Algorithm to Estimate Direction of Arrival with Interpolated Array Elements for Coprime Array Holes

A novel method for adding antennas in the coprime arrays is introduced in this study, in order to solve the problem of the reduced degree of freedom of the array in the hole-existing coprime arrays. 'e minimum number of antennas interpolated in the algorithm maximizes the available degrees of freedom of virtual arrays, and the number of interpolated antennas does not change the original aperture size of the coprime arrays. With the proposed algorithm, the estimate of the direction of arrival is more accurate for a given signal-to-noise ratio. 'e scheme first finds the regular pattern of hole positions in virtual array elements, and then, according to the regular pattern, the position of the hole of the partial virtual array element is interpolated with the array element antenna at the position of the corresponding coprime arrays.'e holes of the virtual array element are filled, giving virtual uniform continuous array elements with maximum degrees of freedom. We use the ESPRIT, and the simulation results show that the proposed algorithm improves the accuracy and resolution of estimates of the direction of arrival.


Introduction
Estimating the direction of arrival (DOA) is an important research topic in the field of array signal processing and is widely applied in radar, communications, electronic countermeasures, and many other fields [1][2][3][4][5][6][7]. Although estimating the DOA can be well addressed by using traditional uniform linear arrays (ULAs), such estimates may be accurate when the number of signal sources exceeds the number of sensors because of constraints imposed on the degrees of freedom by the number of sensors in the array.
Coprime arrays [8][9][10][11] are coming to the fore in academia because provide accurate DOA estimation when signal numbers are greater than the number of array elements. Moreover, virtual arrays derived from a coprime array contain more virtual sensors, thereby allowing for more accurate DOA estimates. Virtual arrays thus derived contain both uniform continuous parts and discrete parts. In 2011, Pal and Vaidyanathan used the continuous part of virtual arrays to estimate DOA by harnessing spatial smoothing algorithms [12] following which several related DOA estimation algorithms were developed [13][14][15].
However, these algorithms did not exploit discrete sensors, resulting in inefficient use of apertures of virtual arrays and reduction in the degree of freedom of the coprime array.
An increased number of virtual sensors must be used to avoid reduction the degree of freedom. Although the sparse signal reconstruction algorithm [16] leverages all virtual sensors, it incurs a high computational cost because it significantly reuses equivalent virtual signals obtained by covariance vectorization of the sample. Boudaher et al. [17] proposed to apply temporal signal coherence to fill partial holes in moving coprime arrays, but precise temporal correlation is difficult to realize in actual applications. In addition, Zhang et al. [18] used sensors around the holes of virtual sensors to fit the virtual array response of the missing position. However, the significant error in the fitted virtual array response due to the presence of continuous holes reduced the accuracy of such estimates.
To resolve these issues, we propose herein an improved approach that is based on the discovery of a regular pattern in the position of the hole in a virtual array element, which allows the interpolation of a small number of array-element antennas at the position of coprime arrays to fill these holes and thereby maximize the degree of freedom of virtual array elements. e results of a simulation and a comparison with the primary parameters from previous work [15,18], including DOA estimates, estimated resolution, and estimated accuracy, clearly show the advantages of the proposed method.

Mathematical Model
e coprime array consists of two groups of a uniform linear array. As shown in Figure 1, the number of physical array elements in the first group is N, the distance between array elements is Md, the number of physical array elements in the second group is 2M, and the corresponding distance between array element is Nd. Given that the first array element is shared, the total number of array elements is 2M + N − 1. Furthermore, P � [P 1 , P 2 , . . . , P (2M+N− 1) ] indicates the position of the physical array elements in the coprime array. Note that M < N, so d is the half wavelength of the signal when arrays are coprime with each other.
For simplicity, we assume K uncorrelated signals with different incident angles θ � [θ 1 , θ 2 , . . . , θ K ] with respect to the receiver array. e actual received data at the t snapshot are where A(θ) � [a(θ 1 ), a(θ 2 ), . . . , a(θ K )] is the array manifold of dimension (2M + N − 1)K, and element k is is the received-signal vector. e direction vector a(θ k ) � [e (− j2π/λ)P 1 sin(θ k ) , . . . , e − j(− j2π/λ)P (2M+N− 1) sin(θ k ) ], and n(t) is the additive white Gaussian noise vector with power σ 2 n and is independent between signals. e theoretical covariance matrix of the received signal x(t) in the coprime array is where E(·) is the expectation, (·) H is the conjugate transposition, is a diagonal matrix of dimension K × K, σ 2 � [σ 2 1 , σ 2 2 , . . . , σ 2 K ] T is the power vector of the signal, I is the unit matrix of dimension (2M + N − 1)(2M + N − 1), and the theoretical covariance matrix R xx is acquired during a single snapshot. In general, the adopted sampled covariance matrix is obtained via the samples at snapshot T:

Construction of Virtual
Array. e virtual array is constructed from coprime arrays. In particular, equivalent virtual signals are obtained from the vectorization of the sample covariance of received signals in the coprime arrays. Explicitly, the position of virtual elements is given as To define the position P in the coprime array, S can be denoted by the union of three subsets: Analysis of the above equations shows that equations (8) and (9) are the set of self-differences corresponding to the received signal, and equation (10) is the set of mutual differences corresponding to the received signal, where S 1 and S 2 are subsets of S 3 , so the position of the virtual array element can be represented as Figure 2 shows the position of the coprime array and the virtual array when M � 3 and N � 5. e virtual array is composed of two parts, namely, the continuous array and the discrete array. According to the position of the virtual elements in equation (11), the continuous array spans from − (MN + N − 1) to MN + M − 1, which is equivalent to the range − 17 d to 17 d, as shown in Figure 2(b). e position of the hole of the discrete array is

Array Element Interpolated in Coprime Array.
is is equivalent to ± 18 d, ± 21 d, ± 23 d, { ± 24 d}, as shown in Figure 2(b). Given the pattern of the hole positions, we infer that all holes may be filled by interpolating the array element antenna at the position of the maximum continuous hole corresponding to the coprime array. In Figure 2 (1), vectorizing the sampled covariance of the received signals gives

DOA Estimate Based on ESPRIT Algorithm. After adding new elements to A(θ) in equation
, the element in a * (θ k ) ⊗ a(θ k ) can be expressed as e − j(2π/λ)(P i − P j )sin(θ k ) , and p i and p j represent antennas i and j in the coprime arrays that generated the equivalent virtual signal for each virtual array element in the virtual array. Liu and Vaidyanathan [19] suggested that the equivalent covariance of virtual array signals can be acquired by rearranging the elements in z in the Toeplitz matrix configuration. Given the symmetry about position 0 between the position of the array elements − p and p, the elements of their equivalent virtual signals in B are conjugate to each other. us, only the equivalent virtual signal must be processed for the continuous part at positions from 0 to p in the virtual array, as shown in Figure 3. e covariance of the virtual array is denoted as where T(y) is the Hermitian Toeplitz matrix, and in the first row, z (i) means the equivalent virtual signal at array element i in the virtual array.
In the uniform linear array of Figure 3, the virtual subarray S Y is assembled from virtual sensors at positions from 1 d to 25 d, where d is the corresponding virtual sensor displacement in each subarray. e signal received by the virtual subarray at time t can be expressed as where A X and A Y are the steering matrices of the virtual subarrays S X and S Y and n X and n Y are the additive white noise in the virtual subarrays. e virtual subarrays S X and S Y are displaced a distance d in a uniform linear array, indicating that subarrays S X and S Y are translation invariant. e received signal S Y (t) from subarray S Y in equation (15) can be expressed by using the steering matrix of the subarray S X as where is the diagonal matrix of dimension K × K, also called the rotation operator. e theoretical covariances of the virtual subarrays S X (t) and S Y (t) are e rotation invariance of the virtual subarrays leads to the rotation invariance of the virtual subarray signals. erefore, the ESPRIT algorithm can be used in the field of the coprime array. In the virtual array, the equivalent virtual signal is obtained by vectorizing the sample covariance of the signal received by the coprime array, whereas the theoretical covariance matrices R S X S X and R S Y S Y are the values at the single snapshot and are not available.

Mathematical Problems in Engineering
Since the virtual subarray is on the virtual uniform linear array, the virtual signal subspace of the subarray is also contained in the signal subspace of the virtual array. e virtual covariance matrix R zz in equation (13) can be decomposed into its eigenvalues as where Λ S is the diagonal matrix with a maximum of K eigenvalues on the diagonal, is the subspace of the signal composed of the corresponding eigenvectors, Λ N is the diagonal matrix consisting of relatively small eigenvalues, and E N is the noise subspace composed of the corresponding eigenvector. Based on the definition of the virtual subspace, the signal subspace E SX and E SY corresponding to subarrays S X and S Y can be obtained by replacing the first and last rows of E s , respectively, with e orthogonality between signal and noise subspaces means that a unique nonsingular matrix T of dimension K × K should exist with

Combining equation (24) with equations (22) and (23) gives
Using the definition F � − Q X Q − 1 Y , the rotation-invariance between signal subspace of subarrays E SX and E SY is e rotation operator F between signal subspaces is where E + SX is the pseudoinverse of E SX . Substituting F into equation (25)fd26 gives where T is invertible and A X is a full rank matrix. is gives erefore, Φ and F are similar matrices with the same eigenvalues. Since the DOA of the incident source in equation (17) is included in Φ, the estimate of K DOAs can be expressed as where F k is eigenvalue k of F and Im(·) gives the imaginary part.

Algorithm
Steps. e steps of the proposed algorithm are as follows: Step 1: after adding the antennas, calculate the sample covariance R xx of the coprime array and vectorize it as per equation (12) Step 2: compute the covariance R zz of the virtual array in equation (13) Step 3: obtain the corresponding signal subspace E SX and E SY of the virtual arrays S X and S Y in equation (21) Step 4: calculate the rotation operator F in equation (27) according to the rotational invariance of the signal subspace Step 5: calculate the eigenvalues F k (k � 1, 2, . . . , K) of the rotation operator F and estimate the DOA by using equation (30)
e results of the simulation results are compared with published results [15,18]. e classical DOA estimation algorithm of Zhou and Zhou [15] leverages the ESPRIT algorithm for signal processing of continuous virtual arrays with unexploited holes. e method proposed by Zhang et al. [18] fits the array response of holes and estimates the DOA by using spatial smoothing for the entire virtual array.

Experiment 1: Feasibility Analysis.
Given the limitations in the degree of freedom of Zhou and Zhou [15], experiment 1 is compared with the results of Zhang et al [18]. We use 21 signal sources evenly distributed over the range [− 60°, 60°]. Assuming that more signals are incident on the sensor than the number of sensors, the signal-to-noise ratio (SNR) is SNR � 0, and the number of snapshots is T � 500. Figure 4 shows the estimate of the DOA by the proposed method as a function of signal source. ese results show that the DOA estimates are accurate for the 21 signal sources. Figure 5 shows the DOA estimated by using the algorithm of Zhang et al. [18], which shows that only 19 DOA estimates are detected.
To validate the increase in array freedom provided by the method proposed herein, we assume that 24 signal sources are evenly distributed over a range [− 60°, 60°], which is the same as used by Zhang et al. [18], e SNR is SNR � 10, and the snapshot number is T � 500. As shown in Figure 6, 24 DOAs are detected, indicating an improvement in the array freedom over the 20 DOA detected by Zhang et al [18].

Experiment 2: Comparison of Estimation Accuracy.
e root-mean-square error (RMSE) of the estimated angle was used to compare the estimation accuracy of DOA which is higher. e RMSE is defined as where K is the number of signal sources, Q is the number of MonteCarlo experiments, θ k,q is the angle estimated for signal k in the qth MonteCarlo experiment, and θ k is the kth DOA. In experiment 2, Q � 500 and K � 13, uniformly distributed over the range [− 60°, 60°]. Figure 7 shows the RMSE of the estimated angle estimation as a function of the SNR. e SNR ranges from − 5 to 25 dB in 5 dB steps, and the number of snapshots is 500. As can be seen in Figure 7, the RSME decreases with increasing SNR. Compared with the published algorithms [15,18], the algorithm proposed herein produces a smaller RSME for the estimated angle and is thus more accurate than the published algorithms. Figure 8 plots the RMSE of the angle estimate as a function of the number of snapshots, where SNR is taken to be 10 dB.
ese results show that, compared with the published algorithms [15,18], the algorithm proposed herein produces a smaller RMSE for estimating the angle. Such as, it is about 0.2 smaller than the algorithm in the       literature [18] and 0.05 smaller than the algorithm in literature [15].

Experiment 3: Comparison of Resolution.
e angles of incidence for the two signal sources tested in our current simulation are taken as 38.22°and 40.32°, where SNR � − 5 dB and the number of snapshots is 500. As shown in Figure 9, when the SNR is low, the proposed method accurately resolves the direction of two nearby incoming waves. In contrast, as shown in Figure 10, the algorithm of Zhang et al. [18] does not satisfactorily resolve the direction of two nearby incoming waves.

Conclusions
We propose herein a method for interpolating a small number of antennas in the position of the coprime array to fill the holes of the virtual array derived from the coprime array.
is method allows us to maintain the maximum degrees of freedom of the coprime array despite the unavailable holes in the virtual array. Meanwhile, the added antennas allow all holes of virtual arrays to be filled without expanding the original aperture of the coprime array, leading to the maximum number of degrees of freedom. In addition, we use the ESPRIT algorithm to estimate the DOA. Finally, comparing the proposed algorithm with simulation results demonstrates that the proposed algorithm increases the number of degrees of freedom of the coprime array and identifies more DOAs. us, the proposed algorithm substantially improves the resolution and accuracy of DOA estimates.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon reasonable request and with permission from the funders.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.