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.
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 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 . Furthermore, indicates the position of the physical array elements in the coprime array. Note that , so 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 with respect to the receiver array. The actual received data at the t snapshot arewhere is the array manifold of dimension , and element k isis the received-signal vector. The direction vector and is the additive white Gaussian noise vector with power and is independent between signals. The theoretical covariance matrix of the received signal in the coprime array iswhere is the expectation, is the conjugate transposition,is a diagonal matrix of dimension , is the power vector of the signal, I is the unit matrix of dimension , and the theoretical covariance matrix is acquired during a single snapshot. In general, the adopted sampled covariance matrix is obtained via the samples at snapshot T:
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
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 and are subsets of , so the position of the virtual array element can be represented as
3.2. Array Element Interpolated in Coprime Array
Figure 2 shows the position of the coprime array and the virtual array when and .
(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 to , which is equivalent to the range to , as shown in Figure 2(b). The position of the hole of the discrete array is , with , where . This is equivalent to , 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 and . Two antennas were added at and in the coprime arrays, so we obtain the positions and for holes based on the difference between the antenna positions at in arrays and the positions of the two added antennas. The positions of holes are obtained from the difference between the position of the added antenna and the position in the arrays. The positions of holes are obtained from the difference between the position of the added antenna and the position in the arrays. Consequently, all holes are filled, so the virtual arrays are continuous from to . 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 to A(θ) in equation (1), vectorizing the sampled covariance of the received signals giveswhere , , the element in can be expressed as , and and represent antennas and 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 in the Toeplitz matrix configuration. Given the symmetry about position 0 between the position of the array elements and , the elements of their equivalent virtual signals in are conjugate to each other. Thus, only the equivalent virtual signal must be processed for the continuous part at positions from 0 to in the virtual array, as shown in Figure 3.
The covariance of the virtual array is denoted aswhere is the Hermitian Toeplitz matrix, and in the first row, means the equivalent virtual signal at array element in the virtual array.
In the uniform linear array of Figure 3, the virtual subarray is assembled from virtual sensors at positions from to , where d is the corresponding virtual sensor displacement in each subarray.
The signal received by the virtual subarray at time t can be expressed aswhere and are the steering matrices of the virtual subarrays and and and are the additive white noise in the virtual subarrays.
The virtual subarrays and are displaced a distance d in a uniform linear array, indicating that subarrays and are translation invariant. The received signal from subarray in equation (15) can be expressed by using the steering matrix of the subarray aswhereis the diagonal matrix of dimension , also called the rotation operator. The theoretical covariances of the virtual subarrays and are
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 matrices and 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 matrix in equation (13) can be decomposed into its eigenvalues aswhere is the diagonal matrix with a maximum of eigenvalues on the diagonal, is the subspace of the signal composed of the corresponding eigenvectors, is the diagonal matrix consisting of relatively small eigenvalues, and is the noise subspace composed of the corresponding eigenvector. Based on the definition of the virtual subspace, the signal subspace and corresponding to subarrays and can be obtained by replacing the first and last rows of , respectively, with
The orthogonality between signal and noise subspaces means that a unique nonsingular matrix of dimension should exist withwhere the definition of the matrix and the rank of the matrix should be because the same subspace is spanned by and . Meanwhile, a full-ranked matrix with dimension should exist and be orthogonal to :
Combining equation (24) with equations (22) and (23) gives
Using the definition , the rotation-invariance between signal subspace of subarrays and is
The rotation operator between signal subspaces iswhere is the pseudoinverse of . Substituting into equation (25)fd26 giveswhere is invertible and is a full rank matrix. This gives
Therefore, and are similar matrices with the same eigenvalues. Since the DOA of the incident source in equation (17) is included in , the estimate of DOAs can be expressed aswhere is eigenvalue of F and 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 covariance of the coprime array and vectorize it as per equation (12) Step 2: compute the covariance of the virtual array in equation (13) Step 3: obtain the corresponding signal subspace and of the virtual arrays and 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 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, , , and the array element positions are . 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 . Assuming that more signals are incident on the sensor than the number of sensors, the signal-to-noise ratio (SNR) is , and the number of snapshots is .
Figure 4 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.
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 , which is the same as used by Zhang et al. [18], The SNR is , and the snapshot number is . 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].
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 aswhere is the number of signal sources, is the number of MonteCarlo experiments, is the angle estimated for signal in the th MonteCarlo experiment, and is the th DOA. In experiment 2, and , uniformly distributed over the range .
Figure 7 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 8 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].
4.3. Experiment 3: Comparison of Resolution
The angles of incidence for the two signal sources tested in our current simulation are taken as and , 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.
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.
Data Availability
The 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
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 61901409 and 61661030).