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2D DOA estimation method based on coprime array MIMO radar
EURASIP Journal on Advances in Signal Processing volumeÂ 2021, ArticleÂ number:Â 72 (2021)
Abstract
Aiming at the problem that traditional direction of arrival (DOA) estimation methods cannot handle multiple sources with high accuracy while increasing the degrees of freedom (DOF), a new method for 2D DOA estimation based on coprime array MIMO radar (SAMIMOCA) is proposed. First of all, in order to ensure the accuracy of multisource estimation when the number of elements is finite, a new coprime array model based on MIMO (MIMOCA) is proposed. This method is based on a new MIMO arraybased coprime array model (MIMOCA), which improves the accuracy of multisource estimation when the number of array elements is limited, and obtains a larger array aperture with a smaller number of array elements, and improves the estimation accuracy of 2D DOA. Finally, the effectiveness and reliability of the proposed SMMIMOCA method in improving the DOF of array and DOA accuracy are verified by experiments.
1 Introduction
The traditional direction of arrival (DOA) method generally uses a uniform linear array, and the number of estimable target sources is less than the number of array elements. Classical methods such as multiple signal classification (MUSIC) method [1,2,3] or estimation of signal parameters via rotational invariance techniques (ESPRIT) method [4, 5] use N array elements to estimate at most N1 target signals, and the degree of freedom of the array is limited. Therefore, in the case of a certain number of array elements, how to optimize the array structure to obtain a larger array aperture to improve the DOA estimation accuracy and multitarget resolution has always been a hot issue for scholars [6,7,8].
In recent years, with the continuous indepth study of the array element structure, domestic and foreign scholars have proposed many nonuniform array structures [9,10,11]. For example, the nested array structure can estimate up to 2N signal sources by using N array elements. The nested array is not only easy to construct, but also easy to obtain the specific position of the array element and higher array freedom. However, because the distance between some elements in the nested array is very small, mutual coupling between the antennas will be caused, thereby affecting the performance of the array DOA estimation. With the appearance of the coprime array structure, the effect of mutual coupling between antennas in nested arrays has been solved by it. At the same time, the degree of freedom of the array is greatly improved by increasing the difference information of the array element position [12,13,14,15]. Although the traditional DOA method can easily be extended to two dimensions to deal with a planar array or a circular array, the traditional DOA method uses a parallel uniform linear array composed of several linear subarrays, which leads to a problem of large computational complexity. In order to improve the accuracy of twodimensional DOA estimation, antiinterference, etc., a multiinput multipleoutput (MIMO) radar is proposed. MIMO radar uses multiple antennas to transmit different waveforms and receive reflected signals from multiple targets. Therefore, it can achieve large degrees of freedom (DOF) based on waveform diversity, thereby improving spatial resolution, enhancing parameter identification, and improving target detection performance [16,17,18,19,20]. The method that Li et al. proposed [21, 22] combines MIMO and coprime array to estimate DOA, which improves the estimation performance of DOA. However, the uniform linear arrays were still used as coprime array to construct effective differential arrays with ideal characteristics, which can only provide onedimensional DOA estimation; as a result, the accuracy of twodimensional DOA estimation is not very high. Therefore, this paper proposes a new coprime array model based on MIMO (MIMOCA). To improve the accuracy of twodimensional DOA estimation, the transmitting array of the array combination is a special irregular array, and the receiving array is a uniform linear array.
Bautista and Buck et al. [23] proposed the use of compressed sensing for sparse matrix processing, which reduces the computational complexity of DOA estimation, but it is not used in the MIMO coprime array structure. So et al. [24] proposed a fast DOA estimation method with parallel uniform linear arrays, which constructs a subarray, but when there are many sources, additional matching is required, and the sensors are not fully utilized. Tayem et al. [25, 26] proposed a new array, that is, each of the three parallel subarrays is consistent, but it is impossible to detect more sources under the same number of array elements. Therefore, on the basis of the new array combination method (MIMOCA) proposed in this paper, it combines the methods of compressed sensing [27,28,29,30] and proposes a new twodimensional DOA method, that is, a new method for 2D DOA estimation based on coprime array MIMO radar (SAMIMOCA). First of all, this method can be realized by constructing an equivalent array of sparse array, that is, using the sparse array topology of virtual array elements to analyze a larger number of twodimensional DOA sources, and can automatically match the corresponding azimuth and elevation angles. Besides, by transforming the twodimensional DOA estimation into two independent onedimensional DOA estimation problems, only one variable can be estimated, thus reducing the computational complexity. Then, when the number of information sources is greater than or equal to the number of array elements, a virtual differential array is established, and sparse reconstruction and least squares operations are performed. The sparse matrix is processed through the compressed sensing method, so that \(M + 4\) array elements can identify \(2M^{2}\) sources. Finally, experiments verify the validity and reliability of SAMIMOCA for 2D DOA estimation.
The content of this paper is mainly structured as follows. SectionÂ 2 gives the array configuration and signal model of the SAMIMOCA method. SectionÂ 3 gives the DOA estimation method based on the sparse array. SectionÂ 4 gives the experimental results and analysis (Table 1).
2 Preliminaries
In this section, the array model and signal model of this article are mainly given.
2.1 Array model
As shown in Fig.Â 1, MIMO targets 4 transmit arrays and M receive arrays, that is, the total number of arrays is Mâ€‰+â€‰4. Due to the nature of the MIMO array model, the array can be virtualized so that the number of virtualized arrays is 4M. The array arrangement of the transmitting array is shown in Fig.Â 1, and the receiving array is M uniform linear arrays with a spacing of 2Md along the xaxis direction, where \(d = \lambda /2\) and \(\lambda\) is wavelength.
Due to the nature of the MIMO radar, the virtual array is shown in Fig.Â 2.
Due to the need to construct a coprime array, the last element of subarray 2 is discarded to form a coprime array, as shown in Fig.Â 3.
As shown in Fig.Â 3, the coprime array consists of three sparse linear uniform arrays. Subarray 1 has 2Â M array elements, and its array element spacing is Md; and subarrays 2 and 3 have M1 and M array elements, and its array element spacing is. The array element spacing d is \(\lambda /2\), where \(\lambda\) is the wavelength of the corresponding carrier frequency. By choosing \(M \in {\mathbb{N}}^{ + }\) and \(2M \in {\mathbb{N}}^{ + }\) to be relatively prime (where \({\mathbb{N}}^{ + }\) is expressed as a set of positive integers), the minimum cell spacing along the yaxis is \(\lambda /2\). This article assumes that the array sensor is located at
where \(2m \in [0,2M  1]\),Â \(m_{1} \in [1,M  1]\),Â \(m_{2} \in [0,M  1]\),Â \(n,m_{1} ,m_{2} \in {\mathbb{N}}^{ + }\), where (x,y) represents the coordinates in the xâ€‰âˆ’â€‰y plane. To make the distinction simple, let 2Mâ€‰=â€‰N. Then the array sensor is located at
2.2 Signal model
For the estimation of the onedimensional wave arrival angle direction, compared with the traditional coprime array, the main difference is that these subarrays are no longer collinear, and are placed in parallel at distances d and Ld, \(\left( {L \in {\mathbb{N}}^{ + } } \right)\), that is, the minimum unit spacing d along the xaxis. As L increases, the aperture of the array also increases, and the resolution also increases. But the larger the aperture, the signal will be correlated, so the value of L should not be very large.
The output is
where \(\theta_{k}\) and \(\varphi_{k}\) in Eq.Â (3) are the azimuth and elevation angles of the kth source, respectively. \(\otimes\) is expressed as the Kronecker product. \(n(l)\) is another noise vector whose elements are independently and evenly distributed in \((i,i,d)\) and obey the Gaussian distribution \(CN\left( {0,\sigma_{n}^{2} I_{{N_{t}^{i} }} } \right)\), where \(i = 1,2,3\). \(a_{t} (\theta_{k} ,\varphi_{k} ) = a_{ty} (\theta_{k} ,\varphi_{k} ) \otimes a_{tx} (\theta_{k} ,\varphi_{k} )\), \(a_{r} (\theta_{k} ,\varphi_{k} ) = a_{ry} (\theta_{k} ,\varphi_{k} ) \otimes a_{rx} (\theta_{k} ,\varphi_{k} )\), where \(a_{ty} (\theta_{k} ,\varphi_{k} )\) and \(a_{tx} (\theta_{k} ,\varphi_{k} )\) are the steering vectors of the transmitting array. At the same time, \(a_{t} (\theta_{k} ,\varphi_{k} ) \otimes a_{r} (\theta_{k} ,\varphi_{k} )\) corresponds to the Kronecker product of the receiving direction vector and the sending direction vector of the kth target. \(a_{ty} (\theta_{k} ,\varphi_{k} ) \otimes a_{tx} (\theta_{k} ,\varphi_{k} )\) and \(a_{ry} (\theta_{k} ,\varphi_{k} ) \otimes a_{rx} (\theta_{k} ,\varphi_{k} )\) are the same. Let \(a_{ti} (\theta_{q} ,\varphi_{q} ) \otimes a_{ri} (\theta_{q} ,\varphi_{q} ) = a_{i} (\theta_{q} ,\phi_{q} )\), suppose the relationship after the virtual is:
where
EquationÂ (5) represents \((\theta_{q} ,\phi_{q} )\) corresponding to the steering vector of the ith subarray, where \(q = 1, \ldots ,Q\),Â \(i = 1,2,3\). \(y_{j}^{i} ,1 \le j \le N_{t}^{i}\) is the y coordinate of the ith sensor. Where \(N_{t}^{i}\) is the total number of sensors in the ith subarray, that is, \(N_{t}^{1} = 2M\),Â \(N_{t}^{2} = M  1\),Â \(N_{t}^{3} = M\).Similarly, \(x_{i}\) represents the position of the ith subarray along the xaxis, and the noise vector element is in the ith subarray \(n_{i} (t)\), where \((i,i,d)\) is independently and uniformly distributed and obeys the Gaussian distribution \(CN\left( {0,\sigma_{n}^{2} I_{{N_{t}^{i} }} } \right)\), where \(i = 1,2,3\). In order to transform the twodimensional DOA estimation problem into two independent onedimensional problems, as shown in Fig.Â 4, \(\alpha_{q} ,\beta_{q} \in [0^{ \circ } ,180^{ \circ } ]\), where qâ€‰=â€‰1, â€¦, Q, respectively, are expressed as the angle between the incident direction and the yaxis and xaxis The relationship between \(\alpha_{q} ,\beta_{q}\) and \(\theta_{q} ,\phi_{q}\) is
Therefore, the data vector received in Eq.Â (4) is
The corresponding steering vector is
Set \(s(t) = [s_{1} (t), \ldots ,s_{Q} (t)]^{T}\) to the signal vector, \(A_{i} = [a_{i} (\alpha_{1} ), \ldots ,a_{i} (\alpha_{Q} )]\) is the array manifold corresponding to the ith subarray, where \(i = 1,2,3\), the data vector of the receiving channel can be written as
The diagonal matrix is expressed as
Although traditional methods can achieve highresolution DOA estimation, the \(Q < N_{t} Q\) conditions must be met to obtain the noise subspace. In application, the problem of detecting information sources with more than the number of array elements has become the focus of research. In this section, an effective method is proposed to achieve the equivalence of differential arrays with a larger number of DOF. In addition, the group sparse array technology is used to improve the estimation accuracy of DOA, and the differential covariance equations of \(x_{i} (t)\) and \(x_{k} (t)\) are constructed.
The crosscovariance matrix of the data vectors accepted by subarrays \(x_{i} (t)\) and \(x_{k} (t),1 \le i,k \le 3\) can be obtained. The crosscovariance matrix is
where \(R_{s} = E[s(t)s^{H} (t)] = {\text{diag}}([\sigma_{1}^{2} , \ldots ,\sigma_{Q}^{2} ])\) is the covariance matrix of the \(Q \times Q\) dimensional signal, and its diagonal term represents the scattered power of the signal. In addition,
When iâ€‰=â€‰k, it becomes the identity matrix.
The matrix \(R_{{x_{ik} }}\) is quantized to obtain the following measurement vector:
where
where \(\overline{a}_{ik} (\alpha_{q} ) = a_{i} (\alpha_{q} ) \otimes a_{k}^{*} (\alpha_{q} )\), \(1 \le q \le Q\), \(\left( \cdot \right)^{*}\) is denoted as conjugate. \(i = {\text{vec}}(I_{{N_{t}^{i} }} )\), using the van der Monte structure of vectors \(a_{i} (\alpha_{q} )\) and \(a_{k} (\alpha_{q} )\), the entry in \(\overline{a}_{ik} (\alpha_{q} )\) retains the \({\text{e}}^{j\pi (Mn  Nm)} \cos (\alpha_{q} )\) factor. Therefore, \(z_{ik}\) can be regarded as a data vector received from a single snapshot signal vector \(b_{ik}\), and the array manifold \(A_{ik}\) corresponds to a virtual array whose virtual elements are located in the selfhysteresis and crosslag between different subarray sets. Due to the relative prime properties of M and N, there are fewer redundant elements in these virtual arrays. Therefore, the degree of freedom in the common array is greatly increased, so that more sources of \(N_{t}\) can be estimated with fewer array elements.
3 Methods
On the basis of the array model and signal model proposed in Sect.Â 2, this section proposes a twodimensional DOA estimation method based on a sparse array to ensure the performance of fine processing of multiple sources while increasing the degree of freedom.
3.1 2D DOA estimation method for sparse array
Based on the MIMOCA array model and signal model mentioned in Sect.Â 2, the signal vector in Eq.Â (14), \(Z_{ik,} 1 \le i,k \le 3\), can be sparsely expressed on the entire discrete angle grid as
where \(\overline{A}^{ \circ }_{ik}\) is defined as the grid \(\alpha_{g} ,g = 1, \ldots ,G_{\alpha }\) where \(\overline{a}_{ik} (\alpha_{g} )\) is located, where \(G_{\alpha } \gg Q\); \(b_{ik}^{ \circ }\) is a sparse vector, and its nonzero entry position corresponds to the DOA estimated by \(\alpha_{q}\), where \(q = 1, \ldots ,Q\). For different subarrays, nonzero items usually have different values, but share the same position when searching. In other words, \(b_{ik}^{ \circ }\) exhibits a set of sparsity on all pairs of subarrays. Therefore, the estimation of \(\alpha_{q} ,q = 1, \ldots ,Q\), which can be solved in the sparse reconstruction framework [23], making full use of all the DOF of mutual lag and cross lag. Many effective methods in the framework of convex optimization [27, 28] and Bayesian sparse learning [29] can be used to solve the sparse reconstruction problem of complexvalued groups [29, 30]. In this paper, the complex multitask Bayesian compressed sensing method is introduced into the SAMIMOCA method, mainly because the method has superior performance and robustness to solve the coherence problem, as follows,
In order to use selflag and crosslag, this paper replanned the vector \(z_{ik}\),
The respective steering matrix of each vector is
The dimensionality of the unknown sparse vector is extended to \(\overline{b}_{ik}^{ \circ }\), and an additional element with a noise power of \(\sigma_{n}^{2}\) is required. In this case, use \(\overline{b}_{ik}^{ \circ }\) to estimate that the first \(G_{\alpha }\) is used to determine \(\alpha_{q}\), and discard the last element. In addition, the error vector \(\varepsilon_{ik}\) is included in (18) to illustrate the difference between the statistical expectation and the sample average when calculating the covariance matrix. The difference is modeled as \(i,i,d\), and since a sufficient number of samples are used in the averaging, a Gaussian complex number is produced.
Suppose the elements in \(\overline{b}_{ik}^{ \circ }\) come from the product of the following zeromean Gaussian distribution
where \(N\left( {xa,b} \right)\) means that the random variable x follows the Gaussian distribution and the complex Gaussian distribution of the mean a and variance b, respectively. \(\overline{b}_{ik}^{ \circ g} = [\tilde{b}_{ik}^{{ \circ g_{R} }} \tilde{b}_{ik}^{{ \circ g_{I} }} ]^{T}\) is a \(2 \times 1\) vector composed of the real part coefficient \(\tilde{b}_{ik}^{{ \circ g_{R} }}\) and the imaginary part coefficient \(\tilde{b}_{ik}^{{ \circ g_{I} }}\), corresponding to the gth grid. It can be easily determined that when \(\gamma_{g}\) is set to 0, \(\overline{b}_{ik}^{ \circ g}\) approaches zero [30,31,32,33]. To achieve the sparsity of \(\overline{b}_{ik}^{ \circ }\), a Gamma prior is set to \(\gamma_{g}^{  1} \sim{\text{Gamma}}(\gamma_{g}^{  1} a,b)\), where \({\text{Gamma}}(x^{  1} a,b) =\Gamma (a)^{  1} b^{a} x^{  (a  1)} {\text{e}}^{{  \frac{b}{x}}}\) and \(\Gamma ( \cdot )\) are \({\text{Gamma}}\) functions. \(a\) and b are hyperparameters. The vector \(\gamma = [\gamma_{1} , \ldots ,\gamma_{G} ]^{T}\) contains \(\overline{b}_{ik}^{ \circ g}\), where \(g = 1, \ldots ,G_{\alpha }\) is shared with all groups to enhance sparsity. Similarly, the Gaussian prior is \(N\left( {0,\xi_{0} I_{2} } \right)\), \(\varepsilon_{ik}\) is set, and the \({\text{Gamma}}\) prior is on \(\xi_{0}^{  1}\), with hyperparameters c and d. Defining the density function of the two \(G_{a} \times 1\) vectors \(\overline{b}_{ik}^{ \circ R} = [b_{ik}^{ \circ 1k} , \ldots ,b_{ik}^{ \circ GR} ]^{T}\) and \(\overline{b}_{ik}^{{ \circ_{I} }} = [b_{ik}^{ \circ 1I} , \ldots ,b_{ik}^{ \circ GI} ]^{T}\) associated as \(\overline{b}_{ik}^{{ \circ_{RI} }} = [(\overline{b}_{ik}^{{ \circ_{R} }} )^{T} ,(\overline{b}_{ik}^{{ \circ_{I} }} )^{T} ]^{T}\) can be evaluated as
where
Obviously, when \(\gamma\) and \(\xi_{0}\) are given, (23) and (24) can be used to derive the mean and variance of each scattering system in \(\overline{b}_{ik}^{ \circ RI}\). On the other hand, the values of \(\gamma\) and \(\xi_{0}\) are determined by maximizing the logarithm of the edge likelihood, which can be achieved by the expectation maximization method to produce:
where \(\mu_{ik,g}\) and \(\mu_{{ik,g + G_{\alpha } }}\) are the gth and \(g + G_{\alpha }\) th elements of the \(\mu_{ik}\) vector, and \(\Sigma_{ik,gg}\) and \(\Sigma_{{ik,(g + G_{\alpha } )(g + G_{\alpha } }} )\) are the \((g,g)\) and \((g + G_{\alpha } ,g + G_{\alpha } )\) th elements in the matrix \(\Sigma_{ik}\), because \(\gamma\) and \(\xi_{0}\) depend on \(\mu_{ik}\) and \(\Sigma_{ik}\). Since CMTBCS is iterative, iterate between (22)â€“(24) and (27)â€“(28) until the convergence criterion is reached. The estimated value \(\hat{\alpha }_{q} ,q = 1, \ldots ,Q\) can obtain the maximum value of Q in \(\sum\nolimits_{i,k = 1}^{3} {\left( {b_{ik}^{ \circ gR} + b_{ik}^{ \circ gI} } \right),g = 1, \ldots ,G}\). Then the \(Q \times 1\) vector in (14), that is, \(b_{ik} ,i \ne k\) can be estimated by least squares fitting, expressed as
where
Therefore, \(\beta_{q} ,q = 1, \ldots ,Q\) estimates to the following form
where \(\hat{b}_{q}\) is the qth element of vector \(\hat{b}_{ik}\), so \(\hat{\beta }_{q}\) automatically matches \(\hat{\alpha }_{q}\). \(\hat{\alpha }_{q}\) can be obtained in the same way. Therefore, according to Eqs.Â (6) and (7), \(\hat{\theta }_{q}\) and \(\hat{\phi }_{q}\) can be obtained.
3.2 Steps of the SAMIMOCA method
The steps of the SAMIMOCA method are as follows:

Step1 Construct the MIMO array model to obtain the MIMO coprime array model in Fig.Â 1.

Step2 To obtain a coprime array, discard the last element of the second row of the virtual array, and get Eq.Â (3).

Step3 Use Eqs.Â (6) and (7) to turn the twodimensional problem into two independent onedimensional DOA estimation problems.

Step4 Use Eq.Â (12) to obtain the crosscovariance matrix \(R_{{x_{ik} }}\) of the signal.

Step5 Use Eq.Â (14) to obtain the quantized measurement vector \(z_{ik}\).

Step6 Use Eq.Â (18) to obtain the replanned \(z_{ik}\).

Step8 The twodimensional DOA estimation of the \(N_{t}\) signals are obtained by matching Eqs.Â (32) and (33) (Fig.Â 5).
4 Results and discussion
This section mainly gives the experimental results and analysis. In this section, first of all, it is verified through experiments that the method in this paper is effective and reliable in improving the degree of freedom. Besides, it is verified through experiments that this article is effective and reliable in improving the performance of 2D DOA estimation.
4.1 Degree of freedom analysis
In the case of onedimensional, the obtained coarray is equivalent to the traditional coprime array, that is, the number of estimated signals can reach: \(Q_{av} = MN\), that is, \(Q_{av} = M^{2}\). For a given number of physical antennas \(N_{t} = 2M + N  1 = 4M  1\), \(Q_{av}\) can be obtained in the following way
Obviously, the effective optimal coprime pair is that 2M and N are as equal as possible, that is, the array in this paper is selected as the optimal number of array elements. In this case, the maximum number of estimated signals \(Q_{av}\) is
As shown in Fig.Â 6, although the value of \(Q_{\max }\) of all methods increases with the increase of \(N_{t}\), it is obvious that the method based on the coprime array (SAMIMOCA method, TDSRCS method) is significantly better than other methods method. When \(N_{t} > 6\), the method based on the relative prime array can resolve more sources than the number of other array sensors. For other methods, the number of resolvable sources is less than the number of sensors.
4.2 2D DOA estimation performance comparison
4.2.1 The relationship between SNR and mean square error
Compare the SAMIMOCA proposed in this article with the TPAUL method, TDULPM method, TDSRCS method and PULRARE method to verify twodimensional DOA estimation performance of SAMIMOCA [26, 34,35,36]. Perform 100 Monte Carlo simulations for each method, and define the root mean square error as
where I is the number of Monte Carlo experiments, and Q is the number of sources. We set Mâ€‰=â€‰4; that is, the array configuration of the \(N_{t} = 4M  1 = 15\) antenna. In addition, let Lâ€‰=â€‰20. Assume that Q farfield sources with the same power are on the elevation plane \((\theta_{q} ,\phi_{q} )\), where \(\theta_{q} \in [0^{ \circ } ,90^{ \circ } ]\), \(\phi_{q} \in [  90^{ \circ } ,90^{ \circ } ]\), \(q = 1, \ldots ,Q\). The grid interval in the angular space is set to \(0.1^{ \circ }\), and the parameter in the Bayesian sparse learning group is set to \(a = b = c = d = 0\).
FigureÂ 7 shows the SAMIMOCA method and TPAUL method, TDULPM method, TDSRCS method, and PULRARE method when the number of sources Qâ€‰=â€‰3 and the number of snapshots Tâ€‰=â€‰500 The estimated performance is compared, and the root mean square error (RMSE) changes of the method under different signaltonoise ratio (SNR) are investigated. At the elevation angle, the performance of SAMIMOCA and PULRARE methods is close, but at the azimuth angle, when the SNR is 0, the performance of SAMIMOCA is improved by about 47.1% compared with the PULRARE method. Compared to the TPAUL method, it has increased by about 61.5%. By comparing RMSE under different SNR, it is concluded that SAMIMOCA has better estimation performance than several other methods under low SNR. Specific data are shown in Tables 2 and 3.
In different SNR comparisons, the lower the RMSE, the higher the resolution. It can be seen from Tables 2 and 3 and Fig.Â 7 that compared with other methods, the SAMIMOCA method has lower RMSE than other methods. Therefore, the resolution of the SAMIMOCA method is better in different SNR situations.
4.2.2 The relationship between angle and number of snapshots
FigureÂ 8 shows that the SAMIMOCA method, TPAUL method, TDULPM method, TDSRCS method and PULRARE method are used to estimate the position when the number of sources Qâ€‰=â€‰3 and \({\text{SNR}} = 0\). The performance comparison of each method, the comparison of RMSE under different snapshots. The overall performance of the SAMIMOCA method is improved by about 44.5% compared to the TPAUL method and about 23.4% compared with the PULRARE method. Experimental results show that SAMIMOCA performs better than TPAUL method and PULRARE method under different snapshots.
In different snapshots comparisons, the lower the RMSE, the higher the resolution. It can be seen from Table 4 and Fig. 8 that compared with other methods, the SAMIMOCA method has lower RMSE than other methods. Therefore, the resolution of the SAMIMOCA method is better in different SNR situations.
4.2.3 Comparison of 2D DOA estimation
In the above two sets of experiments, the performance of the method at low signal sources was tested, and the comparison of different SNR and the mean square error of different snapshots proved the superiority of the method. Next, conduct a multisource experiment. There are Q sources, the number of sources is greater than the number of arrays, the SNR is kept to 0, and the number of snapshots is set to 500, compared with the TDSRCS method, as shown in Fig.Â 9.
FigureÂ 9 shows the DOA estimation performance of each method, each of which represents the SAMIMOCA method in this paper, the actual value and the TDSRCS method when the SNRâ€‰=â€‰0, and the number of snapshots is 500. The DOA estimation result can be intuitively seen from Fig.Â 9 that SAMIMOCA is closer to the actual angle and has better twodimensional DOA estimation performance.
4.2.4 Complexity analysis
Because SAMIMOCA method and TDSRCS method can estimate more sources under the condition of finite array elements, while other methods can estimate less than SAMIMOCA method and TDSRCS method under the same conditions as SAMIMOCA method and TDSRCS method. The complexity of SAMIMOCA method and TDSRCS method is compared. The complexity of TDSRCS method is \(O\left( {NM^{2} + 2M^{3} + \left( {M^{3} + M^{2} } \right)Q + M^{3} Q^{3} + L^{3} } \right)\). The complexity of SAMIMOCA method is \(O\left( {\left( {4M  1} \right)^{2} + 2n^{3} + 9Q^{2} } \right)\). The time required for them is compared as shown in Fig.Â 10, from which it can be seen that SAMIMOCA method requires less time than TDSRCS method.
5 Conclusion
Aiming at the problem that traditional array signal processing methods cannot handle multiple sources with high accuracy while increasing the degree of freedom, this paper proposes a new twodimensional DOA estimation method based on MIMO radar coprime array. This method mainly uses the characteristics of coprime arrays and MIMO radars, which combines the theory of compressed sensing to improve the degree of freedom and accuracy of DOA estimation. Through experimental verification, compared with TPAUL method, TDULPM method, TDSRCS method and PULRARE method, this method can effectively distinguish more signal sources. Whatâ€™s more, it has high twodimensional DOA estimation accuracy and improves the degree of freedom of twodimensional DOA estimation. Compared with TDSRCS method, SAMIMOCA method reduces a certain amount of calculation. In the future, the processing of coherent sources and noncircular signals will be continue studied. The combination of Doppler frequency shift and angle measurement to further reduce the amount of calculation for target positioning will be considered. The applicability to colored noise will be considered.
Availability of data and materials
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Abbreviations
 DOA:

Direction of arrival
 MUSIC:

Multiple signal classification
 ESPRIT:

Estimation of signal parameters via rotational invariance technique
 SNR:

Signaltonoise ratio
 RMSE:

The root mean square error
 MIMO:

Multipleinput multipleoutput
 MIMOCA:

Coprime array model based on MIMO
 SAMIMOCA:

The method of 2D DOA estimation based on coprime array MIMO radar
 TDULPM:

Improved twodimensional DOA estimation algorithm for twoparallel uniform linear arrays using propagator
 TPAUL:

An improved 2D DOA estimation algorithm for threeparallel ULAs
 PULRARE:

A rankreductionbased 2D DOA estimation algorithm for three parallel uniform linear arrays
 TDSRCS:

A SRbased 2D DOA estimation algorithm using coprime array
References
R. Schmidt, Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag. 34(3), 276â€“280 (1986). https://doi.org/10.1109/TAP.1986.1143830
D. Ciuonzo, G. Romano, R. Solimene, Performance analysis of timereversal MUSIC. IEEE Trans. Signal Process. 63(10), 2650â€“2662 (2015). https://doi.org/10.1109/TSP.2015.2417507
D. Ciuonzo, On timereversal imaging by statistical testing. IEEE Signal Process. Lett. 24(7), 1024â€“1028 (2017). https://doi.org/10.1109/LSP.2017.2704612
R. Roy, T. Kailath, ESPRITestimation of signal parameters via rotational in variance techniques. IEEE Trans. Acoust. Speech Signal Process. 37(7), 984â€“995 (1986). https://doi.org/10.1117/12.55606
P. Ma, J. Li, F. Xu, X. Zhang, Holefree coprime array for DOA estimation: augmented uniform coarray. IEEE Signal Process. Lett. 28, 36â€“40 (2021). https://doi.org/10.1109/LSP.2020.3044019
P. Pal, P.P. Vaidyanathan, Nested arrays: a novel approach to array processing with enhanced degrees of freedom. IEEE Trans. Signal Process. 58(8), 4167â€“4181 (2010). https://doi.org/10.1109/TSP.2010.2049264
X. Li, J. Chen, W. Tan, W. Yang, Gridless DOA estimation method for monostatic MIMO array base on covariance matrix reconstruction. Syst. Eng. Electron. 42(5), 969â€“977 (2019)
S. Qin, Y.D. Zhang, M.G. Amin, Improved twodimensional DOA estimation using parallel coprime arrays. Signal Process. 172(Jul.), 107428.1107428.9 (2020). https://doi.org/10.1016/j.sigpro.2019.107428
C. Liu, P.P. Vaidyanathan, Super nested arrays: linear sparse arrays with reduced mutual couplingâ€”part I: fundamentals. IEEE Trans. Signal Process. 64(15), 3997â€“4012 (2016). https://doi.org/10.1109/TSP.2016.2558159
C. Liu, P.P. Vaidyanathan, Super nested arrays: linear sparse arrays with reduced mutual couplingâ€”part II: highorder extensions. IEEE Trans. Signal Process. 64(16), 4203â€“4217 (2016). https://doi.org/10.1109/TSP.2016.2558159
J. Liu, Y. Zhang, Y. Lu, S. Ren, S. Cao, Augmented nested arrays with enhanced DOF and reduced mutual coupling. IEEE Trans. Signal Process. 65(21), 5549â€“5563 (2017). https://doi.org/10.1109/tsp.2017.2736493
G. Wang, Z. Fei, S. Ren, 2D DOA estimation exploiting vertical synthetic planar arrays. IEEE Access 9, 3497â€“3507 (2021). https://doi.org/10.1109/ACCESS.2020.3047686
Z. Zhang, Y. Guo, Y. Huang, P. Zhang, A 2D DOA estimation method with reduced complexity in unfolded coprime Lshaped array. IEEE Syst. J. 15(1), 407â€“410 (2021). https://doi.org/10.1109/JSYST.2019.2948089
J. Li, D. Jiang, X. Zhang, DOA estimation based on combined unitary ESPRIT for coprime MIMO radar. IEEE Commun. Lett. 21(1), 96â€“99 (2017). https://doi.org/10.1109/LCOMM.2016.2618789
Z. Meng, W. Zhou, Robust adaptive beamforming for coprime array with steering vector estimation and covariance matrix reconstruction. IET Commun. 14, 2749â€“2758 (2020). https://doi.org/10.1049/ietcom.2019.1314
G. Li, T. Li, M. Xu et al., Sparse massive MIMOOFDM channel estimation based on compressed sensing over frequency offset environment. EURASIP J. Adv. Signal Process. 2019, 31 (2019). https://doi.org/10.1186/s1363401906273
Y. Zhou, X. Chen, Y. Li et al., A fast STAP method using persymmetry covariance matrix estimation for clutter suppression in airborne MIMO radar. EURASIP J. Adv. Signal Process. 2019, 13 (2019). https://doi.org/10.1186/s136340190610z
J. Li, P. Stoica, MIMO radar with colocated antennas. IEEE Signal Process. Mag. 24(5), 106â€“114 (2007). https://doi.org/10.1109/MSP.2007.904812
F. Wen, Z. Zhang, X. Zhang, CRBs for directionofdeparture and directionofarrival estimation in collocated MIMO radar in the presence of unknown spatially coloured noise. IET Radar Sonar Navig. 13(4), 530â€“537 (2019). https://doi.org/10.1049/ietrsn.2018.5386
F. Wen, Computationally efficient DOA estimation algorithm for MIMO radar with imperfect waveforms. IEEE Commun. Lett. 23(6), 1037â€“1040 (2019). https://doi.org/10.1049/ietrsn.2018.5386
J. Li, L. He, Y. He, X. Zhang, Joint direction of arrival estimation and array calibration for coprime MIMO radar. Digit. Signal Process. 94, 67â€“74 (2019). https://doi.org/10.1016/j.dsp.2019.07.002
W. Zhou, Q. Wang, J. Wang, DOA estimation for monostatic MIMO radar based on unfolded coprime array. J. Nanjing Univ. Posts Telecommun. Nat. Sci. Ed. 39(6), 1â€“8 (2019). https://doi.org/10.1049/el.2016.3818
R. Bautista, J.R. Buck, Detecting Gaussian signals using coprime sensor arrays in spatially correlated Gaussian noise. IEEE Trans. Signal Process. 67(5), 1296â€“1306 (2019). https://doi.org/10.1109/TSP.2018.2887399
W. So, A fast algorithm for 2D directionofarrival estimation. Signal Process. (2003). https://doi.org/10.1016/S01651684(03)00118X
N. Tayem, H.M. Kwon, Azimuth and elevation angle estimation with no failure and no eigen decom position. Signal Process. 86(1), 8â€“16 (2006)
H. Chen, C. Hou, Q. Wang et al., Improved azimuth/elevation angle estimation algorithm for threeparallel uniform linear arrays. IEEE Antennas Wirel. Propag. Lett. 14, 329â€“332 (2015). https://doi.org/10.1109/LAWP.2014.2360419
D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289â€“1306 (2006). https://doi.org/10.1109/TIT.2006.871582
M. Yuan, Model selection and estimation in regression with grouped variables. J. R. Stat. Soc 68(1), 49â€“67 (2006)
S. Ji, D. Dunson, L. Carin, Multitask compressive sensing. IEEE Trans. Signal Process. 57(1), 92â€“106 (2009). https://doi.org/10.1109/TSP.2008.2005866
Q. Wu, Y. D. Zhang, M. G. Amin and B. Himed. Complex multitask Bayesian compressive sensing, in 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, Italy (2014), pp. 3375â€“3379. https://doi.org/10.1109/ICASSP.2014.6854226
G. C. Cawley and N. L. C. Talbot. Sparse Bayesian learning and the relevance multilayer perceptron network, in Proceedings. 2005 IEEE International Joint Conference on Neural Networks, 2005. Montreal, QC, Canada, vol. 2 (2005), pp. 1320â€“1324. https://doi.org/10.1109/IJCNN.2005.1556045.
S. Ji, Y. Xue, L. Carin, Bayesian compressive sensing. IEEE Trans. Signal Process. 56(6), 2346â€“2356 (2008). https://doi.org/10.1109/TSP.2007.914345
D.J.C. MacKay, Bayesian interpolation. Neural Comput. 4(3), 415â€“447 (1992). https://doi.org/10.1162/neco.1992.4.3.415
J. Li, X. Zhang, C. Han, Improved twodimensional DOA estimation algorithm for twoparallel uniform linear arrays using propagator method. Signal Process. 92(12), 3032â€“3038 (2012). https://doi.org/10.1016/j.sigpro.2012.06.010
Z. Yi, X. Xu, Y.A. Sheikh et al., A rankreduction based 2D DOA estimation algorithm for three parallel uniform linear arrays. Signal Process. 120(MAR.), 305â€“310 (2016). https://doi.org/10.1016/j.sigpro.2015.09.019
J. Li, D. Jiang, X. Zhang, Sparse representation based twodimensional direction of arrival estimation using coprime array. Multidimension. Syst. Signal Process. 29, 35â€“47 (2018). https://doi.org/10.1007/s1104501604539
Acknowledgements
The authors would like to thank Dayu Yin for his academic support, and I can finish the thesis successfully.
Funding
This work was supported by the National Natural Science Foundation of China under Grant Nos. 61801170 and 61801435, Jiangsu Overseas Visiting Scholar Program for University Prominent Yong & Middleaged Teachers and Presidents, the Scientific and Technological Key Project of Henan Province under Grant 212102210559 and in part by the Training Program for Young Scholar of Henan Province for Colleges and Universities under Grant No. 2020GGJS172.
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ZZJ analyzed and interpreted the data regarding the SAMIMOCA and was a major contributor in writing the manuscript. JAS and JCT analyzed the theory of compressed sensing and MIMO. ZF and WY provided experimental ideas. All authors read and approved the final manuscript.
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Zhang, F., Zhang, Z., Jin, A. et al. 2D DOA estimation method based on coprime array MIMO radar. EURASIP J. Adv. Signal Process. 2021, 72 (2021). https://doi.org/10.1186/s1363402100783x
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DOI: https://doi.org/10.1186/s1363402100783x