNR for different algorithms. Three uncorrelated narrowband BPSKmodulated signals impinge on the ULA from [ − 10∘, 0∘, 8∘]. From Fig. 3, it is observed that the proposed algorithm provides the optimal performance for the DoA estimation compared with the other algorithms since the proposed algorithm takes full advantage of the DoF of augmented array, which benefit from noncircular signal, and the proposed algorithm exploits the forward/backward spatial smoothing to improve the robust of the algorithm. We also observe that the proposed algorithm outperforms than SW l_{1}SVD when the SNR is less than 0 dB; the reason for the improvement of the DoA estimation is benefited from the aperture extension and subspace weighting.
Figure 4 shows the RMSE versus the snapshots with the different algorithms. Three uncorrelated narrowband BPSKmodulated signals are located [ − 10∘, 0∘, 8∘]. The SNR is fixed to 0 dB, and we vary the number of snapshots from 20 to 200 with the step interval 20. From Fig. 4, it can be seen that the proposed algorithm achieves better performance of angle estimation compared to other algorithms for all snapshots, even in the smallsize snapshot case. This indicates that the proposed algorithm improves the DoA estimation performance for different snapshots.
Figure 5 depicts the RMSE of DoA estimation performance as a function of the angular separation between two targets. Two uncorrelated equal power BPSK sources is located at θ_{1}=−10∘ and θ_{1}=−10∘+△θ, where △θ varies from 2 to 20∘. From Fig. 5, it is shown that the performance of DoA estimation is improved with the angular separation larger compared to MUSIC, NCMUSIC, l_{1}SVD, and SW l_{2,1}SVD. For closely space targets, the proposed algorithm has the minimum RMSE, which indicates that the proposed algorithm can achieve higher spatial angle resolution than the other algorithms.
Figure 6 displays the RMSE of DoA estimation performance as a function of correlation coefficient, where two correlated narrowband BPSKmodulated signals are located [ − 10∘, 8∘], the SNR is fixed to 0 dB, and the snapshot number is 100. As we can see from Fig. 6, the proposed method exhibits the best estimation performance among all the algorithms. The performance of the subspacebased algorithms, such as MUSIC and NCMUSIC, are degrading with the correlation coefficient increasing; the performance of the other sparsebased algorithms like the l_{1}SVD and SW l_{2,1}SVD also are affected by correlation coefficient, but the performance of the sparsebased algorithms are better than that of the subspacebased algorithms. As indicated in Fig. 6, the proposed algorithm have the best estimation performance among all the algorithms. The reason is that the proposed algorithm utilizes the spatialsmoothing process to mitigate the impact of correlated source; in addition, the sparsebased algorithm also is robust to the correlated source.
5 Methods
A weighted subspace mixednorm minimization DoA estimation algorithm is proposed for strictly noncircular signal source. The proposed algorithm employs the noncircular property of signal source to extend the array aperture. Then, the spatial smoothing technique and the unitary transformation are implemented to handle the correlate source and convert the complex value matrix to realvalue matrix. Furthermore, reducing the matrix dimension process is employed by SVD. In the end, the DoA estimation formulated a weighted subspace mixnorm minimization problem. The SOCP is employed to solve the convex optimization problem.
6 Conclusions
A superresolution SSRbased DoA estimation algorithm was proposed for strictly noncircular sources in this paper, which is based on weighted subspace mixednorm minimization. The proposed algorithm increased the degree of array freedom by exploiting the noncircular property of the signal, transformed the array data to realvalue domain by utilizing the centroHermitian of the augmented matrix, and reduced the dimension of matrix by SVD operation which effectively lower the computational complexity. Last but not the least, we exploited the weighted strategy to enhance the liability of the sparse DoA estimation. The results showed that the proposed algorithm achieved the optimal performance of DoA estimation in the angular resolution and estimation accuracy and can well work in the underdetermined case. In the future, we will incorporate some other wireless communication techniques such as [30–34] to further improve the performance of the 5G mobile communication system.
Abbreviations
 5G:

Fifth generation
 BPSK:

binary phase shift keying
 CSCG:

Circularly symmetric complex Gaussian distribution
 DoF:

Degree of freedom
 DoA:

Directionofarrival
 ESPRIT:

Estimation of signal parameters via rotational invariance technique
 MASK:

Mary amplitude shift keying
 NCMUSIC:

Noncircular multiple signal classification
 NCRootMUSIC:

Polynomial rooting NCMUSIC
 NCRootFOMUSIC:

Fourthorder NCRootMUSIC
 NCunitaryESPRIT:

Unitary estimation of ESPRIT for noncircular sources
 PAM:

Binary pulse amplitude modulated
 RMSE:

Rootmeansquare error
 SNR:

Signaltonoise ratio
 SSR:

Sparse signal recovery
 SVD:

Singular value decomposition
 SOCP:

Secondorder cone program
 ULA:

Uniform linear array
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Funding
This work was supported in part by the National Natural Science Foundation of China under grants 61671378 and 61801132, the Natural Science Foundation of Guangdong Province of China under grant 2018A030310338, and the Project of Educational Commission of Guangdong Province of China under grant 2017KQNCX155.
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Mostly, the writing material was extracted from different journals as presented in the references. A MATLAB tool has been used to simulate the concept.
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WT designed the idea, performed the experiments, and wrote the manuscript. XF gave valuable suggestions on the structuring of the paper, XY assisted in processing the data, and LJ assisted in the revising and proofreading of the manuscript. All authors read and approved the final version of the manuscript.
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Tan, W., Feng, X., Ye, X. et al. Directionofarrival of strictly noncircular sources based on weighted mixednorm minimization. J Wireless Com Network 2018, 225 (2018). https://doi.org/10.1186/s136380181234y
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DOI: https://doi.org/10.1186/s136380181234y
Keywords
 DoA estimation
 Uniform linear array
 Subspaceweighted mixednorm minimization
 Strictly noncircular source