Non-circular signal DOA estimation based on coprime array MIMO radar

: Aiming at the problems of low degree of freedom, small array aperture, phase ambiguity and other problems of traditional coprime array direction of arrival estimation methods, a non-circular signal DOA estimation method based on expanded coprime array MIMO radar is proposed. Firstly, this method combines the coprime array and the MIMO radar to form transmitter and receiver array. Secondly, the array is expanded using the non-circular signal characteristics to reconstruct the received signal matrix. Then the dimensionality reduction is performed. The two-dimensional spectral peak search is converted into an optimization problem, and the optimization of the two-dimensional MUSIC algorithm is reconstructed using constraints, and a cost function is constructed to solve the problem. In addition, using the power series of the noise eigenvalues to correct the noise subspace to further improve the accuracy of the algorithm. Finally, the problem of no phase ambiguity in the method in this article is derived. Simulation experiments show that the method in this article can effectively avoid phase ambiguity, greatly improve the degree of freedom, and expand the array aperture. Compared with the traditional MUSIC algorithm and the mutual prime array MUSIC algorithm, it has better resolution and DOA estimation accuracy.


Introduction
methods such as Multiple signal classification (MUSIC) method [14][15] or Estimation of signal parameters via rotational invariance techniques (ESPRIT) method [16][17]. However, because uniform linear array is used as the transmitter and receiver array, the spacing of its array elements is generally half a wavelength, and the array aperture is greatly restricted, which affects the performance of MIMO radar angle resolution and multi-target parameter estimation. Therefore, Sparse arrays [18][19][20][21][22] such as nested arrays, coprime arrays and minimum redundant arrays are proposed. Compared with uniform array, Sparse array can use the special arrangement of the array to obtain a larger number of virtual array elements with the same actual number of array elements, and has higher angular resolution and parameter resolution. Reference [23] proposed the structural model of coprime array, two uniform subarrays form coprime array. The number of the subarray elements is M and N, and the distance between the array elements is Nd and Md (d is half the wavelength), And M, N are mutually prime numbers. Reference [24] proposed a coprime array DOA estimation method based on the joint MUSIC algorithm. This method decomposes the coprime array into two uniform subarrays and performs MUSIC estimation separately, and then combines the estimation results of the two subarrays to obtain the final DOA estimation, thereby eliminating the phase ambiguity problem. but this method decomposes the coprime array into Two subarrays lose part of the received data, thereby reducing the DOA estimation performance. Reference [25] proposed a DOA estimation method based on a coprime array. This method expands the coprime array to obtain a virtual array, and introduces spatial smoothing technology to solve the problem that the virtual array is not linear uniform array. but this method will lose the discontinuous information of the array elements, thereby affecting the degree of freedom of DOA estimation. Reference [26][27] respectively proposed the ESPRIT algorithm and the unitary ESPRIT algorithm for the MIMO radar coprime array to jointly estimate the Direction of Departure (DOD) and the Direction of Arrival (DOA). This method divides the coprime array into two uniform sparse subarrays as transmitter and receiver arrays. The coprime array formed by them increases the aperture of the array, thereby improving the performance of DOA estimation. However, the degree of freedom of the above method is limited by the number of elements of the subarray, and additional calculations are required to eliminate the ambiguity problem. Reference [28] proposed DOA estimation method based on the expanded coprime array MIMO radar MUSIC algorithm. This method uses the expanded coprime array as the transmitter and receiver arrays of the MIMO radar to increase the array aperture, and uses the MUSIC algorithm to estimate the DOA, which effectively improves the DOA estimates performance, but this method requires a lot of calculations.
The above literature does not involve DOA estimation methods for non-circular signals. However, there are a large number of non-circular signals in the actual environment [29][30], such as Minimum Shift Keying (MSK) signals, Binary Phase Shift Keying (BPSK) signals and Amplitude Modulation (AM) signals etc. The unique non-circular characteristics of non-circular signals can further expand the virtual array, increase the array aperture, effectively double the dimension of the receiver array, and improve the performance of target parameter estimation and the resolution of multiple sources. Therefore, based on the literature [28], this article proposes a non-circular signal dimensionality reduction DOA estimation method based on the expanded coprime array MIMO radar (NRC-MIMO MUSIC). This method uses the expanded coprime array MIMO radar model to obtain a larger virtual array when the actual number of array elements is the same. This method first uses the characteristics of noncircular signals to reconstruct the receiver array in the MUSIC algorithm. Then the reconstructed two-dimensional DOA estimation problem is transformed into an optimization problem, and the optimization problem is reconstructed using constraints, and then the Lagrange multiplier method is used to construct the cost function, and the one-dimensional spectral peak search function is obtained. Aiming at the problem of poor accuracy of the method in the environment of low signal-to-noise ratio(SNR) and small number of snapshots, the power series of noise eigenvalues is used to modify the noise subspace to further improve the accuracy of the algorithm. The method proposed in this article uses a coprime array as the transceiver array of the MIMO radar, which greatly increases the array aperture of the virtual array, eliminates the phase ambiguity problem caused by the array element spacing larger than half the wavelength, and significantly improves the DOA estimation performance. Finally, simulation experiments verify the effectiveness of the algorithm. Compared with the classic MUSIC algorithm and the traditional coprime array MUSIC algorithm, the method in this article has better performance in DOA estimation accuracy, successful resolution, and low signal-to-noise ratio (SNR) environments.

Preliminaries
The geometric structure of the expanded coprime array MIMO radar is shown in Figure 1. Both the transmitter array and the receiver array are composed of two uniform sparse sub-arrays, the two sub-arrays are expanded side by side, and the last element of subarray 1 is used as the first element of subarray 2. The sparse uniform subarray 1 and subarray 2 together form the expanded coprime array. Subarray 1 contains M array elements, and the distance between each adjacent array element is Nd (where N is the number of array elements and d is half the wavelength). The other sub-array 2 contains N array elements, and the distance between each adjacent array element is Md (where M is the number of array elements and d is half the wavelength). The two numbers M and N are mutually prime, and the distance between adjacent elements of subarray 1 and subarray 2 is greater than half of the wavelength. The formed coprime array contains M+N-1 elements, as shown in Figure 1. The coprime array takes the first element as the reference point, and the position of the element can be expressed as: Therefore, the array flow matrix of the transmitter array and the receiver array are At and Ar respectively. It can be expressed as: From the above formula (6) and (7), the flow pattern A of the entire virtual array can be obtained as: ,  represents the Kronecker product. The received signal of the coprime array can be expressed as: x t As t n t  (9) where s(t) is the source signal vector (non-circular signal vector in this article), n(t) is the noise vector, and A is the array flow matrix. For the non-circular signal s(t), according to its definition, the non-circular signal is relative to the circular signal. If the signal has the characteristics of rotation invariance, the signal s(t) is called the circular signal. That is, when { ( )} Et s0 , is the noncircular signal. The non-circular signal s(t) can be expressed as: where,  is the Kth non-circular phase of the non-circular signal,

Expanded Coprime Array MIMO Radar Non-circular Signal Dimensionality Reduction DOA Estimation Method
From the above formula (9) and (10), the received signal of the coprime array can be expressed as: Using the non-circular characteristic of the signal s(t), the array flow matrix can be reconstructed, and the received signal can be reconstructed as: where, Then the covariance matrix of the received signal can be obtained from L snapshots. That is Perform eigen decomposition on the covariance matrix A, we can get where, Ds is K×K diagonal matrix, whose diagonal elements are composed of K larger eigenvalues of the covariance matrix. Es is the signal subspace, which is the space formed by the eigenvectors corresponding to the K larger eigenvalues of the covariance matrix. Dn is composed of (M+N-1) 2 -K smaller eigenvalues with smaller diagonal elements. En is the noise subspace, which is the space formed by the eigenvectors corresponding to the (M+N-1) 2 -K smaller eigenvalues of the covariance matrixR . According to the orthogonality between the noise subspace and the direction vector, the following spatial spectrum function is constructed as: After reconstructing the receiving matrix from non-circular signals, the spatial spectrum function is a two-dimensional spectral peak search, which is highly complex, and the following dimensionality reduction processing is performed. Firstly, reconstruct the formula (16), then: where, Define function ( , ) V  : substituting formula (17) into the above formula (21), we can get: Where, let where, the above formula (22) is about the problem of quadratic optimization, to find its optimal solution ( , )  . Firstly, increase the constraint of to eliminate the solution of ( )=0 q  , so as to obtain the optimal solution ( , )  of ( , ) V  . Use constraints to reconstruct the secondary optimization problem and seek the optimal solution, that is: The method of solving the optimal solution using the Lagrange multiplier method, Construct the cost function ( , ) L  , that is: where,  is constant. The partial derivative of formula (25) can be obtained as follows:   (26) where, let the above formula (26) be equal to 0, then we can get: Because of the constraint of , combined with formula (27), we can get: Combining formulas (24) and (28) In the process of searching for the above-mentioned spectral peaks, the number of snapshots is limited due to the actual situation, which affects the accuracy of the noise subspace En, and reduces the DOA estimation performance of the algorithm. For this reason, use the power series of the noise feature to modify the corresponding noise subspace En, and the power series of the noise feature: 2 2 ( 1) where, using formula (31), the above () Q  can be re-expressed as: Where, substituting formula (32) into the above formula (30), the corrected onedimensional peak search function can be obtained: In summary, the implementation steps of the non-circular signal dimensionality reduction DOA estimation method based on the expanded coprime array MIMO radar proposed in this article are shown in Table 1: Table 1 The method steps of this article Step1: The received signal X(t) is constructed from the coprime array model, and the receiving matrix is expanded and reconstructed by formula (10); Step2: Solve the covariance matrix R by formula (12), and perform eigenvalue decomposition on the covariance matrix to obtain the noise subspace En; Step3: Reconstruct in formula (16) and transform it into a secondary optimization problem; Step4: Use formulas (24) and (25) to find the optimal solution and get the corresponding  ; Step5: Formula (30) can be obtained by () Q  , and () Q  can be corrected by formula (31); Step6: Finally, the DOA estimated value is obtained from formula (33).

No phase ambiguity proof
Since the element spacing of the sparse array is greater than half a wavelength, there is a problem of angular ambiguity. But the method in this article adopts the expanded coprime array, which can effectively suppress the phase ambiguity problem. The proof is as follows: Suppose there is phase ambiguity, that is, there is an ambiguity angle θm, which satisfies: where ˆk  is the estimated value of the direction of arrival, and θm is the value of the ambiguity angle. From formula (34), we can get: Substituting formulas (2) and (3) into formula (35) above, we can get: Expand the above formula (36), we can get: where, dl is the element spacing. From the above formula (37), we can get: where, k1∈ (−N, N), k2∈ (−M, M). In the coprime array, M and N are integers that are mutually prime numbers. According to theorem 1 of literature [24], it can be known that there is no M and N in formula (36), that is, there is no blur angle θm that holds true

Results and discussion
In order to verify the effectiveness of the algorithm in this article, the NRC-MIMO MUSIC algorithm in this article is compared with the classic MUSIC algorithm and the traditional coprime array MUSIC algorithm. Define the Root Mean Square Error (RMSE) formula as follows:   The proposed algorithm is simulated on MATLAB R2018b software to verify its performance. Set the transmitting array element and the receiving array element to M=3 and N=4 respectively. The number of sources is 2, the target angle is 10° and 20°, the number of snapshots is 100, and the number of Monte Carlo experiments is 100. Figure  2 shows the estimation results of the proposed algorithm for all targets when the SNR=10 dB. It can be seen that the algorithm in this article can accurately estimate the angle of multiple independent targets at the same time.   Figure 3 shows the DOA of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm under the condition of 100 snapshots under different SNB Estimated performance. It can be seen from Figure 3 that with the gradual improvement of the signal-to-noise ratio of the three algorithms, the root mean square error RMSE is all getting smaller, and the DOA estimation performance is improved. In addition, it can be seen from the above figure that the algorithm proposed in this article is better than the other two algorithms, and the DOA estimation performance is better. It can also be seen that the relatively prime array can significantly improve the DOA estimation performance compared to the uniform linear array. Figure 4 shows the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm under 100 effective computer simulation experiments, the target detection success rate varies with the SNR. Target detection success rate, that is, the ratio of the number of successful DOA estimates to the number of trials. When ˆ-1 KK  (where ˆk  is the estimated value of DOA and k  is the actual value), it is deemed to be a successful estimate of DOA. It can be seen from Figure 4 that the success rate of the three algorithms increases as the SNR increases. When the SNR is equal to 10dB, the success rates of the three algorithms all reach 100%. But when the SNR is less than 10dB, the success rate of this algorithm is better than the other two algorithms. In the case of low SNR, it still maintains a high success rate. This shows that the algorithm in this article is still applicable under the condition of low SNR.   Table 2 shows the specific values of the RMSE of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO non-circular signal dimensionality reduction MUSIC algorithm under different SNR. It can be seen from Table 2 that as the SNR increases, the estimation performance of the three algorithms gradually improves. However, the algorithm proposed in this article is better than the other two algorithms under different SNR, and the estimation accuracy is higher. When the SNR is 15dB, the estimation performance of the algorithm proposed in this article can be equivalent to that of the classic MUSIC when the SNR is 25dB. When the SNR=10dB, the RMSE of the algorithm in this article is reduced by about 64% compared to the classic MUSIC algorithm, and about 39% compared with the traditional coprime array MUSIC algorithm. The RMSE of the traditional coprime array MUSIC algorithm is reduced by about 41% Compared with the classic MUSIC algorithm. Therefore, it can be seen that compared to the other two algorithms, the estimation accuracy of the algorithm in this article is higher, and the coprime array can significantly improve the DOA estimation accuracy of the algorithm compared with the uniform linear array.  Figure 5 shows the variation of the RMSE of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm under different snapshots. It can be seen from Figure 5 that as the number of snapshots increases, the RMSE of the three algorithms gradually decreases, and the estimation performance gradually improves. Among them, the classic MUSIC algorithm has the worst angle estimation performance, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm has the best angle estimation performance and is more stable. The algorithm in this article still has good estimation performance when the number of snapshots is low. Figure 6 shows the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm under 100 effective computer simulation experiments, the target detection success rate varies with the snapshots. It can be seen from Figure 6 that the success rate of the three algorithms increases as the number of snapshots increases. When the number of snapshots is equal to 300, the target detection success rate of the algorithm proposed in this article is almost 100%, and when the number of snapshots is large enough, the target detection success rate of all algorithms can reach 100%. In the case of the same number of snapshots, the algorithm proposed in this article has a higher target detection success rate than the other two algorithms. In the case of a lower number of snapshots, the target detection success rate of the algorithm proposed in this article is significantly higher than that of the classic MUSIC algorithm and traditional coprime array MUSIC algorithm.   Table 3 shows the specific values of the RMSE of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO non-circular signal dimensionality reduction MUSIC algorithm under different snapshots. It can be seen from Table 3 that, when the number of snapshots is low, the angle estimation accuracy of the classic MUSIC algorithm is the lowest, and the angle estimation accuracy of the algorithm proposed in this article is the highest. As the number of snapshots increases, the RMSE of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO non-circular signal dimensionality reduction MUSIC algorithm gradually decreases, and the accuracy of the algorithm is gradually improved. Among them, the angle estimation accuracy of the algorithm proposed in this article is the highest, and the angle estimation accuracy of the classic MUSIC algorithm is the lowest. In the case of 100 snapshots, the RMSE of the proposed algorithm is about 62% lower than that of the classic MUSIC algorithm, and the RMSE of the traditional coprime array MUSIC algorithm is about 41% lower than that of the classic MUSIC algorithm.  Figure 7 shows the variation of the RMSE of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm under different the number of array elements. Set the number of transmitting array elements to 4, and the number of receiving array elements to 3, 5, 7, 9, and 11. The number of sources is 2, the target angle is 10° and 20°, the number of snapshots is 100, and the number of Monte Carlo experiments is 100. The experiment parameters are simulated by MATLAB, and RMSE of the three algorithms decreases with the increase of the number of elements, and the accuracy of angle estimation is improved. Among them, the classic MUSIC algorithm has the lowest angle estimation accuracy, and the algorithm proposed in this article has the highest accuracy. In the case of a small number of transmitted array elements, the algorithm proposed in this article has higher estimation accuracy than the other two algorithms.  Table 4 shows the specific values of the RMSE of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO non-circular signal dimensionality reduction MUSIC algorithm under different the number of array elements. It can be seen from Table 4 that when the number of array elements is small, the angle estimation accuracy of the classic MUSIC algorithm is the lowest, and the angle estimation accuracy of the expanded coprime array MIMO noncircular signal dimensionality reduction MUSIC algorithm is the highest. When the number of array elements is 3, the RMSE of the proposed algorithm is about 62% lower than that of the classic MUSIC, and the RMSE of the traditional coprime array MUSIC algorithm is about 54% lower than that of the classic MUSIC algorithm. As the number of array elements increases, the accuracy of each algorithm is gradually improved. The angle estimation accuracy of the algorithm proposed in this article is the highest, and the angle estimation accuracy of the classic MUSIC algorithm is the lowest.
Under the expanded coprime array MIMO radar signal model, set the number of snapshots to 100, the number of sources to 2, the target angle to 10° and 20°, the number of receiving array elements N remains unchanged, and the number of transmitting array elements M to 3, 4, 7. Carry out 100 Monte Carlo experiments, get the RMSE graph of angle measurement accuracy with SNR under different number of transmitting array elements. It can be seen from Figure 8 that in the case of three groups of different transmitting array elements, the algorithms proposed in this paper all increase as the SNR increases, and the accuracy of angle measurement estimation becomes higher. When the number of transmitting array elements is 3, the measurement accuracy is the lowest, and when the number of transmitting array elements is 7, the angle measurement estimation accuracy is the highest.  Table 5 shows the specific values of the root mean square error (RMSE) of the algorithm proposed in this paper with the change of the SNR under different the number of transmitting array elements. It can be seen from Table 5 that when the SNR is low, the angle estimation accuracy is the lowest when the number of transmission elements is 3. The angle estimation accuracy is the highest when the number of transmission elements is 7. When the SNR is 5dB, the RMSE of the transmitting array element M=5 is about 45% lower than the RMSE of M=3. The RMSE of the transmitting array element M=7 is about 60% lower than the RMSE of M=3. As the signal-to-noise ratio (SNR) increases, the accuracy of the algorithm is gradually improved. Among them, the angle estimation accuracy of the transmitting array element number 3 is the lowest, and the angle estimation accuracy of the transmitting array element number 7 is the highest.

Conclusion
In order to solve the problems of low degree of freedom, small array aperture, and phase ambiguity of traditional coprime array direction of arrival estimation methods, this paper proposes a non-circular signal DOA estimation method based on expanded coprime array MIMO radar. This article first combines the expanded coprime array with the MIMO radar to expand the array aperture, improve the DOA estimation performance and increase the degree of freedom. And by introducing non-circular signals, the array aperture is further increased. Then use the idea of dimensionality reduction to reduce the dimensionality of the two-dimensional MUSIC algorithm to reduce the complexity of the algorithm. The power series of noise eigenvalues is used to correct the noise subspace, which further improves the accuracy of the algorithm. Finally, the effectiveness of the algorithm is verified by simulation experiments. Compared with the classic MUSIC algorithm and the traditional MUSIC algorithm, the algorithm in this paper can better improve the DOA estimation accuracy and successful resolution. And it still maintains superior performance under SNR.

Abbreviations:
MIMO: Multiple Input Multiple Output; DOA: Direction of arrival; MUSIC: Multiple signal classification algorithm; SNR: Signal-to-noise ratio; RMSE: Root mean square error; CA: Coprime Array; NRC-MIMO MUSIC: Non-circular signal dimensionality reduction DOA estimation method based on the expanded coprime array MIMO radar