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

Authors: Guicai Yu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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. The 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. The 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. The 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.


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## Body

## 1. 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 [1–7]. 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 [8–11] 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 [13–15]. 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. The 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.


## 2. Mathematical Model

The coprime array consists of two groups of a uniform linear array. As shown in Figure1, 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=P1,P2,…,P2M+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.Figure 1
Model of array.For simplicity, we assumeK uncorrelated signals with different incident angles θ=θ1,θ2,…,θK with respect to the receiver array. The actual received data at the t snapshot are(1)Xt=Aθst+nt,where Aθ=aθ1,aθ2,…,aθK is the array manifold of dimension 2M+N−1K, and element k is(2)aθk=e−j2π/λP1sinθk,…,e−j2π/λP2M+N−1sinθk,Xt=xP1t,xP2t,…,xP2M+N−1tT,is the received-signal vector. The direction vector aθk=e−j2π/λP1sinθk,…,e−j−j2π/λP2M+N−1sinθk, and nt is the additive white Gaussian noise vector with power σn2 and is independent between signals. The theoretical covariance matrix of the received signal xt in the coprime array is(3)Rxx=ExtxHt=AθΣAHθ+σn2I,where E⋅ is the expectation, ⋅H is the conjugate transposition,(4)Σ=EstsHt=diagσ12,σ22,…,σK2,is a diagonal matrix of dimension K×K, σ2=σ12,σ22,…,σK2T is the power vector of the signal, I is the unit matrix of dimension 2M+N−12M+N−1, and the theoretical covariance matrix Rxx is acquired during a single snapshot. In general, the adopted sampled covariance matrix is obtained via the samples at snapshot T:(5)R¯xx=1T∑t=1TxtxHt.


## 3. DOA Estimation Algorithm Interpolated in the Hole of Virtual Array Element

### 3.1. Construction of Virtual Array

The 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(6)S=Pi−Pji,j=1,2,…,2M+N−1.To define the positionP in the coprime array, S can be denoted by the union of three subsets:(7)S=S1∪S2∪S3,(8)S1=Mni−Mnj,0≤ni,nj≤N−1,(9)S2=Nmi−Nmj,0≤mi,mj≤2M−1,(10)S3=±Nm−Mn,0≤m≤2M−1,0≤n≤N−1.,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 S1 and S2 are subsets of S3, so the position of the virtual array element can be represented as(11)S=±nMd−Nmd,0≤n≤N−1,0≤m≤2M−1.


### 3.2. Array Element Interpolated in Coprime Array

Figure2 shows the position of the coprime array and the virtual array when M=3 and N=5.Figure 2
(a) Coprime array (b) virtual array.
(a)(b)The 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 −17d to 17d, as shown in Figure 2(b). The position of the hole of the discrete array is P0=±MN+aM+bN, with P0<2M−1N, where a≥1,b≥0. This is equivalent to ±18d,±21d,±23d,±24d, 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(b), the maximum continuous position is 23d and 24d. Two antennas were added at 23d and 24d in the coprime arrays, so we obtain the positions ±23d and ±24d for holes based on the difference between the antenna positions at 0d in arrays and the positions of the two added antennas. The positions ±18d of holes are obtained from the difference between the position 24d of the added antenna and the position 6d in the arrays. The positions ±21d of holes are obtained from the difference between the position 24d of the added antenna and the position 3d in the arrays. Consequently, all holes are filled, so the virtual arrays are continuous from −25d to 25d. In addition, because all newly added elements are located within the original aperture, the degree of freedom of the arrays may be increased without expanding the original array aperture.


### 3.3. DOA Estimate Based on ESPRIT Algorithm

After adding new elements toA(θ) in equation (1), vectorizing the sampled covariance of the received signals gives(12)z=vecRxx=Bσ2+σn2i,where i=vecI, B=a∗θ1⊗aθ1,…,a∗θK⊗aθK, the element in a∗θk⊗aθk can be expressed as e−j2π/λPi−Pjsinθk, and pi and pj 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. Thus, 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.Figure 3
Subarray of virtual array.The covariance of the virtual array is denoted as(13)Rzz=Ty=z0z1∗…z25∗z1z0…z24∗⋮⋮⋱⋮z25z24⋯z0,where Ty is the Hermitian Toeplitz matrix, and in the first row, zi means the equivalent virtual signal at array element i in the virtual array.In the uniform linear array of Figure3, the virtual subarray SY is assembled from virtual sensors at positions from 1d to 25d, where d is the corresponding virtual sensor displacement in each subarray.The signal received by the virtual subarray at timet can be expressed as(14)SXt=Axst+nXt,(15)SYt=AYst+nYt,where AX and AY are the steering matrices of the virtual subarrays SX and SY and nX and nY are the additive white noise in the virtual subarrays.The virtual subarraysSX and SY are displaced a distance d in a uniform linear array, indicating that subarrays SX and SY are translation invariant. The received signal SYt from subarray SY in equation (15) can be expressed by using the steering matrix of the subarray SX as(16)SYt=AXΦst+nYt,where(17)Φ=diage−j2π/λdsinθ1,…,e−j2π/λdsinθKis the diagonal matrix of dimension K×K, also called the rotation operator. The theoretical covariances of the virtual subarrays SXt and SYt are(18)RSXSX=AXΣAXH+σn2I,(19)RSYSY=AXΦΣΦHAXY+σn2I.The rotation invariance of the virtual subarrays leads to the rotation invariance of the virtual subarray signals. Therefore, 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 matricesRSXSX and RSYSY are the values at the single snapshot and are not available.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.The virtual covariance matrixRzz in equation (13) can be decomposed into its eigenvalues as(20)Rzz=ESΛSESH+ENΛNENH,where ΛS is the diagonal matrix with a maximum of K eigenvalues on the diagonal, is the subspace of the signal composed of the corresponding eigenvectors, ΛN is the diagonal matrix consisting of relatively small eigenvalues, and EN is the noise subspace composed of the corresponding eigenvector. Based on the definition of the virtual subspace, the signal subspace ESX and ESY corresponding to subarrays SX and SY can be obtained by replacing the first and last rows of Es, respectively, with(21)ES=ES1,:ESX=ESYESend,:.The orthogonality between signal and noise subspaces means that a unique nonsingular matrixT of dimension K×K should exist with(22)ESX=AXT,(23)ESY=AYT=AXΦT,where the definition of the matrix ESXY=ESXESY and the rank of the matrix ESXY should be K because the same subspace is spanned by ESX and ESY. Meanwhile, a full-ranked matrix Q=QX;QY with dimension 2K×K should exist and be orthogonal to ESXY:(24)ESXYQ=ESXQX+ESYQY=0.Combining equation (24) with equations (22) and (23) gives(25)−AXTQXQY−1=AXΦT.Using the definitionF=−QXQY−1, the rotation-invariance between signal subspace of subarrays ESX and ESY is(26)ESY=ESXF.The rotation operatorF between signal subspaces is(27)F=ESX+ESY,where ESX+ is the pseudoinverse of ESX. Substituting F into equation (25)fd26 gives(28)AXTF=AXΦT,where T is invertible and AX is a full rank matrix. This gives(29)Φ=TFT−1.Therefore,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(30)θk¯=arcsin−1πImInFk,where Fk is eigenvalue k of F and Im⋅ gives the imaginary part.


### 3.4. Algorithm Steps

The steps of the proposed algorithm are as follows:Step 1: after adding the antennas, calculate the sample covarianceRxx of the coprime array and vectorize it as per equation (12)Step 2: compute the covarianceRzz of the virtual array in equation (13)Step 3: obtain the corresponding signal subspaceESX and ESY of the virtual arrays SX and SY in equation (21)Step 4: calculate the rotation operatorF in equation (27) according to the rotational invariance of the signal subspaceStep 5: calculate the eigenvaluesFkk=1,2,…,K of the rotation operator F and estimate the DOA by using equation (30)


## 3.1. Construction of Virtual Array

The 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(6)S=Pi−Pji,j=1,2,…,2M+N−1.To define the positionP in the coprime array, S can be denoted by the union of three subsets:(7)S=S1∪S2∪S3,(8)S1=Mni−Mnj,0≤ni,nj≤N−1,(9)S2=Nmi−Nmj,0≤mi,mj≤2M−1,(10)S3=±Nm−Mn,0≤m≤2M−1,0≤n≤N−1.,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 S1 and S2 are subsets of S3, so the position of the virtual array element can be represented as(11)S=±nMd−Nmd,0≤n≤N−1,0≤m≤2M−1.


## 3.2. Array Element Interpolated in Coprime Array

Figure2 shows the position of the coprime array and the virtual array when M=3 and N=5.Figure 2
(a) Coprime array (b) virtual array.
(a)(b)The 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 −17d to 17d, as shown in Figure 2(b). The position of the hole of the discrete array is P0=±MN+aM+bN, with P0<2M−1N, where a≥1,b≥0. This is equivalent to ±18d,±21d,±23d,±24d, 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(b), the maximum continuous position is 23d and 24d. Two antennas were added at 23d and 24d in the coprime arrays, so we obtain the positions ±23d and ±24d for holes based on the difference between the antenna positions at 0d in arrays and the positions of the two added antennas. The positions ±18d of holes are obtained from the difference between the position 24d of the added antenna and the position 6d in the arrays. The positions ±21d of holes are obtained from the difference between the position 24d of the added antenna and the position 3d in the arrays. Consequently, all holes are filled, so the virtual arrays are continuous from −25d to 25d. In addition, because all newly added elements are located within the original aperture, the degree of freedom of the arrays may be increased without expanding the original array aperture.


## 3.3. DOA Estimate Based on ESPRIT Algorithm

After adding new elements toA(θ) in equation (1), vectorizing the sampled covariance of the received signals gives(12)z=vecRxx=Bσ2+σn2i,where i=vecI, B=a∗θ1⊗aθ1,…,a∗θK⊗aθK, the element in a∗θk⊗aθk can be expressed as e−j2π/λPi−Pjsinθk, and pi and pj 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. Thus, 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.Figure 3
Subarray of virtual array.The covariance of the virtual array is denoted as(13)Rzz=Ty=z0z1∗…z25∗z1z0…z24∗⋮⋮⋱⋮z25z24⋯z0,where Ty is the Hermitian Toeplitz matrix, and in the first row, zi means the equivalent virtual signal at array element i in the virtual array.In the uniform linear array of Figure3, the virtual subarray SY is assembled from virtual sensors at positions from 1d to 25d, where d is the corresponding virtual sensor displacement in each subarray.The signal received by the virtual subarray at timet can be expressed as(14)SXt=Axst+nXt,(15)SYt=AYst+nYt,where AX and AY are the steering matrices of the virtual subarrays SX and SY and nX and nY are the additive white noise in the virtual subarrays.The virtual subarraysSX and SY are displaced a distance d in a uniform linear array, indicating that subarrays SX and SY are translation invariant. The received signal SYt from subarray SY in equation (15) can be expressed by using the steering matrix of the subarray SX as(16)SYt=AXΦst+nYt,where(17)Φ=diage−j2π/λdsinθ1,…,e−j2π/λdsinθKis the diagonal matrix of dimension K×K, also called the rotation operator. The theoretical covariances of the virtual subarrays SXt and SYt are(18)RSXSX=AXΣAXH+σn2I,(19)RSYSY=AXΦΣΦHAXY+σn2I.The rotation invariance of the virtual subarrays leads to the rotation invariance of the virtual subarray signals. Therefore, 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 matricesRSXSX and RSYSY are the values at the single snapshot and are not available.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.The virtual covariance matrixRzz in equation (13) can be decomposed into its eigenvalues as(20)Rzz=ESΛSESH+ENΛNENH,where ΛS is the diagonal matrix with a maximum of K eigenvalues on the diagonal, is the subspace of the signal composed of the corresponding eigenvectors, ΛN is the diagonal matrix consisting of relatively small eigenvalues, and EN is the noise subspace composed of the corresponding eigenvector. Based on the definition of the virtual subspace, the signal subspace ESX and ESY corresponding to subarrays SX and SY can be obtained by replacing the first and last rows of Es, respectively, with(21)ES=ES1,:ESX=ESYESend,:.The orthogonality between signal and noise subspaces means that a unique nonsingular matrixT of dimension K×K should exist with(22)ESX=AXT,(23)ESY=AYT=AXΦT,where the definition of the matrix ESXY=ESXESY and the rank of the matrix ESXY should be K because the same subspace is spanned by ESX and ESY. Meanwhile, a full-ranked matrix Q=QX;QY with dimension 2K×K should exist and be orthogonal to ESXY:(24)ESXYQ=ESXQX+ESYQY=0.Combining equation (24) with equations (22) and (23) gives(25)−AXTQXQY−1=AXΦT.Using the definitionF=−QXQY−1, the rotation-invariance between signal subspace of subarrays ESX and ESY is(26)ESY=ESXF.The rotation operatorF between signal subspaces is(27)F=ESX+ESY,where ESX+ is the pseudoinverse of ESX. Substituting F into equation (25)fd26 gives(28)AXTF=AXΦT,where T is invertible and AX is a full rank matrix. This gives(29)Φ=TFT−1.Therefore,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(30)θk¯=arcsin−1πImInFk,where Fk is eigenvalue k of F and Im⋅ gives the imaginary part.


## 3.4. Algorithm Steps

The steps of the proposed algorithm are as follows:Step 1: after adding the antennas, calculate the sample covarianceRxx of the coprime array and vectorize it as per equation (12)Step 2: compute the covarianceRzz of the virtual array in equation (13)Step 3: obtain the corresponding signal subspaceESX and ESY of the virtual arrays SX and SY in equation (21)Step 4: calculate the rotation operatorF in equation (27) according to the rotational invariance of the signal subspaceStep 5: calculate the eigenvaluesFkk=1,2,…,K of the rotation operator F and estimate the DOA by using equation (30)


## 4. Stimulation Results

The performance of the proposed algorithm is verified through a simulation. In the coprime array,M=3, N=5, and the array element positions are 0d,3d,5d,6d,9d,10d,12d,15d,20d,23d,24d,25d. The results of the simulation results are compared with published results [15, 18]. The classical DOA estimation algorithm of Zhou and Zhou [15] leverages the ESPRIT algorithm for signal processing of continuous virtual arrays with unexploited holes. The 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.


### 4.1. 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.Figure4 shows the estimate of the DOA by the proposed method as a function of signal source. These 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.Figure 4
Estimate of DOA by using proposed method as a function of signal sources.Figure 5
DOA estimation of the proposed method by Zhang et al. [18].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], The 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].Figure 6
DOA estimate by proposed method with 24 signal sources.


### 4.2. Experiment 2: Comparison of Estimation Accuracy

The root-mean-square error (RMSE) of the estimated angle was used to compare the estimation accuracy of DOA which is higher. The RMSE is defined as(31)RMSE=1KQ∑k=1K∑q=1Qθk,q−θk2,where K is the number of signal sources, Q is the number of MonteCarlo experiments, θk,q is the angle estimated for signal k in the qth MonteCarlo experiment, and θk is the kth DOA. In experiment 2, Q=500 and K=13, uniformly distributed over the range −60°,60°.Figure7 shows the RMSE of the estimated angle estimation as a function of the SNR. The 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 7
Estimation accuracy of the three algorithms as a function of signal-to-noise ratio.Figure8 plots the RMSE of the angle estimate as a function of the number of snapshots, where SNR is taken to be 10 dB. These 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].Figure 8
Estimation accuracy of the three algorithms as a function of number of snapshots.


### 4.3. Experiment 3: Comparison of Resolution

The angles of incidence for the two signal sources tested in our current simulation are taken as38.22° 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.Figure 9
Resolution of proposed DOA algorithm.Figure 10
Resolution of DOA algorithm proposed by Zhang et al. [18].


## 4.1. 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.Figure4 shows the estimate of the DOA by the proposed method as a function of signal source. These 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.Figure 4
Estimate of DOA by using proposed method as a function of signal sources.Figure 5
DOA estimation of the proposed method by Zhang et al. [18].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], The 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].Figure 6
DOA estimate by proposed method with 24 signal sources.


## 4.2. Experiment 2: Comparison of Estimation Accuracy

The root-mean-square error (RMSE) of the estimated angle was used to compare the estimation accuracy of DOA which is higher. The RMSE is defined as(31)RMSE=1KQ∑k=1K∑q=1Qθk,q−θk2,where K is the number of signal sources, Q is the number of MonteCarlo experiments, θk,q is the angle estimated for signal k in the qth MonteCarlo experiment, and θk is the kth DOA. In experiment 2, Q=500 and K=13, uniformly distributed over the range −60°,60°.Figure7 shows the RMSE of the estimated angle estimation as a function of the SNR. The 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 7
Estimation accuracy of the three algorithms as a function of signal-to-noise ratio.Figure8 plots the RMSE of the angle estimate as a function of the number of snapshots, where SNR is taken to be 10 dB. These 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].Figure 8
Estimation accuracy of the three algorithms as a function of number of snapshots.


## 4.3. Experiment 3: Comparison of Resolution

The angles of incidence for the two signal sources tested in our current simulation are taken as38.22° 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.Figure 9
Resolution of proposed DOA algorithm.Figure 10
Resolution of DOA algorithm proposed by Zhang et al. [18].


## 5. 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. This 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. Thus, the proposed algorithm substantially improves the resolution and accuracy of DOA estimates.



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*Source: 1429628-2020-08-31.xml*